# Mathmatical Formalism¶

This chapter details the mathematical formalism on which GYRE is built.

## Physical Formulation¶

### Fluid Equations¶

The basis for most subsequent equations are the fluid equations, comprising the conservation laws for mass

and momentum

the heat equation

and Poisson’s equation

Here, \(\rho\), \(p\), \(T\), \(S\) and \(\vv\) are the fluid density, pressure, temperature, specific entropy and velocity; while \(\Phi\) is the gravitational potential, \(\epsnuc\) is the specific nuclear energy generation rate and \(\vF\) the energy flux.

The energy flux is the sum of the radiative (\(\vFrad\)) and convective (\(\vFcon\)) fluxes,

the radiative flux is given by the radiative diffusion equation,

where \(\kappa\) is the opacity and \(a\) the radiation constant.

### Thermodynamic Relations¶

The fluid equations are augmented by the thermodynamic relationships between the four state variables (\(p\), \(T\), \(\rho\) and \(S\)). Only two of these are required to uniquely specify the state (we assume that the composition remains fixed over an oscillation cycle). In GYRE, \(p\) and \(S\) are adopted as these primary variables, and the other two are presumed to be derivable from them:

### Equilibrium State¶

In a static equilibrium state the fluid velocity vanishes. The momentum equation then becomes the hydrostatic equilibrium equation

Assume the equilibrium state is spherically symmetric, this simplifies to

Poisson’s equation can be integrated once to yield

and so the hydrostatic equilibrium equation becomes

The heat equation in the equilibrium state is

If the star is in thermal equilibrium then the left-hand side vanishes, and the nuclear heating rate balances the flux divergence term.

### 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

Likewise applying a Lagrangian (fixed mass element, denoted by a \(\delta\)) perturbation to the heat equation and the thermodynamic relations, they linearizes about the equilibrium state as

The absence of either a prime or a \(\delta\) denotes an equilibrium quantity. No \(\vv'\) terms appear because the equilibrium state is static. The thermodynamic partial derivatives are defined as

### Separation¶

With a separation of variables in spherical-polar coordinates \((r,\theta,\phi)\), and assuming an oscillatory time (\(t\)) dependence with angular frequency \(\sigma\), solutions to the linearized fluid equation can be expressed as

Here, \(\xir\) is the radial component of the displacement perturbation vector \(\vxi\), and \(\vxih\) is the corresponding horizontal (polar and azimuthal) part of this vector; \(\nablah\) is the horizontal part of the spherical-polar gradient operator; \(Y^{m}_{\ell}\) is the spherical harmonic with harmonic degree \(\ell\) and azimuthal order \(m\); and \(f\) stands for any perturbable scalar. The displacement perturbation vector is related to the velocity perturbation via

## Oscillation Equations¶

The oscillation equations follow from substituting the above solution forms into the linearized equations: