Co-orbital asteroids of terrestrial planets affected by the von Zeipel–Lidov–Kozai mechanism

Abstract We investigate the dynamical behavior of co-orbital asteroids of Venus, Earth, and Mars potentially affected by the von Zeipel–Lidov–Kozai (ZLK) mechanism. Semi-analytical models of the ZLK dynamics for NEOs in the terrestrial planet regions predict that at high eccentricities and inclinations, corresponding to $$ i_{\max } = \arccos {[\sqrt{1-e^2} \cos {(inc)}]} &gt; 30^{\circ } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>i</mml:mi> <mml:mo>max</mml:mo> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>arccos</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msqrt> <mml:mo>cos</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mi>c</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:msup> <mml:mn>30</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:mrow> </mml:math> , equilibrium points should appear at $$90^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>90</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> and $$270^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>270</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> . At low eccentricities and inclinations in the vicinity of Earth and Venus, perturbations from these two planets are dominant, with two equilibrium points occurring at $$0^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> and $$180^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>180</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> . Using numerical simulations performed with the SWIFT and REBOUND integrators, we classify the Kozai states of known co-orbitals and assess their long-term stability. Our results indicate that no current Venus co-orbital is a robust ZLK resonator, while several Earth co-orbitals and one Mars co-orbital (2017 XG62) exhibit libration around one of the four ZLK equilibrium points. Libration around the $$0^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> and $$180^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>180</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> equilibrium points protects asteroids from close encounters with their perturbing planet, Earth, enhancing long-term stability on timescales of $$\simeq 10^{5}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≃</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>5</mml:mn> </mml:msup> </mml:mrow> </mml:math> years. 2017 XG62 is the only known co-orbital asteroid of a terrestrial planet that is currently librating around the high-inclination $$90^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>90</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> equilibrium point. This configuration prevents this asteroid from experiencing close encounters with other terrestrial planets.