In the degenerate interiors of neutron stars, the equation of state is usually just density (and composition) dependent. You can express the pressure as a polytropic law of the form P∝ρα, where ρ is the density.
A stiff (or hard) equation of state is one where the pressure increases a lot for a given increase in density. Such a material would be harder to compress and offers more support against gravity. Conversely, a soft equation of state produces a smaller increase of pressure for a change in density and is easy to compress.
If we using the Lagrangian density of the nucleon-meson many body system. Solve the equation of state and corresponding using different parameters like FSUGOLD,NL3,NL3*.....etc. I know every parameters have a different symmetry energy and compressibility. But, what is physics behind the stiff or soft equation of state?
A stiff equation of state gives a large compressibility and neutron star mass, as you explained that a stiff EOS would be harder to compress and offer more support against gravity. For example, An EOS with compressibility K=500 Mev at nuclear matter saturation density produces M=3.0 solar mass for the mass of neutron star. A softer equation of state K=200 ~ 300 Mev produces about M=2.0. The production of hyperons inside neutron stars also softens EOS and produce small mass of neutron stars (see, for example, Density-Dependent Properties of Hadronic Matter in anExtended Chiral (σ, π, ω) Mean-Field Model\rq Schun T. Uechi and Hiroshi Uechi, (2015), Open Access Library Journal, 2, e2011. http://www.oalib.com/articles/3151189#.VmPyCriLTIU
Hardon-Quark Hybrid Stars Constructed by the Nonlinearσ-ω-ρ Mean-Field Model and MIT-Bag Model \rq, Schun T. Uechi, Hiroshi Uechi, (2015) Open Access Library Journal, 2, e2012. http://www.oalib.com/articles/3151190#.VmP1uLiLTIU
An equation of state is the variation of pressure with density, which can be expressed as Pressure \propto densitygamma. Higher value of gamma means stiff equation of state and lower value of gamma means soft equation of state. A stiff equation of state gives a larger value of the maximum mass.
Bharat, Here is a nice summary of the astrophysics of what is happening: http://physics.stackexchange.com/questions/171612/what-is-meant-by-a-stiff-or-soft-equation-of-state-wrt-neutron-stars
The physics behind it is whether the "gas" is relativistic or non-relativistic, but ultimately it seems to be the strong nuclear force that has to be modeled into an equation of state formulation to calculate the structure of a neutron star. -Suketu
A stiff equation of state tends to have larger pressure for a given density. This material resists compression and these stars have larger radii.
Naturally, a soft equation of state has smaller pressures, is more easily compressed, and produces stars with smaller radii.
I should probably start by saying that the very Physics behind the occurrence of the "stiff" and "soft" equations of state is quite complicated. You can subdivide the EOS into regions: I. Nuclear saturation density; II. Supra-saturation nuclear density. Now for each case the stiffness/softness comes from two main ingredients: (a) EOS of symmetric nuclear matter; (b) EOS of pure neutron matter --> this one can be replaced by the nuclear symmetry energy. Now we can talk about the Physics behind the stiffness/softness for each case:
I. (a) The stiffness/softness is mostly controlled by the incompressibility parameter that can be more or less constrained by nuclear breathing modes. There are no big variations (for example it is well constrained to be about 200 < K < 260 MeV)
I. (b) The stiffness/softness is due variation of the nuclear symmetry energy. The large uncertainty comes from our poor knowledge on isovector channel of nuclear interaction. Physics of exotic nuclei close to dripline contain this information, however they are not yet readily available. This is a hot topic of research.
II. (a) The stiffness/softness is controlled by the skewness parameter (third derivative of energy per nucleon). The physics of finite nuclei is not sensitive to this variation, and therefore one can have models giving ranges of the EOS with maximum NS mass from 2 to 3 solar masses.
II. (b) Density dependence of the symmetry energy at suprasaturation is basically unknown. That is why we have a huge variations in the EOSs.
I suggest you to further read this and references therein: https://www.researchgate.net/publication/232238087_Constraining_the_High-Density_Behavior_of_Nuclear_Symmetry_Energy_withthe_Tidal_Polarizability_of_Neutron_Stars
Article Constraining the High-Density Behavior of Nuclear Symmetry E...
Huni, all you curves in fig. 4 seem to suggest a maximum mass less than 0.8 Msun but all neutron stars are above this limit. Am I misreading your chart in some way?
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