THE BOLOCAM GALACTIC PLANE SURVEY. II. CATALOG OF THE IMAGE DATA

Objects in the BGPS catalog are identified based on their significance relative to the noise in the map. Owing to variations in the observing conditions and integration time across an individual BGPS field, the underlying noise at each position can vary substantially across a mosaic of several scan blocks. The spatially varying noise rms (σ) is calculated as a function of position using the residual maps from the image production pipeline (Paper I). These maps consist of the actual images less than the source emission model (where the subtraction is performed in the time stream). The residual maps are used in the map reconstruction process and are nominally source-free representations of the noise in the map. We estimate the rms by calculating the median absolute deviation in an 11×11 pixel box (80'' square) around each position in the map. The size of the box is chosen as a compromise between needing an adequate sample for the estimator and the need to reproduce sharp features in the noise map. The median absolute deviation estimates the rms while being less sensitive to outliers relative to the standard deviation. The resulting map of the rms shows significant pixel-to-pixel variation owing to uncertainty in the rms estimator for finite numbers of data. To reduce these fluctuations, we smooth the rms maps with a two-dimensional Savitsky–Golay smoothing filter (Press et al. 1992). The filter is seventh degree and smoothes over a box 12' on a side. The action of the filter is to reproduce large-scale features in the map such as the boundaries between two noisy regions while smoothing noise regions on beam-size scales. An example map of the rms is shown in Figure 2 after all processing (rms estimate and smoothing).

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The statistical noise properties of the map are shown in Figure 3, showing the intensity at each position in the map, I and the signal-to-noise ratio I/σ. In general, generating an error estimate from the residual map tracks the position-to-position variation of the noise quite well. Transforming the data into significance space (i.e., dividing the data image by the noise estimate) yields a normal distribution with some deviations. As seen in the bottom panel of the figure in particular, there is a wing of high significance outliers associated with the signal in the map. There are more negative outliers than expected, and this arises from pixel-to-pixel variations in the rms based on local conditions. Since our method cannot reproduce these variations perfectly, regions with higher true rms than represented in the rms map will manifest as high significance peaks. There are likely regions for which the rms is an overestimate and such regions contribute a slight excess of data near I/σ = 0.0, which is impossible to discern in the face of the large number of data that are expected there. We note that the excess of negative outliers may also result from incomplete removal of astrophysical signal in the model map (see Paper I). In principle, it is possible to track the variations of the noise distribution on sub-beam scales using information in the time domain. We found, however, that the generation of noise statistics from the time stream is unreliable and produces non-normal distributions of significance. However, we do use the time-stream estimate of the noise to identify regions where the residual images are contaminated with astrophysical signal and we replace the rms estimate at positions with spuriously large estimates of the noise with a local average. A Gaussian fit to the distribution of significance values reproduces the shape of the distribution well with the exceptions noted above. Fits to the normal distribution find the central value of the distribution 〈S〉 zero in all cases (−0.02 ≲ 〈I/σ〉 ≲ 0.02). The width of the distribution, which should be identically 1 if σ(ℓ, b) is a perfect estimate of the noise, is usually within 2% of 1.00. Hence, the significance distribution has been transformed to the best approximation of a normal distribution with zero mean and dispersion equal to 1.

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Given an estimate of the significance of the image at every position in the data, emission is identified as positive outlying data that are unlikely to be generated by noise. These data form connected regions of positive deviation above a given threshold. We initially mask the data to all regions above 2σ(ℓ, b). Owing to image artifacts from the Bolocam mapping process, the emission profile shows low-amplitude noise on scales smaller than a beam. As a consequence of this noise, holes in the mask can appear where negative noise fluctuations drive a pixel value below the threshold. Additionally, a set cut at a given threshold tends to produce an artificially jagged source boundary. Finally, the PCA cleaning process creates negative bowls around strong sources. To reduce the influence of these effects, we perform a morphological opening operation on the data with a round structuring element with a diameter of a half a beam-width (see Dougherty 1992, for details). The opening operation minimizes small-scale structure at the edges of the mask on scales less than that of a beam by smoothing the edges of the mask.

Following the opening operation, the remaining regions are expanded to include all connected regions of emission above 1σ(ℓ, b) since regions of marginal significance adjacent to regions of emission are likely to be real. After this rejection, a morphological closing operation is performed with the same structuring element as the previous opening operation. The closing operation smoothes edges and eliminates holes in the mask. This processing minimizes the effects of the sharp cut made in significance on the mask, allowing neighboring, marginal-significance data to be included in the mask. An example of the effects for the opening and closing operations is shown in Figure 4. This results in our final mask containing emission. To assess the effects of the mask processing on our data, we examined the brightness distribution of the pixels in the mask pre- and post-processing by the opening and closing operations. Figure 5 presents the pixel brightness distributions of the regions included in our mask both before and after applying the operators. The eliminated pixels are at low significance and the added pixels include marginal emission as desired. The processed mask also includes some negative intensity pixels in the objects. The added negative pixels are only 5% by number of the pixels added by mask processing and are not included in the calculation of source properties (Section 3.4).

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The emission mask is a binary array with the value 1 where the map likely contains emission and 0 otherwise. Identification of objects occurs only within the emission mask region. We consider each spatially contiguous region of emission separately and identify substructure within the region. We determine whether this is significant substructure within the map by searching for distinct local maxima within the region. If there is only one distinct local maximum, the entire region is assigned to a single object. Otherwise, the emission is divided among the local maxima using the algorithm described below.

To assess whether a region has multiple distinct local maxima, we search for all local maxima in the region requiring a local maxima to be larger than all pixels within 1 beam FWHM of the candidate. For each pair of local maxima, we then identify the largest contour value that contains both local maxima (I_crit) and measure the area uniquely associated with each local maximum. If either area is smaller than 3 pixels (as it would be for a noise spike) then the local maximum with the lowest amplitude is removed from consideration (see Figure 6). Furthermore, if either peak is less than 0.5σ above I_crit then the maximum with the smaller value of I_ν/σ is rejected. This filter dramatically reduces substructure in regions of emission that is due to noise fluctuations. We note that the filter is fairly conservative compared to a by-eye decomposition. As a consequence, compound objects are de-blended less aggressively than a manual decomposition. The rejection criteria are evaluated for every pair of local maxima in a connected region, so every local maximum that remains after decimation is distinct from every other remaining maximum.

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The final set of local maxima are thus well separated from their neighbors, have significant contrast with respect to the larger complex, and have a significant portion of the image associated with them. Using this set of local maxima, we partition the regions of emission into catalog objects. This partitioning is accomplished using a seeded watershed algorithm. Such an algorithm is, in principle, similar to the methods used in Clumpfind (Williams et al. 1994) in molecular line astronomy or Source EXtractor (Bertin & Arnouts 1996) in optical astronomy. To effect the decomposition, the emission is contoured with a large number of levels (500 in this case). For each level beginning with the highest, the algorithm clips the emission at that level and examines all the resulting isolated sub-regions. For any pixels not considered at a higher level, the algorithm assigns those pixels to the local maximum to which they are connected by the shortest path contained entirely within the subregion. If there is only one local maximum within the sub-region, the assignment of the newly examined pixels is trivial. The test contour levels are marched progressively lower until all the pixels in the region are assigned to a local maximum. An example of the partitioning of a region is shown in Figure 7.

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A BGPS catalog object is defined by the assignment algorithm established above which yields a set of pixels with known Galactic coordinates (ℓ_j, b_j) and associated intensities I_j (Jy beam⁻¹). Source properties are determined by emission-weighted moments over the coordinate axes of the BGPS image. Positions in the mask with negative intensity are not included in the calculation of properties. For example, the centroid Galactic longitude of an object is

$$\begin{equation} \ell _{\rm cen} = \frac{\sum _j \ell _j I_j}{\sum _j I_j} \end{equation} \tag{ 1 }$$

with b_cen defined in an analogous fashion. The sizes of the objects are determined from the second moments of the emission along the coordinate axes:

$$\begin{eqnarray} \sigma _\ell ^2 &=& \frac{\sum _j \big(\ell _j-\ell _{\rm cen}\big)^2 I_j}{\sum _j I_j} \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} \sigma _b^2 &=& \frac{\sum _j \big(b_j-b_{\rm cen}\big)^2 I_j}{\sum _j I_j} \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \sigma _{\ell b} &=& \frac{\sum _j \big(\ell _j-\ell _{\rm cen}\big)\big(b_j-b_{\rm cen}\big) I_j}{\sum _j I_j}. \end{eqnarray} \tag{ 4 }$$

The principal axes of the flux density distribution are determined by diagonalizing the tensor

$$\begin{equation} \mathbf {I}=\left[\begin{array}{cc} \sigma _\ell ^2 & \sigma _{\ell b} \\ \sigma _{\ell b} & \sigma _b^2\end{array}\right]. \end{equation} \tag{ 5 }$$

The diagonalization yields the major (σ_maj) and minor (σ_min) axis dispersions as well as the position angle of the source. Since the moments are calculated for an emission mask that has been clipped at a positive significance level, these estimates will slightly underestimate the sizes of the sources leading to size estimates that could be smaller than the beam size. Once projected into the principal axes of the intensity distribution, the angular radius of the object can be calculated as the geometric mean of the deconvolved major and minor axes:

$$\begin{equation} \theta _R= \eta \left[\big(\sigma _{\rm maj}^2-\sigma _{\rm bm}^2\big)\big(\sigma _{\rm min}^2-\sigma _{\rm bm}^2\big)\right]^{1/4}. \end{equation} \tag{ 6 }$$

Here σ_bm is the rms size of the beam ($=\theta _{\mathrm{FWHM}}/\sqrt{8\ln 2}=14^{\prime \prime }$) and η is a factor that relates the rms size of the emission distribution to the angular radius of the object determined. The appropriate value of η to be used depends on the true emission distribution of the object and its size relative to the beam. We have elected to use a value of η = 2.4, which is the median value derived for a range of models consisting of a spherical, emissivity distribution $\j (r)\propto r^{\gamma }$ with −2 < γ < 0, a range of radii relative to the beam 0.1 < θ_R/θ_FWHM < 3 at a range of significance (peak signal to noise ranging from 3 to 100). The range of η for the simulations is large, varying by more than a factor of 2. If a specific source model is required for a follow-up study, the appropriate value of η should be calculated for this model and the catalog values can be rescaled to match.

Uncertainties in the derived quantities are calculated by propagating the pixel-wise estimates of the flux density uncertainties through the definitions of the quantities. The coordinate axes are assumed to have negligible errors. The moment descriptions of position, size, and orientation are exact for Gaussian emission profiles and generate good Gaussian approximations for more complex profiles. An example of the derived shapes of the catalog objects is shown in Figure 7 for comparison with the shapes of the regions from which they are derived. In comparing the images, it is clear that the irregularly shaped catalog objects can produce elliptical sources that vary from what a Gaussian fit would produce. However, in most cases the moment approximations characterize the object shapes well.

Since the extracted objects are often asymmetric, the centroid of the emission does not correspond to the peak of emission in an object. Because of this possible offset, we also report the coordinates of the maximum of emission. To minimize the effect of noise on a sub-beam scale, we replace the map values with the median value in a 5 × 5 pixel (36'') box around the position and determine the position of the maximum value in the smoothed map. While this reduces the positional accuracy of the derived maximum, it dramatically improves the stability of the estimate. Since the peaks of emission are rarely point sources, the loss of resolution is made up for by the increased signal to noise in the map. These coordinates are reported as ℓ_max and b_max. If the catalog is being used to follow-up on bright millimeter sources in the plane these coordinates are preferred over the centroid coordinates. Moreover, since the maxima are less subject to decomposition ambiguities, we name our sources based on these coordinates: Gℓℓℓ.ℓℓℓ ± bb.bbb. The smoothed map is only used for the calculation of the position for the maximum; all other properties are calculated from the original BGPS map.

The flux densities of the extracted objects are determined by measuring the flux density in an aperture and by integrating the flux density associated with the region defined by each object in the watershed extraction. In the catalog, three flux density values for each source are reported, corresponding to summing the intensity in apertures with diameters $d_{\rm{app}}=40,^{\prime \prime } 80^{\prime \prime }$, and 120''. We refer to these flux densities as S₄₀, S₈₀, and S_120, respectively. Because many of the objects are identified in crowded fields, we refrain from a general "sky" correction since inspection of the noise statistics suggests that the average value of the sky is quite close to zero. One notable exception to this case is where there are pronounced negative bowls in the emission around the object. Because large apertures may contain parts of these negative bowls, the aperture flux densities of bright objects are likely underestimated. In addition, we also report the object flux density value (S), which is determined by integrating the flux density from all pixels contributing to the object in the watershed decomposition of the emission:

$$\begin{equation} S\equiv \sum _i I_i \Delta \Omega _i, \end{equation} \tag{ 7 }$$

where ΔΩ_i is the angular size of a pixel. The object flux density cannot, by definition of the regions, contain negative bowls, but the bowls indicate that emission is not fully restored to these values, which are thus likely underestimates of the true flux density for bright objects. Since objects typically have angular radii ∼60'', 120'' apertures do not contain all the flux density. Hence, S ≳ S₁₂₀>S₈₀>S₄₀. Since the surface brightness at each position in the map has an associated uncertainty (Section 3.1), we calculate the uncertainty in the flux density measurements by summing the uncertainties in quadrature. We further account for the uncertainty in the size of the beam owing to the uncertainties of the bolometer positions in the focal plane of the instrument (see Paper I). The latter effect produces a fractional uncertainty in the beam area of δΩ_bm/Ω_bm = 0.06. Thus, the uncertainty in a flux density measurement (either aperture or integrated) is

$$\begin{equation} \delta S \equiv \sqrt{\frac{\Omega _i}{\Omega _{\rm bm}}\sum _i \sigma ^2_i+\left(\frac{\delta \Omega _{\rm bm}}{\Omega _{\rm bm}}S\right)^2}, \end{equation} \tag{ 8 }$$

where the ratio Ω_i/Ω_bm accounts for the multiple pixels per independent resolution element. The flux density errors do not account for any systematic uncertainty in the calibration of the image data which is estimated to be less than 50% (Paper I). We discuss the flux densities in more detail in Section 7.2 below.

The primary test of the algorithm is the degree of completeness of the catalog at various flux density limits. To this end, we have inserted a set of point sources into the time stream with flux densities uniformly distributed between 0.01 Jy and 0.2 Jy arranged in a grid. We then process the simulated data with the source extraction algorithm and compare the input and output source catalog. The results of the completeness study are shown in Figure 8. For this field, the mean noise level is σ = 24 mJy beam⁻¹. The catalog is >99% complete at the 5σ level. Below this level, the completeness fraction rolls off until no sources are detected at 2σ. For this regular distribution of sources, there are no false detections throughout the catalog (even at the 2σ level). The catalog parameters described above are selected to minimize the number of false detections. Even though the thresholding of the map is performed at the relatively low 2.0σ level, the wealth of restrictions placed on the extracted regions combine to raise the effective completeness limit to 5σ. We note that the grid arrangement of test sources means that confusion effects are not accounted for in the flux limit.

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In Figure 9, we present the estimated completeness limit as a function of Galactic longitude for the BGPS. The ℓ = 111° region has a relatively low value for the noise rms with several fields being up to a factor of 4 larger than the completeness test field. We assume that the completeness limit scales with the local rms. The assumption is motivated since all catalog source extraction and decomposition is performed in the significance map rather than in the image units. Thus, the absolute value of the image data should not matter. Over >98% of the cataloged area, the completeness limit is everywhere less than 0.4 Jy, thus the survey, as a whole can be taken to be 98% complete at the 0.4 Jy level.

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In addition to completeness limits, we investigated the fidelity of the algorithm by comparing extracted source properties to the input source distributions. We compared the extracted flux density values to the input flux density values as shown in Figure 10. These simulations compare input and recovered properties for resolved sources with intrinsic FWHMs equal to 2.4 times that of the beam. The flux densities of the sources are varied over a significant range, and we measure the recovered values for the flux density using the 40'', 80'', and 120'' apertures and we also compute the integrated flux density in the object. Since the small apertures only contain a fraction of the total flux density, their average flux density recovery is naturally smaller than the input. As the aperture becomes better matched to the source size, the flux density recovery improves, in agreement with analytic expectations based on the source profile. Since many objects are larger than the 40'' and 80'' apertures, catalog flux density values can be viewed in terms of surface brightnesses within these apertures.

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We completed an additional suite of simulations, including sources with sizes significantly larger than the beam. These simulations are used in Paper I to test the effect of the deconvolution process, but we also examine the results of the source extraction. In Figure 10, we show the recovered source size as a function of input size. In general, smaller sources are better recovered with objects much larger than the beam having a smaller size found. Radii of objects become biased for sources with sizes ≳200'', consistent with the experiments on deconvolution discussed in Paper I. Because of the image processing methods used, objects with sizes larger than this should be treated with caution. However, we note that no sources in the actual catalog are found with sizes >220'' and the mean size is only 60'' so objects of these sizes and their properties are likely to be recovered. The catalog algorithm thus does not introduce any additional bias into the properties of the recovered sources. However, the sizes of objects may not be completely recovered in the mapping process and thus the extracted properties may not accurately reflect the true properties of sources because of limitations in the map making process. These map properties should be viewed with the caveats presented in Paper I.

A primary question of the catalog algorithm is how well it separates individual sources into their appropriate subcomponents. Any approach to source extraction and decomposition will be unable to separate some source pairs accurately. However, the limiting separation for source extraction is determined by the method. Our approach is further complicated by the assignment of emission from every position in the catalog to at most one object. Thus, blended sources are difficult to extract separately unless the individual local maxima that define the sources are apparent according to the criteria in Section 3.3. To assess how the catalog algorithm recovers sources, we generated maps of sources with a random distribution in position and processed the maps with the catalog algorithm. The input sources have elliptical Gaussian profiles with a uniform distribution of major and minor axis sizes between our beam size (33'') and 72'', which represent typical objects seen in the BGPS maps. The input sources are oriented randomly. For each pair of input sources, we determine whether the pair is assigned to the same catalog object. We then measure the fraction of blended sources as a function of separation. The results of the experiment are shown in Figure 11 which indicates that the algorithm resolves most sources that are separated by >75'' or 2 beam FWHM. For point sources, the nominal resolution limit is ≈1 beam FWHM; however the algorithm requires ∼2 FWHM between local maxima so this limit on deblending is consistent with the design. Furthermore, the test objects are resolved, which complicates the analysis. Thus, Figure 11 also indicates the fraction of blended sources for pairs of the smallest sources and for pairs of the largest sources (i.e., both sources are in the bottom 15% of the size distribution or the top 15%, respectively). In separating the data based on size, we can assess the degree to which source structure will complicate deblending. As expected, pairs of small sources are easier to resolve (50% are separated for distances of 60'') and pairs of large sources more difficult to deblend (50% at 85''). The large source curve crosses the population curve at large separations (120'') because of small number statistics rather than any algorithmic feature.

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Figure 15 presents the total flux density extracted from the images in catalog sources as a function of coordinates in the Galactic plane. In total, the catalog sources contain 11.9 kJy of flux density. Difficulties in recovery of large-scale structure notwithstanding, this represents the emission profile of the Galaxy along the longitude and latitude direction.

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The right-hand panel of Figure 15 shows that the distribution of flux density in sources peaks at the flux-weighted mean 〈b〉 = −0095 ± 0001. Such an offset was also seen in the ATLASGAL survey of the 12° < b < −30° region of the Galactic plane at λ = 870 μm (Schuller et al. 2009). They speculated that the offset may result from the Sun being located slightly above the Galactic plane. To further investigate this possibility, we examined the latitudinal flux density distribution of BGPS sources as a function of longitude and compared these results to other tracers of the Galactic disk. In particular, we compared the flux-density-weighted values of the mean latitude for objects in our catalog to the COBE/DIRBE data at λ = 2.2 μm and 200 μm (Hauser et al. 1998) and the integrated ¹²CO (1–0) data of Dame et al. (2001). The results of the comparison are shown in Figure 16. For all data sets, we only compute the flux-density-weighted average latitude over the range −05 < b < 05 which is the latitude range over which the BGPS is spatially complete. The data are averaged in 10° bins. The offset of BGPS sources appears at nearly all Galactic longitudes, but is most pronounced toward the Galactic center (−10° < ℓ < 25°). The CO (1–0) data follow a similar trend with a significant offset toward negative latitudes. We verified that the effects persist in the CO data if the averaging range is expanded to larger latitudes (foreground emission from the Aquila rift affects the averaging for large latitude ranges |b| < 5°). We also note that the emission profiles from the DIRBE data do not follow the offset trends seen in the millimeter continuum or the CO. The DIRBE 2.2 μm data primarily traces emission from stellar photospheres and is affected by dust extinction and the 200 μm traces warm dust emission. Both of these emission features show a more symmetrical distribution around the Galactic midplane than do those tracers associated with the molecular gas. We conclude that the offsets toward negative latitude are primarily associated with the molecular ISM. The significant variations with the mean offset are likely the result of individual star-forming structures rather than a global property of all Galactic components. For example, the large negative excursion in the 40° < ℓ < 50° bin is due to the W51 star formation complex at (495, − 04). At ℓ>60°, the space density of BGPS sources drops sharply, so the effects are dominated by the particular distribution of the dense gas within the molecular complex. Specifically, in the ℓ ∼ 70° region, there are more dense gas structures at b>0° in the Cygnus X and Cygnus OB 7 region than there are at b < 0°, while the lower density molecular gas does not exhibit this small-number effect.

