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A low-mass star maintained by sustained hydrogen burning, rather than the continuing cooling of a brown dwarf. Near the hydrogen-burning minimum mass its radius is of order a Jovian radius. Stars below roughly $0.35M_{\odot }$ are approximately fully convective, with a radiative photosphere; mass, composition and age affect the precise transition.

For a homologous stellar polytrope of index $3/2$, take $P=K{\rho }^{5/3}+\mathcal{R}\rho T/\mu$ and zero surface pressure. The stellar virial theorem, with $\Omega =-6G{M}^2/(7R)$, givesThe normalized density profile fixes $A$ independently of mass and radius at fixed composition. Writing ${\overline{T}}_M=\alpha /R-\delta /R^2$ shows that its maximum is ${\alpha }^2/(4\delta )=3\mu G^2{M}^{4/3}/(49\mathcal{R}A)$, at $R=7A{M}^{-1/3}/(3G)$. Below this radius contraction increases the fraction of support provided by electron degeneracy pressure and lowers the thermal temperature. A fixed normalized profile also fixes the ratio of central to mass-averaged temperature, giving a hydrogen-burning minimum mass.

For a specified age and irradiation, a hydrogen–helium sequence rises at low mass, has radii near Jupiter's across giant planets and brown dwarfs, and declines as electron degeneracy pressure becomes important. Above the hydrogen-burning minimum mass, sustained fusion supports a rising low-mass stellar branch.

The radiative family is a stylized approximation to gas-pressure-supported main sequence stars with mostly radiative interiors and moderately temperature-sensitive hydrogen burning, such as the roughly solar-mass regime. Its opacity exponents are local approximations, not a universal composition-independent opacity law. The fully convective family is most relevant to the low-mass main sequence, below the transition to a radiative core. Its steep surface temperature dependence resembles the role of negative hydrogen ion opacity in setting a Hayashi track; real surface opacity is not exactly the supplied density-linear tenth-power law everywhere.

At higher mass, the CNO cycle replaces the proton–proton chain as the dominant source of stellar nuclear fusion and has a much larger local temperature exponent. This promotes a convective core. Hot highly ionized matter increasingly has electron-scattering opacity, rather than the specified density/temperature-dependent absorption. Radiation pressure becomes important in the equation of state, changing $T_c\propto M/R$ and eventually making the luminosity relation approach an Eddington luminosity scaling. Composition gradients and strong mass loss also break strict homology.

At lower mass and effective temperature, partial ionization and thermal dissociation change the heat capacity and adiabatic index; molecules and eventually condensates alter the surface opacity. Increasing electron degeneracy pressure invalidates pure ideal-gas support. Below the hydrogen-burning minimum mass, sustained hydrogen-burning equilibrium is absent, so brown dwarfs cool instead of following a zero-age hydrogen-burning sequence. These changes explain why neither derived slope should be extrapolated indefinitely.

Retain fixed dimensionless structure constants and fixed $M$. Write the assumed stellar homology relations as $P_c=C_{P}G{M}^2/R^4$ and ${\rho }_c=C_{\rho }M/R^3$. The equation of state giveswhere $A=C_P/C_{\rho }$ and $B=C_{\rho }^{2/3}$. The first term is the temperature required for ordinary thermal pressure support; the second subtracts the support supplied by nonrelativistic electron degeneracy pressure.

As a function of $u=1/R$, this is a concave quadratic polynomial. Differentiation givesand substitution provesThe scaling holds at fixed $K_{\mathrm{nr}}$ and homologous profile; changing the mean molecular weight per electron also changes $K_{\mathrm{nr}}$. At the maximum the ion thermal pressure and electron degeneracy pressure contribute equally. Subsequent contraction lets electron degeneracy pressure carry more of the load without raising the temperature; eventually the assumed positive-temperature branch approaches a cold, degenerate configuration.

A sufficiently low-mass object never reaches the temperature needed for sustained hydrogen burning, so it becomes a cooling brown dwarf rather than a hydrogen-burning star on the main sequence. The scaling explains a hydrogen-burning minimum mass; its numerical value additionally requires the composition, detailed structure and nuclear ignition conditions, which are not specified here.

To derive the nonrelativistic electron-degeneracy pressure in a hydrogen-helium mixture, take fully ionized hydrogen and helium with hydrogen mass fraction $X$, the Electron number density isHere helium nuclei have mass $4m_p$ in the approximation used in the question. Filling the two spin states of each momentum mode up to the Fermi momentum $p_0$ givesThe isotropic momentum flux gives pressure $\frac{1}{3}\int pvdn$. For nonrelativistic Electrons $v=p/m_e$, so the electron degeneracy pressure isSubstitute the composition-dependent $n_e$ and simplify powers of two:Comparing $5/3=1+1/n$ gives the polytropic index $n=3/2$. The result assumes complete Electron degeneracy, $p_0\ll m_{e}c$, and negligible nuclear mass corrections.

