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According to E. Chabanat et al. (1997), the speed of sound in a nuclear medium is defined as

$$\left(\frac{v_s}{c}\right)^2 = \frac{dP}{de}$$

with

$$P = \rho^2 \frac{dE/A}{d\rho}$$ and $$e = \rho\left(mc^2 + \frac{E}{A}\right).$$

  1. First request: I was wondering if someone could explain me how we derive this expression.

  2. Second request: I know that the speed of sound will be huge in neutron star comparing to the speed of sound in air but I don't know why this quantity is interesting in nuclear astrophysics studies.

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    $\begingroup$ I am by no means schooled in nuclear astrophysics but I would think the following: The speed of sound tells us how fast pressure waves propagate. So in the context of some neutron star it would tell us how fast some pressure perturbation travels trough the star and I can easily imagine how this plays a role in nuclear astrophysics... $\endgroup$ – gertian Apr 14 '17 at 14:14
  • $\begingroup$ @gertian also the primordial plasma before nucleosynthesis $\endgroup$ – anna v Apr 14 '17 at 14:20
  • $\begingroup$ Yeah definitely such as the Baryon accoustic peak. I didn't know this was considered nuclear astrophysics tough, its more cosmology. If OP wants I can explain this in an anwser... $\endgroup$ – gertian Apr 14 '17 at 14:22
  • $\begingroup$ Thanks! Your comment has given me a good intuition about the interest of the speed of sound in nuclear astrophysics, plus I just found a good paper on the topic. However, I am still looking for a explanation of the derivation of the expression. $\endgroup$ – T. Auerrac Apr 14 '17 at 14:26
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    $\begingroup$ @T.Auerrac, a derivation of that formula can be found in this thread physics.stackexchange.com/questions/23556/… $\endgroup$ – gertian Apr 14 '17 at 15:29
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The speed of sound is an important property of an equation of state (EoS): depending on the setting for various reasons.

In the context of neutron star (NS)/ nuclear EoS the speed of sound is a measure of stiffness of an EoS: a stiff EoS (an EoS with a high sound velocity) generates a high pressure at given energy density. In order to allow for massive ($M>M_\odot$) NS the EoS needs to be rather stiff: to form massive compact objects the EoS needs to be able to generate large pressures to compensate for the strong gravitational attraction. Lets consider a very simple EoS: an EoS with a constant speed of sound $c_s$: $$\epsilon=\frac{P}{c_s^2}+\epsilon_0 \Longleftrightarrow \frac{d P}{d\epsilon}=c_s^2.$$ I use geometrized units with $c=G=1$ in which the speed of sound is dimensionless and measured in fractions of the speed of sound $c$ so $c_s=1/3$ in SI units is $\sim 10^8 \mathrm{m s^{-1}}$.

The following figure shows mass-radius and mass-central pressure curves for three different EoS. The data points of this figure correspond to solutions of the general relativistic structure equations of hydrostatic equilibrium (TOV equations). I matched those constant speed of sound EoS to a realistic low density curst EoS to get NS with realistic radii.

M-R and M-p_c curves

The crosses denote the maximum mass obtainable with the respective EoS and the dotted lines are unstable configurations. So we can clearly see that we need very stiff/high sound velocity EoS to get stable stars with masses above $2M_\odot$. The red line corresponds to NS with an EoS at the causal limit $c_s=1$. With such a stiff EoS we can get stable NS with masses up to $\sim 3.2 M_\odot$.

A realistic NS EoS for the high density regime needs to be rather stiff to allow for $2M_\odot$ NS but it also needs to be causal so $c_s<1$. For most purely nuclear EoS the speed of sound is not constant but density dependent.

So far I only talked about the impact of the sound speed on masses and radii of NS but it is also important for many other things: Deformability, dynamic stability, quakes, transport properties all strongly depend on the EoS and its speed of sound. It is also a quite interesting parameter for Quark matter EoS and hybrid NS (NS containing hadronic and Quark matter). Maybe one short note on this: the asymptotic freedom of QCD suggest that Quark matter at very high densities behaves like a free ultra relativistic gas with a constant speed of sound of $c_s^2=1/3$.

In terms of a derivation of the expression for $c_s$: it can be derived from the relativistic Euler equation $\nabla_\mu T^{\mu\nu}=0$ and the continuity equation projected into the fluid rest frame. [S. Yoshida, 2011, Dummy’s note (5): Sound speed in relativistic fluid] gives a short derivation of the expression for the speed of sound an the corresponding wave equation.

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  • $\begingroup$ I'm a bit surprised you can get any stable star with $P \propto \rho$. I had thought that adiabatic index needed to be at least 4/3 and was closer to 2 for usual neutron star EOS - i.e. $P \propto \rho^2$. What am I missing? $\endgroup$ – Rob Jeffries Apr 14 '17 at 23:14
  • $\begingroup$ Oh sorry: maybe I should have stressed that in this answer I only talked about pressure $P$ and relativistic total internal energy density $\epsilon$. For those two quantities $P=\epsilon$ is the causal limit. When talking about the classical rest mass density it is another story: for $P\equiv\kappa \rho^\Gamma$ the related energy density is $\epsilon=P/(\Gamma-1)$ so $\Gamma=2$ corresponds to the causal limit. In a Newtonian theory however there is no causality limit on the EoS because $c_s>c$ is not forbidden by classical mechanics. $\endgroup$ – N0va Apr 15 '17 at 0:15
  • $\begingroup$ Not talking about the causal limit. I'm talking about the condition $dM/d\rho > 0$ for stability. For a polytropic EOS, the adiabatic index needs to be $>4/3$; including GR this threshold increases. Thus you cannot make a neutron star from an ideal degenerate neutron EOS, because the star becomes unstable when the neutrons are relativistic. I'm obviously missing something, that you understand clearly. Isn't a constant speed of sound (of any value) equivalent to a polytrope with an adiabatic index of 1? Also for quark matter $P = \epsilon/3$, so $c_s = c/\sqrt{3}$? $\endgroup$ – Rob Jeffries Apr 15 '17 at 17:55
  • $\begingroup$ For quark matter I missed a squared you are right $c_s^2=1/3$. About that $4/3$ limit it holds when you consider "pure" EoS: an ultra relativistic gas with $P=\epsilon/3$ can not generate a stable star. But such a setting is not physical; when approaching the surface the ultra relativistic assumption does no longer hold. In the plots above I used the constant speed of sound EoS only for $P>17~\mathrm{MeV fm^3}$ and below a BPS crust EoS. For a pure gas one could use $\epsilon=3P+ \kappa P^{3/5}$ or a interpolated EoS from the exact ideal gas equations. $\endgroup$ – N0va Apr 16 '17 at 12:08

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