Online Astronomy eText: Orbital Motions
The Synodic Period of Revolution Link for sharing this page on Facebook
(also see Planetary Aspects)
(If you are math-phobic, refer to The Use of Mathematics In My Lecture Classes)
      As the planets move around the Sun they change their positions relative to each other. Since the inner planets, in smaller orbits, move faster around the Sun and have shorter orbital periods, they continually gain on the outer planets and regularly lap them. The time required to do this is referred to as the synodic period of revolution, as opposed to the sidereal period of revolution, which is the time required for a planet to move once around the Sun relative to the stars and is equal to the orbital period. (Note that this is similar to the difference between the sidereal period of rotation, which is the time required for a planet to turn once on its axis relative to the stars and is usually defined as its rotation period, and the synodic period of rotation, which is the time required for a planet to turn once on its axis relative to the Sun and is the length of its day.)

Planetary Aspects
      During a synodic period there are four times when each planet is said to have a certain aspect relative to another planet and the Sun. If as seen from one planet the other one is in the same direction as the Sun it is said to be at conjunction, or in conjunction with the Sun. If, on the other hand, a planet is in the opposite direction from the Sun it is said to be at opposition. If the angle between the Sun and the planet is 90 degrees, or a quarter of a circle, the planet is said to be at quadrature Eastern quadrature if the planet is 90 degrees to the east of the Sun, and Western quadrature if the planet is 90 degrees to the west of the Sun. The time required for a planet to go through a cycle of aspects is the same as the time it takes the inner planet to lap the outer one, which is the same as the synodic period of revolution of the two bodies. As a result the two topics (planetary aspects and synodic periods) are closely related; and the reader should refer to the page on aspects for illustrations of the concepts involved.

How To Calculate Synodic Periods Relative To The Earth
     Let's consider the situation as seen from the Earth, which has an orbital period of 365.256 days, and that of some other planet in our solar system, with orbital period P. Each day the Earth moves about 360/365.256 degrees to the East in its orbit, while the other planet moves 360/P degrees to the East. For inner planets such as Mercury and Venus, for which P is shorter than for the Earth, the other planet moves faster than the Earth, and gains

(360/P - 360/365.256) degrees per day

For the inner planet to gain a full lap or 360 degrees on the Earth would take

S (the synodic period) = 360 / (360/P - 360/365.256) days

If the orbital period P of the inner planet is much less than the Earth's orbital period, 360/P will be much larger than 360/365.256, and the Synodic Period S will be relatively close to the orbital period; so for Mercury, with an orbital period over four times shorter than ours, the Synodic Period is only a few weeks longer than the Orbital Period; but if the orbital period of the inner planet is closer to that of the Earth, then the two numbers in the divisor on the right side of the equation will be close together, and the Synodic Period will be very long. In fact, for a planet just barely inside the orbit of the Earth the Synodic Period approaches infinity. The situation is similar to two runners in a race. If one is much faster than the other, he can leave the slower one in the dust, and lap him in hardly more than one lap; but if they are nearly equally matched, it can take a very long time for the faster one to finally lap the slower.
     For an outer planet the mathematics shown above would be essentially the same, but since the Earth (being the inner planet) moves faster, the terms on the right side of the equation would be reversed:

S (the synodic period) = 360 / (360/365.256 - 360/P) days

As for the inner planets, if the inner planet (in this case the Earth) has an orbital period much less than that of the outer planet (which it does for all but Mars), the Synodic Period will be close to the orbital period of the inner planet (the Earth) and therefore not much more than a year, and the further out the outer planet is (and the slower it moves), the closer to the Synodic Period will be to a year. Whereas if the outer planet's orbital period is not much longer than that of the Earth, so that the Earth slowly gains on the outer planet, the Synodic Period will be much longer, and if there was a planet just outside the orbit of the Earth it would take a very long time for the Earth to lap it.

