THE CHANDRA SOURCE CATALOG

The CSC is intended to be a general purpose virtual science facility, and provides simple access to a carefully selected set of generally useful quantities for individual sources or sets of sources matching user-specified search criteria. The catalog is designed to satisfy the needs of a broad-based group of scientists, including those who may be less familiar with astronomical data analysis in the X-ray regime, while at the same time providing more advanced data products suitable for use by astronomers familiar with Chandra data.

The primary design goals for the CSC are to (1) allow simple and quick access to the best estimates of the X-ray source properties for detected sources, with good scientific fidelity, and directly support scientific analysis using the individual source data; (2) facilitate analysis of a wide range of statistical properties for classes of X-ray sources; (3) provide efficient access to calibrated observational data and ancillary data products for individual X-ray sources, so that users can perform detailed further analysis using existing tools such as those included in the Chandra Interactive Analysis of Observations (CIAO; Fruscione et al. 2006) portable data analysis package; and (4) include all real X-ray sources detected down to a predefined threshold level in all of the public Chandra data sets used to populate the catalog, while maintaining the number of spurious sources at an acceptable level.

To achieve these goals, for each detected X-ray source the catalog records the source position and a detailed set of source properties, including commonly used quantities such as multi-band aperture fluxes, cross-band hardness ratios, spectra, temporal variability information, and source extent estimates. In addition to these traditional elements, the catalog includes file-based data products that can be manipulated interactively by the user. The primary data products are photon event lists (e.g., Conroy 1992), which record measures of the location, time of arrival, and energy of each detected photon event in a tabular format. Additional data products derived from the photon event list include images, light curves, and spectra for each source individually from each observation in which a source is detected. The catalog release process is carefully controlled, and a detailed characterization of the statistical properties of the catalog to a well-defined, high level of reliability accompanies each release. Key properties evaluated as part of the statistical characterization include limiting sensitivity, completeness, false source rates, astrometric and photometric accuracy, and variability information.

Both ACIS and HRC cameras operate in a photon counting mode, and register individual X-ray photon events. For each photon event, the two-dimensional position of the event on the detector is recorded, together with the time of arrival and a measure of the energy of the event. In most operating modes, lists of detected events are recorded over the duration of an observation, typically between 1 ks and 160 ks, and are then telemetered to the ground for subsequent processing.

To minimize the effect of bad detector pixels, and to avoid possible burn-in degradation of the camera by bright X-ray sources, the pointing direction of the telescope is normally constantly dithered in a Lissajous pattern, with a typical scale length of about 20'' on the sky and a period of order 1 ks, while taking data. The motion of the telescope is recorded via an "aspect camera" (Aldcroft et al. 2000) that tracks the motion of a set of (usually five) guide stars as a function of time during the observation. The coordinate transformation needed to remove the motion from the event (photon) positions is computed from the aspect camera data and applied during data processing.

Breaking down the four-dimensional X-ray data hypercube into spatial, spectral, and temporal axes provides a natural focus on the properties that may be of interest to the general user, but also identifies some of the complexities inherent in Chandra data that must be addressed by catalog construction and data analysis algorithms.

Spatially, the Chandra PSF varies significantly with off-axis and azimuthal angle (with the former variation dominating), as well as with incident photon energy (Figure 3). Close to the optical axis of the telescope, the PSF is approximately symmetric with a 50% enclosed energy fraction radius of order 03 over a wide range of energies, but at 15' off-axis the PSF is strongly energy dependent, asymmetric, and significantly extended, with a 50% enclosed energy fraction radius of order 13'' at 1.5 keV.

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For the widely used ACIS detector, the instrumental spectral energy resolution is of order 100–200 eV, and depends on incident photon energy and location on the detector. Because the energy resolution is significantly lower than the typical energy width of the features and absorption edges that define the effective area of the telescope optics (and therefore the quantum efficiency of the telescope plus detector system), a full matrix formulation that considers the redistribution of source X-ray flux into the set of instrumental pulse height analyzer bins must be used when performing spectral analyses. This is in contrast to the more familiar scenario from many other wavebands, where the instrumental resolution is often much higher than the spectral variation of quantum efficiency, enabling the commonly used implicit assumption that the flux redistribution matrix is diagonal (and is therefore not considered explicitly).

We note in passing that Chandra is equipped with a pair of transmission gratings that can be inserted into the optical path, and is therefore capable of performing high spectral resolution (slitless spectroscopy) observations. However, such observations are not included in the current release of the CSC.

Time domain analyses must consider the impact of spacecraft dither within an observation. Strong false variability signatures at the dither frequency can arise because of variations of the quantum efficiency over the detector, or because the source or background region dithers off the detector edge or across a gap between adjacent ACIS CCDs. Corrections for these effects, as well as for cosmic X-ray background flares that can be highly variable over periods of a few kiloseconds, must be applied when computing light curves. The extremely low photon event rates common for many faint X-ray sources typically require time domain statistics to be evaluated using event arrival-time formulations instead of rate-based approaches.

An additional level of complexity occurs because many astronomical sources of interest that will be included in the catalog are extremely faint. Rigorous application of Poisson counting statistics is required when deriving source properties and associated errors, separating X-ray analyses from many other wavebands where Gaussian statistics are typically assumed.

The tabulated properties included in the CSC are organized conceptually into two separate tables, the Source Observations Table and the Master Sources Table. Distinguishing between source detections (as identified within a single observation) and X-ray sources physically present on the sky is necessary because many sources are detected in multiple observations and at different off-axis angles (and therefore have different PSF extents).

Each record included in the Source Observations Table tabulates properties derived from a source detection in a single observation. These entries also include pointers to the associated file-based data products that are included in the catalog, which are all observation specific in the first catalog release. Each record in the Source Observations Table is further split internally into a set of source-specific data and a set of observation-specific, but source-independent, data. The latter are recorded once to avoid duplication. A description of the data columns recorded in the Source Observations Table for each source detection is provided in Table 1.

Because of the dependence of the PSF extent with off-axis angle, multiple distinct sources detected on-axis in one observation may be detected as a single source if located far off-axis in a different observation (Figure 4). During catalog processing, source detections from all observations that overlap the same region of the sky are spatially matched to identify distinct X-ray sources. Estimates of the tabulated properties for each distinct X-ray source are derived by combining the data extracted from all source detections and observations that can be uniquely associated, according to the algorithms described in Section 3. The best estimates of the source properties for each distinct X-ray source are recorded in the Master Sources Table. A description of the data columns recorded in the Master Sources Table for each source is provided in Table 2.

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Each distinct X-ray source is thus conceptually represented in the catalog by a single entry in the Master Sources Table, and one or more associated entries in the Source Observations Table (one for each observation in which the source was detected).

All of the tabulated properties included in both the Master Sources Table and the Source Observations Table can be queried by the user. Bi-directional links between the entries in the two tables are managed transparently by the database, so that the user can access all observation data for a single source seamlessly.

If a source detection included in the Source Observations Table can be related unambiguously to a single X-ray source in the Master Sources Table, then the corresponding table entries will be associated by "unique" linkages. Source detections included in the Source Observations Table that cannot be related uniquely to a single X-ray source in the Master Source Table will have their entries associated by "ambiguous" linkages (Figure 5).

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The data from ambiguous source detections are not used when computing the best estimates of the source properties included in the Master Sources Table. In the case of ACIS observations, source detections for which the estimated photon pile-up fraction (Davis 2007a) exceeds ∼10% will not be used if source detections in other ACIS observations do not exceed this threshold.

Using the linkages between the entries in the two tables, the user will nevertheless be able to identify all of the X-ray sources in the catalog that could be associated with a specific detection in a single observation, and vice versa. These linkages may be important, for example, when identifying candidate targets for follow-up studies based on a data signature that is only visible in the observation data for a confused source.

The primary user tool for querying the CSC is the CSCview Web-browser interface (Zografou et al. 2008), which can be accessed from the public catalog Web site.⁴ The user can directly query any of the tabulated properties included in either the Master Sources Table or the Source Observations Table, display the contents of an arbitrary set of properties for matching sources, and retrieve any of the associated file-based data products for further analysis. CSCview provides a form-based data-mining interface, but also allows users to directly enter queries written using the Astronomical Data Query Language (ADQL; Ortiz et al. 2008) standard. Query results can be directly viewed on the screen, or saved to a data file in multiple formats, including tab-delimited ASCII (which can be read directly by several commonly used astronomical applications) and International Virtual Observatory Alliance⁵ (IVOA) standard formats such as VOTable (Ochsenbein 2009).

Automated access to query the catalog from data analysis applications and scripts running on the user's home platform was identified as being needed for several science use cases. VO standard interfaces, including Simple Cone Search (Williams et al. 2008) and Simple Image Access (Tody & Plante 2009), provide limited query and data access capabilities, while more sophisticated interactions are possible through a direct URL connection. Support for VO workflows using applicable standards will be added in the future as these standards stabilize. An interface that integrates catalog access with a visual sky browser provides a simple mechanism for visualizing the regions of the sky included in the catalog, and may also be particularly beneficial for education and public outreach purposes.

Since Chandra is an ongoing mission, the CSC includes a mechanism to permit newly released observations to be added to the catalog and be made visible to end users, while at the same time providing stable, well-defined, and statistically well-characterized released catalog versions to the community. This is achieved by maintaining a revision history for each database table record, together with flags that establish whether catalog quality assurance and catalog inclusion criteria are met, and using distinct views of the catalog databases that utilize these metadata.

"Catalog release views" provide access to each released version of the catalog, with the latest released version being the default. Catalog releases will be infrequent (no more than of order 1 per year) because of the controls built in to the release process, and because of the requirement that each release be accompanied by a detailed statistical characterization of the included source properties. Once data are included in a catalog release view, then they are frozen in that view, even if the source properties are revised or the source is deleted in a later catalog release. A source may be deleted if the detection is subsequently determined to be an artifact of the data or processing, but the most likely reason that a source is deleted from a later catalog release is that additional observations included in the later release resolve the former detection into multiple distinct sources.

"Database views" provide access to the catalog database, including any new content that may not be present in an existing catalog release. Because on-going processing is continually modifying the catalog database, tabulated data and file-based data products in a database view may be superseded at any time, and the statistical properties of the data are not guaranteed.

We anticipate that users who require a stable, well-characterized data set will choose primarily to access the catalog through the latest catalog release view. On the other hand, users who are interested in searching the latest data to identify sources with specific signatures for further study will likely use the latest database view.

The first release of the CSC includes detected sources whose flux estimates are at least 3 times their estimated 1σ uncertainties, which typically corresponds to about 10 net (source) counts on-axis and roughly 20–30 net counts off-axis, in at least one energy band. In this release, multiple observations of the same field are not combined prior to source detection, so the flux significance criterion applies to each observation separately.

For each source detected in an observation, the catalog includes approximately 120 tabulated properties. Most values have associated lower and upper confidence limits, and many are recorded in multiple energy bands. The total number of columns included in the Source Observations Table (including all values and associated confidence limits for all energy bands) is 599.

Roughly 60 master properties are tabulated for each distinct X-ray source on the sky, generated by combining measurements from multiple observations that include the source. Combining all values and associated confidence limits for all energy bands yields a total of 287 columns included in the Master Sources Table.

The tabulated source properties fall mostly into the following broad categories: source name, source positions and position errors, estimates of the raw (measured) extents of the source and the local PSF, and the deconvolved source extents, aperture photometry fluxes and confidence intervals measured or inferred in several ways, spectral hardness ratios, power-law and thermal blackbody spectral fits for bright (>150 net counts) sources, and several source variability measures (Gregory–Loredo, Kolmogorov–Smirnov, and Kuiper tests).

Also included in the CSC are a number of file-based data products in formats suitable for further analysis in CIAO. These products, described in Table 3, include both full-field data products for each observation, and products specific to each detected observation-specific source region.

The full-field data products include a "white-light" full-field photon event list, and multi-band exposure maps, background images, exposure-corrected and background-subtracted images, and limiting sensitivity maps.

Source-specific data products include a white-light photon event list, the source and background region definitions, a weighted ancillary response file (ARF; the time-averaged product of the combined telescope/instrument effective area and the detector quantum efficiency), multi-band exposure maps, images, model ray-trace PSF images, and optimally binned light curves. Observations obtained using the ACIS instrument additionally include low-resolution (E/ΔE ∼ 10–40, depending on incident photon energy and location on the array) source and background spectra and a weighted detector redistribution matrix file (RMF; the probability matrix that maps photon energy to detector pulse height).

2.5.1. Energy Bands

The energy bands used to derive many CSC properties are defined in Table 4. The energy bands are chosen to optimize the detectability of X-ray sources while simultaneously maximizing the discrimination between different spectral shapes on X-ray color–color diagrams.

Table 4. CSC Energy Bands

Band		Energya	Monochromatica	Integrated Effective Areab
Name	Designation	Range	Energy	ACIS-I	ACIS-S	HRC-I
ACIS energy bands
Ultra-soft	u	0.2–0.5	0.4	7.36–2.24	68.7–23.0	...
Soft	s	0.5–1.2	0.92	216–155	411–274	...
Medium	m	1.2–2.0	1.56	438–401	539–493	...
Hard	h	2.0–7.0	3.8	1590–1580	1680–1670	...
Broad	b	0.5–7.0	2.3	2240–2140	2630–2440	...
HRC energy band
Wide	w	0.1–10	1.5	...	...	605

Notes. ^akeV. ^bkeV cm², computed at the ACIS-I, ACIS-S, and HRC-I aim points. For ACIS energy bands, the pair of values are the integrated effective area with zero focal plane contamination (first number) and with the late 2009 level of focal plane contamination (second number).

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The effective area of the telescope (including both the Chandra High Resolution Mirror Assembly (HRMA) and the detectors) is shown in Figure 6 as a function of energy, together with the average ACIS quiescent backgrounds derived from blank sky observations (Markevitch 2001a). The effective area is measured at the locations of the nominal "ACIS-S" aim point on the ACIS S3 CCD, and the nominal "ACIS-I" aim point on the ACIS I3 CCD.

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Where possible, the energy bands are chosen to avoid large changes of effective area within the central region of the band, since such variations degrade the accuracy of the monochromatic effective energy approximation described below. For example, the M-edge of the iridium coating on the HRMA has significant structure in the ∼2.0–2.5 keV energy range that provides a natural breakpoint between the ACIS medium and hard energy bands. Note however that large effective area variations are unavoidable within the ACIS broad and soft energy bands and the HRC wide energy band.

Weighting the effective area by the source spectral shape and integrating over the bandpass provides an indication of the relative detectability of a source in the energy band. Selecting energy band boundaries so that source detectability is roughly the same in different energy bands more uniformly distributes Poisson errors across the bands, and so enhances detectability in the various bands.

Several different source types were simulated when selecting energy bands. These included absorbed non-thermal (power-law) models with photon index values ranging from 1 to 4, absorbed blackbody models with temperature varying from 20 eV to 2.0 keV, and absorbed, hot, optically thin thermal plasma models (Raymond & Smith 1977) with kT = 0.25–4.0 keV. In all cases, the hydrogen absorbing column was varied over the range 1.0 × 10²⁰–1.0 × 10²² cm⁻². Detected X-ray spectra were simulated using PIMMS (Mukai 2009), and then folded through the bandpasses to construct synthetic X-ray color–color diagrams (see Figure 7, for example, color–color diagrams based on the final band parameters). Energy bands chosen to fill the color–color diagrams maximally provide the best discrimination between different spectral shapes. For detailed X-ray spectral-line modeling, the Raymond & Smith (1977) models have been superseded by more recent X-ray plasma models (e.g., Mewe et al. 1995; Smith et al. 2001). However, since the radiated power of the newer models as a function of temperature is not significantly different from the 1993 versions of the Raymond & Smith (1977) models used here, the latter are entirely adequate for the purpose of evaluating coverage of the X-ray color–color diagrams and the task is greatly simplified because of their availability in PIMMS⁶.

