Vulcanoids: An Examination of the Intra-Mercurial Region

A Thesis

Submitted to the Faculty of Graduate Studies and Research

In Partial Fulfillment of the Requirements

For the Degree of

Master of Science

In

Physics

University of Regina

By

Lowell Francis Peltier

Regina, Saskatchewan

September, 2019

Copyright 2019: L. Peltier

UNIVERSITY OF REGINA

FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Lowell Francis Peltier, candidate for the degree of Master of Science in Physics, has presented a thesis titled, Vulcanoids: An Examination of the Intra-Mercurial Region, in an oral examination held on August 30, 2019. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material.

External Examiner: Dr. Ian Coulson, Department of Geology

Supervisor: Dr. Martin Beech, Department of Physics

Committee Member: Dr. Gwen Grinyer, Department of Physics

Committee Member: Dr. Nader Mobed, Department of Physics

Chair of Defense: Dr. Philippe Mather, Campion College

*Not present at defense

Abstract

A review and discussion of the historical and modern ideas pertaining to the vulcanoid region of our solar system, and the theoretical asteroid population therein is presented here. Current observations constrain the maximum possible vulcanoid object size to somewhere between 5 and 10 km in diameter. Any object larger than this located within the putative vulcanoid zone, the region 0.06 AU to 0.25 AU away from the sun, should have been observed in past observations. With this in mind any modern day vulcanoid population would be small in number and size, but as yet, there is no compelling reason to rule out a population in general. This work explores the physical processes at work in the vulcanoid zone, primarily thermal (sublimation mass loss) and radiative (Poynting-

Robertson drag, The Yarkovsky effect) effects are considered in this analysis. A key insight is the role that the variation in the luminosity of the sun over its lifespan has on the thermal and radiative effects inherent in the vulcanoid zone. In addition, the protective effects of a regolith composition are examined. Additionally, collisional effects, unipolar heating, and radiative rotational effects (YORP effect) are briefly explored. All of these effects will alter the lifespan of an asteroid in the region against destruction. Potential lifespans under various assumed initial conditions are explored and compared. Results show that the existence of vulcanoids in the modern day, and even at the end of the sun’s lifespan, is quite possible. Observational evidence presents a bleaker view of the putative vulcanoid population, but does not rule out the possibility entirely.

Keywords: Vulcanoid Asteroids, Orbital Evolution, Thermal Processing, Radiative

Drag, Variable Luminosity, Regolith, Detection Methods

i Acknowledgements

I would like to thank my supervisor Dr. Martin Beech. His guidance and humor has made this entire process more fun than it had any right to be. Without his direction none of this would have been possible.

I would like to thank my committee members Dr. Gwen Grinyer and Dr. Nader

Mobed. Their astute and helpful suggestions have ensured this work is the highest quality it can be.

I would like to acknowledge the contributions the entire physics department has made to my graduate work and physics education in general. My time teaching labs in the physics department has been an invaluable aspect of my time at the University of

Regina.

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Table of Contents

Abstract ...... i Acknowledgements ...... ii Table of Contents ...... iii List of Tables ...... v List of Figures ...... vi

Chapter 1: Introduction ...... 1 1.1 History and Background ...... 1 1.2 21st Century Perspective ...... 7 1.3 Definition of the Vulcanoid Region ...... 9

Chapter 2: Sun-Grazing Comets ...... 11 2.1 Cometary Impact Modification ...... 11 2.2 Impact Probability and Effects ...... 16 2.3 Conclusions ...... 19

Chapter 3: Solar Radiation Effects ...... 21 3.1 Material Properties ...... 21 3.2 Solar Luminosity Variation ...... 23 3.3 Sublimation ...... 25 3.4 Poynting-Robertson Drag ...... 29 3.5 The Yarkovsky Effect ...... 32 3.6 Induction Heating ...... 42

Chapter 4: Results ...... 46 4.1 Parameters ...... 46 4.2 Obliquity of 45° ...... 48 4.3 Obliquity of 135° ...... 53 4.4 Random Obliquities ...... 59

Chapter 5: Future Work ...... 69 5.1 Material Refinements ...... 69

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5.2 Internal Vulcanoid Structure ...... 69 5.3 Rotational effects ...... 70 5.4 Collisional Evolution ...... 73 5.5 Dust Rings of the Inner Solar System ...... 76 5.6 Exoplanetary Application ...... 77 5.7 Red Giants, White Dwarfs, and Planetary Remnants ...... 78 5.8 Migration ...... 78 5.9 Direct Detection ...... 79

Chapter 6: Conclusions ...... 80

References ...... 83

Appendix: Simulation Code ...... 88

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List of Tables

Table 2.1 – Table of Comets ...... 15 Table 3.1 – Material Properties ...... 22

v

List of Figures

Figure 1.1 – The Vulcanoid Zone ...... 10 Figure 2.1 – Ecliptic Crossing Points ...... 14 Figure 3.1 – Luminosity Variation ...... 24 Figure 3.2 – Variable Luminosity Survival Lifetimes ...... 28 Figure 3.3 – Poynting-Robertson Drag ...... 30 Figure 3.4 – The Yarkovsky Effect ...... 33 Figure 3.5 – Radiative Drag for Various Obliquities ...... 39 Figure 3.6 – Vulcanoid Orbital Position Under Radiative Drag ...... 41 Figure 3.7 – Induction Heating ...... 44 Figure 4.1 – Orbital Position vs Time at 45° ...... 49 Figure 4.2 – Fayalite Lifetime – 45° ...... 51 Figure 4.3 – Regolith Lifetime – 45° ...... 52 Figure 4.4 – Orbital Position and Size at 135° ...... 54 Figure 4.5 – Iron Lifetime – 135° ...... 56 Figure 4.6 – Fayalite Lifetime – 135° ...... 57 Figure 4.7 – Regolith Lifetime – 135° ...... 58 Figure 4.8 – Variability Due to Randomized Obliquity ...... 60 Figure 4.9 – Comparison of Maximal Seasonal Effect to Randomized Obliquity ...... 61 Figure 4.10 – Yarkovsky Effect Strength vs Obliquity Angle ...... 62 Figure 4.11 – Iron Lifetime – Random Obliquity ...... 64 Figure 4.12 – Fayalite Lifetime – Random Obliquity ...... 65 Figure 4.13 – Regolith Lifetime – Random Obliquity ...... 66

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Chapter 1: Introduction

1.1 History and Background

The vulcanoid region of our solar system is one with a varied history. The initial investigations of the region began with problems accounting for the perihelion precession of Mercury’s orbit around the sun. Perihelion being the point in Mercury’s orbit when it is closest to the sun. This phenomenon was first explored by Urbain Le

Verrier in 1859 [1]. At the time it was deduced that Mercury’s orbit advanced by 565 arc seconds per century and all but 38 arc seconds per century could be accounted for, largely due to perturbations from Venus, Earth, and Jupiter. (In the modern day this perihelion precession has been revised to 43 arc seconds per century.) This anomaly in the orbit of Mercury led scholars of the day to question some of the basic precepts of physics including the universal applicability of Newton’s law of universal gravitation [2,

3, 4]. Le Verrier’s response to the Hermian (i.e. pertaining to Mercury) perihelion advancement anomaly was a solution employed numerous times in the history of astronomy. Just as the presence of Neptune was inferred due to the anomalous motions of Uranus [3, 5], Le Verrier proposed a new planet, or numerous “corpuscles” (i.e. a ring of asteroids) in an orbit interior to that of Mercury. This body, or collection of bodies, would need to occupy a specific range of orbital radii and masses in order to perturb

Mercury with the observed perihelion precession. Le Verrier argued that the perturbing object was a single large body. In order to account for the observed orbital anomaly this planet would need to have an, assumed circular, orbital radius of 0.14 AU, and an associated mass of 1/17th that of Mercury. Assuming the same bulk density as Mercury

1 itself would result in an object roughly 1900 kilometers in diameter. Such a result was problematic due to the fact that an object of such mass, and therefore large size, would have been, even on an orbit so close to the sun, readily observable by instruments of the day. Despite this contradictory result, due to confidence in the absolute applicability of

Newton’s inverse square law, and Le Verrier’s earlier success in prediction of the presence of Neptune through the observed orbital residuals in the position of Uranus [5] astronomers accepted La Verrier’s proposed planet. They went so far as to name the postulated new planet, calling it Vulcan after the Roman god of fire, metalwork and the forge. This of course led to the modern appellation of the vulcanoids and the vulcanoid region.

Despite numerous searches and several false detections [4, 6, 7], it became apparent after the 29 July 1878 solar eclipse that no large scale planet existed interior to

Mercury’s orbit. Furthermore, using photographic plates obtained during the Lick

Observatory eclipse expeditions of 1901, 1905 and 1908, in 1926, Russell, Dugan and

Stewart in their widely read text Astronomy [8] concluded that “there are no intra-

Mercurial bodies brighter than the eighth magnitude, that is, more than about 30 miles

[48-km] in diameter”. Charles Young in his A Text-book of General Astronomy [9], first published in 1899, (the book that Russell, Dugan and Stewart, in fact, set out to revise) commented that, “it is extremely probable that there are a number, and perhaps a very great number, of intra-Mercurial asteroids”. In fact, after the turn of the 19th century it was widely accepted that the notion of Vulcan the planet was in the end impossible, however, the existence of vulcanoid asteroids was all but a certainty.

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Ultimately the impetus for invoking the existence of intra-mercurial objects

(whether it was the originally proposed planet Vulcan or a population of smaller asteroid sized objects) was rendered obsolete by Albert Einstein. The publication of his founding papers on general relativity, starting in 1915, demonstrated that the anomalous 43 arc seconds per century of the precession of Mercury’s orbit were accounted for by a relativistic correction to the underlying Newtonian theory [4]. This correction was most significant for Mercury due to its proximity to the sun, with general relativistic effects having a much smaller impact on the other more distant planets. This greater understanding of the physical realities of the situation, granted by Einstein’s work, removed the need for massive vulcanoid objects to account for the precession of

Mercury’s orbit, however this did not put an end to the search for vulcanoid objects.

While no longer required to produce dynamical perturbations, the idea of a population of objects existing in the region persists. The search for objects orbiting close to the sun has persisted into the modern era with numerous investigations undertaken [4, 10, 11, 12,

13]. Despite the work performed in these searches, no definitive vulcanoid object has been identified so far. As mentioned above, in 1926 an upper limit of 48 kilometers was placed on the diameter of any possible vulcanoid. Continued work leading to the modern era has reduced this limit by roughly a factor of 10; one example being Steffle et al., [14] concluding that, “there are presently no vulcanoids larger than 5.7 km in diameter”.

Despite the continued observational efforts to locate vulcanoid asteroids, they have remained stubbornly elusive. Objects located so close to the sun make observations incredibly technically challenging, with the sun’s intense light drowning out the signal of any potential vulcanoid. At the present time there is no direct evidence for the existence

3 of vulcanoids and with current observational constraints, at most, a small population of objects could exist within the vulcanoid stability zone in the present epoch. It is equally possible that no vulcanoids exist at the present time. Furthermore it is unclear if even a primordial population of vulcanoids has ever existed at any point in time. This final possibility could in principle be examined through analysis of the Hermian crater size- frequency-distribution (SFD), but even with the greatly improved dataset provided in

2012 by the Mercury Surface, Space Environment, Geochemistry and Ranging

(MESSENGER) spacecraft, no additional cratering above what is expected based on measurements of the Moon and Mars has been found [15]. The regions of Mercury with the highest cratering concentrations are consistent with numbers expected due to cratering during the Late Heavy Bombardment (LHB) [11]. However volcanic activity has greatly complicated the situation adding enough ambiguity to make it impossible to draw definitive conclusions on the impact vulcanoids may have played in the Hermian cratering record. At this time there is no clear observational evidence that a significant population of vulcanoid asteroids existed in the distant past.

Current research into exoplanetary systems can lead to insights into the structure of our own solar system, both in the present and distant past. A significant number of the exoplanetary systems that have been detected are in what is known as tightly-packed planetary (TPP) systems. These systems can contain a large number of terrestrial planets, as many as 7 or more, in a region of space no larger than the orbit of mercury in our solar system [16]. It has been suggested that such TPP systems are unstable over long periods of time. This instability would inevitably lead to the ejection of some planets and the collisional destruction of others. It is quite possible that our solar system originated in a

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TPP type system before evolving into the configuration that we observe today with fewer planets that have close-in orbits. If such events did occur in our distant past they would have led to a large population of asteroids around the parent star, in this case the Sun.

This provides two significant sources for a primordial vulcanoid population; the initial material left over from planetary growth, and the debris created by the collision and destruction of two or more tightly-packed planets. In addition to the above scenario capture of material from other parts of our solar system could be a source represented in a population of vulcanoid asteroids, specifically the capture of Hermian Trojan asteroids

[17], and the inward evolution of cometary nuclei, near-Earth asteroids and assorted collisional ejecta from the outer solar system [18].

In 1999 Evans and Tabachnik [19] were able to show that interior to Mercury’s orbit there was a large region of gravitational stability. Numerical simulations revealed an annulus of stability between 0.1 and 0.19 AU, which Evans and Tabachnik claimed was “one of the most dynamically stable regions in the entire Solar System”. The outer limit of this region was established by objects in the simulation beyond 0.19 AU which would intersect with Mercury’s orbit on timescales shorter than the age of the solar system, approximately 100Myr. The inner edge reported by Evans and Tabachnik is simply due to the smallest orbit considered by their model. In reality smaller orbits would be stable as well. Orbits as small as ~0.01 AU, corresponding to the Sun’s Roche limit, would be gravitationally stable. Within a distance less than 0.01 AU objects pass what is known as the Roche limit. Within this distance the gravitational attraction of the sun overcomes the forces holding objects together that are made of disparate components and are in turn only held together by their own mutual gravitational attraction. Within the

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Roche limit objects are ripped apart by the tidal forces of the sun. In 2002 Evans and

Tabachnik published a second more detailed analysis [20] which confirmed and refined their preliminary work which expanded their stability zone to a range of between 0.09 and 0.20 AU. In every other gravitationally stable region of our solar system a population of objects has been found to exist. This result provides support for the existence of a population of vulcanoids. There is no reason that the vulcanoid region should be any different. That being said the results of Evans and Tabachnik presented here [19, 20] are based entirely upon standard Newtonian N-body dynamics, and no non- gravitational perturbation effects are considered. Other pertinent perturbation effects, such as sublimation mass loss, collisions and/or radiative interactions, will be explored in greater detail in this thesis.

