Formation of power-law energy spectra in space plasmas by stochastic acceleration due to whistler-mode waves
A non‐relativistic Fokker‐Planck equation for the electron distribution function is formulated incorporating the effects of stochastic acceleration by whistler‐mode waves and Coulomb collisions. The stationary solution f to the equation, subject to a zero‐flux boundary condition, is found to be a ge...
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| Published in | Geophysical research letters Vol. 25; no. 21; pp. 4099 - 4102 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Washington, DC
Blackwell Publishing Ltd
01.11.1998
American Geophysical Union |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0094-8276 1944-8007 |
| DOI | 10.1029/1998GL900108 |
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| Abstract | A non‐relativistic Fokker‐Planck equation for the electron distribution function is formulated incorporating the effects of stochastic acceleration by whistler‐mode waves and Coulomb collisions. The stationary solution f to the equation, subject to a zero‐flux boundary condition, is found to be a generalized Lorentzian (or kappa) distribution, which satisfies f ∝ υ2(κ+1) for large velocity υ where κ is the spectral index. The parameter κ depends strongly on the relative wave intensity R. Taking into account the critical energy required for resonance of electrons with whistlers, we calculate a range of values of R for each of a number of different space plasmas for which kappa distributions can be expected to be formed. This study is one of the first in the literature to provide a theoretical justification for the formation of generalized Lorentzian (or kappa) particle distribution functions in space plasmas. |
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| AbstractList | A non‐relativistic Fokker‐Planck equation for the electron distribution function is formulated incorporating the effects of stochastic acceleration by whistler‐mode waves and Coulomb collisions. The stationary solution f to the equation, subject to a zero‐flux boundary condition, is found to be a generalized Lorentzian (or kappa) distribution, which satisfies f ∝ υ2(κ+1) for large velocity υ where κ is the spectral index. The parameter κ depends strongly on the relative wave intensity R. Taking into account the critical energy required for resonance of electrons with whistlers, we calculate a range of values of R for each of a number of different space plasmas for which kappa distributions can be expected to be formed. This study is one of the first in the literature to provide a theoretical justification for the formation of generalized Lorentzian (or kappa) particle distribution functions in space plasmas. A non‐relativistic Fokker‐Planck equation for the electron distribution function is formulated incorporating the effects of stochastic acceleration by whistler‐mode waves and Coulomb collisions. The stationary solution f to the equation, subject to a zero‐flux boundary condition, is found to be a generalized Lorentzian (or kappa) distribution, which satisfies f ∝ υ 2(κ+1) for large velocity υ where κ is the spectral index. The parameter κ depends strongly on the relative wave intensity R . Taking into account the critical energy required for resonance of electrons with whistlers, we calculate a range of values of R for each of a number of different space plasmas for which kappa distributions can be expected to be formed. This study is one of the first in the literature to provide a theoretical justification for the formation of generalized Lorentzian (or kappa) particle distribution functions in space plasmas. |
