## 3. Quasiclassical case
Inserting this into Eq. (5), and integrating over orientations of neutrino momentum we get , while the electron energy is still given by Eq. (6). If the magnetic field is not too large the transverse electron and
proton momenta, and
, are sampled over a dense grid of
values, corresponding to integer indices where represents the
small- with and . After replacing the double sum by the double integral Eq. (10) can be considerably simplified. Note, that , where is a pitch-angle. Then the energy integral is taken by assuming, that the temperature scale is small and provides the sharpest variations of the integrand. If so, we can set , etc. in all the other functions. In principle, this assumption constraints the validity of the quasiclassical approach. We will come back to this point at the end of this section. Finally, we integrate over the azimuthal angle of the neutron momentum, and over its polar angle to eliminate the momentum conserving delta-function, and obtain: where is the field-free emissivity (1), and the factor describes the effect of the magnetic field. , and is now given by ; for , for . The asymptotic behaviour of (13) depends on the relation between
the argument of the function with a prefactor that is actually independent of where arg is an argument of
complex number This information is sufficient to check, if
reproduces the well-known
step-function in the
limit. If
, we are always in domain (III)
(since the
To investigate this possibility, one has to calculate the emissivity from Eq. (10). We have used two approaches, one of which is essentially quasiclassical and good for a rapid computation, while the other is of quantum nature - more precise and time-consuming. At the moment it is convenient to introduce two parameters,
where is the number of the
Landau levels populated by protons. The most interesting situation,
from practical point of view, occurs if where Ai is the Airy function, defined as in Abramowitz & Stegun (1970), and we have assumed , which implies absence of muons and hyperons in neutron-star matter. At negative arguments, which correspond to domain (II), the Airy function oscillates, while for positive arguments, in domain (III), it decreases exponentially approaching the asymptote Inserting the latter equation into Eq. (14), we find, that at
relatively large At very large On the other hand, under quasiclassical assumptions, we are always
interested in the case of . If so,
we can further simplify Eq. (14) at any At this integral is taken analytically and gives . By inserting Eq. (19), one easily verifies, that Eq. (21) reproduces also the asymptote (20) for . Finally, we have calculated the factor
, using both prescriptions (14) +
(18), and (21) at and
In the quasiclassical approach we have made two approximations:
firstly, we have replaced the sum by the integral at finite In this approach, if one transforms the integration over
to that over
in a straightforward manner, the
integrand becomes singular (the denominators of the form
appear). If
the quantity
diverges at integer
. For nonzero If The results of these calculations are presented in Fig. 1 for , and 400 by solid circles. It is seen that with increasing the quantum factor tends to the quasiclassical one . We have also found the fit expression that describes accurately (Fig. 1) the quantum calculations for and and reproduces the quasiclassical curve in the limit :
To summarize, we remind that the direct Urca reaction in the case is operative if the momentum excess . In the presence of the magnetic field, the condition becomes less stringent, and the reaction becomes quite efficient as long as , i.e., . If, for instance, G, and the density of matter is near the direct Urca threshold, one typically has and . Notice that the field-free direct Urca process can also be allowed
beyond the domain due to the
© European Southern Observatory (ESO) 1999 Online publication: December 22, 1998 |