- Email: [email protected]
Statistical Properties of Soft X-ray Fluxes of Solar Flares
CHINESE ASTRONOMY AND ASTROPHYSICS Chinese Astronomy and Astrophysics 39 (2015) 330–340
Statistical Properties of Soft X-ray Fluxes of Solar Flares† ZHANG Ping1,2,3 1
LIU Si-min1,2
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008
2
Key Laboratory of Dark Matter and Space Astronomy, Chinese Academy of Sciences, Nanjing 210008 3
University of Chinese Academy of Sciences, Beijing 100049
Abstract In order to quantitatively study the statistical properties of the soft X-ray emission in solar flares, an algorithm has been developed to automatically detect flares in a given range of peak fluxes, and to analyze the flares observed by the GOESs (Geostationary Operational Environmental Satellites) from 1980 to 2013 in two soft X-ray bands. This study indicates that the statistical characteristics of the variation of flare soft X-ray flux near the peak time are independent to the magnitude of peak flux: on the average, the rising time of flare’s soft X-ray flux is about half of the decay time, and the rising and decay times in the high-energy channel are shorter than the corresponding times in the low-energy channel, however, all these times will increase with the variation amplitude of flare’s soft X-ray flux. Key words
The Sun: flares—The Sun: X-rays—Methods: statistical 1.
INTRODUCTION
A major uncertainty in the studies of statistical properties of solar flares comes from the identification of flares, especially of small flares[1−5] , meanwhile, considering that the effects of solar activities on the terrestrial environment are closely associated with the magnitudes of solar flares, the studies on the statistical properties of large flares have a greater practical †
Supported by National Natural Science Foundation (11173064, 11233008), and Strategic Priority
Research Program of the Chinese Academy of Sciences (XDB09000000) Received 2014–05–24; revised version 2014–07–17
A translation of Acta Astron. Sin. Vol. 56, No. 1, pp. 35–43, 2015 [email protected], [email protected]
0275-1062/15/$-see front matter © 2015 Elsevier B.V. All rights reserved. doi:10.1016/j.chinastron.2015.07.005
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331
importance. In the statistical studies of solar flares, the analysis on the light-variation features of solar flares, which are rather easy to be acquired, and the analysis on the relevant time scales have important significance. The time scales of light variations of solar flares are closely related to the dominant physical mechanisms during the solar eruptions, and they are also the basis for modeling the space weather prediction[6−8] . A series of environmental GOES satellites are operated in the geosynchronous orbits by the National Environmental Satellite, Data, and Information Service belonging to the National Oceanic and Atmospheric Administration of USA, and their onboard X-ray detectors provide the most complete data of soft X-ray fluxes of solar flares up to now. Since the first GOES satellite was launched in 1974, these X-ray detectors have collected the data almost continuously in nearly 40 years, with the time resolution of 2 ∼ 3 s in two soft X-ray A and 0.5 ∼ 4 ˚ A. The flare classification based on the magnitude of wavebands of 1 ∼ 8 ˚ ˚ flare peak flux at the 1 ∼ 8 A band is the generally accepted standard for measuring the magnitudes of flares. And the previous studies on the statistical properties of soft X-ray fluxes of solar flares are mainly based on the flux data at the 1 ∼ 8 ˚ A band. Many parameters describing the flare features exhibit a power-law frequency distribution without a characteristic scale, this is quite similar to the phenomenon of self-organized criticality[7,9−11] . In this case, the larger and smaller events are completely similar to each other in their statistical properties, thus to ensure the power-law frequency distributions of relevant parameters. However, in the detailed multi-band observation and analysis of solar flares, it is found that the physical mechanism to produce large flares seems to be different from the production mechanism of small flares. For instance, the coronal mass ejections are commonly accompanied with large flares, and the two-ribbon flare loop systems also appear in many large flares. Moreover, according to the rising time of X-ray flux in the impulsive phase of flares, they are classified into two types: the gradual and impulsive flares. Further quantitative analysis is needed in order to make sure whether the statistical properties of the soft X-ray emissions in solar flares are related to the peak flux of soft X-ray emission. In this paper, we have developed a group of programs for the flare identification in a given interval of peak fluxes. This method has avoided the traditional dependence of flare identification on the background flux, it can be used to analyze the variation characteristics of large flares (especially near the peak times). Using this algorithm, we have analyzed the flare data observed by the GOES satellites in several ten years at the two soft X-ray wavebands. We have not only reproduced the dependence of the eruption frequency of solar flares with different classes on the solar cycle, but also comparatively studied the behaviors of the solar flares with different classes near the peak times. In Section 2, we summarize the basic characteristics of GOES data, and discuss the algorithm for flare identification, the analyzed results are given in Section 3, and a brief discussion and summary are given in Section 4.
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2.
