- Email: [email protected]
Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona
Available online at www.sciencedirect.com
ScienceDirect Advances in Space Research xxx (2019) xxx–xxx www.elsevier.com/locate/asr
Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona G.A. Casillas-Pe´rez a,⇑, S. Jeyakumar b, A. Carrillo-Vargas c, H.R. Pe´rez-Enrı´quez d a
Instituto de Geofı´sica, Universidad Nacional Auto´noma de Me´xico, Ciudad Universitaria, C.P. 04510 Ciudad de Me´xico, Mexico b Departamento de Astronomı´a, Universidad de Guanajuato, Me´xico, Guanajuato, Gto. C.P. 36000, Mexico c Instituto de Geofı´sica, Universidad Nacional Auto´noma de Me´xico, Unidad Morelia, Antigua Carretera a Pa´tzcuaro No. 8701, Ex-Hacienda de San Jose´ de la Huerta, C.P. 58190 Morelia Michoaca´n, Mexico d Centro de Geociencias, Universidad Nacional Auto´noma de Me´xico, Campus Juriquilla, C.P. 76230 Juriquilla, Quere´taro, Mexico Received 23 July 2019; received in revised form 25 October 2019; accepted 26 October 2019
Abstract Electrons accelerated in the corona during solar activity give rise to radio emission events that can be observed over a wide range of frequencies. Among different finer-scale structures in the dynamic spectra observed in the radio range, fast transients with extents of some milliseconds known as solar radio spikes are observed accompaning the background continuum emission. Fundamental to the generation of radio spikes is a propagating electron beam and following its evolution allows us to understand the physical processes occurring in the solar corona. With the use of a numerical Fokker–Planck code we follow a previous numerical study to simulate the propagation of an electron beam pulse injected in a small region at the top of a magnetic field and outwards the solar corona under typical flare conditions. It was found that in large ambient densities of 1010 cm3 at the injection point, Coulomb collision effects have an important effect on the propagation of the electrons, causing that the injected electrons thermalize faster in a time of 0:1 and 0:4 s for an electron distribution with a low-energy cut off of 16 and 7 keV respectively and a spectral index of 3. For a tenous ambient medium of density 109 cm3 thermalization occurs only for an electron distribution with smaller low-energy cut off (7 keV) with a duration of 1.5 s, while for a larger low-energy cut off (16 keV) the loss of accelerated electrons is very slow, regardles of the spectral index (3; 7). The electron loss time by Coulomb collisions, which depends on the low boundary ambient density, might be an important parameter that influences the generation of radio spikes due to the formation of instabilities in the corona. Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Solar corona; Solar radio spikes; Fokker–Planck equation; Particle acceleration; Electron beam pulse
1. Introduction Accelerated particles moving through the solar corona have been found to produce instabilities as a result of their interaction with the environment. Particularly, the instabilities produced by a beam of accelerated electrons play an important role in the generation of radio emission in a wide range of the electromagnetic spectrum. The perturbations ⇑ Corresponding author.
E-mail address: gacp@geofisica.unam.mx (G.A. Casillas-Pe´rez).
produced by accelerated electrons lead to the generation of particle-wave and particle–particle interactions that may be decisive for the generation of intense radio emissions, as a result of a coherent emission mechanism (Guedel et al., 1991; Aschwanden and Guedel, 1992; Abalde et al., 2001; Huang and Nakajima, 2005; Kontar et al., 2012). The ambient density parameter plays a crucial role in the formation of instabilities in the lower solar corona, where Coulomb collisions have an important contribution in the electron beam energy losses processes. The character-
https://doi.org/10.1016/j.asr.2019.10.036 0273-1177/Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
2
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
istics of the ambient density are also a determining factor in the propagation of radio emission produced by accelerated electrons inducing plasma oscilations in the solar corona. The emission is produced at a frequency f p that is determined by the local electron density ne according to the relapffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tion f p ðHzÞ ¼ 8980 ne ðcm3 Þ. A consequence of this fact is the generation of radio emission frequency drifts during the propagation of accelerated electrons through the corona. The density of the solar corona has been reported to be a relevant factor that influences the behavior of solar activity during energetic events. For example, variation of the ambient density has been considered to be the responsible of large delays occurring between HXR and microwaves (MacKinnon, 1986). Also, the electron density is regarded as an important factor that may influence the growth of waves, and a minimum density is thought to be required for an efficient wave amplification to occur (Aschwanden and Guedel, 1992). Particularly, an upper and lower density limit covering the range between 109 and 1011 cm3 is thought to be appropriate for the occurrence of solar radio spikes (Fleishman et al., 2003). On the other hand, the role of the ambient density is crucial in the formation of instabilities that give rise to radio emissions via either, electron cyclotron maser emission (ECME) or plasma emision mechanisms. The ECME mechanism is related to strong magnetic fields in losscone instabilities where the electromagnetic waves produced are close to the electron gyrofrequency of the source. Plasma emission, is associated with non-thermal electron accelerations in the propagating shocks or magnetic reconnections, causing the production of Langmuir turbulence that eventually induces the formation of electromagnetic waves. Both mechanisms involve the prescence of accelerated electrons inmersed in the solar corona. Considering the propagation of an accelerated electron beam as a driver that favors the production of radio spikes, the evolution of electrons injected into a solar magnetic loop has been numerically studied inquiring the importance of a beam pulse of the order of milliseconds on the onset of radio emission (Melrose and Dulk, 1982; Aschwanden, 1990; Li et al., 2008; Li et al., 2011; Reid and Kontar, 2013; Ratcliffe et al., 2014; Casillas-Pe´rez et al., 2016). In this work, we follow a previous numerical study relating to the evolution of an electron beam pulse injected in a small region of the solar corona assuming that it is related to the generation of radio spikes (Casillas-Pe´rez et al., 2016). For the study, we use a numerical code in order to solve the time-dependent Fokker–Planck (FP) equation as proposed by the scheme of Hamilton et al. (1990). Several studies have discussed the evolution with time of a distribution of electrons accelerated inwards the solar corona and influenced by some physical processes (Hannah et al., 2009; Holman et al., 2011; Bian et al., 2014). Nevertheless, in order to obtain information and insights of the outer regions of the corona, a study of the evolution of a propagating pulse of electrons is required. In particular, in this
work we investigate the influence of Coluomb collision effects on the evolution of an electron beam pulse injected outwards the solar corona. Based on this study we were able to track the spatial, and temporal evolution of a pulse of short duration injected from the top of a magnetic loop, by considering different low boundary ambient density regions of the solar corona. In Section 2 we describe the model for the evolution of the electron beam distribution function. The results of the simulations and the discussion of results are presented in Section 3. Conclusions are presented in Section 4. 2. Model of the electron beam propagation In order to describe the evolution of the density distribution function f ðE; lÞ of a particle with energy E and pitch angle cosine l, at different times and position from the top of the coronal loop, we consider the time-dependent Fokker–Planck (FP) equation (e.g., Lifshitz and Pitaevskii, 1981; Hamilton et al., 1990). _ Þ eE @f _ Þ @ðEf @f @f @ðlf eE @f þ lcb þ þ l ð1 l2 Þ @t @s @l @E me @v me v @l @ @f Dll ¼ þ SðE; l; s; tÞ ð1Þ @l @l where s is the height position from the top of the coronal loop, and Sðs; E; l; tÞ describes the source function. The variable E is the electric field induced by the electron beam, e and me are the electron charge and mass respectively, c is the speed of light, b ¼ v=c, and v is the velocity. The changes in energy and pitch-angle cosine due to Coulomb collisions and magnetic mirror respectively are given by E_ ¼ 4pncr20 lnK=b and l_ ¼ bcð1 l2 ÞðdlnB=dsÞ=2, while Dll ¼ 4pncr20 lnKð1 l2 Þ=b2 c2 is the diffusion coefficient due to scattering processes (Hamilton et al., 1990). _ E_ and Dll, the Eq. (1) in terms Substituting the terms l; of scaled units has the form, @f @f b d ln B @ ¼ lb þ 1 l2 f @s @n 2 dn @l
