TorusGeometry Class Reference

#include <TorusGeometry.hpp>

Inheritance diagram for TorusGeometry:

Public Member Functions
double	cutoffRadius () const
double	density (double R, double z) const override
double	exponent () const
Position	generatePosition () const override
double	index () const
double	maxRadius () const
double	minRadius () const
double	openingAngle () const
bool	reshapeInnerRadius () const
double	SigmaR () const override
double	SigmaZ () const override
Public Member Functions inherited from AxGeometry
virtual double	density (double R, double z) const =0
double	density (Position bfr) const override
int	dimension () const override
virtual double	SigmaR () const =0
double	SigmaX () const override
double	SigmaY () const override
virtual double	density (Position bfr) const =0
virtual int	dimension () const =0
virtual Position	generatePosition () const =0
virtual double	SigmaX () const =0
virtual double	SigmaY () const =0
virtual double	SigmaZ () const =0
Public Member Functions inherited from SimulationItem
template<class T >
T *	find (bool setup=true) const
template<class T >
T *	interface (int levels=-999999, bool setup=true) const
virtual string	itemName () const
void	setup ()
string	typeAndName () const
Public Member Functions inherited from Item
	Item (const Item &)=delete
virtual	~Item ()
void	addChild (Item *child)
const vector< Item * > &	children () const
virtual void	clearItemListProperty (const PropertyDef *property)
void	destroyChild (Item *child)
virtual bool	getBoolProperty (const PropertyDef *property) const
virtual vector< double >	getDoubleListProperty (const PropertyDef *property) const
virtual double	getDoubleProperty (const PropertyDef *property) const
virtual string	getEnumProperty (const PropertyDef *property) const
virtual int	getIntProperty (const PropertyDef *property) const
virtual vector< Item * >	getItemListProperty (const PropertyDef *property) const
virtual Item *	getItemProperty (const PropertyDef *property) const
virtual string	getStringProperty (const PropertyDef *property) const
int	getUtilityProperty (string name) const
virtual void	insertIntoItemListProperty (const PropertyDef *property, int index, Item *item)
Item &	operator= (const Item &)=delete
Item *	parent () const
virtual void	removeFromItemListProperty (const PropertyDef *property, int index)
virtual void	setBoolProperty (const PropertyDef *property, bool value)
virtual void	setDoubleListProperty (const PropertyDef *property, vector< double > value)
virtual void	setDoubleProperty (const PropertyDef *property, double value)
virtual void	setEnumProperty (const PropertyDef *property, string value)
virtual void	setIntProperty (const PropertyDef *property, int value)
virtual void	setItemProperty (const PropertyDef *property, Item *item)
virtual void	setStringProperty (const PropertyDef *property, string value)
void	setUtilityProperty (string name, int value)
virtual string	type () const

Protected Member Functions
	TorusGeometry ()
void	setupSelfBefore () override
Protected Member Functions inherited from AxGeometry
	AxGeometry ()
Protected Member Functions inherited from Geometry
	Geometry ()
Random *	random () const
void	setupSelfBefore () override
Protected Member Functions inherited from SimulationItem
	SimulationItem ()
virtual bool	offersInterface (const std::type_info &interfaceTypeInfo) const
virtual void	setupSelfAfter ()
virtual void	setupSelfBefore ()
Protected Member Functions inherited from Item
	Item ()

Private Types
using	BaseType = AxGeometry
using	ItemType = TorusGeometry

Private Attributes
double	_A
double	_cutoffRadius
const double &	_Delta
double	_exponent
double	_index
double	_maxRadius
double	_minRadius
double	_openingAngle
const double &	_p
const double &	_q
const bool &	_rani
const double &	_rcut
bool	_reshapeInnerRadius
const double &	_rmax
const double &	_rmin
double	_sdiff
double	_sinDelta
double	_smin
double	_tmax
double	_tmin

Friends
class	ItemRegistry

Detailed Description

The TorusGeometry class is a subclass of the AxGeometry class and describes the geometry of an axisymmetric torus as assumed to be present in the centre of active galactic nuclei (AGN). This geometry is described by a radial power-law density with a finite opening angle; see Stalevski et al. (2012, MNRAS, 420, 2756–2772) and Granato & Danese (1994, MNRAS, 268, 235). In formula, it is most easily expressed in spherical coordinates as

$$\rho (r,\theta )=A{r}^{-p}{\text{e}}^{-q|\cos \theta |}\text{for }{r}_{\text{min}}<r<r_{\text{max}}\text{ and }\frac{\pi }{2}-\mathrm{\Delta }<\theta <\frac{\pi }{2}+\mathrm{\Delta }.$$

There are five free parameters describing this dust geometry: the inner and outer radii $r_{\text{min}}$ and $r_{\text{max}}$ of the torus, the radial power law index $p$, the polar index $q$ and the angle $\mathrm{\Delta }$ describing the opening angle of the torus.

