2
$\begingroup$

Is there a formula that would allow me to skip to a particular time step in a simulation involving planetary motion without having the simulation pass through every time step before?

For instance if I wanted to get to the trillionth time step of a simulation is there a formula that would allow me to get to the trillionth time step without passing through ever time step before the trillionth time step? Alternatively if I wanted to get to the millionth time step in the simulation could I use the same formula to skip to the millionth time step? I would like a formula that could also give the positions and velocities of all the bodies in the simulation at whatever time step I decide to skip to.

$\endgroup$
  • $\begingroup$ Are you considering only, say, one planet orbiting a sun, or are you considering something more complicated, like 100 masses all interacting with each other? $\endgroup$ – knzhou Apr 15 '18 at 9:23
  • $\begingroup$ This formula would just be a solution to the equations of motion, and if you have more than two bodies there is no such formula. $\endgroup$ – Javier Apr 15 '18 at 12:49
  • 1
    $\begingroup$ To clarify what Javier said: what you're after is a closed form for the solution to the equations of motion, which can be taken to mean an expression involving only a finite number of basic terms, where a basic term includes various things like constants, addition, trig functions, roots, powers and so on. Such a thing does not exist for gravitating systems with more than two bodies in general. $\endgroup$ – tfb Apr 15 '18 at 15:53
4
$\begingroup$

Is there a formula that would allow me to skip to a particular time step in a simulation involving planetary motion without having the simulation pass through every time step before?

The answer is yes if you're simulating two point masses that interact via Newtonian gravity and nothing else. There's no need to use numerical integration to simulate such a system. The answer is effectively no if you are simulating one object orbiting another with a non-spherical mass distribution (e.g., a satellite orbiting the Earth), or if you are simulating multiple gravitating objects (e.g., our solar system).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.