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Appendix B: Dispersion Relations in a Maxwellian Plasma

In a Maxwellian electron plasma described by the Vlasov equation, the dispersion equation of Bernstein waves can be put into the implicit form

  eqnarray510

where tex2html_wrap_inline1814 is a modified Bessel function of the first kind, tex2html_wrap_inline1816 and tex2html_wrap_inline1818 are the electron (angular) plasma frequency and gyrofrequency, respectively, and tex2html_wrap_inline1478 is the thermal electron gyroradius. One can deduce from (B1) the two partial derivatives of tex2html_wrap_inline1316 as a function of tex2html_wrap_inline1118 and tex2html_wrap_inline1138 :

  eqnarray537

eqnarray546

  eqnarray564

eqnarray576

Some samples of calculations using these equations are shown in Figure B1.

   figure594
Figure B1: Dispersion characteristics of Bernstein waves. (top) Classical dispersion curves deduced from (B1) in the two first intraharmonic bands for three typical values of tex2html_wrap_inline1134 , involving the tex2html_wrap_inline1184 and tex2html_wrap_inline1126 frequencies in the case of tex2html_wrap_inline1188 . (middle) The six corresponding derivatives with respect to tex2html_wrap_inline1118 computed from (B2). Except for the tex2html_wrap_inline1126 band, these derivatives become vanishingly small when the ratio tex2html_wrap_inline1134 increases. (bottom) The six corresponding derivatives with respect to tex2html_wrap_inline1138 , computed from (B3).

Note that all infinite sums in the preceding equations are rather quickly convergent. A criterion to stop their calculation is that the last nth computed term in (B1) satisfies

eqnarray604

with n>m, where m is the order of the upper limit of the band considered and tex2html_wrap_inline1322 the precision to be achieved. For instance, in this paper, where tex2html_wrap_inline1850 , tex2html_wrap_inline1852 , and tex2html_wrap_inline1854 , such computations need 3<n<17 for tex2html_wrap_inline1856 .

Acknowledgments The URAP experiment is a joint project of NASA GSFC, Observatoire de Paris, CRPE, and the University of Minnesota. The Principal Investigator is R.G. Stone. The French contribution was mainly financed by the Centre National d'Études Spatiales. We thank A. Balogh, Principal investigator, and R.J. Forsyth, Co-Investigator, on the magnetometer experiment for kind permission to use their data. We are very grateful to R. Manning and the team of engineers and technicians of the Département de Recherches Spatiales (Observatoire de Paris), who designed and built the radio receiver whose great performances made possible this work. We sincerely thank J.-L. Steinberg for a careful reading and helpful comments on the manuscript.
The Editor thanks C.P. Paranicas and another referee for their assistance in evaluating this paper.


next up previous
Next: References Up: Dispersion of electrostatic waves Previous: Appendix A: Doppler Effect

Michel Moncuquet
Tue Nov 18 19:18:28 MET 1997