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We use all powerful X-ray flares observations from 1976 up to 2013 to study a possible correlation between the moments of occurrence of flares and a phase of a solar cycle. The time interval considered corresponds to cycles 21, 22 and 23 with maximum heights W21=164.5, W22=158.5 and W23=120.8. We find that: 1) Only two of all of 30 flares considered have arisen on a branch of growth of cycles. The others 28 were observed or near to maxima of cycles, or on a decline branch. Probably it is a consequence of a specific field of velocities in a latitudes interval ± 15° on the Sun, favorable for work of the reconnection mechanism; 2) The number of powerful flares in a cycle does not correlate with its height: cycles 21 and 22 are almost identical on height (W21=164.5, W22=158.5), but the quantity of flares in them differs almost in 2 times (8 flares in 21-st cycle and 14 flares in 22-nd). At the same time cycles 21 and 23, appreciably distinguished on height, gave identical quantity of flares (8 flares in each cycle); 3) In the lowest 23-rd cycle (W23= 120.8) there were the most powerful flares for all period of observations: two flares of class X17, one flare of class X20 and, apparently, unique for last 144 years super flare of class X28+. The conclusion is made, that approximately up to 2019 it is possible to expect occurrence of M-class flares and X-class ares up to X8. More powerful flares are not excluded as well. Probability of the super power flares on the scale of that in 1859 (Carrington event) is low.