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With constant gravity and mean molecular weight, hydrostatic equilibrium gives $P_c=P_s\exp (-H_c/h)$, where $h$ is the atmospheric scale height. A flat exoplanet transmission spectrum alone does not determine $H_c$ or a unique cloud-top pressure.

For homologous, chemically uniform, gas-pressure-supported radiative stars with specific energy generation $ϵ\propto X\rho T^{13}$ and composition-dependent constant opacity, stellar homology gives $L_{\mathrm{nuc}}\propto X{\mu }^{13}{M}^{15}{R}^{-16}$ and $L_{\mathrm{rad}}\propto {\mu }^4{M}^3/\kappa$. Equating them gives the displayed scalings. Hydrogen depletion changes both mean molecular weight and electron-scattering opacity; holding composition fixed gives $R\propto M^{3/4}$ and $L\propto M^3$. The power-law burning approximation applies only while hydrogen fuel remains.

For convective contraction with power-law opacity $\kappa \propto \rho T^4$ at fixed stellar mass and opacity normalization, $T_e\propto {\mu }^{1/2}{R}^{1/8}$ and $L\propto {\mu }^2{R}^{5/2}$. Eliminating $R$ gives $L\propto {\mu }^{-8}{T}_e^{20}$. Thus a larger mean molecular weight moves the track to larger effective temperature at a given luminosity, with $T_e{|}_L\propto {\mu }^{2/5}$. Contraction decreases both temperature and luminosity for this particular opacity exponent.

With negligible heavy-element contribution to the mean molecular weight, $\mu =4/(3+5X)$ and $\kappa \propto 1+X$. At fixed mass, composition-dependent CNO homology gives $L\propto (1+X{)}^{-1}(3+5X{)}^{-4}$ and $R\propto [X(1+X){]}^{1/16}(3+5X{)}^{-9/16}$. The Stefan–Boltzmann law then gives $T_e\propto X^{-1/32}(1+X{)}^{-9/32}(3+5X{)}^{-23/32}$. From the formal $X=1$ reference, declining $X$ initially increases luminosity and effective temperature, unlike the usual redward main-sequence turn-off. A trace CNO catalyst abundance is implicit in the burning coefficient; an exactly metal-free pure-hydrogen star cannot burn through that cycle.

In an optically thick steady spherical stellar wind of constant mean molecular weight ${\mu }_{\mathrm{mol}}$, let $v$ be the outward speed and $a_g^2=k_{B}T/({\mu }_{\mathrm{mol}}{m}_u)$ the isothermal sound speed squared. Mass conservation gives ${\rho }^{\prime }/\rho =-2/r-v^{\prime }/v$, where a prime denotes differentiation with respect to radius $r$. Use $P_{\mathrm{gas}}=\rho a_g^2$ and radiative diffusion to write $P_{\mathrm{rad}}^{\prime }/\rho =-g\Gamma$, where $g=G{M}_r/r^2$ and $\Gamma =L_r/L_{\mathrm{Edd}}$ uses the local Eddington luminosity $L_{\mathrm{Edd}}=4\pi cG{M}_r/\kappa$ and opacity $\kappa$. Substituting into the momentum equation yields the displayed wind equation. Smooth sonic point crossing requires both its denominator and numerator to vanish. The radiation-pressure gradient already supplies radiative acceleration; adding that force separately would count it twice.

For a fixed-mass fully mixed hydrogen burning star with $\mu =4/(3+5X)$ whose radiative stellar homology gives $L\propto {\mu }^7$, put $s=3+5X$. The luminosity is $L=L_0(s_0/s{)}^7$ and fuel conservation gives $M{E}_H\dot{X}=-L$. Hence $d(s^8)/dt=-40L_0{s}_0^7/(M{E}_H)$, producing the displayed feedback law. This applies both to CNO homology with density-dependent inverse-cubic opacity and to radiative homology with fifth-power hydrogen burning and inverse-cubic opacity, at fixed metal mass fraction in the latter. Its physical domain ends at $X=0$, before the formal denominator vanishes. The idealization holds the coefficient normalizations fixed while stellar composition changes through the mean molecular weight and fuel abundance.

