BrokenExpDiskGeometry Class Reference

#include <BrokenExpDiskGeometry.hpp>

Inheritance diagram for BrokenExpDiskGeometry:

Public Member Functions
double	breakRadius () const
double	scaleHeight () const
double	scaleLengthInner () const
double	scaleLengthOuter () const
double	sharpness () 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
	BrokenExpDiskGeometry ()
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 ()
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 = BrokenExpDiskGeometry

Private Member Functions
double	radialDensity (double R) const

Private Attributes
double	_beta
double	_breakRadius
const double &	_hinn
const double &	_hout
const double &	_hz
const double &	_Rb
double	_rho0
Array	_Rv
const double &	_s
double	_scaleHeight
double	_scaleLengthInner
double	_scaleLengthOuter
double	_sharpness
double	_SigmaR
Array	_Xv

Friends
class	ItemRegistry

Detailed Description

The BrokenExpDiskGeometry class is a subclass of the SepAxGeometry class. It describes a particular class of models for discs with a so-called break in the radial profile. More specifically, the radial density profile is a broken exponential profile: an inner exponential profile with scale length $h_{R,\text{inn}}$ and an outer exponential profile with scale length $h_{R,\text{out}}$. There is no limitation in the nature of the break: the inner profile can be shallower than the outer one ( $h_{R,\text{out}}<h_{R,\text{inn}}$) or steeper ( $h_{R,\text{out}}>h_{R,\text{inn}}$). These two asymptotic profiles are joined smoothly at a break radius $R_{\text{b}}$, with a dimensionless parameter $s$ that sets the smoothness of the transition between these two regimes. For large values of the smoothness parameter, $s\gg 1$, the transition is sharp, whereas for small values, $s\sim 1$, the transition is very gradual. In the vertical direction, the density decreases exponentially. No truncation is applied. In formula form,

$$\rho (R,z)={\rho }_0{\text{e}}^{-\frac{R}{h_{\text{inn}}}-\frac{|z|}{h_z}}{(1+{\text{e}}^{\frac{s(R-R_{\text{b}})}{h_{\text{out}}}})}^{\frac{1}{s}(\frac{h_{\text{out}}}{h_{\text{inn}}}-1)}.$$

This geometry is strongly inspired by Erwin et al. (2008, AJ, 135, 20) and Erwin (2015, ApJ, 799, 226), but has one significant difference: in our formulation, the sharpness $s$ is a dimensionless number, whereas in Erwin et al. (2008) the sharpness parameter $\alpha$ is a quantity with dimension length ${}^{-1}$. More specifically,

$$\alpha =\frac{s}{h_{\text{out}}}$$

The model contains five free parameters: the scale length of inner disc $h_{\text{inn}}$, the scale length of outer disc $h_{\text{out}}$, the scale height $h_z$, the break radius $R_{\text{b}}$, and the sharpness of the break $s$. The final parameter that appears in the formula above is the central density ${\rho }_0$; it is not a free parameter, but such that the total mass of the geometry is normalized to one.

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

Constructor & Destructor Documentation

BrokenExpDiskGeometry()

BrokenExpDiskGeometry::BrokenExpDiskGeometry ( )

inlineprotected

Default constructor for concrete Item subclass BrokenExpDiskGeometry : "a broken exponential disk geometry" .

Member Function Documentation

breakRadius()

BrokenExpDiskGeometry::breakRadius ( ) const

inline

This function returns the value of the discoverable double property breakRadius : "the break radius" .

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

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

density()

double BrokenExpDiskGeometry::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.

radialDensity()

double BrokenExpDiskGeometry::radialDensity ( double  R ) const

private

This private member function returns the radial part of the density distribution, i.e.

$${\rho }_R(R)={\text{e}}^{-\frac{R}{h_{\text{inn}}}}{(1+{\text{e}}^{\frac{s(R-R_{\text{b}})}{h_{\text{out}}}})}^{\frac{1}{s}(\frac{h_{\text{out}}}{h_{\text{inn}}}-1)}$$

It is just an auxiliary function.

randomCylRadius()

double BrokenExpDiskGeometry::randomCylRadius ( ) const

overrideprotectedvirtual

This function returns the cylindrical radius $R$ of a random position drawn from the geometry, by picking a uniform deviate $\mathcal{X}$. We just use the vector of cumulative masses stored internally.

Implements SepAxGeometry.

randomZ()

double BrokenExpDiskGeometry::randomZ ( ) const

overrideprotectedvirtual

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

$$\mathcal{X}={\int }_{-\mathrm{\infty }}^z{\rho }_z(z^{\prime })\text{d}{z}^{\prime }$$

for $z$. For the exponential disk geometry, this integration is simple, and the inversion results in

$$z=\{\begin{array}{ll}{h}_z\ln (2\mathcal{X})& \text{if }0<\mathcal{X}<\frac{1}{2}\text{,}\\ -h_z\ln [2(1-\mathcal{X})]& \text{if }\frac{1}{2}<\mathcal{X}<1\text{.}\end{array}$$

Implements SepAxGeometry.

scaleHeight()

BrokenExpDiskGeometry::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" .

scaleLengthInner()

BrokenExpDiskGeometry::scaleLengthInner ( ) const

inline

This function returns the value of the discoverable double property scaleLengthInner : "the scale length of the inner disk" .

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

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

scaleLengthOuter()

BrokenExpDiskGeometry::scaleLengthOuter ( ) const

inline

This function returns the value of the discoverable double property scaleLengthOuter : "the scale length of the outer disk" .

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

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

setupSelfBefore()

void BrokenExpDiskGeometry::setupSelfBefore ( )

overrideprotectedvirtual

This function calculates some frequently used values. The central density ${\rho }_0$ is set by the normalization condition that the total mass equals one. One finds after some elementary calculus

$${\rho }_0=\frac{1}{4\pi h_z{I}_R}$$

where $I_R$ is the integral

$$I_R={\int }_0^{\mathrm{\infty }}{\rho }_R(R)R\text{d}R$$

with ${\rho }_R(R)$ the radial part of the density distribution. This integral is calculated using a simple trapezoidal integration. This routine immediately also stores the cumulative radial distribution

$$X(R)=\frac{1}{I_R}{\int }_0^R{\rho }_R(R^{\prime })R^{\prime }\text{d}{R}^{\prime }$$

in an internal array, so that this can be used later on to generate random positions extracted from this geometry. Finally, also the radial surface density is calculated using a similar numerical integration.

Reimplemented from SimulationItem.

sharpness()

BrokenExpDiskGeometry::sharpness ( ) const

inline

This function returns the value of the discoverable double property sharpness : "the sharpness of the break" .

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

The default value for this property is given by the conditional value expression "3" .

SigmaR()

double BrokenExpDiskGeometry::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.

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

This value is calculated numerically during setup and stored as a data member.

Implements AxGeometry.

SigmaZ()

double BrokenExpDiskGeometry::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,

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

For the BrokenExpDiskGeometry model, one obtains

$${\mathrm{\Sigma }}_Z=2{\rho }_0{z}_0{(1+{\text{e}}^{-s{R}_{break}})}^{\frac{1}{s}(\frac{h_{\text{out}}}{h_{\text{inn}}}-1)}$$

Implements Geometry.

---

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

• BrokenExpDiskGeometry.hpp