Spectral components¶

Constant¶

The GModelSpectralConst class implements the constant function

\[\frac{dN}{dE} = N_0\]

where the parameters in the XML definition have the following mappings:

\(N_0\) = Normalization

The XML format for specifying a constant is:

<spectrum type="Constant">
  <parameter name="Normalization" scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
</spectrum>

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

<spectrum type="ConstantValue">
  <parameter name="Value" scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
</spectrum>

Node function¶

The generalisation of the broken power law is the node function, which is defined by a set of energy and intensity values, the so called nodes, which are connected by power laws.

The XML format for specifying a node function is:

<spectrum type="NodeFunction">
  <node>
    <parameter scale="1.0"   name="Energy"    min="0.1"   max="1.0e20" value="1.0"  free="0"/>
    <parameter scale="1e-07" name="Intensity" min="1e-07" max="1000.0" value="1.0"  free="1"/>
  </node>
  <node>
    <parameter scale="1.0"   name="Energy"    min="0.1"   max="1.0e20" value="10.0" free="0"/>
    <parameter scale="1e-07" name="Intensity" min="1e-07" max="1000.0" value="0.1"  free="1"/>
  </node>
</spectrum>

(in this example there are two nodes; the number of nodes in a node function is arbitrary).

Bin function¶

In order to decorrelate the spectral nodes one may use the spectral bin function which specifies a power law intensity distribution with a fixed spectral sloped within energy bins defined by LowerLimit and UpperLimit values given in units of \({\rm MeV}\). Within an energy bin the intensity follows a power law with spectral index defined by the Index parameter. Intensities are given in units of \({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1}\) and are specified for the logarithmic bin centre.

The XML format for specifying a bin function is:

<spectrum type="BinFunction">
  <parameter name="Index" scale="-1" value="2.48" min="0.0" max="+5.0" free="0"/>
  <bin>
    <parameter scale="1.0"   name="LowerLimit" min="0.1"   max="1.0e20" value="0.75" free="0"/>
    <parameter scale="1.0"   name="UpperLimit" min="0.1"   max="1.0e20" value="1.0"  free="0"/>
    <parameter scale="1e-07" name="Intensity"  min="1e-07" max="1000.0" value="1.0"  free="1"/>
  </bin>
  <bin>
    <parameter scale="1.0"   name="LowerLimit" min="0.1"   max="1.0e20" value="1.0"  free="0"/>
    <parameter scale="1.0"   name="UpperLimit" min="0.1"   max="1.0e20" value="3.0"  free="0"/>
    <parameter scale="1e-07" name="Intensity"  min="1e-07" max="1000.0" value="0.5"  free="1"/>
  </bin>
</spectrum>

(in this example there are two bins; the number of bins in a bin function is arbitrary).

File function¶

A function defined using an input ASCII file with columns of energy and differential flux values. The energy values are assumed to be in units of MeV, the flux values are normally assumed to be in units of \({\rm cm}^{-2} {\rm s}^{-1} {\rm MeV}^{-1}\).

The only parameter of the model is a multiplicative normalization:

\[\frac{dN}{dE} = N_0 \left. \frac{dN}{dE} \right\rvert_{\rm file}\]

where the parameters in the XML definition have the following mappings:

\(N_0\) = Normalization

The XML format for specifying a file function is:

<spectrum type="FileFunction" file="data/filefunction.txt">
  <parameter scale="1.0" name="Normalization" min="0.0" max="1000.0" value="1.0" free="1"/>
</spectrum>

If the file attribute is a relative path, the path is relative to the directory where the XML file resides. Alternatively, an absolute path may be specified. Any environment variable present in the path name will be expanded.

Table model¶

An arbitrary spectral model defined on a M-dimensional grid of parameter values. The spectrum is computed using M-dimensional linear interpolation. The model definition is provided by a FITS file that follows the HEASARC OGIP standard.

The structure of the table model FITS file is shown below. The FITS file contains three binary table extensions after an empty image extension.

