3D-HST EMISSION LINE GALAXIES AT z ∼ 2: DISCREPANCIES IN THE OPTICAL/UV STAR FORMATION RATES

To perform our analysis, we took advantage of the Skelton et al. (2014) photometric catalogs produced by the 3D-HST project (Brammer et al. 2012, GO-11600, 12177, and 12328). Skelton et al. (2014) homogeneously combined 147 distinct ground-based and space-based imaging data sets covering the wavelength range 0.3–8.0 μm in five well-observed legacy fields, including GOODS-N, GOODS-S, and COSMOS. In their analysis, Skelton et al. (2014) obtained and reduced HST/WFC3 images from both the CANDELS (Grogin et al. 2011) and 3D-HST (Brammer et al. 2012) surveys, and created a source catalog using SExtractor (Bertin & Arnouts 1996) on co-added F125W+F140W+F160W images. These catalogs, detection segmentation maps, point-spread functions (PSFs), and flux enclosed in PSF-matched apertures were then used to measure total flux densities in a wide variety of publicly available imaging data sets. For z ∼ 2 systems, these data offer unprecedented access to the rest-frame ultraviolet and allow precision measurements of the slope of each object's rest-frame UV continuum.

Our rest-frame optical emission-line measurements come from 3D-HST, a near-IR grism survey with the HST WFC3 camera. The 3D-HST primary observations with the G141 grism consist of R ∼ 130 slitless spectroscopy between 1.08 μm < λ < 1.68 μm over a 625 arcmin² region of sky, which includes ∼80% of the CANDELS footprint; when combined with accompanying direct images through the F140W filter of WFC3, these data provide full coverage of the rest-frame wavelengths 3700–5020 Å for all 1.90 < z < 2.35 galaxies with unobscured emission line fluxes brighter than 10⁻¹⁷ erg cm⁻² s⁻¹ (at 1σ) which corresponds to unobscured SFRs greater than ∼2 M_☉ yr⁻¹. Included in this wavelength range are the strong emission lines of [O ii] λ3727, [O iii] λλ4959, 5007, [Ne iii] λλ3869, 3960, and hydrogen (Hβ, Hγ and Hδ).

To reduce these data, we began with the pre-processed, calibrated "FLT" files in the HST Data Archive. These are the products of the automated reduction process calwf3,⁵ which uses the latest reference files to measure and subtract the bias, correct for non-linearity, flag saturated pixels, subtract the dark image, divide by the flat field, calculate the gain, and apply the flux conversion. This process was identical for both the direct and the grism images, with the exception of the flat fielding step: the grism data were flat fielded at a later stage using the aXe⁶ software and a master sky flat.

Each grism observation was accompanied by a shallow (∼200 s) F140W exposure, which served to define the position of each object's wavelength zeropoint and trace, and hence facilitate spectral calibrations and extraction. These images were combined using the standard procedures of MultiDrizzle (Fruchter et al. 2009), and then co-added with the deeper CANDELS F125W and F160W frames to match the detection image used in the photometric catalogs of Skelton et al. (2014). These data were processed by SExtractor to produce a master catalog of all objects containing more than five pixels above a 3σ per pixel detection threshold and having a total AB magnitude (Oke & Gunn 1983) brighter than 26. The positions of these sources were then transformed back to the coordinate system of the shallower F140W image to enable two-dimensional (2D) spectral extractions on the grism frames. Positional uncertainties from this process were ≲ 0.5 pixel in the F140W frame. The grism data were reduced using version 2.3 of the program aXe (Kümmel et al. 2009), in a manner similar to that described in the WFC3 Grism Cookbook.⁷ The task AXEPREP was used to subtract the master sky frame⁸ from each image; such a step is critical for the extraction of the faintest targets. We do note that the background of a grism image is variable over time and best fit using a full set of master sky images; however, as we are solely concerned with the detection and measurement of emission lines, large-scale variations in the continuum (of the order of ≲5% of the original background) are not a serious issue.

