No hanging out in neighborhoods of infinity in the three-body problem

Abstract

Consider the spatial Newtonian three-body problem at fixed negative energy and fixed angular momentum. The moment of inertia I provides a measure of the overall size of a three-body system. We will prove that there is a positive number $I_0$ depending on the energy and angular momentum levels as well as the masses such that every solution at these levels passes through $I\le I_0$ at some instant of time. Motivation for this result comes from trying to prove the impossibility of realizing a certain syzygy sequence in the zero angular momentum problem.

Appendix: Comparison to Marchal’s equal mass case

We shall now compare our results to Marchal (1990) by examining the case:

$$\begin{aligned} m_i=\frac{1}{3},~H=-\frac{1}{6},~ |J|=\frac{\sqrt{8}}{9}. \end{aligned}$$

As $I^*$ corresponds to the apocenter of the collinear Euler motion (where $U|_{I=1}$ has a saddle point), we have

$$\begin{aligned} I^*=\frac{32}{27}. \end{aligned}$$

Here Marchal observed in Marchal (1990), p. 468, that $\ddot{\rho }<0$ for $I>I_M$ where

$$\begin{aligned} I_M=\frac{1}{3}(2.709629\ldots )^2=2.447363\ldots ,\end{aligned}$$

so that every orbit will enter the region $I\le I_M$ at some instant (of course excluding or passing through any binary collision). Moreover, it was conjectured that in fact all orbits pass below the minimal inertia of the Henon–Brouke orbit (see Marchal 1990, p. 469), which is approximately 2.402035$\ldots$ and resembles an $e=0$ Earth–Moon–Sun cartoon-type orbit.

Applying our final result (Proposition 5) in this case we obtain an $I_0>I_M$. However, our bound only becomes larger when we apply our perturbation arguments; in this case and in general we obtain a lower pre-perturbation $I^{**}<I_M$ (of Proposition 2). This gives hope that if our perturbation methods are improved (Propositions 3, 4, 5), then the Marchal’s bound of $I_M$ could be lowered. Now we outline how $I^{**}<I_M$ in general.

Note that Marchal’s observation on the negativity of $\ddot{\rho }$ works not just in this equal mass case but lends to a shorter proof of Theorem 1 by using Eq. 10. We set $\mu _i=m_i\mu ^{-1}$, $\lambda =r/\rho$, $\gamma =\angle (\xi _1, \xi _2)$ and rewrite Eq. 10 as

$$\begin{aligned} \rho ^3\ddot{\rho }=c^2-\rho M\phi (\lambda ,\gamma ), \end{aligned}$$

where

$$\begin{aligned} \phi (\lambda ,\gamma )=\mu _1\frac{1+\mu _2\cos (\gamma )\lambda }{(1+2\mu _2\cos (\gamma )\lambda +\mu _2^2\lambda ^2)^{3/2}}+\mu _2\frac{1-\mu _1 \cos (\gamma )\lambda }{(1-2\mu _1\cos (\gamma )\lambda +\mu _1^2\lambda ^2)^{3/2}}. \end{aligned}$$

Note that $\phi \ge \delta >0$ throughout $I\ge I^*$ for a $\delta \le \phi (\lambda , \pi /2)<1$ dependent on the masses. Hence, a $\rho _M$ corresponding to Marchal’s $I_M$ is (in general)

$$\begin{aligned} \rho _M=\frac{c_{J_2}^2}{M\delta }. \end{aligned}$$

However although Eq. (10) leads to a simpler proof, following an orbit to pericenter rather than over the region where $\ddot{\rho }<0$ has the potential to yield lower upper bounds; as for Keplerian orbits we have

$$\begin{aligned} \rho ^{pc}=\frac{c^2}{M(1+e)}\le \frac{c_{J_2}^2}{M}<\rho _M. \end{aligned}$$

Thus, the $I^{**}$ of Proposition 2 satisfies

$$\begin{aligned} I^{**}<I_M. \end{aligned}$$

So our strategy of proof provides hope of lowering the bound toward Marchal’s conjectured Henon–Broucke value in this case and a lower upper bound in general provided the techniques in the perturbation steps Propositions 3, 4, 5 are improved to follow the orbits past the $\ddot{\rho }<0$ regime. Can they be improved? Perhaps in some non-equal mass cases or for some (outer) eccentricity $e>0$ orbits above Henon–Broucke? I am optimistic that taking advantage of the sharper bounds and techniques of the literature they can be improved at least for large classes of orbits. This is especially so as many bounds of the perturbation steps here are not the sharpest (as the original motivation here was merely the existence of some upper bound in general specifically the zero angular momentum case).