Homework Help: Radial oscillations of gravitational star

1. Mar 31, 2014

BlackStar

Consider a spherical star made of N (very large number) particles interacting via gravity.Let the mass of ith particle be mi and position be xi
Let $I= \sum_{i=1}^{N}m_{i}r_{i}$, U be potential energy and K be kinetic energy

1)Show that the virial equation takes the form $\frac{d^2I}{dt^2}=-2U+c$
where c is a constant
2)The star undergoes small oscillations with radial displacement proportional to radial distance(ri).Show the angular frequency of the radial oscillation is
$\omega =\left (\frac{|U_0|}{I_0} \right )^\frac{1}{2}$
where U0 and I0 are equillibrium values.
3) If the mass density of the star varies radially as r,show that
$\omega =\left (\frac{(5-\alpha) GM }{(5-2\alpha)R^3} \right )^\frac{1}{2}$
where M is total mass and R is radius of the star.

I got the first part (straightforward) but not the other two.
Source:Newtonian Dynamics, Richard Fitzpatrick

2. Mar 31, 2014

Staff: Mentor

Looks like a homework problem to me.

Here is a hint: How do U and I scale if the whole star gets smaller/larger by some constant factor?
Can you use this to express U in terms of I, U0 and I0?

For (C): you can calculate U as function of I.

3. Apr 1, 2014

BlackStar

Thanks i got it now
I'd forgotten use the binomial approximation for the small perturbations , so i thought oscillations werent simple.