ParticleKinematics Chapter12

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(1)Chapter 12. Particle Kinematics Particle acceleration, an universal theme in high-energy astrophysics, remains a black box in all astronomical observations, even in solar flares, from our nearest astrophysical laboratory. At this point we are just starting to understand the complicated electromagnetic field dynamics in magnetic reconnection regions, which probably provide the most prolific sources of accelerated particles, but we have no reliable method yet to map out the relevant electric and magnetic fields. We do not know the field strengths nor the exact locations of the accelerating fields. In laboratory accelerators, the electromagnetic field strengths, the geometry of the fields, and the particle trajectories are usually known with high precision by design. The laboratory measurements concentrate mainly on the kinematic reconstruction of collisional products of accelerated particle beams that hit a target, which sometimes leads to discoveries of new particles, whose identities are constrained by the kinematics of their trajectories, momentum, energy, and parity conservation laws. In solar flares, we observe analogous collision experiments, where electrons and ions are accelerated in a coronal, collisionless magnetic reconnection region, which then propagate in the low- plasma along the magnetic field lines to the chromospheric footpoints, corresponding to the targets in laboratory accelerators. Progress has been made over recent years to measure the small time-of-flight differences of energized electrons between the coronal acceleration site and the chromospheric target. Because the time-of-flight is a direct function of the flight distance and the velocity (or kinetic energy) of the electrons, we have a powerful new diagnostic on the so far unlocalized particle accelerators. Because the tiny time-of-flight differms, high-precision timing measurements have ences are only in the order of to be conducted, which generally require a photon statistics of photons per s. This requirement calls for large-area detectors with high sensitivity, which were available in the Burst and Transient Source Experiment (BATSE) on the Compton Gamma Ray Observatory (CGRO), which operated between 1991 and 2000. However, the photon count rate in large flares is sufficiently high that also smaller detectors, such as HXRBS/SMM or RHESSI, can accomplish kinematic measurements. In the following we lay out the fundamentals of particle kinematics applied to solar flares, including the propagation processes during acceleration, free-streaming, injection, trapping, and precipitation. A review on this novel topic can be found in Aschwanden (2002b).. .

(2) . 517.

(3) CHAPTER 12. PARTICLE KINEMATICS. 518. MODEL ASSUMPTIONS ACCELERATION acceleration time tAcc=0.1-1.0 s (spectral soft-hard evolution). INJECTION modulated in pulses tPul=0.1-2.0 s synchronized in energy and pitch angle bifurcation in pitch angle α<α0. α>α0. PROPAGATION free-streaming, no energy loss time-of-flight tTOF(E)=l*E-1/2=0.1-0.3 s. TRAPPING collisional deflection into loss-cone tDefl(E)=E3/2/ne=0.5-10 s. ENERGY LOSS collisions in chromosphere tColl=0.01-0.1 s thick-target HXR emission. Figure 12.1: Conceptual breakdown of the flare kinematics into five different physical processes, described in this section (Aschwanden 1998a).. 12.1 Overview on Particle Kinematics In solar flares, the kinematics of nonthermal particles can generally be characterized by at least five different physical processes, as sketched in Fig. 12.1: (1) acceleration, (2) injection, (3) free-streaming propagation, (4) trapping, and (5) precipitation and energy loss, which we will quantify in turn in the following subsections. The acceleration phase is defined by the time interval in which a thermal particle gains velocity and kinetic energy up to a maximum value. At the end of the acceleration phase, we define an injection mechanism that allows a particle to leave the acceleration region and to enter a magnetic field line where it freely propagates. The third and forth process could happen in parallel, some particles propagate freely (free-streaming) on a field line, while others become trapped and bounce forth and back in a magnetic mirror region, depending on the initial pitch angle at injection. The fifth step entails the phase of energy loss, which inevitably occurs when the particles precipitate towards the chromosphere, where the high density thermalizes all particles very rapidly, causing bremsstrahlung that is detectable in hard X-rays ( 13) and gamma rays ( 14). Trapped particles can be detected by thin-target bremsstrahlung in hard X-rays ( 13) or gyrosynchrotron emission ( 15).

(4) 12.1. OVERVIEW ON PARTICLE KINEMATICS. 519. in radio wavelengths. In principle one can define for each kinematic process a time scale, which strongly depends on the velocity, and thus on the kinetic energy of the particle, so that the total elapsed time is composed of the sum of each time interval, . . .

(5) . . . . !! "$#. . %'& &)(. *,+-/.102.3 *,+-/.142. 3. . 56 76 8. (. . where particles bifurcate into free-streaming propagation for small pitch angles .90 .:3 , or trapping for large pitch angles .;4<./3 . In practice, the acceleration time may not be directly measurable if acceleration occurs continuously, while the injection mechanism may synchronize the release of bunches of particles, as suggested by the common appearance of sub-second pulses detected in hard X-rays. Also the energy loss time may not be measurable in the case of thick-target collisions, because the stopping distance in the chromosphere is much shorter than the propagation distance from the coronal acceleration site down to the chromosphere. In this most likely scenario, the time difference between a synchronized pulsed injection at the “exit” of a coronal magnetic reconnection site and the detection of hard X-rays by chromospheric thick-target bremsstrahlung is only dominated by the propagation time interval, either by free-streaming or trapped particle motion, . . . . ! '= . *,+-/.102.3 *,+-/.142.3. 56 76 57 . 6. The results of recent time delay analysis indicates that these processes dominate the timing of hard X-ray producing electrons, namely electron time-of-flight differences ' of free-streaming electrons, , or trapping times of electrons, . The bimodality in the timing can easily be verified by the energy-dependence, because time 2>@?A8BC>D? of-flight differences have an energy dependence of EGF'H'I (in the non-relativistic limit), while trapping times scale with the collisional deflection time, ' > A8KL , "JH!I . . >. . MEF H!I "J!H!I. *,+7-N.102. 3 *,+7-N.142. 3 6. . 56 76 OM. Although these two processes seem to be dominant, based on the observational analysis, we do not want to bias ourselves to consider this as the only possibility, but will also discuss the timing of the other processes (acceleration, injection, and energy loss), which could dominate in specific flare models. For instance, in large-scale DC electric fields, particles experience acceleration all the way on their trajectory and may never reach a free-streaming orbit phase. Thus, large-scale DC electric fields produce a distinctly different timing than small-scale acceleration regions of any kind. We will see that the timing of particle kinematics depends also strongly on the geometry and topology of the magnetic field, which is different for each flare model, and thus can be used as a powerful method to discriminate competing flare models. We emphasize that such kinematic tests of particle acceleration models are relatively novel, while traditional approaches attempted to extract information on acceleration mechanisms by spectral modeling..

(6) CHAPTER 12. PARTICLE KINEMATICS. 520. Light travel time. Kinetic energy. ε2(v2). Distance perp. to field line. ε2(v1). εX,2=hνX,2. εX,1=hνX,1 Particles HXR photons Time. t_c=l/c t2=l/v2. t1=l/v1. Time-of-flight difference ∆t=t2-t1. v. Pitch angle α Distance parallel to field line. Distance perp. to loop axis. v||. Twist angle θ Distance parallel to loop axis. Flare loop flux tube. Figure 12.2: Geometrical effects that determine the timing in free-streaming particle propagation: Top: Velocity dispersion between particles with different kinetic energies , which translate into a time difference between the detected hard X-ray photon energies ; Middle: Pitch angle of particle, which defines the parallel velocity

(7) ; Bottom: Twist angle of magnetic field line, which defines the projected trajectory length "!# .. 12.2 Kinematics of Free-Streaming Particles 12.2.1 Definition of Time-of-Flight Distance We start with the simplest kinematic case, which is the propagation of free-streaming particles in a collisionless plasma. We consider first just the simplest case of constant B B velocity. Two particles that are injected with different velocities and at the “exit” F I of a coronal acceleration site onto a magnetic field line experience a velocity dispersion.

(8) 12.2. KINEMATICS OF FREE-STREAMING PARTICLES ! . that translates into a time-of-flight difference ? . ! . . B F. . ? B.

(9). I. ?. over a common distance. I I.

