If the Earth's rotation slowed down until there were only 180 days in a year, how would I compare the length of a sidereal and solar day in this new situation?

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The solar day is the one we usually think of as a "day"; it's the time it takes for the apparent position of the Sun in the sky to complete one cycle and return to its original position.

The sidereal day, on the other hand, is the time it takes for the apparent position of the stars to complete one cycle.

Why are these different? Because the Earth is moving around the Sun. The stars are so far away that their position can be effectively taken as fixed; but the Earth's position is definitely not fixed.

As a result, there is always exactly one more sidereal day per year than there are solar days. Relative to the stars, we rotate slightly faster because our total rotation is the sum of our rotation and our orbit around the Sun.

Thus, we have 365 (plus some decimals) solar days in a year, but 366 sidereal days in a year. A solar day is 24 hours, but a sidereal day is 23 hours and 56 minutes.

This might be easiest to see with the very extreme example of tidal locking, in which a planet's rotation is so slow that it aligns perfectly with the planet's orbit, and one side of the planet always faces the Sun. In this situation, the sidereal day is equal to one year---and the solar day is infinite. The Sun never moves from its current position in the sky. A tidally locked planet has 1 sidereal day per year, and 0 solar days per year.

If our rotation were slowed so that we only had 180 days in a year instead of 365, this would mean that our solar day is a bit more than twice as long. It would mean that our sidereal day is also a bit more than twice as long, but it will increase by a smaller ratio because the part of the rotation that is due to our orbital motion is unchanged. We would have 180 solar days in a year, each 48 hours and 40 minutes long. We would have 181 sidereal days in a year, each 48 hours and 24 minutes long. While the solar day expanded by 102.78%, the sidereal day only expanded by 102.23%. A small difference, to be sure, but it adds up.

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