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

\[\pderiv{\rho}{t} + \cdot \nabla \left( \rho \vv \right) = 0\]

and momentum

\[\rho \left( \pderiv{}{t} + \vv \cdot \nabla \right) \vv = -\nabla P - \rho \nabla \Phi;\]

the heat equation

\[\rho T \left( \pderiv{}{t} + \vv \cdot \nabla \right) S = \rho \epsnuc - \nabla \cdot \vF;\]

and Poisson’s equation

\[\nabla^{2} \Phi = 4 \pi G \rho.\]

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,

\[\vF = \vFrad + \vFcon;\]

the radiative flux is given by the radiative diffusion equation,

\[\vFrad = \frac{c}{3\kappa\rho} \nabla (a T^{4}),\]

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:

\[\rho = \rho(p, S), \qquad T = T(p, s).\]

Equilibrium State

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

\[\nabla P = - \rho \nabla \Phi.\]

Assume the equilibrium state is spherically symmetric, this simplifies to

\[\deriv{p}{r} = - \rho \deriv{\Phi}{r}.\]

Poisson’s equation can be integrated once to yield

\[\deriv{\Phi}{r} = \frac{G}{r^{2}} \int 4 \pi \rho r^{2} \, \diff{r} = \frac{G M_{r}}{r^{2}},\]

and so the hydrostatic equilibrium equation becomes

\[\deriv{p}{r} = - \rho \frac{G M_{r}}{r^{2}}.\]

The heat equation in the equilibrium state is

\[\rho T \pderiv{S}{t} = \rho \epsnuc - \nabla \cdot \vF.\]

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

\[\rho' + \nabla \cdot ( \rho \vv' ) = 0,\]
\[\rho \pderiv{\vv'}{t} = - \nabla P' + \frac{\rho'}{\rho} \nabla P - \rho \nabla \Phi',\]
\[\nabla^{2} \Phi' = 4 \pi G \rho'\]

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

\[T \pderiv{\delta S}{t} = \delta \epsnuc - \delta \left( \frac{1}{\rho} \nabla \cdot \vF \right).\]
\[\frac{\delta \rho}{\rho} = \frac[1}{\Gamma_{1}} \frac{\delta p}{p} - \upsilon_{T} \frac{\delta S}{c_{p}}\]
\[\frac{\delta T}{T} = \nabla_{\rm ad} \frac{\delta p}{p} + \frac{\delta S}{c_{p}}\]

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

\[\Gamma_{1} = \left( \pderiv{\ln p}{\ln \rho} \right)_{S} \qquad \upsilon_{T} = \left( \pderiv{\ln \rho}{\ln T} \right)_{p} \qquad c_{p} = \left( \pderiv{S}{\ln T} \right_{p} \qquad \nabla_{\rm ad} = \left( \pderiv{T}{p} \right)_{S}\]

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

\[\xir(r,\theta,\phi;t) = \operatorname{Re} \left[ \sqrt{4\pi} \, \txir(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t) \right],\]
\[\vxih(r,\theta,\phi;t) = \operatorname{Re} \left[ \sqrt{4\pi} \, \txih(r) \, r \nablah Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t) \right],\]
\[f'(r,\theta,\phi;t) = \operatorname{Re} \left[ \sqrt{4\pi} \, \tf'(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t) \right]\]

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

\[\vv' = \pderiv{\vxi}{t}\]

Oscillation Equations

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

\[\trho' + \frac{1}{r^{2}} \deriv{}{r} \left( r^{2} \txir \right) - \frac{\ell(\ell+1)}{r} \rho \txih = 0,\]
\[-\sigma^{2] \rho \txir = - \deriv{\tP'}{r} + \frac{\trho'}{\rho} \deriv{P}{r} - \rho \deriv{\tPhi'}{r},\]
\[-\sigma^{2} \rho r \thxi = - \tP' - \rho \tPhi',\]