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Computing Average Source Intensity for X-ray Sources Observed in Multiple X-ray Images
Using quantities measured in synthetic apertures in X-ray images, it's relatively easy to determine the posterior probability distribution for the intensity of an unresolved x-ray source in a single observation. However, if the same source is observed in multiple observations, and one wishes to combine the data to determine a single average intensity, difficulties can ensue.


Simple X-ray Aperture Photometry Problem: Determine source intensity s and errors for an unresolved (point) x-ray source Know:

· Number of counts, C, in source
aperture (solid green ellipse)

· Number of counts, B, in background
aperture (dashed green elliptical annulus)

· Areas As and Ab of source and
background apertures

· f = psf(x,y)dx dy over source
aperture

· g = psf(x,y)dx dy over background
aperture


Simple X-ray Aperture Photometry Problem Solution (see http://cxc.cfa.harvard.edu/csc/memos/files/Kashyap_xraysrc.pdf for details)

· Statistical Model: C ~ Pois(f s + b); · Solution for non-informative priors:

B ~ Pois(g s + r b) where r = Ab / As

0.004

0.003 P( R | nm ) dR

0.002

0.001

0 0 0.01 0.02 0.03 0.04 0.05

Source Rate R (counts/sec)


A more complicated problem: Combining data from the same source, observed at different times, with distinct pdf 's
0.004

0.003 P( R | nm ) dR

0.002

0.001

0 0 0.01 0.02 0.03 0.04 0.05

Source Rate R (counts/sec)

0.004

0.003 P( R | nm ) dR

0.002

0.001

0 0 0.01 0.02 0.03 0.04 0.05

Source Rate R (counts/sec)


Questions about multi-observation problem:

· · · ·

Can the pdf from one observation be used as a prior for a second? What happens if the single observation pdf 's are distinct? What if there are more than two observations? Need a robust solution that's insensitive to the order in which the data are analyzed, i.e., if observation 1 pdf is used as prior to observation 2, should get the same solution as observation 2 pdf used as prior to observation 1, etc. At what point does it make no sense to combine the data? Is there a quantitative measure that can be used to decide when to keep observations separate? In the limit of large counts, solution should approach that using Gaussian statistics, i.e., error-weighted mean of individual intensities.