COMPARISON OF Hα AND UV STAR FORMATION RATES IN THE LOCAL VOLUME: SYSTEMATIC DISCREPANCIES FOR DWARF GALAXIES

In general, attenuation by dust is the first issue to address when comparing UV/optical observed luminosities with intrinsic model values. For normal spiral galaxies (SFR ≳ 1 M_☉ yr⁻¹) there is broad agreement among previous work that although the UV SFR tends to be lower than the Hα SFR before accounting for internal dust attenuation, there is good consistency between the two after proper corrections have been applied. Empirically, these results suggest that the UV stellar continuum is more affected by attenuation than the Hα nebular emission.²⁴ If the same holds true for lower mass dwarf galaxies, dust corrections would drive the Hα-to-UV flux ratio even lower, which would at first appearance seem to exacerbate rather than help reconcile the discrepancy. To check this for our sample, we estimate the attenuation in the Hα flux with the Balmer decrement, and in the FUV flux using the total infrared-to-UV (TIR/FUV) flux ratio. The two methods are independent, the former being rooted in recombination physics and the latter in stellar synthesis and dust models, although an extinction curve must be assumed in each case. The resulting dust corrections are provided in Table 1.

4.1.1. A_Hα via the Balmer Decrement

The validity of using the Balmer decrement to infer the average amount of nebular extinction in galaxies is supported by studies that have shown that the scatter between infrared-based SFRs and Hα SFRs is decreased when the Hα luminosity is corrected for extinction in this manner (e.g., Dopita et al. 2002; Kewley et al. 2002; Moustakas et al. 2006). Kennicutt et al. (2009) reach similar conclusions from the analysis of about 400 nearby galaxies. As in Lee (2006), we assume an intrinsic Case B recombination ratio of 2.86 for Hα/Hβ, and the Cardelli et al. (1989) Milky Way extinction curve with R_V = 3.1, to express A_Hα in terms of the observed Hα/Hβ ratio as

$$\begin{equation} A_{{\rm H}\alpha }= 5.91\, \log \frac{f_{{\rm H}\alpha }}{f_{{\rm H}\beta }} - 2.70. \end{equation} \tag{ 4 }$$

For ∼20% of the sample, spectroscopic measurements of Hα/Hβ from the literature are available. For those without robust spectroscopic measurements, we follow a strategy similar to that used for estimating [N ii]/Hα above. Using the integrated spectroscopy of Moustakas & Kennicutt (2006) we derive an empirical scaling relation between A_Hα and M_B which is shown in Figure 3. The data exhibit the well-known trend that the more luminous galaxies are more heavily obscured. We fit a piecewise function to the data, described by a constant value at the lowest luminosities and a second-order polynomial for M_B < −15.0. Objects that have A_Hα < 0, a signal to noise less than 10 in the Hβ line, an Hβ equivalent width (EW) <5 Å, or show evidence of active galactic nucleus (AGN) emission are excluded from the fit. The adopted scaling relation is

$$\begin{equation} A_{{\rm H}\alpha }=\left\lbrace \begin{array}{@{}lll} 0.10 \;&{\rm if}\; M_B > -14.5\\ 1.971+0.323M_B+0.0134M_B^2 \;&\;{\rm if}\; M_B \le -14.5.\\ \end{array} \right. \end{equation} \tag{ 5 }$$

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Of course, there is a great deal of scatter from this average relationship which becomes more severe with increasing luminosity. However, although there is a 40% scatter for galaxies with M_B < −18, the scatter is much lower for the relatively transparent dwarf galaxies which dominate our Local Volume sample: for −14.7>M_B > − 18 it is 20% and for the lowest luminosity galaxies, where we have assumed a constant average correction, it drops further to 10%.

4.1.2. A_FUV

The total infrared-to-UV flux ratio (TIR/FUV) is an indicator of the UV attenuation in a galaxy as it represents the total stellar light which has been absorbed and re-radiated by dust, relative to the surviving stellar UV light which is directly observed. Using stellar population models with various dust geometries and extinction curves, it has been shown that for systems with recent star formation, such as the spiral and irregular galaxies in our sample, the mapping between A(FUV) and TIR/FUV is relatively tight (∼20%); that is, the contribution of less massive/older stellar populations to the heating of the dust is either negligible or is in direct proportion to that of the O&B stars which dominate the UV emission (e.g., Buat et al. 2005). The mapping as given by Buat et al. is