Structure of table model FITS file

The PARAMETERS extension contains the definition of the model parameters. Each row defines one model parameter. Each model parameter is defined by a unique NAME. The METHOD column indicates whether the model should be interpolated linarly (value 0) or logarithmically (value 1). GammaLib so far only supports linear interpolation, hence the field is ignored. The INITIAL column indicates the initial parameter value, if the value in the DELTA column is negative the parameter will be fixed, otherwise it will be fitted. The MINIMUM and MAXIMUM columns indicate the range of values for a given parameter, the BOTTOM and TOP columns are ignored by GammaLib. The NUMBVALS column indicates the number of parameter values for which the table model was computed, the VALUE column indicates the specific parameter values.

In the example below there are two parameters named Index and Cutoff, and spectra were computed for 100 index values and 50 cutoff values, hence a total of 5000 spectra are stored in the table model.

../../../../../_images/model_table_parameters.png

Table model parameters extension

The ENERGIES extension contains the energy boundaries for the spectra in the usual OGIP format:

../../../../../_images/model_table_energies.png

Energy boundaries extension

The SPECTRA extension contains the spectra of the table model. It consists of two vector columns. The PARAMVAL column provides the parameter values for which the spectrum was computed. Since there are two parameters in the example the vector column has two entries. The INTPSPEC column provides the spectrum \(\frac{dN(p)}{dE}\) for the specific parameters. Since there are 200 energy bins in this example the vector column has 200 entries.

../../../../../_images/model_table_spectra.png

Spectra extension

The model is defined using:

\[\frac{dN}{dE} = N_0 \left. \frac{dN(p)}{dE} \right\rvert_{\rm file}\]

where the parameters in the XML definition have the following mappings:

\(N_0\) = Normalization
\(p\) = M model parameters (e.g. Index, Cutoff)

The XML format for specifying a table model is:

<spectrum type="TableModel" file="model_table.fits">
  <parameter scale="1.0" name="Normalization" min="0.0" max="1000.0" value="1.0" free="1"/>
</spectrum>

If the file attribute is a relative path, the path is relative to the directory where the XML file resides. Alternatively, an absolute path may be specified. Any environment variable present in the path name will be expanded.

Note that the default parameters of the table model are provided in the FITS file, according to the HEASARC OGIP standard. However, table model parameters may also be specified in the XML file, and these parameters will then overwrite the parameters in the FITS file. For example, for a 2-dimensional table model with an Index and a Cutoff parameter, the XML file may look like

<spectrum type="TableModel" file="model_table.fits">
  <parameter scale="1e-16" name="Normalization" min="1e-07" max="1000" value="5.8"  free="1"/>
  <parameter scale="-1"    name="Index"         min="1.0"   max="3.0"  value="2.4"  free="1"/>
  <parameter scale="1e6"   name="Cutoff"        min="0.1"   max="28.2" value="0.89" free="1"/>
</spectrum>

Power law¶

The GModelSpectralPlaw class implements the power law function

\[\frac{dN}{dE} = k_0 \left( \frac{E}{E_0} \right)^{\gamma}\]

where the parameters in the XML definition have the following mappings:

\(k_0\) = Prefactor
\(\gamma\) = Index
\(E_0\) = PivotEnergy

The XML format for specifying a power law is:

<spectrum type="PowerLaw">
  <parameter name="Prefactor"   scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index"       scale="-1"    value="2.48" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="PivotEnergy" scale="1e6"   value="0.3"  min="0.01"  max="1000.0" free="0"/>
</spectrum>

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

<spectrum type="PowerLaw">
  <parameter name="Prefactor" scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index"     scale="-1"    value="2.48" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="Scale"     scale="1e6"   value="0.3"  min="0.01"  max="1000.0" free="0"/>
</spectrum>

An alternative power law function is defined by the GModelSpectralPlawPhotonFlux class that uses the integral photon flux as parameter rather than the Prefactor:

\[\frac{dN}{dE} = \frac{F_{\rm ph}(\gamma+1)E^{\gamma}} {E_{\rm max}^{\gamma+1} - E_{\rm min}^{\gamma+1}}\]

where the parameters in the XML definition have the following mappings:

\(F_{\rm ph}\) = PhotonFlux
\(\gamma\) = Index
\(E_{\rm min}\) = LowerLimit
\(E_{\rm max}\) = UpperLimit

The XML format for specifying a power law defined by the integral photon flux is:

<spectrum type="PowerLaw">
  <parameter scale="1e-07" name="PhotonFlux" min="1e-07" max="1000.0"    value="1.0"      free="1"/>
  <parameter scale="1.0"   name="Index"      min="-5.0"  max="+5.0"      value="-2.0"     free="1"/>
  <parameter scale="1.0"   name="LowerLimit" min="10.0"  max="1000000.0" value="100.0"    free="0"/>
  <parameter scale="1.0"   name="UpperLimit" min="10.0"  max="1000000.0" value="500000.0" free="0"/>
</spectrum>

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

<spectrum type="PowerLaw2">
  <parameter scale="1e-07" name="Intergal"   min="1e-07" max="1000.0"    value="1.0"      free="1"/>
  <parameter scale="1.0"   name="Index"      min="-5.0"  max="+5.0"      value="-2.0"     free="1"/>
  <parameter scale="1.0"   name="LowerLimit" min="10.0"  max="1000000.0" value="100.0"    free="0"/>
  <parameter scale="1.0"   name="UpperLimit" min="10.0"  max="1000000.0" value="500000.0" free="0"/>
</spectrum>

Note

The UpperLimit and LowerLimit parameters are always treated as fixed and, as should be apparent from this definition, the flux given by the PhotonFlux parameter is over the range [LowerLimit, UpperLimit]. Use of this model allows the errors on the integrated photon flux to be evaluated directly by likelihood, obviating the need to propagate the errors if one is using the PowerLaw form.

Another alternative power law function is defined by the GModelSpectralPlawEnergyFlux class that uses the integral energy flux as parameter rather than the Prefactor:

\[\frac{dN}{dE} = \frac{F_{\rm E}(\gamma+2)E^{\gamma}} {E_{\rm max}^{\gamma+2} - E_{\rm min}^{\gamma+2}}\]

where the parameters in the XML definition have the following mappings:

\(F_{\rm E}\) = EnergyFlux
\(\gamma\) = Index
\(E_{\rm min}\) = LowerLimit
\(E_{\rm max}\) = UpperLimit

The XML format for specifying a power law defined by the integral energy flux is:

<spectrum type="PowerLaw">
  <parameter scale="1e-07" name="EnergyFlux" min="1e-07" max="1000.0"    value="1.0"      free="1"/>
  <parameter scale="1.0"   name="Index"      min="-5.0"  max="+5.0"      value="-2.0"     free="1"/>
  <parameter scale="1.0"   name="LowerLimit" min="10.0"  max="1000000.0" value="100.0"    free="0"/>
  <parameter scale="1.0"   name="UpperLimit" min="10.0"  max="1000000.0" value="500000.0" free="0"/>
</spectrum>

Note

The UpperLimit and LowerLimit parameters are always treated as fixed and, as should be apparent from this definition, the flux given by the EnergyFlux parameter is over the range [LowerLimit, UpperLimit]. Use of this model allows the errors on the integrated energy flux to be evaluated directly by likelihood, obviating the need to propagate the errors if one is using the PowerLaw form.

Exponentially cut-off power law¶

The GModelSpectralExpPlaw class implements the exponentially cut-off power law function

\[\frac{dN}{dE} = k_0 \left( \frac{E}{E_0} \right)^{\gamma} \exp \left( \frac{-E}{E_{\rm cut}} \right)\]

where the parameters in the XML definition have the following mappings:

\(k_0\) = Prefactor
\(\gamma\) = Index
\(E_0\) = PivotEnergy
\(E_{\rm cut}\) = CutoffEnergy

The XML format for specifying an exponentially cut-off power law is:

<spectrum type="ExponentialCutoffPowerLaw">
  <parameter name="Prefactor"    scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index"        scale="-1"    value="2.48" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="CutoffEnergy" scale="1e6"   value="1.0"  min="0.01"  max="1000.0" free="1"/>
  <parameter name="PivotEnergy"  scale="1e6"   value="0.3"  min="0.01"  max="1000.0" free="0"/>
</spectrum>

