# 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

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

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

Likewise applying an Eulerian perturbation to the radiative diffusion equation,

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

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

In these expressions, the thermodynamic partial derivatives are defined as

and the nuclear and opacity partials are

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

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. |