Deutsche Version
Canon of solar eclipses from 2501 BC to 1000 AD
Geometry and main characteristics of a solar eclipse
History of solar eclipse calculations
Introduction
Historical chronolgy is based on a tight relationship between theoretical astronomy and historical events.
The benefit of astronomy for chronology is that it is able to pin down astronomical events that were
recorded in antiquity. Most important in that respect are solar and lunar eclipses, first and last sightings
of the Moon after or before new moon, as well as observations of the planets Mercury, Venus, Mars, Jupiter and
Saturn.
Geometry and main characteristics of a solar eclipse
As seen from the Earth, a solar eclipse occurs when the Moon passes between the Sun and the Earth,
and when the Moon fully or partially covers the Sun. This hence only happens if Sun, Earth and Moon
align along a straight line.
Calculation of solar eclipses
History of solar eclipse calculations
In the 19th century, astronomical problems started to get some focus in classical and
ancient studies. Accounts of lunar, planetary and eclipse observations had been found and waited to
be chronologically assessed. For such purposes, Ginzel published 1899 a canon of solar eclipses
between 900 BC and 600 AD[10];
this work remained a milestone as of today for all studies involving eclipses that have been published
with their main purpose to be useful for chronological aims. Ginzel collected eclipse accounts of ancient
writers and suggested identifications; his book contains maps for the regions of classical antiquity
where the tracks of totality of solar eclipses are printed for each century between 900 BC and 600 AD.
Data and data download
If you download the following data and use them in a publication, please refer to the adress
of this website as the origin of the data. A publication in the journal Klio is planned
(R. Gautschy, Sonnenfinsternisse und ihre chronologische Bedeutung: Ein neuer Sonnenfinsterniskanon
für Altertumswissenschaftler).
Bibliography
Due to their rare appearance at a particular site, solar eclipses have the greatest potential to provide a
reliable date of a historical event, accurate to one day if a total or nearly total coverage of the Sun by the Moon
has been passed down to us.
Features of a considerable eclipse of the Sun are:
Before 1000 AD only occurrences of points 1) and 2) are mentioned in historical sources. Concerning point 2)
one must mention, that ancient writers often reported that stars were visible
although today's calculations show unambigously that this was impossible. The description
that day turned into night may thus be a more reliable hint that an eclipse had been total at a certain
site. The report of the visibility of stars during solar eclipses may more likely be regarded as
standard phrase to emphasise the dramatic event. Strikingly, solar and lunar eclipses are often mentioned
in connection with destructions that occurred during earthquakes, with comets, or with other events that were
regarded as bad auguries - this still happened at times when people already understood what caused
eclipses.
Total solar eclipses are rare events. Although they frequently occur somewhere on Earth they recur at any given
place, on the average, only once every 375 years. If the numbers of annular eclipses are added, the value
decreases to about once every 140 years
[1].
The Moon's orbit around the Earth is inclined at an angle of about 5 degrees to the ecliptic, the plane of
the Earth's orbit around the Sun. Therefore, at the time of a new moon, the Moon will usually
pass above or below the ecliptic. A solar eclipse can occur only if the new moon occurs close to one
of the points where the Moon's orbit crosses the ecliptic, known as nodes. On average, such a
situation recurrs after every six, sometimes already after five new moons. The orbits of the Earth and the Moon
are both elliptical. Thus the distances between the two celestial bodies and therefore also the apparent
sizes vary. This effect leads to different types of solar eclipses. If the axis of the
lunar shadow crosses the Earth, the eclipse is referred to as a central eclipse.
Three different kinds of central eclipses can be distinguished:
A solar eclipse during which Earth is crossed by the Moon's penumbra only is called a
partial solar eclipse (P). Such eclipses occur mainly in the polar regions, nonetheless they
can give rise to coverages of 50%, sometimes even 70% in Mediterranean regions.