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In Figure 17, we present two representations of the flux density distribution for objects in the BGPS catalog. The flux density distribution in 40'' apertures is shown in the left-hand panel and follows a power-law form over nearly three orders of magnitude in flux density from the point source completeness limit to the upper flux density limit in the survey. We have fit a power law to the approximation to the differential flux density spectrum dN/dS ≈ ΔN/ΔS ∝ S^−α and find that an index α = 2.4 ± 0.1 represents the data quite well. Despite the similarity to the Salpeter (Salpeter 1955) index α = 2.35, we suspect that this is coincidence because the BGPS sources are found at a wide variety of distances. This range of distances necessitates a complete treatment of the conversion of flux density to mass. Indeed, the Balloon-borne Large Aperture Submillimeter Telescope (BLAST) survey of the Vela molecular complex at λ = 250, 350, and 500 μm found a marginally steeper distribution of object mass—dN/dM ∝ M^−2.8±0.2 for cold cores—than the BGPS finds for flux density (Netterfield et al. 2009). A careful treatment of the distances to BGPS sources may find the mass distribution to be comparable between BGPS objects and those found in the BLAST survey.

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In the right panel of Figure 17, we plot the differential flux density distribution for the total flux density in the objects S (Equation (7)) and find that a power law is not as good a representation for the catalog. Even though the completeness limits should be the same for both flux density measurements, the power-law behavior for the total flux density only become apparent above a much higher flux density level of 0.7 Jy where α = 1.9 ± 0.1. The difference between these two limits arises entirely because of the extended structure accounted for in the total flux density measurement. A detection near the completeness limit is necessarily similar to a point source. To detect extended objects with any degree of fidelity requires a much higher peak value of the flux density so that connected, extended structure is apparent. Thus, source with flux densities near the completeness limit will not sample the full range of source sizes that brighter objects will. Hence, the detection of extended structure is only apparent in bright sources leading to a higher limit. We also note that a power-law index fit to the distribution above the limit of 0.7 Jy gives a significantly different value compared to the index for flux densities derived for 40'' apertures. This simply reflects the additional effects of the included sizes of the sources. In general, sources with bright peaks (measured in the 40'' apertures) also tend to be large so their total recovered flux density is larger than for smaller sources. This effectively makes the distribution more top-heavy, reducing the magnitude of the exponent.

In Figure 18, we show the value derived for the slope of the flux density distribution as a function of Galactic latitude over the contiguous region of the BGPS. We divide the catalog into groups of 50 sources based on the galactic longitude above the survey completeness limit of 0.4 Jy. The variation of α scatters around the global average of α = 2.4 with marginally significant variation as a function of longitude. The variations are strongest in regions of high source density suggesting that blending may contribute to the derived distribution. Specifically, the algorithm has a native angular scale over which it searches for local maxima (a box with edge length of 2 beam FWHM). Structures tend to be subdivided on scales near this search scale. Large, bright objects in crowded regions of the plane may be subdivided more often than sources in less crowded regions. Indeed, in the ℓ = 60° region, where the source density distribution is lowest, the flux density distribution is the most shallow.

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The measured sizes of sources varies significantly for the BGPS sources. We plot the distribution of source sizes and aspect ratios in Figure 19 for the catalog sources. We show the major axis size (ησ_maj where η = 2.4) determined from the moment method and the deconvolved angular radius of the source (θ_R), which includes both major and minor axes (Equation (6)). Because of the deconvolution, the radius θ_R can be smaller than the beam size. However, we also note that some of the major axis sizes are also smaller than the beam size. This results from applying moment methods without accounting for all emission down to the I = 0 level. The clip at 1σ truncates the low-significance band of emission around the sources. For faint sources, the truncation will clip a significant fraction of the emission resulting in an underestimate of the source size. While this effect can be corrected for, we refrain from doing so since such corrections require extrapolating the behavior of the sources below the noise floor in the images.

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We also present data for the aspect ratios of the catalog sources defined as σ_maj/σ_min. Both the radii and the aspect ratios show a peaked distribution with an exponential decline. The typical source in the BGPS has a major axis size of 60'' and a minor axis size of 45''. However, there are significant numbers of large-aspect-ratio sources which represents asymmetric clump structure and, for faint sources, the prevalence of filamentary structure in the survey.