Advertisement

Mathematical modeling of light curves of RHESSI and AGILE terrestrial gamma-ray flashes

  • Marwan AbukhaledEmail author
  • Nidhal Guessoum
  • Noora Alsaeed
Original Article
  • 57 Downloads

Abstract

We consider different distribution functions as possible fits for the light curves (time profiles) of Terrestrial Gamma-ray Flashes (TGFs): the piecewise Gaussian and the piecewise exponential, which correspond to the assumption of exponential growth and decay of the electrons that emit the radiation, and the inverse Gaussian and Ornstein-Uhlenbeck functions, which correspond to the assumption of an exit-time process. We compute maximum likelihood estimations (MLEs) for each of these four functions for a 100 TGFs recorded by the RHESSI and AGILE satellites, and compared those values with MLEs of the lognormal distribution function, which has been very widely used. The analysis of time profiles of these TGFs shows that all five distributions fit the data equally well. The results obtained in this work, combined with the results obtained for the analysis of BATSE and FERMI TGFs (Abukhaled et al. in J. Geophys. Res. Space Phys. 119:5918–5930, 2014) show that the TGF data as recorded thus far by various satellite gamma-ray detectors, are not powerful enough to discriminate between those various mathematical functions, which, if one function were preferred, would identify some physical features of the phenomenon.

Keywords

Terrestrial Gamma-ray Flash TGF AGILE RHESSI Light curve fitting Maximum likelihood Density functions Stochastic processes Electron cascades Thunderstorms 

Notes

Acknowledgements

The authors acknowledge, with gratitude, the use of the AGILE Terrestrial Gamma-ray Flashes (TGF) catalog available online at http://www.asdc.asi.it/mcaltgfcat/ and the online repository for RHESSI TGF data available at http://scipp.ucsc.edu/_dsmith/tgib public/.

