Separated Equations

Because the tidal potential (14) superposes many different spherical harmonics, the separation of variables (7) applied to the oscillation equations must be replaced by the more-general expressions

(15)\[\begin{split}\begin{aligned} \xir(r,\theta,\phi;t) &= \sum_{\ell,m,k} \txirlmk(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii k \Oorb t), \\ \xit(r,\theta,\phi;t) &= \sum_{\ell,m,k} \txihlmk(r) \, \pderiv{}{\theta} Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \Oorb t), \\ \xip(r,\theta,\phi;t) &= \sum_{\ell,m,k} \txihlmk(r) \, \frac{\ii m}{\sin\theta} Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii k \Oorb t), \\ f'(r,\theta,\phi;t) &= \sum_{\ell,m,k} \tflmk'(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii k \Oorb t) \end{aligned}\end{split}\]

(the notation for the sums has been abbreviated). Substituting these solution forms into the linearized equations, and taking advantage of the orthonormality of the spherical harmonics, leads to a set of differential equations for each combination of \(\ell\), \(m\) and \(k\). A given set resembles the corresponding oscillation equations, with just a couple changes:

  • Rather than being an eigenvalue parameter, the oscillation frequency is set by \(\sigma = k \Oorb\), representing the forcing frequency of the partial tidal potential in an inertial frame.

  • The perturbation \(\tPhi'\) is replaced by \(\tPsi' \equiv \tPhi' + \tPhiT\), representing the radial part of the total (self + tidal) gravitational potential perturbation.