3.2.1 Flux of a Source - Gaussian Statistics Case

This is the most useful case as in general not just a detection, but a measurement of the flux (or upper limit on the flux) of a source is required.

__On-axis,__ the signal to noise ratio is given by:

K_{snr} = N_{src}/var(N_{tot}) |
(5) |

where

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

R_{min}t = K^{2}_{snr}/2
+ K^{2}_{snr}/2
+ K^{2}_{snr}R_{b}t |
(6) |

where
*R*_{min} denotes the minimum detectable source count rate,
*R*_{b} the total background rate in the detection cell,
*t* the observation time, and
*K*_{snr} the desired signal to noise ratio (snr) on the flux measurement.
This is derived by combining the equations defining the snr and
variance, and using the additional relation, counts = rate x time.
The standard estimate of *R*_{b} uses the last entry of Table 5,
1.1 `x` 10^{-6} counts s^{-1} arcsec^{-2}, multiplied by the area of
the detection cell (in
*arcsec*^{2}).

Given a desired s/n and a countrate, one can also solve for the time required to achieve a particular s/n for a particular source:

As an illustrative example (see Figure 28), we show the point source
sensitivity for *K*_{snr} = 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 5
`x` 10^{20} cm^{-2} 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 10^{5} 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
^{-1} = 1.06 `x` 10^{-10} erg cm^{-2} s^{-1} 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.

__Off-axis,__ sensitivity estimates must take into account the
cell size from the off-axis PSF (cf. Section 2.2). The cell size can
be taken as the 50% power radius, i.e., a square region where 50% of
the source counts will fall. The background rates must also be scaled
to the new cell size.

The vignetting correction in Figure 14 (Section 2.3.2) can be used to scale down the number of counts detected from off-axis sources due to the decrease in telescope area; since the HRI field of view is limited, the slight energy dependence of the vignetting correction can usually be neglected for feasibility purposes.

1999-05-25