References

  1. Abukhaled, M., Allen, E.J., Guessoum, N.: Testing pulse density distribution for terrestrial gamma-ray flashes. J. Geophys. Res. Space Phys. 119, 5918–5930 (2014).  https://doi.org/10.1002/2014JA020055 ADSCrossRefGoogle Scholar
  2. Babich, L.P., Kutsyk, I.M., Donskoy, E.N., Kudryavtsev, A.Y.: New data on space and time scales of a relativistic runaway electron avalanche for thunderstorms environment: Monte Carlo calculations. Phys. Lett. A 245, 460–470 (1998).  https://doi.org/10.1016/S0375-9601(98)00268-0 ADSCrossRefGoogle Scholar
  3. Briggs, M.S., Fishman, G.J., Connaughton, V., Baht, P.N., Paciesas, W.S., Preece, R.D., Wilson-Hodge, C., Chaplin, V.L., Kippen, R.M., von Kienlin, A., Meegan, C.A., Bissaldi, E., Dwyer, J.R., Smith, D.M., Holzworth, R.H., Grove, J.E., Chekhtman, A.: First results on terrestrial gamma ray flashes from the Fermi Gamma-ray Burst Monitor. J. Geophys. Res. Space Phys. 115, A07323 (2010).  https://doi.org/10.1029/2009JA015242 ADSCrossRefGoogle Scholar
  4. Briggs, M.S., Xiong, S., Connaughton, V., Tierney, D., Fitzpatrick, G., Foley, S., Grove, J.E., Chekhtman, A., Gibby, M., Fishman, G.J., McBreen, S., Chaplin, V.L., Guiriec, S., Layden, E., Bhat, P.N., Hughes, M., Greiner, J., von Kienlin, A., Kippen, R.M., Meegan, C.A., Paciesas, W.S., Preece, R.D., Wilson-Hodge, C., Holzworth, R.H., Hutchins, M.: Terrestrial gamma-ray flashes in the Fermi era: improved observations and analysis methods. J. Geophys. Res. Space Phys. 118, 3805–3830 (2013).  https://doi.org/10.1002/jgra.50205 ADSCrossRefGoogle Scholar
  5. Carlson, B.E., Lehtinen, N.G., Inan, U.S.: Constraints on terrestrial gamma ray flash production from satellite observation. Geophys. Res. Lett. 34, L08809 (2007).  https://doi.org/10.1029/2006GL029229 ADSCrossRefGoogle Scholar
  6. Celestin, S., Pasko, V.P.: Energy and fluxes of thermal runaway electrons produced by exponential growth of streamers during the stepping of lightning leaders and in transient luminous events. J. Geophys. Res. Space Phys. 116, A03315 (2011).  https://doi.org/10.1029/2010JA016260 ADSCrossRefGoogle Scholar
  7. Celestin, S., Xu, W., Pasko, V.P.: Terrestrial gamma ray flashes with energies up to 100 MeV produced by nonequilibrium acceleration of electrons in lightning. J. Geophys. Res. Space Phys. 117, A05315 (2012).  https://doi.org/10.1029/2012JA017535 ADSCrossRefGoogle Scholar
  8. Chronis, T., Briggs, M.S., Priftis, G., Connaughton, V., Brundell, J., Holzworth, R., Heckman, S., McBreen, S., Fitzpatrick, G., Stanbro, M.: Characteristics of thunderstorms that produce terrestrial gamma ray flashes. Bull. Am. Meteorol. Soc. 97, 639–653 (2016).  https://doi.org/10.1175/BAMS-D-14-00239.1 ADSCrossRefGoogle Scholar
  9. Chuang, Y.: On the first passage time distribution of an Ornstein-Uhlenbeck process. Quant. Finance 10, 957–960 (2010).  https://doi.org/10.1080/14697680903373684 MathSciNetCrossRefzbMATHGoogle Scholar
  10. Collier, A.B., Gjesteland, T., Østgaard, N.: Assessing the power law distribution of TGFs. J. Geophys. Res. Space Phys. 116, A10320 (2011).  https://doi.org/10.1029/2011JA016612 ADSCrossRefGoogle Scholar
  11. Cummer, S.A., Briggs, M.S., Dwyer, J.R., Xiong, S., Connaughton, V., Fishman, G.J., Lu, G., Lyu, F., Solanki, R.: The source altitude, electric current, and intrinsic brightness of terrestrial gamma ray flashes. Geophys. Res. Lett. 41, 8586–8593 (2014).  https://doi.org/10.1002/2014GL062196 ADSCrossRefGoogle Scholar
  12. Di Giovanni, A., Al Qasim, A., AlMannaei, A.: A compact gamma spectrometer for space missions with miniaturized satellites. In: 42nd COSPAR Scientific Assembly (2018) Google Scholar
  13. Dwyer, J.R.: The relativistic feedback discharge model of terrestrial gamma ray flashes. J. Geophys. Res. Space Phys. 117, A02308 (2012).  https://doi.org/10.1029/2011JA017160 ADSCrossRefGoogle Scholar
  14. Dwyer, J.R., Smith, D.M.: A comparison between Monte Carlo simulations of runaway breakdown and terrestrial gamma-ray flash observations. Geophys. Res. Lett. 32, 804 (2005).  https://doi.org/10.1029/2005GL023848 CrossRefGoogle Scholar
  15. Dwyer, J.R., Uman, M.A.: The physics of lightning. Phys. Rep. 534, 147–241 (2014).  https://doi.org/10.1016/j.physrep.2013.09.004 ADSMathSciNetCrossRefGoogle Scholar
  16. Dwyer, J.R., Smith, D.M., Cummer, S.A.: High-energy atmospheric physics: terrestrial gamma-ray flashes and related phenomena. Space Sci. Rev. 173, 133–196 (2012).  https://doi.org/10.1007/s11214-012-9894-0 ADSCrossRefGoogle Scholar
  17. Fishman, G.J., Bhat, P.N., Mallozzi, R., Horack, J.M., Koshut, T., Kouveliotou, C., Pendleton, G.N., Meegan, C.A., Wilson, R.B., Paciesas, W.S., Goodman, S.J., Christian, H.J.: Discovery of intense gamma-ray flashes of atmospheric origin. Science 264, 1313–1316 (1994) ADSCrossRefGoogle Scholar
  18. Fitzpatrick, G., et al.: Compton scattering in terrestrial gamma-ray flashes detected with the Fermi gamma-ray burst monitor. Phys. Rev. D 90(4), 043008 (2014) ADSCrossRefGoogle Scholar
  19. Folks, J.L., Chhikara, R.S.: The inverse Gaussian distribution and its statistical application—a review. J. R. Stat. Soc. B 40(3), 263–289 (1978) MathSciNetzbMATHGoogle Scholar
