Subsections

• 5.10.1 Source and background counts with limits.
• 5.10.2 Background scaling, net counts with limits
• 5.10.3 Source validity/significance
• 5.10.4 Photon Flux
• 5.10.5 Characterizations of the background subtracted OBSERVED spectrum
• 5.10.6 Characterizations of the background subtracted INCIDENT spectrum

5.10 Broad Band Photometry

The multi-ObsId spectra and ARFs are used to perform background-subtracted photometry on the sources using a set of energy bands supplied by the observer. Starting with the following input data (bold variables are arrays with an element for each spectral channel):

• Integer observed PI spectral vectors from the source and background regions, $\mathbf{src\_observed\_counts}$ and $\mathbf{bkg\_observed\_counts}$
• The ARF value in each of the spectral channels, $\mathbf{channel\_arf}$
• The scaling (a single quantity, not energy-dependent) of the extracted background spectrum required to model the background in the source aperture, $AREASCAL_{bkg}$, defined by Equation 5 in §5.6.
• The total exposure, $EXPOSURE$

the following photometry computations are performed. (Capitalized variables below are FITS columns in the photometry output file.)

5.10.1 Source and background counts with limits.

$$SRC\_CNTS = \sum_{band} \mathbf{src\_observed\_counts}$$

$$BKG\_CNTS = \sum_{band} \mathbf{bkg\_observed\_counts}$$

Standard 1-sigma Gaussian confidence intervals for both SRC_CNTS and BKG_CNTS are computed using an analytical approximation to upper and lower confidence intervals of a Poissonian distribution (Gehrels, 1986, equations 7 and 12).

$$src\_cnts\_limit\_up = SRC\_CNTS + 1 + \sqrt{SRC\_CNTS + 0.75}$$

$$bkg\_cnts\_limit\_up = BKG\_CNTS + 1 + \sqrt{BKG\_CNTS + 0.75}$$

$$src\_cnts\_limit\_low = SRC\_CNTS * {( 1 - \frac{1}{9.0*SRC\_CNTS} - \frac{1}{3.0*\sqrt{SRC\_CNTS}} )}^3$$

$$bkg\_cnts\_limit\_low = BKG\_CNTS * {( 1 - \frac{1}{9.0*BKG\_CNTS} - \frac{1}{3.0*\sqrt{BKG\_CNTS}} )}^3$$

5.10.2 Background scaling, net counts with limits

$$\mathbf{bkg\_counts\_in\_src\_region} = \mathbf{bkg\_observed\_counts} / AREASCAL_{bkg}$$

$$\mathbf{net\_counts} = \mathbf{src\_observed\_counts} - \mathbf{bkg\_counts\_in\_src\_region}$$

$$NET\_CNTS = \sum_{band} \mathbf{net\_counts}$$

$$BACKSCAL = \sum_{band} \mathbf{bkg\_observed\_counts} / \sum_{band} \mathbf{bkg\_counts\_in\_src\_region}$$

We propagate the SRC_CNTS and BKG_CNTS 1-sigma confidence intervals through the equation $NET\_CNTS = SRC\_CNTS - BKG\_CNTS/BACKSCAL$ using equation 1.31 in ``A Practical Guide to Data Analysis for Physical Science Students'', L. Lyons, 1991 to get a confidence interval on NET_CNTS, conventionally stated as separate upper and lower 1-sigma ``errors'' on NET_CNTS:

$$\begin{eqnarray*} \lefteqn{NET\_CNTS\_SIGMA\_UP =} \\ & & \sqrt{ ( src\_cnts\_... ...TS)^2 + (\frac{bkg\_cnts\_limit\_up - BKG\_CNTS}{BACKSCAL})^2 } \end{eqnarray*}$$

$$\begin{eqnarray*} \lefteqn{NET\_CNTS\_SIGMA\_LOW =} \\ & & \sqrt{ ( src\_cnts\... ...S)^2 + (\frac{bkg\_cnts\_limit\_low - BKG\_CNTS}{BACKSCAL})^2 } \end{eqnarray*}$$

