  In this paper, we study the distribution functions that arise naturally
during self-similar radial infall of collisionless matter. Such matter may be
thought of either as stars or as dark matter particles. If a rigorous steady
state is assumed, then the system is infinite and is described by a universal
distribution function given the self-similar index. The steady logarithmic
potential case is exceptional and yields the familiar Gaussian for an infinite
system with an inverse-square density profile. We show subsequently that for
time-dependent radial self-similar infall, the logarithmic case is accurately
described by the Fridmann and Polyachenko distribution function. The system in
this case is finite but growing. We are able to embed a central mass in the
universal steady distribution only by iteration, except in the case of massless
particles. The iteration yields logarithmic corrections to the massless
particle case and requires a `renormalization' of the central mass. A central
spherical mass may be accurately embedded in the Fridmann and Polyachenko
growing distribution however. Some speculation is given concerning the
importance of radial collisionless infall in actual galaxy formation.
