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

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Appendix A

For performing exact quantum calculation of the neutrino synchrotron emissivity from Eq. (2) we replace the integrations over [FORMULA] and [FORMULA] by the integrations over initial and final electron energies [FORMULA] and [FORMULA], respectively. In this way we can accurately integrate within the intervals [FORMULA] and [FORMULA], where the Fermi-Dirac distributions are rapidly varying. It is also convenient to set [FORMULA] and replace the integration over [FORMULA] by the integration over [FORMULA]. Then

[EQUATION]

Here, [FORMULA] (s enumerates cyclotron harmonics); [FORMULA] and [FORMULA] correspond to the excitation (de-excitation) thresholds of the initial and final electron states, respectively. The neutrino pair energy and longitudinal momenta are determined from conservation laws: [FORMULA] and [FORMULA]. The sign of [FORMULA] and the domain of integration over [FORMULA] are specified by the minimum longitudinal momentum of the final-state electron for given n, [FORMULA] and s:

[EQUATION]

If [FORMULA], one has [FORMULA] and [FORMULA], where

[EQUATION]

If [FORMULA], then there are two integration domains. The first one is [FORMULA], with [FORMULA]. The second domain is [FORMULA], with [FORMULA].

Appendix B

Consider the [FORMULA] asymptote of the function [FORMULA] given by Eq. (12). The integrand of (12) contains the factor [FORMULA]. Since the neutrino-pair energy [FORMULA] varies near the saddle point, we have

[EQUATION]

where [FORMULA] is the value of [FORMULA] in the saddle point, [FORMULA],

[EQUATION]

[FORMULA], and [FORMULA] is the z -coordinate of the saddle point. The term in (B1), which is linear in [FORMULA], does not contribute to (12) due to integration over [FORMULA].

In the low-z limit (i.e., for [FORMULA]), the Bessel functions [FORMULA] and [FORMULA] in (13) can be expressed through McDonald functions [FORMULA] and [FORMULA] of the argument [FORMULA], where [FORMULA] ; [FORMULA] is the coordinate of the saddle point transverse to the magnetic field. These expressions can be written as (Sokolov & Ternov 1974)

[EQUATION]

KLY took into account the main terms of these asymptotes. Here, we include small corrections (proportional to the factor [FORMULA] in square brackets). This yields

[EQUATION]

Let us substitute (B1) and (B4) into (12). Then

[EQUATION]

In this case, [FORMULA] comes from variation of [FORMULA], with

[EQUATION]

Furthermore, [FORMULA] comes from the corrections to the McDonald functions,

[EQUATION]

Appendix C

Consider the neutrino synchrotron radiation by relativistic degenerate electrons in a very strong magnetic field ([FORMULA], [FORMULA]) in which the bulk of electrons populate the ground Landau level. The electron Fermi momentum is then given by [FORMULA], and the chemical potential is [FORMULA]. The number of electrons on the excited Landau levels is exponentially small, and the major contribution to the neutrino emission comes from the electron transitions from the first excited to the ground Landau level. Equation (3) reduces to

[EQUATION]

where [FORMULA]. Inserting (C1) into (2) we can integrate over [FORMULA]:

[EQUATION]

Since the first Landau level is almost empty, we can set [FORMULA], where [FORMULA] K)). The neutrino emission is greatly suppressed by the combination of Fermi-Dirac distributions [FORMULA]. These distributions determine a very narrow integration domain which contributes to [FORMULA].

Equation (C2) can be further simplified in the two limiting cases. The first case is [FORMULA], or [FORMULA], where [FORMULA] and [FORMULA]. The main contribution into [FORMULA] comes from narrow vicinities of two equivalent saddle points [FORMULA], [FORMULA]. Each point corresponds to the most efficient electron transition in which an initial-state electron descends to the ground Landau level just with the Fermi energy [FORMULA], emitting a neutrino-pair with the energy [FORMULA]. In this case, the energy of the initial-state electron is [FORMULA]. One has [FORMULA] in the vicinities of the saddle points. Expanding [FORMULA], [FORMULA], and [FORMULA] in these vicinities in powers of [FORMULA] and [FORMULA], we obtain from Eq. (C2)

[EQUATION]

where [FORMULA] and [FORMULA].

The second case corresponds to [FORMULA] or [FORMULA]. Now the most efficient electron transitions are those in which the initial-state electron is near the bottom of the first Landau level ([FORMULA], [FORMULA]). Accordingly, [FORMULA] and [FORMULA]. The energy of the final-state electron [FORMULA] varies mostly from the maximum energy [FORMULA] (associated with the minimum allowable neutrino-pair energy [FORMULA] at [FORMULA]) to the minimum energy [FORMULA] allowed by the Pauli principle. One can put [FORMULA] in all smooth functions under the integral (C2), and replace integration over [FORMULA] by integration over [FORMULA] or over u, with [FORMULA] in the integration domain. The integration is then taken analytically (for [FORMULA]) yielding

[EQUATION]

where

[EQUATION]

and [FORMULA]. In the limit of [FORMULA] we have [FORMULA] and [FORMULA]. Then Eq. (C4) reproduces the asymptotic expression for the synchrotron neutrino emissivity of electrons from a non-degenerate electron-positron plasma (Eq. (35) of the paper by Kaminker & Yakovlev 1993). In the opposite limit, [FORMULA], we obtain [FORMULA] and [FORMULA].

Both asymptotes, (C3) and (C4), become invalid in a narrow vicinity [FORMULA] of the point [FORMULA] or [FORMULA]. We propose to extend the asymptotes, somewhat arbitrarily, to the very point [FORMULA] by replacing

[EQUATION]

in Eqs. (C3) and Eq. (C4). These replacements do not affect significantly [FORMULA] outside the vicinity of [FORMULA] but produce physically reasonable interpolation within this vicinity. By matching the modified asymptotes at [FORMULA] and implying again [FORMULA] we get [FORMULA]. In this way we obtain a complete set of equations to describe [FORMULA] at [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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