HIGH-RESOLUTION RADIO CONTINUUM MEASUREMENTS OF THE NUCLEAR DISKS OF Arp 220

In Table 1, we report the flux density of the entire system, each nucleus, and the resulting spectral indices. We used a curve of growth method to derive the flux densities. For the integrated flux density, we progressively (u, v) tapered and re-imaged the data, recording the total flux density above a signal-to-noise of five at each resolution. These flux densities agree with those measured from the imaging of individual arrays (see Section 2). For the flux densities of the individual nuclei, we used CASA's imstat task to place circular apertures around each component, varying their radii. We plotted the flux density against aperture radius and looked for convergence in this curve-of-growth to identify the true flux density. We also independently measured the integrated flux of the southwest component seen at C band (see Figure 1(a)), using an aperture in the CASA viewer, and found that it encloses ∼3% of the total 5.95 GHz flux density. The integrated flux densities at both frequencies agree, within the reported errors, with predicted values based on the modeled integrated SED published in the literature (e.g., Anantharamaiah et al. 2000).

Using these flux densities, we calculated the spectral indices, α for F_ν∝ν^α, of the whole system and each nucleus via

$$\begin{equation} \alpha _{1\hbox{-}2} = \frac{\log F_1 - \log F_2 }{\log \nu _1 - \log \nu _2}, \end{equation} \tag{ 1 }$$

where F₁ and F₂ are the flux densities at frequencies ν₁ and ν₂. Equation (1) will be valid for flux densities F₁and F₂ over matched apertures (or for integrated values over whole systems).

To assess the degree to which the measured sizes are characteristic of the whole system, we compared our maps to known distributions of emission from gas and recent radio supernovae and/or supernova remnants (RSNe/SNRs). In Figure 1(b), we plot C band contours over the CO (3→2) map of Sakamoto et al. (2008; restoring Gaussian beam with FWHM 038 × 028 at a p.a. ≈ 23°).

The RSNe/SNRs trace recent star formation, and they can accelerate cosmic ray (CR) electrons that emit synchrotron radiation. We built a map of RSN/SNR number density from the locations of 49 point sources identified by Lonsdale et al. (2006) from 18 cm VLBI observations. On our 33 GHz astrometric grid, we convolved delta functions with a fixed, fiducial intensity at the positions of the point sources with our 33 GHz beam.¹⁴ In Figures 1(c) and (d), we compare the 33 GHz map to the distribution of recent RSNe/SNRs.

Note that although we use RSNe/SNRs as signposts of recent star formation, the VLBI sources do not contribute significantly to the flux that we observe. For a typical synchrotron spectral index α_{1.7–33 GHz} = −0.7, the Lonsdale et al. (2006) RSNe/SNRs would contribute 1.5 mJy at 33 GHz and 4.9 mJy at 6 GHz. This contribution would only account for ∼2.5% of the total flux density that we observe with the VLA. Even at 18 cm, the Lonsdale et al. (2006) RSNe/SNRs have integrated flux only ∼12 mJy at 18 cm, or ∼4% of the total flux density of Arp 220 at that frequency (Williams & Bower 2010). This contribution is small compared to the 10% expected fraction in normal spiral galaxies like M31 or the Milky Way (Pooley 1969; Ilovaisky & Lequeux 1972). This difference is most likely due to free–free absorption at 18 cm (see Section 4.3.4).

The smooth, ellipsoidal isophotes in Figure 1 suggest a disk-like geometry. We modeled the 3σ clipped image of Arp 220 at 33 GHz (outermost contour in Figure 1(c)) using a two-dimensional (2D) nonlinear least-squares fitting technique. After experimenting with Gaussian, exponential, Sérsic and hybrid profiles, we found that the two disks are reasonably described by thin, tilted exponential disks. We fit both nuclei simultaneously by varying, without constraints, the amplitude, p.a., inclination, center, and scale length of each nucleus. Although the parameters did not have constraints, the starting points were educated guesses of the final parameters. In each case, we construct the model image, convolve it with the synthesized beam of our observations, and compare the model and observed intensities to derive χ². Note that the results for the inclination of the disks represent lower limits because we assume thin disks.

The best-fit parameters from the model fitting, along with associated uncertainties, are reported in Table 2. In addition to deriving formal uncertainties, we gauge the accuracy of our fit by varying our approach among several reasonable methods. For example, we adopt a logarithmic, rather than linear, goodness of fit statistic and we fit the radial profile rather than the image itself. These imply an uncertainty of ≈15% for the scale length and a few percent for p.a. and inclination. The error in the normalization is dominated by our overall uncertainty in the amplitude calibration (≈12%). As another point of comparison, we also report the results of simple Gaussian fitting, although, we emphasize that the residuals are substantially poorer for this approach at low and high radius.

