Two Earth-orbiting satellites move on circular geodesics of radius km (altitude km)…
PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Einstein's field equations
The Geometry of Gravity, Ch Curvature: the Riemann tensor, geodesic deviation and the Einstein tensor
Problem
Two Earth-orbiting satellites move on circular geodesics of radius r=7178 km (altitude 800 km) whose orbital planes are tilted with respect to one another by Δι=0.02∘. In the Newtonian limit the entry of Ri0j0 normal to the orbital plane is GM⊕/(r3c2), so the out-of-plane separation S between the satellites obeys S¨=−r3GM⊕S, with initial condition S=0 at a node (where the planes cross) and growing towards a maximum a quarter-period later. Take GM⊕=3.986×1014m3s−2 (equivalently GM⊕/c2=4.44 mm).
((a)) Find the orbital period Torb.
((b)) Find the tidal rate GM⊕/r3 and confirm it equals the square of the orbital angular frequency.
((c)) Find the amplitude of the out-of-plane separation and the time after the node at which it is reached.