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Orbits of Particles Around Compact Objects: from Newtonian Theory to General Relativity

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Orbits of Particles Around Compact Objects: from Newtonian Theory to General Relativity Orbits of Particles Around Compact Objects: From Newtonian Theory to General Relativity Matthías Ásgeir Jónsson FacultyFaculty of of Physics Physics UniversityUniversity of of Iceland Iceland 20162016 ORBITS OF PARTICLES AROUND COMPACT OBJECTS: FROM NEWTONIAN THEORY TO GENERAL RELATIVITY Matthías Ásgeir Jónsson 16 ECTS thesis submitted in partial fulllment of a Baccalaureus Scientiæ degree in Physics Advisor Gunnlaugur Björnsson Faculty of Physics School of Engineering and Natural Sciences University of Iceland Reykjavik, May 2016 Orbits of Particles Around Compact Objects: From Newtonian Theory to General Rela- tivity 16 ECTS thesis submitted in partial fulllment of a B.Sc. degree in Physics Copyright c 2016 Matthías Ásgeir Jónsson All rights reserved Faculty of Physics School of Engineering and Natural Sciences University of Iceland Hjarðarhagi 2-6 107, Reykjavík Iceland Telephone: 525 4000 Bibliographic information: Matthías Ásgeir Jónsson, 2016, Orbits of Particles Around Compact Objects: From Newtonian Theory to General Relativity. B.Sc. thesis, Faculty of Physics, University of Iceland. Abstract This thesis gives a general overview of orbital trajectories of particles around com- pact objects in Newtonian theory and in general relativity. Only two particle sys- tems are considered, with only gravitational attraction taken into account. Orbits are calculated from a set of initial conditions, using dierent methods such as nding roots of an eective potential, by parameterizing, and solving linear and non-linear dierential equations. v Contents List of Figures ix List of Tables xi 1 Introduction 1 2 The Kepler Problem 3 2.1 Kepler's Laws of Planetary Motion . .3 2.2 Newtonian Mechanics . .5 2.2.1 Kepler's First Law Derived . .7 2.2.2 Kepler's Second Law Derived . .9 2.2.3 Kepler's Third Law Derived . 10 2.3 The Lagrangian . 11 2.3.1 Kepler's rst law derived . 12 2.3.2 The Eective Potential . 15 2.3.3 Open and Closed Orbits . 17 2.3.4 Calculated Orbits . 18 2.3.5 Orbital Velocities . 20 2.3.6 Other Central Forces . 21 2.4 A Pseudo-Newtonian Potential . 24 3 Orbits in the Schwarzschild Metric 29 3.1 The Metric . 29 3.2 Time-Like Geodesics: Massive Particles . 31 3.2.1 Radial Geodesics . 33 3.2.2 Bound Orbits . 35 3.2.3 Unbound Orbits . 47 3.3 Null Geodesics: Massless Particles . 50 3.3.1 Cone of avoidance and General Orbits . 54 4 Orbits in the Kerr Metric 59 4.1 The Metric . 59 4.2 The Ergosphere . 60 4.3 Circular Orbits . 61 4.3.1 Null Geodesics: Massless particles . 63 4.3.2 Time-Like Geodesics: Massive Particles . 63 vii Contents 5 Conclusions 65 Bibliography 67 viii List of Figures 2.1 A logarithmic plot of the cube of the average distance as a function of orbital period squared. Data from Nasa [2]. .5 2 2 2.2 The eective potential, Veff , and its components, −k=r and L =2µr . 16 2.3 An Earth like system. Initial conditions: m1 = M , m2 = 3 × −6 , , and . Resulting in a 10 M r0 = 0:9832AU vθ0 = 6:3853AU=yr nearly circular orbit. 19 2.4 Initial conditions: , , , and m1 = M m2 = 0:1M r0 = 1AU vθ0 = 6AU=yr. Resulting in an elliptical orbit with e = 0:17......... 19 2.5 Initital conditions: , , . Result- m1 = m2 = M r0 = 1 vθ0 = 15AU=yr ing in a hyperbolic orbit with e = 1:848................. 20 2.6 Radial velocity curves due to a particle with mass m2 = 9:54 × −4 10 M = MJupiter placed at r0 = 0:5AU with vθ = 6:35AU=yr, around m1 = M .............................. 21 −10 2.7 Radial velocity curves due to a particle with mass m2 = 3×10 M = M⊕ placed at r0 = 0:5AU with vθ = 6:35AU=yr, around m1 = M .. 21 2.8 Closed generic orbits with C < 0, an attractive force. 23 2.9 Closed generic orbits with C > 0, a repulsive force. 24 2.10 Orbits in the Psuedo-Newtonian potential. The red circle represents the event horizon. a) Initial conditions: x0 = 5:06, vθ = 0:427c0. b) Initial conditions: x0 = 2:48, vθ = 0:775c0, e = 0:59........... 26 3.1 The eective potential for dierent values of L............. 32 ix LIST OF FIGURES 3.2 Initial conditions: , , . The red circle M = 1 r0 = 12 vθ0 = 0:4c represents the event horizon. 41 3.3 Orbits of the second kind for case B. a) Initial conditions: M = 1 and r0 = 5. b) Initial conditions: M = 1 and r0 = 5:6. The red circle represents the event horizon. 42 3.4 Initial conditions: M = 1 and r0 = 20. a) Orbit of the rst kind starting from r0 = 20 approaching r = 4:44. b) Orbit of the second kind starting from r = 4:44 spiraling towards the center. The red circle represents the event horizon. 