Given the Earth's current speed around the sun and current rate & axis of rotation, what is the best way to keep time to avoid a leap year? How many hours should we have in the day and days in a year would keep things balanced to not need to add or remove days from the year? Further, how many minutes per hour and seconds per minute should we have to avoid a leap second?

Leap years exist for two reasons:

- There are not an integer number of days in a year.
- People perceive a need to keep the seasons where they are on the calendar.

Given the above, there is no way to avoid leap years, or something similar. Defining the calendar year as being a fixed number of days (e.g., 365 days) would result in the seasons shifting by one day per four years.

Leap seconds exist for two reasons:

- The length of a day as measured by an atomic clock is not constant.
- People perceive a need to keep midnight at midnight, noon at noon.

Given the above, there is no way to avoid leap seconds, or something similar. Defining the day as being a fixed number of atomic clock seconds (e.g., 86400) would result in your clock and the Sun disagreeing on mean local noon, but by a very small amount.

That said, there are serious proposals to eliminate leap seconds. Some people such as those who use UTC to timestamp financial transactions do not like them. So far, those proposals have been rejected. The standard response is that it's not UTC that's broken; it's using of UTC in a context where it shouldn't be used that is broken. If you need a monotonically increasing time scale, use TAI or GPS time instead.

This doesn't really work the way that you are thinking, at least not in a way that is practical for society at all. They problem is that we define a day to be based on Earth rotations relative to the sun, and a year as a full orbit around the sun, and if you find the number of rotations of the earth in a single orbit, it is not an integer (~365.24 rotations (days) in a year). To avoid a leap year, you would need to define the day such that there are an integer number of days in a year (i.e 365 days exactly). The problem with this is that day and night will drift relative to our clocks, and after 2 years, day and night will be switched. The length of the year is also variable and not fundamental, so in order to keep this exact relationship, you'd have to constantly redefine the length of the day, which is not a practical improvement over having leap years.

The leap second has the same type of problem. We want to define the number of seconds in a day as 86400 seconds/day, but the Earth's rotation is not constant. So, in order to keep clocks from drifting, you have to add leap seconds.

I'm a software engineer, and I can speak about the issue with leap seconds.

They are unpredictable. You don't know far in advance whether you will have one. Code that cares about accurate number of seconds will need some kind of update or feed to continue working correctly.

It's also a step that adds complexity. You have to allow for a minute that contains 61 seconds.

For the first issue, a compromise that keeps reasonable tracking between the Earth's rotation and the time of day would be to allow looser tolerance. Rather than being within one second, correct it on schedule every 10 years. Software doesn't have to worry about year-by-year issues, and the clock stays 7 seconds (or ±4 if you jump ahead) to true.

Given that we already have time zones, the sun will not be exactly at the midnight position at midnight *anyway* but will be half an hour ahead or behind. Astronomers already need a special offset clock.

We not only can avoid leap seconds, that's how it used to work in fact. And there is a common newer system which avoids leap seconds as well.

Before 1960, seconds were defined as 1/86400 of a mean solar day. Then when variations in the earth's rotation caused it to get out of sync, a new mean solar day could be computed and divided by 86400 - changing the length of the second in absolute terms, stretching or shrinking it very slightly.

That was a mess, as you can imagine. So the second was defined in terms of a specific number of atomic oscillations which could be made extremely precise. Instead of shrinking and stretching the second to keep an exact number of them in a day, we keep the second fixed and add or subtract one from the (integer) count when we need to adjust.

Those are pretty much the ways to keep earth rotation timing in sync with our clock time - you need some give somewhere, either by changing the length of the second and keeping the count fixed, or you keep the length fixed and change the count. For somebody just writing a simple program to, say, compute the civil seconds between two UTC timestamps, the old way was easier (a fixed count of seconds between two times is trivial). But if you are doing scientific or engineering calculations or experiments to great precision, it's WAY better to have a very firmly fixed length of a second, not changing it from time to time - much worse than the inconvenience of taking leap seconds into account.

But the way, another approach is to just ignore leap seconds and keep your clocks running continuously. That's how GPS time works - it started in sync with UTC, but has not been adjusted for the leap seconds since then, so they are out of sync by a quarter minute or so (I haven't check in some while). That's nice for GPS orbital calculations that cross leap second adjustment boundaries. In the GPS data packet there is information about the current delta between UTC and GPS time so you can calculate civil time from GPS time, as well as a few months advanced warning when a new leap second is going to be added or omitted.

Another answer suggested queuing up leap seconds and making a multi-second leap every decade. That doesn't really simplify your software much tho - now you have to allow minutes with, say, 67 seconds, every decade. Easier to just deal with leap seconds using a table and meanwhile never be off by even 1 second. (The standard allows for them to added or omitted by the way - you could have a 59 second minute or a 61 second minute when you need an adjustment. It's generally the latter tho.

Oh, one other solution. The organization which really tracked all this was called the International Earth Rotation Service, later renamed to International Earth Rotation and Reference Systems Service (IERS). Imagine the chaos if they stopped being funded and the Earth stopped rotating. Anyway, I suppose you could just ask them to rotate it more consistently. :-)

## 'Leap Second' Tonight Will Cause 61-Second Minute

July will arrive a little late this year – one second late, to be exact.

Time will stand still for one second this evening (June 30) as a "leap second" is added to Coordinated Universal Time (UTC), the time standard by which most clocks are regulated. The International Earth Rotation and Reference Systems Service (IERS), which keeps track of time for the world, has decided that the extra second is needed to deal with Earth's irregular but gradually slowing rotation.