A Note About The Mathematical Formulas

The format I use above for the arithmetic used to calculate synodic periods was deliberately chosen, because it matches the way in which the calculations are shown in the table below. However, it does not match what the reader is likely to find in the usual mathematical discussions of synodic periods, because it could be simplified by canceling the 360's shown on the top and bottom of the terms on the right, and still further simplified by taking the reciprocal of each side of the equation. In addition the orbital period of the Earth, instead of being shown by a number, is usually expressed by the symbol E. Doing that makes the two equations look like this:

For inner planets, 1/S = 1/P - 1/E,

While for outer planets, 1/S = 1/E - 1/P,

thereby placing the larger term in front on the right side of the equation in both cases.

However, although this is more elegant it doesn't make the reasoning involved as obvious, hence my decision to use the more direct method of calculation above and in the table below, and only show the formal equations in this note.

The Synodic Periods of the Planets
     In the table below I have shown the orbital period of each planet, its average daily motion in degrees, the difference between that and the average daily motion of the Earth, and the resulting Synodic Period. As noted above, for Mercury the Synodic Period is relatively close to its own orbital period, but for Venus and Mars the synodic periods are considerably longer than their (and our) orbital periods. For the other outer planets the synodic periods are not much longer than a year, and as we go to more and more distant planets, the difference between their synodic period and our orbital period (a year) becomes very small.

Planet

Orbital Period
in Days
Avg Daily Motion
in Degrees
Daily Gain on
the Earth in Degrees
Synodic Period

Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
87.970
224.70
365.256
686.980
4332.59
10759.22
30685.4
60189
90465
4.0923
1.6021
0.985610
0.524033
0.083091
0.033460
0.011732
0.005981
0.003979
3.1067
0.6165

0.461577
0.902519
0.952150
0.973878
0.979629
0.981631
115.88
583.94

779.935
398.884
378.092
369.656
367.486
366.737
    As shown by the table, Mercury takes less than a month to lap the Earth after completing one trip around the Sun (though that still represents nearly a third of an extra orbit), while Venus takes almost two years to lap us, and the Earth takes more than two years to lap Mars. However, it only takes about 13 months for the Earth to lap Jupiter (just about a month longer than once around the Sun), less than two weeks more than a year to lap Saturn, less and less time to lap the other outer planets, and only a little over a day more than a year to lap Pluto. So as stated above, the closer the orbital period of the other planet is to ours, the longer it takes for one to lap the other, and the longer the synodic period is; whereas if the two orbital periods are very different, the synodic period is relatively close to the orbital period of the inner planet.

An Alternative Method of Calculating Synodic Periods
     Although conceptually simple, the method shown above requires many-digit accuracy for planets with very different orbital periods, to show the relatively small difference betwen the orbital period and the synodic period. But for planets with very different orbital periods we can use a simplified calculation similar to that used to determine the difference between the length of the day and of the rotation period of a planet, when the rotation period is much less than the orbital period. For instance, Jupiter takes almost 12 years to go around the Sun. That means that in one year, it only moves 1/12th of the way around the Sun, and the Earth only has to move about an extra 1/12th of an orbit to lap it. During that time Jupiter does move a little further, but unless very accurate calculations are required, it is obvious that it would only take about one extra month for the Earth to catch up with Jupiter and complete one synodic period (as shown in the table above). Similarly, Pluto's orbit is about 250 years, so in one year it moves less than a degree and a half around the Sun, and when the Earth has completed one orbit, it takes less than a day and a half to catch up and lap Pluto. So for planets in the far outer solar system, this "rough and dirty" method is almost as accurate and far less demanding than the more accurate calculations shown above.

A Diagram Showing The Relationship of Synodic Periods To Orbital Size
Diagram showing the change in synodic periods for planets with different orbital sizes
A diagram showing the relationship between orbital size and synodic periods
On the left, synodic periods for inner planets expanded left to right to show greater detail; on the right, synodic periods for outer planets compressed right to left to cover a larger range of distance from the Sun. For very small orbits the synodic period is not much more than the orbital period, as the planet moves around the Sun so fast that the Earth hasn't had time to move very far; but as the orbital size increases the orbital period also increases, and the inner planet has to go further to catch up with the greater motion of the Earth during the longer period of time. For planets with nearly the same orbital period as the Earth the synodic period would approach infinity. For planets with very large orbits the time required for the Earth to catch up with the other planet isn't much longer than a year (even at 8 AUs it is only a couple of weeks or so longer than the year it takes the Earth to go once around its orbit), but as the outer planet's orbit gets smaller it moves faster and it is harder for the Earth to catch up with it, so the synodic period rapidly increases; and if the size of the orbit is almost the size of the Earth's orbit, the synodic period approaches infinity in the same way as for inner planets (although because of the compression of the right side of the diagram, the nearly vertical part of the curve looks steeper).

