Editing Eclipse cycle (section)

==Properties of eclipses==
The properties of eclipses, such as the timing, the distance or size of the Moon and Sun, or the distance the Moon passes north or south of the line between the Sun and the Earth, depend on the details of the orbits of the Moon and the Earth. There exist formulae for calculating the longitude, latitude, and distance of the Moon and of the Sun using sine and cosine series. The arguments of the sine and cosine functions depend on only four values, the Delaunay arguments:

*D, the mean elongation (angle between the Sun and Moon longitudes)
*F, the mean argument of latitude (the angle between the Moon and the ascending node)
*l, the mean anomaly of the Moon (how far the Moon is from perigee)
*l', the mean anomaly of the Sun (or of the Earth)

These four arguments are basically linear functions of time but with slowly varying higher-order terms. A diagram of inex and saros indices such as the "Panorama" shown above is like a map, and we can consider the values of the Delaunay arguments on it. The mean elongation, D, goes through 360° 223 times when the inex value goes up by 1, and 358 times when the saros value goes up by 1. It is thus equivalent to 0°, by definition, at each combination of solar saros index and inex index, because solar eclipses occur when the elongation is zero. From D one can find the actual elapsed time from some reference time such as [[J2000]], which is like a linear function of inex and saros but with a deviation that grows quadratically with distance from the reference time, amounting to about 19 minutes at a distance of 1000 years. The mean argument of latitude, F, is equivalent to 0° or 180° (depending on whether the saros index is even or odd) along the smooth curve going through the centre of the band of eclipses, where [[Gamma (eclipse)|gamma]] is near zero (around inex series 50 at present). F decreases as we go away from this curve towards higher inex series, and increases on the other side, by about 0.5° per inex series. When the inex value is too far from the centre, the eclipses disappear because the Moon is too far north or south of the Sun. The mean anomaly of the Sun is a smooth function, increasing by about 10° when increasing inex by 1 in a saros series and decreasing by about 20° when increasing saros index by 1 in an inex series. This means it is almost  constant when increasing inex by 1 and saros index by 2 (the "Unidos" interval of 65 years). The above graph showing the time of year of eclipses basically shows the solar anomaly, since the perihelion moves by only one day per century in the Julian calendar, or 1.7 days per century in the Gregorian calendar. The mean anomaly of the Moon is more complicated. If we look at the eclipses whose saros index is divisible by 3, then the mean anomaly is a smooth function of inex and saros values. Contours run at an angle, so that mean anomaly is fairly constant when inex and saros values increase together at a ratio of around 21:24. The function varies slowly, changing by only 7.4° when changing the saros index by 3 at a constant inex value. A similar smooth function obtains for eclipses with saros modulo 3 equal to 1, but shifted by about 120°, and for saros modulo 3 equal to 2, shifted by 120° the other way.<ref name=Simon>{{cite journal|display-authors=etal |last1=Jean-Louis Simon |title=Numerical expressions for precession formulae and mean elements for the Moon and the planets |journal=Astronomy and Astrophysics |date=1994 |volume=282 |page=663 |url=https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?db_key=AST&bibcode=1994A%26A...282..663S&letter=0&classic=YES&defaultprint=YES&whole_paper=YES&page=663&epage=663&send=Send+PDF&filetype=.pdf |bibcode=1994A&A...282..663S}}</ref>
<ref>{{cite journal |last1=T. C. van Flandern & K. F. Pulkkinen |title=Low-precision formulae for planetary positions |journal=Astrophysical Journal Supplement Series |date=1979 |volume=41 |page=391 |doi=10.1086/190623 |url=https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1979ApJS...41..391V&defaultprint=YES&page_ind=0&filetype=.pdf |bibcode=1979ApJS...41..391V}}</ref>

[[File:Solar eclipse time of year for saros index divisible by 3.png|thumb|550px|Time of year for solar eclipses between saros 90 and saros 210, but showing only the saros series whose index is divisible by 3. The time of year is related to the anomaly of the Sun. Two of the four eclipses of the year 2000 are indicated, with a line between them which shows (almost exactly) the slope of simultaneity in this graph.
]]
The upshot is that the properties vary slowly over the diagram in any of the three sets of saros series. The accompanying graph shows just the saros series that have saros index modulo 3 equal to zero. The blue areas are where the mean anomaly of the Moon is near 0°, meaning that the Moon is near perigee at the time of the eclipse, and therefore relatively large, favoring total eclipses. In the red area, the Moon is generally further from the Earth, and the eclipses are annular. We can also see the effect of the Sun's anomaly. Eclipses in July, when the Sun is further from the Earth, are more likely to be total, so the blue area extends over a greater range of inex index than for eclipses in January.

The waviness seen in the graph is also due to the Sun's anomaly. In April the Sun is further east than if its longitude progressed evenly, and in October it is further west, and this means that in April the Moon catches up with the Sun relatively late, and in October relatively early. This in turn means that the argument of latitude at the actual time of the eclipse will be raised higher in April and lowered in October. Eclipses (either partial or not) with low inex index (near the upper edge in the "Panorama" graph) fail to occur in April because [[syzygy (astronomy)|syzygy]] occurs too far to the east of the node, but more eclipses occur at high inex values in April because syzygy is not so far west of the node. The opposite applies to October. It also means that in April ascending-node solar eclipses will cast their shadow further north (such as the [[solar eclipse of April 8, 2024]]), and descending-node eclipses further south. The opposite is the case in October.

