MASTER'S THESIS
Meteors and Celestial Dynamics Association and Numerical analysis
Daniel Kastinen 2016
Master of Science in Engineering Technology Space Engineering
Luleå University of Technology Department of Computer Science, Electrical and Space Engineering Meteors and Celestial Dynamics
Association and Numerical analysis
D. Kastinen
August 9, 2016 D. Kastinen Meteors and Celestial Dynamics
Abstract
We have developed a skeleton version of a new toolbox for statistical small body dynamics in the Solar sys- tem. The propagation parts of the software include perturbations from all major planets, radiation pressure and the PoyntingRobertson effect. Currently, the software is constructed to generate clones of parent bodies taking into account uncertainties in observational parameters and the parent body characteristics. To then sample this distribution in a Monte Carlo fashion. These bodies then release test particles using sublimation models. The parent bodies as well as the particle generation process are described by multivariate proba- bility distributions. In our current usage, the distribution represented orbital elements, critical sublimation radius, density, size and surface activity. The software designed to integrate the released particles over a given time scale and examine close encounters with another body in the solar system. We have examined close encounters with the Earth. We have also created module for calculating orbital similarity functions and to find associations and classifications in data sets. This toolbox is entirely modular enabling the use of every step individually.
Validation is performed by simulating known and observed meteor showers, we have simulated the 1933 and 1946 October Draconids as validation, and extended the simulations to the 2011 and 2012 October Dra- conids. The simulation was performed by ejecting material from comet 21P/GiacobiniZinner during seven perihelion passages between 1866 and 1972 and propagating the material forward in time. Each perihelion passage was sampled with 50 orbital clones that produced meteoroid streams. In total 850 clones were prop- agated. The clones were sampled from a multidimensional Gaussian distribution on the orbital elements with width proportional to the given uncertainties. These orbital clones were then sampled from normal distributions on the bulk density, surface activity factor, cometary mass and critical sublimation distance from the Sun, with characteristic values from measurements of 21P/GiacobiniZinner. Each clone ejected 8,000 particles, each with an individual weight proportional to the mass loss (number of meteoroids) they represented. This generated a total of 6.7 million test particles, out of which 43 thousand entered the Earth’s Hill sphere during 1900-2020 and were considered encounters. Using the simulation we produced the unex- pected and measured deviation of the meteor mass index from a power low in the 2012 October Draconids, a feature not present in the 2011 October Draconids. We also predict a October Draconids outburst in 2018 with peak on the night between October 8 and 9 that should be larger than the 2011 and 2012 outbursts. Lastly we present some analysis as a proof of concept for the future development of this toolbox.
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Preface
This master thesis is the culmination of a row of student projects and internships within the area of celestial mechanics, meteor science, statistics, and numerical integrators. I have in 2013 performed collaborative work on the implementation of a large aperture radar meteor database called the Shigaraki Middle and Upper atmospheric Radar Meteor Head Echo Database (MURMHED). The data will together with 20,000 events observed 2011-2012 be released in the form of an open database containing trajectory and orbit information, hosted at the National Institute of Polar Research (NIPR), Tokyo Japan. The construction of this database was supported by a grant from the Japan Society for the Promotion of Science (JSPS) with P.I. Takuji Nakamura. Daniel Kastinen and Johan Kero developed the database format.
In the F7005T Project in physics course I first reviewed the current field of association of meteoroids, both D-criterions and phase-space metrics, and then introduced a new approach to the metrization of trajectories using the physical trajectories themselves as a basis instead of phase spaces. The method uses Hausdorff distance between the subspaces making up the trajectories. I also researched the theory of the meteoroid complex, statistical analysis on clusters with different metrization functions. The constructed algorithms and the results from the different metrics were compared to each other, the statistical distributions were compared to real world data and an analysis to find new clusters in a head echo database is performed.
During summer/autumn 2014 I had a Swedish Institute of Space Physics (IRF), Kiruna, internship within project 1217 (Johan Kero) titled Meteors and celestial dynamics - association and numerical analysis”. This work was the continuation and completion of the work in the ”F7005T Project in physics course”.
I have participated in the 2015 JSPS Summer Program with an internship at the NIPR in Tokyo. Proper methods of finding meteor showers in databases are an open area of research and the theoretical work sur- rounding statistics and mathematics are far from complete. The purpose of this research is to develop proper statistical methods of pattern recognition and machine learning in combination with advanced simulations of orbital evolution and material ejection from celestial bodies. To execute this research a software platform is being written to perform: simulations of dust material ejection from comets being propagated, generation of Solar system initial condition from ephemerides, generation of parent bodies from debisaed population data, propagation of test particles and close encounter determination using mercury6 software, orbital clone gen- eration from observational errors using OrbFit software, several similarity functions such as D-criterions, grouping methods such as kernel density estimation and hierarchical cluster analysis, association threshold determination methods such as k-fold cross validation and association profile analysis, and analysis of large scale meteor databases. During this summer almost all software described above has been developed with the help of NIPR, Nihon University, Institute of Statistical Mathematics (ISM), and National Astronomical Observatory of Japan (NAOJ). With the cooperation of several researchers in Japan, the software was taken to such a state that the versatility of the developed platform can extend to many more research questions outside the scope of the current work. Also, presentation of this work and surrounding questions has been given at NIPR, NAOJ, and Nihon University. Three research trips were executed during the summer. The first one was a visit Nihon University for discussion and presentations, the second on a visit to the Shigaraki Radar facility for a tour and explanation of the system used to collect data to be analysed. And lastly to Norikura Observatory for a meteor shower observation campaign using optical measurements.
During my research into meteor theory, association analysis, and celestial mechanics I ran into many speed bumps due to unclear methodology and vital information being widely spread among many sources. This resulted in a ”random walk” like experience while finding the required material and writing software. Many of the mathematical concepts in literature where presented without rigour and central concept are sometimes
Page II D. Kastinen Meteors and Celestial Dynamics overlooked by experimentalists. This is the reason so much of this work seems as basic knowledge. I have decided to include all these basics in a simple and direct way keeping as much rigour as possible to create a as a self contained overview of the required material to understand and reconstruct my research. Even though i have applied the techniques and methods described on meteors and small body populations they have a extremely wide range of use. The first half of this work consisting of theoretical background can be used independently of my applications. Other applications will be mentioned when relevant and I will try to give as much reference material on each subject as possible or as seen practical.
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Acknowledgements
I would like to thank everyone that has helped me with my education into the many different areas I have studied. Both in courses and on my free time, as the preliminary studies needed far exceeded my expecta- tions due to the broad field nature of the problem. And also my parents for their unwavering support of me in my goal becoming a researcher. I would also like to thank my supervisor Johan for his constant support and good advice. Without his guiding and integration of me into the research community I would never have completed even a fraction of the work described here.
Lastly would also like to thank my girlfriend for putting up with the many late nights of overtime needed to be able to complete this work. Without her support and love I would not have been able to keep working at the pace needed.
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Contents
Page
1 Introduction 1
I Classical celestial mechanics 3
2 Introduction to classical celestial mechanics 3
3 Hamiltonian mechanics 3 3.1 Classical mechanics ...... 3 3.2 From Lagrangian mechanics ...... 3 3.3 The two body problem ...... 6 3.4 Keplarian elements ...... 7 3.5 Hamiltonian form ...... 10 3.6 Poisson brackets ...... 12 3.7 Liouville’s theorem ...... 13 3.8 Phase space paths ...... 14
4 The problem of non gravitational perturbations 15 4.0.1 Radiation pressure ...... 15 4.0.2 Poynting-Robertson effect ...... 16 4.1 Yarkovski effect ...... 16 4.1.1 Yarkovsky O’Keefe Radzievskii Paddack effect ...... 17 4.1.2 Relevance ...... 17
5 Numerical integration 17 5.1 Symplectic structure ...... 17 5.2 Hamiltonian splitting ...... 19 5.2.1 Differential equation flow ...... 20 5.2.2 Order of a split ...... 20 5.3 Bulirsch-Stoer method ...... 21 5.3.1 Richardson extrapolation ...... 22 5.3.2 Modified midpoint method ...... 22 5.3.3 Deufelhard serie ...... 23 5.3.4 Neville’s Algorithm ...... 23
II Solar system population 24
6 Introduction to Solar system population 24
7 The Sun 24
8 The planets 24
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9 Small bodies 24 9.1 Comets ...... 24
10 Dust 26 10.1 Poynting-Robertson lifetimes ...... 26 10.2 Comets ...... 27 10.2.1 Dust ejection ...... 27 10.2.2 Critical sublimation radius ...... 30 10.3 Asteroids ...... 30 10.4 Main belt ...... 30 10.5 Interstellar dust ...... 30 10.6 Nanodust ...... 31
11 Databases 31 11.1 Pan-STARRS Synthetic Solar System Model ...... 31 11.2 JPL NAIF database ...... 32
III Meteors 34
12 Introduction to meteors 34
13 Observations and mesurnemnts 34 13.1 Coordinate systems ...... 34 13.2 Invariable plan and the ecliptic ...... 34 13.3 Meteors ...... 36 13.3.1 Visual ...... 36 13.3.2 Head echo ...... 37 13.4 Ablation ...... 37
14 Meteor and Meteoroid complex 38 14.1 Stream meteoroids ...... 38 14.2 Sporadic meteoroids ...... 39
15 Databases 40 15.1 MURMHED ...... 40 15.2 IAU meteor shower database ...... 42
IV Associating and classifying meteoroids 43
16 Introduction to associating and classifying meteoroids 43
17 Similarity functions 43 17.1 Metric spaces ...... 44
18 Models of meteoroid complexes 45
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19 D-criterion 45 19.1 Southworth and Hawkins ...... 46 19.2 Drummond ...... 46 19.3 Valsecchi, Jopek, and Froeschl´e ...... 47 19.4 Jopek, Rudawska, and Bartczak ...... 48
20 Metric of phase-spaces 49
V Statistical methods 52
21 Introduction to statistical methods 52
22 Overview 52
23 Monte Carlo 53
24 Principal component analysis 54
25 Cluster analysis 55 25.1 Introduction ...... 55 25.2 Definition ...... 57 25.3 Signal to noise ratio ...... 59 25.4 Previous usage of thresholds ...... 61
VI Software 63
26 Introduction to the software 63
27 Modular development 63
28 Dependancies 64 28.1 Programming language ...... 64 28.1.1 C++ 11 ...... 64 28.2 Boost library ...... 64 28.3 SPICE library ...... 65 28.4 Fortran 77 ...... 66
29 Monte Carlo Association Statistics module 66 29.1 Program flow ...... 66 29.2 Execution time diagnostics ...... 67 29.3 Probability draw ...... 70 29.3.1 Dimensionality reduction ...... 76 29.4 Close encounter ...... 77 29.5 Parent body stream divergence ...... 78 29.6 Stream dissipation ...... 81
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29.7 Data formatting ...... 82 29.7.1 Input files ...... 83 29.7.2 Output files ...... 83 29.8 Mass distributions ...... 90 29.8.1 Gr¨un...... 90 29.8.2 Uniform and logarithmic ...... 91 29.8.3 Custom file ...... 91 29.8.4 Comments ...... 91 29.9 Probability formulation ...... 91 29.10Configuration ...... 99
30 Parent Body Ejector module 101 30.1 Symplectic structure ...... 101 30.2 Propagation ...... 101 30.3 Period syncing ...... 102 30.4 Ejection speed ...... 103 30.5 Ejection direction ...... 105 30.6 Particle mass ...... 106 30.7 Ejection times ...... 107 30.8 Data formatting ...... 109 30.8.1 Input files ...... 109 30.8.2 Output files ...... 111 30.9 Configuration ...... 112
31 Orbital Association Analysis module 113 31.1 Similarity functions ...... 113 31.2 Similarity matrix ...... 116 31.3 Clustering ...... 116 31.4 Parameter sweep ...... 116 31.4.1 Fixed step ...... 117 31.4.2 Adaptive step ...... 117 31.5 Error functions ...... 119 31.6 Next update addition ...... 119 31.7 Data formatting ...... 120 31.7.1 Input files ...... 120 31.7.2 Output files ...... 120 31.8 Configuration ...... 121
32 Orbital Stability Estimation module 124
33 Simulation Merger module 124
34 Statistical Uncertenty Orbital Clones module 124
35 NASA Jet Propulsion Laboratory module 125
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36 Scripting of simulations 127
37 External software 128 37.1 Mercury6 ...... 128 37.2 NASA Jet Propulsion Laboratory ephemeris software ...... 131
38 Data transformation logistics 132 38.1 State vectors and Kepler elements ...... 132 38.1.1 From state vectors to Kepler elements ...... 132 38.1.2 From Kepler elements to state vectors ...... 134 38.2 Geocentric coordinates ...... 134 38.2.1 Ecliptic J2000 ...... 134 38.2.2 Equatorial J2000 ...... 135 38.3 Sun centred versions ...... 138
39 MATLAB visualization scripts 138
40 Future work 140 40.1 Purpose and aims ...... 140 40.2 Future development plan ...... 140 40.3 Significance of future work ...... 141 40.4 Significance of community contributions ...... 142
VII Results 143
41 Introduction to Results 143
42 21P/GiacobiniZinner 143 42.1 Input state ...... 143 42.2 Summary of simulations ...... 146 42.3 Validation: 1933 and 1946 October Draconids ...... 146 42.4 2011 October Draconids ...... 149 42.5 2012 October Draconids ...... 153 42.6 Mass transfer difference 2011-2012 ...... 155 42.7 Probable mass transfer cause ...... 155 42.8 Year intensity overview ...... 157 42.9 Cluster analysis calibration ...... 157 42.10Principal component Analysis ...... 160
43 Pan-STARRS Short Period Comets 163 43.1 Input state 1 ...... 163
44 Concluding remarks 164
References 165
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Appendices 172
A Notation, abreviations, technical terms, and nomenclature 172
B List of figures 173
C List of tables 175
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1 Introduction
Figure 1: The life of a meteoroid encountering the Earth. When caught by the Hill Sphere (the volume in which Earth gravity is dominant), if the Earth gravitational pull directs the meteoroid towards the atmosphere it will enter and begin to heat up. The meteoroid may disintegrate during its ablation phase, or in some cases bounce of the atmosphere. The process of colliding with the atmosphere produces a meteor phenomena as the collision turn small areas of atmosphere into plasma. If however the meteoroid survives atmospheric entry, either due to the entry circumstances or its sheer size, a dark flight phase will begin. Then to finally impact the surface and by doing so turning from meteoroid to meteorite.
The dynamics of small bodies in the Solar system form a theoretical basis to study, for example, meteoroid streams (Soja et al., 2014) and their connected meteor showers (Vaubaillon et al., 2011) is a subject requiring many different numerical tools. Whether it is the sporadic complex (Wiegert et al., 2009) of observed meteors or the environment of newly formed meteoroid streams. Development of numerical simulations in this area is needed to test and validate physical models, ranging from theories of dust ejection and ice sublimation of comets (Ryabova, 2013), to the chaos mechanism of supplying the Solar system with new comets (Torbett, 1989). It is also highly relevant for technical models like the ESA Interplanetary Meteoroid Environment for eXploration (IMEX) project (Soja et al., 2015).
Characterization of dust trails and meteoroid streams are important to assess the dust impact hazard to
Page 1 D. Kastinen Meteors and Celestial Dynamics spacecraft. For example, the ESA communications satellite Olympus lost pointing control in 1993 likely due to the impact of a Perseid meteoroid (Caswell et al., 1995). For comet and asteroid interaction missions, such as the NASA space probe Deep Impact (AHearn et al., 2011), JAXAs Hayabusa 2, and future asteroid deflection missions, accurately estimating debris dynamics will be vital.
In spite of the fact that the underlying orbital dynamics is well understood, it is still an open question how much extraterrestrial material enters the Earth atmosphere. The most obvious missing feature in cur- rent numerical models, which cover different aspects of Solar system small body dynamics, is the ability to easily connect programs with different functionality. We believe this is the main reason why contempo- rary techniques have not yet been combined. These include techniques such as Monte Carlo simulations of meteoroid streams, orbital stability analysis, time tracing of orbital similarity functions to better analyse measurements and neural network recognition of meteor showers. The lack of a comprehensive numerical modelling approach appears to be the main bottleneck. Thus, the main focus of this work is dedicated to developing a prototype version of a software toolbox which attempts to supplement the current research area. We here present a skeleton version containing several of the above-mentioned features and we have performed a feasibility check.
There are many other applications of numerical tools oriented towards statistical numerical integration of bodies in the Solar system than the ones mentioned here. We shall however only focus on a few for com- pression of the thesis, but keep concepts and implementation general enough so that the developed material can have applications in other areas of research.
As this work is oriented towards people with a broad scientific background as well as the expert, every section is provided with a small introduction reviewing what reader benefits the most from said section and how it fits with the whole. For those being exposed to many new concepts while reading this first introduc- tion, a quick illustration of the chain of events leading to the meteor phenomena are shown in figure 1. In addition to this illustration it can be added that if the entry circumstances are right, even small particles can survive atmospheric re-entry as shown by simulations in (Love and Brownlee, 1991). In appendix A we have also listed a collection of abbreviations, technical terms, and commonly used symbols.
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Part I Classical celestial mechanics
2 Introduction to classical celestial mechanics
This part is intended for those who are not comfortable with Hamiltonian mechanics and the mathematical concepts relevant to numerical celestial mechanics, or new to celestial mechanics as a whole. Thus this part covers the basics of Hamiltonian systems and briefly address perturbations to the Hamiltonian system in form of electromagnetic forces, we do not cover proper Hamiltonian Perturbation theory as that is outside the scope of this work. The last section of this part is dedicated to numerical methods implemented in the software developed. If standard numerical concepts within celestial mechanics is a new concept, this is chapter that should not be skipped if the goal is to grasp the full context of the developed software.
3 Hamiltonian mechanics
3.1 Classical mechanics
Celestial mechanics has been of interest for as long as mankind had the ability to see the movements of the planets and wonder what governed their motion and how one could predict their paths. The goal of classical mechanics is to glance at the configuration of a system and determine its movement due to a set of forces, whether it be backwards or forwards in time. The first real mathematical theory for this kind of problem was developed by Isaac Newton and is called calculus, or the mathematics of change. Tracing from this foundation of differential equations, integrals and functions of motion a first reformulation of classical mechanics was made by a mathematician called Joseph-Louis Lagrange in 1788. This refor- mulation is commonly called Lagrangian mechanics and is based upon the principle of stationary action. This concept together with its use of generalized coordinates made it much more versatile than classical mechanics.
3.2 From Lagrangian mechanics
Lagrangian mechanics are however not the preferred way to handle celestial mechanics since a set of mathe- matical tools much more suited to the problem can be found in Hamiltonian mechanics. However, these two theories of mechanics are very closely related and below a hint of their connection is supplied. To begin grasping the underlying idea of Lagrangian and Hamiltonian mechanics we must cover action. The concept of action is a way to talk about the path of a system through space-time. Lets consider a bijective function that takes a path in space-time and transforms that path to a real number. Such a function also needs to be a Lorentz invariant quantity, for it to be a covariant way to express unique paths. If we want to form the basis of a realistic theory. When considering the way classical mechanics is viewed, in deterministic motion, one can quickly realize that for a real system the action has to be constant. The reason is because if the action was not constant, the system could suddenly change to a different space time path. This is in other words simply a way of stating that macroscopic objects does not suddenly teleport in space and time, of course there are deeper meaning behind stationary action but this interpretation is sufficient for our purposes. This transformation of a space-time path to a real number, or action S, is the space-time-integration of the Lagrangian density L. Committing slight notational murder, we can express it as
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Z S = Ld4x. (3.1) all spacetime
This is, for example, what is used in Gauss’s law for gravity and in the Feynman path integral formulation of quantum field theory. Here it is important to make a distinction between the Lagrangian density and the Lagrangian. The La- grangian density is integrated over all space-time to extract the action while the Lagrangian L is simply integrated over time to give the action. The relationship between the Lagrangian L and the Lagrangian density L is
ZZZ L = Ldx1x2x3, (3.2) all space
if the spacial coordinates are (x1, x2, x3) and time x0. We know that this Lagrangian is dependant on the generalized coordinates qi, their time derivativesq ˙i, and time t. We can now write the action as
Z t1 S = Ldt, (3.3) t2
and it is from this expression we will demand that the derivative of the action with respect to a phase- space path disturbance evaluated at zero disturbance be equal to zero. Or in other words for those familiar with variational calculus, we set the first variation, or the functional derivative, to zero and thus find the extremal. We can then, through some manipulation and the cunning use of geometric algebra, as outlined in (D’Orangeville and Lasenby, 2003), derive that the following equation,
d ∂L ∂L − = 0, (3.4) dt ∂q˙i ∂qi must hold for every coordinate for this to be true. This is called the Euler-Lagrange equation and its solutions yields the functions of motion for a system. However to derive the Hamiltonian formulation of classical mechanics rather then the Lagrangian a Legendre transform will transform the Lagrangian, L(q, q,˙ t) into a Hamiltonian
X H(q, p, t) = q˙ipi − L(q, q,˙ t), (3.5) i
To understand what this transformation entails let us first realize that the Lagrangian is a function mapping the configuration space of a system (its spacial coordinates) to the real numbers. The Hamiltonian however is mapping a larger space, the phase space, to the real numbers. The phase space considers both the spacial coordinates and their rate of change, their momentum. This may seem like a trivial difference since in Lagrangian mechanics we use configuration space but we also considered the configuration rate of change in the Lagrangian anyway. Even though this is the case, the fact that we are extending the space instead of extracting the rate of change inside the function, comes with a few advantages. So, for a Hamiltonian system the solution to the equations of motion will not only reside in the configuration space but in phase space, effectively including the dynamics of the bodies momentum into the topology. How a Legendre transform extends the configuration space involves creating a cotangent bundle of the space in question and is a more mathematically rigours question not covered here.
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In the above expression we have denoted the momentum, or weighted time derivative, of the generalized
coordinate as pi, giving us the canonical coordinates (q, p) used to describe a system. Then by examining the total differential of the Lagrangian, we can then some manipulation derive the Hamilton equations as
∂H = −p˙i, (3.6) ∂qi ∂H =q ˙i. (3.7) ∂pi As can be seen from this there will be as many equations as degrees of freedom in the described system. These equations are equivalent to the Euler-Lagrange equations. So far the mentioned mathematics have been general for all sets of mechanical systems, moving closer to celestial mechanics we can express the Hamiltonian for a gravitational potential field in a N-body problem as
N 2 N−1 N X p X X mimj H = i − G , (3.8) 2m r i=1 i i=1 j=i+1 ij where, if we use Cartesian coordinates,
q 2 2 2 rij = (qix − qjx) + (qiy − qjy) + (qiz − qjz) , (3.9)
q 2 2 2 pi = pix + piy + piz. (3.10) Here we have changed the notation by instead of referring to the i’th coordinate we are referring to the i’th particle with the associated three Cartesian coordinates, yielding a total of 6N variables in a system of N bodies. Given that we reside in three dimensional space of course. Inserting the expression for the Hamiltonian in equation 3.8 into equations 3.6 and 3.7 we can derive the following two equations of motion;
N 2 N−1 N ∂H ∂ X p X X mimj = i − G = ∂q ∂q 2m r nk nk i=1 i i=1 j=i+1 ij N−1 N N ∂ X X mimj X ∂ mimn = G = G = ∂qnk rij ∂qnk rin i=1 j=i+1 i=1,i6=n N N X 1 2(qnk − qik) X qnk − qik = Gm m − = Gm m ⇔ i n 2 r3 i n r3 i=1,i6=n in i=1,i6=n in N X qnk − qik ⇔ p˙ = − Gm m , (3.11) nk i n r3 i=1,i6=n in and
N 2 N−1 N ∂H ∂ X p X X mimj = i − G = ∂p ∂p 2m r nk nk i=1 i i=1 j=i+1 ij
N 2 ∂ X p pnk = − i = ⇔ ∂p 2m m nk i=1 i n pnk ⇔ q˙nk = , (3.12) mn
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where n ∈ {1, 2,...,N} and k ∈ {x, y, z}. These equations and methods are the basis for formulating a numerical solution to a N-body problem. We can also combine equations 3.11 and 3.12 to write the equations of motion for a general gravitational Hamiltonian on the form
N X qik − qjk m q¨ = − Gm m ⇔ i ik j i r3 j=1,j6=i ji N X qik − qjk ⇔ q¨ = −G m , (3.13) ik j r3 j=1,j6=i ji or in vector notation as
N X qi − qj ¨ qi = −G mj 3 . (3.14) |qi − qj| j=1,j6=i
All of this can be found in any Celestial Mechanics or Hamiltonian mechanics text book, like (Danby, 1992).
3.3 The two body problem
When working with celestial mechanics one almost always comes across the term Keplerian elements. This term refers to the variables describing a Keplarian orbit. If N in equation 3.14 is set to 2, there will be a analytical solution to the differential equations yielding exactly what is called a Kepler or Keplerian orbit. Even though Keplerian orbits and their propagation in many problems are sufficient one should not forget that this is a very limited description of celestial mechanics. By the reduction of a more complex problem down to just a two body problem, most of the interesting qualities are lost. The phrase Keplarian orbit has several meanings, most often (Danby, 1992), as well as in this work, it means the set of all parametrization of solutions to equations 3.11 and 3.12 when N = 2
q1k − q2k p˙1k = −Gm2m1 3 , (3.15) r21 q2k − q1k p˙2k = −Gm1m2 3 , (3.16) r21 p1k q˙1k = , (3.17) m1 p2k q˙2k = , (3.18) m2
which combined with q˙ m = p, and, rij = rji gives
q1 − q2 m1q¨1 = −Gm2m1 3 , (3.19) r12 q2 − q1 m2q¨2 = −Gm2m1 3 . (3.20) r12
Performing a coordinate transformation into relative coordinates R = q1 − q2 and R = |R1 − R2| gives the usually used form of the equations
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R −R q¨ − q¨ = R¨ = −Gm − −Gm = 1 2 2 R3 1 R3 G(m + m ) = − 1 2 R. (3.21) R3 (3.22)
In a lot of literature the combined gravitational parameter µ = G(m1 + m2) is used, sometimes however it is defined as µ = Gm1 without any explanation. This is always derived through the approximation G(m1 + m2) ≈ Gm1 which is only applicable when the mass of the second body is a lot smaller relative to the first. Even tough not used explicitly in our work, for completeness we need to mention constants of motion. Since the two body system covered here has analytical solutions, or is integrable, it will have a number of so called constants of motion. In fact, as the system is integrable, it will have as many constants of motion as free parameters. A constant of motion is simply a field on phase space that is constant along the flow of the solution. Constants of motion are a useful concept because they can revel properties of the motion without one solving the equations of motion. Some times even the trajectories themselves can be derived. This can be the case since if the field is constant along a phase space path, one can also say that the motion of a body follows intersections of multi dimensional ”level curves” of this field. In most cases the total energy of the system is immediately known as a constant of motion, and we will later prove that for Hamiltonians we can always re-write the system so that this is the case. But the energy need not always be constant. Although conservation of energy is a fundamental law there are many systems where energy is not preserved due to the physics behind the energy conversion not being considered. An example of this is radiation pressure, we do not describe the photons or how the momentum from photons are absorbed by the body and re-emmited in the actual Hamiltonian, instead we ad hoc add in the effect resulting in total energy no longer being preserved. Such a system is called dissipative as energy can move into and out of the system.
3.4 Keplarian elements
In the previus section the two body solution was discussed and called a Keplarian orbit. Kepler however did not derive his laws from a law a gravitation but from observation. But since the solution of the two body problem coincide with Kepler’s laws of celestial motion:
(I) The orbit of a planet is an ellipse with the Sun at its pericenter,
(II) A line joining a planet and the Sun sweeps out equal sector areas during equal time intervals,
(III) The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit, often the term Keplerian elements and Keplarian orbit is used for things relating to a two body solution. Despite their limitation to describing only orbits which are solutions to the two body problem, they can be a valuable tool in examining more complex problems by looking at their deviation in time due to perturbations. More on the explicit calculations of keplarian elements from state vectors can be read about in section 38.
The Keplerian (orbital) elements: are used to uniquely identify a specific orbit with a set of six parameters, the choice of parameters often change but here we will most often adopt using the semi- major axis a or perihelion distance q, the orbital inclination i, the numerical eccentricity e, the argument of perihelion ω, the longitude of ascending node Ω, and if we also want to position the object on the orbit, one of the anomalies, true anomaly ν, eccentric anomaly u or mean anomaly v. Illustration can be found in figure 2 and figure 3.
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Figure 2: Digram illustrating and explaining various terms in relation to Orbits of Celestial bodies. (CC-BY- SA-3.0-MIGRATED; Licensed under the GFDL by the author; Released under the GNU Free Documentation License. http://commons.wikimedia.org/wiki/GNU_Free_Documentation_License)
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Figure 3: Digram illustrating the three different anomalies where E is the eccentric anomaly, M the mean anomaly, and ν the true anomaly. The central body is located at the point s and the orbiting body at point p.
It is also useful to introduce the so called longitude of perihelion as
ω¯ = ω + Ω, (3.23)
and the mean longitude as
λ = M + ω + Ω, (3.24)
where M is the mean anomaly. These new variables are useful since Ω is not defined for zero inclination orbits, and in the same way ω is not defined for zero eccentricity orbits. However, if we simply define these angles as zero in these special cases and instead use the composite angles, we have a continues variable that can still contribute information. When working with the propagation of bodies along orbits a number of anomalies are used, as already mentioned. These variables are different kinds of angles associated with the motion of a body along a elliptic orbit. To describe a specific point on a elliptic orbit we can use the true anomaly,
e · r ν = cos−1 ∀ r · v ≥ 0, (3.25) |e||r| e · r ν = 2π − cos−1 ∀ r · v < 0. (3.26) |e||r|
This anomaly measures the angle from periapsis around the focus point. In the above expression e is the so called eccentricity vector, r is the position relative the central body and the bodies v velocity. The eccentricity vector, or the Laplace-Runge-Lenz vector, is a vector quantity associated with ellipses having length equal to the eccentricity of the ellipse and orientation always pointing towards the pericenter of the ellipse. The construction of this vector is purely a mathematical result of the geometry of oriented ellipses
Page 9 D. Kastinen Meteors and Celestial Dynamics but due to its formulation it is a constant of motion in the two body problem and therefore useful when calculating orbital elements from state vectors, or phase space coordinates. Using the true anomaly one can calculate the so called eccentric anomaly,
e + cos ν E = cos−1 ∀ r · v ≥ 0, (3.27) 1 + e cos ν e + cos ν E = 2π − cos−1 ∀ r · v < 0. (3.28) 1 + e cos ν
The eccentric anomaly is the angle representing a point on a auxiliary circle with radius equal to the semi- major axis of the ellipse. This point is defined as where the line perpendicular to the semi major axis passing through the orbiting body intersects the auxiliary circle. Again using the eccentric anomaly the mean anomaly can be calculated,
M = E − e sin E.. (3.29)
This equation is commonly called the Kepler equation and this anomaly represents the angle required so that the area swept on the auxiliary circle is equal to that swept on the ellipse. This is particularly useful since Keplers second law states that the area swept by a line connecting a body and the ellipse pericenter must be a constant of time, making propagation of this angle easy. Problems arise when this method of propagation is put into action since equation 3.29 is a so called transcendental equation. Transcendental equations are equations containing a transcendental function such that it cannot be solved for one factor in terms of another using algebra (Danby, 1992). More on the specific calculation of all these quantities will be covered in later sections, and algorithms for solving the Kepler equation to constuct numerical propagators is not currently relevant to our scope.
3.5 Hamiltonian form
Now that we are getting more comfortable with the notion of how Hamiltonian mechanics work and what they represent we need to generalize the concepts and properties of Hamiltonian systems. Starting with identifying a Hamiltonian system. A system of ordinary differential equations on the form
dx = F(x) (3.30) dt is on Hamiltonian form if
q1 . . q N x = , (3.31) p1 . . pN and there exists a function H(x, t) such that
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dq ∂H i = ∀ i ∈ 1,...,N, (3.32) dt ∂pi dp ∂H i = − ∀ i ∈ 1,...,N. (3.33) dt ∂qi Then we can write the vector function F as
∂H ∂p1 . . ∂H F = ∂pN . (3.34) − ∂H ∂q1 . . − ∂H ∂qN This is a large class of differential equations, which is one reason why Hamiltonian formalism is so useful. Considering Hamiltonian mechanics from the physicist point of view the Hamiltonian is not just a function transform from phase space to real numbers or the Legendre transform of a Lagrangian, but it represents the total energy of the system, in the same way the Lagrangian represents action by acting on configuration space. Assuming the time tested formula that total energy is represented by kinetic and potential energy
|p|2 H = + U(q), (3.35) 2m and that the force exerted on a particle due to a potential field is given by a second order differential equation as
d2q = −∇U(q). (3.36) dt2 All formulas of this type are of Hamiltonian form. This differential equation hold all dynamics caused by classical gravity, however electromagnetic effects and general relativistic effects play a crucial roll in the long term evolution of the Solar system and should not be dismissed easily. Analysing this particular set of differential equations, it is useful to examine how the function H(x, t) evolves with time. By expanding the full time derivative using the chain rule, denoting the scalar product of two vectors as hu, vi,
dH dx ∂H ∂H = h∇H, i + = h∇H, F(x)i + (3.37) dt dt ∂t ∂t where
∂ ∂q1 . . ∂ ∂qN ∇ = ∂ . (3.38) ∂p1 . . ∂ ∂pN
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We can then see that
h∇H, Fi = 0. (3.39)
If this is not obvious, it becomes clear by combining equations 3.32, 3.33 and 3.34. This shows that
dH ∂H = , (3.40) dt ∂t thereby proving that all autonomous, or independent of time, Hamiltonians are constants of motion, and that the systems total energy is preserved. This is a important result since a consequence of Hamiltonian formalism is that all non-autonomous Hamiltonian can be turned into autonomous Hamiltonians by expanding the phase space. Specifically one performs a transformation from the old phase space into a new one with a additional position and conjugate momentum dimension by embedding the old time variable into the configuration space and introducing a new time instead. If H = H(q, p, t) then introduce new phase space angle with corresponding conjugate momentumτ ˙ = T giving a new Hamiltonian
H0 = T + H(q, p, t) (3.41)
where it is easy to see that
dH0 dτ =τ ˙ = = 1 ⇔ T dt ⇔ τ = t. (3.42)
Since τ(t) = t is a direct substitution t 7→ τ makes the Hamiltonian autonomous, and therefore total energy is a constant of motion yet again. If the simple addition of the conjugate momentum seems strange, the operation can be justified by using equations 3.6 and 3.7. These are some basic facts about the result of defining the Hamiltonian and deriving the first variations of the total energy integral, (Hand and Finch, 1998). We shall not cover the variational calculus proofs here, more information can be found in (Logan, 2013).
3.6 Poisson brackets
To later cover some of the specifics of Hamiltonian integration techniques we will quickly need to cover Poisson brackets. Poisson brackets are a useful tool when working with phase space paths. Consider phase dx space M, differential equation dt = F(x), x ∈ M, and function f on M, we can define the Poisson brackets as
N X ∂f ∂g ∂f ∂g {f(x), g(x)} = − . (3.43) ∂q ∂p ∂p ∂q i=1 i i i i
It is straightforward to prove that the Poisson brackets are linear in their arguments
{α1f1 + α2f2, g} = α1{f1, g} + α2{f2, g}, (3.44)
{f, α1g1 + α2g2} = α1{f, g1} + α2{f, g2}, (3.45)
Page 12 D. Kastinen Meteors and Celestial Dynamics and that they are anti-commuting in their arguments
{f, g} = −{g, f}. (3.46)
We define the skew matrix, using the N-dimensional identity matrix IN , as
0 I J = N . (3.47) −IN 0
Here, using the skew matrix defined in equation 3.47 we can redefine the Possion brackets as the skew scalar product of the gradient of two scalar fields
{f, g} = h∇f, J∇gi. (3.48)
This will be useful later when examining phase space paths.
3.7 Liouville’s theorem
We know from vector space analysis that the divergence of a vector field can also be seen as the normalized evolution of a infinitesimal volume element along the vector field
1 d ∂V = div(F(x)) = h∇, F(x)i. (3.49) ∂V dt This can be interpreted as describing how a infinitesimal volume of initial conditions evolve along the flow of the differential equation. It can then be shown for Hamiltonian systems the area element of position and momentum must always stay constant. This implies that in a way volume in phase space is always preserved when a Hamiltonian system evolves with time, a property also called Liouville’s theorem. From this conclusion we can also imply that Hamiltonian systems does not have attractors. An attracting set for a system is a closed subset of the phase space such that for ”many” initial values the system will evolve towards the attracting set. This plays a very important role in the evolution of dynamical systems and more on attractors can be read about in (Milnor, 1985). From this theorem we mathematically conclude what we earlier conjectured that an N-degree of freedom
Hamiltonian is integrable if it has N independent constants of motion Φ1,..., ΦN such that {Φi, Φj} = 0 ∀ i 6= j. There is no set way to find all constants of motion for a system although it is generally easier to find these constants then to evaluate the integral solution of the system. A integrable Hamiltonian system is a system where we can rewrite the differential equation on form 3.30 into a set of integral equations
Z x(t) 1 Z t dxi = dτ ∀ i ∈ 1,..., 2N, (3.50) x(0) Fi(x) 0 where the left hand side of the equation can be evaluated, alas there is also here no set way to determine if there are solutions to the integrals or not. This makes it hard to distinguish between systems that have complicated integrals that one simply has not managed to solve and systems which are not integrable at all.
