Modifications of thick-target model: re-acceleration of electron beams by static and stochastic electric fields
AAstronomy & Astrophysics manuscript no. varady_et_al c (cid:13)
ESO 2018September 18, 2018
Modifications of thick-target model: re-acceleration of electronbeams by static and stochastic electric fields
M. Varady , M. Karlický , Z. Moravec , and J. Kašparová J.E. Purkynˇe University, Physics Department, ˇCeské mládeže 8, 400 96 Ústí nad Labem, Czech Republic,e-mail: [email protected] Astronomical Institute of the Academy of Sciences of the Czech Republic, v.v.i., 25165 Ondˇrejov, Czech Republice-mail: [email protected]
Received 29 July 2013 / Accepted 21 November 2013
ABSTRACT
Context.
The collisional thick-target model (CTTM) of the impulsive phase of solar flares, together with the famous CSHKP model,presented for many years a “standard” model, which straightforwardly explained many observational aspects of flares. On the otherhand, many critical issues appear when the concept is scrutinised theoretically or with the new generation of hard X-ray (HXR)observations. The famous “electron number problem” or problems related to transport of enormous particle fluxes though the coronarepresent only two of them. To resolve the discrepancies, several modifications of the CTTM appeared.
Aims.
We study two of them based on the global and local re-acceleration of non-thermal electrons by static and stochastic electricfields during their transport from the coronal acceleration site to the thick-target region in the chromosphere. We concentrate on acomparison of the non-thermal electron distribution functions, chromospheric energy deposits, and HXR spectra obtained for bothconsidered modifications with the CTTM itself.
Methods.
The results were obtained using a relativistic test-particle approach. We simulated the transport of non-thermal electronswith a power-law spectrum including the influence of scattering, energy losses, magnetic mirroring, and also the e ff ects of the electricfields corresponding to both modifications of the CTTM. Results.
We show that both modifications of the CTTM change the outcome of the chromospheric bombardment in several aspects.The modifications lead to an increase in chromospheric energy deposit, change of its spatial distribution, and a substantial increase inthe corresponding HXR spectrum intensity.
Conclusions.
The re-acceleration in both models reduces the demands on the e ffi ciency of the primary coronal accelerator, on theelectron fluxes transported from the corona downwards, and on the total number of accelerated coronal electrons during flares. Key words.
Sun: flares – acceleration of particles – Sun: X-rays – Sun: chromosphere
1. Introduction
The CTTM of the impulsive phase of solar flares (Brown 1971)for many years presented a successful tool not only for inter-preting the processes related to the energy deposition and HXRproduction in the footpoint regions of flare loops, but also fornaturally explaining many other observational aspects of flareslike the Neupert e ff ect (Dennis & Zarro 1993), the time correla-tion of footpoint HXR intensity and intensities of chromosphericlines (Radziszewski et al. 2007, 2011), or the radio signaturesof particle transport from the corona towards the chromosphere(Bastian et al. 1998). Nevertheless, especially with the onset ofmodern HXR observations such as Yohkoh / HXT, RHESSI (Ko-sugi et al. 1991; Lin et al. 2002), a continuously growing num-ber of discrepancies with the CTTM were beginning to appear.The most striking one is the old standing problem concerning thevery high electron fluxes required to explain the observed highHXR footpoint intensities. This problem is particularly acutein the context of the “standard” CSHKP flare model when as-suming a single coronal acceleration site (Sturrock 1968; Kopp& Pneuman 1976; Shibata 1996), where enormous numbers ofelectrons involved in the impulsive phase have to be gathered,accelerated, and then transported to the thick-target region lo-
Send o ff print requests to : Varady cated in the chromosphere (Brown & Melrose 1977; Brown et al.2009). Another serious class of problems appears as a conse-quence of enormous electric currents arising from the transportof high electron fluxes through the corona down to the chro-mosphere and the inevitable generation of the neutralising re-turn current (van den Oord 1990; Matthews et al. 1996; Karlický2009; Holman 2012). Also the recent measurements of the verti-cal extent of chromospheric HXR sources (Battaglia et al. 2012)are inconsistent with the values predicted by the CTTM.Generally, it is very di ffi cult to explain energy transport bymeans of electron beams with enormous fluxes from the primarycoronal acceleration sites assumed to be located in highly struc-tured coronal current sheets (Shibata & Tanuma 2001; Bártaet al. 2011a,b) to the thermalisation regions that lie relativelydeep in the atmosphere and that produce the observed intensi-ties of footpoint HXR emission in the frame of classical CTTM.Therefore various modifications of the CTTM have been pro-posed to solve the problems. Fletcher & Hudson (2008) suggesta new mechanism of energy transport from the corona down-wards by Alfvén waves, which in the chromosphere accelerateelectrons to energies for X-ray emission. Furthermore, Karlický& Kontar (2012) have investigated an electron acceleration inthe beam-plasma system. Despite e ffi cient beam energy lossesto the thermal plasma, they have found that a noticeable part of Article number, page 1 of 15 a r X i v : . [ a s t r o - ph . S R ] J a n he electron population is accelerated by Langmuir waves pro-duced in this system. Thus, the electrons accelerated during thebeam propagation downwards to the chromosphere can reducethe beam flux in the beam acceleration site in the corona re-quested for X-ray emission. Another modification of the CTTMis the local re-acceleration thick-target model (LRTTM) that hasbeen suggested by Brown et al. (2009). The model assumes aprimary acceleration of electrons in the corona and their trans-port along the magnetic field lines downwards to the thick-targetregion. Here they are subject to secondary local re-accelerationby stochastic electric fields generated in the stochastic currentsheet cascades (Turkmani et al. 2005, 2006) excited by randomphotospheric motions.Karlický (1995) studied another idea – the global re-acceleration thick-target model (GRTTM). The beam electronsaccelerated in the primary coronal acceleration site are ontheir path from the corona to the chromosphere constantly re-accelerated. Such a re-acceleration is caused by small staticelectric fields generated by the electric currents originating dueto the helicity of the magnetic field lines forming the flare loop(e.g. Gordovskyy & Browning 2011, 2012; Gordovskyy et al.2013). The magnitude of the static electric field reaches its max-imum in the thick-target region owing to the sharp decrease inelectric conductivity in the chromosphere and to the prospectiveconvergence of magnetic field in this region.In this paper we study the e ff ects of the local and global re-acceleration of beam electrons at locations close to the hard X-ray chromospheric sources. Section 2 describes our approxima-tions of LRTTM and GRTTM and their implementation to a rel-ativistic test-particle code. In Section 3 we compare both modifi-cations with CTTM in terms of electron beam distribution func-tions, chromospheric energy deposits, and HXR spectra. Mod-elled HXR spectra are also forward-fitted to obtain beam param-eters under the assumption of pure CTTM regardless of any re-acceleration. The results are summarised and discussed in Sec-tion 4.
