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

\[\pderiv{\rho'}{t} + \nabla \cdot ( \rho \vv' ) = 0,\]
(6)\[\rho \pderiv{\vv'}{t} = - \nabla P' - \rho' \nabla \Phi - \rho \nabla \Phi'.\]

(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, it linearizes about the equilibrium state as

\[T \pderiv{\delta S}{t} = \delta \epsnuc - \delta \left( \frac{1}{\rho} \nabla \cdot \vFrad \right),\]

where the heating term \(\delta (\rho^{-1} \nabla \cdot \vFcon)\) has been dropped[1] due to the continued lack of a workable theory for pulsation-convection coupling. Likewise applying a Lagrangian perturbation to the radiative diffusion equation,

\[\delta \vFrad = \left( 4 \frac{\delta T}{T} - \frac{\delta \rho}{\rho} - \frac{\delta \kappa}{\kappa} \right) \vFrad + \frac{\delta(\nabla \ln T)}{\sderiv{\ln T}{r}} \Fradr.\]

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

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

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

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

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

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

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

\[\begin{split}\begin{gathered} \kapad = \frac{\kaprho}{\Gammi} + \nabad \kapT, \qquad \kapS = -\upsT \kaprho + \kapT, \\ \epsnucad = \frac{\epsnucrho}{\Gammi} + \nabad \epsnucT, \qquad \epsnucS = -\upsT \epsnucrho + \epsnucT. \end{gathered}\end{split}\]