Abstract
The propagation of energetic particles in the solar wind depends in a sensitive way on the pitch-angle scattering of particles in the presence of magnetic turbulence. The well-known quasi-linear theory gives an expression for the pitch-angle scattering rate under the assumption of small turbulence levels, but both in the solar wind and in other astrophysical environments the turbulent magnetic field fluctuations can be large. Therefore, a reliable assessment of the pitch-angle scattering requires an investigation that goes beyond the quasi-linear theory. To this end, we employ a recently developed model of synthetic magnetic turbulence, which allows reproduction of a very long spectrum, while varying the turbulence level and the turbulence intermittency. Test particles representing protons with energies in the range \(70~\mbox{keV}\,\mbox{--}\,1~\mbox{MeV}\) are injected in the turbulence spectrum plus a background magnetic field, and the pitch-angle scattering rate is determined by following the individual particles. Using turbulence and intermittency levels comparable to those observed in the solar wind, we find a broad power-law distribution of pitch-angle scattering times, which encompasses the quasi-linear value but extends to values both much larger and much smaller. We find that the distribution of pitch-angle scattering times also depends on the intermittency level. This finding shows that a description of parallel transport based on a single value of the pitch-angle scattering time is not sufficient. These numerical results are compared with observations of the distribution of magnetic variances at the particle resonant scale, measured in the solar wind by the Ulysses spacecraft.
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12 June 2019
Correction to: Solar Phys (2019) 294:34 https://doi.org/10.1007/s11207-019-1421-y
This article was published with an omission in the Acknowledgement text. Please find in this document the correct Acknowledgement text that should be regarded as the final version by the reader.
Acknowledgement The work by S. Perri has been supported by the Agenzia Spaziale Italiana under the contract ASI-INAF 2015-039-R.O “Missione M4 di ESA: Partecipazione Italiana alla fase di assessment della missione THOR”. The work by F. Pucci has been supported by Fonds Wetenshappelijk Onderzoek—Vlaanderen (FWO) through the postdoctoral fellowship 12X0319N.
References
Amato, E.: 2014, The origen of galactic cosmic rays. Int. J. Mod. Phys. D 23, 1430013. DOI .
Beresnyak, A.: 2013, Asymmetric diffusion of magnetic field lines. Astrophys. J. Lett. 767, L39. DOI .
Bieber, J.W., Smith, C.W., Matthaeus, W.H.: 1988, Cosmic-ray pitch-angle scattering in isotropic turbulence. Astrophys. J. 334, 470. DOI .
Boris, J.P., Shanny, R.A.: 1972, Proceedings: Fourth Conference on Numerical Simulation of Plasmas, November 2, 3, 1970, Naval Research Laboratory.
Carbone, V.: 1994, Scaling exponents of the velocity structure functions in the interplanetary medium. Ann. Geophys. 12, 585. DOI .
Crooker, N.U., Gosling, J.T., Bothmer, V., Forsyth, R.J., Gazis, P.R., Hewish, A., Horbury, T.S., Intriligator, D.S., Jokipii, J.R., Kóta, J., Lazarus, A.J., Lee, M.A., Lucek, E., Marsch, E., Posner, A., Richardson, I.G., Roelof, E.C., Schmidt, J.M., Siscoe, G.L., Tsurutani, B.T., Wimmer-Schweingruber, R.F.: 1999, CIR morphology, turbulence, discontinuities, and energetic particles. Space Sci. Rev. 89, 179. DOI .
Dasso, S., Milano, L.J., Matthaeus, W.H., Smith, C.W.: 2005, Anisotropy in fast and slow solar wind fluctuations. Astrophys. J. Lett. 635, L181. DOI .
Giacalone, J.: 2012, Energetic charged particles associated with strong interplanetary shocks. Astrophys. J. 761, 28. DOI .
Giacalone, J., Jokipii, J.R.: 1999, The transport of cosmic rays across a turbulent magnetic field. Astrophys. J. 520, 204. DOI .
Horbury, T.S., Forman, M., Oughton, S.: 2008, Anisotropic scaling of magnetohydrodynamic turbulence. Phys. Rev. Lett. 101(17), 175005. DOI .
Horbury, T.S., Balogh, A., Forsyth, R.J., Smith, E.J.: 1996, The rate of turbulent evolution over the Sun’s poles. Astron. Astrophys. 316, 333.
Horbury, T.S., Balogh, A., Forsyth, R.J., Smith, E.J.: 1997, ULYSSES observations of intermittent heliospheric turbulence. Adv. Space Res. 19, 847. DOI .
Kennel, C.F., Petschek, H.E.: 1966, Limit on stably trapped particle fluxes. J. Geophys. Res. 71, 1. DOI .
Kirk, J.G., Duffy, P., Gallant, Y.A.: 1996, Stochastic particle acceleration at shocks in the presence of braided magnetic fields. Astron. Astrophys. 314, 1010.
Klafter, J., Blumen, A., Shlesinger, M.F.: 1987, Stochastic pathway to anomalous diffusion. Phys. Rev. A 35, 3081. DOI .
Laitinen, T., Kopp, A., Effenberger, F., Dalla, S., Marsh, M.S.: 2016, Solar energetic particle access to distant longitudes through turbulent field-line meandering. Astron. Astrophys. 591, A18. DOI .
