Stacked Observation Detections Table

From the Position and Position Errors column descriptions page:

The position of each stacked observation detection is defined by the ICRS right ascension and declination of the center of the source region in which the detection is located, which is in-turn determined from the wavdetect and/or mkvtbkg detections, as adjusted by the maximum likelihood estimator (MLE) fits to the observed X-ray event distributions.

From the Position and Position Errors column descriptions page:

The position of each stacked observation detection is defined by the ICRS right ascension and declination of the center of the source region in which the detection is located, which is in-turn determined from the wavdetect and/or mkvtbkg detections, as adjusted by the maximum likelihood estimator (MLE) fits to the observed X-ray event distributions.

From the Position and Position Errors column descriptions page:

The position of each stacked observation detection is defined by the ICRS right ascension and declination of the center of the source region in which the detection is located, which is in-turn determined from the wavdetect and/or mkvtbkg detections, as adjusted by the maximum likelihood estimator (MLE) fits to the observed X-ray event distributions.

From the Position and Position Errors column descriptions page:

The position of each stacked observation detection is defined by the ICRS right ascension and declination of the center of the source region in which the detection is located, which is in-turn determined from the wavdetect and/or mkvtbkg detections, as adjusted by the maximum likelihood estimator (MLE) fits to the observed X-ray event distributions.

From the Position and Position Errors column descriptions page:

The position of each stacked observation detection is defined by the ICRS right ascension and declination of the center of the source region in which the detection is located, which is in-turn determined from the wavdetect and/or mkvtbkg detections, as adjusted by the maximum likelihood estimator (MLE) fits to the observed X-ray event distributions.

From the Position and Position Errors column descriptions page:

The mean source region aperture off-axis angle, θ_mean, computed by averaging the off-axis angles θ from all observations in a stack.

From the Source Significance column descriptions page:

Likelihood, detect significance, and flux significance are reported per band for all sources detected in the valid stack. The likelihood reported is the maximum of the likelihood determined from the MLE fit to all valid stack data, and the likelihoods from each individual observation, per band.

Flux significance is a simple estimate of the ratio of the flux measurement to its average error. The mode of the marginalized probability distribution for photflux_aper is used as the flux measurement and the average error, \(\sigma_{e}\), is defined to be:

which are both used to estimate flux significance.

From the Source Significance column descriptions page:

Likelihood, detect significance, and flux significance are reported per band for all sources detected in the valid stack. The likelihood reported is the maximum of the likelihood determined from the MLE fit to all valid stack data, and the likelihoods from each individual observation, per band.

The fundamental metric used to decide whether a source is included in CSC 2.1 is the likelihood,

where \(P\) is the probability that an MLE fit to a point or extended source model, in a region with no source, would yield a change in fit statistic as large or larger than that observed, when compared to a fit to background only.

The likelihood is closely related to the probability, \(P_{\mathrm{Pois}}\), that a Poisson distribution with a mean background in the source aperture would produce at least the number of counts observed in the aperture. This quantity, called detect_significance, is also reported in CSC 2.1. Smoothed background maps are used to estimate mean background, and detect_significance is expressed in terms of the number of \(\sigma\), \(z\), in a zero-mean, unit standard deviation Gaussian distribution that would yield an upper integral probability \(P_{\mathrm{Gaus}}\), from \(z\) to \(\infty\), equivalent to \(P_{\mathrm{Pois}}\). That is,

where

From the Source Significance column descriptions page:

Likelihood, detect significance, and flux significance are reported per band for all sources detected in the valid stack. The likelihood reported is the maximum of the likelihood determined from the MLE fit to all valid stack data, and the likelihoods from each individual observation, per band.

The fundamental metric used to decide whether a source is included in CSC 2.1 is the likelihood,

where \(P\) is the probability that an MLE fit to a point or extended source model, in a region with no source, would yield a change in fit statistic as large or larger than that observed, when compared to a fit to background only.

The likelihood is closely related to the probability, \(P_{\mathrm{Pois}}\), that a Poisson distribution with a mean background in the source aperture would produce at least the number of counts observed in the aperture. This quantity, called detect_significance, is also reported in CSC 2.1. Smoothed background maps are used to estimate mean background, and detect_significance is expressed in terms of the number of \(\sigma\), \(z\), in a zero-mean, unit standard deviation Gaussian distribution that would yield an upper integral probability \(P_{\mathrm{Gaus}}\), from \(z\) to \(\infty\), equivalent to \(P_{\mathrm{Pois}}\). That is,

where

From the Source Flags column descriptions page:

The confusion code for a compact detection is a 16-bit coded integer that has all bits set to zero if the detection's source and background region ellipses do not overlap another source or background region in any source detection energy band, and the compact detection does not overlay an extended detection. Otherwise, the bits are set as follows:

The confusion code for an extend (convex hull) detection is always NULL.

From the Source Flags column descriptions page:

The dither warning flag for a compact detection is a Boolean that has a value of TRUE if the dither warning flag for any contributing per-observation detection is TRUE. Otherwise, the value is FALSE.

The dither warning flag for an extended (convex hull) source is always NULL.

From the Source Flags column descriptions page:

The edge code for a compact or extended (convex hull) detection is a 16-bit coded integer that has all bits set to zero if the detection's position, source region, and background region do not dither off a chip boundary (the edge of the unmasked area of the active region of the ACIS CCD or HRC microchannel plate segment, as appropriate) during the observation. Otherwise, the bits are set as follows:

Note that an extended (convex hull) detection (or associated background region) that extends across more than one chip by definition must dither off the chip boundary.

From the Source Flags column descriptions page:

The extent code for a compact detection is a 16-bit coded integer that has all bits set to zero if a wavelet transform analysis of counts in the detection's source region ellipse is consistent with a point source at the 90% confidence level in all science energy bands . Otherwise, the bits are set as follows:

The extent code for an extended (convex hull) detection is always set to 256 (Extended detection).

