Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.tesis.lebedev.ru/en/docs/124.pdf Äàòà èçìåíåíèÿ: Wed Apr 4 00:00:00 2007 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:08:23 2012 Êîäèðîâêà: koi8-r Ïîèñêîâûå ñëîâà: 47 ï ï ï ï ï ï

The Astrophysical Journal , 630:561 ­ 572, 2005 September 1
# 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.

THE MOTIONS OF THE HARD X-RAY SOURCES IN SOLAR FLARES: IMAGES AND STATISTICS
Sergey A. Bogachev and Boris V. Somov
Astronomical Institute, Moscow State University, Universitetskij Prospekt 13, Moscow 119992, Russia; bogachev@sai.msu.ru

and Takeo Kosugi and Taro Sakao
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Sagamihara , Kanagawa 229-8510, Japan Received 2004 October 8; accepted 2005 May 11

ABSTRACT On the basis of the Yohkoh Hard X-Ray Telescope ( HXT ) data, we present a statistical study of different types of the hard X-ray ( HXR) source motions during solar flares. A total of 72 flares that occurred from 1991 September to 2001 December have been analyzed. In these flares, we have found 198 intense HXR sources that are presumably the chromospheric footpoints of flare loops. The average velocity V and its uncertainty were determined for these sources. For 80% of them, the ratio of V to 3 is larger than 1, strongly suggesting that (1) the moving sources are usually observed rather than stationary ones and (2) the regular displacements of HXR sources dominate their chaotic motions. After co-alignment of the HXT images with the photospheric magnetograms, we have conducted an additional analysis of 31 flares out of 72 and distinguished between three main types of the footpoint motions. Type I consists of the motions preferentially away from and nearly perpendicular to the neutral line ( NL). About 13% of flares (4 out of 31) show this pattern. In type II, the sources move mainly along the NL in antiparallel directions. Such motions have been found in 26% of flares (8 out of 31). Type III involves a similar pattern as type II, but all the HXR sources move in the same direction along the NL. Flares of this type constitute 35% (11 out of 31). In 26% of flares (8 out of 31) we observed more complicated motions that can be described as a combination of the basic types or some modification of them. For the most interesting flares, the results of analysis are illustrated and interpretation is suggested. Subject headings: acceleration of particles -- Sun: flares -- Sun: magnetic fields -- Sun: X-rays, gamma rays g

1. INTRODUCTION Observations of impulsive solar flares made with the Hard X-Ray Telescope ( HXT; Kosugi et al. 1991) aboard the Yohkoh satellite (Ogawara et al. 1991; Martens & Cauffman 2002) revealed that the bulk of hard X-rays ( HXRs) above 30 keV are emitted from the footpoints ( FPs) of flare magnetic loops by high-energy electrons precipitating into the FPs from an acceleration region in the corona. According to the results of analysis of 28 flares, Sakao (1994) inferred that a double source structure is the most frequent type in an energy range above 30 keV. Sakao et al. (1998) have studied the spatial evolution of 14 impulsive flares around the peaking time of the HXT M2 band (33 ­ 53 keV ) emission. For all the flares selected, the separation between the sources has been analyzed as a function of time. In 7 out of 14 flares, the FPs moved from each other (vsep > 0). The rest of the flares showed decreasing FP separation (vsep < 0) or did not show either increasing or decreasing separation of the FPs (vsep $ 0). These two types of FP motions are related to the two subclasses of impulsive flares (Sakao et al. 1998). The flares with vsep > 0 are less impulsive ( LI ): they have a longer duration in the impulsive phase. They also show the existence of a superhot plasma near or above the top of the soft X-ray (SXR) loop ( Masuda et al. 1994, 1995; Tsuneta et al. 1997). The flares with a decreasing FP separation are more impulsive ( MI ) and do not show a significant amount of superhot plasma. The standard model of a solar flare predicts only an increasing separation motion of the FP sources as new field lines reconnect at higher and higher altitudes. 561

In order to explain both subclasses of impulsive flares, Somov et al. (1998) applied an idea of three-dimensional reconnection in the corona at a separator with a longitudinal magnetic field. According to this model, reconnection in the MI flares proceeds with a decrease of the longitudinal field at the separator and with higher reconnection rate. The reconnected field lines become shorter as such a flare evolves and the distance between two FP sources decreases. In contrast, in the LI flares the longitudinal field at the separator increases in time and the FP separation increases as well. First results of RHESSI observations ( Fletcher & Hudson 2002; Krucker et al. 2003) confirm systematic but more complex FP motions than the standard model predicts. Krucker et al. (2003) studied the HXR source motions in the 2002 July 23 flare. Above 30 keV, at least three sources were observed during the impulsive phase of the flare. One FP source moved along the photospheric neutral line ( NL) at a speed of about 50 km sþ1. This is comparable to the values reported by Sakao et al. (1998). The sources on the other side of the NL did not move systematically. McTiernan et al. (2004) proposed to use nonlinear forcefree field extrapolations from vector magnetograms to interpret the different source patterns and motions seen in RHESSI and TRACE images of solar flares. Liu et al. (2004) presented analysis of a simple X-class flare observed with RHESSI on 2003 November 3. They found that the increase in separation of the two FP sources is consistent with magnetic reconnection models proposed for solar flares. The source motions were relatively slower during the more active phases of HXR emission. The motions of HXR sources revealed by Sakao et al. (1998) are consistent with the motions of bright H kernels in flare


