The Astrophysical Journal, 608:554–561, 2004 June 10 # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
PULSED PARTICLE INJECTION IN A RECONNECTION-DRIVEN DYNAMIC TRAP MODEL IN SOLAR FLARES Markus J. Aschwanden Lockheed Martin Advanced Technology Center, Solar and Astrophysics Laboratory, Organization ADBS, Building 252, 3251 Hanover Street, Palo Alto, CA 94304;
[email protected] Received 2003 November 11; accepted 2004 February 24
ABSTRACT The time structure of hard X-ray emission during solar flares shows subsecond pulses, which have an energydependent timing that is consistent with electron time-of-flight delays. The inferred time-of-flight distances imply an injection height about 50% above the soft X-ray–bright flare loops, where electrons are injected in a synchronized way. No physical injection mechanism is known that can account for the energy synchronization and duration of subsecond pulses. Here we propose a model in terms of dynamic loss cone angle evolution of newly reconnected magnetic field lines that relax from the cusp at the reconnection point into a force-free configuration, which can explain the subsecond pulse structure of injected particles as well as the observed correlation between the pulse duration and flare loop size. This quantitative model predicts that the pulse duration of hard X-ray or radio pulses scales with tw 2LB =vA (s), which is the Alfve´nic transit time through the magnetic outflow region of a Petschek-type X-point with magnetic length scale LB , and thus provides a direct diagnostic of the magnetic reconnection geometry. It also demonstrates that the observed pulse durations are primarily controlled by the injection time rather than by the acceleration timescale. Subject headings: Sun: flares — Sun: magnetic fields — Sun: particle emission — Sun: X-rays, gamma rays
1. INTRODUCTION
from the acceleration timescales (or acceleration times that are much shorter than the observed time structures). The mean pulse durations have been measured with an autocorrelation technique in eight flares jointly observed with CGRO and Yohkoh. They were compared with the flare loop sizes measured in Yohkoh images, and a proportional relation was found (Aschwanden et al. 1996c). These timescales tX of hard X-ray pulses were also measured with a wavelet analysis in a comprehensive set of 46 jointly observed CGRO and Yohkoh events, and the same proportional correlation was confirmed (Aschwanden et al. 1998), r loop s: ð1Þ tX ¼ (0:5 0:3) 9 10 cm
The pulsed time structure of energized particles in solar flares has always been a mystery, and no particle acceleration model could account for the intermittent time structure. The fastest time structures discovered with the Hard X-Ray Burst Spectrometer (HXRBS) on the Solar Maximum Mission (SMM ) spacecraft have been found down to rise and decay times of 20 ms and with pulse widths of P45 ms (Kiplinger et al. 1983). Such fast time structures were not very common in SMM data, because only about 10% of all flares detected with HXRBS have a sufficiently high count rate to detect significant variations on a subsecond timescale (Dennis 1985). With the much more sensitive Burst And Transient Source Experiment (BATSE) detectors on the Compton Gamma Ray Observatory (CGRO), which have a sensitive area of 2025 cm2 each, subsecond time structures were detected in virtually all flares recorded in the flare mode (activated by a burst trigger). A systematic study of 640 BATSE flare events with an automated pulse detection algorithm revealed a total of 5430 individual hard X-ray pulses, with an exponential distribution of pulse durations in the range tX 0:3 1:0 s (Aschwanden et al. 1995). The existence and duration of these pulses has been unexplained so far. Detailed timing analysis of these pulses has revealed that they show an energy-dependent timing that is consistent with electron time-of-flight dispersion, t1;2 / l=v1 l=v2 (Aschwanden et al. 1995, 1996a, 1996b, 1996c), corresponding to the time of flight between a coronal acceleration site at a height of 1.5 times the soft X-ray flare loop and the chromospheric footpoints (Aschwanden et al. 1996b, 1996c, 1999). This energy-dependent delay cannot be explained by standard acceleration mechanisms, because they produce an opposite timing, with the higher energy electrons lagging the lower energy electrons (Aschwanden 1996). This timing requires a synchronized injection that is decoupled
