In archaeoastronomy a possible accuracy of 0.5°
is expected for monuments build around 4000 BCE. On this web page I try
to determine the accuracy achievable with computer programs that
the celestial events. The accuracy of these programs should at least be
around one order better than the presumed accuracy of the monuments; so
the accuracy must be much smaller than 30'.
This page will provide some insight on who to make a choice in
computer programs. This will be done along the following lines:
A few features are important when looking at programs for
work. The following features should be taken into account when deciding
to use a program (in order of preference):
I am using at this moment the following programs:
If people know of other (better) programs for archaeoastronomy, let
- celestial objects
In any program (demo, share or free-ware) all planets, objects and
stars which are visible with the naked eyed need to be available.
an outline of the milky way and the horizon
- it should included all
'normal' theories of calculations
Such as: position (VSOP87, DE405, etc.),
perturbations, precession (Simon, J.L. ),
(Stephenson, etc.), nutation, change of obliquity, lunar
motion (n-dot, ELP2000-85, etc.), proper motion (Hipparcos
and Tycho catalogues), light travel time, refraction, horizon
dip, etc. etc.
- has to have an accuracy of much smaller than
60' for azimuth
between 4000 BCE and 2100 CE. This specification has to be relaxed
for the moon.
- provide an inaccuracy range in every display, based on for
- the possibility to step in time using different periods; like
periods (hours, days, sidereal day, tropical year, etc.) and average
periods (like draconic, synodic, nodal cycle, etc.).
- has to have an automatic date/time increase/decrease repeat.
- the possibility to define a sky window
determine when a celestial object passes along this window.
- information on and changing the actual DT
used by the calculations.
- provide visibility (incl. e.g. extinction
of celestial objects in the sky (depending on twilight conditions).
- integrate it in a virtual world
display usable in VRML and QuickTime
- Beside the above mentioned analythical theories in bullet 2),
important to know which numerical model has been used in the
implementation. The numerical method must have been mapped
a well as possible for the period 4000 BCE to 2100 CE (validity period).
The theories and their accuracy
All topics discussed in the following sections don't look at
it centres on the accuracies that can be obtained from these theories.
I assume implementations
(programs) exist that will implement these theories without
of programs should check this.
I have the following list on accuracies (see what inaccuracies
means for local circumstances (like altitude/azimuth)):
- Position of planets and the Sun (lx)
We can use an ephemeris that is based on VSOP82,
DE200, VSOP87, DE405,
etc. here with accepted accuracy up to say 3000/4000 BCE. I understand
this can be done at a positional accuracy (Dlx)
at less than 5" (the Sun) - 14" (Mercury) - 40" (Mars) for geocentric
around 4000 BCE (values of VSOP82, Bretagnon ,
- Lunar position (lm)
This needs to be done using the ELP2000 models (n-dot around -25.7
+/- 0.2 "/cy2 [Pers.comm, Myles Standish, JPL, 2002]). An
for the geocentric longitude (Dlm)
of some 13' (780") at around 4000 BCE is possible in this case. The
(or tidal or secular) acceleration has a similar effect as the DT.
DT (Terrestrial Dynamical Time minus
Time) change the actual time that an event will take place (and thus
the position due to rotation of the earth). I understood that the DT
is not fully known for times well before 700 BCE, so there seems not to
be many older accurate historical ecliptic record available (perhaps
yet found). The DT needs to be compensated
if different than (-26 "/cy2) lunar/secular
acceleration is used for the lunar positon.
But even for times in the near future DT
is not know to the second! See this
link where it states:
It is impossible to predict future values of DT
accurately. At the beginning of AD 1999, it was 64 seconds. By AD 2100
it will probably be between about 3 and 5 minutes.
The following set of DT formula's
is important (least interdependent and within time frame):
- Astronomical Ephemeris (1960)
- Tuckerman (1962, 1964) & Goldstine (1973)
- Stephenson & Houlden (1986)
- Borkowski (1988)
- JPL Horizons (it could be that this one will change in the near
due to updates of JPL Horizon)
- Stephenson: DT=35*T2-20
[Sec] (T from 1735 CE [century]) or see
table (Stephenson , page
515-516). A more optimized formula has
been derived by author of this website.
