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

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2. Quantum formalism

We will mainly use the units in which [FORMULA], where [FORMULA] 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):

[EQUATION]

Here, [FORMULA]  ergs cm3 is the Fermi weak-coupling constant and [FORMULA] is the dimensionless magnetic field ([FORMULA]  G). Furthermore, n and [FORMULA] 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 [FORMULA]. The primed quantities [FORMULA] and [FORMULA] refer to an electron after the emission; its energy is [FORMULA] ; [FORMULA] is Fermi-Dirac distribution of the initial-state electrons, [FORMULA] is the same for the final-state electrons; [FORMULA] is the electron chemical potential and T is the temperature. The energy and momentum carried away by a neutrino-pair are denoted as [FORMULA] and q, respectively. The z component of q is [FORMULA], while the component of q across the magnetic field is denoted by [FORMULA]. The summation and integration in (2) is over all allowed electron transitions. The integration has to be done over the kinematically allowed domain [FORMULA]. 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 [FORMULA] and vanish after the integration, we can rewrite A in a simple form,

[EQUATION]

Here, [FORMULA] and [FORMULA] are the transverse momenta of the initial-state and final-state electrons, respectively;

[EQUATION]

[FORMULA], [FORMULA], and [FORMULA] is an associated Laguerre polynomial (see, e.g., Kaminker & Yakovlev 1981). Furthermore, in Eq. (3) we introduce [FORMULA] and [FORMULA], where [FORMULA] and [FORMULA] 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 [FORMULA] and [FORMULA], while for the emission of the muonic or tauonic neutrinos (neutral currents only), [FORMULA] and [FORMULA] ; [FORMULA] is the Weinberg angle. Adopting [FORMULA] we obtain [FORMULA] and [FORMULA]. 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 [FORMULA] from Eqs. (2) and (3) for arbitrary plasma parameters [FORMULA], T and B, where [FORMULA] is the mass density and [FORMULA] 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.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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