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3.8 Phase space paths
If f(x) is a function on M and x evolves according to the Hamiltonian flow, equations 3.32 and 3.33, a full time derivative can be expressed by
df dx ∂f = h∇f, i + , (3.51) dt dt ∂t
∂f due to the chain rule. But we assume that ∂t = 0 and from equation 3.48, 3.30, and 3.34 we can then rewrite the derivative to
df = {f, H}. (3.52) dt This is a important result since we now know how to a field evolves along a phase space path due to the Hamiltonian (Gel’fand and Dorfman, 1979). As most other literature is unclear on the derivation of the infinite series describing motion that is later used for order determination in numerical methods, we shall here do our own definition and derivation to clarify the process. Considering a coordinate as a function of time we can write
dq i = {q ,H}, (3.53) dt i dp i = {p ,H}. (3.54) dt i
Leaving one argument open in the Possion bracket of the Hamiltonian {·,H} we can consider it as a differ- df ential operator acting on the function, propagating it by the equation dt = {·,H}f. This operator is often called the Liouville operator. Using this, we can examine a differential equation on Hamiltonian form and integrate its evolution. Assuming that the Liouville operator is not a function of time, which we know due to the earlier proof on autonomous Hamiltonians, we can see that the equations are integrable. We now also assume that we have a initial value at t = 0, then the resulting equation is a regular first order differential equation with constant coefficients as
dx dx = F(x) ⇒ i = F (x) = {x ,H} = {·,H}x ⇔ dt dt i i i t{·,H} ⇔ xi = e xi(0). (3.55)
This relation can be used to describe the evolution of said phase space path. It may seem strange that one can take the exponential of a differential operator but it will become clear by the use of Taylor expansion.
Consider the phase space path a time difference h from our initial time t0 as
d h dt xi(t0 + h) = e xi(t) |t=t0 , (3.56) through the use of the power series definition of exponential functions, or Taylor expansion, we can write
∞ i i h d X (h) d e dt = . (3.57) i! dti i=0
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Thus the exponential of a differential operator is a infinite sum of differential operators. Here we also need to consider what the different orders of time derivatives translate to in Liouville operators, let consider a second order time derivative as
d2 d f = {f, H} = {{f, H},H} = {·,H}2f. (3.58) dt2 dt Applying this concept recursively in the Taylor series we get a expression
∞ i X (t) f(t) = {·,H}if(0), (3.59) i! i=0 that is usually called the Lie series of f under the flow of H (Fass`o,1990), this series is usually denoted as t SH for a flow according to Hamiltonian H for a time t. We can this way propagate any field or phase space path a time h by the formula
h{·,H} f(x(t)) |t=t0+h = e f(x(t)) |t=t0 (3.60) where the differential operator eh{·,H} is acting on f(x). This is the central concept needed to describe the main integration method for Hamiltonian integration software.
4 The problem of non gravitational perturbations
The big strength of Hamiltonians can pose a problem in numerical integrations of celestial mechanics. The fact that Hamiltonian mechanics require total energy to be described and preserved excludes the concept of physical effects not modelled properly. This means that if, for example, electromagnetic perturbations are to be described one also has to model the photons that are absorbed, reflected, and emitted, the thermal energy that is stored upon absorption, and then also keep describing these photons as they leave the body. If new photons are produced even mass energy conservations has to be described, otherwise one will break the basic notion of a constant energy in a system. There are however ways around this, most often one sees the phrase ’modified equations of motion’, this basically means that one has implanted a virtual potential field which the objects can interact with to preserve the energy without actually describing the precise physics behind the interaction, only describing the net effect. One can also model electromagnetic effects with something called dissipative mapping techniques as in (Kehoe et al., 2003) which can be incorporated with some success into already existing Hamiltonian integration schemes. The important concept to grasp here is that after a proper Hamiltonian integration step has been performed, we can apply whatever modifications due to other effects we like, as long as we do not do so inside the Hamiltonian propagation step. This can be justified by the idea that each step takes a new initial condition and propagates it, it does not matter what state we give the propagation, as long as we let the algorithm run without interference, all desired properties shall be preserved. This can create a dissipative mapping, with a symplectic step, which may seem like a contradiction at first.
4.0.1 Radiation pressure
We shall use the classical electromagnetic effect, the Sun’s radiation pressure, as a starting point. This perturbation is due to photons being absorbed and remitted, either in a thermal or a reflective way. Since momentum must be preserved during a process this means that simple radiation pressure will push a orbiting
Page 15 D. Kastinen Meteors and Celestial Dynamics
body outward proportional to the power of the incoming radiation, the pressure from absorbed photons can be described by
W P = cos(α) (4.1) a c where α is the angle between the objects normal vector and the incoming radiation of power W . The pressure from reflected photons can instead be described by
2W P = cos2(α). (4.2) r c This means that, together with a bodies albedo A and topology, the total radiative force acting on body can be calculated. Most often when performing electromagnetic perturbation is celestial mechanics one derives secular perturbation on the Kepler elements of a orbit. This is however a very rough approximation and should be used with care.
4.0.2 Poynting-Robertson effect
The Poynting-Robertson effect, or radiation drag, is a secondary effect from that of radiation pressure due to special relativity. The effect can be explained from the perspective of the body orbiting the Sun, the Sun’s radiation appears to be coming from a slightly forward direction because of aberration of light. The absorption of this radiation leads to a force with a component against the direction of movement. The magnitude of the force is
r v r2L GM F = W = s s , (4.3) PR c2 4c2 R5 where c is the speed of light, W is the power of the incoming radiation, r is the radius of the object being radiated, G is the gravitational constant, Ms is the mass of the Sun, Ls is the Solar luminosity and finally R is the orbital radius. As this force is dependant on the radius of the particle, a critical threshold can be found between the radiation pressure and the radiation drag when the effect either pushes the particle away from the Sun or causes it to spiral inwards. This is a important contribution to the appearance of the sporadic sources in meteor science. One important result from early calculations of the effect on small particles is the time of Solar impact for a circular orbit of initial semi major axis a. The calculations performed on a spherical homogeneous sphere of density ρ and radius r that is absorbing and emitting all incoming Solar radiation isotropically across the entire solid angle was found to have a time of Solar impact of
t = 7.0 · 106rρa2 y. (4.4)
9 This is small compared to the age of the Solar system ts ≈ 4.5·10 y. Thus, the sporadic complex of meteors generated by dust particles entering the Earth atmosphere must be resupplied somehow. It is widely accepted that the major supply mechanism is the sublimation of comets, more on this process will be covered in later sections.
4.1 Yarkovski effect
The Yarkovsky effect is a force acting on a rotating body in space due to the anisotropic emission of thermal photons. The part of this effect due to rotation around its own axis is called the diurnal effect. The diurnal
Page 16 D. Kastinen Meteors and Celestial Dynamics
effect however changes with orbital position around the source of radiation, and is called the seasonal effect. From more detailed descriptions of this effect, one can conclude that the seasonal part always produces a net decrease in semi-major axis while the diurnal effect both increase and decrease the semi-major axis. It will also become evident that this effect vanishes for too large bodies due to the relation between its cross-section and its moment of inertia. However, the effect vanishes for very small bodies as well due to the body’s size being comparable to the penetration depth of the thermal wave. Of course since this is also dependant on radiation flux the effect will be dependant on received radiation and surface conductivity. Under a few assumptions the force can be calculated by the integral
Z 2σ 4 FY = − T n⊥dS(u, v), (4.5) S 3mc
where the surface of the object is S(u, v), is the surface thermal emissivity, σ is the Stefan-Boltzman
constant and n⊥ is the surface normal vector. More on this can be read about in (Bottke et al., 2006).
4.1.1 Yarkovsky O’Keefe Radzievskii Paddack effect
A secondary effect of the Yarkovsky effect is the YORP-effect. This is due to the fact that the reflection and re-emission of photons from an objectss surface can also produce a thermal torque if the object has a irregular shape. Over time, these torques will affect the spin rate and obliquities of small bodies in the Solar System. The change of the internal spin and eccentricity vectors can be expressed by
dω T · e = , (4.6) dt C de T − (T · e)e = (4.7) dt Cω where ω is the rotation velocity of the object, C the objects moment of inertia, e is the eccentricity vector, and T is the torque on the object due to thermal radiation, more on this can be read in (Bottke et al., 2006).
4.1.2 Relevance
From the above discussion we can conclude that only the Poynting-Robertson effect and radiation pressure will be relevant to our integrations and thus we will neglect all other electromagnetic forces.
5 Numerical integration
5.1 Symplectic structure
Symplectic integrators are a scheme for numerically integrating a specific group of differential equations connected to symplectic geometry that is a branch of differential geometry and differential topology. Since this type integrator works on geometrical basis it is excellent for integrating physical problems such as classical mechanics where the phase space of a system can take the form of a symplectic manifold. In our i i case the set of differential equations are the Hamiltonian with the Poincar´etwo-form dpi ∧ dq , where q is a generalized coordinate and pi its momentum. These two elements together create our canonical coordinates, phase space, and the symplectic integrator can be viewed as canonical transformations. This is directly related to the concept in section 3.7 since symplectic integrator preserve the Poincar´etwo-form and thus in numerical integration total energy oscillate rather then diverge.
Page 17 D. Kastinen Meteors and Celestial Dynamics
Figure 4: Pendulum phase map with symplectic and non symplectic euler using the same stepsize.
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Figure 5: Relative error in total energy for different initial conditions for a pendulum.
A example of the difference between preserving symplectic structure and not doing so in a numerical inte- gration can be seen in figure 4 and 5. In these pictures are shown what such a simple modification as to make the method semi implicit can do for a numerical long term integration of a Hamiltonian system. It should however be noted that symplectic integrators are only a special case of what is called geometric inte- grators, there are several more geometric structures that can be preserved other then the Poincar two-form, for example if one wants to preserve phase space volume or perhaps limit volume contraction to a constant another approach to integration should be used. For more information see (McLachlan and Quispel, 2006). For more information on the accuracy of symplectic integrators, see (McLachlan and Atela, 1992).
5.2 Hamiltonian splitting
The next step is to cover the Hamiltonian splitting technique. This technique attempts to separate and isolate parts of the Hamiltonian, of a system with no analytical solution, into smaller parts who on their own are integrable. Such a integrator is symplectic. For example in a usual N-body gravitational problem, there is no solution unless N = 2. We can, loosely speaking, say that the problematic term is the sum of gravitational interactions outside that of the Kepler orbit. Meaning that one can separate this problem into several Kepler problems and the gravitational interaction between the smaller bodies. In other words, if we have
N 2 N−1 N X p X X mimj H = i − G , (5.1) 2m r i=1 i i=1 j=i+1 ij where the central mass possess index 1. We can then, as a example, split the Hamiltonian into (Wisdom and Holman, 1991)
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H = Hkep + Hint (5.2) N 2 X p mim0 H = i − G (5.3) kep 2m r i=2 i i1 N−1 N X X mimj H = G (5.4) int r i=2 j=i+1 ij where Hkep and Hint on their own can be solved but not together. This can be seen if we use equations 3.6 and 3.7 on the Hkep and Hint expressions separately. The importance of this kind of split will be evident in coming chapters.
5.2.1 Differential equation flow
Let us consider the flow, Ψ, of a differential function x through its phase space M,
x(t) = Ψtx(0), (5.5)
Ψt : M 7→ M. (5.6)
This flow is a exact deterministic solution, but if the system has no analytical solutions, this flow is not known. Thus we try to find maps, Φ, to approximate this flow. These maps are called integrators, let us for example define the flow of an one step method on a Hamiltonian phase space x = (q, p), with step size h as
Φh :(qn, pn) 7→ (qn+1, pn+1) (5.7)
this function is called the numerical flow of the integration method.
5.2.2 Order of a split
We shall now outline a systematic way to derive an integration method using a Hamiltonian split and to find its order. Even tough higher order integrators generally perform better, there is no guarantee that they will do so. Also, there is usually a trade off point where the extra computational effort starts outweighing the precession gain by increasing the order of integration. The order is simply a measure of Taylor expansion similarity, in our case we will derive the propagation scheme Φ and Taylor expand this flow. Φ is of order p if its Taylor expansion corresponds to the real flow Ψ to term number p. To show this, let us take the exponential general solution to a phase space path evolution as derived in equation 3.60,
h{·,H} f(q, p) |t=t0+h = e f(q, p) |t=t0 (5.8)
and split the Hamiltonian into two parts H = HA + HB. We can write
h{·,HA+HB } h({·,HA}+{·,HB }) f(q, p) |t=t0+h = e f(q, p) |t=t0 = e f(q, p) |t=t0 . (5.9)
since the Poisson brackets are linear in their arguments. We can not however assume that {·,HA} and h({·,HA}+{·,HB }) {·,HB} commute, in other words {·,HA}{·,HB} 6= {·,HB}{·,HA}. This means that e 6=
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eh{·,HA}eh{·,HB } since this mathematical property of exponential functions require the commutative property of the arguments. We can use this to derive the order of the splitting by expanding both expressions as
∞ i i X h ({·,HA} + {·,HB}) eh({·,HA}+{·,HB }) = = i! i=0 ∞ i i X h ({·,HA} + {·,HB}) = 1 + h({·,H } + {·,H }) + (5.10) A B i! i=2 and ∞ i i ! ∞ i i ! X h {·,HA} X h {·,HB} eh{·,HA}eh{·,HB } = = i! i! i=0 i=0 ∞ i i ! ∞ i i ! X h {·,HA} X h {·,HB} = 1 + h{·,H } + 1 + h{·,H } + . (5.11) A i! B i! i=2 i=2 By subtracting the two expressions we can easily see that all first order terms vanish
eh{·,HA}eh{·,HB } − eh({·,HA}+{·,HB }) =
∞ i i ! ∞ i i ! X h {·,HB} X h {·,HA} = h{·,H } + h{·,H }+ A i! i! B i=2 i=2 ∞ i i ! ∞ i i ! X h {·,HA} X h {·,HB} +h{·,H }h{·,H } + + A B i! i! i=2 i=2 ∞ i i X h ({·,HA} + {·,HB}) − = O(h2) (5.12) i! i=2 this means that
h{·,HA+HB } h{·,HA} h{·,HB } 2 e f(q, p) |t=t0 = e e f(q, p) |t=t0 + O(h ). (5.13)
We can no state that our split into eh{·,HA}eh{·,HB } has an error equal to the difference between all terms after the first order Taylor expansion term. This splitting is called a first order integrator, with error O(h2). If we instead split
h {·,H } h{·,H } h {·,H } e 2 B e A e 2 B (5.14) we can see that the Taylor expansion of this expression compared with the Taylor expansion of the exact solution compares to the second term, making this splitting of second order with error O(h3).
5.3 Bulirsch-Stoer method
The BulirschStoer method combines the three ideas of Richardson extrapolation, rational function extrapola- tion in Richardson-type applications, and the modified midpoint method (Press et al., 1992). This technique is not designed for differential equations containing non-smooth functions and it is not designed for differen- tial equation with singular points inside the integration interval. Thus it is suited for integration Hamiltonian systems with slight electromagnetic perturbations. The Bulirsch-Stoer method is widely recognized as one of the best ways to calculate high accuracy solutions to ordinary differential equation with minimal compu- tational time.
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5.3.1 Richardson extrapolation
Richardson extrapolation, or the deferred approach to the limit, is a method for convergence acceleration. More explicitly, the idea behind Richardson extrapolation is that the value of convergence for a numerical calculation in a limit is a function itself that is dependant on the number of terms in the calculation. If this is the case we can fit a rational function, or a polynomial in our case, to a set of resulting values for different number of terms and then evaluate the resulting function at the desired point. Since the goal is to approximate a exact solution often one evaluates this function where it represents infinitely many terms. By doing this one can quickly get a much better estimation of the convergence value at infinity rather than calculating absurd amounts of terms. Obviously this will not work unless the sum of the terms is a relatively smooth function. However, in celestial mechanics there is not a lot of non-smooth behaviour and as such this will not be a problem. As an example of Richardson extrapolation let us consider the series
N X A = lim ai. (5.15) N7→∞ i=1 let us consider a function
N X f(N) = ai, (5.16) i=1 that will evaluate to limN7→∞ f(N) = A. We cannot practically calculate A unless ai is some special expression. If however we assume that f(N) is a rational function of N, even if we can never find f we can fit a function to it by calculating a set of different series evaluations {f(1), f(2), f(3),... } with corresponding input values {1, 2, 3,... } and fit a function to this data fˆ(N) ≈ f(N). If we then take the ˆ now known approximate function and evaluate limN7→∞ f(N) we have performed Richardson extrapolation ˆ as limN7→∞ f(N) ≈ A.
5.3.2 Modified midpoint method
The specific limit we are going to apply this to is the modified midpoint method with infinitely many steps, or with a step size that tends to zero. The specific modification of the usual midpoint method is made to remove oscillatory terms in the error of the method. This together with the asymptotic expansion of a midpoint method in the square of the step size shows us that we can improve the method by the square of the step with extrapolation instead of just by the step itself. The modified midpoint method can be expressed as
TN+1 − TN hj = (5.17) nj
uN = u(TN ) (5.18)
z0 = uN (5.19)
z1 = uN + hjf(z0,TN ) (5.20)
zm+1 = zm−1 + 2hf(zm,TN + mhj) ∀ m ∈ {1, . . . , nj − 1} (5.21) 1 unj = (z + z + h f(z ,T )) (5.22) N+1 2 nj nj −1 nj nj N+1
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5.3.3 Deufelhard serie
In order to fit a polynomial to Richardson’s deferred approach to the limit of infinitely many sub steps in a modified midpoint method we need to choose data points for the fit. Several series have been proposed but the one we deemed best for our purpose was the one Deufelhard (Kirpekar, 2003) suggested
0 nj = 2(j + 1) ∀ j ∈ N . (5.23)
Since a numerical calculation needs a finite series, let us restrict j by a number k = 8, j < k, giving us a a set of 8 n’s,
n = {2, 4, 6,..., 16}. (5.24)
If at this point we deem that our fit is still not good enough, we instead must shorten the general step length instead of the sub step length.
5.3.4 Neville’s Algorithm
Neville’s Algorithm is designed to build a polynomial that goes through a set of n points by combining lower grade polynomials in sequence until a final expression is derived. For example, if given 5 points, the first set of equations will be 5 constants through all 5 points, then these 5 constants will combine in a way that we have 4 lines going through 2 of the points, then again forming 3 parabolas passing through 3 of the points each, continuing until we have a grade 4 polynomial passing through all five points. The algorithm is described by
nj+1 Pj,j(h) = uN+1 ∀ j < k, (5.25)
(hi − h)Pj,i−1 + (h − hj)Pj+1,i Pj,i(h) = ∀ 0 ≤ i < j < k. (5.26) hi − hj
This is the the algorithm used to fit the Deufelhard series evaluation to a function which we can evaluate at 0 step length.
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Part II Solar system population
6 Introduction to Solar system population
To create a useful simulation the input state must be carefully considered and designed. Unfortunately there are many unknowns in the problem considered in our work, but we will here try to compile what is known, to what accuracy, and to motivate the assumptions of the required parameters that are not known. If the reader is knowledgeable on the constituents of the Solar system this part is not vital to progress through the text, however the databases listed at the end may provide useful information.
7 The Sun
As we have not yet implemented nano-dust interactions with the Solar wind plasma we will not cover the theory of Solar wind plasma to dust interactions at this time. One important factor to consider is the electromagnetic radiation that emanates from the Sun causing the previously mentioned radiation pressure and Poynting-Robertson drag perturbations. From mesurnments of the Solar radiation one can find a mean 26 total luminosity of L0 ≈ 3.846 · 10 W integrated over the entire surface (Mamajek et al., 2015). As the radiation expands at the speed of light in spheres the energy flux density will decrease with the distance from the Sun R as
L Ψ = 0 . (7.1) 4πR2 We consider the Solar luminosity as a constant since cycle variations are small enough not to matter for the current calculations (Willson and Hudson, 1991).
8 The planets
The prominent sources of gravitational perturbation influencing the dynamics of dust in the Solar system are the eight planets. As such to create a appropriate initial state for the planets is crucial for a realistic simulation. This can be done by using observational data gathered by the NASA Jet Propulsion Laboratory. Using these observations one can also calculate the planetary masses and the Solar mass. Other then their states and masses not much information about the planets are needed to perform our simulations.
9 Small bodies
Our goal is to simulate the dynamics of the small bodies in the Solar system in a statistical manner, and thus it will be important to consider their populations and properties. We shall here only cover some basics on the major small mass constituents of the Solar system.
9.1 Comets
A comet is a body considerably smaller then a dwarf planet that is made of rock and ice, typically a few kilometres in diameter, orbiting the Sun. When such a body passes close to the Sun the Solar radiation
Page 24 D. Kastinen Meteors and Celestial Dynamics begins heating the body until sublimation occurs and the comet outgases the previously icy material. This produces a visible atmosphere or coma around the comet that some times produces a tail. In recent times we have even begun to examine the small magnetic environment of the comet as it produces its own small magnetosphere and plasma environment when interacting with the Solar wind and the Solar radiation. Also, due to the jets and streams of outgassing, dust of various sizes and compositions are ejected making the environment of a comet as it closes in on the Sun a very dynamic and unique object to examine. The coma and tail are much larger then the actuall nuclei of the comet itself and can sometimes, if sufficiently bright, be seen from the Earth without a telescope, such as the famous Hale-Bopp comet, formally designated C/1995 O1. This comet could be seen from a clear sky in the recent perihelion passage of 1996 - 1997. Short-period comets originate from the Kuiper belt and its scattered disc of material that lies beyond the orbit of Neptune. It has recently become widely acknowledged that long-period comets originate from the Oort cloud. The Oort Cloud is a nearly spherical cloud of comets, dust, ice, and other derbies, which surrounds the Solar system. Its width extends to heliocentric distances of more than 100 000 AU and is currently thought to inject about 12 comets with perihelion distance closer then 3 AU each year (Wiegert and Tremaine, 1999). Recently work has been made on the dynamics of dust particles released from the Oort cloud comets (Nesvorn`yet al., 2011) that is relevant to our purpose.
Figure 6: Histogram of the different distribution of orbital elements within the Pan-STARRS Solar system synthetic model.
The Pan-STARRS synthetic Solar system model predicts the population of short period comets currently in the Solar system to be distributed as shown in figure 6. The main difference between an asteroid and an comet is the presence of an extended, gravitationally unbound atmosphere surrounding the comet central nucleus. If however, a comet experiences so many perihelion passages that it looses nearly all of its volatile ices it will come to resemble a small asteroid.
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10 Dust
Even tough dust in the Solar system may appear sparse it is a major part of the population. The inter- planetary dust is distributed over the entire Solar system. This dust is also numerous enough to scatter Solar light, called the zodiacal light, and produce measurable thermal emissions (Levasseur-Regourd, 1996). This light becomes a predominant feature of the night sky in the 5 µm to 50 µm wavelength. The total mass influx of dust onto the Earth has been estimated to be between 55 and 270 tones each day, in (Plane, 2012) a comprehensive comparison between 13 different mass influx models and measurements is compiled. This mass influx is a huge number with a equally huge estimated range, and is still today a open question in science. This mass influx has various effects on the Earth. For example the mass influx of meteoroids produces a external input of atmospheric metals in the mesosphere and lower thermosphere as there have been good correlation between amount of metallic ions in the lower thermosphere and meteor showers (Plane, 2003). Some of the effects of this input of extraterrestrial material into our atmosphere is sodium D-Line airglow, persistent meteor trains, sporadic layers of metallic ions with impact on radio communications, and noctilucent clouds. It has also been shown that dust is a impact hazard to spacecraft. For example, the ESA communications satellite Olympus lost pointing control in 1993 likely due to the impact of a Perseid meteoroid (Caswell et al., 1995). During the same year due to the Perseids, NASA postponed its space shuttle launch to avoid the meteor shower maximum, and diverted the pointing of the Hubble Space Telescope to avoid lens damage. Thus it becomes evident that the presence of dust in the Solar system is no negligible matter. But to closer examine the dynamics of dust and their effects one must begin with its origin.
10.1 Poynting-Robertson lifetimes
To find the origin of the dust observed to enter the Earth atmosphere and the general interplanetary dust we must consider the previously mentioned Poynting-Robertson effect in section 4.0.2. A quite logical first guess as to the origin of the interplanetary dust would be left over material from the formation of the Solar system. We know that when the Sun ignited it generated a large wave of radiation pressure that pushed all excess gas out of the Solar system creating a close to clean interplanetary environment. This wave of radiation would however be too weak to push larger particles out of the Solar system and thus they would possibly be left in the Solar system. This is where the effect of radiation drag is of vital importance, as a direct consequence of this effect is that all particles either spirals inwards toward the Sun due to the drag or outwards away from the Sun due to pressure. There is also the third cases where the two forces are kept in equilibrium but due to the amount of gravitational perturbations in the Solar system the particles will scatter and sooner or later collide with another object as they are forced to leave such a equilibrium. As early as 1962 the time of Solar impact was calculated (Guess, 1962) for circular orbit of initial semi major axis a, of a spherical homogeneous sphere of density ρ and radius r that is absorbing and emitting all incoming Solar radiation isotropically across the entire solid angle. The resulting formula
t = 7.0 · 106rρa2 y. (10.1)
9 compared to the age of the Solar system ts ≈ 4.5682 · 10 y (Bouvier and Wadhwa, 2010) imply that all dust present at the formation of the Solar system should today have collided with the Sun. But since we today see a steady influx of mass into the atmosphere in the form of meteoroids and meteorites, we must assume that the population of dust is refreshed by some mechanic. The best explanation to this seemingly steady state meteor complex at the Earth is the continous ejection of dust from comets and asteroids. But as many comets also have relatively short active lifetimes of ejection, the population of comets must also have a mechanism for population renewal. There could be several such mechanisms but the most probable
Page 26 D. Kastinen Meteors and Celestial Dynamics source for long-period and short-period comets are the so called ’kicking’ of comets out of the Oort Cloud and the Kuiper Belt respectively. For example in (Levison et al., 2001) the origin of Halley-type comets are examined by considering injection of comets from the Oort cloud.
10.2 Comets
Figure 7: Ejection speed as a function of the common logarithmic mass of the particle.
Since the most probable major source of dust in the Solar system seems to be comets, we must address their relation to the creation of dust. The feature that has made comets visible to the naked eye and therefore known to mankind for thousands of years is its dust and gas tail. This trail increases as the comet approaches the Sun. The basic reason for this loss of mass was explained by a icy conglomerate model as early as 1950 in (Whipple, 1950). In this model the primary component of the nucleus is assumed to be ice with dust particles embedded within it. Dust here is defined as any substance that is in a solid state at temperatures of a few hundred kelvins. As the ice sublimes due to increasing Solar radiation, these grains are released. Their subsequent motion away from the comet nucleus is due to interaction with the gas outflow, radiation pressure, and gravity. This interaction is of course dependant on the size of the dust grains and this effect forms the well known dust and gas trail. To model reasonable meteoroid streams or even their dissipation into a sporadic complex one must model the ejection from the parent body in a sensible way since this is the initial perturbation causing their trajectories to diverge from the comet itself.
10.2.1 Dust ejection
We previously mentioned the icy conglomerate model by Whipple as presented in (Whipple, 1950) and (Whipple, 1951). We shall here quickly review this model and its evolution. Models like this considers the absorption of sunlight by the cometary nuclei, which in consequence heats up to the point where sublimation of the ices present on the surface of the nucleus occurs. When this ice generates gas a pressure difference is generated and the gas shoots outwards. This is quite a violent process, and since the ice is polluted by dust
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particles they are ejected together with the gas. As such the ejection velocity of the particle is dependant on the gas interaction with the dust, i.e. how much momentum the gas transfers to the particle. We will not cover the entire derivation of the ejection physics but instead summarize some of the key points with the models. The Whipple model assumes that the sublimating gas uniformly expands from the sunlit hemisphere of the nucleus, where the mean thermal velocity of the gas is ∝ r−τ and r is the heliocentric radius. Thus the mean thermal velocity resolves to
8k T 1/2 1 v¯ = B g . (10.2) g πµ rτ
−1/2 It is also assumed that the temperature of the gas is the maximal surface temperature Tg = 300r K. Assuming elastic collision between gas and dust particles Whipple then compensates with a drag coefficient
for the inelastic collisions that was chosen to kd = 26/9. Whipple also assumed a spherical nucleus of mass M giving the mass loss due to absorbing sunlight with total luminosity L0 and subsequently sublimating as
L 1 M˙ = πR2 0 . (10.3) c 4πr2 nH
Where the sublimation energy used was H = 1.88·106 J/kg, approximately the sublimation heat of ammonia, −1 and the coefficient n ≈ 1. Here the nuclei radius Rc is a function of the mass Mc through the spherical constant density assumption. Finally, it is assumed that the thermal velocity relation exponent is τ = 0.25. Combining all these expressions with the differential equation for the acceleration of the dust, including the gravitational attraction of the nucleus and assuming dust grains with shape parameter A and mass m, we find two different equations for the escape velocity. The first one where the dust particles are slow compared to the gas thermal velocity
˙ 2 kd Mv¯g A 2GMc V∞ = − , (10.4) 2 πRc m Rc and the second where the dust is fast compared to the gas velocity,
˙ 2 kd M(¯vg − r˙) A 2GMc V∞ = − . (10.5) 2 πRc m Rc
Where Whipple assumes that µ = 20 · 1.661 · 10−24 g allowing a reduction to the frequently used formula
1 8πG v2 = 43R − ρ R . (10.6) c nsσr2.25 3 · 43 c c
Recently due to discrepancies between observation and the ejection speeds derived by Whipple, (Ma et al., 2002) reinvestigated the problem starting from simple physical principles and derived new formulas for the mass loss of the comet,
2 ˙ Rc L 1 1 Mc = 2 − 2 , (10.7) 4H r rs
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and the ejection speed of meteoroids,
2 WRcL 1 1 2GMc v = 2 − 2 − , (10.8) 8πHαsσ r rs Rc where W is the mean thermal velocity, Rc is the radius of the comet, L is the Solar luminosity, H is the latent heat of vaporization, Mc is the mass of the comet, G is the gravitational constant, α is the fraction of the comet that is active, s is the radius of the assumed spherical dust grain, σ it’s bulk density, and rs is the critical heliocentric radius for sublimation to occur.
26 6 To get an idea of the ejection speed profile we have taken the values L = 4 · 10 J/s, H = 2 · 10 16 3 J/kg, Mc = 1.55 · 10 kg, Rc = 10 km, W for water to be 580 m/s at 273 K, σ = 3700 kg/m , α = 0.15, r = 1 AU, rs = 1.25 AU, and have chosen to vary the mass of the particle. Since we have also assumed spherical particles with bulk density σ we are there thereby varying its radius s. The results of this can be seen in figure 7, where a interesting cut off point where the square root becomes imaginary can be seen. This represents the point where the gravitational attraction of the comet overcomes the gas ejection pressure force and thus the particle never escapes, this has to be accounted for in numerical implementation of a ejection simulation. This model by (Ma et al., 2002) has however been criticized since it appears to contain a few mistakes in its derivation (Ryabova, 2013).
As we can see there is lot more to model then just fixing the ejection speeds of the dust grains. To perform a realistic simulation of the formation of a meteoroid stream one must also consider many other questions. Such as the fraction of the comet that is active, i.e. has areas of ices susceptible to Solar radiation absorption facing the Sun. What the bulk density of the dust grains and the comet is. At which point the ices sublimes at such a rate that gas outflow dominates over gravitation exerted by the nucleus. The electromagnetic close contact bonds, or stickiness, of the particles. What the angular momentum of the comet around its own axis is. The Solar cycle variations, the chemical composition of the comet, the amount of jets due to low possible ejection areas but high absorption rates. Anomalous nuclei geometry, and so on. At some point one has to make simplifications, fortunately the nature of Monte-Carlo simulations allow for many of the simplification errors to be minimized since we can use averages and distributions instead of specific assumptions. Also many of the singular effects are totally erased in the grand statistical scheme, such as the ejection directions due to cometary nucleus angular momentum.
We shall lastly cover the modification of Whipple’s model by Hughes (Hughes, 2000) resulting in a im- proved model. In Whipple the entire surface of the comet is active and undergoing sublimation but due to observed jets and non active areas we know that this is not the case. In other words the term 2π in Whipples formula was corrected to Ψπ, according to Hughes the typical value for Ψ is around 0.2 but can be between 0.05 and 0.5. Since only a part of the nucleus is active and thus only a part is absorbing Solar 2 2 radiation. Thus the term πRc is corrected to 2απRc . The typical value for α is assumed to be around 0.1. In Whipple the drag coefficient was fixed to kd = 26/9, this is substituted by another parameter ζ. This parameter has the value 2 if gas absorbed by the ejected particle is re emitted with the same velocity within a short timespan. If the dust particle however has a higher temperature then that of the ice sublimation then ζ > 2. If the particle absorbs gas molecules for a long timespan ζ < 2. Whipple assumed the radial component out of the comet to be constant but many have pointed out that the gases should be accelerated along the adiabatic pressure gradient as they expand into vacuum. Thus a coefficient θ ≈ 2 is introduced. Lastly Hughes disputes Whipples assumption that the velocity of the sublimated gas changes as a function of heliocentric distance and sets the exponent coefficient τ = 0. This results in a formula where the ejection speed is given by
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˙ 2 2Mc θv¯s A 2GMc V∞ = τ ζ − , (10.9) ΦπRc r m Rc where the mass loss is given as
r 2 1 M˙ = 2gπR2S 0 . (10.10) c c 0 r nH
For a good overview and more detailed description see (Ryabova, 2013) and references therein. Taking the most common models, we can find that the ejection velocities deduced are within the range 10 to 103 m/s, depending on the particle size, density, comet activity, perihelion passage, and so on. Thus, when examining a broad range of mass and comet parameters using a model for the ejection can be vital (Vaubaillon et al., 2005).
10.2.2 Critical sublimation radius
If the entire cometary nucleus is in thermal equilibrium with the Solar radiation, the critical radius at which sublimation starts is located around 1.25 AU. However this strongly disagrees with observations, one reason for this is that if only a fraction of the surface is active, i.e. parts of the comet is not radiating, sublimation can start at much larger distances. Also ices such as CO2 and CO will sublime a lot earlier. Most comets are observed to become active at around 3 AU (Delsemme, 1982) (Spinrad, 1987) and is a standard value we have used in our general comet simulations. Sublimation can however occur at even larger distances and when modelling specific comets, one would rather use observational values for this critical distance.
10.3 Asteroids
It has also been shown that asteroids can have significant dust production, for example 3200 Phaethon is today considered the most likely candidate as the parent body for the Geminids meteor shower (Whipple, 1983). Dust production of near-Earth asteroids can be handled the same as cometary dust and as we have not yet performed simulations on asteroids their more in depth theory will not be covered here.
10.4 Main belt
The main asteroid belt is considered a source of interplanetary dust as well due to its relatively high internal collision rate between asteroids. These collisions break apart the colliding objects and the clouds of dust scatter, being subject to perturbation they can eventally escape the main belt and become part of the interplanetary dust complex. This may be a dominant source of meteoroids near Earth (Dermott et al., 2002). This contribution will be considered in future simulations of the sporadic complex, but is irrelevant in simulations of ejections from comets.
10.5 Interstellar dust
Interstellar dust has also been observed in the heliosphere (Gr¨unet al., 1994). This population however is too small compared to other sources to be taken into account in any kind of simulations. It would also be very difficult to model sources for this population as the cosmic year, or the time it takes the solar system to revolve around the galaxy center, is around 200 million Earth years. As such we cannot perform repeated detection of the same galactic region but instead constantly pass trough new areas. However, in the future such a model can very well be tested by the software developed here.
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10.6 Nanodust
Solid particles that are exposed to the Solar wind and the Solar UV radiation carry an electric surface charge. They are therefore influenced by electromagnetic forces. Mann et al. (2007) showed that for dust particles with sizes of several nanometres, the surface-charge-to-mass ratio is large enough for their dynamics in the Solar wind to become similar to that of heavy ions. Such nanodust can reach a speed of the order of Solar wind speed and move on orbits away from the Sun. An intermittent flux of nanodust near 1 AU, Earth orbit, was indeed measured with the plasma wave instrument onboard the STEREO spacecraft (Meyer-Vernet et al., 2009). While there are still many open questions concerning the detection of the nanodust impacts with field measurements (Zaslavsky et al., 2012), with a similar type of field measurement dust impacts were also measured onboard Cassini between 1 and 5 AU (Schippers et al., 2015). A possible evidence for nanodust has also been found by Carpenter et al. (2007) through impact features on foil experiments at the International Space Station (ISS). The measurements with Stereo, Cassini and at ISS are in general agreement with trajectory calculations and with the suggestion that nanodust forms in dust-dust collisions near the Sun where dust fluxes and collision rates are highest (Mann et al., 2014). Czechowski and Mann (2010) studied the dynamics of nanodust in the region inward from 1 AU through numerical simulations of dust trajectories to estimate the flux rates. They found, in addition to those accelerated by Solar wind, a population of trapped grains within about 0.2 AU from the Sun. The presence of fast-flowing Solar wind, the magnetic field sector structure and the heliospheric current sheet contribute to shaping the trajectories. Both the trapping zone and the acceleration outward depend on parameters like dust surface charge, Solar wind speed and magnetic field at the location of the particle, all of which can induce time-variability. If nanodust is generated by mutual collisions of larger particles, also the stochastic nature of the sources near the Sun can generate a flux variation. Juh´asz and Hor´anyi (2013) have shown that even if the production is constant in time, the nanodust flux near the ecliptic plane can be time-variable as a result of the interaction with the interplanetary magnetic field.
