{'fraction_non_alphanumeric': 0.055526260282640794, 'fraction_numerical': 0.03986500738240877, 'mean_word_length': 4.049254526091587, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The relationships of solar flares with both sunspot and geomagnetic activity

The relationships between solar flare parameters (total importance, time duration, flare index, and flux) and sunspot activity (Rz) as well as those between geomagnetic activity (aa index) and the flare parameters can be well described by an integral response model with the response time scales of about eight and thirteen months, respectively. Compared with linear relationships, the correlation coefficients of the flare parameters with Rz, of aa with the flare parameters, and of aa with Rz based on this model have increased about 6%, 17%, and 47% on average, respectively. The time delays of the flare parameters to Rz, of aa to the flare parameters, and of aa to Rz at their peaks in solar cycle can be predicted in part by this model (82%, 47%, and 78%, respectively). These results may be further improved when using a cosine filter with a wider window. It implies that solar flares are related to the accumulation of solar magnetic energies in the past through a time decay factor. The above results may help to understand the mechanism of the solar cycle and to improve the solar flare prediction.


Introduction
Solar flares are powerful eruptions of solar activity (Özgüç & Atac 1989;Mikic & Linker 1994;Jain et al. 2010;Fang 2011) occurring on time scales of minutes up to a few hours (Chandra et al. 2011) and may produce a series of solar-terrestrial effects, which may be hazardous to both spacecraft and astronauts. Understanding the mechanism of solar flares and forecasting them are important for both solar physics and geophysics. Several mechanisms have been proposed to explain the eruptions of solar flares, such as the photospheric converging and shear motions (Mikic & Linker 1994), flux emergence and cancellation (Gan et al. 1993;Zhang et al. 2001), catastrophe model (Forbes 1990), and Kink instability of coronal flux ropes (Sakurai 1976;Li & Gan 2011). The magnetic reconnection plays an important role in triggering solar flares (Lin et al. 1995;Wheatland & Litvinenko 2001;Forbes et al. 2006;Fang 2010).
To quantify the daily flare activity over 24 hours per day, Kleczek (1952) introduced the 'flare index' defined as where 'i' represents the intensity scale of importance and 't' the duration (in minutes) of the flare (Knoska & Petrasek 1984;Atac &Özgüç 1998). This relationship is assumed to give roughly the total energy emitted by a flare (Kleczek 1952). The solar flare activity is found to be closely correlated with sunspots (Özgüç & Atac 1989;Feminella & Storini 1997). Larger flares appear often near larger and more complex active regions (McIntosh 1990;Bachmann & White 1994;Norquist 2011). Sunspot activity is a striking manifestation of magnetic fields on the Sun, associated with the main sites of solar-activity phenomena (Moradi et al. 2010) and related to the energy supplied into the corona (de Toma et al. 2000;Temmer et al. 2003). Studying the relation-ship between solar flares and sunspot activity is useful to understand and predict the former. The flare frequency of occurrence is often predicted by sunspot groups or numbers (McIntosh 1990;Gallagher et al. 2002;Cui et al. 2006;Yu et al. 2009;Huang et al. 2010) for increasing applications in space weather. Solar activity is well known to be at the origin of geomagnetic activity (Snyder et al. 1963;Crooker et al. 1977). Studying the relationship between solar activity, as represented by the International sunspot number (R z ), and geomagnetic activity, as represented by the aa index (Mayaud 1972), is useful for understanding the formation of the latter and the mechanism of solar cycle (Feynman & Crooker 1978;Legrand & Simon 1989;Du 2011a;Du & Wang 2010, 2011a. Conventionally, the relationship between aa and R z is often analyzed by point-point correspondence. However, some questions are hardly understood such as the significant increase in the aa index over the twentieth century (Feynman & Crooker 1978;Clilverd et al. 1998;Lukianova et al. 2009), and the variations in the correlation between aa and R z (Borello-Filisetti et al. 1992;Echer et al. 2004;Du 2011b). It is found that these phenomena can be well explained by an integral response model recently presented by Du (2011c). The value of aa depends not only on the present R z but also on past values.
This study analyzes the relationships between solar flare parameters (Section 2) and R z as well as the relationships between the aa index and the flare parameters using an integral response model (Du 2011c) in Sections 3.1-3.4. Conclusions are summarized finally in Section 4.

