Base class for axisymmetric stationary compact stars in Quasi-Isotropic coordinates (under development). More...

#include <compobj.h>

Inheritance diagram for Lorene::Star_QI:

Public Member Functions
	Star_QI (Map &mp_i)
	Standard constructor.

	Star_QI (const Star_QI &)
	Copy constructor.

	Star_QI (Map &mp_i, FILE *fich)
	Constructor from a file (see `sauve(FILE*)` ).

virtual	~Star_QI ()
	Destructor.

void	operator= (const Star_QI &)
	Assignment to another `Star_QI`.

const Scalar &	get_logn () const
	Returns the logarithm of the lapse N.

const Scalar &	get_tnphi () const
	Returns the component $\tilde N^\varphi = N^\varphi r\sin\theta$ of the shift vector.

const Scalar &	get_nuf () const
	Returns the part of the Metric potential $\nu = \ln N$ = `logn` generated by the matter terms.

const Scalar &	get_nuq () const
	Returns the Part of the Metric potential $\nu = \ln N$ = `logn` generated by the quadratic terms.

const Scalar &	get_dzeta () const
	Returns the Metric potential $\zeta = \ln(AN)$ .

const Scalar &	get_tggg () const
	Returns the Metric potential $\tilde G = (NB-1) r\sin\theta$ .

const Vector &	get_w_shift () const
	Returns the vector used in the decomposition of `shift` , following Shibata's prescription [Prog.

const Scalar &	get_khi_shift () const
	Returns the scalar $\chi$ used in the decomposition of `shift` following Shibata's prescription [Prog.

virtual void	sauve (FILE *) const
	Save in a file.

virtual double	mass_g () const
	Gravitational mass.

virtual double	angu_mom () const
	Angular momentum.

virtual double	grv2 () const
	Error on the virial identity GRV2.

virtual double	grv3 (ostream *ost=0x0) const
	Error on the virial identity GRV3.

virtual double	mom_quad () const
	Quadrupole moment.

void	update_metric ()
	Computes metric coefficients from known potentials.

void	fait_shift ()
	Computes `shift` from `w_shift` and `khi_shift` according to Shibata's prescription [Prog.

void	fait_nphi ()
	Computes `tnphi` and `nphi` from the Cartesian components of the shift, stored in `shift` .

const Scalar &	get_bbb () const
	Returns the metric factor B.

const Scalar &	get_a_car () const
	Returns the square of the metric factor A.

const Scalar &	get_b_car () const
	Returns the square of the metric factor B.

const Scalar &	get_nphi () const
	Returns the metric coefficient $N^\varphi$ .

const Scalar &	get_ak_car () const
	Returns the scalar $A^2 K_{ij} K^{ij}$ .

void	gyoto_data (const char *file_name) const
	Save in a file for GYOTO.

virtual double	r_isco (int lmin, ostream *ost=0x0) const
	Coordinate r of the innermost stable circular orbit (ISCO).

virtual double	f_isco (int lmin) const
	Orbital frequency at the innermost stable circular orbit (ISCO).

virtual double	espec_isco (int lmin) const
	Energy of a particle at the ISCO.

virtual double	lspec_isco (int lmin) const
	Angular momentum of a particle at the ISCO.

virtual double	r_mb (int lmin, ostream *ost=0x0) const
	Coordinate r of the marginally bound circular orbit (R_mb).

virtual void	extrinsic_curvature ()
	Computes the extrinsic curvature and `ak_car` from `nphi` , `nn` and `b_car` .

Map &	set_mp ()
	Read/write of the mapping.

const Map &	get_mp () const
	Returns the mapping.

const Scalar &	get_nn () const
	Returns the lapse function N .

const Vector &	get_beta () const
	Returns the shift vector $\beta^i$ .

const Metric &	get_gamma () const
	Returns the 3-metric $\gamma_{ij}$ .

const Scalar &	get_ener_euler () const
	Returns the total energy density E in the Eulerian frame.

const Vector &	get_mom_euler () const
	Returns the total 3-momentum density in the Eulerian frame.

const Sym_tensor &	get_stress_euler () const
	Returns the stress tensor $S_{ij}$ with respect to the Eulerian observer.

const Sym_tensor &	get_kk () const
	Returns the extrinsic curvature tensor $K_{ij}$ .

virtual double	adm_mass () const
	ADM mass (computed as a surface integral at spatial infinity)

Static Public Member Functions
static double	lambda_grv2 (const Scalar &sou_m, const Scalar &sou_q)
	Computes the coefficient $\lambda$ which ensures that the GRV2 virial identity is satisfied.

Protected Member Functions
virtual void	del_deriv () const
	Deletes all the derived quantities.

virtual void	set_der_0x0 () const
	Sets to `0x0` all the pointers on derived quantities.

virtual ostream &	operator>> (ostream &) const
	Operator >> (virtual function called by the operator <<).

Protected Attributes
Scalar	logn
	Logarithm of the lapse N .

Scalar	tnphi
	Component $\tilde N^\varphi = N^\varphi r\sin\theta$ of the shift vector.

Scalar	nuf
	Part of the Metric potential $\nu = \ln N$ = `logn` generated by the matter terms.

Scalar	nuq
	Part of the Metric potential $\nu = \ln N$ = `logn` generated by the quadratic terms.

Scalar	dzeta
	Metric potential $\zeta = \ln(AN)$ .

Scalar	tggg
	Metric potential $\tilde G = (NB-1) r\sin\theta$ .

Vector	w_shift
	Vector used in the decomposition of `shift` , following Shibata's prescription [Prog.

