The Physical Properties of Asteroids

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Electronic Theses and Dissertations, 2020-

2020

Leos Pohl University of Central Florida

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STARS Citation Pohl, Leos, "The Physical Properties of Asteroids" (2020). Electronic Theses and Dissertations, 2020-. 269. https://stars.library.ucf.edu/etd2020/269 THE PHYSICAL PROPERTIES OF ASTEROIDS

LEOS POHL B.S. Charles University, Prague, 2011 M.S. Charles University, Prague, 2014

A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Physics in the Department of Physics in the College of Sciences at the University of Central Florida Orlando, Florida

Summer Term 2020

ii ABSTRACT

The Small Bodies of the Solar System are leftover material from the formation of planets. Compared with planetary bodies, they have undergone relatively little transformation. Embedded in their physical properties, are clues to the conditions and processes that took place since the condensation of the Solar Nebula. Furthermore, asteroids are sources of raw materials that have become a topic of signiﬁcant interest. In this dissertation, I explore several properties of asteroid material: strength of asteroids, their shielding properties against high energetic particles and their water content. First, I gather all available data on strength of meteoritic material from original papers, unify them into a single data source. Several sources have suggested to apply the Scale Eﬀect to extrapolate the measurements on meteorites to the strength of asteroid size objects.

I show that such claims are not supported by available measurements and argue that the strength of asteroids is mostly driven by their extreme heterogeneity. Additionally,

I observe inverse relationship between porosity and compressive strength for Ordinary

Chondrites. This is not observed for Carbonaceous Chondrites. Next, I study how well material of carbonaceous chondrites acts to decrease and potentially stop charged particles that are found in Cosmic Galactic Rays and Solar protons. Using relativistic quantum mechanical treatment by Bethe with additional high energy corrections, it is found that phyllosilicate materials with hydroxyl interlayer outperform Aluminium in the ability to

iii slow down charged particles of energies typical for Solar protons and Galactic Cosmic

Rays. Finally, I investigate the loss of water on asteroids on two fronts, experimental and theoretical. I quantify how the major components of Carbonaceous Chondrites dehydrate. Then, I demonstrate the possibility of loss of water due to orbits that are close to the Sun.

iv ACKNOWLEDGMENTS

I am grateful to my advisor, Dr. Daniel Britt, for his support, insight, and fruitful discussions during my study. He is a wonderful mentor who has provided me with invaluable opportunities to present my research in the conferences to the global scientiﬁc community, and has helped me grow both professionally and personally.

I would like to express my thanks to the UCF Physics Department for their support throughout this process; my committee members, and all oﬃce staﬀ, especially Esperanza

Soto for all her assistance and incredible patience.

v TABLE OF CONTENTS

LIST OF FIGURES ...... xii

LIST OF TABLES ...... xxi

LIST OF ACRONYMS ...... xxviii

CHAPTER 1 : INTRODUCTION ...... 1

1.1 Robotic Visits to Asteroids ...... 2

1.2 Sample Return Missions from Asteroids ...... 2

1.3 Observations of Asteroids ...... 3

1.3.1 Asteroid Shapes ...... 4

1.3.2 Orbital Elements ...... 5

1.3.3 Spin Period ...... 6

1.3.4 Spectra ...... 9

1.4 Analyses of Meteorites ...... 10

1.4.1 Measurements on Meteorites ...... 10

1.4.2 Classiﬁcation of Meteorites ...... 32

vi CHAPTER 2 : STRENGTHS OF ASTEROIDS AND METEORITES ...... 36

2.1 Background ...... 36

2.2 Introduction ...... 41

2.2.1 Deﬁnition of Quantities Related to the Measurements of Strength 41

2.2.2 Conditions Aﬀecting the Experimental Results ...... 47

2.2.3 System of Units and Signiﬁcant Figures ...... 49

2.2.4 Extracting Data from Plots ...... 50

2.3 Data Sources ...... 51

2.4 Data and Plots ...... 63

2.4.1 Overview of All Measurements ...... 63

2.4.2 Size Dependence of Strength ...... 66

2.4.3 Relationship Between Strength and Density and Porosity ...... 72

2.4.4 The Final Dataset ...... 80

2.5 Discussion and Conclusions ...... 81

2.5.1 Discussion ...... 81

2.5.2 Conclusions ...... 86

CHAPTER 3 : INTERACTIONS OF HIGH ENERGETIC PARTICLES WITH AS-

TEROIDAL MATTER ...... 88

3.1 Background ...... 88

vii 3.2 Introduction ...... 89

3.2.1 Sources of Energetic Particles in Space ...... 93

3.2.2 Asteroidal Mineralogy ...... 98

3.3 Methodology ...... 103

3.3.1 Analytical Methods ...... 103

3.3.2 Numerical Method ...... 110

3.4 Results and Discussion ...... 111

3.4.1 Benchmarking ...... 111

3.4.2 Asteroidal Materials ...... 116

3.4.3 Asteroidal Minerals ...... 120

3.4.4 Discussion ...... 122

3.5 Conclusion ...... 125

CHAPTER 4 : LOSS OF WATER IN ASTEROIDS BY DEHYDRATRION .... 127

4.1 Background ...... 127

4.2 Introduction ...... 130

4.2.1 Hydrated Minerals ...... 130

4.2.2 Existing Studies ...... 133

4.3 Samples and Techniques ...... 150

4.3.1 Samples ...... 150

viii 4.3.2 Methods ...... 152

4.4 Results ...... 174

4.4.1 Serpentines under Inert Gas Flow ...... 174

4.4.2 Results of Dehydroxylation Viewed by XRD ...... 182

4.4.3 Results of Dehydroxylation Viewed by FTIR ...... 184

4.4.4 Heating Rate Eﬀects ...... 187

4.4.5 Sample Size Eﬀects ...... 189

4.4.6 Eﬀects of Varying Grain Size ...... 190

4.4.7 Serpentines under Vacuum ...... 192

4.5 Discussion ...... 196

4.5.1 Parameters Aﬀecting the Results ...... 196

4.5.2 Dehydroxylation of Serpentine Minerals ...... 198

4.6 Conclusions ...... 202

CHAPTER 5 : LOSS OF WATER BY ASTEROIDS — APPLICATION ...... 204

5.1 Introduction ...... 204

5.2 Methods ...... 205

5.2.1 The Heat Conduction Equation ...... 205

5.2.2 Summary of Methods and Assumptions ...... 208

5.3 Results ...... 212

ix 5.4 Discussion and Conclusions ...... 217

5.4.1 Discussion of the Simplifying Assumptions ...... 217

5.4.2 Discussion of the Results ...... 220

5.4.3 Further Comments ...... 222

5.4.4 Conclusions ...... 224

CHAPTER 6 : SUMMARY AND OUTLOOK ...... 225

6.1 Discussion ...... 225

6.2 Summary...... 234

6.3 Outlook ...... 235

APPENDIX A : COPYRIGHT PERMISSIONS ...... 237

A.1 Excerpt from Copyright Transfer Agreement with Wiley Periodicals, Inc. 239

A.2 Copyright Perission Letter from Elsevier Ltd...... 242

APPENDIX B : TABLE OF STRENGTHS OF METEORITES ...... 245

APPENDIX C : Heat Conduction Equation ...... 250

C.1 Derivation of Heat Conduction Equation ...... 251

C.1.1 Integral Form ...... 252

C.1.2 Diﬀerential Form ...... 253

C.2 Boundary Value Problem ...... 254

x C.3 1-Dimensional Problem ...... 256

C.4 Analytical Solution of 1D Problem ...... 259

APPENDIX D : The Numerical Solution to the Heat Conduction Equation ..... 266

D.1 The Statement of the Problem ...... 267

D.2 The Numerical Implementation to the Problem ...... 268

D.2.1 The Coordinate Systems and Orbit Speciﬁcation ...... 268

D.2.2 The Insolation Function ...... 269

D.2.3 Finite Diﬀerences ...... 270

LIST OF REFERENCES ...... 273

xi LIST OF FIGURES

Figure 1.1 Distribution of asteroids up to 6 AU with determined classes from spectral data. Asteroids with undetermined class are plotted in grey and smaller dots.

Data were taken from JPL Small Body Database...... 6

Figure 1.2 Distribution of asteroids beyond Jupiter. Data source: JPL Small Body

Database ...... 7

Figure 1.3 Distribution of asteroids in proper semimajor axis (a), proper eccentric-

ity (e) and proper inclination (sin(i)) shows asteroidal families. The families are not

perfectly visible in the plot due to the amount of asteroids included and resolution.

The data are numerically calculated proper elements from the AstDyS database. 8

Figure 1.4 Distribution of asteroids spin vs size. Data source: JPL Small Body

Database ...... 9

Figure 1.5 Grain density (ρgr) vs. magnetic susceptibility (log χ) for various classes

of meteorites...... 30

Figure 1.6 Meteorite classiﬁcation after Krot et al. (2014). The colours indicate

which factor is taken into account to split the meteorites among the coloured types.

xii The classiﬁcation into diﬀerentiated and primitive is not very clear from the original text...... 34

Figure 1.7 Classiﬁcation of meteorites after Weisberg et al. (2006). The meteorite clans are not included. Colours indicate the classical division of meteorites into Stony,

Stony-Iron and Iron...... 35

Figure 2.1 All data points that are provided in Tab. B.1 for individual meteorite types obtained from quasi-static tests. The compressive and tensile data points include 35 and 18 distinct meteorites respectively...... 63

Figure 2.2 All data points for dynamic measurements that are provided in Tab. B.1.

Only data for L Chondrites are available. The compressive and tensile data points are from two and one distinct meteorites respectively...... 64

Figure 2.3 Compressive strength against sample size for all the data points with published test specimen size. The references in the legend are abbreviated as in

Tab. B.1 ...... 68

Figure 2.4 Tensile strength against sample size for all the data points with published test specimen size. Note that there is no data from (10−6 to 10−2) cm3 so the x axis is broken over this range of values. The references in the legend are abbreviated as in Tab. B.1 ...... 69

Figure 2.5 Relation between sample size and compressive and tensile strengths for meteorite Tsarev (L) as measured by Zotkin et al. (1987)...... 71

xiii Figure 2.6 Relation between sample size and compressive strength for meteorites

Allende and Tamdakht as measured by Cotto-Figueroa et al. (2016). The mean

values group together measurements of similar sizes and the x-error bar provides the

standard deviation from the mean size of the grouped samples. The three largest

Allende data points represent only a single measurement each (thus no error bars). 73

Figure 2.7 Statistical relation between bulk densities obtained from various sources

and compressive strengths of individual meteorite samples coloured by their group

association...... 76

Figure 2.8 Statistical relation between bulk densities obtained from various sources

and tensile strengths of individual meteorite samples coloured by their group associ-

ation...... 77

Figure 2.9 Statistical relation between porosities obtained from various sources and

compressive strengths of individual meteorite samples coloured by their group asso-

ciation...... 78

Figure 2.10 Statistical relation between porosities obtained from various sources and tensile strengths of individual meteorite samples coloured by their group association. 79

Figure 3.1 The spectrum of GCR for the ﬁrst 8 elements. Data have been obtained from the database of GCR observations (Maurin et al., 2013) and the plot summarizes all the available measurements from this source...... 97

Figure 3.2 The stopping power of asteroidal materials compared to Aluminium. 119

Figure 3.3 The stopping power of asteroidal minerals compared to Aluminium. 121

xiv Figure 4.1 Dehydration of Mg Serpentines as proposed by Ball and Taylor (1963). 136

Figure 4.2 Dehydration of Mg Serpentines as proposed by Brindley and Hayami

(1965)...... 139

Figure 4.3 Decomposition of Calcium Oxalate to verify the performance of Thermal

Gravimetric Analysis (TGA) measurement...... 156

Figure 4.4 Baseline drift of empty beams for SDT Q600 at Advanced Materials

Processing and Analysis Center (AMPAC), University of Central Florida (UCF). 157

Figure 4.5 Decreased performance of empty beams as compared to Fig. 4.4. ... 158

Figure 4.6 Illustration of the concept of onset point on Diﬀerential Scanning Calo-

rymetry (DSC) melting curve of Zinc...... 161

Figure 4.7 An Antigorite sample heated to 1000 ◦C (for clarity, only the range

(400–900) ◦C is displayed) is an example of unclear Diﬀerential Thermal Gravimetry

(DTG) peaks. The presented red DTG curve is calculated with 60 point smoothing

(and displayed with its signum is inverted). The low temperature DTG peak shape is due to noise in the data. The red point at 598 ◦C denotes the peak position if the DTG curve is calculated across 300 points (the two peaks disappear). The green curve is the second derivative of the TGA curve calculated across 120 points. The green point at 605 ◦C suggests the DTG peak position by the zero of the second derivative...... 163

Figure 4.8 DSC curves of Antigorite and Calcium Oxalate (CaOx)...... 166

xv Figure 4.9 The dehydration of Antigorite. The TGA curve depicting the mass

loss (the left y axis) is plotted as a function of temperature in blue solid line. Its

derivative with respect to time (DTG) is plotted in green dot-dot-dashed line (the

right y axis) at corresponding temperature points and is calculated as described in

Sec. 4.3.2.2 using 300 points for smoothing. Heat ﬂow (the DSC curve) is plotted in

red dashed line (the right y axis) as a function of temperature and is oriented as such that endothermal reactions result in valleys and exothermic in peaks. Note that the

DTG and DSC curves have been normalized to unity on a sub-range of the x axis to provide a better view on their characteristics...... 175

Figure 4.10 The dehydration of Lizardite. The TGA curve of the mass loss (the left y axis) is plotted as a function of temperature in blue solid line. Its derivative with respect to time (DTG) is plotted in green dot-dot-dashed line (the right y axis) at corresponding temperature points and is calculated as described in Sec. 4.3.2.2 using

180 points for smoothing. Heat ﬂow (the DSC curve) is plotted in red dashed line (the right y axis) as a function of temperature and is oriented as such that endothermal reactions result in valleys and exothermic in peaks. Note that the DTG and DSC curves have been normalized to unity on a sub-range of the x axis to provide a better view on their characteristics...... 177

Figure 4.11 The dehydration of Cronstedtite. The TGA curve of the mass loss (the left y axis) is plotted as a function of temperature in blue solid line. Its derivative with respect to time (DTG) is plotted in green dot-dot-dashed line (the right y axis) at corresponding temperature points and is calculated as described in Sec. 4.3.2.2

xvi using 120 points for smoothing. Heat ﬂow (the DSC curve) is plotted in red dashed

line (the right y axis) as a function of temperature and is oriented as such that

endothermal reactions result in valleys and exothermic in peaks. Note that the DTG

and DSC curves have been normalized to unity on a sub-range of the x axis to provide

a better view on their characteristics...... 179

Figure 4.12 Comparison of the dehydroxylation of Antigorite, Lizardite and Cron-

stedtite using the TGA curve depicting the mass loss (the left y axis) plotted as a

function of temperature in solid lines and its derivative with respect to time (DTG)

plotted in dot-dot-dashed lines (the right y axis) at corresponding temperature points.

The DTG curves were evaluated as described in the previous plots (see Fig. 4.9, 4.10

and 4.11)...... 181

Figure 4.13 X-Ray Diﬀraction (XRD) patterns of unsieved Antigorite. The curves

represent the samples heated to temperatures 128 ◦C, 650 ◦C, 840 ◦C and 1000 ◦C in ascending order. The curves are oﬀset. The y axis denotes intensity in arbitrary

units...... 183

Figure 4.14 XRD patterns of Cronstedtite sieved to grain size <53 µm. The curves

represent the samples heated to various temperatures. The curves are oﬀset from

each other and the temperature that each curve represents in typed under the curve

on the left side. The arrows indicate the peaks relevant to Cronstedtite...... 184

Figure 4.15 Fourier Transform Infrared spectroscopy (FTIR) spectra of unsieved

Antigorite heated to various temperatures...... 185

xvii Figure 4.16 FTIR reﬂectance spectra Cronstedtite sieved to grain size <53 µm heated to 120 ◦C to remove adsorbed water and another sample heated to 600 ◦C...... 186

Figure 4.17 Comparison of two heating rates on a sample of Antigorite sieved to grain size (53–106) µm. The top panel displays the TGA curves, the middle one the calculated DTG using 60 point smoothing (see Sec. 4.3.2.2). The bottom panel shows the DSC signal. Note that both, the DTG and DSC signals are normalized to unity on a sub-range of the displayed x axis for a better visibility of their features. .... 187

Figure 4.18 Comparison of the eﬀects of three distinct sample sizes of Antigorite sieved to (53–106) µm and heated at 30 ◦C min−1. The top panel displays the TGA curves, the middle one the calculated DTG using 60 point smoothing (see Sec. 4.3.2.2).

The bottom panel shows the DSC signals. Note that both, the DTG and DSC signals are normalized to unity on a sub-range of the displayed x axis for a better visibility of their features...... 189

Figure 4.19 The dehydration of Antigorite sieved to diﬀerent grain sizes heated from

◦ ◦ −1 −1 the room temperature until 1100 C at 20 C min under 100 ml min N2 ﬂow. The top panel displays the TGA curves, the middle one the calculated DTG using varying amounts of smoothing (120–240 points, see Sec. 4.3.2.2). The bottom panel shows the DSC signals. Note that both, the DTG and DSC signals are normalized to unity on a sub-range of the displayed x axis for a better visibility of their features. .... 191

Figure 4.20 The dehydration of Cronstedtite at diﬀerent grain sizes heated from the room temperature to 1000 ◦C at 30 ◦C min−1 and 20 ◦C min−1 at the Kennedy Space

xviii −1 Center (KSC) and the UCF respectively under 100 ml min N2 ﬂow. The left panel

depicts the experiment at the KSC, the right one at the UCF...... 193

Figure 4.21 Comparison of Antigorite in vacuum with other runs under 100 ml min−1

N2 ﬂow. The grain size of Antigorite run under vacuum was <53 µm, one of the others had the grain size range (53–106) µm and the other was unsieved. The heating rate under vacuum was 10 ◦C min−1, the other two were heated using 15 ◦C min−1. ... 195

Figure 5.1 Evolution of the maximum temperature experienced by an equatorial

surface element on a sphere with the spin axis perpendicular to the orbital plane as

a function of the radial distance and various thermal parameters. The x parameter

denotes the depth under the surface (0 is the surface) in m (that is the surface, 5 cm

and 10 cm). Lines connecting the points only serve to guide the eye...... 214

Figure 5.2 Relative time spent above the temperature necessary to de-hydroxylate

4 % of Antigorite and Cronstedtite (corresponding to 400 ◦C and 660 ◦C) as a function

of radial distance for various thermal parameters and depths given denoted as x

in m (x = 0 is the surface and the other are at 5 cm and 10 cm). The thermal

parameters corresponding to Γ as well as the spin period and other parameters are

given in Tab. 5.1. Orbits are Keplerian. Diﬀerent style of dashing corresponds

to diﬀerent depths, varying colours correspond to the diﬀerent thermal parameters.

Lines connecting the points are only to guide the eye...... 215

Figure 5.3 Relative time spent above temperature necessary to de-hydroxylate 5 %

of Antigorite and Cronstedtite (corresponding to 420 ◦C and 680 ◦C) as a function

xix of radial distance for various thermal parameters and depths given denoted as x

in m (x = 0 is the surface and the other are at 5 cm and 10 cm). The thermal

parameters corresponding to Γ as well as the spin period and other parameters are

given in Tab. 5.1. Orbits are Keplerian. Diﬀerent style of dashing corresponds

to diﬀerent depths, varying colours correspond to the diﬀerent thermal parameters.

Lines connecting the points are only to guide the eye...... 216

Figure 6.1 The kinetics experiment with a sample of unsieved Antigorite during

which the sample was left isothermally at various temperatures indicated in the plot

for 4 h before the temperature was raised by 50 ◦C. The red curve indicates mass loss

pertinent to the left y axis and the green dashed curve is the temperature increased with time in steps...... 232

Figure C.1 1-Dimensional approximation of an asteroid by thin slabs...... 257

Figure C.2 Comparison of insolation functions...... 265

xx LIST OF TABLES

Table 1.1 The table provides bulk densities (ρblk), grain densities (ρgr), porosities

and magnetic susceptibilities (as log χ) of various meteorite classes composed from

various sources (see the notes at the end of the table). The data from references

max were statistically processed to produce numbers in the format: avgmin ± σ, where

avg is the average value from the data, min and max are minimum and maximum

values respectively and σ is standard deviation from the average. Where there was only one value or all the values were measured the same, minimum and maximum values are not provided and the deviation is zero. The individual statistics were calculated only from falls where possible, if only data on ﬁnds was published by the authors, ﬁnds were used to provide statistics and each such case is noted in the table. Also note that in the case where there were measurements for multiple samples of a particular meteorite, I disregarded the fact that they were part of the same meteorite (that is they were included in the average etc. as any other meteorite).

In particular, some authors tend to use averages weighted by measurements errors in these cases, however, this is incorrect as the measurements on diﬀerent samples cannot be expected to provide the same “correct” value. The data is sorted by grain density within each reference...... 14

xxi Table 1.2 Qualitative overview of mineral compositions of major meteorite types.

The table list the most abundant minerals in the column major and the column minor lists less abundant and accessory minerals. The division between major and minor is author’s subjective selection but in general minor components are those with

<∼ 10% by weight for meteorites where only 3 mineral components are reported and

<∼ 3% for meteorites with large amount of components. Also, if the source for the data calls certain minerals as accessory or minor, I put them into the minor column regardless. Note that this is only qualitative overview, within groups of meteorite it is possible to ﬁnd samples that diﬀer in composition substantially, e.g. in the amount of phyllosilicates...... 19

Table 1.3 This table summarizes mineral abundances of carbonaceous chondrite taxonomic groups expressed as weight percentages. The numbers in the second row in parentheses are the number of samples in each data set. If there is more than one

max sample in a given asteroid group three numbers are given in the format medianmin where median, minimum and maximum are the appropriate percentage values from each set. These numbers are to illustrate the abundance ranges of individual minerals within each group. The data are based on published meteorite mineralogies by Bland et al. (2004); Howard et al. (2015)...... 22

Table 1.4 Spectral features of major meteorite minerals. The data in the column

“Abs.” denote either the range of the centre of the absorption feature or the approximate centre. The column “signiﬁcance” is the relative signiﬁcance of the absorption feature. In most cases it is my subjective decision based on visually comparing the

xxii absorption depth of the individual features and minor ones are always minor with

respect to the major ones. In some cases, minor feature is such that, although not

visible in the spectra by my eyes, the author mentions the feature in the text as

being present. If the author speciﬁcally designates a feature as minor or major, I

used the author’s point of view. The column “Cause” brieﬂy mentions which process

is responsible for the feature...... 26

Table 2.1 Overview of compressive strength (σc) of meteorite groups from available

max data in Tab. B.1. The values in the middle column are meanmin ± std. dev., where min and max are minima and maxima of each group and std. dev. is the standard deviation from the mean of each group. If some of those values are not available, they are omitted...... 66

Table 2.2 Overview of tensile strength (σt) of meteorite groups from available data

max in Tab. B.1. The values in the middle column are meanmin ± std. dev., where min and max are minima and maxima of each group and std. dev. is the standard deviation from the mean of each group in the case of multiple (> 2) measuremts, or the measurement error provided by the author in the case of a single measurement. If some of those values are not available, they are omitted...... 67

Table 3.1 Most abundant elements in per cents in SEP normalized to Hydrogen, data is based on Reames (1995)...... 96

Table 3.2 This table summarizes mineral abundances of carbonaceous chondrite taxonomic groups expressed as weight percentages. The numbers in the second row in

xxiii parentheses are the number of samples in each data set. If there is more than one

Table 3.3 Comparison of our implementation of Bethe’s approach with Groom et al.

(2001) where more correction terms are included and with the SRIM software package

(denoted “SRIM e”) which implements even more corrections mostly based on ﬁtting experimental data. The table shows stopping powers in units MeV cm2 g−1, i.e. per unit density. We also give the nuclear stopping power calculated by the SRIM package

(denoted “SRIM n”) to make sure that it is negligible compared to the electronic stopping power...... 112

Table 3.4 Comparison of results with two diﬀerent algorithms for δ and with and without the 13% rule. The units are: [S] = MeV cm2 g−1, [δ] = 1 and [I] = eV. .. 115

Table 3.5 The composition of taxonomic groups proxied by the Orgueil CI1 chondrite for CIs and the Murchison CM2 chondrite for CMs. The data are adopted from Bland et al. (2004). We denote Serpentine2 a mixture of Saponite and Serpenine and we assume that it is mostly Serpentine but with a diﬀerent density...... 118

xxiv Table 3.6 Stopping power per unit density of asteroidal material compared to Alu- minium. The stopping power is in units of MeV cm2 g−1. The numbers in parenthesis are percent diﬀerence from Aluminium...... 118

Table 3.7 Stopping power of asteroidal material compared to Aluminium. The stopping power is in units of MeV cm−1. The numbers in parenthesis are percent diﬀerence from Aluminium...... 119

Table 3.8 Stopping power of typical asteroidal minerals compared to Aluminium.

The stopping power is in units of MeV cm2 g−1. The numbers in parenthesis are diﬀerence in per cent from Aluminium...... 120

Table 4.1 This table summarizes phyllosilicate abundances of CC subgroups expressed as weight percentages from Tab. 3.2. The term Serpentine in the table header is used solely to denote Mg Serpentines. The numbers in the second column in the parentheses are the number of samples in each data set. If there is more than one

max sample in the group, three numbers are given in the format medianmin appropriate for each set. These numbers are to illustrate the abundance ranges of phyllosilicates in Carbonaceous Chondrites (CCs). The data are based on published meteorite mineralogies by Bland et al. (2004); Howard et al. (2015)...... 132

Table 4.2 Quantities characterising the dehydradration of Serpentine type minerals. 182

Table 4.3 Comparison of the eﬀects of two heating rates on characteristincs of dehydration of Antigorite. The runs were executed at KSC...... 188

xxv Table 4.4 Comparison of the eﬀects of an increasing sample size on the characteristics of dehydration of Antigorite. The runs were executed at KSC...... 190

Table 4.5 Comparison of the eﬀects of the various grain sizes on the characteristics of dehydration of Antigorite. The runs were executed at UCF. The columns denoted

“Avg” denote average of the values to the left...... 194

Table 5.1 Thermal parameters used for calculations. Units of Γ, α, ρ and c are in the International System of Units (SI) base system. P spin is in days and rmax are in astronomical units. The subscript at rmax denotes mineral, Antigorite (An) or

Cronstedtite(Cr), and the percentage denotes the that the value was determined at for the temperature at which the mineral loses the set amount of mass as determined in Ch. 4...... 213

Table B.1 Summary of the strength measurements from the literature. The columns denote: “From” the data source, “Met” meteorite name, “S” Shock stage, “W”

Weathering, “L” Length (typically along the compression axis), “W/D” is one cross- sectional side for cuboid samples (Width) or Diameter for cylindrical samples, “D” is the other side of a cuboid (Depth), “ρblk” bulk density, “ρgr” grain density, “P” Poros- ity, “Rt” compression or extension rate of the testing machine (or strain rate if units

−1 are s or any other quantity, with appropriate units, aﬀecting the strain rate), “σc” compressive strength, “σt” tensile strength, “Ed” and “Ev” are Young’s moduli measured Directly and from Velocities respectively, “νd” and “νv” are Poisson coeﬃcients measured Directly and from sound Velocities respectively, “vl” velocity of longitudi-

xxvi nal waves, “vt” velocity of transversal waves. The meteorite names followed by “*” denote data extracted from plots as described in Sec. 2.2.4. Letters a, b, c in italic following the meteorite name means measurements along 3 diﬀerent axes on the same sample. The data come from the following sources (abbreviated as suggested): Bald- win and Sheaﬀer (1971) (Ba1971), Buddhue (1942) (Bu1942), Cotto-Figueroa et al.

(2016) (Co2016), Furnish et al. (1995) (Fu1994), Grokhovsky et al. (2013) (Gr2013),

Hogan et al. (2015) (Ho2015), Jenniskens et al. (2009) (Je2009), Jenniskens et al.

(2012) (Je2012), Jenniskens et al. (2014) (Je2014), Kimberley et al. (2010) (Ki2010),

Kimberley and Ramesh (2011) (Ki2011), Kimberley et al. (2015) (Ki2015), Knox

(1970) (Kn1970), Medvedev (1974) (Me1974), Medvedev et al. (1985) (Me1985),

Miura et al. (2008) (Mi2008), Molesky et al. (2015) (Mo2015), Slyuta et al. (2007)

(Sl2007), Slyuta et al. (2008a) (Sl2008), Slyuta et al. (2008b) (Sl2008a) Slyuta et al.

(2009) (Sl2009), Slyuta (2010) (Sl2010), Tsuchiyama et al. (2008) (Ts2008), Voropaev et al. (2017b) (Vo2017), Yavnel (1963) (Ya1963), Zotkin et al. (1987) (Zo1987). . 246

xxvii LIST OF ACRONYMS

AMPAC Advanced Materials Processing and Analysis Center.

CCs Carbonaceous Chondrites.

DSC Diﬀerential Scanning Calorymetry.

DTA Diﬀerential Thermal Analysis.

DTG Diﬀerential Thermal Gravimetry.

EGA Evolved Gas Analysis.

FTIR Fourier Transform Infrared spectroscopy.

Georgia Tech Georgia Institute of Technology.

HCE Heat Conduction Equation.

ISRU In Situ Resource Utilisation.

KSC Kennedy Space Center.

MAS NMR Magic Angle Spinning Nuclear Magnetic Resonance.

xxviii MBAs Main Belt Asteroids.

NEOs Near Earth Objects.

PSA Particle Size Analysis.

QMS Quadrupole Mass Spectrometer.

SEM Secondary Electron Multiplier.

SI International System of Units.

TGA Thermal Gravimetric Analysis.

TPD Temperature Programmed Desorption.

UCF University of Central Florida.

XRD X-Ray Diﬀraction.

XRPD X-Ray Powder Diﬀraction.

YORP Yarkovsky–O’Keefe–Radzievskii–Paddack.

xxix CHAPTER 1 INTRODUCTION

Asteroids along with other minor bodies of our Solar System (comets, planetary satellites, Kuiper Belt objects) are the leftover material from the planet formation process.

Although many of them have undergone various changes (including space weathering, collisions and breakups), these processes have changed them negligibly compared to what happened to the material that ended up making planets. Therefore they provide valuable clues to the early history of the Solar System. Understanding the processes that led to the creation of these bodies, can unravel important information about the evolution of our Solar System, verify our theories and help better understand and predict the observed distant planetary systems as well as constrain the processes in the early stages of formation of solar systems.

Despite their important part in the evolution the Solar System, the knowledge about their physical properties is incomplete at best. Four major paths to understand the properties of asteroids have been pursued:

1. In situ robotic and,

2. sample return missions,

3. analyses of meteorites,

1 4. observations of asteroids,

While the ﬁrst two approches provide the most straightforward and reliable infor-

mation, the lack of reasonable amounts of returned asteroidal material and consequently

lack of statistical data prevents us from obtaining signiﬁcant understanding of asteroidal

properties from in situ robotic and sample return missions.

1.1 Robotic Visits to Asteroids

Robotic missions have the ability to provide further data that are not accessible from telescopic observations, these include better resolution imagery and spectroscopy, density measurements from gravitational perturbations during ﬂybys (e.g. Yeomans et al.

(2000)) and, in the case of landing missions, these can provide us with valuable data on surface regolith properties. Unfortunately, these missions, which have as their main target a sophisticated study of one or two asteroids are very rare and more importantly they only survey the surfaces of small bodies.

1.2 Sample Return Missions from Asteroids

Until now, we only had one single sample return from an asteroid thanks to

Hayabusa mission. There are two missions in progress, Hayabusa 2 and OSIRIS-REx.

2 Yet, neither of these missions will be able to properly investigate the interior of their

target asteroid.

1.3 Observations of Asteroids

Asteroid observation is another quite accessible class of measurements, especially

if we consider a simple positional observation which leads to orbit characterization. We

can then obtain data on asteroid distribution density in our Solar System. The obtained

set of asteroids is biased because it is much easier to “spot” a close, large asteroid which

reﬂects a large portion of incident light than a small, dark object further away. Further

details can be found in the subsection on Orbital Elements.

Besides a simple positional observation, we can also observe and record brightness

of asteroids (the total amount of light reﬂected from the asteroid surface). This pro-

vides information on their apparent magnitude which is partly a measure of their size. If

we record this quantity in time, we obtain a light curve. These time dependent bright-

ness data points provide important constraints on the spin period of the object. Some

additional discussion follows in the subsection on Spin Period.

If we further process the light from the asteroid, we can obtain another important

piece of information, its spectrum. The spectrum gives away some of the asteroid’s surface composition and not just elemental but also mineralogical (see the subsection on Spectra).

3 1.3.1 Asteroid Shapes

Further information on the asteroid shape can be extracted from the time varying apparent magnitude.

As the asteroid rotates, it scatters the light from the Sun incident on its surface.

The “amount of light” that is scattered depends on the shape of the asteroid, its surface properties and irregularities, the observer’s position, the asteroid’s rotation phase, the position of the Sun etc. These determine how much light is scattered into various directions at various time instants as a function of the incident light and its direction.

Therefore, if we are viewing the asteroid from the Earth, we can see a dimming and brightening point. If we formally introduce such a function f which returns the ﬂux of photons Φi from the asteroid into our apparatus during some time instant (ti, ti + ∆t),

~ ~ ~ ~ ~ then we can write: Φi = f(Σ, Π, Ξ, Υ, ti), where Σ contains all parameters related to the asteroid’s spin-orbital configuration, Π~ contains all parameters that describe the relative position of the asteroid, the Sun and the observer, Ξ~ contains all parameters describing the asteroid’s shape and Υ~ contains all parameters which describe the reflection of the incident light from the asteroid’s surface. If we know all the parameters and the function f, it is easy to determine the flux and thus solve the so called direct problem.

It turns out that with certain assumptions, we can also solve the inverse problem, i.e. determine the set of parameters Σ~ , Ξ~ , Υ~ (i.e. the spin-orbital conﬁguration, shape and information about the surface reﬂection) from the knowledge of Φi at some ti. In other words, provided certain assumptions, we can determine the size and shape of the

4 asteroid just from the knowledge of evolution of the apparent magnitude in time. More

details can be found in (Kaasalainen et al., 1992; Ďurech et al., 2010, 2015b,a).

Three important and complementary classiﬁcation schemes for asteroids result

from the notes above. We can classify asteroids based on their:

1. orbital elements,

2. spin period

3. spectra.

1.3.2 Orbital Elements

The roughest classiﬁcation is simply based on semi-major axis, this sequences the

solar system into Near-Earth space, Main belt space and trans Neptunian space and

consequently we have Near Earth, Main belt and trans Neptunian objects. We can add

two more groups, the Greeks and the Trojans (located near the L4 and L5 points of

Jupiter) and the Kentaurs located between the orbits of Jupiter and Neptune. Figs. 1.1

and 1.2 display the distribution of asteroids in the solar system in osculating elements

with spectral classiﬁcation.

Even more detailed look into the structure of the Main belt itself, especially if

we are looking in proper elements, reveals conglomerations of asteroids, the so called asteroidal families, the remnants after the collisions between small, yet large, proto-

5 bodies (see Fig. 1.3). The families thus provide another classiﬁcation of its own type (as we use orbital elements, spectral clues etc. to determine the family associations).

Figure 1.1: Distribution of asteroids up to 6 AU with determined classes from spectral data. Asteroids with undetermined class are plotted in grey and smaller dots. Data were taken from JPL Small Body Database.

1.3.3 Spin Period

The plot of spin period vs. a measure of asteroid size, see Fig. 1.4, reveals that there might be a lower limit on the spin period that is governed by material properties.

This has provided limits on internal structure of asteroids, e.g. Scheeres et al. (2015) and references therein. Based on this data and models, we may think of most asteroids as rubble piles, however, the exact nature of such rubble piles requires detailed under-

6 Figure 1.2: Distribution of asteroids beyond Jupiter. Data source: JPL Small Body Database standing of internal structure. Simple equilibrium between gravitational and centrifugal forces as suggested in Harris (1996) may provide too high rotation period limit if cohesion and friction are taken into account. Many approaches model the interior of asteroid with spheres and even though some of the models do include friction between the spheres, it is obvious that if the shape of the material grains is not spherical but, rather, has sharp edges, not only will the friction be different, there will also be a larger contact area among the particles; this also affects the cohesion forces. Modelling the internal structure of these bodies is very difficult as we have little constraints on their structure. Kohout et al. (2017) has shown that at least some meter sized stony asteroids, which end up as

7 Figure 1.3: Distribution of asteroids in proper semimajor axis (a), proper eccentricity (e) and proper inclination (sin(i)) shows asteroidal families. The families are not perfectly visible in the plot due to the amount of asteroids included and resolution. The data are numerically calculated proper elements from the AstDyS database. meteorites, are likely made of parts with varying strengths with weak zones and coherent parts which however diﬀer in size.

8 Figure 1.4: Distribution of asteroids spin vs size. Data source: JPL Small Body Database

1.3.4 Spectra

I summarize the diagnostic spectral features in visible and near-infrared ranges for major asteroidal minerals in Tab. 1.4. The data in Tab. 1.4 are usually derived under very specific conditions on the Earth. However, one must be very careful because the spectra of asteroids are distorted by several effects: grain size, space weathering, temperature, specifics of the observing geometry, presence of atmosphere for Earth based observations.

Classifying asteroids based on their spectra is done using either DeMeo et al.

(2009) or Tholen and Barucci (1989).Their comparison and many of the issues with spectral classiﬁcations can be found in DeMeo et al. (2015). The spectra provide the primary link between asteroids and meteorites.

9 1.4 Analyses of Meteorites

Meteorites1 are the most physically accessible remnant material from the Solar

System “building” phase, asteroids, or parts of asteroids that, during their orbital evolution, were caught by Earth gravitational ﬁeld, passed through the terrestrial atmosphere and impacted onto the surface.

1.4.1 Measurements on Meteorites

There are several basic measurements that can be done on meteorites which provide the cornerstone of our knowledge about asteroids. These include determination of density and porosity, mineralogy, elemental composition, strength and hardness properties, electric and magnetic properties, thermal properties and optical properties. In this section I will go over existing data about various the above mentioned physical properties of meteorites and discuss their relation to the properties of asteroids.

While the relevant experimental methods have been developed to a very high degree of accuracy, in the case of meteorites, the value of the data is limited by extreme heterogeneity of the samples and as such, the statistical methods of processing measurements have to deal with various issues, such as, what should be a correct measured value on such heterogenous object. In this case, the statistical measures should be treated as indicators of variability rather than in the typical sense of experimental physics.

1A closely related terms are meteoroids, fragments of Solar System bodies, which upon entering the Earth’s atmosphere heat up and emit light, an eﬀect called a meteor.

10 1.4.1.1 Density and Porosity

Density is one of the most basic descriptive macroscopic properties of matter that

is closely related to basic forces such as gravity and necessary in almost all theoretical

models (such as understanding heat conduction or in thermodynamic calculations during

collision). Density is also one of the few properties of asteroids that can be measured

reliably for certain asteroids from the ground. Data is typically measured as bulk and

grain densities.

The bulk density takes into account only the macroscopic shape of the sample and

uses Archimedes methods (immersing the sample into a container with glass beads) or

three dimensional scanning techniques to obtain the total volume of the sample.

The grain density is typically measured with gas pycnometer that makes use of

the fact that small molecules of an inert gas (Helium) can penetrate even the tiny cracks

that connect the interior of the sample with the surface. However, the question arises

about the cracks that are not connected to the surface.

Porosity provides information about miniature cracks within the sample and consequently provides clues on processes that the sample might have undergone in order for these cracks to be created. If one knows the grain density (ρgrain) and bulk density

(ρbulk), the porosity (P ) is deﬁned as P = 1 − ρbulk/ρgrain

For stony meteorites, a very good resource of measurements on many individual

meteorites is Britt and Consolmagno (2003). A large amount of measurements were

undertaken in thesis Macke (2010), a subset of which were also published as Macke

11 et al. (2010, 2011a,b). Further measurements can be found in Wilkison et al. (2003) and

Consolmagno and Britt (1998); Consolmagno et al. (1998, 2006, 2008a,b). I summarize the results from literature on density and porosity in Tab. 1.1 , for further results and discussions, the aforementioned literature provides good starting point.

One thing worth mentioning with the publicly available data is that authors report averages of density and the related values in an inconsistent manner. E.g. Wilkison et al.

(2003) obtains the averages for each meteorite class by weighted average where the weights are measurement errors on individual samples. This is obviously very incorrect as such an approach assumes that all the samples are identical which is not even correct for samples of a single meteorite, much less for a meteorite class, there is simply no single correct value of density or porosity to which all samples tend. Macke et al. (2011b) chose to also report weighted averages, however, he uses sample masses as the weights. While this is correct in the sense that larger samples tend to have smaller measurement errors and he takes care to only provide such weighted averages for samples of a meteorite that are as similar as possible (identical shock stage etc.), it is still not a correct procedure, as such a process does not avoid the problem of heterogeneity (large pieces of the original meteorite are not necessarily more representative of the average density of the original meteorite, they are just bigger, and probably are easier to ﬁnd than the same mass of smaller pieces which might not even have survived). A more appropriate way to report averages of meteorite groups are simple means along with standard deviations from those means. While such an approach assumes that all original measured values have been obtained with the same experimental procedure as well as with the same uncertainties,

12 it provides a notion of “range” or variability of those values which is more appropriate for inhomogeneous objects.

13 Table 1.1: The table provides bulk densities (ρblk), grain densities (ρgr), porosities and magnetic susceptibilities (as log χ) of various meteorite classes composed from various sources (see the notes at the end of the table). The data from references were statistically processed to produce numbers in max the format: avgmin ±σ, where avg is the average value from the data, min and max are minimum and maximum values respectively and σ is standard deviation from the average. Where there was only one value or all the values were measured the same, minimum and maximum values are not provided and the deviation is zero. The individual statistics were calculated only from falls where possible, if only data on finds was published by the authors, finds were used to provide statistics and each such case is noted in the table. Also note that in the case where there were measurements for multiple samples of a particular meteorite, I disregarded the fact that they were part of the same meteorite (that is they were included in the average etc. as any other meteorite). In particular, some authors tend to use averages weighted by measurements errors in these cases, however, this is incorrect as the measurements on different samples cannot be expected to provide the same “correct” value. The data is sorted by grain density within each reference.

−3 −3 g Source Class ρblk [g · cm ] ρgr [g · cm ] Porosity [%] log χ

a 2.43 2.71 13.70 W03 CM 2.392.34 ± 0.05 2.712.70 ± 0.00 11.8510.00 ± 1.85 HED 2.96 ± 0.00 Aubrite 3.15 ± 0.00 3.14 ± 0.00 −0.30 ± 0.00 3.77 3.87 14.40 L 3.343.01 ± 0.16 3.383.03 ± 0.20 1.67−11.90 ± 6.36 CV 2.98 ± 0.00 3.42 ± 0.00 12.80 ± 0.00 3.44 3.92 18.30 LL 3.293.17 ± 0.09 3.563.29 ± 0.21 7.52−2.50 ± 6.69 EL 3.64 ± 0.00 3.62 ± 0.00 −0.50 ± 0.00 3.62 4.57 27.10 H 3.393.04 ± 0.19 3.683.00 ± 0.38 6.93−7.70 ± 9.61 Pallasiteb 4.54 ± 0.00 Mesosiderite 4.58 ± 0.00 5.96 ± 0.00 23.10 ± 0.00

ae 3.40 3.62 13.12 4.61 C06 LL 3.283.15 ± 0.09 3.553.50 ± 0.04 8.043.74 ± 3.20 4.173.52 ± 0.32 3.70 3.69 10.85 5.18 L 3.423.24 ± 0.11 3.593.45 ± 0.05 4.74−5.37 ± 3.87 4.894.68 ± 0.13 3.74 4.15 20.22 5.53 H 3.533.00 ± 0.18 3.813.71 ± 0.09 7.05−0.51 ± 5.48 5.365.13 ± 0.10 C08a CI 1.60 ± 0.03 2.46 ± 0.04 35.00 CM 2.25 ± 0.08 2.90 ± 0.08 23.10 ± 4.70 Eucrite 2.90 ± 0.13 3.18 ± 0.05 9.30 ± 6.20 Howardite 2.97 ± 0.16 3.22 ± 0.16 Nakhlite 3.15 ± 0.07 3.29 ± 0.09 5.70 ± 3.50 CVo 2.79 ± 0.06 3.30 ± 0.15 21.80 ± 1.70 Shergottite 3.03 ± 0.28 3.39 ± 0.03 13.60 ± 9.50 CO 3.03 ± 0.19 3.41 ± 0.23 10.80 ± 9.10 CVrf 0.12 ± 0.25 3.45 ± 0.09 9.70 ± 4.90 Diogenite 3.23 ± 0.15 3.46 ± 0.12 2.50 ± 2.20 LL 3.22 ± 0.22 3.54 ± 0.13 8.20 ± 5.50 L 3.36 ± 0.16 3.56 ± 0.10 5.60 ± 4.70

14 −3 −3 g Source Class ρblk [g · cm ] ρgr [g · cm ] Porosity [%] log χ CK 2.85 ± 0.08 3.58 ± 0.09 21.80 ± 2.20 H 3.42 ± 0.18 3.72 ± 0.12 7.00 ± 4.90 Chassignite 3.40 ± 0.11 3.72 ± 0.04 6.80 ± 2.20

a 4.10 4.17 11.70 5.64 M10 EL 3.553.15 ± 0.22 3.653.45 ± 0.17 2.77−2.40 ± 3.95 5.475.33 ± 0.10 3.65 3.76 11.70 5.63 EH 3.543.25 ± 0.11 3.663.52 ± 0.06 3.29−0.10 ± 3.03 5.485.30 ± 0.09 a b 2.87 3.24 11.41 3.36 M11a Lunar 2.812.73 ± 0.06 2.882.64 ± 0.22 5.081.13 ± 4.52 3.102.62 ± 0.34 3.15 3.29 21.42 4.26 Aubrites 2.892.53 ± 0.19 3.193.14 ± 0.04 9.131.74 ± 5.28 3.522.93 ± 0.34 3.31 3.51 19.59 4.43 HED 2.902.61 ± 0.15 3.252.99 ± 0.11 10.69−2.50 ± 4.77 3.052.56 ± 0.40 3.48 3.73 13.51 3.17 SNC 3.082.83 ± 0.22 3.363.25 ± 0.17 8.252.96 ± 4.08 2.912.79 ± 0.13 b 3.24 3.48 6.79 3.15 Angrites 3.213.18 ± 0.03 3.433.37 ± 0.05 6.235.66 ± 0.57 2.962.77 ± 0.19 3.36 3.47 9.40 5.01 Ureilites 3.223.14 ± 0.10 3.433.40 ± 0.03 6.141.37 ± 3.45 4.994.97 ± 0.02 Primitive 3.34 ± 0.00 3.45 ± 0.00 3.21 ± 0.00 4.50 ± 0.00 Achondriteb Enstatite 3.38 ± 0.00 3.51 ± 0.00 3.83 ± 0.00 5.37 ± 0.00 Achondriteb b 3.90 3.65 15.11 4.29 Brachinites 3.483.10 ± 0.24 3.553.47 ± 0.05 1.52−10.69 ± 8.07 4.063.79 ± 0.18 3.59 3.67 8.10 5.18 Acapulcoites 3.463.33 ± 0.13 3.653.63 ± 0.02 5.072.03 ± 3.04 5.115.04 ± 0.07 Winonaites 3.29 ± 0.00 3.79 ± 0.00 13.01 ± 0.00 5.55 ± 0.00 3.82 4.16 9.14 5.68 Lodranites 3.753.68 ± 0.07 4.114.05 ± 0.06 8.608.05 ± 0.55 5.665.63 ± 0.02

15 −3 −3 g Source Class ρblk [g · cm ] ρgr [g · cm ] Porosity [%] log χ

a c 4.86 M11b CI 1.57 ± 0.00 2.42 ± 0.00 34.90 ± 0.00 4.494.11 ± 0.31 2.43 3.05 36.70 4.77 CM 2.271.88 ± 0.12 2.912.74 ± 0.08 21.9413.70 ± 4.55 3.783.30 ± 0.30 3.19 3.54 25.00 5.17 CR 2.902.29 ± 0.32 3.323.06 ± 0.15 12.743.70 ± 7.77 5.074.89 ± 0.09 2.97 3.65 23.40 4.72 CK 2.852.76 ± 0.06 3.603.51 ± 0.03 21.0716.40 ± 1.91 4.674.59 ± 0.04 d 3.48 3.78 41.30 4.91 CO 2.962.18 ± 0.30 3.633.30 ± 0.10 18.602.60 ± 9.40 4.524.23 ± 0.17 30.00 3.86 27.90 4.49 CV 3.352.59 ± 3.44 3.633.38 ± 0.08 20.05−1.10 ± 5.45 3.783.53 ± 0.28 b 3.84 3.66 −2.30 5.39 CH 3.783.74 ± 0.04 3.653.65 ± 0.00 −3.50−5.10 ± 1.18 5.305.22 ± 0.07 2.80 23.90 4.72 C 2.802.79 ± 0.00 3.66 ± 0.00 23.7523.60 ± 0.15 4.694.67 ± 0.02 b 5.55 5.66 10.80 5.79 CB 5.224.90 ± 0.23 5.655.63 ± 0.02 6.102.00 ± 3.13 5.575.31 ± 0.15 a The references are as follows: W03 stands for Wilkison et al. (2003), C06 for Consolmagno et al. (2006), C08 for Consolmagno et al. (2008a), M10 for Macke et al. (2010), M11a for Macke et al. (2011a) and M11b for Macke et al. (2011b). b The data only includes ﬁnds, no falls were available. c Data come from Consolmagno et al. (1998), magnetic susceptibility from Rochette et al. (2008). d One value for Kainsatz meteorite was ignored in calculation of magnetic susceptibility statistic as log χ was 50, about an order of magnitude than the other values for other samples of the same meteorite. e Only ordinary chondrites are presented from the reference as other meteorites that were measure by Consolmagno et al. (2006) are also included in the work of Macke et al. (2011a). f The average bulk density of 0.12 is published in Consolmagno et al. (2008a). Unfortunately, no individual data were published in the work. g Negative porosity happens due to measurement errors. If the measurement errors are taken in to account, positive value of porosity is then always within the error. This makes statistical averaging of porosity a little dubious but nonetheless statistically valid.

