Energy Dispersion

The energy dispersion information is stored in a FITS file with one required extensions (HDU). The stored quantity is a PDF for the energy migration

\[\mu = \frac{E_{\mathrm{reco}}}{E_{\mathrm{true}}}\]

as a function of true energy and offset. It should be normalized to unity. The migration range covered in the file must be large enough to make this possible (Suggestion: \(1/3 < \mu < 3\))

Transformation

For the purpose of some analysis, for example when extracting an RMF file, it is necessary to calculate the detector response \(R(I,J)\), i.e. the probability to find an energy from within a given true energy bin I of width \(\Delta E_{\mathrm{true}}\) within a certain reconstructed energy bin J of width \(\Delta E_{\mathrm{reco}}\). In order to do so, the following integration has to be performed (for a fixed offset).

\[R(I,J) = \frac{ \int_{\Delta E_{\mathrm{true}}} R(I,E_{\mathrm{true}})\ d E_{\mathrm{true}}}{\Delta E_{\mathrm{true}}},\]

where

\[R(I,E_{\mathrm{true}}) = \int_{\mu(\Delta E_{\mathrm{reco}})} \mathrm{PDF}(E_{\mathrm{true}}, \mu)\ d \mu\]

is the probability to find a given true energy \(E_{\mathrm{true}}\) in the reconstructed energy band J.

edisp_2d format

The energy dispersion information is saved as a BINTABLE HDU with the following required columns.

Columns:

  • MATRIX type: float, dimensions: 3
    • Matrix holding the probability for a given energy migration at a certain true energy and offset.
  • ENERG_LO, ENERG_HI – ndim: 1, unit: TeV
    • Energy axis
  • THETA_LO, THETA_HI – ndim: 1, unit: deg
    • Field of view offset axis
  • MIGRA_LO, MIGRA_HI – ndim: 1, unit: dimensionless
    • Energy migration axis (defined above)
  • MATRIX – ndim: 3, unit: dimensionless
    • Energy dispersion \(dP/d\mu\), see formula above.

Recommended axis order: ENERGY, MIGRA, THETA

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