2. Model description
The simulations presented in this work start with an injection ofan initial electron beam into a closed magnetic loop at its summitpoint using a test-particle approach (Varady et al. 2010). Phys-ically, the initial beam represents a population of non-thermalelectrons generated at the primary acceleration site located in thecorona above the flare loop. Our simulations do not treat the pri-mary acceleration itself. The non-thermal electrons are assumedto obey a single power law in energy, so their initial spectrum (inunits: electrons cm − s − keV − ) is F ( E , z ) = ( δ p − F E (cid:16) EE (cid:17) − δ p , for E ≤ E ≤ E , for other E (1)(Nagai & Emslie 1984). The electron flux at the loop top, whichcorresponds to the column density z =
0, is determined by thetotal energy flux F , the low and high-energy cuto ff s E , E andthe power-law index δ p . All the models presented in this workstart with the same initial beam parameters δ p = E =
10 keVand E =
400 keV. For F we use two values F = × and 10 erg cm − s − , with the latter only as the CTTM ref-erence flux for a comparison with the models of secondary re-acceleration. We study two various cases of initial pitch angle distribution.The pitch angle ϑ determines the angle between the non-thermalelectron velocity component parallel to the magnetic field line (cid:51) (cid:107) and the total electron velocity (cid:51) µ ≡ cos ϑ = (cid:51) (cid:107) (cid:51) . (2)The initial µ -distribution is given by function M ( µ ) and must benormalised. The angularly dependent initial electron flux is then F ( E , µ , z ) = M ( µ ) F ( E , z ) , (cid:90) − M ( µ )d µ = . (3)We consider two extreme cases:1. a fully focussed beam M FF ≡ M ( µ ) = δ ( µ − µ c ) , (4)where δ is the Dirac function and µ c = ±
1; and2. a semi-uniformly distributed beam M SU ≡ M ( µ ) = (cid:40) , µ ∈ ( − , − . ∪ (0 . , , µ ∈ ( − . , . . (5)The initial pitch angle distribution reflects the properties of theprimary coronal accelerator. The first distribution may representan extreme case of an electron beam accelerated in the coro-nal current sheet with an X-point, and the second is close to theoutcome of the acceleration mechanisms involving the plasmawave turbulence in a second-order Fermi process (Winter et al.2011). The electrons with negative µ propagate to the left, withpositive µ to the right half of the loop. Since we study the ef-fects of the electron beam bombardment of the chromosphere,we excluded the population with µ ∈ ( − . , .
5) from the uni-form distribution. This approximation substantially decreasesthe computational cost. The choice of M ( µ ) influences the ini-tial energy flux along magnetic field lines towards a single leftor right footpoint. The parallel fluxes towards individual foot-points are F / M FF and 3 F / M SU , respectively. Thetotal number of non-thermal electrons injected into the loop perunit area and time is ≈ × electrons cm − s − (relevant tothe energy flux F = × erg cm − s − and both pitch angledistributions).We consider a converging magnetic field along the looptowards the photosphere with a constant mirror ratio R m ≡ B / B =
5, where B and B are the magnetic fields at the looptop in the corona and at the base of the loop in the photosphere,respectively. To model the field convergence we adopted the for-mula proposed by Bai (1982), where the magnetic field strength B is only a function of the column density z calculated from theloop top downwards B ( z ) B = (cid:40) + ( R m − z / z m ) , for z ≤ z m R m , for z ≥ z m , (6)where z m = × cm − . For the VAL C atmosphere (Vernazzaet al. 1981) z m is located in the chromosphere – correspondingposition s m = .
36 Mm, temperature T m = n m = × cm − . The adopted configuration of the mag-netic field is shown in Fig. 1. The convergence of the magneticfield in the vicinity of the loop footpoints influences the modelin two aspects. First, only part of the beam electrons with lowpitch angles satisfying the condition sin ϑ ≤ R m passes through Article number, page 2 of 15. Varady et al.: Modifications of thick-target model: re-acceleration of electron beams ...
Fig. 1.
Left:
Hydrogen ionisation (black line) and relative magnetic field strength B / B (blue line), right temperature (red line), and hydrogendensity (black line) in the lower parts of the VAL C atmosphere. The dashed line indicates the lower boundary of the magnetic mirror. the magnetic mirror. Second, the corresponding flux is focussedthanks to the field convergence that results in an increase in theenergy deposit per unit volume in the constricted flux tube. Theremaining beam particles are reflected by the mirror and moveback to the loop top and further to the second part of the loop(Karlický & Henoux 1993).The corresponding energy deposits, non-thermal electrondistribution functions, and the HXR spectra are determined pri-marily by the parameters of the electron beam itself, but also bythe properties of the target atmosphere. The results are obtainedfor the VAL C atmosphere (see Fig. 1) (Vernazza et al. 1981),which was extrapolated to the hot ∼ –10 cm − corona. The length of the whole loop is L =
20 Mm,so the source of the energetic particles (primary coronal acceler-ation site) is located at s =
10 Mm.The hydrodynamic flare models show that a rapid and mas-sive flare energy release in the thick-target region dramaticallychanges the temperature and ionisation structure in the chromo-sphere on very short timescales ≤ The problem of collisional particle transport in a partially ionisedatmosphere in the cold target approximation was analysed byEmslie (1978). Bai (1982) presented a Monte-Carlo method thatis useful for computer implementation of the transport of en-ergetic electrons in a fully ionised hydrogen plasma in a non-uniform magnetic field. It has been shown by MacKinnon &Craig (1991) that the coupled system of stochastic equations pre-sented in Bai (1982) is formally equivalent to the correspondingFokker-Planck (FP) equation, therefore the method proposed byBai (1982) has to give equivalent results as the direct solutionof the FP equation. We modified the approach of Bai (1982) for a partially ionised cold target and developed a relativistic test-particle code. The code follows the motion of a chain of beamelectron clusters, test-particles with a power-law spectrum alonga magnetic field line described with the following equation ofmotiond p e d t = − C e ( (cid:51) e ) + F m − e E , (7)where p e is the momentum of the electron cluster, − C e ( (cid:51) e ) isthe collisional drag also responsible for the e ff ects of scattering, F m is the magnetic mirror force, and the term − e E expresses theforce controlling the secondary acceleration. In the scenario of classical CTTM, the non-thermal electronslose their energy and are scattered by the Coulomb collisionswith the particles of the ambient plasma (see the term − C e ( (cid:51) e )in equation (7)). The energy loss of a non-thermal electron ∆ E with kinetic energy E and velocity (cid:51) caused by Coulomb colli-sions in a partly ionised hydrogen cold target, per time-step ∆ t ,can be approximated by ∆ E = − π e E (cid:2) Λ x + Λ (cid:48) (1 − x ) (cid:3) n (cid:51) ∆ t , (8)where n = n p + n n is the number density of equivalent hydrogenatoms, n p and n n are the proton and hydrogen number densities,respectively, x = n p / n is the hydrogen ionisation, and Λ , Λ (cid:48) arethe Coulomb logarithms (Emslie 1978).The scattering due to Coulomb collisions is simulated usingthe Monte Carlo method. According to Bai (1982), the relationbetween the rms of the scattering angle ∆ ϑ C , the ratio ∆ E / E ,and the Lorentz factor γ L is ∆ ϑ = (cid:32) ∆ EE (cid:33) (cid:32) γ L + (cid:33) , (9)when ∆ ϑ (cid:28) ∆ E / E (cid:28) ∆ ϑ C is given by a Gaussian distribution, the rmsof which is computed by the equation (9).The change in the pitch angle caused by the magnetic force F m , see equation (7), in the region of magnetic field convergenceis ∆ ϑ B = B i + − B i B i tan ϑ i , (10) Article number, page 3 of 15 ig. 2.
