Bounds on the photon mass via the Shapiro effect in the solar system

We study the effects of a finite mass for the photon on its propagation in a weak gravitational field. In particular, we analyze the gravitational time delay, also known as the Shapiro effect. We work in isotropic coordinates in the weak-field limit and find that the mass-dependent corrections enhance the gravitational time delay. Doppler-tracking data from the Cassini mission allow us to set an upper bound on the photon mass, namely ${m}_{\ensuremath{\gamma}}<4.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}\text{ }\text{ }\mathrm{eV}/{\mathrm{c}}^{2}$ at 95% CL. We also discuss next-generation solar-system tests of general relativity that could improve this upper limit, potentially by a factor of ten. Though not competitive with the currently best limits, our bound is at the ballpark of earlier ones based on the gravitational bending of light by the Sun.