Stability analysis of apsidal alignment in double-averaged restricted elliptic three-body problem

Abstract

We are dealing with the averaged model used to study the secular effects in the motion of a body of the negligible mass in the context of a spatial restricted elliptic three-body problem. It admits a two-parameter family of equilibria (stationary solutions) corresponding to the motion of the third body in the plane of primaries motion, so that the apse line of the orbit of this body is aligned with the apse lines of the primaries’ orbits. The aim of our investigation is to analyse the stability of these equilibria. We show that they are stable in the linear approximation. The Arnold–Moser stability theorem provides sufficient conditions under which this means stability in a nonlinear sense too. These conditions are violated for parameters of the problem that belong to a set formed by a finite number of analytic curves in the parameters’ plane. As it turned out, in the system under consideration, violation of these conditions in some cases actually leads to an instability.

Appendices

Appendix 1: Computation of the frequency $\Omega _{2}$

The frequency of the small oscillations around the equilibrium is

$$\begin{aligned} {{\Omega }_{2}}=\sqrt{{{\left. \frac{{{\partial }^{2}}\bar{R}}{\partial p_{2}^{2}}\cdot \frac{{{\partial }^{2}}\bar{R}}{\partial q_{2}^{2}} \right| }_{{{p}_{2}}=p_{2*},{{q}_{2}}=0}}}. \end{aligned}$$

From the definition of the variables $p_{2}$ and $q_{2}$, we have:

$$\begin{aligned} p_{2}^{2}+q_{2}^{2}=2{{a}^{1/2}}\left( 1-\sqrt{1-{{e}^{2}}} \right) \end{aligned}$$

and

$$\begin{aligned} c=\cos {\omega }=\frac{{{p}_{2}}}{{{(p_{2}^{2}+q_{2}^{2})}^{1/2}}}. \end{aligned}$$

Then, we have

$$\begin{aligned} {{\left. \frac{\partial e}{\partial {{p}_{2}}} \right| }_{{{p}_{2}}=p_{2*},{{q}_{2}}=0}}=\frac{p_{2*}}{{{a}^{1/2}}}\frac{\sqrt{1-e_{*}^{2}}}{{{e}_{*}}}, \quad {{\left. \frac{\partial e}{\partial {{q}_{2}}} \right| }_{{{p}_{2}}=p_{2*},{{q}_{2}}=0}}=0 \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&{{\left. \frac{\partial c}{\partial {{q}_{2}}} \right| }_{{{p}_{2}}=p_{2*},{{q}_{2}}=0}}=0,{{\left. \,\,\frac{{{\partial }^{2}}c}{\partial q_{2}^{2}} \right| }_{{{p}_{2}}=p_{2*},{{q}_{2}}=0}}=-\frac{1}{{{(p_{2*})}^{2}}}, \\&{{\left. \frac{\partial c}{\partial {{p}_{2}}} \right| }_{{{p}_{2}}=p_{2*},{{q}_{2}}=0}}={{\left. \frac{{{\partial }^{2}}c}{\partial p_{2}^{2}} \right| }_{{{p}_{2}}=p_{2*},{{q}_{2}}=0}}=0. \\ \end{aligned} \end{aligned}$$

Now it is easy to find

$$\begin{aligned} {{\left. \frac{{{\partial }^{2}}\bar{R}}{\partial q_{2}^{2}} \right| }_{{{p}_{2}}=p_{2*},{{q}_{2}}=0}}={{\left. \,\,-\frac{\partial \bar{R}}{\partial c} \right| }_{e={{e}_{*}},c=1}}\cdot \frac{1}{{{(p_{2*})}^{2}}} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {{\left. \frac{{{\partial }^{2}}\bar{R}}{\partial p_{2}^{2}} \right| }_{{{p}_{2}}=p_{2*},{{q}_{2}}=0}}={{\left. \,\,\frac{{{\partial }^{2}}\bar{R}}{\partial {{e}^{2}}} \right| }_{e={{e}_{*}},c=1}}&\cdot {{\left( {{\left. \,\,\frac{\partial e}{\partial {{p}_{2}}} \right| }_{{{p}_{2}}=p_{2*},{{q}_{2}}=0}} \right) }^{2}} \\&={{\left. \,\,\frac{{{\partial }^{2}}\bar{R}}{\partial {{e}^{2}}} \right| }_{e={{e}_{*}},c=1}}\cdot \frac{{{(p_{2*})}^{2}}}{a}\frac{1-e_{*}^{2}}{e_{*}^{2}}. \end{aligned} \end{aligned}$$

