A solution obtained by reading in rotating neutron star initial data from the RotNS code based on [47] and [48]. More...

#include <RotatingStar.hpp>

Classes
struct	PolytropicConstant
	The polytropic constant of the fluid. More...
struct	RotNsFilename
	The path to the RotNS data file. More...

Public Types
using	options
Public Types inherited from RelativisticEuler::AnalyticSolution< 3 >
using	tags

Public Member Functions
	RotatingStar (const RotatingStar &)
RotatingStar &	operator= (const RotatingStar &)
	RotatingStar (RotatingStar &&)=default
RotatingStar &	operator= (RotatingStar &&)=default
	RotatingStar (std::string rot_ns_filename, std::unique_ptr< EquationsOfState::EquationOfState< true, 1 > > equation_of_state)
	RotatingStar (std::string rot_ns_filename, double polytropic_constant)
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 &p) override
const EquationsOfState::EquationOfState< true, 1 > &	equation_of_state () const
double	equatorial_radius () const
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.
Public Member Functions inherited from hydro::TemperatureInitialization< RotatingStar >
auto	variables (const tnsr::I< DataType, Dim > &x, tmpl::list< hydro::Tags::Temperature< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::Temperature< DataType > >

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

Protected Types
template<typename DataType>
using	DerivLapse
template<typename DataType>
using	DerivShift
template<typename DataType>
using	DerivSpatialMetric

Protected Member Functions
template<typename DataType>
void	interpolate_vars_if_necessary (gsl::not_null< IntermediateVariables< DataType > * > vars) const
template<typename DataType>
void	interpolate_deriv_vars_if_necessary (gsl::not_null< IntermediateVariables< DataType > * > vars) const
template<typename DataType>
Scalar< DataType >	lapse (const DataType &gamma, const DataType &rho) const
template<typename DataType>
tnsr::I< DataType, 3, Frame::Inertial >	shift (const DataType &omega, const DataType &phi, const DataType &radius, const DataType &sin_theta) const
template<typename DataType>
tnsr::ii< DataType, 3, Frame::Inertial >	spatial_metric (const DataType &gamma, const DataType &rho, const DataType &alpha, const DataType &phi) const
template<typename DataType>
tnsr::II< DataType, 3, Frame::Inertial >	inverse_spatial_metric (const DataType &gamma, const DataType &rho, const DataType &alpha, const DataType &phi) const
template<typename DataType>
Scalar< DataType >	sqrt_det_spatial_metric (const DataType &gamma, const DataType &rho, const DataType &alpha) const
template<typename DataType>
auto	make_metric_data (size_t num_points) const -> typename IntermediateVariables< DataType >::MetricData
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::RestMassDensity< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::RestMassDensity< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::ElectronFraction< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::ElectronFraction< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::SpecificEnthalpy< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::SpecificEnthalpy< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::Temperature< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::Temperature< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::Pressure< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::Pressure< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::SpecificInternalEnergy< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::SpecificInternalEnergy< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::SpatialVelocity< DataType, 3 > >) const -> tuples::TaggedTuple< hydro::Tags::SpatialVelocity< DataType, 3 > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::LorentzFactor< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::LorentzFactor< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::MagneticField< DataType, 3 > >) const -> tuples::TaggedTuple< hydro::Tags::MagneticField< DataType, 3 > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::DivergenceCleaningField< DataType > >) const -> tuples::TaggedTuple< hydro::Tags::DivergenceCleaningField< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< gr::Tags::Lapse< DataType > >) const -> tuples::TaggedTuple< gr::Tags::Lapse< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< gr::Tags::Shift< DataType, 3 > >) const -> tuples::TaggedTuple< gr::Tags::Shift< DataType, 3 > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< gr::Tags::SpatialMetric< DataType, 3 > >) const -> tuples::TaggedTuple< gr::Tags::SpatialMetric< DataType, 3 > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< gr::Tags::SqrtDetSpatialMetric< DataType > >) const -> tuples::TaggedTuple< gr::Tags::SqrtDetSpatialMetric< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< gr::Tags::InverseSpatialMetric< DataType, 3 > >) const -> tuples::TaggedTuple< gr::Tags::InverseSpatialMetric< DataType, 3 > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list<::Tags::dt< gr::Tags::Lapse< DataType > > >) const -> tuples::TaggedTuple<::Tags::dt< gr::Tags::Lapse< DataType > > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< DerivLapse< DataType > >) const -> tuples::TaggedTuple< DerivLapse< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list<::Tags::dt< gr::Tags::Shift< DataType, 3 > > >) const -> tuples::TaggedTuple<::Tags::dt< gr::Tags::Shift< DataType, 3 > > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< DerivShift< DataType > >) const -> tuples::TaggedTuple< DerivShift< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list<::Tags::dt< gr::Tags::SpatialMetric< DataType, 3 > > >) const -> tuples::TaggedTuple<::Tags::dt< gr::Tags::SpatialMetric< DataType, 3 > > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< DerivSpatialMetric< DataType > >) const -> tuples::TaggedTuple< DerivSpatialMetric< DataType > >
template<typename DataType>
auto	variables (gsl::not_null< IntermediateVariables< DataType > * > vars, const tnsr::I< DataType, 3 > &x, tmpl::list< gr::Tags::ExtrinsicCurvature< DataType, 3 > >) const -> tuples::TaggedTuple< gr::Tags::ExtrinsicCurvature< DataType, 3 > >

