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 analyse 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 upper bounds on the photon mass at the level of m_γ 3 × 10⁻⁷ \, eV/c². We also discuss next-generation solar-system tests of general relativity that could improve this upper limit, potentially reaching m_γ 8 × 10⁻⁸ \, eV/c². Though not competitive with the currently best limits, our projected bounds are at the ballpark of earlier ones based on the gravitational deflection angle.