562

BOGACHEV ET AL. number of images could exceptionally reach 1000 ­ 2000 in the set. The important part of our study was the co-alignment of HXT images with simultaneous magnetic fields. For the flares that occurred from 1991 to 1996, we used the magnetograms taken from Kitt Peak Observatory ( KPO). These are the full-disk images containing 1788 ; 1788 pixels, with the pixel size equaling 1B14. For the flares after 1996, we used the magnetograms from the Michelson Doppler Imager ( MDI ) on SOHO. 2.2. Methods of Analysis We have developed an automatic procedure to study the motions of HXR sources, which includes the following steps: 1. Removing of sidelobes from the images. 2. Selection of the most intense sources and determination of their coordinates. 3. Identification of the positions of the same source in the successive images. 4. Calculation of the average velocity of the sources. 5. Overlaying the source trajectories on magnetograms. Standard imaging programs from the HXT software package (such as HXTPRO) include background subtraction from the data. The typical nonflare background count rate in the M2 energy band is 1 count sþ1 per subcollimator. However, the synthesized HXT images often have sidelobes that are artifacts from the image deconvolution and make the image unsuitable for automatic processing. The theory of signal processing recommends using Fourier filters or wavelet analysis to remove such features from a data set. The threshold level can be based on the percent of wavelet power or some other method. Suggestions for choosing the threshold are given in Donoho & Johnstone (1994). The routine we have used is more simple and based on the Otsu (1979) thresholding algorithm for gray level histograms. First, we transform the initial 64 ; 64 data matrix I(x, y) to the histogram Ni, where Ni is the number of pixels at level i. Then suppose that the pixels are dichotomized into two classes, which denote pixels with levels ½0; : : : ; k and ½k × 1; : : : ; 255, respectively. The Otsu algorithm determines the threshold by determining the gray level k that maximizes the between-class variance B(k) of the histogram: B Ï k ÷ ¼ ½ !0 Ïk ÷ þ !1 Ïk ÷2 ; !0 Ïk ÷½1 þ !0 Ïk ÷ Ï 1÷

ribbons. Asai et al. (2003) reported an examination of the fine structure inside flare ribbons seen in H during the X2.3 flare on 2001 April 10. They identified the conjugate H kernels in both ribbons and found that the pairs of highly correlated kernels were related to the FPs of the postflare loops seen in the TRACE 171 8 images. As the flare progresses, the loops and pairs of H kernels moved from the strongly sheared to the less sheared configuration. For the two-ribbon ``Bastille day'' flare on 2000 July 14, the motions of bright HXR kernels from strong to weak sheared structure were also observed in the HXR ribbons ( Masuda et al. 2001). Somov et al. (2002) suggested the following interpretation of kernel motions in the Bastille day flare. During 2 days before the flare, the bases of magnetic separatrices were moved by the largescale photospheric flows of two types. First, the shear flows, which are parallel to the NL, increase the length of field lines in the corona and produce an excess of energy related to magnetic shear. Second, the converging flows, i.e., the flows directed to the NL, create preflare current layers in the corona and provide an excess of energy as a magnetic energy of these layers. During the flare, both excesses of magnetic energy are quickly released completely or partially. In order to reveal the observable types of the FP motions, we have analyzed 72 flares, using the imaging data taken with the Yohkoh HXT. The preliminary results of this work are reported in Somov et al. (2005). We summarize these data in the next section and describe the methods of their analysis. In x 3 the HXR images are overlaid on the photospheric magnetograms in order to find basic types of FP motions. A summary of results and discussion are given in x 4. 2. OBSERVATIONS AND DATA ANALYSIS 2.1. Observation Data The HXT aboard Yohkoh provided simultaneous imaging in four X-ray channels, L0 (13.9 ­ 22.7 keV ), M1 (22.7 ­ 32.7 keV ), M2 (32.7 ­ 52.7 keV ), and H (52.7 ­ 92.8 keV ), with the temporal resolution of 0.5 s in the flare mode ( Kosugi et al. 1991). From 1991 September to 2001 December, the HXT observed about 2000 solar flares in an energy range above 30 keV. We selected 72 flares according to the following criteria (the flares are listed in Table 1): 1. The integral photon count of HXRs in the M2 band is greater than 1000 counts per subcollimator. Thus, the HXR flux provides enough information for the synthesis of more than 10 images. 2. An active region is within about 45 of the center of the solar disk, so that the source motions are not seriously distorted by projection effects. For the analysis, we used the M2 band only, since the L0 and M1 bands may have a considerable contribution of the superhot plasma emission. Certainly, in order to study the nonthermal processes in flares, we could also use the H-band data, but the HXR emission was usually too weak in this channel to analyze source motions with proper accuracy. In order to synthesize images, we applied the standard maximum entropy method ( Frieden 1972; Gull & Daniell 1978; Willingale 1981) from the Yohkoh HXT software package. The images contain 64 ; 64 pixels, with the pixel size equaling 2B47. For each flare from Table 1, we synthesized a set of images with the temporal step of about several seconds. The typical number of images was 50 ­ 100 per flare, although for the large events, like the 3B/ X5.7 flare on 2000 July 14 (the Bastille day flare), the

