## EOS_Hybrid

Date

Abstract

EOS_Hybrid. The equation of state used for “simple” core collapse simulations.

### 1 The equations

This equation provides the hybrid polytropic / ideal gas equation of state used by Dimmelmeier et al. [1] for supernova collapse. A thorn wanting to use this needs to use the CactusEOS interface found in EOS_Base.

The equations are

$\begin{array}{rcll}P& =& {P}_{\mathrm{\text{poly}}}+{P}_{\mathrm{\text{th}}}& \text{(1)}\text{}\text{}\\ \frac{\partial P}{\partial \rho }& =& \frac{\partial {P}_{\mathrm{\text{poly}}}}{\partial \rho }+\frac{\partial {P}_{{\mathrm{\text{th}}}_{1}}}{\partial \rho }+\frac{\partial {P}_{{\mathrm{\text{th}}}_{2}}}{\partial \rho }& \text{(2)}\text{}\text{}\\ \frac{\partial P}{\partial 𝜖}& =& \left({\gamma }_{\mathrm{\text{th}}}-1\right)\rho ,& \text{(3)}\text{}\text{}\end{array}$

where

$\begin{array}{rcll}{P}_{\mathrm{\text{poly}}}& =& K{\rho }^{\gamma }& \text{(4)}\text{}\text{}\\ {P}_{\mathrm{\text{th}}}& =& -K\frac{{\gamma }_{\mathrm{\text{th}}}-1}{\gamma -1}{\rho }^{\gamma }+\left({\gamma }_{\mathrm{\text{th}}}-1\right)\rho 𝜖-\left({\gamma }_{\mathrm{\text{th}}}-1\right)\frac{\gamma -{\gamma }_{1}}{{\gamma }_{2}-1}K{\rho }_{\mathrm{\text{nuc}}}^{{\gamma }_{1}-1}\rho & \text{(5)}\text{}\text{}\\ \frac{\partial {P}_{\mathrm{\text{poly}}}}{\partial \rho }& =& \gamma K{\rho }^{\gamma -1}& \text{(6)}\text{}\text{}\\ \frac{\partial {P}_{{\mathrm{\text{th}}}_{1}}}{\partial \rho }& =& -\gamma K\frac{{\gamma }_{\mathrm{\text{th}}}-1}{\gamma -1}{\rho }^{\gamma -1}& \text{(7)}\text{}\text{}\\ \frac{\partial {P}_{{\mathrm{\text{th}}}_{2}}}{\partial \rho }& =& \left(\gamma -1\right)𝜖-\left({\gamma }_{\mathrm{\text{th}}}-1\right)\frac{\gamma -{\gamma }_{1}}{\left({\gamma }_{1}-1\right)\left({\gamma }_{2}-1\right)}K{\rho }_{\mathrm{\text{nuc}}}\left({\gamma }_{1}-1\right).& \text{(8)}\text{}\text{}\end{array}$