A SUBMILLIMETER PERSPECTIVE ON THE GOODS FIELDS. II. THE HIGH RADIO POWER POPULATION IN THE GOODS-N^{* † ‡ §}

The local 20 cm population may be roughly divided into primarily star-forming galaxies at low radio power and primarily active galactic nucleus (AGN) dominated galaxies at high radio power (e.g., Condon 1989; Sadler et al. 2002; Mauch & Sadler 2007; Best & Heckman 2012). Such separations by galaxy type are mostly based on spectroscopic identifications. However, as we move to higher redshifts, many of the radio sources become too faint for spectroscopic identifications. For example, in Barger et al. (2014), we found that an extended tail of microJansky radio sources with faint near-infrared (NIR) counterparts (${K}_{S}\gt 22.5$) make up ∼30% of the 20 cm population in the extremely deep (11.5 μJy at $5\sigma$) Karl G. Jansky Very Large Array (VLA) image (F. Owen 2017, in preparation) of the Chandra Deep Field-north (CDF-N; Alexander et al. 2003). To complicate matters, at high redshifts there are many more galaxies with extreme star formation rates (SFRs). Thus, the high radio power population begins to contain a significant fraction of star-forming galaxies, as well as continuing to be populated by numerous AGNs, resulting in a complex function with redshift of these competing processes (e.g., Cowie et al. 2004a).

In this paper, we develop a method to separate star-forming galaxies—galaxies where the SFRs inferred from the submillimeter observations are consistent with those inferred from the radio power—from AGNs in the high radio power population using a combination of ultradeep 20 cm and 850 μm observations of the CDF-N. Long-wavelength submillimeter observations provide a clean discrimination between AGNs, whose spectral energy distributions (SEDs) drop extremely rapidly above a rest-frame wavelength of about $100\,\mu {\rm{m}}$ (e.g., Mullaney et al. 2011), and galaxies dominated by star formation, whose graybody emission extends smoothly to the submillimeter. Both radio-quiet AGNs and star-forming galaxies can lie on the well-known FIR ($8\mbox{--}1000\,\mu {\rm{m}}$)-radio correlation (e.g., Condon 1992; Morić et al. 2010; Wong et al. 2016); however, because of their very different long-wavelength properties, they can be distinguished using the present technique. We then compare our classifications with Very Long Baseline Interferometry (VLBI) classifications and with the X-ray properties of the sources.

Although submillimeter observations provide a valuable means of picking out extreme star formers in the high radio power population, observing large areas to any significant depth in the submillimeter is very difficult. Thus, we conclude by looking for a simpler diagnostic for separating star-forming galaxies from AGNs based on the radio data alone, namely, the radio size.

Our primary science goals are (1) to look for evidence of a characteristic maximum SFR for galaxies, as proposed by Karim et al. (2013) and Barger et al. (2014), (2) to determine whether locally determined relations between X-ray luminosity and radio power for star-forming galaxies apply at high redshifts and high radio powers, (3) to find whether AGNs in the high-redshift, high radio power population lie on the FIR-radio correlation (de Jong et al. 1985; Helou et al. 1985), as has been observed locally (Condon 1992; Morić et al. 2010; Wong et al. 2016), and (4) to examine whether any AGNs that lie on the correlation are there because star formation in the host galaxy is driving the observed radio emission, as proposed by Condon et al. (2013).

In Section 2, we present the radio, submillimeter, X-ray, and spectroscopic and photometric redshift data sets that we use in our analysis. In Section 3, we develop our classification method using the ratio of submillimeter flux to radio power and we compare our classifications of the high radio power sources with those made using radio-excess measurements, VLBI, and X-rays. In Section 4, we measure the FIR luminosities for the high radio power sources and determine their locations on a FIR luminosity versus radio power plot. We then construct SEDs for the star-forming galaxies and for the AGNs separately and determine whether they are consistent with the idea that star formation in the host galaxies is responsible for putting some AGNs on the FIR-radio correlation. Finally, in Section 5, we explore whether the separation between star-forming galaxies and AGNs could be done from radio sizes alone.

We assume the Wilkinson Microwave Anisotropy Probe cosmology of H₀ = 70.5 km s⁻¹ Mpc⁻¹, ${{\rm{\Omega }}}_{{\rm{M}}}=0.27$, and ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.73$ (Larson et al. 2011) throughout.

