THE STAR FORMATION LAW AT LOW SURFACE DENSITY

The UV light from galaxies traces directly the photospheric emission from massive stars and thus can be used to determine the SFRs of galaxies (e.g., Kennicutt 1998b). In particular, the GALEX FUV band is sensitive to stars with main-sequence lifetimes shorter than ∼10⁸ yr (Martin et al. 2005), and thus, SFRs determined from the UV are effectively averages of the SFR over this timescale.

Based upon the data from the SINGS sample, Kennicutt et al. (2007) and Calzetti et al. (2007) have argued that it is best to combine a measure of the unattenuated SFR (e.g., Hα or UV flux) with a measure of the dust reprocessed light detected in the FIR at 24 μm. For our estimates of the SFRs, we have assumed that low surface brightness galaxies have little dust, and that most of the light from recent star formation in LSB galaxies escapes unimpeded in the UV. We will test this assumption with the small amount of FIR data available for LSB galaxies in the next section.

We determined the Galactic reddening E(B − V) at the position of each galaxy from the maps presented in Schlegel et al. (1998) and these values are listed in Table 1. For the Cardelli et al. (1989) extinction law with R_V = 3.1, the ratio of the extinction to the reddening is A_UV/E(B − V) = 8.2 in both of the GALEX bands (Wyder et al. 2007). In addition, all of the surface brightness measurements in this paper were corrected for the (1 + z)⁴ surface brightness dimming with redshift.

For easy comparison with previous results, we have assumed the same relation between UV luminosity and SFR as given in Kennicutt (1998b):

$$\begin{equation} {\rm SFR} = 1.4 \times 10^{-28} L_{\nu }, \end{equation} \tag{ 3 }$$

where SFR is the SFR in units of M_☉ yr⁻¹ and L_ν is the UV luminosity in erg s⁻¹ Hz⁻¹ measured over the wavelength range 1500–2800 Å, where the UV spectrum from star-forming galaxies is expected to be relatively flat. This relation assumes a constant SFR and a Salpeter (1955) stellar initial mass function (IMF) extending from 0.1 to 100 M_☉. The relation above can be modified to give the SFR surface density as a function of the UV surface brightness:

$$\begin{equation} \log {\Sigma _{\rm SFR}} = 7.413 - 0.4 \mu _{{\rm UV}}, \end{equation} \tag{ 4 }$$

where Σ_SFR is the SFR surface density in M_☉ yr⁻¹ kpc⁻² and μ_UV is the UV surface brightness in mag arcsec⁻². We used the FUV surface brightness to determine Σ_SFR for our sample except for UGC 6614 where we used the NUV measurement and an assumed color of (FUV − NUV) ≈ 0.2 mag due to the lack of FUV data for that galaxy. When computing the FUV surface brightness, we corrected the galaxies to face-on by dividing by the area of a circle with radius given by the semimajor axis of the ellipse used to extract the flux. This is equivalent to assuming that the galaxies are intrinsically circular and are optically thin in the UV. The average SFR surface densities Σ_SFR calculated from Equation (4) are listed in Table 6.

Most recent determinations of the IMF agree with the Salpeter (1955) form at higher masses but have relatively fewer stars below about 1 M_☉ (Kroupa 2002). Thus, the conversion between UV luminosity and SFR for other IMFs would lead in general to lower SFRs than predicted by Equation (3). If the IMF does not vary from galaxy to galaxy, then assuming a different IMF would change all of the SFRs simply by a constant factor.

One of the largest uncertainties in using the UV to measure SFRs is the unknown amount of light absorbed by dust internal to each galaxy. It is usually assumed that LSB galaxies have little dust although there is not very much actual data supporting this assumption. Relying upon number counts of distant galaxies observed through the giant LSB galaxy UGC 6614, Holwerda et al. (2005) concluded that the I-band attenuation in UGC 6614 is consistent with zero, although the uncertainties as a function of radius are ∼±1 mag. When translated to the UV, the uncertainties would be even larger and thus do not place very strong constraints on the amount of UV light absorbed by dust. Matthews & Wood (2001) used a Monte Carlo radiative transfer code to model the amount of dust absorption in the edge-on LSB galaxy UGC 7321 and found that the radial color gradients were consistent with a small amount of dust reddening. According to this model, the reddening would be entirely negligible if this galaxy were viewed closer to face-on.

