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When will the 9 planets stand in one straight line ?

- stephen chiu (08.03.2001)

The answer is: they have never been precisely aligned, and they will not do so in the future! We are referring to the true planetary alignment, not the grouping of planets in a region of sky. If the Nine Planets really align, the inner and outer planets will locate at both sides of the Earth, and will not appear at the same region of sky.

Before continuing, let us play an IQ quiz. If a clock is started from 0:00, when will the hour hand, minute hand and second overlap each other?

Let the second hand and minute hand meet every t minutes,
then,

1 / t = 1 / period of the second hand - 1 / period of the minute hand
= 1 / 1 - 1 / 60
= 59 / 60
t = 60 / 59 1.017 min

Or it can be written as T (Period of overlap) = j t (j must be integers such as 1,2,3K)

So how about the minute hand and hour hand?

1 / t'   = 1 / period of the minute hand - 1 / period of the hour hand
= 1 / 60 - 1 / (1260)
= 11 / 720
t' =  720 / 1165.455 min

Or written as T' = k t ( k must be integers such as 1,2,3K)

If all of them overlap, T = T' 

j t   = k t'
j / k = t' t 
= 720 / 11 59 / 60
= 708 / 11

This happens at the 708th overlap between the second hand and minute hand, or the 11th overlap between the minute hand and hour hand, which is 12 hours later. (Actually it is easy to see this must be one of the answers. But it is a bit surprising that they simply do not overlap at other times.)

Let us try to solve the planetary alignment problem by the same method.
Here we only demonstrate the alignment among the Earth and the farthest three planets, the rest is left for you as an exercise.
We start with the Earth, Uranus and Neptune:

j Synodic period between the Earth and Uranus = k Synodic period between the Earth and Neptune
j 369.66 = k 367.49
j / k = 36749 / 36966

Then we consider the Earth, Neptune, and Pluto:

k Synodic period between the Earth and Neptune = m Synodic period between the Earth and Pluto
k 367.49 = m 366.72
k / m = 36672 / 36749

As the ks in both equations above must be equal, k will be the LCM of 36966 and 36672, i.e. 225,936,192. It means the 225,936,192th oppositions between the Earth and Neptune. For these four planets, it will take 227 million years for them to align. For the Nine Planets, the time for them to align will be even longer than the age of the Solar System!

[Synodic Period of Planets]

Mercury 115.88 days
Venus 583.92 days
Mars 779.94 days
Jupiter 398.88 days
Saturn 378.09 days
Uranus 396.66 days
Neptune  367.49 days
Pluto 366.72 days

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Here we assume planets move in circular motion at constant speed, and the synodic periods given above is just an approximate value. Actually the planetary motion is full of small variants, and the synodic period values are irrational number. The quotient of two irrational numbers cannot be written as a fraction. Therefore, the Nine Planets simply do not align.

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LAU Kai-ip
Asst. Curator, Hong Kong Space Museum
12.11.2001