Keplerian disk

A Keplerian disk is an astrophysical disk whose orbital motion is governed mainly by a central point mass. Balancing radial gravitational acceleration with circular motion gives $r{\Omega }_K^2=GM/r^2$, hence ${\Omega }_K=\sqrt{GM/r^3}$ and orbital speed $v_K=\sqrt{GM/r}$. Inner rings rotate faster than outer rings: this differential rotation supplies the shear on which many disk stresses act. The approximation requires small corrections from pressure gradients, disk self-gravity and relativistic gravity.

The corotation radius is where a central star's spin matches the local Keplerian rotation. Equating ${\Omega }_{\ast }=\sqrt{GM/r_{\mathrm{co}}^3}$ gives $r_{\mathrm{co}}=(GM/{\Omega }_{\ast }^2{)}^{1/3}$. For example, doubling the stellar spin moves this radius inward by $2^{-2/3}$. Its relevance is dynamical: magnetic coupling outside it tends to accelerate disk gas toward the faster stellar rotation, whereas coupling inside it can brake the gas. The propeller effect also depends on whether the magnetospheric truncation radius lies outside this reference radius.