11 Databases
There are several sources of information available for the scientific community today. We will here cover the databases used in our simulations.
11.1 Pan-STARRS Synthetic Solar System Model
Table 1: Data included in the Pan-STARRS Moving Object Processing System Synthetic Solar System Model.
File name Abbreviation Description Sample size S0 NEOs Near-Earth Objects ∼270k S1.01-10 MBOs Main Belt Objects ∼100k S2 MBOs to NEOs - ∼50k S.hilda - Hildas ∼1.8k Sc SPCs Short-Period-Comets ∼28k SH - Hyperbolic objects ∼8.3k SL LPCs Long-period comets ∼9.4k SR - Centaures ∼60k SS - Scattered ∼11k ST TNOs Trans-Neptunian Objects ∼48k St4,5,6,7,8 - Trojan of Mars, Jupiter, Saturn, Uranus, Neptune -
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In this work we will frequently mention the Pan-STARRS Moving Object Processing System (MOPS) Syn- thetic Solar System Model (S3M). This is a attempt at compiling a comprehensive flux-limited model of the majority of the small-body populations in the Solar system. This database was first presented in (Grav et al., 2011) and will be used in, for example, the Pan-STARRS 1 all-sky survey. The model consists of synthetic populations of the objects listed in table 1. The data is compiled in ASCII files with spaces as column separators and each row corresponds to a synthetic object, the current object parameters are in order:
1. MOPS id
2. Semi-major axis (AU)
3. Eeccentricity
4. Inclination (degrees)
5. Longitude of ascending node (degrees)
6. Longitude of perihelion (degrees)
7. Mean anomaly at epoch (degrees)
8. Absolute magnitude
9. MOID
10. Opik collision probability with Earth
11. Epoch of orbits (yyyymmdd)
12. Expected year of discovery after start (years)
We have used this database as a sample for the initial distribution of orbital elements for SPCs to create example simulations and shall in the future conduct several more simulation using this database.
11.2 JPL NAIF database
Table 2: JPL’s measurement database of Solar system major body gravitational parameters
3 2 3 2 Body AU /days GMsun/GMbody km /s MERCURY 0.491248045036476000D-10 6023682.155592 22031.780000 VENUS 0.724345233264412000D-09 408523.718658 324858.592000 EARTH 0.888769244512563400D-09 332946.048834 398600.435436 MARS 0.954954869555077000D-10 3098703.590291 42828.375214 JUPITER 0.282534584083387000D-06 1047.348625 126712764.800000 SATURN 0.845970607324503000D-07 3497.901768 37940585.200000 URANUS 0.129202482578296000D-07 22902.981613 5794548.600000 NEPTUNE 0.152435734788511000D-07 19412.259776 6836527.100580 PLUTO 0.217844105197418000D-11 135836683.768617 977.000000 SUN 0.295912208285591100D-03 1.000000 132712440041.939400 MOON 0.109318945074237400D-10 27068703.241203 4902.800066
The entire JPL database used can be found at http://naif.jpl.nasa.gov/pub/naif/generic_ kernels/ through the NAIF JPL NASA joint public file server. The use of these files, called Kernels,
Page 32 D. Kastinen Meteors and Celestial Dynamics will be covered in section 28.3 describing the SPICE software. Available in this database is also the mass parameter (GM) for the Sun, Moon, and Planets for three different unit systems that we have used in simulations, listed in table 2.
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Part III Meteors
12 Introduction to meteors
This part is dedicated to explaining the meteor phenomena and its different dynamics, to provide a under- standing to those new to the area of meteor science about the measurement object and the various techniques involved. The part requires no prior knowledge and is very general and broad, not diving into specific models more than needed.
13 Observations and mesurnemnts
This section is dedicated to cover the the actual observation and measurements of the meteor phenomena and consequently the meteoroid itself.
13.1 Coordinate systems
The equatorial coordinate system: is a celestial coordinate system that can be implemented in spherical or rectangular coordinates, both have a geocentric origin and a fundamental plane consisting of the projection of the Earth’s equator onto the celestial sphere. The primary direction is oriented towards the vernal equinox, therefore the coordinate system does not rotate with the Earth. A right-handed
convention is followed. In most literature the variables given are the geocentric right ascension αG, declination δG, and geocentric velocity vg. Illustration can be found in figure 8
The ecliptic coordinate system: is a celestial coordinate system that can be implemented in spherical or rectangular coordinates. Both can have either geocentric or heliocentric origin and the fundamental plane consists of the ecliptic, or the apparent path of the Sun on the celestial sphere. The primary direction is oriented towards the vernal equinox, and a right-handed convention is followed. The common variables in a geocentric system are longitude λ, latitude β and distance ∆, or in a heliocentric system with longitude l, latitude b and distance r. Illustration can be found in figure 8.
13.2 Invariable plan and the ecliptic
The invariable plane in a planetary system is the plane normal to the total angular momentum vector of that system and passing through the barycentre. This plane is sometimes refereed to as the Laplace’s invariable plane and can be calculated given any set of state vectors of position and momentum (q, p) and body masses m. Starting with calculating the center of mass as
PN m x X = i=1 i i , (13.1) PN i=1 mi and the total momentum as
N X P = pi, (13.2) i=1
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Figure 8: Diagram of ecliptic coordinates, depicting the ecliptic, equator, ecliptic poles, longitude and latitude, and the vernal equinox direction. (Tfr000 / Licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. http://creativecommons.org/licenses/by-sa/3.0/)
Page 35 D. Kastinen Meteors and Celestial Dynamics the invariable plane is defined by the total angular momentum
N X H = (xi − X) × (pi − P). (13.3) i=1
It would seem as this invariable plane, a fundamental quantity in physics, would be perfect as a reference plane. This is however not so in most cases. Due to tradition we use the ecliptic, the orbital plane of the Earth, system oriented at the J2000.0 equinox as a reference (Beutler, 2005). Here J2000.0 refers to the state of the ecliptic at the fractional year 2000.0 in Julian calendar time. Since the total angular momentum of a system is invariant, hence the name invariable plane, to transform from a ecliptic system to a invariable one can be easily done with knowledge of the state vectors as we just need to find the Earth angular momentum relative the total angular momentum. If we set ˜i to the angle between the ecliptic pole J2000.0 and the pole of the invariable plane. And set Ω˜ as the ecliptic longitude of the intersection of the invariable plane with the ecliptic. Then, with calculations done on the DE405 JPL emphermis, which we will cover in detail later, we find these angles as ˜i = 1.5787◦ and Ω˜ = 107.5822◦ (Souami and Souchay, 2011). This will however not yield the correct results if we do not also rotate the coordinate system, in the ecliptic plane, so that the x axis is aligned towards the vernal equinox.
13.3 Meteors
Dust present in the Solar system may only be a small component of its total mass but this does not reflect its importance to many areas of research. For example, dust dynamics is crucial in cosmological models, in Solar system and planetary formation, modelling of accretion discs, in cometary and asteroidal models and in knowing our own local space environment for the benefit of future space missions. Dust also plays a role in the present Earth dynamics, for example when this dust encounters the Earth and interacts with our atmosphere, producing what is called a meteor, a violent event of ablation takes place causing a series of unusual phenomena to occur due to this extraterrestrial input and dispersion of material into our atmosphere. It is therefore interesting to investigate how dust behaves dynamically in the Solar system due to gravitational and electromagnetic perturbation over long time scales since this will be integral in modelling our Solar system. It is through models that we can for example explain how our meteoroid complex today is resupplied with material from comets and asteroids, and thereby find information on the small body population of the Solar system. However due to the limitations on detecting dust in space due to its sparse nature and small size one instead turns to Earth based observations. As such, to investigate the meteor flux, the meteoroid orbits and their interactions with the upper atmosphere is important but at the same time the method of investigation is limited, especially for precise measurements. High power large aperture radar observation is a recent technique to provide useful information on these matters.
13.3.1 Visual
The first type of meteor observations were visual observation of the plasma trail generated when the mete- oroid interacts with the atmosphere. By examining the trail one can quite accurately calculate a orbit by compensating for Earth’s gravitational interaction and then converting the heading and speed of the object into Keplerian elements. This however needs multiple simultaneous observations to be done. The actual nature of trail and the head of the meteor is however much more complex than a simple ball of plasma. For example, the meteor is not only dependant on the meteoroid but also the density of the atmosphere, the translation speed due to air flow at the point of interaction, and the composition and density of the meteoroid. All these factors and more affect the meteor and therefore the detection. From
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Figure 9: Concept image of a meteoroid head echo visual data a spectral analysis can be performed to extract information on the chemical properties of the meteoroid. This in combination with radar observation can provide additional information on the event.
13.3.2 Head echo
A meteor head echo is caused by radio waves scattered from the dense region of plasma surrounding and co-moving with a meteoroid during atmospheric flight. The received signals Doppler shift and targets range rate can therefore be used to accurately determine meteoroid velocity. This, as with visual observations, can be calculated into Keplarian elements to reconstruct a orbit. A illustration of the detection of a meteor can be seen in figure 9.
13.4 Ablation
Meteoroid ablation is the mass loss of meteoroids due to vaporization, fusion of molten material and frag- mentation (Rogers et al., 2005). As previously explained, the meteor phenomena occur when dust encounters the Earth and interacts with our atmosphere (Ceplecha et al., 1998). A meteor event detected by radar can be split in two different parts; the head echo, which is a signal echo originating from the ball of plasma co-moving with the meteoroid, and the trail echo, which is the echo from the trail of plasma left in the atmo- sphere drifting with the local wind speed (Kero, 2008). Radar systems are usually optimized to detect either head or trail echoes, but both kinds can be detected also with one and the same radar. If one configures the radar for trail detection the amount of head echo’s detected will be lowered and vice versa. The trail echo is traditionally split into two categories, the underdense and overdense trail echos. In an underdense echo each free electron scatters the incident wave independently and the signal becomes the sum of the contributions from all individual electrons. In a overdense echo the electron density is so high that the radio waves scatter as from a cylindrical, metallic surface. Apart from radar observations of ablation processes, one can also perform camera observation, most commonly visual observation, by recording the light emitted during the ablation process and in the ’afterglow’ of the plasma trail. As mentioned, this can yield spectral information
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Figure 10: Contour plot of the logarithm of the total number of events per cell of the Hammer-Aitoff pro- jection, entire MURMHED. The labels are the six sporadic sources: (SA) South apex,(NA) North apex,(ST) South toroidal,(NT) North toroidal,(H) Helion and (AH) Anti-Helion. of the meteoroid useful in determining composition, size and density. However the meteoroids need to be of much larger size then the lower limit for radar detections since the ablation process have to produce sufficient light to be recorded from ground observations, rather than just scatter the radio wave.
14 Meteor and Meteoroid complex
The meteoroid complex is a collection term for small bodies in the Solar System. Traditionally, the meteoroid complex is split into two parts; stream meteoroids and the sporadic meteoroids. There is also the term meteor complex, this is the collection term for all meteor events at the Earth. Thus, the meteor complex is a subset of the meteoroid complex, of all meteoroids that encounter the Earth and produces meteors. Since detection of meteoroids in space is difficult, one usually detects the meteor complex instead and use this to infer properties about the meteoroid complex.
14.1 Stream meteoroids
Shower meteors or stream meteoroids are a concentration of radiants with characteristics similar enough to proclaim that they originate from the same source. Practically all meteor showers are produced by the meteoroid stream generated from a parent body like an asteroid or a comet. In the case of a comet, thermal radiation from the Sun heats the surface, sublimating ice inducing jets of gas and dust being ejected into space. A comet closing in on its perihelion passage will form two tails, one gas tail directed parallel with the radiation pressure and one dust trail following in the wake of the comets orbit. We shall go further into this process later. The particles in this trail will have seemingly random velocities away from the parent body and their relative velocity to the parent body is small in comparison with their orbital speed thus making
Page 38 D. Kastinen Meteors and Celestial Dynamics
Figure 11: Hammer-Aitoff projection of all MURMHED meteor radiants with color encoded geocentric ve- locities with showers Orionids and Geminids marked. it possible to correlate them to the generating object. A dust trail from a comet or a asteroid produces a meteor shower when the Earth orbit and a filament of dust crosses paths in space and the particles become a part of our meteor complex. When considering this scenario we realize that, just like the exhaust fumes from a jet plane, the meteoroid streams will dissipate over time but in the case of meteoroid streams the time for dispersion ranges from hundreds to several thousand years if not more. More information about statistical modelling of meteoroid streams by chi-squared density functions can be found in (Drummond, 2000). Also, a more in depth description on the properties of comets and asteroids can be found in (Jenniskens, 2006).
In figure 11 a Hammer-Aitoff projection of the MURMHED database is shown with apex located at 270◦ and the Sun located at 0◦, this database will be covered in the coming section.
14.2 Sporadic meteoroids
The sporadic meteoroids is the most abundant source, yet it has for a long time been the least under- stood part of the Earth’s meteoroid complex. A Hammer-Aitoff projection of the meteoroid radiants from MURMHED is displayed in figure 10. Figure 10 is a Hammer-Aitoff projection logarithmic radiant density plot which has been zenith distance corrected to represent the correct meteor flux according to (Zvolankova, 1983). As can be seen the sporadic complex seems to be split into several different sources. The sporadic background must be understood for one to separate the stream meteoroids from the sporadic ones. The origin of the sporadic background is quite simply dominated by stream meteoroids that has dissipated and can no longer be traced back to a parent body. It seems today that our sporadic complex is in a fairly steady state, meaning that by calculating the total influx of mass into the atmosphere due to meteoroids one can also calculate the needed production of new particles to maintain the sporadic complex. A good summary of the events is given in (Wiegert et al., 2009), to quote:
”Eventually, perturbations accumulate and disperse the meteoroid stream, the original close orbital relation- ship between individual meteoroids becoming difficult to determine. At this point, the particles have become
Page 39 D. Kastinen Meteors and Celestial Dynamics part of the second component of the meteoroid population, the ’sporadic meteoroids’, which form a more diffuse but far from isotropic background flux of particles.”
15 Databases
15.1 MURMHED
The primary meteor database used in our work is from the Shigaraki Middle and Upper atmospheric Radar Meteor Head Echo Database (MURMHED) (Kastinen et al., 2014). High power large aperture (HPLA) radar observation is a recent technique to provide useful information on meteor influx and orbits, as well as interactions with the atmosphere (Nakamura et al., 2014). Since 2009 orbital data of about 120,000 meteors (Kero et al., 2012a) have been collected using a novel head echo analysis algorithm for the lower VHF band (Kero et al., 2012b). The data was collected using the middle and upper atmosphere radar (MU radar) of Kyoto University at Shigaraki (34.9N, 136.1S). The MU radar is a large atmospheric VHF radar with 46.5 MHz frequency, 1 MW output transmission power and 8330 m2 aperture array antenna. The database currently holds 53 different parameters for each event and a number of associated time series consisting of range, height, radar cross section, signal to noise ratio, radial velocity and meteoroid velocity. The database parameters are listed below.
MJD Ze uncorr[deg] Eccentricity Ze error[deg]
Year[UT] Ze correction[deg] Perihelion dist[AU] Vel error[deg]
Month[UT] Obs initial vel[km/s] Lon of asc node[deg] Semimajor axis error[AU]
Day[UT] Geocentric vel[km/s] Inclination[deg] Eccentricity error
Hour[UT] RCS[dBsm] Arg of periapsis[deg] Perihelion dist error[AU]
Minute[UT] SNR[dB] Period[yr] Inclination error[deg]
Second[UT] Start hgt[km] Heliocentric vel[km/s] Arg of periapsis error[deg]
Duration[s] End hgt[km] Radiant ecl lon[deg] Period error[yr]
RA[deg] Az of start point[deg] Radiant ecl lat[deg] Heliocentric vel error
Dec[deg] Ze of start point[deg] Sol lon[deg] Ecl lon error[deg]
Az[deg] Az of end point[deg] RA error[deg] Ecl lat error[deg]
Ze[deg] Ze of end point[deg] Dec error[deg] Hammer X
Az uncorr[deg] Semimajor axis[AU] Az error[deg] Hammer Y
In figures 12, 14, 13, 15, 16 and 17 a example entry of the times series for a echo is displayed.
Page 40 D. Kastinen Meteors and Celestial Dynamics
Figure 12: Meteoroid velocity time series of head echo Figure 15: Range-time intensity plot of head echo with MJD 55008.9589319839 from MURMHED with MJD 55008.9589319839 from MURMHED
Figure 13: Radial velocity time series of head echo Figure 16: Radar cross section time series of head with MJD 55008.9589319839 from MURMHED echo with MJD 55008.9589319839 from MURMHED
Figure 14: Height time series of of head echo with Figure 17: Signal to noise ratio time series of head MJD 55008.9589319839 from MURMHED echo with MJD 55008.9589319839 from MURMHED
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As any measurement instrument the MU radar has a bias. One of these biases are due to the fact that the antennas are not all seeing around the entire Earth. As they are stationed at a longitude and a latitude certain parts of meteor outburst event may for example fall below the horizon. Not only this however, one cannot assume that all is seen in the view limited by occupying a fixed point on the Earth as the antennas have a gain pattern surrounding its azimuth. This gain pattern, both theoretical and experimental can be read more about in (Kero et al., 2011). This can play an important roll in a statistical analysis. Another detection specific variable is the mass thresholds detectable by the radar. Since different masses and compositions create different targets for the radar echo, some configurations of masses cannot be detected by the MU radar. The meteoroid complex is dynamic in its size and composition distribution so this will also matter when applying theoretical models to a set of recorded data.
15.2 IAU meteor shower database
The IAU Meteor Data Center (MDC) is situated at the Astronomical Institute of the Slovak Academy of Sciences and under the auspices of International Astronomical Union (IAU). The MDC is the international organization that handles the designation of meteor showers and the collection of trajectory observations of meteoroid orbits. They host a list of 697 showers, where 112 are well established and 37 are pro tempore (or ”for the time being”), the rest is not yet fully confirmed to be actual meteor showers. Since our software intends to simulate meteor showers on the Earth by encounter with cometary dust trails such a database will be very useful. The entire database can be found at http://www.astro.amu.edu.pl/˜jopek/ MDC2007/index.php.
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Part IV Associating and classifying meteoroids
16 Introduction to associating and classifying meteoroids
This part is intended towards those interested in classification applications and the mathematical foundation upon which it is formulated. Some of the result analysis relies on understanding of the D-criterions and metrics given here. We also cover some historical progressions of the developed functions and methods.
At first glance the task of finding meteor showers seem trivial, but as ones dives into the subject, more and more difficulties emerge from different areas of physics and mathematics. In this part we shall attempt to create a review of the current area of research concerning association of meteoroids. We shall also later contribute to it by developing new methods and ideas for addressing the classification meteoroids.
A lot of problems in astrophysics and celestial mechanics can be traced back to the comparison of ob- jects, or collections of objects, and the resulting associations that are made. Often, this process involves creating some sort of similarity or dissimilarity measure, like the commonly known D-criterion. Sometimes, this measure can be an representation of distance, i.e. a metric. More often, however, it represents a subjec- tive view of similarity. Using such a measure, a set of objects can be examined for patterns. Most common types of association analysis for meteor measurements that we are aware of are the density function analysis, the wavelet transform, and cluster analysis. Another common method is to manually look at the data. The density function method will bin the space looking for regions of bins with high frequencies of meteors. However, to avoid manual selection, it is often needed to use a critical threshold where one can say that the density is large enough to imply the presence of a pattern. Sometimes this threshold can be derived by long term continuous observations, while this may not always be the case. The same principle applies for the wavelet analysis, with the difference that the specific pattern of the mother wavelet is searched for in addition to an increase in meteor count. Lastly, the cluster analysis approach involves choosing a critical distance, or threshold, at which a connection is made. If two objects are within this threshold they are connected to each other. After a connection has been made, there are several ways to merge the clusters. However, whatever method is used the clusters that are formed depend on the size of the critical threshold. Thus, all these methods rely on knowledge of either the background, i.e. the sporadic meteor complex and measurement uncertainties, or of the critical threshold needed for the shower to be selected.
In a straightforward and integrable system one can use models to exactly propose what restrictions to apply to the data in order to associate elements. If, however, the system is either chaotic or the amount of variables is so large that it appears impossible to predict a definite evolution of the system, statistics and approximations have to be applied. When considering meteoroids or meteors, the reliability of an association heavily depends on applied assumptions of, e.g. the correct value for the critical threshold (Kastinen et al., 2014). This is one of the main reasons we have chosen to apply a statistical approach to the problem, in order to start investigating the general statistics related to meteoroid streams.
17 Similarity functions
The concept of similarity is a highly subjective concept and similarities are viewed very differently depending on context and observer. Similarity functions are used in a multitude of areas such as multimedia retrieval, biometry, pattern recognition, manufacturing industry, and scientific research. This is the reason why it is hard to corner a area of mathematics which can hold all similarity measures. With a similarity measure
Page 43 D. Kastinen Meteors and Celestial Dynamics we are referring to a function mapping the Cartesian product of a set M with itself to the real numbers, M × M 7→ R. A similarity functions purpose is turning our concept of similarity between members of a set into a quantitative measure, often represented by the real number line. This concept closely resembles a area of mathematics called metric spaces, and often similarity functions can be proved to be metrics, but this is also not always the case. A metric may not correctly model the intuitive notion of similarity, for example single objects can be viewed as self-dissimilar while two distinct object can be viewed as identical. The direction of comparison can affect the comparison, for example comparing a orange to a apple may not be the same as comparing a apple to a orange. A similarity function does not even need to be transitive. There are however also advantages to restricting ones functions to metrics as they are a very well studied in mathematics with postulates that support common assumptions on similarity. From a more numerical point of view metric indexing also allows efficient indexing and searching (Zezula et al., 2006).
17.1 Metric spaces
There are many areas within mathematics which use the concept of a metric. For example Topology is the study of shapes and continuous deformations. This area focuses on the properties of a space that are preserved under continuous deformations, like stretching and bending, but transformations, like cutting and pasting or piercing, are not allowed. This field developed from examining geometry and set theory, as such it is closely linked to the metric as this is a geometric measure of a distance. When talking about metrics one should also mention the concept of a measure. A measure is a generalization of the concepts of length, area, volume, and so on. In this thesis we will not go further into the mathematics of these areas but only concern ourself with their application, for a more in depth explanation of measure theory see (Bogachev, 2007).
A metric is a function that defines a distance between elements of a set, this function must follow cer- tain rules. Denoting the set as E, x, y ∈ E, and define a function d : E × E 7→ R, then the pair (E, d) is called a metric space iff
d(x, y) ≥ 0, the separation axiom (17.1) d(x, y) = 0 ⇒ x = y, the coincidence axiom (17.2) d(x, y) = d(y, x), the symmetric relation (17.3) d(x, z) ≤ d(x, y) + d(y, z), the triangle inequality. (17.4)
The concept of a similarity does not need to follow these rules however, one can easily create a artificial metric that is useful for ad hoc tasks, but at this point one should take care. If just creating a function that maps our sense of similarity well in the cases that we thought of, this is fine. However, if we then extend its use to cases that we didnt specifically test, consider, or develop it for its behaviour may be totally unpredictable. A function that does not fulfil these metric requirements does not have all mathematical properties that a metric should have and therefore may not behave as we expect it too.
As with all physics, one ends up in a pros versus cons situation. Should one use a metric as the base of similarity of should one just model the similarity based on our own intuition? Are the more fundamental properties of measuring distance between points superior to our own intuition? To answer these questions one cannot do much else then experiment for the specific situation. It is very important to be aware of the factors that can affect your system, be it physical or mathematical. Often too much thought is put into forcing physical data into some kind of model and the mathematical description of ones conception may suffer for it. As a last note on metric spaces, it can be mentioned that from a single metric, a entire family of metrics can be
Page 44 D. Kastinen Meteors and Celestial Dynamics
found by applying metric preserving functions to the original. Some of these function are covered in (Doboˇs, Qn 1998). One can also create products of metric spaces (Mi, di) so that M = i∈1 Mi (M,N(d1, . . . , dn)) is pPn 2 + a metric space where N(d1, . . . , dn) = i=1 di and N is the euclidean norm defining N : M × M 7→ R + since di : Mi × Mi 7→ R . We shall not go deeper into these concepts since we have not yet implemented numerical version of these concepts in the software. This is however planned for future work, to examine different ways of applying metrics in this research field.
18 Models of meteoroid complexes
To be able to device a good similarity function, or validate a existing one, one needs not only observations but also models of the objects to be examined. This is so that tests can be performed and compared between simulations and observation. Also through synthetic data, calibrations of methods should be performed to eliminate arbitrary ad hoc choices. For example the cluster analysis, that uses a critical threshold to associate points, need a model to calibrate the critical threshold. If one does not use a model to justify this distance, the findings are as good as eye-balling the data. The best complete model we have found on the sporadic complex was developed and presented in (Wiegert et al., 2009). Since then several articles suggesting improvements and different approaches has appeared, like the recent (Nesvorny, 2013) and (Pokorn´yet al., 2014) which certainly could improve a model of the sporadic background.
All these models have one thing in common; they generate a sporadic background by the simulation of parent bodies producing dust to be exposed to long term perturbations before colliding with the Earth gen- erating a subset of the meteor complex. Due to the nature of perturbation dynamics only certain types of parent bodies, or dust producers, can contribute to certain parts of the sporadic complex. The perturbation effects include those of the Sun, the planets, radiation forces and collisions. For example according to simu- lations in (Wiegert et al., 2009) only minor contributions to the helion and anti-helion sporadic sources (see figure 10) are made by Halley-family comets and the main contributor was determined to be the Jupiter- family comets. There can however be high uncertainty in this kind of simulation if one does not consider the affects of chaos and ensure that a very small change of the initial condition does not drastically change the results. If the simulation is stable, it is perfect for examining sporadic statistics and therefore also for finding unknown meteor showers. Alas, this type of simulation is difficult to setup as it contains a lot of free parameters, such as dust production rates, that must be calibrated correctly.
19 D-criterion
Most of the previous attempts at creating a distance function for trajectories in celestial mechanics involve creating a function for the many different phase spaces or configuration spaces of Keplerian orbits. One is the so called D-criterion originally developed and presented in (Southworth and Hawkins, 1963) and then extended on by (Porubcan, 1977), (Drummond, 1980), (Jopek, 1993), (Jopek and Froeschle, 1997), (Valsecchi et al., 1999), and (Jopek et al., 2008). As this is a quite long list of evolution we shall only cover some of the essentials. The D criterion is based on a sum of weighted differences between the orbits dependent variables, we could however not find any literature that indicated that the D-criterion or its modifications satisfies the metric space axioms, with one exception. We have also observed that some of them does not fulfil the triangle inequality and as such the D-criterions should not be regarded as metrics.
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19.1 Southworth and Hawkins
The Southworth and Hawkins D-criterion, DSH (Southworth and Hawkins, 1963), was the first D-criterion developed in 1963 and can be written in familiar elements as
I e + e Π 2 D2 = (e − e )2 + (q − q )2 + (2 sin ab )2 + a b 2 sin ab , (19.1) SH b a b a 2 2 2 where
I i − i Ω − Ω (2 sin ab )2 = (2 sin b a )2 + sin i sin i (2 sin b a )2 (19.2) 2 2 a b 2 is the angle between the orbital planes and
i − i Ω − Ω I Π = ω − ω + 2 arcsin cos b a sin b a sec ab (19.3) ab 2 1 2 2 2 is the difference between the longitudes of perihelion measured from the intersection of the orbits. This function performs remarkably well for being the first of its kind. It is however desirable to gather as much material for comparison and as it is still well used, it will be good to examine it further.
19.2 Drummond
Later in (Drummond, 1980) modified the function to create the DD criterion which was a more correctly normalized version of DSH and can be written in familiar elements as
2 2 2 2 2 eb − ea qb − qa I ea + eb Θ DD = + + ◦ + ◦ , (19.4) eb + ea qb + qa 180 2 180
where
I = arccos (cos ia cos ib + sin ia sin ib cos (Ωb − Ωa)), (19.5)
and
Θ = arccos (sin βa sin βb + cos βa cos βb cos (λb − λa)). (19.6)
The ecliptic latitudes of the perihelion points can be expressed as
( βa = arcsin (sin ia sin ωa), (19.7a)
βb = arcsin (sin ib sin ωb), (19.7b) and the ecliptic longitudes of the perihelion points as
( λa = Ωa + arctan (cos ia tan ωa), (19.8a)
λb = Ωb + arctan (cos ib tan ωb). (19.8b)
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The λ parameter has the additional condition of adding 180◦ if cos ω < 0. This function is included in the
analysis due to its similarity to DSH since a similar statistical result will indicate that simple modifications of distance functions are not desirable to obtain new statistical properties. More on the analysis on this function can be read in (Drummond, 1981).
19.3 Valsecchi, Jopek, and Froeschl´e
In (Valsecchi et al., 1999) introduced yet another D-criterion, DN , but with another approach to the problem. Instead of focusing on Keplerian elements they based their function on geocentric quantities and used the geometric setup of Opik’s¨ theory of close encounters:
2 2 2 2 DN = (Ub − Ua) + w1(cos θb − cos θa) + (∆ξ) , (19.9)
where |U| = U is unperturbed geocentric velocity derived through the Tisserad parameter which can be used if one sets the mass of the Sun and the gravitational constant to unity and assume that the heliocentric velocity of the Earth is also unity. The Tisserad is a reasonable variable to use for a orbital measure as it is a constant of motion in the restricted 3 body problem. θ and φ are the two angles that give its geocentric
anti radiant and w1 is a suitable weight factor. The last component is somewhat more complicated and it can be expressed as the minimum of two elements:
2 2 2 2 2 (∆ξ) = min{w2(∆φI ) + w3(∆λI ) , w2(∆φII ) + w3(∆λII ) }, (19.10)
where w2 and w3 weight the differences between the geocentric anti radiant angle, φ, and the Solar longitude, λ,
φb − φa ∆φI = 2 sin , (19.11a) 2 ◦ 180 + φb − φa ∆φII = 2 sin , (19.11b) 2 λ − λ ∆λ = 2 sin b a , (19.11c) I 2 180◦ + λ − λ ∆λ = 2 sin b a . (19.11d) II 2 The angles of U can be determined by:
U θ = arccos y , (19.12) U U φ = arctan x . (19.13) U
Since U is based upon geocentric quantities it can easily be calculated from the declination, δG, and right ascension, αG, and the geocentric velocity VG of the observed head echo:
U − cos δ cos α x V G G U = R (λ )R () G − cos δ sin α , (19.14) y z y 29.7 G G Uz sin δG
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where λ is the Solar longitude at the observation time and is the inclination of the ecliptic plane to the celestial equator. The vector is described in a heliocentric ecliptic coordinate system making the transfor- mations
cos λ − sin λ 0 Rz(λ ) = sin λ cos λ 0 , (19.15) 0 0 1
cos 0 − sin Ry() = 0 1 0 . (19.16) sin 0 cos
This criterion is novel in its approach and it shows good qualities and will in the future be implemented numerically, more about this criterion can be read about in (Valsecchi et al., 1999),(Jopek et al., 1999) and (Jopek et al., 2003).
19.4 Jopek, Rudawska, and Bartczak
More recently in 2008 (Jopek et al., 2008) introduced the new criterion,
2 2 2 2 DV = wc1(ci1 − cj1) + wc2(ci2 − cj2) + 1.5wc3(ci3 − cj3) + 2 2 2 2 + we1(ei1 − ej1) + we2(ei2 − ej2) + we3(ei3 − ej3) + 2wE(Ei − Ej) , (19.17)
where wc1, wc2, wc3, we1, we2, we3 and wE are suitably chosen weight factors. In (Jopek et al., 2008) they −2 −2 −2 recommend using wck = (2σck) , wek = (2σek) , wE = (2σE) , k = 1, 2, 3 where σ denotes the expected standard deviations of each variable from a stream. This distance function is defined in the domain of the heliocentric orbital elements. Following the direction pointed out by Nesluˇsanthe full set of the vectorial elements are utilized in the metric. A orbit is described by the triplet
ci Oi = ei , (19.18) Ei
where the Laplace-Runge-Lenz vector is
e 1 r˙ × c r e = e = − , (19.19) 2 µ |r| e3
which magnitude is equal to the numerical eccentricity, |e| = e, and the area vector is
c1 c = c2 = r × r˙, (19.20) c3
and the energy is
Page 48 D. Kastinen Meteors and Celestial Dynamics
1 µ E = ˙r2 − , (19.21) 2 |r|
where µ is the standard gravitational constant. The elements used in the metric can be calculated from r = (x, y, z)T and ˙r which are the heliocentric vectors of the position and velocity of the meteoroid. More can be read on this in (Neslusan, 2002).
This is the only D-criterion we have found that fulfils all the metric properties and has been constructed independently by (Kholshevnikov, 2008) and can be made to coincide if the correct weights are chosen.
20 Metric of phase-spaces
Unlike the previously mentioned D-criterion that were developed for the specific task of associating orbits, the functions in this chapter were developed to be proper metrics for the phase space of Keplerian orbits. The question of topological and geometric structures on this space did not receive much attention until 1970 when the brilliant result of Moser (Moser, 1970) showed that the singularities in any system of elements for elliptic orbits is due to the fact that the topology of the space of elliptic Keplerian orbits with fixed negative energy is that of the product of 2-dimensional spheres.
In (Kholshevnikov and Vassiliev, 2004) published their work on the geometric structure of 5-dimensional space of of elliptic Keplerian orbits were they generalized their previous developed Euclidean and Chebyshev metrics with a H¨oldertype metric. The spaces worked on are denoted E and E∗ and a orbit is denoted E. Elements of E are oriented Keplerian ellipses with positive semi major axes and eccentricities, where the case zero eccentricity produces a oriented circle without marked pericenter. Elements of E∗ are the same but the case of a circle instead has its pericenter marked. These two functions respectively
( ) 1 Z 1/p ρ(p, E, E0) = min sp(Q(u),Q0(u + v))du , (20.1) 2π for mapping E × E onto R+ and
1 Z 1/p ρ∗(p, E, E0) = sp(Q(u),Q0(u))du , (20.2) 2π for mapping E∗ × E∗ onto R+, where s(Q(u),Q0(u0)) is the R3 euclidean distance between the two points Q and Q0 on the two ellipses having eccentric anomalies u and u0. In the case of the metric for E the distance is heavily dependant on the lines of apsides even though for example two coplanar orbits with very low eccentricity almost occupy the same set in R3. This is however not the case with the metric of E∗, which is why we have chosen to only examine this function closer.
The normalized squared distance function W (u, u0), following the relation s(Q(u),Q0(u0)) = p2aa0W (u, u0), can be expressed as
0 0 0 W (u, u ) =W0 + W1 cos u + W2 sin u + W3 cos u + W4 sin u + 0 0 0 0 + 2(W5 cos u cos u + W6 cos u sin u + W7 sin u cos u + W8 sin u sin u ). (20.3)
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By using the eccentric and true anomaly, a point on a elliptic Keplarian orbit can be expressed as
r = P(cos u − e) + S sin u, (20.4) a giving the components of W (u, u0) as
0 2 0 02 0 0 4W0 = 2(α + α ) + αe + α e − 4P · P ee , (20.5a) 0 0 W1 = P · P e − αe, (20.5b) W = P0 · Se0, (20.5c) 2 0 0 0 W3 = P · P e − α e , (20.5d) 0 W4 = P · S e, (20.5e) 0 2W5 = −P · P , (20.5f) 0 2W6 = −P · S , (20.5g) 2W = −P0 · S, (20.5h) 7 0 2W8 = −S · S . (20.5i) The P and S vectors can be expressed in familiar elements as
cos ω cos Ω − cos i sin ω sin Ω P = cos ω sin Ω + cos i sin ω cos Ω , (20.6) sin i sin ω − sin ω cos Ω − cos i cos ω sin Ω S = ηQ = η − sin ω sin Ω + cos i cos ω cos Ω , (20.7) sin i cos ω √ 2 a 0 a0 where η = 1 − e , α = a0 and α = a .
