FERMI LARGE AREA TELESCOPE FIRST SOURCE CATALOG

The detection step used the same ideas that were detailed in Abdo et al. (2009m). It was based on the same three energy bands, combining Front and Back events to preserve spatial resolution. The detection does not use events below 200 MeV, which have poor angular resolution. It uses events up to 100 GeV. The full band (6.7 × 10⁶ counts) starts at 200 MeV for Front and 400 MeV for Back events. The medium band (12.0 × 10⁵ counts) starts at 1 GeV for Front and 2 GeV for Back events. The hard band (10.7 × 10⁴ counts) starts at 5 GeV for Front and 10 GeV for Back events.

We used the same partitioning of the sky into 24 planar projections as in the BSL, and the same two wavelet-based detection methods: mr_filter (Starck & Pierre 1998) and PGWave (Damiani et al. 1997; Ciprini et al. 2007). The methods looked for sources on top of the diffuse emission model described in Section 3. For mr_filter the threshold was set in each image using the False Discovery Rate procedure (Benjamini & Hochberg 1995) at 5% of false detections. For PGWave we used a flat threshold at 4σ. For comparison with the BSL, the number of "seed" sources from mr_filter was 857 in the full band, 932 in the medium band, and 331 in the hard band. Contrary to the BSL procedure, we combined the results of those two methods (eliminating duplicates) rather than choosing a baseline method and using the other for comparison. The rationale was to limit the number of missed sources to a minimum, since the later steps do not introduce any additional sources. Duplicates were defined after the first localization (pointfit in Section 4.2, run separately on each list of seeds). If two resulting positions were consistent within the quadratic sum of 95% error radii only one source was kept (that with highest significance estimate). Where pointfit did not converge, the 95% error radius was set to 03, typical for faint sources (Section 4.2).

To that same end we also introduced for 1FGL two other detection methods.

• 1.  
  Point find, a tool that searches for candidate point sources by maximizing the likelihood function for trial point sources at each direction in a HEALPix (Górski et al. 2005) order 9 (pixel size ∼0.1 deg²) tessellation of the sky. The algorithm for evaluating the likelihood is optimized for speed by using energy-dependent binning of the photon data, choosing four energy bands per decade starting at 700 MeV, and a HEALPix order commensurate with the PSF width in each band. A first pass examines the significance of a trial point source at the center of each pixel, on the assumption that the diffuse background is adequately described by the model for Galactic diffuse emission and ignoring any nearby point sources. The likelihood is optimized with respect to the signal fraction (i.e., the source and diffuse intensities are not fit separately) in each energy band, with the total likelihood being the product over all the bands. This makes the result independent of the spectrum of the point source or of the diffuse background. While the search is quite efficient, it produces many false signals, so a second pass is used to optimize a more detailed likelihood function which includes nearby detected sources and fits the test source flux and diffuse background normalization independently. The result of the second pass is a map of TS from which the coordinates of candidate point sources can be derived.
• 2.  
  The minimum spanning tree (Campana et al. 2008) looks for clusters of high-energy events (>4 GeV outside the Galactic plane and >10 GeV at |b| < 15°). It is restricted to high energies because it does not account for structured background, but can efficiently detect very hard sources.

We combined the "seed" positions from those two methods with those from the wavelet-based methods, using the same procedure for removing duplicates as above.

Finally, we introduced external seeds from the BZCAT (Massaro et al. 2009) and WMAP (Wright et al. 2009) catalogs. The BZCAT catalog is not homogeneous but includes the great majority of known, well-characterized blazars. It is a superset of the CGRaBS (Healey et al. 2008) catalog and has broader sky coverage. The WMAP catalog includes mainly bright FSRQ blazars, and was used primarily to try to recover soft-spectrum sources that might have been missed by the source-detection algorithms.

In order to not bias the 1FGL catalog toward those external sources, we used them as seeds only when there was no seed from the detection methods within its 95% error radius. Of the 335 BZCAT seeds introduced, 24 survived as LAT γ-ray sources in this catalog. Of the seven WMAP seeds, three remain in the catalog.

The variety of seeds that we used means that the catalog is not homogeneous. Because of the strong underlying diffuse emission, achieving a truly homogeneous catalog was not possible in any case. Our aim was to provide enough seeds to allow the main maximum likelihood analysis (Section 4.3) to be the defining step of the catalog construction. The total number of seeds was 2433.

The localization of faint or soft sources is more sensitive to the diffuse emission and to nearby sources than for brighter sources, so we proceeded in three steps instead of just one for the bright sources considered in the BSL.

• 1.  
  The first step consisted of localizing the sources before the main maximum likelihood analysis (Section 4.3) as we did for the BSL (using pointfit), treating each source independently but in descending order of significance and incorporating the bright sources into the background for the fainter ones. This is fast and provides a good enough starting point for step 2.
• 2.  
  The second step consisted of improving the localization within the main maximum likelihood analysis (Section 4.3) using the gtfindsrc utility in the Science Tools.⁷³ Again sources are considered in descending order of significance. When localizing one source, the others are fixed in position, but the fluxes and spectral indices of sources within 2° are left free to accommodate the loss of low-energy photons in the model if the source that is being localized moves away. At the end of that step we have a good representation of the location, flux, and spectral shape of the sources over the entire sky, but a single error radius to describe the error box.
• 3.  
  The third step is new and described in more detail below. It uses a similar framework as the first step, but incorporates the results of the main maximum likelihood analysis for all sources other than the one being considered, so it has a good representation of the source's surroundings. It is faster than gtfindsrc and gttsmap and returns a full TS map around each source and an elliptical representation as well as an indicator of the quality of the elliptical fit.

The first and third steps used a likelihood analysis tool (pointfit) that provides speed at little sacrifice of precision by maximizing a specially constructed binned likelihood function. Photons are assigned to 12 energy bands (four per decade from 100 MeV–100 GeV) and HEALpix-based spatial bins for which the size is selected to be small compared with the scale set by the PSF. Since the PSF for Front-converting photons is significantly smaller than that for Back conversions, there are separate spatial bins for Front and Back. Note that the width of the PSF at a given energy is only a weak function of incidence angle. For pointfit the likelihood function is evaluated using the PSF averaged over the full field of view for each energy band. For each band, we define the likelihood as a function of the position and flux of the assumed point source, and adopt as the background the sum of Galactic diffuse, isotropic diffuse (see Section 3), and any nearby (i.e., within 5°), other point sources in the catalog. The flux for each band is then evaluated by maximizing the likelihood of the data given the model using the coordinates defined by gtfindsrc. The overall likelihood function, as a function of the source position, is then the product of the band likelihoods. We define a function of the position p, as 2(log(L_max) − log(L(p)), where L is the likelihood function described above. This function, according to Wilks' theorem (Wilks 1938), is the probability distribution for the coordinates of the point source consistent with the observed data. Note that the width of this distribution is a measure of the uncertainty, and that it scales directly with the width of the PSF.

We then fit the distribution to a two-dimensional quadratic form with five parameters describing the expected elliptical shape: the coordinates (R.A. and decl.) of the center of the ellipse, semimajor and semiminor axis extents (α and β), and the position angle ϕ of the ellipse.⁷⁴ A "quality" factor is evaluated to represent the goodness of the fit: it is the square root of the sum of the squares of the deviations for eight points sampled along the contour where the value is expected to be 4.0, that is, 2σ from the maximum likelihood coordinates of the source.

We quote the parameters of the ellipse that would contain 95% of the probability for the location of the source; for Gaussian errors this would be a radius of 2.45σ. An analysis of the deviations of 396 AGNs at high latitudes from the positions of the nearest LAT point sources indicated that the PSF width is underestimated, on average, by a factor of 1.10 ± 0.05. Thus the final uncertainties reported by pointfit were scaled up by a factor of 1.1. To visually assess the fits, a TS map was made for each source, and these were considered in evaluating the analysis flags that are discussed in Section 4.8.

Twelve sources either did not converge at the third step, converged to a point far away (>1°), or were in crowded regions where the procedure (which does not have free parameters for the fluxes of nearby sources) may not be reliable. Those 12 were left at their gtfindsrc positions. They can be easily identified in the 1FGL catalog because they have identical semimajor and semiminor axes for the source location uncertainty and position angle 0. The LAT-detected pulsars and X-ray binaries, which were placed at the high-precision positions of these identified sources, have null values in the localization parameters.

Figure 2 illustrates the resulting position errors as a function of the TS values obtained in Section 4.3. The relatively large dispersion that is seen at a given TS is in part due to the local conditions (level of diffuse γ-ray emission) but primarily depends upon the source spectrum. Hard-spectrum sources are better localized than soft ones for the same TS (Figure 3) because the PSF is so much narrower at high energy. At our threshold of TS = 25, the typical 95% position error is about 10' and most 95% errors are below 20'.