The extra second will be inserted just before midnight UTC — just before midnight GMT, and just before 8 p.m. EDT. Instead of rolling straight through from 23:59:59 to 00:00:00, UTC will tick over to 23:59:60 for a second. [June 2015 Gets An Extra Second (Video)]

## A Day is not Exactly 24 Hours.

There are also different kinds of days:

** Sidereal Day –**This is the time it takes the Earth to spin one complete 360 degree rotation on its axis, as measured against the background stars. It is 23 hours, 56 minutes, and 4 seconds long.

** Solar Day –**This is how long it takes the sun to track a full 360-degree circuit in the sky, from one meridian crossing to the next. It is 24 hours long.

The reason for the nearly 4-minute difference between a sidereal day and a solar day is that in one day, the Earth travels about 1.5 million miles along its orbit. So it takes an extra 4 minutes of rotation to bring us back in line with the sun as compared with the day before.

## Leap Days Explained!

Photo illustration by Phil Plait. Photo by Shuttertstock/Catalin Petolea.

*This article is a modified and updated version of one I wrote in—oddly enough—2008 and then updated for 2012. Barring a colossal asteroid impact or a Trump presidency, I’ll probably be around to do it in 2020, too. But not 2200. Even if my floating head in a jar is still around, it won’t matter, as you’ll see if you read on.*

*Note: This post has math in it. Quite a bit. But it’s really just arithmetic—decimals and multiplication. If you’re a mathaphobe, then skip to the end, but you’ll have to trust me on the numbers.*

*If you’re a mathophile and a pedant, then you may fret over my ignoring significant digits below. But in this case the mantissa is what’s important, since what we’re doing here is a variation of modulus math the actual fraction of a day left over is what adds up, and it doesn’t matter how many whole days there are once the leap day corrections are applied to the calendar. So, I kept all the numbers to four decimal places (unless they end in 0), and ignored sigfigs. Yes, this leads to some roundoff errors, but over the span of time we’re talking here they don’t matter much.*

When I was a kid, I had a friend whose birthday was on Feb. 29. I used to rib him that he was only 3 years old, and he would visibly restrain himself from punching me. Evidently he heard that joke a lot.

Of course, he was really 12. But since Feb. 29 is a leap day, it only comes once every four years.

But *why* is leap day only a quadrennial event?

Why is anything anything? *Because astronomy!*

OK, maybe I’m biased, but in this case it’s true. We have two basic units of time: the day and the year. Of all the everyday measurements we use, these are the only two based on concrete physical events: the time it takes for the Earth to spin once on its axis, and the time it takes the Earth to go around the Sun. Every other unit of time we use (second, hour, week, month) is rather arbitrary. Convenient, but they are not defined by independent, nonarbitrary events. *

It takes roughly 365 days for the Earth to orbit the Sun once. If it were *exactly* 365 days, we’d be all set! Our calendars would be the same every year, and there’d be no worries.

But that’s *not* the way things are. The length of the day and year are not exact multiples they don’t divide evenly. There are actually about **365.25** days in a year. That extra fraction is critical it adds up. Every year, our calendar is off by about a quarter of a day, an extra 6 hours just sitting there, left over.

After one year the calendar is off by ¼ of a day. After two years it’s a half day off, then ¾, then, after four years, the calendar is off by roughly a whole day:

4 years at 365 (calendar) days/year = **1,460** days, but

4 years at 365.25 (physical) days/year = **1,461** days

So after four years the calendar is *behind* by a day. The Earth has spun one extra time over those four years, and we need to make up for that. So, to balance out the calendar again we add that day back once every four years. February is the shortest month (due to some Caesarian shenanigans), so we stick the day there, call it Feb. 29—leap day—and everyone is happy.

Except there’s still a problem. I lied to you (well, not really, but go with me here). *The year is not exactly 365.25 days long*. If it were, every four years the calendar would catch up to the Earth’s actual spin, and we’d be fine.

But it’s not, and this is where the fun begins.

Our official day is 86,400 seconds long. I won’t go into details on the length of the year itself (you can twist your brain into knots reading about that if you care to), but the year we now use is called a Tropical Year, and it’s 365.2422 days long. This isn’t exact, but let’s round to four decimal places to keep our brains from melting.

Obviously, 365.2422 is a bit short of 365.25 (by about 11 minutes). That hardly matters, right?

Actually, yeah, it does. Over time even that little bit adds up. After four years, for example, we don’t have **1,461** physical days, we have:

4 years at 365.2422 (real) days/year = **1460.9688 days**

That means that when we add a whole day in every four years, we’re adding too much! But I don’t see any easy way to add only 0.9688 days to our calendar, so adding a whole day is understandable.

Where does this leave us? Adding in a leap day every four years gets the calendar way closer to being accurate, but it’s still not *exactly* on the money it’s still just a hair out of whack. This time, it’s *ahead* of the Earth’s physical spin, because we added a whole day, which is too much. How much ahead?

Well, we added one whole day instead of 0.9688 days, which is a difference of **0.0312 days**. That’s 0.7488 hours, which is very close to 45 minutes.

That’s not a big deal, but you can see that eventually we’ll run into trouble again. The calendar gains 45 minutes every four years. After we’ve had 32 leap years (which is 4 x 32 = 128 years of calendar time) we’ll be off by a day again, because 32 x 0.0312 days is very close to a whole day! It’s only off by a couple of minutes, which is pretty good.

So we need to adjust our calendar again. We could just skip leap day one year out of every 128 and the calendar would be very close to accurate. But that’s a pain. Who can remember an interval of 128 years?

So instead it was decided to leave off a leap day every 100 years, which is easier to keep track of. So, every century, we can skip leap day to keep the calendar closer to what the Earth is doing, and everyone’s happy.

