The $\textit{NuSTAR}$ Extragalactic Surveys: Source Catalogs from the Extended $\textit{Chandra}$ Deep Field-South and the $\textit{Chandra}$ Deep Field-North

We present a routinized and reliable method to obtain source catalogs from the $\textit{Nuclear Spectroscopic Telescope Array}$ ($\textit{NuSTAR}$) extragalactic surveys of the Extended $\textit{Chandra}$ Deep Field-South (E-CDF-S) and $\textit{Chandra}$ Deep Field-North (CDF-N). The $\textit{NuSTAR}$ E-CDF-S survey covers a sky area of $\approx30'\times30'$ to a maximum depth of $\sim$ 230 ks corrected for vignetting in the 3--24 keV band, with a total of 58 sources detected in our E-CDF-S catalog; the $\textit{NuSTAR}$ CDF-N survey covers a sky area of $\approx7'\times10'$ to a maximum depth of $\sim$ 440 ks corrected for vignetting in the 3--24 keV band, with a total of 42 sources detected in our CDF-N catalog that is produced for the first time. We verify the reliability of our two catalogs by crossmatching them with the relevant catalogs from the $\textit{Chandra}$ X-ray observatory, and find that the fluxes of our $\textit{NuSTAR}$ sources are generally consistent with that of their $\textit{Chandra}$ counterparts. Our two catalogs are produced following the exactly same method and made publicly available, thereby providing a uniform platform that facilitates further studies involving these two fields. Our source-detection method provides a systematic approach for source cataloging in other $\textit{NuSTAR}$ extragalactic surveys.


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The NuSTAR E-CDF-S and CDF-N Surveys Zhang & Xue line and continuum emission processes, and a high-quality X-ray spectrum is effective to infer physical conditions near the central SMBH.

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The NuSTAR E-CDF-S and CDF-N Surveys Zhang & Xue brevity, the data reduction and source detection is introduced in detail only for the E-CDF-S. We summarize our results in Section 4. We use J2000.0 coordinates and a cosmology of H 0 = 71 km s −1 Mpc −1 , Ω M = 0.27, and Ω Λ = 0.73.

Data Reduction
We collect 33 valid observations from the NuSTAR E-CDF-S survey that cover a sky area of ≈ 30 × 30 , almost each of which has an effective exposure of ≈ 45 ks. The details of these observations are presented in Table 1.

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The NuSTAR E-CDF-S and CDF-N Surveys Zhang & Xue

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The NuSTAR E-CDF-S and CDF-N Surveys Zhang & Xue default parameters. Following M15, full-field lightcurves in the entire energy band (i.e., 3-78 keV) with a bin size of 20 s are produced to inspect the influence of flaring events. The dmgti tool of the Chandra Interactive Analysis of Observations (CIAO) is used to make a user-defined good-time interval (GTI) file to avoid background flaring when the average binned count rate exceeds 1.5 cts s −1 in the light curves. Taking the GTI files into account, we run nupipeline again to obtain the 66 cleaned event files.

Science, exposure, and background mosaics
From the cleaned event files, we produce exposure maps with the NuSTARDAS program nuexpomap. For the effects of vignetting, the same energy correction values as those in M15 are adopted to generate the effective exposure maps, i.e., 5.42, 13.02, and 9.88 keV for the soft, hard, and full bands, respectively. The E-CDF-S reaches a maximum depth of ∼ 230 ks corrected for vignetting in the full band.
Due to the high count-rate backgrounds in the NuSTAR E-CDF-S observations, we generate model background maps using the IDL software nuskybgd [17] . Following the similar strategy adopted by M15, we choose four large (i.e., radius of 3 ) circular regions centered on the four chips of the detector as our background regions. With the user-defined regions, the nuskybgd software can extract and fit the corresponding spectra in XSPEC with the preset models and derive the best-fit parameters. These parameters are used to generate "fake" background images of the observations. Using the FTOOLS task XIMAGE, these simulated images are collected and merged into background mosaics weighted by the corresponding exposure maps; similarly, using XIMAGE, the stacked science mosaics (see Figure 1) are directly produced from the cleaned event files.
We note that in the newest version of nuskybgd, the use of the "nuabs" XSPEC model has been phased out of nuskybgd routines. However, we find that when "nuabs" is removed, the model background counts are significantly lower than what they should

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The NuSTAR E-CDF-S and CDF-N Surveys Zhang & Xue be. Consequently, we turn to using the old version that includes the "nuabs" model in the spectral fitting process.

