On the existence of precession of planets' orbits in Newtonian gravity

In a paper published in the mainstream journal Physics of the Dark Universe (C. Corda, Physics of the Dark Universe 32 (2021) 100834) we have shown that, contrary to a longstanding conviction older than 160 years, the precession of planets' orbits exists in Newtonian gravity if ones correctly analyzes the situation without neglecting the mass of the planet. In any case, the predicted Newtonian result was too large with respect to the observational values, despite it was, coincidently, in good accordance with the observational value of the precession of Mercury's orbit. In this new paper the situation is reanalyzed in Newtonian physics. It will be indeed shown that, despite the orbit's precession does not occur when the reference frame of the Sun is approximated as being fixed with respect to the fixed stars, it occurs, instead, in the real (in Newtonian sense) non-inertial reference frame of the Sun and it is due to the well known fact that, in a Newtonian framework, the distance which is travelled by a body depends on the reference frame in which the motion of the body is analyzed. After reviewing the solution of the problem which analyzes the planet's orbit as a harmonic oscillator, it will be shown that the precession is due to the breakdown of the conservation of the Hamilton vector in the non-inertial reference frame of the Sun. This approach will give a value of the same order of magnitude of previous result, but larger than that one and again without consistency with the observational values. In other words, it will be shown that it is not correct that Newtonian theory cannot predict an anomalous rate of precession of planets' orbits. The real problem is instead that a pure Newtonian prediction is too large to be consistent with the observational values.