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.

K_snr = N_src/ $\displaystyle \left(\vphantom{var(N_{tot})}\right.$ var(N_tot) $\displaystyle \left.\vphantom{var(N_{tot})}\right)^{1/2}_{}$

(5)

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

R_mint = $\displaystyle \left(\vphantom{{K^{2}_{snr}}/{2}}\right.$ K²_snr/2 $\displaystyle \left.\vphantom{{K^{2}_{snr}}/{2}}\right)$ + $\displaystyle \left(\vphantom{ \left( {K^{2}_{snr}}/{2} \right)^{2} +K^{2}_{snr}R_{b}t }\right.$ $\displaystyle \left(\vphantom{ {K^{2}_{snr}}/{2} }\right.$ K²_snr/2 $\displaystyle \left.\vphantom{ {K^{2}_{snr}}/{2} }\right)^{2}_{}$ + K²_snrR_bt $\displaystyle \left.\vphantom{ \left( {K^{2}_{snr}}/{2} \right)^{2} +K^{2}_{snr}R_{b}t }\right)^{1/2}_{}$

(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 arcsec²).

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:

t = K²_snr $\displaystyle \left[\vphantom{{1/R} + {R_{b}/R^{2}}}\right.$ 1/R + R_b/R² $\displaystyle \left.\vphantom{{1/R} + {R_{b}/R^{2}}}\right]$

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.

**Figure 28:** The relationship between R and t for a snr of 5, with the standard background rate, alpha=0.5 and log NH=20.7. (a) a log-log plot; (b) the same, but for a linear scale.
$\includegraphics[angle=270,scale=0.5]{ros1.ps}$ $\includegraphics[angle=270,scale=0.5]{ros3.ps}$

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 10²⁰ 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 10⁵ 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.