It has been established that while the vulcanoid region is gravitationally stable, it is far from dynamically inactive. Asteroids and cometary nuclei are commonly observed to pass through the vulcanoid region in the modern era [17]. This phenomenon will be explored in greater detail in Chapter 2. A search of the JPL (Jet Propulsion Lab) Small-

Body database returns that at the present epoch there are 244 asteroids that cross the orbit of Mercury with perihelion distances located within the vulcanoid zone. While the capture of one of these objects by the vulcanoid region is extremely unlikely, it is far from impossible. By similar considerations an impact on the surface of Mercury would lead to impact ejecta. Some fraction of this material could find its way into a stable orbit within the vulcanoid region. In fact, substantial impacts are believed to have played a large role in the evolution of Mercury. Wieczorek et al., [21] proposed that a significant impact would account for the unanticipated 3:2 spin-orbit resonance of Mercury.

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Additionally, the massive Caloris impact basin, with a diameter of roughly 1450 km, is estimated to be the result of a substantial impact around 3.7 billion years ago. Significant amounts of material would have been ejected by such impacts, and could have easily attained a stable orbit within the vulcanoid zone.

1.2 21st Century Perspective

The most recent event to generate significant interest in the vulcanoid region was the announcement of the Mercury Surface, Space Environment, Geochemistry and

Ranging (MESSENGER). The spacecraft and its wide angle camera was used to image the vulcanoid zone in a search for vulcanoids prior to its insertion into the orbit of

Mercury in March 2011. These observations did not reveal any anomalous objects brighter than a limiting visual magnitude of +8 (This limit corresponds to sizes larger than ~15 km in diameter.) [22]. Further searches of the region have been undertaken by the Solar and Heliospheric Observatory (SOHO) and Solar Terrestrial Relations

Observatory (STEREO) sun-monitoring space-platforms since the year 2000 [12, 23, 24] however no vulcanoid objects have been positively identified. These null results mean one of two things; the vulcanoid region is devoid of any objects, or the size distribution and surface albedo of the vulcanoids result in reflected surface light that is below the threshold for detection by available instrumentation. Analysis of data from the STEREO

Heliospheric Imager by Steffl et al. [14] concludes that no object larger than roughly 6 km across (corresponding to an assumed R-band albedo of 0.05, which is a characteristic value derived for Mercury and the C-type asteroids.) exists in the current vulcanoid population.

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The funding announcement of the MESSENGER mission in 1999 led to the publishing of two important papers on the topic by Stern and Durda [25] and

Vokrouhlicky, Farinella and Bottke [26] in 2000. Stern and Durda focused primarily on the various mechanisms of the collisional evolution of the vulcanoid region, concluding that due to the high collision velocities survival is unlikely for timescales comparable to the age of the solar system. However, this conclusion is examined in further detail in the future work section presented below. Stern and Durda argued that, “it is plausible that the entire region is virtually empty of kilometre-scale and larger objects”. Vokrouhlicky,

Farinella and Bottke took an approach considering the radiative drag forces that act upon objects in the vulcanoid zone. It was their finding that such depletion mechanisms were highly efficient and that objects smaller than approximately a kilometer in diameter would be rapidly cleared from the region. They concluded that “Collisions probably eliminated most larger (D > 1 km) vulcanoids, while Yarkovsky drift removed the smaller objects (D < 1 km). Thus, the sole remaining evidence supporting the existence of the vulcanoid populations at any time is contained in Mercury’s enigmatic crater record”. The general consensus during the 21st century has been against the existence of any substantial vulcanoid population, not just in the present but also at any time in the past.

However, the understanding of asteroid structure and non-gravitational phenomenon, as they pertain to orbital evolution, has advanced significantly since the turn of the century. This work intends to examine these past results with the benefit of a more modern understanding of the forces at play in order to see how prior conclusions stand up to modern scrutiny.

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1.3 Definition of the Vulcanoid Region

While the existence of a population of asteroids in the vulcanoid zone has been in doubt, the region such objects must occupy, should they exist, has been explored in detail. As discussed above Evans and Tabachnik [19, 20] performed numerical simulations that result in a vulcanoid stability zone between 0.09 and 0.20 AU from the sun, with orbital inclinations of about 5° but allowing for inclinations as large as 10°. Of further note these authors find two strong mean-motion clearing-zones at 0.15 AU (due to a 4:1 resonance with Mercury) and at 0.18 AU (due to an 8:1 resonance with Venus and 3:1 resonance with Mercury) which would leave gaps in any population of vulcanoids. Based on dynamic and thermodynamic considerations Campins et al. [27] suggests a region between 0.1 and 0.25 AU from the sun and confined to the ecliptic.

Steffl et al. [24] and Durda et al. [12] put the boundaries at 0.07 and 0.21 AU from the sun with inclinations as high as 15°. Stern and Durda [25] set an interior limit of 0.06

AU based on thermodynamic effects and inclinations up to 16°. Vokrouhlicky, et al. [26] adopts the limits used by Evans and Tabachnik [14]. With the limits imposed by past research in mind, for this analysis the maximum values are adopted to give the maximum vulcanoid region. This region is defined (see figure 1.1) as the region extending from

0.06 to 0.25 AU away from the sun with inclinations as large as 15° from the ecliptic.

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Figure 1.1 – The Vulcanoid Zone This schematic diagram illustrates the maximum extent of the vulcanoid zone. The diagram also highlights clearing zones caused by mean motion resonances with Mercury and Venus, as derived by Evans and Tabachnik [19, 20]. The applicable resonances are the 4:1, 3:1, and 2:1 resonances with Mercury and the 8:1 resonance with Venus.

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Chapter 2: Sun-Grazing Comets

2.1 Cometary Impact Modification

Cometary nuclei have long been known as vast repositories of knowledge about the early solar system [28]. In a similar sense comets can be thought of as probes into regions of the solar system we do not possess the ability to examine through more conventional means. The basic principle we are exploiting is that cometary nuclei can reveal the presence of other bodies in the solar system through collisions and the accompanying observable phenomenon that result. Should a cometary nucleus collide with another body in the solar system it would trigger an outburst activity that could then be readily observed by instruments available to us [29]. Cometary outburst activity could, in theory, indicate the presence of an interplanetary object that intersects a comet’s orbital path. If the orbit of such a comet is well understood the potential locations of these collisional bodies can be traced back.

The discovery of a main belt population of comets [30, 31] allows us insight into a similar phenomenon. This unexpected reservoir of cometary nuclei provides us with an ideal population to examine impact event outbursts, although it must be acknowledged that additional trigger mechanisms besides collisions exist, such as thermal forcing and rotational spin-up. The extraordinary discovery images of 354P/LINEAR demonstrate the effects that such a collision outburst would have for a main-belt cometary nucleus and a multiple-tens-of-meter sized main-belt asteroid [32]. 354P/LINEAR is a main belt asteroid that experienced an impact leading to a large comet-like tail of debris. Other comets, not belonging to the main-belt population, are known to pass through the main-

11 belt region and are known to undergo outburst activity [33]. This activity was likely driven by collisions with meter-sized asteroids.

In general collisions between a cometary nucleus and smaller sized impactors are most likely in locations of long lived stable orbits, such as the main-belt region. For other populations of comets, while less likely, they can undergo collisions at various points in their orbit. Of interest in this thesis are comets with orbital paths crossing the vulcanoid zone. Collisions may occur for both long period and Jupiter family comets, and sungrazers and Halley type comets as they pass through the intra-mercurial zone where vulcanoids may reside. While no vulcanoids have been confirmed to exist in the region, comets passing through are known to undergo outburst activity. While comets are known to undergo outburst activity for a number of reasons [34, 35, 36] one possible mechanism is an impact with another body. The argument can be made that at least some of this outburst activity could be due to a collision with an as yet unobserved vulcanoid asteroid.

A comet passing through the vulcanoid zone could conceivably experience a collision and would then exhibit behaviour modified by that collision. This impact would lead to modified observable behaviour in the comet nuclei, e.g. outburst activity or fragmentation events, or unusual activity in associated meteor showers, e.g. meteor storms or unusually high fireball activity. The largest limitation to this approach is differentiating collisional activity from other forms of nuclear fragmentation. Perihelion is a time when a cometary nucleus is exposed to extreme tidal and thermal effects. These extreme conditions can lead to nuclear fragmentation that, from our perspective, is indistinguishable from an impact event. As a result of these conditions, Belton [37] finds

12 that the disruption rate for a 2 km diameter, active nucleus is of order 5x10-5 probability of disruption per year regardless of whether it passes through the vulcanoid region or not. In reality the odds of a cometary nucleus experiencing an impact as it crosses the vulcanoid zone is extremely small, but it is not necessarily zero.

Figure 2.1 shows the distribution of nodal points for comets known to pass between 0.06 and 0.25 AU from the sun. The data used to construct figure 2.1 comes from the JPL data archive up to and including P/2014 E1 (Larson). This data set is made up of 3261 sets of orbital elements, of which 406 recorded comets are found to have passed through the assumed vulcanoid zone. The earliest recorded comet in the dataset that meets our criteria is C/400 F1, with comet C/2008 J13 (SOHO) being the most recent. Within the comets selected for inclusion in figure 2.1 are numerous sungrazing comets (with the Kreutz, Kracht, Marsden, and Meyer groups all being represented [38]), a number of long period comets and two short period comets. Several comets included in figure 2.1 exhibit behaviour indicative of nuclear fragmentation. These comets are listed, with a brief summary of their properties, in table 2.1. For a more detailed examination of notable comets see the paper by Beech and Peltier [18].

Of the 406 comets identified in this analysis the vast majority, 346 to be exact, were discovered using the coronagraph instrument on the SOHO spacecraft [39]. The typical time these SOHO comets spend passing through the vulcanoid zone is of the order of 1 day with a typical path length of 0.1 AU. With the assumption that the typical comet nucleus is of the size of 1 km, then the total volume swept out through the vulcanoid zone is a few 10-15 AU3. This accounts for a small fraction of the total volume of the vulcanoid zone, 0.0167 AU3 and should be indicative of the extremely low

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Figure 2.1 – Ecliptic Crossing Points This figure shows the nodal-crossing points for comets passing through the vulcanoid zone. The ascending and descending nodes of the Kreutz sungrazing comets can be seen as two broad fan-like structures extending on either side of the sun. Comets highlighted by filled circles can be found described in table 2.1.

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Table 2.1 – Table of Comets Table of comets known to have passed through the vulcanoid zone that display evidence of fragmentation or associated behavior. Column 1 indicates the comet designation, column 2 is the perihelion distance and column 3 is a brief outline of the distinguishing characteristics.

Comet q (au) Behavior / associations Naked-eye comet; tail extended as much as 70o on the sky [40] C/1769 P1 0.124 Potential to produce a meteor shower on Mars [41] Comet Mellish. Associated meteoroid streams – 11th century C/1917 F1 0.190 MON fireballs [42] Potential to produce a meteor shower on Venus [41, 43] C/ 1931 P1 0.045 Potential to produce meteor showers on Mars [44, 29] The “Great Southern Comet” C/ 1947 X1 0.110 Nucleus fragmented post perihelion passage C/ 1953 X1 0.072 Nucleus fragmented C/ 1970 K1 0.009 Nucleus split from a Kreutz sungrazing comet C/ 1975 V1 0.197 Nucleus fragmented post perihelion passage C/ 1975 V2 0.219 Association with the β Tucanid meteoroid stream [29] C/ 1987 W1 0.200 Nucleus fragmented C/ 1991 X2 0.199 Lost post perihelion passage – nucleus disruption (?) Outburst activity in 1986 96P / 0.124 2 small fragments observed ahead of nucleus in 2012 [45] Machholz 1 Associated with Arietid and S. δ Aquarid meteoroid streams [29]

15 probability of a collision. That is not to say that a collision is impossible, merely improbable. Due to the low interaction probability it is unlikely that any of the currently observed comets passing through the vulcanoid zone have experienced a collision.

However the dataset currently available to us is obviously historically incomplete, with only about 12 years of data from SOHO available and used in this analysis. As time goes on and more cometary data is collected the larger the chance that this unlikely event is observed, assuming of course that a population of vulcanoids exists for such a collision to occur.

2.2 Impact Probability and Effects

The probability of a collision P between a comet and a given impactor, such as a vulcanoid, can be found through a formula derived by Kessler [46]. This probability is given by,

= (2.1) 2 (sin )( )( ) 푉푟푟푟휎휎 푃 3 2 휋 푅푅� 푖 푅 − 푞 푄 − 푅 where Vrel is the relative velocity at the time of encounter, σ is the collisional cross- section area, t is the time of potential encounter, a is the semi-major axis of the comet’s orbit, i is the angle between the cometary orbit and the orbit of the impactor, with q = a(1 – e) and Q = a(1 + e) being the perihelion and aphelion distances of the comet respectively, and e is the comet’s orbital eccentricity. It is assumed here that the angle β in Kessler’s original equation, which accounts for the latitude variation in the impactor number density, is zero. The collision cross-section is taken to be the geometric cross-

2 section with σ = π(rc + ra) , where rc and ra are the radii of the cometary nucleus and the

16 impactor respectively. Using Opik’s formula [47] we find the relative velocity at the time of impact is given by,

= 3 2 (1 ) cos (2.2) 2 푅 � 2 2 푉푟푟푟 � − − � − 푒 푖� 푉푐푐�푐 � 푅 where is the circular velocity at radius R. 2 푉푐푐�푐 For the purposes of this exercise it is possible to examine more closely the comet

Mellish, C/1917 F1 referenced in table 2.1, and its orbital characteristics. Comet Mellish is chosen due to the fact that it has experienced no significant alterations in its orbit over the past 40,000 years [48] allowing repeated passes through the vulcanoid zone and interesting meteor shower activity that is likely indicative of a fragmentation event.

