## How often are the planets aligned

The following text was originally published in Heelal (Belgium), Vol. 40, No. 10, pages 270-272 (October 1995).

Sometimes the following question is posed to an astronomer: suppose that at some instant all nine planets (Mercury to Pluto) are exactly aligned with the Sun; after what time interval will they again be situated on a straight line?

More precisely stated: if all planets have the same heliocentric longitude, when will they again have the same heliocentric longitude (not necessarily the same as at the first alignment, of course). In this prospect, the heliocentric latitudes of the planets are not taken into account; only their longitudes are considered here.

Actually, I could restrict myself to the answer "never" and end this chapter here. But presumably the reader would want some more explanation.

Why never ? For the evident reason that even only three planets never can have the same heliocentric longitude. And when I say 'exactly', then I really mean exactly, to an accuracy of one arcsecond, and even to an accuracy of a thousandth of an arcsecond. You might call this an exaggerated form of hairsplitting, but such a precision is needed to state what is really meant.

When a planet catches another one in its revolution around the Sun, then both bodies have again the same heliocentric longitude. Let us suppose that both are then at longitude 123°57'49".08. At that very instant, a third planet never can have exactly that same longitude; its longitude will always differ by at least a bit from the value mentioned (and in most cases much more than 'a bit') — see the 'mathematical approach' further on.

Here we should point out that it occurs often enough that three planets are exactly aligned. This occurs, for instance, when as seen from the Earth there is a conjunction between Venus and Mars. At that instant Mars, Venus and the Earth are situated on a straight line, but this line does not pass through the Sun (see the left drawing of Figure 31. a). A situation such as that shown in the drawing at right in the figure (and it is this what we mean) never can occur because, as we said, three planets never can have exactly the same heliocentric longitude. Although such an alignment is indeed possible theoretically, it never takes place in actual practice because the probability of this occurrence is equal to zero.

If we still want to be accurate, then there is already a problem in the case of only two planets. Consider for instance the case of Jupiter and Saturn. During the period 1940-2020, these planets are five times in heliocentric conjunction, namely on the following dates; for each case, we mention the common heliocentric longitude of the two planets, referred to the mean equinox of the date:

We see that the time intervals are not equal. Between the conjunctions of 1981 and 2000 the interval is 19 years and 2 months, while those of 1940 and 1961 are separated by 20 years and 5 months. These differences are easily explained. The orbits of the planets are not exactly circular; they are ellipses, and a planet describes its elliptical orbit at a variable speed: it is fastest at perihelion, slowest at aphelion. So it is not surprising that the time differences between successive heliocentric conjunctions of two planets are not equal.

For this reason, it is not possible to give an accurate answer to the question "After what time are Jupiter and Saturn in heliocentric conjunction again?" except, of course, in a well-defined case. It is possible to calculate the exact date of the

1940 November 15

1961 April 16 1981 April 16 2000 June 22

2020 November 2

Jupiter-Saturn conjunction following that of A.D. 2020. But generally speaking only a mean value can be given.

To complicate the matter even more: the eccentricities of the planetary orbits slowly vary with time. The longitudes of their perihelia are also subject to variations. Presently, the eccentricity of the orbit of Jupiter is increasing, while that of Saturn decreases. The longitude of Saturn's perihelion increases faster than that of Jupiter by 21 arcminutes per century.

And that is not yet the end of the story! Planets do attract each other, so their motions are somewhat perturbed. For instance, there exists a so-called long-period inequality in the motions of Jupiter and Saturn, with a period of 883 years. During half this period, the speed of Jupiter in its orbit is a bit faster than its mean value, while on the contrary Saturn is somewhat slower than normally. During the other half of the 883-year period, the opposite takes place. It is a sort of periodic exchange of energy between the two giant planets: when one of them accelerates, the other is slowing down. Due to this long-period perturbation, Jupiter can be up to 20', and Saturn up to 49', ahead or behind in its orbit. There is a similar inequality of long period between the motions of Uranus and Neptune; here the period is 4233 years.

At this point, it will be clear to the reader why it is hopeless to give an accurate answer to some questions, such as the one posed at the beginning of this chapter.

### Simplifying the problem

Well, under these circumstances, let us simplify the problem by considering imaginary planets moving around the Sun on circular orbits with constant speeds. From what has been said above, it is evident that even only three planets never can have exactly the same heliocentric longitude. Let us repeat again that it could indeed happen theoretically, but that in practice the probability of the event is zero.

The reader now will reply: "When I ask after what time interval all planets arrive back again on a straight line with the Sun, then I do not mean exactly on a straight line". — Well, then this is a very different problem! But we should know precisely what the real question is.

So, let us suppose that it is asked after what time period all the planets are again inside a very narrow sector, for instance in a heliocentric sector of one degree.

The probability for this occurrence is no longer zero, although it remains a very rare event. As we have seen in the preceding chapter, between the years 0 and 4000 the three outermost giant planets Saturn, Uranus and Neptune come only seven times into a heliocentric sector of 10°. On 1306 September 4 the four giant planets were in a sector of 7°06\ their smallest one during the period 0-4000. That is still a much larger sector than 1°, and it concerns only four planets.

Let us now make a calculation using the mean speeds of the planets, and for a heliocentric sector of one degree. The revolution period of Mercury, referred to the mean equinox of the date, is 87.9684336 days, so that Mercury's mean daily motion is 4.092 3771 degrees. Similarly we find that the mean daily motion of Venus is 1.602 1687 degrees. The difference between the two values is 2.4902084 degrees per day. The mean time interval between two successive heliocentric conjunctions Mercury-Venus is obtained by dividing 360 (degrees) by 2.4902084; we obtain 114.57 days, or 0.3958 year.

The probability that a third planet lies, together with Mercury and Venus, in a sector of one degree is 1/360. Therefore, the three planets will be situated inside a sector of 1° every 0.3958 x 360 years.

For all nine planets, that is, Mercury-Venus + 7 planets, the mean frequency is therefore once every 0.3958 x 3607 years, or 3 x 1017 years.

This reasoning is not quite correct, however. Suppose that Mercury and Venus are in conjunction at longitude 50°00', and that at this instant Neptune is at longitude 51°01\ Then the sector delimited by these three planets is 1 °01or slightly larger than the preset limit of one degree. After approximately six hours, Mercury has overtaken the slow Neptune, and is now too at longitude 51 °01'. But meanwhile Venus is arrived at longitude 50°24', so that the minimum sector of the three planets is actually 0°37', not 1°01\

Nevertheless, the reasoning gives an approximate order of magnitude: a 'period' much longer than the age of our solar system!

Problem: Can the reader find a more accurate value for the period ? Use constant, mean values for the motions of the planets, and calculate the frequency of the occurrence of the nine planets in a heliocentric sector of one degree. What is the mean frequency for a sector of ten degrees ?

### Mathematical approach

Nevertheless, let us try to answer the question posed at the beginning of this chapter. Suppose that at a given instant all planets are exactly aligned with the Sun, so that they have the same heliocentric longitude. After what time will they be aligned again? To avoid the difficulties mentioned earlier, we will consider fictional planets having a constant speed and not subject to perturbations.

Because the new alignment should not occur necessarily at the same heliocentric longitude as the first one, we will work with the synodic periods of revolution, not with the sidereal periods. The synodic period of a planet is the (mean) interval of time between successive conjunctions of that planet and the Earth, as observed from the Sun. And for practical reasons we will not take Pluto into consideration. The synodic periods, rounded to the nearest integer number of days, are as follows:

 Mercury 116 days Venus

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