Spatial components¶

Note

In the following model descriptions, celestial coordinates RA and DEC may be replaced by Galactic coordinates GLON and GLAT.

Point source¶

The PointSource model specifies a source that has no spatial extension. It is defined by its celestial coordinates RA and DEC given in units of degrees.

<source name="Crab" type="PointSource">
  <spatialModel type="PointSource">
    <parameter name="RA"  scale="1.0" value="83.6331" min="-360" max="360" free="1"/>
    <parameter name="DEC" scale="1.0" value="22.0145" min="-90"  max="90"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

where

RA is the Right Ascension (degrees)
DEC is the Declination (degrees)

Note

For compatibility with the Fermi/LAT ScienceTools the model type PointSource can be replaced by SkyDirFunction.

Radial disk¶

The RadialDisk model specifies a uniform circular intensity distribution, defined by the celestial coordinates RA and DEC of the disk centre and the disk Radius. All parameters are given in units of degrees.

<source name="Crab" type="ExtendedSource">
  <spatialModel type="RadialDisk">
    <parameter name="RA"     scale="1.0" value="83.6331" min="-360" max="360" free="1"/>
    <parameter name="DEC"    scale="1.0" value="22.0145" min="-90"  max="90"  free="1"/>
    <parameter name="Radius" scale="1.0" value="0.20"    min="0.01" max="10"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

where

RA is the Right Ascension of the disk centre (degrees)
DEC is the Declination of the disk centre (degrees)
Radius is the disk radius (degrees)

Radial ring¶

The RadialRing model specifies a uniform intensity distribution within a circular ring. The circular ring is defined by the celestial coordinates RA and DEC of the ring centre, the ring inner radius defined by Radius and the ring width, defined by Width. Specifically, the ring outer radius is given by Radius+Width. All parameters are given in units of degrees.

<source name="Crab" type="ExtendedSource">
  <spatialModel type="RadialRing">
    <parameter name="RA"     scale="1.0" value="83.6331" min="-360" max="360" free="1"/>
    <parameter name="DEC"    scale="1.0" value="22.0145" min="-90"  max="90"  free="1"/>
    <parameter name="Radius" scale="1.0" value="0.20"    min="0.01" max="10"  free="1"/>
    <parameter name="Width"  scale="1.0" value="0.15"    min="0.01" max="10"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

where

RA is the Right Ascension of the ring centre (degrees)
DEC is the Declination of the ring centre (degrees)
Radius is the inner ring radius (degrees)
Width is the ring width radius (degrees)

Radial Gaussian¶

The RadialGaussian model specifies a spherical Gaussian intensity distribution, defined by the celestial coordinates RA and DEC of the Gaussian centre and the Gaussian Sigma parameter. All parameters are given in units of degrees.

<source name="Crab" type="ExtendedSource">
  <spatialModel type="RadialGaussian">
    <parameter name="RA"    scale="1.0" value="83.6331" min="-360" max="360" free="1"/>
    <parameter name="DEC"   scale="1.0" value="22.0145" min="-90"  max="90"  free="1"/>
    <parameter name="Sigma" scale="1.0" value="0.20"    min="0.01" max="10"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

and implements

\[M_{\rm spatial}(\theta) = \frac{1}{2 \pi \sigma^2} \exp \left(-\frac{1}{2}\frac{\theta^2}{\sigma^2} \right),\]

where

RA is the Right Ascension of the Gaussian centre (degrees)
DEC is the Declination of the Gaussian centre (degrees)
$\sigma$ = Sigma (degrees)

Radial general Gaussian¶

The RadialGeneralGaussian model specifies a generalised Gaussian intensity distribution, defined by the celestial coordinates RA and DEC of the generalised Gaussian centre, a radius Radius``and a radial index parameter ``R_Index.