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The detection and localization steps provide estimates of source significances. However, since the detection step does not use the energy information and the localization step fits only one source at a time, these estimates are not sufficiently accurate for use in the catalog. To better estimate the source significances, we use a three-dimensional maximum likelihood algorithm (gtlike) in unbinned mode (i.e., the position and energy of each event is considered individually) applied on the full energy range from 100 MeV–100 GeV using the P6_V3 IRFs (see Section 2). This is part of the standard Science Tools software package, currently at version 9r15p5. The tool does not vary the source position, but does adjust the source spectrum. The underlying optimization engine is Minuit.⁷⁵ The code works well with up to ∼30 free parameters, an important consideration for regions where sources are close enough together to partially overlap. The gtlike tool provides the best-fit parameters for each source and the Test Statistic TS = 2Δlog(likelihood) between models with and without the source. The TS associated with each source is a measure of the source significance. Error estimates (and a full covariance matrix) are obtained from Minuit in the quadratic approximation around the best fit. For this stage we modeled the sources with simple power-law spectra. It should be noted that gtlike does not include the energy dispersion in the TS calculation (i.e., it assumes that the measured energy is the true energy). Given the 8% to 10% energy resolution of the LAT over the wide energy bands used in the present analyses, this approximation is justified.

Because the fitted fluxes and spectra of the sources can be very sensitive to even slight errors in the spectral shape of the diffuse emission we allow the Galactic diffuse model (Section 3) to be corrected (i.e., multiplied) locally by a power law in energy with free normalization and spectral slope. The slope varies between 0 and 0.07 (making it harder) in the Galactic plane and the normalization by ±10% (down from 0.15% and 20% for the BSL). The smaller excursions of that corrective slope when compared to the BSL reflect the better fit of the current diffuse model to the data. The normalization of the isotropic component of the diffuse emission (which represents the extragalactic and residual backgrounds) was left free. The three free parameters were separately adjusted in each region of interest (RoI).

We split the sky into overlapping circular RoIs. The parameters are free for sources in the central part of each RoI (RoI radius minus 7°), such that all free sources are well within the RoI even at low energy (7° is larger than r₆₈ at 100 MeV). It is advantageous (for the global convergence over the entire sky) to use large RoIs, but at the same time smaller RoIs allow spectral variations of the diffuse emission relative to the model to be corrected in more detail. We set the RoI sizes so that not more than eight sources are free at a time. Adding three parameters for the diffuse model, the total number of free parameters in each RoI is normally 19 at most. We needed 445 RoIs to cover the 2433 seed positions. The RoI radii range between 9° and 15°.

We proceed iteratively. All RoIs are processed in parallel and a global current model is assembled after each step in which the best-fit parameters for each source are taken from the RoI whose center is closest to the source. The local model for each RoI includes sources up to 7° outside the RoI (which can contribute at low energy due to the broad PSF). Their parameters are fixed to their values in the global model at the previous step. The parameters of the sources inside the RoI but within 7° of the border are also fixed except in two cases (not considered for the BSL analysis).

• 1.  
  Sources within 2° of any source inside the central part, because they can influence the inner source. 2° is chosen to be larger than twice the containment radius at 1 GeV (2 × 08) where the LAT sensitivity peaks (Figure 18). We leave both flux and spectral index free for these.
• 2.  
  Very bright sources contributing more than 5% of the total counts in the RoI because they can influence the diffuse emission parameters. We leave only the flux free for these.

All seed sources start at 0 flux at the first step; the starting point for the slope is 2. We iterate over five steps; the fits change very little after the fourth. To facilitate the convergence the seed sources are not entered all at once. The brightest 10% of the sources are entered at the first step, 30% at the second step, and finally all at the third step. At each step we remove seed sources with low TS, raising the threshold for inclusion into the global model from 10 at the third step to 15 at the fourth and finally 25 at the last step. All seeds are reentered at the fourth step to avoid losing faint sources before the global model has fully converged. We have checked via simulations that removing the faint sources has little impact on the bright ones, much less than changing the diffuse model (Section 4.6). This procedure left 1451 sources above threshold. The variation of the detection threshold across the sky and the dependence of the threshold on source spectrum are discussed in Appendix A.

The TS of each source can be related to the probability that such an excess can be obtained from background fluctuations alone. The probability distribution in such a situation (source over background) is not known precisely (Protassov et al. 2002). However since we consider only positive fluctuations, and each fit involves four degrees of freedom (two for position, plus flux and spectral index), the probability to get at least TS at a given position in the sky is close to 1/2 of the χ² distribution with four degrees of freedom (Mattox et al. 1996), so that TS = 25 corresponds to a false detection probability of 2.5 × 10⁻⁵ or 4.1σ (one sided). For the BSL we considered only two degrees of freedom because the localization was based on a simpler algorithm which did not involve explicit minimization of the same likelihood function.

The sources that we see are best (most strongly) detected around 1 GeV. This is approximately the median of the Pivot_Energy quantity in the catalog, i.e., the energy at which the uncertainties in normalization and spectral index for the power-law fit are uncorrelated. At 1 GeV, the 68% containment radius is approximately r₆₈ = 08. The number of independent elements in the sky (trials factor) is about 4π/(πr²₆₈) in which r₆₈ is converted to radians. This is about 2 × 10⁴ so at a threshold of TS = 25 we expect less than 1 spurious source by chance only. If any, there might be a few very hard spurious sources in the catalog because hard sources have a smaller effective PSF so that the trial factor is larger. The main reason for potentially spurious sources, though, is our imperfect knowledge of the underlying diffuse emission (Section 4.6).

The maximum likelihood method described in Section 4.3 provides good estimates of the source significances and the overall spectral slope, but not very accurate estimates of the fluxes. This is because the spectra of most sources do not follow a single power law over that broad energy range (three decades). Within the two most populous categories, the AGNs often have broken power-law spectra and the pulsars have power-law spectra with an exponential cutoff. In both cases fitting a single power law over the entire range overpredicts the flux in the low-energy region of the spectrum, which contains the majority of the photons from the source, biasing the fluxes high. On the other hand, the effect on the significance is low due to the broad PSF and high background at low energies.

In addition, the significance is mostly obtained from GeV photons (Figure 18) whereas the photon flux in the full range (above 100 MeV) is dominated by lower energy events so that the uncertainty on that flux can be quite large even for highly significant sources. For example, the typical relative uncertainty on the photon flux above 100 MeV is 23% for a TS = 100 source with spectral index 2.2.

To provide better estimates of the source fluxes, we decided to split the range into five energy bands from 100 to 300 MeV, 300 MeV to 1 GeV, 1 to 3 GeV, 3 to 10 GeV, and 10 to 100 GeV (the number of counts does not justify dividing the last decade into two bands). The list of sources remained the same in all bands. It is generally not possible to fit the spectral index in each of those relatively narrow energy bands (and the flux estimate does not depend very much on the index), so we simply froze the spectral index of each source to the best fit over the full interval. The spectral bias to the Galactic diffuse emission (Section 4.3) was also frozen.

The estimate from the sum of the five bands is on average within 30% of the flux obtained from the global power-law fit (as described in Section 2, with excursions up to a factor of 2). We have also compared those estimates with a more precise spectral model for the three bright pulsars (Vela, Geminga, and the Crab). The sum of the five fluxes is within 5% of the more precise flux estimate, whereas the power-law estimate is 25% too high for Vela and Geminga. However because the flux in each band is not constrained globally as in the power-law model, the relative uncertainty of the sum of the five fluxes is even larger than that for the power-law fit, typically 50% for a TS = 100 source with spectral index 2.2. For that reason we do not show this very poorly measured quantity in Table 2. We provide instead the photon flux between 1 and 100 GeV (the sum of the three high energy bands), which is much better defined. The relative uncertainty on this flux is typically 18% for a TS = 100 source with spectral index 2.2.

In contrast, the energy flux over the full band is better defined than the photon flux because it does not depend as much on the poorly measured low-energy fluxes. So, we provide this quantity in Table 2. Here again the sum of the energy fluxes in the five bands provides a more reliable estimate of the overall flux than the power-law fit. The relative uncertainty on the energy flux between 100 MeV and 100 GeV is typically 26% for a TS = 100 source with spectral index 2.2.

An additional difficulty that does not exist when considering the full data is that, because we wish to provide the fluxes in all bands for all sources, we must handle the case of sources that are not significant in one of the bands. Many sources have TS < 10 in one or several bands: 1135 in the 100–300 MeV band, 630 in the 300 MeV to 1 GeV band, 359 in the 1–3 GeV band, 503 in the 3–10 GeV band, and 800 in the 10–100 GeV band. There are even a number of sources which have upper limits in all bands, even though they are formally significant (as defined in Section 4.3 with a power-law model) if the events at all energies are considered together. It is particularly difficult to measure fluxes below 300 MeV because of the large source confusion and the modest effective area of the LAT at those energies with the current event cuts (Section 2).