Because of the relatively narrow tracks of totality only few people are able to observe
a total eclipse of the Sun. Considerably broader than the Moon's
umbra is its penumbra, which has the size of several thousand kilometers and which allows
from more than one fourth of the Earth's surface the observation of a partial coverage of the Sun.
Regarding the nomenclature, unfortunately such a partial coverage during a central eclipse is
also designated a partial solar eclipse.
For historians, the most important characteristics of a solar eclipse is its coverage ratio
and magnitude, both measure the amount of the coverage.
During a solar eclipse, the coverage ratio and the magnitude increase slowly, reach a maximum
value and decrease again.
The calculation of solar eclipses is done with the help of the so called Besselian elements.
The method was developed by Friedrich Wilhelm Bessel in 1842 and was repeatedly refined since then
[2]. The basic idea of the
method is that the Besselian elements describe the motion of the lunar shadow on a suitably chosen,
so called fundamental plane. The fundamental plane crosses the centre of the Earth and is
perpendicular to the axis of the shadow cone. Comparatively few numerical parameters are sufficient to describe
the movement of the shadow in this plane. Based on this, the next step is to
project the shadow cone onto the Earth's surface.
In the geometrical system shown above, the coordinates of the centre of the umbra are x and y in the fundamental plane, the
coordinates of the centre of the Moon are x, y and z. The Besselian elements designate then the following quantities:
The equatorial radius of the Earth is used as unit of legth.
The Besselian elements are time dependent. Thus they must be specified for a timespan of a
few hours to follow a solar eclipse. Two different possibilities exist to publish Besselian elements for solar eclipses:
Either the values of all non-constant Besselian elements are tabulated in intervals
so that interim values can be accurately interpolated; or the Besselian elements are given for a reference
time t0 together with their hourly variations. This allows for the calculation of the values
for other epochs as linear functions of time. Instead of linear functions a polynomial approximation
can be chosen if higher accurracy is required. In that latter case, the polynomial coefficients
up to the 3rd order are usually sufficient.
If one could observe a solar eclipse on the Earth's surface from the Sun, one would observe
the large penumbra of the Moon - grey coloured in the following figure - and the considerably
smaller umbra - shown in black - to move across the Earth:
All points on the curve N-PN1-PS1-S have sunrise. For all points on the N-S meridian it is noon.
All points on the curve N-PN2-PS2-S have sunset. If the just mentioned interception points
are projected onto a geographical map of the Earth, the resulting regions have oval shapes: see PN1-P2-PS1-P1 and
PN2-P4-PS2-P3, respectively, in the figure to the right.
ΔT and its uncertainty
For many centuries, the fundamental unit of time was the rotational period of the Earth with respect
to the Sun. Universal Time (UT), also called Greenwich Mean Time (GMT), is based on the mean solar
time at Greenwich. Unfortunately, UT is not a uniform time scale over historical times because Earth's rotational period
gradually decreases. Therefore, the calculation of local circumstances of solar eclipses
in the far past is subject to uncertainties. As the Earth rotates,
tidal friction, inflicted by the gravitational attraction of the Moon and the Sun is at work. The
result is a transfer of angular momentum from the Earth to the Moon. The Earth loses energy and slows down,
the Moon gains energy and its distance from the Earth increases.
Today, atomic clocks are used for accurate timekeeping.
Terrestrial Dynamical Time (TDT) is such an atomic time scale. Solar eclipse calculations are based
on TDT, the position of the visibility area of an eclipse depends however on UT. Calculations from
TDT have to be converted into UT; therefore the time difference between TDT and UT must be known. This
time difference, called ΔT , which adds up to about 12 hours around 2000 BC and the
uncertainty of ΔT - about 2 hours in 2000 BC - has to be taken into account in the
calculations. For more information about ΔT see e.g. the webpage of
Robert van Gent.
