J. Geophys. Res., 91, 3152 - 3166, 1986.
(Received August 8, 1985; revised November 8, 1985; accepted December 2, 1985.)
Copyright 1986 by the American Geophysical Union.
Paper number 5A8825
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In the introduction to this paper we discussed the importance of relativistic effects on wave dispersion in regions of low plasma density, such as the auroral plasma cavity [Calvert, 1981]. We have emphasized the effect of the hot electrons on the wave dispersion, using both a ring distribution and a shell distribution. The latter showed that the unstable mode for low wave vectors was trapped even when temperature is included, provided that the thermal velocity was lower than the momentum for which the energetic electrons have a peak in their phase space density, i.e., j > 1. In this section we shall address the applicability of the analysis presented in preceding sections to the auroral zone electron distributions.
There are three basic assumptions to be questioned. First, is it reasonable to assume that auroral electron distributions fall into the parameter ranges considered here? In other words, can we justify the assumption that 0.1 n / n 10, and that 0.1 / 10, as used in Figures 3, 4, 5, 6, and 7? The range in / is adequate, because of the presence of the auroral cavity [Calvert, 1981]. Second, what effect does the asymmetry of the observed~distributions have on our analysis? Lastly, how important is the assumption that k = 0?
To determine if the range in plasma parameters is reasonable, we show four theoretical phase space density plots in Figure 8. Strangeway  has considered the data published by Croley et al.  and shown that j 4 may be a reasonable value for a DGH fit to the energetic auroral electrons. The upper two plots in Figure 8 show the phase space density of two j = 4 distributions which are just unstable, one with low / , the other with high / . The corresponding solutions of the shell dispersion relation are shown in Figure 5 , and also given as headers to the phase space density plots. The lower two plots correspond to two weakly unstable modes for different j values shown in Figure 4, one with j = 1, the other with j = 5.
Fig. 8. Representative distribution functions for which relativistic modifications are important. The plasma parameters together with the corresponding solutions of the shell dispersion relation for k = 0 are given as headers to each phase space density plot. The dashed lines give the phase density of each electron species, with the sum being given by the solid line. When obtaining solutions of the dispersion relation, we have assumed that / = 0.01, and as can be seen from the figure / j = 200 (p/mc = 0.1).
Figure 8 shows that there can be considerable variability in the shape of the distribution function, but the mode with frequency below the gyrofrequency is unstable. As discussed before, this mode will be trapped for all save the j = 1 distribution function. One significant difference between the j = 1 distribution function and the other three is the presence of a minimum at p = 0 for the former while the latter have a maximum at p = 0. All the j > 1 distributions have double maxima.
In passing, we note that the distinction between j = 1 and j > 1 may explain the difference in the wave dispersion. It can be shown from (3) that when both ambient and hot electrons are present there will always be a maximum in the phase space density at p = 0 for j > 1. If there is a second maximum at higher momentum, then we might expect to be able to characterize the electrons as two electron species with their own gyrotrequencies. On the other hand, if j = 1, then there can be a minimum at p = 0, and the electrons may be considered to be a single electron species. While there need not necessarily be a minimum at p = 0 for j = 1, that there can be a minimum suggests that j = 1 is a special case, and the wave dispersion can be different than for j > 1.
Another important point concerning the shell distribution is well demonstrated by the phase space density plot shown at upper right in Figure 8 . For this distribution, both ambient and hot electrons have equal number densities, even though the maximum at p = 0 is nearly two orders of magnitude larger than the maximum near p = 0.1mc. This is due to the spherical symmetry of the distribution. On integrating over momentum space, the phase space density must be multiplied by p , weighting the integral to higher moments. This suggests that the presence of a large peak at low momenta need not preclude relativistic effects modifying wave dispersion.
We can attempt to compare the model distribution function with the observed distributions. Unfortunately, the data published by Croley et al.  only measure phase space density down to v 10 km/s, i.e., p/mc 0.033. More recently, Menietti and Burch  have published electron data from the DE-1 spacecraft. Their Figure 3 shows electron phase space density for an altitude of 13,200 km at an invariant latitude of 67.9°. There is strong evidence in the measured distribution for acceleration by a parallel electric field, and this distribution might be a likely source of AKR. The measurements of Menietti and Burch go to much lower energy 1 eV, i.e., p/mc 0.005. Their data indicate that the phase space density may indeed be at least two orders of magnitude larger than the peak at larger momenta. However, Menietti and Burch point out that some of the intense low-energy fluxes are due to photoelectrons. It appears then that distribution functions such as those shown on the right of Figure 8 may reasonably represent the observed distributions, at least in terms of the corresponding densities and thermal widths.
We now come to the second issue, in that the distributions shown in Figure 8 are assumed to be spherically symmetric in phase space. One of the more striking features of the observed distributions is the loss cone in the reflected auroral electrons. In their original exposition of the cyclotron maser instability as a source for AKR, Wu and Lee  used the loss cone for driving the instability. Pritchett and Strangeway  have shown that asymmetric distributions containing a single-sided loss cone plus a hole at low energies are most unstable to waves propagating with angles of 90°-95° (typically) with respect to the magnetic field, when the loss cone is only present in the distribution for pitch angles > 90°. The shift from purely perpendicular waves is due to the asymmetry of the distribution.