There have also been several studies using the ratios of Balmer lines to determine the reddening for H II regions in LSB galaxies. McGaugh (1994) calculated the reddening E(B − V) from the Balmer lines, finding that some LSB galaxies have a reddening consistent with zero although their sample has values ranging up to E(B − V) ∼ 0.6 mag. Fitting the measurements for their entire sample of LSB galaxies, Burkholder et al. (2001) found an average reddening of E(B − V) = 0.3 ± 0.05 mag. A similar range of reddenings from near zero to ∼0.4 (with a tail reaching to even higher values) was found by Bergmann et al. (2003) and de Blok & van der Hulst (1998a) from the Balmer lines as well. There appear to be variations in the amount of dust even among H II regions within the same galaxy. For example, the reddenings among seven individual H II regions within the LSB galaxy LSBC F563-01 lie in the range E(B − V) = 0–1 mag (de Blok & van der Hulst 1998a).

It is not clear how these measurements translate into values for the attenuation at UV wavelengths. Based upon observations of starburst galaxies, Calzetti et al. (1994) found that the reddening in the ionized gas determined from the Balmer lines is about half that for the stellar continuum. Using the starburst reddening law, the ratio of FUV attenuation to the reddening in the gas is A_FUV/E(B − V) = 4.5. Using the Calzetti et al. (1994) ratio, the UV attenuation implied by the Balmer line data would lie in the range from zero to ∼2 mag. Thus, there would appear to be a range of dust content at least within the ionized gas in LSB galaxies. On the other hand, it is not clear that the same relation between the reddening in the ionized gas and the stellar continuum for starburst galaxies would necessarily apply to the much lower density environments in LSB galaxies.

As the UV light absorbed by dust is eventually re-emitted in the FIR, the ratio of FIR to UV flux can be used to estimate the UV attenuation nearly independently of the dust geometry or intrinsic dust properties as well as the metallicity or age of the stellar population except for galaxies with very old populations (Gordon et al. 2000). Recently, there have been FIR measurements of a small sample of five LSB galaxies with the Multiband Imaging Photometer (MIPS) instrument aboard the Spitzer Space Telescope (Hinz et al. 2007; Rahman et al. 2007). Only two of these galaxies, the giant LSB galaxies Malin 1 and UGC 6614, are in common with the sample presented in this paper. However, an additional two LSB galaxies with Spitzer data, namely UGC 6151 and UGC 9024, have also been observed in the UV with GALEX. We have obtained UV fluxes for these two galaxies using the same procedure as for the other LSB galaxies as described in Section 2 except that we chose the axis ratio, position angle, and semimajor axis of the ellipse used to measure the total flux based upon the NUV image.

We used the formulae given in Dale & Helou (2002) and the observations at 24, 70, and 160 μm from Hinz et al. (2007) to calculate the total FIR flux from 8 to 1000 μm. Only UGC 6614 and UGC 6151 have detections in all three MIPS bands and thus we only computed total FIR fluxes for these two galaxies. Malin 1 is only detected at 24 μm while UGC 9024 is detected only at 24 and 70 μm. In these cases, we calculated an upper limit for the total FIR luminosity using the fluxes in the detected bands plus the 3σ upper limits given by Hinz et al. (2007) for the remaining bands. We calculated a UV flux for each galaxy from the observed FUV flux using F_UV = BC_starsνF_ν, where BC_stars is a bolometric correction from the FUV flux to the total unattenuated stellar emission over 912–10000 Å. We assumed a value of BC_stars = 1.68 from Calzetti et al. (2000). For UGC 6614 and UGC 9024, we calculated a FUV flux using the NUV measurement and assuming a color of (FUV − NUV) = 0.2 mag.

The ratio of FIR to UV flux is plotted as a function of the H i to stellar mass ratio M_HI/M_star in Figure 14. For a comparison sample, we plot the positions of a subset of the SINGS sample that has data in all three Spitzer MIPS bands as well as H i masses and K-band measurements (Kennicutt et al. 2003; Dale et al. 2007). To determine stellar masses of the SINGS galaxies, we used the relation between K-band mass to light ratio and (B − R) color from Bell et al. (2003) converted to our assumed IMF. While the SINGS sample is not a statistical sample, it was chosen to span the parameters of normal spiral and irregular galaxies in the local universe. Thus, it provides a useful comparison sample of primarily higher surface brightness spiral and irregular galaxies. To determine the stellar masses for the LSB galaxies, we relied upon the relation between K-band stellar mass to light ratio and (g − r) color from Bell et al. (2003). Two out of the four galaxies with FIR data have (g − r) ≈ 0.7, while the remaining two do not have SDSS imaging. Thus, we assume a color of (g − r) = 0.7 for calculating the mass to light ratio for all four LSB galaxies. In addition, as the LSB galaxies do not have K-band fluxes measured, we converted the 4.5 μm fluxes to K-band values assuming a color of (K − [4.5]) ∼ −1 derived from the mode of the color distribution for the SINGS sample.

