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Introduce the dimensionless variablesUsing $P=\Sigma H^2{\Omega }_z^2$, vertical hydrostatic equilibrium becomes the parameter-free equationwith normalizations $\int \tilde{\rho }d\tilde{z}=1$ and $\int \tilde{p}d\tilde{z}=1$. For an isothermal atmosphere, $p=c_s^2\rho$, and these conventions give the Gaussian distribution

The standard Shakura--Sunyaev thin disk has three qualitative radial zones. Its hot inner part is dominated by radiation pressure and electron-scattering opacity; farther out, gas pressure overtakes radiation pressure while electron scattering can remain the main opacity; in the cool outer zone, gas pressure remains dominant and free-free opacity becomes important. The transition radii vary with $M$, $\dot{M}$, and $\alpha$.

For the inner zone, hold the surface density $\Sigma$ fixed during a local thermal perturbation. Vertical hydrostatic equilibrium givesso $H\propto T_c^4/\Sigma$. The alpha disk prescription then yieldswhereas optically thick radiative diffusion with nearly constant electron-scattering opacity givesAt equilibrium $Q^{+}=Q^{-}=Q_0$. For net cooling $\dot{Q}=Q^{-}-Q^{+}$,Thus the total-pressure alpha prescription predicts thermal instability of a radiation-pressure-dominated alpha disk: a temperature increase makes heating outrun cooling. The absence of ubiquitous, large-amplitude thermal limit cycles in luminous AGN light curves indicates that this local model omits stabilizing effects, plausibly magnetic pressure and stress, vertical advection, winds, or a stress law that does not simply track total pressure.

For a family in stellar homology, dimensionless radial profiles are fixed. Mass conservation and hydrostatic equilibrium then give the central scalingsThe ideal gas equation of state, with fixed mean molecular weight, consequently giveswhere fixed dimensional constants such as $G$ are suppressed in homology relations.

The proton–proton chain law $ϵ\propto \rho T^{7/2}$ gives the nuclear luminosityOn the other hand, radiative diffusion in a star givesUsing the Kramers opacity law ${\kappa }_c\propto {\rho }_c{T}_c^{-7/2}$,Thermal equilibrium requires $L_{\mathrm{nuc}}=L_{\mathrm{rad}}$. Their common mass factor cancels, leaving $R^{-13/2}\propto R^{-1/2}$, and hence

This model captures the gas-pressure support, pp-chain burning, and strongly mass-dependent luminosity of the lower main sequence, including the Sun approximately. Its fully radiative assumption is an idealization: the Sun has a convective envelope, and sufficiently low-mass red dwarfs become largely or fully convective, so real radii are not exactly constant.

Plane-parallel hydrostatic equilibrium gives $dP/dz=-\rho g$, while $d\tau =-\kappa \rho dz$. HenceFor $\kappa ={\kappa }_0{P}^{\alpha -1}{T}^{4-4\beta }$,The grey-atmosphere relation from part (ii) can be writtenIntegrating from the zero-pressure top, where $T=T_0$, gives

The natural matching point to the stellar interior is the photosphere $\tau =2/3$, where $T=T_e$ and $T_e^4=2T_0^4$. Multiplying the preceding expression by ${\kappa }_0{T}^{4-4\beta }/g$ then yields the stellar surface boundary conditionThe local relation $L_r=4\pi r^2\sigma T^4$ identifies this matching temperature with the local effective temperature.

Because $H\ll R$ and $M_{\mathrm{env}}\ll M$, the envelope has nearly constant area $4\pi R^2$ and nearly constant surface gravity of a star $g=GM/R^2$. Plane-parallel hydrostatic equilibrium isSince $d\Sigma /dz=-\rho$, one has $dP/d\Sigma =g$. Taking the pressure to vanish with $\Sigma$ at the surface gives

The spherical stellar energy equation is $d{L}_r/dr=4\pi r^2\rho ϵ$. With $r\simeq R$ and $F=L_r/(4\pi R^2)$,Division by $d\Sigma /dr=-\rho$ givesLikewise, radiative diffusion in a star givesand therefore

At the midplane, radial hydrostatic equilibrium givesIf $d{P}_0/dr<0$, then $\Omega <{\Omega }_K$: the pressure gradient provides part of the inward force and the gas is Sub-Keplerian. For radial pressure scale comparable to $r$,Using $H\sim c_s/{\Omega }_K$ and ${\Omega }_K-\Omega \simeq ({\Omega }_K^2-{\Omega }^2)/(2{\Omega }_K)$,

For $|z|\ll r$, the gravitational potential has the Taylor expansionHence the vertical gravity is $-{\partial }_z\Phi \simeq -{\Omega }_K^{2}z$, where ${\Omega }_K^2=G{M}_{\star }/r^3$. Vertical hydrostatic equilibrium balances a pressure gradient of order $P/H$ against $\rho {\Omega }_K^{2}H$. Since $c_s^2\sim P/\rho$,The thin disk condition $H/r\ll 1$ is therefore equivalent to a highly supersonic orbital speed.