This spatiotemporal correlation was interpreted in terms of a scale-invariant magnetic reconnection geometry (Aschwanden 2002, x 4.3). A suitable reconnection model that can explain such fast time structures was proposed by Petschek (1964) in terms of a compact diffusion region in an X-type reconnection geometry (Fig. 1). Dynamic models of reconnection regions predict that the processes of tearing instability and coalescence of magnetic islands occur iteratively, leading to a scenario known as intermittent or impulsive bursty reconnection (Leboef et al. 1982; Priest 1985; Tajima et al. 1987; Kliem 1989, 1995). Since the timescales of coalescence of magnetic islands scale with the Alfve´n velocity, tcoal ¼ 1
k coal ; vA
¼
ucoal 0:1 1; vA
ð2Þ
where ucoal is the velocity of approaching filaments and k coal is the spatial separation of filaments, this theoretical spatiotemporal relation tcoal / k coal could explain the observed scaling tX / rloop (eq. [1]), if the spatial scale of filaments or magnetic islands is scale-invariant in flare loops, i.e., k coal / rloop . The 554
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Petschek model
s
hock
s Slow δ
∆
Magnetic field |B(h)| [G]
1.0 0.8 0.6 0.4 0.2 -LB
0.0 -4
-2
LB
0 Distance (h-hX) [Mm]
2
4
Fig. 1.—Definition of the magnetic length scale LB at the X-point in the context of a Petschek-type X-point geometry (top), with the height in horizontal direction. The magnetic field B(h hX ) as a function of height near the X-point is shown (bottom) for the model parameters used in Fig. 3. The magnetic length scale LB is defined by the range where the magnetic field increases to half of the asymptotic value far from the X-point.
interpretation of multiple (subsecond) time structures of radio pulsations as a signature of impulsive bursty reconnection was applied to radio data and modeled with numerical MHD simulations in a study by Kliem et al. (2000). In this study we pursue the consequences of impulsive bursty reconnection further and develop a dynamic trap model that relates quantitatively the field-line dynamics to the timescales of pulsed hard X-ray bursts. This model explains the synchronized injection of hard X-ray producing electrons and predicts the scaling of hard X-ray and radio pulse durations as a function of the Alfve´nic reconnection outflow speed and geometry of a Petschek-type reconnection model. A dynamic model with particle acceleration in similar reconnection geometries has been suggested by Somov & Kosugi (1997), the so-called collapsing trap model, in which mirroring particles become energized by first-order Fermi acceleration between the reconnection outflow region and the fast shock front below. That model is also driven by the relaxation of newly reconnected field lines (like ours) and could well serve as acceleration process in our concept, but we include the pitch-angle dynamics of the collapsing field lines to quantify the timescale of the resulting hard X-ray pulses from precipitating electrons. A partly similar model was pursued by Birn et al. (1997, 1998, 2000) to explain particle acceleration and injection in magnetospheric substorms, where the field-line relaxation is
usually referred to as ‘‘dipolarization.’’ While our model is purely analytic, the studies of Birn et al. (1997, 1998, 2000) are based on numerical MHD simulations, which indeed reproduce the observed bursty dynamics of particle injections and thus refute steady-state reconnection models. Our simple analytical model may provide new physical insights and diagnostics of such reconnection-driven particle injection processes, where the rise time of precipitating particle fluxes is controlled by the specific acceleration process and the decay time is controlled by the trapping properties of the relaxing field lines. 2. THEORETICAL MODEL In the standard flare reconnection model (Carmichael 1964; Sturrock 1966; Hirayama 1974; Kopp & Pneuman 1976; Tsuneta 1996; Tsuneta et al. 1997; Shibata et al. 1995), an X-type reconnection occurs in a coronal height hX and the newly reconnected magnetic field lines relax into a force-free configuration, which later becomes (after chromospheric evaporation) the soft X-ray–bright flare loop (Fig. 2). The height ratio of the reconnection point height hX to the flare loop height hL , hX qh ¼ 1:5; ð3Þ hL
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Fig. 2.—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 loss cone angle of the trapped particles gradually opens up and releases more particles from the trap.
