An explanation by Luboš Motl

"What I find bizarre is that people keep on repeating wrong dates of the equinoxes and solstices.

These days, children – at least in Europe – should learn March 20th as the date when the spring begins. Is that the case?

Where does the discrepancy come from? Let's look at the times (in Greenwich mean time i.e. Universal Time of the Spring equinoxes over a decade:

2010: 17:32

2011: 23:21

2012: 05:14

2013: 11:02

2014: 16:57

2015: 22:45

2016: 04:30

2017: 10:28

2018: 16:15

2019: 21:58

2020: 03:50

The date is always March 20th. What is the rule? You may see that every non-leap year, the precise moment is about 5 hours and 50 minutes, plus minus 5 minutes or so (the Earth just isn't as regular as you may expect – the irregularities come from the disordered impact of the Moon, Jupiter, and others), after the moment we saw in the previous year.

On the leap years, the special moment takes place about 18 hours and 10 minutes, plus minus 5 minutes or so, before the moment we remember from the last year. Needless to say, the increments are roughly +6 hours and –18 hours i.e. +1/4 and –3/4 of the solar day. This agrees with the fact that the spacing between two spring equinoxes is about 365.25 solar days while the civic calendar records either 365 or 366 new dates during that interval. The difference between 365.25 on one side and 365 or 366 on the other side manifests itself as those +6 or –18 hours.

Note that 18 hours and 10 minutes above is exactly one day minus 5 hours and 50 minutes; these numbers aren't independent.

But you may see that even after 4 years, the timing isn't quite periodic. For example, in 2016, the equinox will arrive 44 minutes earlier than in 2012. In 2020, it will be 40 minutes earlier than in 2016. In average, you may see that every 4 years, the equinox comes about 42 minutes earlier than 4 years earlier. In average, this drift gives you 10.5 minutes per year i.e. 1,050 minutes per century. Because the equinox will be at 3:50 am in 2020 and every 4 years, we will remove 42 minutes or so, 24 more years – roughly in 2044 – the date will jump to March 19th instead of March 20th, Greenwich Mean Time.

On the contrary, in 2011, the time was 23:21. So 4 years earlier, in 2007, it was probably 42 minutes later or so, in the morning of March 21st. If you return deeper to the 20th century, you may encounter many years in which the equinox occurred on March 21st.

I said that the drift gives you about 1050 minutes per century. That translates to about 18 hours per century. Why is this second-order discrepancy 18 hours? You may notice that it's 3/4 of a solar day again. And again, it's no accident. This new discrepancy arises because the year isn't quite 365.25 solar days. It's a little bit less than that. 365.24219 or so. We deal with this fact by stealing the leap year status from the years that are multiples of 100 (1700, 1800, 1900, 2100), even though they're multiples of four, but we restore the leap year status to multiples of 400 again (most prominently 2000), even though they're multiples of 100 as well.

This combined rule brings the average year to 365.2425 average solar days which is close enough. The remaining discrepancy accumulates to one day each 3,000 years or so,

So those 1050 minutes or 18 hours per century occur exactly because once per century – more precisely 3/4 times per century – we remove one leap year i.e. one day (February 29th) again. The times must adjust themselves so that the timing is pretty much the same as 400 years ago. And this is exactly achieved by making the equinox 42+ minutes earlier than 4 years earlier, each four years, except for years like 2100 when it's 18 hours (minus 42+ minutes) later than four years earlier. These required 42+ minutes in 4 years exactly explain why the non-leap-to-the-following-leap-year annual shift was 18 hours and 10 minutes rather than just 18 hours. All these corrections are ultimately calculable from the number of solar days per year, 365.24219.

At any rate, if you're teaching this stuff to kids, you should tell them that the spring begins on March 20th, summer solstice occurs on June 21st (although 20th will be increasingly more often), autumn equinox is on September 22nd or 23rd (this will be almost 50-50), and winter mostly starts on December 21st. Those days will be "mostly OK" when these kids become adults."

The dating suggested is very well for present day peoples who have expert mathematicians calculating for them day lengths using astronomical data and precision clocks. So agreed.

ReplyDeleteHowever it is not the way that the ancients did it. They did it another way that I have completely evaluated and have prepared for publication.