To reduce computation time and to simplify the calculations the special case of p = 2 was choosen and can, as pointed out in (Kholshevnikov and Vassiliev, 2004), be reduced to
2 0 0 p 2 2 ρ (2,E,E ) = 2aa W0 − (W5 + W8) + (W6 − W7) . (20.8)
Later in 2008 K. V. Kholshevnikov (Kholshevnikov, 2008) further extended on this by presenting even more metric spaces of Keplerian orbits and their subspaces. In (Kholshevnikov, 2008) the two spaces examined were the space H(b), b ≥ 0 of curvilinear orbits and the space H of orbits, both curvilinear and rectilinear. The space H does not however distinguish between ascending or descending orbits while the space of H(b), b ≥ 0 does not allow for retrograde orbits. This modeling will have to be done outside of the metric space function and a post-distance calculation is trivial to perform. The distance function introduced into the space of H(b), b ≥ 0 is
2 1 2 2 % (ε1, ε2) = 2 |c1 − c2| + |e1 − e2| , (20.9) µ L1 and the metric of H is
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2 2 1 2 2 L2 2 %1(ε1, ε2) = 2 |c1 − c2| + |e1 − e2| + 4 (h1 − h2) . (20.10) µ L1 µ where the c and e vectors are the same as in equations 19.20 and 19.19.
These are used to describe the orbit in the H(b), b ≥ 0 space, in the H space a orbit is described by these vector together with the energy constant
r˙ 2 µ h = E = − . (20.11) 2 |r|
However, since the spaces are supposed to describe Keplarian orbits the following restrictions must be met in each space, limiting its dimensionality. In H(b), b ≥ 0
(c · e = 0, (20.12a) |c| > b, (20.12b) and in H
( c · e = 0, (20.13a) 2h|c|2 − µ2(e2 − 1) = 0. (20.13b)
Using these equations one can write the metric for both spaces in familiar terms as
2 1 √ 2 2 % (ε1, ε2) = (p1 + p2 − 2 p1p2 cos ξ) + (e1 + e2 − 2e1e2 cos ζ) (20.14) L1 and
2 2 2 2 L2 1 1 %1(ε1, ε2) = % + − , (20.15) 4 a1 a2 where
cos ξ = cos i1 cos i2 + sin i1 sin i2 cos (Ω1 − Ω2), (20.16)
cos ζ =(cos ω1 cos ω2 + cos i1 cos i2 sin ω1 sin ω2) cos (Ω1 − Ω2)+
+ (cos i2 cos ω1 sin ω2 − cos i1 sin ω1 cos ω2) sin (Ω1 − Ω2)+
+ sin i1 sin i2 sin ω1 sin ω2. (20.17)
Although it has been mentioned in (Todorov, 2012) that the ”spaces of Keplerian curvilinear orbits, all orbits and elliptic orbits with marked pericenter cannot carry a norm, compatible with their standard topology”, this need not affect the usefulness of the derived metrics.
Page 51 D. Kastinen Meteors and Celestial Dynamics
Part V Statistical methods
21 Introduction to statistical methods
This part is dedicated to explaining some basic concepts of statistics that are used in our work. If the reader is unfamiliar with terms such as Monte Carlo methods, this part is highly recommended. We will also discuss cluster analysis and its use in meteor science and how one should approach the method and the problem.
22 Overview
As our software in oriented towards being able to perform the integration of small body dynamics in a statis- tical fashion by Monte Carlo sampling of a distribution, we must cover some basic statistical methods. Our goal is also to build a platform on which we can easily generate synthetic data to test models and methods, or calibrate analysis of observations. To this end we will also cover the concept of association and cluster analysis. For example, this is used for fining head echoes originating from the same parent body. When performing such a search the probability of false association is extremely important.
Statistics are merely functions performed on a entire set, for example the mean value, variance, and corre- lation are all functions defined as
f : B(R) 7→ R. (22.1)
This directly makes all higher mathematical construction on the set unimportant to the characteristics of statistics. For example, consider the travelling salesman problem. This problem asks the question: given a set of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? let us now assume that we are using different functions to determine the distance between the cities, this can obviously affect what route is the most efficient. It does not however affect the fundamental set of nodes and connections itself. So, for example, we can infer things about the map without ever choosing a distance function by, for example, looking at the statistics of connections between nodes. Obviously this is only useful when the amount of nodes and connections are far greater then any human can observe and comprehend. We can also use statistics to map how these different functions affect the distances in general. Taking the mean and variance of all distances and show how these change with the chaining functions will give a fairly good picture of the dependence on the function. The important fact is that the statistical meth- ods are just as valid regardless of the functions used, they are still just mapping a set of data, regardless of how it was derived. One can however question the validity of the function from a physical point of view thanks to statistics as well, such that it should actually represent distance travelled.
Using statistics as probability is a valid approach to many problems when one knows that a phenomena is not subjected to large none-modelled perturbations. Then the actual distribution of event outcomes will become the probability of that outcome, with a given level of reliability. For example, consider that we do not know how a certain coin responds to a toss, it may not be a 50/50 % chance of heads or tails due to irregularities in the coin or external factors. One way to make a prediction would be to start performing tosses with said coin, after many tosses we may find that the distribution of results are instead 25/75 %. We can then use this as our prediction model stating that ”there is a 25% chance of heads this toss”. This kind of empirical method is not applicable if experiments cannot be performed, let us say that we only
Page 52 D. Kastinen Meteors and Celestial Dynamics
have a picture of the coin in question. Another approach would then be to first try to extract the physical parameters from said picture, then create a model from classic mechanics, including air where the coin is the be tossed, and numerically solving this model with different initial conditions. This method introduces a multitude of challenges, firstly the model of the coin and toss must be correct, then choosing a sufficient set of initial conditions is also needed. When all these have been fulfilled we can again perform statistics on this generated data that is not measured but synthetic, and use this as a model of prediction. In the end this method may yield a 28/72 % chance, which is close but not what experiments would have yielded. Often this is due to the fact that the model of the event not exact. Most often it is nearly impossible to distinguish such a error since the event being modelled occurs rarely, is very subtle with a low signal to noise ratio, or has high uncertainties involved. In our example of the coin, assumptions on coin density and dimensions can be wrong, wind speed in the room where the toss is performed is highly unpredictable and may affect outcome, and so on.
What this above example illustrates directly translates to our work. We have measurements of parent bodies and from this we try to derive a distribution as a result of repeated simulations. From the resulting synthetic data we then want to draw conclusions about outcome. And these conclusion need not always be predictions but also aspects of the outcome that we cannot observe, essentially generating a missing piece of the puzzle ourself. As one of our case studies covered in section 42, where we find the mass distribu- tion transfer between a comet and the Earth, something that cannot be measured accurately by current instruments.
23 Monte Carlo
The Monte Carlo method in statistical analysis and mathematics as a whole is more oriented towards experimental mathematicians. This term may be somewhat unfamiliar to most but a clarification of this may be useful for putting this work into the correct context. The core difference between a theoretical mathematician and a experimental mathematician is that the theoretical deduces conclusion from postulates while the experimental one infers conclusions from observations and calculations. The Monte Carlo method is a perfect example of this difference. Monte Carlo methods are now an essential part of statistical analysis in contemporary science (Robert and Casella, 2013), and thus only a brief review is given here to better explain the methodology of the software. One of the features of Monte Carlo methods is that they rely on sampling and on concepts of probability. To illustrate this let us consider the calculation of π using uniform distributions. This may seem very roundabout, but it is a good example. Consider a two dimensional uniform distribution
1 U(x, y) = ∀ x ∈ [−r, r], y ∈ [−r, r], (23.1) A where the A = (2r)2 = 4r2 is the area within which the distribution is located. A circle centred in the origin 2 and with radius r will have area Ac = πr . The ratio of these two areas are
A π c = . (23.2) A 4 Since the probability distribution is uniform, we also know that this ratio is equal to the ratio between the probabilities of a random point ending up inside the circle and a random point ending up in the square, or in other words
Page 53 D. Kastinen Meteors and Celestial Dynamics
Figure 18: Example illustrating the concept of a Monte Carlo simulation calculating the value of π simply from the use of uniform distributions.
r 2π r 2π P R R U(x(ρ, θ), y(ρ, θ))ρdρdθ R R 1 ρdρdθ 1 πr2 π circle = 0 0 = 0 0 A = A = . (23.3) P R r R r R r R r 1 1 2 4 square −r −r U(x, y)dxdy −r −r A dxdy A 4r Thus, as we draw more and more points N from this uniform distribution and label the amount of points p 2 2 that fulfil x + y ≤ r as Nc we know that as we approach infinity the ratio of these two number will approach the ratio of the probabilities and thus the areas as
Nc π lim = ,N ∈ N. (23.4) N7→∞ N 4 We can now use this to approximate π by
N π ≈ 4 c . (23.5) N A example implementation of this is shown in figure 18.
Our application of Monte Carlo is in the sense that our initial distribution is known but it will affect the outgoing distribution in a unknown fashion due to the complexity of the in-between numerical propagation. More on how to interpret the data output of the software will be given in the next part.
24 Principal component analysis
Principal component analysis, or PCA, is a algorithm based on orthogonal transformations to convert a set of data points with possibly correlated variables to a set of points of linearly uncorrelated variables
Page 54 D. Kastinen Meteors and Celestial Dynamics
called principal components. The number of principal components are always equal to or less than the dimensionality of the input data, where the less then part apply if all the data has zero variance in a certain direction. This analysis procedure is constructed in such a way that the first principal component has the largest possible variance, and each succeeding principal component has the largest possible variance while being orthogonal to the preceding components. Normalizing the vectors we will find a new set of orthonormal uncorrelated basis vectors. This can be recognized as the Gram-Schmidt process of finding a orthonormal set of basis vectors. The principal components are orthogonal since they are eigenvectors of the covariance matrix. It is however very apparent that such a analysis is very sensitive to variable scaling. Thus one should, unless all direction have the same scaling, use the principal component analysis on the z-score
ξ − µ z = , (24.1) σ of the data instead, where µ is the mean value and σ is the standard deviation. By doing this the principal components are instead solely based on correlations and not dependant on relative scaling.
The strength of applying such a method is that one can from data with high dimensionally; reduce the number of dimensions in the data and create lower dimensional data-sets that only include the directions with variance, or patterns. Also, reducing data in this way provides a way to produce graphical represen- tations of high dimensional data sets wile loosing minimal information since most of the variations are still included in the representations.
As a example consider a large data set of different apples where there are 20 measurements taken on each apple. All these apples are from the same breed of apple tree but we want to find in-species trends and variance. To find and visualize this data to know what direction the research should take we can apply the PCA algorithm on the z-score of our data set and examine the eigenvalues and directions. We here use the z-score since most of the variables should be in different measurement regimes, let us say that some of the variables are size measurements while other are chemical compositions. Upon examining the eigenvalues we find that only 2 out of 20 dimensions actually have high variance. This indicates that inter-individual vari- ances inside a species only occur on a 2-dimensional manifold, and that we have 18 dimensions of redundant data processing. This would also allow us to accurately visualize this variation since we are not projecting away structure. When we then examine these two directions of most variance, or the plane of most variance, we perhaps find that the basis for this plane can reveal other trends in the data.
In this example we can see that the PCA is not directly a tool for generating scientific results, but rather to indicate in what direction one should focus the research efforts. It is also a great tool for examining if one has enough information in ones data to find patterns. If the plane of most variance only shows a big ball of distributed data without any visible structure, there is probably not enough information in the data to deduce any sort of patterns.
25 Cluster analysis
25.1 Introduction
Cluster analysis or clustering is a method for grouping a set in such a way that objects in the same group are more similar to each other then they are to those in any other group. This measure of similarity is subjective
Page 55 D. Kastinen Meteors and Celestial Dynamics
Figure 19: Example of clustering within 6 nodes in a two dimensional plane where the similarity between nodes are their geometric distance. Clustering is performed using a arbitrary critical distance.
Page 56 D. Kastinen Meteors and Celestial Dynamics and is the reason we covered similarity function in section 17. These similarity function tells the clustering how similar objects are and enables it to perform the grouping. Often one uses cluster analysis in exploratory data mining and statistical data analysis, for example machine learning and pattern recognition. Cluster analysis is not a single algorithm or a single method but rather the idea behind it, clustering can be done in many ways and we shall here cover only a few. One of the first uses was in anthropology as early as 1932 and biometrics (Edwards and Cavalli-Sforza, 1965), as such extensive work has been done in the subject and plenty of material is available, like (Romesburg, 2004). The idea behind clustering is that one cycles through every node in a set of nodes and asks the question: is this node similar enough to any other node? If the answer is yes, then those two nodes become associated and form a cluster. Depending on the rule for which one merges clusters with, different configurations emerge. An example of clustering as shown in figure 19. In this simple example a single link adds nodes to a cluster and the geometric two dimensional distance is the nodes similarity.
25.2 Definition
The purpose of this section is to take the analysis procedure and put it in a abstract context to more clearly extract the important mathematical and statistical choices that are made along the way. Also this will make it easier to propagate choices into effects on the final results. It is highly recommended that readers of this section is well versed in mathematical notation and basic set theory (Jech, 2013). let us consider a space M, a subset of this space X ⊂ M, and a function mapping f : M × M 7→ R+ which we shall call the similarity function.
We can introduce a grouping algorithm to assign each element in the subset X to a group, or essentially breaking it into several smaller subsets. These subsets shall be denoted Xi ⊂ X. There are many grouping methods to choose from, but in all cases there is the need for a additional input of data to create a specific grouping configuration. In the case of cluster analysis and density analysis which we shall call the additional parameter the critical threshold, Dc.
CLUSTERING ALGORITHM
INPUT :(X,Dc)
OUTPUT : {Xi}i∈I < Algorithm statement... > ...
return {Xi}i∈I
Where in the above algorithm I is the indexing set containing all the labels for Xi. For example, if 3 clusters {X1,X2,X3} where formed then the indexing set will be I = {1, 2, 3}. Just considering clustering as our grouping method, there are still several ways to create clusters, so the method will have to be tailored to application. A large group of clustering methods are called hierarchical clustering methods, these depend on a merging algorithm. The starting point of the clustering is that each element in X is in its own cluster, to do this easily since we know that our set has finite cardinality we can introduce the indexing set as I ⊂ N :#I = #X so that
Xi := {ai} : ai ∈ X, i ∈ I.
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S At this point #Xi = 1, and we can see that i Xi = X will always be true. To perform a merge between two clusters we check a linkage criteria between two clusters: if this criteria is fulfilled, they are merged into a new and larger one
∀ (Xi,Xj): i 6= j, i ∈ I, j ∈ I if CRITERA
Xi := Xi ∪ Xj I := I \{j}
This process is repeated until no new merges happen. Using a single linkage method the linkage criteria is min ({f(a, b): a ∈ Xi, b ∈ Xj}) < Dc. In words, this criteria is: If any pair of two nodes from two different groups have a connection, the groups merge. If we use the complete linkage criteria f(a, b) < Dc ∀ a ∈ Xi, b ∈ Xj the clustering dynamics will most often look completely different. This linkage criteria can be put into words as: If all pairs of two nodes from two different groups have connections, the groups merge.
We are now ready to construct a complete algorithm, using single linkage as a example we can write
CLUSTERING ALGORITHM
INPUT :(X,Dc)
OUTPUT : {Xi}i∈I I ⊂ N :#I = #X
Xi := {ai} : ai ∈ X, i ∈ I do
I0 := I
∀ (Xi,Xj): i 6= j, i ∈ I, j ∈ I
if min ({f(a, b): a ∈ Xi, b ∈ Xj}) < Dc
Xi := Xi ∪ Xj I := I \{j}
while I0 6= I
return {Xi}i∈I
This algorithm is very inefficient in computation but instead easy to follow in reasoning. We can now introduce a classification we shall call the association of the clustering. To do this we will be classifying objects into two categories, nodes with a connection and nodes without. The amount of associ- ation after a clustering is performed is the ratio between the nodes with connections and all nodes. Thus, if all nodes have a connection, the entire cluster is associated. This measure is designed to show how many members of the set is considered similar to another member.
Since we cannot know at what value the variable Dc produces the desired clustering we will have to de- termine this somehow. By sweeping from Dc = 0 to Dcmax we can get the complete clustering profile of the input set using the a certain linkage criteria. We define Dcmax as the point where the nodes and connections form a complete graph or where no more cluster merges occur, depending on the analysis type. We will only need to sweep until the entire set X is clustered into a single set #I = 1 to extract all the information on
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the clustering since no other merges can happens after this. One advantage of using a metric function as
similarity function is that this will guarantee a finite value of Dcmax if the subset X has a finite diameter.
Introducing a set of real data into this concept, our goal is to run a clustering algorithm on our real world
measurement data set X˜, with a specific Dc, and achieve a certain configuration of clustering that revels meteor showers. We do not know what the clustering should look like in this particular set so we cannot
simply sweep Dc and pick the right one. We do however know what physical laws and dynamics the system that generated our measurements are governed by. And the governing dynamics of the data can give hints about what critical threshold to pick. This reasoning means that we can create a artificial data set governed by the same dynamics, examine the clustering in this synthetic data, and from the results of clustering calibrate our critical threshold applied on real data. Now it is important to think about the problems involved with this approach; The success of using the
same Dc that created the correct clustering in a simulation on real world data is totally dependant on the accuracy of the simulation and the accuracy of the measurement. If we where to be able to simulate the exact scenario to a very precise degree or measure the objects properties perfectly, we would probably not need clustering methods since there are no ambiguities. Thus we must assume that our knowledge about the real world scenario is limited to at lest a range of possibilities. The best way to concentrate a range of possibilities into a single calibration is a Monte Carlo type simulation. We will thus run through our entire
range of possibilities and every time find a new Dc that produced the desired clustering. The average value of these multiple clustering sweeps will give a good representation of what value to use in the real world data analysis.
25.3 Signal to noise ratio
A signal to noise ratio may seem like a strange concept to introduce in cluster analysis but the translation from regular signal analysis to clustering is straight forward. Consider a clustering configuration, where we in one of the clusters consisting of 100 nodes have 90 nodes that are very similar. But the similarity function is not perfect, and the linkage criteria is not really sufficient either, so 10 of the nodes have been miss-classified. They should actually belong to a different cluster. We can then say that this cluster has a
signal to noise ratio of SNR = 90/10 = 9, or in SNRdB = 10 log10(90/10) = 9.54 dB.
In our application, a signal to noise ratio in cluster analysis can be artificially estimated by simulating a meteor shower, and by simultaneously simulating the sporadic complex. Performing the clustering on both data sets but with the knowledge of what nodes are actually a part of showers and what nodes are not, we can consider two limits:
1. Dc shower is the limit at which meteor showers start clustering
2. Dc sporadic is the limit at which the sporadic complex start clustering
Somewhere between these two limits we can find a good clustering threshold for meteor showers in real world data. If the value of Dc shower is ”close” to the value of Dc sporadic the clustering method chosen is unreliable. If Dc sporadic is lower than Dc shower the method is not able to perform the desired clustering at all. Only if Dc sporadic is larger and sufficiently separated from Dc shower is the method a valid algorithm for finding association in the real world data set.
We have performed this type of analysis on the same example as in figure 19. In figure 20 we see the result of a background to signal count ratio of 1 to 3. The signal is distributed in a Gaussian fashion and the background is uniform. The top graph shows geometric representation of clusters, nodes, and connections
Page 59 D. Kastinen Meteors and Celestial Dynamics
Figure 20: Example of cluster analysis with background data on a two dimensional plane where the similarity between nodes are their geometric distance. Top graph shows geometric representation of clusters, nodes, and connections where red nodes are signal data and blue nodes background data. Bottom graph shows the sweep of critical threshold and the respective association. Background to signal count ratio of 1 to 3.
Figure 21: Background to signal count ratio of 1 to 1.
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Figure 22: Background to signal count ratio of 5 to 1.
where red nodes are signal data and blue nodes background data. Bottom graph shows the sweep of critical threshold and the respective association. In figure 21 we see the same analysis but with a ratio of 1 to 1. And finally in figure 22 we see a background to signal node count ratio of 5 to 1. We can from figure 20 realize that even tough we have good association in our signal and high signal to background count, the cluster is resolved and seems to contain most of the desired data but we still have clusters that form in the background data. This indicates that one must disregard clusters of certain sizes depending on the data configurations. This may also be a indication that single linkage clustering may not be suitable since it does not represent density the same way average linkage does. Thus the samples statistical properties, such as density, variation, size, and so on, will affect the cluster analysis itself and is important to consider before application.
25.4 Previous usage of thresholds
Many of the papers published searching for meteor showers using cluster analysis area does not mention exactly how the critical threshold is derived, if the uncertainty of that choice has been investigated, or what the sample statistical properties are. This can easily lead to misleading results. For example the thresholds that (Southworth and Hawkins, 1963) derived
3601/4 D = 0.2 , (25.1) SHc N and (Lindblad, 1971) derived
0 −1/4 DSHc = 0.8N , (25.2)
Page 61 D. Kastinen Meteors and Celestial Dynamics where N is the sample size. This formula is very hard to evaluate and where derived in a empirical manner without consideration of their bias. Now that computer power has increased to such a degree that we can run proper simulations to generate synthetic data such formulas should not be used any more. In (Welch, 2001) and (Jopek and Froeschle, 1997) a similar statistical analysis to the one developed here is used. However due to lack of information about the generation of the sporadic background in these papers it is difficult to compare the results, also in (Welch, 2001) no reliability level is calculated.
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Part VI Software
26 Introduction to the software
This section is intended to explain the bulk of the work performed and should not be skipped as it is vital to interpret the results correctly. Also this part functions like a instruction manual for the use of the software.
Even though the plethora of applications many of the computational software’s used by scientists are out- dated by modern standards. One such example is the mercury6 software (Chambers, 1999) commonly used today for integrations of the Solar system. This code has been the base of follow up version, Swift, and later Swifter, which recently was ported to C/CUDE as cuSwift to do GPU calculations. This is the most up to date dedicated Newtonian gravitation Hamiltonian integrator software to our knowledge. We are thus disregarding short term integrators not utilizing specialized Hamiltonian techniques such as the ESA NEOprop2, which contains most relevant non-gravitational effects in addition to the gravitational perturbations, but relines on regular integration techniques even though symplectic in nature. Many of these older programs lack the versatility possible with modern programming languages and few are managed as easily assessable open source software. Advances have been made in the subject of symplectic integrators but few have actually been implemented in publicly available software.
The perhaps most missed feature in current models is the ability to easily connect programs with differ- ent functionalities. We believe this is the main reason why many contemporary techniques such as Monte Carlo simulations of meteoroid streams, orbital stability analysis, time tracing of orbital similarity functions to better analyse measurements, neural network recognition of meteor showers, and so on, have not yet been combined. And this was also the bottleneck to perform the research presented in this thesis. Thus we have developed such a software attempting to supplement the current research area.
The recent progress in measurements, like the Rosetta mission, means that a multitude of data is avail- able to use with this software, for example to improve and test models of meteoroid ejections, composition, and propagation through the Solar system. Also, with improved meteor observations, like with the future radar system EISCAT˙3D (McCrea et al., 2015), it will be important to perform statistical simulations to evaluate the meteor data correctly. This software gives an opportunity to use measurements gathered from a variety of ESA missions in previously unrelated procedures.
27 Modular development
One of the most challenging tasks when creating new software is broad functionality without loosing usabil- ity. To maximize the functionality of all the developed software we have chosen to proceed in a modular fashion, where every significant self contained part of the analysis and simulation is made into a independent module. One module acts as a master program that has the ability to call several independent modules. The modules can run independently of the master program. The user can therefore link modules in different orders, choose to distribute data between modules in different ways, and tailor case studies in a powerful fashion. The parent bodies can be hand-picked for case studies, or generated from probability distributions like e.g. the Pan-STARRS Synthetic Solar System Model. In the example simulation of the random stream environment the software was configured so that the the distributions represent orbital elements, critical sublimation radius, density, size, and surface activity. The software then integrated ejected particles over a chosen time and examined close encounters with the Earth. The real power of the software lies in its
Page 63 D. Kastinen Meteors and Celestial Dynamics statistical approach. The ability to choose distributions and then do repeated simulations implies that one can assess the most probable outcomes based on the uncertainty of the input parameters rather than relying on assumptions. The software enables a long-term statistical approach, which is needed to investigate the sporadic meteoroid complex and its origin.
Below a flowchart of the modular nature of the developed software is shown, where the MCAS program is the master program. Currently the Celestial Mechanics Simulator module and the Orbital Stability Es- timation module have not yet been implemented in code, also the Statistical Uncertainty Orbital Clones is currently functional at a bare minimum level. This is due to the fact that the current version is a proof of concept for the full scale project and thus non vital functions and modules where not implemented. OSE Orbital Stability Estimation
OAA MCAS CMS/mercury6 Orbital Association Analysis Monte Carlo Association Statistics Celestial Mechanics Simulator
PBE SUOC JPL Parent Body Ejector Statistical Uncertainty Orbital Clones JPL ephemeris calculator
28 Dependancies
28.1 Programming language
When starting development of this software several languages was considered and some preliminary testing was done using MATLAB integrators. While very high level languages like Python did not provide the com- plexity required to performed efficient scientific calculations and data handling very low level and restricted languages like Fortran are too ridged to provide flexibility during development and future modification. Thus we have chosen the trade off point of C++, where the object oriented programming language can provide enough ”behind the scenes” functionality to make the code easily modifiable and dynamic while allowing complex programming functions closer to the machine hardware to speed up calculations. Thus complex algorithms, like integrators, will be written in C and wrapped in C++. Also C++ is a language with plethora of libraries already available thus lowering the development work needed.
28.1.1 C++ 11
Some of the functions or syntax made available in the C++ 11 has been used to provide easy modification of the source code as many of the functions implemented in this C++ dialect reduces the coding experience needed by the user. However the majority of the code is written according to C++ 98 standard and only small modification is required to compile using this dialect instead.
28.2 Boost library
To provide easy development of file handling and system interaction we have used several of the Boost libraries. Boost provides free peer-reviewed portable C++ source libraries with strong usability. The specific
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libraries and function used will not be covered here, for this we refer to the future release of the code. However to re-compile the developed software the boost libraries must be installed with on the compiling system. For example if a Ubuntu type Linux distribution is used one should install the boost dependencies with:
Listing 1: Example command
1 sudo apt-get install libboost-all-dev
More on boost can be found in (Sch¨aling,2011).
28.3 SPICE library
SPICE is an information system designed to assist scientists in planning and interpreting observations from instruments and to assist engineers in modelling, planning and executing space activities. This tool is developed by The Navigation and Ancillary Information Facility (NAIF), acting under the directions of NASA’s Planetary Science Division. SPICE is designed to use data files refereed to as ”kernels” containing observational data and complementary information such as leap second data to compute the desired parameters referring to a object in the Solar system. The kernels also come with more information about that specific file provided in a ”readme” document included in the folder. The different types of kernels available are:
• DSK - Digital Shape Kernel:
• FK - Frames Kernel
• LSK - Leapseconds Kernel
• PCK - Planetary Constants Kernel
• SPK - Spacecraft and Planet Kernel (ephemeris kernel)
In out case we have used the Toolkit version for C to develop a small module that can compute the geometric positions of the planets and other objects in the desired coordinate system. This toolkit is needed to compile the JPL module and can be found on https://naif.jpl.nasa.gov/naif/toolkit.html, where installation guides and extensive documentation is available. The system currently includes the following aberration corrections to correct for a one way light time and stellar aberration error.
• ”NONE”: Apply no correction, gives the geometric state.
• ”LT”: Apply correction for one way light time using Newtonian formulation.
• ”LT+S”: Apply correction for one way light time and stellar aberration using Newtonian formulation.
• ”CN”: Apply correction using converged Newtonian light time.
• ”CN+S”: Apply correction using converged Newtonian light time and stellar aberration.
In current applications however we are only interested in the geometric state of the planets and in MCAS this has been fixed to ”NONE”.
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28.4 Fortran 77
Since we have substituted the CMS module for mercury6 until development of this module is mature enough for early deployment, a Fortran 77 compiler is required to run the entire system. For example is a Ubuntu type Linux distribution is used one should install the Fortran 77 dependencies with:
Listing 2: Example command
1 sudo apt-get install fort77
29 Monte Carlo Association Statistics module
Monte Carlo Association Statistics or MCAS is developed to manage a simulation of bodies that produce small particles in our Solar system in a Monte Carlo fashion. The parent bodies are generated from probability distributions of some variable or set of variables. In its current stage the available variables to distribute are orbital elements, critical sublimation radius, parent body mass, parent body size, particle mass distribution, and surface activity. The software is then designed to integrate these particles over some time scale and examine close encounters with a major body (planet) in the Solar system. We have concentrated on close encounters with the Earth. As mentioned in the software introduction, a Monte Carlo type simulation is preferable to solve many of the contemporary science questions mentioned. As we have covered the basic basic principles of such a method in section 23 we shall now more closely specify our intentions and applications of this principle in the developed software. The following chapters will specify the implementation specifics of the MCAS module.
29.1 Program flow
Below is a diagram illustrating the flow of the software. The purpose of the diagram is to quickly get a overview of the practical implementation and capabilities of the program.
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Start
SUOC Parent body distribution
Pick random Parent body Will body produce particles? No Yes
JPL PBE Particle mass distribution Simulation initial state Propagate and eject particles
OSE Orbital stabillity check of parent bodies
Save data on all close mercury6 encounters inside Hill Sphere Propagate ejected particles and parent body and their parents
Load time series of close encounters OAA Association analysis on generated showers Monte Carlo done? Examine divergence from parent body No Yes
Done
29.2 Execution time diagnostics
In almost every type of numerical calculation or simulation software there is a bottleneck for the execution time. In Hamiltonian propagation and cluster analysis these bottlenecks all stem from the fact that inter- particle calculations scale as O(N 2) and thus grows beyond normal computing requirements quite quickly. There are however several other pitfalls for the inexperienced programmer, such as faulty balancing of RAM saving versus hard drive write and read speeds. Thus to find these bottlenecks and further optimize a newly developed software execution timers are a excellent way to improve either the software itself or the simulation settings by perhaps lowering resolution in certain bottleneck areas where possible. The current implementation of MCAS has been divided into 9 time sections within which the execution time
Page 67 D. Kastinen Meteors and Celestial Dynamics of that section is saved to a file. The tenth section is the execution of the entire software, the file containing this data is specified in the output section. This is useful both for development and also for post-simulation analysis, the sections functions can be found in the below list as:
1. Parent body generation
2. PBE time
3. mercury6 time
4. Snapshot time
5. close6 time
6. element6 time
7. Stream dissipation time
8. Parent body divergence and file handling
9. Cluster analysis
10. Total time
Figure 23: Execution time for different segments of MCAS, stream dissipation calculation enabled.
As an example of this kind of time execution analysis we have run 2 different simulations, each with the same exact integration parameters and settings, except we have turned on Stream dissipation analysis in one of
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Figure 24: Execution time for different segments of MCAS, stream dissipation calculation disabled.
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the simulations. The output time data can be found in figure 23 for the simulation with stream dissipation analysis and in figure 24 for the simulation without. The simulation was performed with a maximum of 1000 test particles per ejection, one perihelion ejection passage with the parent body picked from the Pan- STARRS cometary database. The particles where integrated for 220 years in mercury6 using a step length of 16 days and the association analysis was performed with 4 similarity function conducting a parameter sweep of single linkage clustering. These graphs indicate the problem of calculating stream dissipation in the current fashion. A solution to this problem is proposed by creating a custom output solution in CMS that creates files allowing the calculation to be more efficient by grouping data in files of time intervals rather than object time series, and additionally by printing larger time steps even though the propagation was performed with smaller steps. This would also require a function estimating the amount of RAM that can be used by the stream dissipation calculation to optimize the amount of data loaded from the CMS output at once.
29.3 Probability draw
To be able to numerically simulate a process dependant on a stochastic variable, or events based on proba- bility, one needs to generate random values with the correct frequencies. In C++ the only random number generator available by default generates a uniform distribution between two numbers. To be able to draw events based on general probability density function we must specify a algorithm to transform uniform dis- tributions to a desired distribution. We shall hereby refer to the action of finding a random value based on a general probability distribution function as a draw from that distribution. The most basic version of such a function is to draw from the one dimensional histogram. This function, given a set of bins and a set probabilities or frequencies for each bin, one can define a algorithm as:
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DRAW FROM 1D DISTRIBUTION INPUT : Histogram bins: X Histogram frequencies: F OUTPUT : Random draw from histogram Save size of histogram vector: s = SIZE(F ) Generate uniform random number between 0 and 1: r = RAND F Normalize the histogram: F0 = SUM(F) Start cumulative adding at 0: H = 0 FOR i FROM 1 TO s Check if we found the selected bin: IF H <= r AND r < (H + F0[i]) return X[i] ELSE Otherwise prepare for the next check: H = H + F0[i] END END
This algorithm will converge to produce exactly the given histogram as one continues to draw. This algorithm is the fastest of the ones implemented in the software but it does not provide a satisfactory result given a more carefully considered input distribution. As the vast majority of the simulation time in the current application of the software is spent on time integration of the Solar system the speed of this algorithm is not important. If however a application is considered where many generations of random initial conditions must be done compared to the amount of time integrations, one can consider this simplified algorithm. The major fault of this algorithm is the discrete return values that are the center of the bin. To rectify this we make the return value a continous distribution by returning, instead of the bin center, a value randomly chosen inside the bin by a uniform distribution:
DRAW FROM 1D DISTRIBUTION . . FOR i FROM 1 TO s Check if we found the selected bin: IF H <= r AND r < (H + F0[i])
Draw the ’in bin’ value: r2 = RAND 1 r return (X[i − 1] + X[i]) + 2 (X[i + 1] − X[i − 1]) 2 2 . .
To show the effect of this improvement in a practical application we have taken some input data for the SUOC module, namely the comet population of the Pan-STARRS Synthetic Solar System Model. In figure 25 we can see the orbital shape parameters, semi major axis and eccentricity, as histograms and as a density
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Figure 25: Comparison between the described first method of drawing a random value from a distribution versus the real multi dimensional distribution
Figure 26: Comparison between the described improved first method of drawing a random value from a distribution versus the real multi dimensional distribution
Page 72 D. Kastinen Meteors and Celestial Dynamics map where the upper row show the actual data and the bottom row show graphs of the first histogram generation algorithm. In figure 26 the improved continuous version of the histogram generation algorithm is shown in the same kind of graphs. We can see the significant improvement in the density map since this shows the smoothing effect of the continuous rather then the discrete return values, however the histograms seems mostly unaffected. This effect is something that one should be very aware of since a histogram may fail to represent important trends in data. Using a one dimensional histogram assumes that variables are uncorrelated, and this simplification can be seen in the figures as the shapes of the two dimensional distributions in the top and bottom row of figure 26 are not the same. To better illustrate the effect we have created a artificial distribution suitable for this, as shown in figure 27. To correctly represent the data we have to assume that the parameters need not be independent, so instead each object is treated as an N dimensional point. A psudo code for the continuous version algorithm for such a multidimensional function is given below:
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DRAW FROM ND DISTRIBUTION INPUT : Histogram bins: X Histogram frequencies: F OUTPUT : Random draw from histogram Save size of histogram vector: s = SIZE(F) Save dimension of histogram volumes: N = DIM(X )
Calculate bin sizes and save: Xs = BIN SIZE(X ) Generate uniform random number between 0 and 1: r = RAND(1) F Normalize the histogram: F0 = SUM(F) Start cumulative adding at 0: H = 0 FOR i FROM 1 TO s Check if we found the selected bin: IF H <= r AND r < (H + F0[i]) Draw the ’in volume’ value: r = RAND(N) − 0.5
return r · Xs[i] + X [i] ELSE Otherwise prepare for the next check: H = H + F0[i] END END
Here X is a list of all N dimensional points in the distribution representing the center of histogram hyper volumes, each housing a frequency listed in F. Here we can see that the non trivial programming does not lie in the drawing function but the creation of the histogram bin list in a sufficient fashion as to give a good
frequency to resolution ratio. As a last note on this example, r·Xs[i] denotes the element wise multiplication, not matrix algebra or scalar product. The result of implementing this on the artificial data and the orbit shape data can be seen in figures 28 and 29. The reason behind the extensive discussion of random point generation is that the method one generates the initial distribution is crucial to understanding the end distribution. In our use of the software many of the variables are picked randomly to, over time, eliminate the possibility of assumption errors. One should however be aware of the fact that the more stochastic space one has to explore the less specific the results become and the more computation time is needed. Currently the variables generated initially from probability distributions are:
• Parent body semi major axis
• Parent body eccentricity
• Parent body inclination
• Parent body argument of perihelion
• Parent body longitude of ascending node
• Parent body epoch of orbital elements
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Figure 27: Comparison between the described improved first method of drawing a random value from a distribution versus the artificially generated multi dimensional distribution
Figure 28: Comparison between the described multi dimensional method of drawing a random value from a distribution versus the Pan-STARRS multi dimensional distribution
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Figure 29: Comparison between the described multi dimensional method of drawing a random value from a distribution versus the artificially generated multi dimensional distribution
• Parent body radius
• Parent body ejected material bulk density
• Parent body surface sublimation activity factor
• Parent body critical heliocentric sublimation radius
• Parent body mass
Then at later stages the following variables are also generated in such a fashion
• Ejected particle mass
• Ejected particle direction
One can thus, by setting some of these distributions to Dirac-delta functions, examine the statistical outcome of any input distribution or any combinations of the above.
29.3.1 Dimensionality reduction
In section 24 we covered the principal component analysis and briefly covered dimensionality reduction. Such a feature is not yet implemented but is planned in future versions. In light of the problem with multidimensional distribution we know that their sampling size increases exponentially with the amount of dimensions they are spread across. This is why such a dimensionality reduction function would be useful, since we cannot assume that the datasets available are of appropriate size to sample the entire space, or that we have enough computing power to cover the entire space.
