Gm. Webb et al., GREEN FORMULA AND VARIATIONAL-PRINCIPLES FOR COSMIC-RAY TRANSPORT WITH APPLICATION TO ROTATING AND SHEARING FLOWS, The Astrophysical journal, 424(1), 1994, pp. 158-180
Green's theorem and Green's formula for the diffusive cosmic-ray trans
port equation in relativistic flows are derived. Green's formula gives
the solution of the transport equation in terms of the Green's functi
on of the adjoint transport equation, and in terms of distributed sour
ces throughout the region R of interest, plus terms involving the part
icle intensity and streaming on the boundary. The adjoint transport eq
uation describes the time-reversed particle transport. An Euler-Lagran
ge variational principle is then obtained for both the mean scattering
frame distribution function f and its adj oint f dagger. Variations o
f the variational functional with respect to f dagger yield the transp
ort equation, whereas variations of f yield the adjoint transport equa
tion. The variational principle, when combined with Noether's theorem,
yields the conservation law associated with Green's theorem. An inves
tigation of the transport equation for steady, azimuthal, rotating flo
ws suggests the introduction of a new independent variable H to replac
e the comoving frame momentum variable p'. For the case of rigid rotat
ing flows, H is conserved and is shown to be analogous to the Hamilton
ian for a bead on a rigidly rotating wire. The variable H corresponds
to a balance between the centrifugal force and the particle inertia in
the rotating frame. The physical interpretation of H includes a discu
ssion of nonrelativistic and special relativistic rotating flows as we
ll as the cases of azimuthal, differentially rotating flows about Schw
arzschild and Kerr black holes. Green's formula is then applied to the
problem of the acceleration of ultra-high-energy cosmic rays by galac
tic rotation. The model for galactic rotation assumes an angular veloc
ity law OMEGA = OMEGA0(OMEGA0bar/omegaBAR), where omegaBAR denotes rad
ial distance from the axis of rotation. Green's functions for the gala
ctic rotation problem are used to investigate the spectrum of accelera
ted particles arising from monoenergetic and truncated power-law sourc
es. We conclude that it is possible to accelerate particles beyond the
knee by galactic rotation but not in sufficient number to adequately
explain the observed spectrum.