| Author | Ma, Chun-yu Summers, Danny |
| Author_xml | – sequence: 1 givenname: Chun-yu surname: Ma fullname: Ma, Chun-yu organization: Department of Mathematics and Statistics, Memorial University of Newfoundland, St John's, Newfoundland, A1C 5S7, Canada – sequence: 2 givenname: Danny surname: Summers fullname: Summers, Danny organization: Department of Mathematics and Statistics, Memorial University of Newfoundland, St John's, Newfoundland, A1C 5S7, Canada |
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| Copyright | 1998 by the Chinese Geophysical Society 1999 INIST-CNRS |
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| Keywords | Energy spectra Astrophysical plasma Theoretical study Particle acceleration Fokker-Planck equation Particle wave interaction Coulomb interaction Power law Steady state solution Whistler wave |
| Language | English |
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| References | Sittler, E. C.Jr.K. W. OgilvieJ. D. Scudder, Survey of low-energy plasma electrons in Saturn's magnetosphere: Voyagers 1 and 2, J. Geophys. Res., 88, 8847-8870, 1983 Christon, S. P.D. G. MitchellD. J. WilliamsL. A. FrankC. Y. HuangT. E. Eastman, Energy spectra of plasma sheet ions and electrons from ≈ 50 eV/e to ≈1 Mev during plasma temperature transitions, J. Geophys. Res., 93, 2562-2572, 1988 Gurevich, A. V., On the amount of accelerated particles in an ionized gas under various accelerating mechanisms, Sov. Phys. JETP, 11, 1150-1157, 1960 Melrose, D. B.Instabilities in Space and Laboratory Plasmas, Cambridge University Press, New York, 1986 Xue, S.R. M. ThorneD. Summers, Electromagnetic ioncyclotron instability in space plasmas, J. Geophys. Res., 98, 17475-17484, 1993 Mace, R. L.M. A. Hellberg, A dispersion function for plasmas containing superthermal particles, Phys. Plasmas, 2, 2098-2109, 1995 Summers, D.S. XueR. M. Thorne, Calculation of the dielectric tensor for a generalized Lorentzian (kappa) distribution function, Phys. Plasmas, 1, 2012-2025, 1994 Kennel, C. F.F. Engelmann, Velocity space diffusion from weak plasma turbulence in a magnetic field, Phys. Fluids, 9, 2377-2388, 1966 Summers, D.R. M. Thorne, A new tool for analyzing microinstabilities in a space plasmas modeled by a generalized Lorentzian (kappa) distribution, J. Geophys. Res., 97, 16827-16832, 1992 Gosling, J. T.J. R. AsbridgeS. J. BameW. C. FeldmanR. D. ZwicklG. PaschmannN. SckopkeR. J. Hynds, Interplanetary ions during an energetic storm particle event: The distribution function from solar wind thermal energies to 1.6 Mev, J. Geophys. Res., 86, 547-554, 1981 Dessler, A. J. (Physics of the Jovian Magnetosphere, Cambridge University Press, New York, 1983 Summers, D.R. M. Thorne, The modified plasma dispersion function, Phys. Fluids, B3, 1835-1847, 1991 Hasegawa, A.K. MimaM. Duong-van, Plasma distribution function in a superthermal radiation field, Phys. Rev. Lett., 54, 2608-2610, 1985 Schlickeiser, R., γ-ray evidence for galactic in situ electron acceleration, Astron. Astrophys., 319, L5-L8, 1997 Collier, M. R., On generating kappa-like distribution functions using velocity space Lévy flights, Geophys. Res. Lett., 20, 1531-1534, 1993 Hinton, F. L.Basic Plasma Physics I, A. A. GaleevR. N. Sudan147, North-Holland Publishing Company, Amsterdam, 1983 Pryadka, J. M.V. Petrosian, Stochastic acceleration of low-energy electrons in cold plasmas, Astrophys. J., 482, 774-781, 1997 Steinacker, J.J. A. Miller, Stochastic gyroresonant