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To illustrate this we shall consider the Pan-STARRS set of comets previously discussed. Using 5 orbital parameters and ignoring the orbit epoch as a parameter we are left with a 5 dimensional space. Let us also consider a set of resolutions of ∆a = 0.5 AU, ∆e = 0.05, ∆i = 5◦,∆ω = 10◦, ∆Ω = 10◦ to be sufficient for representing the data. This set contains 27 501 artificial comets, let us label this N. We can assume the parameters to range from a ∈ [2 AU, 10 AU],e ∈ [0, 1), i ∈ [0◦, 180◦],ω ∈ [0◦, 360◦], Ω ∈ [0◦, 360◦]. We now know the size of the sample and the size of the space it is distributed in, let us now explore the possible resolutions we can find acceptable. For the purpose of the example let us say that we cannot give any statistical significance to a average histogram frequency of less then 100. Let us also assume that the data for now is uniformly distributed in this space. In the case of just simple histograms for each variable, i.e. if all parameters are independent, we would get a set of resolutions of ∆a = 0.03 AU, ∆e = 0.0036, and so on. This is very well below what we defined as acceptable. If however we instead partition the space into hypercubes and calculate the resolutions with 100 samples in each cube we find the resolutions to be ∆a = 2.6 AU,∆e = 0.32,∆i = 58◦,∆ω = 117◦, ∆Ω = 117◦. Thus, it is impossible to reach any reasonable resolution with a sample size of only 27 501 if all the 5 variables are dependant, for this we would need a sample size of around 1 500 000 000. However, let us say that only the eccentricity and inclination where found to be dependant on each other while the rest where totally independent. Then we can perform the same calculation and find a set of multi dimensional resolutions of ∆a = 0.48 AU,∆e = 0.06, while the rest are the one dimensional resolutions. Thus, depending on the sample size, dimensionality reduction can be essential. This is only a problem however if one wants to draw a number of orbits randomly that exceeds the distribution size. If the number of draws is smaller then the sample size one can simply integrate the sample points themselves instead of creating a continuous distribution function.
29.4 Close encounter
When examining interactions between particles and planets, and in our case the meteor phenomena, there is a significant difference between a close encounter and a actual collision. This is due to the fact that even if the meteoroid enters the Earth’s field of superior gravitational influence, its Hill Sphere, the particle may never encounter the atmosphere and undergo ablation. The Hill sphere of a body can be approximated as
m r ≈ a(1 − e) , (29.1) H 3M where rH is the Hill sphere radius, a and e is the familiar orbital shape parameters, and m is the smaller body mass while M is the larger body mass. In our case, m would be the mass of the Earth and M the mass of the Sun. The parameters a and e would be the orbital parameters of the Earth. Even tough there is a option in the configuration of MCAS to change what is regarded as a meteor with regards to the close encounters closest distance we have in our demo simulation chosen this to be 1 Hill radii. The reason behind this simplification is that the probability of encounter for a single particle greatly decreases when instead picking only particles that passes through the atmosphere, requiring magnitudes of more particles and thus a computation time of O(N 2) increase as we add more particles. This is not applicable when performing our kind of statistics based simulations. Thus we have chosen to regard all encounter within 1 Hill radii as meteors. This simplification is not as bad as it seems due the same reason that the approximation was implemented. Since there is not computational power to represent every ejected dust grain from the comet during subli- mation each test particle integrated represents millions and millions of dust particles. All these particles will have small perturbations in their initial conditions and thus we can instead of regarding each test particle as one single meteoroid, we can view them as small swarms of meteoroids. When viewed this way it becomes reasonable that even if the particles only comes within one Hill radii of Earth, some part of this ”swarm”
Page 77 D. Kastinen Meteors and Celestial Dynamics will still produce meteors.
29.5 Parent body stream divergence
One interesting aspect of a meteoroid stream is its age. This is however something that is hard to determine from observations since the only aspect detectable is the meteor showers orbital configuration. Thus it may be interesting to, in simulations, calculate how far from the parent body the orbits have deviated before encounter. Such a calculation may give a good indication of what epoch trail the detected meteor shower was a part of. By this we mean that if the parent body is known, and we have just detected a meteor shower with a certain orbital elements distribution, one can make a fast first assumption on what trail caused the shower by comparing the orbits of the current parent body and the detections. As such, to calculate such a parent body divergence time series we are only interested in meteoroids that actually encountered the Earth and in the simulation we shall only perform this analysis on those. Since there are many orbital similarity function available we have implemented this divergence calculation using the 4 already implemented function in the OAA module. The calculation logistics are straight forward, we load the time series for the parent body, and then in succession we load the time series for each encountered test particle and calculate the time time series of the similarity function. Below we demonstrate some pseudo code to clarify the method for one similarity function:
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PARENT BODY METEOROID DIVERGANCE
INPUT : Particles time series: Oi(t0 + n∆t) ∀ n ∈ [0,N], i ∈ [1,M]
Time series for parent body: OPB(t0 + n∆t) ∀ n ∈ [0,N] Similarity function: F
OUTPUT : Divergance time series: Di(t0 + n∆t) ∀ n ∈ [0,N], i ∈ [1,M] FOR i FROM 1 TO M FOR k FROM 0 TO N
Calculate the orbital similarity: Di(t0 + k∆t) = F (Oi(t0 + k∆t), OPB(t0 + k∆t)) END END
Figure 30: A highly perturbed meteoroid stream encountering Earth over 400 years.
In the above example O(t) are the orbital elements needed for the similarity function at time t, whether it be geocentric quantities, positions and velocity vectors, or Kepler elements. As a example we have chosen two different iterations of the Pan-STARRS simulation. The first is one of a stream that experiences a relatively small amount of perturbation, as can be seen of the shower orbital plot in figure 30. Using the orbital similarity functions one can quantify the difference in the perturbation effect on the orbit compared to the parent body by using the algorithm above, the resulting data for the Earth encountered particles can be seen in figure 32. The second one illustrates a stream in a region with a high amount of perturbations, or low e-folding time, causing the orbits to spread over a large area qickly as can be seen in figure 31. The resulting criterion divergence time series can be found in figure 33.
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Figure 31: A slightly perturbed meteoroid stream encountering Earth over 400 years.
Figure 32: The DSH criterion measure of the stream divergence from its parent body in a highly perturbed meteoroid stream encountering Earth over 400 years.
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Figure 33: The DSH criterion measure of the stream divergence from its parent body in a slightly perturbed meteoroid stream encountering Earth over 400 years.
29.6 Stream dissipation
To quantify stream dissipation is not a easy question as the morphology of a stream may evolve in may different ways depending on the gravitational environment in the general area of the parent body. If the area is dominated by gravitational resonances chaotic effects becomes predominant and the stream will quickly scatter over the Solar system. Previously most stream morphology examination have visually shown cross section of the stream, or snapshots of its evolution at different stages. Or, in its early life, examined its cross section particle density distribution. All these methods however become troublesome when summarizing over many different streams, examining many iterations of the same stream simultaneously, or doing a statistical analysis and thus we cannot utilize the regular methods. To resolve this we will use the previously mentioned orbital similarity functions. This is advantageous since we are no longer relying on regular 3-dimensional distances in space but rather the similarity of the orbits in the stream, and how this similarity evolves with time. This also removes the need to find some sort of ”cross section” of the stream, as the orbit configuration is not dependant on where in the orbit the particle is. It only considered how the particle has been perturbed, i.e. the particles that ”lag behind” the comet does not give a large distance if their orbits still coincides with the other particles. One way to do this would be to calculate the distance matrix
Dij(tn) = D(Oi(tn), Oj(tn)), (29.2) in each time step. Here we have denoted the similarity function as D and a orbits describing parameters at 2 time tn as O(tn). However, the calculation of this matrix has the problem of having a scaling of O(N ). Due to the fashion that mercury6 outputs time series and the inherent O(N 2) scaling of the calculation RAM and hard drive usage will be a problem when using this to estimate stream dissipation. To resolve this we will only use the distance matrix as a intermediate calculation and the actual output will be the mean value
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and standard deviation of this matrix. As we would only want to load a few times series at a time and we cannot load all particle data for one time we will calculate the mean
N 1 X µ = x , (29.3) N N i i=1
and standard deviation
N N !2 1 X 1 X σ2 = x2 − x , (29.4) N N i N i i=1 i=1 with a sample increment formula as given by
µN+1 = µN + ∆µ, (29.5) x − µ ∆µ = N+1 N , (29.6) N + 1 and
2 2 σN+1 = σN + ∆σ, (29.7) x2 − σ2 + Nµ2 ∆σ = N+1 N N − µ2 . (29.8) N + 1 N+1 Or in direct formulas as
µN+1 = µN + A = 1 x = µ 1 − + N+1 , (29.9) N N + 1 N + 1 2 2 σN+1 = σN + A = N ! 1 X 1 x2 = x2 1 − + N+1 − µ2 . (29.10) N i N + 1 N + 1 N+1 i=1
Of course we also need to account for the sample bias, so the output variable will be
N r N s2 = σ2 ⇔ s = σ . (29.11) N N − 1 N N − 1 N
This will be used in such a way that each unique pair of time series are loaded, the pairs similarity function time series are calculated, and added by the above formulas to the output data. A set of preliminary results using different similarity function can be found in the results section.
29.7 Data formatting
Essential in any data analysis of results is to have complete knowledge of the data formatting of the output files. Thus this entire section is dedicated to the specification of all the output and input files of this software.
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29.7.1 Input files
Table 3: family
Row Description Data type Units Notes 1 Comet radius Double m - 2 Bulk density Double kg/m3 - 3 Comet activity fraction Double - Fraction of comet subliming 4 Critical distance for sublimation Double AU Heliocentric 5 Comet mass Double AU Optional row Column Description Data type Units Notes 1 Distribution type Char - U: Uniform, N: Normal distribution 2 Distribution value 1 Double - U: minimum value, N: mean 3 Distribution value 2 Double - U: maximum value, N: standard deviation
Table 4: mass.data Column separator: ’Space’ 0x20 Input File description: Major body masses (Sun and all planets)
Row Description Data type Units Notes 1 Mass Double Several choices, see configuration file -
Table 5: mass_dist.txt Column separator: ’Space’ 0x20 Input File description: Custom mass distribution input file
Row Description Data type Units Notes 1 Bin center value Double grams - 2 Bin probability Double - Does not need to be normalized
29.7.2 Output files
Table 14: error_profile_
Row Description Data type Units Notes Odd Critical value used Double - - Even Error function value Double - -
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Table 6: execute_time.txt Column separator: ’Space’ 0x20 Output File description: Execution time of sections in software simulation. May have unequal number of columns for each row.
Column Description Data type Units Notes All Section iteration Double seconds - Row Description Data type Units Notes 1 Parent body generation section Double seconds - 2 PBE section Double seconds - 3 mercury6 section Double seconds - 4 Snapshot section Double seconds - 5 close6 section Double seconds - 6 element6 section Double seconds - 7 Stream dissipation section Double seconds - 8 PB divergence section Double seconds - 9 Cluster analysis section Double seconds - 10 Entire software Double seconds Only one column
Table 7: snapshot_pos.txt Column separator: ’Space’ 0x20 Output File description: Takes a snapshot of all test particles positions at specified time
Column Description Data type Units Notes 1 Simulation id Integer - Also MC run ID 3 X position Double AU Ecliptic heliocentric J2000.0 equinox 4 Y position Double AU Ecliptic heliocentric J2000.0 equinox 5 Z position Double AU Ecliptic heliocentric J2000.0 equinox
Table 8: snapshot_vel.txt Column separator: ’Space’ 0x20 Output File description: Takes a snapshot of all test particles velocities at specified time
Column Description Data type Units Notes 1 Simulation id Integer - Also MC run ID 3 X velocity Double AU/day Ecliptic heliocentric J2000.0 equinox 4 Y velocity Double AU/day Ecliptic heliocentric J2000.0 equinox 5 Z velocity Double AU/day Ecliptic heliocentric J2000.0 equinox
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Table 9: snapshot_earth.txt Column separator: ’Space’ 0x20 Output File description: Takes a snapshot of 10 or less if too close to integration start Earth positions at specified time
Column Description Data type Units Notes 1 Simulation id Integer - Also MC run ID 2 Relative time from simulation start Double years - 3 X position Double AU Ecliptic heliocentric J2000.0 equinox 4 y position Double AU Ecliptic heliocentric J2000.0 equinox 5 Z position Double AU Ecliptic heliocentric J2000.0 equinox 6 X velocity Double AU/day Ecliptic heliocentric J2000.0 equinox 7 y velocity Double AU/day Ecliptic heliocentric J2000.0 equinox 8 Z velocity Double AU/day Ecliptic heliocentric J2000.0 equinox
Table 10: time_data.txt Column separator: ’Space’ 0x20 Output File description: Simulations Earth time data
Column Description Data type Units Notes 1 Parent body id Integer - Also MC run ID 2 Parent body sync from simulation start Double Days - 3 PBE start time Double JD - 4 PBE stop time Double JD - 5 mercury6 start time Double JD - 6 mercury6 stop time Double JD - 7 PB perihelion passage 1 Double JD - . . 6+n PB perihelion passage n Double JD -
Table 11: PB_kep_dist.txt Column separator: ’Space’ 0x20 Output File description: Histogram
Row Description Data type Units Notes 1 Semi-Major axis Double AU Bin centre 2 Frequency Integer - - 3 Eccentricity Double - Bin centre 4 Frequency Integer - - 5 Inclination Double Degrees Bin centre 6 Frequency Integer - - 7 Argument of perihelion Double Degrees Bin centre 8 Frequency Integer - - 9 Longitude of ascending node Double Degrees Bin centre 10 Frequency Integer - -
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Table 12: function_divergance_
Row Description Data type Units Notes 1st of section Time Double Days Repeated for each simulation ID Rest of section Criterion value Double - - Column Description Data type Units Notes 1 Parent body ID Integer - Designates data section Rest Data - - -
Table 13: stream_dissipation_data_
Row Description Data type Units Notes 1st of section Time Double Days Repeated for each simulation ID 2nd of section Mean criterion Double - On stream internal distances 3rd of section Criterion standard deviation Double - On stream internal distances Column Description Data type Units Notes 1 Parent body ID Integer - Designates data section Rest Data - - -
Table 15: association_profile_
Row Description Data type Units Notes Odd Critical value used Double - - Even Association amount Double - -
Table 16: particle_weight_stat.txt Column separator: ’Space’ 0x20 Output File description: Number of meteoroids each test particle represents for each parent body Note: Only shower generating parent bodies
Column Description Data type Units Notes 1 Number of meteoroids weight Double - -
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Table 17: particle_ejection_v_stat.txt Column separator: ’Space’ 0x20 Output File description: Ejection velocity relative parent body Note:-
Row Description Data type Units Notes Odd Heliocentric distance Double AU - Even Ejection speed Double m/s - Column Description Data type Units Notes 1 Parent body ID Integer - - Rest Data - - -
Table 18: PB_kep_timeS_data.txt Column separator: ’Space’ 0x20 Output File description: Time series data for all shower producing parent bodies
Column Description Data type Units Notes 1 Parent body ID Integer - Also shower ID 2 Time Double Days - 3 Semi-Major axis Double AU - 4 Eccentricity Double - - 5 Inclination Double Degrees - 6 Argument of Perihelion Double Degrees 7 Longitude of Ascending Node Double Degrees - 8 True anomaly Double Degrees - 9 Mass Double kg -
Table 19: earth_encounter_data.txt Column separator: ’Space’ 0x20 Output File description: Earth orbital elements for each encounter
Column Description Data type Units Notes 1 Parent body ID Integer - Also shower ID 2 Time Double Year - 3 Closest distance Double AU - 4 Semi-Major axis Double AU - 5 Eccentricity Double - - 6 Inclination Double Degrees - 7 Longitude of Perihelion Double Radians NOT argument of perihelion 8 Longitude of Ascending Node Double Radians - 9 Mean anomaly Double - Mean longitude if Eccentricity < 1.e-8
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Table 20: met_encounter_data.txt Column separator: ’Space’ 0x20 Output File description: Meteoroid orbital elements for each encounter (at encounter)
Column Description Data type Units Notes 1 Parent body ID Integer - Also shower ID 2 Time Double Year - 3 Closest distance Double AU - 4 Semi-Major axis Double AU - 5 Eccentricity Double - - 6 Inclination Double Degrees - 7 Longitude of Perihelion Double Radians NOT argument of perihelion 8 Longitude of Ascending Node Double Radians - 9 Mean anomaly Double Radians Mean longitude if Eccentricity < 1.e-8 10 Mass Double kg -
Table 21: PB_kep_data.txt Column separator: ’Space’ 0x20 Output File description: All generated parent body Kepler elements
Column Description Data type Units Notes 1 Parent body ID Integer - - 2 Semi-Major axis Double AU - 3 Eccentricity Double - - 4 Inclination Double Degrees - 5 Argument of Perihelion Double Degrees - 6 Longitude of Ascending Node Double Degrees - 9 True anomaly Double Radians -
Table 22: PB_M_stat.txt Column separator: ’Space’ 0x20 Output File description: Shower generating parent body mass change
Row Description Data type Units Notes All Parent body mass Double kg New entry every change in mass, different row length
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Table 23: met_before_encounter_data.txt Column separator: ’Space’ 0x20 Output File description: Meteoroid orbital elements at integration step before each encounter
Column Description Data type Units Notes 1 Parent body ID Integer - Also shower ID 2 Time Double Year - 3 Semi-Major axis Double AU - 4 Eccentricity Double - - 5 Inclination Double Degrees - 6 Argument of Perihelion Double Degrees - 7 Longitude of Ascending Node Double Degrees - 8 Mean anomaly Double Degrees - 9 True anomaly Double Degrees - 10 Mass from mercury Double Solar masses Set to 0
Table 24: ejector_char_data.txt Column separator: ’Space’ 0x20 Output File description: Parent body characteristics Note: Only shower generating parent bodies
Column Description Data type Units Notes 1 Comet radius Double m - 2 Bulk density Double kg/m3 - 3 Comet activity fraction Double - Part of comet subliming 4 Critical distance for sublimation Double m Heliocentric
Table 25: ejector_kep_data.txt Column separator: ’Space’ 0x20 Output File description: Parent body characteristics Note: Only shower generating parent bodies
Column Description Data type Units Notes 1 Semi-Major axis Double AU - 2 Eccentricity Double - - 3 Inclination Double Degrees - 4 Argument of Perihelion Double Degrees - 5 Longitude of Ascending Node Double Degrees - 6 True anomaly Double Degrees At initial state
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Table 26: ejector_n_data.txt Column separator: ’Space’ 0x20 Output File description: Parent body count data Note: Only subliming parent bodies
Column Description Data type Units Notes 1 Parent body ID Integer - Number of generated parent bodies 2 Total particles created Integer - - 2 Particles created this propagation Integer - - 2 Total Earth encountered particles Integer - - 2 Earth encountered particles this propagation Integer - -
29.8 Mass distributions
The PBE software takes a custom input of ejected particle mass distribution. During ejection each particle is assigned a random mass according to the supplied distribution Ψ(m). The MCAS software automatically configures the PBE mass distribution input according to 4 options:
• The Gr¨uninterplanetary flux model
• Uniform distribution
• Common logarithmic uniform distribution
• Custom file
29.8.1 Gr¨un
The perhaps most commonly used meteoroid flux model at Earth is the one developed in (Grun et al., 1985). It is based on function fitting of measurement data and gives the total average meteoroid flux in terms of the integral flux. The flux function is given by
7 F (m) = 3.15576 · 10 (F1(m) + F2(m) + F3(m)) , (29.12) 3 0.306 −4.38 F1(m) = 2.2 · 10 m + 15 , (29.13) −9 11 2 27 4−0.36 F1(m) = 1.3 · 10 m + 10 m + 10 m , (29.14) −16 6 2−0.85 F1(m) = 1.3 · 10 m + 10 m , (29.15) (29.16)
−9 where the function F1(m) refers to particles with mass over 10 g, function F2(m) refers to the mass range −9 −14 between 10 g and 10 , and lastly the function F3(m) refers to the smallest particles with mass under 10−14 g. Usually when this function is used one modulates this flux with a Earth shielding and Gravitational focussing term which we shall neglect due to the fact that the simulations concern particle distributions at the edge of the Earth Hill sphere.
To use this as a viable probability distribution the software converts the particle flux into a histogram type data function and uses this as input to PBE,
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F (m) ΨGRUN(m) = , (29.17) αGRUN Z mmin αGRUN = F (m)dm. (29.18) mmax
29.8.2 Uniform and logarithmic
Given two boundary values mmin and mmax the resulting probability distribution functions will be
1 ΨLIN(m) = , (29.19) mmax − mmin for a uniform representation and
1 ΨLOG(m) = , (29.20) m(log10(mmax) − log10(mmin)) for a common logarithmic uniform representation. The common logarithmic representation is useful when doing specific mass propagation analysis over large ranges of masses spanning several magnitudes.
29.8.3 Custom file
The file input functions on a item probability basis, each column is a value with a respective probability to acquire this item when a readout of the distribution is done. As can be seen in table 5 the input file is constructed like a histogram where the bin center is given with its frequency or probability. Thus generating bin centres that are not equally spaced will distort the probability function as the probabilities are spread over the entire bin.
29.8.4 Comments
Due to the fact that particle propagation calculation has a time complexity of O(N 2) where N is the number of particles it is very hard to get good statistical coverage using a model such as the Gr¨unmodel. This is due to the fact that one type of mass is orders of magnitude more abundant than the rest, thus making the probability of the smaller particles being ejected so small that their resulting statistics are often useless. This is the reason why the logarithmic model was introduced, so that mass transfer over large mass ranges can be examined with statistical coverage. More recently, a technical model called the Interplanetary Meteoroid Environment for eXploration (IMEX) is currently under development on a European Space Agency (ESA) contract for the planning of space missions (Soja et al., 2015). IMEX is designed to produce a static database of all released particles, taking into account an important, but limited, fraction of the Solar system comet population. The model currently uses a static set of 428 short period comets integrated for 200-400 years with a Runge-Kutta integrator. The ejection was done using a static sublimation radius from 3 AU ejecting a mass distribution of 8 masses 100 microns to 1 cm. The model was verified using comet trails comparison with IR images and performing a case study of Leonids in 2001 with Dust from 55P/Tempel Tuttle ejected between 1690 and 1998.
29.9 Probability formulation
We shall now address the actual probability formulation of the simulation. The basic concept is that the dynamics are too vast and difficult to visualize by simply examining the data and the governing equations.
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Also due to uncertainties in measurements in a region where chaos is present, one must instead let the simulation work out how the input distributions change as time passes. This concept can be summarized easily in the below flowchart. Compare and deduce
Initial distribution ξ Simulation MCAS Result distribution χ
Here the initial distribution is sampled in a Monte Carlo fashion and thus generates the resulting distribution. By comparing the distributions before and after the effects of time we can, if used correctly, deduce the answer to our questions. N Let us consider a input probability distribution of orbital elements of a parent body, ξOrb ∈ R . We will propagate that parent body and release particles until the sublimation process is completed. The sublimation M is however dependant on the characteristics of the body, which themselves can be distributions, ξPB ∈ R . 3 1 Lastly we have the distributions of ejection direction, ξDir ∈ R , and of particle mass ξM ∈ R . This can be seen as the complete set of input distributions. After ejections are performed the set of initial conditions for the ejected particles are input to the long term propagator generating our output distribu- tions. Any later calculations done on these parameters, such as cluster analysis or coordinate changes, only transforms the output distributions further. Of course one can set some of the parameter distributions to Dirac-delta function thus eliminating it as a variable but it is still a input distribution even though its constant nature. Even though not essential for our understanding of the results we need to consider the fact that the distri- bution of ejected particles in phase space is directly dependant on the distributions of the parent body. This
can be seen as a function dependant on stochastic variables such as ηParticles itself is a distribution
6 ηParticles(ξOrb, ξPB, ξDir, ξM) ∈ R . (29.21)
The next step is, as mentioned, to compare and deduce. To explain how such a deduction can be done by comparison of distributions we shall consider a few general examples. In the following examples P (statement) denotes the probability of statement, E(distribution) denotes the expected value of a distribution, or the mean of a sample, and # means ”the number of”. In the following examples we will refer to a demo simulation of the software done with the Pan-STARRS cometary database, the simulation was performed using:
• 500 Generated meteor showers
• Ejected in the year 1800
• Propagated until 2200
• Meteoroid passing within 0.01 AU considered a Meteor
• Standard physical characteristics for comets
• Each sublimating comet ejects a maximum of 8000 particles
This simulation is just a demo simulation and was performed while the software was till under construction and is therefore not a complete case study of nay kind, instead it serves as a illustrative example. Let us begin with the simple question: What parent bodies produce meteor showers?. The answer to this question is trivial as we need only select all parent bodies in the output data that ejected a particle which encounters Earth. Using our example simulation we produce the distributions in figure 34. This plot suffers from
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Figure 34: The histograms describing the orbital parameters of generated parent bodies. Blue histogram bars represent the initial distribution while red shows the resulting distribution that generated meteor showers on Earth.
the previously discussed problem of simple histograms not showing the true multidimensional distributions, however in this example the visualization is sufficient. Even simpler questions can give definitive answers such as: What is P (A comet produces a meter shower) given our initial distribution? This can be found approximatively by our simulation as,
# Generated showers P (A comet produces a meter shower) ≈ . (29.22) # Generated comets
In the future the ≈ shall be omitted as we define the numerical probability as the statement dependant on the simulation results. This straight forward reasoning can be extended to several other question. Given our demo simulation, we can find that the results are:
• 500 Generated meteor showers
• 995 Generated meteoroid streams
• 3370 Generated comets
• 7801548 Generated particles
• 29598 Earth encountered particles
We can translate this into numerical probability as
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# Generated streams P (A comet produces a meteoroid stream) = = (29.23) # Generated comets = 29.5%, # Generated showers P (A comet produces a meter shower) = = (29.24) # Generated comets = 14.8%, # Generated showers P (A cometary meteoroid stream encounters earth) = = (29.25) # Generated streams = 50.2%, # Earth encountered particles P (A particle of cometary origin encounters earth) = = (29.26) # Generated particles = 0.38%.
The probability given in equation 29.26 is a useful measure for future simulation runs. Using that fraction determine that to achieve a resolution on Earth encountered particles of 100 000 data points, we will need to generate around 100000/0.0038 = 26315789 particles. Here we should also address the apparently high amount of streams that seem to encounter the Earth and what is considered a ”shower”. In equation 29.25 we can see that 50.2% of all generated meteoroid streams in this example using the Pan-STARRS distribution produced a meteor ”shower” on Earth. This result will need to be considered with four things in mind:
1. Only 3 370 points where drawn from a initial sample of 27 501 comets.
2. There are only short period comets in the initial sample.
3. A ”stream” is here defined as all particles from a single parent body, no matter the dispersion of said stream
4. A meteor ”shower” is defined as a close encounter, regardless of the shower intensity
As we can see here, when calculating such probabilities one should be careful to draw conclusions before the definition bias has been addressed. We can say nothing about the intensity of the stream encounters contributing to the 50.2% as it could as well be a single meteoroid hitting the Earth at one time and it would count as if that stream produced a meteor shower in the simulation software. Also, the stream could be in a region with such low e-folding time, i.e. with high dispersion rate due to chaos, that the stream would after just 100 years be spread over almost the entire Solar system and should not be considered a ”stream” any more but rather sporadic background. All these considerations are useful when defining better and more precise measures of data analysis. Let us consider the two last points concerning the stream and shower definitions and implement them in a improved cometary stream encounter probability calculation as
P (A cometary meteoroid stream produces meteor shower) = # Generated showers with > 10 test particles and mean D < 0.4/century comet divergance = SH = # Generated streams = 20.2%. (29.27)
In equation 29.27 we can see that if we constrained the streams so that they could not dissipate large amounts over a century and if we required at lest 10 test particles out of 8 000 to encounter the Earth we reduced the
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Figure 35: The distribution of stream to parent body mean divergence among the Pan-STARRS distribution of comets.
Page 95 D. Kastinen Meteors and Celestial Dynamics
probability to 20.2%. We have however not yet explained the concept of mean comet divergence measured in
DSH . Here we have calculated the gradient per century of the encountered particles parent body divergence, previously shown in figure 32 and 33, and taken the mean of each curve. This gives a mean divergence per century for each individual particle in a shower. Then we have taken the mean of the particles mean gradient in a single shower and we call that the mean criterion divergence per century of the shower. This is a relativity crude method mostly developed for stream dynamic exploratory reasons. The histogram of all these means are shown in figure 35. There are many more easily calculated results using the generated output data, depending on the application of the software. One general formula for deriving a probability based on the simulation is
# Simulations containing event X P (Event X occuring) = . (29.28) # Simulations This is the reasoning we used to find the expressions for equations 29.23 and 29.23 as each simulation generated one comet.
Figure 36: Probability of encountering a random cometary meteoroid stream ejected in the year 1800 in a given year.
Such a concept is indeed very basic but can be extended to a more sophisticated examination by using the same equation as in 29.28 but distributed over a parameter, like for example time. To illustrate this, we can use the same model simulation as before but instead check what the probability is that a stream ejected from a random comet encounters Earth as a function of time, as can be seen in figure 36. In this figure equation 29.24 was used but on every individual year instead, counting the number of comets that generated a shower that year of all the generated ones. Obviously this is not proper probability distribution function over time but rather a collection of individual probabilities. This is due to the fact that the same shower can be counted for more then one year as it can be a reoccurring shower, if one where to remove the shower when it encounters once, the diagram would represent a proper probability distribution function that integrates to 1.
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Figure 37: The mean time after ejection to encounter for each meteoroid stream.
Figure 38: The variance from the mean of time after ejection to encounter within a each stream.
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As the setup of the software is not a specialized simulation of a particular parent body but instead represents a wide population the results does not predict shower occurance times and to extinguish different showers in the data is not useful. However, the fact that the probability quickly rises to a steady probability of encounter indicates how cometary streams spread in the Solar system. It would seem since around half of every stream, as calculated with 29.25, within 400 years dissipates and encounter Earth one can assume a very different rate of dissipation among different cometary streams. Of course this graph is also affected by the number of particles generated since the entire initial distribution was not sampled with high frequency, only 995 points for a wide high dimensional distribution, but it is enough as a demonstration tool. To find out more about how and why this graphs seems to flat out, we can compare the ejection time with the encounter time. To do this in a meaningful manner one should not mean over every encountered particles arrival time since that would mean mixing particles of different origin. Instead we will mean within one generated shower produced by only one comet, and examine the distribution of means, , illustrated in figure 37, and standard deviations, illustrated in figure 38, of the internal arrival time. From this we can see that the initial distribution of comets can be split into two families focusing on the recurrence of the meteor showers. In figure 38 we can see a clear peak around 0 standard deviation in encounter time internally for showers. We can also observe a much wider distribution centred on 80-100 years. This indicates that there is one group of meteor showers that occur only once, thus having around 0 in time deviation internally, and the other population is of reoccurring meteor showers, either reoccurring over many years or returning few times with large times between shower returns. This result also illustrates the concept of a sporadic complex very well as the streams spread very differently but still over half of all streams encounter the Earth at one point or another, as can be seen from equation 29.25. In the final part, such methods as illustrated here shall be used to arrive at real conclusion regarding a case study. They will be applied on other types of simulations, such as orbital clones sampling the measurement uncertainty of a single comet and deriving the most probable encounter time for the Earth observed meteor shower. To complete this probabilistic picture we need to further also solidify the concept of a mean simulation. Since the Monte-Carlo fashion of sampling here does not draw the entire sample at once but rather draws only one point and runs the numerical methods on that single point, we call this a single simulation. When mentioning a mean simulation we thereby refer to the summarization of all simulations and mean forming over the number of Monte Carlo iterations. This will provide a type of mean simulation. This concept is more suited for distribution representing clones of the same object or a very small population, for example trying to derive a ”mean cometary meteor shower” is not useful since most observed meteor showers are very different in nature, also the initial distribution of comets is wide and thus a mean will not provide useful data. However, what can be interesting to examine it not the specifics of the shower itself but rather the mean dissipation of the meteoroid streams that encounter the Earth, and the mean ”density distribution” of the generated meteor shower. This will give a good idea on what kind of deviations from the sporadic background to look for when searching for new meteor showers in observational data. Also, one should always check the convergence of such a mean simulation. And the convergence should be specified to the application. For example, if simulating a meteor shower in the year 2000 and 2001 at the same time. The convergence of the mean configuration distribution for the 2000 meteor shower could occur well before the convergence for the 2001 meteor shower distribution since particles may not propagate in the same numbers to this year. To conclude, the manner in which one should analyse the data and what comparisons should be done depends entirely on the type of simulation that is performed. However the idea described here are a few of the general probabilistic and statistical methods that apply to all output data, and can be applied on most of the output parameter space.
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29.10 Configuration
To streamline this work we have omitted listing all code implementation choices, however to provide a more complete picture of the constructed software we have below included a sample configuration file. The below file can be used to run the software module as the comments are omitted by the phrasing function. Listing 3: MCAS Configuration example
1 #%%%%%%%%%%%%%%%% CONFIGURATION FILE %%%%%%%%%%%%%%%%
2 # Lines starting with# are ignored
3 #------
4 #Calculation settings
5 #
6 #Integrators:
7 # 0: CMS
8 # 1: mercury6
9 #
10 #SolarsystemIC generator:
11 # 0: JPL(invariable plane, no equanox orientation)
12 # 1: JPL SPICE-c(J200.0)
13 #
14 #Family:
15 # 0 = NEOs (~270k)
16 # 1= MBOs (~100k)
17 # 2= MBOs --> NEOs (~50k)
18 # 3 = Hildas (~1.8k)
19 # 4 = Comets (~28k)
20 # 5 = Hyperbolic (~8.3k)
21 # 6 = LPCs (~9.4k)
22 # 7 = Centaures (~60k)
23 # 8 = Scattered (~11k)
24 # 9 = TNOs (~48k)
25 # 10 = BOTKKE NEO’s
26 # 11 = Statistical uncententy orbital clone generator(SUOC)
27 #
28 #Particle mass distribution:
29 # 0 = GRUN particle model(only min mass)
30 # 1 = Uniform representation, can also be single size if min= max
31 # 2 = 10 logarithmic uniform representation(equal probabillity for each exponent)
32 # 3 = Custom file(mass_dist.txt), first line masses, second line their respective probabillites
33 #
34 #Particle mass distribtuion resolution:
35 # The resolution of the resulting histogram used in generating the random particles
36 #
37 #Encounter end date(Julian days):
38 # If not zero, then ignore integration time and use this date instead allowing variable integration time
39 #
40 #Distribution type:
41 # 0: Pick random bodies from the supplied set
42 # 1: Create histogram over orbital elements and pick random combinations
43 # 2: Create multidimensional map over distribution of supplied set and drawa set of coordinates from this
44 # 3: Reduced multidimensional map using principal component analysis
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45 #
46 #
47 #Planet mass input format:
48 # 0: kg 49 # 1:AU **3/DAY**2 50 #
51 #
52 #X types:
53 # 0: Showers
54 # 1: Clones
55 #------
56 Integrator =1
57 SolarsystemIC generator =1
58 Start date(Julian days) =2424861.2071
59 Integration time for encounter(years) =0
60 Encounter end date(Julian days) =2458849.5
61 Family =11
62 Distribution type =0
63 Calculate clusters =1
64 Calculate stream dissipations =0
65 Particle mass distribution =2
66 Particle mass distribtuion resolution =100
67 Lower mass bound(g) =1e-6
68 Upper mass bound(g) =1e2
69 Memory allocation(Mb) =1e+03
70 X type =1
71 Number ofX to generate =50
72 Close enoucnter treshold( hill radii) =1.2
73 Planet mass input format =1
74 Snapshot date =0
75 #------
76 #OUTPUT settings
77 #
78 #Logfile:
79 # 0 = No file
80 # 1 = Write file
81 #------
82 Logfile output =1
83 #------
84 # mercury6 settings
85 #------
86 #Mercury integrator:
87 # 0 = Hybrid
88 # 1 =BS
89 #
90 #------
91 Mercury integrator =0
92 Time setp(days) =1
93 IncludePR effect =1
94 Ejection distance(AU) =100
95 Hybrid changeover(hill radii) =5
96 Max particles in mercury.inc =10000
97 #------
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98 # SUOC settings
99 #------
100 # Sigma range:
101 # Normal distribution sigma range
102 #
103 # Distribution resolution:
104 # Normal distribution numerical resolutioni.e.
105 # the number of bins.
106 #
107 #------
108 Number of clones =5000
109 Sigma range =5
110 Distribution resolution =5000
111 %%%%%%%%%%%%%%%%%%%% ENDOF FILE %%%%%%%%%%%%%%%%%%%%
30 Parent Body Ejector module
The Parent Body Ejector module is designed to simplify the creation of long term initial condition generation. Often authors will eject particles only at perihelion and in only two directions, as in (Vaubaillon et al., 2011), thus creating a easily scripted generation of initial conditions for propagation. Due to the cumbersome nature of generating more complex initial condition continous ejection in different direction at different speeds are not considered, practically omitting much of the in-stream structure while only looking at the extremes. This module will make generation of such initial conditions easy where several ejection models are already available, parent body parameters are simply input and the output is a simulation state once ejection of particles is complete.