Data
The data used are the time series of monthly mean geomagnetic aa index 1 (Mayaud 1972), the international sunspot number (R z ) 2 , and solar flare parameters based on Geostationary Operational Environmental Satellite (GOES) soft X-ray flares shown as follows 3 .
(ii) T : time duration of flare (in minutes) .
(iv) F : flux from event start to end (in J/m 2 ) .
These parameters are first summed over each day and, then, averaged over each month to obtain the monthly means of the daily integrated quantities. To filter out high frequency variations in the data, the parameters are smoothed with the commonly used 13-month running mean technique. The solar flare parameters since July 1966 are shown in Fig. 1(a) and (b). It is seen that these parameters are well correlated. For example, I is well correlated with T (r = 0.89, Fig. 1(a)), and Q is well correlated with F (r = 0.67, Fig. 1(b)), both being significant at the 99% level of confidence. Figure 1(c) depicts the time series of R z (solid) and aa (dotted) with a correlation coefficient of r = 0.40.

Results
It is well known that solar flares tend to lag behind sunspot activity by several months (Wheatland & Litvinenko 2001;Temmer et al. 2003) or even a few years (Wagner 1988;Aschwanden 1994). To have a better understanding of the relationships and time delays between solar flares and R z , we employ the following integral response model (Du 2011c) to study the relationships between the flare parameters (P = I, T , Q and F ) and R z , where y 0 is a constant, reflecting the part of y = P that is uncorrelated to x = R z (related to other phenomena); D is the 'dynamic response factor' of y to x, representing the initial generation efficiency of y by x (∂y/∂x| t ′ =t ); and τ is the 'response time scale' of y to x, indicating the dependence of the current y(t) on the past x(t ′ ) through a time decay factor e −(t−t ′ )/τ (τ = 0 reflects the point-point correspondence of y to x, i.e., the current y(t) is only related to the current x(t); τ = +∞ represents that y is uncorrelated to x). In application, both y and x are discrete variables. Therefore, we use the second formula in Equation (2) with the summation being taken over from the starting time (t 0 ) of the series (see Fig. 1) to time t. The three parameters (D, τ and y 0 ) are determined by a nonlinear least-square fitting algorithm. Besides, as the geomagnetic activity (aa index) often lags behind solar flares by several months, the relationships between aa and the flare parameters are also analyzed by the same model.

Relationship between R z -I-aa
First, we analyze the relationship between y = I and x = R z since March 1976 (t 0 ) with Equation (2) in the form of (3). Although the correlation coefficient between I and R z (r 0 = 0.87) has not been significantly improved by this model (r f = 0.88), the lag times of I to R z at their peaks (time differences between the peak timings) for Cycles 21-23 (L 1 = 31, 24, 17 with a mean L 1 = 24 months) can be predicted in part by Equation (3) as shown in Fig. 2(a) for the corresponding ones in brackets (L f1 = 4, 8, 5 with a mean L f1 = 6). It implies that the current flares are related to the accumulation of solar magnetic energy in the past through a time decay factor. Active magnetic structures may evolve from the photosphere to upper chromosphere with different speeds and times (Lin et al. 1995;Wheatland & Litvinenko 2001).
The relationship between y = aa and x = I can be fitted by as shown in Fig. 2(b): aa (solid), I (dashed), and the reconstructed series aa f (dotted) by Equation (4). One can see that aa f reflects well the profile of aa. The correlation coefficient between aa and aa f (r f = 0.74) is higher than that between aa and I (r 0 = 0.61). About half of the lag times of aa to I at their peaks for Cycles 21-23 (L 2 = 4, 2, 23 with a mean L 2 = 10) can be predicted by Equation (4) as shown in Fig. 2(b) for the corresponding ones in brackets (L f2 = 5, 4, 8 with a mean L f2 = 6). The relationship between y = aa and x = R z is analyzed by using the following equation, Figure 2(c) illustrates aa (solid), R z (dashed), and the reconstructed series aa f (dotted) by this equation. The correlation coefficient between aa and aa f (r f = 0.73) is much higher than that between aa and R z (r 0 = 0.51). The lag times of aa to R z at their peaks for Cycles 21-23 (L = 35, 26, 40 with a mean L = 34) can be well predicted by Equation (5) as shown in Fig. 2(c) for the corresponding ones in brackets (L f = 27, 29, 28 with a mean L f = 28). The above results are listed in Table 1, in which σ refers to the standard deviation, the last column indicates the relevant averages of fitted/observed lag times at the corresponding peaks over Cycles 21-23 (L f /L), and the last three rows represent the relevant averages of the parameters for the relationships between P -R z , aa-P , and aa-R z , respectively, where P = I, T, Q, F .