where is the mean value of histogram, ! 0(k) is the zeroth-order cumulative moment of the histogram, and ! 1(k) is the first-order cumulative moment of the histogram. In Figure 1 we use the 1991 December 26 flare as an example to illustrate this method. Here I0 ¼ 53. Note that the clip threshold for HXT images does not usually exceed 10% ­ 20% of the maximum intensity. The second step is selection of the most intense sources in the image and determination of their coordinates. We use peak points of the data matrix I(x, y) to preliminarily mark the source positions. If a distance between two peaks is less than 500 , we consider them as a single source with a complex shape. Different methods can be adopted in order to calculate the center of a source. We use here the standard formula for the ``center of gravity'' (see also the Appendix): rc ¼
N 1X Ik rk ; I k ¼1

Ï 2÷


TABLE 1 Summary o f Flare s Start Time (UT) 19:49:12 21:16:04 22:37:04 10:48:07 21:35:43 12:26:10 23:06:54 21:29:11 19:18:17 15:49:27 01:41:05 09:45:09 13:47:28 09:03:34 11:24:51 09:03:18 06:04:39 01:11:19 18:31:12 10:55:47 23:44:17 14:08:05 14:20:15 07:39:51 04:05:29 21:20:24 17:35:14 02:35:10 21:17:29 02:12:18 06:50:41 05:11:16 08:38:04 11:25:47 08:12:39 01:35:20 17:55:40 10:53:33 01:49:45 15:55:40 20:50:40 15:43:04 11:39:23 10:30:33 10:19:44 10:24:16 13:47:37 14:34:51 02:47:01 14:35:03 04:11:32 01:44:53 15:07:38 18:37:38 16:37:37 02:53:12 05:12:58 01:47:31 05:19:12 10:15:34 13:43:23 16:29:11 16:43:52 15:15:36 End Time ( UT ) 19:49:30 21:17:39 22:40:12 10:49:49 21:38:24 12:27:24 23:08:17 21:30:00 19:18:41 15:53:48 01:43:43 09:48:24 13:48:03 09:04:47 11:27:34 09:04:37 06:05:11 01:11:37 18:31:41 11:00:53 23:45:02 14:08:31 14:22:24 07:40:34 04:05:43 21:21:33 17:37:13 02:35:33 21:22:00 02:12:51 07:11:43 05:11:56 08:41:24 11:27:35 08:14:03 01:35:49 18:02:04 10:54:23 01:51:00 15:56:57 20:51:40 15:48:43 11:39:43 10:34:53 10:23:22 10:36:14 13:49:52 14:35:30 02:48:36 14:35:16 04:17:46 01:49:57 15:10:56 18:42:35 16:44:20 02:54:22 05:14:38 01:47:50 05:33:59 10:22:16 13:45:16 16:35:23 16:45:05 15:16:53

Number 1........................................ 2........................................ 3........................................ 4........................................ 5........................................ 6........................................ 7........................................ 8........................................ 9........................................ 10...................................... 11...................................... 12...................................... 13...................................... 14...................................... 15...................................... 16...................................... 17...................................... 18...................................... 19...................................... 20...................................... 21...................................... 22...................................... 23...................................... 24...................................... 25...................................... 26...................................... 27...................................... 28...................................... 29...................................... 30...................................... 31...................................... 32...................................... 33...................................... 34...................................... 35...................................... 36...................................... 37...................................... 38...................................... 39...................................... 40...................................... 41...................................... 42...................................... 43...................................... 44...................................... 45...................................... 46...................................... 47...................................... 48...................................... 49...................................... 50...................................... 51...................................... 52...................................... 53...................................... 54...................................... 55...................................... 56...................................... 57...................................... 58...................................... 59...................................... 60...................................... 61...................................... 62...................................... 63...................................... 64......................................