Examining the conditions present at comet Mellish’s inner most nodal point, the orbital radius of our hypothetical vulcanoid would be R = 0.2 AU which gives Vcirc = 59.6 km/s.

Opik’s formula for the relative velocity additionally gives, Vrel = 57.0 km/s. With the radius of comet Mellish being taken as rc = 1500 meters, the collision probability evaluates to

= (3 × 10 )(1 + ) (2.3) −25 2 푃 푟푎⁄푟푐 �

The encounter time will be of order the time it takes comet Mellish to travel through the vulcanoid zone. Orbital integration gives a time of ≈ 2.2 days. Using this characteristic time we find the impact probability per perihelion passage of comet Mellish with a vulcanoid of radius ra is (6 × 10 )(1 + ) collisions per passage. As −20 2 푎 푐 mentioned before this comet푃 has≈ had an effectively푟 ⁄ 푟unchanged orbit for the past 400

17 centuries which leads to an impact probability in that time of

(2 × 10 )(1 + ) collisions in the past 400 years. This illustrates that over −17 2 푎 푐 푃roughly≈ 275 passes through푟 ⁄푟 the vulcanoid zone the interaction probability with any one vulcanoid is extremely small. This would indicate that in order for a collision to be likely in that period there would need to be approximately 5×1016 objects (i.e. it would take this many objects to reach a probability of 1 for the chance of a collision in this time) which would necessarily be very small objects.

Any vulcanoids present would be part of a collisionally evolved population, and in such a population there are far more small objects than larger ones. The number of objects in a given population, larger than size r, can be approximated by a power law of the form,

( ) = ( ) (2.4) 훼 0 푟 푁 푟 푁 푟 � �푟0� where r0 is a specific reference size and α is a constant of order -3.5 chosen as appropriate for a steady state collisional fragmentation distribution [49]. As will be discussed in more detail later in this work, the Yarkovsky effect is efficient at clearing smaller objects which places a lower limit of the sizes of objects that will persist for any significant length of time in the vulcanoid zone. Upper limits have also been placed on possible vulcanoids due to past non-detections. With these constraints in mind, Steffl et al. [24] estimate that no more than ~ 900 vulcanoids can exist between these limits.

Using this estimate in equation 2.4 means that fragments as small as 5 cm need to be considered to achieve the required 5×1016 objects. The Yarkovsky effect would clear out objects of this size in much less than a million years and as such the vulcanoid zone

18 could not support such a population for any significant length of time. However, with these limitations in mind, it is possible for a collisionally evolved primordial population of vulcanoids to generate significant numbers of smaller fragments as they are broken down. This could lead to a robust population of vulcanoids but likely not one large enough to satisfy the 5×1016 objects threshold.

The exact effect of a collision with comet Mellish is difficult to estimate but it is possible to use results from the Deep Impact mission [50] as a guide. In the Deep Impact mission roughly 106 kg of nuclear material was excavated and released from comet

9P/Tempel 1 following the impact of a 372 kg copper cylinder at a velocity of 10.3 km/s.

In the situation outlined for comet Mellish the impact velocities would be much higher, of order 57 km/s. In order to match the energy present in the Deep Impact cylinder comet

Mellish would only require an impactor of 12 kg, which corresponds to an object of 20 cm in diameter assuming a density of 3300 kg/m3. Extending this calculation to the 5 cm objects evaluated earlier would result in impact energies of only 3.5×108 J which is 57 times smaller than the energies utilized in the Deep Impact mission. This is not an encouraging result, in addition to its being unlikely that a significant population of 5 cm objects exists, any impact with such an object would not possess enough energy to produce an effect that could currently be observed.

2.3 Conclusions

The vulcanoid zone is dynamically active with numerous cometary nuclei passing through. Based on data from the SOHO coronagraph the passage of cometary nuclei is an almost daily occurrence. This is also the point in a comet’s orbit, at perihelion, where

19 it is subjected to the greatest thermal and physical stresses making fragmentation most likely at this point neglecting of the effects of impacts [37, 51]. As such this complicates any attempt to separate fragmentation events into those caused by impacts versus those caused by other destructive events. This work does not seek to claim that all, or even any, perihelion fragmentation events are caused by impacts with vulcanoid asteroids.

The intent of this work is to point out that while the probability of such an interaction is low it cannot be discounted out-of-hand and is far from impossible.

20

Chapter 3: Solar Radiation Effects

3.1 Material Properties

Before the effects that solar radiation would have on vulcanoids can be examined the constituent materials and properties of those materials must be established. This is complicated by the fact that, as yet, no vulcanoid candidate has been observed. This means that the materials such a body would be made out of have not been established.

As such, for this analysis three possible materials are explored to provide a range of values indicative of whatever material may exist. For a full collection of materials and values see Table 3.1.

The first material examined is iron which provides an exemplar of robust metallic materials, possibly remnant cores of larger bodies lost over time. Iron was chosen as the strongest material that would be found in any significant quantity in a vulcanoid population offering a type of upper limit. Analysis demonstrated that iron has an expected survival lifetime much less than was initially assumed. However, an iron rich population cannot be ruled out and so its results are presented here, even still offering a type of limit.

The second material chosen is fayalite which as an archetypal rocky body found in the solar system. While it cannot be effectively established that vulcanoids would be composed of fayalite in examinations of other rocky compositions it was observed that they all broadly possessed similar attributes and provided broadly similar results.

Fayalite was chosen to present here as a common rocky body observed in the solar

21

Table 3.1 – Material Properties Table of material properties for three chosen materials used in the model.

Material Iron Fayalite Regolith Description a a b Material-dependent sublimation parameter A 48354 60377 60377 (experimentally determined) Material-dependent sublimation parameter B 29.2 a 37.7 a 37.7 b (experimentally determined) μ 55.845 a 203.774 a 203.774 b Mean Molecular Weight in amu α 1 a 0.1 a 0.1b Material specific evaporation coefficient ρ 7874c 4390 c 2200d Density of Body in kg m-3 K 80 c 3.85 c 0.001d Surface thermal conductivity in (W m-1 K-1) c c b -1 -1 Cp 500 797.1 797.1 Heat Capacity in (J kg K )

Notes. (a) Values taken from [52] (b) Values assumed to be the same as those of Fayalite. (c) Values taken from [53] (d) Values taken from [54]

22 system and the resulting values in simulations can be taken as general results for a rocky body. If a more promising candidate for the composition of vulcanoids was discovered in a rocky material, then simulations could be rerun with different material properties but the final results would almost certainly remain within an order of magnitude of the fayalite values.

The third and final material chosen is the least well constrained of those chosen and is an examination of a body covered in regolith. As a form of pulverized rock most of the material properties are adopted as the same as those of fayalite. The significant difference and the one most important in this analysis is the greatly reduced thermal conductivity, a value of order 10-3 W/m/K for regolith compared to 3.85 W/m/K for fayalite. As will be discussed in more detail in the chapter on future work there is significant room for improvement on these regolith values. These material properties have not been well explored for regolith in the literature and many refinements could be made. With current understanding even the value for the thermal conductivity can only be given to an order of magnitude with any level of confidence. Furthermore a body composed entirely of regolith is not particularly realistic, however this is the way it is handled by our current modelling. A more complicated model that could handle composite materials would be required to effectively model this behaviour and will be discussed in future work.

3.2 Solar Luminosity Variation

As previously discussed, for the purposes of this analysis the inner edge of the vulcanoid region is set at 0.06 AU and is defined by sublimation effects caused by solar

23

2.0

1.9

1.8 4th order fit: 1.7 Age (Gyr) versus Luminosity (Lsun) 1.6 b[0] = 0.7157248347 1.5 b[1] = 0.0401120615 b[2] = 0.012761239 b[3] = -2.4116691812e-3 1.4 b[4] = 1.9198252799e-4

1.3 r ² = 0.9997

1.2 Luminosity (Lsun) 1.1

1.0

0.9

0.8

0.7

0.6 -1 0 1 2 3 4 5 6 7 8 9 10 11 Age (Gyr)

Figure 3.1 – Luminosity Variation Luminosity (normalized to current solar luminosity) versus age plotted in Gyr for a one solar mass stellar model. The coefficients to the 4th order polynomial fit are listed in the top left of the diagram. Model data is taken from the EZ-web server [55].

24 heating effects. These sublimation effects are related to heating caused by the sun, and are directly proportional to the sun’s luminosity. The majority of past analyses have assumed a constant solar luminosity set at the value measured at the present time. In fact the luminosity has varied significantly over the age of the solar system being roughly 25-

30 percent more luminous now than in the past. This variable luminosity has a significant effect on all of the other solar radiation effects as explored in this chapter.

The exact variation in solar luminosity used in this analysis is modelled using a

4th-order polynomial least-square fit to a solar mass model obtained from the EZ-web

[55] and can be found in figure 3.1.

3.3 Sublimation

Using a similar technique as was applied to α Centauri B in a past work [56], the effects of sublimation mass loss will be explored for our solar system on the vulcanoid asteroid population. While inherently a reasonably small effect, over the age of the solar system and at the extremely close distances vulcanoids can be found from the sun this effect causes significant mass loss up to and including destruction of the body. The primary factor affecting the mass loss rate due to sublimation will be the temperature of the body.

This temperature, given by the black-body temperature [52] is,

(1 ) ( ) = 1 (3.1) 2 × 4 �4 − 퐴 퐿 � 푇 � 2� 휋�휋퐷

25 where T, A, D, σ and ε are the temperature (K), albedo, orbital radius (m), Stefan–

Boltzmann constant (W m-2 K-4), and emissivity of the vulcanoid. The factor of two is chosen to reflect slow rotation of the vulcanoid, in the case of fast rotation a four is used in its place. L(t) is the time dependent luminosity of the sun (W) as discussed in the previous section. This temperature can then be used to find the vapour pressure [52] at the surface of the vulcanoid given by,

( ) = 0.1 ( / ) (3.2) 퐵−퐴 푇 푃푉 푇 푒 where PV(T) is the vapour pressure, A and B are material dependent experimentally derived values [52], and the factor of 0.1 is used to convert dynes/cm2 into Pa. This vapour pressure is then used in what is known as the Langmuir equation to compute a mass loss rate [52] given by,

( ) = ( ) (3.3) 2 휇푚푢 퐽 푇 훼푃푉 푇 � 휋푘퐵푇 where J(T) is the mass loss rate (kg m-2 s-1), α is the evaporation coefficient to parametrize kinetic inhibition of the sublimation process, μ is the molecular weight of the molecule that is sublimating in atomic mass units, mu is the atomic mass unit conversion into kg, and kB is the Boltzmann constant (J/K). This allows us to obtain the following result,

26

( ) = (3.4) 푑� 퐽 푇 푑� 휌 where dR/dt is the change in radius over time of the body (m/s), and ρ is the density of the body (kg/m3).

This result allows the calculation of survival lifetimes of vulcanoid asteroids for various initial conditions. Using this result it is possible to see the effect that variable luminosity has on survival lifetimes against sublimation. Figure 3.2 shows the effects of constant luminosity compared to the effects of time dependent luminosity for both iron and fayalite. The simulation is halted after ten billion years because at that point the sun will have left the main sequence and our luminosity assumptions will no longer be valid.

Furthermore the sun will expand into a red giant at that point and its outer layers will expand and subsume the inner solar system, including the vulcanoid region.

For an iron composition the variable luminosity condition allows for a much longer survival lifetime due to the lower luminosity of the early solar system. Under an assumption of constant luminosity an iron asteroid would need to have an initial radius of much more than 100 km (of order ~300 km) to survive until the current age of the solar system, an unlikely situation. Under the variable luminosity conditions an asteroid of pure iron would need to have an initial radius of less than 80 km.

The situation for fayalite is somewhat more complicated. Variable luminosity leads to longer survival lifetimes until a radius of 40 km is reached. This is because after the simulation reaches the current age of the solar system the luminosity of the sun continues to increase and therefore the sublimation rate increases. With a radius of

27

Figure 3.2 – Variable Luminosity Survival Lifetimes Vulcanoid survival lifetimes for different compositions and luminosity conditions. Blue and red lines represent iron and fayalite respectively. Solid and dashed lines represent variable and constant luminosity respectively. The horizontal dotted line represents the current age of the solar system (t=4.57 Gyr). Lifetimes are presented for a distance of 0.06 AU to demonstrate the most exaggerated situation. Regolith uses the same sublimation values as fayalite and so its results would be identical, as such it is not presented here.

28

40 km the survival lifetime is greater than the age of the solar system and this is the point at which the increasing luminosity overtakes the constant luminosity sublimation rate. As such for initial fayalite asteroid radii greater than 40 km they survive longer under constant luminosity than they would for the variable case. In either situation fayalite asteroids of reasonably small size can survive to the current age of the solar system, in the variable case an initial radius of less than 10 km is all that is required, even in this most extreme sublimation situation at only 0.06 AU from the sun.

3.4 Poynting-Robertson Drag

There are two main solar radiative effects that can modify the orbits of small objects in the solar system, the first of which is known as Poynting-Robertson (PR) drag [57]. PR drag causes small bodies to lose angular momentum and will eventually cause an object to spiral into the sun. See figure 3.3 for a diagram illustrating the effect.