Scalar	khi_shift
	Scalar $\chi$ used in the decomposition of `shift` , following Shibata's prescription [Prog.

Scalar	ssjm1_nuf
	Effective source at the previous step for the resolution of the Poisson equation for `nuf` by means of `Map_et::poisson` .

Scalar	ssjm1_nuq
	Effective source at the previous step for the resolution of the Poisson equation for `nuq` by means of `Map_et::poisson` .

Scalar	ssjm1_dzeta
	Effective source at the previous step for the resolution of the Poisson equation for `dzeta` .

Scalar	ssjm1_tggg
	Effective source at the previous step for the resolution of the Poisson equation for `tggg` .

Scalar	ssjm1_khi
	Effective source at the previous step for the resolution of the Poisson equation for the scalar $\chi$ by means of `Map_et::poisson` .

Vector	ssjm1_wshift
	Effective source at the previous step for the resolution of the vector Poisson equation for .

double *	p_grv2
	Error on the virial identity GRV2.

double *	p_grv3
	Error on the virial identity GRV3.

double *	p_mom_quad
	Quadrupole moment

double *	p_mass_g
	Gravitational mass (ADM mass as a volume integral)

Scalar	a_car
	Square of the metric factor A.

Scalar	bbb
	Metric factor B.

Scalar	b_car
	Square of the metric factor B.

Scalar	nphi
	Metric coefficient $N^\varphi$ .

Scalar	ak_car
	Scalar $A^2 K_{ij} K^{ij}$ .

double *	p_angu_mom
	Angular momentum.

double *	p_r_isco
	Coordinate r of the ISCO.

double *	p_f_isco
	Orbital frequency of the ISCO.

double *	p_espec_isco
	Specific energy of a particle at the ISCO.

double *	p_lspec_isco
	Specific angular momentum of a particle at the ISCO.

double *	p_r_mb
	Coordinate r of the marginally bound orbit.

Map &	mp
	Mapping describing the coordinate system (r,theta,phi)

Scalar	nn
	Lapse function N .

Vector	beta
	Shift vector $\beta^i$ .

Metric	gamma
	3-metric $\gamma_{ij}$

Scalar	ener_euler
	Total energy density E in the Eulerian frame.

Vector	mom_euler
	Total 3-momentum density in the Eulerian frame.

Sym_tensor	stress_euler
	Stress tensor $S_{ij}$ with respect to the Eulerian observer.

Sym_tensor	kk
	Extrinsic curvature tensor $K_{ij}$

double *	p_adm_mass
	ADM mass.

Detailed Description

Base class for axisymmetric stationary compact stars in Quasi-Isotropic coordinates (under development).

()

The time slice $t=\mathrm{const}$ has the topology of $R^3$ and the metric is expressed in Quasi-Isotropic (QI) coordinates :

$ds^2 = - N^2 dt^2 + A^2 (dr^2 + r^2 d\theta^2) + B^2 r^2 \sin^2\theta (d\varphi - N^\varphi dt)^2$

Definition at line 487 of file compobj.h.

Constructor & Destructor Documentation

◆ Star_QI() [1/3]

Lorene::Star_QI::Star_QI ( Map & mp_i )

Standard constructor.

Parameters

mp_i	Mapping on which the star is contructed

Definition at line 68 of file star_QI.C.

References Lorene::Compobj::beta, dzeta, khi_shift, logn, nuf, nuq, set_der_0x0(), Lorene::Tensor::set_etat_zero(), ssjm1_dzeta, ssjm1_khi, ssjm1_nuf, ssjm1_nuq, ssjm1_tggg, ssjm1_wshift, tggg, tnphi, and w_shift.

◆ Star_QI() [2/3]

Lorene::Star_QI::Star_QI ( const Star_QI & st )

Copy constructor.

Definition at line 113 of file star_QI.C.

References set_der_0x0().

◆ Star_QI() [3/3]

Lorene::Star_QI::Star_QI	(	Map &	mp_i,
		FILE *	fich
	)

Constructor from a file (see sauve(FILE*) ).

Parameters

mp_i	Mapping on which the star is constructed
fich	input file (must have been created by the function `Star_QI::sauve` )

Definition at line 137 of file star_QI.C.

References dzeta, fait_nphi(), fait_shift(), Lorene::Map::get_bvect_cart(), Lorene::Map::get_mg(), khi_shift, logn, Lorene::Compobj::mp, nuf, nuq, set_der_0x0(), ssjm1_dzeta, ssjm1_khi, ssjm1_nuf, ssjm1_nuq, ssjm1_tggg, ssjm1_wshift, tggg, update_metric(), and w_shift.

◆ ~Star_QI()

Lorene::Star_QI::~Star_QI ( )

virtual

Destructor.

Definition at line 211 of file star_QI.C.

References del_deriv().

Member Function Documentation

◆ adm_mass()

double Lorene::Compobj::adm_mass ( ) const

virtualinherited

ADM mass (computed as a surface integral at spatial infinity)

Definition at line 310 of file compobj.C.

References Lorene::Metric::cov(), Lorene::Tensor::derive_con(), Lorene::Tensor_sym::derive_con(), Lorene::Vector::flux(), Lorene::Compobj::gamma, Lorene::Tensor::get_triad(), Lorene::Compobj::mp, Lorene::Compobj::p_adm_mass, Lorene::Tensor::trace(), and Lorene::Tensor::up().

◆ angu_mom()

double Lorene::Star_QI::angu_mom ( ) const

virtual

Angular momentum.

Reimplemented from Lorene::Compobj_QI.

Definition at line 96 of file star_QI_global.C.

