DUST-CORRECTED STAR FORMATION RATES OF GALAXIES. I. COMBINATIONS OF Hα AND INFRARED TRACERS

Our basic approach is to compare and calibrate global attenuation estimates for galaxies derived from the combination of Hα and IR luminosities of galaxies with independently derived attenuation measures from integrated optical emission-line spectra. Our samples are drawn from two surveys of nearby galaxies, the SINGS survey (Kennicutt et al. 2003) and a survey of integrated spectrophotometry of 417 galaxies by Moustakas & Kennicutt (2006; hereafter denoted MK06).¹⁰ The SINGS sample is comprised of 75 galaxies with distances less than 30 Mpc, which were chosen to span wide ranges in morphological type, luminosity, and dust opacity. The MK06 survey was designed to include the full range of optical spectral characteristics found in present-day galaxies, and includes a subsample of normal galaxies as well as large subsamples of optically selected starburst galaxies and IR-luminous galaxies. The combined sample includes objects ranging from dwarf irregular galaxies to giant spirals and IR-luminous galaxies (−13.4 ⩾ M_B ⩾ −22.4), with SFRs of ∼0.001–100 M_☉ yr⁻¹, and ∼0.01 <L_IR/L_B<100. This diversity is important for testing the applicability limits for our methods, and for uncovering any second-order dependences of our results on properties of the galaxies or their SEDs. Detailed information about the SINGS and MK06 samples can be found in the respective survey papers.

Our analysis draws on subsets of these samples which satisfy a number of further selection criteria. Galaxies showing no detectable star formation as measured at Hα were excluded; most of those were early-type E and S0 galaxies. Likewise, we required that the galaxies had well measured IR fluxes or strong upper limit fluxes from Spitzer (SINGS sample) and/or IRAS (for both samples), as discussed further in Section 2.4. We also applied a signal/noise requirement on the optical spectra, to ensure an accurate measurement of the Balmer decrement (Hα/Hβ ratio). This was done by a combination of formal signal/noise estimates and visual inspection of the continuum-subtracted spectra at Hβ. This translated to a minimum signal-to-noise ratio (S/N) ∼15 at Hβ. Finally, we separated galaxies with spectra dominated by star formation from those with bright active galactic nuclei (AGN-dominated) or composites of star formation and AGN signatures, so their behavior could be analyzed separately. This separation was performed using the optical emission-line spectra, using the criteria of Kewley et al. (2001) and Kauffmann et al. (2003) criteria as described in Moustakas et al. (2006).

Application of these minimum signal/noise criteria in IR photometry, Hα photometry, and spectroscopic S/N yielded subsamples of 58 SINGS galaxies (including four AGN-dominated galaxies, NGC 3190, NGC 4550, NGC 4569, and NGC 4579), and 147 galaxies from MK06 (including three AGN-dominated galaxies and 31 with composite nuclei). The galaxy population retains the mixture of the parent samples, with normal spiral and irregular galaxies and especially strong representation of UV, blue, and IR-selected starburst galaxies.

This combination of selection criteria (adequate signal/noise in the IR continuum and Balmer emission lines, emission-line spectra dominated by star formation) was applied to provide an accurate calibration of the attenuation-correction methods that are developed in this paper. However, it is important to bear in mind that these criteria tend to favor the selection of galaxies with significant SFRs per unit mass, typically intermediate to late-type spiral galaxies and luminous irregular and starburst galaxies. Objects that tend to be excluded include early-type (E, S0, Sa) galaxies with weak line emission, dwarf irregular galaxies with little or no detectable dust emission, and the most highly obscured IR-luminous and ultraluminous galaxies, which tend to exhibit weak Hβ line emission and (frequently) AGN signatures in their spectra. Some of these selection effects are unimportant, for example, the absence of IR-weak galaxies from the sample is irrelevant because no attenuation correction is required in those cases. However, the very early-type galaxies and extreme starbursts may probe star formation and attenuation regimes that fall outside of the valid range of our calibrations. We discuss the potential effects of these selection effects further in Section 6.

Integrated Hα fluxes and Hα/Hβ flux ratios for the MK06 sample were taken directly from the integrated spectra. The galaxies were observed with the B&C Spectrograph on the Steward Observatory Bok 2.3 m telescope using a long-slit drift-scanning technique, which produced an integrated spectrum over a rectangular aperture that covered most or all of each galaxy. The resolution of the spectra (∼8 Å) is sufficient to deblend Hα from the neighboring [N ii]λλ6548,6584 lines. The underlying continuum and absorption-line spectra were fitted and removed as part of the emission-line flux measurements, which removes the effects of stellar Balmer absorption on the Hα and Hβ measurements (see MK06 for details).

Drift-scanned spectra were also obtained for the SINGS sample, using the same instrument and setup as MK06 for northern galaxies, and the R-C spectrograph on the Cerro Tololo Inter-American Observatory (CTIO) 1.5 m telescope for a handful of galaxies that could not be reached from Kitt Peak. These observations were configured to match the spatial coverage of the mid-IR spectral maps obtained as part of SINGS. Mapping entire galaxies was not practical with the Spitzer Infrared Spectrograph (IRS), so we mapped representative subregions of the galaxies instead. One set of optical observations consisted of drift scans covering an area of 09 × 33 (09 × 7' for CTIO) oriented to coincide with the Spitzer IRS low-resolution maps, with multiple observations laid end to end to extend from the centers of the galaxies to R ⩾ 0.5 R₂₅ (de Vaucouleurs et al. 1991). A summed spectrum was then extracted by integrating over this 09-wide strip. The areal coverage of these drift scan apertures ranged from <10% of the total projected region within R₂₅ for the largest galaxies in the sample (e.g., M81), to nearly 100% for the smallest galaxies in the sample. We also obtained a separate drift-scanned spectra for the optical centers of each galaxy, using 20'' × 20'' apertures, to approximately match the coverage of a set of high spectral resolution Spitzer IRS observations. Again the two-dimensional spectra were collapsed into a single integrated spectrum covering the central aperture. A third set of (pointed) spectra just covering the galactic nuclei were also obtained. Those data are not used in this paper, apart from helping to classify the nuclear emission types, and checking to ensure that AGN emission does not dominate the other, larger-aperture spectra. The processing and emission-line extractions for these data followed the procedures described in MK06 and J. Moustakas et al. (2009, in preparation), and the data can be found in the latter paper.

For some of the SINGS galaxies, the drift-scanned spectra could not be used to reliably estimate the disk-averaged Hα/Hβ ratio, either because the Hβ line was too weak or because the spatial coverage of the spectra was too limited. For 23 of these cases, spectra for individual H ii regions were available from the literature, and we used these to estimate the disk-averaged Hα/Hβ ratio and its uncertainty. We checked the validity of this procedure by comparing the mean Balmer decrements with those derived from drift-scanned spectra when both types of data were available, and they yield consistent results with no systematic bias, though with less precision than from the integrated spectra.

Table 1 lists the adopted Hα/Hβ ratios (and data sources) for the SINGS galaxies, Table 2¹¹ lists the adopted Hα and Hβ fluxes for the MK06 galaxies, and Table 3 lists the Hα and Hβ measurements of the central 20'' × 20'' region of SINGS galaxies.

Integrated Hα fluxes for the MK06 sample were taken directly from the drift-scanned integrated spectra. The same applied to the observations of the central 20'' × 20'' regions in the SINGS galaxies. In many cases, integrated Hα fluxes for the SINGS galaxies had to be obtained from other sources, because of the incomplete spatial coverage of the drift-scanned spectra. Fluxes were compiled from the literature or were measured from Hα narrowband images obtained as part of the SINGS project.¹² Most of these measurements include emission from the neighboring [N ii]λλ6548,6584 lines. We applied [N ii] corrections on a galaxy by galaxy basis, using the [N ii]/Hα ratio measured in our drift-scanned spectra (26 galaxies), or individual measurements of disk H ii regions in the galaxies from the literature (19 galaxies). For an additional 13 galaxies, we applied the mean relation between average [N ii]/Hα and M_B in the MK06 sample, as published in Kennicutt et al. (2008).

The accuracy of the individual Hα fluxes varies considerably, depending on the absolute and relative strength of the line (relative to continuum) and the source of the fluxes. A comparison of MK06 Hα fluxes with high-quality narrowband imaging measurements by Kennicutt et al. (2008) shows that the average uncertainty of the fluxes is approximately ±10%–15%. This uncertainty could increase to as much as ±30% for galaxies with weak emission or with a strong underlying continuum. Table 1 lists the adopted Hα fluxes and [N ii]/Hα ratios for the SINGS sample.

For the SINGS sample, we have used the integrated Spitzer measurements from Dale et al. (2007) at wavelengths of 3.6, 8.0, 24, 70, and 160 μm. We also used the SINGS images (ver. DR5) to measure 8 μm and 24 μm fluxes with the same 20'' × 20'' circumnuclear apertures that were measured spectroscopically (Sections 2.2 and 2.3). We did not measure 70 μm or 160 μm fluxes for the circumnuclear apertures, because the instrumental beam sizes at those wavelengths were comparable to or larger than the apertures. We refer the reader to Dale et al. (2007) for a detailed listing and discussion of the uncertainties in these fluxes. Aperture corrections (including extended source corrections for scattering of diffuse radiation across the IRAC focal plane) were applied to the circumnuclear measurements, and amount to approximately 10% at 8 μm and 25% at 24 μm.

Most of the galaxies in the MK06 sample have not been observed with Spitzer, so we compiled integrated fluxes at 25, 60, and 100 μm from the IRAS survey. Whenever possible we adopted in order of priority the fluxes measured in the IRAS Revised Bright Galaxy Sample (Sanders et al. 2003), followed by IRAS Bright Galaxy Sample (Soifer et al. 1989), then the large optical galaxy catalog (Rice et al. 1988) and finally version 2 of the IRAS Faint Source Catalog (Moshir et al. 1990). For this study, we are mainly interested in the 25 μm band fluxes and the TIR fluxes. Many galaxies in the sample are undetected or have only marginal detections at 25 μm, and we restricted our analysis to galaxies with >3σ detections at that wavelength. As a supplement to the Spitzer measurements, we also compiled the same IRAS data for the SINGS sample. Of the 58 galaxies in our main Spitzer sample, 46 were observed by IRAS and satisfy our 25 μm signal/noise criterion.

