2 Statistical analysis

2.1 X-ray flare data

The data used in this paper come from the collection of the X-ray $M\geq 1$flares (Li & Zhang 1994), which collected all X-ray $M\geq 1$ flare events listed in SGD (Solar Geophysical Data) in the time interval of 1 April 1987 to 30 December 1992 corresponding to the maximum period of the 22nd solar cycle. The table gives the following parameters of each flare event: date, beginning time, maximum time, ending time, X-ray class, and position on the solar disk. The total number of the considered flare events is 2052. The sample does not collect flare events before 1987 and after 1992, because their number was not large enough significant for a statistical analysis.

So we just use the data listed in their table and do not add any data to complete the analysis of an entire solar cycle. Table 1 gives the number of the flares classified by hemisphere and by X-ray class.

 

Table 1: The classification number of $M\geq 1$ X-ray flares at the period of 1987 to 1992

^* The flare events are summed up into the numbers of this row only from 1 April 1987.
^** Number in this column corresponds to the number of the flare events of each year, the half of the sum $\sum No.$

Here, S, N, E, and W stand for the southern, northern, eastern, and western hemispheres respectively; M and X stand for the X-ray flares of M and X classes (or importances) respectively; the last column gives the total number of the annual flare events, and the last row gives the total number in the flare-class within the considered six years.

2.2 Distributions of the flare events on the solar disk

2.2.1 N-S distribution

The spatial distribution of flares in heliographic latitude is usually studied to determine whether there is a N-S spatial asymmetry or not, and whether the asymmetry is dependent on the intensities of the flares, or any other associated characteristics. Figure 1 gives the annual number ratio of the flare events in the southern hemisphere ($N_{\rm S}$ = 1226) to those in the northern hemisphere ($N_{\rm N}$ = 826). There is no flare of X class occurring in 1987, here we let the value $N_{\rm S}/N_{\rm N}$ of this year equal to 1 for the flares of X class. The same is $N_{\rm E}/N_{\rm W}$ in Fig. 3. Figure 1 clearly shows that the flare events in the southern hemisphere are much more numerous than those in the northern hemisphere in five years of the six for M class, X class, and $M \& X$ class respectively. The ratio of the total number of the flare events in the southern hemisphere to that in the northern hemisphere is 1.49 for M class, 1.37 for X class, 1.48 for $M \& X$ class. An overall southern bias apparently prevailed in the solar cycle 22. The above numbers indicate that the flare events of $M \& X$class in the southern hemisphere exceeded the events in the northern hemisphere by an unexpected large percentage $48.6\%$ during the investigated interval. This is a strong N-S asymmetry.

  

Figure 1: The annual number ratio of the flare events in the southern hemisphere ($N_{\rm S}$) to those in the northern hemisphere ($N_{\rm N}$). All flare events of $M \& X$ classes are considered in the dashed curve, those of M class in the dashed and dotted curve, and those of X class in the dotted curve

To be sure that this result cannot be obtained purely by chance we check by using probability laws an compute what would be the ratio of flares in the southern hemisphere to those in the northern hemisphere that we can expect only by chance. Let us consider a distribution of n objects (flares) in 2 classes. The probability that one flare (one object) occurs in one hemisphere (class one) by chance is p=1/2. We use the following binomial formula to derive the probability P(k) of getting k objects in class 1 and n-k objects in class 2.

$$P(k)=\left ( \begin{array} {c} n \\ k \\ \end{array} \right ) p^{k}(1-p)^{n-k}.$$ (1)

In our case for a total number $n=N_{\rm S} +N_{\rm N}$=2052, we have for example:

$$P(k)=\frac{2052!}{(2052-k)! k!} \frac{1}{2^{2052}}\cdot$$

The probability to get more than d objects in one class is:

$$P(\ge d)=\sum_{k=d}^{n}P(k).$$ (2)

We have 1226 (class M + X) events occuring in the southern hemisphere, the probability to get such amount of events or even more by chance is $P(\ge1226)\sim 10^{-8}$. 1137 events occurred in the southern hemisphere within the total 1898 events of M class, $P(\ge1137)\sim 10^{-8}$. 89 events within the total 154 events of X class occurred in the southern hemisphere, $P(\ge 89)=0.026$. The probability to get by chance more than 1226 flares in the southern hemisphere is very weak, more than 1137 M class flares is also weak, to get more than 89 X events is less than 3%. In general, when $P(\ge d) < 5\%$ we have a statistically significant result, and when $P(\ge d) < 1\%$ the result is highly significant. So the numbers 1.49, 1.48 and 1.37 are statistically significant. There really existed an N-S asymmetry for the flares of M class, for the flare events of X class, and for the total flare events of $M \& X$ class respectively.

