TTauriDiskGeometry Class Reference

#include <TTauriDiskGeometry.hpp>

Inheritance diagram for TTauriDiskGeometry:

Public Member Functions
double	density (double R, double z) const override
Position	generatePosition () const override
double	maxRadius () const
double	minRadius () const
double	radialIndex () const
double	scaleHeight () const
double	scaleLength () const
double	SigmaR () const override
double	SigmaZ () const override
double	verticalIndex () const
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
	TTauriDiskGeometry ()
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 = TTauriDiskGeometry

Private Attributes
const double &	_a
double	_a178
const double &	_b
double	_glnInn
double	_glnInnOut
double	_maxRadius
double	_minRadius
double	_radialIndex
const double &	_Rd
double	_rho0
const double &	_Rinn
const double &	_Rout
double	_s0
double	_scaleHeight
double	_scaleLength
double	_verticalIndex
const double &	_zd

Friends
class	ItemRegistry

Detailed Description

The TTauriDiskGeometry class describes the geometry of a typical passive disk around a T Tauri star. The disks are axisymmetric with a central cavity and are characterized by the density distribution

$$\rho (R,z)={\rho }_0{\left(\frac{R}{R_d}\right)}^{-\alpha }\exp \{-b{\left[\frac{z/z_d}{(R/R_d{)}^{9/8}}\right]}^2\}{R}_{\text{inn}}<R<R_{\text{out}}.$$

There are six parameters for this geometry: the exponent indices $\alpha >0$ and $b>0$, the inner and outer radii $R_{\text{inn}}>0$ and $R_{\text{out}}>R_{\text{inn}}$, and the scale length $R_d>0$ and scale height $z_d>0$.

Special cases of this geometry are used in the radiative transfer benchmark problems described by Pascucci et al. (2004, A&A, 417, 793) and Pinte et al. (2009, A&A, 498, 967).

Note: for the implementation of this geometry we use the generalized logarithm defined by SpecialFunctions::gln() and its inverse SpecialFunctions::gexp(). Specifically, we often use the difference function SpecialFunctions::gln2() defined as

$$\text{gln2}(p,x_1,x_2)=\text{gln}(p,x_1)-\text{gln}(p,x_2)={\int }_{x_2}^{x_1}{t}^{-p}\text{d}t.$$

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

TTauriDiskGeometry()

TTauriDiskGeometry::TTauriDiskGeometry ( )

inlineprotected

Default constructor for concrete Item subclass TTauriDiskGeometry : "a T Tauri disk geometry" .

Member Function Documentation

density()

double TTauriDiskGeometry::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.

generatePosition()

Position TTauriDiskGeometry::generatePosition ( ) const

overridevirtual

This function generates a random position from the geometry by drawing a random point from the three-dimensional probability density $p(\mathbf{r})\text{d}\mathbf{r}=\rho (\mathbf{r})\text{d}\mathbf{r}$. In the present case, we accomplish this by picking a random cylindrical radius $R$, a random height $z$, and a random azimuth $\varphi$ from the appropriate one-dimensional probability distribution functions.

We first generate a random radius $R$ from the marginal distribution

$$p(R)\text{d}R=2\pi R\text{d}R{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\rho (R,z)\text{d}z.$$

The cumulative distribution is

$$P(R)=2\pi {\int }_{R_{\text{inn}}}^R{R}^{\prime }\text{d}{R}^{\prime }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\rho (R^{\prime },z)\text{d}z.$$

Using the definitions of gln() and gln2() noted in the class header, the cumulative distribution can be written as

$$P(R)=\frac{\text{gln2}(\alpha -\frac{17}{8},\frac{R}{R_d},\frac{R_{\text{inn}}}{R_d})}{\text{gln2}(\alpha -\frac{17}{8},\frac{R_{\text{out}}}{R_d},\frac{R_{\text{inn}}}{R_d})}=\frac{\text{gln}(\alpha -\frac{17}{8},\frac{R}{R_d},)-\text{gln}(\alpha -\frac{17}{8},\frac{R_{\text{inn}}}{R_d})}{\text{gln2}(\alpha -\frac{17}{8},\frac{R_{\text{out}}}{R_d},\frac{R_{\text{inn}}}{R_d})}.$$

