## 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 $\Gamma =2$):