The effect can be understood from two different perspectives; from the reference frame of the sun, denoted as (a) in figure 3.3, and from the reference frame of the object experiencing the drag, denoted as (b). In the reference frame of the sun, situation (a), all radiation absorbed by the incident object is received in a perfectly radial direction with respect to the sun. As such, at this point, the radiation has not had an effect on the angular momentum of the body, however once the object reemits the radiation it does so in a way that is not isotropic with respect to the sun’s reference frame. This anisotropic emission causes photons to carry angular momentum away from the object. In the reference frame of the body, situation (b), the relative motion between the object and the sun causes the incident radiation to appear to be coming from a slightly forward direction, a phenomenon known as the aberration of light. Due to this slight angle to the

29

Figure 3.3 – Poynting-Robertson Drag Figure a shows the effect from the reference frame of the sun. Figure b shows the effect from the reference frame of the object experiencing the drag. Angles of incident and emitted radiation are exaggerated to illustrate the concept.

30 incident radiation it causes a force that opposes the motion of the body. This angle is extremely small due to the large difference between the speed of light and the speed of the incident object, however in the situation of the vulcanoids even a small force over billions of years can have a significant effect on the orbit of the body.

Regardless of the reference frame the practical effect is the same. This drag force over time circularizes orbits and reduces the semi-major axis of the orbit. In order to quantify this change in orbital radius a term called α must be calculated that is used as an intermediate step for various calculations related to PR drag. This α term [57] is given by,

3 ( ) = (3.5) 16 퐿 � 훼 2 휋�푐 휌 where R and ρ are the radius and density of the body, respectively. This α term depends only on the physical characteristics of the body and the luminosity of the sun L(t), as such in the variable luminosity conditions α too is time dependent. Of note for this work,

α can be used to calculate the rate of change for the semi-major axis of a body in a circular orbit [57]. This is given by,

2 = (3.6) 푑� 훼 − 푑� �

31

where da/dt is the rate of change for the semi-major axis and a is the current semi-major axis. Since α is dependent on the inverse of the radius of the body PR drag is a small effect, even over the age of the solar system, for asteroids of any significant size. On its own PR drag does not have any significant effect; it is only through its interactions with sublimation that together they are able to effect significant changes in orbit and size. As the body drifts inward due to drag it gets closer to the sun and therefore hotter. As it gets hotter it sublimates more quickly making the body smaller. As the body gets smaller PR drag becomes more efficient at moving the body inward. This acts as a sort of feedback loop, with PR drag making sublimation more efficient which in turn makes the drag more efficient.

3.5 The Yarkovsky Effect

In small bodies PR drag is the dominant effect in orbital modification, but for objects of intermediate sizes, roughly 10 cm to 10 km in diameter, what is known as the

Yarkovsky effect or Yarkovsky drift is the dominant mechanism. The Yarkovsky effect affects rotating bodies as they release thermal photons during their rotation. As these photons travel away from the body they carry away momentum, but they do so anisotropically. As the body is heated, for the purposes of this work the heating is done by the sun, the heating is done unevenly with one side of the body, the side exposed to the sun, heated more than the other. The object continues to rotate and reemits this heat in a different direction.

There are two different ways the Yarkovsky effect acts on a body, a diurnal component and a seasonal component. The diurnal effect applies as the body rotates

32

Figure 3.4 – The Yarkovsky Effect As the body is heated by the sun the direction of re- emission of these thermal photons changes over the course of its rotation. The hottest section, depicted here in red, emits the most energy and carries the most momentum away from the body, while the cooler sections, depicted here in blue, emit comparatively less. In the situation depicted here the hottest sections remit in the same direction as the body’s forward motion. This would have the effect of draining its net forward momentum and will cause it to spiral in towards the sun. With rotation in the opposite direction a spiral outward would be observed.

33 about its own axis over the course of its “day” see figure 3.4. As the sun heats one hemisphere of the object there is a “lag” as that heated area rotates with respect to this

“mid-day” position before the majority of this thermal radiation is reradiated back into space. The surface continues to cool and emit fewer and fewer thermal photons until it completes its rotation and is once again irradiated by the sun. Depending on the direction of this rotation with respect to its motion it can either decrease or increase the body’s angular momentum causing it to spiral in or away from the sun, respectively.

The second form the Yarkovsky effect takes is the seasonal effect. These effects are more difficult to visualize and are simplest to understand for a non-rotating body revolving around the sun. For the seasonal effect the heated face of the body is always moving toward the “dusk” edge of the body as it travels around the sun. This invariably leads to a braking force slowing the orbiting body and causing it to spiral in towards the sun. In numerous rotational configurations the seasonal effect can easily be drowned out by the effects of the diurnal component. In situations where rotation is very fast or the body is small enough to be heated all the way through the seasonal component is minimal. The seasonal component is maximized as the obliquity of the body reaches 90°, i.e. the body’s rotation axis is in the same plane as its orbit.

The exact details of the equations governing the Yarkovsky effect have been explored in great depth [54, 58, 59]. Of note for this work are the equations governing secular changes to the semi-major axis of bodies around the sun, and the pertinent equations are repeated here. The first thing necessary for calculating the changes due to the Yarkovsky effect is the solar flux at the distance of the body of interest. This flux is given by,

34

( ) ( ) = 4 (3.7) 퐿 � 퐹 � 2 where F(t) is the time variable solar flux,휋 �this variation is due to the time variable luminosity L(t), and a is the semi-major axis of the body in question. Using this it is possible to find the radiation pressure [54] on the body given by,

( ) = (3.8) 2 휋푅 퐹 � 훷 where is the radiation pressure on the body,푚� R is the radius of the body, m is the mass of the body,훷 and c is the speed of light. From here it becomes necessary to examine the material properties of a vulcanoid that would affect the delay of the reemission of radiation causing the Yarkovsky effect. To do so it is necessary to calculate the thermal inertia of the body [54] given by,

= (3.9)

훤 �휌휌휌 where is the thermal inertia, ρ is the density of the body, K is the surface thermal conductivity훤 (W/m/K), and C is the heat capacity (J/kg/K). From this, what is known as the thermal parameter [54] can be calculated,

= (3.10) 훤√휈 훩 3 𝜀푇 where is the thermal parameter, is the rotational frequency (radians/s) for the body, is the 훩emissivity of the body, 휈 is the Stefan-Boltzmann constant, and T is the� black-body temperature. When used휎 in the seasonal component the value is what is

휈 35 known as the mean orbital motion of the body, is denoted by n, and is found using a straightforward calculation using Kepler’s third law. Knowing its orbital radius allows us to calculate the orbital period and from there the mean motion. The rotational frequency of the diurnal component is simply any given rotational rate, denoted by . The diurnal component introduces significantly more uncertainty. Without an observed휔 vulcanoid the exact rotation rate such a body would take is not well constrained and could in theory range over a wide range of values, from no rotation all the way up to values large enough to cause the destruction of the body. In the absence of any compelling theoretical reason to choose a specific value other observed asteroids in the solar system are used as a guide. Farinella, Vokrouhlickey, and Hartmann [53] explore this issue and decide upon a spin period of 5 hours. Other spin periods arising from more complicated size-rotation rate ratios are also explored but for this analysis they offer no probative value and merely increase complexity. In this work an angular frequency of 5 hours is adopted, with the acknowledgment that it is somewhat arbitrary.

The next value required in the derivation of the full Yarkovsky effect calculation is a parameter known as and its full derivation is extremely complicated and is dependent upon both the thermal훺 parameter , and the radius of the body modified by several other factors. The full derivation is훩 beyond the scope of this work, refer to

Vokrouhlicky [59] for the full analysis. Of interest here, beyond an object size of a few centimeters the size dependencies disappear and the factors they influence in become a factor of ½. Since vulcanoids of sizes of even a few meters would be efficien훺tly cleared from the vulcanoid zone, for the rest of this analysis the simplified are adopted and used. This parameter [54] takes the form, 훺

36

1 = 2 1 (3.11) 1 +− +훩2 where is the required parameter훺 and is the2 thermal parameter. With all of these 훩 훩 assembled훺 components it is now possible 훩to construct the full change in semi-major axis due to the Yarkovsky effect. The diurnal and seasonal expressions [54] are as follows,

8 (1 ) = cos (3.12) 9 푑� 4 (1 − 퐴) 훷 퐷 � �퐷�푢�푢𝑢 =− 훺 sin 훾 (3.13) 푑� 9 푛 푑� − 퐴 훷 2 � � 훺푆 훾 푑� 푆𝑆푆푆𝑆� 푛 where da/dt is the change of semi-major axis per change in time (m/s) for either the diurnal or seasonal component in equations (3.12) and (3.13) respectively, A is the albedo of the body, n is the mean motion of the body, is the obliquity of the body

(defined below). This can also be represented as a single larger훾 expression [54],

4 (1 ) = [ cos 2 sin ] (3.14) 9 푑� − 퐴 훷 2 − 훺퐷 훾 − 훺푆 훾 푑� 푛 where da/dt represents the total change in semi-major axis over time due to both components of the Yarkovsky effect, D is the parameter using values for the diurnal rotation rate, S is the parameter using values훺 for the seasonal rotation rate.

훺 At this point it is important to address the term, the obliquity, of the body. This term can take any value from 0° to 180°, and, as 훾is the usual refrain, there is no strong evidence for vulcanoids having any one orientation over another. Instead, three cases that

37 maximize the Yarkovsky effect in different ways will be examined, 0°, 90°, and 180°.

This is not to suggest that vulcanoids are more likely to be in any of these configurations, it simply allows us to examine the most extreme results possible, knowing that reality will likely lie somewhere between these limits. In reality any random orientation is possible and, assuming the vulcanoid population is subject to collisional activity, will likely change over the lifespan of a vulcanoid. Simulations of randomly varying obliquities show that position over time is highly variable, with a sensitive dependence on initial obliquities. For an obliquity of 0° the seasonal component is supressed and the diurnal component will act to increase the semi-major axis of the body. For an obliquity of 90° the diurnal component is supressed and the seasonal component will act to decrease the semi-major axis of the body. Finally, for an obliquity of 180° the seasonal component is once again supressed but now the diurnal component will act to decrease the semi-major axis of the body.

Figure 3.5 includes all of the various radiative drag effects discussed so far, namely the

Yarkovsky effect and PR drag with sublimation effects neglected. Luminosity, temperatures, and positions are updated in time steps of 1 Myrs. The figure serves to illustrate the maximal effects for the various different obliquity values. The diagram uses a 10 km diameter asteroid composed of fayalite with an initial position of 0.15 AU. The values presented are not intended to have any particular importance, the purpose is merely to delineate the effects of the various obliquity values. Arbitrary values for all of these variables could be adopted, and the relative strengths of the effects would vary somewhat, but the general trend of the figure would remain with smaller values leading to more rapid changes. The largest alteration for other values would be the relative

38

Figure 3.5 – Radiative Drag for Various Obliquities This figure illustrates the maximum effects the Yarkovsky and Poynting-Robertson effects can have on an object. Values are given for an asteroid 10 km in diameter with a fayalite composition, initial position of 0.15 AU. The horizontal dotted lines indicated the edges of the vulcanoid zone and the vertical dotted line indicates the current age of the solar system. The upper edge of the figure is set by Mercury’s semi-major axis, the lower edge by the sun. The blue line represents a 0° obliquity and the maximum effect the diurnal effect can have to increase the body’s semi- major axis. The dashed black line represents a 45° obliquity and the “average” increase effect that a combination of the different aspects of the Yarkovsky and PR drag would have. The green line represents a 90° obliquity and the maximum seasonal Yarkovsky effect acting to decrease the semi-major-axis of the body. The dotted black line represents a 135° obliquity and the “average” decrease effect. The red line represents a 180° obliquity and the maximum effect the diurnal effect can have to decrease the body’s semi-major axis. The solid black line represents a randomized obliquity that varies from 0° to 180° at every timestep (1Myr).

39 strength of the seasonal effects compared to the diurnal effects. As the test body becomes smaller all of the effects have a stronger relative influence and cause it to spiral into the sun or interact with Mercury more quickly. Random variations at every timestep (1 Myr) varying over the full interval demonstrate a jittery, but likely more accurate image of what an orbital path would look like.

Figure 3.6 illustrates the effect radiative pressures have on various initial positions and materials. Test objects were 10 km in diameter and the obliquity was set to

45° to illustrate both the diurnal and seasonal Yarkovsky effects. Smaller objects change the relative values in complicated and interrelated ways, but primarily have the effect of causing test bodies to reach their final states more quickly. Even at this large size and the relatively sedate pace it travels, all three materials have drastically different outcomes.

Regolith is ejected at roughly the same rate regardless of initial position. Fayalite’s path eventually converges, but is always ejected from the vulcanoid zone very quickly. Iron is almost entirely insensitive to initial position. For initial positions on the outer edge it spirals inwards towards the center of the vulcanoid zone, while initial positions on the inner edge cause it to spiral outwards towards the central vulcanoid region. Regardless of initial position iron’s path always converges and spirals toward the outer quarter of the vulcanoid zone remaining there for the duration of the sun’s main sequence lifespan.

Regolith is much more variable than the other two materials, it can either be ejected almost immediately on the outer edge, remain in the vulcanoid zone until around the current age of the solar system when it starts centrally located, or even remain in the vulcanoid zone for the sun’s entire lifespan when it starts on the inner edge.

