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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.

The Rosseland mean opacity is the transport opacity appropriate to an optically thick region close to local thermodynamic equilibrium. If ${\kappa }_{\nu }$ is monochromatic transport opacity and $B_{\nu }(T)$ the Planck function, thenIt is a weighted harmonic mean: low-opacity frequency windows carry much of the diffusive flux. One must combine the monochromatic processes before averaging, rather than adding their separately averaged opacities.

In hot fully ionized material, electron-scattering opacity supplies an almost density- and temperature-independent contribution at fixed composition. In the nonrelativistic Thomson scattering limit, ${\kappa }_{\mathrm{es}}={\sigma }_T{n}_e/\rho \simeq 0.2(1+X)\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$. Free-free opacity, the absorption counterpart of thermal bremsstrahlung, comes from photon absorption during encounters between free electrons and ions. Its approximate mean obeys Kramers' opacity law, ${\kappa }_{\mathrm{ff}}\propto \rho T^{-7/2}$, with composition and quantum correction factors.

When ions retain bound electrons, bound-free absorption is photoionization; edges, occupations and ionization fractions strongly affect its frequency dependence. Bound-bound absorption produces absorption lines. Numerous metal lines can close low-opacity windows and produce important opacity features in partially ionized layers. Line populations depend on excitation as well as ionization, while Doppler broadening, pressure broadening and overlapping lines change the transport windows. Stimulated inverse processes must be included consistently with detailed balance and the local thermal radiation field.

In cooler hydrogen-rich atmospheres, negative hydrogen ion opacity from both bound-free and free-free processes is often important. Its strong temperature dependence helps determine convective surface relations. In still cooler layers, molecular bands, and at sufficiently low temperatures dust absorption/scattering, may be important. These processes cannot be represented by one Kramers power law over the whole density-temperature plane.

An opacity calculation therefore needs the chemical abundances, the equation of state, electron density and bound-state populations. Saha ionization equation and thermal excitation populations give useful dilute LTE estimates; high-density pressure ionization, nonideal effects and altered level populations require corrections. Composition matters even when one writes only $\kappa (\rho ,T)$: hydrogen, helium and the metal abundance pattern are additional fixed inputs.

In dense degenerate cores, thermal conduction by electrons can dominate energy transport. An equivalent conductive opacity may be introduced so that $1/{\kappa }_{\mathrm{eff}}=1/{\kappa }_R+1/{\kappa }_{\mathrm{cond}}$, because the two diffusive fluxes add; conduction is not itself a photon absorption process. Very hot regimes also require departures from the simple nonrelativistic scattering approximation. Near the optically thin photosphere, a Rosseland diffusion description alone is inadequate, and frequency-dependent radiative transfer with appropriate boundary conditions is required. These regime changes are why realistic stellar opacity tables, rather than one global power law, enter stellar structure.

Buoyancy and the convective criterion. Consider a small fluid parcel displaced outward by $\delta r$. Assume that sound waves keep its pressure equal to the ambient pressure, while heat exchange is slow enough that its displacement is an adiabatic process. Composition is uniform, so the parcel and its surroundings have the same mean molecular weight. For an ideal gas, $P\propto \rho T$, and an adiabatic parcel obeys $P\propto {\rho }^{\gamma }$. If $\delta \ln P<0$ is the pressure change on moving outward, thenThe density excess of the parcel at its new location is thereforeIf $\nabla >(\gamma -1)/\gamma$, this excess is negative: the outward-displaced parcel is lighter and accelerates farther outward. A downward displacement similarly continues downward. Conversely, a smaller gradient gives a restoring buoyancy force. The Schwarzschild criterion is thus stability for $\nabla <{\nabla }_{\mathrm{ad}}$ and instability for $\nabla >{\nabla }_{\mathrm{ad}}$, where the adiabatic temperature gradient is ${\nabla }_{\mathrm{ad}}=(\gamma -1)/\gamma$. For a monatomic gas,A composition gradient would require the additional density effect in the Ledoux criterion; it is absent in this calculation.