$$\begin{equation} A({\rm FUV}) = -0.0333 x^3 + 0.3522 x^2 + 1.1960 x^3 +0.4967, \end{equation} \tag{ 6 }$$

where $x=\log\, (\frac{{\rm TIR}}{{\rm FUV}})$. MIPS 24, 70, and 160 μm integrated fluxes from the Spitzer LVL program are available for the majority of galaxies used in this analysis (Dale et al. 2009), and we compute TIR fluxes for those galaxies that have robust detections in all three bands (50% of our sample) using the calibration of Dale & Helou (2002):

$$\begin{equation} {\rm TIR}=1.559 \nu f_{\nu }(24)+ 0.7686 \nu f_{\nu }(70)+ 1.347 \nu f_{\nu }(24). \end{equation} \tag{ 7 }$$

FUV is computed as νf_ν at 1520 Å.

For those galaxies for which both a spectroscopic measurement of Hα/Hβ and MIPS FIR data are available, we compare the Hα and FUV attenuations in the bottom panel of Figure 4 (solid points). There is a good correlation with a slope of 1.8 (solid line). Interestingly, A(FUV)/A(Hα) = 1.8 is what is expected for the Calzetti obscuration curve and differential extinction law (Calzetti 2001), which is based on UV luminous starbursts. This agreement provides some assurance that the attenuation corrections we have derived are reasonable and generally consistent. In Figure 4, we also plot the A(Hα) estimated from M_B using Equation (5) (open circles). As expected, there is more scatter between A(FUV) and the estimated A(Hα), but the values are still correlated with a slope of ∼1.8.

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When FIR data are not available, it is common to resort to methods which estimate A(FUV) directly from some measure of the UV spectral slope (e.g., a UV color). Previously, this method appeared to be reasonable since it is well known that the UV spectral slope β and TIR/FUV are tightly correlated (and thus the UV slope and A(FUV) should also be well correlated), at least for samples of local starbursts (Calzetti et al. 1994; Meurer et al. 1999). It has become clear however that the "IRX-β" correlation for starbursts does not hold for normal star-forming galaxies, which have lower values of TIR/FIR at a given UV color, and more scatter between the quantities. Calibrations and scaling relationships which better describe the correlation for more normal systems have been published (e.g., Kong et al. 2004; Seibert et al. 2005; Cortese et al. 2006; Boissier et al. 2007; Gil de Paz et al. 2007). However, as we show in Dale et al. (2009) and J. C. Lee et al. (2009, in preparation (Paper II)), the majority of galaxies in our local volume sample are low-luminosity dwarf irregulars which are blue (0 ≲ FUV-NUV ≲ 0.5), have low TIR/FUV (≲2), and thus lie in the area of the IRX-β diagram where the UV color is not (strongly) correlated with TIR/FUV and attenuation. Thus, such IRX-β relations cannot be used to infer TIR/FUV and A(FUV) for our sample overall. For example, a relation by Cortese et al. (2006), which is based on normal but more luminous spirals and provides a poor fit for galaxies with TIR/FUV ≲2 results in A(FUV) estimates that are too high when compared to the attenuations based on the Balmer decrement, as illustrated in Figure 4. We thus take another approach for estimating A(FUV) for galaxies with no FIR data or only upper-limit detections. Based on Figure 4, we simply scale the computed A(Hα) values (whether calculated from the Balmer decrement or estimated from M_B) by a factor of 1.8. This value is also adopted when Equations (6) and (7) produce a negative correction (i.e., for TIR/FUV < 0.3). In all, A(FUV) is estimated from A(Hα) for ∼60% of the galaxies in Table 1. Based on comparison with TIR/FUV based attenuations (Figure 4), there is a 1σ uncertainty of ∼17% when estimating A(FUV) from Balmer decrement based A(Hα) values. When A(Hα) is itself estimated (Equation (5)), the average uncertainties increase (26%), and are larger for more luminous objects (M_B < −18; ∼40%) than for the dwarf galaxies that dominate our local volume sample (M_B > − 18; ∼20%).