From the Source Flags column descriptions page:

The multi-chip code for a compact or extended (convex hull) detection is a 16-bit coded integer that has all bits set to zero if the detection's position, source region, and background region do not dither between two or more chips (ACIS CCDs or HRC microchannel plate segment, as appropriate) during the observation. Otherwise, the bits are set as follows:

Note that an extended (convex hull) detection (or associated background region) that extends across more than one chip by definition must dither across the chips.

From the Source Flags column descriptions page:

The pileup warning flag for a compact detection is a Boolean that has a value of TRUE if the pileup fraction exceed ~10% for any contributing ACIS per-observation detections and energy bands. Otherwise, the value is FALSE.

The pileup warning flag for an extended (convex hull) detection is always NULL.

From the Source Flags column descriptions page:

The saturated detection flag is for a compact detection is a Boolean that has a value of TRUE if all contributing observations are ACIS observations and all per-observation detections are significantly piled-up, i.e., sat_src_flag is TRUE for all of the contributing per-observation detections. Detection properties (including the pileup warning flag) are unreliable for all ACIS energy bands. Otherwise, the value is FALSE.

sat_src_flag for an extended (convex hull) source is always NULL.

From the Source Flags column descriptions page:

The streak detection flag for a compact detection is a Boolean that has a value of TRUE if all contributing observations are ACIS observations and all per-observation source regions overlap a defined region enclosing an identified readout streak, i.e., streak_src_flag is TRUE for all of the contributing per-observation detections. Otherwise, the value is FALSE.

The streak source flag for an extended (convex hull) detection is TRUE if any contributing observations are ACIS observations and any per-observation detection source region overlaps a defined region enclosing an identified readout streak . Otherwise, the value is FALSE.

From the Source Flags column descriptions page:

The variability flag is a Boolean that has a value of TRUE if variability is detected (the corresponding var_index value is ≥6) within any single observation or between any pair of observations contributing to the stacked detection, in any science energy band. Otherwise, the value is FALSE.

From the Source Flags column descriptions page:

The inter-observation variable hardness ratio flag for a compact detection is a Boolean that has a value of TRUE if one or more of the hardness ratios computed for any of the contributing observation detections is statistically inconsistent with the corresponding hardness ratios computed for any other contributing observation detections. Otherwise, the values is FALSE.

From the Source Variability column descriptions page:

A Boolean set to FALSE if var_inter_hard_prob is below 0.3 for all three hardness ratios, and set to TRUE otherwise.

From the Source Flags column descriptions page:

The manual detection addition flag for a compact or extended (convex hull) detection is a Boolean that has a value of TRUE if the stacked observation detection was manually added to the catalog by human review. Otherwise, the value is FALSE.

Detections that are manually added must satisfy detection likelihood and other validity checks in order to appear in the final catalog. See also the Manual Source/Detection Inclusion Flag below.

From the Source Flags column descriptions page:

The manual detection inclusion flag for a compact or extended (convex hull) detection is a Boolean that has a value of TRUE if the stacked observation detection was manually included in this catalog by human review. Otherwise, the value is FALSE.

Detections that are manually included are not required to satisfy detection likelihood or other validity checks. Manually included detections may or may not be manually added; if they are manually added then the man_add_flag will also be set to TRUE. See also the Manual Source/Detection Addition Flag above.

From the Source Flags column descriptions page:

The manual detection position flag for a compact detection is a Boolean that has a value of TRUE if the final detection position was manually modified from the fitted position (determined by the maximum likelihood estimator [MLE]) were manually modified by human review. Otherwise, the value is FALSE.

The manual detection position flag for an extended (convex hull) detection is set to TRUE if the final detection position was manually modified from the flux-weighted centroid position by human review. Otherwise, the value is FALSE.

From the Source Flags column descriptions page:

The manual detection region parameters flag for a compact source is a Boolean that has a value of TRUE if any of the detection's region parameters (i.e., the source region ellipse semi-axes and/or rotation angle, and/or position that define the detection region evaluated by the maximum likelihood estimator [MLE]) were manually modified by human review. Otherwise, the value is FALSE.

The manual detection region parameters flag for an extended (convex hull) source is set to TRUE if the shape or position of the defining polygon was manually modified by human review. Otherwise, the value is FALSE.

From the 'Deconvolved Source Extent' section of the Source Extent and Errors column descriptions page:

For stacked observation detections, the deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-deconvolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization represents the best estimate values and associated errors for the \(1\sigma\) radius along the major axis, the \(1\sigma\) radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that has been fitted to the observed source spatial event distribution deconvolved with the ray-trace local PSF at the location of that source event distribution.

Deconvolving the raw source image with the PSF produces the intrinsic shape of the source. In order to determine whether a source is extended, we first need to realize that the intrinsic size of a point source is, by definition, zero. By performing the fitting described above, srcextent estimates the raw sizes and errors of both the source and the PSF. In what follows, we derive the size and error for the intrinsic size of the source, and then we establish a criterion to flag a source as extended.

In principle, one can determine the parameters of the intrinsic source ellipse, \(\{a_{1},a_{2},\phi\}\), by solving a non-linear system of equations involving the PSF parameters, \(\{a_{1},a_{2},\psi\}\), and the measured source parameters, \(\{\sigma_{1},\sigma_{2},\delta\}\). However, because these equations are based on a crude approximation and because the input parameters are often uncertain, such an elaborate calculation seems unjustified.

A much simpler and more robust approach makes use of the identity:

which applies to the convolution of two elliptical Gaussians having arbitrary relative sizes and position angles. Using this identity, one can define a root-sum-square intrinsic source size:

that depends only on the sizes of the relevant ellipses and is independent of their orientations. This expression is analogous to the well-known result for convolution of 1D Gaussians and for convolution of circular Gaussians in 2D.

