Predictions of Eclipses

## Eclipses

To get a solar or lunar eclipse, the Moon must have the proper phase (New Moon for a solar eclipse, or Full Moon for a lunar eclipse), and the Moon must be close enough to the plane of the Earth's orbit around the Sun, where the Earth, the Sun, and the Earth's shadow are. The orbit of the Moon makes an angle of about 5 degrees with the orbit of the Earth, and usually the Moon is too far above or below the orbit of the Earth when it is Full Moon or New Moon, and that's why we don't have an eclipse at every Full Moon or New Moon.

The two places where the orbit of the Moon intersects the plane of the Earth's orbit are called the nodes of the lunar orbit. Because the Sun and the Moon appear quite large in the sky, an eclipse can occur even when the Moon is still a bit above or below the plane of the Earth's orbit during Full or New Moon. If the distance of (the middle of) the Full or New Moon from the nearest node is less than 13.9 degrees, then there will certainly be an eclipse. If the distance is more than 21.0 degrees, then there is certainly no eclipse. If the distance is between those two boundaries, then it depends on other things (such as the distance of the Moon from the Earth) whether there will be an eclipse or not.

Whether an eclipse will occur depends mostly on the coming together of two periodical phenomena, in this case the reaching of the proper lunar phase and the passing through a node. For more generality, we'll investigate the coming together of two arbitrary phenomena A and B. We assume that phenomenon A has a period P`A`, that phenomenon B has period P`B`, and that the lengths of those periods are constant. The ratio

(Eq. 1) γ``` = ```P`A``⁄`P`B`` `

measures the average number of phenomena B that occur for each phenomenon A. For eclipses, A is the proper lunar phase (Full Moon for lunar eclipses, New Moon for solar eclipses), and B is the passage through a node of the lunar orbit. We have P`A` = 29.530588853 days and P`B` = 13.606110408 days, so γ = 2.170391681.

If an A and a B happen at the same time, then we get the combination effect, which we'll call Z. In our example, Z is an eclipse. Often there is a little leeway so A and B can have a small time difference and yet produce a Z. We can find two boundaries d`1` and d`2` so that a Z certainly occurs if the difference between an A and the nearest B is less than d`1`` `P`A`, and a Z certainly does not occur if the time difference is greater than d`2`` `P`A`. For solar and lunar eclipses we have d`1` = 0.077 en d`2` = 0.117.

### Approximation With a Ratio of Whole Numbers

To base predictions of Z on periods, we must approximate γ with a ratio a`⁄`b of positive whole numbers. If we can find such a and b, then we can say that a periods P`B` are almost equal to b periods P`A`, and that hence that much time after a previous Z another Z will occur.

If we know γ, then we can determine the best approximations by ratios of whole numbers. The error in such an approximation is equal to

(Eq. 2) δ``` = ```a` - `b` `γ``` ```

and that is smallest, for a given b, if

(Eq. 3) a` = [`b` `γ```] ```

where `[` `]` indicates rounding to the nearest whole number. With that value of a, the error in the approximation is equal to

(Eq. 4) δ`(`b```) = [```b` `γ```] - ```b` `γ``` ```

which always lies between −1/2 and +1/2. We want δ`(`b`)` as close to zero as possible. If δ`(`b`)` is closer to zero for a certain value of b than for all smaller values of b, then we call that value of b very good. We can find all very good values of b with the following method.

This method works for numbers smaller (in absolute value) than 1/2. We therefore seek very good approximations for ε``` = ```γ` - [`γ`]`. If we have found a very good approximation ai`⁄`bi, then the corresponding very good approximation for γ is equal to `(`ai` + [`γ`] `bi`)⁄`bi.

The first very good approximation is a`1`` = 0, `b`1`` = 1, `δ``` = ```−ε. The second very good approximation is a`2`` = 1, `b`2`` = `` 1⁄`ε` ````, ```δ` = 1⁄`` 1⁄`ε` ```` - ```ε where   indicates the largest whole number less (in the direction of minus infinity) than its argument. Once we've found two very good approximations, we can find the next one:

(Eq. 5) ki` = `` `δ`(`i`-2)``⁄`δ`(`i`-1)`` ```` ```<` 0 `

(Eq. 6) ai` = `a`(`i`-2)`` - `ki` `a`(`i`-1)`` `

(Eq. 7) bi` = `b`(`i`-2)`` - `ki` `b`(`i`-1)`` `

(Eq. 8) δi` = `δ`(`i`-2)`` - `k` `δ`(`i`-1)`` = `ai` - `ε` `bi` `

where   indicates the least whole number greater (in the direction of plus infinity) than its argument.