@f @ f g ln K @ 1 l2 þ 2 þ g ln K @E b @l b c2 @l þ aelb
@f eð1 l2 Þ @f þa þ SðE; l; s; tÞ @E b @l
ð2Þ
here E ¼ c 1 is the kinetic energy expressed in units of mc2 , and the scaled variables are given by the dimensionless variables sc ¼ 1=ð4pn0 cr20 Þ; e ¼ E=E 0 ; a¼
eE 0 sc ; me c
E0 ¼
s ¼ t=sc ;
n ¼ s=ðcsc Þ;
g ¼ n=n0 ;
me c2 2pe3 n0
where ln K is the Coulomb logarithm assumed to be 10; r0 is the radius of the electron, E is the electric field
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
induced by the electron beam, and n0 and n are the unit of ambient density scale and the ambient density respectively. Considering that this study is focused on the behavior of an electron beam affected by Coulomb collision effects, the results reported here do not take into account the magnetic convergence B nor the electric field terms e in the Eq. (2). The evolution of an electron beam is investigated by injecting a short duration pulse of accelerated electrons at a lower boundary of 109 cm above the photosphere (the top of a characteristic solar flare loop). The pulse is followed to a distance of about 3 1011 cm. The distribution function and the parameters used in this study are as those used in Siversky and Zharkova (2009), and has the form ! d1 2 ðE=E0 Þ ð1 lÞ f ðE; lÞ ¼ f n wðsÞ exp ð3Þ dþc Dl2 ðE=E0 Þ þ 1 Table 1 Parameters of the simulation runs. Run
c
E0 (keV)
nb cm3
B1 B2 B3
3 3 7
16 7 16
1:75 107 1:74 107 2:87 107
3
where c ¼ 3; 7 are the power-law indexes of the electron distribution for energies above and below the cut off energy E0 respectively, Dl ¼ 0:2 is the Gaussian width of the pitch angle distribution and d ¼ 10. F n is a normalization coefficient which is normilized for a flux F 0 ¼ 1012 erg cm2 s1 at the lower boundary using Eqs. (14) and (15) of Zharkova and Gordovskyy (2005), and wðsÞ is the temporal profile of the beam, shaped by a rectangular function (Zharkova and Gordovskyy, 2005) with a width of 0:2 103 s. The distribution function is assumed to cause an initial perturbation in a homogenoeus plasma. The initial and upper boundary conditions of the beam distribution are f ðs ¼ 0; n; E; lÞ ¼ 0 and f ðnmax ; E; lÞ ¼ f ðnmax dn; E; lÞ respectively. As an initial estimation, the magnetic convergence and the electric field terms were not included. The former is due to the fact that variatons in the magnetic field are not relevant for the contemplated traveling distances of the electron beam. The latter is because the impact of the induced electric field on the evolution of an electron beam of short duration is small (Siversky and Zharkova, 2009). Simulations were carried out for two low energy cut offs E0 = 16 and 7 keV and energy spectral indexes c ¼ 3 and 7 (Siversky and Zharkova, 2009). The parameters used for
Fig. 1. Total density n ðcm3 Þ as a function of position plotted at different times: t ¼ 0:282 (solid red), 0:785 (dashed green) and 1:968s (dotted blue). The thin and thick lines correspond to model runs with and without Coulomb collisions effects. The upper (left and middle) and bottom panels correspond to the ambient densities 109 ; 5 109 and 1010 cm3 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
4
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
the simulations and the density of the electron beam are given in Table 1 (model runs B1, B2, and B3). Simulations without collisional effect terms were also carried out for comparative analysis (model runs A1, A2, and A3 respectively). All model runs were carried out considering different low boundary electron ambient densities under the range of typical solar flare conditions (109 ; 5 109 and 1010 cm3) (Hamilton et al., 1990; Hamilton and Petrosian, 1992). The simulations were carried out with a resolution of 100, 30 and 30 grids in position, l and energy axes respectively. The form of the ambient density in the corona is given by Ratcliffe (2013) n ¼ n0 exp ððs s0 Þ=H scale Þ
ð4Þ
where n0 ¼ 1010 cm3 is the density at the lower boundary and H scale ¼ 1010 cm is the scale height. 3. Results and discussion The electron beam density as a function of position at different times for model run B1 is plotted in Fig. 1. The upper (left and right) and bottom panels belong accordingly to coronal ambient densities 109 ; 5 109 and 1010
cm3. The thick and thin lines correspond to model runs where Coulomb collisions effects are included and omitted respectively. From the plot, it can be observed that the density of the electron beam decreases as the pulse enlarges with time. This behavior is more evident for the thick line curves where the loss of electrons is more clear. The results suggests that a relation between the evolution of the electron beam population and the ambient density is present, since the population diminishes more rapidly in presence of a denser environment. For the same simulation times, the electron beam density reached at the densest region (bottom panel) is almost two orders of magnitude smaller than that obtained at the least dense region (upper left panel). This is a consequence of a greater interaction among particles due to collisional effects. The electron densities (cm3) and their respective energy at the location where the peak of the beam occurs at different times is plotted in Fig. 2. The plots correspond to three different low boundary ambient densities 109 (upper left), 5 109 (upper right) and 1010 cm3 (bottom). The results indicate that a considerable fraction of the population of accelerated electrons are thermalized. Over time the electron beam density decreases and the spectrum moves towards higher energies, while the low energy electrons are lost by collisions. This is more obvious for a denser
Fig. 2. Electron density (cm3) with its respective energy at the location where the peak of the pulse occurs is plotted for model run B1 at different times: 0:085 (solid red), 1:97 (dashed green) and 7:89 s (dotted blue). The upper (left and middle) and bottom panels correspond to the ambient densities 109 , 5 109 and 1010 cm3 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
background scenario. In that case, the electron beam density decreases almost five orders of magnitude from its initial value (bottom panel). A smaller decrease of the electron beam density occurs at lower coronal densities (upper left and bottom panels). For the lowest ambient density, a decrease of the electron distribution density occurs without a marked shift to a higher energy. In this case the population is dominated by 19 keV electrons which are thermalized when they lose their energy due to collisions with much lower energy electrons in the ambient. For the higher ambient densities the shift of the spectra to higher energies (30 and 40 keV) suggests that part of the whole accelerated electrons retain energy which is used to travel longer distances through the corona before they lose completely their energy) (see Fig. 1). Given that the range of energies considered in this study are > 1 keV, the influence of warm-target effects on the accelerated electrons is not relevant and all the electrons are expected to be thermilized due to Coulomb collisions effects (Kontar et al., 2019). Before their thermalization, on their way through the corona the accelerated particles may cause the formation of instabilities and favor the generation of radio emsission.