4.1.3. Comparison of Dust-corrected SFRs

After accounting for the effects of dust using the best available attenuation value for each galaxy, the Hα-to-FUV flux and SFR ratios are replotted in Figure 5. Data points where more robust corrections are applied (i.e., based on Balmer decrements and/or TIR-to-FUV ratios) are distinguished in red. There are no significant differences between the trends described by galaxies where scaling relationships are used to estimate attenuations and those which have more robust corrections. Binned averages with 1σ scatters are again given in Table 2. For all galaxies, the correction increases the FUV flux relative to the Hα flux (FUV is more attenuated as expected). However, since higher luminosity galaxies tend to suffer from more attenuation than those at lower luminosity (e.g., Figure 3), the Hα-to-FUV flux ratio is depressed by a greater factor at the high-luminosity end. At L(Hα) ∼10⁴¹ erg s⁻¹ (SFR ∼ 1 M_☉ yr⁻¹), the ratio decreases by ∼ 0.2 dex, and as a result falls below the K98 value (solid line). Galaxies with L(Hα) ≲10³⁸ erg s⁻¹ (SFR(Hα)≲ 10⁻³ M_☉ yr⁻¹) are minimally affected. The main consequence is that the slope above SFR(Hα) ∼ 3 × 10⁻² M_☉ yr⁻¹ is flattened (i.e., the dust-corrected ratio in this regime is constant on average). Adopting this lower value as the fiducial expected ratio (instead of the K98 value) would have the effect of mitigating the relative discrepancy at lower luminosities by 0.1 dex. Even in this situation however, factor of 2 offsets in the Hα-to-FUV ratio at SFR(Hα) ∼ 2 × 10⁻³M_☉ yr⁻¹, which increase to factors of ≳10 at SFR(Hα) = 10⁻⁴ M_☉ yr⁻¹, would still remain.

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Differences in stellar evolution and atmosphere models used to calibrate Hα and FUV luminosities as SFR indicators give rise to differences in the respective SFR conversion factors and hence to the expected Hα-to-FUV ratio. While this will not produce systematic trends in the ratio as a function of the luminosity for a given metallicity and IMF, it does define the fiducial from which deviations are measured. In Figure 5, a gray-shaded area is overplotted to indicate the range of ratios based on widely used synthesis models for solar metallicity and a Salpeter IMF with mass limits of 0.1 and 100 M_☉ yr⁻¹. These have been computed by Iglesias-Paramo et al. (2004) and Meurer et al. (2009) for the synthesis codes of Leitherer et al. (1999; Starbust99), Bruzual & Charlot (2003; BC03), and Fioc & Rocca-Volmerange (1997; PEGASE). All of the models primarily adopt stellar evolutionary tracks from the Padova group (e.g., Girardi et al. 1996 and references therein), but differ in their treatment of stellar atmospheres. There are uncertainties of ∼20% in the Hα-to-FUV ratio due to the models alone. The dust-corrected ratios computed in the last section for the more luminous galaxies in our sample are within the range of expected model values. It thus appears reasonable to use galaxies with SFR(Hα)≳ 3 ×10⁻² M_☉ yr⁻¹ to empirically define the fiducial Hα-to-FUV ratio at solar metallicity. Hereafter, we measure deviations from there instead of the K98 value.

Standard SFR conversion recipes (and hence the expected Hα-to-FUV ratio) generally assume solar metallicity populations. However, our sample spans a range of metallicities and is increasingly dominated by metal-poor dwarfs at low luminosities and SFRs. Given this trend in sample properties, it is possible that the variation in the Hα-to-FUV ratio could potentially be driven by systematic variations in metallicity.

Metallicity influences the spectral energy distribution (SED) through its effect on the stellar opacity. Lower metallicity stars of a given mass will have lower opacities, lower pressures, and thus will be relatively smaller and have hotter atmospheres. They produce a larger number of UV photons (both ionizing and non-ionizing), so SFR conversions based on solar metallicity populations will tend to overestimate the true SFR when applied to metal-poor systems (e.g., Lee et al. 2002, 2009; Brinchmann et al. 2004). In the same vein however, metal-poor populations will produce more ionizing flux relative to non-ionizing UV continuum, leading to larger Hα-to-FUV ratios at low metallicity. This produces an effect which is the opposite of that observed, as also previously noted by Sullivan et al. (2000), Bell & Kennicutt (2001) and Meurer et al. (2009), and therefore cannot be the cause of the discrepancy.

