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I'm working on a flintlock fantasy series and I want to figure out how to get three comets with different orbital periods to occasionally all show up at the same time. All three are "great comets" that are visible to the naked eye and have noticeable tails. (Ideas for how to make them distinct from each other would also be nice.) My thinking is that when they do, it does, in fact, mean a disaster is coming - because they cause all kinds of Undead to rise up, but that's not related to their orbital periods, is it? (Though it does relate to why all three are green in color.) This triple apparition only happens once every 1,125 years. Obviously, this means that I need to figure out orbital periods that only overlap once, however, I want each comet to show up at other times between the triple apparitions, which means none of them can have an orbital period of 1,125 years. Comet 1 has an orbital period of 45 years, so it shows up 25 times during the cycle. I'm not sure how to determine the orbital periods for Comet 2 and Comet 3. I know Comet 3 needs to be a long-period comet, but Comet 2 could be either a short-period or long-period comet. I'm okay with having double apparitions as well, but I would need to figure out how often those would occur and which comets would be involved. So, to sum up:

Total Time between Triple Apparitions: 1,125 years

Comet 1 Orbital Period: 45 years

Comet 2 Orbital Period: ??? years

Comet 3 Orbital Period: ??? years

Are double apparitions possible? If so, when and which comets are involved?

Thanks for any help you guys can offer and be sure to have fun! I'd like this cycle to be as interesting as possible, but I'm terrible with numbers, so I really appreciate having assistance from people who can solve puzzles like this.

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Several possibilities.

One combo is 25, 45, & 375.
Another is 45, 125, & 375.

The trick is to find the prime factors of your desired conjunction, and the ways to arrange them so that your starting number is the smallest common multiple of them all.

For your 1125 target, your factors are 3, 3, 5, 5, 5. This limits the number of options because the same factors show up so often. If you were to adjust to use 1050 (2,3,5,5,7) for example you would have many more options because you have fewer cases where one combination is a factor of another.

Edit

Going with 1125, there are 10 unique factor that can make up this number, excluding 1 & 1125 (a comet with same orbital period as the planet is not very interesting, and probably wouldn't count as a comet, while one that only shows up for the triple conjunction might as well be the only one). They are: 3, 5, 9, 15, 25, 45, 75, 125, 225, & 375. A comet with any of these orbital periods will show up every 1125 years, the trick is picking the set which gives the frequency that you want.


You noted 45 as a desired period (call this A), so we can examine the options compared to it...

For periods of 3 - 15 year, they will be in sync with every visit of A (all of these options are factors of 45), again not all that interesting.

However if B has a period of 25 years, then A & B will align every 225 years. That has potential so lets go with it for a moment. Our options for C at this point are 125 & 375, anything else is going to have your triple hitting too frequently. For both of these, A & C will only align at the triple conjunction every 1125 years, also B & C will align every time C shows up. Seems too predictable...

Next option for B would be 75 years, again A & B align every 225 years. C options are still the same, meaning that A & C are still only going to hit together every 1125 years. However, B & C do offer some variety this time around. If C is 125, then B & C will align every 375 years. If C is 375, then we again have B & C in alignment every time C comes around.

Next option for B is 125. Now A & B are only going to sync up every 1125 years. Your options for C 225 & 375. With 225, A & C will align every time C shows up, while B & C will only align at 1125 years. With 375, A & C will align at 1125 while B & C will align every time C comes by. None of those seem very likely to fit what you are after.

Your last option is A=45, B=225, & C=375. In this case, you don't get any double alignments, just the triple every 1125. That doesn't seem to fit either.

In all if you are set on 1125, and 45 for one of the periods, then 75 & 125 seem like your best bets for the other two.


Alternatively, if you adjust to use 1050 as the period of your triple conjunction, then you get 22 unique factors to play with. They are 2, 3, 5, 6, 7, 10, 14, 15, 21, 25, 30, 35, 42, 50, 70, 75, 105, 150, 175, 210, 350, & 525.

Just to pick one that looks interesting, A = 30, B = 35, & C = 50. In this configuration you have between 5 and 30 years between passes of a comet, with an average of ~14 years. A & B align every 210 years, A & C every 150 years, and B & C every 350 years. This gives you doubles somewhere between 30 and 150 years apart, and all 3 come together every 1050 years.

