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BlackStar
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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-α,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
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-α,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