1 Introduction

As ordinary nucleons (neutrons and protons), which are composed of quarks, are squeezed tightly enough at high temperature and pressure, they may turn into a soup of deconfined free quarks. This type of phase transition might be expected in the center of a neutron star which consists mainly of neutron matter. In the conventional picture of neutron stars with quark matter, the neutron (hadron) and quark phases are separated by a sharp boundary with density discontinuity; such stars are called "hybrid'' stars. But, as first considered by Glendenning (1992) for "complex'' systems, it is possible that bulk quark and nuclear matter could coexist over macroscopical distances in neutron stars; such stars are called "mixed'' stars (see, e.g., Heiselberg et al. 2000 for a recent review about hybrid and mixed stars). A further radical view is that the whole neutron star should be phase-converted to be a strange star, which consists of nearly equal numbers of u, d, and s quarks and associated electrons for charge neutrality, if strange quark matter (SQM) is the true ground state of strong interacting matter (Bodmer 1971; Witten 1984; see also, e.g., Madsen 1999, for a recent review of the physics and astrophysics of strange matter and strange stars). Strange stars provide a sharp contrast to neutron stars (e.g. the mass-radius relations and the surface conditions), although both neutron and strange stars may have similar radii ($\sim$10 km) and masses ($\sim$ $1.4~ M_{\odot}$).

Pulsars can be modeled as neutron stars or strange stars. No strong observational evidence known hitherto favors either of the models. It is thus of great importance and interest in current astrophysics to distinguish neutron stars and strange stars observationally. Recently, at least three hopeful ways have been proposed for identifying a strange star: 1, hot strange stars (or neutron stars containing significant quantities of strange matter) may rotate more rapidly since the higher bulk viscosity (Wang & Lu 1984; Madsen 1992) of strange matter can effectively damp away the r-mode instability (Madsen 1998); 2, the approximate mass-radius (M-R) relations of strange stars ( $M\propto R^3$) are in striking contrast to those of neutron stars ( $M\propto R^{-3}$). Comparisons of observation-determined relations with theoretical relations may thus determine whether an object is a neutron star or a strange star (Li et al. 1999); 3, It is possible to distinguish "bare'' polar cap strange stars from neutron stars via pulsar magnetospheric and polar radiation because of the striking differences between the polar surface conditions of the two types of stars (e.g., clear drifting pattern from "antipulsars'', Xu et al. 1999; Xu et al. 2001). In this paper, we shall study possible differences in the processes of magnetic field generation by dynamo action in strange stars and in neutron stars which could eventually lead to observational distinctions between the two types of stars.

Magnetic fields play a key role in pulsar life. Unfortunately, there is still no consensus on the physical origin of strong fields. Naively, it is supposed that pulsars' fields are the result of the conservation of magnetic flux during supernova collapses. This idea faces at least two problems (Thompson & Duncan 1993, here after TD93): 1, dynamo action can occur in the convective episodes of both the iron core and the newborn star; 2, very strong fields (>10¹² G) of pulsars are probably not "fossil fields'' since only a few percent of white dwarfs have fields in excess of 10⁶ G. Based on the Newtonian scalar virial theorem, one can estimate the limiting interior magnetic field $B_{\rm max}\sim 10^{18} M_1 R_6^{-2}$ G of a star with mass $M=M_1~M_\odot$ and radius R=R₆ 10⁶ cm (Lai & Shapiro 1991). Both the estimate based on flux conservation and that based on the virial argument, do not depend on the detailed fluid properties of the stellar interiors. Thompson & Duncan (TD93, Duncan & Thompson 1992) have extensively considered turbulent dynamo amplification in protoneutron stars (PNSs) and in the progenitor stars, and found that multipolar structure fields as strong as 10¹⁶ G ("magnetar'') can be generated by PNS convection. This suggestion seems to have been confirmed by recent observations (e.g., Kouveliotou et al. 1998). Furthermore, there arises a possible way to distinguish strange stars from neutron stars since the dynamo-originated fields depend on the detailed fluid properties of the stellar interiors. One interesting question thus is: Can strange stars create magnetic fields as strong as 10¹⁶ G? (i.e., can strange stars act as magnetars?). If strange stars can, is there any difference between the processes of field generation in PNSs and that in "proto-'' strange stars (PSSs)? Theoretically, there is little work known hitherto on the generation of the magnetic fields of strange stars. It is supposed that spontaneous magnetization could result in the generation of a compact quark star (Tatsumi 2000). Alternatively, we suggest that dynamo-amplification of the magnetic field could also play an important role in PSSs with high temperatures. In this work, we are trying to find whether there are any differences between the dynamo-originated fields of neutron stars and that of strange stars, in order to contribute to the debate on the existence of strange stars. In our discussion, we presume that the strong magnetic fields of strange stars originate in the strange cores, rather than in the crusts with mass $\sim$ $10^{-5}~ M_\odot$, since strange stars produced during supernova explosions cannot have such crusts (Xu et al. 2001).

One difficult issue in the study of dynamo-originated fields of strange stars is to determine whether color superconductivity (CSC) occurs in PSS since the dynamo mechanism may not work effectively in a superconducting plasma. Recent calculations, based on a model where quarks interact via a point-like four-fermion interaction, showed that the energy gap $\Delta$ of zero-temperature strange matter could be 10-100 MeV for plausible values of the coupling (Alford et al. 1999); thus the critical temperature of forming possible quark Cooper pairs $T_{\rm c}\sim \Delta/2 \sim 5{-}50$ MeV. However, $T_{\rm c}$ could be altered if a more realistic quark-quark interaction is used and/or if the the trapping of neutrinos in PSSs is included. It is also hard to sufficiently determine the temperature $T({\rm PSS})$ of PSSs. We therefore simply assume in this paper that $T({\rm PSS})>T_{\rm c}$ in the first few seconds (the time-scale for dynamo action of PSS) after PSS formation.