## 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

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

where

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

These expressions do not include the conversions between cgs and Cactus units that are necessary inside the code.

For more details, a recent paper to start with would be [1].

### References

[1]   H. Dimmelmeier, J. Novak, A. Font, J. M. Ibanez, E. Mueller Combining spectral and shock-capturing methods: A new numerical approach for 3D relativistic core collapse simulations Phys. Rev. D71 064023 (2005)

### 2 Parameters

 eos_gamma_supernuclear Scope: restricted REAL Description: Gamma_2 for EOS Range Default: 1.66666666666666 :

 eos_gamma_th Scope: restricted REAL Description: Gamma_th for EOS Range Default: 1.5 :

 rho_nuc Scope: restricted REAL Description: Nuclear matter density Range Default: 1.e-10 :

 eos_gamma Scope: shared from EOS_2D_POLYTROPE REAL

 eos_k Scope: shared from EOS_2D_POLYTROPE REAL

 gamma_ini Scope: shared from EOS_2D_POLYTROPE REAL

### 3 Interfaces

Implements:

eos_hybrid

Inherits:

hydrobase

eos_2d_polytrope

#### Grid Variables

##### 3.0.1 PRIVATE GROUPS
 Group Names Variable Names Details pressure_contributions compact 0 pressure_poly description Polytropic and thermal contributions to the pressure pressure_th dimensions 3 distribution DEFAULT group type GF tags Prolongation=”None” timelevels 1 variable type REAL

### 4 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinEOS/EOS_Hybrid. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

NONE

#### Scheduled Functions

CCTK_STARTUP

eos_hybrid_startup

setup the eos used by dimmelmeier et al. in core collapse

 After: eos_polytrope_startup Language: fortran Type: function

CCTK_ANALYSIS

check_poly_contrib

output the polytropic and thermal contributions

 Language: fortran Storage: pressure_contributions Triggers: pressure_contributions Type: function