The orbit of 2010 TK7: possible regions of stability for other Earth Trojan asteroids

Abstract

The first Earth Trojan has been observed and found to be on an interesting orbit close to the Lagrange point L4. In the present study, we therefore perform a detailed investigation of the stability of its orbit and moreover extend the study to give an idea of the probability of finding additional Earth Trojans. Our results are derived using three different approaches. In the first, we derive an analytical mapping in the spatial elliptic restricted three-body problem to find the phase space structure of the dynamical problem. We then explore the stability of the asteroid in the context of the phase space geometry, including the indirect influence of the additional planets of our Solar system. In the second approach, we use precise numerical methods to integrate the orbit forward and backward in time in different dynamical models. On the basis of a set of 400 clone orbits, we derive the probability of capture and escape of the Earth Trojan asteroid 2010 TK7. To this end, in the third approach we perform an extensive numerical investigation of the stability region of the Earth's Lagrangian points. We present a detailed parameter study of possible stable tadpole and horseshoe orbits of additional Earth Trojans, i.e. with respect to the semi-major axes and inclinations of thousands of fictitious Trojans. All three approaches lead to the conclusion that the Earth Trojan asteroid 2010 TK7 finds itself in an unstable region on the edge of a stable zone; additional Earth Trojan asteroids may be found in this regime of stability.

Search for L5 Earth Trojans with DECam

“…Numerical integrations indicate that 2010 TK 7 has a highly chaotic and short-lived orbit. Estimated lifetimes vary from ∼7000 years (Connors et al 2011) to about ∼0.25 Myrs (Dvorak et al 2012). Taking into account the Yarkovsky effect, Zhou et al (2018) also concluded that 2010 TK 7 is too small to maintain a long-term stable orbit.…”

“…However, the orbit of this object is not consistent with primordial ETs, which are expected to be long-term stable librators near L4 or L5. In particular, 2010 TK 7 has a large amplitude tadpole orbit, librating between Earth and L3 (the Lagrange point behind the Sun), rather than remaining near L4 (Connors et al 2011;Dvorak et al 2012). We consider such orbits distinct from primordial ETs, though it is possible for these orbits to also be stable on the order of ∼Myrs (Marzari & Scholl 2013;Dvorak et al 2012).…”

Orbital stability of Earth Trojans

“…The dynamical model adopted in our simulations, which is referred to as Ve2Ne hereafter, is consisted of the Sun, all of the planets in our Solar System except Mercury (from Venus to Neptune) and massless fictitious Earth Trojans (test particles). Mercury is excluded because it has negligible influence on the evolution of Earth Trojans in this research while the model including Mercury consumes much longer computation time (Dvorak et al 2012). We adopt the Earth-Moon barycenter instead of the separate Earth and Moon as Dvorak et al (2012) did.…”

“…Note that in accordance with practice, we denote the secular resonance as ν i when g = g i and ν 1i when s = s i , where g (g i ) and s (s i ) represent the precession rate of the perihelion and ascending node of Trojans (planet). Dvorak et al (2012) adopted the maximum eccentricity as the stability indicator and constructed a similar stability diagram. In that investigation, a small stability region appears around i = 50 • , which is unexpected according to our simulations.…”

2004 KV18: a visitor from the scattered disc to the Neptune Trojan population

“…Horner & Evans ). We note, in passing, that such a temporary capture hypothesis has invoked for the origin of the recently discovered Earth Trojan, 2010 TK 7 (Connors, Wiegert & Veillet ; Dvorak, Lhotka & Zhou ).…”