## Saturday, January 09, 2021

### [yiylnxnp] computing tadpole and horseshoe orbits

first, consider a satellite in a circular orbit.

given exact initial conditions and an exact length of time, how much computational effort is required to determine the state (position, velocity) of the satellite at the end of the time interval, to within a given error bound?

not very much: compute orbital period from initial conditions.  divide the time by orbital period.  something something mean anomaly something something pi to get an angle.  then, sine and cosine will provide position, velocity is along the tangent, speed was given, and we are done.  do the arithmetic with enough precision to satisfy the error bound.  the computational effort required is that of doing arithmetic and of computing sine and cosine at the necessary precision.  very roughly, this is polynomial in the number is bits.  the number of bits in the time interval t is O(log(t)).

a binary system of two point masses in elliptic orbits around a common barycenter seems not much more difficult, though it requires Newton's Method to solve Kepler's Equation.

how much computational effort is required to compute tadpole orbits and horseshoe orbits, the quasiperiodic stable solutions to the circular restricted 3 body problem?

the conventional method of solving the 3 body problem is numerical integration.  numerical integration requires computational effort at least linear in the time interval: O(t).  this is already exponentially worse than O(log(t)^n) of the circular case.  things might be worse, much worse, if we need to maintain lots of precision during numerical integration to get a final answer within the desired error bound.  the famous kind of "much worse" situation is a chaotic system.  are tadpole orbits chaotic?  how much precision do we need during numerical integration if we know a priori that the satellite is in a tadpole (or horseshoe) orbit?  although we've posited that initial conditions are given exactly, so it might seem that the sensitivity of chaotic systems to initial conditions might not be relevant, numerical integration computes intermediate system states, say, time step 2, to finite precision.  if the system is sensitive to initial conditions, then it is to conditions at time step 2, so the final answer will be sensitive to the amount of finite precision with which we chose to approximate time step 2.

can we avoid numerical integration and solve tadpole and horseshoe orbits with less computational effort?  maybe precompute something.  maybe Lissajous curve.

systems like the Bunimovich stadium show that very little is necessary for chaos to happen.

chaoticity is not necessarily a problem.  the circular 2 body problem is chaotic in the following sense.  consider satellites in 2 circular orbits, one slightly above the other.  over time, because of their different periods, the satellite positions will diverge from each other more and more.  this demonstrates sensitivity to initial conditions.  (this system is approximately analogous to the "circle map" in dynamical systems.)  nevertheless, we showed above how to solve the circular case without much effort, sidestepping some issues by having the initial conditions specified exactly.  (it took some iterations to define the problem in the way described above, in terms of computational complexity, to avoid these problems.)

Karl Sundman solved the general 3-body problem with a convergent infinite series.  how does this mesh with the fact that the general 3-body problem is chaotic?  Sundman's solution reportedly converges too slowly to be practical.  can it be improved when restricting to tadpole orbits in the circular restricted 3-body problem?

Alexandre G. Van Anderlecht, "Tadpole orbits in the L4/L5 region: Construction and links to other families of periodic orbits".  this provides a good overview of the 3 body problem.  however, its main tool for calculating tadpole orbits is numerical integration, which strongly suggests there isn't anything better.

more difficult: what if we relax the circular restricted 3 body problem to allow elliptic orbits for the two heavier masses?  we keep restricting the infinitesimal lightest mass to a quasiperiodic stable orbit.  when do such orbits exist?  how can we be sure that the lightest mass will never be ejected?

doing everything in 3D also adds complications.  are there tadpole orbits in the CR3BP relaxed such that the lightest mass is not restricted to the plane (ecliptic) of the two heavier masses?  probably yes.