44 3.5 Capture orbits for case E. a) Initial conditions: M = 1, r0 = 10, and . b) Initial conditions: , , and . vθ0 = 0:02c M = 1 r0 = 30 vtheta0 = 0:05c The red circle represents the event horizon. 47 3.6 Initial conditions: , , and . The red circle represents M = 1 r0 = 8 vθ0 the event horizon. 48 3.7 Initial conditions: M = 1 and r0 = 2:8. The red circle represents the event horizon. 49 3.8 Unbound orbits for case C. Parameters from Chandrasekhar (1983). a) Initial conditions: M = 0:3, e = 0:001i, and α = 1. b) Initial conditions: M = 0:3, e = 0:1i, and α = 1. The red circle represents the event horizon. 50 3.9 The photon eective potential. 51 3.10 The cone of avoidance at dierent distances [9]. 55 3.11 Solutions of equation (3.103), with M = 1, r0 = 12, and varying θ... 57 4.1 The positive and negative potentials for a photon with L = 3, around a rotating black hole of mass M = 1, with a = 0:5. The Black dashed line is r+ and the dotted line is r0 at θ = π=2.............. 63 4.2 Circular orbits in the equatorial plane (colored) in the Kerr metric. 64 x List of Tables 2.1 Orbital energy and eccenetricity . 17 xi 1 Introduction Orbits of particles are an extremely important topic in physics, and are at the foundation of astrophysics. Understanding of orbits is vital to elucidate the behavior of astrophysical phenomena such as planetary systems, accretion disks, and black holes. In this paper we focus on orbits of test particles around stars and compact objects such as black holes. The goal is to calculate orbital trajectories of particles from a set of initial conditions. Each chapter gets increasingly more complex as we move towards orbits in general relativity. The aim is then to introduce core concepts in the earlier chapters for use in the later ones. In all discussions, we consider only systems of two particles, i.e the two body problem, which can be reduced to only one orbiting particle. We begin with a discussion on Kepler's laws of planetary motion and then derive them using Newtonian mechanics. From there we introduce an element from rela- tivity into our Newtonian model before the discussion of orbits in general relativity. A large portion is dedicated to orbits in the Schwarzschild metric, and almost all possible orbits there are covered. For the Kerr metric we only cover circular orbits in the equatorial plane. 1 2 The Kepler Problem In this chapter we focus solely on Kepler orbits (or Keplerian orbits). Kepler orbits are solutions of the two body problem and describe the motion of two bodies around a common center of mass due to their gravitational attraction. We treat the orbiting bodies as point particles and neglect other eects such as perturbations from external bodies, radiation pressure, non-uniform gravity, and general relativity. Kepler orbits are thus solutions of a highly idealized version of the two body problem called the Kepler problem. We also assume, in all cases, that our orbits are stable, i.e. will not decay over time. The orbits, as we shall see, can only take on 4 dierent shapes depending on the initial conditions. These shapes (or curves) are known as conic sections. The discussion below follows and is based on Carroll and Ostlie (2014). 2.1 Kepler's Laws of Planetary Motion The rst successful description of planetary motion in agreement with observational data was put forth by the German mathematician Johannes Kepler. After the death of Tycho Brahe, the foremost naked-eye-observer at the time, Kepler inherited his massive amount of data and over a period of about 20 years put forth three laws of planetary motion famously attributed to him [1]. These three laws, known as Kepler's Laws of Planetary Motion, are at the foundation of modern astronomy. They are as follows: Kepler's First Law A planet orbits the Sun in an ellipse, with the sun at one focus of the ellipse. Kepler's Second Law A line connecting a planet to the Sun sweeps out equal areas in equal time intervals. Kepler's Third Law The Harmonic Law The square of the orbital period of a planet is proportional to the cube of the planet's average distance from the Sun. 3 2 The Kepler Problem The laws are described above in a qualitative manner since they were deduced from purely empirical data. We can describe them more quantitatively in mathematical form. Kepler's First Law a(1 − e2) r = ; (2.1) 1 + e cos θ Kepler's Second Law ∆A = constant; (2.2) ∆t Kepler's Third Law The Harmonic Law P 2 = a3: (2.3) Here r is the planet's distance from the sun (at a focal point), a its average distance from the sun (the semi-major axis) and e is the orbit's eccentricity - an indicator of its shape/structure.
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