The Schwarzschild radial equation for a bound timelike orbit

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · Gravitation and central-force motion

The Geometry of Gravity, Ch Orbits and light rays in Schwarzschild spacetime

Problem

The Schwarzschild radial equation for a bound timelike orbit, written in terms of

u\equiv1/r

and

\phi

, is

u''+u=\frac{GM}{L^2}+\frac{3GM}{c^2}u^2,

primes denoting

d/d\phi

, where

L

is the angular momentum per unit rest mass. ((a)) Solve this equation to first order in

GM/rc^2

, treating the last term as a small perturbation on the Newtonian ellipse

u_0=(GM/L^2)(1+e\cos\phi)

. Show that the perihelion of the orbit advances by

\Delta\varpi=6\pi GM/\big(c^2a(1-e^2)\big)

per orbit, with

a

the semi-major axis and

e

the eccentricity. ((b)) The binary pulsar PSR B1913+16 (the Hulse–Taylor pulsar) has orbital period

T_{\rm orb}=7.751

h, eccentricity

e=0.617

and total mass (both neutron stars together)

M=2.828\,M_\odot

. Using Kepler's third law for the relative orbit,

a^3=GMT_{\rm orb}^2/4\pi^2

, find the semi-major axis

a

, and hence the periastron advance rate in degrees per year, using

GM_\odot/c^2=1.48

km. Compare with the measured rate of

4.226\ ^\circ\,{\rm yr}^{-1}