has been determined by electron time-of-flight (TOF) measurements, using pulse background subtraction methods (hX =hL ¼ 1:7 0:4; lTOF =lhalf -loop ¼ 1:4 0:3; Aschwanden et al. 1996c), pulse deconvolution methods (lTOF =lhalf -loop ¼ 1:6 0:6; Aschwanden et al. 1999), or by loop shrinkage measurements (lcusp =lhalf -loop ¼ 1:25; 1:5; Forbes & Acton 1996). The reconnection outflow reaches Alfve´nic speed vA (hX ) ¼ vA0 (see Fig. 8 in Schumacher et al. 2000 for a numerical MHD simulation) and carries the magnetic flux of the newly reconnected field lines. The newly reconnected magnetic field lines experience a strong Lorentz force j < B in the initial cusp shape, which gradually reduces when the field lines relax into a force-free shape, because the curvature force is reciprocal to the curvature radius Rc . If the magnetic field has only components in the plane, as shown in Figure 1 (top), then (Priest 1982, p.102) 2 B 1 B2 1 (B = : )B ¼ en : ð4Þ j < B ¼ 9 þ 4 8 4 Rc Thus, in our dynamic model we assume that the motion of the post-reconnection field line occurs with Alfve´nic speed vA in the outflow region, where the curvature radius is minimal near the cusp, and 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. We approximate the height dependence of the apex of the relaxing field line with an exponential function (Fig. 2) with an e-folding relaxation timescale tR , t h(t) ¼ hL þ (hX hL ) exp ; ð5Þ tR which matches the initial condition h(t ¼ 0) ¼ hX (demarcating the height of the X-point) and final asymptotic limit h(t ¼ 1) ¼ hL (the height of the relaxed flare loop). The speed of the height change is simply the derivative of equation (5), dh(t) (hX hL ) t t ¼ v (t) ¼ exp ¼ vX exp ; dt tR tR tR ð6Þ
which defines the relaxation timescale tR , tR ¼
h X hL ; vX
ð7Þ
so that equation (6) fulfils the initial condition v (t ¼ 0) ¼ vX and the final asymptotic limit v (t ¼ 1) ¼ 0. Reconnection theories predict that the outflows have Alfve´nic speed, so we can set the initial speed of the relaxing field line approximately equal to the reconnection outflow speed, assuming frozen-in flux conditions, Bext vX vA ¼ 2:18 ; 1011 pffiffiffiffiffiffi ; nX
ð8Þ
where Bext denotes the external magnetic field strength on both sides of the outflow region and nX denotes the internal electron density in the X-point. The magnetic field in the X-point is zero in our twodimensional geometry (Fig. 1, top) and gradually increases with increasing distance from the X-point. Correspondingly, the loss cone angle is zero at the X-point and then gradually increases with increasing distance from the X-point, or as a function of time for the progressively relaxing field line. A convenient parameterization for this evolution of the loss cone angle (h½t) is an exponential function, with an e-folding timescale tI that we call ‘‘injection time,’’ (t) ¼
t 1 exp : 2 tI
ð9Þ
The pitch angle is initially zero, (t ¼ 0) ¼ 0, and then opens up to a full half-cone (t 7! 1) ¼ =2 for relaxation times t 3 tI [using the approximation B(h < hL ) ¼ BL ¼ const inside the flare loop]. We define a mirror ratio between the magnetic field B(h) at an arbitrary height h in the cusp and BL at the field-line footpoint, which relates to the loss cone angle (h½t) by R(h½t) ¼
BL 1 ¼ : 2 B(h½t) sin (h½t)
ð10Þ
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With this parameterization, the evolution of the magnetic field strength at the apex of the relaxing field line is (using eqs. [9]–[10]) t 1 exp B(t) ¼ BL sin (t) ¼ BL sin ; 2 tI 2
2
ð11Þ
or, inserting the transformation h(t) from equation (5), we obtain the height dependence of the magnetic field ( " #) h hL tR =tI 2 1 : B(h) ¼ BL sin 2 hX hL
ð12Þ