Source Detection
As shown in Figure 1, traditional source detection methods (e.g., WAVDETECT [18] and ACIS Extract [19] adopted in Xue et al. [8] ; Xue et al. [14] ; Luo et al. [11] ) are invalid due to the heavy background. Following the general strategy adopted for NuSTAR surveys (M15; Masini et al. [20] ), we use the incomplete Gamma (igamma) function (see Georgakakis et al. [21] ) in the Scipy.special package to produce false probability (P false ) maps for source detection: where N Sci and N Bgd represent the photon counts within one region at the same position in the science and background mosaics, respectively. The P false value gives the probability that a signal with N Sci counts is purely due to random fluctuation given the background of N Bgd , which means that the signal is more likely to be real as P false decreases.
We smooth the science and background mosaics with top-hat functions of different radii, with the former (10 ) being smaller for finer structures and the latter (20 ) being larger to decrease the background influence. The P false maps are produced using three methods: (1) P false value at the position (x,y) is directly derived by igamma(Sci(x,y), Bgd(x,y)), where the resulting P false maps are called P r0 maps; (2) At the position (x,y), we perform aperture photometry with a circular region of radius 10 on the mosaics, then calculate the P false value from igamma(Sci 10 (x, y),Bgd 10 (x, y)), where the resulting P false maps are called P r10 maps; (3) The same procedure as method 2 but using a 20 radius aperture is adopted to obtain the P r20 maps. Considering the potential signals residing in the local minimums of the P false maps, we produce the inverse P false maps using log(1/P false ) to identify peaks with the SExtractor source-detection algorithm [22] .
Some modifications on the default SExtractor configuration file default.sex are listed below:

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The NuSTAR E-CDF-S and CDF-N Surveys Zhang & Xue (1) DETECT_MINAREA is set to 1. Because we smooth the input maps, even one pixel in the 1/P false maps can be considered as a potential source.
(2) THRESH_TYPE is set to ABSOLUTE. Under this condition, DETECT_THRESH represents the detection threshold above which a signal in the maps can be considered to be significant.
(3) FILTER and CLEAN are set to N(o). We do not need these processes as what we deal with here are not real observational images.
With proper DETECT_THRESH values, SExtractor is able to detect potential sources in the three standard bands to produce our initial catalogs.

Simulations
We perform a series of simulations to determine DETECT_THRESH. Because the science mosaics are smoothed with the 10 -radius circular top-hat function (see The simulations are repeated for 100 times, in each of which we produce inverse P false maps with three different aperture radii (i.e., 0 , 10 , and 20 ) in the three standard bands. DETECT_THRESH is set to the value above which SExtractor can find no more than N signals in these simulated maps per aperture radius per band, and R = N/100 represents the false detection rate. These thresholds are applied to the real inverse P false maps for source detection.

Final Catalog Production
We first set R = 0.1 for loose thresholds, with which our algorithm is run on the P r0 , P r10 , and P r20 maps to generate three seed lists in each band. These seed lists are produced from different P false maps, thus there might be multiple nearby detections belonging to one same source. To identify every unique source, we merge the seed lists for each band, then run a modified friends-of-friends algorithm (hereafter FOF)

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The NuSTAR E-CDF-S and CDF-N Surveys Zhang & Xue on these merged lists for deduplication.
FOF is common in cosmology for identifying groups in dense fields [23] , which demands that any particle (or galaxy) that finds another one within a distance l (called linking length) should be linked to it to form a group. Sources in the merged lists are split into group sources and isolated sources by FOF with a linking length of 30 . For each group, members are ranked by their inverse P r20 values (even if being initially detected from other P false maps), then compete against any others within a 30 radius, in which a member of higher value can survive and participate in a next competition.
After several rounds of competitions, the final survivals from the group sources are combined with the isolated sources to make up the deduplicated seed lists for each band.
The deduplicated seed lists from different bands are combined into one seed list and then split by FOF in the same way. For each group in the combined list, members are ranked by their inverse P r20 values in the full band (regardless of whether being detected in this band), then participate in competitions against each other. After deduplication, the remaining sources are collected to construct our seed catalog with R = 0.1. The details are presented in Table 2. To produce a reliable final catalog, we then set R = 0.01, which means only 1 false signal being detected in 100 simulations. The thresholds of the inverse P r20 maps are 3.99, 4.36 and 4.33 in the soft, hard, and full bands, respectively, corresponding