From the exponential model, we obtain deconvolved scale lengths of 30 and 21 pc for the east and west nucleus, respectively. Our results for p.a., inclination, and center did not vary significantly with the choice of functional form. While the fits appear to be good descriptions, they are not perfect. From the residual images, we found that the western nucleus showed higher residuals in the disk than the eastern nucleus, while the center of the eastern nucleus had higher residuals than the western nucleus.

Both nuclei are well resolved, showing significant extent compared with the synthesized beam. The implied deconvolved half-light radii, R_50d, are 51 and 35 pc, respectively; that is, if viewed face-on, we would expect half the emission from Arp 220 to come from nuclear disks ∼100 (east) and ∼70 (west) pc across. In Figures 1 and 2, we have also shown that the size measurement agrees with that implied by the RSN/SNR distribution (Lonsdale et al. 2006). In fact, Herrero-Illana et al. (2012) derive scale lengths for the RSN/SNR distribution, from Lonsdale et al. (2006) and Parra et al. (2007), that are consistent with the results shown in Table 2.

The p.a.s of the east and west nuclei agree well with those of the kinematic major axes of the disks measured from sub-millimeter (sub-mm) CO observations (Sakamoto et al. 1999, 2008), H i absorption observations from Mundell et al. (2001), and H53α radio recombination line Rodríguez-Rico et al. (2005). The velocity gradients along these p.a.s on individual nuclei have also been observed in the 2 μm H₂ line (Genzel et al. 2001) which traces hot molecular gas and 2.3 μm CO absorption (the latter traces stellar velocities: Engel et al. 2011).

We performed two checks on the size measurements. First, as a point of comparison, we report in Table 2 a 2D Gaussian fit to each nucleus at 33 GHz. We obtained deconvolved FWHM sizes of 023 × 013 (86 × 46 pc²) for the eastern and 017 × 010 (64 × 38 pc²) for the western nucleus. These sizes agree fairly well with previous, marginally resolved, estimates at other frequencies. Downes & Eckart (2007) found a deconvolved major axis size of 019 = 70 pc for the western nucleus at 1.3 mm. Sakamoto et al. (2008) found major axis sizes of 027 = 100 pc (FWHM, east) and 016 = 59 pc (FWHM, west) at 860 μm. However, we show through radial profiles (Figure 2) that the disks are better described by an exponential morphology. In fact, the deconvolved Gaussian fit would underestimate the deconvolved half-light diameter of the disks by 9%(west) and 15% (east), if we account for inclination effect in the Gaussian fit, i.e., the deconvolved ${\rm FWHM_{{\rm major}}}$ is smaller by ∼9% and 15% when compared to the deconvolved half-light diameter (2 × R_50d).

Second, we calculated the area on the sky containing half of the flux associated with each nucleus. This very basic measure still suffers from beam dilution and inclination effects, but provides a measure of size that is independent of the functional form. We derived this image-based A_50 sky by identifying the isointensity contour that encloses 50% of the total flux density of each nucleus. We summed the area of the pixels (pixel size = 002) enclosed within that contour and estimate the observed radius for i = 554 (east) and i = 491 (west), and ${R_{\rm 50\, sky} = \sqrt{A_{\rm 50\, sky}/(\pi \cos {i})}}$. This is ∼73 pc for the eastern nucleus and ∼46 pc for the western nucleus. We report the observed R_50 sky values in Table 2. As with the Gaussian, the measured area shows broad agreement with the exponential profile fitting, though differing in detail.

We created deprojected, azimuthally averaged radial profiles of 33 GHz intensity to assess the accuracy of our models and compare the structure of the nuclei to that of the RSN/SNR number density map. Assuming a thin tilted ring geometry, we calculated deprojected profiles for the observed emission, the emission in the convolved model, and the RSN/SNR number density map. In each case, the center of the profiles correspond to the highest intensity pixel in the observed image for each nucleus. We plot these profiles in Figure 2. We included only emission above a signal-to-noise ratio of three and then averaged the intensity in a series of inclined, ∼0035 wide (half the clean beam size) rings, adopting the best-fit model inclination and p.a. from our modeling (Table 2). Note that these rings oversample the ∼007 beam, so that adjacent bins in Figure 2 are not independent. We normalized the RSN/SNR radial profile to match the 33 GHz profile at r ≈ 009. The plotted error bars were calculated from the standard deviation of the flux within each annulus divided by the square root of the area in that annulus expressed in units of the beam size (i.e., the number of independent beams).

