How Should I Make a Calendar for Trappist-1

Alright, so I have already asked a few questions about this system, but this is the most important one. Firstly, I am needing to see if I got a relatively simple math problem right, and then I need help figuring out a problem where I completely lack any ability to solve it. Note, the system is going to be ever so slightly modified to make all the math simpler. Assume all orbital ratios are perfect and the orbits are perfect circles and that they are all on exactly the same plan. (magic accounts for the slight tinkering with reality) Also, assume there are no moons or other visible planets.

Firstly, I wanted to figure out what my calendars "month" was. Now, since Trappist-1's seven known worlds have an orbital resonance of 2:3:4:6:9:15:24, it seems to me like these orbits are going to end up in the same position, relative to one another, very often. If I understand this ratio right, every time the outer planet performs two orbits, the next planet will be in the same place it was 2 outer planet orbits ago, and the same is true for every other planet. In order to get this into a number I could easily use for World Building, I took one of the orbits, (the sixth planet, Trappist-1g) divided it into 12 equal units, and called those days. Since there are 12 days every sixth planet orbit, and three sixth planet orbits every cycle, the planets of Trappist-1 should repeat themselves every 36 "days". Since these days are nearly the same size as earth days, this means that this time orbital resonance cycle (which I originally intended to be Trappist's year) would most accurately be called a "month" in English terms. My simple question here is, did I get this math right?

Now, for the complicated question. One of the important parts of the calendar I do not know how to solve is, how do I determine at what days and times the planets will transit one another? I know the Trappist planets have orbital periods (in Trappist "days") of (farthest out first) 18,12,9,6,4,2.4,1.5. However, I still don't know how this can help me determine when these planets transit each-other at various points during the month. Beyond this, I also need to determine what the "sub-transit" point is on each world (the Trappist worlds are not tidally locked in this setting, but each one rotates at such a speed that it has a day night cycle equal in length to the day-night cycle on every other planet, the length of which has been previously described). To solve this, what I think I need is to know a "state zero" where all seven planets are in a particular position (hopefully a real one), and then have a set of equations that lets me figure out (whenever I need them) when one planet eclipses the other. Once I have these, I do plan to put every eclipse on the month-long calendar to help me figure out how the people of those worlds would spend their time. However, what I at least already know is that any larger units of time (like years) are certain to be an arbitrary of months attached to each-other and called something bigger, probably generally reflecting the smaller scale units aforementioned.

• Edit: @skout, to prevent closure of this interesting question, I have highlighted your actual question. And please also take into account Worldbuilding is not about opinions on naming things, like months or weeks.. this is about the underlying math of your calendar. Nov 21, 2021 at 20:53
• I've considered the same question, though I hadn't come up with any numbers yet (I came here to see if anyone else had yet). The problem with the word "day" is that it already means something. Since a tidally locked planet has a year that is the same length as it's day, both of these terms are pretty useless here. So you need a new term (even Mars, which has proper days, uses the word 'sol,' from 'solar day' to avoid confusion with earth days). I like the word 'circad' (from 'circa diem,' meaning 'about a day') used for Darian calendars adapted to the moons of gas Giants. So a circad is the te Dec 12, 2021 at 17:44
• @brandonmack I also like the term circad and I believe I saw the term originate on this very stack exchange. What I didn't mention, and what I probably should have, is that in the world I am working on, all of the planets actually have a 24h day night cycle that evenly divides the orbits of all but the inner two planets. The reason for this is magic, not science, and makes it to where "day" actually makes sense as a unit of time. This unit though is just not quite equal to an earth day. Dec 30, 2021 at 3:59
• @brandonmack the term started here: worldbuilding.stackexchange.com/questions/104210/… by user DPT Dec 30, 2021 at 4:02

Trappist-1's seven known worlds have an orbital resonance of 2:3:4:6:9:15:24... the planets of Trappist-1 should repeat themselves every 36 "days"...

... smallest common multiplier of all the above, which means $$8\cdot 9\cdot 5 = 360$$ time units.

I also have some objection on the use of "days" for rotation round the central star, we usually call the "years", with the "day" of a planet being a revolution around its own axis. With "year" denoting a rotation around the central star, the "same configuration" event will happen every:

• 180 "years" of the innermost planet
• 120 "years" of the second
• 90 "years" of the third
• 60 "years" of the 4th
• 40 "years" pf the 5th
• 24 "years" of the 5th (one before last)
• 15 "years" of the most distant one.

One of the important parts of the calendar I do not know how to solve is, how do I determine at what days and times the planets will transit one another?

This is highly dependent on the "phase" of each planet along its rotation around the central star (i.e. what fraction of their orbit they are at the - arbitrary chosen - "zero" moment).

I really doubt that you will ever see all the planets aligned, that would make the strongest perturbation on their orbit and destabilize their "dance" over long times, especially because of the tides induced in each of them (tides are "dissipative", you won't get back the gravitational energy that transforms in the deformation of the plantes).

Feeling of the math-aware segment of my guts, in the most stable configuration, you may find at most 3 planets aligned, with one of them being on the other side of the start.

(Damn's, yet another like-to-do-someday project on my list, code a program to explore the number/frequency of the peculiar configurations such a system would show)

As for the calendar, I suggest you using the "eclipses" as the "peg dates" of the calendar, they are the easiest to notice from each location (either a star eclipse or a "sibling planet" eclipse).

Unfortunately, their moment will be dependent on the particular of the "initial phase".

See also Syzygy - "a roughly straight-line configuration of three or more celestial bodies in a gravitational system" - which may add other minor "peg times" in the calendar of each planet.

• The outer two align every 120 days, and the next one in will turn up in three different places those days - assuming it wants to spoil our astronomy show, we'll call those places "gauche" and "anti". :) All the others will be the same every time, wherever they are. (Unless the resonance isn't quite perfect, and as I recall ... it isn't) Nov 22, 2021 at 1:16
• Though I would like to agree with the 360 time units, you have afew problems in your post. First off, you misunderstood what I meant by day. A day is defined as the length of time it takes for Trappist-1g to complete 1/12th of it's orbit for all planets. Yes, this does reflect how each planet orbits around their axis, but the amount of time it takes is more important than anything else for this question. 1b's year (week) is 1.5 days, 1c is 2.4, 1d is 4, 1e is 6, 1f is 9, 1g is 12, and 1h is 18 days. Nov 22, 2021 at 2:59
• The numbers you gave me also tell me this resonance period should take anywhere from 270 to 360 days, but each planet gives me a different value when I multiply the number of days by the number of days in a year. Nov 22, 2021 at 3:03
• @skout "The numbers you gave me also tell me this resonance period should take anywhere from 270 to 360 days" What you call days is pretty unclear to me - this is why a switched to the unambiguous "year" as the unit. If you have a clear idea what it is, find the conversion factor between the smallest time unit (which is half of the year of the innermost planet) and your duration-of-the-trappist-day and apply that conversion factor. If you cannot do it, it may be a sign that your definition for a "day" is indeed ambiguous and won't lead you to a well defined calendar. Nov 22, 2021 at 3:11
• @Adrian Colomitchi The funny thing is, if you move each and evey value you gave for years 1 decimal place over, you will get the exact amount of "days" in the year of each planet. In half a year of trappist 1b, there is 0.75 days. Nov 22, 2021 at 4:17