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The NuSTAR E-CDF-S and CDF-N Surveys Zhang & Xue to ≈ 99% reliability [24] . We remove the sources that do not meet any of our final thresholds and construct the final catalog with the remaining ones.
The E-CDF-S final catalog contains 58 sources, each of which is detected in at least one of the three standard bands. Of these 58 sources, 33, 13, and 54 are detected in the soft, hard, and full bands, respectively; 3, 1, 21 are detected only in the soft, hard, and full bands, respectively; no source is detected in exactly the soft and hard bands, 21 in exactly the soft and full bands, and 3 in exactly the hard and full bands; and 9 are detected in all the three standard bands.

Photometry and Deblending
The radius of 90% encircled-energy fraction contour of the NuSTAR point spread function (PSF) is approximately 67.5 , which is relatively large compared to the average distance among sources. We adopt the similar strategy to that of M15 to choose an aperture size of 30 for photometry extraction and assume that the net counts within this aperture are only contaminated by other nearby NuSTAR-detected sources within 90 .
For each source in our catalog, the total and background counts are calculated within a circular region of radius 30 in the science and background mosaics, respectively, while the net counts are derived by subtracting the background counts from the total counts. Following Gehrels [25] , we estimate the upper and lower 1σ confidence limits on the total counts; for those not detected in certain bands, only the upper limits are derived, using: where λ u and λ l represent the upper and lower limits, n is the photon counts, and CL represents the confidence level, respectively.
The background count error can be approximated by σ Bgd = where C Bgd is the background counts and f actor gives the ratio between the total area where the background model is defined and the area for photometry extraction
To deblend the sources in our catalog, the FOF algorithm is applied again to split them into group sources and isolated sources with a different linking length of 90 .
For the isolated sources, we assume that they cannot be contaminated by any other sources (away beyond 90 ); for the group sources, a system of n linear simultaneous equations is established: where C n T is the total net counts of source n, C n D is the deblended net counts of source n, and N (r i,j ) is the normalized function of the separation between the sources i and j (r i,j represents the separation distance, while N (0) = 1), in which several simplifications are proposed to avoid the complications of the nonazimuthally symmetric NuSTAR PSF.
Following the deblending procedure above, we then perform deblending with another aperture of 20 radius, and recalculate the P false of each source after deblending.
The post-deblending P false values are compared with the P r20 thresholds, and 4 of the 58 sources in our catalog become no longer significant. Additionaly, we find 1 source in the area of relatively low exposure (< 40 ks, corresponding to 10% of the maximum survey exposure). All of these 5 sources are detected in the full band only, and we flag but do not remove them (see Figure 1).
To validate the reliability of our catalog (a total of 58 sources), we match it to the previous NuSTAR E-CDF-S catalog (a total of 54 sources) in M15 using a matching radius r m = 30 and find a total of 36 counterpart pairs. We compare their net counts in Figure 2 and find good consistency within 1σ errors. We also compare their aperture-corrected fluxes (see Section 2.3) and find good agreement between each

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The NuSTAR E-CDF-S and CDF-N Surveys Zhang & Xue

other.
A significant fraction of the M15 sources are not detected by our work (and vise versa), mainly due to two facts: the detailed cataloging methodologies are different between our work and M15, and those unmatched sources generally have lower net counts such that they could be too faint to be detected by either work.

