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A disk flares when its disk aspect ratio increases outward. This is stronger than the statement that its height increases: a disk with $H\propto r$ has constant aspect ratio and is not flaring by this definition. The radial temperature and surface density of a disk profiles determine flaring in a specified vertical equilibrium model.

In a star-dominated Keplerian disk with $H\simeq c_s/\Omega$ and characteristic disk mass $M_D\sim \pi r^2\Sigma$, the Toomre stability criterion becomes $Q\sim (H/r)(M_{\ast }/M_D)$. Thus instability can occur at a disk-to-star mass ratio comparable to the small disk aspect ratio. The numerical coefficient depends on whether $M_D$ means a local annular estimate or integrated enclosed mass.

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.

For a local axisymmetric Fourier mode, the Toomre stability criterion balances three contributions to the squared oscillation frequency:The radial epicyclic frequency supplies rotational restoration at long wavelengths; isothermal sound speed and pressure stabilize short wavelengths; disk self-gravity destabilizes intermediate wavelengths. Minimizing over $|k|$ gives $|k|=\pi G{\Sigma }_0/c_s^2$ and ${\omega }_{\min }^2={\kappa }^2(1-Q^{-2})$. Axisymmetric gravitational instability occurs when $Q<1$.

In a centrally dominated Keplerian disk, $\kappa =\Omega$ and vertical hydrostatic equilibrium gives $c_s\simeq H\Omega$. Using ${\Omega }^2=G{M}_{\ast }/r^3$ and a local disk-mass estimate $M_D\sim \pi r^2{\Sigma }_0$,This is the disk mass form of the Toomre criterion. An actual enclosed mass depends on the radial surface density profile and changes an order-one coefficient. In particular, a profile proportional to $r^{-3/2}$ has $M_D(<r)\simeq 4\pi r^2{\Sigma }_0$ when its inner cutoff is negligible.

For the protosolar estimate, take $M_{\ast }\simeq M_{\odot }\simeq 2\times 10^{33}\mathrm{g}$. This solar mass is implicit in identifying the central star with the young Sun. The supplied constant disk aspect ratio givesUsing the printed approximate astronomical unit and gravitational constant, one can also obtain $\Omega (1\mathrm{AU})\simeq 3.7\times 10^{-7}{\mathrm{s}}^{-1}$ and $c_s\simeq 3.7\times 10^5{\mathrm{cm}\mathrm{s}}^{-1}$; they give the same $Q$. The gravitational constant cancels from the mass form.

At one astronomical unit the disk is very stable, with $Q\simeq 640$. Formal extrapolation givesThis is far beyond the planetary region and any plausible extent of the minimum-mass solar nebula. Moreover, the extrapolated enclosed disk mass is already a substantial fraction of a solar mass, so the centrally dominated approximation becomes questionable. The formal radius is not a prediction of a real unstable outer nebula. Direct gas fragmentation by gravitational instability is unlikely to have formed Solar System planets in this model. Core accretion is the more natural route; an earlier substantially more massive or colder disk would be a different model. Even $Q<1$ alone does not guarantee fragmentation, because sufficiently rapid cooling is also needed.

Use a gas-pressure-supported, optically thick alpha disk, with $\mathcal{R}=k/({\mu }_m{m}_p)$ and positive parameters. Here $\sigma$ is the Stefan–Boltzmann constant in the Stefan–Boltzmann law, and $F_{+}$ and $F_{-}$ below are heating and cooling flux estimates per disk face. Hydrostatic equilibrium and the ideal gas relation give $\mathcal{R}T\sim {\Omega }^2{H}^2$, while $\rho \sim \Sigma /H$. Integrating viscous heating over one semi-thickness and estimating radiative diffusion giveHere the stipulated negative hydrogen ion opacity exponent is $\kappa ={\kappa }_0{\rho }^{1/3}{T}^{10}$, which is a local approximation distinct from other empirical opacity fits. Substituting $T\sim {\Omega }^2{H}^2/\mathcal{R}$ and equating heating and cooling givesDropping numerical constants of order one is essential here. This is the negative-hydrogen-opacity alpha-disk thickness scaling. Its angular thickness estimate isFor a central mass $M$, Keplerian rotation gives $\Omega =(GM/r^3{)}^{1/2}$, so $H/r\propto r^{53/82}\Sigma (r{)}^{-7/41}$. The isothermal sound speed $c_s^2=\mathcal{R}T$ isAn adiabatic sound speed differs by the constant factor $\sqrt{\gamma }$ if the adiabatic index is fixed. For $\Sigma \propto r^{-p}$, constant $\alpha$, and fixed composition, the disk aspect ratio grows as $r^{(53+14p)/82}$, so there is disk flaring when $p>-53/14$; $c_s\propto r^{(6+7p)/41}$. The problem supplies no numerical parameters or specified $\Sigma (r)$, so no unique numerical $H/r$ or unconditional radial power can be inferred.

If the additional standard closure of a constant inward accretion rate well outside a zero-torque edge is intended, $\bar{\nu }\sim \alpha \Omega H^2$ and $\bar{\nu }\Sigma \simeq \dot{M}/(3\pi )$ imply $\Sigma \propto (\alpha \Omega H^2{)}^{-1}$. Eliminating it gives $H^9\propto {\Omega }^{-38/3}$ at fixed $\dot{M},\alpha$, hence $H/r\propto r^{10/9}$ and $c_s\propto r^{11/18}$. These stronger powers use that extra steady-flow assumption. All these local scalings cease to apply if the thin disk or the stated opacity regime fails.

For a gas-supported alpha disk with $\kappa \propto {\rho }^{1/3}{T}^{10}$, compare rates at fixed surface density of a disk, radius, $\alpha$ and composition on the thermal timescale of an accretion disk. Assume that the vertical dynamical timescale of a disk is shorter than that thermal time, and that the viscous timescale is longer; the separation is controlled for $\alpha \ll 1$ and small disk aspect ratio. Vertical hydrostatic equilibrium gives $H\propto T^{1/2}$, $\rho \propto T^{-1/2}$, so $\kappa \propto T^{59/6}$. Viscous heating satisfies $Q^{+}\propto T$ and radiative cooling satisfies $Q^{-}\propto T^4/(\kappa \Sigma )\propto T^{-35/6}$. At equilibrium a positive temperature perturbation changes the net heating by $(41/6)(Q_0/T)\delta T>0$. With positive effective heat capacity, it grows rather than relaxes. This is a local thermal statement and uses the fixed-column, rather than fixed-volume-density, comparison.