<source name="Crab" type="ExtendedSource">
  <spatialModel type="RadialGeneralGaussian">
    <parameter name="RA"      scale="1.0" value="83.6331" min="-360" max="360" free="1"/>
    <parameter name="DEC"     scale="1.0" value="22.0145" min="-90"  max="90"  free="1"/>
    <parameter name="Radius"  scale="1.0" value="0.20"    min="0.01" max="10"  free="1"/>
    <parameter name="R_Index" scale="1.0" value="0.5"     min="0.01" max="10"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

and implements

\[M_{\rm spatial}(\theta) = \frac{1}{2 \pi r^2 \eta \Gamma(2\eta)} \exp \left[- \left(\frac{\theta^2}{r^2}\right)^\frac{1}{\eta} \right],\]

where

RA is the Right Ascension of the Gaussian centre (degrees)
DEC is the Declination of the Gaussian centre (degrees)
$r$ = Radius (degrees)
$\eta$ = R_Index

The model normalisation is correct in the small angle approximation and for $\eta$ of the order of unity or smaller.

Radial shell¶

The RadialShell model specifies a 3-dimensional shell projected on the sky. The shell is defined by the celestial coordinates RA and DEC of the shell centre, the inner radius of the shell defined by Radius and the width of the shell, defined by Width. Specifically, the outer radius of the shell is given by Radius+Width. All parameters are given in units of degrees.

<source name="Crab" type="ExtendedSource">
  <spatialModel type="RadialShell">
    <parameter name="RA"     scale="1.0" value="83.6331" min="-360" max="360" free="1"/>
    <parameter name="DEC"    scale="1.0" value="22.0145" min="-90"  max="90"  free="1"/>
    <parameter name="Radius" scale="1.0" value="0.30"    min="0.01" max="10"  free="1"/>
    <parameter name="Width"  scale="1.0" value="0.10"    min="0.01" max="10"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

and implements

\[\begin{split}M_{\rm spatial}(\theta) = n_0 \left \{ \begin{array}{l l} \displaystyle \sqrt{ \theta_{\rm out}^2 - \theta^2 } - \sqrt{ \theta_{\rm in}^2 - \theta^2 } & \mbox{if $\theta \le \theta_{\rm in}$} \\ \\ \displaystyle \sqrt{ \theta_{\rm out}^2 - \theta^2 } & \mbox{if $\theta_{\rm in} < \theta \le \theta_{\rm out}$} \\ \\ \displaystyle 0 & \mbox{if $\theta > \theta_{\rm out}$} \end{array} \right .\end{split}\]

where

RA is the Right Ascension of the shell centre (degrees)
DEC is the Declination of the shell centre (degrees)
$\theta_{\rm out}$ = Radius + Width (degrees)
$\theta_{\rm in}$ = Radius (degrees)

Radial profiles¶

Radial profiles are defined by a arbitrary function of the radial distance from a central position. The following radial profiles exist:

Burkert Dark matter profile¶

<source name="Crab" type="ExtendedSource">
  <spatialModel type="DMBurkertProfile">
    <parameter name="RA"           scale="1.0" value="83.6331" min="-360"    max="360"   free="1"/>
    <parameter name="DEC"          scale="1.0" value="22.0145" min="-90"     max="90"    free="1"/>
    <parameter name="ScaleRadius"  scale="1.0" value="21.5"    min="0.0001"  max="1000"  free="0"/>
    <parameter name="ScaleDensity" scale="1.0" value="0.2"     min="0.0001"  max="1000"  free="0"/>
    <parameter name="HaloDistance" scale="1.0" value="7.94"    min="0.0001"  max="1000"  free="0"/>
    <parameter name="ThetaMin"     scale="1.0" value="1.0e-6"  min="1.0e-10" max="1000"  free="0"/>
    <parameter name="ThetaMax"     scale="1.0" value="180.0"   min="0.0001"  max="1000"  free="0"/>
    <parameter name="CoreRadius"   scale="1.0" value="0.5"     min="0.0001"  max="1000"  free="0"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

