Two planets in orbit

[Frank23]

Imagine two planets in orbit round a star, each moving clockwise, the star static. The outer planet takes 60 years to complete orbit, the inner planet 24 years. In how many years will the two planets next form a straight line with each other and the star. It is a long time since I left school, so can any one help me with this.

[Loosehead]

360 years.

[THECORBYLOON]

If the Planet A takes 24 years to orbit the star, in one year it moves 15 degrees and if Planet B takes 24 years to orbit the star, it moves 6 degrees each year. If they start off next to each other (Planet A, Planet B, Star) they will be directly opposite each other when there are 180 degrees between them which will take 20 years (A will have moved 300 degrees and B 120 degrees)

They will be in line again Planet A, Planet B, Star when there is 360 degrees movement between them and that will take 40 years (A travels 600 degrees and B 240 degrees)

[THECORBYLOON]

Sorry the answer should have read "....if Planet B takes 60 years...."

[Frank23]

Thank you, I will bow to your knowledge. It far exceeds mine.

[Peter Pedant]

The answers above assume that the planets have no action on each other, but are in orbit sort of independently.

In reality [ha!] when they go near each other they sortta serve. This introduces non linearity in to the system and the straightforward versions of Newton equations dont apply. So there is no straight formula to use...and computer simulation is used.

I think Newton was aware of this back in 1687.

[nucleardream]

every 20 years

[mibn2cweus]

The period between two planets passing in their orbits around the sun (revolving in the same direction) can be found by dividing the longer orbital period by the ratio (longer / shorter) of their orbital periods minus one.

Divide the larger (60 years) by the smaller (24 years),
60 / 24 = 2.5
and subtract one.
2.5 - 1 = 1.5
Divide the years of the longer planets orbital period by this to find the period between passes.
60 / 1.5 = 40 years!

This will vary since planets have elliptical orbits and do not maintain a constant orbital velocity.

[Frank23]

Thank you all, I am more than a trifle confused now. Isn't old age grand!!!!!!!!!!!!! Have a good May Day everyone.

[mibn2cweus]

For the sake of understanding we can simplify thus:

Let's look at two planets that have orbital period ratio of two (years) to one (year). When the 1Y planet completes one orbit the 2Y planet would be half way around; so in two years they meet again.
Ratio 2 minus 1 = 1, then 2Y / 1 = 2 years between "meetings".

For a ratio of 3Y to 1Y,
Planet 1Y completes 1.5 orbits while 3Y makes 0.5 orbit, and they meet again.
Ratio 3 minus 1 = 2, then 3Y / 2 = 1.5 years.

Remember to divide the longer period orbit by the (ratio minus one) to get the period between meetings. If your still confused, take consolation in that I made these answers up! Is there a real genius out there who can explain why this works?