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

<spectrum type="ExpCutoff">
  <parameter name="Prefactor" scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index"     scale="-1"    value="2.48" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="Cutoff"    scale="1e6"   value="1.0"  min="0.01"  max="1000.0" free="1"/>
  <parameter name="Scale"     scale="1e6"   value="0.3"  min="0.01"  max="1000.0" free="0"/>
</spectrum>

An alternative exponentially cut-off power law function is defined by the GModelSpectralExpInvPlaw class which makes use of the inverse of the cut-off energy for function parametrisation:

\[\frac{dN}{dE} = k_0 \left( \frac{E}{E_0} \right)^{\gamma} \exp \left( -\lambda E \right)\]

where the parameters in the XML definition have the following mappings:

\(k_0\) = Prefactor
\(\gamma\) = Index
\(E_0\) = PivotEnergy
\(\lambda\) = InverseCutoffEnergy

The XML format for specifying an exponentially cut-off power law using this alternative parametrisation is:

<spectrum type="ExponentialCutoffPowerLaw">
  <parameter name="Prefactor"           scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index"               scale="-1"    value="2.48" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="InverseCutoffEnergy" scale="1e-6"  value="1.0"  min="0.0"   max="100.0"  free="1"/>
  <parameter name="PivotEnergy"         scale="1e6"   value="0.3"  min="0.01"  max="1000.0" free="0"/>
</spectrum>

Super exponentially cut-off power law¶

The GModelSpectralSuperExpPlaw class implements the super exponentially cut-off power law function

\[\frac{dN}{dE} = k_0 \left( \frac{E}{E_0} \right)^{\gamma} \exp \left( -\left( \frac{E}{E_{\rm cut}} \right)^{\alpha} \right)\]

where the parameters in the XML definition have the following mappings:

\(k_0\) = Prefactor
\(\gamma\) = Index1
\(\alpha\) = Index2
\(E_0\) = PivotEnergy
\(E_{\rm cut}\) = CutoffEnergy

<spectrum type="SuperExponentialCutoffPowerLaw">
  <parameter name="Prefactor"    scale="1e-16" value="1.0" min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index1"       scale="-1"    value="2.0" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="CutoffEnergy" scale="1e6"   value="1.0" min="0.01"  max="1000.0" free="1"/>
  <parameter name="Index2"       scale="1.0"   value="1.5" min="0.1"   max="5.0"    free="1"/>
  <parameter name="PivotEnergy"  scale="1e6"   value="1.0" min="0.01"  max="1000.0" free="0"/>
</spectrum>

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

<spectrum type="PLSuperExpCutoff">
  <parameter name="Prefactor"   scale="1e-16" value="1.0" min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index1"      scale="-1"    value="2.0" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="Cutoff"      scale="1e6"   value="1.0" min="0.01"  max="1000.0" free="1"/>
  <parameter name="Index2"      scale="1.0"   value="1.5" min="0.1"   max="5.0"    free="1"/>
  <parameter name="Scale"       scale="1e6"   value="1.0" min="0.01"  max="1000.0" free="0"/>
</spectrum>

Broken power law¶

The GModelSpectralBrokenPlaw class implements the broken power law function

\[\begin{split}\frac{dN}{dE} = k_0 \times \left \{ \begin{eqnarray} \left( \frac{E}{E_b} \right)^{\gamma_1} & {\rm if\,\,} E < E_b \\ \left( \frac{E}{E_b} \right)^{\gamma_2} & {\rm otherwise} \end{eqnarray} \right .\end{split}\]

where the parameters in the XML definition have the following mappings:

\(k_0\) = Prefactor
\(\gamma_1\) = Index1
\(\gamma_2\) = Index2
\(E_b\) = BreakEnergy

The XML format for specifying a broken power law is:

<spectrum type="BrokenPowerLaw">
  <parameter name="Prefactor"   scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index1"      scale="-1"    value="2.48" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="BreakEnergy" scale="1e6"   value="0.3"  min="0.01"  max="1000.0" free="1"/>
  <parameter name="Index2"      scale="-1"    value="2.70" min="0.01"  max="1000.0" free="1"/>
</spectrum>

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

<spectrum type="BrokenPowerLaw">
  <parameter name="Prefactor"  scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index1"     scale="-1"    value="2.48" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="BreakValue" scale="1e6"   value="0.3"  min="0.01"  max="1000.0" free="1"/>
  <parameter name="Index2"     scale="-1"    value="2.70" min="0.01"  max="1000.0" free="1"/>
</spectrum>

