Low Surface Brightness Galaxy catalogue selected from the α.40-SDSS DR7 Survey and Tully–Fisher relation Free

Abstract

We present a catalogue of an H i-selected sample of 1129 low surface brightness galaxies (LSBGs) searched from the α.40-SDSS DR7 survey. This sample, consisting of various types of galaxies in terms of luminosity and morphology, has extended the parameter space covered by the existing LSBG samples. Based on a subsample of 173 LSBGs that are selected from our entire LSBG sample to have the 2-horn shapes of the H i line profiles, minor-to-major axial ratios (b/a) less than 0.6 and signal-to-noise ratio of H i detection greater than 6.5, we investigated the Tully–Fisher relation (TFr) of LSBGs in the optical B, g, and r bands and near-infrared (NIR) J, H, and K bands as well. In optical bands, the LSBG subsample follows the fundamental TFr that was previously defined for normal spiral galaxies. In NIR bands, the TFrs for our LSBG subsample are slightly different from the TFrs for the normal bright galaxies. This might be due to the internal extinction issue. Furthermore, the mass-to-light ratio (M/L), disc scale length (h), and mass surface density (⁠|${\bar{\sigma}}$|⁠) for our LSBG subsample were deduced from the optical TFr results. Compared with high surface brightness galaxies (HSBGs), our LSBGs have higher M/L, larger h, and lower |${\bar{\sigma}}$| than HSBGs.

1 INTRODUCTION

Low surface brightness galaxies (LSBGs) are galaxies that have central surface brightness fainter than the night sky (∼22.5 B mag arcsec⁻²; e.g. Impey & Bothun 1997; Impey, Burkholder & Sprayberry 2001; Ceccarelli et al. 2012). Comparing with their high surface brightness galaxy (HSBG) counterparts, they are gas-rich, metal-poor (≤1/3 solar abundance; McGaugh & Bothun 1994), dust-poor (e.g. Matthews, van Driel & Monnier-Ragaigne 2001), and with diffuse, low-density stellar discs, (e.g. de Blok, McGaugh & van der Hulst 1996; Burkholder et al. 2001; O’Neil et al. 2004; Trachternach et al. 2006), which indicates that they are low in star formation and insufficiently evolved (e.g. Das et al. 2009; Galaz et al. 2011; Lei et al. 2018). Usually, dwarf galaxies can mostly satisfy the definition of LSBGs, so dwarf galaxies make a great contribution to the LSBG family. Meanwhile, some of the giant galaxies have very extended and thus faint disc. LSBGs between dwarf and giant types are more similar to late-type spiral galaxies in general properties (Chung et al. 2002). These main types of LSBGs are proposed to have fundamentally different formation and evolutionary histories from each other.

LSBGs contribute 30–60 per cent to the number density of local galaxies (McGaugh, Bothun & Schombert 1995; McGaugh 1996; Bothun, Impey & McGaugh 1997; O’Neil, Bothun & Schombert 2000; Trachternach et al. 2006; Haberzettl et al. 2007) and ∼20 per cent to the dynamical mass of galaxies (Minchin et al. 2004), making them cosmologically significant. However, due to the observational capability and selection effects, LSBGs are seriously underrepresented in the present-day galaxy catalogues, which inevitably lead a bias (towards HSB Glaxies) to our understanding of the galaxy formation and evolution. More LSBG samples are necessary. The first LSBG samples were mainly composed of LSBGs in the bright end of surface brightness (Schombert et al. 1992; McGaugh & Bothun 1994; de Blok, van der Hulst & Bothun 1995; Impey et al. 1996). Afterwards, Zhong et al. (2008) derived a large sample of 12 282 LSBGs from the main galaxy sample of SDSS DR4 (Adelman-McCarthy et al. 2006), with dwarf galaxies removed. Williams et al. (2016) discovered LSBGs that have fainter surface brightness (μ₀(B) ∼ 24–28 mag arcsec⁻²) using SDSS gri combined images within the GAMA equatorial fields (Driver et al. 2011; Liske et al. 2015). As LSBGs are mostly rich in H i gas, the extragalactic H i surveys, such as the Arecibo Legacy Fast Arecibo L-band Feed Array (ALFALFA; Giovanelli et al. 2005), are good data bases for searching LSBGs. So, we have established a sample of 1129 LSBGs from the combination of 40 |${{\ \rm per\ cent}}$| data release of ALFALFA (α.40; Haynes et al. 2011) and SDSS DR7 (Abazajian et al. 2009) imaging survey. The sample establishment, including the key steps of sky subtraction and surface photometry, have been reported in detail in Du et al. (2015), which is the first paper of a series of our papers on LSBG studies. In this paper, we expect to release the catalogue of the α.40-SDSS DR7 LSBG sample.

Additionally, Tully–Fisher relation (TFr), an empirically linear correlation between the rotational velocity and the total absolute magnitude, was first discovered for HSB spiral galaxies in Tully & Fisher (1977). The TFr has been widely studied for various samples of HSB spiral galaxies in optical bands (Tully & Fisher 1977; Bottinelli et al. 1986; Fouque, Bottinelli & Gouguenheim 1990; Sprayberry, Bernstein & Impey 1995; Courteau et al. 1997; Pizagno et al. 2007). Later, because near-infrared (NIR) luminosity suffers less from Galactic and internal extinction than optical luminosity, and is more sensitive to stellar mass and thus correlates more tightly with the rotational velocity, many TFr studies have been performed in NIR passbands (Burstein 1982; Freedman 1990; Theureau et al. 2007; Liu et al. 2008; Masters et al. 2008; Tiley et al. 2016), in mid-infrared passbands (Sorce, Tully & Courtois 2012a; Sorce et al. 2013), and even in multiwavelength passbands from optical to NIR (Sakai et al. 2000; Bell & de Jong 2001; Ponomareva et al. 2017). Recently, the baryonic TFr, a relation between the rotational velocity and the total baryonic mass (including both stellar and gas mass), has received a lot of attention (McGaugh et al. 2000; McGaugh 2005; Begum et al. 2008; Trachternach et al. 2009; Gurovich et al. 2010; McGaugh 2012; Zaritsky et al. 2014; Papastergis, Adams & van der Hulst 2016). So far, TFr studies for LSBGs are deficient due to the lack of LSBG samples (Sprayberry et al. 1995; Zwaan et al. 1995; O’Neil et al. 2000; Chung et al. 2002). The TFr studies are all performed in optical passbands. Zwaan et al. (1995) gathered 42 LSBGs from literature, and showed that their LSBGs followed the fundamental TFr defined for HSB spiral galaxies. This draws a picture that LSBGs may not be fundamentally different from normal HSBGs. However, based on a homogeneous sample of 43 LSBGs, O’Neil et al. (2000) claimed that their LSBGs did not show any significant correlation between the velocity width and the optical absolute magnitude. This conclusion was later proved by Chung et al. (2002). In this paper, we expect to investigate the TFr of LSBGs based on our own LSBG sample.