Using the equation above, one can derive an analytic expression for the uncertainty in \(a_{\mathrm{rss}}\) in terms of the measurement errors associated with \(\sigma_{i}\) and \(b_{i}\). Because \(\sigma_{i}\) and \(b_{i}\) are non-negative, evaluating the right-hand side of the equation using the corresponding mean values should give a reasonable estimate of the mean value of \(a_{\mathrm{rss}}\). A Taylor series expansion of the right-hand side evaluated at the mean parameter values, therefore, yields the uncertainty:

where \(\left(\Delta X \right)^{2}\) represents the variance in \(X\) and where

A source is extended if its root-sum-square intrinsic size is larger than the root-sum-square error of this size. Namely, if its derived intrinsic size, that depends on the raw source and PSF sizes and errors, is larger than the fluctuations due to the uncertainty in the determination (different from zero). If \(a_{\mathrm{rss}} > f \Delta a_{\mathrm{rss}}\), then the source is extended at the \(f\sigma\) level. In CSC2, we use \(f = 5\), which implies that sources are flagged as extended if the intrinsic size is determined with a significance of \(5\sigma\).

From the 'Deconvolved Source Extent' section of the Source Extent and Errors column descriptions page:

For stacked observation detections, the deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-deconvolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization represents the best estimate values and associated errors for the \(1\sigma\) radius along the major axis, the \(1\sigma\) radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that has been fitted to the observed source spatial event distribution deconvolved with the ray-trace local PSF at the location of that source event distribution.

Deconvolving the raw source image with the PSF produces the intrinsic shape of the source. In order to determine whether a source is extended, we first need to realize that the intrinsic size of a point source is, by definition, zero. By performing the fitting described above, srcextent estimates the raw sizes and errors of both the source and the PSF. In what follows, we derive the size and error for the intrinsic size of the source, and then we establish a criterion to flag a source as extended.

In principle, one can determine the parameters of the intrinsic source ellipse, \(\{a_{1},a_{2},\phi\}\), by solving a non-linear system of equations involving the PSF parameters, \(\{a_{1},a_{2},\psi\}\), and the measured source parameters, \(\{\sigma_{1},\sigma_{2},\delta\}\). However, because these equations are based on a crude approximation and because the input parameters are often uncertain, such an elaborate calculation seems unjustified.

A much simpler and more robust approach makes use of the identity:

which applies to the convolution of two elliptical Gaussians having arbitrary relative sizes and position angles. Using this identity, one can define a root-sum-square intrinsic source size:

that depends only on the sizes of the relevant ellipses and is independent of their orientations. This expression is analogous to the well-known result for convolution of 1D Gaussians and for convolution of circular Gaussians in 2D.

Using the equation above, one can derive an analytic expression for the uncertainty in \(a_{\mathrm{rss}}\) in terms of the measurement errors associated with \(\sigma_{i}\) and \(b_{i}\). Because \(\sigma_{i}\) and \(b_{i}\) are non-negative, evaluating the right-hand side of the equation using the corresponding mean values should give a reasonable estimate of the mean value of \(a_{\mathrm{rss}}\). A Taylor series expansion of the right-hand side evaluated at the mean parameter values, therefore, yields the uncertainty:

where \(\left(\Delta X \right)^{2}\) represents the variance in \(X\) and where

A source is extended if its root-sum-square intrinsic size is larger than the root-sum-square error of this size. Namely, if its derived intrinsic size, that depends on the raw source and PSF sizes and errors, is larger than the fluctuations due to the uncertainty in the determination (different from zero). If \(a_{\mathrm{rss}} > f \Delta a_{\mathrm{rss}}\), then the source is extended at the \(f\sigma\) level. In CSC2, we use \(f = 5\), which implies that sources are flagged as extended if the intrinsic size is determined with a significance of \(5\sigma\).

From the 'Deconvolved Source Extent' section of the Source Extent and Errors column descriptions page:

For stacked observation detections, the deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-deconvolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization represents the best estimate values and associated errors for the \(1\sigma\) radius along the major axis, the \(1\sigma\) radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that has been fitted to the observed source spatial event distribution deconvolved with the ray-trace local PSF at the location of that source event distribution.

Deconvolving the raw source image with the PSF produces the intrinsic shape of the source. In order to determine whether a source is extended, we first need to realize that the intrinsic size of a point source is, by definition, zero. By performing the fitting described above, srcextent estimates the raw sizes and errors of both the source and the PSF. In what follows, we derive the size and error for the intrinsic size of the source, and then we establish a criterion to flag a source as extended.

In principle, one can determine the parameters of the intrinsic source ellipse, \(\{a_{1},a_{2},\phi\}\), by solving a non-linear system of equations involving the PSF parameters, \(\{a_{1},a_{2},\psi\}\), and the measured source parameters, \(\{\sigma_{1},\sigma_{2},\delta\}\). However, because these equations are based on a crude approximation and because the input parameters are often uncertain, such an elaborate calculation seems unjustified.

A much simpler and more robust approach makes use of the identity:

which applies to the convolution of two elliptical Gaussians having arbitrary relative sizes and position angles. Using this identity, one can define a root-sum-square intrinsic source size:

that depends only on the sizes of the relevant ellipses and is independent of their orientations. This expression is analogous to the well-known result for convolution of 1D Gaussians and for convolution of circular Gaussians in 2D.