Opacity and pressure. The supplied opacity is characteristic of a cool hydrogen-rich stellar atmosphere dominated by negative hydrogen ion opacity. Mostly neutral hydrogen can attach electrons supplied by partially ionized metals; bound-free and free-free absorption involving the resulting $H^{-}$ ions produces a steep positive temperature dependence. The weak density dependence and strong temperature sensitivity can be represented by local power-law fits. At fixed composition the given pressure form is equivalent to $\kappa \propto {\rho }^{1/2}{T}^{17/2}$, because $P\propto \rho T$. This is an approximate fit over the relevant cool-atmosphere regime, not an exact opacity law at all temperatures; molecular opacity or hydrogen ionization eventually changes the regime.

For a geometrically thin atmosphere take the surface gravity of a star to be constant, $g_s=GM/R^2$. Combining $dP/dr=-\rho g_s$ with $d\tau /dr=-\kappa \rho$ gives hydrostatic equilibrium in optical depth,The grey atmosphere relation gives $T^8=T_e^8(3\tau +2{)}^2/16$. With negligible external pressure, $P(0)=0$, integration yieldsAn imposed nonzero external pressure would add $P(0{)}^{3/2}$; the boundary assumption is what fixes the integration constant here.

The factor in parentheses is $3\tau /[2(3\tau +2)]$. HenceThe grey-atmosphere convection onset with eighth-power temperature opacity occurs when this reaches $2/5$, soThe radiative continuation below this boundary would be unstable. Efficient convection instead keeps the deeper star approximately adiabatic.

Matching to the convective interior. Write $P=K{\rho }^{5/3}$. Eliminating density using the ideal gas law givesat fixed composition. Since ${\tau }_c$ is a fixed number, the matching temperature $T_b$ is proportional to $T_e$, while the matching pressure satisfies $P_b^{3/2}\propto g_s{T}_e^{-8}$. ConsequentlyThe polytropic mass-radius relation for index $3/2$ gives $K\propto R{M}^{1/3}$. Raising the matching relation to the ninth power and using $g_s\propto M/R^2$ therefore givesFinally the Stefan–Boltzmann law $L=4\pi R^2{\sigma }_{\mathrm{SB}}{T}_e^4$ yields the Hayashi relation with eighth-power temperature opacity,Composition and the opacity normalization are fixed in this comparison. The entropy coefficient $K$ is spatially constant in each convective model, but need not be the same for stars with different masses or radii.

Use a gas-pressure-supported, optically thick alpha disk, with $\mathcal{R}=k/({\mu }_m{m}_p)$ and positive parameters. Here $\sigma$ is the Stefan–Boltzmann constant in the Stefan–Boltzmann law, and $F_{+}$ and $F_{-}$ below are heating and cooling flux estimates per disk face. Hydrostatic equilibrium and the ideal gas relation give $\mathcal{R}T\sim {\Omega }^2{H}^2$, while $\rho \sim \Sigma /H$. Integrating viscous heating over one semi-thickness and estimating radiative diffusion giveHere the stipulated negative hydrogen ion opacity exponent is $\kappa ={\kappa }_0{\rho }^{1/3}{T}^{10}$, which is a local approximation distinct from other empirical opacity fits. Substituting $T\sim {\Omega }^2{H}^2/\mathcal{R}$ and equating heating and cooling givesDropping numerical constants of order one is essential here. This is the negative-hydrogen-opacity alpha-disk thickness scaling. Its angular thickness estimate isFor a central mass $M$, Keplerian rotation gives $\Omega =(GM/r^3{)}^{1/2}$, so $H/r\propto r^{53/82}\Sigma (r{)}^{-7/41}$. The isothermal sound speed $c_s^2=\mathcal{R}T$ isAn adiabatic sound speed differs by the constant factor $\sqrt{\gamma }$ if the adiabatic index is fixed. For $\Sigma \propto r^{-p}$, constant $\alpha$, and fixed composition, the disk aspect ratio grows as $r^{(53+14p)/82}$, so there is disk flaring when $p>-53/14$; $c_s\propto r^{(6+7p)/41}$. The problem supplies no numerical parameters or specified $\Sigma (r)$, so no unique numerical $H/r$ or unconditional radial power can be inferred.

