Accurate light-time correction due to a gravitating mass

This technical paper of mathematical physics arose as an aftermath of the 2002 Cassini experiment (Bertotti et al 2003 Nature 425 374–6), in which the PPN parameter γ was measured with an accuracy σγ = 2.3 × 10−5 and found consistent with the prediction γ = 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 (8) for the gravitational delay Δt that differs from the standard formula (2); this difference is of second order in powers of m—the gravitational radius of the Sun—but in Cassini's case it was much larger than the expected order of magnitude m2/b, where b is the distance of the closest approach of the ray. Since the ODP does not take into account any other second-order terms, it is necessary, also in view of future more accurate experiments, to revisit the whole problem, to systematically evaluate higher order corrections and to determine which terms, and why, are larger than the expected value. We note that 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. This method is closely connected with the much more general approach of Le Poncin-Lafitte et al (2004 Class. Quantum Grav. 21 4463–83), which is based on Synge's world function. 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 required quantities, in particular the delay and the deflection, are easily 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, say of order R, the second-order correction has an enhanced part of order m2R/b2, which corresponds just to the second-order terms introduced in the ODP. Gravitational deflection of the image of a far away source when observed from a finite distance from the mass is obtained up to O(m2).