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Radial oscillations of gravitational star

  1. Mar 31, 2014 #1
    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. jcsd
  3. Mar 31, 2014 #2

    mfb

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    2016 Award

    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.
     
  4. Apr 1, 2014 #3
    Thanks i got it now
    I'd forgotten use the binomial approximation for the small perturbations , so i thought oscillations werent simple.
     
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