A binary star has masses and

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 binary star has masses

M_1 = 2.0\,M_\odot

and

M_2 = 1.0\,M_\odot

, with

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

. The stars attract each other by gravity,

V(r) = -k/r

. ((a)) Find the reduced mass

\mu

of the two-body problem, in kilograms. ((b)) Write down (do not evaluate) the quadrature integral

\varphi(r)

for this orbit, with

V_{\rm eff}(r) = -k/r + \ell^2/2\mu r^2

substituted explicitly, in terms of

E

\ell

\mu

and

k

. ((c)) The gravitational attraction is a power law

F \propto r^{\beta}

with

\beta = -2

. Use

\Delta\varphi = \pi/\sqrt{\beta+3}

to find the apsidal angle, in radians and in degrees.

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