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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Using simple arguments, Strangeway  determined that for distributions like those observed on the polar-orbiting S3-3 spacecraft, the ring distribution given by (1) may be adequate provided there are roughly equal densities of energetic and ambient electrons. Since the discussion was somewhat qualitative, we shall study the effects of temperature in more detail here. The distribution function we shall employ is the spherically symmetric DGH distribution.
where is the Lorentz factor equal to (1 + p / m2 c2 ), K() is the second-order MacDonald function, and is a parameter specifying the thermal spread of the distribution. As , (2) reduces to
where (v) is the gamma function. We therefore identify with with an inverse temperature, = 2mc / p, where p is a thermal momentum, the exponential term in (3) being equivalent to exp (-p / p).
When j 0, the DGH distribution function has a maximum at
As , the peak occurs at p jp. In the previous sections we used p / mc = 0.1 as a representative ring momentum. For the peak to occur . For the peak to occur at the same momentum in (3) we require / j 200.
When j = 0 the distribution function given by (2) reduces to the relativistic Maxwellian. Shkarofsky  has derived the conductivity tensor for electromagnetic waves for the Maxwellian . If we denote the conductivity tensor for f(p) as , then
since the derivation of the conductivity tensor required integration over momentum space and the momentum only enters through the Lorentz factor in (2). We can therefore use the results of Shkarofsky to evaluate the conductivity tensor for the distribution given by (2).
One advantage then of using the distribution function (2) is the ability to relate the conductivity tensor (5) to the previous work of Shkarofsky. A second advantage concerns the application to energetic electron distributions as observed on polar-orbiting spacecraft such as S3-3 and DE-1. In general the observed distributions show features reminiscent of both the ring distribution (1) and the spherically symmetric "shell" distribution (2). The electron data often show a peak in the phase space density at 90° pitch angles with the peak momentum in the keV range, similar to a ring distribution. On the other hand, the distributions are much more shell-like at higher momenta. Furthermore, theoretically we expect the combined effect of an accelerating field-aligned potential drop and the magnetic mirror force to produce an oblate spheroidal boundary in momentum space [Chiu and Schulz, 1978]. At low altitudes, the boundary is almost spherical, and so a shell-like distribution may be a reasonable approximation to the theoretically predicted distribution function, apart from the presence of the loss cone. It has been shown [Croley et al., 1978] that the theoretical model of Chiu and Schulz can be related to the observed distribution, bearing in mind the smoothing inherent in the observations due to both plasma instabilities and instrumental effects.
Strangeway  has argued that the dispersion properties of both ring and shell distributions are likely to be similar. However, the ring distribution is not easily related to the observations. For this reason, a shell-like distribution function is preferable. Unfortunately, including relativistic terms results in a rather unwieldy form of the conductivity tensor for arbitrary wave vectors, even for the Maxwellian [Shkarofsky, 1966]. Hence, the ring distribution is of use for studying oblique propagation. On the other hand, the shell distribution relation is simplified considerably by assuming k = 0. Since we would like to compare both ring and shell distributions, we shall therefore assume that k = 0, which allows us to identify the wave modes and explore their dependence on various plasma parameters. In section 4 we shall address the dependence on wave vector and justify our identification of the wave modes.
From R. J. Strangeway (unpublished manuscript, 1986) where the derivation is described in detail, we find that the dispersion relation for R-X polarized waves is given by
when k = 0, where indicates summation of electron species, is the plasma frequency for a particular species, is the inverse temperature for that species, j is the power of the shell distribution, = ( - ) / , and Z is the plasma dispersion function [Fried and Conte, 1961].
As discussed by R. J. Strangeway (unpublished manuscript, 1986), the introduction of the square root in the argument for the Z function in (6) results in a branch cut from = 0 to infinity. If the branch cut is taken to = +, then the analytic continuation of the function is such that the physically allowable solutions lie on one sheet, which we refer to as the lower sheet. For the lower sheet Im [( )] > 0. The value of D() on the upper sheet can be shown to be simply related to the value on the lower sheet through the analytic continuation of the Z function; hence
except on the branch cut, where the subscripts u and l refer to the upper and lower sheets, respectively, and angle brackets indicate complex conjugation. The relationship (7) indicates that solutions to (6) come in complex conjugate pairs, except when Re () > 0 and Im () = 0 . The restriction of solutions to the lower sheet ensures that a Maxwellian is stable.
Solutions of (6) are plotted in Figure 3 assuming both hot and ambient population are present. The hot population can have different values of j, while the ambient population has j = 0, i.e., Maxwellian. Frequency is plotted in the left-hand column of Figure 3, with the corresponding growth rate in the right-hand column for different values of j. As indicated by the heading to the figure, we fix / j = 200, i.e., p / mc 0.1, the temperatures of the ambient and hot electron distributions are the same, as are the number densities. The numbers to the right of each panel give the j parameter for the hot electron population, with j = corresponding to a delta function. When j = 0, both populations are Maxwellian and we use ph = 200 for the hot electron population. For reference with this and subsequent figures, if we define = / j, then = j, j 0: = , j = 0.
Fig. 3. Solutions of the k = 0 shell dispersion relation (6) for variable /. The left-handed column shows frequency while the right column shows the corresponding growth rates. The integers inside and to the right of each panel give the j parameter of the DGH distribution. For j >1, / j = 200. When j = 0, both distributions are Maxwellian and we set = 200. For the prupose of identifying modes, the solid line corresponds to the "trapped" mode, the dashed line corresponds to the R-X mode, and the dotted line corresponds to a Bernstein mode.