Relationship between R z -T -aa
The relationship between R z -T -aa since March 1976 (t 0 ) can also be analyzed by the technique in the previous section, with the results shown in Fig. 3 and Table 1. The following can be noted. Equation (2) is higher than that of T with R z (r 0 = 0.79).
(ii) The correlation coefficient of aa with the reconstructed series aa f (r f = 0.74) from T by Equation (2) is higher than that of aa with T (r 0 = 0.66).
(iii) The lag times of T to R z (L f1 /L 1 = 25/21), aa to T (L f2 /L 2 = 5/12), and aa to R z (L f /L = 28/34) at their peaks for Cycles 21-23 can be approximately predicted by the model. Figure 4 illustrates the relationship between R z -Q-aa since July 1966 (t 0 ) by using the tech- b Average of the corresponding parameters for the relationships between P (= I, T, Q, F ) and Rz. nique in Section 3.1. One can note the following.

Relationship between R z -Q-aa
(i) The correlation coefficient of Q with the reconstructed series Q f (r f = 0.90) from R z by Equation (2) has not improved in comparison to that of Q with R z (r 0 = 0.90), implying that Q and R z peak nearly at the same time (Kleczek 1952).
(ii) The correlation coefficient of aa with the reconstructed series aa f (r f = 0.58) from Q by Equation (2) is much higher than that of aa with Q (r 0 = 0.37).
(iii) The correlation coefficient of aa with the reconstructed series aa f (r f = 0.64) from R z by Equation (2) is much higher than that of aa with R z (r 0 = 0.32).
(iv) The lag times of Q to R z (L f1 /L 1 = 0/5), aa to Q (L f2 /L 2 = 24/29), and aa to R z (L f /L = 31/34) at their peaks for Cycles 21-23 can be predicted in part by Equation (2). Figure 5 shows the relationship between R z -Faa since July 1997 (t 0 ) using the technique in Section 3.1. One sees the following.

Relationship between R z -F -aa
(i) The correlation coefficient of F with the reconstructed series F f (r f = 0.89) from R z by Equation (2) is higher than that of F with R z (r 0 = 0.79).
(ii) The correlation coefficient of aa with the reconstructed series aa f (r f = 0.75) from F by Equation (2) is equal to that of aa with F (r 0 = 0.75).
(iii) The correlation coefficient of aa with the reconstructed series aa f (r f = 0.79) from R z by Equation (2) is much higher than that of aa with R z (r 0 = 0.62).
(iv) The lag times of F to R z (L f1 /L 1 = 25/17), and aa to R z (L f /L = 26/40) at their peaks for Cycle 23 can be predicted in part by Equation (2). While the lag time of aa to F (L f2 /L 2 = 0/23) at their peaks for Cycle 23 has not been predicted by Equation (2) due to the great fluctuations in both aa and F .
These results imply that solar flares depend not only on the present but also on past solar activities (R z ), reflecting the long-term evolution characteristics of solar magnetic field structures (energy) evolving from the photosphere to upper chromosphere (Donnelly 1987;Zhang et al. 2007;Fig. 5.-Similar to Fig. 2 for the relationship between R z -F -aa. Lin 2009). The correlations between solar flare parameters and R z are not simply due to the time shifts (Bachmann & White 1994). Solar flares may play a role for the formation of geomagnetic activity (aa) from solar (magnetic field) activity (R z ), although the processes are not completely clear (Cliver & Hudson 2002).

