## Tuesday, January 11, 2022

### Planetary Sidereal & Synodic Period Computations In Astronomy

The sidereal period  in astronomy is the time interval referencing a planet's revolution about the Sun,  reckoned against the fixed stars. We photograph a planet at point 'x' in its orbit, and also with the stars determining a specific background format, then again when it reaches the same point x, relative to that background.

Of particular interest also in these computations are what we call the 'mean motions'. Let P1 and P2 be the sidereal periods of revolution of two planets around the Sun, then the mean motions are defined:

n1 = 360 deg/ P1

n2 = 360 deg/P2

where we assume for example, that n1 > n2

Of  additional use here are the configurations for superior and inferior planets as shown in the diagram below:

As indicated by inspection of Fig. 1(a), a superior planet (say Mars)may appear to be 90 degrees away from the Sun in the sky. Therefore, a line from Earth to the Sun makes a right angle with the line extending from Earth (E) to the planet. A planet in this configuration is said to be in quadrature. (The planet rises or sets at noon or midnight).

If the superior planet's on the other side of the Sun from Earth, it's then in the same direction from Earth as the Sun and is said to be in conjunction. (Note: for the inferior planet, e.g. Venus in Fig. 1(b) the same direction implies inferior conjunction).

Another  aspect or configuration is referred to as elongation. This is just the angle formed between the Earth-planet direction and the Earth-Sun direction. In other words, this elongation is the angular distance from the Sun as seen from the Earth. In the case of the inferior planets (such as Venus and Mercury) one will have a maximum western elongation, and maximum eastern elongation, as shown.

Note also that a superior planet at conjunction has an elongation of 0 degrees, and ditto for the inferior planet at inferior conjunction. Meanwhile, the superior planet has an elongation of 180 degrees at opposition, and 90 degrees at quadrature. An inferior planet, as can be discerned from the geometry in 1(b), can never ever be at opposition. It can, however, be at superior conjunction, when it is on the opposite side of the Sun from Earth.

By inspection of Fig. 2, let the alignment Sp1p2 denote the positions of the Sun, Venus (p1)  and Earth (p2)  at a particular time or "epoch" in the parlance of celestial mechanics. Then, it is clear  by the same inspection that one synodic period S will have elapsed by the time the two are in the next alignment Sp1'p2'. So, during the time interval D t 1 (for Venus) its radius vector will have gained 360 degrees on the Earth's.

Now, since Sp1 gains on Sp2 by 0.61646 deg/day then it gains 360 degrees in a time
D t =  S =

360 deg/ (0.61646 deg/day) = 583.9 days

Which is exactly the synodic period (S) for Venus!

(Note that S usually is used in astronomy textbooks to denote the synodic period, but not to be confused in any way with the Sun (S) in diagram!)

Hence, in general :

S  =  360 deg/ (n1 - n2)

Or, by reference to the earlier formulations for the n1, n2 mean motions:

S  ( 360/P1 - 360/ P2) = 360 deg

And:

S  =    360 deg/ [360/P1 - 360/ P2]

Or:

1/ S = 1/P1 - 1/P2

What about computing the sidereal period of Venus from its synodic period?  This can also be done, e.g.

If the length of the Earth's year is 365¼ days, find the sidereal period of Venus:

1/S = 1/P1 - 1/P2

Venus is an inferior planet, so that Earth's period is inserted for P2.

Then:

1/(583.8 d) = 1/P1 - 1/(365¼ d)

Or:

1/P1 = [365¼ + 583.9 d]/ [583.9 x 365¼ ]

yielding P1 = 224.7 days

In most problem -solving sets the planet's period is sought when Earth's is known. The key to the solution lies in determining whether the unknown planet is an inferior one (e.g. interior to Earth's orbit) or is a superior one, e.g. exterior to the Earth's orbit. Thus there emerge two cases:

(a) The planet is inferior (like Venus), then P1 refers to the planet's sidereal period and P2 refers to Earth's.

(b) The planet is superior (e.g. Jupiter) then P1 refers to the Earth's sidereal period and P2 to the planet's.

Suggested Problems:

(1) What would be the sidereal period of an inferior planet that appeared at greatest western elongation exactly once a year?

(2) If the synodic period of Saturn is 1.03513 years, find its sidereal period.

(3) A planet's elongation is measured as 125 degrees. Is it an inferior or superior planet?

(4) If the sidereal period of Mercury is 88 days what is its synodic period?