THE BASIC CHARACTERISTICS OF GOES DATA AND THE CRITERIA FOR FLARE IDENTIFICATIONS
A series of environmental GOES satellites are operated in the geosynchronous orbits by the National Oceanic and Atmospheric Administration of USA, which are mainly used for the weather forecast, the environmental monitoring of terrestrial space, and the meteorological research. The X-ray detectors onboard the GOES satellites have continuously observed the fluxes of solar soft X-ray emission since 1974, and the GOES satellites worked for the different periods are listed in Table 1. In some periods, there are multiple GOES satellites operated simultaneously, and the relevant data can be used to estimate the uncertainties caused by instrumental effects[12−13] . The X-ray detectors mainly monitor the radiations A and 1 ∼8 ˚ A two bands[14] , and according to the from the full solar disk in the 0.5 ∼ 4˚ ˚ band, the flares can be classified into the A, B, magnitude of peak flux at the 1 ∼8 A C, M, and X totally five classes (see the labels on the right vertical axis of the left panel in Fig. 1), which correspond to the intervals of peak fluxes of [10−8 , 10−7 ), [10−7 , 10−6 ), [10−6 , 10−5 ), [10−5 , 10−4 ), and [10−4 , ∞)W · m−2 , respectively. In order to mark the flare class more quantitatively, it is commonly to add the digital parts of peak fluxes to the above characters of flare classes. For example, the M2.5 class represents the flare with the peak flux of 2.5 × 10−5 W · m−2 . ˚ two bands can reach ∼ The observing accuracies in the 0.5 ∼ 4 ˚ A and 1 ∼ 8 A 10−9 W·m−2 and ∼ 10−8 W·m−2 , respectively. When the fluxes are larger than 10−4 W·m−2 , the detectors may be saturated. Due to the effects of saturation, instrumental error, and non-solar signal etc., there are some spoiled points in the GOES data of soft X-ray fluxes, which are marked in the relevant data files, and the occupied proportion is commonly less than 1.0%, as shown by the cross symbols in Fig.1. The satellites are sometimes located in the side far away from the sun, the solar radiation is sheltered by the earth to cause a certain of data missing[2] . The time of data missing is closely related to the satellite orbit, and the occupied proportion in the overall observing time is also about several percent. It is considered that some problems may exist in the earlier data of GOES satellites, hence we only use the GOES data of the low and high two energy channels after 1980 in the following analysis. The data in the low-energy channel of 1.0 ∼ 8.0 ˚ A were mainly used in the previous studies on the statistical properties of flare soft X-ray flux variations, and there were some uncertainties in the determinations of the starting and ending times of flares, owing to the effects of the background radiation and the temporal overlap of adjacent flares, etc.[2,4,15] . It is considered that the solar effects on the solar-terrestrial space become more evident when the fluxes are higher, a kind of program is introduced in the following for the identification of flares in a given interval of peak fluxes. Such a program can be used to study more effectively the statistical properties of large flares and the variation characteristics of flare emissions near the peak times. (1) At first, we use the GOES software in the SSWIDL
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333
Table 1 The service periods and time resolutions of the soft X-ray detectors onboard the GOES satellites GOES-N
Service Period
Time Resolution/s
GOES 91
1-Jul-1974 to 17-Oct-1975
3
GOES 92
5-Feb-1975 to 31-Mar-1978
3
GOES 1
17-Jan-1976 to 1-Jun-1978
3
GOES 2
1-Aug-1977 to 30-Apr-1983
3
GOES 3
5-Jul-1978 to 4-Jan-1980
3
GOES 5
1 to 30-May-1983, 1-Jul-1983 to 31-Jul-1984
3
GOES 6
2 to 30-Jun-1983, 1-Aug-1984 to 18-Aug-1994
3
GOES 7
1-Jan-1994 to 3-Aug-1996
3
GOES 8
21-Mar-1996 to 18-Jun-2003
3
GOES 9
20-Mar-1996 to 24-Jul-1998
3
GOES 10
10-Jul-1998 to 1-Dec-2009
3
GOES 11
21-Jun-2006 to 11-Feb-2008
3
GOES 12
13-Dec-2002 to 8-May-2007
3
GOES 14
2-Dec-2009 to 4-Nov-2010
2
GOES 15
1-Sep-2010 to Present
2
GOES 13
package to remove the spoiled points in the GOES data, and remove the time intervals with data missing. (2) For the given interval [FL , FH ] of fluxes, we determine all the data points with the observed fluxes located in this interval. (3) Considered the removal of spoiled points and the statistical fluctuations of data themselves, we assume that two adjacent effective data points with a time interval less than 30 s are continuous. (4) In order to ensure the significance of flare signals, we only use the time slices with a duration larger than 30 s. (5) Finally, for these time slices, we determine the fluxes Fp , Fs , and Fe corresponding to the peak time (tp ), starting time (ts ), and ending time (te ), respectively. For the time slices with a duration larger than 30 s and the peak fluxes located in the interval of [FL , FH ], the corresponding Fp , Fs , and Fe should satisfy the following relations: Fs − FL ≤ 10%, FL
(1)
Fe − FL ≤ 10%, FL
(2)
Fp − FL ≤ 10%, FL
(3)
here, 10% corresponds to the statistical fluctuations of data. Eq.(1) requires that the flux at the starting time can not exceed the 10% of the lower limit of the interval of peak fluxes,
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which can effectively exclude the time slices corresponding to the decay-phase flares with the peak fluxes larger than the upper limit of the given interval of peak fluxes. Similarly, Eq.(2) can exclude the time slices corresponding to the rise-phase flares with the peak fluxes larger than the upper limit of the given interval of peak fluxes. Eq.(3) requires the flare fluxes have a remarkable enhancement. Fig.1 gives an example to apply this program to the observed data. Though it is not necessary to determine and deduct the background flux for each flare in this method, but when the background flux is higher, the temporal overlap of flares will be serious. In Fig.1 a series of small flares in the low-energy channel are not identified after 13:00, because they are considered as the rising phase of a subsequent larger flare. In principle, the effect of this shortage on the results can be reduced by adjusting the range of flux interval, but when the complexity of flare behaviors is considered, a more complicated program of flare identification is needed to completely overcome this shortage. The right panel of Fig.1 also shows that there is a certain difference between the fluxes observed by two different GOES satellites. When a flare is observed by multiple satellites simultaneously, we can use the arithmetic mean value of the flare parameters observed by the different satellites as the corresponding flare parameter. Flux GOES12 3 sec 10−2 10−3
