On the solution of the continuity equation for precipitating electrons in solar flares
Abstract
Electrons accelerated in solar flares are injected into the surrounding plasma, where they are subjected to the influence of collisional (Coulomb) energy losses. Their evolution is modeled by a partial differential equation describing continuity of electron number. In a recent paper, Dobranskis and Zharkova claim to have found an 'updated exact analytical solution' to this continuity equation. Their solution contains an additional term that drives an exponential decrease in electron density with depth, leading them to assert that the wellknown solution derived by Brown, Syrovatskii and Shmeleva, and many others is invalid. We show that the solution of Dobranskis and Zharkova results from a fundamental error in the application of the method of characteristics and is hence incorrect. Further, their comparison of the 'new' analytical solution with numerical solutions of the FokkerPlanck equation fails to lend support to their result. We conclude that Dobranskis and Zharkova's solution of the universally accepted and wellestablished continuity equation is incorrect, and that their criticism of the correct solution is unfounded. We also demonstrate the formal equivalence of the approaches of Syrovatskii and Shmeleva and Brown, with particular reference to the evolution of the electron flux and number density (both differential in energy) inmore »
 Authors:
 Department of Physics and Astronomy, Western Kentucky University, Bowling Green, KY 42101 (United States)
 Code 671, NASA Goddard Space Flight Center, Greenbelt, MD 20771 (United States)
 Department of Mathematics, University of Waikato, P. B. 3105, Hamilton (New Zealand)
 Publication Date:
 OSTI Identifier:
 22365258
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Astrophysical Journal; Journal Volume: 792; Journal Issue: 1; Other Information: Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ANALYTICAL SOLUTION; CONTINUITY EQUATIONS; COULOMB ENERGY; DENSITY; ELECTRON DENSITY; ELECTRONS; EMISSION; FOKKERPLANCK EQUATION; GAMMA RADIATION; LOSSES; NUMERICAL SOLUTION; PLASMA; PRECIPITATION; SOLAR FLARES; STAR EVOLUTION; SUN; X RADIATION
Citation Formats
Emslie, A. Gordon, Holman, Gordon D., and Litvinenko, Yuri E., Email: emslieg@wku.edu, Email: gordon.d.holman@nasa.gov. On the solution of the continuity equation for precipitating electrons in solar flares. United States: N. p., 2014.
Web. doi:10.1088/0004637X/792/1/5.
Emslie, A. Gordon, Holman, Gordon D., & Litvinenko, Yuri E., Email: emslieg@wku.edu, Email: gordon.d.holman@nasa.gov. On the solution of the continuity equation for precipitating electrons in solar flares. United States. doi:10.1088/0004637X/792/1/5.
Emslie, A. Gordon, Holman, Gordon D., and Litvinenko, Yuri E., Email: emslieg@wku.edu, Email: gordon.d.holman@nasa.gov. Mon .
"On the solution of the continuity equation for precipitating electrons in solar flares". United States.
doi:10.1088/0004637X/792/1/5.
@article{osti_22365258,
title = {On the solution of the continuity equation for precipitating electrons in solar flares},
author = {Emslie, A. Gordon and Holman, Gordon D. and Litvinenko, Yuri E., Email: emslieg@wku.edu, Email: gordon.d.holman@nasa.gov},
abstractNote = {Electrons accelerated in solar flares are injected into the surrounding plasma, where they are subjected to the influence of collisional (Coulomb) energy losses. Their evolution is modeled by a partial differential equation describing continuity of electron number. In a recent paper, Dobranskis and Zharkova claim to have found an 'updated exact analytical solution' to this continuity equation. Their solution contains an additional term that drives an exponential decrease in electron density with depth, leading them to assert that the wellknown solution derived by Brown, Syrovatskii and Shmeleva, and many others is invalid. We show that the solution of Dobranskis and Zharkova results from a fundamental error in the application of the method of characteristics and is hence incorrect. Further, their comparison of the 'new' analytical solution with numerical solutions of the FokkerPlanck equation fails to lend support to their result. We conclude that Dobranskis and Zharkova's solution of the universally accepted and wellestablished continuity equation is incorrect, and that their criticism of the correct solution is unfounded. We also demonstrate the formal equivalence of the approaches of Syrovatskii and Shmeleva and Brown, with particular reference to the evolution of the electron flux and number density (both differential in energy) in a collisional thick target. We strongly urge use of these longestablished, correct solutions in future works.},
doi = {10.1088/0004637X/792/1/5},
journal = {Astrophysical Journal},
number = 1,
volume = 792,
place = {United States},
year = {Mon Sep 01 00:00:00 EDT 2014},
month = {Mon Sep 01 00:00:00 EDT 2014}
}

The original continuity equation (CE) used for the interpretation of the power law energy spectra of beam electrons in flares was written and solved for an electron beam flux while ignoring an additional free term with an electron density. In order to remedy this omission, the original CE for electron flux, considering beam's energy losses in Coulomb collisions, was first differentiated by the two independent variables: depth and energy leading to partial differential equation for an electron beam density instead of flux with the additional free term. The analytical solution of this partial differential continuity equation (PDCE) is obtained bymore »

Continuity and differentiability properties of the solution of the linear transport equation
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