THE STAR FORMATION IN RADIO SURVEY: GBT 33 GHz OBSERVATIONS OF NEARBY GALAXY NUCLEI AND EXTRANUCLEAR STAR-FORMING REGIONS

The SFRS sample comprises nuclear and extranuclear star-forming regions in 56 nearby galaxies (d < 30 Mpc) observed as part of the SINGS (Kennicutt et al. 2003) and KINGFISH (Kennicutt et al. 2011) legacy programs. Each of these nuclear and extranuclear star-forming complexes have ∼1' × 05 sized mid-IR (i.e., low resolution from 5 to 14 μm and high resolution from 10 to 37 μm) spectral mappings carried out by the Infrared Spectrograph instrument on board Spitzer, and 47'' × 47'' sized Herschel/PACS far-IR spectral mappings for a combination of the principal atomic ISM cooling lines of [O i]63 μm, [O iii]88 μm, [N ii]122,205 μm, and [C ii]158 μm. Two galaxies that are exceptions include NGC 5194 and NGC 2403; these galaxies were part of the SINGS sample, but are not formally included in KINGFISH. They were observed with Herschel as part of the Very Nearby Galaxy Survey (VNGS; PI: C. Wilson). Similarly, there are additional KINGFISH galaxies that were not part of SINGS, but have existing Spitzer data: NGC 5457 (M 101), IC 342, NGC 3077, and NGC 2146.

SINGS and KINGFISH galaxies were chosen to cover the full range of integrated properties and ISM conditions found in the local universe, spanning the full range in morphological types, a factor of ∼10⁵ in infrared (IR: 8–1000 μm) luminosity, a factor of ∼10³ in L_IR/L_opt, and a large range in SFR (≲ 10⁻³–10 M_☉ yr⁻¹). Similarly, spectroscopically targeted extranuclear sources included in SINGS and KINGFISH were selected to cover the full range of physical conditions and spectral characteristics found in (bright) IR sources in nearby galaxies, requiring optical and IR selections. Optically selected extranuclear regions were chosen to span a large range in physical properties including extinction-corrected production rate of ionizing photons [Q(H⁰) ∼ 10⁴⁹–10⁵² s⁻¹], metallicity (∼0.1–3 Z_☉), visual extinction (A_V ≲ 4 mag), radiation field intensity (100-fold range), ionizing stellar temperature (T_eff ∼ 3.5–5.5 × 10⁴ K), and local H₂/H i ratios (≲ 0.1– ≳ 10). A sub-sample of IR-selected extranuclear targets were chosen to span a range in f_ν(8 μm)/f_ν(24 μm) and f_Hα/f_ν(8 μm) ratios.

The total sample over the entire sky consists of 118 star-forming complexes (56 nuclei and 62 extranuclear regions), 103 of which (46 nuclei and 57 extranuclear regions; see Tables 1 and 2, respectively) are observable with the GBT (i.e., having δ > −30°). Galaxy morphologies, adopted distances, and optically defined nuclear types are given in Table 1. In Figure 1, we show the location of each target region on Spitzer 24 μm images; the corresponding circle diameters are 25'', which match the FWHM of our lowest resolution data (i.e., the beam of the GBT 33 GHz radio data) for the present multi-wavelength study.

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Observations in the Ka band (26–40 GHz) were taken using the CCB on the 100 m Robert C. Byrd GBT (Jewell & Prestage 2004) over a two year period spanning 2009 March through 2011 January. The CCB simultaneously measures the entire Ka bandwidth over four equally spaced frequency channels (i.e., 27.75, 31.25, 34.75, and 38.25 GHz) and synchronously reads out and demodulates the beam-switched signal to remove atmospheric fluctuation and/or gain variations. The FWHM of the GBT beam across the full Ka band was typically ≈25'' among our sets of observations (see Section 3.1). Given the range of distances to the sample galaxies, this projects to linear scales of 0.37–3.7 kpc. Reference beams used for sky subtraction are measured by nodding 13 away from the source, and their positions are identified on top of Spitzer 24 μm images in Figure 1.

Observations were made using an "On-the-Fly Nod" variant of a symmetric nodding procedure (i.e., double-differencing technique; Readhead et al. 1989) with data being collected continuously through an entire observation, including slews between beams. This procedure alternately places the source of interest in each of the two beams of the Ka-band receiver in an A/B/B/A pattern. Such a sequence is able to cancel means and gradients in the atmospheric or receiver noise with time. Each nodding cycle lasts ≈70 s: an on-source dwell time of 10 s for each of the four phases, along with 10 s spent on the initial position acquisition, and 10 s for the two slews between beams. Thus, ≈40 s is spent on source per nodding cycle. A detailed description on the performance of the CCB receiver, the data reduction pipeline, and error estimates is given in Mason et al. (2009). In addition to any systematic errors, we assign a calibration error of 10% to the flux density measurement at each frequency channel.

Our observing strategy was constructed to make the most efficient use of the telescope. Thus, given the large range in brightness among our targeted regions, we varied the time spent on source based on an estimate of the expected 33 GHz flux density using the Spitzer 24 μm maps. The number of nods spent on each source was chosen to reach a signal-to-noise ratio of ≈10, while never spending ≲5 minutes (i.e., ≈4 nods) nor ≳70 minutes (i.e., ≈60 nods) of total integration time on a given source. Naturally, not all nods were deemed usable given weather conditions and technical difficulties, and sources were revisited if possible. We list the number of nods used and taken, along with the corresponding time spent on source, for the nuclear and extranuclear regions in the Appendix, Tables 6 and 7 respectively.

Galaxy Evolution Explorer (GALEX) far-UV (FUV; 1528 Å) and near-UV (NUV; 2271 Å) data were taken from the GALEX archive (GR6) and will be included in the GALEX Large Galaxy Atlas (M. Seibert et al., in preparation). We also made use of star masks generated by the GALEX team to identify and remove foreground stars for the analysis. The angular resolutions of the FUV and NUV images are 425 and 525, respectively, while the calibration uncertainty for these data is ≈15% in both bands.

