November 27 - 28, 2014
Intensive Workshop on Ancient and Medieval Eclipse Data
NAOJ, Mitaka, Tokyo

Historical Astronomy and Eclipses
TANIKAWA, Kiyotaka and SÔMA, Mitsuru
National Astronomical Observatory of Japan
2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan
Abstract
In this report, we first review the history of the study for the determination of thelong-term variations of the Earth rotation and the accelerations of the lunar motion,
which were once called the solar and lunar accelerations (including the accelerationsof the planetary motions). In the second part of the report, we describe the present
status of the study focusing on our works.

1

Introduction

1.1

∆T

The delay time ∆T of the Earth rotation is defined by∆T = TT − UT
where TT (Terrestrial Time) denotes the homogeneously flowing time, and UT (Universal
Time) is the time measured by the rotation of the Earth. ∆T has been adjusted to zeroin the 19th century, and the rate has been also adjusted then.
∆T increases if we go back to the past. This reflects that the Earth rotation is secularly
slowing down due to the tidal interactions with the moon. The Earth rotation was fasterin the past. We want to determine long-term variations of ∆T .

1.2

Determination of long-term variations of ∆T

Variations of ∆T reflects, on the one hand, the secular decrease of the Earth’s spin angularmomentum due to the conservation of the angular momentum of the Earth–Moon system.
The Moon is receding from the Earth by 3.8cm/year due to the tidal friction, which resultsin the increase of the orbital angular momentum of the Earth–Moon system. Then thespin angular momentum of the Earth decreases. Variations of ∆T reflects, on the other
hand, the irregular variations of the inertial moment of the Earth. The larger the inertial
moment, the smaller the spin rate.
The ancient observational data indicates the existence of the uplift of the sea levelbackwards in time at least until 2700 years ago (see the discussions in Sect. 3.2). In
1

addition shorter variations of ∆T are expected. In order to obtain the shorter variations,
we need to analyze time-dense ancient data with small errors.
We collect ancient data of various kinds from various countries. There are solar andlunar eclipses. There are timed and untimed records. There are total, annular and partial
eclipses. There are records with comments and without comments. There are records ofoccultations of planets and bright stars by the Moon. There are observations of equinoxes,and meridian passages of Mercury and Venus.
There is a long history of studies of lunar and solar accelerations. Let us briefly review
the history of the past studies.

2

Studies of solar and lunar accelerations

2.1

The early interest

Until the early twentieth century, the slowing down of the rotation of the Earth was notknown. Astronomers thought that the Earth was rotating with constant speed. Here, let
us explain what happens to the observations of the motion of heavenly bodies (the Sun,
the Moon and planets) if one assumes that the rotation of the Earth is constant contrary
to the fact that the rotation of the Earth is slowing down secularly. If we measure thetime by the rotation of the Earth, then,
1. The unit of time grows. As a result,
2. The motion of the Sun accelerates.
3. The motion of the Moon accelerates.
4. The motion of planets accelerates.
In this case, the accelerations of celestial bodies should be observed. Already Edmund
Halley was interested in the existence of lunar accelerations (Halley, 1695). He writes,“And I could then pronounce in what Proportion the Moon’s Motion does Accelerate;
which that it does, I think I can demonstrate, and shall (God willing) one day, make it
appear to the Publick.”
Dunthorne (1739) derived the precession of the equinox 11◦ 320 in 810 years, i.e. 5100 .2
per year, and the obliquity 23◦ 350 from the separate observations of Ptolemy and alBattani. After this, Dunthorne became interested in ancient astronomical observations
such as Theon’s solar eclipses in AD 365. In 1749, he found after analyzing solar andlunar eclipses of the latter half of the fifteenth century that the calculated position of theMoon from the eclipses is 50 forward of the tabular values. He concluded that the moon’s
mean motion is swifter than the tabular values. In other words, the motion of the Moonis accelerated.
The size of the solar acceleration is small compared to the lunar acceleration, andtherefore its discovery was long after. Cowell (1905) finally found evidence from ancient
eclipses for a significant solar acceleration.
22.2

