Class for perfect fluid rotating gravastar. More...

#include <gravastar.h>

Inheritance diagram for Lorene::Gravastar:

Public Member Functions
	Gravastar (Map &mp_i, int nzet_i, const Eos &eos_i, const double rho_core_i)
	Standard constructor.
	Gravastar (const Gravastar &)
	Copy constructor.
	Gravastar (Map &mp_i, const Eos &eos_i, FILE *fich)
	Constructor from a file (see `sauve(FILE*)` ).
	~Gravastar ()
	Destructor.
void	operator= (const Gravastar &)
	Assignment to another `Gravastar`.
void	equation_of_state ()
	Allows to computes the proper baryon and energy density, as well as pressure from the enthalpy, in the gravastar's crust only (in the core, p=-rho=cst).
void	equilibrium (double omega0, double fact_omega, int nzadapt, const Tbl &ent_limit, const Itbl &icontrol, const Tbl &control, Tbl &diff, Param *=0x0)
	Computes an equilibrium configuration.
bool	is_relativistic () const
	Returns `true` for a relativistic star, `false` for a Newtonian one.
virtual double	get_omega_c () const
	Returns the central value of the rotation angular velocity ([f_unit] ).
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_tnphi () const
	Returns the component $\tilde N^\varphi = N^\varphi r\sin\theta$ of the shift vector.
const Scalar &	get_uuu () const
	Returns the norm of `u_euler`.
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.
const Sym_tensor &	get_tkij () const
	Returns the tensor ${\tilde K_{ij}}$ related to the extrinsic curvature tensor by ${\tilde K_{ij}} = B^{-2} K_{ij}$ .
const Scalar &	get_ak_car () const
	Returns the scalar $A^2 K_{ij} K^{ij}$ .
virtual void	sauve (FILE *) const
	Save in a file.
virtual void	display_poly (ostream &) const
	Display in polytropic units.
virtual const Itbl &	l_surf () const
	Description of the stellar surface: returns a 2-D `Itbl` containing the values of the domain index l on the surface at the collocation points in $(\theta', \phi')$ .
virtual double	mass_b () const
	Baryon mass.
virtual double	mass_g () const
	Gravitational mass.
virtual double	angu_mom () const
	Angular momentum.
virtual double	tsw () const
	Ratio T/W.
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	r_circ () const
	Circumferential radius.
virtual double	aplat () const
	Flatening r_pole/r_eq.
virtual double	area () const
	Integrated surface area in ${\rm km}^2$ .
virtual double	mean_radius () const
	Mean star radius from the area $r_{\rm mean} = \sqrt{\mathcal{A}} / 4\pi$ .
virtual double	z_eqf () const
	Forward redshift factor at equator.
virtual double	z_eqb () const
	Backward redshift factor at equator.
virtual double	z_pole () const
	Redshift factor at North pole.
virtual double	mom_quad () const
	Quadrupole moment.
virtual double	r_isco (ostream *ost=0x0) const
	Circumferential radius of the innermost stable circular orbit (ISCO).
virtual double	f_isco () const
	Orbital frequency at the innermost stable circular orbit (ISCO).
virtual double	espec_isco () const
	Energy of a particle on the ISCO.
virtual double	lspec_isco () const
	Angular momentum of a particle on the ISCO.
virtual double	f_eq () const
	Orbital frequency at the equator.
virtual void	hydro_euler ()
	Computes the hydrodynamical quantities relative to the Eulerian observer from those in the fluid frame.
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` .
void	extrinsic_curvature ()
	Computes `tkij` and `ak_car` from `shift` , `nnn` and `b_car` .
virtual void	equilibrium (double ent_c, double omega0, double fact_omega, int nzadapt, const Tbl &ent_limit, const Itbl &icontrol, const Tbl &control, double mbar_wanted, double aexp_mass, Tbl &diff, Param *=0x0)
	Computes an equilibrium configuration.
Map &	set_mp ()
	Read/write of the mapping.
void	set_enthalpy (const Scalar &)
	Assignment of the enthalpy field.
virtual void	equilibrium_spher (double ent_c, double precis=1.e-14, const Tbl *pent_limit=0x0)
	Computes a spherical static configuration.
const Map &	get_mp () const
	Returns the mapping.
int	get_nzet () const
	Returns the number of domains occupied by the star.
const Eos &	get_eos () const
	Returns the equation of state.
const Scalar &	get_ent () const
	Returns the enthalpy field.
const Scalar &	get_nbar () const
	Returns the proper baryon density.
const Scalar &	get_ener () const
	Returns the proper total energy density.
const Scalar &	get_press () const
	Returns the fluid pressure.
const Scalar &	get_ener_euler () const
	Returns the total energy density with respect to the Eulerian observer.
const Scalar &	get_s_euler () const
	Returns the trace of the stress tensor in the Eulerian frame.
const Scalar &	get_gam_euler () const
	Returns the Lorentz factor between the fluid and Eulerian observers.
const Vector &	get_u_euler () const
	Returns the fluid 3-velocity with respect to the Eulerian observer.
const Tensor &	get_stress_euler () const
	Returns the spatial part of the stress-energy tensor with respect to the Eulerian observer.
const Scalar &	get_logn () const
	Returns the logarithm of the lapse N.
const Scalar &	get_nn () const
	Returns the lapse function N.
const Vector &	get_beta () const
	Returns the shift vector $\beta^i$ .
const Scalar &	get_lnq () const
const Metric &	get_gamma () const
	Returns the 3-metric $\gamma$ .
double	ray_eq () const
	Coordinate radius at $\phi=0$ , $\theta=\pi/2$ [r_unit].
double	ray_eq_pis2 () const
	Coordinate radius at $\phi=\pi/2$ , $\theta=\pi/2$ [r_unit].
double	ray_eq_pi () const
	Coordinate radius at $\phi=\pi$ , $\theta=\pi/2$ [r_unit].
double	ray_eq_3pis2 () const
	Coordinate radius at $\phi=3\pi/2$ , $\theta=\pi/2$ [r_unit].
double	ray_pole () const
	Coordinate radius at $\theta=0$ [r_unit].
const Tbl &	xi_surf () const
	Description of the stellar surface: returns a 2-D `Tbl` containing the values of the radial coordinate $\xi$ on the surface at the collocation points in $(\theta', \phi')$ .

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 ostream &	operator>> (ostream &) const
	Operator >> (virtual function called by the operator <<).
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 void	del_hydro_euler ()
	Sets to `ETATNONDEF` (undefined state) the hydrodynamical quantities relative to the Eulerian observer.
virtual void	partial_display (ostream &) const
	Printing of some informations, excluding all global quantities.