In this paper, we briefly introduce the establishment of our LSBG sample in Section 2.1 and release the complete catalogue in Section 2.2. We perform a luminosity classification in Section 3.1 and a morphological classification for our LSBG sample by machine learning in Section 3.2. The optical and radio properties are studied in Section 4. In Section 5, we study the optical and NIR TFr for a subsample from our entire LSBG sample. The strength and weakness of this LSBG sample are discussed in Section 6.1. The mass-to-light ratios (M/Ls), disc scale length, and mass surface density deduced from the TFr results on the basis of our LSBG subsample are discussed in Sections 6.2∼6.4. Finally, we summarize the paper in Section 7. Through this paper, the distances that we used to convert angular sizes to physical (kpc) sizes are directly the ones listed in the α.40 catalogue (Haynes et al. 2011), which adopted the Hubble constant to be H₀ = 70 km s⁻¹ Mpc⁻¹. Magnitudes in this paper are all in the AB magnitude system.

2 OUR α.40-SDSS DR7 LSBG SAMPLE

2.1 A brief review of sample establishment

In Du et al. (2015), we have constructed a sample of LSBGs that is searched from the α.40-SDSS DR7 survey combination. We just make a brief introduction of the sample selection in this section because details have already been reported in Du et al. (2015).

First of all, we reconstructed the sky background maps for SDSS DR7 g- and r-band images for each galaxy of the entire α.40-SDSS DR7 sample (12 423 galaxies). The method we used for reconstructing sky background map is to fit the sky pixels on the object-subtracted image row-by-row and column-by-column (Zheng et al. 1999; Wu et al. 2002), which is a better method of constructing sky background for bright galaxies with faint extended outskirts and in particular for LSBGs (Du et al. 2015). After subtracting the sky background of galaxies, we made surface photometry using SE xtractor (Bertin & Arnouts 1996) and made radial profile fitting using Galfit (Peng et al. 2002). Afterwards, on the basis of the photometric and fitting results, the g- and r-band central surface brightness (μ₀) were calculated and then converted to the B-band values (μ₀(B)). Finally, 1129 galaxies that have μ₀(B) fainter than 22.5 mag arcsec⁻² and minor-to-major axial ratios (b/a) in both g and r bands larger than 0.3 were selected to be our LSBG sample. It is an H i-selected LSBG sample, of which the data reduction, sky subtraction, surface photometry, profile fitting, selection criteria, and environment properties have been reported and studied in detail in Du et al. (2015). In this paper, we aim to make the catalogue of this LSBG sample released to the public who might be interested in this sample and also investigate the TFrs of LSBGs in the optical and NIR bands based on this LSBG sample.

2.2 Catalogue

We present the entire catalogue of our LSBG sample in this paper. A short list of the catalogue is shown in Table 1 and the long rest list is shown in Table A1 in Appendix. The column descriptions of both tables are shown next:

Column 1: galaxy name as it appears in the α.40 catalogue.

Column 2: velocity width of the H i line profile measured at the 50 |${{\ \rm per\ cent}}$| level of the peak (stemed from the α.40 catalogue).

Columns 3 |$\&$| 4: g- and r-band magnitudes and errors measured on the sky-subtracted images of galaxies within the automatic aperture (auto) by SE xtractor. As mentioned before, the sky subtraction was performed using a column-by-column and row-by-row fitting method, which was more precise (details in section 3.1 in Du et al. 2015). These magnitudes have already been corrected for the Galactic extinction using the dust maps of Schlegel, Finkbeiner & Davis (1998; SFD98). It is worth noting that SE xtractor has computed several types of magnitudes: isophotal (ISO), corrected isophotal (ISOCOR), fixed-aperture, auto, and petrosian (PETRO) magnitudes. Thereinto, the auto, inspired by Kron’s ‘first moment’ algorithm (see details in Kron 1980), is a flexible and accurate elliptical aperture whose elongation ε and position angle θ are defined by the second-order moments of the object’s light distribution. Then, within this aperture, the characteristic radius r₁ 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 galaxy profiles convolved with Gaussian seeing, more than 90 |${{\ \rm per\ cent}}$| of the flux is expected to lie within a circular aperture of radius kr₁ if k = 2, almost independently of their magnitudes. This picture remains unchanged if they consider an ellipse with εkr₁ and |$\frac{kr_{1}}{ \epsilon }$| as the principal axes. By choosing a larger k = 2.5, more than 96 |${{\ \rm per\ cent}}$| of the flux is captured within the elliptical aperture. So, the auto magnitudes are intended to give the most precise estimate of ‘total magnitudes’, at least for galaxies. More details about the Kron radius and the auto photometry used in SE xtractor have been reported in SE xtractor manual and Kron (1980), Infante (1987), and Bertin & Arnouts (1996). For our automatic photometry using SE xtractor, we keep k = 2.5 that is also the default setting of SE xtractor (Bertin & Arnouts 1996), and our following studies and analysis would all be based on the auto magnitudes in this table.

Columns 6 |$\&$| 7: g-band major-to-minor axis ratio (b/a) and disc scale length (h(g)) that were derived from fitting the sky-subtracted image of the galaxy with a single exponential profile using galfit (Peng et al. 2002). The details about our fitting processes have been reported in section 3.3 in Du et al. (2015). No uncertainties on b/a or h(g) are listed in this table because galfit did not provide any errors for these outputs. galfit gave the disc scale length in angular size, and we converted the angular size to the physical size (kpc) using the distance information from the α.40 catalogue (Haynes et al. 2011).

Column 8: g-band effective radius (R_eff(g)) given by SE xtractor. SE xtractor gave the effective radius in angular size and we then converted the angular size to the physical size using the distance information from the α.40 catalogue. No uncertainties on R_eff(g) are listed in this table because SE xtractor did not provide the uncertainties for this parameter.

Column 9: g-band central surface brightness in mag arcsec⁻², μ_0,g, which is calculated from the combination of g-band auto magnitude ,g (Column 3), major-to-minor axis ratio, b/a (Column 6), and disc scale length, h(g) (Column 7), according to the calculation formula (1b) in Du et al. (2015). As we have no available uncertainties on b/a and h(g), unfortunately we could not provide the uncertainties on |$\mu (0,g)$|⁠. Additionally, the g-band effective radius and central surface brightness in columns 8 |$\&$| 9 are listed in this table only because they might be useful for readers who would like to pick out candidates of ultradiffuse galaxies (UDGs) from this LSBG sample.

