Is the speed of sound almost as high as the speed of light in neutron stars? Have you ever wondered about the elastic properties of neutron stars?
Such stars, being immensely dense, in which neutrons are bound together by the strong nuclear force on top of the strong gravity that “presses” them together, one would think they must have extremely large Young modulus, and the speed of sound could be on a par with the speed of light in the vacuum. 
If we let $c_s$ be the speed of sound, and also assume that the neutron star is isotropic, then using the well known equation for the speed for acoustic waves in solids, we can write the following equation for the crust of the neutron star
$c_s=\sqrt{\frac{E}{\rho}}$ 
For a  neutron star of density $\rho =5.9\times 10^{17}$ Kg m$^{-3}$ and Young modulus of about $E=5.3\times 10^{30}$Pa we get a value for $c_s=3.0\times 10^6$ ms$^{-1}$!  
The Questions are:
1) How can sound travel at such immense speeds inside a neutron star?  
2) Should nuclear interactions, n-n and q-q , dictate the elastic properties of a neutron star, or is it just the gravitational force?
 A: The neutron star crust is a solid and there are indeed elastic waves for which the speed of sound is controlled by the shear modulus. I'm not sure where you got your estimate of the shear modulus from (there is some literature on the subject, see for example http://arxiv.org/abs/1104.0173).
Most of the neutron star is a liquid, and the speed of sound is given by the usual hydrodynamic result
$$c_s^2=\left(\frac{\partial P}{\partial\rho}\right)_{s}$$
In dilute, weakly interacting neutron matter the speed of sound
(in units of the speed of light $c$) is
$$
c_s^2 = \frac{1}{3}\frac{k_F}{\sqrt{k_F^2+m^2}}$$
where the Fermi momentum $k_F$ is determined by the density,
$$
 n = \frac{k_F^3}{3\pi^2}$$
In the high density (relativistic) limit the speed of sound approaches $c/\sqrt{3}$. In the center of a neutron star you get quite close to this. Interaction change this result by factors of order one (indeed, recent  observations of neutron star masses and radii suggest that the speed of sound near the center is close to $c$), but as an order of magnitude estimate these
simple results are quite good. 
A: The equation of state for a neutron star depends on its density.  Under some conditions, the star's support is dominated by the strong nuclear force (such that $P\propto n^2$.) As density increases, pressure increases less strongly as a function of density, as nuclear forces give way to neutron degeneracy.  How exactly this equation of state might communicate pressure fluctuations is not well-defined.  Since degenerate neutrons are essentially frictionless, they may actually transmit such fluctuations quite a bit faster than the $0.01 c$ you have suggested.  (Again, I'm not sure if a Young's modulus is even a meaningful quantity within a neutron star; perhaps it would be in an iron shell around the neutronic core, but that doesn't seem to be what you are interested in).  More information on the subject of neutron stars and equations of state is here.
A: Velocity of sound can never exceed velocity of light in neutron stars.
There are some work available on the elastic properties of crystalline structure of neutron stars, I personally am not on a level to understand it at the moment but just think how sound is produced.
For production of sound waves in any matter, one particle has to send information to the next particle, this transmission of information cannot exceed the velocity of light, because the elementary particles communicate through the transfer of photons at least what we know right now.
Also as the velocity of neutrons or protons in itself cannot be equal to the velocity of light, so it is highly unlikely that even in solid core of neutron stars the velocity of sound will reach to the velocity of light.
But we know something about supersonic velocities and we also know that shock waves can travel faster than sonic velocities in a medium.
Considering the point, even if velocity of sound becomes equal to the velocity of light then there is a huge possibility for superluminal velocities as well just like supersonic, for these shock waves and this is pointless.
There have been some work however done on the maximum velocity of sound in any medium, am not sure if this sonic limit will hold forever, you can check it out here:
https://www.livescience.com/fastest-speed-of-sound-measured.html
A: Bad question- the speed of light should be rephrased as "Speed of electromagnetic radiation" and then specify what wavelength of radiation we're talking about and which part of the neutron star is being discussed.  Nobody has determined the EM properties of the inside of neutron stars so the question is not on target.
Different parts of a neutron star have different properties - see http://en.wikipedia.org/wiki/Neutron_star
Visible light might even travel slower than sound waves at some points in the star -hmmm!
NS. 
