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

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5. The case of superstrong magnetic fields

Using the formalism of Sect. 2 let us outline the main features of the direct Urca reaction produced by electrons and protons occupying the lowest Landau levels (with proton spins aligned with the magnetic field). Notice, however, that one needs superhigh magnetic field, [FORMULA] G, to force all electrons and protons into their ground Landau levels. Let [FORMULA] and [FORMULA] be the number densities of these particles. Contrary to the field-free Fermi momenta valid at not too high fields and used in Sects. 3 and 4, the limiting momenta along the superhigh magnetic field become field-dependent, [FORMULA], [FORMULA]. The distribution of neutrons can be characterized by two Fermi momenta for particles with spins along and against the magnetic field [FORMULA], [FORMULA]. Our starting point is Eq. (8), which reduces to

[EQUATION]

Let us insert it into Eq. (5), neglect [FORMULA] in the momentum-conserving delta function, and integrate over the neutrino orientations and over an azimuthal angle of the neutron momentum. Then we convert the integrals over [FORMULA] and [FORMULA] into the integrals over particle energies, take the standard energy integral, and perform the integration over the neutron pitch-angle. We obtain (the inverse reaction included)

[EQUATION]

where [FORMULA], and [FORMULA] corresponds to two different reaction channels in which the electron and proton momenta along the z-axis are either parallel ([FORMULA]) or antiparallel ([FORMULA]). The step functions indicate that the channels are open if [FORMULA]. The channel [FORMULA] is always open in npe dense matter with the superstrong magnetic field, while the channel [FORMULA] is open only if [FORMULA]. The latter condition is opposite to the familiar condition [FORMULA] in the field-free case. One has [FORMULA] and [FORMULA], but one cannot expect [FORMULA] to be essentially larger than the field-free emissivity [FORMULA] as long as [FORMULA] G. Note also, that Eq. (25) describes the contribution of particles populating the ground Landau levels to the emissivity not only in the limit of superstrong fields. For it to be valid at moderate fields one should substitute number densities of e and p on the ground levels for [FORMULA] and [FORMULA] in the definition of [FORMULA].

Similar result for the superstrong magnetic fields has been obtained recently by Leinson and Pérez (1997). Nevertheless, their expression differs from our in several respects. Most importantly, the authors got [FORMULA] instead of [FORMULA], which substantially overestimates the partial rate in the corresponding channel. Much more different result under the same assumptions was obtained by Bandyopadhyay et al. (1998). In our notations, the latter authors found the emissivity to be proportional to [FORMULA], and obtained the spurious enhancement of the neutrino emission and additional acceleration of the cooling in a superstrong magnetic field. Their condition which opens the reaction channel is opposite to the actual one.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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