A planet of mass

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Adiabatic invariants, canonical perturbation theory and integrable systems

Problem

A planet of mass μ\mu, much smaller than its star's mass MM, orbits inside a cloud of dust of uniform density ρ\rho that extends well beyond the orbit. The dust adds to the Kepler Hamiltonian the small term ϵH1=βr2\epsilon H_1 = \beta r^2, with β=2π3GNρμ\beta = \tfrac{2\pi}{3}G_{\rm N}\rho\mu (an added constant does nothing and is dropped), while k=GNMμk = G_{\rm N}M\mu. The time average of r2r^2 over a Kepler orbit is ⟨r2⟩=a2(1+32e2)\langle r^2\rangle = a^2\left(1+\tfrac32e^2\right). ((a)) Express ⟨r2⟩\langle r^2\rangle in terms of the Kepler actions NN and ℓ\ell. ((b)) Find the pericentre's rate ϖ˙\dot\varpi and its shift Δϖ\Delta\varpi per orbit, and show that Δϖ=−6πβa3k1−e2\Delta\varpi = -\dfrac{6\pi\beta a^3}{k}\sqrt{1-e^2}. Does the pericentre advance or regress? ((c)) For a nearly circular orbit, check this against Chapter 8's apsidal angle π/3+rf′(r)/f(r)\pi/\sqrt{3+rf'(r)/f(r)}, to first order in βa3/k\beta a^3/k. ((d)) Evaluate Δϖ\Delta\varpi per orbit for M=1.989×1030M = 1.989\times10^{30} kg, ρ=1.0×10−15 kg m−3\rho = 1.0\times10^{-15}\,\mathrm{kg\,m^{-3}}, a=1.496×1011a = 1.496\times10^{11} m and e=0.60e = 0.60.

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