A&A 428, 191-197 (2004)
P. Haensel1 - A. Y. Potekhin2,3
1 - N. Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland
2 - Ioffe Physico-Technical Institute, Politekhnicheskaya 26, 194021 St. Petersburg, Russia
3 - Isaac Newton Institute of Chile, St. Petersburg Branch, Russia
Received 23 July 2004 / Accepted 7 August 2004
Analytical representations are derived for two equations of state (EOSs) of neutron-star matter: FPS and SLy. Each of these EOSs is unified, that is, it describes the crust and the core of a neutron star using the same physical model. Two versions of the EOS parametrization are considered. In the first one, pressure and mass density are given as functions of the baryon density. In the second version, pressure, mass density, and baryon density are given as functions of the pseudo-enthalpy, which makes this representation particularly useful for 2-D calculations of stationary rotating configurations of neutron stars.
Key words: equation of state - dense matter - stars: neutron
The equation of state (EOS) of dense matter is a crucial input for the neutron-star structure calculations. Under standard conditions neutron-star matter is strongly degenerate, and therefore the matter pressure is temperature independent; exceptions are the outermost (a few meters thick) envelopes, newly-born neutron stars, and the envelopes of exploding X-ray bursters. At the EOS is not affected by the magnetic field even as strong as 1014 G and by the temperature K. Therefore, except for a thin outer envelope (which for the bottom density of is 20 m thick and contains only of matter), the EOS of neutron-star matter has a one-parameter character. In order to determine the neutron-star structure up to the maximum allowable mass, , one has to know the EOS up to a few times . While the EOS of the neutron-star crust ( ) is rather well known (e.g., Haensel 2001), the EOS of the liquid core at , which is crucial for determining , remains uncertain (e.g., Haensel 2003; Heiselberg & Pandharipande 2000).
The EOS of the crust depends on the crust formation scenario. Two limiting cases are: cold catalyzed matter being at the ground state at a fixed baryon density n, and the accreted crust formed via compression at T<109 K from the thermonuclear ashes of the X-ray bursts in the outer envelopes of accreting neutron stars (Haensel & Zdunik 1990,2003). The EOS of the liquid core does not depend on the formation scenario, and can slightly deviate from the ground state one only when there are deviations from the weak interaction equilibrium - e.g., when rapid matter flow (like in stellar pulsations) is involved. In the present paper we consider the standard case of the ground-state matter.
A "unified EOS'' is obtained in the many-body calculations based on a single effective nuclear Hamiltonian, and is valid in all regions of the neutron star interior. For unified EOSs the transitions between the outer crust and the inner crust, and between the inner crust and the core are obtained as a result of many-body calculation. Alas, up to now only a few models of unified EOSs have been constructed. All other EOSs consist of crust and core segments obtained using different physical models. The crust-core interface has there no physical meaning and both segments are joined using an ad hoc matching procedure. Therefore, neutron-star models based on these EOSs contain a shell with an unphysical EOS. For such "matched EOSs'' it is not possible to study phenomena which are sensitive to the position of the crust-core interface.
In the present paper we consider two unified EOSs, the FPS EOS of Pandharipande & Ravenhall (1989) and the SLy EOS of Douchin & Haensel (2001).
EOSs are usually given in the form of tables. Therefore, in order to use them, one has to employ interpolation between the tabulated points. However, the interpolation procedure is not unique. This introduces ambiguities in the calculated parameters of the neutron star models. Moreover, interpolation should be done respecting exact thermodynamic relations. This turned out to be a particularly serious problem in the high-precision 2-D calculations of models of rapidly rotating neutron stars (Nozawa et al. 1998). In the 3-D calculations of the stationary configurations of a close binary neutron-star system one needs derivatives of pressure with respect to the enthalpy, and tabulated non-polytropic EOSs are not useful in this respect (see, e.g., Gourgoulhon et al. 2001).