40

Figure 3.6 – Vulcanoid Orbital Position Under Radiative Drag The three separate figures represent the radiative pressure effects on the three test materials; blue represents iron, red represents fayalite, and green represents regolith. Within each diagram three lines are plotted representing an initial position on the inner edge, central region, or outer edge of the vulcanoid zone (indicated by the horizontal dotted lines). Current age of the solar system indicated by vertical dotted line. All obliquity values are set at 45° and the test body is 10 km in diameter. Initial position has a dramatic effect on final position of the test objects. Note that iron is always pushed towards the center of the vulcanoid zone, regolith is always pushed out towards the orbit of Mercury before the current age of the solar system, and regolith may be pushed out immediately, survive until the current age of the solar system, or even survive for the entire period of the sun’s main sequence lifetime.

41

3.6 Induction Heating

The final effect to explore is the effect the sun’s magnetic field has on the vulcanoid zone. Before the sun settles into its main sequence steady hydrogen burning it will have gone through a brief, but energetic T Tauri phase. This T Tauri phase is characterized by a strong highly magnetized wind. This phase in stellar development is the end of the solar nebula blasting primordial dust and gas out of the solar system and operates on a time scale of several million years [60]. During this process the strong magnetic wind will result in a unipolar dynamo heating mechanism creating a significant heating effect on the internal structure of bodies subject to that wind [61, 62, 63].

The T Tauri wind is taken to be a fully ionized plasma with velocity v. This plasma will have a magnetic field denoted as B, configured in such a way that the electric field E = 0 within the wind’s rest frame. Any object moving through this wind, such as a vulcanoid, will experience an induced motional electric field given by,

= × (3.15) 푣 퐸푚 − � � 퐵 푐 This electric field will lead to a current within the body and therefore ohmic dissipation which will lead to a heating effect. An upper limit to this heating effect is given by,

= (3.16) 2 훤 휎퐸푚 where is the electrical conductivity of the asteroid material. Under ideal conditions

Menzel휎 and Roberge [63] state that internal heating will vary as,

= / (3.17) 2 2 −1161 푇 42 훤 푣 퐵0 푒

3 where is the internal heating (W/m ), v is the wind velocity (km/s), B0 is the magnetic field (G훤auss), and T is the temperature. This temperature will be consistent with the results of equation 3.1. The magnetic field of the sun is extremely intricate requiring complex magneto-hydrodynamic calculations to find accurate magnetic field results.

Such simulations are beyond the scope of this work. Other methods such as approximating the sun’s magnetic field as a dipole can be employed, but ultimately even that would seem to be needless complication. Menzel and Roberge and their magneto- hydrodynamic simulations [63] argue for a near constant field strength of 0.3 Gauss out to a distance of several AU, much further than is of interest for the vulcanoids. As such, this value is adopted for the magnetic field strength.

This heating effect is plotted as a function of internal heating versus object distance in figure 3.7 for varying wind velocities. The typical velocity is believed to be between 0.1 and 1 km/s with short-lived burst activity with velocities as large as 10 km/s. The narrow box on the far left of the diagram shows the extent of the vulcanoid region. From this we can see the effect of (idealized) induction heating would be between 10-4 and 10-2 W/m3 for typical wind velocities, reaching as high as 1 W/m3 for the 10 km/s outburst activity. Once the sun leaves its T Tauri phase its surface magnetic field drops by a factor of at least 103 and at this point this induction heating mechanism no longer has any appreciable effect on the solar system.

The interior heating of an asteroid can be found by calculating the total energy deposited within. This will be given by ΔQ = Δt and with a given heat capacity C and

훤

43

100

10-1 V= 10 km/s Vulcanoid zone 10-2

) -3 3 10 V= 1 km/s

26Al, t = 0.0 Myr 10-4 Heating (W/m

-5 26 10 V= 0.1 km/s Al, t = 2.0 Myr

26Al, t = 4.0 Myr 10-6

10-7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 R(au)

Figure 3.7 – Induction Heating The induction heating effect (W/m3) during the sun’s short lived T Tauri phase for the inner solar system. Expected wind velocities are between 0.1 and 1 km/s with outburst activity of 10 km/s is expected for shorter periods. Ambient field strength is 02 Gauss. The vulcanoid zone is presented as the narrow red box on the left-hand side of the diagram. Heating effects due to the radioactive decay of Aluminum 26 are presented as a sequence of blue horizontal lines, at different lengths of time from the solar system’s formation. This corresponds to planetesimal formation and then roughly 2 and 4 half-lives later.

44 material density ρ, the resulting temperature change will result in an asteroid temperature of Tf this value is given by,

Q = (3.19)

Δ 휌휌�푇푓 − 푇�� It is possible to calculate the time it would take to drive an asteroid to its melting point using typical values for fayalite, see table 3.1. For fayalite this melting point is ~1800 K.

For a wind velocity of 0.1 and 1 km/s it would take of order 106 and 104 years, respectively, to reach this melting point. It is important to note that these values are derived for an idealized situation, and the true value would be somewhat lower. Despite that, these heating values are still orders of magnitude larger than the values found from the decay of 26Al. This suggests that vulcanoids may very well have a differentiated interior soon after formation. Such structure will have a significant effect on the evolution of vulcanoid populations, discussed in further detail in section 5.2. What this means is that by the time of the Late Heavy Bombardment (at a solar system age of ~800 million years) it is reasonable to expect that the majority of primordial vulcanoids would possess a differentiated interior structure.

45

Chapter 4: Results

4.1 Parameters

There are a number of key parameters that have a significant effect on the lifetime and end state for test vulcanoid bodies. The four most important parameters identified are; initial radius, initial orbital position, obliquity, and composition material.

The initial position and radius is set at time zero, the formation of the solar system. The initial radius and initial orbital position can be simulated over a large interval. For the simulations presented here initial radius varies from 500 m to 50 000 m, and initial orbital radius varies over the entire vulcanoid zone, as has been defined earlier, from

0.06 AU to 0.25 AU. The material and obliquity values are handled differently. As discussed previously three materials have been chosen as indicative of a broader class of possible materials, iron, fayalite, and regolith. Each of these materials was modelled separately and their results will be presented individually for each situation of interest.

Wherever possible the gradients in figures are color coded to the material used; with blue corresponding to iron, red to fayalite, and green to regolith. Finally obliquity, while this parameter can vary over a range from 0° to 180°, this creates an extremely large parameter space that does not provide significant probative value. Instead, three different obliquity situations will be presented. First obliquity fixed to 45°, this provides an

“average” increase to semi-major axis effect balancing the impact of the positive diurnal and negative seasonal Yarkovsky effect. Second obliquity fixed to 135°, this is a situation reflecting the “average” decrease effect to semi-major axis balancing the diurnal and seasonal Yarkovsky effects which are both negative in this configuration.

46

Finally the obliquity is allowed to vary randomly over its full range (0° to 180°) with a new random value chosen at every timestep (1 Myr) to simulate the more realistic situation of orbital realignment due to collisions and other effects over its lifespan. This random approach provides a less “smooth” dataset due to the fact that the initial random values chosen can have a significant impact on the body’s end state. However this approach still provides the most realistic result available.

Simulations were performed to determine the survival lifetime for vulcanoids possessing the various parameters discussed. See appendix for a copy of the code used.

Simulations were run until one of several conditions was reached, indicating the test body had reached the end of its lifetime or our simulations no longer reflect the physical reality of the situation. Simulations were terminated under four end states. First, simulations were terminated after 10 Gyr because after this point the sun leaves the main sequence swelling into a red giant. At this point our luminosity assumptions cease to hold and more importantly the outer envelope of the sun will expand and subsume the entire vulcanoid zone rendering any further analysis pointless. Secondly, simulations were terminated when the test body had sublimated away to zero, or negative, radius, after all once the test body is gone there is nothing left to simulate. In reality several key assumptions made start to break down as the size of the body approaches zero, but once the test body had reached such a small size it would not be capable of surviving for very long regardless with various radiative effects becoming extremely efficient at removing small objects from the vulcanoid region. Zero is chosen as a convenient endpoint rather than an arbitrary larger value. The final two end states are determined by the vulcanoid region; if the semi-major axis of the body exceeds 0.25 AU or decreases below 0.06 AU

47 the simulation is terminated. This is due to the fact that objects will not be able to persist on a stable orbit outside the vulcanoid zone for any significant length of time. Outside of

0.25 AU gravitation interactions with Mercury will dominate orbital characteristics, and inside 0.06 AU objects that close to the sun will sublimate extremely quickly and be subject to other gravitational effects, i.e. tidal forces, which will quickly act to destroy it.

Results will be presented in the next sections divided by obliquity values used in the simulations with plots presenting the lifetime over the full initial position and initial radius parameter space with separate plots presented for each material. The color gradient used in figures changes color steps at 1 Gyr increments. The transition from greyscale to color indicates the center of the survival lifetime, i.e. 5 Gyr. This roughly corresponds to the current age of the solar system of 4.57 Gyr. As such this can be used as a rough guideline. The sections of the figure that are colored correspond to survival lifetimes equal to or less than the current age of the solar system and greyscale sections correspond to survival lifetimes greater than the age of the solar system.

4.2 Obliquity of 45°

When the obliquity is set to this angle the diurnal Yarkovsky effect acts to increase the semi-major axis of the test body, while the seasonal Yarkovsky effect and

Poynting-Robertson drag act to decrease it. In this configuration the diurnal effect is stronger than the negative effects for most conditions leading to a predominant increase in semi-major axis.

48

Figure 4.1 – Orbital Position vs Time at 45° This figure plots the orbital position of several test bodies corresponding to 0.06 AU (solid lines) and 0.25 AU (dashed lines) starting positions for all three materials. The horizontal dotted lines correspond to the edges of the vulcanoid zone. The vertical dotted line corresponds to the current age of the solar system. Note that iron survives regardless of starting position. Regolith survives over the lifespan of the sun when starting on the inner edge. Fayalite does not survive to the current age of the solar system for any initial position in this configuration. Regolith and fayalite on the outer edge is lost immediately.

49

At an obliquity of 45° iron simulations possess a surprising result; iron vulcanoids will survive for over 10 Gyr for all initial positions and sizes. The combined drag effects force the position of the body to a roughly central region of the vulcanoid zone, pushing it inward from the outer edge and outward from the inner edge.

Furthermore it is quickly pushed away from that inner edge before any significant amount of sublimation can take place. The usual three-dimensional surface contour plots that will be common throughout the rest of this chapter would not be appropriate for this configuration. Such a figure would simply be a flat plain, instead figure 4.1 presents the variation in orbital position of test bodies over time. Results are presented for a 5 km radius initial size. Fayalite and regolith are plotted as well to serve as a comparison. Note that iron beginning at either extreme travels towards the center of the vulcanoid region, never leaving over the main sequence lifespan of the sun.

Figures 4.2 and 4.3 present the contour plots for material compositions of fayalite and regolith. In this configuration the most significantly depleted region of the contour plots correspond to initial positions closest to the edges of the vulcanoid zone and initial radii closest to zero. This result should not be surprising since obliquities in this configuration cause fayalite and regolith to drift outwards, which will quickly push bodies on the outer edge into gravitational interactions with Mercury. Bodies with small initial radii are more severely affected by radiative pressure effects and are also quickly pushed out of the vulcanoid zone, for the smallest radii even when initially located on the inner edge.

50

Figure 4.2 – Fayalite Lifetime – 45° This figure represents the survival lifetime of a fayalite vulcanoid over a range of initial radii and initial positions with an obliquity of 45°. Note the upper edge of the bottom figure where initial positions close to the outer edge of the vulcanoid region allow test objects to be rapidly pushed out of the zone and into interactions with Mercury. The left edge of the same figure reflects small radii that allow radiative drag effects to quickly clear the objects.

51

Figure 4.3 – Regolith Lifetime – 45° This figure represents the survival lifetime of a regolith vulcanoid over a range of initial radii and initial positions with an obliquity of 45°. Note the similar shape and features from figure 4.2. The lower thermal conductivity slows radiative drag effects and allows longer lifetimes against clearing for regolith compared to fayalite.

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When the obliquity is 45° simulations show that sublimation is not a significant effect.

Sublimation only occurs to any significant degree on the inner edge of the vulcanoid zone, but this configuration acts to push objects away from this sublimation zone. As such the majority of bodies remain the same size throughout their lifetime. Small objects are very rapidly pushed into the cooler outer regions and as such suffer very little sublimation. Larger bodies remain closer to the sun for longer and experience more sublimation, however they are still pushed into cooler regions over time and the mass that is lost is a relatively small percentage of their initial mass. All of these effects conspire to ensure that no significant sublimation occurs for vulcanoids of any size with an obliquity of 45°.

4.3 Obliquity of 135°

When the obliquity is set to 135° both the diurnal and seasonal components of the

Yarkovsky effect act to decrease the semi-major axis of vulcanoids, this causes them to drift in towards the sun. This configuration is something of a worst case scenario, with all effects acting together to drive objects towards the sun. In reality such a dire situation is not very likely but this serves as a sort of lower limit, illustrating these detrimental effects all working together.

Figure 4.4 shows both the positions and sizes over time for a test body of 5 km initial radius. This figure serves to illustrate the general trends found over the full parameter space. Fayalite has the shortest lifetime of the materials examined. This is largely due to the rapid clearing time fayalite experiences due to the various radiative drag effects. Fayalite test bodies rapidly spiral into the inner vulcanoid zone where they

53

Figure 4.4 – Orbital Position and Size at 135° This figure shows the position and size for various test bodies over time with an initial radius of 5 km. Solid lines indicate an initial position of 0.10 AU, dashed lines 0.18 AU, and dotted lines 0.25 AU. Green, blue, and red represent regolith, iron, and fayalite respectively. The horizontal black dotted lines indicate the edges of the vulcanoid zone and the vertical dotted line indicated the current age of the solar system.