Protected Attributes
double	rho_core
	Energy density in gravastar's core.
bool	relativistic
	Indicator of relativity: `true` for a relativistic star, `false` for a Newtonian one.
double	unsurc2
	: `unsurc2=1` for a relativistic star, 0 for a Newtonian one.
double	omega
	Rotation angular velocity ([f_unit] ).
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	tnphi
	Component $\tilde N^\varphi = N^\varphi r\sin\theta$ of the shift vector.
Scalar	uuu
	Norm of `u_euler`.
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.
Sym_tensor	tkij
	Tensor ${\tilde K_{ij}}$ related to the extrinsic curvature tensor by ${\tilde K_{ij}} = B^{-2} K_{ij}$ .
Scalar	ak_car
	Scalar $A^2 K_{ij} K^{ij}$ .
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_angu_mom
	Angular momentum.
double *	p_tsw
	Ratio T/W.
double *	p_grv2
	Error on the virial identity GRV2.
double *	p_grv3
	Error on the virial identity GRV3.
double *	p_r_circ
	Circumferential radius.
double *	p_aplat
	Flatening r_pole/r_eq.
double *	p_area
	Integrated surface area.
double *	p_z_eqf
	Forward redshift factor at equator.
double *	p_z_eqb
	Backward redshift factor at equator.
double *	p_z_pole
	Redshift factor at North pole.
double *	p_mom_quad
	Quadrupole moment.
double *	p_r_isco
	Circumferential radius of the ISCO.
double *	p_f_isco
	Orbital frequency of the ISCO.
double *	p_espec_isco
	Specific energy of a particle on the ISCO.
double *	p_lspec_isco
	Specific angular momentum of a particle on the ISCO.
double *	p_f_eq
	Orbital frequency at the equator.
Map &	mp
	Mapping associated with the star.
int	nzet
	Number of domains of `*mp` occupied by the star.
const Eos &	eos
	Equation of state of the stellar matter.
Scalar	ent
	Log-enthalpy.
Scalar	nbar
	Baryon density in the fluid frame.
Scalar	ener
	Total energy density in the fluid frame.
Scalar	press
	Fluid pressure.
Scalar	ener_euler
	Total energy density in the Eulerian frame.
Scalar	s_euler
	Trace of the stress scalar in the Eulerian frame.
Scalar	gam_euler
	Lorentz factor between the fluid and Eulerian observers.
Vector	u_euler
	Fluid 3-velocity with respect to the Eulerian observer.
Sym_tensor	stress_euler
	Spatial part of the stress-energy tensor with respect to the Eulerian observer.
Scalar	logn
	Logarithm of the lapse N .
Scalar	nn
	Lapse function N .
Vector	beta
	Shift vector.
Scalar	lnq
Metric	gamma
	3-metric
double *	p_ray_eq
	Coordinate radius at $\phi=0$ , $\theta=\pi/2$ .
double *	p_ray_eq_pis2
	Coordinate radius at $\phi=\pi/2$ , $\theta=\pi/2$ .
double *	p_ray_eq_pi
	Coordinate radius at $\phi=\pi$ , $\theta=\pi/2$ .
double *	p_ray_eq_3pis2
	Coordinate radius at $\phi=3\pi/2$ , $\theta=\pi/2$ .
double *	p_ray_pole
	Coordinate radius at $\theta=0$ .
Itbl *	p_l_surf
	Description of the stellar surface: 2-D `Itbl` containing the values of the domain index l on the surface at the collocation points in $(\theta', \phi')$ .
Tbl *	p_xi_surf
	Description of the stellar surface: 2-D `Tbl` containing the values of the radial coordinate $\xi$ on the surface at the collocation points in $(\theta', \phi')$ .
double *	p_mass_b
	Baryon mass.
double *	p_mass_g
	Gravitational mass.

Detailed Description

Class for perfect fluid rotating gravastar.

()

Definition at line 49 of file gravastar.h.

Constructor & Destructor Documentation

◆ Gravastar() [1/3]

Lorene::Gravastar::Gravastar	(	Map &	mp_i,
		int	nzet_i,
		const Eos &	eos_i,
		const double	rho_core_i )

Standard constructor.

Parameters

mp_i	Mapping on which the gravastar is contructed
nzet_i	Number of domains occupied by the gravastar
eos_i	Equation of state of the crust matter
rho_core_i	Energy density in gravastar's core (constant, =-p_core) NB:pas de choix pour l'EOS en fait : a virer??

Definition at line 51 of file gravastar.C.

References Lorene::Map(), rho_core, and Lorene::Star_rot::Star_rot().

◆ Gravastar() [2/3]

Lorene::Gravastar::Gravastar ( const Gravastar & )

Copy constructor.

References Gravastar().

◆ Gravastar() [3/3]

Lorene::Gravastar::Gravastar	(	Map &	mp_i,
		const Eos &	eos_i,
		FILE *	fich )

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

Parameters

mp_i	Mapping on which the gravastar is constructed
eos_i	Equation of state of the crust matter
fich	input file (must have been created by the function `Gravastar::sauve` )

References Lorene::Map().

◆ ~Gravastar()

Lorene::Gravastar::~Gravastar ( )

Destructor.

Definition at line 59 of file gravastar.C.

References Lorene::Star_rot::del_deriv().

Member Function Documentation

◆ angu_mom()

double Lorene::Star_rot::angu_mom ( ) const

virtualinherited

Angular momentum.

Definition at line 150 of file star_rot_global.C.

References a_car, b_car, Lorene::Star::ener_euler, Lorene::Scalar::get_spectral_va(), Lorene::Scalar::integrale(), Lorene::Scalar::mult_r(), Lorene::Valeur::mult_st(), Lorene::Star::nbar, p_angu_mom, Lorene::Star::press, relativistic, Lorene::Scalar::set_spectral_va(), and uuu.

◆ aplat()

double Lorene::Star_rot::aplat ( ) const

virtualinherited

Flatening r_pole/r_eq.

Definition at line 426 of file star_rot_global.C.

References p_aplat, Lorene::Star::ray_eq(), and Lorene::Star::ray_pole().

◆ area()

double Lorene::Star_rot::area ( ) const

virtualinherited

Integrated surface area in ${\rm km}^2$ .

Definition at line 366 of file star_rot_global.C.

References Lorene::Valeur::annule_hard(), Lorene::Valeur::c_cf, Lorene::Valeur::dsdt(), get_a_car(), Lorene::Mg3d::get_angu(), get_bbb(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nt(), Lorene::Scalar::get_spectral_va(), l_surf(), Lorene::Star::mp, Lorene::Valeur::mult_st(), p_area, Lorene::Valeur::set(), Lorene::sqrt(), Lorene::Valeur::std_base_scal(), Lorene::Valeur::val_point_jk(), Lorene::Map_radial::val_r_jk(), and Lorene::Star::xi_surf().

◆ del_deriv()

void Lorene::Star_rot::del_deriv ( ) const

protectedvirtualinherited

Deletes all the derived quantities.

Reimplemented from Lorene::Star.

Definition at line 297 of file star_rot.C.

References Lorene::Star::del_deriv(), p_angu_mom, p_aplat, p_area, p_espec_isco, p_f_eq, p_f_isco, p_grv2, p_grv3, p_lspec_isco, p_mom_quad, p_r_circ, p_r_isco, p_tsw, p_z_eqb, p_z_eqf, p_z_pole, and set_der_0x0().

◆ del_hydro_euler()

void Lorene::Star_rot::del_hydro_euler ( )

protectedvirtualinherited

Sets to ETATNONDEF (undefined state) the hydrodynamical quantities relative to the Eulerian observer.

Reimplemented from Lorene::Star.

Definition at line 345 of file star_rot.C.

References del_deriv(), and Lorene::Star::del_hydro_euler().

◆ display_poly()

void Lorene::Star_rot::display_poly ( ostream & ost ) const

virtualinherited

Display in polytropic units.

Definition at line 649 of file star_rot.C.

References angu_mom(), Lorene::Star::ener, Lorene::Star::eos, Lorene::Eos_poly::get_gam(), Lorene::Eos_poly::get_kap(), get_omega_c(), mass_b(), mass_g(), mean_radius(), Lorene::Star::nbar, Lorene::pow(), r_circ(), Lorene::Star::ray_eq(), and Lorene::sqrt().

◆ equation_of_state()

void Lorene::Gravastar::equation_of_state ( )

Allows to computes the proper baryon and energy density, as well as pressure from the enthalpy, in the gravastar's crust only (in the core, p=-rho=cst).

Definition at line 67 of file gravastar.C.

