How frequently do these four planets line up with each other?

Question

Earth, Jupiter, Saturn, and Uranus all revolve around the sun. Earth takes 1 year, Jupiter 12 years, Saturn 30 years, and Uranus 84 years.

Answer 1

The planets orbiting our Sun are roughly in the same plane, so it would not be unusual for them to line up from the perspective of the Sun, at times. However the planets do not all log in a single orbit of the Sun in a set amount of time. The planets far from the Sun are orbiting in a much longer path and are moving more slowly on their path. Astronomers spend many dedicated hours working out relationships of celestial bodies so that the eclipse, Moon phase, tides, and other seasonal benchmarks are well known typically way in advnace of the event. Using the orbital periods given, we can calculate how often we will see (more or less) these planets line up in order from the Sun.

In order to determine when four planets will align, We must calculate the synodic period

(P_sync = 1/ (1/T₁ - 1/T₂)) for each pair, where T₁ is the faster (closer to the Sun) and T₂ is the slower. After finding the synodic period for each, the least common multiple of all the periods of the planets will be when they line up.

In our problem, the synodic period is given, 1,12,30, and 84 years. The least common multiple for these periods is 2²X 3 X 5 X 7= 420. So these four planets would appear to be "aligned" with reference to the Sun every 420 years. The actual position in space would follow a curve, not a straight line.