Happy New Year everyone. I hope the changing of the calendar brings about the changes you are hoping for. In celebration of the new year, I wanted to look at some time related trivia.

**Time in a year**

A year is generally defined as the time it takes for the Earth to revolve around the Sun (see video below). This would be the sidereal year and the approximate duration of time this takes is 365.256 days (1). We would have that 2019 would be ushered in at 6:08 AM this morning.

In the video we see the Earth (blue) revolving around the Sun (yellow).

Another way that a year may be defined is the time it takes for the four seasons to complete and start again. The seasons, instead of being determined by location of the Earth relative to the Sun, are determined by the tilt of the Earth relative to the Sun. Therefore, this measure of the year would be equivalent to the period of the ecliptic, that is the path in the sky of the sun during the day. This year. tropical year, is approximately 365.242 days (2). Using this definition, the new year would have been ushered in at 5:49 AM.

Here the line represents the ecliptic over the course of the tropical year.

We can also define a year with respect to months. We think of a month as the time it takes for the moon to complete one revolution around the Earth. This sidereal month is approximately 27.32 days (3). If we then call a year equivalent to 12 months, we would have that a year would be 327.84 days. This would have placed the new year back on October 23 at 8:10 PM.

If we instead take a month to be how long it takes for the phases of the moon to complete a period, that is full moon to full moon, this lunar month would be 29.53 days (4). Taking 12 of these to be a year would give us 354.36 days. Therefore, the new year would have been December 20 at 8:38 AM.

The Gregorian year is the amount of time from 12:00 AM January 1st to the next years 12:00 AM. During a non-leap year, this is exactly 365 days. On a leap year, this is 366 days. The Gregorian calendar is the current calendar and has a leap year every four years. Unless that fourth year is a multiple of 100, then you only have a leap year if the century is a multiple of four. For example 2000 was a leap year, but 1900 was not. Over the 400 year period of the leap years, the average Gregorian year is 365.2425 day.

**How often do these line up?**

With all of these definitions of year proposed, I wanted to take some time to compare them. Rather than just finding the difference, I wanted to know how often the new year would line up at the same time using each of the definitions. To begin with, we will look at some of them pairwise.

We first compare the Gregorian and the tropical year. While they are not exactly the same, they are extremely close. The 26 second difference is close enough that, since we have been using approximation throughout, we can call these the same. Therefore, the Gregorian and tropical calendar, on average line up. However, since the Gregorian calendar changes the number of days in each year to accomplish this average, we would say that new years would correspond whenever the Gregorian calendar has completed a cycle of leap years. As we saw above, this is every 400 years. Therefore, every 400 years the two new years would correspond with each other.

In order to compare the tropical to the sidereal year, we will need to find the least common multiple of 365.242 and 365.256 and 365. By the least common multiple we mean the smallest number \(x\) such that \(m*365.242=x=n*365.256\) for some natural numbers \(m\) and \(n\). In general, for real numbers, such an \(x\) need not exist. For example there is no least common multiple of \(\sqrt{2}\) and \(1\). However, this number will exist for any positive rational number. In this case, we get \(66.7\) million days, which would mean that the new years would coincide every 182 thousand (Gregorian) years.

Instead of continuing for all pairwise definitions of the year, we will instead determine how often all of them would correspond. For this we would find the least common multiple of the days in every type of year. This would be approximately 2.7 quadrillion (\(2.7*10^{15}\)) days. Therefore, all of the above defined years would have a coinciding new year every 7.4 trillion (\(7.4 *10^{12}\) Gregorian) years.

As a note, I was rounding to the nearest thousandth of a day in order to find these periods. That is, when we say that the new years coincide, they do so up to a thousandth of a day, or about 1.4 minutes. If you want them to coincide to the second, you would need to be more precise, but you would follow the same process.

**Conclusion**

I hope you enjoyed the bit of trivia on the first day of the year. It was amusing for me to think about as I watched the countdowns for the new year on television. Trying to line up the start of the year in the face of the ambiguous definition of year seemed daunting. I hope the next year (however you measure it) is great for you.

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