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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 normalized
so that when integrated over the sphere the result is unity. For clarify,
the spatial normalisation is omitted in the formulae below.

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)

The RA and DEC parameters can be remplaced by GLON and GLAT
for Galactic coordinates
<source name="Crab" type="PointSource">
  <spatialModel type="PointSource">
    <parameter name="GLON" scale="1.0" value="184.5575" min="-360" max="360" free="1"/>
    <parameter name="GLAT" scale="1.0" value="-5.7844"  min="-90"  max="90"  free="1"/>
  </spatialModel>
  <spectrum type="...">
    ...
  </spectrum>
</source>

where

GLON is the Galactic longitude (degrees)
GLAT is the Galactic latitude (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 (degrees)
DEC is the Declination (degrees)
Radius is the disk radius (degrees)


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 (degrees)
DEC is the Declination (degrees)
\sigma
 = Sigma (degrees)


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
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 .

where

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


Note
For all radial source models the RA and DEC parameters can be
remplaced by GLON and GLAT for Galactic coordinates.


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.


Note
For all elliptical source models the RA and DEC parameters can be
remplaced by GLON and GLAT for Galactic coordinates.


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.