electron acceleration in a low-beta plasma. I. Interaction with parallel transverse cold plasma waves, Astrophys. J., 393, 764-781, 1992 Divine, N.H. B. Garrett, Charged particle distribution in Jupiter's magnetosphere, J. Geophys. Res., 88, 6889-6903, 1983 Mace, R. L., Whistler instability enhanced by superthermal electrons within the Earth's foreshock, J. Geophys. Res., 103, 14643-14654, 1998 Summers, D.R. M. ThorneH. Matsumoto, Evaluation of the modified plasma dispersion function for half-integral indices, Phys. Plasmas, 3, 2496-2501, 1996 Armstrong, T. P.M. T. PaonessaE. V. Bell IIS. M. Krimigis, Voyager observations of Saturnian ion and electron phase space densities, J. Geophys. Res., 88, 8893-8904, 1983 Vasyliunas, V. M., A survey of low-energy electrons in the evening sector of the magnetosphere with OGO 1 and OGO 3, J. Geophys. Res., 73, 2839-2884, 1968 Kivelson, M. G.C. T. RussellIntroduction to Space Physics, Cambridge University Press, New York, 1995 Dermer, C. D.J. A. MillerH. Li, Stochastic particle acceleration near accretion black holes, Astrophys. J., 456, 106-109, 1996 1960; 11 1992; 393 1966; 9 1997; 482 1980; 2 1993; 98 1993; 20 1994; 294 1986 1995 1983 1998; 103 1996; 456 1995; 2 1997; 319 1991; B3 1992; 97 1994; 1 1968; 73 1988; 93 1985; 54 1996; 3 1981; 86 1983; 88 Gurevich A. V. (e_1_2_1_9_1) 1960; 11 Hinton F. L. (e_1_2_1_11_1) 1983 e_1_2_1_23_1 e_1_2_1_24_1 e_1_2_1_21_1 e_1_2_1_22_1 e_1_2_1_27_1 e_1_2_1_28_1 e_1_2_1_25_1 e_1_2_1_26_1 Petrosian V. (e_1_2_1_18_1) 1994 e_1_2_1_7_1 e_1_2_1_8_1 Schlickeiser R. (e_1_2_1_20_1) 1997; 319 e_1_2_1_5_1 e_1_2_1_6_1 e_1_2_1_3_1 e_1_2_1_12_1 e_1_2_1_4_1 e_1_2_1_13_1 e_1_2_1_10_1 e_1_2_1_2_1 e_1_2_1_16_1 e_1_2_1_17_1 e_1_2_1_14_1 e_1_2_1_15_1 e_1_2_1_19_1 |
| References_xml | – reference: Divine, N.H. B. Garrett, Charged particle distribution in Jupiter's magnetosphere, J. Geophys. Res., 88, 6889-6903, 1983 – reference: Kivelson, M. G.C. T. RussellIntroduction to Space Physics, Cambridge University Press, New York, 1995 – reference: Pryadka, J. M.V. Petrosian, Stochastic acceleration of low-energy electrons in cold plasmas, Astrophys. J., 482, 774-781, 1997 – reference: Summers, D.R. M. Thorne, A new tool for analyzing microinstabilities in a space plasmas modeled by a generalized Lorentzian (kappa) distribution, J. Geophys. Res., 97, 16827-16832, 1992 – reference: Christon, S. P.D. G. MitchellD. J. WilliamsL. A. FrankC. Y. HuangT. E. Eastman, Energy spectra of plasma sheet ions and electrons from ≈ 50 eV/e to ≈1 Mev during plasma temperature transitions, J. Geophys. Res., 93, 2562-2572, 1988 – reference: Summers, D.R. M. ThorneH. Matsumoto, Evaluation of the modified plasma dispersion function for half-integral indices, Phys. Plasmas, 3, 2496-2501, 1996 – reference: Summers, D.R. M. Thorne, The modified plasma dispersion function, Phys. Fluids, B3, 1835-1847, 1991 – reference: Gosling, J. T.J. R. AsbridgeS. J. BameW. C. FeldmanR. D. ZwicklG. PaschmannN. SckopkeR. J. Hynds, Interplanetary ions during an energetic storm particle event: The distribution function from solar wind thermal energies to 1.6 Mev, J. Geophys. Res., 86, 547-554, 1981 – reference: Vasyliunas, V. M., A survey of low-energy electrons in the evening sector of the magnetosphere with OGO 1 and OGO 3, J. Geophys. Res., 73, 2839-2884, 1968 – reference: Steinacker, J.J. A. Miller, Stochastic gyroresonant electron acceleration in a low-beta plasma. I. Interaction with parallel transverse cold plasma waves, Astrophys. J., 393, 764-781, 1992 – reference: Summers, D.S. XueR. M. Thorne, Calculation of the