5
The velocity dispersion of the electron distribution may provoke the propagation of fast electrons at longer distances followed by a bumb of slower electrons at the tail of the distribution. On the other hand, the collision time is calculated using the equation of the deflection time of non-thermal electrons given by Zharkova and Gordovskyy (2005), sc ¼ 1:47 108 E3=2 =ne , where E (keV) and ne (cm3) correspond to the particle energies and the ambient density respectively. The results of the collision time for model run B1 at locations reached by the electron distribution after 0:08 s are 12:3; 2:5 and 1:2 s corresponding to the low boundary ambient densities 109 ; 5 109 and 1010 cm3 respectively. These times are enough for the electrons to provoke instabilities and produce radio emission in the corona. Spectra at the peak (red), leading (blue) and trailing (green) edges of a pulse for a time of 0.489 s are plotted in Fig. 3. The upper (left and middle) and bottom panels correspond to the ambient densities 109 ; 5 109 and 1010 cm3 respectively. From the plots, it can be observed that the leading edge of the pulse is separated from its peak and trailing edge which exhibit comparable densities for
Fig. 3. Spectra at the peak (solid red), leading (dotted blue) and trailing (dashed green) edges of a pulse are plotted for model B1, for a time of 0:489 s. The upper (left and middle) and bottom panels correspond to the ambient densities 109 ; 5 109 and 1010 cm3 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
6
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
energies 620 keV (green and red curves). This separation is larger for a denser ambient, where collision effects contribute significantly to the loss of electrons in the pulse. The results suggest the presence of a leading edge population composed of energetic electrons, followed by a trailing edge formed of a bundle of low-energy electrons. The arising of the electron distribution with two populations on colisional time-scales might be a determining factor for the formation of instabilities that are responsible of the production of radio emisions. For example, a growth of Langmuir waves is suggested to occur as a result of fast moving electrons overtaking slower ones (Kontar et al., 2012), as a result of a bump in tail distribution formed for an unstable distribution where @f =@v > 0 (Aschwanden, 2005). In order to delineate the Coulomb collision effects, the column density of the electron beam in the presence of Coulomb collisions (ncoul ) is compared with that of the beam that includes only positional advection (nadv ). The time variation of the ratio of the two densities ncoul=adv ¼ ncoul =nadv was calculated. This ratio give us information about the fraction of the column density obtained
as a consequence of Coulomb collision effects with respect to the column density of the electron distribution without the presence of these effects. The time variation of the ratio ncoul=adv is shown in the plots of Fig. 4. The panels of the figure correspond to the low boundary ambient densities 109 (upper left), 5 109 (upper right) and 1010 cm3 (bottom). From the plots, we can see that due to collisions the injected electrons are thermalized faster whenever they are injected from a larger ambient density. For the densest background (1010 cm3, bottom panel), the loss of the electron population with a low-energy cut off of 16 keV and spectral index 3 is reached in a very brief time < 0.4 s. A shorter time of 0.2 s is obtained with a spectral index of 7, while the lowest time of 0.1 s is obtained for a lowenergy cut off of 7 keV, which indicates that a distribution with lower low-energy cut off is determining in the loss of electrons in presence of Coulom collision effects. On the other hand, for the least dense medium (109 cm3, upper left panel) the loss of the population is reached at a greater time 1.5 s for an electron distribution with a low-energy cut off of 7 keV and a spectral index of 3, while for a higher low-energy cut off (16 keV) the los of accelerated electrons
Fig. 4. The ratio of column density throughout the simulation domain of model runs B with those of A are plotted against time. The dashed green, solid red and dotted blue curves represent model runs B1, B2, and B3 respectively. The upper (left and middle) and bottom panels correspond to the ambient densities 109 ; 5 109 and 1010 cm3 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
is very slow with an asymptotic fall over an extended duration. This suggests that Coulomb collisions do not seem to have an important influence on the evolution of an electron beam distribution with high low-energy cut off, when the pulse is injected at lower boundary regions with low density regardless of the value of the spectral index (3; 7). For the study we have considered the trapping time to be equal to the acceleration time of the evolution of the electron distribution. The results of the deflecting times for ambient densities ne ¼ 109 1010 cm3 for the model run B1 are in the order of sc 5 12 s for 19 48 keV electrons. Comparision of these values with the corresponding electron beam evolution times of 2 8 s, indicates that sa < sc and suggests a scenario with a weakdiffusion regime. The results obtained for the thermalizing time of the accelerated electrons suggest that the growth of the waves should occur much faster than the damping time due to collisions, in order for a maser mechanism being the origin of radio emission. This is in agreement with the estimated acceleration time of particles inside turbulent regions, which has been reported to be less or equal than the collision time (Hamilton and Petrosian, 1992). Also, it is in accordance with the time scale of the energy loss by collisions at a density of 1010 cm3, which has been estimated to be 61 s for electrons with energy < 10 keV (Leach and Petrosian, 1981) and with growth of waves in times 6 106 s as reported by Melrose and Dulk (1982) and Aschwanden and Benz (1988). In particular, Aschwanden and Benz (1988) have reported growth rates of 102 s and 106 s for plasma/gyrofrequency rates of 1.4 and 0.1 respectively. In the case of radio spikes this time has been reported to be in the range between 5:6 103 s and 8:0 102 s (Kuijpers et al., 1981). On the other hand, for the least dense background (109 cm3) only the model run that considered the lower cut off energy (7 keV) exhibited an important decrease in the density of the electron beam (Fig. 4, upper left panel). The loss of the electron population at the first second of the evolution is > 90% whereas it falls in a range 50 70% for the models with a higher cut off energy (16 keV). The curves in the plot for the electron population in the latter case indicates that the electron population suffers a very slow decrease after the first two seconds of the pulse evolution. 4. Conclusions We studied the evolution of an electron beam injected outwards at the top of the solar corona, and the influence of the ambient density at the low boundary on the evolution of the pulse. It was found that in larger ambient densities, collision effects have more impact on the slower electrons that travel at the trailing edge of the pulse. These electrons are thermalized faster than the energetic electrons propagating at the front edge. This behavior is more
7