(10). ?. 521. . F. . 56 56 8. (. . ,. I. . where and are the relativistic Lorentz factors as defined in Eqs. (11.1.11) and F I (11.1.13). Obviously, the faster particles (with higher kinetic energy) arrive first, preceding the slower ones. Thus, if propagation delays dominate the total energy-dependent timing (Eq. 12.1.1), hard X-ray pulses from electron bremsstrahlung should peak first at the highest energies. The conversion of the kinetic energy of electrons, into photon energies detected in hard X-rays, , can be calculated from the bremsstrahlung cross-section, which we will treat in 13. This conversion factor depends on the energy, the spectral slope of the electron spectrum, and the assumed A bremsstrahlung cross-section, and amounts typically to 6 , since electrons can only produce hard X-ray photons of lower energy. As the diagram illustrates in Fig. 12.2, this conversion factor is needed to evaluate the correct kinetic energies and velocities of the particles from a measured hard X-ray delay . ? The time-of-flight distance evaluated with Eq. (12.2.1) measures the path length of an electron spiraling along the magnetic field trajectory with some pitch angle ?! #" (Fig. 12.2 middle). In order to obtain the length of the magnetic field line, we need B$ to correct for the parallel velocity component, ,. . ? #" . ?. %. B$ (. &%. . 5. (' +*).,+ B. . =6 . -. 56 56 57 . 6. .0/. The numerical value of 1% + =6 .0/ is obtained by averaging the pitch angle over a loop bounce time in the limit of large mirror ratios (see derivation in 12.5). An additional length correction needs to be applied if the magnetic field line is helically twisted (Fig. 12.2, bottom). If we straighten the flare loop to a cylindrical ? % ' geometry with radius 2 , the projected length of the cylinder is related to the ? #" length of the helical field line by

(11). ? %! . ? #". (. &3. . 4' +5)76 . &3. 5. 98. -. K. I. 2. ?. #";:. 6. <*. (. 56 56 OM . K. with being the number of complete twists by 5 - radians. The helicity of coronal loops cannot exceed a few radians before they become magnetically unstable. For instance, a critical twist of 5 6 - is predicted for erupting prominences (Vrsnak et al. 1991). The kink instability is predicted for twists in excess of =/=6 < - (Mikíc et al. 1990). Thus K we might use an estimated twist of 5 - (i.e., for the number of twists) and a loop A"? % ' aspect ratio of 5>2 6 5 (observed for the Masuda flare), which yields a length correction factor of 13 =6 <5 in Eqn. (12.2.3). Combining the two factors we have. . . . ? %' . 4 3. ?. #". 4. % @?. ?. . . 6. /. ?. . (. . 56 56 /. so the length of the particle trajectory projected onto the loop axis is about half of the effective time-of-flight distance..

(12) CHAPTER 12. PARTICLE KINEMATICS. 522 1.5•105. Count rate [cts/s]. 1991 DEC 15, 1832 UT CGRO/BATSE/MER. 1.0•105 44.0 keV 58.6 keV 77.4 keV 5.0•104. 102.1 keV 127.0 keV 166.8 keV 232.5 keV 320.6 keV. 0 1832:00. 1832:30. 1833:00. 1833:30. Time [UT]. Figure 12.3: Hard X-ray observations of the 1991-Dec-15, 18:32 UT flare, recorded with BATSE/CGRO in the Medium Energy Resolution (MER) mode that contains 16 energy channels. The channels #3-10 are shown, with the low energy edges indicated on the left. The channels are incrementally shifted and a lower envelope is indicated, computed from a Fourier filter with a cutoff at a time scale of 4 s. The fine structure (with the envelope subtracted) are cross-correlated in the delay measurements shown in Fig. 12.4 (Aschwanden 1996c). 800. 91 DEC 15 1832 UT _ ms 74 +20. 342. _ 7 ms 74 +. 250. _ 4 ms 63 +. 178. + _ 3 ms + _ 2 ms + _ 2 ms + _ 2 ms + _ 0 ms. 131 103 79 60 46. 600. 400. 200. 0 0. 53 42 31 19 0. 50 100 HXR time delay tX(Emin)-tX(E) [ms]. HXR energy ε[keV]. Electron energy E[keV]. χ2red= 0.59. 150. Figure 12.4: HXT time delay measurements between channel #3 and #4-10 from the data shown in Fig. 12.3. The delays and statistical uncertainties are indicated on the right. The hard X-ray energies on the right axis represent the medians of the count spectra in each channel, while the electron energies on the left axis are computed with the conversion factor

(13)

(14) . The solid line represents a fit of the 1-parameter model given in Eq. (12.2.1), yielding a TOF distance of km, with

(15) (Aschwanden 1996c)..

(16) 12.2. KINEMATICS OF FREE-STREAMING PARTICLES. 523. 12.2.2 Time-of-Flight Measurements The previously described pure time-of-flight (TOF) model, where propagation of freestreaming electrons dominates the energy-dependent timing in Eq. (12.1.1), with all other terms negligible, seems to explain the relative time delays of fast time structures seen in hard X-ray time profiles satisfactorily. The smoothly-varying component in the hard X-ray time profiles usually exhibits a different energy-dependent timing that is attributed to trapped particles ( 12.5). An example is shown in Fig. 12.3, where the smoothly-varying background is subtracted (dashed curves in Fig. 12.3), and the remaining fast time structures are cross-correlated in different energies, leading to the delay curve shown in Fig. 12.4. Taking the conversion factor of photon to electron A energies into account, & =6 , and fitting the TOF model (Eq. 12.2.1), a ? ( ( 5 < I or time-of-flight distance of km is found, with a reduced / 0.59 (Aschwanden 1996c). As Fig. 12.4 shows, the relative time delays between adjacent energy channels are 0 /*. keV) > . keV)= 5 generally very small, for instance ms between the lowest two channels. How is it possible to measure reliably such small time delays beyond the temporal resolution or time binning of the instrument, which is 64 ms for the BATSE/MER data used here? The trick is simply to use a crosscorrelation technique with interpolation at the maximum of the cross-correlation coefficient (CCC), which yields sub-binning accuracy if the photon statistics is sufficiently high. The cross-correlation coefficient between two flux profiles is defined by

(17) , ( F E

(18)

(19) I 56 56 7 I , I . . . . . . F. E. I. where the index runs only over the overlapping time interval in both energy channels to avoid a bias from aliasing effects. Empirically we can measure the accuracy of the delay measurement by repeating the cross-correlation by adding random noise (Aschwanden & Schwartz 1995). If the correlated time profiles contain pulses with gaussian-like shape, characterized by a gaussian width , we can assert the uncertainty A . For of the mean just as for a normal distribution of events, which is . the delay measurement we can use then the law of error propagation,

(20). F. . I. . . 56 56 M Thus, for the data shown in Fig. 12.3 we have roughly count rates of order %$

(21) cts s EF in the lowest channels, and pulses with gaussian widths of ms, so A we estimate an uncertainty of

(22) '& 5 ($

(23) *) ms for a single pulse. Averaging over 12 pulses increases the accuracy by another factor of 5 O6 and . 8. . I. F. :. I. F. I. =8 . I. :. I. !#". I. . F. . 6. . .. I. /. /. . yields the obtained accuracy of 2 ms, as it was evaluated by repeating with added random noise. These measurements where accomplished with the BATSE detectors on CGRO, which had the largest collecting area of 2000 cm I . Other hard X-ray detectors have smaller areas, e.g., HXRBS/SMM with 71 cm I and RHESSI with an effective area of 38 cm I($ 9 detectors $ (=6 I ) (bi-grid transmission) = 85 cm E I , so the count rates are a factor of 5* smaller and the timing accuracy is reduced by a factor of .. . . .