Date 1991 1991 1991 1991 1991 1991 1992 1992 1992 1992 1992 1992 1992 1992 1992 1992 1992 1992 1993 1993 1993 1993 1993 1993 1994 1994 1994 1997 1998 1998 1998 1999 1999 1999 1999 1999 1999 1999 1999 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2001 2001 2001 2001 2001 2001 2001 2001 2001 Oct 30 Nov 13 Nov 15 Dec 26 Dec 26 Dec 28 Feb 14 Feb 15 Apr 24 May 08 Jun 07 Jul 08 Aug 11 Aug 20 Sep 05 Sep 06 Sep 11 Dec 15 Feb 11 Feb 18 Apr 06 Apr 22 Jun 07 Oct 02 Jan 06 Jun 30 Aug 14 Dec 01 May 03 Sep 06 Sep 23 Jun 26 Jun 27 Jun 30 Jul 28 Aug 25 Aug 28 Dec 22 Dec 27 May 18 May 23 Jun 07 Jul 11 Jul 12 Jul 14 Jul 14 Jul 14 Jul 21 Jul 25 Sep 15 Sep 16 Oct 29 Nov 24 Nov 25 Nov 26 Mar 29 Mar 30 Apr 06 Apr 10 Apr 12 Apr 25 Aug 25 Sep 08 Sep 09

Number of Images 31 149 355 150 279 122 159 68 38 497 260 369 54 120 213 125 51 32 46 555 72 46 240 73 20 132 205 40 484 54 2259 46 359 201 143 47 733 84 131 126 100 601 35 495 438 1417 219 63 177 22 697 362 390 585 782 113 143 32 1741 784 199 745 129 139

Motion Type .. . .. . .. . .. . .. . .. . .. . .. . .. . 1+3a 3 .. . 3 .. . .. . 3 .. . .. . .. . .. . .. . 3 1+3 .. . .. . 1+2 1 3 .. . 2 1 .. . 1+2 .. . 1+2 .. . 1+2 .. . .. . 2 .. . 3 2 3 2 .. . 1 .. . .. . 3 2 2 2 3 3 .. . .. . .. . 1+2 1+2 .. . 2 .. . 3

Magnetic Data ... ... ... ... ... ... ... ... ... KPO KPO KPO KPO KPO KPO KPO ... KPO KPO ... KPO KPO KPO KPO KPO KPO KPO MDI MDI KPO KPO MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI KPO MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI MDI


TABLE 1-- Continued Start Time (UT) 09 17 28 19 23 31 04 06 18:40:10 08:21:00 02:02:51 16:22:58 02:15:10 08:05:01 16:05:44 02:58:42 End Time ( UT ) 18:40:41 08:22:01 02:03:20 16:31:52 02:19:16 08:05:54 16:24:50 02:59:21

Number 65................................... 66................................... 67................................... 68................................... 69................................... 70................................... 71................................... 72...................................
a

Date 2001 2001 2001 2001 2001 2001 2001 2001 Sep Sep Sep Oct Oct Oct Nov Nov

Number of Images 48 112 47 986 394 88 2196 60

Motion Type .. .. .. .. .. .. 1 .. . . . . . . .

Magnetic Data MDI MDI MDI MDI MDI MDI MDI MDI

Combination of two types of motion.

Fig. 1.--Removing of sidelobes from an HXT image. (a) Initial I(x, y) data matrix. (b) Calculation of the threshold level I0. (c) Data matrix after the thresholding. (d ) HXR sources in the image.


HARD X-RAY SOURCE MOTION IN SOLAR FLARES The uncertainty in the fit velocity is Vx vx × Vy v ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y ; Vx2 × Vy2 where þ
Fig. 2.--HXR source positions in the first (a, b, c) and in the second (A, B, C) of the two successive HXT images.

565

Ï 7÷

v x ;

v

à
y

¼

n X r i ¼1

! w
i

0:5

where I¼
N X k ¼1

2 n n XX ;4 wi wi ti2 þ
i ¼1 i ¼1

n X i ¼1

!3 2 wi ti 5

þ0:5

:

Ï 8÷

I

k

Ï 3÷

is the integral intensity of the source. The sum in equations (2) 2 and (3) is taken over all the source points. The uncertainties x and 2 with which the components xc and yc of the source poy sition can be measured depend on the angular resolution àr of the image and the signal-to-noise ratio (S/ N ):
2 x ; 2 y

Here r is the ``scattering'' uncertainty that may be estimated from the fit as the standard deviation (see, for example, Bevington 1969): " #0:5 n 1X 2 Ïr0 × V ti þ ri ÷ ; Ï 9÷ r ¼ n þ 2 i ¼1 where



