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Gyoto::Metric::KerrBL Class Reference

Metric around a Kerr black-hole in Boyer-Lindquist coordinates. More...

#include <GyotoKerrBL.h>

Inheritance diagram for Gyoto::Metric::KerrBL:
Gyoto::Metric::Generic Gyoto::SmartPointee Gyoto::Object Gyoto::Hook::Teller

Public Types

typedef Gyoto::SmartPointer
< Gyoto::SmartPointee
Subcontractor_t (Gyoto::FactoryMessenger *)
 A subcontractor builds an object upon order from the Factory. More...
 

Public Member Functions

virtual Property const * getProperties () const
 Get list of properties. More...
 
 KerrBL ()
 Default constructor.
 
virtual KerrBLclone () const
 Virtual copy constructor.
 
void spin (const double spin)
 Set spin.
 
double spin () const
 Returns spin.
 
double difftol () const
 Get difftol_.
 
void difftol (double t)
 Set difftol_.
 
void horizonSecurity (double drhor)
 
double horizonSecurity () const
 
void genericIntegrator (bool)
 
bool genericIntegrator () const
 
virtual double getRms () const
 
virtual double getRmb () const
 
virtual double getSpecificAngularMomentum (double rr) const
 
virtual double getPotential (double pos[4], double l_cst) const
 
void gmunu (double g[4][4], const double *pos) const
 Metric coefficients. More...
 
double gmunu (const double *const x, int mu, int nu) const
 Metric coefficients. More...
 
void gmunu_up (double gup[4][4], const double *pos) const
 gμ,ν
 
double gmunu_up (const double *const x, int mu, int nu) const
 
int christoffel (double dst[4][4][4], const double *pos) const
 
double ScalarProd (const double pos[4], const double u1[4], const double u2[4]) const
 Scalar product. More...
 
void nullifyCoord (double coord[8], double &tdot2) const
 Set tdot (coord[4]) such that coord is light-like and return other possible tdot. More...
 
void nullifyCoord (double coord[8]) const
 Set tdot (coord[4]) such that coord is light-like. Everything is in geometrical units. More...
 
virtual void circularVelocity (double const pos[4], double vel[4], double dir=1.) const
 Yield circular velocity at a given position. More...
 
virtual void MakeCoord (const double coordin[8], const double cst[5], double coordout[8]) const
 Inverse function of MakeMomentumAndCst. More...
 
void MakeMomentum (const double coordin[8], const double cst[5], double coordout[8]) const
 Transforms from Boyer-Lindquist coordinates [t,r,th,phi,tdot,rdot,thdot,phidot] to [t,r,th,phi,pt,pr,pth,pphi] where pt,pr... are generalized momenta.
 
int myrk4 (const double coor[8], const double cst[5], double h, double res[8]) const
 Internal-use RK4 proxy.
 
int myrk4_adaptive (Gyoto::Worldline *line, const double coor[8], double lastnorm, double normref, double coor1[8], double h0, double &h1, double h1max=GYOTO_DEFAULT_DELTA_MAX) const
 Internal-use adaptive RK4 proxy.
 
void setParticleProperties (Worldline *line, const double *coord) const
 Set Metric-specific constants of motion. Used e.g. in KerrBL.
 
virtual int isStopCondition (double const *const coord) const
 Check whether integration should stop. More...
 
void observerTetrad (std::string const obskind, double const pos[4], double fourvel[4], double screen1[4], double screen2[4], double screen3[4]) const
 Computes the orthonormal local tetrad of the observer. More...
 
const std::string kind () const
 Get kind_.
 
int coordKind () const
 Get coordinate kind.
 
int getRefCount ()
 
void mass (const double)
 Set mass used in unitLength()
 
void mass (const double, const std::string &unit)
 Set mass used in unitLength()
 
double mass () const
 Get mass used in unitLength()
 
double mass (const std::string &unit) const
 Get mass used in unitLength()
 
double unitLength () const
 M * G / c^2, M is in kg, unitLength in meters. More...
 
double unitLength (const std::string &unit) const
 unitLength expressed in specified unit
 
double deltaMin () const
 
void deltaMin (double h1)
 
double deltaMax () const
 
virtual double deltaMax (double const pos[8], double delta_max_external) const
 
void deltaMax (double h1)
 
double deltaMaxOverR () const
 Get delta_max_over_r_.
 
