A star of mass ( ) moves in the outer part of a galaxy whose rotation curve is flat at

PHYSICS · Multi-part · 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 of mass

m=1.0\,M_\odot

(

M_\odot=1.989\times10^{30}\,\mathrm{kg}

) moves in the outer part of a galaxy whose rotation curve is flat at

v_{\rm c}=220\,\mathrm{km\,s^{-1}}

. Treat the galaxy's spherically distributed mass as fixed, so the star's reduced mass is

m

itself, and let it exert on the star a central force

F(r)=-C/r

, a power law with

\beta=-1

, with potential

V(r)=C\ln(r/r_{\rm ref})

for a positive constant

C

. The reference radius

r_{\rm ref}

is arbitrary: changing it only adds a constant to

V

. ((a)) Show that every circular orbit in this force has the same speed, and use

v_{\rm c}

to find

C

in joules. ((b)) Write down (do not evaluate) the quadrature integral

\varphi(r)

for this potential, with

V_{\rm eff}(r)=C\ln(r/r_{\rm ref})+\ell^2/2mr^2

substituted, in terms of

E

\ell

m

C

and

r_{\rm ref}

. ((c)) Find the apsidal angle for

\beta=-1

, in radians and in degrees.