(24) CHAPTER 12. PARTICLE KINEMATICS. 524. Velocity v. 1) Small-Scale Stochastic Acceleration v2 v1. Velocity v. 2) Small-Scale Electric DC-field Acceleration v2 v1. Velocity v. 3) Large-Scale Electric DC-field Acceleration v2 v1. Velocity v. 4) DC Acceleration with different start points v2. Velocity v. S1 S2 5) DC Acceleration with different exit points E2 E1. v1 v2 v1. Distance l Figure 12.5: Five different models for the timing of electron acceleration and propagation. The velocity changes of a low-energy ( ) and a high-energy electron ( ) are shown along a 1-dimensional path from the beginning of acceleration (left side) to the thick-target site (right side). Model 1 and 2 characterize small-scale acceleration processes, while Model 3-5 depict scenarios with large-scale acceleration. Model 4 and 5 illustrate different start ( ) and exit positions ( ) for the accelerated electrons (Aschwanden 1996c). . 12.3 Kinematics of Particle Acceleration We refine now our kinematic models by including finite acceleration times and study how it affects the timing of hard X-ray emission, or the arrival times of electrons at the thick-target site. Basically one can distinguish between two opposite scenarios: small-scale and large-scale acceleration processes. In small-scale acceleration processes, one can generally neglect acceleration time !. ! scales compared with propagation time scales, , supposed that the acceleration path is small compared with the free-flight propagation path. Such situations are depicted in Model 1 and 2 (Fig. 12.5). The hard X-ray timing can then be described with the time-of-flight model ( 12.2), which was found to be fully consistent with the data for the flare shown in Fig. 12.3..

(25) 12.3. KINEMATICS OF PARTICLE ACCELERATION Stochastic Acceleration (Miller et al. 1995) tacc+tprop tacc+tprop*1.8. Electron energy E[keV]. 800. 525. tprop. 600. 400. 200. 0 -100. -50 0 50 HXR time delay tX(Emin)-tX(E) [ms]. 100. Figure 12.6: Fit of the stochastic acceleration model of Miller et al. (1996) to the same data as shown in Fig. 12.4. Adding the predicted acceleration time to the propagation time (thick line) yields negative delays (dashed line). Adjusting the propagation time by a factor of 1.8 (thin line) yields an acceptable fit in the 100-200 keV range, but not at higher energies (Aschwanden 1996c).. 12.3.1 Stochastic Acceleration We consider a stochastic acceleration process as it can occur in coronal regions with enhanced wave turbulence (or similarly in shock fronts). Even when the spatial scale of the acceleration region is small compared with the TOF propagation distance (Model 1 in Fig. 12.5), this does not necessarily imply that the acceleration time is also much smaller than the propagation time. In the case of diffusive stochastic acceleration the particles can be bounced around in a turbulent region significantly longer than the travel time through this region. For instance, LaRosa et al. (1995) estimate the bulk energiza . tion time of electrons in a reconnection-driven MHD-turbulent cascade to =20 keV) <O ms, which is comparable with the propagation time inferred in our flare (Fig. 12.4), =20 keV) =l/v =29,000 km /(0.27 c)=360 ms. More specifically, Miller et al. (1996) estimate acceleration times of 70 ms to energize electrons from 4 to 50 keV, or about 180 ms to 511 keV. They specify an energy dependence of. . . . . . . . . L. I. . F'H. =6 /5<. O5. ). (. .

(26) . . 56 O=6 8. to energize electrons by gyroresonant interactions with fast mode waves in an MHDturbulent cascade. We fit this model to the 1991-Dec-15 flare and show the expected hard X-ray timing in Fig. 12.6. First we add the acceleration time to the same prop? ( agation time inferred in Fig. 12.4 (based on a TOF distance of 5 km). The expected hard X-ray delay (dashed curve in Fig. 12.6) becomes negative above 200 keV, meaning that the high-energy electrons arrive later than the low-energy electrons at the chromosphere due to the longer acceleration time. If we perform a fit of the. .

(27) CHAPTER 12. PARTICLE KINEMATICS. 526 . . ! . combined expression . (using Eq.12.2.1 and Eq.12.3.1) we find that the data can be reasonably fitted in the 100-200 keV range with a 1.8 times larger TOF distance (to compensate for the acceleration time), but the hard X-ray delay decreases above 200 keV significantly below the measured values. Thus, the energy-dependent scaling of the acceleration time specified in the stochastic acceleration model of Miller et al. (1996) cannot fit the observed delays over the entire energy range of 80-800 keV for this specific flare. This example illustrates that the observed hard X-ray delays require that the higherenergy electrons arrive earlier than the low-energy electrons, which is a natural outcome for time-of-flight dispersion, but is an opposite trend to most acceleration models, where it takes statistically longer to accelerate to higher energies. This is a strong indication that acceleration times might not be dominant for the observed hard X-ray timing.. 12.3.2 Electric DC-Field Acceleration Acceleration mechanisms employing DC electric fields have been studied by various researchers, e.g., Holman (1985), Tsuneta (1985), or Emslie & Henoux (1995), but there is no detailed comparison of the predicted timing with observations. The models 2-5 shown in Fig. 12.5 depict various scenarios where the acceleration and propagation time scales have different weighting, depending on the spatial location and extent of the DC field. Because the free-flight path of electrons is complementary to the acceleration path length in unidirectional DC fields, in a more direct fashion than in the case of stochastic acceleration, the resulting hard X-ray timing provides a crucial test between different models. The simplest case is given in Model 2 (Fig. 12.5), where the spatial extent of the DC field is small compared with the TOF distance, and necessarily also implies that !. the acceleration time is small compared with the free-flight time ( , and thus can be neglected. The hard X-ray timing can then adequately be described with Eq. (12.2.1), which fits the data satisfactorily for the 1991-Dec-15 flare (Fig. 12.4). Another simple approach is to assume that an electric field extends over the entire flare loop and that electrons are accelerated from one end of the loop to the other. Assuming a constant electric field, the electrons would end up with a monoenergetic spectrum and coincident timing, and thus cannot explain the observed delays (Fig. 12.4). A first variant is to assume that a flare loop consists of a number of current channels with different electric fields. In this scenario electrons are accelerated in separate channels with different electric fields , obeying the force equation (Eq. 11.1.1), . L. . . . . L!B. . ?. . 5 6 O6 5. (. where represents the kinetic energy of an electron gained by the electric field over a ? distance . The acceleration can then be expressed as function of the kinetic electron energy or Lorentz factor (using Eq. 11.1.12) by. . . . . L ? ?. I . 86 . 5 6 O6 O7.

(28) 12.3. KINEMATICS OF PARTICLE ACCELERATION. DC Electric Field Acceleration (Model 3) tacc. 600. 400. 200. 0 -100. -50 0 50 HXR time delay tX(Emin)-tX(E) [ms]. DC Electric Field Acceleration (Model 4) (Model 5) Emax/mec2=2, 20. 800 Electron energy E[keV]. Electron energy E[keV]. 800. 527. 600. 400. 200. 0 -100. 100. -50 0 50 HXR time delay [ms]. 100. Figure 12.7: Fit of the DC electric field Model 3 (left) and Model 4 and 5 (right) for the. km fitted same data as shown in Fig. 12.4. With respect to the TOF distance ,. in Fig. 12.4, the best fit of Model 3 (left) yields an acceleration distance of (for ) and Model 4 is shown for , and Model 5 yields # (for ), (Aschwanden 1996c).. . .

(29) '. The acceleration time as function of the energy can then be derived by integrating the force equation (Eq. 12.3.2) and inserting Eq. (11.1.13), . . Inserting the acceleration. . . . . . . . &. . I. . 6. 56 O=6 . . /. from Eq. (12.3.3) into Eq. (12.3.4) yields then ? . . " 6 56 O=6 7. . . . . In Model 3 (Fig. 12.5), the hard X-ray timing is entirely determined by this energy dependence on the acceleration process, corresponding to the approximation . in the general timing equation (Eq. 12.1.1). We fit this model to the observed hard X-ray timing of the 1991-Dec-15 flare and show the results in Fig. 12.7 (left panel). Interestingly, this model shows a very similar energy dependence as the TOF propagation model (shown in Fig. 12.4) and thus fits the data equally well. The inferred acceleration path length is a factor of 0.44 shorter than the path length in the TOF propagation model, because the average electron speed is about half of the final speed applied in the propagation model (being exactly half in the nonrelativistic limit). Thus, the two models cannot be distinguished from the timing alone, but the inferred distance scale is a factor of 5 different. However, there is no model that explains the existence of many current channels with very different large-scale electric fields in the first place, which have to be switched on and off on sub-second time scales to explain the observed hard X-ray pulses.. . In Model 4 and 5 (Fig. 12.5) we investigate two further variants of electric DC field acceleration, where the accelerated electrons are allowed to enter into (Model 4).

