# What's a good model for the vernal equinox year? Can it increase forever?

Duncan Steel (2002) makes an interesting argument that the notion of year is (1) somewhat arbitrary and, more interestingly (2) one notion of year can increase while another decreases.

His examples are tropical year (which decreases over time, at least in the linear Newcomb model) and the vernal equinox year which seems to be currently increasing.

Actually looking at the data that his point is based on, namely Meeus and Savoie (1992), we can find another measure that is decreasing, namely the solstice year(s), either June or December; the autumnal equinox year is also decreasing: Steel actually uses the fact that the vernal equinox year is increasing as an argument that the Gregorian calendar is "getting better" in a sense that it's closer to hitting its target year length. (The opposite is the case if one consider say either of the solstice years... or the tropical year, which is the point that a lot of other people make, e.g. Moyer (1982).)

What I really want to ask here: assuming a simple linear decrease model for the tropical year (e.g. Newcomb), will the trends of the equinox and solstice years reverse at some point in the future?

(And if it's not too much extra to ask:) What are the implications of the more complex models for the tropical year (e.g. cubic VSOP) on the evolution of the equinox and solstice years?

• No, it can't increase forever, but forever's a long time. ;) How about limiting your question to a somewhat shorter time period? We are talking about some rather long period cycles here, so I guess you want to cover at least $10^5$ years. And of course over that time span the length of the day will increase noticeably. Aug 30, 2018 at 5:44
• @PM2Ring: it seems to me that saying that the trend would reverse but only in tens of thousands of years should be part of the answer, not part of the question...
– Fizz
Aug 30, 2018 at 10:43

The main factors that affect the length of the March equinox year (*1) over time can be divided into three components:

• Changes in the length of the solar day
• Changes in the length of the mean tropical year
• Difference between the mean tropical year and March equinox year

Of these, the change in the length of the solar day is most significant in the long term.

## The length of the solar day

The length of the solar day is continuously increasing at a rate of roughly 1.75 milliseconds per century on average. This means that if we measure any type of year in terms of solar days, its length will decrease simply because our measuring unit is becoming longer; this comes to about 0.64 seconds per century, or 7.4 millionths of a day per century. In particular, calendars count solar days, which means that any calendar with a fixed average year length will inevitably become inaccurate.

## Change in the mean tropical year over time

The change in the mean tropical year over time (as measured in atomic time) is mainly an oscillation of period around 41,000 years and amplitude around 15 seconds, which is linked to the obliquity cycle. The long-term average (over millions of years) is relatively stable in the vicinity of 31,556,924 seconds (365 days 5 hours 48 minutes 44 seconds).

Combined with the above, the mean tropical year when measured in solar days decreases over time at a slightly varying rate.

## Difference between the March equinox year and mean tropical year

The difference between the March equinox year and mean tropical year depends on where the vernal equinox is in Earth's orbit, and is thus affected by the precession of Earth's orbit. In short, when the March equinox is near perihelion the March equinox year will be longer than the mean tropical year, and when it is near aphelion the March equinox year will be shorter (as a reminder, perihelion is when the Earth is closest to the Sun, and aphelion is when the Earth is furthest away). (*2)

Kepler's laws tell us that the Earth moves faster in its orbit when it is nearer to the Sun; this leads to variations of a few days in the lengths of the seasons. Due to apsidal precession, the aphelion and perihelion move slowly over time; this movement causes the variation to show up when computing the March equinox year. Specifically, as the apsides move forward in Earth's orbit over time, a tropical year is slightly less than one orbit relative to the apsides (one such full orbit is known as an anomalistic year). As a result, a small chunk of the orbit is left out, and the duration of that chunk depends on where it is relative to the apsides i.e. shorter near perihelion and longer near aphelion. (*3)

Since perihelion is currently in January and advancing towards the March equinox, the March equinox year is increasing relative to the mean tropical year; the same holds for the June solstice. The rate of this increase will slow down as perihelion approaches the equinox. Once perihelion passes the March equinox in around 6000 CE the difference between the March equinox year and the mean tropical year will decrease; at the same time the June solstice year will be increasing at its fastest rate, and the September equinox year will begin increasing relative to the mean tropical year as the perihelion advances towards that equinox.

Note that the size of this effect will be influenced by the obliquity of Earth's orbit; as the obliquity decreases over the next few tens of thousands of years, the magnitude of these differences will also decrease.

## Results

When we combine these effects, we find that each equinox and solstice has a period of relative stability alternating with a period of fast decrease. In stable periods, the increase due to apsidal precession cancels out the decrease due to lengthening of the day, while in fast decrease periods both effects add together.

It just so happens that the March equinox has been in such a stable period for thousands of years. The March equinox year will stop increasing around 3000 CE, then begin to decrease. By that time, the June solstice year will be increasing, while the September equinox year will continue decreasing for a long time, and the December solstice year will be in the middle of its fast decrease period.

All of this is most easily seen in the graphs on Irv Bromberg's excellent page.

(*1) Of course, the same factors apply for any other year based around a fixed solar longitude, such as the June solstice year or the Besselian year.

(*2) This is largely a reminder for myself -- I mixed them up while writing this post.

(*3) If Earth's orbit was in the opposite direction for some reason, this would be reversed, with a tropical year being longer than an anomalistic year and an extra chunk of orbit being added.