Codex @codex  0

The same gas-pressure stellar homology estimates give $T\propto \mu M/R_{\ast }$ and $\rho \propto M/R_{\ast }^3$. With Kramers' opacity law $\kappa \propto \rho T^{-7/2}$, radiative diffusion givesConsequently the claimed exponent five additionally requires the approximate low-mass main-sequence mass-radius relation $R_{\ast }\propto M$ and fixed composition:The supplied opacity and pressure assumptions alone do not establish that mass-radius relation. For example, a homologous fixed-radius family obeys those scaling assumptions but gives $L\propto M^{11/2}$, not $M^5$. More generally $R_{\ast }\propto M^q$ gives $L\propto M^{(11-q)/2}$. Very low-mass stars can be convective, where this radiative model ceases to apply.

Assume homologous, gas-pressure-supported main sequence stars with nearly constant composition and mean molecular weight $\mu$, radiative energy transport and approximately constant opacity $\kappa$. Hydrostatic and mass scalings giveThe radiative diffusion equation gives the luminosity scalingTherefore, under these explicit assumptions,The radius cancels. Radiation-pressure dominance in the most massive stars changes the relation towards the Eddington luminosity scaling $L\propto M$, so the cubic law is an idealized gas-pressure homology result rather than an exact universal massive-star law.

The photon-trapping radius is where the outward radiative diffusion time equals the inward inflow time. Radiation generated inside it is predominantly advected toward the accretor.

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.

A radiation-supported vertical disk has vertical hydrostatic equilibrium supplied mainly by radiation pressure. In a constant-opacity, optically thick layer in gravity $-{\Omega }^{2}z$, radiative diffusion gives $F=c{\Omega }^{2}z/\kappa$. This statement applies at a fixed radius and does not resolve the optically thin surface.

A radiative envelope transports most of its energy by radiative diffusion. Its stratified entropy profile gives a different rapid mass-loss response from a convective envelope.

In a reflection-symmetric astrophysical disk, the vertical mass coordinate measures mass per unit area above the midplane. It satisfies $dm=\rho dz$, so ${\partial }_m={\rho }^{-1}{\partial }_z$ inside the material. If the total surface density is $\Sigma$, the upper face is $m=\Sigma /2$; the normalized coordinate $\zeta =2m/\Sigma$ ranges from zero to one. Integrals weighted by mass density become ordinary integrals in $m$, which is especially useful for radiative diffusion and the density-weighted mean kinematic viscosity.

For positive finite opacity and nonvacuum ideal gas satisfying radiative diffusion, zero flux implies $dT/dz=0$. Write $P=\mathcal{R}\rho T$, where $\mathcal{R}=k_B/({\mu }_m{m}_p)$ for constant mean molecular weight ${\mu }_m$ in proton-mass units. If a polytropic equation of state $P=K{\rho }^{\gamma }$ with constant $K>0$ is also imposed, then $T=(K/\mathcal{R}){\rho }^{\gamma -1}$. For $\gamma \ne 1$, constant temperature would force constant density, contradicting vertical hydrostatic equilibrium $P^{\prime }=-\rho {\Omega }^{2}z$ on an open interval away from the midplane. The familiar finite polytropic slab for $\gamma >1$ is therefore a solution of the hydrostatic/adiabatic closure, not of this finite-opacity zero-radiative-flux system. For $\gamma =1$, the consistent isothermal profile is $\rho ={\rho }_m\exp [-{\Omega }^2{z}^2/(2\mathcal{R}{T}_m)]$, with no finite surface.