A satellite of 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

A satellite of mass mm, with m≪MEarthm \ll M_{\rm Earth} so that μ≈m\mu \approx m and k=GNMEarthmk = G_{\rm N}M_{\rm Earth}m, orbits the Earth under V(r)=−k/rV(r) = -k/r. At perigee its distance is rp=7.00×106 mr_{\rm p} = 7.00\times10^{6}\,\mathrm{m} and its speed is vp=9.00 km s−1v_{\rm p} = 9.00\,\mathrm{km\,s^{-1}}. Take GNMEarth=3.986×1014 m3 s−2G_{\rm N}M_{\rm Earth} = 3.986\times10^{14}\,\mathrm{m^3\,s^{-2}}. ((a)) Starting from ALRL=p×L−μkr^\mathbf{A}_{\rm LRL} = \mathbf{p}\times\mathbf{L} - \mu k\hat{\mathbf{r}}, derive the polar orbit equation r=porb/(1+ecos⁡φ)r = p_{\rm orb}/(1+e\cos\varphi) with e=A/μke = A/\mu k, without evaluating any integral. ((b)) At perigee, use ∣p×L∣=pL|\mathbf p\times\mathbf L| = pL (since p⊥L\mathbf p\perp\mathbf L automatically and p⊥r\mathbf p\perp\mathbf r at perigee) to find the satellite's eccentricity ee. ((c)) For a potential V(r)V(\mathbf r) homogeneous of degree ν\nu, so that r⋅∇V=νV\mathbf r\cdot\nabla V = \nu V (Euler's theorem), derive 2⟨T⟩=ν⟨V⟩2\langle T\rangle = \nu\langle V\rangle from a long-time average of D˙\dot D with D=p⋅rD = \mathbf p\cdot\mathbf r, stating the condition on the motion this needs. Confirm that gravity (ν=−1\nu=-1) reproduces 2⟨T⟩=−⟨V⟩2\langle T\rangle = -\langle V\rangle, and check that the satellite's orbit meets the stated condition.

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