Are both correct? If only one is, which is it. Either way, why?
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It turns out there are a surprising large number of ways to measure how long a year really is.
The first number you mentioned is called the sidereal year and measure the time until the earth ends up being where it was in respect to the sun and the stars. You look up at the stars in the sky and where the sun is in respect to them and wait until it is back the way it was.
The other figure you mentioned is the tropical year and the one we most commonly use because it keeps track of the seasons.
If you look at the numbers you will notice that the difference is slight only about 20 minutes a year. It would take 25,772 years for this difference to add up to a whole year.
The earth rotates around its own axis (once a day) not unlike a fast spinning top, but if you ever looked at a top you will have noticed that they tend to not just spin but also much more slowly change where the top of the top is pointing in a circular fashion that much slower cycling is called precession.
With the earth it means that very slowly the direction the earth axis points changes over time and for example the pole star we see to day is a different one than the people alive 2000 years ago had as a pole star or the people in a few millennia will have.
Today’s pole star is Polaris but in the year 7575 (if men is still alive) it will be alpha centauri.
This motion of the earths axis means takes 20 minutes away from each star based year to create a season based year.
This whole circle repeats ever 26 millennia, the time it takes for the difference between the tropical and sidereal year to add up to one year.
Weird right?
Of course there are also other ways to define a year like instead of looking at summer and winter or the stars, you look at when earth is closest and farthest from the sun. Currently Earth is closest to the sun in early January and farthest in early July on our calender about a fortnight after the winter and summer solstice.
But since the elliptic orbit of the earth around the sun is actually slowly rotating itself, the year measured based on that called the anomalistic is about 25 minutes longer than the year based on the season and in a few short millennia we will have the earth being closest to the sun when the northern hemisphere enjoys summer.
The coolest named year however is the draconic year. It is based on eclipses and also close to a year, but much shorter.
The point being the stuff that you learn in school where the earth goes around the sun in neat regular circles is all a lie or at least a gross oversimplification. There are a ton of different rotations going on and even the way things rotate rotates itself over long enough time.
If you want to answer the question of “how long is a year” with any real accuracy you first have to decided which of dozen or more different phenomena you actually want to measure.
Of course over the average lifetime of a human most of the difference don’t really matter much, but some people like precision.
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