References Lorene::Compobj_QI::a_car, Lorene::Compobj_QI::b_car, Lorene::Compobj_QI::bbb, Lorene::Map::get_bvect_spher(), Lorene::Scalar::get_spectral_va(), Lorene::Tensor::get_triad(), Lorene::Scalar::integrale(), Lorene::Compobj::mom_euler, Lorene::Compobj::mp, Lorene::Scalar::mult_r(), Lorene::Compobj_QI::p_angu_mom, and Lorene::Scalar::set_spectral_va().

◆ del_deriv()

void Lorene::Star_QI::del_deriv ( ) const

protectedvirtual

Deletes all the derived quantities.

Reimplemented from Lorene::Compobj_QI.

Reimplemented in Lorene::Boson_star.

Definition at line 222 of file star_QI.C.

References Lorene::Compobj_QI::del_deriv(), p_grv2, p_grv3, p_mass_g, p_mom_quad, and set_der_0x0().

◆ espec_isco()

double Lorene::Compobj_QI::espec_isco ( int lmin ) const

virtualinherited

Energy of a particle at the ISCO.

Definition at line 320 of file compobj_QI_global.C.

References Lorene::Compobj_QI::p_espec_isco, and Lorene::Compobj_QI::r_isco().

◆ extrinsic_curvature()

void Lorene::Compobj_QI::extrinsic_curvature ( )

virtualinherited

Computes the extrinsic curvature and ak_car from nphi , nn and b_car .

Reimplemented from Lorene::Compobj.

Reimplemented in Lorene::AltBH_QI.

Definition at line 339 of file compobj_QI.C.

References Lorene::Compobj_QI::ak_car, Lorene::Compobj_QI::b_car, Lorene::Compobj_QI::del_deriv(), Lorene::Scalar::dsdr(), Lorene::Scalar::dsdt(), Lorene::Compobj::extrinsic_curvature(), Lorene::Map::get_mg(), Lorene::Scalar::inc_dzpuis(), Lorene::Compobj::kk, Lorene::Compobj::mp, Lorene::Scalar::mult_rsint(), Lorene::Scalar::mult_sint(), Lorene::Compobj::nn, Lorene::Compobj_QI::nphi, and Lorene::Tensor::set().

◆ f_isco()

double Lorene::Compobj_QI::f_isco ( int lmin ) const

virtualinherited

Orbital frequency at the innermost stable circular orbit (ISCO).

Definition at line 286 of file compobj_QI_global.C.

References Lorene::Compobj_QI::p_f_isco, and Lorene::Compobj_QI::r_isco().

◆ fait_nphi()

void Lorene::Star_QI::fait_nphi ( )

Computes tnphi and nphi from the Cartesian components of the shift, stored in shift .

Definition at line 386 of file star_QI.C.

References Lorene::Compobj::beta, Lorene::Map::comp_p_from_cartesian(), Lorene::Scalar::div_rsint(), Lorene::Compobj::mp, Lorene::Compobj_QI::nphi, and tnphi.

◆ fait_shift()

void Lorene::Star_QI::fait_shift ( )

Computes shift from w_shift and khi_shift according to Shibata's prescription [Prog.

Theor. Phys. 101 , 1199 (1999)] :

$N^i = {7\over 8} W^i - {1\over 8} \left(\nabla^i\chi+\nabla^iW^kx_k\right)$

Definition at line 340 of file star_QI.C.

References Lorene::Compobj::beta, Lorene::contract(), Lorene::Map::cosp, Lorene::Map::cost, Lorene::Tensor::dec_dzpuis(), Lorene::Scalar::derive_con(), Lorene::Tensor::derive_con(), Lorene::Scalar::domain(), Lorene::Map::flat_met_cart(), Lorene::Map::get_bvect_cart(), Lorene::Map::get_mg(), Lorene::Mg3d::get_nzone(), khi_shift, Lorene::Compobj::mp, Lorene::Vector::set(), Lorene::Scalar::set_domain(), Lorene::Scalar::set_dzpuis(), Lorene::Map::sinp, Lorene::Map::sint, Lorene::Vector::std_spectral_base(), w_shift, Lorene::Map::x, Lorene::Map::y, and Lorene::Map::z.

◆ get_a_car()

const Scalar & Lorene::Compobj_QI::get_a_car ( ) const

inlineinherited

Returns the square of the metric factor A.

Definition at line 378 of file compobj.h.

References Lorene::Compobj_QI::a_car.

◆ get_ak_car()

const Scalar & Lorene::Compobj_QI::get_ak_car ( ) const

inlineinherited

Returns the scalar $A^2 K_{ij} K^{ij}$ .

For axisymmetric stars, this quantity is related to the derivatives of $N^\varphi$ by

$A^2 K_{ij} K^{ij} = {B^2 \over 2 N^2} \, r^2\sin^2\theta \, \left[ \left( {\partial N^\varphi \over \partial r} \right) ^2 + {1\over r^2} \left( {\partial N^\varphi \over \partial \theta} \right) ^2 \right] \ .$

In particular it is related to the quantities $k_1$ and $k_2$ introduced by Eqs. (3.7) and (3.8) of Bonazzola et al. Astron. Astrophys. 278 , 421 (1993) by

$A^2 K_{ij} K^{ij} = 2 A^2 (k_1^2 + k_2^2) \ .$

Definition at line 404 of file compobj.h.

References Lorene::Compobj_QI::ak_car.

◆ get_b_car()

const Scalar & Lorene::Compobj_QI::get_b_car ( ) const

inlineinherited

Returns the square of the metric factor B.

Definition at line 381 of file compobj.h.

References Lorene::Compobj_QI::b_car.

