MICROWAVE BURSTS WITH FINE STRUCTURES IN THE DECAY PHASE OF A SOLAR FLARE

Transcription

1 C The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi: / x/745/2/186 MICROWAVE BURSTS WITH FINE STRUCTURES IN THE DECAY PHASE OF A SOLAR FLARE Jing Huang and Baolin Tan Key Laboratory of Solar Activities, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, , China; huangj@nao.cas.cn, bltan@nao.cas.cn Received 2011 June 3; accepted 2011 November 4; published 2012 January 17 ABSTRACT This paper presents the microwave bursts with fine structures (FSs) at GHz in the decay phase of a solar flare observed by the Chinese Solar Broadband Radio Spectrometer in Huairou, which show a peak-to-peak correlation with kev hard X-ray (HXR) bursts observed by RHESSI. In the microwave spectra, we have identified stripe-like bursts such as lace bursts, fiber structures, zebra patterns (ZPs), and quasi-periodic pulsations. We also have detected short narrowband bursts such as dots, type III, and spikes. The lace bursts had rarely been reported, but in this event they are observed to occur frequently in the decay phase of the flare. The similarity between 25 and 50 kev HXR light curve and microwave time profiles at GHz suggests that these microwave FSs are related to the properties of electron acceleration. The electron velocity inferred from the frequency drift rates in short narrowband bursts is in the range of 0.13c 0.53c and the corresponding energy is about kev, which is close to the energy of HXR-emitting electrons. From the Alfvén soliton model of fiber structures, the double plasma resonance model of ZPs, and the Bernstein model of the lace bursts, we derived a similar magnetic field strength in the range of G. Additionally, the physical conditions of the source regions such as height, width, and velocity are estimated. Key words: Sun: corona Sun: flares Sun: radio radiation 1. INTRODUCTION The flare-related microwave emission includes a wide variety of emission processes and is a useful tool for probing the associated energy release, electron accelerations, and a series of parameters in the magnetized plasma source region (Bastian et al. 1998). While incoherent radiations such as gyrosynchrotron and free free emissions appear in the microwave wavelengths, the coherent plasma emission plays a dominant role at meter and decimeter wavelengths, which may be associated with the flare primary energy-releasing sites (Karlicky et al. 2001). Arzner & Benz (2005) have investigated the temporal relations between hard X-ray (HXR) peaks and microwave fine structures (FSs) in metric and decimetric range of solar flares. They found that solar microwave type III bursts often coincide with enhanced HXR emission, which is generally interpreted as the signature of electron beams propagating along the magnetic field. Huang et al. (2008) have investigated the distribution of FS on a broadband microwave dynamic spectrum of GHz around solar flares. They found that most FSs, such as type III bursts, type U bursts, slow-drifting bursts, zebra patterns (ZPs), quasiperiodic pulsations (QPPs), spike bursts, dot bursts, patches, and continuum, occurred before the maximum of the soft X-ray (SXR) emission, i.e., in the rising phase of flares. According to the Neupert effect (Neupert 1968), the integral of the microwave and HXR emission is roughly proportional to the SXR intensity. Therefore, the microwave and HXR bursts tend to occur in the rising phase of the solar flare. The double plasma resonance (DPR) model is usually used to explain the formation of stripe-like FS structures (Zheleznyakov & Zlotnik 1975a). Some radio FSs such as the tadpoles (Zheleznyakov & Zlotnik 1975b), ZPs (Zheleznyakov & Zlotnik 1975c; Chernov 1976, 1996; Ledenev et al. 2001), lace bursts (Karlicky et al. 2001), and dot-like bursts (Karishan et al. 2003; Meszarosova et al. 2008) are modeled by the DPR mechanism. The DPR is excited by a nonequilibrium distribution of electrons passing through the ambient plasma in the inhomogeneous plasma loop when the following resonance condition is fulfilled (Zheleznyakov & Zlotnik 1975a): f uh = ( f 2 pe + f 2 ce) 1/2 = sfce, (1) where f uh is the upper hybrid frequency, f pe is the electron plasma frequency, f ce is the cyclotron frequency, and s is the harmonic number. The source can be the loss-cone distribution function of superthermal electrons or an electron beam anisotropic in temperature with T > T, where T and T are the temperatures of energetic electrons across and along the magnetic field, respectively. The anisotropic temperature beam can be formed by the expansion of the hot plasma into the cold plasma along the magnetic field lines. The generated upper hybrid waves are then transformed into the