In present paper, the effect of relativistic hot electron beam for field aligned Whistler mode waves has been studied theoretically in the presence of AC electric field perpendicular to magnetic field. Studies have been performed using perturbative approach along with the method of characteristic solutions and are valid for comparatively small ambient magnetic field of Uranus, of the order of nano Tesla, as observed by Voyager 2. The detailed derivation and calculations has been done for dispersion relation and growth rate for magnetosphere of Uranus. Analyses are done by changing various plasma parameters which are explained in result and discussions section of this paper. Extensive study of wave-particle interactions and numerical calculations concludes that in case of injection of a distribution of particles having a positive slope in v⊥, temperature anisotropy remains the main source of free energy. It is seen that other effective parameters for the growth of whistler mode waves are AC frequency and higher number density of hot electrons. We also learn that even the minimal presence of such energetic particles having a positive slope of distribution function and increasing power of perpendicular thermal velocity can increase the growth rate significantly in the magnetosphere of Uranus. The present work is basically based upon the theoretical investigation and mathematical analysis of the magnetosphere of Uranus, supported by satellite data.
2. Sagdeev, R. Z. and A. A. Galeev, Non-linear Plasma Theory, W. A. Benjamin, Inc. Book Company, New York City, NY, 1969.
3. Scarf, F. L. and D. A. Gurnett, "A plasma wave investigation for the Voyager mission," Space Sci. Rev., Vol. 21, 289-308, 1977.
4. Gurnett, D. A. and F. L. Scarf, Physics of Jovian Magnetosphere, A. J. Dessler (ed.), 285, Cambridge University Press, Cambridge, 1983.
5. Lin, N., P. J. Kellogg, J. P. Thiessen, D. Lengyel-Frey, B. T. Tsurutani, and J. L. Phillips, "Whistler mode waves in the Jovian magnetosheath," J. Geophys. Res., Vol. 99, No. A12, 23527-23540, 1994.
6. Hobara, Y., S. Kanemaru, M. Hayakawa, and D. A. Gurnett, "On estimating the amplitude of Jovian whistlers observed by Voyager 1 and implications concerning lightning," J. Geophys. Res., Vol. 102, No. A4, 7115-7125, 1997.
7. Kurth, W. S., D. A. Gurnett, A. M. Persoon, A. Roux, S. J. Bolton, and C. J. Alexander, "The plasma wave environment of Europa," Planet. Space Sci., Vol. 49, No. 3-4, 345-363, 2001.
8. Kurth, W. S., D. A. Gurnett, and F. L. Scarf, "Sporadic narrowband radio emissions from Uranus," J. Geophys. Res., Vol. 91, No. 11, 11958-11964, 1986.
9. Orlowski, D. S. and C. T. Russell, "Comparison of properties of upstream whistlers at different planets," Adv. Space Res., Vol. 16, 137-141, 1995.
10. Gurnett, D. A., F. L. Scarf, W. S. Kurth, and R. L. Poynter, "First plasma wave observation at Uranus," Science, Vol. 233, No. 4759, 106-109, 1986.
11. Ness, N. F., M. H. Acuna, K. W. Behannon, L. F. Burlaga, J. E. P. Connerney, R. P. Lepping, and F. M. Neubauer, "Magnetic fields at Uranus," Science, Vol. 233, No. 4759, 85-89, 1986.
12. Hudson, M. K., J. T. Clark, and J. A. Warren, "Ionospheric dynamo theory for production of far ultraviolet emissions on Uranus," J. Geophys. Res., Vol. 94, 6517-6522, 1989.
13. Clarke, J. T., M. K. Hudson, and Y. L. Yung, "The excitation of the far ultraviolet eclectroglow emission on Uranus, Saturn and Jupiter," J. Geophys. Res., Vol. 92, No. A13, 15139-15147, 1987.
14. Pandey, R. P., K. M. Singh, and R. S. Pandey, "A theoretical study of the whistler mode instability at the Uranian bow shock," Earth, Moon and Planets, Vol. 87, No. 2, 59-71, 2001.
15. Wygant, J. R., M. Bensadoum, and F. S. Mozer, "Electric field measurements at subcritical oblique bow shock crossings," J. Geophys. Res., Vol. 92, No. A10, 11109-11121, 1987.
16. Winckler, J. R., P. R. Malcolm, R. L. Arnoldy, W. J. Burke, K. N. Erickson, J. Ernstmeyer, R. C. Franz, T. J. Hallinan, P. J. Kellogg, S. J. Monson, K. A. Lynch, G. Murphy, and R. J. Nemzek, "ECHO 7: An electron beam experiment in magnetosphere," Eos, Transactions American Geophysical Union, Vol. 70, No. 25, 657-668, 1989.
17. Winckler, J. R., "The application of arti¯cial electron beams to magnetospheric research," Reviews of Geophysics, Vol. 18, No. 3, 659-682, 1980.
18. Winglee, R. M. and P. J. Kellogg, "Electron beam injection during active experiments: Electromagnetic wave emissions," J. Geophys. Res., Vol. 95, No. A5, 6167-6190, 1990.
19. Pandey, R. P., S. M. Karim, K. M. Singh, and R. S. Pandey, "Effect of cold plasma injection on whistler mode instability triggered by perpendicular AC electric field at Uranus," Earth, Moon and Planets, Vol. 91, No. 4, 195-207, 2003.
20. Misra, K. D. and R. S. Pandey, "Generation of whistler emissions by injection of hot electrons in the presence of a perpendicular as electric field," J. Geophys. Res., Vol. 100, No. A10, 19405-19411, 1995.
21. Sazhin, S. S., "Oblique whistler mode growth and damping in a hot anisotropic plasma," Planet Space Science, Vol. 36, 663-667, 1988.
22. Chu, K. R. and J. L. Hirshfield, "Comparitive study of the axial and azimuthal bunching mechanism in electromagnetic cyclotron instability," Phys. Fluid, Vol. 21, 461, 1978.
23. Pandey, R. S. and R. Kaur, "Generation of low frequency electromagnetic wave by injection of cold electron for relativistic and non-relativistic subtracted bi-Maxwellian distribution with perpendicular AC electric field for magnetosphere of Uranus," Progress In Electromagnetic Research B, Vol. 45, 337-352, 2012.
24. Pandey, R. S., K. Rajbir, S. Kumar, and K. Mukesh, "Study of VLF mode instability with AC electric field for subtracted bi-Maxwellian in the magnetosphere of Uranus," J. Emerging Trends in Eng. and Applied Sci., Vol. 4, No. 2, 201-206, 2013.
25. Misra, K. D. and T. Haile, "Effect of AC electric field on the whistler mode instability in the magnetosphere," J. Geophys. Res., Vol. 98, No. A6, 9297-9305, 1993.
26. Bret, A., M. C. Firpo, and C. Deutsch, "Electromagnetic instabilities for relativistic beam-plasma interaction in whole k space: Nonrelativistic beam and plasma temperature effects," Phys. Review E, Vol. 7, 016403, 2005.