I see that some graphs in the article need clarification. I will try to give a brief explanation here.

To visualize the change in the azimuths of the Old Kingdom pyramids with time, we can use two-dimentional graph (the azimuths of the pyramids along the Y-axis and the years along the X-axis). Azimuths are measurable values, therefore, they are known exactly, however years are calculated values, which are calculated on the basis of the chosen chronology. The durations of the kings' reigns for the Old Kingdom are calculated by historians more or less accurately (of course, some inaccuracies still exist) on the basis of King Lists and records on regnal years of individual rulers. Durations of reigns allow us, for example, to find out how many years separate the beginning of reign of Snofru and the beginning of the reign of Menkaure, etc.; however, it is impossible to find out the exact age of the Old Kingdom, by summing the durations of reigns from some known time point (for example, some astronomical synchronism) to the past, since there is great uncertainty in the duration of the reign of 9-10 dynasties:
- Beckerath, based on his own assumptions, estimates the duration to be 130 years, suggesting that about 100 years of them the Herakleopolitan dynasty ruled in parallel with the 11th Thebean dynasty, that is, the net chronological duration of the reign of 9-10 dynasties is estimated by him as 30 years. All currently accepted chronologies include this assumption.
- Breasted estimates the duration of the Herakleopolitan dynasty to be 280 years, based on Manetho's data. As I already mentioned, some modern chronologists consider Manetho's data correct.
Thus, the net chronological duration of reign of 9-10 dynasties is estimated as 30-280 years, giving an uncertainity of 250 years, which directly affects the age of the Old Kingdom.

Since the age of the Old Kingdom contains great uncertainty, we can use the durations of the kings' reigns to build a graph, avoiding exact dates for the X-axis, by summing the total duration from some conditional point (for example, the beginning of the 3rd dynasty as I do in the article). For the durations of the reigns, I use the data by Shaw's chronology, as the most famous at present (if we take another chronology, for example by Hornung, Krauss and Warburton, nothing will change much).

We get the following graph where the error margins in the azimuths of 3rd dynasty pyramids and pyramid of Teti characterize absense of exact data on azimuths but not the inaccuracies in orientation of these buildings:



I want to note that all data on azimuths are taken from reliable sources, the links to which are given in the article for each value separately. I did not find any other accurate data on the azimuths of the pyramids of the Old Kingdom.

The data on the graph can be divided into visually identifiable groups (for example, a group indicated in blue. Spence's work is devoted to the analysis of only this group).
For each of the groups, we can draw an approximating line, which makes it possible to assess the trend:



Interestingly, all 4 approximating straight lines have almost the same angle of inclination relative to the X-axis (approximately y = 0.33x).

Spence suggested that the observed gradient of the approximating line for the blue group is caused by the precessional shift of the reference pair of stars.
Obviously, if observations are made in different positions of the sky, then the stars will shift in different ways due to the precession:
- from west to east;
- from east to west;
- will not change the azimuth for a long time.
The rate of change of the azimuth for stellar objects in the circumpolar region ranges from -40'/century (corresponds to the moving of the object from east to west; the line on the graph for such an object will directed (roughly) from the upper left corner to the lower right corner) to +40'/century (corresponds to the moving of the object from west to east; the line will directed (roughly) from the lower left to the upper right corner) and depends on the position of the sky during observations.

Spence was able to find such a pair of stars (Mitsar-Kochab) for which the rate of precessional shift (31'/century) was very close to the rate of change of the azimuths of the 4th dynasty pyramids (in other words, if we draw the change in the azimuth of this star-pair with time on our graph, then the inclination of the resulting line relative to the X-axis will be approximately the same as for the blue approximating line).

Since the data on the azimuths on the graph fit well on the 4 approximating lines, this mean that 4 different stellar targets were used for 4 different groups of pyramids. And since the gradient of the approximating lines is almost the same, this means that the orientation procedure was always carried out in the same position of the sky (I assumed that this was the upper culmination of Meskhetiu, because firstly this idea has confirmations in iconography, and secondly, the direction and speed of the precessional shift of stars in this position of the sky is in good agreement with the graph on azimuths).

It is no longer difficult to find which stars correspond to which groups of pyramids, since only 9 stars are considered (7 stars of Meskhetiu, Thuban and 10 Dra) and the position of the sky is known. Having found the correspondence of the azimuths of stars and azimuths of the pyramids, we get the absolute dates of laying the foundations of the pyramids. This is the astronomical synchronism found, which in my opinion exhaustively resolves the question on the orientation of all the measured pyramids of the Old Kingdom.





Edited 9 time(s). Last edit at 02/11/2020 08:13AM by keeperzz.