The Hα imaging used in the analysis is taken from Leroy et al. (2012), where details about the data quality and preparation (e.g., correction for [N ii] emission) can be found. Like the UV data, Hα images were corrected for foreground stars. The typical resolution of the Hα images is ≈2'', and the calibration uncertainty among these maps is taken to be ≈20%.

Archival Spitzer 24 μm data were largely taken from the SINGS and Local Volume Legacy (LVL) legacy programs, and have a calibration uncertainty of ≈5%. Details on the associated observation strategies and data reduction steps can be found in Dale et al. (2007) and Dale et al. (2009), respectively. Two galaxies, IC 342 and NGC 2146, were not a part of SINGS or LVL; their 24 μm imaging comes from Engelbracht et al. (2008).

We additionally made use of Herschel 70, 100, 160, and 250 μm data from KINGFISH (see Kennicutt et al. 2011). Observations with the PACS instrument (Poglitsch et al. 2010) were carried out at 70, 100, and 160 μm in the Scan-Map mode and reduced by the Scanamorphos data reduction pipeline, version 12.5 (Roussel 2012). The goal of the Scanamorphos reduction is to preserve low surface brightness emission. The 250 μm observations were made using the SPIRE instrument (Griffin et al. 2010) and reduced using the HIPE version spire-5.0.1894. For NGC 2403 and NGC 5194, we make use of Herschel 70, 160, and 250 μm imaging obtained as part of the VNGS program. Information on the observational strategy and data processing can be found in Bendo et al. (2012) for NGC 2403 and Mentuch Cooper et al. (2012) for NGC 5194.

Ancillary radio data at 1.365 and 1.697 GHz for 24 of the sample galaxies are available from the Westerbork Synthesis Radio Telescope SINGS survey (Braun et al. 2007). The intrinsic FWHM of the radio beams is approximately 11'' east–west by 11/sin δ'' north–south at 1.5 GHz and scales as wavelength, where δ is the source declination. For five galaxies (i.e., NGC 628, NGC 3627, NGC 4254, NGC 4321, and NGC 4569), the north–south axis of the WSRT-SINGS beam at 1.7 GHz is larger than median FWHM of the 33 GHz GBT beam (including the 1σ scatter), ranging between 365 and 44''. Similarly, for seven galaxies (i.e., NGC 628, NGC 3627, NGC 4254, NGC 4321, NGC 4569, and NGC 4725, and NGC 4826), the north–south axis of the WSRT-SINGS beam at 1.4 GHz is larger than the median +1σ scatter on the 33 GHz GBT beam, ranging between 29'' and 555. Since this will affect the accuracy of the matched photometry with the GBT 33 GHz data, we do not include these sources in the radio spectral index analysis. The flux density calibration of the radio maps is better than ≈5%.

Among all sets of observations, we measure a median beam width over all channels of 2507 with a dispersion of 31. However, given that we are targeting resolved sources, and the beam size varies over the full Ka band, we apply a correction to the flux densities at each frequency channel as if their beam were at the nominal 25''. We use the scaling factors given in Murphy et al. (2010), since the distance to NGC 6946 is close to the median distance among the entire SFRS sample. These scale factors were derived by computing the photometry for NGC 6946 on the 8.5 GHz map using the beam sizes for each frequency channel averaged over that entire observing session, and are 0.89, 0.95, 1.00, and 1.04 at 27.75, 31.25, 34.75, and 38.25 GHz, respectively. While the exact values of these corrections factors will change depending on source geometry/morphology, the final 33 GHz photometry, averaged over the entire band, should be robust. The combination of resolved sources and the changing beam size across the Ka band will complicate the interpretation of in-band radio spectral indices, and is therefore not presented.

While the unblocked aperture and active surface of the GBT mitigate the significance of sidelobes, we inspected calibration scans from each observing session to determine if there were instances of days exhibiting sidelobes with large amplitudes. The added power may cause an overestimation of the 33 GHz flux density. We find that sidelobes, when measurable, have an amplitude that is ≲2% of the beam peak, on average, over the entire 2 year observing campaign. At this level, our results should not be significantly affected by the presence of sidelobes given that Murphy et al. (2010) found negligible differences in their ancillary radio, submillimeter, and IR photometry using a model beam having sidelobes at 5% of the beam peak.

The averaged Ka-band flux densities, weighted by the errors from each channel over the full band, are given in Tables 1 and 2 along with uncertainties; the corresponding effective frequency is given for each source in the Appendix, Tables 6 and 7, and is ≈33 GHz for each position (i.e., a median of 32.92 GHz with a dispersion of 0.79 GHz, corresponding to a ≈2% scatter in flux densities assuming a spectral index near 33 GHz of ∼0.5). For sources not detected at the 3σ level, we list the 3σ upper limit. Flux densities of the individual channels, before applying any corrections, are given with 1σ errors in the Appendix, Tables 6 and 7, for the galaxy nuclei and extranuclear regions, respectively. We do not give upper limits for the individual channels, but rather list the actual measured values and estimated errors.

Since the reference beam throw is only 13, real signal in our reference positions is a concern; having reference positions on the galaxy may result in an underestimation of the 33 GHz flux density (see Figure 1). For NGC 6946, we note that the positions presented here correct those originally given in Murphy et al. (2010), however this slight correction does not impact their results. To quantitatively assess the severity of having real signal in the reference beam for our measurement, we estimate oversubtractions using the 24 μm data by comparing the flux densities measured at the reference positions compared to the on-source position. To correct for these losses, we add back to the 33 GHz flux densities the average flux densities measured among the reference positions using modeled 33 GHz maps by scaling the 24 μm data using Equation (2) of Murphy et al. (2006b). The minimum flux density of the reference beam positions for each target was then taken as the local sky and subtracted.

