Solar flares, the most powerful magnetic explosions in our solar system, are often accompanied by intense radio emission explosions known as solar radio explosions type III. These bursts provide valuable clues about the acceleration and transport of energy electrons in the atmosphere of the sun and beyond. Understanding these explosions is not only crucial for solar physics, but also to predict spatial weather events that can interrupt our technological infrastructure. A key question that surrounds the bursts of type III has been the beams of energy electrons, which are believed to be responsible for the radio emission, they travel huge distances from the sun (for example, Ginzburg and Zhelezniakov, 1958). The relaxation of the electron beam in the short distance in contrast with large travel distances observed is often known as Sturrock dilemma (Sturrock, 1964).
A new research work of Kontart et al 2024 It incorporates a crucial element often overlooked in previous studies: the finite spatial size of the electron cloud and the consequent spatial variability of the quasylinean relaxation process. The standard approach assumes that this quasylinean relaxation is uniformly rapid throughout the space (for example,. Ryutov and Sagdeev, 1970, Melnik, vn et al, 1999). However, this assumption is too simplistic. Kontart et al 2024 Point out that the quasylinean relaxation rate depends on the density of the electron beam itself. Since the density of the beam varies naturally through its length, the relaxation process will also vary.
Non -linear diffusion
This spatial variation in relaxation leads to a new concept in the model: non -linear diffusion. The diffusion, in general, describes the spread of particles from a region of greater concentration to a region of lower concentration. In the context of the propagation electrons, it refers to the spatial expansion of the lightning as it travels outside the sun. The “non -linear” aspect comes from the fact that the diffusion coefficient, which quantifies the speed of this propagation, is not constant, but depends on the local density of the electron beam. Specifically, diffusion is faster in the regions where the density of the beam is lower and slower in the regions where the density is greater. This is because a lower electron density leads to a slower quasylinear relaxation and, therefore, a faster diffusion, while a higher electron density leads to faster relaxation and a slower diffusion. You can write the non -linal dissemination equation with $ N (x, t) $ standard with $ N_B $
$$ \ fraud {\ partial n} {\ partial t}+\ fraud x} d_ {xx}^0 \ fraud {n_ {b}} {n} \ fraid {\ partial n} {\ partial x} = 0 \ ,, $$
where the non -linear dependence of $ d_ {xx} $ at $ n (x, t) $ is explicitly highlighted when introducing $ d_ {xx} = d_ {xx}^0 \ fraud {n_ {b}}} {n} $.
Figure 1: AI-generated imagemy inspired by him Non-linear-infusion monting model.
Super-Diffusion: Ballistic expansion of the electron beam
The solution to this equation reveals that the electron beam suffers a super diffusion. The evolution of an electron beam given by the initial condition
$$ n (x, t = 0) = n_b \ delta \ left (x/d \ right) \ ,, $$
Where $ n_b $ is the density of the electron beam and $ d $ is the characteristic size, you can write
$$ n (x, t) = \ left (\ fraud N__ {b} t} +\ fraud {2 \ pi^2} {n_bd^2} d^0_ {xx} t \ right)^{-1} $$
This means that the beam expands much faster than would be under standard diffusion. Instead of the width of the beam, it increases with the square root of time (as in normal diffusion), it increases linearly over time, a behavior known as ballistic expansion.
This super diffusion is a direct consequence of the non -linear nature of the diffusion. The fastest diffusion in the regions of lower density of the beam causes it to extend more quickly than expected. The electron density and the spectral energy density of Langmuu waves tend to decrease with the distance of the sun. To validate the model, we compare the analytical solution with the results of the numerical simulations of the complete kinetic equations, carried out in previous studies (Figure 2).
Figure 2: Distribution of simulated electrons ($ f (v, x, t) $) (left), spectral energy density ($ w (v, x, t) $) (center) and electron beam density ($ n (x (x , t) $) (correct) at the three moments of time $ t = 0.5, 3, $ 6 s for the following ray-plasma parameters $ n_b = 120 $ cm $^{-3} $, $ n_p = 6 \ times 10^8 $ cm $^{-3} $ (that is, $ f_ {pe} \ Simeq 220 $ mHz) and $ v_0 = 10^{10} $ cm/s, $ V _ {\ min} = 0.1v_0 $, $ d = 3 \ Times10^$ $ cm. The discontinuous line shows the decrease in maximum density.
Conclusions
The solution allows to investigate the large -scale evolution of electron beam waves and langmuir and quantifiably explains the results of the numerical simulation. In the application to the bursting of solar radius type III, the spectral energy density of the emission of plasma through the Langmuir waves depends on the density of the beam and would decrease inversely with the distance, which is required to explain the variations of Radial Radial Rapaga Flow Type III (((Krupar et al. 2014). The spatial expansion of the beam is also qualitatively better for the time width of the bursts of type III (Reid and Kontar 2018).
Based on the recent article by Eduard P. Kontar, Francesco Azzollini and Olena Lyubchyk, The Astrophysical Journal, 976 233 (2024). DOI: 10.3847/1538-4357/AD8560
References:
Ginzburg, VL and Zhelezniakov, VV 1958, SVA, 2, 653
Kontart et al, 2024, APJ, 976, 233
Krupar et al, 2014, Solar Physics, 289, 3121
Melnik, vn et al, 1999, Solar Physics, 184, 353
Reid and Kontar, 2018, Astronomy & Astrophysics, 614, A69
Ryutov and Sagdeev, 1970 Jetp, 31,396
Sturrock, pa, 1964, Proceedings of the AAS-NASA Symposium, P357
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