Conjunctions, inequalities

Synodic period

Periods of motion

Basic orbital periods of bodies are given with regard to stars.  These are so called sidereal periods.

In the next computations we will use mean periods according to VSOP87 (Bretagnon, Variations Seculaires des Orbites Planetaires).

Conjunction

Conjunction is a close apparition (join, alignment) of two or more bodies. For simplicity we will consider only alignment of bodies in plane perpendicular to approximate plane of planetary motion (i.e. conjunction in longitude). Exact alignments in one line are sporadic and they are studied in connection with other phenomena (eclipses of Sun and Moon, transits of Mercury and Venus through solar disc…)

If bodies are observed from the Sun, we speak about heliocentric conjunction (conjunction with the Sun).
E.g. if bodies ordered Sun-Venus-Earth-Mars are in a straight line, we say Venus-Earth, Earth-Mars, Venus-Mars and Venus-Earth-Mars are in conjunction (seen from the Sun).

Practical astronomy understands by conjunction usually geocentric conjunction. If Sun-Venus-Earth-Mars are aligned, it is said, Venus is in conjunction and Mars in opposition (with the Sun, seen from the Earth).

Synodic period of two bodies

Mean period, with which (helio)centric conjunctions of two bodies repeat, is called synodic period.
Synodic (relative) period of two periods P,Q is period:

We designate synodic period with round brackets ().

For any periods A,B and constant k it holds:

·         (A,B) = -(B,A)

·         (k∙A,k∙B) = k∙(A,B)

·         ((A,M),(B,M)) = (A,B).

In practical astronomy it is implied, that one of the period is orbital period of the Earth. E.g. synodic period of Jupiter is determined to be c. 399 days. It is period of Jupiter with regard to Earth: (E,J) = (365.256,4332.59) = 398.9 days.
Se synodickou periodou se postupně rozvírá (a pak zase přivírá) úhel P-S-Q; S je centrum, bod (těžiště soustavy) okolo kterého tělesa P,Q obíhají.
With synodic period the angle P-S-Q gradually opens (and then closes); S is the point (centre), around which motion of bodies P,Q happens.

Pair synodic periods of planets

Synodic day

Synodic day is rotational period Tr measured with regard to orbital period T, i.e. synodic period (Tr,T).

In case of  planet of the Slunce we speak about “solar day”,  e.g. if Earth as rotational period  Tr =1 stellar (sidereal) day, is its solar day equal to (1.0, 365.256) = 1.0027 stellar days. We divide solar day to 24 hours.

Reversing of polarity

Let us have a number of small periods Pi: (P0, P1,..., Pn), and greater period Q. If periods Pi have common multiple P, which is approximately equal to Q, it could appear, all the system has period P.
But evidently it is not true. Deviations of periods (P-Q) will gradually accumulate; during synodic period (P,Q).

Such accumulation of  deviations can appear as transformation of cycle P, as change of its polarity.

E.g. periods 3,4 and 13 years makes common multiple c. 12-13 years.  It can be well approximated by period P=12 years. But after a longer time we will register beats with period c. (12,13)=156 years.

In solar system periods (U,N)= 171.44 years and 9∙(J,S)=178.730 differ by more than 7 years. Though they are usually covered by so called 180-years period. Deviations of periods oscilates with period c. (9∙(J,S),(U,N))=(178.7, 171.4)=4200 years.

I.Charvatova has found period c. 4400 years (resp. its aliquots 2200 years and 1100 years) in motion of the Solar system gravity centre. According to I.Charvatova is the basic interval made of c. 55 conjunctions (J,S), i.e. 1100 years. Observed deviations of motional characteristics are in turns positive and negative (intervals in years): (-2200,-1100) +; (-1100,0) -; (0,+1100) +; (+1100,+2200) -.

Distortion

Mentioned definition of conjunctions is not physically correct, it is only geometrical construction. We neglect finite speed of light as well as time-space relations and so on.

E.g. to compare of conjunctions V-E (period 1.6 years) and conjunctions J-N (period 12.8 years),  we have to consider space distance about 29 AU.  (Light run this distance c. 4.3∙10¹² m during c. 14300 seconds, i.e. c. 4 hours…).

Inequalities

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, i.e. P/Q=p/q, where p,q are whole numbers.

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

Period I is called period of inequality (or inequality period, inequality):

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.

Big trigon

Conjunction of planets J and S appears 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 approximately (J,S)/S *360° = 0.67416*360 = 242.698°.

Because 240°=(2/3)*360°, conjunction places make equilateral triangle, so called "big trigon". During 19.859 years this trigon moves 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 years, 46∙(J,S) = 913.5 years, 49∙(J,S) = 973.1 years

Change of planetary speed 

J.H.Lambert has noted, that mean speed of Saturn increased compared to speed from Galileo’s measurements. This deviation was later make clear by Laplace with help of effect of small denominators.

Great inequality

Value of inequality Jupiter-Saturn (so called "great inequality", long-period inequality, Laplace's 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 years, S=29.457158) we have: I = (J/2,S/5) = -883.3 years.

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

View from the Earth

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

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 (i.e. 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).

(Some values in the table, e.g. 3.4994, 14.9870, come out near to integer fraction of  terrestrial year. Value (J/3,S/4)=8.5412 let is equal to ecliptic year Y).

Synchronization of perihelia

Mayan calendar round (52 years) and computation of tuns (1-18) makes period 9 Aztec centuries i.e. 9*102 = 936 years.

With the same period perihelia of Jupiter and Saturn synchronize. Value (J_a/2,S_a/5) computed from anomalistic period J_a,S_a  makes c. 938-939 years.

Laplace's cyklus

Planetary interactions