EOS_Polytrope

Ian Hawke

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 {eqnarray} \label {eq:eosformulas} P & = & K \rho ^{\Gamma } \\ \epsilon & = & \frac {K \rho ^{\Gamma -1}}{\Gamma - 1} \\ \rho & = & \frac {P}{(\Gamma - 1) \epsilon } \\ \frac {\partial P}{\partial \rho } & = & K \Gamma \rho ^{\Gamma -1} \\ \frac {\partial P}{\partial \epsilon } & = & 0. \end {eqnarray}

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: \begin {equation} \rho _{\mbox {\tiny Cactus}}=\frac {G^3M_{\odot }^2}{c^6}\cdot \rho \approx 1.6167\cdot 10^{-21}\frac {\mbox {m}^3}{\mbox {kg}}\cdot \rho = 1.6167\cdot 10^{-18}\frac {\mbox {cm}^3}{\mbox {g}}\cdot \rho \end {equation} and one example for calculating \(K\) (for \(\Gamma =2\)): \begin {equation} K_{\mbox {\tiny Cactus}}=\frac {c^4}{G^3M_{\odot }^2}\cdot K \approx 6.8824\cdot 10^{-11}\frac {\mbox {m}^5}{\mbox {kg}\cdot \mbox {s}^2}\cdot K= 6.8824\cdot 10^{-4}\frac {\mbox {cm}^5}{\mbox {g}\cdot \mbox {s}^2}\cdot K \end {equation}

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

3 Interfaces

General

Implements:

eos_2d_polytrope

Inherits:

eos_base

Uses header:

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.

Storage

NONE

Scheduled Functions

CCTK_STARTUP

  eos_polytrope_startup

  setup the polytropic eos

 

 Language:fortran
  Type: function