References Lorene::Param::add_int(), Lorene::Scalar::allocate_all(), Lorene::Star_rot::del_deriv(), Lorene::Star::ener, Lorene::Star::ent, Lorene::Star::eos, Lorene::Mg3d::get_grille3d(), Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Star::mp, Lorene::Star::nbar, Lorene::Star::nzet, Lorene::Star::press, rho_core, Lorene::Mtbl::set(), Lorene::Scalar::set_domain(), Lorene::Mtbl::set_etat_qcq(), Lorene::Tbl::set_etat_qcq(), Lorene::Scalar::std_spectral_base(), Lorene::Mtbl::t, and Lorene::Grille3d::x.

◆ equilibrium() [1/2]

void Lorene::Gravastar::equilibrium	(	double	omega0,
		double	fact_omega,
		int	nzadapt,
		const Tbl &	ent_limit,
		const Itbl &	icontrol,
		const Tbl &	control,
		Tbl &	diff,
		Param *	= 0x0 )

Computes an equilibrium configuration.

Parameters

omega0	[input] Requested angular velocity (if `fact_omega=1`. )
fact_omega	[input] 1.01 = search for the Keplerian frequency, = otherwise.
nzadapt	[input] Number of (inner) domains where the mapping adaptation to an iso-enthalpy surface should be performed
ent_limit	[input] 1-D `Tbl` of dimension `nzet` which defines the enthalpy at the outer boundary of each domain
icontrol	[input] Set of integer parameters (stored as a 1-D `Itbl` of size 8) to control the iteration: `icontrol(0)` = mer_max : maximum number of steps `icontrol(1)` = mer_rot : step at which the rotation is switched on `icontrol(2)` = mer_change_omega : step at which the rotation velocity is changed to reach the final one `icontrol(3)` = mer_fix_omega : step at which the final rotation velocity must have been reached `icontrol(4)` = mer_mass : the absolute value of `mer_mass` is the step from which the baryon mass is forced to converge, by varying the central enthalpy (`mer_mass>0` ) or the angular velocity (`mer_mass<0` ) `icontrol(5)` = mermax_poisson : maximum number of steps in `Map_et::poisson` `icontrol(6)` = mer_triax : step at which the 3-D perturbation is switched on `icontrol(7)` = delta_mer_kep : number of steps after `mer_fix_omega` when `omega` starts to be increased by `fact_omega` to search for the Keplerian velocity
control	[input] Set of parameters (stored as a 1-D `Tbl` of size 7) to control the iteration: `control(0)` = precis : threshold on the enthalpy relative change for ending the computation `control(1)` = omega_ini : initial angular velocity, switched on only if `mer_rot<0` , otherwise 0 is used `control(2)` = relax : relaxation factor in the main iteration `control(3)` = relax_poisson : relaxation factor in `Map_et::poisson` `control(4)` = thres_adapt : threshold on dH/dr for freezing the adaptation of the mapping `control(5)` = ampli_triax : relative amplitude of the 3-D perturbation `control(6)` = precis_adapt : precision for `Map_et::adapt`
diff	[output] 1-D `Tbl` of size 7 for the storage of some error indicators : `diff(0)` : Relative change in the enthalpy field between two successive steps `diff(1)` : Relative error in the resolution of the Poisson equation for `nuf` `diff(2)` : Relative error in the resolution of the Poisson equation for `nuq` `diff(3)` : Relative error in the resolution of the Poisson equation for `dzeta` `diff(4)` : Relative error in the resolution of the Poisson equation for `tggg` `diff(5)` : Relative error in the resolution of the equation for `shift` (x comp.) `diff(6)` : Relative error in the resolution of the equation for `shift` (y comp.)

Definition at line 47 of file gravastar_equil.C.

◆ equilibrium() [2/2]

void Lorene::Star_rot::equilibrium	(	double	ent_c,
		double	omega0,
		double	fact_omega,
		int	nzadapt,
		const Tbl &	ent_limit,
		const Itbl &	icontrol,
		const Tbl &	control,
		double	mbar_wanted,
		double	aexp_mass,
		Tbl &	diff,
		Param *	= 0x0 )

virtualinherited

Computes an equilibrium configuration.

Parameters

ent_c	[input] Central enthalpy
omega0	[input] Requested angular velocity (if `fact_omega=1`. )
fact_omega	[input] 1.01 = search for the Keplerian frequency, = otherwise.
nzadapt	[input] Number of (inner) domains where the mapping adaptation to an iso-enthalpy surface should be performed
ent_limit	[input] 1-D `Tbl` of dimension `nzet` which defines the enthalpy at the outer boundary of each domain
icontrol	[input] Set of integer parameters (stored as a 1-D `Itbl` of size 8) to control the iteration: `icontrol(0)` = mer_max : maximum number of steps `icontrol(1)` = mer_rot : step at which the rotation is switched on `icontrol(2)` = mer_change_omega : step at which the rotation velocity is changed to reach the final one `icontrol(3)` = mer_fix_omega : step at which the final rotation velocity must have been reached `icontrol(4)` = mer_mass : the absolute value of `mer_mass` is the step from which the baryon mass is forced to converge, by varying the central enthalpy (`mer_mass>0` ) or the angular velocity (`mer_mass<0` ) `icontrol(5)` = mermax_poisson : maximum number of steps in `Map_et::poisson` `icontrol(6)` = mer_triax : step at which the 3-D perturbation is switched on `icontrol(7)` = delta_mer_kep : number of steps after `mer_fix_omega` when `omega` starts to be increased by `fact_omega` to search for the Keplerian velocity
control	[input] Set of parameters (stored as a 1-D `Tbl` of size 7) to control the iteration: `control(0)` = precis : threshold on the enthalpy relative change for ending the computation `control(1)` = omega_ini : initial angular velocity, switched on only if `mer_rot<0` , otherwise 0 is used `control(2)` = relax : relaxation factor in the main iteration `control(3)` = relax_poisson : relaxation factor in `Map_et::poisson` `control(4)` = thres_adapt : threshold on dH/dr for freezing the adaptation of the mapping `control(5)` = ampli_triax : relative amplitude of the 3-D perturbation `control(6)` = precis_adapt : precision for `Map_et::adapt`
mbar_wanted	[input] Requested baryon mass (effective only if `mer_mass` > `mer_max` )
aexp_mass	[input] Exponent for the increase factor of the central enthalpy to converge to the requested baryon mass
diff	[output] 1-D `Tbl` of size 7 for the storage of some error indicators : `diff(0)` : Relative change in the enthalpy field between two successive steps `diff(1)` : Relative error in the resolution of the Poisson equation for `nuf` `diff(2)` : Relative error in the resolution of the Poisson equation for `nuq` `diff(3)` : Relative error in the resolution of the Poisson equation for `dzeta` `diff(4)` : Relative error in the resolution of the Poisson equation for `tggg` `diff(5)` : Relative error in the resolution of the equation for `shift` (x comp.) `diff(6)` : Relative error in the resolution of the equation for `shift` (y comp.)

Definition at line 71 of file star_rot_equil.C.