void deltaMaxOverR (double t)
 Set delta_max_over_r_.
 
bool keplerian () const
 Get keplerian_.
 
void keplerian (bool)
 Set keplerian_.
 
virtual void cartesianVelocity (double const coord[8], double vel[3])
 Compute xprime, yprime and zprime from 8-coordinates.
 
virtual double SysPrimeToTdot (const double coord[4], const double v[3]) const
 Compute tdot as a function of dr/dt, dtheta/dt and dphi/dt. Everything is in geometrical units. More...
 
virtual double Norm3D (double *pos) const
 not clear
 
virtual double christoffel (const double coord[8], const int alpha, const int mu, const int nu) const
 Chistoffel symbol. More...
 
virtual int christoffel (double dst[4][4][4], const double coord[8]) const
 Chistoffel symbol. More...
 
virtual int diff (const double y[8], double res[8]) const
 F function such as dy/dtau=F(y,cst)
 
void incRefCount ()
 Increment the reference counter. Warning: Don't mess with the counter.
 
int decRefCount ()
 Decrement the reference counter and return current value. Warning: Don't mess with the counter.
 
void set (Property const &p, Value val)
 Set Value of a Property.
 
void set (Property const &p, Value val, std::string const &unit)
 Set Value (expressed in unit) of a Property.
 
void set (std::string const &pname, Value val)
 Set Value of a Property.
 
void set (std::string const &pname, Value val, std::string const &unit)
 Set Value (expressed in unit) of a Property.
 
Value get (Property const &p) const
 Get Value of a Property.
 
Value get (std::string const &pname) const
 Get Value of a Property.
 
Value get (Property const &p, std::string const &unit) const
 Get Value of a Property, converted to unit.
 
Value get (std::string const &pname, std::string const &unit) const
 Get Value of a Property, converted to unit.
 
Property const * property (std::string const pname) const
 Find property by name. More...
 
virtual void fillProperty (Gyoto::FactoryMessenger *fmp, Property const &p) const
 Output a single Property to XML. More...
 
virtual void fillElement (Gyoto::FactoryMessenger *fmp) const
 Fill the XML element for this Object. More...
 
virtual void setParameters (Gyoto::FactoryMessenger *fmp)
 Main loop for parsing Properties from XML description. More...
 
virtual int setParameter (std::string name, std::string content, std::string unit)
 Set parameter by name. More...
 
virtual void setParameter (Gyoto::Property const &p, std::string const &name, std::string const &content, std::string const &unit)
 Set parameter by Property (and name) More...
 
virtual void hook (Listener *listener)
 Start listening. More...
 
virtual void unhook (Listener *listener)
 Stop listening. More...
 

Static Public Attributes

static GYOTO_OBJECT Property const properties []
 

Protected Member Functions

int myrk4 (Worldline *line, const double coordin[8], double h, double res[8]) const
 RK4 integrator.
 
void kind (const std::string)
 Set kind_. More...
 
void coordKind (int coordkind)
 Set coordkind_. More...
 
virtual void tellListeners ()
 Call tell() on each hooked Listener. More...
 

Protected Attributes

double spin_
 Angular momentum parameter.
 
double a2_
 spin_*spin_
 
double a3_
 a2_*spin_
 
double a4_
 a2_*a2_
 
double difftol_
 Numerical tuning parameter. More...
 
double rsink_
 numerical horizon
 
double drhor_
 horizon security
 
bool generic_integrator_
 which integrator to use
 
double delta_min_
 Minimum integration step for the adaptive integrator.
 
double delta_max_
 Maximum integration step for the adaptive integrator.
 
double delta_max_over_r_
 Numerical tuning parameter. More...
 
bool keplerian_
 1 if circularVelocity should return the Newtonian Keplerian velocity, in r^-3/2
 
std::string kind_
 The "kind" that is output in the XML entity. More...
 

Private Member Functions

int CheckCons (const double coor_init[8], const double cst[5], double coor_fin[8]) const
 Ensure conservation of the constants of motion. More...
 
void Normalize4v (double coord[8], const double part_mass) const
 Normalize 4-velocity. More...
 
virtual int diff (const double y[8], const double cst[5], double res[8]) const
 Used in RK4 proxies.
 
virtual void computeCst (const double coord[8], double cst[5]) const
 

Friends

class Gyoto::SmartPointer< Gyoto::Metric::KerrBL >
 

Detailed Description

Metric around a Kerr black-hole in Boyer-Lindquist coordinates.

