We have computed the lunar and solar eclipses for the time period -800 to 1650. The events are computed using modern models Elp2000/82 for the Sun (Bretagnon 1986) and VSOP87d for the Moon (Chapront-Touze 1983), and the ancient models from Ptolemy’s Almagest (Pedersen 2011). Often in discussions of ancient astronomy it is helpful to be able to look up directly the time and circumstances of specific eclipses as seen from a specific geographical location for both modern models and the Almagest models, so that is what our tables provide.⬈#p1

Our calculations of lunar eclipses using the modern models are straightforward. One first finds the moment of mean opposition and then, using a simple bracket search, the moment of true opposition. Then using modern values (Meeus 1998, Chapter 55) for the size and distance of the Sun and Moon and the radius σ of the Earth’s shadow at the distance of the Moon, one searches for, in turn, the moment when the Moon first touches the umbra, then when the Moon is totally within the umbra, then when the Moon is at minimum distance 𝜌_o from the shadow center, then when the Moon first exits the umbra, then when the Moon last touches the umbra. The eclipse magnitude is then given by

m = \frac{σ - ρ_{0} + r_{M}}{2 r_{M}}

where r_M is the radius of the Moon.

The calculation of lunar eclipses using the Almagest models is similar. The moment t̅ of mean opposition is related to the moment t of true opposition by t = t̅ + δ_t where

δ t = \frac{q_{S} (a_{S} (\overline{t})) - q_{M} (a_{M} (\overline{t}))}{η + ω_{a} {\frac{d q_{M}}{d α_{M}} |}_{t = \overline{t}} - ω_{S} {\frac{d q_{S}}{d α_{S}} |}_{t = \overline{t}}}

q_s and q_M are the equations of center with arguments a_S and a_M for the Sun and Moon, η and ω_α are the mean motion in elongation and anomaly of the Moon, and ω_S is the mean motion in longitude of Sun.

The calculation of solar eclipses is similar to the calculation of lunar eclipses except that (a) we look for mean and true conjunction instead of opposition, and (b) we must correct the coordinates of the Moon and Sun for parallax. For the modern models, the distance to the Sun is so large that the solar parallax can be completely neglected. For the Almagest models the solar parallax is small, but not entirely negligible, so following Ptolemy we include it. For the modern models we use analytic formulas for the parallax in ecliptical longitude and latitude (Meeus 1998, 282). For the Almagest models we use formulas that Ptolemy attributes to Hipparchus

\begin{array}{c} π_{λ} = - π_{h} \cos ψ \\ π_{β} = - π_{h} \sin ψ \end{array}

where π_h is the horizontal parallax of the Moon and ψ is the angle of the ecliptic with the altitude circle that passes through the Moon at geocentric altitude h:

\cos ψ = \frac{- 1}{\tan (90^{\circ} - h) \tan (λ_{H} - λ_{M})}

where λ_H is the degree of the ecliptic rising at the time of the eclipse, and is given by

\tan λ_{H} = \frac{- \sin ϵ \tan ϕ - \cos ϵ \sin θ}{\cos θ}

where ϴ is the local sidereal time and ε is the obliquity of the ecliptic.

It is true that π_λ and π_β as just computed are only approximations, but contrary to what Ptolemy says in the Almagest they are pretty good approximations. In some circumstances it happens that due to the approximations cosψ is greater than 1 or less than -1, and this leads to the occasional NaN (not a number) entries in columns BE-BM.⬈#p5

So for solar eclipses one computes the (approximate) time of conjunction as the moment when

λ_{S} = λ_{M}^{'} = λ_{M} + π_{λ}

and then the magnitude of the eclipse is given by

λ_{S} = λ_{M}^{'} = λ_{M} + π_{λ}

where β'_M = β_M + π_β is the apparent latitude of the Moon at the latitude PHI of the observer.