30.1 Symplectic structure
Symplectic integrators are a scheme for numerically integrating a specific group of differential equations con- nected to symplectic geometry that is a branch of differential geometry and differential topology. Since this type integrator works on geometrical basis it is excellent for integrating physical problems such as classical mechanics where the phase space of a system can take the form of a symplectic manifold. In our case the set i i of differential equations are the Hamiltonian with the two-form dpi ∧dq , where q is a generalized coordinate and pi its momentum. These two elements together create our canonical coordinates, phase space, and the symplectic integrator can be viewed as canonical transformations.
A example of the difference between preserving symplectic structure and not doing so in a numerical inte- gration can be seen in figure 4 and 5. In these pictures are shown what such a simple modification as to make the method semi implicit can do for a numerical long term integration of a Hamiltonian system.
30.2 Propagation
As only relatively short time spans should be integrated using this module we have so far implemented only two different integration methods, the Bulirsch-Stoer method described in section 5.3, and a symplectic 8 part Hamiltonian split method. The split concept and order determination was described in section 5.2.2. The particular symplectic split implemented originates from a newly discovered family of splits (Blanes et al.,
2013), where we have picked the 8 order H = Ha + Hb splitting, where Ha is the kinetic energy and Hb is the potential. This split can be written as
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a1 b1 a2 b2 a3 b3 a4 b4 a5 b4 a4 b3 a3 b2 a2 b1 a1 Ha Hb Ha Hb Ha Hb Ha Hb Ha Hb Ha Hb Ha Hb Ha Hb Ha (30.1)
with numerical values of the coefficients as (Blanes et al., 2013)
a1 = 0.03809449742241219545697532230863756534060 a2 = 0.1452987161169137492940200726606637497442 a = 0.2076276957255412507162056113249882065158 3 a = 0.4359097036515261592231548624010651844006 4 a5 = −0.6538612258327867093807117373907094120024 (30.2) b = 0.09585888083707521061077150377145884776921 1 b = 0.2044461531429987806805077839164344779763 2 b3 = 0.2170703479789911017143385924306336714532 b4 = −0.01737538195906509300561788011852699719871
This means that we shall, in each step, apply the separate Hamiltonians Ha and Hb each with the total time step modified by the above coefficients. In other words, we apply the effect of kinetic energy by a time
a1Tstep, then we apply the effect of the gravitational pull of all bodies by a time b1Tstep, and so on. Both of these methods have electromagnetic corrections implemented following the form from (Burns et al., 2014),
! dv SAQ v • Sˆ v m = pr (1 − )Sˆ − , (30.3) p dt c c c where mp is the particle rest mass, A the geometrical cross-section of the particle, Qpr the radiation efficiency factor, and S is the energy flux density in the radiation beam. In our implementation we have set the Qpr coefficient to 1 as that is a reasonable assumption and used standard mean luminosity coefficients for the Sun as given by (Mamajek et al., 2015).
30.3 Period syncing
When designing a propagator one cannot assume all inputs to be initially time synchronized, thus we shall consider two possibilities; that the initial state of the Solar system is given at a different time than that of the parent body, and that the initial state of the parent body is inside the sublimation sphere. To account for the first possibility we will simply employ a method that propagates the Solar system forwards or backwards to match the time of the parent body. To resolve the second problem is somewhat more difficult due to the fact that we cannot know the parent body propagated backwards in time will have a stable orbit. As such, one should be able to chose between two different algorithms. One that propagates backwards until periapsis for short term stable orbits, and one that simply propagates backwards until the parent body is outside the sublimation radius, with some termination condition if the propagation goes on for too long for general trajectories. There is also the possibility that the orbit never leaves the sublimation sphere, if that is the case the algorithm should stop at a apsis point. The first kind can be summarized by:
FIND APHELION
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INPUT : Simulation state at t1
OUTPUT : Simulation state at t0 < t1 s 3 aPB Find approximate orbital period of PB: TPB = 2π µSun Current time simulation time is t
Begin backwards propagation: WHILE t − t1 < TPB Propagate the state one step : t 7→ t − ∆t
Check if the orbital motion has turned : IF (Previous |rPB • vPB|)(Current |rPB • vPB|) < 0 Break while loop END IF END WHILE
return Simulation state at: t0
And the second can be written as
FIND APHELION
INPUT : Simulation state at t1 Termination condition Γ
OUTPUT : Simulation state at t0 < t1 Current time simulation time is t Begin backwards propagation: WHILE Γ Propagate the state one step : t 7→ t − ∆t
Check if outside sublimation sphere : IF |xPB − xSun| > Rsublim Break while loop END IF END WHILE
return Simulation state at: t0
In the above algorithm the termination condition Γ will vary depending on the application. It can be a max
time Γ = t − t1 < Tmax, or it can be some orbital element condition Γ = aPB > amax, thus this will have to be changed from application to application. This dynamic change has not yet been implemented so currently
Γ = t − t1 < 0.75TPB.
30.4 Ejection speed
To proceed further with the ejection of particles, we need to calculate the ejection speed of the particles from the parent body. In section 10.2.1 we have defined several models and more thoroughly described their derivation, thus here we shall only give the equations implemented in the software:
• (I) Model from (Whipple, 1951)
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• (II) Model from (Hughes, 2000)
• (III) Model from (Ma et al., 2002)
• (IV) Linear heliocentric model
The formula for (I) is given by,
˙ 2 kd Mcv¯g A 2GMc V∞ = − , (30.4) 2 πRc m Rc where the comet mass loss in the case of a spherical body is
r 2 1 M˙ = πR2S 0 . (30.5) c c 0 r nH
6 The used coefficient values are, kd = 26/9 for the particle gas drag, sublimation heat of H = 1.88 · 10 J/kg, andv ¯g is derived from the mean thermal velocity of gas
8k T 1/2 1 v¯ = B g (30.6) g πµ rτ
−24 −1/2 where we assume µ = 20 · 1.661 · 10 g, Tg = 300r K, and τ = 0.25. The values of Mc and Rc are inputs of the parent body characteristics For the ejection model in (II), winch is a modification of (I), the ejection speed is given by
˙ 2 2Mc θv¯s A 2GMc V∞ = τ ξ − , (30.7) ΦπRc r m Rc where the mass loss is given as
r 2 1 M˙ = 2gπR2S 0 , (30.8) c c 0 r nH and the thermal velocity of the gas is
8k T 1/2 v¯ = B g . (30.9) g πµ
Using this model we have chosen static coefficients for the surface absorption factor g = 0.1, the free drag coefficient ξ = 2.0, the adiabatic acceleration θ = 2, τ = 0. The coefficients n and H are as in model (I). The coefficient Ψ, or the surface activity factor, is a input of the parent body characteristics and A and m are given by the input bulk density and mass distributions. The ejection speed formula used in model (III) is given by
2 WRcL 1 1 2GMc v = 2 − 2 − , (30.10) 8πHαsσ r rs Rc
Page 104 D. Kastinen Meteors and Celestial Dynamics and the cometary mass loss is given by
2 ˙ Rc L 1 1 Mc = 2 − 2 . (30.11) 4H r rs
Here the mean thermal velocity is set to W = 580 m/s, the Solar luminosity L is chosen as previously, the latent heat of vaporization H = 2 · 106 J/kg. In this formula the input fraction of the comet that is active is
α. s is the radius of the assumed spherical dust grain, σ it’s bulk density, and rs is the critical heliocentric radius for sublimation to occur, all of which are dominated by input data. In all the above formulas G is the gravitational constant. All these parameters and the models are discussed in (Ryabova, 2013). Lastly the formula for (IV) is simply a increase in ejection velocity which is linear in the heliocentric distance, starting from 0 m/s at the input rs and peak at a input parameter at perihelion. In the future we plan to implement full control over the model coefficients by creating configurations files for each model and also to implement more models. The ability to chose these coefficients as distributions is also planned for future versions. This can be very useful since one can examine how subtle differences propagate through time, such as the value of the sublimation. A variable that is dependant on what ices are present on the comet.
30.5 Ejection direction
Figure 39: Two spherical distributions, the left illustrating a spatial angle uniform distribution, the right illustrating a true spherically uniform distribution.
The question of how ejection points on the sphere representing the parent body is picked is a relevant one due to several factors. The main reason is that there is a sensitivity in the resulting deviation from the parent orbit to the direction of the initial perturbations. For example in cometary ejections due to sublimation classical dust stream simulations (Vaubaillon et al., 2011) are commonly set-up with only two perturbation directions, both tangent to the orbit and in the orbital plane (Sato, 2003).
However there are many advantages to use a random uniform distribution on the sphere. One such is
Page 105 D. Kastinen Meteors and Celestial Dynamics the ”filling” of the dust stream such that the entire range of possible perturbations are sampled, and not only the extreme perturbation cases. It may be argued that a uniform distribution is unrealistic due to jets and non active areas of cometary nuclei. The individual shape and activity of the individual simulated comet can however be ignored if repeated simulations are performed on a wide population. The activity configuration can also be ignored due to the spin of the comet. There are many mechanisms that can alter the spin state of a nucleus, most are stochastic in nature or have large time scales. However the out gassing torque is the primary mechanism of spin-state change (Samarasinha et al., 2004). Since out gassing is the main effect for producing dust streams we know that this effect is present in simulations of cometary dust streams.
One cannot simply create a uniform distribution by picking the polar coordinates θ and φ from uniform distributions since the spherical area element dΩ = sin(φ)dθdφ is itself dependant on φ. Instead one can use the rules of quaternion transformation to transform uniform distributions U(a, b) to a uniform US(r) spherical distribution (Marsaglia et al., 1972). This can be done by using the four dimensional stochastic vector
ξ0 ξ1 x = , (30.12) ξ2 ξ3
ξi ∈ U(−1, 1) ∀ i ∈ [0, 3], (30.13) and restricting this distribution to only
x • x < 1. (30.14)
Then the resulting distribution that is left can be transformed to the surface of the sphere by
2(ξ ξ + ξ ξ ) x = 1 3 0 2 , (30.15) x • x 2(ξ ξ − ξ ξ ) y = 2 3 0 1 , (30.16) x • x ξ2 + ξ2 − ξ2 − ξ2 z = 0 3 1 2 , (30.17) x • x x Y = y ∈ US(1) (30.18) z
The resulting distribution can be seen in figure 39 together with the non uniform distribution defined by uniformly distributing the polar angles.
30.6 Particle mass
In section 29.8 we covered the types of input mass, let us now address how these distributions are used. First we need to find to define a mean particle mass, to be used in the mass timing ejection calculation. Given a vector with masses m and a vector of the same size with the corresponding probabilities Pm for each mass to be ejected. If the probability vector is normalized we can write the mean mass as a simple scalar product,
Page 106 D. Kastinen Meteors and Celestial Dynamics
µm = m • Pm. (30.19)
As we draw masses from Pm the mean of the masses will converge to µm with sufficient high draw counts. This indicates that with a sufficient amount of particles per ejection, N, this can estimate the amount of mass ejected as Nµm.
30.7 Ejection times
Now that we have defined at what speed the particles are to be ejected, and in what directions, finally we have to address at what times the ejections are to be. This is less straight forward then expected as depending on the amount of test particles one cannot eject at each discrete time step of the simulation, since one sometimes want very small accurate time steps while ejecting few particles. Thus we have chosen to implement a type of cumulative ejection, where we have as user input a maximum amount of ejected test particles and a maximum amount of ejection instances. This provides relatively large ejection configuration control. However, since the mass loss of the comet increases as it approaches the Sun it will eject more meteoroids with higher velocities. Thus one would in most application want more precision in ejections around the area of highest mass loss and largest ejection velocities. To fulfil these requirements we will first propagate the synchronized system once without ejection particles. During this propagation we only calculate the mass loss of the comet in each step to estimate the total mass loss of the comet. Since few body propagation for short time frames such as one or two orbits is very fast with modern computers this can be done with little computational cost. Using this total mass loss of the comet we shall, given the input ejection instances, choose a mass per ejection limit. When the second propagation starts we calculate the mass loss in each step, and add it to a cumulative value. Once this value exceeded the mass per ejection limit, a set of particles are ejected. The amount of particles in each ejection depend on the maximum number of particles and the maximum number of ejection instances. Why we refer to this as a ”maximum” number is due to the fact that the first integration was an estimation, and some ejections may fail. Also since the particles are randomly generated from mass distributions it may occur that small variations in mass ejections may occur if too few test particles are chosen. Lastly, a possible variation in ejection particles and instances can occur if the particles generated are unable to escape the comets gravitational field. Since the mass distribution input allows for any mass there can be, depending on the ejection model, some masses that are not accelerated enough by the outflow of the gas to reach escape velocity from the comet. If this happens, the simulation will find that the particle could not escape and draw a new random mass for this particle. If this happens too often, the simulation will be slowed down too much and it will skip that specific ejection moment. But it will not completely omit the mass loss that the comet has experienced, it will instead propagate one step, increase the cumulative mass loss, and attempt to eject particles again with the new increased mass loss. Thus we shall not lose mass loss representation if the particle mass range cannot be expressed computationally efficiently at a point but instead eject more particles once we have statistically enough ejection of that particle mass to represent the ejection given the number of allowed particles. Below are psudo code algorithms for the discussed procedures.
ESTIMATE MASS LOSS
Page 107 D. Kastinen Meteors and Celestial Dynamics
INPUT : Initial state:(qi, pi)|t=t0 ∀ i ∈ [1,N] Integration step: ∆t
Max particles: Pmax
Max ejections: Emax Parent body data: G
OUTPUT : Mass loss per ejection: Me
Particles per ejection: Pe Initialize total mass loss: ∆M = 0
Initialize propagation: t = t0
Estimate mass loss: WHILE t < t1 Propagate one step: t 7→ t + ∆t
Calculate mass loss: M˙ = SUBLIM(qPB, pPB, G) ∆M = ∆M + M˙ END WHILE ∆M Estimate mass loss per ejection: Me = Emax Pmax Particles per ejection: Pe = Emax
CUMULATIVE EJECTION
INPUT : Initial state:(qi, pi)|t=t0 ∀ i ∈ [1,N] Integration step: ∆t
Mass loss per ejection: Me
Particles per ejection: Pe Parent body data: G
OUTPUT : System state after ejection: (qi, pi)|t=t1 ∀ i ∈ [1,N + NP ]
Page 108 D. Kastinen Meteors and Celestial Dynamics
Initialize cumulative mass loss: ∆M = 0
Initialize total mass ejected: Mt = 0
Initialize propagation: t = t0
Estimate mass loss: WHILE t < t1 Propagate one step: t 7→ t + ∆t
Calculate mass loss: M˙ = SUBLIM(qPB, pPB, G) ∆M = ∆M + M˙
IF ∆M > Me
Eject particles: EJECT(Pe) Check if signifficant ejection possible: IF Enough ejections sucseed Add ejected particles to simulation state: ADD PARTICLES
Add ejected mass to total: Mt = Mt + ∆M Reset cumulative mass loss: ∆M = 0 END IF END IF END WHILE
Lastly, we can estimate the particle weight Wp as
Mt Wp = , (30.20) NP µm where NP is the number of generated particles, Mt is the total ejected mass in particles, and µm is the mean particle weight.
30.8 Data formatting
30.8.1 Input files
Table 27: mass.data, Column separator: ’Space’ 0x20 Input File description: Simulation state mass vector
Row Description Data type Units Notes 1 Object mass Double kg -
Table 28: pos.data, Column separator: ’Space’ 0x20 Input File description: Simulation state position vector, one row for each object
Column Description Data type Units Notes 1 x-component Double m - 2 y-component Double m - 3 z-component Double m -
Page 109 D. Kastinen Meteors and Celestial Dynamics
Table 29: vel.data, Column separator: ’Space’ 0x20 Input File description: Simulation state velocity vector, one row for each object
Column Description Data type Units Notes 1 x-component Double m/s - 2 y-component Double m/s - 3 z-component Double m/s -
Table 30: mass_dist.data Column separator: ’Space’ 0x20 Input File description: Ejection mass distribution
Row Description Data type Units Notes 1 Bin center value Double grams - 2 Bin probability Double - Does not need to be normalized
Table 31: body_data.data Column separator: ’Space’ 0x20 Input File description: Parent body data
Row Description Data type Units Notes 1 Position - - - Column Description Data type Units Notes 1 x-component Double m - 2 y-component Double m - 3 z-component Double m - Row Description Data type Units Notes 2 Velocity - - - Column Description Data type Units Notes 1 x-component Double m/s - 2 y-component Double m/s - 2 z-component Double m/s - Row Description Data type Units Notes 3 Properties - - - Column Description Data type Units Notes 3 Nucleus radius Double m - 3 Bulk density Double kg/m3 - 3 Nucleus activity factor Double 1 - 3 Critical sublimation radius Double m heliocentric 3 Nucleus mass Double kg - 3 Perihelion passage difference Double days Relative system initial state
Page 110 D. Kastinen Meteors and Celestial Dynamics
30.8.2 Output files
Table 32: particle_q_data.data, Column separator: ’Space’ 0x20 Output File description: Simulation state position vectors, one row for each object
Column Description Data type Units Notes 1 x-component Double m If equatorial J2000.0 equinox: Vernal equinox component 2 y-component Double m Perpendicular to equinox in plane 2 z-component Double m Perpendicular to reference plane
Table 33: particle_v_data.data, Column separator: ’Space’ 0x20 Output File description: Simulation state velocity vectors, one row for each object
Column Description Data type Units Notes 1 x-component Double m/s If equatorial J2000.0 equinox: Vernal equinox component 2 y-component Double m/s Perpendicular to equinox in plane 2 z-component Double m/s Perpendicular to reference plane
Table 34: particle_m_data.data, Column separator: ’Space’ 0x20 Output File description: Simulation state mass vector, one column for each object
Row Description Data type Units Notes 1 Object mass Double kg -
Table 35: particle_t_data.data, Column separator: ’Space’ 0x20 Output File description: Simulation ejection time vector, one column for each object
Row Description Data type Units Notes 1 Time of ejection Double s Relative simulation start
Table 36: particle_abs_v_data.data, Column separator: ’Space’ 0x20 Output File description: Simulation ejection velocity vector, one column for each particle
Row Description Data type Units Notes 1 Heliocentric distance Double AU - 2 Ejection velocity relative parent body Double m/s -
Page 111 D. Kastinen Meteors and Celestial Dynamics
Table 37: particle_kep_data.data, Column separator: ’Space’ 0x20 Output File description: Kepler data after ejection, one row for each object
Column Description Data type Units Notes 1 Semi-Major axis Double AU - 2 Eccentricity Double - - 3 Inclination Double rad - 4 Argument of Perihelion Double rad - 5 Longitude of Ascending Node Double rad - 6 True anomaly Double rad - 7 Mean anomaly Double rad - 8 Argument of Latitude Double rad - 9 True Longitude Double rad - 10 Longitude of Perihelion Double rad - 11 Semilatus Rectum Double m -
Table 38: body_m_data.data, Column separator: ’Space’ 0x20 Output File description: Simulation state parent body mass, one row for each instance
Column Description Data type Units Notes 1 Time Double days Relative simulation start 2 Mass Double kg Change varies with sublimation model
Table 39: sim_data.data, Column separator: ’Space’ 0x20 Output File description: Simulation summary data
Column Description Data type Units Notes 1 Particle representation weight Double 1 Same for all particles from this simulation 1 Total simulations time Double s - 1 Bulk density Double kg/m3 - 1 Parent body end mass Double kg - 1 Time synchronization required Double s From input start to PBE start 1 Orbits simulated for ejection Double 1 -
30.9 Configuration
To streamline this work we have omitted listing all code implementation choices, however to provide a more complete picture of the constructed software we have below included a sample configuration file. The below file can be used to run the software module as the comments are omitted by the phrasing function.
Listing 4: PBE Configuration example
1 #%%%%%%%%%%%%%%%% CONFIGURATION FILE %%%%%%%%%%%%%%%%%
2 #------
3 #Simulation settings
4 #------
Page 112 D. Kastinen Meteors and Celestial Dynamics
5 #Ejection model:
6 # 0 = Ma et al. 2002
7 # 1 = Hughes et al. 2000
8 # 2 = Whipple 1951
9 # 3 = from0 to input at perihelion
10 # 4 = Only at perihelion
11 #
12 #------
13 Ejection model =1
14 Time step(days) =16
15 Number of orbits =1
16 Max Particles =8e+03
17 Maximum number of ejection instances =20
18 Print precision =20
19 Output type(helio0/bary 1) =0
20 Max velocity for model3&4[m/s] =30
21 %%%%%%%%%%%%%%%%%%%% ENDOF FILE %%%%%%%%%%%%%%%%%%%%
31 Orbital Association Analysis module
The Orbital Association Analysis module is designed to implement a multitude of methods to associate orbits and analyse the results. One common approach we have implemented is the use of orbital similarity function to create associations and cluster analysis to examine the made associations. In the future this will become a platform for testing novel association and analysis methods. Such novel methods could be topological similarity functions rather than function based on orbital element differences, or machine learning methods rather then static density analysis or wavelet coefficient thresholds. In this section we will cover the current implementation state of the OAA module and define what methods are implemented and in what fashion.
31.1 Similarity functions
We have chosen to define similarity functions such that 0 indicate complete similarity and the maximum value of the function indicates complete dissimilarity. In the current version we have implemented 4 similarity functions:
• Standard D-criterion: DSH
• Normalized D-criterion: DD
• H¨olderp=2 Metric: ρ2
• Euclidean composite Metric: %1
The actual implementation of the functions are described below. The variables q, a, e, i, ω, Ω refer to the standard orbital elements unless otherwise stated.
The Southworth and Hawkins D-criterion, DSH (Southworth and Hawkins, 1963), is defined as;
I e + e Π 2 D2 = (e − e )2 + (q − q )2 + (2 sin ab )2 + a b 2 sin ab , (31.1) SH b a b a 2 2 2
Page 113 D. Kastinen Meteors and Celestial Dynamics
where
I i − i Ω − Ω (2 sin ab )2 = (2 sin b a )2 + sin i sin i (2 sin b a )2 (31.2) 2 2 a b 2 is the angle between the orbital planes and
i − i Ω − Ω I Π = ω − ω + 2 arcsin cos b a sin b a sec ab (31.3) ab 2 1 2 2 2 is the difference between the longitudes of perihelion measured from the intersection of the orbits. The
Drummond D-criterion, DD (Drummond, 1980), a more correctly normalized version of DSH ;
2 2 2 2 2 eb − ea qb − qa I ea + eb Θ DD = + + ◦ + ◦ , (31.4) eb + ea qb + qa 180 2 180
where
I = arccos (cos ia cos ib + sin ia sin ib cos (Ωb − Ωa)), (31.5)
and
Θ = arccos (sin βa sin βb + cos βa cos βb cos (λb − λa)). (31.6)
The ecliptic latitudes of the perihelion points can be expressed as
( βa = arcsin (sin ia sin ωa), (31.7a)
βb = arcsin (sin ib sin ωb), (31.7b) and the ecliptic longitudes of the perihelion points as
( λa = Ωa + arctan (cos ia tan ωa), (31.8a)
λb = Ωb + arctan (cos ib tan ωb). (31.8b) The λ parameter has the additional condition of adding 180◦ if cos ω < 0. Since the only difference between
DSH and DD is a normalization procedure, it is interesting to examine the different association patters due to such a change of the similarity function. Then we also implemented a H¨oldertype metric acting on the space of oriented Keplerian ellipses with positive semi major axes and eccentricities E, where the case zero eccentricity produces a oriented circle without marked pericenter. This metric was constructed (Kholshevnikov and Vassiliev, 2004) and is described by
( ) 1 Z 1/p ρ(p, E, E0) = min sp(Q(u),Q0(u + v))du , (31.9) 2π for mapping E × E onto R+. In the special case of p = 2 that we have chosen to implement; the metric integration reduces to
Page 114 D. Kastinen Meteors and Celestial Dynamics
2 0 0 p 2 2 ρ (2,E,E ) = 2aa W0 − (W5 + W8) + (W6 − W7) , (31.10) where the W coefficients are found by
0 2 0 02 0 0 4W0 = 2(α + α ) + αe + α e − 4P · P ee , (31.11a) 0 0 W1 = P · P e − αe, (31.11b) W = P0 · Se0, (31.11c) 2 0 0 0 W3 = P · P e − α e , (31.11d) 0 W4 = P · S e, (31.11e) 0 2W5 = −P · P , (31.11f) 0 2W6 = −P · S , (31.11g) 2W = −P0 · S, (31.11h) 7 0 2W8 = −S · S . (31.11i) The P and S vectors for each orbit is found by
cos ω cos Ω − cos i sin ω sin Ω P = cos ω sin Ω + cos i sin ω cos Ω , (31.12) sin i sin ω − sin ω cos Ω − cos i cos ω sin Ω S = ηQ = η − sin ω sin Ω + cos i cos ω cos Ω , (31.13) sin i cos ω √ 2 a 0 a0 where η = 1 − e , α = a0 and α = a and the prime denotes the second input orbit. This is a interesting metric to implement since it function on a different principle, the integrated total orbital distance, than the D-criterions. The fourth and last function implemented so far is another type of metric on the space H of orbits, both curvilinear and rectilinear. The space H does not however distinguish between ascending or descending orbits but instead acts on fundamental orbital quantities. This function was presented in (Kholshevnikov, 2008) and its definition is
2 2 1 2 2 L2 2 %1(ε1, ε2) = 2 |c1 − c2| + |e1 − e2| + 4 (h1 − h2) , (31.14) µ L1 µ where the c and e vectors are the area vector, or the specific angular momentum vector, and the Laplace- Runge-Lenz vector respectively. This metric can be reduced to the software implemented version of
2 2 2 1 √ 2 2 L2 1 1 %1(ε1, ε2) = (p1 + p2 − 2 p1p2 cos ξ) + (e1 + e2 − 2e1e2 cos ζ) + − . (31.15) L1 4 a1 a2
Where L1 and L2 are scaling constants and ξ and ζ are calculated by
cos ξ = cos i1 cos i2 + sin i1 sin i2 cos (Ω1 − Ω2), (31.16)
Page 115 D. Kastinen Meteors and Celestial Dynamics
cos ζ =(cos ω1 cos ω2 + cos i1 cos i2 sin ω1 sin ω2) cos (Ω1 − Ω2)+
+ (cos i2 cos ω1 sin ω2 − cos i1 sin ω1 cos ω2) sin (Ω1 − Ω2)+
+ sin i1 sin i2 sin ω1 sin ω2. (31.17)
By comparing these four function we can perform a number of comparisons. For example the difference between two metric approaches, the integral orbital distance versus orbital element distances, and the effect of normalization in similarity functions.
31.2 Similarity matrix
Given a general similarity function f(xi, xj) acting on two objects xi and xj from a set of N objects {xi : 1 ≤ i ≤ N} we can define the similarity matrix of that set as
Dij = f(xi, xj) ∀ 1 ≤ i ≤ N 1 ≤ j ≤ N. (31.18)
When f is a metric, the diagonal will always be zero due to the metric properties but with similarity functions this need not be the case as objects can be non self similar, i.e. have a non zero distance to itself. This has a calculation complexity of O(N 2) but the amount of calculations can be reduced from N 2 to (N − 1)N/2 if the function is commutative, if the self similarity value is not of interest the amount of calculations can be reduced to (N − 2)(N − 1)/2, which is still O(N 2) but more effective.
31.3 Clustering
Currently only one cluster merging function is implemented, the single linkage clustering rule. This rule can be written as
min ({f(a, b): a ∈ Xi, b ∈ Xj}) < Dc, (31.19) where the two sets that are being checked for merging is Xi and Xj, the critical clustering threshold is Dc and the similarity function is f. If this statement is resolved as true, the two clusters being checked are merged to a single one. To implement this in a C++ algorithm we have constructed a recursive node walk function. Given a set of N nodes {xi : 1 ≤ i ≤ N}, we can consider a function that, given a node; finds all other nodes within a critical distance Dc of the input node. If this function then calls itself on all found nodes, we easily see that given a node, this function will always explore the entire cluster of the given node and noting else. Thus calling this recursive function once for each node, and having the function skip all nodes already visited once by it, will search the entire node network. The clusters are found by having this function record every node it visits with each calling, since one calling will completely explore one cluster and disable all nodes in that cluster so that the function skips them, the algorithm will only visit each node once and reduce computing time.
To speed up calculations the function finds nodes that are within Dc by a boolean look-up matrix. This matrix is the boolean value of the inequality posed on the similarity matrix Gij = Dij < Dc = f(xi, xj) < Dc ∀ i 6= j. Thus, if the input node is xk the function will call itself with node xj as input if Gkj is true.
31.4 Parameter sweep
In the parameter sweep we call the clustering function while increasing the critical threshold Dc from zero, where in a set of N nodes there will be N clusters, to the point of termination. Currently there are 2 types of termination points:
Page 116 D. Kastinen Meteors and Celestial Dynamics
1. Association maximum
2. Error function minimum
For the first termination criterion, as previously, we have here defined the amount of association in a set as the amount of node that has a connection. For example, let us say we have 10 nodes, and 5 clusters, each with 2 nodes in the cluster. This way all nodes are in some way associated with another node and we have defined the association to be at its maximum, even though the clustering process is not complete, i.e. when all nodes form one big cluster. If the first termination point is chosen, the parameter sweep will terminate once every node has at least one connection. In the case of single linkage clustering there is also another state to consider, one where the set of nodes and their connections form a complete graph. A complete graph is when every node in a N set has N − 1 connections to every other node. As the level of association to be examined can vary between different research and the time to perform the parameter sweep is dependant on the size of the sample and the similarity function, there may be the desire to implement ones own definition of a correct clustering. Thus we have also implemented the use of custom error function.
31.4.1 Fixed step
As the parameter sweep searches the value space of the critical threshold it will have to step through this space in some manner. Depending on the structure of the examined association algorithm and the similarity function using a fixed step may be advantageous to adaptive step methods and vice versa, thus both are included in the implementation. Currently one can set a fixed step that the algorithm steps with until the termination criteria has been fulfilled. There is currently a hard coded upper threshold on the number of steps that the sweep can perform without any change in association state, once this limit has been passed the sweep self terminates. In future implementations this will not be a hard coded value but instead a input option. This safety measure is in place since we have found that certain data configurations using some function can create vast differences for some nodes. For example in the case of the orbital similarity functions, using a function where one of the dependant variables are the semi major axis instead of the perihelion distance, on a set of orbits all with small semi major axis seems fine. However, in some cases, if a gravitational slingshot effect occurs during the simulation sending one body out of the Solar system at very high speeds, if the Solar system escape velocity is not exceeded and the orbit does not turn hyperbolic, the semi major axis of the object can be any value between 0 and ∞. A fixed step would in this case essentially step forever trying to associate this one last point if the termination criterion requires all nodes to have at least one connection. This threshold is currently set to 1000 steps without any state change.
31.4.2 Adaptive step
The adaptive step algorithm function on the notion of state change of the evaluation function. Currently, the evaluation function is the same as the termination criterion function, i.e. the error function or association amount. If the state of the evaluation function does not change, i.e. no new important association are made, the step size in increased according to some rule. The complete algorithm for step increase can be found below.
ADAPTIVE STEP INCREASE
Page 117 D. Kastinen Meteors and Celestial Dynamics
1.Calculate association state using current Dc 2.Calculate evaluation function using current association state 3.WHILE Next step is not OK
3.1. Step Dc by current step toDc2
3.2. Calculate association state using Dc2 3.3. Calculate evaluation function using next association state 3.4. IF Next evaluation step differs from current 3.4.1. Next step is OK ELSE 3.4.2. Increase current step size by rule END IF END WHILE
The above pseudo code may seem more ineffective than a normal fixed step but considering that the step size increment rule can be dependant on the step size, we can find many scenarios when this is more effective than a fixed step. For example a multiplicative function that doubles the step size each time the next step check is performed and fails, would quickly traverse huge gaps in the parameter space. However we immediately find a drawback; the more the step size is increased the less resolution we have, thus we need a counter to this effect with a adaptive step decrease algorithm, described below.
ADAPTIVE STEP DECREASE
1.Calculate association state using current Dc 2.Calculate evaluation function using current association state 3.WHILE Next step is not OK
3.1. Step Dc by current step toDc2
3.2. Calculate association state using Dc2 3.3. Calculate evaluation function using next association state 3.4. IF Next step differ too much from current 3.4.1. Decrease current step size by rule ELSE 3.4.2. Next step is OK END IF END WHILE
Here, the definition of the association states differing too much is very ambiguous and can be chosen very differently depending on the features that one want to examine. We have chosen this to be defined as when more than one cluster merging occurs. This means that if for example, after a large value of no state changes, we in one step suddenly associate our entire sample, then the step size will continue to decrease until we can resolve every single cluster merge. Thus we have combined both of the above algorithms to create our adaptive step. This step algorithm is only faster in certain situations but it has the advantage of using less saved data as only steps that changes
Page 118 D. Kastinen Meteors and Celestial Dynamics the state evaluation function are saved. However due to possible computational problems the adaptive step algorithm uses a minimum and maximum step size as well to limit run-away behaviour. If the step is changed to this maximum or minimum value, a new step is forced instead of a step change. The reason for this can be illustrated easily if one considers a case where one cluster in the middle of two other is about to merge the three clusters into a single one. If the merging distance between the middle cluster and the side ones are infinitesimally close to each other the adaptive step decrease will change the step size forever to resolve the two different merges. Thus a minimum step size limit has to be set.
31.5 Error functions
The use of a error function is commonly used in areas such as pattern recognition and machine learning. Its purpose is to represent the amount of error in the performed algorithm. These will be tailored to the research to be performed and currently only two error functions are implemented. These function are to take the output of the association as input and quantify the performance of the algorithm where a low value is the best. The functions in the case of cluster analysis take the clusters as input and are currently defined as
# number of nodes F = , (31.20) err # of nodes in largest cluster # of clusters F = . (31.21) err # number of nodes
As can be seen both of these function are at a minimum when all nodes are in a single cluster, however they emphasise very different properties when examining their behaviour during a parameter sweep. The function in equation 31.20 always has a minimum of 1, but instead has a maximum that is the number of nodes. This function also decrees in error rapidly if there is a single large cluster and thus emphasises the formation of large connected structures over many small ones. However the function in equation 31.21 has a fixed maximum error of 1, and a minimum error of 1 divided by the number of nodes. This function does not care about the relative node distribution among the clusters, only how many clusters there are. For example, let us consider the case of 50 nodes in two clusters. If the node distribution in this example was so that one cluster contained 49 of the nodes and the other cluster only contained the last node, function 31.20 would only be one step from its minimum, and so would the function in 31.21. However, if the node distribution instead was so that each of the two clusters had 25 nodes, then 31.21 would have the exactly same error value, but now 31.20 would only be half way between its maximum and its minimum. This example illustrates how important it is to consider the emphasis of ones association interpretation when considering if the algorithm did well or not. One could for example imagine a error function that counts the number of total connections in the set and compares this to the number of possible connections. This would instead emphasize interconnectivity of nodes and thus the error function would decrease if a new connection between two nodes already in the same cluster is made, even if no new clusters are merged. The error function is a tool to examine what algorithm configurations produce desirable results, it is also a tool to examine how the change of the input configuration changes the algorithm performance. For example a very steep error function rapidly falling in value after a certain value may indicate some finer structure of the data or of the similarity function that was not considered before.
31.6 Next update addition
In the next step of implementation we have planned to implement 3 new similarity functions:
• Valsecchi geocentric D-criterion DN
Page 119 D. Kastinen Meteors and Celestial Dynamics
• Hausdorff metric
• Custom time dependant similarity function
We also plan to implement four new cluster merge functions,
• Complete linkage: f(a, b) < Dc ∀ a ∈ Xi, b ∈ Xj
1 1 P P • Average linkage: f(a, b) < Dc #Xi #Xj a∈Xi b∈Xj
• Centroid linkage, if ci is centroid of Xi: f(ci, cj) < Dc.
• Ward’s linkage method
Many of these cluster merge function are more similar to density analysis since they emphasize the in-cluster variation much more then single linkage. The advantage to clustering methods versus density analysis is that as phase space dimensions increase the density field calculation increases exponentially while the cluster analysis does not and is thus a much faster algorithm in essence. Some preliminary plans have also been prepared for implementation of Big data versions of similarity matrices and merging functions. These big data function will use compressed data files saved to the hard drive as temporary storage of calculation data that is not to be used immediately to reduce RAM usage.
31.7 Data formatting
31.7.1 Input files
Table 40: OAA_input.txt, Column separator: ’Space’ 0x20 Input File description: Node data, each row corresponds to one node Currently only Kepler element node data input
Column Description Data type Units Notes 1 Semi-Major axis Double AU - 2 Eccentricity Double - - 3 Inclination Double Degrees - 4 Longitude of Perihelion Double Degrees NOT argument of perihelion 5 Longitude of Ascending Node Double Degrees -
31.7.2 Output files
Table 41: clusters.txt, Column separator: ’Space’ 0x20 Output File description: Cluster state where each row is one cluster This file has variable row sizes as it lists all indexes for cluster members on one row
Column Description Data type Units Notes n Cluster member n node ID Integer - -
Page 120 D. Kastinen Meteors and Celestial Dynamics
Table 42: error_profile_
Row Description Data type Units Notes 1 Critical value used Double - - 2 Error function value Double - -
Table 43: association_profile_
Row Description Data type Units Notes 1 Critical value used Double - - 2 Association amount Double - -
31.8 Configuration
To streamline this work we have omitted listing all code implementation choices, however to provide a more complete picture of the constructed software we have below included a sample configuration file. The below file can be used to run the software module as the comments are omitted by the phrasing function.
