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From Newtonian mechanics, the precession of Mercury can be calculated by taking into account the gravitational pull of other planets. From that, I assume that in the absence of external planets, Newtonian mechanics predicts that the elliptical orbit does not precess at all.

From General Relativity, it is known from the Schwarzschild metric that there is an additional relativistic term which has to be taken into account in order to calculate the precession exactly. However, does this precession still occur in the absence of other planets? (so just a two body system with the Sun and Mercury). If so, how far off would this result be from the real result (when ignoring other planets)?

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As tabulated e.g. at this Astronomy.SE question, Mercury's perihelion is observed to precess by about 580 arcseconds per century. The majority of this precession is due to three- and four-body interactions involving the other major planets. About 10% of the effect, 43 arcseconds per century, was unexplained until the publication of general relativity.

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    $\begingroup$ There had been an attempt to explain the precession by postulating a hypothetical planet Vulcan, but this was never found. So maybe"unexplained" could be "not successfully explaind" Since the precession of Mercury is not the only experimental verification of General Relativity, planet Vulcan has suffered a more ignominious demotion than Pluto, to non-existent. $\endgroup$
    – Simon Crase
    Commented Sep 22 at 1:22
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The inter webs https://en.wikipedia.org/wiki/Tests_of_general_relativity answers the stats of this question, and seems to have some equations too.

But: the other planets completely dominate the process (532 "/C).

$J2_{sol}^{\dagger}$ is 0.03"/C.

The geodetic effect is the famous 42"/C (really 43)

Frame dragging is a pathetic -0.002 "/C.

[$\dagger$] $J_2$ is the oblateness, and does anyone know how to do the planet symbols in LaTex? (It's a little late for me, as I am back to LEO, GEO, and HEO).

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    $\begingroup$ Hmm, so this does suggest that the influence of other planets is actually the dominant effect here, and the geodetic effect is a smaller correction.. $\endgroup$
    – Stallmp
    Commented Sep 19 at 12:50
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    $\begingroup$ Yup, I was disappointed when I first heard that--just because. But it's hard measurement. Most meaningful astrometry is hard, but has had a huge impact on science--like finding Uranus. $\endgroup$
    – JEB
    Commented Sep 19 at 12:52
  • $\begingroup$ I don't think the planet symbols are available in Mathjax, but in Latex you want this Q&A on the Tex SE. To be honest I've always found the planet symbols confuse most people, but if that's what you want, go for it. $\endgroup$
    – StephenG - Help Ukraine
    Commented Sep 19 at 15:04
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    $\begingroup$ It's common to use \odot for the Sun symbol, eg $J_{2\odot}$ $\endgroup$
    – PM 2Ring
    Commented Sep 20 at 6:07
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The thing I think you’re talking about is real, but it has little to do with the other planets.

There are actually two relativistic terms: the geodetic precession term and the frame-dragging precession, a.k.a. the Lense-Thirring effect. The first is due to the presence of the Sun’s spacetime curvature, and the second is due to the effect the Sun’s rotation has on spacetime.

Both effects cause orbital precessions (mostly the geodetic effect, though), in effect because spacetime geometry (and thus space, time, and gravity) are all slightly different at aphelion versus perihelion, which leads to orbits and gyroscopes precessing through space. Both of these effects were confirmed to be real in large part by observing the precession of Mercury’s line of nodes. This is mentioned in later copies of Relativity (mine is the 100th anniversary one, which includes some addenda about later confirmations of relativity). It was confirmed more recently, and to greater precision, by the Gravity Probe B experiment, which very carefully measured both forms of precession around the Earth, with much less mass and angular momentum than the Sun.

Neither form of relativistic precession has anything to do with the other planets. I’d imagine that there would be some relativistic perturbation to Mercury’s orbit due to Venus and maybe Earth, but the other planets are certainly too far away to have a meaningful effect in that way. You don’t need general relativity to account for these perturbations; the planets’ gravitational fields are so weak that the Newtonian approximation for their interactions is essentially valid, and you can calculate the precessions they cause with simple Newtonian mechanics. In the case of the Sun, relativity gets too complex to linearize, and considering the full Kerr metric for the Sun you see geodetic and frame-dragging terms that, combined with Newtonian perturbations from the other planets, closely match what is observed.

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    $\begingroup$ Thanks a lot for your answer! I thought that Newtonian gravity gave quite a significant non-zero result for the precession (of course still smaller than the one using GR) which was attributed to other planets? Or is that in NG purely due to frame-dragging precession as you mentioned? $\endgroup$
    – Stallmp
    Commented Sep 19 at 12:42
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    $\begingroup$ FD precession is a result of GR; perturbations from other planets is still considered in NG, but I’ve never found it to be useful compared to the result from the geodetic effect. $\endgroup$
    – controlgroup
    Commented Sep 19 at 12:44
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    $\begingroup$ The admitted guess that other planets contribute neglibly to the precession, so that the precession is mostly from relativity, is just plain wrong. $\endgroup$
    – rob ♦
    Commented Sep 19 at 12:53
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    $\begingroup$ I didn’t write what I meant: I was referring to particularly relativistic precession because of the other planet’s gravity wells (I misread the question it seems). $\endgroup$
    – controlgroup
    Commented Sep 19 at 12:56
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    $\begingroup$ Oh, I see that you might have been trying to say that there isn't a need for a relativistic correction from the other planets. In that case, I suggest you rephrase. $\endgroup$
    – rob ♦
    Commented Sep 19 at 12:58

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