Column 10: B-band central surface brightness in mag arcsec⁻², μ_0,B. This value was converted from μ_0,g (Column 9) and μ_0,r according to the transformation formula in Smith et al. (2002), which was also listed as formula (1c) in Du et al. (2015) for convenience. μ_0,r was calculated using the same method as μ_0,g in Column 9. As we have no available uncertainties on both μ_0,g and μ_0,r, we cannot provide any uncertainties on μ_0,B.

Columns 11∼14: Besides the central surface brightness based on the auto magnitudes (Column 10), we also list the B-band central surface brightness based on the ISO, ISOCOR, PETRO magnitudes, and the average of the four (AVE) for each galaxy. As we have difficulty in providing the errors on these central surface brightness of each type, we wish that readers could make their own impression and constraints on the uncertainties of the central surface brightness by comparing μ₀ based on different magnitude types.

Column 15: morphological type by decision trees with random forest of machine learning (see details in Section 3.2)

Column 16: luminosity type in terms of B-band absolute magnitude. D, G, and M denote dwarf LSBGs, giant LSBGs, and moderate-luminosity LSBGs (see details in Section 3.1), respectively.

More useful parameters, such as RA, Dec., distance, radial velocity, and the H i mass, can be easily accessed from the α.40 catalogue (Haynes et al. 2011) or the entire ALFALFA catalogue (Haynes et al. 2018).

3 CLASSIFICATION OF THE LSBG SAMPLE

3.1 Luminosity classification

The surface brightness of galaxies shows a continuous distribution, so the traditional definition of an LSBG (Impey & Bothun 1997) based on a simple threshold in central surface brightness is in fact unphysical. This broad definition means that the existing LSBG catalogues must include LSBGs of different types. To date, three main types have been discovered in the LSBG regime. They are dwarf LSBGs (Sandage & Binggeli 1984; Impey, Bothun & Malin 1988), giant LSBGs (Bothun et al. 1987, 1990; Sprayberry et al. 1993), and we call the LSBGs with luminosities between dwarfs and giants the ’moderate-luminosity’ LSBGs. Majority of existing LSBG catalogues include a large fraction of moderate-luminosity LSBGs, such as Impey et al. (1996) and Zhong et al. (2008):

1. dwarf LSBGs: Generally, galaxies with M_B > −17 mag are considered as dwarf galaxies (Dunn 2010 and references therein), which can easily satisfy the definition for LSBGs. Among the dwarf LSBGs, a population of UDG (van Dokkum et al. 2015) is special because they have faint (μ_g(0) > 24 mag arcsec⁻²) and very extended discs (effective radius, R_eff > 1.5 kpc).

2. giant LSBGs: They are bright (M_B > −19 mag; Sabatini, Roberts & Davies 2003), but their disc surface brightness are quite faint because their discs are very extended. For example, the disc of Malin-1, an extreme giant LSB galaxy, has extended to ∼160 kpc in diameter (Galaz et al. 2015; Boissier et al. 2016). Compared with the dwarf LSBGs, giant LSBGs are often redder in colour.

3. moderate-luminosity LSBGs: LSBGs with moderate luminosity (−19 < M_B < −17 mag) are more similar to late-type spiral galaxies in appearance and general properties, except for the overall low surface brightness and surface density. Up to now, it is proposed that different types of LSBGs may have distinct formation and evolutionary histories from each other (Matthews et al. 2001).

In Fig. 1(a), We classified the galaxies in our LSBG sample in terms of B-band absolute magnitude. Of the total 1129 galaxies, 39 galaxies are giants (M_B ≤ −19.0 mag), 607 galaxies are dwarfs (M_B ≥ −17.0 mag), among which 28 galaxies are UDG candidates with μ_g(0) > 24 mag arcsec⁻² and R_g,e > 1.5 kpc, and the rest 483 galaxies are moderate-luminosity LSBGs (−19.0 < M_B < −17.0 mag) . Additionally, in B – V colour, our sample is dominated by blue galaxies since it has 909 blue (B – V ≤ 0.6 mag) and 220 red galaxies (B − V > 0.6 mag (Fig. 1b).

3.2 Morphological classification by machine learning

For a statistical information of the morphologies of our LSBG sample, we made morphological classification for our sample using machine learning technique. Among various machine learning classifiers, we chose the classifier of decision trees with random forests that has been proved to have the highest accuracy for classification (Dobrycheva et al. 2017).

Above all, we should derive a training sample to train the classifier. There is a catalogue of 14 034 SDSS galaxies with detailed visual classifications (Nair & Abraham 2010), covering morphological types from T = −5 to 10 (E to Im) in de Vaucouleurs system. We randomly selected 80 |${{\ \rm per\ cent}}$| of galaxies from this catalogue as the training sample and the rest 20 |${{\ \rm per\ cent}}$| as the test sample.

First of all, the attributes of redshift (z), g-band magnitude (g), and absolute magnitude (M_g), r-band magnitude(r), g – r colour, and the concentration index (R₉₀/R₅₀) of galaxies in the training sample were adopted to train the random forests Classifier model. The trained Classifier was then fed into the test sample to predict the morphological types and test the accuracy. The accuracy of the trained classifier was 82.5 |${{\ \rm per\ cent}}$| for the total test sample and 88.8 |${{\ \rm per\ cent}}$| for E type (E, S0), 82.8 |${{\ \rm per\ cent}}$| for S type (Sa, Sab, Sb, Sbc, Sc, Scd), 31.4 |${{\ \rm per\ cent}}$| for L type (Sd, Sm, Im), and 12.8 |${{\ \rm per\ cent}}$| for U type (unknown) of the test sample. Secondly, we adopted this trained random forests classifier to predict the morphological types (E, S, L, or U) of our LSBG sample. The results from the classifier show that of the total 1129 galaxies in our sample, 950 galaxies (84.1 |${{\ \rm per\ cent}}$|⁠) are classified into L type, 151 galaxies (13.4 |${{\ \rm per\ cent}}$|⁠) are classified into S type, 2 galaxies (0.2 |${{\ \rm per\ cent}}$|⁠) are classified into E type, and 26 galaxies (2.3 |${{\ \rm per\ cent}}$|⁠) are classified into U type, which indicates that our LSBG sample is dominated by the late-type galaxies in morphology. For a general review, the morphological classification results are listed in the column of T_M in Table 1.

Additionally, in Fig. 2, we checked the fraction of galaxies of E, S, and L types for our LSBGs of different luminosity types of giant, moderate-luminosity and dwarf LSBGs classified in Section 3.1. For giant LSBGs (red), the morphological types of galaxies are dominated by earlier morphological types (S). For moderate-luminosity LSBGs (green), the morphological types of galaxies are composed of less S and more L types. For dwarf LSBGs (blue), the morphological types of galaxies are absolutely dominated by very late morphological type (L). Therefore, for our LSBG sample of different luminosity types from giant to dwarf, the dominance of morphological types have changed from earlier types (S) to later types (L). So, the morphological classifications by machine learning technique agree statistically with the luminosity classification for the three luminosity types of LSBGs in our sample.

