A&A 380, 151-167 (2001)
DOI: 10.1051/0004-6361:20011402
F. Douchin ^{1,2} - P. Haensel^{3}
1 - Department of Physics, University of Illinois at Urbana-Champaign,
Urbana, Illinois 61801, USA
2 - Centre de Recherche Astronomique de Lyon, ENS de Lyon, 69364 Lyon, France
3 -
N. Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland
Received 31 July 2001 / Accepted 10 September 2001
Abstract
An equation of state (EOS) of neutron star matter, describing
both the neutron star crust and the liquid core, is calculated.
It is based on the effective nuclear interaction SLy of the
Skyrme type, which is particularly suitable for the application
to the calculation of the properties of very neutron rich matter
(Chabanat et al. 1997, 1998). The structure of the crust, and its
EOS, is calculated in the T=0 approximation, and under the assumption
of the ground state composition. The crust-core
transition is a very weakly first-order phase transition, with
relative density jump of about one percent. The EOS of the liquid core
is calculated assuming (minimal) npe
composition.
Parameters of static neutron stars are calculated and compared with
existing observational data on neutron stars.
The minimum and
maximum masses of static neutron stars are
and
,
respectively. Effects of rotation on the minimum
and the maximum mass of neutron stars are briefly discussed.
Key words: dense matter - equation of state - stars: neutron
The EOS is predominantly determined by the nuclear (strong) interaction between elementary constituents of dense matter. Even in the neutron star crust, with density below normal nuclear density (corresponding to baryon density ), nuclear interactions are responsible for the properties (and actually - for the very existence!) of neutron rich nuclei, crucial for the crust EOS. The knowledge of these interactions is particularly important for the structure of the inner neutron star crust, where nuclei are immersed in a neutron gas, and even more so for the EOS of the liquid core. Nuclear interactions are actually responsible for a dramatic lifting of from , obtained when interactions are switched-off (Oppenheimer & Volkoff 1939), above measured of PSR B1913+16, and maybe even above , as suggested by some models of the kHz quasi periodic oscillations in 4U 1820-30 (Zhang et al. 1997; Miller et al. 1998; Kluzniak 1998).
The outer envelope of a neutron star with contains the same elementary constituents as ordinary (e.g. terrestrial) matter, i.e., protons, neutrons, and electrons. Unfortunately, even within this subnuclear density envelope, calculation of the EOS starting from an experimentally determined bare nucleon-nucleon (NN) interaction in vacuum, supplemented with a three-nucleon (NNN) force (which is necessary to fit the properties of and simultaneously with the two body data), is not feasible. This is due to the prohibitive complexity of the many-body problem to be solved in the case of heavy nuclei (more generally: for nuclear structures - spheres, rods, plates etc. Lorenz et al. 1993) immersed in a neutron gas. To make a calculation feasible, one uses a mean field approximation with an effective NN interaction, an approach used with great success in terrestrial nuclear physics. The most ambitious application of this approach to the determination of the structure and EOS of the neutron star crust remains the classical work of Negele & Vautherin (1973). Other authors, who treated this problem, used additional approximations of the quantum mean-field scheme (see: Oyamatsu 1993; Lorenz et al. 1993; Sumiyoshi et al. 1995; Cheng et al. 1997; Douchin & Haensel 2000, and references therein).
It is clear that in order to describe in a physically (in particular, thermodynamically) consistent way both the crust, the liquid core, and the transition between them, one has to use the same many-body model and the same effective NN interaction, on both sides of the crust-core interface. The mean-field scheme can also be applied for the description of the spatially uniform npe liquid provided one uses appropriate effective NN interaction. Note that in this case the calculation of the ground state of nucleon matter can be done also, with rather high precision (at not too high density), starting with bare nuclear Hamiltonian (resulting from bare NN and NNN interactions) (Wiringa et al. 1988; Akmal et al. 1998). The calculated ground state energy (here is the actual ground state wave function, which includes nucleon correlations, and minimizes the energy of the system) has then to be approximated, as well as possible, by , where is the Hartree-Fock wave function, and is effective nuclear Hamiltonian.
Some authors formulated the nuclear many-body problem, relevant for neutron star matter, within relativistic mean-field models, in which nuclear interactions are described by a phenomenological Lagrangian involving coupling of the nucleon fields to the meson fields (Sumiyoshi et al. 1995; Cheng et al. 1997). While such an approach has an obvious advantage at very high density (it yields causal EOS, by construction), its meaning at lower densities is not clear (see, e.g., Heiselberg & Pandharipande 2000). In the present paper we restrict ourselves to the non-relativistic approach.
Once the many-body approximation was fixed, the input consists of the effective NN interaction, which has to reproduce a wealth of experimental data on atomic nuclei, especially those with high neutron excess, as well as reproduce the most reliable numerical results concerning the ground state of dense homogeneous neutron rich nucleon matter. In the case of the calculation of the EOS of neutron star matter, the latter condition may be reduced to the limiting case of pure neutron matter; as it turns out, in such a case many-body calculations with a bare nucleon Hamiltonian are particularly precise, mostly because of the less important role played by the tensor forces (Wiringa et al. 1988).
