ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 9, с. 1217-1225
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
FROM THE CRUST TO THE CORE OF NEUTRON STARS ON A MICROSCOPIC BASIS
©2014 M. Baldo1)*, G. F. Burgio1^ M. Centelles2), B. K. Sharma2), X. Viñas2)
Received June 29, 2013
Within a microscopic approach the structure of Neutron Stars is usually studied by modelling the homogeneous nuclear matter of the core by a suitable Equation of State, based on a many-body theory, and the crust by a functional based on a more phenomenological approach. We present the first calculation of Neutron Star overall structure by adopting for the core an Equation of State derived from the Brueckner— Hartree—Fock theory and for the crust, including the pasta phase, an Energy Density Functional based on the same Equation of State, and which is able to describe accurately the binding energy of nuclei throughout the mass table. Comparison with other approaches is discussed. The relevance of the crust Equation of State for the Neutron Star radius is particularly emphasised.
DOI: 10.7868/S0044002714080030
1. INTRODUCTION
A convergent effort of experimental and theoretical nuclear physics has been developing along several years to determine the structure and properties of Neutron Stars (NS). These studies are expected to reveal the properties and composition of neutron-rich nuclear matter at high density and at the same time the properties of exotic nuclei that are present in the crust and cannot be produced in laboratory because of the large asymmetry. The interpretation of the signals coming from the astrophysical observations on the processes and phenomena that occur in NS requires reliable theoretical inputs. The interplay between the observations and the theoretical predictions has stimulated an impressive progress in the field, and it is expected to help answering many fundamental questions on the properties of matter under extreme conditions and the corresponding elementary processes that can occur. Among others, we mention the maximum mass beyond which a NS collapses to a black hole, the baryon composition of matter at high density, and the properties of extremely asymmetric matter. It is therefore of great interest to have a sound theoretical background for the development of the field, in order to reduce the uncertainty on the possible conclusions one can draw from these studies. In particular, it can be of great help to
Sezione di Catania, and Dipartimento di Fisica e Astronomia, Universita di Catania, Italy.
2)Departament d'Estructura i Constituentes de la Materia and Institut de Ciencies del Cosmos, Facultat de Fisica, Universitat de Barcelona, Spain. E-mail: baldo@ct.infn.it
develop a unified theory which is able to describe on a microscopic level the overall structure of NS, from the crust to the inner core. This is not a simple task, since the methods developed for homogeneous nuclear matter cannot be easily extended to nuclei and to the non-homogeneous matter present in NS crust. Recently [1—3], an Energy Density Functional (EDF) to describe finite nuclei has been developed, based on the nuclear matter Equation of State (EOS) derived from the Brueckner—Hartree—Fock (BHF) scheme. We employ this EOS and the corresponding EDFto describe the whole NS structure and compare with the results obtained with the few other methods that encompass the whole NS structure. We compare also with the few semi-microscopic approaches, where the crust and the core can be also described within the same theoretical scheme. In this paper we limit the treatment to only nucleon degrees of freedom, neglecting the possibility of the appearance of exotic components like hyperons and quarks, for which the uncertainty is too large to perform a fruitful comparison with other approaches.
The present article is organized as follows. In Section 2 we introduce the models of the EOS considered in this work. Section 3 is devoted to the description of our calculations oftheNS crust, whereas in Section 4 we describe the calculation of the EOS of the NS core. In Section 5 we discuss the results for the mass and radius of a NS. Finally, our concluding remarks are presented in Section 6.
2. EOS OF NUCLEAR MATTER
In this section we first remind briefly the BHF method for the nuclear matter EOS. This theo-
retical scheme is based on the Brueckner— Bethe— Goldstone (BBG) many-body theory, which is the linked cluster expansion of the energy per nucleon of nuclear matter (see chapter 1 of [4] and references therein). In this many-body approach one systematically replaces the bare nucleon—nucleon (Nn) v interaction by the Brueckner reaction matrix G, which is the solution of the Bethe—Goldstone equation
G[p; u] = v + Jy
ka kb
\kgkb)Q{kakb\ co - e(ka) - e(kb)
G[p; u], (1)
where p is the nucleon number density, and u is the starting energy. The single-particle energy e(k),
fr2k2
e(k)=e(k]P) = — + U(k]P), (2)
and the Pauli operator Q constrains the intermediate baryon pairs to momenta above the Fermi momentum. The BHF approximation for the single-particle potential U(k; p) using the continuous choice is
U (k; p)= (3)
= Re ^ {kk'\G[p; e(k)+ e(k')]\kk')a, k'<kF
where the subscript "a" indicates anti-symmetriza-tion of the matrix element. Due to the occurrence of U(k) in Eq. (2), these equations constitute a coupled system that has to be solved in a self-consistent manner for several momenta of the particles involved, at the considered densities. In the BHF approximation the energy per nucleon is
E A
3 h2k2 5 2m
+
(4)
+ 2A E (kk'\G[p-,e(k)+e(k')]\kk')a.