We illustrate these effects in Figure 6, which presents the SFR conversion factors and expected Hα-to-FUV ratio, based on the Bruzual & Charlot (2003) stellar population synthesis models with Salpeter IMF and "Padova 1994" tracks. Values ranging from Z_☉/50 (characteristic of the nebular oxygen abundance of the most metal poor local galaxies known; Kunth & Ostlin 2000) to 2.5 Z_☉ are shown. Metallicity has less of an effect on the FUV than the Hα (note that the vertical scales on all panels span the same range), resulting in larger Hα-to-FUV ratios at low metallicity. Based on these models, the difference in the expected ratios at sub-solar metallicities relative to solar is small, <0.07 dex, so this should minimally impact the observed trends in Figure 5. We check on the magnitude of the effect by first estimating metallicities from the luminosity–metallicity relationship found locally for dwarf irregular galaxies (Lee et al. 2003), and then computing corrections for the SFRs of the individual galaxies in the sample. Replotting the metallicity-corrected SFRs confirms that the effect on the magnitude of the overall systematic decrease in the Hα-to-FUV ratio is negligible. It should be kept in mind, however, that the evolution of massive stars at sub-solar metallicity is currently ill constrained due to uncertainties in stellar rotation and mixing which may amplify this effect (e.g., Gallart et al. 2005; Leitherer 2008 and references therein).

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In the calibration used to convert the Hα luminosity to the SFR, it is assumed that every Lyman continuum photon emitted results in the ionization of a hydrogen atom. However, if there is leakage of Lyman continuum photons into the intergalactic medium (IGM), the observed flux would be lower than expected, and the true SFR would be underestimated from Hα. In order for leakage alone to explain the observed trend in the Hα-to-FUV ratio, the fraction of ionizing photons escaping the galaxies would have to increase with decreasing luminosity and SFR, and at least half of the ionizing photons would have to be lost in the typical system with log L(Hα) ∼38.4. While the gas column densities in dwarf galaxies tend to be lower (e.g., Bigiel et al. 2008), and there could be more leakage from the individual H ii regions (the fraction of diffuse ionized gas appears to be nominally higher in dwarfs; e.g., Oey et al. 2007), star-forming dwarf galaxies are often embedded in large envelopes of H i (e.g., Begum et al. 2008) which makes it unlikely that the Lyman continuum photons find their way completely out of a galaxy and into the IGM. Moreover, observations which have attempted to directly detect escaping Lyman continuum photons have all found upper limits ≲10% (e.g., Leitherer et al. 1995; Bergvall et al. 2006; Grimes et al. 2007; Siana et al. 2007). These detection experiments have been performed on starbursting galaxies where leakage is thought to most likely occur. In contrast, the majority of the dwarfs in the sample are relatively normal in their current star formation activity (Lee et al. 2009). It thus seems unlikely that leakage can be the cause for the trend.

Ionizing photons can also be "lost" in a second way, via absorption of Lyman continuum photons by dust (i.e., dust absorption that Balmer decrement based attenuation corrections cannot account for since the ionizing photons encounter dust before they are able to ionize hydrogen). However, it seems even more unlikely that this effect can cause the observed trend in the Hα-to-UV ratio since the absorption would need to be larger in relatively transparent dwarf galaxies than more luminous, dustier systems.²⁵

Previous work tentatively attributed possible trends in the Hα-to-FUV ratio to a systematic increase in the prevalence of bursts in the recent SFHs of low-mass galaxies (e.g., Bell & Kennicutt 2001). The fiducial expected ratio is an "equilibrium" value which is calculated assuming that the SFR has been constant over a time period that is long enough for the birth of stars responsible for the FUV and Hα emission to balance their deaths. Variations in the SFR over timescales ∼100 Myr would disrupt this equilibrium. In the time following a burst of star formation, a deficiency of ionizing stars develops as they expire relative to lower-mass, longer-lived stars that also contribute to the UV emission, and the Hα-to-FUV flux ratio will thus be lower than expected (e.g., Sullivan et al. 2000, 2004; Iglesias-Paramo et al. 2004).

