How long for a complete solar system cycle
Considering the solar system as a closed system, how long until all the planets pass twice by the same position relative to each other? I would say that you'd have to get the length of the year in each planet and find the lowest number that can be divided by them, is that correct or are there more things to take into account?
Furthermore, can one do this also considering the moons of the planets?
The planets should be in "exactly" the same position after at most t = Π Pᵢ years, where Pᵢ is the period (in years) of the i'th planet, and Π denotes the product. It's only inexact dues to very slight disturbances of moons and other smaller celestials bodies. I get 1,108,001 years and 5 months.
By "position relative", do you mean relative to the Sun or to the distant stars? For example, Mercury's solar orbit "laps" Venus's solar orbit once every ~144.5 days: Mercury completes ~1.64 orbits in that time and Venus completes ~0.64 orbits in that time, so they are in roughly the same position with respect to each other, using the Sun as a central point. However, they are 0.64*360 degrees or ~230 degrees away from where they last lined up in relation to the distant stars.
It depends how precisely you mean the same position.
Take 2 since my previous attempt was ... optimistic.
In the late 1970's, the outer planets were moving into a configuration that made a Planetary Grand Tour much cheaper and quicker. They will be in a similar configuration in about 175 years.
But when will the planets be in the exact configuration? Never, well, not before the Sun turns into a red giant and swallows the inner planets.
But from time to time the planets will all be on the same side of the Sun. If the planets were all in a line, then after 495 years they would all be on the same side of the Sun, then again at 500 years, 522 years, 523 years (these last two are, of course 88 days apart, the period of Mercury), 995 years, 997 years, 1006, 1183 years, etc.
Considering the solar system as a closed system, how long until all the planets pass twice by the same position relative to each other?
If that means when will the bodies in the solar system simultaneously repeat their positions from some time long ago, the answer almost certainly is never.
The only way to get a repeated position amongst a pair of planets is if the ratio of their orbital periods is a rational number, and if perturbations are also somehow related rationally. The only way to get a repeated position amongst all of the planets is for this to be the case amongst all pairs of planets.
The planets migrated from where they initially formed to their present positions. That the current orbits would somehow be representable exactly in terms of the rationals would be quite the long shot. It's a space of measure zero.
By assuming the orbital periods of planets around sun and treating the solar system as closed and considering only planets Mercury,Venus Mars, Jupiter, Saturn , the least common multiple of the periods may give an idea of repeatability of particular position of planets around sun.The phases of moon is not considered here.The LCM shows the repeatability of a configuration is about
1,87,500 solar years .The planetary periods are corrected to near integer for easy estimate of the order
Hello, This is an older question, The LCM isn't the right method here, since the length of time of an orbit is not an integer. You should round, because there is no justificaiton in rounding to one day, one month, one hour... The choice would give immensely different values. Try looking at some newer questions, or older questions that don't have answers yet! (minor point, in English we don't divide numbers in crores, so the way of putting commas in numbers is different)
If you found, you will need to make clear how and where. There is no natural period to orbital periods to make it clear without express worrying