RelativisticEuler::Solutions::TovStar Class Reference

A static spherically symmetric star. More...

#include <TovStar.hpp>

Classes
struct	CentralDensity
	The central density of the star. More...
struct	Coordinates
	Areal (Schwarzschild) or isotropic coordinates. More...

Public Types
using	equation_of_state_type = EquationsOfState::EquationOfState< true, 1 >
using	options = implementation defined
template<typename DataType >
using	tags = implementation defined
Public Types inherited from RelativisticEuler::AnalyticSolution< 3 >
using	tags = implementation defined

Public Member Functions
	TovStar (const TovStar &)
TovStar &	operator= (const TovStar &)
	TovStar (TovStar &&)=default
TovStar &	operator= (TovStar &&)=default
	TovStar (double central_rest_mass_density, std::unique_ptr< EquationsOfState::EquationOfState< true, 1 > > equation_of_state, const RelativisticEuler::Solutions::TovCoordinates coordinate_system=RelativisticEuler::Solutions::TovCoordinates::Schwarzschild)
auto	get_clone () const -> std::unique_ptr< evolution::initial_data::InitialData > override
template<typename DataType , typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataType, 3 > &x, const double, tmpl::list< Tags... >) const
	Retrieve a collection of variables at (x, t)
void	pup (PUP::er &) override
	NOLINTNEXTLINE(google-runtime-references)
const EquationsOfState::EquationOfState< true, 1 > &	equation_of_state () const
const RelativisticEuler::Solutions::TovSolution &	radial_solution () const
	The radial profile of the star.
virtual auto	get_clone () const -> std::unique_ptr< InitialData >=0

Static Public Attributes
static constexpr size_t	volume_dim = 3_st
static constexpr Options::String	help
Static Public Attributes inherited from RelativisticEuler::AnalyticSolution< 3 >
static constexpr size_t	volume_dim

Protected Member Functions
template<template< class, tov_detail::StarRegion > class VarsComputer, typename DataType , typename... Tags, typename... VarsComputerArgs>
tuples::TaggedTuple< Tags... >	variables_impl (const tnsr::I< DataType, 3 > &x, tmpl::list< Tags... >, VarsComputerArgs &&... vars_computer_args) const

Friends
bool	operator== (const TovStar &lhs, const TovStar &rhs)

Detailed Description

A static spherically symmetric star.

An analytic solution for a static, spherically-symmetric star found by solving the Tolman-Oppenheimer-Volkoff (TOV) equations. The equation of state is assumed to be that of a polytropic fluid.

If the spherically symmetric metric is written as

$$d{s}^2=-e^{2{\mathrm{\Phi }}_t}d{t}^2+e^{2{\mathrm{\Phi }}_r}d{r}^2+e^{2{\mathrm{\Phi }}_{\mathrm{\Omega }}}{r}^{2}d{\mathrm{\Omega }}^2$$

where $r={\delta }_{mn}{x}^m{x}^n$ is the radial coordinate and ${\mathrm{\Phi }}_t$, ${\mathrm{\Phi }}_r$, and ${\mathrm{\Phi }}_{\mathrm{\Omega }}$ are the metric potentials, then the lapse, shift, and spatial metric in Cartesian coordinates are

$$\begin{array}{rl}\alpha & =e^{{\mathrm{\Phi }}_t}\\ {\beta }^i& =0\\ {\gamma }_{ij}& ={\delta }_{ij}{e}^{2{\mathrm{\Phi }}_{\mathrm{\Omega }}}+{\delta }_{im}{\delta }_{jn}\frac{x^m{x}^n}{r^2}(e^{2{\mathrm{\Phi }}_r}-e^{2{\mathrm{\Phi }}_{\mathrm{\Omega }}})\end{array}$$

We solve the TOV equations with the method implemented in RelativisticEuler::Solutions::TovSolution. It provides the areal mass-over-radius $m(r)/r$ and the log of the specific enthalpy $\log h$. In areal (Schwarzschild) coordinates the spatial metric potentials are

$$\begin{array}{}\text{(1)}& e^{{\mathrm{\Phi }}_r}& ={(1-\frac{2m}{r})}^{-1/2}\text{(2)}& e^{{\mathrm{\Phi }}_{\mathrm{\Omega }}}& =1\end{array}$$

In isotropic coordinates the spatial metric potentials are

$$\begin{array}{}\text{(3)}& e^{2{\mathrm{\Phi }}_r}=e^{2{\mathrm{\Phi }}_{\mathrm{\Omega }}}={\psi }^4\text{,}\end{array}$$

where $\psi =\sqrt{r/\overline{r}}$ is the conformal factor, $r$ is the areal (Schwarzschild) radius and $\overline{r}$ is the isotropic radius. See RelativisticEuler::Solutions::TovSolution for details.

Warning

Isotropic coordinates should be used because the metric derivatives are smooth. Otherwise the grid will over-compensate with finite difference cells.

Member Function Documentation

get_clone()

auto RelativisticEuler::Solutions::TovStar::get_clone ( ) const -> std::unique_ptr< evolution::initial_data::InitialData >

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

help

constexpr Options::String RelativisticEuler::Solutions::TovStar::help

staticconstexpr

Initial value:

= {
      "A static, spherically-symmetric star found by solving the \n"
      "Tolman-Oppenheimer-Volkoff (TOV) equations, with a given central \n"
      "density and equation of state."}

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The documentation for this class was generated from the following file:

• src/PointwiseFunctions/AnalyticSolutions/RelativisticEuler/TovStar.hpp