dielectric tensor for a generalized Lorentzian (kappa) distribution function, Phys. Plasmas, 1, 2012-2025, 1994 – reference: Schlickeiser, R., γ-ray evidence for galactic in situ electron acceleration, Astron. Astrophys., 319, L5-L8, 1997 – reference: Dessler, A. J. (Physics of the Jovian Magnetosphere, Cambridge University Press, New York, 1983 – reference: Dermer, C. D.J. A. MillerH. Li, Stochastic particle acceleration near accretion black holes, Astrophys. J., 456, 106-109, 1996 – reference: Collier, M. R., On generating kappa-like distribution functions using velocity space Lévy flights, Geophys. Res. Lett., 20, 1531-1534, 1993 – reference: Mace, R. L., Whistler instability enhanced by superthermal electrons within the Earth's foreshock, J. Geophys. Res., 103, 14643-14654, 1998 – reference: Hasegawa, A.K. MimaM. Duong-van, Plasma distribution function in a superthermal radiation field, Phys. Rev. Lett., 54, 2608-2610, 1985 – reference: Hinton, F. L.Basic Plasma Physics I, A. A. GaleevR. N. Sudan147, North-Holland Publishing Company, Amsterdam, 1983 – reference: Melrose, D. B.Instabilities in Space and Laboratory Plasmas, Cambridge University Press, New York, 1986 – reference: Gurevich, A. V., On the amount of accelerated particles in an ionized gas under various accelerating mechanisms, Sov. Phys. JETP, 11, 1150-1157, 1960 – reference: Sittler, E. C.Jr.K. W. OgilvieJ. D. Scudder, Survey of low-energy plasma electrons in Saturn's magnetosphere: Voyagers 1 and 2, J. Geophys. Res., 88, 8847-8870, 1983 – reference: Xue, S.R. M. ThorneD. Summers, Electromagnetic ioncyclotron instability in space plasmas, J. Geophys. Res., 98, 17475-17484, 1993 – reference: Mace, R. L.M. A. Hellberg, A dispersion function for plasmas containing superthermal particles, Phys. Plasmas, 2, 2098-2109, 1995 – reference: Armstrong, T. P.M. T. PaonessaE. V. Bell IIS. M. Krimigis, Voyager observations of Saturnian ion and electron phase space densities, J. Geophys. Res., 88, 8893-8904, 1983 – reference: Kennel, C. F.F. Engelmann, Velocity space diffusion from weak plasma turbulence in a magnetic field, Phys. Fluids, 9, 2377-2388, 1966 – year: 1983 – volume: 93 start-page: 2562 year: 1988 end-page: 2572 article-title: Energy spectra of plasma sheet ions and electrons from ≈ 50 eV/e to ≈1 Mev during plasma temperature transitions publication-title: J. Geophys. Res. – volume: 98 start-page: 17475 year: 1993 end-page: 17484 article-title: Electromagnetic ioncyclotron instability in space plasmas publication-title: J. Geophys. Res. – volume: 3 start-page: 2496 year: 1996 end-page: 2501 article-title: Evaluation of the modified plasma dispersion function for half‐integral indices publication-title: Phys. Plasmas – volume: 54 start-page: 2608 year: 1985 end-page: 2610 article-title: Plasma distribution function in a superthermal radiation field publication-title: Phys. Rev. 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| Snippet | A non‐relativistic Fokker‐Planck equation for the electron distribution function is formulated incorporating the effects of stochastic acceleration by... |
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| SubjectTerms | Astronomy Earth, ocean, space Exact sciences and technology Fundamental aspects of astrophysics Fundamental astronomy and astrophysics. Instrumentation, techniques, and astronomical observations Magnetohydrodynamics and plasmas |
| Title | Formation of power-law energy spectra in space plasmas by stochastic acceleration due to whistler-mode waves |
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