= Solution
A <mass-radius curve of solar-composition substellar objects> must specify age, irradiation and composition. For a mature, weakly irradiated <hydrogen>–<helium> sequence, the qualitative <mass-radius relation> has three main trends:
* At low giant-planet masses, weak compression allows a rough $R\propto M^{1/3}$ scaling at nearly fixed mean <mass density>. Radius rises toward a broad maximum around one to a few Jupiter masses.
* Through massive gas giants and <brown dwarfs>, radii remain near Jupiter's and eventually decline as compression and <electron degeneracy pressure> become important. A nonrelativistic degenerate $n=3/2$ <stellar polytrope> gives the approximate <polytropic mass-radius relation> $R\propto M^{-1/3}$; finite thermal pressure and real equations of state flatten and modify this idealized scaling.
* Above the <hydrogen-burning minimum mass>, sustained <hydrogen burning> supplies the energy that supports a rising low-mass main-sequence branch. For near-solar composition, the threshold is around $80$ Jupiter masses (about $0.075$ solar masses), with composition-dependent variation. <Red dwarfs> then have radii increasing with mass, approximately linearly over part of the low-mass sequence.
The conventional <deuterium-burning mass> near $13$ Jupiter masses marks a commonly used giant-planet/<brown dwarf> classification scale, not a sharp discontinuity in radius or a universal formation boundary. Young <brown dwarfs> are larger and cool by contraction; irradiation, a heavy-element core and <hot-Jupiter radius inflation> shift planets away from a single solar-composition curve.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-315-mass-radius.png]
{title=Schematic mass-radius trends for solar-composition bodies}
The drawing is a qualitative mature sequence, not an interpolation of a specified evolutionary model grid. Three relevant observations, using representative historical measurements, are:
* \b[Inflated radii:] <HD 209458 b> has a radius around $1.35R_J$ at a mass around $0.7M_J$, illustrating <hot-Jupiter radius inflation> relative to a simple mature unirradiated sequence. https://arxiv.org/abs/astro-ph/0101336[Early transit photometry] measured its unusually large radius.
* \b[Compact metal-enriched planets:] <HD 149026 b> was measured near $0.36M_J$ and $0.73R_J$. Its small radius for its mass and irradiation implies substantial heavy-element enrichment in interior models, showing the importance of composition in the <planetary mass-radius relation>; see https://arxiv.org/abs/astro-ph/0507009[the discovery analysis].
* \b[Jupiter-like sizes at substellar masses:] <CoRoT-3b> was measured near $22M_J$ but only $1.0R_J$, supporting the broad near-constant-radius region extending into <brown dwarf> masses. https://arxiv.org/abs/0810.0919[Its measured mass and radius] also illustrate why mass alone does not establish a formation history.
\b[Gas-giant radius growth gives way to a nearly Jupiter-sized compressed branch, followed by rising radii in hydrogen-burning stars.]
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