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Could an Earth-like planet have a cycle of seasons changing length without changing the length of the year? Having long summers and short winters at the start of the cycle and shifting towards long winters and short summers, then returning to long summers when the cycle repeats?

Example: Temperate northern-hemisphere, an 8 year cycle

Year 1 - Summer is 25% longer than Winter; Year 2 - Summer is 50% longer than Winter; Year 3 - Summer is 25% longer than Winter; Year 4 - Summer and Winter are even length; Year 5 - Winter is 25% longer than Summer; Year 6 - Winter is 50% longer than Summer; Year 7 - Winter is 25% longer than Summer; Year 8 - Winter and Summer are even length;

Is there a reason something like this could happpen without resorting to magic? What other effects would it have on the planet?

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I don't know if planets can form this way, but the conditions are easy enough to describe. You state that the length of the year is fixed, so the orbital period isn't shifting. This means the only thing that can change is the direction of the angular momentum vector.

One piece of the puzzle is going to have to be precession. In precession, the direction of the angular momentum vector shifts over time. This happens to real planets. The schedule of the zodiacs has shifted over the last few thousand years because the tilt of the planet has changed direction. But this on its own is not sufficient to cause what you want.

But what if we had a more elliptical orbit? In an elliptical orbit, a planet moves faster near the sun and slower when it is further away. This means whatever season occurs further away lasts longer. And, if the planet precesses, the seasons will shift along the orbit! This means each season will get it's turn to be longer on the cycle of the precession.

Two details for you to work out. First is whether your planetary creation can support such fast precession. You're looking for something a thousand times faster that the precession of the Earth. Second is whether the secondary climate effects of that elliptical orbit are acceptable. Earth's orbit is circular enough that the distance from the sun plays a much smaller effect in seasonal variation than our tilt. In a highly elliptical orbit needed to make this seasonal length variation a reality, the difference between apoapsis and periapsis could have a larger effect on temperature. But regardless, the seasons (as defined by the cycle of solstices and equinoxes) will vary as you seek.

We can, and the very least, rough out some calculations. In a rough way, we can equate the length of a season to time the planet spends near apoapsis and periapsis. It's probably not exact, but it should be close. And it's easy to calculate. The mean anomaly of a planet is an angular rate that sweeps out equal areas in equal times. It is also proportional to time, so a 50% increase in time spent in part of an orbit corresponds exactly to a 50% increase in the change in mean anomaly.

And this is why it was convenient to look just at the most extreme moment in the orbit, rather than the whole season. Calculating these percentages for a season requires integration, but the rates at the extremes can be calculated directly. The instantaneous area swept at any point is proportional to the distance to the central body (the sun). So to get a 50% variation in length of season, you need a 50% difference between altitude at apoapsis and periapsis (furthest from the sun and closest to the sun, respectively). This means that if the close part of the orbit is at Earth distances, the far part is all the way out at Mars' orbit.

The precession is harder to calculate, because there are many more degrees of freedom. Faster rotation (shorter days) increases precession. A more asymmetric planet or a rotational axis that is further from a symmetric axis will speed up precession. You will definitely want a rocky planet though. Any fluid motion will form vortices that damp precession. A planet that suffered a giant collision is probably your best bet to achieve these conditions.

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