*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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Fig. 1 Wave dispersion surfaces for the ring dispersion. Four models are present when both a hot and cold electron component are included.

Fig. 2. Concatenated dispersion surfaces with their corresponding growth rates. The four dispersion surfaces shown in Figure 1 have been merged into a single display. We have truncated the surface identified with the "cold intermediate" mode for reasons of clarity. An unstable mode is present where two stable modes coalesce, and the corresponding growth rates are shown by the contour plot.

Fig. 3. Solutions of thek= 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 thejparameter of the DGH distribution. Forj>1, /j= 200. Whenj= 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 theR-Xmode, and the dotted line corresponds to a Bernstein mode.

Fig. 4. Solutions of thek= 0 shell dispersion relation (6) for variablen/n. Similar to Figure 3.

Fig. 5. Solutions of thek=0 shell dispersion relation (6) for variable / . Similar to Figure 3.

Fig. 6. Solutions of the approximate shell dispersion relation (10) for finitekand low ambient electron temperature. Similar in format to Figure 3.

Fig. 7. Solutions of the approximate shell dispersion relation (10) for equal temperature distributions. Similar in format to Figure 3. Note the expanded frequency scale.

Fig. 8. Representative distribution functions for which relativistic modifications are important. The plasma parameters together with the corresponding solutions of the shell dispersion relation fork= 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).

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