evident for denser coronal environments with low boundary densities of 5 109 and 1010 cm3 where a faster decrease of the electron population occurs within a time 0:2 1:0 s and 0:1 0:4 s respectively. A similar behavior presenting a notable population decrease, in a less dense environment of 109 cm3 occurs, but with smaller lowenergy cut off of 7 keV, while for a higher low-energy cut off (16 keV) a very slow decrease of the electron pulse density was found. Furthermore, it was found that a predominant energy of 19 keV prevails during the evolution of an electron beam distribution injected at a low boundary with a density of 109 cm3, regardless of the low-energy cut off (16; 7) keV and the spectral index (3; 7) of the electron distribution function. The accelerated electrons suffer an energy loss and are completely thermalized by Coulomb collisions after their travel through the corona. This behaviour does not occur for an electron distribution injected at higher low boundary densities of 5 109 and 109 cm3. In these cases, electrons with higher energies ( 30 and 48 keV respectively) prevails as part of the non-thermal population, which have enough energy to travel longer distances than those reached by a distribution injected at a less dense background (109 cm-3). Considering all the aforementioned, if we assume that the bunch of thermalized electrons induce the appropriate instability conditions for the production of radio spikes, then the characteristics of the ambient density would be determinant in the estimated properties of the created spikes. Electrons travelling shorter distances through lower densities of the corona are expected to produce radio spikes with smaller bandwidth and frequency drift rates than those produced by accelerated electrons with larger energies traveling longer distances. Although the density of the electron population that arrives at longer distances is smaller, the electrons possess higher energy which may result in the production of more intense radio spikes, even though with a less number of ocurrence. On the other hand, since the time that the electron beam requires to get thermalized, and the duration of the instabilities produced by the accelerated electrons of the pulse depends on the low boundary ambient density, the duration of the radio spikes generated due to the instabilities is also expected to be influenced by the low boundary ambient density. The interaction of an electron beam with the solar corona may have different effects that depend on the ambient density. An appropriate environment density may be favorable in order that a coherent mechanism occurs and subsequently, a variety of solar fast transients be originated. With the appropriate environment density the instabilities generated by the electron beam may favor the production of solar fast transient events as suggested by Aschwanden and Guedel (1992) and Casillas-Pe´rez et al. (2019). The formation of instabilities and its evolution into efficient growth rates depends on the way that the electron
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
8
distribution evolves with time and space. This evolution may depends on the form and estimate of some parameters of the electron distribution function, such as the spectral index and the cut off energy of the electrons. Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgements We appreciate the comments and valuable suggestions of the anonymous referees that helped to improve the content of the article. Gilberto A. Casillas-Pe´rez expresses his gratitude to Miguel A. Garcı´a-Palacios from the IGFUNAM for technical work provided. References Abalde, J.R., Borotto, F.A., Chian, A.C.-L., 2001. Plasma emission mechanism for radio spikes from solar flares. Adv. Space Res. 28, 741– 746. Aschwanden, M.J., 1990. Relaxation of the loss-cone by quasi-linear diffusion of the electron-cyclotron maser instability in the solar corona. A&AS 85, 1141–1177. Aschwanden, M.J., 2005. Physics of the Solar Corona. An Introduction with Problems and Solutions, 2nd ed. Springer-Verlag, Berlin. Aschwanden, M.J., Benz, A.O., 1988. On the electron-cyclotron maser instability. I - Quasi-linear diffusion in the loss cone. II - Pulsations in the quasi-stationary state. Apj 332, 447–475. Aschwanden, M.J., Guedel, M., 1992. The coevolution of decimetric millisecond spikes and hard X-ray emission during solar flares. Apj 401, 736–753. 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 (2), 142. Casillas-Pe´rez, G.A., Jeyakumar, S., Carrillo-Vargas, A., Pe´rez-Enrı´quez, H.R., 2019. Study of solar radio spikes and their relation to energetic solar events. Sol. Phys. 294, 10. Casillas-Pe´rez, G.A., Jeyakumar, S., Pe´rez-Enrı´quez, H.R., Trinidad, M. A., 2016. Evolution of electron beam pulses of short duration in the solar corona. Adv. Space Res. 58, 1991–1996. Fleishman, G.D., Gary, D.E., Nita, G.M., 2003. Decimetric spike bursts versus microwave continuum. ApJ 593, 571–580. Guedel, M., Benz, A.O., Aschwanden, M.J., 1991. The association of solar millisecond radio spikes with hard X-ray emission. A&A 251, 285–297.
Hamilton, R.J., Lu, E.T., Petrosian, V., 1990. Numerical solution of the time-dependent kinetic equation for electrons in magnetized plasma. ApJ 354, 726–734. Hamilton, R.J., Petrosian, V., 1992. Stochastic acceleration of electrons. I - Effects of collisions in solar flares. ApJ 398, 350–358. Hannah, I.G., Kontar, E.P., Sirenko, O.K., 2009. The effect of waveparticle interactions on low-energy cutoffs in solar flare electron spectra. Astrophys. J. 707 (1), L45–L50. Holman, G.D., Aschwanden, M.J., Aurass, H., Battaglia, M., Grigis, P. C., Kontar, E.P., Liu, W., Saint-Hilaire, P., Zharkova, V.V., 2011. Implications of X-ray observations for electron acceleration and propagation in solar flares. Space Sci. Rev. 159 (1–4), 107–166. Huang, G., Nakajima, H., 2005. Location and parameters of a microwave millisecond spike event. Astrophys. Space Sci. 295 (4), 423–442. Kontar, E.P., Jeffrey, N.L.S., Emslie, A.G., 2019. Determination of the total accelerated electron rate and power using solar flare hard X-ray spectra. ApJ 871 (2), 225. Kontar, E.P., Ratcliffe, H., Bian, N.H., 2012. Wave-particle interactions in non-uniform plasma and the interpretation of hard X-ray spectra in solar flares. A&A 539, A43. Kuijpers, J., van der Post, P., Slottje, C., 1981. Runaway acceleration in a radio flare. A&A 103, 331–338. Leach, J., Petrosian, V., 1981. Impulsive phase of solar flares. I Characteristics of high energy electrons. ApJ 251, 781–791. Li, B., Cairns, I.H., Robinson, P.A., 2008. Simulations of coronal type III solar radio bursts: 1. Simulation model. J. Geophys. Res. (Space Phys.) 113, A06104. Li, B., Cairns, I.H., Yan, Y.H., Robinson, P.A., 2011. Decimetric Type III bursts: generation and propagation. ApJL 738, L9. Lifshitz, E.H., Pitaevskii, L.P., 1981. Physical Kinetics. Pergamon Press, Oxford. MacKinnon, A.L., 1986. Time-dependent density in the electron-trap model of solar flare hard X-ray bursts. A&A 163, 239–245. Melrose, D.B., Dulk, G.A., 1982. Electron-cyclotron masers as the source of certain solar and stellar radio bursts. ApJ 259, 844–858. Ratcliffe, H., 2013. Electron beam evolution and radio emission in the inhomogeneous solar corona. Ph.D. thesis. University of Glasgow, United Kingdom. Ratcliffe, H., Kontar, E.P., Reid, H.A.S., 2014. Large-scale simulations of solar type III radio bursts: flux density, drift rate, duration, and bandwidth. A&A 572, A111. Reid, H.A.S., Kontar, E.P., 2013. Evolution of the Solar flare energetic electrons in the inhomogeneous inner heliosphere. Sol. Phys. 285, 217– 232. Siversky, T.V., Zharkova, V.V., 2009. Stationary and impulsive injection of electron beams in converging magnetic field. A&A 504, 1057–1070. Zharkova, V.V., Gordovskyy, M., 2005. The kinetic effects of electron beam precipitation and resulting hard X-ray intensity in solar flares. A&A 432, 1033–1047.
