A&A 387, 710-713 (2002)
DOI: 10.1051/0004-6361:20020395
J. Hu^{1} - R. X. Xu^{2,1}
1 - Center for Astrophysics and Physics Department,
Tsinghua University, Beijing 100084, PR China
2 -
School of Physics, Peking University, Beijing 100871, PR China
Received 7 January 2002 / Accepted 4 March 2002
Abstract
The surface electric characteristics of bare strange stars are
investigated with the inclusion of boundary effects. The thickness
of the electron layer where pairs can be spontaneously created is
calculated as a function of the bag parameters. We find that
previous estimates are representative for bag parameters within a
rather wide range, and therefore our results favor the thermal
radiation mechanism of bare strange stars advanced by V. V. Usov.
Key words: radiation mechanisms: thermal - elementary particles - stars: neutron
It is of great importance to identify strange stars; a new window to distinguish neutron stars and bare strange stars (BSSs) has been proposed recently based on their sharp differences in surface conditions (Xu et al. 2001; Usov 2001a). It is therefore essential to study the surface properties of BSSs in detail, e.g., the degree of the thermal luminosity of a hot BSS.
The surface electric field should be very strong (10^{17} V/cm) near the bare quark surface of a strange star because of the mass difference of the strange quark and the up (or down) quark (Alcock et al. 1986), which could play an important role in producing the thermal emission of BSSs by the Usov mechanism (Usov 1998; Usov 2001b), because strange quark matter is a poor radiator of thermal photons at frequencies lower than its plasma frequency (20 MeV) (Alcock et al. 1986). The basic idea of the Usov mechanism is that e pairs are created rapidly in a few empty quantum states with energy ( is the Fermi energy, m the electron mass) due to the very strong electric field in an electron layer (with a height of 500 fm above quark surface); the pairs subsequently annihilate into photons which are then thermalized in the electron layer^{} around a BSS. This radiative mechanism recently has been applied tentatively to soft ray repeaters (Usov 2001c; Usov 2001d). In addition, the strong electric field plays an essential role in forming a possible crust around a strange star, which has been investigated extensively by many authors (e.g., Martemyanov 1992; Kettner et al. 1995; Huang & Lu 1997; Phukon 2000; see Zdunik et al. 2001 for the recent developments with the inclusion of rotating and general-relativistic effects). Also it should be noted that this electric field has some important implications on pulsar radio emission mechanisms (Xu & Qiao 1998; Xu et al. 1999; Xu et al. 2001).
In fact the Usov mechanism of pair production depends on many parameters; it is therefore imperative to study the dependence of the process on these parameters. With some typical parameters chosen by Usov in his calculations, the resultant thickness of the electron layer with the electric field V/cm (the critical field necessary for pair production), , is 500 fm. However the proper determination of the thickness should be done with the dynamical theory of the strange quark matter. Because of the intractable nature of quantum chromodynamics, some phenomenological models, i.e., the MIT bag model (e.g., Jensen & Madsen 1996), the quark mass-density-dependent model (e.g., Lugones & Benvenuto 1995), and the quark potential model (e.g., Dey et al. 1998), have been applied to the descriptions of the strange quark matter. In the bag model, is a function of (the coupling constant for strong interactions), (the strange quark mass), and B (the bag constant). Also it should be noticed that the quark number densities, (i = u, d, s for up, down and strange quarks, respectively), are assumed to be uniform below the quark surface, and therefore the quark charge density is constant near the surface. The potential is usually chosen as 20 MeV for typical cases. However, the quark number densities should not be uniform near the surface if boundary effects are included, since charge neutrality is broken there.
In this paper we improve the calculation of the electric field in the vicinity of a BSS surface, using the popular bag model and the Thomas-Fermi model. We investigate the electric characteristics of BSSs for different values of . Initially we use typical parameters for MeV and MeV. Then the thickness, , of the electron layer where pairs can be created is computed as functions of B, , and . It is found that Usov's estimates are representative for bag-model parameters within a rather wide range. Boundary effect are also considered in this calculation.
The interesting electric properties of strange stars were first noted by Alcock et al. (1986), who presented numerical calculations of the electric potentials of strange stars with or without crusts. Analytical solutions of the electron number density and the electric field were also formulated, which could be helpful in dealing with some physical processes near the quark surfaces of strange stars (Xu & Qiao 1999). In this section, the bag model is applied to calculate the quark charge density; the Thomas-Fermi Model is employed to find dependence on , , and B, with the inclusion of boundary effects.