¼

N N Ïà r ÷2 X 2 1X Ik × 2 Ïrk þ rc ÷2 Ik : I k ¼1 I 2 k ¼1

Ï 4÷ r0 ¼

The S/ N is defined here as the standard deviation of the count rate due to Poisson statistics. Two parts of equation (4) are related to angular and S/ N components of the uncertainty. ffiThey pffiffiffi both decrease with the growth of N approximately as 1/ N . The S/ N part depends also onpffiffi source intensity. For more bright the sources, it decreases as 1/ I . Although we usually see only one or two sources in HXRs, the large flares often exhibit complex HXR structures with a large number of emission centers. The kernels in such structures may disappear, change position, or combine with other sources as the flare evolves. As a consequence, the difficult question is how to identify the same source in the next image. We demonstrate our approach to this problem in Figure 2. At each time t ¼ ti (i ¼ 1; 2; 3; : : : ), we plot all the possible connections between the source centers (a, b, c) in the current image and in the next one (A, B, C, D). Then we select a pair with the minimal distance between the sources ( for example, a and A) and exclude this pair from the next iteration. Thus, we find the second pair (say, b and B). Applying this method to the configuration in Figure 2 step by step, we obtain a ! A, b ! B, and c ! C. Source D does not have an analog in the first image and seems to have appeared recently. When we have finished with formation of the data line ri ¼ ½x(ti ); y(ti ), we apply the method of least squares to derive the regular velocity V of the source (r ¼ r0 × V t ). In order to propagate the uncertainties given by equation (4) through V , we used weighted fitting ! n n n n X X XX wi wi ti ri þ w i ti wi ri V¼ 2 ! 3þ1 2 n n n XX X ;4 wi wi ti2 þ wi ti 5 ;
i ¼1 i ¼1 i ¼1 i ¼1 i ¼1 i ¼1 i ¼1

n n 1X 1X ri þ V ti : n i ¼1 n i ¼1

Ï10÷

The factor 1/(n þ 2) arises because 2 degrees of freedom have been lost in the two-parameter fit to the data. The last step of the program is overlaying the trajectories of the sources on the SOHO MDI and KPO magnetograms. All the magnetic field data were previously transferred to the HXT coordinate system and to the time of the HXT observations. We used the differential rotation function from Howard et al. (1990). 3. THE TYPES OF HXR SOURCE MOTIONS In order to distinguish types of source motions with respect to magnetic field, we characterize the field by a photospheric NL or, more exactly, a simplified NL (Gorbachev & Somov 1989). We found the magnetic data for 61 out of the 72 flares. In most of

Ï 5÷

where 2 2 wi ¼ 1=x ; 1=y : Ï 6÷
Fig. 3.-- Order of the magnetic line reconnection (1-2-3) in the standard two-dimensional model of a solar flare and HXR source motions in the chromosphere.


566

BOGACHEV ET AL.

Vol. 630

Fig. 4.--Type I motion of HXR sources in flares.

them, the HXR sources systematically moved during flare. However, only in 31 flares were the magnetic configuration and the structure of HXR sources simple enough to (1) draw the simplified NL and (2) compare the source motions with the direction of the NL. Our statistics are based only on these 31 flares. Using them, we conditionally distinguish three types of FP motions. 3.1. Type I Motions In general, the direction of FP motions in a flare depends mainly on the large-scale magnetic field configuration in an active region.

The simplest example of such a configuration, a standard twodimensional model, is sketched in Figure 3. We assumed here that, before a flare, the reconnecting current layer was induced in the corona. During a flare, this layer provides powerful fluxes of energy along the newly reconnected field lines. As the flare progresses, the FPs of reconnected lines move away from the NL with a velocity roughly proportional to the rate of reconnection. This is the well-known prediction of the standard model, explaining the effect of the increasing separation between flare ribbons.

Fig. 5.--Motion of HXR sources more complicated than type I.


No. 1, 2005

HARD X-RAY SOURCE MOTION IN SOLAR FLARES

567

field component, which is parallel to the NL. As a consequence, the other types of FP motions dominate in flares. 3.2. The Motions along the Neutral Line In many flares, we do not see systematic motions of the HXR sources away from the NL. The apparent displacements of FPs in these flares are directed mainly along the NL, which is inexplicable by the standard model. There are two types of such longitudinal motions: the FP sources move in antiparallel direction (type II ) or they move in the same direction (type III ). Type II is well known from HXT and RHESSI observations. We have found such motions of FPs in 8 out of 31 flares (26%). We think that the antiparallel displacements of HXR sources represent the effect of relaxation of the nonpotential shear component of the magnetic field (Somov 2000). In contrast to the standard model discussed above, such configurations accumulate an amount of energy in the form of magnetic energy of a sheared field. During the impulsive phase, this amount converts into kinetic and thermal energies of accelerated particles and superhot plasma. Figure 6 presents a cartoon of highly sheared configuration. The apparent motions of HXR sources are determined here by the order of reconnection from line 1 to 3 and are directed along the NL. How are such highly sheared three-dimensional structures formed? We think that large-scale photospheric flows of different types play a leading role in this process and illustrate this in Figure 7. These flows can deform the photospheric NL in such way that it acquires the shape of the letter S. Gorbachev & Somov (1988, 1990) proved that such distortion leads to the separator appearance in the corona above the NL, and the necessary condition for reconnection in the solar atmosphere becomes satisfied. Developing this idea, we assume that a causal connection may exist between the type II flares and the S-shaped bend of the NL. Figure 8 shows some flares of type II. Note that these examples do not allow us to distinguish the flares with increasing

Fig. 6.-- Order of the magnetic line reconnection (1-2-3) in a highly sheared magnetic configuration and HXR source motions in the chromosphere.