After subtracting the sky background, we began the process of extracting the 2D spectrum of every object in the master SExtractor catalog. This was done using AXECORE, which defines each source's extraction geometry, flat fields the region containing the spectral information, applies the wavelength calibration, and determines the contamination from overlapping spectra. Each object was traced with a variable aperture based on its size on the direct image (±4 times the projected width of the source in the direction perpendicular to the spectral trace). Objects present on multiple frames were processed by DRZPREP and AXEDRIZZLE, which rejected the cosmic rays, drizzled the data to a common system (Fruchter et al. 2009), and co-added the images into one higher signal-to-noise ratio 2D spectrum. Finally, the optimal extraction method discussed by Kümmel et al. (2009) was employed to create a one-dimensional (1D) spectrum for each object that includes flux density, error on the flux density, and a contamination fraction in units of flux density.

To conclude our extraction process, we created a Web site that combined the 2D grism images with the 1D extracted spectra in a visually effective format. This step was performed with the program aXe2web,⁹ which was used to convert an input catalog and the aXe output files into a summary of the full reduction. Each object was displayed on a separate row with its magnitude, (x, y) position, equatorial coordinates, direct image cutout, grism image cutout, and its 1D extracted spectrum in counts and flux. This Web site format provides an easy and efficient way to view a summary of the reductions, maintain quality control, and select subsamples of objects for science purposes.

The Hβ line luminosities were determined by fitting the continuum of each z ∼ 2 spectrum with a fourth-order polynomial, while simultaneously fitting Gaussians of a common width and redshift to the emission lines of [O ii] λ3727, [Ne iii] λ3869, Hγ, Hβ, [O iii] λ4959, and [O iii] λ5007. The fourth-order polynomial was used due to the possible presence of a 4000 Å break. However, we also fit first-order polynomials and Gaussians in small wavelength windows for [O ii], [Ne iii], Hγ, and the combination of Hβ and [O iii] due to their proximity. Both methods yielded consistent results. Example fits around Hβ and the [O iii] doublet are shown in Figure 1. These line fluxes were then increased by 5% to compensate for the fact that our grism extraction apertures (which were typically 2''–4'' in diameter) enclosed only 93% to 97% of the spectral flux for a point source.¹¹ The galaxies in our sample are not point sources but are small compared to the extraction aperture, with typical half-light radii of 0.25–050 (A. Hagen et al. 2014, in preparation), indicating that our correction for extraction aperture flux loss is appropriate. Finally, these total Hβ fluxes were converted to luminosity using the standard cosmology stated in the introduction.

Note that we do not correct for underlying stellar absorption, which can affect determinations of the SFR (Moustakas et al. 2006). In the local universe, typical corrections for Balmer absorption are ∼4 Å in equivalent width (EW), but this number is a function of both the stellar population's age and IMF (Groves et al. 2012). Moreover, as seen in Figure 3, 4 Å is relatively small compared to the measured EWs of our objects. Indeed, to verify that the effect is minor, we repeated all of our analyses while statistically adding 4 Å EW to each of our Hβ measurements. This has the effect of increasing all of our Hβ fluxes by an average of ∼10%, and increasing the significance of our findings.

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In addition to providing measurements of total luminosity, our Gaussian fits to [O ii], [Ne iii], Hβ, and [O iii] also allow the measurement of every system's gas phase metallicity, 12 + log (O/H). The details of this analysis can be found in H. Gebhardt et al. (2014, in preparation) including a catalog of the sources in this work, but in brief, we used the observed flux ratios and the polynomial relations in Table 4 of Maiolino et al. (2008) to estimate metallicity via the abundance-sensitive diagnostics of [Ne iii] to [O ii], [O iii] to [O ii], and R₂₃ (([O iii] + [O ii])/Hβ) (Zaritsky et al. 1994). These estimates should be relatively reliable, as they match the T_e "direct" methods, which are applicable to systems with 12 + log (O/H) < 8.35, to the photo-ionization models of Kewley & Dopita (2002), which are useful for 12 + log (O/H) > 8.35. Some of these measures are relatively insensitive to extinction, while others are highly affected by reddening. To account for this effect, we adopted a Calzetti et al. (2000) extinction curve and computed gas-phase metallicity likelihood functions for fixed E(B − V) = 0.2. The most likely system metallicity was adopted for our analysis. Some abundance indicators, such as R₂₃, are double-valued, and many of our likelihood curves have two local maxima. Fortunately, the use of the other diagnostics, such as [Ne iii] to [O ii] and [O iii] to [O ii], helped split this degeneracy, and usually led to the preference of one solution over the other. To verify that our fixed extinction value did not affect our results, we repeated our following analysis using all possible values of reddening (0 < E(B − V) < 2), and then marginalized over this uniform reddening distribution to derive the gas-phase metallicity likelihood function. The most likely system metallicities for all reddenings did not change our conclusions, only increased our metallicity error bars. The distribution of our best-fit metallicities is shown in on the right side of Figure 3.