observable electromagnetic waves. In this model, the different stripes can be generated from different flux tubes. The DPR model is probably the most developed heterogeneous one to explain the ZPs (Pearlstein et al. 1966; Zheleznyakov & Zlotnik 1975a, 1975c; Berney & Benz 1978; Winglee & Dulk 1986; Zlotnik et al. 2003; Yasnov & Karlicky 2004; Kuznetsov & Tsap 2007). In this model, the frequency separation between the adjacent zebra stripes is dominated not only by the electron gyrofrequency but also by the variations of the plasma density and magnetic field. When the radio emission is generated from the coalescence of an excited plasma wave and a low-frequency electrostatic wave, the emission is strongly polarized and the frequency is f f pe sf ce. The frequency separation between the adjacent zebra stripes is Δf = sf ce H b / sh b (s +1)H p. Here, H b = f B (df B /dr) 1 = B(dB/dr) 1 and H p = f pe (df pe /dr) 1 = 2n e (dn e /dr) 1 = 2H n. H b and H n are the scale lengths of the magnetic field B and the plasma density n e around the coronal source region, respectively. For f ce f pe and s 1, the 1

2 frequency separation is (Zheleznyakov & Zlotnik 1975c) H b Δf H b 2H n f ce. (2) Besides the DPR model, several other models have been developed for the formation of ZPs (Rosenberg 1972; Zaitsev & Stepannov 1983; Ledenev et al. 2001; Chernov 2010; Tan2010). In the Bernstein mode (BM) model, the coupling between two Bernstein modes, or a Bernstein mode and another electrostatic upper hybrid wave, can explain the microwave ZP with a few equidistant stripes. The emission is a result of nonlinear coalescence of these waves into electromagnetic emission at frequency (Zheleznyakov & Zlotnik 1975a) f = f uh + sf ce f pe + sf ce. (3) The frequency separation of the adjacent zebra stripes is just equal to the electron gyrofrequency: Δf = f B (Zheleznyakov & Zlotnik 1975c). The whistler wave (WW) model indicates that the coupling of a plasma wave and whistlers can operate in different conditions. When whistlers are generated at the normal Doppler cyclotron resonance, they can escape along the magnetic loop and yield fiber bursts. When whistlers are generated at the anomalous Doppler cyclotron resonance under large angles to the magnetic field, they form standing wave packets in front of the shock wave, and when the group velocity of whistlers is approximated to the velocity of the shock, a ZP structure with slow oscillating frequency drift will appear. The WW group velocity peaks at the whistler frequency f w 0.25f ce. The frequency separation Δf between zebra stripes is about two times the whistler frequency: Δf 2f w 0.5f ce. The Alfvén soliton model is also used to interpret the origin of fiber bursts. The source of the fiber bursts is regarded as a duct of solitons in the magnetic loss-cone configurations (Bernold & Treumann 1983; Wang & Zhong 2006). The nonlinear coupling between the WW trapped in solitons and the high-frequency electrostatic waves will produce the fiber burst. The soliton will decay before reaching the downward branch of the loop, which results in the negative frequency drifting rate. The spikes and the narrowband type III bursts are always regarded as the direct signals of the energetic electron beams, which are generated in coronal loops with sufficiently strong magnetic field inhomogeneities by the electron cyclotron maser mechanism (Melrose & Dulk 1982; Fleishman & Melnikov 1999). The magnetic field inhomogeneities are a consequence of the primary energy release. In some regions of the loop the distribution is anisotropic enough to produce the electron cyclotron maser instability, i.e., micro-traps where individual spikes form. It is known to all that, during the post-flares, the process of particle acceleration is still occurring. Although the microwave FSs occurring during the post-flare are not as abundant as those of the impulsive and peak phase, we could also extract some diagnostic information on the processes of the post-flare energy release, the characteristics of the energetic particles, and the plasma features from them. Fortunately, in the decay phase of the flare event on 2004 December 1, we found several microwave FSs, such as lace bursts, dot bursts, type III bursts, spike bursts, fiber bursts, ZPs, and QPP structures, from the observation at the Chinese Solar Broadband Radio Spectrometer in Huairou (SBRS/Huairou). In this work, we focus on the spectral analysis of these FSs and drawing out some physical parameters of the post-flare. Section 2 introduces the main features of the microwave bursts with FSs that occurred in the decay phase of the flare. The physical parameters of the source regions deduced from the microwave FSs are presented in Section 3. Section 4 provides the main conclusions. 