So we have

$$\begin{aligned} \Omega _{2}^{2}={{\left. {{\left. \,\,-\frac{1}{a}\frac{\partial \bar{R}}{\partial c} \right| }_{e={{e}_{*}},c=1}}\,\,\cdot \frac{{{\partial }^{2}}\bar{R}}{\partial {{e}^{2}}} \right| }_{e={{e}_{*}},c=1}}\cdot \frac{1-e_{*}^{2}}{e_{*}^{2}}. \end{aligned}$$

Appendix 2: Stability condition for 1:1 resonance

Sufficient stability and instability conditions for 1:1 resonance are established in Sokolskii (1974). We reproduce here, with minor additions,² results of this paper for the case when the fourth-order Birkhoff normal form of the Hamiltonian near the considered equilibrium is

$$\begin{aligned} \begin{aligned} H&=-\left[ {{\Omega }_{2}}{{\rho }_{2}}-{{\Omega }_{3}}{{\rho }_{3}} +\frac{1}{2}\left( {{a}_{11}}{\rho }_{2}^{2}+2{{a}_{12}}{{\rho }_{2}}{\rho }_{3}+{{a}_{22}}{\rho }_{3}^{2} \right) \right. \\&\quad \left. +\,b{{\rho }_{2}}{{\rho }_{3}}\cos 2({{\varphi }_{2}}+{{\varphi }_{3}}+\varkappa )\right] . \end{aligned} \end{aligned}$$

(16)

Here the sign “-” before the square bracket is used for compatibility of notation with the main text, where the Hamiltonian is $H=-\bar{V}$, and $\mu \bar{V}$ is the double-averaged force function of the planet. We assume that the frequencies are close to 1:1 resonance: $\Omega _{3}=\Omega _{2}+\delta$ with a small $\delta$.

We make a canonical transformation of variables

$$\begin{aligned} \left( \rho _{2},\rho _{3}, \varphi _{2},\varphi _{3} \right) \mapsto \left( I_{2},I_{3}, \gamma ,\chi \right) , \end{aligned}$$

with the generating function

$$\begin{aligned} S=\left( \varphi _{2}+\varphi _{3}+\varkappa / 2 \right) \,I_{2}+\varphi _{3}\,I_{3}. \end{aligned}$$

New and old variables are related as follows:

$$\begin{aligned} \gamma =\varphi _{2}+\varphi _{3}+\varkappa / 2,\quad \chi =\varphi _{3}, \quad \rho _{2}=I_{2}, \quad \rho _{3}=I_{2}+I_{3}. \end{aligned}$$

Then, we have

$$\begin{aligned} \begin{aligned} H&=-[\Omega _{2} I_{2}-(\Omega _{2}+\delta )(I_{2}+I_{3}) \qquad \qquad \qquad \qquad \\&\qquad +\frac{1}{2}\left( a_{11} I_{2}^{2}+2 a_{12} I_{2}\left( I_{2}+I_{3}\right) +a_{22}\left( I_{2}+I_{3}\right) ^{2}\right) +b I_{2}\left( I_{2}+I_{3}\right) \cos 2 \gamma ]. \end{aligned} \end{aligned}$$

(17)

Hamiltonian in the new variables does not depend on the angle $\chi$. For the canonically conjugate to $\chi$ variable $I_3$, we have $\dot{I}_3=0$. Thus, $I_3=\mathrm{const}$. For $I_2,\gamma$, we get a Hamiltonian system with one degree of freedom depending on a parameter $I_3$.

