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I read that in the Ptolemaic model (or the geocentric system), for the retrogade motion to occur at opposition, the radii of the outer planets' epicycles have to be aligned with the earth-sun radius. For that reason the epicycles of Mars Saturn and Jupiter had to be exactly one solar year, so that it would align with the direction of the earth-sun radius. See for example here. This is one of the anomalies that couldn't be explained until Copernicus came along (namely, why the epicycles of the outer planets share the same time period as opposed to the inner planets), see here.

But I'm not sure I really understand why this is so. Can anyone explain this to me in simple terms (no math please)? A diagram would really help, but I wasn't able to find a diagram or a video that would specifically illustrate this. Most diagrams would just show the general alignement of the radii vectors of the outer planets to the sun's, but would not show how this applies to when Mars is at opposition, or why this wouldn't work if they weren't aligned (I do understand why the inner planets' deferents had to be a solar year though).

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It’s a little difficult to explain, so I have created an external web page that features animations you can play to see why this is the case. There is no math to understand; it’s all visual.

The page is at https://ecliptiqc.ca/Ptolemy.php

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  • $\begingroup$ Wow. Thank you so much!! I didn't expect an entire page with an animation! It would be hlepful though if I was able to replay the animation a few times, as it is now, you can't replay animation 1, so I have to referesh the page every single time, which is a bit annoying. I like the fact that it stops at opposition, so you realize what's going on. In any case, thank you for your animation, this will help me and others. You may also want to include a word on the fact that Ptolemy made the outer planets epicycles a solar year for this reason, so that they can be parellel to the earth sun radius. $\endgroup$
    – Bach
    May 2 at 14:29
  • $\begingroup$ Your (first) wish was my command: The animations may now be restarted without having to reload the page. As for the epicycles being a solar year, they are not. They are the difference between the planet’s synodic period and the solar year. For example, Saturn moves almost a degree per day on its epicycle, whereas Mars moves at not quite a degree per day on its epicycle. $\endgroup$
    – Pierre Paquette
    May 3 at 23:31
  • $\begingroup$ But if that was the case, wouldn't the alignment of the radii with the earth-sun radius eventually get misaligned? I thought the epicycles had to be exactly one solar year (to ensure that it stayed aligned with the sun-earth radius), but the deferent of Mars was 1.88 and longer for the other planets, that's why opposition of Mars wouldn't happen only once in roughly two years. $\endgroup$
    – Bach
    May 4 at 16:29
  • $\begingroup$ It depends on your point of view. With respect to the stars, yes, the epicycle radius does have a period of exactly one Earth year (neglecting precession), but not with respect to the line that joins the Earth to the center of the epicycle; otherwise, we would get an opposition every Earth year. This line’s direction changes as the epicycle moves on the deferent, so the epicyclic period is not a solar year with respect to it. $\endgroup$
    – Pierre Paquette
    May 4 at 22:58

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