  20. Gjesteland, T.: Properties of terrestrial gamma ray flashes, modelling and analysis of BATSE and RHESSI data. PhD thesis, University of Bergen (2012) Google Scholar
  21. Gjesteland, T., Østgaard, N., Connell, P.H., Stadsnes, J., Fishman, G.J.: Effects of dead time losses on terrestrial gamma ray flash measurements with the Burst and Transient Source Experiment. J. Geophys. Res. Space Phys. 115, A00E21 (2010).  https://doi.org/10.1029/2009JA014578 ADSCrossRefGoogle Scholar
  22. Gjesteland, T., Østgaard, N., Collier, A.B., Carlson, B.E., Eyles, C., Smith, D.M.: A new method reveals more TGFs in the RHESSI data. Geophys. Res. Lett. 39, L05102 (2012).  https://doi.org/10.1029/2012GL050899 ADSCrossRefGoogle Scholar
  23. Gjesteland, T., Østgaard, N., Bitzer, P., Christian, H.J.: On the timing between terrestrial gamma ray flashes, radio atmospherics, and optical lightning emission. J. Geophys. Res. Space Phys. 122, 7734–7741 (2017).  https://doi.org/10.1002/2017JA024285 ADSCrossRefGoogle Scholar
  24. Gurevich, A.V., Milikh, G.M., Roussel-Dupre, R.: Runaway electron mechanism of air breakdown and preconditioning during a thunderstorm. Phys. Lett. A 165, 463–468 (1992) ADSCrossRefGoogle Scholar
  25. Lehtinen, N.G., Walt, M., Inan, U.S., Bell, T.F., Pasko, V.P.: \(\gamma\)-ray emission produced by a relativistic beam of runaway electrons accelerated by quasi-electrostatic thundercloud fields. Geophys. Res. Lett. 23(19), 2645–2648 (1996) ADSCrossRefGoogle Scholar
  26. Lehtinen, N.G., Bell, T.F., Inan, U.S.: Monte Carlo simulation of runaway MeV electron breakdown with application to red sprites and terrestrial gamma ray flashes. J. Geophys. Res. Space Phys. 104(A11), 24699–24712 (1999) ADSCrossRefGoogle Scholar
  27. Lemeshko, B.Y., Lemeshko, S.B., Akushkina, K.A., Nikulin, M.S., Saaidia, N.: Inverse Gaussian model and its applications in reliability and survival analysis. In: Mathematical and Statistical Models and Methods in Reliability Statistics for Industry and Technology, pp. 433–453 (2010).  https://doi.org/10.1007/978-0-8176-4971-5_33 CrossRefzbMATHGoogle Scholar
  28. Marisaldi, M., et al.: Detection of terrestrial gamma ray flashes up to 40 MeV by the AGILE satellite. J. Geophys. Res. Space Phys. 115, A00E13 (2010).  https://doi.org/10.1029/2009JA014502 CrossRefGoogle Scholar
  29. Marisaldi, M., et al.: Enhanced detection of terrestrial gamma-ray flashes by AGILE. Geophys. Res. Lett. 42(21), 9481–9487 (2015) ADSCrossRefGoogle Scholar
  30. Marisaldi, M., Ursi, A., Argan, A., Tavani, M., Labanti, C., Fuschino, F.: One year of AGILE Terrestrial Gamma-ray Flashes detection in the enhanced configuration. In: EGU General Assembly Conference Abstracts, vol. 18, p. 12031 (2016) Google Scholar
  31. McBreen, B., Jurley, K.J., Long, R., Metcalfe, L.: Lognormal distribution in gamma-ray bursts and cosmic lightning. Mon. Not. R. Astron. Soc. 271, 662–666 (1994) ADSCrossRefGoogle Scholar
  32. Østgaard, N., Gjesteland, T., Stadsnes, J., Connell, P.H., Carlson, B.: Production altitude and time delays of the terrestrial gamma flashes: revisiting the Burst and Transient Source Experiment spectra. J. Geophys. Res. Space Phys. 113, A02307 (2008).  https://doi.org/10.1029/2007JA012618 ADSCrossRefGoogle Scholar
  33. Østgaard, N., Albrecthsen, K.H., Gjesteland, T., Collier, A.: A new population of terrestrial gamma-ray flashes in the RHESSI data. Geophys. Res. Lett. 42, 10,937–10,942 (2015).  https://doi.org/10.1002/2015GL067064 CrossRefGoogle Scholar
  34. Rutjes, C., Diniz, G., Ferreira, I.S., Ebert, U.: TGF afterglows: a new radiation mechanism from thunderstorms. Geophys. Res. Lett. 44, 10702–10712 (2017).  https://doi.org/10.1002/2017GL075552 ADSCrossRefGoogle Scholar
  35. Smith, D.M., Lopez, L.I., Lin, R.P., Barrington-Leigh, C.P.: Terrestrial gamma-ray flashes observed up to 20 MeV. Science 307, 1085–1088 (2005) ADSCrossRefGoogle Scholar
  36. Tierney, D., et al.: Fluence distribution of terrestrial gamma ray flashes observed by the Fermi Gamma-Ray Burst Monitor. J. Geophys. Res. Space Phys. 118, 6644 (2013).  https://doi.org/10.1002/jgra.50580 ADSCrossRefGoogle Scholar
  37. Törnqvist, L., Vartia, P., Vartia, Y.O.: How should relative changes be measured? Am. Stat. 39(1), 43–46 (1985) Google Scholar
  38. Ursi, A., Guidorzi, C., Marisaldi, M., Sarria, D., Frontera, F.: Terrestrial gamma-ray flashes in the BeppoSAX data archive. J. Atmos. Sol.-Terr. Phys. 156, 50–56 (2017).  https://doi.org/10.1016/j.jastp.2017.02.014 ADSCrossRefGoogle Scholar
  39. Wilson, C.T.R.: The electric field of a thundercloud and some of its effects. Proc. Phys. Soc. Lond. Ser. D 37, 32 (1925) CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Marwan Abukhaled
    • 1
    Email author
  • Nidhal Guessoum
    • 2
  • Noora Alsaeed
    • 3
  1. 1.Department of Mathematics and StatisticsAmerican University of SharjahSharjahUAE
  2. 2.Department of PhysicsAmerican University of SharjahSharjahUAE
  3. 3.Department of Astrophysical and Planetary SciencesUniversity of Colorado BoulderBoulderUSA

Personalised recommendations