## Appendix A: shear viscosity in a clumpy disk at elastic/inelastic encountersThe shear viscosity in a differentially rotating clumpy disk, with
accounting for gravitational interactions between the clumps of mass
where , and is frequency of clump-clump collisions given by Eq. (A2) corresponds to Eq. (57) of SK when one puts because in a flat disk the long-range gravitational interactions could be neglected compared to the short-range ones. The value of is related to the characteristic time of gravitational encounters, , given by Eq. (2.3) simply by . By substituting Eq. (A2) into Eq. (A1), the latter could be written in the form: where for a Keplerian disk, for a solid-body rotating disk, and . The viscosity coefficient for clump-clump collisions, when the clump gravity is negligible [Goldreich & Tremaine 1978, Eq. (46)], can be written in the form analogous to Eq. (A3): Eq. (2.1) unifies the representation of viscosity both in the case
when interactions are inelastic, while gravitation is negligible [Eq.
(A4)] ## Appendix B: character of perturbations in a differentially rotating vs. a solid-body rotating diskLet us consider an incompressible, differentially rotating, liquid cylinder. The linearized equations of motion take the form: Substituting a perturbation in the form of the wave packet As always and therefore the condition of instability is 0, which corresponds to the growth of the leading spirals. Under this condition the instability is monotonous without any oscillations with epicyclic frequency. If the rotation is solid-body, one obtains = 0, i.e. the perturbations are stationary. The above solution has a drawback since the Note that if one would consider absolutely compressible case, i.e. in the above equations, the continuity equation takes the form: and we immediately obtain The contrast to other work is illuminating. Julian & Toomre
(1966) considered perturbations in a very thin collisionless
self-gravitating stellar disk. In fact, for such perturbations a
collisionless ## Appendix C: Velocity perturbations in a plane shear layerWe start from linearized equations for short-wave, low-frequency perturbations in a barotropic gaseous disk in the corotating frame of reference (for derivation, see Fridman 1989): Here = const is the angular velocity of the
corotating frame of reference; This system has an exact solution in the form (Fridman, 1989): For small For a plane shear layer, a similar set of equations can be derived.
The simpliest way to do this would be to put in the above system
0, Here Obviously, in a plane layer there are no any Coriolis forces at all. Nevertheless, the solution (C3) has the same form as (C1). What is relevant to the real physical meaning of our solution is conservation of vorticity (Eq. C2), which has been emphasized here. © European Southern Observatory (ESO) 1998 Online publication: August 6, 1998 |