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Newtonian gravitation with radially varying velocity-dependent mass

A new extension to Newtonian celestial mechanics is examined. We focus on the scenario of a point-like body with negligible mass orbiting a spherically symmetric massive body. We take the implicitly time-dependent mass of electrodynamics one step further. We let the mass of the orbiting body vary not only with the velocity, but also with the position within the gravitational field. We find a family of expressions for the gravitational acceleration that explains the anomalous precession of perihelion of the planets and in the strong field limit results in orbits in close agreement with the predictions of the Schwarzschild solution. Regarding the orbital velocity of a body in circular orbit and the acceleration of a body at rest, the new theory gives the same results as classically. This is not the case with the post-Newtonian expansion even if terms at the third post-Newtonian, 3PN, level are included. Arguably, the major benefit of the new theory is that it presents a method that is much less intricate and more practical to deal with than general relativity, while reproducing most of its results, at least in the spherically symmetric case. While the differences between the final expression and the corresponding expression from the post-Newtonian expansion are small and subtle, the new theory gives results that in several ways are closer to both the classical results and to what the Schwarzschild solution predicts.