I figured out that the osculating semi-major axis of the planet orbits in Horizons is calculated from the observed/computed state vectors by assuming the kinetic energy as given by the observed velocity vector (i,e. by the disturbed orbit), but the potential energy as given by the gravitational field of the sun only i.e. the undisturbed orbit. The osculating semi-major axis is then calculated from the total energy (kinetic + potential energy) via the classical relationship for the gravitational 2-body problem.
Does anybody know what the philosophy behind this is? Would it not be more meaningful and consistent to calculate the potential energy including the effect of the other bodies in the solar system as well?
Of course, one can in principle 'encode' the observations through any scheme one likes, but this may then have no more value than using Ptolemy's theory of epicycles to represent the orbits of the planets. The point is that the Keplerian elements are still nowadays frequently used in quantitative astronomy, especially when discussing secular (long-tern) changes of orbits. And even when averaging over long times, the gravitational potential in the solar system will be different from that of the sun alone, so the orbits won't be given by the classical 2-body equations anymore.
To further clarify my point:
My question is about the way the osculating elements (and the mean orbital elements derived from this) are calculated in Horizons, namely, as I found, by considering only the main mass (e.g. the sun) when computing the gravitational potential (from the measured x,y.x positions) but ignoring the gravitational potential of the other masses (e.g. other planets).
Including instead also the gravitational potential of the disturbing masses would (negatively) increase the overall gravitational potential energy of the system and therefore also the total energy. This in turn would reduce the semi-major axis and orbital period by a significant amount. Considering that the semi-major axis is identical to the average distance between the masses in the two-body problem, it should, like the orbital period, be however an objective and unique quantity, and there should therefore be only one unique way of calculating these from the state vectors.
So in my view it is more than just a question of being useful or not.