You may not have recognised it, but the examples you have exercised so far have neglected the impact of energy dispersion on the analysis. In reality, however, the reconstructed event energy will differ from the true photon energy, and this effect will become particularily important at low energies. There are therefore cases where you want to take the effects of energy dispersion into account, and you can do this in all ctools that make use of the instrument response functions by setting the hidden edisp parameter to yes.
Warning
The maximum likelihood computations including energy dispersion are relatively time consuming, and in many situations the impact of the energy dispersion on the analysis results will be very small. For this reason, energy dispersion is by default disabled in ctools.
To simulate events taking into account the energy dispersion you should run the ctobssim tool with edisp=yes parameter:
$ ctobssim edisp=yes
RA of pointing (degrees) (0-360) [83.63]
Dec of pointing (degrees) (-90-90) [22.01]
Radius of FOV (degrees) (0-180) [5.0]
Start time (MET in s) [0.0]
End time (MET in s) [1800.0]
Lower energy limit (TeV) [0.1]
Upper energy limit (TeV) [100.0]
Calibration database [prod2]
Instrument response function [South_0.5h]
Input model XML file [$CTOOLS/share/models/crab.xml]
Output event data file or observation definition XML file [events.fits]
You may then select events as before using the ctselect tool (since this tool does not need the instrument response function it has no hidden edisp parameter):
$ ctselect
Input event list or observation definition XML file [events.fits]
RA for ROI centre (degrees) (0-360) [83.63]
Dec for ROI centre (degrees) (-90-90) [22.01]
Radius of ROI (degrees) (0-180) [3.0]
Start time (CTA MET in seconds) [0.0]
End time (CTA MET in seconds) [0.0]
Lower energy limit (TeV) [0.1]
Upper energy limit (TeV) [100.0]
Output event list or observation definition XML file [selected_events.fits]
You then perform a maximum likelihood fit using the ctlike tool with edisp=yes so that energy dispersion is correctly taken into account in the likelihood computation:
$ ctlike edisp=yes
Input event list, counts cube or observation definition XML file [selected_events.fits]
Calibration database [prod2]
Instrument response function [South_0.5h]
Input model XML file [$CTOOLS/share/models/crab.xml]
Output model XML file [crab_results.xml]
Below you see the output from the ctlike.log file.
2016-06-29T20:33:36: +=================================+
2016-06-29T20:33:36: | Maximum likelihood optimisation |
2016-06-29T20:33:36: +=================================+
2016-06-29T20:33:39: >Iteration 0: -logL=133777.933, Lambda=1.0e-03
2016-06-29T20:33:42: >Iteration 1: -logL=133772.125, Lambda=1.0e-03, delta=5.808, max(|grad|)=23.221525 [Index:7]
2016-06-29T20:33:45: >Iteration 2: -logL=133772.111, Lambda=1.0e-04, delta=0.013, max(|grad|)=-0.042654 [Prefactor:6]
2016-06-29T20:33:48: >Iteration 3: -logL=133772.111, Lambda=1.0e-05, delta=0.000, max(|grad|)=-0.000611 [Index:3]
...
2016-06-29T20:33:50: +=========================================+
2016-06-29T20:33:50: | Maximum likelihood optimisation results |
2016-06-29T20:33:50: +=========================================+
2016-06-29T20:33:50: === GOptimizerLM ===
2016-06-29T20:33:50: Optimized function value ..: 133772.111
2016-06-29T20:33:50: Absolute precision ........: 0.005
2016-06-29T20:33:50: Acceptable value decrease .: 2
2016-06-29T20:33:50: Optimization status .......: converged
2016-06-29T20:33:50: Number of parameters ......: 10
2016-06-29T20:33:50: Number of free parameters .: 4
2016-06-29T20:33:50: Number of iterations ......: 3
2016-06-29T20:33:50: Lambda ....................: 1e-06
2016-06-29T20:33:50: Maximum log likelihood ....: -133772.111
2016-06-29T20:33:50: Observed events (Nobs) ...: 20952.000
2016-06-29T20:33:50: Predicted events (Npred) ..: 20952.000 (Nobs - Npred = 1.01277e-06)
2016-06-29T20:33:50: === GModels ===
2016-06-29T20:33:50: Number of models ..........: 2
2016-06-29T20:33:50: Number of parameters ......: 10
2016-06-29T20:33:50: === GModelSky ===
2016-06-29T20:33:50: Name ......................: Crab
2016-06-29T20:33:50: Instruments ...............: all
2016-06-29T20:33:50: Instrument scale factors ..: unity
2016-06-29T20:33:50: Observation identifiers ...: all
2016-06-29T20:33:50: Model type ................: PointSource
2016-06-29T20:33:50: Model components ..........: "PointSource" * "PowerLaw" * "Constant"
2016-06-29T20:33:50: Number of parameters ......: 6
2016-06-29T20:33:50: Number of spatial par's ...: 2
2016-06-29T20:33:50: RA .......................: 83.6331 [-360,360] deg (fixed,scale=1)
2016-06-29T20:33:50: DEC ......................: 22.0145 [-90,90] deg (fixed,scale=1)
2016-06-29T20:33:50: Number of spectral par's ..: 3
2016-06-29T20:33:50: Prefactor ................: 5.74029e-16 +/- 1.04025e-17 [1e-23,1e-13] ph/cm2/s/MeV (free,scale=1e-16,gradient)
2016-06-29T20:33:50: Index ....................: -2.49899 +/- 0.015621 [-0,-5] (free,scale=-1,gradient)
2016-06-29T20:33:50: PivotEnergy ..............: 300000 [10000,1e+09] MeV (fixed,scale=1e+06,gradient)
2016-06-29T20:33:50: Number of temporal par's ..: 1
2016-06-29T20:33:50: Normalization ............: 1 (relative value) (fixed,scale=1,gradient)
2016-06-29T20:33:50: === GCTAModelIrfBackground ===
2016-06-29T20:33:50: Name ......................: CTABackgroundModel
2016-06-29T20:33:50: Instruments ...............: CTA
2016-06-29T20:33:50: Instrument scale factors ..: unity
2016-06-29T20:33:50: Observation identifiers ...: all
2016-06-29T20:33:50: Model type ................: "PowerLaw" * "Constant"
2016-06-29T20:33:50: Number of parameters ......: 4
2016-06-29T20:33:50: Number of spectral par's ..: 3
2016-06-29T20:33:50: Prefactor ................: 0.959074 +/- 0.0134496 [0.001,1000] ph/cm2/s/MeV (free,scale=1,gradient)
2016-06-29T20:33:50: Index ....................: -0.0162199 +/- 0.00850751 [-5,5] (free,scale=1,gradient)
2016-06-29T20:33:50: PivotEnergy ..............: 1e+06 [10000,1e+09] MeV (fixed,scale=1e+06,gradient)
2016-06-29T20:33:50: Number of temporal par's ..: 1
2016-06-29T20:33:50: Normalization ............: 1 (relative value) (fixed,scale=1,gradient)