Seasons on a 4D Planet

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Seasons on a 4D Planet

Postby PatrickPowers » Thu Dec 13, 2018 5:40 am

So I did the calculations for height of the sun in the sky at various locations on the planet. The periods of rotation were different between two planes, the wx plane and the yz plane, with one being four times that of the other.

Conceptually the run rotates around the Earth during the year. I chose the wz plane for the plane of this rotation.

I placed the sun 23.26 degrees north of the equator in the wzplane. Then rotated the sun around the planet with the rotation matrix

| cos(u) -sin(u) 0 0 |
| sin(u) cos(u) 0 0 |
| 0 0 cos(u) -sin(u) |
| 0 0 sin(u) cos(u) |


Next did the same with sun initially 23.26 degrees south of the equator. The results were quite different. In one case the wx rotation dominates, in the other the yz rotation.

The cycle of seasons lasts half of the year. That it, the planet goes through the cycle of seasons twice in one year. I suspected as much, but it is nice to see it confirmed. The two cycles are slightly different, but this might be insignificant.

The reason I suspected this is that after half a year the Sun is over the antipode of the place it began. The antipode is at the same latitude as the original location. So the season should be more or less the same every half year.

One of the poles is hotter than the other, though not greatly so. I suppose that if the obliquity of the ecliptic were zero they would be the same. The climate seems to be temperate everywhere. The length of days doesn't change at the poles. They are moving in great circles, so that makes sense.

To truly get the answer it would be necessary to do integration to see how much solar energy a spot gets during a certain time. This is kind of unusual because it's necessary to truncate the integration of the dot product. A spot gets zero energy during the night, not negative energy. So this will have to wait until I get a more powerful mathematical tool.

But I find the details unpredictable. I can't imagine the geometry.
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Re: Seasons on a 4D Planet

Postby PatrickPowers » Sat Feb 23, 2019 12:34 pm

4D planets with isoclinic rotations have what I call phase zones. Locations with the same phase have more or less the same climate, with some additional influence due to latitude.

The phase of a location is the difference of its two longitudes mod 360.

The question I had was, how do the phase zones move over time? I thought they would cycle around the planet once a year, but not. They oscillate twice a year. That is, the zone moves east for three months then west for three months. How far they move varies greatly depending on the plane of the orbit of the planet around the sun. There is actually a reasonable chance that the zones hardly move at all. The very hot places stay very hot and the very cold places very cold and there is no change during the year (except for the smaller effect due to latitude). This would be truly extreme. There would be a place on such a planet where the sun never leaves the horizon. It just circles around there all year. This would not be particularly unusual on a planet with isoclinic rotation.

At the other extreme the phase zones can move a full 180 degrees during their cycle, showing a somewhat Earthlike season. I suppose 90 degrees of phase zone movement is more or less typical. In such a case the hot places stay hot all year and the cold places stay cold, but at least they have significant seasons.

For these phase zones to affect the climate in a seasonal way the rotation has to be quite close to isoclinic, within perhaps 2%. Let's say that the ratio between the two orbits is 1.05. Then the phase zones move with a period of 20 days. That would be too rapid for the mass of the planet's surface to change much in temperate, so there wouldn't be a seasonal effect.

Isoclinic planets can have extreme seasons or none at all. The same happens in our solar system. Uranus has truly extreme seasons and Jupiter hardly any at all. It depends on the obliquity of the ecliptic. Well, the sunlight is so faint on Uranus that seasons don't matter that much. If you look at it that way, you could say that Mercury has the most extreme seasons. But it would be more accurate to say that Mercury has very long days, not seasons.
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Re: Seasons on a 4D Planet

Postby PatrickPowers » Thu Feb 28, 2019 1:02 pm

Here's a map of the average solar energy per day for various areas on a 4D planet orbiting a sun. The planet has an isoclinic rotation.

There are 5 latitude tori flattened into cylinders and nested. It is rather sphere-shaped. Add more tori and this becomes even more clear. But then the tori obscure one another and less information is visible. It's better this way.

The zones of equal phase are visible as regions of equal solar energy for that day.

For some reason it is necessary to click on this link to view the image.
Obliquity[33,-14] Phase=0.png
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Re: Seasons on a 4D Planet

Postby PatrickPowers » Fri Mar 01, 2019 8:28 am

Here is what I call the croissant view of a 4D planet. It has 5 nested latitude tori unrolled into cylinders. It shows the areas of the tori without distortion.

Croissant View.jpg
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The center cylinder is the equator. It is a square rolled into a cylinder. Or a 4D torus unrolled into a cylinder. Same thing. So the cylinder is 2pi wider than its radius.
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