We present an analytical unified representation for 22 equations of state (EoS) of dense matter in neutron stars. Such analytical representations can be useful for modeling neutron star structure in modified theories of gravity with high order derivatives.
The different possible equations of state (EoS) of dense matter inside neutron stars (NSs) are often given as tabulated data which are fed into the solution of the hydrostatic equilibrium equations via an interpolation technique. While this method works well in the case of general relativity (GR) for which the hydrostatic equilibrium is described by the TOV equations (Tolman 1939;Oppenheimer & Volkoff 1939), it leads to problems in modified theories of gravity, e.g. f (R) theories (de Felice & Tsujikawa 2010) in which the equations describing the hydrostatic equilibrium contain second derivatives of P(ρ) (see e.g. Arapoglu et al. (2011)) where P is the pressure and ρ is the density. In such cases the order of the polynomial used in interpolation technique may effect the results if it is not sufficiently high and the linear interpolation technique will certainly fail. It is then necessary to employ analytical representations of the EoS that are sufficiently many times differentiable. In this work, we provide an accurate unified analytical representation for 22 EoS.
A unified analytical representation for two EoS, FPS and SLY, were provided by Haensel & Potekhin (2004) (hereafter HP04) where the authors provide a relation between ζ = log(P/dyn cm -2 ) and ξ = log(ρ/g cm -3 ) with 18 free parameters. The unified representation we provide here is an extension of HP04 with 23 parameters 12 of which are fixed to represent low density regimes of BPS (Baym et al. 1971) and NV (Negele & Vautherin 1973). The rest 11 free parameters are used to fit the different EoS at high density regimes and to match them. The function we use to represent EoS for NS is
where f 0 (x) = 1/(1 + exp x) is the matching function (also used in HP04). Here
and ζ high = (a 3 + a 4 ξ) f 0 (a 5 (a 6 -ξ)) + (a 7 + a 8 ξ + a 9 ξ 2 ) f 0 (a 10 (a 11 -ξ))
describe the low and high density regimes, respectively. We provide the values of the fit parameters c i and a i for ξ > 5 in Table 1 andTable 2.
In Figure 1, we show our results for fitting the EoS data of AP4 with Equation (1). In Figure 2, we compare the M-R relations obtained by the analytical representation with that obtained by feeding the EoS data via interpolation technique. The maximum relative error is ∼ 0.05% near the maximum mass.
The unified analytical expression presented in this work is an accurate representation of many EoS suggested for NSs. The analytical representation is preferable for solving the structure of NSs in modified theories of gravity where hydrostatic equilibrium equations are of 4th order. In such cases the usual interpolation technique will fail because the high order derivatives may not be continuous if the order of the polynomial used for interpolation is not sufficiently high. 9695 -68.4757 -194.2320 -379.9680 -377.0480 -380.2820 -381.1690 15.3263 -398.4300
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