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XMM-Newton Science Analysis System


sensitivity (esensitivity-1.8) [xmmsas_20080701_1801-8.0.0]

Meta Index / Home Page / Description

Theory

The task eboxdetect takes as input $N$ images of the same part of the sky in $N$ separate energy bands, and (in `map' mode) $N$ images of the estimated background counts per pixel. The task applies a sliding box to these stacked images. At each pixel, then, the task has available $N$ values $C_i$ of the actual number of events found within the box and $N$ values $\langle B_i \rangle$ of the expected number of background events within the same box. Considering for a moment just the values from a single band in isolation, if there were no sources in the field, then the probability $p(C)$ of finding $C$ counts in this box would be given by the familiar Poisson expression

$$p(C) = \frac{\langle B\rangle^C \ e^{-\langle B\rangle}}{C!}.$$

The probability $P_{\rm {null}}(C)$ that the observed count $C$ is due only to background is therefore the sum of $p(x)$ for all values of $x$ from $C$ to infinity. This can be shown to be equal to

$$P_{\rm {null}}(C) = 1-Q(C, \langle B\rangle)$$

where $Q$ is the incomplete gamma function, defined by

$$Q(C, \langle B\rangle) = \frac{\int_{\langle B\rangle}^{\in... ... \, e^{-t} t^{C-1}}{\int_{0}^{\infty} \ dt \, e^{-t} t^{C-1}}.$$

eboxdetect calculates the value of the LIKE column for the $i$th energy band from the likelihood $L_i = -ln(P_{\rm {null}})$. The LIKE value $L_{\rm {summ}}$ for the summary row is calculated by eboxdetect as follows. First, the values of LIKE for the individual bands are summed. This sum can be shown itself to follow a Poisson-like probability distribution. Leaving out the details, the probability $P_{\rm {null}}(\bar{C})$ that the vector $\bar{C}$ of counts in the $N$ bands is due solely to background can be shown to be approximately given by

$$P_{\rm {null}}(\bar{C}) = Q\Big[N, N-\sum_{i=1}^{N} Q\big(C_i, \langle B_i\rangle\big) \Big].$$

The summary-row value of LIKE is thus calculated from

$$L_{\rm {summ}} = -ln\Big\{Q\Big[N, N-\sum_{i=1}^{N} Q\big(C_i, \langle B_i\rangle\big) \Big]\Big\}.$$ (1)

If there is no source at that location then clearly $C_i = B_i$. Suppose however that there is actually a source which contributes, on average, $\langle S_i\rangle$ counts to band $i$ within the box in question. Equation 1 becomes in this case

$$L_{\rm {summ}} = -ln\Big\{Q\Big[N, N-\sum_{i=1}^{N} Q\big(B_i+S_i, \langle B_i\rangle\big) \Big]\Big\}.$$

The question which sensitivity should answer is, how small can the $S_i$ become while retaining $L_{\rm {summ}} \ge L_{\rm {cutoff}}$? The first step in answering this is to reduce the degrees of freedom in the problem. For this purpose it is convenient to express each $\langle S_i\rangle$ as a product of two factors: a purely spectral factor $\mu_i=\langle S_i/\sum S_i\rangle$, which will be held fixed, and a pure flux factor $\langle S\rangle=\langle \sum S_i\rangle$, which we vary. Let us find that $\langle S_{\rm {min}}\rangle$ which is the solution to

$$L_{\rm {cutoff}} = -ln\Big\{Q\Big[N, N-\sum_{i=1}^{N} Q\big... ...gle S_{\rm {min}}\rangle, \langle B_i\rangle\big) \Big]\Big\}.$$ (2)

This value of $\langle S\rangle$ represents the expectation value of counts within the detection box of a source of spectrum $\mu_i$ which will, on average, result in a source detection against the given background.

This is exactly what sensitivity does: solve equation 2. All that remains after that is to calculate $\langle S_{i,\rm {min}}\rangle = \mu_i \langle S_{\rm {min}}\rangle$ for all $i$, then convert these counts values to count rates by multiplication by the box exposure at that pixel, supplied via the parameter psfexpmapsets.