For the sources with poorly measured fluxes (where TS < 10 or the nominal uncertainty of the flux is larger than half the flux itself), we replace the flux value from the likelihood analysis by a 2σ upper limit, indicating the upper limit by a 0 in the flux uncertainty column of Table 3; the corresponding columns of the FITS version of the 1FGL catalog are described in Appendix D. The upper limit is obtained by looking for 2Δlog(likelihood) = 4 when increasing the flux from the maximum likelihood value. When the maximum likelihood value is very close to 0 (i.e., the flux that maximizes the likelihood would be negative), solving 2Δlog(likelihood) = 4 tends to underestimate the upper limit. We evaluate flux upper limits for the cases where the maximum likelihood source flux would be negative because negative fluxes are not physical and the likelihood function cannot be evaluated if the flux is negative enough to correspond to a negative photon probability density. The absence of negative fluctuations is accounted for in evaluating the significances of the sources (e.g., Mattox et al. 1996). Whenever TS < 1 we switch to the Bayesian method proposed by Helene (1983). We do not use that method throughout because it is about five times slower to compute.

Table 3. First LAT Catalog: Spectral Information

				100 MeV–300 MeV			300 MeV–1 GeV			1 GeV–3 GeV			3 GeV–10 GeV			10 GeV–100 GeV
Name 1FGL	Γ	ΔΓ	Curv.	F1a	ΔF1a	$\sqrt{{\rm TS}_{\rm 1}}$	F2a	ΔF2a	$\sqrt{{\rm TS}_{\rm 2}}$	F3b	ΔF3b	$\sqrt{{\rm TS}_{\rm 3}}$	F4c	ΔF4c	$\sqrt{{\rm TS}_{\rm 4}}$	F5c	ΔF5c	$\sqrt{{\rm TS}_{\rm 5}}$
J0000.8+6600c	2.60	0.09	...	8.3	0.0	2.6	1.9	0.3	6.4	2.6	0.6	5.2	3.7	1.5	4.1	0.8	0.0	0.0
J0000.9−0745	2.41	0.20	...	2.9	0.0	2.5	0.5	0.0	3.0	0.7	0.0	2.2	3.8	0.0	3.7	1.5	0.0	2.4
J0001.9−4158	1.92	0.25	...	2.1	0.0	1.8	0.2	0.0	1.1	0.6	0.0	2.2	2.9	1.1	6.0	1.7	0.0	0.0
J0003.1+6227	2.53	0.10	T	3.0	0.0	0.0	1.7	0.3	6.8	2.0	0.5	5.0	4.3	0.0	1.4	1.4	0.0	1.9
J0004.3+2207	2.35	0.21	...	1.8	0.0	0.9	0.4	0.0	1.8	0.4	0.2	3.2	1.9	0.9	3.8	0.9	0.0	0.0
J0004.7−4737	2.56	0.17	...	2.4	0.8	3.3	0.3	0.1	3.6	0.8	0.2	5.0	2.0	0.0	2.0	1.5	0.0	0.2
J0005.1+6829	2.58	0.12	...	3.9	0.0	0.6	1.3	0.3	5.3	2.2	0.0	3.1	4.9	0.0	2.0	1.0	0.0	0.0
J0005.7+3815	2.86	0.13	...	3.3	0.9	3.7	0.5	0.1	4.4	1.1	0.0	3.1	2.5	0.0	1.6	1.1	0.0	0.0
J0006.9+4652	2.55	0.11	...	2.9	0.9	3.3	0.7	0.1	6.4	0.8	0.3	3.7	3.0	1.3	4.3	1.5	0.0	2.1
J0007.0+7303	1.97	0.01	T	20.9	1.2	19.8	12.0	0.3	60.5	49.0	1.3	82.3	135.6	6.5	57.6	8.5	1.6	15.3
J0008.3+1452	2.00	0.21	...	0.9	0.0	0.0	0.2	0.0	0.5	0.6	0.2	3.8	2.0	0.9	4.5	1.2	0.0	0.0
J0008.9+0635	2.28	0.22	...	1.6	0.0	0.3	0.5	0.0	3.1	0.5	0.0	1.0	2.2	1.0	4.0	1.5	0.0	2.9
J0009.1+5031	2.41	0.13	...	4.0	0.0	2.3	0.5	0.1	4.6	0.9	0.3	4.6	3.5	1.2	5.1	1.4	0.0	2.0
J0011.1+0050	2.51	0.15	...	1.2	0.0	0.0	0.4	0.1	4.8	0.5	0.2	4.1	1.7	0.0	0.3	0.9	0.0	0.0
J0013.1−3952	2.09	0.22	...	1.2	0.0	0.1	0.3	0.0	1.4	0.9	0.0	2.9	2.3	0.0	1.4	2.1	0.0	4.3

Notes. ^aIn units of 10⁻⁸ photons cm⁻² s⁻¹. ^bIn units of 10⁻⁹ photons cm⁻² s⁻¹. ^cIn units of 10⁻¹⁰ photons cm⁻² s⁻¹.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The five fluxes provide a rough spectrum, allowing departures from a power law to be judged. This is the main advantage over the BSL scheme which involved only two bands. Examples of those rough spectra are given in Figures 4 and 5 for a bright pulsar (Vela) and a bright blazar (3C 454.3). In order to quantify departures from a power-law shape, we introduce a Curvature_Index

$$\begin{equation} C = \sum _i \frac{\big(F_i - F_i^{\rm PL}\big)^2}{\sigma _i^2 + \big(f_i^{\rm rel} F_i\big)^2}, \end{equation} \tag{ 1 }$$

where i runs over all bands and F^PL_i is the flux predicted in that band from the global power-law fit. f^rel_i reflects the relative systematic uncertainty on effective area described in Section 4.6. It is set to 10%, 5%, 10%, 15%, and 20% in the bands [0.1,0.3], [0.3,1], [1, 3], [3, 10], and [10, 100] GeV, respectively. Note that this systematic uncertainty on the effective area is not included in the uncertainties reported in Table 3 (or in the FITS file), because this systematic factor cancels when comparing each of the band fluxes between different sources. We use for F_i and σ_i the best-fit and 1σ estimates even when the values are reported as upper limits in the table, both for computing the Curvature_Index and the sums (photon flux and energy flux).

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Since the power-law fit involves two parameters (normalization and spectral index), C would be expected to follow a χ² distribution with 5 − 2 = 3 degrees of freedom if the power-law hypothesis was true. At the 99% confidence level, the spectral shape is significantly different from a power law if C>11.34. That condition is met by 225 sources (at 99% confidence, we expect 15 false positives). The curvature index is by no means an estimate of curvature itself, just a statistical indicator. A faint source with a strongly curved spectrum can have the same curvature index as a bright source with a slightly curved spectrum. Since the relative uncertainties on the fluxes in each band are quite different and depend on the spectral index itself, it is difficult to build a curvature indicator similar to the fractional variability for the light curves. The curvature index is also not exclusively an indicator of curvature. Any kind of deviation from the best-fit power law can trigger that index, although curvature is by far the most common.

Variability is very common at γ-ray energies (particularly among accreting sources) and it is useful to estimate it. To that end we derive a variability index for each source by splitting the LAT data into a number of time intervals and deriving a flux for each source in each interval, using the same energy range as in Section 4.3 (100 MeV to 100 GeV). We split the full 11 month interval into N_int = 11 intervals of about 1 month each (2624 ks or 30.37 days). This is much more than the week used in the BSL, in order to preserve some statistical precision for the majority of faint sources we are dealing with here. It is also far enough from half the precession period of the orbit (≈0.5  ×   53.4 = 26.7 days) that we do not expect possible systematic effects as a function of off-axis angle to be coherent with those intervals.

To avoid ending up with too large error bars in relatively short time intervals, we froze the spectral index of each source to the best fit over the full interval. Sources do vary in spectral shape as well as in flux, of course, but we do not aim at characterizing source variability here, just detecting it. It is very unlikely that a true variability in shape will be such that it will not show up in flux at all. In addition, little spectral variability was found in bright AGN where it would be detectable if present (Abdo et al. 2010n). Because we do not expect the diffuse emission to vary, we freeze the spectral adjustment of the Galactic diffuse component to the local (in the same RoI) best-fit index from the full interval. So the fitting procedure is the same as in Section 4.4 with all spectral shape parameters frozen.

The variability index is defined as a simple χ² criterion:

$$\begin{eqnarray} &&w_i = \frac{1}{\sigma _i^2 + (f_{\rm rel} F_i)^2} \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&F_{\rm wt} = \frac{\sum _i w_i F_i}{\sum _i w_i} \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &&V = \sum _i w_i (F_i - F_{\rm wt})^2, \end{eqnarray} \tag{ 4 }$$

where i runs over the 11 intervals and σ_i is the statistical uncertainty in F_i. As for the BSL we have added in quadrature a fraction f_rel = 3% of the flux for each interval F_i to the statistical error estimates σ_i (for each 1 month time interval) used to compute the variability index.⁷⁶ Since the weighted average flux F_wt is not known a priori, V is expected, in the absence of variability, to follow a χ² distribution with 10 (= N_int− 1) degrees of freedom. At the 99% confidence level, the light curve is significantly different from a flat one if V>23.21. That condition is met by 241 sources (at 99% confidence, we expect 15 false positives). For those sources we provide directly in the FITS version of the table the maximum monthly flux (Peak_Flux) and its uncertainty, as well as the time when it occurred (Time_Peak); see Table 11 for the column specifications.