Using the equation above, one can derive an analytic expression for the uncertainty in \(a_{\mathrm{rss}}\) in terms of the measurement errors associated with \(\sigma_{i}\) and \(b_{i}\). Because \(\sigma_{i}\) and \(b_{i}\) are non-negative, evaluating the right-hand side of the equation using the corresponding mean values should give a reasonable estimate of the mean value of \(a_{\mathrm{rss}}\). A Taylor series expansion of the right-hand side evaluated at the mean parameter values, therefore, yields the uncertainty:

where \(\left(\Delta X \right)^{2}\) represents the variance in \(X\) and where

A source is extended if its root-sum-square intrinsic size is larger than the root-sum-square error of this size. Namely, if its derived intrinsic size, that depends on the raw source and PSF sizes and errors, is larger than the fluctuations due to the uncertainty in the determination (different from zero). If \(a_{\mathrm{rss}} > f \Delta a_{\mathrm{rss}}\), then the source is extended at the \(f\sigma\) level. In CSC2, we use \(f = 5\), which implies that sources are flagged as extended if the intrinsic size is determined with a significance of \(5\sigma\).

From the 'Deconvolved Source Extent' section of the Source Extent and Errors column descriptions page:

For stacked observation detections, the deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-deconvolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization represents the best estimate values and associated errors for the \(1\sigma\) radius along the major axis, the \(1\sigma\) radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that has been fitted to the observed source spatial event distribution deconvolved with the ray-trace local PSF at the location of that source event distribution.

Deconvolving the raw source image with the PSF produces the intrinsic shape of the source. In order to determine whether a source is extended, we first need to realize that the intrinsic size of a point source is, by definition, zero. By performing the fitting described above, srcextent estimates the raw sizes and errors of both the source and the PSF. In what follows, we derive the size and error for the intrinsic size of the source, and then we establish a criterion to flag a source as extended.

In principle, one can determine the parameters of the intrinsic source ellipse, \(\{a_{1},a_{2},\phi\}\), by solving a non-linear system of equations involving the PSF parameters, \(\{a_{1},a_{2},\psi\}\), and the measured source parameters, \(\{\sigma_{1},\sigma_{2},\delta\}\). However, because these equations are based on a crude approximation and because the input parameters are often uncertain, such an elaborate calculation seems unjustified.

A much simpler and more robust approach makes use of the identity:

which applies to the convolution of two elliptical Gaussians having arbitrary relative sizes and position angles. Using this identity, one can define a root-sum-square intrinsic source size:

that depends only on the sizes of the relevant ellipses and is independent of their orientations. This expression is analogous to the well-known result for convolution of 1D Gaussians and for convolution of circular Gaussians in 2D.

Using the equation above, one can derive an analytic expression for the uncertainty in \(a_{\mathrm{rss}}\) in terms of the measurement errors associated with \(\sigma_{i}\) and \(b_{i}\). Because \(\sigma_{i}\) and \(b_{i}\) are non-negative, evaluating the right-hand side of the equation using the corresponding mean values should give a reasonable estimate of the mean value of \(a_{\mathrm{rss}}\). A Taylor series expansion of the right-hand side evaluated at the mean parameter values, therefore, yields the uncertainty:

where \(\left(\Delta X \right)^{2}\) represents the variance in \(X\) and where

A source is extended if its root-sum-square intrinsic size is larger than the root-sum-square error of this size. Namely, if its derived intrinsic size, that depends on the raw source and PSF sizes and errors, is larger than the fluctuations due to the uncertainty in the determination (different from zero). If \(a_{\mathrm{rss}} > f \Delta a_{\mathrm{rss}}\), then the source is extended at the \(f\sigma\) level. In CSC2, we use \(f = 5\), which implies that sources are flagged as extended if the intrinsic size is determined with a significance of \(5\sigma\).

From the 'Deconvolved Source Extent' section of the Source Extent and Errors column descriptions page:

For stacked observation detections, the deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-deconvolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization represents the best estimate values and associated errors for the \(1\sigma\) radius along the major axis, the \(1\sigma\) radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that has been fitted to the observed source spatial event distribution deconvolved with the ray-trace local PSF at the location of that source event distribution.

Deconvolving the raw source image with the PSF produces the intrinsic shape of the source. In order to determine whether a source is extended, we first need to realize that the intrinsic size of a point source is, by definition, zero. By performing the fitting described above, srcextent estimates the raw sizes and errors of both the source and the PSF. In what follows, we derive the size and error for the intrinsic size of the source, and then we establish a criterion to flag a source as extended.

In principle, one can determine the parameters of the intrinsic source ellipse, \(\{a_{1},a_{2},\phi\}\), by solving a non-linear system of equations involving the PSF parameters, \(\{a_{1},a_{2},\psi\}\), and the measured source parameters, \(\{\sigma_{1},\sigma_{2},\delta\}\). However, because these equations are based on a crude approximation and because the input parameters are often uncertain, such an elaborate calculation seems unjustified.

A much simpler and more robust approach makes use of the identity:

which applies to the convolution of two elliptical Gaussians having arbitrary relative sizes and position angles. Using this identity, one can define a root-sum-square intrinsic source size:

that depends only on the sizes of the relevant ellipses and is independent of their orientations. This expression is analogous to the well-known result for convolution of 1D Gaussians and for convolution of circular Gaussians in 2D.

Using the equation above, one can derive an analytic expression for the uncertainty in \(a_{\mathrm{rss}}\) in terms of the measurement errors associated with \(\sigma_{i}\) and \(b_{i}\). Because \(\sigma_{i}\) and \(b_{i}\) are non-negative, evaluating the right-hand side of the equation using the corresponding mean values should give a reasonable estimate of the mean value of \(a_{\mathrm{rss}}\). A Taylor series expansion of the right-hand side evaluated at the mean parameter values, therefore, yields the uncertainty:

where \(\left(\Delta X \right)^{2}\) represents the variance in \(X\) and where

A source is extended if its root-sum-square intrinsic size is larger than the root-sum-square error of this size. Namely, if its derived intrinsic size, that depends on the raw source and PSF sizes and errors, is larger than the fluctuations due to the uncertainty in the determination (different from zero). If \(a_{\mathrm{rss}} > f \Delta a_{\mathrm{rss}}\), then the source is extended at the \(f\sigma\) level. In CSC2, we use \(f = 5\), which implies that sources are flagged as extended if the intrinsic size is determined with a significance of \(5\sigma\).