Flux GOES 10/12 3 sec
8×10−6
(Fp,Tp)
GOES archive: SDAC Cleaned 1.0 – 8.0 Å 0.5 – 4.0 Å
Flux/(W·m−2)
6×10−6 10−4
X
10−5
M
10−6
C
10−7
B
10−8 00:00
04:00 08:00 12:00 16:00 20:00 Start Time (04-Nov-2003 00:00:00)
4×10−6
A 2×10−6
(Fe,Te) (Fs,Ts)
04:15 04:30 04:45 05:00 Start Time (04-Nov-2003 04:01:51)
05:15
Fig. 1 An example of flare detection. Left panel: the cross signs stand for the spoiled points. The horizontal solid and dashed lines correspond to the flux limits of the two soft X-ray channels at 1 ∼ 8 ˚ A ˚, respectively. The downward arrows mark the peak times of the detected flares. Right: an and 0.5 ∼ 4 A enlarged view of the box in the left panel. The higher fluxes come from the GOES 10 data, and the flare parameters are the arithmetic means of the corresponding parameters obtained from the data of two satellites.
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3.
335
RESULTS
Refer to the frequency analysis on the flare eruptions in the past 33 years (1980-2012) made by Aschwanden et al.[2] , Fig.2 gives the results obtained by using our program of flare identification. Here, similar to the definition of flare classes in the low-energy channel, we have divided the flares of high-energy channel into the a, b, c, m, and x five classes, which correspond to the intervals of peak fluxes of [10−9 , 10−8 ), [10−8 , 10−7 ), [10−7 , 10−6 ), [10−6 , 10−5 ), and [10−5 , ∞) W· m−2 , respectively. It is noticed that this definition of flare classes does not one-by-one correspond to the definition of flare classes in the low-energy channel. This implies that the M-class flare in the low-energy channel does not certainly correspond to the m-class flare in the high-energy channel. Combined with the our understanding on the GOES data as mentioned above, the background noise may be dominant when the peak flux is smaller (≤ 10−8 W· m−2 ), while the detectors may be saturated when the peak flux is larger (≥ 10−4 W· m−2 ), hence we directly use the reliable data in continuous flux intervals to perform the flare identification.
Fig. 2 The monthly flare numbers over the past 33 years. Different colors denote the different flare classes. The left panel corresponds to the low-energy channel, and the right one corresponds to the high-energy channel. The inverse correlation of the monthly flare numbers between the (B, C) class flares and the other flares is caused by the limitation of our flare identification code due to the background effect.
It can be found from Fig.2 that though there is some difference of flare number between the low-energy and high-energy channels, an evident solar cycle of 11 years appears totally in the monthly eruption rates of these two channels. Furthermore, although the flare numbers of the low-energy and high-energy channels are comparable in the interval of highest peak fluxes, the flare number of the low-energy channel for the next two intervals of peak fluxes is more than 2 times that of the high-energy channel. The frequency distributions of flares
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exhibit mostly the power-law distribution[4] , it can be seen from Fig.2 that the powerlaw index in the high-energy channel is smaller than that in the low-energy channel. For instance, in the frequency distribution, the flare number of the low-energy channel varies about 15 times from the interval (M,X) to the interval (X,X10), but the flare number of the high-energy channel varies about 8 times from the interval (m3,x3) to the interval (x3,x30), considered that the selected flux intervals have the same flux span in the two energy channels, hence, the flare numbers of different classes in the low-energy and high-energy channels have certain comparability. With the decreasing peak flux, the frequency distribution of flares deviates from the power-law, which is mainly caused by the limitation of our flare identification code as mentioned above. Because under the effect of background, the lowenergy channel has a higher possibility of missing small flares, the monthly flare numbers of the (B,C) class flares exhibit an inverse correlation with the solar activity. In order to analyze the variation characteristics of flare fluxes around the peak times, we select an interval of flare peak values for the every half order of magnitude, and define the rising time tR and decay time tD as follows: tR = tp − ts ,
(4)
tD = te − tp .
(5)
Fig.3 gives the correlation of these two times. Except the differences of flare numbers of different classes, the correlation seems to be independent of the absolute magnitude of flare peak flux. We also give the linearly fitting results of correlations in this figure. The errors of relevant parameters represent the corresponding statistical errors. The right panels of Fig.3 give the relevant results obtained by analyzing the 10-year observations from 2003 to 2012, and it is found that in comparison with the left panels, the relevant fitting parameters vary greatly with the sample. This also implies that the systematic errors of these parameters are much larger than the related statistical errors. Considered these systematic errors, we believe that the correlation between the two times does not depend on the absolute magnitude of flare peak flux. In addition, we have noticed that the fitting slope is smaller for the flares with smaller peak fluxes, which may be caused by the limitation of our flare identification code.
