Accurate light-time correction due to a gravitating mass

This work arose as an aftermath of Cassini's 2002 experiment \cite{bblipt03}, in which the PPN parameter \(\gamma\) was measured with an accuracy \(\sigma_\gamma = 2.3\times 10^{-5}\) and found consistent with the prediction \(\gamma =1\) of general relativity. The Orbit Determination Program (ODP) of NASA's Jet Propulsion Laboratory, which was used in the data analysis, is based on an expression for the gravitational delay which differs from the standard formula; this difference is of second order in powers of \(m\) -- the sun's gravitational radius -- but in Cassini's case it was much larger than the expected order of magnitude \(m^2/b\), where \(b\) is the ray's closest approach distance. Since the ODP does not account for any other second-order terms, it is necessary, also in view of future more accurate experiments, to systematically evaluate higher order corrections and to determine which terms are significant. Light propagation in a static spacetime is equivalent to a problem in ordinary geometrical optics; Fermat's action functional at its minimum is just the light-time between the two end points A and B. A new and powerful formulation is thus obtained. Asymptotic power series are necessary to provide a safe and automatic way of selecting which terms to keep at each order. Higher order approximations to the delay and the deflection are obtained. We also show that in a close superior conjunction, when \(b\) is much smaller than the distances of A and B from the Sun, of order \(R\), say, the second-order correction has an \emph{enhanced} part of order \(m^2R/b^2\), which corresponds just to the second-order terms introduced in the ODP. Gravitational deflection of the image of a far away source, observed from a finite distance from the mass, is obtained to \(O(m^2)\).