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  • $\begingroup$ So, if the time between the triple apparition is 1,050 years, what would be a setup that would feature, say, at least one double apparition for each possible pairing of the comets? (1 & 2, 2 & 3, and 1 & 3.) I want these things to be a serious pain in the ass even when they don't all show up at the same time. They're also one of the reasons astronomy developed the way it did on the planet: The people needed to determine when they'd be showing up again so they'd be ready to deal with the Undead becoming more active during the apparitions. (Huh. Undead. Apparitions. I just noticed that.) $\endgroup$ – Patrick-Leigh Dec 6 '18 at 5:23
  • $\begingroup$ I did this off the cuff last night, give me a bit and I can give you a break down of your options... Also note that I am plotting for them to be "in alignment" on schedule, there may be other near passes where they come one after another, but don't quite coincide. $\endgroup$ – Rozwel Dec 6 '18 at 13:49
  • $\begingroup$ Oooh, this is very nice! Thank you so much! If I do the 30, 35, 50 setup, it means that the comets would be frequent events, but individually they wouldn't be that big a problem. It's those double apparitions that are more of a hassle and finally the triple whammy. And, best of all, it's still not exactly 1,000 years, which is so cliche it's not even funny. 1,050 misses the mark by a small amount, sure, but it's enough to satisfy me. Very much appreciated! Thanks! $\endgroup$ – Patrick-Leigh Dec 7 '18 at 4:16
  • $\begingroup$ As a side note, working through the orbits on this one has me wanting to go play Kerbal again... $\endgroup$ – Rozwel Dec 8 '18 at 0:44
  • $\begingroup$ Yeah, I can see why this topic would have that effect! $\endgroup$ – Patrick-Leigh Dec 8 '18 at 2:29
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The term is least common multiple. For example, 45, 75, and 125 have a least common multiple of 1125.

Appearances: 45, 75, 90, 125, 135, 150, 180, 225 (45 and 75), 250, 270, 300, 315, 360, 375 (75 and 125), 405, 450 (45 and 75), 495, 500, 525, 540, 585, 600, 625, 630, 675 (45 and 75), 720, 750 (75 and 125), 765, 810, 825, 855, 875, 900 (45 and 75), 945, 975, 990, 1000, 1035, 1050, 1080, 1125.

  • Every 1125 years, all three.
  • Every 375 years, the 75 and 125. The 1125 will be the third conjunction of those two in the period.
  • Every 225 years, the 45 and 75. The 1125 will be the fifth conjunction of those two in the period.
  • The 45 and 125 only appear at the same time in the triple conjunction, because 1125 is the least common multiple of 45 and 125.

As previously noted, there are other options. However, unless you want one of the comets to only appear in conjunction with another comet, you want to avoid options where one period directly divides another. For example, 25 and 75 or 25 and 125. With either of those, the 75 or the 125 would only appear together with the 25. Because the least common multiple would be 75 or 125 (25 divides either).

If you wanted all possible pairs to have dual conjunctions, you would need a period that factored into more than two prime factors ($3^2\cdot 5^3 = 1125$). Essentially you'd want each comet's period to include its own prime factor. With 1125, we're basically limited to some combination of threes and fives.

$$45 = 3^2\cdot 5$$ $$75 = 3 \cdot 5^2$$ $$125 = 5^3$$

And since you already chose 45, we can't use 3, 5, 9, or 15 without 45 always appearing with whichever of those. If you allowed something other than 45, we could have 9, 75, 125 or similar. To get the triple conjunction at 125, we need a multiple of 9 and a multiple of 125. Because 9 and 125 have 1125 as the least common multiple.

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  • $\begingroup$ This is perfect! The comets are a regular enough occurrence to be a mild problem when they show up individually while the dual conjunctions would pose more of a problem, yet do not occur with as much regularity. Then you get the triple whammy, which is unique in that it is the only time where 45 and 125 get to be together. So, it's like a really horrible family reunion. Maybe I should play with that kind of theme with how they're named. "The Twisted Sisters" or something like that, maybe? Anyway, thank you so much. This is a big, big help! $\endgroup$ – Patrick-Leigh Dec 7 '18 at 2:39

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