A key SFR index in our analysis is the wavelength-integrated TIR luminosity. The IRAS and Spitzer photometry do not cover enough wavelengths to uniquely define this flux integral, with the emission longward of 100 μm and 160 um, respectively, being especially poorly constrained. As a result, there are numerous prescriptions in the literature for computing a TIR flux from the IRAS and Spitzer band fluxes. For the sake of consistency, we have adopted the definition of TIR flux from Dale & Helou (2002), which is the bolometric IR flux over the wavelength range 3–1100 μm. We also adopt the semiempirical prescriptions from that paper for estimating f(TIR) from weighted sums of MIPS 24, 70, and 160 μm fluxes (Equation (4) of their paper), and from IRAS 25, 60, and 100 μm fluxes (Equation (5) of their paper).

It is important to test the consistency of the IRAS and Spitzer flux scales, and since the SINGS sample was measured with both sets of instruments they provide a direct standard for this comparison. We first compared the consistency of the Spitzer MIPS photometry at 24 μm with the IRAS 25 μm fluxes, using the common definition of monochromatic flux f(λ) = νf_ν. The left panel of Figure 1 shows the ratio of MIPS/IRAS fluxes as a function of 24 μm flux. Overall the flux scales are in excellent agreement, with an average ratio f(MIPS)/f(IRAS) = 0.98 ± 0.06 after the 3σ outliers were excluded. The three outliers are faint and/or low surface brightness IR emitters with single published IRAS flux measurements. Apart from these isolated examples, we find that the MIPS 24 μm and IRAS 25 μm fluxes can be used interchangeably. For the sake of simplicity, we shall use the terms "24 μm fluxes" and "24 μm luminosities" to refer to measurements made either at 24 μm or 25 μm.

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The right-hand panel of Figure 1 shows the ratio of MIPS and IRAS TIR fluxes. Both sets of fluxes were computed using the prescriptions of Dale & Helou (2002). Here, the deviations are larger, with a scatter of ±23%  and a systematic offset of 24% in flux scales (MIPS larger). This offset is considerably larger than the uncertainty given by Dale & Helou (2002). To better understand this discrepancy, we compared our flux ratios f(TIR)_MIPS/f(TIR)_IRAS as a function of FIR color f_ν(60 μm)/f_ν(100 μm), as shown in Figure 2. For galaxies with warm IR colors (high f_ν(60 μm)/f_ν(100 μm) ratios), the SED peaks near or shortward of 100 μm, so the 25, 60, and 100 μm IRAS fluxes provide a relatively reliable estimate of the integrated TIR flux. However, for galaxies with colder colors (f_ν(60 μm)/f_ν(100 μm) < 0.4), the peak of the SED is longward of 70 μm, so there is more of a danger of systematic uncertainties affecting the TIR flux estimates. The solid line in Figure 2 shows the approximate magnitude of this error predicted by the Dale & Helou (2002) SED models. The actual comparison for the SINGS galaxies displays the sense of this predicted trend, but the systematic differences between MIPS and IRAS measurements is more severe. This comes about because the SINGS sample included a number of galaxies with colder dust than any of the galaxies in the template reference sample explored by Dale & Helou (2002). For that reason, we analyze the Spitzer and IRAS TIR data separately in the analyses that follow.

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In Section 5.3, we explore whether the integrated (mainly nonthermal) radio continuum fluxes of galaxies can be combined in the same way with optical emission-line fluxes to provide attenuation-corrected SFRs. Radio continuum fluxes at 1.4 GHz for the SINGS sample were taken from the compilation of Dale et al. (2007). To obtain a matching set of radio data for the MK06 sample, we cross-correlated the galaxies with the NRAO VLA Sky Survey (NVSS; Condon et al. 1998). After experimenting with several choices of matching radii, we used a radius of 20'', which provided the optimal yield of matched sources without introducing significant numbers of spurious matches with background sources. This produced radio fluxes for 100 of the 113 star-forming galaxies in the MK06 sample. We also used pointed observations of 32 galaxies from Condon (1987), and adopted them in preference to the NVSS when both sets of measurements were available, to minimize problems with missing extended flux in the NVSS data (Yun et al. 2001).

Our method uses the IR luminosities of galaxies to estimate attenuation corrections for their optical emission-line (or UV continuum) luminosities. Its physical basis is a simple energy balance argument (see Inoue et al. 2001 and Hirashita et al. 2003 for similar approaches to this problem). We assume that on average the attenuated luminosity in Hα is re-radiated in the IR, with a scaling factor that is calibrated empirically, and will differ depending on the IR emission component used and secondarily on the nature of the dust-heating stellar population. Following Kennicutt et al. (2007), we can construct a linear combination of the observed Hα and IR luminosities that reproduces the true unattenuated Hα luminosity

$$\begin{equation} L({\rm H\alpha })_{\rm corr} = L({\rm H\alpha })_{\rm obs} + a_{\lambda } L({\rm IR}), \end{equation} \tag{ 1 }$$

where L(Hα)_corr and L(Hα)_obs denote the attenuation-corrected and observed Hα luminosities, respectively, L(IR) represents the IR luminosity over a given wavelength bandpass, and a_λ is the appropriate scaling coefficient for that wavelength band and dust-heating population. The equivalent attenuation correction for Hα is simply

$$\begin{equation} A({\rm H\alpha }) = 2.5 \log \bigg[{1 + {{a_{\lambda } L({\rm IR})} \over {L({\rm H\alpha })_{\rm obs}}}}\bigg]. \end{equation} \tag{ 2 }$$

Equations (1) and (2) represent simple approximations to a much more complicated dust radiative transfer process in galaxies (e.g., Witt & Gordon 2000; Charlot & Fall 2000), but their physical motivation can be readily understood as follows. We first define a scaling factor η that represents the fraction of the bolometric luminosity of a stellar population that is reprocessed as Hα emission in a surrounding ionization-bounded H ii region

$$\begin{equation} L({\rm H\alpha })_{\rm corr} = \eta L_{\rm bol}, \end{equation} \tag{ 3 }$$

where as before L(Hα)_corr represents the intrinsic (extinction-free) Hα luminosity and L_bol represents the bolometric stellar luminosity of the same region. Considering now explicitly the attenuation of the emission line, its attenuated luminosity can be expressed as

$$\begin{equation} L({\rm H\alpha })_{\rm obs} = L({\rm H\alpha })_{\rm corr}\, {e}^{-{\tau _{\lambda }}}. \end{equation} \tag{ 4 }$$

The attenuation in magnitudes A(λ) = 1.086 τ_λ. The IR luminosity for the region (consider, for example, the total wavelength-integrated IR luminosity) likewise can be expressed as

$$\begin{equation} L({\rm TIR}) = L_{\rm bol}\, (1 - {e}^{{-\overline{\tau }}}), \end{equation} \tag{ 5 }$$

where the relevant opacity in this case is the effective (absorption and luminosity-weighted) opacity of the starlight heating the dust.

$$\begin{equation} \bar{\tau }= - \ln \frac{\int L_\lambda e^{-\tau _\lambda } d\lambda }{\int L_\lambda d\lambda }. \end{equation} \tag{ 6 }$$

This mean opacity depends on the wavelength dependence of the attenuation, the SEDs of the stars that heat the dust, and the dust covering factors for these stellar populations. In general, it will not necessarily be the same as the dust opacity τ_λ for the emission line of interest. We can substitute Equations (4) and (5) into Equation (3) to eliminate L_bol and express L(Hα)_corr in terms of the observable luminosities L(Hα)_obs and L(TIR) and the opacities. It is convenient to introduce a scaling parameter β, the ratio of the emission-line opacity to effective mean stellar opacity for the stars heating the dust

$$\begin{equation} \beta \equiv {{\tau _\lambda } \over {\overline{\tau }}} \end{equation} \tag{ 7 }$$

then Equation (1) takes the modified form

$$\begin{equation} L({\rm H\alpha })_{\rm corr} = L({\rm H\alpha })_{\rm obs} + \eta L({\rm TIR})\, {{(1 - {e}^{-\beta \overline{\tau }}}) \over {(1 - {e}^{-\overline{\tau }}})}. \end{equation} \tag{ 8 }$$

In detail, the opacity index β depends on several parameters, most importantly the wavelength of the optical (or UV) SFR tracer, but also on the overall stellar population mix and the distribution of dust opacities for different age populations, all which will influence $\overline{\tau }$.

These relations take a much simpler form in the special situation where $\tau _{{\rm H\alpha }} \simeq \overline{\tau }$ (i.e., β ≃ 1). In that case, the opacity term in Equation (8) drops out, and we are left with the simple result

$$\begin{equation} L({\rm H\alpha })_{\rm corr} = L({\rm H\alpha })_{\rm obs} + \eta L({\rm TIR}) \end{equation} \tag{ 9 }$$

the same as Equation (1) with coefficient a_λ = η, the bolometric correction term for L(Hα). If this approximation holds then the attenuation-corrected emission-line luminosity can be derived from a linear combination of the observed Hα luminosities and the TIR (or other IR) luminosity.

Thanks to fortuitous physical circumstances this approximation is valid for the Hα emission line. Although the line is emitted in the red (0.6563 μm), observations show that the dust attenuation of galaxies in the Balmer lines is approximately 2.3 times higher than the corresponding attenuation in the stellar continuum (see Calzetti 2001 and references therein). As a result, the attenuation of the Hα line is comparable to that of the stellar continuum in the 0.3–0.4 μm range, close to where the peak contribution to dust heating occurs in most galaxies (Calzetti 2001). Below in Section 3.3, we use the SEDs of the galaxies in our sample to confirm this quantitatively.

We can also use the results presented above to explore the validity of applying a linear combination of emission-line and IR fluxes (Equation (1)) in the more general case when the line attenuation is systematically larger or smaller than the mean opacity to the dust-heating starlight. This would be the case for emission lines at much shorter or longer wavelengths than Hα, or even for Hα itself, if the dust-heating starlight was dominated by extremely young (i.e., blue) or old (i.e., red) stellar populations. In such cases the re-emitted starlight in the dust continuum will tend to undercompensate or overcompensate for the flux attenuated in the emission line. The magnitude of this effect can be calculated using Equation (8). Figure 3 plots the ratio of the attenuation-corrected luminosity estimated from naive application of Equation (1) to the actual intrinsic luminosity, for six values of β in the range 0.25–2 (the solid line at unity is the case for β = 1). The systematic error reaches its maximum value for optical depths of 0.7–2 (approximately the same range in magnitudes). For most combinations of β and opacity, the systematic errors are small, of order 10%–20%, when compared to other systematic uncertainties in the determination of the SFRs. The mismatch in opacities is also unimportant for low optical depths, because in those situations the emission contribution from the IR is negligible. The values of β shown in Figure 3 cover a wide range of potential emission lines from Paα at 1.89 μm (β ∼ 0.25) to [O ii]λ3727 (β ∼ 2). In this paper, we shall apply the linear relations in Equations (1) and (2), and use comparisons to independent measurements of attenuation-corrected SFRs and attenuations to evaluate the efficacy of the method.