A random $R$ is generated by picking a uniform deviate $\mathcal{X}$ and setting $\mathcal{X}=P(R)$. After resolving for $R$, and using the inverse generalized logarithm gexp(), we obtain

$$R=R_d\text{gexp}[\alpha -\frac{17}{8},\text{gln}(\alpha -\frac{17}{8},\frac{R_{\text{inn}}}{R_d})+\mathcal{X}\text{gln2}(\alpha -\frac{17}{8},\frac{R_{\text{out}}}{R_d},\frac{R_{\text{inn}}}{R_d})].$$

We then generate a random height from the conditional distribution function

$$p(z)\text{d}z={\displaystyle \frac{\rho (R,z)\text{d}z}{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\rho (R,z^{\prime })\text{d}{z}^{\prime }}},$$

where $R$ is the random cylindrical radius generated before. One easily finds that this distribution is a Gaussian distribution with mean zero and dispersion

$$\sigma (R)=\frac{1}{\sqrt{2b}}{z}_d{\left(\frac{R}{R_d}\right)}^{9/8}.$$

Generating a random $z$ from this distribution is easy as the Random class contains a gaussian random number generating function.

Finally, we simply generate the azimuth from a uniform distribution between 0 and $2\pi$.

Implements Geometry.

maxRadius()

TTauriDiskGeometry::maxRadius ( ) const

inline

This function returns the value of the discoverable double property maxRadius : "the outer radius of the disk" .

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

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

minRadius()

TTauriDiskGeometry::minRadius ( ) const

inline

This function returns the value of the discoverable double property minRadius : "the inner radius of the disk" .

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

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

radialIndex()

TTauriDiskGeometry::radialIndex ( ) const

inline

This function returns the value of the discoverable double property radialIndex : "the radial exponent index α" .

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

The default value for this property is given by the conditional value expression "2.5" .

scaleHeight()

TTauriDiskGeometry::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()

TTauriDiskGeometry::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 TTauriDiskGeometry::setupSelfBefore ( )

overrideprotectedvirtual

This function verifies that the parameters are valid and it calculates the normalization constant ${\rho }_0$ determined by requiring that total mass equals one:

$$1=2\pi {\int }_{R_{\text{inn}}}^{R_{\text{out}}}R\text{d}R{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\rho (R,z)\text{d}z.$$

Given that

$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\exp (-a^2{x}^2)\text{d}x=\frac{\sqrt{\pi }}{a},$$

and using the definition of gln2() noted in the class header, we find after some algebra that

$${\rho }_0={\left[2{\pi }^{3/2}{b}^{-1/2}{R}_d^2{z}_d\text{gln2}(\alpha -\frac{17}{8},\frac{R_{\text{out}}}{R_d},\frac{R_{\text{inn}}}{R_d})\right]}^{-1}.$$

Reimplemented from Geometry.

SigmaR()

double TTauriDiskGeometry::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,

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

Using the definition of gln2() noted in the class header, we easily find for the T Tauri disk geometry that

$${\mathrm{\Sigma }}_R={\rho }_0{R}_d\text{gln2}(\alpha ,\frac{R_{\text{out}}}{R_d},\frac{R_{\text{inn}}}{R_d}).$$

Implements AxGeometry.

SigmaZ()

double TTauriDiskGeometry::SigmaZ ( ) const

overridevirtual

This function returns the vertical surface density, 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 T Tauri disk geometry with its central cylindrical cavity, this integral is zero.

Implements Geometry.

verticalIndex()

TTauriDiskGeometry::verticalIndex ( ) const

inline

This function returns the value of the discoverable double property verticalIndex : "the vertical exponent index b" .

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

The default value for this property is given by the conditional value expression "0.5" .

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

• TTauriDiskGeometry.hpp