5.10.3 Source validity/significance

We compute a traditional ``significance'' as the photometric signal to noise ratio:

$$SRC\_SIGNIF = \frac{NET\_CNTS}{NET\_CNTS\_SIGMA\_UP}$$

We also compute the ``significance'' of the observed SRC_CNTS as a disproof of the ``null hypothesis'' which is that there is no actual source, i.e. that all the observed counts are background. Weisskopf et al. (2007, Appendix A2) showed that the probability of observing SRC_CNTS or more counts in the source aperture and observing BKG_CNTS in the background region, conditioned on the assumption that there is no source (i.e. both apertures are sampling the same background population) can be found by integrating a binomial distrubution. This calculation can be performed with the following call to IDLs binomial function:

$$PROB\_NO\_SOURCE = binomial( SRC\_CNTS, SRC\_CNTS + BKG\_CNTS, \frac{1}{1 + BACKSCAL} )$$ (10)

Since this expression takes into account the uncertainty one has in estimating the background level (i.e. the Poisson nature of BKG_CNTS), PROB_NO_SOURCE will increase as BKG_CNTS decreases, i.e. a source will be less significant (more consistent with the background) when the background is poorly estimated.

As one might expect, when the background is accurately estimated (BKG_CNTS is large) this expression approaches a simple integral of the Poisson distribution over the interval [SRC_CNTS,$\infty$] (Weisskopf et al., 2007, § A2), which is equal to:

$$1 - \sum_{i=0}^{SRC\_CNTS - 1} Poisson(i;bkg\_counts\_in\_src\_region)$$ (11)

Important Note: Changing the extraction region will usually change the source significances defined above since both SRC_CNTS and BKG_CNTS will change. The maximal significance is NOT obtained with the largest extraction region (PSF Fraction) because as you enlarge the extraction region you eventually start adding more background than signal and significance goes down.

Some observers choose to use these AE significance statistics in their source detection process--a liberal set of source candidates are nominated by various means, those candidates are extracted by AE, and the catalog is pruned by thresholding on the AE significance statistics. One might choose this approach based on the hope/expectation that AE's significance statistics are more accurate than those computed by source detection codes because AE has a more complex process for estimating the background for each source (critical for these statistics). Or, one might choose this approach because some source candidates are nominated via processes that do not offer significance statistics (e.g. sources detected by-eye, sources detected in reconstructed images, or sources identified from other wavebands).

Note that this sort of procedure has strong analogies to the classic ``cell detection'' algorithm. Viewed in this way, AE can be seen to be evaluating the source significance statistic on only a limited number of cells--the AE extraction regions associated with the list of proposed sources. The procedure for nominating sources can be seen to be merely an efficiency technique--we avoid running AE at the many potential source locations where we have a low expectation that the AE significance statistics would be interesting.

Please note that it is not at all clear whether SRC_SIGNIF or PROB_NO_SOURCE threshold values used to prune the catalog can be used to analytically estimate the number of spurious sources produced by this hybrid procedure. Even though PROB_NO_SOURCE is a classic confidence level, there is no obvious ``number of independent trials'' that can be identified in the hybrid detection process outlined above. Laborious simulations would seem to offer the only hope of quantifying false detection rates for such a detection process. Such simulations may be nearly impossible when the source nomination process is not simple/automated.

Again viewed as a form of cell detection, the hybrid detection process outlined above might be expected to produce spurious sources which are spatially non-uniform because the cell size follows the Chandra 90% PSF size. Since the density of possible cells (extraction regions) is higher on-axis one might expect more spurious sources there (as observed in simulations with CIAOs celldetect). I think observers should be very cautious about performing studies of source spatial distribution that include weak sources near the limits of detection for ANY detection method that is used.