Lazarian, A., Yan, H.: 2014, Superdiffusion of cosmic rays: Implications for cosmic ray acceleration. Astrophys. J. 784, 38.
Malara, F., Di Mare, F., Nigro, G., Sorriso-Valvo, L.: 2016, Fast algorithm for a three-dimensional synthetic model of intermittent turbulence. Phys. Rev. E 94(5), 053109. DOI .
Palmer, I.D.: 1982, Transport coefficients of low-energy cosmic rays in interplanetary space. Rev. Geophys. Space Phys. 20, 335. DOI .
Perri, S.: 2018, Superdiffusion of relativistic electrons at supernova remnant shocks. Plasma Phys. Control. Fusion 60(1), 014005. DOI .
Perri, S., Zimbardo, G.: 2007, Evidence of superdiffusive transport of electrons accelerated at interplanetary shocks. Astrophys. J. Lett. 671, L177. DOI .
Perri, S., Zimbardo, G.: 2008, Superdiffusive transport of electrons accelerated at corotating interaction regions. J. Geophys. Res. 113, A03107. DOI .
Perri, S., Zimbardo, G.: 2009, Ion superdiffusion at the solar wind termination shock. Astrophys. J. Lett. 693, L118. DOI .
Perri, S., Zimbardo, G.: 2012a, Magnetic variances and pitch-angle scattering times upstream of interplanetary shocks. Astrophys. J. 754, 8. DOI .
Perri, S., Zimbardo, G.: 2012b, Superdiffusive shock acceleration. Astrophys. J. 750, 87. DOI .
Perri, S., Zimbardo, G.: 2015, Short acceleration times from superdiffusive shock acceleration in the heliosphere. Astrophys. J. 815, 75. DOI .
Perri, S., Amato, E., Zimbardo, G.: 2016, Transport of relativistic electrons at shocks in shell-type supernova remnants: diffusive and superdiffusive regimes. Astron. Astrophys. 596, A34. DOI .
Perri, S., Zimbardo, G., Effenberger, F., Fichtner, H.: 2015, Parameter estimation of superdiffusive motion of energetic particles upstream of heliospheric shocks. Astron. Astrophys. 578, A2. DOI .
Pucci, F., Malara, F., Perri, S., Zimbardo, G., Sorriso-Valvo, L., Valentini, F.: 2016, Energetic particle transport in the presence of magnetic turbulence: influence of spectral extension and intermittency. Mon. Not. Roy. Astron. Soc. 459, 3395. DOI .
Reames, D.V.: 1999, Particle acceleration at the Sun and in the heliosphere. Space Sci. Rev. 90, 413. DOI .
Saul, L., Möbius, E., Isenberg, P., Bochsler, P.: 2007, On pitch-angle scattering rates of interstellar pickup ions as determined by in situ measurement of velocity distributions. Astrophys. J. 655, 672. DOI .
Sorriso-Valvo, L., Carbone, F., Perri, S., Greco, A., Marino, R., Bruno, R.: 2018, On the statistical properties of turbulent energy transfer rate in the inner heliosphere. Solar Phys. 293, 10. DOI .
Sugiyama, T., Shiota, D.: 2011, Sign for super-diffusive transport of energetic ions associated with a coronal-mass-ejection-driven interplanetary shock. Astrophys. J. Lett. 731, L34. DOI .
Wang, Y., Qin, G., Zhang, M., Dalla, S.: 2014, A numerical simulation of solar energetic particle dropouts during impulsive events. Astrophys. J. 789, 157. DOI .
Xu, S., Yan, H.: 2013, Cosmic-ray parallel and perpendicular transport in turbulent magnetic fields. Astrophys. J. 779, 140. DOI .
Yordanova, E., Balogh, A., Noullez, A., von Steiger, R.: 2009, Turbulence and intermittency in the heliospheric magnetic field in fast and slow solar wind. J. Geophys. Res. 114, A08101. DOI .
Zimbardo, G., Perri, S.: 2013, From Lévy walks to superdiffusive shock acceleration. Astrophys. J. 778, 35. DOI .
Zimbardo, G., Perri, S.: 2017, Superdiffusive shock acceleration at galaxy cluster shocks. Nat. Astron. 1, 0163. DOI .
Zimbardo, G., Perri, S.: 2018, Understanding the radio spectral indices of galaxy cluster relics by superdiffusive shock acceleration. Mon. Not. Roy. Astron. Soc. 478, 4922. DOI .
Acknowledgement
The work by S. Perri has been supported by the Agenzia Spaziale Italiana under the contract ASI-INAF 2015-039-R.O “Missione M4 di ESA: Partecipazione Italiana alla fase di assessment della missione THOR”.
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Solar Wind at the Dawn of the Parker Solar Probe and Solar Orbiter Era
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Perri, S., Pucci, F., Malara, F. et al. On the Power-Law Distribution of Pitch-Angle Scattering Times in Solar Wind Turbulence. Sol Phys 294, 34 (2019). https://doi.org/10.1007/s11207-019-1421-y
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DOI: https://doi.org/10.1007/s11207-019-1421-y