In our calculation, the quarks and electrons near the surface of
the strange star keep chemical equilibrium locally; the relation
between charge density and electric potential is described by the
classical Poisson equation. The thermodynamic potentials
as functions of chemical potentials
(i = u, d, s, e),
and
can be found in
the literature (Alcock et al. 1986). In the chemical equilibrium
of weak interaction,
The quark number densities drop to zero on the surface, but they are not uniform below the surface if the boundary effects are considered. The chemical potential can be determined by the condition that the pressure on the quark surface is zero, P|_{z=0}=0. The pressure is (Alcock et al. 1986).
The kinetic energy of electrons is equal to the electric
potential,
,
in the Thomas-Fermi model. The electron
number density reads (Alcock et al. 1986)
The boundary conditions for Eq. (6) are
0 | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 | |
B^{1/4}/MeV | 146 | 144 | 139 | 133 | 127 | 120 |
2.86 | 2.71 | 2.41 | 2.10 | 1.80 | 1.48 | |
3.95 | 3.68 | 3.14 | 2.60 | 2.07 | 1.54 | |
1.77 | 1.74 | 1.68 | 1.61 | 1.52 | 1.42 | |
13.9 | 11.9 | 7.77 | 4.03 | 1.21 | 0.30 | |
31.65 | 29.99 | 26.05 | 20.93 | 14.01 | 4.08 | |
30.69 | 29.11 | 25.34 | 20.44 | 13.77 | 4.06 |
By integrating Eq. (6) in
and
respectively with the boundary conditions, we get the
expression of the electric field
,
where
.
Obviously
is a function of
,
which can be obtained from the continuity of Eat z=0. Substituting
into the equations of E and
then integrating them, we get the expression of V. For ,
we find
(7) | |||
(8) | |||
(9) |
Figure 1: Electric potential as a function of z with , 0.3, 0.5, 0.7, 0.9. | |
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Figure 2: Electric field changes slowly above the quark surface. The coupling constant is chosen to be 0, 0.7, 0.9, 0.94. | |
Open with DEXTER |
Figure 3: Electric field varies sharply below the quark surface. , 0.5, 0.9. | |
Open with DEXTER |
Figure 4: A set of calculated thickness , as functions of bag parameters of B and , for different coupling constants , 0.3, 0.6, 0.9. Indicated numbers denote the correspondence values of . | |
Open with DEXTER |
Secondly, we investigate numerically the thickness, , of the electron layer where pairs can be created, as functions of the bag parameters , B, and . The results are shown in Fig. 4. It is found that the Usov mechanism can work for bag parameters within a rather wide range. The thickness could be large enough (i.e., fm) as long as (1) B is not too large and is not too small, or (2) B is not too small and is not too large. Increasing has an adverse influence on the Usov mechanism. For instance in Fig. 2, the field E cannot exceed the critical field V/cm (necessary for pair production) if . Nevertheless previous dynamical calculations show that is less possible for stable strange quark matter. In conclusion we could expect that the pair emission process proposed by Usov might appear in nature for acceptable bag parameters.
We have improved the calculation of the electric characteristics of bare strange stars with the inclusion of boundary effects (i.e., the effects of non-local neutrality near and below the quark surface). From our calculation, we find that the Usov mechanism can work for bag parameters within a rather wide range.
As shown in Table 1, both and V_{0}, as well as their very small difference, decrease as increases. Our results on the electric potential V(z) for z<0 is quite different to the previous calculation given by Alcock et al. (1986) where the boundary effects were not included (see Fig. 1). It is shown that the strong electric field resides only about 10 fm below the quark surface (see Fig. 3), rather than 10^{2} fm obtained by Alcock et al. (1986).
We can proof that E is almost an exponential function of zbelow the quark surface. We denote the right hand side of Eq. (6) as f(V). As , namely , we have . Approximating f(V)as f'(V_{0})(V-V_{0}), we can obtain and for z < 0.
Acknowledgements
This work is supported by National Nature Sciences Foundation of China (10173002) and the Special Funds for Major State Basic Research Projects of China (G2000077602). The authors sincerely thank Dr. Shuangnan Zhang for his comments and the improvement of the language.