Transport of energetic charged particles in a radial magnetic field. Part 1. Large-angle scattering

A new approach, the propagating-source method, is introduced to solve the t ime-dependent Boltzmann equation. The method relies on the decomposition of the particle distribution function into scattered and unscattered particle s. It is assumed in this paper that the particles are transported in a cons tant-velocity spherically expanding supersonic flow (such as the solar wind ) in the presence of a radial magnetic field. Attention too has been restri cted to very fast particles. The present paper addresses only large-angle s cattering, which is modelled as a BGK relaxation time operator. A subsequen t paper (Part 2) will apply the propagating-source method to a small-angle quasilinear scattering operator. Initially, we consider the simplest form o f the BGK Boltzmann equation, which omits both adiabatic deceleration and f ocusing, to re-derive the well-known telegrapher equation for particle tran sport. However, the derivation based on the propagating-source method yield s an inhomogeneous form of the telegrapher equation; a form for which the w ell-known problem of coherent pulse solutions is absent. Furthermore, the i nhomogeneous telegrapher equation is valid for times t much smaller than th e 'scattering time' T, i.e. for times t much less than T, as well as for t >T. More complicated forms of BGK Boltzmann equation that now include focus ing and adiabatic deceleration are solved. The basic results to emerge from this new approach to solving the BGK Boltzmann equation are the following. (i) Low-order polynomial expansions can be used to investigate particle pr opagation and transport at arbitrarily small times in a scattering medium. (ii) The theory of characteristics for linear hyperbolic equations illumina tes the role of causality in the expanded integro-differential Fokker-Planc k equation. (iii) The propagating-source approach is not restricted to isot ropic initial data, but instead arbitrarily anisotropic initial data can be investigated. Examples using different ring-beam distributions are present ed. (iv) Finally, the numerical scheme can include both small-angle and lar ge-angle particle scattering operators (Part 2). A detailed discussion of t he results for the various Boltzmann-equation models is given. In general, it is found that particle beams that experience scattering by, for example, interplanetary fluctuation are likely to remain highly anisotropic for man y scattering times. This makes the use of the diffusion approximation for c harged-particle transport particularly dangerous under many reasonable sola r-wind conditions, especially in the inner heliosphere.