Ginzel 1884, Fotherigham 1920, and Schoch 1926

In the late 19th century Hansen’s Lunar Tables (1857) were used to calculate the Moon’s
positions, but it was known that the lunar acceleration obtained from ancient solar eclipses
was diﬀerent from the theoretical one. The coeﬃcient of the T 2 term in the Moon’s mean
longitude with respect to the fixed stars obtained from ancient solar eclipses was fairly
close to 1000 (e.g. Celoria, 1877; Ginzel, 1884, 1918) but its theoretical value was about
600 . Here T is the time counted in centuries. At that time it was not established yet what
the cause of this discrepancy between the observations and the theory and it was being
discussed if there existed errors in the lunar theory.
Fotheringham (1921) appreciated the publication of Tables of the Motion of the Moon
by Brown (1919), which was resulted from Brown’s stupendous work on the theory of the
motion of the Moon. Brown (1915) confirmed the coeﬃcient of the T 2 term in the Moon’s
mean longitude with respect to the fixed stars as 600 .03. His theory was the most precise
gravitational theory of the motion of the Moon, but even using this newest theory, thediscrepancy in the lunar acceleration between the observations and the theory was not
explained. Brown was confident that his gravitational theory was correct and therefore hefelt no necessity of introducing artificial accelerating terms for the motion of the Moon.
The diminution in the eccentricity of the Earth’s orbit, which Laplace discovered, andthe gravitational eﬀect of the oblateness of the Earth’s figure, which Stockwell discovered,
were not enough to explain the observed lunar acceleration.
For the purpose of obtaining a precise acceleration of the Moon modern astronomical
observations are of little value because the time span is too short to obtain the accelerationand moreover the Moon’s motion is subject to unexplainable fluctuations, from which it
is impossible to disentangle the eﬀect of the acceleration.
Fotheringham (1920) analyzed the following 11 ancient solar eclipses and using also
some ancient observations of lunar eclipses, occultations and equinoxes, he obtained thevalues of the coeﬃcients of the T 2 terms in the mean longitudes with respect to the fixedstars as +1000 .8 and +100 .5 for the Moon and for the Sun, respectively.

1
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The eclipse of Babylon−1062 7 31
The eponym canon eclipse −762 6 15
The eclipse of Archilochus−647 4 6
The eclipse of Thales−584 5 28
The eclipse of Pindar−462 4 30
The eclipse of Thucydides−430 8 3
The eclipse of Agathocles−309 8 15
The eclipse of Hipparchus−128 11 20
The eclipse of Phlegon+29 11 24
The eclipse of Plutarch+71 3 20
The eclipse of Theon+364 6 16
3

Schoch (1926) examined many ancient solar eclipses, as was cited by Fotheringham(1920), but he used only one observation of an appulse of Spica with the Moon to derive
the coeﬃcient of the T 2 term in the Moon’s mean longitude with respect to the fixed stars
as 1100 .09. The observation he used was the one made by Timocharis at Alexandria on−282 Nov. 9, when Spica in the morning touched the north cusp of the Moon after theMoon had risen over the horizon (Schoch, 1928). Schoch’s value was important because
his expression for the Moon’s longitude was adopted by Neugebauer (1929) and was usedin the tables by Tuckerman (1964), and the tables were used by some historians to examine
the places where ancient eclipses were seen.

2.3

de Sitter 1927 and Spencer Jones 1939

De Sitter (1927) analyzed the observed positions of the Moon, the Sun, Mercury and
Venus since the 17th century and showed that the observed fluctuations in their positions
were proportional to their mean motions. The fluctuations in their observed positions
could therefore be attributed to a variation of the adopted unit of time provided by therotation of the Earth.
Spencer Jones (1939) reanalyzed the observed positions of the Moon, the Sun, Mercury,
and Venus and established the evidence that the fluctuations were proportional to theirmean motions and were therefore attributed to variation of the rotation of the Earth. He
obtained expressions for the corrections to the mean longitudes of the Moon, the Sun,
Mercury, and Venus, from which Clemence (1948) deduced the tidal acceleration term in
the Moon’s mean longitude to be −1100 .22T 2 , which was not obtained by gravitational
theory. Clemence also obtained an expression for the corrections to the Moon’s mean
longitude based on Spencer Jones’s results in order to obtain a strictly gravitational lunar
ephemeris expressed in the same measure of time as defined by Newcomb’s Tables of theSun. Clemence called the time Newtonian time, which was later named Ephemeris Time.
Since then obtaining the values of ∆T = ET − UT for various epochs and the value ofthe tidal acceleration coeﬃcient of the Moon from observations has become one of theimportant purposes in positional astronomy. Note that ET means Ephemeris Time, which
is a uniform measure of time, and UT means Universal Time, which is determined by therotation of the Earth. Currently ET is replaced by TT, viz. Terrestrial Time, which isdefined by TAI + 32.184s where TAI is International Atomic Time. TT is considered tobe continuous both in the rate and in the value with ET.