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Grimm et al. (2009) compared broadband X-ray photometry with accurate ACIS spectral fits and found that model-independent fluxes could be derived from the photometry measurements to an accuracy of about 50% or better for a broad range of plausible spectra. They used similar but not identical energy bands to those adopted for the CSC, but did not use the method of deriving fluxes from individual photon energies employed herein.

Combining all of these considerations (McCollough 2007) yields the following selection of energy bands for the CSC.

The ACIS soft (s) energy band spans the energy range 0.5–1.2 keV. The lower bound is a compromise that is set by several considerations. ACIS calibration uncertainties increase rapidly below 0.5 keV, so this establishes a fairly hard lower limit to avoid degrading source measurements in the energy band. As shown in Figure 8, below about 0.6 keV the background count rate begins to increase rapidly, while the integrated effective area rises very slowly, resulting in few additional source counts. While pushing the band edge to higher energy will result in a lower background, the integrated effective area drops rapidly if the lower bound is raised above ∼0.8 keV, reducing the number of source counts collected in the band. We choose to set the lower bound equal to 0.5 keV since doing so enhances the detectability of super-soft sources, while not noticeably impacting measurements of other sources. The upper cutoff for the soft energy band is set equal to 1.2 keV, which balances the preference for uniform integrated effective areas amongst the energy bands with the desire to maximize the area of X-ray color–color plot parameter space spanned by the simulations.

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The lower bound of the ACIS medium (m) energy band matches the upper bound of the soft energy band. We locate the upper band cutoff at 2.0 keV since this value tends to maximize the coverage of the X-ray color–color diagram. This value also moves the iridium M-edge out of the sensitive medium band, and instead placing it immediately above the lower boundary of the ACIS hard (h) energy band.

The high energy boundary of the latter band is set to 7.0 keV. This cutoff provides a good compromise between maximizing integrated effective area and minimizing total background counts (Figure 8). Above 7.0 keV, the background rate increases rapidly at the ACIS-S, while below this energy the integrated effective area decreases rapidly at the ACIS-I aim point. Placing the hard energy band cutoff at 7.0 keV also has the advantage that the Fe Kα line is included in the band, allowing intense Fe line sources to be detected without compromising the measurement quality for typical catalog sources.

The ACIS broad (b) band covers the same energy range as the combined soft, medium, and hard bands, and therefore spans the energy range 0.5–7.0 keV.

Simulations indicate that an additional energy band extending below 0.5 keV is beneficial for discriminating super-soft X-ray sources in color–color plots. The ACIS front-illuminated CCDs have minimal quantum efficiency below 0.3 keV, while the response of the back-illuminated CCDs extends down to ∼0.1 keV. Hydrocarbon contamination is present on both the HRMA optics (Jerius 2005) and the ACIS optical blocking filter (Marshall et al. 2004). The latter reduces the effective area at low energies, and enhances the depth of the carbon K-edge. An ACIS ultra-soft (u) band covering 0.2–0.5 keV is added to provide better discrimination of super-soft sources. Source detection is not performed in this energy band, because of the typical lower overall signal-to-noise ratio (S/N) and the resulting enhanced false-source rate.

Finally, since the HRC (particularly HRC-I) has minimal spectral resolution, a single wide (w) band that includes essentially the entire pulse height spectrum (specifically, PI values 0:254), roughly equivalent to 0.1–10 keV, is used for HRC observations.

While bands in these general energy ranges give the best balance of count rate and spectral discrimination, our simulations indicate that the exact choice of band boundary energies is not critical at the 10% level.

2.5.2. Band Effective Energies

In principle, the variations of HRMA effective area, detector quantum efficiency, and (for ACIS) focal plane contamination, with energy imply that energy-dependent data products such as exposure maps or PSFs should be constructed by integrating the source spectrum over the energy band. This approach would be both extremely time consuming, and require knowledge of the source spectrum that is typically not available a priori. In practice, a monochromatic effective energy is chosen for each energy band to be used to construct energy-dependent data products (McCollough 2007).

The monochromatic effective energy for each band is determined using the relation

$$\begin{eqnarray} E_{\rm eff} &=& \int\! dE E A(E) Q(E) C(E) S(E)/\nonumber\\ &&\int\! dE \, A(E) Q(E) C(E) S(E), \end{eqnarray} \tag{ 1 }$$

where E is the energy, A is the effective area of the HRMA, Q is the detector quantum efficiency, C is the reduction in transmission due to focal plane contamination, S is a power-law spectral weighting function of the form (E/E₀)^−α, and the integral is performed over the energy band.

The monochromatic effective energies for each energy band were calculated for sources located at the ACIS-I and ACIS-S aim points, and also for the nominal aim point on the HRC-I detector. Since the CSC is constructed from observations acquired throughout the Chandra mission, ACIS focal plane contamination models with both zero contamination (appropriate for observations obtained early in the mission) and the contamination level current as of late 2009 were employed. Power-law spectral weighting functions with α varying from 0.0 to 2.0 were used. Setting α = 1 gives a spectral weighting function that approximates an absorbed Γ = 1.7 power-law spectrum, and the limits for α were chosen to span the typical range of values determined from fits to a canonical subset of Chandra data sets. The remaining parameters in Equation (1) are extracted from the Chandra calibration database (CalDB; George & Corcoran 2005; Graessle et al. 2006). The monochromatic effective energies for ACIS were chosen to be the approximate arithmetic means of the α = 1 values derived for the ACIS-I and ACIS-S aim points, with zero and late 2009 focal plane contamination. For ACIS energy bands other than the broad band, the monochromatic effective energies computed for a single value of α all agree within ≲0.1 keV. The dependence on α is similarly small, except for the hard energy band, where varying α from 0.0 to 2.0 changes the monochromatic effective energy from ∼4.2 keV to ∼3.4 keV. For the ACIS broad energy band, the agreement between the different models for a single value of α is ∼±0.3 keV. However, for this band the dependence on α is more significant, varying from ∼3.3 keV for α = 0.0 to ∼1.6 keV for α = 2.0. The monochromatic effective energies used to construct the CSC are reported in Table 4.

Although the use of a single monochromatic effective energy for each energy band simplifies data analysis by removing the dependence on the source spectrum, some error will be introduced for sources that have either extremely soft or extremely hard spectra compared to the canonical α = 1.0 power-law spectral weighting function. Knowledge of the expected magnitude of the error that may be introduced is helpful when evaluating catalog properties.

For both the ACIS medium and hard energy bands, neither extremely soft nor extremely hard source spectra induce variations in exposure map levels that are greater than ∼10%, so photometric errors due to source spectral shape should not exceed this value. In the ACIS soft energy band, very soft spectra may produce deviations of order 5%–20%, with the largest excursions expected for the front-illuminated CCDs. These differences increase to ∼15%–35% for the ACIS ultra-soft energy band, with the largest values once again associated with the front-illuminated CCDs. For all of the ACIS narrow energy bands, the errors induced by extremely hard spectra are much smaller than those caused by extremely soft spectra. The presence of the iridium edge and the large energy ranges included in the ACIS broad and HRC wide energy bands may produce significantly larger variations for extreme spectral shapes. Very soft spectra can alter exposure map values by ∼65%–90% in the ACIS broad energy band, although there is little impact in the HRC wide energy band. Conversely, extremely hard spectra may induce changes up to ∼70% in the HRC wide energy band, and ∼25%–30% in the ACIS broad energy band. As described in Section 4.4, model-based statistical characterization of CSC source fluxes (F. A. Primini et al. 2010, in preparation) produces results that are generally consistent with these expectation, with the exception that flux errors in the ACIS broad energy band appear to be ∼10% for most sources.

When computing fluxes for point sources, an aperture correction is applied to compensate for the fraction of the PSF that is not included in the aperture. Since the extent of the Chandra PSF varies with energy, using a monochromatic effective energy can introduce a flux error because the energy dependence of the PSF fraction is not considered. This error can be bounded by a post facto comparison of PSF fractions for catalog source detections in the five ACIS energy bands. The majority of variations between energy bands fall in the range 4%–8%, with 90% of source detections showing <10% differences. These values represent an upper bound on the error introduced within an energy band by the use of a monochromatic effective energy.

2.5.3. Coordinate Systems and Image Binning

While the CSC ultimately aims to be a comprehensive catalog of X-ray sources detected by Chandra, all of the functionality required to achieve that goal are not included in the release 1 processing system. A set of pre-filters is used to limit the data content to the set of observations that the catalog processing system is capable of handling.

For release 1, only public ACIS "timed-exposure" readout mode imaging observations obtained using either the "faint," "very faint," or "faint with bias" data modes are included. ACIS observations that are obtained using CCD subarrays with ⩽128 rows are also excluded, because there are too few rows to ensure that source-free regions can be reliably identified when constructing the high spatial frequency background map. HRC-I imaging mode observations are not included in release 1 of the catalog, but are included in incremental release 1.1. HRC-S observations are excluded because of the presence of background features associated with the edges of "T"-shaped energy-suppression filter regions that form part of the UV/ion-shield. Observations of solar system objects are not included in the CSC.

All observations included in the CSC must have been processed using the standard data processing pipelines included in version 7.6.7, or later, of the CXCDS. This version of the data system was used to perform the most recent bulk reprocessing of Chandra data, and includes revisions to the pipelines that compute the aspect solution that is used to correct for the spacecraft dither motion and register the source events on the sky. Observations must have successfully passed the "validation and verification" (quality assurance) checks that are performed upon completion of standard data processing.

The largest scale lengths used to detect sources to be included in the CSC have angular extents ∼30''. Sources with apparent sizes greater than this are either not detected, or may be incorrectly detected as multiple close sources. Prior to catalog construction, all observations are inspected visually for the presence of extended sources that may be incorrectly detected, and such observations are excluded from catalog processing. For ACIS observations, if the presence of any spatially extended emission is restricted to a single CCD only, then the data from that CCD are dropped, and sources detected on any remaining CCDs are typically included in the catalog. The latter rule allows many sources surrounding bright, extended cores of galaxies to be included in the catalog, rather than having the entire observation rejected outright.

While the visual inspection and rejection process is inherently subjective in nature, an attempt was made to calibrate the method by constructing a "training set" of several hundred observations that were processed through a test version of the catalog pipelines. The training set observations included a wide variety of point, compact, and extended sources, with differing exposures and S/N, which were classified as accept/reject based on the actual results of running the pipeline source detection and source property extraction steps. These observations and classifications were then used to train the personnel who performed the visual inspection process.

Although all observations included in the CSC have been processed through the CXCDS standard data processing pipelines, we nevertheless re-run the instrument-specific calibration steps as the first step in catalog construction. One reason for reapplying the instrumental calibrations is that they are subject to continuous improvements, and may have been revised since the last time the observations were processed or reprocessed. A second reason is to ensure that a single set of calibrations are applied to all data sets, so that the resulting catalog will be calibrated as homogeneously as possible.

For ACIS, the principal instrument-specific calibrations that are re-applied are the (time-dependent) gain calibration and the correction for CCD charge transfer inefficiency (CTI). The former calibration maps the measured pulse height for each detected X-ray event into a measurement of the energy of the corresponding incident X-ray photon. CTI correction attempts to account for charge lost to traps in the CCD substrate when the charge is being read out. This effect is considerably larger than anticipated prior to launch because of damage to the ACIS CCDs caused by the spacecraft's radiation environment. Additionally, observation-specific bad pixels and hot pixels are flagged for removal, as are "streak" events on CCD S4 (ACIS-8). The latter apparently result from a flaw in the serial readout electronics (Houck 2000). Pixel afterglow events, which arise because of energy deposited into the CCD substrate by cosmic ray charged particles, are removed using the acis_run_hotpix tool that is also included in CIAO. Although this program can miss some real faint afterglows, such events are very unlikely to exceed the flux significance threshold required for inclusion in the catalog. The default 0.5 pixel event position randomization in chip coordinates is used when the calibrations are reapplied.

The main instrument-specific calibrations for HRC data relate to the "degapping" correction that is applied to the raw X-ray event positions to compensate for distortions introduced by the HRC detector readout hardware. Several additional calibrations compensate for effects introduced by amplifier range switching and ringing in the HRC electronics, and a number of validity tests are performed to flag X-ray event positions that cannot be properly corrected due to amplifier saturation and other effects.

Since data are recorded continuously during an observation, a "Mission Time Line" is constructed during standard data processing that records the values of key spacecraft and instrument parameters as a function of time. These parameters are compared with a set of criteria that define acceptable values, and "Good Time Intervals" (GTIs) that include scientifically valid data are computed for the observation. The GTI filter from standard data processing is reapplied without change as part of the recalibration process.

Background event screening performed as part of catalog data recalibration is somewhat more aggressive than that performed as part of standard data processing, typically reducing the non-X-ray background. For a 10 ks observation, the median catalog background rate is roughly 80% of the nominal field background rates (Chandra X-ray Center 2009), although there is considerable scatter. F. A. Primini et al. (2010, in preparation) include a detailed statistical analysis of the improvements to the non-X-ray background afforded by this screening.

The reduction of the background event rate is achieved by removing time intervals containing strong background flares. These time intervals are identified separately for each chip. First, the background regions of the image are identified by constructing a histogram of the event data, determining the mean and standard deviations of the histogram values, and rejecting all pixels that have values more than three standard deviations above the mean. An optimally binned light curve of the background pixels is then created using the Gregory–Loredo algorithm (see Section 3.12.1). Time bins for which the count rate exceeds 10× the minimum light curve value are identified. The corresponding intervals are considered to be background flares, and the GTIs are revised to exclude those periods.

We emphasize that the objective of this procedure is to remove only the most intense background flares, which occur relatively infrequently. Time intervals that include moderately enhanced background rates are not rejected by this process, since their contributions increase the overall S/N. The aggregate loss of good exposure time exceeds 25% for less than 1.5% of the observations included in the catalog; the loss is greater than 10% for 3% of the observations, and greater than 5% for 5% of the observations.

For each observation included in the CSC, the recalibrated photon event list is archived, together with several additional full-field data products. These include multi-resolution exposure maps computed at the monochromatic effective energies of each energy band and the associated ancillary data products (aspect histogram, bad pixel map, and field-of-view region definition), used to construct them (see Table 3).

For the first release of the CSC, background maps are used for automated source detection. They are created directly from each individual observation with the necessary accuracy. The general observation background is assumed to vary smoothly with position, and is modeled using a single low spatial frequency component. Although this assumption is in general satisfied across the fields of view included in this catalog release, there may be localized regions where the background intensity has a strong spatial dependence, and therefore where the detectability of sources may be reduced. Several different approaches were considered for constructing the low spatial frequency background component, including spatial transforms, low-pass filters, and data smoothing. However, the most effective and physically meaningful technique is a modified form of a Poisson mean. This method, described below, estimates the local background from the peak of the Poisson count distribution included in a defined sampling area. The dimensions of the sampling area act effectively as a spatial low-pass filter that determines the minimum angular size that contributes to the background.

High spatial frequency linear features, commonly referred to as "readout streaks," result when bright X-ray sources are observed with ACIS. These streaks arise from source photons that are detected during the CCD readout frame transfer interval (∼40 μs per row) following each exposure (∼3.2 s per exposure for a typical observation). All pixels along a given readout column are effectively exposed to all points on the sky that lie along that column during the frame transfer interval, so that columns including bright X-ray sources have enhanced count rates along their length. Unless accounted for by the source detection step, the increased counts in the bright readout streak are detected as multiple sources. Although readout streaks are comprised of mislocated source photons, we choose to model them as a background component.