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Fig. 3 The correlation between the rising and decay times near the peak for the different class flares. The left panel in the first row corresponds to the case of the low-energy channel in a 33-year period (1980∼2012), and the right one corresponds to the case of the low-energy channel in a 10-year period (2003∼2012); the left panel in the second row corresponds to the case of the high-energy channel in the 33-year period, and the right panel in the second row corresponds to the case of the high-energy channel in the 10-year period.
Fig.4 gives the correlations of these flare peak fluxes with the corresponding flare durations (tDURAT ION ), rising times, and decay times, in which tDURAT ION = tR + tD .
(6)
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Fig. 4 The correlations of the flare peak fluxes of different classes with the corresponding durations, rising and decay times. The first row shows the results of the low-energy channel, and the second row shows the results of the high-energy channel.
All these times increase statistically with the increasing peak flux, but the dispersions of these times themselves are rather large, and the degree of dispersion seems to be independent of both the variation amplitude and absolute magnitude of peak flux. In order to analyze the statistical properties of these times more quantitatively, Fig.5 gives the mean values and variances of the logarithms of the rising time, decay time and their ratio for the solar flares of different classes, and the error bars represent the statistical variances of relevant parameters. The frequency distribution for many characteristic parameters of flares is a broad powerlaw or double power-law distribution, thus the directly obtained mean values of relevant parameters can not represent the characters of the system, and it is more reasonable to derive their mean values in the logarithmic space in order to know the variation tendency of these characteristic parameters of the flares. In Fig.5, the logarithm of the ratio between the rising time and decay time does not depend on the flare class (see the third row of the first column), and the mean value of the logarithm of this ratio is distributed around 0.3. This implies that on the average, the decay time is about the two folds of the rising time. The mean values of the logarithms of rising and decay times at the high-energy channel are about 2.1 and 2.4, respectively (see the first and second rows of the first column), and the corresponding variances are around 0.37 and 0.44, respectively (see the first and second rows of the second column), and all these values do not depend on the flare class. The rising and decay times in the low-energy channel are longer than the corresponding times
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in the high-energy channel, while the mean values and variances of the logarithms of these times seem to decrease with the increasing flare peak flux. Especially, the variance of the logarithmic ratios between the decay times and rising times exhibits an evident dependence on the flare class. Considered that this tendency obviously appears in small flares, we believe that this tendency is caused by the limitation of our flare identification code, which does not represent the intrinsic properties of small flares. When the flare peak fluxes are lower, due to the effect of background fluxes and the temporal overlap of flares, the obtained rising and decay times do not definitely correspond to individual flare events, and therefore these times will be more scattered.
Fig. 5 The statistical properties of rise time, decay time, and their ratio for different class flares. The triangles stand for the case of high-energy channel, and the diamonds stand for the case of low-energy channel. For the purpose of illustration, the data points for the high-energy channel have been shifted to the left by 0.1.
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4.
DISCUSSIONS AND CONCLUSIONS
Through developing a group of programs for the flare identification in a given interval of peak fluxes, we have analyzed the 33-year data of soft X-ray fluxes observed by the GOES satellites at two energy channels, and reproduced the 11-year cycle in the frequency distribution of flare eruptions with different classes. These results simultaneously show the different frequency distributions of peak fluxes at the high and low two energy channels, in comparison with the low-energy channel often studied by people, the peak fluxes at the high-energy channel are distributed in a broader range. With further analysis on the statistical features of the rising and decay times around the peak times of flares, we have found that the rising and decay times are correlated very well, and the decay time is averagely about twice the rising time. Though both the rising and decay times will increase with the increasing amplitude of flux variation around the peak time, but the increments are much smaller in comparison with the intrinsic dispersions of these times for a given amplitude of flux variation. In general, the rising and decay times at the high-energy channel are a little shorter than the corresponding times at the low-energy channel, but these times do not vary with the absolute magnitude of flare peak flux. This implies that the variation characteristics of large flares near the peak times are similar to those of small flares. This to a certain extent supports the explanation of the model of self-organized criticality on the statistical properties of flares, but taking the difference in the frequency distributions of flare peak fluxes at the low-energy and high-energy channels into consideration, we have to consider the related radiation mechanisms of soft X-ray emissions and the relevant physical processes in order to obtain a more quantitative explanation on these results. We also have not yet found any evidence for the existence of the impulsive and gradual two kinds of flares. References 1
Li Y. P., Gan W. Q., Feng L., et al., RAA, 2013, 13, 1482
2
Aschwanden M. J., Freeland S. L., ApJ, 2012, 754, 112
3
