A&A 366, 532-537 (2001)
DOI: 10.1051/0004-6361:20000090
Z. X. Ma^{1} - Z. G. Dai^{1,2} - T. Lu^{1,2}
1 - Department of Astronomy, Nanjing University, Nanjing 210093,
PR China
2 -
IHEP, Chinese Academy of Sciences, Beijing 100039, PR China
Received 3 July 2000 / Accepted 30 October 2000
Abstract
We study the equation of state (EOS) of the dense matter in the core of
neutron stars with hyperons included and the star structure based on the
Zimanyi & Moszkowski (ZM) model in the relativistic mean-field theory with
a set of recent satisfactory parameters. The relation between hyperon
abundances and baryon number densities is calculated and the distribution of
baryons in the core of a typical neutron star of 1.4
is presented.
Our results satisfy the requirements from observations of the mass of binary
radio pulsars, and the ratio,
,
of the crustal momentum
of inertia to the total one. The actual surface
thermal radiation detected seems to indicate that baryons in the core of
neutron stars should pair to form a superfluid phase, if hyperons appear in
the core of neutron stars.
Key words: stars: neutron - equations of state - elementary particles
The investigation of properties of hypernuclear matter plays an important role in the knowledge of neutron stars (Balberg et al. 1999). The existence of stable matter at hypernuclear densities is unique to neutron stars, and the macroscopic properties of neutron stars, including some observable quantities, may reveal some physical properties of hypernuclear matter. A notable characteristic of hypernuclear matter is the appearance of new hadronic degrees of freedom in addition to neutrons and protons. One degree of freedom is the formation of hyperons - strange baryons - which is the main subject of this paper. Other possibilities include meson condensation and a deconfined quark phase. The existence of hyperons in neutron stars was first proposed by Ambartsumyan & Saakyan (1960) and has been examined by many authors. For a review, see Glendenning (1996) and Prakash et al. (1997).
To determine the equation of state (EOS) for dense matter through the many body theory of interacting baryons, numerous approaches have been proposed. In recent years, these studies are mainly implemented in the framework of the field theory models, among which the model proposed first by Walecka (1974) is of growing interest. The model is one kind of relativistic mean field theory. In the standard model of Walecka (1974) the incompressibility of nuclear matter is overestimated. There are two ways to solve this problem. Boguta & Bodmer (1977, hereafter BB) introduced cubic and quartic terms for the scalar field into the Lagrangian and reproduced reasonable incompressibility values in comparison with empirical data. Along this direction, many authors (Kapusta & Olive 1990; Ellis et al. 1991; Sumiyoshi et al. 1992; Sumiyoshi & Toki 1994; Sumiyoshi et al. 1995; Cheng et al. 1996; Schaffner & Mishustin 1996) studied the EOSs for dense matter and the properties of neutron stars. Zimanyi & Moszkowski (1990, hereafter ZM) proposed another nonlinear model, in which the nonlinearity is contained in connection between the effective nucleon mass and the scalar field. The ZM model also gives reasonable incompressibility values. This model has been employed to study the properties of neutron stars (Cheng et al. 1996) and of supernova cores (Dai & Cheng 1998). These theories can be considered as phenomenological, since the coupling constants and meson masses of the effective meson-nucleon Lagrangian are taken as free parameters which are adjusted to fit the properties of nuclear matter and finite nuclei. The ZM model has no extra terms, and consequently deals with fewer parameters as compared with the BB model.
In this paper, we use the ZM model to calculate the relative fractions of the equilibrium composition of dense matter with hyperons as a function of baryon density and the EOSs. The properties of neutron stars are then discussed. This paper is arranged as follows. We describe the framework based on the ZM model in Sect. 2, and present our numerical results in Sect. 3. Section 4 contains conclusions and discussions.
