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

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3. Quasiclassical treatment

In this section, we develop quasiclassical description of the neutrino synchrotron emission from a degenerate, ultrarelativistic electron gas in a strong magnetic field [FORMULA] -1014  G. Therefore we assume that [FORMULA] and [FORMULA], where [FORMULA] is the degeneracy temperature. We mainly analyze the case in which the electrons populate many Landau levels. This is so if the magnetic field is nonquantizing or weakly quantizing (e.g., Kaminker & Yakovlev, 1994), i.e., if [FORMULA]. The latter condition is realized at sufficiently high densities [FORMULA], where [FORMULA]  g cm [FORMULA]  g cm-3, and [FORMULA]. At these densities, [FORMULA] is nearly the same as without magnetic field, [FORMULA], where [FORMULA] is the field-free Fermi momentum. The formulated conditions are typical for the neutron star crusts.

In the case of many populated Landau levels, we can replace the summation over n by the integration over [FORMULA] in Eq. (2). The remaining summation over [FORMULA] can be conveniently replaced by the summation over discrete cyclotron harmonics [FORMULA] =1, 2, 3, ... Furthermore, an initial-state electron can be described by its quasiclassical momentum p and pitch-angle [FORMULA] ([FORMULA], [FORMULA]). Since we consider strongly degenerate electrons we can set [FORMULA] and [FORMULA] in all smooth functions under the integrals. Then the integration over p is done analytically. In this way we transform the rigorous quantum formalism of Sect. 2 to the quasiclassical approximation used by KLY (see their Eqs. (3), (4) and (8)).

According to KLY the quasiclassical neutrino synchrotron emission is different in three temperature domains [FORMULA], [FORMULA], and [FORMULA] separated by two typical temperatures [FORMULA] and [FORMULA]:

[EQUATION]

Here, [FORMULA] is the relativistic parameter ([FORMULA], in our case), and [FORMULA] 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 [FORMULA], where [FORMULA].

Figure 1 demonstrates the main parameter domains [FORMULA], [FORMULA] and [FORMULA] for the ground-state (cold catalyzed) matter of NS crusts. Thermal effects on the nuclear composition are neglected which is justified as long as [FORMULA] 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 ([FORMULA] g cm-3), is described using the results by Haensel & Pichon (1994) based on new laboratory measurements of masses of neutron-rich nuclei. At higher densities, in the inner NS crust, we use the results of Negele & Vautherin (1973) derived on the basis of a modified Hartree-Fock method. Small discontinuities of the curves in Fig. 1 are due to the jumps of the nuclear composition. Notice, that the properties of the neutrino synchrotron emission vary rather smoothly in the transition regions from domain [FORMULA] to [FORMULA] and from [FORMULA] to [FORMULA].

[FIGURE] Fig. 1. Density-temperature domains of different neutrino synchrotron regimes. [FORMULA] is temperature (5) below which (domain [FORMULA]) the emission goes via the ground cyclotron harmonics [FORMULA] (for [FORMULA], 1013 and 1014  G, [FORMULA] shown in parenthesis). [FORMULA] is temperature (5) below which (in domain [FORMULA]) the Pauli principle restricts the number of cyclotron harmonics (see text); it is shown only for [FORMULA]  G because [FORMULA] becomes higher than the electron degeneracy temperature [FORMULA] for [FORMULA] and [FORMULA]  G. Domain [FORMULA] ([FORMULA]) is realized only for [FORMULA]  G; [FORMULA] is independent of B, for displayed parameters.

The high-temperature domain [FORMULA] is defined as [FORMULA] ; it is realized for not too high densities and magnetic fields where [FORMULA]. In Fig. 1, this domain exists only for [FORMULA]  G at [FORMULA]  g cm-3. In domain [FORMULA], the degenerate electrons emit neutrinos through many cyclotron harmonics; typical harmonics is [FORMULA]. Corresponding neutrino energies [FORMULA] are not restricted by the Pauli principle. The quasiclassical approach of KLY yields

[EQUATION]

where [FORMULA]. 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 moderate-temperature domain [FORMULA] is defined as [FORMULA] and [FORMULA]. It covers wide temperature and density ranges (Fig. 1) most important for applications. In this domain, neutrinos are again emitted through many cyclotron harmonics [FORMULA], but their spectrum is restricted by the Pauli principle, and typical neutrino energies are [FORMULA]. As shown by KLY, in this case the neutrino emissivity is remarkably independent of the electron number density:

[EQUATION]

where [FORMULA] is the value of the Riemann zeta function.

