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Suppose that you deposit an astronomer, armed with our current knowledge of orbital mechanics, on a dome on the far side of the Moon, so that the Earth is perpetually hidden from them.

(And, of course, assume that this person has no specific knowledge about the system they're in beyond what they can glean from observations. If you will, imagine that they learned all our modern orbital mechanics and related physics in alpha centauri, and then got teleported to our Moon.)

Now, it is reasonable to expect that this person should be able to deduce from observations of the sky that the body they are on is one half of a binary system, and they should be able to measure the orbital characteristics (semi-major axis, ellipticity, inclination) as well as the position of the barycentre (much closer to the other body, corresponding to a much more massive partner). What observations are needed to deduce this? What level of observational accuracy is needed for those observations, and to what historical epoch does it correspond to? (I.e. would Tycho Brahe's kit have been sufficient? Would Galileo's? Would the ancient Greeks'? Or would this require a late-19th-century (or even later) observatory?)

(As pointed out in MartinV's answer, our astronomer might find it hard to distinguish between situations with an orbiting pair vs one single huge body. Thus, if convenient, you can assume that, via short ~100km forays from the dome, our astronomer is able to measure the lunar radius by measuring solar inclinations at different points with known distances between them, à la Erathostenes.)

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  • $\begingroup$ Stellar parallaxe would definitively help a long way, and that's 19th century. $\endgroup$
    – user18466
    Sep 5, 2017 at 13:26
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    $\begingroup$ @LucJ.Bourhis Stellar parallax, or at least its leading component, goes with the orbital radius around the Sun, and the lunar-orbit components would be much smaller than that, so that looks like a non-obvious solution to me (and it's not obvious that 19th-century observations would make the needed accuracy, either). I suspect the likeliest candidate is the parallax of the Sun w.r.t. the stellar background (or, equivalently, the stars' positions w.r.t. a clock synchronized with the lunar day), but I would like to know what accuracy (compared to historical references) you'd need for that. $\endgroup$
    – Emilio Pisanty
    Sep 5, 2017 at 13:32
  • $\begingroup$ Sure! This is not what I had in mind. Historically, the fact that the Moon is observable from Earth was instrumental in measuring the distance Earth-Sun from which a lot hinges afaiu. I was thinking of using parallax to get a handle on that distance. $\endgroup$
    – user18466
    Sep 5, 2017 at 13:44
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    $\begingroup$ Still the half-month parallax might be useful for something like Mars. The Moon's orbital diameter is ~0.0026 AU, Mars can be ~1AU out, so that angle is ~0.0052 radians or 0.3 degrees, no? Not sure how those compare to stellar parallax observations over the years but it seems like that might shift Mars' position relative to distant stars in an observable way. $\endgroup$
    – CR Drost
    Sep 11, 2017 at 2:35
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    $\begingroup$ I wonder if the Moon's 5 degree inclination would be enough. (5.14 degrees), that works out to about a 9% of it's orbital distance up and down every 29 days) or 1/6th of 1 degree relative to the sun. Relative to Mars on close pass, a bit less. 14 days observing Mars move up or down but not in a consistent way, as in, sometimes up, sometimes down, might be the most noticeable. $\endgroup$
    – userLTK
    Sep 15, 2017 at 5:10

3 Answers 3

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A body tide seismometer on the lunar far side would pick up both the solar tide, and the 20 inch body distortion produced by the earth. While "tidally locked", the moon is not in a perfectly circular orbit, and also wobbles a bit; libration. Your seismometer should pick up both effects.

Watching the parallax of mars cycle every 28 days, as suggested in comments above, might be a simpler way to go.

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  • $\begingroup$ This is an interesting answer. The Earth will indeed raise a small "tide" on the Moon and libration will cause it to move around. The effect is small and subtle - on its own, would it allow a clever scientist to deduce the existence of an unknown and invisible body in space? $\endgroup$
    – MartinV
    Sep 15, 2017 at 15:34
  • $\begingroup$ Seismic effects are an interesting, though it's not clear to me how you could actually measure them. And yes, I agree that parallax is the likeliest answer, but I was looking for something rather more quantitative and detailed. $\endgroup$
    – Emilio Pisanty
    Sep 15, 2017 at 15:46
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This is a really good question - and quite subtle.

TL;DR;

Earliest opportunity might be that inter-month changes in the stellar parallax of the Sun might lead the observer to conclude that either i) the Moon is a single, very large, rotating body or ii) it is part of a multi-body co-rotating system. However i) would seem to be inconsistent with a near and strongly curved horizon.

If not then, certainly when we develop a quantitative model of orbital mechanics involving mass and gravity

I don't think stellar parallax would directly help us as it (in the modern day) just tells us that we are in orbit around the sun and little about the Earth-Moon system itself.

