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[1] 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, the thermodynamic partial derivatives are defined as

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

and the nuclear and opacity partials are

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

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

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

Footnotes

[1]This is known as the frozen convection approximation. GYRE offers multiple ways to freeze convection (see the Oscillation Parameters section); the one here is the default.