Member Typedef Documentation

typedef Gyoto::SmartPointer<Gyoto::SmartPointee> Gyoto::SmartPointee::Subcontractor_t(Gyoto::FactoryMessenger *)
inherited

A subcontractor builds an object upon order from the Factory.

Various classes need to provide a subcontractor to be able to instanciate themselves upon order from the Factory. A subcontractor is a function (often a static member function) which accepts a pointer to a FactoryMessenger as unique parameter, communicates with the Factory using this messenger to read an XML description of the object to build, and returns this objet. SmartPointee::Subcontractor_t* is just generic enough a typedef to cast to and from other subcontractor types: Astrobj::Subcontractor_t, Metric::Subcontractor_t, Spectrum::Subcontractor_t. A subcontractor needs to be registered using the relevant Register() function: Astrobj::Register(), Metric::Register(), Spectrum::Register().

Member Function Documentation

int Gyoto::Metric::KerrBL::CheckCons ( const double  coor_init[8],
const double  cst[5],
double  coor_fin[8] 
) const
private

Ensure conservation of the constants of motion.

Tweak thetadot if necessary.

virtual double Gyoto::Metric::Generic::christoffel ( const double  coord[8],
const int  alpha,
const int  mu,
const int  nu 
) const
virtualinherited

Chistoffel symbol.

Value of Christoffel symbol Γαμν at point (x1, x2, x3).

Reimplemented in Gyoto::Metric::NumericalMetricLorene, Gyoto::Metric::RotStar3_1, and Gyoto::Metric::Minkowski.

virtual int Gyoto::Metric::Generic::christoffel ( double  dst[4][4][4],
const double  coord[8] 
) const
virtualinherited

Chistoffel symbol.

Value of Christoffel symbol Γαμν at point (x1, x2, x3).

virtual void Gyoto::Metric::KerrBL::circularVelocity ( double const  pos[4],
double  vel[4],
double  dir = 1. 
) const
virtual

Yield circular velocity at a given position.

Give the velocity of a massive particle in circular orbit at the given position projected onto the equatorial plane. Such a velocity may not exist everywhere (or anywhere) for a given metric. This method is intended to be used by Astrobj classes such as Torus or ThinDisk.

If keplerian_ is set to true, this method should return the Keplerian velcity instead (derived classes should ensure this, see KerrBL::circularVelocity() for instance).

The default implementation throws an error if keplerian_ is set to false.

Parameters
posinput: position,
veloutput: velocity,
dir1 for corotating, -1 for counterrotating.

Reimplemented from Gyoto::Metric::Generic.

virtual void Gyoto::Metric::KerrBL::computeCst ( const double  coord[8],
double  cst[5] 
) const
privatevirtual

Integrator. Computes the evolution of y (initcond=y(0)).

void Gyoto::Metric::Generic::coordKind ( int  coordkind)
protectedinherited

Set coordkind_.

coordkind(int coordkind) is protected because, for most Metrics, it should not be changed in runtime.Set coordinate kind

double Gyoto::Metric::Generic::deltaMax ( ) const
inherited

Get delta_max_

virtual double Gyoto::Metric::Generic::deltaMax ( double const  pos[8],
double  delta_max_external 
) const
virtualinherited

Get delta max at a given position

Parameters
pos4-position
optional]delta_max_external external constraint on delta_max
Returns
the smallest value between delta_max_, delta_max_external, and R*delta_max_over_r_ where R is pos[1] in spherical coordinates and max(x1, x2, x3) in Cartesian coordinates.
void Gyoto::Metric::Generic::deltaMax ( double  h1)
inherited

Set delta_max_

double Gyoto::Metric::Generic::deltaMin ( ) const
inherited

Get delta_min_

void Gyoto::Metric::Generic::deltaMin ( double  h1)
inherited

Set delta_min_

virtual void Gyoto::Object::fillElement ( Gyoto::FactoryMessenger *  fmp) const
virtualinherited

Fill the XML element for this Object.

The base implementation simply calls fillProperty() for each Property defined for the Object.