These corrections are typically small, having a median value of ≈2% among all sources. We conservatively apply corrections to only those 22 sources estimated to be missing >15% of the actual flux density based on the 24 μm photometry alone. The median correction to the 33 GHz flux density for these sources is ≈24%. The uncorrected 33 GHz flux densities are given in the final columns of Tables 1 and 2. We note that for the case of NGC 6946, using 8.5 GHz derived 33 GHz maps resulted in corrections that agreed to those based on the 24 μm derived values to within a few percent (Murphy et al. 2010).

The photometry was carried out on the ancillary data sets after matching their resolution, cropping each image to a common field of view, and re-gridding to a common pixel scale. To accurately match the photometry from these images to the GBT measurements, maps were convolved to the resolution of the GBT beam in the Ka band (i.e., 25'') following the image registration method of Aniano et al. (2011). Using the combination of all IR data, total IR emission and uncertainty maps were constructed using the models of Draine & Li (2007) as described in Aniano et al. (2012).

For each data set, the flux density at the position of each GBT pointing is the surface brightness multiplied by the beam solid angle. Uncertainties on the photometry are estimated as a combination of the calibration and beam-size uncertainties; we assign a 17% uncertainty to account for the dispersion in the beam areas among our sets of GBT observations as given in Section 3.1.

In the case of the UV and Hα photometry, we correct each region for Milky Way extinction using Schlegel et al. (1998) assuming A_V/E(B − V) = 3.1 and the modeled extinction curves of Weingartner & Draine (2001) and Draine (2003). The multi-wavelength photometry is given in Tables 3 and 4, along with the corresponding 1σ errors, for the nuclei and extranuclear regions, respectively.

For the non-AGN sources in the sample, we can estimate SFRs using the new GBT 33 GHz data. In this section, we present the calibrations given in Murphy et al. (2011), where more details about their derivations can be found. These calibrations, which update those found in Kennicutt (1998) and have been adopted by Kennicutt & Evans (2012), were calculated using Starburst99 (Leitherer et al. 1999) for a common initial mass function (IMF) so that each diagnostic can be compared fairly. A summary of these relations can be found in the Appendix, Table 8.

We choose a Kroupa (Kroupa 2001) IMF, having a slope of −1.3 for stellar masses between 0.1 and 0.5 M_☉ and −2.3 for stellar masses ranging between 0.5 and 100 M_☉. Assuming solar metallicity and a continuous, constant SFR over ∼100 Myr, Starburst99 stellar population models yield the following relation between the SFR and production rate of ionizing photons, Q(H⁰):

$$\begin{equation} \left(\frac{\rm SFR}{M_{\odot }\,{\rm yr^{-1}}}\right) = 7.29\times 10^{-54}\left[\frac{Q(H^{0})}{\rm s^{-1}}\right]. \end{equation} \tag{ 1 }$$

Since the ionizing flux comes from very massive stars with lifetimes ≲10 Myr, we note that the coefficient in Equation (1) is nearly independent of starburst age under the assumption of continuous star formation so long as it is ≳10 Myr. Accordingly, such measurements sample the current (i.e., ∼10 Myr) star formation activity. The integrated UV spectrum is also dominated by young stars, however, it is sensitive to a significantly longer timescale of recent (∼10–100 Myr; Kennicutt 1998; Calzetti et al. 2005; Salim et al. 2007) star formation activity. We convolve the output Starburst99 spectrum with the GALEX FUV transmission curve to obtain the following conversion between SFR and (extinction-corrected) FUV luminosity:

$$\begin{equation} \left(\frac{{\rm SFR}_{{\rm FUV}}}{M_{\odot }\,{\rm yr^{-1}}}\right) = 4.42\times 10^{-44}\left(\frac{L_{\rm FUV}}{\rm erg \,s^{-1}}\right). \end{equation} \tag{ 2 }$$

Similarly, we can write such an expression for the GALEX NUV band such that

$$\begin{equation} \left(\frac{{\rm SFR}_{{\rm NUV}}}{M_{\odot }\,{\rm yr^{-1}}}\right) = 7.15\times 10^{-44}\left(\frac{L_{\rm NUV}}{\rm erg \,s^{-1}}\right). \end{equation} \tag{ 3 }$$

The ionizing photon rate can of course be expressed as an (extinction-corrected) H recombination line flux, such that for Case B recombination, and assuming an electron temperature T_e = 10⁴ K, the Hα recombination line strength is related to the SFR by

$$\begin{equation} \left(\frac{\rm SFR_{H\alpha }}{M_{\odot } \,{\rm yr^{-1}}}\right) = 5.37\times 10^{-42}\left(\frac{L_{{\rm H}\alpha }}{\rm erg \,s^{-1}}\right). \end{equation} \tag{ 4 }$$

The above equation indicates that the SFR is directly proportional to the Hα line luminosity, assuming that a constant fraction (i.e., ≈45%) of the ionized H atoms will emit an Hα photon as they recombine, and that the extinction correction is accurate. However, if a significant fraction of ionizing photons are absorbed by dust, the above equation will underestimate the SFR, and the = sign should be replaced by ⩾.

Similarly, at high radio frequencies, where τ ≪ 1, the ionizing photon rate is directly proportional to the thermal spectral luminosity, L^T_ν, varying only weakly with electron temperature T_e (Rubin 1968), such that

$$\begin{eqnarray} \left[\frac{Q(H^{0})}{{\rm s}^{-1}}\right] &=& 6.3\times 10^{25} \left(\frac{T_{\rm e}}{10^{4}\,{\rm K}}\right)^{-0.45} \left(\frac{\nu }{\rm GHz}\right)^{0.1} \nonumber\\ &&\times \left(\frac{L_{\nu }^{\rm T}}{\rm erg \,s^{-1} \,Hz^{-1}}\right). \end{eqnarray} \tag{ 5 }$$

By combining Equations (1) and (5), one can derive a relation between the SFR and thermal radio emission:

$$\begin{eqnarray} \left(\frac{\rm SFR_{\nu }^{T}}{M_{\odot }\,{\rm yr^{-1}}}\right) &=& 4.6\times 10^{-28} \left(\frac{T_{\rm e}}{10^{4}\,{\rm K}}\right)^{-0.45} \left(\frac{\nu }{\rm GHz}\right)^{0.1} \nonumber\\ &&\times \left(\frac{L_{\nu }^{\rm T}}{\rm erg \,s^{-1} \,Hz^{-1}}\right). \end{eqnarray} \tag{ 6 }$$

As with the H recombination line fluxes, Q(H⁰), and consequently the SFR, may in fact be underestimated by the free–free emission if a significant fraction of ionizing photons are absorbed by dust; in this case the = sign in the above equation should be replaced by ⩾. However, it is worth noting that unlike free–free emission, which arises directly from the ionized gas, optical/NIR H recombination line fluxes may also suffer extinction internal to the H ii region itself, resulting in an even larger deficit. For example, the extinction internal to H ii regions in the starbursting dwarf galaxy NGC  5253 is measured to be quite large (A_V = 16–18 mag; Turner et al. 2003).