2. OBSERVATIONS AND DATA ANALYSIS 2.1. Nonthermal Bursts in the Flare Decay Phase On 2004 December 1, an M1.1 solar flare occurred in the active region AR starting at 07:00 UT and ending at 07:41 UT, with the peak at 07:20 in the GOES SXR flux. Several instruments observed this flare, especially RHESSI and SBRS/Huairou. RHESSI registers the HXR emission with high cadence and high sensitivity (Lin et al. 2002). SBRS/Huairou provides the microwave broadband dynamic spectra of the burst at a frequency of GHz (cadence of 1.25 ms and spectral resolution of 4 MHz), GHz (cadence of 8 ms and spectral resolution of 10 MHz), and GHz (cadence of 5 ms and spectral resolution of 20 MHz) with dual circular polarization (left and right circular polarization). The observation sensitivity is S/S 2%, where S is quiet solar background emission (Fu et al. 2004). Figure 1 shows the profiles of the SXR flux from GOES and the radio dynamic spectra at GHz, GHz, and GHz recorded by SBRS/Huairou. The microwave temporal profiles at 1.20, 3.05, and 6.10 GHz are overplotted. From the microwave spectra, it can be seen that the microwave burst at low frequency is much more complex than that at higher frequency bands. The main microwave enhancement occurs in the rising and peak phases of the GOES SXR flare. Huang et al. (2007) investigated the microwave bursts with FSs at a frequency of GHz in the impulsive phase of the flare and found that there are two segments of ZPs (Z2 at 07:09:05 UT and Z3 at 07:09:11 UT) and one complex structure denoted N1 in Figure 1(b). At GHz, there is only one pulse in the initial phase. During the whole burst process, the bursts at higher frequency ( GHz and GHz) present continua in the spectra with no microwave FS. In the decay phase of the flare, especially from 07:24 UT to 07:40 UT, the microwave emissions at the higher frequencies of GHz and GHz have only smooth decays, but at the lower frequency of GHz, it has four remarkable bright parts in the dynamic spectrum, including many FSs. Figure 2 presents the comparison between the microwave temporal profile at 1.20 GHz (solid line) and the HXR flux (in 1/3 intensity) at kev (dotted line) from RHESSI during 07:20 07:40 UT, after the GOES SXR flare peak. The HXR flux has many remarkable bursts in the flare decay phase superimposed on a smoothly varying component. The HXR smooth component is possibly produced by the secondly precipitating electrons as a result of the trapping process. The pulsing part is produced by the injected nonthermal electrons, which directly reflect the acceleration process (Aschwanden 2002). The microwave profile has several pulses, which have a peak-to-peak connection with the HXR flux by removing its smooth component. The similarity of their temporal profiles indicates that both emissions are possibly produced by the same population electrons, although they are produced by a different emission mechanism. Hence, we could find some diagnostic information on electron acceleration, transport and the local plasma density, and the magnetic field from the microwave 2

3 Figure 1. Panel (a) is the temporal profile of the GOES SXR flux in an M1.1 flare event on 2004 December 1. Panels (b), (c), and (d) are the dynamic spectra of the microwave bursts at GHz, GHz, and GHz, respectively, with the temporal profiles at 1.20, 3.05, and 6.10 GHz observed at SBRS/Huairou overplotted. bursts with many FSs. We focus on the analysis of the microwave bursts at GHz with many FSs recorded. From the dynamic spectrum recorded by SBRS/Huairou at GHz, the lace bursts, dot bursts, type III bursts, spike bursts, ZPs, fiber bursts, and QPPs are identified in the flare decay phase. These FSs can be roughly classified into two classes according to their patterns in the spectrum. One is the stripe-like burst, which comprises several stripes with small or moderate frequency drifting rate. Sometimes, the frequency drifting rate may change drastically between negative and positive. It includes lace bursts, ZPs, fiber bursts, and QPPs. Here, we regard QPP as a stripe-like burst, because the frequency drifting features in the whole QPP chain are similar to those of the above stripe-like bursts. The other one is the short narrowband burst, which is constituted of bursts of very short duration, narrow frequency band, large frequency drifting rate, and appearing in great clusters stochastically on the dynamic spectrum. It includes dot bursts, spike bursts, and narrowband type III bursts Stripe-like Bursts Lace Bursts The so-called lace bursts show a rapid frequency variation in the dynamic spectrum that appears as one or more stripelike necklaces. The lace bursts have rarely been reported in 3

