Two stars of masses and orbit their common centre of mass under their mutual…
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Lagrangian mechanics
Action and Symmetry, Ch The Kepler problem redone: reduction, quadrature and the Runge-Lenz vector
Problem
Two stars of masses m1=2.0×1030kg and m2=1.0×1030kg orbit their common centre of mass under their mutual gravitational attraction, V(r)=−GNm1m2/r, on a bound elliptical orbit of total energy (for the reduced one-body problem) E=−3.0×1033J. All other bodies and all dissipation are ignored.
((a)) Remove the cyclic centre-of-mass coordinate to reduce the pair to one body of reduced mass μ moving in V(r) about a fixed centre. State μ numerically.
((b)) Mechanical similarity applied to a potential V∝rn relates the long-time averages ⟨T⟩ and ⟨V⟩ of a bound orbit through the exponent n alone. State that relation for n=−1, and say what the scaling argument does and does not establish about why such an average exists.
((c)) The heavy symmetric top's Lagrangian has two cyclic angles, precession ϕ and spin ψ, and removing them leaves a single effective potential for the nutation angle θ. The two-body reduction removes the three components of the cyclic Rcm, uses the conserved direction of L to fix the plane of the orbit, and then removes the cyclic polar angle φ. Count the coordinates left after each step, and say which of the steps has a counterpart in the top's reduction and which does not.
((d)) Derive the relation of part (b) directly from the virial theorem: define D=p⋅r for the reduced one-body problem, find D˙, state the condition on the orbit that makes ⟨D˙⟩→0 over a long time, and confirm the bound orbit meets it. Then use E=⟨T⟩+⟨V⟩ to find ⟨T⟩ and ⟨V⟩ numerically.