54 sublimate away very quickly. Iron has intermediate survival lifetimes. It is better able to resist radiative drag compared to fayalite, but it is more vulnerable to the effects of sublimation. As such while it takes longer to reach the inner area of the vulcanoid zone it sublimates more quickly than fayalite at the same orbital radius. However the greater drag forces acting on fayalite cause it to move to closer, and hotter, regions much more quickly leading to a longer relative lifespan for iron. Regolith has the longest survival lifetime by a significant margin. As can be seen in figure 4.4, the innermost initial position for regolith (0.10 AU) has a longer lifespan than fayalite starting on the exterior edge of the vulcanoid zone. Furthermore regolith starting at 0.10 AU has a longer lifespan than all except iron originating from the outer reaches of the vulcanoid zone.

This long lifespan is largely because of regolith’s resistance to the Yarkovsky effect due to its extremely low surface thermal conductivity. This acts to delay test bodies travelling into the inner vulcanoid zone where sublimation effects take over. Regolith is able to survive until the current age of the solar system, and frequently over the lifespan of the sun, over most of the parameter space.

Figures 4.5, 4.6, and 4.7 present vulcanoid survival lifetimes for iron, fayalite, and regolith respectively for an obliquity of 135° over the full parameter space. As can be seen even under the condition of a 135° obliquity there are many situations for which vulcanoids can survive until the current age of the solar system, and even over the sun’s entire lifespan. Iron, for example, can survive in the 0.25 AU region until the current age of the solar system for all initial radii above ~4.5 km. The situation is somewhat bleaker for fayalite. It requires large initial radii and large initial distances in order to survive. It is only possible for a fayalite vulcanoid to survive until the current age of the solar

55

Figure 4.5 – Iron Lifetime – 135° This figure represents the survival lifetime of an iron vulcanoid over a range of initial radii and initial positions with an obliquity of 135°. Clearing effects are not as significant for iron, but an iron composition is much more sensitive to sublimation. As such it does not travel to the inner vulcanoid zone as quickly as fayalite, but it sublimates more quickly once it is there. This leads to an intermediate survival lifetime compared to fayalite and regolith.

56

Figure 4.6 – Fayalite Lifetime – 135° This figure represents the survival lifetime of a fayalite vulcanoid over a range of initial radii and initial positions with an obliquity of 135°. Fayalite rapidly spirals into the inner vulcanoid zone where it quickly sublimates away. It has the shortest lifespan of the three materials.

57

Figure 4.7 – Regolith Lifetime – 135° This figure represents the survival lifetime of a regolith vulcanoid over a range of initial radii and initial positions with an obliquity of 135°. Due to its resistance to radiative drag effects regolith takes the most time to approach the inner vulcanoid zone, as such it has a significantly longer lifespan than the other two materials examined.

58 system if it originated from the outer half of the vulcanoid zone and even then only for large initial radii. Regolith has a long lifespan over most of the parameter space. Only the smallest and nearest vulcanoids are lost before the current age of the solar system.

Referring to figure 4.7, the main losses come from the lower and left-most edges of the contour plot. Vulcanoids of those extremely small sizes and extremely close positions to the sun are lost regardless of the resistance of regolith. Even in this grim situation the vast majority of regolith vulcanoids could survive for the entire lifespan of the sun.

4.4 Random Obliquities

Randomly varying obliquities is the most realistic of the situations presented in this chapter. At every time step in the simulation (1 Myr) a new obliquity is generated somewhere between 0° and 180° in order to simulate reorientation of the rotation axis due to collisions or other disruptive activity. End states are highly variable depending on obliquities generated and this leads to a certain amount of “jitter” to the data received, see figure 4.8. Fayalite suffers the largest amount of variation in this random approach due to its sensitivity to the Yarkovsky effect. In just the three simulations used to generate figure 4.8 the lifespan of the fayalite test body varied by ~1 Gyrs. Iron by comparison varied by ~100 Myrs.

An interesting effect of this random approach is that the shape of the position over time graphs closely mirror those that are obtained from setting the obliquity to 90°, an angle where the diurnal Yarkovsky effect is shut down and only seasonal effects persist. This is reasonable considering that the average angle between 0° and 180° would be 90°. However as figure 4.9 illustrates the effect is not exactly the same. The random

59

Figure 4.8 – Variability Due to Randomized Obliquity This figure plots the position and size of test bodies over time. All test bodies use the same initial position, 0.15 AU, and initial object radius of 5 km. At each time step a new obliquity is randomly generated. For each material three different simulations were run and their results plotted here using the same color. As can be seen the greatest variation between simulations is found for fayalite with a range of lifetimes varying over ~1 Gyrs. Iron’s lifetime varies over ~100 Myrs. Regolith remains in the vulcanoid zone over the lifespan of the sun with very little change in its semi-major axis.

60

Figure 4.9 – Comparison of Maximal Seasonal Effect to Randomized Obliquity This figure plots the position and size of test bodies over time. All test bodies use the same initial position, 0.15 AU, and initial object radius of 5 km. This figure compares the solid lines with randomly varying obliquities and the dashed lines where the test bodies had a set obliquity of 90°. The random variations have longer survival lifetimes but possess similar relative relationships with the other materials of the same obliquity. Iron has the shortest lifespan, followed by fayalite, and finally regolith survives for the full age of the solar system.

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Figure 4.10 – Yarkovsky Effect Strength vs Obliquity Angle This figure plots the relative strengths of the two forms of the Yarkovsky effect on all three materials at a heliocentric distance of 0.15 AU. Diurnal effects are plotted with a solid line, Seasonal effects with a dashed line. Note that the diurnal component depends of cos( ) while the seasonal component is a sin2( ) dependency, where is the obliquity angle. Note that 훾 the diurnal component for fayalite is much stronger than the other two materials. 훾 훾 Similarly the seasonal component for regolith, while non-zero, is small enough that it cannot be seen at the same scale as all other components. Also note that the effect is symmetric about 180°.

62 variation serves to extend the lifespan of all materials compared to the 90° simulations.

This is an interesting effect and is due to the positive and negative portions of the diurnal

Yarkovsky effect, on average, cancelling one another out. At the same time while the diurnal forces are working against one another they are decreasing the impact of the seasonal effect whenever the angle moves away from 90°. This causes an object to follow a similar path that it would take when it is fixed at an obliquity angle of 90°, while taking longer to make the journey due to the periods when the seasonal effect cannot act. See figure 4.10 for a comparison of the relative strengths of the Yarkovsky effect. It is also interesting to note that the survival times for iron are lower than the other materials contrary to what is seen in the previous two sections. This is because the seasonal effect has a greater negative impact on iron compared to the other two materials, because the net effect of the diurnal components cancels out on average in the random simulations the seasonal component becomes dominant. It is also important to note the extremely large amplitude of the diurnal effect on fayalite. While the net effect is canceled out on average, the large sudden swings in semi-major axis lead to the extreme “jitteriness” seen in the fayalite simulations.

All of this allows us to generate graphs examining the full parameter space in a similar manner to earlier sections. Figures 4.11, 4.12, and 4.13 show the survival lifetimes for vulcanoids over the entire parameter space. Note the undulations seen in all of these figures. This is due to the random variations already discussed. As such the contour plots do not have well established borders to transition between colors, and therefore survival lifetimes. Enough iteration of random simulations could provide a more exact “average” value, but even with the “fuzziness” of the edges they are clear

63

Figure 4.11 – Iron Lifetime – Random Obliquity This figure represents the survival lifetime of an iron vulcanoid over a range of initial radii and initial positions with an obliquity that varies randomly at every time step in the simulation. Note that iron has the poorest survival prospects of all materials for random obliquities. This is due to iron’s susceptibility to the seasonal Yarkovsky effect that will dominate the average change in semi-major axis. This causes iron to spiral towards the sun, and greater sublimation rates, compared to the other materials.

64

Figure 4.12 – Fayalite Lifetime – Random Obliquity This figure represents the survival lifetime of a fayalite vulcanoid over a range of initial radii and initial positions with an obliquity that varies randomly at every time step in the simulation. Note the rapid variation in lifetime from point to point leading to undulations throughout the graph. This is due to the large swings in semi-major axis inherent in a fayalite composition subject to the diurnal Yarkovsky effect.

65

Figure 4.13 – Regolith Lifetime – Random Obliquity This figure represents the survival lifetime of a regolith vulcanoid over a range of initial radii and initial positions with an obliquity that varies randomly at every time step in the simulation. Note that nearly the entire parameter space is able to survive for the lifespan of the sun. The seemingly low lifespan on the upper edge of the figure is due to boundary effects in the simulation. Regolith is only weakly affected by the seasonal Yarkovsky effect, but much more strongly affected by diurnal variations. As such even after numerous obliquities maximizing the seasonal effect it may only take a single value increasing the diurnal component to terminate the simulation.

66 enough to illustrate the general trends to be discussed. Another feature to discuss is the upper edge of the contour plots, along the 0.25 AU axis. There are various points along this axis that have a survival lifetime of near 0 years that show up as a series of colored dots in the plot. This is due to random variations along that upper edge. If one of the first obliquity values generated acts to increase the body’s semi-major axis it can immediately leave the vulcanoid zone and that simulation would be terminated. This is not an effect actually reflecting physical reality, but instead an interaction of the simulations with the imposed boundary conditions.

These random simulations provide the most optimistic picture for the vulcanoid zone of all the situations presented thus far. For all materials the majority of the parameter space allows vulcanoids to survive not just to the current age of the solar system, but for the entire lifespan of the sun. Under the assumption of random obliquity variation, iron possesses the shortest survival lifespan. This is due to the previously discussed effect of the greater impact the seasonal component of the Yarkovsky effect has on iron compared to the other materials. Fayalite falls somewhere in the middle for survival times, with significant variation due to the strong effect the diurnal component has on fayalite. Finally regolith possesses an incredible survivability for nearly all initial conditions.

A regolith asteroid with an initial radius as small as ~250 m and ~500 m would be able to survive to the current age of the solar system and for the entire main sequence lifespan of the sun, respectively, for all but the smallest initial positions. Similarly, a regolith asteroid with an initial orbital position of just ~0.065 AU and ~0.08 AU would be able to survive to the current age of the solar system and for the entire main sequence

67 lifespan of the sun, respectively, for all but the smallest initial radii. Combining limiting factors in both size and position, a vulcanoid with an initial radius of only ~2 km and an initial position of ~0.075 AU could survive until the current age of the solar system.

Similarly a body of initial radius of ~3.5 km and initial position of ~0.095 AU would be able to survive until the sun swells into a red giant and subsumes the vulcanoid zone.

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Chapter 5: Future Work

5.1 Material Refinements

The chosen materials modelled were chosen as exemplars of a broad class of materials, specifically iron, rock, and regolith, but in the presence of new evidence in the composition of the vulcanoids simulations could be rerun with more specificity. Even for the materials chosen refinements can be made in the various material specific parameters evaluated in other research, such as refining numbers for the specific temperature ranges found in the vulcanoid zone.

The largest area for improvement comes from the values used for regolith. Many assumptions needed to be made for the inclusion of regolith and these assumptions are largely untested. Even the values that can be found in existing literature are poorly defined, only offering values to an order of magnitude. Such measurements would be difficult to perform and require experimental analysis of regolith samples under various conditions. Such an experiment would provide the single greatest refinement to this work.

5.2 Internal Vulcanoid Structure

Vulcanoids are modelled using a simple single material spherical body in the simulations in this thesis. In the real world vulcanoids would likely have a more complicated structure. As discussed in the induction heating section, section 3.6, there is the distinct possibility that vulcanoids would have a differentiated internal structure possessing something like an iron core surrounded by rock. Another use for a composite

69 structure would apply to the modelling of regolith. The current model uses a body made purely of regolith which is not a practical structure in reality. Any body that is covered in regolith would possess a solid core likely composed of a stony material.

As it stands modelling such structure would add significant computation complexity and offer little probative value. While it is likely that internal structure of some variety exists, exactly what form it would take is almost entirely unknown.

Modelling such structures would be little more than speculation at this point, creating an extremely large parameter space. With current results in mind, the composite structure should fall within the limits imposed by the pure compositions already explored. In future, with a more complete understanding of vulcanoid structure, simulations could be developed for any compelling internal structure.

5.3 Rotational effects

As discussed in section 3.5 model results make an assumption of a rotation rate of 5 hours. In reality the details surrounding rotation rate are much more complicated and can be explored in more detail.

The first phenomenon that could be explored is rotational bursting behaviour. As the spin rate of an object increases it will eventually pass some critical value past which its structural integrity fails and it rips itself apart, creating a large debris cloud. In the situation of the vulcanoids this offers a potential detection mechanism, such destruction would create a large debris cloud in a large sunlight reflecting volume of space [18]. In the limit of a strengthless body, forces can be analysed where the force of gravitational

70 attraction is balanced by the force caused by centripetal acceleration. This critical priod is given by,

3 = (5.1) 휋 푃퐶𝐶퐶 � 퐺� where PCrit is the critical period with smaller periods causing bursting behaviour, G is the gravitational constant, and ρ is the density. Critical periods for iron and fayalite compositions are 1.2 and 1.6 hours, respectively. The spin up timescale [64] can be found for a non-symmetric body from,

= = (5.2) / 푃 퐼� 푇푆푆𝑆 푈� 푑� 푑� 푁 where TSpin Up is the spin up timescale, I is the moment of inertia, is the angular velocity ( = 2π / P), and N is the mass loss induced torque. Assuming훺 the spun up object is a훺 sphere with I = 2 M R2/5 gives the result of,

8 = (5.3) 9 ( 4/ ) 휋 훺�푅 푇푆푆𝑆 푈� � � 푉�푇 푑푑 푑� where KT is the dimensionless moment arm of the torque (KT = 0 for radial mass loss, and KT = 1 is extreme tangential mass loss.) and V is the sublimation outflow velocity.