4 Figure 2. Temporal profiles of the microwave flux at 1.20 GHz (solid line) from SBRS/Huairou and HXR at kev from RHESSI (dotted line, in 1/3 intensity) during the flare decay phase from 07:20 UT to 07:40 UT. the previous observations. The most famous event of lace burst is reported by Karlicky et al. (2001). They found that, during the 9 years of observation from the Ondrejov radiospectrograph, out of the 681 recorded events only 3 lace bursts were observed. Jiricka et al. (2001) also reported lace bursts with a duration of several minutes in the GHz frequency range (Brazilian Solar Spectroscope). Karlicky et al. (2001) and Jiricka et al. (2001) proposed that lace bursts are similar to ZPs and explained them by using a DPR model in which lace bursts are generated from the turbulent plasma associated with the magnetic reconnection outflow. According to this model, the rapid variation of the high-frequency boundary of the lace bursts is due to the corresponding changes in the magnetic field and plasma parameters within the microwave source region. The lace bursts are the most obvious fine structure in the decay phase of the M1.1 flare on 2004 December 1. There are seven remarkable segments of lace bursts observed from SBRS/Huairou. Figure 3 presents two examples of lace bursts that occurred at 07:25:48 07:25:52 UT and 07:33:18 07:33:25 UT. It is found that sometimes there are several lace stripes that occur in groups similar to the ZP structure within several seconds (left panel), and the frequency separation between adjacent stripes is about MHz. But at other times, the lace bursts only have one single stripe (right panel). The most prominent feature of the lace burst is the rapid variation of the frequency drifting rate of each stripe. As shown in Figure 3, the stripes of lace bursts represent a wave with the frequency drifting rate changing from positive to negative, and vice versa. For each stripe, there is an inflexion at high frequency in the observed band. We distinguish 48 stripe segments with positive frequency drifting rate and 42 stripe segments with negative frequency drifting rate from these lace bursts. The positively frequency drifting stripes have a rate mainly covering the range of MHz s 1 withamean value of MHz s 1, and the negatively frequency drifting stripes have a mean value of MHz s 1. Figure 4(a) is the dynamic spectrum of a segment of lace burst, which shows an asymmetric distribution of the emission in frequency. Figure 4(b) shows the spectral profiles at two different times, marked with dashed and dotted lines in Figure 4(a), respectively. These profiles have a gradual increase from the low-frequency side and a rapid decrease to high frequency. The peak flux at lower frequency is larger than that at higher frequency. We select a lace stripe with positive frequency drifting (from high frequency to low frequency) and a lace stripe with negative frequency drifting (from low frequency to high frequency) from these lace bursts to investigate the evolution of their peak flux, peak frequency, and half-power bandwidth. Figure 5 presents the relationships between the peak flux, the half-power bandwidth, and the emission frequency in a stripe with positive frequency drifting rate (a) and a stripe with negative frequency drifting rate (b). It indicates that the peak flux is decreasing with the frequency. At the same time, the half-power bandwidth has a random distribution with an average value of 38 MHz in the negative drifting stripes and 28 MHz in the positive drifting stripes. From Figure 3, we find that almost all the lace bursts are strongly left-handed circularly polarized. The degree of polarization (DoP) can be calculated as follows: Pol = (L R)/(L + R), where L and R are the emission fluxes in left- and right-handed circular polarization, respectively. Here, the background emission is subtracted from the flux. The bottom panel of Figure 5 presents the DoP of the lace stripes selected in (a) via frequency. It is found that the DoP covers from 70% to about 100% and the value has a slightly decreasing trend toward the increasing frequency, although this trend is not significant Fiber Structures There are several clusters of fiber bursts in the flare decay phase. The left panel of Figure 6 presents an example of fiber structure that occurred during 07:36:06 07:36:18 UT. It is composed of several slantwise unbent stripes with intermediate frequency drifting rate and random frequency separation 4

5 Figure 3. Dynamic spectra of two segments of lace bursts in decay phase of the M1.1 flare event observed at SBRS/Huairou at 07:25:48 07:25:52 UT and 07:33:18 07:33:25 UT on 2004 December 1. Figure 4. Example of the stripe segments in the lace burst. (a) The dynamic spectrum. (b) The spectral profiles at two different times, shown by dashed and dotted lines in the spectrum (a), respectively. 5

