  In this paper, we consider the dynamics of a tagged point particle in a gas
of moving hard-spheres that are non-interacting among each other. This model is
known as the ideal Rayleigh gas. We add to this model the possibility of
annihilation (ideal Rayleigh gas with annihilation), requiring that each
obstacle is either annihilating or elastic, which determines whether the tagged
particle is elastically reflected or removed from the system. We provide a
rigorous derivation of a linear Boltzmann equation with annihilation from this
particle model in the Boltzmann-Grad limit. Moreover, we give explicit
estimates for the error in the kinetic limit by estimating the contributions of
the configurations which prevent the Markovianity. The estimates show that the
system can be approximated by the Boltzmann equation on an algebraically long
time scale in the scaling parameter.
