# Why is there one fewer solar day per year than there are sidereal days?

solar day = time between solar noons

sidereal day = period of Earth's spin

Wikipedia says "relative to the stars, the Sun appears to move around Earth once per year. Therefore, there is one fewer solar day per year than there are sidereal days."

Shouldn't it be relative to the Earth instead of the relative to the stars? I'm having trouble following this argument. Can someone please explain it in more detail?

• For the record, "sidereal" means "relative to the stars", just as "solar" means "relative to the sun". Moreover, @somebody's point about "relative to the Earth" meaning the Earth is considered fixed and unmoving applies. Commented Dec 15, 2021 at 1:25

Shouldn't it be relative to the Earth instead of the relative to the stars?

We need some reference background to plot the "movement" of the Sun. If we could see the stars during the day, and we were to go to a fixed point on the equator and mark the location of the Sun each day at noon on a star chart, this point would move in a circle through the stars once per year. The Sun rotates around the Earth more slowly than the stars do, so the number of solar rotations is one fewer than the number of sidereal rotations.

Imagine walking counterclockwise around a circular track, facing North the whole time. Suppose there's a light in the middle of the track. If you start out in the Eastern part of the track, the light will start out on your left. Once you get to the Northern part of the track, the light will be at your back. When you get to the Western part, it will be on your right. At the Southern part, it will be in front of you. So the light will appear to rotate around you counterclockwise.

So if the Earth didn't rotate at all, the Sun would appear to rise and set once over the course of the year. This one circuit due to the revolution around the Sun cancels out one of the 366 circuits due to the rotation of the Earth, leaving only 365 solar cycles.

• 2nd "Eastern" in the 2nd para should be "Western", shouldn't it? Commented Dec 14, 2021 at 12:31
• Will not the light rotate to the opposite direction that means clockwise? Commented Sep 4, 2022 at 16:07
• @ChrisH No, it will not. Commented Sep 4, 2022 at 16:09

From this youtube video comes this frame:

Relative to distant stars, Earth takes 23 hours and 56 minutes to spin once around its axis. During that time, it also orbited a little bit on its way around the Sun. Thus, to catch up, it has to spin a few extra minutes such that it's again noon at a given location. That's the 24 hours between noon and noon.

I find that Wikipedia sentence confusing too, but if you read the paragraph leading to it, I think it is rather clear how this works.

• A simpler explanation may involve noting that, over a year, the line marking solar noon itself makes one revolution Commented Dec 14, 2021 at 15:40
• Then the 23h56min rotation tine would be constant at all points in the orbit around the sun. But the 4 minutes extra between sunsets, wouldn't this be different at different places in the orbit due to the eccentricity? I would expect this to be constant across the orbit only for a circular orbit
– Jojo
Commented Dec 14, 2021 at 17:47
• @Joe, what you write is correct, but Earth's orbit is almost circular, so the difference is really tiny.
– rfl
Commented Dec 14, 2021 at 20:24
• Do you know what order of magnitude the difference is between opposite points on the orbit?
– Jojo
Commented Dec 15, 2021 at 8:14
• I just checked using my favorite planetarium software Stellarium (it's free and pretty :) ) and counted 15 seconds difference for the solar day noon-noon on January 4th versus July 4th next year, the days of Ap- and Perihelion.
– rfl
Commented Dec 15, 2021 at 9:28

### On the specific wording in question

Relative to the Earth would normally mean the Earth is fixed in place. So days relative to the Earth doesn't exactly make sense.

Better, quoting the exact section linked:

The stars are so far away that Earth's movement along its orbit makes nearly no difference to their apparent direction

If it helps, it's basically the same as saying "relative to the universe".

### Explanation

(similar explanation to the Wikipedia article)

In the frame of reference of a solar day, the Sun and Earth are always in a fixed place - say, the Sun directly above the Earth.

On the other hand, in the frame of reference of the stars (universe), the Earth orbits the Sun once per year. Equivalently, the Sun may seem to orbit the Earth once per year.

If this isn't intuitive, pretend there's someone walking around you in a circle, and you keep in eye contact for a whole revolution. They're moving around you, so you're definitely turning a full revolution - and they're always looking in the opposite direction (= at you) - so they must have also turned a full revolution - and since they're always facing you, you must have appeared to made one orbit around them from their perspective.

