Currently reading the following document - Moyer (1971). Excellent read if anyone is interested.
I note that in Eqs. $(12)-(20)$, Moyer uses the coordinate time as the affine parameter in the Euler - Lagrange equations of motion. I am terribly confused about this. Normally, for a timelike geodesic the affine parameter is the proper time. Later in the document the independent variable will be the geocentric coordinate time or the Barycentric coordinate time. Does anyone have a simple explanation why they don't use the proper time?
The notation is the same in the IERS technical note for Eq. $(10.12)$.
Answer
The reason is simple: The goal in that technical report is to establish a relativistically-correct framework via which an N-body problem can be solved numerically, where N is smallish, but is larger than two. In particular, the goal is to describe the solar system.
General relativistic effects in the solar system are small, considerably smaller than are the perturbations that result from the Newtonian gravitational interactions amongst planets. This means it makes much more sense to use coordinate time rather than proper time. (This raises the question: Proper time of what? This is an N-body problem, after all.)
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