|
The energy equation writes: |
|
We consider an equation for the energy density of the thermal radiation $`E_r`$ and of the internal energy $`e=\rho c_v T`$, where $`\rho`$
|
|
\ No newline at end of file |
|
and $`T`$ are the gas volume density and the gas temperature and
|
|
|
|
$`c_v`$ is the specific heat at constant volume.
|
|
|
|
|
|
|
|
## Disk with full radiation treatement (no Stellar heating)
|
|
|
|
|
|
|
|
We follow the evolution of both quantities using the so called two temperature approach (Commercon 2011):
|
|
|
|
|
|
|
|
|
|
|
|
$`\left\lbrace \begin{array}{lll}
|
|
|
|
\frac{\partial E_{\rm r}}{\partial t} - \nabla \cdot \vec F & = &
|
|
|
|
\rho \kappa_p(a_rT^4 - cE_{\rm r}) \\
|
|
|
|
\frac{\partial e}{\partial t} + \nabla \cdot(e\vec v) &=
|
|
|
|
&
|
|
|
|
-P \nabla \cdot \vec v -\rho \kappa_p(a_rT^4 - cE_{\rm r}) + Q^+
|
|
|
|
\end{array} \right.`$
|
|
|
|
|
|
|
|
where $`\vec F = \frac{c\lambda}{\rho \kappa_r} \nabla E_{\rm r}`$ is the radiation flux vector calculated in the flux limited diffusion approximation (FLD, see Levermore and Pomraning 1981) with $`\lambda`$ the flux limiter.
|
|
|
|
We indicate with $`\kappa_p`$ and $`\kappa_r`$ respectively the Planck and the Rosseland mean opacities (here assumed equal, see Bitsch et al 2013), $`\sigma`$ is the Stefan-Boltzmann constant, $`c`$ is the speed of light.
|
|
|
|
|
|
|
|
We consider an ideal gas of pressure $`P`$ with equation of state:
|
|
|
|
$`P = (\gamma-1)e`$ for a gas with adiabatic index $`\gamma`$.
|
|
|
|
The terms $`P \nabla \cdot \vec v `$ and $`Q^+`$ are respectively the compressional heating and the viscous heating ( see Mihalas and Mihalas 1984). We do not include here the heating from the central star.
|
|
|
|
|
|
|
|
## The contribution of the Stellar heating |
|
|
|
\ No newline at end of file |