STATISTICS OF FLUCTUATIONS IN THE SOLAR SOFT X-RAY EMISSION AND

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1 THE ASTROPHYSICAL JOURNAL, 484:920È926, 1997 August 1 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. STATISTICS OF FLUCTUATIONS IN THE SOLAR SOFT X-RAY EMISSION S. UENO1 AND S. MINESHIGE Department of Astronomy, Faculty of Science, Kyoto University, Sakyo-ku, Kyoto , Japan H. NEGORO Institute of Space and Astronautical Science, Yoshinodai, Sagamihara, Kanagawa 226, Japan K. SHIBATA Solar Physics Division, National Astronomical Observatory, Mitaka 181, Japan AND H. S. HUDSON Solar Physics Research Corporation, Institute of Space and Astronautical Science, Yoshinodai, Sagamihara, Kanagawa 226, Japan Received 1996 September 23; accepted 1997 February 28 ABSTRACT X-ray emission from the Sun Ñuctuates as a result of occasional Ñare events. We have calculated the power spectra of the solar soft X-ray variations using the photometric data of the GOES 6 satellite. The data cover the period 1991 September to 1994 April, about 32 months in total, and we have worked with 10 minute averages. We Ðnd that the total power spectral densities (PSDs) have three distinct components; a Ñat or slightly decreasing low-frequency section, a 1/f-like moderate decline mediumfrequency section, and steep decline high-frequency parts. The break frequencies separating three parts are f ^ 10~4.7 Hz and 10~3.8 Hz, respectively. Such downward breaks are expected from the shot- break noise (Ñarelike) character of solar X-ray emission, and we interpret the lower break frequency as indicating an upper limit on Ñare timescales. These break frequencies do not vary appreciably with activity level. This suggests the existence of a universal mechanism for triggering Ñares in the solar corona. Moreover, the power-law index (where we assume PSD P f ~b, f being frequency) of each part does not vary appreciably with the level of activity either; its average is b ^ 0.45, 0.95, and 1.5, respectively. The overall shape of the PSD is quite similar to those of other astrophysical objects such as black hole candidate stars and active galactic nuclei, albeit on a vastly di erent scale. Subject headings: Sun: activity È Sun: Ñares È Sun: X-rays, gamma rays 1. INTRODUCTION It is widely believed that solar Ñares are caused by a release of magnetic energy stored in the solar coronal Ðelds. If so, we may deðne basic time and spatial scales, such as the Alfve n transit time over a pressure scale height. One might then expect that the pattern of Ñare occurrence would reñect these typical scales. Interestingly, however, this is not the case: total energy outputs and peak intensities of Ñares are distributed rather smoothly. A power-law dependence of solar Ñares on peak intensity has been established through a number of observations (Drake 1971; Datlowe, Elcan, & Hudson 1974; Lin et al. 1984; Dennis 1985). The essence is that, although bigger Ñares occur less frequently than smaller Ñares, there is basically no preferential energy. We speculate that the occurrence of Ñares may not be the result of simple local physics but is somehow controlled globally. To investigate this possibility, thorough statistical studies of solar Ñares are indispensable. Many studies concerning the statistics of solar Ñares have been made so far, but most of them have treated solar Ñares as individual, independent events. To derive useful information, such as periodicities and noise properties, the technique of power spectral densities (PSDs) is often used. In fact, there are some studies on long-term periodicities of solar Ñare occurrence (Rieger et al. 1984; Kile & Cliver 1991; Bai & Sturrock 1991, 1993). However, there have been few attempts to investigate high-frequency noise properties of these patterns. In the present study, we calcu- 1 ueno=kwasan.kyoto-u.ac.jp. late the PSD, using GOES 6 data, and discuss the statistics of solar Ñares from viewpoints that di er from previous ones. We shall describe the GOES 6 data in 2. The overall light curves and the PSD will be presented in 3. Solar activity signiðcantly decreased during this period. To see how the statistical nature changes with time, we divide the entire data set into 16 subsets, each with a duration of 2 months, and study the light curves and power spectra of each individual subset. The variation of each spectral parameter will also be displayed in 3. In 4 we focus on the long-timescale periodicity of Ñare occurrence. The Ðnal section is devoted to summary and discussion on the statistics of solar Ñares and its relation to X-ray Ñuctuation from other objects, such as black hole systems. 2. DATA We use the soft X-ray light curves from 1991 September to 1994 April obtained with GOES 6. The overall light curves are displayed in Figure 1. Clearly, the corona was rather active, producing large X-ray variations due to major Ñares only until the Ðrst half of 1993, and the number of large peaks decreased rapidly afterward. The GOES 6 database contains soft X-ray Ñuxes (about 2È10 kev) sampled at 3 s intervals. Because of range changes in the logarithmic readout, the noise properties of a GOES light curve are complicated. In the concrete, the light curve with full time resolution shows steplike proðles. To minimize this problem we have arbitrarily smoothed the data into 10 minute bins, a total of 133,622 samples in the 32 month observation period. 920