6 Figure 5. Relation between the peak flux, the half-power bandwidth, and the emission frequency of one positive frequency drifting lace stripe (a) and one negative frequency drifting lace stripe (b). The bottom panel presents the relation between the DoP of the lace stripes in (a) and the emission frequency. between the adjacent stripes. The frequency drifting rate of the fiber stripes is between 58 and 73 MHz s 1, and the mean value is about 65 MHz s 1. The central emission frequency is about 1220 MHz. The frequency bandwidth is in the range of MHz. These fibers also appear in the left-handed polarization, and the DoP is almost 100% Zebra Pattern Structures The middle panel of Figure 6 presents a ZP that occurred in the flare decay phase during 07:31:32 07:31:35 UT. It has four stripes with a frequency drifting rate around 145 MHz s 1. The frequency separation between the adjacent stripes Δf is about MHz, which is much broader than that of Z2 (24 MHz) and Z3 (16 MHz) in the impulsive phase of the flare. The frequency bandwidth of the ZP is above 240 MHz. It extends to the lower frequency below 1.10 GHz and to the higher frequency beyond 1.34 GHz in the spectrum. Similar to the other FSs, the DoP of the ZP structure is also approximated to about 100%. Another prominent feature of the ZP is that all the stripes are composed of spike bursts. The frequency bandwidth of the spikes is about MHz, and the frequency drifting rate of each spike is about 3.70 GHz s 1. This ZP could thus be regarded as a spiky ZP Quasi-periodic Pulsation The right panel of Figure 6 shows a QPP structure in the decay phase of the flare, which is also left-hand polarized. The wavelet analysis indicates that the QPP structure is mainly composed of 6

7 The Astrophysical Journal, 745:186 (10pp), 2012 February 1 Figure 6. Dynamic spectra of the stripe-like bursts that occurred in the decay phase of the M1.1 flare event observed by SBRS/Huairou. The left panel is the fiber burst, the middle panel is the ZPs, and the right panel is the QPP. Figure 7. Dynamic spectra of the short narrowband burst in the decay phase of the M1.1 flare event observed by SBRS/Huairou. The left panel is the dot bursts, the middle panel is the type III bursts, and the right panel is the spike bursts. dynamic spectrum is random. To study the spectral characters of the individual burst, 70 samples are selected randomly. The statistical results of the dot burst show that the duration is mainly in the range of ms, the frequency bandwidth ranges from 24 to 30 MHz, and their mean values are 20.7 ms and 27.7 MHz, respectively. Additionally, similar to the other FSs, all of the dot bursts are strongly left-handed circularly polarized with a DoP above 90%. The frequency drifting rate ranges from 4.10 to 8.60 GHz s 1, and the mean value is about 4.95 GHz s 1. two QPP components. One component has a period of about 70 ms, and the other has a period of about ms. Both of them belong to the very short period pulsation (Tan et al. 2007, 2010). Each QPP pulse drifts from high frequency to low frequency at a rate of 3.70 to 7.30 GHz s 1, with a mean value of 5.40 GHz s Short Narrowband Bursts Dot Bursts The dot bursts appear as a group of individual bursts looking like a dot with short duration and narrow frequency bandwidth. During the decay phase, there are seven clusters of dot bursts appearing on the dynamic spectrum. The left panel of Figure 7 presents an example of the dot bursts that occurred at 07:28:29 07:28:30 UT. The distribution of the dot bursts on the Narrowband Type III Bursts Around the end of the flare, there are several clusters of type III bursts. The middle panel of Figure 7 presents a cluster of type III bursts. It occurred at 07:36: :36:50.4 UT, lasting for about 0.6 s, and was composed of more than 22 narrowband 7

8 Table 1 Main Features of the Microwave Bursts Associated with the Decay Phase of the M1.1 Flare Class Duration Δf DR DoP v Δr (s) (MHz) (MHz s 1 ) (%) (kms 1 ) (km) Lace Fiber Zebra > >4000 QPP-spike Type III Spike Dot Notes. Δf is the frequency bandwidth, DR is the frequency drift rate, DoP is the degree of polarization, and Δr is the estimated width of the source regions. type III bursts. The mean duration of an individual type III burst is about 16 ms, and the frequency bandwidth is in the range of MHz, with an average value of about 110 MHz. They are also strongly left-handed circularly polarized with a DoP close to 100%. The emission of each individual type III burst drifts from low frequency to high frequency. The frequency drift rate ranges from 3.20 to 9.60 GHz s 1, and the mean value is about 4.81 