The first couple of very good approximations that we find for eclipses are listed in the following table. The very good approximations are a`⁄`b. The corresponding period of prediction and great period (to be explained later) are y (in years) en c (in years). The number of successful predictions in a row to be expected is between n`1` and n`2`. The fraction of successful predictions of further eclipses based on earlier eclipses and the prediction period is equal to P. Very good approximation number 12 has the unusably large prediction period of 12393.4 years.

 i b a y c n`1` n`2` P name 1 1 2 0.08 0.5 0 2 0.114 2 5 11 0.4 2.7 1 2 0.230 3 6 13 0.5 22 6 11 0.872 semester 4 41 89 3.3 238 11 17 0.912 hepton 5 47 102 3.8 452 18 27 0.933 octon 6 88 191 7.1 1290 27 42 0.953 7 135 293 10.9 3790 53 79 0.962 tritos 8 223 484 18.0 6790 58 86 0.987 saros 9 358 777 28.9 130200 694 1024 0.966 inex 10 4161 9031 336.4 1564600 715 1055 0.912 11 4519 9808 365.4 56739300 23916 35254 0.921

Notice that ai` `b`(`i`-1)`` - `a`(`i`-1)`` `bi alternates between +1 and −1, and that

(Eq. 9) an`⁄`bn` = `a`1`` + ``(`i`=2)`n``` (-1)```i` 1⁄(`b`(`i`-1)`` `bi`) `

for example, a`8``⁄`b`8`` = 484⁄223 = 2 + 1⁄(1``∗``5) - 1⁄(5``∗``6) + 1⁄(6``∗``41) - 1⁄(41``∗``47) + 1⁄(47``∗``88) - 1⁄(88``∗``135) + 1⁄(135``∗``223)`.

### The Great Period

Eventually we use some approximation

(Eq. 10) γ```' = ```a`⁄`b` `

for γ with whole numbers a and b. These numbers do not have to be determined using the method described above, and don't even need to give a particularly good approximation to γ. We assume (without loss of generality) that a and b have no divisors in common.

This approximation corresponds to the assumption that b periods P`A` are equal to a periods P`B` and that that much time after a previous Z there will be a next Z. We'll refer to this period of time as the period` `of` `prediction and will indicate it as y. If the approximation is not perfect (so

(Eq. 11) δ``` = |```γ` - `γ```'| ```

is not equal to zero), then the correspondence between an A and a B will be slightly different with each next predicted Z, and in the course of time A and B will get out of step so much that Z no longer happens when it was predicted. After about

(Eq. 12) n` = 1⁄`δ``` ```

periods of prediction, the small deviations add up to a whole period of prediction, and then the correspondence is back where it started. We'll refer to this period of time as the great period. Its length is estimated as

(Eq. 13) c` = `n` `y` `

The number of predictions of Z that will come true in a row lies between

(Eq. 14) n`1`` = `` `d`1`` `n` `` `

and

(Eq. 15) n`2`` = `` `d`2`` `n` `` `

and these numbers correspond to periods of

(Eq. 16) c`1`` = `n`1`` `y` ```` ```d`1`` `c` `

and

(Eq. 17) c`2`` = `n`2`` `y` ```` ```d`2`` `c` `

A couple of these values are listed in the preceding table.

### Labelling the Series of Predictions

With the adopted approximation γ```' = ```a`⁄`b there are b different series of predictions. At any given moment, between d`1`` `b and d`2`` `b of these series are active, which means that predictions of phenomena Z in those series have a non-zero chance around that time of coming true. Because of the error δ a given series will change from active to inactive after a while, and later back to active. At any given moment there is one series for which the phenomena A and B are closest together for each Z. We call that series the best series. A best series is the best for a period of time equal to about c`⁄`b, and after that some other series becomes the best one. We label each series with a number such that at any moment the best series has a number that is one greater than the number of the series that was the best one previously.

First, we label all phenomena A in the order in which they occur. We assign the number 0 to a specific A. Later phenomena A have positive numbers, and earlier ones negative numbers. We use k here to denote the number of a phenomenon A. A suitable formula for calculating the number of the prediction series that contains k is then

(Eq. 18) s` = `a` `k` + `s`0`` mod `b` `

in which the `mod `b means "except for multiples of 2[1]3[1]b4[1]5[1]", and the s`0` is the number of the series that contains the A with k = 0. If you want to know which k belong in series number s, then you can use:

(Eq. 19) k` = `l`∗``(`s` - `s`0``) mod `b` `

in which l is a number such that a` `l``` = 1 mod ```b. Generally,

(Eq. 20) l` = `a`(`φ`(`b`) - 1)`` mod `b` `

with φ`(`b`)` the count of non-negative whole numbers less than b that is relatively prime to b. If the b belongs to a very good approximation, then

(Eq. 21) li` = `bj` `

in which j is the greatest odd number less than i.