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
ScienceDirect Advances in Space Research xxx (2019) xxx–xxx www.elsevier.com/locate/asr
Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona G.A. Casillas-Pe´rez a,⇑, S. Jeyakumar b, A. Carrillo-Vargas c, H.R. Pe´rez-Enrı´quez d a
Instituto de Geofı´sica, Universidad Nacional Auto´noma de Me´xico, Ciudad Universitaria, C.P. 04510 Ciudad de Me´xico, Mexico b Departamento de Astronomı´a, Universidad de Guanajuato, Me´xico, Guanajuato, Gto. C.P. 36000, Mexico c Instituto de Geofı´sica, Universidad Nacional Auto´noma de Me´xico, Unidad Morelia, Antigua Carretera a Pa´tzcuaro No. 8701, Ex-Hacienda de San Jose´ de la Huerta, C.P. 58190 Morelia Michoaca´n, Mexico d Centro de Geociencias, Universidad Nacional Auto´noma de Me´xico, Campus Juriquilla, C.P. 76230 Juriquilla, Quere´taro, Mexico Received 23 July 2019; received in revised form 25 October 2019; accepted 26 October 2019
Abstract Electrons accelerated in the corona during solar activity give rise to radio emission events that can be observed over a wide range of frequencies. Among different finer-scale structures in the dynamic spectra observed in the radio range, fast transients with extents of some milliseconds known as solar radio spikes are observed accompaning the background continuum emission. Fundamental to the generation of radio spikes is a propagating electron beam and following its evolution allows us to understand the physical processes occurring in the solar corona. With the use of a numerical Fokker–Planck code we follow a previous numerical study to simulate the propagation of an electron beam pulse injected in a small region at the top of a magnetic field and outwards the solar corona under typical flare conditions. It was found that in large ambient densities of 1010 cm3 at the injection point, Coulomb collision effects have an important effect on the propagation of the electrons, causing that the injected electrons thermalize faster in a time of 0:1 and 0:4 s for an electron distribution with a low-energy cut off of 16 and 7 keV respectively and a spectral index of 3. For a tenous ambient medium of density 109 cm3 thermalization occurs only for an electron distribution with smaller low-energy cut off (7 keV) with a duration of 1.5 s, while for a larger low-energy cut off (16 keV) the loss of accelerated electrons is very slow, regardles of the spectral index (3; 7). The electron loss time by Coulomb collisions, which depends on the low boundary ambient density, might be an important parameter that influences the generation of radio spikes due to the formation of instabilities in the corona. Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Solar corona; Solar radio spikes; Fokker–Planck equation; Particle acceleration; Electron beam pulse
1. Introduction Accelerated particles moving through the solar corona have been found to produce instabilities as a result of their interaction with the environment. Particularly, the instabilities produced by a beam of accelerated electrons play an important role in the generation of radio emission in a wide range of the electromagnetic spectrum. The perturbations ⇑ Corresponding author.
E-mail address: gacp@geofisica.unam.mx (G.A. Casillas-Pe´rez).
produced by accelerated electrons lead to the generation of particle-wave and particle–particle interactions that may be decisive for the generation of intense radio emissions, as a result of a coherent emission mechanism (Guedel et al., 1991; Aschwanden and Guedel, 1992; Abalde et al., 2001; Huang and Nakajima, 2005; Kontar et al., 2012). The ambient density parameter plays a crucial role in the formation of instabilities in the lower solar corona, where Coulomb collisions have an important contribution in the electron beam energy losses processes. The character-
https://doi.org/10.1016/j.asr.2019.10.036 0273-1177/Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
2
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
istics of the ambient density are also a determining factor in the propagation of radio emission produced by accelerated electrons inducing plasma oscilations in the solar corona. The emission is produced at a frequency f p that is determined by the local electron density ne according to the relapffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tion f p ðHzÞ ¼ 8980 ne ðcm3 Þ. A consequence of this fact is the generation of radio emission frequency drifts during the propagation of accelerated electrons through the corona. The density of the solar corona has been reported to be a relevant factor that influences the behavior of solar activity during energetic events. For example, variation of the ambient density has been considered to be the responsible of large delays occurring between HXR and microwaves (MacKinnon, 1986). Also, the electron density is regarded as an important factor that may influence the growth of waves, and a minimum density is thought to be required for an efficient wave amplification to occur (Aschwanden and Guedel, 1992). Particularly, an upper and lower density limit covering the range between 109 and 1011 cm3 is thought to be appropriate for the occurrence of solar radio spikes (Fleishman et al., 2003). On the other hand, the role of the ambient density is crucial in the formation of instabilities that give rise to radio emissions via either, electron cyclotron maser emission (ECME) or plasma emision mechanisms. The ECME mechanism is related to strong magnetic fields in losscone instabilities where the electromagnetic waves produced are close to the electron gyrofrequency of the source. Plasma emission, is associated with non-thermal electron accelerations in the propagating shocks or magnetic reconnections, causing the production of Langmuir turbulence that eventually induces the formation of electromagnetic waves. Both mechanisms involve the prescence of accelerated electrons inmersed in the solar corona. Considering the propagation of an accelerated electron beam as a driver that favors the production of radio spikes, the evolution of electrons injected into a solar magnetic loop has been numerically studied inquiring the importance of a beam pulse of the order of milliseconds on the onset of radio emission (Melrose and Dulk, 1982; Aschwanden, 1990; Li et al., 2008; Li et al., 2011; Reid and Kontar, 2013; Ratcliffe et al., 2014; Casillas-Pe´rez et al., 2016). In this work, we follow a previous numerical study relating to the evolution of an electron beam pulse injected in a small region of the solar corona assuming that it is related to the generation of radio spikes (Casillas-Pe´rez et al., 2016). For the study, we use a numerical code in order to solve the time-dependent Fokker–Planck (FP) equation as proposed by the scheme of Hamilton et al. (1990). Several studies have discussed the evolution with time of a distribution of electrons accelerated inwards the solar corona and influenced by some physical processes (Hannah et al., 2009; Holman et al., 2011; Bian et al., 2014). Nevertheless, in order to obtain information and insights of the outer regions of the corona, a study of the evolution of a propagating pulse of electrons is required. In particular, in this
work we investigate the influence of Coluomb collision effects on the evolution of an electron beam pulse injected outwards the solar corona. Based on this study we were able to track the spatial, and temporal evolution of a pulse of short duration injected from the top of a magnetic loop, by considering different low boundary ambient density regions of the solar corona. In Section 2 we describe the model for the evolution of the electron beam distribution function. The results of the simulations and the discussion of results are presented in Section 3. Conclusions are presented in Section 4. 2. Model of the electron beam propagation In order to describe the evolution of the density distribution function f ðE; lÞ of a particle with energy E and pitch angle cosine l, at different times and position from the top of the coronal loop, we consider the time-dependent Fokker–Planck (FP) equation (e.g., Lifshitz and Pitaevskii, 1981; Hamilton et al., 1990). _ Þ eE @f _ Þ @ðEf @f @f @ðlf eE @f þ lcb þ þ l ð1 l2 Þ @t @s @l @E me @v me v @l @ @f Dll ¼ þ SðE; l; s; tÞ ð1Þ @l @l where s is the height position from the top of the coronal loop, and Sðs; E; l; tÞ describes the source function. The variable E is the electric field induced by the electron beam, e and me are the electron charge and mass respectively, c is the speed of light, b ¼ v=c, and v is the velocity. The changes in energy and pitch-angle cosine due to Coulomb collisions and magnetic mirror respectively are given by E_ ¼ 4pncr20 lnK=b and l_ ¼ bcð1 l2 ÞðdlnB=dsÞ=2, while Dll ¼ 4pncr20 lnKð1 l2 Þ=b2 c2 is the diffusion coefficient due to scattering processes (Hamilton et al., 1990). _ E_ and Dll, the Eq. (1) in terms Substituting the terms l; of scaled units has the form, @f @f b d ln B @ ¼ lb þ 1 l2 f @s @n 2 dn @l
