5.9 ARFs

Construction of an ancillary response file (ARF) for each source in single observation consists of several steps.

• The CIAO tool mkarf is called to construct an ARF for each of the CCDs on which the source is observed. When the source spans multiple CCDs the ARFs are summed. The observer supplies appropriate aspect histogram files--these are typically obtained as a byproduct of exposure map construction tool (e.g. via the tool ae_make_emap in §7.20).

• The source extraction region is applied to each monochromatic PSF available (§5.1), and a PSF fraction is computed at each energy. The ARF is reduced by this PSF fraction curve (interpolated between sampled energies) to account for the source events that fell outside the extraction region.

• If desired, the observer can apply an externally-computed correction curve to the ARFs, e.g. a correction for a filter transmission or QE effect not yet modeled by the CALDB.

In a multi-observation reduction, a multi-ObsId ARF is constructed using the FTOOL addarf, using the observation's EXPOSURE keywords as the weights:

$$ARF = \frac{\sum_{i} EXPOSURE_i \times ARF_i}{\sum_{i} EXPOSURE_i}$$ (6)

A multi-ObsId source spectrum is constructed by summing the extracted spectra (eq. 1), and a multi-ObsId EXPOSURE value is defined as:

$$EXPOSURE = \sum_{i} EXPOSURE_i$$ (7)

To see that this is the appropriate way to combine the ARFs consider what happens inside the fitting program (e.g. XSPEC). The source model ( $photons/s/cm^2$) is multiplied by the multi-ObsId ARF ( $cm^2 counts/photon$) and by EXPOSURE (s) to produce a model spectrum (counts), shown on the left-hand side of eq. 9.

$$model \times ARF \times EXPOSURE = model \times \frac{\sum_{i} EXPOSURE_i \times ARF_i}{\sum_{i} EXPOSURE_i} \times EXPOSURE$$ (8)

$$= \sum_{i} model \times EXPOSURE_i \times ARF_i$$ (9)

By substituting the definition of the multi-ObsId ARF (eq. 6) and the definition of the multi-ObsId EXPOSURE (eq. 7) we can see that the model spectrum is precisely the sum of the data each ObsId is expected to produce (right-hand side of eq. 9).

It is interesting to consider the case where one of the extractions being merged has an ARF that is much smaller than the others (e.g. because it is in a chip gap, or because it has a very low PSF fraction). From equations 6 and 7 it's clear that including this sort of insigificant extraction in the merge can significantly change both the multi-ObsId ARF and the multi-ObsId EXPOSURE time. This seems very wrong at first glance, however, as shown in eq. 9, the two effects cancel each other within the fitting process.