Except there’s still *still* a problem. Since we do this every 100 years, we’re still not making the right adjustment. We’ve added that 0.0312 days in 25 times, not 32 times, and that’s not enough.

To be precise, after a century the calendar will be ahead by:

25 x 0.0312 days = **0.7800 days**

That’s close to a whole day. Of course, seeing what we’ve already gone through, you would be forgiven for a sense of foreboding that this won’t work out perfectly. And you’d be right. We’ll get to that.

But first, here’s another way to think about all this that I’ll throw in just to check the math. After 100 years, we’ll have had 25 leap years, and 75 non leap years. That’s a total of:

(25 leap years x 366 days/leap year) + (75 years x 365 days/year) = **36,525** calendar days

But in reality we’ve had 100 years of 365.2422 days, or 36,524.22 days. So now we’re off by:

36,525 - 36524.22 = **.78 days**

which, within roundoff errors, is the same number I got above. Woohoo. The math works.

Where was I? Oh, right. So, after 100 years the calendar has gained more than ¾ of a day on the physical number of days in a year when we add in a whole day every four years. That means we have to stop the calendar and let the spin of the Earth catch up. To do this, once per century we *don’t* add in a leap day.

To make it simpler (because yegads we need to), we only do this in years divisible by 100. So the years 1700, 1800, and 1900 were *not* leap years. We didn’t add an extra day, and the calendar edged that much closer to matching reality.

But notice, he says chuckling evilly, that I didn’t mention the year 2000. Why not?

Because as I said a moment ago, even this latest step isn’t quite enough. Remember, after 100 years, the calendar still isn’t off by a whole number. It’s ahead by 0.7800 days. So when we subtract a day by not having leap year every century, we’re overcompensating *we’re subtracting too much*. We’re *behind* now, by:

1 - 0.7800 days = **0.2200 days**

Arg! So every 100 years, the calendar lags behind by 0.22 days. If you’re ahead of me here (and really, I can barely keep up with myself at this point), you might say, “Hey! That number, if multiplied by 5, is very close to a whole day! So we should put the leap day back *in* every 500 years, and then the calendar will be very close to being right again!”

What can I say? You are clearly very smart and a logical thinker. Sadly, the people in charge of calendars are not you. They went a different route.

How? Instead of adding a leap day back in every 500 years, they decided to add it in every **400** years! Why? Well, in general, if there’s a more difficult way to do something, that’s how it’ll be done.

So, after 400 years, we’ve messed up the calendar by 0.22 days four times (once every 100 years for 400 years), and after four centuries the calendar is behind by

4 x 0.22 days = **0.88 days**

That’s close to a whole day, so let’s run with it. That means every 400 years we can add Feb. 29 magically back into the calendar, and once again the calendar is marginally closer to being accurate.

As a check, let’s do the math a different way. Right up until February of the last year in a 400 year cycle, we’ve had 303 non-leap years, and 96 leap years (remember, we’re not counting the 400 th year just yet).

(96 leap years x 366 days/leap year) + (303 years x 365 days/year) = **145,731** calendar days

If we then *don’t* make the 400 th year a leap year, we add in 365 more days to get a total of **146,096 days**.

400 x 365.2422 days = **146,096.88** days

So I was right! After 400 years we’re *behind* by 0.88 days, so we break the “every 100 years” rule to *add* in a whole day every 400 years, and the calendar is much closer to being on schedule.

We can see the remainder is 0.88 days, which checks with the previous calculation, and so I’m confident I’ve done this right. (Phew!)

But I can’t let this go. I have to point out that even after all this the calendar’s still not *completely* accurate at this point, because now we’re *ahead* again. We’ve added a whole day every 400 years, when we should have added only 0.88 days, so we’re ahead now by:

1 - 0.88 days = **0.12 days**.

The funny thing is, *no one worries about that*. There is no official rule for leap days with cycles bigger than 400 years. I think this is extremely ironic, because if we took one more step we can make the calendar extremely accurate. How?

## Why do we have leap days?

**Note 1**: Tomorrow is leap day! 29 February, 2020. And I am nothing if not frugal (or at least marginally lazy): This article is a slightly edited version of the same one I posted in 2008, 2012, and 2016. You may notice a pattern. I expect I will continue to do so up until 2200, for reasons that will become apparent as you read, assuming I'm still alive and not locked in a stasis pod somewhere.

**Note 2**: This post has math in it. Quite a bit. But it's really just arithmetic decimals and multiplication. If you're a numerophobe, then skip to the end, but you'll have to trust me on the numbers.

*If you're a numerophile and a pedant, then you may fret over my somewhat contemptuous handling of significant digits below. But in this case the mantissa (the collected numbers to the right of the decimal point) is what's important, since those are what cause all the leap day grief in the first place. If I carried that out too far it would make this whole mess quite a bit messier, so I kept all the numbers to four decimal places (unless they end in 0), and ignore sigfigs. Yes, this leads to some roundoff errors, and I recognize that, in one form or another, that's ironically part of the whole leap day problem in the first place. Happily, though, over the span of time we're talking here they really don't matter much.*

*OK, ready? Let's make some math!*

When I was a kid, I had a friend whose birthday was on February 29. I used to rib him that he was only 3 years old, and he would visibly restrain himself from punching me. Evidently he heard that joke a lot.

Of course, he was really 12. But since February 29 is a leap day, it only comes once every four years.

But *why* is leap day only a quadrennial event?

Why is anything anything? *Because astronomy!*

OK, maybe I'm biased, but in this case it's true. We have two basic units of time: the day and the year. Of all the everyday measurements we use, these are the only two based on concrete physical events: the time it takes for the Earth to spin once on its axis, and the time it takes the Earth to go around the Sun. Every other unit of time we use (second, hour, week, month) is rather arbitrary. Convenient, but they are not defined by independent, non-arbitrary events * .