@f @ f g ln K @ 1 l2 þ 2 þ g ln K @E b @l b c2 @l þ aelb
@f eð1 l2 Þ @f þa þ SðE; l; s; tÞ @E b @l
ð2Þ
here E ¼ c 1 is the kinetic energy expressed in units of mc2 , and the scaled variables are given by the dimensionless variables sc ¼ 1=ð4pn0 cr20 Þ; e ¼ E=E 0 ; a¼
eE 0 sc ; me c
E0 ¼
s ¼ t=sc ;
n ¼ s=ðcsc Þ;
g ¼ n=n0 ;
me c2 2pe3 n0
where ln K is the Coulomb logarithm assumed to be 10; r0 is the radius of the electron, E is the electric field
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
induced by the electron beam, and n0 and n are the unit of ambient density scale and the ambient density respectively. Considering that this study is focused on the behavior of an electron beam affected by Coulomb collision effects, the results reported here do not take into account the magnetic convergence B nor the electric field terms e in the Eq. (2). The evolution of an electron beam is investigated by injecting a short duration pulse of accelerated electrons at a lower boundary of 109 cm above the photosphere (the top of a characteristic solar flare loop). The pulse is followed to a distance of about 3 1011 cm. The distribution function and the parameters used in this study are as those used in Siversky and Zharkova (2009), and has the form ! d1 2 ðE=E0 Þ ð1 lÞ f ðE; lÞ ¼ f n wðsÞ exp ð3Þ dþc Dl2 ðE=E0 Þ þ 1 Table 1 Parameters of the simulation runs. Run
c
E0 (keV)
nb cm3
B1 B2 B3
3 3 7
16 7 16
1:75 107 1:74 107 2:87 107
3
where c ¼ 3; 7 are the power-law indexes of the electron distribution for energies above and below the cut off energy E0 respectively, Dl ¼ 0:2 is the Gaussian width of the pitch angle distribution and d ¼ 10. F n is a normalization coefficient which is normilized for a flux F 0 ¼ 1012 erg cm2 s1 at the lower boundary using Eqs. (14) and (15) of Zharkova and Gordovskyy (2005), and wðsÞ is the temporal profile of the beam, shaped by a rectangular function (Zharkova and Gordovskyy, 2005) with a width of 0:2 103 s. The distribution function is assumed to cause an initial perturbation in a homogenoeus plasma. The initial and upper boundary conditions of the beam distribution are f ðs ¼ 0; n; E; lÞ ¼ 0 and f ðnmax ; E; lÞ ¼ f ðnmax dn; E; lÞ respectively. As an initial estimation, the magnetic convergence and the electric field terms were not included. The former is due to the fact that variatons in the magnetic field are not relevant for the contemplated traveling distances of the electron beam. The latter is because the impact of the induced electric field on the evolution of an electron beam of short duration is small (Siversky and Zharkova, 2009). Simulations were carried out for two low energy cut offs E0 = 16 and 7 keV and energy spectral indexes c ¼ 3 and 7 (Siversky and Zharkova, 2009). The parameters used for
Fig. 1. Total density n ðcm3 Þ as a function of position plotted at different times: t ¼ 0:282 (solid red), 0:785 (dashed green) and 1:968s (dotted blue). The thin and thick lines correspond to model runs with and without Coulomb collisions effects. The upper (left and middle) and bottom panels correspond to the ambient densities 109 ; 5 109 and 1010 cm3 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
4
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
the simulations and the density of the electron beam are given in Table 1 (model runs B1, B2, and B3). Simulations without collisional effect terms were also carried out for comparative analysis (model runs A1, A2, and A3 respectively). All model runs were carried out considering different low boundary electron ambient densities under the range of typical solar flare conditions (109 ; 5 109 and 1010 cm3) (Hamilton et al., 1990; Hamilton and Petrosian, 1992). The simulations were carried out with a resolution of 100, 30 and 30 grids in position, l and energy axes respectively. The form of the ambient density in the corona is given by Ratcliffe (2013) n ¼ n0 exp ððs s0 Þ=H scale Þ
ð4Þ
where n0 ¼ 1010 cm3 is the density at the lower boundary and H scale ¼ 1010 cm is the scale height. 3. Results and discussion The electron beam density as a function of position at different times for model run B1 is plotted in Fig. 1. The upper (left and right) and bottom panels belong accordingly to coronal ambient densities 109 ; 5 109 and 1010
cm3. The thick and thin lines correspond to model runs where Coulomb collisions effects are included and omitted respectively. From the plot, it can be observed that the density of the electron beam decreases as the pulse enlarges with time. This behavior is more evident for the thick line curves where the loss of electrons is more clear. The results suggests that a relation between the evolution of the electron beam population and the ambient density is present, since the population diminishes more rapidly in presence of a denser environment. For the same simulation times, the electron beam density reached at the densest region (bottom panel) is almost two orders of magnitude smaller than that obtained at the least dense region (upper left panel). This is a consequence of a greater interaction among particles due to collisional effects. The electron densities (cm3) and their respective energy at the location where the peak of the beam occurs at different times is plotted in Fig. 2. The plots correspond to three different low boundary ambient densities 109 (upper left), 5 109 (upper right) and 1010 cm3 (bottom). The results indicate that a considerable fraction of the population of accelerated electrons are thermalized. Over time the electron beam density decreases and the spectrum moves towards higher energies, while the low energy electrons are lost by collisions. This is more obvious for a denser
Fig. 2. Electron density (cm3) with its respective energy at the location where the peak of the pulse occurs is plotted for model run B1 at different times: 0:085 (solid red), 1:97 (dashed green) and 7:89 s (dotted blue). The upper (left and middle) and bottom panels correspond to the ambient densities 109 , 5 109 and 1010 cm3 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
background scenario. In that case, the electron beam density decreases almost five orders of magnitude from its initial value (bottom panel). A smaller decrease of the electron beam density occurs at lower coronal densities (upper left and bottom panels). For the lowest ambient density, a decrease of the electron distribution density occurs without a marked shift to a higher energy. In this case the population is dominated by 19 keV electrons which are thermalized when they lose their energy due to collisions with much lower energy electrons in the ambient. For the higher ambient densities the shift of the spectra to higher energies (30 and 40 keV) suggests that part of the whole accelerated electrons retain energy which is used to travel longer distances through the corona before they lose completely their energy) (see Fig. 1). Given that the range of energies considered in this study are > 1 keV, the influence of warm-target effects on the accelerated electrons is not relevant and all the electrons are expected to be thermilized due to Coulomb collisions effects (Kontar et al., 2019). Before their thermalization, on their way through the corona the accelerated particles may cause the formation of instabilities and favor the generation of radio emsission.
