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This comment under an answer to Path of Mercury and general relativity mentions that the Messenger spacecraft was used to measure the precession of Mercury's orbit to such accuracy that the tiny component due to the Sun's $J_2$ could be resolved from other components including relativistic effects and the gravitational pull of other solar-system bodies. The value of this component is given there as "about 0.03 arcseconds/century".

That rate corresponds to a motion of the apses of about 500 meters per year, which I roughly estimate translates to an error in Mercury's position that accumulates at about 100 meters per year.

That seems to be within the a range that could be doable with an extended series of radar measurements. For comparison, in the 1960's Venus' diameter was measured to +/- 3000 meters way back in the 1960's and a precision measurement has such importance to general relativity validation.

Questions:

  1. How was the Messenger spacecraft used to measure Mercury's orbital precession to such accuracy?
  2. Why couldn't this have been done earlier using a series of radar observations, or could it have?
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The measurements discussed weren't done by Messenger, they were done by round-trip delay-doppler radar ranging using the coherent transponder of Messenger's Radio_science_subsystem.

Park et al. (2017) discuss separating the effects of the solar quadrupole moment from the partially degenerate effects of the post-Newtonian $\beta$ and $\gamma$ parameters (where $\beta=\gamma=1$ in General Relativity). The "general" General Relativistic correction to the perhelion prececssion itself is given by: $$\Delta \dot{\omega} \propto (2 - \beta + 2\gamma) $$ and the effect caused by the solar quadrupole moment is only 0.07% the size of this. Therefore just observing the precession does not yield an accurate value of $J_2$ without assuming the value of $\beta$ and $\gamma$, and even then would need precisons in position measurements of hundreds of metres over decades.

Park et al. instead show that separating $J_2$ from $\beta$ and $\gamma$ requires a precise measurement of a periodic modulation (i.e. it isn't an accumulating error) in the argument of perhelion caused by the General Relativistic perturbations, which translates to modulations of order of 5-10 km in position. These also depend linearly on $\beta$ and $\gamma$ only, and if the amplitudes of these modulations can be measured to better than $\sim 0.07$% ($\simeq 3.5-7$ m) then $J_2$ could be determined independently of $\beta$ and $\gamma$.

Given that the position of the spacecraft can be tracked to a lttle better than 1 metre, this is possible. $J_2$ is indeed measured to about 4%, and half of this uncertainty is due to remaining degeneracy with the $\beta$ parameter and uncertainty in the adopted $\gamma$ parameter.

This suggests that precisions of the order of a few metres were required for this experiment and I don't believe this is possible by radar ranging to the planet itself.

Indeed on p.181 of the 1993 version of "Theory and Experiment in Gravitational Physics by Will, it is said that

Unfortunately, measurements of the orbit of Mercury alone are incapable at present of separating the effects of relativistic gravity and of solar quadrupole moment in the determination of [the change in the perihelion precession rate]

and then on p.183

The accuracy required for such measurements would necessitate tracking of a spacecraft in orbit around Mercury

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