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Astron. Astrophys. 342, 192-200 (1999)

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1. Introduction

The presence or absence of the direct Urca process ([FORMULA], [FORMULA]) in the core of a neutron star is the most important issue of the stellar cooling. If operative, it dominates the cooling at the neutrino stage (age [FORMULA] - [FORMULA] yr) being several orders of magnitude more efficient than any other neutrino emission process (e.g., Pethick, 1992). However, the direct Urca can occur under stringent conditions: one requires quite high fraction of protons [[FORMULA], where [FORMULA] is a Fermi momentum, and [FORMULA] is a number density of particles of species [FORMULA]] in order to conserve momentum in the reaction. Nevertheless, some equations of state (EOSs) allow for that (Lattimer et al., 1991). On the other hand, most realistic EOS of dense matter (Wiringa et al., 1988) predicts too small fraction of protons, which decreases with growing density; consequently, the direct Urca is forbidden in the entire neutron star core.

In this paper we study the possibility for the direct Urca to be open in the presence of a magnetic field B, if the proton fraction is too low to open the process at [FORMULA]. The beta-decay and related reactions in strong magnetic fields have been studied since late 1960's (e.g., Canuto & Chiu 1971, Dorofeev et al. 1985, Lai & Shapiro 1991, and references therein). However, these results have been obtained under various simplified assumptions (constant matrix elements, non-degenerate nucleons, etc.) and do not give the emissivity of the direct Urca reaction in the neutron star cores. Several works on the subject have appeared most recently. Leinson & Pérez (1997) considered the case of superstrong fields ([FORMULA] G), in which electrons and protons occupy only the lowest Landau levels. They found that such fields relaxed the requirement of high proton fraction, and the direct Urca was always permitted leading to a rapid cooling of a neutron star. The case of superstrong fields was examined also by Bandyopadhyay et al. (1998). These authors found, by contrast, that the condition "[FORMULA]" (quotation implies that one should be careful with definition of the Fermi momentum in superstrong fields) still determined the direct Urca threshold, and that the fields enhanced the neutrino emissivity by 1 - 2 orders of magnitude in the permitted regime compared to the standard value (Lattimer et al., 1991)


Here [FORMULA] K, [FORMULA] is the electron density, [FORMULA] fm-3, [FORMULA] and [FORMULA] are nucleon effective masses in dense matter, [FORMULA] and [FORMULA] are their bare masses.

The present paper is organized as follows.

In Sect. 2 we obtain a general expression for the neutrino emissivity [FORMULA] of the direct Urca process.

In Sect. 3 we concentrate on a realistic case, in which the magnetic field is not extremely high (although still high: [FORMULA]), and charged particles populate many Landau levels. We show that the field keeps the direct Urca open slightly outside the region where [FORMULA] (the standard, [FORMULA], definition of [FORMULA] applies).

In Sect. 4 we illustrate this result by a series of cooling simulations of magnetized neutron stars with no superfluid in the inner cores and neutron superfluid in the outer cores.

Finally, in Sect. 5 we treat briefly the case of super strong magnetic fields. We show that the results by Leinson & Pérez (1997) are basically correct, although not very accurate in details, while those reported by Bandyopadhyay et al. (1998) are inaccurate.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998