An effective nucleon Hamiltonian contains a number of parameters which are usually fixed by fitting experimental data on saturation properties of bulk nuclear matter and experimental properties of selected atomic nuclei. The parameters of are also constrained by some general condition, e.g., of spin stability (Kutschera & Wójcik 1994). Most of the existing effective interactions were fitted to the properties of laboratory atomic nuclei, with (N-Z)/A < 0.3, while in the bottom layers of the neutron-star crust, and even more so in the liquid core, one expects . In view of this, application of these effective nuclear interactions to neutron star interior involves a rather risky extrapolation to strongly asymmetric nucleon matter. In order to remove a part of this uncertainty, modifications of effective nuclear forces, to make them consistent with available (and possibly reliable) results of microscopic calculations of neutron matter, have been applied. Such a procedure was used in the seventies to obtain the Sk force (Lattimer & Ravenhall 1978), via a rather ad hoc modification of the Sk1 force constructed originally by Vautherin and Brink (Vautherin & Brink 1970) to describe terrestrial nuclei. In this way, Sk became consistent with energy per nucleon of neutron matter calculated by Siemens & Pandharipande (1971). Later, generalized types of the Skyrme interaction, FPS (Pandharipande & Ravenhall 1989) and FPS21 (Pethick et al. 1995), with a larger number of fitted parameters and more general density dependence, were derived by fitting the temperature and baryon density dependent energies per baryon of nuclear and neutron matter obtained in microscopic calculations of Friedman & Pandharipande (1981).
A new set of the Skyrme-type effective N-N interactions has been derived recently, based on an approach which may be more appropriate, as far as the applications to a very neutron rich matter are concerned (Chabanat et al. 1997; Chabanat et al. 1998). While being of a two-body type, this effective interaction contains, in the spirit of the Skyrme model, a term resulting from averaging of an original three-body component. Relevant additional experimental items concerning neutron rich nuclei (including isovector effective masses), constraints of spin stability and requirement of consistency with the UV14+VIII equation of state (EOS) of dense neutron matter of Wiringa et al. (1988) for were combined with the general procedure of fitting the properties of doubly magic nuclei. This procedure led to a set of the SLy (Skyrme Lyon) effective nucleon-nucleon interactions which - due to the emphasis put on their neutron-excess dependence - seem to be particularly suitable for the calculations of the properties of neutron-star interiors. The FPS force was constructed as a generalized Skyrme model, by fitting the properties of asymmetric dense, cold and hot, nucleon matter, calculated by Friedman & Pandhripande (1981); fitting of the ground state properties of laboratory nuclei was not included in their derivation. Luckily, the FPS force turned out to reproduce rather well, without additional adjustement, the ground state energies of eight doubly closed-shell nuclei ranging from to (Lorenz et al. 1993).
The SLy forces have been constructed so as to be consistent with the UV14+UVII model of Wiringa et al. (1988) of neutron matter above n_{0} (Chabanat et al. 1997; Chabanat et al. 1998). It is therefore of interest to check how well these effective N-N interactions reproduce the UV14+UVII equation of state of neutron matter at subnuclear densities. This feature is quite important for the correct calculation of the equation of state of the bottom layers of neutron star crust and of the liquid core, which contain only a few percent of protons. We do not have direct access to the "experimental equation of state'' of pure neutron matter at subnuclear densities. However, results of the best numerical many-body calculations of the ground state of neutron matter with realistic seems to be sufficiently precise at subnuclear densities to be used as an ersatz of experimental data (Pethick et al. 1995). The SLy effective interaction passes this test very well, in contrast to most of other models of (Douchin & Haensel 2000). In what follows, by the SLy interaction we will mean the basic SLy4 model of Chabanat et al. (1998).
After our unified EOS was constructed, a new state-of-the-art microscopic calculation of the EOS of dense matter (Akmal et al. 1998), which in many respect is superior to the ten years older models of Wiringa et al. (1988), became available. The nuclear Hamiltonian of Akmal et al. (1998) is based on a new Argonne two-nucleon interaction AV18, takes into account relativistic boost corrections to the two-nucleon interaction, and includes new Urbana model of three-nucleon interaction, UIX. In what follows, the most complete models of the EOS of dense cold catalyzed matter, , calculated by Akmal et al. (1998), will be referred to as APR (Akmal Pandharipande Ravenhall). It should be stressed, that in contrast to the FPS and SLy EOS, the APR EOS describes only the liquid core of neutron star, and therefore is not a "unified EOS'' of the neutron star interior. As we will show, neutron star models based on the APR EOS of the liquid core, supplemented with our EOS of the crust, are not very different from the stellar models calculated using our complete, unified EOS.
In the present paper we calculate the unified EOS for neutron star matter using the SLy effective NN interaction. Nuclei in the crust are decribed using the Compressible Liquid Drop Model, with parameters calculated using the many-body methods presented in Douchin et al. (2000) and Douchin & Haensel (2000). The calculation of the EOS is continued to higher densities, characteristic of the liquid core of neutron star. Using our EOS, we then calculate neutron star models and compare their parameters with those obtained using older FPS effective NN interaction. We consider also effects of rotation on neutron star structure. Our neutron star models are then confronted with observations of neutron stars.