Veronig A. M., Temmer M., Hanslmeier A., SoPh, 2004, 219, 125
4
Veronig A., Temmer M., Hanslmeier A., et al., A&A, 2002, 382, 1070
5
Temmer M., Veronig A., Hanslmeier A., et al., A&A, 2001, 375, 1049
6
Veronig A., Vrˇsnak B., Dennis B. R., et al., A&A, 2002, 392, 699
7
Aschwanden M. J., SoPh, 2011, 274, 119
8
Wheatland M. S., Litvinenko Y. E., SoPh, 2002, 211, 255
9
Lu E. T., Hamilton R. J., ApJ, 1991, 380, L89
10
Aschwanden M. J., SoPh, 2011, 274, 99
11
Aschwanden M. J., A&A, 2012, 539, A2
12
Han F. R., Liu S. M., 2012, Acta Astronomy Sinica, 53, 482
13
Han F. R., Liu S. M., ChA&A, 2013, 37, 277
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Garcia H. A., SoPh, 1994, 154, 275
15
Ryan D. F., Milligan R. O., Gallagher P. T, et al., ApJS, 2012, 202, 11
Statistical Properties of Soft X-ray Fluxes of Solar Flares† ZHANG Ping1,2,3 1
LIU Si-min1,2
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008
2
Key Laboratory of Dark Matter and Space Astronomy, Chinese Academy of Sciences, Nanjing 210008 3
University of Chinese Academy of Sciences, Beijing 100049
Abstract In order to quantitatively study the statistical properties of the soft X-ray emission in solar flares, an algorithm has been developed to automatically detect flares in a given range of peak fluxes, and to analyze the flares observed by the GOESs (Geostationary Operational Environmental Satellites) from 1980 to 2013 in two soft X-ray bands. This study indicates that the statistical characteristics of the variation of flare soft X-ray flux near the peak time are independent to the magnitude of peak flux: on the average, the rising time of flare’s soft X-ray flux is about half of the decay time, and the rising and decay times in the high-energy channel are shorter than the corresponding times in the low-energy channel, however, all these times will increase with the variation amplitude of flare’s soft X-ray flux. Key words
The Sun: flares—The Sun: X-rays—Methods: statistical 1.
INTRODUCTION
A major uncertainty in the studies of statistical properties of solar flares comes from the identification of flares, especially of small flares[1−5] , meanwhile, considering that the effects of solar activities on the terrestrial environment are closely associated with the magnitudes of solar flares, the studies on the statistical properties of large flares have a greater practical †
Supported by National Natural Science Foundation (11173064, 11233008), and Strategic Priority
Research Program of the Chinese Academy of Sciences (XDB09000000) Received 2014–05–24; revised version 2014–07–17
A translation of Acta Astron. Sin. Vol. 56, No. 1, pp. 35–43, 2015 [email protected], [email protected]
0275-1062/15/$-see front matter © 2015 Elsevier B.V. All rights reserved. doi:10.1016/j.chinastron.2015.07.005
ZHANG Ping and LIU Si-min / Chinese Astronomy and Astrophysics 39 (2015) 330–340
331
importance. In the statistical studies of solar flares, the analysis on the light-variation features of solar flares, which are rather easy to be acquired, and the analysis on the relevant time scales have important significance. The time scales of light variations of solar flares are closely related to the dominant physical mechanisms during the solar eruptions, and they are also the basis for modeling the space weather prediction[6−8] . A series of environmental GOES satellites are operated in the geosynchronous orbits by the National Environmental Satellite, Data, and Information Service belonging to the National Oceanic and Atmospheric Administration of USA, and their onboard X-ray detectors provide the most complete data of soft X-ray fluxes of solar flares up to now. Since the first GOES satellite was launched in 1974, these X-ray detectors have collected the data almost continuously in nearly 40 years, with the time resolution of 2 ∼ 3 s in two soft X-ray A and 0.5 ∼ 4 ˚ A. The flare classification based on the magnitude of wavebands of 1 ∼ 8 ˚ ˚ flare peak flux at the 1 ∼ 8 A band is the generally accepted standard for measuring the magnitudes of flares. And the previous studies on the statistical properties of soft X-ray fluxes of solar flares are mainly based on the flux data at the 1 ∼ 8 ˚ A band. Many parameters describing the flare features exhibit a power-law frequency distribution without a characteristic scale, this is quite similar to the phenomenon of self-organized criticality[7,9−11] . In this case, the larger and smaller events are completely similar to each other in their statistical properties, thus to ensure the power-law frequency distributions of relevant parameters. However, in the detailed multi-band observation and analysis of solar flares, it is found that the physical mechanism to produce large flares seems to be different from the production mechanism of small flares. For instance, the coronal mass ejections are commonly accompanied with large flares, and the two-ribbon flare loop systems also appear in many large flares. Moreover, according to the rising time of X-ray flux in the impulsive phase of flares, they are classified into two types: the gradual and impulsive flares. Further quantitative analysis is needed in order to make sure whether the statistical properties of the soft X-ray emissions in solar flares are related to the peak flux of soft X-ray emission. In this paper, we have developed a group of programs for the flare identification in a given interval of peak fluxes. This method has avoided the traditional dependence of flare identification on the background flux, it can be used to analyze the variation characteristics of large flares (especially near the peak times). Using this algorithm, we have analyzed the flare data observed by the GOES satellites in several ten years at the two soft X-ray wavebands. We have not only reproduced the dependence of the eruption frequency of solar flares with different classes on the solar cycle, but also comparatively studied the behaviors of the solar flares with different classes near the peak times. In Section 2, we summarize the basic characteristics of GOES data, and discuss the algorithm for flare identification, the analyzed results are given in Section 3, and a brief discussion and summary are given in Section 4.