For the delta function we find that there are two modes. The higher- frequency mode, as indicated by the dashed line, increases in frequency as / is increased. The frequency of this mode is greater than the gyrofrequency. The corresponding growth rate, shown in the right-hand column, is zero, and the mode is marginally stable. We therefore identify this mode with the R-X mode in Figure 1. The second mode, shown by the solid line, lies below the gyrofrequency and has a positive growth rate. We identify this mode with the unstable trapped mode in Figure 1. In this and subsequent figures, we use dashed lines to represent the R-X mode, while solid lines are used to indicate the "trapped" mode. For the high j values this identification is unambiguous, the trapped mode is the lower-frequency mode, and this mode can be unstable. However, as we shall show later, the identification of the modes becomes more difficult for low j values.
A third mode is also plotted in Figure 3. For the purposes of identifying the mode, we consider the solutions shown in the bottom panels for j = 0. These solutions correspond to a single Maxwellian. At low / only one mode (the R-X mode) is shown, and that mode is damped. As / is increased, the mode becomes marginally stable and a second mode is also found. This has been called a "Bernstein" mode by Pritchett [1984b], and we indicate this mode by dotted lines. The Bernstein mode shown in Figure 3 is not present when j = . This is because there is no branch cut for the delta function dispersion relation. When j , both the R-X mode and the Bernstein mode lie on the branch cut described earlier, and therefore are only marginally stable for > . On the lower sheet the R-X mode is found as 0 from above, while the Bernstein mode is found for 0 from below. As the frequency decreases so that , both modes merge, and the solutions are found for complex conjugate frequencies, one on each sheet. We only show the solution for the lower sheet and that solution is damped for j = 0.
When j is large, i.e., low-temperature distributions, the solutions are very similar to thej = case, apart from the additional Bernstein mode. However, including temperature restricts the range in instability for the trapped mode. At low / the mode is damped as pointed out by Winglee  for a loss cone type distribution. For lower j values, the trapped mode is damped for all values of / .
One last point to be made from Figure 3 concerns the form of the solutions for j = 1. The R-X mode looks very similar. but the trapped mode behavior is rather different than that found for j > 1. Unlike the other cases, the trapped mode couples to the Bernstein mode at high / . At low / . the mode we have called the trapped mode lies on the branch cut with > and 0 from below. It appears then that the labelling of the modes is not so obvious for j = 1.
Before investigating the dependence on other plasma parameters we shall summarize the properties of the wave modes. The highest-frequency mode has been identified with the R-X mode, with the implication that this mode approaches the light cone as wave vector increases. The mode is either stable or damped and when marginally stable ( > ), the R-X mode lies on the branch cut with 0 from above. This mode is present even when j = . As frequency decreases, the R-X mode couples to a second mode which lies on the branch cut with 0 from below. We have identified this mode with a Bernstein mode, which implies that the frequency of this mode is nearly constant for all wave vectors. The Bernstein mode and the R-X mode merge and are damped for < . The last mode found is the "trapped" mode. This mode usually has the lowest frequency, and can be unstable. By implication, this mode cannot propagate for finite wave vector. In Figure 3, the trapped mode is separate from both the R-X and Bernstein modes, except for j = 1. We shall verify the inferences made concerning the dispersion properties of the waves in section 4.
In Figure 4 we fix / and vary the number density ratio. We have again assumed equal temperature electron populations. At low hot electron densities, the three modes are very distinct for j 2. The R-X mode and Bernstein are both marginally stable, merging to form a damped mode at sufficiently high n / n (the Bernstein mode is again not present for j = ). The trapped mode is damped at low n / n and becomes unstable at high n / n. As might be expected, the required density ratio for instability decreases as j increases.
Fig. 4. Solutions of the k= 0 shell dispersion relation (6) for variable n / n. Similar to Figure 3.
For j = 1 the trapped mode behavior shown in Figure 4 is again very different. For the particular choice of parameters, the trapped mode is always marginally stable, lying on the branch cut. On the other hand, the R-X mode merges with the Bernstein mode, as found for j > 1, but the R-X mode can also be unstable for sufficiently high n / n.
The parameter / is varied in Figure 5, the other parameters being held fixed. If / is large, then the background plasma has a low temperature in comparison to the hot electrons. When this is the case, the three modes are very distinct for 1 j < , the high-frequency R-X mode and the Bernstein mode are both marginally stable, while the low- frequency trapped mode is unstable provided the energetic electrons are sufficiently peaked. As the ambient electron temperature is increased (i.e., / decreases), the three modes approach one another. At sufficiently low energetic electron temperatures, the low-frequency mode is again unstable, while the R-X mode is damped when j 2. For j = 1, the R-X mode also becomes damped. However, the trapped mode shows a much more complicated structure. Near / = 1, the trapped mode changes from being strongly damped ( / < -0.005, below the lower limit of the plots) to marginally stable. At the same time the frequency varies considerably, reaching a maximum as 0 for the trapped mode. When the frequency of the trapped mode maximizes, both the Bernstein mode and the R-X mode merge and form a single damped mode.
Fig. 5. Solutions of the k=0 shell dispersion relation (6) for variable / . Similar to Figure 3.
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