<p>The measurements discussed weren't done <em>by</em> Messenger, they were done by round-trip delay-doppler radar ranging <em>using</em> the coherent transponder of Messenger's <a href="https://en.wikipedia.org/wiki/Radio_science_subsystem" rel="nofollow noreferrer">Radio_science_subsystem</a>.</p>

<p><a href="https://core.ac.uk/download/pdf/83235115.pdf" rel="nofollow noreferrer">Park et al. (2017)</a> discuss separating the effects of the solar quadrupole moment from the partially degenerate effects of the post-Newtonian <span class="math-container">$\beta$</span> and <span class="math-container">$\gamma$</span> parameters (where <span class="math-container">$\beta=\gamma=1$</span> in General Relativity). The "general" General Relativistic correction to the perhelion prececssion itself is given by:
<span class="math-container">$$\Delta \dot{\omega} \propto (2 - \beta + 2\gamma) $$</span>
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 <span class="math-container">$J_2$</span> without assuming the value of <span class="math-container">$\beta$</span> and <span class="math-container">$\gamma$</span>, and even then would need precisons in position measurements of hundreds of metres over decades.</p>

<p>Park et al. instead show that separating <span class="math-container">$J_2$</span> from <span class="math-container">$\beta$</span> and <span class="math-container">$\gamma$</span>
 requires a precise measurement of a <em>periodic</em> 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 <span class="math-container">$\beta$</span> and <span class="math-container">$\gamma$</span> only, and if the amplitudes of these modulations can be measured to better than <span class="math-container">$\sim 0.07$</span>% (<span class="math-container">$\simeq 3.5-7$</span> m) then <span class="math-container">$J_2$</span> could be determined independently of <span class="math-container">$\beta$</span> and <span class="math-container">$\gamma$</span>.</p>

<p>Given that the position of the spacecraft can be tracked to a lttle better than 1 metre, this is possible. <span class="math-container">$J_2$</span> is indeed measured to about 4%, and half of this uncertainty is due to remaining degeneracy with the <span class="math-container">$\beta$</span> parameter and uncertainty in the adopted <span class="math-container">$\gamma$</span> parameter.</p>

<p>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. </p>

<p>Indeed on p.181 of the 1993 version of "<a href="http://202.38.64.11/~jmy/documents/publications/Clifford%20M.%20Will%20Theory%20and%20Experiment%20in%20Gravitational%20Physics%20%201993.pdf" rel="nofollow noreferrer">Theory and Experiment in Gravitational Physics</a> by Will, it is said that </p>

<blockquote>
  <p>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]</p>
</blockquote>

<p>and then on p.183</p>

<blockquote>
  <p>The accuracy required for such measurements would necessitate tracking of a spacecraft in orbit around Mercury</p>
</blockquote>

CC BY-SA 4.0
2022-01-07T14:20:07.290Z
>"Given that the position of the spacecraft can be tracked to a lttle better than 1 metre" ... does that 1m accuracy just apply to radial distance from Earth-based observer - or does it also apply to transverse position (which is a function of direction)?
steveOw
CC BY-SA 4.0
2020-05-10T11:41:33.617Z
2020-05-10T13:02:45.300Z
2020-05-10T13:02:45.300Z
ProfRob
<p>This <a href="https://astronomy.stackexchange.com/questions/34846/path-of-mercury-and-general-relativity/34847?noredirect=1#comment67030_34847">comment</a> under an answer to <em>Path of Mercury and general relativity</em> 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 <span class="math-container">$J_2$</span> 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". </p>

<p>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.</p>

<p>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 <a href="https://space.stackexchange.com/q/30430/12102">measured to +/- 3000 meters way back in the 1960's</a> and a precision measurement has such importance to general relativity validation.</p>

<p><strong>Questions:</strong></p>

<ol>
<li>How was the Messenger spacecraft used to measure Mercury's orbital precession to such accuracy? </li>
<li>Why couldn't this have been done earlier using a series of radar observations, or could it have?</li>
</ol>