To obtain the rest-frame UV flux densities of our sources, we used the photometric catalogs produced by Skelton et al. (2014). For star-forming populations, the wavelength range between 1250 Å < λ < 2600 Å samples the Rayleigh-Jeans portion of the hot stars' spectral energy distributions. Consequently, in the absence of reddening, the spectral slope across this region should be relatively constant, i.e.,

$$\begin{equation} F(\lambda) \propto \lambda ^\beta , \end{equation} \tag{ 1 }$$

where β ∼ −2.25 for systems which have been forming stars at a constant rate for more than ∼10⁸ yr (Calzetti 2001). Values of β larger than −2.25 can be attributed to the effects of internal extinction, and, if the law of Calzetti (2001) holds, A₁₆₀₀ ∼ 2.31 Δβ.

As Figure 4 illustrates, the photometry covering our program's fields is quite extensive (Skelton et al. 2014). In the COSMOS field, broadband and intermediate-band measurements constitute a set of ∼19 data points which can be fit for β and the observed flux density at 1600 Å. These bands include g, r, and i from the Canada–France–Hawaii Telescope (Erben et al. 2009; Hildebrandt et al. 2009), B_J, V_J, r+, i+, and 11 intermediate bands from Subaru (Taniguchi et al. 2007; Ilbert et al. 2009), and F606W from HST/ACS (Grogin et al. 2011; Koekemoer et al. 2011). In the GOODS-S field, there are ∼20 data points covering the wavelength range 1250 Å < λ_rest < 2600 Å. These include B, V, R_c, and I from the Wide Field Imager (WFI) 2.2 m (Erben et al. 2005; Hildebrandt et al. 2006), R from VLT/VIMOS (Nonino et al. 2009), 12 intermediate bands from Subaru (Cardamone et al. 2010), and F435W, F606W, and F775W from HST/ACS (Giavalisco et al. 2004; Grogin et al. 2011; Koekemoer et al. 2011). Photometry in the GOODS-N field is not nearly as comprehensive, but it does include ∼9 data points in the z ∼ 2 rest-frame UV. These include G and R_S from Keck/LRIS (Steidel et al. 2003), B_J, V_J, R, and i from Subaru (Capak et al. 2004), and F435W, F606W, and F775W from HST/ACS (Giavalisco et al. 2004; Grogin et al. 2011; Koekemoer et al. 2011). Since the PSF-matched apertures of the Skelton et al. (2014) photometric catalog are corrected for flux losses based on high-resolution imaging (HST/WFC3 F160W or F140W) at roughly the same wavelength as our grism observations, they serve as a good match to the total Hβ fluxes provided by our grism measurements.

While complications may arise if the reddening curves contain a Milky Way-type bump at ∼2175 Å, a careful examination of the COSMOS and GOODS-S photometry reveals no evidence for such a feature. This result is consistent with previous analyses, which have shown the bump to be less pronounced or non-existent in high EW objects such as those being studied (e.g., Kriek & Conroy 2013).

Strong emission lines may be excited by the ionizing photons of hot stars, shocks in the interstellar medium (ISM), and/or AGNs. In the local universe, diagnostic line ratios work quite well for discriminating between these mechanisms (e.g., Baldwin et al. 1981; Kewley et al. 2013), but at z ∼ 2, key lines such as Hα and [S ii] λλ6716, 6731 shift out of the range of the WFC3 grism. Fortunately, there are medium and deep X-ray data over all three of our program fields (Elvis et al. 2009; Alexander et al. 2003; Xue et al. 2011). At the redshifts considered here (1.90 < z < 2.35), the X-rays associated with normal star formation are well below the limits of these surveys (Lehmer et al. 2010). Consequently, any emission-line galaxy whose position lies coincident with an X-ray source is likely powered by an AGN.

