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ExpDiskGeometry Class Reference

#include <ExpDiskGeometry.hpp>

Inheritance diagram for ExpDiskGeometry:
Inheritance graph
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Public Member Functions

double maxRadius () const
 
double maxZ () const
 
double minRadius () const
 
double scaleHeight () const
 
double scaleLength () 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 ItemgetItemProperty (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)
 
Itemoperator= (const Item &)=delete
 
Itemparent () 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

 ExpDiskGeometry ()
 
double density (double R, double z) const override
 
double randomCylRadius () const override
 
double randomZ () const override
 
void setupSelfBefore () override
 
double SigmaR () const override
 
double SigmaZ () const override
 
- Protected Member Functions inherited from SepAxGeometry
 SepAxGeometry ()
 
- Protected Member Functions inherited from AxGeometry
 AxGeometry ()
 
- Protected Member Functions inherited from Geometry
 Geometry ()
 
Randomrandom () 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 = ExpDiskGeometry
 

Private Attributes

const double & _hR
 
const double & _hz
 
double _maxRadius
 
double _maxZ
 
double _minRadius
 
double _rho0
 
const double & _Rmax
 
const double & _Rmin
 
double _scaleHeight
 
double _scaleLength
 
const double & _zmax
 

Friends

class ItemRegistry
 

Detailed Description

The ExpDiskGeometry class is a subclass of the SepAxGeometry class, and describes axisymmetric geometries characterized by a double-exponential profile, in which the density decreases exponentially in the radial and the vertical directions; see van der Kruit (1986, A&A, 157, 230–244). A truncation can be applied in both the radial and vertical direction, and an inner cylindrical hole can be included. In formula form

\[ \rho(R,z) = \rho_0\, {\text{e}}^{-\frac{R}{h_R}-\frac{|z|}{h_z}}, \]

for \(R_{\text{min}} \leq R \leq R_{\text{max}}\) and \(|z|\leq z_{\text{max}}\). The model contains five free parameters: the scale length \(h_R\), the vertical scale height \(h_z\), the radial truncation radius \(R_{\text{max}}\), the vertical truncation radius \(z_{\text{max}}\), and the inner radius \(R_{\text{min}}\).

Constructor & Destructor Documentation

◆ ExpDiskGeometry()

ExpDiskGeometry::ExpDiskGeometry ( )
inlineprotected

Default constructor for concrete Item subclass ExpDiskGeometry : "an exponential disk geometry" .

Member Function Documentation

◆ density()

double ExpDiskGeometry::density ( double  R,
double  z 
) const
overrideprotectedvirtual

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

Implements AxGeometry.

◆ maxRadius()

ExpDiskGeometry::maxRadius ( ) const
inline

This function returns the value of the discoverable double property maxRadius : "the truncation radius (zero means no truncation)" .

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

◆ maxZ()

ExpDiskGeometry::maxZ ( ) const
inline

This function returns the value of the discoverable double property maxZ : "the truncation height (zero means no truncation)" .

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

◆ minRadius()

ExpDiskGeometry::minRadius ( ) const
inline

This function returns the value of the discoverable double property minRadius : "the radius of the central cavity" .

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" .

◆ randomCylRadius()

double ExpDiskGeometry::randomCylRadius ( ) const
overrideprotectedvirtual

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 \(R\). Substituting the exponential radial profile (without truncation) into this equation, we obtain

\[ {\cal{X}} = 1 - \left( 1+\frac{R}{h_R} \right) \exp \left( -\frac{R}{h_R} \right). \]

This equation can be solved by means of the Lambert function of order \(-1\), yielding

\[ R = h_R \left[ -1-W_{-1} \left( \frac{ {\cal{X}}-1}{\text{e}} \right) \right]. \]

The Lambert function \(W_{-1}(z)\) is implemented in the function SpecialFunctions::LambertW1. The truncation and the inner hole are taken into account by rejecting values larger than \(R_{\text{max}}\) or smaller than \(R_{\text{min}}\).

Implements SepAxGeometry.

◆ randomZ()

double ExpDiskGeometry::randomZ ( ) const
overrideprotectedvirtual

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 exponential disk geometry, this integration is simple, and the inversion results in

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

The truncation is taken into account by rejecting values \(|z|\) larger than \(z_{\text{max}}\).

Implements SepAxGeometry.

◆ scaleHeight()

ExpDiskGeometry::scaleHeight ( ) const
inline

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

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

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

◆ scaleLength()

ExpDiskGeometry::scaleLength ( ) const
inline

This function returns the value of the discoverable double property scaleLength : "the scale length" .

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

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

◆ setupSelfBefore()

void ExpDiskGeometry::setupSelfBefore ( )
overrideprotectedvirtual

This function verifies the validity of the parameters. The central density \(\rho_0\) is set by the normalization condition that the total mass equals one. One finds after some elementary calculus

\[ \frac{1}{\rho_0} = 4\pi\, h_R^2\, h_z \left( 1 - {\text{e}}^{-z_{\text{max}}/h_z} \right) \left[ \left( 1+\frac{R_{\text{min}}}{h_R} \right) {\text{e}}^{-R_{\text{min}}/h_R}- \left( 1+\frac{R_{\text{max}}}{h_R} \right) {\text{e}}^{-R_{\text{max}}/h_R} \right] . \]

In case there is no truncation in either radial or vertical directions and no inner hole, this reduces to

\[ \rho_0 = \frac{1}{ 4\pi\, h_R^2\, h_z }. \]

Reimplemented from SimulationItem.

◆ SigmaR()

double ExpDiskGeometry::SigmaR ( ) const
overrideprotectedvirtual

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 exponential disc geometry we find

\[ \Sigma_R = \rho_0 h_R \left( {\text{e}}^{-R_{\text{min}}/h_R} - {\text{e}}^{-R_{\text{max}}/h_R} \right), \]

which reduces to \( \Sigma_R = \rho_0 h_R \) if there is no radial truncation and no inner hole.

Implements AxGeometry.

◆ SigmaZ()

double ExpDiskGeometry::SigmaZ ( ) const
overrideprotectedvirtual

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 exponential disc geometry we find

\[ \Sigma_Z = 2\,\rho_0 h_Z \left( 1 - {\text{e}}^{-z_{\text{max}}/h_z} \right), \]

which reduces to \( \Sigma_Z = 2\,\rho_0 h_z \) if there is no vertical truncation. If there is an inner hole, obviously \(\Sigma_Z=0\).

Implements Geometry.


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