Background maps computed for ACIS observations include contributions from both components, while HRC background maps include only the low spatial frequency component.

The reader should note that background maps are not used when deriving source properties such as aperture photometry. Instead, a local background value determined in an annular aperture surrounding the source is used, as described in Section 3.4.1. Significant spatial variations of the observed X-ray flux on the scale of the background aperture will increase the background local variance, thus reducing the significance of the source detection, perhaps below the threshold required for inclusion of the source in the catalog. This effect is seen in some galaxy cores, where the unresolved emission contributes X-ray flux to the annular background apertures surrounding each source.

3.3.1. ACIS High Spatial Frequency Background

The algorithm described here is a refinement of the method used by McCollough & Rots (2005) to address the impact of readout streaks on source detection. The streak map is computed at single pixel resolution independently for each ACIS CCD and energy band. The first step is to identify the bright-source-free regions on the detector. For ease of computation the orientation of the X-axis is defined to be along the chip rows (perpendicular to the readout direction) and the Y-axis is defined to be along the direction of the readout columns. To identify the source-free regions, the photon event totals, X_sum summed along the X-axis are constructed, and the median ($\tilde{X}$), mode ($\hat{X}$), and standard deviation (σ_X) of the distribution of the X_sum values are computed. These values provide a basic characterization of the background. From an examination of many data histograms, the maximum value of X_sum which can still considered background dominated is given by

$$\begin{equation*} X_{\rm sum} ({\rm max}) = \min [\tilde{X} + n \, \sigma _X, 2 \, {\hat X}], \end{equation*}$$

where n is set to 1. Rows for which X_sum ≫ X_sum(max) include a substantial bright source contribution. All rows with X_sum ⩽ X_sum(max) (excluding off-chip and dither regions) are considered to comprise the source-free regions and are used to calculate the streak map.

The average number of events per pixel is calculated separately for each readout column (Y-axis direction) from all of the rows in the source-free regions. These values are replicated across each CCD row to create an image that includes the sum of the readout streak contribution and the mean one-dimensional low spatial frequency background component. The latter must be accounted for when combining the high spatial frequency readout streak map with the two-dimensional low spatial frequency background map.

For the algorithm to obtain a good measure of the background, of order 100 bright-source-free rows are required. This condition is satisfied for most observations. Observations with too few source-free rows poorly sample the background. This can lead to erroneously low intensities for bright readout streaks in the resulting background map, which may enhance the false source rate along these streaks. Faint sources that fall in the source-free rows will be considered to be part of the background, which can lead to similar results. Nevertheless, the algorithm is remarkably effective, even in crowded regions such as the Orion complex and the Galactic center fields. An example broadband ACIS streak map, created for observation 00735 (M81), is shown in Figure 11 (left).

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3.3.2. Low Spatial Frequency Background

For each observation, a low spatial frequency background map is constructed separately for each energy band and image-blocking factor (see Section 3.4). McCollough & Rots (2008) provide an initial discussion of this algorithm and general background map creation.

As described above, for ACIS observations the high spatial frequency background map includes a component that represents the one-dimensional average of the low-frequency background over the rows used to create the streak map. This component, as well as the high spatial frequency background, are removed by subtracting the streak map from the original image from which it was created. For each image-blocking factor, the difference image is constructed by subtracting the appropriately regridded streak map from the corresponding blocked original image.

For each pixel in the resulting difference image, a centered sampling region with dimensions n × n pixels is defined. Spatial scales smaller than ∼n pixels are attenuated. The sampling regions are truncated at the edges of the images, and so some higher frequency information may propagate into the background map. However this effect has not been found to have any significant impact on the utility of the resulting map.

A histogram of the count distribution is constructed from the pixels included in the sampling region associated with each image pixel. The first histogram bin will typically span the count range from −0.5 to +0.5 for ACIS observations, since the readout streak map has been subtracted and there will be some negative pixels. The low spatial frequency background at this image pixel location is computed using a modified form of a Poisson mean

$$\begin{equation*} b_{\rm lf} = \mbox{\rm mean}[h(a) \cup h(b) \cup h(c)], \end{equation*}$$

where h(x) is the number of counts in histogram bin x, a is the bin with the maximum number of histogram counts, and b and c are the lower and higher bins immediately adjacent to a. The low spatial frequency background map is formed by computing b_lf for each pixel location in the image. For ACIS observations, n = 129 pixels, corresponding to a spatial scale of order 1' for images blocked at single pixel resolution. Figure 11 (right) displays the ACIS broad band low spatial frequency map for observation 00735 (M81) that corresponds to the streak map shown in the left-hand panel of the figure.

3.3.3. Total Background Map

The first step in creating the total background map is to correct the readout streak map (for ACIS observations only) for the effects of reduced exposure near the edges of the observation that arise due to the spacecraft dither, by dividing by the appropriate band-specific normalized exposure map. Similarly, the low spatial frequency background map is corrected by dividing by the smoothed, band-specific normalized exposure map. The smoothing that is applied to the normalized exposure map in the latter case matches the smoothing applied when constructing the low spatial frequency background map. Finally, the two background components for each energy band are summed to produce the total exposure-normalized background map that is required for source detection.

Figure 12 displays the central region of the broad band ACIS image of M81 (observation 00735), with source detections overlaid. The source detections shown in the left-hand panel are those that result if the background is modeled internally by wavdetect (see Section 3.4, below); the panel on the right shows the source detections resulting from using the total background from Figure 11. Using the background map has eliminated the false sources detected on the readout streak.

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The total background maps for each energy band are also archived and accessible through the catalog. These maps differ from those used for source detection in that they have been multiplied by the normalized band-specific exposure map, and are therefore recorded in units of counts. For the convenience of the user, we also store multi-resolution photon-flux images for the full field of each observation, created by filtering the photon event list by energy band, binning to the appropriate image resolution, subtracting the total background map appropriate to the energy band, and dividing by the corresponding exposure map.

Candidate sources for inclusion in the CSC are identified using the CIAO wavdetect wavelet-based source detection algorithm (Freeman et al. 2002). wavdetect has been used successfully with Chandra data by a number of authors (e.g., Brandt et al. 2001; Giaconni et al. 2002; Lehmer et al. 2005; Kim et al. 2007; Muno et al. 2009), and its capabilities and limitations are well known (e.g., Valtchanov et al. 2001).

Early in the catalog processing pipeline development cycle, several different methods for detecting sources were evaluated. In addition to wavdetect, these included the CIAO implementations of the sliding cell (Harnden et al. 1984; Calderwood et al. 2001) and Voronoi tessellation and percolation (Ebeling & Wiedenmann 1993) algorithms, and a version of the SExtractor package (Bertin & Arnouts 1996) modified locally to use Poisson errors in the low count regime.

The Voronoi tessellation and percolation algorithm was quickly discarded because of the significant computational requirements and complexities for automated use. A series of simulations was used to compare the performance of the remaining methods with respect to source detection efficiency for isolated point sources, the efficiency with which close, equally bright pairs of point sources with 2'' and 4'' separations are resolved, and false source detection rate (A. Dobrzycki 2002, private communication; Hain et al. 2004). The first two properties were evaluated for point sources containing 10, 30, 100, and 2000 counts, with off-axis angles 0'–10' with 1' spacing, and nominal background rates for exposure times of 3, 10, 30, and 100 ks. The false source rate was evaluated as a function of off-axis angle for the same exposure times.

All three detection algorithms performed reliably for bright, isolated sources located close to the optical axis. Compared to the remaining methods, wavdetect had better source detection efficiency for faint sources located several arcminutes off-axis, and was able to resolve close pairs of sources more reliably than the sliding cell technique. The locally modified version of SExtractor provided inconsistent results, in some cases detecting large numbers of spurious sources.

These simulations were performed early in the catalog processing pipeline development cycle, as an aid in selecting the source detection algorithm to be used for catalog construction. They did not make use of the background maps described in the previous section. The actual performance of the source detection process used to construct the CSC is established from more detailed and robust simulations, as described in Section 4 and references therein.

Based on the results of the simulations, wavdetect was selected as the source detection method of choice for the CSC.

The wavdetect algorithm does not require a uniform PSF over the field of view, and is effective in detecting compact sources in moderately crowded fields with variable exposure and Poisson background statistics. To detect candidate sources in a two-dimensional image D, wavdetect repeatedly constructs the two-dimensional correlation integral

$$\begin{equation} C(x, y; {\boldsymbol{\alpha }}) = \int \!\!\int dx^\prime \,dy^\prime \, W(x-x^\prime, y-y^\prime ; {\boldsymbol{\alpha }})\, D(x^\prime, y^\prime) \end{equation} \tag{ 2 }$$

for a set of Marr ("Mexican Hat") wavelet functions, W, with scale sizes that are appropriate to the source dimensions to be detected. The elliptical form of the Marr wavelet may be written in the dimensionless form

$$\begin{equation} W(x, y; {\boldsymbol{\alpha }}) = (2 - \rho ^2)\exp (-\rho ^2/2), \end{equation} \tag{ 3 }$$

where

$$\begin{equation*} \rho ^2 = \frac{1}{a_1^2}(x\cos \phi + y\sin \phi)^2 + \frac{1}{a_2^2}(-x\sin \phi + y\cos \phi)^2 \end{equation*}$$

and the parameters ${\boldsymbol{\alpha }}= (a_1, a_2, \phi)$ define the semi-major and semi-minor radii and rotation angle of the Mexican Hat.

A localized clump of counts in the image D will produce a local maximum of C if the scale sizes defined by ${\boldsymbol{\alpha }}$ are approximately the same as, or larger than, the dimension of the clump. To determine whether a local maximum of C is due to the presence of a source, the detection significance, S_i,j, in each image pixel (i, j) is determined from

$$\begin{equation*} S_{i, j} = \int _{C_{i, j}}^\infty dC\,p(C|n_{B, i, j}), \end{equation*}$$

where n_B,i,j is the number of background counts within the limited spatial extent of W, and p(C|n_B,i,j) is the probability of C given the background B. If S_i,j ⩽ S₀, where S₀ is a defined limiting significance level, then pixel (i, j) is identified as a source pixel.

The limiting significance level used to generate the CSC is set to S₀ = 2.5 × 10⁻⁷. This formally corresponds to ∼1 false source due to random fluctuations per 2048 × 2048 pixel image, although due to the heuristics of the algorithm, the actual number of false sources may be lower. The situation is further complicated in our case because the final candidate source list output from the CSC source detection pipeline is a combination of several wavdetect runs in different energy bands (see below). We note that reliable quantitative estimates of the false source rates and detection efficiency can only be provided through simulations, as discussed in Section 4. As described in Section 2.5, we impose an additional restriction on the flux significance of a source. To ensure that the flux significance requirement is the defining criterion for a source to be included in the catalog, we have verified that the flux significances of sources that pass our wavdetect threshold extend well below that required to satisfy the flux significance rule (see Figure 13). We estimate that roughly ∼1/3 of all the sources detected by wavdetect fall below this threshold.

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Source detection is performed recursively by applying wavdetect to multiply-blocked sky images constructed as described in Section 2.5.3. The use of a constant blocked image size maintains algorithm efficiency while not compromising detection efficiency in the outer areas of the field of view where the PSF size is significantly larger than a single pixel.

Applied to the CSC, wavelets with scales a_i = 1, 2, 4, 8, and 16 (blocked) pixels are computed for each image-blocking factor and each energy band except for the ACIS ultra-soft band. This combination of wavelet scales and image-blocking factors provides good sensitivity for detection of sources with observed angular extents ≲30''. Some point sources with extreme off-axis angles, θ>20', may not be detected because the size of the local PSF exceeds the largest wavelet scale/blocking factor combination. F. A. Primini et al. (2010, in preparation) calibrate this effect statistically.

Source detection is not performed in the ACIS ultra-soft energy band. This band is impacted heavily both by increased background and by decreased effective area because of ACIS focal plane contamination (the ratio of integrated background to effective area is 1–2 orders of magnitude larger for the u band when compared to the other ACIS energy bands). Under these circumstances we are limited by the accuracy of the background map determination; small errors in the background map result in an unacceptable fraction of spurious source detections.

The wavdetect algorithm incorporates steps to compare nearby correlation maxima identified at multiple wavelet scales to ensure that each source is counted only once. After duplicates are eliminated, a source cell that includes the pixels containing the majority of the source flux is constructed. Although a source cell may have an arbitrary shape, for simplicity an elliptical representation of the source region is used throughout the CSC. The lengths of the semi-axes of this source region ellipse are set equal to the 3σ orthogonal deviations of the distribution of the counts in the source cell.

Source region ellipses for candidate sources detected within a single observation from images with different blocking factors or in different energy bands are combined outside of wavdetect to produce a single merged source list. This step rejects any detections that have rms radii smaller than the 50% enclosed counts fraction (ECF) radius of the local PSF, calculated at the monochromatic effective energy of the band in which the source is detected. Such detections are likely artifacts arising from cosmic ray impacts. Candidate source detections whose centroids are closer than the local PSF radius, or that are closer than 3/4 of the mean detected source ellipse radii, are deemed to be duplicates. If any duplicates are identified, then the detection from the image with the smallest blocking factor is kept, and if the image-blocking factors are equal, then the detection with the highest significance is used. This approach ensures that data from the highest spatial resolution blocked image will be used to detect point and compact sources. However, knotty emission that is located on top of extended structures will tend to be identified as distinct compact sources, while the extended emission is not recorded.

3.4.1. Source Apertures

Numerous source-specific catalog properties are evaluated within defined apertures. We define the "PSF 90% ECF aperture" for each source to be the ellipse that encloses 90% of the total counts in a model PSF centered on the source position. Because the size of the PSF is energy dependent, the dimensions of the PSF 90% ECF aperture vary with energy band.

We define the "source region aperture" for each source to be equal to the corresponding 3σ source region ellipse included in the merged source list, scaled by a factor of 1.5×. Like the PSF 90% ECF aperture, the source region aperture is also centered on the source position, but the dimensions of the aperture are independent of energy band. Evaluation of model PSFs with off-axis angles ≲10' demonstrates that the dimensions of the source region aperture correspond approximately to the dimensions of the PSF 90% ECF ellipses for the ACIS broad energy band. This is confirmed a posteriori by examining the distribution of PSF aperture fractions in source and background (see below) region apertures of all individual catalog sources with ACIS broad band flux significance ⩾3.0. Figure 14 demonstrates that the source region apertures typically include ∼90%–95% of the PSF, while the background region apertures contain ≲5%–10%. We emphasize that while these fractions are typical, the actual PSF fractions, determined by integrating the model PSF over the source and background region apertures and excluding regions from contaminating sources, are used for the actual determination of source fluxes (see Section 3.7).

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Comparison of the source fluxes within the PSF 90% ECF aperture and the source region aperture provide a crude indication of whether a source is extended. If the flux in the source region aperture is significantly greater than the flux in the PSF 90% ECF aperture, then the source region determined by wavdetect is considerably larger than the local PSF, and the source is likely extended.

Both the PSF 90% ECF aperture and the source region aperture are surrounded by corresponding background region annular apertures. In both cases, the inner edge of the annulus is set equal to the outer edge of the corresponding source aperture, while the radius of the outer edge of the annulus is set equal to 5× the inner radius of the source region aperture. Although the background region apertures defined in this manner include ∼5%–10% of the X-rays from the source, this contamination is accounted for explicitly when computing aperture photometry fluxes.

Overlapping sources could contaminate any measurements obtained through the source and background apertures. To avoid this, both types of apertures are modified to exclude areas that are included in any overlapping source region apertures, or that fall off the detector. Areas surrounding ACIS readout streaks are also excluded from the modified background apertures. Aperture-specific catalog quantities are derived from the event data in the appropriate modified aperture. The fractions of the local model PSF counts that are included in the modified apertures are recorded in the catalog for each source, and are used to apply aperture corrections when computing fluxes, under the assumption that the source is well modeled by the PSF.