5
The velocity dispersion of the electron distribution may provoke the propagation of fast electrons at longer distances followed by a bumb of slower electrons at the tail of the distribution. On the other hand, the collision time is calculated using the equation of the deflection time of non-thermal electrons given by Zharkova and Gordovskyy (2005), sc ¼ 1:47 108 E3=2 =ne , where E (keV) and ne (cm3) correspond to the particle energies and the ambient density respectively. The results of the collision time for model run B1 at locations reached by the electron distribution after 0:08 s are 12:3; 2:5 and 1:2 s corresponding to the low boundary ambient densities 109 ; 5 109 and 1010 cm3 respectively. These times are enough for the electrons to provoke instabilities and produce radio emission in the corona. Spectra at the peak (red), leading (blue) and trailing (green) edges of a pulse for a time of 0.489 s are plotted in Fig. 3. The upper (left and middle) and bottom panels correspond to the ambient densities 109 ; 5 109 and 1010 cm3 respectively. From the plots, it can be observed that the leading edge of the pulse is separated from its peak and trailing edge which exhibit comparable densities for
Fig. 3. Spectra at the peak (solid red), leading (dotted blue) and trailing (dashed green) edges of a pulse are plotted for model B1, for a time of 0:489 s. The upper (left and middle) and bottom panels correspond to the ambient densities 109 ; 5 109 and 1010 cm3 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
6
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
energies 620 keV (green and red curves). This separation is larger for a denser ambient, where collision effects contribute significantly to the loss of electrons in the pulse. The results suggest the presence of a leading edge population composed of energetic electrons, followed by a trailing edge formed of a bundle of low-energy electrons. The arising of the electron distribution with two populations on colisional time-scales might be a determining factor for the formation of instabilities that are responsible of the production of radio emisions. For example, a growth of Langmuir waves is suggested to occur as a result of fast moving electrons overtaking slower ones (Kontar et al., 2012), as a result of a bump in tail distribution formed for an unstable distribution where @f =@v > 0 (Aschwanden, 2005). In order to delineate the Coulomb collision effects, the column density of the electron beam in the presence of Coulomb collisions (ncoul ) is compared with that of the beam that includes only positional advection (nadv ). The time variation of the ratio of the two densities ncoul=adv ¼ ncoul =nadv was calculated. This ratio give us information about the fraction of the column density obtained
as a consequence of Coulomb collision effects with respect to the column density of the electron distribution without the presence of these effects. The time variation of the ratio ncoul=adv is shown in the plots of Fig. 4. The panels of the figure correspond to the low boundary ambient densities 109 (upper left), 5 109 (upper right) and 1010 cm3 (bottom). From the plots, we can see that due to collisions the injected electrons are thermalized faster whenever they are injected from a larger ambient density. For the densest background (1010 cm3, bottom panel), the loss of the electron population with a low-energy cut off of 16 keV and spectral index 3 is reached in a very brief time < 0.4 s. A shorter time of 0.2 s is obtained with a spectral index of 7, while the lowest time of 0.1 s is obtained for a lowenergy cut off of 7 keV, which indicates that a distribution with lower low-energy cut off is determining in the loss of electrons in presence of Coulom collision effects. On the other hand, for the least dense medium (109 cm3, upper left panel) the loss of the population is reached at a greater time 1.5 s for an electron distribution with a low-energy cut off of 7 keV and a spectral index of 3, while for a higher low-energy cut off (16 keV) the los of accelerated electrons
Fig. 4. The ratio of column density throughout the simulation domain of model runs B with those of A are plotted against time. The dashed green, solid red and dotted blue curves represent model runs B1, B2, and B3 respectively. The upper (left and middle) and bottom panels correspond to the ambient densities 109 ; 5 109 and 1010 cm3 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
is very slow with an asymptotic fall over an extended duration. This suggests that Coulomb collisions do not seem to have an important influence on the evolution of an electron beam distribution with high low-energy cut off, when the pulse is injected at lower boundary regions with low density regardless of the value of the spectral index (3; 7). For the study we have considered the trapping time to be equal to the acceleration time of the evolution of the electron distribution. The results of the deflecting times for ambient densities ne ¼ 109 1010 cm3 for the model run B1 are in the order of sc 5 12 s for 19 48 keV electrons. Comparision of these values with the corresponding electron beam evolution times of 2 8 s, indicates that sa < sc and suggests a scenario with a weakdiffusion regime. The results obtained for the thermalizing time of the accelerated electrons suggest that the growth of the waves should occur much faster than the damping time due to collisions, in order for a maser mechanism being the origin of radio emission. This is in agreement with the estimated acceleration time of particles inside turbulent regions, which has been reported to be less or equal than the collision time (Hamilton and Petrosian, 1992). Also, it is in accordance with the time scale of the energy loss by collisions at a density of 1010 cm3, which has been estimated to be 61 s for electrons with energy < 10 keV (Leach and Petrosian, 1981) and with growth of waves in times 6 106 s as reported by Melrose and Dulk (1982) and Aschwanden and Benz (1988). In particular, Aschwanden and Benz (1988) have reported growth rates of 102 s and 106 s for plasma/gyrofrequency rates of 1.4 and 0.1 respectively. In the case of radio spikes this time has been reported to be in the range between 5:6 103 s and 8:0 102 s (Kuijpers et al., 1981). On the other hand, for the least dense background (109 cm3) only the model run that considered the lower cut off energy (7 keV) exhibited an important decrease in the density of the electron beam (Fig. 4, upper left panel). The loss of the electron population at the first second of the evolution is > 90% whereas it falls in a range 50 70% for the models with a higher cut off energy (16 keV). The curves in the plot for the electron population in the latter case indicates that the electron population suffers a very slow decrease after the first two seconds of the pulse evolution. 4. Conclusions We studied the evolution of an electron beam injected outwards at the top of the solar corona, and the influence of the ambient density at the low boundary on the evolution of the pulse. It was found that in larger ambient densities, collision effects have more impact on the slower electrons that travel at the trailing edge of the pulse. These electrons are thermalized faster than the energetic electrons propagating at the front edge. This behavior is more
7
