Star vibration frequency due to gravitation I found the following problem on a Classical Mechanics MIT problem set, which is intended to be solved by dimensional analysis:

Derive an expression for the vibration frequency of a star of mass M and
  radius R, if that vibration is caused by gravitational instabilities.

Assuming the only relevant constant is $G$, it's easy to conjecture $\frac{1}{M}\sqrt{\frac{G}{R^3}}$ using only dimensional analysis. I would like to know if this matches the actual answer to the problem.
 A: There are many different types of stellar oscillations in astroseismology. If the only restoring force is gravity, then we are talking about g-modes (buoyancy-driven modes).
The frequency of buoyancy-driven oscillations is the Brunt–Väisälä frequency
$$ N = \sqrt{-\frac{g}{\rho}\frac{\mathrm{d}\rho}{\mathrm{d}r}}, $$
where $r$ is the radial coordinate, $\rho$ is the density, and $g$ is the gravitational acceleration.
Replacing $g$ with its value at the surface, using the background state of hydrostatic equilibrium, and assuming some reasonable equation of state (say $p \propto \rho^\Gamma$ for constant $\Gamma$), some quick and dirty manipulation gets
$$ N \approx \sqrt{\frac{GM}{\Gamma R^3}}, $$
where at this level of approximation we may as well approximate $\Gamma$ as $1$. For the Sun this comes out to a frequency of about $0.1\ \mathrm{mHz}$, right about what helioseismologists predict using more sophisticated numerical treatments.
Note, though, that Solar g-modes are small enough in amplitude to be almost undetectable, so this is more a theoretical exercise. It's worth mentioning that Brunt–Väisälä frequency has more of a use than just getting at the frequency of such oscillations. If it is calculated to be imaginary in some part of a star, displacements from equilibrium grow rather than shrink, and that region of the star is linearly unstable to convection.