To identify the AGN, we therefore cross-correlated our z ∼ 2 object catalog with the list of X-ray sources found in the COSMOS, GOODS-N, and GOODS-S regions. Only four of our emission-line galaxies lie within 25 of an X-ray source; these objects have been removed from our sample.

While we cannot exclude the possibility that low-luminosity AGNs are contributing flux to our survey, we can place limits on their effect. To do this, we converted the flux limits of the three X-ray surveys covering our fields (Elvis et al. 2009; Alexander et al. 2003; Xue et al. 2011) into a 2 KeV luminosity density (L_2 KeV) using a power-law index of 1.9 and the online conversion tool, PIMMS.¹² For AGNs, there is a strong correlation between L_2 KeV and the luminosity density at 2500 Å, L₂₅₀₀. We therefore used Equation (6) in Lusso et al. (2010) to convert the L_2 KeV limits into the maximum contribution "normal" AGNs make to L₂₅₀₀. We then assumed a power-law slope of α = −0.5 (i.e., L_ν∝ν^α; Vanden Berk et al. 2001) to convert L₂₅₀₀ to L₁₆₀₀ as well as L₄₈₆₁. With an extra assumption that for AGNs the Hβ EW is typically ∼100 Å (Binette et al. 1993), we were able to estimate the maximal contribution AGNs have to the observed UV and Hβ luminosities. This can be seen in Figure 5. For GOODS-N and GOODS-S, AGNs have very little effect on either luminosity measurement and can be neglected. For COSMOS, the maximal AGN contribution to the UV and Hβ luminosities is less than or roughly equal to the median observed values, and the ratio of the contribution to L_Hβ to L₁₆₀₀ is roughly what is expected from normal star-forming galaxies. After excluding the individual X-ray sources from our analysis, it is clear that the remaining AGNs have very little effect on the observed UV and Hβ luminosities, and thus do not change our following SFR analysis.

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Two of the most common tracers of star formation are the UV continuum and the hydrogen recombination lines (e.g., Hβ). Both quantities are sensitive to the existence of short-lived, massive stars, but their systematics are different. The ultraviolet continuum at 1600 Å measures the photospheric emission of stars with masses greater than a few M_☉, and hence the method records star formation over a timescale of ∼100 Myr. In contrast, the recombination lines of hydrogen are powered by photons shortward of 13.6 eV, with

$$\begin{equation} L_{{\rm H}\beta } = Q({\rm H}^0) \left(\frac{{\alpha }^{\rm eff}_{\rm H\beta }}{{\alpha }_{B}}\right) h{\nu }_{{\rm H}\beta } \approx 4.81 \times 10^{-13}\ Q({\rm H}^0)\ {\rm erg\ s^{-1}}, \end{equation} \tag{ 2 }$$

where α_B and ${\alpha }^{\rm eff}_{\rm H\beta }$ are the recombination coefficients for Case B and Hβ, respectively (Pengelly 1964; Osterbrock & Ferland 2006). This number is virtually independent of temperature, density, and metallicity; the luminosity of Hβ only depends on the production rate of ionizing photons (Q) and the assumption that the interstellar medium is optically thick in the Lyman continuum. Since the stars that produce these ionizing photons have higher masses (M ≳ 15 M_☉) and shorter lifetimes (t ≲ 10 Myr) than the stars traced by the rest-frame UV, the exact relationship between the two SFR indicators can be complicated. In particular, variables such as the IMF, the stellar metallicity, and the history of star formation can all affect the observed ratio of the indicators.