The modified source region and background region aperture definitions are recorded as FITS files using the spatial region file convention (Rots & McDowell 2008). CIAO (Fruscione et al. 2006) can be used to apply these regions as spatial filters to extract the photon event data for the source (or background) from the archived photon event list. To simplify access to file-based data products (see Table 3) for individual sources, we also separately store the source region photon event list, per-band exposure maps, and per-band source region images. These products include data from the rectangular region of the sky that is oriented north–south/east–west and that bounds the background region.

3.4.2. Matching Source Detections from Multiple Observations

Each source record in the CSC Master Sources Table is constructed by combining source detections included in the Source Observations Table from one or more observations. A necessary first step in this process requires matching the source detections from all of the observations that include the same region of the sky. Cross-matching algorithms (e.g., Devereux et al. 2005; Gray et al. 2006) are often focused on efficiently matching large catalogs, and typically use criteria on the position difference distribution, or cross-correlation approach techniques, for identifying matches. In many cases, these approaches assume (often implicitly) that the source PSF is at least approximately spatially uniform across the field of view, and comparable between the data sets being matched.

However, when matching source detections across multiple Chandra observations, the strong dependence of the PSF size with off-axis angle must be considered explicitly, since source detections that are well off-axis in one observation are often resolved into multiple sources close to the optical axis in other observations. Under these circumstances the source positions determined by wavdetect are not comparable, and cannot be used for source matching. Instead, the source matching approach used for the CSC is based on the overlaps between the PSF 90% ECF apertures of the source detections from the individual observations. Although empirical in nature, this algorithm works well for matching compact source detections between Chandra observations.

The detailed algorithm is described in Appendix A. The method identifies the overlap fractions between the PSF 90% ECF apertures of overlapping source detections from the observations, and separates them into three different categories.

The first category is the simplest, where the source detections from the various observations have apertures that all mutually overlap (Figure 15, left). This is the most common situation, and corresponds to the case where the source detections all uniquely match a single source on the sky. Roughly 90% of the ∼18,000 sources in the Master Sources Table that are linked to more than one source detection in the Source Observations Table fall into this category. Each of the matching entries in the latter table will be associated with the corresponding Master Sources Table entry with a "unique" linkage, as described in Section 2.3.

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In the second category, the aperture associated with a source detection in one observation overlaps the apertures associated with multiple distinct source detections from other observations. This circumstance typically arises because source detections from a single observation are always assumed to be distinct; this assumption can fail very far off-axis (θ ≳ 20'), where the PSF size exceeds the maximum wavdetect wavelet scale/image-blocking factor combination. This category is illustrated in Figure 15 (center), and arises most often because a source detection in one observation is resolved into multiple sources by one or more of the overlapping observations. The unresolved source detection in the Source Observations Table will be connected to all Master Sources Table entries associated with the matching resolved source detections via "ambiguous" linkages, and the detection will be flagged as confused. The X-ray photon events associated with the unresolved detection cannot be distributed across the matching resolved sources. In release 1 of the CSC, source properties derived from the detection are not used to compute the source properties included in the Master Sources Table. Upper limits for photometric quantities could in principle be extracted from the unresolved source detection, and these would be quite valuable for variability studies. This capability will be included in a future release of the CSC.

In a few cases, a set of aperture overlaps cannot be automatically resolved using the current algorithm. This third category typically occurs when there are multiple overlapping, confused source detections. In this case, the source detections are flagged for review by a human, who is then responsible for resolving the matches. Only 415 (out of 94,676) master sources include source detections that required manual review, and the majority of these were readily resolved as confused "pairs of pairs." The latter case, which is illustrated in Figure 15 (right), commonly occurs because a pair of source detections included in a single observation both overlap a pair of source detections in another observation. Each of the source detections in the first observation overlaps both source detections in the second observation, making the detections confused, and vice versa. If a manual review is required to complete a match for a specific source, then a flag is set in the catalog to indicate this fact.

The actual processing required to perform the matches is complex. Whenever new overlapping source detections are identified during catalog processing, the set of matches is recomputed using all of the observations processed so far. The algorithm then queries the prior state of the catalog, and determines the set of updates that are necessary to migrate from that state to the newly determined state. This procedure works regardless of the order in which observations processed, and also permits an already-processed observation to be reprocessed should an error have occurred.

3.4.3. Source Naming

Within a single observation, the detected source positions are those assigned by the wavdetect algorithm. Their accuracy can be estimated by evaluating the mean (signed) coordinate differences between the wavdetect positions and the positions of matching sources in the seventh data release (DR7) of the Sloan Digital Sky Survey (SDSS; Abazajian et al. 2009). The latter are extracted from the CSC/SDSS Cross-Match Catalog.⁸ As shown in Figure 16, the mean coordinate differences demonstrate good position agreement between the CSC and SDSS for sources with θ ≲ 8' (where we have restricted the comparison to include only individual observations of CSC sources with at least 500 net counts to minimize statistical errors).

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For larger off-axis angles, Figure 16 suggests that there may be a systematic offset between the wavdetect source positions and the SDSS source positions. The measured mean position difference is ≲03 for θ ≲ 15', but appears to increase with off-axis angle. The exact cause of this offset is uncertain. Some authors (e.g., Alexander et al. 2003; Lehmer et al. 2005; Luo et al. 2008) have reported a similar effect, which they attribute to centroiding errors introduced by of the asymmetric nature of the Chandra PSF at large off-axis angles.

High-quality PSF simulations generated using the SAOTrace (formerly SAOsac) ray-trace code (Jerius et al. 1995, 2004) confirm that the asymmetry can displace the measured centroid from the requested location of the simulated PSF on a uniform pixel grid. We designate the vector orientation from the measured centroid position to the requested location of the PSF as ${\boldsymbol{\xi }}$. As shown in Figure 16, the expected centroid displacement along ${\boldsymbol{\xi }}$ computed from the simulations is not a good measure of the actual mean coordinate difference, which is a factor of order 2 times smaller than predicted by the models. One possible reason for the disagreement between the model and actual measurements is that the Chandra plate scale calibration was derived from observations of NGC 2516 and LMC X-1 using source centroid measurements that were not corrected for the asymmetry of the PSF (Markevitch 2001b).

As described in Section 2.5.3, the celestial coordinates of a source observed by Chandra are computed by applying a series of transforms to the measured position of the source on the detector. The final step in this process applies the measured plate scale calibration to the difference between the position of the source on the virtual sky pixel plane and a known fiducial point, which is typically the telescope optical axis. If all of the star–star baselines used to calibrate the plate scale were oriented parallel to ${\boldsymbol{\xi }}$, then the lack of correction for the PSF asymmetry would to first order compensate for the linear component of the systematic position offset when measuring real sources whose locations are fixed in world coordinates rather than virtual pixel plane coordinates. Since not all star–star baselines were so aligned, some residual systematic position offset may be expected, but at an undetermined level that is less than predicted from the PSF simulations.

Systematic position offsets can also arise from uncertainties in the detector geometry. As an example, consider imaging observations that use the nominal ACIS-I aim point. Sources with θ ≲ 8' will be located on the same CCD array as the aim point. As θ increases, an increasing fractions of sources will instead be positioned on ACIS-S array CCDs, until for θ ≳ 11' all sources will be located on the ACIS-S array. Uncertainties in the relative positions and tilts of the ACIS-I and ACIS-S arrays would therefore introduce systematic position offsets for sources with large off-axis angles, while not impacting sources that fall on the same CCD array as the aim point. This signature is consistent with the absence of mean position differences measured for θ ≲ 8'.

Because of the small magnitude of the mean position differences measured for θ ≲ 15', we have chosen not to apply an uncertain correction to source positions in release 1 of the catalog. We plan to investigate in detail the cause of the systematic position offsets at large off-axis angles, and adjust source positions in future catalog releases if appropriate.

3.5.1. Source Position Uncertainty

In addition to reporting measured source positions, wavdetect also reports positional errors associated with each source detection. The reported errors are based on a statistical moments analysis, and do not consider instrumental effects such as pixelization, aspect-induced blur, or asymmetrical PSF structure that may contribute to the total positional uncertainties. Simulations that compare the reference positions of artificially generated sources with their positions determined by wavdetect indicate that the positional uncertainties computed by wavdetect are underestimated for sources with large off-axis angles. The simulation results, which quantify the dependence of positional uncertainties of simulated sources on off-axis angle, were found to be consistent with the more extensive simulations used to construct the Chandra Multiwavelength Project (ChaMP) X-ray point source catalog (Kim et al. 2007).

In the first release of the CSC, source position error ellipses are substituted by error circles computed using the ChaMP positional uncertainty relations

$$\begin{equation} \log P = \left\lbrace \begin{array}{@{}l} 0.1145\theta - 0.4957 \log S_w + 0.1932 \\ \ms \quad 0.0000 < \log S_w \le 2.1393\\ \ms 0.0968\theta - 0.2064 \log S_w - 0.4260 \\ \ms \quad 2.1393 < \log S_w \le 3.3000. \end{array} \right. \end{equation} \tag{ 4 }$$

In these equations, P is the positional uncertainty in arcseconds, θ is the off-axis angle in arcminutes, and S_w is the source net counts reported by wavdetect. These relations were derived to characterize the positional uncertainties at the 95% confidence level of X-ray point sources in the ChaMP X-ray point source catalog, which includes ∼6800 X-ray sources detected in 149 Chandra observations. The values of log P computed using Equations (4) are not equal at the boundary where log S_w= 2.1393 (roughly 138 net counts). However, this error is negligible for θ ≲ 10'.

Although HRC observations are not included in the first release of the CSC, we have used a series of simulations to derive an improved positional uncertainty relation that is appropriate for sources detected in HRC-I observations. The simulations include ∼6000 point sources spanning 0 < θ < 22' and 9 < S_w < 3600. The best-fit surface for the 95% position uncertainty quantile is

$$\begin{eqnarray} \log P &=& 0.752569 + 0.216985\theta + 0.000242\theta ^2\nonumber\\ && - 1.142476 \log S_w + 0.172132\log ^2 S_w \nonumber\\ &&- 0.040549\theta \log S_w. \end{eqnarray} \tag{ 5 }$$

The simulations do not sample the region with θ>20' and S_w < ∼50, and so we impose an upper bound of log P = 2.128393 on this relation, to cap the uncertainty in this regime.

In release 1.1 of the CSC, positional uncertainties for sources detected in ACIS observations are computed using Equation (4) while positional uncertainties for sources detected in HRC-I observations are computed using Equation (5).

The positional uncertainties from Equations (4) and (5) provide a good measure of the statistical uncertainty of the location of the source in the frame of the observation, but do not consider potential sources of error that are external to the observation. These include the error in the mean aspect solution for the observation, the astrometric errors in the AXAF (Chandra) Guide and Acquisition Star Catalog (Schmidt & Green 2003), and the calibration of the geometry of the spacecraft and focal plane. As described in Section 4.3, Rots (2009) has recently used the CSC/SDSS Cross-Match Catalog to calibrate the combined external error by analyzing the statistical distribution of the measured separations of CSC point source detections from individual observations with their counterparts in SDSS DR7 (Abazajian et al. 2009).

The resulting external astrometric error is 016 ± 00.01 (1 σ), or 039 (95% confidence). The latter must be added in quadrature to the position uncertainties from Equations (4) and (5) to compute the absolute position error for CSC sources. In release 1 of the catalog, the positional error reported in the catalog tables is taken directly from Equation (4), so the quadrature addition of the external astrometric error component must be performed by the user. Release 1.1 of the CSC will include this error component directly in the tabulated values.

A post facto histogram of the 95% confidence positional uncertainties, including the external astrometric error, for source detections included in release 1 of the CSC is presented in Figure 17. The figure demonstrates that statistical errors due to lack of net source counts do dominate the positional uncertainty for all but the brightest sources. However, for bright sources detected at small off-axis angles, the external astrometric error limits the accuracy of the derived source positions. Any error introduced by using uncorrected source centroid positions from wavdetect is negligible for the overwhelming majority of source detections.

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3.5.2. Combining Source Positions from Multiple Observations

Improved estimates of the position and positional uncertainty of each X-ray source are determined from the statistically independent source detections included in the set of individual observations using a multivariate optimal weighting formalism. This technique is effective in cases where simple averaging fails, for example, when the area defining the source position varies significantly from observation to observation. We express the uncertainties of the estimates in the form of error ellipses centered upon the estimated source positions. An equivalent approach has been used for weapons targeting (Orechovesky 1996). To our knowledge the usage here is the first documented application to astrophysical data.

In the multivariate optimal weighting formalism, given a set of estimates, X_i, of the mean of some two-dimensional quantity, and the 2 × 2 covariance matrices, σ²_i, associated with these estimates, an improved estimate, X, of the mean, and the associated covariance matrix, σ², are (e.g., Davis 2007b)

$$\begin{equation} X = \sigma ^2 \sum _i \frac{X_i}{\sigma _i^2} ; \sigma ^2 = \left[ \sum _i \frac{1}{\sigma _i^2} \right]^{-1}. \end{equation} \tag{ 6 }$$

For the application described here, we take X_i to be the ith estimate of the source position, projected onto a common tangent plane (which is constructed at the mean position of the ellipse centers). The corresponding covariance matrix is

$$\begin{equation} \sigma _i^2 = \left(\begin{array}{@{}cc@{}} \sigma ^{\prime 2}_{1,i}\cos ^2 \vartheta _i + \sigma ^{\prime 2}_{2,i} \sin ^2 \vartheta _i & \big(\sigma ^{\prime 2}_{2,i} - \sigma ^{\prime 2}_{1,i}\big) \cos \vartheta _i \sin \vartheta _i \\ \ms \big(\sigma ^{\prime 2}_{2,i} - \sigma ^{\prime 2}_{1,i}\big) \cos \vartheta _i \sin \vartheta _i & \sigma ^{\prime 2}_{1,i}\sin ^2 \vartheta _i + \sigma ^{\prime 2}_{2,i} \cos ^2 \vartheta _i \end{array} \right), \end{equation} \tag{ 7 }$$

where σ'_1,i and σ'_2,i are the lengths of the semi-minor and semi-major axes, respectively, of the ith error ellipse projected onto the common tangent plane, and ϑ_i is the angle that the major axis of the ith error ellipse makes with respect to the tangent plane y-axis. The derivation of Equation (7) is presented in Appendix B.

Once the covariance matrices corresponding to the error ellipses for each individual source observation, Equation (7), are computed, the error ellipses are combined using Equation (6). This yields the optimally weighted source position and position error ellipse for the combined set of observations, on the common tangent plane. Mapping these back to the celestial sphere provides the combined source position and error ellipse estimates.

The observed spatial extent of a source is estimated using a rotated elliptical Gaussian parameterization of the form

$$\begin{equation} S(x, y; \balpha) = \frac{s_0}{\sigma _1 \sigma _2} \exp [-\pi ({\newmathcal A}\mathbf {x})^2], \end{equation} \tag{ 8 }$$

where

$$\begin{eqnarray*} {\newmathcal A} = \left({\begin{array}{@{}cc@{}} \sigma _1^{-1} & 0 \\ \ms 0 & \sigma _2^{-1} \end{array}}\right) \left({\begin{array}{@{}cc@{}} \cos \phi _0 & \sin \phi _0 \\ \ms -\sin \phi _0 & \cos \phi _0 \end{array}}\right) &;& \mathbf {x} = \left({\begin{array}{@{}c@{}} x \\ \ms y\end{array}}\right), \end{eqnarray*}$$

where (x, y) is the Cartesian center location of the Gaussian, and the parameters $\balpha = (\sigma _1, \sigma _2, \phi _0)$ are the 1σ radii along the major and minor ellipse axes, and the position angle of the major axis of the ellipse, respectively.

