Solar Physics

, 294:29 | Cite as

Growth Rates of the Upper-Hybrid Waves for Power-Law and Kappa Distributions with a Loss-Cone Anisotropy

  • L. V. Yasnov
  • J. BenáčekEmail author
  • M. Karlický


Fine structures of radio bursts play an important role in the diagnostics of the solar flare plasma. Among them the zebras, which are prevalently assumed to be generated by the double-plasma resonance instability, belong to the most important ones. In this paper we compute the growth rate of this instability for two types of the electron distribution: a) for the power-law distribution and b) for the kappa distribution, in both cases with the loss-cone type anisotropy. We find that the growth rate of the upper-hybrid waves for the power-law momentum distribution strongly depends on the pitch-angle boundary. The maximum growth rate is found for the pitch angle \(\theta_{\mathrm{c}} \approx 50^{\circ}\). For small angles the growth rate profile is very flat and for high pitch angles the wave absorption occurs. Furthermore, analyzing the growth rate of the upper-hybrid waves for the kappa momentum distribution we find that a decrease of the characteristic momentum \(p_{\kappa}\) shifts the maximum of the growth rate to lower values of the ratio of the electron-plasma and electron-cyclotron frequencies, and the frequency widths of the growth rate peaks are very broad. But if we consider the kappa distribution which is isotropic up to some large momentum \(p_{\mathrm{m}}\) and anisotropic with loss-cone above this momentum then distinct peaks of the growth rate appear and thus distinct zebra stripes can be generated. It means that the restriction of small momenta for the anisotropic part of distributions is of principal importance for the zebra stripe generation. Finally, for the zebra stripes observed on 1 August 2010, the growth rates in dependence on the radio frequency are computed. It is shown that in this case the growth rate peaks are more distinct than in usually presented dependencies of growth rates on the ratio of the plasma and cyclotron frequencies.


Sun: corona Sun: flares Sun: radio radiation 



We thank an anonymous referee for valuable comments. M. Karlický acknowledges support from Grants 17-16447S and 18-09072S of the Grant Agency of the Czech Republic. L.V. Yasnov acknowledge support from Grant 18-29-21016-mk and partly from Grant 18-02-00045 of the Russian Foundation for Basic Research. This work was supported by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project IT4Innovations National Supercomputing Center LM2015070.

Disclosure of Potential Conflicts of Interest

No potential conflict of interest was reported by the authors.