- Stephenson & Morrison: DT=32*T2-20
[Sec] (T from 1820 CE [century]) or see
table (Stephenson, Morrison ,
Because the D
formula's have common basis
eclipse data, the above values in some way will flatten the inaccuracy
The accuracy of D
T) is around 4000 BCE some ± 250
around 3000 BCE: ± 150 [min] and around 480 CE: ± 10
This accuracy is determine by half the spread between min. and max.
values from above formula, which results into a somewhat higher value
the sigma of calculated values.
In some literature (Bretagnon [1986
page 5) the accuracy of D
is quoted around 4000 BCE at some ± 120 [min] (interpolated this
gives some 85 [min] in 3000 BCE and 10 [min] in 484 CE).
Uncertainty values of Stephenson and Houlden (1986) are also quoted here
In my following evaluations, I take the maximum of these two methods;
so I use the values of half the spread of calculated values. Comments
this choice? Let me
and Tycho catalogue
have a location inaccuracy due to proper motion
of resp. around 15" and 180" in declination (Dds)
times at 4000 BCE (based on information from Michael Perryman, ESA,
Comm. ). Be aware that this inaccuracy increases for fast-moving/close-by
stars (sections 1.2, 1.5 and table 1.2.3)
- Obliquity (e)
According to Bretagnon (, page
6) the error in obliquity is around 0.1" (De)
at 4000 BCE.
- Luni-solar precession (Press.)
According to "Simon, J.-L., Bretagnon, P., Chapront, J.,
M., Francou, G., Laskar, J.: 1994, Numerical expressions for
formulae and mean elements for the moon and the planets. Astron.
282, 663" the accuracy of the precession in longitude is around 1"
(DPress.) for precession angle looking at
Accuracy determination in azimuth and
rules, VSOP82, Monte Carlo analysis (some
runs), geographical latitude of 52° and without taking into account
refraction (in case of altitude).
Values in bold are not reaching the specified
accuracy range of much smaller than 1800".
This gives the following Ddx for
the Sun, Mercury, Mars and the Moon around 4000 BCE: 4", 12", 36" and
Error on HA (Hour Angle):
This gives the following DHAx
the Sun, Mercury, Mars and the Moon around 4000 BCE: 7", 7", 17" and
Using this we determine Dazix
and Daltx (Duffett-Smith ,
f = geographical latitude
azix + Dazix=acos((sin(dx+Ddx+Press.+DPress.)-sin(f)*sin(altx
The errors in azimuth and altitude vary of course due to actual values
of declination (dx) and hour
(HAx). When using Monte Carlo analysis, the maximum errors
in azimuth and altitude are around the same value as the Dlx.
So this gives the following Daltx and
for the Sun, Mercury, Mars and the Moon around 4000 BCE: 10", 14", 40"
Dazis = (see above depending on ds,
Dalts = (see above depending
on ds, Dds,
The errors in azimuth and altitude vary of course due to actual values
of declination (ds) and hour
(HAs). When using Monte Carlo analysis, the errors found in
azimuth and altitude are some 2-3 times bigger than the Dds.
So this gives the following Dalts and
for stars from Hypparcos and Tycos catelogue around 4000 BCE: 35" and
(the error can be bigger, like when AH=0 and the declination has the
value as the geographical latitude).
Accuracy's of occultation events
- Solar eclipse
Due to the DT and lunar acceleration it
seems not to be possible to predict an eclipse with an
in DDT better than a few hours local
due the lack of enough history. Beside the local time (mostly due to DT)
also the location (mostly due to n-dot and DT)
where the eclipse will take place, can vary (if an solar eclipse is
of course also depends if the Sun is above the horizon).
The following sensitivity analysis on solar eclipses is done:
- Have solar eclipses in remote times, e.g. the present benchmark
(total eclipse on Jan. 14th, 484 CE, Athens, Greece) and an
annular eclipse at 2997 BCE (Jan. 27th). The two eclipse
give hopefully a picture how accuracy of DT
behaves over a period until 4000 BCE. The eclipse of 484 CE is well documented
in history and the eclipse of 2997 BCE is one of the last full
that can be calculated with DE406 (which only models until 3001 BCE).