Listing 5: OAA Configuration example
1 #%%%%%%%%%%%%%%%% CONFIGURATION FILE %%%%%%%%%%%%%%%%
2 # Lines starting with# are ignored
3 #------
4 #Main Calculation settings
5 #
6 # INPUT FORMAT (0 index): MUMRHED
7 # VAR: COL:
8 #a 26
9 #e 27
10 #i 30
11 # omega 31
12 # Omega 29
13 # ra9
14 # dec 10
15 # v_g 17
16 # lambda 36
17 #
18 # INPUT FORMAT (0 index): MCAS
19 # VAR: COL:
20 #a3
21 #e4
22 #i5
23 # omega6
24 # Omega7
25 #Analysis type:
26 # 0 = Entire database: Analysis is performed on entire input file at once.
27 # 1 = BIG DATA entire database: Special matrix partition functions are used to allow computing extreamly large databases.
Page 121 D. Kastinen Meteors and Celestial Dynamics
28 #
29 #Input format:
30 # 0 = MURMHED
31 # 1 = MCAS
32 #
33 #Cluster analysis:
34 # 0 = Do not use cluster analysis
35 # 1 = Use cluster analysis
36 #
37 #Cluster analysis type:
38 # 0 = Single linkage cluster analysis
39 # 1 = Mean linkage cluster analysis
40 # 2 = ...
41 #
42 #Parameter sweep:
43 # 0 = No parameter sweep, will use the static settings in the association type configurations
44 # 1 = Will sweep the critial parameters to finda sucsess function maxima, using settings from association type
45 # configurations.
46 #
47 #Parameter sweep adaptive step:
48 # 0 = Disable adaptive step
49 # 1 = Enable adaptive step
50 #
51 #Parameter sweep termination:
52 # 0 = Error function is minimum
53 # 1 = All nodes are clustered
54 # 2 = Only one cluster exists
55 #
56 #Error function:
57 # 0 = ERR_FUNC is# number of nodes/# of nodes in largest cluster
58 # 1 = ERR_FUNC is# of clusters/# number of nodes
59 #
60 #------
61 Analysis type =0
62 Input format =0
63 Relative object row =1
64 Cluster analysis =1
65 Cluster analysis type =0
66 Parameter sweep =1
67 Parameter sweep adaptive step =0
68 Adaptive step enable limit(fraction) =0.2
69 Adaptive step multiplyer =1.7
70 Parameter sweep termination =0
71 Error function =1
72 Memory allocation(Mb) =1e+03
73 #
74 #------
75 #OUTPUT settings
76 #
77 #Logfile:
78 # 0 = No file
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79 # 1 = Write file
80 #------
81 Logfile output =1
82 #
83 #------
84 #D-criterion options
85 #------
86 #
87 #Criterion selection:
88 # 0 = Disable criterion
89 # 1 = Enable criterion
90 # D_SH = Southwork& Hawkins/cite
91 # D_D = Drummond/cite
92 # D_J = Jopek/cite
93 # D_N = ...
94 #
95 #Static value: If no parameter sweep is to be performed use
96 # this single critical value for the analysis.
97 #
98 #
99 #------
100 D_SH criterion =1
101 D_SH static value =0.1
102 D_SH step =0.0001
103 D_D criterion =1
104 D_D static value =0.2
105 D_D step =0.001
106 #
107 #------
108 #Metric and psudo-metric options
109 #------
110 #
111 #Metric selection:
112 # 0 = Disable metric
113 # 1 = Enable metric
114 # rho2 = Kholshevnikov cite
115 # varrho2 = again cite
116 #
117 #Static value: If no parameter sweep is to be performed use
118 # this single critical value for the analysis.
119 #
120 #
121 #------
122 rho2 metric =1
123 rho2 static value =0.2
124 rho2 step =0.0001
125 varrho1 metric =1
126 varrho1 static value =0.2
127 varrho1 step =0.0001
128 %%%%%%%%%%%%%%%%%%%% ENDOF FILE %%%%%%%%%%%%%%%%%%%%
Page 123 D. Kastinen Meteors and Celestial Dynamics
32 Orbital Stability Estimation module
The orbital stability estimation module will function on integration of the variational equation to estimate the Lyapunov exponent of a orbit, this module is however not yet ready for early deployment or documentation. One of the very useful applications of such a function is to, before Monte Carlo sampling of the initial distri- butions, to examine the phase space dissipation, or e-folding time, of a certain initial condition to regulate the sampling so that sufficient resolution can be used and that the simulation can omit certain propagations. Changing the uncertainties sampling and the integration time would be based on the dissipation of the end state distribution due to initial conditions with low or high e-folding times compared to the integration time. Some research into the interplay between Monte-Carlo methods and chaotic system has been done. Such as (Leit˜aoet al., 2014) where the effect of different complexities of chaotic phase spaces on the efficiency of importance sampling in Monte Carlo simulations is examined.
33 Simulation Merger module
The simulation merger module currently only performs index-preserving merges of simulation data. This lets us, for example, take two parallel simulations that each produce 10 meteor showers and merge them together. In each of the simulation the index of each shower will range from 1 to 10, and the generated parent bodies will range from 1 to whatever number of clones was necessary to produce 10 showers. When performing data analysis it may be desirable to combine simulation data, if for example one is examining the joint effect of two different bodies on Earth at the same time or the effect of one parent body at one certain time for two different ejection epochs. Thus, the data merger software will instead of duplicating simulation indexes, create one long unique index list, where in the merged simulation the showers would range from 1 to 20 in index. It does this with all index dependant output files, as described in the data output section. A example syntax is shown below:
Listing 6: SM module execution syntax
1 ./SM simulation_folder_1/ simulation_folder_2/ merged_simulation_folder/
This allows for quick set up of multi computer and multi core simulations.
34 Statistical Uncertenty Orbital Clones module
One of the fundamental principles of our software is to use distributions rather then single assumptions, and as such it is very relevant how the initial distributions are created. Thus we have dedicated an entire module to the task of, from different sources of data such as samples or observations or pure uncertainties, creating a appropriate distribution to transfer to the main module. Currently the module only allows for input of Kepler orbital elements distributed as a multivariate normal distribution with diagonal covariance matrix. Given a space x ∈ RN we can define the multivariate normal distributions density function as
Σij = Cov(xi, xj), (34.1) 1 1 f(x) = exp − (x − µ)T Σ−1(x − µ) (34.2) p(2π)N Det(Σ) 2
where Σ is the symmetric positive definite covariance matrix and µ is the multi dimensional expected value. The input file currently only supports a diagonal covariance matrix
Page 124 D. Kastinen Meteors and Celestial Dynamics
Σij = δijσi, (34.3) in orbital elements, input format can be found in 44. Some preliminary considerations on Markov Chain Monte Carlo (MCMC) methods (Oszkiewicz et al., 2009) and measurement uncertainty (Weisman et al., 2014) have been conducted and we are planning on adding such functionality in a soon coming update.
Table 44: orb_in.data, in the case of Kepler element input configuration Column separator: ’Space’ 0x20 Input File description: Data required to generate orbital clones
Row Description Data type Units Notes 1 Mean µ Double - - 2 Standard deviation σ Double - - Column Description Data type Units Notes 1 Semi-Major axis Double AU - 2 Eccentricity Double - - 3 Inclination Double Degrees - 4 Argument of Perihelion Double Degrees - 5 Longitude of Ascending Node Double Degrees - 6 True Anomaly Double Degrees Optional
35 NASA Jet Propulsion Laboratory module
Currently the JPL module only encompasses the most basic of SPICE functionalities; it loads a set of input kernels and returns the J2000.0 Sun centric ecliptic state vectors of the desired object given the corrections stated. Currently, we have included the following SPICE kernels:
• Leap second file: naif0011.tls
• JPL planetary and lunar ephemeris DE430: de430.bsp
• JPL planetary and lunar ephemeris DE431 part 1: de431 part-1.bsp
• JPL planetary and lunar ephemeris DE431 part 2: de431 part-2.bsp
Allowing for the software to extract state vectors between 1549 December 31 00:00:00.000 2650 January 25 00:00:00.000 for:
• MERCURY BARYCENTER relative SOLAR SYSTEM BARYCENTER
• VENUS BARYCENTER relative SOLAR SYSTEM BARYCENTER
• EARTH BARYCENTER relative SOLAR SYSTEM BARYCENTER
• MARS BARYCENTER relative SOLAR SYSTEM BARYCENTER
• JUPITER BARYCENTER relative SOLAR SYSTEM BARYCENTER
• SATURN BARYCENTER relative SOLAR SYSTEM BARYCENTER
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• URANUS BARYCENTER relative SOLAR SYSTEM BARYCENTER
• NEPTUNE BARYCENTER relative SOLAR SYSTEM BARYCENTER
• PLUTO BARYCENTER relative SOLAR SYSTEM BARYCENTER
• SUN relative SOLAR SYSTEM BARYCENTER
• MERCURY relative MERCURY BARYCENTER
• VENUS relative VENUS BARYCENTER
• MOON relative EARTH BARYCENTER
• EARTH relative EARTH BARYCENTER
From these bodies all internal relative distances can be found. These values can be found with added aberration corrections, covered in section 28. The time for the desired state vectors is given in seconds past J2000 TDB and currently, all positions returned are relative to the Sun. Now that we have covered the functionally of the program we shall address the calling arguments. The first argument is the target body, the second specifies if the returned data is the barycentre of the body or the geometric center. The third argument specifies the time. The fourth argument the target folder for the output files containing the state vectors. The rest of the arguments specifies all kernels to be loaded. The software will create the output file if not present, and if a file is present the state vectors are appended to the end of the file. The file names for the position vector is pos.data and for the velocity vector vel.data. A example call of the module to extract the state vectors for the Neptune barycentre relative to the Sun at 1900 November 26 12:00:00 UT without any aberration corrections is given below:
Listing 7: JPL module execution example
1 ./JPL_SPICE NEPTUNE BARYCENTER -3.127248e+09 OUTPUT/ NONE/de430.bsp /de431_part-1.bsp/de431_part-2.bsp/naif0011.tls
As can be seen the time is given as -3.127248e+09, this value can be found by the Julian date of the J2000 TDB time of 2451545.0 and calculating the number of seconds past this date
tJ2000s = 86400(JD − 2451545.0). (35.1)
A Julian date, or JD, is calculate by a Julian day number, or JDN, and time of day by
hour − 12 minute second JD = JDN + + + . (35.2) 24 1440 86400 And lastly, to convert between a common day Gregorian calendar date to a Julian day number we need to calculate three coefficients
14 − month a = , (35.3) 12 y = year + 4800 − a, (35.4) m = month + 12a − 3, (35.5)
where bxc denotes the floor function. Using these coefficients we can find the JDN as
Page 126 D. Kastinen Meteors and Celestial Dynamics
153m + 2 jy k j y k j y k JDN = day + + 365y + − + − 32045. (35.6) 5 4 100 400
As practical calculations example we can calculate the JD and then the tJ2000s of 1900 November 26 12:00:00 in 10 steps:
14−11 1. a = 12 = 0 2. y = 1900 + 4800 − a = 6700
3. m = 11 + 12a − 3 = 8
153m+2 4. 5 = 245 y 5. 4 = 1675 y 6. 100 = 67 y 7. 400 = 16 8. JDN = 26 + 245 + 365 · 6700 + 1675 − 67 + 16 − 32045 = 2415350
12−12 0 0 9. JD = 2415350 + 24 + 1440 + 86400 = 2415350.0
10. tJ2000s = 86400(2415350.0 − 2451545.0) = −3127248000
Currently the coordinate system the state vectores are to be returned in is fixed as the ecliptic J2000.0 reference. This is a inertial frame, in other words it is, non-rotating with respect to stars, it has a non- accelerating origin, and it typicly has zero velocity. The inertial frames suported by spice at the moment is J2000 (also called ICRF or EME2000) and ECLIPJ2000. The J2000 frame is based on the Earths equator and equinox while ECLIPJ2000 is based on the Earth’s orbital plane and equinox.
36 Scripting of simulations
It can sometimes be very useful to script runs of multiple simulations running on different cores and config- urations. Thus below we have included a small bash script example that was used to handle the simulation logistics of the runs performed in this work. Here we had prepared a set of different configuration files and orbital element files for each simulation. We then ran the simulations with a bash script that took two input numbers. The first number represented the simulation core number and the second the simulation run, this avoids manual input of configuration files in large simulation batches.
Listing 8: Example bash script for use with MCAS
1 #!/bin/bash
2
3 echo ’Setting up simulation core’$1 ’with simulation id’$2;
4 rm -v MCAS_v2_sim$1/SETTINGS/MCAS_config.cfg;
5 rm -v MCAS_v2_sim$1/SUOC/orb_in.data;
6 cp -v SIMSETTINGS/sim$2/MCAS_config.cfg MCAS_v2_sim$1/SETTINGS/MCAS_config.cfg;
7 cp -v SIMSETTINGS/sim$2/orb_in.data MCAS_v2_sim$1/SUOC/orb_in.data;
8 echo ’Simulation starting’;
9 cd MCAS_v2_sim$1/;
10 ./MCAS;
Page 127 D. Kastinen Meteors and Celestial Dynamics
37 External software
37.1 Mercury6
Until development of the CMS module is complete the mercury6 software package (Chambers, 1999) will be used to perform the long term N-body integrations. mercury6 is written in Fortran 77 and currently includes the following N-body algorithms:
• a second-order mixed-variable symplectic algorithm incorporating simple symplectic correctors
• a general Bulirsch-Stoer
• conservative Bulirsch-Stoer
• Everhart’s RA15
• Hybrid symplectic/Bulirsch-Stoer integrator
The standard release version of mercury6, the 6.2 version, includes only the effects of Newtonian gravi- tational forces between point masses and not electromagnetic perturbation. However the code contains a empty function able to include the effects of other forces. By modifying this subroutine in the source code, for example the non-gravitational forces described in (Marsden et al., 1973) for comets can be implemented in mercury6. The most relevant non-gravitational force for the purposes of our current work is Radiation pressure and the Poynting-Robertson effect, previously covered in section 4.0.2. There has been some con- fusion over the implementation of electromagnetic effects previusly, however as per the re-consideration in (Burns et al., 2014) where the process was discussed in a straight forward manner, we have used
! dv SAQ v • Sˆ v m = pr (1 − )Sˆ − . (37.1) p dt c c c
In the above equation mp is the particle mass, v is the particle velocity, Sˆ is the unit vector in the direction of the incoming radiation beam, S is the Solar radiation flux density at the position of the particle, Qpr is the radiation pressure efficiency factor, A is the cross-section of the particle, and c is the speed of light. If in heliocentric coordinates the radiation source vector can be written as
x Sˆ = (37.2) |x| and the Solar radiation flux density can be written as
L S = 0 , (37.3) 4π|x|2 where L0 is the Solar total luminosity. We have also assumed spherical grains thus the parameter A will be
A = πr2, (37.4) where r is the particle radius. Given the particle mass we can find the radius by the bulk density as
Page 128 D. Kastinen Meteors and Celestial Dynamics
1 4πr3 3m 3 m = ρV = ρ ⇔ r = p . (37.5) p 3 4πρ
Inserting these into equation 37.1 we find
dv 1 L0 AQpr v • x x v = 2 (1 − ) − = dt mp 4π|x| c c|x| |x| c 2 3 3mp L0Qpr v • x x v = 2 (1 − ) − . (37.6) 4πρ 4|x| cmp c|x| |x| c
26 We can assume Qpr ≈ 1 (Burns et al., 1979) and L0 ≈ 3.846 · 10 W as mentioned in section 7. However, the mercury6 code does not allow for additional body parameters to be passed to the user force function. This creates the problem that we cannot have particles with different bulk density and size without recompiling the entire software. And since every comet in the current application of the software will have varied properties these properties will have to be input parameters. Thus we have modified the mercury6 code to allow for these parameters to be passed to the program through the so called small.in file. The complete modification of the mercury6 code will be made available when testing is complete. Also, units internally in mercury6 is in AU for length, days for time, and Solar masses for mass, and thus all the SI input units will have to be converted inside the function. The complete implementation can be found below.
Listing 9: MFO USER.FOR subroutine
1 c Applies an arbitrary force, defined by the user.
2 c
3 c If using with the symplectic algorithm MAL_MVS, the force should be
4 c small compared with the force from the central object.
5 c If using with the conservative Bulirsch-Stoer algorithm MAL_BS2, the
6 c force should not be a function of the velocities.
7 c
8 c N.B. All coordinates and velocities must be with respect to central body
9 c ===
10 c------
11 c
12 subroutine mfo_user (time,jcen,nbod,nbig,m,x,v,a,params,
13 % output_flag,global_params)
14 c
15 implicit none
16 include ’mercury.inc’
17 c
18 c Input/Output
19 integer nbod, nbig, output_flag 20 real*8 time,jcen(3),m(nbod),x(3,nbod),v(3,nbod),a(3,nbod) 21 real*8 params(3,nbod),global_params(4) 22 c
23 c Local
24 integer j 25 real*8 xt,yt,zt,R2,R 26 real*8 vxt,vyt,vzt 27 real*8 L0,Q_pr,c0,m_p,rho 28 real*8 coef1,coef2,VX_dot 29 c
Page 129 D. Kastinen Meteors and Celestial Dynamics
30 c------
31 c
32 c default accelerations are zero
33 do j = 1, nbig
34 a(1,j) = 0.d0
35 a(2,j) = 0.d0
36 a(3,j) = 0.d0
37 end do
38
39 c Solar LUM
40 L0=5.5739511D-12
41 c PI is already defined
42 c Speed of light
43 c0=173.144632327D0
44 c Radiation presure efficiency factor
45 Q_pr=1.0D0
46 do j = nbig+1, nbod
47 if (params(1,j).NE.0.AND.params(2,j).NE.0) then
48 xt=x(1,j)
49 yt=x(2,j)
50 zt=x(3,j)
51 vxt=v(1,j)
52 vyt=v(2,j)
53 vzt=v(3,j)
54
55 rho=params(2,j)*1683.60312587D0 56 m_p=params(1,j)/1.98855D30
57
58 R2 = xt*xt + yt*yt + zt*zt 59 R = R2**0.5D0
60
61 coef1=((m_p*0.75D0/(PI*rho))**(2.D0/3.D0)) 62 coef2=L0*Q_pr/(4*R2*c0*m_p) 63 VX_dot = (vxt*xt + vyt*yt + vzt*zt)/R
64
65 a(1,j) = coef1*coef2*((1 - VX_dot/c0)*(xt/R) - vxt/c0) 66 a(2,j) = coef1*coef2*((1 - VX_dot/c0)*(yt/R) - vyt/c0) 67 a(3,j) = coef1*coef2*((1 - VX_dot/c0)*(zt/R) - vzt/c0) 68 else
69 a(1,j) = 0.d0
70 a(2,j) = 0.d0
71 a(3,j) = 0.d0
72 end if
73 end do
74 c
75 c------
76 c
77 return
78 end
79 c
Page 130 D. Kastinen Meteors and Celestial Dynamics
37.2 NASA Jet Propulsion Laboratory ephemeris software
As a early development alternative to SPICE the current version of MCAS supports use of a modified version of the JPL ephemeris software (Hardy, 2004). Even though this is to be phased out in later development stages we shall here go over the modifications and its use. This software library is designed to read and write NASA’s Jet Propulsion Laboratory planetary ephemeris files, convert ephemeris file types, and calculate positions and velocities from ephemeris. The software itself was originally written in Fortran but later ported to C. We have used the C version that was made available for free under the Lesser Gnu Public License. The library contains the following functions,
• append: Joins binary ephemeris files.
• convert: Creates binary ephemeris files.
• extract: Extracts segment from binary ephemeris files.
• print header: Extracts header data from binary ephemeris files.
• read record: Reads one record of binary ephemeris file data.
• scan records: Lists start/stop times of ephemeris file records.
• rdeph: Calculate positions and velocities of a planet at the supplied time.
We have only modified rdeph by making the program also write the positions and velocities to a file instead of just printing the data to the screen. We shall now detail the step by step requirements to to set up a ephemeris file for use in our software.
• (I): Download the appropriate set of ephemeris files from (JPL, 2016) in ASCII format, we have used version 430 from the year 1550 to 2550 in this example.
• (II): Change the defined EPHEMERIS variable value found in file ephem˙types.h to the version downloaded, in this cased 430. Then also change the elif if clause to the same version so that error does not occur.
• (III): Compile the convert and the append programs.
• (IV): Convert all the ASCII files to binary, example command for one of the files: ./convert header.430˙572 ascp1550.430 bin1550.430
• (V): Create the combined binary base to append to using the chronologically first file, example com- mand: cp bin1550.430 bin1550to2550.430
• (VI): Append ALL binary files in chronological order to the combined file using the append program, example command for FIRST append: ./append bin1550to2550.430 bin1650.430
• (VI): Place the completed bin1550to2550.430 file in the JPL/ folder in the MCAS structure and rename it to ephBin.000 the modified JPL software can now generate initial conditions for the Solar system within the time frame available in the binary file. The file distributed with the complete MCAS software is the one described in the example using the 430 ephemeris (Folkner et al., 2014).
Page 131 D. Kastinen Meteors and Celestial Dynamics
38 Data transformation logistics
As many authors present their results in different coordinate basis it is preferable to specify how exactly the transformation from default output to other representations is performed. Here we shall define a few basic transformations.
38.1 State vectors and Kepler elements
The concept of a set of state vectors comes from the formulation of a Hamiltonian system. As mentioned before a set of state vectors for a object describes its state in phase space, we will denote state vectors as a pair of three dimensional vectors representing position and velocity (r, v) together with the gravitational parameter of the object and the central mass as covered in section 3.3. Kepler elements are defined as
described in section 3.4. We will denote the orthonormal coordinate base as {ex, ey, ez} and the projection of a vector onto this base as u • ex = ux.
38.1.1 From state vectors to Kepler elements
Given the body state (r, v) and its gravitational parameter µ = G(m + M ) we find the Kepler elements systemically by first introducing
r = |x|, (38.1) v = |v|, (38.2) r v = • v. (38.3) r r The semi major axis can then be calculated for elliptic orbits as
µ a = − , (38.4) 2 and for hyperbolic as
µ a = , (38.5) 2 where
v2 µ = − . (38.6) 2 r We then calculate the specific angular momentum as
h = r × v (38.7) h = |h|. (38.8)
As this quantity is perpendicular to the orbital plane we can calculate the inclination of the orbit as
h i = cos−1 z . (38.9) h
Page 132 D. Kastinen Meteors and Celestial Dynamics
Here we can see that the inclination runs from 0 to π where an inclination over π/2 represents a retrograde orbit. We then calculate the vector pointing towards the ascending node
n = ez × h, (38.10) n = |n|, (38.11)
and use it to calculate the longitude of the ascending node as
n Ω = cos−1 x ∀ n ≥ 0, (38.12) n y n Ω = 2π − cos−1 x ∀ n < 0. (38.13) n y
We notice that if h only has a ez component the vector pointing towards the ascending node will be the zero vector. This is due to the fact that if a orbit lies in the reference plane, it will never ascend or descend into that plane as it is always in the plane. Thus we define the longitude of the ascending node to be zero in this case
Ω = 0 ∀ i = 0. (38.14)
This is where one uses the longitude of perihelion rather than the longitude of the ascending node
ω¯ = ω + Ω. (38.15)
Continuing on to the eccentricity of the orbit. This can be calculated by the use of the eccentricity vector which can be expressed by
1 µ e = (v2 − )r − rv v , (38.16) µ r r e = |e|. (38.17)
The argument of periapsis can now be calculated by
n • e ω = cos−1 ∀ e ≥ 0 (38.18) ne z n • e ω = 2π − cos−1 ∀ e < 0. (38.19) ne z Here we also notice a degenerate case where the orbit will not have a argument of periapsis in a perfect circular orbit as all points are equally close the central focus point. Thus also this is defined as zero when the eccentricity is zero
ω = 0 ∀ e = 0. (38.20)
A consequence of this is that all circular orbits with inclination seems to have a periapsis in the ascending node, and one should be aware of this. Lastly we need to calculate the position on the orbit to complete the transformation,
Page 133 D. Kastinen Meteors and Celestial Dynamics
e • r ν = cos−1 ∀ v ≥ 0 (38.21) er r e • r ν = 2π − cos−1 ∀ v < 0. (38.22) er r
38.1.2 From Kepler elements to state vectors
Starting with a set of 6 Kepler elements we first define the co-planar orbit equations as
r cos(ν) cos(ν) 1 − e2 o = r sin(ν) = a sin(ν) (38.23) 1 + e cos(π + ν) 0 0 − sin(ν) r µ o˙ = cos(ν) − e . (38.24) a(1 − e) 0
The position dependant radius is found by
1 − e2 r = a , (38.25) 1 + e cos(π + ν)
as we have defined the apoapsis to point towards the reference direction ex. We can then find the orbit in the reference frame in which the orbital elements are given by the rotations
r = Rz(−Ω)Rx(−i)Rz(−ω)o, (38.26)
v = Rz(−Ω)Rx(−i)Rz(−ω)o˙ , (38.27)
where the rotation matrecies are defined as
cos(θ) − sin(θ) 0 1 0 0 Rz(θ) = sin(θ) cos(θ) 0 ,Rx(θ) = 0 cos(θ) − sin(θ) . (38.28) 0 0 1 0 sin(θ) cos(θ)
38.2 Geocentric coordinates
38.2.1 Ecliptic J2000
The concept of a ecliptic geocentric coordinate system when examining meteor radiants is to have a general reference system for the direction from which the meteor phenomena is originating. Thus the point describing a single meteor will not represent the meteoroid position, or its destination, but rather the direction from which it enters the Earth gravitational field before gravitational focusing applies. The reference is the vernal equinox, i.e the x-axis of the system is aligned in the equinox direction, and the reference plane is the ecliptic, i.e the y-axis lies in the ecliptic and the z axis is almost parallel to the Earth orbital momentum vector.
Let us consider a meteoroid with orbital parameters {an, en, in, ωn, Ωn, νn} where ν is the true anomaly at the point right before gravitational focusing has become predominant. Let us also consider the Earth
with orbital parameters{aE, eE, iE, ωE, ΩE, νE} where νE is the Earth true anomaly at the same time as the meteoroid has true anomaly νn.
Page 134 D. Kastinen Meteors and Celestial Dynamics
To find the ecliptic radiant we need both the velocity vectors of the meteoroid and Earth, thus we begin with transforming the orbital elements to state vectors, as described in section 38.1. The end result will be a meteoroid state (xn, vn) , and a Earth state (xE, vE). The inclination is defined such that it is the inclination to the reference plane at the ascending node and thus a orbit with inclination below 90◦ is prograde and a orbit with i > 90◦ is retrograde. In this definition the co-planar, apsis aligned orbit has focus in origin and apoapsis pointing in the direction of positive x and counter clockwise movement as prograde motion. One common mistake is to define the system so that the periapsis and not the apoapsis is pointing towards the equinox before spatial rotations has been performed. The first matrix rotates the orbit in the orbital plane around the orbital plane normal, the second matrix then inclines the orbit. If we incline along the x axis the ascending node will point towards vernal equinox when the longitude of the ascending node is 0◦. Another common mistake to incline along the y axis, resulting in a orbit that will initially have a longitude of the ascending node relative equinox of −90◦ and a increased angle to the argument of periapsis of 90◦. Lastly we will rotate the ascending node away from the reference direction and thus find our J2000.0 ecliptic equinox oriented orbit given the 5 orbital elements and the orbital position. Now that we have the velocity vectors calculated for both the Earth and the meteoroid at the point of encounter we will find the geocentric velocity vector for the meteoroid as
0 vn = vn − vE. (38.29)
This represents the direction the meteoroid is travelling relative the Earth, we are interested in pointing to the apparent direction the meteor is coming from, and thus we shall define
0 u = −vn, (38.30) u = |u|. (38.31)
We can proceed to calculate the relevant geocentric ecliptic coordinates as
u λ = tan−1 x , (38.32) uy u β = 180◦ − cos−1 z , (38.33) u where λ is the ecliptic longitude, β is the ecliptic latitude, and u is the geocentric velocity.
38.2.2 Equatorial J2000
To transform coordinates from the ecliptic coordinate system to the equatorial we need to know the obliquity of the ecliptic. This is the term used to describe the inclination of Earth’s equator with respect to the ecliptic. Its mean value is currently around 23.4◦, however it is also decreasing by around 0.013◦ ever hundred years due to planetary perturbations. To transform between the ecliptic and the equatorial coordinate systems is easy if the coordinate system used is aligned towards the vernal equinox. The vernal equinox is the direction of intersection between the ecliptic plane and the equatorial, and thus, one only needs to rotate around the vernal equinox direction to swhich between the too coordinate systems. However, since the obliquity of the ecliptic changes due to planetary perturbation, we need to account for this change. This can be done by a Taylor series expansion and numerical integrations as done by (Laskar, 1986). There the mean obliquity of the ecliptic was calculated to
Page 135 D. Kastinen Meteors and Celestial Dynamics
00 2 3 4 5 0 (T ) = − 4680.93T − 1.55T + 1999.25T − 51.38T − 249.67T + − 39.05T 6 + 7.12T 7 + 27.87T 8 − 5.79T 9 − 2.45T 10, (38.34) in arc seconds, where T is defined as hundreds of Julian centuries from J2000.0, or
1 JD − 2451545.0 JD − 2451545.0 T = = . (38.35) 100 36525 3652500
This formula has a probable accuracy of around 0.02 arc seconds, or 5.56 · 10−6 degrees, after 1000 years and accurate within a few arc seconds up to 10 000 years. To convert from arc seconds to degrees we simply take
00(T ) (T ) = 0 (38.36) 0 3600 This is however the mean obliquity of the ecliptic, to find the instantaneous obliquity we must include the nutation of the equator ∆,
(T ) = 0(T ) + ∆(T ) (38.37)
The nutation of the obliquity has been defined in the International Astronomical Unions (IAU’s) 2000B theory section, which is based on (McCarthy and Luzum, 2003). Here the nutation in longitude is defined as
78 X 0 00 ∆Ψ(T ) = (Ai + 100AiT ) sin(θ) + Ai cos(θ), (38.38) i=1
and the nutation in obliquity is defined as
78 X 0 00 ∆(T ) = (Bi + 100BiT ) cos(θ) + Bi sin(θ), (38.39) i=1
where
5 X θ = NiFi. (38.40) i=1
Here the coefficients Ni are integers and Fi the five fundamental arguments
1. Mean anomaly of the Moon
2. Mean anomaly of the Sun
3. Mean longitude of the Moon
4. Mean elongation of the Moon from the Sun
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5. Mean longitude of the ascending node of the lunar orbit
The coefficient lists A and B can be found at the IAU website. Once the nutation and the mean obliquities has been calculated one can proceed to find the total nutation matrix defined as
N (T ) = Rx(−)Rz(−∆Ψ)Rx(0). (38.41)
Thus to find a true equinox of date position x from a mean equinox of date position x0 one simply applys the nutation matrix
x = N x0.
Figure 40: Calculation of the variations in the obliquity of the ecliptic for two different time scales, ± 10 years from J2000.0 and ± 10 000 years from J2000.0.
These variations are however on the order of a few tenths of arc seconds and can be disregarded due to observational uncertainties, at lest in meteor studies. In figure 40 we can see the variations of the obliquities of the ecliptic for two different time frames, using the above methods. To calculate the coordinates in a equatorial we simply rotate the geocentric velocity in equation 38.29 from the ecliptic as
0 0 Rx()vn = Vn, (38.42) and again define the direction of origin as
0 U = −Vn, (38.43) U = |U|. (38.44)
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We can proceed to calculate the relevant geocentric equatorial coordinates as
U α = tan−1 x , (38.45) Uy U δ = 180◦ − cos−1 z , (38.46) U where α is the right ascension, δ is the declination, and U is the geocentric velocity.
38.3 Sun centred versions
One can define Sun centred version of these coordinate systems by accounting for the longitude of the Sun, defined as the angular distance along the Earth’s orbit measured from the intersection of the ecliptic and the celestial equator. In other words relative vernal equinox. The Sun is furthermore defined to move from south to north. A Solar longitude coordinate gives the position of the Earth on its orbit and can be calculated from the date, but not vice versa unless given the year, due to time corrections. In such a coordinate system the Sun, and the Earth apex will be stationary making it a convenient representation when visualizing the sporadic meteoroid complex apparent sources.
39 MATLAB visualization scripts
Figure 41: The graphical user interface used to call all the data analysis scripts developed.
One must always complement the computational power of the developed software with sensible visualization and data summation tools since the output data is generally too large for manual evaluation. Thus a extensive library of plotting and data summarization scripts with a accompanying GUI was developed in MATLAB.A
Page 138 D. Kastinen Meteors and Celestial Dynamics example of the GUI can be seen in figure 41. This group of scripts is the one used to generate all graphs included in the results part of this work. Currently we have developed scripts to visualize
• Statistics on mass ejection
• Statistics on ejection speeds
• Statistics on parent bodies
• Statistics on meteoroids
• Statistics on test particles
• Total simulation statistics
• Similarity function divergence from parent bodies for shower causing particles
• Orbit configurations
• Meteor association parameter sweeps
• Ejected mass distribution
• Encountered mass distribution
• Snapshot of simulation state
• Specific data filters
• Mass transfer functions
• Trail transfer functions
• Principal component analysis
• Calculation of geocentric coordinates
• Stream dissipation analysis
• Execution time diagnostics
• Statistics on Earth at encounter times
• Mass encounter difference between 2 years
• Check of electromagnetic coefficient for perturbation
• Mass distribution as a function of trail origin
• Yearly clustering configurations
• Orbit criterion evolution in trail
• Control of statistical convergence of Monte Carlo sampling
• Time dependant initial distribution dissipation plots
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40 Future work
40.1 Purpose and aims
At the moment we are planning a continuation project that would address three major questions within Solar system small body dynamics:
(I) What is the mass propagation of material from parent bodies to Earth?
(II) What are the dominant effects on the fluxes of Solar system nanometer-sized dust?
(III) What is the influence of chaos on meteoroids streams and when does it become predominant?
In spite of the fact that the underlying orbital dynamics is well understood, it is still an open question how much extraterrestrial material enters the Earth atmosphere. The most obvious missing feature in current numerical models, which cover different aspects of Solar system small body dynamics, is the ability to easily connect programs with different functionality. We believe this is the main reason why contemporary techniques have not yet been combined. These include techniques such as Monte Carlo simulations of meteoroid streams, orbital stability analysis, time tracing of orbital similarity functions to better analyse measurements and neural network recognition of meteor showers. The lack of a comprehensive numerical modelling approach appears to be the main bottleneck. Thus, the main focus of the future project is dedicated to developing a software toolbox which attempts to supplement the current research area. The work presented here works as a proof of concept providing a solid basis for full development.
40.2 Future development plan
The major workload in this proposed project would be software development and performing the simulations needed to address the science questions. Reviewed numerical software all seems to have been designed with a specific task in mind, thus requiring extensive programming skills and development time for its re usage. To avoid such a development process, every module described below will be designed with re usability and ease-of-modification in mind. Below, the proposed work is divided into four Work Packages (WP), the first three aimed towards one science question each. WP I contains development of the Celestial Mechanics Simulator (CMS), Statistical Uncertainty Orbital Clones (SUOC), and Parent Body Ejector (PBE) modules needed to address mass propagation of material from parent bodies.
• I.1: Implement Hamiltonian integration methods in CMS with electromagnetic effects, close encoun- ters, and time zone integration, replacing mercury6. Modify MCAS to utilize all the functionality of CMS in the statistical calculation.
• I.2: Using statistical software libraries like STAN, implement in SUOC Bayesian statistical inference and other methods needed to procure all input probability distributions. Modify MCAS to utilize the entity of SUOC’s functionality.
• I.3: Review Rosetta results and implement additional features in PBE: e.g. non-uniform spherical ejection, jet ejection, size-dependent particle density.
• I.4: Perform case studies addressing mass propagation from parent bodies.
WP II contains development of modules needed to address the flux of nanodust.
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• II.1: Modify MCAS so that various nanodust distributions can be used as input instead of ejection of dust from comets with PBE. Add Solar wind conditions and interaction corrections to CMS.
• II.2: Perform simulations to investigate which effects dominate the fluxes of nanodust.
• II.3: Investigate the time variable flux at Earth and other parts of the Solar system.
WP III includes a new module called Orbital Stability Estimation (OSE) needed to address chaos and orbital stability.
• III.1: Implement numerical integration of the variational equations needed to estimate chaos effects and information dissipation rates in OSE. Modify MCAS and CMS to utilize this integration module.
• III.2: Derive a method for characterizing the rate of information loss of a certain initial orbit using the Lyapunov Characteristic Exponents.
