2.1 Basic Equations

The hydrodynamics equations are expressed as conservation relations plus source terms for

$$\rho, \; \rho v{\rm 1}, \rho v{\rm 2}, \rho v{\rm 3}, \; \rho e{\rm ikg} \enspace ,$$ (1)

the mass density, the three mass fluxes and the total energy density (per volume), respectively. Each quantity $q$ has its corresponding flux $\mbox{f}(q)$ and possibly source term $\mbox{s}(q)$. For convenience,

$$\rho, \; v{\rm 1}, v{\rm 2}, v{\rm 3}, \; e{\rm i}$$ (2)

are chosen as independent quantities. The conserved quantities are purely algebraic combinations of these.

The 3D hydrodynamics equations, including source terms due to gravity, are the mass conservation equation

$$\frac{\partial \rho}{\partial t} + \frac{\partial \; \rho \... ...frac{\partial \; \rho \; v{\rm 3}}{\partial x3} = 0 \enspace ,$$ (3)

the momentum equation

$$\frac{\partial }{\partial t} \left( \! \begin{array}{c} ... ...\rho \; g{\rm 2} \\ \rho \; g{\rm 3} \end{array} \! \right)$$ (4)

and the energy equation including radiative heating term $Q_{\rm rad}$

$$\frac{\partial \rho e{\rm ik}}{\partial t} + \frac{\partial... ... \; v{\rm 2} + g{\rm 3} \; v{\rm 3} ) + Q_{\rm rad} \enspace .$$ (5)

The pressure $P$ is computed from density $\rho$ and internal energy $e{\rm i}$ via a (tabulated) equation of state

$$P = P (\rho, e{\rm i}) \enspace .$$ (6)

For local models the gravity field is simply given by

$$\vec{g} = \left( \begin{array}{r} 0 \\ 0 \\ -g \\ \end{array} \right) \enspace .$$ (7)

For global models it is given by

$$\vec{g} = \left( \begin{array}{c} g{\rm 1} \\ g{\rm 2}... ...\\ \frac{\partial }{\partial x3} \end{array} \right) \Phi$$ (8)

with

$$\Phi = -\frac{G M_\ast}{\left( r_0^4+r^4/\sqrt{1+(r/r_1)^8} \right)^{1/4}} \enspace .$$ (9)

Here, $M_\ast$ is the mass of the star to be modeled; $r_0$ and $r_1$ are free smoothing parameters.

In addition, there are equations for the non-local radiation transport. If grey opacity tables are used, the opacities $\kappa$ are a simple function of e.g. temperature $T$ and pressure $P$

$$\kappa = \kappa ( T, P )$$ (10)

and the source function $S$ is given by

$$S= \frac{\sigma}{\pi} T^4 \enspace .$$ (11)

The change in optical depth $\Delta \tau$ along a path with length $\Delta x$ is than

$$\Delta \tau = \kappa \rho \Delta x \enspace .$$ (12)

The variation of the intensity $I$ with optical depth $\tau$ along a ray with orientation $(\theta, \varphi)$ can be described by the simple differential equation

$$\frac{{\rm d} I}{{\rm d} \tau} = - I+ S \enspace .$$ (13)

The radiative energy flux is given by

$$F_{\rm rad} = \int_{0}^{2\pi} \int_{0}^{\pi} I\cos \theta \sin \theta \;{\rm d}\theta \;{\rm d}\varphi \enspace .$$ (14)

The energy change can than be computed from the flux divergence with

$$\frac{\partial \rho e{\rm i}}{\partial t} = Q_{\rm rad} = ... ...ac{\partial F_{x3,\rm rad} }{\partial x3} \right) \enspace .$$ (15)