It takes roughly 365 days for the Earth to orbit the Sun once. If it were *exactly* 365 days, we'd be all set! Our calendars would be the same every year, and there'd be no worries.

But that's not the way things are. The length of the day and year are not exact multiples they don't divide evenly. There are actually about **365.25** days in a year. That extra fraction is critical it adds up. Every year, our calendar is off by about a quarter of a day, an extra 6 hours just sitting there, left over.

After one year the calendar is off by 1/4 of a day. After two years it's a half a day off, then 3/4, then, after four years, the calendar is off by roughly a whole day:

4 years at 365 (calendar) days/year = **1460 days**, but

4 years at 365.25 (physical) days/year = **1461 days**.

So after four years the calendar is *behind* by a day. The Earth has spun one extra time over those four years, and we need to make up for that. So, to balance out the calendar again we add that day back once every four years. February is the shortest month (due to some Caesarian shenanigans), so we stick the day there, call it February 29 — Leap Day — and everyone is happy.

And that's why we have Leap Day every four years. Done and done.

Except not so much. I lied to you earlier (well, not really, but go with me here). *The year is not exactly 365.25 days long*. If it were, every four years the calendar would catch up to the Earth's actual spin and we'd be fine.

But it's not, and this is where the fun begins.

Personally, I don’t think it’s that bad. Credit: The Internet it’s a meme after all

Our official day is 86,400 seconds long. I won't go into details on the length of the year itself (you twist your brain into knots reading about that if you care to), but the year we now use is called a Tropical Year, and it's 365.2422 days long. This isn't exact, but let's round to four decimal places to keep our brains from melting.

Obviously, 365.2422 is a bit short of 365.25 (by about 11 minutes). That hardly matters, right?

Actually, yeah, it does. Over time even that little bit adds up. After four years, for example, we don't have **1461** physical days, we have

4 years at 365.2422 days/ (tropical) year = **1460.9688 days**.

That means that when we add a whole day in every four years, we're adding too much! It's pretty close, sure, but when we add a whole day to the calendar every four years instead of 0.9688 days it's still off.

Where does this leave us? Well, we're closer, but still not *exactly* on the money it's still just a hair out of whack. This time, the calendar is *ahead* of the Earth's physical spin. Let's see how much ahead.

Well, we added one whole day instead of 0.9688 days, which is a difference of **0.0312 days**. That's 0.7488 hours, which is very close to 45 minutes.

That's not a big deal, but you can see that eventually we'll run into trouble again. The calendar gains 45 minutes every four years. After we've had 32 leap years (which is 4 x 32 = 128 years of calendar time) we'll be off by a day again, because 32 x 0.0312 days is very close to a whole day! It's only off by a couple of minutes, which is pretty good.

So we need to adjust our calendar again. We could just skip leap day one year out of every 128 and the calendar would be very close to accurate. But that's a pain. Who can remember an interval of 128 years?

So instead it was decided to leave off a leap day every 100 years, which is easier to keep track of. So, every century, we can skip leap day to keep the calendar closer to what the Earth is doing, and everyone's happy.

Except there's still *still* a problem. Since we do this every 100 years we're still not making the right adjustment. We've added that 0.0312 days in 25 times, not 32 times, and that's not enough.

To be precise, after a century the calendar will be ahead by

25 x 0.0312 days = **0.7800 days**.

That's close to a whole day. Of course, seeing what we've already gone through, you would be forgiven your sense of foreboding that this won't work out perfectly. And you'd be right. We'll get to that.

But first, here's another way to think about all this that I'll throw in just to check the math. After 100 years, we'll have had 25 leap years, and 75 non leap years. That's a total of

(25 leap years x 366 days/leap year) + (75 years x 365 days/year) = **36,525 calendar days**.

But in reality we've had 100 years of 365.2422 days, or 36,524.22 days. So now we're off by

36,525 - 36524.22 = **.78 days**

which, within roundoff errors, is the same number I got above. Woohoo. The math works. (duh)

The phase of the Moon on 29 February, 2020. Why? Because it's pretty, and I figured this would be a good break from the math. Credit: NASA's Scientific Visualization Studio

Where was I? Oh, right. So, after 100 years the calendar has gained over 3/4 of a day on the physical number of days in a year when we add in a whole day every four years. That means we have to stop the calendar and let the spin of the Earth catch up. To do this, once per century we *don't* add in a leap day.

To make it simpler (because yegads we need to), we only do this in years divisible by 100. So the years 1700, 1800, and 1900 were *not* leap years. We didn't add an extra day, and the calendar edged that much closer to matching reality.

But notice, he says chuckling evilly, that I didn't mention the year 2000. Why not?

Because as I said a moment ago, even this latest step isn't quite enough. Remember, after 100 years, the calendar still isn't off by a whole number. It's ahead by 0.7800 days. So when we subtract a day by not having leap year every century, we're overcompensating *we're subtracting too much*. We're *behind* now, by

1 - 0.7800 days = **0.2200 days**.

Arg! So every 100 years, the calendar lags behind by 0.22 days. If you're ahead of me here (and really, I can barely keep up with myself at this point), you might say "Hey! That number, if multiplied by 5, is very close to a whole day! So we should put the leap day back *in* every 500 years, and then the calendar will be very close to being right again!"

What can I say? You are clearly very smart and a logical thinker. Sadly, the people in charge of calendars are not you. They went a different route.

How? Instead of adding a leap day back in every 500 years, they decided to add it in every 400 years! Why? Well, in general, if there's a more difficult way to do something, that's how it'll be done. I don't have a better answer than that, but it does seem to be true quite often.