GHz s 1. The high-frequency cutoff of the type III burst cluster is modulated as one or more inverse U type, which presents similarly to the lace-like pattern. Additionally, we find another important feature that the narrowband type III bursts have some periodicity on the pattern and the period is about ms. Type III burst groups may be regarded as drifting narrowband QPPs Spike Bursts Similar to the dot bursts, the spike bursts are also in random distribution. There are several clusters of spike bursts that occurred in the flare decay phase (right panel in Figure 7). The frequency bandwidth of spike bursts is in the range of MHz, with an average value of about 54 MHz, which is narrower than that of narrowband type III bursts and wider than that of dot bursts. In contrast to the narrowband type III bursts, almost all the spike bursts are drifting from high frequency to lower frequency, the frequency drifting rate ranges from 2.31 to 9.66 GHz s 1, and the mean value is about 5.17 GHz. The duration of the spikes is in the range of ms, with an average value of about 22 ms. Moreover, all the spike bursts are strongly left-handed circularly polarized with a DoP above 80%. Karishan et al. (2003) observed the dot-like burst in the frequency range of GHz in some regular patterns, such as strings of beads in positive or negative frequency drift rate, or inverted-u chains on definite trajectories like a parabolic arch. However, in this event, the dot bursts differ greatly from the regime of Karishan et al. (2003), that is, the dot bursts occurred in large numbers of groups in an irregular pattern. Additionally, it is found that some dot bursts are interspersed among the spike clusters and the frequency drifting rates of dot bursts and spike bursts, including the spikes found in ZPs and QPPs, are very close to each other. These facts may imply that dot bursts and spikes are generated from a similar mechanism. Both Narrowband type III bursts and spikes or dot bursts present a fast frequency drifting rate, but with opposite signs. The former have a regular distribution while the latter show a random distribution on the spectra. Both differences indicate that they may be produced from different mechanisms. Narrowband type III bursts may be generated from the energetic electrons formed at some accelerated sites and propagating downward to the denser plasma loops with the emitting frequency increasing. Spikes and dot bursts are proposed to be emitted by the electrons that accelerated in the magnetic islands of the current-carrying plasma loops, propagating to the ambient dilute plasma region with a decreased emitting frequency (Tan & Huang 2006; Tan 2010). Table 1 is a brief summary of the main features of microwave FSs associated with the flare decay phase. Listed parameters are explained in the next section. 3. PHYSICAL ANALYSIS 3.1. Estimation of Magnetic Field Strength Both ZPs and fiber bursts can be used to estimate the magnetic field strength in the source region, even though such estimation depends on the theoretical models. From the Alfvén soliton model, the fiber burst is regarded as a duct of solitons in the magnetic loss-cone configurations (Bernold & Treumann 1983; Wang & Zhong 2006). The nonlinear coupling between the WW trapped in solitons and the high-frequency electrostatic waves will produce the fiber burst. The soliton will decay before reaching the downward branch of the loop, which results in a negative frequency drifting rate. The magnetic field is deduced as (Bernold & Treumann 1983) B 4πL n mi m e df df L n (G). (4) ec dt dt Here, m i is the mass of ions, L n is the density scale length in the source region, and we can assume L n 10 7 m empirically (Bernold & Treumann 1983). Then, from the frequency drifting rate of the fiber burst in our observation, we get a magnetic field in the source region of about G. Using the BM model of ZPs, we can estimate the magnetic field of the source region by measuring the frequency separation between the adjacent zebra stripes according to Equation (3): B 2πm e Δf Δf, (5) e where B is in Gauss. Substituting the frequency separation (60 70 MHz) into the above expression, we may get the magnetic field: B G. From the WW model, we also get an estimation of magnetic field strength as B 2(2πm e /e)δf Δf. In the same way, we obtain the magnetic field around G. In the DPR model, as the DoP of ZPs is close to 100%, the emission may be generated from the coalescence of an excited plasma wave and a lowfrequency electrostatic wave, and its frequency is very close to 8

