  In the core accretion hypothesis, giant planets form by gas accretion onto
solid protoplanetary cores. The minimum (or critical) core mass to form a gas
giant is typically quoted as 10 Earth masses. The actual value depends on
several factors: the location in the protoplanetary disk, atmospheric opacity,
and the accretion rate of solids. Motivated by ongoing direct imaging searches
for giant planets, this study investigates core mass requirements in the outer
disk. To determine the fastest allowed rates of gas accretion, we consider
solid cores that no longer accrete planetesimals, as this would heat the
gaseous envelope. Our spherical, two-layer atmospheric cooling model includes
an inner convective region and an outer radiative zone that matches onto the
disk. We determine the minimum core mass for a giant planet to form within a
typical disk lifetime of 3 Myr. The minimum core mass declines with disk
radius, from ~8.5 Earth masses at 5 AU to ~3.5 Earth masses at 100 AU, with
standard interstellar grain opacities. Lower temperatures in the outer disk
explain this trend, while variations in disk density are less influential. At
all distances, a lower dust opacity or higher mean molecular weight reduces the
critical core mass. Our non-self-gravitating, analytic cooling model reveals
that self-gravity significantly affects early atmospheric evolution, starting
when the atmosphere is only ~10% as massive as the core.