The method of the calculation of the EOS for the crust and the liquid core of neutron star is described in Sect. 2. Results for the structure and the EOS of the crust are given in Sect. 3, and those for the liquid core in Sect. 4. Models of neutron stars are reviewed in Sect. 6. Effects of rotation on neutron star structure are briefly discussed in the two last subsections of Sect. 6. Comparison with observations of neutron stars is presented in Sect. 7. Finally, Sect. 8 contains summary and conclusion of our paper.
In order to calculate , we used the Wigner-Seitz approximation. In the case of spheres, bubbles, rods and tubes, Wigner-Seitz cells were approximated by spheres and cylinders, of radius . In the case of slabs, Wigner-Seitz cells were bounded by planes, with being defined as the half-distance between plane boundaries of the cell. At given average nucleon (baryon) density, , and for an assumed shape of nuclear structures, the energy density was minimized with respect to thermodynamic variables, under the condition of an average charge neutrality. Spherical nuclei are energetically preferred over other nuclear shapes, and also over homogeneous npe matter, down to . Within our set of possible nuclear shapes therefore, the ground state of neutron-star crust contains spherical nuclei only. A detailed study of the bottom layers of the inner crust, including the determination of its bottom edge, is presented in Douchin & Haensel (2000).
Z | A | u | |||||
(fm) | (fm) | (fm) | (%) | ||||
1.2126 E-4 |
42.198 | 130.076 | 0.0000 | 5.451 | 5.915 | 63.503 | 0.063 |
1.6241 E-4 | 42.698 | 135.750 | 0.0000 | 5.518 | 6.016 | 58.440 | 0.084 |
1.9772 E-4 | 43.019 | 139.956 | 0.0000 | 5.565 | 6.089 | 55.287 | 0.102 |
2.0905 E-4 | 43.106 | 141.564 | 0.0000 | 5.578 | 6.111 | 54.470 | 0.107 |
2.2059 E-4 | 43.140 | 142.161 | 0.0247 | 5.585 | 6.122 | 54.032 | 0.110 |
2.3114 E-4 | 43.163 | 142.562 | 0.0513 | 5.590 | 6.128 | 53.745 | 0.113 |
2.6426 E-4 | 43.215 | 143.530 | 0.1299 | 5.601 | 6.145 | 53.020 | 0.118 |
3.0533 E-4 | 43.265 | 144.490 | 0.2107 | 5.612 | 6.162 | 52.312 | 0.123 |
3.5331 E-4 | 43.313 | 145.444 | 0.2853 | 5.623 | 6.179 | 51.617 | 0.129 |
4.0764 E-4 | 43.359 | 146.398 | 0.3512 | 5.634 | 6.195 | 50.937 | 0.135 |
4.6800 E-4 | 43.404 | 147.351 | 0.4082 | 5.645 | 6.212 | 50.269 | 0.142 |
5.3414 E-4 | 43.447 | 148.306 | 0.4573 | 5.656 | 6.228 | 49.615 | 0.148 |
6.0594 E-4 | 43.490 | 149.263 | 0.4994 | 5.667 | 6.245 | 48.974 | 0.155 |
7.6608 E-4 | 43.571 | 151.184 | 0.5669 | 5.690 | 6.278 | 47.736 | 0.169 |
1.0471 E-3 | 43.685 | 154.094 | 0.6384 | 5.725 | 6.328 | 45.972 | 0.193 |
1.2616 E-3 | 43.755 | 156.055 | 0.6727 | 5.748 | 6.362 | 44.847 | 0.211 |
1.6246 E-3 | 43.851 | 159.030 | 0.7111 | 5.784 | 6.413 | 43.245 | 0.239 |
2.0384 E-3 | 43.935 | 162.051 | 0.7389 | 5.821 | 6.465 | 41.732 | 0.271 |
2.6726 E-3 | 44.030 | 166.150 | 0.7652 | 5.871 | 6.535 | 39.835 | 0.320 |
3.4064 E-3 | 44.101 | 170.333 | 0.7836 | 5.923 | 6.606 | 38.068 | 0.377 |
4.4746 E-3 | 44.155 | 175.678 | 0.7994 | 5.989 | 6.698 | 36.012 | 0.460 |
5.7260 E-3 | 44.164 | 181.144 | 0.8099 | 6.059 | 6.792 | 34.122 | 0.560 |
7.4963 E-3 | 44.108 | 187.838 | 0.8179 | 6.146 | 6.908 | 32.030 | 0.706 |
Z | A | u | |||||
(fm) | (fm) | (fm) | (%) | ||||
9.9795 E-3 |
43.939 | 195.775 | 0.8231 | 6.253 | 7.048 | 29.806 | 0.923 |
1.2513 E-2 | 43.691 | 202.614 | 0.8250 | 6.350 | 7.171 | 28.060 | 1.159 |
1.6547 E-2 | 43.198 | 211.641 | 0.8249 | 6.488 | 7.341 | 25.932 | 1.566 |
2.1405 E-2 | 42.506 | 220.400 | 0.8222 | 6.637 | 7.516 | 24.000 | 2.115 |
2.4157 E-2 | 42.089 | 224.660 | 0.8200 | 6.718 | 7.606 | 23.106 | 2.458 |
2.7894 E-2 | 41.507 | 229.922 | 0.8164 | 6.825 | 7.721 | 22.046 | 2.967 |
3.1941 E-2 | 40.876 | 235.253 | 0.8116 | 6.942 | 7.840 | 21.053 | 3.585 |
3.6264 E-2 | 40.219 | 240.924 | 0.8055 | 7.072 | 7.967 | 20.128 | 4.337 |