As in Section 4.4 it often happens that a source is not significant in all intervals. To preserve the variability index (Equation (4)) we keep the best-fit value and its estimated error even when the source is not significant. This does not work, however, when the best fit is close to zero because in that case the log(likelihood) as a function of flux is very asymmetric. Whenever TS < 10 or the nominal flux uncertainty is larger than half the flux itself we compute the 2σ upper limit and replace the error estimate for that interval (σ_i) with half the difference between that upper limit and the best fit. This is an estimate of the error on the positive side only. Because the parabolic extrapolation often exceeds the log(likelihood) profile at 2σ this is more conservative than computing the 1σ upper limit directly. The best fit itself is retained. Note that this error estimate can be a large overestimate of the error on the negative side, particularly in the deep Poisson regime at high energy. This explains why σ_i/F_i can be as high as 1 even when TS is 4 or so in that interval. As in Section 4.4 we switch to the Bayesian method whenever TS < 1.

Examples of light curves are given in Figures 6 and 7 for a bright constant source (the Vela pulsar) and a bright variable source (the blazar 3C 454.3). With the 3% systematic relative uncertainty no pulsar is found to be variable. The very brightest pulsars (Vela and Geminga) appear to have observed variability below 3%, so this may be overly conservative. It is not a critical parameter though, as it affects only the very brightest sources.

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The fractional variability of the sources is defined from the excess variance on top of the statistical and systematic fluctuations:

$$\begin{equation} \delta F/F = \sqrt{\frac{\sum _i (F_i - F_{\rm av})^2}{(N_{\rm int}-1) F_{\rm av}^2} - \frac{\sum _i \sigma _i^2}{N_{\rm int} F_{\rm av}^2} - f_{\rm rel}^2}. \end{equation} \tag{ 5 }$$

The typical fractional variability is 50%, with only a few strongly variable sources beyond δF/F = 1. This is qualitatively similar to what was reported in Figure 8 of Abdo et al. (2009m). The criterion we use is not sensitive to relative variations smaller than 60% at TS = 100. That limit goes down to 20% as TS increases to 1000. We are certainly missing many variable AGNs below TS = 100 and up to TS = 1000. There is no indication that fainter sources are less variable than brighter ones; we simply cannot measure their variability.

Both the curvature index and the variability index highlight certain types of sources. This is best illustrated in Figure 8 in which one is plotted against the other for the main types of identified or associated sources (from the association procedure described in Section 6). One can clearly separate the pulsar branch at large curvature and small variability from the blazar branch at large variability and smaller curvature.

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In this work we did not test for or account for source extension. All sources are assumed to be point-like. This is true for the major source populations in the GeV range (blazars, pulsars). On the other hand, the TeV instruments have detected many extended sources in the Galactic plane, mostly pulsar wind nebulae (PWNs) and supernova remnants (SNRs), (e.g., Aharonian et al. 2005) and the LAT has already started detecting extended sources (e.g., Abdo et al. 2009i). Because measuring extension over a PSF which varies so much with energy is delicate, we are not yet ready to address this matter systematically across all the sources in a large catalog such as this.

We have addressed the issue of systematics for localization in Section 4.2. Another related limitation is that of source confusion. This is of course strong in the inner Galaxy (Section 4.7) but it is also a significant issue elsewhere. The average distance between sources outside the Galactic plane is 3° in 1FGL, to be compared with a per photon containment radius r₆₈ = 08 at 1 GeV where the sensitivity is best. The ratio between both numbers is not large enough that confusion can be neglected. The simplest way to quantify this is to look at the distribution of distances between each source and its nearest neighbor (D_n) in the area of the sky where the source density is approximately uniform, i.e., outside the Galactic plane. This is shown in Figure 9. The source concentration in the Galactic plane is very narrow (less than 1°) but we need to make sure that those sources do not get chosen as nearest neighbors so we select |b|>10°. The histogram of D_n (after taking out the geometric factor as in Figure 9) should follow

$$\begin{equation} H(D_n) = N_{\rm true} \, \rho _{\rm src} \, \exp \left(- \pi D_n^2 \rho _{\rm src} \right), \end{equation} \tag{ 6 }$$

where ρ_src is the source density (the number of sources per square degree) and N_true is the true number of sources (after correcting for missed sources due to confusion). The exponential term is the probability that no nearest source exists. It is apparent that, contrary to expectations, the histogram falls off toward D_n = 0. This indicates that confusion is important, even in the extragalactic sky. The effect disappears only at distances larger than 15. To get N_true, one may solve for the number of observed sources at distances beyond 15. Since ρ_src = N_true/A_tot in which A_tot is the sky area at |b|>10°, this amounts to solving

$$\begin{equation} N_{\rm obs}(>1.5\mbox{$^\circ $}) = N_{\rm true} \, \exp \left(- N_{\rm true} A_0 / A_{\rm tot} \right) \end{equation} \tag{ 7 }$$

in which A₀ is the area up to 15. This results in N_true − N_obs = 80 missed sources on top of the N_obs = 1043 sources observed at |b|>10°. Those missed sources are probably the reason for some of the asymmetries in the TS maps discussed in Section 4.2. The conclusion is that globally we missed nearly 10% of the extragalactic sources. But because of the worse PSF at low energy, soft sources are comparatively more affected than hard sources. This is approximately indicated by the difference between the full and the dashed lines in Figure 20.

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Another important issue is the systematic uncertainties on the effective area of the instrument. At the time of the BSL we used pre-launch calibration and cautioned that there were indications that our effective area was reduced in flight due to pileup. Since then, the pileup effect has been integrated in the simulation of the instrument (Rando et al. 2009) and many tests have shown that the resulting calibration (P6_V3) is consistent with the data. The estimate of the remaining systematic uncertainty is 10% at 100 MeV, 5% at 500 MeV, rising to 20% at 10 GeV and above. This uncertainty applies uniformly to all sources. Our relative errors (comparing one source to another or the same source as a function of time) are much smaller, as indicated in Section 4.5. The fluxes resulting from this new calibration are systematically higher than the BSL fluxes. For example, the fluxes of the three brightest pulsars (Vela, Geminga, and Crab) are about 30% larger in 1FGL than in the BSL. The differences are more pronounced for soft sources than hard ones. This implies also that the 1FGL fluxes are significantly larger than the EGRET fluxes in the 3EG catalog (Hartman et al. 1999) which happened to be close to the BSL fluxes. As shown by diffuse (Abdo et al. 2009g) and point source (Abdo et al. 2009h, 2010e) observations, the LAT data produce spectra systematically steeper than those reported in EGRET analysis. LAT fluxes are greater at energies below 200 MeV and less at energies above a few GeV.

The model of diffuse emission is the other important source of uncertainties. Contrary to the effective area, it does not affect all sources equally: its effects are smaller outside the Galactic plane (|b|>10°) where the diffuse emission is faint and varying on large angular scales. It is also less of a problem in the high energy bands (>3 GeV) where the PSF is sharp enough that the sources dominate the background under the PSF. But it is a serious issue inside the Galactic plane (|b| < 10°) in the low energy bands (<1 GeV) and particularly inside the Galactic ridge (|l| < 60°) where the diffuse emission is strongest and very structured, following the molecular cloud distribution. It is not easy to assess precisely how large the uncertainty is, for lack of a proper reference model. We discuss the Galactic ridge more specifically in Section 4.7. For an automatic assessment we have tried re-extracting the source fluxes assuming a different diffuse model, derived from GALPROP (as we did for the BSL) but with protons and electrons adjusted to the data (globally). The model reference is 54_87Xexph7S. The results show that the systematic uncertainty more or less follows the statistical one (i.e., it is larger for fainter sources in relative terms) and is of the same order. More precisely, the dispersion is 0.7σ on flux and 0.5σ on spectral index at |b|>10°, and 1.8σ on flux and 1.2σ on spectral index at |b| < 10°. We have not increased the errors accordingly, though, because this alternative model does not fit the data as well as the reference model. From that point of view we may expect this estimate to be an upper limit. On the other hand, both models rely on nearly the same set of H i and CO maps of the gas in the interstellar medium, which we know are an imperfect representation of the mass. That is, potentially large systematic uncertainties are not accounted for by the comparison. So we present the figures as qualitative estimates.

Figure 10 shows an example of the striking, and physically unlikely, correspondence between the 1FGL sources and tracers of the column density of interstellar gas, in this case E(B − V) reddening. The sources in Orion appear to be tightly associated with the regions with greatest column densities. Yet no particular classes of γ-ray emitters are known to be associated with interstellar cloud complexes. Young SNRs would be resolved in the radio and in γ-rays in the nearby clouds outside the Galactic plane. Even if radio-quiet pulsars were the sources, they would not be expected to be aligned so closely with the regions of highest column densities. The implication is that peak column densities are being systematically underestimated in the model for the Galactic diffuse emission used in the analysis. E(B − V) is not directly used in the model; as described in Section 3 the column densities are derived from surveys of H i and CO line emission, the latter as a tracer of molecular hydrogen. An E(B − V) "residual" map E(B − V)_res, representing interstellar reddening that is not correlated with N(H i) or W(CO), is included in the model. So the peak column densities would need to be underestimated both in CO and E(B − V). We are studying the effect and strategies for validating the model for Galactic diffuse emission at high column densities.

