Temporal model components

The following sections present the temporal model components that are available in ctools.

Warning

Recall that the source model is factorised according to

\[M(p,E,t) = M_{\rm spatial}(p|E) \times M_{\rm spectral}(E) \times M_{\rm temporal}(t)\]

which implies that the temporal model component is multiplied with the spectral model component.

Constant

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

This temporal model component implements a constant source

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

where

• \(N_0\) = Normalization

Light Curve

<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>

This temporal model component implements a light curve \(r(t)\)

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

where

• \(N_0\) = Normalization

The light curve is defined by nodes in a FITS file that specify the relative flux normalization as function of time (file model_temporal_lightcurve.fits in the example above). 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.

Structure of light curve FITS file

Warning

Fitting of light curves only makes sense for an unbinned maximum likelihood analysis, since in a binned or stacked analysis the times of individual events are dropped.

Phase Curve

<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>

This temporal model component implements a phase curve \(r(\Phi(t))\)

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

where the phase as function of time is computed using

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

and

• \(N_0\) = Normalization

• \(t_0\) = MJD

• \(\Phi_0\) = Phase

• \(f\) = F0

• \(\dot{f}\) = F1

• \(\ddot{f}\) = F2

The phase curve is defined by nodes in a FITS file that specify the relative flux normalization as function of phase (file model_temporal_phasecurve.fits in the example above). 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.

Structure of phase curve FITS file

By default, the NORM values are recomputed internally so that the phase-averaged normalisation is one, i.e.

\[\int_0^1 r(\Phi) d\Phi = 1\]

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.

Warning

Fitting of phase curves only makes sense for an unbinned maximum likelihood analysis, since in a binned or stacked analysis the times of individual events are dropped.

Warning

Fitting of phase curve parameters may not properly work for pulsar frequencies.