Tidal Power for Inclined and Eccentric Orbits¶
Starting from eq. 21 of Lai (2012)
˙E=−∫d3xρ(r)∂ξ(r,t)∂t⋅∇U∗(r,t)=∫d3x∇⋅(ρ(r)∂ξ(r,t)∂t)U∗(r,t)
Where the surface term is zero because the density is zero at the surface.
˙E=−(GM′ω0a3)2Ω∑m,m′,μ,μ′Um,m′Uμ,μ′im′exp(i(μ′−m′)Ωt+iΔm,m′)∫d3xδˉρm,m′(r)r2Y∗2,μ(θ,ϕ)
Since δˉρm,m′(r)∝expimϕ we must have m=μ, further, averaging over an orbit we must have the time dependence term in the exponent vanish, ⇒μ′=m′:
˙E=−(GM′ω0a3)2Ω∑m,m′U2m,m′im′exp(iΔm,m′)∫d3xδˉρm,m′(r)r2Y∗2,m(θ,ϕ)=−T0Ω∑m,m′U2m,m′im′exp(iΔm,m′)κm,m′
So taking the real part:
˙E=T0Ω∑m,m′U2m,m′m′sin(Δm,m′)κm,m′