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In addition to the concerns about the accuracy of column densities toward the peaks of interstellar clouds, self-absorption of H i at low latitudes can introduce small angular-scale underestimates of the column densities and intensities of the diffuse emission. The current, half-degree binned model of the interstellar emission used for the source analysis also cannot capture structure on smaller angular scales. Bright structure on finer scales could be detected as unresolved point sources.

Figure 10 illustrates another, better understood issue with the model for Galactic diffuse emission. In the Orion nebula (near l, b ∼ 209°, −195), a massive star-forming region that is extremely bright in the infrared, the infrared color corrections used to evaluate E(B − V) are inaccurate and the column densities inferred from E(B − V) are underestimated; the depression in E(B − V) in the nebula does not correspond to a decreased column density of gas. The model of Galactic diffuse emission was constructed with E(B − V)_res allowed to be a signed correction for the column densities inferred from H i and CO lines. This made essential improvements in large regions, but in discrete directions toward massive star-forming regions, negative E(B − V)_res can introduce deep depressions in the predicted diffuse emission. Figure 11 illustrates the depression around the S225 star-forming region, and its close correspondence with a 1FGL source.

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Similar considerations relate to the sources at low latitudes in the inner Galaxy. The density of unassociated sources in the Galactic ridge (300° < l < 60°, |b| < 1°) is very high (Figure 12), and their latitude distribution is exceedingly narrow (Figure 13). If these 1FGL sources are true γ-ray emitters they must have a very small scale height in the Milky Way, like that of the youngest massive star-forming regions, traced by ultracompact H ii regions (∼25' FWHM; e.g., Giveon et al. 2005), or be quite distant and hence very luminous. The 1FGL sources do not have an obvious correspondence with the ultracompact H ii regions, and the latter are not plausible γ-ray sources, but owing to the effects described above, embedded star-forming regions can influence tracers of gas and dust and thereby potentially introduce small-scale errors in the model of Galactic diffuse emission. The inferred luminosities of the 1FGL sources in the Galactic ridge would be quite high if the scale height of their distribution is the characteristic of most tracers of Population I objects. For a relatively narrow dispersion of 40 pc, the characteristic distances of these sources are ∼11 kpc (i.e., more distant than the Galactic center) and the γ-ray luminosities exceed 10³⁶ erg s⁻¹ (i.e., more than an order of magnitude more luminous than the Vela pulsar, Abdo et al. 2009h). For broader dispersions about the plane, the distances and luminosities would increase correspondingly.

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The 1FGL sources toward the peaks of local interstellar clouds and the Galactic ridge all have analysis flags set (Section 4.8) in the catalog. We have also added a designator "c" to their names to indicate that they are to be considered as potentially confused with interstellar diffuse emission or perhaps spurious. In addition, the "c" designator is used for unidentified 1FGL sources in crowded regions of high source density outside the Galactic ridge, as a caution about the complications due to PSF overlaps. The "c" designator, thus applied to 161 of the 1FGL sources, is a warning that the existence of the source or its measured properties (location, flux, spectrum) may not be reliable.

We have identified a number of conditions that can shed doubt on a source. They are described in Table 4. As noted, setting of flags 4 and 5 depends on the energy band in which a source is detected. Both flags are more easily set at low energy, as a result of the narrower PSF at high energy, so the flags are set on the basis of the highest band in which a source is significant. Flag 5 signals confusion and depends on a reference distance θ_ref. Because statistics are better at low energy (enough events to sample the core of the PSF), θ_ref is set to the FWHM there (minimum distance to have two peaks with a local minimum in between in the counts map). At high energy, there are fewer events so θ_ref is set to the larger value 2r₆₈. In the intermediate band, we interpolate between FWHM and 2r₆₈. In the FITS version of the catalog, these flags are summarized in a single integer column (Flags; see Appendix D). Each condition is indicated by one bit among the 16 bits forming Flags. The bit is raised (set to 1) in the dubious case, so that good sources have Flags = 0.

Table 4. Definitions of the Analysis Flags

Flaga	Meaning
1	Source with TS>35 which went to TS < 25 when changing
	the diffuse model (Section 4.6). Note that sources with TS < 35
	are not flagged with this bit because normal statistical
	fluctuations can push them to TS < 25.
2	Moved beyond its 95% error ellipse when changing the diffuse
	model.
3	Flux or spectral index changed by more than 3σ when
	changing the diffuse model. Requires also that the flux change
	by more than 35% (to not flag strong sources).
4	Source-to-background ratio less than 30% in the highest band in
	which TS>25. The background is integrated over πr268
	or 1 deg2, whichever is smaller.
5	Closer than θref from a brighter neighbor. θref is defined in the
	highest energy band in which source TS>25. θref is set to
	26 (FWHM) below 300 MeV, 152 between 300 MeV and
	1 GeV, 084 between 1 GeV and 3 GeV, and 2 r68 above 3 GeV.
6	On top of an interstellar gas clump or small-scale defect in the
	model of diffuse emission.
7	Unstable position determination; result from gtfindsrc outside
	the 95% ellipse from pointlike (see Section 4.2).
8	pointlike did not converge. Position from gtfindsrc.
9	Elliptical quality >10 in pointlike (i.e., TS contour does not
	look elliptical).

Note. ^aIn the FITS version the values are encoded in a single column, with Flag n having value 2⁽ⁿ⁻¹⁾. For information about the FITS version of the table see Appendix D and Section 5.

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The idea for a protocol for population testing in γ-ray source data such as represented by the Fermi 1st-year catalog was introduced by Torres & Reimer (2005), a paper to which we refer for general background. The test was devised to provide high levels of confidence in population classification with small number statistics. Essential to this protocol is the use of Feldman & Cousins (1998) confidence level intervals in a priori, physically selected samples of plausible γ-ray emitters.

Thus, we consider spatial correlations between the 1st-year catalog sources (1451 detections) and a priori selected (on physical grounds or earlier hints in γ-ray data of previous missions) sets of astrophysical sources, details on which are given below. To test for spatial correlations, each of the Fermi sources is described by a centroid position and an uncertainty. The latter is described by an ellipse, with the semiminor and semimajor axes and a position angle; with all these values being taken from the catalog at 95% CL. Those sources that are identified beyond doubt by timing (all γ-ray pulsars and three X-ray binaries) have their positions assigned to be coincident with the corresponding astrophysical sources (as located in other wavelengths or by LAT pulsar timing; Smith et al. 2008). However, for population searches, we maintain the uncertainty of their detections around these positions. In order to test systematics we also analyze the case in which all 1FGL sources have their positional uncertainties enlarged by increasing their corresponding semiminor and semimajor axes by 20%. This is done consistently both for the real set of 1FGL sources, and for each of the simulated sets generated by Monte Carlo (see below), and is meant as a check of the results obtained with nominal uncertainty values.

Let ${\cal C}(A)$ represent the number of coincidences between candidate counterparts for population A and LAT sources. In case of finding the same Fermi detections spatially coincident with several astrophysical objects of the same class, we count all of the coincidences as one. An example of this is seen for millisecond pulsars (MSPs) pertaining to the same globular cluster, e.g., 47 Tuc. To determine the number of excess coincidences above the noise level, i.e., the sources that are expected to correlate by chance, we subtract the background b caused by these random coincidences, given the number and distribution of LAT sources and members of the testing population.

To obtain this latter number, we produce Monte Carlo simulations shuffling the orientation of the elliptical position uncertainty (but not its size) and the centroid position of each of the actual 1FGL sources, thus generating sets of fake LAT source detections. Each simulated 1FGL catalog is constructed to have the total number of sources (1451 detections), and both the longitude and latitude distributions of the real set. In order to obtain the most conservative results we proceed to simulate fake LAT source catalogs by maintaining the latitude (longitude) histogram of the real set with 5 and 10 (30, 60) degrees binning, and subsequently take the largest value for the expected average number of random positional coincidences among them (and this is called b).

The number of excess coincidences above chance associations is then ${\cal E}(A)={\cal C}(A)-b(A)$. This number is used to test the null hypothesis: Population A is not γ-ray emitting at flux-levels detectable by the 1st-year LAT catalog. The predicted number of source coincidences for this hypothesis is equal to 0, and the total expected events if this hypothesis is valid is equal to b. The greater the excess of the real number of correlations over the corresponding b-value for that population, the easier it is to rule out the null hypothesis.