From the 'Deconvolved Source Extent' section of the Source Extent and Errors column descriptions page:

For stacked observation detections, the deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-deconvolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization represents the best estimate values and associated errors for the \(1\sigma\) radius along the major axis, the \(1\sigma\) radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that has been fitted to the observed source spatial event distribution deconvolved with the ray-trace local PSF at the location of that source event distribution.

Deconvolving the raw source image with the PSF produces the intrinsic shape of the source. In order to determine whether a source is extended, we first need to realize that the intrinsic size of a point source is, by definition, zero. By performing the fitting described above, srcextent estimates the raw sizes and errors of both the source and the PSF. In what follows, we derive the size and error for the intrinsic size of the source, and then we establish a criterion to flag a source as extended.

In principle, one can determine the parameters of the intrinsic source ellipse, \(\{a_{1},a_{2},\phi\}\), by solving a non-linear system of equations involving the PSF parameters, \(\{a_{1},a_{2},\psi\}\), and the measured source parameters, \(\{\sigma_{1},\sigma_{2},\delta\}\). However, because these equations are based on a crude approximation and because the input parameters are often uncertain, such an elaborate calculation seems unjustified.

A much simpler and more robust approach makes use of the identity:

which applies to the convolution of two elliptical Gaussians having arbitrary relative sizes and position angles. Using this identity, one can define a root-sum-square intrinsic source size:

that depends only on the sizes of the relevant ellipses and is independent of their orientations. This expression is analogous to the well-known result for convolution of 1D Gaussians and for convolution of circular Gaussians in 2D.

Using the equation above, one can derive an analytic expression for the uncertainty in \(a_{\mathrm{rss}}\) in terms of the measurement errors associated with \(\sigma_{i}\) and \(b_{i}\). Because \(\sigma_{i}\) and \(b_{i}\) are non-negative, evaluating the right-hand side of the equation using the corresponding mean values should give a reasonable estimate of the mean value of \(a_{\mathrm{rss}}\). A Taylor series expansion of the right-hand side evaluated at the mean parameter values, therefore, yields the uncertainty:

where \(\left(\Delta X \right)^{2}\) represents the variance in \(X\) and where

A source is extended if its root-sum-square intrinsic size is larger than the root-sum-square error of this size. Namely, if its derived intrinsic size, that depends on the raw source and PSF sizes and errors, is larger than the fluctuations due to the uncertainty in the determination (different from zero). If \(a_{\mathrm{rss}} > f \Delta a_{\mathrm{rss}}\), then the source is extended at the \(f\sigma\) level. In CSC2, we use \(f = 5\), which implies that sources are flagged as extended if the intrinsic size is determined with a significance of \(5\sigma\).

From the 'Deconvolved Source Extent' section of the Source Extent and Errors column descriptions page:

For stacked observation detections, the deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-deconvolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization represents the best estimate values and associated errors for the \(1\sigma\) radius along the major axis, the \(1\sigma\) radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that has been fitted to the observed source spatial event distribution deconvolved with the ray-trace local PSF at the location of that source event distribution.

Deconvolving the raw source image with the PSF produces the intrinsic shape of the source. In order to determine whether a source is extended, we first need to realize that the intrinsic size of a point source is, by definition, zero. By performing the fitting described above, srcextent estimates the raw sizes and errors of both the source and the PSF. In what follows, we derive the size and error for the intrinsic size of the source, and then we establish a criterion to flag a source as extended.

In principle, one can determine the parameters of the intrinsic source ellipse, \(\{a_{1},a_{2},\phi\}\), by solving a non-linear system of equations involving the PSF parameters, \(\{a_{1},a_{2},\psi\}\), and the measured source parameters, \(\{\sigma_{1},\sigma_{2},\delta\}\). However, because these equations are based on a crude approximation and because the input parameters are often uncertain, such an elaborate calculation seems unjustified.

A much simpler and more robust approach makes use of the identity:

which applies to the convolution of two elliptical Gaussians having arbitrary relative sizes and position angles. Using this identity, one can define a root-sum-square intrinsic source size:

that depends only on the sizes of the relevant ellipses and is independent of their orientations. This expression is analogous to the well-known result for convolution of 1D Gaussians and for convolution of circular Gaussians in 2D.

Using the equation above, one can derive an analytic expression for the uncertainty in \(a_{\mathrm{rss}}\) in terms of the measurement errors associated with \(\sigma_{i}\) and \(b_{i}\). Because \(\sigma_{i}\) and \(b_{i}\) are non-negative, evaluating the right-hand side of the equation using the corresponding mean values should give a reasonable estimate of the mean value of \(a_{\mathrm{rss}}\). A Taylor series expansion of the right-hand side evaluated at the mean parameter values, therefore, yields the uncertainty:

where \(\left(\Delta X \right)^{2}\) represents the variance in \(X\) and where

A source is extended if its root-sum-square intrinsic size is larger than the root-sum-square error of this size. Namely, if its derived intrinsic size, that depends on the raw source and PSF sizes and errors, is larger than the fluctuations due to the uncertainty in the determination (different from zero). If \(a_{\mathrm{rss}} > f \Delta a_{\mathrm{rss}}\), then the source is extended at the \(f\sigma\) level. In CSC2, we use \(f = 5\), which implies that sources are flagged as extended if the intrinsic size is determined with a significance of \(5\sigma\).

From the 'Deconvolved Source Extent' section of the Source Extent and Errors column descriptions page:

For stacked observation detections, the deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-deconvolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization represents the best estimate values and associated errors for the \(1\sigma\) radius along the major axis, the \(1\sigma\) radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that has been fitted to the observed source spatial event distribution deconvolved with the ray-trace local PSF at the location of that source event distribution.

