On the possibility of measuring the solar oblateness and some relativistic effects from planetary ranging

In this paper we first calculate the post-Newtonian gravitoelectric secular rate of the mean anomaly of a test particle freely orbiting a spherically symmetric central mass. Then, we propose a novel approach to suitably combine the presently available planetary ranging data to Mercury, Venus and Mars in order to determine, simultaneously and independently of each other, the Sun's quadrupole mass moment $J_{2\odot}$ and the secular advances of the perihelion and the mean anomaly. This would also allow to obtain the PPN parameters γ and β independently. We propose to analyze the time series of three linear combinations of the observational residuals of the rates of the nodes $\dot\Omega$, the longitudes of perihelia $\dot\pi$ and mean anomalies $\dot{\mathcal M}$ of Mercury, Venus and Mars suitably built up in order to absorb the secular precessions induced by the solar oblateness and the post-Newtonian gravitoelectric forces. The values of the three investigated parameters can be obtained by fitting the expected linear trends with straight lines, determining their slopes in arcseconds per century and suitably normalizing them. According to the present-day EPM2000 and DE405 ephemerides accuracy, the obtainable precision would be of the order of 10-3–10-4 for the PPN parameters and, more interestingly, of 10-9 for $J_{2\odot}$. It must be pointed out that the future BepiColombo mission should improve the knowledge of the Mercury's orbit perhaps by one order of magnitude