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Calculate Synodical Periods


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1. Synodical periods seen from Earth ... 1.1. Synodical periods between all planets

The sidereal period of a planet is the period in which the planet orbits once around the Sun, as seen from the Sun and measured relative to the stars. In other words: the period after which, as seen from the Sun, the planet returns to the same star in the sky.

A synodical period is, in general, a period after which some phenomenon repeats itself that depends on the motion of more than one thing. The synodical period of a planet is (if no second moving thing is mentioned) the period after which the phenomenons of the planet relative to the Sun (such as opposition, conjunction, or greatest elongation) repeat themselves, as seen from Earth.

1. Synodical periods seen from Earth

Given the sidereal period \( P_\text{sid} \) of a planet, measured in sidereal years, you can calculate the synodical period \( P_\text{syn} \), also measured in sidereal years, as follows:

\begin{equation} P_\text{syn} = \frac{1}{\left| \frac{1}{P_\text{sid}} - 1 \right|} = \frac{P_\text{sid}}{|P_\text{sid} - 1|} \end{equation}

Here is the formula for calculating the sidereal period of a superior planet from its synodical period:

\begin{equation} P_\text{sid} = \frac{1}{\frac{1}{P_\text{syn}} - 1} = \frac{P_\text{syn}}{P_\text{syn} - 1} \end{equation}

and here is the formula for an inferior planet:

\begin{equation} P_\text{sid} = \frac{1}{\frac{1}{P_\text{syn}} + 1} = \frac{P_\text{syn}}{P_\text{syn} + 1} \end{equation}

It was assumed here that the planet orbits around the Sun in the same direction as the Earth does; otherwise, the minus sign in equations 1 and 2 should be replaced with a plus sign, and the plus sign in equation 3 with a minus sign.

Here is a list with some values for the distance from the Sun (in AU), the sidereal and corresponding synodical periods (in sidereal years and in days), and the name of the planet for which those values hold (if such a planet exists):

Table 1: Synodical Periods Versus Distance

Distance Sidereal Synodical Planet
AU years days years days
0.39 0.24 88 0.32 116 Mercury
0.48 0.33 122 0.50 182
0.63 0.50 182 1.00 365
0.71 0.60 219 1.50 548
0.72 0.62 225 1.60 584 Venus
0.76 0.67 244 2.00 730
0.83 0.75 274 3.00 1096
1.00 1.00 365 Earth
1.31 1.50 548 3.00 1096
1.52 1.88 687 2.14 780 Mars
1.59 2.00 730 2.00 730
2.08 3.00 1096 1.50 548
5.20 11.87 4335 1.09 399 Jupiter
9.54 29.5 1.04 378 Saturn
19.22 84.3 1.012 370 Uranus
30.11 165.2 1.006 367 Neptune
30 1.006 367 Pluto

For Pluto, the shortest distance to the Sun is listed, which that planet is close to right now. For the other planets, the semimajor axis is listed.

Only objects whose orbit around the Sun has a semimajor axis of less than 0.63 AU have a synodical period of less than a year. Only objects whose semimajor axis lies between 0.83 and 1.31 AU have a synodical period of at least three years. The closer the orbit of an object is to that of Earth (as far as the semimajor axis is concerned), the longer the synodical period of that object is. For very distant objects, the synodical period is just a bit greater than one year.

1.1. Synodical periods between all planets

You can also calculate synodical periods for phenomenons relative to the Sun as seen from a planet other than the Earth. If the sidereal period of the first planet is \( P_\text{sid1} \) and of the second planet \( P_\text{sid2} \) (both measured in the same units, for example days or years), then the synodical period \( P_\text{syn} \) (measured in the same units again) of the first planet as seen from the second planet is equal to

\begin{equation} P_\text{syn} = \frac{1}{\left| \frac{1}{P_\text{sid1}} - \frac{1}{P_\text{sid2}} \right|} \end{equation}

Synodical periods are symmetrical with regards to the two moving planets: The synodical period of Mars as seen from Earth is 781 days, so the synodical period of Earth as seen from Mars is also 781 days.

Here is a table that shows the average mutual synodical periods of all planet pairs, in days if there is no decimal point in the number, and otherwise in years:

Table 2: Synodical Periods Between All Planets (I)

Me Ve Ea Ma Ju Sa Ur Ne Pl
Mercury 145 116 101 90 89 88 88 88
Venus 145 584 334 237 229 226 226 225
Earth 116 584 780 399 378 370 367 367
Mars 101 334 780 816 734 703 695 692
Jupiter 90 237 399 816 19.8 13.8 12.8 12.5
Saturn 89 229 378 734 19.8 45.5 36.0 33.5
Uranus 88 226 370 703 13.8 45.5 171.9 127.8
Neptune 88 226 367 695 12.8 36.0 171.9 498.2
Pluto 88 225 367 692 12.5 33.5 127.8 498.2

Here is a table that shows the synodical periods from small to large:

Table 3: Synodical Periods Between All Planets (II)

P1 P2 Syn
|
P1 P2 Syn
me pl 88.09 d
|
ea ju 399 d
me ne 88.13 d
|
ea ve 584 d
me ur 88.3 d
|
ma pl 692 d
me sa 88.7 d
|
ma ne 695 d
me ju 90 d
|
ma ur 703 d
me ma 101 d
|
ma sa 734 d
me ea 116 d
|
ma ea 780 d
me ve 145 d
|
ma ju 816 d
ve pl 225.2 d
|
ju pl 12.5 y
ve ne 225.5 d
|
ju ne 12.8 y
ve ur 226.3 d
|
ju ur 13.8 y
ve sa 229 d
|
ju sa 19.8 y
ve ju 237 d
|
sa pl 33.5 y
ve ma 334 d
|
sa ne 36.0 y
ea pl 366.7 d
|
sa ur 45.5 y
ea ne 367.5 d
|
ur pl 128 y
ea ur 370 d
|
ur ne 172 y
ea sa 378 d
|
ne pl 498 y

The mutual synodical periods of pairs of planets that do not include the Earth are still interesting for observations made from the Earth, because the period after which phenomenons involving two planets (such as their conjunction in the Earth's sky) repeat themselves is on average equal to their mutual synodical period. For example, Jupiter and Saturn are close together in the Earth's sky on average once every 19.8 years.



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Last updated: 2016-02-07