In this problem we will perform a linear stability analysis of the fluid equations of motion for gravitational stability, with the effect of rotation included. Consider a rotating, self-gravitating cylinder, rotating with uniform angular velocity Nez, where e, is the unit vector in the z direction. Consider an adiabatic equation of state, so that 8P/8p=0", where a is the sound speed. The equation of continuity is, of course, ap/at + V. (pu) = 0
The equation for linear momentum, including the centrifugal and Coriolis forces, is now
au/at + 1/2Vu^2 + (V x u) x u = -VP/p + V + Rπ^2er - 2ex x u
and the gravitational potential satisfies Poisson's equation
V^2 = 4πGp
The first item is to characterise the equilibrium state. Assume po = constant. Po = constant, uo = 0 Show that the equilibrium is satisfied if
0 = 1/2R^2^2 + constant
and = 2πGp0.
Let us now consider linear perturbations to these equations with a specific frequency and wavenumber. That means we replace a/at with w and each a/ax with kx and similarly for y, x.
First consider the case for perturbations along the rotation axis, z, such that u1 = (0,0,u1). In this case, show that the perturbations must satisfy a dispersion relation w2 = k2a2 - 4πGp0. Derive the condition for instability and compare it to the Jeans mass we discussed in class.
Secondly, consider perturbation perpendicular to the rotation axis. In this case we can consider a Cartesian xy system. Show that the dispersion relation is now w2 = k2a2 - 4πGpo + 4π2. Does the addition of rotation make the system more stable or less stable?