13 ms·

> I would imagine that instead of an n body calculation it's lot simpler to calculate the influence from one body plus one unchanging vector, but I've never tr

> I would imagine that instead of an n body calculation it's lot simpler to calculate the influence from one body plus one unchanging vector, but I've never tried it. I think what you describe could either be Euler's three-body problem (two fixed point masses and a particle) [0], or the restricted three-body problem (two point masses and a particle) in a rotating/pulsating reference frame. The former does have exact solutions, and I don't believe the latter does, though I'm admittedly not familiar with the literature. I'm also not sure how easy/hard it is to evaluate the exact solution, and how the difficulty compares to proper n-body integration. That being said, I think using Euler's three-body problem would result in losing some potentially useful n-body effects. For example, centrifugal/centripetal forces would be missing compared to a restricted three-body problem in a rotating reference frame, so Lagrange points might not be present. There might be other effects I'm not aware of as well. > Not a problem because if you're not close to anything then your orbit won't be chaotic. I'm not sure I understand why being farther away from something would result in less chaotic trajectories? If anything, I'd expect more interesting orbits due to the lack of one dominating influence. [0]: https://en.wikipedia.org/wiki/Euler%27s_three-body_problem https://en.wikipedia.org/wiki/Euler%27s_three-body_problem

Dylan16807 6y ago
> I think what you describe could either be Euler's three-body problem (two fixed point masses and a particle) [0], or the restricted three-body problem (two point masses and a particle) in a rotating/pulsating reference frame. Even simpler, though, because only one of the masses needs to have a location. The other one is effectively at a fixed direction and distance, far enough away that you can ignore relative motion. > I'm not sure I understand why being farther away from something would result in less chaotic trajectories? If anything, I'd expect more interesting orbits due to the lack of one dominating influence. I'll rephrase. The gravitational vector on the craft won't be shifting very fast, so you can get away with a quite big timestep.
- aw1621107 6y ago
  > Even simpler, though, because only one of the masses needs to have a location. The other one is effectively at a fixed direction and distance, far enough away that you can ignore relative motion. This still sounds like precisely what I described. Euler's three-body problem is two fixed point masses, so there's no relative motion by construction, and the restricted three-body problem in a rotating/pulsating reference frame has mathematical transformations applied so the two bodies are "effectively fixed" relative to each other in that reference frame (while preserving effects due to rotations, such as centripetal/centrifugal forces). The question in such a case becomes whether such a thing is substantially better than a regular n-body integrator. Euler's three-body problem may lose some useful n-body effects such as Lagrange points, which partially defeats the purpose of moving away from Keplerian orbits, and the restricted three-body problem arguably isn't simplified enough compared to full-blown n-body integration to be worth it. > I'll rephrase. The gravitational vector on the craft won't be shifting very fast, so you can get away with a quite big timestep. Ah, that makes more sense. IIRC Principia has an adaptive timestep, so it already does that, though that's with full n-body calculations. Swapping between the approximation and proper n-body depending on position relative to other bodies seems like a rather complex scheme, though, and I'm not sure whether that'd be the best approach.
  - Dylan16807 6y ago
    > This still sounds like precisely what I described. Euler's three-body problem is two fixed point masses, so there's no relative motion by construction, and the restricted three-body problem in a rotating/pulsating reference frame has mathematical transformations applied so the two bodies are "effectively fixed" relative to each other in that reference frame (while preserving effects due to rotations, such as centripetal/centrifugal forces). You still have to care about the where the particle is relative to both masses. The whole point of the calculation is figuring out which way the particle goes, and the simplification I'm suggesting removes a lot of that math. Instead of two masses providing a continuously varying force in both direction and magnitude, you have one mass providing a continuously varying force plus a static offset. This removes multiple degrees of freedom from the problem. > Swapping between the approximation and proper n-body depending on position relative to other bodies seems like a rather complex scheme, though, and I'm not sure whether that'd be the best approach. I wasn't suggesting swapping between the methods. If you're in the middle of nowhere, then while the Keplerian portion of the model will be a smaller factor, it won't harm anything.

Dylan16807 6y ago

> I think what you describe could either be Euler's three-body problem (two fixed point masses and a particle) [0], or the restricted three-body problem (two point masses and a particle) in a rotating/pulsating reference frame. Even simpler, though, because only one of the masses needs to have a location. The other one is effectively at a fixed direction and distance, far enough away that you can ignore relative motion. > I'm not sure I understand why being farther away from something would result in less chaotic trajectories? If anything, I'd expect more interesting orbits due to the lack of one dominating influence. I'll rephrase. The gravitational vector on the craft won't be shifting very fast, so you can get away with a quite big timestep.

aw1621107 6y ago
> Even simpler, though, because only one of the masses needs to have a location. The other one is effectively at a fixed direction and distance, far enough away that you can ignore relative motion. This still sounds like precisely what I described. Euler's three-body problem is two fixed point masses, so there's no relative motion by construction, and the restricted three-body problem in a rotating/pulsating reference frame has mathematical transformations applied so the two bodies are "effectively fixed" relative to each other in that reference frame (while preserving effects due to rotations, such as centripetal/centrifugal forces). The question in such a case becomes whether such a thing is substantially better than a regular n-body integrator. Euler's three-body problem may lose some useful n-body effects such as Lagrange points, which partially defeats the purpose of moving away from Keplerian orbits, and the restricted three-body problem arguably isn't simplified enough compared to full-blown n-body integration to be worth it. > I'll rephrase. The gravitational vector on the craft won't be shifting very fast, so you can get away with a quite big timestep. Ah, that makes more sense. IIRC Principia has an adaptive timestep, so it already does that, though that's with full n-body calculations. Swapping between the approximation and proper n-body depending on position relative to other bodies seems like a rather complex scheme, though, and I'm not sure whether that'd be the best approach.
- Dylan16807 6y ago
  > This still sounds like precisely what I described. Euler's three-body problem is two fixed point masses, so there's no relative motion by construction, and the restricted three-body problem in a rotating/pulsating reference frame has mathematical transformations applied so the two bodies are "effectively fixed" relative to each other in that reference frame (while preserving effects due to rotations, such as centripetal/centrifugal forces). You still have to care about the where the particle is relative to both masses. The whole point of the calculation is figuring out which way the particle goes, and the simplification I'm suggesting removes a lot of that math. Instead of two masses providing a continuously varying force in both direction and magnitude, you have one mass providing a continuously varying force plus a static offset. This removes multiple degrees of freedom from the problem. > Swapping between the approximation and proper n-body depending on position relative to other bodies seems like a rather complex scheme, though, and I'm not sure whether that'd be the best approach. I wasn't suggesting swapping between the methods. If you're in the middle of nowhere, then while the Keplerian portion of the model will be a smaller factor, it won't harm anything.