## 3. Quasiclassical treatmentIn this section, we develop quasiclassical description of the
neutrino synchrotron emission from a degenerate, ultrarelativistic
electron gas in a strong magnetic field
-10 In the case of many populated Landau levels, we can replace the
summation over According to KLY the quasiclassical neutrino synchrotron emission is different in three temperature domains , , and separated by two typical temperatures and : Here, is the relativistic parameter (, in our case), and is the electron gyrofrequency at the Fermi surface. Notice, that while applying our results to the neutron star crusts, one can use a simplified expression , where . Figure 1 demonstrates the main parameter domains
, and
for the ground-state (cold catalyzed) matter of
NS crusts. Thermal effects on the nuclear composition are neglected
which is justified as long as K (e.g., Haensel
et al. 1996). It is assumed that nuclei of one species are available
at any fixed density (pressure). Then the increase of density
(pressure) is accompanied by jumps of nuclear composition (e.g.,
Haensel et al. 1996). The ground-state matter in the outer NS crust,
at densities below the neutron drip density ( g
cm
The where . Here and in what follows, numerical factors in the practical expressions are slightly different from those presented by KLY because now we use more accurate value of the Fermi constant (see Eq. (2)). The where is the value of the Riemann zeta function. The third, The emissivity in the combined domain , including a smooth transition from to at , was calculated accurately in KLY. The results (Eqs. (13), (15) and (18) in KLY), valid at , can be conveniently rewritten as where the analytic fits to the functions read In this case , , , , , , , , , , , , . Now let us study the neutrino emissivity in the combined domain . Using the quasiclassical expressions of KLY in the ultrarelativistic limit () we obtain Here, is a Bessel function of argument , and . The neutrino-pair energy can be expressed as . The integration should be done over the kinematically allowed domain: , , . One can easily see that depends on the only
parameter At , the neutrino emissivity, determined by
in Eq. (12), comes from many cyclotron
harmonics (see KLY, for details). A Bessel function
and its derivative can be replaced by McDonald
functions. The main contribution to the integrals (12) comes from a
narrow vicinity , and
of the saddle point and
, in which the neutrino-pair energy is nearly
constant, . Adopting these approximations and
replacing the sum over In the opposite limit of , one can keep the
contribution from the first harmonics , and
replace ,
in Eq. (12) as described in KLY. At any
kinematically allowed and
, the neutrino-pair emission is suppressed by a
small factor . As shown in KLY, the most
efficient neutrino synchrotron radiation occurs from a small vicinity
of the allowed region, where has minimum and
the neutrino energy is most strongly reduced
by the quantum recoil effect. This region corresponds to the backward
electron scattering. The integration over is
then done analytically, and we are left with a two-fold integration
over and . In the limit
of very high The convergence of this asymptote is very slow, and we present the second-order correction term which improves the convergence considerably. For instance, at the two-term asymptote gives an error of about 6.5%, while the one-term asymptote gives an error of about 36%. Notice, that the emissivity given by Eq. (20) in KLY in the limit of is 4 times smaller than the correct emissivity presented here (due to a simple omission in evaluating made by KLY). Thus Eqs. (20) and (21) in KLY are inaccurate at . In addition to analyzing the asymptotes, we have calculated
numerically from Eq. (12) in the
quasiclassical approximation at intermediate where , . The fit
reproduces both the low- Now, we can easily combine Eqs. (8) and (11) and obtain a general fit expression for the neutrino synchrotron emissivity which is valid everywhere in domains , , (, ), where the electrons are degenerate, relativistic and populate many Landau levels: Here, is given by Eq. (7), while and are defined by Eqs. (9), (10) and (15). The neutrino synchrotron emission in domains
(in our notations), has been reconsidered
recently by Vidaurre et al. (1995). Their results are compared with
ours in Fig. 2 which shows the neutrino synchrotron emissivity as
a function of
© European Southern Observatory (ESO) 1997 Online publication: March 24, 1998 |