3.9888 E-2 | 39.699 | 245.999 | 0.7994 | 7.187 | 8.077 | 19.433 | 5.058 |
4.4578 E-2 | 39.094 | 253.566 | 0.7900 | 7.352 | 8.231 | 18.630 | 6.146 |
4.8425 E-2 | 38.686 | 261.185 | 0.7806 | 7.505 | 8.372 | 18.038 | 7.202 |
5.2327 E-2 | 38.393 | 270.963 | 0.7693 | 7.685 | 8.538 | 17.499 | 8.470 |
5.6264 E-2 | 38.281 | 283.993 | 0.7553 | 7.900 | 8.737 | 17.014 | 10.011 |
6.0219 E-2 | 38.458 | 302.074 | 0.7381 | 8.167 | 8.987 | 16.598 | 11.914 |
6.4183 E-2 | 39.116 | 328.489 | 0.7163 | 8.513 | 9.315 | 16.271 | 14.323 |
6.7163 E-2 | 40.154 | 357.685 | 0.6958 | 8.853 | 9.642 | 16.107 | 16.606 |
7.0154 E-2 | 42.051 | 401.652 | 0.6699 | 9.312 | 10.088 | 16.058 | 19.501 |
7.3174 E-2 | 45.719 | 476.253 | 0.6354 | 9.990 | 10.753 | 16.213 | 23.393 |
7.5226 E-2 | 50.492 | 566.654 | 0.6038 | 10.701 | 11.456 | 16.557 | 26.996 |
7.5959 E-2 | 53.162 | 615.840 | 0.5898 | 11.051 | 11.803 | 16.772 | 28.603 |
P | P | ||||||
( ) | ( ) | ( ) | ( ) | ( ) | ( ) | ||
2.0905 E-4 | 3.4951 E11 | 6.2150 E29 | 1.177 | 9.9795 E-3 | 1.6774 E13 | 3.0720 E31 | 1.342 |
2.2059 E-4 | 3.6883 E11 | 6.4304 E29 | 0.527 | 1.2513 E-2 | 2.1042 E13 | 4.1574 E31 | 1.332 |
2.3114 E-4 | 3.8650 E11 | 6.5813 E29 | 0.476 | 1.6547 E-2 | 2.7844 E13 | 6.0234 E31 | 1.322 |
2.6426 E-4 | 4.4199 E11 | 6.9945 E29 | 0.447 | 2.1405 E-2 | 3.6043 E13 | 8.4613 E31 | 1.320 |
3.0533 E-4 | 5.1080 E11 | 7.4685 E29 | 0.466 | 2.4157 E-2 | 4.0688 E13 | 9.9286 E31 | 1.325 |
3.5331 E-4 | 5.9119 E11 | 8.0149 E29 | 0.504 | 2.7894 E-2 | 4.7001 E13 | 1.2023 E32 | 1.338 |
4.0764 E-4 | 6.8224 E11 | 8.6443 E29 | 0.554 | 3.1941 E-2 | 5.3843 E13 | 1.4430 E32 | 1.358 |
4.6800 E-4 | 7.8339 E11 | 9.3667 E29 | 0.610 | 3.6264 E-2 | 6.1153 E13 | 1.7175 E32 | 1.387 |
5.3414 E-4 | 8.9426 E11 | 1.0191 E30 | 0.668 | 3.9888 E-2 | 6.7284 E13 | 1.9626 E32 | 1.416 |
6.0594 E-4 | 1.0146 E12 | 1.1128 E30 | 0.726 | 4.4578 E-2 | 7.5224 E13 | 2.3024 E32 | 1.458 |
7.6608 E-4 | 1.2831 E12 | 1.3370 E30 | 0.840 | 4.8425 E-2 | 8.1738 E13 | 2.6018 E32 | 1.496 |
1.0471 E-3 | 1.7543 E12 | 1.7792 E30 | 0.987 | 5.2327 E-2 | 8.8350 E13 | 2.9261 E32 | 1.536 |
1.2616 E-3 | 2.1141 E12 | 2.1547 E30 | 1.067 | 5.6264 E-2 | 9.5022 E13 | 3.2756 E32 | 1.576 |
1.6246 E-3 | 2.7232 E12 | 2.8565 E30 | 1.160 | 6.0219 E-2 | 1.0173 E14 | 3.6505 E32 | 1.615 |
2.0384 E-3 | 3.4178 E12 | 3.7461 E30 | 1.227 | 6.4183 E-2 | 1.0845 E14 | 4.0509 E32 | 1.650 |
2.6726 E-3 | 4.4827 E12 | 5.2679 E30 | 1.286 | 6.7163 E-2 | 1.1351 E14 | 4.3681 E32 | 1.672 |
3.4064 E-3 | 5.7153 E12 | 7.2304 E30 | 1.322 | 7.0154 E-2 | 1.1859 E14 | 4.6998 E32 | 1.686 |
4.4746 E-3 | 7.5106 E12 | 1.0405 E31 | 1.344 | 7.3174 E-2 | 1.2372 E14 | 5.0462 E32 | 1.685 |
5.7260 E-3 | 9.6148 E12 | 1.4513 E31 | 1.353 | 7.5226 E-2 | 1.2720 E14 | 5.2856 E32 | 1.662 |
7.4963 E-3 | 1.2593 E13 | 2.0894 E31 | 1.351 | 7.5959 E-2 | 1.2845 E14 | 5.3739 E32 | 1.644 |
One has to be aware of the simplications and approximations inherent to the CLDM. While the parameters of this model are determined in quantum-mechanical many-body calculation, the model itself is par excellence classical. It does not exhibit therefore the shell effects corresponding to the closure of proton or neutron shells in nuclei or the effect of neutron or proton pairing. Shell effects imply particularly strong binding of nuclei with "magic numbers'' of Z=28 and N=50, 82. Consequently (except at lowest density) nuclei present in the ground state of outer crust are expected to have Z=28 (at lower density) or N=50, 82 (at higher density) (Baym et al. 1971b; Haensel & Pichon 1994). This feature is absent in the CLDM EOS, which additionally treats Z and A as continuous variables. As a result, CLDM EOS at is softer and has lower value of neutron drip density (real nuclei are stabilized against neutron drip by the pairing and the shell effects for neutrons) than that based on experimental nuclear masses.