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The largest ratio among the LSB galaxies is F_FIR/F_UV = 0.42 ± 0.06 for UGC 6614 whereas the other measurement and two upper limits are 0.2 or lower. A ratio of one would imply equal amounts of light being emitted in the UV and FIR. The fact that the ratio is significantly less than one in the LSB galaxies means that the majority of UV light from massive stars is not absorbed by dust. This stands in contrast to the majority of the SINGS galaxies which in general have FIR to UV ratios larger than one. LSB galaxies in general are relatively gas rich and this is consistent with the higher M_HI/M_star ratios compared to many of the higher surface brightness galaxies in the SINGS sample. Based upon the relatively low FIR fluxes for these four galaxies, we assumed for the purposes of this paper that the UV attenuation in our sample of LSB galaxies is negligible. However, it is important to bear in mind the relative paucity of data on the dust content of LSB galaxies.

These FIR and UV fluxes refer to the average UV attenuation within each galaxy. There are almost certainly more local variations in the FIR/UV ratio within each galaxy. For instance, comparing the UV image of UGC 6614 in Figure 5 with the 24 μm image in Hinz et al. (2007), FIR emission is only detected in the center of the galaxy and in the innermost star-forming ring, while the UV emission extends to much larger radii. It seems likely that the FIR/UV ratio would decrease with radius, an effect not uncommon among higher surface brightness galaxies (Popescu et al. 2005; Boissier et al. 2007).

Another important assumption we have made in order to use the UV fluxes to infer SFRs for the LSB galaxies is that the UV light is coming only from massive young stars and that the SFR in the LSB galaxies has been relatively constant, or at least constant on the timescales sampled by stars in the UV. In this section, we use the colors of the LSB galaxies to place some constraints on their stellar populations and star formation histories.

Since the UV is an indicator of the recent SFR and the optical r band is more representative of the total stellar mass, the (NUV − r) color provides information about the SFR divided by the stellar mass M_*, or the specific SFR, and hence the average age of the stars in a galaxy (e.g., Salim et al. 2005). While the relationship between (NUV − r) color and SFR/M_* is complicated by the effects of dust in high surface brightness spiral galaxies, the likely smaller amount of dust attenuation present in LSB galaxies (see Section 3.2), renders interpretation of the color in terms of SFR/M_* more robust.

The location of the LSB galaxies in the (NUV − r) versus M_r galaxy color–magnitude diagram is shown in Figure 15 as the red points with errors bars. For comparison we plot as contours the volume density of galaxies in the local universe from Wyder et al. (2007) based upon matching the SDSS main galaxy sample with the GALEX Medium Imaging Survey catalog. While the SDSS main galaxy sample includes an explicit cut on the r-band half-light surface brightness of μ_r,1/2 < 24.5 mag arcsec⁻² for spectroscopic target selection (Strauss et al. 2002), the completeness of the SDSS sample is greater than 50% only for μ_r,1/2 < 23.4 mag arcsec⁻² (Blanton et al. 2005). Since some of the LSB galaxies in our sample have half-light surface brightnesses brighter than this limit (see Table 5), there is some overlap in surface brightness between our sample and that used to generate the contours in Figure 15. Nevertheless, this comparison sample is dominated by higher surface brightness galaxies and can serve as a useful reference with which to compare the LSB sample.

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Galaxies tend to exhibit a bimodal distribution in color and this is reflected in the blue and red sequences visible in the color–magnitude diagram. Our sample of LSB galaxies lies along the same blue sequence as defined by the higher surface brightness galaxies. While there can be UV flux from evolved low-mass stars, these galaxies have (NUV − r)>4 mag, as is evident from the location of the red sequence in Figure 15. Thus, the colors of the LSB galaxies would imply that their UV emission is dominated by star formation and not by older stars. There does appear to be a tendency, especially for the LSB galaxies with −20 < M_r < −17, to be somewhat bluer than the peak of the blue sequence but the small sample size and heterogeneous selection criteria of our sample preclude us from making stronger inferences about the variation in color with surface brightness. The somewhat bluer colors would be consistent with the lower dust content in LSB galaxies. The fact that the LSB galaxies lie along the blue sequence would imply that their stellar populations are on average similar to high surface brightness galaxies of similar luminosity. Galaxies that are in the process of quenching their star formation and transitioning to the red sequence would be found at intermediate colors around (NUV − r) ∼ 4 mag (Martin et al. 2007). With the exception perhaps of Malin 1, the LSB galaxies are not found in this transition region of the color–magnitude diagram and would argue that their low surface brightness is not due to fading of a once higher surface brightness galaxy that has already begun to shut off its star formation.