The most common transformations between luminosity and SFR are those given by Kennicutt (1998) and updated by Kennicutt & Evans (2012). These conversions, which were originally tabulated in Hao et al. (2011) and Murphy et al. (2011), are based on results from the STARBURST99 population synthesis code (Leitherer et al. 1999; Vázquez & Leitherer 2005), and assume a constant star formation history, solar metallicity, a Kroupa (2001) IMF, and a stellar population age of 10⁸ yr. Using these relations, along with the assumption of Case B recombination with an intrinsic Hα/Hβ ratio of 2.86 (Brocklehurst 1971; Osterbrock & Ferland 2006), a galaxy's SFR can be inferred from

$$\begin{equation} \log {\rm SFR}_{\rm H\beta } = \log L_{\rm H\beta } - 40.81 \,(M_{\odot }\, {\rm yr}^{-1}), \end{equation} \tag{ 3 }$$

$$\begin{equation} \log {\rm SFR}_{\rm UV} = \log L_{{\rm UV}} - 43.35 \,(M_{\odot }\, {\rm yr}^{-1}), \end{equation} \tag{ 4 }$$

where L_Hβ and L_UV represent the total luminosities of Hβ and the UV continuum, after correcting for interstellar extinction.

This last issue can be problematic. According to Calzetti (2001), in the local universe, the total extinction (in magnitudes) at 1600 Å is related to the slope of the UV continuum via

$$\begin{equation} A_{1600} = \kappa _{\beta } \times (\beta - {\beta }_0), \end{equation} \tag{ 5 }$$

where κ_β = 2.31 and β₀ = −2.25 for populations that have been forming stars for more than ∼10⁸ yr. The connection between β and the extinction of the Hβ emission line is more tenuous, but again from Calzetti (2001)

$$\begin{equation} A_{\rm H\beta } = \zeta _{{\rm H}\beta } \, A_{1600} \end{equation} \tag{ 6 }$$

with ζ_Hβ = 0.83. We begin by adopting these coefficients in our analysis, and then test for variations by examining the systematics of the inferred SFR.

Figure 6 compares the Hβ and UV SFRs for our sample of 260 1.90 < z < 2.35 galaxies in the COSMOS, GOODS-N, and GOODS-S regions. From the figure, it is clear that the relations summarized in Kennicutt & Evans (2012), which typically produce a one-to-one correspondence in the z ∼ 0 universe (at least for SFR ≳ 10⁻² M_☉ yr⁻¹; see Lee et al. 2009), do not work as well at z ∼ 2. On average, Hβ SFRs are ∼1.8 times higher than those derived from the flux of the rest-frame UV. Moreover, this effect cannot simply be attributed to extinction, as there is no correlation between the Hβ/UV SFR ratio and the value of β. To identify the source of the discrepancy, we need to examine the effects that various assumptions have on the derived SFRs.

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To model the systematics of the Balmer line and UV SFR indicators, we began with the assumptions that no ionizing photons are escaping into the intergalactic medium, and that dust in the H ii regions has, at most, a minor effect on the conversion of far-UV radiation to Hβ. The former assertion is probably quite good: at z ∼ 3, the escape fraction of Lyman continuum photons is certainly less than 20%, and probably below ∼5% (Chen et al. 2007; Iwata et al. 2009; Vanzella et al. 2010; Mostardi et al. 2013), and observations in the local universe suggest f_esc < 1% (Adams et al. 2011). The latter assumption is also justifiable. Enshrouding dust exists for only a short period of time before O stars evaporate or evacuate it (Lada & Lada 2003), and the dust that does survive likely affects the UV and the Lyman continuum in roughly equal proportions. In the metal-rich H ii regions of the Milky Way, ∼25% of far-UV photons may be absorbed by dust (see Table 3 of McKee & Williams 1997), but in the metal-poor systems of the z ∼ 2 universe, this fraction is likely to be much less. We therefore believe that this is not a large effect, and we can proceed to translate stellar emission into the observables of Hβ and L₁₆₀₀.

We examined the response of L_Hβ and L_UV to changes in stellar population by first adopting as the baseline model of Hao et al. (2011) and Murphy et al. (2011) (summarized in Kennicutt & Evans 2012) SFR calibration, which uses solar metallicity, a Kroupa (2001) IMF, an upper mass cutoff of 100 M_☉, and a constant star formation history. We then varied these parameters one at a time, using Version 6.0.4 of the STARBURST99 (Leitherer et al. 1999; Vázquez & Leitherer 2005; Leitherer et al. 2010) population synthesis code with its Padova isochrones. Figure 7 displays the results, which we discuss below. This procedure is similar and consistent with previous works, albeit for Hα to UV, which investigated the ratio produced by different IMFs (e.g., Meurer et al. 2009), stellar metallicities (e.g., Lee et al. 2009), and star formation histories (e.g., Sullivan et al. 2000).