Einasto Dark matter profile¶

<source name="Crab" type="ExtendedSource">
  <spatialModel type="DMEinastoProfile">
    <parameter name="RA"           scale="1.0" value="83.6331" min="-360"    max="360"   free="1"/>
    <parameter name="DEC"          scale="1.0" value="22.0145" min="-90"     max="90"    free="1"/>
    <parameter name="ScaleRadius"  scale="1.0" value="21.5"    min="0.0001"  max="1000"  free="0"/>
    <parameter name="ScaleDensity" scale="1.0" value="0.2"     min="0.0001"  max="1000"  free="0"/>
    <parameter name="HaloDistance" scale="1.0" value="7.94"    min="0.0001"  max="1000"  free="0"/>
    <parameter name="Alpha"        scale="1.0" value="0.17"    min="0.0001"  max="1000"  free="0"/>
    <parameter name="ThetaMin"     scale="1.0" value="1.0e-6"  min="1.0e-10" max="1000"  free="0"/>
    <parameter name="ThetaMax"     scale="1.0" value="180.0"   min="0.0001"  max="1000"  free="0"/>
    <parameter name="CoreRadius"   scale="1.0" value="0.5"     min="0.0001"  max="1000"  free="0"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

Zhao Dark matter profile¶

<source name="Crab" type="ExtendedSource">
  <spatialModel type="DMZhaoProfile">
    <parameter name="RA"           scale="1.0" value="83.6331" min="-360"    max="360"   free="1"/>
    <parameter name="DEC"          scale="1.0" value="22.0145" min="-90"     max="90"    free="1"/>
    <parameter name="ScaleRadius"  scale="1.0" value="21.5"    min="0.0001"  max="1000"  free="0"/>
    <parameter name="ScaleDensity" scale="1.0" value="0.2"     min="0.0001"  max="1000"  free="0"/>
    <parameter name="HaloDistance" scale="1.0" value="7.94"    min="0.0001"  max="1000"  free="0"/>
    <parameter name="Alpha"        scale="1.0" value="0.17"    min="0.0001"  max="1000"  free="0"/>
    <parameter name="Beta"         scale="1.0" value="3.00"    min="0.0001"  max="1000"  free="0"/>
    <parameter name="Gamma"        scale="1.0" value="1.00"    min="0.0001"  max="1000"  free="0"/>
    <parameter name="ThetaMin"     scale="1.0" value="1.0e-6"  min="1.0e-10" max="1000"  free="0"/>
    <parameter name="ThetaMax"     scale="1.0" value="180.0"   min="0.0001"  max="1000"  free="0"/>
    <parameter name="CoreRadius"   scale="1.0" value="0.5"     min="0.0001"  max="1000"  free="0"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

Gaussian profile¶

This profile is equivalent to RadialGaussian.

<source name="Crab" type="ExtendedSource">
  <spatialModel type="GaussianProfile">
    <parameter name="RA"    scale="1.0" value="83.6331" min="-360" max="360" free="1"/>
    <parameter name="DEC"   scale="1.0" value="22.0145" min="-90"  max="90"  free="1"/>
    <parameter name="Sigma" scale="1.0" value="0.45"    min="0.01" max="10"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

Elliptical disk¶

The EllipticalDisk model specifies a uniform elliptical intensity distribution, defined by the celestial coordinates RA and DEC of the centre of the ellipse, the minor and major radii MinorRadius and MajorRadius of the ellipse, and the position angle PA that is counted counter-clockwise from celestial North. All parameters are given in units of degrees.

<source name="Crab" type="ExtendedSource">
  <spatialModel type="EllipticalDisk">
    <parameter name="RA"          scale="1.0" value="83.6331" min="-360"  max="360" free="1"/>
    <parameter name="DEC"         scale="1.0" value="22.0145" min="-90"   max="90"  free="1"/>
    <parameter name="PA"          scale="1.0" value="45.0"    min="-360"  max="360" free="1"/>
    <parameter name="MinorRadius" scale="1.0" value="0.5"     min="0.001" max="10"  free="1"/>
    <parameter name="MajorRadius" scale="1.0" value="2.0"     min="0.001" max="10"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

where

RA is the Right Ascension (degrees)
DEC is the Declination (degrees)
PA is the position angle, counted counterclockwise from North (degrees)
MinorRadius is the minor radius of the ellipse (degrees)
MajorRadius is the major radius of the ellipse (degrees)

Elliptical Gaussian¶

The EllipticalGaussian model specifies an elliptical Gaussian intensity distribution, defined by the celestial coordinates RA and DEC of the centre of the ellipse, the minor and major sigma parameter MinorRadius and MajorRadius of the ellipse, and the position angle PA that is counted counter-clockwise from celestial North. All parameters are given in units of degrees.