4 OPTICAL AND RADIO PROPERTIES

We show the optical and radio properties of the LSBG sample in Fig. 3. In each panel, dwarf, moderate-luminosity, and giant LSBGs in our sample are represented by the cyan, green, and red circles. Additionally, we compare our LSBG sample with the 21 HSBGs (purple stars) that have optical band photometry from Ponomareva et al. (2017) and Ponomareva et al. (2017) and also with the available LSBGs from McGaugh & Bothun (1994; open triangles), O’Neil et al. (2000; open squares), and de Blok et al. (1995)(asterisks) in the figure.

We show the g- and r-band auto magnitude distributions for both our LSBG sample (black) and the HSBGs (purple) in panels (a) and (b). In order to show the distribution of the HSBGs clearly, the zoom-in windows are also plotted within each panel. The g-band magnitude of our LSBGs peaks at 17.57 mag and has a Gaussian FWHM of 2.05 mag in g band. The r-band magnitude of our LSBGs peaks at 17.26 mag and has a Gaussian FWHM of 2.17 mag in r band. Comparing with the HSBGs, our LSBGs are entirely much fainter.

In panels (c) and (d), both our LSBG sample and the HSBGs are shown in the size – central surface brightness plane. Comparing with our LSBGs, the HSBGs are systematically brighter in central surface brightness and larger in apparent size. As the point spread function (PSF) widths of the SDSS g- and r-band images are statistically 1.0 ∼ 2.0 arcsec, galaxies with very small measured disc scale length (below h = 1.0 arcsec line) are better to be considered to be spurious. In order to show the distribution of LSBGs and the h = 1.0 arcsec line clearly, we overplot the zoom-in windows within panels (c) and (d). Fortunately, no galaxies in our sample have measured disc scale lengths less than 1.0 arcsec. For the SDSS survey, Kniazev et al. (2004) found that the 3σ limiting surface brightness goes down to ∼26.4 mag arcsec⁻² in g band and goes down to ∼26.2 mag arcsec⁻² in r band in a circular aperture of a radius of 12 arcsec. Therefore, we should have lower confidence in our measured μ₀ values for those galaxies that have measured μ_0,g fainter than 26.4 mag arcsec⁻² or μ_0,r fainter than 26.2 mag arcsec⁻², but the disc scale lengths are measured to be smaller than 12 arcsec. Furthermore, for those galaxies that are extended to have measured disc scale lengths larger than 12 arcsec, it is logically reasonable for them to have μ_0,g fainter than 26.4 mag arcsec⁻² or μ_0,r fainter than 26.2 mag arcsec⁻². It mainly depends on the sizes of those galaxies that how faint their central surface brightness could reach. From the literature, Pohlen & Trujillo (2006) found that the 3σ limiting surface brightness of the SDSS images could go down to ∼27.0 mag arcsec⁻² at a very large circular radius of 150 arcsec in both g and r bands. Therefore, the central surface brightness are not expected to be measured fainter than ∼27.0 mag arcsec⁻² for our LSBGs that have disc scale length measured to be far less than 150 arcsec. So, for our LSBGs that have disc scale length measured to be larger than 12 arcsec but smaller than 150 arcsec, the μ₀ measurements for these galaxies that are measured to be fainter than or around ∼27.0 mag arcsec⁻² might be unreliable. For a reminder, all of the seven galaxies with such unreliable measurements of μ₀ were marked (as black filled circles) to the right of the 3σ limiting line (red dashed lines) that was extracted by (Kniazev et al. 2004) for the SDSS imaging survey in Figs 3(c) and (d). For emphasis, they were still marked in the following panels. For further reminders, in Table 1, the AGC names of these seven galaxies are labelled by adding a star symbol. It is necessary to note ahead of time that these seven galaxies that have unreliable measurements of central surface brightness would be excluded in the TFr studies later in Section 5.

In panel (e), it shows that LSBGs and HSBGs almost cover the similar range in physical size. For the LSBG sample itself, it shows that LSBGs that have higher luminosities tend to be larger in physical size. Additionally, there are 24 dwarf LSBGs (overplotted with black plus) that can be candidates of UDGs because they have μ₀(g) > 24.0 mag arcsec⁻² and effective radius R_eff(g) > 1.5 kpc (van Dokkum et al. 2015).

In panels (f) and (g), HSBGs (purple stars) are at the bottom edge or below the LSBGs (circles), implying that LSBGs usually have larger physical size and have higher H i mass than the HSBGs with the same level of luminosity. For the LSBG sample itself, it shows that our LSBGs that have fainter luminosities tend to be lower in H i mass and smaller in size. This phenomenon for our LSBGs agrees with that for other LSBG samples from O’Neil et al. (2000; black open squares) and McGaugh & Bothun (1994; black open triangles). Comparatively, for our sample, giant LSBGs (red circles) have the highest H i masses and largest sizes, then the moderate-luminosity LSBGs (green open circles), and the dwarf LSBGs have the lowest H i masses and smallest sizes. It is worth noting that in panel (g), an outlier with disc scale length smaller than ∼0.1 kpc (the black dashed line that is a widely accepted lower size cut-off for a galaxy) deviates from most of the LSBGs in size. We checked this outlier (AGC 748778) and found that it is very nearby (radial velocity = 258 km s⁻¹), has a very low H i mass (M(H i) = 10^6.36 M_⊙) and a very narrow width of the H i line profile, and appears to be extremely faint, irregular, and small in SDSS gri combined images. Such a galaxy image was not expected to be well fitted by an exponential profile model using galfit. So, the disc scale length measured for this outlier (AGC 748778) might be unreliable. However, it will not seriously affect the statistical trend of the sample distribution in the disc scale length – luminosity plane. For a reminder, in Table 1, the AGC name, 748778, that might have unreliable measurements for disc scale length value is labelled by adding an asterisk symbol. Additionally, this galaxy would also be excluded in the TFr fitting later in this paper.

In panel (h), the HSBGs are systematically brighter in both central surface brightness and luminosity than the LSBGs. By comparison, there are obviously no LSBGs with μ₀(B) < 22.5 mag arcsec⁻². This is due to our selection threshold. It is interesting that in the μ₀(B) – M(B) plane, there is a void at the lower left corner for our sample and another sample of McGaugh & Bothun (1994; open triangles). This void space should have been populated by very low surface brightness, but intrinsically very luminous and large galaxies, which are in fact the giant LSBGs. This void might be either due to the low number density of giant LSBGs or that our sample actually lacks for LSBGs with bulges due to our selection method (see more in section 6.1 or in Du et al. 2015).