Deconvolving the raw source image with the PSF produces the intrinsic shape of the source. In order to determine whether a source is extended, we first need to realize that the intrinsic size of a point source is, by definition, zero. By performing the fitting described above, srcextent estimates the raw sizes and errors of both the source and the PSF. In what follows, we derive the size and error for the intrinsic size of the source, and then we establish a criterion to flag a source as extended.

In principle, one can determine the parameters of the intrinsic source ellipse, \(\{a_{1},a_{2},\phi\}\), by solving a non-linear system of equations involving the PSF parameters, \(\{a_{1},a_{2},\psi\}\), and the measured source parameters, \(\{\sigma_{1},\sigma_{2},\delta\}\). However, because these equations are based on a crude approximation and because the input parameters are often uncertain, such an elaborate calculation seems unjustified.

A much simpler and more robust approach makes use of the identity:

which applies to the convolution of two elliptical Gaussians having arbitrary relative sizes and position angles. Using this identity, one can define a root-sum-square intrinsic source size:

that depends only on the sizes of the relevant ellipses and is independent of their orientations. This expression is analogous to the well-known result for convolution of 1D Gaussians and for convolution of circular Gaussians in 2D.

Using the equation above, one can derive an analytic expression for the uncertainty in \(a_{\mathrm{rss}}\) in terms of the measurement errors associated with \(\sigma_{i}\) and \(b_{i}\). Because \(\sigma_{i}\) and \(b_{i}\) are non-negative, evaluating the right-hand side of the equation using the corresponding mean values should give a reasonable estimate of the mean value of \(a_{\mathrm{rss}}\). A Taylor series expansion of the right-hand side evaluated at the mean parameter values, therefore, yields the uncertainty:

where \(\left(\Delta X \right)^{2}\) represents the variance in \(X\) and where

A source is extended if its root-sum-square intrinsic size is larger than the root-sum-square error of this size. Namely, if its derived intrinsic size, that depends on the raw source and PSF sizes and errors, is larger than the fluctuations due to the uncertainty in the determination (different from zero). If \(a_{\mathrm{rss}} > f \Delta a_{\mathrm{rss}}\), then the source is extended at the \(f\sigma\) level. In CSC2, we use \(f = 5\), which implies that sources are flagged as extended if the intrinsic size is determined with a significance of \(5\sigma\).

From the 'Deconvolved Source Extent' section of the Source Extent and Errors column descriptions page:

For stacked observation detections, the deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-deconvolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization represents the best estimate values and associated errors for the \(1\sigma\) radius along the major axis, the \(1\sigma\) radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that has been fitted to the observed source spatial event distribution deconvolved with the ray-trace local PSF at the location of that source event distribution.

Deconvolving the raw source image with the PSF produces the intrinsic shape of the source. In order to determine whether a source is extended, we first need to realize that the intrinsic size of a point source is, by definition, zero. By performing the fitting described above, srcextent estimates the raw sizes and errors of both the source and the PSF. In what follows, we derive the size and error for the intrinsic size of the source, and then we establish a criterion to flag a source as extended.

In principle, one can determine the parameters of the intrinsic source ellipse, \(\{a_{1},a_{2},\phi\}\), by solving a non-linear system of equations involving the PSF parameters, \(\{a_{1},a_{2},\psi\}\), and the measured source parameters, \(\{\sigma_{1},\sigma_{2},\delta\}\). However, because these equations are based on a crude approximation and because the input parameters are often uncertain, such an elaborate calculation seems unjustified.

A much simpler and more robust approach makes use of the identity:

which applies to the convolution of two elliptical Gaussians having arbitrary relative sizes and position angles. Using this identity, one can define a root-sum-square intrinsic source size:

that depends only on the sizes of the relevant ellipses and is independent of their orientations. This expression is analogous to the well-known result for convolution of 1D Gaussians and for convolution of circular Gaussians in 2D.

Using the equation above, one can derive an analytic expression for the uncertainty in \(a_{\mathrm{rss}}\) in terms of the measurement errors associated with \(\sigma_{i}\) and \(b_{i}\). Because \(\sigma_{i}\) and \(b_{i}\) are non-negative, evaluating the right-hand side of the equation using the corresponding mean values should give a reasonable estimate of the mean value of \(a_{\mathrm{rss}}\). A Taylor series expansion of the right-hand side evaluated at the mean parameter values, therefore, yields the uncertainty:

where \(\left(\Delta X \right)^{2}\) represents the variance in \(X\) and where

A source is extended if its root-sum-square intrinsic size is larger than the root-sum-square error of this size. Namely, if its derived intrinsic size, that depends on the raw source and PSF sizes and errors, is larger than the fluctuations due to the uncertainty in the determination (different from zero). If \(a_{\mathrm{rss}} > f \Delta a_{\mathrm{rss}}\), then the source is extended at the \(f\sigma\) level. In CSC2, we use \(f = 5\), which implies that sources are flagged as extended if the intrinsic size is determined with a significance of \(5\sigma\).