Discussions and Conclusions
In this study, we investigated the relationships between the solar flare parameters (P = I, T , Q and F ) and sunspot activity (R z ), and between geomagnetic activity (aa) and the flare parameters via the integral response model (Equation (2)). The results indicate that (i) the correlation coefficients between the flare parameters and R z have increased about 6% from r 0 = 0.84 to r f = 0.89 on average when using Equation (2) and the time delays at their peaks for Cycles 21-23 can be well predicted by this model, L f1 /L 1 = 14/17 = 82%; (ii) the correlation coefficients between aa and the flare parameters have increased about 17% from r 0 = 0.60 to r f = 0.70 on average when using Equation (2) and half of the time delays at their peaks can be predicted by this model, L f2 /L 2 = 9/19 = 47%; and (iii) the correlation coefficient between aa and R z has increased about 47% from r 0 = 0.49 to r f = 0.72 on average when using Equation (2) and the time delays at their peaks can be well predicted by this model, L f /L = 28/36 = 78%. This model might be used to improve the solar flare prediction, which should  (6)).
be studied in future.
It is seen in Fig. 2(a) and Table 1 that the time delays between I and R z at their peaks for Cycles 21-23 have not been well predicted by the model (6/24 = 25%). This is due to the large fluctuations in the data. To suppress further the fluctuations, we introduce a cosine filter with the weights given by for b = 24 months. Since b −b π 4b cos( π∆t 2b ) = 1, the weights W C (∆t) are normalized. Using the series smoothed by this filter, we re-analyze the results in Fig. 2, as shown in Fig. 6.
In Equation (2), the output y depends on the past values of input x (τ > 0) rather than only the current value (τ = 0). The stronger the input (x), the more it contributes to the output (y), and the longer the lag time of y to x (Du 2011c). Therefore, solar flares are related to the accumulation of solar magnetic energy in the past rather than the simple time shifts of occurrences (Bachmann & White 1994). The average response time scale of flare parameters to R z in this model (τ = 8) is close to the coronal response time (∼ 10 months) derived from a model for dynamical energy balance in the flaring solar corona (Wheatland & Litvinenko 2001;Litvinenko & Wheatland 2004).
There are various types of active regions in a solar cycle. Small active regions of simple magnetic structure are short-lived and produce minor solar flares, while large active regions of complex magnetic structure are long-lived and produce major solar flares (and hard X-ray flares). It is seen in Fig. 1(a) that T is well correlated with I (r = 0.89), with the regression equation given by T = 67.9 ± 2.2 + (1.94 ± 0.05)I.
According to the above discussions, minor (lowenergy) solar flares lag behind the input R z shorter times with shorter durations while major solar flares lag behind R z longer times with longer durations. Therefore, (i) the time delays between flare activities and sunspot activity come mainly from the major flares rather than the weak ones; (ii) major flares tend to have longer durations and may occur until quite late in the decay phase of a solar cycle (Temmer et al. 2003;Tan 2011); and (iii) the upper chromospheric activity indices (Donnelly 1987;Bachmann & White 1994) and the solar flares (Wheatland & Litvinenko 2001;Temmer et al. 2003) tend to lag behind the sunspot number by several months in a hierarchy manner (Bachmann & White 1994).
Although it is unclear how solar flares affect geomagnetic activities (Gosling 1993;Cliver & Hudson 2002), it is apparent that geomagnetic activities are well correlated with the solar flares. For example, aa is well correlated with F (r 0 = 0.75). As flares are unable to travel to 1 A.U., streams of matter emanating from large flares were considered as the prime cause of geomagnetic storms (Hale 1931;Chapman 1950;Pudovkin et al. 1977). However, Gosling (1993) argued that CMEs, not flares, were the critical element for large geomagnetic storms, interplanetary shocks, and major solar energetic particle (SEP) events. In fact, solar flares may affect geomagnetic activities via different processes related to the flare brightening, erupting, particle ejections, and other unknown effects (Cliver & Hudson 2002). Therefore, the relationships between geomagnetic activity (aa) and solar flares can also be well described by Equation (2). Since geomagnetic activity (aa) can be resulted from various activity phenomena (Legrand & Simon 1989;Tsurutani et al. 1995), the geomagnetic activity is the integral of the effects of all these phenomena, including solar winds, CMEs, solar flares and others. The lag time of aa to solar flare has not been well predicted by the model (9/19) due to the additional effects of other activities. While the lag times of both solar flare and aa to R z at their peaks have been well predicted by the model (14/17, 28/36) because the solar magnetic field activity is the main source of them.
The main conclusions can be drawn as follows, (i) The relationships between the flare parameters (P = I, T, Q, F ) and sunspot activity (R z ) can be well described by an integral response model (r = 0.89) with a mean response time scale of about eight months. The time delays between the flare parameters and R z at their peaks can be well predicted by this model (82%).
(ii) The relationships between geomagnetic activity (aa) and the flare parameters can be better described by this model (r = 0.70) with a mean response time scale of thirteen months than by a linear dependence (r = 0.60). The time delays between aa and the flare parameters at their peaks can be predicted in half by this model (47%).
(iii) The relationship between aa and R z can be much better described by this model (r = 0.72) with a mean response time scale of about twenty-six months than by a linear dependence (r = 0.49). The time delay between aa and R z at their peaks can be predicted in part by this model (78%).