In panels (i)∼(l), comparing with LSBGs, the HSBGs are systematically redder in colour. For the LSBGs, while our sample has extended the parameter space covered by the previous samples towards lower luminosity and lower H i mass, the previous samples also cover parameter space that is not covered by our sample, specifically towards redder B – V colour and higher H i mass-to-luminosity ratio. The main reason might be that our sample lacks for LSBGs with bulges that are generally redder than disc and dwarf galaxies due to the selection method (see more discussions in section 6.1 or in Du et al. 2015), although five LSBGs (overplotted with the open diamonds) in our sample appear to be very red (B – V > 1.0 mag), low-luminosity (M(B) > −17.0 mag) (panel i) and gas-rich (M|$_\rm {H\, {{\small I}}}$|/L_B > 1.0; panels k and l). In panel (l), a clear trend between absolute magnitude and gas-to-star ratio appears to exist for the entire LSBG sample, showing that fainter LSBGs would be larger in gas-to-star ratio, however, it shows no colour–absolute magnitude relation (panel i), colour–gas content correlation (panel j) nor colour–gas-to-star relation (panel k).

5 TULLY–FISHER RELATION

The TFr has been well defined between the luminosity and the maximal rotational velocity for late-type spiral galaxies (with high surface brightness; Tully & Fisher 1977).

There are two main methods to measure the maximal rotational velocities, V_max, of spiral galaxies. First, V_max could be measured from the width of the global H i line profile, as the observed width of the H i line profile, in km s⁻¹, gives the observed Doppler broadening due principally to the galaxy’s rotation. In the literature, the width of the H i line profile at 50 |${{\ \rm per\ cent}}$| of peak flux, W₅₀, has been greatly used as a good measure of the V_max of a galaxy to study the TFr of spiral galaxies in the absolute magnitude – W₅₀ plane (Zwaan et al. 1995; O’Neil et al. 2000; Chung et al. 2002). Secondly, V_max could be measured from the high-quality rotation curves that can be derived from spatially resolved H i velocity maps of galaxies. It is clearly demonstrated in Ponomareva et al. (2017) that the shape of the global H i line profile may hint at the shape of the rotation curve of a galaxy in several cases. Generally, the global H i line profile can be observationally obtained much faster than the high-quality H i velocity map for a galaxy, so the first method that uses the line width to measure the maximal rotation velocity of a spiral galaxy is feasible for large samples of galaxies. A detailed comparison in fig. 6 in Ponomareva et al. (2017) demonstrated that the rotational velocity measured from W₅₀ will be underestimated in comparison with V_max derived from the high-quality rotation curve for those dwarf galaxies that mostly have slowly rising rotation curves or for those massive spiral galaxies that mostly have declining rotation curves. However, for those galaxies that have the classical double peak (2-horn) profiles, the rotational velocity measured from W₅₀ are generally consistent with V_max derived from the rotation curves because the 2-horn line profile gives an indication that the rotation curve of a galaxy will reach its flat part. In view of this, our entire sample that is composed of a variety of galaxy types from dwarf to giant in luminosity and from disc-like to irregular in morphology, corresponding to various rotation curves, should be carefully refined in terms of H i line profile for better study of TFr.

In order to make sure that the W₅₀ is a good measure of V_max of the galaxy, first of all, with the examples of good 2-horn H i profiles given in Courtois et al. (2009) as standards, we manually inspected the H i line profile for every LSBG of our entire sample and selected those galaxies that have 2-horn H i line profiles. For example, we show the 2-horn H i line profiles of several of our LSBGs to give an idea of our identification for the 2-horn profiles in Fig. 4. These example LSBGs were chosen to have their signal-to-noise ratio (SNR) of H i detection ranging from high to low. Secondly, we are recommended by the α.40 release paper (Haynes et al. 2011) to have those galaxies with the SNR of the H i line less than 6.5 excluded because it is claimed in Haynes et al. (2011) that the detection of H i with SNR > 6.5 by ALFALFA is a real detection. Thirdly, large uncertainties might be inherent in inclination correction for W₅₀ for less-inclined galaxies (nearly face-on, with large minor-to-major axis ratio, b/a). This problem could be largely mitigated for largely inclined galaxies (nearly edge-on, with small b/a), in particular for edge-on galaxies, so we decided to further have those less-inclined galaxies (g-band b/a > 0.6) excluded. This step is only to mitigate the large uncertainties inherent in inclination correction for W₅₀ of less-inclined galaxies.

After the above criteria of H i-line profile shape, the SNR of H i line and the b/a, a subsample of 175 LSBGs were selected from our entire sample. This subsample is appropriate for the TFr studies because the key point is that, for this subsample galaxies, the W₅₀ of the H i line can accurately measure the maximal rotational velocities. To make the results more reliable, we removed the galaxies that are members of the seven galaxies that were mentioned in Section 4 to have unreliable measurements of central surface brightness and also removed the galaxy AGC 748778 that was also mentioned in Section 4 to have very small disc scale length measurements from this subsample. After such removal, our final subsample for TFr studies contains 173 LSBGs. We list these 173 LSBGs in Table 2 that includes the AGC name, inclination angle, i, calculated following equation (3), corrected W₅₀ calculated by equation (2), distance from the released entire ALFALFA catalogue, and the B-band absolute magnitude that has been corrected for Galactic and internal extinction following equation (4), the NIR-band absolute magnitudes that have been corrected only for Galactic extinction. For those galaxies with no UKIRT Infrared Deep Sky Survey (UKIDSS) observations, we just leave their corrected J-, H-, and K-band absolute magnitudes blank in Table 2.

Although this subsample is only 15 |${{\ \rm per\ cent}}$| of the entire sample in size, it includes all of the galaxies in the entire sample that are better suitable for TFr studies because the maximum rotation velocity of these galaxies could be properly measured by the width of the H i line, and these galaxies also suffered less from the uncertainties induced by the inclination correction for the H i line width.

To investigate the TFr, the W50 of the H i line should be first of all corrected for inclination effect. For the subsample galaxies, we have the minor-to-major axis ratios, b/a (q in Table 1), which were estimated by us fitting the g-band galaxy images with a single exponential profile model. As galaxies in this subsample are visually round, the b/a ratios derived from a single exponential model fitting for this subsample galaxies are more appropriate.

