The inverse Lidov-Kozai resonance for an outer test particle due to an eccentric perturber. (arXiv:1904.12062v1 [astro-ph.EP])
<a href="http://arxiv.org/find/astro-ph/1/au:+Elia_G/0/1/0/all/0/1">Gonzalo Carlos de Elía</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Zanardi_M/0/1/0/all/0/1">Macarena Zanardi</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Dugaro_A/0/1/0/all/0/1">Agustín Dugaro</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Naoz_S/0/1/0/all/0/1">Smadar Naoz</a>
We analyze the behavior of the argument of pericenter $omega_2$ of an outer
particle in the elliptical restricted three-body problem, focusing on the
inverse Lidov-Kozai resonance. First, we calculate the contribution of the
terms of quadrupole, octupole, and hexadecapolar order of the secular
approximation of the potential to the outer particle’s $omega_2$ precession
rate $(domega_2/dtau)$. Then, we derive analytical criteria that determine
the vanishing of the $omega_2$ quadrupole precession rate
$(domega_2/dtau)_{text{quad}}$ for different values of the inner perturber’s
eccentricity $e_1$. Finally, we use such analytical considerations and describe
the behavior of $omega_2$ of outer particles extracted from N-body
simulations. Our analytical study indicates that the values of the inclination
$i_2$ and the ascending node longitude $Omega_2$ associated with the outer
particle that vanish $(domega_2/dtau)_{text{quad}}$ strongly depend on the
eccentricity $e_1$ of the inner perturber. In fact, if $e_1 <$ 0.25 ($>$
0.40825), $(domega_2/dtau)_{text{quad}}$ is only vanished for particles
whose $Omega_2$ circulates (librates). For $e_1$ between 0.25 and 0.40825,
$(domega_2/dtau)_{text{quad}}$ can be vanished for any particle for a
suitable selection of pairs ($Omega_2$, $i_2$). Our analysis of the N-body
simulations shows that the inverse Lidov-Kozai resonance is possible for small,
moderate and high values of $e_1$. Moreover, such a resonance produces
distinctive features in the evolution of a particle in the ($Omega_2$, $i_2$)
plane. In fact, if $omega_2$ librates and $Omega_2$ circulates, the extremes
of $i_2$ at $Omega_2 =$ 90$^{circ}$ and 270$^{circ}$ do not reach the same
value, while if $omega_2$ and $Omega_2$ librate, the evolutionary trajectory
of the particle in the ($Omega_2$, $i_2$) plane evidences an asymmetry respect
to $i_2 =$ 90$^{circ}$.
We analyze the behavior of the argument of pericenter $omega_2$ of an outer
particle in the elliptical restricted three-body problem, focusing on the
inverse Lidov-Kozai resonance. First, we calculate the contribution of the
terms of quadrupole, octupole, and hexadecapolar order of the secular
approximation of the potential to the outer particle’s $omega_2$ precession
rate $(domega_2/dtau)$. Then, we derive analytical criteria that determine
the vanishing of the $omega_2$ quadrupole precession rate
$(domega_2/dtau)_{text{quad}}$ for different values of the inner perturber’s
eccentricity $e_1$. Finally, we use such analytical considerations and describe
the behavior of $omega_2$ of outer particles extracted from N-body
simulations. Our analytical study indicates that the values of the inclination
$i_2$ and the ascending node longitude $Omega_2$ associated with the outer
particle that vanish $(domega_2/dtau)_{text{quad}}$ strongly depend on the
eccentricity $e_1$ of the inner perturber. In fact, if $e_1 <$ 0.25 ($>$
0.40825), $(domega_2/dtau)_{text{quad}}$ is only vanished for particles
whose $Omega_2$ circulates (librates). For $e_1$ between 0.25 and 0.40825,
$(domega_2/dtau)_{text{quad}}$ can be vanished for any particle for a
suitable selection of pairs ($Omega_2$, $i_2$). Our analysis of the N-body
simulations shows that the inverse Lidov-Kozai resonance is possible for small,
moderate and high values of $e_1$. Moreover, such a resonance produces
distinctive features in the evolution of a particle in the ($Omega_2$, $i_2$)
plane. In fact, if $omega_2$ librates and $Omega_2$ circulates, the extremes
of $i_2$ at $Omega_2 =$ 90$^{circ}$ and 270$^{circ}$ do not reach the same
value, while if $omega_2$ and $Omega_2$ librate, the evolutionary trajectory
of the particle in the ($Omega_2$, $i_2$) plane evidences an asymmetry respect
to $i_2 =$ 90$^{circ}$.
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