So, after 400 years, we've messed up the calendar by 0.22 days four times (once every 100 years for 400 years), and after four centuries the calendar is behind by

4 x 0.22 days = **0.88 days**.

That's close to a whole day, so let's run with it. That means every 400 years we can add February 29 magically back into the calendar, and once again the calendar is marginally closer to being accurate.

As a check, let's do the math again a different way. Right up until February of the last year in a 400 year cycle, we've had 303 non-leap years, and 96 leap years (remember, we're not counting the 400th year just yet).

(96 leap years x 366 days/leap year) + (303 years x 365 days/year) = **145,731 calendar days**.

If we then don't make the 400th year a leap year, we add in 365 more days to get a total of 146,096 days.

400 x 365.2422 days = **146,096.88 days**.

So I was right! After 400 years we're behind by 0.88 days, so we break the "every 100 years" rule to *add* in a whole day every 400 years, and the calendar is much closer to being on schedule.

We can see the remainder is 0.88 days, which checks with the previous calculation, and so I'm confident I've done this right. (phew)

If you prefer graphics and my voice telling you all this, then watch this video.

But I can't let this go. I have to point out that even after all this the calendar's still not *completely* accurate at this point, because now we're *ahead* again. We've added a whole day every 400 years, when we should have added only 0.88 days, so we're ahead now by

1 - 0.88 days = **0.12 days**.

The funny thing is, *no one worries about that*. There is no official rule for leap days with cycles bigger than 400 years. I think this is extremely ironic, because if we took one more step we can make the calendar *extremely* accurate. How?

The amount we are off every 400 years is almost exactly 1/8th of a day! So after 3200 years, we've had 8 of those 400 year cycles, so we're ahead by

8 x 0.12 days = **0.96 days**.

If we then left leap day off the calendars again every 3200 years, we'd only be behind by 0.04 days! That's way better than any other adjustment we've made so far (it's good to less than a minute). I can't believe we stopped making fixes at the 400-year cycle.

But, still, yay, we're done! We can now, *finally*, see how the Leap Year Rule works.

**What to do to figure out if it's a leap year or not:**

We add a leap day every 4 years, except for every 100 years, except for every 400 years.

If the year is divisible by 4, then it's a leap year, **UNLESS**

it's also divisible by 100, then it's *not* a leap year, **UNLESS FURTHER**

the year is divisible by 400, then it *is* a leap year.

So 1996 was a leap year, but 1997, 1998, and 1999 were not. 2000 was a leap year, because even though it is divisible by 100 it's *also* divisible by 400.

1700, 1800, and 1900 were not leap years, but 2000 was. 2100 won't be, nor 2200, nor 2300. But 2400 will be.

This whole 400-year thingy was started in the year 1582 by Pope Gregory XIII. That's close enough to the year 1600 (which was a leap year!), so in my book, the year 4800 should *not* be a leap year, and then the calendar will be off by less than a minute compared to Earth's spin. That's impressive.

But who listens to me? If you've made it this far without frying your cerebrum, then I guess *you* listen to me. All this is fun, in my opinion, and if you're still with me here then you know as much about leap years as I do.

Which is probably too much. All you really need to know is that this year, 2020, is a leap year, and we'll have plenty more for some time. You can go through my math and check me if you'd like.

Or you can just believe me. Call it a leap of faith.

** Yes, the month is based on the cycles of the Moon, but there is no real definition for "month" which is one reason they're all over the place in terms of length.*

## Adding Time

Leap seconds don't come on a regular schedule because Earth's rotation varies, says Demetrios Matsakis, chief scientist for time services with the __United States Naval Observatory__ in Washington D.C. __Our planet is slowing down__, but it does so in unpredictable ways. So some periods require more leap seconds than others.

The __International Earth Rotation and Reference Systems Service__ continuously monitors our planet and will recommend adding leap seconds to the __International Telecommunications Union__ (ITU). The ITU makes the ultimate decision on whether to add a leap second or not.

The last leap second was added in 2012, but in the early 1980s, time scientists were adding them every year, Levine explains.

## The Shortest Day

As I write this post, it’s completely dark outside and it’s only 5 o’clock in the afternoon. Today is 4 December, and most people I come across think that it will continue to get dark earlier and earlier in the afternoons until 21 December, the shortest day of the year (at least for those of us in the northern hemisphere). This, however, is not the case. The evenings in fact start to draw out a week or so ** before** December 21, although it does not start to get lighter in the mornings until early in the new year.

This post aims to explain this interesting phenomenon.

**Sunrise and Sunset in December**

The table below shows the sunrise and sunset times for London in December 2014.

In the table the **daylight interval** column shows the number of hours, minutes and seconds between sunrise and sunset. This clearly shows that December 21 has the shortest period of daylight. However, the time of sunrise continues to get later and later throughout the whole of December, whereas the time of sunset starts getting later after December 12. This is good news for Mrs Geek, who walks home from work in the dark at this time of year.

The column in the table showing the **solar noon** gives the time of day that the Sun is at its highest in the sky or, to put it another way, the middle of the day at the mid-point between the times of sunrise and sunset. The table shows that, during December, the solar noon moves later and later by about 30 seconds each day.

**Why does the solar noon shift ?**

A **solar day** is not always exactly 24 hours. In fact, it is 24 hours only four times a year, and never in December. The definition of a solar day is the period of time between solar noon on one day and solar noon on the next day. It is at its shortest, around 23 hours 59 mins 38 seconds, in mid September and at its longest, around 24 hours 30 seconds around Christmas Day.

As you can imagine, it would be complete chaos if our clocks and watches had to cope with days of different lengths, so we use 24 hours, the average over the whole year, for all timekeeping purposes *(See Note 1).*

So, as I mentioned before, the solar days in December are on average 24 hours and 30 seconds, while our clocks and watches are still assuming that each day is exactly 24 hours. This causes the day to shift about 30 seconds later each day, as shown in the diagram below. This explains why the evenings start drawing out before the shortest day, but it continues to get darker in the mornings until the new year.

**Sundials and the “equation of time”**

Before the invention of accurate clocks sundials were widely used to keep time.

*Sundial in Harrogate in the North of England*

As the length of a solar day varies over the course of a year, the solar time, which is the time given by a sundial, will not be the same as the time measured by a clock which assumes that all days are exactly 24 hours long. Before the invention of accurate clocks in the 17th century because the variation is so small virtually everyone in the world, apart from a very small number of astronomers, would have been unaware of this.

However, in the eighteenth and nineteenth century as mechanical clocks started to take over timekeeping from sundials, the difference between the time measured by an accurate clock which is called **mean time** and solar time became an issue for everyday life. Astronomers call this difference ‘the equation of time’. It was first calculated and measured by the British astronomer John Flamsteed (1646-1713) in 1673.

Incidentally Flamsteed was appointed by the king as the first British Astronomer Royal in 1675, for which he was given the allowance of £100 per year. He also set up the Royal Observatory at Greenwich, shown below.

The diagram below shows how the equation of time varies throughout the year.

As you can see from the diagram, if we were to use a sundial to measure time

- from 15 Apr to 13 June and 1 Sept to 25 December to the sundial would be fast
- from 25 December to 15 Apr and 13 June to 1 September the sundial would be slow.

The days when the differences are greatest are

- November 3/4 when at 11:44 am, a sundial in London would be showing a time of 12 noon
- February 11/12 when at 12:14 pm a sundial in London would be showing a time of 12 noon.
*See Note 2*

**Why does the length of a solar day vary ?**

The reason why the length of the solar day varies is due to two different factors.

- The fact that the Earth moves in an elliptical (oval-shaped) orbit around the Sun and its speed varies, being faster in earlier January, when it is closer to the Sun and slower in early July, when it is further away.
- The fact that the axis of the Earth’s rotation is tilted.

The combination of these two factors gives the equation of time shown in the picture above. How these factors affect the length of the day and thus the equation of time is little too complicated to cover in a blog such as mine, which is aimed at the non-scientist. If you want to find out more the equation of time page on the Royal Greenwich Observatory website gives a lot of further information. To view it click here

**What about the southern hemisphere ?**

In the southern hemisphere the** longest day** is around December 21. What happens is that the Sun starts rising later before December 21, but it doesn’t start getting dark earlier in the evening until well after December 21. This is illustrated in the table below, which shows the sunrise and sunset times for December for Wellington in New Zealand, which lies at a latitude of roughly 41 degrees South.

The Earth’s rotation is slowing down, causing the length of a day to get gradually longer. In the year 1900 a mean solar day was 24 hours long. Now, in the early 21st century, a mean solar day is actually 24 hours 0.002 seconds long. To prevent the day we measure using accurate clocks from drifting away from the “natural day” we need to add an second called a leap second ever few years. For more information on this see my post: The Days are Getting Longer.

For most places in the world the Sun isn’t at its highest in the sky at 12 noon. This is because, rather than each area having its own local time, the world is divided into time zones, which are normally a whole number of hours ahead of or behind Greenwich Mean Time (GMT). For example, Manchester, where Mrs Geek and I live, is roughly 2.5 degrees West of Greenwich, but is on the same time zone. Because it is further West, the Sun rises and sets later than it does in Greenwich. From late Oct to late March, in Manchester the Sun is at its highest in the sky at 12:10 pm (on average) compared to 12-noon at Greenwich. At the end of March the UK puts its clocks forward by one hour, so from late March to late October the Sun will be (on average) at its highest in the sky at 1:10 pm in Manchester.

## To eliminate the need for an extra day every four years, why couldn't we just redefine the second?

Since we need (approximately) one extra day every four years to keep our calendar in sync with our orbit, why couldn't we just multiply (four years +1/ four years) 1461 / 1460 = 1.0006849315. to each existing second? After four years' time weɽ be enough later to eliminate the need for the leap day, right? Obviously all GPS satellites/world clocks/etc would have to be adjusted but that seems like something that could be done. I know this is missing something but that why I'm here!

This would mean a day is not 24 hours any more. So each day, "midnight" would slip a little bit. After 2 years, "midnight" would be in the middle of the day!

The fundamental issue that can't be solved here is there are not an integer number of days in a year.

There's not a whole number of days in the year, yet a day will always begin when the sun comes up and end when it goes down. So instead of slipping in the extra time into each day, we save up a day's worth and use it all at once every four years.

Not to mention physicists everywhere and everywhen collectively losing their goddamn minds.

Then just redefine the seconds minutes hours and days to accommodate for the extra time. It's like using imperial units instead of just converting to metric.

Maybe in the future they will use controlled asteroid slingshots to adjust Earth's rotational speed to solve this problem once and for all

The ratio of Earth's orbital period to its rotational period is not an integer. That is why we have leap years, not because of units. If you don't do this, the solstices and equinoxes start moving through the calendar, and we try to avoid that.

In simplest terms, the earth will still rotate

365.25 times (days) per full orbit around the sun (year), no matter how you divide up each day into arbitrary length seconds.

Actually the "1.0006849315s as one second" thing solves the exact problem. With "1.0006849315s as one second" the solstices would not be moving through the calendar.(Roughtly)

The problem is that if we use that we will have the number of days wrong. We would have 1460 days but 1461 sunrises/sunsets in 4 years. I.E. the time will be moving through the day.

