  We investigate the gravitational radiation from binary systems in conformal
gravity (CG) and massive conformal gravity (MCG). CG might explain observed
galaxy rotation curves without dark matter, and both models are of interest in
the context of quantum gravity. Here we show that gravitational radiation
emitted by compact binaries allows us to strongly constrain both models. We
work in Weyl gauge, which fixes the rescaling invariance of the models, and
derive the linearized fourth-order equation of motion for the metric, which
describes massless and massive modes of propagation. In the limit of a large
graviton mass, MCG reduces to general relativity (GR), whereas CG does not.
Coordinates are fixed by Teyssandier gauge to show that for a conserved
energy-momentum tensor the gravitational radiation is due to the time-dependent
quadrupole moment of a non-relativistic source and we derive the gravitational
energy-momentum tensor for both models. We apply our findings to the case of
close binaries on circular orbits, which have been used to indirectly infer the
existence of gravitational radiation prior to the direct observation of
gravitational waves. As an example, we analyze the binary system PSR
J1012+5307, chosen for its small eccentricity. When fixing the graviton mass in
CG such that observed galaxy rotation curves could be explained without dark
matter, the gravitational radiation of a binary system is much smaller than in
GR. The same holds for MCG for small masses of the graviton. Thus gravitational
radiation cannot explain the orbital decay of binary systems and replace dark
matter simultaneously. We also analyse MCG for large graviton masses and
conclude that MCG can describe the orbital periods of compact binaries in
agreement with data, as it reduces to GR in that limit.
