dimensionless units. Finally, we give our conclusions in section 5,

The transition to Radial Flow
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We start from a standard thin accretion disk where material orbits on quasi-circular Keplerian orbits around a central star of mass M. Assuming axial symmetry and cylindrical coordinates, the total angular momentum of a ring at radius $R$ of width $\Delta R$ will be:

$$L=2 \pi R^{3} \Delta R \Sigma(R) \Omega(R)$$

where $\Sigma$ and $\Omega$ are the surface density and orbital frequency profiles of the disk, respectively. If the breaking torque on a given ring of the disk, due to its exterior ring is denoted by $\tau(R)$, the total breaking torque on the ring at radius $R$ will be:

$$\tau_{B} = \tau(R+ \Delta R) - \tau(R).$$

The rate of change of the angular momentum of the ring in question can now be calculated as:

$$\dot{L}=~\tau_{B}=~\frac{d\tau}{dR} \Delta R.$$

If at any radius a substantial fraction of the angular momentum of the ring is lost through viscous torques over an orbital period, the assumption of quasi-circular orbits will break down, and the disk will make a transition to mostly radial orbits. This condition can be stated as:

$$\frac{L}{2 \dot{L}}< \frac{2 \pi}{\Omega}.$$

Substitution of equations (1) and 