To examine whether variations in the SFR can account for the decline of the Hα-to-FUV ratio of the magnitude that is observed, synthesis modeling is needed. Here the work of Iglesias-Paramo et al. (2004) can be directly applied. Iglesias-Paramo et al. (2004) sought to use the Hα-to-FUV flux ratio to constrain the effects of the cluster environment on the recent SFH of spiral galaxies, and computed a grid of Starburst99 models for a range of burst amplitudes and durations, assuming a Salpeter IMF. The bursts were superimposed on 13 Gyr old models which were evolved assuming an exponentially declining SFR, with decay timescales typical of values normally used to fit star-forming galaxies (from 3 to 15 Gyr). Variations in the predicted Hα-to-FUV flux ratio were found to be insensitive to differences in decay timescale values within this range. The results of their modeling are tabulated, and it is shown that in order to produce Hα-to-FUV flux ratios that are on average about a factor of 2 lower than the fiducial equilibrium value in a given sample, bursts which elevate the SFR of the galaxies by a factor of 100 for a 100 Myr duration are required. However, such large amplitude bursts do not appear to commonly occur in the overall dwarf galaxy population. The most direct constraints on burst amplitudes and durations in dwarf galaxies are provided by studies which reconstruct SFHs from resolved observations of stellar populations in the nearest low-mass systems. Most recently, Weisz et al. (2008) and McQuinn et al. (2009) have analyzed Hubble Space Telescope (HST)/Advanced Camera for Surveys (ACS) imaging of dwarf galaxies in the M81 group and a small sample of low-mass starburst systems, respectively. Typical burst amplitudes range from a few to ∼10, an order of magnitude smaller than the factor of 100 bursts required to depress the Hα-to-FUV flux ratios by a factor of 2. Further, the 11HUGS sample itself provides a statistical constraint on the average dwarf galaxy starburst amplitude via the ratio of fraction of star formation (as traced by Hα) concentrated in starbursts to their number fraction (Lee et al. 2009). The high degree of statistical completeness of 11HUGS makes this calculation possible for galaxies with M_B < −15, and again, the burst amplitude is found to be relatively modest (∼4), at least for this population of dwarfs. From the above arguments it appears that modest starbursts by themselves, even if they are common in low-mass systems, cannot explain the observed deviation in the Hα-to-FUV flux ratios from the expected value.

An implicit assumption in all stellar population synthesis models (and the resulting calibrations based upon them) is that statistical numbers of stars spanning a full range of masses, typically to a limit of M_* = 100 M_☉, are produced. In the regime of ultra-low SFRs where only a handful of O-stars are formed in a given system over timescales comparable to their lifetimes (i.e., a few million years), this assumption is clearly violated. Under these circumstances, Hα emission can appear deficient, or even absent, although lower mass star formation, which dominates the total mass formed, may take place and the underlying IMF is invariant. Such statistical effects on the Hα luminosity has been previously explored for example by Boissier et al. (2007) and Thilker et al. (2007) in the context of extended UV disks. More detailed work has been done by Cerviño & Valls-Gabaud (2003), Cerviño et al. (2003), and Cerviño & Luridiana (2004) for modeling properties of low-mass stellar clusters and other simple, single-age stellar populations. To evaluate whether such effects can account for our observations, we first perform a simple back-of-the-envelope calculation, and then examine results from the Monte Carlo simulations of C. L. Tremonti et al. (2009, in preparation).

We first estimate when Poisson fluctuations should begin to affect the Hα output by computing the SFR at which the number of O-stars on the main sequence at any given time will be less than 10. Assuming a Salpeter IMF with standard mass limits of 0.1–100 M_☉ yr⁻¹, and considering that only stars with masses >18 M_☉ significantly contribute to the ionizing flux (e.g., Hunter & Elmegreen 2004, appendix B), simple integrals show that there are 10 ionizing stars for a total stellar mass of 4.3 × 10³ M_☉ formed. To compute the corresponding SFR above which the Hα flux should be robust against stochasticity in the formation of ionizing stars, we divide this mass by the 3 Myr lifetime of a 100 M_☉ star (Schaerer et al. 1993), which yields 1.4 × 10⁻³ M_☉ yr⁻¹ (log SFR = −2.8). This is a conservative limit compared to calculations based on the formation of single O-stars (e.g., Lee et al. 2009; Meurer et al. 2009).²⁶ However, Figures 1 and 5 show that systematic deficiencies in the Hα-to-FUV ratio begin at SFRs that are almost an order of magnitude higher (log SFR ∼−2.0). At log SFR = −2.8 the ratio is not just beginning to deviate from the expected value; it is already more than a factor of 2 lower. This back-of-the envelope calculation suggests that stochasticity in the sampling of an invariant Salpeter IMF alone is unlikely to give rise to the observed trends. The same holds true for the Kroupa (2001) and Chabrier (2003) IMFs, for which the computed SFR limit would be 1.5 times lower. In all cases, the FUV flux is negligibly affected.

