I wouldn't know how to mathematically solve it *formally*, but I think you can kind of do a little educated guess work.

Let's say planet x has an orbital period of 1 year, planet y of 3 years and planet z of 5 years.

The synodic period of planet x and y is 1.5 years, according to the wikipedia link aphh gave:

T(syn) = 1/|1/1 - 1/3| = 1.5

Now, every 1.5 years when the planets line up, they displace an angle of 0.5 orbits (planet x moves a distance of 1.5 orbits, planet y a distance of 0.5 orbits, common decimal so -> displacement of 0.5 orbits).

Every 1.5 years, planet z displaces 0.3 orbits.

Now, when all three planets are aligned, they must have moved some distance each, where each distance is separated by an integer number of orbits:

0.5n - 0.3n = p

where n,p are part of I

(n is amount of 1.5 year iterations, p is the integer difference in orbits)

And so we can informally solve for when n is the lowest it can be:

0.2n = p -> n=5

When n is 5, that means 5 1.5 year iterations have passed or 7.5 years, and so every 7.5 years, planet x, y and z (with periods 1, 3 and 5 years respectively) align, given they were aligned in the past (I'm not sure if it is a certainty they must be aligned given all possible starting points).

This can be verified by checking the angle each has gone through in 7.5 years - planet x at 7.5 orbits, planet y at 2.5 orbits and planet z at 1.5 orbits. All have displaced 0.5 orbits, and so are aligned.

Using Earth, Mars and Jupiter, relative to Earth period:

Earth: 1.0000 years

Mars: 1.8808 years

Jupiter: 11.8590 years

Earth and Mars (E/M) have a synodic period:

T(syn[E/M]) = 1/|1/1 - 1/1.8808| = 2.1353 years

Every one of these periods, the E/M alignment goes through a displacement:

distance = 2.1353 / T(earth) = 2.1353 -> displacement of 0.1353 orbits

Every 2.1353 years, Jupiter goes through a displacement:

distance = 2.1353 / T(Jupiter) = 0.1801 orbits

So 0.1353 orbtis and 0.1801 orbits must align at some point:

0.1801n - 0.1353n = x; n,x part of I

0.04473n = x

n = 313

You'll notice that 0.04473*313 doesn't exactly yield an integer, but it does come very close - and as in physics we deal with approximate measurements, any number that is very close can essentially be called an alignment. How did I figure this was the smallest number that would give an integer to 2 decimal places? Just iterated through on my calculator. :lol: I really have no idea how to formally solve the problem.

313 sets of 2.1353 year orbits yields an alignment period of 668.36 years.

We can verify this by testing how many orbits each would have gone through after 668.36 years:

Earth: 668.36

Mars: 355.36

Jupiter: 56.36

Therefore alignment after every 668.36 years.

This of course assumes that they were aligned in the past...I have no idea if it is a requirement that planets go through all possible spatial arrangements in that time. It also obviously assumes circular orbits...no idea how ellipses change things.

I guess you could repeat this same thing for all planets.

Venus: 0.61519 years

displacement made by Earth/Mars/Jupiter alignment (EMJ) every 668.3588 years = 0.3588 orbits

distance made by Venus every 668.3588 years = 1086.4266 orbits -> displacement = 0.4266 orbits

0.4266n - 0.3588n = x; n,x part of I

0.06781n = x

n = 59

With n = 59, amount of years per alignment = n*668.3588 = 39433.167

Check:

Earth: 39433.17 orbits

Mars: 20966.17 orbits

Jupiter: 3325.17 orbits

Venus: 64099.17 orbits

Therefore there is alignment of Venus, Earth, Mars and Jupiter every 39433.167 years, where alignment is considered alignment when planets are within an angle of 1/100 of an orbit (3.6 degrees).

I'll stop there for now. I hope that's actually right, if anyone finds errors then please specifiy so I can know exactly what made this an unfortunate waste of time. :lol: