LOW SURFACE BRIGHTNESS GALAXIES SELECTED FROM THE 40% SKY AREA OF THE ALFALFA H i SURVEY. I. SAMPLE AND STATISTICAL PROPERTIES

The Arecibo L-band Feed Array (ALFA), an L-band (1.4 GHz) receiver in Arecibo, the world's most sensitive radio telescope at Arecibo Observatory in Puerto Rico, enables such an extragalactic H i survey as ALFALFA (the Arecibo Legacy Fast ALFA survey; Giovanelli et al. 2005a; Giovanelli 2007; Haynes 2007). Exploiting the large collecting area of the Arecibo antenna and its relatively small beam size (∼35), ALFALFA is a very wide area (7000 deg² of the sky at high Galactic latitudes) blind extragalactic H i survey, which aims to catalog all gas-bearing extragalactic objects in the local universe and has conducted a deep and precise census of the local H i universe over a cosmologically significant column. Initiated in 2005 February and completed in 2012, ALFALFA has obtained a Hi-line spectral database of more than 30,000 extragalactic Hi-line sources, covering the redshift range between −1600 and 18,000 km s⁻¹ with a resolution of ∼5 km s¹ (Giovanelli et al. 2005b).

A catalog of H i detections covering about 40% of the full ALFALFA survey sky area has now been publicly released as the α.40 catalog (Haynes et al. 2011). The sky areas contained in the α.40 catalog are regions 07^h30$^{{\rm m}}\lt$ R.A. < 16^h30^m, +04${}^\circ \lt$ decl. < +16°, and +24${}^\circ \lt$ decl. < +28° (the "spring" region) and 22^h00$^{{\rm m}}\lt$ R.A. < 03^h00^m, +14${}^\circ \lt$ decl. < +16°, and +24${}^\circ \lt$ decl. < +32° (the "fall" region). The α.40 catalog consists of 15,855 Hi detections, 15,041 of which are certainly extragalactic objects; the remaining 814 are more likely to be Galactic high-velocity clouds. LSB galaxies mostly contain large reservoirs of gas (Hii) (Schombert et al. 2013); therefore the α.40 catalog provides us with an excellent database in which to search for a large number of LSB galaxies in the local universe.

Mapping one quarter of the entire sky, the SDSS (Gunn et al. 1998; York et al. 2000; Lupton et al. 2001; Stoughton et al. 2002; Strauss et al. 2002) aims to obtain CCD imaging in five broad bands (u, g, r, i, and z) and spectroscopy from 3800 to 9200 Å of millions of galaxies, quasars, and stars, using a dedicated wide-field 2.5 m telescope (Gunn et al. 2006) at Apache Point Observatory in New Mexico. DR7 (Abazajian et al. 2009) is the seventh major data release and provides images, spectra, and scientific catalogs. It is the final data release of SDSS-II, an extension of the original SDSS, which shares some footprints with the α.40 data set.

By cross-referencing the α.40 and SDSS DR7 photometric data sets where the two share footprints, the ALFALFA team provides the cross-identifications of α.40 H i sources with the photometric and spectroscopic catalogs associated with the SDSS DR7 in the α.40 catalog. Of the total 15,041 extragalactic objects from the α.40 catalog, there are 12,468 sources having optical counterparts (OCs) in the SDSS DR7 photometric data set (Haynes et al. 2011). Here, we have to introduce two SDSS photometric catalogs, the PhotoObjAll and the PhotoPrimary catalogs. The PhotoObjAll catalog contains all measured parameters for all photometric objects of SDSS. As an object may be observed two or more times in SDSS imaging due to the overlaps at many levels of the imaging (runs and stripes), the PhotoPrimary catalog has been created to only contain the best observation of an object. Usually, we should take objects in the PhotoPrimary catalog for convincing scientific studies. Among the 12,468 ALFALFA sources with SDSS DR7 OCs, there are 12,423 sources belonging to the PhotoPrimary catalog. Therefore, for a convincing scientific study in this paper, we will only regard these 12,423 PhotoPrimary sources as our parent sample.

Here we give a brief description of this parent sample. As shown in Figure 1, the parent sample covers a distance range from 0 to 260 Mpc, with 84% between 50 and 220 Mpc; a H i mass range from 10$^{6.11}$ to 10$^{10.85}$ ${{M}_{\odot }}$, with only 1.6% as low H i mass sources (M(H i) < 10$^{7.7}$ ${{M}_{\odot }}$; Huang et al. 2012); a heliocentric radial velocity range from −400 to 18,000 km s⁻¹, with 75.7% between 5000 and 15,000 km s^−1; a Hi-line width range from 9 to 885 km s^−1, with 99.7% below 600 km s⁻¹; an r-band magnitude range from 9.73 to 21.88 mag, with 4.5% fainter than 18 mag; and a g–r color range from −0.96 to 1.65 mag. Here, the magnitudes of the g and r bands and the color of g–r are all derived from the optical surface photometry done by us, with elliptical apertures obtained using the SExtractor software (Bertin & Arnouts 1996), the process of which will be described in detail in the next section.

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SDSS analyzes the raw image data, including bias subtraction, sky and PSF determination, flat-fielding, and finding and measuring the properties of objects by its Photometric Pipeline (PHOTO). For subtraction of the sky background of images, PHOTO performs a strategy of a little simplicity. It at first estimates an initial global sky, which is taken from the median value of every pixel in the image, clipped at 2.32σ. Second, PHOTO proceeds to find all the bright objects, which are typically those pixels with values more than 51 times of the initial global sky estimation, and then mask the bright object pixels. Third, once the bright objects have been masked, PHOTO determines the same clipped median sky value locally within each 256 × 256 pixel box on a grid with 128 pixels spacing and then bilinearly interpolates this sky value to each pixel to construct a sky background image. Finally, the sky image is subtracted from the image.

This strategy of sky background estimation used in the SDSS PHOTO has its disadvantage, which is a high risk of considering the large extended outskirts of the bright objects as part of the sky background, and thus overestimates the sky background of the bright objects, inevitably causing underestimation of the luminosity and radius of the bright objects (Lauer et al. 2007; Liu et al. 2008; Hyde & Bernardi 2009; He et al. 2013). This sky subtraction problem has also been recognized by the SDSS team (Adelman-McCarthy et al. 2006, 2008), and they quantified the underestimation of the brightnesses of galaxies of large angular extent due to poor sky subtraction to exceed 0.2 mag for galaxies brighter than r = 14 mag (Adelman-McCarthy et al. 2008). This bias was proved to be even larger in Lauer et al. (2007), who found that for the brightest cluster galaxies (BCGs) in their sample of luminous galaxies, the SDSS luminosities and radius are strongly biased to low values by excessive sky subtraction. For the BCGs with large effective radiuses (${{R}_{e}}\;\gt \;60$ arcsec), the underestimation discrepancies of the SDSS r-band magnitudes resulting from the excessive sky subtraction are even larger than 1.0 mag. Liu et al. (2008) performed accurate sky background subtractions and surface photometry for a carefully selected sample of 85 BCGs in the SDSS r band. The comparison of their photometric results with those of the SDSS demonstrated that the SDSS pipeline underestimated the sizes and luminosities of BCGs, and the discrepancies would become larger if the sizes of BCGs were larger. Hyde & Bernardi (2009) compared the SDSS photometric reductions with those of their own code for a subset of their full sample of early-type galaxies (ETGs) and, as expected, found that the SDSS underestimated the sizes and brightnesses for large objects because of the sky subtraction problems of the SDSS PHOTO pipeline. He et al. (2013) performed accurate sky subtraction and surface photometry for a complete and homogeneous sample of bright ETGs and compared their measurements with those from the SDSS DR7. They found that the SDSS measurements are on average 0.16 mag and at maximum 0.8 mag lower than their own measurements in Petrosian magnitude due to the overestimations of the sky background by the SDSS PHOTO pipeline and smaller in effective radius. Such underestimations of the luminosities and sizes of the brightest ETGs also led to underestimations of the luminosity density and stellar mass density for bright ETGs. This recognized issue of the sky background subtraction in the SDSS data releases (DRs) from DR1 to DR7 has been significantly improved by reprocessing all SDSS-imaging data, using a more sophisticated algorithm for sky background subtraction in the SDSS DR8 (Aihara et al. 2011). However, compared with the measurements of He et al. (2013), it still underestimates the luminosities of ETGs by about 0.12 mag on average, due to the overestimations of the sky background subtraction in SDSS DR8.

Low-luminosity galaxies tend to have lower surface brightness than average, and the flux of such a galaxy with LSB is significantly reduced because the sky subtraction determination subtracts a substantial fraction of the galactic light; see Strauss et al. (2002) and Blanton et al. (2005). For galaxies with extended LSB outskirts, which are easily considered as the sky light by the SDSS photometric pipeline, underestimations for luminosities are still serious, and the bias can even reach as high as 0.5 mag (Lisker et al. 2007). Therefore, there are limits of the SDSS sky background subtraction algorithm, not the SDSS data, but deriving a sample of LSB galaxies from photometry alone requires accurate sky background subtraction because an overestimation of background such as that in the SDSS PHOTO pipeline would bias the number of true LSB galaxies toward a higher value. The photometric pipeline of the SDSS was not optimized for finding LSB objects (Blanton et al. 2005). Therefore, if we expect to derive a reliable LSB galactic sample from the α.40 SDSS DR7 parent sample by optical photometry, we have to carefully estimate the sky background images and do the surface photometry ourselves for the SDSS DR7 images of all of the 12,423 galaxies in our parent sample.

In preparation, we derive the FITS image files of the corrected frames (fpC-images) with 2048 × 1489 pixels in both the g (4686 Å) and r (6165 Å) bands from the SDSS DR7 database for all of the 12,423 galaxies in our parent sample. The fpC-images are the images that have been preprocessed by the SDSS photometric pipeline, including bias subtraction, flat-fielding, purge of bright stars, and corrections for bad pixels (bad columns, bleed trails, and those corrupted by cosmic rays) by interpolated values. We then adopted a more precise method, which was developed by Zheng et al. (1999) and Wu et al. (2002), to estimate the sky background image for every fpC-image in our parent sample.

First, we produce a smoothed version of each fpc-image by filtering it with a Gaussian function of FWHM = 8 pixels to make the area of each object a bit more extended in the image (the smoothed fpC-image).

Second, we use the software SExtractor (Bertin & Arnouts 1996) to automatically detect all objects with higher peak flux, 2.0σ above a global sky background value, which is simply estimated by the SExtractor itself in the fpC-image. Subsequently, we mask out all the objects detected by SExtractor in the fpC-image to produce an object-masking image. We then carefully check this object-masking image by eye and, unfortunately, find that the wings of bright stars or the faint stellar halos of galaxies have not been well masked. This would lead to overestimation for the sky background and then underestimation for the luminosity of objects. Through trial and error, we find that if the original fpC-image is directly used for SExtractor to detect objects, it may be hard to derive a good object-masking image that has the wings of the bright objects or the faint stellar halos of galaxies completely masked. Instead, the smoothed fpC-image generated in the last step is an optimal choice for SExtractor to produce a good object-masking image by detecting objects well. It is worth noting that empirically, through trial and error, the Gaussian function with FWHM = 8 seems to be the best choice because it can produce the well-smoothed fpC-image, which can be used by SExtractor to detect well not only the bright inner regions of objects but also the wings of bright objects and the faint stellar halos of galaxies and can then generate a complete (good) object-masking file. (e.g., Figure 2(b) is the complete masking file for the original fpC-image of Figure 2(a)).

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Third, we subtracted all of the objects from the fpC-image, according to the masked areas defined in the complete object-masking file. The object-subtracted image would leave us only but sufficient sky pixels, from which we could attempt to precisely determine a reliable sky background.

Last, we used the object-subtracted image to model the sky background. We performed a least-squares polynomial fit of low order to the sky pixels of each row of the object-subtracted image and replaced the masked pixels of this row by the fitted values. Here, the reason why we restrict the fits to low-order polynomials is that this can avoid introducing spurious fluctuations to the masked regions in the fitted sky by interpolations. This fitting process was performed first row by row and then column by column. The individually derived row-fitted and column-fitted sky images were then averaged, and this averaged image was later smoothed with a median filtering box 31 × 31 pixels in size to eliminate any small artifacts from the modeling process. This smoothed sky image is finally adopted as the sky background (Figure 2(c)) and is subtracted from the original fpC-image (Figure 2(d) is the sky-subtracted image of the original fpC-image, shown as Figure 2(a)). Note that the sky background (Figure 2(c)) shows a gradient across the frame, as expected. Compared with a straight 2D background fit, which has been found to systematically underfit or overfit certain regions of the images (Zheng et al. 1999), the method of fitting the sky piecewise in a row-by-row and column-by-column fashion using low-order polynomials can ensure that we predict the sky background underneath the object-masked regions in a mutually orthogonal manner, which not only permits reasonable interpolations of sky under the galactic region but also fits the inherent lumpiness of the sky at LSB (Zheng et al. 1999). Additionally, this bidirectional fitting method has already been checked and successfully used in many papers (Zheng et al. 1999; Wu et al. 2002; Lin et al. 2003; Liu et al. 2005; Duan 2006; Cao & Wu 2007; Li et al. 2007; Liu et al. 2008; Chonis et al. 2011; Mao et al. 2014).

Following the specific steps listed above, we have accurately estimated sky background maps and then subtracted the sky backgrounds from the fpC-images for all galaxies in our parent sample in both the g and r bands. To check the quality of our sky subtraction, we have made the necessary tests. As an example, for the galaxy presented in Figures 2 and 3 shows distributions of counts in the sky-subtracted frame (solid black) for all unmasked pixels of the whole frame (global; left panel) and of the local vicinity around our galaxy (local; right panel). The surrounding local vicinity is defined by the region between the boundaries of the two square boxes sized 250 × 250 pixels and 500 × 500 pixels from the galactic center, respectively. If our sky background model is successful, the counts are expected to follow a Gaussian distribution with a mean close to zero both globally and locally. This is indeed the case, as can be seen from the distribution represented in solid black in Figure 3. The count distributions for the whole frame and the local region in the SDSS r band are both well fitted by the Gaussian functions, with the mean values very close to 0 ADU. The dispersions of the Gaussian distributions are 6.83 ADU for the whole frame and 6.74 ADU for the local region, respectively. Additionally, the distributions represented in dashed gray in Figure 3 are count distributions in the frame without sky background subtraction by our accurate row-by-row and column-by-column fitting method, but only with the simple global or local mean value subtraction for all unmasked pixels of the whole frame and of the local region. These dashed gray distributions are plotted for a further demonstration of the goodness of our sky subtraction. It is clear that the mean values of the Gaussian distributions for the frame with our sky background subtraction (solid black) are much closer to zero than for the frame with no accurate sky background subtraction, but only with the simple global mean value subtraction (the dashed gray curve) both globally and locally (0 versus −1.63 ADU for the whole frame and 0.03 versus 0.09 ADU for the local region). Evidently, this row-by-row and column-by-column method of sky background subtraction can well recover the sky background model, but the most noticeable problem of this method, compared to a straight 2D fitting, might be that it is a little more time-consuming as a result of its relatively complicated fitting algorithm.

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In the system of the SDSS Petrosian photometry, Petrosian circular apertures are used (Strauss et al. 2002). Although the circular aperture can work well for objects with small angular extent or spherical-like morphology, it is not an optimal choice for galaxies such as most of the galaxies in our parent sample shown in Figure 4, which usually have large angular extent, irregular morphology, or edge-on shape. For objects such as those shown in Figure 4, the circular apertures are too small in size to include all the inherent intrinsic light from the objects that we are interested in; or for the objects shown in Figure 5, the circular apertures are so large that they inevitably involve the light from the adjacent objects. Large numbers of LSB galaxies are morphologically similar to late-type (Sc and later) spiral galaxies with amorphous or fragmentary and faint spiral patterns, or morphologically similar to irregular galaxies (McGaugh et al. 1995). Therefore, the circular apertures are clearly not the best choices for LSB galaxies. In this paper, we will do surface photometry with the elliptical apertures using the SExtractor software (Bertin & Arnouts 1996) for the sky-subtracted frames of every galaxy in our parent sample .

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The SExtrator package is a source-detection and photometry routine. For photometry, there are six different approaches (isophotal, corrected-isophotal, automatic, best, fixed-aperture, and Petrosian) in the SExtractor software. Therefore, the automatic aperture magnitudes (AUTO), inspired by Kron's "first moment" algorithm (see details in Kron 1980), are intended to give the most precise estimate of "total magnitudes," at least for galaxies. SExtractor's automatic aperture is a flexible and accurate elliptical aperture whose elongation, , and position angle, θ, are defined by the second-order moments of the object's light distribution. Then, within this aperture, the characteristic radius, r_1, is defined as that weighted by the light distribution function (r₁ = $\frac{\sum rI(r)}{\sum I(r)}$). Kron (1980) and Infante (1987) have verified that for stars and galactic profiles convolved with Gaussian seeing, more than 90% of the flux is expected to lie within a circular aperture of radius, kr_1, if k = 2, almost independently of their magnitudes. This picture remains unchanged if they consider an ellipse with $\epsilon k{{r}_{1}}$ and $\frac{k{{r}_{1}}}{\epsilon }$ as the principal axes. By choosing a larger k = 2.5, more than 96% of the flux is captured within the elliptical aperture. Therefore, we keep k = 2.5, which is also the default setting of SExtractor during the automatic elliptical aperture photometry (Bertin & Arnouts 1996). More details about the Kron radius and the automatic aperture photometry used in SExtractor are derived in Kron (1980), Infante (1987), and Bertin & Arnouts (1996).

Therefore, we would perform the photometry by SExtractor for the sky-subtracted fpC-images in both the g and r bands of every galaxy in our parent sample. First, the sky-subtracted fpC-images in the r band of all the galaxies in our parent sample were fed into the SExtractor routine (Bertin & Arnouts 1996) to derive the r-band magnitude and galactic central coordinate of each galaxy. Then, we used the same galactic center and aperture as those of the r-band image to measure the g-band sky-subtracted fpC-images and derived the g-band magnitude of each galaxy. We show examples of the SExtractor AUTO elliptical aperture at the SExtractor-determined galactic center (white ellipses in Figure 4), comparing them with the SDSS circular Petrosian aperture at the SDSS-determined galactic center (red circles in Figure 4) for four galaxies in our sample. Clearly, the AUTO photometry appears indeed to be more appropriate for those galaxies because the AUTO ellipses could capture all of the light from the objective sources themselves but exclude the light from other nearby objects. However, the SDSS Petrosian circles adopted in the SDSS Petrosian magnitudes seem to have excluded amounts of light from the objective sources themselves or even included light from adjacent sources if the objective galaxy was to be wholly included in a circular aperture.

The photometry for all the galaxies using SExtractor was automatically done in batch mode, and then we made a visual check on the AUTO aperture photometric results and redid the AUTO photometry by correcting the inappropriate apertures. Finally, the resultant g-band magnitude and g–r color are previously shown in Figures 1(c) and (d).

Although besides photometry, some interesting fitted parameters can also be made by SExtractor currently, these fitted results are only the rough guess and estimate of the parameters because the advantage of SExtractor is its photometry instead of geometric fitting, by which SExtractor only aims to give a quick look for the interesting fitted parameters. Therefore, we choose to use the Galfit software (Peng et al. 2002), which is good at galactic fitting. By setting a single radial profile function or a combination of a number of functions, e.g., the Sérsic, exponential, and Nuker models and others that are allowed in most literature, and artificially setting the initial values for a set of input parameters, Galfit starts a nonlinear least-squares fitting. During the fitting, minimization of residuals between model and image is done by using the Levenberg–Marquardt downhill-gradient method. The process of minimization iterates until convergence is achieved. The solution in the case of convergence should then be regarded as the optimum solution in the parameter space.

Therefore, after surface photometry using the SExtractor software, all of the sky-subtracted images in the g and r bands of the 12,423 galaxies of our parent sample were fed into the galaxy-fitting procedure, Galfit (Peng et al. 2002), in a batch mode. As the majority of LSB galaxies lack strong bulges (de Blok et al. 1995; Beijersbergen et al. 1999), a decomposition into bulge and disk is not essential for LSB galaxies, especially for the more disk-dominated LSB galaxies, which are preferred in our future science goals. Therefore, we fit each galaxy in our parent sample only with an exponential profile function and set the parametric results from the SExtractor to be the initial values for the set of input parameters (galactic magnitude, disk-scale length, b/a ratio, and position angle) for Galfit, which would finally determine the best-fit values for the interesting parameters, including the disk-scale length in pixels (α), axis ratio (q), and inclination angle (i) for each galaxy of our parent sample, and would generate a triplet of thumbnails, including the original galaxy, the exponential model fit, and the residual image (see Figure 6).

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Although the best-fit results are derived in the case of convergence and ${{\chi }^{2}}$ minimization, problems with local minima and numerical degeneracies between some parameters may be present, as the parameter space is large (Peng et al. 2002). To reinforce our confidence in the parameter optima, we use Monte Carlo simulations to search the parameter space for each galaxy of our parent sample. We make 200 Monte Carlo simulations for each galaxy of our parent sample by taking the following steps.

• 1.  
  Set the exponential function as the galactic radial profile function for Galfit.
• 2.  
  Randomly select a set of parameters (magnitude, disk-scale length, b/a ratio, and position angle), drawing from distributions of these parameter values centered on the best-fit values previously determined by Galfit fitting.
• 3.  
  Using these randomly selected parameters as initial guesses for the set of input parameters, minimize the ${{\chi }^{2}}$ until the convergence is reached by Galfit.
• 4.  
  Repeat steps 1–3 200 times to see whether they return the same optimized solution as that derived by using the SExtractor results for the initial guesses for Galfit or other equally plausible ones.

For convenience, we use the term "best-fit solution" to denote the best-fit solution from Galfit by using the SExtractor results as the initial guesses and the term "simulated solutions" to denote the solution from the Monte Carlo simulations by Galfit which use randomly selected parameters as the initial guesses. We compare the "best-fit solution" with the "simulated solution" for galaxies of our parent sample and find good agreement between the two solutions for every galaxy, which strongly verifies the validity of our best-fit solutions for the useful galactic parameters. For example, in Figure 7, we take the same galaxy (AGC 572) as that shown in Figure 6, which shows the galaxy-fitting process of Galfit to present the comparison between the "best-fit" and the "simulated" solutions. In Figure 7, comparisons between the "best-fit solution" (the solid line) and the "simulated solutions" from 200 Monte Carlo simulations (the dots) are shown for the two useful parameters in this paper of the disk-scale length and axis ratio from top to bottom, respectively. For each panel from this figure, the "simulated solutions" from 200 Monte Carlo simulations strongly converge at their mean value (denoted as "Simulation mean and sigma" in Figure 7), and the mean value of the "simulated solutions" is well consistent with the "best-fit" solution from Galfit (denoted as "Galfit" in Figure 7). It is worth noting that in the bottom panel of Figure 7, it looks like the axis ratio b/a values from the simulations are 0.8108, 0.8109, and 0.8110 only, with nothing in between. Actually, this is because the Galfit software has rounded its final output results to only four decimal places, so any difference between the b/a values after four decimal places cannot be apparent to us. This further strengthens the consistency of the b/a values from the 200 Monte Carlo simulations, as the differences are at least  after three decimal places.

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The typically used photometric parameter to separate the high and low surface brightness regime of galaxies is the central surface brightness of the disk in the B band, ${{\mu }_{0}}$(B) (Freeman 1970; Impey et al. 1996; O'Neil et al. 1997a; Zhong et al. 2008). Therefore in this subsection, we will calculate ${{\mu }_{0}}$(B) for each galaxy in our parent sample. Deduced from the exponential function of the radial profile of the disk component of a galaxy, the central disk surface brightness can be expressed as Equation (1a), where ${{\mu }_{0}}$ refers to the central disk surface brightness in mag arcsec⁻², m refers to the apparent magnitude in mag, α represents the disk-scale length in pixels, and A is the area of one pixel. For the SDSS image, A equals 0.396 × 0.396 in arcsec²/pixel. Following O'Neil et al. (1997a), Trachternach et al. (2006), and Zhong et al. (2008), the central surface brightness should furthermore be corrected by inclination and cosmological dimming effects, so it can be finally expressed as Equation (1b), where q and z refer to the axis ratio and redshift, respectively.

$$\begin{eqnarray}&&{{\mu }_{0}}=m+2.5{{{\rm log} }_{10}}(2\pi {{\alpha }^{2}}A)\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&{{\mu }_{0}}=m+2.5{{{\rm log} }_{10}}(2\pi {{\alpha }^{2}}Aq)-10{{{\rm log} }_{10}}(1+z)\end{eqnarray} \tag{ 1b }$$

$$\begin{eqnarray}&&{{\mu }_{0}}(B)={{\mu }_{0}}(g)+0.47({{\mu }_{0}}(g)-{{\mu }_{0}}(r))+0.17.\end{eqnarray} \tag{ 1c }$$

Using the magnitude measured with the AUTO aperture by SExtractor and subsequently Galactic-extinction-corrected by us as m, the disk-scale length in pixels (α), the axis ratio (q) from the exponential profile fittings by Galfit, the redshift (z) deduced from the H i velocity in the α.40 parametric catalog, and the pixel area of A = 0.396 × 0.396 in arcsec²/pixel for the SDSS image, we first calculated the central disk surface brightness in both the g and r bands, ${{\mu }_{0}}$(g) and ${{\mu }_{0}}$(r), in units of mag arcsec⁻², according to Equation (1b). Then, the B-band central disk surface brightness was calculated from ${{\mu }_{0}}$(g) and ${{\mu }_{0}}$(r) by using (Equation (1c)), which stems from the filter transformation relation between the g, r, and B bands (Equation (2a)) derived from Smith et al. (2002).

LSB galaxies are commonly defined as a population of diffuse galaxies whose central surface brightness in the B band, ${{\mu }_{0}}$(B), falls below a specific threshold value, which is 21.65 ± 0.30 in Freeman (1970), 22.0 in O'Neil et al. (1997a), and 23.0 in Impey & Bothun (1997). In general, the most common threshold values found in the literature are between 21.5 and 23.0 mag arcsec². In this paper, we adopted a threshold of ${{\mu }_{0}}$(B) $\geqslant$ 22.5 mag arcsec⁻² for the B-band central surface brightness and put additional constraints on the axis ratio (b/a $\geqslant$ 0.3) to select a sample of LSB galaxies with no edge-on galaxies. With these two constraints, we finally constructed a sample of LSB galaxies consisting of 1129 non-edge-on LSB galaxies from the α.40 SDSS DR7 survey. This sample is a relatively unbiased sample of the disk-dominated LSB galaxies, and it is complete both in H i observation and in optical magnitude within the magnitude limit of SDSS photometric observations.

4.2.1. Optical Properties

Distributions of the B-band central surface brightness (${{\mu }_{0}}$(B)), absolute magnitude (M(B)), the B–V color (B–V), and the scatter of ${{\mu }_{0}}$(B) against the B-band apparent magnitude (B) for all galaxies in our LSB galactic sample are shown in Figure 8(a)∼(d).

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The B-band central surface brightnesses of all these LSB galaxies (Figure 8(a)) distribute from 22.5 to 28.3 mag arcsec⁻², with 53% (597/1129) between 22.5 and 23.0 mag arcsec⁻², 34% (388/1129) between 23.0 and 24.0 mag arcsec⁻², 9% (103/1129) between 24.0 and 25.0 mag arcsec⁻², and the remaining 4% (41/1129) with the B-band central surface brightness fainter than 25.0 mag arcsec⁻².