Protected Attributes
std::string	rot_ns_filename_ {}
detail::CstSolution	cst_solution_ {}
double	polytropic_constant_ = std::numeric_limits<double>::signaling_NaN()
double	polytropic_exponent_ = std::numeric_limits<double>::signaling_NaN()
bool	is_polytrope_ {}
std::unique_ptr< EquationsOfState::EquationOfState< true, 1 > >	equation_of_state_

Static Protected Attributes
static constexpr double	atmosphere_floor_ = 1.e-50

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

Detailed Description

A solution obtained by reading in rotating neutron star initial data from the RotNS code based on [47] and [48].

The code that generates the initial data is part of a private SXS repository called RotNS.

The metric in spherical coordinates is given by [47]

\begin{align} ds^2=-e^{\gamma+\rho}dt^2+e^{2\alpha}(dr^2+r^2d\theta^2) +e^{\gamma-\rho}r^2\sin^2(\theta)(d\phi-\omega dt)^2. \end{align}

We use rotation about the \(z\)-axis. That is,

\begin{align} g_{tt} &=-e^{\gamma+\rho} + e^{\gamma-\rho}r^2\sin^2(\theta)\omega^2 \\ g_{rr} &=e^{2\alpha} \\ g_{\theta\theta} &=e^{2\alpha}r^2 \\ g_{\phi\phi} &=e^{\gamma-\rho}r^2\sin^2(\theta) \\ g_{t\phi} &=-e^{\gamma-\rho}r^2\sin^2(\theta)\omega. \end{align}

We can transform from spherical to Cartesian coordinates using

\begin{align} \label{eq:Jacobian} \frac{\partial (r,\theta,\phi)}{\partial (x,y,z)}= \begin{pmatrix} \cos(\phi) \sin(\theta) & \sin(\theta)\sin(\phi) & \cos(\theta) \\ \tfrac{\cos(\phi)\cos(\theta)}{r} & \tfrac{\sin(\phi)\cos(\theta)}{r} & -\tfrac{\sin(\theta)}{r} \\ -\tfrac{\sin(\phi)}{r\sin(\theta)} & \tfrac{\cos(\phi)}{r\sin(\theta)} & 0 \end{pmatrix} \end{align}

and

\begin{align} \frac{\partial (x,y,z)}{\partial (r,\theta,\phi)}= \begin{pmatrix} \cos(\phi) \sin(\theta) & r\cos(\phi)\cos(\theta) & -r\sin(\theta)\sin(\phi) \\ \sin(\theta)\sin(\phi) & r\sin(\phi)\cos(\theta) & r\cos(\phi)\sin(\theta) \\ \cos(\theta) & -r\sin(\theta) & 0 \end{pmatrix} \end{align}

We denote the lapse as \(N\) since \(\alpha\) is already being used,

\begin{align} N = e^{(\gamma+\rho)/2}. \end{align}

The shift is

\begin{align} \beta^\phi =& -\omega \end{align}

so

\begin{align} \beta^x &= \partial_\phi x \beta^\phi = \sin(\phi)\omega r \sin(\theta), \\ \beta^y &= \partial_\phi y \beta^\phi = -\cos(\phi)\omega r \sin(\theta), \\ \beta^z &= 0. \end{align}

