A Kustaanheimo-Stiefel regularization of the elliptic restricted three-body problem and the detection of close encounters with fast Lyapunov indicators

Abstract

We present the Kustaanheimo-Stiefel (KS) regularization of the elliptic restricted three-body problem (ER3BP) at the secondary body $P_2$, and discuss its use to study a category of transits through its Hill's sphere (fast close encounters). Starting from the Hamiltonian representation of the problem using the synodic rotating-pulsating reference frame and the true anomaly of $P_2$ as independent variable, we perform the regularization at the secondary body analogous to the circular case by applying the classical KS transformation and the iso-energetic reduction in an extended 10-dimensional phase-space. Using such regularized Hamiltonian we recover a definition of fast close encounters in the ER3BP for small values of the mass parameter $μ$ (while we do not require a smallness condition on the eccentricity of the primaries), and we show that for these encounters the solutions of the variational equations are characterized by an exponential growth during the fast transits through the Hill's sphere. Thus, for small $μ$, we justify the effectiveness of the regularized fast Lyapunov indicators (RFLIs) to detect orbits with multiple fast close encounters. Finally, we provide numerical demonstrations and show the benefits of the regularization in terms of the computational cost.

Figures (6)

• Figure 1: Physical orbit (reported in convenient aspect ratio for visual clarity) integrated backward in true anomaly up to $f=-2\pi$ and then forward up to $f=2\pi$ following the arrow heads for $x_0=1-\mu+ 1.921451079855507\cdot10^{-3}$ ($\approx 0.01$ AU of altitude), $y_0=z_0=f_0=0$, $p_{x,0}=0.2$, $p_{y,0}=1.8$, $p_{z,0}=0.6$. The corresponding KS-transformed quadruple precision initial data are: u_1=0.0438343595807618585658005372351908591\;,U_1=0.0175337438323047538346610707549189101\;,U_2=0.0702185800222737827036567637151165400\;,U_3=0.0526012314969142580345362603111425415\;,\Phi=-1.38220656687993415599045818608844111\;,u_2=u_3=u_4=\phi=U_4=0\;.The yellow dot symbolizes the Sun, whereas the black thin style curve represents Jupiter's elliptic motion. Top panel: Cartesian version (backward in blue overlapping the forward in red) traced in the rotating-pulsating frame $Oxyz$. Left bottom panel: Cartesian backward (blue) and forward (red) trajectory in the inertial barycentric frame $OXYZ$ (Appendix \ref{['subsecapp:xyztoXYZ']}) with osculating heliocentric ellipses belonging to mutually inclined planes. Right bottom panel: KS integration of the inertial trajectory in the forward case. The close encounter is of hyperbolic type (fast): $\Gamma_0=1.4282186\le\Gamma_s\le\Gamma_s\vert_{\min d_2}<(3/2)\Gamma_0$ in $B(\mu^{\frac{1}{3}})$.
• Figure 2: Projections on the coordinate planes of the right bottom panel of Fig. \ref{['fig:backforw']} on equal axis aspect ratio.
• Figure 3: Reference orbit for mFLI/RFLI computations in the Sun-Earth system. Left panel: Cartesian orbit in the inertial barycentric frame $OXYZ$ (green) approaching the Earth (black thin) at $f_c\approx106^{\circ}$ from $f_0=0.9862623425908257$. The initial orbital elements of the test particle are: $a_P(f_0)=1.3103706971044482$ (semi-major axis), $e_P(f_0)=0.6$ (eccentricity), $f_P(f_0)=0.22823102675215523$ (true anomaly), $i_P(f_0)=\omega_P(f_0)=\Omega_P(f_0)=0$ (inclination, argument of pericenter, longitude of the ascending node). The yellow dot represents the Sun. Right panel: trend of $\Gamma_s>0$ throughout the fast close encounter (note, indeed, that $\Gamma_s\approx\Gamma_0\in[\Gamma_0/2,3\Gamma_0/2]$).
• Figure 4: mFLI$_{\chi}$ charts of the Sun-Earth ER3BP over $1000\times1000$ regularly spaced initial conditions close to those of Fig. \ref{['fig:reforb']}. Here $\lambda=r_H$ in $\chi$ and $w_0\in\mathbb{S}^7$. Top panel: planar section with three magnifications of the close encounter lobes emerging diagonally in the figure. In the central magnification we discern three main lobes (middle, lower, upper), on which sample orbits $\ell_M$, $\ell_L$, $\ell_U$ are taken, respectively. Bottom panel: spatial sections merged to corresponding above planar magnifications of the main lobes. The effect is a continuation of the close encounter structures along the $z'$-axis. Sample orbits $\ell_M^+$, $\ell_U^+$ are taken on the positive $z'$ extensions of the main lobes (left), while sample orbits $\ell_M^-$, $\ell_L^-$ are taken on the negative $z'$ extensions of the main lobes (right).
• Figure 5: RFLI, mFLI, $\Gamma_s$ and $d_2$ as functions of $s$ for the sample orbits of Fig. \ref{['fig:mFLIEarth']}. Top left panel: RFLI$(s)$. Top right panel: mFLI$(s)$. Bottom left panel: $\Gamma_s$ along the whole propagation (dashed) and in $B(\mu^{\frac{1}{3}})$ (solid colored). Bottom right panel: $d_2(s)$ in $B(\mu^{\frac{1}{3}})$ (solid colored) and graph continuations outside $B(\mu^{\frac{1}{3}})$ (dashed).

Theorems & Definitions (10)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Remark 1
• Theorem 1
• proof
• Remark 2