16 1.4.1.2 Bulk Composition

Bulk elemental composition is one of the main criteria for meteorite classiﬁcation.

Elemental composition was in depth reviewed by Wasson and Kallemeyn (1988).

1.4.1.3 Mineralogical Analyses

Mineralogical composition is usually studied using X-Ray Diﬀraction methods, e.g. Bland et al. (2004); Alexander et al. (2013); Howard et al. (2015) have provided a relatively large number of measurements on carbonaceous Chondrites. A simple qualitative overview of mineral composition of various meteorite groups is presented in Tab. 1.2.

The table shows the most significant minerals contained in meteorites in each taxonomic group. It is important to keep in mind that the table is only qualitative and it is possible to find members within each group which have either different proportions of individual minerals or even members that are missing some of the major minerals. This is due to the fact that meteorite groups are not strictly organized by mineral composition but rather by bulk elemental composition and also other factors. However, it is in particular the mineral composition that provides the link to asteroids through spectral absorption lines.

Upon closer inspection of Tab. 1.2, we can very roughly distinguish that meteorites are either “dry” silicate oxides or hydrated (“wet”) silicate oxides which vary in metal content signiﬁcantly or they are metal dominated. Metal content is very important because the absorption bands due to electronic transitions in Fe are probably the most

17 important bridge between the asteroids and meteorites as these bands occur in visible and near infra-red parts of the reﬂected spectrum and thus are observable on asteroids.

Further observable bands are those due to phyllosilicates (in particular due to OH). Yet again, the standard classiﬁcation of meteorites mostly ignores the mineralogy (other than through correspondence with bulk elemental content).

Tab. 1.2 also suggests that the only “wet” group are carbonaceous Chondrites.

A more detailed view of the composition of carbonaceous Chondrites can be found in

Tab. 1.4.1.3. The table has been composed from mineral abundances determined by X-

Ray Diﬀraction (XRD) by Bland et al. (2004); Howard et al. (2015) and it also summarizes the total phyllosilicate content ranges of each meteorite group. The total phyllosilicate content can vary signiﬁcantly within each taxonomic group as evidenced by CM2 data

(from ∼ 56% to ∼ 85% by mass) or even more by CR2 data (from ∼ 7% to ∼ 60%). The phyllosilicate variations within other groups cannot be judged because of limited data on their mineralogy.

18 Table 1.2: Qualitative overview of mineral compositions of major meteorite types. The table list the most abundant minerals in the column major and the column minor lists less abundant and accessory minerals. The division between major and minor is author’s subjective selection but in general minor components are those with <∼ 10% by weight for meteorites where only 3 mineral components are reported and <∼ 3% for meteorites with large amount of components. Also, if the source for the data calls certain minerals as accessory or minor, I put them into the minor column regardless. Note that this is only qualitative overview, within groups of meteorite it is possible to ﬁnd samples that diﬀer in composition substantially, e.g. in the amount of phyllosilicates.

Type Major Minor Primitive Acapulcoites olivine, orthopyroxene, plagioclase, achondrites Fe-Ni metal, Troilite Lodranites olivine, orthopyroxene plagioclase, Fe-Ni metal Brachinites olivine clinopyroxene, orthopyroxene, plagioclase, metal Winonaites pyroxene Mg-rich olivine, troilite, De-Ni metal Ureilites olivine pyroxene (pigeonite), graphite, diamond, Fe-Ni metal, troilite HED Eucrites pyroxene (pigeonite), plagioclase silica, chromite, troilite, Fe-Ni metal achondrites (anorthite) Diogenites orthopyroxene (Mg rich) olivine, plagioclase Howardites mixture of eucritic and diogentic pieces Other Angrites pyroxene (fassaite), plagioclase olivine, kirschsteinite achondrites (anorthite) Aubrites orthopyroxene (enstatite) olivine, Fe-Ni metal, troilite Lunar Anorthositic plagioclase (anorthite) pyroxene, olivine achondrites Highland Rocks Mare basalts olivine, augite, plagioclase, pyroxene chromite, ilmenite, apatite, troilite, minor Fe-Ni metal

19 Type Major Minor Mare gabbro plagioclase, pyroxene Fe-Ti oxides, sulfides Lunar Norites olivine, pyroxene, plagioclase, feldspar chromite, ilmenite, troilite, Fe-Ni (alkali) metal Martian Basaltic clinopyroxene (pigeonite), augite plagioclase (maskelynite), olivine, Meteorites shergottites orthopyroxene, oxides, sulfides, phosphates Lherzolitic olivine, chromite, orthopyroxene plagioclase (maskelynite), shergottites clinopyroxene, oxides, phosphates Nakhlites clinopyroxene (augite), plagioclase, olivine, amphibole, clay minerals feldspar (alkali), pyrone, Fe-Ti oxides, (iddingsite, smectite), carbonate and sulfides, phosphates sulfate salts Chassignites olivine (Fe), clinopyroxene, chromite, sulfate salts plagioclasse Orthopyroxenites orthopyroxene (Mg) plagioclase (maskelynite), chromite, carbonate Stony-Iron Main Pallasites Olivine (Mg), Fe-Ni matrix troilite, schreibersite, chromite Eagle Station Olivine (Fe), Fe-Ni matrix troilite, schreibersite, chromite Pallasites Pyroxene pyroxene, Fe-Ni matrix clinopyroxene, olivine Pallasites Ungrouped olivine, Fe-Ni matrix Pallasites Mesosiderites Fe-Ni metal, pyroxene, plagioclase Iron Fe-Ni metal troilite, graphite, schreibersite, cohenite Ordinary H Fe and Ni metal, Olivine, Fe-Ni oxides chondrites Orthopyroxene (bronzite),

20 Type Major Minor L Fe and Ni metal, Olivine, Orthopyroxene (hypersthene) LL Olivine, Fe metal Enstatite Enstatite Fe metal chondrites Carbonaceous CI Serpentine, Saponite, Magnetite Olivine, Troilite, Pyrrhotite, amino chondrites acids, PAHs CM Serpentine, Constedtite, Olivine Pyroxene, Pyrrhotite, Calcite, Magnetite, organics CV Olivine, Pentlandite, CAI (Ca, Al, Magnetite, Fe metal Ti), Enstatite CO CK Olivine, Pyroxene, Magnetite CR Olivine, Pyroxene, Serpentine, Cronstedtite, Fe metal Magnetite, Fe sulphide CH Fe-Ni metal Phyllosilicates, CAI CB Fe-Ni metal, silicates C ungrouped

21 Table 1.3: This table summarizes mineral abundances of carbonaceous chondrite taxonomic groups expressed as weight percentages. The numbers in the second row in parentheses are the number of samples in each data set. If there is more than one sample in a given asteroid group three numbers are given max in the format medianmin where median, minimum and maximum are the appropriate percentage values from each set. These numbers are to illustrate the abundance ranges of individual minerals within each group. The data are based on published meteorite mineralogies by Bland et al. (2004); Howard et al. (2015). C2-ung CI1 CM1 CM2 CM2/1 CR1 CR2 CR3 CV3

(4) (1) (2) (23) (1) (1) (7) (2) (1)

14.8 7.7 23.1 43.7 36.9 Olivine 13.27.9 7.2 7.06.3 12.33.3 8.7 5.2 34.012.2 36.836.7 81.6 2.9 23.2 36.8 35.9 Pyroxene 1.80.0 -- 5.60.0 0.7 2.2 31.68.4 33.030.1 - 1.7 1.9 4.2 Calcite 0.40.0 - 1.81.7 1.10.0 1.2 5.7 --- 9.3 Gypsum - - - 0.00.0 ----- 11.6 2.7 5.4 7.1 5.3 Sulﬁde 3.70.0 - 2.01.2 1.80.0 1.0 9.3 4.73.2 4.43.5 - 1 0.4 7.1 5.6 Metals --- 0.00.0 -- 3.60.2 4.94.2 0.2 11.4 26.3 58.5 Cronstedtite 0.00.0 - 24.021.6 26.80.0 24.2 9.5 --- 2 74.5 66.0 82.8 60.0 1.5 Serpentine 66.955.1 71.5 63.661.2 42.822.2 62.4 57.9 15.37.1 1.41.3 - 3 8.3 1.8 8.4 10.9 6.2 Magnetite 5.94.5 9.7 1.71.6 1.90.3 1.7 10.0 6.22.0 5.13.9 0.3 Plagioclase ------0.9 0.6 0.5 Pentlandite 0.00.0 -- 0.00.0 ---- 11.1 2.2 Enstatite - - - 0.00.0 ---- 5.9 8.5 2.9 Pyrrhotite 0.00.0 4.5 - 0.00.0 ----- 4 14.4 Carbonate 0.00.0 ------Troilite - 2.1 ------Ferrihydrite - 5.0 ------Ilmenite ------18.5 24.0 Agglutinates ------10.80.0 19.114.1 - 74.5 87.6 84.5 60.0 1.5 Phyllosilicates 70.060.3 71.5 87.587.5 74.156.3 86.6 67.4 15.37.1 1.41.3 -

1 Metals include Fe and Ni. 2 Serpentine including Serpentine-Saponite mixture. 3 Magnetite including Magnetite mixed with rust. 4 Fe-Mg Carbonate.

22 1.4.1.4 Optical Properties

Optical properties of meteorites are almost uniquely covered by analyses of their absorption spectra. Spectra of meteorites and constituent minerals have been analysed in depth by many authors, the important overviews are Gaﬀey (1976), Pieters and McFad- den (1994), speciﬁcally in detail, Olivine have been studied by Burns (1970, 1974), pyrox- enes by Cloutis (2002), iron meteorites by Cloutis et al. (2010), carbonaceous Chondrites in a series of 8 papers by Cloutis et al. (2011a,b, 2012a,b,f,e,c,d), CM and CI Chondrites also by Trigo-Rodríguez et al. (2013). Spectrum of meteorites is very important since, as mentioned above, it provides an important link between asteroids and meteorites.

However, due to the limitation of the observable spectral range of asteroids, the diagnostic features that can provide a connection between a given asteroid and a meteorite in hand are not unique or are diﬃcult to resolve and cover only a limited spectral range.

Most of the absorption features in visible and near infra-red for typical minerals found in meteorites are summarized in Tab. 1.4.

The two basic diagnostic features of mineral composition in the visible and near infra-red reflectance spectra are the absorption bands (their position, depth and shape) and the slope of the spectrum. In the visible and near infrared spectroscopy of silicate minerals (which meteorites mostly are), two effects play the important role: crystal field transitions and charge transfer transitions Gaffey (1976).

The crystal ﬁeld transition only occurs in metals with their 3d electron shell partly

ﬁlled (and Copper), that is Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and is due to directionality

23 of various d-orbitals, some may be oriented toward the surrounding anions (ligands) and thus closer to them than others and consequently experience stronger repulsion.

This leads to splitting of d-orbitals in energy (orbitals closer to surrounding anions have higher energy). But then, upon an absorption of a photon, an electron can jump between the split orbitals. The energy diﬀerence between the split levels depends on the cations, the general geometry of the structure and symmetry and thus on the particularities that usually deﬁne a mineral. For example, if we replace a cation by a “larger” one, the interaction of d-orbitals that face the ligands directly increases (more than with the other orbitals), leading to larger energy split which in turn shifts the absorption to higher frequency and thus to lower wavelength and vice versa (e.g. replacing Fe2+ by Mg2+ makes the interaction smaller and thus shifts the absorption to higher wavelengths). The oxidation state plays an important role as, typically, a higher oxidation state results in bigger interaction between the negative ligands and the more positive metal which leads to increasing the energy gap between the d-orbitals.

The charge transfer absorption occurs when a photon is absorbed. Then an electron can be transferred to a neighbouring anion or another cation (however, this in only allowed between e.g. Fe2+ and Fe3+ or Fe2+ and Ti4+, that is metals which can exist in multiple oxidation states and are in a correct spin electron conﬁguration). This typically involves larger energies (that is, it produces a stronger UV absorption which weakens towards the visible region, creating what is called a red slope — less absorption toward red end of spectrum compared to blue end).

24 Further diagnostic features are typically related to the spring like nature of molecules and their degrees of freedom. These are mostly applicable in visible and near infrared only to water molecules and CO type structures.

To conclude, the major diagnostic features in silicate minerals in the visible and near infrared reflectance spectra are due to metals, importantly due to Fe2+ and Fe3+ and due to water molecules (H2O or OH). Their presence in various minerals results in different absorption bands and Iron also affects the slope of the spectrum. To complicate matters significantly, there are many other, composition unrelated, effects on the position of the bands or slope, such as: grain size, temperature, vacuum (or lack of thereof), weathering etc.

25 Table 1.4: Spectral features of major meteorite minerals. The data in the column “Abs.” denote either the range of the centre of the absorption feature or the approximate centre. The column “significance” is the relative significance of the absorption feature. In most cases it is my subjective decision based on visually comparing the absorption depth of the individual features and minor ones are always minor with respect to the major ones. In some cases, minor feature is such that, although not visible in the spectra by my eyes, the author mentions the feature in the text as being present. If the author specifically designates a feature as minor or major, I used the author’s point of view. The column “Cause” briefly mentions which process is responsible for the feature.

Mineral Abs. [µm] Significance Cause CAI5 ∼ 0.55 minor Fe2+ ∼ 2.1 major broad Fe2+ Carbonates6 ∼ 2.3 major C-O stretching ≥ 2.5 minor C-O ∼ 1 major Fe2+ crystal field transition ∼ 1.3 minor Fe2+ crystal field transition Feldspar2 ∼ 1.1 − 1.3 major electronic transition in Fe2+ 6 Ferrihydrite . 0.5 major Fe-O charge transfer ∼ 0.85 − 0.90 major Fe3+ spin-forbidden ∼ 1.4 major OH ∼ 1.9 major OH/H2O 6 Hexahydrite/epsomite ∼ 1.45 major H2O ∼ 1.95 major H2O ∼ 2.5 major H2O Magnetite6 ∼ 0.48 minor Fe3+ spin-forbidden + ∼ 1.0 − 1.3 minor Octahedral Fe2 Olivine5 ∼ 0.62 less major Fe2+ Olivine1,7 ∼ 0.85 major Fe2+ in M1, e− ∼ 1.05 major Fe2+ in M2, e− ∼ 1.25 major Fe2+ in M1, e− Pentlandite5 ∼ 1 minor Pyroxene (Cr ∼ 0.455 major Cr3+ in M1 octahedral, – HCP)4 crystal field transition ∼ 0.64 broad or minor Cr3+ in M1 octahedral, crystal field transition ∼ 0.66 broad or minor Cr3+ in M1 octahedral, crystal field transition

26 Mineral Abs. [µm] Significance Cause ∼ 0.69 broad or minor Cr3+ in M1 octahedral, crystal field transition ∼ 0.45 minor Cr3+ in M1 octahedral, crystal field transition ∼ 0.6 minor Cr3+ in M1 octahedral, crystal field transition ∼ 0.67 minor Cr3+ in M1 octahedral, crystal field transition Pyroxene ∼ 0.46 minor Fe2+-Ti4+ charge transfer (Ti)4 ∼ 0.75 minor Fe2+-Fe3+ charge transfer ∼ 1 major Fe2+ ∼ 2.3 major Fe2+ Pyroxene ∼ 0.445 major V3+ in M1, crystal field (V)4 transition ∼ 0.68 major V3+ in M1, crystal field transition ∼ 1 minor Fe ∼ 2 minor Fe Pyroxene ∼ 0.345 minor Mn3+ in M1, crystal field (Mn)4 transition ∼ 0.36 minor Mn3+ in M1, crystal field transition ∼ 0.41 major Mn3+ in M1, crystal field transition ∼ 0.42 minor Mn3+ in M1, crystal field transition ∼ 0.44 minor Mn3+ in M1, crystal field transition ∼ 0.505 minor Fe2+, spin-forbidden ∼ 0.54 major Mn3+ in M1, crystal field transition ∼ 1.5 major Mn3+ in M1, crystal field transition Pyroxene ∼ 0.425 minor Fe2+ in M2, spin-forbidden (Fe)4 ∼ 0.445 minor Fe2+ in M2, spin-forbidden ∼ 0.48 minor Fe2+ in M2, spin-forbidden ∼ 0.505 minor Fe2+ in M2, spin-forbidden ∼ 0.545 minor Fe2+ in M2, spin-forbidden

27 Mineral Abs. [µm] Significance Cause Pyroxene (Fe ∼ 0.69 minor Fe2+-Fe3+ charge transfer – LCP)4 Pyroxene (Fe ∼ 0.75 − 0.80 major Fe2+-Fe3+ charge transfer – HCP)4 Pyrrhotite5 − Saponite6 ∼ 0.65 − 0.68 minor Fe2+-Fe3+ charge transfer ∼ 0.90 − 0.94 major Octahedral Fe2+ ∼ 1.10 − 1.15 minor Octahedral Fe2+ ∼ 1.4 major OH ∼ 1.9 major OH/H2O ∼ 2.3 − 2.4 major Mg-OH Serpentine ∼ 0.65 − 0.66 minor Fe2+-Fe3+ charge transfer (Fe)6 ∼ 0.70 − 0.75 minor Fe2+-Fe3+ charge transfer ∼ 0.90 − 0.98 minor Fe2+ crystal field transitions ∼ 1.10 − 1.15 minor Fe2+ crystal field transitions ∼ 1.4 minor OH ∼ 1.9 major OH or H2O Serpentine ∼ 1.4 major OH (Mg)6 ∼ 2.3 major Mg-OH Tochilinite3 ∼ 1 minor probably Fe2+ ∼ 2.7 major OH ∼ 2.8 major OH 1 Burns (1970), 2 Adams and Goullaud (1978), 3 Moroz et al. (1997), 4 Cloutis (2002), 5 Cloutis et al. (2011b), 6 Cloutis et al. (2011a), 7 Isaacson et al. (2014)

28 1.4.1.5 Magnetic Properties

Magnetic susceptibility measurements characterize the state of magnetization of the parent body. Also, along with grain density, they provide a means to classify ordinary Chondrites into H, L, LL groups Consolmagno et al. (2006) and also to distinguish between EH and EL Chondrites Macke et al. (2010). Results of measurements found in literature (Consolmagno et al., 2006; Macke et al., 2010, 2011a,b) are summarized in

Tab. 1.1.

To illustrate some of the ideas, I plot the data on grain density vs. magnetic susceptibility in Fig. 1.5. The plot illustrates the idea of Consolmagno et al. (2006) that ordinary Chondrites can be distinguished in the grain density and magnetic susceptibility space.

For ordinary Chondrites, one can notice that Finds tend to lower grain density and lower susceptibility. Another quite obvious point is clustering by meteorite class not only for ordinary Chondrites. There are several exceptions to clustering, ﬁrst, CVs cluster at high density and medium susceptibility and there are several data points of CV ﬁnds towards the lower density and susceptibility as expected. But there is another cluster of

CV falls that is more consistent with LL Chondrites. Further there are several CV ﬁnds that are above the second CV cluster in susceptibility. The ﬁrst two clusters of falls and

finds correspond to CVos and the second couple to CVrs. There is also one CV find that is more consistent with HEDs. There are a few CM falls that are significantly shifted to higher susceptibilities. Surprisingly, CM finds tend to higher densities. HEDs display

29 no difference between Falls and Finds in density-susceptibility space. The same seems to hold for CKs as well. Finally, EHs and ELs would plot along with H Falls without any significant difference between falls and finds.

6.0 H Falls H Finds L Falls L Finds LL Falls 5.5 CK Falls CK Finds CM Falls CM Finds CV Falls 5.0 CV Finds HED Falls HED Finds

4.5 [1]

¡ 4.0 log

3.5

3.0

2.5

2.0 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 -3 ¢gr [g/cm ]

Figure 1.5: Grain density (ρgr) vs. magnetic susceptibility (log χ) for various classes of meteorites.

1.4.1.6 Thermal Properties

Thermal parameters are heat capacity and heat conductivity. These quantities are temperature dependent and it is necessary to measure them as a function of temperature relevant for asteroids. For this reason, we only summarize the relevant literature as

30 references rather than collect the data. Opeil et al. (2010, 2012) provides measurements of heat conductivities between approximately (5–300) K on several ordinary Chondrites,

2 HEDs, EH and EL, an iron meteorites and CM and CK samples. Macke et al. (2014a,b) measured heat capacity at 175 K for HEDs and ordinary Chondrites. Macke et al. (2016) provided data on heat capacities of ordinary Chondrites in the range (75–300) K. Opeil and Britt (2016) measured heat capacities and thermal conductivities of CM Chondrites between approximately (5–300) K.

Besides conductivity and capacity, heat expansion is an important eﬀect for asteroidal bodies as it can be responsible for grain size distribution and porosity. Recently,

Opeil and Britt (2016) measured the linear heat expansion coeﬃcient of some carbonaceous Chondrites as a function of temperature (between approximately (5–300) K). Also,

Medvedev et al. (1985) measured linear heat expansion coeﬃcient at 150 K and at 400 K for some ordinary Chondrites.

1.4.1.7 Strength Properties

For a detailed overview of strength properties which resulted in a publication, please see Ch. 2.

31 1.4.2 Classiﬁcation of Meteorites

Meteorites are classiﬁed into groups and subgroups based on various physical characteristics. A quick overview of the meteorite classiﬁcation can be found in DeMeo et al.

(2015). A very well and legibly written paper on the classiﬁcation is by Krot et al.

(2014). I redrew the classification after Krot et al. (2014) in Fig. 1.6. Sadly, even Krot et al. (2014) in his nice paper does not avoid getting into some confusion when it comes to classification of non-Chondritic meteorites, specifically, his text contradicts itself in several places on the classification into differentiated and undifferentiated meteorites. As this issue seems to be prevalent in literature, it might seem that differentiation and melting is not a good grouping characteristic. Another, a little different, classification is after

Weisberg et al. (2006) which is presented in Fig. 1.7. Comparing the two classifications, one can notice that while the classification of Chondritic material is well established, the major differences in classification are within non-Chondritic materials. Fig. 1.6 also contains a few notes on classification factors for a given group. From this, one can see, that the major classification factors are: bulk elemental composition, isotopic fractions

(mainly Oxygen), oxidation states and a “look under a microscope”. The presented figures omit completely further classification based on petrological type and shock. Despite the complex and exhaustive nature of current meteorite classification, they systematically omit one of the most important pieces of information — the required link to asteroids.

Meteorites are rarely classiﬁed by their mineralogy which is the cornerstone for providing

32 the link between asteroids and meteorites due to the available measurements on asteroids

(see later in the section).

33 Chondrites Carbonaceous

CI CM CO CR CB CH CV CK (Ivuna) (Mighei) (Ornans) (Renazzo) (Bencubbin) (Allan Hills) (Vigarano) (Karoonda)

CV-oxA CV-oxB CBa CBb CV-red (Allende) (Bali)

Ordinary Bulk composition and texture

refractory lithophile/Si abu., O isotopes, refractory H L LL inclusion abu., matrix/chondrules abundance

Enstatite Bulk Iron Content

Metallic Iron Content EH EL

Fe-Ni Metal Content Others

R K Degree of melting (Rumuruti) (Kakangari) Chemical composition (Ga, Ge content, Ir, Au and P content)

Non-Chondrites Achondrites Asteroidal Primitive

Acapodranites Winonaite Brachinite

Acapulcoite Lodranite

Differentiated

Angrite Aubrite Ureilite HED

Howardite Eucrite Diogenite

Planetary Lunar

Martian

Shergottites Nakhlites Chassignites Orthopyroxenite Stony-Iron Mesosiderite

Pallasites

Main Eagle station Pyroxene

Iron

IAB IC IIAB IIC IID IIE IIG IIIAB IIICD IIIE IIIF IVA IVB

Figure 1.6: Meteorite classification after Krot et al. (2014). The colours indicate which factor is taken into account to split the meteorites among the coloured types. The classification into differentiated and primitive is not very clear from the original text.

34 Undifferentiated Carbonaceous Chondrites

CI CM CO CR CB CH CV CK (Ivuna) (Mighei) (Ornans) (Renazzo) (Bencubbin) (Allan Hills) (Vigarano) (Karoonda)

CV-oxA CV-oxB CBa CBb CV-red (Allende) (Bali) Ordinary

H L LL

Enstatite

EH EL

Stony Others

Stony-Iron R K (Rumuruti) (Kakangari) Iron

Primitive Achondrites

Ureilite Brachinite Acapulcoite Lodranite Winonaite IAB IIICD

Differentiated Angrite Achondrites

Aubrite

HED

Howardite Eucrite Diogenite Mesosiderite

Pallasites

Main Eagle station Pyroxene

IC IIAB IIC IID IIE IIG IIIAB IIIE IIIF IVA IVB Lunar

Martian

Shergottites Nakhlites Chassignites Orthopyroxenite

Figure 1.7: Classiﬁcation of meteorites after Weisberg et al. (2006). The meteorite clans are not included. Colours indicate the classical division of meteorites into Stony, Stony– Iron and Iron.

35 CHAPTER 2 STRENGTHS OF ASTEROIDS AND METEORITES

2.1 Background

Strength is another physical property of meteorites that can be measured directly and related to asteroids. During the preparation of this work, I have found that not only the data are scarce but vast amount of them is not generally available as they were originally published in Russian language literature and never translated. Such data are only available through various reviews, usually, in some statistically processed form.

Such processing of data led in several cases to errors. One such source of error is due to the units in which the original authors reported the data and which varied over time and geographical place. Another source of issues were roundings and misprints. Besides the obvious errors in print, I encountered two other prevalent issues one is related to fundamentally diﬀerent measurement methods and the other relates to the “quality” of published data. One of the basic principles of experimental physics is that every

(good) experiment needs to be repeatable and this requires knowledge of experimental techniques as well as all relevant conditions that can aﬀect the outcome of the experiment.

Unfortunately, some authors in this area fail to mention even the basic details on their setup. Since several substantially diﬀerent methods are used to measure strength of

36 materials, not knowing the experimental details, makes it quite hard to understand the published numbers in context.

For the reasons mentioned in the previous paragraph, I had decided to collect all strength data that I was able to ﬁnd and publish them as they were originally provided by the authors with minimum processing (aside from unit standardisation to the SI) along with the experimental details related to those data.

There have been several notable reviews on this topic, speciﬁcally: Tsvetkov and

Skripnik (1991) in Russian, Svetsov et al. (1995); Popova et al. (2011), Flynn et al.

(2018) and Ostrowski and Bryson (2019). All the existing references are papers discussing strength of meteorites in some context but they are not systematic overviews of strength properties only and typically contain a subset of data presented in this chapter. Further, none of the aforementioned references goes into any detail on experimental methods.

Consequently, the main goal of this chapter is to provide a single data source with all, up to date, measurements of strengths of meteorites without any statistical processing, supplemented with descriptions of the methodology of individual authors so that the reader can asses the quality of the presented data. I also feel necessary to mention that all the information comes from the original publications and the references therein, rather than having been recycled from the existing reviews, and has been verified multiple times so that it provides error-free resource. In constructing the final table of all data, one of the intentions was ease of use for interested researchers. For that reason the table is also provided as a data file in a spreadsheet format in the relevant publication.

37 Since the major interest for studying meteoritic properties is to understand the

related properties of asteroids, it is necessary to understand the relationship between the

measured strength of a meteorite and its implication for the parent object. As indirectly

inferred in Sec. 1.4.2, the parent bodies of meteorites are not necessarily only asteroids

and as such, to be able to comment on the strength of asteroids, we need only be interested

in the strength properties of meteoritic groups that originate from asteroids.

Further, there are several models by which we understand asteroids as objects. We

can think of an them as of solid or monolithic bodies at one extreme or as conglomerates of

dust at the other extreme. Most of the asteroids are somewhere between these two states

and can be described as rubble piles. With suﬃcient pressure or with suﬃcient amount of

collisions one can become the other and vice versa. Intuition based on physical processes

during the evolution of the Solar System dictates that the smaller the body, the more

likely (or the more larger part of it) it is going to be a monolith and the larger the body,

the more likely it is a conglomerate of fragments. This idea immediately implies (without

any need of any measurement on meteorites) that there are necessarily several levels of

strength within such a body.

The from the lowest strength on the order of units of Pa governing the loose structure to the order of ∼ 100 MPa governing the monolithic structures. Such “strength structure” of an asteroid is purely due to the processes that the body has undergone.

Additionally, the individual monolithic structures are quite likely to differ significantly in their properties as it is quite feasible that they have undergone different history or

38 are made of diﬀerent materials. And even within each given monolith, the lack of its homogeneity can dictate extra strength levels.

As the result, the strength properties of asteroids are, in the crudest approxi- mations, going to be governed by gravitational, collisional and thermal histories of the objects which in turn determine their structural assortment as well as the homogeneity

(or lack of thereof) of the monolithic parts. As such, measurements on meteorites can only provide information strongest components of an asteroid since the sample had to survive the atmospheric passage as well as an impact on the ground. On the other hand, impact and subsequent Earth’s weathering can also introduce weak points in the material and consequently lower the strength of the sample. Finally the measurement method itself can also skew the result into either direction.

The aforementioned processes and ideas are inevitable when dealing with meteorites and it is important to keep those ideas in mind when trying to extend the measured values on small samples of meteorites (which represent only a very selective part of the parent object) to large asteroidal bodies. Unfortunately, almost none of the references for this work even mention the outlined issues, except for Yavnel (1963) who goes into detail on the eﬀects of atmospheric passage and impact on the ground onto the sample, his discussions are only pertinent to iron meteorites.

The text of this chapter is sectioned as follows. The ﬁrst section contains introductory remarks where I discuss various quantities that appear in the publications and methods of their measurement. It also contains discussion the eﬀects of strain rate and

39 sample size. Since some of the data were extracted from plots, the section also mentions the technique used in such case. Second section is devoted to comments on methodology and provided experimental details of individual references. Third section contains results of this work, mainly, the table with data points, tables with compressive and tensile strengths of various groups of meteorites and multiple plots showing the variability of strength in within individual meteoritic groups as well as relations between sample size and strength and density and porosity and strength. Finally, the last section reviews some of the results observable from the uniﬁed dataset and discusses besides others the relationship of the size and strength (so called scale eﬀect).

Ever since the ﬁrst measurements or ideas about the strength of meteorites there has been the unavoidable question how to relate those measurements to the small bodies of the Solar System. One aspect is related to scaling from small bodies to larger ones, however, the discussion of this important issue is not the topic of this paper. Second, it is obvious that measurements on meteorites give only a partial information on the strength of the parent objects, either asteroids or small meteoroids. In many ways these properties are aﬀected by the passage through Earth’s atmosphere and the impact onto the ground itself, not mentioning the preparation of the test samples and other issues that skew the data on strength of the original body. We found very few discussions of these problems in the literature. Some quality discussion can be found in Yavnel (1963) but it is primarily focused on iron meteorites, not so much on Chondrites.

Some materials in the following sections are reproduced from Pohl and Britt (2020) with the permission of Wiley Periodicals, Inc (see Sec. A.1) who is being acknowledged

40 thus. All plots and tables are reproduced in their unchanged form, as well as Sec. 2.3.

The rest of the material has been adjusted for this work both in lexicographic style as well as in focus on certain details.

2.2 Introduction

2.2.1 Deﬁnition of Quantities Related to the Measurements of Strength

Several physical quantities are used to talk about the state or behaviour of matter under applied forces. This intention of this subsection is to deﬁne those quantities and describe typical experimental methods that are in use to measure them.

2.2.1.1 Stress

For a body of material with an oriented surface speciﬁed by the surface normal at some point, stress at that point is the force per unit area that the side from which the normal points away acts on the part to which the normal points to. In the experiments considered in this work, stress is typically very simply measured as the applied force (read out from the testing machine) per area of the sample on which it acts.

41 2.2.1.2 Strain

In literature, one ﬁnds two principal measures of strain. The Cauchy (or engi-

neering) strain for which the changed quantities are related to their initial values, thus

for change of length, one has εc = ∆L/L0 where ∆L is the total change in length due to applied force and L0 is the initial length before deformation. The true strain can be best understood if one imagines the deformation taking place in discrete time steps and at each time step, one measures the Cauchy strain δLn/Ln where δLn is the length change at

the n-th step and Ln is the initial length before n-th step deformation. “Summing” these strains gives the true strain, thus one has for true strain ε = R L dL = ln L/L where t L0 L 0

L0 is the initial length and L is the ﬁnal length. And the relation between the strains is

εt = ln(εc + 1). Few papers directly state which measurement technique of strain is used.

2.2.1.3 Sound Wave Velocities

The longitudinal and transverse sound wave velocities (vl and vt) are measured from generated ultrasonic waves, e.g. as in Medvedev et al. (1985). They can then be related to both Young’s modulus and the Poisson’s coeﬃcient.

42 2.2.1.4 Young’s Modulus

The Young modulus (E) can be measured directly from compression testing by determining the linear part of the stress-strain curve using Hook’s law. It can also be

2 2 2 4−3vl /vt determined from the sound velocities in the material using the relation E = ρ vt 2 2 , 1−vl /vt where vl and vt are the longitudinal and transversal (or p and s) sound wave velocities and ρ is the density of the material.

2.2.1.5 Poisson Ratio

The Poisson ratio (ν) measurement is very similar to Young’s modulus, but the direct measurement is typically more diﬃcult as it requires measurements of displacement both in the direction along the axis of compression and in the lateral direction, Poisson ratio is then ν = εl , where ε is the strain in the lateral direction and ε is the strain εc l c along the axis of compression. From sound velocities, one can make use of the relation

2 2 2−vl /vt ν = 2 2 . It should be noted that both Young’s modulus and Poisson ratio measured 2(1−vl /vt ) directly and from sound velocities can diﬀer by over 10%. This was noted by Medvedev

(1974); Medvedev et al. (1985).

43 2.2.1.6 Compressive Strength

Strength of a material can be thought of as the ability of the material to resist applied forces. Such definition is not of much use though, since it does not provide any details on the types of forces and the meaning of the word resist. Throughout the literature on material properties, one finds that by resist, it is usually meant “until a complete failure of the material”. However, this is still more qualitative definition than quantitative and leaves room for each experimentalist to choose the point when the material fails. Generally, the following approaches to failure are found in the literature:

1. forces that act on the sample at the appearance of the ﬁrst visible crack, e.g. Yavnel

(1963),

2. forces that act on the sample at the time of complete destruction of the sample

(however this is still very subjective),

3. the maximum of the stress/strain plot.

The compressive strength (σc) is uniquely measured by applying force along a certain axis until a predefined state of the sample (mentioned above) is obtained. The force along the axis can be exerted on the sample either while perpendicular to this axis, the material is not confined — uniaxial compression, or the sample can be confined in the perpendicular directions as well — non-uniaxial (triaxial) compression. Almost all available data is on uniaxial compressive strength with the only exception being Voropaev et al. (2017b) who measured the triaxial compressive strength as well. Samples are

44 typically of cubic, cuboidal or cylindrical shape. Extremely small samples (∼ µm) in the form of grains can also be measured (Tsuchiyama et al., 2008).

2.2.1.7 Tensile Strength

To measure the tensile strength (σt), three major techniques are employed in the literature. The standard method is inverted compressive measurement where the sample is stretched along a particular axis (in this case, only uniaxial measurements are used).

Usually samples require more processing (and more processing carries the risk of a more biased measurement, i.e. only stronger samples survive processing to be measured) since, unlike for compressive measurement, there is a need for a grip on the sample and one needs to make sure that the rupture of the sample is not due to processing itself (sample should not fail at the point of the “grip”).

Another technique that is sometimes used is splitting the sample by a wedge. The basic idea is to transform compressive force to tensile force through a wedge (Grebjonkin and Gavrish, 2004). The Russian testing norm GOST21153.3-85 (1986) suggests that a

ﬂat plate 2 cm thick and at least 10 cm by 10 cm in cross-section be cut from a meteorite sample. A square mesh is then drawn onto the cross-sectional area with a square side of

2 cm. The sample is put into the compression machine between two right angled wedges parallel along a mesh line. The sample is broken in cuboidal beams while denoting force at which the sample is broken and the length of the breakage line. The beams are then broken into cubes which should have resulting side approximately 2 cm. These cubes can

45 then be used for compressive strength measurements. From the geometry, the tensile force

produced by the wedges can be calculated as σt = F/(2l h), F being the compressive force

at the time of fracture, l is the measured length of fracture and h is the thickness of the

sample. The detailed description of the process can be found in the norm GOST21153.3-

85 (1986). The importance of the technique is that it can provide valuable statistics on

both the tensile and compressive strengths of a given meteorite sample and it can also

provide data on anisotropy of strength properties (Slyuta et al., 2009).

Finally, a technique called Brazilian disk test is also used. The technique com-

presses a thin disk along its diameter which leads to development of tensile stresses in

the disk and thus the method can be used to indirectly measure tensile strength. The

parameters in this test are the diameter of the disk and the ratio of the disk thickness

to the diameter. GOST21153.3-85 (1986) suggests the diameter of the disk should be

between (3–6) cm and the ratio of thickness to diameter between 0.7 and 1.1. Many

issues exist with this type of test and are mostly related to the boundary conditions

between the sample and the compressing machine. Two options for boundary conditions

are suggested by GOST21153.3-85 (1986), either using steel plates or wedges. Based on

the boundary condition, the resulting tensile strength can be determined by a simple

F relation: σt = K d w , where F is the machine induced force at the time of breakage, d is the disk diameter, w its width and K depends on the boundary conditions, for example, . K = 2/π = 0.637 for steel plates (Andreev, 1995; GOST21153.3-85, 1986).

46 2.2.2 Conditions Aﬀecting the Experimental Results

2.2.2.1 Sample Size

Change in strength properties with the size of the sample is manifested as a scale eﬀect. The fact that strength related properties change with the size of the sample being tested have been known for some time in the mechanics of rocks and engineering literature. In the area of meteorites and asteroids, the scale eﬀect is often introduced via the argument of Weibull (1939), for example, Popova et al. (2011); Cotto-Figueroa et al.

(2016) who interpret the ideas of Weibull (1939) into the statement: “the strength of a body in nature tends to decrease as body size increases” (Popova et al., 2011). Despite an overwhelming simpliﬁcation of material behaviour under applied forces as well as extrapolation over many orders of magnitude that are contained in such statements, sample size (and possibly shape) do have an eﬀect on the obtained data in strength related experiments.

2.2.2.2 Temperature

The temperature under which the testing experiments are conducted does play a role to a certain level. While there seems to be data supporting that the sensitivity of the results to the environmental temperature is very low, the strength of a material does depend on the its thermal processing. For example for granite, Török and Török (2015);

47 Wang et al. (2019) showed decrease of both tensile and compressive strength with heating

of the sample in the range between room temperature and 800 ◦C, similarly for concretes,

Mundhada and Pofale (2015) show fall of compressive strength by about 50% within this range; Sygała et al. (2013) show decrease in compressive strength of limestone by almost

50% between room temperature and 100 ◦C and for Serpentinite the compressive strength

drops by a factor of ∼ 5 between room temperature and 500 ◦C.

2.2.2.3 Strain Rate

As inferred from the paragraphs above, the compressive and tensile experiments

are conducted by increase in the applied force on the sample until some deﬁned state

(failure) of the sample is attained. The velocity of such strength increase is projected

into time derivative of strain, typically one uses the Cauchy strain to determine the strain

rate: ε˙ ≡ dε(t) = d L(t)−L0 = L˙ and as a results of this, the physical dimension of this dt dt L0 L0

quantity is [ε ˙] = s−1.

Two strain rate regimes are prevalent in the literature. The quasi-static strain rates

−1 −2 −1 cover the range ε˙ . 1 s (most experiments are conducted with ε˙ . 10 s ). These

“slow” experiments can simulate the increase of stress during the passage of a meteor

through the Earth atmosphere. To simulate forces applied during the impact phase, the

−1 experiments with dynamic strain rates ε˙ & 10 s are necessary (most experiments are

2 −1 conducted with ε˙ & 10 s though).

48 To illustrate the diﬀerences between the two regimes, one can consider, for example, the meteorite MAC 88118 for which the compressive strength (∼ 200 MPa) in dynamic regime (∼ 103 s−1) is larger by about factor of 4 compared to quasi-static regime

(∼ 50 MPa at ε˙ ∼ 10−3 s−1) (Kimberley and Ramesh, 2011). Another study by Hogan et al. (2015) on the meteorite GRO 85209 determined that the dynamic regime compressive strength (∼ 300 MPa at ε˙ ∼ 103 s−1) is about 5 times larger that in the quasi-static case (∼ 60 MPa at ε˙ ∼ 10−3 s−1) and the factor was almost 10 for the tests of tensile strength in favour of the dynamic regime (∼ 9 MPa at ε˙ ∼ 10−3 s−1 and ∼ 90 MPa at

ε˙ ∼ 450 s−1).

2.2.3 System of Units and Signiﬁcant Figures

Throughout this work, I use only SI and SI derived units. Notably, for strength the unit of Pascal is used.

Significant figures are treated as follows. For the data that are published in SI units, the significant figures of the quantity and its uncertainty are preserved as in the paper with the assumption that all zeros preceding the decimal separator are significant

(in the case where the data is reported with incorrect amount of significant figures, such as (3.51 ± 0.0021), I still preserve what was reported by the authors). If the uncertainty is reported as a coefficient of variation, the absolute error was calculated and rounded to the same amount of significant figures as the reported quantity.

49 When other than SI units are used — either kilogram-force (kg m−2) or pound-

force (lb in−2) — the conversion to the SI system is carried out using 1 lb = 0.453 592 37 kg and 1 in = 25.4 mm and the quantity is output with the same amount of signiﬁcant ﬁgures as original one (with the same assumption on zeros as for the SI system).

2.2.4 Extracting Data from Plots

Several authors published data only as plots and provided only their summary in textual or tabular form. Since the individual data points from the plots are quite valuable,

I decided to user WebPlotDigitizer1 and Engauge Digitizer2 to obtain numerical values

from plots. In every case, I ﬁrst tested the process on some tabulated set. This showed

that the biggest source of error was caused by incorrect determination of the plotted

point centre. To quantify the error, I use half the linear size of the plotted data mark (in

the case of a circular marker, the radius, in the case of a square shaped marker, half its

side). Such a process however, leads to incorrect determination of error for logarithmic

scales where the such error need to be asymmetrical. However, since the size of the point

was suﬃciently small in all cases, such linearisation of logarithm did aﬀect the value

signiﬁcantly. All data determined in the described manner are denoted with an asterisk

after the meteorite name in the ﬁnal table. 1https://automeris.io/WebPlotDigitizer 2https://markummitchell.github.io/engauge-digitizer/

50 2.3 Data Sources

This section is devoted to discussing each of the used references in constructing the ﬁnal dataset. The resources are organized in the same order as the data in Tab. B.1.

The purpose of this part is to provide maximum information on the experimental design and relevant conditions. However, the original goal of this section proved quite diﬃcult as some authors fail to mention even the basic information on their setup while other have very detailed description of their methods. In addition to that, several authors reference norms to describe their procedures. The end product is then such that for generally available papers (contemporary journals in English language), I only mention the very basic information of the setup and conditions or merely state that the authors fail to mention them. In the case of the Russian language papers from either Meteoritika journal or publications by the Soviet Academy of Sciences, I purposely devote more space to the details provided in the papers, in cases, also citing non-strength related data (such as thermal measurements) which I feel might be of use to the general audience.

Baldwin and Sheaffer (1971) measured both tensile and compressive strengths on two samples of Seminole meteorite but provide no details on how the measurements were made or how they define strength. The authors provide the sample sizes as masses (we were unable to find any information on the density of Seminole meteorite): 0.432 g for the sample in tensile test and 4.39 g for the sample in the compressive test.

51 Buddhue (1942) made compression tests using an Olson universal testing machine with a speciﬁed compression rate of 1.27 mm min−1. The pressure was noted at the time of sample failure but the failure itself is not well deﬁned by the author. There is no information on sizes of the samples.

Cotto-Figueroa et al. (2016) did compression tests at room temperature with a well deﬁned displacement rate. Strain was measured using non-contact image system by observing position of dots which were sprayed on the sample. Further they also used extensometer to monitor the strain. The authors do not explicitly mention their deﬁnition of strength but from the text one can infer that they took the value of stress at the maximum on the stress-strain plot. They also determined the velocity of longitudinal and transverse elastic waves. These measurements were done for each side of the sample, but the authors do not explicitly state whether they observed any anisotropy (see e.g.

Slyuta et al. (2009); Voropaev et al. (2017a)). Further, authors calculated the Young’s modulus ratio from the longitudinal and transverse speeds of sound. They also provide the Young’s modulus values derived from the linear portion of the stress-strain relation obtained by the non-contact strain measurement method. We calculated the values of bulk density from the sample mass and dimensions provided in the paper.

Furnish et al. (1995) provide results of quasi static compressive measurements on

3 iron meteorites, however, no details about these quasi static measurements are given.

The strengths provided are yield strengths, that is the maximum attained by the linear

52 part of the stress strain curve before plastic deformation occurs. While the paper focuses on high-strain rates, no data are given for high strain rates related to impact collisions.