References a_car, Lorene::abs(), Lorene::Param::add_cmp_mod(), Lorene::Param::add_double(), Lorene::Param::add_double_mod(), Lorene::Param::add_int(), Lorene::Param::add_int_mod(), Lorene::Param::add_tbl(), Lorene::Param::add_tenseur_mod(), ak_car, Lorene::Tensor::annule_domain(), bbb, Lorene::Star::beta, Lorene::Valeur::c_cf, Lorene::Vector::change_triad(), Lorene::Valeur::coef(), Lorene::contract(), Lorene::cos(), Lorene::Scalar::derive_con(), Lorene::Scalar::derive_cov(), Lorene::diffrel(), dzeta, Lorene::Star::ener_euler, Lorene::Star::ent, Lorene::Star::equation_of_state(), fait_nphi(), Lorene::Star::gam_euler, Lorene::Valeur::get_base(), 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_t(), grv2(), Lorene::Map_et::homothetie(), hydro_euler(), khi_shift, Lorene::log(), Lorene::log10(), Lorene::Star::logn, mass_b(), mass_g(), Lorene::Star::mp, Lorene::Scalar::mult_rsint(), Lorene::Star::nbar, Lorene::Star::nn, nphi, nuf, nuq, Lorene::Star::nzet, omega, partial_display(), Lorene::phi, Lorene::Scalar::poisson(), Lorene::Tenseur::poisson_vect(), Lorene::pow(), Lorene::Star::press, Lorene::Star::ray_eq(), Lorene::Star::ray_pole(), relativistic, Lorene::Star::s_euler, Lorene::Tbl::set(), Lorene::Tenseur::set(), Lorene::Vector::set(), Lorene::Tbl::set_etat_qcq(), Lorene::Tenseur::set_etat_qcq(), Lorene::Cmp::set_etat_zero(), Lorene::sint, Lorene::sqrt(), ssjm1_khi, ssjm1_nuf, ssjm1_nuq, ssjm1_tggg, ssjm1_wshift, Lorene::Scalar::std_spectral_base(), Lorene::Scalar::test_poisson(), tggg, tkij, Lorene::Star::u_euler, Lorene::Tensor::up(), update_metric(), uuu, Lorene::Scalar::val_grid_point(), and w_shift.

◆ equilibrium_spher()

void Lorene::Star::equilibrium_spher	(	double	ent_c,
		double	precis = 1.e-14,
		const Tbl *	pent_limit = 0x0 )

virtualinherited

Computes a spherical static configuration.

Parameters

ent_c	[input] central value of the enthalpy
precis	[input] threshold in the relative difference between the enthalpy fields of two consecutive steps to stop the iterative procedure (default value: 1.e-14)
ent_limit	[input] : array of enthalpy values to be set at the boundaries between the domains; if set to 0x0 (default), the initial values will be kept.

Definition at line 95 of file star_equil_spher.C.

References Lorene::Map_et::adapt(), Lorene::Param::add_double(), Lorene::Param::add_int(), Lorene::Param::add_int_mod(), Lorene::Param::add_tbl(), Lorene::diffrel(), Lorene::Map_af::dsdr(), Lorene::Scalar::dsdr(), ener, ener_euler, ent, equation_of_state(), Lorene::exp(), gam_euler, gamma, Lorene::Map_af::get_alpha(), Lorene::Map_et::get_alpha(), Lorene::Map_af::get_beta(), Lorene::Map_et::get_beta(), Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_nzone(), Lorene::Map_af::homothetie(), Lorene::Scalar::integrale(), logn, mass_b(), mass_g(), mp, nn, Lorene::norme(), nzet, Lorene::Map_af::poisson(), press, s_euler, Lorene::Tensor::set(), Lorene::Map_af::set_alpha(), Lorene::Map_af::set_beta(), Lorene::Scalar::set_dzpuis(), Lorene::Cmp::set_etat_qcq(), Lorene::Tensor::set_etat_zero(), Lorene::sqrt(), Lorene::Scalar::std_spectral_base(), u_euler, and Lorene::Scalar::val_grid_point().

◆ espec_isco()

double Lorene::Star_rot::espec_isco ( ) const

virtualinherited

Energy of a particle on the ISCO.

Definition at line 289 of file star_rot_isco.C.

References p_espec_isco, and r_isco().

◆ extrinsic_curvature()

void Lorene::Star_rot::extrinsic_curvature ( )

inherited

Computes tkij and ak_car from shift , nnn and b_car .

Definition at line 110 of file star_rot_upmetr.C.

References ak_car, b_car, Lorene::Star::beta, Lorene::contract(), Lorene::Scalar::get_etat(), Lorene::Scalar::get_spectral_va(), Lorene::Star::mp, Lorene::Valeur::mult_ct(), Lorene::Scalar::mult_rsint(), Lorene::Valeur::mult_st(), Lorene::Star::nn, nphi, Lorene::Scalar::set_spectral_va(), and tkij.

◆ f_eq()

double Lorene::Star_rot::f_eq ( ) const

virtualinherited

Orbital frequency at the equator.

Definition at line 307 of file star_rot_isco.C.

References p_f_eq, and r_isco().

◆ f_isco()

double Lorene::Star_rot::f_isco ( ) const

virtualinherited

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

Definition at line 255 of file star_rot_isco.C.

References p_f_isco, and r_isco().

◆ fait_nphi()

void Lorene::Star_rot::fait_nphi ( )

inherited

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

Definition at line 748 of file star_rot.C.

References Lorene::Star::beta, Lorene::Star::mp, nphi, and tnphi.

◆ fait_shift()

void Lorene::Star_rot::fait_shift ( )

inherited

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 703 of file star_rot.C.

References Lorene::Star::beta, Lorene::contract(), Lorene::cost, Lorene::Tensor::dec_dzpuis(), Lorene::Scalar::domain(), khi_shift, Lorene::Star::mp, Lorene::Vector::set(), Lorene::Scalar::set_domain(), Lorene::Scalar::set_dzpuis(), Lorene::Vector::std_spectral_base(), and w_shift.

◆ get_a_car()

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

inlineinherited

Returns the square of the metric factor A.

Definition at line 319 of file star_rot.h.

References a_car.

◆ get_ak_car()

const Scalar & Lorene::Star_rot::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 403 of file star_rot.h.

References ak_car.

◆ get_b_car()

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

inlineinherited

Returns the square of the metric factor B.

Definition at line 322 of file star_rot.h.

References b_car.

◆ get_bbb()

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

inlineinherited

Returns the metric factor B.

Definition at line 316 of file star_rot.h.

References bbb.

◆ get_beta()

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

inlineinherited

Returns the shift vector $\beta^i$ .

Definition at line 402 of file star.h.

References beta.

◆ get_dzeta()

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

inlineinherited

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

Definition at line 346 of file star_rot.h.

References dzeta.

◆ get_ener()

const Scalar & Lorene::Star::get_ener ( ) const

inlineinherited

Returns the proper total energy density.

Definition at line 370 of file star.h.

References ener.

◆ get_ener_euler()

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

inlineinherited

Returns the total energy density with respect to the Eulerian observer.

Definition at line 376 of file star.h.

References ener_euler.

◆ get_ent()

const Scalar & Lorene::Star::get_ent ( ) const

inlineinherited

Returns the enthalpy field.

Definition at line 364 of file star.h.

References ent.

◆ get_eos()

const Eos & Lorene::Star::get_eos ( ) const

inlineinherited

Returns the equation of state.

Definition at line 361 of file star.h.

References eos.

◆ get_gam_euler()

const Scalar & Lorene::Star::get_gam_euler ( ) const

inlineinherited

Returns the Lorentz factor between the fluid and Eulerian observers.

Definition at line 382 of file star.h.

References gam_euler.

◆ get_gamma()

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

inlineinherited

Returns the 3-metric $\gamma$ .

Definition at line 409 of file star.h.

References gamma.

◆ get_khi_shift()

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

inlineinherited

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 377 of file star_rot.h.

References khi_shift.

◆ get_lnq()

const Scalar & Lorene::Star::get_lnq ( ) const

inlineinherited

Definition at line 406 of file star.h.

◆ get_logn()

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

inlineinherited

Returns the logarithm of the lapse N.

In the Newtonian case, this is the Newtonian gravitational potential (in units of $c^2$ ).

Definition at line 396 of file star.h.

References logn.

◆ get_mp()

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

inlineinherited

Returns the mapping.

Definition at line 355 of file star.h.

