.. _osc-linear-eqns:

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

.. math::

   \pderiv{\rho'}{t} + \nabla \cdot ( \rho \vv' ) = 0,

.. math::
   :label: e:osc-lin-mom

   \rho \pderiv{\vv'}{t} = - \nabla P' - \rho' \nabla \Phi - \rho \nabla \Phi'.

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

.. math::

   \nabla^{2} \Phi' = 4 \pi G \rho'

Applying a Lagrangian (fixed mass element, denoted by a
:math:`\delta`) perturbation to the heat equation, it linearizes about
the equilibrium state as

.. math::

   T \pderiv{\delta S}{t} = \delta \epsnuc - 
   \delta \left( \frac{1}{\rho} \nabla \cdot \vFrad \right),

where the heating term :math:`\delta (\rho^{-1} \nabla \cdot \vFcon)`
has been dropped\ [#freeze]_ due to the continued lack of a workable theory for
pulsation-convection coupling. Likewise applying a
Lagrangian perturbation to the radiative diffusion equation,

.. math::

   \delta \vFrad =
   \left( 4 \frac{\delta T}{T} - \frac{\delta \rho}{\rho} - \frac{\delta \kappa}{\kappa} \right) \vFrad +
   \frac{\delta(\nabla \ln T)}{\sderiv{\ln T}{r}} \Fradr.

The thermodynamic relations linearize to

.. math::

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

.. math::

   \begin{gathered}
   \frac{\delta \epsnuc}{\epsnuc} = \epsnucrho \frac{\delta \rho}{\rho} + \epsnucT \frac{\delta T}{T} = \epsnucad \frac{\delta P}{P} + \epsnucS \frac{\delta S}{\cP},\\
   \frac{\delta \kappa}{\kappa} = \kaprho \frac{\delta \rho}{\rho} + \kapT \frac{\delta T}{T} = \kapad \frac{\delta P}{P} + \kapS \frac{\delta S}{\cP}.
   \end{gathered}

In these expressions, Eulerian and Lagrangian perturbations to any
scalar quantity :math:`f` are related via

.. math::
   :label: e:osc-eul-lag

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

Moreover, the thermodynamic partial derivatives are defined as

.. math::

   \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

.. math::

   \begin{gathered}
   \epsnucrho = \left( \pderiv{\ln \epsnuc}{\ln \rho} \right)_{T}, \quad
   \epsnucT = \left( \pderiv{\ln \epsnuc}{\ln T} \right)_{\rho}, \quad
   \epsnucad = \left( \pderiv{\ln \epsnuc}{\ln P} \right)_{\rm ad}, \quad
   \epsnucS = \cP \left( \pderiv{\ln \epsnuc}{S} \right)_{P}, \\
   \kaprho = \left( \pderiv{\ln \kappa}{\ln \rho} \right)_{T}, \quad
   \kapT = \left( \pderiv{\ln \kappa}{\ln T} \right)_{\rho}, \qquad
   \kapad = \left( \pderiv{\ln \kappa}{\ln P} \right)_{\rm ad}, \quad
   \kapS = \cP \left( \pderiv{\ln \kappa}{S} \right)_{P}.
   \end{gathered}

The :math:`(\rho,T)` and :math:`(P,S)` pairs of partials are related by

.. math::

   \begin{gathered}
   \epsnucad = \frac{\epsnucrho}{\Gammi} + \nabad \epsnucT, \qquad
   \epsnucS = -\upsT \epsnucrho + \epsnucT, \\
   \kapad = \frac{\kaprho}{\Gammi} + \nabad \kapT, \qquad
   \kapS = -\upsT \kaprho + \kapT.
   \end{gathered}

.. rubric:: Footnotes

.. [#freeze] This is known as a *frozen convection*
             approximation. GYRE offers multiple ways to freeze
             convection; see the :ref:`osc-conv` section for further
             details.