Correct extraction of a diffuse source differs from extraction of a point source in a number of important ways.
Observers commonly need to express the observed strength of a diffuse source in terms of surface brightness, for example normalizing a luminosity calculated via XSPEC by some measure of the size of the source on the sky. If the response of the observatory was constant within the selected extraction region, then the appropriate size normalization would simply be the geometric area of the region. However, in a typical ACIS observation the response varies strongly across the extraction region in several ways:
When a diffuse source is extracted in CIAO, this spatially-varying response is abstracted/averaged into a single set of response files (ARF and RMF). Obviously, the appropriate region size normalization depends on how this average response is calculated, since in the denominator of the final surface brightness expression, e.g. , the response of the observatory and the size normalization are degenerate (i.e. are multiplied together). Now, if one averaged the observatory response over the region in the SKY coordinate system, , (including the effects of bad columns, chip gaps, detector edges, and point source masks) then that multi-ObsId response would account for everything, and the appropriate size normalization would simply be the geometric area of the region. However, it is important to understand that this is not the algorithm employed by the tool mkwarf. Instead, mkwarf (through its WMAP input) forms a weighted average of the response of the observatory within a set of cells on the detector. In this process there is no concept of reduced exposure time arising from dithering over unobserved parts of the focal plane, and there is no concept of point source masking.
The good news is that we have on hand a data product that does represent the ``depth'' of the observation (exposure time effective area) everywhere, namely the exposure map
Given that the ARF produced by mkwarf is the only convenient representation we have for the energy-dependence of the response, a reasonable approach would seem to be to choose any scaling for that ARF and/or for the EXPOSURE time and/or for the geometric region area such that in the end our extracted spectrum is normalized by at energy . AE chooses to scale mkwarf's ARF to produce a ``surface brightness ARF'' (designated below by the subscript SB) as follows:
The units of the ARF are thus changed from to . All ``flux'' quantities derived from XSPEC should then be understood to be surface brightness quantities with arcsec appended to the units. Actual integrated fluxes over the entire diffuse region are then estimated by multiplying inferred surface brightnesses by the geometric area of the region.
The conversion of each extraction's calibration (ARF) to surface brightness units, described above, is very convenient when multiple observations are to be combined. Each extraction of a diffuse region will generally have different sub-regions that are unobserved in that ObsID, and thus different normalizations for mkwarf's ARF. Once all the extractions are calibrated in surface brightness units, they can be straightforwardly merged in the same way that point sources are merged (§5.6). There is a clear analogy between this practice and the way AE handles PSF fractions when multiple point source extractions are merged; in that case since each extraction can have a different PSF fraction, AE chooses to scale each observation's ARF by its PSF fraction prior to merging.