C. L. Tremonti et al. (2009, in preparation) have preformed detailed Monte Carlo simulations to confirm these conclusions. Such calculations not only allow for a more robust prediction of the mean deviations due to stochasticity in the production of massive ionizing stars, but also provide quantitative predictions of the accompanying increase in scatter. In brief, we generate a pool of stars which have a mass distribution given by the Salpeter IMF, and assign each star a mass-dependent main-sequence lifetime, ionizing photon production rate, and FUV luminosity. We randomly draw stars from the pool at a rate proportional to the desired SFR and follow the ensemble population over time. SFRs are calculated from the FUV and Hα luminosities at t > 1 Gyr. We have run simulations for a large grid of SFRs; the results are overplotted on the data in Figure 7. The median-predicted values (solid line) as well as the values at the 2.5 and 97.5 percentile points (dotted lines) are shown. The median flux ratio begins to significantly deviate from the fiducial value only for log SFR <−3, which is consistent with expectations based on the back-of-the-envelope estimate given above.

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We note that the results of our calculations agree with the work of Cerviño et al. (2003) and Cerviño & Luridiana (2004), who have used statistical formalism to compute values for the "Lowest Luminosity Limit" (LLL), the limiting stellar mass formed in a single event whose properties can be modeled without accounting for stochastic effects. According to their calculations, ∼3 × 10³ M_☉ is the mass associated with the LLL in the production of Lyman continuum photons. Again, taking the O-star lifetime to be ∼3 Myr as above, this limiting mass would correspond to log SFR = −3. However, as Cerviño et al. (2003) discuss, statistical effects may be present in formation events involving up to ten times the LLL mass. Based upon the Monte Carlo simulations, these effects manifest as an increase in scatter in the Hα-to-FUV ratio beginning at log SFR ∼−2, although the median value of the ratio remains stable until log SFR ∼−3.

Therefore, to summarize, although stochasticity in the sampling of an invariant IMF does result in a declining Hα-to-FUV ratio with SFR, we find that the mean effect is not large enough to account for the observations. One important caveat regarding this conclusion however is that our calculations assume that the SFR is constant. As discussed in Section 4.5, moderate variations of the SFR with time are seen in the recent SFHs reconstructed from resolved stellar population data of nearby dwarf galaxies (e.g., Dohm-Palmer et al. 1998; Weisz et al. 2008), and stochasticity may amplify the effect of such variations on the Hα-to-FUV ratio. As illustrated in Figure 7 however, an initial analysis where bursts with 100 Myr duration and a factor of 4 amplitude are included in our Monte Carlo simulations show that this is unlikely to produce significant changes in the predictions based on a constant SFR alone. However, more comprehensive modeling covering a range of SFHs is needed, and these issues are further examined in C. L. Tremonti et al. (2009, in preparation). In particular, histories where the SFR shows short timescale (∼10 Myr) fluctuations of factors of 10 or greater may be considered. Such histories may be the characteristic of the most extreme dwarfs in our sample, in which only one or two H ii regions are visible, and where such fluctuations are not necessarily indicative of starburst activity, but rather of Poisson noise in the formation of individual star clusters.

Finally, we examine the basal assumption required for the calibration of all star formation indicators, the form of the IMF. Naturally, indicators which trace highly luminous but relatively rarer massive stars are particularly sensitive to its form. However, the IMF is generally assumed to be invariant, and if so, the Hα-to-FUV flux ratio should be constant, modulo the other factors discussed above. Uncertainties in a universal IMF therefore cannot themselves produce trends in the Hα-to-FUV flux ratio. Rather, systematic variations which result in a deficiency of ionizing stars in dwarf galaxies would be needed to explain the observations. This possibility should be considered, given that we cannot explain the magnitude of the Hα-to-FUV offset with the other parameters that have been explored thus far.

Evaluating the plausibility of this scenario in present day galaxies has been an issue of much recent work. For example, Hoversten & Glazebrook (2008; HG08) examined the Hα EW and optical colors of a large sample of galaxies taken from the Sloan Digital Sky Survey (SDSS). They find that the Hα EWs are systematically low for the optical colors in low-luminosity galaxies given expectations based on continuous star formation. This is consistent with our results for a more limited sample drawn from the Local Volume (Lee 2006; Lee et al. 2008). However, whereas Lee (2006) and Lee et al. (2008) concluded that such discrepancies could be explained by invoking starbursts in the recent past histories of dwarfs (most systems would then be in a post-burst state where the ionizing stars from the burst have died off, but the intermediate-mass stars continue to contribute to the continuum flux density at Hα, thus depressing the Hα EW), the analysis of HG08 appears to have ruled this out. The much larger SDSS sample of HG08 enabled more detailed modeling which probed the temporal variation of the EWs and colors over a burst cycle. HG08 found that fine tuning to coordinate burst times was necessary to reproduce the data, and that a more likely explanation was that low-luminosity galaxies have an IMF deficient in massive stars.

