# How did Meeus calculate equinox and solstice dates?

In Astronomical Algorithms (2nd ed, ch. 27, 2009 corrected printing) Jean Meeus gives expressions to calculate the date and time (dynamical time, equivalent to Terrestrial Time) of equinoxes and solstices from the year -1000 to the year +3000. The expressions are accurate to 51 seconds or better for the years 1951-2050. First what Meeus calls the "instant of the 'mean" equinox or solstice" is calculated using a fourth degree polynomial; there are 8 expressions. There are different expressions for each solstice or equinox, and different expressions for the year ranges -1000 to 1000 vs. 1000 to 3000. Then two corrections are applied; the corrections are calculated the same way no mater which time period or equinox or solstice is being corrected. The first step is to calculate:

$$T = \frac{(\text{mean JD of event} - 2451545.0)}{36525}$$

$$W = 35999.373°T - 2.47°$$

$$\Delta \lambda = 1 + 0.334 \cos W + 0.007 \cos 2 W$$

Next, an additional correction is computed involving 24 periodic terms with various periods.

Can anyone explain, in general terms, how Meeus derived these expressions? I'm especially interested in understanding what the "mean" value represents?

• iausofa.org/publications/sofa_iau_coll_180.html is probably unhelpful, but it's the official set of algorithms IAU uses. Meeus' calculations are simplified versions of these (as noted on the page above). ilrs.gsfc.nasa.gov/docs/2014/196C.pdf may also (not) be helpful: it talks about the Earth's "zonal harmonics" and "potential Love numbers", both of which affect equinoxes and solstices slightly.
– user21
Jun 4, 2015 at 0:15
• thetropicalevents.com may or may not be helpful.
– user21
Jan 8, 2016 at 6:53

The mean instant of an equinox JDE_{0} (or solstice) is the statistical mean calculated over a sample of equinox's instants (since they do not happen exactly in the same instant/day every year).

The first eight equations are derived by interpolation and the corrections (the 24 terms) are derived as a truncated iteration, I suppose from the JDE{N}-JDE{0,N}.

The truth is that Meeus does not explain anything at all (though he mentions interpolating the apparent longitude of the sun in 3 dates for higher accuracy). Without an explanation in the book, it is pretty difficult to guess how he did it.

You can find some more details about the iterative method here: