Long-Term Cycling of Kozai-Lidov Cycles: Extreme Eccentricities and Inclinations Excited by a Distant Eccentric Perturber

The very long-term evolution of the hierarchical restricted three-body problem is calculated analytically for high inclinations. The Kozai-Lidov Cycles (KLCs) slowly evolve due to the octupole term in the perturber’s potential and exhibit striking features, including extremely high eccentricities and the generation of retrograde orbits with respect to the perturber. These features were found in recent numerical experiments of the nonrestricted three-body problem and were attributed inaccurately to the comparable and low masses of the two orbiting companions. Our calculation is done by averaging for the first time the double averaged secular equations of motion over the KLCs and finding a new constant of the motion. These very long-term effects are likely to be important in various astrophysical systems thought to involve KLCs, such as hot Jupiters, irregular moons of planets, and many others.

Figure 1

Upper panel: $⟨f_j⟩$ (solid) and $⟨f_{\Omega }⟩$ (dashed) from Eqs. (13). Lower panel: $F$ from Eq. (15).

Figure 2

Results of numerical integrations for ${ϵ}_{\mathrm{Oct}}=0.01$, with initial conditions ${\omega }_0=0$, ${\Omega }_0=\pi$, $i_0=80°$, $e_0=0.1$ where for $\omega =0$ $e_{\min }=e_0$. The blue solid lines are the result of the integration of the full secular equations, Eqs. (4), while the red dashed lines are the result of the integration of the averaged equations, Eqs. (11, 13, 6), and using the Kozai-Lidov relations to determine $e_{\min },e_{\max }$ and $i_{\min },i_{\max }$. The two green horizontal lines in the top panel represent $\pm j_{\max }$, given by Eq. (10).

Figure 3

Results of numerical integrations for ${ϵ}_{\mathrm{Oct}}=0.01$, with initial conditions ${\omega }_0=0$, ${\Omega }_0=0$, $i_0=88°$, $e_0=0.1$. The line styles are as in Fig. 2.

Figure 4

The thick black line is the analytical threshold for a flip, Eq. (16), at $e_0\to 0$ (this reduces to (17) for $i_0>61.7°$). The results from numerical integrations of the full secular equations are shown as red circles (cases that had a flip) and blue circles (cases that did not flip). All simulations had ${\omega }_0=0$, $e_0=0.001$ and $\Omega$ scanned between 0 to $2\pi$, covering the rotating KLCs initial conditions. Note that $e_{\min }=e_0$ when ${\omega }_0=0$. The thin black lines are the approximate analytical thresholds resulting from Eq. (16) for $e_0=0.2,0.335(=e_{\min ,m}),0.5$ (left to right at high ${ϵ}_c$).