A new approach, the propagating-source method, is introduced to
solve the time-dependent Boltzmann equation. The method relies on the
decomposition of the particle distribution function into scattered and
unscattered particles. It is assumed in this paper that the particles are
transported in a constant-velocity spherically expanding supersonic flow (such
as the solar wind) in the presence of a radial magnetic field. Attention too has
been restricted to very fast particles. The present paper addresses only large-angle scattering, which is modelled as a BGK relaxation time operator. A
subsequent paper (Part 2) will apply the propagating-source method to a small-angle quasilinear scattering operator. Initially, we consider the simplest form of
the BGK Boltzmann equation, which omits both adiabatic deceleration and
focusing, to re-derive the well-known telegrapher equation for particle
transport. However, the derivation based on the propagating-source method
yields an inhomogeneous form of the telegrapher equation; a form for which
the well-known problem of coherent pulse solutions is absent. Furthermore, the
inhomogeneous telegrapher equation is valid for times t much smaller than the
‘scattering time’ τ, i.e. for times t [Lt ] τ, as
well as for t > τ. More complicated
forms of the BGK Boltzmann equation that now include focusing 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 propagation and
transport at arbitrarily small times in a scattering medium. (ii) The theory of
characteristics for linear hyperbolic equations illuminates the role of causality
in the expanded integro-differential Fokker–Planck equation. (iii) The
propagating-source approach is not restricted to isotropic initial data, but
instead arbitrarily anisotropic initial data can be investigated. Examples using
different ring-beam distributions are presented. (iv) Finally, the numerical
scheme can include both small-angle and large-angle particle scattering
operators (Part 2). A detailed discussion of the results for the various
Boltzmann-equation models is given. In general, it is found that particle beams
that experience scattering by, for example, interplanetary fluctuations are
likely to remain highly anisotropic for many scattering times. This makes the
use of the diffusion approximation for charged-particle transport particularly
dangerous under many reasonable solar-wind conditions, especially in the inner
heliosphere.
Photospheric magnetic field of an eroded-by-solar-wind coronal mass ejection