The testing power of a sample of finite size is limited: if using the same set of data, claiming the discovery of one population affects the level of confidence by which one can claim the discovery of a second. In order to control the reliability of our results, we require that the combination of all of our claims be bounded by a probability of 10⁻⁵, which then becomes the total budget ${\cal B}$. This low probability provides an overall significance of about 5σ, which implies individual claims of populations must arise with higher confidence. The total budget can then be divided into individual ones for each population, P_A, P_B, etc., such that $\sum _i P_i={\cal B}.$ Then, population A will be claimed as detected with x% CL if and only if

• 1.  
  the Poissonian cumulative probability (CP) for obtaining the real number of spatial coincidences (or more) as a result of chance coincidence is less than the a priori assigned budget P_A (as opposed to being less than only the larger, total budget) and
• 2.  
  the number of excesses ${\cal E}(A)$ is beyond the upper limit of the corresponding confidence interval for x% CL, from Feldman & Cousins (1998).

The latter value, x%, for each population is then the confidence level obtained using the tables in Feldman & Cousins (1998) for which the upper end of the interval equals the real number of excesses. (With number of events b, background b, recall that in the null hypothesis, there are 0 expected events above background.) The values of P_i for each population are chosen very conservatively: 0.01% B to each of the only two populations that were unambiguously identified in the EGRET catalog, pulsars, and blazars; 0.1% B to MSPs and EGRET-coincident SNRs, which were hinted at in the EGRET catalog; and the rest of the budget equally distributed into those classes where no high-energy γ-ray emission was previously reported.

Finally, we provide brief notes on the populations selected for the test. The selections were made before the start of Fermi operations, as described in the internal LAT-document AM-09067,⁷⁷ which lists each specific source selected in each class.

• 1.  
  Blazars: the selected blazars are a subset of the CGRABS catalog (Healey et al. 2008), which is a uniform all-sky survey of EGRET-like blazars, selected by their figure-of-merit (FOM; see Healey et al. 2008). In total, the CGRABS catalog includes 1625 sources. To assemble the blazar test list we cut the CGRABS catalog at the smallest FoM value for which all high-confidence EGRET blazars of the 3EG catalog (Hartman et al. 1999), which are also "Mattox-blazars" (Mattox et al. 2001), are included (see Sowards-Emmerd et al. 2003, for a comparison of 3EG and "Mattox"-blazars). This cut is at FoM = 0.111, and results in a total of 215 sources and constitutes the blazar list used for the application of the protocol.
• 2.  
  Misaligned jet sources: we start with the compilation of extragalactic jet sources (Liu & Zhang 2002) which has 661 entries collected from the literature (as of 2000 December). This list contains radio galaxies, radio quasars, BL Lac objects, and Seyfert galaxies. For a redshift cut at z < 0.032 (chosen to select a moderate number of the closest members of this class) the list features 51 sources: 34 are classified as radio galaxies (including all γ-ray radio galaxies detected with high confidence so far), 16 as Seyfert galaxies, and one as BL Lac object (not yet detected in γ-rays and not included in the CGRABS list). 3C 111 and 3C 120 are added on the basis of evidence for their detection by EGRET.
• 3.  
  Starbursts: this list includes all starburst galaxies for which in the study of Torres et al. (2004) the combination of gas content, cosmic-ray density, and distance indicated a flux detectable by the LAT in one year.
• 4.  
  Halo dwarf galaxies: the list consists of presently known objects in this category up to 1000 kly. This sample of Milky Way satellite galaxies is useful for probing localized γ-ray excesses due to annihilation of dark matter particles. Dwarf spheroidal galaxies (dSph) may be manifestations of the largest clumps predicted by the CDM scenario.
• 5.  
  Galaxy clusters: we restrict this sample to 30 clusters with the highest values of mass-to-distance-squared, M/d², from the HIFLUCS sample of the Reiprich & Böhringer (2002) sample, complemented by clusters that are interesting individually (Bullet cluster, RX J1347.5), as well as those reported to have radio haloes. Galaxy clusters with reported hard X-ray emission and those predicted from large-scale structure formation simulations to be detectable by the LAT in one year (Pfrommer 2008) are implicitly included in the sample.
• 6.  
  Pulsars: these are selected from the ATNF catalog (Manchester et al. 2005), with a cut on $\dot{E} > 10^{34}$ erg s⁻¹.
• 7.  
  MSPs: these are also selected from the ATNF catalog to have periods less than 10 ms, with the same cut as above in $\dot{E}$.
• 8.  
  Magnetars: we include all 13 known SGRs and AXPs.
• 9.  
  EGRET-SNRs: these are all SNRs that were found to be spatially coincident with an EGRET source, as discussed by Torres et al. (2003).
• 10.  
  TeV shell-type SNRs: these were intended to be tested separately to answer the question whether these objects also emit GeV γ-rays on at least the level of the LAT 1 yr sensitivity. The sample comprised four sources (RX J1713.7−3946, RX J0852−460, RCW 86, and Cas A).
• 11.  
  Star-star binaries: these are from the VIIth Wolf–Rayet (WR) catalog of van der Hucht (2001), requiring 0.001 E_kin,tot/(πd²_L)>10, resulting in a sample of 41 sources. The factor 0.001 is motivated by a reasonable 10% acceleration efficiency, i.e., wind-energy-to-relativistic-particle-energy-conversion efficiency, and 1% radiative efficiency in these environments, while the factor 10 relates to the LAT sensitivity anticipated at the time of protocol application.
• 12.  
  Star-compact object binaries: these are all known γ-ray binaries and microquasars (MQs).
• 13.  
  Binary pulsars: we include all objects known in this category.
• 14.  
  Globular clusters: these are known to contain a relatively large number of MSPs whose individual and collective emission in the X-ray and γ-ray energy bands may be detectable by the LAT. Here the aim is to test for the collective emission from MSPs, since MSPs are principally able to accelerate leptons at the shock waves originated in collisions of the pulsar winds and/or inside pulsar magnetospheres, and inside a globular cluster these are subsequently able to Comptonize stellar and microwave background radiation. The globular clusters in this list are restricted to be closer than 6 kpc.

6.1.1. Protocol Results

Table 8 summarizes the results of the application of the protocol for population identification to the 1FGL catalog. The first column shows the name of the test population. The second column (N_p) shows the number of astrophysical objects included in the test of the population. The third column shows the expected number of random coincidences (or background, b(A)) between the 1451 LAT sources in the 1FGL catalog, and the N_p candidates in each population as explained above. The fourth column (${\cal C}$) gives the actual number of positional coincidences between the members of each population and the 1FGL catalog. The fifth column shows the CP of obtaining ${\cal C}$ or more positional coincidences between 1FGL sources and the astrophysical candidates purely by chance. The sixth column shows the a priori assigned probability (P(A)) for discovery of each population, the sum of all P giving the total budget for population discovery. The seventh column answers yes or no to the question of whether the actual random probability for an equal or a larger number of excesses to occur by chance (CP) is larger than the a priori assigned budget for this population. For the cases in which this answer is yes, the last column gives the confidence level of the detection of the corresponding population; alternatively stated, this is the confidence with which we are ruling out the hypothesis that such a population is not present among the LAT detections (the null hypothesis).

Table 8. Results from Application of the Population Protocol

Test Population	Np	b	${\cal C}$	CP	P	(CP <P)?	CL
Galactic populations
Pulsars	215	1.440	30	5.3 × 10−29	1.0 × 10−9	Yes	>99.999%
Millisecond pulsars	23	0.050	7	1.5 × 10−13	1.0 × 10−8	Yes	99.89%
EGRET SNRs	23	1.590	13	1.5 × 10−8	1.0 × 10−8	No	...
TeV SNRs	4	0.920	3	6.6 × 10−2	9.9 × 10−7	No	...
Magnetars	13	0.120	0	...	9.9 × 10−7	...	...
WR binaries	41	0.260	0	...	9.9 × 10−7	...	...
MQ/γ-ray bin.	17	0.140	3	4.1 × 10−4	9.9 × 10−7	No	...
Binary pulsars	10	0.040	0	...	9.9 × 10−7	...	...
Globular clusters	29	0.240	4	1.1 × 10−4	9.9 × 10−7	No	...
Extragalactic populations
Blazars	215	0.480	61	0.0	1.0 × 10−9	Yes	>99.999%
Misaligned jet sources	53	0.150	5	5.5 × 10−7	9.9 × 10−7	Yes	99.25%
Starbursts	15	0.050	4	2.5 × 10−7	9.9 × 10−7	Yes	97.89%
Galaxy clusters	48	0.150	0	...	9.9 × 10−7	...	...
Dwarf spheriodals	18	0.070	0	...	9.9 × 10−7	...	...

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Not surprisingly, the source populations already conclusively identified in the EGRET era are found with the highest confidence in the investigated 1FGL coincidences even when very strict thresholds were chosen for associations to be claimed, justifying a posteriori the very low budget assigned to these populations.

To study the sensitivity of the results to the sizes of the error ellipses for the sources, we also evaluated the coincidences and chance probabilities when the extents of the ellipses are increased by (very conservative) 20%. The findings for misaligned jet sources and EGRET SNR populations were not robust against this change. When the sizes of the source location regions are increased the probabilities of chance associations necessarily increase for any given source. For the misaligned jet source population, this was enough to push the population below the a priori probability budget. For the EGRET SNRs, enlarging the source location regions had the offsetting effect of adding one more coincidence with 1FGL sources so the net result was a chance probability below the a priori threshold.

