Predictions of 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 PA, that phenomenon B has period PB, and that the lengths of those periods are constant. The
ratio
(Eq. 1) γ
= PA⁄PB
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 PA = 29.530588853 days and PB = 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 d1 and d2 so that a Z certainly occurs if the difference between an
A and the nearest B is less than d1 PA, and a Z certainly does not occur if the time difference
is greater than d2 PA. For solar
and lunar eclipses we have d1 = 0.077 en d2 = 0.117.
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 PB are almost equal to b periods PA, 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 a1 = 0, b1 = 1, δ
= −ε. The second very good approximation is a2 = 1, b2 = ⌊ 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 n1 and n2. 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 |
n |
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 = a1 + ∑(i=2)n
(-1)i 1⁄(b(i-1) bi)
for example, a8⁄b8 = 484⁄223 = 2 + 1⁄(1∗5) - 1⁄(5∗6) + 1⁄(6∗41) - 1⁄(41∗47) + 1⁄(47∗88) - 1⁄(88∗135) + 1⁄(135∗223).
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 PA are equal to a periods PB 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) n1 = ⌊ d1 n ⌋
and
(Eq. 15) n2 = ⌈ d2 n ⌉
and these numbers correspond to periods of
(Eq. 16) c1 = n1 y ≈
d1 c
and
(Eq. 17) c2 = n2 y ≈
d2 c
A couple of these values are listed in the
preceding table.
With the adopted approximation γ'
= a⁄b there are b different series of
predictions. At any given moment, between d1 b and d2 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 + s0 mod b
in which the mod b means "except for multiples of 2[1]3[1]b4[1]5[1]", and the s0 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 - s0) 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 s0 = 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 + s0 - b S'k
we find that
(Eq. 26) s'∗ = s0 - 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 = S0 + γ
k
(Eq. 28) s∗ = a k + s0 - 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 + s0 - 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.
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 + σ0 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 σ0. 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) σ0 = s0 + [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) σ0 = s0 + 242.
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
= γ0 + p k, then equation 27 becomes
(Eq. 36) Sk = S0 + γ
k = S0 + γ0 k + p k2
As long as the deviation p k2 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 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, click here 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]
http://www.astro.uu.nl/~strous/AA/en/reken/saros.html
Lunar Eclipses:
2001 - 2008
The table below lists every lunar eclipse from 2001 through
2008.
|
Lunar Eclipses: 2001 - 2008 |
||||
|
Date |
Eclipse |
Umbral |
Total |
Geographic Region of |
|
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 |
Eclipse |
Central |
Geographic Region of |
|
Total |
1.050 |
04m57s |
e S. America,
Africa |
|
|
Annular |
0.968 |
03m53s |
N. & C. America,
nw S. America |
|
|
Annular |
0.996 |
00m23s |
e Asia,
Australia, w N. America |
|
|
Total |
1.024 |
02m04s |
s Africa,
Antarctica, Indonesia, Australia |
|
|
Annular |
0.938 |
03m37s |
||
|
Total |
1.038 |
01m57s |
Australia, N.
Z., Antarctica, s S. America |
|
|
Partial |
0.736 |
- |
Antarctica, s Africa |
|
|
Partial |
0.927 |
- |
ne Asia, Hawaii, Alaska |
|
|
Hybrid |
1.007 |
00m42s |
N. Zealand, N. & S. America |
|
|
Annular |
0.958 |
04m32s |
Europe, Africa, s Asia |
|
|
Total |
1.052 |
04m07s |
Africa, Europe, w Asia |
|
|
Annular |
0.935 |
07m09s |
S. America, w Africa, Antarctica
|
|
|
Partial |
0.874 |
- |
Asia, Alaska |
|
|
Partial |
0.749 |
- |
S. America, Antarctica |
|
|
Annular |
0.965 |
02m12s |
Antarctica, e Australia, N.
Zealand |
|
|
Total |
1.039 |
02m27s |
ne N. America, Europe, Asia |
|
|
|
|
|
|
|
Geographic abbreviations: n = north, s = south, e = east, w
= west, c = central
