Two asteroids of comparable mass

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

Two asteroids of comparable mass,

m_1 = 5.0\times10^{13}\,\mathrm{kg}

and

m_2 = 8.0\times10^{13}\,\mathrm{kg}

, orbit their common centre of mass under a mutual attraction that is slightly steeper than gravity,

F(r) = -Cr^{\beta}

with

\beta = -2.1

and

C \gt 0

, so that

V(r) = -\dfrac{C}{1.1}r^{-1.1}

. ((a)) Find the reduced mass

\mu

of the pair, in kilograms. ((b)) Write down (do not evaluate) the quadrature integral

\varphi(r)

for this potential, with

V_{\rm eff}(r)

substituted, in terms of

E

\ell

\mu

and

C

. ((c)) Find the apsidal angle for

\beta = -2.1

, in radians and in degrees, and state by how much the pericentre advances on each radial period.

Browse the problem library