JWST MIRI detections of H$\alpha$ and [O III] and direct metallicity measurement of the $z=10.17$ lensed galaxy MACS0647$-$JD

JWST spectroscopy has revolutionized our understanding of galaxies in the early universe. Covering wavelengths up to $5.3\,{\rm \mu m}$, NIRSpec can detect rest-frame optical emission lines H$\alpha$ out to $z = 7$ and [O III] to $z = 9.5$. Observing these lines in more distant galaxies requires longer wavelength spectroscopy with MIRI. Here we present MIRI MRS IFU observations of the lensed galaxy merger MACS0647$-$JD at $z = 10.165$. With exposure times of 4.2 hours in each of two bands, we detect H$\alpha$ at $9\sigma$, [O III]$\,\lambda5008$ at $11\sigma$, and [O III]$\,\lambda4960$ at $3\sigma$. Combined with previously reported NIRSpec spectroscopy that yields seven emission lines including the auroral line [O III]$\,\lambda4363$, we present the first direct metallicity measurement of a $z > 10$ galaxy: $12+{\rm log(O/H)}= 7.79\pm0.09$, or $0.13^{+0.02}_{-0.03}\,Z_{\odot}$. This is similar to galaxies at $z \sim 4 - 9$ with direct metallicity measurements, though higher than expected given the high specific star formation rate ${\rm log(sSFR / yr^{-1})} = -7.4 \pm 0.3$. We further constrain the ionization parameter ${\rm log}(U)$ = $-1.9 \pm 0.1$, ionizing photon production efficiency ${\rm log}(\xi_{\rm ion})$ = $25.3\pm0.1$, and star formation rate $5.0\pm0.6\,M_{\odot}/{\rm yr}$ within the past $10\,{\rm Myr}$. These observations demonstrate the combined power of JWST NIRSpec and MIRI for studying galaxies in the first $500$ million years.