◆ get_bbb()

const Scalar & Lorene::Compobj_QI::get_bbb ( ) const

inlineinherited

Returns the metric factor B.

Definition at line 375 of file compobj.h.

References Lorene::Compobj_QI::bbb.

◆ get_beta()

const Vector & Lorene::Compobj::get_beta ( ) const

inlineinherited

Returns the shift vector $\beta^i$ .

Definition at line 213 of file compobj.h.

References Lorene::Compobj::beta.

◆ get_dzeta()

const Scalar & Lorene::Star_QI::get_dzeta ( ) const

inline

Returns the Metric potential $\zeta = \ln(AN)$ .

Definition at line 655 of file compobj.h.

References dzeta.

◆ get_ener_euler()

const Scalar & Lorene::Compobj::get_ener_euler ( ) const

inlineinherited

Returns the total energy density E in the Eulerian frame.

Definition at line 219 of file compobj.h.

References Lorene::Compobj::ener_euler.

◆ get_gamma()

const Metric & Lorene::Compobj::get_gamma ( ) const

inlineinherited

Returns the 3-metric $\gamma_{ij}$ .

Definition at line 216 of file compobj.h.

References Lorene::Compobj::gamma.

◆ get_khi_shift()

const Scalar & Lorene::Star_QI::get_khi_shift ( ) const

inline

Returns the scalar $\chi$ used in the decomposition of shift
following Shibata's prescription [Prog.

Theor. Phys. 101 , 1199 (1999)] :

$N^i = {7\over 8} W^i - {1\over 8} \left(\nabla^i\chi+\nabla^iW^kx_k\right)$

NB: w_shift contains the components of $W^i$ with respect to the Cartesian triad associated with the mapping mp .

Definition at line 686 of file compobj.h.

References khi_shift.

◆ get_kk()

const Sym_tensor & Lorene::Compobj::get_kk ( ) const

inlineinherited

Returns the extrinsic curvature tensor $K_{ij}$ .

Definition at line 228 of file compobj.h.

References Lorene::Compobj::kk.

◆ get_logn()

const Scalar & Lorene::Star_QI::get_logn ( ) const

inline

Returns the logarithm of the lapse N.

Definition at line 636 of file compobj.h.

References logn.

◆ get_mom_euler()

const Vector & Lorene::Compobj::get_mom_euler ( ) const

inlineinherited

Returns the total 3-momentum density $P^i$ in the Eulerian frame.

Definition at line 222 of file compobj.h.

References Lorene::Compobj::mom_euler.

◆ get_mp()

const Map & Lorene::Compobj::get_mp ( ) const

inlineinherited

Returns the mapping.

Definition at line 207 of file compobj.h.

References Lorene::Compobj::mp.

◆ get_nn()

const Scalar & Lorene::Compobj::get_nn ( ) const

inlineinherited

Returns the lapse function N .

Definition at line 210 of file compobj.h.

References Lorene::Compobj::nn.

◆ get_nphi()

const Scalar & Lorene::Compobj_QI::get_nphi ( ) const

inlineinherited

Returns the metric coefficient $N^\varphi$ .

Definition at line 384 of file compobj.h.

References Lorene::Compobj_QI::nphi.

◆ get_nuf()

const Scalar & Lorene::Star_QI::get_nuf ( ) const

inline

Returns the part of the Metric potential $\nu = \ln N$ = logn generated by the matter terms.

Definition at line 647 of file compobj.h.

References nuf.

◆ get_nuq()

const Scalar & Lorene::Star_QI::get_nuq ( ) const

inline

Returns the Part of the Metric potential $\nu = \ln N$ = logn generated by the quadratic terms.

Definition at line 652 of file compobj.h.

References nuq.

◆ get_stress_euler()

const Sym_tensor & Lorene::Compobj::get_stress_euler ( ) const

inlineinherited

Returns the stress tensor $S_{ij}$ with respect to the Eulerian observer.

Definition at line 225 of file compobj.h.

References Lorene::Compobj::stress_euler.

◆ get_tggg()

const Scalar & Lorene::Star_QI::get_tggg ( ) const

inline

Returns the Metric potential $\tilde G = (NB-1) r\sin\theta$ .

Definition at line 658 of file compobj.h.

References tggg.

◆ get_tnphi()

const Scalar & Lorene::Star_QI::get_tnphi ( ) const

inline

Returns the component $\tilde N^\varphi = N^\varphi r\sin\theta$ of the shift vector.

Definition at line 642 of file compobj.h.

References tnphi.

◆ get_w_shift()

const Vector & Lorene::Star_QI::get_w_shift ( ) const

inline

Returns the vector $W^i$ used in the decomposition of shift , following Shibata's prescription [Prog.

Theor. Phys. 101 , 1199 (1999)] :

$N^i = {7\over 8} W^i - {1\over 8} \left(\nabla^i\chi+\nabla^iW^kx_k\right)$

NB: w_shift contains the components of $W^i$ with respect to the Cartesian triad associated with the mapping mp .

Definition at line 672 of file compobj.h.

References w_shift.

◆ grv2()

double Lorene::Star_QI::grv2 ( ) const

virtual

Error on the virial identity GRV2.

This indicator is only valid for relativistic computations.

Definition at line 123 of file star_QI_global.C.

References Lorene::Compobj_QI::a_car, Lorene::Compobj_QI::ak_car, Lorene::Compobj_QI::b_car, Lorene::Scalar::derive_cov(), Lorene::Map::flat_met_spher(), Lorene::Map::get_bvect_spher(), Lorene::Tensor::get_triad(), lambda_grv2(), logn, Lorene::Compobj::mp, p_grv2, and Lorene::Compobj::stress_euler.

