[Alan McDougall (OP)]

It was proved by Bruns and Poincaré  in 1887 that the general case does not have an analytical solution. If you take three bodies of unknown masses, starting in general orbital positions, then you don't have a solution.

Maybe I did not word the problem correctly, if so please help.

Any thoughts?

Thanks!

Alan

[jeffreyH]

If you cannot estimate the mass of the objects with enough precision then any solutions are unlikely to match observation. You may still approximate observations very well and be able to obtain more precision by feeding observational corrections back into the function used to determine the motions.

[evan_au]

If you take three bodies of unknown masses

It's actually a harder problem than you suggest.

Even if you know the masses of all three objects precisely, along with their exact positions and velocities, there is no formula that will predict the exact positions at all points of time in the future (or the past).

Lagrange discovered 5 special cases which have a stable solution, but they require that one of the masses be very small (almost reducing it to a 2-body problem). But this doesn't help the general case.

Mathematically, the restriction is on "closed form" expressions like the sine function, which has a defined value at all points in time.

The three-body problem is a chaotic system; even the smallest error in the initial conditions can produce arbitrarily large errors in the future. Chaotic systems can be simulated on a computer, but any errors in the initial conditions or rounding errors in the calculations will cause errors to accumulate rapidly, quickly overwhelming the actual answer.
See: https://en.wikipedia.org/wiki/Three-body_problem

PS: Many real planetary systems have far more than 3 planets!

[Alan McDougall (OP)]

If you take three bodies of unknown masses

It's actually a harder problem than you suggest.

Even if you know the masses of all three objects precisely, along with their exact positions and velocities, there is no formula that will predict the exact positions at all points of time in the future (or the past).

Lagrange discovered 5 special cases which have a stable solution, but they require that one of the masses be very small (almost reducing it to a 2-body problem). But this doesn't help the general case.

Mathematically, the restriction is on "closed form" expressions like the sine function, which has a defined value at all points in time.

The three-body problem is a chaotic system; even the smallest error in the initial conditions can produce arbitrarily large errors in the future. Chaotic systems can be simulated on a computer, but any errors in the initial conditions or rounding errors in the calculations will cause errors to accumulate rapidly, quickly overwhelming the actual answer.
See: https://en.wikipedia.org/wiki/Three-body_problem

PS: Many real planetary systems have far more than 3 planets!

The problem I posed involved just three planets orbiting around each other, with no other object to compare them with, "in a hypothetical otherwise empty universe".

"Sorry if I  worded the problem incorrectly"! Maybe we could consider the problem where the exact masses of the three planets are a known quality?

[chiralSPO]

If I understand correctly, the question is: three planets of unknown mass are interacting with each other gravitationally. Given their orbital trajectories and relative positions, can we determine the mass of each of the planets?

My suspicion is that it could be determined directly (theoretically) by spectroscopically determining the gravitational red shift of light emitted from the surface of each of the planets. I don't know whether we have the capability of doing this (and somehow doubt it, otherwise we would be doing this already), but my understanding of the relevant physics indicates it would be potentially possible, especially for very dense planets...

[jeffreyH]

Having investigated the mathematics of Leonhard Euler I have come across his version of the three body problem which does have solutions since it is more like a two body problem. Where the third body is very much smaller than the other two. This is not only useful when studying celestial motions but also the motions of particles within fields whose force follows the inverse square law. The historical background as well as the technical details are available in the wikipedia article.

https://en.m.wikipedia.org/wiki/Euler%27s_three-body_problem