Skip to main content
SpringerLink
Log in

Energetic particle distributions in space physics represented by the Pearson differential equation for the kappa distribution

  • Original Article
  • Published:
Astrophysics and Space Science Aims and scope Submit manuscript

Abstract

The kappa distribution has been used for decades as a numerical fitting function to represent the energetic particle distributions measured with satellite instruments. The distributions are generally either a three dimensional distribution in the particle velocity or the one dimensional distribution in a velocity component in a particular direction. The theoretical basis for the origin of the kappa functional form is incomplete. The current paper considers published data for one dimensional distributions and provides a unique method of analysis to determine whether the measured distributions are well represented by the kappa distribution. The analysis is based on the Pearson differential equation which arises from the steady distribution of a Fokker-Planck equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data Availability

There are no original data.

Code Availability

The Matlab codes used are original and basic.

References

  • Abdul, R.F., Mace, R.L.: A method to generate kappa distributed random deviates for particle-in-cell simulations. Comput. Phys. Commun. 185, 2383 (2014)

    Article  ADS  Google Scholar 

  • Bian, N.H., Emslie, A.G., Stackhouse, D.J., Kontar, E.P.: The formation of kappa-distribution accelerated electron populations in solar flares. Astrophys. J. 796, 142 (2014)

    Article  ADS  Google Scholar 

  • Eu, B.C.: Nonequilibrium Statistical Mechanics. Springer, Dordrecht (1998)

    Book  MATH  Google Scholar 

  • Fahr, F.J., Shizgal, B.: Modern exospheric theories and their observational relevance. Rev. Geophys. Space Phys. 21, 75 (1983)

    Article  ADS  Google Scholar 

  • Feldman, W.C., Asbridge, J.R., Bame, S.J., Montgomery, M.D., Gary, S.P.: Solar wind electrons. J. Geophys. Res. 80, 4181 (1975)

    Article  ADS  Google Scholar 

  • Guo, R.: Stationary states of polytorpic plasmas. Phys. Plasmas 27, 122104 (2020)

    Article  ADS  Google Scholar 

  • Helberg, M.A., Mace, R.L., Baluku, T.K., Kourakis, I., Saini, N.S.: Comment on “Mathematical and physical aspects of kappa velocity distribution” [Phys. Plasmas 14, 110702 (2007)]. Phys. Plasmas 16, 094701 (2009)

    Article  ADS  Google Scholar 

  • Kowari, K.-I., Leung, K., Shizgal, B.D.: The coupling of electron thermalization and electron attachment in CCl4/Ar and CCl4/Ne mixtures. J. Chem. Phys. 108, 1587 (1998)

    Article  ADS  Google Scholar 

  • Lazar, M., Pierrard, V., Shaaban, S.M., Fichtner, H., Poedts, S.: Dual Maxwellian-kappa modeling of the solar wind electrons: new clues on the temperature of kappa populations. Astron. Astrophys. 602, A44 (2017)

    Article  ADS  Google Scholar 

  • Leubner, M.P.: Fundamental issues on kappa-distributions in space plasmas and interplanetary proton distributions. Phys. Plasmas 11, 1308 (2004)

    Article  ADS  Google Scholar 

  • Leubner, M.P., Vörös, Z.: A nonextensive entropy approach to solar wind intermittency. Astrophys. J. 618, 547 (2005)

    Article  ADS  Google Scholar 

  • Livadiotis, G.: Kappa Distributions. Elsevier, Amsterdam (2017)

    Book  MATH  Google Scholar 

  • Livadiotis, G.: Derivation of the entropic formula for the statistical mechanics of space plasmas. Nonlinear Process. Geophys. 25, 77 (2018)

    Article  ADS  Google Scholar 

  • Livadiotis, G., McComas, D.J.: Beyond kappa distributions: exploiting Tsallis statistical mechanics in space plasmas. J. Geophys. Res. 114, A11105 (2009)

    ADS  Google Scholar 

  • Livadiotis, G., McComas, D.J.: Understanding kappa distributions: a toolbox for space science and astrophysics. Space Sci. Rev. 175, 183 (2013)

    Article  ADS  Google Scholar 

  • Maksimovic, M., Pierrard, V., Lemaire, J.F.: A kinetic model of the solar wind with kappa distribution functions in the corona. Astron. Astrophys. 324, 725 (1997)

    ADS  Google Scholar 

  • Maksimovic, M., Zouganelis, I., Chaufray, J.-Y., et al.: Radial evolution of the electron distribution functions in the fast solar wind between 0.3 and 1.5 AU. J. Geophys. Res. 110, A09104 (2005)

    ADS  Google Scholar 

  • Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2006)

    MATH  Google Scholar 

  • Oka, M., Birn, J., Battaglia, M., et al.: Electron power-law spectra in solar and space plasmas. Space Sci. Rev. 214, 82 (2018)

    Article  ADS  Google Scholar 

  • Palmroth, M., Ganse, U., Plau-Kempf, Y., et al.: Vlasov methods in space physics and astrophysics. Living Rev. Comput. Astrophys. 4, 1 (2018)