◆ grv3()

double Lorene::Star_QI::grv3 ( ostream * ost = 0x0 ) const

virtual

Error on the virial identity GRV3.

The error is computed as the integral defined by Eq. (43) of [Gourgoulhon and Bonazzola, Class. Quantum Grav. 11, 443 (1994)] divided by the integral of the matter terms.

Parameters

ost	output stream to give details of the computation; if set to 0x0 [default value], no details will be given.

Definition at line 148 of file star_QI_global.C.

References Lorene::Compobj_QI::a_car, Lorene::Compobj_QI::ak_car, Lorene::Compobj_QI::bbb, Lorene::Scalar::derive_cov(), Lorene::Scalar::dsdr(), dzeta, Lorene::Map::flat_met_spher(), Lorene::Compobj::gamma, Lorene::Scalar::get_dzpuis(), Lorene::Scalar::get_etat(), Lorene::Scalar::integrale(), Lorene::log(), logn, Lorene::Compobj::mp, Lorene::Valeur::mult_ct(), p_grv3, Lorene::Scalar::set_dzpuis(), Lorene::Scalar::set_spectral_va(), Lorene::Scalar::srdsdt(), Lorene::Valeur::ssint(), Lorene::Scalar::std_spectral_base(), Lorene::Compobj::stress_euler, Lorene::Valeur::sx(), Lorene::Tensor::trace(), and Lorene::Map_radial::xsr.

◆ gyoto_data()

void Lorene::Compobj_QI::gyoto_data ( const char * file_name ) const

inherited

Save in a file for GYOTO.

Definition at line 232 of file compobj_QI.C.

References Lorene::Compobj::beta, Lorene::Metric::con(), Lorene::Metric::cov(), Lorene::fwrite_be(), Lorene::Compobj::gamma, Lorene::Map::get_mg(), Lorene::Compobj::kk, Lorene::Compobj::mp, Lorene::Compobj::nn, Lorene::Compobj_QI::r_isco(), Lorene::Compobj_QI::r_mb(), Lorene::Map::sauve(), Lorene::Scalar::sauve(), Lorene::Tensor::sauve(), Lorene::Tensor_sym::sauve(), and Lorene::Mg3d::sauve().

◆ lambda_grv2()

double Lorene::Star_QI::lambda_grv2	(	const Scalar &	sou_m,
		const Scalar &	sou_q
	)

static

Computes the coefficient $\lambda$ which ensures that the GRV2 virial identity is satisfied.

$\lambda$ is the coefficient by which one must multiply the quadratic source term $\sigma_q$ of the 2-D Poisson equation

$\Delta_2 u = \sigma_m + \sigma_q$

in order that the total source does not contain any monopolar term, i.e. in order that

$\int_0^{2\pi} \int_0^{+\infty} \sigma(r, \theta) \, r \, dr \, d\theta = 0 \ ,$

where $\sigma = \sigma_m + \sigma_q$ . $\lambda$ is computed according to the formula

$\lambda = - { \int_0^{2\pi} \int_0^{+\infty} \sigma_m(r, \theta) \, r \, dr \, d\theta \over \int_0^{2\pi} \int_0^{+\infty} \sigma_q(r, \theta) \, r \, dr \, d\theta } \ .$

Then, by construction, the new source $\sigma' = \sigma_m + \lambda \sigma_q$ has a vanishing monopolar term.

Parameters

sou_m	[input] matter source term $\sigma_m$
sou_q	[input] quadratic source term $\sigma_q$

Returns: value of $\lambda$

Definition at line 321 of file star_QI_global.C.

References Lorene::Valeur::c, Lorene::Scalar::check_dzpuis(), Lorene::Valeur::coef_i(), Lorene::Map_radial::dxdr, Lorene::Map_af::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Scalar::get_etat(), Lorene::Tbl::get_etat(), Lorene::Valeur::get_etat(), Lorene::Mg3d::get_grille3d(), Lorene::Map::get_mg(), Lorene::Tensor::get_mp(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_va(), Lorene::Mg3d::get_type_r(), Lorene::Map_af::set_alpha(), Lorene::Map_af::set_beta(), Lorene::Mtbl::t, Lorene::Tbl::t, Lorene::Map::val_r(), Lorene::Grille3d::x, and Lorene::Map_radial::xsr.

◆ lspec_isco()

double Lorene::Compobj_QI::lspec_isco ( int lmin ) const

virtualinherited

Angular momentum of a particle at the ISCO.

Definition at line 303 of file compobj_QI_global.C.

References Lorene::Compobj_QI::p_lspec_isco, and Lorene::Compobj_QI::r_isco().

◆ mass_g()

double Lorene::Star_QI::mass_g ( ) const

virtual

Gravitational mass.

Definition at line 64 of file star_QI_global.C.

References Lorene::Compobj_QI::a_car, Lorene::Compobj_QI::b_car, Lorene::Compobj_QI::bbb, Lorene::Compobj::ener_euler, Lorene::Compobj::gamma, Lorene::Scalar::integrale(), Lorene::Compobj::mom_euler, Lorene::Compobj::nn, p_mass_g, Lorene::Scalar::std_spectral_base(), Lorene::Compobj::stress_euler, tnphi, and Lorene::Tensor::trace().

◆ mom_quad()

double Lorene::Star_QI::mom_quad ( ) const

virtual

Quadrupole moment.

