## EOS_Polytrope

22/4/2002

Abstract

EOS_Polytrope

### 1 The equations

This equation provides a polytropic equation of state to thorns using the CactusEOS interface found in EOS_Base. As such it’s a fake, as EOS_Base assumes that, e.g., the pressure is a function of both density and speciﬁc internal energy. Here the pressure is just a function of the density, and is set appropriately (the speciﬁc internal energy is always ignored).

The two ﬂuid constants are $K$ (eos_k) and $\Gamma$ (eos_gamma), which default to 100 and 2 respectively. The formulas that are applied under the appropriate EOS_Base function calls are

$\begin{array}{rcll}P& =& K{\rho }^{\Gamma }& \text{(1)}\text{}\text{}\\ 𝜖& =& \frac{K{\rho }^{\Gamma -1}}{\Gamma -1}& \text{(2)}\text{}\text{}\\ \rho & =& \frac{P}{\left(\Gamma -1\right)𝜖}& \text{(3)}\text{}\text{}\\ \frac{\partial P}{\partial \rho }& =& K\Gamma {\rho }^{\Gamma -1}& \text{(4)}\text{}\text{}\\ \frac{\partial P}{\partial 𝜖}& =& 0.& \text{(5)}\text{}\text{}\end{array}$

To calculate the units of the Cactus quantities and back, remember that $G=c={M}_{\odot }=1$ in Cactus.
Here is one example how to calculate densities:

 ${\rho }_{\text{Cactus}}=\frac{{G}^{3}{M}_{\odot }^{2}}{{c}^{6}}\cdot \rho \approx 1.6167\cdot 1{0}^{-21}\frac{{\text{m}}^{3}}{\text{kg}}\cdot \rho =1.6167\cdot 1{0}^{-18}\frac{{\text{cm}}^{3}}{\text{g}}\cdot \rho$ (6)

and one example for calculating $K$ (for $\Gamma =2$):

 ${K}_{\text{Cactus}}=\frac{{c}^{4}}{{G}^{3}{M}_{\odot }^{2}}\cdot K\approx 6.8824\cdot 1{0}^{-11}\frac{{\text{m}}^{5}}{\text{kg}\cdot {\text{s}}^{2}}\cdot K=6.8824\cdot 1{0}^{-4}\frac{{\text{cm}}^{5}}{\text{g}\cdot {\text{s}}^{2}}\cdot K$ (7)

### 2 Parameters

 eos_gamma Scope: restricted REAL Description: Adiabatic Index for Ideal Fluid Range Default: 2.0 :

 eos_k Scope: restricted REAL Description: Polytropic constant Range Default: 80.0 :

 gamma_ini Scope: restricted REAL Description: Polytropic Gamma used for the initial model (e.g., by RNSID) Range Default: 2.0 :

 use_cgs Scope: restricted BOOLEAN Description: Use the CGS units Default: no

Implements:

eos_2d_polytrope

Inherits:

eos_base

EOS_Base.h

EOS_Base.inc

### 4 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinEOS/EOS_Polytrope. 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_polytrope_startup

setup the polytropic eos

 Language: fortran Type: function