Tidal Power for Inclined and Eccentric Orbits

Starting from eq. 21 of Lai (2012)

˙E=d3xρ(r)ξ(r,t)tU(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=(GMω0a3)2Ωm,m,μ,μUm,mUμ,μimexp(i(μm)Ωt+iΔm,m)d3xδˉρm,m(r)r2Y2,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=(GMω0a3)2Ωm,mU2m,mimexp(iΔm,m)d3xδˉρm,m(r)r2Y2,m(θ,ϕ)=T0Ωm,mU2m,mimexp(iΔm,m)κm,m

So taking the real part:

˙E=T0Ωm,mU2m,mmsin(Δm,m)κm,m