5.10.4 Photon Flux

AE provides two estimates for the flux of photons incident on Chandra, which has units of photon /cm**2 /s.

$$\mathbf{net\_flux} = \mathbf{net\_counts} / \mathbf{channel\_arf} / EXPOSURE$$

$$FLUX1 = \sum_{band} \mathbf{net\_flux} \ \ \ (photon /cm**2 /s)$$

Note that for weak sources the FLUX1 estimate can suffer from large Poisson errors with respect to true incident flux because events (either source or background) at energies where the ARF is very small have a large effect on FLUX1. For example, one observed event at 8 keV, where the ARF is tiny, makes a much larger contribution to FLUX1 than one event at 2 keV where the ARF is large.

$$MEAN\_ARF = \frac{\sum_{band} \mathbf{channel\_arf}}{\sum_{band} 1}$$

$$FLUX2 = NET\_CNTS / MEAN\_ARF / EXPOSURE \ \ \ (photon /cm**2 /s)$$

Note that the FLUX2 estimate suffers from a systematic error with respect to true incident flux because the scalar MEAN_ARF (a simple mean) is the correct effective area normalization only if the incident spectrum is flat. Some observers have found that summing the FLUX2 values computed over several adjacent narrow energy bands will produce a more accurate result than a single wide-band FLUX2 value.

Obviously spectral fitting should produce the most accurate flux estimates.

5.10.5 Characterizations of the background subtracted OBSERVED spectrum

AE provides estimates for the median (50% quantile), 25% quantile, and 75% quantile of the observed event energies over a variety of bands. These statistics are background-corrected, meaning they seek to characterize the observed spectrum of the astrophysical source if background were not present. AE implements an intuitive and straightforward background correction for standard quartiles, based on the observed cumulative distribution of the net spectrum, as shown in Figure 4. This method appears to be equivalent to that described by Hong et al. (2004, Appendix C), which was developed independently. AE does not yet attempt to estimate individual confidence intervals for these background-corrected quantiles; we expect that resampling techniques would be the best approach to that task.

$$ENERG\_PCT25\_OBSERVED = 25^{th} \mbox{percentile energy}$$

$$ENERG\_PCT50\_OBSERVED = 50^{th} \mbox{percentile energy}$$

$$ENERG\_PCT75\_OBSERVED = 75^{th} \mbox{percentile energy}$$

These are the energies below which 25%, 50%, and 75% of net_counts are found. The 50% percentile could be called a ``median observed energy corrected for background''. The mean observered event energy, corrected for background, is also estimated.

$$ENERG\_MEAN\_OBSERVED = \sum_{band} [\mathbf{channel\_energy} * \mathbf{net\_counts}] / \sum_{band} \mathbf{net\_counts}$$

Due to background subtraction, ENERG_MEAN_OBSERVED can fall outside the energy band!

Figure 4: Example calculation of the background-corrected median energy statistic. Each large upward jump in the cumulative distribution of net counts (black) represents the 20 counts observed in a source aperture; each small downward jump represents the 100 counts observed in a background region, scaled down to match the source aperture size. The lowest energy (red line) and highest energy (green line) at which the 50th percentile (blue line) is reached are averaged to produce an estimate of the median energy of the parent source.

5.10.6 Characterizations of the background subtracted INCIDENT spectrum

Similarly, AE provides estimates for the median (50% quantile), 25% quantile, and 75% quantile of the energies of photons incident on Chandra over a variety of bands.

$$ENERG\_PCT25\_INCIDENT = 25^{th} \mbox{percentile energy}$$

$$ENERG\_PCT50\_INCIDENT = 50^{th} \mbox{percentile energy}$$

$$ENERG\_PCT75\_INCIDENT = 75^{th} \mbox{percentile energy}$$

These are the energies below which 25%, 50%, and 75% of net_flux (§5.10.4) are found. The 50% percentile could be called a ``median incident energy corrected for background''. All the _INCIDENT statistics can suffer from large Poisson errors as discussed for FLUX1 above. The mean incident event energy, corrected for background, is also estimated.

$$ENERG\_MEAN\_INCIDENT = \sum_{band} [\mathbf{channel\_energy} * \mathbf{net\_flux}] / \sum_{band} \mathbf{net\_flux}$$

Due to background subtraction, ENERG_MEAN_INCIDENT can fall outside the energy band!