Based upon N-body simulations of LSB galaxies and their ISM, Gerritesen & de Blok (1999) found that the instantaneous SFRs in LSB galaxies fluctuate about their average value over a timescale of 20 Myr and with an amplitude of ∼0.1 M_☉ yr⁻¹. These fluctuations are due to the finite size of the gas and star particles in their model, presumed to correspond in real galaxies to the formation of individual clusters or OB associations. If the average SFR is less than 0.1 M_☉ yr⁻¹, these fluctuations would translate into a rather large spread in the colors of LSB galaxies. The integrated SFRs for our sample are listed in Table 6. 14 out of the 19 galaxies in our sample have SFRs greater than 0.1 M_☉ yr⁻¹ at a level where the stochastic fluctuations in the SFR would have much less of an effect on the integrated colors. Thus, our sample more closely resembles the high SFR model of Gerritesen & de Blok (1999) which has a SFR that decreases only slightly over the 3 Gyr of time followed by their simulation. In addition, our use of the FUV band to measure the SFRs would tend to average over any fluctuations in the SFR on timescales substantially less than 100 Myr.

Since the NUV band is sensitive to stars with a somewhat larger range of ages than the FUV band (Martin et al. 2005), the (FUV − NUV) color can provide some constraints on the very recent star formation history (≲1 Gyr). In Figure 16, we plot in red the (FUV − NUV) colors of the LSB galaxies as a function of the H i mass. For comparison we also plot the locations of the SINGS sample as the black pluses. With a couple of exceptions, the LSB galaxies have colors (FUV − NUV) ∼ 0.2 mag, similar to most of the galaxies in the SINGS sample.

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Based upon a sample of 15 LSB galaxies with GALEX FUV and NUV data, Boissier et al. (2008) found that the (FUV − NUV) colors become redder above an H i mass of 10¹⁰ M_☉. On average the (FUV − NUV) colors for the LSB galaxies in this paper are not as red as those presented in Boissier et al. (2008). The largest difference is for Malin 1, for which Boissier et al. (2008) measured a color of 0.84 ±  0.10. In contrast, we find a color of 0.24 ± 0.2. Our measurements of the total NUV magnitude agree while our FUV flux is brighter than that measured by Boissier et al. (2008). This difference is due in part to the difference in the GALEX calibration between the GR1 and GR3 data releases as well as differences in the choice of aperture and the precise sky background level. As most of the galaxies in Boissier et al. (2008) do not have resolved H i maps, these were not included in our sample. While there does appear to be some LSB galaxies with redder UV colors, particularly at higher masses, this does not appear to be universally true.

The (FUV − NUV) colors of galaxies can be affected by several factors including the recent star formation history, the metallicity, and reddening due to dust. We have argued in Section 3.2 that LSB galaxies likely have low amounts of UV attenuation from dust and therefore, dust probably does not affect the (FUV − NUV) colors of our sample. There is some disagreement among stellar population models about the intrinsic (FUV − NUV) colors of galaxies. For models with a constant SFR, no dust, solar metallicity, and a Kroupa et al. (1993) stellar IMF reaching to 100 M_☉, Boissier et al. (2008) predict a color of (FUV − NUV) ≈ 0.2 mag after about 1 Gyr. Models with lower metallicities yield slightly bluer colors. On the other hand, the models of Bruzual & Charlot (2003) for a similar set of parameters predict a color of (FUV − NUV) ≈ 0.0 mag for ages greater than 1 Gyr. With only a couple of exceptions, the (FUV − NUV) colors of all of our LSB galaxies are consistent to within the errors with the Boissier et al. models and somewhat redder than that predicted by Bruzual & Charlot. Given the errors on the (FUV − NUV) color and the disagreement among models, we do not find any strong evidence from the colors of the LSB galaxies for either variable star formation histories or a nonstandard IMF. If this had been the case, then we would be underestimating the SFRs in the LSB galaxies using the standard conversion factor between UV luminosity and SFR in Equation (3).

We have used the H i radial surface density profiles from van der Hulst et al. (1993), de Blok et al. (1996), and Pickering et al. (1997) to measure the average gas surface densities for our sample of LSB galaxies within the same aperture used to measure the total UV flux. In order to be consistent with measurements from Kennicutt (1998a), we did not correct the gas densities for helium or other heavy elements.

The total gas surface density should include both the atomic and molecular gas. Only a few LSB galaxies have molecular gas detected from radio observations of the CO lines while most remain undetected (de Blok & van der Hulst 1998b; O'Neil et al. 2000, 2003; Matthews & Gao 2001; O'Neil & Schinnerer 2004; Matthews et al. 2005; Das et al. 2006; Schombert et al. 1990; Braine et al. 2000). The few detections and many upper limits correspond to very low molecular fractions in the range 1%–10% for most LSB galaxies, assuming a Galactic CO to H₂ conversion factor (O'Neil et al. 2003). Among the galaxies with molecular gas detected, CO maps of the giant LSB galaxies LSBC F568-06 and UGC 6614 show molecular gas clearly offset from the nucleus and only detected at certain locations, indicating that what little molecular gas they do have is irregularly distributed (Das et al. 2006).