Let's look at how a Ptolemy equivalent on the Moon (call him Moon-Ptolemy) might see it. He would have no way of distinguishing the Earth-Moon system from his assumption that he is just sitting on a solid object at the center of creation. Of course he wouldn't see a "moon" in orbit around him, but he would see the Sun, stars and major planets. Stellar parallax (to him, the Sun "moving through the Zodiac) would just tell him that Sun is rotating around his Moon, as are the planets. The existence of planetary epicycles would be a curiosity required to make his model work - but it does work and he has no notion of the Earth

Moon-Galileo might (or might not) be able to develop the heliocentric model - he misses out on one key insight that Earth-Galileo had: that the Earth wasn't special because other planets also had moons. Moon-Galileo would find the orbital system of Jupiter interesting but not a key insight, so he might not develop the new model. Even so, someone else would.

Nevertheless, in a qualitative scientific world, there would still be nothing to help the Moon observer deduce the existence of the Earth behind the horizon.

I suspect the truth would become unavoidable when orbital mechanics developed sufficiently to incorporate mass and gravity into calculation. It might have been around the time of Moon-Kepler.

I'm not sure I agree with the comments looking at observations of the planets - I don't see how they help distinguish between an Earth-Moon system as opposed to a simple, very large, rotating Moon body with no co-orbiter (which would be a natural assumption to make). Even the monthly changes in parallax caused by the rotation of the Moon around the Earth might be waved away by suggesting the simple rotation of a much larger Moon body - though our hero might certainly question the compatibility of this with the apparent curvature and distance to their Moon horizon.

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    $\begingroup$ It seems I miscommunicated the intent of the question. You can assume full modern knowledge of orbital mechanics if required ─ when doing historical comparisons, I'm mostly interested in experimental technique, not conceptual advances. If you will, you can think of the situation as an astronomer that learned all of modern orbital mechanics on alpha centauri, and then got teleported to the surface of our Moon. So, they have full knowledge of how gravity and mechanics work, they just don't have any pre-observation knowledge of the specific system they're in. $\endgroup$
    – Emilio Pisanty
    Sep 15, 2017 at 15:10
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    $\begingroup$ But as far as your Moon-Ptolemy goes, wouldn't he need to include an epicycle for the Sun? That's the stellar-parallax observation that would get translated into the Earth-Moon orbital radius once you switched to a heliocentric perspective. But how important would it be, and how hard would it be to measure? $\endgroup$
    – Emilio Pisanty
    Sep 15, 2017 at 15:13
  • $\begingroup$ Thanks Emilio - on the solar epicycle, I think it would likely be too small given the equipment available to Moon-Ptolemy. On your other comment - no miscommunication at all; it was your question, so I think I misunderstood! Given an expert with modern equipment transplanted to the Moon, I think they'd figure it out rather quickly - a combination of the wobbles in parallax combined with the apparent size of the Moon would certainly raise the question. In fact, the expert would look at the small, rocky Moon itself and immediately ask the question "what is this orbiting around?" $\endgroup$
    – MartinV
    Sep 15, 2017 at 15:55
  • $\begingroup$ Yes, my expectation is that they'd figure it out rather quickly; the question was how, and what equipment they would need. (On your last sentence, I don't think it'd be a natural pre-expectation, given how little we know of the prevalence of rocky exoplanets. But from just looking at the surface, the differences between the Moon and Ceres are minimal, so the surface does not necessarily suggest that you're on a satellite. Instead, given the radius and surface gravity, the rocky surface and lack of atmosphere might be quite natural features.) $\endgroup$
    – Emilio Pisanty
    Sep 15, 2017 at 16:11
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An observer on the far side of the moon would have hard time to explain it stands on a single planet, because of the movement of the most noticeable thing in the sky: the sun!

Indeed, because of the eccentricity of the moon's orbit around earth, the day length, i.e. the sun's "speed in the sky", depends on where you stand in your lunar orbit.

And from observations it can do, e.g. other planets that are nearly perfectly round in the solar system (and for well known reasons), she should be forced to rule out the "I'm standing on an elliptical single celestial body" hypothesis.

I cannot calculate the variation of day length on the far side of the moon in a reasonable amount of time, sorry about that.

Another effect I'll try to illustrate with Wikipedia images: the sun trajectory's elevation in the sky would change year after year (cycle: between 8 and 9 earthlings years), due to moon's apsidal precession and its tilted orbit plan:

Moon apsidal precession.png
By Rfassbind - Own work., Public Domain, Link

Lunar perturbation.jpg
By Geologician, Homunculus 2 - from English Wikipedia, CC BY 3.0, Link

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