The quadrupole moment Q is defined according to Eq. (7) of [Salgado, Bonazzola, Gourgoulhon and Haensel, Astron. Astrophys. 291 , 155 (1994)]. At the Newtonian limit it is related to the component ${\bar I}_{zz}$ of the MTW (1973) reduced quadrupole moment ${\bar I}_{ij}$ by: $Q = -3/2 {\bar I}_{zz}$ . Note that Q is the negative of the quadrupole moment defined by Laarakkers and Poisson, Astrophys. J. 512 , 282 (1999).

Definition at line 250 of file star_QI_global.C.

References Lorene::Compobj_QI::a_car, Lorene::Compobj_QI::ak_car, Lorene::Compobj_QI::bbb, Lorene::Scalar::check_dzpuis(), Lorene::Scalar::derive_cov(), Lorene::Compobj::ener_euler, Lorene::Map::flat_met_spher(), Lorene::Compobj::gamma, Lorene::Scalar::get_etat(), Lorene::Scalar::inc_dzpuis(), Lorene::Scalar::integrale(), Lorene::log(), logn, Lorene::Compobj::mp, Lorene::Valeur::mult_ct(), Lorene::Scalar::mult_r(), p_mom_quad, Lorene::Scalar::set_spectral_va(), Lorene::Scalar::std_spectral_base(), Lorene::Compobj::stress_euler, and Lorene::Tensor::trace().

◆ operator=()

void Lorene::Star_QI::operator= ( const Star_QI & st )

Assignment to another Star_QI.

Definition at line 250 of file star_QI.C.

References del_deriv(), dzeta, khi_shift, logn, nuf, nuq, Lorene::Compobj_QI::operator=(), ssjm1_dzeta, ssjm1_khi, ssjm1_nuf, ssjm1_nuq, ssjm1_tggg, ssjm1_wshift, tggg, tnphi, and w_shift.

◆ operator>>()

ostream & Lorene::Star_QI::operator>> ( ostream & ost ) const

protectedvirtual

Operator >> (virtual function called by the operator <<).

Reimplemented from Lorene::Compobj_QI.

Reimplemented in Lorene::Boson_star.

Definition at line 300 of file star_QI.C.

References angu_mom(), dzeta, grv2(), grv3(), logn, mass_g(), mom_quad(), nuf, nuq, Lorene::Compobj_QI::operator>>(), Lorene::pow(), and Lorene::Scalar::val_grid_point().

◆ r_isco()

double Lorene::Compobj_QI::r_isco	(	int	lmin,
		ostream *	ost = `0x0`
	)		const

virtualinherited

Coordinate r of the innermost stable circular orbit (ISCO).

Parameters

lmin	index of the innermost domain in which the ISCO is searched: the ISCO is searched inwards from the last but one domain to the domain of index lmin.
ost	output stream to give details of the computation; if set to 0x0 [default value], no details will be given.

Definition at line 108 of file compobj_QI_global.C.

References Lorene::Param::add_int(), Lorene::Param::add_scalar(), Lorene::Tensor::annule_domain(), Lorene::Compobj_QI::bbb, Lorene::Scalar::dsdr(), Lorene::Map::get_mg(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_va(), Lorene::Compobj::mp, Lorene::Compobj::nn, Lorene::Compobj_QI::nphi, Lorene::Compobj_QI::p_espec_isco, Lorene::Compobj_QI::p_f_isco, Lorene::Compobj_QI::p_lspec_isco, Lorene::Compobj_QI::p_r_isco, Lorene::Map::r, Lorene::sqrt(), Lorene::Scalar::std_spectral_base(), Lorene::Valeur::val_point(), Lorene::Map::val_r(), and Lorene::zerosec().

◆ r_mb()

double Lorene::Compobj_QI::r_mb	(	int	lmin,
		ostream *	ost = `0x0`
	)		const

virtualinherited

Coordinate r of the marginally bound circular orbit (R_mb).

Definition at line 341 of file compobj_QI_global.C.

References Lorene::Param::add_int(), Lorene::Param::add_scalar(), Lorene::Tensor::annule_domain(), Lorene::Compobj_QI::bbb, Lorene::Scalar::dsdr(), Lorene::Map::get_mg(), Lorene::Mg3d::get_nzone(), Lorene::Scalar::get_spectral_va(), Lorene::Compobj::mp, Lorene::Compobj::nn, Lorene::Compobj_QI::nphi, Lorene::Compobj_QI::p_r_mb, Lorene::Map::r, Lorene::Compobj_QI::r_isco(), Lorene::sqrt(), Lorene::Scalar::std_spectral_base(), Lorene::Valeur::val_point(), Lorene::Map::val_r(), and Lorene::zerosec().

◆ sauve()

void Lorene::Star_QI::sauve ( FILE * fich ) const

virtual

Save in a file.

Reimplemented from Lorene::Compobj_QI.

Reimplemented in Lorene::Boson_star.

Definition at line 279 of file star_QI.C.

References dzeta, khi_shift, nuf, nuq, Lorene::Scalar::sauve(), Lorene::Tensor::sauve(), ssjm1_dzeta, ssjm1_khi, ssjm1_nuf, ssjm1_nuq, ssjm1_tggg, ssjm1_wshift, tggg, and w_shift.

◆ set_der_0x0()

void Lorene::Star_QI::set_der_0x0 ( ) const

protectedvirtual

Sets to 0x0 all the pointers on derived quantities.

Reimplemented in Lorene::Boson_star.

Definition at line 235 of file star_QI.C.

References p_grv2, p_grv3, p_mass_g, and p_mom_quad.

◆ set_mp()

Map & Lorene::Compobj::set_mp ( )

inlineinherited

Read/write of the mapping.

Definition at line 200 of file compobj.h.

References Lorene::Compobj::mp.

