A&A 380, 151-167 (2001)
DOI: 10.1051/0004-6361:20011402
F. Douchin 1,2 - P. Haensel3
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 n0 (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).
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Z | A | ![]() |
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u |
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(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 |
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Z | A | ![]() |
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u |
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(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 |
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P | ![]() |
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P | ![]() |
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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.
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(%) | (%) | (%) |
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(%) | (%) | (%) |
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 |
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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
![]() |
Open with DEXTER |
EOS | M | R | ![]() |
![]() |
![]() |
A |
![]() |
![]() |
I |
![]() |
[km] |
![]() |
![]() |
![]() |
[1057] |
![]() |
![]() |
||
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![]() ![]() |
Open with DEXTER |
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
![]() ![]() |
Open with DEXTER |
![]() |
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
![]() ![]() |
Open with DEXTER |
![]() |
Figure 7:
Surface redshift
![]() ![]() ![]() |
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![]() |
Figure 8:
Binding energy relative to dispersed
![]() |
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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, ![]() ![]() |
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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.