Appendix 2


R. G. Bauval

Precession calculation is a vital tool for the historian to help him understand ancient man, whose religion was often directed to the ‘sky gods’ and thus based on observations of the sky – what today we would call naked-eye observational astronomy. It can be thus understood that ancient man built religious monuments, temples and, more prolifically, tombs which made use of geometrical astronomy to express astronomical alignments and other phenomena of the sky using symbolic architecture. It further follows that if an architectural feature of a monument is suspected to have been aligned towards a specific star, then, with the use of precession, it is possible to work out the date of such a monument to a fairly good level of precision. By also ‘re-creating’ the sky for the given epoch, we can see what they saw and hence understand further the religious importance of their observations through the design and symbolic expression of the monument.

Before electronic scientific calculators and computers became household equipment, precessional calculations had to be done long-hand. These were not only complex but tedious, especially because the formulas combined spherical geometry and trigonometry through several steps of computations. If only one or two calculations were required, this was not so bad, but if several stars and dates had to be verified, the calculations might take all day. Happily for us today, a good personal computer does this for us: anyone with a PC can not only perform precessional calculation with a few touches of the keyboard but also actually see on the screen the effects precession has on an artificial sky globe.1 But what exactly is precession?

24. Precession

The sun and moon exert a gravitational pull on the earth’s equatorial bulge, causing the planet to ‘wobble’ in a very slow cycle known as precession. The simplest way to think of precession is to imagine the earth as being a spinning top which also has a slow ‘wobble’ of just under 26,000 years’ cycle. The extended axis through the poles thus performs a slow almost circular motion against the background of the starry sky and returns to the same place every 26,000 years. Every half-cycle of precession i.e. 13,000 years, a star finds itself in opposite direction on the precessional cycle, such that if it is observed at the high-point (highest declination) on the precessional cycle then 13,000 years later (or earlier) its position would be at its low-point (lowest declination) on the cycle.

The precessional effect is most noticeable at the meridian. Taking Orion’s Belt as an example, in c.AD2550 it will be at highest declination (c. –0.8 degrees) very near the celestial equator. Thus it was at lowest declination (c. –48 degrees declination) in c.10450BC. During the Pyramid Age, c. 2500BC, it was at c. –15 degrees declination.

The length of the precession cycle, however, is not absolutely constant but changes slightly from epoch to epoch. It is generally accepted, however, that it lies between 25,800 and 26,000 years. We have taken the value 26,000 years throughout The Orion Mystery.It must be noted that there is another shorter complex motion called nutation which takes 18.6 years.2 This causes little ‘hiccups’ every 18.6 years on the otherwise smooth circular motion of precession. Nutation is generally ignored in precessional calculations for distant epochs since it is not possible to determine whether a hiccup was occurring at the date considered.

Both precession and nutation, of course, are not proper motions of the stars themselves, but are due to the movements of our own planet, producing the apparent motions of the stars. All stars, however, do have their own proper motions, i.e., they move in space. The closer the star, the greater visual effect its proper motion has over a given time. The farther the star, the smaller the visual effect. Proper motion is measured in angular change as a combination of declination and right ascension, these being the given co-ordinates of the stars on a sky map. Sirius is among the closest stars to our planet, at about 8.4 light-years away. The angular change due to its proper motion is given as –1.21 arcseconds per year. Over thousands of years this is quite noticeable and thus must be taken into account when precessional calculations are made. On the other hand the stars of Orion’s Belt are very far indeed, about 1400 light-years, and generally no proper motion is registered.3 Some researchers prefer to allocate a very small proper motion if a distant epoch is considered, but the resulting effect comes to well below the one-arcminute level for the epoch of the Pyramid Age. This cannot be perceived with the naked eye and thus proper motion is assumed to be negligible in such a case.4

When considering precession for relatively short periods of time, say fifty to one hundred years, the first approach is a simple rule of thumb where the sun appears to move against the background of the stars near the ecliptic (the path of the sun) by about 50.3 arcseconds per year. For 100 years this is about 1 degree 23 minutes and very noticeable indeed to a keen observer. Not all stars, however, are near the ecliptic and this rule of thumb cannot be simply applied to them. Nor does it show the effect of precession on declination. Mathematically, this is obtained by using the formula:

Change in Right Ascension (RA) = 3.07″ + 1.34″ sinRA tand then, Change in declination (d) = 20.0″ cosRA

In the case of very long periods of time, however, such as several millennia, a much more rigorous approach must be taken. In Sky Catalogue 2000.0, vol. I, the Rigorous Formula for Precession is given. Three auxiliary constants, A, B and C, are determined by the selection of the dates of the initial epoch (taken as AD2000) and the final epoch considered. These are given as:

A = 2305.647″ T + 0.302″ T2 + 0.018″ T3
B = A + 0.791″ T2
C = 2003.829″ T - 0.426″ T2 - 0.042″ T3

The first thing to do is to correct the position of the star for proper motion, given as (u)RA and (u)d for Right Ascension and declination respectively for one year, where the values of u are in arcseconds. This is done by multiplying (u)RA and (u)d by the number of years. The values are negative if before AD2000 and positive if after AD2000. The value (u) is the proper motion taken from tables. The result is added (forward in time) or subtracted (backward in time) from the Right Ascension and declination of the selected star’s coordinates at the start of epoch AD2000. The new declination is given as d (0) and the new RA is given as RA(0). This therefore accounts for proper motion. The Rigorous Formula for Precession is then applied as follows:

cosd(RA - B) = cosdo sin[RA(o) + A]

cosd cos(RA - B) = cosC cosdo cos[RA(o) + A] – sinC sind(o)

sind = cosC sind(o) + sinC cosd(o) cos[RA(o) + A]

A good scientific pocket calculator will do these operations quite easily. We have seen that there are other corrections than proper motion to be considered, such as nutation, and visual aberrations such as stellar parallax and refraction of light through the layers of gases in the atmosphere, but these are generally ignored. Allowing for their assumed effect may actually distort rather than improve the result by not knowing the exact value to consider, i.e., we have no way of knowing what was the density and clarity of the atmosphere on a given day in a given epoch. It is thus generally acceptable to ignore these effects, and assume that the plus and minus effects of nutation, aberration, parallax and refraction more or less cancel each other out.