The parameters of S are determined using a wavelet-based approach that is similar to that used for source detection. The methods differ in their details and assumptions, however.

For source detection, the choice of wavelet scales used by wavdetect to detect sources is determined a priori. Because of the strong variation of PSF size with off-axis angle, multiple wavelet scales and input image-blocking factors are required to search for sources at all off-axis angles, as described in Section 3.4. Stepping between the discrete wavelet scales and image-blocking factors as a function of off-axis angle introduces small but systematic biases in the derived dimensions of the source region ellipses (and therefore the source region apertures; see Figure 18).

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Photometric, spectral, and temporal properties determined from the X-ray events included in the aperture are not impacted by these effects, since the aperture dimensions are sufficiently large that they typically enclose ∼90% of the PSF counts, and a correction factor is applied for the fraction of the PSF that falls outside of the aperture. However, these biases render the source region aperture dimensions unsuitable for use as a measure of the source extent.

The wavelet-based approach used for estimating the source extent determines the optimal wavelet scale size directly from the data under the assumption that a source exists at approximately the location determined by wavdetect. The raw source extent estimates derived using this approach vary smoothly with off-axis angle (Figure 18).

The two-dimensional correlation integral

$$\begin{equation} C(x, y; {\boldsymbol{\alpha }}) = \int _{-X}^X\int _{-Y}^Y dx^\prime \,dy^\prime \, W(x-x^\prime, y-y^\prime ; {\boldsymbol{\alpha }})\,S(x^\prime, y^\prime ; {\boldsymbol{\alpha }}),\vspace*{3pt} \end{equation} \tag{ 9 }$$

where the region of interest is |x'| ⩽ X and |y'| ⩽ Y, and W is again specified by Equation (3), is computed first.

We choose a coordinate system in which the peak of S is centered at the origin. The quantity $\psi (x, y; {\boldsymbol{\alpha }}) = C(x, y; {\boldsymbol{\alpha }})/(a_1a_2)^{1/2}$ has a maximum value at the origin when $a_i=\sigma _i\sqrt{3}$ and ϕ = ϕ₀ (Damiani et al. 1997). The source parameters are determined by maximizing ψ.

In practice, Equation (9) is evaluated as a discrete sum over the pixels of the image. Although integration of Equation (3) does not yield a simple closed-form solution, and numerical integration is computationally expensive, for the purpose of optimizing $\psi _0 = C(0, 0; {\boldsymbol{\alpha }})/(a_1a_2)^{1/2}$, a rectangular approximation for the integral over each pixel is sufficient. In this approximation,

$$\begin{equation*} W_{mn}(x_i, y_j; {\boldsymbol{\alpha }}) \approx W(x_m - x_i, y_n - y_j; {\boldsymbol{\alpha }}) \Delta x \Delta y, \end{equation*}$$

where $W(x, y; {\boldsymbol{\alpha }})$ is evaluated at the center of each pixel and the pixel area is ΔxΔy.

A small sub-image of the source is extracted centered on the source position determined by wavdetect. The accuracy of this source position is refined by searching the center of the sub-image for the coordinates (x₀, y₀) that maximize ψ₀(x, y; a, a, 0). A new sub-image is then extracted using the improved source position.

Finally, the size and orientation of the elliptical Gaussian source parameterization are derived by maximizing $\psi (x_0, y_0; {\boldsymbol{\alpha }})$. Choosing good initial values for a_i helps to ensure that this optimization step converges reliably. We set the initial values a₁ = a₂ = max [(d₁d₂)^1/2, a_grid], where d_i are the ellipse semi-axes derived by wavdetect, and the value of a_grid is obtained by examining ψ₀(x₀, y₀; a, a, 0) on a grid of a values spanning the half-width of the source image; a_grid is usually the smallest a that corresponds to either a local maximum or an inflection point. This choice is motivated by the observation that when a single source is present, the location of a local maximum provides a good estimate of the source size (see Figure 19). Similarly, when the source of interest is blended with other nearby sources, an inflection point where ∂²_aψ₀ = 0 often occurs near the "edge" of the central source. When the first occurrence of ∂_aψ₀ = 0 occurs at a local minimum, a_grid is the smallest value of a on the pixel grid.

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3.6.1. PSF Extent

3.6.2. Intrinsic Source Extent

Using the rotated elliptical Gaussian parameterizations derived above, the observed source extent, S(x, y; σ₁, σ₂, ϕ₀), can be treated as the convolution of the local PSF, p(x, y; b₁, b₂, ψ) with the intrinsic source extent, s(x, y; a₁, a₂, ϕ), where we parameterize the latter similarly to Equation (8). In general, ϕ₀ ≠ ϕ, since the PSF-convolved ellipse need not have the same orientation as the intrinsic source ellipse.

In principle, one can determine the parameters, (a₁, a₂, ϕ), of the intrinsic source ellipse by solving a nonlinear system of equations involving the PSF parameters, (b₁, b₂, ψ), and the observed source parameters, (σ₁, σ₂, ϕ₀). However, because these equations are based on simple assumptions regarding the source and PSF profiles, 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

$$\begin{equation*} \sigma _1^2 + \sigma _2^2 = a_1^2 + a_2^2 + b_1^2 + b_2^2, \end{equation*}$$

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,

$$\begin{eqnarray} a_{\rm rss} &=& \frac{1}{\sqrt{2}}\big(a_1^2+a_2^2\big)^{1/2}\nonumber\\ &=& \frac{1}{\sqrt{2}}\max \big[0, \big(\sigma _1^2+\sigma _2^2\big) - \big(b_1^2+b_2^2\big)\big]^{1/2}, \end{eqnarray} \tag{ 10 }$$

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 one-dimensional Gaussians and for convolution of circular Gaussians in two dimensions. The factors of $1{/}\sqrt{2}$ ensure that the statistic value gives the radius of the source image when applied to circular source images.

Using Equation (10), one can derive an analytic expression for the uncertainty in a_rss in terms of the measurement errors associated with σ_i and b_i. Because σ_i and b_i are non-negative, evaluating the right-hand side of Equation (10) using the corresponding mean values should give a reasonable estimate of the mean value of a_rss. A Taylor series expansion of the right-hand side of Equation (10) evaluated at the mean parameter values therefore yields the uncertainty

$$\begin{equation} \Delta a_{\rm rss} = \frac{1}{\sqrt{2}a}\big[\sigma _1^2(\Delta \sigma _1)^2 + \sigma _2^2(\Delta \sigma _2)^2 + b_1^2(\Delta b_1)^2 + b_2^2(\Delta b_2)^2\big]^{1/2}\!\!, \end{equation} \tag{ 11 }$$

where (ΔX)² represents the variance in X, and where

$$\begin{equation*} a = \left\lbrace \begin{array}{@{}l l} \dsty a_{\rm rss} & \quad a_{\rm rss} > 0 \\ \ms \dsty \sqrt{b_1^2+b_2^2} & \quad a_{\rm rss} = 0. \end{array} \right. \end{equation*}$$

3.6.3. Combining Intrinsic Source Extent Estimates from Multiple Observations

Measurements of the mean intrinsic source extent derived from multiple independent observations, a_rss,i ± Δa_rss,i, are combined using the multivariate optimal weighting formalism, Equations (6). The minimum variance estimator of the intrinsic source size is the variance-weighted mean,

$$\begin{equation*} \overline{a_{\rm rss}} = {\rm Var}[\overline{a_{\rm rss}}]\sum _i{\rm Var}[a_{{\rm rss},i}]^{-1}a_{{\rm rss},i}, \end{equation*}$$

where Var[a_rss,i] = (Δa_rss,i)². The variance in $\overline{a_{\rm rss}}$ is

$$\begin{equation*} {\rm Var}[a_{\rm rss}] = \left[\sum _i{\rm Var}[a_{{\rm rss},i}]^{-1}\right]^{-1}. \end{equation*}$$

Net source counts, count rates, and photon and energy fluxes for point sources are computed from counts and exposure data accumulated in independent source and background apertures, R_s and R_b. Typically, these apertures are simple elliptical regions and surrounding elliptical annuli, but arbitrary areas from either aperture may be excluded to avoid contamination from nearby sources, or missing data due to detector edges. The net aperture areas, A_s and A_b, and the fractions, α and β, of source counts expected in both apertures are determined from

$$\begin{eqnarray*} A_s=\int _{R_s}dx dy &;& A_b =\int _{R_b}dx\,dy \end{eqnarray*}$$

and

$$\begin{eqnarray*} \alpha = \int _{R_s}dx\,dy\,{\it{PSF}}(x,y);& \beta = \int _{R_b}dx\,dy\,{\it{PSF}}(x,y). \end{eqnarray*}$$

We use the PSFs described in Section 3.6.1 to estimate α and β. Although the finite number of PSF counts leads to some uncertainty in the estimate for β, the effect of this uncertainty on the derived net counts, count rates, and fluxes is small (typically ≪1%).

If a uniform background over the scale of R_s and R_b is assumed, then the net source counts with aperture corrections applied, S, can be determined by solving the simultaneous set of linear equations

$$\begin{eqnarray} C = \alpha S + b;& B = \beta S + r b, \end{eqnarray} \tag{ 12 }$$

where C and B are the total counts in R_s and R_b, respectively, b represents the background in R_s, and r = A_b/A_s. The solution is

$$\begin{eqnarray*} &&S = (rC-B)/(r\alpha -\beta). \end{eqnarray*}$$

In general, selecting a background aperture so that β → 0 is difficult, since the inner radii of such annuli could range from ∼25'' to >1000'', depending on θ and energy. Such large background apertures would be subject to errors due to intrinsic background variations, diffuse source emission, and background contributions from multiple detector chips. We choose rather to use smaller apertures, containing ∼5%–10% of the source flux (see Figure 14), whose effects can be modeled more accurately.

To apply a consistent statistical approach in determining confidence bounds for all photometric quantities (see below), we assume that the generic photometric quantity S (whether counts, count rate, photon flux, or energy flux) can be converted to counts by multiplying by appropriate generic conversion factors f and g defined below, averaged over R_s, R_b, and generalize Equations (12) to include these terms,

$$\begin{eqnarray} C = f S + b;& B = g S + r b. \end{eqnarray} \tag{ 13 }$$

For example, if S represents a count rate, f = α〈T_s〉 and g = β〈T_b〉, where 〈T_s〉 and 〈T_b〉 represent average exposure times in R_s and R_b, respectively. The corresponding definitions for photon flux are f = α〈E_s〉 and g = β〈E_b〉, where 〈E_s〉 and 〈E_b〉 are the average exposure map values (in cm² s), computed at the monochromatic effective energy of the band, in R_s and R_b, respectively. For energy flux, f = α/〈F_s〉 and g = β/〈F_b〉, where 〈F_s〉 and 〈F_b〉 represent average fluxes (in erg cm^-2 s⁻¹) in R_s and R_b, respectively. The values 〈F_s〉 and 〈F_b〉 are determined by applying quantum efficiency and effective area corrections to individual event energies in R_s and R_b, and computing the averages of the resulting quantities.

Finally, we relax the assumption of uniform background over R_s and R_b by defining r = A_b〈T_b〉/A_s〈T_s〉 for source rate, r = A_b〈E_b〉/A_s〈E_s〉 for photon flux, and r = A_b〈F_s〉/A_s〈F_b〉 for energy flux. With these definitions, the general solution for S may be written as

$$\begin{eqnarray} &&S = (rC-B)/(rf-g). \end{eqnarray} \tag{ 14 }$$

To determine confidence bounds for S, the background marginalized posterior probability density is computed first, with the assumption that C and B are Poisson-distributed random variables whose means are θ = fS + b and ϕ = gS + rb, respectively. The posterior probability density for S may then be written as

$$\begin{equation*} p(S|CB) = \int _0^\infty db\,p(Sb|CB). \end{equation*}$$

To determine p(Sb|CB), we use Bayes' theorem to write the joint posterior probability density for p(θϕ|CB), taking advantage of the fact that R_s and R_b are independent:

$$\begin{equation*} p(\theta \phi |CB) = \frac{p(\theta)p(C|\theta)p(\phi)p(B|\phi)}{p(CB)}. \end{equation*}$$

The likelihoods are simple Poisson probabilities,

$$\begin{eqnarray*} p(C|\theta) = \frac{\theta ^Ce^{-\theta }}{\Gamma (C+1)} &;& p(B|\phi) = \frac{\phi ^Be^{-\phi }}{\Gamma (B+1)}, \end{eqnarray*}$$

and we use generalized γ-priors for p(θ) and p(ϕ):

$$\begin{eqnarray*} p(\theta) = \frac{\rho _S^{\pi _S}\theta ^{\pi _S-1}e^{-\rho _S\theta }}{\Gamma (\pi _S)} ;& p(\phi) = \frac{\rho _B^{\pi _B}\phi ^{\pi _B-1}e^{-\rho _B\phi }}{\Gamma (\pi _B)}, \end{eqnarray*}$$

where the parameters π_S, ρ_S, π_B, and ρ_B define the function shapes, and P(CB) is determined through normalization of p(θϕ|CB). Once p(θϕ|CB) is known, p(Sb|CB) may be found from simple substitution of variables,

$$\begin{eqnarray*} p(\theta \phi |CB)d\theta d\phi & = & p(\theta (S,b)\phi (S,b) |CB)\left|\frac{\partial (\theta,\phi)}{\partial (S,b)}\right|dS db\\ & = & p(Sb|CB)(rf-g)dSdb. \end{eqnarray*}$$

Details of the derivation may be found in V. Kashyap & F. A. Primini (2010, in preparation), but here we merely cite the final results under the additional assumption of non-informative priors π_S = π_B = 1 and ρ_S = ρ_B = 0:

$$\begin{eqnarray*} p(S|CB)dS &=&dS(rf-g)\sum _{k=0}^C\sum _{j=0}^B\frac{(fS)^ke^{-fS}}{\Gamma (k+1)}\frac{(gS)^je^{-gS }}{\Gamma (j+1)}\\ &&\times e^{(B-j)\ln (r)+\ln (\Gamma (C+B-k-j+1))} \nonumber \\ &&\times e^{-\ln (\Gamma (C-k+1))-\ln (\Gamma (B-j+1))-(C+B-k-j+1)\ln (1+r)}\nonumber. \end{eqnarray*}$$

Because of the computationally intensive nature of this expression, p(S|CB)dS is approximated with an equivalent Gaussian distribution when C + B>50 counts. The validity of this approximation is verified through simulations. Examples of p(S|CB) for three CSC sources are shown in Figure 20.

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By using the different definitions for f and g as described above, probability densities for net counts, rates, and photon and energy fluxes can then be computed.

Confidence bounds are determined by numerically integrating p(S|CB) in alternating steps above and below its mode until the desired confidence level is achieved. The values of S at these points then determine the confidence bounds. If the value of S = 0 is reached before summation is complete, or if the mode itself is 0, integration continues for S above the mode, and the resulting bound is considered an upper limit.

We note that in our approach, photon flux and energy flux are determined somewhat differently. While average exposure map values are used when computing photon flux, in the case of energy flux the values 〈F_s〉 and 〈F_b〉 are determined by applying quantum efficiency and effective area corrections to individual event energies. For sources with few counts in either the source or background region, a single photon detected at an uncharacteristically low- or high energy (where the Chandra effective area is small) can make a dominant contribution to the estimated energy flux. In such cases, the true uncertainty will be significantly larger than our estimated errors. A post facto comparison of energy flux estimates computed in this manner with energy fluxes calculated using an assumed canonical power-law spectral model (see Section 3.10, below), indicates that fewer than 1% of ACIS broad energy band fluxes are affected by this problem. A more detailed analysis of the statistical accuracy of the energy flux determinations is provided by F. A. Primini et al. (2010, in preparation).