The spatial metric is

\begin{align} \gamma_{xx} &= \sin^2(\theta)\cos^2(\phi) e^{(2 \alpha)} + \cos^2(\theta)\cos^2(\phi) e^{(2 \alpha)} + \sin^2(\phi) e^{(\gamma-\rho)} \notag \\ &=\cos^2(\phi)e^{2\alpha} + \sin^2(\phi) e^{(\gamma-\rho)} \\ \gamma_{xy} &= \sin^2(\theta)\cos(\phi)\sin(\phi) e^{(2 \alpha)} + \cos^2(\theta)\cos(\phi)\sin(\phi) e^{(2 \alpha)} - \sin(\phi)\cos(\phi) e^{(\gamma-\rho)} \notag \\ &=\cos(\phi)\sin(\phi)e^{2\alpha} - \sin(\phi)\cos(\phi) e^{(\gamma-\rho)} \\ \gamma_{yy} &= \sin^2(\theta)\sin^2(\phi) e^{(2 \alpha)} + \cos^2(\theta)\sin^2(\phi) e^{(2 \alpha)} + \cos^2(\phi) e^{(\gamma-\rho)} \notag \\ &=\sin^2(\phi)e^{2\alpha} + \cos^2(\phi) e^{(\gamma-\rho)} \\ \gamma_{xz} &= \sin(\theta)\cos(\phi)\cos(\theta) e^{2\alpha} -\cos(\theta)\cos(\phi)\sin(\theta)e^{2\alpha} = 0 \\ \gamma_{yz} &= \sin(\theta)\sin(\phi)\cos(\theta) e^{2\alpha} -\cos(\theta)\sin(\phi)\sin(\theta)e^{2\alpha} = 0 \\ \gamma_{zz} &=\cos^2(\theta)e^{2\alpha} + \sin^2(\theta) e^{2\alpha} = e^{2\alpha} \end{align}

and its determinant is

\begin{align} \gamma = e^{4\alpha + (\gamma-\rho)} = e^{4\alpha}e^{(\gamma-\rho)}. \end{align}

At \(r=0\) we have \(2\alpha=\gamma-\rho\) and so the \(\gamma_{xx}=\gamma_{yy}=\gamma_{zz} = e^{2\alpha}\) and all other components are zero. The inverse spatial metric is given by

\begin{align} \gamma^{xx} &= \frac{\gamma_{yy}}{e^{2\alpha}e^{(\gamma-\rho)}} = \left[\sin^2(\phi)e^{2\alpha} + \cos^2(\phi) e^{(\gamma-\rho)}\right] e^{-2\alpha} e^{-(\gamma-\rho)} \notag \\ &=\sin^2(\phi)e^{-(\gamma-\rho)} + \cos^2(\phi) e^{-2\alpha} \\ \gamma^{yy} &= \frac{\gamma_{xx}}{e^{2\alpha}e^{(\gamma-\rho)}} = \left[\cos^2(\phi)e^{2\alpha} + \sin^2(\phi) e^{(\gamma-\rho)}\right] e^{-2\alpha} e^{-(\gamma-\rho)} \notag \\ &=\cos^2(\phi) e^{-(\gamma-\rho)} + \sin^2(\phi) e^{-2\alpha} \\ \gamma^{xy} &=\frac{-\gamma_{xy}}{e^{2\alpha}e^{(\gamma-\rho)}} = -\left[\cos(\phi)\sin(\phi)e^{2\alpha} - \sin(\phi)\cos(\phi) e^{(\gamma-\rho)}\right] e^{-2\alpha} e^{-(\gamma-\rho)} \notag \\ &=-\cos(\phi)\sin(\phi)e^{-(\gamma-\rho)} - \sin(\phi)\cos(\phi)e^{-2\alpha} \notag \\ &=\cos(\phi)\sin(\phi) \left[e^{-2\alpha} - e^{-(\gamma-\rho)}\right] \\ \gamma^{xz} &= 0 \\ \gamma^{yz} &= 0 \\ \gamma^{zz} &= e^{-2\alpha}. \end{align}

The 4-velocity in spherical coordinates is given by

\begin{align} u^{\bar{a}}=\frac{e^{-(\rho+\gamma)/2}}{\sqrt{1-v^2}} \left[1,0,0,\Omega\right], \end{align}

where

\begin{align} v=(\Omega-\omega)r\sin(\theta)e^{-\rho}. \end{align}

Transforming to Cartesian coordinates we have

\begin{align} u^t &=\frac{e^{-(\rho+\gamma)/2}}{\sqrt{1-v^2}} \\ u^x &=\partial_\phi x u^\phi = -r\sin(\theta)\sin(\phi) u^t\Omega \\ u^y &=\partial_\phi y u^\phi = r\sin(\theta)\cos(\phi) u^t\Omega \\ u^z &= 0. \end{align}

The Lorentz factor is given by

\begin{align} W &=Nu^t=e^{(\gamma+\rho)/2}\frac{e^{-(\rho+\gamma)/2}}{\sqrt{1-v^2}} \notag \\ &=\frac{1}{\sqrt{1-v^2}}. \end{align}

Using

\begin{align} v^i = \frac{1}{N}\left(\frac{u^i}{u^t} + \beta^i\right) \end{align}

we get

\begin{align} v^x &= -e^{-(\gamma+\rho)/2}r\sin(\theta)\sin(\phi)(\Omega-\omega) =-e^{-(\gamma-\rho)/2}\sin(\phi) v\\ v^y &= e^{-(\gamma+\rho)/2}r\sin(\theta)\cos(\phi)(\Omega-\omega) = e^{-(\gamma-\rho)/2}\cos(\phi)v \\ v^z&=0. \end{align}