References Lorene::Map(), and mp.

◆ get_nbar()

const Scalar & Lorene::Star::get_nbar ( ) const

inlineinherited

Returns the proper baryon density.

Definition at line 367 of file star.h.

References nbar.

◆ get_nn()

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

inlineinherited

Returns the lapse function N.

Definition at line 399 of file star.h.

References nn.

◆ get_nphi()

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

inlineinherited

Returns the metric coefficient $N^\varphi$ .

Definition at line 325 of file star_rot.h.

References nphi.

◆ get_nuf()

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

inlineinherited

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

Definition at line 338 of file star_rot.h.

References nuf.

◆ get_nuq()

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

inlineinherited

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

Definition at line 343 of file star_rot.h.

References nuq.

◆ get_nzet()

int Lorene::Star::get_nzet ( ) const

inlineinherited

Returns the number of domains occupied by the star.

Definition at line 358 of file star.h.

References nzet.

◆ get_omega_c()

double Lorene::Star_rot::get_omega_c ( ) const

virtualinherited

Returns the central value of the rotation angular velocity ([f_unit] ).

Definition at line 640 of file star_rot.C.

References omega.

◆ get_press()

const Scalar & Lorene::Star::get_press ( ) const

inlineinherited

Returns the fluid pressure.

Definition at line 373 of file star.h.

References press.

◆ get_s_euler()

const Scalar & Lorene::Star::get_s_euler ( ) const

inlineinherited

Returns the trace of the stress tensor in the Eulerian frame.

Definition at line 379 of file star.h.

References s_euler.

◆ get_stress_euler()

const Tensor & Lorene::Star::get_stress_euler ( ) const

inlineinherited

Returns the spatial part of the stress-energy tensor with respect to the Eulerian observer.

Definition at line 390 of file star.h.

References stress_euler.

◆ get_tggg()

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

inlineinherited

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

Definition at line 349 of file star_rot.h.

References tggg.

◆ get_tkij()

const Sym_tensor & Lorene::Star_rot::get_tkij ( ) const

inlineinherited

Returns the tensor ${\tilde K_{ij}}$ related to the extrinsic curvature tensor by ${\tilde K_{ij}} = B^{-2} K_{ij}$ .

tkij contains the Cartesian components of ${\tilde K_{ij}}$ .

Definition at line 384 of file star_rot.h.

References tkij.

◆ get_tnphi()

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

inlineinherited

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

Definition at line 330 of file star_rot.h.

References tnphi.

◆ get_u_euler()

const Vector & Lorene::Star::get_u_euler ( ) const

inlineinherited

Returns the fluid 3-velocity with respect to the Eulerian observer.

Definition at line 385 of file star.h.

References u_euler.

◆ get_uuu()

const Scalar & Lorene::Star_rot::get_uuu ( ) const

inlineinherited

Returns the norm of u_euler.

Definition at line 333 of file star_rot.h.

References uuu.

◆ get_w_shift()

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

inlineinherited

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 363 of file star_rot.h.

References w_shift.

◆ grv2()

double Lorene::Star_rot::grv2 ( ) const

virtualinherited

Error on the virial identity GRV2.

This indicator is only valid for relativistic computations.

Definition at line 204 of file star_rot_global.C.

References a_car, ak_car, Lorene::Star::ener_euler, lambda_grv2(), Lorene::Star::logn, Lorene::Star::mp, Lorene::Star::nbar, p_grv2, Lorene::Star::press, relativistic, and uuu.

◆ grv3()

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

virtualinherited

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 235 of file star_rot_global.C.

References a_car, ak_car, bbb, Lorene::Scalar::derive_cov(), Lorene::Scalar::dsdr(), dzeta, Lorene::Scalar::get_dzpuis(), Lorene::Scalar::get_etat(), Lorene::Scalar::integrale(), Lorene::log(), Lorene::Star::logn, Lorene::Star::mp, Lorene::Valeur::mult_ct(), Lorene::Star::nbar, p_grv3, Lorene::Star::press, relativistic, Lorene::Star::s_euler, Lorene::Scalar::set_dzpuis(), Lorene::Scalar::set_spectral_va(), Lorene::Scalar::srdsdt(), Lorene::Valeur::ssint(), Lorene::Scalar::std_spectral_base(), Lorene::Valeur::sx(), uuu, and Lorene::Map_radial::xsr.

◆ hydro_euler()

void Lorene::Star_rot::hydro_euler ( )

virtualinherited

Computes the hydrodynamical quantities relative to the Eulerian observer from those in the fluid frame.

The calculation is performed starting from the quantities ent , ener , press , and a_car ,
which are supposed to be up to date.
From these, the following fields are updated: gam_euler , u_euler , ener_euler , s_euler .

Reimplemented from Lorene::Star.

Definition at line 57 of file star_rot_hydro.C.

References a_car, b_car, Lorene::Star::beta, Lorene::Vector::change_triad(), del_deriv(), Lorene::Star::ener, Lorene::Star::ener_euler, Lorene::Star::gam_euler, Lorene::Star::mp, Lorene::Star::nn, omega, Lorene::Star::press, Lorene::Star::s_euler, Lorene::sqrt(), Lorene::Star::u_euler, unsurc2, uuu, Lorene::x, and Lorene::y.

◆ is_relativistic()

bool Lorene::Star_rot::is_relativistic ( ) const

inlineinherited

Returns true for a relativistic star, false for a Newtonian one.

Definition at line 308 of file star_rot.h.

References relativistic.

◆ l_surf()

const Itbl & Lorene::Star_rot::l_surf ( ) const

virtualinherited

Description of the stellar surface: returns a 2-D Itbl containing the values of the domain index l on the surface at the collocation points in $(\theta', \phi')$ .

The stellar surface is defined as the location where the enthalpy (member ent ) vanishes.

Reimplemented from Lorene::Star.

Definition at line 68 of file star_rot_global.C.

References Lorene::Star::ent, Lorene::Star::mp, Lorene::Star::nzet, Lorene::Star::p_l_surf, and Lorene::Star::p_xi_surf.

◆ lambda_grv2()

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

staticinherited

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 62 of file star_rot_lambda_grv2.C.

References 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::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::Grille3d::x, and Lorene::Map_radial::xsr.

◆ lspec_isco()

double Lorene::Star_rot::lspec_isco ( ) const

virtualinherited

Angular momentum of a particle on the ISCO.

Definition at line 272 of file star_rot_isco.C.

References p_lspec_isco, and r_isco().

◆ mass_b()

double Lorene::Star_rot::mass_b ( ) const

virtualinherited

Baryon mass.

Implements Lorene::Star.

Definition at line 98 of file star_rot_global.C.

References a_car, bbb, Lorene::Star::gam_euler, Lorene::Scalar::integrale(), Lorene::Star::nbar, Lorene::Star::p_mass_b, and relativistic.

◆ mass_g()

double Lorene::Star_rot::mass_g ( ) const

virtualinherited

Gravitational mass.

Implements Lorene::Star.

Definition at line 124 of file star_rot_global.C.

References a_car, bbb, Lorene::Star::ener_euler, Lorene::Scalar::integrale(), mass_b(), Lorene::Star::nn, Lorene::Star::p_mass_g, Lorene::Star::press, relativistic, Lorene::Star::s_euler, Lorene::Scalar::std_spectral_base(), tnphi, and uuu.

◆ mean_radius()

double Lorene::Star_rot::mean_radius ( ) const

virtualinherited

Mean star radius from the area $r_{\rm mean} = \sqrt{\mathcal{A}} / 4\pi$ .

Definition at line 416 of file star_rot_global.C.

References area(), and Lorene::sqrt().

