1. Introduction

[2]  The time profile of particle fluxes in the solar events has a typical shape with the more or less quick rise, the maximum, and much slower decay towards the level before the flare. The phase of decay contains important information on the physical processes to which particles are subjected in the interplanetary space. Different mechanisms of particle propagation lead to different laws describing the decrease of fluxes at the late stage of the event. Sometimes this picture can be presented in the diffusion approximation. Then, within the elementary diffusion model under the assumption of a pulsed source of particles and the boundary condition J(r) = 0 at r = ( J is the flux of particles and r is the distance from the source), the time profile of fluxes at the decay phase of the event has a power law shape and it is proportional to t^-3/2, where t is the time from the moment of the injection of particles. In diffusion models with the absorbing boundary situated at a finite distance from the source, the decline of the intensity can be described by the exponential function with the standard value of the characteristic decay time which does not depend on the parameters of the surrounding plasma and the spectrum of particles.

[3]  However, in the interplanetary space, the solar wind providing the convective transport of particles and their adiabatic cooling is usually present. If such processes prevail in comparison with the diffusion, the fluxes decrease exponentially. Power law intensity decrease can frequently be observed for high energies (>50 MeV), while for smaller energies 10 MeV, convective transport and adiabatic cooling begin to play substantially greater role and the decline of particle fluxes becomes exponential [Lee, 2000]. Thus, the very shape of the decay contains certain information about the processes of particle propagation in the interplanetary space. Moreover, the dependence of the decay rate on different parameters clarifies the role of three main mechanisms (diffusion, convection, and adiabatic cooling) during the propagation from the source to the point of observation. We discuss interrelation of the above mechanisms in section 4. Our task is not the description of particular solar energetic particle (SEP) events but the elucidation of statistical regularities that characterize this phase of the event.

[4]  It was shown in our previous papers that in about 90% of events the fluxes of protons with low energies ( <10 MeV) have an exponential decay while for particles with high energies ( >30-60 MeV) exponential decay is detected much more seldom. We discussed in detail the mean values of τ, as well as the peculiarities of the characteristic decay time as a function of the size of the event (distributions of τ for protons with energies E >4 MeV are practically the same for events with J _max > 100 (cm² s sr)^-1 and with J _max > 2-3 (cm² s sr)^-1 [Daibog et al., 2003a]), parameters of the surrounding plasma (the solar wind velocity and the magnetic field intensity [Daibog et al., 2005a, 2005b]), the angular distance between the source and the point of observations, etc., and also variations of τ during the solar activity cycle [Daibog et al., 2003b; Kecskeméty et al., 2003].

[5]  One can assume that if the decay of particle fluxes in the solar event does not change its character for a long time (of the order of a day or more), then the nearest interplanetary space (IS) is homogeneous and quasi-stationary. This guarantees the constancy of τ in the case of the exponential decline. The statement that the IS is quasi-stationary assumes the invariance of the whole complex of its properties. They are the following: (1) gradient of the particle density in the vicinity of the observation point and the velocity (determined by the diffusion coefficient of particles and their convective outflow) with which particles leave the given region of space, (2) adiabatic cooling of particles in the process of propagation, and (3) possible acceleration of particles in the vicinity of the point of observation. The relative contribution of each of these processes into the formation of a time profile varies in different events. That is why we have to discuss the whole complex of processes what results in the exponential decrease of particle fluxes during an extended period (sometimes up to several days). One can describe this decrease rather strictly by the exponential law with a constant characteristic time τ. The numerical value of τ is the generalized characteristic of the action of space on the time profile of particle fluxes. The investigation of the role of each component of such action is one of the tasks of the interplanetary space exploration. In this paper we investigate the dependences of the characteristic time of proton fluxes decay τ in the events with the exponential decrease as functions of the solar wind velocity, the index of the energetic spectrum, and also the energy of particles. These dependences make it possible to observe the action of the main mechanisms that determine to a considerable extent the value of τ.

[6]  It follows from the most general considerations that if the convective transport of particles by the solar wind takes place, τ should decrease with a growth of the solar wind speed V. If the adiabatic cooling of particles takes place, the rate of decline should increase due to the falling particle spectrum J(E) E^-γ and τ diminishes with a growth of the spectral index γ. With the increase of the distance r from the source, τ grows because it requires more time for particles localized in a larger volume to outflow from this volume.

[7]  Forman [1970] and Jokipii [1972] showed that if the convective transport and adiabatic cooling dominate over the diffusion during the decay stage, the temporal profile of particle fluxes is described by the dependence J(t) exp(-t/τ). They obtained analytically an expression for the characteristic decay time τ taking into account the dependences on all three parameters r, V, and γ:

(1)

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