THE FERMI HAZE: A GAMMA-RAY COUNTERPART TO THE MICROWAVE HAZE

There are three well-known mechanisms for generating gamma rays at the energies observed by Fermi. First, at low (∼1 GeV) energies, gamma-ray emission is dominated by photons produced by the decay of π⁰ particles generated in the collisions of cosmic ray protons (which have been accelerated by SNe) with gas and dust in the ISM. Second, relativistic electrons colliding with nuclei (mostly protons) in the ISM produce bremsstrahlung radiation. Finally, those same electrons interact with the ISRF and IC scatter CMB, infrared, and optical photons up to gamma-ray energies. A schematic of the relative importance of these emission mechanisms in the Galactic plane (|ℓ| ⩽ 30, |b| ⩽ 5) generated by the GALPROP code, version 50p (Strong et al. 2000; Porter & Strong 2005; Strong et al. 2007), is shown in Figure 2.

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Since π⁰ gammas and bremsstrahlung are produced by interactions of protons and electrons (respectively) with the ISM, these emission mechanisms should be morphologically correlated with other tracers of the ISM, such as the Schlegel, Finkbeiner, & Davis (SFD) dust map based on 100 μm far IR data (Schlegel et al. 1998). The π⁰ gamma-ray intensity scales with the ISM volume density times the proton CR density, integrated along the line of sight. In the limit where the proton cosmic ray spectrum and density is spatially uniform, the ISM column density is a good tracer of π⁰ emission. Likewise, for a uniform electron spectrum, it is a good tracer of bremsstrahlung. Because our analysis is limited to |b|>5°, much of the emission we see is within a few kpc, so the assumption of uniform CR density is more valid than it would be for the entire Galaxy, particularly for protons, which have much larger propagation lengths than electrons.

Since our goal is to search for an IC emission component with a morphology which roughly matches the microwave haze (i.e., centered on the GC, roughly spherical, and about 20°–40° in radius), we now attempt to remove the π⁰ emission from the maps shown in Figure 1, using the same template fitting technique used in Finkbeiner (2004a) and Dobler & Finkbeiner (2008). We perform multiple types of template fits. Type 1 uses only the Fermi map itself at 1–2 GeV which roughly traces π⁰ emission because the gamma-ray sky at those energies is dominated by π⁰ gammas (with subdominant contributions from bremsstrahlung and IC). Type 2 uses only the SFD dust map which is a direct tracer of ISM density and so should roughly map where the π⁰ gammas and much of the bremsstrahlung are produced—again, up to some uncertainty involving the line-of-sight distribution. In each case, a uniform background is included in the fit, making our results insensitive to zero-point offsets in the maps.

We model the Fermi map at energy E, F(E), as a linear combination of template maps, F_model = Tc_T, where F_model is a column vector of N_pix unmasked pixel values, T is the N_template × N_pix template matrix, and the correlation coefficients c_T are chosen to minimize the mean squared residual,

$$\begin{equation} \langle (F-F_{\rm model})^2 \rangle = \langle (F-T c_T)^2 \rangle, \end{equation} \tag{ 1 }$$

averaging over pixels. The least-squares solution,

$$\begin{equation} c_{T}(E) = (T^{\rm T}T) ^{-1} \times (T^{\rm T}F(E)), \end{equation} \tag{ 2 }$$

yields the template correlation coefficients at each energy. In this fit, we mask out the Fermi three-month point-source catalog as well as the LMC, SMC, Orion-Barnard's Loop, and Cen A (see Appendix B for details). We also mask all pixels with Galactic latitude |b| < 5°. Cross-correlations are done over unmasked pixels and for several different ranges in ℓ: eight longitudinal slices that have |ℓ| ⩾ 75° (to avoid the GC) and width 30° (regions 1–7) as well as one region with 75° < ℓ < 285° (region 8). Note that region 8 is the union of regions 1–7; regions 1–7 are fit individually to show the variation in the fit spectrum, and region 8 is fit to obtain the mean outer Galaxy signal. With these correlation coefficients, we define the residual map to be

$$\begin{equation} R_{T}(E) = F(E) - c_{T}(E) \times T. \end{equation} \tag{ 3 }$$

To the extent that the templates in T match the morphology of the π⁰ and bremsstrahlung gammas, the residual map will include only IC emission.