5.1 Optical TFr

In Fig. 5(a), we plot our LSBG subsample (black dots) in the plane of corrected H i line width versus corrected absolute magnitude (TF plane) in B band. The linear regression line (black solid line) of the TFr and the corresponding 95 per cent confidence bands (black dashed hyperbolae) of the regression line are overplotted. The function we used for regressing the TFr line is the MPFITEXY function in IDL. This function takes into account errors in both axes and performs an inverse least-squares regression, which can resolve the Malmquist bias (Willick 1994) that affects the slope of the TFr. Furthermore, in order to estimate the confidence intervals for the coefficients of the regression lines, we performed pair bootstrap resampling for 1000 times. Quantitatively, the vertical scatter and the tightness of the TFr fit are also measured following the method of Ponomareva et al. (2017). The coefficients, confidence intervals, scatter, and tightness of the TFr fit are also quantitatively listed in Table 3.

For comparison with the TFr of the LSBG sample of Zwaan et al. (1995), we overplotted the B-band TFr line, which is followed by the LSBG sample of Zwaan et al. (1995) as the red line in Fig. 5(a). Apparently in Fig. 5(a), the slope of the TFr line (black solid line) for our LSBG subsample seems to be well consistent with the slope of the TFr line (red solid line) for the LSBG sample of Zwaan et al. (1995). Importantly, this blue TFr line was in Zwaan et al. (1995) to be well consistent with the TFr of the HSBGs compiled by Broeils (1992). Therefore, we could indirectly deduce that the slope of the TFr line for our LSBG subsample is also in good agreement with the slope of the TFr line for HSBGs, albeit our LSBG sample has a larger scatter than the HSBGs (σ = 1.14 mag versus 0.77 mag) in B band.

Conclusively, in the optical B, g, and r bands, the TFr lines for our LSBG subsample agree with the TFr lines for HSBGs in slope. This result is consistent with the previous optical TFr results for LSBGs from Zwaan et al. (1995) and Sprayberry et al. (1995), which confirmed that the LSBGs follow the same TFr as the normal spiral galaxies (HSBGs) in the optical bands. However, compared with the HSBGs in Fig. 5, our LSBGs have relatively larger scatter and are less tighter around our TFr lines, which might be due to the variations in M/L among our LSBGs.

5.2 NIR TFr

In the subsample of 173 LSBGs that are appropriate for studying TFr, 99 galaxies have been observed in J, H, and K bands by the Large Area Survey of UKIDSS. In this subsection, we investigate the TFrs for these 99 LSBGs in J, H, and K bands.

First of all, we subtracted the sky background for NIR band images of galaxies in J, H, and K bands, using the same method as described in Zheng et al. (1999), Wu et al. (2002), and Du et al. (2015). Then, the magnitudes were measured on the sky-subtracted images within the auto aperture by SE xtractor. The measured magnitudes were then corrected for Galactic extinction using the prescription of Schlafly & Finkbeiner (2011). The internal extinction of LSBGs is very difficult to estimate. Tully et al. (1998) found that the internal extinction was unmeasurably small for faint galaxies with M_B ≥ −17 mag, which is the case for most of the galaxies in our LSBG subsample (see Fig. 5a). Most LSBGs are thought to be poor in dust based on the statistically inferred low internal extinction (e.g. Tully et al. 1998; Masters et al. 2008). Therefore, we did not make any internal extinction correction for the NIR-band magnitudes of our LSBG subsample, so as to avoid large uncertainties that might be introduced in the magnitudes due to internal correction. The details on data reduction including sky subtraction and surface photometry in the NIR bands will be reported in our forthcoming paper that will be on spectral energy distributions of our entire LSBG sample. Then, the NIR-band magnitudes were converted to absolute magnitude using the distance from the ALFALFA catalogue.

Based on the corrected W50 of H i and the absolute magnitudes without internal extinction correction, we show these 99 LSBGs (black dots) in the TFr planes in J, H, and K bands, respectively, and the corresponding regression TFr lines (black solid lines; by the MPFITEXY function in IDL) in Fig. 6. The coefficients, confidence bands, scatter, and tightness of the regression lines (black lines in Fig. 6) for J, H, and K bands were tabulated in the middle three lines in Table 3. For comparison, we overplotted the TFr lines of the three NIR bands for the HSBGs of Ponomareva et al. (2017) as the blue solid lines in Fig. 6.

Compared with optical bands, the slopes of the NIR-band TFr lines (black solid lines) for our LSBG subsample do not appear to be well consistent with the slopes of the NIR-band TFr lines (blue solid lines) for the HSBGs of Ponomareva et al. (2017). In each panel of Fig. 6, the offset in magnitude between the two TFr lines for LSBGs (black solid line) and HSBGs (blue solid line) increases with the line width. We think this might be caused by the effect of internal extinction correction for magnitudes. As LSBGs are well known to have little internal extinction, especially in the NIR bands, we did not do any internal extinction correction for magnitudes in J, H, and K bands for our LSBGs to avoid large uncertainties that might be introduced into the magnitudes. However, the internal extinction is a significant effect on magnitudes for HSBGs, so Ponomareva et al. (2017) corrected the magnitudes in J, H, and K bands for internal extinction for their HSBGs. The prescription they used for correcting internal extinction was from Tully et al. (1998), Sorce, Courtois & Tully (2012b), and Ponomareva et al. (2017), which calibrated the internal extinction value as an increasing function of the width of the H i line. This may account for the increasing magnitude offset between the TFr lines of our LSBGs (black solid lines in Fig. 6) and the HSBGs (blue solid lines in Fig. 6) from Ponomareva et al. (2017) with increasing line width. To check the effect of the internal extinction correction, we attempted to correct the internal extinction for J-, H-, and K-band absolute magnitudes for our LSBG subsample, using the same prescription as was used by Ponomareva et al. (2017) and listed in detail in Ponomareva et al. (2017). Based on the corrected absolute magnitudes, we further studied the NIR-band TFr regressions (black solid lines) in Fig. 7. The coefficients of the new TFr lines based on the corrected magnitudes in J, H, and K bands are tabulated in the last three lines in Table 3. In order to better compare the NIR TFr lines based on magnitudes before and after internal extinction correction, the TFr lines that were previously regressed before the internal extinction correction for NIR-band magnitudes were overplotted as the green lines in Fig. 7, which are exactly the black solid lines in Fig. 6. By comparison, no big difference was found between the TFr lines regressed before and after correcting the internal extinction for the magnitudes in J, H, and K bands, suggesting that the internal extinction correction is not an important factor that caused the difference between the TFr lines of LSBGs and HSBGs. This result further strengthens that the dust internal extinction is very little for LSBGs.

