Introduction
Were the builders of Stonehenge aware of the several motions of the moon, lunar orbit and the inclination of lunar orbit turning in space? Do the 30 sarsen stones represent their astronomy knowledge? This page combines several blog posts, first one written while an error in a modern astronomer's formula obfuscated possible evidence, then one recounting my discovery of the error and the evidence thereafter revealed. The posts follow below in their proper chronology, with the summation of evidence in the final section, Stonehenge and Lunar Motions.
Stonehenge and Astronomy
2012.01.21  Recently in online fora I interjected the topic of tallies, simply counting intervals; days, rotations, lunar orbits, nodal periods, full moons, eclipses, years and solar orbits. Newgrange and Knowth kerbstone numbers led to the discussion of Stonehenge numerology and the possible import of the number of sarsen stones, thirty. While following these fora topics I considered the numbers discussed in relation to astronomy constants and the ratios of astronomy tallies. This post focuses on the 30 sarsens at Stonehenge and the question posed in a forum, "Does 30 have an explanation in astronomy?"
I want to begin this discussion with more context before my main focus, Stonehenge astronomy. There is considerable disparity of opinion and variety of interpretation regarding ancient astronomy. Naysayers are calling Neolithic culture "primitive" and their art "scrawlings" while questioning the ability of humans of that era to count and record astronomy periods. I suggest they view a few Stonehenge photos and videos for more context on what the "primitives" accomplished. The enthusiasts are seeking meanings and records in the details of ancient construction and artwork. I interjected astronomy tallies in the hope of clarifying how easy it may have been for ancient observers to arrive at astronomical knowledge. If sufficient cycles and periods are counted concurrently from the same zero moment, in due time accurate ratios of astronomical motions become apparent. Somehow this simple and obvious fact seems to have gone unnoticed in modern academic discourse.
Today, in our "advanced" world, popular cosmovision seems very primitive. Do your own sampling by asking a few people, "In what direction is the earth turning?" If they know that easy answer, ask in what direction the moon orbits the earth or the earth orbits the sun. Even the very educated generally have a problem with the most basic of astronomy questions. In general and with anthropology and archaeology paradigms, the tendency to resist thinking of people in the past or in other cultures as more knowledgable than we are impedes understanding the past. What tidbits of the history and prehistory of astronomy we possess are often interpreted in a context of ignorance of both the past and of astronomy, and often with correlated assumptions that others must be at least equally as ignorant as we are. Ignorance of one's own assumptions plays a role and seems to keep the naysayers and enthusiasts entrenched in their own views.
Ancient astronomy was brought to popular attention with the publication of Stonehenge Decoded by Gerald S. Hawkins in 1965. Hawkins used a computer to analyze the alignments of many points in the monument to many possible astronomical events. He concluded the monument was a Neolithic computer. His work was properly criticized on probability grounds—given enough points, of course there will be alignments, especially when close is defined as good enough. The most conservative consensus today is that Stonehenge has one alignment, the Avenue roughly aligns with the summer solstice sunrise in one direction and winter solstice sunset in the other. The Avenue could also be a simple geometric subdivision of space, and, it has to point somewhere. Meanwhile, numerous hypothetical alignments continue to be proposed. Alexander Thom correctly ignored alignments between features too close together for reliable conclusions.
 Astronomy at Stonehenge? An expert discusses the controversial question of whether Stonehenge was an astronomical observatory. Audio.
 NOVA Secrets of Stonehenge. New archeological finds shed light on the most misunderstood monument of the ancient world. Video.
One analytic approach is counting the stones, such as the 30 sarsens. In 2006, I wrote, "Keeping a lunisolar calendar reveals the facts. Observing the moon and sun moving in sidereal terms quickly reveals the ratio of days and rotations per orbit and the concurrent lunar synodic and sidereal ratio. There is only one way that the numbers fit. Without an intervening dogma/belief system, it is simple logic and math."