Smoothly broken power law¶

The GModelSpectralSmoothBrokenPlaw class implements the smoothly broken power law function

\[\frac{dN}{dE} = k_0 \left( \frac{E}{E_0} \right)^{\gamma_1} \left[ 1 + \left( \frac{E}{E_b} \right)^{\frac{\gamma_1 - \gamma_2}{\beta}} \right]^{-\beta}\]

where the parameters in the XML definition have the following mappings:

\(k_0\) = Prefactor
\(\gamma_1\) = Index1
\(E_0\) = PivotEnergy
\(\gamma_2\) = Index2
\(E_b\) = BreakEnergy
\(\beta\) = BreakSmoothness

The XML format for specifying a smoothly broken power law is:

<spectrum type="SmoothBrokenPowerLaw">
  <parameter name="Prefactor"       scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index1"          scale="-1"    value="2.48" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="PivotEnergy"     scale="1e6"   value="1.0"  min="0.01"  max="1000.0" free="0"/>
  <parameter name="Index2"          scale="-1"    value="2.70" min="0.01"  max="+5.0"   free="1"/>
  <parameter name="BreakEnergy"     scale="1e6"   value="0.3"  min="0.01"  max="1000.0" free="1"/>
  <parameter name="BreakSmoothness" scale="1.0"   value="0.2"  min="0.01"  max="10.0"   free="0"/>
</spectrum>

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

<spectrum type="SmoothBrokenPowerLaw">
  <parameter name="Prefactor"   scale="1e-16" value="5.7"  min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index1"      scale="-1"    value="2.48" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="Scale"       scale="1e6"   value="1.0"  min="0.01"  max="1000.0" free="0"/>
  <parameter name="Index2"      scale="-1"    value="2.70" min="0.01"  max="+5.0"   free="1"/>
  <parameter name="BreakValue"  scale="1e6"   value="0.3"  min="0.01"  max="1000.0" free="1"/>
  <parameter name="Beta"        scale="1.0"   value="0.2"  min="0.01"  max="10.0"   free="0"/>
</spectrum>

Gaussian¶

The GModelSpectralGauss class implements the gaussian function

\[\frac{dN}{dE} = \frac{N_0}{\sqrt{2\pi}\sigma} \exp \left( \frac{-(E-\bar{E})^2}{2 \sigma^2} \right)\]

where the parameters in the XML definition have the following mappings:

\(N_0\) = Normalization
\(\bar{E}\) = Mean
\(\sigma\) = Sigma

The XML format for specifying a Gaussian is:

<spectrum type="Gaussian">
  <parameter name="Normalization" scale="1e-10" value="1.0"  min="1e-07" max="1000.0" free="1"/>
  <parameter name="Mean"          scale="1e6"   value="5.0"  min="0.01"  max="100.0"  free="1"/>
  <parameter name="Sigma"         scale="1e6"   value="1.0"  min="0.01"  max="100.0"  free="1"/>
</spectrum>

Log parabola¶

The GModelSpectralLogParabola class implements the log parabola function

\[\frac{dN}{dE} = k_0 \left( \frac{E}{E_0} \right)^{\gamma+\eta \ln(E/E_0)}\]

where the parameters in the XML definition have the following mappings:

\(k_0\) = Prefactor
\(\gamma\) = Index
\(\eta\) = Curvature
\(E_0\) = PivotEnergy

The XML format for specifying a log parabola spectrum is:

<spectrum type="LogParabola">
  <parameter name="Prefactor"   scale="1e-17" value="5.878"   min="1e-07" max="1000.0" free="1"/>
  <parameter name="Index"       scale="-1"    value="2.32473" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="Curvature"   scale="-1"    value="0.074"   min="-5.0"  max="+5.0"   free="1"/>
  <parameter name="PivotEnergy" scale="1e6"   value="1.0"     min="0.01"  max="1000.0" free="0"/>
</spectrum>