Derived classes should avoid overriding fillElement(). It may make sense occasionally, e.g. to make sure that the metric is output first.

To customize how a given Property is rendered, it is better to override fillProperty().

If this method is overridden, the implementation should in general call fillElement() on the direct base.

virtual void Gyoto::Object::fillProperty ( Gyoto::FactoryMessenger *  fmp,
Property const &  p 
) const
virtualinherited

Output a single Property to XML.

The base implementation decides what to do based on the p.type. The format matches how setParameters() an setParameter() would interpret the XML descition.

Overriding this method should be avoided, but makes sense in some cases (for instance Screen::fillProperty() selects a different unit for Distance based on its magnitude, so that stellar sizes are expressed in solar radii while smaller sizes can be expressed in meters and larger sizes in parsecs).

Overriding implementation should fall-back on calling the implementation in the direct parent class:

class A: public Object {};
class B: public A {
using B::setParameter;
virtual void fillProperty(Gyoto::FactoryMessenger *fmp,
Property const &p) const ;
};
void B::fillProperty(Gyoto::FactoryMessenger *fmp,
Property const &p) const {
if (name=="Duff") fmp->doSomething();
else A::fillProperty(fmp, p);
}

Reimplemented in Gyoto::Screen, Gyoto::Astrobj::Star, Gyoto::Spectrometer::Uniform, Gyoto::Astrobj::Disk3D, Gyoto::Astrobj::PatternDisk, Gyoto::Astrobj::PolishDoughnut, and Gyoto::Astrobj::DirectionalDisk.

virtual double Gyoto::Metric::KerrBL::getPotential ( double  pos[4],
double  l_cst 
) const
virtual

Returns potential W=-ln(|u_t|) for a cst specific angular momentum l_cst Should be implemented in derived classes if useful If called on the base class, returns an error

Reimplemented from Gyoto::Metric::Generic.

virtual Property const* Gyoto::Metric::KerrBL::getProperties ( ) const
virtual

Get list of properties.

This method is declared automatically by the GYOTO_OBJECT macro and defined automatically by the GYOTO_PROPERTY_END macro.

Reimplemented from Gyoto::Metric::Generic.

virtual double Gyoto::Metric::KerrBL::getRmb ( ) const
virtual

Returns the marginally bound radius Should be implemented in derived classes if useful If called on the base class, returns an error

Reimplemented from Gyoto::Metric::Generic.

virtual double Gyoto::Metric::KerrBL::getRms ( ) const
virtual

Returns the marginally stable (ISCO) radius Should be implemented in derived classes if useful If called on the base class, returns an error

Reimplemented from Gyoto::Metric::Generic.

virtual double Gyoto::Metric::KerrBL::getSpecificAngularMomentum ( double  rr) const
virtual

Returns the specific angular momentum l=-u_phi/u_t Should be implemented in derived classes if useful If called on the base class, returns an error

Reimplemented from Gyoto::Metric::Generic.

void Gyoto::Metric::KerrBL::gmunu ( double  g[4][4],
const double *  pos 
) const
virtual

Metric coefficients.

The default implementation calls double gmunu(const double * x, int mu, int nu) const.

Parameters
[out]g4x4 array to store the coeefficients
[in]x4-position at which to compute the coefficients;
Returns
Metric coefficient gμ,ν at point x

Reimplemented from Gyoto::Metric::Generic.

double Gyoto::Metric::KerrBL::gmunu ( const double *const  x,
int  mu,
int  nu 
) const
virtual

Metric coefficients.

The default implementation calls Metric:: gmunu(double g[4][4], const double * pos) const

Parameters
x4-position at which to compute the coefficient;
mu1st index of coefficient, 0≤μ≤3;
nu2nd index of coefficient, 0≤ν≤3;
Returns
Metric coefficient gμ,ν at point x

Reimplemented from Gyoto::Metric::Generic.

virtual void Gyoto::Hook::Teller::hook ( Listener listener)
virtualinherited

Start listening.

Use from a Hook::Listener object method:

teller->hook(this)

where "this" is a Listener and "teller" is a Teller.

Use unhook() later to stop listening to a given Teller.

Parameters
listenerpointer to the new listener
virtual int Gyoto::Metric::KerrBL::isStopCondition ( double const *const  coord) const
virtual

Check whether integration should stop.

