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A story I wanted to think about has a society centred on the Moon in some colony, and for some reason didn´t have telescopes of their own. Why is not important.

The Moon orbits the Sun in almost the same way the Earth does. It should be possible to construct a lunar-centric solar system as accurate in its predictions as a geocentric one would be, which is nearly perfect for figuring out where things are in the sky at any given point and should be virtually impossible to disprove without telescopes with reasonably precisely engineered mirrors (though the lack of atmospheric haze on the Moon would mean some differences in measurement of things).

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    $\begingroup$ What kind of thing are you thinking about? A "well you can do mechanics in an accelerating and rotating frame" or "With enough epicycles you can fit just about anything"? $\endgroup$
    – James K
    Commented Jun 25, 2023 at 12:11
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    $\begingroup$ If your colony is on the near side, it will be fairly obvious to them that the Earth's motion is very different to all the other celestial bodies. $\endgroup$
    – PM 2Ring
    Commented Jun 25, 2023 at 13:19
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    $\begingroup$ FWIW, as a scifi reader, I'd find it hard to accept that they have the technology to thrive on the Moon & view the stars, but they have no telescopes. $\endgroup$
    – PM 2Ring
    Commented Jun 25, 2023 at 13:23
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    $\begingroup$ BTW, you can make Moon-centred orbit plots with my Horizons script at the end of this answer. astronomy.stackexchange.com/a/49823/16685 By eye, they're indistinguishable from Earth-centred plots. $\endgroup$
    – PM 2Ring
    Commented Jun 25, 2023 at 13:33
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    $\begingroup$ I’m voting to close this question because this is more suited to [worldbuild.se] where this sort of subject is better explored. $\endgroup$
    – StephenG - Help Ukraine
    Commented Jun 25, 2023 at 13:33

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Epicycles are circles moving on circles. And since the position of a point on a circle can be expressed in terms of sin and cos, the position of a point moving along an epicycle is a sum of sin and cos. This is essentially a Fourier series.

And Fourier series can approximate nearly anything.

So by fitting enough epicycles, you can describe pretty nearly any motion, to as much accuracy as you need.

From the moon, the stars would move over the sky every 27.5 days. The sun and distant planets would have a motion that could be described by epicycles, and the Earth would have a strange wobbling motion in the sky, this too could be described by epicycles, if so desired.

Parallax would make the position of Mars and Venus measurably different, relative to the stars, compared with the position from Earth, but again an epicycle or two could account for this motion as well. Clever use the the equant and deferent method (perhaps taken to a couple of levels) could save you some epicycles.

The basic way you can describe the motion of a planet is to look at it's most significant motion, approximate that as a circle, subtract that circle from it's motion and see how it deviates, then add an epicycle to describe the deviation, and repeat.

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