Now adopt the stated approximate partially degenerate equation of state, keeping composition and hence $K,\mu$ fixed during contraction. Set $q=m/M$ and define the mass-averaged temperature ${\overline{T}}_M={\int }_0^{1}Tdq$. The stellar virial theorem from part (a) givesFor an index-$3/2$ stellar polytrope, the permitted gravitational-energy formula is $\Omega =-6G{M}^2/(7R)$. Also, using $\bar{\rho }=3M/(4\pi R^3)$,Thus, with $A=3K{\int }_0^1[3\rho /(4\pi \bar{\rho }){]}^{2/3}dq$,The normalized density $\rho /\bar{\rho }$ is a fixed function of $q$ for this homologous polytropic structure. Therefore its integral and $K$ fix $A$, which is independent of $M$ and $R$ for a common composition. The mean here is mass-weighted, rather than a volume average.

For the mass-averaged temperature maximum in a partially degenerate polytrope, put $\alpha =2\mu GM/(7\mathcal{R})$ and $\delta =\mu A{M}^{2/3}/(3\mathcal{R})$, both positive. As a function of $s=1/R$ the temperature is the concave quadraticIt reaches its maximum when $R=2\delta /\alpha =7A{M}^{-1/3}/(3G)$, withAt large radius, contraction increases the temperature as in a thermally supported star. Once Electron degeneracy provides a sufficiently large fraction of the support, further contraction can lower the thermal temperature. The formal zero-temperature radius is $R_{\mathrm{deg}}=\delta /\alpha =R_{\max }/2$, with $R_{\mathrm{deg}}\propto M^{-1/3}$. The continuation to negative temperatures would be unphysical; the adopted sequence only describes its positive-temperature part.

The contraction-limited central temperature has the same mass scaling. Indeed, if the total polytropic pressure is $P=K_{\mathrm{tot}}{\rho }^{5/3}$, the approximate equation of state gives $T=(\mu /\mathcal{R})(K_{\mathrm{tot}}-K){\rho }^{2/3}$. The fixed density profile therefore fixes $T_c/{\overline{T}}_M$. A sufficiently small mass never reaches the central temperature required for sustained hydrogen burning, producing a hydrogen-burning minimum mass. Such an object becomes a cooling, degeneracy-supported brown dwarf instead of settling onto the main sequence. More massive contracting objects can ignite hydrogen before reaching this degeneracy-limited maximum. This conclusion uses the nonrelativistic, fixed-composition model; it does not require extrapolation into the relativistic regime.

Keep the fixed stellar homology profile, so that ${\rho }_c=C_{\rho }M/R^3$ and $P_c=C_{P}G{M}^2/R^4$. Solving the partially degenerate equation of state for the central temperature giveswhere $A=C_P/C_{\rho }$ and $B=C_{\rho }^{2/3}$ are positive constants. The ion mean molecular weight is distinct from the total-particle mean molecular weight used in part (a).

Set $y=1/R$. The expression in parentheses is a concave quadratic in $y$. Its maximum occurs atSubstitution gives the contraction-limited central temperatureAt this point electron degeneracy pressure supplies half the required central pressure. Further contraction increasingly goes into the density-dependent degeneracy term rather than heating the ions. The formula applies along the positive-temperature part of the idealized homologous sequence; a negative value beyond it is not a physical temperature.

For sufficiently small mass, this maximum never reaches the temperature needed for sustained hydrogen burning. Such an object stays below the hydrogen-burning minimum mass and becomes a cooling brown dwarf, rather than an ordinary main sequence star. Some such objects burn their initial deuterium, but that finite reservoir does not provide lasting hydrogen burning.

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:

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.

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.

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:

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.

The drawing is a qualitative mature sequence, not an interpolation of a specified evolutionary model grid. Three relevant observations, using representative historical measurements, are:

On a planetary mass-radius relation, compressed rocky planets grow sublinearly, approximately $R\propto M^{0.25-0.3}$. Adding a hydrogen-helium envelope produces a rapid radius increase toward sub-Neptunes and gas giants. Around a few Jupiter masses the radius is nearly constant and then decreases as electron degeneracy pressure becomes important, approximately approaching the nonrelativistic degenerate scaling $R\propto M^{-1/3}$.

Brown dwarfs occupy roughly $13$ to $75$--$80$ Jupiter masses, with deuterium burning near the lower conventional boundary and sustained hydrogen burning beginning at the hydrogen-burning minimum mass. Low-mass main-sequence stars then have radii that increase with mass. Thus an isolated-body sketch has a rising rocky branch, a broad giant-planet/brown-dwarf radius maximum and decline, followed by a rising stellar branch.

Hot-Jupiter radius inflation places strongly irradiated hot Jupiters above the isolated giant-planet sequence. Irradiation retards cooling and contraction; additional proposed contributions include tidal heating, Ohmic dissipation, atmospheric circulation depositing energy at depth, enhanced opacity, and residual youth.