A star orbits in a galaxy whose rotation curve is flat

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Astrophysics & Cosmology · Gravitation and central-force motion

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

Problem

A star orbits in a galaxy whose rotation curve is flat, so the gravitational force on it behaves as F(r)∝1/rF(r)\propto 1/r (that is, β=−1\beta=-1 in F∝rβF\propto r^\beta), and its orbit is bound (E<0E<0). Someone argues: "The star's orbit is bound, so after one full revolution about the galactic centre it must return to its starting point, tracing out a closed curve, just as a planet does around the Sun." Find the error in this argument, using the apsidal angle for β=−1\beta=-1.

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