Application To Other Planets
     Although all the calculations above were based on a comparison of the Earth to other planets, we could calculate the synodic period of Venus as seen from Mars (and vice-versa, as the two values would be exactly the same), or of Saturn as seen from Jupiter (and vice-versa). All we'd have to do is replace the orbital period of the Earth by that of one of the two planets for which we wanted to calculate the synodic period, and go through the arithmetic. So, using the numbers in the table above, Jupiter would gain 0.083091 - 0.033460 degrees per day on Saturn (= 0.049631 degrees per day), and would therefore take 360/0.049631 = 7253.53 days, or 19.86 years to lap Saturn, making the synodic period of each planet (relative to the other) almost 20 years.
     Some Notes About "Commensurability": It is interesting to note that the orbital period of Saturn is almost exactly 2 1/2 times the orbital period of Jupiter and 1 1/2 times the two planets' synodic period, so that in 60 years Saturn goes around the Sun twice, Jupiter five times, and Jupiter laps Saturn three times. This "small whole number" (5/2) relationship between the two planets' orbital periods is referred to as "commensurability", and indicates that the orbital periods of the two planets are "locked together", so that their relative orbital positions should remain fixed for all time. Commensurability relationships occur throughout the solar system, and are an important result of the gravitational interactions of the planets with each other, because although the Sun plays the primary role in establishing the planets' basic orbital motions, the way in which their orbits are spaced can be strongly affected by their mutual gravitational interactions. As a further example, in the outer solar system there are a large number of objects whose orbital periods are controlled by Neptune, of which the most well-known is Pluto. Pluto's gravity has a negligible effect on Neptune's orbit, but Neptune's gravity has a relatively strong effect on Pluto's orbit, so that Pluto's orbital period averages exactly 3/2 the length of Neptune's orbital period, and whenever Neptune laps Pluto, Pluto is nearly as far from the Sun as possible and about 1.5 billion miles away from Neptune, ensuring that there is no chance of ever being very close together, despite the fact that at perihelion Pluto is actually closer to the Sun than Neptune ever gets. The host of smaller bodies in the outer solar system that are also locked to the orbit of Neptune have orbital period ratios of 4/3, 5/2 or 2/1 relative to Neptune, and since Pluto was the first of them to be discovered and is by far the largest, such objects are called Plutinos.

Caveats For Orbits That Are Relatively Eccentric
     All the calculations above assume that the planetary orbits are circular, so that the motion around the Sun each day is a constant; but none of them are truly circular, and some are very non-circular, or noticeably elliptical. This is particularly important in comparing the motion of Mercury to that of the Earth. Whenever Mercury laps the Earth it has two parts to its motion: (1) its orbital period, which is always the same (almost exactly 88 days), plus (2) the extra time required to catch up with the motion the Earth had during that orbital period. The motion of the Earth varies by only a few percent, but the motion of Mercury varies by nearly 50%, being much faster than usual at perihelion, when it is closest to the Sun and moving faster, and much slower at aphelion, when it is furthest from the Sun and moving slower. If the portion of Mercury's orbit that it has to move through to catch up with the Earth is near aphelion, it will take longer than usual to catch up with us, and that particular synodic period will be a week or two longer than the average synodic period shown in the table above; whereas if the portion of Mercury's orbit that it has to move through to catch up with the Earth is near aphelion, it will take less time than usual to catch up with us, and that particular synodic period will be a week or two shorter than the average synodic period. So there can be a difference of several weeks in how long it can take Mercury to complete a synodic period (and move through one cycle of its planetary aspects), depending upon whether it happens to be near perihelion, near aphelion, or somewhere in between during the extra month (or so) that it takes to lap us.

Note to self: Add diagram from Mercury Project part 2c to show the variation in its synodic period