2.1. Radio and Submillimeter

2.2. X-Ray

The X-ray data are the 2 Ms X-ray image of Alexander et al. (2003), which they aligned with the Richards (2000) radio image. Near the aim point, the X-ray catalog reaches a limiting flux of ${f}_{0.5-2\mathrm{keV}}\approx 1.5\times {10}^{-17}$ erg cm⁻² s⁻¹. Matching X-ray counterparts from the Alexander et al. (2003) catalogs to the radio sources is not critically dependent on the choice of match radius, as can be seen from Figure 7 of Alexander et al. (2003), which shows the positional offset between the X-ray and radio sources versus off-axis angle. Following Barger et al. (2007), we use a 15 search radius. Of the radio sources in the CDF-N sample, 142 have X-ray counterparts. (There are a further 138 X-ray sources in the region that do not have radio counterparts.) When a radio source does not have an X-ray counterpart in the catalog, we measure the X-ray counts at the radio position from the X-ray image using a $3^{\prime\prime}$ diameter and convert them to fluxes assuming a fixed photon index of ${\rm{\Gamma }}=1.8$. For all of the sources, we compute the rest-frame $2\mbox{--}8\,\mathrm{keV}$ luminosities, L_X, from the $0.5\mbox{--}2\,\mathrm{keV}$ fluxes with no absorption correction and ${\rm{\Gamma }}=1.8$ using

$$\begin{eqnarray}&&{L}_{{\rm{X}}}=4\pi {d}_{L}^{2}{f}_{0.5\mbox{--}2\mathrm{keV}}{((1+z)/4)}^{{\rm{\Gamma }}-2}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 1 }$$

We take ${L}_{{\rm{X}}}\gt {10}^{44}$ erg s⁻¹ as the threshold for a source to be classified as an X-ray quasar.

2.3. Redshifts

2.4. High Radio Power Sources

For all of the sources with either spectroscopic or photometric redshifts, we compute the rest-frame radio power using the equation

$$\begin{eqnarray}&&{P}_{20\mathrm{cm}}=4\pi {{d}_{L}}^{2}{f}_{20\mathrm{cm}}{10}^{-29}{(1+z)}^{\alpha -1}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1},\end{eqnarray} \tag{ 2 }$$

where d_L is the luminosity distance (cm) and f_{20 cm} is the 20 cm flux in units of μJy. This equation assumes ${S}_{\nu }\propto {\nu }^{-\alpha }$, where we adopt a radio spectral index of $\alpha =0.8$ (Condon 1992; Ibar et al. 2010). The choice of α may not be appropriate for AGNs and also may be problematic for high-redshift sources. However, it should be accurate enough for the present purposes.

There are 46 high radio power sources (defined here as ${P}_{20\mathrm{cm}}\gt {10}^{31}$ erg s⁻¹ Hz⁻¹) with spectroscopic or photometric redshifts in the 124 arcmin² area. All of these lie at $z\gt 0.8$. The lowest redshift is z = 0.847, and the highest redshift is z = 5.18 (see Figure 1). We return to the sources with no redshift identifications at the end of Section 3.

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In Figure 1, we show the distribution of redshifts versus radio power, distinguishing sources with submillimeter detections ($\gt 3\sigma ;$ red circles) from those without (black circles). Most of the sources lie in the redshift range $z=1\mbox{--}3$. The spectroscopic and photometric redshift measurements may introduce biases in the identifications. In particular, the highest redshift ($z\gt 4$) sources are all based on CO measurements; thus, all are submillimeter-detected sources. However, this does not affect our analysis, since we compare only identified sources, and such a comparison does not depend on the completeness of the sample.

2.5. The FIR-radio Correlation

The local FIR-radio correlation (de Jong et al. 1985; Helou et al. 1985) is well characterized by a linear fit over five orders of magnitude. It is parameterized by the quantity

$$\begin{eqnarray}&&q=\mathrm{log}\left(\displaystyle \frac{{L}_{8\mbox{--}1000\mu {\rm{m}}}}{3.75\times {10}^{12}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right)-\mathrm{log}\left(\displaystyle \frac{{P}_{20\mathrm{cm}}}{\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}}\right).\end{eqnarray} \tag{ 3 }$$

The average local value is $q=2.52\pm 0.01$ for an $8\mbox{--}1000\,\mu {\rm{m}}$ luminosity (Yun et al. 2001; this is after applying their correction of 1.5 from their measured $42.5\mbox{--}122.5\,\mu {\rm{m}}$ luminosity; see also Bell 2003). This is very similar to the values measured at high redshifts from submillimeter samples with accurate positions from submillimeter interferometric observations (e.g., average $q=2.51\pm 0.01$, Barger et al. 2012; median $q=2.56\pm 0.05$, Thomson et al. 2014).

Barger et al. (2014) noticed in their plot of 850 μm flux (their 850 μm rms error was $\lt 1.5$ mJy) versus radio power that submillimeter-detected ($\gt 4\sigma$) radio sources (submillimeter-bright) bifurcate from other radio sources (submillimeter-blank) above ${P}_{20\mathrm{cm}}\approx {10}^{31}$ erg s⁻¹ Hz⁻¹. Based on a small number (five) of such sources that also had high-resolution radio observations (Chapman et al. 2004; Muxlow et al. 2005; Momjian et al. 2010; Guidetti et al. 2013), Barger et al. found that the submillimeter-bright sources appeared to be extended and star-formation dominated, while the submillimeter-blank sources appeared to be compact.

Here we explore this observed bifurcation further using our deeper submillimeter data. In Figure 2, we plot 850 μm flux versus radio power for the high radio power sources with spectroscopic or photometric redshifts (black circles). Because the negative K-correction in the FIR/submillimeter closely offsets the dimming effects of distance for $z\gt 1$ (Blain et al. 2002; Casey et al. 2014), the observed-frame submillimeter flux is a proxy for FIR luminosity for star-forming galaxies. (Note that the precise conversion of submillimeter flux to FIR luminosity depends on the SED of the galaxy.) Thus, we can compare the submillimeter flux with the 20 cm power, which should also measure the SFR if it is dominated by diffuse synchrotron emission produced by relativistic electrons accelerated in supernovae remnants. We denote the sources that also have SMA detections with blue circles. Using Equation (3), we also plot submillimeter flux versus radio power (blue curve) assuming q = 2.52 and adopting the mean conversion

$$\begin{eqnarray}&&\mathrm{log}\,{L}_{8\mbox{--}1000\mu {\rm{m}}}\ (\mathrm{erg}\,{{\rm{s}}}^{-1})=\mathrm{log}\,{S}_{850\mu {\rm{m}}}(\mathrm{mJy})+45.60\pm 0.05\end{eqnarray} \tag{ 4 }$$

determined by Cowie et al. (2016; their Equation (3), which was based on 26 SCUBA-2 galaxies in the GOODS-Herschel (Elbaz et al. 2011) region with accurate positions and spectroscopic redshifts).

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The bifurcation between submillimeter-bright (star-forming galaxies) and submillimeter-blank (AGNs) is clear, with only a few sources lying in a region that may indicate they are composites (where the radio power has contributions from both star formation and AGN activity). We mark with enclosing red squares the X-ray quasars in the field, all of which lie on the submillimeter-blank track.

Before proceeding to a comparison of our classifications with radio-excess, VLBI, and X-ray classifications, it is important to note that our analysis does not include the radio sources without spectroscopic or photometric redshifts in the area. These sources are faint in the optical and NIR. Based on the K–z relation, they are likely to lie at high redshifts (Barger et al. 2014). The higher redshifts (which shifts emission lines out of the observable spectral range) and faintness make them hard to identify with optical and NIR spectroscopy. However, they appear to contain a similar fraction of submillimeter galaxies as the high radio power sources with redshifts; 34 ± 9% of those without redshifts have 850 μm detections above a $3\sigma$ threshold, compared to 46 ± 9% of those with redshifts and a radio power above 10³¹ erg s⁻¹ Hz⁻¹. Thus, while the unidentified sources may comprise the higher redshift tail of the high radio power sample, the submillimeter detections suggest they may be similar to the identified sources in other regards.

3.1. Radio Excess

3.2. VLBI

Next, we compare our classifications with those made from VLBI 20 cm observations. First, in Figure 3, we show the information presented in Figure 2 (but restricted to sources that match our high radio power definition of ${P}_{20\mathrm{cm}}\,\gt {10}^{31}$ erg s⁻¹ Hz⁻¹) as a ratio plot, which allows us to illustrate the full dynamic range of the data. We plot the ratio of $850\,\mu {\rm{m}}$ flux to 20 cm power versus 20 cm power. We show the radio sources for which the submillimeter flux measurement at the radio position has a signal-to-noise ratio (S/N) $\geqslant \,2\sigma$ as circles with $1\sigma$ uncertainties, and we show S/N $\lt \,2\sigma$ as circles at the $2\sigma$ limit with downward pointing arrows. We use colors to denote the sources classified as either compact (Chi et al. 2013; cyan) or extended (Momjian et al. 2010; red) from VLBI data. Finally, we denote with green shading the factor of two range found by Barger et al. (2014) over which we expect the SFRs derived from the submillimeter fluxes to be consistent with the SFRs derived from the radio power, assuming the FIR-radio correlation with the radio emission dominantly powered by star formation. This multiplicative factor of two is the systematic error in the individual SFRs determined from the submillimeter fluxes based on the variations in their SEDs (see also Cowie et al. 2016).

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All of the radio sources within the green shaded region have S/N $\geqslant \,2\sigma$ submillimeter flux measurements, while only a handful of the radio sources that lie below the green shaded region do. The sources lying below the green shaded region have submillimeter flux measurements or upper limits that are well below what would be expected if the radio power were produced by star formation. Thus, the radio power in these sources must be dominated by AGN contributions. All of the VLBI compact sources lie below the green shaded region, while the one VLBI extended source lies within the green shaded region.

Perhaps most striking, above ${P}_{20\mathrm{cm}}\approx 5\times {10}^{31}$erg s⁻¹ Hz⁻¹ there are no sources in the green shaded region, which is consistent with the claim by Karim et al. (2013) and Barger et al. (2014) that there is a characteristic maximum SFR for star-forming galaxies above which the number of galaxies drops extremely rapidly. We note that although more luminous star-forming galaxies, such as the submillimeter galaxy GN20 (Pope et al. 2005; Daddi et al. 2009b), which lies just outside the 124 arcmin² area studied here, do exist, they are rare. GN20, which has ${P}_{20\mathrm{cm}}=7.5\times {10}^{31}$ erg s⁻¹ Hz⁻¹, is one of only two sources with 850 μm fluxes above 15 mJy in the more extended 400 arcmin² region of the CDF-N covered by the SCUBA-2 image to an rms of 1.5 mJy (Cowie et al. 2016). Based on the submillimeter number densities, the surface density of these more luminous star-forming galaxies with ${P}_{20\mathrm{cm}}\gt 5\times {10}^{31}$ erg s⁻¹ Hz⁻¹ is about 0.01 times the surface density of star-forming galaxies with ${P}_{20\mathrm{cm}}=(1\mbox{--}5)\times {10}^{31}$ erg s⁻¹ Hz⁻¹ (the $\pm 1\sigma$ range is 0.006 to 0.037).

Hereafter, we classify the sources that lie inside the green shaded region as star-forming galaxies, the small number of sources that lie below that region but have $\geqslant 3\sigma$ submillimeter detections as composites, and the remaining sources as AGN dominated.

3.3. X-Ray

We now compare our classifications with the X-ray properties of the sample. In Figure 4, we show rest-frame $2\mbox{--}8\,\mathrm{keV}$ luminosity (computed from the observed-frame $0.5\mbox{--}2\,\mathrm{keV}$ flux, as described in Section 2.2, to take advantage of the higher sensitivity in this energy band) versus the 20 cm power for the radio sources with spectroscopic or photometric redshifts and ${P}_{20\mathrm{cm}}\gt 1.4\times {10}^{31}$ erg s⁻¹ Hz⁻¹. The use of the $0.5\mbox{--}2\,\mathrm{keV}$ flux to compute the $2\mbox{--}8\,\mathrm{keV}$ luminosity also minimizes any uncertainty in the K-corrections computed from the X-ray spectral slope with the relation being exact at z = 3. We chose this radio power threshold to provide a clean separation in Figure 3 between star-forming galaxies and AGNs.

