| References: | 1. The Distribution in X-Ray Class of Solar Flares, R. Thompson (1993) |
| 2. The different types of solar flares, J. Janssens (2002) | |
| 3. Predictions on the start and amplitude of new solar cycle 24, J. Janssens (2005) |
Introduction - In his article [1], Richard Thompson discusses some equations that give an idea on the total number of flares that can be expected for a certain yearly Wolfnumber, as well as the strength of these flares. It is the purpose of the current article to update these equations by using data up to February 2005, as well as incorporating the less energetic C-class and the superenergetic flares. The three solar cycles will be compared, and a prediction made for the upcoming SC24.
The relation Flares - Wolfnumber – To forecasters of spaceweather, the number of energetic flares the sun can produce and their strength is an important given. This kind of data is being used by lots of organisations like radio and satellite operators, electricity companies, the oil industry (pipe lines!), and so on. As a first step, for each year the average Wolfnumber is determined, as well as the number of solar flares. In determining the equations, the number of C1- and C2-flares was not taken into account. This is because during active periods, the solar X-ray background radiation regularly exceeds the C2-level, resulting in weaker flares not being detected. Flares in the C3- and C4-classes suffer a little bit of this phenomenon. The established relationships are based on data from January 1976 till December 2004, using only flares that are linked to sunspotgroups. Table underneath provides for each type of X-ray flare (see [2]) the formulae (linear & power) with corresponding correlation (r2) and standard deviation, and the last column contains the number of flares on which the equations are based. Both the formulae and the standard deviations are expressed as a function of the Wolfnumber R.
| Type | Linear | r2 | St.Dev. | Power | r2 | St.Dev. | #Flares |
|---|---|---|---|---|---|---|---|
| All>C3 | 8,54 . R - 127,10 | 0,893 | 1,81 . R | 0,71 . R1,49 | 0,952 | 1,29 . R | 15464 |
| C>C3 | 5,78 . R - 79,84 | 0,904 | 0,79 . R | 0,45 . R1,51 | 0,953 | 0,82 . R | 10658 |
| M | 2,53 . R - 43,60 | 0,827 | 0,54 . R | 0,21 . R1,47 | 0,938 | 0,50 . R | 4419 |
| X | 0,22 . R - 3,66 | 0,566 | 0,09 . R | 0,11 . R1,07 | 0,727 | 0,08 . R | 387 |
Assuming an average Wolfnumber R of 25 for the year 2005, according to the power-formula we may expect 58 (+/- 21) C-flares, 24 (+/- 13) M-flares and (still) 3 (+/-2) X-flares. According to the linear formula, these numbers would be respectively 65 (+/- 20), 20 (+/- 14) and 2 (+/- 2). These are of course only statistics. It may very well be that e.g. there are no X-flares at all this year, or -on the contrary- double as many: the standard deviations are quite large. So, the calculated numbers are indicative only. Moreover, all these eruptions are not neatly distributed throughout the year, but they come in bunches according to the rythm with which active sunspotgroups manifest themselves. As an example, the three big "Halloween"-groups that appeared in the October-November 2003 timeframe produced in just a few weeks 11 of the 19 X-flares for the whole year!
Flare-distribution – As soon as one has an idea on the number of eruptions that can be expected in a certain period, one can use the next equation to calculate the number of flares N per X-ray class (see [2]) that can be expected:
N = Z . DELTA . 4,55 . Y-2,13
Z represents the total number of flares (> C3) expected to occur in 2005. DELTA = 1 for C3, 1 for C4,... , 10 for M1, 10 for M2,... , 100 for X1,... , 1000 for X10 and 1000 for X20. Y represents the X-ray class. E.g. for the class C1 (containing all flares C1,0 to C1,9), Y is the average of the classes C1-C2, or 1,5. Similarly for the class M2 (Y=25) and X4 (Y=450). Also here, the equation did not account for the number of C1- and C2-flares. Graph underneath shows there exists a good correlation between the data points and the calculated equation. However, one should not forget that the axes are based on logaritms, meaning small deviations actually represent significant differences in absolute values (e.g. with the number C3- and C4-flares).
Some practical applications. How many C8-flares can be expected in 2005? From the previous calculations, we know that Z = 86 (+/-32). Because in this case it concerns a flare from the C-class, DELTA is 1. For the class C8, Y=8,5. From this, it follows that
N = 86 . 1 . 4,55 . (8,5)-2,13 = 4
So, according to statistics, this year may see 4 (+/-1,5) C8-flares. How many X5-flares could there be? Because it concerns a flare of type X, DELTA = 100. For the class X5, Y=550. Hence, we have
N = 86 . 100 . 4,55 . (550)-2,13 = 0,06
In practice, this means no X5-flares are expected, but because N is NOT zero, there is always a remote possibility on a X5-flare. The longer the period that is investigated, the more accurate the results will be.
Comparison of SC21, 22 and 23 – The previous calculation was repeated for each solar cycle separately. Z was replaced by the total number of flares that occured per cycle. For SC23, it obviously concerns only partial results, but in the remaining 18 to 24 months, no spectacular number of flares are expected. The formulae are:
| SC21: | N = 5496 . DELTA . 4,22 . Y-2,10 |
| SC22: | N = 5675 . DELTA . 3,88 . Y-2,08 |
| SC23: | N = 4360 . DELTA . 4,10 . Y-2,12 |
These can also be shown in graphical form, together with the different data points:
At first sight, both the formulae as the graphic show very little differences, but because of the logaritmic scale, things are not entirely what they seem to be. SC23 was indeed less productive in energetic flares than its predecessors. There were already signs in the much lower total number of flares, and the fact that the power (for SC23: -2,12) was the most negative of the three cycles: the number of flares decreases the fastest with increasing flare class. SC23 does get some good points for producing flares in each flare class: SC21 lacked an X20, and SC22 did not produce a X7 and X8. Moreover, SC23 is the only cycle having produced 2 X20-flares, including the most powerful flare since the beginning of satellite observations (X28).
Prediction for solar cycle 24 – In a previous work [3], the first cautious predictions for the next solar cycle were being made. A maximum Wolfnumber of 100 +/-20 is expected in 2011. For the calculation of the yearly Wolfnumbers and the resulting number of flares, SC10 (Rmax = 98,5, duration = 135 maanden) was taken as a base. The solar minimum was set in 2007, and the duration at 11 years (weaker cycles normally have a longer duration). From the derived yearly Wolfnumbers, the total number of flares (> C3) can be determined using previous formulae. Linear formulae were used for most years, except during cycle minimum (2005-2007, 2016-2017). The results are shown in the next diagram.
During the years of solar cycle maximum, SC24 is expected to produce about 670 (+/- 170) flares (> C3), of which 200 (+/- 50) M- and 17 (+/- 9) X- flares. Statistically, SC24 should contain about 3500 flares (sommation from 2007 till 2017), which is a bit lower than derived from a regression analysis with previous cycles (for SC23, the total was estimated to be 4600). With 3500, it is statistically still possible for SC24 to produce three X10- and even one X20-flare. Despite the moderate amplitude of SC24, the sun seems to guarantee excitement for the next 10 years!