In terms of the growth rate, the asymmetry is therefore important. However, it is not clear that an asymmetric distribution will have significantly different wave dispersion than a symmetric distribution. For the purpose of discussion, we shall assume that the phase space density is given by
This distribution is a function of p only for all pitch angles less than - . It can be shown that the conductivity tensor for such a distribution contains terms such as
where J is the Bessel function of the first kind of order n, with argument , = k p / m , and the summation is carried out over all n. The magnetic field defines the parallel or z direction and k defines the x direction. We have neglected the terms introduced by the discontinuity in f(p) at = - . In terms of cold plasma dispersion, the parallel electric field corresponds to an O mode wave, while the perpendicular fields give X mode waves. If and vanish in the conductivity tensor, then the X mode and O mode dispersion relations are separable. This occurs if k = 0 end f(p) is a symmetric function in p ( is of a form similar to with nJ / being replaced by -iJJ' ). To determine the importance of asymmetry , we shall assume k = 0 and 1, in which case the n = 1 term is dominant in (13a)and (13b) and
We can therefore estimate that
when p is the characteristic momentum of the distribution. For the present study, kp / m 0.1. More importantly, (15) depends strongly on . For example, if = 35°, then 0.02kp / m.
The asymmetry of the observed distribution is therefore not likely to significantly alter the dispersive properties of the plasma near 90°, since the X mode remains decoupled from the O mode. However, the asymmetry may result in moving the region of maximum growth away from 90°, and we must determine the range in k for which the shell dispersion relation is still valid.
We must therefore address our last assumption, k = 0. From a mathematical point of view, this was a useful assumption , since the conductivity tensor (5) could be calculated analytically. However, the inhomogeneity of the auroral field line suggests that even if the most unstable mode is found for k = 0, refractive effects will result in k 0. In addition, we have already noted that an asymmetric distribution will probably have k 0 for maximum growth. p> One way of estimating the range of validity for assuming k = 0 is by inspection of the resonance condition given by the denominator in (13a) and (13b), - k p / m - n. The assumption k = 0 ensures that the correction due to the Lorentz factor is much larger than the Doppler shift. This will still be the case provided ck / < p / 2mc where p is some characteristic momentum, for example, a thermal momentum. However, as we discuss below, the constraint on k is more likely to be ck / p / 2mc where p is the peak momentum of the DGH distribution.
The resonance condition describes an ellipse in momentum space, and as pointed out by Omidi et al. , the ellipse encloses the origin provided < n . The equation for the resonance ellipse can be written as
Because we have used momentum rather than velocity to determine the resonance condition, we find that the major axis of the ellipse lies on the p axis. For the present analysis where , and ck , the center of the resonance ellipse is given by p / mc ck / . Since the resonance ellipse encloses the origin for the trapped mode (for which < ), a major part of the resonance ellipse lies in regions of momentum space between the peaks of the distribution function at p = 0 and p = p if we restrict the center of the resonance ellipse to p p / 2, i.e., ck / p / 2mc.
Obviously, the condition given above does not ensure the presence of instability, but is only indicative of the range in k for which instability is likely. Nevertheless, we note that for ck / 1, and p / mc 0.1, we might expect instability provided k / k 0.05, i.e., 87° 93°. The range in propagation angles is consistent with the results of Pritchett and Strangeway , bearing in mind the asymmetry of the distribution used in the simulation.
While the growth rate is likely to be sensitive to the size of k , the dispersive properties of the plasma are less dependent on k , as we found for the effect of asymmetry. To show this, we consider the conductivity tensor for the relativistic Maxwellian, as given by Shkarofsky . Employing similar arguments to those used when discussing the effects of asymmetry, we find that
is an integration variable, varying from 0 . If we assume ck and that ck / then R and
For the spherical shell distribution, we can replace by mc / p = 100 when estimating the coupling between X and O mode waves due to modifications of the conductivity tensor. In addition, ck / 1, hence / < 0.01k / k.
In summary, we have shown that the DGH distribution is useful in determining the nature of the wave dispersion in a plasma containing both hot and cool electrons. While the growth rate will depend on such features as the asymmetry of the distribution and the value of k, we have found that the dispersive properties of the plasma are not radically altered for propagation within a few degrees of perpendicular. We have also estimated that the dispersive properties are essentially unchanged even if the distribution has a 35° loss cone. Lastly, we have argued that the published auroral electron distribution may be consistent with distributions such as those shown on the right of Figure 8.
As a last remark on the applicability of the present analysis to auroral electron distributions, we note that the presence of a "trapped" mode appears to be a general consequence of any "two-species" electron distribution. For example, Winglee  found such a mode was present using a "loss cone" type of distribution and a cold ambient plasma. The possible existence of the trapped mode may be important for auroral electron distributions since this mode occurs for < and can therefore be driven unstable by features in both the upgoing and downgoing distribution, similar to the Z mode as discussed by Omidi et al. .
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