# Linearized Equations

Applying an Eulerian (fixed position, denoted by a prime) perturbation
to the mass and momentum conservation equations, they linearize about
the static equilibrium state as

\[\rho' + \nabla \cdot ( \rho \vv' ) = 0,\]

\[\rho \pderiv{\vv'}{t} = - \nabla P' - \rho' \nabla P - \rho \nabla \Phi'.\]

(in these expressions, the absence of a prime denotes an
equilibrium quantity). Likewise, Poisson’s equation becomes

\[\nabla^{2} \Phi' = 4 \pi G \rho'\]

Applying a Lagrangian (fixed mass element, denoted by a
\(\delta\)) perturbation to the heat equation, and neglecting the
convective heating term \(\delta (\rho^{-1} \nabla \cdot
\vFcon)\), it linearizes about the equilibrium state as

\[T \pderiv{\delta S}{t} = \delta \epsnuc -
\frac{1}{\rho} \nabla \cdot \left[ \vFrad' + \vxi (\nabla \cdot \vFrad) \right].\]

Likewise applying an Eulerian perturbation to the radiative diffusion equation,

\[\vFrad' = \Fradr \left[
\left( - \frac{\kappa'}{\kappa} - \frac{\rho'}{\rho} + 4 \frac{T'}{T} \right) \ver
+ \frac{\nabla (T'/T)}{\sderiv{\ln T}{r}} \right]\]

where \(\ver\) is the radial unit vector. The thermodynamic
relations linearize to

\[\frac{\delta \rho}{\rho} = \frac{1}{\Gammi} \frac{\delta P}{P} - \upsT \frac{\delta S}{\cP},
\qquad
\frac{\delta T}{T} = \nabad \frac{\delta P}{P} + \frac{\delta S}{\cP},\]

and the peturbations to the nuclear energy generation rate and opacity can be expressed as

\[\frac{\delta \epsnuc}{\epsnuc} = \epsad \frac{\delta P}{P} + \epsS \frac{\delta S}{\cP},
\qquad
\frac{\delta \kappa}{\kappa} = \kapad \frac{\delta P}{P} + \kapS \frac{\delta S}{\cP}.\]

In these expressions, Eulerian and Lagrangian perturbations to any
scalar quantity \(f\) are related via

\[\frac{\delta f}{f} = \frac{f'}{f} + \frac{\xir}{r} \deriv{\ln f}{\ln r}.\]

Moreover, the thermodynamic partial derivatives are defined as

\[\Gammi = \left( \pderiv{\ln P}{\ln \rho} \right)_{S}, \quad
\upsT = \left( \pderiv{\ln \rho}{\ln T} \right)_{P}, \quad
\cP = \left( \pderiv{S}{\ln T} \right)_{P}, \quad
\nabad = \left( \pderiv{\ln T}{\ln P} \right)_{S},\]

and the nuclear and opacity partials are

\[\epsad = \left( \pderiv{\ln \epsnuc}{\ln P} \right)_{\rm ad}, \quad
\epsS = \cP \left( \pderiv{\ln \epsnuc}{S} \right)_{P}, \quad
\kapad = \left( \pderiv{\ln \kappa}{\ln P} \right)_{\rm ad}, \quad
\kapS = \cP \left( \pderiv{\ln \kappa}{S} \right)_{P}.\]

The latter can be calculated from corresponding density and
temperature partials via

\[\begin{split}\begin{gathered}
\kapad = \frac{\kaprho}{\Gammi} + \nabad \kapT, \qquad
\kapS = -\upsT \kaprho + \kapT, \\
\epsad = \frac{\epsrho}{\Gammi} + \nabad \epsT, \qquad
\epsS = -\upsT \epsrho + \epsT.
\end{gathered}\end{split}\]

Footnotes