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

(Abridged) Our sensitive (${\sigma }_{\mathrm{n}}=572{\mathrm{n}\mathrm{J}\mathrm{y}\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}^{-1}$), high-resolution (FWHM $\theta_{1/2} = 220,{\rm mas} \approx2$\mathrm{k}\mathrm{p}\mathrm{c}\mathrm{a}\mathrm{t}$z\gtrsim1$)10$,$GHz image covering a single Karl G.~Jansky Very Large Array (VLA) primary beam (FWHM ${\mathrm{\Theta }}_{1\mathrm{/}2}={4.25}^{\mathrm{\prime }}$) 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 $\langle\theta_{\rm M} \rangle=167\pm32,{\rm mas} \approx 1.2\pm0.28,{\rm kpc}$withrmsscatter91was$\approx$ 0.79 kpc. In units of the effective radius $r_{\mathrm{e}}$ that encloses half their flux, these radio sizes are $\langle r_{\rm e}\rangle = 69\pm13{\rm mas} \approx \pm114$\mathrm{p}\mathrm{c}$$andhavermsscatter$38$\mathrm{m}\mathrm{a}\mathrm{s}$\approx324$\mathrm{p}\mathrm{c}$$. 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 (${\theta }_{1\mathrm{/}2}=1.0\mathrm{"}$) image with $\approx 10\times$ better brightness sensitivity to detect extended sources and measure matched-resolution spectral indices ${\alpha }_{1.4}^{10}$. It contains 6 new sources with $S_{\mathrm{p}}\gtrsim 3.9\mu {\mathrm{J}\mathrm{y}\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}^{-1}$ and OIR counterparts. The median redshift of all 38 sources is $1.24\pm 0.15$. The 19 sources with 1.4$$GHz counterparts have median spectral index $-0.74\pm 0.10$ with rms scatter $0.35$. Including upper limits on $\alpha$ for sources not detected at 1.4$$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%.