Grokhovsky et al. (2013) provide initial measurements of strength on the Chelyabinsk meteorite as well as additional data on the well-studied Tsarev meteorite. The compressive strength measurements were determined using INSTRON 8801 machine with a well deﬁned compression rate. The values of strength are deﬁned as “rupture loads”. Samples sizes are not provided. This conference abstract also provides measurements on thermal conductivity at various temperatures, heat capacity.

Hogan et al. (2015) measured uniaxial compressive and tensile strength both under quasi static conditions as well as for high strain rates. Uniaxial compressive strength was measured on cuboidal samples that were 5.3 mm in length and had a base of 4 mm ×

3.5 mm. The Brazilian tests used disks 10 mm in diameter and 1.5 mm thick. The quasi- static uniaxial measurements were performed using MTS servo-hydraulic machine, high strain rate experiments were performed using Kolsky bar (Kolsky, 1949). The tensile strength in Brazilian disk method is defined as the maximum stress in the stress-time plot. Although, not explicitly stated by the authors, one can infer from the data that the compressive strength is defined in the same way, that is the maximum in the stress-time plot. The Young’s modulus (linear fit to the stress-strain curve) measured during quasi- static tests is used in determination of the strain rate during the Brazilian disk test. The data are only tabulated for 2 values of strain rate, the rest is published in a plot. We

53 used the procedure described in Sec. 2.2.4 to extract the data and multiplied them by constants σ0 and ˙0 (since the original data is normalized) provided in Tab. 1 of their paper.

Jenniskens et al. (2009, 2012, 2014) provide information on tensile strength of

Almahata Sitta and Novato and compressive strength of Sutter’s Mill meteorites, however, no information beyond the numbers are provided (including how they determined the error and whether it reflects sample to sample variation or whether it is the error of the method itself). In the case of Jenniskens et al. (2012), the densities and porosities were calculated on a different sample than the one for which strength was measured, despite that, we decided to provide them. In the case of Jenniskens et al. (2014) we omit the published densities since none were provided for the sample with measured strength and the density sample-to-sample variation was significant.

Kimberley et al. (2010); Kimberley and Ramesh (2011) provide measurements on MacAlpine Hills 88118 with varying strain rates. The uniaxial quasi-static ((10−3–

10−1) s−1) measurements were done using MTS machine as described for Hogan et al.

(2015). The measurements for high strain ((102–103) s−1) rates were done using Kolsky bar. Both types of test were done on cubic samples with well deﬁned size. The Young’s modulus was determined from the linear portion of the stress strain plot. And the strength is deﬁned as the maximum of the stress in the stress-strain plot. The data for various

54 strain rates are not tabulated in the paper and only provided in a plot, we therefore used

the procedure described in Sec. 2.2.4 to extract the data.

Kimberley et al. (2015) provides a measurement of uniaxial compressive strength of

GRO 85209 determined as the maximum stress in the stress-time plot using a constant

stress rate 26 MPa µs−1 on an MTS machine such as used by Kimberley et al. (2010).

We assume that the sample or samples used are the same ones as in Hogan et al. (2015).

Information on the sample size is not provided.

Knox (1970) measured yield strength of Canyon Diablo, Brenham and Odessa (iron) meteorites. There is no information on the strain rate but, given the procedures used, one can assume a quasi static compression. The sizes of the samples are provided. The samples of Canyon Diablo were used in the previous investigation by the author (Knox,

1954) which included heating to various temperatures, the highest of which was ∼ 1200 ◦C.

Further, the author did several measurements on the same sample, therefore any plastic

deformation added up and their published data are averages over several identical ex-

periments on the same sample. The Canyon Diablo sample denoted as 1A was annealed

(heated to ∼ 870 ◦C) prior to testing. The author used an Instron machine with an “F”

compression cell, the strain was measured using linear variable diﬀerential transformer

and data were plotted on a chart. The authors publish the data as oﬀset yield strength

which is the value of the stress that is obtained from the intersection of the stress-strain

curve and a line parallel to the elastic part of the stress-strain curve which is oﬀset by

55 a certain amount towards larger strain. The authors provide data for several oﬀsets. In our data table, we use the data for 0.2% oﬀset. Further, the authors publish data from

2 diﬀerent charting systems, we only reproduce the Moseley chart data.

Medvedev (1974) provides measurements on sample number 1723 of the Kunashak meteorite and sample number 1831 of the Elenovka meteorite. The author measured uniaxial compression using semi-regular specimens (the size of the samples was not provided) and tensile strength using the wedge separation technique. He also measured longitudinal and transverse elastic waves using ultrasonic pulses. These sound velocities were then used to calculate Young’s modulus and Poisson’s ratio. We have to note that the values presented by the authors diﬀer from the ones obtained from the formula for

Young’s modulus and Poisson’s ratio using the presented p and s wave velocities and density that the authors measured. We were not able to determine the source of this discrepancy (but in Medvedev et al. (1985) the presented relation between the longitudinal velocity and Young’s modulus and Poisson’s coeﬃcient is incorrect, which might be the source of this discrepancy) and if one wants to use the data, recalculation is suggested.

Young’s modulus was also measured directly from deformation. For the sake of completeness and as this paper is not generally available, it is worth noting that the author also made thermal measurements on these two meteorite samples: thermal diﬀusivity, conductivity and heat capacity were determined to be 9.2 × 10−7 m2 s−1, 2.2 W m−1 K−1 and 0.68 kJ kg−1 K−1 respectively for the Kunashak meteorite and for Elenovka the authors provide value of heat capacity of 0.76 kJ kg−1 K−1 only, determined from one of

56 their references; thermal coeﬃcients of linear expansion at 150 K and 400 K were determined as 8.8 × 10−6 K−1 and 10.1 × 10−6 K−1 respectively for the Kunashak sample and

6.6 × 10−6 K−1 and 8.2 × 10−6 K−1 respectively for the Elenovka sample.

Medvedev et al. (1985) is an extension of the previous paper to additional meteorites. Authors measured longitudinal and transverse elastic sound waves using an ultrasonic generator. Tensile strength was measured using the wedge separation method on plates of material 10 mm wide which produced cubes which were most likely subsequently used for compressive measurements (as described in GOST21153.3-85 (1986)). Young’s modulus and Poisson ratio were calculated from the sound velocities (longitudinal and transverse) and from densities which were determined using hydrostatic weighing (and as in the previous paper Medvedev (1974), we were unable to reproduce the calculated results). The authors also discovered anisotropy in sound wave propagation through a sample of Tsarev meteorite which was later conﬁrmed and measured in terms of strength anisotropy by Slyuta et al. (2009). They also observed dependence of strength on the degree of terrestrial alteration (decrease in strength with increasing weathering). They note that depending on the level of terrestrial alteration, the 3 samples of Tsarev meteorite vary signiﬁcantly. The most altered sample was missing part of Fe-Ni component and had compressive and tensile strengths 157 MPa and 18 MPa respectively. The in- termediate weathered sample had the respective strengths 222 MPa and 26 MPa. The least weathered was the strongest sample of the meteorite and was determined to have the strengths 450 MPa and 55 MPa. Further, several measurements on the same sample

57 of the Tsarev meteorite were performed and signiﬁcant variations of 25% in compressive strength and of 21% in tensile strength were found within those measurements. Besides strength measurements, a full suite of thermal measurements was performed. For the meteorites: Krymka 1705, Tsarev 15380a, Tsarev 15384a, Tsarev 15384b, Tsarev 15391,

Kunashak 1723, Kyushu 2157, Putlusk 544, respectively, the values of thermal conductivity in W m−1 K−1 are: 2.32, 3.68, 3.71, 3.89, 2.76, 2.89, 2.3, 3.05, respectively; the values of thermal diﬀusivity in m2 s−1 are: 1.05, 1.13, 1.2, 1.14, 1.17, 1.04, 1.04, 1.03, respectively; and the values of speciﬁc heat capacity in kJ kg−1 K−1 are: 0.68, 0.92, 0.87,

0.99, 0.73, 0.78, 0.57, 0.83, respectively. It was also observed that, in the case of the

Tsarev meteorite, the impact metamorphism caused hardening of the matrix and a sig- niﬁcant decrease of heat conductivity. The linear expansion coeﬃcient was on average

(6–9) × 10−6 K−1 at 150 K and (8–10) × 10−6 K−1 at 400 K. The authors also performed a suite of 107 measurements of density on samples of Tsarev meteorite showing a bimodal distribution of density with the peaks at 3.3 and 3.5 g/cm3.

Miura et al. (2008) determined compressive strength of Murchison carbonaceous

Chondrite as well as the strength of L Chondrite La Criolla on cylindrical samples of well defined sizes which were compressed with the help of a metallic disk placed on each end at a well defined rate. The authors provide 2 measurements on each sample. The compressive strength is defined as the maximum stress before the failure of the sample, however, the definition of failure is not clear.

58 Molesky et al. (2015) measured density and compressive strength of North West

Africa 869, however, provided little detail on the experimental setup and samples.

Slyuta et al. (2007, 2008a,b, 2009); Slyuta (2010) provide measurements of strength on Tsarev, Sayh al Uhaymir and Ghubara meteorites. The measurements are according to the GOST21153.3-85 (1986) norm, that is, tensile strength is measured using the wedge splitting method and the resulting cubical samples are used for compressive tests. Density is determined using hydrostatic weighing. Young’s modulus was calculated from stress-strain relation and the compressive strength was taken the maximum in the stress-strain plot. Slyuta et al. (2007) measured Sayh al Uhaymir 001: density is average of 24 measurements, speed of longitudinal waves is average of 216 measurements, tensile and compressive strengths are determined from 27 and 28 measurements respectively,

Young modulus comes from 45 measurements and Poisson coeﬃcient comes from 39 measurements. Slyuta et al. (2008a) adds data for Ghubara (density from 3 measurements, compressive and tensile strengths from 5 measurements and longitudinal waves from 21 measurements). Slyuta et al. (2008b, 2009) provide important results on anisotropy of

Sayh Al Uhaymir 001 and 2 samples of Tsarev meteorite. Averages from the anisotropy measurements on Tsarev and Sayh al Uhaymir were taken from Slyuta (2010). The paper also confirms findings by Medvedev et al. (1985) about anisotropy of sound waves. The results are that in one direction the compressive strength is about 1.6 times larger than in other two perpendicular directions. Tensile strength shows significantly lower anisotropy.

59 Tsuchiyama et al. (2008) provides tensile strength data on carbonaceous Chondrites, the measurements were performed on 100 µm sized grains. The authors used Shimadzu

MCT-W500 machine with a ﬂat diamond indenter (500 µm diameter).

Voropaev et al. (2017b) provide measurements on Chelyabinsk meteorite on two distinct types of samples, those with Chondritic texture and those with shock-melted material, both using cylindrical shaped samples with well deﬁned sizes. Besides standard uniaxial compression and tensile measurements (direct extension of the sample), authors also provide triaxial compression measurements. These are not provided in Tab. B.1

(since this is the only relevant non-uniaxial measurement that we found in literature) and for the Chondritic material the strength was 142 MPa at lateral pressure 10 MPa and for the shocked material the strength was 124 MPa at 5 MPa lateral pressure (the lateral pressure is constant until the sample breaks down). Young’s modulus was determined from the linear part of stress-strain plot and Poisson’s coeﬃcient from axial and lateral strains. Since the authors call the obtained strengths “ultimate”, we assume that these are obtained as maxima of the stress in the would be stress-time plot.

Yavnel (1963) the author provides a very extensive study of mechanical properties of the Sikhote-Alin iron meteorite. Importantly, he provides measurements on several types of samples from the meteorite: mono crystal samples which are mostly made of the major mineral of the meteorite (Kamacite) and poly crystal samples which include Ni-Fe inclusions and other materials. Mono crystal samples are divided into two groups. The

60 samples that were cut from the meteorites without external deformation. These were cut

out in a way to exclude anisotropy in the measurements. These are denoted “T1” in the

data table. Both tensile and compressive tests are done with a press of the Gagarin type

with cylindrical samples of well deﬁned sizes and at well deﬁned strain rates. Besides

tensile strength, the author also determines elongation and decrease in cross-sectional

area during the tensile measurements. He deﬁnes the strength (both in compression

and tensile) as the pressure induced by the machine per unit area of the sample at the

moment when the ﬁrst crack appears. The author also notes that the monocrystalline

samples experienced brittle failure during tensile testing. The yield strength increases

with decreasing temperature such that σY = 133.6 MPa, σY = 228 MPa, σY = 303 MPa

for temperatures 300 K, 77 K and 4.2 K respectively. The samples in the second group

(denoted “T2” in the data table) were annealed to remove displacements of the matrix,

distortions due to Rhabdite inclusions and due to Neumann lines. For annealed samples,

only tensile data is available since the author states that he was not able to destroy the

samples during compression tests (due to a very plastic behaviour). During the tensile

tests on polycrystalline samples of the meteorite, the failure was also of the brittle type.

The compression test was done on two samples, however for each sample the author

used a diﬀerent strain rate which he did not specify. Resistance to bending was also

measured for a prism shaped sample with a square cross-sectional area of width 40 mm

and length 150 mm and the distance between supports was 130 mm. The resistance

2 to bending was calculated using the formula Rb = 3P L/(2bh ), where P is the load,

L is the distance between supports, b is the width of the sample and h is the height

61 of the sample. The obtained magnitude was Rb = (127 ± 16) MPa. The important

conclusion is that the tensile strength of monocrystalline samples is almost 10 times the

strength of polycrystalline samples. The polycrystalline samples are about 25% weaker

than monocrystalline ones in compression.

Zotkin et al. (1987) provide analysis of multiple samples of the Tsarev meteorite.

Most importantly this work provides measurements of both tensile and compressive

strengths based on the size of the sample. The authors used cubic shaped samples with

dimensions from 10 to 100 mm. Compression tests were done according to standard tech-

niques used at the time (by which the author most likely refers to the GOST standard

for compressive measurements) and tensile strength was determined by wedge splitting

method (GOST21153.3-85, 1986). The strain rate was so low such that the test could be

considered static (but no speciﬁc strain rate is provided). The authors found the max-

imum compression strength for 40 mm cubes and maximum tensile strength for 15 mm cubes; the strength falls oﬀ for larger specimen but also for smaller ones. The authors also measured dependence of Young’s modulus, Poisson’s ratio (both measured directly) and the velocity of longitudinal and transverse elastic waves on sample size. The data on those characteristics are only published as plots, we thus used the procedure described in Sec. 2.2.4 to extract the data. The error of the estimation of the sample size from the plot (as described in Sec. 2.2.4) was 0.1 cm.

62 2.4 Data and Plots

2.4.1 Overview of All Measurements

All available data collected from sources described in the previous section are displayed separately for quasi-static and dynamic (see Sec. 2.2.2.3) experiments in Figs. 2.1 and 2.2. While the data in quasi-static testing cover several meteorite types and 35 and

18 distinct meteorites for compressive and tensile test respectively, the dynamic measurements cover only two and a single L Chondrites in compressive and tensile measurements respectively.

Compressive Tensile

10000 10000 Achondrite C CI CM CV Fe H L LL Pallasite

1000 1000

100 100 Strength [MPa] Strength 10 10

1 1

Achondrite C CI CM CV Fe H L LL Pallasite 0.1 0.1

Figure 2.1: All data points that are provided in Tab. B.1 for individual meteorite types obtained from quasi-static tests. The compressive and tensile data points include 35 and 18 distinct meteorites respectively.

63 Compressive Tensile

10000 10000 L

1000 1000

100 100 Strength [MPa] Strength 10 10

1 1

L 0.1 0.1

Figure 2.2: All data points for dynamic measurements that are provided in Tab. B.1. Only data for L Chondrites are available. The compressive and tensile data points are from two and one distinct meteorites respectively.

64 A glance at the plots reveals several pieces of information:

1. the data exhibit large variability within meteorite groups, in the case of ordinary

Chondrites, typically almost over one magnitude,

2. there is about an order of magnitude diﬀerence between the compressive and tensile

strengths in ordinary Chondrites, the diﬀerence becomes smaller with increasing

iron content,

3. iron meteorites exhibit the largest strength properties of all groups and small diﬀer-

ence in the compressive and tensile test. Larger amount of iron extends the range

of elastic and plastic behaviours of the material while on the other hand, less metal

leads to a more brittle behaviour and such materials fail more easily under tensile

stress,

4. the compressive strength of carbonaceous Chondrites is about an order of magnitude

lower than for ordinary Chondrites, there is not enough data on the tensile strength

of carbonaceous Chondrites but the single data points suggest a similar diﬀerence

between the respective tensile strengths,

5. in the case of dynamic compressive and tensile strengths, the variability in L Chon-

drite seems much lower than in the case of quasi-static testing, however, this is

probably related to the fact that the quasi-static measurements cover more mete-

orites, while in the case of the dynamic compressive and tensile measurements, the

data points originate from two and a single meteorites respectively.

65 Table 2.1: Overview of compressive strength (σc) of meteorite groups from available data max in Tab. B.1. The values in the middle column are meanmin ± std. dev., where min and max are minima and maxima of each group and std. dev. is the standard deviation from the mean of each group. If some of those values are not available, they are omitted.

Group σc [MPa] # measurements 82 CM 6650 2 58 CV 3523 ± 10 10 700 Fe 42471 ± 143 15 321 H 14926 ± 76 19 1100 L 2996 ± 164 82 160 LL 8645 ± 51 4 Pallasite 432 1

The results in the Fig. 2.2 should not be compared with Fig. 2.1 to asses the results and claims in Kimberley and Ramesh (2011) and Hogan et al. (2015) regarding the diﬀerence in strength in those two regimes as it is necessary to compare those results on the same sample or with appropriate statistics.

The data in the Figs. 2.1 and 2.2 can be also summarized by simple statistical values such as mean, the maximum strength within the meteorite group and the respective minimum. Such view is provided in Tabs. 2.1 and 2.2.

2.4.2 Size Dependence of Strength

I have brieﬂy noted on the emergence of ideas trying to extrapolate the measured strengths from either meteorite measurements or from ram pressure on a passing meteor and its brightness onto asteroids in Sec. 2.2.2.1. These ideas are primarily based on a monotonic and continuous exponential relationship expressed in Weibull (1939). Thus,

66 Table 2.2: Overview of tensile strength (σt) of meteorite groups from available data in max Tab. B.1. The values in the middle column are meanmin ± std. dev., where min and max are minima and maxima of each group and std. dev. is the standard deviation from the mean of each group in the case of multiple (> 2) measuremts, or the measurement error provided by the author in the case of a single measurement. If some of those values are not available, they are omitted.

Group σt[MPa] # measurements Achondrite 56 ± 26 1 6.7 C 3.80.8 2 2.8 CI 1.80.7 2 8.8 CM 5.42.0 2 483 Fe 34043 ± 129 11 31 H 2723 2 79 L 422.0 ± 17 62 22 LL 9.01.7 ± 11 3 it is interesting to try and observe such relationship in the meteoritic samples, since, if such ideas are to be valid as extrapolation, they necessarily need to be valid within some range of measured values.

If we consider all data points where the authors also published the sizes of the samples, we obtain Figs. 2.3 and 2.4 relating the sample size and compressive and tensile strength respectively. The plots depict the sample volume on the horizontal axis and compressive and tensile strength on the vertical axis respectively. There is no obvious relationship between the sample size and strength at ﬁrst glance in either case despite the size range of 5 and 9 orders of magnitude in the compressive and tensile strengths respectively.

A more detailed analysis can be done speciﬁcally for Tsarev (L) meteorite where one may use the data by Zotkin et al. (1987). The relationship is displayed in Fig. 2.5 where the orange dots with sticks represent mean and deviation of the measured values

67 10-2 10-1 100 101 102 103 1000 1000

100 100 Strength [MPa] Strength

CM Murchison (Mi2008) CV Allende (Co2016) Fe Canyon Diablo (Kn1970) Fe Odessa (iron) (Kn1970) Fe Sikhote-Alin (Ya1963) H Tamdakht (Co2016) L GRO 85209 (Ho2015) L La Criolla (Mi2008) L MacAlpine Hills (Ki2011) 10 L Tsarev (Zo1987) 10 LL Chelyabinsk (Vo2017) Pallasite Brenham (Kn1970)

10-2 10-1 100 101 102 103 Size [cm3]

Figure 2.3: Compressive strength against sample size for all the data points with published test specimen size. The references in the legend are abbreviated as in Tab. B.1

68 10-7 10-6 10-2 10-1 100 101 102 1000 1000

100 100

10 10 Strength [MPa] Strength

1 C Tagish Lake (Ts2008) 1 CI Ivuna (Ts2008) CI Orgueil (Ts2008) CM Murchison (Ts2008) CM Murray (Ts2008) Fe Sikhote-Alin - monoc. (Ya1963) Fe Sikhote-Alin - polyc. (Ya1963) L GRO 85209 (Ho2015) L Tsarev (Zo1987) LL Chelyabinsk (Vo2017) 0.1 0.1 10-7 10-6 10-2 10-1 100 101 102 Size [cm3]

Figure 2.4: Tensile strength against sample size for all the data points with published test specimen size. Note that there is no data from (10−6 to 10−2) cm3 so the x axis is broken over this range of values. The references in the legend are abbreviated as in Tab. B.1

69 for a given sample volume, the blue triangles are the actual data. Within the statistical indicators, no size eﬀect can be observed within the data range (three orders of magnitude). Neglecting the deviation and only focusing on means, three main features can be described:

1. For cubic samples with side between (1–1.5) cm ((1–3.375) cm3), there is a decline

of strength with size,

2. for cubic samples with side between (1.5–4) cm ((3.375–64) cm3), there is an increase

in strength with size,

3. for cubic samples with side between (4–10) cm ((64–1000) cm3, there is a decrease

of strength with size.

While the two border tails in Fig. 2.5 can be considered as “monotonic continuous exponential”, the middle section cannot and certainly goes against the predictions based on Weibull (1939) and in general against any monotonic scale effect. However, the points above neglect the statistical significance of the data points. Further, there are only two measurements on the largest specimen (cube with size 10 cm) and as such the significance of the low mean strength measured for this specimen size of data point is very low. Con- sidering the data deviation, the data can be interpolated with a horizontal line without any difference (from statistical point of view) and hence there is no apparent statistically significant scale effect in the data by Zotkin et al. (1987).

Another complete dataset that can be used to evaluate scale eﬀect was measured on Allende (CV) and Tamdakht (H) meteorites by Cotto-Figueroa et al. (2016). The

70 1 10 100 1000 600 600 Compressive Data 550 Compressive Mean and Deviation 550

500 500

450 450

400 400

350 350 Strength [MPa] Strength

300 300

250 250

200 200

80 Tensile Data 80 Tensile Mean and Deviation

70 70

60 60

50 50 Strength [MPa] Strength 40 40

30 30

20 20 1 10 100 1000 Size [cm3]

Figure 2.5: Relation between sample size and compressive and tensile strengths for meteorite Tsarev (L) as measured by Zotkin et al. (1987).

71 compressive strength of those meteorites as function of size are displayed in Fig. 2.6, where the blue ﬁlled squares with sticks are means with deviations for Tamdakht, blue triangles are the respective measured values and orange disks with sticks are means with deviations for Allende, orange triangles being the respective measured values (note that in this case, the samples were of various sizes and measurements for similar sizes were grouped together and mean was size with deviation was calculated as well as the mean strength with its deviation, thus the horizontal sticks). Also note that the three largest samples of Allende (CV) are each a single measurement. The following points can be made about from the plot (Fig. 2.6):

1. Allende (CV) data present no scale eﬀect, respectively, the data is insuﬃcient for

its evaluation,

2. Tamdakht (H) does not present any scale eﬀect when the deviation is considered,

3. neglecting deviations for Tamdakht (H), one might see a situation similar to Tsarev

(L) in Fig. 2.5 with ﬁrst decrease in strength with size and then increase, how-

ever, there are too few data points and as such the data do not support any such

conclusions.

2.4.3 Relationship Between Strength and Density and Porosity

An intuition might give the idea that the higher density material is probable to withstand higher stresses and thus have higher strength. Studying this proposition

72 0.1 1 10 100 250 250 Allende Data Allende Mean and Deviation Tamdakht Data Tamdakht Mean and Deviation

200 200

150 150 Strength [MPa] Strength 100 100

50 50

0 0 0.1 1 10 100 Size [cm3]

Figure 2.6: Relation between sample size and compressive strength for meteorites Al- lende and Tamdakht as measured by Cotto-Figueroa et al. (2016). The mean values group together measurements of similar sizes and the x-error bar provides the standard deviation from the mean size of the grouped samples. The three largest Allende data points represent only a single measurement each (thus no error bars).

73 directly from the dataset in Tab. B.1 is not possible since the authors rarely measured the density. Since the statistics on meteoritic densities is vast (see Sec. 1.4.1.1), we may attempt a statistical description. Analogously, one may expect the meteorites with higher porosity to be in general weaker.

For the purpose described above, I have gathered the density (and porosity) data in the following manner:

1. In the case of meteorites where the authors also measured density (porosity) along-

side with strength, this data is used,

2. for Tagish Lake (C) bulk density and porosity were taken from Ralchenko et al.

(2014),

3. for Almahata Sitta (Ureilite), the data from Kohout et al. (2011) were used,

4. the bulk density of Tsarev (L) meteorite is from the papers by: Zotkin (1982);

Guskova (1982); Zotkin and Tsvetkov (1984); Zotkin et al. (1987),

5. the bulk densities of Covert (H), La Lande (L), Morland (H) and porosities of Bren-

ham (Pallasite), Kunashak (L), and Ivuna (CI) are from Britt and Consolmagno

(2003),

6. the bulk density of Novato (L) is from Jenniskens et al. (2014),

7. porosities of Canyon Diablo (Iron) and Odessa (Iron) are from Consolmagno and

Britt (1998),

74 8. all the remaining data that were not available in the above sources are from Macke

(2010).

Using this extended dataset, I calculated, for each meteorite, the mean strength

(compressive and tensile), mean bulk density and mean porosity and the respective standard deviations (provided the data was available, i.e. for some meteorites it was only possible to evaluate compressive strength and bulk density etc.). From these values the relationships between bulk density and compressive and tensile strengths are then graphi- cally displayed in Figs. 2.7 and 2.8; Figs. 2.9 and 2.10 present the analogous data between porosity and strength (compressive and tensile). Each ﬁgure depicts on its horizontal axis either density or porosity and on the vertical axis either compressive or tensile strength.

Each data point with horizontal and vertical sticks represents the mean value of density

(porosity) and strength and the appropriate standard deviations of a single meteorite, each colour and point shape correspond to a meteorite group described in the legend.

The plots depicting the relationship of density and strength (Figs. 2.7 and 2.8) do not display any strong support for the intuition introduced above in the case of compressive strength, however, in the case of tensile strength, the relationship that the more dense meteorites are stronger is visually quite obvious. In the case of compressive strength, the large spread of strength in ordinary Chondrites hinders the visual assessment (speciﬁcally the Novato (L) meteorite with its enormous strength of 1.1 GPa).

In the case of the plots of porosity and strength (Figs. 2.9 and 2.10), there would be a nice visual trend of decreasing strength with increasing porosity if not for the most

75 porous Sutter’s Mill (CV) with 31% porosity and 82 MPa compressive strength. For the tensile strength, there is too wide a spread at low porosities and very few data points at larger porosities to be able to comment on monotonic behaviour of tensile strength with porosity.

2 3 4 5 6 7 8

1000 1000

100 100 Strength [MPa] Strength

CM CV Fe 10 H 10 L LL Pallasite

2 3 4 5 6 7 8 Density ×10-3 [kg m-3]

Figure 2.7: Statistical relation between bulk densities obtained from various sources and compressive strengths of individual meteorite samples coloured by their group association.

76 2 3 4 5 6 7 8

100 100

10 10 Strength [MPa] Strength

1 1 Achondrite C CI CM Fe H L LL 0.1 0.1 2 3 4 5 6 7 8 Density ×10-3 [kg m-3]

Figure 2.8: Statistical relation between bulk densities obtained from various sources and tensile strengths of individual meteorite samples coloured by their group association.

77 0 5 10 15 20 25 30 35

CM CV Fe H L LL Pallasite

100 100 Strength [MPa] Strength

10 10

0 5 10 15 20 25 30 35 Porosity [%]

Figure 2.9: Statistical relation between porosities obtained from various sources and compressive strengths of individual meteorite samples coloured by their group association.

78 0 5 10 15 20 25 30 35 40

Achondrite C CI CM Fe H L LL 100 100

10 10 Strength [MPa] Strength

1 1

0 5 10 15 20 25 30 35 40 Porosity [%]

Figure 2.10: Statistical relation between porosities obtained from various sources and tensile strengths of individual meteorite samples coloured by their group association.

79 2.4.4 The Final Dataset

The collected data are presented as a single large table in App. B, Tab. B.1. All the data in the table come from the original papers with the only exception that the meteorite type was taken from the Meteoritical Bulletin Database3. While it might be possible to

ﬁll in other data such as density or porosity in a similar way as they were gathered for the purpose of presenting the relationship between the density and porosity and strength in subsection 2.4.3, but that would not be in accordance with the primary idea of this work to obtain reliable original data (since e.g. the strength of a particular sample and density on another sample are not necessarily representing the same material).

3https://www.lpi.usra.edu/meteor/

80 2.5 Discussion and Conclusions

2.5.1 Discussion

2.5.1.1 Quality of data

The data provided in the previous section cannot be treated as equal vis-à-vis their quality and reliability. For that reason, section 2.3 is included in this work so that the reader can estimate the quality of each reference. The resulting strength data depend on the used methods, sample size but also on shape of the sample as well as its preparation. They also weakly depend on (within each strain rate mode — quasi- static or dynamic) the strain rate as mentioned in 2.2.2.3. As mentioned by Medvedev et al. (1985), the diﬀerence between the sound velocities measured directly and those calculated from measured Young’s modulus and Poisson’s ratio diﬀer by about 13%.

While the author considers this to be acceptable, however, given the inconsistencies in the calculated moduli described in appropriate paragraph of section 2.3, it may be larger or smaller. Nevertheless, the diﬀerence in these two approaches exists (as I conﬁrmed when checking the results of calculated Young’s modulus). Other experiment related issues that one has to be aware of are boundary conditions such as in the Brazilian disk method described in subsection 2.2.1 and discussed in depth in Andreev (1995).

81 2.5.1.2 Scale Eﬀect

I devoted a signiﬁcant part of the previous chapter to detection of possible dependence of strength on sample size in the available data (see subsection 2.4.2). It is of interest to attempt a discussion of the eﬀects observed in subsection 2.4.2, more specifically in Fig. 2.5. An attempt to explain the observed behaviour may follow as such, starting from the maximum in the compressive strength in Fig 2.5 which occurs for cubes with side 4 cm (64 cm3):

1. As we decrease the sample volume, the ratio of surface area to volume increases and

since for a failure to occur, it is necessary to connect to two parts of the surface from

some initial crack within the sample volume, the creates a more “opportunities” to

do so.

2. At the minimum (which occurs for 1.5 cm sided cubes — 3.375 cm3), the volume of

the material is so small that it is eﬀectively a diﬀerent material than it was in the

range above and as the volume decreases further, the failure is dominated by the

amount of pre-existing weaknesses within the volume.

3. Another eﬀect to describe the previous point is that at some size, the material

stops being representative of the previous range (e.g. due to less inhomogeneities)

and thus “jumps” to higher strength regime where its size-strength relation by a

monotonic scale eﬀect.

82 4. The observed dependence can also arise in the case when the processing of the sam-

ples (or the meteorite entry and its breakup) preferentially introduces more weak

points in smaller volumes (for example, if the smaller samples require diﬀerent han-

dling and preparation). However, this stands against the eﬀect that any processing

also removes weaknesses and thus the result is not obvious.

Another important remark is that in general, strength scale eﬀect does no manifest as a declining function of size, even that it does not necessarily be purely monotonic. For example, for rock salt Andreev (1995) shows inverse scale eﬀect in the linear size range of the samples (5–30) cm and even in some samples of concrete the plots in Andreev (1995) show similar behaviour as in Fig. 2.5.

To conclude the critical evaluation of the scale eﬀect, it is in place to note that the paper by Weibull (1939) was intended for engineering applications and as such to sizes that are used in those applications, I do not believe that the author ever intended extrapolation of measurements over many orders of magnitude (e.g. in a case of (100–

1000) m asteroids, that would mean extrapolation over 3–4 orders of magnitude in linear size). Speciﬁcally this can be seen from the derivation of the exponential law presented in

Weibull (1939) where the assumptions are very strong and signiﬁcantly limit the possible extrapolation4. Additionally, from the fact that the treatment of scale eﬀect is intended

4The paper makes the so called weakest link assumption in the manner that is best described on a thin long slab of material. If one assumes that this slab is likely to break break at a certain applied stress then if one makes a material of two such slabs and applies the same stress and in the case that one of those slabs breaks, the second slab will suddenly be experiencing twice that stress and it would be almost certain that it breaks. One can immediately see that if the number of such slabs increases more and more, they no longer carry twice the stress but the stress will distribute among them and the more the slabs, the smaller the increase of the stress on the remaining slabs. Further, since the approach does not take into account any crack dynamics, it is limited in the way that as the size increases more and

83 for engineering applications, it also results another hidden assumption, that of homogeneous materials. The materials for which such approach is intended necessarily need to be homogeneous in composition, such as cement, it cannot be used for conglomerates of materials where the heterogeneity depends on size. Thus the strength properties of materials such as meteorites (and consequently asteroids) are governed by heterogeneity

(many meteorites do not ﬁt well even the heterogeneity category and should be described as conglomerates) and its size dependence rather than by scale eﬀect theories such as

Weibull (1939).

There are plenty of issues associated with scale eﬀect, especially with the one which is based on Weibull (1939). However, the detailed discussion of those issues is beyond the scope of this paper.

2.5.1.3 Density and Porosity

The relationship between density and compressive strength is not obvious from

Fig. 2.7 due to high spread of strength values of L Chondrites. But in general the relationship is not as obvious as intuition would suggest. The lower density is usually connected also to higher porosity and in if there is porosity, the reaction of the material to compression is to ﬁll the void spaces which reconnects several of the pre-existing weaknesses which may help resisting the increased stress. more, the probability of failure becomes constant, however, the “law” derived under the assumption by the paper leads to exponential strength decrease, not non-zero asymptotic.

84 On the other hand, the relation between tensile strength and density given by

Fig. 2.8 displays surprisingly clear monotonic behaviour. However, in this case, there are only 16 data points (as compared to the density-compressive strength plot with 29 data points). The easiest explanation would be due to porosity, since in the case of tensile stress, porosity significantly weakens the material. However, the relationship between porosity and tensile strength in Fig. 2.10 does not support this claim. Admittedly, the strength-porosity plots suffer significantly from low amount of data and thus it is impossible to use them for to form any solid points.

Finally, the relationship between compressive strength and porosity displays an interesting feature. For L Chondrites, there is quire clear monotonic behaviour that with increasing porosity from ∼ 3% to ∼ 15%, the strength drops by an order of magnitude, however, for carbonaceous Chondrites porosity does not have any signiﬁcant eﬀect on strength (and if some, with increasing porosity, the material becomes a little stronger).

Even if some of the plots and data views do not provide clear relationship between various quantities and strength, it is due to large variability of strength properties which is clearly visible in all the mentioned ﬁgures in Sec. 2.4 even within individual groups of meteorites. Among other issues, this provides a simple reason why a monotonic exponential scale should not be expected to play any signiﬁcant role when attempting to use available strength measurements on meteorites to describe the strength properties of asteroids, that is, levels of heterogeneities and conglomeration dictate their strength responses to applied forces.

85 2.5.2 Conclusions

In this work, I have reviewed all available original papers that deal with strength properties of meteorites and extracted data which contain measurements of strength and related quantities. The data is provided “as were” originally published with the exception of unit standardisation. From each reference, from which I obtained the data, I also extracted as many additional information on the experimental methods, set-ups and conditions as the original source provided and brieﬂy discussed them. The introduction section also contains basic description of the relevant physical quantities vis-à-vis their meaning and methods of determination with the intention to provide necessary references as well as to create a self suﬃcient work so that the reader is not required additional reading and searching through journals. As such, this compilation is the most up-to-date overview and data source of meteorite strengths measurements published in literature.

I have determined the following based on the data:

1. The diﬀerence in compressive and tensile quasi-static strength typically is:

(a) an order of magnitude for ordinary Chondrites (40 MPa . σc . 600 MPa and

10 MPa . σt . 80 MPa, excluding outliers) and the diﬀerence decreases with

increasing Iron content;

(b) within the same order of magnitude for Iron meteorites (300 MPa . σc .

700 MPa and 300 MPa . σt . 500 MPa, excluding outliers) with data tending

towards higher values in the range;

86 (c) an order of magnitude or more (σc ∼ 10 MPa and σt ∼ 1 MPa), in the case of

carbonaceous Chondrites.

2. The data variation is usually within the same order of magnitude except for ordinary

Chondrites where the measurements vary over an order of magnitude or more.

3. The compressive strength of carbonaceous Chondrites is about an order of magni-

tude below that of ordinary Chondrites.

4. There is no obvious monotonic relationship between density and compressive strength.

The relationship between the tensile strength and density is positive.

5. There is no general trend between porosity and either compressive or tensile strength.

6. If one restricts attention to Ordinary Chondrites, there is inverse relationship be-

tween porosity and compressive strength. In the case of carbonaceous Chondrites

there is a suggestion of either constant or positive relationship between porosity

and compressive strength.

7. There is no proof of scale eﬀect in the data in such a way that there is no statistically

signiﬁcant relationship between size of the sample and its strength (either tensile

or compressive).

The obtained data show that meteorites vary tremendously in their strength properties within their groups, even within samples of the same meteorite. This is governed by their mineralogy, shock history and the complex evolution path of their parent bodies.

87 CHAPTER 3 INTERACTIONS OF HIGH ENERGETIC PARTICLES WITH ASTEROIDAL MATTER

3.1 Background

Economic aspects of space exploration present a severely limiting factor in large scale in situ research. One of the largest obstacles is the Earth itself with its deep gravitational well. At the same time, its magnetic ﬁeld provides protection from invisible particles of high energies that are lethal to living cells. Space exploration thus has to deal with contradicting requirements, get into space with as little mass as possible to keep the costs minimal, however, as will be shown in this chapter, in order to protect either the equipment or the human crew, the more mass the better the protection.

Since there is already signiﬁcant mass of material in space in the form of asteroids, the question presents itself, if such matter could be used to provide the necessary protection against the cosmic radiation while keeping the cost low since the material itself does not need to be heaved from the Earth’s surface. Similar ideas were explored earlier when it was hoped for a Lunar base. However, the material that was explored was of Lunar origin and as such quite diﬀerent to the material found on certain types of asteroids.

88 The following sections are reproduced from the Author Accepted Manuscript version of Pohl and Britt (2017) with the permission of Elsevier Ltd (see Sec. A.2) who is being acknowledged thus.

3.2 Introduction

Space radiation is a signiﬁcant obstacle in deep space human exploration. This radiation can be understood as a ﬂux of energetic particles which deposit energy in matter and subsequently represent a serious risk for both living organisms as well as sensitive apparatuses.

In general, all energetic particles (atoms, atomic nuclei, protons, neutrons, electrons, positrons, pions, photons, etc.) can interact when traversing through matter. The major diﬀerences are in the type of interaction and in the probability of such an interaction.

Protons can interact through Coulomb interaction (which is true for all charged particles) during which they transfer part of their kinetic energy, usually to atomic electrons, but the energy can also be transferred to atomic nuclei. At high energy

(& 1 GeV) protons (actually this is true for any hadron) strongly interact after having traversed a certain distance (the nuclear (hadronic) interaction length, which for solids is ∼ (10 − −100) cm (Tavernier, 2010)). At even higher energies, protons can undergo multiple nuclear interactions while being slowed down by Coulomb interactions along

89 their path. The nuclei in the target material can thus be broken into several nuclear fragments which are unstable (usually due to excess of neutrons). The fragments stabilize by emitting excess neutrons. This can be accompanied by other decays producing protons, electrons and neutrinos. Further, incident protons can also interact with the constituent protons or neutrons of the nucleus; this results in further hadrons (due to energy limitations these are mostly pions). If these secondary hadrons have suﬃcient energy, the whole process of nuclear interactions can continue. These processes are responsible for the so called secondary particle production (or secondary showers).

Neutrons interact strongly with atomic nuclei by either scattering (elastic or inelastic) or absorption. The more interesting type of scattering is inelastic because then the nucleus can be excited which is followed by electromagnetic emission; the neutron itself is slowed down. Absorption of neutrons can have several results: the nucleus can get into an excited state and subsequently release a photon or several photons; it can re-emit a neutron or emit several of them; it can emit charged particles (usually protons,

α-particles); or it can undergo a ﬁssion event accompanied by several neutrons. The type of interaction of neutrons is energy dependent. High energy neutrons (& 1 GeV) interact in a similar way to high energy protons. For neutrons with energies in the range from

∼ (0.1−−10) MeV, the most probable interaction is elastic scattering (the neutrons loses kinetic energy and slows down until in thermal equilibrium with the target material).

Above ∼ 1 MeV inelastic scattering probability with atomic nuclei is not negligible, leading to excitation of the nuclei and subsequent photon emission. In this energy range,

90 neutron capture is also possible, followed the processes described above. For low energy

neutrons (below ∼ 10 eV) both elastic scattering and neutron capture are possible.

Interaction of photons is only electromagnetic and can take take several forms.

The photon can be absorbed by the atom which results in an excited state, either raising

the energy of one of its electrons, or even ejecting an electron completely. In the latter

case the vacancy is ﬁlled with an electron usually from the outermost shell, resulting in

photon emission (usually in the X-Ray spectrum) which is in turn absorbed by one of

the neighbouring atoms exciting it as well or removing one of its electrons. This process

is called the photoelectric eﬀect and it is the most probable interaction for photons with

energies between ∼ (1 − −100) keV. Compton scattering is a result of a collision between a charged particle (usually Compton scattering refers to only scattering on electrons) and a photon in which it transfers some of its kinetic energy to the electron which recoils and can be ejected from the atom or gets excited in the atom (with all the subsequent eﬀects already discussed). This interaction is most probable for photons between ∼

(0.1 − −1) MeV. Under the electromagnetic ﬁeld of the nucleus (or electron as well), pair production can occur in which the photon transforms into an electron-positron pair and it is the most probable interaction for photons above ∼ 1 MeV. Finally, nuclear interaction can take place (for the sake of completeness, it is necessary to note that strictly speaking, this interaction is mediated by a quark-antiquark pair production which then strongly interacts with the nucleus) which leads to the excitation of the nucleus and subsequent emission of either a neutron, a proton or ﬁssion. The probability of nuclear interaction is negligible for photons with energies below ∼ 10 MeV.

91 Ultimately, the eﬀects on target material are either an atomic excitation and subsequent de-excitation accompanied by emission of a photon (and its interactions as described above) or an atomic ionization (ejection of an electron) or a change of constituents.

In living organisms, specifically in cells, ionization of target material results in a production of radicals. Without ionization, atoms and molecules in cells are in stable state, that is their reactivity is low, similarly to salt dissolved in water into Chlorine anion and Sodium cation; despite both being electrically charged, they are rather non-reactive as their electron shells are “closed”. However, as ionization removes electrons or scatters protons (as is the case of neutron flux), it creates charged particles which have “gaps” in their electron shells and atoms have tendency to close those shells. These ions are called radicals — highly reactive ions. These radicals, as they chemically interact with nearby atoms and molecules, break the existing chemical bonds. Oxides formed from the event help to spread the radiation damage to neighbouring initially unaffected cells (Azzam et al., 2012). The most dangerous effect is on Deoxyribonucleic acid (DNA) within cells.

When both of the strands of DNA are broken, the cell attempts to repair them which may result in an incorrect “reconnection” of the strands and either a mutation or a chromosome abnormality occurs or the cell dies (if only a single strand of DNA is broken, the other can serve as a template for the repair and the risk of erroneous recombination is smaller).

The detrimental eﬀect of the various energetic particles is not the same. One of the quantifying parameters of the eﬀect is how much energy is released per unit traversed

92 length — Linear Energy Transfer (LET). The eﬀect also depends on the type of material.

For example, water has a large probability of interacting with incident neutrons and thus neutrons have significant effect on cells and therefore neutrons have high LET in cellular material. It is also worth mentioning that particles with high LET will cause more concentrated damage (on a small area) the repair of which is more difficult than the rather widespread but dilute damage caused by low LET particles.

Because of these eﬀects, a shield of a suﬃcient mass and a proper composition encompassing a spacecraft is necessary to provide protection during deep space missions.

However, the costs of transportation of the required mass of such a shield from the Earth surface are very high. Another option is to use material that is in a less deep potential well

(the Moon) or material already in space (asteroids). The feasibility of using the lunar regolith as a shielding material against high energy particles was analysed by several authors, for instance Miller et al. (2009) who measured dose reduction properties of lunar soil on soil samples or Pham and El-Genk (2009) who used the Monte Carlo radiation transport code to calculate the eﬀects of lunar soil on the attenuation of Solar Energetic

Particle events. In this work, we focus on the shielding properties of asteroidal material.

3.2.1 Sources of Energetic Particles in Space

Energetic particles permeating the interplanetary space are supplied by Solar En- ergetic Particle (SEP) events and Galactic Cosmic Rays (GCR). Mewaldt et al. (2005) notes that GCRs present the greatest threat (measured as radiation dose) to astronauts

93 devoid of the protection of the Earth’s magnetic ﬁeld. The particles from these two sources primarily diﬀer in spectrum as well as in their temporal and spatial variability.

3.2.1.1 Solar Energetic Particles

Solar Energetic Particles (SEPs) are ions and electrons which originate in the solar ﬂare regions (and sometimes neutrons which are the product of proton nuclear interactions at the Sun and have enough kinetic energy to reach 1 AU (Ryan et al.,

2000)). The spectrum of SEPs varies both in time and space signiﬁcantly as it strongly depends on time variation of the Sun’s activity. Two main warning signs of an imminent

SEP event can be observed.

The first warning sign is the gradual increase in particle flux. Although in many cases the intensity of particle flux rises gradually (usually it takes a few hours before the maximum intensity is attained (Mewaldt et al., 2005)), the events on 20th January

2005 and in February 1956 showed that a rapid increase in flux can happen relatively unexpectedly (e.g. for the 2005 event, the peak flux of protons with energies > 100 MeV was reached within ∼ 30 min) (Mewaldt et al., 2005; Meyer et al., 1956) and there is very little time for taking precautions (e.g. for astronauts outside of a shielded area to get to safety of a shielded habitat). Both events were characterized by a very steep increase in the proton flux and relatively high intensities of the incident particles. The intensity of protons from a large SEP event can become over 105× greater than the intensity of

GCRs (Mewaldt et al., 2007). Although SEPs consist of protons as well as of heavier

94 ionized nuclei (up to iron), due to their abundance, protons present about ∼ 90% of the

health risk (attributed to SEPs) to humans (Mewaldt et al., 2007). Table 3.1 provides

the most abundant elements in SEP events normalized to proton abundance, data are

based on Reames (1995).

Another warning sign of incoming energetic particles from a solar event (besides

a gradual increase in proton intensity) can be the increase in X-Ray and radio emissions;

both types of radiation are generated during a solar ﬂare by hot plasma (Miroshnichenko,

2014). While it takes only about 8 minutes 20 seconds for electromagnetic radiation to

reach the Earth’s orbit, 10 MeV protons need about 80 minutes, however, 100 MeV protons

only about 27 minutes and 500 MeV protons only ∼ 14 minutes.

In large SEP events the energies of the solar protons can reach up to several ∼ GeV,

e.g. for the February 23rd 1956 SEP event, the integral ﬂux of protons > 1 GeV was on

the order ∼ 106 protons · m2 sr−1 s−1 (Miroshnichenko, 2014). Other large SEP events

(e.g. August 1972, October 1989, July 14th 2000, October 28th 2003, January 20th 2005)

have proton integral ﬂuxes with energy > 100 MeV greater than 108 protons · m2 sr−1 s−1

(Mewaldt et al., 2005).

