A ship sails a loxodrome (rhumb line): a path on the Earth's surface

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Tensors and differential geometry

Invariance, Ch The metric, the connection and geodesics

Problem

A ship sails a loxodrome (rhumb line): a path on the Earth's surface, treated as a sphere of radius R⊕=6371 kmR_\oplus = 6371\,\mathrm{km}, that makes a constant angle β=60∘\beta = 60^\circ with every meridian it crosses. In colatitude θ\theta (measured from the north pole) the spherical metric is ds2=R⊕2(dθ2+sin⁡2θ dϕ2)\mathrm{d}s^2 = R_\oplus^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2), and the northward length of a step ds\mathrm{d}s along the path, R⊕∣dθ∣R_\oplus|\mathrm{d}\theta|, equals dscos⁡β\mathrm{d}s\cos\beta. The ship starts at latitude 10∘N10^\circ\mathrm{N} and finishes at latitude 40∘N40^\circ\mathrm{N}, sailing generally northward. Find the length of the loxodrome.

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