### To several decimal places, how many days are in one year?

• This morning I was interested in finding out how many days are in one year on average, to several decimal places (because I am a mathematician, so I like to be accurate). Unfortunately, Googling the phrase "days in a year" results in several answers that differ as soon as the third decimal place (taking into account rounding).

A second question in addition to the title question: Does the duration of earth's orbit increase and decrease, or change only monotonically over time?

You probably saw several different answers because there are several different ways to define the number of days in a year, e.g., a sidereal year or a tropical year. These two at least differ on the order of \$10^{-3}\$ days which is precisely what you reported. You need to pick a type of definition for the year first (and also a definition for a day), then figure out how many days are in that year.

Small quip, but studies suggest that Jesus was born perhaps 4 years before, not 4 years after year 0.

@userLTK I could have been mistaken. I'll edit my post.

In the Gregorian calendar, there are 365.2425 days in a year. http://hpiers.obspm.fr/eop-pc/models/constants.html gives a more exact answer, but it depends if day means 86400s or time it takes Earth to rotate with reference to the mean sun (currently a little longer than 86400s, and increasing slowly as the moon's tidal friction reduces the rotation angular speed).

Please, one question per question. Can you edit to remove the last three paragraphs? I don't know what you mean by "the leap year cycle will be broken", and I have no idea what you are getting at by asking about "year 0"

@JamesK I have removed my last two questions about leap years and when the beginning of year CE0 was. I think that, some time in the future, there will be a once-off eight-year gap between two leap years.

@barrycarter are you talking about a rotation that is about a degree bigger than 360°? Assuming that the earth is sort of rolling around the sun instead of grinding against it like a bowling ball.

@ahorn - The next year that is divisible by four that will not be a leap year will be 2100. The years that mark the turn of a century are leap years only if the year number is divisible by 400.

Phil Plait had an excellent blog post on this last year. I believe it was while he was still on Slate.

5 years ago

A mathematician would probably find the imprecision of precise orbital time a little frustrating.

"the duration of 9 192 631 770 periods of the radiation corresponding
to the transition between the two hyperfine levels of the ground state
of the caesium 133 atom"

A 24 hour day is 86 400 of those, well, more or less.

A problem that early time-keepers ran into was that the length of the day varied based on the time of year. Earth does a complete rotation in about 23 hours, 56 minutes, so, the rotation really doesn't factor into it directly, but because Earth orbits the sun about 1 degree of arc every 24 hours, an apparent solar day is has been defined as 24 hours long for centuries. Days are shorter than that when the earth is closer to the sun, at perihelion, (by about 20 seconds) and days are longer than the 24 hour average by about 30 seconds at aphelion. Not enough for anyone to notice unless they were measuring solar time very precisely.

So a day has to be averaged out, called a mean solar day, and that's what 24 hours was based on. The problem is, Earth's rotation is slowing down. The Moon is very gradually slowing down the rotation of the Earth by tidal interaction, so an average day is currently about 86 400.002 seconds. Leap seconds are added almost every year and in the future, leap seconds will be added more often. Officially a day is still 86 400 seconds, and when a leap-second is added, the "leap-day" is 86 401 seconds. But you could use either number for number of days in a year, average day (currently 86 400.002) or SI day* (86 400). As I understand it, a year is usually measured in 86 400 second days or SI days, not "average" days, but it's important to be specific about which one you're using. I've seen the occasional article that says that 200 million years ago a year was about 400 days long. That's clearly using average says, not SI days.

And if you live in high altitude, your clock will run even faster and a day will run a tiny bit longer thanks to relativity but . . . lets not even go there for the calculations).

A solar year, based on the position of the sun and the precise time of the solstice of your choice, which is a precise and momentary alignment. The length of a solar year changes a bit on a year-to-year basis. The position of the moon and gravitational perturbations from other planets, affect Earth's period of rotation a little bit, so like the day, the year should be averaged out also. Modern leap-year calendars adjust days to years quote well, but not to the accuracy you're asking about.