In view of the deficiencies and ambiguities inherent in the tabulated EOSs, which are particularly serious in the case of matched EOSs, it is of great interest to derive analytical representations of the EOSs. They have two important advantages over the tabulated ones. First, there is no ambiguity of interpolation and the derivatives can be precisely calculated. Second, these representations can be constructed fulfilling exactly the thermodynamic relations. In this way, analytical EOSs can allow for a very high precision of neutron star structure calculation in the 2-D and 3-D cases.
In the present paper we derive analytical representations of the unified FPS and SLy EOSs. In Sect. 2 we discuss general properties of the EOS used for the calculations of non-rotating and rotating neutron-star models. Analytical representations of the EOSs for non-rotating stars are described in Sect. 3, while Sect. 4 is devoted to the case of rotating stars. Discussion and conclusion are presented in Sect. 5.
A given EOS is usually presented in the form of a table containing a grid of calculated values of matter density (full energy density including the rest energies of the matter constituents divided by c2, i.e., ), baryon (number) density n, and pressure P. The EOS ( ) is then interpolated between the tabulated points so as to get the one-parameter EOS in the form P=P(n), . The interpolation involves some degree of indeterminacy (there are many ways of interpolating) and this itself implies some ambiguity as far as the calculated values of the neutron-star parameters (for example, the value of ) are concerned.
High precision determination of the baryon number A and gravitational mass M of an equilibrium configuration requires condition of thermodynamic consistency of functions and P(n) to be strictly fulfilled. The first law of thermodynamics in the T=0 limit implies relation
An example of an interpolation recipe which respects
Eqs. (1)-(3) and at the same
time yields highly smooth functions P(n) and
presented by Swesty (1996). Only a thermodynamically consistent
interpolation yields neutron-star models which strictly satisfy
the relation connecting the baryon chemical potential
and the metric function
a given circumferential radius r.
This strict relation stems from the equation for the
For a thermodynamically consistent EOS, Eq. (5)
implies the baryon density profile within a static neutron star,
Rotation breaks the spherical symmetry of the equilibrium configuration.
Stationary configurations of a rigidly rotating star
of ideal liquid are solutions of 2-D axially symmetric partial differential
hydrostatic equilibrium in coordinates r and
relativistic stars is reviewed by Stergioulas 2003). As far as the EOS is
concerned, it is suitable to parametrize it in terms of a dimensionless
There are three qualitatively different domains of the interior of a
neutron star, separated by phase transition points:
the outer crust (consisting of the electrons and atomic nuclei),
crust (consisting of the electrons, nuclei, and dripped neutrons), and
which contains the electrons, neutrons, protons, -mesons,
and possibly - and K-mesons, some hyperons, or
quark matter. The latter species are contained in the innermost stellar
domain called the inner core. In addition, there can be density
discontinuities at the interfaces between layers containing different
nuclei in the crust.
In the fitting, we neglect these small
discontinuities and approximate the EOS by fully analytical functions.
However, the different character of the EOS in the different domains
is reflected by the complexity of the fit, which consists of
several fractional-polynomial parts, matched together by virtue of the
|Figure 1: Neutron-star EOS for non-rotational configurations: BPS (triangles), Haensel & Pichon (1994) (HP94, stars), SLy (dots), OPAL at T=106, 107, and 108 K (dashed lines), the fit (14) (solid line) and the fit modified at low (dotted line).|
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For non-rotating configurations,
we have parametrized the pressure as
function of density.
Let us denote
Then the parametrization reads
Figure 1 shows against for a tabulated EOS (symbols) and the corresponding fit (solid line). Triangles correspond to BPS, stars to HP94, and dots to SLy data. By construction, the fit is accurate at g cm-3. As stated above, at lower density the EOS becomes temperature-dependent (for T values typical for neutron star envelopes). This is illustrated by the dashed lines, that show the OPAL EOS (Rogers et al. 1996) for T=106, 107, and 108 K. However, a reasonable continuation of the fit to lower densities can be constructed by a simple interpolation. For instance, the dotted line in Fig. 1 corresponds to , where is given by Eq. (14) (where we always assume ), and approximates the OPAL EOS near at T=107 K (here P is in dyn cm-2 and in g cm-3).