At lower radio frequencies, which are typically dominated by non-thermal synchrotron emission, calibrations between the supernova rate, and thus the SFR, have been developed. From the output of Starburst99, which assumed a supernova cutoff mass of 8 M_☉, we find that the total core-collapse supernova rate, $\dot{N}_{\rm SN}$, is related to the SFR by

$$\begin{equation} \left(\frac{\rm SFR}{M_{\odot }\,{\rm yr^{-1}}}\right) = 86.3 \left(\frac{\dot{N}_{\rm SN}}{\rm yr^{-1}}\right). \end{equation} \tag{ 7 }$$

Work comparing the non-thermal spectral luminosity with the supernova rate in the Galaxy has yielded an empirical calibration such that

$$\begin{equation} \left(\frac{L_{\nu }^{\rm NT}}{\rm erg \,s^{-1} \,Hz^{-1}}\right) = 1.3\times 10^{30} \left(\frac{\dot{N}_{\rm SN}}{\rm yr^{-1}}\right) \left(\frac{\nu }{\rm GHz}\right)^{-\alpha ^{\rm NT}}, \end{equation} \tag{ 8 }$$

where α^NT is the non-thermal radio spectral index (Tammann 1982; Condon & Yin 1990). By combining Equations (7) and (8), we can express the SFR as a function of the non-thermal radio emission, where

$$\begin{eqnarray} &&\left(\frac{\rm SFR_{\nu }^{NT}}{M_{\odot }\,{\rm yr^{-1}}}\right) =6.64\times 10^{-29} \left(\frac{\nu }{\rm GHz}\right)^{\alpha ^{\rm NT}}\left(\frac{L_{\nu }^{\rm NT}}{\rm erg \,s^{-1} \,Hz^{-1}}\right). \nonumber\\ \end{eqnarray} \tag{ 9 }$$

Since it will take ∼30 Myr for 8 M_☉ stars to go supernova, and the radiating lifetimes of CR electrons can be on the order of tens of Myr in normal galaxies, the non-thermal emission is sensitive to slightly longer timescales than free–free or H recombination line emission.

The observed radio continuum emission comprises both free–free and synchrotron emission. Therefore, we can combine Equations (6) and (9) to construct a single expression for the SFR from the total radio continuum emission at a given frequency such that

$$\begin{eqnarray} \left(\frac{\rm SFR_{\nu }}{M_{\odot }\,{\rm yr^{-1}}}\right) &=& 10^{-27} \left[2.18 \left(\frac{T_{\rm e}}{10^{4}\,{\rm K}}\right)^{0.45} \left(\frac{\nu }{\rm GHz}\right)^{-0.1}\right. \nonumber\\ &&+ \left.15.1 \left(\frac{\nu }{\rm GHz}\right)^{-\alpha ^{\rm NT}}\right]^{-1} \left(\frac{L_{\nu }}{\rm erg \,s^{-1} \,Hz^{-1}}\right). \nonumber\\ \end{eqnarray} \tag{ 10 }$$

This equation essentially weights the observed radio continuum luminosity based on the expected thermal fraction at a given frequency. As pointed out above, the thermal and non-thermal emission timescales are mismatched, with free–free emission being sensitive to massive stars with ages ≲10 Myr and the non-thermal emission being most sensitive to the lowest mass (i.e., ≳8 M_☉) supernova progenitors having lifetimes of ≲30 Myr. However, Equation (10) should hold under the assumption of continuous star formation on timescales ≳30 Myr.

In our analysis, we have assumed an electron temperature of T_e = 10⁴ K when calculating SFRs using Equations (6) and (10). However, variations in the actual electron temperatures will inject scatter into our observed trends. For instance, assuming a value of T_e as low as 5000 K will result in SFRs that are 37% and 21% larger than assuming T_e = 10⁴ K using Equations (6) and (10), respectively. Thus, we expect any scatter introduced by the assumption of a constant electron temperature to be smaller than these values.

In Figure 2, the radio spectral indices measured between 1.7 and 33 GHz are shown for all 53 sources having data at 1.7 GHz with a resolution that is equal to, or higher than, the GBT data. Active galactic nuclei (AGNs) are identified among the galaxy nuclei, typically having radio spectral indices that are steeper than the rest of the sources. Using the spectral index information over this rather long lever arm, we can estimate the thermal emission fraction at 33 GHz if we assume a fixed non-thermal spectral index. We assume a constant non-thermal radio spectral index of α^NT = 0.85 given that this was the average non-thermal spectral index found among the 10 star-forming regions studied in NGC 6946 by Murphy et al. (2011). Furthermore, this is very close to the value found by Niklas et al. (1997, i.e., α^NT = 0.83 with a scatter of $\sigma _{\alpha ^{\rm NT}} = 0.13$) for a sample of 74 nearby galaxies. We do not estimate the thermal fractions for galaxy nuclei containing AGNs since there is likely to be more variation in their intrinsic non-thermal slope. Following Klein et al. (1984), and assuming that the free–free emission is optically thin, we estimate the thermal fraction such that

$$\begin{equation} f_{\rm T}^{\nu _{1}} = \frac{\left(\frac{\nu _{2}}{\nu _{1}}\right)^{-\alpha } - \left(\frac{\nu _{2}}{\nu _{1}}\right)^{-\alpha ^{\rm NT}}}{\left(\frac{\nu _{2}}{\nu _{1}}\right)^{-0.1} - \left(\frac{\nu _{2}}{\nu _{1}}\right)^{-\alpha ^{\rm NT}}}, \end{equation} \tag{ 11 }$$

where, for our specific case, ν₁ = 33 GHz, ν₂ = 1.7 GHz, α is the observed radio spectral index between ν₁ and ν₂, and we fix the thermal radio spectral index to 0.1. We find a median 33 GHz thermal fraction of ≈76% among all sources, with a lower and upper quartile of 64% and 87%, respectively (see Figure 3). While this method has been shown to overestimate thermal fractions for star-forming regions in a high-resolution radio study of M 33 (Tabatabaei et al. 2007), those authors assumed a much steeper non-thermal index (i.e., α^NT = 1.0) than adopted here. Non-thermal spectral indices have been found to be significantly flatter than 1.0 around giant H ii regions based on comparisons between multifrequency radio and de-reddened Hα observations in NGC 6946 (Tabatabaei et al. 2012). Assuming α^NT = 1.0 would increase our thermal fraction estimates to 〈f^33 GHz_T〉 ≈ 85% with a dispersion of 15%.

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In Figure 4, we plot SFRs for all non-AGN sources having low-frequency radio data, allowing us to estimate the 33 GHz thermal fraction, in two ways. We first compute 33 GHz SFRs using only the thermal fraction at 33 GHz and Equation (6). In the top panel of Figure 4, these are plotted against 33 GHz SFRs using the total 33 GHz spectral luminosity with Equation (10), again assuming a non-thermal spectral index of α^NT = 0.85. There are six sources (i.e., NGC 3938 Enuc. 1, NGC 4631, NGC 4631 Enuc. 2, NGC 5194 Enuc. 8, NGC 5194 Enuc. 10, and NGC 5194 Enuc. 11) with SFR estimates that are discrepant by more than a factor of ∼2. This discrepancy may arise from our assumption for the non-thermal spectral index being too flat; these are the only sources shown in Figure 2 for which α^33 GHz_1.7 GHz > 0.70, and are also the only sources in Figure 3 having thermal fractions <40%. For the two observations toward NGC 4631, we might expect a steeper spectral index as the galaxy is observed to be edge-on, and has a very prominent non-thermal radio component. Observing through the disk of this source could lead to a significant amount of diffuse non-thermal emission in the beam, steepening the observed spectrum. However, for the remaining four sources, NGC 3938 Enuc. 1, NGC 5194 Enuc. 10, and NGC 5194 Enuc. 11, it is worth noting that the 33 GHz photometry required a correction for badly placed off-nod positions; underestimating the corrections will result in artificially steep spectral index estimates. In the top panel of Figure 4, we also plot a one-to-one line, showing that these two estimates are generally consistent with one another. The dispersion about an ordinary least-squares fit line is less than ≈0.05 dex (i.e., 12%).

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In the bottom panel of Figure 4, we plot the ratio of these two SFR estimates, illustrating the effect for different assumptions of α^NT. We allow α^NT to range between 0.6 and 1.05 (left to right) in increments of 0.05. Except for the six discrepant sources identified above, the two estimates appear generally consistent within errors when varying the assumption for the non-thermal radio spectral index. Given the extremely good agreement between these two estimates, which may not be terribly surprising given that the 33 GHz flux densities are generally dominated by thermal emission, SFRs estimated using Equation (10) are assumed as our reference value for comparisons throughout the paper. By using Equation (10), we are able to calculate SFRs for the entire sample (i.e., lower frequency radio data are not necessarily required). Before moving on, it is worth investigating how the variations in the physical size of the GBT beam may affect the SFR estimates given that Equation (10) does not take into account morphological differences between the (compact) free–free and (diffuse) non-thermal emission in galaxies.

4.1.1. Distance Effects

The distances to the galaxies being investigated vary by a factor of ∼10. Consequently, the physical areas our observations average over span a factor of ∼100. By averaging over larger pieces of each galaxy, more diffuse emission is introduced into each beam relative to additional emission from compact H ii regions and, for the 33 GHz observations, likely introduces more non-thermal emission. To investigate the consequence of this effect, we plot the 33 GHz thermal fraction estimates as a function of the projected diameter of the region being measured in Figure 3. We find large (i.e., ≈93⁺⁷_{− 10} %) thermal fractions for all sources having sizes ≲0.5 kpc, and a larger mix of thermal fractions for larger diameters. Excluding the six sources having thermal fractions ≲40%, as they may be artificially low (see above), there does appear to be a weak trend of decreasing thermal fraction with increasing projected area. However, it is hard to reliably quantify any trend given that there are so few sources with lower frequency radio data at the large distance end of the sample.

To look at the effect of distance in another way, we plot the 33 GHz SFRs, normalized by the projected area of the beam, versus distance in Figure 5 for all detected, non-AGN sources. We calculate SFRs using Equation (10), with α^NT = 0.85, since this expression takes both thermal and non-thermal emission components into account. If the SFR calibration is being affected by the inclusion of a substantial amount of non-thermal emission for the more distant galaxies, we would expect a trend of increasing SFR per unit area as the beam size increases. Perhaps there is a weak trend; the median SFR surface density at distances below and above 15 Mpc is ≈0.03 and ≈0.08 M_☉ yr⁻¹ kpc², respectively. However, the lack of a meaningful number of sources, especially extranuclear regions, at d > 15 Mpc, makes it difficult to quantify any trend. Thus, it appears that larger beams do not significantly affect our 33 GHz SFR estimates.