2 FLUCTUATIONS IN SOLAR SOFT X-RAY EMISSION 921 FIG. 1a FIG. 1.ÈAll of the soft X-ray light curves from the whole Sun obtained with GOES 6. The vertical axes are (a) on the linear scale and (b) on the logarithmic scale. FIG. 1b 3. POWER SPECTRUM OF SOLAR SOFT X-RAY VARIATIONS 3.1. PSDs of Entire L ight Curves We calculate the PSD in the following way. At Ðrst, we transform the time of each data point, t ( j \ 1ÈJ), into the j corresponding phase, /, for a given frequency, f (i \ 1È i,j i N). Next, we plot the normalized X-ray Ñux, L (t ) ( j \ 1ÈJ), j against x \ sin (/ ) and Ðt these J data points by a straight i,j line: L \ a x ] b. We also plot L (t ) against y \ cos (/ ) i i j i,j and Ðt the data again by a straight line: L \ c y ] d. i i Finally, we sum the two squares of these declines, a2]c2, i i which corresponds to the PSD (except for a proportionality constant) at a frequency f. i The PSD of the entire time series is depicted in Figure 2. Here power spectra are averaged in frequency over equal intervals on the logarithmic scale; that is, we averaged over the intervals from f to f (i \ 1toN[ 1), where f \ f i i`1 i`1 i ] \ 1.12f (since ^ 1.12). The peaks at low fre- i quencies reñect structure in solar Ñare occurrence on the long timescale. At high frequencies, larger than D10~5 Hz, we can see a steepening of the spectrum to a 1/f-like decline. At Ðrst, we focus on the noise features at high frequencies. In order to increase the signal-to-noise ratio, we Ðrst calculated PSDs for each of 16 subsets with 2 month durations (see Table 1 for the deðnition of subsets). We then averaged each PSD with frequency bins from f to f, where f \ f i i`1 i`1 i ] , and Ðnally averaged the 16 PSDs. Figure 3 shows the averaged PSDs obtained in this way. Here the error bars show p/n1@2, where p is the standard deviation of 16 power spectra of each data subset and n is the number of data subsets, i.e., n \ 16. We Ðnd that the solar PSDs appear to consist of two and possibly three distinct spectral domains. Break frequencies separating the three distinct parts with di erent slopes are found at around f ^ 10~4.7 Hz and f ^ 10~3.8 Hz, respectively. These Ðxed frequencies were then used as limits of each domain for Ðts to powerlaw functions proportional to f~b, with b being a Ðtting parameter. The results of Ðtting are summarized in Table 2. Errors represent deviations of each PSD from the best-ðt curve. It is interesting that these power-law indices are similar to those of a representative black hole candidate, Cyg X-1 (cf. Negoro et al. 1995; see inset in Fig. 3 and see Table 2). What physical processes do these frequencies imply? Generally, if a single-ñare light curve can be well represent- TABLE 1 CONTENTS OF SUBSET DATA Subset Start Time End Time Number (108 s) (108 s) Year and Month FIG. 2.ÈPower spectral density of the whole time series of solar soft X-rays depicted in Fig. 1. Power density is binned in the frequency direction over the intervals from f to f, where f \ f ] \ 1.12f. i i`1 i`1 i i Sep, Oct Nov, Dec Jan, Feb Mar, Apr May, Jun Jul, Aug Sep, Oct Nov, Dec Jan, Feb Mar, Apr May, Jun Jul, Aug Sep, Oct Nov, Dec Jan, Feb Mar, Apr

3 922 UENO ET AL. Vol. 484 FIG. 3.ÈPower spectral density for the 32 month data, which are averaged over time and frequency. The power spectrum of Cyg X-1 obtained bynegoro (1992) is also shown in the inset. ed by an exponential function of time [P exp ([t/q), q being a constant], its power spectrum has a Ñat-top proðle with a steep decline (Pf ~2) at high frequencies (Lorentzian proðle): (2nqf )2 S ( f ) P q 1 ] (2nqf )2. (1) In this case, the break frequency corresponds to f \ (2nq)~1. To see the range of typical decay time q, we tried to Ðt each Ñare light curve with an exponential function for 10 big Ñares and 20 small Ñares present in the GOES data. Two such examples are displayed in Figure 4. One has TABLE 2 POWER-LAW INDEX OF TOTAL POWER SPECTRUM Frequency Range Object (Hz) Power-Law Index b Sun... \10~ ^ ~4.7 to 10~ ^ 0.03 [10~ ^ 0.02 Cyg X-1... \10~1 D0.0 10~1 to 100 D0.9 [100 D1.5 the longest decay time with q D 8300 s, whereas another long has the shortest, q D 550 s, among the Ðtted data. Note, short however, that any Ñuctuations with even shorter duration (q\550 s) have been suppressed by Ðnite time resolution of our rebinning of the data (10 minutes). The corresponding break frequencies are f ^ 10~4.7 and f ^ 10~3.5, respectively. These values long agree well with the short two break frequencies seen in Figure 3. The departure from the Lorentzian 1/f 2 fallo to high frequencies above 10~4.7 Hz could therefore be plausibly explained by a broad distribution of Ñare decay times (Drake 1971). This means that we may be able to reproduce Figure 3 by superposing S ( f ) with 550 \q\8300. q In order to conðrm this prediction, we performed PSD analyses of simulated Ñare light curves for the following two cases. In the Ðrst case, we adopted 20,000 steps for the total observed time, and 5 and 70 steps for the shortest and longest durations of Ñares, respectively. In the second case, we again adopted 20,000 steps for the total time, but 2 and 20 steps for the shortest and longest ones, respectively. In both simulations, the peak intensity distribution of Ñares was set to be N P P~1.8, which is consistent with previous observational studies. Time intervals between neighboring Ñares were simply assumed to be random, and the mean

4 No. 2, 1997 FLUCTUATIONS IN SOLAR SOFT X-RAY EMISSION 923 FIG. 4.ÈSoft X-ray time proðles of the solar Ñares (white squares) and the best-ðt curve (solid line) for (a) the longest duration Ñare (1992 May 3; 22 hr 48 minutes) and (b) the shortest duration Ñare (1992 May 28; 13 hr 26 minutes) among the 32 month data. interval equals 10 and 9.1 steps in the two cases, respectively. Moreover, Ñare durations were also taken to be random between the shortest and longest durations. We depict these two simulated light curves in Figure 5 and smoothed power spectral densities in Figure 6. Apparently, the simulated light curves and PSDs are very similar to those observed. Quantitatively, we can predict that f \ short 10~1.50 Hz and f \ 10~2.64 Hz in the Ðrst case, and long f \ 10~1.10 Hz and f \ 10~2.10 Hz in the second short long case, by estimation from the shortest and the longest durations, respectively. In fact, these values are found in simulated PSDs (Fig. 6), with good agreement. We thus conðrmed our interpretation of the shape of PSDs to be correct. The values of the upper break frequency, near 10~3.8 Hz, are too close to the Nyquist frequency of our binning to be signiðcant in this study. More quantitative discussion will be given in our next paper. Finally, we also examine periodicities in the occurrence of Ñares. In the PSDs we could Ðnd the apparent periodicities of 2.6, 17.7, 18.4, 24.8, 26.4, 30.5, 46.3, 51.8, 61.3, 105, and 123 days. Bai & Sturrock (1991) also reported the periodicities 25.5, 51.0, 76.5, 102.0, 127.5, and days before solar cycle 21. We have conðrmed the periodicities of D26, 51, 102, and 125 days, but none with 76.5 and 154 days. Instead, we found the strong periodicity of 61.3 days. These results are, however, very preliminary, and we need more detailed analysis to conðrm our Ðndings in a future work L ong-term Changes of PSDs In the previous subsection, we were concerned with the PSDs for the entire light curve. It is still open to question whether the power-law indices (or the break frequencies) of the PSDs remain the same all the time regardless of the di erent magnitudes of Ñare activity. We thus next investigate the individual PSDs for each subset, Ðnding a variety of PSDs with di erent shapes and magnitudes (see Table 3 and Fig. 5). To summarize, most PSDs show the three distinct components. If we denote the power-law indices of the components from lower to higher frequencies as b, b, and b, we usually Ðnd b \b \b ; that is, the PSD is composed TABLE 3 TIME VARIATION OF POWER-LAW INDEX FIG. 5.ÈSimulated light curves of Ñares. (a) Durations of Ñares are adopted as 5 and 70 steps, respectively, for the shortest and the longest ones. (b) Durations are set as 2 and 20 steps, respectively, for the shortest and the longest ones. These durations are taken to be at random between the shortest and longest durations. Time intervals between neighboring Ñares are also assumed to be random. On the other hand, the peak intensity distribution of Ñares was set to be N P P~1.8. Subset Number b 0 b 1 log ( f ) break (Hz) b 2 log ( f ) break (Hz) b [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [0.13 [ [ [ [ [ [ [ [

