4 Any differences of dynamo-originated fields?

As discussed in the previous section, magnetic dipole fields stronger than 10¹¹ G could be generated by dynamos in PSSs. However, if pulsars are born as strange stars, an acute question is how to explain the fields of millisecond pulsars with $\sim$10⁸ G since fields don't decay in strange stars with CSC. Millisecond pulsars are supposed to be spun-up ("recycled'') as the result of accretion from companion stars in their histories. They could be strange stars with $\sim$ $10^{-5}~ M_\odot$ crusts, although a lot of ordinary pulsars might be strange stars with bare polar caps and much thinner crusts (Xu et al. 2001). We suggest that accretion may reduce the dipole field strength from $\sim$10¹² G to $\sim$10⁸ G of strange stars in four possible ways: (1) the accreted matter may screen or bury the original fields (Bisnovati-Kogan & Komberg 1974; Zhang et al. 1994; Zhang 2000); (2) field decays in the accretion-heated crust (Konar & Bhattacharya 1997); (3) the large scale fields (e.g., dipole structure) could be changed significantly because the plasma accreted onto the polar caps would squeeze the frozen magnetic fields toward the equator (Cheng & Zhang 1998), and the dipole field thus appears to decay; (4) magnetic fields frozen in the bottom crust may dissipate and be annihilated during the combustion process of the bottom matter into SQM when the crust becomes too heavy to be supported by the Coulomb barrier.

A notable difference between strange stars and neutron stars is that the magnetic field of a strange core is stable and does not decay due to CSC, while the field of a neutron star should be expelled from the interior to the crust where the Ohmic dissipation occurs. In present models for neutron star magnetism, the fields permeate either the whole star or only the crust (see, e.g., a short review by Mitra et al. 1999). It is found by observational and statistical analyses that a pulsar's field can only decrease substantially in the accretion-phase but does not decay significantly during the pulsar lifetime (Hartman et al. 1997). New calculations of the timescale for Ohmic dissipation in the crust have shown that the field can persist for more than 10¹⁰ years (Sang & Chanmugam 1987). It thus may be difficult to distinguish the field evolutions of strange stars and neutron stars in the radio pulsar-phase. Nevertheless, the field decay modes in the accretion-phase could be quite different for neutron stars and strange stars. In fact the items (1)-(3) in the above proposals for field decay are relevant to both neutron stars and strange stars, but item (4) can only apply to strange stars. Furthermore, items (1)-(3) could result in different physical processes for strange star and neutron star because they have very different structures. For example, as matter accretes, the radius of a strange star increases, while the radius of a neutron star decreases. Also, the material in neutron star crust moves continuously, and the movement just pushes and squeezes the original field. However for strange stars, a SQM phase-transition occurs when the accreted material moves across the Coulomb barrier with thickness of $\sim$200 fm. Actually, this phase-transition process has been included in the explanation of some burst phenomena, such as bursting X-ray pulsar GRO J1744-28 (Cheng et al. 1998), but the field evolution in accreting strange star has not yet been discussed extensively. We propose here that strange stars and neutron stars may be distinguished by their field decays during the accretion-phases. Future investigations of this issue will be of interest.

As addressed in the previous sections, many fluid parameters for the large-scale convection scenario in PSSs and PNSs are similar. Both kind of stars have convective layers with thickness $\sim$10⁵ cm and flow velocities $\sim$10⁸ cm s^-1. The scale length of turbulent eddies of both PSS and PNS is $\sim$2 m since neutrino viscosity damps flow only on scale greater than $l\sim l^{\rm N}\sim 2$ m. The neutrino viscosities and the magnetic Reynolds numbers in PSSs and PNSs are also not quite different: $\nu \sim 10^{10}$ cm²/s, $\nu^{\rm N} \sim 10^8$ cm²/s; ${\cal R}_{\rm m}\sim 10^{16}$, ${\cal R}_{\rm m}^{\rm N}\sim 10^{17}$. Therefore, the general configurations and strength of dynamo-generated magnetic fields in PSSs and PNSs may be similar. However, the fluid properties calculated in this paper for SQM are rather uncertain since our knowledge of SQM fluid is less than that of nuclear matter. Detailed studies of the fluid properties in PSSs, such as the neutrino fraction and viscosity, are necessary and may help us to see differences in the dynamo-originated fields. In addition, the timescale available for dynamo action in PSSs may be significantly smaller than that in PNSs since the energy gap for proton superconductivity is order of 1 MeV (rather than 10-100 MeV in SQM). This property may also affect the structure of the fields.