3Modern determinations of ∆T

3.1

R.R. Newton 1970

Newton (1970) used ancient astronomical observations to determine the tidal accelerationin the Moon’s orbital motion and the acceleration of the Earth’s spin rate. The types ofobservation he used include equinoxes, solstices, places of totality or near-totality of solar
eclipses, times of various phases of solar and lunar eclipses, magnitudes of partial solar
and lunar eclipses, conjunctions, and occultations. Among them, records of solar eclipses
with large magnitudes occupy a large fraction of those he analyzed and the numbers of4

such solar eclipses are listed in Table 3.1. He assigned a reliability (weight) to each eclipse
record in the form of numbers from 0 to 1 for his analyses.
Table 3.1. Large solar eclipses used by Newton
Places
No. Periods in AD years
British Isles
7
664–1191
Babylonia + Assyria
2−1062, −762
China20
−708–1221
Europe13−50–1241
Mediterranean countries
27−688–590

3.2

F.R. Stephenson, 1995, 1997

F.R. Stephenson collected many ancient observation data from various sources. In particular, he brought East Asian data into analyses. The numbers of data until 1600 he used
are given in Table 3.2.
Table 3.2. The numbers of solar and lunar eclipses used
PlacesSolar Ecl.
Babylonia + Assyria 14
China
70
Korea
2
Japan
2
Europe + Alexandria 47
Arab25

Lunar Ecl.
74
25
0
0
8
29

There are a lot of observations in Asia other than China, but much of them were notincorporated. The analysis of the reliability and incorporation of these data are left to
future authors.
The main result of Stephenson (1997) is represented in Fig. 1. Here, the abscissa
denotes years and the ordinate stands for ∆T . In the figure, the solid curve is the expectedvariation of ∆T due to the angular momentum conservation of the Earth-Moon system.
Various point marks represent timed lunar and solar eclipses from diﬀerent periods anddiﬀerent countries. The dotted curve is the cubic spline ∆T curve. Interpretation due to
Stephenson is that the scatters of the data points are very large, so that no short-term
variations can be derived. The long-term variations of ∆T , then, should be smooth.
The scattered points are not around the solid curve but around the dotted curve.
This provides evidence that there is a non-tidal component in the variation of the Earth’s
rotation. A possible our interpretation for it is that the sea level was higher and theinertial moment of the Earth was larger than those expected from neglecting the longterm variations of the global climates as we go back to the past. The observationaltendency is explainable if ices and glaciers in the Arctic and the Antarctic regions weremelted because of the warm climate toward a few thousand years ago.
5

Figure 1: ∆T variations. The solid curve represents the ∆T curve predicted by conservation of angular momentum of the Earth-Moon system. The dotted curve is the cubic
spline fitting to ∆T due to Stephenson (1997). Timed lunar and solar eclipses are highly
weighted.

4

Tanikawa and Sôma since 2001

Tanikawa and Sôma started their career of study of the ∆T determinations in 2001 almost
just after the publication of the celebrated textbook of Stepheson (1997). Sôma had beeninterested in this subject since his master’s thesis in 1980. Our main purpose is to findshort periodic variations of ∆T of the order of a few hundred years or less.
There were two motivations. The first one is that Stephenson’s cubic spline curve may
not reflect the reality because it is heavily smoothed. His result is strongly dependent
on the timed lunar and solar eclipses. Partial eclipses are used because they are timed.
Babylonian timed lunar eclipses give so dispersed points in ∆T . This seems to us thatthe accuracy of Babylonian clocks are very bad. The second motivation is that Asian
total solar eclipses are well oﬀ the curve. This suggests that there are fluctuations of ∆T
which the spline curve misses.
For brevity of description, we summarize the Guideline of our study. Most importantly, we determine the range of ∆T using contemporaneous plural records. According
to the method of analysis of Stephenson (1997), it is clear that Stephenson regards eachobservational data as having large intrinsic uncertainty which sometimes amounts to a
few thousand time seconds. Parameters to be determined from these records are ∆T andthe acceleration ṅ of the lunar motion where n is the lunar mean motion. These twoparameters are considered to change slowly. The ∆T values may change seculatly by atmost 10 seconds each year even in two thousand years ago. The change of ṅ is consideredsmaller. Then, it is reasonable that we can regard these parameters are constant within
6ten years with uncertainty of, say, 100 seconds.
We, in principle, do not use lunar eclipse data, because we do not know the accuracyof ancient clocks. The accuracy of ancient clocks should be determined after we obtain
good ∆T variations.
Now, we incorporate solar eclipse and lunar occultation data as many as possible.
We try to use total and near total eclipses, eclipses at sunrise or sunset, lunar grazing
occultations, and lunar occultations at moonrise or moonset. We use data observed atplural sites; we use contemporaneous observations; we analyze the data with the Sômadiagram which will be introduced in section 4.2.

