= Solution
For an old, approximately solar-composition H/He object with no strong irradiation or ongoing heat input, the cold <mass-radius relation> reaches a broad maximum of \b[order one Jupiter radius, nominally about $1$--$1.2R_J$, at a few Jupiter masses]. This is a property of that composition and low-entropy sequence, not a composition-independent theorem from <hydrostatic equilibrium> alone. Young isolated substellar objects can be larger while cooling; strict thermal steady state of a non-burning isolated body would require its internal luminosity to vanish.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-315-mass-radius.png]
{title=Schematic mass-radius sequence from ice giants to low-mass stars with composition and degeneracy slope guides}
At low mass, an incompressible fixed-composition approximation gives $M\simeq4\pi\bar\rho R^3/3$, so \b[$R\propto M^{1/3}$]. Ice giants contain dense heavy material and modest H/He envelopes, so real radii depend strongly on composition and are not obtained by extending a pure-hydrogen curve through them.
Gas giants approach a regime in which Coulomb interactions, compression and partial <electron degeneracy pressure> compete. An approximate index-one <stellar polytrope>, $P=K\rho^2$, gives \b[$R\propto M^0$] at fixed $K$, explaining the broad Jupiter-sized plateau. In the nonrelativistic degenerate limit, $P=K\rho^{5/3}$ is an index-$3/2$ <stellar polytrope>, giving \b[$R\propto M^{-1/3}$] at fixed composition. This is a limiting slope; finite-temperature <brown dwarfs> do not follow that exact power law over their entire observed mass range.
The general <polytropic mass-radius relation> displays both limits:
$$
R\propto K^{n/(3-n)}M^{(1-n)/(3-n)}.
$$
At roughly $13M_J$, deuterium burning becomes possible, with a composition-dependent threshold. This is a conventional planet/brown-dwarf division rather than a unique formation definition. At roughly $0.075$--$0.08M_\odot$, about $75$--$85M_J$, sustained hydrogen burning can support a low-mass star. Its nuclear energy supply maintains a <specific entropy> that prevents indefinite cooling onto the cold substellar sequence; its approximate main-sequence relation is $R\propto M$ over a limited low-mass interval.
Ice giants chiefly lose formation heat and some gravitational/radiogenic energy, with <convection> in fluid regions and <thermal conduction> or inhibited <convection> where composition stratifies the interior. Gas giants and non-burning <brown dwarfs> radiate stored heat and contraction energy; their deep interiors are largely convective, with radiative <photospheres>. <Helium> separation can supply extra gravitational energy in mature giant planets, while <brown dwarfs> above the deuterium threshold have a temporary nuclear contribution. Low-mass stars are powered primarily by hydrogen fusion and are largely or fully convective at the lowest stellar masses. The transition to sustained fusion is the essential change from a cooling body to a long-lived thermal balance.
Back to article page