- Determine min. and max. for DT at
with the help of the above mentioned DT
minimum and maximum DT [min] for 2997
and 484 CE (between brackets the formula that gave
|2997 BCE min
|2997 BCE max
|484 CE min
|484 CE max
- Determine with these DT values
acceleration parameter belonging to the specific ephemeris; the
in time and location of maximum conjunction of the Moon and the Sun.
The measure of location accuracy is measured by half the angular
between the eclipse paths of min. and max. DT
at the same Terrestrial Dynamical Time (this distance is thus a
of change in geographical longitude and latitude).
In the sensitivity analysis I am not able to change the n-dot (due
to fixed ephemeris and its fix use of an n-dot), furthermore I assume
the sensitivity analysis that the resulting location difference due to
the error in n-dot is small compared to the resulting difference due to
error in the DT.
The following computer programs are used in this analysis:
- The Digital Universe (DE200).
The location accuracy at max. eclipse is for 2997 BCE around ±
32° and for 484 CE around ± 5°.
animation by Andrew Sinclair (DE406)
more visual information (see below animated pictures, © Andrew
The location accuracy at max. eclipse is for 2997 BCE around ±
57° and for 484 CE around ± 4°.
Jan 27th, 2997 BCE, DTmin=1196
Jan 27th, 2997 BCE, DTmax=1492
Jan 14th, 484 CE, DTmin=77
Jan 14th, 484 CE, DTmax=95
So combining the two, it looks like the max. error in location is for
CE is around 5° and for 2997 BCE is around 57° (this error is
depending on the path the eclipse takes, in the above examples the
error was in the geographical longitude)
- Espenak's Google Map
This one can also provide comparable pictures as Andrew Sinclair
(except they are static), where one can change the DT
by changing an URL argument. Here is a link for the Jan. 14th 484 CE
ecplise with DT=77 [min] and here for DT=95 [min]
- Occultation of moon/planet with star
This is depending on accuracy of planet/moon position and
of the star. The lunar/planet position has already their errors, so
this with position of star (stellar motion) it will become a little bit
more inaccurate. One can add the accuracy's of both together: so (Ddo)2
= (Dax)2+(Das)2 +
This becomes thus for 4000 BCE:
19" (Mercury), 39" (Mars) and 707" (the Moon)
19" (Mercury), 25" (Mars) and 317" (the Moon)
180" (Mercury), 184" (Mars) and 730" (the Moon)
180" (Mercury), 181" (Mars) and 364" (the Moon)
Accuracy for set/rise events
Important: In the below section
the influence of DT is not included (DT has considerable influence on the time of the
set/rise [not much on the azimuth], but this is left
for another time, perhaps I can help you in a personal e-mail: let me
The error in the azimuth near the horizon is mainly determined by
and refraction uncertainties (assuming an accurate altitude of the
The variation can be for parallax between
and 1.0° (Dpar) and for refraction a
(1s) of 30% of nominal value is assumed
, page 126).
Other errors are the error (1s) in
of the celestial object of 0.20° for
Moon and 0.001° for The Sun (Ddx).
latitude at 52°, horizon apparent altitude of 0° and around
limits this gives:
Dazim = 0.56° or Dazis
= 0.29° (these values are calculated with Monte Carlo analysis,
some 4500 runs)
Around the equinoxes the errors are around Dazim
= 0.86° or Dazis = 0.22°
The azimuth error for the Moon and the Sun if the type of set/rise
is not know (top or bottom limb; thus an extra uncertainty (Dsize)
between 0° and 0.52°), but no error in declination of
object (so actual observation) is:
Dazim = 0.46° or Dazis
= 0.38° (these values are calculated with Monte Carlo analysis,
Around the equinoxes the errors are around Dazim
= 0.82° or Dazis = 0.29°
For solar and lunar alignments, the found errors are of the same order
as the 0.5° of Ruggles (,
ix) or Schaefer (, page 126).
Benchmark for archaeoastronomy software
A few things should be important for benchmarking:
At this moment several of these items are done (so looking at solar,
lunar and deltaT related issues). Future work will
this to more aspects of benchmarking (like for instance for star
- determine if a computer program supports the specifications
- determine which theories are being used in
- benchmarking them against a few known celestial
accepted implemented standard(s) (like tried with JPL
The proposed benchmarks are primarily for looking at the
implementation of available formula/emphemeris into programming code
(so is its
more or less about debugging;-):
- solar eclipses as discussed below, like total solar eclipse on
Jan. 14th, 484 and Aug. 11th, 1999
Will give some better idea of the implementation of DeltaT, the lunar
n-dot and the emphemeris of the moon and sun.
between stars and planets (perhaps Venus has a priority because of its
mention in many archaeoastronomy texts), like Regulus conjuction with
Venus on Oct. 1st, 2044.
