When should I care about the background?
The background should be taken into account for all sources, but most especially for soft sources and diffuse sources. The exposure and vignetting corrections are energy-dependent, so extended/diffuse sources will be most affected. The detector artifacts (window support structure, for example) are energy-dependent, so they will hit the data in different ways. If your sources are photon-limited, then the analysis will depend upon how you treat the background.
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