How Do You Calculate the Period of Radial Oscillations in a Neutron Star?

[June_cosmo]

Assuming a neutron star is a uniformly dense sphere of radius 10km and mass =1.4 mass of sun, derive the period of radial oscillations.First use hydrostatic equilibrium to calculate p, then the velocity of sound is $$v= \sqrt{ \gamma p / \rho}$$, so the period of pulsation is time it takes from r=0 to r=R and come back.

I first used hydrostatic equilibrium:$$ \frac{dp}{dr}=- \frac{GM(r)\rho}{r^2}$$
and $$M(r)= \frac{4}{3} \pi r^3 \rho$$ so that $$ p(r)=- \frac {2}{3} \pi \rho^2r^2$$,
so question 1: how does there is a negative value?
question 2:how do I calculate time from r=0 to r=R and back?

 

[mfb]

Don't forget the integration constant. What is the boundary condition for pressure?

question 2:how do I calculate time from r=0 to r=R and back?

That follows from the velocity.

 

[June_cosmo]

Don't forget the integration constant. What is the boundary condition for pressure?
That follows from the velocity.

Thanks for your answer. No the problem doesn't say the boundary condition. How do I know that?

 

[mfb]

No the problem doesn't say the boundary condition. How do I know that?

What do you expect as pressure at the surface?

 

[June_cosmo]

What do you expect as pressure at the surface?

Because P(r) decrease along r it would be 0? so I can derive the constant?

 

[mfb]

It is zero, as particles would move outwards otherwise. Yes, this allows to find the constant.