Will give some better idea of
the implementation of luni-solar precession and the emphemeris of the
- occultations between stars and the moon
Will give some better idea of the implementation of luni-solar
precession and the emphemeris of the moon.
- some related to possible alignments mentioned in literature:
- Orion stars (Zeta, Epsilon and Delta) at Giza, Eqypt in 2500
- Reappearing sun at Maeshowe, Orkney in 2800 BCE (Reijs )
- Moon at major standstill limit at Dowth, Ireland on Feb. 25th,
2983 BCE (Coffey )
Total eclipse of January 14th, 484
Within the mailing list HASTRO-L (in 1996), software was discussed and
some programs were checked against the eclipse of 14 January 484 CE
Athens (38° 0' latitude, 23° 44' longitude). According to
this should be around that place and time.
Mr. Dearborn made an overview of the results and I (VR) have attached
new information obtained since 1996.
A general question asked in any area of research is how dependable
are your sources (data). In archaeoastronomy, many students depend on
software for calculations of events and orientations in the distant
Even when you are writing your own software, it is a fairly complex
to determine how uncertainties in the approximations of various
propagate through a calculation to the answers that you seek. In
on the History of Astronomy List server (HASTRO-L), Leigh Palmer, from
San Francisco State University, proposed a test for such software. As a
test of the long term accuracy; How well it represents the eclipse of
January 484 (Julian Calendar)?
A. Fletcher, in Schove's (1984, p. 81)
of Eclipses and Comets 1 - 1000 AD", quotes from Marinus' Life of the
philosopher Proclus as follows:
"Portents occurred a year before his death, such as the solar
which was so considerable that night occurred in the daytime. For there
was deep darkness and stars were seen. This happened in Capricorn near
the rising point (of the Sun)."
Totality is clearly implicit in this, but nothing in what Fletcher
cites identifies from where the observation of totality was made.
reports discussion by F.K. Ginzel (1899) and Neugebauer (1931) wherein
they "argue respectively for totality at, and only near Athens". He
cites Stephenson and Clark (1978) as saying that "this is probably the
most reliable of all solar eclipses reported in the Classics"
Before presenting the results, we wish to reiterate that these
perform many functions, and that the accuracy of a program in
a single eclipse is at best suggestive of its ability to represent
eclipses near that epoch. There are sensitive geometric effects for
observers, and there are genuinely poorly known variables (like the
between ephemeris time and universal time; DT).
In the following list, the reported results are summarized. Because
styles are not identical, and there is some variation in exactly what
reported. Robert Oliver, on the performance of Dance 2.71 and Total
1.5. Cary James sent results from the DOS based programs, EZCosmos 3.0
and PEEP 1.02 (Planetary Event and Eclipse (Predictor). Richard Johnson
sent output from EZCosmos 4.0 and Eclipse Complete 2.0. Jim Fuchs
input on MyStars!. Peter Jones provided input on Cartes du Ciel.
gillies macbain provided input for Voyager. Guus Gilein provided
feedback on Guide and Redshift 5. Rob van Gent provided feebback on
Redshift 3. Vladimir Pakhomov provided
input on SkyChart III. David Herald provided input on winOccult. Sourav
Maiti for SwissEph and Eclipse Finder. Victor
Reijs composed the rest.
presentation using Google maps can be seen on Espenak's
The below programs (in alphabetic order) have been tested. Bold
printed program names have an eclipse moment of around 5:48 +/- 0:30
(3s) UTC on Jan 14th, 484
Ephemeris and error in DT
as reference; remember that the precise
is not known from historical accounts!).
A good link on planetarium programs is here
(MacOS, MS Windows, DOS, X, Palm, OS/2 WARP).
