Editing Eclipse cycle (section)

==Long-term trends==
The lengths of the synodic, draconic, and anomalistic months, the length of the day, and the length of the anomalistic year are all slowly changing. The synodic and draconic months, the day, and the anomalistic year (at least at present) are getting longer, whereas the anomalistic month is getting shorter. The eccentricity of the Earth's orbit is presently decreasing at about one percent per 300 years, thus decreasing the effect of the sun's anomaly. Formulae for the Delaunay arguments show that the lengthening of the synodic month means that eclipses tend to occur later than they would otherwise proportionally to the square of the time separation from now, by about 0.32 hours per millennium squared. The other Delaunay arguments (mean anomaly of the Moon and of the sun and the argument of latitude) will all be increased because of this, but on the other hand the Delaunay arguments are also affected by the fact that the lengths of the draconic month and anomalistic month and year are changing. The net results are:

*the mean argument of latitude is decreased by 0.16° per millennium squared, corresponding to 0.00045 draconic months
*the mean anomaly of the Moon is increased by 1.1° per millennium squared, corresponding to 0.0030 anomalistic months
*the mean anomaly of the sun is decreased by 0.002° per millennium squared, which is fairly negligible.

As an example, from the solar eclipse of April, 1688 BC, to that of April, AD 1623, is 110 inex plus 7 saros (equivalent to a "Palaea-Horologia" plus a "tritrix", 3310.09 Julian years). According to the table above, the Delaunay arguments should change by:

*40941 synodic months, 44429.003 draconic months, 43877.032 anomalistic months, 3310.007 anomalistic years, resp.

But because of the changing lengths of these, they actually changed by:<ref name=Simon/>

*40940.998 synodic months, 44429.006 draconic months, 43876.990 anomalistic months, 3310.007 anomalistic years, resp.

Note that in this example, in terms of anomaly (position with respect to perigee) the moon returns to within 1% of an orbit (about 3.4°), rather than 3.2% as predicted using today's values of month lengths.

The fact that the day is getting longer means there are more revolutions of the Earth since some point in the past than what one might calculate from the time and date, and fewer from now to some future time. This effect means eclipses occur earlier in the day or calendar, going in the opposite direction relative to the effect of the lengthening synodic month already mentioned. This effect is known as [[ΔT (timekeeping)|ΔT]]. It cannot be calculated exactly but amounts to around 50 minutes per millennium squared.<ref>Based on the length of a century increasing by 62 seconds per century (see [[ΔT (timekeeping)]]).</ref> In our example above, this means that although the eclipse in 1688 BC was centred on March 16 at 00:15:31 in [[Dynamic time]], it actually occurred before midnight and therefore on March 15 (using time based on the location of present-day Greenwich, and using the [[proleptic Julian calendar]]).<ref>{{cite web |last1=Fred Espenak |title=Five Millennium Catalog of Solar Eclipses -1699 to -1600 ( 1700 BCE to 1601 BCE ) |url=https://eclipse.gsfc.nasa.gov/SEcat5/SE-1699--1600.html |website=NASA Eclipse Web Site}} The [[Universal Time]] of given moments of an eclipse are different from the [[Ephemeris Time]] given by calculations{{dash}}see p. 59 of Schaefer, B. E. (March 1990). [https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1990QJRAS..31...53S&defaultprint=YES&filetype=.pdf Lunar visibility and the crucifixion]. Royal Astronomical Society Quarterly Journal, 31(1), 53–67.</ref>

The fact that the argument of latitude is decreased explains why one sees a curvature in the "Panorama" above. Central eclipses in the past and in the future are higher in the graph (lower inex number) than what one would expect from a linear extrapolation. This is because the ratio of the length of a synodic month to the length of a draconic month is getting smaller. Although both are getting longer, the draconic month is doing so more quickly because the rate at which the node moves west is decreasing.<ref name=Simon/>