3.7.1. Determining Flux Significance

Significances for all aperture photometry quantities are determined directly from the probability densities p(S|CB). Our goal is to provide a simple statistic that is robust to calculate, easily interpretable by non-expert users, and consistent with the classical S/N definition for high count sources. To this end, we compute the FWHM of p(S|CB), since the latter has a well-defined width even for low-significance sources in the catalog, as shown in Figure 20. If S = 0 is reached before the half-maximum point below the mode is found, the HWHM is computed from values above the mode and FWHM is set equal to 2 × HWHM. The FWHM is then used to compute the "equivalent σ" for a Gaussian probability density,

$$\begin{equation*} \sigma _e = \frac{\mbox{\rm FWHM}}{2\sqrt{2\ln 2}}. \end{equation*}$$

The flux significance value that is reported in the catalog for a source is defined to be S/N = S/σ_e, where S is determined from Equation (14). This value must be at least 3.0 in at least one energy band for an observation of a source to be included in the first release of the catalog.

The flux significance threshold that we use imposes a conservative limit on sources included in the CSC, which we deem necessary to reduce the number of spurious sources at low count levels to an acceptable value. Comparing our results to those of other large Chandra surveys whose source lists are derived from wavdetect, but whose detection procedures differ, is useful. In Figure 21, we compare the distribution of net counts for CSC sources detected in the ACIS broad (0.5–7.0 keV) energy band with distributions of similar quantities for four other Chandra catalogs derived from a range of ACIS exposures comparable to those in the CSC: AEGIS-X (Laird et al. 2009, 0.5–7.0 keV), the Galactic Center catalog (Muno et al. 2009, 0.5–8.0 keV), C-COSMOS (Elvis et al. 2009, 0.5–7.0 keV), and ChaMP (Kim et al. 2007, 0.5–8.0 keV). We note that while these other catalogs do include sources with fewer net counts than the CSC, the additions are in general not large, comprising ∼5%, ∼9%, ∼9%, and ∼24%, for AEGIS-X, the Galactic Center catalog, C-COSMOS, and ChaMP, respectively. We attribute the larger percentage in ChaMP to the restricted fields of view and the careful manual screening of source detections used when constructing that catalog. The CSC appears to fare worse in comparison to the XBootes survey (Kenter et al. 2005, 0.5–7.0 keV), most of whose sources have fewer than 10 net counts. However, the XBootes survey is composed of many 5 ks non-overlapping observations for which the very low ACIS background allows a lower count threshold. In contrast, the CSC is constructed from observations comprising a wide range of exposures, ∼70% of which are greater than 5 ks and ∼10% of which are greater than 50 ks. Finally, as mentioned in Section 3.4, ∼1/3 of all sources detected by wavdetect in the ACIS broad energy band fall below the flux significance threshold. However, we expect that a substantial fraction of these sources are spurious.

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3.7.2. Combining Aperture Photometry from Multiple Observations

Ideally, one should compute aperture photometry quantities for combined observations by computing a joint probability density, using p(S|CB) from one observation as the prior for the next. However, this approach is difficult to implement and computationally expensive, especially when probability densities from individual observations do not overlap significantly. We have therefore chosen simply to combine aperture data from various observations and compute a single p(S|CB) from those data. We compute

$$\begin{eqnarray*} \sum _i C_i = S \sum _i f_i + \sum _i b_i &;& \sum _i B_i = S \sum _i g_i + \sum _i r_i b_i. \end{eqnarray*}$$

To cast these in the same form as Equations (13), we define r' = ∑_ir_ib_i/∑_ib_i, where the b_i are determined from the solutions to Equations (13) for individual observations. One can then write

$$\begin{eqnarray*} \sum _i C_i = S \sum _i f_i + \sum _i b_i;& \sum _i B_i = S \sum _i g_i + r^\prime \sum _i b_i. \end{eqnarray*}$$

which are identical in form to Equations (13). The combined aperture photometry quantities and bounds can then be determined as described earlier.

For the purposes of the catalog, "limiting sensitivity" is defined to be the flux of a point source that meets but does not exceed the flux significance threshold for inclusion in the catalog. Limiting sensitivity is a function of source position, background, and the algorithm used to calculate flux and flux significance. At any point within the field of view of an observation, the limiting sensitivity can be used as a simple X-ray flux limit for individual sources detected at other wavelengths. For the catalog, a full-field sensitivity map is provided for each observation and energy band as a file-based data product (see Table 3). These data are also required to calculate sky coverage histograms (solid angle surveyed as a function of limiting flux), which are themselves needed to calculate luminosity functions and source surface brightness versus number density relationship.

As described above, the flux significance of a catalog source is defined to be the ratio of the source flux to the equivalent σ determined from the width of the flux's posterior probability density. There is no equivalent quantity for sensitivity, and for simplicity and ease of computation, we use a technique similar to that developed by Maccacaro et al. (1982) for the Einstein Observatory Medium Sensitivity Survey, namely, we approximate significance using the aperture photometry relations of section 3.7, under the assumption of Gaussian statistics, and use the model background maps, randomized to provide statistics appropriate to the observation in question, to determine aperture counts.

Recall from Equation (14) that the flux may be written as

$$\begin{equation} S = (rC-B)/(rf-g). \end{equation} \tag{ 15 }$$

Since C and B are independent random variables, the variance on S may be written as

$$\begin{eqnarray*} \sigma _S^2 &=& \frac{r^2\sigma _C^2 + \sigma _B^2}{(rf-g)^2} = \frac{r^2C + B}{(rf-g)^2}, \end{eqnarray*}$$

assuming Gaussian statistics. The significance, S/σ_S may then be written as

$$\begin{equation} S/\sigma _S = \frac{(rC-B)}{\sqrt{r^2C + B}}. \end{equation} \tag{ 16 }$$

The limiting sensitivity is found by determining the minimum number of counts C_min in the source aperture that yields the flux significance threshold S/N_min in Equation (16),

$$\begin{equation*} \mathit {{\rm S/N}}_{\rm min} = \frac{(rC_{\rm min}-B)}{\sqrt{r^2C_{\rm min} + B}}, \end{equation*}$$

whose solution is

$$\begin{equation*} rC_{\rm min} = B + \frac{r \mathit ({{\rm S/N}}_{\rm min})^2}{2} \left\lbrace 1+\sqrt{1+\frac{4B}{r \mathit ({{\rm S/N}}_{\rm min})^2}\left(1+\frac{1}{r}\right)}\right\rbrace, \end{equation*}$$

and the limiting sensitivity for that aperture is then given by Equation (15),

$$\begin{eqnarray} S_{\rm min} &=& (rC_{\rm min}-B)/(rf-g) \nonumber \\ &=& \frac{r \mathit ({{\rm S/N}}_{\rm min})^2}{2} \left\lbrace\! 1\,{+} \, \sqrt{1 \, {+}\, \frac{4B}{r \mathit ({{\rm S/N}}_{\rm min})^2}\left(1 \, {+} \, \frac{1}{r}\right)}\!\right\rbrace\! (rf \, {-} \, g)^{-1} \nonumber \\ &=& \frac{\mathit ({{\rm S/N}}_{\rm min})^2}{2f} \left\lbrace 1+\sqrt{1+\frac{4B}{r \mathit ({{\rm S/N}}_{\rm min})^2}\left(1+\frac{1}{r}\right)}\right\rbrace, \end{eqnarray} \tag{ 17 }$$

where we have approximated

$$\begin{equation*} (rf-g)^{-1} \approx (rf)^{-1}\left\lbrace 1+\frac{g}{rf}\right\rbrace \approx (rf)^{-1}. \end{equation*}$$

Since the limiting sensitivity maps are computed from background maps with no real sources, information about real source apertures is unavailable. Rather, for each element in the map, circular source and annular background apertures appropriate to the 90% ECF source aperture at that location are constructed, and used to determine B, r, and f for use in Equation (17).

The assumption of Gaussian statistics, and the subsequent simplification in the algorithm, is made of necessity, since limiting sensitivity must be computed not for each source but for each pixel in each of five energy band images. We have, however, verified the performance of the algorithm by comparing detected source fluxes with values of limiting sensitivity at the source locations, for thousands of catalog sources in all energy bands (F. A. Primini et al. 2010, in preparation), and find good agreement.

For observations of sources with at least 150 net counts in the energy band 0.5–7 keV obtained using the ACIS detector, we further characterize the intrinsic source properties by attempting to fit the observed counts spectrum with both an absorbed blackbody spectral model and an absorbed power-law spectral model. These two models represent basic spectral shapes of thermal and non-thermal X-ray emission.

The standard forward fitting method used in X-ray spectral analysis computes the predicted counts produced by the spectral model with the observed counts in the detector channel space, and iteratively refines the model parameters to improve the quality of the fit.

Instrumental response functions (Davis 2001a) define the mappings between physical (source) space and detector space. George et al. (2007) describe two of these calibration files, the detector RMF and the ARF. The former specifies the energy dispersion relation $R(E^\prime, \hat{p}^\prime ; E, \hat{p}, t)$ that defines the probability that a photon of actual energy E, location $\hat{p}$, and arrival time t will be observed with an apparent energy E' and location $\hat{p}^\prime$, while the instrumental effective area $A(\hat{p}^\prime ; E, \hat{p}, t)$ is recorded in the latter. The final dispersion relation is the photon spatial dispersion $P(\hat{p}^\prime ; E, \hat{p}, t)$ transfer function due to the instrumental PSF.

With these definitions, the model $M(E^\prime, \hat{p}^\prime, t)$ that describes the expected distribution of counts arriving at the detector is then

$$\begin{eqnarray} M(E^\prime, \hat{p}^\prime, t) &=& \int dE \, d\hat{p} \, R(E^\prime ; E, \hat{p}, t) \, P(\hat{p}^\prime ; E, \hat{p}, t) \nonumber\\ &&\times A(E, \hat{p}^\prime, t) \, S(E,\hat{p}, t), \end{eqnarray} \tag{ 18 }$$

where $S(E, \hat{p}, t)$ is the physical model that defines the physical energy spectrum, spatial morphology, and temporal variability of the source.

We follow standard practice by ignoring the dependency on photon arrival time, and instead consider only the total number of photons that arrived during the observation in the forward fitting process. The source position and shape are taken as known, and we assume that the source photons are collected from the detector area containing an entire source region of interest. The latter assumption is valid provided that sources are spatially separated on scales of order the size of the PSF or larger. In crowded fields, or for sources that have a complex diffuse structure, the contribution from the other sources are important. With the assumptions listed above, Equation (18) reduces to

$$\begin{equation*} M(E^\prime) = \int dE \, R(E^\prime ; E) \, A(E) \, S(E), \end{equation*}$$

where the source emitted spectrum S(E) depends on the source physics. The forward fitting procedure solves for the best-fit parameters for S(E), assuming a pre-defined fit statistic. Since spectral fitting is only performed for sources with a minimum of 150 net counts, a χ² fit statistic is used, but note that this assumes a Gaussian distribution for the source counts.

For all sources observed using the ACIS detector (i.e., not just those with at least 150 net counts in the broad energy band), the catalog processing pipelines extract the observed energy spectra of the photons included in the source and background regions of each detected source and store these in a standard format (PHA file; Arnaud & George 2009). An appropriate associated ARF and RMF are computed by weighting the instrumental responses based on the history of how the source and background regions move over the surface of the detector due to the spacecraft dither motion. The extracted spectra, and associated ARF and RMF are stored as file-based data products (see Table 3) and can be retrieved by the user for further analysis such as low-count spectral fitting or spectral stacking.

To fit the background subtracted data, each PHA spectrum is grouped to a minimum of 16 counts per channel bin, and the source model parameters are varied to minimize the χ² statistic (assuming data variance, σ²_i = N_i,S + (A_S/A_B)²N_i,B). Two models are applied to the data in order to evaluate source properties: (1) an absorbed blackbody model $f(E) = \exp ^{-N_{\rm H} \sigma _E}\, A (E^2/(\exp ^{E/kT}-1))$; and (2) an absorbed power-law model $f(E) = \exp ^{-N_{\rm H}\sigma _E}\, A\, E^{-\Gamma }$. In these models, N_H is the equivalent hydrogen column density, σ_E is the photo-electric cross-section based on Balucinska-Church & McCammon (1992) and metal abundances from Anders & Grevesse (1989), A is the model normalization at E = 1 keV, kT is the blackbody temperature, and Γ is the power-law photon index. Forward fitting is performed using the Sherpa fitting engine (Freeman et al. 2001; Doe et al. 2007). Sherpa finds the best-fit model parameters and calculates two-sided confidence intervals for each significant parameter. The model flux for the best-fit parameters over the energy range 0.5–7 keV is also calculated.

The 68%(1σ) confidence limits for each parameter are calculated using the "projection" method in Sherpa. This method finds the two-sided confidence bounds independently for each parameter. The algorithm assumes that the current model has been fitted, and that all of the parameters are at the values corresponding to a best fit which is at the minimum of the fit statistic (χ²_min). For each parameter of interest, the search for the lower or upper bound starts at the best-fit value, which is then varied along the parameter axis. At each new value, the parameter of interest is frozen and a new best-fit model is determined by minimizing χ² over the remaining thawed parameters. The new χ² statistic, χ²_new, is determined and the difference between the new and the minimum statistics, Δχ² = χ²_new − χ²_min, is calculated. A change in Δχ² equal to 1 corresponds to a 68% confidence bound (Avni 1976), so the parameter of interest is varied until this value of Δχ² is obtained.

We note here that energy-dependent aperture corrections are not applied when performing the spectral model fits. Since the Chandra PSF is somewhat more extended at higher energies, the lack of correction has the effect of slightly softening the calculated spectral slope. The correction in Γ is approximately 0.03–0.05 for power-law spectra with a wide range of spectral indices. For sources included in release 1 of the CSC for which spectral fits have been performed, the error in Γ introduced by not applying energy-dependent aperture corrections is roughly 6 times smaller than the median computed 1σ confidence limits. About 2.5% of sources with spectral fits have computed confidence limits ≲0.05, and in these cases appropriate caution should be exercised when using the spectral fit properties.

For most source properties, values recorded in the Master Sources Table are computed by combining the relevant data from the set of observations in which the source is detected. However, for simplicity, the spectral model fit parameter values recorded in the Master Sources Table are taken directly from the single observation of the source that has the highest significance, Equation (16). In this case, data from multiple observations are not combined to compute the Master Source Table spectral fit properties.

Spectral model fits are not performed for sources with <150 net counts. However, for all sources we estimate energy fluxes using canonical absorbed power-law and blackbody spectral models.

For a canonical source model S(E) whose integral over the energy band is S', a corresponding band count rate, C' in counts s⁻¹, can be computed from the effective area calibration, A(E), and the RMF, R(E', E). The count rate is ∫dE R(E'; E) A(E) S(E), where the integral is performed over the energy band. For HRC observations, a diagonal RMF is assumed. The actual flux of a source can be estimated from S' by scaling the latter by the ratio of the measured and modeled source aperture count rates in the energy band. Since the only free parameter in this case is the normalization of the model, the calculation can be performed for sources with too few counts for a reliable spectral fit.

The canonical power-law spectral model has a fixed photon index Γ = 1.7, which falls in the range of values (Γ ∼ 1.5–2.5) that are typical of active galactic nucleus (AGN) spectra (Ishibashi & Courvoisier 2010). The value chosen matches the photon index used to convert count rates to energy fluxes in the second XMM-Newton serendipitous source catalog (2XMM; Watson et al. 2008), to simplify comparison of CSC and 2XMM source fluxes. Since we anticipate that the majority of sources with spectra that are best fit by a power-law model are AGN, we fix the total neutral hydrogen absorbing column N_H equal to the Galactic column, N_H(Gal), under the assumption that this represents a lower limit to the true column density.

