  We extend the perturbative classical double copy to the analysis of bound
systems. We first obtain the leading order perturbative gluon radiation field
sourced by a system of interacting color charges in arbitrary time dependent
orbits, and test its validity by taking relativistic bremsstrahlung and
non-relativistic bound state limits. By generalizing the color to kinematic
replacement rules recently used in the context of classical bremsstrahlung, we
map the gluon emission amplitude to the radiation fields of dilaton gravity
sourced by interacting particles in generic (self-consistent) orbits. As an
application, we reproduce the leading post-Newtonian radiation fields and
energy flux for point masses in non-relativistic orbits from the double copy of
gauge theory.
