Three Dollar Quill

(This is getting away from the category name a bit, but it's in theme with the other entries, so... I'm just going to put off thinking about how to "fix" this until I forget that it's a thing.)

Okay, I'm doing not-great, because it hurts to talk, but I'm still thinking about stuff, so let's get into it.

I basically understand how to do year-related calculations for an Earth-like planet, in particular, one with a low orbital eccentricity, and nothing else in the vicinity strongly interfering with its orbit. Anomalistic year combines with apsidal precession, giving mean sidereal year, which combines with axial precession to give the familiar tropical year.

Suppose we have a planet with extreme orbital eccentricity. In that case, the anomalistic year would be of primary concern (because part of the planet would be much warmer at pericenter), with the tropical year being secondary or irrelevan­t. If the apsidal and axial precessions did not cancel, then there would also be a long-term cycle of libration of the warmest latitude at pericenter, oscillating between the tropics. Things get confusing if the seasonal cycle is still relevant, because being near pericenter speeds up the sidereal and tropical years.

What if the orbit weren't a straightforward ellipse? What about tadpole orbits and horseshoe orbits? Fair warning, I'm shooting from the hip here, because my attention span is not at my best right now. The first thing I note is that "period in the rotating reference frame" takes the place of the anomalistic year, and "rotation of the reference frame" takes the place of apsidal precession. For planets of comparable mass in a horseshoe orbit, this should be all that's needed for a decent mean sidereal year, which can then plug into everything else. This is because the inner and outer orbits will have orbital periods within, say, two percent of each other, by Kepler's laws. Although, given that I'm imagining someone on this hypothetical planet caring enough to get a calendar right, at some point someone will realize that the day length changes by like five seconds every thirty years or so, and this will bother them.

"Yes, yes, the other planet grew to the size of a second moon in the sky, then receded, but I'm more concerned about how all of my clocks are off by the same amount, and more every day. Wait, what if my clocks are right and the day is wrong?"

Thank you.

Anyway, now I'm trying to figure out what it would look like to have a very precise calendar with comparable masses, or a doesn't-get-extreme-anomaly calendar with different masses. The first idea that comes to mind for me is to maintain two tropical calendars, one for each range of orbital distance, and somehow interpolate between. them. However, since the switchover doesn't change the current season, there could be a "unified" calendar with a system of intercalation. This would then leave the problem of how to determine when to switch, and how to keep the year right over long periods of time.

However, I'm about done thinking about this for tonight, so I'll just let that question hang, and wrap things up for now.

Good night.