Separated Equations

The linearized equations are fully separable in spherical-polar coordinates \((r,\theta,\phi)\) and time \(t\). Assuming that all perturbations have an oscillatory time dependence \(\propto \exp(-\ii \sigma t)\), where \(\sigma\) is the angular frequency, this seperability can be exploited through the following ansatzes:

The Eulerian perturbations to any scalar quantity \(f\) and vector quantity \(\va\) take the forms

(8)\[\begin{split}\begin{aligned} f'(r,\theta,\phi;t) &= \tf'(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t), \\ \va'(r,\theta,\phi;t) &= \left[ \tar'(r) \, \ver + \tah'(r) \, r \, \nablah \right] Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t), \end{aligned}\end{split}\]

respectively. Here, the functions \(\tf'\), \(\tar'\) and \(\tah'\) describe the radial dependence of the perturbations; \(Y^{m}_{\ell}\) is a spherical harmonic with harmonic degree \(\ell\) and azimuthal order \(m\);

\[\nablah \equiv \frac{1}{r} \left[ \vet \pderiv{}{\theta} + \frac{\vep}{\sin\theta} \pderiv{}{\phi} \right]\]

is the horizontal part of the gradient operator; and \((\ver,\vet,\vep)\) are the unit basis vectors in the \((r,\theta,\phi)\) directions.
The velocity perturbation vector can be expressed as

\[\vv' = \pderiv{\vxi}{t},\]

where the displacement perturbation vector \(\vxi\) describes the spatial displacement of a fluid element from its equilibrium position. As a specific application of eqn. (8), \(\vxi\) takes the form

(10)\[\vxi(r,\theta,\phi;t) = \left[ \txir(r) \, \ver + \txih(r) \, r \, \nablah \right] Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t),\]

where \(\txir\) and \(\txih\) describe the radial dependence of the radial and horizontal displacement perturbations.
Lagrangian perturbations to scalar and vector quantities can be derived by applying eqn. (7) to the expressions in eqn. (8). Specifically, the Lagrangian perturbations to any scalar quantity \(f\) and vector quantity \(\va\) take the forms

\[\begin{split}\begin{aligned} \delta f(r,\theta,\phi;t) &= \delta \tf(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t), \\ \delta \va(r,\theta,\phi;t) &= \left[ \delta \tar(r) \, \ver + \delta \tah(r) \, r \, \nablah \right] \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t), \end{aligned}\end{split}\]

respectively. Assuming a spherically symmetric equilibrium state, the functions \(\delta \tf\), \(\delta \tar\) and \(\delta \tah\) are related to their Lagrangian counterparts by

\[\begin{split}\begin{aligned} \delta \tf &= \tf' + \txir \deriv{f}{r}, \\ \delta \tar &= \tar' + \txir \deriv{a_{r}}{r}, \\ \delta \tah &= \tah' + \frac{\txih}{r} a_{r}. \end{aligned}\end{split}\]

Substituting the solution forms dictated by these anzatzes into the linearized equations leads to a system of ordinary differential equations for the radial functions \(\txir\), \(\txih\), \(\tP'\) etc. The mechanical (mass and momentum conservation) equations become

is the horizontal part of the gradient operator; and \((\ver,\vet,\vep)\) are the unit basis vectors in the \((r,\theta,\phi)\) directions.

The velocity perturbation vector can be expressed as

\[\vv' = \pderiv{\vxi}{t},\]

where the displacement perturbation vector \(\vxi\) describes the spatial displacement of a fluid element from its equilibrium position. As a specific application of eqn. (8), \(\vxi\) takes the form

