# 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 neglecting1 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

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.