Spectral model components¶
The following sections present the spectral model components that are available in ctools.
Warning
Source intensities are generally given in units of \({\rm photons}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1}\).
An exception to this rule exists for the DiffuseMapCube spatial
model where intensities are unitless and the spectral model presents a
relative scaling of the diffuse model cube values.
If spectral models are used in combination with RadialAcceptance CTA
background models, intensity units are given in
\({\rm events}\,\,{\rm s}^{-1}\,{\rm MeV}^{-1}\,{\rm sr}^{-1}\)
and correspond to the on-axis count rate.
For other CTA background models intensities are unitless and the spectral model presents a relative scaling of the background model values.
Constant¶
<spectrum type="Constant"> <parameter name="Normalization" scale="1e-16" value="5.7" min="1e-07" max="1000.0" free="1"/> </spectrum>This spectral model component implements the constant function
\[M_{\rm spectral}(E) = N_0\]where
\(N_0\) =
Normalization\(({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1})\)Note
For compatibility with the Fermi/LAT ScienceTools the model type
Constantcan be replaced byConstantValueand the parameterNormalizationbyValue.
Power law¶
<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>This spectral model component implements the power law function
\[M_{\rm spectral}(E) = k_0 \left( \frac{E}{E_0} \right)^{\gamma}\]where
\(k_0\) =
Prefactor\(({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1})\)\(\gamma\) =
Index\(E_0\) =
PivotEnergy\(({\rm MeV})\)Warning
The
PivotEnergyparameter is not intended to be fitted.Note
For compatibility with the Fermi/LAT ScienceTools the parameter
PivotEnergycan be replaced byScale.An alternative power law function that uses the integral photon flux as parameter rather than the Prefactor is specified by
<spectrum type="PowerLaw"> <parameter name="PhotonFlux" scale="1e-07" value="1.0" min="1e-07" max="1000.0" free="1"/> <parameter name="Index" scale="1.0" value="-2.0" min="-5.0" max="+5.0" free="1"/> <parameter name="LowerLimit" scale="1.0" value="100.0" min="10.0" max="1000000.0" free="0"/> <parameter name="UpperLimit" scale="1.0" value="500000.0" min="10.0" max="1000000.0" free="0"/> </spectrum>This spectral model component implements the power law function
\[M_{\rm spectral}(E) = \frac{N(\gamma+1)E^{\gamma}} {E_{\rm max}^{\gamma+1} - E_{\rm min}^{\gamma+1}}\]where
\(N\) =
PhotonFlux\(({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1})\)\(\gamma\) =
Index\(E_{\rm min}\) =
LowerLimit\(({\rm MeV})\)\(E_{\rm max}\) =
UpperLimit\(({\rm MeV})\)Warning
The
LowerLimitandUpperLimitparameters are always treated as fixed and the flux given by thePhotonFluxparameter is computed over the range set by these two parameters. Use of this model allows the errors on the integral flux to be evaluated directly by ctlike.Note
For compatibility with the Fermi/LAT ScienceTools the model type
PowerLawcan be replaced byPowerLaw2and the parameterPhotonFluxbyIntegral.
Exponentially cut-off power law¶
<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>This spectral model component implements the exponentially cut-off power law function
\[M_{\rm spectral}(E) = k_0 \left( \frac{E}{E_0} \right)^{\gamma} \exp \left( \frac{-E}{E_{\rm cut}} \right)\]where
\(k_0\) =
Prefactor\(({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1})\)\(\gamma\) =
Index\(E_0\) =
PivotEnergy\(({\rm MeV})\)\(E_{\rm cut}\) =
CutoffEnergy\(({\rm MeV})\)Warning
The
PivotEnergyparameter is not intended to be fitted.Note
For compatibility with the Fermi/LAT ScienceTools the model type
ExponentialCutoffPowerLawcan be replaced byExpCutoffand the parametersCutoffEnergybyCutoffandPivotEnergybyScale.
Super exponentially cut-off power law¶
<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>This spectral model component implements the super exponentially cut-off power law function
\[M_{\rm spectral}(E) = k_0 \left( \frac{E}{E_0} \right)^{\gamma} \exp \left( -\left( \frac{E}{E_{\rm cut}} \right)^{\alpha} \right)\]where
\(k_0\) =
Prefactor\(({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1})\)\(\gamma\) =
Index1\(\alpha\) =
Index2\(E_0\) =
PivotEnergy\(({\rm MeV})\)\(E_{\rm cut}\) =
CutoffEnergy\(({\rm MeV})\)Warning
The
PivotEnergyparameter is not intended to be fitted.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¶
<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>This spectral model component implements the broken power law function
\[\begin{split}M_{\rm spectral}(E) = k_0 \times \left \{ \begin{eqnarray} \left( \frac{E}{E_b} \right)^{\gamma_1} & {\rm if\,\,} E < E_b \\ \left( \frac{E}{E_b} \right)^{\gamma_2} & {\rm otherwise} \end{eqnarray} \right .\end{split}\]where
\(k_0\) =
Prefactor\(({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1})\)\(\gamma_1\) =
Index1\(\gamma_2\) =
Index2\(E_b\) =
BreakEnergy\(({\rm MeV})\)Warning
Note that the
BreakEnergyparameter may be poorly constrained if there is no clear spectral cut-off in the spectrum. This model may lead to complications in the maximum likelihood fitting.Note
For compatibility with the Fermi/LAT ScienceTools the parameters
BreakEnergycan be replaced byBreakValue.