◆ mom_quad()

double Lorene::Star_rot::mom_quad ( ) const

virtualinherited

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 521 of file star_rot_global.C.

References a_car, ak_car, bbb, Lorene::Scalar::check_dzpuis(), Lorene::Scalar::derive_cov(), Lorene::Star::ener_euler, Lorene::Scalar::get_etat(), Lorene::Scalar::inc_dzpuis(), Lorene::Scalar::integrale(), Lorene::log(), Lorene::Star::logn, Lorene::Star::mp, Lorene::Valeur::mult_ct(), Lorene::Scalar::mult_r(), Lorene::Star::nbar, p_mom_quad, relativistic, Lorene::Star::s_euler, Lorene::Scalar::set_spectral_va(), and Lorene::Scalar::std_spectral_base().

◆ operator=()

void Lorene::Gravastar::operator= ( const Gravastar & )

Assignment to another Gravastar.

References Gravastar().

◆ operator>>()

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

protectedvirtual

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

Reimplemented from Lorene::Star_rot.

Definition at line 185 of file gravastar.C.

References Lorene::Star_rot::a_car, Lorene::Star_rot::angu_mom(), Lorene::Star_rot::aplat(), Lorene::Star_rot::b_car, Lorene::diffrel(), Lorene::Star_rot::dzeta, Lorene::Star::ener, Lorene::Star::ent, Lorene::Star::eos, Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Star_rot::get_omega_c(), Lorene::Tbl::get_taille(), Lorene::Star_rot::grv2(), Lorene::Star_rot::grv3(), Lorene::Star_rot::khi_shift, Lorene::Star::logn, Lorene::Star_rot::mom_quad(), Lorene::Star::mp, Lorene::Star::nn, Lorene::Star_rot::nphi, Lorene::Star::nzet, Lorene::Star_rot::omega, Lorene::pow(), Lorene::Star::press, Lorene::Star_rot::r_circ(), Lorene::Star::ray_eq(), Lorene::Star::ray_eq_pi(), Lorene::Star::ray_eq_pis2(), Lorene::Star::ray_pole(), Lorene::Star_rot::tsw(), Lorene::Star_rot::uuu, Lorene::Star_rot::w_shift, Lorene::Star_rot::z_eqb(), Lorene::Star_rot::z_eqf(), and Lorene::Star_rot::z_pole().

◆ partial_display()

void Lorene::Star_rot::partial_display ( ostream & ost ) const

protectedvirtualinherited

Printing of some informations, excluding all global quantities.

Definition at line 590 of file star_rot.C.

References a_car, aplat(), b_car, dzeta, Lorene::Star::ener, Lorene::Star::ent, get_omega_c(), Lorene::Star::logn, Lorene::Star::mp, Lorene::Star::nbar, Lorene::Star::nn, nphi, Lorene::Star::nzet, Lorene::Star::press, Lorene::Star::ray_eq(), and uuu.

◆ r_circ()

double Lorene::Star_rot::r_circ ( ) const

virtualinherited

Circumferential radius.

Definition at line 342 of file star_rot_global.C.

References bbb, Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_type_t(), Lorene::Star::mp, Lorene::Star::nzet, p_r_circ, and Lorene::Star::ray_eq().

◆ r_isco()

double Lorene::Star_rot::r_isco ( ostream * ost = 0x0 ) const

virtualinherited

Circumferential radius of the innermost stable circular orbit (ISCO).

Parameters

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

Definition at line 71 of file star_rot_isco.C.

References Lorene::Param::add_int(), Lorene::Param::add_scalar(), Lorene::Tensor::annule_domain(), bbb, Lorene::Scalar::dsdr(), Lorene::Scalar::get_spectral_va(), Lorene::Star::mp, Lorene::Star::nn, nphi, Lorene::Star::nzet, p_espec_isco, p_f_eq, p_f_isco, p_lspec_isco, p_r_isco, Lorene::r, Lorene::Star::ray_eq(), Lorene::sqrt(), Lorene::Scalar::std_spectral_base(), Lorene::Valeur::val_point(), and Lorene::zerosec().

◆ ray_eq()

double Lorene::Star::ray_eq ( ) const

inherited

Coordinate radius at $\phi=0$ , $\theta=\pi/2$ [r_unit].

Definition at line 108 of file star_global.C.

References Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_type_p(), Lorene::Mg3d::get_type_t(), l_surf(), mp, p_ray_eq, Lorene::phi, and xi_surf().

◆ ray_eq_3pis2()

double Lorene::Star::ray_eq_3pis2 ( ) const

inherited

Coordinate radius at $\phi=3\pi/2$ , $\theta=\pi/2$ [r_unit].

Definition at line 233 of file star_global.C.

References Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_type_p(), Lorene::Mg3d::get_type_t(), l_surf(), mp, p_ray_eq_3pis2, Lorene::phi, ray_eq_pis2(), and xi_surf().

◆ ray_eq_pi()

double Lorene::Star::ray_eq_pi ( ) const

inherited

Coordinate radius at $\phi=\pi$ , $\theta=\pi/2$ [r_unit].

Definition at line 186 of file star_global.C.

References Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_type_p(), Lorene::Mg3d::get_type_t(), l_surf(), mp, p_ray_eq_pi, Lorene::phi, ray_eq(), and xi_surf().

◆ ray_eq_pis2()

double Lorene::Star::ray_eq_pis2 ( ) const

inherited

Coordinate radius at $\phi=\pi/2$ , $\theta=\pi/2$ [r_unit].

Definition at line 138 of file star_global.C.

References Lorene::Mg3d::get_np(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_type_p(), Lorene::Mg3d::get_type_t(), l_surf(), mp, p_ray_eq_pis2, Lorene::phi, and xi_surf().

◆ ray_pole()

double Lorene::Star::ray_pole ( ) const

inherited

Coordinate radius at $\theta=0$ [r_unit].

Definition at line 278 of file star_global.C.

References Lorene::Mg3d::get_type_t(), l_surf(), mp, p_ray_pole, Lorene::phi, and xi_surf().

◆ sauve()

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

virtualinherited

Save in a file.

Reimplemented from Lorene::Star.

Definition at line 401 of file star_rot.C.

References dzeta, Lorene::fwrite_be(), khi_shift, nuf, nuq, omega, relativistic, Lorene::Star::sauve(), ssjm1_dzeta, ssjm1_khi, ssjm1_nuf, ssjm1_nuq, ssjm1_tggg, ssjm1_wshift, tggg, and w_shift.

◆ set_der_0x0()

void Lorene::Star_rot::set_der_0x0 ( ) const

protectedvirtualinherited

Sets to 0x0 all the pointers on derived quantities.

Reimplemented from Lorene::Star.

Definition at line 322 of file star_rot.C.

References p_angu_mom, p_aplat, p_area, p_espec_isco, p_f_eq, p_f_isco, p_grv2, p_grv3, p_lspec_isco, p_mom_quad, p_r_circ, p_r_isco, p_tsw, p_z_eqb, p_z_eqf, p_z_pole, and Lorene::Star::set_der_0x0().

◆ set_enthalpy()

void Lorene::Star::set_enthalpy ( const Scalar & ent_i )

inherited

Assignment of the enthalpy field.

Definition at line 379 of file star.C.

References del_deriv(), ent, and equation_of_state().

◆ set_mp()

Map & Lorene::Star::set_mp ( )

inlineinherited

Read/write of the mapping.

Definition at line 322 of file star.h.

References Lorene::Map(), and mp.

◆ tsw()

double Lorene::Star_rot::tsw ( ) const

virtualinherited

Ratio T/W.

Definition at line 176 of file star_rot_global.C.