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Unlike the far-IR continuum emission, for which a non-negligible fraction may be powered by an older stellar population (e.g., Sauvage & Thuan 1992; Walterbos & Greenawalt 1996; Bendo et al. 2010), warm dust emission appears to be more tightly correlated to the current star formation activity in the disks of galaxies (e.g., Helou et al. 2004; Murphy et al. 2006a). Consequently, a number of calibrations relating the warm dust emission at 24 μm to SFRs have been introduced in the literature (e.g., Wu et al. 2005; Pérez-González et al. 2006; Alonso-Herrero et al. 2006; Calzetti et al. 2007; Relaño et al. 2007; Zhu et al. 2008; Rieke et al. 2009); a detailed comparison among each of these relations can be found in Calzetti et al. (2010).

In Figure 6, we plot 24 μm flux densities against corresponding (corrected) 33 GHz flux densities for the entire sample of 103 galaxy nuclei and extranuclear regions. Apart from the galaxy nuclei that are known to harbor AGNs, a number of which are radio loud at 33 GHz relative to the observed 24 μm emission, there is a fairly tight correlation between the 24 μm and 33 GHz flux densities. The only other clearly outlying source is NGC 1377, which has a large 24 μm flux and remains undetected at 33 GHz. This galaxy is thought to be experiencing a nascent starburst, for which it is within a few Myr of the onset of an intense star formation episode after being quiescent for at least ∼100 Myr (e.g., Roussel et al. 2003, 2006). This starburst, while heating the dust, has yet to produce detectable free–free emission, nor observable signatures of CRs. Overplotted in Figure 6 are both a one-to-one line, scaled by the median 24 μm to 33 GHz flux density ratio 〈f_ν(24 μm)/S_33 GHz〉 = 142, and an ordinary least-squares fit to all non-AGN sources detected at 33 GHz given by

$$\begin{equation} \left[\frac{f_{\nu }(24\,\mu {\rm m})}{\rm mJy}\right] = 140\left(\frac{S_{\rm 33\,GHz}}{\rm mJy}\right)^{0.94}, \end{equation} \tag{ 12 }$$

which are consistent within errors. The scatter about the ordinary least-squares fit is ∼0.26 dex.

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After converting the 24 μm flux densities into luminosities, we relate these to corresponding 33 GHz SFR estimates to come up with the following relation among all non-AGN detected sources:

$$\begin{equation} \left(\frac{\rm SFR_{24\,{\mu {\rm m}}}}{M_{\odot }\,{\rm yr^{-1}}}\right) = 3.1\times 10^{-38} \left[\frac{\nu L_{\nu }(24\,{\mu {\rm m}})}{\rm erg \,s^{-1}} \right]^{0.88}. \end{equation} \tag{ 13 }$$

The fitting coefficients are consistent with others in the literature (for a detailed compilation see Calzetti et al. 2010). In Figure 7, we plot the ratio of the 24 μm to 33 GHz SFR against the 33 GHz SFRs finding that the scatter about this relation is ≈0.25 dex. The square identifies the location of the anomalous dust detection in NGC 6946 (Murphy et al. 2010; Scaife et al. 2010), which may also explain why other sources lie significantly below the unity line.

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We might expect different relations between warm 24 μm dust emission per unit SFR among the extranuclear regions and nuclei due to additional old stellar population heating of grains by galaxy bulges. To investigate this, we can separate our sample, and measure the median ratio of the 24 μm spectral luminosity to 33 GHz SFR for the nuclei and extranuclear regions. The average ratios, along with the associated dispersions, are given in Table 5. Within the scatter, which is just under a factor of ∼2, the median ratios are consistent with one another, as well as with the median ratio for all non-AGN detected sources. Comparing the ratio of the total IR luminosity to 33 GHz SFR for nuclei, extranuclear regions, and all non-AGN detected sources, we again find that the median values are consistent. Accordingly, we can use these average ratios to write empirical relations between the 24 μm spectral luminosity and total IR luminosity with the SFR that should be reliable to within a factor of ∼2 such that

$$\begin{equation} \left(\frac{\rm SFR^{\rm Avg.}_{24\,{\mu {\rm m}}}}{M_{\odot }\,{\rm yr^{-1}}}\right) = 2.45\times 10^{-43} \left[\frac{\nu L_{\nu }(24\,{\mu {\rm m}})}{\rm erg \,s^{-1}} \right] \end{equation} \tag{ 14 }$$

and

$$\begin{equation} \left(\frac{\rm SFR^{\rm Avg.}_{\rm IR}}{M_{\odot }\,{\rm yr^{-1}}}\right) = 3.15\times 10^{-44} \left(\frac{L_{\rm IR} }{\rm erg \,s^{-1}} \right). \end{equation} \tag{ 15 }$$

We note that the coefficient equating the total IR luminosity with SFR derived in Murphy et al. (2011) is ≈23% larger than this empirically measured coefficient, and thus consistent given the nearly factor of ∼2 scatter.

Given that not all of the UV/optical photons will be absorbed and re-radiated by dust, a series of new empirical calibrations based on the linear combination of observed 24 μm (obscured star formation) and Hα (unobscured star formation) luminosities have been developed (e.g., Calzetti et al. 2007; Kennicutt et al. 2007, 2009; Zhu et al. 2008). These empirical star formation recipes usually take the form of

$$\begin{equation} \left(\frac{\rm SFR_{\rm mix}}{M_{\odot }\,{\rm yr^{-1}}}\right) = 5.37\times 10^{-42} \left[\frac{L_{\rm H\alpha } + \xi _{\rm 24}\nu L_{\nu }(24\,{\mu {\rm m}})}{\rm erg \,s^{-1}}\right], \end{equation} \tag{ 16 }$$

where the coefficient ξ₂₄, scaling the warm 24 μm dust emission, has been found to vary depending on the physical scale being investigated. For instance, on the scale of individual H ii regions, ξ₂₄ ≈ 0.031 (Calzetti et al. 2007), where as for whole galaxies ξ₂₄ ≈ 0.020 (Kennicutt et al. 2009). A luminosity dependence has also been found, with ξ₂₄ increasing with the 24 μm luminosity of a galaxy (Calzetti et al. 2010).