Edit: Also note that a "Day" does not actually represent the earth rotation cycle. A "Year" does not actually represent the earth orbiting cycle.

Why do we have "day"? Because the ancient people noticed that we have a day/night cycle.

Why do we have "year"? Because the ancient people noticed that we have a "weather cycle"(seasons).

So they developed time/calendar to match these cycles. These cycles and the rotation/orbiting cycle are not exatly the same.

You are missing the fact that after one year your clock would be skewed by 6 hours, after two years you would have that midnight on you clock would really be noon outside.

Only after 4 years the time on the clock would be same as the time of the day.

So the answer is "yes, it will fix the leap day problem" but with the side-effect that it will create an even worse problem.

This solution ignores the fact that the day is not an arbitrary unit of time. The second is a man-made unit that comes from dividing the day (a natural phenomenon).

Changing the duration of the second would change the length of a day (on the clock), which creates a new problem. The clock would drift away from the natural cycle of sunrise and sunset (the 'real' day) Eventually youɽ have sunrise at midnight, etc.

The original problem is that the year does not divide into an even number of days. Uncorrected, this would allow our calendar to get out of sync with the seasons. OP's proposal would allow the clock day to get out of sync with the natural day.

Under the Leap Day system, we allow the fractional days to build up into a whole day, then correct the accumulated error all at once. This has the major benefit of keeping the clock in sync with the natural day, at all times. An extra day every fourth year* is considered less disruptive than letting clocks get out of sync with the sunrise and sunset.

*Bonus info: Technically, Leap Years occur every fourth year, except on years divisible by 100, unless that year is also divisible by 400. So, 2000 was a Leap Year, but only by exception to the exception. 2100 will not be a Leap Year.

The second actually *used* to be defined in terms of the earth's rotation and orbit, it was 1/60x60x24 th of a mean solar day. It was after it became apparent that this measure wasn't very good that we redefined the second in terms of the oscillations of the cesium atom.

This is generally how units work - they're defined in terms of something convenient until measurements need to be more precise than the current definition allows, at which point it is given a more precise redefinition.

For example, the earth's day and orbit may naively seem to be very fixed things, but at the precision needed for modern units, they fall short. The earth's orbit is constantly perturbed by the gravity of other planets. The length of the earth's day is slowed by the moon's gravity. Earthquakes moves mass in the crust, which changes the length of the day in the same way that a spinning ballerina pulling her arms in will spin faster. In fact, we have to add leap seconds to keep our clocks synced up with the earth! This isn't because our clocks are bad at keeping time, it's because the earth is!

By defining the second (and thus, all time) in terms of oscillations of the cesium atom we have picked a timekeeper that is absolute and independent of any artifact or happenstance of nature. We have picked a definition for time that can be reproduced anywhere without loss of generality. The consequence? We need leap years and leap seconds every so often. I'll take it.

## Is There A Perfect Calendar?

The simple answer is no. None of the calendar systems currently in use around the world perfectly reflect the length of a tropical year. However, there are calendar systems that are more accurate than the Gregorian calendar we use today.

The table shows how accurately the different systems reflect the length of a tropical year, sorted from most to least accurate. Calendars that are designed to reflect time spans other than the tropical year are not listed. This includes the Islamic, Buddhist, and Hindu calendar systems.

### Calendar Accuracy Comparison

* There is no 365-day calendar system currently in use for civil purposes. Past examples include the ancient civil Egyptian calendar, the Maya Haab' calendar, and the Aztec Xiuhpohualli calendar.

## Time Systems in Astronomy

Local time is the time we use in ordinary life. For example, this class is scheduled to meet four days a week at 1:00 PM local time.

Local time is designed to follow the Sun. The goal is to arrange our clocks so that the Sun rises around 6 AM, reaches its peak around noon, and sets around 6 PM. It's convenient from a practical point of view: if you are travelling to a foreign country and know that your plane will arrive at 23:00 local time, then you know most people will probably be at home and sleeping when you touch down. You'll probably have to call a cab .

One annoying feature of local time is that (in most states of the US) it shifts by one hour twice a year. During Daylight Savings Time, which starts April 3 this year, clocks are shifted forward one hour: what used to be 7 AM is now 8 AM. Sometime in the autumn (this year on October 30), clocks are shifted back one hour to Standard Time.

This shift can make it difficult to calculate the interval between two events, if they span the start or end of Daylight Savings Time.

#### Universal Time

When people living in different parts of the world want to coordinate their actions, the differences between time zones become a big hassle.

- Greenwich Time, or Greenwich Mean Time (GMT)
- Zulu Time (a military designation)
- Universal Time (*)
- Coordinate Universal Time (UTC) (*)

(*) There are actually small differences between several varieties of Universal-ish time. For our purposes, they may be treated the same, but if you are interested in accuracy at the sub-second level, you'll need to know the differences. See the references at the end of today's lecture.

#### Atomic Time

- 1990 Jan 1, 16:57:00 UT: the Sun reaches its highest point in the sky over Rochester
- 24 hours later: the Sun is again at its highest point
- another 24 hours later: the Sun is again at its highest point

These very small drifts between the Earth's rotation and the ticking of a perfect clock lead to leap seconds: extra seconds which are added (or subtracted) from the official civil time to keep the civil clocks in line with the Earth's rotation (and, hence, with the Sun).

To avoid the complications of leap seconds, scientists have devised a time system based on clocks which are more consistent than the Earth's rotation. International Atomic Time (TAI) is defined by the action of a set of atomic clocks. You can find tables which tell the difference between Universal Time and TAI over a range of dates.

The Global Positioning System (GPS) satellites use something like TAI: a system which runs more precisely than the Earth's rotation, and does not include leap seconds.