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At lower redshifts ($z\lt 1.3$) and lower radio powers, Mineo et al. (2014) compared X-ray luminosity with radio power for a sample of galaxies from the CDF fields that they considered to be star-formation dominated based on their colors and morphologies. In Figure 4(a), we show their CDF-N sample with ${P}_{20\mathrm{cm}}\gt {10}^{30}$ erg s⁻¹ Hz⁻¹ (blue squares) after converting their $0.5\mbox{--}8\,\mathrm{keV}$ luminosities to $2\mbox{--}8\,\mathrm{keV}$ luminosities using a factor of 0.56. We also show their measurements for a sample of local ULIRGs (golden triangles). Mineo et al. fitted their data over four orders of magnitude in radio power with the linear relation

$$\begin{eqnarray}&&{L}_{2\mbox{--}8\mathrm{keV}}\ (\mathrm{erg}\,{{\rm{s}}}^{-1})=1.2\pm 0.1\times {10}^{11}{P}_{20\mathrm{cm}}\,(\mathrm{erg}\,{\mathrm{Hz}}^{-1})\end{eqnarray} \tag{ 5 }$$

(black line in Figure 4). Lehmer et al. (2016) discuss the dependence of the Mineo et al. relation on the specific SFRs, but for very high specific SFRs, they reproduce the Mineo et al. equation, which should be appropriate for the present objects.

The X-ray luminosities of the star-forming galaxies in Figure 4(a) are consistent with an extrapolation of the Mineo et al. (2014) equation to higher radio powers. The mean $2\mbox{--}8\,\mathrm{keV}$ luminosity to radio power ratio of our star-forming galaxies is $1.22\pm 0.43\times {10}^{11}$ Hz, where we have computed the error using the jackknife resampling statistic. This is almost identical to the Mineo et al. normalization, extending the applicability of their relation to galaxies with SFRs $\gt 1000\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. The extremely high SFRs of the present sample result in some of the X-ray luminosities being even greater than the ${L}_{2\mbox{--}8\mathrm{keV}}={10}^{42}$ erg s⁻¹ value often used to identify sources as clear AGNs (e.g., Hornschemeier et al. 2001; Barger et al. 2002; Bauer et al. 2004; Szokoly et al. 2004) based on maximal local star-forming galaxy X-ray luminosities. Laird et al. (2010) similarly concluded that very high SFRs were responsible for the high X-ray luminosities of most of the sources in their submillimeter-selected sample in the same field.

The AGN-classified sources in Figure 4(b) show a much wider spread in ${L}_{2\mbox{--}8\mathrm{keV}}$, ranging from near quasar X-ray luminosities to nearly X-ray undetected sources. However, there are a number of sources with luminosities close to those of the star-forming galaxies, and there is clearly not a one-to-one correlation between our classifications and X-ray luminosity.

Radio power is often used along with the FIR-radio correlation as a measure of the SFR (e.g., Cram et al. 1998; Hopkins et al. 2003; Netzer et al. 2007; Mineo et al. 2014; Mushotzky et al. 2014). However, this could be risky when no other information is available on whether a source is AGN dominated or star-formation dominated. After all, it is well known that some local radio-quiet AGNs lie on the FIR-radio correlation (e.g., Condon 1992; Morić et al. 2010; Wong et al. 2016), even if it is not understood why.

In Figure 5, we plot (a) $8\mbox{--}1000\,\mu {\rm{m}}$ luminosity and (b) $42.5\mbox{--}122.5\,\mu {\rm{m}}$ luminosity versus radio power for the radio sources with spectroscopic or photometric redshifts and ${P}_{20\mathrm{cm}}\gt 1.4\times {10}^{31}$ erg s⁻¹ Hz⁻¹. We use colors to denote the sources classified as star-forming galaxies (green circles), composites (blue circles), and AGNs (red circles). In (a), we also mark the sources that Chi et al. (2013) found to be AGNs using VLBI observations with surrounding black circles. All of these sources lie below the FIR-radio correlation. The star-forming galaxies in (a) have a mean $q=2.42\pm 0.06$, where we have computed the error with the jackknife resampling statistic; all lie within a multiplicative factor of three of the median value (q = 2.35), which we denote by shading. Five AGNs and four composites also lie in the region.

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In Figure 6(a), we show the SEDs of 12 of the star-forming galaxies that lie on the FIR-radio correlation, as defined by the shaded region of Figure 5(a). We excluded the highest redshift source HDF850.1, because the short wavelength data are not well defined. These star-forming galaxies show the characteristic shape of luminous starburst galaxies, as we illustrate by superimposing the Arp 220 SED from Silva et al. (1998; red curve).

