Transient Terrestrial Trojans: Comparative Short-term Dynamical Evolution of 2010 TK₇ and 2020 XL₅

Until 2010, we had no observational evidence on the existence of Trojan companions of our planet; an early survey carried out between 1978 and 1982 using the 1.2 m Palomar Schmidt telescope failed to find any, but emphasized that the viewing geometry to observe putative Earth Trojan asteroids from the ground is rather unfavorable (Dunbar & Helin 1983). However, solid numerical evidence for the presence of such a Trojan asteroid population had been available for decades (see e.g., Hollabaugh & Everhart 1973; Dunbar 1980; Mikkola & Innanen 1990). Dunbar (1980) first argued that the 13:8 mean motion resonance between Venus and Earth made the tadpole orbits of Earth Trojans inherently unstable. In marked contrast, the Trojan asteroids of Mars (see e.g., Christou 2013; de La Fuente Marcos & de La Fuente Marcos 2013; de la Fuente Marcos & de la Fuente Marcos 2021) and Jupiter (see e.g., Dvorak & Schwarz 2005; Di Sisto et al. 2014) are long-term stable. It is therefore not surprising that the first Trojan of our planet, 2010 TK₇, was discovered by the NEOWISE team (Gilmore et al. 2010) using NASA's Wide-field Infrared Survey Explorer, a space telescope, and that its confirmed Trojan resonant state was found to be transient (Connors et al. 2011). Here, we provide a preliminary assessment of the current dynamical status and short-term orbital evolution of 2020 XL₅, a recently discovered near-Earth asteroid that might be the second known representative of this elusive population.

The main parameter to classify Earth Trojans is the value of the relative mean longitude, which is the difference between the mean longitude of the minor body and that of Earth. The mean longitude is given by λ = Ω + ω + M, where Ω is the longitude of the ascending node, ω is the argument of perihelion, and M is the mean anomaly. Therefore, the critical angle is λ_r = λ − λ_E. If the value of λ_r oscillates around 60°, the object is called an L₄ Trojan and leads Earth in its orbit, when it librates around −60°, it is an L₅ Trojan and it trails our planet (see e.g., Murray & Dermott 1999). Numerical simulations are necessary to confirm any suspected resonant behavior. Here, we use N-body simulations carried out as described by de la Fuente Marcos & de la Fuente Marcos (2012) and publicly available input data from Jet Propulsion Laboratory's (JPL) Small-Body Database (SBDB) ³ and HORIZONS ⁴ on-line solar system data and ephemeris computation service (Giorgini 2015). Asteroid 2020 XL₅ was discovered on 2020 December 12 by the Pan-STARRS 1 telescope system at Haleakala. ⁵ The orbit determination of this Apollo asteroid is based on 22 observations spanning a data-arc of 43 d. Its absolute magnitude is H = 20.2 ± 0.5 mag (assumed G = 0.15), which suggests a diameter in the range ∼150–1500 m for an assumed albedo in the range 0.60–0.01 (a size of 300–400 m is likely). As of 2021 January 11, its heliocentric orbit determination is: semimajor axis, a = 1.0009 ± 0.0002 au, eccentricity, e = 0.38729 ± 0.00008, inclination, i = 13849 ± 0002, Ω = 153608 ± 0008, and ω = 8796 ± 002. ⁶ Currently, this object experiences regular close encounters with Venus, sometimes as close as 0.03 au from a planet that has a Hill radius of 0.0067 au. We also collected data for the previously known Earth Trojan, 2010 TK₇—with heliocentric orbit determination based on 56 observations spanning a data-arc of 2586 days as of 2017 October 31, a = 0.999273055 ± 0.000000008 au, e = 0.1904803 ± 0.0000005, i = 2089793 ± 000005, Ω = 9646358 ± 000012, and ω = 4596835 ± 000011 (same epoch as 2020 XL₅)—in order to explore the comparative short-term dynamical evolution of the candidate and the Trojan.

Figure 1 focuses on the evolution of λ_r and summarizes our findings for representative orbits, the nominal one and those with Cartesian vectors separated ±0.5σ and ±1σ (for 2020 XL₅) and ±3σ and ±9σ (for 2010 TK₇) from the nominal values. Figure 1, top panel, shows that although the evolution of the nominal orbit of 2020 XL₅ is consistent with L₄ Trojan behavior, orbits close to the nominal one lead to fully non-resonant evolution. With the current orbit, less than 50% of the control orbits lead to L₄ Trojan behavior. In stark contrast, the evolution of all the control orbits of 2010 TK₇ within ±9σ of the nominal one (not ±1σ as in the case of 2020 XL₅) is consistent with L₄ Trojan behavior within about 2000 yr of the current epoch (symmetric). Our calculations show that the current orbit determination of 2020 XL₅ is still too uncertain and its orbital evolution too chaotic—as it may experience close encounters with Venus and eventually with the Earth–Moon system—to confirm a current Trojan engagement with Earth, although the nominal orbit shows such a behavior. More observations are required to provide a conclusive answer. As for 2010 TK₇, we confirm the results in Connors et al. (2011), including its relative stability on timescales of 10⁵ yr and its probable jumping nature as it can jump from librating around L₄ to librating around L₅.

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