Accurate light-time correction due to a gravitating mass
Neil Ashby, Bruno Bertotti
This technical paper of mathematical physics arose as an aftermath of the 2002 Cassini experiment, in which the PPN parameter γ was measured with an accuracy ςγ = 2.3 × 10-5 and found consistent with the rediction 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 for the gravitational delay which differs from the standard formula; 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 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 Poncin-Kafutte (2004), 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, of order R, say, the second-order correction has an enhanced part of order m2R/b2 which corresponds just to the second-order terms introduced in the ODP.
and Bertotti, B.
Accurate light-time correction due to a gravitating mass, Classical and Quantum Gravity, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=905436
(Accessed December 3, 2023)