- Astro Meeuws (tested in 1996)
The program found it at 08.01 UTC Jan14th, 484 CE (full eclipse) in
(now called AstroSeeker) from Zephyr
(date exe-file: 31/7/1989)
Some Zephyr products can find celestial objects within a skywindow
The eclipse happened, according to the
function of that program, on: 6:06 UTC.
Series96 from the Bureau des Longitudes between 1900 and 2100. Plan404
based on DE404 by Steve
Moshier between -3000 and +3000. ELP2000 with a truncation for all
terms smaller than 10E-8. Double-parabolic fit by Stephenson 1986
Has features: 1, 2 (not earlier than 3001
BCE), 3, 5, 6, 8
Total eclipse happens at 05:57 UTC at Jan 14, 484 CE in Athens
The path of totality passes north of Athens. In Athens, an 84%
eclipsed sun rises at about 6:03 UTC.
Andrew Sinclair eclipse animation (tested in 2002)
This animation uses JPL DE406 (n-dot: -25.7 ["/cy2]), and
one can input ones own DT.
Has features: 2 (not earlier than 3001
BCE), 3, 6, 8
Total eclipse at Jan 14, 484 CE in Athens around
- Eclipse Complete 2.0 (tested in 1996)
From Athens, the maximum eclipse was about 85%, occurring at 6:40 UTC,
nearly an hour after sunrise.
eclipse predictions (tested in 2007)
Espenak (see his eclipse
web site) using the following ephemeris: for the Sun VSOP87, for
the Moon ELP-2000/82. The value
for the Moon's n-dot is -25.858 ["/cy2] and DT from
own polynomal description).
Has features: 2 (not earlier then 2000
BCE), 8 (one can't change DT)
Total eclipse happens at around
UTC at Jan 14, 484 CE in Athens
V3.0 and V4.0
The eclipse begins about 30 minutes after sunrise, reaching 95% at
7:30 UTC. The altitude at this time was 15.62 degrees above the
VSOP87/DE200, a truncated version of the ELP-2000 for the moon and
and Stephenson for DT (1984, n-dot around
For the stars it uses the Hipparcos, ACT and GSC catalogs
Has features: 1, 2, 3, 8 (one can change DT
The Sun rises in eclipse (about one minute after the moon) at 5:31
UTC. By 6:05 UTC the eclipse reaches approximately totality.
HORIZON Ephemeris Version 3.15
The is a web based planetarium program (based on DE405/DE406 (n-dot:
–25.7 ["/cy2]) and their own DT
formula related to Stephenson  (n-dot: -25.7 ["/cy2]))
(Pers. comm. Giorgini ).
Has features: 1 (no Milky Way contour),
2 (not earlier than 3001 BCE), 3
Total eclipse happens at 5:48 UTC on Jan 14th, 484 CE in Athens
- Mobile Panjika (checked Dec.
Here is another high precise archaeoastronomy software can for mobile
phone calculate eclipse timings accurately from 3000 BCE to 3000 CE.
The heart is SwissEphemeris, but ported for mobile phones and the value
used for the Moon's n-dot is -25.858 ["/cy2] and DT
from Morrison/Stephenson 
The time of max eclipse at Athens (14 jan 484 CE): 05:43 UTC
Gives a clear view of the phases and visibility of the moon.
Eclipse happens at 05:38 UTC at Jan 14, 484 CE in Athens, 98% and the
Sun 1 degree below horizon
- PEEP V1.02
The eclipse was total at 5:56 UTC, just after sunrise.
- Planetary, Lunar and Stellar Visibility
The eclipse at Athens happens at 6:03 UT on Jan 14th, 484 CE in Athens
(around 100% eclipse)
Maximum eclipse (no total) happens at 06.38, sun 0.5 degrees below
horizon. It shows just the tiniest smidgen
short of totality at Athens with the Sun and the Moon just in the
- RedShift 3
Maximum eclipse at 5:48 UTC on Jan 14th, 484 CE in Athens, on the verge
of a total eclipse
Has features: 1, 2 (except horizon dip,
change in obliquity), 5 (lunar synodic and day), 6
Eclipse happens at 11:05 UTC on Jan 14th, 484 CE in Athens (around
This program is based on VSOP87,
ELP2000-82B and Stephenson & Houlden (1986) (is under study by
author (pers. comm Marriott ) and n-dot =-26 ["/cy2] adapted to
Has features: 1, 2 (except horizon dip),
3, 7 (manual with Annotation: camera frame), 8
Eclipse happens at 6:05 UTC on Jan 14th, 484 CE in Athens (almost full
Has features: 1, 2, 6, 7, 8, 9
The partially eclipse happened, according to the
finding-minimum-angular-separation function of that program, 6:30
UTC on Jan 14th, 484 CE in Athens
based program. Stand alone version has full VSOP87 and ELP 2000-82
precision and Chapront, Chapront-Touzé & Francou (1997) for DT.
