= Solution
\b[<Core accretion> builds a solid core and then a massive gaseous envelope before <protoplanetary disk dispersal>.] The solid and <gas> stages overlap; this is not direct collapse of a whole disk patch.
In a <protoplanetary disk>, refractory grains and, beyond a <snow line>, condensed ices provide solids. Settling and <protoplanetary dust coagulation> build aggregates, but fragmentation, bouncing and <radial drift of solids in a gas disk> can interrupt simple sticking. Concentration processes such as the <streaming instability> can produce dense particle structures that collapse into <planetesimals>. Subsequent collisions build <planetary embryos>; alternatively, embryos can efficiently capture drifting particles through <gas>-assisted <pebble accretion>. Neither the initial planetesimal population nor unrestricted growth through every size should be assumed without a mechanism.
For <gravitationally focused growth of a planetary core>, the two-body capture cross-section is
$$
\sigma_{\rm coll}=\pi R_c^2\left(1+\frac{v_{\rm esc}^2}{v_\infty^2}\right),\qquad v_{\rm esc}^2=\frac{2GM_c}{R_c}.
$$
An illustrative thin-disk estimate is $\dot M_c\sim\Sigma_s\Omega\sigma_{\rm coll}$, up to geometry and velocity-distribution factors. At fixed solid <density> and random speed, strong <gravitational focusing> gives $\dot M_c\propto M_c^{4/3}$: <runaway planetesimal growth>. As large embryos stir the smaller bodies, encounter speeds increase and focusing weakens, giving <oligarchic growth>. Consuming the local <feeding zone> leads toward an <isolation mass>; radial supply, migration and mergers can extend growth. <Snow lines> can increase the solid reservoir, but a distant core also grows more slowly because orbital times are longer.<Lecture notes on the formation and early evolution of planetary systems by Philip J. Armitage>[Version1, sectionIII.B, pages27–31: <gravitational focusing> and planetesimal growth]{ref}
A growing core binds <gas> over a scale limited by the smaller of its <Bondi radius> and <Hill radius>. The envelope approximately satisfies
$$
\frac{dP}{dr}=-\frac{G[M_c+m(r)]\rho}{r^2},\qquad\frac{dm}{dr}=4\pi r^2\rho.
$$
An <equation of state>, disk boundary conditions, and radiative or convective energy transport close the structure. In an optically thick radiative region,
$$
\frac{dT}{dr}=-\frac{3\kappa\rho L}{64\pi\sigma_{\rm SB}r^2T^3}.
$$
Here $\kappa$ is <opacity>; an unstable radiative gradient instead drives <convection>. Solid accretion releases roughly $L_s\sim GM_c\dot M_c/R_c$, heating and supporting the envelope. <Gas> capture consequently depends on cooling, not just whether the potential can bind <gas> instantaneously.
The envelope loses energy on a <Kelvin-Helmholtz cooling time>, $t_{\rm KH}\sim|E_{\rm env}|/L$. Lower grain <opacity>, higher core <mass>, or reduced solid-accretion heating can accelerate contraction. The <critical core mass for gas accretion> is therefore model dependent; “of order ten Earth masses” is a common scale, not a universal switch. As envelope self-<gravity> becomes important and the <gas> and solid masses become comparable, contraction permits still more <gas> capture, producing <runaway gas accretion>. The classical sequence has a rapid solid-growth phase, a slower envelope-growth phase, and accelerated <gas> accumulation.<Formation of the giant planets by concurrent accretion of solids and gas by James B. Pollack and others>[1996 model: coupled solid and <gas> growth, three phases and crossover]{ref}
\Image[/past-exam-of-the-exoplanets-and-planetary-systems-course-of-the-university-of-cambridge/2015/paper-1-core-accretion.png]
{title=Core accretion requires both solid growth and envelope cooling while the disk still supplies gas}
{source=paper-1-core-accretion.py}
\b[The decisive race is $t_{\rm core}+t_{\rm envelope}\lesssim t_{\rm disk}$.] Failure to cool or obtain a sufficiently large core before <gas> removal leaves a smaller solid or ice-rich <planet>. Successful growth is ultimately limited by disk supply, <gap opening by a planet>, and dispersal; a gap does not completely seal off accretion. Angular-momentum exchange drives <planetary migration>, so the observed orbit need not be the formation <radius>. Accretion-shock radiation and subsequent contraction set the young giant's entropy and luminosity; final core <mass>, envelope enrichment and orbital location depend on the whole growth history, not on one fixed threshold. The necessary cooling, supply and orbital constraints make the model testable rather than an assertion that every embryo must become a <gas giant>.
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