Hydrostatic equilibrium requiresFor the polytrope $p=K{\rho }^{1+1/n}$, the specific enthalpy isThus $h+\Phi$ is constant. Since $\rho =p=0$ at $r=R$, this constant is ${\Omega }^2{R}^2/2$, andfor $0\le r\le R$, with both fields zero outside the model star.

For a homologous star, hydrostatic equilibrium, mass conservation, and the ideal gas equation give the central scalingswhere $\mu$ is the mean molecular weight. Integrating the nuclear energy-generation law over a fixed homologous profile givesRadiative stellar structure gives independentlyEquating the two luminosities yieldsAt fixed zero-age composition, the stars are therefore homologous with

The effective temperature satisfies $L=4\pi R^2\sigma T_e^4$, so $T_e\propto M^{3/8}$. The zero-age main sequence consequently hasIt is a steep line rising toward high luminosity and high temperature on a Hertzsprung-Russell diagram.

For fully ionized hydrogen and helium with $Y=1-X$,while $\kappa \propto 1+X$. At fixed mass,and the corresponding radius relation is

At the pure-hydrogen zero-age point $X=1$,whereasHenceandAs hydrogen is consumed, $X$ falls, so both $L$ and $T_e$ rise. In the usual diagram with temperature increasing leftward, the evolutionary track initially moves upward and leftward from the zero-age main sequence.

Let $c_s^2\sim P_g/\rho$. The assumed magnetic pressure givesso the plasma beta obeys $\boxed{\beta =P_g/P_m\sim {\mathrm{Ma}}^{-1}}$. In the magnetically dominated regime, vertical hydrostatic equilibrium gives $P_m/\rho \sim {\Omega }^2{H}^2$. Combining this with $P_m/\rho \sim r\Omega c_s$ yields

Let $p=K{\rho }^{1+1/m}$. Hydrostatic equilibrium in the uniform gravitational field givesSubstitution and integration with $\rho =0$ at $z=0$ giveswhere $H$ fixes the normalization. Integrating the hydrostatic equation from $z$ to the free surface gives the pressure directly:This indeed has $p\propto {\rho }^{1+1/m}$ and $p(0)=0$, as required for a polytropic atmosphere.

A stellar polytrope of index $n$ obeysCombining hydrostatic equilibrium, $dP/dr=-Gm\rho /r^2$, with mass conservation, $dm/dr=4\pi r^2\rho$, gives the Lane-Emden equationRegularity and normalization at the stellar centre requireIntegrating the Lane-Emden equation from the centre then gives the enclosed mass

For $n=5$, direct substitution into the equation findsso $\beta =1/3$ and $\gamma =-1/2$. This profile has no finite first zero and hence has infinite radius, but its total mass converges:Its mean density over the full, infinite configuration is consequently zero.

For a perfect gas, $T\propto P/\rho \propto {\rho }^{1/5}$, so $T/T_c=\theta$. The luminosity from CNO cycle burning with $ϵ={ϵ}_0\rho T^{11}$ is thereforewhereThus $A$ is of order unity. The model is physically poor because the $n=5$ polytrope has infinite radius and zero mean density, while strongly temperature-sensitive CNO burning changes the thermal gradient and commonly creates convection. Real cores also have evolving composition, non-polytropic opacity and energy transport, and boundaries supplied by the surrounding star.

In stellar homology, two stars have identical dimensionless density, pressure, temperature, and luminosity profiles. Mass conservation, hydrostatic equilibrium, and the perfect-gas equation of state then giveThe chemical composition is initially fixed here, so $\mu$ is constant.

For opacity $\kappa ={\kappa }_0{\rho }^a{T}^b$, radiative diffusion gives the homology scalingNuclear burning with $ϵ={ϵ}_0\rho T^n$ instead givesEquating the two luminosities yieldsand the mass-luminosity relationFor usual main-sequence opacity and burning laws, $\eta >1$, so massive stars are much more luminous and exhaust a fuel supply proportional to $M$ in a time $t_{\ast }\propto M/L\propto M^{1-\eta }$.