CC BY-SA 4.0
2020-05-10T08:11:37.867Z
You should consider the possibility that it wasn't done for resources efficiency reasons: it might have been done with radar, but wasn't as we had the probe in orbit. Deep sky antenna time is extremely valuable
planetmaker
CC BY-SA 4.0
2020-05-10T08:40:25.590Z
@planetmaker Planetary science and tests of general relativity are also valuable. Radar measurements of Mercury and Venus had value for several reasons. I don't know as much about Mercury and it's a much more challenging target, but Venus has been looked at in radar quite a bit! Rather than consider, let's just get to an answer here.
uhoh
CC BY-SA 4.0
2020-05-10T08:40:38.057Z
@planetmaker Some early radar work on venus: [How was the radius of Venus measured so accurately (± 3 km) via radar in the mid 1960's, before Venera 4 and Mariner 5?](https://space.stackexchange.com/q/30430/12102) and [When did planetary scientists realize Venus' surface pressure was almost 100x that on Earth? How did they find out?](https://space.stackexchange.com/q/30346/12102)
uhoh
CC BY-SA 4.0
2020-05-10T08:41:27.657Z
@planetmaker [**Radar observations in 1965** proved that the planet (Mercury) has a 3:2 spin–orbit resonance, rotating three times for every two revolutions around the Sun. The eccentricity of Mercury's orbit makes this resonance stable—at perihelion, when the solar tide is strongest, the Sun is nearly still in Mercury's sky.](https://en.wikipedia.org/wiki/Mercury_(planet)#Spin%E2%80%93orbit_resonance)
uhoh
<general-relativity><history><mercury><astrometry><radar>
How was the Messenger spacecraft used to measure Mercury's orbital precession to such accuracy? Could this have been done using radar?
uhoh
# How was the Messenger spacecraft used to measure Mercury's orbital precession to such accuracy? Could this have been done using radar?

(Asked by: username_0 on 2020-01-28)

This [comment](https://astronomy.stackexchange.com/questions/34846/path-of-mercury-and-general-relativity/34847?noredirect=1#comment67030_34847) 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](https://space.stackexchange.com/q/30430/12102) 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?

**Tags:** general-relativity, history, mercury, astrometry, radar
> **Comments:**
> **username_1**: You should consider the possibility that it wasn't done for resources efficiency reasons: it might have been done with radar, but wasn't as we had the probe in orbit. Deep sky antenna time is extremely valuable  
> **username_0**: @planetmaker Planetary science and tests of general relativity are also valuable. Radar measurements of Mercury and Venus had value for several reasons. I don't know as much about Mercury and it's a much more challenging target, but Venus has been looked at in radar quite a bit! Rather than consider, let's just get to an answer here.  
> **username_0**: @planetmaker Some early radar work on venus: [How was the radius of Venus measured so accurately (± 3 km) via radar in the mid 1960's, before Venera 4 and Mariner 5?](https://space.stackexchange.com/q/30430/12102) and [When did planetary scientists realize Venus' surface pressure was almost 100x that on Earth? How did they find out?](https://space.stackexchange.com/q/30346/12102)  
> **username_0**: @planetmaker [**Radar observations in 1965** proved that the planet (Mercury) has a 3:2 spin–orbit resonance, rotating three times for every two revolutions around the Sun. The eccentricity of Mercury's orbit makes this resonance stable—at perihelion, when the solar tide is strongest, the Sun is nearly still in Mercury's sky.](https://en.wikipedia.org/wiki/Mercury_(planet)#Spin%E2%80%93orbit_resonance)  

---

## 1 answers

### Answer by username_2 on 2020-05-10. Score: 3 | Accepted answer

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](https://en.wikipedia.org/wiki/Radio_science_subsystem).

[Park et al. (2017)](https://core.ac.uk/download/pdf/83235115.pdf) 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](http://202.38.64.11/~jmy/documents/publications/Clifford%20M.%20Will%20Theory%20and%20Experiment%20in%20Gravitational%20Physics%20%201993.pdf) 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

> **Comments:**
> **username_3**: >"Given that the position of the spacecraft can be tracked to a lttle better than 1 metre" ... does that 1m accuracy just apply to radial distance from Earth-based observer - or does it also apply to transverse position (which is a function of direction)?  

---