The canonical blackbody spectral model has a fixed temperature kT = 1.0 keV and total neutral hydrogen column density N_H = 3 × 10²⁰ cm⁻². The latter value matches the median column density identified by Saxton (2003), and also corresponds to the typical column density found within 1 kpc of the Sun (Liszt 1983). As discussed by McCollough (2010), the choice of blackbody temperature is a compromise between the possible ranges of values for different classes of thermal X-ray emitters. Sources for which a thermal model best represents the data will likely lie in our galaxy, and so in this case setting N_H = N_H(Gal) would overestimate the total neutral hydrogen absorbing column.

Similar to spectral model fits, master source spectral model energy fluxes are taken directly from the single observation of the source that has the highest significance.

While the spectral model fits described in Section 3.9 provide detailed information about a source's spectral properties, only about 10% of the source observations included in the CSC have sufficient net counts to perform the fitting process. As an aid to characterizing the spectral properties of the remaining catalog sources, hardness ratios are computed between the hard, medium, and soft energy bands for all sources observed with the ACIS detector.

The spectral hardness ratio for the pair of energy bands x and y is defined as

$$\begin{equation} {\newmathcal{HR}}_{xy} = \frac{F_x - F_y}{F_b}, \end{equation} \tag{ 19 }$$

where F_x and F_y are the photon fluxes measured in the energy bands x and y respectively (x is always the higher-energy band of the pair), and F_b is the photon flux in the ACIS broad energy band, F_b = F_h + F_m + F_s.

A catalog source may be readily detected in one or more energy bands, but remain undetected or include very few total counts in other bands. Since hardness ratios are cross-band measures, a technique that applies rigorous statistical methods in the Poisson regime is required to compute these values and their associated confidence limits robustly. The hardness ratios included in the CSC are computed using a Bayesian approach developed by Park et al. (2006), which should be consulted for a detailed description of the algorithm.

To ensure that the Poisson errors are propagated correctly, the conversion between counts and photon flux for each energy band is modeled as a linear process, with a scale factor that is determined from the effective area of the telescope/instrument combination computed at the monochromatic effective energy of the band. This implies that the photon fluxes in Equation (19) may not match exactly the aperture photometry fluxes derived in Section 3.7.

Specifically, we model the observed total and background counts, C_x and B_x, in the hard, medium, and soft ACIS energy bands as

$$\begin{eqnarray*} C_x &\sim & {\rm Poisson}[e_x(\lambda _x+\xi _x)], \\ B_x &\sim & {\rm Poisson}[r e_x\xi _x], \end{eqnarray*}$$

where x represents the energy band (one of h, m, or s); λ_x and ξ_x are the expected source and background counts intensities, respectively; e_x is the band's conversion factor that scales counts to photon fluxes; and r is the ratio of the background aperture area to the source aperture area.

With these definitions, the spectral hardness ratio for the pair of energy bands x and y is determined by computing the expectation value

$$\begin{equation*} {\newmathcal{HR}}_{xy} = \frac{\lambda _x-\lambda _y}{\lambda _h+\lambda _m+\lambda _s}. \end{equation*}$$

Following the lead of Park et al. (2006), the joint posterior probability distribution can be written as

$$\begin{eqnarray*} &&p(\lambda _s, \lambda _m, \lambda _h|C_s, C_m, C_h, B_s, B_m, B_h) \nonumber\\ &&\qquad = p(\lambda _s|C_s, B_s) p(\lambda _m|C_m, B_m) p(\lambda _h|C_h, B_h), \end{eqnarray*}$$

where we have made use of the fact that the λ_x are independent. Marginalizing over nuisance variables yields the posterior distribution for the hardness ratios (equivalent to Equation (14) of Park et al. 2006):

$$\begin{eqnarray*} &&p({\newmathcal{HR}}_{xy}|C_s, C_m, C_h, B_s, B_m, B_h)\, d {\newmathcal{HR}}_{xy} = d {\newmathcal {HR}}_{xy}\\ &&\quad\times \int _{\psi, \omega }\!\! d\psi \,d\omega \left(\frac{2}{\omega }\right)p({\newmathcal{HR}}_{xy}, \psi, \omega |C_s, C_m, C_h, B_s, B_m, B_h), \end{eqnarray*}$$

where ψ = λ_x + λ_y and ω = λ_s + λ_m + λ_h.

The spectral hardness ratios that are included in the CSC are determined separately for each observation in which a source is detected, and also from the ensemble of all observations of the source. The former quantities are recorded in the Source Observations Table, while the latter are recorded in the Master Sources Table. The prior probability distributions used to derive the Bayesian posterior probabilities are computed differently in these two cases.

For a single observation, non-informative conjugate γ-prior distributions (van Dyk et al. 2001) are used for the source and background intensities. These distributions ensure that the posterior probabilities conjugate to the expected Poisson distributions of counts with no other prior information. When multiple observations are combined, the ensemble hardness ratios are computed by stepping through all of the observations of a source in order of increasing net broad band source counts. The posterior probability distribution computed from each observation is used as the prior probability distribution for the subsequent step. If the propagated prior probability distribution is not consistent with the observed counts in any step, then a conjugate γ-prior is used instead, and a catalog flag is set to indicate that the source spectrum is variable.

The CSC includes estimates of the probability that the flux from a source is temporally variable both within a single observation and between two or more observations in which the source was detected. These estimates are distinguished not only by the fact that they measure variability on different timescales, but also because their definitions differ fundamentally. Within an observation, we measure the probability that the source flux is not consistent with a constant level during a (largely) continuous observation, which is equivalent to estimating the probability that the source is variable, and is a positive statement with respect to variability. The inter-observation variability estimates measure the probability that the average flux levels during the different observations are consistent with a uniform source intensity. This provides only a lower limit to the probability of the source being variable, since we have no information about the source's behavior during the gaps between the observations, which are often widely separated.

The intent of the various variability measures included in the CSC is to provide users a means to easily select potentially variable sources. The individual source light curves or event lists should be assessed to reveal the true nature of the source's temporal characteristics. Moderately intense background flares that are not rejected as part of the enhanced background event screening (see Section 3.2) may cause sources to be incorrectly identified as variable. This possibility can be evaluated by comparing the structure of the source and background light curves.

3.12.1. Intra-observation Variability

The probability that a source is variable is estimated separately in each energy band using the Gregory–Loredo and Kolmogorov–Smirnov (K-S) algorithms, and Kuiper's variation on the latter. A brief description of each of the three algorithms is provided below. Gregory–Loredo probabilities are used to construct intra-observation variability indices that provide a shorthand measure of variability.

All three algorithms directly use the photon event arrival times to compute the variability probabilities, and apply corrections for variations of the geometric areas of the source and background region apertures due to the spacecraft dither-induced motion during the observation. The latter corrections are necessary since a source region that is moving across the edge of the detector or over a bad detector region might otherwise be erroneously classified as variable. Optimal resolution light curves are generated as by-products of the Gregory–Loredo test, and their power spectra are evaluated for the presence of the fundamental spacecraft pitch and yaw dither frequencies or associated beat frequencies. If there is a peak in the power spectrum at one of these frequencies that is at least 5× the rms value, then a catalog warning flag is set for the source observation to indicate that the intra-observation variability properties are unreliable.

The K-S test (Massey 1951) is a familiar and well-established robust test for comparing two distributions that are a function of a single variable. In the simplest case, we compare the cumulative sum of photon events, as a function of time, against a linearly increasing function that represents a constant flux. This null-hypothesis function is modified as necessary to account for data gaps and variations in effective area.

For an observation with N events, let S_N(t) be the cumulative sum of detected events as a function of time t, and P(t) the cumulative function that represents a constant flux. The K-S statistic D_N is defined as

$$\begin{equation*} D_N = \sup _t{|S_N(t) - P(t)|}. \end{equation*}$$

The K-S derived probability that the two distributions S_N(t) and P(t) do not belong to the same population, and therefore that the source is variable, is given by

$$\begin{equation} p_{\rm var} = Q_{\rm KS}(\sqrt{N} D_N) \end{equation} \tag{ 20 }$$

where

$$\begin{equation*} Q_{\rm KS}(\lambda) = 2 \sum _{j=1}^\infty {(-1)^{j-1}\, e^{-2j^2\lambda ^2}}. \end{equation*}$$

Equation (20) is strictly valid only in the asymptotic limit as N → ∞. In practice N ≳ 20 is "large enough," especially if conservative significance levels ≲0.01 are required (e.g., Press et al. 1986).

Kuiper (1962) proposed a variation on the K-S test that involves replacing the expression for D_N by the difference between the largest positive and negative deviations,

$$\begin{equation*} D_N = \sup _t[S_N(t) - P(t)] - \inf _t[S_N(t) - P(t)]. \end{equation*}$$

Folding this expression into Equation (20) yields the Kuiper derived probability that the source is variable. While the K-S test is primarily sensitive to differences between the median values of the cumulative distribution functions, the Kuiper test statistic is as sensitive to differences in the tails of the distributions. In many cases, this makes the Kuiper test a more robust variation of the traditional K-S test for evaluating the probability that a source is variable.

The Gregory–Loredo test (Gregory & Loredo 1992) is based on a Bayesian approach to detecting variability. The method works very well on photon event data and is capable of dealing with data gaps. We have incorporated the capability to include temporal variations in effective area. Although the algorithm was developed for detecting periodic signals, it is a perfectly suitable method for detecting random variability by forcing the period to equal the length of the observation.

Briefly, the Gregory–Loredo algorithm bins the N observed photon events into a series of histograms containing m bins, where m runs from 2 to m_max. If the observed distribution of events across the m histogram bins is n₁, n₂, ..., n_m, then the probability that this distribution came about by chance can be determined from the ratio of the multiplicity of the distribution, N!/(n₁! · n₂! ⋅ ⋅ ⋅ n_m!), to the total number, m^N, of possible distributions. The inverse of this ratio is a measure of the significance of the distribution. Following Gregory & Loredo (1992), we calculate an odds ratio O_m for m bins versus a flat light curve as

$$\begin{equation} O_m = T\; \frac{N!\; (m-1)!}{ (N+m-1)!}\ \frac{S_m\; m^N}{W_m}, \end{equation} \tag{ 21 }$$

where we have rewritten the multiplicity of the distribution, W_m, as

$$\begin{equation*} W_m = \frac{N!}{\prod _{j=1}^{m}{n_j!}}. \end{equation*}$$

Data gaps are accounted for through the binning factor, S_m, which is (Appendix B of Gregory & Loredo 1992)

$$\begin{equation*} S_m = \prod _{j=1}^{m}{s_j}^{-n_j}, \end{equation*}$$

where

$$\begin{equation*} s_j = \frac{t_j}{T/m}, \end{equation*}$$

t_j is the amount of good exposure time in bin j, and T is the total good exposure time for the observation. The odds are summed over all values of m ⩾ 2 to determine the odds that the source is time variable. m_max is chosen for each case in such a way that the odds ratios corresponding to higher values of m contribute negligibly to the total. The probability, p_m, of a particular binning, m, is simply

$$\begin{equation*} p_m = O_m / \sum _{j=1}^{m_{\rm max}} O_j. \end{equation*}$$

Summing over bins m ⩾ 2 corresponding to a non-constant source flux yields the Gregory–Loredo variability probability

$$\begin{eqnarray} p_{\rm GL} &=& \sum _{j=2}^{m_{\rm max}} p_j\\ &=& \frac{O}{1+O}, \end{eqnarray} \tag{ 22 }$$

where $O = \sum _{j=2}^{m_{\rm max}} O_j$, and we have made use of the fact that O₁ = 1.

The Gregory–Loredo algorithm bins the events into a series of light curves of varying resolution, corresponding to the number of bins, m, in the range 2 to m_max. Using the definitions above, the bins that comprise the normalized light curve, h_m, associated with a specific value of m are

$$\begin{equation*} h_{j,m} = \frac{n_j}{s_j N}, \end{equation*}$$

and the corresponding standard deviations derived from the posterior distribution are

$$\begin{equation*} \sigma _{j,m} = \frac{1}{s_j} \sqrt{\frac{s_j h_{j,m} (1 - s_j h_{j,m})}{N+m+1}}. \end{equation*}$$

As described by Gregory & Loredo (1992), an optimal resolution, light curve, h, can be obtained by combining the individual light curves, h_m, weighted by the probabilities, p_m:

$$\begin{equation*} h = (1 - p_{\rm GL})h_1 + \sum _{j=2}^{m_{\rm max}}p_mh_m. \end{equation*}$$

The optimal resolution light curve computed from the events included in the source region aperture for each source is recorded as a file-based data product (see Table 3) that is accessible through the catalog. As well as the light curve, h, this data product includes the uncertainty, σ, and upper and lower confidence intervals, h − 3 σ and h + 3 σ, respectively. To allow the users to verify the significance of features that may be present in the light curve, the file also includes the corresponding quantities derived from the events extracted from the background region aperture, using the same binning.

Careful judgment should be applied when assessing the reliability of source variability indicators using the source and background light curves. Since the background region aperture may contain up to ∼10% of the source flux (see Section 3.4.1), the background and source light curves may appear similar for very bright sources. The PSF wings of unrelated but nearby strongly variable sources may contaminate both the source and background region apertures of the source being investigated. An observation may have experienced background variations intense enough to be noticeable when compared to the target source's count rate, but not strong enough to have been removed by background screening during observation recalibration. In the first case the source is truly variable, but this is not necessarily so in the latter two examples. A helpful, though not definitive, test is to scale the amplitude of the flux variations in the source and background region apertures by their respective areas (recorded in the FITS keyword APERTURE). If the variations of the two scaled amplitudes are similar, then there is a good chance that a background problem is responsible. If the source region scaled amplitude is considerably larger than the background region scaled amplitude and the source is strong, then one is likely to have a truly variable source.

The Gregory–Loredo test appears to provide a more uniform and reliable measure of variability than either the K-S or Kuiper tests, although the Gregory–Loredo algorithm is more "conservative" than the other tests. In cases where the K-S and/or Kuiper tests detect variability, but the Gregory–Loredo test does not, close inspection of the light curve often, but not always, demonstrated that the level of variability does not exceed the 3σ bounds on the light curve. In cases where there are considerable data gaps, Gregory–Loredo is not always be able to detect variability on timescales comparable to those gaps.

To provide the user with a short-hand measure of variability that allows selection of sources on different degrees of confidence, the CSC includes a set of integer "variability index" values in the range [0, 10]. These indices are based on a combination of the Gregory–Loredo probability, p_GL, the logarithm of the odds ratio, O, and a secondary criterion that addresses the overall deviation of the light curve from the mean value. The latter criterion is based on the parameters f₃ and f₅, which are the fractions of the light curve that falls within 3σ and 5σ of the average rate, respectively. Table 6 defines the mapping of the test parameters to variability index values.

Table 6. Intra-observation Variability Indices

Variability	Conditiona	Meaning
Index
0	pGL ⩽ 0.5	Definitely not variable
1	0.5 < pGL < 0.667 AND f3>0.997 AND f5 = 1.0	Not considered variable
2	0.667 ⩽ pGL < 0.9 AND f3>0.997 AND f5 = 1.0	Probably not variable
3	0.5 ⩽ pGL < 0.6 AND (f3 ⩽ 0.997 OR f5 < 1.0)	May be variable
4	0.6 ⩽ pGL < 0.667 AND (f3 ⩽ 0.997 OR f5 < 1.0)	Likely to be variable
5	0.667 ⩽ pGL < 0.9 AND (f3 ⩽ 0.997 OR f5 < 1.0)	Considered variable
6	0.9 ⩽ pGL AND O < 2.0	Definitely variable
7	2.0 ⩽ O < 4.0	Definitely variable
8	4.0 ⩽ O < 10.0	Definitely variable
9	10.0 ⩽ O < 30.0	Definitely variable
10	30.0 ⩽ O	Definitely variable

Note. ^ap_GL is the Gregory–Loredo variability probability, Equation (22); f₃ and f₅ are the fractions of the light curve that fall within 3σ and 5σ of the average rate, respectively; and $O = \sum _{j=2}^{m_{\rm max}} O_j$ is the sum of the odds-ratios, Equation (21), for two or more bins.