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In Figure 6(b), we show the five AGNs from Figure 5(a) that lie on the FIR-radio correlation. They show typical torus SEDs and are undetected at wavelengths $\gtrsim 60$ μm in the rest frame. We superimpose the quasar type I (blue curve) and type II (green curve) templates from the SWIRE template library⁶ (Polletta et al. 2006, 2007). Symeonidis et al. (2016) give a very similar SED for the type I quasars based on a carefully selected and analyzed sample.

The template quasar SEDs are broadly similar to our observations, though our observed SEDs appear to fall more sharply at the longer wavelengths. For these AGNs, the FIR luminosity and the radio power are not a measure of the SFR in the host galaxies. Indeed, the fact that these AGNs lie on the FIR-radio correlation seems to be a coincidence. From Figure 5(b), we see that when we use the $42.5\mbox{--}122.5\,\mu {\rm{m}}$ luminosity instead of the $8\mbox{--}1000\,\mu {\rm{m}}$ luminosity, the locations of the star-forming galaxies remain about the same, while the AGNs drop substantially.

In previous works, radio morphologies and/or radio spectral indices have been used, together with mid-infrared data, to pick out star-forming galaxies and then measure their radio sizes (Muxlow et al. 2005; Guidetti et al. 2013). Muxlow et al. found sizes from 02–30 with a median size of 10. They also found that most of the AGNs were compact. In other works, radio sizes of submillimeter and millimeter selected galaxies have been measured (Chapman et al. 2004; Biggs & Ivison 2008; Miettinen et al. 2015). The advantage of the present data is that we can compare the radio sizes of the classified star-forming galaxies and AGNs in our purely radio selected sample without any selection dependence on the radio morphologies. We could, in principle, use the submillimeter sizes from the SMA data to carry out a similar exercise, but the resolution and signal-to-noise is too low for this to be useful.

In Figure 7, we show radio size versus the ratio of 850 μm flux to radio power for the ${f}_{20\mathrm{cm}}\gt 40\,\mu \mathrm{Jy}$ sources with spectroscopic or photometric redshifts and ${P}_{20\mathrm{cm}}\gt 1.4\,\times {10}^{31}$ erg s⁻¹ Hz⁻¹. We apply the radio flux limit to ensure high enough S/Ns for accurate deconvolution to be able to measure the radio sizes. This limit was based on model tests within the current data set. We show the unresolved sources with downward pointing arrows. We again use colors to denote the sources classified as star-forming galaxies (green), composites (blue), or AGNs (red). We use larger black circles to highlight the two double-lobed radio sources in our sample, which are very extended in the radio.

All but one of the star-forming galaxies are resolved in the 20 cm image, and all but one of the resolved star-forming galaxies have sizes in the rather narrow range 042 to 136 (physical sizes from 4 to 12 kpc). The one larger source is the heavily studied source SMG123707+621408. This source has two separated components, each of which is individually similar to the remaining star-forming galaxies. However, since the CO map of Tacconi et al. (2006) appears to join the two components, we have used the full extent of the two components in measuring the radio size rather than splitting them. We note, however, that the submillimeter continuum flux arises from only one of the components. Deciding what size to allocate to such sources is difficult; fortunately, though, such sources are not common, with only this one example lying in our observed region.

We show the distributions of the sizes in the AGN (top) and star-forming galaxy (bottom) populations in Figure 8, where we have put the unresolved sources in at the upper limits on their sizes (open portions of histograms). The median size of the star-forming galaxies is 10 ± 0.3, while that of the AGNs (excluding the double-lobed sources) is 045 ± 0.05. (The value of the AGNs drops to 04 if we put the unresolved sources in at zero.)

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There are only two composites, one of which is unresolved. If we conservatively put the three unresolved AGNs at their upper limits and include both composites in a combined AGN/composite category (putting the unresolved composite at its upper limit), then the differences in the size distributions between the star-forming galaxies and the AGNs/composites are highly significant, with a Mann–Whitney test giving only a 0.002 probability (two sided) that the two distributions are consistent.

Even with the current 20 cm resolution, we can pick out most of the star-forming galaxies based on radio size and morphology. The double-lobed radio sources are easily classified as AGNs on the basis of radio morphology, while for the remaining sources, we find that choosing those that are resolved and greater than 05 (blue dotted line in Figure 7) would correctly classify the bulk of the star-forming galaxies, while avoiding contamination from the resolved AGNs. However, it should be noted that the most compact star-forming galaxies have sizes that are hard to distinguish from the AGNs using the present spatial resolution.