To our knowledge, in the ZM model, only nucleons and leptons are included
in previous works. In order to describe the properties of dense matter, it
is necessary to take hyperons into account. We follow the notations of
Walecka (1974). The Lagrangian density of the system is given by
= | |||
(1) |
(2) |
(3) |
(4) |
Specices | Masses (MeV) | Charges | I_{3} |
p | 938 | 1 | |
n | 938 | 0 | |
1116 | 0 | 0 | |
1193 | 1 | 1 | |
1193 | 0 | 0 | |
1193 | -1 | -1 | |
1318 | 0 | ||
1318 | -1 |
(5) |
From Eq. (1), we can derive the Euler-Lagrange equations. The Dirac equation
for baryons is given by
(6) |
(7) |
(8) |
(9) |
= | |||
= | (10) |
(11) |
(12) |
= | |||
(13) |
P | = | ||
(14) |
(15) |
(16) |
(17) |
(18) |
(19) |
(20) |
(21) |
(22) |
(23) |
(24) |
(25) |
The number densities at which hyperons appear and the abundances of hyperons are related to their isospins, electric charges, effective masses and coupling constants. Since nuclear matter has an excess of positive charge and negative isospin, the appearance of hyperons with negative charge and positive isospin is favorable. The relative fractions of the equilibrium composition of dense matter as a function of baryon density are shown in Fig. 1 corresponding to UC, and in Fig. 2 corresponding to MC. Some qualitative properties of hyperon formation in dense matter can be deduced from Figs. 1 and 2.
Figure 1: Relative fractions of neutron star matter as a function of baryon density for UC | |
Open with DEXTER |
Figure 2: Relative fractions of neutron star matter as a function of baryon density for MC | |
Open with DEXTER |
(1) The density at which hyperons appear is 0.38 fm^{-3} for both MC and UC, that is well consistent with resent works (Balberg et al. 1999; Schaffner & Mishustin 1996; Glendenning 1985, 1996) which predict that hyperons will form at a density of 2 ( is the nuclear saturation density);
(2) Companying with hyperon accumulation, the immediate deleptonization, as a common characteristic of dense matter including hyperons (Balberg et al. 1999), is clearly exhibited in Figs. 1 and 2. The relative electron fraction reaches its maximum value of 2.4% for UC and 2.2% for MC both at baryon density 0.38 fm^{-3} (at which begins to appear), respectively, and then decreases with increasing of baryon density. The muon populations vanish when the electron fraction drops blow 0.5%;
(3) The first hyperon species that appears is followed by . However, the formation of is quickly moderated due to its negative isospin, then the abundances exceed the at baryon density 1.19 fm^{-3} for MC and 2.68 fm^{-3} for UC, respectively. For MC, the abundances exceed the proton and neutron abundances at total hadron number density 1.2 fm^{-3} and 1.72 fm^{-3}, respectively;
(4) It is worthwhile to point out that compared to other relativistic field models (Knorren 1995; Schaffner & Mishustin 1996; Glendenning 1996; Balberg 1997), the ZM model yields somewhat smaller lepton and hyperon fractions and high densities at which , , and form this is due to the low energy per nucleon predicted by the ZM model.
In the ZM model, Dai & Cheng (1998) studied the properties of nuclear matter with the parameters given in Tables 1 and 2, but their work doesn't include hyperons. The incompressibility, saturation density, and binding energy per nucleon obtained by Dai & Cheng are 225 MeV, 0.16 fm^{-1}, and -16.0 MeV, respectively. These values are very close to those derived from nuclear experiments. In Figs. 3 and 4, we plot the EOSs based on the ZM model with UC and MC. As a comparison, we also show the EOS based on the ZM model without any hyperon. The inclusion of hyperons softens the EOSs, since hyperons offer new degree of freedom for baryon matter and decrease the degenerate pressure, which has been noticed by some authors (e.g. Glendenning 1996).
