BackWhat I have referred to on the previous page as the interval between successive oppositions is actually one example of what is called a "Synodic Period", the more general definition of which is the time it takes for a set of (but usually just three) celestial objects to regain a given configuration. For a planetary opposition (the correct name of which is actually a superior heliocentric conjunction!), the configuration in question is that the Sun, the Earth and the planet should all lie on a straight line (in that order). Another example of a synodic period is the interval between successive Full Moons: in this case the configuration is that the Sun, the Earth and the Moon should all lie on a straight line (or at least in the same plane at right angles to the Earth's orbit - if the configuration is truly a straight line a lunar eclipse will result). Note that the definition takes no account of which body is orbiting round which, merely their geometric orientation, and that the line/plane does not have to have the same position in space each time.
The basic difference between a Sidereal Period (the orbital period relative to "the distant stars") and a Synodic Period is that the former exists "in isolation", and is thus an absolute period, but the latter arises from the combined motion of two bodies relative to a third and so is an apparent period. A simple example would be a slowly rotating playground roundabout. If the roundabout is rotating once per minute from the point of view of a stationary observer, they will see a given spot come round every 60 seconds. If the observer now walks round the roundabout at a rate of once per three minutes they will see the given spot only once every 90 seconds, if they are walking in the same direction as the roundabout is rotating, or once every 45 seconds if walking in the opposite direction. These periodicities are apparent, as the absolute rotation rate clearly remains at once per minute, and only exist relative to the moving observer.
Finding a formula for the value of a synodic period is not entirely straightforward! Consider the case of two bodies orbiting at different distances from a third body. Simple orbital dynamics tells us that the orbital period of the body further out will be greater than that of the nearer-in one so, for ease of calculation, let's make them 10 years and 1 year respectively. Now - once the initial conjunction has been established, the faster-moving body will get back to the same position in space again after exactly one year. However, there will be no conjunction yet as the other body has also moved along its orbit: one-tenth of the way round, in fact. So, for the conjunction to be re-established the faster-moving body must also go round another tenth of an orbit. The second conjunction will thus occur one-and-a-tenth years after the first i.e. after 1.1 years.
While this is a reasonable "first-order" estimate, there is a problem - while the faster-moving body is traversing the extra tenth of an orbit the slower-moving body is still on the move. Consequently, when the faster-moving body has got to what seemed to be the correct position for the conjunction the slower-moving body will be one tenth of one tenth (i.e. one hundredth) of an orbit further on. The second conjunction will thus actually occur one-and-a-tenth-plus-a-hundredth years after the first i.e. after 1.11 years. But wait! The logic of this process can clearly keep going, with the next value being one plus a tenth plus a hundredth plus a thousandth i.e. 1.111 years, and so on for ever. Consequently, the faster-moving body never catches up with the slower-moving one and so there can be no second conjunction!!
This is clearly ridiculous and is known as Zeno's Paradox, after the Greek philosopher Zeno of Elea who first thought of it. The resolution of the Paradox lies in the fact that the orbiting bodies do not move in discrete steps but continuously. The faster-moving body will therefore inevitably overtake the slower-moving one, creating a further conjunction as it does so, before moving on round its orbit. However, while the above "thought experiment" is thus not an accurate model of the true physical situation, it does give us a strong clue as to how the instant of conjunction can be calculated. This does involve a bit of maths though!
The sum we were creating can be expressed as 1 plus one-tenth plus one-tenth squared plus one-tenth cubed .... etc. If we now represent the orbital period of the slower-moving body as P, instead of giving it an actual numerical value, the sum becomes 1 + (1/P) + (1/P)^2 + (1/P)^3 .... etc. (where the symbol ^ means "to the power"). It would seem impossible to find the result of this infinite sum as we'd have to keep adding smaller and smaller increments for ever - just like in Zeno's Paradox. However! There is a "mathematical trick" we can apply. Consider what happens if we multiply the entire sum by P: the result is P + (1) + (1/P) + (1/P)^2 ..... etc. If we now subtract the first sum from the second the "1" and almost all the terms in (1/P) cancel out, leaving us with just P together with a term "one over P to the power infinity", which is the last term of the first (infinite) series. As (1/P) is less than 1 this term is clearly zero (well, alright, "infinitesimally small" if you insist!), leaving us with just P. In other words, if we represent the value of the original infinite sum by S we now know that (P * S) - S = P. Re-arranging the left-hand side to S * (P - 1) and dividing both sides by (P - 1) we find that S = P / (P - 1), which is the final result we have been looking for. Applying this to our example, S = 10 / (10 - 1) which is "ten ninths" or 1.1111111 ..... etc. It can be seen that this is the value we would have got had we continued our original example to infinity, as the process was adding a "1" at the end of the value each time. QED, therefore!