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

<spectrum type="LogParabola">
  <parameter name="norm"  scale="1e-17" value="5.878"   min="1e-07" max="1000.0" free="1"/>
  <parameter name="alpha" scale="1"     value="2.32473" min="0.0"   max="+5.0"   free="1"/>
  <parameter name="beta"  scale="1"     value="0.074"   min="-5.0"  max="+5.0"   free="1"/>
  <parameter name="Eb"    scale="1e6"   value="1.0"     min="0.01"  max="1000.0" free="0"/>
</spectrum>

where

alpha = -Index
beta = -Curvature

Composite model¶

Spectral model components can be combined into a single model using the GModelSpectralComposite class. The class computes

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

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

The XML format for specifying a composite spectral model is:

<spectrum type="Composite">
  <spectrum type="PowerLaw" component="SoftComponent">
    <parameter name="Prefactor"   scale="1e-17" value="3"  min="1e-07" max="1000.0" free="1"/>
    <parameter name="Index"       scale="-1"    value="3.5" min="0.0"   max="+5.0"   free="1"/>
    <parameter name="PivotEnergy" scale="1e6"   value="1"  min="0.01"  max="1000.0" free="0"/>
  </spectrum>
  <spectrum type="PowerLaw" component="HardComponent">
    <parameter name="Prefactor"   scale="1e-17" value="5"  min="1e-07" max="1000.0" free="1"/>
    <parameter name="Index"       scale="-1"    value="2.0" min="0.0"   max="+5.0"   free="1"/>
    <parameter name="PivotEnergy" scale="1e6"   value="1"  min="0.01"  max="1000.0" free="0"/>
  </spectrum>
</spectrum>

Multiplicative model¶

Another composite spectral model is the multiplicative spectral model that is implemented by the GModelSpectralMultiplicative class. The class computes

\[M_{\rm spectral}(E | t) = \prod_{i=0}^{N-1} M_{\rm spectral}^{(i)}(E | t)\]

where \(M_{\rm spectral}^{(i)}(E | t)\) is any spectral model component (including another composite or multiplicative model), and \(N\) is the number of model components that are multiplied. This model can for example be used to model any kind of gamma-ray absorption.

The XML format for specifying a multiplicative spectral model is:

<spectrum type="Multiplicative">
  <spectrum type="PowerLaw" component="PowerLawComponent">
    <parameter name="Prefactor"   scale="1e-17" value="1.0"  min="1e-07" max="1000.0" free="1"/>
    <parameter name="Index"       scale="-1"    value="2.48" min="0.0"   max="+5.0"   free="1"/>
    <parameter name="PivotEnergy" scale="1e6"   value="1.0"  min="0.01"  max="1000.0" free="0"/>
  </spectrum>
  <spectrum type="ExponentialCutoffPowerLaw" component="CutoffComponent">
    <parameter name="Prefactor"    scale="1.0" value="1.0" min="1e-07" max="1000.0" free="0"/>
    <parameter name="Index"        scale="1.0" value="0.0" min="-2.0"  max="+2.0"   free="0"/>
    <parameter name="CutoffEnergy" scale="1e6" value="1.0" min="0.01"  max="1000.0" free="1"/>
    <parameter name="PivotEnergy"  scale="1e6" value="1.0" min="0.01"  max="1000.0" free="0"/>
  </spectrum>
</spectrum>

Exponential model¶

Yet another composite model is the exponential model that is implemented by the GModelSpectralExponential class. The class computes the exponential of a spectral model

\[M_{\rm spectral}(E | t) = \exp \left( M_{\rm spectral}(E | t) \right)\]

where \(M_{\rm spectral}(E | t)\) is any spectral model component.

The XML format for specifying an exponential spectral model is:

<spectrum type="Exponential">
  <spectrum type="Constant">
    <parameter name="Normalization" scale="-1.0" value="3.5" min="0.0" max="1000." free="1"/>
  </spectrum>
</spectrum>

Table Of Contents

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Spectral components¶

Constant¶

Node function¶

Bin function¶

File function¶

Table model¶

Power law¶

Exponentially cut-off power law¶

Super exponentially cut-off power law¶

Broken power law¶

Smoothly broken power law¶

Gaussian¶

Log parabola¶

Composite model¶

Multiplicative model¶

Exponential model¶