3.2.1.2 Galactic Cosmic Rays

Galactic Cosmic Rays (GCRs) are energetic particles that originate outside our

Solar system. They consist of both ionized nuclei (usually fully), electrons and neutrons.

95 Table 3.1: Most abundant elements in per cents in SEP normalized to Hydrogen, data is based on Reames (1995). Element Abundance H 100.0 He 3.631 C 0.030 O 0.064 Mg 0.012 Ne 0.010 N 0.008

While composition of GCRs closely follows that of Interstellar Matter (IM), the observed

diﬀerences in GCRs as compared to interstellar medium are in larger relative abundances

of elements from Li to Fe, lower content of Hydrogen (about 79% of all GCRs are Hydro-

gen nuclei compared to ≈ 91% in IM), higher content of He (≈ 15% in GCRs compared to ≈ 9% in IS) and lower ratio of H/He (about 5.4 for GCRs compared to ≈ 10.0 in IM)

(Olive and Particle Data Group, 2014; Ferrière, 2001).

The observable energy spectrum is limited from below by two eﬀects: due to

ionization losses to interstellar matter, the lower limit for protons is ∼ 50 MeV; due

to the magnetic ﬁeld carried by the solar wind, our Sun provides yet another inhibitor

to the observable energies of GCRs, increasing the lower observable limit to about (200–

300) MeV. GCRs are thus also aﬀected by the Solar cycle and therefore the ﬂux of GCRs is

time and place dependent in the similar way as for SEPs, however, the relation is inverse

— the larger solar activity (and thus stronger particle ﬂux from the Sun), the larger

the inhibiting magnetic ﬁeld and the more attenuated the GCRs. From about 10 GeV

for protons (or more generally from ∼ 10 GeV/nucleon) (Biermann and Sigl, 2001) the

magnetic ﬁeld has little eﬀect on the particles and the spectrum is well represented by a

96 piecewise power law. The known spectra of GCRs extend up to ∼ 3 × 1020 eV with very low ﬂux ∼ 1 particle · sr−1 · km−1 · (100 yr)−1 (Biermann and Sigl, 2001) (particles with energies above ∼ 3 × 1018 eV/nucleon originate from extra-galactic space as evidenced by their Larmor radius of about ∼ 0.3 kpc (i.e. the diameter of the orbit is on the limit of the thickness of the Milky Way disk which is ∼ (0.4 − −0.6) kpc (Ferrière, 2001)) as well as by diﬀerent composition and a change in the slope of energy spectrum — the ankle).

1 GeV protons have ﬂux of about 103 particles · m−2 sr−1 s−1. We present the observed

GCR spectra of the ﬁrst 8 elements in Fig. 3.1.

105 ● ●●●●● ●●● ● ●● ● ●●●● H 104 ● ● ●● ● ●●●●●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 3 ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 10 ■● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● He ● ■●■■●■●■●■●■●●■●●■●●■●■●●■●●●●■●●●■●●■●●●●●●●●●●■●■●●●●■●●●●●●●●■●●●●■●●●■●●●●●●● ■ ●■●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●■●●■●●■●●●■●■●■●●■●●■●●●●●●■●●●●●●● 2 ●●●● ●●● ●●●●●●● ■● ●■●■■■■■●●●●●●●●●●● 10 ● ●●●●●■●■■■●■■■●■■■■●■■■●■■■■■■■■■■■●■■■■■■■■■■■■●■■■■■■■■■■■●■■■■●●●●●●●●●●●●● ● ■●●●■■■■■■●■■■■■■■■■■■■■■■■■■■■■●■■■■■■■■■■■■■■■■■■■■■■■■■■●■■ ●●●●●●● Li ■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■●■■■■■■■■■■■●●●●●● ◆ 1 ■ ■ ■■■■■■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ●●●● 10 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■●●●● ■ ■■■■■■■■■ ■■■■■■■■■■■■■●●●●●●● ■■■■◇◇■ ◇○ ○◇○◇○◇◇ ■■■■■■■■●●●●● 0 ○ ○ ◇ ◇ ■ ■■■■■●●●● Be 10 ○○○○○ ■○ ■■■■■●■●■●■●●●●● ▲ ○○○○○◇○◇○○◇○○◇○◇◇○◇◇○◇○◇○◇○○◇○○○○○○◇○○○○○◇○ ■■■■■●■●■●●●● ○◆○◇○◇○◇○▼◇▼◇▼○▼◇◇○◇▼○ ▼◇▲○◇▼▼◇◇○◇▼▼◇▼○◇○◇○◇▼○◇○◇○◇○◇○○◇▼◇◇◇ ■■■■●■●■●● -1 ◇ ▼▼▼○▼◆▼◆▼○□◆▼▼◇○◆□◆○▼◆○□◇□□□▼◆◇□▼◇▼▼◇▼▼□▼▼◇□▼○▼▼▼□□◇○□◇○◇◇○◇○◇○◇ ■■●■●■●●● 10 ○◆□◆▼◆○▼◆□○◆▼▲□○◆▼○□▲◇○□◇□◇◇□◇□◇◇□◆◇◆◆▲◆◆□◆◆□▲◆□▲□□▼□▼□□▼▼□□▼◇▼□▼□▼▼□▼□◇◇□◇○○◇○○◇○◇◇○ ■■■■●■●● B ▼○ ◆▲◆□▼◇▲▼▲◆▼▲▼▲▼▲▼▲▲□▼□◆▲▲□▲▼▲▲▲▲▲□○▲▲▼◆▲○◆▲◆○◆▼□□▼◇○□▼□▼□▼□◇▼□○▼□□○◇○◇○○◇○○◇○◇◇■■■■■●■●●● ▼ ◇◆▼▼▲▼□▲▼▲□□□□□□ ▲ ▲▲◇▲▲▼□◇▲▲◆▲▼▲□◆▼□▼▼◇□▼□▼▼◇○◇○◇○◇○○◇◇■■●■●■●■●● -2 ◆ ▲ ▼ ▲▼▲◆▼◆▼□▼▼□□▼□◇○◇○◇ ■■■●● 10 ▼ □ ▼□□□□▲◆▲◆□▲▲□◆▼▼□▼□▼▼□◇○○◇■■■■●●■●● ] ▲▲▲▲ ▲▲▲▲ ▼□▼□□○○○◇ ■■●■●● ▲ ▲ ▲▲◆▲ ▼□▼□○○◇○◇○■■■●■●■● C sr ▲◆▼□□ ○◇◇◇■■■●■■●●● ○ / ▼▼ ■●■● -3 ▲ ▼□□○○ ■■■●■ s 10 ▲◆▼▼□○○○◇◇◇■■●■● / ▲ ▼▼▼□□○○ ■■■●■● 2 ● ▲◆▼ □○○◇◇○■■●■●● -4 ▲ ▼▼▼□□○○◇◇■■●●■ N m ▼ ○○◇ ■ □ / 10 ▲ ▼▼□ ◇○◇ ● ▲◆▼▼□□□○◇■■■●●■ ◆◆ ▼▼▼ □○○◇○◇◇●■ -5 ▲ ▼▼□ ○○◇◇■ ●■■ 10 ◆ ▼ ○○◇■◇●● O GeV ▲◆ ▼□□○◇○○□ ■● ◇ Particle flux ▼▼ ◇ ■●■ ▲ □○○○◇◇ ■●■●■ -6 ▼ ▼◆□□ ○ ■●■● [#/ 10 ▼▼ ▼ ○◇○◇◇ ■●■ ▲◆ ▲◆▼▼□□ ◇○◇ ●■● □□○ ◇●■●■■● -7 ▲ ▼ ○ ●●●■ 10 ▲ □ ◇◇◇●●■●■ ▼ □ ○○ ●●●●■ □ ◇■◇◇●■●■● -8 ○◇■●■●●■●■ 10 ◇ ■●■●■ □ ○◇○◇○●■●◇■●●■ -9 ○◇◇ ■● 10 ◇○■●●●■ ▼ ◇◇●●■●■ ■ ●◇■■ 10-10 ○● ● ◇◇■●●■■ ● -11 ●◇ ■ 10 ■■● ● 10-12 ■● 10-13

10-3 10-2 10-1 100 101 102 103 104 105 106 Particle energy [GeV]

Figure 3.1: The spectrum of GCR for the ﬁrst 8 elements. Data have been obtained from the database of GCR observations (Maurin et al., 2013) and the plot summarizes all the available measurements from this source.

97 3.2.2 Asteroidal Mineralogy

Composition of asteroids is different from that of lunar regolith to a degree that it is not a priori possible to conclude that asteroid material has the same shielding properties as lunar regolith and it is not possible to reuse the already available results. The major difference is in phyllosilicate content. These layered hydrated sheets of silicates are abundantly found in carbonaceous chondrites, specifically in CM, CR and CI groups while missing in the lunar regolith (Heiken et al., 1991).

The signiﬁcance of phyllosilicates for shielding against high energy particles lies in their water and hydroxyl content1 which can be up to 12.5% of their weight (Alexander et al., 2013). Water has very good properties not only for slowing down (shielding) charged energetic particles (GCR and SEP) but it also has a large cross section for interaction with neutrons which result from interactions of the incident particles with the matter that makes up the shield.

The difference in composition between asteroidal material and lunar regolith can be illustrated by several works, Bland et al. (2004) determined from X-Ray diffraction that Serpentine, a typical phyllosilicate, makes 22.8 weight per cent of Murchison CM2 chondrite, Cronstedtite makes 58.5% while Olivine as the only significant anhydrous mineral makes only 11.6%; for Tagish Lake C2 chondrite, the Saponite-Serpentine clay makes 60.3 weight per cent, anhydrous minerals Pyrrhotite, Magnetite and Olivine make together only about 39%; and for Orgueil CI1, Saponite, Serpentine and Ferrihydrite make

1throughout the work we use the words water and hydrated to suggest both water and hydroxyl molecules

98 almost 77 weight per cent, the rest being Olivine, Troilite, Pyrrhotite and Magnetite.

Hutchison (2004) provides mean water contents of various chondrite groups, e.g. for CI

18% or CM 12.6%.

Drever et al. (1970) observed traces of hydrated silicates in Apollo 11 samples, the amount in the tested samples was negligible for the purposes of particle shielding.

Zolensky (1997) studied a carbonaceous chondrite brought by Apollo 12 from the Bench

Crater and identiﬁed hydrated phyllosilicates and proposed more abundant presence of hydrated minerals on the Moon and also suggests that about “1% of the lunar regolith consists of phyllosilicates or their dehydration products”. McCord et al. (2011) concluded that the probability of survival of the meteorite supplied phyllosilicates is low. Although the current knowledge rebuts the original notion of complete absence of water and hydrated minerals on the Moon, the amount of these hydrated minerals on the surface, while still not exactly known, is very likely low; especially when compared to the respective abundances in asteroids.

Studies of various Apollo samples have been conducted by X-Ray spectroscopy.

The results for instance by Taylor et al. (2001, 2010) show that besides agglutinate glass which can make up to 72% of the lunar soil, the dominant minerals present in Apollo samples (from either the highlands or maria) are Plagioclase with abundance up to 62% and Pyroxene with abundance up to 20%, where their ratios vary based on the location.

Other signiﬁcant minerals determined by X-Ray spectroscopy are Olivine and Ilmenite

(Taylor et al., 2001, 2010). Only Olivine and Pyroxene are usually found in abundance

99 in CM, CR and CI meteorite samples. CV meteorites are mostly Olivine and Enstatite

with little or no Plagioclase.

Water in the hydrated minerals usually occurs in the form of hydroxyl group (OH)

which is bound in the crystal structure as alternating layers of silicates, metal cations,

and hydroxyl. Several important phyllosilicates can be identiﬁed in meteorite samples.

Serpentine2 is commonly created by mixing water with either Forsterite3 or Enstatite4.

The actual result of hydration depends on both temperature and pH conditions (Oze and

Sharma, 2005; Ohnishi and Tomeoka, 2007). Serpentine is abundant in CM meteorites and can also be found mixed with Saponite5 which is another important phyllosilicate found in CI meteorite samples (Bland et al., 2004). Usually Serpentine and Saponite phases are intermixed so that it is not possible to report them individually (King et al.,

2015a).

Cronstedtite6 is also a common phyllosilicate found usually as Mg-Fe phase7 (Zega and Buseck, 2003). It is abundantly found in CM meteorites (Dyl et al., 2010) making about 59% of Murchison meteorite (Bland et al., 2004) but its content in other CMs can be as low as 10% (e.g. DOM 08013) (Howard et al., 2015).

Other carbonaceous chondrites are not easily classiﬁed by phyllosilicate abundance. Recently, Howard et al. (2015) proposed grouping of carbonaceous chondrites based on their phyllosilicate content. This is a preferred meteorite classiﬁcation with

2 Usually denoted as Mg2Si2O5(OH)4 3 2 Mg2SiO4 + 3 H2O −−→ Mg3Si2O5(OH)4 + Mg(OH)2 (Klein et al., 2008) 4 6 MgSiO3 + 3 H2O −−→ Mg3Si2O5(OH)4 + Mg3Si4O10(OH)2 (Mottl, 2015) 5 2+ generally (Ca0.5|Na)0.3(Mg|Fe )3(Si|Al)4O10(OH)2 · 4 H2O (Anthony et al., 2015) 6 2+ 3+ Fe2 Fe2 SiO5(OH)4 7 2+ 3+ Fe0.4 Mg1.6Fe2 SiO5(OH)4

100 respect to stopping properties of asteroid material. In standard classiﬁcation, in general,

we ﬁnd that the CV group contains 1.9 − 4.2% phyllosilicates with uncertain individual mineral structure (Howard et al., 2009); in the CR group, it is possible to ﬁnd both samples abundant in phyllosilicates (e.g. Al Rais with phyllosilicates totalling 60%) as well as

samples with little phyllosilicate content (e.g. QUE 99177 with 1.5% phyllosilicates and

abundant in Olivine and Pyroxene which make up about 73% and the rest being metal,

sulphides or amorphous) (Howard et al., 2015; Cloutis et al., 2012a; Howard et al., 2011).

We present summary of mineral content in various carbonaceous chondrite me-

teoritic groups in Tab. 3.2. The table has been composed from mineral abundances

determined by X-Ray Diﬀraction (XRD) by Bland et al. (2004); Howard et al. (2015).

The table also summarizes the total phyllosilicate content ranges of each meteorite group.

The total phyllosilicate content can vary signiﬁcantly within each taxonomic group as ev-

idenced by CM2 data (from ∼ 56% to ∼ 85% by mass) or even more by CR2 data (from

∼ 7% to ∼ 60%). The phyllosilicate variations within other groups cannot be judged

because of limited data on their mineralogy.

101 Table 3.2: This table summarizes mineral abundances of carbonaceous chondrite taxonomic groups expressed as weight percentages. The numbers in the second row in parentheses are the number of samples in each data set. If there is more than one sam- max ple in a given asteroid group three numbers are given in the format medianmin where median, minimum and maximum are the appropriate percentage values from each set. These numbers are to illustrate the abundance ranges of individual minerals within each group. The data are based on published meteorite mineralogies by Bland et al. (2004); Howard et al. (2015). C2-ung CI1 CM1 CM2 CM2/1 CR1 CR2 CR3 CV3 (4) (1) (2) (23) (1) (1) (7) (2) (1)

102 3.3 Methodology

3.3.1 Analytical Methods

3.3.1.1 Bethe-Bloch Equation

In the previous section we have mentioned that GCRs and SEPs consist of ionized hydrogen or heavier elements. When such a particle enters target material, nuclear interactions with atomic nuclei and Coulomb interactions with electrons and atomic nuclei take place as was described in the introduction. Nuclear reactions are responsible for cascade processes and the production of secondary particles (neutrons, electrons, photons

— various types of radiation, etc.). Because the mass of the incident proton (or a heavier nucleus) is comparable (or a bit smaller) than the mass of atomic nuclei in the target material, the Coulomb interactions of ions with the nuclei lead primarily to almost elastic scattering (with very small angles (Jackson, 1998)). The Coulomb interactions with the target electrons are responsible for the most signiﬁcant loss of the kinetic energy of the incident particle. This results in an almost straight path of the incident particle in the material.

The total energy lost by an incident particle per unit traversed path is called the stopping power or more correctly the stopping force. Its units are usually given in

[MeV cm−1]8. The total distance that the particle has travelled in the matter until it has

8Although the proper name as suggested by the units is stopping force, the name stopping power is used more often for historical reasons (Sigmund, 2006).

103 lost all of its kinetic energy is called the range (integral of reciprocal function of stopping force) which has a unit of length.

Considering the above, we take into account only the Coulomb interactions of the incident particle with the target electrons to analyse the eﬀectiveness of stopping of charged incident particles. Quantum mechanical and relativistic treatments of the problem were done by Bethe in 1930s (Bethe, 1930, 1932; Møller, 1932; Bloch, 1933).

The theory was further extended to include more detailed eﬀects described by Fermi

(1940), Fano (1963) and others. Summaries of the available theoretical knowledge can be found in Bohr (1948), Fano (1963) and Sigmund (2006). It is not our intention to present theoretical details of the approach, which can be found in the aforementioned references, however, it is necessary to present some of the theory in order to discuss the limitations of the approach and possible ways to mitigate them.

A generalized formula for stopping power that is based on the results of Bethe

(1930); Bloch (1933); Fano (1963) can be written as follows:

4 2 " 2 2 2 # NAe Zt Zp 1 2mec β γ ∆Tmax 2 δ C S(β) = ρ 2 2 2 ln 2 − β − − + b(β) + ... . (3.1) 4πε0mec At β 2 hIi 2 Z2

The notation is as follows: ρ is the density of the target material; NA is Avogadro’s

constant; e is the elementary charge; ε0 is the vacuum permittivity; Zt and At are the

atomic and mass numbers of the target nuclei; Zp is the atomic number of the projectile

v nucleus; me is the rest mass of an electron; c is the speed of light in vacuum; β = c ;

1 γ = √ ; ∆Tmax is the maximum energy transfer in a single collision; hIi is the mean 1−β2

104 excitation energy; δ is a density correction term; C is a shell correction term; b(β) is a 2 Z2

Barkas correction term; and “... ” stands for all other corrections that are not included.

The maximum energy transfer in a single collision can be found as (Olive and

Particle Data Group, 2014):

2 2 2 2mec β γ ∆Tmax = 2 , (3.2) 1 + 2 me γ + me mi mi where mi is the mass of the incident projectile.

The original Bethe’s derivation used perturbation approach and thus the correction terms. The assumptions of the theory presented here are:

1. Only interactions with target electrons are considered.

This assumption was justiﬁed by Bethe (1930) who showed that the transfer of

energy to nucleus is smaller by at least a factor of me/mtarget which is less than

0.1% of the energy loss.

2. mi me.

This is justiﬁed well if we consider protons and heavier nuclei as the incident par-

ticles.

3. The incident particle is fully stripped of electrons.

This reﬂects the fact that electrons in projectile can further shield the electric ﬁeld.

For more details on the case when the projectile is not fully ionized, see e.g. Sigmund

(2014). Because the ions in SEPs and GCRs are almost always fully stripped of all

electrons, this assumption is reasonable.

105 4. The incident particle does not pick up electrons while traversing the material.

Constant charge of the projectile is a basic requirement of the analytical approach.

See the next assumption.

5. The velocity of the incident particle is much larger than the electron velocity in the

target material.

Slow projectile ions can start picking up electrons and thus partly neutralize their

charge. Because of the previous assumptions, this inevitably provides a lower limit

where the theory can be applied. Ziegler et al. (2008) notes that the theory holds

for energies above 1 MeV u−1. In the case of SEPs and GCRs this is usually well

satisﬁed. Moreover, the range of projectiles below 1 MeV u−1 is very short (about

18 mm in air).

Another eﬀect related to these assumptions is that the theory does not take into

account the internal motion of electrons in the target and their bonds to the nucleus.

This is the reason for the shell correction term. For detailed description of this term,

the reader can consult Fano (1963); Sigmund (2006). Fano (1963) mentions that

this correction plays role up to 100 MeV for protons, similarly Ziegler et al. (2008)

states that the correction is signiﬁcant for protons between 1 MeV and 100 MeV

with the maximum correction to stopping force being ∼ 6%.

6. The target material is made of non-interacting atoms of a single kind.

This assumption raises two problems:

106 Collective phenomena in dense media is handled by the density correction term in

Eq. (3.1). Detailed description can be found in Fermi (1940); Fano (1963); Stern-

heimer et al. (1982); Sigmund (2006). The relative signiﬁcance of this corrective

term increases with kinetic energy of the incident projectile. Ziegler et al. (2008)

suggests that this term is signiﬁcant when the kinetic energy of the projectile ex-

−2 ceeds its rest mass (the rest mass of proton m0 ≈ 938 MeV c0 ).

We will discuss the issue of compounds in a separate paragraph.

Barkas Correction The ﬁrst order perturbation that is used in the derivation of

2 Eq. (3.1) leads to the dependence of stopping force on Z1 (square of projectile atomic

number). Therefore the predicted stopping force on positively and negatively charged

projectiles is the same. However, as a positively charged projectile enters the target, it

tends to attract electrons and increase the local electron density and thus increase the

stopping force; on the other hand a negatively charged projectile tends to repel electrons

in the target and thus decrease the local density. Analogously, projectile with larger

charge attracts or repels electrons more strongly and thus increase or decrease the local

density compared to a projectile with smaller charge. This eﬀect is observable as long

as the projectile moves slowly enough for the target electrons to reposition into the new

conﬁguration. Andersen et al. (1969) observed the diﬀerence in stopping force and sug-

3 gested that the dependence of this term goes as Z1 . According to Ziegler et al. (2008)

the eﬀect is most signiﬁcant around 1 MeV u−1 and becomes negligible for energies above

∼ 10 MeV u−1.

107 Mean Excitation Energy The method for analytical calculation of the mean excitation energy was provided by Bethe (1930). Details can also be found in Ahlen (1980).

Usually, this parameter is determined from range measurements, e.g. Sternheimer et al.

(1984).

Stopping Force in Compounds The most common approach is to idealize the target compound as if made of layers of elemental materials (Olive and Particle Data Group,

2014). Eﬀectively, one assumes that the stopping force of a compound is a linear combination of the stopping forces of its elemental constituents (this approach is called Bragg additivity):

N X S = wi Si , (3.3) i=1 where N is the number of elemental constituents, wi and Si are the weight fraction and the stopping force of the ith elemental constituent. The validity of this approach is limited by the fact that the energy transfer to the electrons depends on the details of the electronic structure of the target material (the strength of the electronic bonds, their orbital conﬁguration etc.). Other approaches are possible that incorporate the internal structure and motion of electrons, we refer to Ziegler et al. (2008) and the references therein, in particular, Sigmund (1982).

Bragg Additivity The Bragg additivity as suggested by Eq. 3.3 can be equiva- lently written (if we neglect all the corrective terms for now) (Seltzer and Berger, 1982a):

4 2 NAe Zt Zp 1 2 2 2 2 S(β) = ρ 2 2 2 ln 2mec β γ ∆Tmax − ln hIi − β , (3.4) 4πε0mec At β 2

108 where:

N Z X Zi = wi , (3.5) A i=1 Ai PN Zi i=1 wi ln Ii hIi = exp Ai , (3.6) D Z E A where the indexed variables are the respective values for individual elemental constituents

Z and hIi can be regarded as the mean excitation energy of the compound and h A i can be regarded as the eﬀective value for the compound material. Olive and Particle Data

Group (2014) suggests using experimentally determined values for the excitation energy hIi. If these are not available, value determined from Eq. (3.6) should be used. Seltzer and Berger (1982a) recommends to increase the value of the excitation energy of certain elements that make up condensed compounds by 13%.

Density Correction in Compounds To obtain the proper density correction in compounds, we cannot use the analogy of Eq. (3.6) because the density correction needs to reﬂect the density of electrons (Olive and Particle Data Group, 2014). To evaluate the density correction in compounds, we use the algorithm suggested in Groom et al. (2001) and the parameters published in the same paper.

3.3.1.2 Implementation of Bethe-Bloch Equation

In our implementation of Bethe’s theory, we use Eq. (3.4) with the mean excitation energy determined from available experimental data for compounds. If the data is not

109 available we use Eq. (3.6) with the 13% rule applied. The density correction is according to Groom et al. (2001).

4 2 " # NAe Zt Zp 1 2 2 2 2 δ S(β) = ρ 2 2 2 ln 2mec β γ ∆Tmax − ln hIi − β − . (3.7) 4πε0mec At β 2 2

As we deal with energies & 50 MeV, we neglect both the Barkas and Shell correction terms.

The calculation procedure is as follows:

1. we choose a material (asteroidal composition from available data, e.g. Bland et al.

(2004) or a given asteroidal mineral),

2. for each mineral in the material, we determine its structure,

3. if experimental data is available for some of the compounds that make up the

mineral (e.g. water or hydroxyl group), we use that data for mean excitation energy

of the compound as a whole, if not, we use the methods described above,

4. we calculate the density correction term in an analogous way as we calculate the

mean excitation energy,

5. we compute the stopping force in units MeV cm2 g−1 according to Eq. (3.7).

3.3.2 Numerical Method

To verify our results using the simpliﬁed Bethe equation (i.e. without the low

energy correction terms and detailed structure of electronic conﬁguration of the target

110 material), we make use of the software package SRIM (Ziegler et al., 2008). We also use this method to determine range. We refer the reader to the book Ziegler et al. (2008) for all the details about the package. However, we stress that the SRIM software package assumes that the target is amorphous and thus its application on crystal structures is yet another approximation besides the program’s own simpliﬁcations.

3.4 Results and Discussion

In this section, we present the results of our calculations according to the procedure described in Sec. 3.3.1.2 together with discussions of those results.

3.4.1 Benchmarking

As the ﬁrst step, we wanted to verify the correctness of our calculations by comparison with existing results. For this, we used the results published by Groom et al.

(2001) and we also employed the software package SRIM (Ziegler et al., 2008). We chose two materials for this comparison, pure elemental iron and liquid water. The mean excitation energy as well as parameters for calculating the density correction are readily available for both of them. The comparison at selected energies can be found in Tab. 3.3.

Despite neglecting low energy corrections and some high energy ones as well, our results reproduce those of Groom et al. (2001) and those obtained by the SRIM package well

111 with the exception of the lowest energy in iron. However, at 10 MeV we should still expect some low energy corrections to be non-negligible.

Table 3.3: Comparison of our implementation of Bethe’s approach with Groom et al. (2001) where more correction terms are included and with the SRIM software package (denoted “SRIM e”) which implements even more corrections mostly based on ﬁtting experimental data. The table shows stopping powers in units MeV cm2 g−1, i.e. per unit density. We also give the nuclear stopping power calculated by the SRIM package (denoted “SRIM n”) to make sure that it is negligible compared to the electronic stopping power. 10 MeV 100 MeV 1 GeV 10 GeV Iron Ours 29.41 5.060 1.575 1.603 Groom 28.54 5.045 1.575 1.603 SRIM e 28.74 5.055 1.575 1.602 SRIM n 1.437 × 10−2 1.835 × 10−3 2.231 × 10−4 2.626 × 10−5 Water Ours 45.95 7.290 2.210 2.132 Groom 45.94 7.290 2.210 2.132 SRIM e 46.57 7.372 2.230 2.319 SRIM n 2.7 × 10−2 3.307 × 10−3 3.913 × 10−4 4.519 × 10−5

At this moment, we would like to mention that our primary goal is comparison of asteroidal materials to Aluminium rather than providing the most accurate stopping power of asteroidal materials with as many correction terms as possible. The mean excitation energy and the parameters for density correction are rarely available for the compounds found in asteroidal materials. It is therefore necessary to calculate the mean excitation energy from Eq. (3.6) and the density correction from a general algorithm

(Sternheimer and Peierls, 1971). On the other hand, for Aluminium these parameters are readily available. The treatment of other correction terms in compounds is usually even less clear.

112 Based on the presented results, we can conclude that from ∼ 10 MeV to ∼ 10 GeV our implementation provides good conformance to other published results.

Before continuing with the full results, we should explicitly note that although the density correction (high energy) is included in our calculations, the comparison with

Aluminium can be tainted to some degree as the algorithms to compute the correction term for Al and for most asteroidal materials are diﬀerent. In the case of Al, there is a set of parameters that can be readily used to calculate the density correction according to Sternheimer et al. (1982) and the mean excitation energy is also available from experimental data. For many compounds these data are not available, in which case we use the algorithm given by Sternheimer and Peierls (1971) and worked out explicitly in Groom et al. (2001) to determine the parameters necessary to calculate the density correction and we also use the Bragg additivity, Eq. (3.6) to determine the excitation energy (I). Further, we have mentioned in Sec. 3.3.1.1 that if the compound contains certain elements, better results of I are achieved by increasing the elemental Ii by 13%.

To determine the effect of each of those algorithms, we ran further benchmarking tests aimed specifically at the density correction (δ) and the effect of the 13% rule. The

ﬁrst algorithm (denoted a1 ) calculates δ according to Sternheimer and Peierls (1971) and uses a general prescription to determine the necessary parameters that are required to calculate δ — useful for compounds where parameters for calculation of the density correction are not available; it also calculates the mean excitation energy (I) by assuming

Bragg additivity without the 13% rule (note that I is also used in calculation of δ so that the error in I propagates into δ as well). The second algorithm (a2 ) is based on

113 Sternheimer et al. (1982) and makes use of published values of parameters for calculation of δ as well as published values of I. Material parameters for our calculations of δ and I values are based on Groom et al. (2001). The third algorithm (a3 ) applies the 13% rule when calculating the mean excitation energy as described in Sec. 3.3.1.1, otherwise it is analogous to a1. The results are summarized in Tab. 3.4 for one elemental material and three compound materials. Algorithm a3 is only used where the 13% rule is applicable and thus diﬀerent I is achieved.

By studying Tab. 3.4 we gather the following information. The differences in δ are huge up to around 1 GeV. This is due to the fact that the density correction does not start at the same energies in algorithms a1 and a2 and it rises very quickly — this is visible in the case of Fe; however, the numerical value of the correction itself is rather small compared to the first order term and the maximum resulting difference in the stopping power is only about 2% in Fe at 1 GeV. The situation with the difference in the density correction term is similar for both Fe and H2O. In both these materials, 13% rule is not applicable. The mean excitation energy for Fe is the same in a1 and a2 but in the case of water there is about 8% difference between the excitation energy of the H2O determined from experiment and the one from Bragg additivity determined as 2 H + O.

In the case of Calcium Carbonate, the use of 13% rule results in a better I value (from

≈ 10% difference from the experimental value, it decreased to about ≈ 6%, the difference between a1 and a2 was 2% and inclusion of the 13% rule in the I calculation resulted in only a 1% maximum difference from a2 (i.e. the difference a3 from a2 ). Similarly, for Quartz, the difference in the I dropped from 10% to 4% which lead to a decrease in

114 Table 3.4: Comparison of results with two diﬀerent algorithms for δ and with and without the 13% rule. The units are: [S] = MeV cm2 g−1, [δ] = 1 and [I] = eV. Mat. Val. Alg. 10 MeV 100 MeV 1 GeV 10 GeV Fe S a1 29.42 5.071 1.600 1.617 a2 29.41 5.060 1.575 1.603 a3 0.000 0.000 0.000 0.000 δ a1 0.000 0.000 0.061 1.726 a2 0.003 0.027 0.322 1.914 I a2 286.0

H2O S a1 46.62 7.368 2.226 2.156 a2 45.95 7.290 2.210 2.132 a3 46.62 7.368 2.226 2.156 δ a1 0.000 0.000 0.040 1.908 a2 0.000 0.000 0.024 2.023 a3 0.000 0.000 0.040 1.908 I a1 69.0 a2 75.0

CaCO3 S a1 37.75 6.149 1.882 1.828 a2 36.98 6.061 1.852 1.828 a3 37.40 6.109 1.873 1.825 δ a1 0.000 0.000 0.105 2.214 a2 0.000 0.000 0.188 1.999 a3 0.000 0.000 0.099 2.149 I a1 122.7 a2 136.4 a3 128.8

SiO2 S a1 37.56 6.126 1.878 1.838 a2 36.81 6.041 1.859 1.845 a3 37.14 6.078 1.867 1.835 δ a1 0.000 0.000 0.087 2.020 a2 0.000 0.000 0.072 1.721 a3 0.000 0.000 0.080 1.944 I a1 125.7 a2 139.2 a3 133.0

115 diﬀerence in stopping power from 2% maximum between (a1 and a2 ) to 1% between (a3 and a2 ).

To conclude, we have shown that the 13% rule can improve the stopping power value for compounds. Although there is no solid theoretical basis for using this particular rule, this rule is generally recommended, e.g. Seltzer and Berger (1982b). We have veriﬁed that it can improve the results on 2 compounds. We have also shown that even though the general algorithms a1 and a3 respectively do not use experimentally derived parameters for speciﬁc compounds, the resulting stopping power is within 2% of the results obtained using the experimentally derived parameters. The errors generally decrease with increasing kinetic energy of the incident proton, however, beyond 10 GeV we may expect some further errors growing and thus we limit our calculations to the

p region of proton kinetic energy Ek ∈ (10 MeV, 10 GeV).

3.4.2 Asteroidal Materials

In this section, we present the results of calculations of stopping power of protons in asteroidal materials. From the introductory Sec. 3.2.2 we know that phyllosilicates are important minerals in asteroidal material and that they usually contain water either as OH or H2O. In our calculations, we treat H2O as a single atom for which we use

I from literature and this value of I enters the formula Eq. (3.6) as a single Ii. Note that in this case the density correction is calculated by the algorithm a3. There are no analogous experimental data available on mean the excitation energy of hydroxyl.

116 However, because the bonding of OH is similar to H2O, we decided to approximate the

hydroxyl mean excitation energy by the mean excitation energy of H2O vapour. The

only other experimental data that are available for minerals in meteoritic material are

for Calcite for which we use the a2 algorithm.

We present the results for two asteroid taxonomic groups: the CM and CI chon-

drites. Because the mineralogy of these groups is not ﬁxed, we use the Murchison CM2

chondrite as a proxy for the CM group and the Orgueil CI1 chondrite as a proxy for

CI taxonomy group. The exact composition that is used in the calculation is shown

in Tab. 3.5 and is based on the results from X-Ray Diﬀraction (XRD) and Mössbauer

spectroscopy by Bland et al. (2004).

We compare the stopping power of these two asteroid proxies with pure Alu-

minium. The results are depicted in Fig 3.2. We can see that the stopping power is very

similar for all three materials. The tabulated results at distinct energies are presented

in Tab. 3.6. From the results, we can conclude that the stopping power of the asteroidal

materials proxied by CI Orgueil and CM Murchison is better than Aluminium across the

entire energy range of our calculation.

From the results presented in Fig. 3.2 and in Tab. 3.6, we see that the diﬀerence

in stopping power is low (starting at ≈ 7% in favour of Orgueil at 10 MeV and dropping to ≈ 3% at 10 GeV. The units of the results are MeV cm2 g−1 because the results are per

unit bulk density. When we include the density, we get the units of MeV cm−1. These

117 Table 3.5: The composition of taxonomic groups proxied by the Orgueil CI1 chondrite for CIs and the Murchison CM2 chondrite for CMs. The data are adopted from Bland et al. (2004). We denote Serpentine2 a mixture of Saponite and Serpenine and we assume that it is mostly Serpentine but with a diﬀerent density. Group Mineral Formula Mass Density Fraction [kg m−3]

CI Fo100 Mg2SiO4 2.4% 3220 Fo80 Fe0.3Mg1.7O4Si1 3.3% 3460 Fo60 Fe0.8Mg1.2O4Si1 1.5% 3690 Troilite FeS 2.1% 4700 Pyrrhotite Fe0.95S 4.5% 4580 Magnetite Fe3O4 9.7% 5200 Serpentine Mg3Si2O5(OH)4 7.3% 2600 Serpentine2 Mg3Si2O5(OH)4 64.2% 2550 Ferrihydrite Fe2O3(H2O)0.5 5.0% 3800

CM Fo100 Mg2SiO4 7.4% 3220 Fo80 Fe0.3Mg1.7O4Si1 2.2% 3460 Fo50 FeMg1O4Si1 5.0% 3810 Clinoenstatite Mg2Si2O6 2.2% 3300 Pyrrhotite Fe0.95S 2.9% 4580 Pentlandite Fe4.5Ni4.5S8 0.5% 5080 Magnetite Fe3O4 0.4% 5200 Serpentine Mg3Si2O5(OH)4 22.8% 2550 Cronstedtite Fe4SiO5(OH)4 58.5% 2950 Calcite CaCO3 1.1% 2720

Table 3.6: Stopping power per unit density of asteroidal material compared to Aluminium. The stopping power is in units of MeV cm2 g−1. The numbers in parenthesis are percent diﬀerence from Aluminium. 10 MeV 100 MeV 1 GeV 10 GeV CI Orgueil 36.58 6.012 1.850 1.815 (6.7)(5.8)(5.5)(2.8) CM Murchison 35.00 5.805 1.796 1.781 (2.1)(2.2)(2.4)(0.9) Aluminium 34.27 5.682 1.753 1.766

118 CI Orgueil

CM Murchison

Al ]

1 1

- 10 g · 2 cm · MeV [ Stopping power

101 102 103 104 Proton Kinetic Energy [MeV]

Figure 3.2: The stopping power of asteroidal materials compared to Aluminium. results are presented in Tab 3.7. We can see that inclusion of density in this case leads to larger diﬀerences in per cents.

Table 3.7: Stopping power of asteroidal material compared to Aluminium. The stopping power is in units of MeV cm−1. The numbers in parenthesis are percent diﬀerence from Aluminium. 10 MeV 100 MeV 1 GeV 10 GeV CI Orgueil 106.4 17.57 5.420 5.340 (15.0)(14.5)(14.5)(12.0) CM Murchison 103.8 17.23 5.332 5.293 (12.1)(12.3)(12.6)(11.0) Aluminium 92.54 15.34 4.734 4.767

Conclusion We have found that asteroidal materials have very similar shielding properties against high energy protons as Aluminium. This can also be enhanced by increasing the density of the material. In this case the results suggest that CM and CI chondrites

119 have over 10% superior stopping properties against protons with energies from ∼ 10 MeV to ∼ 10 GeV.

3.4.3 Asteroidal Minerals

In this section, we analyse the stopping properties of primary asteroidal materials which were discussed in Sec. 3.2.2. Based on the results of the previous section, we can expect the results to be relatively close to each other with diﬀerences on the order of per cents.

The results are presented in Fig. 3.3 which compares the stopping power of Ser- pentine, Saponite, Cronstedtite and Olivine to Aluminium. We also present these results plus also the results for Fayalite in Tab. 3.8 at discrete energies of incident protons.

Table 3.8: Stopping power of typical asteroidal minerals compared to Aluminium. The stopping power is in units of MeV cm2 g−1. The numbers in parenthesis are diﬀerence in per cent from Aluminium. 10 MeV 100 MeV 1 GeV 10 GeV Serpentine 38.13 6.214 1.902 1.851 (11.3)(9.4)(8.5)(4.8) Saponite 37.80 6.175 1.894 1.853 (10.3)(8.7)(8.0)(5.0) Cronstedtite 33.69 5.634 1.751 1.749 (−1.7)(−0.8)(−0.1)(−1.0) Forsterite 36.87 6.041 1.854 1.807 (7.6)(6.3)(5.8)(2.4) Fayalite 32.85 5.520 1.719 1.715 (−4.2)(−2.9)(−1.9)(−2.9) Aluminium 34.27 5.682 1.753 1.766

120 Serpentine

Saponite

Cronstedtite

Forsterite ]

1 1 Al - 10 g · 2 cm · MeV [ Stopping power

101 102 103 104 Proton Kinetic Energy [MeV]

Figure 3.3: The stopping power of asteroidal minerals compared to Aluminium.

Based on the presented results, we can conclude that Serpentine and Saponite

have signiﬁcantly higher stopping properties. This is mainly due to the presence of

signiﬁcant amount of water (hydroxyl) compared to other elements. It may be surprising

that Cronstedtite has a lot lower stopping power than Serpentine despite the similar

D Z E formula but from Eq. (3.4) we see that the zero order of stopping power ∼ A which for

Serpentine is ≈ 4% larger than for Cronstedtite and from Eq. (3.6) we see that the mean excitation energy of a compound is also proportional to this ratio and thus tending to reduce the mean excitation energy. We can also understand the relatively high stopping power of Forsterite compared to Fayalite for the similar reasons. For reasons explained later in the discussion, we only provide these results in per unit density format.

121 3.4.4 Discussion

If we look back at Eq. (3.4), we can see that for the stopping power, in units

MeV cm−1, S ∼ ρ and thus the stopping power of certain compounds with density included can be larger than other compounds for which it was larger per unit density.

This is for example the case of Fayalite where if the we calculate the stopping power in

MeV cm−1 we get larger values than for Serpentine, unlike if we calculate it per unit density. However, the measured density of the individual minerals, as presented in Tab. 3.5, is called the grain density (density of only the solid material in a mineral). However, when we consider asteroidal materials, we need to keep in mind that they are also characterized by the bulk density (the mass divided by the bulk volume) and that the grain density may not be representative of the bulk density which may be signiﬁcantly lower due to porosity of asteroidal material. Further, even grains are porous to some degree due to cracks on the order of µm but the ranges of these micro-porosities that we observe in meteorites is limited (Britt et al., 2002). This implies that presenting calculations as

“lost energy per unit length” introduces serious uncertainties.

Further uncertainties result from the method that we used, besides the approxi- mations already discussed in Sec. 3.3.1.1, it is also necessary to take into account that we consider the material as homogeneous layers of elemental materials. In reality, a meteorite is a mixture, which on microscopic level, is not very homogeneous due to diﬀerences in the composition of matrix and chondrules as well as due to vacancies, impurities and other crystal lattice defects. Further, our calculations neglect the eﬀects of the crystal

122 lattice altogether. And despite the fact that we have shown how various ways to determine the stopping power affect its value, we cannot claim that the uncertainties derived in Sec. 3.4.1 are, in general, applicable to all materials that make up asteroids. Also, the formulas given for the minerals in Tab. 3.5 may not reflect the reality. Minerals do not usually have a single chemical formula, rather they form solid solutions between various end-member states and thus have a range of compositions (e.g. as we have seen, the amount of Fe in the crystal structure can affect the stopping power to a non-negligible degree). In addition, minerals are quite often found intermixed on the micron scale (e.g.

Serpentine and Saponite in CI Orgueil). This has an important implication: it is difficult to a priori select an asteroid that would provide a specified stopping power per unit mass. The causes are two fold: first, there is significant uncertainty in the composition of asteroids; second, even if we have the exact mineralogical composition, the chemical composition of minerals varies and the degree of this variation within the material is uncertain a priori. And even a posteriori there are uncertainties: to provide consistent and reliable stopping power properties, the mined material has to be homogeneous, variations, for instance, in the iron content or density can be detrimental for the reliability of the shielding properties. One way these uncertainties can be mitigated is benefication or extraction of only required minerals and their processing to some normalized elemental content as well as homogenisation of the bulk density. We have already shown that removal Fe from Olivine can have a significant boost in stopping power.

An important issue that we have not treated relates to the processes inside the incident nucleus. The most important one being neutron production due to processes

123 described in the introduction section. As neutrons have no charge, they penetrate deeper

into the material, and even though neutrons are not contained in the GCR or SEP (free

neutrons have mean lifetime ≈ 14.7 min) their flux can become significant as protons and heavier charged nuclei traverse the medium. In general, the longer the path the charged particles have to traverse, the less the charged particle flux beyond the material and the higher the neutron flux. The protection against neutron flux depends on their kinetic energy; the terrestrial experience is typically with neutrons up to ∼ 20 MeV which are

produced in ﬁssion (Terrell, 1959) and fusion (the spectrum of neutrons in fusion reaction

is (1.5–20) MeV (Källne et al., 2004)) reactions. Typically, low energy neutrons (up to

. 10 eV) are eﬀectively stopped by Cadmium and Boron. Above these thermal energies, the absorption cross-section drops rapidly and the usual approach in reactor design is to

ﬁrst moderate the high energy neutrons through scattering in low mass number materials

(typically water). However, moderation itself can cause troubles as it can result in neutron capture and γ rays on the order ∼ 1 MeV; these are absorbed by high atomic number materials, e.g. lead.

For energies up to ∼ 20 MeV we have a relatively good understanding of neutron interactions in matter, including organic matter and the consequential dose. We can limit the dose by a clever combination of various materials as described above. Beyond

20 MeV we have little experimental data yet these energies are readily achievable from

GCR protons interactions in matter. The consequences of neutron radiation at these energies for living matter are uncertain. In calculations, the interactions of neutrons are mostly treated using Monte Carlo techniques. One problem these techniques have to

124 face in asteroidal material results from uncertain composition. The number of possible interactions of neutrons in matter is large and care needs to be taken to analyse what eﬀect a change in composition has on the interactions and slowing down of neutrons.

Also most of these techniques require the speciﬁcation of the geometry of the absorber.

In these Monte Carlo techniques, usually the problem is treated as a whole — stopping of charged particles and their nuclear interactions and consequently the interactions of neutrons.

3.5 Conclusion

We have veriﬁed that our calculations of stopping power of protons on several materials reproduce results published in literature. We have discussed the eﬀects of algorithms recommended for calculation of stopping power in materials where no experimentally derived parameters exist. We have calculated stopping power of protons in two proxy materials of CM and CI carbonaceous chondrites — the CM Murchison and the CI Orgueil — in Tab. 3.6. We have concluded that these materials, as speci-

ﬁed in Tab. 3.5, have better shielding properties against protons with energies between

10 MeV and 10 GeV than the typical space construction material Aluminium. Because the asteroidal materials would probably have inconsistent shielding properties, we have calculated the stopping power of major minerals found in meteorites. The results suggest that separation into individual minerals and depleting some those minerals of high atomic number elements may be beneﬁcial for their shielding properties. According to

125 the results presented in Tab. 3.8, typical phyllosilicates found in meteoritic material have higher stopping power than Aluminium; this is also true for the Magnesium end member of Olivine — Forsterite. On the other hand, the iron end member — Fayalite — has a lower stopping power than Aluminium. We have also discussed the uncertainties in our calculations as well as touched on the problem of secondary particles, namely neutrons.

This work shows that asteroidal material has a potential in providing radiation shielding comparable or better to Aluminium and that modest processing along with shield designs driven by our theoretical understanding of particle interactions can yield more eﬃcient shielding.

126 CHAPTER 4 LOSS OF WATER IN ASTEROIDS BY DEHYDRATRION

4.1 Background

The previous chapter dealt primarily with phyllosilicate rich materials that make up asteroids. It has been shown in Sec. 3.2.2 that certain CCs, are very rich in certain members of the Serpentine mineral group which in turn contain OH– embedded in their crystal lattices. When combined with a proton, they form molecular water.

The OH related absorption feature in (2.7–2.8) µm, often called the 3 µm feature, have been observed remotely in reﬂectance spectra of several Main Belt Asteroids

(MBAs) (Rivkin et al., 2015b) and Near Earth Objects (NEOs) (e.g. Rivkin et al. (2013,

2018)). This absorption is caused by stretching of the bond between the O and H atoms.

Since many phyllosilicates have both Fe2+ and Fe3+, which can exchange electron upon absorbing a photon of energy equivalent to approximately (0.6–0.7) µm, these also have observable absorption feature in this range. Rivkin et al. (2015a) suggest that whenever there is an OH stretching in the above mentioned range, there is also present an absorption relating to the electron transfer between the irons. And since Ch asteroid class is deﬁned by the iron electron exchange absorption, Rivkin and DeMeo (2019) suggest that every Ch asteroid is necessarily hydrated. They also claim that the presence of the elec-

127 tron transfer absorption in the Ch asteroids and most CM Chondrites is strong enough

evidence to link the Ch asteroid class with the CM Chondritic meteorites.