Calculations in this canon
For the calculation of the lunar and solar positions between 2500 BC and 1000 AD, the DE406 ephemeris
of the Jet Propulsion Laboratory have been used, they allow for the accurate calculation of positions of the Sun,
the Moon and of all the planets between 3001 BC and 3000 AD
[3]. For a detailed description of the ephemeris see
here.
For ΔT, the polynomial expressions of Espenak[4]
have been used, for an estimation of the uncertainty of these values, the formula of
Huber[5] was applied. All calculations
use a value for the Moon's secular acceleration of
year
ΔT
uncertainty (ΔT)
-3000
20h 31m
±2h 30m
-2500
16h 30m
±1h 42m
-2000
12h 54m
±1h 02m
-1500
9h 44m
±32m
-1000
7h 01m
±11m
-500
4h 45m
±7m
0
2h 35m
±5m
500
1h 55m
±2m20s
1000
26m 10s
±55s
The calculation of solar eclipses and their local circumstances for selected places were
carried out in three steps.
This canon contains all solar eclipses between 2500 BC and 1000 AD which were potentially noticeable
at any place in the geographical region between (20°N, 5°O), (20°N, 50°O), (50°N, 5°O) and (50°N, 50°O)
and thus are candidates for identifications with eclipse accounts even if the eclipse
has not been predicted. Detailed eclipse predictions are found for the first time in Babylon after 300 BC.
Since then, eclipse accounts can be found on a more regular basis that mention solar eclipses that reached
magnitudes of less than 0.5.
Example: -2000
mean ΔT: 12h 54m
upper ΔT: mean ΔT - uncertainty = 12h 54m
- 1h 02m = 11h 52m
lower ΔT: mean ΔT + uncertainty = 12h 54m
+ 1h 02m = 13h 56m
This canon of solar eclipses is designed tp provide information about historical eclipses, thus the Besselian elements
are calculated for a reference time t0 and their hourly variations are given. This linear form
of calculation of the elements is sufficiently accurate because the uncertainty of the ΔT value
during the timespan in question is much larger than the differences between linear and
3rd-order polynomial formulation. However, Besselian
elements in polynomial form, for all solar eclipses between 2000 BC and 3000 AD, calculated
with the mean ΔT-value and slightly older lunar and solar ephemeris can be found at the
NASA Eclipse Web Site.
In 1986, Stephenson & Houlden published a canon of solar eclipses for East Asia covering the timespan
between 1500 BC and 1900 AD[11].
For the calculation of the position of the Moon, the J=2 lunar ephemeris of the International Astronomical
Union (1968) were used[12],
although adapted to a more up-to-date value of the lunar acceleration of -26"/cy2. Their
atlas contains maps showing the tracks of totality for each eclipse and for selected longitudes the
corresponding latitudes, the local time, and the altitude of the Sun.
As of now, the last canon of eclipses that was designed for the particular needs of historians was the one
published by Salvo de Meis[13] in 2002.
His canon contains calculations and maps of the areas of visibility for solar and lunar eclipses,
which are mentioned in ancient sources, starting in 763 BC and ending in 1740 AD. For each eclipse
the citation in English translation, and the time and the magnitude are given for the site(s) of observation.
The calculations in the de Meis canon are based on the
VSOP87[14] (for the Sun)
and ELP2000-ephemeris[15]
(for the Moon) of the Bureau des Longitudes in Paris.
The babylonian solar and lunar eclipse accounts and calculations that are passed down were published
by Huber & de Meis[16] in 2004.
The calculations of the ephemeris were based on computer programs that used the tables of
Tuckerman[17] and
Goldstine[18].
Huber included a few more perturbation terms, however. The maps with the areas of visibility were made
by S. de Meis, based on the same ephemeris as in his
canon[13]. For each identifiable
eclipse Huber specifies: the date in the babylonian calendar, universal time, instant
of onset and the corresponding altitude, instant of maximum eclipse and the corresponding altitude, end
of the eclipse and the corresponding altitude, the magnitude, duration of a possible totality, as
well as the sunrise and sunset times.
