### 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?

http://www.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). http://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.

http://www.thetropicalevents.com/ may or may not be helpful.

• 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:

I hope this helps :)

Thanks. I learned about iterative methods in university, using the textbook "Numerical Methods with Fortran IV Case Studies". Assignments were carried out with punched cards. I am more interested in what Meeus means by "mean" equinox; he is doing something to smooth out the year-to-year variations of apparent equinoxes.

• I obtained a more recent (2002) book by Meeus, More Mathematical Astronomy Morsels, published by Willmann-Bell. Chapter 63 is about the Gregorian calendar and the vernal equinox. It contains expressions from J. L. Simon et al. "Numerical expressions for precession formulae and mean elements for the Moon and planets".

One can use the expressions to find the orbital elements of the Earth for a given date near an equinox or solstice, and then solve Kepler's equation to find the "true" longitude of the Earth. The longitude is true in the sense that the elliptical orbit is used rather than the fictitious circular orbit that is used for mean mean longitude, but the periodic perturbations from the Moon and other planets are smoothed out. One iterates until dates are found where the longitude is 0, 90, 180 and 270 in years either side of the year of interest. The intervals between solstices or equinoxes are found, and divided by 2.

I tried this for year 0 and year 2000 and it agreed with the full precision that Meeus published in 2002, six decimal places, or about 0.1 second of time. On the other hand, around 2000, the results were different by about 8 minutes from the results from the book in the question. The 2002 book and 1992 paper were about finding the mean interval between successive equinoxes or solstices, in a discussion of how "tropical year" has been defined throughout history. I suspect the methods in the book in the first question were not used for this purpose.