Classical electric conductivity σ in the lower VAL C atmo-sphere according to Kubát & Karlický (1986) and the magnitude of thecorresponding E G for various current densities j . providing ( ∆ ϑ B ) (cid:28) B i and B i + are the magnetic field strengths at the beginning and end of theparticle path, and ϑ i is the initial pitch angle. The total change ofthe pitch angle in a single time-step due to collisions and mag-netic field non-uniformity is ∆ ϑ = ∆ ϑ C + ∆ ϑ B , and the new elec-tron pitch angle ϑ is then obtained using the cosine rule from thespherical trigonometrycos ϑ = cos ϑ i cos ∆ ϑ + sin ϑ i sin ∆ ϑ cos ϕ , (11)where ϕ is the azimuthal angle given by a uniform distribution0 ≤ ϕ < π . More details concerning computer implementationcan be found in Varady et al. (2005, 2010) and Kašparová et al.(2009). To include the secondary acceleration mechanisms, we added ei-ther the static or stochastic electric fields that re-accelerate ordecelerate the test-particles with respect to the mutual direc-tions of the electric field and instantaneous test-particle veloc-ities. The interaction of the non-thermal particles with the re-accelerating electric field, the − e E term in equation (7), is cal-culated using the Boris relativistic algorithm (see Peratt 1992,Sect. 8.5.2). The e ff ects of the return current are not consid-ered. Relatively low electron fluxes transported from the corona( F / = . × erg cm − s − towards each footpoint) partiallyjustify this negligence. We now consider a situation where electric currents flow in theflare loop before and during the flare impulsive phase due to thenon-zero helicity of the pre-flare magnetic field (Karlický 1995).Furthermore, at the very beginning of the flare, the current-carrying loops are unstable to the kink and tearing-mode insta-bilities, which produce filamented electric currents in a naturalway (Kuijpers et al. 1981; Karlický & Kliem 2010; Kliem et al.2010; Gordovskyy & Browning 2011). If electrons are accel-erated in the coronal part of the individual current thread, theypropagate along it and interact with the corresponding global re-acceleration resistive static electric field E G driving the current.The field corresponding to the current density j is E G = j /σ , (12) where σ is the plasma electric conductivity. The general formulafor plasma conductivity is σ = ω πν e , (13)where ω = π e n e / m e is the electron plasma frequency, and ν e the electron collisional rate. In case electric currents propagatein plasma free of any plasma waves, the collisional frequencycorresponds to the classical value ν e = . × − n e T / Λ , (14)in the SI units, where T e is the electron temperature. On the otherhand, the presence of plasma waves can increase the collisionalfrequency to anomalous values: for the anomalous resistivity seeHeyvaerts (1981).To assess the influence of static electric field on the outcomeof the chromospheric bombardment by non-thermal electrons,we assume a single thread of constant current density with mag-nitude below any current instability thresholds. Then we cal-culate the magnitude of corresponding direct field E G along thethread using the classical isotropic electric conductivity obtainedby Kubát & Karlický (1986). The conductivity was calculatedusing the updated values of proton–hydrogen scattering cross-section for the quiet VAL C atmosphere (see Fig. 2). Owing totemperature dependence of σ and the convergence of magneticfield in the chromosphere, contributing to the increase in the lo-cal current density, the resulting E G grows rather quickly in thechromosphere (see Fig. 2). Furthermore, E G tends to acceleratethe beam electrons towards one footpoint and to decelerate themtowards the second one, providing an asymmetric flare heating ofthe individual thread footpoints. From now on, we refer to the in-dividual footpoints as the primary and the secondary footpoints,respectively and to this model as the global re-accelerating thick-target model (GRTTM).The steep increase in E G , hence the high e ffi ciency ofGRTTM, is essentially linked with the decrease in temperaturein the chromosphere. In contrast, we have already pointed outthat chromospheric plasma in flares is heated to temperatures upto 10 K on the timescales ≤ σ ∝ T / ) in the corresponding region, and by the samefactor it decreases the electric field E G , so the flare heating ofthe chromosphere should basically cease the re-acceleration inthe thick-target region very early after the start of the impulsivephase. On the other hand, under the flare conditions, genera-tion of a high anomalous resistivity could be expected due toplasma instabilities, so the accelerating mechanism could con-tinue working. Inspired by Brown et al. (2009) and Turkmani & Brown(2012), we produced a simplified local re-acceleration thick-target model (LRTTM). To approximate the distribution of elec-tric fields arising as a consequence of a current sheet cascadein the randomly stressed magnetic fields (Turkmani et al. 2005,2006), we assume a region (between 1 – 2 Mm) of stochas-tic re-acceleration electric field E L , spatially modulated by thefunction shown in Fig. 3 (bottom). The position of the local re-acceleration region is one of the free parameters of the model. Itroughly corresponds to the chromosphere and encompasses the Article number, page 4 of 15. Varady et al.: Modifications of thick-target model: re-acceleration of electron beams ... regions of magnetic field convergence and the rapid change ofhydrogen ionisation (see Fig. 1).The stochastic electric fields E L are generated only in thedirections parallel and anti-parallel relative to the loop axis, andtheir distribution corresponds to Gaussians with various meanvalues E L and variances var( E L ) = E − E L2 . We examine twotypes of E L : E L -I. A stochastic electric field with zero mean value E L = , var( E L ) > . (15) E L -II. A combination of spatially localised static electric fieldwith a stochastic component (see Fig. 3) E L (cid:44) , var( E L ) ≥ . (16)In the case of E L -II, the sign of the static component E L alwaysassures acceleration of the non-thermal electrons towards thenearest footpoint. This field type can develop in the thick-targetregion if stochastic fields are present in a globally twisted mag-netic loop. In comparison with the GRTTM, the LRTTM is char-acterised by abrupt changes in magnitude and orientation of theaccelerating or decelerating electric fields representing the indi-vidual current sheets in the thick-target region (Turkmani et al.2006) (compare Figs. 2 and 3).The integration of motion of individual beam electron clus-ters for the LRTTM is performed in the following way. In eachtime-step (corresponding to ∆ t = × − s), we generate a ran-dom value of E L for each particle within the acceleration region.In this way we model the situation where the beam electronsare moving in the stochastic electric fields, whose configurationtemporally changes. Therefore, the electrons only have a neg-ligible chance of passing through exactly the same configura-tion of current sheets and of experiencing the same acceleration(deceleration) sequence. The time-step basically determines thespatial extent of the individual current sheets. In order to keepthe size independent of particle velocities, we weight E L usinga factor (cid:51) / (cid:51) , where (cid:51) and (cid:51) are the velocities corresponding tothe low-energy cuto ff and to the particular particle, respectively.The time-step ∆ t = × − s thus corresponds to the currentsheet size ∼ E L we relativistically movethe electron from the old to the new position. Then we calculatethe energy loss and scattering due to the passage of the particlethrough the corresponding column of plasma and the e ff ects ofconverging magnetic field. This is done repeatedly for the wholepopulation of test-particles. The corresponding total energy de-posit and HXR spectrum are then calculated. The intensity I ( (cid:15), s ) [photons cm − s − keV − ] of HXRbremsstrahlung observed on energy (cid:15) , emitted by plasmaat a position s along the flare loop, detected in the vicinity of theEarth, was calculated using the formula (Brown 1971) I ( (cid:15), s ) = n p ( s ) V ( s )4 π R (cid:90) ∞ (cid:15) Q ( E , (cid:15) ) (cid:51) ( E ) n ( E , s ) d E . (17)Here, n p ( s ) V ( s ) is the total number of protons in the emittingplasma volume V ( s ) at a position s , distance R = (cid:51) ( E )is the electron velocity calculated relativistically from the elec-tron energy, and n ( E , s ) is the number density of non-thermal Fig. 3.