The idea, hypothetically, is that if Stonehenge served an astronomy function, wouldn't the number of sarsens, 30 stones in a perfect circle capped with carefullyleveled lintels, also have some relationship to astronomy. While 30 might just have been a convenient number to create a circular structure, a logical approach for investigating an interpretation asks if independent data in another domain supports the supposition. The question was posed on a discussion board with numerous history of astronomy experts. Noone seems to have the answer yet, or at least noone has offered it to the group. Interestingly, I posted the answer shortly before the question arose, and noone has noted recognizing that solution. It seems that paradigms serve as blinders and assumptions do not allow considering solutions beyond presentday astronomy methods. Simply counting intervals is not part of the current astronomy tool kit.
The tally 30 does have astronomical significance. Thirty turns of lunar orbit axis (t = 30.0) is the motion tally I propose. Of course, knowing about the lunar orbit turning in fixed space and knowing its interval is advanced heliocentric knowledge. Ask a "modern" astronomer how long it takes lunar orbit to turn thirty times and in what direction it is turning, and you might only get a perplexed expression. If you lack awareness of a motion, you don't expect ancient astronomers to know it. Why did geocentrism persist for so long? Not knowing about rotation and solar orbit implies a lack of the tally variables used to solve the puzzle of cosmic order. Are we still blinded by engrained assumptions of a geocentric universe? I also noted the following (now known to be incorrect as detailed below):
 Lunar orbits and nodal periods commensurate at 3.0 t, with 746 l : 749 n
 746.0 lunar orbits equals 749.000455 nodal periods (1.0 : 1.000 000 6)
So, if you have one skeptical bone (or even if your dog has one), by now you should have asked, "How do/can we determine if the builders of Stonehenge knew all this?" We can trust the astronomy. However, the intention of the builders and their degree of understanding astronomy cannot be so easily determined. At least hypothetically, long ago a civilzation somewhere could have counted the intervals and determined the obvious pattern. Just because astronomers today seem to not recognize the import of the number 30 in astronomy tallies is not proof that someone many thousands of years ago would not do so. After all, they were not encumbered by our postDark Age assumptions and the waning influences of a theocratic, geocentric world view.
Assigning this knowledge to the builders of Stonehenge is a very high hurdle relative to our understanding of their knowledge base. Understanding this information is even a high hurdle for many students of archaeoastronomy and ancient cosmology today! Nonetheless, given the data the question is posed, "Did knowledge of astronomy 3,500 years ago surpass, in some ways, what is commonly understood today?"
As I've said before, the telescope is no substitute for counting and thinking. It is important to understand we often are blind to our own assumptions. The past is not limited to our imagining. Stonehenge is awesomely impressive, certainly not the work of "primitives" lacking geometry and counting, the two simple ingredients to solving the seemingly complex puzzle of cosmology. If the builders did know all this astronomy and incorporated 30 sarsens because of it, and we now find that to be incredibe and impressive, does that say more about our ignorance than it does about their abilities?
You can test these ideas and other numbers yourself with my applets. Downloads are available on my Astronomy Page. For further reading related to these ideas, see also: Eclipses, Cosmic Clockwork of the Ancients.
Correcting Modern Astronomy Formulas
2012.08.14  A return to this page is necessary because I discovered an error in an astronomy formula, one resulting in the error above—lunar orbits and nodal periods do NOT commensurate at 3.0 t : 746 l : 749 n. Three turns of lunar orbit equals instead 3.0 t : 745.9 l : 748.9 n. What remains constant is lunar orbits plus the number of turns of lunar orbit equals the number of nodal periods (l + t = n), be that three, 30, or any number or fraction.
Interestingly perhaps, the error arose in the same online forum, the one populated by leading academics in astronomy, the history of astronomy, and ancient astronomy. The error only became apparent to me when i started analyzing large numbers found in a Maya astronomical context. The formula in question was for the number of days per lunar orbit turn, and the error was in the temporal change. So, the further back in time, the greater the disparity became. Equivalently, the longer the duration of the periods considered, the greater the disparity also.
