# Finding the Right Initial Conditions for a Three-Body Problem Periodic Solution

The planar restricted circular three-body problem is about computing the trajectory of a small mass, usually a speccraft, that is affected by two larger masses, Earth and the Moon. The differential equation for this problem is given as

$$y_1'' = y_1 + 2y_2' - \mu_2 \frac{y_1+\mu_1}{D_1} - \mu_1 \frac{y_1-\mu_2}{D_2}$$

$$y_2'' = y_2 - 2y_1' - \mu_2 \frac{y_2}{D_1} -\mu_1 \frac{y_2}{D_2}$$

where

$$\mu_1 = \frac{m_1}{m_1+m_2}$$

$$\mu_2 = 1-\mu_1$$

$$D_1 = ((y_1+\mu_1)^2 + y_2^2 )^{3/2}$$

$$D_1 = ((y_1-\mu_2)^2 + y_2^2 )^{3/2}$$

For these equations for a certain initial conditions a so-called Arenstorf orbit can be computed which is periodic. The initial conditions are given as (for example in Solving Ordinary Differential Equations I, Nonstiff Problems, pg. 130),

$y_1(0) = 0.994$, $y_1'(0)=0$, $y_2'(0) = -2.0015851063790825224053786222$

Every textbook, lecture note repeats these values, which produces, after integration, a nice orbit around earth and moon. But noone seems to know where they come from. Were they computed analytically? Or a trial and error approach was used to find them, or a combination of both?

If someone has code that can compute these values it would be much appreciated.

Thanks,

So, what we can do to reduce the trial and error to reasonable amounts is to try to find the fixed points and their linearized behavior, approximate the separatrices to get a partitioned space, then choose some sample IC's within each partition to try and determine the behavior of the system in that regime. You may stumble upon some points which generally move to fixed points, some which move to periodic orbits, some which move to quasiperiodic orbits, and some which move to chaos. Within the regime where a cycle happens, there must be some fixed point. You can transform that fixed point to the origin, then get the amplitude/phase system by assuming $y_i = r_i \cos(t+\psi_i)$ for $i=1,2$, then use variation of parameters and averaging to get an approximation of the cycle.
For the linear stability analysis, see any undergraduate book on Linear and Nonlinear ODE's. Approximation methods for invariant manifolds usually come in the form of some power series methods, but you can search for those. Once you've determined some approximate partitioning, just pick a bunch of initial conditions, then numerically solve the equations (using, say, ode45 with MATLAB) for each of the IC's and plot the results on the $(y_1,y_2)$-plane. For the variation of parameters and averaging, you can look at some books on Nonlinear Oscillations or Perturbation Methods, for example by Ali Nayfeh.
Based on this, I did a search, and found this paper. How to solve the boundary problem between point A and B? The author says guess an initial velocity $v_i$, and since starting position known, all initial value are known. Integrate, if the result is far from point B, not within alloted time, update $v_i$ according to a gradient formule, integrate again.