Precession of planets' orbits between Newtonian gravity and Einstein's general relativity

The recent result that, contrary to a longstanding conviction older than 160 years, the precession of Mercury perihelion can be achieved in Newtonian gravity with a very high precision by correctly analyzing the situation without neglecting Mercury's mass is confirmed. Orbit's precession does not occur in an inertial (in Newtonian sense) frame of reference, but it occurs in the non-inertial frame of reference of the Sun because in Newtonian theory, the distance which is travelled by a body depends on the frame of reference in which the motion of the body is analyzed. After reviewing a previous solution, the precession is directly obtained from the orbit equation in the non-inertial frame of reference of the Sun. The Newtonian formula of the precession of planets' perihelion breaks down for the other planets because the predicted Newtonian result is indeed too large for Venus and Earth. In a previous paper, it was shown that corrections due to gravitational and rotational time dilation, in an intermediate framework (a particular post-Newtonian approximation of general relativity) which analyzes gravity between Newton and Einstein, solve the problem in accordance with some other results in the literature. By adding such corrections, a result consistent with the one of general relativity was indeed obtained. Those corrections also are re-analyzed in this paper by stressing the difference between general relativity and Newtonian physics by evoking the Einstein Equivalence Principle, and the third postulate of relativity. This framework of gravity between Newton and Einstein seems consistent with global covariance. Finally, it is important to stress that a better understanding of gravitational effects in an intermediate framework between Newtonian theory and general relativity, which is one of the goals of this paper, could, in principle, be crucial for a better understanding of the famous Dark Matter and Dark Energy problems.