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2.
THE BASIC CHARACTERISTICS OF GOES DATA AND THE CRITERIA FOR FLARE IDENTIFICATIONS
A series of environmental GOES satellites are operated in the geosynchronous orbits by the National Oceanic and Atmospheric Administration of USA, which are mainly used for the weather forecast, the environmental monitoring of terrestrial space, and the meteorological research. The X-ray detectors onboard the GOES satellites have continuously observed the fluxes of solar soft X-ray emission since 1974, and the GOES satellites worked for the different periods are listed in Table 1. In some periods, there are multiple GOES satellites operated simultaneously, and the relevant data can be used to estimate the uncertainties caused by instrumental effects[12−13] . The X-ray detectors mainly monitor the radiations A and 1 ∼8 ˚ A two bands[14] , and according to the from the full solar disk in the 0.5 ∼ 4˚ ˚ band, the flares can be classified into the A, B, magnitude of peak flux at the 1 ∼8 A C, M, and X totally five classes (see the labels on the right vertical axis of the left panel in Fig. 1), which correspond to the intervals of peak fluxes of [10−8 , 10−7 ), [10−7 , 10−6 ), [10−6 , 10−5 ), [10−5 , 10−4 ), and [10−4 , ∞)W · m−2 , respectively. In order to mark the flare class more quantitatively, it is commonly to add the digital parts of peak fluxes to the above characters of flare classes. For example, the M2.5 class represents the flare with the peak flux of 2.5 × 10−5 W · m−2 . ˚ two bands can reach ∼ The observing accuracies in the 0.5 ∼ 4 ˚ A and 1 ∼ 8 A 10−9 W·m−2 and ∼ 10−8 W·m−2 , respectively. When the fluxes are larger than 10−4 W·m−2 , the detectors may be saturated. Due to the effects of saturation, instrumental error, and non-solar signal etc., there are some spoiled points in the GOES data of soft X-ray fluxes, which are marked in the relevant data files, and the occupied proportion is commonly less than 1.0%, as shown by the cross symbols in Fig.1. The satellites are sometimes located in the side far away from the sun, the solar radiation is sheltered by the earth to cause a certain of data missing[2] . The time of data missing is closely related to the satellite orbit, and the occupied proportion in the overall observing time is also about several percent. It is considered that some problems may exist in the earlier data of GOES satellites, hence we only use the GOES data of the low and high two energy channels after 1980 in the following analysis. The data in the low-energy channel of 1.0 ∼ 8.0 ˚ A were mainly used in the previous studies on the statistical properties of flare soft X-ray flux variations, and there were some uncertainties in the determinations of the starting and ending times of flares, owing to the effects of the background radiation and the temporal overlap of adjacent flares, etc.[2,4,15] . It is considered that the solar effects on the solar-terrestrial space become more evident when the fluxes are higher, a kind of program is introduced in the following for the identification of flares in a given interval of peak fluxes. Such a program can be used to study more effectively the statistical properties of large flares and the variation characteristics of flare emissions near the peak times. (1) At first, we use the GOES software in the SSWIDL
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Table 1 The service periods and time resolutions of the soft X-ray detectors onboard the GOES satellites GOES-N
Service Period
Time Resolution/s
GOES 91
1-Jul-1974 to 17-Oct-1975
3
GOES 92
5-Feb-1975 to 31-Mar-1978
3
GOES 1
17-Jan-1976 to 1-Jun-1978
3
GOES 2
1-Aug-1977 to 30-Apr-1983
3
GOES 3
5-Jul-1978 to 4-Jan-1980
3
GOES 5
1 to 30-May-1983, 1-Jul-1983 to 31-Jul-1984
3
GOES 6
2 to 30-Jun-1983, 1-Aug-1984 to 18-Aug-1994
3
GOES 7
1-Jan-1994 to 3-Aug-1996
3
GOES 8
21-Mar-1996 to 18-Jun-2003
3
GOES 9
20-Mar-1996 to 24-Jul-1998
3
GOES 10
10-Jul-1998 to 1-Dec-2009
3
GOES 11
21-Jun-2006 to 11-Feb-2008
3
GOES 12
13-Dec-2002 to 8-May-2007
3
GOES 14
2-Dec-2009 to 4-Nov-2010
2
GOES 15
1-Sep-2010 to Present
2
GOES 13
package to remove the spoiled points in the GOES data, and remove the time intervals with data missing. (2) For the given interval [FL , FH ] of fluxes, we determine all the data points with the observed fluxes located in this interval. (3) Considered the removal of spoiled points and the statistical fluctuations of data themselves, we assume that two adjacent effective data points with a time interval less than 30 s are continuous. (4) In order to ensure the significance of flare signals, we only use the time slices with a duration larger than 30 s. (5) Finally, for these time slices, we determine the fluxes Fp , Fs , and Fe corresponding to the peak time (tp ), starting time (ts ), and ending time (te ), respectively. For the time slices with a duration larger than 30 s and the peak fluxes located in the interval of [FL , FH ], the corresponding Fp , Fs , and Fe should satisfy the following relations: Fs − FL ≤ 10%, FL
(1)
Fe − FL ≤ 10%, FL
(2)
Fp − FL ≤ 10%, FL
(3)
here, 10% corresponds to the statistical fluctuations of data. Eq.(1) requires that the flux at the starting time can not exceed the 10% of the lower limit of the interval of peak fluxes,