9 the fundamental plasma waves. Then from Equations (1) and (2), we get the expression of the magnetic field: B 2πm e e 2H n H b H b Δf. (6) Here, we adopt the Newkirk model as the template of the solar atmospheric plasma density: n(r) M /r. Here, M is a constant and its magnitude does not affect our estimates, and r is the distance from the center of the Sun to the source region. The scale length of the plasma density around the source region can be deduced as H n r 2. The coronal magnetic field above the active region can be modeled by B(r) = 0.5/(r 1) 1.5 (1.02 r 10) (Dulk & McLean 1978), and the scale length of the magnetic field around the source region can be deduced: H b (2/3)(r 1). Here, r, H n, and H b are in the units of the solar optical radius: R = m. Then (2H n H b )/H b = ( r 2 r +1)/(r 1). Here, the parameter r is a key factor. However, we do not know the exact r directly from the observations. We have tried a range of r to fit the observed spectrum and found the best fit for r = (about 66 Mm above the solar photosphere), and the magnetic field strength is about G. The similarity between the lace burst clusters and ZPs implies that the lace bursts could also be applied to estimate the magnetic field strength by measuring their frequency separation between the adjacent stripes. However, as we mentioned in Section 2, the frequency separation between the adjacent stripes in lace burst clusters ( MHz) is much greater than that in ZPs (60 70 MHz). It is possible that lace bursts and ZPs are generated from different mechanisms. According to Karlicky et al. (2001), the lace bursts may be generated in turbulent plasmas associated with the plasma outflows from the magnetic reconnection sites. If we adopt the BM model to explain the generation of lace bursts, the stripes are generated from the coupling between two Bernstein modes or a Bernstein mode and another electrostatic upper hybrid wave, and the frequency separation between the adjacent stripes is just equal to the electron gyrofrequency f B. Then by adopting Equation (5), the magnetic field strength of the source region of lace bursts is estimated to be in the range of G, which just falls into the range obtained from the Alfvén soliton model of fiber bursts Estimation of the Plasma Density The generation mechanisms of most FSs are typically related to the coupling between upper hybrid waves and Bernstein waves, or WW, or other electrostatic waves, and their emission frequencies are roughly around the plasma frequency or the second harmonics. Then we obtain an estimation of the plasma density in the source region (Tan 2008): n e f 2 81s. (7) 2 Here s is the harmonic number, s = 1 corresponds to the fundamental wave, and s = 2 corresponds to the second harmonic wave. As the DoP of almost all the FSs is very large (Table 1), we suppose that the emission is possibly fundamental emission. Then we get an estimate of the plasma density in the source region of about ( ) cm 3 with an emission frequency around GHz. Such plasma density always appears around the primary energy-release sites of the flare Estimation of the Nonthermal Electron Energy or the Motion of the Source Region The change of the emission frequency during the burst of microwave FSs is normally produced by the propagating nonthermal electrons or the plasma flows. Hence, the frequency drifting could be used to estimate the nonthermal electron energy or the motion of the source region. From the plasma emission mechanism (Equation (7)), the relationship between the frequency drifting rate df /dt and the velocity of the energetic electrons v = dr/dt is df dt f 2L n v. (8) With the above assumption L n 10 7 m and the frequency drifting rate of the microwave FSs in our observation, the velocities of the sources are obtained and shown in Table 1. It is found that the velocity of the short narrowband bursts (spike, type III, and the spikes in QPP) is about 0.13c 0.54c (c is the velocity of light), which is comparable to the velocity of nonthermal electrons with energy of about kev. It is reasonable to indicate that short narrowband bursts are connected to the nonthermal electron beams with energy of about kev. The peak-to-peak relationship between the HXR bursts at kev and the microwave bursts at a frequency of GHz suggests that they may be produced from the same population of nonthermal electrons. Similarly, the motion of the source region of the stripelike bursts (lace, fiber, and zebra) could be deduced from Equation (8) with their frequency drifting rate, which are in the range from several hundreds to several thousands of kilometers per second. As a comparison, we make an estimation of the Alfvénic speed as follows. By supposing a magnetic field in the source region of about 100 G, the Alfvénic speed is about 2000 km s 1, which is just the same order of the velocity of the stripe-like bursts. This implies that the frequency drifting rate of the stripe-like bursts may reflect the motion of plasma flows Estimation of the Width of the Source Region From the plasma emission mechanism and the observed instantaneous frequency bandwidth (Δf ), we make an approximate estimation of the width (Δr) of the source regions as. Δr = 2L n Δf f. (9) It is found that the source regions of the stripe-like bursts are several thousands of kilometers, while those of the short narrowband bursts (type III bursts, spikes, and dot bursts) are as small as several hundreds of kilometers. Especially for the dot bursts and spike bursts, the source regions are about 400 km, which is smaller than 1 arcsec. 