◆ update_metric()

void Lorene::Star_QI::update_metric ( )

virtual

Computes metric coefficients from known potentials.

The calculation is performed starting from the quantities logn , dzeta , tggg and shift , which are supposed to be up to date.
From these, the following fields are updated: nnn , a_car , bbb and b_car, as well as the 3-metric gamma.

Reimplemented from Lorene::Compobj_QI.

Definition at line 410 of file star_QI.C.

References Lorene::Compobj_QI::a_car, Lorene::Compobj_QI::b_car, Lorene::Compobj_QI::bbb, del_deriv(), Lorene::Scalar::div_rsint(), dzeta, Lorene::exp(), logn, Lorene::Compobj::nn, Lorene::Scalar::std_spectral_base(), tggg, and Lorene::Compobj_QI::update_metric().

Member Data Documentation

◆ a_car

Scalar Lorene::Compobj_QI::a_car

protectedinherited

Square of the metric factor A.

Definition at line 287 of file compobj.h.

◆ ak_car

Scalar Lorene::Compobj_QI::ak_car

protectedinherited

Scalar $A^2 K_{ij} K^{ij}$ .

For axisymmetric stars, this quantity is related to the derivatives of $N^\varphi$ by

$A^2 K_{ij} K^{ij} = {B^2 \over 2 N^2} \, r^2\sin^2\theta \, \left[ \left( {\partial N^\varphi \over \partial r} \right) ^2 + {1\over r^2} \left( {\partial N^\varphi \over \partial \theta} \right) ^2 \right] \ .$

In particular it is related to the quantities $k_1$ and $k_2$ introduced by Eqs.~(3.7) and (3.8) of Bonazzola et al. Astron. Astrophys. 278 , 421 (1993) by

$A^2 K_{ij} K^{ij} = 2 A^2 (k_1^2 + k_2^2) \ .$

Definition at line 315 of file compobj.h.

◆ b_car

Scalar Lorene::Compobj_QI::b_car

protectedinherited

Square of the metric factor B.

Definition at line 293 of file compobj.h.

◆ bbb

Scalar Lorene::Compobj_QI::bbb

protectedinherited

Metric factor B.

Definition at line 290 of file compobj.h.

◆ beta

Vector Lorene::Compobj::beta

protectedinherited

Shift vector $\beta^i$ .

Definition at line 138 of file compobj.h.

◆ dzeta

Scalar Lorene::Star_QI::dzeta

protected

Metric potential $\zeta = \ln(AN)$ .

Definition at line 513 of file compobj.h.

◆ ener_euler

Scalar Lorene::Compobj::ener_euler

protectedinherited

Total energy density E in the Eulerian frame.

Definition at line 144 of file compobj.h.

◆ gamma

Metric Lorene::Compobj::gamma

protectedinherited

3-metric $\gamma_{ij}$

Definition at line 141 of file compobj.h.

◆ khi_shift

Scalar Lorene::Star_QI::khi_shift

protected

Scalar $\chi$ used in the decomposition of shift , following Shibata's prescription [Prog.

Theor. Phys. 101 , 1199 (1999)] :

$N^i = {7\over 8} W^i - {1\over 8} \left(\nabla^i\chi+\nabla^iW^kx_k\right)$

Definition at line 539 of file compobj.h.

◆ kk

Sym_tensor Lorene::Compobj::kk

protectedinherited

Extrinsic curvature tensor $K_{ij}$

Definition at line 153 of file compobj.h.

◆ logn

Scalar Lorene::Star_QI::logn

protected

Logarithm of the lapse N .

Definition at line 495 of file compobj.h.

◆ mom_euler

Vector Lorene::Compobj::mom_euler

protectedinherited

Total 3-momentum density $P^i$ in the Eulerian frame.

Definition at line 147 of file compobj.h.

◆ mp

Map& Lorene::Compobj::mp

protectedinherited

Mapping describing the coordinate system (r,theta,phi)

Definition at line 132 of file compobj.h.

◆ nn

Scalar Lorene::Compobj::nn

protectedinherited

Lapse function N .

Definition at line 135 of file compobj.h.

◆ nphi

Scalar Lorene::Compobj_QI::nphi

protectedinherited

Metric coefficient $N^\varphi$ .

Definition at line 296 of file compobj.h.

◆ nuf

Scalar Lorene::Star_QI::nuf

protected

Part of the Metric potential $\nu = \ln N$ = logn generated by the matter terms.

Definition at line 505 of file compobj.h.

◆ nuq

Scalar Lorene::Star_QI::nuq

protected

Part of the Metric potential $\nu = \ln N$ = logn generated by the quadratic terms.

Definition at line 510 of file compobj.h.

◆ p_adm_mass

double* Lorene::Compobj::p_adm_mass

mutableprotectedinherited

ADM mass.

Definition at line 158 of file compobj.h.

◆ p_angu_mom

double* Lorene::Compobj_QI::p_angu_mom

mutableprotectedinherited

Angular momentum.

Definition at line 321 of file compobj.h.

◆ p_espec_isco

double* Lorene::Compobj_QI::p_espec_isco

mutableprotectedinherited

Specific energy of a particle at the ISCO.

Definition at line 325 of file compobj.h.

◆ p_f_isco

double* Lorene::Compobj_QI::p_f_isco

mutableprotectedinherited

Orbital frequency of the ISCO.

Definition at line 323 of file compobj.h.

◆ p_grv2

double* Lorene::Star_QI::p_grv2

mutableprotected

Error on the virial identity GRV2.

Definition at line 585 of file compobj.h.

◆ p_grv3

double* Lorene::Star_QI::p_grv3

mutableprotected

Error on the virial identity GRV3.