The magnetic field strength in the X-point is zero in the twodimensional approximation, BX ¼ B(h ¼ hX ) ¼ 0, and then increases beyond the outflow region to the ambient value, which we denote with BL ¼ B(h ¼ hL ) at the height of the flare loop (at the position after final relaxation). So the magnetic field monotonically increases from the X-point toward the flare loop top. In summary, we implicitly made three assumptions in the derivation of equations (9)–(12): (1) twodimensional magnetic field geometry (yielding a vanishing field BX ¼ 0 in the X-point), (2) uniform magnetic field in the flare loop (yielding a fully open loss cone after field-line relaxation), and (3) no time dependence in the pitch-angle distribution of the accelerated particle distribution. The latter assumption represents a zero-order approximation that may break down in the case of time-varying magnetic fields during the field-line relaxation process. For the spatial variation of the magnetic field near the X-point (eq. [12]), it is actually more instructive to express the injection time tI in terms of a spatial scale. We define a magnetic length scale LB where the magnetic field increases from zero at the X-point toward half of the value at infinite, B(h ¼ hX LB ) ¼ BL =2 (Fig. 1, bottom). This can be calculated straightforwardly from equation (12), and we find the relation tI ¼ tR
ln (1 qB ) tR tan qB tR qB ; ln (1=2) 2 2
ð13Þ
where qB expresses the fraction of the magnetic length scale LB to the cusp distance (hX hL ), qB ¼
LB : (hX hL )
ð14Þ
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 initially see a very small loss cone angle and are thus fully trapped, while the gradual opening forced by the relaxation of the field line untraps them as a 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 N ( ) d / 2 sin d :
ð15Þ
The assumption of isotropic acceleration is, of course, a simple approximation justified only by the lack of theoretical consensus. Theoretical and numerical work show everything
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from preferentially parallel acceleration (by DC electric fields or by first-order Fermi acceleration; e.g., Somov & Kosugi 1997) to preferentially perpendicular acceleration (by stochastic gyroresonant acceleration or betatron acceleration; e.g., Birn et al. 1997). Adopting the isotropic acceleration approach (eq. [15]), the precipitation rate dNP (t)=dt is then proportional to the time derivative, where the loss cone angle has the time dependence given in equation (9), dNP (t) dN ( ) d(t) d (t) ¼ / cos ½ (t) dt d dt dt t t 1 exp / cos exp : 2 tI tI
ð16Þ
This implies that the opening speed of the loss cone angle is fastest at beginning and slows down with time. The time profile of the precipitation rate dNP (t)=dt (eq. [16]) can be approximated with an exponential function, dNP (t) t exp ; ð17Þ dt tP for which we find numerically that the time constant of the precipitating flux tP depends on the injection time as (in the limit that the loss cone opens up to ¼ =2), tP 0:493tI :
ð18Þ
To account for a finite acceleration time, we assume that the number of particles accelerated to an arbitrary energy increases with some power a of the trapping time t, so the fraction of accelerated particles increases as NA (t) / t a :
ð19Þ
The flux of precipitating accelerated particles then has the time dependence dNP (t) t / t a exp F(t) ¼ NA (t) : ð20Þ dt tP For a given acceleration mechanism, the parameter a is a constant, and thus the pulse profiles are scale-invariant if scaled by the normalized time t=tP (eq. [17]) or t=tI (eq. [16]). Therefore, the FWHM of the flux profile, tw , has a unique relation to the timescale tI , which we calculate numerically for different acceleration power indices, a ¼ 0:0, 0:5, 1, 1:5, and 2. The ratio qa of the FWHM to the injection time tI is found, 8 0:493 for a ¼ 0:0; > > > > > > 1:014 for a ¼ 0:5; tw < ð21Þ qa ¼ ¼ 1:284 for a ¼ 1:0; > tI > > 1:503 for a ¼ 1:5; > > > : 1:697 for a ¼ 2:0: We show the time evolution of the various parameters in Figure 3: the height h(t)=hL , relaxation velocity v (t)=vX , magnetic field at apex of the relaxing field line B(h)=BL , loss cone angle (t)=(=2), magnetic mirror ratio R(t), and precipitation flux F(t)= max (F ).
