4 Equations of state

4.1 Relativistic polytropes

We use the following relation as a polytropic equation of state (Tooper 1965):

$$\begin{eqnarray} \varepsilon & = & K {\rho^\gamma \over \gamma - 1} + \rho \, c^... ... p & = & K\, \rho^\gamma \ , \\ \gamma & = & 1 + {1 \over N} \ , \end{eqnarray}$$ (23) (24) (25)

where K and N are the polytropic constant and polytropic index, respectively, while $\rho$ is the rest mass density.

It should be noted that this equation of state includes the limiting case of $\varepsilon = \rho c^2 =$ constant, when $\gamma = \infty$(N=0). The constant density models are also called homogeneous models. For polytropes of index N<1.0, the density does not go to zero smoothly at the surface and the first derivatives of the density across the surface are discontinuous. This kind of discontinuity may become the cause of unfavorable behavior of solutions, unless it is treated carefully. For constant density models, the situation is even worse, since the density itself is discontinuous across the surface.

Although polytropic EOSs are not as realistic as tabulated EOSs (but one can reproduce neutron star bulk properties with an $N \simeq 1.0$ polytrope), they are helpful to check numerical codes. Since the hydrostationary equation can be analytically integrated and no additional numerical errors arise in solving it.

4.2 Short description of realistic equations of state

As discussed in the introduction, the main uncertainties about neutron star properties are related to the unknown interactions of the neutron star matter at high density regions. In the last decades, many equations of state have been proposed by considering different kinds of interactions into account. A large collection of representative equations of state were compiled by Arnett & Bowers (1977), who constructed nonrotating neutron star models and obtained physical quantities for slowly rotating neutron stars. We will choose three equations of state of Arnett & Bowers' compilation, i.e. equations C, G and L according to their notation. Equations C, G and L are those derived by Bethe & Johnson (1974), Canuto & Chitre (1974) and Pandharipande & Smith (1975) (see also Pandharipande et al. 1976), respectively. Those equations of state were also used by Friedman et al. (1986) for constructing rapidly rotating relativistic neutron stars models.

In addition to these equations of state we also employ the WFF3 (UV₁₄+TNI) equation of state by Wiringa et al. (1988), the FPS equation of state by Lorenz et al. (1993), and the equation of state which represents a causal limit (CLES).

Some characteristic features of each equation of state can be summarized as follows.

Bethe - Johnson I (C):

EOS C is of intermediate stiffness. The maximum gravitational mass of a spherical neutron star for this EOS is $1.85\ M_{\odot}$. The density range is from $1.71 \ 10^{14}$ g cm^-3 to $3.23 \ 10^{15}$ g cm^-3. Hyperons as well as nucleons are taken into account. The interaction is assumed non-relativistic and represented by the modified Reid soft core potential with non-integer parameters. To include the many-body theory, the constrained variational principle is employed. This equation of state is joined to the composite BBP($\varepsilon/c^2 \gt 4.3 \ 10^{11}$g cm^-3) - BPS (10⁴ g cm$^{-3} < \varepsilon/c^2 < 4.3 \ 10^{11}$g cm^-3) - FMT($\varepsilon /c^2 <$ 10⁴ g cm^-3). Here BBP, BPS and FMT denote equations of state by Baym et al. (1971a), Baym et al. (1971b) and Feynman et al. (1949), respectively.

Canuto - Chitre (G):

EOS G an extremely soft equation of state. The maximum gravitational mass of a spherical neutron star for this equation of state is $1.36\ M_{\odot}$, so this EOS is not acceptable as a realistic candidate for the true EOS of neutron star matter. It is used in this comparison, because it is close to the softest possible realistic EOS consistent with observational constraints. The density range is from $2.37 \ 10^{15}$ g cm^-3 to 7.23 10¹⁵ g cm^-3. Crystallization of neutrons is included. The interaction is non-relativistic and represented by the modified Reid soft core potential. This equation of state is joined to the composite PC($7 \ 10^{14}$ g cm$^{-3} < \varepsilon/c^2 < 2.4$ 10¹⁵ g cm^-3) - BBP($4.3 \ 10^{11}$ g cm$^{-3} < \varepsilon/c^2 < 7$ 10¹⁴ g cm^-3) - BPS(10⁴ g cm$^{-3} < \varepsilon/c^2 < 4.3$ 10¹¹g cm^-3) - FMT($\varepsilon/c^2 < 10^4$ g cm^-3). Here PC denotes the equation of state by Pandharipande (1971).

Pandharipande - Smith (L):

EOS L is an extremely stiff EOS. The maximum gravitational mass of a spherical neutron star for this equation of state is $2.70\ M_{\odot}$. The density range is larger than $4.386 \ 10^{11}$ g cm^-3. Compositions consist of neutrons. The interaction is assumed non-relativistic and is represented by the nuclear attraction due to scalar particle exchange. This equation of state is joined to the BPS (10⁴ g cm$^{-3} < \varepsilon/c^2 < 4.4$ 10¹¹ g cm^-3) - FMT($\varepsilon/c^2 < 10^4$ g cm^-3).

Wiringa - Fiks - Farbrocini (WFF3):

EOS WFF3 (Wiringa et al. 1988) is of intermediate stiffness. At present, the WFF3 equation of state is regarded as one of the best candidates for the high density region. This EOS is an improved version of the equation of state by Friedman & Pandharipande (1981). The nucleon-nucleon interaction described by a two-body Urbana UV₁₄ potential and the phenomenological three-nucleon TNI interaction are taken into account. Compositions are considered to be neutrons. The maximum gravitational mass of a spherical neutron star for this equation of state is $1.84\ M_{\odot}$.Although the usual WFF3 EOS is joined to EOS NV (Negele & Vautherin 1973), we will also use a different version, in which it is joined to the more modern FPS EOS (Lorenz et al. 1993).

Lorenz - Ravenhall - Pethick (FPS):

This EOS is also a modern version of the equation of state by Friedman & Pandharipande (1981). The nucleon-nucleon interaction described by a two-body Urbana UV₁₄ potential and the phenomenological three-nucleon TNI interaction are taken into account. In the FPS equation of state the Skyrme model is used, where the effective interaction has the spatial character of a two-body delta function plus derivatives. The FPS equation of state can be considered to be an improved version of the BBP equation of state in the region of the lower density.

Causal limit equation of state (CLES):

As an extreme case, we consider an equation of state which consists of a causal limit EOS ($\varepsilon = p +$ constant) for $\varepsilon/c^2 \gt 1.66\break 10^{14}$ g cm^-3 and the FPS EOS below that density. The causal limit EOS has the property that, in the interior of the star, the phase velocity of sound is equal to the velocity of light in vacuo, i.e. $v_{\rm s} = \sqrt{{\rm d}p/{\rm d} \varepsilon}=c$.