In Figure 2, we show that the exponential model matches the data well for both nuclei, matching slightly better for the western nucleus. Meanwhile, the Gaussian profile is not as good as the exponential profile when compared to the observed data, being particularly poorer in the outer parts of the disks. The linearity of the semilog profiles also confirms (and motivates) our adoption of an exponential functional form. The RSN/SNR radial profiles mostly follow the integrated radio emission profiles (and thus also the model). The agreement is better in the western nucleus. The eastern nucleus lacks a bright central peak and shows a somewhat more scattered distribution, which could be caused by stochasticity and timescale effects; i.e., there just may not be enough RSNe/SNRs visible to give a smooth appearance (compare to Figure 1(d) to see the clumpy nature of the SNe distribution). The same idea applies for the outskirts of the western nucleus. In the rest of the paper, we will consider that the RSN/SNR number density distribution follow the continuum emission observed at 33 GHz closely enough that we can take our measured 33 GHz sizes as indicative of the distribution of active star formation in Arp 220.

The brightness temperature, T_b, can be used to constrain the emission mechanism and energy source, and may give clues regarding the optical depth. With well resolved sizes, we can circumvent beam dilution that often confuses estimates of T_b. We calculated T_b using the Rayleigh–Jeans approximation via

$$\begin{equation} T_{b} = \left(\frac{S_{\nu }}{\Omega _{{\rm source}}}\right)\frac{c^{2}}{2 k_{B} \nu ^{2}}, \end{equation} \tag{ 2 }$$

with S_ν the flux density at frequency ν and Ω_source the area subtended by the source.

We report average Rayleigh–Jeans brightness temperatures, T_b, in Table 3. From our model, we take the area ${ A_{50d} \equiv \pi R_{50d}^2}$ that we expect to enclose half the emission if the system were viewed face on (see the "deconvolved" column in Table 2). Assuming this to be the true area of the disk at all frequencies, we derive average T_b over the half-light region. This means we used half of the observed flux density for S_ν and A_50d for ${\rm \Omega _{{\rm source}}}$ in Equation (2). Our rationale for this assumption is that the 33 GHz image appear to be optically thin, high-resolution tracer of the distribution of recent star formation. Assuming that this structure is common across wavelength regimes allows us to use a "true" size in place of a size observed with a much coarser beam. We also calculated the peak T_b at each band from the peak flux density and the area of the clean beam at each frequency. This peak T_b is higher at 33 GHz than at 6 GHz; this simply reflects that the area of Arp 220 at 33 GHz is smaller than the beam size at 6 GHz. For exactly this reason—the small size of Arp 220 and the variable resolution at different frequencies—the peak measurement has limited utility and we only report the "average" version.

The average T_b is from 10^4.4 K at 6 GHz to 10^2.5 (≈300) K at 33 GHz, for the east nucleus and 10^4.8 K at 6 GHz to 10^2.9 (≈800) K at 33 GHz, for the west nucleus.

Synchrotron radiation appears to produce most of the continuum emission at both 6 and 33 GHz. The high brightness temperature of a few × 10⁴ K, inferred at 6 GHz by using the 33 GHz nuclear sizes, argues in this direction. This high brightness temperature cannot come from Hii regions, even if they are completely opaque, because in a purely thermal environment, the electron temperature of such regions should not exceed 10⁴ K. If we combined the high brightness temperature with the observed internal C band spectral index of the total system, α_{4.7–7.2 GHz} = −0.61 ± 0.04, we infer that most of the emission at 6 GHz is synchrotron.

The spectral index between 6 and 33 GHz, α_6–33 GHz = −0.69 ± 0.07, matches the internal C band α of the total system within the errors; the same is true for the two nuclei separately (see Table 1). The similarity between α_{4.7–7.2 GHz} and α_6–33 GHz indicates no significant spectral flattening between 6 and 33 GHz, suggesting that synchrotron dominates the emission across this range of frequencies. Reinforcing this point, our total flux density and spectral index agree with the predictions made by Anantharamaiah et al. (2000, see their Figure 10(b)). They found synchrotron emission to dominate below ∼60 GHz, and estimated the thermal fraction at 6 GHz and 33 GHz to be ∼15% and 35%, respectively. If we assume a thermal fraction of 35% at 33 GHz, a nonthermal spectral index of −0.76 (see Table 9 in Anantharamaiah et al. 2000), and a typical thermal spectra index of −0.1, we obtain α_6–33 GHz ≈ −0.60 and α_{4.7–7.2 GHz} ≈ −0.66, which deviate ∼1.3σ from the observed values. The difference between the predicted spectral indices is 0.06, while for the observed values it is closer to −0.08  ±  0.11. If we assume the observed values are the true ones, this slight discrepancy could indicate potential opacity effects between 4.7 and 7.2 GHz (see Section 4.3.4), or the presence of a steeper nonthermal spectral index between 6 and 33 GHz. However, this discrepancy does not affect our interpretation of nonthermal emission dominating at 33 GHz. In fact, a higher thermal fraction at 33 GHz will only make this discrepancy worse (see below). Overall, our results are consistent with previous results showing lower thermal fractions at 33 GHz for merging starbursts compared to normal galaxies (Murphy 2013).