Figure 3 shows the resultant residual maps using the SFD map as a morphological tracer of π⁰ emission for the region 8 fits. The most striking feature of the difference maps is the extended emission centered around the GC and extending roughly 40° in b. The morphological correlation between the WMAP synchrotron and the R_SFD(5–10 GeV) is striking as is shown in Figure 4. Here the 41 GHz synchrotron (haze plus Haslam-correlated emission, see Haslam et al. 1982; Dobler & Finkbeiner 2008) is shown side by side with the 5–10 GeV Fermi residual map with the mask used in the Dobler & Finkbeiner (2008) microwave analysis overlaid for visualization.

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Figure 5 shows residual maps using the 1–2 GeV Fermi maps as a template for the π⁰ emission. This sort of "internal linear combination" has the advantage that π⁰ emission cancels out as long as the shape of the proton CR spectrum is the same everywhere—it does not rely on the proton CR density to be uniform. The residual maps look largely similar to the case with the SFD template regressed out, but there are some notable differences. In particular, using this 1–2 GeV template, the IC haze has a slightly "taller" appearance. This seems to be due to a disk-like component that is present in the 1–2 GeV maps but not in the dust map. This is probably because the 1–2 GeV Fermi map is not entirely π⁰ emission, but also contains bremsstrahlung and IC components generated by electrons accelerated by SNe in the disk. The result is that when this lower energy map is regressed out (i.e., cross-correlated and subtracted) from higher energy maps, this emission component is subtracted along with the dust-correlated emission. Conversely, the SFD map contains only emission from dust grains and not relativistic electrons; while the bremsstrahlung from those electrons largely traces the gas distribution, the IC does not, and so is not regressed out when using the SFD template.

Despite the significant shot noise in the Fermi map, Figure 4 shows that there is a clear morphological correlation between the microwave haze and the gamma-ray haze. Of course, we do not expect the morphologies to agree perfectly since the microwave haze is generated by interactions of electrons with the magnetic field while this IC haze is due to interactions of the electrons with the ISRF. Nevertheless, this is evidence that the microwave haze seen in the WMAP data is indeed synchrotron and is not some other component such as free–free emission or spinning dust.

Because the solution to Equation (2) minimizes the variance of the (zero mean) residual map R(E), the presence of the IC haze affects the coefficients c_T(E), since there is non-zero spatial correlation between the templates and the IC haze. To relax the stress on the fit, we expand our template analysis with a third type. Type 3 uses four templates in T: the SFD map to trace π⁰ and bremsstrahlung emission, the 408 MHz Haslam et al. (1982) map which is dominated by radio wavelength synchrotron and thus roughly traces soft spectrum electrons which produce soft IC and bremsstrahlung, and a bivariate Gaussian of width σ_ℓ = 15° and σ_b = 25°. We note that this template is chosen to roughly match the morphology in Figure 5 and has no other physical motivation. We also use a uniform template to fit out the isotropic background signal in the maps, again, making our results insensitive to zero points. Lastly, for this fit we use all values in ℓ (region 9).

Note that since the bremsstrahlung originates from interactions of the electrons with the ISM, its spatial distribution depends on both the gas density and the cosmic ray electron density; consequently, some contribution from bremsstrahlung will be present in both the SFD-correlated and Haslam-correlated emission.

The previous fits were done with uniform weighting and assuming Gaussian errors, minimizing χ². For the Type 3 fit we do a more careful regression, maximizing the Poisson likelihood of the four-template model in order to weight the Fermi data properly. In other words, for each set of model parameters, we compute the log likelihood

$$\begin{equation} \ln {\mathcal L} = \sum _i k_i\ln \mu _i - \mu _i - \ln (k_i!), \end{equation} \tag{ 4 }$$

where μ_i is the synthetic map (i.e., linear combination of templates) at pixel i, and k is the map of observed counts. Note that the last term does not depend on the model parameters. It may appear strange at first to compute a Poisson likelihood on smoothed maps, however, the smoothing is necessary to match PSFs at different energies and with various templates (some of which have lower resolution than Fermi in the energy range of interest). The smoothing itself does not pose any problems for relative likelihoods, as we show in Appendix C.

However, we must keep in mind that the uncertainties derived in this way are the formal errors corresponding to $\Delta \ln {\mathcal L} = 1/2$, which would be 1σ in the case of Gaussian errors. The error bars plotted are simply the square root of the diagonals of the covariance matrix. This estimate of the uncertainty should be accurate at high energies, where photon Poisson noise dominates. At low energies, although the formal errors properly reflect the uncertainty in the fit coefficients for this simple model, the true uncertainty is dominated by the fact that the four-template model is not an adequate representation of the data.

Figure 6 shows the skymaps and best-fit solution including the residual map at 10–20 GeV while Figure 7 shows residual maps at other energies. It is clear from these residuals that the template fitting produces a relatively good approximation of the gamma-ray data over large areas of sky. Furthermore, Figure 7 shows that not including the bivariate Gaussian template for the IC haze yields a statistically significant residual toward the center indicating that a model including an IC haze is a better match to the data then one without. The prominent North Polar spur feature in the Haslam map, which is thought to originate from synchrotron emission from electrons in Loop I (Large et al. 1962), is oversubtracted in each case, because the North Polar spur is brighter in the Haslam map than in the gamma-ray maps (i.e., the ratio of synchrotron microwaves to IC gamma rays in the North Polar spur is larger than in the rest of the Haslam map).¹⁰ This may be due to different ISRF, B-field, and ISM density values in Loop I relative to the inner Galaxy (since Loop I is thought to be quite nearby), or may be due to a softer-than-usual electron spectrum in Loop I, since the electrons producing the synchrotron measured in the Haslam map are much lower energy than those producing IC gamma rays at energies measured by Fermi.

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Figure 8 shows these same residual maps, but with a smoothing of 10° which is on the order of the scale of the haze emission. With this large smoothing, smaller scale variations (due to individual photons at high energies) are smoothed over and the residual maps clearly show that the haze is a robust feature at all energies.

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In Figure 9, we show the results of a four-template fit using the 1–2 GeV map instead of SFD to trace the π⁰ emission. This figure shows that the haze is not due to the SFD template being an imperfect tracer of π⁰ emission. If the proton cosmic ray density is higher toward the GC, then SFD may systematically underestimate the π⁰ emission there and perhaps the gamma-ray haze is the result. However, Figure 9 shows that this is clearly not the case. The 1–2 GeV template includes the effects of proton cosmic ray density variations (as well as line-of-sight gas density effects) and the haze remains as a robust residual. That is, the haze is not due to imperfect templates as suggested by Linden & Profumo (2010).

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In Figure 10, we show the haze amplitude (residual maps from the Type 3 fit plus the correlation-coefficient-weighted bivariate Gaussian template) as a function of Galactic latitude and for the longitudinal bin: −15° < ℓ < 15°. Although the data are noisy, the figure shows that the Fermi haze dies off by roughly b ∼ 50° in all energy bands. For comparison, we show the same plot for just the residual map. The figure also shows the amplitude as a function of longitude for the latitudinal bin −20° < b < −10°. The more rapid falloff of the haze emission with ℓ compared to b indicates a haze morphology elongated in the b direction.

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There are two important features to note about Figures 3 through 7. First, the IC haze has a spectrum which is harder than the other IC in the Galaxy. This is evidenced by the fact that using the 1–2 GeV Fermi map as a template removes the IC emission from disk electrons but the haze IC fades more slowly with energy. Second, the IC haze is morphologically distinct from either the π⁰ emission or IC and bremsstrahlung from the disk electrons. These two facts taken together strongly suggested that the electrons responsible for the microwave and gamma-ray haze are from a separate component with a harder spectrum than SN shock-accelerated electrons.

So far, we have only done template fits of maps of data to other maps of data. Using the SFD dust map as a tracer of π⁰ emission is equivalent to assuming that the proton CR spectrum and density are spatially uniform, and the dust/gas ratio is constant. Using the Haslam map to estimate IC emission is equivalent to assuming that the B-field and ISRF have similar spatial variation, and neglecting the anisotropies in both IC and synchrotron emission. Consequently, there have been concerns (Linden & Profumo 2010) that this template-based analysis might introduce spurious large-scale residuals. To address these concerns, we investigate whether the Fermi diffuse model contains a structure similar to the haze emission.

The Fermi team provides a model of diffuse gamma-ray emission consisting of maps sampled at 30 energy bins from 50 MeV to 100 GeV.¹¹ These maps are based on template fits to the gamma-ray data and also include an IC component generated by the GALPROP cosmic ray propagation code.

The difference between the model and data is shown in Figure 11. We have interpolated the model to the energy ranges of interest and performed the simple one-template fit to the data (analogous to our Type 1 and 2 fits described above). This fit allows for any error in the normalization of the diffuse model; the fit coefficients are within a few percent of unity in every case.

