User:JimLowdermilk/Sandbox

The Sothic cycle is a specific example of two cycles of differing length interacting to cycle together, here called a tertiary cycle. This is mathematically defined by the formula 1/a + 1/b = 1/t or half the harmonic mean. In the case of the Sothic cycle the two cycles are a 365d Egyptian calendar year and a "Sothic" year. Other tertiary cycles occur with turn signals on two different cars, two pumps filling a swimming poll, or the hands on an analog clock passing each other. It is the time required for a faster car to get one lap ahead of a slower car traveling on a race track.

The Sothic year is the length of time for the star Sirius/Sothis to visually return to the same position in relation to the sun. Star years measured in this way vary due to precession, the movement of the earth's axis in relation to the sun. The length of time for a star to make a yearly path can be marked when it rises to a defined altitude above a local horizon at the time of sunrise. This altitude does not have to be the altitude of first possible visibility. Throughout the year the star will rise approximately four minutes earlier each successive sunrise. Eventually the star will return to its same relative location at sunrise. This length of time can be called an observational year.

Stars that reside close to the ecliptic or the ecliptic meridian will on average exhibit observational years close to the sidereal year of 365.2564d. The ecliptic and the meridian cut the sky into four quadrants. The axis of the earth wobbles around slowly moving the observer and changing the observation of the event. If the axis swings the observer closer to the event its observational year will be shortened. Likewise, the observational year can be lengthened when the axis swings away from the observer. This depends upon which quadrant of the sky the phenomenon is observed.

The Sothic year is remarkable because its average duration was exactly 365.25d at some point in the centuries well before the unification of Egypt. The slow rate of change from this value is also of note. If observations and records could have been maintained during predynastic times the Sothic rise would optimally return to the same calendar day after 1461 calendar years. This value would drop to about 1456 calendar years by the Middle Kingdom. The 1461 value could also be maintained if the date of the Sothic rise were artificially maintained by moving the feast in celebration of this event one day every fourth year instead of rarely adjusting it according to observation.

It has been noticed, and the Sothic cycle confirms, that Sirius does not move retrograde across the sky like other stars, a phenomenon widely known as the precession of the equinox. This gives Sirius the unusual characteristic of appearing to stay stable relative to the equinox and solstices, and for the same reason, the helical rising (or zenith) of Sirius does not slip through the calendar (at the precession rate of about one day per 71.6 years), as other stars do. This remarkable stability within the solar year may be one reason that the Egyptians used it as a basis for their calendar whereas no other star would have sufficed.

According to Jed Buchwald, professor of the history of science at Caltech, in his article Egyptian Stars Under Paris Skies "Sirius remains about the same distance from the equinoxes—and so from the solstices—throughout these many centuries, despite precession."

The lunisolar theory of precession requires that the earth wobble enough to lose one complete rotation on its axis and one revolution around the sun (relative to the fixed stars) per precession cycle. Modern astronomers now measure the rate of precession via radio telescopes fixed on distant quasars and a process known as Very Long Baseline Interferometry (VLBI) confirms the earth changes orientation to the stars at about 50.3 arc seconds p/y, equating to one complete precession of the equinox in about 25,700 years. Nonetheless, Sirius, due to its unusual characteristics, remains practically stationary making it the ideal marker for ancient Egyptian planning purposes.