Jewitt [65] suggests that KT is of order 0.05, and V is assumed to be ~200 m/s. With an

71

4 initial spin period of 5 hours, the spin up timescale is TSpin Up (yr) ~ 10 / (dM/dt), where the mass loss is expressed in kilograms per second. Assuming a helocentric distance of

0.06 AU, what has been defined as the inner edge of the vulcanoid zone, and through numeric integration of equation 3.3 a mass loss rate of ~10-3 and ~10-5 kg/s for iron and fayalite, respectively, this corresponds to a spin up time of ~107 years for iron and ~109 years for fayalite. As such, it can be concluded that catastrophic bursting is a possible end state for vulcanoids on the internal edge of the vulcanoid zone. Granvik et al. [66] has suggested that the fewer than model expected near earth asteroids with perihelia less than 0.1 AU may be due to this catastrophic bursting.

The mass loss induced torque is only a significant effect at extremely small heliocentric distances, as bodies inhabit the more distant reaches of the vulcanoid zone thermal photon spin up becomes a more significant effect. This thermal photon effect is known as the Yarkovsky–O'Keefe–Radzievskii–Paddack (YORP) effect [54, 64, 67] and is a more complicated version of the simpler Yarkovsky effect already discussed in section 3.5. For the Yarkovsky effect a spherical symmetry is assumed and as such all force is directed radially with respect to the body’s center of mass. This has the effect of altering the position of the body relative to the sun but it does not have any effect on the rotation rate of the body. The YORP effect considers objects with more complicated geometry. The reemission of thermal photons for an extended object do not necessarily have force vectors passing through the center of mass of the body. This leads to a net torque on the body causing a change to its rotation rate. As with the Yarkovsky effect over short periods of time the YORP effect would have little impact; it is only over extended amounts of time that the YORP effect would have any significant effect.

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Rubincam [67] observes that for slow rotators the YORP effect can induce random tumbling modes that would shut down the Yarkovsky and YORP effect due to the rapidly changing obliquity and could have important consequences for the orbital evolution of objects located near the outer edge of the vulcanoid zone. Due to the lack of any specific vulcanoid shape model the YORP effect has not been examined in any great detail, but with new developments this is a promising area of future work.

The final cause of rotational modification would be due to collisional activity between bodies within the vulcanoid zone. Collisions could have the effect of randomly varying rotational characteristics which would have a significant impact on orbital evolution. Collisions will be discussed in more detail in the following section.

5.4 Collisional Evolution

For any population of objects possessing similar initial orbits collisions are a likely component of their evolution. These collisions have been examined in some detail in the past [25]. Additionally vulcanoids may be subject to collisions involving cometary nuclei with small perihelion distances [18, 66]. In an earlier epoch it is also possible the vulcanoids underwent collisional evolution due to material associated with the Late

Heavy Bombardment and, if our solar system experienced evolution similar to observed exoplanetary systems known as Tightly-Packed Planetary (TPP) systems, then material ejected in this process could have played an important collisional role.

Sturn and Durda [25] give the characteristic encounter velocity between two vulcanoids as,

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0.1 ( ) = 180 < > (5.4)

Where V is the encounter푉𝑒� velocity,푘�⁄푠 is the푒 average� orbital eccentricity, and a is the enc � semi-major axis. For objects in the vulcanoid zone (0.06 AU < a < 0.25 AU) it can be seen that collisional velocities are going to be extremely high, of order 2 km/s on the inner edge and 1 km/s on the outer edge. These encounter velocities are much larger than the escape velocities for objects of this size. Escape velocities are given by,

2 = (5.5) 퐺 � 푉𝑒� � 푅

For a 10 km asteroid (assuming spherical volume and uniform density this gives a mass of 4.1×1015 kg and 2.3×1015 kg for iron and fayalite, respectively) escape velocities are

~10 m/s and ~8 m/s, respectively, as such it is expected that collisions will be highly erosive, with little chance for ejected material to fall back to the impact surface.

This would seem to argue against vulcanoids developing extensive regolith on their surface, however collisions are not the only mechanism to produce regolith.

Specifically thermal fatigue and fragmentation would be a highly efficient mechanic within the vulcanoid zone due to high temperatures and large rapid changes in temperature. Delbo et al. [68] have argued that at 1 AU from the sun that the survival lifetime of a 10 cm boulder against thermal cracking could be as short as only one thousand years. Such a regolith layer has an important effect on Yarkovsky drift as has been discussed in section 3.5, but this regolith also has a significant effect upon collisional evolution.

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Sturns and Durda [25] have performed sophisticated numerical simulations modelling collisional activity of the vulcanoids over the age of the solar system.

Assuming initial populations and material strength properties demonstrated that collisions were highly efficient at clearing the vulcanoid zone of the majority of objects, however it is important to note that all of their results showed a remaining population of a few kilometer sized vulcanoids in the present day. Advancements have been made in the years since this analysis was performed, specifically in our understanding of the role things like regolith play in collisions. It is now understood that objects possessing deep surface regolith and a high degree of internal porosity can survive impacts of much higher energies than objects possessing a monolithic structure. A notable example of this phenomenon is the analysis performed by Syal et al. [69] examining the survival of the

22km moon of Mars, Phobos, against the impact of its largest crater, the 9 km diameter crater Stickney. In summary, they find that the relatively high porosity allowed Phobos to survive complete destruction from what would otherwise have been a catastrophic impact.

The situation for vulcanoids becomes somewhat more complicated due to their proximity to the sun. While the inner edge of the vulcanoid zone, 0.06 AU, is outside of the Roche limit of the sun, the sun’s tidal forces will have an impact on objects that close. Granvik et al. [66] argue that a possible explanation for the small numbers of low- albedo near-earth asteroids is due to catastrophic destruction due to tidal forces once their perihelion distances drift below 0.06 AU. This catastrophic tidal disruption may be an important effect on fragmentary bodies located near the inner edge of the vulcanoid

75 zone. This effect would only become more pronounced with the rapid spin-up timescales found on this inner edge, recalling section 5.3.

With this understanding of the deleterious effects of collisional activity the earlier presented vulcanoid lifetimes are almost certainly longer than the true values.

Sublimation will not be the only effect causing mass loss, collisions will be another highly efficient mass loss mechanism. The greater mass loss will necessarily lead to shorter survival lifetimes. However the exact role collisions will play is extremely poorly constrained at present, it will decrease the lifespan but the exact amount would require further analysis.

In summary, many of the prior analyses examining the collisional history of the vulcanoids could be re-examined with the greater modern understanding possessed today especially in regard to the role regolith would play in any such collisions. In general the full collisional details are now understood to be a more complicated situation than was examined in the past. Even in these more dire conclusions reached by past results they have not conclusively ruled out the survival of vulcanoids, even if they limited them to a mere handful of asteroids. Reanalysis would likely leave the possibility of a more extensive, but still relatively small vulcanoid population. This would be a promising, but computationally intensive, avenue to explore in future research.

5.5 Dust Rings of the Inner Solar System

The presence of a ring of dust in the path of the earth’s orbit has been known for some 20 years, but it is only within the past few years that similar structures have been found and examined for the orbits of Venus [70] and more recently Mercury [71]. These

76 dust populations, as a whole, are likely long lived structures, however individual dust grains would have a rather short survival lifespan. As has been examined in this thesis effects like Poynting Robertson drag and the Yarkovsky effect, among others, are extremely efficient clearing mechanisms for small bodies in the inner reaches of the solar system. As such in order for the dust populations that we observe in the modern day to exist there must necessarily be some mechanism to replenish the dust therein. While there are other possible responsible mechanisms, it cannot be ruled out that the vulcanoids and mass loss from them do not play a role. An area of future study would be an examination of the effect and possible role the vulcanoids could play in this emerging research, these rings of dust.

5.6 Exoplanetary Application

In our solar system there are no large bodies close enough to the sun that sublimation mass loss would have a significant effect. One of the most active areas of astrophysics research today is the search for exoplanets. As our knowledge of exoplanetary systems grows, more diverse planetary arrangements are discovered. Many exoplanetary systems known today include planets closer to their parent star than the vulcanoid zone is located relative to our sun. Previous work has been done on this topic and a paper has been published by this author [56] examining sublimation mass loss for

Alpha Centauri Bb. While the existence of this specific planet has since been disproven, the techniques applied in the sublimation analysis are still valid. Similar examinations could be performed on numerous exoplanetary systems of interest.

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5.7 Red Giants, White Dwarfs, and Planetary Remnants

In a manner similar to techniques discussed in the previous section about exoplanetary research, work could also be done examining remnant planets orbiting white dwarf stars. Research has been done looking for planetary remnants orbiting white dwarfs [72, 73, 74]. These planetary remnants would have survived the volatile conditions and high temperatures found during the parent star’s red giant phase, the transition between main sequence star and its end state as a white dwarf. If a planet were to survive such conditions it would likely undergo significant sublimation mass loss.

Techniques similar to those used in this analysis could be adapted to examine the survivability of planets under such extremely harsh conditions. Possible remnants could include things like metallic cores of terrestrial planets or even cores of more massive gas and ice giants. Such an analysis would allow us to effectively turn back the clock and examine what such a system would have looked like before the catastrophic end stages of its lifespan. Perhaps giving us insight into what our solar system may look like once our own sun evolves into a white dwarf.

5.8 Migration

Up to this point in this analysis any vulcanoids have been assumed to originate from a primordial source at the formation of the solar system. There is no strict reason that vulcanoids must only originate from such time. The vulcanoid region is stable and would be capable of capturing other bodies within the solar system as they migrated inwards. Such migratory bodies could be the only objects currently in the vulcanoid

78 region, or they could supplement a primordial population. Once in the vulcanoid zone such bodies would be subject to the same radiative forces and thermal effects presented earlier. Such an analysis could be adapted to latter inclusions to model their behaviour as well. It must be acknowledged that such a capture is rather unlikely, but cannot be ruled out entirely. On timescales of billions of years, even unlikely things happen occasionally.

5.9 Direct Detection

Ultimately the single largest advancement in research regarding vulcanoids would come from the direct detection of an actual asteroid in the vulcanoid zone. Past work has been able to put upper limits on the size of vulcanoids today, but have not detected an actual vulcanoid [4, 10, 11, 12, 13, 25]. Such a detection would allow several of the extremely broad parameters used in this research to be narrowed considerably, refining results in numerous ways. Work is ongoing using existing and proposed orbital platforms to attempt to directly measure vulcanoids [75, 76].

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Chapter 6: Conclusions

The search for Vulcan and then the vulcanoids has been ongoing for roughly 150 years without any detection. While this is not an encouraging result, it is not entirely surprising. The vulcanoids, should they exist, inhabit an area of the solar system notoriously difficult to examine. In fact, outside of rare events like solar eclipses, historically the region was completely unexplored. As such, while no definitive result has been found so far, it is equally true that no result rules out the vulcanoids completely.

Current observations place strong constraints on the region that limit the possibilities.

Should there be asteroids present in the region they could not be numerous and observations indicate there could not be more than a handful of approximately 10 km sized asteroids at the present epoch. This still leaves a significant range of possibilities where vulcanoids could yet be hiding, waiting to be discovered.

While it is true that past work has demonstrated that collisions would have devastating effects on theoretical vulcanoid populations that does not tell the whole story. Many effects discussed in this work such as radiative drag and thermal effects play a key role in the survival or destruction of any body within the vulcanoid zone.

Furthermore, more modern analysis of collisional impacts reflects the importance of considering a porous structure for a body involved in a collision, e.g. Phobos. Past work has largely modelled vulcanoids as simple rocky bodies when reality could be much more complicated, with structures composed of iron and other metals, extensive regolith layers, which would act to cushion impacts and delay Yarkovsky drift, or even a

80 combination of all of these things with differentiated structures combining metallic cores with rocky mantles covered in regolith.

One of the largest unexplored areas of research, as it pertains to the vulcanoids, is the exact role regolith plays. In large part even the values used for the various physical constants relating to regolith are poorly understood or are largely assumed. A key to future research in this area will be refinement of our knowledge of regolith and the various thermal and collisional properties that regolith possesses. Regolith and the mechanisms it enacts are a vital part of the behaviour of bodies in the vulcanoid zone and until more research is performed our understanding of the vulcanoid zone will remain incomplete.

The unfortunate reality is that there is more unknown about the vulcanoid zone than there is known. At the present time, there are neither good models nor observational data to guide a researcher when it comes to the spin-evolution of vulcanoids as it relates to analysis of the Yarkovsky effect, nor the even more complicated YORP effect.

Furthermore, there is little to no data regarding material strength, chemical composition, or internal structure that could constrain models in order to answer numerous questions about vulcanoids. All of this means a great deal needs to be assumed and alterations to these fundamental assumptions could lead to significant alterations of resultant modelling. In the analysis presented here three primary materials were discussed. This was not due to any evidence that vulcanoids are composed of these materials; instead these materials serve as exemplars of a broad class of materials, namely metallic, rocky, and fragmentary bodies. Whatever a vulcanoid would be composed of in reality would

81 be similar to one of these materials, but knowledge of true material properties would provide much more refined results.

One of the crucial insights presented in this analysis is the role that the variable luminosity of the sun over its lifespan plays in the evolution of the vulcanoids. The vast majority of analyses conducted in the past assume a constant solar luminosity fixed at the present day value, however in reality the sun was as much as 30% less luminous in its early life. This dramatic shift over time has profound effects on bodies in the vulcanoid zone which are so dependent on radiative drag and thermal effects, themselves wholly dependent upon the luminosity of the sun. With the inclusion of this variable luminosity, the survival lifetime against sublimation for many vulcanoids was increased dramatically over past estimates with far more dire predictions.

In conclusion, nothing in this analysis can state definitively whether vulcanoids exist at the present epoch, nor at any point in the past. However neither does anything presented rule out the possibility either. Analysis relating to regolith and the variable luminosity of the sun paints an encouraging picture for the survival of a broad range of objects. While the upper limit on vulcanoids seems to shrink with every passing year perhaps fate will present a result before that limit reaches zero.