One critical assumption that went into determining these low molecular fractions is that the standard Galactic conversion factor between CO luminosity and H₂ mass applies to LSB galaxies. Since LSB galaxies have on average oxygen abundances below the Solar value (Burkholder et al. 2001; McGaugh 1994), the ratio of CO to H₂ would be expected to be lower simply due to the overall lower metallicity. On the other hand, observations of individual molecular clouds in nearby low metallicity dwarf galaxies are consistent with the standard Galactic CO-to-H₂ conversion factor (Leroy et al. 2006; Bolatto et al. 2008). In addition, the low dust content in LSB galaxies would be expected to lower the CO/H₂ ratio because the dust can act as a catalyst for the formation of CO as well as shielding the molecules from potentially damaging UV radiation (Mihos et al. 1999). Despite these uncertainties, we assumed for the purposes of this paper that the gas mass in LSB galaxies is dominated by the atomic gas.

We plot the SFR surface density as a function of the gas surface density in Figure 17. The green circles, red triangles, and blue stars are the galaxies with H i data from de Blok et al. (1996), Pickering et al. (1997), and van der Hulst et al. (1993), respectively. For comparison, we also plot the sample of spiral and starburst galaxies from Kennicutt (1998a) as the black pluses. The solid line is the power-law fit to the high surface brightness sample of the form of Equation (1). Kennicutt (1998a) found best-fit values of A = (2.5 ± 0.7) × 10⁻⁴ and N = 1.4 ± 0.15 for Σ_SFR in M_☉ yr⁻¹ kpc⁻² and Σ_gas in M_☉ pc⁻². The LSB galaxies lie below the extrapolation of this fit to the higher surface density galaxies. The median offset between the LSB galaxies and the extrapolation of the power law is about 0.7 dex, or a factor of 5 in SFR surface density.

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For reference the dotted lines in Figure 17 indicate lines of constant star formation efficiency of 1%, 10%, and 100% as labeled in the figure assuming a timescale for star formation of 10⁸ yr. The choice of 10⁸ yr is somewhat arbitrary although it does correspond to typical orbital timescales in galaxies. These curves would imply that the average star formation efficiency in LSB galaxies is only at most a few percent, significantly lower than in higher surface density galaxies.

Note that we assumed that the LSB galaxies have negligible amounts of molecular gas when computing Σ_gas. If in fact there were more molecular gas than would be indicated by the available CO measurements, then the LSB galaxies would shift to the right, or even further from the fit to the higher surface brightness galaxies. As described above, we have not included any correction for internal dust in the LSB galaxies. Any correction for dust would tend to push the points back closer to the fit to the higher surface brightness galaxies. However, the average correction necessary would correspond to an internal extinction of about 1.8 mag, significantly larger than the UV attenuation of ≲0.4 mag implied by the FIR/UV ratios in Figure 14.

Another critical assumption used to calculate the SFRs in Figure 17 is that the stellar IMF is the same at low and high surface densities. Recently it has been suggested that the IMF at low density should have fewer high-mass stars than at higher densities (Krumholz & McKee 2008). If this were to be the case, then the conversion between UV luminosity and SFR in Equation (3) would tend to underestimate the true SFR. Indeed, the extreme outer disks of some galaxies have an apparent edge in the Hα images but no apparent edge in the UV. Since only the very most massive stars are capable of ionizing hydrogen and producing significant Hα-emitting regions, a relative lack of massive stars would help explain the low Hα/UV ratio in the outer regions of some disk galaxies (Thilker et al. 2005; Boissier et al. 2007). The surface densities probed in the extended disks of high surface brightness galaxies are in general in the regime below ∼10 M_☉ pc⁻², similar to the LSB galaxies in our sample. Unfortunately, there are very few integrated Hα fluxes available for LSB galaxies so that we cannot check whether the UV/Hα ratio is similar to that in the extended low density outer regions of high surface brightness galaxies. Nevertheless, variations in the high-mass IMF would affect SFRs determined in the UV less than those from Hα because of the wider mass range of stars contributing to the UV luminosity.