(30) CHAPTER 12. PARTICLE KINEMATICS. 528. or exit from (Model 5) an electric field channel at different locations. In both models the resulting electron energy is proportional to the acceleration path length, assuming a constant (mean) electric field in all current channels. Model 4 is a natural situation in the sense that all electrons in a current channel experience acceleration once the electric field is turned on. In this scenario, the acceleration path length of each electron is defined by the distance between its start position and the loop footpoint. Defining L the acceleration by the maximum electron energy I obtained ? by the electric field over the loop length , (or an electron energy gained over a . proportionally smaller distance ),. . . . L ? . . . 5 6 O6 7 . 5 6 O6 )". (. L . . .. we find the following timing for electrons in Model 4: .

(31) . . . . &. I. ?. . . . . &. I . 8. . 6. . . The resulting timing in Model 4 is shown in Fig. 12.7 (right panel) for the same TOF ? ( ) distance 5 km used in Fig. 12.4 and for 2 5 (or MeV). The high-energy electrons arrive later at the thick-target site than the low-energy electrons for every parameter combination, and thus, cannot fit the data. Therefore, Model 4 can clearly be rejected for all flares where the high-energy electrons arrive first. In Model 5 the electrons are allowed to exit a current channel with an accelerating electric field at an arbitrary location. Because electron spectra have always negative slopes, this means that more electrons leave the current channel after a short distance than after longer distances. This model may mimic a realistic situation when the current channels are relatively thin (for most extreme aspect ratios see Emslie & Henoux 1995), so that electrons exit a current channel by cross-field drifts. The length of the acceleration path can then be determined from the final electron energy using Eq. (12.3.6), . . ?. . . ? . . 8 . . 5 6 O6 7 . 6. . <. The resulting timing of electrons arriving at the thick-target site is then composed of . . the sum of the acceleration time over the acceleration path length and the ? . free-flight propagation time over the remaining path length , . . . . . . ?. . . &. . . . . . . . &. & . I. . EG

(32) F EF E I. I . . ?. 6. . . . . . . . 5 6 O6 7. The fit of the timing Model 5 is also shown in Fig. 12.7 (right panel) for two different parameter combinations ( MeV and 10 MeV). The essential result is that .

(33) 12.4. KINEMATICS OF PARTICLE INJECTION. 529. Model 5 fits the data the better the smaller the acceleration time is relative to the propagation time, a situation that approaches asymptotically Model 2 for high electric field strengths. Consequently, the best fit is consistent with a small-scale acceleration region like in Model 2. Acceleration by field-aligned DC electric fields increase only the parallel velocity of a particle, while the perpendicular velocity given by the gyromotion, approxiB B7 L mately with thermal speed, , remains unaffected. Thus the pitch angle of an L accelerated particle would change as function of the gained energy I 8 as, B$. .. . . . -. ' ). . B. 8. B. . . :. 6. 56 O=6

(34) 7 . For every electron that is accelerated out of a thermal distribution to relativistic energies, this would produce very small pitch angles in the order of a few degrees, to that all accelerated electrons would immediately precipitate and no trapping is possible, in contradiction to the ubiquitous trapping observed in virtually all flares. In summary, we find that all large-scale DC electric field acceleration models have some problems to accomodate the observed energy-dependent hard X-ray time delays. The only model that yields a consistent timing (Model 3) requires the existence of many current channels with different electric fields, which need to be switched on and off on subsecond time scales in synchronization. Such a requirement is difficult to realize, since the currents cannot be switched off faster than with Alfvénic speed (Melrose 1992). The problems go all away for small-scale electric fields, either Super-Dreicer DC fields in reconnection regions as envisioned by Litvinenko (1996b) or convective electric fields in coalescing magnetic islands (Kliem 1994), as long as the acceleration . times are substantially smaller than the propagation times to the chromosphere, . In this case, however, our time delay measurements are not sensitive anymore .

(35) to any particular acceleration time dependence , except that we can place upper limits.. 12.4 Kinematics of Particle Injection 12.4.1 Scenarios of Synchronized Injection In the previous sections we essentially concluded that the observed energy-dependent hard X-ray delays (where the lower energies of pulses are delayed with respect to the higher energies) are not sensitive to acceleration models, because the models generally predict the opposite timing, i.e., longer acceleration times for high-energy particles than for low-energy particles. On the other side, acceleration times are estimated to be comparable or longer than propagation times, e.g., for stochastic acceleration (Eq. 12.3.1) or for sub-Dreicer electric fields in large current sheets. The insensitivity of the observed energy-dependent timing to these acceleration time scales can thus only be reconciled by an intermediate injection mechanism that synchronizes the injection of accelerated particles at the “exit” of the acceleration region (Fig. 12.1). In other.

(36) 530. CHAPTER 12. PARTICLE KINEMATICS. Figure 12.8: Accelerated particles are trapped in the cusp region while they undergo first-order Fermi acceleration. Particles leave the acceleration-plus-trap region once they become injected onto magnetic field lines that connect to the footpoints. The injection mechanism could be related to magnetic island dissipations in the reconnection region, fast shock waves, or the pitch angle evolution (Somov & Kosugi, 1997).. words, the acceleration region could be a black box that accelerates particles continuously or with an arbitrary time cadence, while an injection mechanism opens a gateway in a pulsed fashion onto magnetic field lines that connect to the chromospheric footpoints of flare loops. The start time of particle propagation is then decoupled from the acceleration time. The physical nature of the postulated particle injection mechanism is little explored and may be different for various acceleration and flare models. For parallel DC electric field acceleration, the particle leaves the acceleration path directly and continues propagation (e.g. Model 5 in the previous section) without any intervening delay, so there is no injection mechanism present. Also for perpendicular DC electric field acceleration (Fig. 11.3, right panel; Litvinenko 1996b), there is no time delay between the acceleration inside the current sheet and subsequent propagation once the particle exits the current sheet sideward. In contrast, for DC field acceleration in magnetic islands (Figs. 11.5 and 11.6; Kliem 1994), for stochastic acceleration (e.g. Miller et.

(37) 12.4. KINEMATICS OF PARTICLE INJECTION. 531. al. 1996), for shock acceleration (Fig. 11.19; Tsuneta & Naito 1998), in particular in collapsing traps (Figs. 11.18 and 12.8; Somov & Kosugi, 1997), particles are temporarily trapped during acceleration and do not escape onto a magnetic field line (leading to the chromospheric footpoints) before an injection mechanism deflects the particle out of the acceleration trap. This injection mechanism could be the dynamics of a macroscopic structure, e.g. the coalescence of two magnetic islands into a single one, which changes the locations of the separatrix surfaces that divide trapped from free-streaming electrons. Alternatively, particles can also leave the acceleration region by microscopic changes, such as by changing the pitch angles or by a transit near an X-point. Generally, when particles are accelerated in parallel direction to the magnetic field, their pitch angles become smaller and the probability to be mirrored in converging magnetic bottles becomes smaller. So every parallel acceleration mechanism controls also the pitch-angle evolution of particles in such a way that they automatically escape after a finite time from the magnetic trap. However, the observation of subsecond pulses in hard X-rays and radio bursts that appear to be strictly synchronized at all (non-thermal) energies (Fig. 12.3) suggest a macroscopic mechanism rather than a microscopic kinetic effect, because microscopic effects are statistical and not synchronized between particles of different energies.. 12.4.2 Model of Particle Injection During Magnetic Reconnection Let us derive a quantitative model (Aschwanden 2004) that explains particle injection in solar flares in the context of a Petschek-type magnetic reconnection scenario. In the standard flare reconnection model ( 10.5.1; Carmichael 1964; Sturrock 1966; Hirayama 1974; Kopp & Pneuman 1976; Tsuneta 1996a; Tsuneta et al. 1997; Shibata 1995), an X-type reconnection occurs in a coronal height and the newlyreconnected magnetic field lines relax into a force-free configuration which becomes later (after chromospheric evaporation) the soft-X-ray bright flare loop. The height ratio of the reconnection point height to the flare loop height , . &3. 8. . :. . 6. (. . . . . 56 6 8 /. has been determined by electron time-of-flight measurements, using pulse background A ? A ? % % ' subtraction methods ( 6 )# =6 / ; 3 76 / 6 O ; Aschwanden ? A"? % E %! 6 . =6 . ; Aschwanden et al. 1996c), pulse deconvolution methods ( 3 ? & A E ? % %' ( 6 50 6 ; Forbes et al. 1999d), or by loop shrinkage measurements ( 3 E & Acton 1996). B) and carries The reconnection outflow has initially an Alfvénic speed the magnetic flux of the newly-reconnected field lines. The newly-reconnected mag$ in the initial cusp shape, which netic field lines experience a strong Lorentz force . gradually reduces when the field line relaxes into a force-free shape, because the cur%! vature force is reciprocal to the curvature radius 2 , if the magnetic field has only a parallel component (Eqs. 6.2.16-18),. $ .