It has also been shown that MBAs can be injected into the inner solar system by

resonances working in conjunction with the Yarkovsky eﬀect (e.g. Bottke et al. (2002)).

Once in the inner Solar System, their perihelia can become small enough that the expe-

rienced temperatures might lead to disintegration of the hydrated minerals and the loss

– + of water e.g. by recombination OH + H −−→ H2O.

Being able to distinguish those asteroids that are anhydrous, whether they ever contained hydrated minerals or the extent to which the loss of water occurred, might provide additional restrictions on their history.

Recently, asteroids have also become objects of interest for their economic value, particularly as a resource of raw materials that have become increasingly costly to obtain from the interior of the gravitational well of the Earth. But also, as a means for In Situ

Resource Utilisation (ISRU), e.g. as a source for producing fuel in space or obtaining water without having to bring it from the Earth. In this area, understanding whether a given object contains water in some form is crucial. As has been mentioned above, remote tools oﬀer only very limited options to ascertain the hydration, videlicet the diﬃculty to observe the OH stretching feature due to terrestrial atmosphere as a direct proof of hydration (the claim by Rivkin et al. (2015a) does not imply the opposite, id est that the lack of the Fe charge transfer absorption implies the lack of hydration). Also, spectroscopy only probes the negligibly thin surface shell. It is reasonable to assume that

128 proﬁt seeking entities, as well as space agencies doing risk assessment, require at least certain type of probabilistic description of whether a given object has some water worth accessing or not.

To be able to begin to assess the probabilities of hydration of a given asteroid, one needs to understand its history ﬁrst, or the orbital probabilities. These can be estimated based on large number of numerical integrations of the evolution of MBAs such as by Bottke et al. (2002). Once orbits are available, in the second step, one can easily calculate the temperature distribution within the body, provided one knows well the thermal parameters of the object and, ideally, its shape. In the third step, once one knows the distribution of temperature for various orbits, one can evaluate whether, and to what extent, the temperature leads to a loss of water. Since initially, one described the orbits with probabilities, this would lead to the probabilities of dehydration.

Similarly to the process described above, Delbo and Michel (2011) analysed the probability of dehydration of asteroid (101955) Bennu. They concluded that dehydration is improbable due to a requirement of an extreme proximity of the asteroid to the Sun during its orbital evolution.

However simple the above description may look, it leads to very complex problems.

For example, if a temperature on the surface leads to the loss of water, and then the object comes close to a planet whose gravitation can distort its shape and lead to material restructuring such that the hydrated material comes to the surface. In principle, this

129 could lead to signiﬁcant dehydration within the volume (beyond the reach of thermal wave), if such event repeats.

Another set of problems are related to proper knowledge of thermal parameters which are not well constrained, especially since many astronomic observations report those packed into a single quantity — thermal inertia.

Yet another issue is that understanding of dehydration process is very limited, only recently several authors (Garenne et al., 2014; King et al., 2015b) started analysing thermal decomposition of meteoritic material. While dehydration of some components of hydrated meteorites (namely Mg Serpentines) has been studied under higher pressures relevant at core mantle boundary (Ulmer and Trommsdorﬀ, 1995). Studies under atmospheric pressure focused on the dehydration have been less common and experiments under vacuum, relevant to the outer space environment, almost non-existent.

The lack of data in the area of behaviour of water rich components of CCs moti- vates this chapter.

4.2 Introduction

4.2.1 Hydrated Minerals

Based on the studies by Bland et al. (2004) and Howard et al. (2015), the major

Serpentine species contained in CCs are Mg members (typically represented by Antig-

130 orite) and an Fe member, Cronstedtite. Other signiﬁcant Mg Serpentine species are

Lizardite and Chrysotile, however, the above studies did not diﬀerentiate between Mg

Serpentines (e.g. Bland et al. (2004) assumed Mg Serpentine to be given stoichiometrically as Mg2.9Fe0.1Si2O5(OH)4). Besides Serpentines, Bland et al. (2004) also mentions

Saponite which they list as combined with Mg Serpentine.

Tab. 4.1 overviews the available data from literature where the authors obtained mineralogy of various meteorites by XRD measurements and were able to distinguish between Fe and Mg members of the Serpentine mineral group (the table is composed from Tab. 3.2). It is possible to observe: 1. Hydrated minerals are abundant across several of the measured CCs groups. Speciﬁcally, the measured CV sample is devoid of any Phyllosilicates. 2. The CM Chondrites are speciﬁcally rich in Cronstedtite1, in some cases Cronstedtite is the most dominant component of the measured CM sample by weight. The data in the table, together with the delivery of asteroids to the near Earth space, imply the necessity of studying the process of dehydration of the aforementioned minerals. 1Cronstedtite is quite scarce on the Earth and its abundance in the CM Chondrites is an interesting question by itself.

131 Table 4.1: This table summarizes phyllosilicate abundances of CC subgroups expressed as weight percentages from Tab. 3.2. The term Serpentine in the table header is used solely to denote Mg Serpentines. The numbers in the second column in the parentheses are the number of samples in each data set. If there is more than one sample in the max group, three numbers are given in the format medianmin appropriate for each set. These numbers are to illustrate the abundance ranges of phyllosilicates in CCs. The data are based on published meteorite mineralogies by Bland et al. (2004); Howard et al. (2015). Cronstedtite Serpentine Phyllosilicates

11.4 74.5 74.5 C2-ung (4) 0.00.0 66.955.1 70.060.3

CI1 (1) - 71.5 71.5

26.3 66.0 87.6 CM1 (2) 24.021.6 63.661.2 87.587.5 58.5 82.8 84.5 CM2 (23) 26.80.0 42.822.2 74.156.3

CM2/1 (1) 24.2 62.4 86.6

CR1 (1) 9.5 57.9 67.4

60.0 60.0 CR2 (7) - 15.37.1 15.37.1 1.5 1.5 CR3 (2) - 1.41.3 1.41.3

CV3 (1) ---

132 4.2.2 Existing Studies

This section provides an in-depth overview of the literature related to the dehydration of phyllosilicates and CCs, since the process of dehydration is complex and still not perfectly understood.

4.2.2.1 Studies on Dehydration of Phyllosilicates

General changes in the crystal structure of various layered hydrated silicates due to heating were studied by Brindley (1961) who proposed two paths how OH can turn into H2O and consequently be liberated from the structure.

– – 2– (i) (OH) + (OH) −−→ H2O + O2 ,

+ – (ii)H + (OH) −−→ H2O.

In the first case, two neighbouring OH interact (as suggested by the process (i)) and the resulting molecular water can either escape from the free surface or, if this happens inside the structure, it has to diffuse to the surface, first through interstitials and then through micro cracks. Such process would leave the structure in high state of disorder (Brindley, 1961). At the free surfaces or in micro cracks, where protons can migrate readily, the process (ii) above can occur. However, in order to keep the structure electrically neutral, it is necessary to compensate the migrating charge of the protons

133 by cation diﬀusion while anions are mostly stationary. These general ideas were further

extended in the subsequent works overviewed below.

Speciﬁcally for Serpentine group minerals, based on studies by Brindley and Zuss-

man (1957); Steadman (1957), Brindley (1961) suggested the dehydration scheme for Mg

Serpentine: 2 Mg3Si2O5(OH)4 −−→ 3 Mg2SiO4 + SiO2 + 4 H2O (i.e. serpentine turns into

Forsterite, Silica and water). Further he provides a relationship between unit crystal cells

of Mg serpentine and Forsterite such that 8 unit cell volumes of Serpentine turn into 9

unit cell volumes of Forsterite.

In the previous study, aimed speciﬁcally at Serpentines, Brindley and Zussman

(1957) examined specimens that were heated isothermally at various temperatures (for

up to 12 hours) in air atmosphere by X-Ray Powder Diﬀraction (XRPD). The work pri-

marily focuses on relationship between crystal structures of Serpentines and the resulting

Forsterite, but they also provide quantitative thermal results. In the case of Antigorite,

they found that Forsterite appears between (625–650) ◦C, while Antigorite disappears be-

tween (650–700) ◦C. In the case of Lizardite, Forsterite becomes visible in XRPD in the

range (575–600) ◦C and the Lizardite signal disappears between (575–600) ◦C. The data provided by the authors suggest non-negligible variability based on the crystal packing

(Antigorite, Lizardite, Chrysotile) as well as within the same crystal packing (e.g. nature of Antigorite). The authors determined that the conﬁguration of Oxygen atoms in

Serpentine and the resulting Forsterite are similar suggesting that reorganisation occurs mainly in Si and Mg atoms.

134 Ball and Taylor (1963) studied orientations of dehydrated products of Lizardite

and Chrysotile with respect to the orientation of the original Serpentines. The exper-

iments were done mostly under water vapour pressure of ∼ 60 MPa but also under air

conditions. They proposed a complete dehydration scheme of Mg Serpentines regardless

whether the reactions proceed in dry air or in water vapour pressurized atmosphere. Their

idea postulates that during the temperature rise, donor and acceptor regions appear in

the structure between which cations migrate, protons from the acceptor region ﬂow into

– the donor region and bind with OH to form neutral H2O which leaves the system. This

also liberates Si4+ and Mg2+ cations which ﬂow in the opposite direction to the proton

ﬂow to maintain charge neutrality of the structure. The donor region is depleted and

pores are left in place. The process continues but in the acceptor regions, the Si4+ and

Mg2+ cations also have to ﬂow in opposite directions to preserve neutrality which leads to formation of Si rich and Mg rich regions. Oxygen anions are assumed stationary. In the Mg regions, Forsterite gradually nucleates. Finally, as the temperature increases to around 1000 ◦C, Enstatite forms in Si rich regions (or Talc in the case of hydrothermal dehydration). The authors also describe in lengthy detail the changes in atomic packing and cations arrangement in order to produce Forsterite. Schematically, I present the process in Fig. 4.1. Note that the stages described above mostly overlap both in time

(temperature) and space.

Brindley and Hayami (1963) examined Serpentine (no further explanation of what type of Serpentine is provided) dehydration under air atmosphere. From XRPD, they observed at around 600 ◦C a range of about 30 ◦C where the Serpentine becomes amorphous

135 Chrysotile 9Mg3Si2O5(OH)4

6Mg2+, 4Si4+ Acceptor region Donor regions 7Mg3Si2O5(OH)4 2Mg3Si2O5(OH)4 28H+

Anhydrite 600 °C pores 18H2O 9Mg3Si2O7

Mg-rich Si Si-rich region Mg region

Forsterite 800 °C

Enstatite 1000 °C

Figure 4.1: Dehydration of Mg Serpentines as proposed by Ball and Taylor (1963).

136 before Forsterite nucleates. The exact temperature ranges were found to be dependent on the grain size and heating rate. This process supports the idea presented in Ball and

Taylor (1963). From their TGA data, Brindley and Hayami (1963) conclude that the

finer grained samples dehydrate faster. Further, they claim that to fully dehydrate their samples, it sufficed to heat them at 570 ◦C for 18 hr. On further heating, they observed that Forsterite developed faster in the coarse grained specimens. If the transformation to Forsterite had been initiated on the grain surface, an opposite relationship would have been expected. Next, upon heating samples isothermally at various temperatures, they observed that formation of Forsterite proceeds very rapidly in the first (2–4) hr (e.g. at the temperature of 800 ◦C, almost 70% of Forsterite by weight is formed) followed by a very slow additional crystallisation within the following 100 hr. They argue (based on the

XRPD data) that fast dehydration leads to a greater disorder in the structure and thus to a slower recrystallization. Since the lack of order increases with the dehydration rate, they suggest that with the increasing rate of dehydration, the more volume of the crystal experiences the process (i) mentioned above which inhibits recrystallisation by providing a less ordered dehydrated structure rather than the proton migration suggested by the process (ii) which has a tendency to preserve the structure and ease recrystallization.

Weber and Greer (1965), whose experiments were done under water vapour pressure, found that Serpentines with higher Ni content have higher characteristic temperature of dehydration. Unfortunately, this work contradicts itself when the author claims that the lower stability of Serpentine is due to positional mismatch between layers which can be improved by wave-like or tubular structures (which are characteristic for Antig-

137 orite and Chrysotile respectively) however, in the conclusions they claim the opposite, that wave-like and tubular structures are less thermally stable which is not even supported by their own data (they determined average temperatures of dehydration at ∼ 103 kPa of water vapour pressure: 700 ◦C, 664 ◦C and 635 ◦C for Antigorite, Chrysotile and Lizardite respectively).

Brindley and Hayami (1965) further studied the dehydration of Mg Serpentines by XRPD to confirm the processes put forward by Ball and Taylor (1963). While they could confirm the first stage where the loss of water occurs and an amorphous phase is formed, based on a strong correlation between crystallographic orientations of the original

Serpentine and the resulting Forsterite, they suggest that the amorphous phase cannot be too disordered. Since in their previous study (Brindley and Hayami, 1963), they found, as mentioned above, that Forsterite nucleates and makes up to 80% of the product by weight as they increase the temperature and extend the time under that temperature and at the same time, no Enstatite was observed, they concluded that this requires that all MgO ends up in Forsterite which contradicts the creation of Si and Mg rich regions as indicated by Ball and Taylor (1963) (see Fig. 4.1). The crystallographic orientation relationship between the Serpentine and Forsterite as well as the previously determined fact that 8 unit cells of Serpentine turn into 9 unit cells of Forsterite restrict the amount of cations that can be exchanged. These indication led them to propose an alternate dehydration and recrystallization scheme to the one by Ball and Taylor (1963) where the donor regions become highly disorganized patches of silica while the acceptor region is very close in composition to Forsterite which, with increasing temperature, slowly nucleates taking all

138 the existing Mg. A thermally induced reaction between Forsterite and excess SiO2 then accounts for the Enstatite at higher temperatures. This is schematically illustrated in

Fig. 4.2.

Serpentine 9Mg3Si2O5(OH)4

6Mg2+, 1Si4+ Acceptor region Donor regions 4Mg3Si2O5(OH)4 2Mg3Si2O5(OH)4 16H+

Disorganized Forsterite 600 °C 3SiO2 12H2O 9Mg2SiO4

Crystallized Forsterite 800 °C 9Mg2SiO4

Forsterite Si, Mg 1000 °C Silica

Forsterite Enstatite

Figure 4.2: Dehydration of Mg Serpentines as proposed by Brindley and Hayami (1965).

In another study, Martin (1977) conﬁrmed the schema of Fig. 4.2 by Brindley and Hayami (1965) for temperatures below 800 ◦C, however, above 800 ◦C the opposite migration of Si4+ and Mg2+ proceeds fast and Forsterite and Enstatite spontaneously crystallise. Martin (1977) attributed an observed exothermic peak around 810 ◦C in

139 Diﬀerential Thermal Analysis (DTA) data to this crystallization. However, this picture

is in contrast with the multiple observations that Enstatite does not crystallize until

1000 ◦C.

More light on the processes that occur in Chrysotile at higher temperatures was shed in the study by MacKenzie and Meinhold (1994) who used Magic Angle Spinning Nu- clear Magnetic Resonance (MAS NMR) to investigate the phases that arise in Chrysotile due to heating. Their results indicate that two diﬀerent types of dehydrated material are formed, which are termed Dehydroxylate I and II. Dehydroxylate I forms ﬁrst and preserves well the atomic distances as well as the coordination numbers of Mg from the parent Chrysotile; this phase shows as amorphous in XRPD and gives rise to Forsterite.

While Forsterite nucleates from Dehydroxylate I, Dehydroxylate II forms and it is seen again as amorphous in XRPD, however, based on MAS NMR data, this phase has a well deﬁned Si positions. This amorphous phase is still stable above 800 ◦C, unlike Dehydrox-

ylate I. Further, the authors comment that above 800 ◦C the amount of Forsterite further

increases as well as the ﬁrst SiO2 islands appear which gain atoms from Dehydroxylate II.

It is concluded that the above mentioned exothermic peak in the DTA data is due to the

destruction of O layers and the emergence of SiO2. As the heating continues, Enstatite

appears in larger quantities. Above 1150 ◦C the reaction between Forsterite and Silica,

according to Brindley and Hayami (1965), enhances the crystallization of Enstatite. In

their conclusions, the authors suggest a very intriguing idea as to the existence of the

two amorphous phases. Dehydroxylate I is thought of as the result of dehydration of

the surface of the grain (or, in the case of Chrysotile, ﬁbres), water escapes immediately.

140 However, it had been shown by Ball and Taylor (1963) that under water vapour pres-

sure, the structure of Si and oxygen is not destroyed, while Mg takes on most of the

changes, which is consistent with Dehydroxylate II. As such, Dehydroxylate I reﬂects

the dehydration of the surface areas of the grains (ﬁbres) and Dehydroxylate II reﬂects

the dehydration occurring inside the volume of the grain (ﬁbre) where conditions can

approximate those of the dehydration under water-vapour pressure.

Studies by Hršak et al. (2005, 2008) focused more on the dehydration process

itself rather than on a detailed study of the crystallographic changes. They determined

that, in their Antigorite samples, the endothermic lattice breakdown occurs at 663 ◦C

and the loss of 13% of weight is assumed to be due to the loss of OH induced water

as well as due to the adsorbed molecular water. They observe exothermic formation of

Forsterite at 819 ◦C. In the Hršak et al. (2008) study they also measure the dilatation of

the material and they found that this is a two phase process, the ﬁrst contraction starts

at around 600 ◦C coinciding with the onset of the dehydration. The second contraction starts at about 860 ◦C, the approximate temperature of Forsterite crystallization. The

contraction due to the dehydration is about 0.45% and the contraction after the Forsterite

crystallization amounted to 9.17%.

The most recent study that focuses on all assortments of Mg Serpentines is by Viti

(2010). His study encompasses Antigorite, Lizardite, Chrysotile and polygonal Serpen-

tine and provides full TGA, DTG (diﬀerentiated TGA signal with respect to time) and

DTA data on dehydration of the aforementioned minerals along with Evolved Gas Anal-

ysis (EGA) and subsequent XRPD measurements. The author reports the total mass

141 losses measured at 990 ◦C 12%, 13%, 14% and 14 for Antigorite, Lizardite, polygonal

Serpentine and Chrysotile respectively. The Chrysotile sample with Ca impurities was

measured to lose 16% of the total mass. He also observed a doublet in DTA signal between

(550–800) ◦C for all the samples as well as the exothermic signal. Their exact positions and shapes were observed to vary with the Serpentine subtype, the DTA peaks occurred

(704–712) ◦C (main) and (747–761) ◦C (secondary), (649–652) ◦C (secondary) and 714 ◦C

(main), 653 ◦C (secondary) and 691 ◦C (main), 654 ◦C in Antigorite, Lizardite, polygonal

Serpentine and Chrysotile respectively (the ranges are due to multiple measurements).

The Forsterite crystallization endothermic peak is apparent between (820–826) ◦C. Antig-

orite samples exhibit a very weak additional exotherm around (880–889) ◦C. The author reports that varying grain size or heating rate only slightly aﬀect the positions of the peaks (ﬁner grain sizes tend to have peaks at lower temperature and in the case of faster heating, the peaks occur at higher temperatures). XRPD data show Antigorite up to

740 ◦C and no longer present in the 800 ◦C heated sample. Lizardite up to 775 ◦C and no longer present at 875 ◦C. Forsterite in the Antigorite sample appears from 740 ◦C and Enstatite from 1000 ◦C. However, in the Lizardite samples, while Forsterite appears at 775 ◦C and Enstatite is already visible from 875 ◦C. The author concludes from the

XRPD data that the exothermic reaction around 820 ◦C is due to Enstatite crystallization. However, his XRPD data are typically taken 100 ◦C apart and to support this, it would be useful to have the XRPD determined more densely.

All of the above studies focused solely on Mg Serpentines and most speciﬁcally, on Chrysotile, or Asbestos. Fe Serpentines were studied only rarely. Steadman (1957)

142 studied dehydration of Cronstedtite and published his ﬁndings in a single page paper.

The authors found that the iron in the lattice oxidises at 275 ◦C which is accompanied by a shrinking of the lattice. Further heating lead to additional changes in the unit cell dimensions. They observed emergence of a Spinel-like structure (however, not in all samples) around 700 ◦C. From XRPD they claim that the Oxygen atoms did not change their positions and are the same in both Cronstedtite and the Spinel phase. They found no traces of free SiO2. They suggest that the Fe ion in the Spinel phase was likely replaced by Si. On additional heating, Haematite forms above 750 ◦C. Even further heating leads to the formation of Cristobalite.

Brindley (1961) suggests that Cronstedtite turns into Fe3+, an Si Spinel phase, water, Haematite, and Cristobalite.

4.2.2.2 Studies on Dehydration of Meteorites

Dehydration studies of meteoritic samples are focused mostly on CI and CM Chon- drites as they contain the largest amounts of phyllosilicates (see Tab. 4.1). Garenne et al.

(2014) used TGA to study water content in 26 samples of CM and 7 samples of CR Chon- drites. Further, the authors also measured certain pure minerals, including Chrysotile,

Cronstedtite, Greenalite, Goethite, Ferrihydrite, Smectite and Calcite, however, they only publish the resulting TGA and DTG (they use derivative with respect to the temperature which is not ideal since it assumes that during the whole experiment, the temperature

143 rises linearly) in the appendix without any analysis or comments. Based on the data on the minerals they segment the dehydration process in 4 ranges, based on the temperature:

1. (25–200) ◦C:

• The loss of mass is due to the adsorbed water and the water in (2–50) nm

pores (mesopores).

• This typically presents itself as two peaks in DTG data.

• This range is most easily contaminated by terrestrial water.

2. (200–400) ◦C:

• The loss of mass is due to oxide-hydroxide minerals.

• Since they observed Ferrihydrite and Goethite to lose mass in this range, they

conclude that those are the minerals that aﬀect the mass loss in this stage in

the meteorites.

• They are unable to discern whether the minerals are products of terrestrial

weathering or whether they were formed extra-terrestrially.

3. (400–770) ◦C:

• The loss of mass is due to the decomposition of phyllosilicate minerals.

• The authors claim that in this region, Pentlandite and Pyrrhotite undergo a

decomposition which could aﬀect the mass loss.

4. (770–900) ◦C:

144 • The loss of the is mass due to CO2 emitted from the decomposing Calcium

Carbonates.

Garenne et al. (2014) report the total mass loss for CM Chondrites between (12.9–

16.9) %, which distributes into the four stages as: (200–400) ◦C, (1–4.3) %; (400–770) ◦C,

(5.8–12.9) %; (770–900) ◦C, (0–2.3) %. For CR Chondrites, the data showed a signiﬁcant mass gain which starts from about 400 ◦C and continues until the heating is completed.

The authors do not comment on the reason for the mass gain besides showing that it happens mostly due to the chondrules, by testing a pure part of the matrix which only showed some mass gain above 800 ◦C. The sample of GRO 95577 (CR1) was the only representative of CR Chondrites that did not take on a signiﬁcant amount of mass and its total mass loss amounted to 19.1 %. The sample of Orgueil meteorite (CI) lost in total 27.5 % of its mass due to heating up to 900 ◦C and the mass loss was distributed as: (0–200) ◦C, 7.4 %; (200–400) ◦C, 5.2 %; (400–770) ◦C, 12.2 %; (770–900) ◦C, 2.7 %.

They also analysed absorbance spectra of the meteoritic samples between (2.5–25) µm, however, they only analysed samples heated to 300 ◦C. Finally, the authors claim ﬁrst, that the water abundance follows abundance of phyllosilicates determined by XRD by

Howard et al. (2009). They also suggest that the positions of peaks in DTG can be used to determine mineralogy.

In another study, Beck et al. (2014) heated specimens of Sutter’s Mill CM Chon- drite and two other CM Chondrites. The two samples of Sutter’s Mill exhibited 6.7 % and

145 10.5 % total mass loss, where the range (400–770) ◦C contributed with 3.2 % and 3.7 %, respectively.

CI Chondrites were studied by King et al. (2015b), namely Ivuna (CI), Orgueil

(CI), Yamato 82162 and 980115 (CI-like) using TGA and transmission spectroscopy ((2–

25) µm). They also dehydrated several minerals that they assumed should constitute CI

Chondrites — phyllosilicates (including Cronstedtite), Fe oxide-hydroxides and carbonates. However, no details, besides DTG (with respect to the temperature) are provided.

Similarly to Garenne et al. (2014), they partition the temperature range into 4 regions, the ﬁrst two regions are the same as in Garenne et al. (2014) but they deﬁne the range for phyllosilicates as (300–800) ◦C, partly overlapping the range for oxide-hydroxides, and (800–1000) ◦C for carbonates. They found the total mass loss (average) between

(25–1000) ◦C in Ivuna samples to be 28.7 %, distributed as: (200–400) ◦C, 5.1 %; (300–

800) ◦C, 15.4 ; and (800–1000) ◦C, 1.8 %. For Orgueil, the total mass loss was 30.3 %:

(200–400) ◦C, 5.6 %; (300–800) ◦C, 14.8 ; and (800–1000) ◦C, 2.1 %. For Yamato 82162,

the total mass loss amounted to 14.1 %: (200–400) ◦C, 2.2 %; (300–800) ◦C, 4.8 ; and (800–

1000) ◦C, 2.0 %. And for Yamato 980115, 18.8 % in total and distributed: (200–400) ◦C,

3.2 %; (300–800) ◦C, 6.3 ; and (800–1000) ◦C, 1.6 %. They suggest that the mass loss in

Ivuna in the oxide-hydroxide range is due to Ferrihydrite based on their mineral data and

the mass loss in the phyllosilicate range is due to Serpentine and Smectite. Further they

also admit that the phyllosilicate range overlaps with the decomposition of Carbonates

◦ which release CO2 between (600–800) C. They suggest that the mass loss in the range

(800–1000) ◦C is caused by Troilite.

146 King et al. (2015b) conclude that the water contents of Ivuna, Orgueil, Yamato

82162 and 980115 are 18.7 %, 18.3 %, 6.1 % and 8.1 % respectively. They also admit signiﬁcant discrepancy in their results if they compare them with data from bulk modal mineralogy and the TGA results of pure minerals. If they take into account the amount of Antigorite and Saponite and Ferrihydrite determined from XRD and the amount of water lost by those minerals in the analogous TGA experiments, the total amount of water should be ∼ 6.5 % for Ivuna or Orgueil which is by a factor of three inconsistent with their results obtained through dehydration of the meteorites themselves. The similar issue is probably in Garenne et al. (2014) as noted by King et al. (2015b).

To conclude the part on dehydration of meteoritic materials, it is unavoidable to add a few notes of caution when relying on these results. While the results obtained by

Garenne et al. (2014); Beck et al. (2014) are important, there are several shortcomings of their studies.

Firstly, as mentioned above, they suggest that it is possible to use DTG peaks to determine mineralogy. While they admit that there might be an issue since the composition of terrestrial minerals and extra-terrestrial mineral components of meteorites do not match, probably a more signiﬁcant issue is due to their experimental setup. In their TGA experiment, they put the sample into a pan which they cover with a pierced lid. Such setup allows pressures up to 0.2 MPa to develop inside the pan, depending on the size of the hole in the lid. Increased pressures will inhibit dehydration by exerting pressure on the surface of the grains and thus preventing water molecules to leave the system. Further, the thermodynamics of the system is very diﬀerent to the one with a

147 free surface. This not only leads to changes in positions of endothermic and exothermic peaks but also to the changes in the slopes of mass loss curves and consequently changes in DTG curves.

Secondly, their attempt was to characterize CCs by their mass loss, yet, as the authors also admit, the range (0–200) ◦C can be tainted by terrestrial contamination to various degrees. But the percentages of mass loss mentioned in their paper are calculated as the mass lost in a given range divided by the total mass of the sample before heating, and because this mass also includes the varying amounts of terrestrial water in the range

(0–200) ◦C, the percentages, e.g. in the range (400–770) % are also aﬀected by the amount of terrestrial water in the sample and as such do not characterize the water content of the non-terrestrial water. This issue also aﬀects the results by King et al. (2015b). A little less skewed results could have been gained by subtracting the amount of water lost in the (0–200) ◦C from the total mass of the sample and calculate the percentages from this corrected amount.

4.2.2.3 This Study

As evident from the above literature overview, I have been unable to ﬁnd a study that would systematically analyse the dehydration of individual components of CCs which would allow to describe the dehydration of any particular asteroidal matter whose mineralogical make-up is known. As such, the primary scope of this work is to provide input data for calculations illustrated in Sec. 4.1. Additional data serve to extend the knowl-

148 edge about the processes that occur within Serpentine group minerals upon heating and which are relevant to the small bodies in our Solar System.

More speciﬁcally, the aim is to describe the dehydration of Antigorite, Lizardite and Cronstedtite (Chrysotile was excluded from the study to ease the mind of Health and

Safety Oﬃcers), ﬁrst, under an inert atmosphere, to establish a baseline comparable with existing works (especially the ones described in Subsection Studies on Dehydration of

Phyllosilicates). Studies in atmosphere are important for another reason. They enable to study various parameters affecting the dehydration (both the onset temperature as well as the speed of dehydration) easily. It has proved very difficult to do TGA experiments under vacuum (even weak vacuum). Although several TGA devices are claimed to be vacuum ready by their respective manufacturers, it is often only for the purposes of evacuating the heating oven initially and then flood it with inert gas to minimize any oxidizing reactions.

Some of the dehydration experiments are limited by the amounts of sample that was available, speciﬁcally, Cronstedtite, which is a rather rare mineral on the Earth and its availability prohibits any large volume experiments.

Additional data related to dehydration, such as spectral evolution with temperature, or mineralogy are not part of this study although such data are planned to be published in a later work.

149 4.3 Samples and Techniques

4.3.1 Samples

Based on the mineral abundances of CCs in Tab. 3.2 (or Tab. 4.1), the Serpentine mineral group represents most of the hydrated minerals both by weight as well as by volume (Howard et al., 2015). Current data however, do not distinguish in more detail between the various members of the Serpentine group except between Cronstedtite and the other Mg Serpentines. Since the most common members of Mg Serpentines are

Antigorite, Lizardite and Chrysotile, the experiments are focused on those minerals with the exception of Chrysotile.

Chrysotile is one of the most common types of asbestos (hence its thermal decomposition has been the most studied) and studying such materials has become more difficult recently due to increasing demands from health and safety institutions. Further, its tubular and fibrous nature make analysis very difficult since it is almost impossible to grind it into powder (and grinding would require very complex measures to prevent asbestos contamination of a laboratory). A way to reduce its size might be using scissors but then, it is difficult to compare the TGA of the other, more finely ground, Serpentines.

For these reasons, I decided not to pursue any investigation of a thermal decomposition of Chrysotile.

150 4.3.1.1 Antigorite

Antigorite samples were obtained from Northﬁl’s Matheson Processing Facility in Canada in a pre-ground form. The material was sieved into four grain size ranges:

< 53 µm, (53–106) µm, (106–212) µm and > 212 µm. All these sizes were used in TGA analyses. However, since both XRD and FTIR are sensitive to grain sizes and require as random orientations of the grain surfaces as possible, only < 53 µm grain size was used for those analyses.

4.3.1.2 Lizardite

Lizardite samples were also in a pre-ground form. The Particle Size Analysis

(PSA) analysis showed that it was ground to a signiﬁcantly smaller grain size than Antig- orite and it was not feasible to sieve it to the same ranges as Antigorite, speciﬁcally, there was not enough larger grains to supply the ranges > 106 µm. Therefore, Lizardite was used unsieved.

4.3.1.3 Cronstedtite

Approximately 13 g of Fe Serpentine, Cronstedtite, was procured from the Col- orado School of Mines in the accessory form on a larger sample of Pyrite. Cronstedtite crystals were picked by hand using tweezers and a binocular microscope. From this

151 lot, I hand picked the purest crystals without any visible contamination and carefully

ground them using a pestle and mortar. This resulted in approximately 4 g of ground

Cronstedtite which I sieved into the same particle size range as Antigorite.

4.3.2 Methods

4.3.2.1 TGA

The initial TGA analyses were performed at the KSC using TA Instruments

SDT Q600 capable of simultaneous DSC and TGA measurements in a single experiment.

The instrument uses two parallel beams, one for a sample, the other as a reference; each

works as scales and a thermocouple to measure the diﬀerence in the mass and the tem-

perature between the sample and the reference. The sample is put in an Aluminium

oxide crucible and positioned at the end of the beams on the Platinum thermocouple.

The oven part of the machine then slides onto the thermocouples. The device has the

option of purge gas connection to minimize any oxidising reactions.

Subsequent experiments were carried out at the UCF at the AMPAC using the

same type of instrument (SDT Q600). The reason for switching to this location was

easier accessibility of the instrument a better control over the operations.

The sensitivity of the balance mechanism for both instruments is 0.1 µg and ac-

curacy ±1 %. The accuracy and precision of the thermocouples are ±1 ◦C and ±0.5 ◦C respectively and the sensitivity to the temperature diﬀerence between them is 0.001 ◦C.

152 The accuracy and precision of the heat flow measurements are better than ±2 %. The preceding values are the best achievable and provided as the machine specification by the manufacturer. To achieve these parameters, precise calibration of the instrument is required with respect to all measured quantities — mass, temperature, temperature difference and heat flow. With time these values degrade, most often due to the beams which provide measurement signals. Other issues are related to pores and cracks in the furnace and improper purge gas.

Calibration Since the instrument at UCF AMPAC showed signiﬁcant drifts of the heat ﬂow baseline2 (almost 100 mW) as compared to the instrument at KSC (∼ 10 mW), calibration of the instrument had to be executed.

Weight calibration consists of two runs, one with empty beams and second with manufacturer supplied standard masses. The major reason to calibrate is a diﬀerent responsiveness of the beams material as the temperature increases and, as the inert gas changes density with temperature, to account for a changing buoyancy of the beams in the inert gas ﬂow.

Temperature calibration is carried out by melting a known standard and determining the position of DTA peak. Initially we used both Tin and Zinc. However, after several calibrations, the temperature was only calibrated with a newly obtained Tin standard.

2Ideally, an instrument without any sample should show zero heat ﬂow as the temperature increases, since the amount of heat to the sample and reference beams should be the same, however, this is never true due to the asymmetry of the position of the beams inside the oven. Baseline drift is either a continuous increase and/or decrease of heat ﬂow with temperature when the experiment is run with empty beams. High quality DSC measurements are typically burdened by drifts on the order of µW.

153 Finally, the heat ﬂow, or DSC operation of the instrument, was calibrated with a

Sapphire disk and using known values of its heat capacity from the literature. The data was supplied also by melting Zinc and comparing the calculated heat of fusion with the one from literature.

Addressing the heat ﬂow situation by recalibration, discussions with the manufacturer and testing resulted in replacement of the beams and subsequent recalibration of the instrument and implementation of new experimental protocols. This improved the heat ﬂow drift to approximately 60 mW, which despite still quite high, became stable across experiments.

The new protocols were speciﬁcally:

1. Each experimental day had to be started with a preheat run with empty pans.

This was to address a more uniform initial state of the oven. It was also helpful in

monitoring the quality of the heat ﬂow and it possible deviations.

2. Both the sample and the reference crucibles were marked with a tiny dot using a

graphite pencil. This enabled reproducibility of the position of the sample crucible

during calibration and individual experiments.

3. Before tarring the scales, which precedes each run, (15–20) min wait time was intro-

duced to make sure that the beams are stable both due to vibrations after putting

the sample crucible onto the beam as well as the vibrations induced by closing of

the furnace and introduction of inert gas ﬂow. Similar procedure was followed after

putting the sample into the crucible and the crucible onto the beam. After closing

154 the oven, another (15–20) min wait time was observed to stabilize the scales again

before starting the procedure.

4. On a periodical basis, the oven was cleaned by heating it to maximum temperature

and keeping it there for 20 min to remove all possible products that might have

volatilised from other experiments and were stuck on the sides of the oven assembly.

In order to evaluate the performance of the instrument at AMPAC, UCF, several types of tests were undertaken. Calcium Oxalate (CaC2O4 · H2O) is not a recognized

TGA calibration standard, however, its thermal decomposition has been mapped very

well and it is often used as a compound to verify machine’s precision as well as accuracy

of the results. It has three decomposition steps: 1. CaC2O4 · H2O −−→ CaC2O4 + H2O,

2. CaC2O4 −−→ CaCO3 + CO, 3. CaCO3 −−→ CaO + CO2, which occur at distinct tem-

peratures. The veriﬁcation is based on stoichiometric losses of individual decomposition

steps which are, respectively: 1. 12.3 %, 2. 19.2 %, 3. 30.1 %.

The decomposition of Calcium Oxalate on SDT Q600 at AMPAC, UCF after the

ﬁrst batch of calibrations (before the change of the beams) is displayed in Fig. 4.3. The

results are very close to the stoichiometric values (note that they depend also on the

purity of the provided sample).

As mentioned above, another performance test applicable for SDT Q600 is the

baseline drift, speciﬁcally, the mass baseline and the heat ﬂow baseline. Both baselines

should be done with empty beams according to the manufacturer. The baseline drift test

from the same day as the above test with Calcium Oxalate is shown in Fig. 4.4. The

155 159.1 °C 61.7 % 100 12.3% H2O 474.1 °C 90

194.8 °C 80 18.8% CO

) 710.6 °C % ( 70 t h g i

e 522.3 °C

W 60

30.7% CO2 50

40 784.6 °C 30 0 100 200 300 400 500 600 700 800 900 1000

Temperature T (°C)

Figure 4.3: Decomposition of Calcium Oxalate to verify the performance of TGA measurement. mass drifts is about 6 µg which is a very good performance. The DSC baseline drift is about 50 mW which is not so good.

The performance significantly changed about 5 months after the verification which was only discovered in a posteriori data analysis. The decreased performance in mass baseline drift is plotted in Fig. 4.5. The baseline drift of mass increased by over an order of magnitude and due to this, all runs done on the instrument at AMPAC, UCF, which are not preceded by an empty beam preheat run, are calculated with an error that is based on mass drift 100 µg. The excessive drift was confirmed in several subsequent runs that were executed before at the start of an experimental day.

156 4 10

2 20

2 Weight (µg)

40 Heat Flow (mW)

50 6

8 60 0 200 400 600 800 1000 Temperature (°C)

Figure 4.4: Baseline drift of empty beams for SDT Q600 at AMPAC, UCF.

157 0 20

20 Weight (µg) Heat Flow (mW)

60 60 0 200 400 600 800 1000 1200 Temperature (°C)

Figure 4.5: Decreased performance of empty beams as compared to Fig. 4.4.

158 4.3.2.2 Methods of Analysis of TGA and DSC Data

In order to achieve consistency and be able to compare data among various ex-

periments, procedures, described below, were used.

Terminology

Derivatives Small changes in measured values are sometimes diﬃcult to see.

To better analyse and understand the data, derivatives of the measured values can be

used. A DTG curve (the ﬁrst derivative of the mass-temperature curve) is often used to

describe the mass loss event; second order derivatives can also be useful. In this work,

all numerical ﬁrst order derivatives are determined by taking a point on the curve, where

the derivative is supposed to be determined, and the n/2 preceding and ensuing points and fitting them with a line using the least squares method (id est, the derivative curve is smoothed using n points). This window then moves to the following point and the process is repeated. The second derivatives are calculated using a similar method but by fitting a second order polynomial. The amount of smoothing points was typically 60 for the first derivative and 120 for the second derivative. However, in several cases it was necessary to increase the amount of points up to 300 for the first derivative, the second derivative was then smoothed using also at most 300 points. Since the data record rate in all the experiments is 2 data points per second, this results in smoothing by 10 ◦C and 20 ◦C for the first and second derivatives respectively based on 60 and 120 point smoothing.

This approach is necessary to remove high frequency oscillations brought about by the

159 discrete nature of the data. This method introduces several artiﬁcial problems, e.g.

the point where the derivative attains an extremum or crosses zero is, to some extent,

dependent on the smoothing.

An Onset Point An extrapolated location on the x-axis where a certain event

starts can be used to determine a starting moment of a transformation which is depicted

as a peak in a heat ﬂow curve plotted against temperature. An onset point is deﬁned as

an x coordinate of an intersection of a tangent to the curve before the transformation with

the tangent at the inﬂection point on the left portion of the peak of the transition. This

concept is illustrated in Fig. 4.6 on the melting heat ﬂow curve of Zinc. The tangents are

determined from a given point by linear regression through surrounding points (similar

to the way the derivatives are determined, however, fewer points are used in this case,

typically 36 points corresponding to a temperature range of 6 ◦C for the commonly used heating rate of 20 ◦C min−1). The inﬂection point is determined by the zero of the second derivative calculated from the given curve. The onset point is a standard method to determine melting points in DSC curves. Unlike the DSC peaks, it is more impervious to shifting to higher temperatures with increased heating rates. The ﬁrst tangent is sensitive to selecting the point at which it should be determined. This will be discussed below.

An Endset Point An analogy of the onset point for the right side of the peak.

Analysis Methods The aim of this study is to obtain information on temperatures at

which the minerals of CCs start losing water, how much water they lose and the kinetics

160 0

Onset x: 419.527 °C

) W

m -5 (

w o l F

t a e H -10

-15 400 405 410 415 420 425 430 435 440 445

Exo Up Temperature T (°C)

Figure 4.6: Illustration of the concept of onset point on DSC melting curve of Zinc.

161 of the process. Determining the content is based on a simple calculation directly from the TGA data. One should take the difference between the mass of the sample before transformation takes place and after it has finished. In a TGA curve this would ideally be delimited by the portions of the curve which are horizontal. However, several steps of dehydration often overlap each other and are not separated by a horizontal portion of the mass loss curve. In this case, the mass loss of a given steps strongly depends on the points that are selected as start and end points of that transformation. Since the transformations are not clearly separated, it is never possible to specify this exactly. In this case, a consistent approach should be selected. There are several options how to demarcate the temperatures which determine the change in mass: 1. as onset and endset of the DSC peak, 2. as onset and endset of the DTG peak, 3. based on points where the second derivative crosses zero (inflection points). If the start or end part of the transition is on the “flat” portion of the TGA curve, the first two methods will typically provide temperature at which the loss has already started or has not ended yet (which can lead to significant amount being neglected).

The first method also requires quality DSC data without baseline drift to be able to identify the peaks onset and endset. In the case of data for this study, this is rarely the case or not always satisfied for all the peaks (weak peaks tend to be less visible due to heat flow drift).

The peaks on the DTG curve are often observed with superimposed small peaks or their shape is distorted to a larger or a smaller temperature by either numerical artefacts or noise in the original data. Fig. 4.7 illustrates such a case. Neither of the two DTG

162 peaks can be determined from the signal maximum. One option is to further smooth the

ﬁrst derivative or use the more smooth second derivative as done in the Fig. 4.7. One

can also see in Fig. 4.7 that the distance in temperature of the superimposed smaller

peaks peaks is about 40 ◦C. These issues may be due to either two transitions occurring

at similar temperature with similar mass losses or due to the sample behaviour as the

water makes its way out, it can also be noise. Consequently, these eﬀects cause signiﬁcant

(higher than those due to the measurement) uncertainties in determining of transition

temperatures from calculated DTG signals.

100 2

-3 n 1x10 d

D D

98 0.08 e e r r i v . i v .

W W e

96 e i g i 605 °C g h h

0.06 t t

0 d ) 0x10 ( W 2 %

94 ( ( W e

i t g e h h i g g t i

h ) e 0.04 t

92 / )

W d

/ (

d T ( ) -3 T

(

-1x10 )

90 598 °C % 2

(

0.02 / %

° C

) ° C 88 ² )

0.00 86 -2x10 -3 400 450 500 550 600 650 700 750 800 850 900

Temperature T (°C)

Figure 4.7: An Antigorite sample heated to 1000 ◦C (for clarity, only the range (400–900) ◦C is displayed) is an example of unclear DTG peaks. The presented red DTG curve is calculated with 60 point smoothing (and displayed with its signum is inverted). The low temperature DTG peak shape is due to noise in the data. The red point at 598 ◦C denotes the peak position if the DTG curve is calculated across 300 points (the two peaks disappear). The green curve is the second derivative of the TGA curve calculated across 120 points. The green point at 605 ◦C suggests the DTG peak position by the zero of the second derivative.

163 Given the issues illustrated in the previous paragraph, the start temperature of the mass loss is determined as the position on the x axis (temperature) where the first derivative of the TGA curve crosses zero the last time before the event, if it does not, the zero of the second derivative is used, and where the second derivative of the TGA curve is zero as well. A little subjective judgement is also used when the data contain significant noise (such as the derivatives oscillating before the event, crossing zero several times). If two events are overlapped and difficult to distinguish in this way, since where the second event begins, the first one is still losing mass causing a slope of the TGA curve — the

ﬁrst derivative never crosses zero; the point is determined from the zero of the second derivative).

In the cases where the DTG peak is determined, its position is estimated as follows: in the case of a visually symmetric peak (the smoothing is gradually increased until 300 points are reached), its position is determined from its maximum (or minimum). In other cases, such as illustrated in Fig. 4.7, its position is determined from the zero of the second derivative of the TGA curve.

The point which describes the start of the dehydroxylation is determined as the onset point on the TGA curve. The point from which the ﬁrst tangent is calculated is the same as the point described above for the start of the mass loss (determining the tangent using the 36 point method described above, in the cases of extremely steep portions of

DTG curve, only 12 points are used). It is important to note that the onset point is not really the point where the mass loss starts, e.g. in the case of Calcium Oxalate depicted in Fig. 4.3 the mass loss between the point at which the ﬁrst tangent is evaluated and

164 the onset point is 3.1 % which is already 25 % mass loss in that transition step. It serves the purpose of a measure.

The best way to describe a phase transition is from DSC curve peaks. If the obtained data have “nice” DSC curve without baseline drift, which makes DSC analysis difficult, the onset point is also obtained from the DSC curve by the method described above. Comparison of three curves is shown in Fig. 4.8. All curves are significantly affected by baseline drift. The major drift for the green curve occurs before the transitions.

The ﬁrst peak around 600 ◦C is still within the drift and also its intensity is small and

the point of the ﬁrst tangent, for the onset point, would have to be determined based on

zero of the second derivative. The two following peaks are strong enough to provide good

data. Both the red and blue curves show signiﬁcant drift. But since the endothermic

reaction of Calcium Oxalate is strong, so that the peaks present well, the data can be

used (the ﬁrst peak is in the region where the drift was quite small). The ﬁrst transition

in the blue curve is very diﬃcult to estimate using the onset point. The second transition

can be estimated but the third one is almost invisible in the curve.

4.3.2.3 Parameters Aﬀecting the TGA and DSC

There are several external parameters that aﬀect the results of the experiments

which are either experiment related (heating rate or isothermal experiment, inert gas

ﬂow, mass of the sample) or sample related (grain size, adsorbed water content, oxidation

potential).

165 -2 Antigorite (UCF) Antigorite (KSC) -3

CaOx (UCF) ) g / W (

Q -4

) d z e i l a -5 m r o N (

w o l -6 F

t a e H -7

-8 100 200 300 400 500 600 700 800 900 1000

Exo Up Temperature T (°C)

Figure 4.8: DSC curves of Antigorite and Calcium Oxalate (CaOx).

Grain Size Literature on the eﬀect of the grain size in TGA is rather controversial.

Carthew (1955) showed that the area under the DTA curve of Kaolinite does not depend on the grain size if various grain sizes are prepared using the sedimentation method.