4.1

Eclipses observed at plural sites

The number of solar eclipses observed at plural sites is small. However, these are more
useful when observing sites are far apart. Deep eclipses are desirable. But this condition
is not indispensable if the sites are far apart. We list them in Table 4.1. We show themaps of eclipse bands for the eclipses in BC188, AD873, and AD1415 in Fig. 2.

Table 4.1. Deep solar eclipses observed at plural sites.
Year Month Day
BC 648.04.06
BC 188.07.17
AD 873.07.28
AD 968.12.22
AD 1133.08.02
AD 1183.11.17
AD 1241.10.06
AD 1245.07.25
AD 1415.06.07

4.2

Magnitudes and places
Total at Thasos or Paros; Sunset at Qufu
Total at Rome; nearly total at Chang-an
Annular at Nishapur; Annular at Kyoto
Total at Constantinople; Total at Farfa
Total at many cities in Europe
Deep at Mizushima, Kaesong, and Lin-an
Total at Stade, Reichersburg, and Nile Delta
Annular at Kaesong; Deep at Kyoto
Total at Aizu (Japan), Neider Alteich,Prague, Wroclaw, Kobrin, and Mytho;
Partial at Kyoto

Contemporaneous observations

There can be a lot of groups of contemporaneous observations. Here, we list representative
examples in Table 4.2. Let us describe how to use these contemporaneous observations.
In general, we fix the latitude and longitude of the observing site. For the fixed value ofṅ, we get the upper and lower bounds of ∆T for which the site is inside the totality or
annularity band of the given eclipse. Then, we change the value of ṅ by some amount andcalculate the upper and lower bounds of ∆T for which the site is again inside the totality
or annularity band. We repeat this process several times. After this process, we obtainnearly parallel curves in the ṅ-∆T -plane, which we call the Sôma diagram. The region
sandwiched by these two curves is where the eclipse is total or annular at the observingsite. As a starting value of ṅ we take −2500 .826/cy, which is the value intrinsic to the JPL
7

-187

7 17

TT - UT = 12600.0 sec

873

7 28

TT - UT = 3500.0 sec
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10

Figure 2: Eclipse bands of the three eclipses in BC 188, AD 873, and AD 1415. The
eclipse of BC 188 was observed at Rome and Chang-an (髟螳). The eclipse of AD 873 wasobserved at Nishapur and Kyoto (莠驛). The eclipse of AD 1415 was observed at Aizu(莨壽), Neider Alteich, Prague, and Wroclaw as total. In Kyoto (莠驛), it was observed
as partial.

8lunar and planetary ephemerides DE405 and DE406 (Chapront et al. 2002). This value
is consistent with the values of the more recent ephemerides DE430 (−2500 .82 ± 000 .03/cy)
and DE431 (−2500 .80 ± 000 .03/cy), and agrees well with the value −2500 .858 ± 000 .003/cy
obtained from analyses of LLR observations (Chapront et al. 2002).
In the case of lunar grazing occultations of planets or bright stars, we generally determine the one boundary for which these objects are hidden by the lunar disk. So, we geteither the upper ot lower boundary of ∆T . For the sunset or sunrise eclipses, there canbe two boundary curves. In fact, one boundary corresponds to the situation such thatthe sun is hidden by the moon, whereas the other boundary is such that eclipse itself isinterrupted by the sunset or sunrise.

