# How to predict future position of body in solar system (without Kepler's equations, N-body problem)?

I have a solar system where each body interacts with each other by force $$F = \frac{Gm_1m_2}{R^2}$$

Is it possible to predict future position of some random body after t seconds since bodies velocity and position measurement?

I guess, it is possible only with numerical methods, but how to to it fast without calculating system state for each future frame? I don't need a big precise, just a rough estimation to target into sphere of influence of planet.

• If you only have the planets, then numerically solving the force equations in a straightforward way is pretty fast. The time it takes to simulate T units of time forward scales as O(T*N^2) - when N increases,it becomes less effective, but I doubt you want to include every minor body. If you have a long T the real problem may be precision instead. Oct 7 '21 at 16:07
• > If you have a long T T is small enought to fly from one position of SS to the given body Oct 7 '21 at 16:39
• If you don't want to integrate the equations of motion, then you really do need to approximate the orbits using Kepler's equations (which isn't hard to do), but that's only useful when you can pretend that each orbiting body only interacts gravitationally with its primary (and not with the other bodies). Oct 7 '21 at 23:12
• You selected the wrong answer. Give people time before selecting an answer. Oct 9 '21 at 7:19
• For our solar system, assuming moons orbit elliptically around planets and planets orbit elliptically around the sun works fairly well. You may also want to look at astronomy.stackexchange.com/questions/13488 which is mostly about our solar system, but a couple of bullet points mention arbitrary systems. physics.stackexchange.com/questions/25241 may also be helpful. Oct 11 '21 at 17:17

You can't.

An N-body system with n>2 is (in general) chaotic. This means that any inaccuracy in the initial state of the system will grow exponentially.

You can't get a rough estimate of any planets position at a future time.

So you can't predict roughly where a body will be a long way into the future, even if you numerically predict frame by frame.

In particular situations it may be possible to make an estimate over short enough time periods. For example if the central "sun" is massive enough and the planets never approach close to each other (as is the case in our solar system) you can approximate with Keplerian orbits (perhaps with perturbations). If the orbits are roughly circular you might be able to get an acceptable short term estimate using simple circular motion and Kepler's third law. But even our solar system is chaotic in the longer term.

• This answer is nonsense. If it was true, the various space agencies would be unable to send vehicles to other solar system bodies. While it is true the solar system is stably chaotic, the time span for than chaos is in the several tens of millions of years. Whether the solar system is unstable over the course of billions of years is an open question. Oct 9 '21 at 7:19
• It is correct and it answers the question. The solar system can be modelled very well by numerical methods or Kepler's laws. But the question is about a hypothetical n-body system and for a method of estimating the location of the planets without using Numerical methods or Kepler's law. I state that is impossible. In general, such a system is chaotic. But over a short enough time span it may be possible to use circular motion as an approximation to Keplers laws. Oct 9 '21 at 13:04
• The question asks about the solar system. I'm perhaps being a bit over pedantic here, but there is exactly one solar system in the entire universe. Planet-like objects that orbit stars other than the Sun: Those are properly called star systems rather than solar systems. Oct 9 '21 at 13:10
• No. It asks about "a solar system" not "the solar system". I see a significant difference there. What the OP is calling "a solar system" you might call "a star system". You may want to clarify that with the OP, but I'm clear that they are asking about a hypothetical system of bodies around a star. Oct 9 '21 at 13:39
• The phrase "I have a solar system" suggests world building and hence I'd agree that it is about a hypothetical system rather than ours. Oct 10 '21 at 8:52