  The scattering of small bodies by planets is an important dynamical process
in planetary systems. We present an analytical model to describe this process
using the simplifying assumption that each particle's dynamics is dominated by
a single planet at a time. As such the scattering process can be considered as
a series of three body problems during each of which the Tisserand parameter
with respect to the relevant planet is conserved. This constrains the orbital
parameter space into which a particle can be scattered. Such arguments have
previously been applied to the process by which comets are scattered to the
inner Solar System from the Kuiper belt. Our analysis generalises this for an
arbitrary planetary system. For particles scattered from an outer belt directly
along a chain of planets, based on the initial value of the Tisserand
parameter, we find that it is possible to (i) determine which planets can eject
the particles from the system, (ii) define a minimum stellar distance to which
particles can be scattered, and (iii) constrain range of particle inclinations
(and hence the disc height) at different distances. Applying this to the Solar
System, we determine that the planets are close to optimally separated for
scattering particles between them. Concerning warm dust found around stars that
also have Kuiper belt analogues, we show that, if there is to be a dynamical
link between the outer and inner regions, then certain architectures for the
intervening planetary system are incapable of producing the observations.
Furthermore we show that for certain planetary systems, comets can be scattered
from an outer belt, or with fewer constraints, from an Oort cloud analogue,
onto star-grazing orbits, in support of a planetary origin to the metal
pollution and dustiness of some nearby white dwarfs.
