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Astron. Astrophys. 328, 409-418 (1997)

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4. Discussion and results

Figures 5-8 display density dependence of the neutrino synchrotron emissivity [FORMULA] calculated from Eq. (16) for the magnetic fields [FORMULA], 1013, and 1014  G at four temperatures [FORMULA], [FORMULA], [FORMULA], and [FORMULA]  K, respectively. We adopt the ground-state model of matter in the NS crust (see Sect. 3). Various neutrino-emission regimes can be understood by comparison with Fig. 1.

The high-density (horizontal) parts of the synchrotron curves correspond to domain [FORMULA] (Eq. (7)), where [FORMULA] is density independent. The low-density bends are associated with transitions either into domain [FORMULA] (where [FORMULA] according to Eq. (6)) or into domain [FORMULA] (where [FORMULA] decreases exponentially due to cyclotron harmonics suppression, Eq. (14)). Domain [FORMULA] is realized only for [FORMULA]  G and [FORMULA]  K in Figs. 5 and 6. The low-density bends of [FORMULA] in domain [FORMULA] are much steeper than those in domain [FORMULA]. These bends are more pronounced at highest [FORMULA]  G, at which domain [FORMULA] extends to higher T and [FORMULA] (Figs. 7 and 8).

For comparison, we also plot the emissivities produced by other neutrino generation mechanisms: the electron-positron pair annihilation into neutrino pairs, electron-nucleus bremsstrahlung, plasmon decay and photon decay. The pair annihilation in a magnetized plasma has been considered by Kaminker et al. (1992a, b), and Kaminker & Yakovlev (1994). For the parameters of study, the emissivity appears to be weakly dependent on the magnetic field. At [FORMULA] G it is very close to the zero-field emissivity (Itoh et al. 1989, 1996). As seen from Figs. 5 and 6, the pair-annihilation emissivity differs slightly from the zero-field one only in a not too hot plasma ([FORMULA]  K) at [FORMULA]  G. The neutrino bremsstrahlung curves are plotted neglecting the influence of the magnetic field. The effect of the field on the bremsstrahlung has not been studied so far but it is expected to be weak, for the parameters in Figs. 5-8 . We use the results of Haensel et al. (1996) to describe the neutrino pair due to Coulomb scattering of electrons by atomic nuclei in the liquid phase of matter. In the solid phase, similar process is known to consist of two parts: the phonon and static lattice contributions. We use the results by Yakovlev & Kaminker (1996) to evaluate the phonon contribution. As for the static lattice contribution, we employ the most recent theory by Pethick & Thorsson (1996) and perform numerical calculation from Eqs. (28) and (29) of their paper (adopting the Debye-Waller factor and the nuclear form-factor which were used by Yakovlev & Kaminker 1996). Numerous jumps of the bremsstrahlung curves in Figs. 5 - 8 are associated either with jump-like changes of nuclear composition of cold-catalyzed matter or with solid-liquid phase transitions (see Haensel et al. 1996 for details). The neutrino emissivities from other processes are determined by the electron and positron number densities which are nearly continuous function of the density. Therefore, all other curves are smooth. The neutrino generation due to plasmon and photon decays in a magnetic field has not been considered in the literature, and we present the field-free results of Itoh et al. (1989, 1996), for illustration.

In the case of zero magnetic field, the bremsstrahlung process dominates completely in most dense layers of the NS crust at not too high temperatures [FORMULA]  K. Plasmon decay, photon decay, and pair annihilation are significant at high temperatures, [FORMULA]  K, but their emissivities become negligible very soon as temperature decreases.

The synchrotron emissivity is, to some extent, similar to the bremsstrahlung, for it persists over the wide temperature and density ranges. In the presence of the strong magnetic field [FORMULA]  G, the synchrotron emission is seen to be important and even dominant for any T in Figs. 5 -8 . In a hot plasma (Fig. 5), the synchrotron emission is significant at comparatively low densities, [FORMULA] - [FORMULA]  g cm-3. With decreasing T the neutrino synchrotron emission becomes more important at higher densities. At [FORMULA]  K, only the bremsstrahlung and synchrotron emissions actually survive (Fig. 8); if [FORMULA]  G, the synchrotron emission dominates over the bremsstrahlung one in a wide density range, [FORMULA]  g cm-3.

[FIGURE] Fig. 5. Density dependence of neutrino emissivities from ground-state matter of the NS crust due to various mechanisms at [FORMULA]  K. Curves `syn' `(12)', `(13)', and `(14)' refer to the synchrotron mechanism at [FORMULA] 1012, 1013 and 1014  G, respectively. Curve `pairs' corresponds to neutrino emission due to annihilation of electron-positron pairs; it is almost independent of B at given T. Other curves are for [FORMULA]: `brems' - the total electron-nucleus bremsstrahlung; `plasma' - plasmon decay; `photo' - photoneutrino process (see text).
[FIGURE] Fig. 6. Same as in Fig. 5 but at [FORMULA]  K. Pair annihilation depends noticeably on B.

[FIGURE] Fig. 7. Same as in Figs. 5 and 6 but at [FORMULA]  K. Pair annihilation and photoneutrino processes become negligible.
[FIGURE] Fig. 8. Same as in Figs. 5 - 7 but at [FORMULA]  K. Plasmon decay becomes negligible.
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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998