Spatial source model components¶

The following sections present the spatial model components that are available in ctools for gamma-ray sources.

Note

Except of the DiffuseMapCube model, all spatial models are normalised so that when integrated over the sphere the result is unity.

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 describes a point source
<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 source¶

RadialDisk¶

The RadialDisk model describes a uniform intensity distribution within a given radius

<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)

RadialRing¶

The RadialRing model specifies a uniform intensity distribution within a circular ring
<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)

Note

Specifying the inner ring radius and ring width guarantees that both parameters are well defined. The ring outer radius is given by Radius+Width.

RadialGaussian¶

The RadialGaussian model describes a Gaussian intensity distribution
<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)

RadialGeneralGaussian¶

The RadialGeneralGaussian model describes a generalised Gaussian intensity distribution, with the radial profile index as a model parameter.
<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.

RadialShell¶

The RadialShell model describes a spherical shell projected on the sky
<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 source¶

EllipticalDisk¶

The EllipticalDisk model describes a uniform intensity distribution within an elliptical circumference:

<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)

EllipticalGaussian¶

The EllipticalGaussian model describes a Gaussian intensity distribution
<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.

Diffuse source¶

DiffuseIsotropic¶

The DiffuseIsotropic model describes an isotropic intensity distribution
<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>
where

Value is isotropic intensity

Note

For compatibility with the Fermi/LAT ScienceTools the model type DiffuseIsotropic can be replaced by ConstantValue.

DiffuseMap¶

The DiffuseMap model describes an arbitrary intensity distribution in form of a sky map
<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>
where

Normalization is a normalization value

and the file attribute specifies a sky map FITS file name. If a file name without path is specified it is assumed that the FITS file resides in the same directory as the model definition XML file.

Note

For compatibility with the Fermi/LAT ScienceTools the model type DiffuseMap can be replaced by SpatialMap and the parameter Normalization can be replaced by Prefactor.
Note

By default, the diffuse map is normalised so that

\[\int_{\Omega} M_{\rm spatial}(p|E) \, d\Omega = 1\]

which means that the units of the spatial model component are $[M_{\rm spatial}] = {\rm sr}^{-1}$. To avoid the normalisation you may add the normalize="0" attribute to the spatial model tag.
<source name="Crab" type="DiffuseSource">
  <spatialModel type="DiffuseMap" file="map.fits" normalize="0">
    <parameter name="Normalization" scale="1" value="1" min="0.001" max="1000.0" free="0"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>
In that case, generally

\[\int_{\Omega} M_{\rm spatial}(p|E) \, d\Omega \neq 1\]

and the spectral component cannot be directly interpreted as a physical source intensity.

DiffuseMapCube¶

The DiffuseMapCube model describes an arbitrary energy-dependent intensity distribution in form of a map cube
<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>
where

Normalization is a normalization value

Note that the map cube is not normalised to unit, hence generally

\[\int_{\Omega} M_{\rm spatial}(p|E) \, d\Omega \neq 1\]

To compute the flux in a given energy band for a DiffuseMapCube model you have to integrated both the spatial and spectral components, i.e.

\[\Phi = \int_{\Omega} \int_{E} M_{\rm spatial}(p|E) \times M_{\rm spectral}(E)\, d\Omega \, dE\]

Note

For compatibility with the Fermi/LAT ScienceTools the model type DiffuseMapCube can be replaced by MapCubeFunction and the parameter Normalization can be replaced by Value.

Composite model¶

The Composite model implements a composite model that is the sum of an arbitrary number of spatial models

<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>

which implements

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

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

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