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. Page I D. Kastinen Meteors and Celestial Dynamics 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. Page III D. Kastinen Meteors and Celestial Dynamics 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. Page IV D. Kastinen Meteors and Celestial Dynamics 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 .