Constraining the Preferred-Frame $\alpha_1$, $\alpha_2$ parameters from Solar System planetary precessions

Analytical expressions for the orbital precessions affecting the relative motion of the components of a local binary system induced by Lorentz-violating Preferred Frame Effects (PFE) are explicitly computed in terms of the PPN parameters $\alpha_1$, $\alpha_2$. A linear combination of the supplementary perihelion precessions of all the inner planets of the Solar System, able to remove the a-priori bias of unmodelled/mismodelled standard effects such as the general relativistic Lense-Thirring precessions and the classical rates due to the Sun's oblateness $J_2$, allows to infer $|\alpha_1| \leq 6\times 10^{-6}, |\alpha_2| \leq 3.5\times 10^{-5}$. Such bounds should be improved in the near future after processing the data that are being collected by the MESSENGER spacecraft, currently orbiting Mercury. Further improvements may come in the mid-future from the approved BepiColombo mission to Mercury. The constraint $|\alpha_2|\leq 10^{-7}$ existing in the literature is critically discussed (Abridged).