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.