Smoothly broken power law¶
<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>This spectral model component implements the smoothly broken power law function
\[M_{\rm spectral}(E) = k_0 \left( \frac{E}{E_0} \right)^{\gamma_1} \left[ 1 + \left( \frac{E}{E_b} \right)^{\frac{\gamma_1 - \gamma_2}{\beta}} \right]^{-\beta}\]where
\(k_0\) =
Prefactor\(({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1})\)\(\gamma_1\) =
Index1\(E_0\) =
PivotEnergy\(\gamma_2\) =
Index2\(E_b\) =
BreakEnergy\(({\rm MeV})\)\(\beta\) =
BreakSmoothnessWarning
The pivot energy should be set far away from the expected break energy value.
Warning
When the two indices are close together, the \(\beta\) parameter becomes poorly constrained. Since the \(\beta\) parameter also scales the indices, this can cause very large errors in the estimates of the various spectral parameters. In this case, consider fixing \(\beta\).
Note
For compatibility with the Fermi/LAT ScienceTools the parameters
PivotEnergycan be replaced byScale,BreakEnergybyBreakValueandBreakSmoothnessbyBeta.
Log parabola¶
<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>This spectral model component implements the log parabola function
\[M_{\rm spectral}(E) = k_0 \left( \frac{E}{E_0} \right)^{\gamma+\eta \ln(E/E_0)}\]where
\(k_0\) =
Prefactor\(({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1})\)\(\gamma\) =
Index\(\eta\) =
Curvature\(E_0\) =
PivotEnergy\(({\rm MeV})\)Warning
The
PivotEnergyparameter is not intended to be fitted.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
Gaussian¶
<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>This spectral model component implements the gaussian function
\[M_{\rm spectral}(E) = \frac{N_0}{\sqrt{2\pi}\sigma} \exp \left( \frac{-(E-\bar{E})^2}{2 \sigma^2} \right)\]where
\(N_0\) =
Normalization\(({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1})\)\(\bar{E}\) =
Mean\(({\rm MeV})\)\(\sigma\) =
Sigma\(({\rm MeV})\)
File function¶
<spectrum type="FileFunction" file="data/filefunction.txt"> <parameter name="Normalization" scale="1.0" value="1.0" min="0.0" max="1000.0" free="1"/> </spectrum>This spectral model component implements an arbitrary function that is defined by intensity values at specific energies. The energy and intensity values are defined using an ASCII file with columns of energy and differential flux values. Energies are given in units of \({\rm MeV}\), intensities are given in units of \({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1}\). The only parameter is a multiplicative normalization:
\[M_{\rm spectral}(E) = N_0 \left. \frac{dN}{dE} \right\rvert_{\rm file}\]where
\(N_0\) =
NormalizationWarning
If the file name is given without a path it is expected that the file resides in the same directory than the XML file. If the file resides in a different directory, an absolute path name should be specified. Any environment variable present in the path name will be expanded.
Node function¶
<spectrum type="NodeFunction"> <node> <parameter name="Energy" scale="1.0" value="1.0" min="0.1" max="1.0e20" free="0"/> <parameter name="Intensity" scale="1e-07" value="1.0" min="1e-07" max="1000.0" free="1"/> </node> <node> <parameter name="Energy" scale="10.0" value="1.0" min="0.1" max="1.0e20" free="0"/> <parameter name="Intensity" scale="1e-08" value="1.0" min="1e-07" max="1000.0" free="1"/> </node> </spectrum>This spectral model component implements a generalised broken power law which is defined by a set of energy and intensity values (the so called nodes) that are piecewise connected by power laws. Energies are given in units of \({\rm MeV}\), intensities are given in units of \({\rm ph}\,\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm MeV}^{-1}\).
Warning
An arbitrary number of energy-intensity nodes can be defined in a node function. The nodes need to be sorted by increasing energy. Although the fitting of the
Energyparameters is formally possible it may lead to numerical complications. IfEnergyparameters are to be fitted make sure that theminandmaxattributes are set in a way that avoids inversion of the energy ordering.