There is evidence that the outer low-density regions of higher surface brightness galaxies also tend to show a steeper slope in the Schmidt–Kennicutt law. In particular, Kennicutt (1989, 1998a) used radial Hα and gas density profiles to show that galaxies tend to have well-defined edges to their star-forming disks below a gas density threshold that varies from galaxy to galaxy in the range 3–10 M_☉ pc⁻². Much below the threshold no Hα emission is detected while near the threshold the correlation of SFR surface density with gas density is steeper than the 1.4 power law at higher densities. Kennicutt (1989) showed that the gas density thresholds could be explained as the radius inside of which the gas is unstable to gravitational instabilities as predicted by the Toomre criterion (Toomre 1964). LSB galaxies have gas densities below the critical density defined by the Toomre criterion throughout much of their disks (van der Hulst et al. 1993), similar to the outer regions of high surface brightness galaxies.

We have used the H i and UV radial profiles to investigate the form of the star formation law locally within the LSB galaxies. The azimuthally averaged gas and SFR surface densities for the LSB galaxies are plotted in Figure 18. A solid line connects all of the data for a particular galaxy. The colors correspond to the source of the H i data as indicated in the figure. With a couple of exceptions, the azimuthally averaged data follow the same trends as seen in the galaxy-wide averaged data. The radial profiles indicate that the star formation law is steeper at low density than the canonical power law fit to the higher density points. The two red curves that deviate the most from the average trend are the giant LSB galaxies LSBC F568-06 and UGC 6614. Both of these galaxies have central minima in their H i profiles even though their UV surface brightness profiles continue to increase towards their centers. Both of these galaxies have an active nucleus exhibiting broad emission lines (Bothun et al. 1990; Schombert 1998) and thus at least some of the UV emission in their centers is probably not due to star formation. The lack of H i in the center could be an indication that there is a significant amount of molecular gas in the inner regions. Although the CO emission detected in these two galaxies by Das et al. (2006) is in the central regions of both systems, the relative amount of gas detected would only contribute ∼0.2–0.3 M_☉ pc⁻² when azimuthally averaged over the radii covered by the CO observations and hence would not significantly affect the total gas surface density.

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Despite the downturn observed in the radial profiles in Figure 18, the UV and H i data are still of low enough resolution such that we are averaging over fairly large regions within each galaxy. It could be that there is a constant gas density threshold and that the filling factor of gas clouds that lie above the threshold is simply lower in LSB galaxies. This would lead to an effectively lower star formation efficiency when averaging over the entire galaxy. Higher resolution UV and H i data would be necessary to investigate this possibility.

Although the correlation of Σ_SFR with Σ_gas has been one of the most widely used star formation prescriptions, there are alternative formulations. In particular, Kennicutt (1998a) showed that a relation of the form of Equation (2) fits the data for spiral and starburst galaxies as well as the power law in Equation (1). We have used the published H i rotation curves for our galaxies to determine the rotational velocity V_rot at a radius 0.7 times the radius used to determine both Σ_SFR and Σ_gas. This choice of radius was made to make our measurements as consistent as possible with those for the higher surface brightness galaxies where the dynamical time was determined at a radius of 0.7R₂₅ (R. Kennicutt 2008, private communication). For many of the galaxies in our sample, this radius occurs in the flat part of the rotation curve, and thus V_rot is fairly well defined. Following Kennicutt (1998a), we defined the dynamical time as τ_dyn = 2πR/V_rot. The values for V_rot and τ_dyn are listed in Table 6.

This alternative star formation law is plotted in Figure 19. Similar to Figure 17, the LSB galaxies are plotted as the colored points with the color indicating the source of the H i data as noted in the legend. The black pluses are the high surface brightness spiral and starburst sample from Kennicutt (1998a), while the solid black line is the fit to these points of the form of Equation (2) with B = 0.11. Similar to the first version of the star formation law, the LSB galaxies lie below the extrapolation of the fit to the higher surface brightness sample. The median offset is 0.5 dex below although there is more scatter in the LSB SFR surface densities versus Σ_gas/τ_dyn as compared to that seen in Figure 17 versus Σ_gas alone.

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As noted by Kennicutt (1998a), one interpretation of the fit to the high surface brightness galaxies shown in Figure 19 is that a constant fraction of the available gas is transformed into stars per orbit. The zero point of the relation plotted in Figure 19 corresponds to approximately 11% of the gas transformed into stars per orbit. This could in principle arise if star formation were triggered by passages through spiral arms in which case the SFR would naturally correlate with the amount of gas available and vary inversely with the orbital time. In this interpretation, while the star formation efficiency is approximately a constant among the high surface brightness galaxies, the efficiency would be somewhat lower in LSB systems.

