= Solution
For a local axisymmetric <Fourier mode>, the <Toomre stability criterion> balances three contributions to the squared oscillation <frequency>:
$$
\omega^2=\kappa^2+c_s^2k^2-2\pi G\Sigma_0|k|.
$$
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 $\omega_{\min}^2=\kappa^2(1-Q^{-2})$. \b[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 $\Omega^2=GM_*/r^3$ and a local disk-mass estimate $M_D\sim\pi r^2\Sigma_0$,
$$
\boxed{Q\sim\frac Hr\frac{M_*}{M_D},\qquad\text{instability when }\frac{M_D}{M_*}\gtrsim\frac Hr.}
$$
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 $M_D(<r)\simeq4\pi r^2\Sigma_0$ when its inner cutoff is negligible.
For the protosolar estimate, take $M_*\simeq M_\odot\simeq2\times10^{33}\,\mathrm g$. This <solar mass> is implicit in identifying the central star with the young Sun. The supplied constant <disk aspect ratio> gives
$$
Q(r)=\frac{0.1M_\odot}{\pi\Sigma_0(r)r^2}\simeq\frac{2\times10^{32}}{\pi\times10^3\times10^{26}}\left(\frac r{\mathrm{AU}}\right)^{-1/2}
\simeq6.4\times10^2\left(\frac r{\mathrm{AU}}\right)^{-1/2}.
$$
Using the printed approximate <astronomical unit> and <gravitational constant>, one can also obtain $\Omega(1\,\mathrm{AU})\simeq3.7\times10^{-7}\,\mathrm{s}^{-1}$ and $c_s\simeq3.7\times10^5\,\mathrm{cm\,s}^{-1}$; they give the same $Q$. The <gravitational constant> cancels from the mass form.
\b[At one astronomical unit the disk is very stable], with $Q\simeq640$. Formal extrapolation gives
$$
\boxed{r_{Q=1}\simeq(6.4\times10^2)^2\,\mathrm{AU}\simeq4\times10^5\,\mathrm{AU}.}
$$
This 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. \b[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.
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