Definition at line 586 of file compobj.h.

◆ p_lspec_isco

double* Lorene::Compobj_QI::p_lspec_isco

mutableprotectedinherited

Specific angular momentum of a particle at the ISCO.

Definition at line 327 of file compobj.h.

◆ p_mass_g

double* Lorene::Star_QI::p_mass_g

mutableprotected

Gravitational mass (ADM mass as a volume integral)

Definition at line 588 of file compobj.h.

◆ p_mom_quad

double* Lorene::Star_QI::p_mom_quad

mutableprotected

Quadrupole moment

Definition at line 587 of file compobj.h.

◆ p_r_isco

double* Lorene::Compobj_QI::p_r_isco

mutableprotectedinherited

Coordinate r of the ISCO.

Definition at line 322 of file compobj.h.

◆ p_r_mb

double* Lorene::Compobj_QI::p_r_mb

mutableprotectedinherited

Coordinate r of the marginally bound orbit.

Definition at line 328 of file compobj.h.

◆ ssjm1_dzeta

Scalar Lorene::Star_QI::ssjm1_dzeta

protected

Effective source at the previous step for the resolution of the Poisson equation for dzeta .

Definition at line 556 of file compobj.h.

◆ ssjm1_khi

Scalar Lorene::Star_QI::ssjm1_khi

protected

Effective source at the previous step for the resolution of the Poisson equation for the scalar $\chi$ by means of Map_et::poisson .

$\chi$ is an intermediate quantity for the resolution of the elliptic equation for the shift vector $N^i$

Definition at line 569 of file compobj.h.

◆ ssjm1_nuf

Scalar Lorene::Star_QI::ssjm1_nuf

protected

Effective source at the previous step for the resolution of the Poisson equation for nuf by means of Map_et::poisson .

Definition at line 545 of file compobj.h.

◆ ssjm1_nuq

Scalar Lorene::Star_QI::ssjm1_nuq

protected

Effective source at the previous step for the resolution of the Poisson equation for nuq by means of Map_et::poisson .

Definition at line 551 of file compobj.h.

◆ ssjm1_tggg

Scalar Lorene::Star_QI::ssjm1_tggg

protected

Effective source at the previous step for the resolution of the Poisson equation for tggg .

Definition at line 561 of file compobj.h.

◆ ssjm1_wshift

Vector Lorene::Star_QI::ssjm1_wshift

protected

Effective source at the previous step for the resolution of the vector Poisson equation for $W^i$ .

$W^i$ is an intermediate quantity for the resolution of the elliptic equation for the shift vector $N^i$ (Components with respect to the Cartesian triad associated with the mapping mp )

Definition at line 578 of file compobj.h.

◆ stress_euler

Sym_tensor Lorene::Compobj::stress_euler

protectedinherited

Stress tensor $S_{ij}$ with respect to the Eulerian observer.

Definition at line 150 of file compobj.h.

◆ tggg

Scalar Lorene::Star_QI::tggg

protected

Metric potential $\tilde G = (NB-1) r\sin\theta$ .

Definition at line 516 of file compobj.h.

◆ tnphi

Scalar Lorene::Star_QI::tnphi

protected

Component $\tilde N^\varphi = N^\varphi r\sin\theta$ of the shift vector.

Definition at line 500 of file compobj.h.

◆ w_shift

Vector Lorene::Star_QI::w_shift

protected

Vector $W^i$ used in the decomposition of shift , following Shibata's prescription [Prog.

Theor. Phys. 101 , 1199 (1999)] :

$N^i = {7\over 8} W^i - {1\over 8} \left(\nabla^i\chi+\nabla^iW^kx_k\right)$

NB: w_shift contains the components of $W^i$ with respect to the Cartesian triad associated with the mapping mp .

Definition at line 529 of file compobj.h.

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

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ Star_QI() [1/3]

◆ Star_QI() [2/3]

◆ Star_QI() [3/3]

◆ ~Star_QI()

Member Function Documentation

◆ adm_mass()

◆ angu_mom()

◆ del_deriv()

◆ espec_isco()

◆ extrinsic_curvature()

◆ f_isco()

◆ fait_nphi()

◆ fait_shift()

◆ get_a_car()

◆ get_ak_car()

◆ get_b_car()

◆ get_bbb()

◆ get_beta()

◆ get_dzeta()

◆ get_ener_euler()

◆ get_gamma()

◆ get_khi_shift()

◆ get_kk()

◆ get_logn()

◆ get_mom_euler()

◆ get_mp()

◆ get_nn()

◆ get_nphi()

◆ get_nuf()

◆ get_nuq()

◆ get_stress_euler()

◆ get_tggg()

◆ get_tnphi()

◆ get_w_shift()

◆ grv2()

◆ grv3()

◆ gyoto_data()

◆ lambda_grv2()

◆ lspec_isco()

◆ mass_g()

◆ mom_quad()

◆ operator=()

◆ operator>>()

◆ r_isco()

◆ r_mb()

◆ sauve()

◆ set_der_0x0()

◆ set_mp()

◆ update_metric()

Member Data Documentation

◆ a_car

◆ ak_car

◆ b_car

◆ bbb

◆ beta

◆ dzeta

◆ ener_euler

◆ gamma

◆ khi_shift

◆ kk

◆ logn

◆ mom_euler

◆ mp

◆ nn

◆ nphi

◆ nuf

◆ nuq

◆ p_adm_mass

◆ p_angu_mom

◆ p_espec_isco

◆ p_f_isco

◆ p_grv2

◆ p_grv3

◆ p_lspec_isco