One possible explanation for the decreased star formation efficiency at low density is the low fraction of molecular gas observed in environments where Σ_gas < 10 M_☉ pc². Blitz & Rosolowsky (2006) have argued that the molecular fraction in disk galaxies is correlated with the hydrostatic pressure in the ISM. In this model, the pressure is a function of the stellar surface density, the gas surface density, and the gas velocity dispersion. Thus, the LSB galaxies would be expected to have lower ISM pressure and thus low molecular fractions. If it is really just the molecular gas which is relevant for star formation in galaxies, then the apparently low star formation efficiency in the total gas in LSB galaxies is simply a reflection of their low molecular content.

Krumholz & McKee (2005) have developed a model in which star formation in galaxies occurs in the densest subregions of molecular clouds that are supersonically turbulent with a log-normal density distribution. They coupled this model with the correlation between the molecular fraction and the ISM pressure from Blitz & Rosolowsky (2006) to derive an analytical prediction for both forms of the star formation law. For the first form of the law as a function of Σ_gas, the Krumholz & McKee (2005) model prediction is given by

$$\begin{equation} \Sigma _{\rm SFR} = 3.9 \times 10^{-4} \phi _{\bar{P}}^{0.34} Q^{-1.32} f_{{\rm GMC}} \Sigma _{\rm gas}^{1.33}\; M_{\odot }\; {\rm yr^{-1}\; kpc^{-2}}, \end{equation} \tag{ 5 }$$

where Σ_gas is the total (atomic plus molecular) gas surface density in units of M_☉ pc⁻², f_GMC is the fraction of gas in giant molecular clouds, $\phi _{\bar{P}}$ is the ratio of the pressure inside a molecular cloud to the pressure at its surface, and Q is the stability parameter as defined by Toomre (1964). We adopt Q = 1.5, a value typical of spiral galaxies (Martin & Kennicutt 2001). Krumholz and McKee argued that $\phi _{\bar{P}} \approx 10-8 f_{{\rm GMC}}$ and we adopt $\phi _{\bar{P}} = 9 f_{{\rm GMC}}$. Finally, f_GMC is given by

$$\begin{equation} f_{{\rm GMC}} = \big(1 + 250 \Sigma _{\rm gas}^{-2} \big)^{-1}. \end{equation} \tag{ 6 }$$

Similarly, the Krumholz and McKee prediction for the second form of the star formation law in terms of the dynamical time is given by

$$\begin{equation} \Sigma _{\rm SFR} = 5.5 \times 10^4 \phi _{\bar{P}}^{0.34} Q^{-1.32} f_{{\rm GMC}} \left(\frac{\Sigma _{\rm gas}}{\tau _{\rm dyn}}\right)^{0.89}\;M_{\odot }\; {\rm yr^{-1}\; kpc^{-2}}, \end{equation} \tag{ 7 }$$

where τ_dyn is in units of yr and f_GMC is given as a function of Σ_gas/τ_dyn by

$$\begin{equation} f_{{\rm GMC}} = \left[ 1 + 2.0 \times 10^{-10} \left(\frac{\Sigma _{\rm gas}}{\tau _{\rm dyn}}\right)^{-1.34} \right]^{-1}. \end{equation} \tag{ 8 }$$

Note that Equation (8) was derived from Equation (6) and an empirical relation between Σ_gas and τ_dyn.

In Figure 20, we plot the prediction from Equations (5) and (6) as a function of Σ_gas as the dashed line in the left panel. The corresponding prediction from Equations (7) and (8) as a function of Σ_gas/τ_dyn is shown in the right panel. For both forms of the star formation law, we additionally shifted the curves to the left (i.e., lower gas densities) by a factor of 1.37 to account for the fact that the observations do not include a correction for helium or other heavy elements. In both forms of the star formation law the downturn in Σ_SFR at low density is due to the declining molecular fraction as a function of density, similar to what is observed in the LSB galaxies. In both panels, the LSB data appear to be a bit above the theoretical prediction. However, given the uncertainties in the molecular content of the LSB galaxies as well as the uncertainties inherent in deriving SFRs, the models are in reasonable agreement with the observations.

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To further test the idea that it is the amount of molecular gas which determines the SFRs in galaxies rather than the total gas density, we would require measurements of the molecular gas content in LSB galaxies. Unfortunately, only one of the galaxies in our sample has a CO detection while an additional seven galaxies have only upper limits to their CO flux (Das et al. 2006; Schombert et al. 1990; Braine et al. 2000; de Blok & van der Hulst 1998b). The CO data for these galaxies typically come from single dish measurements that sample the central regions with diameters in the range of 22''–55'', or significantly smaller than the UV and H i sizes listed in Table 3. Therefore, we have remeasured the FUV surface brightness using the same aperture as used for the CO data. The corresponding values for Σ_SFR determined from Equation (4) are listed in Table 7. We have converted the detection and upper limits in the CO brightness temperature into molecular hydrogen surface densities using the Galactic conversion factor 4.5 M_☉ pc⁻² (K km s⁻¹); Bloemen et al. 1986), and these values for $\Sigma _{{\rm H}_2}$ are listed in Table 7 as well. The sole detection of CO flux is for UGC 6614, which when averaged over the 100'' diameter region sampled by the CO data, corresponds to a surface density of $\Sigma _{{\rm H}_2} = 0.2$ M_☉ pc⁻² (Das et al. 2006).¹¹

