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the signal to noise ratio is given by:


where tex2html_wrap_inline17621 is the number of source counts, and tex2html_wrap_inline17623 is the variance in the total counts (source + background) and is typically given by tex2html_wrap_inline17625 , which assumes no uncertainty in the background.

The minimum detectable source count rate corresponding to a particular signal to noise ratio is given by the relationship


where tex2html_wrap_inline17629 denotes the minimum detectable source count rate, tex2html_wrap_inline17631 the total background rate in the detection cell (about tex2html_wrap_inline17633 counts s tex2html_wrap_inline17635 ), t the observation time, and tex2html_wrap_inline17639 the signal to noise ratio (snr) threshold for detection.

This is derived by combining the equations defining the snr and variance, and using the additional relation, counts = rate x time.

As an illustrative example, we show in Figure 11.3 in Appendix F, the point source sensitivity for tex2html_wrap_inline17641 = 5 versus observing time for a source with a power law spectrum. That is, we assume a point source with a power law spectrum of energy index 0.5 similar to the extragalactic X-ray background. A hydrogen column density of tex2html_wrap_inline17643  cm tex2html_wrap_inline17645 is used. Notice that the sensitivity increases linearly with observing time for short observations where there are virtually no background counts in the detection cell. For observations that are long enough for background to be significant, longer than about tex2html_wrap_inline17647  s, the sensitivity increases only as the square root of the observation time.

In general, the conversion from HRI counts to energy flux depends on the spectrum of the X-ray source. For the power law spectrum characterized by the parameters given above, this conversion is given by: 1 count s tex2html_wrap_inline17649  erg cm tex2html_wrap_inline17651  s tex2html_wrap_inline17653 in the energy band 0.1 to 2.4 keV. The following section discusses the conversion from energy flux to count rate in more detail.

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