Top:
Example of E L -II type stochastic electric field with E L = . − and var( E L ) = . − corresponding to the distributionfunction in Fig. 13. Bottom:
The the spatial modulation of E L . electrons per energy in the emitting volume having kinetic en-ergy E . The cross section Q ( E , (cid:15) ) for bremsstrahlung was cal-culated using a semi-relativistic formula given by (Haug 1997),multiplied by the Elwert factor (Elwert 1939), considering thelimit case when the entire electron kinetic energy is emitted. Theprecision of the method should be better than 1 % for energies ≤
300 keV (Haug 1997). To calculate the emitting volume V ( s )we assume a circular cross section of the converging loop witha radius r ( s ) = . √ B / B ( s ) Mm. The HXR spectra are calcu-lated on a spatial (height) grid ( s , s +∆ s ). The individual emittingvolumes along the grid are then V ( s ) = π r ( s ) ∆ s .
3. Results
We now concentrate on a comparison of outcomes of chromo-spheric bombardment for two modifications of CTTM with theCTTM itself. In this section we present the non-thermal elec-tron distribution functions in the vicinity of footpoints and sev-eral properties of the corresponding energy deposits and HXRintensities and spectra. The quantitative results for the CTTM,GRTTM, and both considered types of LRTTM are summarisedin Figs. 8, 11, and 12 and Tables 1, 2, and 3, respectively (Ta-bles 2 and 3 available only in the online version). Here, the factor F R / F gives the ratio of the reflected (due to the magnetic mir-roring, re-acceleration, and backscattering) to the original non-thermal electron energy flux coming from the corona at position s = t = . Article number, page 5 of 15 ig. 4.
CTTM time evolution of distribution functions of non-thermal electron energies versus positions with a colour coded M ( µ ) in the VAL Catmosphere. Left: M FF , right: M SU . From top to bottom: individual snapshots at t = . , . , . F / = . × erg cm − s − . The dotted horizontal linesindicate the bottom boundary of the magnetic mirror. Only the lower part of the loop and one footpoint are displayed. posits for the individual models, we calculate the total energydeposited into the chromosphere along a magnetic flux tube as E ch = (cid:90) chromosphere E dep ( s )d V ( s ) = S B . (cid:90) E dep ( s ) B ( s ) d s (18)and give the position of the energy deposit maximum s max inthe atmosphere. The factor B / B ( s ) in integral (18) accounts forthe convergence of the magnetic field, E dep ( s ) is the local energydeposit in units [erg cm − s − ], and the limits of integration cor-respond to the upper and lower boundaries of the chromosphere.The lower limit lies far below the stopping depths of the beamelectrons for all the studied models. When all the beam energyis deposited into the chromosphere and S = , the value of E ch in units [erg s − ] corresponds to the value of the initial flux F . For HXR we give the intensity I
25 keV and the power-law in-dex γ
25 keV measured at energy 25 keV. Furthermore, we appliedthe RHESSI spectral analysis software (OSPEX) to modelled http://hesperia.gsfc.nasa.gov/rhessi2/home/software/spectroscopy/spectral-analysis-software/ total X-ray spectra to imitate common spectral analysis. We as-sumed that these spectra were incident on RHESSI detectors andforward-fitted the “detected” count spectra. In the fitting we usedthe OSPEX thick-target model and a single power-law injectedelectron spectrum. In this way we obtained the fitted electronbeam parameters. To account for the non-uniform ionisationstructure of the X-ray emitting atmosphere, the fitting function f_thick_nui in the step-function mode was chosen. When thefitted parameters of f_thick_nui were unrealistic and the X-ray emission was formed deep in the layers of almost neutralplasma, f_thick with neutral energy loss term was used. Also,we modified the standard OSPEX energy loss term and the ratioof Coulomb logarithms to be consistent with relations used inthe test-particle code. The results of this analysis, the fitted en-ergy flux F (cid:48) , the power-law index δ (cid:48) p , and the low-energy cuto ff E (cid:48) are listed in Tables
1, 2, and 3 and displayed in Figs. 8, 11,and 12. Tables 2 and 3 are online only.Article number, page 6 of 15. Varady et al.: Modifications of thick-target model: re-acceleration of electron beams ...
Fig. 5.
Left:
CTTM instantaneous energy deposits into the VAL C atmosphere at t = F / = × erg cm − s − (redlines) and F / = . × erg cm − s − (black lines). The dotted vertical line indicates the bottom boundary of the magnetic mirror. Right:
TheHXR spectra integrated over one half of the loop. In both panels the solid lines represent M FF , the dotted lines M SU case. To produce a basis for comparison we present results for theclassical CTTM in a converging magnetic field. The informa-tion on kinematics of non-thermal electrons for both initial µ -distributions we considered is incorporated into Fig. 4. We firstconcentrate on the left-hands panels showing the time depen-dent distributions for M FF case. The top panel for t = . . . ff and the bottom bound-ary of the magnetic mirror is dominated by red, so a vast ma-jority of particles move downwards with µ ≈
1. At low ener-gies ( E <
20 keV), a low-energy tail of particles starts to formin the region under the lower boundary of the magnetic mirror.It consists of particles with originally higher energies that lostpart of their energy owing to their interactions with the targetplasma. The tail is rich in particles with µ ≈ µ ≈ − <
20 keV)with − ≤ µ < t = .
15 s, when even the particles with lowest energies reachedthe thick-target region, shows the proceeding thermalisation ofbeam electrons in this region and increase in particle numberwith µ ≤ µ ≈ − t = . E <
20 keV is dominated by particleswith µ ≈
0. The reflected energy flux propagating upwards isapproximately 4% of the original flux F for the M FF case (seeTable 1).The distribution functions corresponding to M SU are shownin Fig. 4 (right). The overall behaviour of the beam electrons is quite similar to the previously discussed case. The most obviousdi ff erence is the enhancement of the particle populations with µ < F ) and µ ≈ E <
40 keV) lo-calised above the bottom boundary of the magnetic mirror. Thedi ff erences between the M FF and M SU cases naturally influencethe resulting energy deposits and properties of the correspond-ing HXR emission (see Figs. 4, 5). The CTTM in the adoptedarrangement gives identical results for both footpoints. There-fore for t > . t ≤ . ff ects of the µ -distribution and magneticfield convergence, Table 1 also lists the characteristics of CTTMfor the case of no magnetic mirror, i.e. R m =
1. It shows thatit is the magnetic field convergence that significantly influences F R / F and E ch in the case of M SU .A comparison of energy deposits for both considered initial µ -distributions is shown in Fig. 5 (left). Because the adopted en-ergy flux for both models considering secondary re-acceleration F / = . × erg cm − s − is unrealistically low in thecontext of CTTM and flare physics, we also plot energy de-posits for the much higher and more realistic value F / = × erg cm − s − . The results corresponding to this flux willbe used as a basis for comparison with the energy deposits andHXR spectra obtained from the models involving the secondaryacceleration mechanisms. The chromospheric energy deposit E ch scales linearly with F (see Table 1), and the positions of en-ergy deposit maxima are almost identical for all the consideredcases approximately corresponding to the placement of the lowerboundary of the magnetic mirror s max = .