The formula was posted on the History of Astronomy discussion board, but noone seemed to have noticed the error, or at least noone offered such to the group. I started the new discussion thread with the following:
Subject: 
Lunar orbit formulations 
Date: 
Mon, 6 Dec 2010 
Following on the discussions by Göran Henriksson, Tom Peters, et. al., a summation of formulas might be useful (at least to me). I'd like to see any competing formulas for lunar orbit and the lunar nodal period. Also, how do such reformulations impact other formulas? I currently calculate 18.60004 years for lunar nodal regression of one orbit. The recent discussion centers on lunar orbit, not the nodes. ... 
Tom Peters replied with, in pertinent parts:
I replied in part:
… Wikipedia has the following firstorder linear approximations for the epoch J2000.0 (1 January 2000 12:00 TT) in Terrestrial Time with days of 86,400 SI seconds where Y is Julian years of 365.25 days (?):
sidereal orbit 27.321661547 + 0.000000001857 × Y days
nodal period 27.212220817 + 0.000000003833 × Y days
synodic month 29.530588853 + 0.000000002162 × Y days
I am attempting reformulating with a logical, unified, and fully integrated method. This seems to require reverse engineering modern astronomy from synodic observations to their determinative sidereal fundamentals. I am trying to reformulate such that:
1.) solar and lunar orbit period formulas determine lunar synodic,
2.) solar orbit and precession determines the year period,
3.) solar orbit, lunar orbit, and rotation of the lunar node determine eclipse nodal period and the lunar orbit ecliptic cycle, and
4.) solar orbit, lunar orbit, rotation of the lunar node, and equatorial precession determine the lunar standstill period, and I expect equality to be stable temporally. In other words, I expect the geometric theorem [ x +/ 1 = y ] to be adhered to when I change the date in my research application. Examples (with rounding):
x  1 = y
x = 366.25636 rotations per orbit
y = 365.25636 days per orbit
x = 13.36875 lunar orbits per solar orbit
y = 12.36875 lunar synodic periods per solar orbit
x = 12.73767 nodal periods per eclipse nodal period
y = 11.73767 moons per eclipse nodal period
x = 19.613 eclipse nodal periods per lunar standstill
y = 18.613 years per lunar standstill
x + 1 = y
x = 25,770.17 orbits per precession cycle
y = 25,771.17 years per precession cycle
I want periodicity formulas of solar and lunar orbits to define lunar
synodic ( x  1 = y ), etc., rather than have the derivations
define the fundamentals. Without inserting the Wikipedia formulas above, it seems obvious they will not adhere to this geometric theorem in a temporal sense, thus a unified formulation from fundamental motions still escapes me. ... 
Tom Peters replied, 14 Dec 2010, in parts (with formulas containing several errors):
I've made contributions to that, the present values may be from Meeus and derived from the ELP200082.
From J.Chapront, M.ChaprontTouse, & G.Francou, Astron.Astrophys 387, 700..709 (2002), p.704 Table 4, I derive...
sidereal speed = (1732559343.3328  13.7400*t + 0.019812*t**2  0.00012676*t**3)"/cy >
sidereal month = 27.321661553560 + 0.000000216673*t  0.0000000014478*t**2 + 0.0000000000019989*t**3 days
days nodal motion = (6967919.8851 + 12.7186*t + 0.022875*t**2  0.00014344*t**3)"/cy >
nodal period = 6793.4765010  0.0124002*t  0.000022302*t**2 + 0.00000013985*t**3 days
EarthMoon barycenter speed = (129597742.3016  0.0404*t + 0.000027*t**2 + 0.00000060t**3)"/cy >
sidereal year = 365.2563629530 + 0.0000001139*t  0.000000000076*t**2  0.00000000000169*t**3 days
Note: these are measured from the fixed ICRS aequinox of J2000 ... 