Recently, Meurer et al. (2009) have examined correlations in the Hα-to-FUV flux ratio with the Hα and R-band surface brightnesses, using the H i-selected samples of the Survey of Ionization in Neutral Gas Galaxies (SINGG; Meurer et al. 2006) and the Survey of UV emission in Neutral Gas Galaxies (SUNGG). Meurer et al. find that the Hα-to-FUV flux ratio decreases with decreasing surface brightness. After ruling out the more likely causes of the systematic in an analysis parallel to the one presented here, they have concluded that it is evidence for a non-uniform IMF. Their study is complementary to ours since they have examined correlations in the Hα-to-FUV flux ratio with the Hα and R-band surface brightness instead of integrated luminosities, and use an independent data set. The galaxy sample however is weighted more heavily toward more luminous star-forming systems (e.g., Paper I, Figures 7 and 8); there are only ∼10 galaxies with dust-corrected SFR(Hα) < 0.01 M_☉ yr⁻¹ and this precludes a robust determination of the observational trend with integrated luminosity as presented here (M. Seibert 2009, private communication).

On the theoretical side, Kroupa & Weidner (2003) and Weidner & Kroupa (2005, 2006) have formulated a model based on both statistical arguments and observational constraints which produces IMFs that appear to be steeper for galaxies with lower SFRs. Several critical assumptions are made. The first is that all stars are born in stellar clusters, which universally form stars according to the Kroupa (2001) IMF. The formation of stellar clusters themselves is also governed by a power law. While the convolution of cluster and stellar IMFs by itself does not produce effects beyond simple random sampling of the stellar IMF alone, the model also assumes that (1) the maximum mass of a cluster formed in a given galaxy, m^max_cl, depends on its SFR, and (2) the maximum mass of a star formed in a given cluster, m^max_*, depends on the total cluster mass M_cl, and is lower than the strict limit where m^max_* = M_cl. The claim is that both m^max_cl(SFR) and m^max_*(M_cl) are both robustly constrained by observations. Further, it is shown that m^max_*(M_cl) is also well modeled by the statistical condition:

$$\begin{equation} \int _{m^{\rm max}_{\ast }}^{150\;M_{\odot }}\xi (m)dm=1 \;\; \bigwedge \;\; M_{\rm cl}=\int _{m^{\rm min}_{\ast }}^{m^{\rm max}_{\ast }}m\xi (m)dm. \end{equation} \tag{ 8 }$$

All these effects taken together can be expressed as

$$\begin{eqnarray} \xi _{\rm IGIMF}(m,t)&=&\int ^{m^{\rm max}_{\rm cl}({\rm SFR}(t))}_{m^{\rm min}_{\rm cl}}\xi _{\ast }(m\le m^{\rm max}_{\ast }(M_{\rm cl})) \;\nonumber\\ &&\times\xi _{\rm cl}(M_{\rm cl})\;\;dM_{\rm cl}, \end{eqnarray} \tag{ 9 }$$

where ξ_IGIMF is referred to as the "Integrated Galaxial IMF."

Recently, Pflamm-Altenburg et al. (2007, 2009) have used this model to generate predictions for trends in FUV and Hα properties. They consider the case where the cluster mass function ξ_cl has a Salpeter slope (referred to as the "standard model") and also examine the consequences of assuming one which has a higher ratio of high-mass to low-mass clusters, i.e., α_cl = 1.0 for 5 M_☉ ⩽ M_cl < 50 M_☉ and α_cl = 2.0 for 50 M_☉ ⩽ M_cl ⩽ m^max_cl(SFR). Adopting this ξ_cl results in smaller relative deviations from a universal Kroupa IMF and is referred to as the "minimal1" model. The predictions are overplotted on the data in Figure 8, where the open red squares represent the standard model, while the filled red squares represent the minimal1 model. Both are in good agreement with the data at the faint end, while the data for the more luminous galaxies in the sample prefer the minimal1 model. Therefore, the IGIMF model is able to account for the declining Hα-to-FUV ratio observed in our Local Volume galaxy sample.