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Our method in effect uses the ratio of IR to Hα fluxes of galaxies to derive attenuated-corrected Hα luminosities and SFRs. These need to be calibrated using independent measurements of the Hα attenuation corrections. For the latter, we have chosen to use the stellar-absorption-corrected Hα/Hβ ratios from our integrated spectra, converted to Hα attenuations using a Galactic extinction curve. Specifically, we assume an intrinsic Hα/Hβ ratios for Case B recombination (I(Hα)/I(Hβ) = 2.86) at electron temperature T_e = 10,000 K and density N_e = 100 cm⁻³ (Hummer & Storey 1987). The dust attenuation curve of O'Donnell (1994) was used to convert the observed reddenings to attenuation values at Hα (R_V = 3.1). The latter is equivalent to adopting a Galactic dust extinction curve, and assuming a foreground dust screen approximation for the attenuation correction in the Balmer lines. We discuss the possible limitations of these assumptions below in Sections 3.3 and 6.3.

As emphasized earlier, our simple energy balance method is an idealized prescription that approximates a much more complicated radiative transfer problem in galaxies, and it is important to bear in mind and quantify any systematic errors and limitations that may be introduced by these approximations. The most important of these approximations are (1) assumption of comparable extinction in the optical emission-line tracer and in the dust-heating continuum, (2) assumption of isotropic dust geometry, (3) adoption of an average dust-heating stellar population mix across the sample, and (4) assumption of a reliable set of reference SFRs and attenuation measurements, which are free of dust-dependent systematic errors. We defer a full discussion of these assumptions and any associated uncertainties to Section 6, after the main results have been presented. However, we briefly address the points here, to offer reassurance that the assumptions are reasonable for the current application.

As shown earlier, the use of linear combination of Hα and IR SFR tracers to derive attenuation-corrected SFRs is strictly valid in the case where the averaged dust attenuations in Hα and the dust-heating stellar continuum are comparable. The attenuation laws of Calzetti (2001 and references therein) show that this is a plausible assumption, but we can test it directly. We combined measurements of the galaxies in MK06 sample from the Galaxy Evolution Explorer (GALEX; Martin et al. 2005), the Two Micron All Sky Survey, and the Sloan Digital Sky Survey (SDSS; Adelman-McCarthy et al. 2006) to derive the bolometric luminosities of the galaxies, inclusive of the re-emitted dust luminosities. The ratio of these luminosities to the TIR luminosities was then used to derive an estimate of $\overline{\tau }$ for each galaxy, using Equation (5). These values were then compared in turn to measurements of τ_Hα from the Balmer decrements. Figure 4 shows a histogram of the resulting values of $\beta = {\tau _{\rm H\alpha }}/{\overline{\tau }}$. The median value of β = 1.08, which is consistent with unity within the uncertainties in the optical depths. Much of the dispersion in values is physical, and reflects differences in SED shape within the sample (B. D. Johnson et al. 2009, in preparation). The result justifies the use of the linear approximation, at least for combinations of Hα and IR fluxes, and of course for combinations of UV continuum and IR fluxes (Paper II).

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Variations in dust geometry will be another source of uncertainty in the IR-based attenuation corrections. Our model assumes an isotropic distribution of dust around the stars, but we expect few regions to adhere to this idealized geometry. So long as there is no systematic bias toward foreground or background dust in galaxies, we do not expect this to introduce significant systematic errors, but geometry variations may introduce a significant random error into individual determinations of attenuation. When C07 applied our method to 220 bright H ii regions in 33 SINGS galaxies, they found excellent consistency between the derived Hα attenuations and those derived from Paα/Hα ratios, but with a dispersion of ±0.3 dex for individual objects. This factor-of-2 dispersion is consistent with the combined effects of varying dust geometry and stellar population in the H ii regions. We might well expect the dispersions to be lower for galaxies, where we are integrating over hundreds to thousands of individual star-forming regions.

As will be discussed further in Section 6, variations in dust-heating stellar populations, the longstanding "IR cirrus" problem, is probably the largest source of systematic uncertainty in this approach. The dust heating in galaxies arises from a much larger range of stellar ages (and dust-heating geometries) than in the H ii regions and starbursts studied by C07, so the calibration coefficients in Equation (1) derived by C07 are not expected to apply to the measurements of galaxies as a whole. Comparing the results for galaxies and H ii regions will provide an indirect measure of the relative contributions of heating from very young (<10 Myr old) and older stars, and some indication of the uncertainties in SFRs when different types of galaxies are compared.

Finally, any systematic error in the reference attenuations derived from Balmer decrements will propagate with full weight into our calibrations. Our approximation to dust with a Galactic extinction curve in the foreground screen limit is the most common convention for analyses of both H ii region spectra and integrated spectra of galaxies (see Calzetti 2001 and references therein), but its use raises some legitimate questions. A large body of theoretical studies of dust radiative transfer in galaxies (e.g., Witt & Gordon 2000; Charlot & Fall 2000; Charlot & Longhetti 2001; Tuffs et al. 2004; Jonsson et al. 2009) shows that the actual relation between the reddening of an integrated spectrum and its dust attenuation can vary considerably, depending on the dust geometry and the relation between extinction and stellar age. In galaxies with severe dust extinction, such as in many ULIRGs, these geometric effects will cause the Balmer decrement to be biased to the least obscured regions, leading to a systematic underestimate of the actual attenuation. The predictions for more typical disk galaxies are inconclusive, with some studies suggesting a mild bias in attenuations estimated from the foreground dust screen approximation (e.g., Charlot & Fall 2000), but with other studies showing little or no bias (e.g., Jonsson et al. 2009). However, observations at other wavelengths (Paα, thermal radio continuum, and UV+IR) allow us to independently derive Hα attenuation estimates for subsets of our sample, and test for systematic errors in the zero point of our Balmer decrement based attenuation scale. As shown in Section 6.3, these tests confirm the reliability of the Balmer attenuation scale at the ±10%–20% level, which is comparable to other systematic errors in our SFR scales. On that basis, we have chosen to use the Balmer decrements with the most commonly applied attenuation correction procedures. We also publish the observed Balmer-line ratios so other workers may apply alternative attenuation corrections in the future.

As an introduction, it is instructive to compare the consistency of Hα and IR SFR measures before any corrections for attenuation are applied. Figure 5 compares the observed Hα luminosities of the SINGS and MK06 galaxies with their corresponding 24 μm (νf_ν) luminosities. When converting fluxes to luminosities, we have used distances listed for SINGS galaxies in Table 1, and from MK06 in Table 2. Both sets of distances assume H₀ = 70 km s⁻¹ Mpc⁻¹ with local flow corrections. In Figure 5 (only), we do not distinguish SINGS galaxies from MK06 galaxies and we code the points by spectral type, with open round points representing galaxies dominated by star formation, crosses representing galaxies with strong AGN signatures in their spectra, and circles with embedded crosses denoting composite AGN and star formation dominated spectra. The axis label along the top of the plot shows the corresponding SFR (uncorrected for attenuation), using the calibration of K98. Finally, the solid line shows a linear (unity slope) relation for reference, with the zero-point set to match the mean relation between L(24 μm) and L(Hα)_corr found by Zhu et al. (2008) for nearby galaxies in the Spitzer Wide-Area Infrared Extragalactic (SWIRE) survey (Lonsdale et al. 2003).

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The Hα and 24 μm luminosities show a broad correlation, as is often the case when absolute luminosities are compared. Upon closer examination, however, important departures are apparent. The rms dispersion around the linear relation is ±0.5 dex, more than a factor of 3. Part of this dispersion is caused by AGN contributions to the dust emission. The galaxies with significant AGN contributions clearly are displaced in the diagram, reflecting in part the additional dust heating by the active nuclei (e.g., Soifer et al. 1987). Since we are solely interested in calibrating SFR tracers, we shall exclude galaxies with strong AGN signatures from most of the subsequent analysis in this paper, but clearly the issue of AGN contamination is always an important one when applying any SFR tracer. Even among the star-forming galaxies, however, the dispersion about the linear relation is large, approximately ±0.4 dex. The dispersion among these galaxies is caused almost entirely by variations in dust attenuation; the observed Hα flux underestimates the SFR in dusty galaxies, while the IR emission underestimates the SFR in less dusty galaxies. Moreover, the relation between Hα and IR emission clearly is nonlinear, with the mean ratio of IR to Hα luminosities increasing by a factor of 30 from the faintest to the most luminous galaxies in the sample. This nonlinearity is a manifestation of the well established relation between attenuation and SFR (e.g., Wang & Heckman 1996). Higher SFRs are associated with regions of higher gas surface density (Kennicutt 1998b; C07) and hence higher dust column densities. As a result, galaxies with the highest SFRs tend to suffer heavy attenuation, and the observed Hα flux severely underrepresents the actual SFR, moving points in Figure 5 to the left. Likewise, many (but not all) of the galaxies with low SFRs tend to be low-mass galaxies with lower dust contents and column densities. For those objects, the attenuation at Hα is low and the emission line provides an accurate measure of the SFR, but then the dust emission severely underrepresents the SFR, shifting points down in Figure 5.

We can remove the part of the nonlinearity in Figure 5 that is produced by attenuation of Hα by comparing instead the 24 μm luminosities with the attenuation-corrected Hα luminosities, as derived from the Balmer decrements in the integrated spectra (Section 3.2). This comparison is shown in the top panel of Figure 6(now with the AGN-dominated and composite spectrum galaxies removed). Here and throughout the paper, integrated measurements of SINGS galaxies are shown as open circles, and those for the MK06 sample are shown as solid circles. Applying the reddening corrections tightens the correlation with 24 μm emission considerably, but the dispersion about the mean relation remains substantial at ±0.3 dex, and the nonlinearity remains. Nearly all of the scatter and nonlinearity in this relation are caused by variations in the fraction of young starlight that is reprocessed by dust.