References a_car, angu_mom(), bbb, Lorene::Star::ener, Lorene::Star::gam_euler, Lorene::Scalar::integrale(), Lorene::Star::logn, mass_g(), Lorene::Star::nbar, omega, p_tsw, and relativistic.

◆ update_metric()

void Lorene::Star_rot::update_metric ( )

inherited

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.

Definition at line 51 of file star_rot_upmetr.C.

References a_car, b_car, bbb, del_deriv(), Lorene::Scalar::div_rsint(), dzeta, Lorene::exp(), extrinsic_curvature(), Lorene::Star::gamma, Lorene::Star::logn, Lorene::Star::mp, Lorene::Star::nn, Lorene::Tensor::set(), tggg, and unsurc2.

◆ xi_surf()

const Tbl & Lorene::Star::xi_surf ( ) const

inherited

Description of the stellar surface: returns a 2-D Tbl containing the values of the radial coordinate $\xi$ on the surface at the collocation points in $(\theta', \phi')$ .

The stellar surface is defined as the location where the enthalpy (member ent) vanishes.

Definition at line 89 of file star_global.C.

References l_surf(), p_l_surf, and p_xi_surf.

◆ z_eqb()

double Lorene::Star_rot::z_eqb ( ) const

virtualinherited

Backward redshift factor at equator.

Definition at line 471 of file star_rot_global.C.

References Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_type_t(), Lorene::Star::mp, Lorene::Star::nn, nphi, Lorene::Star::nzet, p_z_eqb, r_circ(), Lorene::sqrt(), and uuu.

◆ z_eqf()

double Lorene::Star_rot::z_eqf ( ) const

virtualinherited

Forward redshift factor at equator.

Definition at line 443 of file star_rot_global.C.

References Lorene::Mg3d::get_nr(), Lorene::Mg3d::get_nt(), Lorene::Mg3d::get_type_t(), Lorene::Star::mp, Lorene::Star::nn, nphi, Lorene::Star::nzet, p_z_eqf, r_circ(), Lorene::sqrt(), and uuu.

◆ z_pole()

double Lorene::Star_rot::z_pole ( ) const

virtualinherited

Redshift factor at North pole.

Definition at line 502 of file star_rot_global.C.

References Lorene::Star::nn, p_z_pole, and Lorene::Star::ray_pole().

Member Data Documentation

◆ a_car

Scalar Lorene::Star_rot::a_car

protectedinherited

Square of the metric factor A.

Definition at line 104 of file star_rot.h.

◆ ak_car

Scalar Lorene::Star_rot::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 186 of file star_rot.h.

◆ b_car

Scalar Lorene::Star_rot::b_car

protectedinherited

Square of the metric factor B.

Definition at line 110 of file star_rot.h.

◆ bbb

Scalar Lorene::Star_rot::bbb

protectedinherited

Metric factor B.

Definition at line 107 of file star_rot.h.

◆ beta

Vector Lorene::Star::beta

protectedinherited

Shift vector.

Definition at line 228 of file star.h.

◆ dzeta

Scalar Lorene::Star_rot::dzeta

protectedinherited

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

Definition at line 134 of file star_rot.h.

◆ ener

Scalar Lorene::Star::ener

protectedinherited

Total energy density in the fluid frame.

Definition at line 193 of file star.h.

◆ ener_euler

Scalar Lorene::Star::ener_euler

protectedinherited

Total energy density in the Eulerian frame.

Definition at line 198 of file star.h.

◆ ent

Scalar Lorene::Star::ent

protectedinherited

Log-enthalpy.

Definition at line 190 of file star.h.

◆ eos

const Eos& Lorene::Star::eos

protectedinherited

Equation of state of the stellar matter.

Definition at line 185 of file star.h.

◆ gam_euler

Scalar Lorene::Star::gam_euler

protectedinherited

Lorentz factor between the fluid and Eulerian observers.

Definition at line 204 of file star.h.

◆ gamma

Metric Lorene::Star::gamma

protectedinherited

3-metric

Definition at line 235 of file star.h.

◆ khi_shift

Scalar Lorene::Star_rot::khi_shift

protectedinherited

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 160 of file star_rot.h.

◆ lnq

Scalar Lorene::Star::lnq

protectedinherited

Definition at line 232 of file star.h.

◆ logn

Scalar Lorene::Star::logn

protectedinherited

Logarithm of the lapse N .

In the Newtonian case, this is the Newtonian gravitational potential (in units of $c^2$ ).

Definition at line 222 of file star.h.

◆ mp

Map& Lorene::Star::mp

protectedinherited

Mapping associated with the star.

Definition at line 180 of file star.h.

◆ nbar

Scalar Lorene::Star::nbar

protectedinherited

Baryon density in the fluid frame.

Definition at line 192 of file star.h.

◆ nn

Scalar Lorene::Star::nn

protectedinherited

Lapse function N .

Definition at line 225 of file star.h.

◆ nphi

Scalar Lorene::Star_rot::nphi

protectedinherited

Metric coefficient $N^\varphi$ .

Definition at line 113 of file star_rot.h.

◆ nuf

Scalar Lorene::Star_rot::nuf

protectedinherited

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

Definition at line 126 of file star_rot.h.

◆ nuq

Scalar Lorene::Star_rot::nuq

protectedinherited

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

Definition at line 131 of file star_rot.h.

◆ nzet

int Lorene::Star::nzet

protectedinherited

Number of domains of *mp occupied by the star.

Definition at line 183 of file star.h.

◆ omega

double Lorene::Star_rot::omega

protectedinherited

Rotation angular velocity ([f_unit] ).

Definition at line 101 of file star_rot.h.

◆ p_angu_mom

double* Lorene::Star_rot::p_angu_mom

mutableprotectedinherited

Angular momentum.

Definition at line 232 of file star_rot.h.

◆ p_aplat

double* Lorene::Star_rot::p_aplat

mutableprotectedinherited

Flatening r_pole/r_eq.

Definition at line 237 of file star_rot.h.

◆ p_area

double* Lorene::Star_rot::p_area

mutableprotectedinherited

Integrated surface area.

Definition at line 238 of file star_rot.h.

◆ p_espec_isco

double* Lorene::Star_rot::p_espec_isco

mutableprotectedinherited

Specific energy of a particle on the ISCO.

Definition at line 246 of file star_rot.h.

◆ p_f_eq

double* Lorene::Star_rot::p_f_eq

mutableprotectedinherited

Orbital frequency at the equator.

Definition at line 249 of file star_rot.h.

◆ p_f_isco

double* Lorene::Star_rot::p_f_isco

mutableprotectedinherited

Orbital frequency of the ISCO.

Definition at line 244 of file star_rot.h.

◆ p_grv2

double* Lorene::Star_rot::p_grv2

mutableprotectedinherited

Error on the virial identity GRV2.

Definition at line 234 of file star_rot.h.

◆ p_grv3

double* Lorene::Star_rot::p_grv3

mutableprotectedinherited

Error on the virial identity GRV3.

Definition at line 235 of file star_rot.h.

◆ p_l_surf

Itbl* Lorene::Star::p_l_surf

mutableprotectedinherited

Description of the stellar surface: 2-D Itbl containing the values of the domain index l on the surface at the collocation points in $(\theta', \phi')$ .

Definition at line 260 of file star.h.

◆ p_lspec_isco

double* Lorene::Star_rot::p_lspec_isco

mutableprotectedinherited

Specific angular momentum of a particle on the ISCO.

Definition at line 248 of file star_rot.h.

◆ p_mass_b

double* Lorene::Star::p_mass_b

mutableprotectedinherited

Baryon mass.

Definition at line 268 of file star.h.

◆ p_mass_g

double* Lorene::Star::p_mass_g

mutableprotectedinherited

Gravitational mass.

Definition at line 269 of file star.h.

