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The SKIRT project
advanced radiative transfer for astrophysics
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The SKIRT project
advanced radiative transfer for astrophysics
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#include <DonutGeometry.hpp>
Public Member Functions | |
| double | density (double R, double z) const override |
| Position | generatePosition () const override |
| double | largeRadius () const |
| double | SigmaR () const override |
| double | SigmaZ () const override |
| double | smallRadius () const |
| Public Member Functions inherited from AxGeometry | |
| double | density (Position bfr) const override |
| int | dimension () const override |
| double | SigmaX () const override |
| double | SigmaY () const override |
| Public Member Functions inherited from SimulationItem | |
| template<class T> | |
| T * | find (bool setup=true) const |
| template<class T> | |
| T * | interface (int upLevels, int downLevels, bool setup) 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 | |
| DonutGeometry () | |
| 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 () |
| Protected Member Functions inherited from Item | |
| Item () | |
Private Types | |
| using | BaseType |
| using | ItemType |
Private Attributes | |
| double | _A |
| double | _largeRadius |
| const double & | _R0 |
| const double & | _r0 |
| double | _r02 |
| Array | _Rv |
| double | _smallRadius |
| Array | _Xv |
Friends | |
| class | ItemRegistry |
The DonutGeometry class is a subclass of the AxGeometry class and describes the geometry of an axisymmetric ring torus (i.e. a doughnut geometry) of uniform density. This ring torus geometry is often used to model the dusty gas in the equatorial plane of active galactic nuclei (AGN), to explain the observed variety of AGN types as a distribution of observed system orientations relative to this obscuring torus of gas and dust; see Antonucci (1993, ARA&A, 31, 473), Urry & Padovani (1995, PASP, 107, 803) and Netzer (2015, ARA&A, 53, 365).
The ring torus geometry is obtained by revolving a circle of radius \(r_{\text{small}}\) about an axis that is co-planar with this circle, at a radius \(r_{\text{large}}\). Inside this surface of revolution the density has a constant value, and outside this surface the density is zero. In formula, this is easily expressed in cylindrical coordinates as
\[ \rho(R, z) = A\ \quad\text{for } \sqrt{(R-r_{\text{large}})^2 + z^2}<r_{\text{small}}. \]
There are two free parameters describing this torus geometry: the torus radius in the azimuthal plane \(r_{\text{small}}\) and the torus radius in the equatorial plane \(r_{\text{large}}\).
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inlineprotected |
Default constructor for concrete Item subclass DonutGeometry: "a donut torus geometry".
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overridevirtual |
This function returns the density \(\rho(R,z)\) at the cylindrical radius \(R\) and height \(z\). It just implements the geometry definition.
Implements AxGeometry.
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overridevirtual |
This function generates a random position from the uniform ring torus geometry. Because the cylindrical volume element \(R\,{\text{d}}R\,{\text{d}}\varphi\,{\text{d}}z\) depends on \(R\), sampling a position uniformly within the meridional cross-section circle and revolving it around the z-axis would not produce a position that is uniform by volume within the torus. Instead, the function first draws a random cylindrical radius \(R\) from the marginal probability density \(p(R)\,{\text{d}}R \propto R\, \sqrt{r_{\text{small}}^2-(R-r_{\text{large}})^2}\,{\text{d}}R\) tabulated by setupSelfBefore(), using linear interpolation of its cumulative distribution (the cumulative distribution of this \(p(R)\) does not invert analytically, because its antiderivative mixes an arcsine term with algebraic terms in \(R\)). Because the density is uniform inside the torus, the height \(z\) then has a uniform conditional distribution given this \(R\), over the range \(|z| \le \sqrt{r_{\text{small}}^2-(R-r_{\text{large}})^2}\) set by the torus tube at that radius, so a random \(z\) is drawn directly by choosing a random deviate \({\cal{X}}_1\) and setting \(z = \sqrt{r_{\text{small}}^2-(R-r_{\text{large}})^2}\,(2{\cal{X}}_1-1)\). Finally, a random azimuth angle \(\phi\) is found by choosing a random deviate \({\cal{X}}_2\) and setting \(\phi = 2\pi {\cal{X}}_2\).
Implements Geometry.
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inline |
This function returns the value of the discoverable double property largeRadius: "the radius of the torus circle in the equatorial plane".
This property represents a physical quantity of type "length".
The minimum value for this property is "]0".
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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.
\[ A = \frac{1}{V} = \frac{1}{2 \pi^2 r_{\text{large}} r_{\text{small}}^2}. \]
It also tabulates, using trapezoidal integration at a large number of radii spanning \([r_{\text{large}} - r_{\text{small}}, r_{\text{large}} + r_{\text{small}}]\), the cumulative distribution of the marginal radial probability density
\[ p(R)\,{\text{d}}R \propto R\, \sqrt{r_{\text{small}}^2 - (R-r_{\text{large}})^2}\,{\text{d}}R, \]
so that generatePosition() can generate a random radius through numerical inversion.
Reimplemented from SimulationItem.
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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,
\[\Sigma_R = \int_0^\infty \rho(R,0)\,{\text{d}}R. \]
For this geometry,
\[ \Sigma_R = A\,2 r_{\text{small}}. \]
Implements AxGeometry.
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overridevirtual |
This function returns the Z-axis surface density, i.e. the integration of the density along the entire Z-axis,
\[ \Sigma_Z = \int_{-\infty}^\infty \rho(0,z)\, {\text{d}}z. \]
For this geometry, the integral is simply zero.
Implements Geometry.
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inline |
This function returns the value of the discoverable double property smallRadius: "the radius of the torus circle in the azimuthal plane".
This property represents a physical quantity of type "length".
The minimum value for this property is "]0".