The ideal gas law is the equation of state relating the absolute pressure $P$, volume $V$, particle count $N$, and absolute thermodynamic temperature $T$ of a classical ideal gas:Here $k_B$ is the Boltzmann constant, $N_{\mathrm{A}}$ is the Avogadro constant, $R$ is the molar gas constant, and $n_{\mathrm{mol}}$ is the amount of substance in moles. The distinct symbol $n_{\mathrm{mol}}$ avoids confusing amount of substance with the number density $n=N/V$, measured in ${\mathrm{m}}^{-3}$: the same law can be written $P=n{k}_{B}T$. Use absolute pressure, relative to a vacuum, rather than an unconverted gauge pressure. ^[ref]

The exact SI definitions of the Avogadro constant and Boltzmann constant giveThis numerical value is exact; the applicability of the ideal gas model to a real gas is an approximation. ^{[ref] [ref]}

For $n_{\mathrm{mol}}=1\mathrm{mol}$, $T=300\mathrm{K}$, and $V=0.025{\mathrm{m}}^3$, direct substitution givesAt fixed temperature and particle count, doubling the volume to $0.050{\mathrm{m}}^3$ halves the pressure to approximately $49.887\mathrm{kPa}$: this is Boyle's law. At fixed volume and particle count, doubling the temperature from $300\mathrm{K}$ to $600\mathrm{K}$ doubles the pressure to approximately $199.547\mathrm{kPa}$. These ratios use kelvins, not readings in the degree Celsius scale: $300\mathrm{K}$ is $26.85{}^{\circ }\mathrm{C}$, and $600\mathrm{K}$ is $326.85{}^{\circ }\mathrm{C}$.

The curved slices show Boyle's law at three temperatures; the straight slice shows proportionality to absolute temperature at one fixed volume. They are relations among equilibrium states, rather than a prescribed process history. Compression could hold the temperature fixed through heat transfer, or raise it during adiabatic compression. To calculate a process's heat or work, supplement the ideal gas law with the process constraint, the first law of thermodynamics, and the relevant internal energy or heat capacity information. The graph's extrapolation toward $T=0$ is mathematical; it does not assert that a real gas remains classical or gaseous there.

For a stellar mixture with fixed mean molecular weight, the ideal gas law can be written $P=\mathcal{R}\rho T/\mu$, where $\rho$ is mass density and $\mathcal{R}$ is the stellar gas constant. It assumes negligible interaction corrections and a nondegenerate gas; excitation or ionization can change the particle count and thermodynamic response.

In his 1834 memoir on the motive power of heat, Émile Clapeyron combined gas laws into a relation between pressure, volume, and temperature while analysing heat engines (printed page 164). The equation above uses modern SI notation and absolute thermodynamic temperature; it is not a transcription of his historical temperature convention. ^[ref]

For a gas-pressure-supported polytrope of index five with uniform mean molecular weight, $T/T_c=\theta$. A specific stellar energy-generation rate $ϵ={ϵ}_0\rho T^q$ therefore integrates to $4\pi {ϵ}_0{\alpha }^3{\rho }_c^2{T}_c^q{\int }_0^{\infty }{\xi }^2{\theta }^{10+q}d\xi$. Substituting $u=\xi /\sqrt{3+{\xi }^2}$ gives the displayed formula, convergent for $q>-7$. At $q=11$ its dimensionless coefficient is approximately $1.02942$. The scale factor is a volume, ${\alpha }^3$, rather than an area.

The mean molecular weight per ion counts nuclei rather than nuclei plus free electrons: $\rho /({\mu }_i{m}_u)$ is the ion number density. For species with mass fractions $X_j$ and mass numbers $A_j$, ${\mu }_i^{-1}={\sum }_j{X}_j/A_j$. It is useful when ions provide a thermal ideal gas pressure but electron degeneracy pressure supplies a separate contribution.

Let $\mathcal{R}=k_B/({\mu }_m{m}_p)$ for constant mean molecular weight ${\mu }_m$ in proton-mass units, and let $\sigma$ be the Stefan–Boltzmann constant in the Stefan–Boltzmann law. The gas sound speed used here satisfies $c_s^2=\mathcal{R}T$, and $\Omega$ is the local orbital frequency. For the local opacity approximation $\kappa ={\kappa }_0{\rho }^{1/3}{T}^{10}$, a gas-supported optically thick alpha disk has $\rho \sim \Sigma /H$ and $\mathcal{R}T\sim {\Omega }^2{H}^2$. Integrated viscous heating is $F_{+}\sim \alpha \Sigma {\Omega }^3{H}^2$. Radiative diffusion gives $F_{-}\sim (\sigma /{\kappa }_0)T^{-6}{\Sigma }^{-4/3}{H}^{1/3}\sim (\sigma /{\kappa }_0){\mathcal{R}}^6{\Omega }^{-12}{\Sigma }^{-4/3}{H}^{-35/3}$. Equating the fluxes proves the thickness scaling. At fixed composition and $\alpha$, it gives $H/r\propto r^{53/82}{\Sigma }^{-7/41}$ and $c_s\propto r^{6/41}{\Sigma }^{-7/41}$ in Keplerian rotation. Radial powers therefore require a surface-density profile; they are not determined by opacity alone. The formula is local to the stated opacity regime and small disk aspect ratio.

