  We present new Karl G. Jansky Very Large Array radio continuum images of the
nuclei of Arp 220, the nearest ultra-luminous infrared galaxy. These images
have both the angular resolution to study detailed morphologies of the two
nuclei that power the system and sensitivity to a wide range of spatial scales.
At 33 GHz, and with a resolution of 0".081 x 0".063 (29.9 x 23.3 pc), we
resolve the emission surrounding both nuclei and conclude that is mostly
synchrotron in nature. The spatial distributions of radio emission in both
nuclei are well described by exponential profiles. These have deconvolved
half-light radii of 51 and 35 pc for the eastern and western nuclei, and they
match the number density profile of radio supernovae observed with very long
baseline interferometry. This similarity might be due to the fast cooling of
cosmic rays electrons caused by the presence of a strong (~ mG) magnetic field
in this system. We estimate high luminosity surface densities of
$\mathrm{\Sigma_{IR} \sim 4.2^{+1.6}_{-0.7} \times 10^{13}}$ (east) and
$\mathrm{\sim 9.7^{+3.7}_{-2.4} \times 10^{13}~(west)~L_{\odot}~kpc^{-2}}$, and
star formation rate surface densities of $\mathrm{\Sigma_{SFR} \sim
10^{3.7\pm0.1}}$ (east) and $\mathrm{\sim
10^{4.1\pm0.1}~(west)~M_{\odot}~yr^{-1}~kpc^{-2}}$. These values, especially
for the western nucleus are, to our knowledge, the highest luminosity and star
formation rate surface densities measured for any star-forming system. Despite
these high values, the nuclei lie below the dusty Eddington limit in which
radiation pressure is balanced only by self-gravity. The small measured sizes
also imply that the nuclei of Arp 220 are only transparent in the frequency
range ~ 5 to 350 GHz. Our results offer no clear evidence that an active
galactic nucleus dominates the emission from either nucleus at 33 GHz.