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3.12.2. Inter-observation Variability

Inter-observation variability is based on comparison of source region aperture photon fluxes, and their confidence intervals, from multiple observations in which the source is detected. The catalog provides a probability that the data are not consistent with a constant-flux source, as well as an inter-observation variability index that is similar to the index defined for intra-observation variability. These measures of variability are assessed for each spectral energy band independently, and consequently no cross-instrument comparison is performed. In the first release of the CSC, observations that cover the same region of the sky, but in which the source is not detected, are not considered when computing inter-observation variability. These observations should enter into the variability assessment as flux upper limits, since they could conceivably be inconsistent with a constant source flux. A future release of the catalog will address this limitation.

As mentioned above, the inter-observation variability probability must be interpreted differently from the intra-observation variability probability. Whereas the light curve can be used to declare a source to be variable or non-variable within the time range of a single observation, one can never conclude that a source does not vary between multiple observations. If inter-observation variability is detected, then the source is definitely variable; however the converse is not true.

The inter-observation variability probability is simply based on the reduced χ² of the distribution of the source region aperture photon fluxes of the individual observations and their confidence intervals. For a source detected in n separate observations, we first use the source region aperture photon flux, S_i, and the associated lower and upper 1σ confidence limits, S⁻_i and S⁺_i, respectively, to compute an initial estimate of the variance-weighted mean source region aperture photon flux

$$\begin{equation*} S_0 = \sum _{i=1}^n\frac{S_i}{\sigma _{0,i}^2}\bigg/\sum _{i=1}^n\frac{1}{\sigma _{0,i}^2}, \end{equation*}$$

where we take σ_0,i = (S⁺_i − S⁻_i)/2. Using this estimate of the mean flux, we define the "effective σ" for the ith observation of the source as

$$\begin{equation*} \sigma _i = \left\lbrace \begin{array}{@{}l l} S_i - S_i^- & \quad S_i > S_0 \\ \ms S_i^+ - S_i & \quad S_i < S_0 \\ \ms (S_i^+ - S_i^-) / 2 & \quad S_i = S_0. \end{array} \right. \end{equation*}$$

A refined estimate of the variance-weighted mean flux is then given by

$$\begin{equation*} S = \sum _{i=1}^n\frac{S_i}{\sigma _i^2}\bigg/\sum _{i=1}^n\frac{1}{\sigma _i^2}. \end{equation*}$$

and the reduced χ² is

$$\begin{equation*} \chi ^2 = \frac{1}{n-1}\sum _{i=1}^n\frac{(S_i-S)^2}{\sigma _i^2}. \end{equation*}$$

The inter-observation variability index is assigned on the basis of the reduced χ², according to Table 7. Note that the values 1, 2, 9, and 10 are not used.

Table 7. Inter-observation Variability Indices

Variability	Reduced χ2
Index	2 Observations		>2 Observations
0		<0.4		<0.8
3	⩾0.4	<0.7	⩾0.8	<1.0
4	⩾0.7	<1.0	⩾1.0	<1.15
5	⩾1.0	<2.7	⩾1.15	<2.1
6	⩾2.7	<7.0	⩾2.1	<3.8
7	⩾7.0	<12.0	⩾3.8	<5.5
8	⩾12.0		⩾5.5

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Each entry in both the Master Sources Table and the Source Observations Table includes several source-specific flags and codes that identify specific circumstances that may be of relevance to the catalog user. Some flags and codes are used to encode source properties that are commonly searched for by users, as an aid to simplify catalog queries. However, in most cases flags and codes are intended to warn the user of conditions that may degrade the quality of measured source properties, or that may limit the usefulness of the source detection for some investigations.

The codes and flags included in the Source Observations Table are defined in Table 1. Flags are Boolean quantities that describe "yes/no" or "true/false" properties, whereas codes are multi-bit data values that encode several levels of information. Translations of the bit-encodings can be found in Table 8.

The extent and variability codes require additional explanation. The former encodes a conservative estimate of whether the intrinsic extent of a source, a_rss (Equation (10)), is inconsistent with the extent of the local PSF in each energy band. Specifically, a source is considered extended in an energy band if a_rss>5 Δa_rss in that energy band, where Δa_rss is the uncertainty in a_rss, given by Equation (11). The variability code bit corresponding to a specific energy band is set if the intra-observation variability index (Table 6) ⩾3. A zero code therefore implies that the source is either definitely not variable, not considered variable, or probably not variable, depending on the value of the variability index. Similarly, a non-zero code implies that the source either may be variable, is likely to be variable, is considered variable, or is definitely variable.

The remaining codes and flags all warn of conditions that may affect derived source properties to some extent. The streak source flag, if set, indicates that the source detection is located on an ACIS readout streak. If the readout streak is associated with a bright source, then there is a significant probability that the source properties may be compromised. This is particularly true of aperture photometry values. If the streak is especially intense, then the source detection may not be real. If the saturated source flag is set, then the source is definitely real, but is so bright that photon pile-up has eroded the core of the source image so that a single source has a "cratered" appearance (Figure 22). All source properties are compromised.

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The Master Sources Table includes only flags, and these are defined in Table 2. In most cases, Master Sources Table flags summarize the corresponding Source Observations Table flags and codes for all of the source detections that have "unique" linkages to the master source. The confusion flag is an exception to this, in that it is set for a master source if the confusion codes for any of the associated source detections indicate that multiple sources are present in the source region or that the source region overlaps another source region, or if there are any source detections that have "ambiguous" linkages to the current master source.

The master extent and variability flags are set if the corresponding codes for any uniquely matched source detections indicate that the source is extended or variable (as appropriate) in any energy band. The remaining master source flags are set only if the corresponding Source Observations Table flags are set for all uniquely associated source detections, indicating that the corresponding warning criteria are violated in all observations of the source.

The scientific integrity of the CSC is guaranteed through a set of quality assurance steps that are performed as part of the catalog construction process (Evans et al. 2008). Many of these analyses are executed automatically at the completion of each stage of catalog pipeline processing, so that any issues can be identified and corrected before they can affect downstream processing. These mechanisms detect pipeline processing errors, and identify potential data quality issues by comparing key diagnostic output products with predefined standards. Each standard that is violated will either trigger a human review to determine how to proceed, or will initiate one or more automated actions. The latter typically result in termination of the processing thread for a subset of the input data.

The vast majority of violations that occur because of data quality issues address the reality of detected sources, and are typically resolved without human intervention. Following the source detection step, detected source regions that are either significantly smaller than the dimensions of the local PSF or significantly larger than the maximum expected source size, or which exceed a maximum ellipticity threshold, are deemed to be artifacts, and the processing thread for the source region is terminated immediately to avoid evaluating source properties unnecessarily. Sources that have too few counts, or that have a detection significance that is too low to pass the catalog S/N threshold are similarly discarded.

The cores of sources observed with ACIS that are sufficiently bright can be eroded by photon pile-up. The source detection algorithm incorrectly detects bright spots on the ring surrounding the dark center of the image as distinct sources. Saturated sources are identified using a sliding matched filter algorithm. The source detections are manually adjusted so that a single source centered on the crater is included in the Source Observations Table for the source, and the source properties are flagged as having been manually modified, so that the user can exclude such sources if they so wish. The tabulated source position errors are unreliable for sources whose regions have been manually modified.

A more detailed comparison of the source dimensions with the local PSF is performed after the source properties are computed, to identify sources that are statistically smaller than the PSF in all energy bands. The fraction of the local PSF that is included within the modified source region aperture (as defined in Section 3.4.1) must be sufficiently large that the source location and aperture photometry can be computed from the fraction of the aperture that is not contaminated by overlapping sources. Finally, the S/N of the source is evaluated and compared with the minimum required for inclusion in the catalog.

Human review is primarily required to address the occurrence of unexpected pipeline warnings or errors. Although an automated process performs the laborious task of scanning the log files associated with each processing pipeline to identify problems, the wide diversity of possible error conditions require human intelligence to assess the reason for the failure and determine how to proceed. The typical response is to terminate the current processing thread, perform any needed repairs, and initiate reprocessing of the thread.

Other conditions that trigger a human review are precautionary in nature, and include cases where the local spatial density of detected sources is too high, or the total number of sources detected in the field of view exceeds a predefined threshold. Although these conditions most likely arise because of field crowding, they could indicate an error in the source detection process. Errors that generate a large number of sources would require substantial cleanup if processing was allowed to continue incorrectly.

To estimate false source rates, a series of blank-sky simulations with exposure times of ∼10, 30, 60, and 120 ks were constructed for typical ACIS imaging CCD configurations. For each simulation, a template background event list for each active CCD was used to define the overall spatial variation of the background, and the total number of background events was determined from the nominal field background rates (Chandra X-ray Center 2009) and the simulated exposure time. For all CCDs except chip S4 (ACIS-8) the template background event lists recorded in the instrumental calibration database were used. Chip S4 is significantly affected by a variable pattern of linear streaks that appear to be caused by a flaw in the serial readout which randomly deposits significant amounts of charge along pixel rows as they are read out (Houck 2000). Because of this issue, no adequate template is available for chip S4, and so one was constructed by combining several CSC event lists that do not include bright sources on that CCD. Each simulated blank-sky event list was then processed through the CSC pipeline source detection steps. The false source rates derived from these simulations are reported in Table 9. From these data we derive a simple linear relation for the number of false sources per field as a function of livetime, namely

$$\begin{equation*} \log (R_{\rm fs})= -3.345 + 1.6\times \log (t_{\rm live}), \end{equation*}$$

where R_fs is the false source rate, and t_live is the exposure livetime in units of kiloseconds. Using this relation, we estimate that ∼370 sources (∼0.4%) included in the catalog are spurious.

As can be seen from the table, the false source rate is appreciable only for exposures longer than ∼50 ks. There is some evidence for a clustering of false source detections near chip edges and at the boundaries between the back- and front-illuminated CCDs. This should not be surprising since the low spatial frequency background is poorly constrained or changing rapidly in these locations. To investigate these effects further, the false source rates near the chip edges and interfaces were examined separately for the longest simulated exposures. Figure 23 demonstrates that false source rates are enhanced in these regions for the 125 ks simulation.

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Source detection efficiency is characterized using point source simulations. A spatially random distribution of point sources is added to the blank sky simulations described above using the MARX simulator (Wise et al. 2003) to generate the incident X-ray photons. Separate simulations were generated for non-thermal sources with a power-law spectral distribution F_E ∝ E^−1.7, and for thermal blackbody sources with temperature kT = 1.0 keV, spectra. A neutral hydrogen absorbing column N_H = 3 × 10²⁰ cm⁻² was assumed for all sources. Source fluxes were drawn from a power-law N>S distribution with index 1.5. The overall normalization was adjusted to yield a few hundred detectable sources per simulation, a compromise aimed at reducing source confusion while limiting the total number of simulations required to obtain good statistics. The effects of photon pile-up (Davis 2001b) and observation-specific bad pixels were included by post-processing each simulation with marxpileup and acis_process_events, respectively. The source events from the MARX simulations were then merged with the appropriate simulated blank-sky event lists, keeping only MARX-simulated source events that fell on active CCDs for the observation. As with the blank-sky simulations, simulated event lists were then processed through the CSC pipeline source detection and source properties extraction steps, and the resulting sources that would have been included in the catalog were tabulated. Finally, these sources were cross-referenced with the input source lists to allow a source-by-source comparison of input and derived properties.

Source detection efficiency is determined by comparing the measured N>S and input N>S distributions. The ratio of these two distributions represents the fraction of input sources of a given incident flux that are actually detected. Results of the comparison for the ACIS broad energy band detections from the shortest and longest ACIS-012367 power-law spectral distribution simulation sets are shown in Figure 24. The standard CSC processing pipeline further combines ACIS source detections from the broad, soft, medium, and hard energy bands to construct the final detected source list. This step was not performed as part of these simulations. However, since the simulated source spectra are homogeneous and well detected in the broad energy band, the difference is not significant in this case.

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As mentioned in Section 3.5.1, the absolute astrometric accuracy of release 1 of the CSC was evaluated post facto by cross-matching catalog sources with their counterparts from the SDSS DR7 (Abazajian et al. 2009). Like the CSC, the SDSS DR7 is referenced to the International Celestial Reference System (ICRS; Arias et al. 1995), and has statistical positional uncertainties ∼45 milliarcseconds (mas) rms per coordinate for bright stars, with systematic errors <20 mas (Abazajian et al. 2009). Only CSC-SDSS source pairs with more than 90% match probability, evaluated according to the Bayesian probabilistic formalism described by Budavári & Szalay (2008), were evaluated, resulting in 6310 source pairs associated with 9476 sources detected in individual observations. Full details of the analysis and results are presented by Rots (2009) and F. A. Primini et al. (2010, in preparation). Here we summarize the main result.

For each matching CSC-SDSS source pair, the separation, ρ, and the total 1σ position error are computed, summing in quadrature the CSC and SDSS errors (and remembering that CSC position errors are reported as 95% uncertainties). We then examine the value of reduced χ² = ∑(ρ/σ_tot)²/(n − 1) for bins in σ_tot covering the range ∼01–2''. The value of the reduced χ² is reasonably close to 1, except for σ_tot ≲ 03 (indicating that the errors are underestimated in that range). Adding a systematic astrometric error component of 016 ± 00.01 to σ_tot yields reduced χ² near 1 for all values of σ_tot. We therefore adopt that value as the systematic astrometric error present in release 1 of the CSC.

The distribution of the normalized separations for the CSC-SDSS source match pairs is shown in the left panel of Figure 25, together with the theoretical Rayleigh distribution for the same number of sources. The overall shape of the curve agrees with the Rayleigh distribution, although there is a slight deficit at high values of normalized separation, suggesting that the overall error may be overestimated for sources at large off-axis angles. In the right panel of Figure 25, we present the average CSC-SDSS separation as a function of off-axis angle, with the systematic astrometric error included. The average CSC 1σ positional error ranges from 02 on-axis to ∼35 at ∼14' off-axis.

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To assess the accuracy of CSC source fluxes, the measured source region aperture photon fluxes are compared with the input photon fluxes of the simulated point sources. Figure 26 presents the comparison of ACIS broad energy band photon fluxes for simulated sources with a power-law spectrum. Inspection of the figure reveals good agreement for sources with off-axis angles within 10' of the aim point. For sources beyond 10', photon fluxes appear to be systematically overestimated by a factor of ∼2 for sources fainter than ∼3 × 10⁻⁶ photons cm⁻² s⁻¹.

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The systematic error in the faint flux bins is more prominent in the ACIS soft energy band, in which the measured fluxes appear underestimated for all simulated input flux levels. Further investigation of this effect will be reported by F. A. Primini et al. (2010, in preparation). Preliminary analysis suggests that the effect results from the use of a monochromatic exposure map (computed at the effective energy of the band) when determining source fluxes. Models based on this assumption reproduce the general features of the apparent systematic errors, and for the assumed model power-law spectrum the error is ∼10% in the broad, medium, and hard energy bands, ∼20%–30% in the soft energy band, and ∼30% in the ultra-soft energy band.