A fraction of such flares appears to be very small: only 4 out of 31 flares, 2 of which are shown in Figure 4. Moreover, for the flare on 1998 September 23, there is some uncertainty of how to draw a simplified NL. Presumably this event does not represent a clear example of type I flares. In addition, we have found eight flares, where the motions away from the NL were mixed with the other components. In these flares, examples of which are shown in Figure 5, the projection of the motion vector on the simplified NL is not small. We see that solar flares are usually not so simple as the standard model predicts. Under actual conditions in the solar atmosphere, reconnection always occurs in a more complicated magnetic configuration, at least in the presence of the magnetic

Fig. 7.--Formation of highly sheared three-dimensional magnetic structures by large-scale photospheric flows of two different types: (a) differential rotation and (b) sheared vortex.


568

BOGACHEV ET AL.

Vol. 630

Fig. 8.--Type II motion of HXR sources in solar flares.

separation from the flares with the decreasing one. Both schemes can be found in the same flare. In the onset of a flare, the HXR sources begin to move one to another and the distance between them decreases. Then they pass through a ``critical point.'' At this moment, the line connecting the sources becomes nearly perpendicular to the NL. Finally, the sources move one from another with increasing separation between them. Such a motion pattern is close to the B configuration in Figure 7 and was assumed by Somov et al. (2002) for the Bastille day flare.

Contrary to type II flares, in type III the HXR sources move along the NL in parallel direction. Both source velocities are usually comparable, and the distance between the sources does not change considerably as shown in Figure 9. We have found these motions in 11 out of 31 flares (35%). The flare on 2000 November 26 is especially interesting because the whole loop is well seen in HXRs in the disk projection. Three HXR sources (one loop-top source and two FP sources) move in the same direction with comparable velocities. In terms of reconnection, it


No. 1, 2005

HARD X-RAY SOURCE MOTION IN SOLAR FLARES

569

Fig. 9.--Type III motion of HXR sources in flares.

means that the acceleration region also moves in the corona along the separator. 4. DISCUSSION AND CONCLUSIONS Using the Yohkoh HXT data in an energy range above 30 keV, we have examined in detail the HXR source motions during 72 solar flares selected with two criteria: (1) the flare occurred within 45 of the center of the disk and (2) the integral photon count in the HXT M2 band was greater than 1000 counts per subcollimator. For all these flares, the average velocity V of sources and its uncertainty were determined by the method of

least squares. We have found and analyzed 198 sources. For 80% of them, the ratio of V to 3 is larger than 1. Thus, the regularly moving HXR sources are usually observed in flares rather than chaotically moving or stationary ones. We consider the HXR source motions during the impulsive phase of a flare as a chromospheric signature of the progressive reconnection in the corona. Since the FPs of newly reconnected field lines move from those of previously reconnected lines, the places of electron precipitation into the chromosphere change their position during the flare. In order to study the relationship between the direction of motions and the configuration of the


570

BOGACHEV ET AL.

Vol. 630

magnetic field in the active region, we have co-aligned HXT images in 61 out of the 72 flares listed in Table 1 with MDI and KPO magnetograms translated to the time of HXT observations. For 31 of them, we were successful in the simplified NL drawing. Using these 31 flares, we have inferred that there are three main types of FP motions in flares. Type I represents the motions of HXR sources away from and nearly perpendicular to the NL, in agreement with the standard model of a flare. However, only 4 out of 31 (13%) flares showed this pattern of motions. Thus, the standard model is a strong oversimplification that cannot explain main features of actual flares. In type II, the sources on both sides of the NL move along the NL in antiparallel direction rather than perpendicular to the NL. Such motions were found in 8 out of 31 flares (26%). Assuming the HXR sources to be the FPs of newly reconnected loops, this motion indicates that the reconnected field lines are highly sheared and the shear angle changes as the flare evolves. The difference between numbers of flares of type I and of type II means that the highly sheared magnetic structures are much more favorable for flare production than simple two-dimensional configurations without the shear flows in the photosphere. Type III is similar to type II except that both sources move in the same direction along the NL. This happens in about 35% of flares (11 out of 31). The parallel motions of the FPs cannot be explained as a relaxation of the magnetic field from strong to weak sheared configuration. We believe that such motions are the chromospheric signature of a displacement of the particle acceleration region during a flare. The H observations by Wang et al. (2003) indicate that an electric field in the corona is not uniform along the reconnecting current layer at the separator. The peak point of the electric field (related to a region of the most powerful particle acceleration) may change its position during the