Eclipses that occur when the earth is near perihelion (sun anomaly near zero) are in saros series in which the gamma value changes little every 18.03 years.  The reason for this is that from one eclipse to the next in the saros series, the day in the year advances by about 11 days, but the Sun's position moves eastward by more than what it does for that change of day in year at other times. This means the Sun's position relative to the node doesn't change as much as for saros series giving eclipses at other times of the year. In the first half of the 21st century, solar saros series showing this slow rate of change of gamma include 122 (giving an eclipse on January 6, 2019), 132 (January 5, 2038), 141 (January 15, 2010), and 151 (January 4, 2011). Sometimes this phenomenon leads to a saros series giving a large number of central eclipses, for example solar saros 128 gave 20 eclipses with |γ|<0.75 between 1615 and 1958, whereas series 135 gave only nine, between 1872 and 2016.<ref name=Duke/>

[[File:Length of semester.png|thumb|500px|Length of "semester" interval. The length varies considerably, depending on the lunar and solar anomalies of the two eclipses.]]
[[File:Length of Hipparchic period zoomed out.png|thumb|500px|Length of Hipparchic intervals ending in 2001-2050 on the same scale as above. The interval is quite constant because it is close to a whole number of anomalistic months (4573.002) and to a whole number of anomalistic years (344.979).]]

The time interval between two eclipses in an eclipse cycle is variable. The time of an eclipse can be advanced or delayed by up to ten hours due to the eccentricity of the Moon's orbit{{dash}}the eclipse will be early when the Moon is going from perigee to apogee, and late when it is going from apogee toward perigee. The time is also delayed because of the eccentricity of the Earth's orbit. Eclipses occur about four hours later in April and four hours earlier in October. This means that the delay varies from eclipse to eclipse in a series. The delay is the sum of two sine-like functions, one based on the time in the anomalistic year and one on the time in the anomalistic month. The periods of these two waves depends on how close the nominal interval between two eclipses in the series is to a whole number of anomalistic years and anomalistic months. In series like the "Immobilis" or the "Accuratissima", which are near whole numbers of both, the delay varies very slowly, so the interval is quite constant. In series like the octon, the Moon's anomaly changes considerably at least twice every three intervals, so the intervals vary considerably.

The "Panorama" can also be related to where on the Earth the shadow of the Moon falls at the central time of the eclipse. If this "maximum eclipse" for a given eclipse is at a particular location, eclipses three saros later will be at a similar latitude (because the saros is close to a whole number of draconic months) and longitude (because a period of three saros is always within a couple hours of being 19755.96 days long, which would change the longitude by about 13° eastward). If instead we increase the saros index at a constant inex index, the intervals are quite variable because the number of anomalistic months or years is not very close to a whole number. This means that although the latitude will be similar (but changing sign), the longitude change can vary by more than 180°. Moving by six inex (a de la Hire cycle) preserves the latitude fairly well but the longitude change is very variable because of the variation of the solar anomaly.

[[File:Solar eclipse anomaly 2001-2040.png|thumb|400px|Cosine of mean anomaly of moon at solar eclipses, 2001 through 2040. The curves connect eclipses that are 12 synodic months apart, but do not represent the anomaly between the eclipses. In each such series of four eclipses, the mean anomaly follows a sine wave. The moon is largest when the cosine of the anomaly is 1. On average every 3 years there is a "super moon" eclipse, with anomaly near zero.]]
Both the angular size of the Moon in the sky at eclipses at the ascending node and the size of the Sun at those eclipses vary in a sort of sine wave. The sizes at the descending node vary in the same way, but 180° out of phase. The Moon is large at an ascending-node eclipse when its perigee is near the ascending node, so the period for the size of the Moon is the time it takes for the angle between the node and the perigee to go through 360°, or

:<math>\frac 1{1/\text{period of node}+1/\text{period of perigee}}=\frac 1{1/18.60+1/8.85}=5.997</math> years

(Note that a plus sign is used because the perigee moves eastward whereas the node moves westward.) A maximum of this is in 2024 (September), explaining why the ascending-node [[solar eclipse of April 8, 2024]], is near perigee and total and the descending-node [[solar eclipse of October 2, 2024]], is near apogee and annular. Although this cycle is about a day less than six years, [[Super moon|super-moon]] eclipses actually occur every three years on average, because there are also the ones at the descending node that occur in between the ones at the ascending node.
At lunar eclipses the size of the Moon is 180° out of phase with its size at solar eclipses.

The Sun is large at an ascending-node eclipse when its perigee (the direction toward the Sun when it is closest to the Earth) is near the ascending node, so the period for the size of the Sun is

:<math>\frac 1{1/\text{period of node}-1/\text{period of perigee}}=\frac 1{1/18.60+1/41\text{ million}}=18.60</math> years

In terms of Delaunay arguments, the Sun is biggest at ascending-node solar eclipses and smallest at descending-node solar eclipses around when l'+D=F (modulo 360°), such as June, 2010. It is smallest at descending-node solar eclipses and biggest at ascending-node solar eclipses 9.3 years later, such as September, 2019.