Codex @codex  0

Assume stellar homology: fixed dimensionless profiles, uniform fixed composition, negligible radiation pressure, ideal gas support, and radiative transport throughout the model. Mass conservation gives ${\rho }_c\propto M/R^3$. The hydrostatic pressure support equation gives $P_c\propto G{M}^2/R^4$; combining with $P\propto \rho T$ gives $T_c\propto M/R$.

Integrating specific proton–proton chain energy generation over mass givesFor radiative diffusion in a star, $T_c/R\propto {\kappa }_c{\rho }_{c}L/(R^2{T}_c^3)$, hence $L_{\mathrm{rad}}\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. With the Kramers' opacity law ${\kappa }_c\propto {\rho }_c{T}_c^{-7/2}$,Equating generated and transported luminosity gives $M^{1/2}\propto R^{13/2}$, so the radiative homology with proton-proton burning and Kramers opacity relation isThe proportionality constants are fixed only within the adopted homologous family, not for arbitrary evolving stars.

Retain the same stellar homology and gas-pressure assumptions, but use the specified CNO cycle law $ϵ\propto \rho T^{16}$ and constant electron-scattering opacity. The nuclear luminosity now scales asThe radiative diffusion in a star scaling becomesEquating the two yields $R^{19}\propto M^{15}$. Therefore radiative homology with CNO burning and electron scattering givesSince $R\propto L^{5/19}$, the Stefan–Boltzmann law givesThis is a model comparison using the exponent 16 prescribed here. Actual massive stars may have appreciable radiation pressure and convective cores, which violate the assumptions of this purely radiative gas-pressure family.

The radiative diffusion in a star equation gives the homology scalingUsing the central scalings from part (i) and the electron-scattering opacity $\kappa \propto 1+X$ yieldsThe radius cancels because an electron-scattering opacity is independent of density and temperature.

Radiative diffusion in a star can be writtenDividing by hydrostatic equilibrium giveswhich is constant by assumption. Since both $P_{\mathrm{rad}}$ and $P$ vanish at the surface, integration gives $P_{\mathrm{rad}}=(1-\beta )P$ with constant $\beta$.

Eliminating $T$ between the gas and radiation equations of state now givesThus the star is an $n=3$ stellar polytrope. PutThe structure equations become the Lane-Emden equationThe surface is the first zero ${\xi }_1\simeq 6.89685$. Writingthe Lane-Emden mass formula givesThereforewhereThe additional radiative-equilibrium relation is $1-\beta =\nu L/(4\pi cGM)$.

Letbe the local isothermal sound speed squared. Steady spherical mass conservation givesThe momentum equation, with radiative acceleration represented by the radiation-pressure gradient, isRadiative diffusion in a star givesUsing $P_{\mathrm{rad}}/P_g=(1-\beta )/\beta$ andthis becomesSubstitution of $d{P}_g/dr=a_i^{2}d\rho /dr+\rho a_i^{2}d\log T/dr$ and the continuity equation yields the wind equation

Its topology is that of the Parker wind equation. The coefficient of $dv/dr$ vanishes at the sonic line $v=a_i$. A smooth transonic solution must pass through a critical point where the right-hand side also vanishes; generic subsonic solutions are breezes or turn back, while generic supersonic branches cannot be joined smoothly to a quasi-static stellar atmosphere.

At the critical point,The assumed inequality $a_i^2\ll G{m}_r/r$ makes the second factor positive and close to one. Since $0<\beta <1$,Thus acceleration through a regular sonic point requires radiation to cancel most, but not all, of the effective gravity and in particularIf the local luminosity reached or exceeded $L_{\mathrm{crit}}$ in this diffusion model, the numerator would have the wrong sign for the subsonic branch to cross the sonic line smoothly under the cold-wind assumption.

At $x=1/2$, defineDifferentiating the temperature profile givesSince the luminosity is already constant there, the radiative diffusion in a star equation yields

For the Kramers' opacity law $\kappa ={\kappa }_0\rho T^{-7/2}$, substitution giveswhereThus $\alpha =11/2$ and $\beta =-1/2$ in the notation of the question.

For constant electron-scattering opacity ${\kappa }_{\mathrm{es}}$,whereHence $\delta =3$ and $\gamma =0$: the radius cancels, and the mass dependence is shallower than under Kramers opacity.

