Results are presented from the first cometary observations using the Atacama Large Millimeter/Submillimeter Array (ALMA), including measurements of the spatially-resolved distributions of HCN, HNC, H$_2$CO and dust within the comae of two comets: C/2012 F6 (Lemmon) and C/2012 S1 (ISON), observed at heliocentric distances of 1.5 AU and 0.54 AU, respectively. These observations (with angular resolution $approx0.5$), reveal an unprecedented level of detail in the distributions of these fundamental cometary molecules, and demonstrate the power of ALMA for quantitative measurements of the distributions of molecules and dust in the inner comae of typical bright comets. In both comets, HCN is found to originate from (or within a few hundred km of) the nucleus, with a spatial distribution largely consistent with spherically-symmetric, uniform outflow. By contrast, the HNC distributions are clumpy and asymmetrical, with peaks at cometocentric radii $sim$500-1000~km, consistent with release of HNC in collimated outflow(s). Compared to HCN, the H$_2$CO distribution in comet Lemmon is very extended. The interferometric visibility amplitudes are consistent with coma production of H$_2$CO and HNC from unidentified precursor material(s) in both comets. Adopting a Haser model, the H$_2$CO parent scale-length is found to be a few thousand km in Lemmon and only a few hundred km in ISON, consistent with destruction of the precursor by photolysis or thermal degradation at a rate which scales in proportion to the Solar radiation flux.
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