Calculating the times of equinoxes

[Dominic Ford]

I’m posting this on the off-chance there’s anyone here who’s done their own calculations of the times and dates of equinoxes.

Last month we had an enquiry from somebody who wanted to know whether the equinox was on 22 Sept or 23 Sept. I directed them towards a chart of the time of the equinox versus year, which shows the four-year saw-tooth pattern it does as a result of leap years (see here https://in-the-sky.org/news.php?id=20170922_07_100#exact_time ). This year’s equinox was actually on 22 Sept.

However, there are some interesting anomalies here. That chart is based on my own calculations (details below), but compared to other sources, some equinoxes / solstices are out by a few minutes, up to about 90 minutes in some cases (e.g. the Sept 2020 equinox).

Of course, not many people care about that level of precision, but it bothered me as most of the calculations I do (e.g. planetary oppositions, and the phases of the Moon) agree with other sources to better than 100 seconds.

The source that everybody on the internet seems to use for the times of equinoxes is the US Naval Observatory’s almanac (cited by <https://en.wikipedia.org/wiki/Equinox&gt;, for example, and Google), although the relevant part of the USNO’s own website has been offline for several days now… http://aa.usno.navy.mil/data/docs/EarthSeasons.php

I’d be interested to know if anybody knows what method the USNO use to calculate the times they publish?

My own calculations are based on the NASA DE405 ephemeris, and unless I’ve mucked something up, they ought to be very accurate. I use the position of the Sun from DE405, the position of the Earth-Moon barycentre from DE405, and the Earth-to-Moon-vector from DE405. I then compute the position of the Earth (corrected for the Moon’s pull) via:

Earth_pos = Earth_Moon_barycentre_pos – (Moon_mass) / (Moon_mass + Earth_mass) * Earth_Moon_vector

I figure out the direction of the Earth-Sun line in the coordinate system DE405 uses (ICRF), and transform that from J2000.0 coordinates into instantaneous celestial coordinates for the epoch of observation (using Jean Meeus’s Astronomical Algorithms), and look for extrema and zero-crossings in the Sun’s declination.

I’ve probably mucked something up, but I’ve checked it pretty thoroughly over the past 10 days. I’m curious to know whether the USNO define the equinox in some curious way…?

[James Lancashire]

Jean Meeus sets the standard for such celestial calculations – and anomalies. I would think he has an article somewhere for centuries of predictions.
There is also a discussion at https://astronomy.stackexchange.com/questions/10901/how-did-meeus-calculate-equinox-and-solstice-dates

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