Take homologous profiles and hold composition, hence mean molecular weight, fixed. The stellar mass conservation equation and stellar hydrostatic equation give ${\rho }_c\propto M/R^3$ and $P_c\propto G{M}^2/R^4$. With ideal gas pressure support, $T_c\propto M/R$. Integrating the specific nuclear rate through homologous profiles then givesThe radiative diffusion in a star equation gives $L_{\mathrm{rad}}\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. Here ${\kappa }_c\propto {\rho }_c^{1/2}{T}_c^{-5/2}\propto M^{-2}R$, soEquality of generated and transported luminosity implies $M^7{R}^{-8}\propto M^5{R}^{-1}$, whenceThis is the radiative homology with density-half inverse-five-halves opacity scaling.

The Stefan–Boltzmann law defines the effective temperature by $L=4\pi R^2{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4$. Therefore $T_{\mathrm{eff}}\propto M^{29/28}$. In theoretical Hertzsprung-Russell diagram coordinates, the slope isThe usual graphical convention puts increasing temperature to the left, so the line descends as one moves right. If the horizontal coordinate is $-\log T_{\mathrm{eff}}$, its plotted slope is $-132/29$.

The envelope is light and geometrically thin, so neglect its self-gravity and replace $m(r)$ and $r$ by the core's fixed $M,R$. The stellar hydrostatic equation becomes $dP/dz=-\rho g$, where $g=GM/R^2$. Since $d\Sigma /dz=-\rho$, it gives $dP/d\Sigma =g$. Neglecting the surface pressure yields $\boxed{P=g\Sigma }$.

With $F=L_r/(4\pi R^2)$, the luminosity equation gives $dF/dz=\rho ϵ$, and radiative diffusion in a star gives $dT/dz=-3\kappa \rho F/(4ac{T}^3)$. Dividing each by $d\Sigma /dz$ givesThe signs reflect that column mass increases inward: the temperature rises inward while outward flux decreases toward its source-free inner boundary.

Let $B=\mu g/\mathcal{R}$ for constant mean molecular weight. The gas equation of state and $P=g\Sigma$ give $\rho =B\Sigma /T$. With the supplied opacity and heating laws, put $y=T^7$ and $x={\Sigma }^2/2$. ThenDefine $A=21{\kappa }_{0}B/(4ac)>0$. Since $dx/d\Sigma =\Sigma$, the equations become $dy/dx=AF$ and $dF/dx=-{ϵ}_{0}B{y}^2$. Thus the inverse-square-opacity burning envelope satisfies

At the outer boundary $x=0$, the radiative-zero approximation neglects photospheric temperature relative to the burning layer, so $y=0$ and $y^{\prime }=A{F}_s=AL/(4\pi R^2)$. This is an approximation, not an exactly zero physical photospheric temperature. At the base $x_b={\Sigma }_0^2/2$, $y=y_b=T_0^7$. There is no luminosity entering from the helium core because all the luminosity is generated in the envelope, so $F(x_b)=0$ and $y^{\prime }(x_b)=0$.

Multiply the second-order equation by $y^{\prime }$ and integrate using the base conditions:Therefore the base temperature obeysThe physical branch has $y^{\prime }\ge 0$. A second integration givesThe integral is finite because its endpoint singularity is proportional to $(1-t{)}^{-1/2}$. Hence $y_b\propto x_b^{-2}\propto {\Sigma }_0^{-4}$ and $F_s\propto y_b^{3/2}\propto {\Sigma }_0^{-6}$. More explicitly,This envelope-mass comparison holds the core and the composition/opacity/heating normalizations fixed; the coefficient depends on $g$.

The steady solutions are susceptible to thin-shell instability. At fixed overlying column, base pressure is constrained by $g\Sigma$, so $\rho \propto T^{-1}$. The nuclear heating per unit mass scales as $T^{14}$, whereas the radiative cooling through a fixed-column profile scales as $T^7$: the opacity decreases as $T^{-3}$ and diffusion adds the fourth-power temperature dependence. A positive temperature perturbation therefore increases heating faster than cooling; thin-layer expansion does not supply the strong pressure relief of a whole-star expansion. The inverse relation $F_s\propto {\Sigma }_0^{-6}$ also means consuming the envelope raises the corresponding steady luminosity rather than damping the burning. This idealized model predicts runaway or flashes, despite nondegenerate gas. It is not a claim that all accreting white dwarfs are unstable: sufficiently thick burning layers, altered reaction regimes or other transport/accretion conditions can stabilize burning outside these assumptions.

Let ${\rho }_g(r)$ be the gas density and $M(<r)$ the total gravitating mass. For a spherically symmetric gas in hydrostatic equilibrium, radial force balance isUsing the ideal gas law $P={\rho }_g{k}_{B}T/(\mu m_p)$, with constant mean molecular weight $\mu$, differentiate the product rather than setting $T$ constant:Substitution and cancellation of ${\rho }_g$ give the cluster hydrostatic mass estimatorHere $k_B$ is the Boltzmann constant, denoted $K$ in the question. The prefactor has units of mass; a pressure decreasing outward makes the bracket negative and the inferred mass positive. For an isothermal gas with ${\rho }_g\propto r^{-\beta }$, this reduces to $M(<r)=\beta k_{B}Tr/(\mu m_{p}G)$. The density in the logarithmic derivative is the gas density, not the total density. A varying $\mu$ or nonthermal pressure would require the corresponding extra terms.

A stellar polytrope has a spatially constant coefficient $K$ in the relation $P=K{\rho }^{1+1/n}$; $n$ is its polytropic index. This relates pressure to mass density and need not describe an adiabatic gas. Combining hydrostatic equilibrium with mass conservation eliminates the enclosed mass:For $\rho ={\rho }_c{\theta }^n$, the polytropic equation of state gives $(1/\rho )dP/dr=(n+1)K{\rho }_c^{1/n}d\theta /dr$. Choose the Lane-Emden variables for a stellar polytrope withSubstitution then gives the dimensionless Lane-Emden equation:A regular stellar centre has $\theta (0)=1$, ${\theta }^{\prime }(0)=0$ and $m(0)=0$. In particular the regular central expansion is $\theta =1-{\xi }^2/6+O({\xi }^4)$. For an isolated finite-radius stellar polytrope with negligible external pressure, the surface is the first zero ${\xi }_1$ of $\theta$, and $R=\alpha {\xi }_1$. A core embedded in an envelope can instead end at a positive matching pressure, before this zero.

The integrated Lane-Emden equation is ${\xi }^2{\theta }^{\prime }=-{\int }_0^{\xi }{s}^2\theta (s{)}^{n}ds$. Thus mass conservation gives the Lane-Emden mass formula directly:For the proposed functional form, differentiation givesChoosing $\gamma =-1/2$ removes the extra polynomial factor; equality to $-{\theta }^5$ then requires $3\beta =1$. The regular polytrope of index five is thereforeIts enclosed mass and enclosed mean mass density areThere is no finite zero: its isolated radius is infinite. Nevertheless its total mass converges, since the outer mass density falls as $r^{-5}$:The zero mean mass density is a limiting volume average, not a statement that the material has zero mass density at finite radius.

To turn this density model into a CNO cycle burning model, use gas-dominated pressure and uniform mean molecular weight. The ideal gas law and the polytropic equation of state then imply $T/T_c=\theta$. Without this thermal assumption a polytropic equation of state for total pressure does not alone fix the temperature profile. Since the stellar energy-generation rate is power per unit mass, its contribution to luminosity is $ϵdm$, giving the luminosity integral of an index-five polytrope:Put $u=\xi /\sqrt{3+{\xi }^2}$. Then $\xi =\sqrt{3}u/\sqrt{1-u^2}$ and the dimensionless integral becomes $3\sqrt{3}{\int }_0^1{u}^2(1-u^2{)}^{8}du$. Evaluation givesThe PDF's power of $\alpha$ in the proposed luminosity scaling is a genuine dimensional error: it must be three, rather than two. The volume element supplies ${\alpha }^3$; replacing it by ${\alpha }^2$ would give power divided by length, and could not be repaired by a dimensionless numerical constant.

