The gyroscope precession outside a slowly rotating body of angular momentum is = G…

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

The gyroscope precession outside a slowly rotating body of angular momentum

\vec J

\vec\Omega = \frac{G}{c^2r^3}\left[3(\vec J\cdot\hat r)\hat r - \vec J\right].

For the Earth (

J = 5.85\times10^{33}\ \mathrm{kg\,m^2\,s^{-1}}

) averaged over a circular polar orbit of radius

r_{\rm orb}

this gives

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

, which evaluates to

41

mas/yr for the Gravity Probe B orbit,

r_{\rm orb} = 7.01\times10^6

m. ((a)) A satellite instead follows a circular equatorial orbit (its orbital plane coinciding with the Earth's equatorial plane) at the same radius

r_{\rm orb} = 7.01\times10^6

m. Find the gyroscope precession rate for this orbit, in mas/yr, and state how its magnitude and sense (relative to

\vec J

) compare with the polar-orbit average quoted above. ((b)) A different satellite carries a gyroscope in a circular polar orbit at GPS altitude,

r_{\rm orb}' = 2.656\times10^7

m. Using the

r_{\rm orb}^{-3}

scaling of the polar-orbit average, find its precession rate in mas/yr, and the factor by which it is smaller than the Gravity Probe B rate of

41

mas/yr.