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Previously several groups have investigated the correlation between 24 μm IR emission and attenuation-corrected Hα and Paα emission, and used them to calibrate the 24 μm emission as a SFR measure (Wu et al. 2005; Alonso-Herrero et al. 2006; Relaño et al. 2007; C07). The relations from Wu et al. (2005), Relaño et al. (2007), and C07 can be directly compared to our data, as shown in the top panel of Figure 6. Apart from slight deviations at the extremes in luminosity, our data generally follow these relations as well. Nevertheless, it is dangerous to use the IR luminosity by itself as a quantitative SFR tracer, because in any galaxy other than an extremely dusty starburst the dust reprocesses only a fraction of the young starlight, and any linear scaling of a SFR calibration based on dusty starburst galaxies will tend to systematically underestimate the SFR in lower-luminosity normal galaxies. This is shown by the dotted line in the top panel of Figure 6, which shows a linear (slope unity) relation fitted to the data. The calibrations of Wu et al. (2005) and Relaño et al. (2007) mitigate this effect by fitting a nonlinear relation between L(24) and L(Hα), which effectively builds in a decrease in mean attenuation with decreasing SFR. However, it represents a crude, unphysical correction at best. While most present-day galaxies with low Hα luminosities tend to be dwarf galaxies with low dust contents and optical depths, this same range of fluxes is occupied by dusty galaxies with low SFRs (mostly early-type spirals); attenuation does not correlate monotonically with the SFR itself. Thus, while nonlinear calibrations of the 24 μm luminosity may provide crude statistical measures of the SFR, the uncertainties associated with measurements of individual galaxies remain very high. One risks even larger, systematic errors if one were to apply such relations at high redshift, because there is no reason to expect that the relations between mean attenuation and SFR observed at z = 0 necessarily apply in the early universe.

However, we can mitigate the effects of variable attenuation between the galaxies by applying Equation (1), using a linear combination of the observed (uncorrected) Hα and mid-IR luminosities to estimate the attenuation-corrected Hα luminosity. The result is shown in the bottom panel of Figure 6, which compares the corrected Hα luminosities derived from the ratio L(24)/L(Hα)_obs with those derived from the Balmer decrements. We fitted for the value of the scaling factor a in Equation (1) that provides the best overall agreement with the luminosities corrected from the Balmer decrements (a = 0.020 ± 0.001_r ± 0.005_s), where the first error term lists the formal (random) fitting uncertainty, and the second term includes possible systematic errors in the calibration zero point including any uncertainty in the reference attenuation scale based on Balmer decrement measurements (see Section 6.3). We shall adopt this convention of citing both random and systematic errors throughout the remainder of this paper.

The consistency between the two independent sets of attenuation-corrected Hα luminosities is striking, as is the decrease in the dispersion between the two SFR estimates, relative to the comparisons of 24 μm versus Hα luminosities alone in the top panel of Figure 6. This is especially apparent for the galaxies with L(Hα) < 10⁴² erg s⁻¹. The galaxies with the lowest luminosities tend to have very little dust, and hence nearly no reddening and only weak IR emission. In such galaxies, the Hα luminosity dominates the sum in Equation (1). On the other hand, most of the galaxies with L(Hα)>10⁴² erg s⁻¹ in Figure 6 are LIRGs, with very high attenuations. In such cases, it is the IR term in Equation (1) that dominates the sum.

Our fitted value of the coefficient a in Equation (1) is 35% lower than the value a = 0.031 ± 0.006 derived by C07 based on measurements of H ii regions in SINGS galaxies. However, our value is in excellent agreement with a = 0.022 derived by Zhu et al. (2008) for a sample of luminous star-forming galaxies observed in common between the SWIRE survey and the SDSS. We believe that the difference in scaling coefficients is consistent with expectation given the different stellar age distributions in H ii regions and galaxies, as discussed in Section 6.

Before examining the implications of Figure 6 in more depth, we can make the same comparison, but combining instead the Hα and TIR luminosities of the galaxies. The results are shown in Figure 7. As before the top panels show the TIR luminosity as a function of L(Hα)_corr, while the bottom plots show the linear combination of TIR and Hα luminosities (in this case with the fitted a = 0.0024 ± 0.0001_r ± 0.0006_s). Because of the systematic differences between MIPS and IRAS TIR scales, we separate these data in Figure 7, with the left panels showing Spitzer MIPS-based TIR luminosities (SINGS sample) and the right panels showing IRAS-based luminosities (SINGS and MK06 samples). Since MIPS observations are not available for most of the MK06 sample most of the TIR data used in the remainder of this paper will be based on IRAS luminosities, to maximize the uniformity of the data. However, the more sensitive MIPS data allow us to extend the range of luminosities covered by nearly an order of magnitude.

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The most gratifying result shown in the bottom panel of Figure 7 is not the linearity of the relation (which is dictated by the fitting of a) but rather the tightness of the relation. To illustrate this better, Figure 8 shows the residuals from the fits in Figure 6 (left panels, 24 μm and Hα) and Figure 7 (right panels, IRAS TIR and Hα). The combination of Hα and IR flux dramatically improves the consistency and tightness of both relations. In both cases, these dispersions are less than half of the rms (logarithmic) deviations when the IR luminosities by themselves are compared to the Balmer-corrected luminosities. We suspect that the tighter residuals in the Hα + TIR index (±0.09 dex rms versus±0.12 dex for Hα + 24 μm) reflect the higher quality of the longer wavelength IRAS measurements as well as lower sensitivity to local variations in dust heating (see Section 6 for further discussion). The lack of any systematic residual with total luminosity (and thus SFR) is also significant, and argues against any large systematic error in the derived values of the fitting coefficients a.

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Figures 5–7 compare absolute luminosities in various wavelengths, and it is well known that such plots can sometimes mask underlying physical trends, because of the large range in galaxy masses that underlies any comparison of absolute quantities. These effects can be removed by analyzing mean surface brightnesses (which scale with the SFR per unit disk area) instead of luminosities. For this comparison, we defined the disk area as the deprojected area of the spectroscopic aperture for the MK06 galaxies, and area within the R₂₅ radius (de Vaucouleurs et al. 1991) for the SINGS galaxies. These provide only approximate measures of the radii of the star-forming disks (e.g., Kennicutt 1989, 1998b), but they suffice for this statistical comparison. The top panels of Figure 9 show that the same trend we observed in disk opacity functions of luminosity are also seen in SFR per unit area. This is not surprising, because it is well known that the SFR per unit area correlates strongly with the average surface density of gas (e.g., Kennicutt 1998b), and it stands to reason that it would correlate with the mean column density of dust as well. The bottom two panels show that our prescriptions for correcting the Hα emission for attenuation using IR observations are in excellent agreement with the Balmer-derived attenuations over the full range (more than a factor of 1000) in Hα surface brightness.

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Finally, Figure 10 compares the Hα attenuation estimates themselves. The left and right panels compare the Hα attenuations derived using Hα + 24 μm and Hα + TIR luminosities, respectively, with those derived from the spectroscopic Hα/Hβ attenuation values. The solid line in each panel shows the line of equality, while the dotted lines on either side contain 68% of the points (∼1σ; the overall dispersions are identical to those in the previous figure). Overall there is excellent consistency between the attenuation estimates. The outliers tend to be SINGS galaxies where spatially undersampled spectroscopic strips do not provide a representative measure of the global reddening, along with a few galaxies with deeply embedded central star-forming regions. Overall, however, the attenuation estimates, especially those derived from the combination of Hα and TIR luminosities, show impressive consistency with the spectroscopic reddening-derived attenuations, given that the methods are entirely different.

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The consistency between the IR-derived and Balmer-derived attenuations for the integrated measurements of the SINGS and MK06 galaxies is also much tighter than for comparable measurements of individual giant H ii regions. C07 applied the same methodology to Spitzer 24 μm and Hα measurements of 220 H ii regions and star-forming complexes in 33 SINGS galaxies, compared these to attenuations derived from Paα/Hα ratios, and derived a mean dispersion of ±0.3 dex (a factor of 2) about the mean relation. A similar comparison of a homogeneous set of data for 42 H ii regions in M51 by Kennicutt et al. (2007) yielded a dispersion of ±0.25 dex. For the integrated measurements in this paper, we find a corresponding dispersion of ±0.12 dex between 24 μm + Hα and Hα/Hβ measurements, and even less for the TIR + Hα and Balmer-derived attenuations. Zhu et al. (2008) made a similar comparison of attenuated-corrected luminosities based on 24 μm + Hα and Hα/Hβ measurements for their SWIRE+SDSS sample. They do not quote a dispersion, but based on their Figure 5 we estimate that it is approximately ±0.14 dex, somewhat larger than our result, but understandable in view of the uncertainties in the large-aperture corrections that were required to match the SDSS fiber fluxes to the Spitzer 24 μm observations. The larger residuals in the H ii region results may be caused in part by larger measuring uncertainties in the Paα and Hα photometry of the H ii regions; however, we would also expect physical effects such as variations in attenuation geometry and age variations in the H ii regions to produce larger residuals. These latter effects tend to average out when integrated over an entire galaxy's population of star-forming regions.

The correlations between the IR-based and optically based attenuation measurements are so tight that it is worthwhile investigating whether the residuals from these relations vary systematically with the SEDs or physical properties of the galaxies. As with any quantitative star formation tracer, it is important to establish the limiting range of SFRs and galaxy types over which these methods can be applied reliably, and assess whether they can be extrapolated to galaxy types outside of our sample, for example, the extremely luminous star-forming galaxies observed at high redshift.

We tested for systematic residuals against eight parameters, including SFR (Figure 8), SFR per unit area (cf.  Figure 9), and six other parameters shown in Figures 11 and 13. Figure 11 shows the logarithmic residuals of attenuation-corrected Hα luminosities derived from Hα + 24 μm (left) and Hα + TIR (right), relative to those derived using the Balmer decrement, plotted as functions of FIR 60 μm to 100 μm flux ratio (top panel), Balmer-derived attenuation (middle panel), and axial ratio or inclination (bottom panel). The only significant trends seen are a systematic residual in Hα + 24 μm based SFRs with f_ν(60 μm)/f_ν(100 μm) ratio and Balmer attenuation. This trend with IR color is a by-product of the systematic variation in IR SED shapes of galaxies; galaxies with higher 60–100 μm flux ratios also show higher L(24 μm)/L(TIR) (or L(25 μm)/L(TIR)) ratios. This well known trend is illustrated for our sample in Figure 12, with the sequence of SED models by Dale & Helou (2002) superimposed. As a result, we believe that the trend in residuals in the upper left-hand panel of Figure 11 is an artifact of the variation in SED shapes of galaxies, and this contributes considerably to the larger scatter in this index relative to a linear combination of Hα and TIR luminosities. Interestingly, the latter residuals show no systematic trend with galaxy IR SED shape.

