Glossary O

The change in the obliquity of the ecliptic over 20,000 years as plotted

by Laskar, J. (1986: 'Secular terms of classical planetary theories using

the results of general theory'. Astronomy and Astrophysics. 157, 59–70).

 Image: Wikimedia Commons.

Obliquity. More fully, the obliquity of the ecliptic refers to the angle between the ecliptic and the celestial equator. In modern terms this is the angle between the Earth's plane of rotation with the plane of its orbit around the Sun. Owing to gravitational effects in the solar system this varies slightly over long timescales in a periodic way. Currently it is 23.439° but it has been declining for the past 8000 years at a fairly steady rate. In the time of Hipparchos (c. 140 BCE), for example, modern calculations show that it was 23.712°.
     The early C6th BCE philosopher Anaximander is said to have first recognised that the path of the Sun lies obliquely to the plane of the Earth (Pliny the Elder, Natural History 2.8). This is possible (in a sense), even with his cyclindrical conception of the Earth and his system of rings for the celestial bodies, since it would just imply that the flat upper surface of the Earth is inclined to the plane of the ring that contains the Sun. In the C4th BCE, Aristotle's student, Eudemos, attributed its discovery to Oinopides who lived in the middle of the C5th BCE. This makes more sense since Oinopides probably grasped that the Earth was a sphere and was equipped to make some sort of measurement of the size of the angle.
     All the calculations on this website use the VCW (2011) model for the values of precession which is given by the formula (VCW equation 11):

ε = 84283.175915  −  0.4436568T  +  0.00000146 T² + 151×10⁻⁹ T³  +  Σ_ω

where ε is the value in seconds of arc, T is the time in Julian centuries (Epoch J2000.0) and Σ_ω is the summation of tabulated cosine and sine terms (VCW Table 4).

Octaeteris. Literally 'eight years', this is a period of 8 solar years at the end of which the same phases of the Moon recur on approximately the same day. It is less accurate than the Metonic Cycle of 19 years and therefore was devised earlier, probably by Kleostratos (late C6th BCE), for which there is evidence in the writings of the C3rd Roman writer Censorinus.
     The length of the octaeteris is 8 solar years and thus 2922 days, using 1 year = 365.25 days, the standard year length as determined by the C4th BCE. This coincides approximately with 99 lunations of 29.5 days each, since this period occupies 2920.5 days, just one and a half days short. In modern terms, we have 8 tropical years = 2921.9375 days, whereas 99 synodic months = 2923.5282 days, a discrepancy of 1.5907 days.
      The octaeteris was used to calculate the dates for the quadrennial Olympic Games. One period of four years was 50 lunations was followed by a period of 49 lunations. There is a dial for this on the Antikythera Mechanism. Interestingly, the octaeteris also coincides with 5 cycles of visibility of the planet Venus (2920 days), so that if Venus is near the Moon at the beginning of the cycle and in a certain phase then it is near it and in the same phase at the end of the cycle.