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I want to know the role of polytropic index gamma in equation of state. If different polytropes give different equation of state then how the polytropic behaviour vary in the 4 layers of neutron star that are

Outer crust

Inner crust

Outer core

Inner core

What appropriate densities and polytropes should be chosen for the best model of equation Of state?

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3 Answers 3

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As @CountTo10 pointed out: getting a "realistic" equation of state (EoS) is difficult: to get neutron stars with realistic masses, radii and gradients one would need an EoS containing different polytropes for the different regions. Finding the right ones and constructing a proper, thermodynamic consistent, EoS with them is difficult and the result will be not unique.

That being said if one is interested in a simple EoS consisting of only one polytrope for testing numerics or just playing around I can recommend the ones presented by for example Bonazzola et al. 1993:

\begin{align} \epsilon(n)&=m_Bn+\frac{\kappa\epsilon_0}{\gamma-1}\left(\frac{n}{n_0}\right)^\gamma,\\ p(n)&=\kappa\epsilon_0\left(\frac{n}{n_0}\right)^\gamma. \end{align}

Where $n$ is the baryon number density, $\epsilon$ the energy density and $p$ the pressure. $\kappa$ and $\gamma$ are dimensionless parameters, $m_B$ is the mean baryon mass $m_B=931.2\,\mathrm{MeV}$ and $n_0$ and $\epsilon_0$ are arbitrary number and energy densities. For neutron star matter values around nuclear density are appropriate so for example $n_0=0.1\, \mathrm{fm}^{-3}$ and $\epsilon_0=m_B n_0=93.1\, \mathrm{MeV}\mathrm{fm}^{-3}$. With those values suitable $\kappa$'s are around $0.05$ and $\gamma\sim2$. The above relations can be inverted to give an expression $\epsilon(p)$: $$ \epsilon(p)=\frac{m_B}{n_0} (\frac{p}{\kappa\epsilon_0})^{1/\gamma}+ \frac{p}{\gamma-1}. $$ Using such an equation of state one can solve the General relativistic stellar structure equations (TOV equations) to get masses and radii in dependence of EoS and central pressure (or equivalent central density):

MR and Mnbar plots for different polytropes

On the left mass over radius and on the right mass over central baryon density. The crosses mark stars with maximum mass, the dots stars with specific compactnesses and the dotted lines unstable configurations. The three curves correspond to different polytropic EoS with $(\kappa|\gamma)$.

As you can see the configurations are rather sensible to small changes in $\kappa$ and $\gamma$ since both affect the overall stiffness of the EoS: $\gamma$ especially in the high pressure regime and $\kappa$ especially in the low pressure regime.

All three shown here work ok: masses and radii are on typical neutron star scales and density and pressure gradients do not look bad either. But the outer layers of the neutron stars and the radii in detail are not realistic, for that one would need an EoS for the crust; either a different polytrope or a realistic, tabulated EoS like BPS or NV. But for testing numerics and playing around those polytropes are well suited.

In my computations I use the EoS with medium stiffness with $\kappa=0.05$ and $\gamma=2$ since it has a nice maximum mass of 2.233 solar masses and nice intermediate radii around 15 km. One could certainly play a bit around to get to 2 solar masses with smaller radii by adjusting $\kappa$ and $\gamma$ slightly.

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  • $\begingroup$ +1 there are answers and then there are real answers. I learnt a lot from yours, thanks for posting it. $\endgroup$
    – user108787
    Nov 8, 2016 at 12:38
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I note that you have recently received an answer on a related question Pressure and Density of a Neutron Star.

I can only add this excerpt and images from a Wikipage you may have already read, Wikipedia Polytropes and suggest (inadequately, I appreciate) that you interpolate appropriately between the (unfortunately) small range of different models provided.

enter image description here

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Example models by polytropic index

Density (normalized to average density) versus radius (normalized to external radius) for a polytrope with index n=3.

Neutron stars are well modeled by polytropes with index about in the range between = 0.5 and n = 1.

A polytrope with index n = 1.5 is a good model for fully convective star cores (like those of red giants), brown dwarfs, giant gaseous planets (like Jupiter), or even for rocky planets.

