= Solution
In <core accretion>, solid material grows from dust and aggregates into planetesimals or rapidly accreted pebbles. A sufficiently large core binds an atmosphere; beyond the snow line, additional condensed volatiles aid solid growth. Cooling allows the envelope to contract and acquire more <gas>. Once <gas> self-gravity becomes important, rapid <gas> accretion can build a giant <planet> until disk supply, gap formation or disk dispersal limits growth. Migration and interactions with other bodies can alter its final orbit.
Heavy-element enrichment in giant <planets>, evidence for substantial interior heavy material, the prevalence of smaller solid <planets> and the correlation of giant-planet occurrence with host-star metallicity all support an important role for solid accumulation. They do not uniquely establish one growth pathway. A classic difficulty is forming cores and cooling envelopes before a <gas> disk disperses, especially far from the <star>; migration, envelope <opacity>, solid-delivery rates and fragmentation complicate the calculation. Pebble-assisted growth can change the timescale, and disk gravitational instability remains a distinct possible route in some environments.
<Planetary luminosity> is the total emitted <power>. For the Kelvin-Helmholtz budget one needs the \b[intrinsic] <luminosity>, excluding reradiated absorbed starlight. With orbital distance $r_p$, <Bond albedo> $A_B$, solar <luminosity> $L_\odot$ and present-like <radius> $R_J$,
$$
P_{\rm abs}=(1-A_B)\frac{L_\odot R_J^2}{4r_p^2}.
$$
The given ratio of emitted to absorbed <power> is $1.67$, so $L_{\rm int}=0.67P_{\rm abs}$. Using the supplied <radius>, $r_p=5.2$ AU and a solar <luminosity> around $3.8\times10^{26}$ W gives
$$
\boxed{P_{\rm abs}\simeq5.1\times10^{17}\,\mathrm W,\qquad
L_{\rm int}\simeq3.4\times10^{17}\,\mathrm W,}
$$
consistent with the quoted $3.42\times10^{17}$ W. Taking $1.67P_{\rm abs}$ as intrinsic <power> would incorrectly include reradiated sunlight.
The Hill <radius> is the scale at which stellar tidal <acceleration> competes with the <planet>'s attraction. It bounds the approximate region in which <gas> can remain gravitationally associated with the <planet> and helps set feeding and envelope boundary scales. With a Jupiter <mass>,
$$
\boxed{R_H\simeq5.2\,\mathrm{AU}\left(\frac{M_J}{3M_\odot}\right)^{1/3}
\simeq0.355\,\mathrm{AU}\simeq5.3\times10^{10}\,\mathrm m.}
$$
Using this as the stipulated extended proto-planet <radius> and the present-like intrinsic <luminosity> gives
$$
\boxed{t_{\rm KH}\sim\frac{GM_J^2}{R_HL_{\rm int}}
\simeq4.2\times10^8\,\mathrm{yr}.}
$$
Using the compact present <radius> instead would give an even longer value, of order $3\times10^{11}$ years at that same <luminosity>. Neither number is a complete formation <time>: <mass>, <radius>, <luminosity> and envelope structure evolve together. The Hill <radius> is not automatically the <radius> holding most of the binding <energy>.
A constant present-like <luminosity> with this simple <energy> budget is too slow compared with a gas-disk lifetime of a few million years. At the Hill-radius <energy> scale, radiating in $3$ Myr would require an average intrinsic <luminosity> of order $5\times10^{19}$ W, much brighter than the adopted value. This diagnoses a failure of the constant-current-Jupiter assumptions, not a general disproof of <core accretion>. Formation models must compute a much more luminous, evolving accretion and contraction history.
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