Suppose first that the gas-pressure fraction $\beta =P_g/P$ is spatially constant, with $0<\beta <1$. This is a sufficient condition for a stellar polytrope. Sincethe ratio $P_{\mathrm{rad}}/P_g=(1-\beta )/\beta$ givesThereforewithFor an ordinary positive polytropic index one also assumes $0\le s<3$.

The radiative diffusion in a star equation can be written asBecause $P_{\mathrm{rad}}=(1-\beta )P$ and $\beta$ is constant, hydrostatic equilibrium givesEquating them and using $\eta =(L_r/L)/(m_r/M)$ yields

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.

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

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.

In the optically thick radiative diffusion in a star approximation, the heat flux is $\mathbf{F}=-\chi \nabla T$. The conductivity depends on opacity, mass density and temperature. It relates a material temperature gradient to energy transport and must not be confused with the emergent effective temperature.

A stellar core in which radiative diffusion in a star carries most of the energy, rather than convection. Its formation in a pre-main-sequence star breaks the fully convective assumption behind an idealized Hayashi track.

For gas-pressure-supported stellar homology with $\kappa \propto {\rho }^{1/2}{T}^{-5/2}$ and specific nuclear heating $ϵ\propto \rho T^5$, the central scalings give $L_{\mathrm{nuc}}\propto M^7{R}^{-8}$ and radiative diffusion in a star gives $L_{\mathrm{rad}}\propto M^5{R}^{-1}$. Equating them yields the displayed laws. The Stefan–Boltzmann law gives $T_{\mathrm{eff}}\propto M^{29/28}$, hence $d\log L/d\log T_{\mathrm{eff}}=132/29$. Composition and dimensionless profiles are held fixed.

For gas-pressure-supported stellar homology with opacity $\kappa ={\kappa }_{0}Z\rho T^{-3}$ and stellar energy-generation rate $ϵ={ϵ}_0{X}^2\rho T^5$, radiative diffusion in a star gives $L\propto R{T}_c^7/(Z{\rho }_c^2)\propto {\mu }^7{M}^5/Z$. Integrating the heating rate instead gives $L\propto X^2{\mu }^5{M}^7/R^8$. Equality in stellar thermal equilibrium yields the displayed radius law and $T_c\propto {\mu }^{5/4}{M}^{3/4}{X}^{-1/4}{Z}^{-1/8}$. This fixes the dimensionless profiles and coefficients, and neglects radiation pressure; it is a model comparison rather than a universal law for all main sequence stars. At fixed mass and metal mass fraction, homogeneous hydrogen burning gives the homogeneous fuel-depletion luminosity feedback through $L\propto {\mu }^7$.

For a chemically uniform, ideal-gas-supported stellar homology sequence with nuclear energy generation $ϵ={ϵ}_0{X}^2\rho T^4$ and opacity $\kappa ={\kappa }_{0}Z(1+X)\rho T^{-4}$, radiative diffusion in a star fixes $LRZ(1+X)/({\mu }^8{M}^6)$ and nuclear heating fixes $X^2{\mu }^4{M}^6/(R^{7}L)$. Eliminating $L$ gives $R^6\propto X^{2}Z(1+X){\mu }^{-4}$, henceThe absence of $M$ in the radius law is a consequence of these particular power laws; it is not a universal relation for real stars. At fixed initial $X$ and $\mu$, the Stefan–Boltzmann law gives $T_e\propto Z^{-1/12}$ at fixed luminosity.

In a negligible-mass white dwarf envelope, take enclosed mass and luminosity constant, neglect radiation pressure, and use an ideal gas with the Kramers' opacity law. Dividing hydrostatic pressure support equation by radiative diffusion in a star gives $PdP\propto (M/L)T^{15/2}dT$. Neglecting the surface integration constant produces the displayed pressure and mass density laws. It is a deep-envelope approximation, not a finite-temperature photospheric boundary condition.

The radiative temperature gradient needed to carry a spherical stellar luminosity follows by dividing radiative diffusion in a star by the hydrostatic pressure support equation. It is a logarithmic pressure gradient, not $dT/dr$ itself. Comparing it with the adiabatic temperature gradient by the Schwarzschild criterion determines whether purely radiative transport is locally unstable at uniform composition.