For example, the solar eclipse of 5 February 2000 belongs, according to an oft-used scheme, to saros number 150. If we assign k = 2 to that New Moon (so that the first New Moon of the year 2000 has number 1), then s`0` = 74 and the formula to calculate the number of the saros (with a = 484 and b = 223 according to the table)

(Eq. 22) s``` = 484 ```k` + 74 = 38 `k``` + 74 mod 223 ```

Likewise, the k that belong to a particular saros can be found using

(Eq. 23) k``` = 135 (```s` - 74) = 135 `s``` + 45 mod 223 ```

The condition of phenomena B at the moment of an A is, according to our approximation, indicated by

(Eq. 24) S`'`k``` = ```S`'0 + `γ```' ```k` `

If S`'`k is a whole number, then we have landed exactly on a B, so then we're also in the middle of a Z. S`'0` is the condition of B at the moment that corresponds to k = 0. With the definition

(Eq. 25) s`'``∗`` = `a` `k` + `s`0`` - `b` `S`'`k``` ```

we find that

(Eq. 26) s`'``∗`` = `s`0`` - `b` `S```'0 ```

and that is a constant. The whole number closest to s`'``∗` is the number of the best series (except for multiples of b) according to the approximation.

Similar to S`'`k and s`'``∗` we also define

(Eq. 27) Sk` = `S`0`` + `γ``` ```k` `

(Eq. 28) s`∗`` = `a` `k` + `s`0`` - `b` `Sk` `

and s`∗` shows which series is really the best one at that time. This s`∗` is not a constant, but increases by one about every c`⁄`b years, because it is based on γ rather than γ`'`. For the saros (for solar eclipses) at the beginning of 2000, s`∗` was equal to 135.8998, so saros series 136 was then the best one, and shall remain the best one until midway through 2018. The number of the best saros series increases by one about every 31 years.

A problem of equation 18 is that it provides the number of the saros series except for multiples of b. Which multiple should we choose? If you wish to give every distinct group of successive active phenomena from a particular prediction series a unique number, then the following formula is best:

(Eq. 29) sk` = `a` `k` + `s`0`` - `b` [`Sk`] `

In that case the next set of active phenomena from a particular prediction series gets a number that is b greater than for the previous set of active phenomena, so they are different and yet equal except for multiples of b.

If you'd rather link all phenomena of a particular prediction series with the same number, then the original equation <%ref sarosnummer> is best, but only if you interpret `mod `b as the remainder after division by b.

For predictions of solar eclipses around the year 2000 using the saros we can use the following rules of thumb: certainly a solar eclipse if sk is between 108 and 164, and certainly no solar eclipse if sk is smaller than 92 or greater than 180. About every 30 years these boundaries shift up by one.

### Related Phenomena

A phenomenon Z occurs when we're sufficiently close to a B when there is an A. In some cases there is a phenomenon ζ related to Z when we're sufficiently close to a B if we're a certain fixed part along of the distance between two successive As and have thus ended up at a phenomenon α related to A. For example, if A is a New Moon and B is the passage through a node of the lunar orbit, then Z is a solar eclipse. To get a lunar eclipse ζ we must be sufficiently close to a node (B) when we're midway between two New Moons (A) at a Full Moon (α).

For such a related phenomenon ζ we find the exact same formulas as for Z, except that the constants may be different. If the α occurs at a fraction φ of the distance between successive As, and if we identify the α with the k of the preceding A, then we find that the Σ`'`k, which is the S`'`k of the α, is equal to

(Eq. 30) Σ`'`k``` = ```S`'0 + `γ`'(`k` + `φ```) = (```S`'0 + `γ`'`φ```) + ```γ`'`k` = `Σ```'0 + ```γ`'`k` `

We can also immediately write down the formula for labelling the prediction series of ζ, similar to equation <%ref sarosnummer>:

(Eq. 31) σk` = `a` `k` + `σ`0`` mod `b` `

and a formula for the best prediction series for ζ, similar to equation 25:

(Eq. 32) σ`'``∗`` = `σk` - `b` `Σ`'`k``` ```

We are still free in our choice of σ`0`. We can, for instance, select to put s`'``∗` and σ`'``∗` as close together as possible, so that for Z and ζ series with mostly the same numbers are active. For that case we find

(Eq. 33) σ`0`` = `s`0`` + [`a` `φ```] ```

For lunar eclipses we then find, with φ = 0.5 compared to solar eclipses,

(Eq. 34) Σ`'`k``` = ```S`'`k``` + 242⁄223 ```` `S`'`k``` + 1,085196 ```

(Eq. 35) σ`0`` = `s`0`` + 242. `

### Changing Periods

The prediction of phenomena based on periods assumes that the periods, and γ, are fixed, but that is often not the case in practice. If the variation of γ with k around k = 0 can be reasonably approximated with γ``` = ```γ`0`` + `p` `k, then equation 27 becomes

(Eq. 36) Sk` = `S`0`` + `γ``` ```k` = `S`0`` + `γ`0`` `k` + `p` `k`2`` `

As long as the deviation p` `k`2` is much smaller than 1, equation 27 with p = 0 is a reasonable approximation. If we take 1/10 as the greatest acceptable deviation, then the equations with assumed-constant γ`'` can be used for `|`k`|` less than about `1⁄`√p. For the saros, p` ```` 6````∗``10(-12)`, so the assumption of constant γ is reasonable for `|`k`| `<``` 400,000```. The formulas that are suitable for today will therefore remain reasonable for roughly the next 33,000 years.

## The Saros and Solar Eclipses

The saros is the best-known period for predicting solar and lunar eclipses. The saros is based on the near-equality of 223 synodical months (6585.321 days) and 242 draconitic months (6585.357 days). This period corresponds to 18 years and about 10 1/3 days (with 5 leap days in the period) or 11 1/3 days (with 4 leap days). Coincidentally, this period is also almost equal to 239 anomalistic months (6585.537 days). Of all periods that can be used to predict solar eclipses, up to at least 12,000 years, the saros is the one that yields the greatest chances of success: If you predict that there will be another eclipse one saros after a previous one, then you'll be right about 98.7% of the time (based on information about all solar eclipses between the years −1999 and 3000). Refer to eclipse cycles for elaboration on saros. Also, to read about controversy of ancient prediction related to the Stonehenge

For solar eclipses we find

(Eq. 37) sk` = 484 `k` + 74 - 223 [0.892838 + 2.1703916819 `k```] = 38 ```k` + 74 - 223 [0.892838 + 0.1703916819 `k`] `

if k counts New Moons since the beginning of the year 2000. For lunar eclipses we find

(Eq. 38) σk` = 484 `k` + 316 - 223 [1.978040 + 2.1703916819 `k```] = 38 ```k` + 93 - 223 [0.978040 + 0.1703916819 `k`] `

Lunar Eclipses: 2001 - 2008

The table below lists every lunar eclipse from 2001 through 2008.

 Lunar Eclipses: 2001 - 2008 Date Eclipse Type Umbral Magnitude Total Duration Geographic Region of Eclipse Visibility Total 1.195 62m Partial 0.499 - Penumbral -0.110 - Penumbral -0.283 - Penumbral -0.788 - Penumbral -0.222 - Total 1.134 53m Total 1.022 24m Total 1.309 01h16m S. America, Europe, Africa, Asia, Aus. Total 1.313 01h21m Americas, Europe, Africa, c Asia Penumbral -0.139 - e Asia, Aus., Pacific, Americas Partial 0.068 00h58m Asia, Aus., Pacific, North America Penumbral -0.055 - Americas, Europe, Africa, Asia Partial 0.189 - Europe, Africa, Asia, Aus. Total 1.238 01h14m Americas, Europe, Africa, Asia Total 1.481 01h31m e Asia, Aus., Pacific, Americas Total 1.111 00h51m c Pacific, Americas, Europe, Africa Partial 0.813 - S. America, Europe, Africa, Asia, Aus.

Geographic abbreviations (used above): n = north, s = south, e = east, w = west, c = central

Solar Eclipses: 2001 - 2008

The table below lists every solar eclipse from 2001 through 2008.

 Solar Eclipses: 2001 - 2008 Date Eclipse Type Eclipse Magnitude Central Duration Geographic Region of Eclipse Visibility Total 1.050 04m57s Annular 0.968 03m53s Annular 0.996 00m23s Total 1.024 02m04s Annular 0.938 03m37s Total 1.038 01m57s Partial 0.736 - Antarctica, s Africa Partial 0.927 - ne Asia, Hawaii, Alaska Hybrid 1.007 00m42s N. Zealand, N. & S. America [Hybrid: s Pacific, Panama, Colombia, Venezuela] Annular 0.958 04m32s Europe, Africa, s Asia [Annular: Portugal, Spain, Libia, Sudan, Kenya] Total 1.052 04m07s Africa, Europe, w Asia [Total: c Africa, Turkey, Russia] Annular 0.935 07m09s S. America, w Africa, Antarctica [Annular: Guyana, Suriname, F. Guiana, s Atlantic] Partial 0.874 - Asia, Alaska Partial 0.749 - S. America, Antarctica Annular 0.965 02m12s Antarctica, e Australia, N. Zealand [Annular: Antarctica] Total 1.039 02m27s ne N. America, Europe, Asia [Total: n Canada, Greenland, Siberia, Mongolia, China]

Geographic abbreviations: n = north, s = south, e = east, w = west, c = central

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