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*Astron. Astrophys. 328, 409-418 (1997)*
## 2. Quantum formalism
We will mainly use the units in which , where
is the Boltzmann constant. We will return to
ordinary physical units whenever necessary. The general expression for
the neutrino synchrotron energy loss rate (emissivity, ergs
s^{-1} cm^{-3}) from an electron gas of any degeneracy
and relativity, immersed in a quantizing magnetic field, was obtained
by Kaminker et al. (1992a):
Here, ergs cm^{3} is the
Fermi weak-coupling constant and is the
dimensionless magnetic field ( G).
Furthermore, *n* and are, respectively, the
Landau level number and the momentum along the magnetic field for an
electron before a neutrino-pair emission; the energy of this electron
is . The primed quantities
and refer to an electron after the emission;
its energy is ; is
Fermi-Dirac distribution of the initial-state electrons,
is the same for the final-state electrons;
is the electron chemical potential and *T*
is the temperature. The energy and momentum carried away by a
neutrino-pair are denoted as and **q**,
respectively. The *z* component of **q** is
, while the component of **q** across the
magnetic field is denoted by . The summation and
integration in (2) is over all allowed electron transitions. The
integration has to be done over the kinematically allowed domain
. The differential transition rate *A*
(summed over initial and final electron spin states) is given by
Eq. (17) of Kaminker et al. (1992a). Taking into account that
some terms are odd functions of and vanish after
the integration, we can rewrite *A* in a simple form,
Here, and are the
transverse momenta of the initial-state and final-state electrons,
respectively;
, , and
is an associated Laguerre polynomial (see,
e.g., Kaminker & Yakovlev 1981). Furthermore, in Eq. (3) we
introduce and , where
and are the vector and
axial vector weak interaction constants, respectively, and summation
is over all neutrino flavors. For the emission of the electron
neutrinos (charged + neutral currents), one has
and , while for the emission of the muonic or
tauonic neutrinos (neutral currents only), and
; is the Weinberg angle.
Adopting we obtain and
. A comparison of the neutrino synchrotron
emission by electrons with the familiar electromagnetic synchrotron
emission has been done by KLY. Although electromagnetic radiation is
much more intense it does not emerge from deep neutron star layers
(neutron star interior is opaque to photons) while neutrino emission
escapes freely from stellar interior, producing an efficient internal
cooling.
We have composed a computer code which calculates
from Eqs. (2) and (3) for arbitrary plasma
parameters , *T* and *B*, where
is the mass density and
is the number of electrons per baryon. Technical details are presented
in Appendix A, the results are illustrated in Sect. 4, and
the case of very strong magnetic field, in which the bulk of electrons
occupy the ground Landau level, is considered in Appendix C.
© European Southern Observatory (ESO) 1997
Online publication: March 24, 1998
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