Many precision tests of general relativity are associated with binary pulsars in elliptical orbits.

(a) Verify that the radius of the relative orbit of the pulsars can be written as r = p/(1 + e cos ∅), where p is the semi-latus rectum, e is the eccentricity, and d∅/dt = (Mp)^1/2/r² with M the total mass.

(b) Show that the traceless mass quadrupole moment, Eq. (27.53), in suitable coordinates, is [cf. Eq. (27.70d)]

(c) Use computer algebra to evaluate the second and third time derivatives of this tensor, and then use Eqs. (27.61) and (27.62) to calculate the orbit-averaged energy and angular momentum emitted in gravitational waves, per orbit Your answer should be

(d) Combine the results from part (c) with Kepler’s laws to calculate an expression for the rate of increase of pulse period and decrease of the eccentricity.

(e) Consider a parabolic encounter between two stars, and show that the energy and angular momentum radiated are, respectively,

Equations.

Ijk = p (1 + e cos 6) cos cos sin 0 - cos sin sin 0 0 0 1|3 (27.77a)