Lada (2006), Hughes et al. (2009). This leads one to expect the transition to radial flow to occur in many of the stellar accretion disks, interior to radii given by equation (10).

Finally, we can explore the consequences of introducing an $\alpha$ prescription (Shakura & Sunyaev 1973) into condition (9), $\nu=\alpha c H$, where $c$ and $H$ are respectively the sound speed and height of the disk, with $\alpha$ a dimensionless number, yielding the dimensionless condition:

$${\mathcal M}  \left( \frac{R}{H} \right) < 6 \pi \alpha (n-1/2),$$

where ${\mathcal M}$ is the ratio of the Keplerian orbital velocity in the disk to the sound speed in the disk. Although the above condition is only valid for constant $\nu$, it illustrates the equivalence between the assumption of quasi-circular orbits for the material in the disk, and the assumption that the disk is thin. We see that the breakdown of the assumption of quasi-circular orbits (condition 9), corresponds to the point where disk is fat.

Writing ${\mathcal M}$ in astrophysical units as:

$${\mathcal M} = 1.8 \left( \frac{200 K}{T} \right)^{1/2}  \left( \frac{50 AU}{R} \right)^{1/2}  
\left( \frac{M}{0.5 M_{\odot}} \right)^{1/2},$$

we see that for values typical of what is observationally inferred for the stellar mass 