Figure 3: The pressure vs. baryon number desity for UC, MC and N, N refers to nuclear matter without hyperon | |
Open with DEXTER |
Figure 4: The pressure vs. energy density for UC, MC and N, N refers to nuclear matter without hyperon | |
Open with DEXTER |
After the EOSs are available, we can calculate the hydrostatic structure of
a neutron star by solving the Tolman-Oppenheimer-Volkoff equation:
(26) |
(27) |
The structure of neutron stars is displayed in Figs. 5-8. Figures 5 and 6 show the hyperon abundances as a function of radius for UC and MC, respectively. Figure 7 shows the total mass as a function of radius. For UC, the maximum mass of a neutron star is 1.55 with the central density of 3.47 10^{15} gcm^{-3} and the radius R=9.44 km; while for MC, the maximum mass is 1.4 with the central density of 3.56 10^{15} g cm^{-3} and the radius of 9.39 km. Here it is clear that compared to other relativistic field model with hyperons the ZM model yields low neutron star maximum mass, which indicates relatively soft equations of state, especially for MC. The neutron star maximum masses yielded by other relativistic field model mainly line in 1.5 (Balberg 1999; Glendenning 1996, 1985; Schaffner & Mishutin 1995). Figure 8 shows the ratio of the crustal moment of inertia to the total one as a function of stellar mass. The moment of inertia of the neutron stars is calculated in the slow rotation approximation, following Arnett & Bowers (1977). For a typical neutron star of 1.4 , the ratio is 0.012 and 0.021 for MC and UC, respectively.
Figure 5: Relative populations vs. radius in a neutron star of 1.4 for UC | |
Open with DEXTER |
Figure 6: Relative populations vs. radius in a neutron star of 1.4 for MC | |
Open with DEXTER |
Figure 7: Gravitational mass (in units of ) versus radius for UC and MC | |
Open with DEXTER |
Figure 8: The ratio of crustal momentum of inertia to the total one as a function of stellar mass for UC and MC | |
Open with DEXTER |
By using the ZM model with two sets of coupling constants for interactions between baryons, we have calculated the EOS for dense matter and the properties of neutron stars. Now we discuss astrophysical implications of our results.
First, early observations of binary radio pulsars gave neutron star masses of (Joss & Rappaport 1984; Taylor & Weinberg 1989) and recent estimations by Thorsett & Chakrabarty (1999) found from double neutron star binaries and from neutron star and white dwarf binaries. These observational results are satisfied by our EOSs.
Second, pulsar glitch phenomena have been suggested as a probe of neutron
star properties (Link et al. 1992). The postglitch behavior of the pulsar
indicates a change in the spindown rate,
,
ranging from a fraction of one percent (Crab) to a few percent (2.4%
of Vela, Alpar 1993). According to the "two component model" (Shapiro &
Teukolsky 1983),
(28) |
Third, the implications of hyperon formation for neutron star cooling have been discussed in several studies (Prakash et al. 1992; Prakash 1994; Haensel & Gnedin 1994; Lattimer et al. 1994; Schaab et al. 1996). According to current point of view hyperons will provide direct Urca processes. For more details about the threshold conditions at which hyperon - related direct Urca processes occur see Balberg et al. (1999). If neutron stars cool through direct Urca processes, their temperature should drop too rapidly to be detectable within less than 100 yr after their birth. But in recent years, there is strong evidence that actual surface thermal radiation has been detected for some pulsars, i.e., PSR 0833-45 (Ögelman et al. 1995), PSR 0656+14 (Finley et al. 1992), PSR 0630+178 (Halpern & Holt 1992), PSR 1005-52 (Ögelman & Finley 1993), which indicates that direct Urca processes are suppressed in the core through most of the thermal evolution stages. Fortunately, it was pointed out that the direct Urca processes may be suppressed if the participating baryons pair into a superfluid state, for more details see Schaab et al. (1998). So we are apt to the conclusion that baryons populating in the core of neutron stars should exist in a superfluid state.
Acknowledgements
This work was partly supported by the National Natural Science Foundation of China, grants 19773007, 19973003 and 19825109, and the National 973 Project on Fundamental Researches.