flare. Consequently, three HXR sources (the loop-top source and two FP sources) should move in the same direction along the NL. The study reveals a more complex situation for motions of HXR sources (and, probably, H kernels) than the previous study of Sakao et al. (1998) did, where the separation of the conjugate FPs has been investigated only from a view of the distance to each other. We examine the individual motions of the FPs with respect to the smoothed NL and see that both of Sakao's types ( LI and MI ) can be explained as variants of the flares of type II. Both kinds of separation (decreasing and increasing) may be observed in the same flare as consecutive stages of its evolution, a new result inconsistent with the older classification. The most important modification of Sakao's scheme is the introduction of type III motions. They are not accompanied by changes of distance between the sources and for this reason were not in the previous classification. We found such flares in HXR and are very interested in confirmation of this result in H . In general, we expect that a detailed study of H kernel motions will reveal the same situation for FP motions as the current study shows. The other motion patterns in first approximation can be described as a combination of three basic types, leading us to the idea that types I ­ III are really the three fundamental components of any FP motion. This seems to be reasonable because of the following relationships. Type I velocity depends on the rate of the reconnection in the corona. Type II motion indicates the shear relaxation during the flare. And type III is presumably related with the motion of the acceleration region in the corona along the separator. The question remains of what are the reasons of the apparent prevalence of one or two components over the other in different flares. We hope to find an explanation in different topological and physical conditions; we expect that this will help reveal the underlying physics.

APPENDIX UNCERTAINTY IN THE CENTER OF GRAVITY OF THE SOURCE Assume that the source is given by the set of values Ii determined in the points xi, yi. In this case, the gravity center x0, y0 of the source can be found as Pn Pn I i yi i ¼ 1 Ii xi i x0 ¼ P n ; y0 ¼ P ¼ 1 : ÏA1÷ n Ii i ¼1 i ¼ 1 Ii Now let the points xi and yi have uncertainties àxi and àyi , respectively. These uncertainties are mainly due to the angular resolution of the image. The uncertainty of the signal Ii depends on the S/ N in the point i and equals à Ii: xi ¼ xi ô à xi ; yi ¼ yi ô à yi ; Ii ¼ Ii ô à Ii ; S=Ni ¼ Ii =à Ii : Our purpose here is to propagate these errors in x0 and y0: x0 ¼ x0 ô à x0 ; y0 ¼ y0 ô à y0 : If the measured variables xi , yi , and Ii are independent and the errors à xi , àyi , and à Ii are random, then Ï à x0 ÷ 2 ¼ 2 X 2 n n X @ x0 @ x0 à xk × à Ik : @ xk @ Ik k ¼1 k ¼1 ÏA8÷ ÏA6÷ ÏA7÷ ÏA2÷ ÏA3÷ ÏA4÷ ÏA5÷


No. 1, 2005

HARD X-RAY SOURCE MOTION IN SOLAR FLARES

571

We use here the index k in order to distinguish it from the index i in equation (A1). From equation (A1), P @ x0 Ï@=@ xk ÷ n¼ 1 Ii xi Ik à xk Pn i à xk ¼ à x k ¼ Pn : @ xk Ii i ¼1 i ¼ 1 Ii Similarly, @ x0 à Ik ¼ @ Ik ¼ þP
n i ¼ 1 Ii

ÏA9÷

à

Ï@=@ Ik ÷ P

P

x

k

P

þ Pn n i ¼ 1 I i xi þ i ¼ 1 Ii xi þ Pn à2 i ¼ 1 Ii

à

Ï@=@ Ik ÷

Pn

i ¼ 1 Ii

àI

k

n n i ¼ 1 Ii þ i ¼ 1 I i xi þ Pn à2 i ¼ 1 Ii

àI à I k ¼ Ï x k þ x 0 ÷ Pn

k

i ¼ 1 Ii

:

ÏA10÷

From equations (A8), (A9), and (A10), the uncertainties in x0 are 2 X 2 X n n n X @ x0 @ x0 I k à xk 2 Pn Ï à x0 ÷ ¼ à xk × à Ik ¼ @ xk @ Ik i ¼ 1 Ii k ¼1 k ¼1 k ¼1 ! 2 Pn 2 Pn 2 n X à Ik k ¼ 1 Ï I k à xk ÷ × k ¼ 1 ½ Ï xk þ x0 ÷ à I k × Ï x k þ x 0 ÷ Pn ¼ : þ Pn à2 Ii i ¼ 1 Ii k ¼1
2 i ¼1