At sufficiently high mass, radiation pressure and radiative acceleration become important. Hydrostatic balance requires the luminosity to remain below the Eddington luminosityThe upper envelope is linear in $M$, so the relation must flatten toward slope one; the corresponding nuclear lifetime approaches a weakly mass-dependent or roughly constant value rather than continuing the steep decline predicted by gas-pressure homology.

During core hydrogen burning, conversion of hydrogen into helium reduces the number of free particles per unit mass and raises the mean molecular weight. The core contracts and heats to retain pressure support, while the envelope expands and the luminosity generally rises. Homology ultimately fails because composition becomes strongly nonuniform, an inert helium core and hydrogen-burning shell appear, and the core and envelope acquire qualitatively different equations of state, transport regimes, and radial scales as the star leaves the main sequence.

The first equation is vertical hydrostatic equilibrium. In a thin disk, stellar gravity has vertical component $-{\Omega }^{2}z$, giving $dP/dz=-\rho {\Omega }^{2}z$. The second is local energy conservation: Keplerian shear has $rd\Omega /dr=-3\Omega /2$, so viscous dissipation per unit volume isThe third equation is radiative diffusion. With radiation energy density $a_{\mathrm{R}}{T}^4=4\sigma T^4/c$, an optically thick medium carrieswhich rearranges to the stated gradient. Thomson scattering makes $\kappa$ approximately independent of density and temperature. The final equation adds perfect-gas pressure $k\rho T/({\mu }_m{m}_p)$ and radiation pressure $a_{\mathrm{R}}{T}^4/3=4\sigma T^4/(3c)$.

Photoionization can heat a shallow halo's gas well above its virial temperature. If the heated sound speed exceeds the escape-speed scale, the previous hydrostatic equilibrium is lost and a pressure-driven outflow removes gas. A crossing time gives a useful estimate, but photon supply, the gas profile, shielding and recombinations can delay the process. This particularly affects haloes whose virial temperatures are below the usual atomic cooling threshold and reduces their subsequent star formation.

In the exact constant-opacity, pure-radiation pressure diffusion model, dynamic viscosity is height-independent. Imposing an additional exact pointwise relation $\mu \propto p(z)$ forces constant pressure, incompatible with nonzero vertical hydrostatic equilibrium away from the midplane. The characteristic-pressure alpha disk prescription can still be used for order-of-magnitude scalings; it is not an exact simultaneous vertical solution.

A stellar polytrope with polytropic index one has $\nabla \Phi =-2K\nabla \rho$ in hydrostatic equilibrium. Combining this with the Poisson equation for Newtonian gravity gives the Helmholtz equationThe spherical solution regular at the origin is $\rho (r)={\rho }_c\sin (kr)/(kr)$. Its first zero gives $R=\pi /k=\sqrt{\pi K/(2G)}$, independent of central mass density, and $\bar{\rho }/{\rho }_c=3/{\pi }^2$.

A thin radiative envelope with constant enclosed mass, luminosity and mean molecular weight, ideal gas pressure and opacity $\kappa ={\kappa }_0{\rho }^n{T}^{-s}$ has $a=n+1$, $b=n+s+4$, and $C=16\pi a_{\mathrm{rad}}cGM/[3{\kappa }_0(\mu m_u/k_B{)}^{n}L]$. Combining radiative diffusion in a star with hydrostatic equilibrium gives ${\nabla }_{\mathrm{rad}}=P^a/(C{T}^b)$, so $P^a-P_0^a=(aC/b)(T^b-T_0^b)$ for nonzero exponents. Zero exponents give logarithmic limits. The model must be stopped or modified if the Schwarzschild criterion predicts convection.

A pressure scale height is the local distance over which pressure would change by a factor $e$ if its logarithmic gradient were constant. In hydrostatic equilibrium, $H_P=P/(\rho g)$; for an ideal gas it is $k_{B}T/(\mu m_{u}g)$. A small logarithmic pressure interval has depth approximately $H_P\Delta \log P$.

The inner Shakura--Sunyaev thin disk is supported mainly by radiation pressure and has nearly constant electron-scattering opacity. Equating radiative diffusion with local dissipation and imposing vertical hydrostatic equilibrium gives the characteristic half-thicknessThe radiation-supported height approaches a plateau away from the inner boundary. The formal zero at that boundary is outside the reliable radiation-dominated approximation. At an Eddington accretion rate defined using efficiency $0.1$, this height can make the inner flow only marginally thin.