These results are complimentary to several ones published recently. Oliver & Ballester (1994) and Atac & Ozguc (1996) stressed the dominance of southern hemisphere during most Solar Cycle 22 using sunspot areas and solar flare index, respectively. Such a result points out a possible trend in the behaviour of the N-S asymmetry.

If globally it is in good agreement with previous studies, we did not find any variation with the flare class and any relation with the magnetic field flux obtained by some of these studies. Garcia (1990) found that the degree of N-S asymmetry apparently is a function of the intensity of the studied events, and the most intense events show the largest amount of N-S asymmetry. This statement was valid for the solar Cycle 21 but according to our results it does not seem to be confirmed for Solar Cycle 22. Our statistical results show that the degree ($36.9\%$) of N-S asymmetry for the flares of X class is smaller than the degree ($49.4\%$) for the flares of M class. Howard (1974) ever studied solar magnetic flux data for the period 1967 to 1973. He found that the total flux in the northern hemisphere exceeded that in the southern hemisphere, but only by a small percentage 7%. The discrepancy between our results and the Heras's results could be due to the fact that in the upper latitude the N-S asymmetry flux is less pronounced than in the lower ones or more difficult to establish because of the weak statistics.

Further investigation of the relation between degree of N-S asymmetry and intensity of studied events is needed in future. Heliographic latitude of each flare event is plotted in Fig. 2 with respect to the date of occurrence. Southern heliographic latitudes take the negative sign in the figure. Several features and peculiarities are immediately apparent. At the beginning of the 22nd solar cycle, the X-ray $M\geq 1$ flares occurred in the relatively high latitudes (above $20^{\circ}$), then, as the cycle is progressing, there was an obvious shift in the statistical centroïd position to the lower latitudes. The majority of the flare events occurred in the latitudes between $8^{\circ}$ and $35^{\circ}$. Only 2 flares occurred in the latitudes over $40^{\circ}$, and no flares occurred in the latitudes over $45^{\circ}$. Garcia (1990) gave the distribution of $M\geq 1$ flares from 1969 to 1984 with respect to the heliographic latitude in Figs. 2 and 3 of his paper and find similar results. The distribution in latitude of the flares is comparable to the sunspot distribution.

  

Figure 2: Distribution of X-ray $M\geq 1$ flares from 1987 to 1992, with respect to heliographic latitude

2.2.2 W-E distribution

The annual number ratio of the flare events in the eastern hemisphere ($N_{\rm E}$) to those in the western hemisphere ($N_{\rm W}$) is plotted in Fig. 3. The flare events in the eastern hemisphere are more numerous than those in the western hemisphere in four years, which correspond to the maximum period. The ratio of the total number of flares in the eastern hemisphere to that in the western hemispheres is 969/929(=1.043) for M class, 80/74(=1.083) for X class, 1049/1003(=1.046) for $M \& X$ classses, and the corresponding actual probabilities $P(\ge d)$ are 0.179, 0.314, and 0.155, respectively. In general, when $P(\ge d) \gt 10\%$ it implies a statistically insignificant result.

  

Figure 3: The annual number ratio of the flare events in the eastern hemisphere ($N_{\rm E}$) to those in the western hemisphere ($N_{\rm W}$). All flare events of M and X classes are considered in the dashed curve, those of M class considered in the dashed and dotted curve, and those of X class considered in the dotted curve

So the slight W-E asymmetry that we have found in the distribution of the X-ray $M\ge 1$ flare events during the period of 1987 to 1992 cannot be considered significant in terms of W-E asymmetry. Heliographic longitude of each flare event is plotted in Fig. 4 with respect to the date of occurrence. Western heliographic longitudes take the negative sign in the figure. The events are spread in all the longitudes.

  

Figure 4: Distribution of X-ray $M\geq 1$ flares from 1987 to 1992, with respect to heliographic longitude

The $\chi^{2}$ test can be used to check the null-hypothesis whether the longitude distribution of the events is due quite by chance. In order to check this point we divide firstly the events into nine longitude ranges, and the number f_i of the events in each range is listed in Table 2. Then we calculate the value of $\chi^{2}$:

$$\chi^{2}=\sum_{i=1}^{n}{{(f_{i}-\bar{f})^{2}}\over{\bar{f}}}$$ (3)

where $\bar{f}={{\sum_{i=1}^{n}f_{i}}\over{n}}$ and n=9 . According to Eq. (3) and Table 2 we have $\chi^{2}=33.27$. Looking at the tables of critical $\chi^{2}$ with n-1 degree of freedom corresponding to our null hypothesis we find for example that the critical value of $\chi^{2}$ distribution $\chi^{2}_{0.005}(8)=21.9$, which is less than our value $\chi^{2}$. It means that the probability to have the observed distribution by chance is less than 5/1000. So, we can consider that the events are not uniformly distributed in the longitude.

 

Table 2: Number of the flare events in different longitude ranges

^* The symbol [means "including the value" of the number in the bracket, the symbol ("excluding the value".