#### Barycentric (or heliocentric) time

Suppose that there is a distant celestial source which undergoes some regular variation: for example, an eclipsing binary star which has a period of exactly 5 hours.

By a nice coincidence, today the Earth happens to be at the point in its orbit which is closest to this distant binary star.

Measurements made last night indicate that the next set of eclipses will occur at these times:

And . they do occur at these times. Good!

But over the course of several months, the Earth will move around the Sun in its orbit. About two months later, the Earth will be located here:

When we observe the binary star from this location, we see that the eclipses are occurring a little bit later than predicted:

So, if one monitors a celestial source over the course of one year, one will see small variations in the time of (what should be) regular events.

The small variations accumulate over the course of an entire year to impressive amounts: the time it takes light to travel across the distance of the Earth's orbit, from one side of the Sun to the other, is .

- "heliocentric" means "the time measured by an observer sitting at the center of the Sun
- "barycentric" means "the time measured by an observer sitting at the center of mass of the Solar System"

Because the Sun doesn't sit perfectly still, but does move in a very small orbit due to the gravitational pull of the planets (mostly Jupiter), the best choice is barycentric. The official abbreviation is TDB, which stands for Barycentric Dynamical Time.

Using barycentric time is not always necessary. It does appear frequently in the timing of eclipses of binary stars: scientists searching for changes of just a few seconds in the period of a binary star's orbit (which may reveal properties of the interior stellar structure) must avoid any systematic errors which are minutes in size.

#### Julian Date

- How many days between 2012 March 1 and 2012 March 7?
- How many days between 2012 Jan 1 and 2012 March 7?
- How many days between 2011 Jan 1 and 2012 March 7?
- How many days between 1911 Jan 1 and 2012 March 7?
- How many days between 1711 Jan 1 and 2012 March 7?

The first few are easy, but as one considers longer and longer stretches of time, another complication appears: **leap days**.

Leap days are extra days inserted into the modern calendar as February 29. We need these extra days to keep the calendar in sync with the Sun. The problem is that the Earth takes about 365 **and a quarter** days to orbit the Sun. If we simply count 365 days as one year, then the calendar will start running ahead of the Sun: the shortest day of the year might be Dec 21 in 2005, but then Dec 22 in 2009, Dec 23 in 2013, and so on. After a century, the shortest day of the year would move to roughly January 15!

- if year is not divisible by 4, it is not a leap year
- if year IS divisible by 4, then
- if divisible by 400, IS leap year, else
- if divisible by 100, IS NOT leap year, else
- IS leap year

Scaliger named his system after his father, Julius Caesar Scaliger. We call dates in this system

**Julian Dates**, or JD for short.Now (2012 March 13 at 9:20 AM Eastern Daylight Time) is Julian Date 2,456,000.0555. It gets a bit awkward to keep typing all those digits, and hard to keep track of them all, so astronomers frequently subtract away a big round number. For example,

This sort of modification is handy when plotting light curves of variable stars, since otherwise the labels on the time axis can be so long that they run into each other.

If you ever decide to cut off a portion of the long JD number, please be explicit about exactly what you are subtracting.

#### Local Sidereal Time

Finally, there is one time system which is designed specifically for observers out in the field. Local Sidereal Time (or LST for short), like civil time or Universal Time, is based on the Earth's rotation but, unlike them, it does not account for the Earth's orbital motion around the Sun.

The Earth takes about 23 hours and 56 minutes to rotate once around its axis. That means that if you go outside and watch the motion of one particular star, there will be 23 hours and 56 minutes between successive star-rises or successive star-sets. Because the Earth moves a small distance around the Sun its orbit, the Sun takes an additional 4 minutes to "catch up" to the Earth: there are 24 hours between successive sunrises or successive sunsets. This introduces a gradual shift in the positions of stars from night to night (click on the image to see the nightly shift over the course of one week).

The shift is obvious if we skip ahead by 30 days at a time. You've undoubtedly noticed that stars visible early in the evening during one month aren't visible at the same time several months later.

Local Sidereal Time (LST) treats the stars in your sky like a big clock: the LST is equal to the Right Ascension of the stars which are currently overhead (or as high in the sky as they get). Tonight, for example, at 10:00 PM, the sky will look like this:

Note that Procyon, at RA = 07:39, is slightly to the west of the

**meridian**(the line running North-South through overhead) that means that the LST is later than 07:39. On the other hand, Dubhe, at RA = 11:03, is still rising up from the East, so the LST must be earlier than 11:03.Let's focus on the stars in the southern part of the sky tonight. What is the LST when the sky looks like this?

One hour later, the sky looks like this. What is the LST now?

One hour later, the sky looks like this. What is the LST now?

#### Homework

- What is the Local Sideral Time tonight at 11:00 PM EDT? Provide your answer to within one minute.
- The first manned spaceflight occurred on April 12, 1961.
- What was the most recent manned flight into space? Who was aboard this flight?
- How many days between the first and most recent flights?

#### For more information

- A brief listing of various time systems based on a posting to sci.astro by Paul Schlyter.
- The US Naval Observatory's list of time systems
- The Chandra Observatory's list of time systems
- Two Self-Consistent FORTRAN Subroutines for the Computation of the Earth's Motion, a paper by P. Stumpff published in 1980, describes how to make the barycentric correction to observed times of celestial events.
- A discussion of leap years thanks to Steffen Thornsen.
- Notes on Julian Dates from the sci.astro FAQs
- A list of the leap seconds appears on Wikipedia's entry for "leap second".

*Last modified by MWR 3/8/2005*Copyright © Michael Richmond. This work is licensed under a Creative Commons License.