The overall SFR of the system provides an alternate way to estimate the expected thermal radio continuum emission. Beginning with the IR (8–1000 μm) luminosity of Arp 220, we estimate the expected thermal luminosity of the system if the IR is all due to star formation by following SFR conversions from Table 8 in Murphy et al. (2012)¹⁵ and assuming an electron temperature of 7500 K (Anantharamaiah et al. 2000). This approach predicts a thermal fraction of ∼55% at 33 GHz and ∼20% at 6 GHz. This is in good agreement with the thermal fraction at 33 GHz obtained in Condon (1992) for a prototypical starburst, M82. However, if we derive the expected spectral index as we did at the end of the previous paragraph (assuming 55% of thermal fraction at 33 GHz), we obtain α_6–33 GHz ≈ −0.49 and α_{4.7–7.2 GHz} ≈ −0.57, which deviates considerably from what we observe between 6 and 33 GHz. Note that α_{4.7–7.2 GHz} does not vary significantly from what we observe, this is due to the small thermal fraction expected at this frequency range.

The easiest explanation for the lower-than-expected thermal flux is that a significant fraction of the ionizing photons produced by young stars are absorbed by dust before they produce ionizations. This would lower the free–free estimate in the calculation. We estimate that to match our observations, we would require that at least 20% of the ionizing photons be absorbed by dust. This number seems plausible for an environment as dust embedded as Arp 220 (see Section 4.3.4) and is consistent with some of the arguments made when considering the apparent deficit of IR cooling line emission (Díaz-Santos et al. 2013, and references therein). Alternatively, an initial mass function (IMF) that produces more bolometric light (and thus IR and likely SNe) relative to ionizing photons could resolve the discrepancy. That is, we could invoke an "intermediate-heavy" IMF compared to that used in Murphy et al. (2012). We could also reconcile our two estimates if the synchrotron spectral slope decays drastically between 6 and 33 GHz, so that the apparent 33–6 GHz index is a combination of very steep, curving synchrotron and emerging thermal emission. However, this would require that thermal emission make up most of the SED at higher frequencies, which is not observed (see Anantharamaiah et al. 2000; Clemens et al. 2010).

Our best interpretation of the data is that the 33 GHz emission is mostly synchrotron, in mild contrast with a typical starburst galaxy (see Figure 1 in Condon 1992). We suggest that the most likely cause is the suppression of thermal radio emission as dust absorbs ionizing photons.

In Figure 1(b), we show the 6 GHz emission is largely co-spatial with CO emission. The 6 GHz emission is our more sensitive band, with a beam nearly matched to the CO, and—as just discussed—we expect that it traces the same synchrotron emission as the 33 GHz. The CO and 6 GHz emission cover roughly the same area, have broadly coincident peaks, and both show an extended faint feature to the southwest (Mazzarella et al. 1992, note a similar coincidence between 18 cm emission and the Arp 220 starburst traced in the near-IR). The distributions of CO and 6 GHz emission do significantly differ in detail. The ratio of fluxes for the two nuclei is 1:2 (east:west) for CO and almost 1:1 for continuum. In the west nucleus, the morphology is more centrally concentrated at 6 GHz compared to the CO map. Some of these differences may reflect real differences between the current gas reservoir and recent star formation, but they may also reflect temperature and optical depth effects. The CO (3→2) emission in this region shows good evidence for optical thickness (Sakamoto et al. 2008), and the densities are high enough that the gas temperature will be likely coupled to the dust (∼100 K). Therefore, making a straightforward interpretation of the CO in terms of column density is challenging. We draw the broad conclusion from Figure 1(b) that the synchrotron originates from the same region as, and in very rough proportion to, the molecular gas supply.

