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
template<typename DataType>
using	tags
Public Types inherited from RelativisticEuler::AnalyticSolution< 3 >
using	tags

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.
Public Member Functions inherited from evolution::initial_data::InitialData
virtual const InitialData &	unwrap () const
	Unwrap wrapper types (e.g. WithNoise, which wraps analytic initial data and adds random noise) to reach the underlying analytic solution or data.

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

\[ds^2 = - e^{2 \Phi_t} dt^2 + e^{2 \Phi_r} dr^2 + e^{2 \Phi_\Omega} r^2 d\Omega^2 \]

where \(r = \delta_{mn} x^m x^n\) is the radial coordinate and \(\Phi_t\), \(\Phi_r\), and \(\Phi_\Omega\) are the metric potentials, then the lapse, shift, and spatial metric in Cartesian coordinates are

\begin{align*}\alpha &= e^{\Phi_t} \\ \beta^i &= 0 \\ \gamma_{ij} &= \delta_{ij} e^{2 \Phi_\Omega} + \delta_{im} \delta_{jn} \frac{x^m x^n}{r^2} \left( e^{2 \Phi_r} - e^{2 \Phi_\Omega} \right) \end{align*}

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{align}e^{\Phi_r} &= \left(1 - \frac{2m}{r}\right)^{-1/2} \\ e^{\Phi_\Omega} &= 1 \end{align}

In isotropic coordinates the spatial metric potentials are

\begin{equation}e^{2\Phi_r} = e^{2\Phi_\Omega} = \psi^4 \text{,} \end{equation}

where \(\psi = \sqrt{r / \bar{r}}\) is the conformal factor, \(r\) is the areal (Schwarzschild) radius and \(\bar{r}\) 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 Typedef Documentation

options

using RelativisticEuler::Solutions::TovStar::options

Initial value:

 
      tmpl::list<CentralDensity,
                 hydro::OptionTags::InitialDataEquationOfState<true, 1>,
                 Coordinates>

tags

template<typename DataType>

using RelativisticEuler::Solutions::TovStar::tags

Initial value:

 tmpl::list_difference<
      typename tov_detail::TovVariablesCache<DataType>::tags_list,
      tmpl::list<
          
          tov_detail::Tags::MassOverRadius<DataType>,
          tov_detail::Tags::LogSpecificEnthalpy<DataType>,
          tov_detail::Tags::ConformalFactor<DataType>,
          tov_detail::Tags::DrConformalFactor<DataType>,
          tov_detail::Tags::ArealRadius<DataType>,
          tov_detail::Tags::DrArealRadius<DataType>,
          tov_detail::Tags::DrPressure<DataType>,
          tov_detail::Tags::MetricTimePotential<DataType>,
          tov_detail::Tags::DrMetricTimePotential<DataType>,
          tov_detail::Tags::MetricRadialPotential<DataType>,
          tov_detail::Tags::DrMetricRadialPotential<DataType>,
          tov_detail::Tags::MetricAngularPotential<DataType>,
          tov_detail::Tags::DrMetricAngularPotential<DataType>>>

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

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."}

---

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

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