(%) | (%) | (%) | (%) | (%) | (%) | ||
0.0771 | 3.516 | 3.516 | 0.000 | 0.490 | 7.516 | 4.960 | 2.556 |
0.0800 | 3.592 | 3.592 | 0.000 | 0.520 | 7.587 | 4.954 | 2.634 |
0.0850 | 3.717 | 3.717 | 0.000 | 0.550 | 7.660 | 4.952 | 2.708 |
0.0900 | 3.833 | 3.833 | 0.000 | 0.580 | 7.736 | 4.955 | 2.781 |
0.1000 | 4.046 | 4.046 | 0.000 | 0.610 | 7.818 | 4.964 | 2.854 |
0.1100 | 4.233 | 4.233 | 0.000 | 0.640 | 7.907 | 4.979 | 2.927 |
0.1200 | 4.403 | 4.398 | 0.005 | 0.670 | 8.003 | 5.001 | 3.002 |
0.1300 | 4.622 | 4.521 | 0.101 | 0.700 | 8.109 | 5.030 | 3.079 |
0.1600 | 5.270 | 4.760 | 0.510 | 0.750 | 8.309 | 5.094 | 3.215 |
0.1900 | 5.791 | 4.896 | 0.895 | 0.800 | 8.539 | 5.178 | 3.361 |
0.2200 | 6.192 | 4.973 | 1.219 | 0.850 | 8.803 | 5.284 | 3.519 |
0.2500 | 6.499 | 5.014 | 1.485 | 0.900 | 9.102 | 5.410 | 3.692 |
0.2800 | 6.736 | 5.031 | 1.705 | 0.950 | 9.437 | 5.557 | 3.880 |
0.3100 | 6.920 | 5.034 | 1.887 | 1.000 | 9.808 | 5.726 | 4.083 |
0.3400 | 7.066 | 5.026 | 2.040 | 1.100 | 10.663 | 6.124 | 4.539 |
0.3700 | 7.185 | 5.014 | 2.170 | 1.200 | 11.661 | 6.602 | 5.060 |
0.4000 | 7.283 | 4.999 | 2.283 | 1.300 | 12.794 | 7.151 | 5.643 |
0.4300 | 7.368 | 4.984 | 2.383 | 1.400 | 14.043 | 7.762 | 6.281 |
0.4600 | 7.444 | 4.971 | 2.473 | 1.500 | 15.389 | 8.424 | 6.965 |
P | P | ||||||
( ) | ( ) | ( ) | ( ) | ( ) | ( ) | ||
0.0771 | 1.3038 E14 | 5.3739 E32 | 2.159 | 0.4900 | 8.8509 E14 | 1.0315 E35 | 2.953 |
0.0800 | 1.3531 E14 | 5.8260 E32 | 2.217 | 0.5200 | 9.4695 E14 | 1.2289 E35 | 2.943 |
0.0850 | 1.4381 E14 | 6.6828 E32 | 2.309 | 0.5500 | 1.0102 E15 | 1.4491 E35 | 2.933 |
0.0900 | 1.5232 E14 | 7.6443 E32 | 2.394 | 0.5800 | 1.0748 E15 | 1.6930 E35 | 2.924 |
0.1000 | 1.6935 E14 | 9.9146 E32 | 2.539 | 0.6100 | 1.1408 E15 | 1.9616 E35 | 2.916 |
0.1100 | 1.8641 E14 | 1.2701 E33 | 2.655 | 0.6400 | 1.2085 E15 | 2.2559 E35 | 2.908 |
0.1200 | 2.0350 E14 | 1.6063 E33 | 2.708 | 0.6700 | 1.2777 E15 | 2.5769 E35 | 2.900 |
0.1300 | 2.2063 E14 | 1.9971 E33 | 2.746 | 0.7000 | 1.3486 E15 | 2.9255 E35 | 2.893 |
0.1600 | 2.7223 E14 | 3.5927 E33 | 2.905 | 0.7500 | 1.4706 E15 | 3.5702 E35 | 2.881 |
0.1900 | 3.2424 E14 | 5.9667 E33 | 2.990 | 0.8000 | 1.5977 E15 | 4.2981 E35 | 2.869 |
0.2200 | 3.7675 E14 | 9.2766 E33 | 3.025 | 0.8500 | 1.7302 E15 | 5.1129 E35 | 2.858 |
0.2500 | 4.2983 E14 | 1.3668 E34 | 3.035 | 0.9000 | 1.8683 E15 | 6.0183 E35 | 2.847 |
0.2800 | 4.8358 E14 | 1.9277 E34 | 3.032 | 0.9500 | 2.0123 E15 | 7.0176 E35 | 2.836 |
0.3100 | 5.3808 E14 | 2.6235 E34 | 3.023 | 1.0000 | 2.1624 E15 | 8.1139 E35 | 2.824 |
0.3400 | 5.9340 E14 | 3.4670 E34 | 3.012 | 1.1000 | 2.4820 E15 | 1.0609 E36 | 2.801 |
0.3700 | 6.4963 E14 | 4.4702 E34 | 2.999 | 1.2000 | 2.8289 E15 | 1.3524 E36 | 2.778 |
0.4000 | 7.0684 E14 | 5.6451 E34 | 2.987 | 1.3000 | 3.2048 E15 | 1.6876 E36 | 2.754 |
0.4300 | 7.6510 E14 | 7.0033 E34 | 2.975 | 1.4000 | 3.6113 E15 | 2.0679 E36 | 2.731 |
0.4600 | 8.2450 E14 | 8.5561 E34 | 2.964 | 1.5000 | 4.0498 E15 | 2.4947 E36 | 2.708 |
As we see in Tables 1 and 2, the number of nucleons in a nucleus, A, grows monotonically with increasing density and reaches about 600 at the edge of the crust. However, the number of protons changes rather weakly, from near neutron drip, to near the edge of the crust. Our results for Z of spherical nuclei are similar to those obtained in Ravenhall et al. (1972) and Oyamatsu (1993), but are somewhat higher than those obtained using a relativistic mean-field model in Sumiyoshi et al. (1995). The problem of stability of nuclei in the bottom layer of the inner crust with respect to fission was discussed in Douchin & Haensel (2000).