Main sequence stars like our Sun and relativistic degenerate cores like those of white dwarfs are usually modeled by a polytrope with index n = 3, corresponding to the Eddington standard model of stellar structure.

A polytrope with index n = 5 has an infinite radius. It corresponds to the simplest plausible model of a self-consistent stellar system, first studied by A. Schuster in 1883.

A polytrope with index n = ∞ corresponds to what is called an isothermal sphere, that is an isothermal self-gravitating sphere of gas, whose structure is identical to the structure of a collisionless system of stars like a globular cluster.

In general as the polytropic index increases, the density distribution is more heavily weighted toward the center (r = 0) of the body.

A more detailed look at the appropriate index to be used is covered in this book: Neutron Stars: Equations of State:

A typical example of the details contained in this book and the methodology used to determine equations of state is contained below:

If $T <<T_F$, the electrons are nearly free (§ 4.1.2), and the magnetic field can be regarded as constant (and, hence, force-free) in a local part of the thin outer envelope, Eq. (6.62) is valid for any field strength B. Thus, the structure of the envelope is again described by a self similar solution. However, the character of the solution at $\rho < \rho _B$ is different. In particular, in Eq. (6.63) one should replace the nonmagnetic electron relativity parameter $x_r$ by the parameter $x_B$ that is appropriate for a strongly quantizing magnetic field. In this case, a given geometrical depth z corresponds to a higher density p. We plot the $\rho (z)$ and $\Delta M(z)$ profiles in the neutron star envelope for $B = 10^{12}$ G and $10^{13}$ G. For instance, the magnetic field $B = 10^13$ G strongly affects the density distribution in the layer $\rho < \rho_B$ N $10^6 g cm^{-3}$, located at $z < 3$ m. According to Eq. (4.30), the density at the bottom of this layer scales as $B^{3/2}$. Equation (4.32) shows that the EOS in this layer is polytropic, but the polytropic index differs from that at $B = 0$; now it is equal to $n = 2(y = 3)$ or $n = 1 (y = 2)$ for the non-relativistic or ultrarelativistic electron gases, respectively.

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  • $\begingroup$ How can the x-axis on the plot have values greater then 1 if it is normalised to the external radius? It implies the density is $>0$ outside the star? Also, the density is clearly normalised to the central density, not the average density. $\endgroup$
    – ProfRob
    Aug 3, 2023 at 8:24
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Here is (an example) of exactly what you need. This is a plot of adiabatic index (defined as $\gamma = (\rho + P/c^2) dP/d\rho$), versus density, provided by Douchin & Haensel (2001), along with the corresponding equation of state (cgs units).

Adiabatic index

Equation of state

There is so much beautiful physics on display here.

The outer crust has $\gamma \simeq 4/3$ dominated by ideal EDP from relativistically degenerate electrons. The adiabatic index falls below this as the density increase because of inverse beta decay. At $\rho \sim 4\times 10^{11}$ g/cm$^3$ there is an abrupt softening as neutrons drip out of the nuclei in the inner crust (outer and inner crust are separated by the dotted line). Then as the density increases, so does the neutron density and the value of $\gamma$ approaches that of non-relativistic degeneracy pressure for the neutrons (5/3). The dashed line marks the transition from the solid inner crust to a liquid core. The adiabatic index then grows further as the neutrons come close enough to feel the repulsive strong nuclear force. A small notch in the adiabatic index marks the appearance of muons. Finally there is a softening at extremely high densities caused by the increasing proton fraction.

It is of course worth stressing that the values to the right of the dashed line are theoretically quite contentious and uncertain. In this equation of state there appears to be no allowance made for the (possible) appearance of extra hadronic degrees of freedom (e.g. hyperons) or a transition to quark matter. This particular flavour of equation of state predicts a maximum mass for a neutron stars of just over $2.05 M_{\odot}$, so is just compatible with current observational constraints.

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  • $\begingroup$ Hmm its really helps me.its means my statistics is much larger than I expect for outer core. I also follow article of feryal ozel. $\endgroup$
    – umar khan
    Nov 9, 2016 at 6:34

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