<source name="Crab" type="ExtendedSource">
  <spatialModel type="EllipticalGaussian">
    <parameter name="RA"          scale="1.0" value="83.6331" min="-360"  max="360" free="1"/>
    <parameter name="DEC"         scale="1.0" value="22.0145" min="-90"   max="90"  free="1"/>
    <parameter name="PA"          scale="1.0" value="45.0"    min="-360"  max="360" free="1"/>
    <parameter name="MinorRadius" scale="1.0" value="0.5"     min="0.001" max="10"  free="1"/>
    <parameter name="MajorRadius" scale="1.0" value="2.0"     min="0.001" max="10"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

and implements

\[M_{\rm spatial}(\theta, \phi) = \exp \left( -\frac{\theta^2}{2 r_\mathrm{eff}^2} \right),\]

with

\[r_\mathrm{eff} = \frac{ab} {\sqrt{\left( a \sin (\phi - \phi_0) \right)^2 + \sqrt{\left( b \cos (\phi - \phi_0) \right)^2}}}\]

where

RA is the Right Ascension (degrees)
DEC is the Declination (degrees)
PA is the position angle, counted counterclockwise from North (degrees)
$a$ = MinorRadius (degrees)
$b$ = MajorRadius (degrees)
$\phi_0$ is the position angle of the ellipse, counted counterclockwise from North
$\phi$ is the azimuth angle with respect to North.

EllipticalGeneralGaussian¶

The EllipticalGeneralGaussian model describes a Gaussian intensity distribution

<source name="Crab" type="ExtendedSource">
  <spatialModel type="EllipticalGeneralGaussian">
    <parameter name="RA"          scale="1.0" value="83.6331" min="-360"  max="360" free="1"/>
    <parameter name="DEC"         scale="1.0" value="22.0145" min="-90"   max="90"  free="1"/>
    <parameter name="PA"          scale="1.0" value="45.0"    min="-360"  max="360" free="1"/>
    <parameter name="MinorRadius" scale="1.0" value="0.5"     min="0.001" max="10"  free="1"/>
    <parameter name="MajorRadius" scale="1.0" value="2.0"     min="0.001" max="10"  free="1"/>
    <parameter name="R_Index"     scale="1.0" value="0.5"     min="0.01"  max="10"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

and implements

\[M_{\rm spatial}(\theta, \phi) = \frac{1}{2 \pi r^2 \eta \Gamma(2\eta)} \exp \left[ -\left(\frac{\theta^2}{2 r_\mathrm{eff}^2}\right)^\frac{1}{\eta} \right],\]

with

\[r_\mathrm{eff} = \frac{ab} {\sqrt{\left( a \sin (\phi - \phi_0) \right)^2 + \sqrt{\left( b \cos (\phi - \phi_0) \right)^2}}}\]

where

RA is the Right Ascension (degrees)
DEC is the Declination (degrees)
PA is the position angle, counted counterclockwise from North (degrees)
$a$ = MinorRadius (degrees)
$b$ = MajorRadius (degrees)
$\phi_0$ is the position angle of the ellipse, counted counterclockwise from North
$\phi$ is the azimuth angle with respect to North
$\eta$ = R_Index

The model normalisation is correct in the small angle approximation and for $\eta$ of the order of unity or smaller.

Isotropic source¶

The DiffuseIsotropic model specifies an isotropic intensity distribution. The only parameter of the model is a normalisation factor, specified by the parameter Value.