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At each energy range, the haze is clearly visible in the residual maps. GALPROP uses the standard inhomogeneous ISRF model (Porter & Strong 2005), making it unlikely that the observed residual is due to the expected spatial variation of the ISRF. Furthermore, the diffuse model includes a disk-like injection of primary electrons and estimates for the line-of-sight density variations of the ISM. The fact that the haze residual remains suggests that, given the propagation model included in GALPROP, disk-like sources will not produce the observed IC emission, nor will expected variations in the gas density and proton CR density along the line of sight.

While the morphology of the gamma-ray haze is indicative of IC emission from the microwave haze electrons, we now attempt to estimate the spectrum of this emission. This is difficult for several reasons. First, π⁰ emission is dominant (or nearly so) at most energies in Fermi's energy range. Thus, in a given region, we must estimate the spectral shape and amplitude of the π⁰ emission in order to subtract it from the total. Second, the total number of photons measured by Fermi decreases rapidly with increasing energy. For example, in the inner Galaxy (|ℓ| ⩽ 30, |b| ⩽ 5), there are only ∼4300 photons between 10 and 100 GeV, resulting in substantial Poisson errors. Lastly, there is also significant isotropic background due to extra-galactic emission and charged particle contamination, including heavy nuclei at high energies. We will show that the isotropic backgrounds are manageable, and present two types of spectra: template coefficients c_T(E) and total fluxes dN/dE.

The Fermi data contain gamma rays from an unresolved extragalactic signal with dN/dE ∼ E^−2.45 (Ackermann 2010) as well as particle contamination. We make no attempt to separate the extragalactic gamma-ray signal from the particle contamination. Instead, we measure the spectrum of this (nearly) isotropic background in eight regions at high latitude and test the assumption of isotropy. Specifically, we take combinations of longitude ranges of −180° < ℓ < −90°, −90° < ℓ < 0°, 0° < ℓ < 90°, and 90° < ℓ < 180° together with latitude ranges of −90° < b < −60° and 60° < b < 90° for the 4 × 2 = 8 regions. The regions are chosen to be far from the plane to avoid contamination from Galactic emissions. We use the point-source masked maps described above, binned in 12 energy bins from 0.3 to 300 GeV. The energy bins, mean background, etc. are given in Table 1 and the eight spectra, along with the inverse-variance weighted mean, are plotted in Figure 12. We note that the error bars are the standard deviation of the eight regions, not the standard deviation of the mean. To be conservative, we use this standard deviation as the uncertainty in the background in the remaining stages of the analysis. Future work by the Fermi team to understand both the particle contamination and the gamma-ray sky will likely reduce this error in the background substantially.

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Figure 13 shows c_T(E) × 〈T〉 for the two templates and regions 1–7 used in the Type 1 and Type 2 fits along with the model π⁰ spectrum from GALPROP. It is clear from the figure that the cross-correlation technique produces π⁰ spectra that are remarkably similar to the model spectrum at low energies, while at high energies the cross-correlation spectrum is slightly higher than the model spectrum. This could be due to a number of reasons such as non-zero spatial correlation between the templates and the harder spectrum haze IC, contamination from background events such as heavy nuclei, or uncertainties in the π⁰ emission model. Of these, the first is most likely since the cross-correlation between the templates and a nearly isotropic background is likely small and since the spectrum of π⁰ gammas is quite well known.

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Template-correlated spectra for the Type 3 template fit are shown in Figure 14. Here the correlation coefficients are weighted by the mean of each template in the "haze" region (see Table 2). As shown in the figure, the spectra for the SFD and Haslam maps reasonably match the model expectations in that region.¹² That is, the SFD-correlated emission roughly follows the model π⁰ spectrum while the Haslam-correlated spectrum resembles a combination of IC and bremsstrahlung emission. However, the haze-correlated emission is clearly significantly harder than either of these components. This fact coupled with the distinct spatial morphology of the haze indicates that the IC haze is generated by a separate electron component.

While the template-correlated spectra and residual maps are useful for identifying separate components, for the purposes of comparing the map intensities to a model for the physical mechanisms, we now generate total intensity spectra in several regions of interest. We define three key regions: a "haze region" south of the GC,¹³ the "four corners" region from Cholis et al. (2009b), and a Galactic plane region used by Porter (2009).

