Spirograph Patterns of Star Clusters (UMAP)

We develop and analyze a model describing the spirographic motion of a young star within its birthplace, an embedded cluster. We begin by first introducing the mathematical formalism of the classical two-body problem. We thenextend this formalismfor the caseof a star moving through a spherically symmetric extended mass distribution. Finally, we undertake an analytical treatment of the resulting spirographic orbits and provide the necessary tools to explore the beauty of these spirographic orbits numerically.

Table of Contents

1. INTRODUCTION

2. THE STANDARD TWO-BODY PROBLEM
2.1 Standard Formalism
2.2 Potential Formalism: Analytical Treatment of the Problem
2.3 Numerical Analysis Using Mathematica

3. ANALYTICAL TREATMENT
3.1 Hernquist Density Profile
3.2 Hernquist Potential Function
3.3 Nondimensionalized Potential Formalism

4. NUMERICAL SIMULATIONS

5. SOLUTIONS TO THE EXERCISES

ACKNOWLEDGMENTS

APPENDIX A: ELLIPTIC ORBITS

APPENDIX B: SPIROGRAPHIC ORBITS

REFERENCES

ABOUT THE AUTHOR