Guide to the lunar eclipse spreadsheet columns

Column(s)	Content
A-C	year, month, day for the opposition of the Moon and Sun
D	ET for the modern models or local time for the Almagest models, for the opposition
E-F	mean anomaly α of the Sun and Moon
G	mean distance of the Moon from its ascending node
H-I	the equation q = λ - λ̅ of the Sun and Moon (true minus mean longitude)
J-M	the equatorial horizontal parallax π and semidiameter r of the Sun and Moon
N	the shadow radius (in degrees) 𝜎 = 1.01π_M - r_s + π_s at the distance of the Moon
O-P	the derivative dq / dα of the Sun and Moon
Q	the speed of the Moon relative to the Sun at opposition
R	ET at opposition
S	ET at the approximate moment of maximum eclipse
T	the eclipse magnitude m = (𝜎 - 𝜌₀ + r_M) / 2r_M
U	𝜌₀ is the distance of the Moon’s center from the shadow center
V-W	𝜌₁ = 𝜎 + r_M and 𝜌₂ = 𝜎 - r_M
X	the moment when the Moon first touches the umbra
Y	the moment when the Moon is totally within the umbra
Z	the moment when the Moon is at minimum distance from the shadow center
AA	the moment when the Moon first exits the umbra
AB	the moment when the Moon last touches the umbra
AC-AE	the longitude, latitude, and distance of the Moon at the moment of maximum eclipse
AF	𝛥T = ET - UT

Guide to the solar eclipse spreadsheet columns

Column(s)	Content
A-C	year, month, and day for the conjunction of the Moon and Sun
D-E	geographical longitude and latitude of the observer
F	UT for the conjunction
G-H	mean anomaly α of the Sun and the local sidereal time Θ
I-J	mean anomaly of the Moon and the mean distance from its ascending node
K-L	the equation q = λ - λ̅ of the Sun and Moon (true minus mean longitude)
M-R	the equatorial horizontal parallax π and the true and parallax corrected semidiameter r of the Sun and Moon
S-T	the derivative dq / dα of the Sun and Moon
U	the speed of the Moon relative to the Sun at opposition
V	the eclipse magnitude
W	𝜌₀ is the distance of the Moon’s center from the shadow center
X-Y	𝜌₁ = 𝜎 + r_ᴍ and 𝜌₂ = 𝜎 - r_ᴍ
Z	the moment when the Moon first touches the umbra
AA	the moment when the Moon is totally within the umbra
AB	the moment when the Moon is at minimum distance from the shadow center
AC	the moment when the Moon first exits the umbra
AD	the moment when the Moon last touches the umbra
AE-AK	the true longitude, latitude, distance, right ascension, declination, azimuth, and altitude of the Sun at the moment of maximum eclipse
AL-AQ	as above for the Sun but parallax corrected
AR-AX	the longitude, latitude, distance from Earth, right ascension, declination, azimuth, and altitude of the Moon at the moment of maximum eclipse
AY-BD	as above for the Moon but parallax corrected
BE-BF	the parallax corrected longitude and latitude of the Moon using Hipparchus’ approximation.
BG-BM	the degree of the ecliptic λ_ʜ rising at the time of the eclipse, the angle Ψ of the ecliptic with the altitude circle that passes through the Moon, all at the time of maximum eclipse
BN-BO	the parallax corrected longitude and latitude of the Sun using Hipparchus’ approximation.
BP-BV	the degree of the ecliptic λ_ʜ rising at the time of the eclipse, the angle Ψ of the ecliptic with the altitude circle that passes through the Sun, all at the time of maximum eclipse
BW-CB	the true parallax corrections in longitude, latitude, right ascension, declination, azimuth and altitude for the Moon
CC	𝛥T = ET - UT for the modern models

References

Pierre Bretagnon, Francou G., "Planetary Theories in rectangular and spherical variables: VSOP87 solution", Astron. Astrophys., vol. 202, no. 309 (1988).⬈#reference-1

Chapront-Touze, M. and Chapront, J., "The Lunar Ephemeris ELP 2000", Astron. and Astrophys. vol. 124, no. 1, pp 50-62 (1983).⬈#reference-2

Jean Meeus 1998, Astronomical Algorithms, 2^nd edition, Willmann-Bell, Richmond.⬈#reference-3

Noel Swerdlow 2004, Planetary, Stellar, and Lunar Visibility, v 3.1, www.alcyone.de, by Rainer Lange and Noel Swerdlow.⬈#reference-4

Olaf Pederson 2011, A Survey of the Almagest, with Annotation and New Commentary by Alexander Jones, Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York.⬈#reference-5

Carl Schoch 1924, The “Arcus Visionis” of the Planets in the Babylonian Observations, Monthly Notices of the Royal Astronomical Society, Volume 84, Issue 9, 14 July 731-734.⬈#reference-15

Gerald J. Toomer 1984, Ptolemy’s Almagest, translated and Annotated, Duckworth, London.⬈#reference-15

Tables of Lunar and Solar Eclipses from -800 to 1650 Using Modern and Almagest Models

Archival Version of Spradsheets of Computed Data for Lunar and Solar eclipses

Guide to the lunar eclipse spreadsheet columns

Guide to the solar eclipse spreadsheet columns

References