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A next critical test of the IGIMF model would involve computing the expected scatter in the relationship given plausible SFHs, since it is not clear whether the IGIMF will allow for Hα-to-UV ratios higher than the predicted curve (as seen in the data, modulo extinction), given the deterministic upper-mass limits on the IMF in the low activity regime. Monte Carlo simulations analogous to those described in the previous section should also be run with the IGIMF as the input IMF to check that the combination of stochasticity, variations in the SFH, and the IGIMF do not lead to larger declines in the Hα-to-UV ratio than is seen in the data. Finally, further work is still also required to better connect the IGIMF model to the physics of the fragmentation and collapse of gas into stars.

Whether or not IMF variations (or other unidentified issues) are the true culprits underlying the systematic variation in the Hα-to-FUV ratio, it seems reasonable to surmise that the FUV luminosity is the more robust SFR indicator in individual galaxies with low total SFRs and low dust attenuations. This is also supported by the finding that the increase in scatter in the Hα EW (which traces the SFR per unit stellar mass; i.e., the specific SFR, hereafter SSFR) at low galaxy luminosities (Lee et al. 2007), is mitigated when the SSFR is instead based upon the FUV. Figure 9 illustrates the variation of the SSFR with M_B, where the SSFR has been dust corrected and is based on the Hα (left panel) and FUV luminosities (right panel). The SFRs and attenuation corrections are identical to those used throughout this paper, while the stellar masses are based upon M_B and color-dependent mass-to-light ratios as described in Bothwell et al. (2009). These plots are analogous to those presented in Lee et al. (2007). The previously reported increase in scatter near M_B ∼ −15 is apparent in both panels. However, whereas the scatter in SSFR(Hα) increases toward the lowest luminosities by ∼60%, the increase is only ∼25% for SSFR(FUV). Given the longer lifetimes and greater numbers of stars which dominate the UV output of a galaxy, this result is reasonable as the UV will be less affected by purely stochastic variations in the formation of high-mass stars. This relative reduction in the increase in scatter is also generally consistent with the predictions of our Monte Carlo simulations for SFR ≲ 0.01 M_☉ yr⁻¹ as reported in C. L. Tremonti et al. (2009, in preparation) and discussed in Section 4.6.

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Therefore, the results of our analysis suggest that when possible, the SFR should be based on the FUV luminosity when SFR ≲ 0.01 M_☉ yr⁻¹. Hα is a less reliable indicator when the integrated, galaxy-wide log L(Hα) (erg s⁻¹) ≲39.4. However, for those instances where only an Hα measurement is available, we provide an empirical re-calibration of SFR(Hα), using the extinction-corrected K98 SFR(FUV) as the reference value. For an observed, non-dust corrected, integrated L(Hα) <2.5 × 10³⁹ erg s⁻¹:

$$\begin{eqnarray} \log\, ({\rm SFR}\,({M}_{\odot }\;{\rm yr}^{-1})) &=& 0.62\; \log\, (7.9 \times 10^{-42}\nonumber\\ &&\times L(H\alpha)({\rm erg s}^{-1}))-0.47. \end{eqnarray} \tag{ 10 }$$

This relation is the result of a linear least squares fit of SFR(FUV), where L(FUV) has been corrected for internal dust extinction, as a function of SFR(Hα), which has not been dust corrected. Thus, the SFR calculated using this expression will (1) represent intrinsic, dust unattenuated values as appropriate for the dust content typical of local galaxies, (2) trace the activity averaged over the timescale of the FUV emission (∼100 Myr) instead of the native Hα instantaneous timescale (∼3 Myr), and (3) carries the same assumptions (fiducial stellar models) as the K98 UV SFR calibration. The random error on the SFR, which will increase with decreasing L(Hα), can be estimated from the 1σ scatter in SFR(Hα)/SFR(FUV) as listed in Table 2.

As with any SFR prescription, it is important to be aware of the ranges of applicability of Equation (10) and its limitations. The relation has been calibrated with integrated SFRs of local dwarf irregular galaxies for which log L(Hα) ≳ 37, so application to other systems with low total SFRs (e.g., massive elliptical galaxies, or galaxies with log L(Hα) (erg s⁻¹) < 37) may not be warranted.²⁷ We also note that higher order terms may be required to more accurately describe the systematic underestimation of the SFR in this regime, although the current data set does not adequately constrain such terms. For example, Pflamm-Altenburg et al. (2007) offer a calibration based on the IGIMF model that is described by a fifth-order polynomial. In Figure 10, our empirical relation is compared with the IGIMF prescriptions. The IGIMF "minimal1" model is consistent for log L(Hα) ≳ 38, but predicts up to 0.2 dex more star formation in the regime where there is sufficient data available to calibrate our relation.

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