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which can effectively exclude the time slices corresponding to the decay-phase flares with the peak fluxes larger than the upper limit of the given interval of peak fluxes. Similarly, Eq.(2) can exclude the time slices corresponding to the rise-phase flares with the peak fluxes larger than the upper limit of the given interval of peak fluxes. Eq.(3) requires the flare fluxes have a remarkable enhancement. Fig.1 gives an example to apply this program to the observed data. Though it is not necessary to determine and deduct the background flux for each flare in this method, but when the background flux is higher, the temporal overlap of flares will be serious. In Fig.1 a series of small flares in the low-energy channel are not identified after 13:00, because they are considered as the rising phase of a subsequent larger flare. In principle, the effect of this shortage on the results can be reduced by adjusting the range of flux interval, but when the complexity of flare behaviors is considered, a more complicated program of flare identification is needed to completely overcome this shortage. The right panel of Fig.1 also shows that there is a certain difference between the fluxes observed by two different GOES satellites. When a flare is observed by multiple satellites simultaneously, we can use the arithmetic mean value of the flare parameters observed by the different satellites as the corresponding flare parameter. Flux GOES12 3 sec 10−2 10−3
Flux GOES 10/12 3 sec
8×10−6
(Fp,Tp)
GOES archive: SDAC Cleaned 1.0 – 8.0 Å 0.5 – 4.0 Å
Flux/(W·m−2)
6×10−6 10−4
X
10−5
M
10−6
C
10−7
B
10−8 00:00
04:00 08:00 12:00 16:00 20:00 Start Time (04-Nov-2003 00:00:00)
4×10−6
A 2×10−6
(Fe,Te) (Fs,Ts)
04:15 04:30 04:45 05:00 Start Time (04-Nov-2003 04:01:51)
05:15
Fig. 1 An example of flare detection. Left panel: the cross signs stand for the spoiled points. The horizontal solid and dashed lines correspond to the flux limits of the two soft X-ray channels at 1 ∼ 8 ˚ A ˚, respectively. The downward arrows mark the peak times of the detected flares. Right: an and 0.5 ∼ 4 A enlarged view of the box in the left panel. The higher fluxes come from the GOES 10 data, and the flare parameters are the arithmetic means of the corresponding parameters obtained from the data of two satellites.
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RESULTS
Refer to the frequency analysis on the flare eruptions in the past 33 years (1980-2012) made by Aschwanden et al.[2] , Fig.2 gives the results obtained by using our program of flare identification. Here, similar to the definition of flare classes in the low-energy channel, we have divided the flares of high-energy channel into the a, b, c, m, and x five classes, which correspond to the intervals of peak fluxes of [10−9 , 10−8 ), [10−8 , 10−7 ), [10−7 , 10−6 ), [10−6 , 10−5 ), and [10−5 , ∞) W· m−2 , respectively. It is noticed that this definition of flare classes does not one-by-one correspond to the definition of flare classes in the low-energy channel. This implies that the M-class flare in the low-energy channel does not certainly correspond to the m-class flare in the high-energy channel. Combined with the our understanding on the GOES data as mentioned above, the background noise may be dominant when the peak flux is smaller (≤ 10−8 W· m−2 ), while the detectors may be saturated when the peak flux is larger (≥ 10−4 W· m−2 ), hence we directly use the reliable data in continuous flux intervals to perform the flare identification.
Fig. 2 The monthly flare numbers over the past 33 years. Different colors denote the different flare classes. The left panel corresponds to the low-energy channel, and the right one corresponds to the high-energy channel. The inverse correlation of the monthly flare numbers between the (B, C) class flares and the other flares is caused by the limitation of our flare identification code due to the background effect.
It can be found from Fig.2 that though there is some difference of flare number between the low-energy and high-energy channels, an evident solar cycle of 11 years appears totally in the monthly eruption rates of these two channels. Furthermore, although the flare numbers of the low-energy and high-energy channels are comparable in the interval of highest peak fluxes, the flare number of the low-energy channel for the next two intervals of peak fluxes is more than 2 times that of the high-energy channel. The frequency distributions of flares
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exhibit mostly the power-law distribution[4] , it can be seen from Fig.2 that the powerlaw index in the high-energy channel is smaller than that in the low-energy channel. For instance, in the frequency distribution, the flare number of the low-energy channel varies about 15 times from the interval (M,X) to the interval (X,X10), but the flare number of the high-energy channel varies about 8 times from the interval (m3,x3) to the interval (x3,x30), considered that the selected flux intervals have the same flux span in the two energy channels, hence, the flare numbers of different classes in the low-energy and high-energy channels have certain comparability. With the decreasing peak flux, the frequency distribution of flares deviates from the power-law, which is mainly caused by the limitation of our flare identification code as mentioned above. Because under the effect of background, the lowenergy channel has a higher possibility of missing small flares, the monthly flare numbers of the (B,C) class flares exhibit an inverse correlation with the solar activity. In order to analyze the variation characteristics of flare fluxes around the peak times, we select an interval of flare peak values for the every half order of magnitude, and define the rising time tR and decay time tD as follows: tR = tp − ts ,
(4)
tD = te − tp .