He also claims that below 2 µm, the heat of dehydration does not depend on the grain size. Bayliss (1964) conﬁrmed this and pushed the argument much further, he argued that grinding damages crystal structure, decreases crystallinity of the sample and eases dehydroxylation which is the reason why other researches observed a lower temperature of the transformation with a smaller grain size. On the other hand, research by Brindley and Hayami (1963), clearly shows diﬀerent kinetics of dehydration of Serpentine between

ﬁne and coarse grained samples. Further more, the discoveries of Ball and Taylor (1963);

Brindley and Hayami (1965); Martin (1977); MacKenzie and Meinhold (1994) which describe the process of dehydroxylation and their results converged into a very reasonable

166 model (see Sec. Studies on Dehydration of Phyllosilicates), while not directly contradicting the work of Carthew (1955), oppose the conclusions of Bayliss (1964) since, it would be difficult to reconcile the same temperature of dehydration and a changing surface to volume ratio and as such, different methods of dehydration. Also, Földvári (2011) in his handbook notes that the grain size affects the observed dehydration temperature.

Another reason for the grain size eﬀect on the observed temperature of dehydration and in general the shapes of TGA and DSC curves can be due to thermal coupling between the grains and the crucible and other grains. As such, the experimental setup can play role and for this reason I decided to run several experiments with varying grain sizes, to determine their eﬀect on the TGA and DSC curves.

Adsorbed Water Several works on the dehydration of meteoritic material, mentioned in Sec. Studies on Dehydration of Meteorites, show a rather pronounced initial loss of mass due to adsorbed water (e.g. Garenne et al. (2014) in his dehydration of the sample of ALH 84033 in Fig.1 shows approximately 5 % between (0–200) ◦C). The initial experiments on Antigorite showed no signiﬁcant mass loss that would correspond to adsorbed water. However, several runs on Cronstedtite displayed a non-negligible amount of mass loss between the 50 ◦C and the start of the main dehydroxylation event, typically, the mass amounted to at most 1 % for the smallest grain size. Several of the experiments thus included varying time of isothermal heating around 130 ◦C with the attempt to remove the adsorbed water. See later for more information.

167 Oxidation In several cases, oxidation reactions were noticeable despite N2 gas ﬂow.

Interestingly, those runs were almost uniquely done at KSC suggesting that the purity of

N2 may have an eﬀect. There were not many possibilities to avoid these cases.

Inert Gas Flow In all experiments, the inert gas ﬂow was N2 and its ﬂow rate was set to 100 ml min−1. Since the ideal goal was to achieve results under vacuum conditions, there was little reason to study variation in this parameter on the results.

Mass of the Sample The larger the mass of the sample, the smaller the resolution but the better the sensitivity to a smaller mass loss. The DSC peak inevitably shifts to higher temperatures with size of the sample. The kinetics (the slope) of the TGA curve can also be expected to change. Since I planned to run subsequent experiments such as

XRD and FTIR where the more the sample, the better, tests to quantify the eﬀect of mass of the sample between (5–50) mg were undertaken.

Heating Rate Two types of experiments can be performed with SDT Q600. In ramp mode, the temperature increases linearly at a preset rate. In isothermal mode, the instrument heats up to a preset temperature and keeps the temperature for a predefined time. Combination of both is possible as well. Ramp rate affects both the resolution as well as the sensitivity, a faster heating rate increases the sensitivity and decreases the resolution. They also affect the position of the DSC peaks, faster heating rates shift the

DSC and DTG peaks to a higher temperature. For that reason, various heating rates were tested to determine the best ones for Serpentine dehydration. Further, combination

168 of ramp and isothermal modes were also run in order to determine the dynamics of dehydroxylation.

4.3.2.4 Temperature Programmed Desorption (TPD)

In TPD, one observes the rate of water molecules leaving the sample as it is heated. Its major advantage is that the method is naturally designed to operate under very hard vacuum (∼ 10−10 torr). However, the method is not directly comparable to the results obtained by TGA. The TPD runs were done by Alexandr Aleksandrov at

Georgia Institute of Technology (Georgia Tech), School of Chemistry. The method is brieﬂy described by Hibbitts et al. (2011) on Lunar samples and by Poston et al. (2013) on asteroid material simulants. Since the method was adjusted for this project a little, a short overview follows.

1 mg of sample was put on a resistive heating strip made of stainless steel with the size 20 mm × 5 mm × 0.025 mm. 0.002 ml of water was added to create a slurry which was spread over the area of the heating strip and dried under 50 ◦C to form a layer with good thermal coupling to the strip. The strip was placed on a horizontal sample holder inside a vacuum chamber. The vacuum chamber was then pumped for three days while being kept at 127 ◦C to remove any adsorbed water. The vacuum chamber was evacuated to 5 × 10−10 torr prior to increasing the temperature further. The vacuum chamber was connected to PrismaPlus Quadrupole Mass Spectrometer (QMS) with electron impact

169 ionizer providing ﬂux of 70 eV electrons to ionise H2O molecules which were selected by

their molecular mass of 18 u upon impacting the detector.

Due to the large temperature range of the experiment ((30–880) ◦C) it was neces-

sary to employ two diﬀerent detectors which forced the experiment to be split into two

phases. During the low temperature regime, when the rate of water molecules leaving

the sample is low (and thus the associated equilibrium pressure in the chamber), a more

sensitive Secondary Electron Multiplier (SEM) was used. At higher rates, a Faraday cup

was used. Thus, in the ﬁrst run, the sample was heated (30–500) ◦C with the SEM as the

detector. Then the sample was let cool to about 30 ◦C and the second run followed with

the Faraday cup as the QMS detector. The heating rate in both phases was 20 ◦C min−1.

The above two phase design results in data with incomparable units over the range

(30–880) ◦C. I decided to compile the data with the assumption that at the splitting

temperature, when the Faraday cup starts to be used, its intensity should correspond to

the intensity of the SEM detector. Thus, I scaled the data from the Faraday cup based

on their values at 500 ◦C to the data from the SEM. This requires several assumptions:

1. the responsiveness to the increase in pressure of water molecules of both detectors is the same, 2. the sample starts cooling immediately when then end temperature of the

ﬁrst run is reached (i.e. there is no inertia), 3. once the sample starts to cool down, water molecules stop leaving its surface. Most of these assumptions can be considered satisﬁed. The experiment design is such that water molecules should always end up in the QMS detector and as the thermal velocity of a room temperature water molecule in vacuum is ∼ 600 m s−1, the delay between the release of the molecule and its capture by

170 a QMS detector should be in the range (0.1–10) ms even after colliding with the wall of the vacuum chamber. Due to a very low mass of the sample, any inertia should not be an issue. The last assumption is not possible to circumvent other than by very fast cooling which might damage the thermal coupling and I assumed that the water molecules leaving the system after the heating is stopped do not aﬀect the result signiﬁcantly, if they do, this might shift any data obtained with the Faraday cup to higher temperatures.

The results obtained by TPD under ultra high vacuum are not directly comparable with those acquired by TGA under inert gas ﬂow. In theory, if the signal from the

QMS detectors was proportional to the counts of water molecules, one might be able to construct mass loss from the area under the curve. This would require careful calibration of the instrument and testing. TPD only takes into account water lost from the system, while TGA takes into account all mass lost. While it is feasible to assume that most of the mass lost, observed in a TGA signal of Serpentine group minerals, is due to OH, quantification at what point do the non-water volatiles significantly affect the difference in those signals, would be necessary. As mentioned above, the mass affects the diffusion of OH and consequently H2O from the system, as such, TPD is designed so that diffusion was minimised. Such design in TGA would lead to the loss of sensitivity and significant increase of error. Note that under the best performance TGA instrument SDT Q600, which was used in this study, is between (25–45) µg according to the manufacturer. This results in an error of (2.5–4.5) % in the case of measuring 1 mg sample. However, in our instrument, the drift was observed to reach up to 100 µg which would result in an error of

171 10 % in 1 mg sample. The error of the TPD method used by Georgia Tech is not known at all.

The idea that the area under the TPD curve should be proportional to the amount of water lost during the TPD experiment suggests a possible link to connect the data with TGA results. If the area below the TPD curve is proportional to the total mass loss during heating at a constant rate, then TPD curve is proportional to the derivative of mass with respect to time (or temperature since heating rate is constant) and, as such, to the DTG curve which can be calculated from the TGA data under the same conditions (that is the same level of vacuum, the same heating rate and the same mass of the sample). This connection is not without possible lapses, for example, due to the sample mounting, the water might need more time to diﬀuse from a TGA sample than from TPD sample. This area has not been well studied and there might be issues that are not immediately clear. However, without any better tool at my disposal, I will make connections between TGA and TPD by connecting DTG with the TPD curve.

4.3.2.5 XRD

XRD was used to characterize the mineralogy of the samples. Even though samples were either characterized under microscope (Cronstedtite) or obtained as individual mineral species (Antigorite and Lizardite), XRD would provide more insight into additional phases present in the sample.

172 Two diﬀerent XRD instruments were used. Rigaku D-Max B X-Ray diﬀractometer with Cu–Kα radiation with 1.5406 Å wavelength in Bragg-Brentano θ-2θ geometry at the department of Chemistry at UCF was used. Samples were always mounted on a zero background plate.

PANalytical Empyrean with Cu–Kα X-Ray tube with 1.540 598 0 Å wavelength

(tension 45 kV and current 40 mA) in Bragg-Brentano θ-θ geometry and X’Celerator detector at the UCF AMPAC was used as well. Both standard sample holders as well as zero background holders were used.

4.3.2.6 FTIR

For spectroscopic analyses, Nicolet iS50 spectrometer was used at UCF Physics department. The sample compartments were purged with N2 gas. Samples were analysed for their reﬂectance spectra using Pike Technologies EasiDiﬀ accessory with DTGS detector, KBr beam splitter with aperture set to 43 and optical velocity to 0.4747.

It was not possible to analyse pure samples since their amount was insuﬃcient.

For that reason the heated sample was mixed with Potassium Bromide to act as a buﬀer.

Potassium Bromide was then used as background for the analysis. Spectra were taken from 1.25 µm to 20 µm. All experiments were done using 128 scans with resolution set to

2 (thus resulting in data spacing of 0.241 cm−1).

173 4.4 Results

4.4.1 Serpentines under Inert Gas Flow

Antigorite sieved to grain size < 53 µm and heated to 1200 ◦C at 20 ◦C min−1 under

100 ml N2 ﬂow was analysed in triplicate. Fig. 4.9 depicts the process on one of the tested samples (mass 12 mg). The ﬁgure displays three signals — TGA (solid blue), DTG (dot- dot-dashed green) and DSC (dashed red). Only TGA y axis values provide information.

The other two signals were normalized to unity, each on a certain sub-range of the x axis, to exaggerate their peaks.

The DTG curve clearly shows two valleys centred at 592 ◦C and 727 ◦C demarcating two dehydroxylation steps. The DSC curve shows the two peaks as well although the ﬁrst peak cannot be exactly determined. It shows a further change in slope between (750–

850) ◦C suggesting a possible further reaction. However, based on the analysis of DSC performance, the DSC signal exhibits a very strong negative drift beyond 600 ◦C. This might also be a reason that no crystallisation peaks are visible for the temperatures above

800 ◦C where recrystallisation of Forsterite can be expected (see Sec. 4.2.2.1). Finally, a small tooth can be observed in the TGA curve around 650 ◦C and 850 ◦C which also shows on the DSC curve. Its nature will be discussed later. It is not observed in the DTG data due to the nature of smoothing described in Sec. 4.3.2.2 (in this case 300 points were used).

174 100 200 300 400 500 600 700 800 900 1000 100.00 1

0.8

95.00

0.6

TGA Antigorite m)/dt [1]

DTG Antigorite ¡ DSC Antigorite Mass [%]

0.4 Q [1], d (

90.00

0.2

85.00 0 100 200 300 400 500 600 700 800 900 1000 T [C]

Figure 4.9: The dehydration of Antigorite. The TGA curve depicting the mass loss (the left y axis) is plotted as a function of temperature in blue solid line. Its derivative with respect to time (DTG) is plotted in green dot-dot-dashed line (the right y axis) at corresponding temperature points and is calculated as described in Sec. 4.3.2.2 using 300 points for smoothing. Heat ﬂow (the DSC curve) is plotted in red dashed line (the right y axis) as a function of temperature and is oriented as such that endothermal reactions result in valleys and exothermic in peaks. Note that the DTG and DSC curves have been normalized to unity on a sub-range of the x axis to provide a better view on their characteristics.

175 Calculations3 show (see Sec. 4.3.2.2 for description) the onset, end-set temperatures for the ﬁrst stage (548 ± 5) ◦C and (615 ± 6) ◦C and the mass loss of approximately 3 %. The second stage is characterised by the onset and endset temperatures of

(688 ± 1) ◦C and (774 ± 4) ◦C and the mass loss of (9.6 ± 0.3) %. The DTG peaks for

stage 1 and 2 appear at (592 ± 4) ◦C and (714 ± 12) ◦C. The higher variability of the second DTG peak can be due to the high smoothing of the derivatives which was necessary because of the tooth-like features mentioned above. The total mass loss of Antigorite due to the loss of bound OH is on average from the three runs (12.8 ± 0.3) %. The significantly different mass loss in the two steps suggests that the first step is the loss of water in the surface layers. Adsorbed water was very low ((0.35 ± 0.02) %) and was determined as the mass loss from 50 ◦C until the point where the first tangent for calculation of the onset temperature was determined (see 4.3.2.2 for details). The numbers are summarised in Tab. 4.2.

The dehydration of approximately 23 mg of unsieved Lizardite heated to 1200 ◦C

◦ −1 at 20 C min under 100 ml N2 is depicted in Fig. 4.10. The ﬁgure displays three signals

— TGA (solid blue), DTG (dot-dot-dashed green) and DSC (dashed red). Only TGA y axis values provide information. The other two signals were normalized to unity, each on a certain sub-range of x axis, to exaggerate their peaks.

The DTG curve shows two valleys very clearly and also a suggestion of a possible third valley close to 800 ◦C. The DSC curve shows three clear peaks (although the ﬁrst

3Since only three measurements are involved, the descriptive statistics are merely to depict how the results may vary, however, one should keep in mind that it results from only three data points.

176 100 200 300 400 500 600 700 800 900 1000 100.00 1

0.8

95.00

0.6

TGA Lizardite m)/dt [1]

DTG Lizardite ¡ DSC Lizardite Mass [%]

0.4 Q [1], d (

90.00

0.2

85.00 0 100 200 300 400 500 600 700 800 900 1000 T [C]

Figure 4.10: The dehydration of Lizardite. The TGA curve of the mass loss (the left y axis) is plotted as a function of temperature in blue solid line. Its derivative with respect to time (DTG) is plotted in green dot-dot-dashed line (the right y axis) at corresponding temperature points and is calculated as described in Sec. 4.3.2.2 using 180 points for smoothing. Heat ﬂow (the DSC curve) is plotted in red dashed line (the right y axis) as a function of temperature and is oriented as such that endothermal reactions result in valleys and exothermic in peaks. Note that the DTG and DSC curves have been normalized to unity on a sub-range of the x axis to provide a better view on their characteristics.

177 one is tainted by the tooth in the TGA curve and the associated step change in the heat

capacity). In this case the endothermic reactions were strong enough to overpower the

negative baseline drift in DSC. No crystallisation peaks are observed, however.

The onset and endset temperatures for the ﬁrst stage are determined as 533 ◦C

and 592 ◦C with the DTG peak centred at 592 ◦C. The second stage is characterized

by 693 ◦C, 762 ◦C and 723 ◦C the onset, endset and DTG peak temperatures. The total

mass lost in steps 1 and 2 is 3.1 % and 9.5 % respectively making the total mass lost

12.5 %. It is interesting to note that while the onset temperature of the ﬁrst step is about

15 ◦C lower than for Antigorite, the onset of the second phase starts a little later than for Antigorite. However, this might be related to the grain size of the sample or its mass.

These eﬀects are discussed later in this section. The adsorbed water amounted to 0.35 %

(determined as described for Antigorite).

The process of dehydration of Cronstedtite, sieved to grain size <53 µm, is depicted in Fig. 4.11. The mass of the sample was approximately 11 mg and it was heated up to

◦ ◦ −1 1000 C at 20 C min under 100 ml N2 ﬂow. The plot displays the same type of curves as for Lizardite and Antigorite and the same description applies with respect to their colour and normalisation.

The DTG curve shows two clear valleys at 472 ◦C and 535 ◦C and a suggestion of another valley in between these two. The DSC curve is strongly drifting to negative heat ﬂow and the ﬁrst DTG valley is only depicted as a change in slope. However, the valley that is only suggested in the DTG data is clearly visible in DSC at 525 ◦C (the

178 100 200 300 400 500 600 700 100.00 1

95.00 0.8

90.00 0.6

TGA Cronstedtite m)/dt [1]

DTG Cronstedtite ¡ DSC Cronstedtite Mass [%]

85.00 0.4 Q [1], d (

80.00 0.2

75.00 0 100 200 300 400 500 600 700 T [C]

Figure 4.11: The dehydration of Cronstedtite. The TGA curve of the mass loss (the left y axis) is plotted as a function of temperature in blue solid line. Its derivative with respect to time (DTG) is plotted in green dot-dot-dashed line (the right y axis) at corresponding temperature points and is calculated as described in Sec. 4.3.2.2 using 120 points for smoothing. Heat ﬂow (the DSC curve) is plotted in red dashed line (the right y axis) as a function of temperature and is oriented as such that endothermal reactions result in valleys and exothermic in peaks. Note that the DTG and DSC curves have been normalized to unity on a sub-range of the x axis to provide a better view on their characteristics.

179 next DSC peak that corresponds to the DTG peak at 535 ◦C, is at 541 ◦C suggesting that the poorly visible DTG peak should occur around 519 ◦C, assuming a linear relationship

between the DTG and DSC peak positions within this short range). No crystallization

is observed in this run probably due to a signiﬁcant baseline drift. The valley occurring

just before 200 ◦C is due to an isothermal stage at 130 ◦C to remove adsorbed water.

The onset and endset temperatures of the ﬁrst stage were determined as 422 ◦C

and 480 ◦C and for the second stage 517 ◦C and 547 ◦C and the mass losses are 11.5 ◦C,

9.5 ◦C and 20.2 ◦C for the ﬁrst and second phases and in total. The adsorbed water amounted to 1.1 %, out of that approximately 0.6 % was removed during the isothermal

phase and 0.4 mass loss occurred between (130–231) ◦C (the isothermal water removal

and start of the main dehydroxylation). It has to be noted that the dehydration depicted

in Fig. 4.11 is one of a few runs where it is possible to discriminate the two stages based

on the DTG curve. In most runs the two stages overlap too much (this depends on the

initial mass size of the sample and on the heating rate). Also it is important to notice

the amount of mass loss in the ﬁrst stage is larger than in the second one, suggesting

that the dehydroxylation process of Cronstedtite might be more complex than that of

Mg Serpentines. Also the amount of mass loss is peculiar.

The comparison using the TGA and DTG curves can be observed in Fig. 4.12.

The comparison of the descriptive values of the curves can be found in Tab. 4.2.

180 100 200 300 400 500 600 700 800 900 1000 100.00 TGA Antigorite TGA Cronstedtite TGA Lizardite DTG Antigorite 4 DTG Cronstedtite DTG Lizardite

95.00

90.00

2 Mass [%]

m)/dt [%/min] ¡

85.00 d (

80.00

75.00 100 200 300 400 500 600 700 800 900 1000 T [C]

Figure 4.12: Comparison of the dehydroxylation of Antigorite, Lizardite and Cronstedtite using the TGA curve depicting the mass loss (the left y axis) plotted as a function of temperature in solid lines and its derivative with respect to time (DTG) plotted in dot-dot-dashed lines (the right y axis) at corresponding temperature points. The DTG curves were evaluated as described in the previous plots (see Fig. 4.9, 4.10 and 4.11).

181 Table 4.2: Quantities characterising the dehydradration of Serpentine type minerals. Sample Antigorite Lizardite Cronstedtite grain size <53 x¯ ± σ unsieved <53 Mass [mg] 9 7 12 9 ± 3 26 11 Adsorbed H2O [%] 0.34 0.37 0.33 0.35 ± 0.02 0.35 1.1 Step 1 Onset T [C] 544 554 546 548 ± 5 533 422 Endset T [C] 620 608 618 615 ± 6 592 480 DTG peak [C] 588 596 592 592 ± 4 573 472 ∆m1 [%] 3.1 3.2 3.2 3.1 ± 0.1 3.1 11.5 Step 2 Onset T [C] 687 688 689 688 ± 1 693 517 Endset T [C] 773 771 779 774 ± 4 762 547 DTG peak [C] 723 716 727 715 ± 12 723 535 ∆m2 [%] 10.0 9.3 9.6 9.6 ± 0.3 9.5 9.5 ∆m [%] 13.1 12.4 12.8 12.8 ± 0.3 12.5 20.2

4.4.2 Results of Dehydroxylation Viewed by XRD

XRD was used in order to analyse what happened with the mineral structure as well as to conﬁrm that Antigorite and Cronstedtite minerals have been de-hydroxylated.

Since the de-hydroxylation is accompanied by the destruction of the crystal lattice described in detail in Sec. 4.2.2.1, it should be expected that the reﬂections of crystal planes should cease to appear as the samples are heated to higher temperatures.

The samples of Antigorite (unsieved) and Cronstedtite (sieved to <53 µm) were were ﬁrst heated to 130 ◦C and kept isothermally for about 5 minutes for a quick removal of any adsorbed water and then heated at 20 ◦C min−1 to various temperatures selected based on the observed features in TGA studies (see Sec. 4.4.1). At those temperatures they were kept isothermally for 10 min. After the experiment, XRD was performed. All the results in this section were obtained using the Rigaku D-Max B X-Ray diﬀractometer

182 (see Sec. 4.3.2.5). For Antigorite, I present the results in Fig. 4.13. From the ﬁgure, one

can see that the Antigorite signal (denoted in the ﬁgure as “Serp”) completely disappears

between (650–840) ◦C.

35 Per Qtz Lim Fo

Dic 25 Kao/Serp, Zeo?

Chl 20 Serp

Alb? Serp Cal

Intensity (offset) 15 Serp

Dol 10

10 20 30 40 50 60 70 ¡2 theta

Figure 4.13: XRD patterns of unsieved Antigorite. The curves represent the samples heated to temperatures 128 ◦C, 650 ◦C, 840 ◦C and 1000 ◦C in ascending order. The curves are oﬀset. The y axis denotes intensity in arbitrary units.

For Cronstedtite, I present the results of XRD analysis in Fig. 4.14. The analysis was done in several steps. The major Cronstedtite relevant peaks are denoted with arrows.

From the plot, it can be seen that the major transition occurs between (330–470) ◦C where the Cronstedtite peak either completely disappears or the broad peak visible in the yellow

183 curve (demarcated as "470b") is a type of transition structure. The data at 470 ◦C were analysed twice because of high signal to noise which is visible in the orange curve.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

880°C

720°C

590°C

540°C

470b°C Intensity (arbitrary)

470°C

330°C

280°C

150°C

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 Angle (2θ)

Figure 4.14: XRD patterns of Cronstedtite sieved to grain size <53 µm. The curves represent the samples heated to various temperatures. The curves are oﬀset from each other and the temperature that each curve represents in typed under the curve on the left side. The arrows indicate the peaks relevant to Cronstedtite.

4.4.3 Results of Dehydroxylation Viewed by FTIR

Another analysis to conﬁrm dehydroxylation and also to observe its eﬀects on

reﬂectance spectra was done using FTIR.

184 The near infrared spectra of Antigorite samples analysed in Sec. 4.4.2 are presented

in Fig. 4.15. The focus is on the region (1.5–6) µm. The stretching OH absorption band is centred at 2.7 µm. Heating to 650 ◦C removes only the broad shoulder but the intensity

of the negative sharp peak is unchanged. Only after heating to 840 ◦C, does the valley

change its depth. Further heating to 1000 ◦C changes little. The remaining broad valley

might be related to the fact that the Antigorite sample was diluted in KBr salt which

is very hygroscopic and this type of absorption is prevalent in KBr often even after

signiﬁcant heating. This is, however, beyond the scope of this work.

Wavenumber [cm-1] 8000 7000 6000 5000 4000 3000 2000 90

ectance [%] ¡ 50 Re

Antigorite 128°C Antigorite 650°C Antigorite 840°C Antigorite 1000°C 20 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

¢ [µm]

Figure 4.15: FTIR spectra of unsieved Antigorite heated to various temperatures.

185 The reflectance spectrum of Cronstedtite is displayed in Fig. 4.16. The spectrum has a very different profile than that of Antigorite. Specifically, the absorption around

3 µm is very broad and does not display features similar to Fig. 4.15. Upon heating to

600 ◦C, all features disappear and the spectrum is completely featureless with signiﬁcantly lower overall reﬂectance.

Wavenumber [cm-1] 8000 7000 6000 5000 4000 3000 2000 70

ectance [%] ¡ Re

Cronstedtite 120°C Cronstedtite 600°C 10 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

¢ [µm]

Figure 4.16: FTIR reﬂectance spectra Cronstedtite sieved to grain size <53 µm heated to 120 ◦C to remove adsorbed water and another sample heated to 600 ◦C.

186 4.4.4 Heating Rate Eﬀects

Analysis of eﬀects of varying heating rates on the results provided by the TGA,

DTG and DSC curves were studied at the KSC laboratory. The analysis was done on

Antigorite sample sieved to grain size range (53–106) µm heated from the room temper-

◦ −1 ◦ −1 ◦ −1 ature to 1000 C under 100 ml min of N2 ﬂow at 15 C min and 30 C min heating rates. The two runs are depicted in Fig. 4.17.

400 450 500 550 600 650 700 750 800 850 900

100.00 100

95.00 TGA 30 C/min 95 TGA 15 C/min Mass [%]

90.00 90

DTG 30 C/min DTG 15 C/min

m)/dt [1] ¡ d (

DSC 30 C/min DSC 15 C/min Q [1]

400 450 500 550 600 650 700 750 800 850 900 T [C]

Figure 4.17: Comparison of two heating rates on a sample of Antigorite sieved to grain size (53–106) µm. The top panel displays the TGA curves, the middle one the calculated DTG using 60 point smoothing (see Sec. 4.3.2.2). The bottom panel shows the DSC signal. Note that both, the DTG and DSC signals are normalized to unity on a sub-range of the displayed x axis for a better visibility of their features.

187 The plot shows that the onset temperatures are shifted to higher values by about

30 ◦C for the faster heating rate. Similarly the DTG peak is shifted by 23 ◦C and 30 ◦C for the first and second phases respectively. In the case of the second transition, it was also possible to determine the DSC peaks which are offset in the second phase by 24 ◦C in a similar manner. The slower heating rate (and probably also due to a little lower mass) along with the omnipresent DSC drift, caused the lack of sensitivity and thus the peak in the first phase is not visible sufficiently. Further, due to the shape of the slower heating rate DSC curve, it is not possible to determine the onset temperature from the this curve for comparison with faster heating rate. The parameters are summarised in Tab. 4.3.

Table 4.3: Comparison of the eﬀects of two heating rates on characteristincs of dehydration of Antigorite. The runs were executed at KSC. Antigorite grain size 53-106 Mass [mg] 6.6 9.7 Heating Rate [C/min] 15.0 30.0 Step 1 Onset T [C] 565 597 Endset T [C] 618 645 DTG peak [C] 600 623 DSC peak [C] NA 641 ∆m [%] 2.8 3.1 Step 2 Onset T [C] 688 720 Endset T [C] 765 786 DTG peak [C] 724 754 DSC peak [C] 735 758 ∆m [%] 9.8 9.8 ∆m [%] 12.5 12.7

188 4.4.5 Sample Size Eﬀects

Analysis of eﬀects of various mass sizes of a sample was also studied initially

at the KSC. Analyses were performed on Antigorite samples sieved to (53–106) µm of

◦ −1 approximate masses 10 mg, 20 mg and 30 mg heated to 1000 C under 100 ml min N2

ﬂow at the rate of 30 ◦C min−1.

500 550 600 650 700 750 800 850 900

100.00 TGA 10 mg 100 TGA 20 mg TGA 30 mg

95.00 95 Mass [%]

90.00 90

DTG 10 mg DTG 20 mg

m)/dt [1] DTG 30 mg ¡ d (

DSC 10 mg DSC 20 mg DSC 30 mg Q [1]

500 550 600 650 700 750 800 850 900 T [C]

Figure 4.18: Comparison of the eﬀects of three distinct sample sizes of Antigorite sieved to (53–106) µm and heated at 30 ◦C min−1. The top panel displays the TGA curves, the middle one the calculated DTG using 60 point smoothing (see Sec. 4.3.2.2). The bottom panel shows the DSC signals. Note that both, the DTG and DSC signals are normalized to unity on a sub-range of the displayed x axis for a better visibility of their features.

The runs show that with an increasing mass, all peaks shift to higher temperatures, especially the peaks in the second dehydroxylation stage. The TGA data show a little

189 different profile of the 10 mg sample compared to the other ones. The data are quantified

in Tab. 4.4.

Table 4.4: Comparison of the eﬀects of an increasing sample size on the characteristics of dehydration of Antigorite. The runs were executed at KSC. Sample Antigorite Grain Size 53-106 Mass 10.0 19.0 29.7 on ◦ Step 1 T1 C 590 596 594 593 ± 3 end ◦ T1 C 640 641 641 641 ± 1 DTG ◦ T1 C 613 621 620 618 ± 4 DSC ◦ T1 C 638 636 635 636 ± 2 DSC−on ◦ T1 C 579 586 591 585 ± 6 ∆m1 % 3.1 3.0 3.1 3.05 ± 0.05 on ◦ Step 2 T2 C 715 722 727 722 ± 6 end ◦ T2 C 784 791 796 790 ± 6 DTG ◦ T2 C 750 757 770 759 ± 10 DSC ◦ T2 C 754 768 778 767 ± 12 DSC−on ◦ T2 C 694 702 704 700 ± 6 ∆m2 % 9.3 9.4 9.5 9.4 ± 0.1

4.4.6 Eﬀects of Varying Grain Size

The eﬀects of varying grain size were studied on both Antigorite and Cronstedtite.

The analysis is presented for Antigorite. Fig. 4.19 presents all the runs that were carried

out at the UCF AMPAC laboratory to test the grain size of Antigorite. Note that the

plots were focused only onto the range of temperatures (500–850) ◦C to better see the diﬀerences. The plot presents 2 runs of grain sizes <53 µm and (53–106) µm and 3 runs on the grain size >212 µm. No tests were performed on the grain size (53–106) ◦C. One of

190 the three runs on the >212 µm grain size is rather anomalous with respect to the amount of the lost material. It was not possible to determine the origin of such discrepancy but among all runs such discrepancy was unique. Also note that the discrepancy appears only above 725 ◦C. The runs with grain size (106–212) µm yielded very similar results.

Also the two runs with grain size <53 µm were very varying.

500 550 600 650 700 750 800 850

100.00 100

95.00 TGA >212 um 95 TGA 106-212 um TGA <53 um Mass [%] 90.00 90

DTG >212 um DTG 106-212 um

m)/dt [1] DTG <53 um ¡ d (

DSC >212 um DSC 106-212 um

Q [1] DSC <53 um

500 550 600 650 700 750 800 850 T [C]

Figure 4.19: The dehydration of Antigorite sieved to diﬀerent grain sizes heated from ◦ ◦ −1 −1 the room temperature until 1100 C at 20 C min under 100 ml min N2 ﬂow. The top panel displays the TGA curves, the middle one the calculated DTG using varying amounts of smoothing (120–240 points, see Sec. 4.3.2.2). The bottom panel shows the DSC signals. Note that both, the DTG and DSC signals are normalized to unity on a sub-range of the displayed x axis for a better visibility of their features.

All of the samples presented negligible amount of “adsorbed water” (by which I mean all mass lost before the main event). Despite the variations among individual runs

191 of the largest and smallest grain size, several trends can be observed. Speciﬁcally, if one focuses only on the mass lost in the individual phases, the mass lost in the ﬁrst stage decreases with increasing grain size from about 3.6 % for <53 µm to about 2.2 % lost for

>212 µm. The relationship of mass lost with grain size in the second phase is not clear.

The obtained data are summarised in Tab. 4.5.

The experiments with different grains of Cronstedtite done at the KSC as well as those carried out later at the UCF AMPAC are presented in Fig. 4.20. Both plots have similar features although the variability is significant. First, the smallest grain size tends to lose more mass in total. The data with faster heating rate show that there seems to be a similar first stage as in Antigorite which has not been seen in the data in Sec. 4.4.1.

Further, assuming that the step in >212 µm in the KSC data signifies loss of material from the pan, the TGA curve would probably end at even less total mass loss. The nature of the difference in the total mass loss between the two datasets is unknown and is significant (an order of units of percentages, surpassing any errors due to a baseline drift). Beside the feature similar to Antigorite (smaller grain sized material loses more water), another feature, not observed in Antigorite samples, are more significant amounts of adsorbed water for the small grains rather than for the larger grain sizes.

4.4.7 Serpentines under Vacuum

Two experimental methods were employed to study dehydroxylation behaviour under vacuum conditions. Initially, the TPD method was used. Later, it has become

192 KSC UCF 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900

(< 53) ¡ m 100 100

(53-106) ¡ m

(106-212) ¡ m

(> 212) ¡ m

95 95

90 90 Mass [%]

85 85

80 80

75 75 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 800 900 T [C] T [C]

Figure 4.20: The dehydration of Cronstedtite at diﬀerent grain sizes heated from the room temperature to 1000 ◦C at 30 ◦C min−1 and 20 ◦C min−1 at the KSC and the UCF −1 respectively under 100 ml min N2 ﬂow. The left panel depicts the experiment at the KSC, the right one at the UCF.

193 Table 4.5: Comparison of the eﬀects of the various grain sizes on the characteristics of dehydration of Antigorite. The runs were executed at UCF. The columns denoted “Avg” denote average of the values to the left. Sample Antigorite Grain Size <53 Avg 106-212 Avg >212 Avg Mass 12.0 10.1 11.1 11.9 10.6 11.3 13.2 13.6 11.6 12.8

mH2O % 0.16 0.06 0.11 0.15 0.31 0.23 0.01 0.05 0.09 0.05 on ◦ Step 1 T1 C 534 561 547 571 579 575 562 581 562 568 end ◦ T1 C 599 626 613 632 628 630 635 641 645 641 DTG ◦ T1 C 574 603 588 613 609 611 610 624 612 615 ∆m1 % 3.4 3.8 3.6 3.2 3.2 3.2 2.5 2.2 2.0 2.2 on ◦ Step 2 T2 C 685 701 693 701 703 702 710 711 718 713 end ◦ T2 C 751 767 759 777 777 777 798 796 802 799 DTG ◦ T2 C 718 735 727 733 733 733 749 758 769 758 ∆m2 % 9.8 9.9 9.9 10.1 10.3 10.2 10.4 12.3 9.9 10.9 on ◦ Ttot C 647 663 655 667 669 668 674 687 689 683 end ◦ Ttot C 751 767 759 777 777 777 798 796 802 799 ∆m % 13.5 13.8 13.7 13.5 13.5 13.5 12.8 14.5 12.0 13.1 possible to test a vacuum TGA setup as well. It supplies much higher pressures (∼

10−6 torr). Here I report on the results obtained so far.

While the TPD method can provide data under very low pressures (∼ 10−10 torr) it provides data in a form that is not well comparable with the TGA method. Because of this as well as due to the nature of the experiment (reheating of the sample with diﬀerent detector mentioned in the description of the method in Sec. 4.3.2.4) as well as the fact that only 1 mg was used in the method, I decided to only provide the results of the vacuum TGA which can be immediately compared to the other results in this section and with which I am more familiar.

The result of the experiment with Antigorite sieved to <53 µm and heated to

800 ◦C under vacuum at 10 ◦C min−1 is compared to two runs of Antigorite, one sieved to

194 (53–106) µm and another unsieved, which were heated to 1000 ◦C at 15 ◦C min−1 under

−1 100 ml min N2 ﬂow, is depicted in Fig. 4.21.

100 200 300 400 500 600 700 800 900

Vacuum Inert gas 100

95 Mass [%]

85 100 200 300 400 500 600 700 800 900 T [C]

−1 Figure 4.21: Comparison of Antigorite in vacuum with other runs under 100 ml min N2 ﬂow. The grain size of Antigorite run under vacuum was <53 µm, one of the others had the grain size range (53–106) µm and the other was unsieved. The heating rate under vacuum was 10 ◦C min−1, the other two were heated using 15 ◦C min−1.

The plot shows significantly faster loss of mass in vacuum and the slope of the respective TGA curve is mostly constant during the whole range. The different look of the curves makes correct analysis using onset temperatures and DTG peaks difficult (not to mention that the DTG data need to be calculated at very high smoothing, this will be discussed later). Yet, one can compare the temperatures at which the material has lost a certain amount of mass. For vacuum experiment, the sample has lost 5 % of its

195 initial mass by reaching 622 ◦C while the two inert ﬂow samples needed to reach 698 ◦C.

Since it has been shown that heating rate aﬀects some features of the observed curves,

and because one of the runs presented in Fig. 4.21 is from the analysis of heating rate

effects in Sec. 4.4.4, the comparison with the run at 30 ◦C min−1 from Sec. 4.4.4 shows that doubling heating rate only offsets the point by 25 ◦C, id est the run at 30 ◦C min−1 loses 5 % of mass by 725 ◦C which is far less than about 86 ◦C difference between the points for vacuum and inert gas runs mentioned above.

4.5 Discussion

4.5.1 Parameters Aﬀecting the Results

The tests on two heating rates revealed that increasing the heating rate from

15 ◦C min−1 to 30 ◦C min−1 oﬀsets the characteristic points of the transitions by about

30 ◦C to higher values. Typically, onset temperatures are used to avoid these effects in DSC curves. However, if one looks at the DSC data of the runs from Sec. 4.4.4, a similar offset is visible in the second dehydroxylation stage (in the first one the onset point cannot be determined). The reason is probably related to the mass loss since, for a simple melting transition, no mass loss is observed and the heat only serves as latent energy while in dehydration it also has to impart kinetic energy to water molecules. Thus, there is no good tool for analysis that would be immune to the effects of heating rate. It was also observed in several runs that the faster heating rate provides a more pronounced

196 DSC and DTG peaks for weak transitions, at the same time it tends to hide transitions

that are close in temperature.

Mass size only visibly aﬀected the second dehydration stage in Antigorite which

supports the idea that this stage is the loss of water from deep inside the grains and its

progress is related to the volume. Another analysis, which was not published in Sec. 4.4.5,

done on samples of unsieved Antigorite with masses 25 mg and 40 mg, showed the char-

acteristics of the ﬁrst transition identical within the margin of error due to the selection

of tangents and the machine accuracy in the temperature domain. The characteristics

of the second transition were oﬀset by about 6 ◦C suggesting a very small eﬀect of the sample size on the resulting quantities.

The experiments to determine the dehydration of various grain sizes resulted in rather varying data. The range (106–212) µm was very consistent in both runs where

the parameters were within uncertainties typically implied by the process of selecting the

tangents or by the signal noise and the implied variable smoothing of derivatives. On the

other hand both <53 µm and >212 µm showed some types of variations. One anomalous run with the largest grain size presented 14.5 % mass loss which has not been observed in any other runs with Antigorite. The suspected reasons can be either contamination of the crucible or improperly tarred scales or even improper positioning of the crucible on the thermocouple. This run should probably be disregarded. The variations between the other two runs with this grain size happen only at second transition and they are most likely related to the grain size distribution (since one sample might have contained larger grains and a few more in one run would be enough). Also upon a closer examination,

197 one of those two runs might have lost some material from the pan which in part could account for the difference in the total mass loss. Finally, the larger grain sizes might suffer from a worse thermal coupling. For small grain sizes, the heat conduction is more prominent than for larger grain sizes where convection and radiation might play more significant roles. The anomaly, and about 30 ◦C lower onset temperature, of one of the

<53 µm samples is signiﬁcant. No more runs are available for proper statistics and this should be investigated further.

Runs on Cronstedtite samples, while exhibiting several inconsistencies, show a similar trend as the Antigorite ones. Namely, the first stage of dehydroxylation starts at very similar temperatures but results in significantly different relative mass losses. The second stage starts a little later for the larger grain sizes but results in comparable relative mass losses (probably a little larger for the larger grain sizes but this is tainted by the variability of the <53 µm grain size. This strongly favours the idea that the first stage removes the OH– from the layers of the grains closest to the surface and the second stage is the dehydration of the interior of the grains which, with the larger size of grains, is more difficult to proceed.

4.5.2 Dehydroxylation of Serpentine Minerals

Based on a stoichiometrically pure Antigorite (Mg3Si2O5(OH)4), the process in

Fig. 4.2 dictates that the dehydroxylation should lead to 13 % total mass loss (due to the

198 loss of OH– ). The results presented in Tab. 4.2 are very close to this value suggesting a very pure Antigorite.

The resulting data for Cronstedtite suggest that the onset temperature of the ﬁrst phase is about 122 ◦C below that of Antigorite and the DTG peak about 101 ◦C below

the one for Antigorite. Based on other analyses as well as upon a detailed inspection of

Fig. 4.11, it is very likely, that Cronstedtite undergoes in total 4 transitions. Yet, until a

detailed XRD and FTIR analyses are performed, there is no assurance that the samples

were not contaminated by some other product.

Further, all the dehydration stages of Cronstedtite span signiﬁcantly narrower

temperature ranges than Antigorite. While the ﬁrst stage (which in Cronstedtite most

likely encompasses 2 stages) is similarly wide (58 ◦C, 67 ◦C and 59 ◦C for Cronstedtite,

Antigorite and Lizardite), the second stage (which for Cronstedtite again consists of 2

stages but it is likely that for Antigorite as well, based on some runs) the ranges are 30 ◦C,

86 ◦C and 69 ◦C. The best characterisation is given by taking the onset temperature of

the ﬁrst phase and the endset of the second phase which encompass the whole process and

yield 125 ◦C, 226 ◦C and 190 ◦C for Cronstedtite, Antigorite and Lizardite respectively.

Even a more striking diﬀerence can be shown when one considers at what tem-

perature the minerals have lost e.g. 5 % of their initial mass. This analysis provides the

ﬁgures 422 ◦C for Cronstedtite, 682 ◦C for Antigorite (average of the measurements on the samples from Sec. 4.4.1) and 677 ◦C for Lizardite, that is a diﬀerence of 260 ◦C between

Cronstedtite and Antigorite. One might argue that there is a more pronounced mass

199 loss before the main dehydration event starts in Cronstedtite, possibly from adsorbed

water, but when one calculates the 5 % mass loss with respect to 228 ◦C, the point where

I calculate the initial tangent of the onset temperature and from when I also determine

the mass loss in the ﬁrst dehydroxylation step, one gets 437 ◦C instead of the mentioned

422 ◦C, thus still a large diﬀerence from Antigorite and Lizardite.

Another interesting feature of Cronstedtite samples was that they seem to adsorb water more easily as in most of the runs the “adsorbed water” (in the sense that I have mentioned earlier that this is the amount of mass lost before the main event) in the grain size <53 µm amounted to about 1 % compared to Antigorite where the amount was often on the verge of detectability with 100 µg baseline drift and amounted at most to 0.3 %.

However, I have to admit that this initial mass loss by Cronstedtite may also be due to

contamination of the sample.

An interesting feature was visible in several plots, most notably in the left panel

of Fig. 4.20 for the grain size >212 µg. These step losses of mass can be noted in many

runs (for Antigorite as well), in most runs they are very small but in some, they can be

noticed better. In some runs, the step seems to jump back after some time, while not

in others. The TGA mechanism of SDT Q600 is very sensitive and can detect e.g. the

door closing in the laboratory and other vibrations, such effects have different profile and

are rarely noticeable. The fact that in most cases, the TGA curve does not return (does

not step back up) suggests that a loss of mass occurred from the crucible. Further, these

eﬀects seem to be more common in larger grain sizes which suggests that the grains are

“jumping” out of the crucible (which is 4 mm high). This might be caused by “cracking”

200 of the grains due to a build up of pressure caused by water molecules attempting to leave the sample and gaining more energy with heating. This is an interesting eﬀect worth exploring further.

Finally, even if our Cronstedtite samples were not pure and had some admixtures, this does not change the low dehydroxylation temperatures conclusion.

Similar conclusions to those from TGA analyses was reached after analysing the de- hydroxylated samples using XRD. Both XRD analyses shown in Figs. 4.13 and 4.14 suffer from relatively poor data quality which is due to a very low amount of material. However, the data is good enough to observe the decomposition of the crystal lattice. Also, the resolution was insufficient, and data which are more closely separated in temperature would be needed. This is especially true for Cronstedtite where the de-hydroxylation seems to be almost finished by 470 ◦C, but there is no information on what happens between (330–470) ◦C. Also the Cronstedtite XRD curves obtained on the sample heated to 470 ◦C suffer from poor quality and it is not immediately obvious, whether the broad peak between approximately (3–17) ◦C is an artefact of the low material and a reflection from background or whether it is still some Cronstedtite in a certain transition stage.

The results of XRD were conﬁrmed with FTIR on the OH absorption band around

3 µm. Furthermore, the analysis shows a diﬀerent appearance of the valleys in Cronst- edtite. This might be an artefact of the method in which the sample was diluted in

KBr which has a broad peak also around this range and which is also probably visible in the Antigorite spectra heated to 1000 ◦C. However, the consequences of using KBr as a

201 diluting material for reﬂectance spectra around 3 µm are beyond the scope of this work.

Another interesting feature of Cronstedtite spectrum is that after heating to 600 ◦C it becomes, unlike Antigorite, rather featureless.

The currently only a single run of Antigorite under vacuum conditions showed a very significant differences in the dehydroxylation stages. As mentioned before, if one compares at which temperature the sample loses 5 % of mass, the vacuum offsets this point for Antigorite by about 76 ◦C to a lower value. High vacuum TGA is not available and the only other experimental method is TPD which cannot be compared to TGA and as such is not of much use.

4.6 Conclusions

This chapter focused on the experimental analysis of the dehydration process of the Serpentine group minerals which make up signiﬁcant parts of certain subgroups of

CCs.

The experiments used mainly data obtained from TGA and some from DSC to determine the dehydroxylation properties of Antigorite, Lizardite and Cronstedtite. A signiﬁcant part of the chapter aimed at overviewing the existing studies not merely for the reason of motivation but also to provide context for the experiments.

A large part of the experiemental work dealt with perfecting and understanding the obtained data to avoid any misinterpretation and improper conclusions. Analysis

202 of various specimen and method related eﬀects was introduced theoretically as well as

veriﬁed on our samples. This led to the conclusion that only samples with the same

heating rate and grain size should be compared. It was shown that the mass of the

sample is not too a signiﬁcant factor.

Analysis of the Fe and Mg Serpentines showed a very diﬀerent thermal decompo-

sition behaviour such that Cronstedtite (Fe Serpentine) decomposes at over (100–260) ◦C

lower temperatures (based on which descriptive technique one employs) than Antigorite

and Lizardite. Further, the data show that Cronstedtite is more susceptible to adsorption

of molecular water (or contains some impurities). The results on the Antigorite samples

show mass loss very close to the stoichiometric theoretical values.

The vacuum dehydration of Antigorite shows a faster dehydroxylation process and

a shift to lower temperatures by approximately 75 ◦C (based on temperature at which

5 % mass loss occurs).

203 CHAPTER 5 LOSS OF WATER BY ASTEROIDS — APPLICATION

5.1 Introduction

The data, obtained in the previous chapter have provided details on temperatures at which the components of CCs decompose and, as a result, lose water that they store in the form of hydroxyl molecules. Previously Delbo and Michel (2011) analysed the thermal history of the asteroid 101955 Bennu and concluded that the asteroid most likely lost any molecular interlayer water on its surface and in sub-surface, however, they could not expect a loss of hydroxyl water from the phyllosilicates. The reason is that as a model for dehydroxylation, they used data for Antigorite, which, as has been shown in the previous chapter, needs to be heated to almost 550 ◦C to attain at least the onset temperature of the ﬁrst dehydroxylation step (the surface of the grains) and does not achieve 5 % mass loss until almost 700 ◦C and only at such high temperatures, the second dehydroxylation phase starts.

