Two geodesics (meridians) leave the north pole of a spherical planet of radius with a…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Tensors and differential geometry

Invariance, Ch Curvature

Problem

Two geodesics (meridians) leave the north pole of a spherical planet of radius a=3390 kma = 3390\ \text{km} with a small angle δϕ=0.0200 rad\delta\phi = 0.0200\ \text{rad} between them. Their separation ξ(s)\xi(s), as a function of arc length ss along the meridians, obeys the geodesic deviation equation ξ¨+Kξ=0\ddot\xi + K\xi = 0 with K=1/a2K = 1/a^2. ((a)) Solve the equation with the conditions that hold at the pole, and find the separation at s=πa/4s = \pi a/4. ((b)) Find the arc length ss at which the two meridians meet again.

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