evident for denser coronal environments with low boundary densities of 5 109 and 1010 cm3 where a faster decrease of the electron population occurs within a time 0:2 1:0 s and 0:1 0:4 s respectively. A similar behavior presenting a notable population decrease, in a less dense environment of 109 cm3 occurs, but with smaller lowenergy cut off of 7 keV, while for a higher low-energy cut off (16 keV) a very slow decrease of the electron pulse density was found. Furthermore, it was found that a predominant energy of 19 keV prevails during the evolution of an electron beam distribution injected at a low boundary with a density of 109 cm3, regardless of the low-energy cut off (16; 7) keV and the spectral index (3; 7) of the electron distribution function. The accelerated electrons suffer an energy loss and are completely thermalized by Coulomb collisions after their travel through the corona. This behaviour does not occur for an electron distribution injected at higher low boundary densities of 5 109 and 109 cm3. In these cases, electrons with higher energies ( 30 and 48 keV respectively) prevails as part of the non-thermal population, which have enough energy to travel longer distances than those reached by a distribution injected at a less dense background (109 cm-3). Considering all the aforementioned, if we assume that the bunch of thermalized electrons induce the appropriate instability conditions for the production of radio spikes, then the characteristics of the ambient density would be determinant in the estimated properties of the created spikes. Electrons travelling shorter distances through lower densities of the corona are expected to produce radio spikes with smaller bandwidth and frequency drift rates than those produced by accelerated electrons with larger energies traveling longer distances. Although the density of the electron population that arrives at longer distances is smaller, the electrons possess higher energy which may result in the production of more intense radio spikes, even though with a less number of ocurrence. On the other hand, since the time that the electron beam requires to get thermalized, and the duration of the instabilities produced by the accelerated electrons of the pulse depends on the low boundary ambient density, the duration of the radio spikes generated due to the instabilities is also expected to be influenced by the low boundary ambient density. The interaction of an electron beam with the solar corona may have different effects that depend on the ambient density. An appropriate environment density may be favorable in order that a coherent mechanism occurs and subsequently, a variety of solar fast transients be originated. With the appropriate environment density the instabilities generated by the electron beam may favor the production of solar fast transient events as suggested by Aschwanden and Guedel (1992) and Casillas-Pe´rez et al. (2019). The formation of instabilities and its evolution into efficient growth rates depends on the way that the electron
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036
G.A. Casillas-Pe´rez et al. / Advances in Space Research xxx (2019) xxx–xxx
8
distribution evolves with time and space. This evolution may depends on the form and estimate of some parameters of the electron distribution function, such as the spectral index and the cut off energy of the electrons. Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgements We appreciate the comments and valuable suggestions of the anonymous referees that helped to improve the content of the article. Gilberto A. Casillas-Pe´rez expresses his gratitude to Miguel A. Garcı´a-Palacios from the IGFUNAM for technical work provided. References Abalde, J.R., Borotto, F.A., Chian, A.C.-L., 2001. Plasma emission mechanism for radio spikes from solar flares. Adv. Space Res. 28, 741– 746. Aschwanden, M.J., 1990. Relaxation of the loss-cone by quasi-linear diffusion of the electron-cyclotron maser instability in the solar corona. A&AS 85, 1141–1177. Aschwanden, M.J., 2005. Physics of the Solar Corona. An Introduction with Problems and Solutions, 2nd ed. Springer-Verlag, Berlin. Aschwanden, M.J., Benz, A.O., 1988. On the electron-cyclotron maser instability. I - Quasi-linear diffusion in the loss cone. II - Pulsations in the quasi-stationary state. Apj 332, 447–475. Aschwanden, M.J., Guedel, M., 1992. The coevolution of decimetric millisecond spikes and hard X-ray emission during solar flares. Apj 401, 736–753. 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 (2), 142. Casillas-Pe´rez, G.A., Jeyakumar, S., Carrillo-Vargas, A., Pe´rez-Enrı´quez, H.R., 2019. Study of solar radio spikes and their relation to energetic solar events. Sol. Phys. 294, 10. Casillas-Pe´rez, G.A., Jeyakumar, S., Pe´rez-Enrı´quez, H.R., Trinidad, M. A., 2016. Evolution of electron beam pulses of short duration in the solar corona. Adv. Space Res. 58, 1991–1996. Fleishman, G.D., Gary, D.E., Nita, G.M., 2003. Decimetric spike bursts versus microwave continuum. ApJ 593, 571–580. Guedel, M., Benz, A.O., Aschwanden, M.J., 1991. The association of solar millisecond radio spikes with hard X-ray emission. A&A 251, 285–297.
Hamilton, R.J., Lu, E.T., Petrosian, V., 1990. Numerical solution of the time-dependent kinetic equation for electrons in magnetized plasma. ApJ 354, 726–734. Hamilton, R.J., Petrosian, V., 1992. Stochastic acceleration of electrons. I - Effects of collisions in solar flares. ApJ 398, 350–358. Hannah, I.G., Kontar, E.P., Sirenko, O.K., 2009. The effect of waveparticle interactions on low-energy cutoffs in solar flare electron spectra. Astrophys. J. 707 (1), L45–L50. Holman, G.D., Aschwanden, M.J., Aurass, H., Battaglia, M., Grigis, P. C., Kontar, E.P., Liu, W., Saint-Hilaire, P., Zharkova, V.V., 2011. Implications of X-ray observations for electron acceleration and propagation in solar flares. Space Sci. Rev. 159 (1–4), 107–166. Huang, G., Nakajima, H., 2005. Location and parameters of a microwave millisecond spike event. Astrophys. Space Sci. 295 (4), 423–442. Kontar, E.P., Jeffrey, N.L.S., Emslie, A.G., 2019. Determination of the total accelerated electron rate and power using solar flare hard X-ray spectra. ApJ 871 (2), 225. Kontar, E.P., Ratcliffe, H., Bian, N.H., 2012. Wave-particle interactions in non-uniform plasma and the interpretation of hard X-ray spectra in solar flares. A&A 539, A43. Kuijpers, J., van der Post, P., Slottje, C., 1981. Runaway acceleration in a radio flare. A&A 103, 331–338. Leach, J., Petrosian, V., 1981. Impulsive phase of solar flares. I Characteristics of high energy electrons. ApJ 251, 781–791. Li, B., Cairns, I.H., Robinson, P.A., 2008. Simulations of coronal type III solar radio bursts: 1. Simulation model. J. Geophys. Res. (Space Phys.) 113, A06104. Li, B., Cairns, I.H., Yan, Y.H., Robinson, P.A., 2011. Decimetric Type III bursts: generation and propagation. ApJL 738, L9. Lifshitz, E.H., Pitaevskii, L.P., 1981. Physical Kinetics. Pergamon Press, Oxford. MacKinnon, A.L., 1986. Time-dependent density in the electron-trap model of solar flare hard X-ray bursts. A&A 163, 239–245. Melrose, D.B., Dulk, G.A., 1982. Electron-cyclotron masers as the source of certain solar and stellar radio bursts. ApJ 259, 844–858. Ratcliffe, H., 2013. Electron beam evolution and radio emission in the inhomogeneous solar corona. Ph.D. thesis. University of Glasgow, United Kingdom. Ratcliffe, H., Kontar, E.P., Reid, H.A.S., 2014. Large-scale simulations of solar type III radio bursts: flux density, drift rate, duration, and bandwidth. A&A 572, A111. Reid, H.A.S., Kontar, E.P., 2013. Evolution of the Solar flare energetic electrons in the inhomogeneous inner heliosphere. Sol. Phys. 285, 217– 232. Siversky, T.V., Zharkova, V.V., 2009. Stationary and impulsive injection of electron beams in converging magnetic field. A&A 504, 1057–1070. Zharkova, V.V., Gordovskyy, M., 2005. The kinetic effects of electron beam precipitation and resulting hard X-ray intensity in solar flares. A&A 432, 1033–1047.
Please cite this article as: G. A. Casillas-Pe´rez, S. Jeyakumar, A. Carrillo-Vargas et al., Evolution of an electron beam pulse influenced by coulomb collision effects in the solar corona, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.036