(38) . 8. <. I -. :. /. . -. . .

(39) . . . /. I 2. %!. 6. . . . 56 /. 6 57.

(40) CHAPTER 12. PARTICLE KINEMATICS. 532. t=0.5 tR. t=1.0 tR. t=1.5 tR. t=2.0 tR. t=2.5 tR. t=3.0 tR. Figure 12.9: Scenario of field line relaxation after an X-point reconnection: The apex height of the field line relaxes exponentially into a force-free state from the initial cusp shape to the final quasi-circular geometry. The losscone angle of the trapped particles gradually opens up and releases more particles from the trap (Aschwanden 2004).. Thus, in our dynamic model we assume that the motion of the post-reconnection field B line occurs initially with Alfvénic speed , when the curvature radius is minimal in the cusp, and then decreases asymptotically to zero at the end of the relaxation phase, when the force-free field line reaches the maximum curvature radius in a quasi-circular flare loop geometry (Fig. 12.9). We approximate the height dependence of the apex of the relaxing field line with an exponential function (Fig. 12.9) with an e-folding relaxation time scale , . 4. . . . . . . . 8. . . 5 6 =6 O7. ( :. /. which matches the initial condition 7 (demarcating the height of the X-point) and final asymptotic limit (the height of the relaxed flare loop). The speed of the height change is simply the derivative of Eq. (12.4.3), BG . . . . . . . . . . . 8. which defines the relaxation time scale , . . . . . . :. . . B. . 8. . :. . 5 6 =6 M . 5 6 =6 . (. /. (. . . BG . . B. . /. / . B. 7 so that Eq. (12.4.4) fulfils the initial condition and the final asympBG, totic limit . Reconnection theories predict that the outflows have Alfvénic speed, so we can set the initial value of the relaxing field line approximately equal to the reconnection outflow speed, assuming frozen-in flux conditions, B . B. . 5 6. . &<. $. F'F. . K. L ( . . 5 6 =6 7 /. ..

(41) 12.4. KINEMATICS OF PARTICLE INJECTION. . 533. L. denotes the external magnetic field strength on both sides of the outflow where K region and denotes the internal electron density in the X-point. The magnetic field in the X-point is zero by definition, and gradually increases with increasing distance from the X-point. Correspondingly, the losscone angle . is zero at the X-point, and then gradually increases with increasing distance from the X-point, or as function of time for the progressively relaxing field line. A convenient parameterization for this evolution of the loss-cone angle . is an exponential function, with an e-folding time scale that we call injection time,. . . . . 56 6 ) 6 7 and then opens up to a full half cone . -. .. . 5 , . :. 8. /. The pitch angle is initially zero, . , . - A . . We define a mirror ratio between the 5 for relaxation times magnetic field at the flare loop top, at an arbitrary height in the cusp and which relates to the loss-cone angle . , by, . . . . ,. . . . . . . ,. ). . I .. . . 56. . . 6. ,. 6 <M /. With this parameterization, the evolution of the magnetic field strength at the apex of the relaxing field line is (using Eqs. 12.4.7-8),. . . . . ). I .. . . . or, inserting the transformation dence of the magnetic field. . . . ). . ). , . . -. I. . . 8. 5. . 8. : 7: . (. . 56 /. 6 M. from Eq. (12.4.3), we obtain the height depen-. -. I. . 5

(42). 8. . . . . :. . . H. 56 6

(43) 7 . 6 . . . . /. . The magnetic field strength in the X-point is zero by definition, , and then increases in the reconnection outflow region to the ambient value, which we 4 at the height of the flare loop (at the position after final denote with relaxation). So the magnetic field monotonically increases from the X-point towards the flare loop top. For the spatial variation of the magnetic field near the X-point (Eq. 12.4.10) it is actually more instructive to express the injection time in terms of a spatial scale. We define a magnetic length scale where the magnetic field increases from zero at the A center of the X-point towards half of the value at infinite, ) 5 (Fig. 12.10, bottom panel). This can be calculated straightforwardly from Eq. (12.4.10) and we find the relation,. . . . where . . . . . . . . . . A 5 . . . . -. "! . 5. . -. 5. . expresses the fraction of the magnetic length scale. ,. . . . 56 6 . (. /. to the cusp distance. . . . . . . G . 6. . 56 6 8 5 /.

(44) CHAPTER 12. PARTICLE KINEMATICS. 534. Petschek model. Slow. δ. Magnetic field B(h) [G]. ks. shoc. ∆. 15 10 5 -LB -4. -2. LB. 0 Distance (h-hX) [Mm]. 2. 4. Figure 12.10: Definition of the magnetic length scale at the X-point in the context of a Petschek-type X-point geometry (top panel), with the height in horizontal direction. The magnetic field

(45) as function of height near the X-point is shown (bottom panel) for the model parameters used in Fig. 12.11. The magnetic length scale is defined by the range where the magnetic field increases to half of the asymptotic value far away from the X-point (Aschwanden 2004). The relaxation of the field line has the effect that the narrow loss-cone at the reconnection point opens up and allows particles with gradually larger pitch angles to precipitate. In other words, the particles that are accelerated near the X-point see initially a very small losscone angle and are thus fully trapped, while the gradual opening forced by the relaxation of the field line untraps them as function of the relaxation time. If we assume that the accelerated particles have all been accelerated near the X-point and have an isotropic pitch angle distribution, their initial distribution is . . . . > .. 5. -. ). .. .. 85 6 =6

(46) OM. 6. /. , A The precipitation rate is then proportional to the time derivative, where the losscone angle has the time dependence given in Eq. (12.4.7), . . . , . . . . . .. . . . .. . >. + ) ' *. .. . ,. . . , .. . . . . . -. 8. 5. . 8. . . :7:. . . 8. 85 6 =6 . . /. @/. This implies that the opening speed of the losscone angle if fastest at beginning and slows down with time. To account for a finite acceleration time, we assume that the number of particles accelerated to an arbitrary energy increases with some power of the trapping time , so the fraction of accelerated particles increases as, . . . >$. . 6. . 85 6 =6 7 /. . . :. 6.

(47) 12.4. KINEMATICS OF PARTICLE INJECTION. 535. 2.0. Evolution of parameters [normalized]. 1.5 h(t)/hL. R(t) 1.0. B(t)/BL α(t)/(π/2). tp. v(t)/vX. 0.5. tF 0.0 0. F(t)/max(F) 1. 2. 3. 4. 5. Time t[s]. . Figure 12.11: The dynamic evolution of the parameters is shown for the field-line relaxation model ( 12.4.2). The parameters are: height of X-point Mm, height of relaxed flare loop Mm, density cm , magnetic field at apex of flare loop G, and magnetic scale length in reconnection region . The curves show the evolution of the height , relaxation velocity with km s , magnetic field at apex of relaxing field line , losscone angle , magnetic mirror ratio , and precipitation flux . The relaxation time is s, the injection time scale s, and the FWHM pulse duration is s (Aschwanden 2004).. . .

(48). .

(49) .

(50) .

(51) .

(52). . .

(53)

(54) . .

(55) . . . . The flux of precipitating accelerated particles has then the time dependence. ) , ! . . . >$. . . -. 8. . 5. . 8. . . : 7:. 8. . . :. 6. . 56 6 7 /. .. For a given acceleration mechanism, the parameter is a constant, and thus the pulse ,'A profiles are scale-invariant if scaled by the normalized time . Therefore, the A FWHM of the flux profile, , is also scale-invariant, i.e., the ratio . We calcu , A late this scale-invariant ratio numerically for different acceleration power. . . .