Spatial coincidences found between LAT sources and individual millisecond PSRs (MSPs) and those tested via their globular cluster environment are related in the following sense. Whereas for an isolated MSP there are not many alternative scenarios for producing detectable γ-ray emission, and in fact several of them have been detected and identified by their γ-ray periodicity (Abdo et al. 2009b), in the case of coincidences with globular clusters (e.g., 47 Tuc; Abdo et al. 2009d), an ambiguity exists as long as no pulsed emission from one of the MSPs in the globular cluster is found. Thus, the aim was to search not only for the existence of a population of MSPs, but also to distinguish between environments in which MSPs are found as populations as well. The fact that the class of globular clusters is not detected as a population but the MSPs are can be interpreted as related to the size of the population of MSPs in a globular cluster. Based on the membership of the 1FGL catalog apparently only those clusters hosting a large number of MSPs were found coincident with catalog sources (e.g., Ter 5, 47 Tuc, M28), too small a number to claim them as a population of γ-ray sources in the framework of this test.

The previously reported relation between EGRET sources and SNRs (e.g., Sturner & Dermer 1995; Torres et al. 2003) cannot be confirmed with 1FGL sources using the present test. The chance probability of the number of coincidences seen with 1FGL sources is very small, but the a priori probability budget assigned for this population was smaller still (Table 8). We note also that pulsars have been detected by the LAT in some of the SNRs in our sample.

The fact that the misaligned jet sources are found as a population supports the individual 1FGL source association findings, although the test did not distinguish between radio galaxies and Seyferts, and does not show a significant correlation with ranking via radio core flux. It is to be noted that this class is found only when the uncertainties in the positions of the LAT sources are not enlarged, when the CP is low, but found to be just below the a priori assigned budget for the class.

Starburst galaxies are detected as a population in LAT data, in numbers that add up to the noted individual detections of M82 and NGC 253, and suggests the potential for future discoveries.

Particularly interesting is the non-detection of galaxy clusters, which is not only a statistically significant result, but even a case of zero coincidences. With the absence of even a single galaxy cluster coincidence from the tested sample, we can conclude that X-ray bright nearby galaxy clusters and those exhibiting a radio halo do not constitute a source population above the 1FGL sensitivity limit: the models that we used to select the candidate galaxy clusters overestimate the energy conversation into particle acceleration at GeV and greater energies.

Dwarf spheroidals are not found coincident with LAT sources either. The result is compatible with the null hypothesis that this population does not exist above the sensitivity limit of the catalog, although in some dark-matter-based models of γ-ray production they would have been expected to be seen at the sensitivity level of the 1FGL catalog. The Galactic populations of magnetars, binary pulsars, and WR binaries are similarly not detected. The latter is particularly interesting; whereas positional associations of WR stars with LAT sources can be found using the extensive list of known WR stars, those having the largest wind energies do not present any correlation and thus we disregard WRs as γ-ray emitters at the sensitivity limit of the 1FGL catalog.

Finally, we note that whereas we found three sources correlated with MQ-γ-ray binaries (all of them secured by timing) the population as such is not detected. We can conclude that the binaries identified in the 1FGL catalog are certainly special objects, but probably not archetypal for a population of similar objects. Similarly, the same occurs for TeV SNRs: we find some correlated objects, but not enough to claim a population discovery due to the large expected background (b) for this population.

Our approach to automated source association follows closely that used for the BSL, although we enlarged our database of catalogs of potential counterparts and improved our calibration scheme to control more precisely the expected number of false associations. The association procedure, which follows essentially the ideas developed by Mattox et al. (1997) for the identification of EGRET sources with flat-spectrum radio sources, is described in detail in Appendices B and C. Here we only summarize the essential steps of the automated source association procedure.

The automated source association is based on a list of catalogs that contain potential counterparts of LAT sources. This list has been compiled based either on prior knowledge about classes of high-energy γ-ray emitters or on theoretical expectations. In total, 32 catalogs (some of which are subselections from 24 primary catalogs; see Table 9) have been searched for counterparts covering AGNs (and in particular blazars), nearby and starburst galaxies, pulsars and their nebulae, massive stars and star clusters, and X-ray binaries. For the BSL analysis, only 14 catalogs were searched. In addition, some of the catalogs have been enlarged somewhat (e.g., BZCAT, CRATES, and SNR total; cf. Table 9 with Table 1 of the BSL). Furthermore, since MSPs have now been established as sources of γ-rays, we split the ATNF catalog into normal pulsars and MSPs. For the latter we require the pulse period P< 0.1 s and the period derivative $\dot{P} < 10^{-17}$ s/s. This results in 139 objects. The remaining 1545 objects are considered as normal pulsars. We divided those into high and low $\dot{E}/d^2$ categories, the latter being defined as $\dot{E}/d^2 \le 5 \times 10^{33}$ erg kpc⁻² s⁻¹. This results in 84 high $\dot{E}/d^2$ pulsars and 1461 low $\dot{E}/d^2$ pulsars. Note that this energy-flux selecting is unrelated to the selection on spin-down luminosity applied for the population study in Section 6.1. For SNRs, we divided the Green catalog into two lists, one containing all objects that can be considered as point-like for the LAT (157 objects) and one containing extended SNRs (117 objects, diameters greater than 20'); these subsets are denoted small and large, respectively. We search for counterparts at radio frequencies using the VLBA Calibrator Survey and at TeV energies using the TeVCat catalog. For the latter we divide the TeVCat catalog into small and large angular size subsets at 40'. We also search for coincidences between 1FGL sources and BSL (0FGL), AGILE, and EGRET sources. The complete list of catalogs, the numbers of objects they contain, and the references are presented in Table 9.

Table 9. Catalogs Used for the Automatic Source Association and Results

Name	Objects	Pprior	Nass	Nfalse	$\langle \hat{N}_{\rm false} \rangle$	Ref.
LAT pulsars	56	0.1	56	n.a.	0.4	1
High $\dot{E}/d^2$ pulsars	84	0.024	24	n.a.	0.6	2
Low $\dot{E}/d^2$ pulsars	1461	0.011	1	n.a.	0.3	2
Millisecond pulsars	139	0.278	20	n.a.	1.0	2
Pulsar wind nebulae	69	0.049	27	0.3	0.9	1
High-mass X-ray binaries	114	0.010	3	n.a.	0.3	3
Low-mass X-ray binaries	187	0.050	8	0.4	0.5	4
Small (<20') SNRs	157	0.021	11	0.7	0.7	5
O stars	378	0.015	1	<0.1	<0.1	6
WR stars	226	0.013	11	0.3	0.2	7
LBV stars	35	0.026	2	0.3	0.6	8
Open clusters	1689	0.013	1	0.1	0.4	9
Globular clusters	147	0.272	8	<0.1	0.5	10
Nearby galaxies	276	0.066	5	0.4	0.4	11
Starburst galaxies	14	0.5	2	<0.1	<0.1	12
Blazars (BZCAT)	2837	0.308	487	8.9	6.8	13
Blazars (CGRaBS)	1625	0.238	282	4.7	4.1	14
Blazars (CRATES)	11499	0.333	490	17.2	17.8	15
BL Lac	1122	0.224	218	2.8	2.8	16
AGN	21727	0.021	11	0.7	0.8	16
QSO	85221	0.166	147	7.3	4.9	16
Seyfert galaxies	16343	0.041	24	2.0	1.6	16
Radio-loud Seyfert galaxies	29	0.1	4	<0.1	<0.1	1
VLBA calibrator survey	4558	0.266	484	11.5	10.1	17
Small (<40') TeV sources	92	0.037	42	0.6	0.8	18
Large (>40') TeV sourcesb	11	n.a.	13	n.a.	7.5	18
Large (>20') SNRsb	117	n.a.	48	n.a.	18.1	5
Dwarf galaxiesb	14	n.a.	7	n.a.	2.1	1
1st AGILE cataloga	47	n.a.	52	n.a.	18.6	19
3rd EGRET cataloga	271	n.a.	107	n.a.	25.4	20
EGR cataloga	189	n.a.	66	n.a.	9.1	21
Bright source list (0FGL)	205	n.a.	195c	n.a.	3.9	22

References. (1) Collaboration internal; (2) Manchester et al. 2005; (3) Liu et al. 2006; (4) Liu et al. 2007; (5) Green 2009; (6) Maíz-Apellániz et al. 2004; (7) van der Hucht 2001; (8) Clark et al. 2005; (9) Dias et al. 2002; (10) Harris 1996; (11) Schmidt et al. 1993; (12) Thompson et al. 2007; (13) Massaro et al. 2009; (14) Healey et al. 2007; (15) Healey et al. 2008; (16) Véron-Cetty & Véron 2006; (17) Kovalev 2009b and ref. therein, http://astrogeo.org/vlbi/solutions/2009/c_astro/; (18) http://tevcat.uchicago.edu/ ("Default" and "Newly Announced" categories, version 3.100); (19) Pittori et al. 2009; (20) Hartman et al. 1999; (21) Casandjian & Grenier 2008; (22) Abdo et al. 2009m. ^aCatalog for which the location uncertainties of the counterparts are greater than the location uncertainties for the 1FGL sources. ^bCatalog for which the counterparts are spatially extended sources. ^cSee Section 6.2.10.