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The trend in residuals with Balmer attenuation (middle left panel of Figure 11) is less straightforward to interpret. Interestingly, the sense of the trend is that galaxies with the highest Balmer attenuations have weaker 24 μm emission than expected, which is in the opposite sense of what one might expect if the Balmer decrements systematically underestimated the true attenuation in more dusty environments. The same trend is present in the Hα + TIR residuals (middle right panel), but is considerably weaker. This suggests that much of the trend in the middle left panel may be the by-product of a second-order correlation between average dust temperature (i.e., mix of warm star-forming regions and background cirrus emission) and characteristic opacity of those regions (Dale et al. 2007), but this should be regarded as speculation rather than an explanation.

Interestingly, neither set of IR-based attenuation and SFR estimates shows any systematic residual with Balmer attenuation/SFR as a function of axial ratio. One would not expect the Hα + IR (24 μm or TIR) indices to show any systematic effects with disk inclination, but one might imagine that the Balmer-based measurements would suffer from such a systematic effect. We do know that the Hα surface brightnesses of disks do decrease systematically with increasing inclination (Young et al. 1996), and our results suggest that this increase in attenuation is largely accounted for in both the IR-based and spectroscopic attenuation corrections. This offers some support for the robustness of the Balmer-based attenuation measurements.

Figure 13 shows the same residuals as Figure 11, but plotted this time as functions of three measures of the stellar populations in the galaxies, the integrated Hα equivalent width (EW) of the galaxies, the 4000 Å spectral break strength (Bruzual 1983; as applied by Balogh et al. 1999), and the gas-phase oxygen abundance (12 + log(O/H)) estimated from the nebular lines (Moustakas et al. 2006; J. Moustakas et al. 2009, in preparation). The Hα EWs (top panels) roughly scale with the SFR per unit stellar mass (specific SFR). The residuals show no systematic change with EW over a very large range (5–230 Å), which includes galaxies with Hubble types Sb–Irr (K98). The bottom panels of Figure 13 show that there is no significant trend in attenuation residuals with metallicity.

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The D_n(4000) break provides a reddening-insensitive measure of the relative age of the stellar population in the galaxies (e.g., MacArthur 2005; Johnson et al. 2007b), and we might expect it to correlate with the attenuation residuals, if variations in heating of dust from evolved stars is important. A trend is seen in the Hα + 24 μm residuals (left middle panel of Figure 13), but we believe that this is mainly a second-order consequence of the IR SED dependence discussed earlier. There is no discernible trend in Hα + TIR attenuation residuals with D_n(4000). One might have expected some deviation for the reddest galaxies with larger values of D_n(4000), as the result of dust being heated primarily by evolved stars. Evidence for such deviations has been seen in comparisons analyses of UV and IR-based SFR tracers (e.g., B. D. Johnson et al. 2009, in preparation; Cortese et al. 2008). We believe that two effects contribute to this negative result. One is the limited range in stellar population probed by this sample (D_n(4000) ⩽ 1.4). The other reason is a compensation between two physical effects; in galaxies with redder stellar populations there will be a stronger radiation component from evolved stars, but the radiation will also be shifted to longer wavelengths where the dust opacity is lower. In terms of Equation (8) in Section 3.1, there are changes in the coefficients η and β which tend to compensate for each other.

An alternate way to examine the residuals is to calculate for each individual galaxy a value of the IR/Hα scaling factor in Equation (1) that forces the derived reddening to match that derived from the Balmer decrement

$$\begin{equation} a_{\lambda } = {{L({\rm H\alpha }) \over {L_{\lambda }({\rm IR})}}} \ {[{10^{{\rm Balmer}({\rm H\alpha })} - 1}]}. \end{equation} \tag{ 10 }$$

We expect these individual values of a to show a considerable dispersion, especially in galaxies with relatively low attenuation, where the uncertainties in single attenuation estimates may be comparable to the estimates themselves. Figure 14 shows histograms of a₂₄ and a_TIR for the integrated measurements of SINGS and MK06 samples, with the adopted average fits for the combined sample shown by vertical lines. There are a few prominent outliers in both cases, and these provide valuable examples of when our attenuation measures break down. For the outliers with anomalously high values (a₂₄>0.05 and a_TIR>0.004), the IR emission is considerably weaker than expected from the Balmer decrement. These tend to be either galaxies with low attenuation and IR emission, where the predicted IR emission is sensitive to small uncertainties in the Balmer decrement, or cases where the limited spatial sampling of the SINGS spectroscopy does not provide an accurate measure of the global reddening. The points with anomalously low values of a include metal-poor galaxies with low attenuations, objects with deeply embedded star-forming regions, or other cases where the SINGS spectra probably underestimate the global reddening of the galaxy, because the optical emission does not penetrate into the main IR-emitting region.

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With the advent of large-scale galaxy surveys with Spitzer, there is growing interest in calibrating the rest 8 μm mid-IR luminosities of galaxies as quantitative SFR tracers. Most galaxies exhibit strong emission in the rest 8 μm band, and this wavelength region redshifts into the Spitzer 24 μm band at z ∼ 2. In most galaxies, the primary emission mechanism at 8 μm is molecular band emission from aromatic "PAH" grain species, as distinct from the thermal grain emission that dominates in the FIR, so its reliability as a quantitative SFR tracer must be tested empirically. In massive galaxies with high SFRs, the 8 μm emission correlates reasonably well with the TIR emission sufficiently so that it (or neighboring regions observed with the Infrared Space Observatory (ISO)) has been calibrated as an IR SFR tracer of its own (e.g., Roussel et al. 2001; Boselli et al. 2004; Förster Schreiber et al. 2004; Wu et al. 2005; Farrah et al. 2007; Zhu et al. 2008). However, the dispersion in 8 μm luminosities at a fixed SFR tends to be higher than for longer wavelength tracers such as the 24 μm emission, with systematic dependences of the emissivity on metal abundance, and local radiation field strength and hardness (e.g., Madden 2000; Peeters et al. 2004; Engelbracht et al. 2005, 2008; Dale et al. 2005; Wu et al. 2006; Smith et al. 2007; C07). The large dispersion in PAH strength (more than an order of magnitude across all galaxy types and luminosities) means that the rest 8 μm luminosity of a galaxy provides a crude measure at best of its SFR. Can these effects be mitigated by constructing a composite Hα + 8 μm SFR index?

To carry out this test, we compiled 8 μm fluxes of the SINGS galaxies, both integrated fluxes and those with 20'' central apertures as described in Section 2. The emission in this bandpass is a composite of a (normally) dominant PAH band emission as well as an underlying continuum from dust and evolved stars. The pure dust emission at 8 μm was obtained by subtracting a scaled 3.6 μm emission from the measured 8 μm flux. A scale factor of 0.255 was used here, following C07. To provide a meaningful standard of comparison, we also compiled MIPS 24 μm fluxes for the same objects. Stellar continuum in the 24 μm band (apart from foreground stars, which were removed in the Dale et al. 2007 data) is negligible, so no attempt was made to remove it.

The top panels in Figure 15 show the correlation between 8 μm (left) and 24 μm luminosities and Balmer-corrected Hα luminosities for the sample of star-forming SINGS galaxies. Open circles show integrated measurements of the galaxies, while open squares show measurements of the central 20'' × 20'' star-forming regions. We have also overplotted in small dots the H ii regions from the SINGS sample measured by C07.

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The correlation between 24 μm and corrected Hα luminosities in the top right panel of Figure 15 is the same as seen earlier (Figure 6), but with a clear offset between the relations for entire galaxies and the H ii regions, which we will return to later. The scatter in the relation between 8 μm emission and Hα emission is much higher, confirming trends seen in the H ii regions by C07. Most of this difference reflects the presence of galaxies (and H ii regions) with 8 μm luminosities up to 30 times weaker than the main relation; most of these are metal-poor dwarf galaxies, which are already known to have strongly suppressed PAH emission. The top left panel of Figure 15 aptly illustrates the perils of applying the 8 μm emission of galaxies indiscriminately as a quantitative SFR tracer (cf. Smith et al. 2007; Engelbracht et al. 2008, and references therein).

The two bottom panels of Figure 15 show the results of constructing composite Hα + 8 μm and Hα + 24 μm estimates of the attenuation-corrected Hα luminosity, compared as before with the spectroscopically corrected Hα luminosities. For the 24 μm index, we used the same scaling factor a as for the integrated measurements earlier, to compare how well the central 20'' measurements and H ii region measurements are fitted by the integrated relation. For the 8 μm + Hα composite, we derived the value of a = 0.011 ±  0.001_r ±  0.003_s (integrated and central 20'' data combined). For 24 μm, the calibration from the SINGS and MK06 integrated measurements provides an excellent fit to the SINGS central aperture measurements as well, which is not surprising. What may be more surprising is the excellent consistency of the attenuation-corrected Hα luminosities derived from Hα + 8 μm with those derived from the Balmer decrement attenuation corrections. The metal-poor outliers in the upper left panel of Figure 15 join the main relation in the weighted sum of Hα + 8 μm (lower left panel), because these galaxies have very low reddening and very low 8 μm emission, so both measured quantities in the plot are essentially (and identically) the observed Hα luminosity.

Interestingly, the residuals using the 8 μm fluxes are lower than those derived using 24 μm fluxes, ±0.11 dex versus ±0.14 dex, respectively. This can be understood if the 8 μm luminosity is more tightly coupled to the TIR dust luminosities of galaxies than the 24 μm emission, as shown, for example, by Mattila et al. (1999), Haas et al. (2002), Boselli et al. (2004), and Bendo et al. (2008). Upcoming high angular resolution observations of the FIR continuum of the SINGS sample, currently planned with the Herschel Space Observatory, will allow us to test the coupling of the various IR emission components with higher spatial resolution and sensitivity.

In principle, our approach can be applied to estimate dust attenuation corrections for any optical emission line that is used as a SFR diagnostic. The most commonly applied visible-wavelength line, especially for observations of galaxies at intermediate redshift, is the [O ii]λ3727 forbidden line doublet. The chief advantage of this feature is that it is accessible to ground-based telescopes and CCD spectrometers out to redshifts z ∼ 1.7, whereas Hα redshifts beyond the optical window above z ∼ 0.5. As a result, [O ii]-based SFR estimates are available for tens of thousands of galaxies at z∼ 0.1–1.5 (e.g., Franzetti et al. 2007; Ly et al. 2007; Cooper et al. 2008).