The third, low-temperature domain [FORMULA] corresponds to temperatures [FORMULA] at which the main contribution into the neutrino synchrotron emission comes from a few lower cyclotron harmonics s =1, 2, ...    If [FORMULA], even the first harmonics [FORMULA] appears to be exponentially suppressed as discussed by KLY. A more detailed analysis will be given below.

The emissivity [FORMULA] in the combined domain [FORMULA], including a smooth transition from [FORMULA] to [FORMULA] at [FORMULA], was calculated accurately in KLY. The results (Eqs. (13), (15) and (18) in KLY), valid at [FORMULA], can be conveniently rewritten as

[EQUATION]

where the analytic fits to the functions [FORMULA] read

[EQUATION]

In this case [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA].

Now let us study the neutrino emissivity in the combined domain [FORMULA]. Using the quasiclassical expressions of KLY in the ultrarelativistic limit ([FORMULA]) we obtain

[EQUATION]

where

[EQUATION]

with

[EQUATION]

Here, [FORMULA] is a Bessel function of argument [FORMULA], and [FORMULA]. The neutrino-pair energy can be expressed as [FORMULA]. The integration should be done over the kinematically allowed domain: [FORMULA], [FORMULA], [FORMULA].

One can easily see that [FORMULA] depends on the only parameter z. The domain [FORMULA] corresponds to [FORMULA] while the domain [FORMULA] corresponds to [FORMULA]. Equation (12), in which we set [FORMULA], does not reproduce the high-temperature domain [FORMULA]. However, we have already described the transition from [FORMULA] to [FORMULA] by Eqs. (8- 10).

At [FORMULA], the neutrino emissivity, determined by [FORMULA] in Eq. (12), comes from many cyclotron harmonics (see KLY, for details). A Bessel function [FORMULA] and its derivative can be replaced by McDonald functions. The main contribution to the integrals (12) comes from a narrow vicinity [FORMULA], and [FORMULA] of the saddle point [FORMULA] and [FORMULA], in which the neutrino-pair energy is nearly constant, [FORMULA]. Adopting these approximations and replacing the sum over s by the integral we reproduce evidently the result of KLY, [FORMULA]. However if we take into account small corrections in the expression of a Bessel function through McDonald functions (Sokolov & Ternov 1974), and weak variation of [FORMULA] near the saddle-point, we obtain a more accurate asymptote [FORMULA]. The derivation is outlined in Appendix B.

In the opposite limit of [FORMULA], one can keep the contribution from the first harmonics [FORMULA], and replace [FORMULA],    [FORMULA] in Eq. (12) as described in KLY. At any kinematically allowed [FORMULA] and [FORMULA], the neutrino-pair emission is suppressed by a small factor [FORMULA]. As shown in KLY, the most efficient neutrino synchrotron radiation occurs from a small vicinity of the allowed region, where [FORMULA] has minimum and the neutrino energy [FORMULA] is most strongly reduced by the quantum recoil effect. This region corresponds to the backward electron scattering. The integration over [FORMULA] is then done analytically, and we are left with a two-fold integration over [FORMULA] and [FORMULA]. In the limit of very high z, it gives

[EQUATION]

The convergence of this asymptote is very slow, and we present the second-order correction term [FORMULA] which improves the convergence considerably. For instance, at [FORMULA] 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 [FORMULA] is 4 times smaller than the correct emissivity presented here (due to a simple omission in evaluating [FORMULA] made by KLY). Thus Eqs. (20) and (21) in KLY are inaccurate at [FORMULA].

In addition to analyzing the asymptotes, we have calculated [FORMULA] numerically from Eq. (12) in the quasiclassical approximation at intermediate z. The results are fitted by the analytic expression

[EQUATION]

where [FORMULA], [FORMULA]. The fit reproduces both the low-z and the high-z asymptotes. The rms fit error at [FORMULA] is about 1.6%, and the maximum error is 5% at [FORMULA].

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 [FORMULA], [FORMULA], [FORMULA] ([FORMULA], [FORMULA]), where the electrons are degenerate, relativistic and populate many Landau levels:

[EQUATION]

Here, [FORMULA] is given by Eq. (7), while [FORMULA] and [FORMULA] are defined by Eqs. (9), (10) and (15).