Matching to the Chandra E-CDF-S and CDF-S Catalogs
We first match our catalog to the Chandra 250 ks E-CDF-S catalog (X16) using r m = 30 , and find 51 of the 58 sources to have at least one Chandra counterpart. In these matches, 20, 24, 5, and 2 NuSTAR sources have 1, 2, 3, and 4 Chandra counterparts, respectively; no NuSTAR sources have more than 4 Chandra counterparts.  Figure 1) which might be spurious. Because we aim to find as many sources as possible using our algorithm alone (without manual intervention), these 6 sources are conserved and flagged in our final catalog.
To compare the fluxes of these matched sources, the observed deblended fluxes of the NuSTAR sources are derived following the same approach as Alexander et al. [16] .
For the sources detected both in the soft and hard bands, we calculate their hardness ratios, HR=(H−S)/(H+S), using the Bayesian estimation of hardness ratios method [26] ; for other sources, an HR value corresponding to the power-law spectral photon index of Γ = 1.8 is assumed. Using the derived HRs, the same parameters as in M15 are then adopted to convert count rates to observed fluxes (see Section 2.  Figure 3, which indicates general agreement within a factor of 3 for the majority of the sources. However, the NuSTAR fluxes appear to be systematically lower than the Chandra fluxes, which is mainly due to that the NuSTAR measured/assumed photon index may be different from the Chandra measured/assumed photon index (in the case of 1-to-1 match) or photon indexes (in the case of 1-to-multiple match).

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The NuSTAR E-CDF-S and CDF-N Surveys Zhang & Xue

Data Reduction and Source Detection
We collect 12 valid observations from the NuSTAR CDF-N survey that cover a sky area of ≈ 7 × 10 , almost each of which has an effective exposure of ≈50 ks. We summarize the information of these observations in Table 3. For data reduction, we apply the exactly same procedures as in Section 2.1, therefore the technical details are introduced briefly in this section.  Table 3 continued    Table 4.

Matching to the Chandra CDF-N Catalog
We match our catalog to the Chandra 2 Ms CDF-N catalog (X16) using r m = 30 , and find 33 of the 42 sources to have at least one Chandra counterpart. In these matches, 14, 11, 4, and 3 NuSTAR sources have 1, 2, 3, and 4 Chandra counterparts, respectively, and 1 NuSTAR source has more than 4 Chandra counterparts.
For the NuSTAR sources without Chandra counterparts, we inspect their positions and properties and find that almost all of them (8/9; XIDs=2, 3, 12, 32, 34, 37, 38, 40) reside in or near the edges of the NuSTAR CDF-N field and the remaining one (XID=25) is only detected in the hard band which might be too "hard" to be detected by Chandra (see Figure 4).
We compare the fluxes of these matched sources in Figure 5, which also indicates general agreement within a factor of 3 for the majority of the sources. The normalized flux histograms of the NuSTAR CDF-N and E-CDF-S sources are compared in   In this work, we collect the original observations from the NuSTAR E-CDF-S and CDF-N surveys and produce cleaned event files. Simulated background mosaics are generated using the IDL software nuskybgd, and then processed along with science mosaics to produce P false maps for source detection.
For the NuSTAR E-CDF-S survey, the main results are as follows: (1) The E-CDF-S catalog consists of 58 sources that are detected using our algorithm without manual intervention, with 4 of them being not significant after deblending.
(2) We compare our catalog with the previous NuSTAR E-CDF-S catalog (M15) using r m = 30 and find a total of 36 matches, the net counts of which agree well within 1σ errors.
(3) We compare our catalog with the Chandra E-CDF-S and CDF-S catalogs (X16 and L17), and find a total of 51 matches, the fluxes of which agree well above the detection limit. All of the 4 sources being not significant after deblending have counterparts in the Chandra catalogs; and the 7 unmatched sources are flagged as being spurious but still conserved in the catalog.
For the NuSTAR CDF-N survey, the main results are as follows: (1) The CDF-N catalog, produced for the first time by this work, consists of 42 sources that are detected using our algorithm without manual intervention, with 6 of them being not significant after deblending.
(2) We compare our catalog with the Chandra CDF-N catalog (X16), and find a total of 33 matches, the fluxes of which agree well above the detection limit.
We flag the 9 unmatched sources as being spurious but conserve them in the catalog.
(3) The flux limits are significantly lower in the NuSTAR CDF-N field (having deeper exposures) than that in the NuSTAR E-CDF-S field.
Our source-detection method provides a systematic approach for source cataloging in other NuSTAR extragalactic surveys. We make our NuSTAR E-CDF-S and CDF-N source catalogs publicly available (see Appendix A for catalog description), which

A Catalog Description
The NuSTAR E-CDF-S and CDF-N source catalogs have the same 51 columns that are summarized in Table 5, with the details described below.
(1) Column 1 gives the source sequence number (i.e., XID). We list the sources in the order of decreasing right ascension.
(2) Columns 2 and 3 give the right ascension and declination of the NuSTAR source, respectively.