The conservation of the Hamiltonian and of the variable $I_3$ gives for the level of the Hamiltonian $H=h$

$$\begin{aligned} \left[ P+b\, \cos 2 \gamma \right] I_{2}^{2}+\left[ -\delta +\left( a_{12}+a_{22}+b \cos 2 \gamma \right) I_3\right] I_{2}=c, \end{aligned}$$

where constants P and c are defined by formulas

$$\begin{aligned} P=\frac{1}{2}\left( {{a}_{11}}+2{{a}_{12}}+{{a}_{22}}\right) ,\ c=-h+\delta I_3-a_{22}I_3^2. \end{aligned}$$

Denote $U=-\delta +(a_{12}+a_{22})I_{3}$, $W=b\,I_{3}$. Then, we have

$$\begin{aligned} \left[ P+b\,\cos 2 \gamma \right] \,I_{2}^{2}+\left( U+W\,\cos 2 \gamma \right) \,I_{2}-c=0. \end{aligned}$$

(18)

Solving this relation for $I_2$, we get expressions for phase curves of the considered system:

$$\begin{aligned} I_{2}=\frac{-(U+W \cos 2 \gamma ) \pm \sqrt{(U+W \cos 2 \gamma )^{2}+4 c(P+b \cos 2 \gamma )}}{2(P+b \cos 2 \gamma )}. \end{aligned}$$

If $|P|>|b|$, then, for small enough $\delta , I_3$, in the plane with the polar coordinates $I_2, \gamma$ there is a neighbourhood of the coordinate origin filled by closed phase curves of Hamiltonian (17). This implies the stability of the equilibrium at $\rho _2=0, \rho _3=0$ of Hamiltonian (16). Then, the KAM theory implies the stability of this equilibrium for any Hamiltonian for which (16) is the fourth-order normal form.

Consider now the case of exact resonance, $\delta =0$, for $0<|P|<|b|$. Put $I_3=0$. Then $U=W=0$, and dynamics on the integral level $I_3=0$ is described by the Hamiltonian

$$\begin{aligned} H=-\left[ P+b\,\cos 2 \gamma \right] \,I_{2}^{2}. \end{aligned}$$

This system has an invariant ray $\gamma =\gamma _*=\mathrm{const},\ \cos 2\gamma _*=-P/b, \ \sin 2\gamma _*<0$. On this ray $\dot{I}_2=-2b \sin 2\gamma _*\, I_{2}^{2}$. The solution with the initial condition $I_2=I_{2,0}$ at time $t=0$ is:

$$\begin{aligned} I_2(t)=\frac{I_{2,0}}{1+(2b \sin 2\gamma _*\, I_{2,0})\, t}. \end{aligned}$$

This solution tends to the equilibrium at the origin as $t\rightarrow -\infty$ and is unbounded. Thus, the equilibrium of the normal form Hamiltonian is unstable. One can show that this equilibrium is unstable for the original Hamiltonian (but one cannot claim the existence of unbounded trajectory in the original system for which normal form (12) is constructed).

If $\delta \ne 0$, $|\delta |\ll 1$ and $0<|P|<|b|$, then the equilibrium is stable according to the Arnold–Moser theorem. However, the stability region shrinks to 0 as $\delta \rightarrow 0$. Indeed, the integral relation (18) for $I_3=0, h=0$ reads as:

$$\begin{aligned} \left[ P+b\,\cos 2 \gamma \right] \,I_{2}^{2}-\delta I_{2}=0. \end{aligned}$$

Thus, in this integral level there is the equilibrium at the origin $I_2=0$ and unbounded trajectories

$$\begin{aligned} I_2=\frac{\delta }{P+b\,\cos 2 \gamma }. \end{aligned}$$

Initial conditions for unbounded trajectories tend to the equilibrium as $\delta \rightarrow 0$.