◆ p_mom_quad

double* Lorene::Star_rot::p_mom_quad

mutableprotectedinherited

Quadrupole moment.

Definition at line 242 of file star_rot.h.

◆ p_r_circ

double* Lorene::Star_rot::p_r_circ

mutableprotectedinherited

Circumferential radius.

Definition at line 236 of file star_rot.h.

◆ p_r_isco

double* Lorene::Star_rot::p_r_isco

mutableprotectedinherited

Circumferential radius of the ISCO.

Definition at line 243 of file star_rot.h.

◆ p_ray_eq

double* Lorene::Star::p_ray_eq

mutableprotectedinherited

Coordinate radius at $\phi=0$ , $\theta=\pi/2$ .

Definition at line 242 of file star.h.

◆ p_ray_eq_3pis2

double* Lorene::Star::p_ray_eq_3pis2

mutableprotectedinherited

Coordinate radius at $\phi=3\pi/2$ , $\theta=\pi/2$ .

Definition at line 251 of file star.h.

◆ p_ray_eq_pi

double* Lorene::Star::p_ray_eq_pi

mutableprotectedinherited

Coordinate radius at $\phi=\pi$ , $\theta=\pi/2$ .

Definition at line 248 of file star.h.

◆ p_ray_eq_pis2

double* Lorene::Star::p_ray_eq_pis2

mutableprotectedinherited

Coordinate radius at $\phi=\pi/2$ , $\theta=\pi/2$ .

Definition at line 245 of file star.h.

◆ p_ray_pole

double* Lorene::Star::p_ray_pole

mutableprotectedinherited

Coordinate radius at $\theta=0$ .

Definition at line 254 of file star.h.

◆ p_tsw

double* Lorene::Star_rot::p_tsw

mutableprotectedinherited

Ratio T/W.

Definition at line 233 of file star_rot.h.

◆ p_xi_surf

Tbl* Lorene::Star::p_xi_surf

mutableprotectedinherited

Description of the stellar surface: 2-D Tbl containing the values of the radial coordinate $\xi$ on the surface at the collocation points in $(\theta', \phi')$ .

Definition at line 266 of file star.h.

◆ p_z_eqb

double* Lorene::Star_rot::p_z_eqb

mutableprotectedinherited

Backward redshift factor at equator.

Definition at line 240 of file star_rot.h.

◆ p_z_eqf

double* Lorene::Star_rot::p_z_eqf

mutableprotectedinherited

Forward redshift factor at equator.

Definition at line 239 of file star_rot.h.

◆ p_z_pole

double* Lorene::Star_rot::p_z_pole

mutableprotectedinherited

Redshift factor at North pole.

Definition at line 241 of file star_rot.h.

◆ press

Scalar Lorene::Star::press

protectedinherited

Fluid pressure.

Definition at line 194 of file star.h.

◆ relativistic

bool Lorene::Star_rot::relativistic

protectedinherited

Indicator of relativity: true for a relativistic star, false for a Newtonian one.

Definition at line 94 of file star_rot.h.

◆ rho_core

double Lorene::Gravastar::rho_core

protected

Energy density in gravastar's core.

Definition at line 57 of file gravastar.h.

◆ s_euler

Scalar Lorene::Star::s_euler

protectedinherited

Trace of the stress scalar in the Eulerian frame.

Definition at line 201 of file star.h.

◆ ssjm1_dzeta

Scalar Lorene::Star_rot::ssjm1_dzeta

protectedinherited

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

Definition at line 203 of file star_rot.h.

◆ ssjm1_khi

Scalar Lorene::Star_rot::ssjm1_khi

protectedinherited

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 216 of file star_rot.h.

◆ ssjm1_nuf

Scalar Lorene::Star_rot::ssjm1_nuf

protectedinherited

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

Definition at line 192 of file star_rot.h.

◆ ssjm1_nuq

Scalar Lorene::Star_rot::ssjm1_nuq

protectedinherited

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

Definition at line 198 of file star_rot.h.

◆ ssjm1_tggg

Scalar Lorene::Star_rot::ssjm1_tggg

protectedinherited

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

Definition at line 208 of file star_rot.h.

◆ ssjm1_wshift

Vector Lorene::Star_rot::ssjm1_wshift

protectedinherited

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 225 of file star_rot.h.

◆ stress_euler

Sym_tensor Lorene::Star::stress_euler

protectedinherited

Spatial part of the stress-energy tensor with respect to the Eulerian observer.

Definition at line 212 of file star.h.

◆ tggg

Scalar Lorene::Star_rot::tggg

protectedinherited

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

Definition at line 137 of file star_rot.h.

◆ tkij

Sym_tensor Lorene::Star_rot::tkij

protectedinherited

Tensor ${\tilde K_{ij}}$ related to the extrinsic curvature tensor by ${\tilde K_{ij}} = B^{-2} K_{ij}$ .

tkij contains the Cartesian components of ${\tilde K_{ij}}$ .

Definition at line 167 of file star_rot.h.

◆ tnphi

Scalar Lorene::Star_rot::tnphi

protectedinherited

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

Definition at line 118 of file star_rot.h.

◆ u_euler

Vector Lorene::Star::u_euler

protectedinherited

Fluid 3-velocity with respect to the Eulerian observer.

Definition at line 207 of file star.h.

◆ unsurc2

double Lorene::Star_rot::unsurc2

protectedinherited

$1/c^2$ : unsurc2=1 for a relativistic star, 0 for a Newtonian one.

Definition at line 99 of file star_rot.h.

◆ uuu

Scalar Lorene::Star_rot::uuu

protectedinherited

Norm of u_euler.

Definition at line 121 of file star_rot.h.

◆ w_shift

Vector Lorene::Star_rot::w_shift

protectedinherited

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 150 of file star_rot.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

◆ Gravastar() [1/3]

◆ Gravastar() [2/3]

◆ Gravastar() [3/3]

◆ ~Gravastar()

Member Function Documentation

◆ angu_mom()

◆ aplat()

◆ area()

◆ del_deriv()

◆ del_hydro_euler()

◆ display_poly()

◆ equation_of_state()

◆ equilibrium() [1/2]

◆ equilibrium() [2/2]

◆ equilibrium_spher()

◆ espec_isco()

◆ extrinsic_curvature()

◆ f_eq()

◆ f_isco()

◆ fait_nphi()

◆ fait_shift()

◆ get_a_car()

◆ get_ak_car()

◆ get_b_car()

◆ get_bbb()

◆ get_beta()

◆ get_dzeta()

◆ get_ener()

◆ get_ener_euler()

◆ get_ent()

◆ get_eos()

◆ get_gam_euler()

◆ get_gamma()

◆ get_khi_shift()

◆ get_lnq()

◆ get_logn()

◆ get_mp()

◆ get_nbar()

◆ get_nn()

◆ get_nphi()

◆ get_nuf()

◆ get_nuq()

◆ get_nzet()

◆ get_omega_c()

◆ get_press()

◆ get_s_euler()

◆ get_stress_euler()

◆ get_tggg()

◆ get_tkij()

◆ get_tnphi()

◆ get_u_euler()

◆ get_uuu()

◆ get_w_shift()

◆ grv2()

◆ grv3()

◆ hydro_euler()

◆ is_relativistic()

◆ l_surf()

◆ lambda_grv2()

◆ lspec_isco()

◆ mass_b()

◆ mass_g()

◆ mean_radius()

◆ mom_quad()

◆ operator=()

◆ operator>>()

◆ partial_display()

◆ r_circ()

◆ r_isco()

◆ ray_eq()

◆ ray_eq_3pis2()

◆ ray_eq_pi()

◆ ray_eq_pis2()

◆ ray_pole()