A hypothetical small satellite orbits

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 hypothetical small satellite orbits, in a circular polar orbit of radius rorb=20r_{\rm orb}=20 km, a millisecond pulsar of mass M=1.4 M⊙M = 1.4\,M_\odot, radius R⋆=12R_\star = 12 km, spin period P=1.4P=1.4 ms, and moment of inertia I=1.6×1038 kg m2I = 1.6\times10^{38}\ \mathrm{kg\,m^2}. Take GM⊙/c2=1.48GM_\odot/c^2 = 1.48 km. ((a)) Estimate the frame-dragging precession rate ΩLT≈GJ/(2c2rorb3)\Omega_{\rm LT}\approx GJ/(2c^2r_{\rm orb}^3) of a gyroscope on this satellite, using J=IωJ=I\omega with ω=2π/P\omega=2\pi/P. ((b)) Compute GM/(rorbc2)GM/(r_{\rm orb}c^2) for this orbit and use it to decide whether the linear weak-field formula used in part (a) can be trusted as more than an order-of-magnitude estimate.

Browse the problem library