The calibration and reliability of the [O ii] feature as a quantitative SFR tracer has been discussed by several authors (e.g., Hopkins et al. 2003; Kewley et al. 2004; Moustakas et al. 2006). Unlike the Balmer lines, the luminosity of the collisionally excited [O ii] doublet is not fundamentally coupled to the ionizing flux, so its accuracy is limited by excitation variations, which in turn are systematically correlated with the metal abundance and ionization of the gas. However, the typical variations in intrinsic [O ii]/Hα are of order a factor of 2 or less over a wide range of abundances and galaxy environments, so the index can be useful, especially in applications to large samples. Dust attenuation, however, is a much more severe problem, with typical attenuations in normal galaxies of nearly an order of magnitude, and large variations between objects. Kewley et al. (2004) and Moustakas et al. (2006) provide empirical schemes for correcting for this attenuation as functions of [O ii] luminosity and B-band luminosity, respectively, but these are crude approximations at best. Here, we investigate whether the combination of [O ii] and IR luminosities can provide more robust attenuation-corrected SFRs.

Our results are summarized in Figure 16. The upper left panel compares the observed [O ii] luminosities and reddening-corrected Hα luminosities for the MK06 sample (solid circles) and the inner 20'' × 20'' regions of the SINGS galaxies (open squares). The integrated measurements of the SINGS sample are not plotted because we do not have full-galaxy [O ii] luminosities for that sample. This shows the worst case of applying the [O ii] luminosity with no attenuation correction at all. By coincidence, the mean attenuation-corrected luminosity of [O ii] in the MK06 sample is nearly identical to that of Hα (〈[O ii]/Hα〉 = 0.98), so any difference in luminosities translates identically to a deficit in the estimated SFR. On average, the [O ii] luminosities are suppressed by about 0.6 dex, with a range (excluding the two outliers) of 0.0–1.7 dex (a factor 50 at worst).

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In the other three panels of Figure 16, we apply a weighted sum of [O ii] luminosity and TIR, 24 μm, and 8 μm luminosities. As before, we derived the coefficients a_λ from Equation (1) which best fit the mean relations in Figure 16. Combining observed [O ii] luminosities of galaxies with any of the three IR luminosities can provide a credible attenuation correction to the [O ii] luminosities, with dispersions that are comparable to their Hα + IR counterparts, when differences in sample are taken into account.

In Section 3.1, we pointed out that our linear combination method will be subject to modest systematic errors if the effective attenuation in the emission line of interest deviates significantly from the mean dust opacity of the stellar continuum radiation that heats the dust. In most normal galaxies, the mean emission-line attenuation at [O ii] is significantly higher than in the mean dust-heating stellar continuum (Calzetti 2001), so we might expect our residuals to show a mild dependence on attenuation, as illustrated theoretically in Figure 3. We examined the residuals for such an effect, and observe a qualitative trend in the expected sense, but we cannot reliably separate this from trends introduced by systematic variations in the intrinsic strength of [O ii] relative to the Balmer lines. However, there is other evidence for this effect from the fact that the best-fitting values of a in Figure 16 are about 40%–50% higher than the corresponding values for Hα, even though the intrinsic luminosities of [O ii] and Hα are nearly identical. This difference in fitting coefficients (see Table 4) is a direct result of the higher dust attenuation at [O ii].

All of the composite SFR methods described until now require flux measurements in at least one IR band. It is well known that star-forming galaxies show a tight, linear correlation between their FIR and radio continuum luminosities (e.g., Condon et al. 1991; Yun et al. 2001), so it should be possible to use the radio continuum emission in combination with optical and UV star formation tracers to derive attenuation-corrected SFRs. In this section, we extend our tests to 1.4 GHz radio continuum fluxes, to evaluate the reliability of such hybrid SFR measures.

The results of our comparison are shown in Figure 17. The top panel shows the relationship between observed 1.4 GHz (21 cm) radio continuum luminosity and Balmer-corrected Hα luminosity, while the bottom panel shows the correlation between a weighted sum of uncorrected Hα and radio luminosity with the same Balmer-corrected Hα luminosities. These can be compared directly to Figures 6 and 7, which show similar comparisons but with 24 μm and TIR luminosities, respectively. Qualitatively the radio continuum fluxes and composite SFR indices show that same behavior as we found for their IR counterparts. The 1.4 GHz radio luminosities by themselves show a nonlinear dependence on the attenuation-corrected Hα luminosities (Bell 2003), with a power-law slope of 1.28, even steeper than that seen in TIR luminosity (1.10) and 24 μm luminosity (1.19). However, combining the Hα and radio luminosities removes most of the nonlinearity and much of the dispersion, confirming that the radio luminosities can be used to correct the optical lines for attenuation as well. The mean dispersions around the fits (±0.10 dex for Hα + 1.4 GHz and ±0.12 dex for [O ii] + 1.4 GHz) are 11%–33% larger than the corresponding Hα + TIR and [O ii] + TIR indices, but again these dispersions are small when compared to the random and systematic errors in SFRs derived from any of the individual SFR tracers by themselves.

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It is interesting that the sense of the nonlinearity in the radio–FIR relation is in the opposite sense that one might naively expect. At 1.4 GHz, the radio continuum is dominated by nonthermal (synchrotron) emission, which presumably originates ultimately from supernova events; this interpretation forms the physical basis for using the nonthermal radio emission as a SFR tracer (e.g., Condon 1992 and references therein). If this scaling of radio continuum luminosity with the SFR strictly held over all types and luminosities of star-forming galaxies, then one would expect the correlation in the upper panel of Figure 17 to be strictly linear, while the slope of the radio versus TIR correlation shown in Figure 18 would be considerably shallower than a linear relation, because we already have seen that the dust emission systematically underestimates the SFR in low-luminosity, low-opacity galaxies. Instead, Figure 17 shows a nonlinear dependence of radio emission on attenuation-corrected SFR (also see Bell 2003), and the slope of the radio versus TIR correlation in Figure 18 is steeper than a linear relation. Apparently, the radio emissivity of galaxies declines even more steeply at low luminosity than even the dust emission, producing the residual nonlinearities in Figures 17 and 18. A discussion of the physical explanation for this result is beyond the scope of this paper. One possibility is a systematic change in the cosmic ray lifetimes and/or the magnetic field strengths of galaxies with changing mass and SFR. Another possibility is the increasing role of radio emission from a nuclear accretion disk in more massive galaxies. Whatever the explanation, our results demonstrate that the radio continuum luminosities of galaxies can be combined with emission-line luminosities to provide reasonable measurements of the attenuation-corrected SFRs.

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Throughout this analysis, we have compared the different SFR tracers referenced to the total attenuation-corrected Hα luminosity, rather than the SFR itself. We chose this convention to anchor our results firmly in terms of observable quantities, and to circumvent the additional systematic effects that enter into the conversion of Hα luminosities into SFRs. However, one can readily use the coefficients listed in Table 4 to construct SFR calibrations using these composite indicators. Apart from their dependence on the coefficients in Table 4, these absolute SFR calibrations also scale with the zero point of the reference SFR versus L(Hα) calibration. The latter is dependent on the assumed slope and mass limits of the initial mass function (IMF), and on the stellar synthesis models used in deriving this calibration. For convenience, we provide two such calibrations, one on the zero point of the widely applied calibration of K98 and the other using a more realistic "Kroupa" IMF (Kroupa & Weidner 2003), as used in the current version of the Starburst99 synthesis models (Leitherer et al. 1999).

The calibration of K98 assumed for simplicity a single Salpeter (1955) slope power law, with ξ(m) ∝ m^−α with α = 2.35 between 0.1 and 100 M_☉, where ξ(m) ≡ dN/dm is the number of stars with masses between m and m+dm. Ionizing luminosities for that calibration were taken from Kennicutt et al. (1994). For this zero point, the composite SFR calibrations take the form

$$\begin{equation} {\rm SFR [K98]} (M_\odot \,{\rm yr}^{-1}) = 7.9 \times 10^{-42} [L({\rm H\alpha })_{\rm obs} + a_{\lambda }\,L_{\lambda }] \ ({\rm erg\,s^{-1})}, \end{equation} \tag{ 11 }$$

where L(Hα)_obs is the observed Hα luminosity without correction for internal dust attenuation, a_λ is taken from Table 4 for the IR or radio luminosity of interest (8 μm, 24 um, TIR, or 1.4 GHz), and L_λ is the luminosity in the respective wavelength band. As an example, the calibration for the combination of Hα and 24 μm luminosities is

$$\begin{eqnarray} {\rm SFR [K98]} (M_\odot \,{\rm yr}^{-1}) &=& 7.9 \times 10^{-42} [L({\rm H\alpha })_{\rm obs}\nonumber\\ &&+ 0.020 L(24)] \ ({\rm erg\,s^{-1})}, \end{eqnarray} \tag{ 12 }$$

where L(24) ≡ λ L_λ at 24 μm (or 25 μm). For this example, the dust attenuation at Hα would be given by

$$\begin{equation} {A(\rm H\alpha) (mag)} = 2.5 \log \bigg[1 + {{0.020\,L(24)} \over {L({\rm H\alpha })_{\rm obs}}}\bigg]. \end{equation} \tag{ 13 }$$

The analogous calibration for indices using measurements of the [O ii]λ3727 doublet is

$$\begin{eqnarray} {\rm SFR [K98]} (M_\odot \,{\rm yr}^{-1}) &=& 8.1 \times 10^{-42} [L([{\rm O\,{\mathsc{ii}}}])_{\rm obs}\nonumber\\ && + a{^{\prime }_{\lambda }}\,L_{\lambda }] \ ({\rm erg\,s^{-1})}. \end{eqnarray} \tag{ 14 }$$

Here, we have designated the scaling coefficients a'_λ with a prime symbol to emphasize that these coefficients are different from those derived for Hα. Again, as an example, the corresponding calibration for combining [O ii] and 24 μm luminosities is

$$\begin{eqnarray} {\rm SFR [K98]} (M_\odot \,{\rm yr}^{-1}) &=& 8.1 \times 10^{-42} [L([{\rm O\,{\mathsc{ii}}}])_{\rm obs}\nonumber\\ && + 0.029 L(24)] \ ({\rm erg\,s^{-1})}. \end{eqnarray} \tag{ 15 }$$

The second IMF, which we refer to for convenience as a "Kroupa" IMF, has the slope α = 2.3 for stellar masses 0.5–100 M_☉, and a shallower slope α = 1.3 for the mass range 0.1–0.5 M_☉. The Kroupa IMF is more consistent with recent observations of the Galactic field IMF (e.g., Chabrier 2003; Kroupa & Weidner 2003). We also have recalibrated the ionization rate from this stellar population using version 5.1 of Starburst99.¹³ For these models, the zero point of the SFR is lower than for the K98 calibration by a factor of 1.44. As a result, the calibrations in Equations (11) and (14) become

$$\begin{eqnarray} {\rm SFR [Kroupa]} (M_\odot \,{\rm yr}^{-1}) &=& 5.5 \times 10^{-42} [L({\rm H\alpha })_{\rm obs}\nonumber\\ && + a_{\lambda }\,L_{\lambda }] \ ({\rm erg\,s^{-1})}, \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} {\rm SFR [Kroupa]} (M_\odot \,{\rm yr}^{-1}) &=& 5.6 \times 10^{-42} [L([{\rm O\,{\mathsc{ii}}}])_{\rm obs}\nonumber\\ && + a{^{\prime }_{\lambda }}\,L_{\lambda }] \ ({\rm erg\,s^{-1})}. \end{eqnarray} \tag{ 17 }$$

As with any empirical "toolbox" of SFR calibrations, these composite indicators have been calibrated over limited ranges of galaxy properties, SFRs, and physical environments, and it is important to understand both the range of observations beyond which the methods are untested, and any systematic errors that may have been built into the zero points of the methods. We refer readers to the review in K98 for a discussion of the assumptions and systematic uncertainties underlying the individual SFR tracers that have been considered in this paper. However, there are a few additional cautions and caveats that apply to this new set of composite indicators.