Then, we have to find other possible reasons that might cause the difference between the TFr lines of our LSBGs and the HSBGs from Ponomareva et al. (2017). Compared with the bright disc of HSBGs, the outer disc of LSBGs are fainter and thus more difficult to be detected in NIR bands than HSBGs. Generally, the surface brightness profiles of LSBGs are systematically 3.5 mag fainter than normal HSBGs in J band (Galaz et al. 2002). Therefore, with the shallow surface brightness limiting of UKIDSS, the disc of the extended LSBG could only be detected out to a small radius in NIR bands, compared to the outmost edge of the optical disc. However, for HSBGs in Ponomareva et al. (2017), even although their NIR-band magnitudes are measured on 2MASS images that are even shallower than UKIDSS, the HSBG discs can still be detected out to a larger radius close to the outmost edge of the optical disc. So, this would cause more serious underestimation of the total disc luminosity for LSBGs than for HSBGs in NIR bands, which can then cause the magnitude offset between the TFr lines for LSBGs and HSBGs. This might be one possible reason for the difference between the TFr lines of LSBGs and HSBGs in NIR bands, and more possible reasons needs to be explored in the further work.

6 DISCUSSION

6.1 Strength and weakness of our LSBG sample

As for strength, first, we need the redshift information to correct the cosmological dimming of the central surface brightness of galaxies. However, SDSS has spectroscopic redshifts only for those galaxies that are brighter than 17.77 mag in r band and called the main Galaxy sample of the SDSS survey. As most of LSBGs are fainter in optical band, it means that a large fraction of true LSBGs could not be defined by only using the optical survey of SDSS due to the lack of accurate redshifts from spectroscopy. For example, as shown in Fig. 3(b), 31 |${{\ \rm per\ cent}}$| of galaxies in our LSBG sample are fainter than 17.77 mag in r band. So, if we were selecting our LSBG sample only on the basis of the optical SDSS survey, those 31 |${{\ \rm per\ cent}}$| faint LSBGs would have been lost from our final sample. However, our LSBG sample is an H i-selected sample the radial velocities of which, especially the faint LSBGs, are all available from the α.40 H i catalogue. So, our H i-selected LSBG sample is not limited by the magnitude of the SDSS main Galaxy sample. Instead, it includes more faint LSBGs (31 |${{\ \rm per\ cent}}$| fainter than r = 17.77 mag) that could reach down to the surface brightness depth of the SDSS photometric survey but could not be confirmed by only using the SDSS data due to lack of the spectroscopic redshift.

Secondly, LSBG selections are very sensitive to the subtraction of sky background of images. However, as explained in detail in section 3.1 in Du et al. (2015), the SDSS photometric pipeline has a high risk of considering the large extended outskirts of the objects as part of the sky background, and thus overestimated the sky background of galaxies that further leads to the underestimation of the luminosity of galaxies. The underestimation in luminosity is ∼0.2 mag for bright galaxies with extended outskirts and is up to ∼0.5 mag for LSBGs. Therefore, we reconstructed the sky background map for all galaxy images, using a more precise method which were described and testified in detail in section 3.1 in Du et al. (2015). Thus, we corrected the underestimation of luminosity for galaxies in our LSBG sample.

As for the weakness, on one hand, our LSBG sample was selected based on the central disc surface brightness that was derived by fitting the galaxy image with a single exponential profile model with galfit. So, for galaxies that have light concentration or bulges in the centres, the central disc surface brightness of the disc must have been overestimated (bias to brighter values). However, we made some tests in Du et al. (2015) by fitting the LSBGs with two components (Sersic + Exponential) and found that the overestimation for the disc central surface brightness was systematically very small and less than 0.2 mag arcsec⁻². Additionally, the part of bulge-dominated galaxies (only 5.8 |${{\ \rm per\ cent}}$|⁠) is not dominant in our LSBG sample. In the case of overestimation, the true central disc surface brightness for those galaxies should be even fainter than currently, so these galaxies currently in our LSBG sample should still be members of the LSBG sample. However, it is poor in bulge-dominated LSB galaxies due to our fitting method of only using a single exponential profile precluding the detection of LSBGs with bulges. But this fitting method is fast and valid for the fast majority of the LSBGs. On the other hand, as selected from the H i survey, our LSBG sample is composed of more gas-rich galaxies, but should have been deficient in the true gas-poor LSBGs. If there was no such effects explained above, our LSBG sample would on one hand include more galaxies with large central light concentration or bulges. These galaxies would have large rotational velocities and bright luminosities so that our LSBG sample would be enlarged and stretched to the ends of larger W50 and brighter absolute magnitude in the TF plane in Section 5. On the other hand, our LSBG sample would have more gas-poor galaxies that would be more likely to enlarge or stretch our sample to the end of smaller W50 and lower absolute magnitude in the TF plane in Section 5. With those possibly added LSBGs in the TF plane, the TFr fit in Section 5 would probably be better constrained.

6.2 Mass-to-light ratios

As we concluded in Section 5 that our LSBG subsample follows a normal TFr that is followed by the normal HSBGs, we could attempt to explore the clues for the total M/L of these LSBGs by statistically comparing with the M/L of the Broeil’s HSBGs next, following the comparison method of Zwaan et al. (1995). In Fig. 5(a), the B-band slope of TFr (black solid line) for our LSBG subsample is −6.43 ± 0.51 (Table 3), which is in good agreement with the B-band slope of TFr (red line) for the HSB spiral galaxy sample of Broeils (1992; hereafter Broeils spiral sample), which is found to be −6.59. The intercept of the black line and red line are, respectively, −3.04 ± 1.07 (Table 3) and −3.73 ± 0.77 for the Broeils spiral sample (O’Neil et al. 2000). So, albeit the consistent slope, there is a systematical offset of ∼0.7 mag in absolute magnitude between the two TFr lines. This means that our LSBGs are systematically ∼0.7 mag fainter than the Broeils HSBGs in B-band magnitude, which can be expressed as M_LSBG–M_HSBG = 0.7 mag. As L_LSBG/L_HSBG = 10.0|$^{-0.4\times (M_{\mathrm{ LSBG}}-M_{\mathrm{ HSBG}})}$|⁠, the luminosity of our LSBGs is then systematically ∼0.5 times (=10.0^{−0.4 × 0.7}) the luminosity of the Broeils HSBG sample in B band at a fixed H i line width. As for central surface brightness, the mean B-band central surface brightness is 23.1 mag arcsec⁻² for our LSBG subsample and 21.6 mag arcsec⁻² for the Broeils HSBGs, so our LSBG subsample is 1.5 mag fainter than, or 1/4 (=10^{−0.4 × 1.5}) times, the typical central surface brightness value of the Broeils HSBGs. Therefore, according to equation (7), in order to account for the factors of differences in Σ₀ and L at the fixed line velocity, it can be deduced that M/L of our LSBGs must be ∼2.8 times of the M/L of Broeils HSB spiral galaxy sample.

