I am interested in finding pressure of neutron star! So. Please could any tell me how to choose central density for the inner and outer core of neutron Star. What numeric value should me in both core. Also what should be the polytropic index gamma for outer core.
[Physics] the pressure and density of a neutron star
astrophysicsdensityneutron-starspressure
Related Solutions
In an $n,p,e$ gas the ratio of neutrons to protons decreases with density. For ideal degenerate gases, the Fermi energies are related by $E_{F,n} = E_{F,p} + E_{F,e}$. In this situation, the largest the proton to neutron ratio can become is 1/8 when all the particles are ultra-relativistic at infinite density.
To conserve momentum in the direct URCA process $p_{F,n} \leq p_{F,p} + p_{F,e} = 2p_{F,p}$ (by charge neutrality). Hence because $p_F$ is proportional to $n^{1/3}$, then $n_p \geq n_n/8$.
But I previously showed that $n_p < n_n /8$, hence in completely degenerate, ideal $n,p,e$ gases, the direct URCA process is blocked.
At densities above about $8\times 10^{17}$ kg/m$^3$, the Fermi energy of the electrons equals the rest-mass energy of muons (105 MeV) and muons can be produced. In more realistic models that include nucleon interactions, this occurs at about $3\times 10^{17}$ kg/m$^3$. The charge neutrality equation changes and the net result is that the number of protons in the gas is able to increase slightly - it could get above 10% of the neutron numbers, or even higher if negatively charged Hyperons are produced at even higher densities. This opens up a channel for the direct URCA process to occur.
The image below (taken from a thesis by Stephen Portillo) illustrates the proton (gold), electron (purple) and muon (blue) fractions as a function of baryonic density (in units of the nuclear saturation density $2.8\times10^{17}$ kg/m$^3$). This has been calculated incorporating nucleon interactions. The muons appear at about the saturation density and the proton fraction exceeds 1/8 at about $10^{18}$ kg/m$^3$.
These reactions must involve neutrons, protons and electrons that are close to their Fermi energies (within about $kT$). Therefore this requirement on each species introduces an extra dependence of $\sim kT/E_F \ll 1$ per species into the reaction rate.
In the modified URCA process "bystander particles" are available to balance the momentum and energy equations - i.e. their role is just to provide sources/sinks of energy/momentum. However, the requirement for two more bystander nucleons (also within $kT$ of their Fermi energies) suppresses the reaction rate considerably (two more factors of $kT/E_F$ or about $10^6$ at typical densities). So, even though the modified process dominates over suppressed direct URCA at low densities, if densities are high enough for direct URCA to be possible, it dominates.
There are a few standard textbooks on neutron star.
For interior structure and nuclear physics side two books by Glendenning are good.
http://www-nsdth.lbl.gov/~nkg/description.html
For more general relativity side Shapiro and Teukolsky has been a standard texk book for many years.
http://www.amazon.com/Black-Holes-White-Dwarfs-Neutron/dp/0471873160
Finally, if you seek for real rigor, a new book by Friedman & Stergioulas is must.
http://www.amazon.com/Rotating-Relativistic-Stergioulas-Cambridge-Monographs/dp/0521872545
There are several review papers including two in Living Review.
http://relativity.livingreviews.org/Articles/lrr-2003-3/
http://relativity.livingreviews.org/Articles/lrr-2007-1/
Several by Lattimer and Prakash are also good starting point. For example,
Best Answer
The central density of a neutron star is unknown, but probably lies in the region of $\sim 10^{18}$ kg/m$^3$, depending on the details of the composition, the equation of state and of course the mass of the neutron star.
As an order of magnitude estimate, at these densities, the pressure approaches the maximum for a relativistically degenerate ideal fermion gas of $P \simeq \rho c^2/3 = 3\times10^{34}$ Pa.
By "outer core" I assume you mean the neutron fluid region. Here the density could reach somewhere between $3\times 10^{17}$ and $\sim 10^{18}$ kg/m$^3$ before something may (or may not) cause a phase change (hyperons, quark matter etc.). The pressure at the upper end of this range would approach that which I gave above. A more formal calculation could calculate the equilibrium composition of an n,p,e gas and then use the appropriate formula for an ideal Fermi gas to estimate the total pressure. Such a calculation, at a range of densities, also gives you an estimate of the adiabatic index. I find that the pressure of an ideal n,p,e gas is $7\times 10^{32}$ Pa at $3\times 10^{17}$ kg/m$^3$ (completely dominanted by non-relativistic neutrons), rising to $5\times 10^{33}$ Pa at $10^{18}$ kg/m$^3$. The adiabatic index $\gamma$, (where I define $\gamma$ in terms of $P \propto \rho^{\gamma}$), is approximately 1.6 over this range.
You have to realise though that this is an unrealistic equation of state at these densities. As the neutrons get closer together ($\leq 10^{-15}$ m) they must interact and this hardens the equation of state such that $\gamma \geq 2$, otherwise neutron stars of mass up to $2M_{\odot}$ couldn't exist! Indeed, one definition of the "inner core" might be where this hardening happens.
A recent paper by Hebeler et al. (2013) includes the following plot (in cgs units) which reviews the many possible equations of state in the density range I have discussed above. For reasons discussed in that paper, the authors favour the AP3 or AP4 equations of state which give higher pressures tan the simple n,p,e approximation. The slope of these graphs gives the the value of $\gamma \simeq 3.3$ for AP3 and AP4 for $\rho > 5 \times 10^{17}$ kg/m$^3$.