36 Mm. The peak inthe energy deposits at s max and their steep decrease above it (seeFig. 5, left) are caused by the constricted magnetic flux tube.The influence of the initial µ -distribution is obvious. For the M FF case, particles have a greater chance of passing through the mag-netic mirror and thus of depositing their energy into the deeperlayers. In the M SU case, when the particles reach the thick-targetregion and the region of strongly converging field, their pitch an-gles are generally higher: compare the left-hand and right-handpanels of Fig. 4. Therefore the probability that an electron passesthrough the magnetic mirror is strongly reduced. This naturallyexplains the systematic enhancements in the energy deposits for M SU in the layers above and the decrease in the layers below the Article number, page 7 of 15 able 1.
Summary of results for the CTTM. F / × F R / F E ch / s max I
25 keV γ
25 keV F (cid:48) / × δ (cid:48) p E (cid:48) [erg cm − s − ] [%] [erg s − ] [Mm] [cm − s − keV − ] [erg cm − s − ] [keV] R m = R m = Notes. F – the initial energy flux, F R / F – ratio of reflected to initial energy flux at s = t = . E ch – integrated chromosphericenergy deposit, s max – position of energy deposit maximum, I , γ – HXR intensity and power-law index measured at energy 25 keV, F (cid:48) , δ (cid:48) p , and E (cid:48) – the fitted values of energy flux, power-law index, and low-energy cuto ff , respectively. The non-parenthetical and parenthetical valuesare for the M FF and M SU cases of M ( µ ), respectively. Applies to further online tables. Fig. 6.
GRTTM distribution functions of non-thermal electron energies versus positions with a colour coded M ( µ ) corresponding to the currentdensity j = − in the VAL C atmosphere at time t = . Top: primary footpoint, bottom: secondary footpoint, left: M FF , right: M SU . The solid lines indicate the instantaneous energy deposits corresponding to F / = . × erg cm − s − , the dotted horizontal lines the bottom boundary of the magnetic mirror and the blue ellipses labelled L and H denote tails in theparticle distribution function. Only the vicinity of the footpoints are displayed. lower boundary of the magnetic mirror in comparison with the M FF case.The corresponding HXR spectra are shown in Fig. 5 (right),and their parameters are summarised in Table 1. As expected,the HXR intensity I
25 keV scales linearly with the chromosphericdeposit E ch or the energy flux F . Majority of the total X-rayemission, i.e. summed over the whole loop, comes from the re-gions below the bottom boundary of the magnetic mirror. Asexplained above, the number of particles passing through themagnetic mirror is lower in the M SU case than for M FF , there- fore the HXR emission corresponding to M FF is more intensethan the emission of M SU .HXR spectra are steeper in the M SU case owing to presenceof magnetic field convergence – compare R m = E ch , whereas δ (cid:48) p and E (cid:48) are the same as those of the in-jected power law. An exception is the larger δ (cid:48) p in the M SU case,which corresponds to the mentioned HXR spectral behaviourand the fact that the spectral fitting does not take the scatteringinduced by change in B into account. Article number, page 8 of 15. Varady et al.: Modifications of thick-target model: re-acceleration of electron beams ...
Fig. 7.
GRTTM instantaneous energy deposits ( left ) and HXR spectra ( right ) for the primary ( top ) and secondary ( bottom ) footpoints and theVAL C atmosphere at t = . M FF ) and dotted ( M SU ) lines correspond to the current densities j = , , − ,respectively, and to the energy flux F / = . × erg cm − s − . The grey dashed and solid lines correspond to the CTTM with F / = . × and 5 × erg cm − s − , respectively. The dotted straight vertical line indicates the bottom boundary of the magnetic mirror. The HXR spectraare integrated over one half of the loop. The e ff ects of static (global) electric field E G was studied forcurrent densities in the range from 1 A m − to 6 A m − . The dis-tribution functions of non-thermal electrons for current density j = − and time t = . M FF case. A faint low-energy tail at energies E <
20 keV,located above the bottom boundary of the magnetic mirror, ispredominantly formed of particles with µ ≤ ff ectsof particle scattering and magnetic field convergence, comparewith Fig. 4 (left). This tail becomes more apparent for distribu-tions that correspond to lower j (see Fig. 4). On the other hand,a prominent high-energy tail, on energies from 20 to 300 keVstretching from 1.7 to 0.5 Mm (see the regions labelled H inFig. 6), does not have any counterpart in Fig. 4 for the CTTM.The tail is formed of re-accelerated and relatively focussed par-ticles with µ ≈
1. Another obvious e ff ects of E G are the increasein beam penetration depth with growing j and a weakening of thepopulation of reflected and back-scattered particles propagating towards the secondary footpoint that corresponds to 0.7% of theinitial beam flux only, see Fig. 8 (left).Figure 6 (top right, M SU case) exhibits essentially the samefeatures. The most apparent distinctions between the two dis-tributions are a much richer population of particles in the low-energy tail located above the bottom boundary of the magneticmirror and the existence of a relatively rich population of re-flected and back-scattered particles with µ < ff erences between the distributions corresponding to M FF and M SU cases are solely e ff ects of the initial µ -distribution.The situation at the secondary footpoint is shown in Fig. 6(bottom). In addition to the e ff ect of Coulomb collisions, thefield E G constantly decreases the parallel velocity component ofthe particles propagating towards the secondary footpoint. Thisresults in the formation of an enhanced low-energy tail in theparticle distribution functions located above the bottom bound-ary of the magnetic mirror. Another obvious feature is a richpopulation of reflected or back-scattered particles correspondingapproximately to 15% and 54% of the initial beam flux for the M FF and M SU cases, respectively (see Fig. 8, bottom left). Theseparticles are accelerated by the global field E G back, towards theprimary footpoint. Article number, page 9 of 15 ig. 9.
LRTTM E L -I type distribution functions of the non-thermal electron energies versus positions with a colour-coded M ( µ ) correspondingto E L = − and var( E L ) = − in the VAL C atmosphere at time t = . Left: M FF ; right: M SU . The solid lines indicate the instantaneous energy deposits corresponding to F / = . × erg cm − s − , the dotted horizontal linesthe bottom boundary of the magnetic mirror, the grey area the secondary re-acceleration region, and the blue ellipses labelled L and H denote tailsin the particle distribution function. Only the vicinity of the footpoints is displayed. Fig. 8.