ÏA11÷

This is the general equation for the uncertainty of the center of gravity. If à xk is due to the angular resolution of the image, then it must be constant for all the pixels k in the image: à xk à x $ pixel size of the image: The S/ N (Ik /à Ik) can be defined here as the standard deviation of the count rate Ik due to Poisson statistics: à Ik pffiffiffiffi Ik : ÏA13÷ ÏA12÷

Propagating equations (A12) and (A13) into equation (A11), we have Ï à x0 ÷ 2 ¼ where I¼
n X i ¼1 n n Ï à x÷ 2 X 2 1 X Ik × 2 Ïxk þ x0 ÷2 Ik ; 2 I k ¼1 I k ¼1

ÏA14÷

I

i

ÏA15÷

is the integral intensity of the source. How does this uncertainty depend on the number of pixels n and the intensity I of the source? We can roughly estimate it if we assume that the signal level Ik is the same for all the points k of the source: Ik ¼ I0 ¼ const: In this case I¼ and we have Ï xk þ x0 ÷ 2 : ÏA18÷ I0 n2 pffiffiffi We see that both parts of the uncertainty decrease with the growth of n as 1/ n. The S/ N part of the uncertainty also depends on the pffiffiffiffi source intensity I0. For more bright sources, it decreases as 1/ I0 .
2 n X i ¼1

ÏA16÷

Ii ¼ I0 n

ÏA17÷

Ï à x÷ 2 × Ï à x0 ÷ ¼ n

P

n k ¼1


572

BOGACHEV ET AL.

REFERENCES Asai, A., Ishii, T. T., Kurokawa, H., Yokoyama, T., & Shimojo, M. 2003, ApJ, Masuda, S., Kosugi, T., Hara, H., Tsuneta, S., & Ogawara, Y. 1994, Nature, 586, 624 371, 495 Bevington, P. R. 1969, Data Reduction and Error Analysis for the Physical Masuda, S., Kosugi, T., & Hudson, H. S. 2001, Sol. Phys., 204, 55 Sciences ( McGraw-Hill: New York) McTiernan, J. M., Hudson, H. S., & Metcalf, T. R. 2004, BAAS, 204, 0204 Donoho, D. L., & Johnstone, I. M. 1994, Biometrika, 81, 425 Ogawara, Y., Takano, T., Kato, T., Kosugi, T., Tsuneta, S., Watanabe, T., Fletcher, L., & Hudson, H. S. 2002, Sol. Phys., 210, 307 Kondo, I., & Uchida, Y. 1991, Sol. Phys., 136, 1 Frieden, B. R. 1972, J. Opt. Soc. Am., 62, 511 Otsu, N. 1979, IEEE Trans. Sys., Man, Cybern., 9(1), 62 Gorbachev, V. S., & Somov, B. V. 1988, Sol. Phys., 117, 77 Sakao, T. 1994, Ph.D. thesis, Univ. Tokyo ------. 1989, Soviet Astron., 33, 57 Sakao, T., Kosugi, T., & Masuda, S. 1998, in Observational Plasma Astrophysics: ------. 1990, Adv. Space Res., 10(9), 105 Five Years of Yohkoh and Beyond, ed. T. Watanabe, T. Kosugi, & A. C. Gull, S. F., & Daniell, G. J. 1978, Nature, 272, 686 Sterling ( Dordrecht: Kluwer), 273 Howard, R. F., Harvey, J. W., & Forgach, S. 1990, Sol. Phys., 130, 295 Somov, B. V. 2000, Cosmic Plasma Physics ( Dordrecht: Kluwer) Kosugi, T., et al. 1991, Sol. Phys., 136, 17 Somov, B. V., Kosugi, T., Bogachev, S. A., Sakao, T., & Masuda, S. 2005, Adv. Krucker, S., Hurford, G. J., & Lin, R. P. 2003, ApJ, 595, L103 Space Res., in press Liu, W., Jiang, Y. W., Liu, S., & Petrosian, V. 2004, ApJ, 611, L53 Somov, B. V., Kosugi, T., Hudson, H. S., Sakao, T., & Masuda, S. 2002, ApJ, Martens, P. C. H., & Cauffman, D. 2002, Multi-Wavelength Observations of 579, 863 Coronal Structure and Dynamics, Yohkoh 10th Anniversary Meeting ( London: Somov, B. V., Kosugi, T., & Sakao, T. 1998, ApJ, 497, 943 Pergamon) Tsuneta, S., Masuda, S., Kosugi, T., & Sato, J. 1997, ApJ, 478, 787 Masuda, S., Kosugi, T., Hara, H., Sakao, T., Shibata, K., & Tsuneta, S. 1995, Wang, H., Qiu, J., Jing, J., & Zhang, H. 2003, ApJ, 593, 564 PASJ, 47, 677 Willingale, R. 1981, MNRAS, 194, 359