Close-in super-Earths having radii 1-4 R ⊕ may possess hydrogen atmospheres comprising a few percent by mass of their rocky cores. We determine the conditions under which such atmospheres can be accreted by cores from their parent circumstellar disks. Accretion from the nebula is problematic because it is too efficient: we find that 10 M ⊕ cores embedded in solar metallicity disks tend to undergo runaway gas accretion and explode into Jupiters, irrespective of orbital location. The threat of runaway is especially dire at ~0.1?AU, where solids may coagulate on timescales orders of magnitude shorter than gas clearing times; thus nascent atmospheres on close-in orbits are unlikely to be supported against collapse by planetesimal accretion. The time to runaway accretion is well approximated by the cooling time of the atmosphere's innermost convective zone, whose extent is controlled by where H2 dissociates. Insofar as the temperatures characterizing H2 dissociation are universal, timescales for core instability tend not to vary with orbital distance—and to be alarmingly short for 10 M ⊕ cores. Nevertheless, in the thicket of parameter space, we identify two scenarios, not mutually exclusive, that can reproduce the preponderance of percent-by-mass atmospheres for super-Earths at ~0.1?AU, while still ensuring the formation of Jupiters at 1?AU. Scenario (a): planets form in disks with dust-to-gas ratios that range from ~20× solar at 0.1?AU to ~2× solar at 5?AU. Scenario (b): the final assembly of super-Earth cores from mergers of proto-cores—a process that completes quickly at ~0.1?AU once begun—is delayed by gas dynamical friction until just before disk gas dissipates completely. Both scenarios predict that the occurrence rate for super-Earths versus orbital distance, and the corresponding rate for Jupiters, should trend in opposite directions, as the former population is transformed into the latter: as gas giants become more frequent from ~1 to 10?AU, super-Earths should become more rare.
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