GRTTM summary of calculated parameters of chromosphericbombardment for various current densities j . The solid lines and as-terisks denote the M FF , the dashed lines and triangles denote the M SU . Left: chromospheric energy deposit E ch (lines) and fitted energy flux F (cid:48) (symbols) (top) , the ratio F R / F (bottom) for the primary (blue) andsecondary (green) footpoints. Right: position of energy deposit maxi-mum s max and HXR intensity I ( top ), HXR spectral index γ ,fitted electron beam spectral index δ (cid:48) p and low-energy cuto ff E (cid:48) ( bottom )only for the primary footpoint. The instantaneous energy deposits and HXR spectra for boththe primary and secondary footpoints and various current den-sities are shown in Fig. 7, and the quantitative results, some ofthem only for the primary footpoint, are summarised in Fig. 8(see Table 2 for complete results). The magnitudes and spatialdistributions of energy deposits in the atmosphere, as well asthe production of HXR photons, are extremely sensitive to thecurrent densities in the threads. According to our simulations,the current density j = − increases E ch at the primaryfootpoint of one order and I
25 keV of approximately two orders (see Fig. 8). Moreover, this HXR spectrum is more intense thanthe spectrum of pure CTTM with F / = × erg cm − s − (see Fig. 7, top right). The presence of j also considerablychanges the distribution of the energy deposit in the thick-targetregion. The maximum of the energy deposit s max is substantiallyshifted towards the photosphere (compare the results for j = j > j = − , E ch is comparable to F / = × erg cm − s − of pure CTTM, however the spatial distribution is completelydi ff erent.HXR emission of the primary footpoint comes predomi-nantly from regions well below the bottom of the magnetic mir-ror, close to temperature minimum for j (cid:38) − and photonenergies (cid:38)
50 keV. As j increases, HXR spectra get more intenseand flatter at deka-keV energies, and the maximum photon en-ergy is shifted to higher energies. This is all consistent with thepresence of the high-energy electrons accelerated by E G belowthe magnetic mirror. Although the HXR power-law index γ
25 keV tends to harden as j increases, the fitted CTTM injected elec-tron power-law index δ (cid:48) p becomes steeper. However, at the sametime, the low-energy cuto ff E (cid:48) rises to deka-keV values, caus-ing decrease in γ
25 keV – see fitted parameters in Fig. 8 (bottomright).The model of j = − is similar to the CTTM situation;i.e. similar formation heights of HXR, spectral shape of photonspectrum (Fig. 7, left), and fitted electron distribution (Fig. 8,bottom right). In the case of j = − , the HXR spectra areextremely flat below ∼
40 keV with E (cid:48) ∼
100 keV. Such low-energy cuto ff s are not found from observations, therefore thiscase could represent a limit of possible j in flare loops.The situation at the secondary footpoint is di ff erent (seeFig. 7, bottom). Because a part of energy carried by non-thermalparticles is drained due to the actuation of E G , the resulting chro-mospheric energy deposits for a particular j are smaller than atthe primary footpoint. As expected, this behaviour steeply in-creases with j . Although the HXR spectra of the secondary foot-point are less intense than the spectrum of pure CTTM, the over-all spectral shape is not changed significantly. Consequently, thefitted injected electron beam parameters show only a decrease of F (cid:48) consistent with lower E ch (see Fig. 8, top left) and Table 2. Article number, page 10 of 15. Varady et al.: Modifications of thick-target model: re-acceleration of electron beams ...
Fig. 10.
LRTTM E L -I type instantaneous energy deposits ( left ) and HXR spectra ( right ) for the VAL C atmosphere at t = . M FF )and dotted ( M SU ) blue, green, red, yellow, and orange lines correspond to E L = − and var( E L ) = . , . , , , − , respectively, andenergy flux F / = . × erg cm − s − . The dashed and solid grey lines correspond to the CTTM with F / = . × and 5 × erg cm − s − ,respectively. The dotted straight vertical line indicates the bottom boundary of the magnetic mirror, the grey area the secondary re-accelerationregion. The HXR spectra are integrated over one half of the loop. E L -I type The non-thermal electron distribution functions for the stochas-tic field with E L = − and var( E L ) = − in the VAL Catmosphere and time t = . E L (normally distributed) of parallel oranti-parallel orientation relative to µ =
1. The net accelerationin this type of electric field is a consequence of inverse propor-tionality between the electron collisional energy loss and energyd E / d z ∝ / E , z being the column density (Emslie 1978). Theenergy gain of re-accelerated electrons increases with var( E L )similar to the fuzziness of the high-energy tails and the fluxesof backwards moving electrons (with µ < F R / F corresponding to var( E L ) = − is approximately 31% and120% for the M FF and M SU cases, respectively; i.e., in the lattercase the backward energy flux exceeds the initial flux propagat-ing downwards from the corona (see Fig. 11, right). Anothere ff ect of growing var( E L ) is a decrease in the electron popula-tion having µ distinct from 1 or −
1. Ultimately, for high val-ues of var( E L ), only particles with µ either close to 1 or − µ -distribution then copies thedirectional distribution of the re-accelerating field.The inconspicuous low-energy tail spreads from the top ofthe re-acceleration region to the lower boundary of the magneticmirror, and it is formed of particles of all possible pitch angleswith energies under 20 keV. It is shifted higher into the chromo-sphere in comparison to the low-energy tail in the CTTM case(see Fig. 4). As var( E L ) increases, the low-energy tail becomesless distinct and its location is shifted higher towards the upperboundary of the re-acceleration region. The low-energy tail isformed by concerted actuation of Coulomb collisions and alter-nating stochastic field. The energy deposits and HXR spectra corresponding to vari-ous values of var( E L ) in the range from 0.1 to 5 V m − are shownin Fig. 10 and their main parameters E ch , s max , I
25 keV , and γ
25 keV are displayed in left-hand panels of Figs. 11 and 12 and sum-marised in Table 3.The behaviour of the energy deposits is similar to theGRTTM of the primary footpoint. They increase with var( E L ), s max are shifted to the deeper layers, and the energy is depositedinto an even narrower chromospheric region. For the loweststudied value var( E L ) = . − , we obtained practically nochange in all followed parameters relative to the CTTM with anidentical initial flux (see Figs. 10, 11 and 12).On the other hand, for the maximum value var( E L ) = − there is half an order increase in E ch and a substantialshift of s max towards the photosphere ( ∼
750 km) for both initial µ -distributions. The value of I
25 keV increases considerably (28 × for the M FF and 10 × for the M SU case) relative to the CTTMwith an identical initial flux (see Fig. 12, left).Again, hard X-ray emission comes from the regions belowthe magnetic mirror. As for GRTTM case, as var( E L ) increases,the LRTTM hard X-ray spectra at ∼
25 keV become flatter (see γ
25 keV in Fig. 12, left). Values of var( E L ) ≥ − result inextremely flat photon spectra. On the other hand, the LRTTM X-ray spectra exhibit a double break or a local sudden decrease; seee.g. the spectrum in the ∼
50 – 100 keV range corresponding tovar( E L ) = . − in Fig. 10, right. Such spectral shapes a ff ectthe fitted CTTM electron distributions and result in high valuesof E (cid:48) (located approximately at the energy of a double break)and higher values of δ (cid:48) (see Fig. 12, bottom left). As var( E L )rises, E (cid:48) still increases but δ (cid:48) stays almost constant, i.e. 4 – 5.The model of var( E L ) = − presents a limit, and the hardX-ray spectrum is consistent with a rather flat electron flux spec-trum of high E (cid:48) . Although the spectrum is more intense than thespectrum of pure CTTM with F / = × erg cm − s − (i.e.20 × higher than the initial flux used in this model), owing to thehigh value of E (cid:48) , the fitted electron flux is lower and consistentwith the energy deposit in the chromosphere E ch (see Fig. 11,left). Article number, page 11 of 15 ig. 11.
