A satellite is placed

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 satellite is placed, for the purpose of this estimate, in a circular polar orbit at the altitude of the International Space Station, 400 km above the Earth's surface. Take the Earth's radius as R⊕=6371R_\oplus = 6371 km, its angular momentum as J=5.85×1033 kg m2 s−1J = 5.85\times10^{33}\ \mathrm{kg\,m^2\,s^{-1}}, and GM⊕/c2=4.44GM_\oplus/c^2 = 4.44 mm. ((a)) Estimate the frame-dragging precession of a gyroscope carried on this satellite, averaged over the orbit, using ΩLT≈GJ/(2c2rorb3)\Omega_{\rm LT} \approx GJ/(2c^2 r_{\rm orb}^3), and express the result in milliarcseconds per year. ((b)) Compute GM⊕/(rorbc2)GM_\oplus/(r_{\rm orb}c^2) for this orbit and use it to decide whether the linear weak-field treatment used to derive ΩLT\Omega_{\rm LT} is trustworthy here.

Browse the problem library