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4.2.1. Initial Mass Function

4.2.2. Metallicity

The metallicity of a stellar population affects the ratio of Hβ to UV luminosity through the opacity and line blanketing in higher mass stars. The lower the metallicity, the bluer the stellar population and the higher the ratio of L_Hβ to L₁₆₀₀. This effect can be seen in the bottom left panel of Figure 7: as we change the metallicity of the population from 0.02 solar to twice solar, Hβ becomes enhanced relative to the UV continuum. As demonstrated by H. Gebhardt et al. (2014, in preparation), the gas-phase oxygen abundances for our sample of z ∼ 2 galaxies range between 7.1 < 12 + log (O/H) < 8.7 (i.e., 0.025 Z_☉ < Z < Z_☉ with the solar calibration of Asplund et al. 2009), and the median value of the sample is 12 + log (O/H) = 8.06 (Z ∼ 0.2 Z_☉). Thus, for star formation timescales larger than ∼100 Myr, we should expect a ∼30% increase in the L_Hβ/L₁₆₀₀ ratio compared to that given by Hao et al. (2011) and Murphy et al. (2011).

Figure 8 displays the observed ratio of Hβ to UV continuum luminosity as a function of β, with the galaxies color-coded by their best-fit gas-phase metallicity. An inspection of the figure suggests the existence of a strong positive correlation between galactic extinction (as measured by the slope of the UV continuum) and oxygen abundance. Indeed, a Spearman test confirms this trend, as it rejects the null hypothesis that the two variables are uncorrelated with 99.9999% confidence. Of course, a correlation between extinction and oxygen abundance makes sense, as the formation of dust should be tied to the presence of metals in the ISM (e.g., Garn & Best 2010; Reddy et al. 2010). However, one would also expect a strong anti-correlation between metallicity and L_Hβ/L₁₆₀₀. Since both the UV and Hβ are powered by the energy emitted from young stars, and the L_Hβ/L₁₆₀₀ ratio is sensitive to metallicity, one would expect the two parameters to vary inversely with each other. Indeed, the Spearman test rejects the null hypothesis with 99.8% confidence.

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4.2.3. Star Formation History

Since Hβ and the UV continuum are sensitive to different ranges of stellar mass, the age of the stellar population and the assumed star formation history are important for our analysis. In their calibration of SFR, Hao et al. (2011) and Murphy et al. (2011) assumed a constant SFR history over a timescale greater than 10⁸ Myr. To explore if this assumption is sufficient for our sample of galaxies, we examined the relationship between two proxies of specific star formation rate (sSFR): EW_Hβ and log L₁₆₀₀ − log L_1.45μm. The former should be proportional to the mass-to-light ratio at 4861 Å times the sSFR (see Equation (8) in Zeimann et al. 2013) while the latter weighs the newly formed population relative to the older, longer-lived stars. As shown in Figure 9, a constant star formation history over a suite of ages from 7 × 10⁶ yr to 10⁹ yr cover the range of extinction-corrected, rest-frame EW_Hβ as well as the extinction-corrected UV to IR color.

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In the literature, models with a bursty star formation history have been explored extensively in order to explain the scatter or offsets in Balmer line to UV SFR ratios (e.g., Sullivan et al. 2000; Iglesias-Páramo et al. 2004; Erb et al. 2006; Meurer et al. 2009). The scatter in Figure 6, however, can be explained simply by the combination of measurement error and the uncertainty in the extinction-correction. In other words, we need not invoke bursts to explain the diagram. Moreover, the offset may be more simply explained by a range of ages for a constant star formation history. The relation between Hβ and UV SFR asymptotes for ages larger than 10⁸ yr, while for ages younger than that the Hβ SFR will appear enhanced. Our selection method may preferentially select young or newly formed galaxies, some of which may have ages less than 10⁸ yr. If we equate our extinction-corrected, rest-frame EW_Hβ to an age using the models in Figure 9, we find that our sample selection biases the Hβ/UV SFR ratio by at most ∼10%.