So no matter how many days are in a year, since the Sun appears to make a full revolution around the Earth, it essentially "catches up" with one spin of the Earth, which is why there is one fewer solar day than sidereal days in a year.

• i may have missed seein the answer that describes essentially the same explanation... if this answer counts as a duplicate and should be deleted, please let me know Commented Dec 14, 2021 at 15:42

Work it out for the limiting case of a planet rotation-locked to its sun (as the moon is to the Earth). There is one sidereal day per orbit, but the solar day (or night, to see the stars!) lasts forever. Now imagine a planet rotating once on its axis for each orbit around its sun. Draw the situation at four quadrants. If you need to, then draw two rotations per orbit, drawing the situation at every sixty degrees of orbit. You'll soon understand (in a true-understanding way, that can't be arrived at with mere words).

If this sounds condescending, that's not my intent. It's what Feynman always recommended, that an intuitive understanding of a concrete model is important, even when (unlike this one) the concrete model is only an approximation.

Taking the perspective of a person who believes they are the center of the Universe, the stars circle the The sky once per day. The sun kind of does too. But since (gasp) Earth actually orbits the sun, each year the sun makes one fewer “transit” across our sky compared to all the other stars. Is that correct?

An easy way to picture it is imagining the case if the Earth was tidally locked. In a year, it would rotate once from a galactic perspective, but the Sun would remain fixed in the sky...0 solar days, but 1 sidereal day.

Take an example of a clock having minute and hour hand to understand why there is a day missing in a sidereal year to number of sidereal days in a sidereal year.

In a day of 12 hours, hour and minute hand align at zero hour and again align after 12 hours respective to same position in background. Now the time taken by minute hand is 1/12 days per rotation and by hour hand is 1 day per rotation, therefore the difference of time is, 1 - 1/12 = 11/12 days per rotation This is relative time measure by an hour hand of a minute hand. So in duration of 12 hours, total time taken by a minute hand seeing from hour hand is, 11/12*12 = 11 hours per rotation.

There is one hour missing from a complete rotation of an hour hand which is of 12 hours. So an hour hand counts a day of 11 rotations of minute hand which is 11 hours as 1 rotation of a minute hand to the fixed position of the background is 1 hour.

So when earth or ecliptic completes there 360 cycles, sun or earth complete 1 cycles. From point of view of year, there is one day or cycle less. Here rises one question that, is a day fundamental unit or a year. Answer is a year, which is divided into parts as day because we are observing from year's point of view. It also questions heliocentric view, from it only earth is moving for both day and year, so there should no two different perspective, but it are.

• In a simple language, hour hand completes 1 circle when minute hand completes 12 circles. So hour hand can watch only 11 circles. Commented Sep 4, 2022 at 17:19

Take two coins. Place one above the other. Now, roll one coin along the surface of the other coin until it completes one revolution. You might guess that it will only take one rotation since the circumferences are the same, but it takes two. This is only from your perspective, but from the perspective of the coin, it still only one rotation, since the coin has only completed half a rotation relative to the other coin by the time it completed a full rotation from the perspective of the observer. You can see an example here. If the radius of the stationary coin was 2 times the rotating coin, the coin would rotate 3 times from the perspective of the observer but only two times from the perspective of the coin.

In another way of thinking, this is true because rotation + translation = rotation about an external point (Reference: YouTube: Rotation + Translation = Rotation. Animated proof | #SoME3). So the hidden rotation which manifests as a revolution does not appear to the coins themselves, but the external observer can see it.

Note that in this example, the velocity of revolution is the same as the velocity of rotation, that is the time taken to rotate a distance of x is the same time taken to revolve a distance x. In case of the Earth and the sun, the velocity of revolution is different from the velocity of rotation. But this does not really affect the situation, since it is just the relative number of rotations that matter, and not the distances. So even if the coin rotated faster than it revolved, due to slipping, the result would be that the stationary coin would observe n rotations while an external observer would observe (n+1) rotations. The extra rotation being the revolution along the circumference itself (because rotation along an external point = rotation + translation, and a full revolution is the same as a 360 degree rotation at the same place). Also note that if you keep the revolving coin in the same orientation throughout the revolution, the stationary coin will see it rotate in the other direction, conserving the rotational motion.

So from the perspective of the distant stars (sidereal days), the Earth completes 366 rotations per revolution, but from the perspective of the Sun (solar days), there are only 365 rotations.