Temporal components

Constant

The GModelTemporalConst class implements a temporal model that is constant in time and defined by

\[M_{\rm temporal}(t) = N_0\]

where the parameter in the XML definition has the following mapping:

  • \(N_0\) = Normalization

The XML format for specifying a constant temporal model is:

<temporal type="Constant">
  <parameter name="Normalization" scale="1.0" value="1.0" min="0.1" max="10.0" free="0"/>
</temporal>

Light Curve

The GModelTemporalLightCurve class implements a temporal model that defines an arbitrary light curve, defined by a function in a FITS file.

The only parameter of the model is a multiplicative normalization:

\[M_{\rm temporal}(t) = N_0 r(t)\]

where the parameter in the XML definition has the following mapping:

  • \(N_0\) = Normalization

The XML format for specifying a light curve is:

<temporal type="LightCurve" file="model_temporal_lightcurve.fits">
  <parameter name="Normalization" scale="1" value="1.0" min="0.0" max="1000.0" free="0"/>
</temporal>

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.

The structure of the light curve FITS file is shown in the figure below. The light curve is defined in the first extension of the FITS file and consists of a binary table with the columns TIME and NORM. Times in the TIME columns are given in seconds and are counted with respect to a time reference that is defined in the header of the binary table. Times need to be specified in ascending order. The values in the NORM column specify \(r(t)\) at times \(t\), and should be comprised between 0 and 1.

../../../../../_images/model_lightcurve.png

Structure of light curve FITS file

Phase Curve

The GModelTemporalPhaseCurve class implements a temporal model that defines a temporal variation based on an arbitrary phase \(\Phi(t)\) that is computed using

\[\Phi(t) = \Phi_0 + f(t-t_0) + \frac{1}{2}\dot{f} (t-t_0)^2 + \frac{1}{6}\ddot{f} (t-t_0)^3\]

where

  • \(t_0\) is a reference time,
  • \(\Phi_0\) is the phase at the reference time,
  • \(f\) is the variation frequency at the reference time,
  • \(\dot{f}\) is the first derivative of the variation frequency at the reference time, and
  • \(\ddot{f}\) is the second derivative of the variation frequency at the reference time.

The temporal variation is computed using

\[M_{\rm temporal}(t) = N_0 r(\Phi(t))\]

The parameters in the XML definition have the following mappings:

  • \(N_0\) = Normalization
  • \(t_0\) = MJD
  • \(\Phi_0\) = Phase
  • \(f\) = F0
  • \(\dot{f}\) = F1
  • \(\ddot{f}\) = F2

The XML format for specifying a phase curve is:

<temporal type="PhaseCurve" file="model_temporal_phasecurve.fits">
  <parameter name="Normalization" scale="1" value="1.0"     min="0.0" max="1000.0"   free="0"/>
  <parameter name="MJD"           scale="1" value="51544.5" min="0.0" max="100000.0" free="0"/>
  <parameter name="Phase"         scale="1" value="0.0"     min="0.0" max="1.0"      free="0"/>
  <parameter name="F0"            scale="1" value="1.0"     min="0.0" max="1000.0"   free="0"/>
  <parameter name="F1"            scale="1" value="0.1"     min="0.0" max="1000.0"   free="0"/>
  <parameter name="F2"            scale="1" value="0.01"    min="0.0" max="1000.0"   free="0"/>
</temporal>

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.

The structure of the phase curve FITS file is shown in the figure below. The phase curve is defined in the first extension of the FITS file and consists of a binary table with the columns PHASE and NORM. Phase values in the PHASE column need to be comprised between 0 and 1 and need to be given in ascending order. The values in the NORM column specify \(r(\Phi(t))\) at phases \(\Phi(t)\), and should be comprised between 0 and 1.

../../../../../_images/model_phasecurve.png

Structure of phase curve FITS file

By default, the NORM values are recomputed internally so that the phase-averaged normalisation is one. In that case, the spectral component corresponds to the phase-averaged spectrum. If the internal normalisation should be disabled the normalize="0" attribute needs to be added to the temporal tag, i.e.

<temporal type="PhaseCurve" file="model_temporal_phasecurve.fits" normalize="0">

In that case the NORM values are directly multiplied with the spectral component.