Separated Equations
With a separation of variables in spherical-polar coordinates
\((r,\theta,\phi)\), and assuming an oscillatory time (\(t\))
dependence with angular frequency \(\sigma\), solutions to the
linearized fluid equations can be expressed as
(6)\[\begin{split}\begin{aligned}
\xir(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \txir(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t) \right], \\
\xit(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \txih(r) \, \pderiv{}{\theta} Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t) \right], \\
\xip(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \txih(r) \, \frac{\ii m}{\sin\theta} Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t) \right], \\
f'(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \tf'(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t) \right].
\end{aligned}\end{split}\]
Here, \(\xir\), \(\xit\) and \(\xip\) are the radial,
polar and azimuthal components of the displacement perturbation vector
\(\vxi\); \(Y^{m}_{\ell}\) is the spherical harmonic with
harmonic degree \(\ell\) and azimuthal order \(m\); and again
\(f\) stands for any perturbable scalar. The displacement
perturbation vector is related to the velocity perturbation via
\[\vv' = \pderiv{\vxi}{t}.\]
Substituting the above solution forms into the linearized
equations, the mechanical (mass and momentum
conservation) equations become
\[\trho' + \frac{1}{r^{2}} \deriv{}{r} \left( \rho r^{2} \txir \right) - \frac{\ell(\ell+1)}{r} \rho \txih = 0,\]
\[-\sigma^{2} \rho \txir = - \deriv{\tP'}{r} - \frac{\trho'}{\rho} \deriv{\Phi}{r} - \rho \deriv{\tPhi'}{r},\]
\[-\sigma^{2} \rho r \txih = - \tP' - \rho \tPhi'.\]
Likewise, Poisson’s equation becomes
\[\frac{1}{r^{2}} \deriv{}{r} \left( r^{2} \deriv{\tPhi'}{r} \right) - \frac{\ell(\ell+1)}{r^{2}} \txih = 4 \pi G \trho'\]
and the heat equation becomes
\[-\ii \sigma T \delta \tS = \delta \tepsnuc
- \deriv{\delta \tLrad}{M_{r}} + \frac{\ell(\ell+1)}{\sderiv{\ln T}{r}} \frac{\Fradr}{\rho} \frac{\tT'}{T} +
\ell(\ell + 1) \frac{\txih}{r} \deriv{\Lrad}{M_{r}},\]
where
\[\delta \tLrad \equiv 4 \pi r^{2} \left( \delta \tFradr + 2 \frac{\txir}{r} \Fradr \right)\]
is the Lagrangian perturbation to the radiative luminosity. The radial part of the radiative diffusion equation becomes
\[\tFradr' = \Fradr \left[
-\frac{\tkappa'}{\kappa} - \frac{\trho'}{\rho} + 4 \frac{\tT'}{T}
+ \frac{\sderiv{(\tT'/T)}{r}}{\sderiv{\ln T}{r}} \right];\]
after a fair bit of algebra, this can be translated into an equivalent Lagrangian expression,
\[\delta\tFradr = \Fradr \left[
-\frac{\delta\tkappa}{\kappa} + 2 \frac{\txir}{r} - \ell(\ell+1) \frac{\txih}{r} + 4 \frac{\delta \tT}{T} +
\frac{\sderiv{(\delta \tT/T)}{\ln r}}{\sderiv{\ln T}{\ln r}} \right].\]
Finally, the thermodynamic, nuclear and opacity relations become
\[\frac{\delta \trho}{\rho} = \frac{1}{\Gammi} \frac{\delta \tP}{P} - \upsT \frac{\delta \tS}{\cP},
\qquad
\frac{\delta \tT}{T} = \nabla_{\rm ad} \frac{\delta \tP}{P} + \frac{\delta \tS}{\cP},\]
\[\frac{\delta \tepsnuc}{\epsnuc} = \epsad \frac{\delta \tP}{P} + \epsS \frac{\delta \tS}{\cP},
\qquad
\frac{\delta \tkappa}{\kappa} = \kapad \frac{\delta \tP}{P} + \kapS \frac{\delta \tS}{\cP}.\]