5 924 UENO ET AL. Vol. 484 FIG. 7.ÈTime variations in the statistical properties of solar Ñares during the 32 month data. (a) Variation of the power-law index at medium frequency. (b) Variation of the two break frequencies. The lower line (with lower frequencies) may show the longest duration Ñare and the upper line (with higher frequencies) may show the shortest duration Ñares. (c) Variation of the solar activity. Values on the vertical axis show the soft X-ray Ñux averaged over 2 months. FIG. 6.ÈPower spectral density diagrams of simulated light curves of Ñares (Fig. 5). (a) PSD for the light curve with longer durations. (b) PSD for the light curve with shorter durations. The di erences of break frequencies can be found. of a Ñat part with b D 0, a moderate decline part with b D f ~1, and a steep 1 decline part with b D f~1.5 to f~2 from 2 low to high frequencies, although these 3 indices change from time to time (Table 3). There are not a few PSDs having a 1/f-like decline. Note again that the presence of the steep decline part may be a ected by the Ðnite time resolution. There are some exceptions: In the seventh subset, the relation among the power-law indices is b \b \b. In the 16th subset, we Ðnd b \b. In the 1 13th 3 subset, 2 the PSD appears to have four 2 components. 1^b 3 A steep decline also appears at low frequencies (b \b \b \b ) On the other hand, the break frequencies did not change greatly during the 32 months, although the mean Ñuctua- tion amplitudes decreased appreciably. The variations in the shapes of power spectra and the break frequencies are rather independent of the solar activity. In Figure 7 we depict the time variations of power-law indices, break frequencies, and Ñare activity. The independence of the spectral parameters with time shows that the pattern of Ñare occurrence does not depend on the level of activity. This may suggest the existence of a universal mechanism for triggering Ñares in the solar corona. 4. SUMMARY AND DISCUSSION 4.1. Summary of Solar X-Ray Fluctuation Soft X-ray Ñuxes from the Sun Ñuctuate because large or small Ñare events occur occasionally. By constructing power spectra, we have found the following statistical features of Ñare occurrence. A 1/f-like Ñuctuation appears at medium-frequency regimes ( f \ 10~4.7 to 10~3.8 Hz) in the PSD. The break frequencies, which seem to correspond to inverses of the maximum and minimum duration of Ñares, are independent of the solar activity and do not change much with time. This 1/f-like feature is quite reminiscent of X-ray variability from other astronomical objects, such as Cyg X-1 (black hole candidate) and the nuclei of active galaxies. We have also found that although the normalization of peak intensities changes, the overall Ñuctuation properties (such as the

6 No. 2, 1997 FLUCTUATIONS IN SOLAR SOFT X-RAY EMISSION 925 shape of the PSD) are more or less insensitive to the solar activity. From these features, we may conclude that the mechanism responsible for individual solar Ñares is insensitive to the level of solar activity. Maybe the di erent magnitudes of solar activity can be explained simply by the variation of the strength of helicity added to magnetic Ñux tubes in the solar convection zone. To connect this power-law distribution of solar Ñares to the physics of individual Ñares, Lu & Hamilton (1991) and Lu et al. (1993) propose that the solar coronal magnetic Ðeld is in a self-organized critical state (SOC state), a critical state realized in a nonequilibrium open system (Bak, Tang, & Wiesenfeld 1988). The most outstanding feature of their model is that the system, following cellular automaton rules, spontaneously evolves to and steadily stays at an SOC state. In this state energy stored in magnetic Ðeld would be barely below its critical value, over which an energy-release event begins and leads to a Ñare. The occurrence rate then shows a power-law dependence on size and lifetime of the event. We can expect 1/f-like Ñuctuations on the energy dissipation in such a case, although the energy is injected randomly in time and space. If this is the case, solar Ñares are like avalanches consisting of many small energy-release events, such as magnetic reconnections. To summarize what we have learned through the present studies, the SOC model can naturally explain some of the observed statistical features, since the results are basically insensitive to the parameters of the physical system. If this model is usable, all Ñarelike events, including nanoñares and microñares, can be understood in term of a common physical process. Although the present paper only presents the results of the PSD analyses, complementary information may be derived from the distributions of peak intensities of individual Ñares and intervals of adjacent Ñares. It might be noted that the previous studies have shown