A neutron star has mass and radius km

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Einstein's field equations · Einstein's field equations

The Geometry of Gravity, Ch Linearised gravity: weak fields, gravitomagnetism and the wave equation

Problem

A neutron star has mass

M = 1.4\,M_\odot

and radius

R_\star = 12

km, and spins with period

P = 5.0

ms. For the purpose of estimating its angular momentum, model it as a uniform sphere,

I = \tfrac{2}{5}MR_\star^2

, so that

J = I\omega

with

\omega = 2\pi/P

. A gyroscope is carried in a circular polar orbit of radius

r_{\rm orb} = 3R_\star

around the star. The orbit is treated as Newtonian for the purpose of finding its period. Take

G = 6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}

c = 2.998\times10^8\ \mathrm{m\,s^{-1}}

M_\odot = 1.989\times10^{30}

kg, and

GM_\odot/c^2 = 1.48

km. ((a)) Estimate the star's angular momentum

J

. ((b)) Using the polar-orbit-averaged frame-dragging precession rate

\Omega_{\rm avg} = GJ/(2c^2r_{\rm orb}^3)

and the Newtonian orbital period

T_{\rm orb} = 2\pi\sqrt{r_{\rm orb}^3/GM}

, estimate the precession angle accumulated by the gyroscope in one orbital period. ((c)) Evaluate

GM/(r_{\rm orb}c^2)

and, using the fact that

GM/(R_\star c^2) = 0.17

at the star's surface, decide how much confidence should be placed in the linear estimate above.