Combining the 33 GHz SFRs with the available Hα and 24 μm photometry, we can estimate ξ₂₄ for the star-forming (i.e., non-AGN) regions in our sample. These values are plotted in the top panel of Figure 8 against the 33 GHz SFRs, showing a large scatter. The median and dispersion are ξ₂₄ = 0.018 ± 0.004 and $\sigma _{\xi _{\rm 24}} = 0.031$, respectively. Thus, on ∼kpc scales, which likely average over many non-coeval H ii regions, the scaling coefficient appears to be similar to that for entire galaxies. It is also worth noting that the scatter is quite large, which most likely arises from both the large range in physical sizes being covered by the GBT beam among the entire sample, as well as the difficulty in matching the GBT and imaging photometry.

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Similarly, by combining the SFR estimates from the total IR and (observed) UV emission, one can account for the obscured and unobscured emission contributing to the total (bolometric) SFR, which we define as

$$\begin{equation} {{\rm SFR}_{{\rm tot}}} = {{\rm SFR}_{{\rm FUV}}} + {{\rm SFR}_{{\rm IR}}}. \end{equation} \tag{ 17 }$$

This diagnostic is often used to characterize SFRs from galaxies in both low- (e.g., Iglesias-Páramo et al. 2006; Buat et al. 2007, 2011) and high-z (e.g., Iglesias-Páramo et al. 2004; Elbaz et al. 2007; Daddi et al. 2007; Reddy et al. 2010) studies.

Using the above calibration for the FUV SFR (i.e., Equation (2)), this relation can be expressed as a linear combination of the UV and total IR emission such that,

$$\begin{equation} \left(\frac{\rm SFR_{\rm tot}}{M_{\odot }\,{\rm yr^{-1}}}\right) = 4.42\times 10^{-44} \left(\frac{L_{\rm FUV} + \xi _{\rm IR}L_{\rm IR}}{\rm erg \,s^{-1}} \right), \end{equation} \tag{ 18 }$$

where the coefficient ξ_IR, scaling the total IR emission, has been found to be ∼0.46 for normal star-forming galaxies (Hao et al. 2011) and as large as ∼0.6 for starburst galaxies (e.g., Meurer et al. 1999; Calzetti 2001). These empirical values are significantly smaller than the value of 0.88 reported in Murphy et al. (2011), which assumed that the entire Balmer continuum was absorbed and re-radiated in the IR for their derived IR SFR calibration of individual star-forming complexes.

Using our 33 GHz SFRs, we estimate ξ_IR for the star-forming regions in our sample and plot them in the bottom panel of Figure 8 against the 33 GHz SFRs. The median and dispersion are ξ_IR = 0.50 ± 0.05 and $\sigma _{\xi _{\rm IR}} = 0.47$, respectively. The scatter is even larger than that for ξ₂₄, which may arise from the fact that a calibration such as this depends on the range of ages among individual star formation sites in each star-forming complex covered by the GBT 33 GHz beam. Thus, we expect increasing uncertainties in such a calibration when applied to a single star-forming region depending on how well the IR and UV are measuring emission from recent star formation, not to mention the fact that the IR and UV are sensitive to star formation on different timescales (i.e., ∼10 Myr versus ∼10–100 Myr, respectively).

Using the 33 GHz SFRs as a reference, we have investigated a number of dust-sensitive SFR indicators used in the literature. In Section 4.2, we find a relation between the 33 GHz SFRs and 24 μm luminosity that is consistent with others in the literature. However, while similar, we do find significantly larger scatter. We believe that the increased scatter most likely arises due to a combination of several effects. First, properly matching the single-beam GBT photometry to that from the ancillary images of resolved sources is difficult, and the associated uncertainties will increase the dispersion of any trends. Additionally, the area covered by GBT beam covers a range of physical sizes in the sample, which will also inject scatter into any observed correlations. And, of course, the corrections applied in cases of potential over subtraction of sky due to reference nods landing on bright regions of the targeted galaxy add another source of scatter.

In looking at the relation between the 33 GHz SFRs and 24 μm spectral luminosities for galaxy nuclei and extranuclear regions independently, we find them to be consistent. This is also true for the relation between 33 GHz SFRs and total IR luminosities. Thus, on average, there does not appear to be excess IR emission per unit star formation for galaxy nuclei relative to the extranuclear star-forming complexes on the physical scales being investigated.

In Sections 4.3 and 4.4, we investigate two hybrid star formation diagnostics that attempt to account for the unobscured and obscured star formation components (i.e., Hα + 24 μm and UV+IR). While we find scaling coefficients for the 24 μm and total IR luminosities that are similar to those reported in the literature, albeit with a large scatter most likely due to the reasons mentioned above, it is worth noting that the values reported are similar to those found from measurements of entire galaxies (e.g., Kennicutt et al. 2009; Hao et al. 2011). This agreement most likely arises because of the large area covered by the GBT beam at the distances of the sample galaxies, being roughly ∼1 kpc, on average. Thus, on these physical scales, and by not including a local background subtraction, it appears that the star formation diagnostics presented here are applicable for globally integrated measurements of galaxies.

We summarize the final relations for each of these four dust-dependent SFR estimates, along with all theoretically motivated calibrations discussed in Section 3.3, in the Appendix, Table 8.

In Figure 9, we plot the ratio of the 33 GHz to total IR fluxes as a function of the radio spectral index measured between 1.7 and 33 GHz. There is a clear trend of increasing flux ratio with flatter spectral index. This trend does not seem to arise due to differences in the projected physical area of the 33 GHz beam among the sources (see Figure 10). In fact, this trend is the opposite of the expectation for distance effects: as the physical area subtended by the GBT beam increases, one would expect a larger contribution of diffuse non-thermal emission from a galaxy disk relative to additional free–free emission, thus resulting in a steeper spectral index and an increase in 33 GHz to IR flux ratios.