Has features: 1 (no Milky Way contour),
2 (except horizon dip), 3, 5 (some options)
Eclipse happens at 6:32 UTC on Jan 14th, 484 CE in Athens, Greece
- StarCalc 5.72
Eclipse happens at 6:06 UTC on Jan 14th, 484 CE in Athens, Greece
It uses use for the Moon, the Sun and the Earth position Meeus (VSOP
87) and Chapront moon (ELP2000/82). Uses their own DT
formula (DT can be viewed in the Pro-version
of the program), but planning to use Stephenson in the near future
comm. Braganca ).
It shows the eclipse is at 6:02 UTC at an apparent altitude of 2°
(checked on Jan. 7th, 2005)
Discussing with author about results gotten for the eclipse (looks far
off) and present times. The author is debugging this now due to my
DT from Stephenson&Morrison 
and JPL DE406
for Sun and Moon ephemeris.
Has features: 1, 2, 3
Total eclipse happens at 5:43:55 UTC
Digital Universe V2.0
This program (based on DE200 (n-dot: -23.8946 ["/cy2]),
ELP2000-85 and Stephenson & Houlden (1986; n-dot: -26 ["/cy2]))
looks to have almost all of the wanted features. It is only available
Mac at this moment.
Has features: 1 (in demo version: no
or Milky Way), 2, 3, 5, 6, 8
It could be that features 4 and 7 will
perhaps also be implemented (Pers. comm. Charrois ).
Total eclipse happens at 5:57 UTC at Jan 14, 484 CE in Athens (the
full eclipse happens at 6:06 UTC when using an n-dot of -23.89)
Astronomical algorithms from Jean Meeus (VSOP 87???).
Has features: 1, 5, 6, 8 (one can't
This planetarium program show the Sun eclipsed on 14 Jan. 484 CE at
7:30 UTC (DT = 78.5 [min]). The DT looks to
be oke, so something else in the emphemeris must to be wrong.
A near total eclipse (99.4%) occurs at 5:44 UTC. At this time the Sun
would appear to be rising on a level horizon (altitude -0.6) The path
totality north slightly of Athens.
- Voyager III 3.21 Mac
It uses Stephenson & Houlden (1986) for DT.
The Sun and The
Moon have almost the same coordinates at 5:57 UTC
Athens Jan. 14th 484 CE
- winOccult 3.1
Program build around occulations of the moon. Maximum eclipse at Athens
at 5:57 UTC, at an altitude of 1 degree. The eclipse is not total, with
a magnitude of .968. The calculation is based on a deltaT of 79 [min]
(using Stephenson & Houlden ). Sun: VSOP - which is a fit to
DE200. Moon: Chapront ELP-82B - also a fit to DE200.
Has features: 1, 2, 3, 8 (but not in eclipse module!)
Sky (checked Dec. 22, 2004)
The max. eclipse coverage is at 8:10 UTC. But it is certainly not full.
know if you tested one!
Investigating the precision of a tool is fundamental to basic
and no numerical simulation should be considered to be absolutely
The ground track of eclipses, particularly near the limb of the earth,
are sensitively dependent on the precise value of a number of difficult
to determine variables. So, it is understandable how a program may do
on one eclipse and not so well on another of the same epoch.
The testing discussed here is a good beginning, but confidence
be based on more that one test. In addition, tests should be collected
on events other than eclipses.
I would like to thank the following people for their help and
feedback: Dan Charrois, David Dearborn, Robert van Gent, Michael
Steven Hope, Michael Perryman, Thomas Schmidt, Andrew Sinclair, Myles
and all the people who provided feedback on specific computer programs.
Any remaining errors in methodology or results are my responsibility of
course!!! If you want to provide constructive feedback, let me
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Last content related changes: Aug. 23, 2002