• III.3: Perform simulations to characterize chaotic and stable orbit families.
WP IV contains an additional module called Orbital Association Analysis (OAA) in order to develop a systematic method of meteor shower detection in observational data for validation purposes of all three science questions.
• IV.1: Add several clustering methods, density approximations, wavelet transforms and pattern recog- nition methods to OAA.
• IV.2: Perform case studies to characterize strengths and weaknesses of the different methods.
• IV.3: Construct systematic methods for calibrating meteor searches in databases.
• IV.4: Using calibration results from the IV.3 results, construct a long-term sporadic model, or use an existing one, and derive portable definitions for SNR.
The only module in continuous development for the duration of the project is the MCAS module. This allows for disjointed completion of the different WP-s. Also, possible programming issues and software updates within one WP will not adversely affect the other ones.
40.3 Significance of future work
Each module will be developed with the general needs of the science community in mind. All software will be published as open source and designed for reuse. This will enable collaborative case studies with a variety of research groups. Here we only cover some of the possible future uses in addition to our planned investigations. The mass influx onto Earth as a function of mass, time and place is one of the outstanding questions in meteor science. We would statistically investigate the mass transfer from parent bodies to Earth with numerical modelling and compare with measurements and published results. Our investigation would shed light on how to combine the observations, leading to an improved estimation. The reusable approach of the modules will allow for additional studies such as the mass transfer efficiency to Earth from yet-to-be discovered parent bodies. Also, accurately estimating debris dynamics will be vital for specific events such as comet and asteroid interaction missions, e.g. the NASA space probe Deep Impact, JAXAs Hayabusa 2, and future asteroid deflection missions, and this will be possible to address. We will extend the basic model of dust trajectories used by (Czechowski and Mann, 2010) by including time variations in the nanodust surface charge and the effect of Solar wind fluctuations on the motion of charged dust grains in order to estimate fluxes. The importance of these effects on fluxes of nanodust have not been addressed in previous investigations (Mann et al., 2014).
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Our work on chaos and orbital stability will provide a method to estimate the range of time that simulations and calculates are valid, or in other words, the rate of information loss in certain orbital regions. Such estimations are useful for determining propagation limits in parent body searches and better estimations of the reliability of results. In this context we will also investigate the effects that simplifying assumptions has on model outcomes, and assess improvements. One such example is the frequent assumption of parent bodies ejecting particles only at a single point on their orbit at perihelion (Vaubaillon et al., 2011). Determining which factors are important to consider will enable significantly improved meteor shower predictions. The detection of meteor showers in data can be seen as the separation of the influx rates between new showers versus the sporadic complex (old showers). An SNR definition, or significance estimation, is a basic requirement for machine learning, which has had wide success in many areas and should be implemented also in the systematic search for meteor showers. The next generation atmosphere and geospace research radar EISCAT˙3D will be able to measure up to hundreds of thousands of meteoroid orbits per day with unprecedented accuracy (Pellinen-Wannberg et al., 2016a). Such vast amounts of orbital data will benefit from the development of advanced orbit association analysis methods.
40.4 Significance of community contributions
As the software modules near a state where they can be released as open source to the scientific community, the importance of community contributions to the development will gradually increase. As the developer of the software has specific research goals to fulfil with the software, many dynamic aspect will be lost or overlooked and it is in this regard that the community can contribute immensely. By adding features allowing other types of research be performed or simply informing that the next update should introduce a new feature or algorithm modification will quickly increase the software quality in terms of usefulness to the scientific community.
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Part VII Results
41 Introduction to Results
We have chosen to perform two initial simulations to both showcase the capabilities of the described software and also to contribute to the scientific community. The first case study is a multiple trail simulation of 21P/GiacobiniZinner, a comet which was recognized as the parent body for the October Draconids meteor shower (Davidson, 1915) as early as 1915. This comet has been extensively examined throughout the years after its discovery on December 20, 1900 by Michel Giacobini. The second observation in 1913 by Zinner resulted in its current naming. The reason behind its extensive observations and fast meteor shower connection is its relatively short orbital period of six years and that its perihelion distance is very close to that of the Earth. Thus making the produced dust stream a ideal candidate for being responsible for a strong meteor shower on Earth. And as it happens, in the years 1933 and 1946 a set of spectacular meteor showers came to pass with hourly rates of thousands of meteors. Recently simulations of the meteor shower has been performed by (Vaubaillon et al., 2011) and (Ye et al., 2013b) to explain the recent meteor shower outbursts of 2011 and 2012 respectively. Both of these meteor showers have also been observed by the MU radar in (Kero et al., 2012c) and (Fujiwara et al., 2016). However, when comparing the two observations it was discovered that there was a significant mass difference detected between the two years. This conclusion was drawn by using simultaneous radar and visual observations. As visual observation are only generated by larger grains such simultaneous observations can give some insight into the mass distribution. To explain the actual observed deviation from standard models we must explain the mass index for a meteor shower. The mass index can be calculated using either the underdense echos or the overdens echos covered in section 13. Both these calculations rely on assuming that the cumulative number of echos is proportional to a simple power law. In the case of underdense echoes the cumulative number of echoes N with amplitude greater than A is proportional to A1−s (McIntosh, 1968). For overdense echoes however the cumulative number N is instead proportional to the duration T rather than the amplitude with a relation T 0.75(1−s). In these formulas the aforementioned mass index is the number s, generally assumed to be between 1 and 2. During the 2012 October Draconids visual observers yielded a much lower rate of around 200 ZHR compared to radar observations, indicating that the magnitude dependant cumulative number is not represented by the power law that was just explained. Thus we have performed simulations to determine the actual mass transfer of GiacobiniZinner to the Earth for these meteor showers in a attempt to further our understanding of this difference. The second simulation is a demo simulation designed to showcase the capabilities of the developed software and also as a platform for developing the data analysis procedures needed. The simulations where performed with the previously, in section 11.1, covered Pan-STARRS synthetic Solar system model population of short period comets as a initial distribution.
42 21P/GiacobiniZinner
42.1 Input state
To generate the data needed a set of 18 simulations was set up parallel to each other. One simulation considered one perihelion passage of 21P/GiacobiniZinner each, as per the list:
1. 1866
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2. 1873
3. 1880
4. 1887
5. 1894
6. 1900 Nov. 28
7. 1907 May 19
8. 1913 Nov. 2
9. 1920 May 18
10. 1926 Dec. 11
11. 1933 July 15
12. 1940 Feb. 17
13. 1946 Sept.18
14. 1953 Apr. 16
15. 1959 Oct. 26
16. 1966 Mar. 28
17. 1972 Aug. 4
The simulations after and including the passage of 1900 used orbital elements from JPL small body database, these orbital elements are calculated from observations. There where however two passages not observed, the years 1920 and 1953, and thus these passages must be numerically integrated from the closest observation. The same is true for the passages prior to 1900, thus we have calculated them ourself using the built in time synchronization function in the Parent Body Ejector module previously described in section 30. Due to small variations in the orbital elements during 1900, when propagated backwards through the large perturbation by Jupiter, the variation on the passage dates blow up from hours to days and in extreme cases months, which is why no month and date is provided for passages prior to 1900 in the list. Each simulation was set to produce 50 clones of GiacobiniZinner and then propagate the resulting meteoroid stream until 2020. The ejection model used was the one described as (II) in 10.2.1 implemented from (Hughes, 2000). We considered all passages of particles within the Earth hill sphere as close encounter and regarded them as meteors with a number weight calculated from the sublimation simulation. The Cluster analysis was configured to performed a non-adaptive parameter sweep of a single linkage cluster algorithm using all currently implemented similarity functions. The time step mercury6 was set to 1 day and the step of PBE was set to 16 days. PBE utilized a Bulirsch- Stoer algorithm with multiple sub steps according to the previously described series, and mercury6 was configured to use the hybrid symplectic integrator, both including electromagnetic forces. Finally, the mass distribution for ejection of particles was set to a logarithmic uniform distribution between 10−9 kg to 10−1 kg.
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Figure 42: Histogram of the orbital element distribution for orbital clones of 21P/GiacobiniZinner.
Figure 43: Histogram of the orbital element distribution for encountered test particles ejected from the clones of 21P/GiacobiniZinner.
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42.2 Summary of simulations
The resulting simulation yielded:
• 850 clones of 21P
• All 850 clones underwent sublimation
• 821 of the 850 meteoroid trails encountered Earth before 2020
• On average 52 test particles from each stream encountered Earth
• Using the mass loss weight, this corresponds to on average 7.69 · 109 meteors per trail.
• In total 6 714 499 test particles where propagated.
• Out of them 43 637 encountered the Earth.
In figure 42 we can see the orbital distribution of the 850 generated 21P clones, and in figure 43 we can see the collective orbital distribution of all test particle encounters.
42.3 Validation: 1933 and 1946 October Draconids
Figure 44: Histogram of Probability of encounter and mean shower intensity for the 1933 October Draconids.
A common method to validate software is to make back-predictions on already established data, both nu- merically and observationally. The meteor shower of 1946 October Draconids has been extensively simulated and there are also observations available, making it a very good candidate for software validation. As done in (Vaubaillon et al., 2011) the simulations of the October Draconids should show a meteor shower on Earth during 1933 October 9 and 1946 October 10, both at ≈ 197◦ Solar longitude. Our results for the encounter probability distributed along Solar longitude for the year 1933 can be found in figure 44, the mean flux trail contribution distribution can be found in figure 45, and the trail encounter probability can be found in figure 46. In figures 47, 48, and 49 the respective plots for the 1946 meteor showers can be seen.
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Figure 45: Density maps for the mean trail meteoroid flux distribution for the 1933 October Draconids.
Figure 46: Density maps for the trail distribution probability for the 1933 October Draconids.
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Figure 47: Histogram of Probability of encounter and mean shower intensity for the 1946 October Draconids.
Figure 48: Density maps for the mean trail meteoroid flux distribution for the 1946 October Draconids.
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Figure 49: Density maps for the trail distribution probability for the 1946 October Draconids.
Here the discussed probability and mean simulation formulations in section 29 come into use when examining meteoroid trail encounter events. In figures 44 and 47 the left side of the plots refer to the probability of the encounter occurring, i.e. how many of the simulations that contains meteor events at that Solar longitude bin compared to all performed simulations. The right side however refers to the amount of encountered test particles weighted by the sublimation calculation weight of how many real meteoroid that test particle approximately represents. In the pure histogram versions, the difference is negligible between probability and mean flux. However when looking at the trail distribution in figures 48 and 49 we see a clear difference between probability of meteors occurring and the their mean numbers. The most prominent feature is that there is high probability of the Earth encountering meteoroids from the trails from 1926 to 1940 in the year 1946 but even if such a encounter occurs, the majority of the flux present will be from the 1933 trail alone. These simulated data corresponds well with previous simulation and we regard the validation as accomplished.
42.4 2011 October Draconids
One of the science questions these simulations where to address where the difference between the 2011 and 2012 October Draconids. In figures 50, 51, and 52 the same types of encounter probability and trail distribution plots can be seen as for the 1933 and 1946 October Draconids. One interesting feature we see here is that there is a high flux coming from the trail formed during the year 1972, however the probability of this flux occurring is relatively low compared to other trails. This highlights one of the key points of the future work proposed here. As there are no uncertainties given for the orbital elements of the 1972 epoch on the JPL Small body database that was used for the input configuration, we chose an arbitrary Gaussian distribution around the given elements. That the probability is low, indicates that in many of the simulations the flux does not occur at all since this measure does not reflect the amount of particles as explained earlier. As such one would assume that a part of the input Gaussian distribution is untrue, because if the flux is observed then the parts of the distribution that did not generate it cannot contain the true parent body and vice versa. Also, this flux is around 4 to 8 hours after the main October Draconids peak and should
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Figure 50: Histogram of Probability of encounter and mean shower intensity for the 2011 October Draconids.
Figure 51: Density maps for the mean trail meteoroid flux distribution for the 2011 October Draconids.
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Figure 52: Density maps for the trail distribution probability for the 2011 October Draconids.
Figure 53: Density maps for the mean meteoroid flux mass distribution for the 2011 October Draconids.
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Figure 54: Density maps for the mass distribution probability for the 2011 October Draconids. be distinguishable from the other trails in observational data. Unfortunately the radiant for the dedicated meteor radar Canadian Meteor Orbit Radar (CMOR) that did detailed observations (Ye et al., 2013a) descended below the horizons and could not observe at the time where this intensity increase would have appeared. In (Kero et al., 2012c) there is however one trailing observation with high uncertainty indicating a second peak, and also in the International Meteor Organization (IMO) visual data quicklook website http://www.imo.net/live/draconids2011/, we can see a trailing single observation at 195.382◦ Solar longitude with high ZHR but also large uncertainties. This may be indication that this event actually occurred but no real conclusions can be drawn from the current data. This kind of event is a good example of why the probability distribution approach is a useful analysis type, if we would have had the completed SUOC module and was able to do a MCMC raining (Oszkiewicz et al., 2009) of the ephemeris the probability distribution may have altered the uncertainty of this extra intensity peak drastically. Moving on to the next part of the analysis, we will here introduce a new type of density map, the mass transfer map, as illustrated in figures 53 and 54. Here we have used the same analysis type as with the perihelion passage distribution but instead with particle masses versus time,or Solar longitude. In figure 53 we see the mean mass flux configuration and in figure 54 we see the probability that a certain mass type will propagate to the Earth, not considering the amount of that mass that is propagated. Since the input mass distribution of all iterations are a logarithmic uniform function a equal amount of particles in each mass bin is ejected. This means, that we, by looking at the total amount of particles that propagates to Earth we can find the mass transfer efficiency of a certain parent body. However, if we instead of looking at the total amount of mass transferred to Earth, instead look at the deviation from a uniform distribution observed at Earth, we can infer information about how well different mass propagates from different times and parent bodies to specific years, on Earth. This can be seen as a decoding chart, where if we observe one certain distribution at Earth and de-bias, we can actually make estimations on the ejected mass distribution at the comet itself.
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42.5 2012 October Draconids
Figure 55: Density maps for the mean trail meteoroid flux distribution for the 2012 October Draconids.
Figure 56: Density maps for the trail distribution probability for the 2012 October Draconids.
To complete the comparison between the 2011 and 2012 events we only need to examine the 2012 October
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Figure 57: Density maps for the mean meteoroid flux mass distribution for the 2012 October Draconids.
Figure 58: Density maps for the mass distribution probability for the 2012 October Draconids.
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Draconids as well. Their respective plots can be found in figures 55, 56, 57, and 58. These simulations also correspond well to measurements (Fujiwara et al., 2016).
42.6 Mass transfer difference 2011-2012
Figure 59: The normalized difference in mass distribution between the October Draconids of 2011 and 2012.
As the difference between the overall number of meteors each outburst is less interesting we instead focus on the relation between the masses internally. To accomplish this, we first select all the simulated Earth encounters in one of the years, create a mass distribution plot for this year and normalize it. Then we take the difference between the normalized mass distributions from both years, resulting in figure 59. As we can see here there is a clear difference between the two years in mass propagation efficiency from 21P to the Earth. In 2011 the visual magnitude meteoroids in the mass rage between 10−6 kg to 10−4 kg is up to 30-40% more numerous than in 2012, when compared to the other rates for 2011. We now also have to consider the fact that our input distribution was a logarithmically uniform one, which is not what one expects to find ejected from a comet in real life. As such, even if here, the 0.1 to 0.01 kg meteoroids are relatively more numerous in 2012 than in 2011, these are so rarely ejected that they probably would still not probably have been observed even if they should have occurred in greater numbers.
42.7 Probable mass transfer cause
Having reproduced the observed mass relation difference in a simulation we are able to probe the cause of this difference. One way to do this is to create a subset of simulations, in this case the passages of:
1. 1900 Nov. 28
2. 1907 May 19
3. 1913 Nov. 2
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Figure 60: The normalized difference in mass distribution between the October Draconids of 2011 and 2012 using only a subset of perihelion passages.
Figure 61: The normalized difference in mass distribution between the October Draconids of 2011 and 2012 using only a subset of perihelion passages and propagation without electromagnetic perturbations.
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4. 1920 May 18
5. 1959 Oct. 26
6. 1966 Mar. 28
7. 1972 Aug. 4
We select and use only this data from the above simulation and performed the same mass transfer difference analysis, the result showing in figure 60. We have selected these perihelion passages as they are the most probable, from our simulations, to contribute mass to the 2011 and 2012 October Draconids.
Then we re-run the set of simulation with exactly the same settings, but with one difference: we turn of electromagnetic perturbations. We only do this in the long term propagation however, not in the ejection propagation. The resulting graph can be seen in figure 61. Thus, due to this lack of electromagnetic perturbation, we can see that the mass distribution situation has changed. The range of visual meteors are now much more similar while two differences of equal size has emerged in the radar meteor range. Since exact meteoroid mass determination from observations of meteors is difficult, this kind of di-pole type diffence of the rates in nearby meteoroid mass bins may be undetectable in radar data. Even so, we have just shown that such a mass difference is not what occurs when a more complex model is used. This di-pole difference is most likely due to the fact that a continous ejection model with radiation pressure and sublimation momentum transfer was used rather than set initial velocity. This effectively creates a initial filament structure due to particle mass. Since the gravitational perturbation is ∝ R−2, if there is a filament structure in the correct direction when, for example, Jupiter makes a close passage of the stream, this filament feature is enhanced by the square. The reason the ejection model was not simplified further was that we had not predicted in the software development that we needed to reduce the models complexity and thus there was not time to set up such a simulation, it is however planned for the future.
42.8 Year intensity overview
As we have selected 4 specific years to plot and examine amongst our data, a overall overview of the mean yearly flux distribution is a good point to start from when choosing years to performed specific studies of, as done here. Such a plot can be found in figure 62 together with markers showing the showers that are examined here. From this plot we can see that we have covered some of the major peaks in the activity of the October Draconids in our analysis, and also that in the year 2018, we predict a major outburst of the October Draconids. From examining the Solar longitudes of the 2018 October Draconids simulation we predict the shower to peak in intensity at the night between October 8 and October 9, corresponding to 195.4◦.
42.9 Cluster analysis calibration
When searching for October Draconids in RADAR data one must use a calibrated set-up, as discussed in previous chapters. We shall here try to give some first indications on how such a calibration process can start by using a cluster analysis parameter sweep on the meteor showers generated in our simulation. To do this we will introduce two new plots, the association function plot the error function plot. We have performed the clustering with 4 similarity function, as described in section 31. Since the clustering the meteors are very dependant on how long they have been perturbed in space, it would be misleading to perform a association sweep that disregards time. As such we have produced a year by year version of the analysis procedure for this case where a specific parent body is being repeatedly simulated. Thus we take the total configuration
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Figure 62: Bar plot of the weighted amount of encountered particles per year for all simulated streams of 21P/GiacobiniZinner.
Figure 63: Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the association as a function of critical threshold for all the simulated versions of 2011 October Draconids.
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Figure 64: Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the error as a function of critical threshold for all the simulated versions of 2011 October Draconids.
Figure 65: Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the association and error as a function of critical threshold for the MURMHED observations of 2011 October Draconids.
Page 159 D. Kastinen Meteors and Celestial Dynamics of a meteor storm in one specific year resulting from all perihelion passages, and performed the parameter sweep on this data. In figure 63 we can see the plot of the association functions for the year 2011. In this plot the black lines represent the association function of all the different variations of the meteor shower simulated. The circles represent when the maximum amount of association is reached and the green line is the fraction of shower simulations that has reached maximum. The similar plot for the error function can be seen in figure 64 where we have used the error function in equation 31.21. Here the the blue lines represent the error function as the critical threshold is varied and the circles when each simulated shower reaches its minimum error. The red line is the fraction of showers that has reached minimum. To compare we have performed the cluster analysis on the 13 selected meteors observed during 2011 that was assumed to be a part of the October Draconids (Kero et al., 2012c). Their orbital element data was calculated from the head echo observation and the resulting curves are shown in figure 65. We must however consider several factors here: that the observations are biased as where our simulation is not, how the selection was made in (Kero et al., 2012c), and how clustering is affected by sample size. The last point was discussed in length in section 25, and we can see how a larger particle sample can affect the association curve, however in this case it should not matter very much as the density of the sample has not changed dramatically. The bias correction calculation and measurement uncertainty correction is outside the scope of this work at the moment and shall be ignored, such calculations is proposed in future research. We then have the question of selection left, as observation originating from the expected October Draconids radiant where quite sparse the observation selection was made by simply taking all head echoes inside a box centred on the expected radiant. This may very well, either miss some of the shower signal or include sporadic meteors in the selection. Comparing figures 63 and 64 with 65 we can see that the association quickly rises to a acceptable level within the observations in the same size range as with the simulated association levels. However the critical threshold almost doubles before the clustering process has minimized error and maximized association in the observational data compared with the simulations. This may indicate that a few of the selected meteors from the measurement have very high uncertainties. As when we more closely examine the events listed in (Kero et al., 2012c) we see that two events have high errors in the orbital elements. In fact, they have so high uncertainties that they cover almost the entire predicted shower volume in parameter space. These two events are also, upon closer examination, the reason for the doubling before association maximum. Thus one should aim to both include this measurement uncertainty in the analysis in a better way, and to better fit the simulated data to the likeness of a real observation by including bias calculations as well. In the future, when the next generation of measurements become available (Pellinen- Wannberg et al., 2016b), this kind of analysis can pre-calibrate meteor radars to detect specific meteor showers and performed the selection automatically with a before hand calculated high reliability level, based on also sporadic simulation.
42.10 Principal component Analysis
Lastly we have, as a proof of concept, performed two sets of principal component analysis, as described in section 24. Firstly on all the encountered test particles, regardless of encounter time where the base space was the Kepler elements. The resulting plane of most standardized variance can be seen in figure 66, with the principal component representation shown in figure 67. In the plane of most variance we can see some structure and patterns, finding such structure using PCA is a first indication that pattern recognition methods can work on meteor data. We have in the figure also color coded the perihelion passage that the particle was ejected from, i.e. the trail origin. We can see a certain pattern in the data where a drift seem to occur as later passages produces new particles but this pattern is not distinguished enough to implement a direct application on observational data. It may however be more interesting to only examine a single event or observation campaign, thus we instead restrict our data to only 2011 and calculate the principal components and the new basis coordinates. We can see the result in figure 68 and the principal components in figure 69. As there is some structure difference regarding the 1972 trail encounter in this coordinate basis
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Figure 66: Plane of most standardized variance as extracted with principal component analysis, performed on the set of all encountered particles ejected from 21P.
Figure 67: Principal component representation in the Kepler orbital element basis.
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Figure 68: Plane of most standardized variance as extracted with principal component analysis, performed on the set of encountered particles during the 2011 October Draconids.
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Figure 69: Principal component representation in the Kepler orbital element basis for the 2011 October Draconids analysis.
it would be interesting to examine the PCA of a Kernel Density Estimation (KDE) of the observed meteors. One could in such a KDE use measurement uncertainties and instead extract the regions of most cumulative probability of meteors and overlay with the simulation data. This can give a indication of trail origin of the measurements. Unfortunately the measurement uncertainties are too large and cover several classifications of trail origin simultaneously, this will however be possible with the next generation of measurement instruments (Pellinen-Wannberg et al., 2016b). There was not enough time to implement the KDE functionality in the MATLAB analysis scripts and as such we cannot showcase the results of such a analysis. For the future we plan to implement Kernel Density Estimation techniques and many other machine learning methods to complement this kind of analysis (Murphy, 2012).
43 Pan-STARRS Short Period Comets
43.1 Input state 1
As many of the analysis procedures and algorithms explained in part VI needed visual examples and a general case we designed a simple and easy to explain simulation of the Pan-STARRS short period comets synthetic model. The sampling was sampled using only one simulation, set to generate 500 streams that encountered the Earth within 400 years of ejection. To simplify simulation setup and data analysis we set all perihelion passages to occur at the same date during the year 1800 and integrated to the year 2200. Close encounters with a closest distance under one Earth Hill radii where considered meteor producing test particles. The time step mercury6 was set to 8 day and the step of PBE was set to 16 days. PBE utilized a Bulirsch-Stoer algorithm with multiple substeps according to the previously described series, and mercury6 was configured to use the hybrid symplectic integrator, both including electromagnetic forces. The ejection model used was the one described as (III) in 10.2.1 implemented from (Ma et al., 2002), from with also meteoroid weights
Page 163 D. Kastinen Meteors and Celestial Dynamics where calculated. Finally, the mass distribution for ejection of particles was set to a logarithmic uniform distribution between 10−9 kg to 10−1 kg. As most of the results from this simulation where covered as explanations and examples we will not evaluate further on the results from this simulation at this time. This is due to the fact that any further analysis of showers and stream would be unreliable as the synthetic model comes with a set perihelion passage date that we have disturbed by setting all passages to the year 1800, thus a more correct simulation needs to be setup to extract the real resulting distributions.
44 Concluding remarks
We can conclude that the software developed is a promising simulation tool as all the above meteor shower simulations and the derived conclusions where performed on two personal computers over a 1 week period, using minimal setup preparations and moderate data analysis work. This can allow for precise simulation of almost any situation resulting in a meteoroid stream or large set of small particles and estimation of the results. When the proposed additions in section 40 is completed the software toolbox will be highly versatile and we intend to continuously, through future development, apply it to many other case studies to further the standard of numerical simulations in meteor science.
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Page 171 D. Kastinen Meteors and Celestial Dynamics Appendices
A Notation, abreviations, technical terms, and nomenclature
Table 45: Technical terms and nomenclature
Term Description Meteor The phenomena in the atmosphere of a dust particle ablating Ablation Removal of material an object by vaporization, chip- ping, or other erosive processes Meteoroid Dust particle in the Solar system Meteorite Meteoroid that has survived ablation and lands on the surface of the Earth Sublimation The phase transition from solid to gas
Table 46: Common notation
Symbol Description a Semi-major axis e Eccentricity i Inclination ω Argument of pericentre/perihelion Ω Longitude of the Ascending Node q Perihelion distance
Table 47: Abbreviations
Abreviation Description ESA European Space Agency NASA National Aeronautics and Space Administration PCA Principal Component Analysis KDE Kernel Density Estimation IAU International Astronomical Union IMO International Meteor Organization MU radar Middle and Upper atmospheric radar MURMHED MU Radar Meteor Head Echo Database JPL Jet Propulsion Laboratory NAIF Navigation and Ancillary Information Facility Pan-STARRS Panoramic Survey Telescope and Rapid Response System MC Monte-Carlo MC Markov-Chain MCMC Markov-Chain Monte-Carlo PDF Probability Density Function
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B List of figures
List of Figures
1 The life of a meteoroid encountering the Earth. When caught by the Hill Sphere (the volume in which Earth gravity is dominant), if the Earth gravitational pull directs the meteoroid towards the atmosphere it will enter and begin to heat up. The meteoroid may disintegrate during its ablation phase, or in some cases bounce of the atmosphere. The process of colliding with the atmosphere produces a meteor phenomena as the collision turn small areas of atmosphere into plasma. If however the meteoroid survives atmospheric entry, either due to the entry circumstances or its sheer size, a dark flight phase will begin. Then to finally impact the surface and by doing so turning from meteoroid to meteorite...... 1 2 Digram illustrating and explaining various terms in relation to Orbits of Celestial bodies. (CC- BY-SA-3.0-MIGRATED; Licensed under the GFDL by the author; Released under the GNU Free Documentation License. http://commons.wikimedia.org/wiki/GNU_Free_Documentation_ License)...... 8 3 Digram illustrating the three different anomalies where E is the eccentric anomaly, M the mean anomaly, and ν the true anomaly. The central body is located at the point s and the orbiting body at point p...... 9 4 Pendulum phase map with symplectic and non symplectic euler using the same stepsize. ... 18 5 Relative error in total energy for different initial conditions for a pendulum...... 19 6 Histogram of the different distribution of orbital elements within the Pan-STARRS Solar sys- tem synthetic model...... 25 7 Ejection speed as a function of the common logarithmic mass of the particle...... 27 8 Diagram of ecliptic coordinates, depicting the ecliptic, equator, ecliptic poles, longitude and latitude, and the vernal equinox direction. (Tfr000 / Licensed under the Creative Com- mons Attribution-Share Alike 3.0 Unported license. http://creativecommons.org/ licenses/by-sa/3.0/)...... 35 9 Concept image of a meteoroid head echo ...... 37 10 Contour plot of the logarithm of the total number of events per cell of the Hammer-Aitoff projection, entire MURMHED. The labels are the six sporadic sources: (SA) South apex,(NA) North apex,(ST) South toroidal,(NT) North toroidal,(H) Helion and (AH) Anti-Helion. ... 38 11 Hammer-Aitoff projection of all MURMHED meteor radiants with color encoded geocentric velocities with showers Orionids and Geminids marked...... 39 12 Meteoroid velocity time series of head echo with MJD 55008.9589319839 from MURMHED . 41 13 Radial velocity time series of head echo with MJD 55008.9589319839 from MURMHED ... 41 14 Height time series of of head echo with MJD 55008.9589319839 from MURMHED ...... 41 15 Range-time intensity plot of head echo with MJD 55008.9589319839 from MURMHED .... 41 16 Radar cross section time series of head echo with MJD 55008.9589319839 from MURMHED 41 17 Signal to noise ratio time series of head echo with MJD 55008.9589319839 from MURMHED 41 18 Example illustrating the concept of a Monte Carlo simulation calculating the value of π simply from the use of uniform distributions...... 54 19 Example of clustering within 6 nodes in a two dimensional plane where the similarity between nodes are their geometric distance. Clustering is performed using a arbitrary critical distance. 56
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20 Example of cluster analysis with background data on a two dimensional plane where the sim- ilarity between nodes are their geometric distance. Top graph shows geometric representation of clusters, nodes, and connections where red nodes are signal data and blue nodes background data. Bottom graph shows the sweep of critical threshold and the respective association. Back- ground to signal count ratio of 1 to 3...... 60 21 Background to signal count ratio of 1 to 1...... 60 22 Background to signal count ratio of 5 to 1...... 61 23 Execution time for different segments of MCAS, stream dissipation calculation enabled. .... 68 24 Execution time for different segments of MCAS, stream dissipation calculation disabled. ... 69 25 Comparison between the described first method of drawing a random value from a distribution versus the real multi dimensional distribution ...... 72 26 Comparison between the described improved first method of drawing a random value from a distribution versus the real multi dimensional distribution ...... 72 27 Comparison between the described improved first method of drawing a random value from a distribution versus the artificially generated multi dimensional distribution ...... 75 28 Comparison between the described multi dimensional method of drawing a random value from a distribution versus the Pan-STARRS multi dimensional distribution ...... 75 29 Comparison between the described multi dimensional method of drawing a random value from a distribution versus the artificially generated multi dimensional distribution ...... 76 30 A highly perturbed meteoroid stream encountering Earth over 400 years...... 79 31 A slightly perturbed meteoroid stream encountering Earth over 400 years...... 80
32 The DSH criterion measure of the stream divergence from its parent body in a highly perturbed meteoroid stream encountering Earth over 400 years...... 80
33 The DSH criterion measure of the stream divergence from its parent body in a slightly perturbed meteoroid stream encountering Earth over 400 years...... 81 34 The histograms describing the orbital parameters of generated parent bodies. Blue histogram bars represent the initial distribution while red shows the resulting distribution that generated meteor showers on Earth...... 93 35 The distribution of stream to parent body mean divergence among the Pan-STARRS distribu- tion of comets...... 95 36 Probability of encountering a random cometary meteoroid stream ejected in the year 1800 in a given year...... 96 37 The mean time after ejection to encounter for each meteoroid stream...... 97 38 The variance from the mean of time after ejection to encounter within a each stream...... 97 39 Two spherical distributions, the left illustrating a spatial angle uniform distribution, the right illustrating a true spherically uniform distribution...... 105 40 Calculation of the variations in the obliquity of the ecliptic for two different time scales, ± 10 years from J2000.0 and ± 10 000 years from J2000.0...... 137 41 The graphical user interface used to call all the data analysis scripts developed...... 138 42 Histogram of the orbital element distribution for orbital clones of 21P/GiacobiniZinner. ... 145 43 Histogram of the orbital element distribution for encountered test particles ejected from the clones of 21P/GiacobiniZinner...... 145 44 Histogram of Probability of encounter and mean shower intensity for the 1933 October Draconids.146 45 Density maps for the mean trail meteoroid flux distribution for the 1933 October Draconids. . 147 46 Density maps for the trail distribution probability for the 1933 October Draconids...... 147
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47 Histogram of Probability of encounter and mean shower intensity for the 1946 October Draconids.148 48 Density maps for the mean trail meteoroid flux distribution for the 1946 October Draconids. . 148 49 Density maps for the trail distribution probability for the 1946 October Draconids...... 149 50 Histogram of Probability of encounter and mean shower intensity for the 2011 October Draconids.150 51 Density maps for the mean trail meteoroid flux distribution for the 2011 October Draconids. . 150 52 Density maps for the trail distribution probability for the 2011 October Draconids...... 151 53 Density maps for the mean meteoroid flux mass distribution for the 2011 October Draconids. 151 54 Density maps for the mass distribution probability for the 2011 October Draconids...... 152 55 Density maps for the mean trail meteoroid flux distribution for the 2012 October Draconids. . 153 56 Density maps for the trail distribution probability for the 2012 October Draconids...... 153 57 Density maps for the mean meteoroid flux mass distribution for the 2012 October Draconids. 154 58 Density maps for the mass distribution probability for the 2012 October Draconids...... 154 59 The normalized difference in mass distribution between the October Draconids of 2011 and 2012.155 60 The normalized difference in mass distribution between the October Draconids of 2011 and 2012 using only a subset of perihelion passages...... 156 61 The normalized difference in mass distribution between the October Draconids of 2011 and 2012 using only a subset of perihelion passages and propagation without electromagnetic per- turbations...... 156 62 Bar plot of the weighted amount of encountered particles per year for all simulated streams of 21P/GiacobiniZinner...... 158 63 Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the associa- tion as a function of critical threshold for all the simulated versions of 2011 October Draconids...... 158 64 Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the error as a function of critical threshold for all the simulated versions of 2011 October Draconids. .. 159 65 Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the associ- ation and error as a function of critical threshold for the MURMHED observations of 2011 October Draconids...... 159 66 Plane of most standardized variance as extracted with principal component analysis, performed on the set of all encountered particles ejected from 21P...... 161 67 Principal component representation in the Kepler orbital element basis...... 161 68 Plane of most standardized variance as extracted with principal component analysis, performed on the set of encountered particles during the 2011 October Draconids...... 162 69 Principal component representation in the Kepler orbital element basis for the 2011 October Draconids analysis...... 163
C List of tables
List of Tables
1 Data included in the Pan-STARRS Moving Object Processing System Synthetic Solar System Model...... 31 2 JPL’s measurement database of Solar system major body gravitational parameters ...... 32
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3 family
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24 ejector_char_data.txt Column separator: ’Space’ 0x20 Output File description: Parent body characteristics Note: Only shower generating parent bodies ...... 89 25 ejector_kep_data.txt Column separator: ’Space’ 0x20 Output File description: Parent body characteristics Note: Only shower generating parent bodies ...... 89 26 ejector_n_data.txt Column separator: ’Space’ 0x20 Output File description: Parent body count data Note: Only subliming parent bodies ...... 90 27 mass.data, Column separator: ’Space’ 0x20 Input File description: Simulation state mass vector ...... 109 28 pos.data, Column separator: ’Space’ 0x20 Input File description: Simulation state position vector, one row for each object ...... 109 29 vel.data, Column separator: ’Space’ 0x20 Input File description: Simulation state velocity vector, one row for each object ...... 110 30 mass_dist.data Column separator: ’Space’ 0x20 Input File description: Ejection mass distribution ...... 110 31 body_data.data Column separator: ’Space’ 0x20 Input File description: Parent body data 110 32 particle_q_data.data, Column separator: ’Space’ 0x20 Output File description: Simu- lation state position vectors, one row for each object ...... 111 33 particle_v_data.data, Column separator: ’Space’ 0x20 Output File description: Simu- lation state velocity vectors, one row for each object ...... 111 34 particle_m_data.data, Column separator: ’Space’ 0x20 Output File description: Simu- lation state mass vector, one column for each object ...... 111 35 particle_t_data.data, Column separator: ’Space’ 0x20 Output File description: Simu- lation ejection time vector, one column for each object ...... 111 36 particle_abs_v_data.data, Column separator: ’Space’ 0x20 Output File description: Simulation ejection velocity vector, one column for each particle ...... 111 37 particle_kep_data.data, Column separator: ’Space’ 0x20 Output File description: Ke- pler data after ejection, one row for each object ...... 112 38 body_m_data.data, Column separator: ’Space’ 0x20 Output File description: Simulation state parent body mass, one row for each instance ...... 112 39 sim_data.data, Column separator: ’Space’ 0x20 Output File description: Simulation sum- mary data ...... 112 40 OAA_input.txt, Column separator: ’Space’ 0x20 Input File description: Node data, each row corresponds to one node Currently only Kepler element node data input ...... 120 41 clusters.txt, Column separator: ’Space’ 0x20 Output File description: Cluster state where each row is one cluster This file has variable row sizes as it lists all indexes for cluster members on one row ...... 120 42 error_profile_
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47 Abbreviations ...... 172
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