If the additional standard closure of a constant inward accretion rate well outside a zero-torque edge is intended, $\bar{\nu }\sim \alpha \Omega H^2$ and $\bar{\nu }\Sigma \simeq \dot{M}/(3\pi )$ imply $\Sigma \propto (\alpha \Omega H^2{)}^{-1}$. Eliminating it gives $H^9\propto {\Omega }^{-38/3}$ at fixed $\dot{M},\alpha$, hence $H/r\propto r^{10/9}$ and $c_s\propto r^{11/18}$. These stronger powers use that extra steady-flow assumption. All these local scalings cease to apply if the thin disk or the stated opacity regime fails.

Assume the vertical dynamical timescale of a disk is shorter than the thermal timescale of an accretion disk, which is itself shorter than the viscous timescale. This ordering is controlled in a thin disk with $\alpha \ll 1$: $t_{\mathrm{dyn}}\sim {\Omega }^{-1}$, $t_{\mathrm{th}}\sim (\alpha \Omega {)}^{-1}$ and $t_{\nu }\sim (r/H{)}^2{t}_{\mathrm{th}}$. Therefore compare heating and cooling at fixed surface density of a disk, radius, $\alpha$ and composition. Vertical hydrostatic equilibrium adjusts on the vertical dynamical timescale of a disk, giving $H\propto T^{1/2}$ and $\rho \propto T^{-1/2}$. Consequently the stipulated negative hydrogen ion opacity varies as $\kappa \propto T^{10-1/6}=T^{59/6}$. The rates per disk face then scale asThus heating increases while cooling decreases after a positive temperature perturbation. At equilibrium $Q^{+}=Q^{-}=Q_0$, a small change gives $\delta (Q^{+}-Q^{-})=(41/6)(Q_0/T)\delta T$, which drives the perturbation further from equilibrium for positive effective heat capacity. A negative temperature perturbation likewise increases the net cooling. This proves thermal instability from negative hydrogen ion opacity within the local closure; it is the temperature-slope test of thermal stability of an accretion disk. Holding volume density rather than surface density of a disk fixed would give a different cooling exponent, but still the same qualitative instability; the fixed-column comparison is the relevant thin-disk thermal test.

For negative hydrogen ion opacity, $n=1/2$ and $s=-9$, so the exponents of the power-law opacity radiative envelope are $a=3/2$ and $b=-9/2$. Dividing the integrated pressure relation by $C{T}^b$ givesSince $a/b=-1/3$ and ${\nabla }_{\mathrm{phot}}=1/8$,It starts below the monatomic adiabatic temperature gradient and rises as the temperature increases inward.

For a monatomic ideal gas of fixed composition, the adiabatic temperature gradient is ${\nabla }_{\mathrm{ad}}=2/5$. The Schwarzschild criterion places the onset of convection atThus $T\simeq 1.11T_{\mathrm{eff}}$. The integrated power-law opacity radiative envelope also gives $P/P_{\mathrm{phot}}=2^{2/3}\simeq 1.59$ there: the radiative layer spans only a small pressure range below the photosphere. Its geometrical depth is of order $H_P\log (1.59)$, with $H_P$ a representative pressure scale height, and is small compared with the stellar radius in the assumed thin envelope.

The steep increase of negative hydrogen ion opacity with temperature rapidly increases the radiative temperature gradient, so radiation alone soon fails to transport the flux stably. Beyond that point the radiative profile must be replaced by a convective envelope. Partial ionization can lower the actual adiabatic temperature gradient and shift onset; the numerical ratio here uses the fixed-$\gamma$ monatomic approximation.