(10)\[\vxi(r,\theta,\phi;t) = \left[ \txir(r) \, \ver + \txih(r) \, r \, \nablah \right] Y^{m}_{\ell}(\theta,\phi) \, \exp(\ii \sigma t),\]

where \(\txir\) and \(\txih\) describe the radial dependence of the radial and horizontal displacement perturbations.
Lagrangian perturbations to scalar and vector quantities can be derived by applying eqn. (7) to the expressions in eqn. (8). Specifically, the Lagrangian perturbations to any scalar quantity \(f\) and vector quantity \(\va\) take the forms

\[\begin{split}\begin{aligned} \delta f(r,\theta,\phi;t) &= \delta \tf(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t), \\ \delta \va(r,\theta,\phi;t) &= \left[ \delta \tar(r) \, \ver + \delta \tah(r) \, r \, \nablah \right] \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t), \end{aligned}\end{split}\]

respectively. Assuming a spherically symmetric equilibrium state, the functions \(\delta \tf\), \(\delta \tar\) and \(\delta \tah\) are related to their Lagrangian counterparts by

\[\begin{split}\begin{aligned} \delta \tf &= \tf' + \txir \deriv{f}{r}, \\ \delta \tar &= \tar' + \txir \deriv{a_{r}}{r}, \\ \delta \tah &= \tah' + \frac{\txih}{r} a_{r}. \end{aligned}\end{split}\]

Substituting the solution forms dictated by these anzatzes into the linearized equations leads to a system of ordinary differential equations for the radial functions \(\txir\), \(\txih\), \(\tP'\) etc. The mechanical (mass and momentum conservation) equations become

(11)\[\trho' + \frac{1}{r^{2}} \deriv{}{r} \left( \rho r^{2} \txir \right) - \frac{\lambda}{r} \rho \txih = 0,\]

(12)\[-\sigma^{2} \rho \txih = - \frac{1}{r} \left( \tP' + \rho \tPhi' \right).\]

Likewise, Poisson’s equation becomes

(13)\[\frac{1}{r^{2}} \deriv{}{r} \left( r^{2} \deriv{\tPhi'}{r} \right) - \frac{\lambda}{r^{2}} \tPhi' = 4 \pi G \trho'\]

and the heat equation becomes

\[-\ii \sigma T \delta \tS = \delta \tepsnuc - \deriv{\delta \tLrad}{M_{r}} + \lambda \frac{\txih}{r} \deriv{\Lrad}{M_{r}} + \frac{\lambda}{\rho r} \left( \delta \tFradh - \frac{\txih}{r} \right),\]

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 radiative diffusion equation becomes

\[\begin{split}\begin{aligned} \delta\tFradr &= \left[ 4 \frac{\delta \tT}{T} - \frac{\delta\tkappa}{\kappa} + 2 \frac{\txir}{r} - \lambda \frac{\txih}{r} + \frac{\sderiv{(\delta \tT/T)}{\ln r}}{\sderiv{\ln T}{\ln r}} \right] \Fradr, \\ \delta\tFradh &= \left[ \frac{1}{\sderiv{\ln T}{\ln r}} \frac{\delta \tT}{T} - \frac{\txir}{r} + \frac{\txih}{r} \right] \Fradr. \end{aligned}\end{split}\]

Finally, the thermodynamic, nuclear and opacity relations become

\[\frac{\delta \trho}{\rho} = \frac{1}{\Gammi} \frac{\delta \tP}{P} - \upsT \frac{\delta \tS}{\cP}, \qquad \frac{\delta \tT}{T} = \nabla_{\rm ad} \frac{\delta \tP}{P} + \frac{\delta \tS}{\cP},\]

\[\frac{\delta \tepsnuc}{\epsnuc} = \epsnucad \frac{\delta \tP}{P} + \epsnucS \frac{\delta \tS}{\cP}, \qquad \frac{\delta \tkappa}{\kappa} = \kapad \frac{\delta \tP}{P} + \kapS \frac{\delta \tS}{\cP}.\]

In these equations,

(14)\[\lambda = \ell(\ell+1)\]

is the eigenvalue of the angular parts of the oscillation equations, which enters here into the radial parts as a separation constant. It is related to the local horizontal wavenumber by

\[k_{\rm h}^{2} = \frac{\lambda}{r^{2}}.\]