#include <RingGeometry.hpp>

Inheritance diagram for RingGeometry:

[legend]

Public Member Functions
double	density (double R, double z) const override

double	height () const

double	randomCylRadius () const override

double	randomZ () const override

double	ringRadius () const

double	SigmaR () const override

double	SigmaZ () const override

double	width () const

Public Member Functions inherited from SepAxGeometry
Position	generatePosition () const override

virtual double	randomCylRadius () const =0

virtual double	randomZ () const =0

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
	RingGeometry ()

void	setupSelfBefore () override

Protected Member Functions inherited from SepAxGeometry
	SepAxGeometry ()

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 = SepAxGeometry

using	ItemType = RingGeometry

Private Attributes
double	_A

double	_height

const double &	_hz

const double &	_R0

double	_ringRadius

Array	_Rv

const double &	_w

double	_width

Array	_Xv

Friends
class	ItemRegistry

Detailed Description

The RingGeometry class is a subclass of the SepAxGeometry class and describes the geometry of a ring as one would consider in dust-lane early-type galaxies. The geometry is characterized by a gaussian profile in the radial direction and an exponential fall-off in the vertical direction, resulting in the axisymmetric density profile

\[ \rho(R,z) = A\, \exp\left[ -\frac{(R-R_0)^2}{2w^2} \right] \exp\left( -\frac{|z|}{h_z} \right). \]

There are three free parameters in this class: the radius $R_0$ of the ring, the radial dispersion $w$ of the ring and the vertical scale height $h_z$.

Constructor & Destructor Documentation

◆ RingGeometry()

RingGeometry::RingGeometry ( )

inlineprotected

Default constructor for concrete Item subclass RingGeometry : "a ring geometry" .

Member Function Documentation

◆ density()

double RingGeometry::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.

◆ height()

RingGeometry::height ( ) const

inline

This function returns the value of the discoverable double property height : "the scale height of the ring" .

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

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

◆ randomCylRadius()

double RingGeometry::randomCylRadius ( ) const

overridevirtual

This function returns the cylindrical radius $R$ of a random position drawn from the geometry, by picking a uniform deviate ${\cal{X}}$ and solving the equation

\[ {\cal{X}} = 2\pi \int_0^R \rho_R(R')\, R'\, {\text{d}}R' \]

for $R$. We just use the vector of cumulative masses stored internally.

Implements SepAxGeometry.

◆ randomZ()

double RingGeometry::randomZ ( ) const

overridevirtual

This function returns the height $z$ of a random position drawn from the geometry, by picking a uniform deviate ${\cal{X}}$ and solving the equation

\[ {\cal{X}} = \int_{-\infty}^z \rho_z(z')\, {\text{d}}z' \]

for $z$. For the ring geometry with its exponential vertical profile, this integration is simple, and the inversion results in

\[ z = \begin{cases} \; h_z\,\ln(2{\cal{X}}) & \text{if $0<{\cal{X}}<\tfrac{1}{2}$,} \\ \;-h_z\,\ln[2(1-{\cal{X}})] & \text{if $\tfrac{1}{2}<{\cal{X}}<1$.} \end{cases} \]

Implements SepAxGeometry.

◆ ringRadius()

RingGeometry::ringRadius ( ) const

inline

This function returns the value of the discoverable double property ringRadius : "the radius of the ring" .

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

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

◆ setupSelfBefore()

void RingGeometry::setupSelfBefore ( )

overrideprotectedvirtual

This function verifies the validity of the parameters of the geometry. The density normalization parameter $A$ is set by the normalization condition that the total mass equals one. We find

\[ A = \frac{1}{4\pi\,w^2\,h_z \left[ {\text{e}}^{-t^2} + \sqrt{\pi}\,t\,(1 + {\text{erf}}\,t) \right]}, \]

with $t = R_0/\sqrt{2}w$. The routine also creates a vector with the cumulative distribution $X(r)$ corresponding to the radial part of the density, i.e. mass

\[ X(R) = \int_0^R \rho_R(R')\, R'\, {\text{d}}R' \]

at a large number of radii. For the ring geometry,

\[ \rho_R(R') = 4\pi\,A\,h_z \exp\left[ -\frac{(R-R_0)^2}{2w^2} \right], \]

and thus

\[ X(R) = 4\pi\,A\,w^2\,h_z \left[ \left({\text{e}}^{-t^2}-{\text{e}}^{-u^2}\right) + \sqrt{\pi}\,t\,({\text{erf}}\,t - {\text{erf}}\,u) \right], \]

with $t$ as defined before and $u = (R_0-R)/\sqrt{2}w$.

Reimplemented from Geometry.

◆ SigmaR()

double RingGeometry::SigmaR ( ) const

overridevirtual

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

\[ \Sigma_R = \int_0^infty \rho(R,0)\,{\text{d}}R. \]

For the ring geometry we obtain

\[ \Sigma_R = \sqrt{\frac{\pi}{2}}\,A\,w \left[ 1 + {\text{erf}} \left( \frac{R_0}{\sqrt{2} w} \right) \right]. \]

Implements AxGeometry.

◆ SigmaZ()

double RingGeometry::SigmaZ ( ) const

overridevirtual

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

\[ \Sigma_Z = \int_{-\infty}^\infty \rho(0,0,z)\, {\text{d}}z.\]

For the ring geometry we find

\[ \Sigma_{\text{f}} = 2A\,h_z\, \exp\left(-\frac{R_0^2}{2w^2}\right). \]

Implements Geometry.

◆ width()

RingGeometry::width ( ) const

inline

This function returns the value of the discoverable double property width : "the radial width of the ring" .

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

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

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

RingGeometry.hpp

Public Member Functions

Protected Member Functions

Private Types

Private Attributes

Friends

Detailed Description

Constructor & Destructor Documentation

◆ RingGeometry()

Member Function Documentation

◆ density()

◆ height()

◆ randomCylRadius()

◆ randomZ()

◆ ringRadius()

◆ setupSelfBefore()

◆ SigmaR()

◆ SigmaZ()

◆ width()