Inequalities

• Period of inequality
• Inequality Jupiter-Saturn
• Rotation of the big trigon
• Great inequality and Babylonian period
• Meton's system

Period of inequality

Two bodies P and Q repeat their positions (e.g. in conjunction at the same place), if q periods P is equal to p periods Q, so if:
q*P = p*Q, resp. P/Q=p/q,
where p,q are whole numbers.
In the Solar system there exist several such pairs, which satisfy the relation approximately. But hardly one pair satisfies the relation exactly.

Let
q/Q -p/P = 1/I.

period I is called period of inequality (or inequality period, inequality):
I = (Q/q,P/p) = P*Q/(q*P-p*Q).
Usually I is on order of greater then P and Q (I>>P, I>>Q).

The place, where planets repeat their positions move with period I.
Therefore, inequality I is orbital period.

Inequality Jupiter-Saturn

Value of inequality Jupiter-Saturn (so called "great inequality", Laplacean period, ...) is not known with a good precision. It is assumed, the period is "about 900 years" (840-960 years?).

From Bretagnon data (J=11.861983 let, S=29.457158) we have:
I = (J/2,S/5) = -883.3 years.

	S/1	S/2	S/3	S/4	S/5	S/6	S/7	S/8
J/1	19.859	60.947	57.013	19.422	11.705	8.376	6.522	5.340
J/2	7.426	9.929	14.978	30.474	883.27	28.507	14.487	9.711
J/3	4.567	5.405	6.620	8.538	12.024	20.316	65.464	53.556
J/4	3.298	3.713	4.249	4.965	5.971	7.489	10.042	15.237
J/5	2.580	2.828	3.128	3.500	3.972	4.591	5.438	6.670
J/6	2.119	2.284	2.475	2.703	2.976	3.310	3.729	4.269

According to Ptolemaio values (J=11.862923, S=29.465040) is:
I = (J/2,S/5) = -909.0 years.

Rotation of the big trigon

Conjunction of planet J and S appear on an average every 19.859 years. During this time Jupiter get approximately:
(J,S)/J *360° = 1.67416*360 = 360+242.698°. And Saturn
(J,S)/S *360° = 0.67416*360 = 242.698°.

Conjunction places make equilateral triangle, so called "big trigon". During 19.859 years trigon move by
((J,S)/S - 2/3)*360° = 242.698-240 = 2.698°.

After 120°/2.698 * 19.859 y, i.e. c. 900 years (great inequality) the second apex appear at the starting point, and after c. 1800 years the third apex. The whole triangle returns to its original position after c. 2700 years.

If there was no rotation, conjunction line would be oriented in the same direction every 3*(J,S). So, also after 42*(J,S), 45*(J,S) and 48*(J,S). During this period (c. 900 years) trigon takes approximately 120° forward. Therefore conjunction line is oriented in the same direction every c. 43, 46, and 49 conjunctions.

43*(J,S) = 853.9 let
46*(J,S) = 913.5 let
49*(J,S) = 973.1 let

Big trigon seen from the Earth

Earth swings with a period P, c. 25500-26000 years. Therefore motion of planet appear to be out of focus. Corresponding "distorted" periods are called tropical periods. Let P=25750 years. Then
J' = (11.861983, 25750) = 11.85652 y and
S' = (29.457158, 25750) = 29.42350 y.

During 19.859 years trigon move by
((J',S')/S' - 2/3)*360° = 242.976°-240° = 2.976°.

After 120°/2.976 * 19.859 y, i.e. c. 800 years (great inequality seen from the Earth) the second apex appear at the starting point, and after c. 1600 years the third apex. The whole triangle returns to its original position after c. 2400 years.

Seen from the Earth tropical periods seem to be the only true and correct periods. But we should be careful while computing derived periods. A slight difference of tropical and sidereal period causes in our example a quite different results (2400 y vs. 2700 y, see above).

Great inequality and Babylonian period

Let us assume
I= 2*B = 72*J = 29*S' = 43*(J,S')
Then from J=11.861983 we have:
I=854.06 y, B=427.03 y, S' = 29.450441 y
Synodic period (J,S') = 19.861925 y.

If great inequality J-S had value just I= 2*B= 854.06 let, than derived periods would differ from the Bretagnon periods by ratio:
S/S'=29.457158/29.450441 = 1.000228; (J,S')/(J,S)=19.861925/19.8588709= 1.000154.

See Babylonian period.

Meton's system

In the table above period of beats (J/2,S/5) is highlighted. Now, let us note period (J,S/3).

	S/1	S/2	S/3	S/4	S/5	S/6	S/7	S/8
J/1	19.859	60.947	57.013	19.422	11.705	8.376	6.522	5.340
J/2	7.426	9.929	14.978	30.474	883.27	28.507	14.487	9.711

Meton's, Exeligmos's a Callippos's cycle of lunar phases are multiples of 19-year period.

For period P=(J,S/3)=57.013 years it holds: (2*P,[J,S]) = (J,S).
It is treble of 19.0043 years and third of 171.039 years (i.e. (U,N)?; [U,N]= 111.29 y= 2*55.65 y).
Also it holds approximately: P= (5*(V,E), Ln/2)= (8.0 y, 18.61 y/2).