Calculations made for me in 1987 by astronomer Dr John O’Byrne of the University of Sydney revealed that for the three stars in Orion’s Belt – Zeta (Al Nitak), Epsilon (Al Nilam) and Delta (Al Mintaka), no proper motion correction was considered necessary for the epoch 2500BC. Even by assuming a small value for proper motion effect, the correction needed would be about 65 arcseconds, which Dr O’Byrne felt would be ‘unrealistically large’. Short-term effects such as nutation and aberration were ignored for the reasons given above.

For the star Sirius, a proper-motion adjustment of –1.21 arcseconds per year was required for declination. Going in negative time to the epoch of the Pyramid Age, this meant an adjustment of about +1 degree 33 minutes for epochs around 2500BC to precession had to be made.

For the purpose of The Orion Mystery we have used the Skyglobe version 3.5. This program has the advantage of providing very quickly a visual effect of precession and readings on the screen which give the declination, right ascension, azimuth, altitude and magnitude of a given star for a range of epochs for plus or minus 13,000 years. We found Skyglobe to be a very well-made program and quite accurate for the work covered in The Orion Mystery. Its accuracy is also very acceptable for the discussions. The star’s co-ordinates, however, must be manually adjusted for proper motion. This was generally necessary for Sirius, whose proper motion is significant. We have, however, put the letter c. (circa) before dates signifying ‘approximate’. In principle, precessional calculations dictate that the farther away the epoch under consideration, the greater the margin of error is for proper motion adjustments. No doubt professional astronomers, with more powerful means at their disposal, will find some hairs to split in the data provided in The Orion Mystery. Any refinement would, of course, be welcome. It must always be remembered that observations, for Ancient Egyptians, were made with the unaided eye and with the help of very basic sighting instruments. Values below the 20 arcminute level are not easily perceived with the naked eye. It is widely accepted that the Ancient Egyptians used a sighting instrument they called Maskhet: this was a wooden staff with a slit at one end, the latter used as a collimator to aim at stars. They also used a simple plumb-line to measure the vertical.5 With such sighting rods and plumb-lines, the altitude of a star at the meridian, or its azimuth at rising, can be measured with a very good degree of accuracy, certainly within the 20 arcminute level. Could the Ancient Egyptians have measured precession?

We have seen that precessional shift for, say, Zeta Orionis, which was then some 15 degrees south of the celestial equator, varied as much as 28 arcminutes in one century – equal to the apparent size of the moon. It is generally accepted by Egyptologists that the formative years or religious ideas predate the Pyramid Age by at least 500 years, and possibly much more. Thus over 500 years of observations, a variation of declination for Zeta Orionis between 2950BC and 2450BC would have registered about 2 degrees 16 minutes. This gives a rate of about 27 minutes per century for the change in declination. Having noticed that precession provided a uniform motion ‘eastwards’ of the sun along the ecliptic of some 1 degree 23 minutes per century relative to a given constellation or star,6 it was not difficult for the Ancient Egyptians to deduce that a full cycle would take about 26,000 years to return to the same place relative to the constellation or star. Whether they worked out this value is debatable: what is more likely is that they realised that precession was a cycle (it has a start and an end), and then repeated the cycle for ever.

It is not known exactly when the Ancient Egyptians had developed a calendar, but it is generally accepted that this may have occurred well before the Pyramid Age.7 In the calendar system used by the Egyptians, the year was divided into 12 months each having three decans of 10 days, thus 30 days in the month and 36 decans in a year. This gives a year of 360 days to which 5 extra or epagomenal days were added; these were called ‘the 5 days upon the year’. It was during the 5 epagomenal days that the ‘neters’ or gods were born, who included Osiris and Isis. We thus have a situation where a 360-day year is linked to a 365-day year by the gods. The difference was, to them, caused by the birth of the gods who were said to be the four children of Nut (the sky goddess) Osiris, Isis, Seth and Nephthys with the fifth god being Horus, son of Osiris and Isis.8

In religious terminology, it was thus the gods who turned the 360-day year into a 365-day year. These gods, as we have seen, were of course the stars. In this respect we must consider the question whether the Ancient Egyptians divided the apparent circular motion of the sun’s ecliptic path around the earth into ‘degrees’ and, if so, was the division 360 units. It is a fact that the Egyptians divided the year into 12 months each of 30 days, giving the numerical total 360 days. They also divided the sky into 36 ‘decans’ each of 10 days, also giving the numerical total 360 days. This implies that they divided the ecliptic path of the sun into 360 units or ‘degrees’ to define a day. But the correct numerical division should be 365 units, which they also had computed by adding the 5 days upon the year.

If you find an error please notify us in the comments. Thank you!