A similar situation is also observed on smaller spatial scales by the comparison of the RSN/SNR number density map to the 33 GHz map (see Figure 1(d)). The distributions are co-spatial, but the continuum map appears smoother than the map made from individual RSN/SNR. This is particularly evident in the eastern nucleus, where the covering factor of point sources observed with the Very Long Baseline Array (VLBA) is small—perhaps a result of stochasticity in the rate and lifetime of SN visible using VLBI measurements. The radial profiles in Figure 2 highlight the quantitative agreement between the continuum and the RSNe/SNRs distribution even more. After azimuthal averaging, the VLBA point source maps are a fairly close match to the 33 GHz continuum. This is also supported by the agreement between the scale lengths reported in Table 2 and those found by Herrero-Illana et al. (2012) based on the radial profiles of the RSNe/SNRs distribution observed by Lonsdale et al. (2006) and Parra et al. (2007).

Such a close match between the 33 GHz continuum extent and the RSN/SNR number density map may not be too surprising: if synchrotron radiation arises from CR electrons accelerated by SN shocks, then the 33 GHz continuum emission might be expected to resemble a "puffed up" version of the RSN/SNR distribution due to the diffusion of CR electrons. Instead the distributions match quite well, consistent with most of the 33 GHz emission coming from very close to the original RSN/SNR and little diffusion or secondary CR electron production. This lack of significant propagation could be explained by the cooling timescales being much smaller than the diffusion time. This is expected in compact starbursts with magnetic fields of the order of a mG (see measurements from Robishaw et al. 2008; McBride et al. 2014, based on Zeeman splitting of OH megamaser emission), like Arp 220 (see Figure 1 in Murphy 2009), though not in normal galaxies (Murphy et al. 2006). For Arp 220, the cooling time of CR electrons at 33 GHz is ∼10³ yr, which is a combination of synchrotron, bremsstrahlung, ionization, and inverse Compton losses. We make use of Equation (7) in Murphy (2009) for CR electrons with energies greater than 1 GeV to estimate that the synchrotron emitting electrons at 33 GHz only have time to propagate about 5 pc, which is about 1/10th of the size that we have measured for the nuclei. This short diffusion scale yields a synchrotron image that looks very similar to the sites of original CR production (the RSN/SNR) and thus the sites of active star formation. The advantage of the VLA continuum in this case is that in exchange for coarser native resolution, we achieve sensitivity to most of the flux and spatial scales of interest (and potentially still probe a longer timescale).

The similarity of 6 GHz, 33 GHz, CO surface brightness, and the RSN/SNR number density distributions lead us to view our 33 GHz measurement as indicative of the true size of the main disks of star formation and, presumably, gas and hot dust. These morphologies are also consistent with the nuclear morphologies measured in mid-IR with the Keck Telescope (Soifer et al. 1999). Perhaps surprisingly, the two disks appear fairly similar in terms of profile, scale length, and observed flux. The western nucleus appears hotter and more compact but the differences are small factors, not an order of magnitude. The physical interpretation of such similarities is unclear. Possible explanations include a similarity in the progenitors, or some "loss of memory" during the process of funneling gas to the center of the galaxies during the ongoing interaction.

4.3.1. Gas Surface Densities

4.3.2. Infrared Surface Densities and Star Formation Rates

By following the same approach, we assume the IR emission in Arp 220 is coincident within our measured radio distribution and explore the implications. Conventionally, the IR luminosity surface density, Σ_IR, is defined as the luminosity per unit area of the system. We calculate Σ_IR by assuming that half of the IR luminosity (from 8 to 1000 μm) is generated within the deconvolved, face-on, half-light area (A₅₀), i.e., we calculate an average IR luminosity surface density, within the half-light area, via

$$\begin{equation} \Sigma _{\rm IR} = \left(\frac{0.5 \times L_{\rm IR}[8\hbox{-}1000\,\mu \rm m]}{A_{50}}\right) = \left(\frac{L_{50}}{A_{50}}\right). \end{equation} \tag{ 3 }$$