The coverage of the B-band absolute magnitude (Figure 8(b)) of all the galaxies in the LSB galactic sample is from −27.0 to −12.3 mag. There are 43 galaxies in our LSB galactic sample that are intrinsically very faint (M(B) > −17.3 mag; Poggianti et al. 2001). We present the distribution of the B–V color of the LSB galactic sample in Figure 8(c), which reveals that our sample covers from −0.22 to 1.88 mag in B–V color and has 98.4% galaxies bluer than B–V = 0.75 mag. This strongly proves that this sample is a bluish sample of LSB galaxies. Additionally, we depicted the distribution of the B-band central surface brightness against the apparent magnitude in Figure 8(d), from which we can see a trend that fainter galaxies tend to have fainter central surface brightness.

4.2.2.  The 21 cm H i Properties

4.2.3. Environmental Properties

Environment is a critical factor that affects the star formation and evolution of galaxies. In this subsection, we made a simple study on the environment where galaxies in our LSB galactic sample live.

In order to have an overview, we provide graphic illustrations of distributions of our LSB galaxies overplotted on the large-scale structure plot (Figures 5 and 6 in Haynes et al. 2011) of the ALFALFA survey in the local universe. As an example, Figure 9 in our paper shows cone diagrams of a four-degree-wide slice of the LSB and α.40 galaxies in the ALFALFA spring and fall sky centered on decl. = +26°, including the full ALFALFA bandpass redshift range, cz < 18,000 km s⁻¹. Blue open circles mark the locations of galaxies detected by the ALFALFA survey, whereas red open circles denote galaxies in our LSB galactic sample.

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To study the environment of galaxies in our LSB galactic sample quantitatively, we follow the method of finding clusters given by Wen et al. (2009). For each galaxy in our LSB galactic sample at a given redshift, z, we counted the number of all galaxies detected by the photometric observation of SDSS DR7 within a radius of 1.0 Mpc from the center represented by the galactic position and a photometric redshift gap between $z\pm 0.04(1+z)$. Fortunately, the SDSS provides two useful functions for users. One is the fGetNearbyObjAllEq function, which returns a table of all objects detected by SDSS within a radius in arcmin of a given equatorial point, and the other is the fCosmoDa function, which returns the angular diameter distance at a given redshift. Therefore, with the help with these functions, we can easily derive the number of all objects within that radius and redshift space detected by the SDSS photometric survey. We show the simple environmental properties of our LSB galactic sample in Figure 10, which shows the distributions of the number counts of galaxies within a radius of 0.5 Mpc in panel (a), 1.0 Mpc in panel (b), and 3.0 Mpc in panel (c), as well as the scatter distribution of the detectable galactic number count with a radius of 1.0 Mpc against the H i mass for this LSB galactic sample in panel (d). Clearly, in Figure 10(b), about 92% out of the total galaxies in our LSB galactic sample have less than eight galaxies within about the 1.0 Mpc radius and $z\pm 0.04(1+z)$ redshift space. It strongly distinguishes from the count distribution of the member galactic candidates of clusters within 1.0 Mpc in Figure 4 in Wen et al. (2009), which distributes totally from greater than 8 (gray dashed line) to 50, with the peak at 16 (gray solid line), compared to the peak of ~2–3 for our LSB galactic sample. Even if the radius is increased to 3.0 Mpc, the portion of galaxies having less than eight detectable neighbors is still as high as 65%. This definitely indicates that LSB galaxies are more likely to reside in the field environment. Additionally, from Figures 10(a), (b), and (c), the parent sample has nearly the same number of neighbors as the LSB sample. This is large because the parent sample itself is a gas-rich galactic sample from the ALFALFA H i survey, as galaxies rich in gas generally favor a low-density environment. In Figure 10(d), we show the distribution of the neighboring galactic counts against the H i mass for our LSB galactic sample as black dots. As mentioned before, the H i mass of our LSB galactic sample ranges from 6.5 to 11.0 dex in log M(H i)/M⊙. For a clear understanding of the probable trend between the count of neighboring galaxies and the H i mass, we divided the H i mass range of the LSB galactic sample into nine bins with a bin size of 0.5 dex in log M(H i)/M⊙, and investigated the relation (the gray broken line in Figure 10(d)) between the mean H i masses and mean number counts of neighboring galaxies of the nine mass bins, which does not give a visible relation between the mass and mean counts of neighboring galaxies, but shows that the mean counts of neighboring galaxies in all mass bins appear lower than five.

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