Equations of state

2.6 Equations of state

2.6.1 Relativistic polytropes

An analytic equation of state that is commonly used to model relativistic stars is the adiabatic, relativistic polytropic EOS of Tooper [311 ]:

where $K$ and $Γ$ are the polytropic constant and polytropic exponent, respectively. Notice that the above definition is different from the form $P = K 𝜀Γ$ (also due to Tooper [310 ]) that has also been used as a generalization of the Newtonian polytropic EOS. Instead of $Γ$ , one often uses the polytropic index $N$ , defined through

For the above equation of state, the quantity $∘ ------------ c(Γ − 2)∕(Γ − 1) K1 ∕(Γ −1)∕G$ has units of length. In gravitational units ( $c = G = 1$ ), one can thus use $KN ∕2$ as a fundamental length scale to define dimensionless quantities. Equilibrium models are then characterized by the polytropic index $N$ and their dimensionless central energy density. Equilibrium properties can be scaled to different dimensional values, using appropriate values for $K$ . For $N < 1.0$ ( $N > 1.0$ ) one obtains stiff (soft) models, while for $N ∼ 0.5 –1.0$ , one obtains models with bulk properties that are comparable to those of observed neutron star radii and masses.

Notice that for the above polytropic EOS, the polytropic index $Γ$ coincides with the adiabatic index of a relativistic isentropic fluid

This is not the case for the polytropic equation of state $P = K 𝜀Γ$ , which satisfies (25

) only in the Newtonian limit.

2.6.2 Hadronic equations of state

The true equation of state that describes the interior of compact stars is, still, largely unknown. This comes as a consequence of our inability to verify experimentally the different theories that describe the strong interactions between baryons and the many-body theories of dense matter, at densities larger than about twice the nuclear density (i.e., at densities larger than about 5 × 10¹⁴ g cm^–3).

Many different so-called realistic EOSs have been proposed to date that all produce neutron star models that satisfy the currently available observational constraints. The two most accurate constraints are that the EOS must admit nonrotating neutron stars with gravitational mass of at least $1.44 M ⊙$ and allow rotational periods at least as small as 1.56 ms (see [242 , 186 ]). Recently, the first direct determination of the gravitational redshift of spectral lines produced in the neutron star photosphere has been obtained [74 ]. This determination (in the case of the low-mass X-ray binary EXO 0748-676) yielded a redshift of z = 0.35 at the surface of the neutron star, corresponding to a mass to radius ratio of $M ∕R = 0.23$ (in gravitational units), which is compatible with most normal nuclear matter EOSs and incompatible with some exotic matter EOSs.

The theoretically proposed EOSs are qualitatively and quantitatively very different from each other. Some are based on relativistic many-body theories while others use nonrelativistic theories with baryon-baryon interaction potentials. A classic collection of early proposed EOSs was compiled by Arnett and Bowers [20 ], while recent EOSs are used in Salgado et al. [260 ] and in [84 ]. A review of many modern EOSs can be found in a recent article by Haensel [138 ]. Detailed descriptions and tables of several modern EOSs, especially EOSs with phase transitions, can be found in Glendenning’s book [125 ].

High density equations of state with pion condensation have been proposed by Migdal [227 ] and Sawyer and Scalapino [263 ]. The possibility of kaon condensation is discussed by Brown and Bethe [51 ] (but see also Pandharipande et al. [240 ]). Properties of rotating Skyrmion stars have been computed in [236 ].

The realistic EOSs are supplied in the form of an energy density vs. pressure table and intermediate values are interpolated. This results in some loss of accuracy because the usual interpolation methods do not preserve thermodynamical consistency. Swesty [300 ] devised a cubic Hermite interpolation scheme that does preserve thermodynamical consistency and the scheme has been shown to indeed produce higher accuracy neutron star models in Nozawa et al. [235 ].

Usually, the interior of compact stars is modeled as a one-component ideal fluid. When neutron stars cool below the superfluid transition temperature, the part of the star that becomes superfluid can be described by a two-fluid model and new effects arise. Andersson and Comer [9 ] have recently used such a description in a detailed study of slowly rotating superfluid neutron stars in general relativity, while the first rapidly rotating models are presented in [247 ].

2.6.3 Strange quark equations of state

Strange quark stars are likely to exist, if the ground state of matter at large atomic number is in the form of a quark fluid, which would then be composed of about equal numbers of up, down, and strange quarks together with electrons, which give overall charge neutrality [38 , 98 ]. The strangeness per unit baryon number is $≃ − 1$ . The first relativistic models of stars composed of quark matter were computed by Ipser, Kislinger, and Morley [157 ] and by Brecher and Caporaso [50 ], while the first extensive study of strange quark star properties is due to Witten [325 ].

The strange quark matter equation of state can be represented by the following linear relation between pressure and energy density:

where $𝜀0$ is the energy density at the surface of a bare strange star (neglecting a possible thin crust of normal matter). The MIT bag model of strange quark matter involves three parameters, the bag constant, $ℬ = 𝜀 ∕4 0$ , the mass of the strange quark, $m s$ , and the QCD coupling constant, $α c$ . The constant $a$ in (26

) is equal to 1/3 if one neglects the mass of the strange quark, while it takes the value of $a = 0.289$ for $ms = 250 MeV$ . When measured in units of $−3 ℬ60 = ℬ ∕(60 MeV fm )$ , the constant $B$ is restricted to be in the range

assuming $ms = 0$ . The lower limit is set by the requirement of stability of neutrons with respect to a spontaneous fusion into strangelets, while the upper limit is determined by the energy per baryon of ⁵⁶Fe at zero pressure (930.4 MeV). For other values of $m s$ the above limits are modified somewhat.

A more recent attempt to describe deconfined strange quark matter is the Dey et al. EOS [87 ], which has asymptotic freedom built in. It describes deconfined quarks at high densities and confinement at zero pressure. The Dey et al. EOS can be approximated by a linear relation of the same form as the MIT bag model strange star EOS (26 ). In such a linear approximation, typical values of the constant a are 0.45 – 0.46 [128 ].

Going further A review of strange quark star properties can be found in [320 ]. Hybrid stars that have a mixed-phase region of quark and hadronic matter, have also been proposed (see, e.g., [125 ]). A study of the relaxation effect in dissipative relativistic fluid theories is presented in [199 ].