Table 1: Parameters of the fit (14).
|Figure 2: Comparison of the data and fits for SLy and FPS EOSs for non-rotational configurations. Upper panel: rarefied tabular data (symbols) and the fit (14) (lines); lower panel: relative difference between the data and fit. Filled dots and solid line: SLy; open circles and dot-dashed line: FPS (triangles and stars on the upper panel are BPS and HP94 data at g cm-3).|
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Table 2: Parameters of the fits (15) and (16).
In Fig. 2 we compare the FPS and SLy EOSs. Symbols on the upper panel show the data (triangles, stars, dots, and open circles for BPS, HP94, SLy, and FPS, respectively) and lines show the fit (solid for SLy and dot-dashed for FPS). In order to make the differences between the data and fits and between SLy and FPS EOSs visible, we plot the difference , where P is in dyn cm-2 and in g cm-3. The lower panel shows the relative difference between the tabulated and fitted EOSs (solid and dot-dashed lines for SLy and FPS, respectively). It illustrates the accuracy of the fit (14).
Now, can be easily obtained from Eq. (3). Doing this and substituting in the integrand of Eq. (3) from Eq. (14), we recover the original tabular values with maximum difference <0.4% for FPS and <0.12% for SLy.
In some applications, it may be convenient to use n as independent
variable, and treat
and P as functions of n. For this purpose
one can use the following fit:
|Figure 3: Dependence of on n. Upper panel: rarefied data (symbols) and the fit (15) (lines); lower panel: relative difference between the data and fit. Filled dots and solid line: SLy; open circles and dot-dashed line: FPS.|
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It should be stressed that thermodynamics requires Eq. (1) to be satisfied exactly. To achieve this, one should not totally rely on the fits (15) and (16); otherwise thermodynamic consistency will be violated on the scale of these fits' errors (a fraction of percent). So, if is used as an input, then should be calculated from Eq. (3). Alternatively, if the input is n, then, having calculated from Eq. (15) and from Eq. (14), one should refine using relation (2).
As explained in Sect. 2.2, for rotating stars
it is most useful to parametrize
density and pressure as functions of the pseudo-enthalpy H,
which can be written in terms of the enthalpy per baryon h.
Let us define
In view of relation (8), the function which we
intend to parametrize, ,
is not independent of the function
parametrized by Eq. (14).
In order to fulfill Eq. (8) as closely as possible,
we first calculate
using Eqs. (14) and (8),
and then find the inverse fit .
The best fit reads:
Table 3: Parameters of the fit (17).
|Figure 4: SLy and FPS EOS for rotational configurations. Upper panel: rarefied data to be fitted, calculated according to Eq. (14) (symbols) and the fit (17) (lines); lower panel: relative difference between the data and fit. Filled dots and solid line: SLy; open circles and dot-dashed line: FPS.|
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When used in combination, the fits (14) and (17), together with Eq. (3) or Eq. (16) give the parametrizations of , P(H), and n(H) needed for calculations of the stationary rotating configurations. In this case, the function obtained using Eqs. (17) and (14), reproduces the tabular values with a typical discrepancy of 1-2%, with a maximum within 10% near the crust-core boundary.
The remark on the thermodynamic consistency, made at the end of Sect. 3.1, applies also here: one should refine either n or fitted values, using the exact relations (3) or (2).