Looking at the time of recent summer solstices (I've copied them down below for the June solstice).

2010-06-21 11:28

2011-06-21 17:16

2012-06-20 23:09

2013-06-21 05:04

2014-06-21 10:51

2015-06-21 16:38

2016-06-20 22:34

2017-06-21 04:24

2018-06-21 10:07

2019-06-21 15:54

2020-06-20 21:44

There's as much as 15 minutes variation in the length of a year over this 11 year sample. That's not because Earth's orbit is permanently changing. These are mostly fluctuations due to the Moon and nearby planets.

An "average year", or mean tropical year, is currently about 365.2422 86 400 second days and if you want to get more accurate, 365.2421897 days on 1 January 2000. Formula in the link for adjusting to different years.

I suspect, but I'm not 100% certain that the Earth's mean tropical year moves back and forth with some of Earth's Milankovich cycles. It's not slowing consistently in the sense that a day is slowing due to the Moon's gravity and inverse tidal force.

According to Kepler's laws, the orbital period is based mostly on Earth's semi-major axis. The problem is that Kepler's laws aren't 100% accurate. Relativity is a factor when you try to pinpoint a year precisely to the second and variation in eccentricity, which affects Earth's orbital speed and the precise timing of an average year.

What I know about orbital stability, is that Earth's semi-major axis changes very little, even during the Milankovich cycles, and orbital projections say, that it will probably change little even over time-frames of millions of years.

The Sun is also losing mass, which should slow the Earth's orbit over time, but that effect is also very tiny. Over millions of years, it's thought that the length of a year doesn't change very much. The length of a day, over time-periods that long, changes quite a bit more rapidly. An unanticipated orbital change is always possible, but not thought to be likely. Orbits don't leave much in the way of track-able footprints, so they're hard to pinpoint over long periods of time and are mostly estimated by mathematical models so, take with a grain of salt, but models do suggest that a year is pretty much constant even over periods of millions of years.

For further reading, maybe start here: https://en.wikipedia.org/wiki/Equation_of_time

Hope that helps. Some of the math gets kinda tricky.

• Thanks zephyr. I wasn't familiar with that term.

Great answer. Just for reference, a day defined as 86,400 seconds is also known as an SI day.

(1/3) If solar days are too long, wouldn't it make sense for a leap day to be 86 399 seconds? This is my favourite sentence out of this whole answer: "These are mostly fluctuations due to the moon and nearby planets." With regards to the mean tropical year being 365.2421897 SI days on 1 Jan 2000, I wonder how many years that average takes into account.

(2/3) According to the formula you linked from Wikipedia (I'm not sure how many years into the future it is accurate for), the mean tropical year will get shorter until 2089, and then start to get longer. The thing which sparked my question was wondering what the average length of a month is. For completeness' sake, that's 30.44 days (to two decimal places), using a year of length 365.242086145121 days in 2017.

(3/3) In response to "An unanticipated orbital change is always possible, but not thought to be likely, " I like the idea that we're not 100% certain what will happen to the orbit in millions of years' time. That makes sense. It's also interesting that you say the sun is losing mass. Space must be dynamic. Lastly, at perihelion, doesn't the earth move faster, along a greater arc, and then the day will be longer than at aphelion? Thank you for taking the time to create this answer.

@ahorn Yes, (3/3) you're right. At Perihelion (closest part of the orbit around the sun) the object moves fastest. I'll read through and fix that. I'm always getting those mixed up.

@ahorn (1/3) I'm not sure how you're working out "if solar days are too long" shouldn't a leap-day be 83,999 seconds. Because seconds are constant, as days grow longer they have to add a second to one day every year or two. (very big earthquakes can also affect Earth's rotation speed, as can melting ice-age glaciers), but the Moon is the consistent slowing effect, so the frequency of leap-days will increase. As to your question on the average year in 2000, I have no idea how many years they looked at or how they worked it out. It's a lot of decimal places for something that variable.

I understand leap days now.