Might the 2PN perihelion precession of Mercury become measurable in the next future?

The Hermean average perihelion rate $\dotω^\mathrm{2PN}$, calculated to the second post-Newtonian (2PN) order with the Gauss perturbing equations and the osculating Keplerian orbital elements, ranges from $-18$ to $-4$ microarcseconds per century $\left(μ\mathrm{as\,cty}^{-1}\right)$, depending on the true anomaly at epoch $f_0$. It is the sum of four contributions: one of them is the direct consequence of the 2PN acceleration entering the equations of motion, while the other three are indirect effects of the 1PN component of the Sun's gravitational field. An evaluation of the merely formal uncertainty of the experimental Mercury's perihelion rate $\dotω_\mathrm{exp}$ recently published by the present author, based on 51 years of radiotechnical data processed with the EPM2017 planetary ephemerides by the astronomers E.V. Pitjeva and N.P. Pitjev, is $σ_{\dotω_\mathrm{exp}}\simeq 8\,μ\mathrm{as\,cty}^{-1}$, corresponding to a relative accuracy of $2\times 10^{-7}$ for the combination $\left(2 + 2γ- β\right)/3$ of the PPN parameters $β$ and $γ$ scaling the well known 1PN perihelion precession. In fact, the realistic uncertainty may be up to $\simeq 10-50$ times larger, despite reprocessing the now available raw data of the former MESSENGER mission with a recent improved solar corona model should ameliorate our knowledge of the Hermean orbit. The BepiColombo spacecraft, currently en route to Mercury, might reach a $\simeq 10^{-7}$ accuracy level in constraining $β$ and $γ$ in an extended mission, despite $\simeq 10^{-6}$ seems more likely according to most of the simulations currently available in the literature. Thus, it might be that in the not too distant future it will be necessary to include the 2PN acceleration in the Solar System's dynamics as well.

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