(56) CHAPTER 12. PARTICLE KINEMATICS. 536. hHXR = 22100 km Precipitating electrons. HXR. Acceleration and magnetic trapping in cusp region. SXR. hSXR = 12500 km. Precipitating electrons. HXR. Figure 12.12: Temporary trapping occurs in the acceleration region in the cusp region below the reconnection point in our model (right panel), which can explain the coronal hard X-ray emission observed during the Masuda flare (left panel; Masuda et al. 1994). The observations in the left panel show a Yohkoh/HXT 23-33 keV image (thick contours) and Be119 SXT image (thin contours) of the 92-Jan-13, 1728 UT flare (Aschwanden 1998b).. indices. . =6 ( 6 66 ( 56 , . . . . . 6 O 6 65 6 O 6 ) . . . @/. . . ><>/. . . . . . . *,+*,+*,+*,+*,+-. /. . .. . . . . . . . . 6 6 6 6. 85 6 =6 ). . . . /. . 95 6. , A. We show the time evolution of the various parameters in Fig. 12.11: the height , A BG, A B , magnetic field at apex of the relaxing field line relaxation velocity , . , A - A . losscone angle . , and precipitation flux 5 , magnetic mirror ratio We can now express the FWHM pulse duration only as function of the indepen and , using Eqs. (12.4.11-12) and (12.4.17), dent parameters. . . . . 8. . . . A 5 . . . ( :. . -. . B. 5. . . 5 B . 6. 85 6 =6 M /. &<. The right-hand approximation expresses most succintly the relation of the pulse duration to the underlying physical parameters: the pulse duration is proportional to the B B Alfvénic transit time (with velocity ) through the magnetic length scale of the X-point region. The fastest time structures are produced by the smallest X-point regions (and presumably for the smallest flare loops). For instance, an X-point region B with a spatial extent of 4 km, an Alfvén velocity of km s EF , yields 5 pulses with time scales of typically 6 6 50 s, depending on the acceleration power index . If the Alfvén speed is known, the pulse duration can be used to estimate. . . . .

(57) 12.5. KINEMATICS OF PARTICLE TRAPPING. 537. Injection point (s=0) Pitch angle α(s=0)=α0 Magnetic field B(s=0)=B0. Mirror point (s=sM) Pitch angle α(s=sM)=900 Magnetic field B(s=sM)=BM. . . is defined by the ratio of the magnetic field strengths between the mirror point, and the injection point, , which define the loss cone angle . Particles injected with a larger pitch angle are mirrored, while particles with smaller pitch angles precipitate through the losscone.. Figure 12.13: The magnetic mirror ratio. . . . the magnetic length scale of a Petschek-type X-point (Fig. 12.10), which entails the diffusion region of a magnetic reconnection process. Thus, this model provides a direct diagnostic of the magnetic reconnection geometry. This model provides also a natural explanation for the existence of above-the-loop-top hard X-ray emission (Fig. 12.12), as discovered by Masuda et al. (1994) and reviewed in Fletcher (1999). The model demonstrates also that the observed pulse durations are controlled by the injection time rather than by the acceleration time scale.. 12.5 Kinematics of Particle Trapping 12.5.1 Magnetic Mirroring After a magnetic reconnection process in a flare, the closed field lines always relax into a more force-free state, which corresponds to dipole-like field geometries with the strongest magnetic fields at the footpoints and a weaker magnetic field in the coronal segments inbetween (Fig. 9.4). This implies that each closed magnetic field line forms naturally a magnetic trap (Fig. 12.13), where particles mirror forth and back as long as the trapped plasma is collisionless and adiabatic particle motion is ensured. In the collisionless limit, the particle motion is adiabatic and the magnetic moment

(58) is conserved along the loop coordinate ,.

(59) . I. F . . L!B . . I . . . I. F . LB. . I ). . . . I .. . . . K . The pitch angle . changes as function of the magnetic field line, while the velocity v is constant for adiabatic motion. . . . . 6 . . 56 6 8 . along the field.

(60) CHAPTER 12. PARTICLE KINEMATICS. 538. To understand the basic kinematics in magnetic traps we assume a dipole-like magnetic field, which in lowest order can be approximated by a parabolic equation (e.g., Trottet et al. 1979),. . . . . . . . 3. . 8 I I. . . . 5 6. ( . . 6 5. 7 represents the (minimum) magnetic field strength at the loop where 3 , and is the length from the loop top to the mirror point. The location top at of the mirror point can be defined at the interface between the (coronal) collisionless and (chromospheric) collisional regime, which usually conincides with the zone of largest magnetic divergence, i.e., the canopy structure in the transition region (Fig. 4.25). The magnetic mirror ratio R is defined by the ratio of the magnetic field strengths at the mirror point and looptop,. . . . . . . . . 3. . ). 5 6 . 6 I . 3 . . 6 O7. The losscone angle . 3 is defined by the critical pitch angle . 3 of a particle at the looptop that decides whether a particle is mirrored and trapped (if it has a larger pitch . 3 ), or whether it escapes through the losscone and becomes untrapped angle, . (if it has a smaller pitch angle, . 0<. 3 ). This losscone angle simply follows from comparing the magnetic moment (Eq. 12.5.1) at the loop top, . 7 . 3 , with A - 5, that of the mirror point, . . . . .:3 . -. . ' ). ". . 5 6 . 6. . 6 /M. We will see later on ( 13) that the mirror ratio in flare loops can be determined from measurements of the losscone angle . 3 , using the ratios of directly-precipitating to trapped electrons from the corresponding hard X-ray fluxes. An important quantity is also the bounce time of a mirroring particle forth and back a mirror trap, which is, for our parabolic field (Eq. 12.5.2) and loop half length , . . . (/ 3. B$

(61) . . . . . /. . The ratio of the travel time 3 (/ is, %. . & 3. . B . A. . . I. B I. B. . . -. 5. . F. 5 6 . 6. 6 . EF. v along the magnetic field line to the bounce time . . . . 3 . -. 5 . . F H!I. (. . . 5 6. . 6 .7. which approaches the value 1% =6 .0/ for large mirror ratios . This ratio &% yields also the relevant pitch angle correction factor to convert an electron-time-of ? #""A ? A8 3 flight distance into the length of a magnetic field line, i.e. % (Eq. 12.2.2)..

(62) 12.5. KINEMATICS OF PARTICLE TRAPPING. 539. Velocity distribution f(v||,v⊥) at injection site Injection site (x=0), Binj Pitch angle: α > α0. v⊥ α=(π/2). α<α0. α0. Loss-cone. Mirror point Loss-cone site Bloss. α=0 v||. Trapped electrons (α>α0) Directly precipitating electrons (α<α0). Chromospheric HXR emission site (x=l). . Figure 12.14: The velocity distribution of a trapped particle distribution forms a losscone with ) precipitate a critical pitch angle (right frame). Electrons with small pitch angles ( ) are intermediately directly to the chromosphere, while electrons with large pitch angles ( trapped and precipitate after they become scattered into the losscone (Aschwanden 1998a).. . . . 12.5.2 Bifurcation of Trapping and Precipitation The criterion for the bifurcation of particles trajectories, whether they propagate freestreaming directly to the footpoints or become trapped (Fig. 12.1, 12.14), is controlled by the initial pitch angle . 3 at the injection site. Those particles that have a pitch . 3 ) become trapped, until they are angle larger than the critical losscone angle (. pitch-angle scattered into the losscone (. 0 . 3 ) after an energy-dependent trapping !! " . They subsequently escape from the trap and precipitate (in the so-called time trap-plus-precipitation model). In the simplest model without energy loss in the trap, ' ( for a -like injection, the number of electrons (with kinetic energy ) in the trap decreases exponentially, with an e-folding time constant that corresponds to the ' trapping time (Melrose and Brown 1976), i.e.. . . !' (' !' ( 7 'A !' G6 56 6 ) K ' ( !' (' , is defined by the time derivative of , i.e. The precipitation rate, !! (' K ' ( K ' ( M 'A8 ' ( 56 6 M K ' ( with M ' ( 7 A8 ' . It ( is consequently also exponentially de , the precipitation rate from the trap, creasing. For a general injection function KG ' ( ' !' , can be described by a convolution with the trapping time , , K ' ( ( ' ' 6 56 6 M 3. . . . . <. . . . .

