For performing exact quantum calculation of the neutrino synchrotron emissivity from Eq. (2) we replace the integrations over and by the integrations over initial and final electron energies and , respectively. In this way we can accurately integrate within the intervals and , where the Fermi-Dirac distributions are rapidly varying. It is also convenient to set and replace the integration over by the integration over . Then
Here, (s enumerates cyclotron harmonics); and 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: and . The sign of and the domain of integration over are specified by the minimum longitudinal momentum of the final-state electron for given n, and s:
If , one has and , where
If , then there are two integration domains. The first one is , with . The second domain is , with .
, and is the z -coordinate of the saddle point. The term in (B1), which is linear in , does not contribute to (12) due to integration over .
In the low-z limit (i.e., for ), the Bessel functions and in (13) can be expressed through McDonald functions and of the argument , where ; is the coordinate of the saddle point transverse to the magnetic field. These expressions can be written as (Sokolov & Ternov 1974)
Let us substitute (B1) and (B4) into (12). Then
Consider the neutrino synchrotron radiation by relativistic degenerate electrons in a very strong magnetic field (, ) in which the bulk of electrons populate the ground Landau level. The electron Fermi momentum is then given by , and the chemical potential is . 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
Since the first Landau level is almost empty, we can set , where K)). The neutrino emission is greatly suppressed by the combination of Fermi-Dirac distributions . These distributions determine a very narrow integration domain which contributes to .
Equation (C2) can be further simplified in the two limiting cases. The first case is , or , where and . The main contribution into comes from narrow vicinities of two equivalent saddle points , . 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 , emitting a neutrino-pair with the energy . In this case, the energy of the initial-state electron is . One has in the vicinities of the saddle points. Expanding , , and in these vicinities in powers of and , we obtain from Eq. (C2)
where and .
The second case corresponds to or . Now the most efficient electron transitions are those in which the initial-state electron is near the bottom of the first Landau level (, ). Accordingly, and . The energy of the final-state electron varies mostly from the maximum energy (associated with the minimum allowable neutrino-pair energy at ) to the minimum energy allowed by the Pauli principle. One can put in all smooth functions under the integral (C2), and replace integration over by integration over or over u, with in the integration domain. The integration is then taken analytically (for ) yielding
and . In the limit of we have and . 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, , we obtain and .
in Eqs. (C3) and Eq. (C4). These replacements do not affect significantly outside the vicinity of but produce physically reasonable interpolation within this vicinity. By matching the modified asymptotes at and implying again we get . In this way we obtain a complete set of equations to describe at .
© European Southern Observatory (ESO) 1997
Online publication: March 24, 1998