The parent SINGS and MK06 galaxy samples cover essentially the full range of galaxy types found in the local universe, ranging from virtually dust-free dwarf galaxies to ULIRGs and luminous AGN hosts. However, we needed to restrict our analysis to star-forming galaxies with high signal/noise multiwavelength data, and this vetting process narrowed the coverage of galaxy properties covered by these calibrations. In particular, our galaxies cover restricted ranges in Hα attenuation (0–2.5 mag) and corresponding observed L(TIR)/L(Hα) ratio (45–3150), as well as in underlying stellar population and age (D_n(4000) ⩽ 1.4), corresponding roughly to the transition between lenticular (S0) and spiral galaxies. The highest IR luminosities in our final sample were log L(TIR)/L_☉ ∼ 11.9, just below the threshold for a ULIRG, and the highest attenuation-corrected SFRs are ∼100 M_☉ yr⁻¹. This range readily encompasses all normal galaxies in the present-day universe and most star-forming galaxies out to redshifts z ∼ 1. However, it does not encompass the most luminous star-forming galaxies found at high redshift, or the most dust-obscured LIRGs or ULIRGs found in the present-day universe. At the other extreme, our methods can break down for early-type red galaxies with very low SFRs, where the dominant IR dust emission can arise from evolved stars (e.g., Cortese et al. 2008; B. D. Johnson et al. 2009, in preparation). Both red dusty galaxies and very dusty starburst galaxies will manifest themselves by very red colors in the visible and/or by anomalously large $L({\rm IR})/L(\rm {H\alpha })$ ratios, and application of the limits above can flag such potentially problematic cases. We still believe that these composite methods provide more robust SFRs than any of the single-wavelength methods, but one should attach larger systematic errors (up to a factor of 2) for systems that lie outside of the bounds of our calibrations.

As an additional cautionary note, we emphasize that the relations in Table 4 were derived from integrated measurements of entire galaxies, or of regions covering several square kiloparsecs in area in the centers of galaxies, and as such the SFR calibrations given here only apply in regions that encompass a representative sampling of the integrated light of a galaxy. In particular, the coefficients a differ significantly from those of individual H ii regions and H ii region complexes. When we compare the relation between Hα and 24 μm luminosities of SINGS H ii regions (C07) with that for the integrated measurements of galaxies, we observe a significant offset (see Figure 15), with a best-fitting value a₂₄ = 0.031 ± 0.006 for the H ii regions, compared to a₂₄ = 0.020 ± 0.001_r ± 0.005_s for the galaxies. This difference is almost certainly due to the different stellar age distributions in the H ii regions and the galaxies as a whole. The ionizing lifetimes of O-stars are nearly all less than 5 Myr, so in H ii regions the massive early-type stars dominate both the gas ionization and the heating of the dust that produces the 24 μm dust emission. On the other hand, when one measures entire galaxies that ionized gas emission is still dominated by massive O-stars, but the dust heating includes an additional component from stars older than 5 Myr.

We can estimate the importance of this change in stellar populations quantitatively, by comparing the evolution in ionizing UV, far-UV, and bolometric luminosities of star-forming populations with age. We used version 5.1 of the Starburst99 package of Leitherer et al. (1999) to trace the evolution of Hα luminosity, 1500 Å UV continuum luminosity, and bolometric luminosity for continuously star-forming populations with ages of 5 Myr (appropriate to an H ii region) and 0.1–1 Gyr (appropriate for galaxies, the ratios do not change significantly at larger ages). For either a Salpeter or Kroupa IMF as defined in Section 6.1, the ratio of bolometric luminosity to Hα luminosity increases by a factor of ∼2.0 between ages of 5 Myr and 100 Myr (and ∼2.5 when compared to age 1 Gyr). We expect the ratio of dust luminosity to ionizing luminosity to be less sensitive to these age differences, however, because the attenuation of a given population decreases with increasing stellar age (e.g., Zaritsky et al. 2002). If we apply the prescriptions of Charlot & Fall (2000) or Calzetti et al. (2000), we expect the H ii regions to experience ∼2–3 times the dust attenuation as the average starlight in a starburst population. Thus, the additional component of UV/bolometric luminosity in an older population should increase the dust emission (for a given fixed ionizing luminosity) by a factor 1.4–1.6, similar to the difference in 24 μm/Hα ratios we observe between the H ii regions and the galaxies.

This "older" dust emission component can be directly observed in our Spitzer 24 μm images of the SINGS galaxies. Figure 19 compares 24 μm and Hα images of M81, a nearby, well resolved spiral galaxy that clearly illustrates the different IR emission components. Virtually, all of the 24 μm point sources are associated with bright optical H ii regions (Prescott et al. 2007), which validates our association of the 24 μm emission with the dust attenuation in the H ii regions. However, upon closer examination a diffuse IR emission component can also be observed, which extends between the H ii regions and exhibits a more filamentary morphology, roughly tracing that of the HI gas and the diffuse 8 μm PAH emission (e.g., Gordon et al. 2004; Pérez-González et al. 2006). This diffuse component is the "IR cirrus" component of the dust emission and has been identified in many other galaxies observed with ISO and Spitzer (e.g., Hippelein et al. 2003; Hinz et al. 2004; Popescu et al. 2005). According to the measurements of Dale et al. (2007), this diffuse component contributes ∼50% of the total 24 μm emission in M81, similar to the average fraction in the SINGS sample as a whole. These observations tend to support the suggestion by Zhu et al. (2008) that the differences in the 24 μm versus Hα correlations between galaxies and H ii regions are due to the presence of diffuse IR emission.

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These results should serve as a stern warning that the composite SFR indicators calibrated in this paper, C07, and Zhu et al. (2008) cannot be applied to map the spatially resolved SFR in galaxies without the risk of introducing significant and possibly large systematic errors in the resulting SFR maps. Our results show that these methods can be applied reliably to individual H ii regions and to galaxies as a whole (albeit with different scaling factors between optical and IR emission). However, when observed at high spatial resolution galaxies show diffuse components of both Hα and IR emission which may be completely unassociated with any star formation at the same position. The diffuse IR component is especially problematic, because this dust may be only partially heated by young stars, or possibly not by young stars at all. As a result blind application of the relations in Table 4 to multiwavelength images of galaxies will tend to produce large regions with spuriously high "SFRs" where little or no star formation actually is taking place. Methods incorporating other information will be needed to extend this type of analysis to make reliable spatially resolved maps of star formation in galaxies.

We conclude by re-examining the possibility of systematic errors in our attenuation scales (Section 3.3), now informed by the results presented in the previous sections. The most important assumptions are the adoption of reference attenuation corrections based on Balmer decrements with a Galactic extinction curve and a foreground screen dust geometry, and the assumption of a constant intrinsic ratio of ionizing luminosity to bolometric luminosity.

One of the surprises in the results (to us) has been the relatively tightness of the relations between the attenuation-corrected Hα luminosities, surface brightnesses, and attenuations themselves derived from linear combinations of Hα and IR (or radio) luminosities, when compared to the same corrected quantities derived from Balmer decrement measurements. The agreement in the mean attenuations is not significant at all, of course, because we calibrated the scales to match, but the relatively low dispersions in the comparisons shown in Figures 6–13, along with the absence of significant nonlinearities suggests that any systematic effects from variations in dust geometry and stellar population variations are likely to be of secondary importance.

Although the consistency of our results is encouraging, it does not rule out the possibility of a systematic error that affects the entire attenuation and SFR scales, or a systematic error that scales smoothly with the SFR itself. Perhaps the most vulnerable assumption is the adoption of the stellar-absorption-corrected Balmer decrement in the integrated spectrum (using a foreground dust screen approximation) as the reference for calibrating our composite SFR and attenuation scales. Galaxies clearly display point-to-point variations in line-of-sight attenuation, with the integrated spectrum representing a flux-weighted average, including effects of scattered light. Although recent models suggest that the foreground dust screen approximation does not introduce significant systematic errors when applied to normal star-forming galaxies (Jonsson et al. 2009), it is important to place limits on the magnitude of any systematic errors in our calibrations and attenuation scales resulting from these simplifying assumptions.

Ideally, one would use as references measurements of ionizing fluxes of galaxies that are less susceptible to dust attenuation, such as the Paschen or Brackett emission lines, or the free–free thermal radio continuum, which scales directly with the ionizing flux. Unfortunately, such measurements are not available for large samples of galaxies, or contain such large uncertainties that they cannot be applied individually to galaxies. That is why we have chosen to base our analysis on Balmer decrement measurements. However, some of these measurements are available for subsets of our sample, and we can use these to estimate Hα attenuations and compare them to those we derived from the Balmer decrements.