The spectrum of each region is shown (Figure 15), with the background spectrum from Figure 12 subtracted from the other spectra to remove any isotropic component. The inner Galaxy region clearly shows the low-energy behavior characteristic of π⁰ emission (though it is important to note that there may be significant emission from unresolved point sources such as pulsars in this region which have a similar spectral shape; see Appendix A and Abdo et al. 2009b). At high energies however, there is a significant excess which is not expected for π⁰ emission (cf. Figure 2). The spectrum of this excess is similar to the spectrum derived for the haze template from Figure 14 and is consistent with an IC signal from a hard electron population with energies >10 GeV. The evidence for this excess is more pronounced in the haze and four corners region. Lastly, we take the template estimate of π⁰ emission in the haze region from the Type 3 fits and subtract it from the total emission. This reveals two clear features: a bump centered on roughly 1–2 GeV that is likely due to either bremsstrahlung from a low-energy electron component or emission from unresolved pulsars, and a hard tail above ∼10 GeV that is the IC signal from the haze electrons.

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Both the morphology and the relatively hard spectrum of the gamma-ray haze motivate a common physical origin with the WMAP haze. We now provide a simple estimate of the microwave and gamma-ray signals from a population of hard electrons in the inner Galaxy, to demonstrate that the magnitudes and spectral indices of the two signals are consistent for reasonable parameter values.

We consider a steady-state electron spectrum described by a power law, dN/dE ∝ E^−α, with a high-energy cutoff at 1 TeV (here the cutoff is implemented as a step function, not an exponential falloff; of course this is only an approximation to the true spectrum). This choice is motivated by the local measurement of the cosmic ray electron spectrum by Fermi (Abdo et al. 2010). We consider a region ∼2 kpc above the GC, as an example (and since both hazes are reasonably well measured there), and employ the model for the ISRF used in GALPROP (Porter & Strong 2005) at 2 kpc above the center. We normalize the synchrotron to the approximate value measured by WMAP in the 23 GHz K band (Hooper et al. 2007), ∼15° below the Galactic plane, and compute the corresponding synchrotron and IC spectra. The WMAP haze was estimated to have a spectrum I_ν ∝ ν^−β, β = 0.39–0.67 (Dobler & Finkbeiner 2008), corresponding approximately to an electron spectral index of α ≈ 1.8–2.4; Figure 16 shows our results for a magnetic field of 10 μG, and electron spectral indices α = 2–3. This field strength is appropriate for an exponential model for the Galactic B-field intensity, $|B| = |B_0| e^{-z/z_s}$, with B₀ ≈ 20–30 μG (which seems reasonable; see, e.g., Ferriere 2009 and references therein) and scale height z_s ≈ 2 kpc (the default value in GALPROP). We find good agreement in the case of α ≈ 2–2.5, consistent with the spectrum of the WMAP haze.

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Although a detailed analysis of the possible sources of the Fermi haze are beyond the scope of this paper, a few simple comments are in order. First, the profile is not well described by a disk source. While quantifying this is a subtle task, the success of the template makes this point clear. There are many possibilities to explain the oblong shape, should that persist in future data. For instance, active galactic nucleus (AGN) jets and triaxial DM profiles could both produce signals of this shape. Even approximately spherical injection could yield such a signal, should the diffusion be considerably anisotropic. That said, while one might have attempted to invoke, e.g., rapid and significant longitudinal variation of the magnetic field to explain the microwave haze with a disk-like electron injection profile, such approaches are no longer tenable (and moreover already had significant tension with an understanding of the Haslam synchrotron maps). The presence of this feature in both gamma rays and microwaves demonstrates that the electrons themselves do not follow a disk-like injection profile. Furthermore, we emphasize that the angular scale of the haze is very large, roughly 25°–40° in b. This region of the sky is far from the Galactic disk or the very GC where complications from the central black hole, other point sources, or disk-like pulsars could complicate the analysis.

The LAT is a pair-conversion telescope, in which incoming photons strike layers of tungsten and convert to e⁺e⁻ pairs, which are then tracked (to determine direction) on the way to a calorimeter (to determine energy). The first year P6_V3_DIFFUSE data file contains records of 15,878,650 events, providing the time, energy, arrival direction (with respect to the spacecraft, and in celestial coordinates), and zenith angle for each event. Events are divided into three classes, using a number of cuts to separate photon events from particle background. "Class 3" rejects the largest fraction of background contamination, and these events are used to study diffuse emission. We also require the zenith angle be less than 105° to exclude most atmospheric gammas. By choosing to use these cuts, we study the events most likely to be real gamma rays, at the expense of a smaller effective area. These cuts discard roughly 3/4 of the signal, but vastly reduce the noise.

