Alternatively (as a companion to my other thread on a more generic solution) , I've got a specific situation I'd like to get a handle on. I've got a planet with a 10° axial tilt, in an orbit with a semimajor axis of 10.363 AU and an eccentricity of 0.129. It takes 22.4 years to orbit its star (a 2.2 solar mass star at 46 solar luminosities), and the planet's rotation period is 8.7 hours. As far as I can figure, surface temperatures at the 30-40° latitude range between about -35°C at aphelion and 0°C at perihelion, and the atmosphere is about 90% CO2 and 10% N2, and slightly thinner than Earth's.
It seems obvious to me that the major seasonal effects are going to be caused by the orbital eccentricity rather than the axial tilt, since its orbital distance varies between 9.026 AU and 11.7 AU - the orbit also takes it within the star's habitable zone for very close to 6 years of its 22.4 year long orbit, otherwise it is beyond it (in the zone equivalent to where Mars is in our own system).
What I'm not sure about is how the day length varies. On Earth, the day (or night) can be 24 hours long at locations north of the arctic circle (or south of the southern equivalent), located at close to 67°N and S (90° - earth's axial tilt of 23°). At this latitude, the 24-hour long day occurs at the summer solstice and the 24-hour night occurs at the winter solstice. At the poles themselves the sun sets and rises once per year.
What would happen on my fictional planet, with its much longer year, faster rotation, and smaller axial tilt? I figure that the arctic/antarctic circles would be smaller (at 80° N and S). At higher latitudes, the star can be above or below the horizon for the full 8.7 hour rotational period. The time period between solstices is much longer than on Earth however. The poles can be in total darkness for 11.2 years or in full daylight for 11.2 years - at the equator the length of the day is the same as the length of the night (8.7 hours total) for the 22.4 year orbit.
I think one thing that is confusing me is how to consider the seasons. On Earth, we have seasons largely because of the axial tilt of the planet, causing the sun to more (or less) directly illuminate parts of the surface; but they can also be defined by equinoxes and solstices, which are characteristics of the slightly eccentric orbit of the Earth.
One other thing I'm not sure about - are equinoxes generally defined by being exactly halfway between the solstices during the year (regardless of the orbital eccentricity)? Or would the equinoxes be displaced towards (or away from) one of the solstices if the orbital eccentricity was higher?
On my fictional world, the seasons would be largely determined by the variation in orbit distance over the long year, but the axial tilt would still have some influence. If there was no axial tilt, then there'd be no polar day or night because there wouldn't be an arctic or antarctic circle. But with a 10° tilt, the highest latitudes can have a polar day/night.
Figure 6i-2 (in the middle of the page) seems useful. I think I should be able to replace the y-axis with numbers going from 0 to 8.7 hours, and the x-axis with a 22.4 year long orbital period, and then slightly tweak the graph lines to get something that works for my planet.
At the 0° latitude, the day and night would both be exactly 4.35 hours long throughout the year. So between 0° and 80° we'll have some variability in the day length over the 22.4 year long orbital period, though I'm not sure how to calculate that exactly. I can surmise the following though:
At 90° latitude we'd have the 11.2 year long day and night, with the sun rising and setting at the equinoxes.
At 85° latitude (or at some point between 90° and 80°), there will be an 8 (earth) year long night, followed by a period of a couple of years when there is a day/night cycle that starts with long nights/short days and end with short nights/long days, followed by an 8 year long day, followed by a couple of years of day/night cycle starting with long days/short nights and ending with short days/long nights, and then it's back to the 8 year night.
At 80° latitude, we'd have a sinusoidal line (assuming the equinoxes are still exactly halfway between the solstices), touching the 8.7 hour line at the summer solstice and touching the 0 hour line at the winter solstice. There will be at least one 8.7 hour long day and one 8.7 hour long night during the 22.4 year long orbit at this latitude.
I guess that the lines on the graph would be flattened somewhat at the intermediate latitudes on my planet, since the 80° line there is where the 67° line would be on Earth - which would make sense since the tilt of my planet is lower, so the axial effects should be less than on Earth. So at low latitudes (within 30° N/S), the day length probably isn't going to be all that different to its duration at the equator itself - maybe it'd be about 5 hours at most, and 3 hours at the least over the year. On Earth, even at 60° - close to the polar circles the day length - the day length can still be several hours long. On my fictional planet, maybe the 70° latitude corresponds to this, so the day length here would vary between about 6.5 hours and 2 hours over the course of the year.
I'm brainstorming a bit here and I don't know if that makes much sense... but if anyone spots any obvious errors or has any thoughts about the implications of this then please let me know! The very long polar days and nights intrigue me - the planet doesn't usually get very warm and I doubt that and 11.2 year long day at the poles would really melt the polar ice or anything... but I'm wondering what the effect of an 11.2 year long night at the poles would be - could it get cold enough there to freeze CO2 out of the atmosphere?