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For each catalog in the list, we make use of Bayes' theorem to compute the posterior probabilities P_ik that an object i from the catalog is the correct association of the LAT source k:

$$\begin{equation} P_{ik} = \left(1 + \frac{1-P_{\rm prior}}{P_{\rm prior}} 2 \pi \rho _k a_k b_k e^{\Delta _k} \right)^{-1} \,. \end{equation} \tag{ 8 }$$

P_prior is the prior probability that counterpart i is detectable by the LAT, a_k and b_k are the axes of the ellipse at 1σ, smaller than the semimajor and semiminor axes at 95% confidence by a factor $\sqrt{-2\log (0.05)} = 2.45$, ρ_k is the local counterpart density around source k, and

$$\begin{equation} \Delta _k = \frac{r^2}{2} \left(\frac{\cos ^2(\phi -\phi _k)}{a_k^2} + \frac{\sin ^2(\phi -\phi _k)}{b_k^2} \right) \end{equation} \tag{ 9 }$$

for a given position angle ϕ between LAT source k and the counterpart i, ϕ_k being the position angle of the error ellipse, and r being the angular separation between LAT source k and counterpart i (see Appendix B). For each catalog, prior probabilities P_prior are assigned so that

$$\begin{equation} \mbox{$N_{\rm false}$}= \sum _{P_{ik} \ge P_{\rm thr}} (1-P_{ik}) \end{equation} \tag{ 10 }$$

gives the expected number of false associations that have posterior probabilities above the threshold P_thr (see Appendix C). The corresponding prior probabilities are quoted in Table 9 (Column 3) for all catalogs.

For the automated association of the 1FGL catalog we set P_thr = 0.8, which means that each individual association has a ⩽20% chance of being spurious. This is different from the approach we took for the BSL paper where we constrained the expected number of false associations for each catalog to N_false ⩽ 1, which imposed a relatively tight constraint on source classes with large numbers of associations (such as blazars and pulsars) while source classes with only few associations had a relatively loose constraint. Now, each individual association stands on an equal footing by having a well-defined probability for being spurious. We note that the First LAT AGN Catalog (Abdo et al. 2010l) applies the same association method as we use here but includes associations for AGNs down to P_thr = 0.1.

For a number of catalogs in our list the Bayesian method cannot be applied since either (1) the location uncertainty of the counterpart is larger than the location uncertainty for the 1FGL source, or (2) the counterpart is an extended source; see notes to Table 9. In the first case, we consider all objects i as associations for which the separation to the LAT source k is less than the quadratic sum of the 95% confidence error radius of counterpart i and the semimajor axis α_k. For the bright EGRET pulsars Crab, Geminga, Vela, and PSR J1709−4429, we also list associations, even though in many cases the EGRET-measured locations are formally inconsistent with the positions of the pulsars; there is no doubt that EGRET detected these pulsars. In the second case, we assume that the counterparts have a circular extension and consider all objects i as associations for which the circular extension overlaps with a circle of radius α_k around the LAT source k.

6.2.1. Automated Association Summary

The results of the automated association procedure for each of the external catalogs are summarized in Table 9. For each catalog, we give the number

$$\begin{equation} \mbox{$N_{\rm ass}$}= \sum _{P_{ik} \ge P_{\rm thr}} 1 \end{equation} \tag{ 11 }$$

of LAT sources that have been associated with objects in a given catalog (Column 4). Furthermore, we compute the expected number of false associations N_false using Equation (10) for those catalogs which have been associated with the Bayesian method (Column 5). We cannot give meaningful results for pulsars and high-mass X-ray binaries since, for identified objects, the positions have been fixed in the catalog to their high-precision locations (Section 4), and consequently, their posterior probabilities are by definition 1. However, we alternatively estimated the expected number of false associations using Monte Carlo simulations of 100 realizations of fake LAT catalogs, for which no physical associations with counterpart catalog objects are expected (see Appendix C). We quote the resulting estimates $\langle \hat{N}_{\rm false} \rangle$ in Column 6 of Table 9. We find $\mbox{$N_{\rm false}$}\approx \mbox{$\langle \hat{N}_{\rm false} \rangle $}$ which confirms that the posterior probabilities computed by the automatic association procedure are accurate (otherwise Equation (10) would not hold).

In total we find that 821 of the 1451 sources in the 1FGL catalog (56%) have been associated with a least one non-γ-ray counterpart by the automated procedure at the 80% confidence level. Seven hundred and seventy nine 1FGL sources (54%) have been associated using the Bayesian method while the remaining 42 sources are spatial coincidences based on overlap of the error regions or source extents. From simulations we expect that 57.3 among the 821 sources (7%) are associated spuriously. Considering only the Bayesian associations, 37.5 among the 779 sources (5%) are expected to be spurious. In the following we discuss the automated association results in some detail. Associations with TeV sources are discussed in Section 7.

6.2.2. Blazars

6.2.3. Other AGNs

6.2.4. Normal Galaxies

6.2.5. Pulsars, Pulsar Wind Nebulae, and Globular Clusters

6.2.6. Supernova Remnants

6.2.7. Association of 1FGL J1745.6−2900c with the Galactic Center

6.2.8. X-ray Binaries

6.2.9. O Stars, Wolf–Rayet Stars, Luminous Blue Variable Stars, and Open Clusters

6.2.10. Associations with Bright Source List Sources

Table 9 reports 195 associations between the 205 BSL (0FGL) sources and those of the 1FGL catalog. Strictly applied, the automated association procedure found 186 associations. We have manually adopted nine more for BSL sources confidently associated with blazars or pulsars. Regarding the latter, the 1FGL positions are formally 0, indicating that the source positions were assigned to the more-accurately known positions of the pulsars (see Section 4), affecting the statistical relationship between the 0FGL and 1FGL positions for these.

The 10 BSL sources that do not have clear counterparts in the 1FGL catalog are listed in Table 10. Each is in the Galactic ridge, where for the 1FGL catalog analysis we have recognized difficulties detecting and characterizing sources (Section 4.7).

Table 10. Bright Source List (0FGL) Sources Without 1FGL Counterparts

Name 0FGL	l	b
J1106.4−6055	29052	−060
J1115.8−6108	291.66	−0.38
J1615.6−5049	332.35	−0.01
J1622.4−4945	333.87	−0.01
J1634.9−4737	336.84	−0.03
J1741.4−3046	357.96	−0.19
J1821.4−1444	16.43	−0.22
J1836.1−0727	24.56	−0.03
J1900.0+0356	37.42	−0.11
J1024.0−5754	284.35	−0.45

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6.2.11. Associations with EGRET and AGILE Sources

Firm identifications, indicated in the main table by capitals in the Class column, require more than a high-probability positional association. The strongest test for identification is time variability, either periodicity or correlation with variability seen at another wavelength. The 56 pulsars that have class PSR all show high-confidence (statistical probability of chance occurrence less than 10⁻⁶) periodicity caused by the rotation of the neutron star (Abdo et al. 2010b, 2010c, 2010m; Saz Parkinson et al. 2010). Similar confidence levels apply to the three X-ray binary systems whose orbital periods are detected in the LAT data: LSI +61 303 (Abdo et al. 2009j), LS5039 (Abdo et al. 2009n), and Cygnus X-3 (Abdo et al. 2009p).

With the large number of blazars detected by the LAT and the significant variability seen in many of these, the search for correlated variability that can provide firm identifications is a major effort that has not yet been carried out systematically for the LAT data. We have therefore chosen to list as firm identifications only those blazars for which publications exist showing such variability. These are just 3C 273 (Abdo et al. 2010f), 3C 279 (Iafrate et al. 2009), 3C 454.3 (Abdo et al. 2009e), PKS 1502+106 (Abdo et al. 2010k), and PKS 1510−08 (e.g., Tramacere 2008). Additional studies will undoubtedly expand this list.

Another approach to firm identification, slightly less robust than time variability, is morphology: finding spatial extent in a γ-ray source that matches resolved emission at other wavelengths. Some SNRs have measurable spatial extents in the LAT data and can be considered firm identifications; here we cite W44 (Abdo et al. 2010h), W51C (Abdo et al. 2009i), and IC 443 (Abdo et al. 2010i). The analyses for the 1FGL catalog assume point-like emission, and so the positions and fluxes are not as well characterized as they would be in analyses that take into account the finite angular extents of these sources. A special case is 1FGL J1322.0−4515, which appears to be part of one of the lobes of the emission from radio galaxy Centaurus A (Abdo et al. 2010d). Studies of source morphology are ongoing.

The LMC is an extreme example (Abdo et al. 2010j) of the issue of analyzing an extended source with tools designed to find point sources. As described in Section 6.2.4, the catalog contains five sources that are likely to be related to the diffuse γ-ray emission of the LMC. The LMC can be considered a firmly identified LAT source, but it is not a point source or an ensemble of point sources.