A hypothetical attractive central force has magnitude ( )

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Gravitation and central-force motion

Action and Symmetry, Ch The Kepler problem redone: reduction, quadrature and the Runge-Lenz vector

Problem

A hypothetical attractive central force has magnitude F(r)=Cr−2.5F(r) = Cr^{-2.5} (C>0C>0), directed towards the centre, so β=−2.5\beta=-2.5 in F∝rβF\propto r^{\beta}. ((a)) Find the apsidal angle Δφ\Delta\varphi of a nearly circular orbit in this force, in radians and in degrees. ((b)) Without doing any further calculation, state whether Bertrand's theorem allows every bound orbit in this force to be closed, and say what feature of your answer to part (a) already rules this out for the nearly circular orbits themselves.

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