A particle of reduced mass and angular momentum moves in with and small

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion

Action and Symmetry, Ch Perturbed orbits, precession and the three-body problem

Problem

A particle of reduced mass μ\mu and angular momentum ℓ\ell moves in V(r)=−k/r−γ3/r3V(r) = -k/r - \gamma_3/r^3 with k,γ3>0k,\gamma_3>0 and γ3\gamma_3 small. The Binet orbit equation for u=1/ru=1/r is d2udφ2+u=−μℓ2u2F(1/u)\dfrac{\mathrm d^2u}{\mathrm d\varphi^2} + u = -\dfrac{\mu}{\ell^2u^2}F(1/u), with F(r)=−dV/drF(r)=-\mathrm dV/\mathrm dr the radial force. Substitute VV, keep only terms up to first order in γ3\gamma_3, and show that the pericentre of an orbit of eccentricity ee and semi-latus rectum porb=ℓ2/(μk)p_{\rm orb}=\ell^2/(\mu k) advances by Δωp=6πγ3/(kporb2)\Delta\omega_{\rm p} = 6\pi\gamma_3/(kp_{\rm orb}^2) per orbit, to first order in γ3\gamma_3, for any ee.

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