If the dusty system under consideration is in the vicinity of an AGN central engine or another source which is luminous enough to heat the dust up to sublimation temperature, the inner radius should correspond to sublimation radius and scale as $r_{\text{min}}\propto L(\theta {)}^{0.5}$ (Barvainis, 1987, ApJ, 320, 537, eq (5)). If the primary source assumes anisotropic emission, the inner radius must follow the same dependence as the distribution of the primary source luminosity. Otherwise, dust temperature on the inner boundary of geometry is very likely to be under- or over-estimated. Thus, if the NetzerAccretionDiskGeometry distribution is chosen to describe primary source emission, it is recommended to turn on the anisotropic inner radius option for the torus geometry. The inner radius will then be set by the following formula:

$$r_{\text{min}}\propto (\cos \theta (2\cos \theta +1){)}^{0.5}.$$

This should allow dust to approach all the way to the primary central source in the equatorial plane. However, due to the finite resolution of dust cells, it may happen that some of the innermost cells end up with unphysically high temperatures. For this reason, there is an additional input parameter, the cutoff radius $r_{\text{cut}}$. The value of the cutoff radius is usually found after a few trial-and-error experiments by inspecting temperature distribution maps, until the inner wall of the geometry is at the expected sublimation temperature for a given dust population.

The total dust mass of the model corresponds to the mass of the original geometry, before the inner wall is reshaped to account for anisotropy; the difference is usually rather small.

Constructor & Destructor Documentation

TorusGeometry()

TorusGeometry::TorusGeometry ( )

inlineprotected

Default constructor for concrete Item subclass TorusGeometry : "a torus geometry" .

Member Function Documentation

cutoffRadius()

TorusGeometry::cutoffRadius ( ) const

inline

This function returns the value of the discoverable double property cutoffRadius : "the inner cutoff radius of the torus" .

This property represents a physical quantity of type "length" .

The minimum value for this property is "[0" .

The default value for this property is given by the conditional value expression "0" .

This property is relevant only if the Boolean expression "reshapeInnerRadius" evaluates to true after replacing the names by true or false depending on their presence.

density()

double TorusGeometry::density ( double  R, double  z  ) const

overridevirtual

This function returns the density $\rho (R,z)$ at the cylindrical radius $R$ and height $z$. It just implements the analytical formula.

Implements AxGeometry.

exponent()

TorusGeometry::exponent ( ) const

inline

This function returns the value of the discoverable double property exponent : "the radial powerlaw exponent p of the torus" .

The minimum value for this property is "[0" .

generatePosition()

Position TorusGeometry::generatePosition ( ) const

overridevirtual

This function generates a random position from the torus geometry, by drawing a random point from the three-dimensional probability density $p(\mathbf{r})\text{d}\mathbf{r}=\rho (\mathbf{r})\text{d}\mathbf{r}$. For the torus geometry, the density is a separable function of $r$ and $\theta$, so that a random position can hence be constructed by combining random spherical coordinates, each chosen from their own probability distributions. A random azimuth $\varphi$ is readily found by chosing a random deviate $\mathcal{X}$ and setting $\varphi =2\pi \mathcal{X}$.

For the radial coordinate, the appropriate probability distribution is $p(r)\text{d}r\propto r^{2-p}\text{d}r$. A random radius is generated by picking a new uniform deviate $\mathcal{X}$, and solving the equation

$$\mathcal{X}={\int }_{r_{\text{min}}}^{r}p(r^{\prime })\text{d}{r}^{\prime }$$

for $r$. For $p\ne 3$ we find

$$\mathcal{X}=\frac{r^{3-p}-r_{\text{min}}^{3-p}}{r_{\text{max}}^{3-p}-r_{\text{min}}^{3-p}}.$$

Inverting this results in

$$r={[(1-\mathcal{X})r_{\text{min}}^{3-p}+\mathcal{X}{r}_{\text{max}}^{3-p}]}^{\frac{1}{3-p}}.$$

For $p=3$ this expression does not hold, and for $p\approx 3$ it breaks down numerically. So for $p\approx 3$ we can write the general expression

$$r={\text{gexp}}_{p-2}[{\text{gln}}_{p-2}{r}_{\text{min}}+\mathcal{X}({\text{gln}}_{p-2}{r}_{\text{max}}-{\text{gln}}_{p-2}{r}_{\text{min}})].$$

In this expression, ${\text{gln}}_{p}x$ and ${\text{gexp}}_{p}x$ are the generalized logarithm and exponential functions defined in SpecialFunctions::gln and SpecialFunctions::gexp respectively.

