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

\[-\ii \sigma T \delta \tS = \delta \tepsnuc - \deriv{\delta \tLrad}{M_{r}} + \frac{\ell(\ell+1)}{\sderiv{\ln T}{r}} \frac{\Fradr}{\rho} \frac{\tT'}{T} + \ell(\ell + 1) \frac{\txih}{r} \deriv{\Lrad}{M_{r}},\]


\[\delta \tLrad \equiv 4 \pi r^{2} \left( \delta \tFradr + 2 \frac{\txir}{r} \Fradr \right)\]

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

\[\frac{\delta \trho}{\rho} = \frac{1}{\Gammi} \frac{\delta \tP}{P} - \upsT \frac{\delta \tS}{\cP}, \qquad \frac{\delta \tT}{T} = \nabla_{\rm ad} \frac{\delta \tP}{P} + \frac{\delta \tS}{\cP},\]
\[\frac{\delta \tepsnuc}{\epsnuc} = \epsad \frac{\delta \tP}{P} + \epsS \frac{\delta \tS}{\cP}, \qquad \frac{\delta \tkappa}{\kappa} = \kapad \frac{\delta \tP}{P} + \kapS \frac{\delta \tS}{\cP}.\]