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Appendix: Simulation Code

The code used to generate the survival lifetime plots in chapter 4 is presented below. The code uses the C++ programming language. Other plots were generated using variations on this code.

#include "stdafx.h" #include #include #include #include #include #include using namespace std; using std::cout;

// Subprograms void Luminosity(); //Finds Solar luminosity as a function of time void Flux(); //Find Solar Flux void Temperature(); //Procedure to calculate Temperature void Motion(); //Calculates values for the motion of the body, Period, Rotation, Obliquity void Material(); //Assigns chooses material and assigns material values to the appropriate variables void Mass_Loss(); //find the mass loss due to sublimation void Poynt_Rob(); //calcualte effects of Poynting-Robertson Drag void Yarkovsky(); //Calculates impact of the Yarkovsky effect

// Variables

//Data Type //Value //Definition //Units

double D; //diameter of the body m double R; //Radius of the body m double rad; //orbital radius m double age; //index to advance time through simulation years double Ls; //Luminosity of the sun (W) kg m^2 s^-3 double Fs; //Flux of the sun at a given distance(W/m^2) kg s^-3 double Temp; //Blackbody temp of asteroid K double Period; //Period of revolution s double nuS; //Mean orbital motion (frequency of revolution) (S for Seasonal) radians s^-1 double Rot; //Period of rotation s double nuD; //rotational frequency (D for Diurnal) radians s^-1 double ang; //Obliquity of Asteroid radians double mass; //mass of the asteroid calculated from radius and density assuming a sphere kg double pv; //Vapour Pressure (Pa) kg m^-1 s^-2 double mflux; //Mass loss flux from the surface kg m^-2 s^-1 double drdt; //Mass loss rate m s^-1 double Rlost; //Total radius lost m

88 double Drad; //delta orbital radius, total change in orbital radius m int Rind; //index for looping radius int radind; //index for looping orbital radius double alpha; //accounts for outwardly directed radiation pressure, intermediate calculation m^2 s^-1 double dadt; //change in radius over change in time due to Poynting Robertson m s^-1 double Gamma; //Thermal inertia of the body kg s^-5/2 K^-1 double Phi; //intermediate calculation value, radiation pressure on a spherical body m s^-2 double ThetaS; //Thermal Parameter Seasonal kg^2 s^-9 K^-8 double ThetaD; //Thermal Parameter Diurnal kg^2 s^-9 K^-8 double OmegaS; //intermediate calculation step double OmegaD; //intermediate calculation step double YarkS; //Change in orbital radius over change in time for Seasonal Yarkovsky effect m s^-1 double YarkD; //Change in orbital radius over change in time for Diurnal Yarkovsky effect m s^-1

// Physical and Numerical Constants

//Data Type //Value //Definition //Units const double amu = 1.660539040e-27; //Atomic Mass Units kg const double AU = 1.495978707E+11; //Number of Meters in an Astronomical Unit m const double c = 2.99792458E+08; //Speed of Light m s^-1 const double G = 6.67408E-11; //Universal Graviational Constant G m^3 kg^-1 s^-2 const double kb = 1.38064852e-23; //Boltzmann Constant (J K^-1) m^2 kg s^-2 K^-1 const double Ls0 = 3.828E+26; //Current Luminosity of the Sun (W) kg m^2 s^-3 const double pi = 3.14159265358979; //pi const double stbo = 5.670367E-08; //Stefan-Boltzmann Constant (W m^-2 K^-4) kg s^-3 K^-4 const double year = 3.15576E+07; //Number of Seconds in a Year s

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// Set Constants const double Alb = 0.3; //Albedo set at 0.3 const double eps = 0.9; //Emisivity set to 0.9 const double D0 = 100000; //Initial Asteroid Diameter const double rad0 = 0.25*AU; //Initial Asteroid Position const double dt = 1.0e6; //Time Step set to 1 Myr

//Solar Evolution Constants const double b0 = 7.157248347E-01; const double b1 = 4.01120615E-02; const double b2 = 1.2761239E-02; const double b3 = -2.4116691812E-03; const double b4 = 1.9198252799E-04;

// Material Constants (R. van Lieshout, A&A 572, A76 (2014) table 3)

// 1 - Iron // 2 - Fayalite // 3 - Regolith const double A1 = 48354; //Experimental material property A const double B1 = 29.2; //Experimental material property B const double Mu1 = 55.845; //Mean Molecular Weight in amu const double Al1 = 1.0; //Material specific evaporation coefficient const double rho1 = 7874; //Density of Body in kg m^-3 const double ThCon1 = 80; //Surface thermal conductivity in (W m^-1 K^-1) kg m s^-3 K^-1 const double Cp1 = 500; //Heat Capacity in (J kg^-1 K^-1) m^2 s^-2 K^-1

const double A2 = 60377; //Experimental material property A const double B2 = 37.7; //Experimental material property B const double Mu2 = 203.774;//Mean Molecular Weight in amu const double Al2 = 0.1; //Material specific evaporation coefficient const double rho2 = 4390; //Density of Body in kg m^-3 const double ThCon2 = 3.85;//Surface thermal conductivity in (W m^-1 K^-1) kg m s^-3 K^-1 const double Cp2 = 797.1; //Heat Capacity in (J kg^-1 K^-1) m^2 s^-2 K^-1

const double A3 = 60377; //Experimental material property A const double B3 = 37.7; //Experimental material property B const double Mu3 = 203.774;//Mean Molecular Weight in amu const double Al3 = 0.1; //Material specific evaporation coefficient const double rho3 = 2200; //Density of Body in kg m^-3 const double ThCon3 = 1e-3;//Surface thermal conductivity in (W m^-1 K^-1) kg m s^-3 K^-1 const double Cp3 = 797.1; //Heat Capacity in (J kg^-1 K^-1) m^2 s^-2 K^-1

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//assign general variables to be used for each material selection double A; //Experimental material property A double B; //Experimental material property B double Mu; //Mean Molecular Weight in amu double Al; //Alpha - Material specific evaporation coefficient double rho; //Density of Body in kg m^-3 double ThCon; //Surface thermal conductivity in (W m^-1 K^-1) kg m s^-3 K^-1 double Cp; //Heat Capacity in (J kg^-1 K^-1) m^2 s^-2 K^-1

int Mat; //Material number

//set file label variables int L1; //Obliquity label int L2; //Position label int L3; //Size label int L4; //Material label int Label; //Label int Main() // Main looping program { // construct a trivial random generator engine from a time-based seed: unsigned seed = std::chrono::system_clock::now().time_since_epoch().count();

std::default_random_engine generator(seed); std::uniform_real_distribution distribution(0.0, 2 * pi);

//assign initial values Rind = D0 / 2; radind = rad0;

Material(); //chooses material before looping

L1 = (int)(ang * 180 / (pi)); L2 = (int)(100 * rad0 / AU); L3 = (int)(D0 / 200); L4 = Mat;

Label = /*L1 * 10 +*/ L4; //creates label for filename

//open data file ofstream data;

//assigns unique filename data.open("surfaceR" + std::to_string(Label) + ".csv");

//data file header data << "lifetime vs initial radius" << " Material " << Mat << " Initial Radius " << D0 / 2 << " Initial Position " << rad0 / AU << " Obliquity " << ang * 180 / (pi) << endl;

91 data << "Initial Radius (km)" << "," << "Initial Position (AU)" << "," << "Survival Time (Gyr)" << "," << "Final Radius (km)" << "," << "Final Position (AU)" << "," << "Obliquity (degrees)" << endl; do { //begins size loop do{ //begins orbital radius loop // reinitialize variable each loop age = 0; Rlost = 0; Drad = 0;

R = Rind; rad = radind;

do { //begin main loop //run subroutines to update variables Luminosity(); Flux(); Temperature(); Motion(); Mass_Loss(); Poynt_Rob(); Yarkovsky();

//generates random number between 0 and 2*pi to reflect random changes due to collisional evolution ang = distribution(generator);

//update primary variables R = R - drdt*dt*year; rad = rad + (dadt + YarkD + YarkS)*dt*year; age = age + dt;

} while ((age < 10e9) && (R > 0) && (rad >= 0.06*AU) && (rad <= 0.25*AU)); //end main loop

//print values to data file data << Rind/1000 << "," << radind/AU << "," << age/1e9 << "," << R/1000 << "," << rad/AU << "," << ang*180/pi << endl;

//increment orbital radius looping variable radind = radind - 0.005*AU;

}while ((radind >= 0.06*AU) && (radind <= 0.25*AU)); //end orbital radius loop

//reinitialize orbital radius increment variable for next loop radind = rad0;

//incriment size looping variable with condition for finer values at the low end if (Rind > 500) { Rind = Rind - 500; } else

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{ Rind = Rind - 25; }

} while (Rind > 0); //end size loop

data.close(); //closes data file

return 0; //closes program } void Material() //Assigns chooses material and assigns material values to the appropriate variables { do //sets material variables to chosen material { cout << "Select Asteroid Material:" << endl; cout << "1 - Iron" << endl; cout << "2 - Fayalite" << endl; cout << "3 - Regolith" << endl; cout << "0 - Exit" << endl; cin >> Mat; //inputs material selection

switch (Mat) { case 0: cout << "Exiting Program." << endl; exit(EXIT_SUCCESS); break; case 1: //sets variables to Iron values cout << "Material selected is Iron." << endl; A = A1; B = B1; Mu = Mu1; Al = Al1; rho = rho1; ThCon = ThCon1; Cp = Cp1; break;

case 2: //sets variables to Fayalite values cout << "Material selected is Fayalite." << endl; A = A2; B = B2; Mu = Mu2; Al = Al2; rho = rho2; ThCon = ThCon2; Cp = Cp2; break;

case 3: //sets variables to Regolith values cout << "Material selected is Regolith." << endl; A = A3; B = B3; Mu = Mu3; Al = Al3; rho = rho3;

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ThCon = ThCon3; Cp = Cp3; break;

default: //gives error for invalid selection cout << "Error: Invalid Selection." << endl << endl; } } while ((Mat != 1) && (Mat != 2) && (Mat != 3)); //loops until exit or valid material selected } void Luminosity() //finds solar Luminosity at different times over the age of the solar system { double t; //define variable to convert time into Gyrs t = age / 1e9;//convert time to Gyrs Ls = (b0 + t*b1 + pow(t, 2)*b2 + pow(t, 3)*b3 + pow(t, 4)*b4)*Ls0; //calculates luminosity } void Flux() //finds solar flux { Fs = Ls / (4 * pi*pow((rad), 2)); //calculates solar flux at given distance } void Temperature() //finds effective temperature of the body { Temp = pow(((1 - Alb)*Ls / (2 * 4 * pi*stbo*eps*pow((rad), 2))), 0.25); //calculates effective blackbody temperature (factor of 2 is due to slow rotation, increase to 4 for fast rotation) } void Motion() //finds various values related to motion {

Period = year*pow((rad) / AU, 3 / 2); //finds the orbital period from Kepler's 3rd Law. Need to convert to and from AU and Years (s) nuS = 2 * pi / Period; //Calculates mean orbital motion (frequency of revolution) from the period of revolution (radians s^-1)

Rot = 5 * 3600; //choice in roation period is somewhat arbitrary but 5 hours is a characteristic value for similar bodies of this size.

nuD = 2 * pi / Rot; //Calculates frequency of rotation from the period of rotation (radians s^-1) }

94 void Mass_Loss() //calculates mass loss rates and associated variables (R. van Lieshout, M. Min, and C. Dominik, A&A 572, A76 (2014)) { pv = 0.1*exp(B - A / Temp); //finds vapour pressure (eq. 13) (0.1 in the unit conversion from dynes/cm^2 into pascals) mflux = Al*pv*sqrt(Mu*amu / (2 * pi*kb*Temp)); //finds the mass loss flux (eq. 12) drdt = mflux / rho; //find the change of radius per change in time (m s^-1)

mass = (4 / 3)*pi*pow((R), 3)*rho; //calculates mass of the asteroid assuming constant density and spherical volume } void Poynt_Rob() //calcualte effects of Poynting-Robertson Drag (Wyatt, S. P., and Whipple, F. L. 1950. The Poynting-Robertson effect on meteor streams.Astron.J. 111, 134 - 141.) { alpha = 3 * Ls / (16 * pi*(R)*pow(c, 2)*rho); //comes from outwardly directed radiation pressure, intermediate calculation step dadt = -2 * alpha / (rad); //change in radius over change in time due to Poynting Robertson in m s^-1 } void Yarkovsky() //finds change in semi-major axis due to the Yarkovsky Effect { Gamma = sqrt(rho*ThCon*Cp); //Thermal inertia of the body Phi = pi*pow((R), 2)*Fs / (mass*c); //radiation pressure on a spherical body

ThetaS = Gamma*sqrt(nuS) / (eps*stbo*pow(Temp, 3)); //Thermal Parameter Seasonal ThetaD = Gamma*sqrt(nuD) / (eps*stbo*pow(Temp, 3)); //Thermal Parameter Diurnal

OmegaS = -(0.5*ThetaS) / (1 + ThetaS + 0.5*pow(ThetaS, 2)); //this term is dependent on several material and size dependent factors. OmegaD = -(0.5*ThetaD) / (1 + ThetaD + 0.5*pow(ThetaD, 2)); //In the limit of large particles > ~cm these all reduce to 1/2. (See Vokrouhlicky Diurnal Yarkovsky effect A&A (1998))

YarkS = 4 * (1 - Alb)*Phi / (9 * nuS)*OmegaS*pow(sin(ang), 2); //change in orbital radius over change in time due to seasonal Yarkovsky effect in meters per second YarkD = -8 * (1 - Alb)*Phi / (9 * nuS)*OmegaD*cos(ang); //change in orbital radius over change in time due to diurnal Yarkovsky effect in meters per second }

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