An understanding of the solar modulation of cosmic rays (CRs) contributes to the solution of various problems in heliospheric physics and astrophysics. On the one hand, since modulation is tightly connected to the physics of the solar wind expansion and that of the turbulence being present in the wind plasma, it is used as valuable diagnostic of the large- and small-scale heliospheric structure, respectively. On the other hand, a successful and reliable so-called de-modulation of heliospheric CR spectra yields the unmodulated interstellar CR spectra.
The CRs of interest for solar modulation studies can be divided into two populations, namely Galactic and Anomalous Cosmic Rays (GCRs and ACRs). GCRs are accelerated somewhere in the Galaxy and arriving with their interstellar spectra at the heliosphere. According to the present believe based on ideas by Fisk et al. (1974) and Pesses et al. (1981), ACRs are locally accelerated at the solar wind termination shock, for a recent overview see Fisk et al. (1998). Both GCRs upon arrival at the outer boundary of the heliosphere and ACRs after local acceleration are entering the heliospheric region enclosed by the termination shock. Their transport to the inner heliosphere is mainly determined by spatial diffusion, convection with the solar wind background flow and adiabatic cooling.
It was Parker (1965) who first derived the fundamental CR transport equation, now known as the Parker equation, that takes into account all of these processes:
In this equation, written here for spherical symmetry already, is the quasi-isotropic phase space distribution function of CRs with r and p denoting heliocentric distance and particle momentum, respectively. The solar wind speed is given by V and the coefficient of spatial diffusion by . On the right-hand-side, indicates a source function.
Later, Jokipii et al. (1977) recognized the importance of large-scale drifts of CRs in the heliospheric magnetic field and supplemented Eq. (1) by an appropriate term. While being indispensable for a description of multi-dimensional large-scale modulation, it has been demonstrated that drifts effects do not have to be included in all cases to reproduce observations (see, e.g., Reinecke et al. 1993; le Roux & Fichtner 1997).
Already Parker (1965) has given a variety of solutions of Eq. (1) for simplified cases to explore the effects of spatial diffusion, convection and adiabatic energy loss. In the past 34 years many more analytical or semi-analytical solutions were presented in the context of solar modulation (see, e.g., Fisk & Axford 1969; Gleeson & Webb 1974; Cowsik & Lee 1977; Zhang 1999). Although these solutions describe the basic effects of solar modulation, they are not exact in a strict sense as they are employing various approximations and asymptotic expansions or assumptions about source functions.
Exact solutions for the Green's function of Eq. (1), i.e. for the Parker propagator, were presented by several authors in a different context, namely the acceleration of particles in accretion flows assuming a power law dependence of both the flow speed and the coefficient of spatial diffusion on the phase space coordinates:
These previously published solutions for the Green's function required (see Schneider & Bogdan (1989) and Becker (1992), but note that they used a different sign of ) and can be classified using the auxiliary parameter
The solution derived by Schneider & Bogdan (1989) was valid for , i.e. (see their Eq. (2.6)). Noticing this, and generalizing to an arbitrary dependence of on momentum, Becker (1992) presented the solution for and (see his Eq. (A7) and his comments following it).
For the case of solar modulation (see the discussion in Sect. 4.1. below), we have indeed , but , i.e. . This, however, means that the previously published solutions are not applicable to the case of solar modulation.
It is the purpose of this paper to present an exact solution for the Parker propagator of Parker's equation for the case of solar modulation. This analytical representation of the Parker propagator is then used to determine the modulated spectra of both ACRs and GCRs for three turbulence models. These models are formulated from the plasma wave viewpoint and embrace a slab turbulence with Alfvén waves, an isotropic turbulence with fast magnetosonic waves and a mixture of these two cases.
© European Southern Observatory (ESO) 2000
Online publication: June 26, 2000