Docfile semexcat.htm

SOLAR ECLIPSES in Mesoamerica,
AD 1 to AD 1600

revised and extended November 2003

Please send comments to Felix Verbelen

Several of the Mesoamerican Codices contain elements that are most probably related to solar or lunar eclipses.
Some of these are not fully understood yet.
In order to facilitate further investigations the present list is presented.

Area considered
The list includes the solar eclipses that were theoretically observable from a number of major locations in territories known to have been the cultural area of the Mayas and Aztecs.
In order to get a sufficient geographical and cultural spread, data for the following locations were calculated:


Location	western longitude degrees.decimals	northern latitude degrees.decimals
CHICHEN ITZA	88.60	20.67
CHOLULA	98.30	19.07
COPAN	89.13	14.85
LA VENTA	93.99	18.13
MONTE ALBAN	96.77	17.03
PALENQUE	92.02	17.48
PIEDRAS NEGRAS	91.25	17.13
QUIRIGUA	89.08	15.32
TENOCHTITLAN	99.10	19.45
TEOTIHUACAN	98.87	19.68
TIKAL	89.63	17.22
TULA	99.35	20.05
UXMAL	89.77	20.37
YAXCHILAN	90.97	16.90

These coordinates have been taken from the (U.S.) National Imagery and Mapping Agency and from other sources.

Calendar and period
The list covers the period AD 1 to AD 1600.

Till October 4th 1582, dates are according to the Julian Calendar.
After that date we use to the Gregorian Calendar.

Time indications
All times are expressed in Universal Time (UT).
In a number of countries UT is still referred to as GMT (Greenwich Mean Time).

To derive the local mean solar time, dependent on the location, the following corrections should be added to the given times :


Location	correction (hours and minutes)
CHICHEN ITZA	- 5 h 54 m
CHOLULA	- 6 h 33 m
COPAN	- 5 h 57 m
LA VENTA	- 6 h 16 m
MONTE ALBAN	- 6 h 27 m
PALENQUE	- 6 h 08 m
PIEDRAS NEGRAS	- 6 h 05 m
QUIRIGUA	- 5 h 56 m
TENOCHTITLAN	- 6 h 36 m
TEOTIHUACAN	- 6 h 35 m
TIKAL	- 5 h 59 m
TULA	- 6 h 37 m
UXMAL	- 5 h 59 m
YAXCHILAN	- 6 h 04m

     Example:
      According to our list, a partial solar eclipse occurred at Tikal on April 13th, AD 804.
     Maximum eclipse at Tikal was reached at 22 h 23 m (UT).
     At that moment local mean solar time at Tikal was

           22 h 23 m - 5 h 59 m = 16 h 24 m

Method of calculation and accuracy
To obtain the solar and lunar coordinates, Besselian elements and local circumstances we started from the classical theories [1 to 7], not taking into account the smallest periodical variations.
We neglected these small periodical variations because it is fairly irrelevant to perform lengthy calculations in view to obtain very precise results with respect to the positions of the Sun and the Moon, if at the same time the uncertainties with respect to the Earth's rotation are rather important.
It should indeed be noted that, although the present astronomical theories for the calculation of the presented data are sufficient to obtain high accuracy with respect to the Earth's centre, at the same time the irregularities of the Earth's rotation (DeltaT) prevent a precise calculation for a given location on the Earth's surface, since these irregularities are still insufficiently known for the historical period we considered.

To take into account the general slowing down of the Earth's rotation and longterm periodic irregularities of this rotation, i.e. to convert Terrestrial Time (TT) to Universal Time (UT), several equations have been proposed, among others by Spencer Jones [7], L. V. Morrison and F. R. Stephenson [8], F. R. Stephenson and L. V. Morrison [9], F. R. Stephenson and M. A. Houlden [10], and F. R. Stephenson [11].

We used the following equations  [10] :

     till AD 948:
     DeltaT = 1830 - 405*E +46.5*E^2
     where E = Julian centuries since AD 948

     AD 948 to AD1600:
     DeltaT = 22.5 t^2
     where t = Julian centuries since AD 1850

Datafile
The Datafile lists the solar eclipses that where observable from the different locations indicated above, per calendar date and per location, with the following details :