k,k'<kF
The nuclear EOS can be calculated with good accuracy in the Brueckner two-hole-line approximation with the continuous choice for the single-particle potential, since the results in this scheme are quite close to the calculations which include also the three-hole-line contribution [5]. However, as it is well known, the non-relativistic calculations, based on purely two-body interactions, fail to reproduce the correct saturation point of symmetric nuclear matter and one needs to introduce three-body forces (TBFs). In our approach the TBF is reduced to a density-dependent two-body force by averaging over the position of the third particle. In this work we will illustrate results based on the so-called Urbana model [6]. The nuclear matter EOS was calculated in previous works [1] for both symmetric matter and neutron matter. The TBF produce in symmetric matter a shift of the saturation point of about +1 MeV in energy. For computational
purpose, on the calculated symmetric matter EOS an educated polynomial fit was performed with a fine tuning of the two parameters contained in the TBF, as in references [7, 8], in order to get an optimal saturation point (the minimum), E/A = —0.16 MeV and po = 0.16 fm"3. The higher density EOS, above 0.62 fm"3, is quite smooth and the calculated points have been interpolated numerically without any polynomial fit.
This EOS was used to construct an EDF for describing finite nuclei as explained in [1]. The nuclear matter EOS is assumed to be the bulk part of the functional. One then needs few additional phe-nomenological parameters to fit a large set of nuclear binding energies throughout the nuclear mass table. The additional ingredients of the functional are a surface energy term, necessary to describe correctly the nuclear surface, which is absent in nuclear matter, the single-particle spin—orbit interaction, typical of finite nuclei, and a pairing contribution to describe open-shell nuclei. The bulk symmetry energy is directly taken from the nuclear matter EOS, by a quadratic interpolation between pure neutron matter and symmetric nuclear matter. This approximation is a good one, as it has been checked by comparing with calculations in asymmetric nuclear matter. As explained in [1], no explicit surface symmetry energy was introduced. With a set of three parameters it was possible to get an overall fit of the absolute nuclear binding of 579 even—even nuclei [1] of known experimental masses with an average quadratic deviation of 1.58 MeV. The deviation for the charge radii of 313 even—even nuclei [1], which were not included in the fit, turns out to be of 0.027 fm. Both deviations compete with the best functionals present in the literature. This gives confidence to the use of this functional for the study of the NS crust, where very asymmetric nuclei appear. More details on this EDF, called BCPM (Barcelona—Catania—Paris—Madrid) hereafter, can be found in [1].
There are only few nuclear EOS that have been devised and used to cover the whole NS structure. A partly phenomenological approach, based on the compressible liquid-drop model, has been developed [9] by Lattimer and Swesty (LS). It can cover the whole range of density, including the crust and the pasta phase, and it gives a complete description of the NS matter structure. It also includes the temperature dependence of the EOS, but for our purpose we will employ it only in the zero temperature limit. There are different versions of this EOS, each one corresponding to a different incompressibility. This EOS is derived from a macroscopic functional and it is compatible with an accurate mass formula throughout the nuclear table. Therefore, it belongs to
the set of EOS within which we intend to make a comparison.
Another approach that has the same characteristics has been developed [10] by Shen et al. (SH). It is based on the relativistic mean-field formalism and the Thomas—Fermi (TF) scheme with trial densities. Also in this case we will employ this EOS in the zero-temperature limit. It does not include the pasta phase, but we think that this will not affect the relevance of the comparison. We will not describe in details the LS and SH EOS, since they are available directly on the web in functional and tabular forms, both for LS [11] and SH [12] cases.
Special mention must be given to the works where Skyrme forces are used to calcul
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