<source name="Crab" type="DiffuseSource">
  <spatialModel type="DiffuseIsotropic">
    <parameter name="Value" scale="1" value="1" min="1"  max="1" free="0"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

An alternative XML format is supported for compatibility with the Fermi/LAT XML format:

<source name="Crab" type="DiffuseSource">
  <spatialModel type="ConstantValue">
    <parameter name="Value" scale="1" value="1" min="1"  max="1" free="0"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

Diffuse map¶

The DiffuseMap model specifies an intensity distribution that is represented by a FITS image. The name of the FITS file is specified using the file attribute of the spatialModel tag. If there are several image in the FITS file, the first image will be extracted for the diffuse map. Alternatively, the name of the relevant image extension or the extension number can be specified in square brackets to select a specific image from the FITS file.

The only parameter of the model is a normalisation factor, specified by the parameter Normalization.

<source name="Crab" type="DiffuseSource">
  <spatialModel type="DiffuseMap" file="map.fits">
    <parameter name="Normalization" scale="1" value="1" min="0.001" max="1000.0" free="0"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

An alternative XML format is supported for compatibility with the Fermi/LAT XML format:

<source name="Crab" type="DiffuseSource">
  <spatialModel type="SpatialMap" file="map.fits">
    <parameter name="Prefactor" scale="1" value="1" min="0.001" max="1000.0" free="0"/>
  </spatialModel>
  <spectrum type="...">
   ...
  </spectrum>
</source>

Diffuse map cube¶

The DiffuseMapCube model specifies an energy-dependent intensity distribution that is represented by a FITS file. The name of the FITS file is specified using the file attribute of the spatialModel tag. The model expects a 3-dimensional FITS image plus an extension with the name ENERGIES that specifies the energy for every layer of the FITS image. The number of energies must correspond to the length of the 3rd image axis.

The only parameter of the model is a normalisation factor, specified by the parameter Normalization.

<source name="Crab" type="DiffuseSource">
  <spatialModel type="DiffuseMapCube" file="map_cube.fits">
    <parameter name="Normalization" scale="1" value="1" min="0.001" max="1000.0" free="0"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

An alternative XML format is supported for compatibility with the Fermi/LAT XML format:

<source name="Crab" type="DiffuseSource">
  <spatialModel type="MapCubeFunction" file="map_cube.fits">
    <parameter name="Value" scale="1" value="1" min="0.001" max="1000.0" free="0"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

Composite model¶

Spatial model components can be combined into a single model using the GModelSpatialComposite class. The class computes

\[M_{\rm spatial}(p|E,t) = \frac{1}{N} \sum_{i=0}^{N-1} M_{\rm spatial}^{(i)}(p|E,t)\]

where $M_{\rm spatial}^{(i)}(p|E,t)$ is any spatial model component (including another composite model), and $N$ is the number of model components that are combined.

An example of an XML file for a composite spatial model is shown below. In this example, a point source is added to a radial Gaussian source to form a composite spatial model. All spatial parameters of the composite model are fitted.

<source name="Crab" type="CompositeSource">
  <spatialModel type="Composite">
    <spatialModel type="PointSource" component="PointSource">
      <parameter name="RA"    scale="1.0" value="83.6331" min="-360" max="360" free="1"/>
      <parameter name="DEC"   scale="1.0" value="22.0145" min="-90"  max="90"  free="1"/>
    </spatialModel>
    <spatialModel type="RadialGaussian">
      <parameter name="RA"    scale="1.0" value="83.6331" min="-360" max="360" free="1"/>
      <parameter name="DEC"   scale="1.0" value="22.0145" min="-90"  max="90"  free="1"/>
      <parameter name="Sigma" scale="1.0" value="0.20"    min="0.01" max="10"  free="1"/>
    </spatialModel>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

Table Of Contents

Previous topic

Next topic

Spatial components¶

Point source¶

Radial disk¶

Radial ring¶

Radial Gaussian¶

Radial general Gaussian¶

Radial shell¶

Radial profiles¶

Burkert Dark matter profile¶

Einasto Dark matter profile¶

Zhao Dark matter profile¶

Gaussian profile¶

Elliptical disk¶

Elliptical Gaussian¶

EllipticalGeneralGaussian¶

Isotropic source¶

Diffuse map¶

Diffuse map cube¶

Composite model¶