    Article  ADS  Google Scholar 

  • Patel, H., Shizgal, B.: Pseudospectral solutions of the Fokker-Planck equation for Pearson diffusion that yields a kappa distribution; the associated SUSY Schrodinger equation. Comput. Theor. Chem. 1194, 113059 (2021)

    Article  Google Scholar 

  • Pierrard, V., Lazar, M.: Kappa distributions: theory and applications in space plasmas. Sol. Phys. 267, 153 (2010)

    Article  ADS  Google Scholar 

  • Pierrard, V., Lazar, M., Schlickeiser, R.: Evolution of the electron distribution function in the whistler wave turbulence of the solar wind. Sol. Phys. 269, 421 (2011)

    Article  ADS  Google Scholar 

  • Pitchford, L.C., Phelps, A.V.: Comparative calculations of electron-swarm properties in N2 at moderate E/N values. Phys. Rev. A 25, 540 (1982)

    Article  ADS  Google Scholar 

  • Prigogine, I.: In: Nonequilibrium Statistical Mechanics, Dover (2017)

    MATH  Google Scholar 

  • Rehman, A., Ahmad, M., Shahzad, M.A.: Revisiting some analytical and numerical interpretations of Cairns and kappa Cairns distribution functions. Phys. Plasmas 27, 100901 (2020)

    Article  Google Scholar 

  • Risken, H.: The Fokker-Planck Equation, Methods of Solution and Applications, 2nd edn. Springer, New York (1996)

    MATH  Google Scholar 

  • Schuster, P.: Stochasticity in Processes, Fundamentals and Applications to Chemistry and Biology. Springer, Switzerland (2016)

    Book  MATH  Google Scholar 

  • Shizgal, B.D.: Coulomb collisional processes in space plasma; relaxation of suprathermal particle distributions. Planet. Space Sci. 52, 923 (2004)

    Article  ADS  Google Scholar 

  • Shizgal, B.D.: Suprathermal particle distributions in space physics: kappa distributions and entropy. Astrophys. Space Sci. 312, 227 (2007)

    Article  ADS  MATH  Google Scholar 

  • Shizgal, B.D.: Kappa and other nonequilibrium distributions from the Fokker-Planck equation and the relationship to Tsallis entropy. Phys. Rev. E 97, 052144 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  • Shizgal, B., Arkos, G.: Nonthermal escape of the atmospheres of Venus, Earth, and Mars. Rev. Geophys. 34, 483 (1996)

    Article  ADS  Google Scholar 

  • Shizgal, B., Barrett, J.C.: Time dependent nucleation. J. Chem. Phys. 91, 6505 (1989)

    Article  ADS  Google Scholar 

  • Shizgal, B., Karplus, M.: Nonequilibrium contributions to the rate of reaction. I. Perturbation of the velocity distribution function. J. Chem. Phys. 52, 4262 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  • Shizgal, B.D., Kowari, K.: Electron attachment kinetics coupled to electron thermalization in SF6/Ar mixtures. J. Phys. D, Appl. Phys. 35, 973 (2002)

    Article  ADS  Google Scholar 

  • Stverak, A., Maksimovic, M., Travnicek, P.M., et al.: Radial evolution of nonthermal electron populations in the low-latitude solar wind: helios, cluster, and Ulysses observations. J. Geophys. Res. 114, A05104 (2009)

    ADS  Google Scholar 

  • Summers, D., Thorne, R.M.: The modified plasma dispersion function. Phys. Fluids B 3, 1835 (1991)

    Article  ADS  Google Scholar 

  • Tsallis, C.: Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys. 52, 479 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Tsallis, C.: The nonadditive entropy \({S}_{q}\) and its application in physics and elsewhere: some remarks. Entropy 13, 1765 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam (1992)

    MATH  Google Scholar 

  • Vasyliunas, Y.M.: Low energy electrons on the day side of the magnetosphere. J. Geophys. Res. 73, 7519 (1968)

    Article  ADS  Google Scholar 

  • Viehland, L.A.: Uniform moment theory for charged particle motion in gases. 2. Second approximation. J. Stat. Phys. 163, 175 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Vocks, C., Salem, C., Lin, R.P., Mann, G.: Electron halo and strahl formation in the solar wind by resonant interaction with whistler waves. Astrophys. J. 269, 421 (2011)

    Google Scholar 

  • Zhang, W., Shizgal, B.D.: Fokker-Planck equation for Coulomb relaxation and wave-particle diffusion: spectral solution and the stability of the kappa distribution to Coulomb collisions. Phys. Rev. E 102, 062103 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  • Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford Univ. Press, London (2001)

    MATH  Google Scholar 

Download references

Funding

This research is supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

Author information

Authors and Affiliations

  1. Department of Chemistry, University of British Columbia, Vancouver, British Columbia, Canada

    Bernie D. Shizgal

Authors
  1. Bernie D. Shizgal
    View author publications

    You can also search for this author in PubMed Google Scholar

Ethics declarations

Conflicts of interest/Competing interests

There are no conflicts of interests or competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shizgal, B.D. Energetic particle distributions in space physics represented by the Pearson differential equation for the kappa distribution. Astrophys Space Sci 366, 50 (2021). https://doi.org/10.1007/s10509-021-03956-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10509-021-03956-0

Keywords

Search

Navigation