Radial oscillations of gravitational star

In summary, the virial equation for a spherical star with N particles interacting via gravity takes the form ##\frac{d^2I}{dt^2}=-2U+c##, where c is a constant. The angular frequency of the radial oscillation is ##\omega =\left (\frac{|U_0|}{I_0} \right )^\frac{1}{2}##, where U0 and I0 are equilibrium values. If the mass density of the star varies radially as r-α, the angular frequency is given by ##\omega =\left (\frac{(5-\alpha) GM }{(5-2\alpha)R^3} \right )^\frac{
  • #1
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,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
 
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  • #2
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.
 
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  • #3
Thanks i got it now
I'd forgotten use the binomial approximation for the small perturbations , so i thought oscillations weren't simple.
 

FAQ: Radial oscillations of gravitational star

1. What are radial oscillations of gravitational stars?

Radial oscillations of gravitational stars refer to the periodic expansion and contraction of a star's outer layers in response to changes in its gravitational force. These oscillations are caused by variations in the star's internal pressure and density, and can be observed through changes in its brightness and spectral characteristics.

2. What causes radial oscillations in gravitational stars?

Radial oscillations in gravitational stars are primarily caused by the star's internal pressure and density. As these factors change, the gravitational force on the outer layers of the star also changes, causing the star to expand and contract in a cyclical manner. Other factors, such as magnetic fields and rotation, can also contribute to these oscillations.

3. How are radial oscillations of gravitational stars studied?

Radial oscillations of gravitational stars are studied through observations of the star's brightness and spectral characteristics over time. These observations can be made using telescopes and other instruments, and can provide valuable information about the star's internal structure and dynamics. Theoretical models and simulations are also used to better understand the behavior of these oscillations.

4. Are all stars subject to radial oscillations?

No, not all stars exhibit radial oscillations. These oscillations are most commonly observed in pulsating stars, which have varying internal pressure and density due to their unstable fusion processes. Other types of stars, such as stable main sequence stars, may also experience radial oscillations on a smaller scale.

5. What can we learn from studying radial oscillations of gravitational stars?

Studying radial oscillations of gravitational stars can provide valuable insights into the internal structure and dynamics of these celestial bodies. By analyzing the patterns and characteristics of these oscillations, scientists can better understand the composition, evolution, and lifecycle of stars. This knowledge can also help us to improve our understanding of the universe and its physical laws.

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