Following our measurements above, we use $A_{50d} \equiv \pi (R_{50d,{\rm east}}^2+R_{50d,{\rm west}}^2)$ to derive a total (face-on) IR luminosity surface density of $\Sigma _{\rm IR} \sim 6.0^{+2.3}_{-1.5} \times 10^{13} L_{\odot }\,{\rm kpc^{-2}}$.¹⁷ If we further assume that the ratio of the fluxes between the east and west nuclei at 33 GHz (∼1:1) holds at IR wavelengths, then using the derived radio ${ A_{50d} \equiv \pi R_{50d}^2}$ for the individual disks, we obtain $\Sigma _{\rm IR} \sim 4.2^{+1.6}_{-0.7} \times 10^{13} L_{\odot }\, {\rm kpc^{-2}}$ and $\Sigma _{\rm IR} \sim 9.7^{+3.7}_{-2.4} \times 10^{13} \;L_{\odot }\, {\rm kpc^{-2}}$ for the east and west nucleus, respectively. These values are more than an order of magnitude higher than those for the central 0.3 pc of the Orion Nebula complex and M 82 (∼2 × 10¹² L_☉ kpc⁻² and ∼9 × 10¹¹ L_☉ kpc⁻², respectively (Soifer et al. 2000)), but are closer to those found in the brightest clusters within starburst galaxies (∼5 × 10¹³ L_☉ kpc⁻²; Meurer et al. 1997). Our estimated surface densities are consistent with Soifer et al. (2000), who estimated IR luminosity surface densities of 1–6 × 10¹³ L_☉ kpc⁻² based on mid-IR Keck observations and radio data from Condon et al. (1991).

The deprojected SFR surface density (defined as the SFR per unit area in the disk), Σ_SFR, is a close corollary of the IR luminosity surface density. We estimate this quantity within the deconvolved, face-on, A_50d for each nucleus at 33 GHz, using a 1:1 ratio between east and west, the radio luminosity to SFR conversion from Table 8 in Murphy et al. (2012), and an electron temperature and nonthermal spectral index of 7500 K and 0.76 (the spectral index in Murphy et al. 2012 is defined with the opposite sign compared to our definition), respectively, from Anantharamaiah et al. (2000). We obtain a Σ_SFR of ∼10^{3.7 ± 0.1} and 10^{4.1 ± 0.1} M_☉ yr⁻¹kpc⁻² within the half-light of the eastern and western nuclei, respectively (divide these numbers by two to take into account both sides of the disks). The total SFR calculated from L_IR, 180 M_☉ yr⁻¹, and from the total radio flux density at 33 GHz (L_Radio), 195 M_☉ yr⁻¹, differ by only ∼10%, consistent with Arp 220 lying on the (33 GHz) radio-to-far IR correlation (and meaning that we would obtain essentially the same Σ_SFR for either luminosity). The radio SFR value differs by 20% from that derived from Anantharamaiah et al. (2000), which we consider to be within the uncertainties of such calculations.

Σ_IR tells us coarsely about the density of IR luminosity per unit area, but not necessarily the flux at the surface of the source, which may have important implications for feedback and depends on the detailed geometry of the system. A spherical geometry provides a useful limit on the flux at the surface of the system. In this case, $F_{{\rm sphere}} = L_{50} / (4 \pi R_{50d}^2)$ will be the flux at the surface of a sphere of radius R_50d with the luminosity of Arp 220. Using the measured radio sizes, $F_{{\rm sphere}} \sim 1.5^{+0.6}_{-0.4} \times 10^{13} \,L_{\odot }\, {\rm kpc^{-2}}$ for the entire system, ${\sim } 1.1^{+0.4}_{-0.2} \times 10^{13} \,L_{\odot }\,{\rm kpc^{-2}}$ for the eastern nucleus and ${\sim } 2.4^{+0.9}_{-0.6} \times 10^{13} \,L_{\odot }\, {\rm kpc^{-2}}$ for the western nucleus. A less extreme case, one that may well apply to Arp 220, is a two-sided disk. With one half of the luminosity emergent from each side, we have $F_{{\rm disk}} = L_{50} / (2 \pi R_{50d}^2)$, twice the spherical case. As one would expect, these values are lower than the simple Σ_IR, but they also differ from one another, reinforcing the importance of geometry to the physics of the source.¹⁸

Yet another subtlety arises in the specific case where one wishes to calculate the flux through an area very close to, but just above one side of a disk. This quantity is relevant to the often-discussed case of radiation pressure on dust (see Section 4.3.3) but because of projection effects it is not identical to any of the above quantities. In Appendix A, we show that this one-sided flux perpendicular to the disk, which we call F_near is equal to $L_{\rm IR} /(8 \pi R_{50d}^2)$ (Equation (A7)) in general. This value is further divided by an extra factor of two for the case of the two nuclei of Arp 220 (because L_IR combines the light from the two nuclei). From this calculation, we obtain $F_{{\rm near}}\sim 1.5^{+0.6}_{-0.4} \times 10^{13} \,L_{\odot }\, {\rm kpc^{-2}}$ for the entire system (the same as for F_sphere, though not for the same reason), ${\sim } 1.1^{+0.4}_{-0.2} \times 10^{13} \,L_{\odot }\, {\rm kpc^{-2}}$ for the east nucleus and ${\sim } 2.4^{+1.0}_{-0.6} \times 10^{13} \,L_{\odot }\, {\rm kpc^{-2}}$ for the west nucleus. As discussed in Appendix A, these should be the most appropriate fluxes to consider when assessing the impact of pressure from radiation perpendicular to the disk.