(63) CHAPTER 12. PARTICLE KINEMATICS. 540. Electron injection function n(ε,t+tTOF(ε),x=l) nprec(ε,t+tTOF(ε),x=l) ntrap(ε,t+tTOF(ε),x=l). n(ε,t,x=0). ε1 ε2 ε3. tTOF(ε). ε4. ttrap(ε).

(64) . at the injection site and at the hard X-ray emission site ), delayed by an energy-dependent electron time-of-flight interval , schematically shown for 4 different energies . The injection function at the hard X-ray emission site (thick curve) is broken down into a directly-precipitating component (thin curve) and the trap-precipitating component (hatched curve). Note that the (e-folding) trapping time increases with energy , while the timeof-flight delay decreases (Aschwanden 1998a)..

(65)

(66). . Figure 12.15: Temporal relation of the electron injection function.

(67).

(68).

(69). If the acceleration mechanism produces a wide range of pitch angles, there is always a mixture of small and large pitch angles, which produces then a mixture of directly-precipitating and trap-precipitating particles. This seems generally to be the case based on the data analysis of observed hard X-ray delays ( 13). This dichotomy is also present in the morphological structure of hard X-ray time profiles, where fast pulses indicate the fraction of directly-precipitating electrons, which preserve the pulsed time profile of the injection mechanism, while the smoothly-varying lower envelope represents a measure of the trap-precipitating electrons, which smear out the time profiles of the injection mechanism (Fig. 12.16). Therefore, the total elecK (' tron precipitation rate is composed of a combination of the two components !L. (Fig. 12.15): (1) a fraction of electrons that precipitates directly, and (2) the com-.

(70) 12.5. KINEMATICS OF PARTICLE TRAPPING. 541. HXR Flux. Pulsed HXR fine structure (Direct precipitation). Trapped electrons Smoothly-varying HXR flux (Trap+precipitation). Directly precipitating electrons Chromospheric Thick-target HXR emission. Time. Figure 12.16: Relation between electron propagation and observed hard X-ray time structures: Electrons with small pitch angles (injected near the flare loop top) precipitate directly and produce rapidly-varying hard X-ray pulses with time-of-flight delays, while electrons with large pitch angles become trapped and produce a smoothly-varying hard X-ray flux when they eventually precipitate, with a timing that corresponds to trapping time scales (Aschwanden 1998b). plementary fraction. . . K ( . . 4. !L. . ( . that precipitates after some temporary trapping, !L. ( . . L. . !! L . ( . . ". 3. . . ('. . . L K . '= ( . , !' 6 . . 56 6

(71) 7 . The dichotomy of the direct-precipitating and trap-precipitating electrons can be deconvolved from observed hard X-ray time profiles, for instance by forward-fitting of the convolution function (Eq. 12.5.10), (Aschwanden 1998a).. 12.5.3 Trapping Times Theoretical treatments of trapping models can be found for solar flares (e.g., Benz 1993; 8) or for magnetospheric applications (e.g., Baumjohann & Treumann 1997; 3). Let us briefly review the theoretical development of solar flare models. Trapping times of electrons were estimated from various pitch-angle scattering mechanisms into the loss-cone, e.g. by Coulomb collisional deflection (Benz & Gold 1971), by quasilinear diffusion (via resonant wave-particle interactions) induced by electrostatic (hydrodynamic) waves (Wentzel 1961; Berney & Benz 1978), whistler waves (Kennel & Petschek 1966; Wentzel 1976; Berney & Benz 1978; Kawamura et al. 1981; Chernov 1989), lower-hybrid waves (Benz 1980), electron-cyclotron maser (Aschwanden & Benz 1988a,b; Aschwanden et al. 1990a), or plasma turbulence (e.g.review by Ramaty & Mandzhavidze 1994; Petrosian 1996)..

(72) CHAPTER 12. PARTICLE KINEMATICS. 542 300 250 Electron energy [keV]. 213.5 keV. -3. 200. cm. 11. .0. 165.8 keV 150. ne. 10. =2. 126.1 keV. -3. cm. 11. 100 96.9 keV 76.7 keV. ne =. 5. 10. 1.2. 50 0 0. . 2. 4 6 8 Collisional deflection time [s]. 10. . Figure 12.17: Collisional deflection times are shown for electron densities in the range of.

(73) cm , roughly fitting the observed time delays of the lower envelopes of the hard X-ray time profiles observed during the Masuda flare on 1992-Jan-13, 17:29 UT, and thus are interpreted as trapping time differences (Aschwanden et al. 1996c).. The Coulomb collisional deflection time is considered to be an upper limit for trap ' L

(74) % + ping times, (also called weak-diffusion limit). The trapping time (in L

(75) % the weak-diffusion limit) is given by the electron collisional deflection time (Trubnikov 1965; Spitzer 1967; Schmidt 1979; Benz 1993) ' . where . . . . L

(76) %. . L

(77) 6

(78) KL J!H!I 5 *. . is the Coulomb logarithm,. . . . <6. . L K L. EF H!I . . . (. L. /. 65. . . 85 6 6 8 . 85 6 6 57. (. 6. . . The observed time delays between the slowly-varying components of hard X-ray time profiles ( 13) generally fit the energy-dependence of collisional deflection times (Eq. 12.5.11) K L F'F cm E J (Fig. 12.17), which are found for reasonable plasma densities, i.e., to be consistent with densities measured independently from the emission measures of soft X-ray flare loops (Aschwanden et al. 1997). The trapping in solar flare loops thus seems to be controlled by collisions, i.e., in the weak-diffusion limit. Lower limits of trapping times are controlled by the diffusion rate into the loss. cone (with angle . ), which can occur as fast as every bounce time (the so-called strong-diffusion limit), i.e. (Kennel 1969),. . '.

(79) . 4. 5 . I. 6. . 85 6 6

(80) OM . The temporal dynamics of trap-plus-precipitation models, which includes the temporal and spectral evolution of the trapped and escaping particle distributions and the related hard X-ray fluxes has been analytically described in a number of papers (Melrose &.

(81) 12.6. KINEMATICS OF PARTICLE PRECIPITATION. 543. Brown 1976; Bai 1982a; MacKinnon et al. 1983; Zweibel & Haber 1983; Craig et al. 1985; Vilmer et al. 1986; Ryan 1986; MacKinnon 1986; 1988; Alexander 1990; McClements 1990a,b; 1992; McClements & Baynes 1991; Ryan & Lee 1991; Hamilton et al. 1990; Lu & Petrosian 1988; 1990; Hamilton & Petrosian 1990), as well as simulated numerically (MacKinnon & Craig 1991). Particle dynamics in traps have been modeled to study asymmetries of hard X-ray sources (Melrose & White 1979), or trajectories of radio type N bursts (Hillaris et al. 1988). Besides purely magnetic traps, models with additional electrostatic potentials have also been proposed for solar flares (Spicer & Emslie 1988), similar to models of Auroral Kilometric Radiation (AKR) (e.g. Louarn et al. 1990).. 12.6 Kinematics of Particle Precipitation In the following we consider particle precipitation first in symmetric traps (with symmetric magnetic field; 12.6.1), for sake of simplicity, as well as for asymmetric traps (with asymmetric field; 12.6.2), which occur more frequently in solar flares.. 12.6.1 Symmetric Traps L. The fraction of directly-precipitating electrons can be self-consistently related to the critical losscone angle . 3 . For sake of simplicity, we may consider first the case of an isotropic pitch-angle distribution at the injection site and for symmetric loop geometries, % . !L

(82) . 3. .3 . ). . 3 H!I. .. ). .. .. .. . . . + ).:3 ' *. 56 =6 8. . 6. .. This case corresponds to a double-sided losscone distribution (Fig. 12.14 right panel), as it occurs in symmetric flare loops. With a deconvolution method we can measure L. directly. The corresponding losscone angle ./3 is then (for isotropic pitch angle distributions and symmetric loops), L. .3. . . . -. . . + ) '@' *. . L. . . . 56 =6 57. (. .. leading to the mirror ratio .N3 defined in Eq. (12.5.3). A 3 , i.e., the ratio of the magThe inference of the magnetic mirror ratio netic field ( ) at the mirror point to that at the injection point at 3 ), together with the (projected) time-of-flight distance the loop top ( 9 3 ? between the injection site and chromospheric energy loss site (which is presumably close to the losscone site), yields a measure of the magnetic scale height . Defining the scale height by an exponential model. . 3 . . . . . . 3. . . . . . ?. . . . . and using the definition of the mirror ratio . A. . . . . . . 6. . . ? . . . (. . 56 =6 OM .. 3 , we obtain . 56 =6 .. /.