It has been further shown that, unlike Antigorite, Cronstedtite has a signiﬁcantly lower dehydroxylation temperature. In this chapter, I investigate and discuss, what orbits are necessary for dehydroxylation of either Cronstedtite and Antigorite. As an example, the asteroid (341843) 2008 EV5 is used.

204 5.2 Methods

5.2.1 The Heat Conduction Equation

In order to be able to analyse how big a part of an asteroid has experienced temperatures leading to dehydroxylation, it is necessary to solve the Heat Conduction

Equation (HCE). In App. C.1, I derive the HCE which for constant material parameters

(that is parameters that are independent of temperature, and the region, where the equation is applied, is assumed to be homogeneous and isotropic) can be written:

∂T α (t, ~x) − 4T (t, ~x) = 0 , (5.1) ∂t ρc where T (t, ~x) is the temperature at some time instant t and position given by ~x ∈ V,

V denoting the volume occupied by the asteroid, α denotes the heat conductivity in

WK−1 m−1, ρ the material density in kg m−3, c is the speciﬁc heat capacity of the material in JK−1 kg−1 and 4 is the Laplacian operator (∇ · ∇). In order to be able to obtain a unique solution, this equation needs to be supplied with a boundary condition (along with an initial one). As shown in App. C.2, the full boundary value problem for an asteroid orbiting the Sun is (the equations necessary to be solved inside the asteroid, on the surface and the initial condition):

∂T α (t, ~x) − 4T (t, ~x) = 0 , x ∈ Int V , (5.2) ∂t ρc

4 ε (t)(1 − A) = εσT (t, ~x) + α∇T (t, ~x) · ~n, x ∈ ∂V ,

T (0, ~x) = h(~x) , x ∈ V¯ .

205 Due to the non-linear boundary condition containing the term T 4 both analytical and numerical solutions have their pitfalls (this stems from the problem of finding a root of a polynomial of fourth order both analytically or numerically). Vokrouhlický (1999) provided analytical approximative solution of the HCE on a sphere. However, full three dimensional solutions of HCE on bodies of non-spherical shape are not known. Numer- ical approaches that comprise methods such as finite differences, finite volumes, finite elements, etc. are both complex to implement (e.g. dealing with boundary conditions in 3 dimensional finite differences) and very demanding to computational power (finite elements). Methods such as finite elements are only useful on small objects such as boulders on asteroids, but for large bodies of asteroid size they are hardly applicable without significant computational resources (investigation of varying parameters is simply not possible).

Under certain assumptions, it is possible to reduce the problem of Eq. 5.2 to one dimension (see App C.3 for the full exposition including the discussion of the assumptions). This leads to a system of 1 dimensional slabs that cover an asteroid in a similar fashion as the needles covering the body of a hedgehog. On each slab, one then solves the system:

∂ α ∂2 T (t, x) − T (t, x) = 0 , x ∈ (x , x ), t ≥ 0 , (5.3) ∂t ρc ∂x2 0 1

T (0, x) = h(x) , x ∈ (x0, x1) ,

T (t, x1) = g(t) , t ≥ 0 ,

∂T εσT 4(t, x) + α (t, x) = ε (t)(1 − A) , t ≥ 0 . ∂x x=x0 x=x0

206 which constitute the HCE inside the slab, the initial condition, and two boundary con-

ditions, since in the 1 dimensional case, there are two boundaries and one also needs to

specify the conditions on the inner boundary (assumed to be deep inside the asteroid

unaﬀected by the insolation, or lack of thereof, of the surface). In the case where the

inner boundary condition is of Dirichlet type (see App. C.2), the constant is typically

determined from the average heat ﬂux over the orbit, I call it the equilibrium temperature.

5.2.1.1 Analytical Solution of the Heat Conduction Equation

One can further simplify the problem if one assumes that the insolation of the

surface is harmonic (has a constant part and a time dependent part). Then, assuming

that the variations in the temperature are small compared to the equilibrium temperature,

one can obtain an analytical solution to the HCE. This is demonstrated in App. C.4. The

solution is then given by:

√ πfρc √ i 2πft+ϕ− x πfρc ε1(1 − A) 1 α − x T (t, x) = √ e e α + T , (5.4) 3 2 eq 4σTeq 1 + 2Θ + 2Θ ε (1 − A) 1 T (t, 0) = 1 √ ei(2πft+ϕ) + T , (5.5) 3 2 eq 4σTeq 1 + 2Θ + 2Θ √ πfαρc Θ = 3 , (5.6) 4σTeq Θ tan ϕ = − . (5.7) 1 + Θ

Here, ε1 is the amplitude of the harmonic part of the insolation, Teq is the equilibrium temperature, f is the motion frequency (orbital or spin), is the thermal emissivity, A

207 bond albedo, σ the Stefan-Boltzmann constant. This solution has the disadvantage of only working with an asteroid that is either only spinning or only orbiting.

The above solution is a very rough approximation and the bigger the variations of diurnal temperature on the surface, the worse the approximation. One then has to commit oneself to a numerical solution.

5.2.1.2 Numerical Solution to the Heat Conduction Equation

The method of choice for the numerical solution of the system of equations 5.3 is the ﬁnite diﬀerences. The implemented solution is described in depth in App. D.

5.2.2 Summary of Methods and Assumptions

5.2.2.1 Assumptions Leading to 1D Heat Conduction Equation

The goal of this chapter is to illustrate, in orbital terms, the eﬀect of the lower dehydroxylation temperature of Cronstedtite. Temperature distribution within an asteroid depends on its orbital parameters, shape, spin and composition. These comprise too many parameters to investigate such dependence numerically and one is then forced to simplify the reality.

208 First, we assume that the asteroid is a solid body where the heat transport is only due to conduction and that it contains no internal heat sources.

Second, since the composition is typically unknown as a function of position within the asteroid, the parameters such as density, thermal conductivity and capacity are assumed independent of the position within the body.

Third, it is assumed that thermal parameters do not depend on temperature.

Fourth, the surface boundary condition is assumed to be given by energy balance between the incident ﬂux, the re-radiated ﬂux according to the Boltzmann law and the heat conducted into the subsurface according to the Fourier law.

These four assumptions enable to write the HCE in the form of system 5.2.

Fifth, I assume that the asteroid is large and homogeneous enough and that the heat conduction in non-radial direction can be neglected. Further, I assume that for each place on the surface there is a depth below which the changes of the solar insolation have no eﬀect on the temperatures below that depth, that is that the body is isothermal from certain depth. This enables the one dimensional HCE in the form of system 5.3 to be used.

5.2.2.2 Estimating the Orbits

To be able to determine the eﬀects of the lower de-hydroxylation temperature of

Cronstedtite on the possible de-hydroxylation orbits of asteroids containing Serpentine

209 group phyllosilicates, it is necessary to determine the relationship between the orbital and physical parameters and the temperature distribution within the asteroid. This requires large amount of calculations. However, it is possible to attempt to only estimate the maximum orbits beyond which no de-hydroxylation is possible. However, since this encompasses various values of semimajor axis (a) and eccentricity (e), I do this in steps from the most simple methods to more involved.

In order to obtain the estimate of maximum orbits which can still provide enough solar ﬂux to cause de-hydroxylation of the minerals discussed in Ch. 4, I can make use of the following fact. If I have an elliptical orbit with perihelion distance q and a circular orbit with radial distance q, then a given object receives more ﬂux at any given moment on the circular orbit. And thus for a given perihelion distance, the object on a circular orbit will necessarily experience higher temperatures than that on an elliptical one. This means that a circular orbit with a radial distance q maximises the temperature for any combinations of a and e that result in the perihelion q.

This makes it possible to limit oneself only to circular trajectories and search for radial distances from the Sun at which the one dimensional slabs will no longer experience temperatures above a certain threshold, given by the results of Ch. 4.

Next, if the asteroid’s spin axis is perpendicular to the orbital plane, the equatorial part of a spherical asteroid will receive the most ﬂux at any given moment and, as such will, maximise the temperatures on an arbitrarily shaped body with spin axis perpendicular to the orbital plane. If the asteroid’s spin axis is not perpendicular, this

210 does not need to hold. But, if one assumes that equator of the asteroid has the largest cir- cumference then the 1-D slabs on equator of an equivalent radial asteroid’s with spin axis perpendicular to the orbital plane maximise the temperature of the non-perpendicular axis asteroid. Note that this also tacitly assumes principal axis rotations and convex shapes of asteroids. This assumption is not always correct, since in the case of large axial tilts, one would have more insolation onto the surfaces near the poles.

This enables further restriction of the circular problem from studying a distribution of 1-D slabs in various directions to only a single 1-D slab on the equator.

Next, I assume that the largest temperature, which the object can experience occurs on the surface and that their periodicity if the same as the objects spin period.

This enables me to use the analytical expression from Eq. 5.4 to estimate the maximum temperature. As is apparent from Fig. C.2 which shows the diﬀerence between the real insolation of an equatorial surface element and the insolation employed in the simplistic analytical solution, the analytical solution uses signiﬁcantly higher average insolation which, in this case, is favourable since it will maximise the temperature of any asteroidal body on an elliptical orbit with perihelion distance equal to the radial distance in the analytical solution.

With the assumptions stated, I can start with the simplistic analytical solution to estimate the maximum perihelion distances for which Cronstedtite and Antigorite rich asteroid can experience de-hydroxylation temperatures. Within this limit, I can precise the distance with a more realistic numerical solution on a circular trajectory. I

211 can then verify for thus determined perihelion distance, the behaviour of the maximum temperatures with varying eccentricity. This is still under the assumption of spin axis perpendicular to the orbital plane and equatorial 1-D slab on a spherical asteroid.

Starting with the analytical estimations, I can use Eq. 5.5 which attains maximum if the harmonic part is unit and the form is:

(1 − A) 1 Tmax(r) = Teq(r) + ε1(r) 3 q , (5.8) 4εσTeq(r) 1 + 2Θ(r) + 2Θ2(r) where the above assumptions remove the time and x dependences and I explicitly mark the variables that depend on the distance from the Sun. This can be used to determine r as function of a predeﬁned temperature Tmax.

I use the Eq. 5.8 to determine the maximum circular radii at which the surface temperature, for a given material conﬁguration of an asteroid, is suﬃcient to de-hydroxylate either Cronstedtite or Antigorite.

Once these limits are determined, I use numerical solution to the HCE on a circular orbit to browse various perihelion distances. This can then help inform about how the diﬀerent de-hydroxylation temperatures aﬀect the maximum perihelion distance from the

Sun at which an asteroidal body can experience them.

5.3 Results

Initially, I select three material conﬁgurations noted in Tab. 5.1. The conﬁguration denoted C1 is based on Alí-Lagoa, V. et al. (2014), C2 on the thermal parameters of

212 Cold Bokkeveld (CM2) Chondrite by Opeil et al. (2010) and conﬁguration C3 is also

from Opeil et al. (2010) based on the measurements on NWA 5515 (CK4) Chondrite.

Using the analytical solution of the HCE, I estimate the maximum distances at which

the minimum temperature, necessary for a given mineral to lose 4 % or 5 % of initial mass

due to dehydroxylation, can be attained by an equatorial surface element for the given

conﬁguration, on a circular trajectory, as described in the previous section by Eq. 5.8.

The results are presented in Tab. 5.1 as the distances r (e.g. rmax is the maximum Cr4 % distance where the temperature on the surface can be such that it would cause loss of

4 % of Cronstedtite).

Table 5.1: Thermal parameters used for calculations. Units of Γ, α, ρ and c are in the SI base system. P spin is in days and rmax are in astronomical units. The subscript at rmax denotes mineral, Antigorite (An) or Cronstedtite(Cr), and the percentage denotes the that the value was determined at for the temperature at which the mineral loses the set amount of mass as determined in Ch. 4. C1 C2 C3 Γ 451 645 1407 α 0.1166 0.5 1.48 ρ 3110 1662 2675 c 560 500 500 0.9 A 0.1 P spin 3.725 rmax 0.352 0.344 0.323 Cr4 % rmax 0.334 0.327 0.307 Cr5 % rmax 0.191 0.189 0.182 An4 % rmax 0.183 0.181 0.175 An5 %

Next, using a numerical calculation, I determine, for the conﬁgurations in Tab. 5.1,

the maximum temperatures experienced by an equatorial surface element (the spin axis of

the asteroid is perpendicular to the orbital plane and the orbital period for the calculation

213 1600

¡ =451, x=0.00

¡ =451, x=0.05

¡ =451, x=0.10

¡ =645, x=0.00 1400

¡ =645, x=0.05

¡ =645, x=0.10

¡ =1407, x=0.00

¡ =1407, x=0.05

1200 ¡ =1407, x=0.10

1000

800 [C] max T 600

400

200

-200 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Distance [AU]

Figure 5.1: Evolution of the maximum temperature experienced by an equatorial surface element on a sphere with the spin axis perpendicular to the orbital plane as a function of the radial distance and various thermal parameters. The x parameter denotes the depth under the surface (0 is the surface) in m (that is the surface, 5 cm and 10 cm). Lines connecting the points only serve to guide the eye. is Keplerian) at various distances and various depths. The results are printed in Fig. 5.1.

The plot suggests distances (to the right of the curve) beyond which a given temperature cannot be attained at a given depth and for a given material conﬁguration.

From the data, I can also extract the time that a surface element has spent above a particular temperature. This is simply an average calculated from the total time that the surface element spent above a certain temperature during the calculation divided by the time of the calculation which typically extended at least through 2 orbital periods.

214 1 ¡ =451, x=0.00 Time spent at T > 660 C

¡ =451, x=0.05

¡ =451, x=0.10

0.8 ¡ =645, x=0.00

¡ =645, x=0.05

¡ =645, x=0.10

¡ =1407, x=0.00 0.6 ¡ =1407, x=0.05

¡ =1407, x=0.10

0.4 t (T > 660 C) [1]

0.2

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Distance [AU]

1 ¡ =451, x=0.00 Time spent at T > 400 C

¡ =451, x=0.05

¡ =451, x=0.10

0.8 ¡ =645, x=0.00

¡ =645, x=0.05

¡ =645, x=0.10

¡ =1407, x=0.00 0.6 ¡ =1407, x=0.05

¡ =1407, x=0.10

0.4 t (T > 400 C) [1]

0.2

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Distance [AU]

Figure 5.2: Relative time spent above the temperature necessary to de-hydroxylate 4 % of Antigorite and Cronstedtite (corresponding to 400 ◦C and 660 ◦C) as a function of radial distance for various thermal parameters and depths given denoted as x in m (x = 0 is the surface and the other are at 5 cm and 10 cm). The thermal parameters corresponding to Γ as well as the spin period and other parameters are given in Tab. 5.1. Orbits are Keplerian. Different style of dashing corresponds to different depths, varying colours correspond to the different thermal parameters. Lines connecting the points are only to guide the eye.

For the equatorial surface element, the results are displayed in Figs. 5.2 and 5.3. The

figures display the relative time spent by the surface element at temperatures that are sufficient to de-hydroxylate 4 % and 5 % respectively of Antigorite (660 ◦C and 680 ◦C) and Cronstedtite (400 ◦C and 420 ◦C), as a function of distance and for the configurations from Tab. 5.1 and at various depths.

215 1 ¡ =451, x=0.00 Time spent at T > 680 C

¡ =451, x=0.05

¡ =451, x=0.10

0.8 ¡ =645, x=0.00

¡ =645, x=0.05

¡ =645, x=0.10

¡ =1407, x=0.00 0.6 ¡ =1407, x=0.05

¡ =1407, x=0.10

0.4 t (T > 680 C) [1]

0.2

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Distance [AU]

1 ¡ =451, x=0.00 Time spent at T > 420 C

¡ =451, x=0.05

¡ =451, x=0.10

0.8 ¡ =645, x=0.00

¡ =645, x=0.05

¡ =645, x=0.10

¡ =1407, x=0.00 0.6 ¡ =1407, x=0.05

¡ =1407, x=0.10

0.4 t (T > 420 C) [1]

0.2

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Distance [AU]

Figure 5.3: Relative time spent above temperature necessary to de-hydroxylate 5 % of Antigorite and Cronstedtite (corresponding to 420 ◦C and 680 ◦C) as a function of radial distance for various thermal parameters and depths given denoted as x in m (x = 0 is the surface and the other are at 5 cm and 10 cm). The thermal parameters corresponding to Γ as well as the spin period and other parameters are given in Tab. 5.1. Orbits are Keplerian. Different style of dashing corresponds to different depths, varying colours correspond to the different thermal parameters. Lines connecting the points are only to guide the eye.

216 5.4 Discussion and Conclusions

5.4.1 Discussion of the Simplifying Assumptions

The ﬁrst assumption dictated that the asteroid is a solid body. It can also be assumed that it is a mix of small number of solid bodies. However, it is impossible to account for thermal interaction of billions of small grains which thermally interact by conduction and radiation and whose shape and surface contacts are not known. The issues are computational (too many boundary conditions) and physical (unknown distribution and surface contacts). Since we do not know the interior details of an asteroid, any radiative heat transfers inside the asteroid’s volume are unknown and thus impossible to determine. We thus assume only conductive heat transfer inside the asteroid. It is possible to assume various materials inside the asteroid which are fully fused and exchange heat only through conduction. For the same reason, current models are not capable of properly handling any regolith like structures on the surface of asteroid since it would unavoidably involve accounting for conductive and radiative transfers of billions of grains with various contact surfaces which cannot be known. It is possible to simulate a regolith layer by assuming that, as a bulk, it has certain physical properties (density, heat conductivity and capacity) as some solid and then employing a layer of such a solid material in our calculation but it is important to keep in mind that the physical parameters of such solid are unphysical in the sense that if we took a grain of the regolith material and analysed its density, thermal conductivity and capacity, it would have nothing in common with

217 those parameters of the bulk of that regolith. Thus, if one wants to simulate, in thermal calculations, a layer of regolith, one necessarily has to use some proxy solid material which has the same thermal properties as a layer of regolith (typically of unknown depth) which is arranged in an unknown manner and constitutes of material for which the thermal parameters have not been measured.

The second assumption dealt with the constant thermal parameters. This is obviously incorrect since already in Ch. 2, I state that asteroids are largely inhomogeneous bodies. However, since the internal composition is unknown, such an approximation is unavoidable unless one wants to use thermal models to simulate some possible scenarios of possible material distribution.

The third assumption states that the thermal parameters do not depend on the temperature. However, this assumption is incorrect as shown by e.g. Opeil and Britt

(2016). Yet, this assumption is employed for two reasons. First is numerical solution performance. Thermal parameters are typically non-linearly dependent on temperature and such dependence would transform the set D.4 into a non-linear one and a solution would have to implement analogous steps as for the boundary condition D.5 which would signiﬁcantly aﬀect the performance. Second, the data measured by Opeil and Britt

(2016) were only measured to temperatures below 300 K which are not applicable to the temperatures necessary for de-hydroxylation of Cronstedtite or Antigorite.

The fourth assumption tacitly contains that no heat is used to increase the internal energy of the surface. Deﬁnition of surface temperature itself poses some challenges since

218 any temperature can only be defined in a volume rather than on a surface. As such, care should be taken in interpreting the values of surface temperature and their meaning since in the problem it only serves the purpose of providing a boundary condition. There is also the obvious problem that the first assumption above requires a perfectly flat surface which is never true. As such, a better information may be carried by temperature very close to the surface, e.g. 1 mm. Further, this point also contains the assumption that that the heat radiated from other parts of the asteroid is not incident on the surface element. While this assumption is incorrect on real bodies, for convex shaped asteroid, this assumption is satisfied. And since many of the existing shapes of asteroids are deduced using the lighcurve inversion technique (see e.g. Ďurech et al. (2015a)) which is based on the convex shape assumption, this assumption can be used on many shapes without any issues. Obviously the real world asteroids not only contain craters but also rocks and boulders of various sizes which shade each other and reflect light on each other. A quick estimation of such neglecting is possible. At 0.3 au the solar flux is 15 203 W m−2. The heat flux emitted from 1 m2 surface area heated to 400 K (or 673 ◦C) at a radial distance of 1 m away from the surface is 208 W m−2 which corresponds to solar insolation incident on the unit surface area at 0.3 au under the angle of 89.2°. And as such, this can safely be neglected for larger distant structures. Re-radiation between smaller rocks close to each other may be much larger. However, since such calculations would require very good knowledge of the geometry as well as full three dimensional treatment, I consider it negligible for the purpose of this chapter.

219 The fifth assumption enables the simplification of the three dimensional HCE over the volume of the asteroid to solving a system of one dimensional slabs covering the asteroid. These slabs extend radially from some finite depth towards the surface and the surface normal is equal to the local normal of the three dimensional object. It is important to realise that if the penetration depth of the temperature changes imposed by the variable insolation are very small so that the surface area of the surface element subtended by some solid angle and the area, subtended by the same solid angle, of the subsurface element where the isothermal body assumption holds are similar, one does not need to use the radial part of the HCE in spherical coordinates since the only effect of the spherical coordinates would be to account for the change of the surface area, subtended by that solid angle, through which the heat flows from the centre as the radial distance increases. Implementation of a full three dimensional solution on asteroid sized bodies also requires immense computing power and is not feasible unless for a very specific problem of a given asteroid with very well specified parameters.

5.4.2 Discussion of the Results

In Tab. 5.1, I presented guesses of the maximum distance from the Sun where

Antigorite and Cronstedtite materials would experience such temperatures that would lead to loss 4 % and 5 % of their mass due to dehydroxylation. From Ch. 4.4.1, it is known that those should correspond to temperatures of 660 ◦C and 680 ◦C for Antigorite and 400 ◦C and 420 ◦C for Cronstedtite. Visual inspection of the Fig. 5.1 suggests that

220 the analytically determined ranges in Tab. 5.1 provide a good upper guess, under the assumption of a circular trajectory. Note that there are two major diﬀerences between the analytical and numerical methods as suggested also in App. C. The analytical calculation only considered spinning around the axis and an insolation that is rather diﬀerent from the real one (see Fig. C.2 and the discussion in App. C.4).

Several important features can be seen in the Figs. 5.2 and 5.3. The lower dehydroxylation temperature of Cronstedtite translates into about 0.15 au of additional solar distance during which the object can de-hydroxylate.

Another aspect is related to the fact that very close to the Sun, the model suggests that all of the interior is constantly above the temperature necessary to de-hydroxylate

Cronstedtite. This stems from the way the equilibrium temperature is calculated. How- ever, if a body gets to such a distance, especially if it is rather large, the temperature inside it will certainly not be equal to the equilibrium one which corresponds to the circular trajectory. As such, short excursions of an object to such proximity of the Sun are likely to behave diﬀerently, they should experience lower temperatures and the amount of time spent above the given temperature should be lower.

Accordingly with the previous paragraph, objects on an elliptical orbit, especially those with high eccentricity, will necessarily have a lower equilibrium temperature. With this in mind, the calculations illustrated in the previous section are the limiting situations providing the best dehydroxylation results.

221 Material properties can signiﬁcantly aﬀect the situation in the subsurface but on the surface, the material thermal properties play lesser role.

5.4.3 Further Comments

The aim of this chapter is to quantify, how the lower de-hydroxylation temperature of Cronstedtite can aﬀect the orbits on which an asteroid can experience high enough temperatures so that the de-hydroxylation occurs. To this end, the selected method of analysis is to determine the maximum orbits, in terms of perihelia maxima, beyond which the object can no longer receive suﬃcient energy from the Sun to remove hydroxyl from the crystal lattices of phyllosilicate minerals.

This is done in two steps, ﬁrst a very rough model, based on analytical solution of the HCE, is presented. Its major advantage is simplicity and computational speed.

It can be used to quickly exclude trajectories from any further consideration whether de-hydroxylation could have occurred and it can also be used to set an upper limit for more detailed study.

Then, a numerical solution was used to ﬁnd a more precise upper limit. One limit of this study is the use of only a circular trajectory, but as argued above, this is actually necessary to provide an upper limit. A more serious issue can be seen with the assumption of zero axial tilt along with the assumption of a spherical body. This can be seen if one considers some extreme cases such as 90° obliquity which would obviously

222 result in signiﬁcantly higher temperatures on the Sun facing pole. It can be seen that for the case of 90° axial tilt the polar region would actually maximise all other geometrical conﬁgurations but these are not very typical.

The biggest issue considering the attempts to determine temperature distribution within an asteroid, at least close to the surface relates to the issue with the surface and its properties. The issue to discuss is two fold. The first one relates to the absence of regolith in the above model. This is the biggest simplification and downside. However, given the amount of unknown information regarding the thermal properties, including the total depth of granular material on asteroids, there is little that can be done. I admit that one can resort to the approximation with some solid material which has thermal properties as the bulk layer of regolith, however, the material has quite a lot of unknown or poorly restricted parameters which are difficult to estimate even with measurements on ground asteroidal material. The issues were recently illustrated by the Ryugu and

Bennu missions which discovered a rather unexpected rock distribution on the surfaces of the asteroids rather than finely grained material. Such surface pose even more difficulties to determination of thermal states since, locally, various effects that are often neglected, can start playing significant role. A more important question is how these larger rocks affect the global thermal solution, since one might be inclined to neglect some local self- heating or lateral heat conduction. Since in the case of rocks as found on Bennu, one can expect very poor thermal coupling to material beneath them, be it another layer of rocks covered with finer particles or just the finer particles. In this case, the rocky surface on

223 average may actually act like a solid rock and the approximation of a solid body might provide better results than if there was ﬁne grained regolith present instead.

5.4.4 Conclusions

In this chapter, I have shown, how the dehydroxylation temperatures of Cronst- edtite and Antigorite, as the major components of several CCs subgroups, experimentally derived in Ch. 4, translate into the distances where the dehydroxylation can occur in the

Solar System.

I ﬁnd that the temperature diﬀerence in dehydroxylation of Cronstedtite and

Antigorite translates into approximately 0.15 au farther away from the Sun, where Antig- orite no longer can be dehydrated, while Cronstedtite can.

I have shown that even a simple analytical solution to HCE can provide a good guess where the boundaries of the dehydroxylation region may lie. I then used numerical

HCE solver to determine the time spent above the temperatures critical for dehydroxylation on a circular orbit.

224 CHAPTER 6 SUMMARY AND OUTLOOK

6.1 Discussion

In Sec. 4.5.2, I have discussed the feature in Fig. 4.20 suggestive of step mass loss

during the temperature increase. If the suggested cause of the observed features is due

to cracking, one can relate those observations to the data collected in Ch. 2 where it was

noted that the tensile force of CCs varies on the order of MPa and the compressive force

on the order of 10 MPa. An order of magnitude estimate for the pressures exerted by

hydroxyls or water within the crystal lattice of Antigorite can be done. One can make

use of the unit crystal cell dimensions provided by Capitani and Mellini (2004) which

3 lead to the volume of the unit cell of 2922 Å . The size across the layers was determined

to be 7.263 Å. Carmignano et al. (2019) provides the size of the space between the

Silica and Magnesium cations, where the hydroxyls are contained, to be approximately

3 Å leading to volume of approximately 1208 Å. A simple estimation for a perfect gas then leads to the pressures between approximately (0.15–0.71) MPa K−1 ﬁrst case being

the pressure per unit temperature of water molecules occupying the whole unit cell,

the other being the pressure of hydroxyl molecules occupying just their space between

the cations. For a temperature of about 470 ◦C, this would lead to pressures between

approximately (30–140) MPa. At these temperatures, one also should account for the

225 material volumetric expansion and increased elastic and plastic responses, however, as an order of magnitude estimate, this shows that temperature induced pressure of de- hydroxylating water can reach levels that may surpass the tensile or even compressive strengths of various meteoritic materials discussed in Ch. 2. The subsequent release of energy due to the rupture might be large enough to provide kinetic energy to parts of either the rupturing material or the surrounding material. If this interpretation of

Fig. 4.20 is correct, as the rupture of a grain followed by its ejection due to surplus of kinetic energy as a result of accumulated water pressure inside the grain, it would provide another process for fracturing of asteroidal boulders besides thermal fatigue demonstrated by Delbo et al. (2014). Further, since, unlike the thermal fatigue, this eﬀect does not beneﬁt from multiple cycles but, rather, happens during one cycle of temperature increase, it would provide a faster way to rupture or at least produce cracks in the hydrated material.

A closely related discussion revolves the effect of de-hydroxylation process on the strength of the material. Several processes need to be considered. First, the heating of the material leads to material volumetric expansion. Second, since most of the asteroidal material containing the Serpentine mineral group can be expected to be highly heterogeneous and the mineral phases to be intermixed (e.g. Mg Serpentine and Saponite), in such material, the different coefficients of volumetric expansion will result in emergence of differential stresses inside the material. Third, as the material is heated and water or

OH molecules gain more kinetic energy (be it either in still some form of crystal structure between the phyllosilicate layers or in the amorphous ordering), it tends to exert pres-

226 sure, as suggested in the previous paragraph, this results in additional stress within the material. Fourth, with more heat, the elastic and plastic behaviour can be expected to extend over a larger range of stresses. Processes two and three will lead to introduction of weaknesses in the material, processes one and four will lead to the ability of the material to survive the former two effects without introduction of the weaknesses (instead of e.g. creation of a crack, material will be able to withstand increased the local stresses due to water pressure still within its elastic range). There is a fifth processes that needs to be considered. If the processes increasing the stress (differential volumetric expansion and water pressure) result in a sudden disruption of the material, this effectively removes many of the pre-existing weak points and the two (or more) resulting parts will exhibit stronger properties. Finally a sixth process can be considered and is due to the crystallisation of post-dehydroxylation stages. Such crystallisation in the case of Antigorite results in Forsterite, thermally very stable mineral with melting temperature at approximately

1900 ◦C. Such crystallisation can be also assumed to remove weaknesses in the structure and thus restore the strength of the material. As such, one can consider that the level of heating as well as the amount of exposed time to those temperatures can play role in the final effect of full or partial de-hydroxylation or recrystallisation. One might then assume that as the material starts to de-hydroxylate, its strength is being decreased, any sudden rupture during this regime will result in release of immediate stress state inside the material and a sudden jump to higher strength values, once a transition amorphous phase, devoid of water, is attained, only stresses due to differential volumetric expansion prevail which are probably offset by increased elasticity and plasticity, which is then

227 also manifested in formation of a discernible crystal structure. Obviously at those temperatures, its behaviour can be expected to be highly elastic and plastic, however, upon cooling the material is unlikely to contain weaknesses due to the de-hydroxylation process while cooling from the temperatures below the crystallisation of the post-dehydroxylation minerals might result in a structure with decreased strength.

Further point in the discussion of the results presented in Ch. 2 on the strengths of meteoritic material needs to be made regarding the extension of the results to asteroidal bodies, either the asteroids themselves or boulders found on their surface or possibly the subsurface. While the data presented in Ch. 2 and their discussion provide a strong and founded critique of applicability of a monotonic type of scale effect based on statistical theories to scale from meteorites to asteroids or boulders on them, as a part of the broader discussion, it seems fit to discuss the problem in more depth. The aforementioned scale effect has been suggested to be applied through the work of Weibull (1939) where the author develops the probabilistic description of rupture in the form:

R n(σ,~x)d V P (σ) = 1 − e V , (6.1) where P (σ) is the probability of rupture at the applied stress σ and n(σ, ~x) is a material dependent function integrated over the volume V of the body whose probability of rupture is investigated. This leads to the mean ultimate strength of the material as

Z ∞ − R n(σ,~x)d V σmean = e V d σ . (6.2) 0

228 m The material function is often taken as a power law in the form n(σ) = k σ leading σ0

to the probability of rupture and the mean ultimate strength:

m −V σ P (σ) = 1 − e σ0 , (6.3)

Z ∞ σ0 −xm σmean = e d x . (6.4) V 1/m 0

In order to obtain the Eq. 6.1, one needs to assume that for rupture of any large volume

V the rupture of any sub-volume ∆V is sufficient, or the so called weakest link. The assumption when going to Eq. 6.3 requires a perfectly homogeneous and isotropic material so that one can proceed with the integration over the volume of the material as suggested in Eq. 6.1. This is not satisfied for meteoritic or asteroidal materials and such approach is incorrect which presents itself in the data measured e.g. by Zotkin et al. (1987). It needs to be stipulated that this is not equivalent to a claim that a scale effect be non- existent. From the arguments given by Weibull (1939) one can see that such a statistical description can be manifested (as documented by numerous measurements), however, it is absolutely necessary that either the material is homogeneous or the heterogeneity and lack of isotropy do not change with increasing volume and thus changing the pre-existing distribution of weak points in the material which is the central assumption to the validity of the weakest link assumption.

Another scale related eﬀect can be considered. It has been noted in Ch. 2 that the heterogeneity drives the strength and so one may argue that the decrease of strength with size is related to increase in the heterogeneity of the material with size but this is based on the assumption that a more heterogeneous material is always weaker. This is

229 not necessarily true using a steel wire enforced concrete as a tangent example. In the case of Iron meteorites, the work of Yavnel (1963) is of special importance since the author goes into high level of detail and analysed both the mono-crystalline samples made of

Kamacite and poly-crystalline samples which included also various inclusions of various minerals between Fe-Ni crystals. It can be seen from Tab. B.1 that the poly-crystalline samples are about 25 % weaker in compression and almost an order of magnitude weaker in tensile testing. Since study of similar extent has not been carried out on other types of meteoritic material, the eﬀect of increased heterogeneity in the case of CCs or ordinary

Chondrites is not obvious.

The fact that scale eﬀect does not relate strength with size in an easy and pre- dictable manner on small bodies in space is suggested in the paper by Delbo et al. (2014) in their Fig. 1, the authors show size dependent time-scale which is necessary to destroy boulders on various asteroids at various distances by either micro-meteorite bombardment or by thermal cycling. Once can see from their calculation that the larger the boulder, the less time it takes to break it by thermal cycling but the more time to break it by micro-meteorite bombardment. This can be understood as two types of scale eﬀects inverse to each other.

The extrapolation of the results on meteorites to asteroids is not possible due to their nature as a mixture of various materials, including dust particles that are created during impact events. Also, it is not immediately clear what one should understand by the strength of such a huge body with size on the order of km. The measurements on meteorites can however be a starting point in our understanding of strengths of rocks

230 and probably boulders found on asteroids. These strengths can be modiﬁed by various

processes, such as thermal cycling suggested by Delbo et al. (2014), micro-meteorite

impacts, exposure to energetic particles and, in the case of close approaches to the Sun,

by de-hydration and de-hydroxylation. Statistical theories cannot reproduce these eﬀects

since there is no guarantee that the tested meteorites actually carry a representative

distribution of the weakening eﬀects of these processes and they also assume that these

processes do no depend on the size of the object.

Another point necessary to discuss in more depth is related to the results presented

in Ch. 4 where I provided some experimental data on de-hydroxylation of minerals of the

Serpentine group. All the samples tested had granular form of grains with sizes at most

0.5 mm. With heating rate 20 ◦C min−1 it was shown that it is possible to de-hydroxylate

the material fairly fast (e.g. within 15 min for Antigorite). However, this result was obtained in an experiment with increasing temperature and so with increasing supply of heat. However, as was exposed in the introductory section as well as in the discussion, the two step (in the case of Antigorite) de-hydroxylation most likely represents the dehydroxylation of the surface of the grains and the interior. Since the grain does not

“shed” mass during de-hydroxylation, it is increasingly diﬃcult for the water from the interior of the grain to get outside. This is also suggested by an experiment at which the Antigorite sample was left at a given temperature for 4 h and then the temperature

was increased by 50 ◦C and the sample again left isothermally for 4 h. This experiment is

displayed in Fig. 6.1. It can be seen that the speed of de-hydroxylation slows down with

231 time and only after the temperature is increased does it “speed-up” again but with time slows down and so on.

100.00 850 C

800 C

750 C

95.00

700 C

650 C Mass [%]

600 C

90.00

550 C

500 C

85.00 200 400 600 800 1000 1200 1400 1600 1800 t [min]

Figure 6.1: The kinetics experiment with a sample of unsieved Antigorite during which the sample was left isothermally at various temperatures indicated in the plot for 4 h before the temperature was raised by 50 ◦C. The red curve indicates mass loss pertinent to the left y axis and the green dashed curve is the temperature increased with time in steps.

These points are very closely related to the true surfaces of asteroids 101955 Bennu and 162173 Ryugu where the current missions in progress have provided detailed views of the surface covered with rocks and large boulders rather than with ﬁne regolith. If the surfaces of asteroids targeted to provide hydrated minerals should look alike, the results provided in Ch. 4 might need veriﬁed using much larger samples than grains. Given the

232 results presented in Ch. 4 related to the de-hydroxylation of the surface and grain interior

along with hints provided in Fig. 6.1 and also with the fact the infrared spectroscopy sur-

veys only the thin surface layers, it is feasible to consider that the necessary temperature

to remove water might be somewhat higher in order to provide suﬃcient energy for the

water molecules to escape. This is also related to the ideas discussed above. If the kinetic

energy is not provided to the molecules they might exert suﬃcient pressure to create

ﬁssures and cracks to escape. A detailed study of heat conduction, similar to those for

studying Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) eﬀects (full 3 dimensional

model), in the case of the meter and more sized boulders is necessary to understand their

hydration state.

Another point related to the hydration states of asteroids concerns the possibility

to observe de-hydroxylated asteroids. In Ch. 4, I have mentioned that Serpentine min-

erals have an absorption band in near infrared spectrum between around 2.8 µm due to

stretching of the O–H bond. Rivkin et al. (2015a) connected this absorption with the

absorption due to electron exchange between Fe2+ and Fe3+ between (0.6–0.7) µm. I have shown that the OH stretching band is removed due to de-hydroxylation. However, the question remains whether the Fe related band remains and would be a suﬃcient proof of the previously hydrated state of the surface of the body.

233 6.2 Summary

In this dissertation I investigated several physical properties of asteroid material.

In the introduction I overviewed several physical properties of meteoritic material as a link to the asteroid material. One of the areas that was found interesting was data on strength of meteorites. Aside compiling the most up-to-date database of strength measurements, the data showed high variations even within the same samples of measured material.

Another side product of this compendium of measurements was no observable Scale Eﬀect in the data. I suggest that instead of exponential Scale Eﬀect, it is heterogeneity which governs the strength of the material.

As another interesting aspect of asteroid material, inspired mostly by the possibility of resource utilisation, its interactions with charged particles were investigated. I ﬁnd that phyllosilicate rich materials are particularly good at slowing down Galactic Cosmic

Rays and Solar Protons, better than Aluminium. The conclusions of the research may also have some implications for investigations of meteorite ages by studying their Cosmic

Ray Exposure times. Since the penetration depth of Cosmic Particles depends on the material, the material that is under phyllosilicates will receive less exposure.

Next I investigated how asteroids, dominated by phyllosilicates, lose water. This question was approach at two fronts. First, it was necessary to experimentally determine the temperatures at which the loss of water starts and the kinetics of such process. This was studied using TGA in both inert gas ﬂow and under vacuum conditions. It was found that speciﬁcally, Fe member of Serpentine mineral group, Cronstedtite, de-hydroxylated

234 at signiﬁcantly lower temperatures than other Mg members of the group and as such, makes dehydration by close approach of an asteroid to the Sun feasible.

In the second part, I used the experimentally determine data to illustrate what eﬀect the lower temperature of dehydroxylation has on trajectories of asteroids.

6.3 Outlook

Several additional projects can be identiﬁed as a logically following from this work.

Firstly, in Ch. 4, I determined that Cronstedtite undergoes major thermal transformation between (330–470) ◦C. Using XRD and FTIR detailed studies accompanied possibly by Scanning Electron Microscopy and Transmission Electron Microscopy in diﬀraction mode, one should could hope to learn more details about this transformation. Especially interesting is a more precise determination of the de-hydroxylation temperature and the associated spectral changes since those could be related to observation data.

Another extension of the project from Ch. 4 is related to more detailed vacuum studies. While mild effect of vacuum was observed, more data are necessary to be able to describe the variability as well as the effects on Cronstedtite. Further, asteroids live in a very hard vacuum conditions. Studies under high vacuum conditions, would provide better insight and higher level of confidence.

Another issue related to the studies in Ch. 4 stem from the recent space missions to asteroids 101955 Bennu and 162173 Ryugu which have shown signiﬁcant number of large

235 rocks and boulders on the surface. All our studies were performed on small grains. Heat-

ing experiments on larger rocks might result in a little diﬀerent type of de-hydroxylation

kinetics and also might show a more obvious cracking as the result of accumulated water

pressure.

The newly determined results in Ch. 4 suggest that in some cases, the de-hydroxylation

probabilities of asteroids might need to be revised. As shown in Ch. 5, there is a non-

negligible increase in the perihelion distances at which the possibility of de-hydroxylation

must be studied in detail, including more options for subsurface to de-hydroxylate. E.g.

for the asteroid 101955 Bennu, based on the work by Delbo et al. (2014) there is about

10 % probability that it has spent some time on orbits with a perihelion below 0.3 au which, as shown in Ch. 5, is the perihelion where the Cronstedtite de-hydroxylation temperatures are possible.

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244 APPENDIX B TABLE OF STRENGTHS OF METEORITES

245 Table B.1: Summary of the strength measurements from the literature. The columns denote: “From” the data source, “Met” meteorite name, “S” Shock stage, “W” Weathering, “L” Length (typically along the compression axis), “W/D” is one cross-sectional side for cuboid samples (Width) or Diameter for cylindrical samples, “D” is the other side of a cuboid (Depth), “ρblk” bulk density, “ρgr ” grain density, “P” Porosity, “Rt” compression or extension rate of the testing machine (or strain rate if units are −1 s or any other quantity, with appropriate units, affecting the strain rate), “σc” compressive strength, “σt” tensile strength, “Ed” and “Ev ” are Young’s moduli measured Directly and from Velocities respectively, “νd” and “νv ” are Poisson coefficients measured Directly and from sound Velocities respectively, “vl” velocity of longitudinal waves, “vt” velocity of transversal waves. The meteorite names followed by “*” denote data extracted from plots as described in Sec. 2.2.4. Letters a, b, c in italic following the meteorite name means measurements along 3 different axes on the same sample. The data come from the following sources (abbreviated as suggested): Baldwin and Sheaffer (1971) (Ba1971), Buddhue (1942) (Bu1942), Cotto-Figueroa et al. (2016) (Co2016), Furnish et al. (1995) (Fu1994), Grokhovsky et al. (2013) (Gr2013), Hogan et al. (2015) (Ho2015), Jenniskens et al. (2009) (Je2009), Jenniskens et al. (2012) (Je2012), Jenniskens et al. (2014) (Je2014), Kimberley et al. (2010) (Ki2010), Kimberley and Ramesh (2011) (Ki2011), Kimberley et al. (2015) (Ki2015), Knox (1970) (Kn1970), Medvedev (1974) (Me1974), Medvedev et al. (1985) (Me1985), Miura et al. (2008) (Mi2008), Molesky et al. (2015) (Mo2015), Slyuta et al. (2007) (Sl2007), Slyuta et al. (2008a) (Sl2008), Slyuta et al. (2008b) (Sl2008a) Slyuta et al. (2009) (Sl2009), Slyuta (2010) (Sl2010), Tsuchiyama et al. (2008) (Ts2008), Voropaev et al. (2017b) (Vo2017), Yavnel (1963) (Ya1963), Zotkin et al. (1987) (Zo1987).