The ratio of the flux densities between the east and west nuclei varies with frequency. Some other ratios for the east:west relation found in the literature include 1:4 at mid-IR (Soifer et al. 1999), 1:3 at 18 cm (if we consider only the contribution of the point sources from Lonsdale et al. 2006) and 1:2 at sub-mm wavelengths (Sakamoto et al. 2008). However, most of these observations do not offer high enough spatial resolution to truly isolate the contribution of each nucleus. If we ignore this issue and we assume a ratio of 1:4, and we use Equation (A7), we obtain $F_{\rm near, east} \sim 4.5^{+1.7}_{-1.1} \times 10^{12} \,L_{\odot }\, {\rm kpc^{-2}}$ and $F_{\rm near, west} \sim 3.7^{+1.4}_{-0.9} \times 10^{13} \,L_{\odot }\, {\rm kpc^{-2}}$. In every case, we obtain F_near ≳ 10¹³ L_☉ kpc⁻² for the west nucleus, which is always the higher intensity nucleus.

Note that the same geometric issues discussed here raise a caveat regarding the luminosity of the source, which was derived under the assumption of isotropic emission (Sanders et al. 2003; Sanders & Mirabel 1996). An optically thick thin disk is not an isotropic emitter. However, neither do our observations constrain the geometry of the IR photosphere, which is the relevant surface for this calculation. We discuss this issue in Appendix A. Lacking information, we have assumed the isotropic luminosity throughout this paper, but note the uncertainty (see also Downes & Eckart 2007; Wilson et al. 2014). Note that high angular resolution IR (from 8 to 1000 μm) observations are needed in order to better constrain the true morphology of the IR photosphere of Arp 220.

4.3.3. Radiation Pressure and Maximal Starburst Models

Scoville (2003) and Thompson et al. (2005) argued that for optically thick, dense starburst galaxies, the critical feedback mechanism acting against gravitational collapse, and thus star formation, could be radiation pressure on dust. Although there is ongoing debate about whether or not radiation pressure on dust represents the dominant feedback mechanism in compact starbursts (see Krumholz & Thompson 2013; Socrates & Sironi 2013; Davis et al. 2014, for further discussion), the maximal starburst model of Scoville (2003) and Thompson et al. (2005) represents an interesting point of comparison for our present work.¹⁹ In Figure 3, which closely follows Figure 4 of Thompson et al. (2005), we present our new measurements for the flux near the surface of the source (assuming a thin disk geometry, see Section 4.3.2 and Appendix A) in the context of literature observations and predictions by Thompson et al. (2005). The literature observations show a sample of ULIRGs with sizes based on 8.44 GHz radio maps by (Condon et al. 1991) and luminosities from IRAS. Following the approach presented in Appendix A, we calculate F_near following Equation (A7) assuming half of the total IR luminosity to be enclosed within $A_{50} = \pi R_{50}^{2}$, where R₅₀ = b_maj/2. We used the deconvolved FWHM major axis, b_maj, from Condon et al. (1991) in order to account for inclination effects (we only include resolved sources). Following the discussion in the previous section, the fluxes for the points in Figure 3 differs by a factor of eight compared to those in Thompson et al. (2005),²⁰ for reasons discussed in Appendix A.

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The error bars of our Arp 220 values (filled points in Figure 3) correspond to a combination of the uncertainties in R_50d, and uncertainty in the contribution of each nucleus to the IR luminosity, with the lower/upper limit assuming a ratio of 1:4 between east and west (Soifer et al. 1999). To be conservative, we also assume a 20% uncertainty in the assumption that half of the total IR luminosity is coming from A₅₀. The plotted values for the flux correspond to $F_{\rm near, east} \sim 1.1^{+0.4}_{-0.8} \times 10^{13} \,L_{\odot }\, {\rm kpc^{-2}}$ and $F_{\rm near, west} \sim 2.4^{+2.7}_{-0.9} \times 10^{13} \,L_{\odot }\, {\rm kpc^{-2}}$. In this figure, we show the "Eddington" values for radiation pressure on dust as solid and dashed lines. These represent an envelope for which radiation pressure on dust balances self-gravity. As a result, no equilibrium star-forming system is expected to exist above this line, hence the "Eddington limit" analogy.