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source counts represent the combined net number of background-subtracted source counts in the modified source region (src_cnts_aper) and in the modified elliptical aperture (src_cnts_aper90), corrected by the appropriate PSF aperture fractions, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source counts represent the combined net number of background-subtracted source counts in the modified source region (src_cnts_aper) and in the modified elliptical aperture (src_cnts_aper90), corrected by the appropriate PSF aperture fractions, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source counts represent the combined net number of background-subtracted source counts in the modified source region (src_cnts_aper) and in the modified elliptical aperture (src_cnts_aper90), corrected by the appropriate PSF aperture fractions, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source count rates and associated two-sided confidence limits are defined as the average background-subtracted source count rates in the modified source region (src_rate_aper) and in the modified elliptical aperture (src_rate_aper90), corrected by the appropriate PSF aperture fractions and livetime, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source count rates and associated two-sided confidence limits are defined as the average background-subtracted source count rates in the modified source region (src_rate_aper) and in the modified elliptical aperture (src_rate_aper90), corrected by the appropriate PSF aperture fractions and livetime, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source count rates and associated two-sided confidence limits are defined as the average background-subtracted source count rates in the modified source region (src_rate_aper) and in the modified elliptical aperture (src_rate_aper90), corrected by the appropriate PSF aperture fractions and livetime, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source counts represent the combined net number of background-subtracted source counts in the modified source region (src_cnts_aper) and in the modified elliptical aperture (src_cnts_aper90), corrected by the appropriate PSF aperture fractions, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source counts represent the combined net number of background-subtracted source counts in the modified source region (src_cnts_aper) and in the modified elliptical aperture (src_cnts_aper90), corrected by the appropriate PSF aperture fractions, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source counts represent the combined net number of background-subtracted source counts in the modified source region (src_cnts_aper) and in the modified elliptical aperture (src_cnts_aper90), corrected by the appropriate PSF aperture fractions, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source count rates and associated two-sided confidence limits are defined as the average background-subtracted source count rates in the modified source region (src_rate_aper) and in the modified elliptical aperture (src_rate_aper90), corrected by the appropriate PSF aperture fractions and livetime, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source count rates and associated two-sided confidence limits are defined as the average background-subtracted source count rates in the modified source region (src_rate_aper) and in the modified elliptical aperture (src_rate_aper90), corrected by the appropriate PSF aperture fractions and livetime, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source count rates and associated two-sided confidence limits are defined as the average background-subtracted source count rates in the modified source region (src_rate_aper) and in the modified elliptical aperture (src_rate_aper90), corrected by the appropriate PSF aperture fractions and livetime, for all valid source observations in the stack.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Aperture Photometry Fluxes' section of the Source Fluxes column descriptions page:

Aperture photometry quantities are derived from counts in source regions or elliptical apertures, with background estimated from counts in surrounding background regions. Corrections are made for PSF aperture fractions, livetime, and exposure. In the case of energy fluxes, the conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon. For all aperture photometry quantities, a Bayesian statistical analysis is performed to determine background-marginalized posterior probability distribution for the flux quantity, and the mode and 68% percentiles of the distribution are reported as the flux value and confidence limits.

Fluxes are determined for each per-observation detection, for each stack, and for the master source. At the stack level, aperture data from all valid source observations in the stack are combined. At the master source level, a Bayesian Blocks analysis is performed to determine the sets of source observations consistent with a constant source flux. Aperture data from the set with the largest total exposure are then combined to determine master source 'best estimate' fluxes and confidence limits. In addition, aperture data from all source observations in which the master source was detected or in the field of view are combined to determine master source average fluxes and confidence limits.

The aperture source photon and energy fluxes and associated two-sided confidence limits represent the average background-subtracted fluxes in the modified source region (photflux_aper, flux_aper) and in the modified elliptical aperture (photflux_aper90, flux_aper90), corrected by the appropriate PSF aperture fractions, livetime, and exposure for all valid observations in the stack. The conversion from photons s^-1 cm^-2 to ergs s^-1 cm^-2 is performed by summing the photon energies for each incident source photon and scaling by the local value of the ARF at the location of the incident photon.

From the 'Source Region' section of the Source Extent and Errors column descriptions page:

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

From the 'Source Region' section of the Source Extent and Errors column descriptions page:

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

From the 'Source Region' section of the Source Extent and Errors column descriptions page:

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

From the 'Source Region' section of the Source Extent and Errors column descriptions page:

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

From the 'Source Region' section of the Source Extent and Errors column descriptions page:

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

From the 'Source Region' section of the Source Extent and Errors column descriptions page:

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

From the 'Source Region' section of the Source Extent and Errors column descriptions page:

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

From the 'Source Region' section of the Source Extent and Errors column descriptions page:

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

From the 'Source Region' section of the Source Extent and Errors column descriptions page:

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

From the 'Source Region' section of the Source Extent and Errors column descriptions page:

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

From the Source Variability column descriptions page:

The inter-observation spectral variability probability (var_inter_hard_prob) is a value that records the probability that the source region hardness ratios varied between the contributing observations, based on the hypothesis rejection test described in the hardness ratios and variability memo. The definition of this probability is identical to that of the inter-observation source variability (var_inter_prob), and also utilizes the same hypothesis rejection test, but based on the probability distributions (PDFs) for the hardness ratios, rather than the probability distributions for the fluxes. The definition of the hardness ratio PDFs can be found in the memo, and also in the hardness ratios columns page. High values of var_inter_hard_prob indicate that the source is spectrally variable in the corresponding combination of bands.

From the Source Variability column descriptions page:

Similarly to var_inter_sigma, the inter-observation hardness ratio variability parameter (var_inter_hard_sigma) is the absolute value of the difference between the error weighted mean of the source region hardness ratio PDF when a single hardness ratio is assumed, and the mean of the source region hardness ratio PDF for the individual observation that maximizes the absolute value of the difference:

Of all contributing observations, the observation that yields the highest value for this equation, is used in computing this value, which is recorded in var_inter_hard_sigma. Intuitively, this quantity can be interpreted as the variance of the individual observation hardness ratios.

From the Source Variability column descriptions page:

The inter-observation spectral variability probability (var_inter_hard_prob) is a value that records the probability that the source region hardness ratios varied between the contributing observations, based on the hypothesis rejection test described in the hardness ratios and variability memo. The definition of this probability is identical to that of the inter-observation source variability (var_inter_prob), and also utilizes the same hypothesis rejection test, but based on the probability distributions (PDFs) for the hardness ratios, rather than the probability distributions for the fluxes. The definition of the hardness ratio PDFs can be found in the memo, and also in the hardness ratios columns page. High values of var_inter_hard_prob indicate that the source is spectrally variable in the corresponding combination of bands.