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3.1. Scaling of Pulse Duration with Flare Loop Size In a comprehensive statistical study of 647 solar flares observed with the CGRO at energies 25 keV and with a time resolution of 64 ms over more than 4 minutes duration, a wavelet analysis was employed to analyze the time structures of the hard X-ray time profiles (Aschwanden et al. 1998). In strong flares, which have the best photon statistics and sensitivity to fast time structures, pulses with timescales in the range Tmin 0:1 0:7 s were found. In a subset of 46 flares, simultaneous Yohkoh SXT observations were available and the flare loop size could be measured, yielding flare loop radii in the range r ¼ 3 25 Mm. Moreover, the minimum timescales Tmin of the hard X-ray pulses were found to be correlated with the flare loop radii r (Fig. 4, left), according to the proportionality relation given in equation (1). Using our model prediction for pulse durations, tw qa (=2) LB =vX (eq. [22]), the scaling ratio for X-point heights, qh ¼ (hX =hL ) 1:5 (eq. [3]), and the definition of the Alfve´n velocity vA (eq. [8]), we expect a scaling of 1 1=2 hL B ne tw ¼ 0:50 30 G 10 Mm 109 cm3 qh 1 qa qB ; s: 0:5 1:3 0:1
Fig. 3.—Dynamic evolution of the parameters for the field-line relaxation model described in x 2. The parameters are height of X-point hX ¼ 30 Mm, height of relaxed flare loop hL ¼ 20 Mm, density nX ¼ 109 cm3, magnetic field at apex of flare loop BL ¼ 20 G, and magnetic scale length in reconnection region LB =(hX hL ) ¼ 0:1. The curves show the evolution of the height h(t)=hL , relaxation velocity v(t)=vX with vX ¼ 2070 km s1, magnetic field at apex of relaxing field line B(h)=BL , loss cone angle (t)=(=2), magnetic mirror ratio R(t), and precipitation flux F(t)=max (F ). The relaxation time is tR ¼ 4:84 s, the injection timescale is tI ¼ 1:00 s, the power index is a ¼ 1 (eq. 20), and the FWHM pulse duration is tw ¼ 1:28 s.
We can now express the FWHM pulse duration tw only as a function of the independent parameters tR and qB , using equations (21), (13), and (14), ln (1 qB ) LB : ð22Þ tw ¼ tR qa qa ln (1=2) 2 vX The right-hand approximation expresses most succinctly the relation of the pulse duration to the underlying physical parameters: the pulse duration is proportional to the Alfve´nic transit time (with velocity vX vA ) through the magnetic length scale LB of the X-point region. Therefore, the fastest time structures are produced by the smallest magnetic length scales LB (and presumably by the smallest flare loops). For instance, an X-point region with a spatial extent of LB ¼ 500 km and an Alfve´n velocity of vA ¼ 2000 km s1, yields pulses with timescales of typically tw 0:5 0:25 s, depending on the acceleration power index a. If the Alfve´n speed is known, the pulse duration can be used to estimate the size LB of the outflow region in the X-point. 3. APPLICATION TO OBSERVATIONS We now apply our theoretical model to a number of flare observations and extract the physical parameters that are constrained by the model.
ð23Þ
Thus, our theory reproduces the observed correlation of tw 0:5(hL =10 Mm) s (eq. [1] and Fig. 4) for an average magnetic field strength B 30 G, electron density ne 109 cm3 (above flare loops), cusp height ratio qh ¼ hX =hL 1:5, and X-point length scale qB ¼ LB =(hX hL ) 0:1. The minimum timescales analyzed in the same study also obey another constraint, being always larger than the collisional deflection timescales in the flare loops, based on the soft X-ray electron density (Fig. 4, right panel ). This fact is consistent with the interpretation that the density in the cusp region 109 cm3) of the relaxing field line is always lower (ncusp e than in the soft X-ray–emitting flare loops that have been filled 1010 1012 cm3 (Fig. 4, right with evaporated plasma, nSXR e panel ), so the potential trapping time in the cusp is always longer than the trapping time in the soft X-ray flare loop. 3.2. Distribution of Pulse Durations The diagrams in Figure 4 show that the pulse durations lay in the range tw 0:1 0:7 s, averaged in 46 flares. A more comprehensive study of the time profiles of 5430 hard X-ray pulses observed during 640 flares yielded an exponential distribution of pulse durations in the range tw 0:3 1:0 s (Aschwanden et al. 1995), N (tw ) /
expðtw =0:41 sÞ
for " > 25 keV
expðtw =0:44 sÞ
for " > 50 keV
; ð24Þ
which are almost identical in different energies (Fig. 5). Thus, the average value is of order htw i 0:5 s, as in the wavelet analysis study (Aschwanden et al. 1998), and a spread by a factor of 2 on both sides can easily be accommodated by our model specified in equation (23). The question is why there is an exponential distribution, with the most frequent pulses seen at the shortest timescales of tw k 0:3 s near the measurement cutoff. The reason lies in the reciprocal relation of the pulse
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Tmin [s] =0.49_ +0.28 (r/109 cm)
2.0 N=46
Collisional Deflection Time Collisional Loss Time
1.5 Minimum time scale Tmin [s]
Minimum time scale Tmin [s]
1.5
1.0
0.5
0.0 0
559
1.0
0.5
10 20 Flare loop radius r [Mm]
30
0.0 0.01
0.10 1.00 10.00 Electron density ne [1011 cm-3]
100.00
Fig. 4.—Correlation of the minimum timescale Tmin with the flare loop radius r (left) and trap electron density ne (right) for a set of 46 flares simultaneously observed with BATSE CGRO and Yohkoh. The symbol size of the data points is proportional to the logarithm of the count rate. The mean ratio Tmin =r is indicated (left, solid line), and collisional timescales for 25 keV electrons are shown (right, dashed and dotted lines) (Aschwanden et al. 1998).