Finally, for the polar angle, the appropriate distribution function is

$$p(\theta )\text{d}\theta \propto e^{-q|\cos \theta |}\sin \theta \text{d}\theta .$$

A random polar angle is generated by picking a new uniform deviate $\mathcal{X}$, and solving the equation

$$\mathcal{X}={\int }_0^{\theta }p({\theta }^{\prime })\text{d}{\theta }^{\prime }$$

for $\theta$. We obtain after some calculation

$$\mathcal{X}=\{\begin{array}{ll}{\displaystyle \frac{1}{2}}(1-{\displaystyle \frac{1-{\text{e}}^{-q\cos \theta }}{1-{\text{e}}^{-q\sin \mathrm{\Delta }}}})& \text{for }\frac{\pi }{2}-\mathrm{\Delta }<\theta <\frac{\pi }{2}\\ {\displaystyle \frac{1}{2}}(1+{\displaystyle \frac{1-{\text{e}}^{q\cos \theta }}{1-{\text{e}}^{-q\sin \mathrm{\Delta }}}})& \text{for }\frac{\pi }{2}<\theta <\frac{\pi }{2}+\mathrm{\Delta }\end{array}$$

Inverting this gives

$$\cos \theta =\{\begin{array}{ll}-{\displaystyle \frac{1}{q}}\ln [1-(1-{\text{e}}^{-q\sin \mathrm{\Delta }})(1-2\mathcal{X})]& \text{if }0<\mathcal{X}<\frac{1}{2}\\ {\displaystyle \frac{1}{q}}\ln [1-(1-{\text{e}}^{-q\sin \mathrm{\Delta }})(2\mathcal{X}-1)]& \text{if }\frac{1}{2}<\mathcal{X}<1\end{array}.$$

Implements Geometry.

index()

TorusGeometry::index ( ) const

inline

This function returns the value of the discoverable double property index : "the polar index q of the torus" .

The minimum value for this property is "[0" .

maxRadius()

TorusGeometry::maxRadius ( ) const

inline

This function returns the value of the discoverable double property maxRadius : "the maximum radius of the torus" .

This property represents a physical quantity of type "length" .

The minimum value for this property is "]0" .

minRadius()

TorusGeometry::minRadius ( ) const

inline

This function returns the value of the discoverable double property minRadius : "the minimum radius of the torus" .

This property represents a physical quantity of type "length" .

The minimum value for this property is "]0" .

openingAngle()

TorusGeometry::openingAngle ( ) const

inline

This function returns the value of the discoverable double property openingAngle : "the half opening angle of the torus" .

This property represents a physical quantity of type "posangle" .

The minimum value for this property is "[0 deg" .

The maximum value for this property is "90 deg]" .

reshapeInnerRadius()

TorusGeometry::reshapeInnerRadius ( ) const

inline

This function returns the value of the discoverable Boolean property reshapeInnerRadius : "reshape the inner radius according to the Netzer luminosity profile" .

The default value for this property is given by the conditional value expression "false" .

This property is displayed only if the Boolean expression "Level2" evaluates to true after replacing the names by true or false depending on their presence.

setupSelfBefore()

void TorusGeometry::setupSelfBefore ( )

overrideprotectedvirtual

This function calculates some frequently used values. The normalization parameter $A$ is set by the normalization condition that total mass equals one, i.e.

$$1=2\pi A{\int }_{\pi /2-\mathrm{\Delta }}^{\pi /2+\mathrm{\Delta }}{e}^{-q|\cos \theta |}\sin \theta \text{d}\theta {\int }_{r_{\text{min}}}^{r_{\text{max}}}{r}^{2-p}\text{d}r.$$

This results in

$$A=\frac{q}{4\pi (1-{\text{e}}^{-q\sin \mathrm{\Delta }})}\frac{1}{{\text{gln}}_{p-2}{r}_{\text{max}}-{\text{gln}}_{p-2}{r}_{\text{min}}},$$

with ${\text{gln}}_{p}x$ the generalized logarithm defined in SpecialFunctions::gln. If $q=0$, this expression reduces to

$$A=\frac{1}{4\pi \sin \mathrm{\Delta }({\text{gln}}_{p-2}{r}_{\text{max}}-{\text{gln}}_{p-2}{r}_{\text{min}})}.$$

Reimplemented from Geometry.

SigmaR()

double TorusGeometry::SigmaR ( ) const

overridevirtual

This function returns the radial surface density, i.e. the integration of the density along a line in the equatorial plane starting at the centre of the coordinate system,

$${\mathrm{\Sigma }}_R={\int }_0^{\mathrm{\infty }}\rho (R,0)\text{d}R.$$

For the torus geometry,

$${\mathrm{\Sigma }}_R=A({\text{gln}}_p{r}_{\text{max}}-{\text{gln}}_p{r}_{\text{min}})$$

with ${\text{gln}}_{p}x$ the generalized logarithm defined in SpecialFunctions::gln.

Implements AxGeometry.

SigmaZ()

double TorusGeometry::SigmaZ ( ) const

overridevirtual

This function returns the Z-axis surface density, i.e. the integration of the density along the entire Z-axis,

$${\mathrm{\Sigma }}_Z={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\rho (0,0,z)\text{d}z.$$

For the torus geometry this integral is simply zero (we exclude the special limiting case where $\mathrm{\Delta }=\frac{\pi }{2}$).

Implements Geometry.

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

• TorusGeometry.hpp