Now, this is not your average online discussion group, rather the creme de la creme of academics in history of astronomy and anciernt astronomy. In my thread, Dr. Kim Malville even commented about another aspect of Tom's post:
Dear Tom,
As one of the editors of the Journal of Cosmology I would really welcome receiving your paper. My email is .... Many thanks,
Kim Malville
Professor Emeritus Astrophysical and Planetary Sciences
University of Colorado 
Thus the error went unnoticed and entered my applet and now I need to amend this page. The lesson in this is do your own homework, learn what you must to ascertain that the information you rely on is reliable.
Given how this page began—discussing the same forum and modern astronomy perspectives, this seems a proper place to detail the error. Serendipity has reinforced my comments above! I'd like to say I report all this for its pedagological utility, but really, in science we have to go back and correct our errors. Tom did so recently, albeit it took a long time to convince him of the problem. First he sent an alternate formula for days per lunar nodal period, and I had to demonstrate that his later formulation contradicted the former. He then discovered more errors than the one I noted.
From: Tom Peters
Tue, 31 Jul 2012
heeft Tom Peters het volgende geschreven: From J.Chapront ... I derive...
James Q. Jacobs pointed out that I made a few errors. For the record, the correct formulae are:...
sidereal month = 27.321661553560 + 0.000000216673*t  0.00000000031243*t**2 + 0.0000000000019989*t**3 ...
nodal period = 6793.4765010  0.0124002*t  0.000022325*t**2 + 0.00000013985*t**3 days
... tidal acceleration in lunar longitude is 0.5*25.858"/cy**2.s .... 
Stonehenge and Lunar Motions
2012.08.15  Now back on topic again, on to how my initial results have changed. Given the formula correction, I find that 30 turns of lunar orbit coincides with integer lunar orbits and integer lunar nodal periods during a time close to when the Sarsen Stones were erected. Each turn in fixed space of the inclination of lunar orbit produces an integer difference between the tally of lunar orbits and lunar nodal periods (l + t = n). In 30 turns of lunar orbit, there will always be 30 more nodal periods than lunar orbits.
What I find significant is that during the period that Stonehenge was in use, the number of lunar orbits and nodal periods was gradually approaching integers, numbers without decimal remainders. What has changed after correcting the error is that 30, instead of three, is now the lowest number of lunar orbit turns to equate to integer lunar orbits (and thus also to integral nodal periods). The equation to three turns, discussed above, is a result produced by the formula errors. To further refine accuracy, I also formulated a time conversion to convert the lunar periods to UT days, the number of days the Stonehenge builders would hypothetically have counted.
The formula errors erroneously indicated the ratio 3.0 : 7,46.0 : 7,49.0 with integer accuracy about 1,000 years earlier, before the sarsens were erected. Ancient astronomers counting nodal crossings and lunar orbits would have observed, during a span of nearly 558 years centered on 1523 B.C.E., the simultaneous completion of lunar orbits and nodal periods precisely as their difference corresponded to 30.0 turns of lunar orbit (Table 1).
Table 1. Lunar orbits and Lunar Orbit turns.

Constant 
Multiple 
Value 1523 B.C.E. 
Days (UT1) 
Lunar Orbit (l) 
7,459.0 
dl = 27.3216534 
203,792.35 
Lunar Orbit Turn (t) 
30.0 
dt = 6,793.07377 
203,792.35 
Lunar Nodal Period (n) 
7,489.0 
dn = 27.2122063 
203,792.35 
The hypothesis that the 30 Sarsen Stones are connected to astronomy seems even more plausible given the formula correction. Now, with a better fit to the Stonehenge timeline, the question is simply reframed with new numbers including the precise number of sarsens, "Did the builders of Stonehenge erect 30 sarsens because they understood the 30.0 : 7,459.0 : 7,489.0 integral astronomical ratio of the two lunar motions?"
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Stonehenge and Pi