The polytrope of index five is unlikely to model a real burning stellar core well. An embedded core has a finite boundary, and a complete isolated star cannot have this infinite radius. More importantly, strongly temperature-sensitive CNO cycle heating concentrates the luminosity centrally and tends to produce a convection zone there. A well-mixed gas-pressure-dominated convective core is closer to an adiabatic stellar polytrope of index $3/2$ than to index five. A finite truncation would also change the integral defining $A$; its value above pertains to the full mathematical model.

In stellar homology, write $r=Rx$, $m=Mf(x)$ and $\rho ={\rho }_{c}h(x)$, with common dimensionless profiles. Integrating hydrostatic equilibrium from the surface, where the pressure is negligible, givesThe profile integral is common to the sequence. Similarly, mass conservation gives $M=4\pi {\rho }_c{R}^3{\int }_0^1{x}^{2}h(x)dx$. These prove the homology scaling of central stellar pressure:The ideal gas law consequently gives $T_c\propto \mu M/R$, with the gravitational and gas constants absorbed into the coefficient.

The radiative diffusion in a star equation has a characteristic gradient $T_c/R$. Hence $L\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. Using Kramers' opacity law, and comparing fixed-composition homologous models, this becomesThe volume integral of the stellar energy-generation rate $ϵ={ϵ}_0\rho T^{\nu }$ instead givesThe dimensionless integrals behind these relations do not depend on $M$ along a homologous sequence. In stellar thermal equilibrium the two luminosities agree, proving Kramers homology with a variable nuclear exponent:For the specified proton–proton chain exponent $\nu =7/2$, the mass factor cancels. ThusThe model captures two important features of the lower main sequence: hydrogen burning primarily uses the proton–proton chain, and gas rather than radiation pressure supplies the principal support. A radiative interior and a Kramers' opacity law approximation are useful near the Sun. It is a simplified local model: the Sun has a convective envelope, the coolest main sequence stars become fully convective, and real opacities and burning exponents vary. In particular a literally constant radius is not a realistic law over the entire lower main sequence.

On this proton–proton chain branch $T_c\propto M$, because $R$ and mean molecular weight are fixed. Normalizing to the Sun gives the upper mass estimateFor the specified CNO cycle exponent $\nu =23/2$, the homology relation instead reads $R^{14}\propto M^8$. Substituting into the radiative luminosity relation givesThis branch retains the deliberately idealized fully radiative assumptions, even though real CNO cycle cores are often convective.

For a Hertzsprung-Russell diagram, use the Stefan–Boltzmann law, $L=4\pi R^2\sigma T_{\mathrm{eff}}^4$. On the proton–proton chain branch $T_{\mathrm{eff}}\propto M^{11/8}$, so $L\propto T_{\mathrm{eff}}^4$. On the CNO cycle branch $T_{\mathrm{eff}}\propto M^{57/56}$, giving $L\propto T_{\mathrm{eff}}^{292/57}$. Thus the upper branch rises slightly more steeply toward the hot, luminous upper left in a logarithmic Hertzsprung-Russell diagram. The sketch below matches the approximate branches continuously at their respective burning crossovers; the true crossover has both heating terms and is smooth.

The proton–proton chain begins with reactions between hydrogen nuclei, so its specific heating scales as $X_H^2\rho T^{7/2}$ in this approximation. The catalytic CNO cycle involves hydrogen and CNO nuclei, and scales as $X_H{Z}_{\mathrm{CNO}}\rho T^{23/2}$. Here $X_H$ is the hydrogen mass fraction and $Z_{\mathrm{CNO}}$ the CNO mass fraction, not the total abundance of every metal. ConsequentlyKeeping $X_H$ fixed while dividing $Z_{\mathrm{CNO}}$ by 256 gives the CNO crossover temperatureThe mass estimate uses the unchanged proton–proton chain branch below crossover, together with the stipulated unchanged opacity and mean molecular weight.

The low-CNO proton–proton chain sequence therefore lies on the same locus, but extends to twice the transition mass. Beyond its later crossover the low-CNO CNO cycle sequence has the same logarithmic slope $292/57$ but a different normalization. At a fixed mass where both sequences are CNO-dominated, lowering the nuclear coefficient by 256 givesThus those low-CNO models are smaller, hotter and modestly more luminous at the same mass. At the same effective temperature, their CNO branch is lower in luminosity than the solar-composition CNO branch; these comparisons hold different quantities fixed.