The integrating loop will ask this the Metric through this method whether or not it is happy to conitnue the integration. Typically, the Metric should answer 0 when everything is fine, 1 when too close to the event horizon, inside the BH...

Parameters
coord8-coordinate vector to check.

Reimplemented from Gyoto::Metric::Generic.

void Gyoto::Metric::Generic::kind ( const std::string  )
protectedinherited

Set kind_.

kind(const std::string) is protected because, for most Metrics, it should not be changed in runtime.Set kind_

virtual void Gyoto::Metric::KerrBL::MakeCoord ( const double  coordin[8],
const double  cst[5],
double  coordout[8] 
) const
virtual

Inverse function of MakeMomentumAndCst.

Computes pr, ptheta, E and L from rdot, thetadot, phidot, tdot

void Gyoto::Metric::KerrBL::Normalize4v ( double  coord[8],
const double  part_mass 
) const
private

Normalize 4-velocity.

To 0 or -1. Changes rdot to allow norm conservation.

void Gyoto::Metric::KerrBL::nullifyCoord ( double  coord[8],
double &  tdot2 
) const
virtual

Set tdot (coord[4]) such that coord is light-like and return other possible tdot.

Set coord[4] so that the 4-velocity coord[4:7] is lightlike, i.e. of norm 0. There may be up to two solutions. coord[4] is set to the hightest. The lowest can be retrieved in tdot2. Everything is expressed in geometrical units.

Parameters
[in,out]coord8-position, coord[4] will be set according to the other elements;
[out]tdot2will be set to the smallest solution

Reimplemented from Gyoto::Metric::Generic.

void Gyoto::Metric::KerrBL::nullifyCoord ( double  coord[8]) const
virtual

Set tdot (coord[4]) such that coord is light-like. Everything is in geometrical units.

Set coord[4] so that the 4-velocity coord[4:7] is lightlike, i.e. of norm 0. There may be up to two solutions. coord[4] is set to the hightest. The lowest can be retrieved using nullifyCoord(double coord[8], double& tdot2) const. Everything is expressed in geometrical units.

Parameters
[in,out]coord8-position, coord[4] will be set according to the other elements;

Reimplemented from Gyoto::Metric::Generic.

void Gyoto::Metric::KerrBL::observerTetrad ( std::string const  obskind,
double const  pos[4],
double  fourvel[4],
double  screen1[4],
double  screen2[4],
double  screen3[4] 
) const
virtual

Computes the orthonormal local tetrad of the observer.

Parameters
obskindinput: kind of observer (eg: "ZAMO","KeplerianObserver"...)
posinput: position,
fourveloutput: observer 4-velocity (norm -1)
screen1output: first vector in the screen plane
screen2output: second vector in the screen plane
screen3output: vector normal to the screen

Reimplemented from Gyoto::Metric::Generic.

Property const* Gyoto::Object::property ( std::string const  pname) const
inherited

Find property by name.

Look into the Property list for a Property whose name (or name_false, for a boolean Property) is pname. Return a const pointer to the first such property found, or NULL if none is found.

double Gyoto::Metric::KerrBL::ScalarProd ( const double  pos[4],
const double  u1[4],
const double  u2[4] 
) const
virtual

Scalar product.

Compute the scalarproduct of the two quadrivectors u1 and u2 in this Metric, at point pos expressed in coordinate system sys.

Parameters
pos4-position;
u11st quadrivector;
u22nd quadrivector;
Returns
u1*u2

Reimplemented from Gyoto::Metric::Generic.

virtual int Gyoto::Object::setParameter ( std::string  name,
std::string  content,
std::string  unit 
)
virtualinherited

Set parameter by name.

This function is used when parsing an XML description, if no Property of this name is found. Overriding implementation should fall-back on calling the direct's parent implementation:

class A: public Object {};
class B: public A {
using B::setParameter;
virtual int setParameter(std::string name,
std::string content,
std::string unit);
};
int B::setParameter(std::string name,
std::string content,
std::string unit) {
if (name=="Duff") doSomething(content, unit);
else return A::setParameter(name, content, unit);
return 0; // name was known
}
Parameters
nameXML name of the parameter (XML entity)
contentstring representation of the value
unitstring representation of the unit
Returns
0 if this parameter is known, 1 if it is not.

