The GOODS-N Jansky VLA 10 GHz Pilot Survey: Sizes of Star-Forming $mu$Jy Radio Sources

(Abridged) Our sensitive ($sigma_{rm n} = 572,{rm nJy,beam}^{-1}$), high-resolution (FWHM $theta_{1/2} = 220,{rm mas} approx2mathrm{,kpc~at~}zgtrsim1$) 10$,$GHz image covering a single Karl G.~Jansky Very Large Array (VLA) primary beam (FWHM $Theta_{1/2} = 4.25$) in the GOODS-N field contains 32 sources with $S_{rm p}gtrsim2,mu{rm Jy~beam}^{-1}$ and optical and/or near-infrared (OIR) counterparts. Most are about as large as the star-forming regions that power them. Their median FWHM major axis is $langletheta_{rm M} rangle=167pm32,{rm mas} approx 1.2pm0.28,{rm kpc}$ with rms scatter 91 was $approx$ 0.79 kpc. In units of the effective radius $r_{rm e}$ that encloses half their flux, these radio sizes are $langle r_{rm e}rangle = 69pm13{rm mas} approx pm114mathrm{,pc}$ and have rms scatter $38mathrm{,mas}approx324mathrm{,pc}$. These sizes are smaller than those measured at lower radio frequencies, but agree with dust emission sizes measured at mm/sub-mm wavelengths and extinction-corrected H$alpha$ sizes. We made a low-resolution ($theta_{1/2}=1.0$) image with $approx10times$ better brightness sensitivity to detect extended sources and measure matched-resolution spectral indices $alpha_{1.4}^{10}$. It contains 6 new sources with $S_{rm p}gtrsim3.9,mu{rm Jy~beam}^{-1}$ and OIR counterparts. The median redshift of all 38 sources is $1.24pm0.15$. The 19 sources with 1.4$,$GHz counterparts have median spectral index $-0.74pm0.10$ with rms scatter $0.35$. Including upper limits on $alpha$ for sources not detected at 1.4$,$GHz flattens the median to $gtrsim-0.61$, suggesting that the $mu$Jy radio sources at higher redshifts, and hence selected at higher rest-frame frequencies, may have flatter spectra. If the non-thermal spectral index is -0.85, the median thermal fraction at rest-frame frequency 20$,$GHz is $gtrsim$48%.