From the Source Variability column descriptions page:

Similarly to var_inter_sigma, the inter-observation hardness ratio variability parameter (var_inter_hard_sigma) is the absolute value of the difference between the error weighted mean of the source region hardness ratio PDF when a single hardness ratio is assumed, and the mean of the source region hardness ratio PDF for the individual observation that maximizes the absolute value of the difference:

Of all contributing observations, the observation that yields the highest value for this equation, is used in computing this value, which is recorded in var_inter_hard_sigma. Intuitively, this quantity can be interpreted as the variance of the individual observation hardness ratios.

From the Source Variability column descriptions page:

The inter-observation spectral variability probability (var_inter_hard_prob) is a value that records the probability that the source region hardness ratios varied between the contributing observations, based on the hypothesis rejection test described in the hardness ratios and variability memo. The definition of this probability is identical to that of the inter-observation source variability (var_inter_prob), and also utilizes the same hypothesis rejection test, but based on the probability distributions (PDFs) for the hardness ratios, rather than the probability distributions for the fluxes. The definition of the hardness ratio PDFs can be found in the memo, and also in the hardness ratios columns page. High values of var_inter_hard_prob indicate that the source is spectrally variable in the corresponding combination of bands.

From the Source Variability column descriptions page:

Similarly to var_inter_sigma, the inter-observation hardness ratio variability parameter (var_inter_hard_sigma) is the absolute value of the difference between the error weighted mean of the source region hardness ratio PDF when a single hardness ratio is assumed, and the mean of the source region hardness ratio PDF for the individual observation that maximizes the absolute value of the difference:

Of all contributing observations, the observation that yields the highest value for this equation, is used in computing this value, which is recorded in var_inter_hard_sigma. Intuitively, this quantity can be interpreted as the variance of the individual observation hardness ratios.

The intra-observation variability index (var_intra_index) represents the highest value of the variability indices (var_index) calculated for each of the contributing observations.

The Gregory-Loredo, Kolmogorov-Smirnov (K-S) test, and Kuiper's test intra-observation variability probabilities represent the highest values of the variability probabilities (var_prob, ks_prob, kp_prob) calculated for each of the contributing observations (i.e., the highest level of variability among the observations contributing to the master source entry).

The Gregory-Loredo, Kolmogorov-Smirnov (K-S) test, and Kuiper's test intra-observation variability probabilities represent the highest values of the variability probabilities (var_prob, ks_prob, kp_prob) calculated for each of the contributing observations (i.e., the highest level of variability among the observations contributing to the master source entry).

The Gregory-Loredo, Kolmogorov-Smirnov (K-S) test, and Kuiper's test intra-observation variability probabilities represent the highest values of the variability probabilities (var_prob, ks_prob, kp_prob) calculated for each of the contributing observations (i.e., the highest level of variability among the observations contributing to the master source entry).

From the Source Variability column descriptions page:

The inter-observation variability index (var_inter_index) is an integer value in the range \([0,8]\) that is derived according to the estimated value of the quantity \(D/(N-1)\) defined above. It is used to evaluate whether the source region photon flux is constant between the observations. The degree of confidence in variability expressed by this index is similar to that of the intra-observation variability index. Below we tabulate the association between the value of \(D/(N-1)\) and inter-observation variability index.

The inter-observation variability probability (var_inter_prob) is a value that records the probability that the source region photon flux varied between the contributing observations, based on the hypothesis rejection test described in the hardness ratios and variability memo. Given the \(N\) individual Bayesian probability distribution of the aperture fluxes for the same source in \(N\) different observations (their means and standard deviations), we estimate for each band the maximum likelihood \(\mathcal{L}_{1}^{\mathrm{max}}\) and the corresponding maximizing arguments \(F_{\left\langle band \right\rangle}^{i,\mathrm{max}}\), of the observed fluxes assuming a different flux for each observation, as well as the maximum likelihood \(\mathcal{L}_{2}^{\mathrm{max}}\) and the corresponding maximizing argument \(F_{\left\langle band \right\rangle}^{\mathrm{max}}\) of the observed fluxes assuming a single flux (the latter is the null hypothesis of no variability). As per Wilks' theorem, the quantity:

follows \(\chi^{2}\) distribution with \(N-1\) degrees of freedom, under the null hypothesis. Therefore, the null hypothesis (non-variability) is rejected with a probability proportional to the cumulative distribution of the \(\chi^{2}\) statistic for values smaller than the estimated \(D\). The quantity var_inter_prob represents this cumulative probability, and therefore gives the probability that the source is variable.

The reason for this careful definition is that the probabilities for intra-observation and inter-observation variability are, by necessity, of a different nature. Whereas one can say with reasonable certainty whether a source was variable during an observation covering a contiguous time interval, when comparing measured fluxes from different observations one knows nothing about the source's behavior during the intervening interval(s). Consequently, when the inter-observation variability probability is high (e.g., var_inter_prob > 0.7), one can confidently state that the source is variable on longer time scales, but when the probability is low, all one can say is that the observations are consistent with a constant flux.

From the Source Variability column descriptions page:

The inter-observation flux variability (var_inter_sigma) is the absolute value of the difference between the error weighted mean of the source region photon flux density PDF when a single flux is assumed \(\left( F_{\left\langle band \right\rangle}^{\mathrm{max}} \right)\), and the mean of the source region photon flux density PDF for the individual observation that maximizes the absolute value of the difference \(\left( F_{\left\langle band \right\rangle}^{i,\mathrm{max}} \right)\):

Of all the contributing observations, the observation that yields the highest value for this equation, is used in computing this value, which is recorded in var_inter_sigma. Intuitively, this quantity can be interpreted as the variance of the individual observation fluxes.