Reimplemented in Gyoto::Astrobj::Star, and Gyoto::Metric::RotStar3_1.

virtual void Gyoto::Object::setParameter ( Gyoto::Property const &  p,
std::string const &  name,
std::string const &  content,
std::string const &  unit 
)
virtualinherited

Set parameter by Property (and name)

This function is used when parsing an XML description, if Property (p) of this name is found (i.e. either p.name or p.name_false is equal to name). Implementation should fall-back on calling the direct's parent implementation:

class A: public Object {};
class B: public A {
using B::setParameter;
virtual void setParameter(Gyoto::Property const &p,
std::string name,
std::string content,
std::string unit);
};
void B::setParameter(Gyoto::Property const &p,
std::string name,
std::string content,
std::string unit) {
if (name=="Duff") doSomething(content, unit);
else A::setParameter(p, name, content, unit);
}
Parameters
pProperty that matches name (p.name == name or p.name_false == name)
nameXML name of the parameter (XML entity)
contentstring representation of the value
unitstring representation of the unit

Reimplemented in Gyoto::Astrobj::PolishDoughnut.

virtual void Gyoto::Object::setParameters ( Gyoto::FactoryMessenger *  fmp)
virtualinherited

Main loop for parsing Properties from XML description.

This function queries the FactoryMessenger for elements to parse, and tries to matche each element to a Property to set it accordingly.

Any class that tries to be buildable from XML must supply a subcontractor (for base classes such as Metric, Astrobj, Spectrum and Spectrometer, it is done as a template that must be specialized for each class).

This subcontractor typically looks somewhat like this:

SmartPointer<Metric::Generic>
Gyoto::Metric::MyKind::Subcontractor(FactoryMessenger* fmp) {
SmartPointer<MyKind> gg = new MyKind();
gg -> setParameters(fmp);
return gg;
}
 Although this is discouraged, it is possible to override the
 following functions to customize how XML entities are parsed:
   - setParameters() if low-level access to the
     FactoryMessenger is required;
   - setParameter(std::string name,
                  std::string content,
                  std::string unit)
     to interpret an entity that does not match a Property
     (e.g. alternative name);
   - setParameter(Gyoto::Property const &p,
                  std::string const &name,
                  std::string const &content,
                  std::string const &unit)
     to change how a Property is interpreted.
virtual double Gyoto::Metric::Generic::SysPrimeToTdot ( const double  coord[4],
const double  v[3] 
) const
virtualinherited

Compute tdot as a function of dr/dt, dtheta/dt and dphi/dt. Everything is in geometrical units.

Parameters
coord4-position (geometrical units);
v3-velocity dx1/dx0, dx2/dx0, dx3/dx0;
Returns
tdot = dx0/dtau.
virtual void Gyoto::Hook::Teller::tellListeners ( )
protectedvirtualinherited

Call tell() on each hooked Listener.

Whenever a Teller mutates, it should warn any Listener hooked to it using tellListeners().

virtual void Gyoto::Hook::Teller::unhook ( Listener listener)
virtualinherited

Stop listening.

Use from a Hook::Listener object method:

teller->unhook(this)

where "this" is a Listener, "teller" is a Teller, and "this" has called teller->hook(this) previously.

Parameters
listenerpointer to the listener
double Gyoto::Metric::Generic::unitLength ( ) const
inherited

M * G / c^2, M is in kg, unitLength in meters.

Metrics implementations are free to express lengths and distances in whatever unit they see fit (presumably most often geometrical units). This function returns this unit in SI (meters).

Member Data Documentation

double Gyoto::Metric::Generic::delta_max_over_r_
protectedinherited

Numerical tuning parameter.

Ensure that delta (the numerical integration step) is never larger than a fraction of the distance between the current location and the center of the coordinate system.

For investigations close to the event horizon, 0.5 is usually fine. If high accuracy is needed long after deflection (weak lensing), then this must be smaller. A good test is to look at a MinDistance map for a FixedStar: it must be smooth.

double Gyoto::Metric::KerrBL::difftol_
protected

Numerical tuning parameter.

Small values yield more accurate integration at the expanse of computing time.

std::string Gyoto::Object::kind_
protectedinherited

The "kind" that is output in the XML entity.

E.g. for an Astrobj, fillElement() will ensure

<Astrobj kind="kind_">...</Astrobj>

is written.


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