An important dimensionless parameter characterizing the stiffness of
the EOS at given density is the adiabatic index, defined by
|Figure 5: Adiabatic index for SLy and FPS EOSs. Solid line: analytical approximation (SLy); dotted line: precise values (SLy); dot-dashed line: analytical approximation (FPS).|
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In the outer crust, the value of depends quite weakly on density. At , this value would be if A,Z values were fixed, because in this case P is mainly determined by the pressure of ultrarelativistic electron gas which behaves as . For example, within each shell with constant A and Z of the EOS derived by Haensel & Pichon (1994). However, the compressible liquid drop model used by Douchin & Haensel (2001) effectively smoothes the discontinuities caused by transitions from one to another (A,Z)-species with increasing density, which leads to an effective continuous increase of the A/Z ratio and corresponding decrease of , seen in Fig. 5.
A dramatic drop in occurs at neutron drip threshold, which corresponds to strong softening of the EOS. The analytical expression (19) somewhat smoothes this drop. The behavior of in the inner crust results from an interplay of several factors, with stiffening due to interaction between dripped neutrons, a softening effect of neutron-gas - nuclear-matter coexistence, and the softening Coulomb contribution.
At the crust-core interface, matter strongly stiffens, and jumps from 1.7 to 2.2, which results from disappearance of nuclei. The analytical approximation smoothes this jump also, though reflects the stiffening. This approximation is also smooth across a small discontinuous drop of at , where muons start to replace a part of ultrarelativistic electrons. However, the electrons and muons give only minor contribution to the pressure (and therefore behavior of ) in the core, because the main contribution comes from interactions between nucleons.
Analytical representations of the EOSs used in the modern 2-D and 3-D simulations of neutron star dynamics have many important advantages over the tabulated EOSs. Analytical EOSs do not require any interpolations, are thermodynamically consistent, and allow for a very high precision of calculations. In the present paper we constructed analytical representations, in terms of the continuous and differentiable functions of a single chosen variable, of the SLy and FPS EOSs. For these analytical representations, the thermodynamic relations are exactly satisfied at any point. Two choices of the independent variable were considered. The first one is ; function given by Eq. (14) fits the original tables in the density interval within typical error of 1-2%. Function can be calculated either from Eq. (3) to satisfy exactly the first law of thermodynamics, or from the fit (16) with a typical error 0.1%. A variant which ensures the same accuracy is to choose nas an independent variable and calculate from the fit (15) and from Eq. (14). The other choice of the independent variable is to use the pseudo-enthalpy H. This choice is particularly advantageous for applications to 2-D and 3-D numerical simulations of neutron star dynamics, such as rotation and inspiraling stage of the evolution of relativistic neutron-star - neutron-star binary. We represented both EOSs by the continuous and differentiable functions P(H), , and n(H), where is given by Eq. (17) with typical accuracy within a few percent, while P(H) and n(H)are calculated from the functions and , respectively. Differentiation of then yields analytical representations of the adiabatic index for the SLy and FPS EOSs; this quantity is important, for example, for numerical simulations of dynamics of the neutron-star - neutron-star system at the inspiral phase.
The quality of our analytical representations of the EOSs was tested by evaluation of the virial identities GRV2 (Bonazzola 1973; Bonazzola & Gourgoulhon 1994) and GRV3 (Gourgoulhon & Bonazzola 1994) in the numerical simulations of the 2-D stationary rotation of neutron stars. GRV2 and GRV3 are integral identities which must be satisfied by a stationary solution of the Einstein equations and which are not imposed in the numerical procedure (see Nozawa et al. 1998 for the details of computation of GRV2 and GRV3). For rotating configurations we get , which is excellent.
The subroutines for the numerical applications of analytical EOSs (Fortran and C++ versions) can be downloaded from the public domain http://www.ioffe.ru/astro/NSG/NSEOS/
We are very grateful to M. Bejger for performing the 2-D calculations for rapidly rotating neutron stars with our EOSs, and to M. Shibata for pointing out to significant misprints in Eqs. (15) and (16) of the preliminary version of this paper. The work of P.H. was partially supported by the KBN grant No. 1-P03D-008-27. The work of A.P. was partially supported by the RFBR grants 02-02-17668 and 03-07-90200 and the Russian Leading Scientific Schools grant 1115.2003.2.