This formula is clearly only correct if the orbital period of the faster-moving body is 1 year however, and so it must be generalised. If we let the orbital period of the faster-moving body be Q, the fraction of its own orbit that the slower-moving body will complete for each full orbit of the faster-moving one will now be Q/P rather than 1/P. The infinite sum thus becomes 1 + (Q/P) + (Q/P)^2 + (Q/P)^3 .... etc. The mathematical trick can be repeated if we multiply by P/Q rather than P, to give the result after simplification that S = (P/Q) / (P/Q - 1). However, this is the synodic period in terms of orbits, not years. To convert to years we must multiply by Q. In addition, we can simplify (P/Q - 1) to (P - Q)/Q. The end result is that S = (P * Q) / (P - Q), which clearly reduces to our earlier result if Q=1. Never forget though that P must always be the longer orbital period and Q the shorter. This is important when finding the synodic period for conjunctions of (faster-moving) planets closer to the Sun than the Earth: this is called an inferior heliocentric conjunction and in this case it is P which is 1, not Q.
By dint of a further bout of algebraic manipulation (which I shall leave as an exercise for the reader!), one can convert the above simple result into the more complicated form in which the synodic period is usually quoted: S = 1 / (1/Q - 1/P). This is more popular probably because the equivalent form 1/S = 1/Q - 1/P may be used not only to calculate a synodic period when the orbital periods are known, but (by re-arranging the formula) to calculate any third value when both of the other two are known [as we shall see later]. From a theoretical perspective, it also illustrates rather better than the "compact" form how the result is derived.
I used a basically mathematical argument to derive the compact form but an alternative argument leading to the "popular" form would be to say that if the faster-moving body traverses its orbit at X degrees per year and the slower-moving one at Y deg/yr, the faster-moving body is "pulling ahead" by (X - Y) deg/yr and will therefore overtake the slower-moving body, creating the next conjunction, in 360 / (X - Y) years. Now of course X = 360/Q and Y = 360/P and so S = 360 / (360/Q - 360/P). The factor of 360 can be taken out of the denominator and cancelled with that in the numerator, giving S = 1 / (1/Q - 1/P) which is the relation we require. It can be seen that this form, and more particularly the re-arranged 1/S form, retains the "difference of two quotients" structure illustrative of the derivation.
I said on the previous page that the fact that, for Saturn, 28 opposition periods is almost exactly 29yrs is not a coincidence but a purely mathematical result. Given the above calculations, we can see why this might be so and to what extent it is actually true.
We found that when the Earth is the faster-moving body the synodic period is equal to P/(P-1). It is therefore very likely that taking the nearest integer to the orbital period and dividing it by the integer which is one smaller will generate a value which is close to the true synodic period. How close it is will depend on how close the planet's orbital period is to an integer in the first place, and also on how large it is in absolute terms. This second condition arises because the result of dividing one large number by another does not change a lot when both numbers are varied by a small amount e.g. from a decimal value to the nearest integer, as in this case. This is not true for small numbers, however, as the same "slight change" (+/-0.5) is, proportionally, much greater. In the case of Saturn (orbital period 29.424yrs), 29/28 is equal to 1.0357 while the true synodic period is 1.0352 - pretty close, so despite the orbital period being far from an integer it is clearly "large enough" for the principle to work effectively. In the case of Jupiter (orbital period 11.862yrs), 12/11 is 1.0909 while the true synodic period is 1.0921 - also pretty close, as the period is near to being an integer despite not being large. Uranus (84/83) and Neptune (165/164) give even better results - both are exact to 4 decimal places, mainly because of the large numbers involved but also because their orbital periods are very (in the case of Uranus) and reasonably (in the case of Neptune) close to an integer. However, the result for Mars is poor as, despite being quite close to an integer, its orbital period is small - just 1.881yrs: the integer fraction (2/1) gives a result of 2.0000 while the true synodic period is 2.1351. Even so, this is still only a 6.3% error. Thus, while the "N/N-1" principle is indeed a mathematical consequence, how accurate an answer it gives is dependent on the individual circumstances.
The typical cases of a synodic period I quoted at the start of this page were the interval between successive oppositions of a planet and the interval between Full Moons. I have explored the former pretty thoroughly already, so let's try the Full Moon example.
In this case, the synodic period in question is the apparent orbital period of the Moon as seen by an observer on the Earth, and arises from the rotation of the Moon round the Earth and the orbit of the Earth round the Sun (both measured relative to the stars). Using the terminology established above, P is therefore equal to 365.25636 and Q to 27.32166 - the sidereal orbit periods of Earth and Moon respectively. The compact formula tells us that S = 9979.41045 / 337.93470, which equals 29.53059 days: the length of a Lunar Month.
A similar example concerns the length of a sidereal day. In this case we already know the synodic period - this is the solar day of exactly 24hrs (by definition) which arises from the rotation of the Earth and its orbit round the Sun. Expressing everything in solar days, P is therefore equal to 365.25636 again, S is equal to 1 and Q is what we are trying to find. The easiest way to do so is to use the alternative form of the "quotient" formula: 1/S = 1/Q - 1/P. Re-arranging, we find that 1/Q = 1/1 + 1/365.25636 = 1.00274 and so Q=0.99727. Multiplying by 24 to convert to "solar hours" gives a sidereal day of 23.93447hrs or 23hrs 56mins 4.1sec. Note that in this case the third "celestial body" (in addition to the Sun and the Earth) is actually the observer and the configuration that repeats each synodic period is that of local noon.