Problem 8.4

AP E3200x

Orbital Period (Kepler’s Third Law)

(* dt = *) dθ μ r^2/l /. r -> α/(1 + ϵ Cos[θ])

Integrate[(( (α^2) μ)/(l (((1+ϵ Cos[θ]))^2))), {θ, 0, 2 π}, Assumptions-> 0 < ϵ < 1 && α > 0]
% /. ϵ -> Sqrt[1 - α/a]
% /. α -> l^2/μ/k
% // PowerExpand
(* period *) τ = % // Simplify

Time-Averaged Potential Energy

(* dt (-k/r) = *) dθ (-k/r)μ r^2/l /. r -> α/(1 + ϵ Cos[θ])

Integrate[-(( k α μ)/(l (1+ϵ Cos[θ]))), {θ, 0, 2 π}, Assumptions-> 0 < ϵ < 1 && α > 0 && l > 0]
% / τ
% /. ϵ -> Sqrt[1 - α/a]
% /. α -> l^2/μ/k
(* time average potential *)% // PowerExpand

Time-Averaged Kinetic Energy

(* dt (μ/2)(dr^2 + r^2 dθ^) = *) dθ (μ r^2/l)(l^2/2/μ)(ϵ^2 Sin[θ]^2/α^2 + 1/r^2) /. r -> α/(1 + ϵ Cos[θ])
% // Simplify

Integrate[(( l (1+(ϵ^2)+2 ϵ Cos[θ]))/(2 (((1+ϵ Cos[θ]))^2))), {θ, 0, 2 π}, Assumptions-> 0 < ϵ < 1 && α > 0 && l > 0]
% / τ
% /. ϵ -> Sqrt[1 - α/a]
% /. α -> l^2/μ/k
(* time average kinetic energy *)% // PowerExpand