Apart from Ginzel's canon, all the above mentioned works contain exclusively maps of eclipses that are
documented in written form. On the contrary, the present electronic canon of solar eclipses is set up
very similar to Ginzel's canon. It contains maps of all solar eclipses that were potentially
noticeable in the chosen geographical area, i.e. which reached a magnitude of at least 0.5, between
2500 BC and 1000 AD.
The following excerpt from the compiled tables illustrates how the canon is organised:
For each solar eclipse the following quantities are given:
For the cities Athens, Rome, Babylon, Memphis, Theben, Alexandria and Knossos the following information
is provided:
If a solar eclipse is documented in written form, the information about the source follows, which
is directly clickable - the citation is then shown in the original language (as far as possible)
and in English translation:
For solar eclipses after 100 AD only one single citation was selected - usually the
one giving the most details about the eclipse - even if more than one source mentions the
corresponding eclipse. If the citations are not shown properly or if the file is not opened at
the correct position (this may be browser dependent!), two files containing all eclipse
citations can be opened and downloaded here:
babfinst.pdf and
eclipsecitations.pdf.
A catalogue containing numerous citations for individual eclipses was compiled by
Gorodezkii Michail Leonidovich.
For each solar eclipse, a map of the geographical area between (20°N, 5°O), (20°N, 50°O),
(50°N, 5°O) and (50°N, 50°O) is provided. If an eclipse is documented in written form,
further detail plots for Italy, Greece and Asia Minor, Egypt and the Near East are provided. The maps
are further discussed in the section area of visibility of a solar
eclipse.
Besides the above mentioned publications about solar eclipses whose main focus lies on
historical eclipses, numerous other purely astronomical publications do exist which list
all solar eclipses that are visible somewhere on Earth within certain timespans:
The publications of Oppolzer[19],
Mucke & Meeus[20],
and Espenak[21]
belong into this category.
The following table contains catalogues for the cities Athens, Rome, Babylon, Memphis, Theben,
Alexandria and Knossos showing all solar eclipses for which the Sun was covered by the Moon for
50%, 60%, 70%, 80%, 90% or 100%; these lists contain the following quantities:
Sometimes a maximum eclipse occured before sunrise or after sunset. Therefore, the two columns
"maximum magnitude" and "instant of maximum coverage" contain two values, each seperated by a
diagonal slash. The first magnitude value gives the actually observable maximum magnitude in
addition to the hypothetical maximum values after the slash.
site
50% coverage
60% coverage
70% coverage
80% coverage
90% coverage
100% coverage
Athens
here
here
here
here
here
here
Rome
here
here
here
here
here
here
Babylon
here
here
here
here
here
here
Memphis
here
here
here
here
here
here
Theben
here
here
here
here
here
here
Alexandria
here
here
here
here
here
here
Knossos
here
here
here
here
here
here
Area of visibility of a solar eclipse
This figure shows the area of visibility of the annular-total solar eclipse of May 12th 361 BC, which
is documented in written form in Plutarch, Vita Dionis XIX.4. The Moon's shadow hits the Earth's surface for the first
time at the green point P1: The solar eclipse starts. The two red points PN1 and PS1 designate the northernmost and
southernmost point where the solar eclipse ceases to be observable. For all points on the blue curve
between PN1, P1, and PS1, the solar eclipse starts at sunrise, i.e. the whole eclipse is observable. For all
points along the yellow curve, between PN1, C1, and PS1, the maximum coverage takes place at sunrise, i.e. only the
second half of the eclipse can be observed. For all points on the blue curve between PN1, P2, and PS1, the solar
eclipse ends at sunrise, i.e. no part of the eclipse is observable. Starting on the yellow curve PN1-C1-PS1 the
locus of the zone of totality (red lines) as well as the loci of isomagnitudes (yellow and various green
colour shades) are plotted. For the zone of totality, the northern limit, the central line and the southern
limit is drawn. The green isomagnitudes show where the Sun was covered up to 90%, 80%, 70%, 50%, the yellow isomagnitudes
show where 0% of the Sun was eclipsed. At the green point P4, the Moon's shadow leaves the Earth's surface: The solar
eclipse ends. For all points on the blue curve between PN2, P4, and PS2, the eclipse ends exactely at sunset. For all
points on the yellow curve between PN2, C2, and PS2, the maximum coverage takes place at sunset, i.e. only the first
half of the eclipse can be observed. For all points on the blue curve between PN2, P3, and PS2, the eclipse starts
at sunset, i.e. no part of the eclipse is observable.
