3. Quasiclassical case
(a) General treatment and limit . First of all consider the most realistic case of not too high magnetic fields, in which electrons and protons populate many Landau levels. In this case the transverse wavelengths of electrons and protons are much smaller than their Larmor radii. Thus the situation may be referred to as quasiclassical, and corresponding techniques apply. If the main contribution comes from large n and , the difference between and can be neglected. Moreover, we can neglect the contributions of magnetic momenta of particles to their energies. Thus, we can pull all the functions of energy out of the sum over and , and evaluate the latter sum explicitly:
, 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 n and . Thus, the double sum in Eq. (10) tends to the double integral over and ,
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 F and its indices. In the small-b case one can distinguish three domains of these parameters: (I) ; (II) ; and (III) . In domains (I) and (III) the asymptotes in question decay exponentially when departs from the domain boundaries (which are the turning points of corresponding quasiclassical equation, e.g., Kaminker & Yakovlev, 1981). In both cases the exponents are inversely proportional to b, and both asymptotes tend to zero as , although nonuniformly in the vicinities of the turning points. In domain (II), the asymptote of the expression (13) oscillates according to
where arg is an argument of complex number z which lies in the range . For integer indices, this formula coincides with Eq. (30) in Kaminker & Yakovlev (1981), and it provides an accurate extension of to non-integer indices. One can easily show that a natural continuation of to this case is given by the analytical function , where is the Whittaker function, , , and . Eqs. (15) and (16) are derived using accurate small-b asymptotes of , which, in turn, could be derived from the integral representation of this function by the saddle-point method.
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 z-momentum is conserved), and . If, on the contrary, the integration in (14) covers all three domains, and the boundary of domain (II) corresponds to vanishing square root in Eq. (15). The contribution from domains (I) and (III) is again zero, while the integration over domain (II) yields exactly 1. To verify this, one can put the rapidly oscillating equal to in Eq. (15), and change the integration variables: , . In all cases, theta-function in Eq. (14) plays no role.
(b) Effect of the magnetic field in the forbidden domain (). In a finite magnetic field, the border between the open and closed direct Urca regimes is expected to be smeared out over some scale depending on the field strength. This can be important for neutron star cooling, since the direct Urca process can remain the dominant energy loss mechanism even if it is suppressed exponentially by several orders of magnitude.
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.
where is the number of the Landau levels populated by protons. The most interesting situation, from practical point of view, occurs if x is positive and not very large (say ), since whenever significantly exceeds , the reaction is suppressed too strongly. The first approach is based on Eq. (14). The main contribution to this integral comes from the vicinity of the line , corresponding to the least distance from to domain (II). Since this distance should not be large, we may use in (14) the small-b asymptotic form of (13) in the neighbourhood of the right turning point, i.e., for . It is given, for instance, by Kaminker & Yakovlev (1981) and reads
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
At very large x, however, this asymptote should not be taken too literally because Eq. (18) ceases to be valid far from the turning point.
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 x using (18), and noting that all the functions can be expanded around the line . Then we obtain
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 x from 0 to 20. In both these cases we have obtained identical results. This indicates that for such combinations of x and y, the y-dependence of is insignificant. These results are plotted in Fig. 1 by open circles. The short-dashed curve represents the asymptote (20).
In the quasiclassical approach we have made two approximations: firstly, we have replaced the sum by the integral at finite b, and, secondly, we substituted the asymptote in the form of the Airy function for the function F. To assess the quality of both assumptions we have used a quantum approach based directly on Eq. (10).
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 T this quantity remains convergent but oscillates as a function of B and/or density. These oscillations are quite familiar and appear in many studies (magnetization, electrical and thermal conductivities, etc., Landau and Lifshitz, 1986). They are associated with population of the Landau levels by electrons and protons due to variation of plasma parameters.
If T is larger than the energy spacing between the Landau levels the oscillations are washed out, and a smooth curve emerges. This regime requires very accurate integration over particle momenta in order to include thermal effect. We were able to perform it only for rather small and do not report these results here. In a more important case of lower temperatures the summation over Landau levels and energy integration are independent. The actual neutrino emissivity does oscillate but the quantity of practical significance is the emissivity averaged over the oscillations (a smooth curve again). We call this approach quantum since it involves the summation over Landau levels explicitly. In this way we have calculated the emissivity and smoothed the oscillations artificially by two different methods. We have checked that both methods yield nearly identical smoothed . Simple consideration (see the end of this section) shows that this (nonthermal quantum) approach is valid for , or, equivalently, , where is the proton gyrofrequency and is the proton chemical potential.
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 :
(c) Effect of the magnetic field in the permitted domain (). Since at , one may expect that the magnetic field has a non-trivial effect on the neutrino emissivity in the permitted domain. This appears to be true. Applying the quasiclassical approach in the form of Eq. (21) at negative x, we obtain for an oscillating curve, shown in Fig. 2 by the open circles. Using the more accurate quantum approach we get a series of the curves for , and 400, the curve with highest being again the closest to the quasiclassical result. These oscillations are of quasiclassical nature and have nothing in common with the quantum oscillations discussed above. From the practical point of view, they are not very important, as they hardly have any noticeable effect on the neutron star cooling. Note, that the results for and are accurately fitted by the expression (solid curve in Fig. 2)
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 thermal smearing of the Fermi surface. If and the reaction rate can be written as , where may be expected to be . Thus the thermal effect extends the reaction to the domain where . The smearing is clearly determined by the proton degeneracy parameter, , which is typically about 300 for K. The magnetic field effect is more important than the thermal effect if that can often be the case in the inner cores of neutron stars.
© European Southern Observatory (ESO) 1999
Online publication: December 22, 1998