−3 −3 From Met Type S W L W/D D ρblk × 10 ρgr × 10 P Rt σc σt Ed Ev νd νv vl vt [cm] [cm] [cm] [kg m−3] [kg m−3] [%] [mm/min] [MPa] [MPa] GPa GPa [1] [1] [m s−1] [m s−1]

Ba1971 Seminole H4 ------173.6 22.5 ------Bu1942 Holbrook, Arizona L/LL6 ------1.27 6.21 ------Bu1942 Covert, Kansas H5 ------1.27 75.387 ------Bu1942 Ness County, Kansas L6 ------1.27 82.909 ------Bu1942 Morland, Kansas H6 ------1.27 159.61 ------Bu1942 Alamogordo, New Mexico H5 ------1.27 269.59 ------Bu1942 Kimble County, Texas H6 ------1.27 321.30 ------Bu1942 Arapahoe, Colorado L5 ------1.27 351.70 ------Bu1942 La Lande, New Mexico L5 ------1.27 374.04 ------Bu1942 Descubridora (fragm. of IIIA-Om ------1.27 372.32 ------Charcas), Mexico Co2016 Allende CV3 0.702 0.687 0.684 2.9 -- 0.25 24.7 - 16.27 12.49 -- 2101 1566 Co2016 Allende CV3 0.707 0.713 0.717 2.96 -- 0.25 39.6 - 17.31 13.94 -- 2171 1539 Co2016 Allende CV3 0.731 0.721 0.712 2.88 -- 0.25 31.7 - 17.46 14.70 -- 2261 1615 Co2016 Allende CV3 0.962 0.981 0.984 2.88 -- 0.25 27.6 - 17.21 16.05 -- 2361 1667 Co2016 Allende CV3 1.038 0.982 0.973 2.90 -- 0.25 39.2 - 27.40 28.70 -- 3156 2159 Co2016 Allende CV3 1.016 1.051 0.977 2.91 -- 0.25 36.5 - 15.66 14.94 -- 2267 1630 Co2016 Allende CV3 0.986 1.064 1.006 2.85 -- 0.25 22.5 - 16.26 20.36 -- 2781 1740 Co2016 Allende CV3 1.73 1.703 1.694 2.92 -- 0.25 58.4 - 7.57 14.41 -- 2268 1478 Co2016 Allende CV3 2.535 2.585 2.658 2.899 -- 0.25 32.3 - 12.77 ----- Co2016 Allende CV3 4.408 4.394 4.323 2.922 -- 0.25 41.1 - 8.17 14.38 -- 2262 1476 Co2016 Tamdakht H5 S3 1.285 1.275 1.313 3.482 -- 0.25 83.6 -- 23.02 -- 2644 1694 Co2016 Tamdakht H5 S3 1.290 1.260 1.305 3.493 -- 0.25 120.5 - 23.31 21.63 -- 2624 1606 Co2016 Tamdakht H5 S3 1.284 1.270 1.334 3.524 -- 0.25 130.7 - 31.05 29.76 -- 3415 1796 Co2016 Tamdakht H5 S3 1.294 1.284 1.297 3.543 -- 0.25 84.3 - 21.85 22.58 -- 2813 1586 Co2016 Tamdakht H5 S3 1.266 1.252 1.300 3.527 -- 0.25 186 - 16.89 31.17 -- 2977 2063

246 Co2016 Tamdakht H5 S3 1.295 1.277 1.300 3.402 -- 0.25 25.9 - 9.9 10.97 -- 1800 1240 Co2016 Tamdakht H5 S3 1.291 1.265 1.311 3.530 -- 0.25 247.4 - 23.07 26.54 -- 2742 1946 Co2016 Tamdakht H5 S3 1.284 1.267 1.312 3.500 -- 0.25 120.5 - 16.93 16.74 -- 2781 1330 Co2016 Tamdakht H5 S3 2.155 2.037 1.964 3.472 -- 0.25 160.2 - 12.18 14.69 -- 2086 1380 Co2016 Tamdakht H5 S3 1.944 2.151 2.094 3.510 -- 0.25 76.0 - 15.90 14.96 -- 2140 1350 Co2016 Tamdakht H5 S3 1.949 2.105 2.065 3.481 -- 0.25 99.6 - 14.68 13.08 -- 1976 1290 Co2016 Tamdakht H5 S3 3.130 3.060 3.104 3.376 -- 0.25 97.0 - 19.36 18.15 -- 2459 1491 Co2016 Tamdakht H5 S3 3.125 3.027 3.048 3.498 -- 0.25 183.0 - 25.57 29.82 -- 2983 1938 Fu1995 Hoba IVB-D ------0.001 s−1 700 ------Fu1995 Henbury IIIA-Om ------0.001 s−1 350 ------Fu1995 Coahuila IIAB-H ------0.001 s−1 350 ------Gr2013 Chelyabinsk LL5 ------0.5 64.0 ------Gr2013 Tsarev L5 ------0.5 132.5 ------Ho2015 GRO 85209 L6 S1 0.53 0.35 0.4 3.350 - 7 Quasi-static - - 14 ----- Ho2015 GRO 85209 L6 S1 - 1 0.15 3.350 -- 114 s−1 - 36 ------Ho2015 GRO 85209 L6 S1 0.53 0.35 0.4 3.350 -- 1000 s−1 294 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (0.0010 ± 0.0001) s−1 62 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (0.0010 ± 0.0001) s−1 96 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (0.0010 ± 0.0001) s−1 104 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (0.0010 ± 0.0001) s−1 136 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (1.0 ± 0.1) s−1 68 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (1.0 ± 0.1) s−1 76 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (1.0 ± 0.1) s−1 122 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (100 ± 13) s−1 227 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (100 ± 12) s−1 280 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (120 ± 15) s−1 238 ± 5 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (161 ± 19) s−1 244 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (1001 ± 122) s−1 280 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (1001 ± 120) s−1 293 ± 6 ------Ho2015 GRO 85209* L6 S1 0.53 0.35 0.4 3.350 -- (1001 ± 115) s−1 318 ± 6 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (0.0010 ± 0.0001) s−1 - 9 ± 1 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (1.0 ± 0.1) s−1 - 9 ± 1 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (64 ± 7) s−1 - 31 ± 1 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (90 ± 10) s−1 - 29 ± 1 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (99 ± 11) s−1 - 40 ± 1 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (113 ± 13) s−1 - 36 ± 1 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (157 ± 17) s−1 - 45 ± 1 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (253 ± 29) s−1 - 34 ± 1 ------

246 −3 −3 From Met Type S W L W/D D ρblk × 10 ρgr × 10 P Rt σc σt Ed Ev νd νv vl vt [cm] [cm] [cm] [kg m−3] [kg m−3] [%] [mm/min] [MPa] [MPa] GPa GPa [1] [1] [m s−1] [m s−1]

Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (341 ± 39) s−1 - 65 ± 1 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (380 ± 43) s−1 - 45 ± 1 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (400 ± 47) s−1 - 55 ± 1 ------Ho2015 GRO 85209* L6 S1 - 1 0.15 3.350 -- (447 ± 49) s−1 - 87 ± 1 ------Je2009 Almahata Sitta Ureilite ------56 ± 26 ------Je2012 Sutter’s Mill CM2 - - - 2.31 ± 0.04 3.34 ± 0.02 31 ± 1.4 - 82 ± 6 ------Je2014 Novato N01 L6 S4 ------1100 ± 250 ------Ki2010 MacAlpine Hills 88118 L5 S1 0.5 0.5 0.5 3.240 3.720 13 500 s−1 161 - 8.5 ----- Ki2011 MacAlpine Hills 88118 L5 S1 0.5 0.5 0.5 3.240 3.720 13 0.001 s−1 50 - 3.2 ----- Ki2011 MacAlpine Hills 88118* L5 S1 0.5 0.5 0.5 3.240 3.720 13 (0.0010 ± 0.0001) s−1 49 ± 2 ------Ki2011 MacAlpine Hills 88118* L5 S1 0.5 0.5 0.5 3.240 3.720 13 (0.0010 ± 0.0001) s−1 53 ± 2 ------Ki2011 MacAlpine Hills 88118* L5 S1 0.5 0.5 0.5 3.240 3.720 13 (0.10 ± 0.01) s−1 59 ± 2 ------Ki2011 MacAlpine Hills 88118* L5 S1 0.5 0.5 0.5 3.240 3.720 13 (0.10 ± 0.01) s−1 60 ± 2 ------Ki2011 MacAlpine Hills 88118* L5 S1 0.5 0.5 0.5 3.240 3.720 13 (148 ± 20) s−1 138 ± 2 ------Ki2011 MacAlpine Hills 88118* L5 S1 0.5 0.5 0.5 3.240 3.720 13 (444 ± 64) s−1 161 ± 2 ------Ki2011 MacAlpine Hills 88118* L5 S1 0.5 0.5 0.5 3.240 3.720 13 (493 ± 68) s−1 143 ± 2 ------Ki2011 MacAlpine Hills 88118* L5 S1 0.5 0.5 0.5 3.240 3.720 13 (1001 ± 133) s−1 175 ± 2 ------Ki2011 MacAlpine Hills 88118* L5 S1 0.5 0.5 0.5 3.240 3.720 13 (1398 ± 193) s−1 180 ± 2 ------Ki2011 MacAlpine Hills 88118* L5 S1 0.5 0.5 0.5 3.240 3.720 13 (1247 ± 173) s−1 194 ± 2 ------Ki2015 GRO 85209 L6 S1 ------225 ------Kn1970 Canyon Diablo-1 IAB-MG 0.4 0.2 0.2 ---- 420 ------Kn1970 Canyon Diablo-1A IAB-MG 0.4 0.2 0.2 ---- 414 ------Kn1970 Canyon Diablo-2 IAB-MG 0.4 0.2 0.2 ---- 447 ------Kn1970 Brenham PMG-an 0.4 0.2 0.2 ---- 432 ------Kn1970 Odessa (iron) IAB-MG 0.8 0.4 0.4 ---- 321 ------Me1974 Kunashak L6 - - - 3.54 --- 265 ± 50 49 ± 4 66 81 - 0.17 5000 ± 40 3290 Me1974 Elenovka L5 S2 W0 - - - 3.50 --- 20 ± 15 2 ± 1 3.3 40 − 50 -- 4500 - Me1985 Krymka, 1705 LL3 - - - 3.25 --- 160 22 - 78 - 0.15 4900 3140 Me1985 Elenovka, 1831 L5 - - - 3.50 --- 20 2 - 56 - 0.24 4320 2490 Me1985 Tsarev, 15380a L5 - - - 3.52 --- 222 26 - 158 - 0.19 6990 4300 Me1985 Tsarev, 15384a L5 - - - 3.55 ------161 - 0.19 7000 4350 Me1985 Tsarev, 15384b L5 - - - 3.43 --- 450 54.6 - 190 - 0.29 6970 3770 Me1985 Tsarev, 15391 L5 - - - 3.24 --- 157 16 - 101 - 0.28 6240 3430

247 Me1985 Kunashak, 1723 L6 - - - 3.54 --- 265 49 - 87 - 0.26 5440 3090 Me1985 Kyushu 2157 L6 - - - 3.90 --- 98 11 - 52 - 0.26 3990 2290 Me1985 Pultusk 544 H5 - - - 3.56 --- 213 31 - 76 - 0.27 5150 2860 Mi2008 Murchison CM2 1 0.5 ---- 0.06 50 ------Mi2008 La Criolla L6 1 0.5 ---- 0.06 98 ------Mo2015 Nortwest Africa 869 L3-6 - - - 3.51 ± 0.0021 --- 98.42 ± 13.35 ------Sl2007 Sayh al Uhaymir 001 L5 S2 W1 - - - 3.45 ± 0.02 --- 112.44 ------Sl2007 Sayh al Uhaymir 001 L5 S2 W1 2 − 3 2 − 3 2 − 3 3.45 ± 0.02 --- 101.6 ± 31.6 ------Sl2007 Sayh al Uhaymir 001 L5 S2 W1 - - - 3.45 ± 0.02 ---- 16.12 ± 5.24 ------Sl2007 Sayh al Uhaymir 001 L5 S2 W1 - - - 3.45 ± 0.02 ----- 17.1 ± 2.1 - 0.33 ± 0.07 - 3121.9 ± 636.9 - Sl2009 Sayh al Uhaymir 001 a L5 S2 W1 ------143 ± 29 18 ± 5 ------Sl2009 Sayh al Uhaymir 001 b L5 S2 W1 ------94 ± 27 17 ± 4 ------Sl2009 Sayh al Uhaymir 001 c L5 S2 W1 ------91 ± 21 18 ± 5 ------Sl2010 Sayh al Uhaymir 001 L5 S2 W1 ------105 ± 33 18 ± 5 ------Sl2010 Tsarev, 15390,9 L5 - - - 3.40 --- 203 ± 71 29 ± 10 ------Sl2010 Tsarev, 15384,1 L5 - - - 3.41 --- 194 ± 58 31 ± 11 ------Sl2008 Ghubara L5 - - - 3.45 --- 72.22 ± 22.16 ------Sl2008 Ghubara L5 - - - 3.45 ---- 23.55 ± 7.19 ---- 2989.9 ± 633.6 - Sl2008 Ghubara L5 - - - 3.45 ------Sl008b Tsarev, 15390,9 a L5 ------262 ± 50 28 ± 9 ------Sl008b Tsarev, 15390,9 b L5 ------168 ± 62 34 ± 12 ------Sl008b Tsarev, 15390,9 c L5 ------160 ± 46 27 ± 8 ------Sl008b Tsarev, 15384,1 a L5 ------223 ± 65 31 ± 10 ------Sl008b Tsarev, 15384,1 b L5 ------182 ± 46 34 ± 10 ------Sl008b Tsarev, 15384,1 c L5 ------174 ± 50 29 ± 12 ------Ts2008 Murchison CM2 - 0.01 ------2.0 ± 1.5 ------Ts2008 Murray CM2 - 0.01 ------8.8 ± 4.8 ------Ts2008 Ivuna CI1 - 0.01 ------0.7 ± 0.2 ------Ts2008 Orgueil CI1 - 0.01 ------2.8 ± 1.9 ------Ts2008 Tagish Lake – carb. rich C2-u - 0.01 ------6.7 ± 9.8 ------Ts2008 Tagish Lake – carb. poor C2-u - 0.01 ------0.8 ± 0.3 ------Vo2017 Chelyabinsk - chondritic LL5 S4 5 2.5 - 3.22 --- 45.2 3.4 8.621 - 0.2 --- Vo2017 Chelyabinsk - shock melt LL5 S4 5 2.5 - 3.31 --- 73.66 1.68 8.776 - 0.21 --- Ya1963 Sikhote-Alin – monoc. T1 IIAB-Ogg 1 0.2 ---- 0.2 - 162 ------Ya1963 Sikhote-Alin – monoc. T1 IIAB-Ogg 1 0.2 ---- 0.2 - 426 ------Ya1963 Sikhote-Alin – monoc. T1 IIAB-Ogg 1 0.2 ---- 0.2 - 426 ------Ya1963 Sikhote-Alin – monoc. T1 IIAB-Ogg 1 0.2 ---- 0.2 - 426 ------Ya1963 Sikhote-Alin – monoc. T1 IIAB-Ogg 1 0.2 ---- 0.2 - 483 ------Ya1963 Sikhote-Alin - monoc. T1 IIAB-Ogg 0.6 0.4 ---- 0.2 71 ------Ya1963 Sikhote-Alin - monoc. T1 IIAB-Ogg 0.6 0.4 ---- 0.2 623 ------Ya1963 Sikhote-Alin - monoc. T1 IIAB-Ogg 0.6 0.4 ---- 0.2 481 ------Ya1963 Sikhote-Alin - monoc. T1 IIAB-Ogg 0.6 0.4 ---- 0.2 510 ------

247 −3 −3 From Met Type S W L W/D D ρblk × 10 ρgr × 10 P Rt σc σt Ed Ev νd νv vl vt [cm] [cm] [cm] [kg m−3] [kg m−3] [%] [mm/min] [MPa] [MPa] GPa GPa [1] [1] [m s−1] [m s−1]

Ya1963 Sikhote-Alin - monoc. T1 IIAB-Ogg 0.6 0.4 ---- 0.2 500 ------Ya1963 Sikhote-Alin – monoc. T2 IIAB-Ogg 1 0.2 ---- 0.2 - 370 ------Ya1963 Sikhote-Alin – monoc. T2 IIAB-Ogg 1 0.2 ---- 0.2 - 400 ------Ya1963 Sikhote-Alin – monoc. T2 IIAB-Ogg 1 0.2 ---- 0.2 - 345 ------Ya1963 Sikhote-Alin – monoc. T2 IIAB-Ogg 1 0.2 ---- 0.2 - 332 ------Ya1963 Sikhote-Alin – monoc. T2 IIAB-Ogg 1 0.2 ---- 0.2 - 327 ------Ya1963 Sikhote-Alin – polyc. IIAB-Ogg 16 3 3 ----- 43 ------Ya1963 Sikhote-Alin - polyc. IIAB-Ogg 4 4 4 ---- 368 ------Ya1963 Sikhote-Alin - polyc. IIAB-Ogg 4 4 4 ---- 429 ------Zo1987 Tsarev L5 10.0 10.0 10.0 ---- 220.6 ------Zo1987 Tsarev L5 10.0 10.0 10.0 ---- 281.5 ------Zo1987 Tsarev L5 7.0 7.0 7.0 ---- 330.5 ------Zo1987 Tsarev L5 7.0 7.0 7.0 ---- 347.2 ------Zo1987 Tsarev L5 7.0 7.0 7.0 ---- 352.1 ------Zo1987 Tsarev L5 7.0 7.0 7.0 ---- 357.9 ------Zo1987 Tsarev L5 5.0 5.0 5.0 ---- 260.9 ------Zo1987 Tsarev L5 5.0 5.0 5.0 ---- 398.1 ------Zo1987 Tsarev L5 5.0 5.0 5.0 ---- 416.8 ------Zo1987 Tsarev L5 5.0 5.0 5.0 ---- 427.6 ------Zo1987 Tsarev L5 4.0 4.0 4.0 ---- 411.9 ------Zo1987 Tsarev L5 4.0 4.0 4.0 ---- 443.3 ------Zo1987 Tsarev L5 4.0 4.0 4.0 ---- 512.9 ------Zo1987 Tsarev L5 4.0 4.0 4.0 ---- 527.6 ------Zo1987 Tsarev L5 4.0 4.0 4.0 ---- 550.2 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ---- 261.8 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ---- 319.7 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ---- 390.3 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ---- 408.9 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ---- 437.4 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ---- 459.0 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ---- 307.9 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ---- 328.5 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ---- 357.9 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ---- 372.7 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ---- 457.0 ------

248 Zo1987 Tsarev L5 2.0 2.0 2.0 ---- 459.9 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ---- 259.9 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ---- 290.3 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ---- 312.8 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ---- 319.7 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ---- 348.1 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ---- 423.6 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ---- 291.3 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ---- 303.0 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ---- 306.0 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ---- 307.9 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ---- 397.2 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ---- 453.1 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ---- 495.2 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ---- 343.2 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ---- 360.9 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ---- 368.7 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ---- 437.4 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ---- 445.2 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ---- 452.1 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ---- 469.7 ------Zo1987 Tsarev L5 5.0 5.0 5.0 ----- 35.3 ------Zo1987 Tsarev L5 5.0 5.0 5.0 ----- 42.2 ------Zo1987 Tsarev L5 5.0 5.0 5.0 ----- 50.0 ------Zo1987 Tsarev L5 4.0 4.0 4.0 ----- 42.2 ------Zo1987 Tsarev L5 4.0 4.0 4.0 ----- 42.2 ------Zo1987 Tsarev L5 4.0 4.0 4.0 ----- 48.1 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ----- 47.1 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ----- 48.1 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ----- 51.0 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ----- 53.0 ------Zo1987 Tsarev L5 2.5 2.5 2.5 ----- 55.9 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ----- 24.5 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ----- 43.1 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ----- 46.1 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ----- 52.0 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ----- 54.9 ------Zo1987 Tsarev L5 2.0 2.0 2.0 ----- 59.8 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ----- 46.1 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ----- 51.0 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ----- 54.9 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ----- 56.9 ------

248 −3 −3 From Met Type S W L W/D D ρblk × 10 ρgr × 10 P Rt σc σt Ed Ev νd νv vl vt [cm] [cm] [cm] [kg m−3] [kg m−3] [%] [mm/min] [MPa] [MPa] GPa GPa [1] [1] [m s−1] [m s−1]

Zo1987 Tsarev L5 1.5 1.5 1.5 ----- 57.9 ------Zo1987 Tsarev L5 1.5 1.5 1.5 ----- 64.7 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ----- 32.4 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ----- 40.2 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ----- 41.2 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ----- 43.1 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ----- 45.1 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ----- 48.1 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ----- 54.9 ------Zo1987 Tsarev L5 1.2 1.2 1.2 ----- 65.7 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ----- 51.0 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ----- 52.0 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ----- 53.0 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ----- 54.9 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ----- 57.9 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ----- 66.7 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ----- 69.6 ------Zo1987 Tsarev L5 1.0 1.0 1.0 ----- 78.5 ------Zo1987 Tsarev* L5 1.0 1.0 1.0 ------107 ± 1 ----- Zo1987 Tsarev* L5 1.0 1.0 1.0 ------108 ± 1 ----- Zo1987 Tsarev* L5 1.3 1.3 1.3 ------95 ± 1 ----- Zo1987 Tsarev* L5 1.5 1.5 1.5 ------115 ± 1 ----- Zo1987 Tsarev* L5 2.1 2.1 2.1 ------130 ± 1 ----- Zo1987 Tsarev* L5 2.6 2.6 2.6 ------130 ± 1 ----- Zo1987 Tsarev* L5 4.0 4.0 4.0 ------136 ± 1 ----- Zo1987 Tsarev* L5 4.1 4.1 4.1 ------132 ± 1 ----- Zo1987 Tsarev* L5 5.1 5.1 5.1 ------141 ± 1 ----- Zo1987 Tsarev* L5 7.0 7.0 7.0 ------141 ± 1 ----- Zo1987 Tsarev* L5 7.1 7.1 7.1 ------142 ± 1 ----- Zo1987 Tsarev* L5 7.1 7.1 7.1 ------138 ± 1 ----- Zo1987 Tsarev* L5 7.1 7.1 7.1 ------141 ± 1 ----- Zo1987 Tsarev* L5 10.0 10.0 10.0 ------142 ± 1 ----- Zo1987 Tsarev* L5 1.0 1.0 1.0 ------0.231 ± 0.005 --- Zo1987 Tsarev* L5 1.1 1.1 1.1 ------0.242 ± 0.006 --- Zo1987 Tsarev* L5 1.2 1.2 1.2 ------0.272 ± 0.006 ---

249 Zo1987 Tsarev* L5 1.5 1.5 1.5 ------0.290 ± 0.006 --- Zo1987 Tsarev* L5 2.0 2.0 2.0 ------0.339 ± 0.006 --- Zo1987 Tsarev* L5 2.5 2.5 2.5 ------0.345 ± 0.006 --- Zo1987 Tsarev* L5 4.0 4.0 4.0 ------0.389 ± 0.006 --- Zo1987 Tsarev* L5 4.0 4.0 4.0 ------0.352 ± 0.006 --- Zo1987 Tsarev* L5 5.0 5.0 5.0 ------0.361 ± 0.006 --- Zo1987 Tsarev* L5 7.0 7.0 7.0 ------0.343 ± 0.006 --- Zo1987 Tsarev* L5 7.0 7.0 7.0 ------0.354 ± 0.006 --- Zo1987 Tsarev* L5 7.1 7.1 7.1 ------0.342 ± 0.006 --- Zo1987 Tsarev* L5 10.0 10.0 10.0 ------0.339 ± 0.006 --- Zo1987 Tsarev* L5 0.96 0.96 0.96 ------7314 ± 52 - Zo1987 Tsarev* L5 0.96 0.96 0.96 ------7455 ± 54 - Zo1987 Tsarev* L5 1.2 1.2 1.2 ------6296 ± 63 - Zo1987 Tsarev* L5 1.5 1.5 1.5 ------6820 ± 58 - Zo1987 Tsarev* L5 2.0 2.0 2.0 ------6581 ± 61 - Zo1987 Tsarev* L5 2.5 2.5 2.5 ------6923 ± 63 - Zo1987 Tsarev* L5 3.9 3.9 3.9 ------6529 ± 59 - Zo1987 Tsarev* L5 4.0 4.0 4.0 ------6373 ± 63 - Zo1987 Tsarev* L5 5.0 5.0 5.0 ------6751 ± 63 - Zo1987 Tsarev* L5 6.9 6.9 6.9 ------6431 ± 66 - Zo1987 Tsarev* L5 10.0 10.0 10.0 ------6378 ± 67 - Zo1987 Tsarev* L5 1.0 1.0 1.0 ------3315 ± 61 Zo1987 Tsarev* L5 1.2 1.2 1.2 ------3136 ± 58 Zo1987 Tsarev* L5 1.4 1.4 1.4 ------3511 ± 63 Zo1987 Tsarev* L5 2.0 2.0 2.0 ------3763 ± 56 Zo1987 Tsarev* L5 2.5 2.5 2.5 ------3807 ± 68 Zo1987 Tsarev* L5 3.9 3.9 3.9 ------3958 ± 63 Zo1987 Tsarev* L5 5.0 5.0 5.0 ------4001 ± 66 Zo1987 Tsarev* L5 7.0 7.0 7.0 ------4016 ± 65 Zo1987 Tsarev* L5 10.0 10.0 10.0 ------4063 ± 63

249 APPENDIX C HEAT CONDUCTION EQUATION

250 Please note that in this chapter, I use symbols T and u interchangeably to denote temperature.

C.1 Derivation of Heat Conduction Equation

Let Ω ⊂ R3 be a suﬃciently huge bounded set and let V be a compact connected subset of Ω. We would like to examine space-time distribution of temperature in V. Let’s assume that the following functions are deﬁned on Ω:

ρ(t, x) . . . material density in kg · m−3, f(t, x) . . . heat source density in J · s−1 · kg−1, c(t, x) . . . speciﬁc heat capacity of the material in J · K−1 · kg−1,

α(t, x) . . . heat conductivity of the material in W · K−1 · m−1.

Let us determine energy equilibrium (considering only heat energy) of the subset

V, we have to consider the following processes during a period of t1 − t2:

1. heat can be generated by a heat source inside V,

2. heat can ﬂow through the boundary of V,

3. heat can be transformed to internal energy (causing the temperature to rise).

251 C.1.1 Integral Form

Let’s write down the various energy sources and sinks in the integral form. The

heat generated by a heat source inside V during the period t2 − t1 is:

Z t2 Z ρ(τ, ~x)f(τ, ~x) d~x dτ, ~x ∈ V . (C.1) t1 V

The ﬂow of heat through the boundary is assumed to obey the Fourier’s law:

~q(t, ~x) = −α(t, ~x)∇u(t, ~x) , (C.2)

~q(t, ~x) being the local heat ﬂux in W m−2, u(t, ~x) being the local temperature in K,

∇u(t, ~x) its gradient in K m−1 and α(t, ~x) being the thermal conductivity of the material

in W m−1 K−1. Therefore, the total ﬂux through the boundary ∂V during the period

t2 − t1 can be obtained by:

Z t2 Z −α(t, ~x)∇u(t, ~x) · ~n dS dτ , (C.3) t1 ∂V

~n being outward oriented surface normal. The selection of normal implies that Eq. (C.3)

determines the ﬂow out of V. Finally the change of internal energy of O can be expressed

as:

Z Z (ρ(t2, ~x)c(t2, ~x)u(t2, ~x)) d~x − (ρ(t1, ~x)c(t1, ~x)u(t1, ~x)) d~x. (C.4) V V

It is important to note that the internal energy is a state function, independent of the

path the system took from one state to another and thus the missing time integral. The

252 energy conservation implies the integral form of the heat conduction equation:

Z t2 Z Z ρ(τ, ~x)f(τ, ~x) d~x dτ = (ρ(t1, , ~x)c(t2, ~x)u(t2, ~x)) d~x − t1 V V Z − (ρ(t1, ~x)c(t1, ~x)u(t1, ~x)) d~x − V

Z t2 Z − α(t, ~x)∇u(t, ~x) · ~n dS dτ , t1 ∂V

∀t1 < t2, ∀V ⊂ Ω . (C.5)

C.1.2 Diﬀerential Form

The last term in C.5 can be transformed to an integral over the volume V using the Divergence theorem:

Z Z [α(t, ~x)∇u(t, ~x)] · ~n dS = ∇ · [α(t, ~x)∇u(t, ~x)] dV. (C.6) ∂V V

If we assume that all the functions are smooth enough, that we can switch the integral

1 1 sign and limit, we can multiply Eq. (C.5) by and by , followed by limt →t and t2−t1 |V| 2 1 lim|V|→0+ , |V| being a measure of the set V. This way we obtain the conservation law in a diﬀerential form. For x ∈ V:

∂ ρ(t, ~x)f(t, ~x) = [ρ(t, ~x)c(t, ~x)u(t, ~x)] − ∇ · [α(t, ~x)∇u(t, ~x)] . (C.7) ∂t

If we assume that ρ(t, x), α(t, x) and c(t, x) are constant1 and that ρ 6= 0, c 6= 0, dividing

Eq. (C.7) by ρc, results in:

f(t, ~x) ∂u α = (t, ~x) − 4u(t, ~x) . (C.8) c ∂t ρc 1This assumption requires homogeneous and isotropic material that does not change in time in terms of substance.

253 α The quantity ρc is called thermal diﬀusivity.

C.2 Boundary Value Problem

Eq. C.8 is a linear second order parabolic partial diﬀerential equation for tem-

perature as a function of ~x and t. Since Eq. C.8 is of ﬁrst order in time variable, it is

necessary to supply one initial condition stating T (0, ~x). In order to ensure uniqueness

of the solution, a boundary condition is also necessary. One may typically choose from:

1. The Dirichlet boundary condition speciﬁes: T (t, ~x) = g(t, ~x), for t > 0, ~x ∈ ∂V,

V being the boundary (surface) of the volume (asteroid) where one is solving the

problem.

∂T 2. The Neumann boundary condition speciﬁes: ∂~n (t, ~x) = h(t, ~x), for t > 0, ~x ∈ ∂V

∂T and ∂~n (t, ~x) is a normal derivative, i.e., it speciﬁes ﬂux across the boundary.

3. A combination of the two above.

Note that all of the above boundary conditions are linear (with respect to the temperature). However, on an asteroid, the boundary condition problem is more complex.

First, let us assume a real three dimensional object. Then the only boundary, one has to be concerned with, is the surface. The idea behind the boundary condition on the surface is the same as for derivation of the HCE, without considering the change in internal energy, that is, our energy budget is:

254 1. The heat ﬂux from the Sun incident onto the surface element d~S as our energy

source. It can be determined as:   ε (t)(1 − A) d~S · ~n (t) sgn d~S · ~n (t) > 0 ,  (C.9)   0 else ,

where A is Bond albedo of the asteroid, ε (t) is time-dependent energy ﬂux from

the Sun, d~S is the oriented surface element with magnitude of the surface area and

direction parallel to its normal and ~n (t) is the unit vector aiming towards the sun.

2. Part of this energy is radiated back into space due to non-zero temperature. This

can be determined from the Stefan-Boltzmann law:

P = σ|dS~|T 4 , (C.10)

being thermal emissivity, σ the Stefan-Boltzmann constant, |dS~| the area of the

radiation and T the temperature.

3. Part of the energy ﬂows into the asteroid which can be determined from the Fourier’s

law as α∇T · d~S.

The above energy ideas provide the boundary condition on the surface of the form:

4 ε (t)(1 − A) = σT (t, ~x) + α∇T (t, ~x) · ~n, (C.11)

t > 0 ,

x ∈ ∂V .

255 Giving:

∂T α 1 (t, ~x) − 4T (t, ~x) = f(t, ~x) , x ∈ Int V , (C.12) ∂t ρc c

4 ε (t)(1 − A) = σT (t, ~x) + α∇T (t, ~x) · ~n, x ∈ ∂V ,

T (0, ~x) = h(~x) , x ∈ V¯ , where additionally to the symbols defined above, h(~x) is the initial condition, Int V stands for interior of the volume V and V¯ ≡ Int V ∪ ∂V. The boundary condition renders this system of equations non-linear posing a significant difficulty to both analytical and numerical solutions.

C.3 1-Dimensional Problem

If we imagine an asteroid and take a close look at its surface so that the curvature is negligible, such as in Fig. C.1, an idea might come to mind to try and approximate the asteroid with 1 dimensional slabs, as shown in Fig. C.1. Moreover, if the asteroid is not spherical, it is possible to adjust the surface angle of the slab to correspond to a real surface element orientation. In eﬀect, we can imagine the asteroid as a hedgehog.

The above approximation necessitates accepting several assumptions:

1. Such simpliﬁcation completely neglects any heat transfer in directions perpendicular

to the slab.

2. It requires an additional boundary condition speciﬁed at the other end of the slab.

256 Surface

1-D slab

Center

Figure C.1: 1-Dimensional approximation of an asteroid by thin slabs.

3. The penetration depth of the heat ﬂux has to be small compared with the size of

the asteroid.

4. The surface area of both ends of the slab needs to be the same.

The ﬁrst assumption can only be reasonable if there is no internal heat source and if the material is at least radially isotropic (by which I mean that only changes in material properties in radial direction are allowed, others would lead to temperature gradients in the lateral direction).

The requirement 4 needs to be satisﬁed otherwise the energy density would simply change because of area change with radius. This can be mitigated by transformation into the spherical coordinate system. The requirement 4 can only be satisﬁed if the penetration depth is small compared with the characteristic radius of the object.

Finally, since we now have to sides of the slab, we need 2 boundary conditions.

One is the previously mentioned surface condition governed by the energy conservation between the solar ﬂux, the thermal emission and heat conduction inside the body. Spec- ifying the boundary condition on the other side along with requirements 4 and 3 leads to

257 the necessity of having either a constant temperature or constant heat ﬂux on the interior

end of the slab. This can be reasonable if the asteroid is suﬃciently large given its orbit,

that is, its interior is a suﬃciently large thermal bath so that the energy stored inside

can supply heat to the top layers without itself being aﬀected (the problem can arise for

smaller bodies on highly eccentric orbits or for very close approaches).

The constant temperature inside the asteroid can be estimated from the average

ﬂux over the asteroid’s orbit. Or one can enforce zero heat ﬂux. It can be shown that both

approaches yield very similar results with the only diﬀerence, if one chooses inappropriate

integration depth, the constant temperature is more robust.

The above simpliﬁed problem (assuming no heat generation inside the asteroid)

of Eq. C.12 in 1 dimension reads:

∂ α ∂2 T (t, x) − T (t, x) = 0 , x ∈ (x , x ), t ≥ 0 , (C.13) ∂t ρc ∂x2 0 1

T (0, x) = h(x) , x ∈ (x0, x1) ,

T (t, x1) = g(t) , t ≥ 0 ,

∂T σT 4(t, x) + α (t, x) = ε (t)(1 − A) , t ≥ 0 . ∂x x=x0 x=x0

Here, x0 stands for coordinates of the surface and x1 for the interior limit, h(x) is the initial condition and g(t) is the internal boundary condition.

The system of equations C.13 can be solved analytically in some special cases or numerically.

258 C.4 Analytical Solution of 1D Problem

In one dimension, an analytical solution to the HCE can be derived easily in case we perform several simpliﬁcations. We shall assume that the insolation is a harmonic

iδt function of the form ε(t) = ε0 + ε1e , δ being angular frequency. It is then feasible to expect the resulting steady-state temperature to be a harmonic function as well. We shall assume the temperature (which will be our Ansatz in (C.13)) to be in the form u(t, x) = χ(x)eiδt + C, C being a constant. Let’s plug this Ansatz to our HCE:

∂tu(t, x) − D∂xxu(t, x) = 0 , α D def= ρc

We obtain:

iδt 00 iδt ∂tu(t, x) = iδχ(x)e , ∂xxu(t, x) = χ (x)e ,

00 δ χ (x) − i χ(x) = 0 . D

A general solution to this second order diﬀerential equation for χ(x) is simple:

√ √ √ √ 2 (1+i) δ x − 2 (1+i) δ x χ(x) = Ae 2 D + Be 2 D .

Thus the temperature has the form:

√ √ √ √ 2 (1+i) δ x iδt − 2 (1+i) δ x iδt u(t, x) = Ae 2 D e + Be 2 D e + C.

One can see that either A or B must equal to zero otherwise the temperature amplitude would diverge as we approach plus or minus inﬁnity depending on our choice of x-axis.

259 We shall choose the axis such that x = 0 is the surface and the higher positive values

of x, the larger the depth below the surface. With this choice, it is obvious that A = 0

otherwise the temperature deep inside the asteroid (x → ∞) would approach unphysical

values (u → ∞). However, our boundary condition for temperature inside the asteroid

(C.13) states there is a ﬁnite temperature that in the centre of the asteroid. Thus we get:

√ √ − 2 (1+i) δ x iδt u(t, x) = Be 2 D e + C.

It is feasible to assume that without the presence of the time-dependent ﬂux ε (t), only the constant term C is present in the above expression. At the same time, we assumed that this temperature is equal to Teq in Eq. (C.13). This leads to the following expression for the temperature:

√ √ − 2 (1+i) δ x iδt u(t, x) = Be 2 D e + Teq . (C.14)

Let’s now consider the energy conservation on the surface:

∂u 4 ε (t)(1 − A) = σu (t, x) − α (t, x) . (C.15) x=0 ∂x x=0

Yet, another assumption we shall make is that the temperature varies from Teq by a

2 relatively small amount — u(t, 0) = Teq + δu(t), i.e. we assume that (δu(t)/Teq) 1.

Thus we will be able to linearise the fourth power of the temperature in the above

expression. According to the binomial theorem:

4 4 u (t, 0) = [Teq + δu(t)]  !2 !3 !4 4 δu(t) δu(t) δu(t) δu(t) = Teq 1 + 4 + 6 + 4 +  . Teq Teq Teq Teq

260 Due to the above assumption, we can neglect all terms of higher order than linear which

gives:

4 4 3 4 3 4 u (t, 0) ≈ Teq + 4δu(t)Teq = Teq + 4u(t, 0)Teq − 4Teq .

4 We employed δu(t) = u(t, 0) − Teq. Replacing u (t, x) in Eq. (C.15) with the above x=0 linearisation and using the assumption that the insolation function is harmonic — ε(t) =

iδt ε0 + ε1e , results in:

! ∂u iδt 4 3 ε0 + ε1e (1 − A) = σTeq + 4σTeq u(t, x) − Teq − α (t, x) . (C.16) x=0 ∂x x=0

Next, we calculate the last (gradient) term from Eq. (C.14):

√ √ − 2 (1+i) δ x iδt ∂xu(t, x) = ∂x Be 2 D e + Teq = x=0 x=0 √ s 2! δ = B − (1 + i) eiδt . 2 D

Inserting this into Eq. (C.16):

√ ! s 2 δ ε + ε eiδt (1 − A) = σT 4 + 4σT 3 (u(t, 0) − T ) − αB − (1 + i) eiδt . 0 1 eq eq eq 2 D

(C.17)

∂u(t,x) Let’s look back at Eq. (C.16). Deﬁnitely, the gradient term is zero ( ∂x = 0) in the steady state that exists before we “turn on” our insolation function ε(t) as in this steady state the asteroid’s temperature is constant (equal to Teq) on its surface and everywhere in the interior. The same holds for the term u(t, x) : u(t, x) ≡ Teq in x=0 x=0

4 this case. This leaves only one term on the right hand side in (C.16) — εσTeq. We also

tacitly assume that in this state the asteroid is insolated by a constant ﬂux ε0. All these

261 lead to the following important equality which is independent of time:

4 εS0(1 − A) = σTeq .

So we can recast Eq. (C.17):

√ s 2! δ ε eiδt(1 − A) = 4σT 3 (u(t, 0) − T ) − αB − (1 + i) eiδt . 1 eq eq 2 D

We plug in u(t, x) from Eq. (C.14) at x = 0:

u(t,0) √ z }| { ! s iδt 3 iδt 2 δ iδt ε e (1 − A) = 4σT Be + T−T − αB − (1 + i) e . 1 eq eq eq 2 D

Cancelling all the terms as suggested results in:

√ s 2! δ ε (1 − A) = 4BσT 3 + αB (1 + i) . (C.18) 1 eq 2 D

Finally, solving for B:

ε1(1 − A) B = √ q . (C.19) 3 2 δ 4σTeq + α 2 (1 + i) D

α Next, we use the deﬁnition of D ≡ ρc and transfer the complex unit from the denominator.

This operation results in:

√ √ √ √ ε (1 − A) 4σT 3 + 2 αρcδ − i 2 αρcδ B = 1 e√q √2 2 . (C.20) 2 2 6 √ 2 16 σ Teq 2 αρcδ 1 2αρcδ 1 + + 3 3 2 4σTeq 2 2σTeq | {z } Θ Introducing parameter Θ: √ √ def 2 αρcδ Θ = 3 , (C.21) 2 2σTeq

262 the expression for B coeﬃcient can be rewritten:

ε (1 − A) 1 + 1 (1 − i)Θ B = 1 2 . 3 1 2 4σTeq 1 + Θ + 2 Θ

def 1 We replace the angular frequency δ by the frequency f deﬁned as f = P , P being the

rotation period. Therefore, we have to replace δ → 2πf in all our expressions. This

results in the following expressions:

√ √ ε1(1 − A) 1 + (1 − i)Θ 2 2πfρc 2πift − 2 (1+i) α x u(t, x) = 3 2 e e + Teq , (C.22) 4σTeq 1 + 2Θ + 2Θ

ε1(1 − A) 1 + (1 − i)Θ 2πift u(t, 0) = 3 2 e + Teq , (C.23) 4σTeq 1 + 2Θ + 2Θ √ πfαρc Θ = 3 . (C.24) 4σTeq

def Θ We can do some more algebra if we deﬁne an angle ϕ as tan ϕ = − 1+Θ . This deﬁnition

implies that sin ϕ = −Θ and cos ϕ = 1 + Θ. A closer look at the nominator in Eq. (C.22)

q reveals that 1 + (1 − i)Θ = cos ϕ + i sin ϕ = reiϕ where r = cos2 ϕ + sin2 ϕ which √ transforms to r = 1 + 2Θ + 2Θ2. So we ﬁnally get: √ πfρc √ i 2πft+ϕ− x πfρc ε1(1 − A) 1 α − x u(t, x) = √ e e α + T , (C.25) 3 2 eq 4σTeq 1 + 2Θ + 2Θ ε (1 − A) 1 u(t, 0) = 1 √ ei(2πft+ϕ) + T , (C.26) 3 2 eq 4σTeq 1 + 2Θ + 2Θ √ πfαρc Θ = 3 , (C.27) 4σTeq Θ tan ϕ = − . (C.28) 1 + Θ

We can now see that it was indeed useful to employ the angle ϕ. The angle suggests a

phase shift or delay between the temperature function u(t, x) and the insolation function

ε(t). The angle is a negative descending function of Θ (Θ ≥ 0) and Θ ∈ [0, ∞) thus

263 π ϕ ∈ (− 2 , 0], i.e. we get a bigger phase shift for materials with bigger bulk density and

heat capacity (we need more heat to increase the temperature, i.e. more time to provide

that heat to the asteroid).

We also have a bigger phase shift with increasing heat conductivity. This is due

to the fact that the higher conductivity means that more of the supplied heat traverses

further into the interior of the asteroid therefore more heat is required to increase the tem-

perature at a given point. These are all properties that make inhomogeneous distribution

of temperature possible inside the asteroid.

Asteroids with higher rotation frequency also have a bigger phase shift for the

same reason as before (there is less time for an element on the surface to receive heat

which would propagate into the interior of the asteroid). The maximum phase shift is

◦ −45 . On the other hand, the higher the equilibrium temperature Teq, the smaller the

phase shift.

We can also see that the temperature variations from Teq are exponentially sup-

πfρc 1 2 pressed with depth; ( α ) is then an inverse of characteristic depth at which these

variations are suppressed by a factor e−1, i.e. at this depth the variations of temperature

from Teq are about a third of the variations on the surface.

We note that the temperature behaves as expected. It propagates into the depth

of the asteroid more easily for more conductive materials (higher α) and its propagation

from the surface is somewhat suppressed by high bulk density (ρ) and high heat capacity

264 elwrdfnto si i.C.2. Fig. in is function real-world asteroid. the of interior the into deep propagate not do surface the on temperature ( material asteroid the of Insolation h oprsno h noainfnto rmteaayia ouinadthe and solution analytical the from function insolation the of comparison The iueC2 oprsno noainfunctions. insolation of Comparison C.2: Figure c .As fteatri oae ucl,tecagso the of changes the quickly, rotates asteroid the if Also ). 265 Time Numerical Analytical APPENDIX D THE NUMERICAL SOLUTION TO THE HEAT CONDUCTION EQUATION

266 Ch. 5 makes use of a numerical solution to the HCE. This appendix provides details of the numerical implementation.

D.1 The Statement of the Problem

In App. C, the 1 dimensional HCE for an asteroid insolated by solar heat ﬂux was derived and presented in the form of Eq. C.13. As mentioned in one can assume the interior of the asteroid has constant temperature given by average heat ﬂux over an orbit.

This assumption is sound provided that the asteroid is large enough. This temperature can then be used as the boundary condition on the internal end of the 1 dimensional slab introduced in App. C. While one can also use the Neumann’s boundary condition and prescribe zero heat ﬂux across the inner boundary, I have previously shown (Pohl, 2014) that both methods lead to almost identical results, diﬀering only in the case when one selects an inappropriate depth for the solution (see later in this chapter for explanation).

This internal constant temperature is from here on denoted as equilibrium temperature

(Teq). This temperature can also be used as the initial condition. Other methods to determine the initial condition are discussed later. This leads to the system of equations:

∂ α ∂2 T (t, x) − T (t, x) = 0 , x ∈ (−L, 0), t ≥ 0 , (D.1) ∂t ρc ∂x2

T (0, x) = Teq , x ∈ [−L, 0] ,

T (t, L) = Teq , t ≥ 0 ,

∂T 4 σT (t, x) + α (t, x) = ε (t)(1 − A) , t ≥ 0 . x=0 ∂x x=0

267 Here, L denotes the depth under the surface at which it is assumed that the material

becomes isothermal, that is where there are no changes in temperature with asteroids

spinning and orbital motions. If this depth is too shallow, the calculations will be incorrect

and also the solution will exhibit diﬀerences between the Dirichlet’s and Neumann’s

internal boundary conditions.

D.2 The Numerical Implementation to the Problem

D.2.1 The Coordinate Systems and Orbit Speciﬁcation

All calculations are carried out in the Cartesian system of reference of the Sun.

The Sun is located at the centre of the coordinate system. The x positive direction is the

direction from the Sun to the perihelion. The z positive direction is in the direction of the orbital angular velocity vector (that is z axis is perpendicular to the orbital plane with the asteroid orbiting as the right handed screw). The y axis is then selected to complete the right-handed orthogonal system (aiming along the direction of the velocity vector of the asteroid when in perihelion).

All direction and normal vectors are always normalised to unity. The unit vector in along the z axis is denoted ~ez, the unit vector in the direction of the asteroid spin axis is denoted ~eω, the surface normals are denoted ~n(t) and the unit vector denoting the direction from the Sun to the asteroid ~er(t).

268 In the case of a circular orbit, the orbital parameters are the constant radial

distance of the asteroid from the Sun (r), the orbital period (To) which provides the

momentary (at time t) angle of orbital rotation θ(t) = 2π t and the spin period (T ) which To s provides the momentary angle of spin ϕ(t) = 2π t. In the case of an elliptical orbit, the Ts parameters are given by semi-major axis (a), eccentricity (e), the orbital period (To) and

the spin period (Ts). The Kepler equation is then solved to obtain the radial distance r(t) and true anomaly f(t) at ﬁxed intervals (a day or less). The distances and true anomalies within the intervals where they were determined from the Kepler equation are determined by linear interpolation.

D.2.2 The Insolation Function

Insolation function ε (t) is calculated as:

L ε (t) = |~e (t) · ~n(t)| Θ(−~e (t) · ~n(t)) , (D.2) 4πr2(t) r r

where the Θ(x) is the Heaviside step function deﬁned with Θ(0) = 0, L is the Solar

26 luminosity (L = 3.839 × 10 W) and r(t) is the momentary distance from the Sun, constant in the case of a circular orbit, or calculated from Kepler equation in the case of an elliptical orbit. The vector ~er(t) is obtained by rotating the initial direction from the

Sun to the asteroid (~er(0), which is typically along the x axis) about the z axis (or ~ez) by the angle θ(t) in the case of a circular orbit or by the angle given by the true anomaly

(f(t)) for an elliptical orbit. The vector ~n(t) is obtained by the rotation of the original

269 surface normal (n(0)) by the angle ϕ(t) about the spin axis (~eω) of the asteroid (note

that these are in the Sun’s frame of reference). Thus, the scalar product ~er(t) · ~n(t) gives

the cosine of the angle between the surface normal and the direction towards the sun, if

the Sun is visible or zero otherwise (the minus sign in the Heaviside function is necessary

since the surface element is visible to the Sun when the projection of the normal along

the vector from the Sun to the asteroid is in the opposite direction).

D.2.3 Finite Diﬀerences

Finite diﬀerence methods are implemented to solve the system D.1. The preferred

method to solve the system is by using central diﬀerences in both space and temporal

dimensions which give rise to the Crank-Nicholson ﬁnite diﬀerence scheme and turn the

ﬁrst equation in the system D.1 to:

α ∆t ! α ∆t α ∆t 2 + 2 un+1 − un+1 − un+1 = ρc ∆x2 j ρc ∆x2 j−1 ρc ∆x2 j+1 α ∆t ! α ∆t α ∆t = 2 − 2 un + un + un . (D.3) ρc ∆x2 j ρc ∆x2 j−1 ρc ∆x2 j+1

Here, ∆t is the distance between two points in the temporal grid in s, ∆x is the

n distance of the between two points in the spatial grid in m and uj is the temperature in

K at the j-th spatial grid point and n-th temporal point. The grid is selected such that n = 0 is the beginning of the calculation when all grid spatial points have the temperature equal to the equilibrium one and n = N is the end of the calculation. j = 0 is the point deep inside the asteroid (at depth L) always kept at the equilibrium temperature and

270 α ∆t j = J is the surface point. Denoting b = ρc and r = ∆x2 , this can be recast in a matrix form:

   n+1 2 + 2br −br 0 0 0 ... 0 u1        n+1  −br 2 + 2br −br 0 0 ... 0  u2         n+1  0 −br 2 + 2br −br 0 ... 0  u3    ×   =  ......   .   . . . . .   .         n+1  0 ... 0 0 −br 2 + 2br −br  uJ−2    n+1 0 ... 0 0 0 −br 2 + 2br uJ−1  n n n n+1  bru0 + (2 − 2br)u1 + bru2 + bru0    n n n   bru1 + (2 − 2br)u2 + bru3     n n n   bru2 + (2 − 2br)u3 + bru4    (D.4)  .   .     n n n   bruJ−3 + (2 − 2br)uJ−2 + bruJ−1   n n n n+1 bruJ−2 + (2 − 2br)uJ−1 + bruJ + bruJ

n+1 The issue arises with the term uJ on the right hand side of the equation which connects the boundary condition on the surface and transforms the system to a non- linear one. To avoid that, the surface temperature is ﬁrst approximated by one from

n+1 n the previous time step that is uJ = uJ and the system D.4 is solved using specialized

n+1 algorithms from Intel MKL library. This results in uJ−1 which is used in the boundary condition:

α α σ(un+1)4 + un+1 − εn+1(1 − A) − un+1 = 0 , (D.5) J ∆x J ∆x J−1

n+1 n+1 to obtain uJ which is then plugged in the system D.4 to solve for uJ−1 and the iterations

n+1 are continued until a suﬃcient convergence is achieved. Here, ε is the insolation function from Eq. D.2 evaluated at the time grid point n + 1.

271 The calculation is carried out for several orbital periods. As the problem is periodic, the temperatures should be periodic functions of the orbital period. To achieve this stationary evolution in the numerical solution described above, it is necessary to run the code typically for two full orbital periods. From the third orbital period, the solution becomes very stationary with respect to the orbital period and the temperatures can be used to describe the thermal evolution.

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