Astron. Astrophys. 355, 1009-1014 (2000)

1. Introduction

The true ground state of the hadrons may be strange matter not (Bodmer 1971;Witten 1984; Farhi and Jaffe 1984). If it is true, then there would be a possibility of the existence of the strange star (Alcock et al. 1986; Alcock and Olinto 1988). If strange star exists, it opens up a new family of distinguishable compact astrophysical objects like neutron stars and white dwarfs. There is also a possibility that all neutron stars might have been converted into strange stars because the universe is very likely to be contaminated by strangelets and some of the strangelets might be in contact with the neutron stars (Witten 1984; Glendenning 1990; Madsen and Olesen 1991). It is very likely that the pressure at the center of a neutron star is so large that in that region quark matter will be formed via phase transition from hadronic form of matter (Baym and Chin 1976; Freedman and McLerran 1978). Once a strangelet or quark matter component is present in a neutron star, it will quickly develop the equilibrium strangeness content via weak interactions, such as . The energy will be lowered as strange quarks are created one by one until equilibrium is reached. The conversion of nuclear matter to quark matter may alter the moment of inertia and the epoch over which this conversion takes place may be visible as the spin-down of the pulsar (Glendenning et al. 1997). A new class of strange white dwarfs may also exist in nature which has a central density about times denser than in average normal white dwarf (Glendenning et al. 1995)

Inspite of these theoretical understandings of possible existence of strange matter class stellar objects, the observational searches of these objects have not yielded much positive results. It is due to the fact that it is very difficult to make a distinction between a strange star and a neutron star. The maximum mass and radius of both classes of stars are similar to each other. But the mass-radius relation of a strange star is different from that of a neutron star. It is calculated that the stable neutron star mass generally decreases with radius to a minimum value, while the strange star mass increases in the same range and there is no minimum mass. Such distinguishability feature is taken into account in the calculations made by Li et al. (1995), who determined mass-radius relation semi-empirically from the observations of spin variation and cyclotron spectral line of X -ray source, Her X-1. Comparing the results with theoretical models of neutron star and of a strange star respectively, they came to the conclusions that Her X-1 might be a candidate of strange stars. The another distinguishability feature between a neutron star and strange star is the rotation period. Recently, Madsen (1998) have shown that a young strange star can be distinguished from that of a neutron star by studying the r-mode instability which slows rapidly rotating, hot neutron star via gravitational radiation. Strange stars which are not subjected to this instability might have rotation period below 5 - 10 ms (Madsen 1998). To date no pulsars with the range of periods have been detected, therefore future observations will determine whether such young strange pulsars are existing or not.

Previous studies (Alcock et al. 1986; Huang and Lu 1997; Martemyanov 1994) have shown that a strange star may contain nuclear crust. It is still not clear that a strange star with nuclear crust can explain all phenomenon associated with pulsar glitching such as healing times and recurrence rates. A strange star with a very thin nuclear crust possibly rules out the existence of strange pulsar in nature. The evolution of the strange star soon after the formation depends on the crustal thickness. A strange star with thick nuclear crust may have different cooling time and surface temperature from that of a strange star with thin crust. Thus the study of the nuclear crust in the strange star is important for future observations of strange pulsar.

Alcock et al. (1986) first examined the stability of nuclear crust of a cold strange star and calculated the crust mass. Their studies have shown that a strange star of mass contains crust mass . Kettner et al. (1995) extended the model to include the effect of finite temperature on the crust thickness and found that the temperature lead to a considerable reduction of the electrostatic potential at the surface of the star which holds the crust. Glendenning and Weber (1992) included rotation of the star in the framework of general theory of relativity and calculated the mass, thickness and moment of inertia of the nuclear crust. They found that the general relativistic rotation of the strange star can increase the moment of inertia sufficiently to explain the observed magnitude of pulsar glitches. The electrostatic potential of electrons inside and in the close vicinity of the quark surface is of decisive importance for the existence of the nuclear crust on the quark surface of strange star. This is due to the fact that the strong positive Coulomb barrier prevents atomic nuclei bound in the nuclear crust from comimg into direct contact with strange matter core, where atomic matter would be converted into strange matter. On the other hand neutrons can easily penetrate the Coulomb barrier and are readily absorbed. Therefore, any material which contains free neutrons will not be stable in contact with strange matter. The outer layer of a neutron star is a solid lattice of neutron-rich nuclei neutralized by electrons. This layer can be stable in contact with strange star if gap of sufficient width exists between the crust and the quark matter. The electric field which opens the gap must be capable of supporting some normal material (i.e. ions and electrons) crust.

Neutron stars are strongly magnetized. Observational data indicate that most isolated pulsars have magnetic fields Gauss while most pulsars in binaries have lower magnetic field strength, going down to Gauss. From the observed cyclotron lines in the massive X-ray binary, the strength of the surface magnetic field of neutron star is found to be Gauss. The structure of the interior field is different from that of the surface. This is due to the fact the interior field is expected to be carried by fluxoids in the superconducting core of the neutron star (Bhattacharya and Srinivasan 1995). These fluxoids have cores of size cm consisting of normal proton fluid, and the magnetic field strength at the core of these fluxoids can reach Gauss. It has been argued that (Duncan and Thompson 1992) dynamo action may lead to an amplification of the magnetic field in a collapsing star like type II supernovae. The neutron star which is formed as a remnant of the explosion might have magnetic field of strength upto Gauss.

In a strong magnetic field, the electron motion perpendicular to the field lines is quantized to give discrete Landau Orbitals (Landau and Lifshitz 1965) and the electron behaves as a one-dimensional gas rather than a three-dimensional gas. The energy of a charged particle change significantly in the quantum limit if the magnetic field Gauss, where and are the mass and charge respectively. For electrons, Gauss, while for u and d quarks Gauss and for s quark it is Gauss. Therefore, the quantum-mechanical effect of the magnetic field on a neutron star cannot be neglected.

In this paper we attempt a model to study the effect of the magnetic field on electrostatic potential of electrons which hold the nuclear crust of a non-rotating strange star. Our studies are based on a perturbation expansion of pressure, baryon density and energy density from zero magnetic field values at zero temperature by a Taylor series.

In the next section, we describe the effect of the surface magnetic field on electron pressure and electrostatic potential. In the last section, we present our conclusions.

© European Southern Observatory (ESO) 2000

Online publication: March 21, 2000
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