The Class 3 events are then binned in energy and into spatial pixels to produce counts maps. We use the hierarchical equal-area isolatitude pixelization (HEALPix), a convenient iso-latitude equal-area full-sky pixelization widely used in the CMB community.¹⁶

Because the exposure on the sky is non-uniform, we generate an exposure map using the gtexpcube tool developed by the LAT team.¹⁷ The exposure for each pixel is the LAT effective area at each θ (angle with respect to the LAT axis) summed over the livetime of the LAT at that θ and has units of cm² s. The exposure map spatially modulates the signal ±20% from raw photon counts and is slightly energy dependent for photon energies E>1 GeV. To avoid systematic errors in generating the exposure maps we use the same energy grid used for the spectral plots, which is given in Table 1. The default spatial binning of 1° in latitude and longitude is adequate, as neighboring bins have exposure differences of <0.5%. The LAT contains 12 thin layers of tungsten designated "front" and 4 thicker layers designated "back." Photons may convert to an e⁺e⁻ pair in any of the layers, and events are labeled "front" or "back" accordingly. The effective area as a function of energy is different for front and back events, so we use the "P6_V3_DIFFUSE::FRONT" and "P6_V3_DIFFUSE::BACK" instrumental response functions, respectively. Maps of counts are divided by the exposure and pixel solid angle to produce intensity maps (counts s⁻¹ cm⁻² sr⁻¹). The "combined" maps are simply an exposure-weighted linear combination of the front and back intensity maps. The full-sky Fermi maps are displayed in Figure 1 along with the exposure map and mask.

In our analysis we wish to compare Fermi maps at different energies to each other, and to other templates. In order to match the PSFs of all these maps, we smooth each map by an appropriate kernel. On average, the front and back converting events have different PSFs (by a factor of ∼2), so we smooth the intensity maps of each to a common PSF (usually 2°) to produce "front" and "back" smoothed maps (which can be averaged to obtain "combined" smoothed maps as above).

For the PSF, we use a simple fit to the radius of 68% flux containment (in degrees):

$$\begin{equation} r_{68} = \big((c_1 E^\beta)^\alpha + c_2^\alpha\big)^{1/\alpha }, \end{equation} \tag{ B1 }$$

where E is in GeV, α = 1.2, β = −0.83, and (c₁, c₂) = (0.50, 0.04) and (0.90, 0.09) for front- and back-converting events, respectively. This yields r₆₈ = c₁E^β at low E and r₆₈ = c₂ at high E. For simplicity, we assume that the PSF is a Gaussian so that the FWHM, f = 2r₅₀ ≈ 1.56r₆₈. The "raw" FWHM of a counts map is then taken to be Equation (B1) evaluated at $E_{\rm mean} = \sqrt{E_0E_1}$, where E₀ < E < E₁ for the events in the map. In order to smooth to the target PSF (usually f_targ = 2°), a Gaussian smoothing kernel of size f_kern is used such that

$$\begin{equation} f_{\rm kern} = \sqrt{f_{\rm targ}^2 - f_{\rm raw}^2}. \end{equation} \tag{ B2 }$$

For the lowest energy maps (E < 1 GeV), where f_raw is large, we take f_targ = 3° or 4°.

The point-source mask (Figure 1) contains the 3-month 10σ point-source catalog, plus the LMC, SMC, Orion, and Cen A. The mask radius for point sources is taken to be the 95% containment radius for the lowest energy event in the energy bin. For unsmoothed maps, both the counts map and the exposure map are multiplied by the mask. For smoothed maps, counts and exposure are multiplied by the mask, then smoothed, and then divided. Pixels where the smoothed masked exposure drops below 25% of the mean unmasked exposure are replaced with zeros. In cases where the mask radius is much smaller than the smoothing radius, the mask is "smoothed away." In other cases, masked pixels are visible in the smoothed maps. Because of the energy dependence of the mask radius, the effective mask changes with energy, so care must be taken in cross comparisons between energy bins, especially within a few degrees of the Galactic plane.

All LAT maps described in this section are available in the HEALPix pixelization on the Web.¹⁸ Maps of front- and back-converting events are available, as well as the combined maps. All three are available smoothed and unsmoothed, and with and without point-source masking, for a total of 12 maps at each energy. Both the imbin and specbin energy binnings are available, for a total of 240 FITS files. Gray scale and color jpegs are also available to provide a quick overview. Software used to make the maps is available on request.