Our first empirical test comes from Paα and Hα measurements for 29 of the SINGS galaxies in our sample (C07). The Paα observations were made with the Near-Infrared Camera and Multi-Object Spectrometer (NICMOS) on the Hubble Space Telescope, and are restricted to the central 50'' × 50'' regions of the galaxies (and a 144'' field for M51). We used the Paα fluxes to derive attenuation-corrected Hα luminosities, using the theoretical recombination ratio as described in C07, and using the observed ratio of Paα/Hα fluxes to correct for the (weak) attenuation in Paα (λ = 1.89 μm). We do not have optical spectra with matching coverage to these regions, but we can measure Hα and 24 μm fluxes over the same regions, and derive attenuation-corrected Hα luminosities, using Equation (1) with a = 0.020, as calibrated from our Balmer decrement measurements. Figure 20 compares the resulting Hα surface brightnesses (cf. Figure 9) using the two methods. There is a considerable scatter in the comparison, which probably reflects the measuring uncertainties in the Paα photometry. The two scales are in good agreement, with an average difference of 0.024 ± 0.036 dex (attenuation from Paα/Hα higher by 0.06 ± 0.09 mag) and no evidence for nonlinearity, over a full range of 3 mag in A(Hα).

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Estimated thermal fractions of radio continuum fluxes are available for 27 galaxies in our sample from the study of Niklas et al. (1997). The thermal radio component arises from free–free emission of thermal electrons in the ionized gas, and its brightness scales directly with the unattenuated Hα emission, with a weak dependence on electron temperature (Niklas et al. 1997)

$$\begin{eqnarray} S_{\rm th} ({\rm mJy}) &=& { \bigg({{{2.24 \times 10^9}\,S_{\rm H\alpha }} \over {{\rm erg\,s}^{-1}\,{\rm cm^{-2}}}}}\bigg) {\bigg({T_e \over K}\bigg) ^{0.42}}\bigg(\ln {\bigg({0.05 \over {\nu /{\rm GHz}}}\bigg)}\nonumber\\ && + 1.5\,\ln {(T_e/{\rm K})}\bigg). \end{eqnarray} \tag{ 18 }$$

Since the dust extinction in the radio is negligible the ratio of thermal radio to Hα flux provides a measure of the Hα attenuation that is unaffected by any geometric effects in the Balmer extinction. For most galaxies, the thermal component of the radio emission represents only a small fraction of the total emission at centimeter wavelengths, with the dominant contribution arising from nonthermal synchrotron emission. Niklas et al. (1997) obtained integrated radio fluxes at several (typically 6–7) wavelengths, in order to decompose the thermal and nonthermal components. The resulting thermal radio fractions and fluxes for individual galaxies have an average uncertainties of ±0.33 dex (0.83 mag), but this is sufficient to test for a systematic difference in Hα flux scales. Figure 21 shows a histogram of the differences between the Hα attenuations estimated from the radio/Hα ratios using Equation (18) under the assumption of T_e = 10⁴ K, and those derived from the Hα/Hβ ratios. The median radio continuum attenuation (more reliable than the mean given the large scatter in measurements) is 0.02 ± 0.13 mag lower than that estimated from the Balmer decrements. In view of the large uncertainty in the radio measurements, we cannot rule out a small systematic bias in the Balmer decrement measurements, but this test, as with the Paα measurements, appears to rule out a significant systematic error.

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For our final empirical test, we compared the attenuations estimated from the Balmer decrements with those which can be derived by comparing the resulting Hα-based SFRs with attenuation-corrected SFRs from UV and IR measurements. This test is less direct than those based on radio continuum and Paα measurements described above, because the intrinsic ratio of Hα and UV continuum luminosities of a galaxy is dependent on other factors including the star formation history and the slope and upper mass limit of the IMF. We address this in detail in Paper II, but apply a simple test here to place limits on any systematic errors in the Balmer decrement based attenuation measurements.

Several authors have published prescriptions for using the combination of UV and IR fluxes of galaxies to derive attenuation-corrected UV luminosities and SFRs, based on energy-balance arguments that are similar to those presented in this paper (e.g., Gordon et al. 2000; Bell 2003; Hirashita et al. 2003; Kong et al. 2004; Buat et al. 2005; Iglesias-Páramo et al. 2006; Cortese et al. 2008). We have applied the prescriptions of Kong et al. (2004) and Buat et al. (2005), which lie in the mid-range of published calibrations, to FUV and TIR luminosities of the SINGS and MK06 galaxies (Paper II), to derive attenuated-corrected FUV luminosities, and converted these to SFRs using the most recent Starburst99 models and the Kroupa IMF described in Section 6.1. These yield

$$\begin{equation} {\rm SFR ({\it M_\odot }\,{\rm yr}^{-1})} = 8.8 \times 10^{-29} L_\nu ({\rm erg\,s}^{-1}\,{\rm Hz}^{-1}). \end{equation} \tag{ 19 }$$

We then compared these SFRs to those derived from the observed (uncorrected) Hα luminosities, for the same synthesis models and IMF

$$\begin{equation} {\rm SFR ({\it M_\odot }\,{\rm yr}^{-1}}) = 5.5 \times 10^{-42} L({\rm H\alpha }) ({\rm erg\,s}^{-1}). \end{equation} \tag{ 20 }$$

The ratio of the attenuated-corrected FUV-based SFR to the uncorrected Hα-based SFR provides an indirect estimate of the attenuation in Hα. We find that these attenuation estimates are slightly higher than those derived from the Balmer decrements in the foreground screen approximation, by 0.08 mag in the median for the Kong et al. (2004) calibration and by 0.18 mag for the Buat et al. (2005) calibration. Since the average Hα extinction for the sample is 0.8 mag, the UV-based attenuation estimates are higher by 10% and 23%, respectively. Our own calibration in Paper II (which does not depend explicitly on the Balmer attenuation scale) yields attenuations that are close to those derived by Kong et al. (2004). As a check, one can also use the UV and IR fluxes to derive the FUV attenuation directly using the published calibrations by those authors, and apply the Calzetti (2001) attenuation law to estimate A_Hα from A_FUV. This yields Hα attenuations which are 0.10, 0.15, and 0.07 mag higher than the Balmer-derived attenuations, for the calibrations of Kong et al. (2004), Buat et al. (2005), and Paper II, respectively. These are very close to the values derived from a comparison of FUV and Hα-based SFRs above. Based on the dependence of these results on the calibration used, as well as the systematic dependences on stellar age mix and IMF, we conservatively estimate in the uncertainties in the comparison of UV+IR and Balmer-line attenuation estimates to be ±20%. Therefore, if we average the results based on Kong et al. (2004) and Buat et al. (2005), we find a difference in attenuations of 0.13 ± 0.16 mag.

Taken together, the Paα, thermal radio continuum, and UV+IR-based estimates of Hα attenuations give average values that are 0.06 ± 0.09 mag higher, 0.02 ± 0.13 mag lower, and 0.13 ± 0.16 mag higher than those derived from the Balmer decrements. This may hint at a slight tendency for the Balmer decrements to underestimate the actual Hα attenuations, but within the uncertainties the scales are consistent. It is also notable that other widely adopted schemes such as the modified Charlot–Fall attenuation law (Wild et al. 2007; da Cunha et al. 2008) produce attenuations that are within 0.1 mag on average of those derived here. These results rule out large systematic errors in our attenuation scale, with a very conservative upper limit of 20%.

The other important assumption underlying our analysis is the implicit adoption of a constant dust-heating stellar population for all of the galaxies in our sample. This is the same "IR cirrus" problem that has bedeviled efforts to calibrate the IR emission of galaxies as a quantitative SFR tracer (e.g., Lonsdale Persson & Helou 1987; Sauvage & Thuan 1992). We already have seen evidence for the effects of varying stellar population in the comparison of calibrations of Equation (1) between disk H ii regions and integrated measurements of galaxies (Section 6.2). The results suggest that although the coefficients a derived in this paper appear to apply over a wide range of normal galaxy populations, they may systematically underestimate the SFR by as much as 60% in systems that are dominated by young stars, such as emission-line starburst galaxies and possibly luminous and ultraluminous starburst galaxies with young dust-heating stellar populations.

This difference in calibrations helps to account for a curious inconsistency between our results and previous calibrations of SFRs for starbursts by K98 and other authors. The relationship from that paper

$$\begin{equation} {\rm SFR ({\it M}_\odot \,{\rm yr}^{-1})} = 4.5 \times 10^{-44} L_{\rm bol} ({\rm erg}\,{\rm s}^{-1}) \end{equation} \tag{ 21 }$$

yields SFRs that can be up to 2.2 times higher than those given by Equation (11) and the coefficients in Table 4, when applied to very dusty galaxies, where the TIR term in Equation (11) dominates. Most of this difference can be readily accounted for by the different assumptions underlying the two calibrations. The K98 relation is entirely theoretical, and it assumes dust heating by stars with ages less than 30 Myr, whereas the relations in this paper were calibrated empirically, and automatically incorporate any contributions to dust heating from stars older than 30 Myr. If one adopts instead a bolometric based SFR calibration using a dust-heating continuum that is appropriate for normal spiral galaxies (continuous star formation over 1–10 Gyr) the inconsistencies with the empirical calibrations in this paper largely disappear. Taken at face value, this would suggest that approximately half of the dust heating in the galaxies in our sample arises from stars older than 30 Myr (Cortese et al. 2008; B. D. Johnson et al. 2009, in preparation).

To investigate this quantitatively, we have applied the simple dust radiative transfer model of C07 to calculate approximately the expected TIR emission from a disk galaxy with a constant star formation history over the past 10 Gyr. The C07 models were parameterized in terms of the SFR per unit area, and we converted these approximately to luminosities by using the median area of the Hα disks of the SINGS galaxies (200 kpc²). The model relation is plotted with the observations in the right panel of Figure 22, for the fitted value a = 0.0024, while the left panel shows for comparison the expected relation if only stars younger than 30 Myr heated the dust. The consistency of the model with the fitted attenuation corrections suggests that any bias in the SFR scales from systematic errors in the Balmer-based attenuation corrections is small.

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Our focus on possible systematic effects should not obscure the immense improvements in the reliability of SFRs that are afforded by these composite tracers. Many current measurements of SFRs are based on observations at a single-wavelength region, with perhaps a crude statistical correction for dust attenuation. As shown in the upper panels of Figures 6–8, the random uncertainties in SFRs produced by these single-tracer methods is typically of order a factor of 2, and systematic errors can easily reach a factor of 10 or higher. The methods developed here reduce the random errors in the attenuation corrections from factors of a few to of order ±15%–30%, and systematic errors to of order ±10% in most galaxies. As discussed above, systematic errors of up to a factor of 2 may be present in the worst cases, but those errors are due to uncertainties in stellar populations in the galaxies, not in the dust attenuation scales. Other factors such as IMF variations may introduce larger errors in the SFR scales than discussed here, but at least one should be able to remove dust attenuation as the dominant source of uncertainty in the extragalactic SFR scale.