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.