Actually, under conditions of thermodynamic equilibrium, transition from the crust to the uniform liquid takes place at a constant pressure, and is accompanied by a density jump (first order phase transition). Using Maxwell construction, we find that the edge of the crust has density , and coexists there with uniform npe matter of the density higher by . Crust-liquid core transition is therefore a very weak first-order phase transition; it takes place at .
The equation of state of the liquid core is given in Table 5. Its properties will be discussed in Sect. 5. Here we will restrict ourselves to a comment referring to its practical use in neutron star calculations. The tiny density jump between core and crust is not relevant for the applications to calculations of the neutron star structure (although it can play a role in neutron star dynamics). One can remove the first line of Table 5 and then match the resulting EOS of the core to that of the inner crust, given in Table 3. However, one can also remove the last line of Table 3, and then match the EOS of the inner crust to that of the inner core, given in Table 5. In practice, the difference in neutron star structure, resulting from the difference in these two prescriptions, is negligibly small.
Neutron drip at implies a dramatic drop in , which corresponds to strong softening of the EOS. Density stays continuous at the neutron drip point, with low-density dripped neutrons contributing to and , but exerting a very small pressure, and moreover being in phase equilibrium with nuclear matter of nuclei. Consequently, drops by more than a factor of two, a sizable part of this drop occurring via discontinuous drop at , characteristic of a second-order phase transition. After this initial dramatic drop, matter stiffens, because pressure of neutron gas inscreases. The actual value of results from an interplay of several factors, with stiffening due to Fermi motion and neutron-neutron repulsion in dripped non-relativistic neutron gas and, countering this, softening Coulomb (lattice) contribution, a rather soft contribution of ultrarelativistic electron gas, and a softening effect of neutron gas - nuclear matter coexistence. As one sees in Fig. 2, reaches the value of about 1.6 near the bottom edge of the inner crust, only slightly lower than 5/3 characteristic of a non-relativistic free Fermi gas.
At the crust-core interface, matter strongly stiffens, and increases discontinuously, by 0.5, to about 2.2. This jump results from the disappearence of nuclei: a two-phase nucleon system changes into a single-phase one, and repulsive nucleon-nucleon interaction is no longer countered by softening effects resulting from the presence of nuclear structure and neutron gas - nuclear matter phase coexistence. With increasing density, grows above 3 at , due to increasing contribution of repulsive nucleon interactions.
A tiny notch appears at the muon threshold, at which undergoes small, but clearly visible, discontinuous drop. It is due to the appearance of new fermions - muons, which replace high-energy electrons (electron Fermi energy MeV). Replacing rapidly moving electrons by slowly moving muons leads to a drop in the sound velocity (and ) just after the threshold. Because lepton contribution to pressure is at this density very small, the overall effect is small. A discontinuous drop in at muon threshold is characteristic of a second-order phase transition at which density is continuous but compressibility is not.
At higher densities, , decreases slowly, which results from the interplay of the density dependence of nuclear interactions and of increasing proton fraction.