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The ordinary least-squares fit to the non-AGN sources (solid line) appears generally consistent with, albeit slightly flatter than, the expected trend assuming a fixed ratio of νS_1.7 GHz/F_IR; the dotted line is given by scaling the median ratio νS_1.7 GHz/F_IR by (33/1.7)^{1 − α}. By only having radio data at two well-spaced frequencies, it is not possible to distinguish whether the spectral flattening is being driven by an increase in the thermal fraction or flattening of the non-thermal radio spectral indices. If we are to assume that this trend is in fact dominated by an increase in the thermal fraction, we can use the scatter about the ordinary least-squares fit to the data to set the maximum dispersion for the non-thermal spectral indices among the non-AGN sources, which is found to be $\sigma _{\alpha ^{\rm NT}} \lesssim 0.13$. Interestingly, this is similar to the dispersion in global non-thermal spectral indices measured by Niklas et al. (1997; $\sigma _{\alpha ^{\rm NT}} = 0.13$).

Again, assuming that the trend is in fact driven by a change in thermal fraction among the sample, suggests that the non-thermal emission component may be quite similar among each star-forming region, with a nearly uniform CR electron injection spectrum and similar spectral steepening from associated energy losses (e.g., synchrotron, inverse Compton, bremsstrahlung, and ionization processes). More specifically, there must only be moderate variability in the ratio of synchrotron to total CR electron cooling processes among each of the star-forming complexes being investigated.

Cosmic microwave background experiments were the first to discover the presence of an "anomalous" dust-correlated emission component at frequencies between ∼10 and 90 GHz (e.g., Leitch et al. 1997). In the original GBT pilot study of NGC 6946, Murphy et al. (2010) identified an extranuclear star-forming complex that had a significant amount of excess emission at 33 GHz relative to what was expected by extrapolating from multifrequency measurements at lower frequencies (i.e., <10 GHz). While anomalous dust emission has been observed in excess of synchrotron and free–free components in Galactic H ii regions (e.g., Dickinson et al. 2009), this observation of excess 33 GHz emission toward a star-forming complex in NGC 6946 is identified as the first detection of anomalous dust emission outside of the Milky Way. Now, using the full sample data, we can look for other potential extragalactic anomalous dust-emitting candidates.

As stated above, Figure 9 shows a clear trend in which the ratio of 33 GHz to total IR fluxes increases as the radio spectral index measured between 1.7 and 33 GHz flattens. In Figure 9, we also show the location of NGC 6946 Enuc. 4, identified as containing anomalous dust emission (Murphy et al. 2010; Scaife et al. 2010) that accounts for ≈50% of the observed 33 GHz flux density. However, this source does not appear to be significantly discrepant from the expected trend in this plot. It therefore appears that identifying anomalous dust candidates with such coarse radio spectral resolution may not be possible, and a much finer sampling (i.e., better than a factor of two in frequency steps) is necessary.

Besides NGC 6946 Enuc. 4, there are a number of other sources that both show flat radio spectral indices with large 33 GHz to IR flux ratios; sources having flux ratios νS_33 GHz/F_IR > 4 × 10⁻⁶ and observed radio spectral indices α^33 GHz_1.7 GHz < 0.5 have been highlighted as potential anomalous dust candidates in Figure 9. However, a detailed investigation of these regions to confirm or deny the presence of anomalous dust emission is outside the scope of this paper, and really requires additional radio data at finely spaced frequencies much closer to 33 GHz (e.g., between ∼15 and 90 GHz) to confirm a peaked spectrum.

The archetypal nascent starburst NGC 1377, which is thought to be within a few Myr of the onset of an intense star formation episode after being quiescent for at least ∼100 Myr, remains undetected in the radio even at the depth of our 33 GHz observations. Given its measured flux density at 24 μm and the 3σ upper limit of 0.44 mJy at 33 GHz, we would have expected a ∼70σ detection. As has already been shown by Roussel et al. (2003), this non-detection in the radio most likely does not arise from the source being optically thick. Using the 33 GHz upper limit, we can set even more stringent constraints on this. Assuming an electron temperature of ∼10⁴ K and τ_33 GHz ∼ 1 sets the expected brightness temperature and emission measure of the source to be ∼6000 K and ∼5 × 10⁹ pc cm⁻⁶, respectively. The corresponding size of the source would therefore have to be <1 pc with an electron density of >7 × 10⁵ cm⁻³. These values are significantly more extreme than the size and electron density limits of <28 pc and >2700 cm⁻³ inferred by Roussel et al. (2003).

Given the apparent absence of AGN activity in NGC 1377, any free–free emission associated with ongoing star formation powering the bright dust continuum appears significantly suppressed. If the nascent starburst scenario suggested by Roussel et al. (2003) is true, the lack of free–free emission detected at 33 GHz could very well arise from having extremely young, deeply embedding star formation in which the dust is absorbing ≳95% of the ionizing photons. The absorption of such a high fraction of ionizing photons by dust would then lead to SFRs that are overestimated by a factor of ∼2 by using calibrations relying on dust emission.

Here, we present the more detailed results from the GBT observations, as well as summarize all SFR equations either utilized or empirically derived in the present analysis. For the GBT observations, we give the flux densities and errors reported from the four individual CCB ports, and provide details on how much time was spent on each source in Tables 6 and 7. We again note that the flux densities of the individual channels reported here do not include any corrections. Upper limits for the individual channels are not given, but rather actual measured values and estimated errors are given.

In Table 8, we provide a summary of the theoretical and empirically derived SFR calibrations from this paper. A thorough discussion of the theoretical derivations can be found in Section 3.3 and in Murphy et al. (2011), while details on the empirically derived, dust-dependent SFR estimates are given in Sections 4.2 and 4.3. We also provide the timescale for the emission of each SFR diagnostic to fade to 50% and 5% of its peak value after having had 100 Myr of continuous star formation come to a halt. These timescales, which are sensitive to the exact star formation history, were estimated using Starburst99 (Leitherer et al. 1999) following the same assumptions that went into deriving the corresponding theoretical relations as described in Section 3.3 and in the table caption. Additional considerations for the radiating times of CR electrons were included when calculating the fading time for the non-thermal radio continuum emission, the details of which can be found in Murphy (2009).