LRTTM E L -I (blue) and E L -II (green) (for E L = . − )chromospheric energy deposits E ch (lines) and fitted energy flux F (cid:48) (symbols) ( left ) and the ratio F R / F ( right ) for various var( E L ). Solidlines and asterisks denote M FF ; dashed lines and triangles denote M SU . E L -II type The e ff ects of local re-acceleration due to the stochastic field E L with E L (cid:44) E L = . − and var( E L ) = . − (see the distribution functions for M FF and M SU cases in Fig. 13). The re-acceleration process againresults in formation of fuzzy high-energy tail of particles situ-ated in the secondary acceleration region and covering the en-ergy range from 10 to 100 keV approximately (see the regionslabelled H). The mean energy reached by the re-accelerated elec-trons at the lower boundary of the re-acceleration region steeplyincreases with E L , and at the same time the maximum of energydeposit shifts towards the deeper layers. The mean value of E L also has a strong focussing e ff ect on the re-accelerated electrons.The latter e ff ect reduces the ratio of backscattered and reflectedparticle flux to the initial flux F R / F to less than 1% for the M FF and to 37% for the M SU case, respectively: compare values of F R / F for the individual field types and parameters of E L dis-played in Fig. 11 (right). The value of var( E L ) plays a similarrole to what is described above for the E L -I type. In comparisonwith the e ff ects of E L , it only weakly influences the energy gainof electrons at the lower boundary of the re-acceleration region,it increases the fuzziness of the high-energy tail and the flux ofbackwards moving electrons (with µ < E L ) we also see a decrease in electrons having µ other thanclose to 1 and −
1, which is again the e ff ect of imprint of the di-rectional distribution of E L on the electron µ -distribution, whichwas also found for the stochastic field type E L -I.The stochastic field of E L = . − and var( E L ) = . − (see Fig. 13) practically ceases the formation of thelow-energy tail of particles located in the region between the up-per boundary of the re-acceleration region and the lower bound-ary of the magnetic mirror found in the distribution functionscorresponding to the CTTM, GRTTM, and LRTTM E L -I type(see Figs. 4, 6, and 9). It forms either for lower values of E L ,which is too small to compensate for the collisional energy lossesof the electrons in the region above the lower boundary of themagnetic mirror, or for greater values of var( E L ), when the in-teractions of beam electrons with the stochastic component of E L lead to its formation. On the other hand, a new tail of par-ticles is formed on energies from approximately 1 to 100 keVin the region under the lower boundary of the re-acceleration re-gion where the re-accelerated particles are quickly thermalised(see the regions labelled L).The energy deposits and HXR spectra for E L = . − and various values of var( E L ) from 0 to 5 V m − are plotted in Fig. 12.
LRTTM E L -I ( left ) and E L -II ( right ) (for E L = . − )summary of calculated and fitted parameters of chromospheric bom-bardment for various values of var( E L ). Top: position of energy depositmaximum s max and HXR intensity I . Bottom:
HXR spectral index γ and fitted electron beam spectral index δ (cid:48) p and low-energy cuto ff E (cid:48) . The solid and dashed lines denote M FF and M SU , respectively. Fig. 14, and the parameters E ch , s max , I
25 keV , and γ
25 keV are dis-played in the left-hand and right-hand panels of Figs. 11 and 12,respectively, and summarised in Table 3. The general behaviourof E ch and s max is similar to the GRTTM of primary footpointand LRTTM E L -I type. They are very sensitive to the static com-ponent E L of the stochastic field and only moderately sensitiveto the stochastic component var( E L ). Even for var( E L ) = E L = . − , there is an appreciable increase of E ch (3 . × for the M FF and 5 . × for the M SU case) and a shift of s max ofapproximately 450 km towards the photosphere and substantialgrowth in HXR production ( I
25 keV increases of by an order ofmagnitude for both initial µ -distributions relative to the CTTMwith an identical initial flux). For the identical value of E L andthe maximum value of var( E L ) = − , the increase in E ch is5 . × for the M FF and 10 × for the M SU case, the shift of s max to-wards the photosphere of approximately 750 km (for both initial µ -distributions), and a substantial increase in I
25 keV (35 × for the M FF and almost 130 × for the M SU case) relative to the CTTMwith an identical initial flux. The power-law index γ
25 keV tendsto harden with increasing var( E L ).HXR spectra corresponding to the E L -II type are distinctfrom the previous ones. Here, two re-accelerating processes areinvolved. The static component causes a significant increase ofspectra at deka-keV energies, up to ∼
40 keV, and a steep dou-ble break at energies above. Therefore, the corresponding fit-ted electron flux spectrum assuming pure CTTM shows quite asteep δ (cid:48) (see Fig. 12, bottom right). Such a steep double breakis a consequence of a re-acceleration by a constant electric field.The energy at which it appears is related to the length of there-acceleration region, i.e. the current sheet size. The largerthe size, the steeper the double break and the higher energies atwhich it is located. The presence of the stochastic componentintroduces another shift of the double break to higher energies,likewise for the type I; as var( E L ) increases, the double breakis less prominent. Consequently, E (cid:48) increases and δ (cid:48) decreases Article number, page 12 of 15. Varady et al.: Modifications of thick-target model: re-acceleration of electron beams ...
Fig. 13.
LRTTM E L -II type distribution functions of the non-thermal electron energies versus positions with a colour coded M ( µ ) correspondingto E L = . − and var( E L ) = . − in the VAL C atmosphere at time t = . Left: M FF , right: M SU . The solid lines indicate the instantaneous energy deposits corresponding to F / = . × erg cm − s − , the dotted horizontal linesthe bottom boundary of the magnetic mirror, the grey area the secondary re-acceleration region, and the blue ellipses labelled L and H denote tailsin the particle distribution function. Only the vicinity of the footpoints is displayed. Fig. 14.
LRTTM E L -II type instantaneous energy deposits ( left ) and HXR spectra ( right ) for the VAL C atmosphere at t = . M FF )and dotted ( M SU ) blue, green, red, yellow, and orange lines correspond to E L = . − and var( E L ) = . , . , , , − , respectively andenergy flux F / = . × erg cm − s − . The dashed and solid grey lines correspond to the CTTM with F / = . × and 5 × erg cm − s − ,respectively. The dotted straight vertical line indicates the bottom boundary of the magnetic mirror, the grey area the secondary re-accelerationregion. The HXR spectra are integrated over one half of the loop. (see Figs. 12 and 14). When the stochastic component prevails,i.e. var( E L ) ≥ − , the hard X-ray spectra are of similarspectral shape to the E L -I model but more intense.