Another consideration, as demonstrated by Maraston et al. (2010), is that galaxies at z ∼ 2 are better modeled with an increasing SFR history, and this has a small effect on the predicted value of L_Hβ/L₁₆₀₀. As illustrated in the bottom right panel of Figure 7, an exponentially increasing SFR with an e-folding timescale of τ = 300 Myr will, over the course of a Gyr, increase L_Hβ/L₁₆₀₀ by ∼20% over that of a constant SFR system. Similarly, if star formation has only recently ignited, Hβ will again be boosted relative to the UV continuum. Consequently, unless galaxies at z ∼ 2 already have declining SFR, the Hβ to UV ratio predicted by the Kennicutt & Evans (2012) calibration should be a lower bound to the true value.

As nearly all stars form in clusters, the massive stars responsible for Hβ and UV emission are initially co-located and enshrouded in high optical depth molecular clouds (Lada & Lada 2003). As the stellar population ages, the most massive stars, which are primarily responsible for Hβ emission, go supernovae and evacuate much of their surrounding interstellar material, leaving the longer-lived B stars relatively unobscured. Consequently, as noted many times in the literature (e.g., Charlot & Fall 2000), there can be a systematic difference between the extinction that affects Hβ and that which reddens the UV continuum. Moreover, this offset can be a function of age, star formation history, galactic orientation, and dust composition.

Charlot & Fall (2000) used a simple model of two separate environments, a birth cloud and a global ISM, to estimate the differential extinction seen by nebular emission and longer-lived stars. Meanwhile, Calzetti et al. (2000) inferred an empirical relation between stellar and nebular extinction using observations of eight nearby starburst systems. Both studies reached the same conclusion: in most systems, A_V for the gas should be roughly twice that of the stars.

For z ∼ 2 systems, the wavelength coverage of the 3D-HST survey does not extend out to Hα, and the wavelength separation between Hβ and the higher-order Balmer lines is insufficient to obtain a robust measure of extinction. However, we can measure the amount of extinction affecting the stars via the slope of the rest-frame UV continuum. STARBURST99 models confirm the results of Calzetti (2001) that stellar populations dominated by a roughly steady-state number of young stars will have values of β between −2.4 and −2.2, depending on the timescale for on-going star formation. Significantly, this number has little dependence on the IMF and stellar metallicity; any deviations from this intrinsic slope must either arise from dust attenuation or, to a smaller extent, the SFR history of the stellar population.

In fact, the relationship between the observed ratio of L_Hβ/L₁₆₀₀ and β can reveal more than just the extinction law. The data and axes of Figure 10 are identical to those of Figure 8, i.e., the figure plots the observed luminosities of Hβ relative to the UV continuum, uncorrected for extinction. The best-fit linear relation between this ratio and the slope of the UV continuum (as determined by unweighted least squares) is shown in red. The intercept of this line with β ∼ −2.3 (i.e., where reddening should be minimal) provides information about the parameters of the underlying stellar population. Conversely, the slope of the line, m, constrains the ratio of nebular to UV extinction through

$$\begin{equation} \log {(L_{\rm H\beta } / L_{{\rm UV}})} = \log {(L_{\rm H\beta } / L_{{\rm UV}})_{\rm int}} + m (\beta - \beta _0), \end{equation} \tag{ 7 }$$

where

$$\begin{equation} m = 0.4 \times (1 - \zeta _{{\rm H}\beta }) \kappa _{\beta }. \end{equation} \tag{ 8 }$$

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As is illustrated in the figure, the intercept of the line with β ∼ −2.3 is inconsistent with the SFR calibrations of Hao et al. (2011) and Murphy et al. (2011), as summarized in Kennicutt & Evans (2012), at the 99.9995% confidence level. More specifically, (L_Hβ/L_UV)_int is $1.84^{+0.17}_{-0.17}$ larger than in the local universe, where the uncertainties represent 68% confidence intervals. Conversely, the slope of the relation, m = 0.155 ± 0.043 is perfectly consistent with the value of 0.162 expected from a Calzetti (2001) extinction law. It therefore appears that at z ∼ 2, A_V for the gas is still approximately twice that of the stars.