The maps provided by this canon of solar eclipses are restricted to the geographical area between
(20°N, 5°O), (20°N, 50°O), (50°N, 5°O) and (50°N, 50°O); this means that the whole area of
visibility of the solar eclipse is never shown such as in the figure above.
The two red continuous lines show the zone of totality if the calculations are done with the
mean ΔT-value. The green isomagnitudes which designate the 90%, 80%, 70% or 50% eclipse of the Sun
by the Moon, are usually calculated using the mean ΔT-value. The same applies
for the blue sunrise and sunset curves as well as for the yellow curve delineating the "maximum
in the horizon" curve. Thus the red continuous lines end at the yellow "maximum in the horizon"
line, while the red dashed and red dot-dashed lines end slightly behind or before the
yellow line. The red dashed curves show the zone of totality if the calculations are done with
the lower ΔT-value, the red dot-dashed ones if the upper ΔT-value is used. The deviations
of the dashed and dot-dashed curves from the continuous lines illustrate the uncertainties of the
calculations, which are due to the uncertainty of the ΔT-value. For the adopted example, this
means that with certainty the zone of totality ran over Sicily and Crete, but whether
the Sun was covered 100% or only 96% in Sparta (number 6) cannot be claimed with certainty. The black line
is the central line, calculated with the mean ΔT-value. Black tickmarks on the central line
show in constant time intervals of 15 minutes the Greenwich mean time; these times apply exclusively
to the central line and ought to serve as approximate temporal orientation.
If in the defined geographical region a limiting magnitude of 0.5 is not reached when calculating
with the mean ΔT value, but if this is the case when using the upper or lower ΔT-value,
then all sunrise and sunset curves, as well as the isomagnitudes are drawn with dot-dashed or dashed
lines, respectively.
If the solar eclipse is partial, as shown in the above map, the isomagnitudes are drawn in different shades of purple.
For each solar eclipse, numerous sites which were important during the corresponding period, are marked with small blue circles;
the names and geographical coordinates can be found in the
sites register. Ten important sites
are drawn in red, tagged with a number and the names of the sites are labelled on the right of the map.
For those solar eclipses in the regions Italy, Greece and Asia Minor, Egypt and the Near East for which written
accounts are preserved, additional more detailed plots are available. The following figures show examples
for each region:
The smooth division of the area in steps of 1° should permit to find sites easily that are missing in the
sites register. The sites being labelled with numbers vary depending on the period in question, their
names can be found in the sites register.
Due to the size of the files, the canon of solar eclipses listed in the following table is subdivided into centuries. The
link in the first column leads to a table containing all potentially noticeable solar eclipses during the respective
century. In the last column, .zip-files containing the calculated data can be downloaded. The last row leads to the list
of solar eclipses that are documented in written form but which did not reach the limiting magnitude of 0.5 in the
chosen geographical region.
This work was supported by a Marie Heim-Vögtlin grant of the Swiss National Science Foundation.
I thank Alfred Gautschy, Bill Gray, Peter J. Huber, Kurt Locher, Salvo de Meis and Robert Nufer who contributed
valuable comments at various stages of this project.
Created by
Rita Gautschy, version 2.0, January 2012
