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This evening I have been thinking about the anxiety caused in 1982 by predictions in John Gribbin and Stephen Plagemann 1974 book The Jupiter Effect.

I was a teenager in 1982. The planets aligning somewhat together on one side of the Sun did not cause any problems on Earth.

I am curious to know how frequently the same 5 planetary positions recur, say Mercury in Aries, Venus in Scorpio etc., when we restrict ourselves to the 5 naked eye planets. (By the way, I do not believe in astrology; I am just trying to find the date of an ancient text that quotes planetary positions).

Say, if the orbital periods of Mercury, Venus, Mars, Jupiter and Saturn are 1/4, 2/3, 2, 12 and 30 years, then the least common multiple (LCM) is 60 years. So, the same planetary positions will repeat every 60 years. Is my logic correct, though my orbital periods are not exact? I know very well that the 5 visible planets do not have the resonant orbital periods that I have mentioned above, so I am sure that the same planetary positions do not recur every 60 years. If I used the correct orbital periods, will I get the correct answer to my question, whatever the correct answer may be? The above Wikipedia article says that the 1982-type planetary alignment occured in 1128 AD. One of the article's references says that all planets more or less line up behind each other on the same sector of the sky once every 500 years.

Thus, my real question is: Is there a simple method to calculate (without using astronomy software) how frequently the same combination of planetary positions recur?

The method does not have to be too accurate.

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The inner planets are not resonance orbits with each other, this means that the ratio Earth-year:Martian-year is irrational. And the exact postions of the planets don't recur.

The positions do approximately recur, but the time between approximate recurrences depends on how accurate you want the recurrence to be. "accurate to 90 degrees?" "15 degrees?" "1 degree?" Are you only counting right-ascension, or are you also looking at the declension too (the planets don't orbit in exactly the same plane) Are you willing to make Keplarian approximations, or do you want to include perturbations.

The motions of the planets are simple enough physics, but not simple ratios. So predicting a "recurrence to a particular accuracy" needs physics to forecast.

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  • $\begingroup$ I am looking for an accuracy of + or - 15 degrees. Can assume that all planets are in the ecliptic, so declension does not matter. No need to include perturbations. $\endgroup$
    – doctorsundar
    Commented Oct 5, 2020 at 20:29
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Thus, my real question is: Is there a simple method to calculate (without using astronomy software) how frequently the same combination of planetary positions recur?

The method does not have to be too accurate.

Yes indeed!

You have the right idea. If the periods of "Mercury, Venus, Mars, Jupiter and Saturn (were) 1/4, 2/3, 2, 12 and 30 years" and since they all to into 60 an integral number of times (240, 90, 30, 5, and 3) then the configuration will repeat over and over.

And as the OP seems to indicate that they know or suspect, if the relationships were very close, the configuration from one repeat to the next one 60 years later might be similar.

However as they probably also suspect if Mercury's period is off by only 1 part per thousand and it orbits 240 times, it's going to be roughly a quarter-orbit off.

So this method has poor "dynamic range" in that with any imperfection it breaks down quickly if the shortest period is much smaller than the LCM.

This line of thought is what every Orrery builder goes through when they choose how many teeth to put on their gears.

Of course if you want to use off-the shelf gears with standard tooth numbers instead of DIY, or you want to keep max to min sizes within a restricted range (it's hard to do 240:1 all at once) then you have to get more clever!

Results of Zeamon's Orrery gear ratios:

Object       Actual      Orrery      Orrery
              days        days        years
Mercury      87.969      88.418      0.242071
Venus       224.701     224.870      0.61565
Earth       365.256     365.256      1
Moon         27.322      27.3322     0.07483
Mars        686.980     688.6454     1.885377
Jupiter    4333.036    4314.8143    11.81312
Saturn    10755.704   10807.1021    29.58775
Century   36525.636   36621.5429   100.2627

enter image description here

File:Orrery_small.jpg File:Frederiksborg_slot_-_Museum_20090818_28.JPG

Source and Source

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  • $\begingroup$ Many thanks for your long and detailed reply. In your orrery, what is the time interval in years between any 2 repetitions of the same 5 planetary positions, ignoring the moon? I realize that the second time around the 5 planetary positions may be a little off compared to where they were initially, $\endgroup$
    – doctorsundar
    Commented Oct 5, 2020 at 20:40
  • $\begingroup$ @doctorsundar That's an interesting question! I've included links and discussion to orrerys just to give context to why approximations that will eventually, inevitably repeat are still of interest to some people, even thought they don't reflect reality. I wanted to add some balance to the other slightly dismissive answer. It's not "my" orrery :-) although I wish it were! I don't know the answer to that question right now. $\endgroup$
    – uhoh
    Commented Oct 6, 2020 at 1:51
  • $\begingroup$ @doctorsundar These are based on meshing gears, each with an integer numbers of teeth, and connected by rigid shafts. Object's cycles may be "a little off" from each other one orbit to the next, but after running long enough this system must come back to its previous position because it will always have a LCM as you mention in your question. But it could be very large if enough shafts are used. Imagine a 100:101 pair with a shaft going to a second 101:102 pair, and in parallel a 103:104 pair with a shaft going to a second 104:105 pair. $\endgroup$
    – uhoh
    Commented Oct 6, 2020 at 1:56
  • $\begingroup$ @doctorsundar I don't know how to do the math exactly, but those hypothetical planets must require thousands or perhaps tens of thousands of slowly de-phasing orbits before they return to their original configuration with respect to each other, but eventually they must. However it may take millions of orbits before they return to the same configuration together with a third orrery object. That might be the basis of a new and interesting question! If carefully written, it might be better asked (and better received) in Math SE than here. $\endgroup$
    – uhoh
    Commented Oct 6, 2020 at 1:59
  • $\begingroup$ @doctorsundar if you do ask there, definitely include a link back to your question here and mention what information isn't yet available in answers there. They are a kind and helpful community, and as long as you describe a specific mathematical problem you often get several good answers, even if eventually closed :-) $\endgroup$
    – uhoh
    Commented Oct 6, 2020 at 2:02

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