Ultimate Spectrum of Solar/Stellar Cosmic Rays

Abstract

We propose a physical approach to reconstruct the ultimate spectrum of solar/stellar cosmic rays (SCR) in a given point in the heliosphere (stellar sphere) basing on maximal value of magnetic field strength in active region and its characteristic linear dimension. An accelerator of given dimensions and magnetic field strength may accelerate to some finite energy for a given time (a maximal energy of SCR). We use the spectrum of SCR proposed by Syrovatsky (1961) for relativistic and non-relativistic energies normalizing it to galactic cosmic ray (GCR) intensity at the maximal SCR energy. Maximal values of SCR flux propagating in the heliosphere are determined by equilibrium between pressure of interplanetary magnetic field and dynamic pressure of SCR (Freier and Webber, 1963). We use the characteristic time of fluence accumulation obtained from the observations to derive the ultimate fluence of a single solar proton event (SPE). Our estimates are consisted with the observational upper limits on the strength of SPEs on the timescale of tens of millennia (Usoskin and Kovaltsov, 2012) and during the most active interval of 1945 -1995 during the last 2000 years . The obtained numbers of >30 MeV and >200 MeV solar protons for a single proton event is not enough to explain the extreme particle event occurred in about 775 AD, a sequence of about twenty proton events may explain it. An extension of the model to cases of stellar flares is discussed.