The neutrino synchrotron emission in domains [FORMULA] (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 B for the same plasma parameters ([FORMULA], [FORMULA]  K) as in Fig. 4 by Vidaurre et al. (1995). For the parameters chosen (due to very high [FORMULA]), domain [FORMULA] is not realized, while the transition from domain [FORMULA] to [FORMULA] occurs at fairly high B (for instance, [FORMULA] corresponds to [FORMULA]  G). Our quasiclassical calculation, analytic fit (16) and quantum calculation are so close that yield the same (solid) line. Numerical calculation of Vidaurre et al. (1995) is shown by dashes, and their analytical approximation (claimed to be accurate at intermediate z) by dots. In our notations, the latter approximation reads [FORMULA]. We see that such an approximation reproduces neither low-z nor high-z asymptote, and is, in fact, inaccurate at intermediate z. It also disagrees with numerical curve of Vidaurre et al. (1995) and with our curve. The numerical results by Vidaurre et al. (1995) are considerably different from ours except at [FORMULA]  G. At lower and higher B the disagreement becomes substantial. In domain [FORMULA], Vidaurre et al. (1995) present another analytic expression that differs from Eq. (7) by a numerical factor [FORMULA]. The difference comes from two inaccuracies made by Vidaurre et al. (1995). First, while deriving the emissivity, they use the asymptote of the McDonald function at large argument (their Eq. (19)) instead of exact expression for the McDonald function (as was done by KLY). One can easily verify that this inaccuracy yields an extra factor [FORMULA]. Secondly, Vidaurre et al. (1995) calculated inaccurately an integral over [FORMULA] (their Eq. (22)) as if the integrand were independent of [FORMULA]. This yields the second extra factor 3/2. Thus we can conclude that the results by Vidaurre et al. (1995) are rather inaccurate in a wide parameter range. We have performed extensive comparison of the results obtained with our quantum code and with the quasiclassical approach. We have found very good agreement (within 3-5 %) in all the cases in which the quasiclassical approach can be used (see above). This statement is illustrated in Figs. 3 and 4. Figure 3 shows the quantum and quasiclassical synchrotron emissivities from a plasma with [FORMULA]  g cm-3 and [FORMULA]  K as a function of B. Figure 4displays the ratio of the quantum to quasiclassical emissivities in more detail. The quasiclassical approach is valid as long as [FORMULA] (as long as electrons populate many Landau levels) which corresponds to [FORMULA]  G for our particular parameters. For these magnetic fields, the quantum and quasiclassical results are seen to coincide quite well. Low-amplitude oscillations of the curves in Fig. 4 reflect oscillations of the synchrotron emissivity as calculated with the quantum code. The oscillations are produced by depopulation of higher Landau levels with increasing B. They represent quantum effect associated with square-root singularities of the Landau states. As seen from Fig. 4 the oscillations are smeared out with increasing T by thermal broadening of the square-root singularities (cf. Yakovlev & Kaminker 1994). In a higher field [FORMULA]  G, the electrons populate the ground Landau level alone and the electron chemical potential is reduced by the magnetic field. Very high B, not shown in Fig. 3, remove the electron degeneracy. Note that at high temperatures, one should take into account the synchrotron neutrino emission by positrons (Kaminker & Yakovlev 1994). All these conditions are described by the quantum code while the quasiclassical approach is no longer valid. In Appendix C, we obtain simple asymptotic expressions for [FORMULA] in the case in which the electrons populate the ground Landau level alone. Corresponding curves are shown by dashed curves in Fig. 3, and they are seen to reproduce the exact quantum curves quite accurately.

[FIGURE] Fig. 2. Neutrino synchrotron emissivity vs B from a plasma with [FORMULA]  g cm-3 and [FORMULA]  K. Solid curve shows our quasiclassical result, dash-curve - calculation of Vidaurre et al. (1995), dots - analytic Eq. (26) of these authors.
[FIGURE] Fig. 3. Neutrino synchrotron emissivity versus B from a plasma with [FORMULA]  g cm-3 at [FORMULA], [FORMULA] and [FORMULA]  K. Solid curves are calculated with the quantum code, long-dash-curves are obtained using the quasiclassical calculation valid at [FORMULA]  G, dash-curves show the quantum asymptotic expression (Appendix C) for the case in which the ground Landau level is populated alone, [FORMULA]  G.
[FIGURE] Fig. 4. Ratios of the quantum to quasiclassical neutrino synchrotron emissivities presented in Fig. 3.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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