Lowering with the spatial metric we get

\begin{align} v_x &=\gamma_{xx} v^x + \gamma_{xy} v^y \notag \\ &=-\left[\cos^2(\phi)e^{2\alpha}+\sin^2(\phi)e^{\gamma-\rho}\right] e^{-(\gamma-\rho)/2}\sin(\phi) v \notag \\ &+\left[\cos(\phi)\sin(\phi)e^{2\alpha} - \sin(\phi)\cos(\phi)e^{\gamma-\rho}\right] e^{-(\gamma-\rho)/2}\cos(\phi)v \notag \\ &=-e^{-(\gamma-\rho)/2}v\sin(\phi)e^{\gamma-\rho} \left[\sin^2(\phi)+\cos^2(\phi)\right] \notag \\ &=-e^{(\gamma-\rho)/2}v\sin(\phi) \\ v_y &=\gamma_{yx} v^x + \gamma_{yy} v^y \notag \\ &=-\left[\cos(\phi)\sin(\phi)e^{2\alpha} - \sin(\phi)\cos(\phi) e^{(\gamma-\rho)}\right] e^{-(\gamma-\rho)/2}\sin(\phi) v \notag \\ &+\left[\sin^2(\phi)e^{2\alpha} + \cos^2(\phi) e^{(\gamma-\rho)}\right] e^{-(\gamma-\rho)/2}\cos(\phi)v \notag \\ &=e^{(\gamma-\rho)/2}v\cos(\phi) \\ v_z &= 0. \end{align}

This is consistent with the Lorentz factor read off from \(u^t\) since \(v^iv_i=v^2\). For completeness, \(u_i=Wv_i\) so

\begin{align} u_x &=-\frac{e^{(\gamma-\rho)/2}v\sin(\phi)}{\sqrt{1-v^2}} \\ u_y &=\frac{e^{(\gamma-\rho)/2}v\cos(\phi)}{\sqrt{1-v^2}} \\ u_z&=0. \end{align}

Warning: Near (within 1e-2) \(r=0\) the numerical errors from interpolation and computing the metric derivatives by finite difference no longer cancel out and so the tilde_s time derivative only vanishes to roughly 1e-8 rather than machine precision. Computing the Cartesian derivatives from analytic differentiation of the radial and angular polynomial fits might improve the situation but is a decent about of work to implement.

Member Typedef Documentation

◆ DerivLapse

template<typename DataType>

using RelativisticEuler::Solutions::RotatingStar::DerivLapse

protected

Initial value:

::Tags::deriv<gr::Tags::Lapse<DataType>, tmpl::size_t<3>,

Frame::Inertial>

Frame::Inertial

Definition IndexType.hpp:46

Tags::deriv

Prefix indicating spatial derivatives.

Definition Prefixes.hpp:47

◆ DerivShift

template<typename DataType>

using RelativisticEuler::Solutions::RotatingStar::DerivShift

protected

Initial value:

::Tags::deriv<gr::Tags::Shift<DataType, 3>,

tmpl::size_t<3>, Frame::Inertial>

◆ DerivSpatialMetric

template<typename DataType>

using RelativisticEuler::Solutions::RotatingStar::DerivSpatialMetric

protected

Initial value:

::Tags::deriv<gr::Tags::SpatialMetric<DataType, 3>,

tmpl::size_t<3>, Frame::Inertial>

◆ options

using RelativisticEuler::Solutions::RotatingStar::options

Initial value:

 tmpl::list<Options::Alternatives<
      tmpl::list<RotNsFilename, PolytropicConstant>,
      tmpl::list<RotNsFilename,
                 hydro::OptionTags::InitialDataEquationOfState<true, 1>>>>

Member Function Documentation

◆ get_clone()

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

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

◆ help

Options::String RelativisticEuler::Solutions::RotatingStar::help

staticconstexpr

Initial value:

= {
      "Rotating neutron star initial data solved by the RotNS solver. The data "
      "is read in from disk. If RotNS used a polytropic equation of state, use "
      "the PolytropicConstant option, so that the polytropic index may be read "
      "directly from the RotNS output. Otherwise, set the equation of state to "
      "whichever one you would like to use to load the initial data. Note that "
      "if a polytrope is put into the EquationOfState option rather than using "
      "the PolytropicConstant option, the generic unit conversion will be used "
      "instead of the specific polytrope one."}

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

src/PointwiseFunctions/AnalyticSolutions/RelativisticEuler/RotatingStar.hpp

Classes

Public Types

Public Member Functions

Static Public Attributes

Protected Types

Protected Member Functions

Protected Attributes

Static Protected Attributes

Friends

Detailed Description

Member Typedef Documentation

◆ DerivLapse

◆ DerivShift

◆ DerivSpatialMetric

◆ options

Member Function Documentation

◆ get_clone()

Member Data Documentation

◆ help