5 Conclusion and discussion

Pulsars could be neutron stars and/or strange stars. The origin of neutron star magnetic fields has been discussed in literature, while few papers are concerned with the origin of fields in strange stars. In this paper, we have investigated the dynamo action in PSSs, and have suggested that strange stars can have magnetic fields of dynamo origin similar to those in neutron stars. Our main conclusions are as follows.

1. A significant fraction of gravitational energy has to be converted to differential rotation energy if the angular momentum of each mass element is conserved in the core collapse. Assuming an initial rotating core to be approximated by the model of Bruenn (1985), we have calculated the rotation velocity, the velocity derivative, and the differential rotation energy of a nascent strange star.

2. It is found that PSSs may have a convective layer of thickness $\sim$2 km in the outer part. The large-scale convection has a velocity $\sim$10⁸ cm s^-1, while local turbulent eddies have a scale of $\sim$1 m and a velocity of $\sim$10⁵ cm s^-1.

3. The energy density of differential rotation could be larger than the turbulent energy density if the pulsar initial period ms. Assuming that most of the differential rotation energy is converted to magnetic energy by dynamo action, we obtain the dipole field strength as a function of P and the pulsar period evolution due to magnetic dipole radiation.

4. The fields ( 10¹²-10¹⁸ G) amplified by fast dynamo action are concentrated in filaments with initial radii $\sim$0.01 mm - $\sim$1 cm and with growth times 10^-5 s to 10 s.

5. Strange stars and neutron stars are expected to have different accretion-induced field decay processes, which could be used to distinguish them in the future.

6. Convection with the large scale L is less likely to exist in PSSs than in PNSs.

It is currently believed that anomalous X-ray pulsars (AXPs) and soft gamma repeaters (SGRs) are magnetism-powered in the "magnetar'' model (Duncan & Thompson 1992; Thompson & Duncan 1995). The magnetic reconnection near the surface causes the emission in AXP and SGR. Duncan & Thompson (1992, TD93) suggested that the key parameter that determines whether a PNS becomes an ordinary pulsar or a magnetar is the Rossby number. If magnetars are strange stars, we propose that, besides this key parameter, the initial temperature and the density of trapped neutrinos can also affect the formation of a magnetar since dynamo action could not be very effective when CSC appears. Strange stars with very strong magnetic field may act as "magnetars'' since they could have similar differentiated structure as neutron stars (Benvenuto et al. 1990). However the neutron magnetar model faces a crisis indicated by Pérez Martinez et al. (2000) in that neutron stars may undergo a transverse collapse if their fields exceed a critical value. But for strange magnetars, a relatively small magnetic momentum and large chemical potential of free quarks may favor the formation of very high fields, although a further investigation is needed to find the critical field strength beyond which a strange star cannot be sustained against transverse collapse.

This paper may have two implications for the studies of the r-mode instability (Anderson 1998; Madsen 1998, 2000; Lindblom et al. 1998). (1) As discussed in Sect. 2.1, PSSs and PNSs should rotate strongly differentially (particularly for ms). But the numerical studies that have appeared in the literature are for stars with uniform rotation. Certainly, it is reasonable to assume that the result for uniform rotation is qualitatively representative also for the case of differential rotation. (2) The dynamo-originated strong magnetic fields inside strange stars or neutron stars should be included in investigations of the r-mode instabilities in pulsars. Although a very short period P will be favourable for the occurrence of r-mode instability, a smaller P would also be favourable for the generation of a stronger magnetic field by dynamo action, thus exerting a stabilizing influence. As the temperature decreases, the bulk viscosity also decreases and the r-mode instability may occur if there are no other strong dissipative effects. Previously, Rezzolla et al. (2000) have shown that the interaction of r-mode oscillations with the magnetic field could be important and that the oscillations will be inhibited if the field is of the order $\sim$ $10^{16} (\Omega/\Omega_{\rm B})$ G, where $\Omega$ is the angular velocity of the star and $\Omega_{\rm B}$ is the value at which mass shedding occurs. Since the fields of PSSs (or PNSs) depend on the nature of turbulence and on the rotation frequency, further study is needed to see whether dynamo-generated fields can affect significantly the r-mode oscillation.

Acknowledgements

This research was supported by National Nature Sciences Foundation of China (19803001), and by the Special Funds for Major State Basic Research Projects of China. We thank an anonymous referee for helpful suggestions.