duration tw to the magnetic field strength B in equation (23). If we assume a flat distribution for each parameter over some range, the reciprocal relation to the magnetic field introduces an asymmetry in the distribution of pulse durations. If the distribution of magnetic field strengths is flat, N (B) dB ¼ const, then the distribution of the pulse durations is N (tw )dtw ¼ N (B½tw )
dB(tw ) dtw / tw2 ; dtw
ð25Þ
since N (B½tw ) ¼ const and B(tw ) / tw1 . This is a distribution that falls off as steeply as the observed exponential distribution (eq. [24]). The slope of the distribution of the pulse durations varies depending on the broadness of each parameter distribution. We performed a Monte Carlo simulation by assuming a normal distribution for each of the five parameters in equation (23) and varied the Gaussian widths x . For a Gaussian width ratio of x =hxi ¼ 0:4, we could reproduce the same distribution (Fig. 6) as the observed one (Fig. 5) with the same
Fig. 5.—Observed distribution of hard X-ray pulse durations detected in 25–50 keV and 50–100 keV channels from CGRO BATSE during 647 solar flares. The cutoff of detected pulse widths at P0.3 s is caused by the Fourier (FFT) filter used for structure detection (Aschwanden et al. 1995).
Fig. 6.—Monte Carlo simulation of the distribution of pulse durations according to our model specified by eq. (23). The best fit with observations N (tw ) / exp ( tw =0:4) s (Fig. 5) is achieved by assuming a mean parameter spread of x =hxi ¼ 0:4 for the normal distributions of each parameter x ¼ hL ; B; ne ; qh 1; and qB .
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lines relax into a force-free state (Fig. 7). For a given geometry, magnetic field, and density, our model predicts specific trapping times that are constrained by the dynamics of the relaxing field lines and could be used to model the hard X-ray yield of the accelerated particles (by thin-target bremsstrahlung). Of course, the hard X-ray emissivity depends strongly on the particle density in the cusp region, which could be compressed by shocks in the outflow region by factors of P4. 4. CONCLUSIONS
Fig. 7.—Temporary trapping occurs in the acceleration region in the cusp region below the reconnection point in our model (right), which can explain the coronal hard X-ray emission observed during the Masuda flare (left). Left: Observations show a Yohkoh HXT 23–33 keV image (thick contours) and Be119 SXT image (thin contours) of the 1992 January 13 1728 UT flare (Masuda et al. 1994).