Table 7. Molecular Star Formation Law Data

Galaxy	$\Sigma _{{\rm H}_2}$ (M☉ pc−2)	ΣSFR (10−4 M☉ yr−1 kpc−2)	CO ref.
LSBC F561-01	<1.8	8.8	1
LSBC F563-01	<1.6	8.3	1
LSBC F571-V02	<0.9	6.4	1
LSBC F583-01	<2.2	8.7	1
LSBC F563-V01	<1.3	6.6	2
LSBC F568-V01	<1.4	17	2
Malin 1	<0.7	0.6	3
UGC 6614	0.2	28	4

Notes. The molecular gas surface densities and upper limits assume a conversion factor between CO brightness temperature and molecular gas surface density of 4.5 M_☉ pc⁻² (K km s⁻¹)⁻¹. The UV surface brightnesses were measured in the same aperture as used for the CO observations. References. (1) Schombert et al. 1990; (2) de Blok & van der Hulst 1998b; (3) Braine et al. 2000; (4) Das et al. 2006.

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We plot Σ_SFR as a function of the molecular gas surface density $\Sigma _{{\rm H}_2}$ in Figure 21. The sole galaxy with a molecular gas detection, UGC 6614, is plotted as a red circle while the red arrows indicate upper limits for the remaining seven LSB galaxies with CO observations but no detected molecular gas. As in the previous figures, we also plot the spiral and starburst sample from Kennicutt (1998a). For reference, we plot the same 1.4 power-law fit to Σ_SFR versus Σ_gas as shown in Figure 17. While this power law fits the starburst galaxies with the highest surface densities in which the gas is assumed to be all molecular, the spiral and LSB galaxies tend to scatter to the left of this line. As noted by Kennicutt, the scatter in Σ_SFR versus $\Sigma _{{\rm H}_2}$ is larger than when plotting Σ_SFR versus the total atomic plus molecular gas density. It is also apparent from this plot that a single power law does not appear to provide a very good fit to the combined sample. For reference, we plot in Figure 21 lines of constant star formation efficiency per 10⁸ yr. The implied star formation efficiency for UGC 6614 is nearly 100% per 10⁸ yr, while the lower limits for the CO nondetections lie in the range of 1%–10%. Part of the increased scatter in this diagram may be due to variations in the CO to H₂ conversion factor. Clearly more data on the variation in this conversion factor among galaxies are needed before concluding that the scatter is induced by a real variation in star formation efficiency.

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Finally, we note that most theoretical explanations for the star formation law indicate that the fundamental relationship is between the SFR and gas volume densities whereas we can only measure surface densities. The downturn in both forms of the star formation law could be due to low surface brightness galaxies being on average thicker than higher surface brightness galaxies. Indeed, in their hydrodynamical simulations of star formation in galaxies Robertson & Kravtsov (2008) found that flaring of the disk in the outer regions of their simulated galaxies leads to deviations from the canonical power-law star formation relation with exponent of 1.4. For a disk in which the gas volume density falls off exponentially with height above the plane with scale height h, the gas surface density Σ_gas and the midplane volume density ρ_gas,0 are related by ρ_0,gas = Σ_gas/(2h). This would imply that the LSB galaxies have vertical gas scale heights a factor of 5 above that for higher surface brightness galaxies if the downturn were due entirely to this effect. Optical images of the edge-on LBS galaxy UGC 7321 show that the vertical scale height of the disk is only 140 pc, a value smaller than in most higher surface brightness galaxies (Matthews 2000). If this is representative of LSB galaxies generally, then an increased vertical scale height would not be able to account for the low gas surface densities in LSB galaxies. Based upon N-body-SPH simulations Kaufmann et al. (2007) have argued that dwarf galaxies with rotational velocities of ∼40 km s⁻¹ are formed with significantly thicker disks than higher mass galaxies. This increased puffiness would lead to an effectively lower star formation efficiency in dwarf galaxies. While a few of the LSB galaxies in this paper have rotational velocities in this regime, the galaxies in our sample range in rotational velocity up to nearly 300 km s⁻¹. While it seems unlikely that all LSB galaxies have disks a factor of 5 more extended than high surface brightness galaxies, variations in the disk thickness could account for some of the scatter observed at low density in both forms of the star formation law.