4. CONCLUSION In this work, we made a detailed analysis of the microwave FSs during the decay phase of an M1.1 flare on 2004 December 1 by using the microwave observations of SBRS/Huairou ranging from 1.10 to 7.60 GHz and the HXR observations from RHESSI. We find the following. 1. At GHz, the microwave bursts contain plenty of FSs. At higher frequency bands ( GHz and GHz), the microwave flux only decays smoothly 9

10 without any FSs. The FSs at a frequency of GHz are classified into two kinds: the stripe-like bursts (including lace bursts, fiber structures, ZPs, and QPPs) and the short narrowband bursts (including dot bursts, narrowband type III bursts, and spikes). Almost all of the FSs are strongly left-handed circularly polarized, and the DoP is above 70%. All FSs occurred in clusters. In particular, the short narrowband bursts occurred in great clusters with irregular distribution. 2. The frequency drift rates of dot bursts, spike bursts, and the spikes constituting ZPs and QPPs are very close to each other and have a mean value around 5.0 GHz s 1. The frequency drift rates of narrowband type III bursts have an opposite sign to that of dot bursts and spikes, and the mean value is about 4.81 GHz s 1. It is indicated that the narrowband type III bursts may be generated by the energetic electrons originated from the accelerated site and propagating downward to the plasma loops, while the dot bursts and spike bursts are generated possibly from the energetic electrons accelerated at some magnetic islands in current-carrying plasma loops. 3. The similarity of a kev HXR light curve with microwave time profiles at GHz suggests that these microwave FSs are related to electron acceleration. The electron velocity inferred from the frequency drift rates in short narrowband bursts is in the range of 0.13c 0.54c, and the corresponding energy is about kev, which is close in magnitude to the energy of HXR-emitting electrons. Such estimations depend on the density scale length L n around the source region. 4. The physical conditions of the source regions could be obtained from the dynamic characters of microwave FSs and their proposed mechanisms. From the Alfvén soliton model of fiber bursts, the BM model of lace bursts, and the DPR model of ZP structures, the magnetic field strength of the source region can be estimated, yielding similar results around G. The spatial width of the source regions of the stripe-like bursts is in several thousands of kilometers, and that of the short narrowband bursts is only in several hundreds of kilometers. From the stripe-like bursts, the velocity of the source regions ranges from several hundreds to several thousands of kilometers per second. The height of source regions could be obtained from the DPR model of ZPs, which is about 66 Mm. As the emission frequency is approximately equal to the plasma frequency, the source plasma density is about ( ) cm 3. However, without imaging observations at the corresponding frequency, it is not possible to make a definite conclusion about the acceleration site. When the Chinese Spectral Radioheliograph (CSRH, GHz; Yan et al. 2009) and the Frequency Agile Solar Radiotelescope of the United States (FASR, 50 MHz 20 GHz; Bastian 2003) are fully constructed and go into operation, we can get more insights on the electron acceleration during the solar flare decay phase. The authors thank the anonymous referee for the helpful and valuable comments on this paper and thank the GOES, RHESSI, and SBRS/Huairou teams for providing observation data. The SBRS/Huairou is supported by CAS. This work was supported by NSFC grants , , , , and ; the National Basic Research Program of MOST (grant 2011CB811401); and the Young Researcher Grant of National Astronomical Observatories, CAS (grant ). REFERENCES Arzner, K., & Benz, A. O. 2005, Sol. Phys., 231, 117 Aschwanden, M. J. 2002, Space Sci. Rev., 101, 1 Bastian, T. 2003, Proc. SPIE, 4583, 98 Bastian, T. S., Benz, A. O., & Gary, D. E. 1998, ARA&A, 36, 131 Berney, M., & Benz, A. O. 1978, A&A, 65, 369 Bernold, T. 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