The radio sizes for the star-forming galaxies are comparable to (though slightly larger than) the radio sizes measured for samples of galaxies with submillimeter detections by Chapman et al. (2004) (083 ± 0.14) and Biggs & Ivison (2008) (064 ± 0.14). However, all of these radio sizes are generally larger than those measured from submillimeter continuum observations (typically about 03; Simpson et al. 2015) by factors of two to three. It is unclear what the explanation for this is when we have found here that the radio power is a good tracer of the FIR for the high radio power selected star-forming galaxy sample (see Figure 5). Simpson et al. suggested that it may be a consequence of cosmic ray diffusion, but they then argued that this is unlikely for the strong submillimeter galaxies. (See also Miettinen et al. 2015's Section 6.3 for a discussion of this issue based on a comparison of their 10 cm sizes with the millimeter continuum sizes from Ikarashi et al. 2015.) The origin of the effect therefore remains to be determined.

In this paper, we used ultradeep radio and submillimeter data on the GOODS-N/CDF-N to develop a classification method based on the ratio of the submillimeter flux to the radio power to separate star-forming galaxies from AGNs in the high radio power population. We found that our classifications agreed with classifications made from high-resolution VLBI observations. However, we also found that there was not a one-to-one correlation of our classifications with those made from ultradeep X-ray data on the field. The star-forming galaxies agreed well with an extrapolation of a local relation between X-ray luminosity and radio power. In contrast, the AGNs showed a much wider spread in hard X-ray luminosity, including some that had luminosities close to those of the star-forming galaxies. These results suggest that some high-redshift, extremely high star formation sources may have been incorrectly classified as AGNs if a simple $2\mbox{--}8\,\mathrm{keV}$ luminosity threshold of 10⁴² erg s⁻¹ was used.

We examined the FIR-radio correlation for the classified sources and found that the star-forming galaxies were all within a multiplicative factor of three of the median FIR-radio correlation measured from the star-forming galaxies. However, five AGNs also lay within this region. We constructed the SEDs for both the star-forming galaxies and these five AGNs and found that the star-forming galaxies show the characteristic shape of luminous starburst galaxies, while the AGNs show typical torus SEDs and are undetected at wavelengths $\gtrsim 60\,\mu {\rm{m}}$. For these AGNs, we are not measuring SFR from either the FIR luminosity or the radio power, and the fact that they lie on the FIR-radio correlation seems to be mere coincidence. Thus, at least at these high radio powers, one must be cautious in measuring SFRs for radio sources using the FIR-radio correlation without other information available on whether they are AGN dominated or star-formation dominated.

We find that the number of star-forming galaxies drops rapidly above a radio power of ${P}_{20\mathrm{cm}}\approx 5\times {10}^{31}$ erg s⁻¹ Hz⁻¹. The surface density of these ${P}_{20\mathrm{cm}}\gt 5\times {10}^{31}$ erg s⁻¹ Hz⁻¹ sources is about two orders of magnitude lower than that of star-forming galaxies with ${P}_{20\mathrm{cm}}=(1\mbox{--}5)\times {10}^{31}$ erg s⁻¹ Hz⁻¹. The $5\times {10}^{31}$ erg s⁻¹ Hz⁻¹ bound corresponds to a SFR of just over a thousand solar masses per year using the Murphy et al. (2011) conversion for a Kroupa (2001) initial mass function.

Finally, since obtaining wide-field, deep submillimeter images is not easy, we used our classifications to investigate whether radio sources could be separated into star-forming galaxies and AGNs based on radio sizes alone. We found that even with the current 20 cm resolution, we were able to put most of the sources into the correct class using a size of 05 as the separation point (i.e., AGNs were smaller than this, and star-forming galaxies were larger). The radio sizes of our submillimeter-detected radio sources are generally larger than those measured for submillimeter-selected galaxies from submillimeter continuum data, but the explanation for this is not yet clear.

We thank the anonymous referee for a careful report that helped us to improve the manuscript. We thank P. Barthel for discussions on the VLBI results. We gratefully acknowledge support from NSF grants AST-1313309 (L.L.C.) and AST-1313150 (A.J.B.), and from the John Simon Memorial Guggenheim Foundation and the Trustees of the William F. Vilas Estate (A.J.B.). The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.