exponential fit exp ( tw =0:4 s) above a cutoff of tb > 0:3 s. Thus, making the minimal assumption that each parameter has the same relative scatter x =hxi 0:4, we find that the following distribution of parameters is most consistent with the observations: hL ¼ 10 4 Mm, B ¼ 30 12 G, ne ¼ (1:0 0:4) ; 109 cm3, qh ¼ 1:5 0:2, and qB ¼ 0:10 0:04. The latter ratio translates in absolute units into a range LB ¼ qB hL (qh 1) ¼ 500 400 km for the magnetic length scale of the reconnection outflow region. The fact that the pulse durations are identical at different energies, e.g., at 25 and 50 keV (Aschwanden et al. 1995), or that their distributions are identical (eq. [24]), is also a prediction of our model, confirmed by the observations. In our model, the pulse decay time is only determined by Alfve´nic transit time through the magnetic length scale LB in the outflow region but is decoupled from the acceleration time and thus from the energy of the accelerated particle. In models of stochastic and shock acceleration, for first-order Fermi as well as second-order Fermi acceleration theories, higher energies require statistically longer acceleration times than lower energies. This time dependence of the acceleration mechanism can be accommodated in our model (with eq. [19]), but if the number of particles grows with a time dependence of / t a at every energy, the model predicts that the duration of the precipitating flux is independent of the particle energy. Thus, our model and the agreement with the observations (Figs. 4 and 5) suggest that the pulse decay time (which is nearly proportional to the pulse duration) is determined by the injection mechanism in the dynamic trap rather than by the acceleration time. 3.3. Above-the-Loop-Top Hard X-Ray Sources Masuda et al. (1994) discovered hard X-ray sources above the soft X-ray flare loops, which have been reviewed in Fletcher (1999). Physical mechanisms that have been proposed for this hard X-ray emission are (1) high-density regions in the cusp, (2) trapping by shocks, (3) trapping by converging magnetic fields, or (4) trapping by wave turbulence (Fletcher 1999). In our model we find a natural explanation for temporary trapping in the cusp as a consequence of the low magnetic field near the X-type reconnection diffusion region, which confines the particles accelerated by wave turbulence or by shocks in the outflow region of the reconnection point and releases the accelerated particles gradually once the newly reconnected field
Static trap-plus-precipitation models (e.g., Melrose & Brown 1976) have been applied to solar flares in the past to explain extended hard X-ray and radio emission beyond the impulsive flare phase when particle acceleration is believed to be most intense. Here we develop a dynamic trap-plusprecipitation model that naturally evolves in the intermittent or impulsive bursty reconnection regime. In such a scenario, a multitude of highly localized magnetic reconnection processes occur in a magnetically unstable coronal region, which are spatially highly fragmented because of local tearing instabilities that form magnetic islands and the subsequent coalescence of magnetic islands. We consider the dynamic evolution of newly reconnected magnetic field lines, which relax from the initial cusp shape to a force-free state. The low magnetic field in the outflow region of a Petschek-type X-point initially provides a perfect trap for particles accelerated in the diffusion region or reconnection outflow, because of the high magnetic mirror ratio. With progressive relaxation of the field line, the magnetic mirror ratio reduces and opens up the loss cone, so that the trapped energetic particles are released and injected into a free-streaming orbit toward the chromospheric footpoints. This injection mechanism synchronizes the release of energized particles, decoupled from the acceleration timescale. This model makes a number of testable predictions. 1. The time characteristics of energized particles, and thus the temporal profiles of hard X-ray and radio bursts, are controlled by the injection mechanism rather than by the acceleration mechanism. The duration of the injection pulse is energy-independent and only controlled by physical parameters of the cusp region. 2. The duration of an injection pulse scales with tw 2LB =vA , where tw is the Alfve´nic transit time through the magnetic reconnection region, with LB the magnetic length scale where the magnetic field increases from zero to half of its asymptotic value far from the X-point, in a Petschek-type reconnection geometry. 3. The observed distribution of hard X-ray pulse durations in the range tw 0:3 1:0 s are consistent with the following model parameters in the cusp region: flare loop size hL ¼ 10 4 Mm, magnetic field B ¼ 30 12 G, electron density ne ¼ (1:0 0:4) ; 109 cm3, cusp height ratio qh ¼ hX =hL ¼ 1:5 0:2, and magnetic length scale of outflow region LB ¼ qB hL (qh 1) ¼ 500 400 km or qB ¼ LB =(hX hL ) ¼ 0:10 0:04. This model provides a diagnostic of the outflow region in a Petschek-type reconnection model. 4. The temporary trapping in the cusp region provides a natural explanation for the above-the-loop-top sources discovered by Masuda et al. (1994). 5. The number of hard X-ray pulses observed during a flare provides a diagnostic of the number of intermittent reconnection events, which can be mapped to individual post-flare loops, and this way provides a spatiotemporal mapping of magnetic reconnection events.
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This study represents the first quantitative dynamic trap model that explains the fast time structures of hard X-ray– emitting particles in solar flares. The presented model attempts to relate the spatiotemporal trap evolution controlled by intermittent magnetic reconnection to commonly observed time structures in hard X-rays and radio. It can be considered as a basic framework for further quantitative modeling of hard
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X-ray pulses and radio bursts, and may ultimately lead to better diagnostic tools of magnetic reconnection processes in astrophysical plasmas. Part of this work was supported by NASA contracts NAS538099 (TRACE ) and NAS8-00119 (SXT).
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