However, although the NIR K-band slope of ∼4 (⁠|$L \propto V_{\mathrm{ max}}^{4}$|⁠) is recognized as the universal TFr (The slope k is corresponding to ∼10 when the luminosity is expressed in absolute magnitudes, M, and the TFr is in the form of M = −klog(V_max) + b). It should be noticed that the derivation of the TFr has been questioned by recent papers (e.g. Courteau et al. 2007; Ponomareva et al. 2017), which obtained shallower slopes (α < 4) in agreement with their own data in NIR K band. For our LSBG subsample, we also obtained a shallower TFr. So, insisting on using the theoretical equation (7) that reflects a universal TFr (⁠|$L \propto V_{\mathrm{ max}}^{4}$|⁠), we might have overestimated the luminosity, L, of our LSBG subsample and thus overestimated the luminosity ratios between our LSBG subsample and the Broeil HSBG sample at a fixed rotational velocity, and finally underestimated the M/L of our LSBG subsample. So, the ratio between the total M/L of our LSBG subsample and the Broeils HSBG sample should be larger than ∼2.8 that was deduced above based on a universal TFr of the slope of 4.

6.3 Disc scale lengths

Since |$M/L \propto V_{\mathrm{ max}}^{2}h/L$|⁠, where h is the disc scale length, M/L depends on h/L at a fixed line width. In Section 6.2, using equation (7), we derived that the M/L of our LSBG subsample was 2.8 times the M/L of the Broeils HSB spiral galaxy sample, and L (luminosity) of our LSBG subsample was 0.5 times the luminosity of the Broeils HSB spiral galaxy sample at a fixed line width. So, the disc scale length of our LSBG subsample should be 1.4 times of the disc scale length of the Broeils HSB spiral galaxy sample.

In Fig. 8, we plot the disc scale length versus the corrected H i line width for our LSBG subsample in g band. For comparison, HSBGs of the Broeils sample are overplotted in Fig. 8. It is worth noting that the disc scale length for HSBGs of the Broeils sample in Fig. 8 are in B band because only the B-band disc scale lengths are available for Broeils sample. This would not seriously affect the results about disc scale length later since galaxy size in the B-band image appears very similar to that in g band.

Apparently in Fig. 8, a trend exists between disc scale length and line width for both the LSBGs and the HSBGs. To make the trend as clear as possible, we made a linear regression between line width and disc scale length for both the LSBG subsample (black solid line) and the Broeils’s HSBG sample (red dashed line). It obviously appears that our LSBG subsample is systematically larger than the HSBG sample in disc scale length. Observationally, a minor difference between the slopes (1.05 for LSBGs versus 0.81 for Broeils’s HSBGs) of both lines is present, and the offset between the two fitting lines is 0.24 dex at W50 ∼ 40 km s⁻¹ (the minimal corrected W50 for our LSBG subsample) and 0.45 dex at W50 ∼ 300 km s⁻¹ (the maximal corrected W50 for our LSBG subsample). The offsets imply that our LSBGs have disc scale lengths that are 1.75(=10^0.24) times larger than the HSBGs of Broeils sample at the minimal corrected W50 and 2.82(=10^0.45) times larger than the HSBGs of Broeils sample at the maximal corrected W50. Therefore, our LSBGs are 1.75∼2.54 times larger in size than the HSBGs from the Broeils sample at the fixed line width. This factor range from observation is slightly higher than the factor of 1.4 deduced from the M/L above using equation (7). This discrepancy might be because that our LSBG subsample observationally follows a shallower TFr so that the M/L of our LSBGs has been underestimated by equation (7), which is based on a universal TFr.

6.4 Mass surface density

7 SUMMARY

We present a catalogue of our LSBG sample that is selected from the α.40-SDSS DR7 survey combination and extends the parameter space of the previous LSBG samples. This sample is an H i-selected sample that is composed of 1129 LSBGs of various types of luminosity and morphology.

Based on the width of the H i line and the absolute magnitude, we investigate the TFrs for a subsample from our entire LSBG sample in the optical B, g, and r and NIR J, H, and K bands. The TFrs in the NIR bands are tighter than those in the optical bands for our LSBG subsample. This is because NIR magnitudes are less affected by internal attenuation and recent episodes of star formation than optical bands. In the optical bands, the LSBG subsample follows the fundamental TFr that was defined for the normal spiral galaxies. In the NIR bands, the slopes of the TFr lines for our LSBG subsample are slightly different from those for the bright and nearby galaxies, but are generally consistent with the TFrs for the bright and nearby galaxies within the uncertainties. This conclusions agree with those from the previous TFr studies for LSBGs (Sprayberry et al. 1995; Zwaan et al. 1995; Verheijen 1997).

On the basis of the optical TFr results for our subsample, we further estimated the M/L, disc scale length, and mass surface density of our LSBG subsample. Compared with the normal spiral galaxies with the same H i line width. the LSBGs have higher M/L, higher disc scale length, and lower mass surface brightness than the HSB spiral galaxies, implying that LSBGs are mostly dominated by dark matter and they have low star formation at present.

ACKNOWLEDGEMENTS

We appreciate the anonymous referee for his/her constructive comments, which helped us strengthen this paper. We would like to gratefully thank Prof. Albert Bosma from Aix Marseille University, France, who had face-to-face discussions with us to give us very helpful and suggestive comments on the Tully–Fisher relation part of this paper. We thank Dr. Tao Hong from National Astronomical Observatories, CAS for helpful and constructive discussions and comments. We thank Dr. Anastasis Ponomareva from Australian National University who had private communications and shared her method of intrinsic extinction correction and also PhD thesis with us, which is very helpful for us. We also thank Prof. Galaz Gaspar from Pontificia Universidad Catolica de Chile, who also gave us suggestive comments to help us improve this paper.

WD and HW are supported by the National Natural Science Foundation of China (NSFC) grant Nos.11733006 and 11403037, and the National Key R&D Program of China grant No. 2017YFA0402704. CC is supported by the NSFC grant No. 11803044 and is also supported in part by the Chinese Academy of Sciences (CAS), through a grant to the CAS South America Center for Astronomy (CASSACA) in Santiago, Chile. MZ is supported by the National Key R&D Program of China grant No. 2017YFA0402600, NSFC grant No. U1531246, and the Major Applied Basic Research Program of Guizhou Province (No. JZ[2014]2001-05). YW is supported by the NSFC grant Nos. 11390372 and 11773034. Both WD and CC are also supported by the Young Researcher Grant of National Astronomical Observatories, Chinese Academy of Sciences (NAOC), and the Key Laboratory of Optical Astronomy, NAOC.

REFERENCES

APPENDIX A: THE α.40-SDSS DR7 LSBG CATALOGUE

The rest catalogue of our LSBG sample is shown in Table A1.