For fully ionized hydrogen and helium with mass fractions $X$ and $Y=1-X$, the inverse mean molecular weight counts nuclei plus free electrons:Thus ${\mu }_1=8/13$ for $X_1=0.7$, ${\mu }_2=4/7$ for $X_2=0.8$, and ${\mu }_1/{\mu }_2=14/13$. Keep the same $M,{\kappa }_0$ and dimensionless structure constants when comparing these toy sequences. From part (a),Eliminating $R$ gives the composition shift of a convective pre-main-sequence track:Consequently the tracks in a Hertzsprung-Russell diagram are almost vertical, with $d\log L/d\log T_e=20$. At equal luminosity, $T_{e,1}/T_{e,2}=(14/13{)}^{2/5}\simeq 1.030$: the $X=0.7$ track lies to the hotter, left-hand side of the $X=0.8$ track. Since contraction decreases $R$, both $L$ and $T_e$ decrease in this particular opacity model. Both arrows therefore point downward and slightly rightward on the conventional hot-left diagram.

The plotted reference scales are arbitrary; the horizontal separation is the predicted $\Delta {\log }_{10}{T}_e=(2/5){\log }_{10}(14/13)$. They are schematic Hayashi tracks under the specified atmospheric opacity law, rather than calibrated stellar models.

A protostar leaves this contraction sequence when its assumptions fail. Increasing central stellar temperature makes stellar nuclear fusion significant: burning of deuterium can temporarily slow contraction, and sustained hydrogen burning brings the star to the main sequence. In a sufficiently massive pre-main-sequence star, a radiative core develops, the interior ceases to share one adiabat, and evolution can proceed along a hotter Henyey track. Changing atmospheric opacity, partial ionization, or substantial continuing accretion also changes the predicted path. For sufficiently low mass, electron degeneracy pressure instead limits the increase of central temperature, as in part (c).

After complete ionization, each hydrogen nucleus supplies one ion and one Electron, while each helium-4 nucleus supplies one ion and two Electrons. Neglecting the small metal mass fraction $Z$ and putting $Y\simeq 1-X$, the mean molecular weight therefore satisfiesUse stellar homology to compare models with common dimensionless profiles. The stellar mass conservation equation, stellar hydrostatic equation and ideal gas law giveThe radiative diffusion in a star equation then gives $L_{\mathrm{rad}}\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. Inserting the specified opacity yields $L_{\mathrm{rad}}\propto R{T}_c^7/(Z{\rho }_c^2)\propto {\mu }^7{M}^5/Z$. Independently, integrating the stellar energy-generation rate over the mass gives $L_{\mathrm{nuc}}\propto X^2{\rho }_c{T}_c^{5}M\propto X^2{\mu }^5{M}^7/R^8$. Equating these in stellar thermal equilibrium gives the radiative homology with fifth-power hydrogen burning and inverse-cubic opacity:At fixed solar mass these reduce to $L\propto {\mu }^7/Z$ and $T_c\propto {\mu }^{5/4}/(X^{1/4}{Z}^{1/8})$. The stellar homology approximation, rather than merely the local power laws, is what permits these relations between whole stars.

Equal luminosities imply ${\mu }_2/{\mu }_1=(Z_2/Z_1{)}^{1/7}=2^{-1/7}$. Thusso $X_2=0.8$ to one significant figure. Retaining the unrounded hydrogen mass fraction when comparing central temperatures givesThe first model has the higher central temperature.

For homogeneous hydrogen burning at fixed $M,Z$, fuel conservation gives $M{E}_0\dot{X}=-L$. Since $d\mu /dX=-5{\mu }^2/4$ and $L=L_0(\mu /{\mu }_0{)}^7$, the homogeneous fuel-depletion luminosity feedback obeysIntegration with the initial mean molecular weight ${\mu }_0$ givesThis idealized stellar evolution law stops when $X=0$; it does not predict a physically attainable infinite luminosity. Indeed $\mu =4/3$ at fuel exhaustion, giving $t_H=M{E}_0[1-(3{\mu }_0/4{)}^8]/(10{\mu }_0{L}_0)$, strictly before the formal singularity.

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.