The precise value of the limit depends on the size, gas fraction (f_g) stellar velocity dispersion (σ), Rosseland mean opacity (κ), and dust-to-gas ratio of the system, leading to the large spread in the model lines seen in the figure. Our measurements of Arp 220 appear as solid points in Figure 3. There, the west nucleus of Arp 220 appears among the highest brightness systems. However, the west nucleus does not clearly stand out from the other ULIRGs with respect to this value, partially because the more compact size means that the "maximal" value is larger for Arp 220 than for larger systems. Overall, all the systems plotted in Figure 3 lie roughly around the Eddington limit for a f_g = 0.1 and σ = 200 km s⁻¹ disk in Thompson et al. (2005), which is indicated by the blue solid line.

If we adopt f_g = 1 and σ ≈ 200 km s⁻¹ (e.g., Genzel et al. 2001), indicated by the green line in Figure 3, then we calculate a conservative Eddington limit of ∼9 × 10¹³L_☉ kpc⁻² for the west nucleus and ∼7 × 10¹³L_☉ kpc⁻² for the east nucleus. Refined measurements of the geometry, gas fraction, opacity, dust-to-gas ratio, and kinematics are needed to specify the models more precisely. For most plausible assumed disk properties, we can say that both Arp 220 nuclei lie well below the Thompson et al. (2005) Eddington-limited starburst value, with the western nucleus being the brightest system among the local ULIRGs. If improved measurements demonstrate one or both nuclei to lie significantly above this value, one would need to consider luminosity sources other than star formation (presumably an active galactic nucleus (AGN); Iwasawa et al. 2005; Downes & Eckart 2007; Rangwala et al. 2011) or question the basic assumptions about geometry and equilibrium embedded in the model, but at present little such tension appears to exist.

Another way to assess the role of radiation pressure in Arp 220 is to assume that radiation pressure does represent the dominant force acting against gravity (see Appendix B) and to calculate the required gas opacity, κ, of the system in order to be in hydrostatic equilibrium. By following Equation (B2), and using the derived values for the gas surface density and the flux for each nucleus (see Sections 4.3.1 and 4.3.2), we find that κ_east ≈ 300 cm² g⁻¹ and κ_west ≈ 80 cm² g⁻¹ would be required for radiation pressure to balance gravity. Compared to the models from Semenov et al. (2003), which were used in the models from Thompson et al. (2005), these appear to be unrealistically high values for the gas opacity. We interpret these high values as a reinforcement of our previous findings that the nuclei of Arp 220 lie below the dusty Eddington limit for f_g ∼ 1 and σ = 200 km s⁻¹ by a factor of ∼10, and then are not radiation pressure supported.

4.3.4. Optical Depth

In our observations, the western nucleus is more compact with a higher T_b than the eastern nucleus. However, consistent with previous VLBI observations, we observe no significant central excess in either Arp 220 images or radial profiles (Figures 1 and 2). Parra et al. (2007) discuss the possibility that one of three VLBI point sources showing a flat spectrum (α > −0.5), could be an AGN. However, that is one of several possibilities that could explain the shape of their spectrum.

Most of the 33 GHz emission that we observe comes from the compact, but still resolved, disks around the nuclei. Specifically, the nuclear beam of the west nucleus contains 20% of the total flux of the nucleus, while the other 80% arises from the more extended star-forming regions (Table 1). Our measured R_50d contain similar information (see Table 2). We cannot rule out an AGN in the western nucleus, but if one is present it does not make a dominant, point-like contribution to the overall 33 GHz emission on scales of ≈30 pc. Similarly, Arp 220 does not exceed the "Eddington" value that might eliminate star formation as a viable power source (see Section 4.3.3). Smith et al. (1998) show that the SN rate and luminosity of Arp 220 are broadly consistent with emission only generated by star formation, though uncertainty in the IMF, SN rate, and SFR certainly would still allow an AGN contribution.

No high brightness temperature radio core indicative of an AGN is present. However, given that most AGN are radio-quiet and have weak core emission (e.g., Kellermann et al. 1989; Blundell & Beasley 1998), the absence of a radio core does not rule out the presence of an AGN. Indeed, several studies at other wavelengths have presented evidence of a possible AGN in Arp 220 (e.g., Iwasawa et al. 2005; Downes & Eckart 2007; Rangwala et al. 2011; Imanishi & Saito 2014; Wilson et al. 2014), but to date there is no clear evidence that the putative AGN makes a significant contribution to the bolometric luminosity. If our estimate of N(H) ∼10²⁵ cm⁻² in Section 4.3.1 is correct, it would explain why evidence for AGN in Arp 220 has been so elusive. With such high column densities any AGN would be Compton thick and undetectable by standard AGN diagnostic tools.