(5)
Fig.3 gives the correlation of these two times. Except the differences of flare numbers of different classes, the correlation seems to be independent of the absolute magnitude of flare peak flux. We also give the linearly fitting results of correlations in this figure. The errors of relevant parameters represent the corresponding statistical errors. The right panels of Fig.3 give the relevant results obtained by analyzing the 10-year observations from 2003 to 2012, and it is found that in comparison with the left panels, the relevant fitting parameters vary greatly with the sample. This also implies that the systematic errors of these parameters are much larger than the related statistical errors. Considered these systematic errors, we believe that the correlation between the two times does not depend on the absolute magnitude of flare peak flux. In addition, we have noticed that the fitting slope is smaller for the flares with smaller peak fluxes, which may be caused by the limitation of our flare identification code.
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Fig. 3 The correlation between the rising and decay times near the peak for the different class flares. The left panel in the first row corresponds to the case of the low-energy channel in a 33-year period (1980∼2012), and the right one corresponds to the case of the low-energy channel in a 10-year period (2003∼2012); the left panel in the second row corresponds to the case of the high-energy channel in the 33-year period, and the right panel in the second row corresponds to the case of the high-energy channel in the 10-year period.
Fig.4 gives the correlations of these flare peak fluxes with the corresponding flare durations (tDURAT ION ), rising times, and decay times, in which tDURAT ION = tR + tD .
(6)
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Fig. 4 The correlations of the flare peak fluxes of different classes with the corresponding durations, rising and decay times. The first row shows the results of the low-energy channel, and the second row shows the results of the high-energy channel.
All these times increase statistically with the increasing peak flux, but the dispersions of these times themselves are rather large, and the degree of dispersion seems to be independent of both the variation amplitude and absolute magnitude of peak flux. In order to analyze the statistical properties of these times more quantitatively, Fig.5 gives the mean values and variances of the logarithms of the rising time, decay time and their ratio for the solar flares of different classes, and the error bars represent the statistical variances of relevant parameters. The frequency distribution for many characteristic parameters of flares is a broad powerlaw or double power-law distribution, thus the directly obtained mean values of relevant parameters can not represent the characters of the system, and it is more reasonable to derive their mean values in the logarithmic space in order to know the variation tendency of these characteristic parameters of the flares. In Fig.5, the logarithm of the ratio between the rising time and decay time does not depend on the flare class (see the third row of the first column), and the mean value of the logarithm of this ratio is distributed around 0.3. This implies that on the average, the decay time is about the two folds of the rising time. The mean values of the logarithms of rising and decay times at the high-energy channel are about 2.1 and 2.4, respectively (see the first and second rows of the first column), and the corresponding variances are around 0.37 and 0.44, respectively (see the first and second rows of the second column), and all these values do not depend on the flare class. The rising and decay times in the low-energy channel are longer than the corresponding times
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in the high-energy channel, while the mean values and variances of the logarithms of these times seem to decrease with the increasing flare peak flux. Especially, the variance of the logarithmic ratios between the decay times and rising times exhibits an evident dependence on the flare class. Considered that this tendency obviously appears in small flares, we believe that this tendency is caused by the limitation of our flare identification code, which does not represent the intrinsic properties of small flares. When the flare peak fluxes are lower, due to the effect of background fluxes and the temporal overlap of flares, the obtained rising and decay times do not definitely correspond to individual flare events, and therefore these times will be more scattered.
Fig. 5 The statistical properties of rise time, decay time, and their ratio for different class flares. The triangles stand for the case of high-energy channel, and the diamonds stand for the case of low-energy channel. For the purpose of illustration, the data points for the high-energy channel have been shifted to the left by 0.1.
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4.
DISCUSSIONS AND CONCLUSIONS
Through developing a group of programs for the flare identification in a given interval of peak fluxes, we have analyzed the 33-year data of soft X-ray fluxes observed by the GOES satellites at two energy channels, and reproduced the 11-year cycle in the frequency distribution of flare eruptions with different classes. These results simultaneously show the different frequency distributions of peak fluxes at the high and low two energy channels, in comparison with the low-energy channel often studied by people, the peak fluxes at the high-energy channel are distributed in a broader range. With further analysis on the statistical features of the rising and decay times around the peak times of flares, we have found that the rising and decay times are correlated very well, and the decay time is averagely about twice the rising time. Though both the rising and decay times will increase with the increasing amplitude of flux variation around the peak time, but the increments are much smaller in comparison with the intrinsic dispersions of these times for a given amplitude of flux variation. In general, the rising and decay times at the high-energy channel are a little shorter than the corresponding times at the low-energy channel, but these times do not vary with the absolute magnitude of flare peak flux. This implies that the variation characteristics of large flares near the peak times are similar to those of small flares. This to a certain extent supports the explanation of the model of self-organized criticality on the statistical properties of flares, but taking the difference in the frequency distributions of flare peak fluxes at the low-energy and high-energy channels into consideration, we have to consider the related radiation mechanisms of soft X-ray emissions and the relevant physical processes in order to obtain a more quantitative explanation on these results. We also have not yet found any evidence for the existence of the impulsive and gradual two kinds of flares. References 1
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