A Method of Osculating Orbits in Schwarzschild Adam Pound University of Guelph

1. A Method of Osculating Orbits in Schwarzschild Adam Pound University of Guelph

2. True orbit Osculation with ellipse 2 at 2 Osculation with ellipse 1 at 1 • Motivation: Self-force Problems • self-force is calculated on a geodesic; true orbit never follows a geodesic path • we need to determine true path using force on a geodesic • Method: Osculating Orbits • we assume the true orbit x () is tangent to a geodesic z () at each , allowing us to use the force f  on that geodesic

3. Mathematics of Osculating Orbits • specify a geodesic z with orbital elements IA and parametrize it with parameter . The osculation conditions state • insert these conditions into the equations of motion to find • invert to find evolution equations for IA

4. Bound Eccentric Geodesic Orbits in Schwarzschild • similar to precessing elliptical orbits • can be parametrized with a parameter  running from 0 to 2 over one period of radial motion: • can be characterized by analogues of traditional orbital elements in celestial mechanics: IA = (p, e, w, T, ) • principal elements: p = semi-latus rectum, e = eccentricity • positional elements: w =  at periapsis, T = t at periapis,  =  at periapsis ( = w in Keplerian orbits)

5. The Parametrization in Full (in Schwarzschild coordinates) • p and e are related to the orbital energy and angular momentum:

6. Choice of Phase Space • restricting orbits to a plane, we have a 5D phase space corresponding to initial coordinates and velocities (minus one due to normalization) • rather than using {IA} as our phase space, we use • {p(), e(), w(), t(), ()} • T and  can be recovered from t and  if we need initial conditions on the tangential geodesic • using our geodesic parametrization, we invert • to find equations for p, e, and w:

8. Sample Problem: • a massive particle orbiting a BH in the post-Newtonian regime • the particle’s mass causes a gravitational self-force • we use the hybrid equations of motion presented in Kidder, Will & Wiseman ’93: • these equations reduce to geodesic motion for  = 0 • the self-force is derived from the finite- terms

9. Radiation-Reaction Approximation • the self-force has conservative corrections at 1PN and 2PN, and a dissipative correction at 2.5PN • radiation-reaction approximation uses only 2.5PN correction • we have shown this approximation fails for electromagnetic self-force (gr-qc/0509122) • its accuracy has been studied by Ajith et al. (gr-qc/0503124) in post-Newtonian gravitational case • we test it here using our method of osculating orbits

10. p0 = 100 e0 = 0.9 /M = 0.1 true orbit radiation- reaction approximation Comparison of orbits with and without conservative corrections in self-force

11. 0.5x106 1.0x106 1.5x106 Dephasing of True and Approximate Orbits (same initial conditions as above) • phase difference  ~30 rad after p has decreased by 0.4%

12. 0.5x106 1.0x106 1.5x106 0.5x106 1.0x106 1.5x106 • change in principal elements is roughly correct in radiation-reaction approximation • no change in positional elements in radiation-reaction approximation

13. slight improvement if we match initial χ-averaged elements (e.g. ) • phase difference  ~30 rad after p has decreased by 0.8%

14. more improvement if we match initial t-averaged elements (e.g. ) • phase difference  ~30 rad after p has decreased by 5%

15. total dephasing after p0 0.955p0 (matching non-averaged initial conditions)

16. Conclusion • osculating orbits are ideal for analyzing self-force problems • our method has been successful in a simple problem • we have verified the importance of conservative terms in the gravitational self-force • future applications: • - orbits in the spacetime of a tidally-distorted black hole • - orbits of a self-accelerated charge or mass in the fully relativistic case in Schwarzschild • - generalization of our method to orbits in Kerr?