that the distributions of peak intensities of Ñares can be well represented by power laws with indices c \ 1.8È1.9 (cf. Kakinuma, Yamashita, & Enome 1969; Drake 1971; Datlowe et al. 1974; Dennis 1985, 1988; Shimizu 1995). These distributions may contain important physical information as follows: If this index is less than 2.0, some additional heating mechanisms, such as nanoñares (Parker 1988), are required to account for solar coronal heating (Hudson 1991), while otherwise coronal heating can be explained solely by various sizes of Ñares. Moreover, the distributions of intervals between adjacent peaks can be used to check whether the occurrence of Ñares is random or whether accumulation e ects may exist. Detailed analyses using the GOES data will be presented in our next paper Relationship to X-Ray Fluctuations from Black Hole Objects It is also interesting to discuss the relationship between solar coronae and accretion disk coronae (cf. Galeev, Rosner, & Vaiana 1979; Takahara 1979). As mentioned already, 1/f-like Ñuctuations are rather generally observed in X-ray radiation from stellar black hole candidates like Cyg X-1, neutron star X-ray binaries, and active galactic nuclei (see reviews by Makishima 1988 and Pounds & McHardy 1988; for more recent analysis see Negoro et al and references therein). The X-rayÈemitting regions of these objects seem to be optically thin accretion disks dominated by magnetic pressure, where magnetic Ñares similar to solar Ñares occasionally occur, giving rise to substantial Ñuctuations in X-ray emission (see Shibata, Tajima, & Matsumoto 1990; Mineshige, Kusunose, & Matsumoto 1995, and references therein). It is thus of great importance to study the statistics of solar X-ray Ñuctuations for considering possible physical links between solar coronae and hot accretion disks. In particular, from PSDs the statistical features of solar and nonsolar X-ray Ñuctuations are rather similar to each other. However, the absolute X-ray Ñuxes from accretion disks are larger by many orders of magnitude than those from the solar corona, and the shapes of the Ñare light curves are also rather di erent from those of the X-ray shots in nonsolar sources. In the latter, mass accretion is another important factor in determining the shape of X-ray shot light curves. Manmoto et al. (1996), in fact, have recently succeeded in reproducing X-ray shot proðles by timedependent simulations of the accretion disk. When we consider these facts, it seems plausible that the mechanisms of creating solar Ñares and X-rayÈemitting shots in accretion disks could be related (such as in an SOC model including magnetic reconnection; see, e.g., Takeuchi, Mineshige, & Negoro 1995). In the solar interior, convection gives energy to the magnetic Ðeld. In accretion disks, in contrast, accreting gas Ðrst acquires energy from gravitational potential well. The disk gas can then give energy to the magnetic Ðeld through ampliðcation by the shearing motion of the gas and MHD instabilities associated with rotating disks. When the magnetic energy becomes comparable to internal energy of the disk gas, the Ðeld will eventually dissipate its energy via reconnection. The di erence in shot characteristics, such as intensities, shot proðles, and interval distributions, between the solar corona and an accretion disk source may arise from the di erent physics involved. For accretion disks we mainly observe the accretion process of X-rayÈemitting blobs (powered by gravitational potential energy release due probably to magnetic reconnection), and such a source of energy is not available in the solar case. Furthermore, the solar soft X-ray radiation we analyze here does not represent the fundamental (direct) energy release in a solar Ñare; it might be more appropriate to analyze the shot-noise properties of solar hard X-ray emission as a proxy for the fundamental energyrelease mechanism. Finally, we need to o er an explanation of the dimensional constant we appear to have discovered, namely, the lower break frequency 10~4.7 Hz. It exceeds the Alfve n transit time of any basic Ñare structure, but can be related to typical radiative cooling times for Ñares limited in size scale to about 1 supergranule diameter (about 50,000 km). It is possible that this rough agreement suggests that the characteristic time is basically hydrodynamic in origin and does not reñect the energy-release timescale itself. We are grateful to J. Kubota and T. Kato for useful comments and advice for analysis. This work was supported in part by grants-in-aid from the Ministry of Education, Science, Sports, and Culture of Japan, , (S. M.), and by NASA, NAS (H. S. H ).

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