Bin function¶
<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>This spectral model component implements energy bins defined by
LowerLimitandUpperLimitvalues given in units of \({\rm MeV}\). Within an energy bin the intensity follows a power law with spectral index defined by theIndexparameter. 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.
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
PARAMETERSextension contains the definition of the model parameters. Each row defines one model parameter. Each model parameter is defined by a uniqueNAME. TheMETHODcolumn indicates whether the model should be interpolated linarly (value0) or logarithmically (value1). So far only linear interpolation is supported, hence the field is ignored. TheINITIALcolumn indicates the initial parameter value, if the value in theDELTAcolumn is negative the parameter will be fixed, otherwise it will be fitted. TheMINIMUMandMAXIMUMcolumns indicate the range of values for a given parameter, theBOTTOMandTOPcolumns are ignored. The``NUMBVALS`` column indicates the number of parameter values for which the table model was computed, theVALUEcolumn indicates the specific parameter values.In the example below there are two parameters named
IndexandCutoff, and spectra were computed for 100 index values and 50 cutoff values, hence a total of 5000 spectra are stored in the table model.
Table model parameters extension¶
The
ENERGIESextension contains the energy boundaries for the spectra in the usual OGIP format:
Energy boundaries extension¶
The
SPECTRAextension contains the spectra of the table model. It consists of two vector columns. ThePARAMVALcolumn provides the parameter values for which the spectrum was computed. Since there are two parameters in the example the vector column has two entries. TheINTPSPECcolumn 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.
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 name="Normalization" scale="1.0" value="1.0" min="0.0" max="1000.0" free="1"/> </spectrum>Warning
If the file name is given without a path it is expected that the file resides in the same directory than the XML file. If the file resides in a different directory, an absolute path name should 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
Indexand aCutoffparameter, the XML file may look like<spectrum type="TableModel" file="model_table.fits"> <parameter name="Normalization" scale="1e-16" value="5.8" min="1e-07" max="1000" free="1"/> <parameter name="Index" scale="-1" value="2.4" min="1.0" max="3.0" free="1"/> <parameter name="Cutoff" scale="1e6" value="0.89" min="0.1" max="28.2" free="1"/> </spectrum>
Composite model¶
<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>This spectral model component implements a composite model that is the sum of an arbitrary number of spectral models, computed using
\[M_{\rm spectral}(E) = \sum_{i=0}^{N-1} M_{\rm spectral}^{(i)}(E)\]where \(M_{\rm spectral}^{(i)}(E)\) is any spectral model component (including another composite model), and \(N\) is the number of model components that are combined.
Multiplicative model¶
<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>This spectral model component implements a composite model that is the product of an arbitrary number of spectral models, computed using
\[M_{\rm spectral}(E) = \prod_{i=0}^{N-1} M_{\rm spectral}^{(i)}(E)\]where \(M_{\rm spectral}^{(i)}(E)\) is any spectral model component (including another composite model), and \(N\) is the number of model components that are multiplied.
Exponential model¶
<spectrum type="Exponential"> <spectrum type="FileFunction" file="opacity.txt"> <parameter name="Normalization" scale="-1.0" value="1.0" min="0.0" max="100.0" free="1"/> </spectrum> </spectrum>This spectral model component implements the exponential of an arbitrary spectral model and computes
\[M_{\rm spectral}(E) = \exp \left( \alpha M_{\rm spectral}(E) \right)\]where
\(M_{\rm spectral}(E)\) is any spectral model component
\(\alpha\) =
NormalizationThe model can be used to describe a spectrum with EBL absorption based on a tabulated model of opacity as a function of photon energy. The corresponding XML file structure for such a model is shown below:
<spectrum type="Multiplicative"> <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> <spectrum type="Exponential"> <spectrum type="FileFunction" file="opacity.txt"> <parameter name="Normalization" scale="-1.0" value="1.0" min="0.0" max="100.0" free="1"/> </spectrum> </spectrum> </spectrum>This example corresponds to the function
\[M_{\rm spectral}(E) = k_0 \left( \frac{E}{E_0} \right)^{\gamma} \times \exp\left( -\alpha \, \tau(E) \right)\]where
the first block/factor corresponds to a power law;
the second block/factor models EBL absorption, and it points to an ASCII file with two columns containing energy in \({\rm MeV}\) as first column and opacity \(\tau\) as second column, respectively;
the parameter \(\alpha\) =
Normalizationrepresents an opacity scaling factor.Note
The
Exponentialmodel implements the function \(y=\exp(x)\), hence in the example thescaleattribute of theNormalizationparameter was set to-1to implement the form \(y=\exp(-x)\).