= Solution
The <mass-radius curve of solar-composition substellar objects> reflects the transition from weak compression to pressure ionization and <electron degeneracy pressure>, followed by sustained <hydrogen burning>. A schematic joining representative object classes is:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-59-mass-radius.png]
{title=Schematic mass-radius sequence from ice giants through gas giants and brown dwarfs to low-mass stars}
{height=459}
The illustration is not an age-specific numerical evolutionary model. In particular, <ice giants> contain much more heavy material than a solar-composition giant, so a single uniform-composition equation of state does not describe the entire joined curve.
* At the low-mass, weak-compression end, a fixed-density or fixed-composition approximation gives \b[$R\propto M^{1/3}$]. <Ice giants> such as <Neptune> and <Uranus> have substantial <water>/rock-rich interiors and modest H/He envelopes; changing envelope fraction changes the radius markedly. Their heat comes from retained formation energy, contraction and <radiogenic heating> of heavy material. Fluid interiors generally convect, while composition stratification can impede mixing; outer radiative layers release the heat.
* Ordinary <gas giants> reach radii of order $R_J$ over a broad range around Jovian masses: an effective $n\simeq1$ <polytrope> explains the approximate \b[$R\propto M^0$] segment. Increased mass compresses material enough to offset the added volume. Cooling and <Kelvin-Helmholtz contraction>, with additional differentiation energy such as helium settling in <Saturn>, supply the intrinsic luminosity. Their deep envelopes are usually convective, with radiative photospheres.
* More massive <brown dwarfs> become increasingly supported by <electron degeneracy pressure>. The cold nonrelativistic $n=3/2$ limit gives \b[$R\propto M^{-1/3}$], but finite <entropy> and Coulomb effects flatten actual giant/brown-dwarf curves and their radii depend on age. They cool and contract; temporary deuterium fusion occurs above a composition-dependent <deuterium-burning mass> near $13M_J$. This threshold does not cause a sharp structural kink or permanent stellar luminosity. Interiors are largely convective and surface emission is radiative.
* Near the <hydrogen-burning minimum mass>, roughly $0.075$–$0.08M_\odot$ or $75$–$85M_J$ for near-solar composition, sustained fusion prevents indefinite cooling into a degenerate object. The <low-mass main-sequence star> branch turns upward, with approximately \b[$R\propto M$] over the illustrative interval. Its <entropy> is not constant across masses, so this branch is compatible with an approximately $n=3/2$ internal profile. Hydrogen fusion through the <proton–proton chain> provides energy; the lowest-mass main-sequence stars are fully convective, capped by radiative atmospheres.
Planet/brown-dwarf naming conventions and deuterium burning do not define a universal discontinuity in the <mass-radius relation>. Composition, age and irradiation move the curves; the hydrogen-burning transition changes the long-term energy source more fundamentally.
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