Both and in the liquid interior are shown in Fig. 3. Freezing the composition stiffens neutron star matter, , the effect being of the order of a few percent. Another effect of the composition freezing is removing of a softening just after the appearence of muons, because of the slowness of processes in which they are produced or absorbed.
The characteristic period of sound waves excited in the liquid core can
be estimated as
ms.
Therefore,
is much shorter than
the timescale of beta processes, so that in this case
Figure 1: The SLy EOS of the ground-state neutron star matter. Dotted vertical line corresponds to the neutron drip and the dashed one to the crust-liquid core interface. | |
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Figure 2: Adiabatic index of the Sly EOS versus matter density. Matter is assumed to be in full thermodynamic equilibrium. Dotted vertical line: neutron drip. Dashed vertical line: crust - liquid core interface. Dashed horizontal line in the region of the outer crust is obtained using empirical and semi-empirical (i.e., from mass formulae) masses of nuclei and removing the points corresponding to the density jumps between shells with different nuclei (Baym et al. 1971b; Haensel & Pichon 1994). | |
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EOS | M | R | A | I | |||||
[km] | [10^{57}] | ||||||||
SLy | 2.05 | 9.99 | 1.21 | 2.86 | 1.38 | 2.91 | 0.594 | 6.79 | 1.91 |
FPS | 1.80 | 9.27 | 1.46 | 3.40 | 1.37 | 2.52 | 0.531 | 5.37 | 1.36 |
Figure 3: Adiabatic index of the SLy EOS of the liquid core. Solid line: beta equilibrium of npe matter (equilibrium composition during compression or decompression). Dashed line: vanishing rate of weak processes in npe matter (frozen composition during compression or decompression). | |
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Models of cold, static neutron stars form a one-parameter
family. They can be labeled by their central pressure,
,
or equivalently by their central
density,
.
Figure 4: Gravitational mass M versus central density , for the SLy, FPS, and APR EOS of dense matter. Maximum on the mass-central density curves is indicated by a filled circle. On the APR curve, configurations to the right of the asterisk contain a central core with . Configurations to the right of the maxima are unstable with respect to small radial perturbations, and are denoted by a dotted line. The shaded band corresponds to the range of precisely measured masses of binary radio pulsars. | |
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Figure 5: Gravitational mass versus central density, in the vicinity of the minimum mass, for static neutron stars. Dotted lines - configurations unstable with respect to small radial perturbations. Minimum mass configuration is indicated by a filled circle. | |
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Figure 6: Neutron star radius R versus gravitational mass M, with notation as in Fig. 4. Doubly hatched area is prohibited by general relativity, because it corresponds to (for a general proof, see Weinberg 1972). All hatched triangle (double and single hatched) is prohibited by the general relativity and condition (necessary but not sufficient for respecting causality, Olsson 2000) combined. The shaded band corresponds to the range of precisely measured masses of binary radio pulsars. | |
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Figure 7: Surface redshift versus gravitational mass M. Hatched area is prohibited for EOSs with . Shaded vertical band corresponds to the range of precisely measured masses of binary radio pulsars. The band limited by two dashed horizontal lines corresponds to the estimate of from the measured spectrum of the gamma-ray burst GB 790305b. | |
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Figure 8: Binding energy relative to dispersed versus gravitational mass. The shaded rectangle corresponds to the estimates of the total energy of the neutrino burst in SN 1987A, and to the estimates of the mass of neutron star formed in this event. | |
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Figure 9: Moment of inertia for slow, rigid rotation versus gravitational mass. The configuration with maximum I is indicated by a filled triangle, and that of maximum mass - by a filled circle. A shaded band corresponds to the range of precisely measured masses of binary radio pulsars. | |
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Central density of the maximum allowable mass configuration is the maximum one which can be reached within static neutron stars. Models with have . They are therefore unstable with respect to small radial perturbations and collapse into black holes (see, e.g., Shapiro & Teukolsky 1983). The maximum central density for static stable neutron stars is, for our EOS, , to be compared with for the FPS EOS. Corresponding maximum value of baryon density is , to be compared with obtained for the FPS EOS. A complete set of parameters of configuration with maximum allowable mass for our EOS is presented in Table 6, where the corresponding parameters obtained for the FPS EOS are also given for comparison.
Comparison with the APR EOS is also of interest, and therefore we show the curve for this EOS. The curve obtained for our EOS is quite close to the APR one, especially for . It should be mentioned, that for the APR neutron star models contain a central core with , and should therefore be taken with a grain of salt. Such a problem does not arise for our EOS, for which within all stable neutron star models.
Precisely measured masses of radio pulsars in binaries with another neutron star span the range 1.34-1.44 (Thorsett & Chakrabarty 1999), visualized in Fig. 4 by a shaded band. For neutron stars of such masses, central density is about , slightly below ; this result is nearly the same as for the APR EOS. For the FPS EOS, neutron star of such a mass has higher central density, about .
EOS | R | ||||
[km] | [km] | ||||
SLy | 0.094 | 1.6 | 270 | 0.02 | 3.8 |
FPS | 0.088 | 2.2 | 220 | 0.03 | 4.2 |
The value of for the SLy EOS and the FPS EOS are quite similar: in both cases . Since the SLy EOS is stiffer than the FPS one in the vicinity of the crust-core interface, its configuration is less dense and has larger radius. In both cases it has a small central liquid core, containing of mass in the case of the SLy EOS and of star mass in the case of a softer FPS EOS.