4. Conclusions
We studied modifications of the CTTM by considering two typesof secondary particle acceleration: GRTTM and LRTTM. Inboth cases the re-acceleration takes place during the transportof non-thermal particles, which are primarily accelerated in thecorona. According to Brown et al. (2009), such a re-accelerationgenerally reduces collisional energy loss and Coulomb scatteringand increases the life-time and penetration depth of particles.In the case of GRTTM, the spatially varying direct electricfield spreading along the whole magnetic strand from first tosecond footpoint re-accelerates the beam electrons towards theprimary footpoint and decelerates them towards the secondaryfootpoint, thus producing an asymmetric heating of footpoints.The low electric plasma conductivity and increased current den-sity due to magnetic field convergence are the key constraints forthe functionality of this mechanism. The model was studied forthe mirror ratio R m = j ≤ − . Sig- nificant re-acceleration is present for j (cid:38) − , and for lower j the model is similar to CTTM. However, a question arises as towhether such current densities are realistic. Although the currentdensities derived from magnetic field observations are two ordersof magnitude lower (Guo et al. 2013), in the magnetic rope, es-pecially in their unstable phase at the beginning of the flare, thecurrent density in some filaments could reach these values: seethe processes studied in Gordovskyy & Browning (2011, 2012);Gordovskyy et al. (2013). On the other hand, a current filamenta-tion also means a decrease in the area where this re-accelerationcan operate e ff ectively. Finally, the GRTTM model inherentlyintroduces an asymmetry on opposite sites of the magnetic rope.More observations are needed to check that some asymmetricalX-ray sources are caused by this e ff ect.Two types of electric field were considered for LRTTM: apurely stochastic field var( E L ) ≤ − ( E L -I type) and acombination of var( E L ) and a static component E L = . − ( E L -II type). It has been shown that both types of electric fieldsproduce a substantial secondary re-acceleration ( E L -I type forvar( E L ) (cid:38) . − , E L -II type for all considered field param-eters due to the static field component) with dominant energypropagating towards the photosphere. Article number, page 13 of 15 enerally in all presented models, HXR spectra gets flat-ter below ∼
30 keV and more intense on all energies as re-accelerating fields increase. The flattening then corresponds toan increase in the low-energy cuto ff E (cid:48) of the fitted electron dis-tribution. The e ff ect of flattening of HXR spectra below the low-energy cuto ff can be seen in Brown et al. (2008, Fig. 1e). Ex-tremely flat HXR spectra (related to E (cid:48) (cid:38)
50 keV) were obtainedfor GRTTM of j = − and LRTTM var( E L ) ≥ − ( E L -I type). Such flat spectra or high values of E (cid:48) are not re-ported from the observation, therefore those j and var( E L ) couldrepresent limiting values. In addition, prominent double breaksat keV energies, present in the E L -II cases, are not observed inHXR spectra. This suggests that our model of a constant re-accelerating field over a larger spatial scale, ∼ × higher initial energyflux. GRTTM gives a comparable total chromospheric energydeposit. For the LRTTM the total energy deposits reach onlyabout 30% of the latter value. The re-acceleration also leads tospatial redistribution of the chromospheric energy deposit withthe bulk energy being deposited much deeper into the chromo-sphere and into a narrower layer in comparison to the CTTM.The heights of the energy-deposit maxima are thus substantiallyshifted towards the photosphere (of ≈
800 km for both models).It is a consequence of the re-accelerating fields pushing the non-thermal electrons under the magnetic mirror and under the beam-stopping depth corresponding to the CTTM. The height abovethe photosphere decreases with both the current density for theGRTTM and with the mean value and variance of the stochasticfield for the LRTTM. For the upper values of model parame-ters, we obtained the heights of energy-deposit maxima as onlyapproximately 600 km. This is not far from the upper limitson heights of the flare white-light sources (305 ±
170 km and195 ±
70 km) found from observations (Martínez Oliveros et al.2012).To demonstrate how the secondary accelerating processesmay lead to artificially high CTTM input energy fluxes, we fol-lowed a standard forward-fitting procedure for determining theinjected electron spectrum from an observed X-ray spectrum.Although the spectral fitting does not take any re-accelerationinto account, the fitted F (cid:48) agrees well (within 30%) with E ch inall simulations. This value can di ff er substantially from the in-jected total energy flux, therefore the fitted total energy flux (un-der assumption of pure CTTM) is related more to the energy de-posit of re-accelerated particles than to the injected energy flux.In general, both the considered models with secondary re-acceleration, GRTTM and LRTTM, allow loosening the require-ments on the e ffi ciency of coronal accelerator, thus decreasingthe total number of particles involved in the impulsive phase offlares and the magnitude of the electron flux transported fromthe corona towards the photosphere, as needed to explain theobserved HXR footpoint intensities. These findings agree withthe results obtained by Brown et al. (2009) and Turkmani et al.(2006, 2005). Acknowledgements.
This work was supported by grants P209 / / / / References
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Article number, page 14 of 15 &A–varady_et_al,
Online Material p 15
Table 2.
Summary of results for the GRTTM with F / = . × erg cm − s − . Footpoint j F R / F E ch / s max I
25 keV γ
25 keV F (cid:48) / × , δ (cid:48) p , E (cid:48) [A m − ] [%] [erg s − ] [Mm] [cm − s − keV − ] [erg cm − s − ], , [keV]1.0 3.1 (37) 2.8 (1.7) 1.2 (1.4) 0.71 (0.18) 2.4 (2.7) 3.3, 3.0, 12 (1.6, 3.4, 11)2.0 2.2 (36) 4.2 (2.1) 1.1 (1.1) 1.2 (0.29) 2.4 (2.7) 4.5, 3.0, 15 (2.2, 3.5, 13)Primary 3.0 1.6 (33) 5.5 (3.0) 0.98 (1.1) 2.2 (0.56) 2.5 (2.9) 6.7, 3.1, 20 (3.2, 3.6, 17)4.0 1.5 (33) 7.7 (4.7) 0.87 (0.94) 5.0 (1.5) 2.40 (2.9) 10, 3.3, 30 (4.7, 3.7, 25)5.0 0.92 (32) 15 (7.7) 0.80 (0.83) 13 (4.5) 2.0 (2.3) 17, 3.5, 48 (7.7, 3.9, 39)6.0 0.69 (31) 30 (18) 0.60 (0.63) 38 (17) 1.7 (1.7) 35, 4.5, 100 (17, 4.8, 88)1.0 5.1 (42) 1.9 (1.2) 1.4 (1.4) 0.31 (0.086) 2.4 (2.7) 2.0, 3.0, 9 (1.0, 3.5, 10)2.0 6.6 (43) 1.5 (1.0) 1.4 (1.4) 0.22 (0.066) 2.4 (2.7) 1.6, 3.1, 9 (0.82, 3.5, 10)Secondary 3.0 8.6 (47) 1.4 (0.89) 1.4 (1.6) 0.16 (0.053) 2.4 (2.7) 1.3, 3.1, 8 (0.70, 3.5, 10)4.0 11 (51) 1.2 (0.82) 1.4 (1.6) 0.13 (0.044) 2.4 (2.7) 1.0, 3.1, 8 (0.61, 3.6, 10)5.0 12 (55) 1.1 (0.71) 1.4 (1.7) 0.10 (0.034) 2.4 (2.8) 0.87, 3.1, 8 (0.52, 3.6, 10)6.0 15 (54) 0.93 (0.64) 1.4 (1.7) 0.081 (0.032) 2.5 (2.8) 0.74, 3.2, 8 (0.47, 3.6, 10) Table 3.
Summary of results for the LRTTM with F / = . × erg cm − s − . E L var( E L ) F R / F E ch / s max I
25 keV γ
25 keV F (cid:48) / × , δ (cid:48) p , E (cid:48) [V m − ] [V m − ] [%] [erg s − ] [Mm] [cm − s − keV − ] [erg cm − s −1