A navigation satellite moves on an eccentric orbit with perigee radius and apogee radius

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · The equivalence principle and gravity as geometry

The Geometry of Gravity, Ch Redshift, local inertial frames and the failure of flat spacetime

Problem

A navigation satellite moves on an eccentric orbit with perigee radius

r_{\rm peri}=2.60\times10^{7}\ \mathrm{m}

and apogee radius

r_{\rm apo}=2.70\times10^{7}\ \mathrm{m}

. Take

GM_\oplus=3.986\times10^{14}\ \mathrm{m^3\,s^{-2}}

. Along the orbit, the clock's rate against a ground clock is a fixed constant plus

-2GM_\oplus/(rc^2)

. ((a)) Find the swing in the fractional rate between perigee and apogee. ((b)) Treat the varying part of the rate as a sinusoid in time, with half that swing as its amplitude and a period equal to the orbital period

T=2\pi\sqrt{a^3/GM_\oplus}

, where

a=(r_{\rm peri}+r_{\rm apo})/2

. Find the amplitude of the periodic time offset it produces, in nanoseconds. ((c)) Convert that amplitude into metres of range.

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