  1. Bárta, M., Karlický, M.: 2006, Interference patterns in solar radio spectra: high-resolution structural analysis of the corona. Astron. Astrophys. 450, 359. DOI. ADS. ADSCrossRefGoogle Scholar
  2. Benáček, J., Karlický, M.: 2018, Double plasma resonance instability as a source of solar zebra emission. Astron. Astrophys. 611, A60. DOI. ADS. ADSCrossRefGoogle Scholar
  3. Benáček, J., Karlický, M., Yasnov, L.: 2017, Temperature dependent growth rates of the upper-hybrid waves and solar radio zebra patterns. Astron. Astrophys. 598, A106. DOI. ADS. ADSCrossRefGoogle Scholar
  4. Bian, N.H., Emslie, A.G., Stackhouse, D.J., Kontar, E.P.: 2014, The formation of Kappa-distribution accelerated electron populations in solar flares. Astrophys. J. 796, 142. DOI. ADS. ADSCrossRefGoogle Scholar
  5. Chernov, G.P.: 1976, Microstructure in the continuous radiation of type IV meter bursts. Observations and model of the source. Soviet Astron. 20, 449. ADS. ADSGoogle Scholar
  6. Chernov, G.P.: 1990, Whistlers in the solar corona and their relevance to fine structures of type IV radio emission. Solar Phys. 130, 75. DOI. ADS. ADSCrossRefGoogle Scholar
  7. Chernov, G.P.: 2010, Recent results of zebra patterns in solar radio bursts. Res. Astron. Astrophys. 10, 821. DOI. ADS. ADSCrossRefGoogle Scholar
  8. Chernov, G.P., Yan, Y.-H., Fu, Q.-J.: 2014, The importance of source positions during radio fine structure observations. Res. Astron. Astrophys. 14, 831. DOI. ADS. ADSCrossRefGoogle Scholar
  9. Chernov, G.P., Sych, R.A., Meshalkina, N.S., Yan, Y., Tan, C.: 2012, Spectral and spatial observations of microwave spikes and zebra structure in the short radio burst of May 29, 2003. Astron. Astrophys. 538, A53. DOI. ADS. ADSCrossRefGoogle Scholar
  10. Dory, R.A., Guest, G.E., Harris, E.G.: 1965, Unstable electrostatic plasma waves propagating perpendicular to a magnetic field. Phys. Rev. Lett. 14, 131. DOI. ADS. ADSCrossRefGoogle Scholar
  11. Holman, G.D., Sui, L., Schwartz, R.A., Emslie, A.G.: 2003, Electron bremsstrahlung hard X-ray spectra, electron distributions, and energetics in the 2002 July 23 solar flare. Astrophys. J. Lett. 595, L97. DOI. ADS. ADSCrossRefGoogle Scholar
  12. Karlický, M.: 2013, Radio continua modulated by waves: zebra patterns in solar and pulsar radio spectra? Astron. Astrophys. 552, A90. DOI. ADS. ADSCrossRefGoogle Scholar
  13. Kašparová, J., Karlický, M.: 2009, Kappa distribution and hard X-ray emission of solar flares. Astron. Astrophys. 497, L13. DOI. ADS. ADSCrossRefzbMATHGoogle Scholar
  14. Kontar, E.P., Dickson, E., Kašparová, J.: 2008, Low-energy cutoffs in electron spectra of solar flares: statistical survey. Solar Phys. 252, 139. DOI. ADS. ADSCrossRefGoogle Scholar
  15. Kuijpers, J.: 1974, A coherent radiation mechanism for type IV DM radio bursts. Solar Phys. 36, 157. DOI. ADS. ADSCrossRefGoogle Scholar
  16. Kuijpers, J.: 1975, A unified explanation of solar type IV DM continua and ZEBRA patterns. Astron. Astrophys. 40, 405. ADS. ADSGoogle Scholar
  17. Kuznetsov, A.A., Tsap, Y.T.: 2007, Loss-cone instability and formation of zebra patterns in type IV solar radio bursts. Solar Phys. 241, 127. DOI. ADS. ADSCrossRefGoogle Scholar
  18. LaBelle, J., Treumann, R.A., Yoon, P.H., Karlický, M.: 2003, A model of zebra emission in solar type IV radio bursts. Astrophys. J. 593, 1195. DOI. ADS. ADSCrossRefGoogle Scholar
  19. Laptukhov, A.I., Chernov, G.P.: 2009, Concerning mechanisms for the zebra pattern formation in the solar radio emission. Plasma Phys. Rep. 35, 160. DOI. ADS. ADSCrossRefGoogle Scholar
  20. Ledenev, V.G., Yan, Y., Fu, Q.: 2006, Interference mechanism of “zebra-pattern” formation in solar radio emission. Solar Phys. 233, 129. DOI. ADS. ADSCrossRefGoogle Scholar
  21. Oka, M., Ishikawa, S., Saint-Hilaire, P., Krucker, S., Lin, R.P.: 2013, Kappa distribution model for hard X-ray coronal sources of solar flares. Astrophys. J. 764, 6. DOI. ADS. ADSCrossRefGoogle Scholar
  22. Oka, M., Krucker, S., Hudson, H.S., Saint-Hilaire, P.: 2015, Electron energy partition in the above-the-looptop solar hard X-ray sources. Astrophys. J. 799, 129. DOI. ADS. ADSCrossRefGoogle Scholar
  23. Rosenberg, H., Tarnstrom, G.: 1972, Frequency separation in structure of solar continuum radio bursts. Solar Phys. 24, 210. DOI. ADS. ADSCrossRefGoogle Scholar
  24. Saint-Hilaire, P., Benz, A.O.: 2005, Thermal and non-thermal energies of solar flares. Astron. Astrophys. 435, 743. DOI. ADS. ADSCrossRefGoogle Scholar
  25. Slottje, C.: 1972, Peculiar absorption and emission microstructures in the type IV solar radio outburst of March 2, 1970. Solar Phys. 25, 210. DOI. ADS. ADSCrossRefGoogle Scholar
  26. Stepanov, A.V.: 1974, A mechanism for generating type IV solar radio bursts. Soviet Astron. 17, 781. ADS. ADSGoogle Scholar
  27. Tan, B.: 2010, A physical explanation of solar microwave zebra pattern with the current-carrying plasma loop model. Astrophys. Space Sci. 325, 251. DOI. ADS. ADSCrossRefGoogle Scholar
  28. Tan, B., Yan, Y., Tan, C., Sych, R., Gao, G.: 2012, Microwave zebra pattern structures in the X2.2 solar flare on 2011 February 15. Astrophys. J. 744, 166. DOI. ADS. ADSCrossRefGoogle Scholar
  29. Tan, B., Tan, C., Zhang, Y., Mészárosová, H., Karlický, M.: 2014, Statistics and classification of the microwave zebra patterns associated with solar flares. Astrophys. J. 780, 129. DOI. ADS. ADSCrossRefGoogle Scholar
  30. White, S.M., Melrose, D.B., Dulk, G.A.: 1983, Electron cyclotron masers during solar flares. Proc. Astron. Soc. Aust. 5, 188. ADS. ADSCrossRefGoogle Scholar
  31. Winglee, R.M., Dulk, G.A.: 1986, The electron-cyclotron maser instability as a source of plasma radiation. Astrophys. J. 307, 808. DOI. ADS. ADSCrossRefGoogle Scholar
  32. Yasnov, L.V., Benáček, J., Karlický, M.: 2017, Brightness temperature of radio zebras and wave energy densities in their sources. Solar Phys. 292, 163. DOI. ADS. ADSCrossRefGoogle Scholar
  33. Yasnov, L.V., Karlický, M.: 2004, The growth rate of upper-hybrid waves and dynamics of microwave zebra structures. Solar Phys. 219, 289. DOI. ADS. ADSCrossRefGoogle Scholar
  34. Yasnov, L.V., Karlický, M., Stupishin, A.G.: 2016, Physical conditions in the source region of a zebra structure. Solar Phys. 291, 2037. DOI. ADS. ADSCrossRefGoogle Scholar
  35. Zheleznyakov, V.V., Zlotnik, E.Y.: 1975, Cyclotron wave instability in the corona and origin of solar radio emission with fine structure. III. Origin of zebra-pattern. Solar Phys. 44, 461. DOI. ADS. ADSCrossRefGoogle Scholar
  36. Zheleznyakov, V.V., Zlotnik, E.Y., Zaitsev, V.V., Shaposhnikov, V.E.: 2016, Double plasma resonance and its manifestations in radio astronomy. Phys. Usp. 59, 997. DOI. ADS. ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.St.-Petersburg State UniversitySt.-PetersburgRussia
  2. 2.St.-Petersburg branch of Special Astrophysical ObservatorySt.-PetersburgRussia
  3. 3.Department of Theoretical Physics and AstrophysicsMasaryk UniversityBrnoCzech Republic
  4. 4.Astronomical InstituteAcademy of Sciences of the Czech RepublicOndřejovCzech Republic

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