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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.

Solved by gpt-5.6-sol high.

Insert the self-similar ansatz into the height-integrated equations. Mass conservation is already satisfied because $H\sim c_s/{\Omega }_K\propto R$ and $\rho RH{u}_R$ is constant. The angular-momentum and energy equations reduce towhile radial momentum givesDefineSolving the quadratic gives the exact advection-dominated accretion flow coefficientsFor ${\alpha }^2\ll 1$,and therefore

Efficient cooling means $f\to 0$ and hence ${ϵ}^{\prime }=ϵ/f\to \infty$ for fixed $\gamma <5/3$. ThenThe flow is therefore cold, nearly Keplerian, slowly accreting, and geometrically thin: the standard thin-disk limit.

For significant advection, $f=O(1)$ and all three deviations are explicit:The gas is hot and thick, pressure supplies part of the radial support, rotation is sub-Keplerian, and dissipated entropy is carried inward. As $\gamma \to 5/3$, ${ϵ}^{\prime }\to 0$, so $\Omega \to 0$, $c_s^2\to (2/5)v_K^2$, and $u_R\to -(3\alpha /5)v_K$: the self-similar rotating solution approaches a hot Bondi-like inflow. Sagittarius A* is the standard supermassive example: its luminosity is tiny compared with its Eddington luminosity despite an available gas supply, and its hot optically thin spectrum and low radiative efficiency are described by an ADAF or the broader radiatively inefficient accretion-flow family.

Solved by gpt-5.6-sol high.