(83) CHAPTER 12. PARTICLE KINEMATICS. 544. The magnetic field can be approximated with a dipole field, parametrized with the dipole depth and the photospheric field 3 ,. . . . 8 3. . . . E J. :. to which the magnetic mirror ratio can be related by resulting magnetic scale height is then. . . . . .

(84). . . . 1 O. . . . . . 5 6 6 . (. . 3. A. . . . . A. . . . 3. . The. 5 6 6 7 . 6. .. .. Typical dipole depths are of order Mm, leading to magnetic scale heights of. 3 . 6 . , losscone angles of . 3 Mm, magnetic mirror ratios of , and L. 6 / , for symmetric traps (Aschwanden et al. 1997). precipitation ratios of . . . . . 12.6.2 Asymmetric Traps The timing analysis of CGRO data, based on the total hard X-ray flux without spatial information, provides a global electron trapping time scale. The spatial structure, however, can often be described by two magnetically conjugate footpoint sources, which often have asymmetric hard X-ray fluxes according to the Yohkoh/HXT images. These unequal double footpoint sources indicate electron precipitation sites in a flare loop with asymmetric magnetic field geometry. We need therefore to develop an asymmetric trap model to relate the trapping time information from CGRO data to the asymmetric double footpoint sources seen in Yohkoh/HXT data. In order to mimic an asymmetric trap model we rotate the reference system of a symmetric dipole-like magnetic field by an angle , as shown with three examples in Fig. 12.18 (top): the symmetric case with (left), a weakly asymmetric case 3 O where the dipole coil is rotated by (middle), and strongly asymmetric case 3 where the coil is rotated by . . (A spherically symmetric with a ”unipo sunspot 3 lar” vertical field would correspond to the extreme case of .) The acceleration or injection site into the trap is assumed to be midway (with a magnetic field ) between the two footpoints (with magnetic fields and ). These three mag ( ( F I netic field values are decisive what fraction of electrons are trapped or F I precipitate to the two footpoints. Because of conservation of the magnetic moment,

(85) C IF L vI A > ) . I A K , the critical pitch angles that separate precipitating from trapped particles at the two footpoints are defined by the magnetic mirror ratios . ( F 5 6 .6 " ) . . F. . I. . . . . I. . . . ). ). . . . .. F. I. . I I. 5 6 6 7 is stronger (. F F . 6. . .. <. For positive rotation angles the magnetic field at footpoint (. ), and the critical angle, also called losscone angle, is smaller (. 0 . ) I F I F I than at the opposite footpoint 2. The asymmetric magnetic field along a flare.

(86) 12.6. KINEMATICS OF PARTICLE PRECIPITATION. ACC. 545. ACC. BA. ACC. BA. ψ= 00 B1. BA. B0. ψ=300 B2. ψ=600. B1. B2. B0. B1. B2. 600. 600 B1. 600. 0 -1.5 -1.0 -0.5 0.0 0.5 Loop length s/r0. 40. TWO-SIDED PRECIPITATION. 20. 0 -1.5 -1.0 -0.5 0.0 0.5 Loop length s/r0. 1.0. 1.5. v_| α1. 1 2 1. 1 2. 60. 40. 20. α1 2 v ||. 1.0. TWO-SIDED PRECIPITATION. 0 -1.5 -1.0 -0.5 0.0 0.5 Loop length s/r0. α2 1 α1 2 2 1. v_|. 1.5. 2. 1.0. 1.5. TRAPPING. 60. 40. ONE-SIDED TWO-SIDED PRECIPITATION. 0 -1.5 -1.0 -0.5 0.0 0.5 Loop length s/r0. 1.0. 1.5. v_|. α2. 1 2. B2. 80. 20. 1.0. BA. 0 -1.5 -1.0 -0.5 0.0 0.5 Loop length s/r0. 1.5. ONE-SIDED. Accelerator. 200. TRAPPING. 80 Pitch angle α(s). Accelerator. 60. BA. 0 -1.5 -1.0 -0.5 0.0 0.5 Loop length s/r0. 1.5. TRAPPING. 80 Pitch angle α(s). 1.0. 200. 400. Accelerator. BA. B2. Pitch angle α(s). 200. 400. Accelerator. B2. Accelerator. 400 B1. Magnetic field B(s). 800. Magnetic field B(s). 800. Accelerator. Magnetic field B(s). B1 800. α1 v||. Maximum Trapping. 2 1. 2. v||. No Trapping. Figure 12.18: Three model scenarios for a symmetric (left column), a slightly asymmetric (middle column), and strongly asymmetric magnetic trap (right column). The spatial configuration of a buried dipole and the resulting pitch-angle motion of trapped and/or precipitating electrons is sketched (in top row), the magnetic field is parametrized as function of the loop coordinate (second row), the pitch angle variation as function of the loop coordinate and three pitch angle regimes are shown (third row), and the corresponding pitch-angle regimes in velocity space (v v (bottom row). The numbers 1 and 2 correspond to the left and right losscone site, with the stronger magnetic field located at footpoint 1 (Aschwanden et al. 1999d).. . . .

(87) CHAPTER 12. PARTICLE KINEMATICS. 546. . loop is visualized in Fig. 12.18 (second row) with a quadratic model. Note that in the 3 0 case with strong asymmetry ( . ), trapping is not possible because , I. 0 corresponding to a mirror ratio . I The pitch angle variation . along the loop according to conservation of the magnetic moment is shown in Fig. 12.18 (3rd row). Generally, three regimes can be distinguished in the . -s plane: (1) a direct precipitation regime for initial pitch angles 0 . M 0 . , (2) a secondary precipitation regime after one mirror bouncing F for initial pitch angles . 0 . M 0 . , and (3) a trapping regime for initial pitch F - A I M 0 5 . These three regimes are clearly discernible in the case angles . 09. I 3 with weak asymmetry ( O , Fig. 12.18 middle column). For the symmetric case, the secondary precipitation regime collapses to zero because . . (Fig. 12.18 left F I 3 column). For the strongly asymmetric case ( . ) no trapping is possible because. - A 0 5 with there is no solution for . 0 . I I After we have a quantitative description of the pitch angle ranges that contribute to trapping and precipitation at both footpoints, we can now calculate the relative fractions of precipitating electrons at the two footpoints and obtain quantitative expressions for the hard X-ray flux asymmetry . We visualize the pitch angle regimes in velocity $( space (v v (Fig. 12.18 bottom) and label the different regimes with the footpoint numbers 1 and 2, to which the electrons precipitate, either directly, after one mirror bounce, or after intermediate trapping. We determine now the relative fractions of precipitating electrons by integration over the corresponding pitch angle ranges in velocity space. Here and in the following we assume an isotropic pitch-angle distribution at the K acceleration/injection site [ . ]. The fraction of directly precipitating F electrons at footpoint 1, which has the smaller losscone (. + . ), is F I . ) . . % ' +*). F 6 E 4 5 6 .6 7 . . . . . . . . . . F. F. 3. . . . . . . ). . .. . 5. The fraction of directly precipitating electrons at footpoint 2, which has the larger losscone angle ( . ), includes not only those electrons which precipitate without bouncing I ( ) but also those that bounce once at the mirror site 1 ( ) and precipitate then I I . 02. 7 0 . , at footpoint 2, i.e. with initial pitch angles of I F

(88) %

(89) % . ) . . . ) . . E % 3 4 E I. I. . . I. . ). + ). ,' * F. . 5. . 3. 5. . . . . + )=. ' *. . . I. ). . . . .. 85 6 6 M . 6. .. From the spatially unresolved data (e.g., from CGRO), only the combined fraction L. of directly-precipitating electrons at both footpoints can be measured, which is equivalent to the sum of both footpoint components and , . !L. (. . F. . I. . . . F. + ). ' 5 I. I. 6. . 85 6 6 8 .. The total fraction of trapped electrons is determined by the pitch angle range of the . , larger losscone, i.e. . 0 . 7 0 I

(90) I

(91) % E . ) . . L. % 4' +5). 6 85 6 .6 57 3. . . . . ). . . . .. I. . .

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