At the same value of M between and , the radius of the SLy neutron star is some km larger than that obtained for the FPS EOS; the difference increases with increasing M, and reaches 2 km at . It is of interest to compare R(M) curve for our EOS also with that obtained for the APR EOS. For both curves are quite similar. Note that highest-mass segment (to the right of the asterisk) of the APR curve should be treated with caution, because stellar models contain there a central core with . It is due to this unphysical feature that the APR R(M) curve approaches so closely the prohibited hatched region of the R-M plane.
For a static neutron star, general relativity predicts that the circular Keplerian orbits (for test particles) with are stable, and those with are unstable, where the gravitational radius km (see, e.g., Shapiro & Teukolsky 1983).
The radius of the marginally stable orbit, which separates these two classes of orbits, is therefore km. As we see in Fig. 6, for we have , and therefore for such neutron stars the innermost stable circular orbit (ISCO) is separated from the stellar surface by a gap. A similar situation holds also for the FPS EOS. Note that the existence of a gap between the ISCO and neutron star surface might be important for the interpretation of the spectra of the kiloherhz Quasi Periodic Oscillations observed in the X-ray radiation of some Low Mass X-ray Binaries (van der Klis 2000).
For , the curve for our EOS is quite similar to the APR one.
The scaling argument is much less precise in the case of comparison of maximum for our EOS with the APR one. However, let us remind that the APR curve above the asterisk should be treated with caution. For measured masses of binary pulsars, we get for our EOS erg; corresponding values for the FPS EOS are some erg higher.
Figure 10: Apparent radius of neutron star, , versus gravitational mass, M, Long-dash-dot straight line corresponds to minimum at a given M (see the text). | |
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The maximum mass for rigidly rotating neutron stars is , 18% higher than the maximum mass for static configurations. For the FPS EOS, one obtains (Cook et al. 1994), also 18% higher than for the static configuration. Such an increase of due to a maximal uniform rotation is characteristic of subluminal realistic baryonic EOS (Lasota et al. 1996).
The apparent (or radiation) radius,
measured by a distant observer, ,
is related to
R by
A strict lower bound on at a given M results from the very definition of , and does not depend on any physical constraint: km (Lattimer & Prakash 2001, see also Haensel 2001). As one sees in Fig. 10, the values of at are extremely close (within less than 1%) to . This property has been explained in (Haensel 2001).
The central value of for Geminga, obtained by Golden & Shearer (1999) using the best-fit model atmosphere spectra, cannot be explained by our EOS, and actually - by none of existing baryonic EOS of dense matter (it could be modeled by a small mass strange star covered by a normal matter layer, to produce the observed photon spectrum). However, the uncertainty in the extracted value of is large: it stems mainly from the uncertainty in the distance to Geminga (assumed to be d=159 pc), but a poor knowledge of the photon spectrum plays also an important role. One therefore might argue, that because of these uncertainties the measured value of cannot exclude our and other baryonic EOS of dense matter at a reasonably high confidence level of 95%.
In the case of RX J185635-3754 contradiction between extracted value of , and theoretical models of neutron stars based on our EOS (and on other available models of dense matter) is even more dramatic (Pons et al. 2001)^{}. The central best-fit value of is 8.2 km (Fe atmosphere) and 7.8 km (Si-ash atmosphere), at assumed distance d=61 pc. Non-uniformity of surface temperature, consistent with observational constraints, does not allow to remove this conflict between theory and observations. Unfortunately, proper inclusion of effects of surface magnetic field is not possible because of non-availability of magnetized heavy-metal atmosphere models (Pons et al. 2001). One may only hope, that the problem of the conflict between theoretical and measured of closeby isolated neutron stars will be solved in the future studies.
Very recently, Rutledge et al. (2001) proposed a method of measuring of neutron stars, observed as X-transients in globular clusters. They studied transient X-ray source CXOU 132619.7-472910.8 in NGC 5139. Fitting its photon spectrum with H-atmosphere model, they obtained, at 90% confidence level, km, which is consistent with our EOS, and with FPS, APR and many other available EOS of dense matter. This method of measuring seems to be very promising, because both distance and interstellar hydrogen column density are relatively well known for globular clusters.
Our model of matter at supranuclear densities is the simplest possible, and is based on experimental nuclear physics and relatively precise many-body calculations of dense neutron matter. We did not consider possible dense matter constituents, for which strong interactions are poorly known (hyperons), or which are hypothetical (pion and kaon condensates, quark matter). Such a model as that proposed in the present paper may seem very simple - as compared to a rich spectrum of possibilities considered in the literature on the constitution of dense neutron star cores. However, it has the virtue of giving a unified description of all the interior of a neutron star, and is firmly based on the most solid sector of our knowledge of nuclear interactions.
Acknowledgements
We express our gratitude to A. Potekhin for reading the manuscript, and for remarks and comments which helped to improve the present paper. We are also grateful to him for his precious help in the preparation of figures. This research was partially supported by the KBN grant No. 5P03D.020.20 and by the CNRS/PAN Jumelage Astrophysique Program.