Table 4.2 Examples of contemporaneous observations.
BC 600.09.20
BC 584.05.28
BC 198.08.07
BC 188.07.17
BC 188.07.17
BC 181.03.04
AD 503.08.05
AD 513.08.22
AD 516.04.18
AD 522.06.10
AD 1124.08.11
AD 1133.08.02
AD 1135.01.16
AD 1239.06.03
AD 1241.10.06

Total solar eclipse at Qufu
Total solar eclipse at Qufu
Annular solar eclipse at Chang-an
Deep solar eclipse at Chang-an
Total solar eclipse at Rome
Total solar eclipse at Chang-an
Occultation of Venus at Luoyang
Occultation of Saturn at Luoyang
Annular solar eclipse at Jiankang
Total solar eclipse at Jiankang
Total solar eclipse at Novgorod
Total solar eclipse at seven cities in Europe
Partial solar eclipse at Lin-an
Total solar eclipse at eight cities in Europe
Total solar eclipse at two cities in Europe
and a city in Egypt
AD 1245.07.25 Annular solar eclipse at Kaesong
Deep at Kyoto

We first show the usage of the Sôma diagram with four deep solar eclipses at aroundBC 188 observed at Chang-an and Rome (see Table 4.2). As shown in Fig 3, there are
four pairs of curves corresponding to three observations at Chang-an and one observation
at Rome. The meaning of the abscissa is that 0.0 corresponds to ṅ0 = −2500 .826/cy2 and−2.0 corresponds to ṅ = ṅ0 − 200 .0/cy2 . Thus, the abscissa values are corrections to thecurrent value of ṅ.
The second example of the usage of the Sôma diagram is shown with lunar occultations(see the third group of the observations in Table 4.2). The first two data are the lunar
occultation of Venus on Aug. 5, AD 503 at Luoyang, and the lunar occultation of Saturnon Aug. 22, AD 513 also at Luoyang. The remaining two events are the solar eclipse on
9

Figure 3: The Sôma diagram for the years BC 198, 188, and 181.

Apr. 18, 516 at Jiankang, and the solar eclipse on June 10, 522 also at Jiankang. The
maps of the events are shown in Fig. 4. The two maps at the top show that the lunar
occultations took place in the early morning and in the late evening, respectively. Thesesituations give us the upper and lower boundaries of ∆T for which the lunar occultations
were seen above the horizon. The bottom two figures are for the total and annular solar
eclipses, respectively, and therefore the meaning of the maps is clear.
The Sôma diagram for these events is shown in Fig. 5. The two curves with upward
arrows and downward arrows are for the two lunar occultations. This figure shows thatthe result from two lunar occultations is perfectly consistent with that from two solar
eclipses. This fact shows that occultation data are useful in some cases for our research.

4.3

Some results

We have determined the values of ∆T for some selected periods (Tanikawa & Sôma 2004,
2015; Tanikawa et al. 2010). We plot the results in Fig. 6. We enlarge of the boxed areaof the above figure in Fig. 7.
Let us talk about the warm and cold climates deduced from the long-term variationsof ∆T as shown in Fig. 7. In the warm periods, ice-caps in the Arctic and Antarctic
regions melt, water comes to the equatorial regions, the inertial moment of the Earthincreases, and the angular velocity of the Earth rotatio decreases. In the periods of theslow rotation, the ∆T curve becomes (relatively) flat. In the opposite case, that is, the∆T curve is steep, the climate is cold.
We will talk about the meaning of Fig. 7 more precisely in a separate paper.
10Occultation of Venus on 503

85Occultation of Saturn on 513

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Figure 4: The occultation of Venus on AD 503.08.05 at Luoyang (豢幃區). The occultationof Saturn on AD 513.08.22 at Luoyang (豢幃區). ∆T = 4500 sec. Moonrise. Eclipses on
516.04.18 at Jiankang (蟒蠎). Eclipses on 522.06.10 at Jiankang (蟒蠎). ∆T = 4500 sec.

11

Figure 5: The range of ∆T from the Sôma Diagram for the years AD 503, 513, 516, and522.

Figure 6: ∆T variations. The dotted curve corresponds to the conservation of angular momentum of the Earth–Moon system. Solid curve: the cubic spline fitting due to
Stephenson (1997). Vertical bars show our results.

12

Figure 7: ∆T variations and Climate.

5Concluding remarks

Figs. 6 and 7 suggest that short-term variations, shorter than a few hundred years, of ∆T
exist. In order to draw the figures we have used reliable contemporaneous records. The
figures show that shorter variations in the ∆T curve do exist.

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