Column	details
year	calendar year
m	calendar month
d	calendar day
JD	julian day number (with 2 decimals) Like all other time indications, JD is according to Universal Time.
dT	assumed value for DeltaT (seconds)
h1.m1	time of first contact; This is the moment at which the limbs of Sun and Moon touch for the first time. It is the beginning of the (partial phase of the) eclipse.
se1	elevation of the centre of the solar disk above the horizon at the time of first contact (h1.m1) .
hm.mm	time of maximum eclipse at the observer's location
sem	elevation of the centre of the solar disk above the horizon at the time of maximum eclipse (hm.mm)
t	type of solar eclipse at the considered location, at the time of maximum eclipse: T = total eclipse A = annular eclipse p = partial eclipse
magn	maximum size of the eclipse, this means the fraction of solar diameter eclipsed by the Moon at the time of maximum (hm.mm) eclipse. If "magn" is equal or greater than 1, the eclipse is total.
h4.m4	time of last contact between the lunar and solar limbs. It is the end of the eclipse at the considered location.
se4	elevation of the centre of the solar disk above the horizon at the time of last contact (h4.m4)
hh.mm	right ascension of the Sun at the time of maximum eclipse (expressed in hours and minutes)
dd.dd	declination of the Sun at the time of maximum eclipse (expressed in degrees.decimals)
obs	location for which the eclipse was calculated. (first 3 characters)

Remarks

The times h1.m1, hm.mm and h4.m4 are expressed in hours and minutes Universal Time (UT).
The solar elevations se1, sem and se4 are referred to the centre of the solar disk and expressed in degrees (rounded to the nearest unit).
The given elevations do not take into account atmospheric refraction. When the centre of the solar disk coincides with the geometrical horizon (elevation = 0°), an upward refraction of some 34' is on average to be expected [12]. This means that at that moment the lower solar limb will apparantly still be half a solar diameter above the horizon (supposed perfectly flat). Because of this atmospheric refraction, the upper limb is seen to disappear at the horizon when the centre of the real solar disk is already nearly 1° below the horizon. This is the time of actual sunset.
For the same reason, the Sun's upper limb appears above the horizon at the moment the Sun's centre is still a little less than 1° below the geometrical horizon.
Refraction should rarely cause problems in the evaluation of ancient observations, except perhaps in those cases where the solar eclipse ends within minutes of sunrise or starts at the moment of sundown. In both cases however, the solar eclipse would be barely perceptible with the naked eye, the eclipsed section of the Sun being just too marginal. The only circumstance where the atmospheric refraction becomes an important element is when totality or annularity occurs within seconds after local sunrise or before local sunset.
It is also important to realize that the theoretical horizon will rarely coincide with the real geographical horizon because of distant hills, trees, buildings, clouds, haze and the like, or the elevated location of the observation site. All of these factors could affect the perception of solar eclipses of which the start, maximum or ending occurs near the horizon.
For critical cases we strongly recommend a careful calculation, case by case, taking into consideration the particular features at the rising or setting point of the eclipsed sun at the observer's true horizon.
In some cases se1, sem or se4 have a negative value and thus the Sun's centre was below the theoretical horizon (see also previous remarks) at the given times.
An eclipse is mentioned in the list if se1 or sem or se4 where positive at the given place at least at one of the three calculated moments (beginning, maximum or end).
The data given for the instants when the Sun was below the horizon are to be considered as additional information.
It is however useful to note that the effect of important partial (and surely annular or total) eclipses is well perceivable if this happens not too far below the local horizon. For a detailed description of such an eclipse, please see our account of the predawn solar eclipse of May 31st, 2003 (pdf-file).
The given right ascensions and declinations of the Sun at the time of maximum eclipse are referred to the ecliptic and the mean equinox of the date.

Downloading the datafile
There are 2 ways to download the datafile:

as textfile (671 Kb)
as zip-file (140 Kb)

Bibliography
[1] Improved Lunar Ephemeris - (Washington, 1954)
[2] Explanatory Supplement to the Astronomical Ephemeris - (HMSO, London, 1961)
[3] SMART W.M. - Textbook on Spherical Astronomy - (Cambridge University Press, 1977)
[4] MEEUS Jean - Tables of Moon and Sun (Kessel-Lo, 1962)
[5] MEEUS Jean - Astronomical Formulae for Calculators - (Urania, Hove / VVS, Brussel, 1978)
[6] Mc NALLY D. - Positional Astronomy (Muller, 1974)
[7] DANJON A. - Astronomie Générale (Blanchart, 1980)
[8] MORRISON L.V. and STEPHENSON F.R. - Sun and Planetary Systems - Vol.96 (Reidel, 1982)
[9] STEPHENSON F.R and MORRISON L.V - Long-Term changes in the rotation of the Earth
- Phil.Trans.Royal Soc. - Vol.313 (1984)
[10] STEPHENSON F.R and HOULDEN M.A. - Atlas of Historical Eclipse Maps - Cambridge Univ.Press. (1986)
[11] STEPHENSON F.R. - Historical Eclipses and Earth's Rotation - Cambridge Univ. Press. (1997)
[12] Explanatory Supplement to the Astronomical Almanac - (U.S. Naval Observatory, 1992)

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SOLAR ECLIPSES in Mesoamerica, AD 1 to AD 1600

SOLAR ECLIPSES in Mesoamerica,
AD 1 to AD 1600