Third-order stability of the long-period Trojan librations

Rabe, Eugene

doi:10.1086/110196

Cited by 25 publications

(22 citation statements)

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“…the studies of Rabe (1967); Bien & Schubart (1984); Lhotka et al (2008) and Érdi et al (2009) as well as many others. Ever since, extensive numerical studies have been undertaken to find the extension of the stability regions around the equilibrium points of the planets, e.g.…”

Section: Introductionmentioning

confidence: 84%

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

Dvořák

Lhotka

Zhou

2012

A&A

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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.

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Section: Introductionmentioning

confidence: 84%

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

Dvořák

Lhotka

Zhou

2012

A&A

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“…The first comprehensive works aimed at understanding the dynamical properties of the Trojan asteroids in the frame of the restricted three-body problem were developed by Szebehely (1967) and Rabe (1967). In fact, Szebehely (1967) performed a detailed description of such problem including regularization, equilibrium points, periodic orbits and their stability, Hill curves and their implications.…”

Section: Introductionmentioning

confidence: 99%

Collisional and dynamical evolution of theL$\mathsf{_{4}}$ Trojan asteroids

Elía¹,

Brunini²

2007

A&A

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Aims.In this paper, we analyze the collisional and dynamical evolution of the population of L 4 Jovian Trojans. Methods. To do this, we test different collisional parameters and include a dynamical treatment, taking into account the stability and instability zones of the L 4 Trojan swarm. This procedure allows us to estimate the size distribution of the L 4 Trojans, to study their mean collisional lifetimes, to analyze the formation of families, to obtain ejection rates of Trojan fragments and to discuss their possible contribution to the current populations of Centaurs and Jupiter-family comets. Results. Our estimates of the L 4 Trojan cumulative size distribution show waves that propagate from diameters of ∼0.1 to ∼80 km around the values derived from optical surveys. On the other hand, the mean collisional lifetimes obtained from our simulations indicate that the large Trojan asteroids have likely survived without being catastrophically fragmented over the age of the Solar System. With regards to the Trojan removal, we calculate a maximum ejection rate of Trojan fragments from L 4 of ∼50 objects larger than 1 km of diameter per Myr, which results to be significantly smaller than values previously published. Such estimates allow us to infer that the contribution of the Trojan asteroids to the current populations of Centaurs and Jupiter-family comets is negligible. In addition, our results are in agreement with the formation of few Trojan families in the L 4 swarm. On the other hand, we infer that the current orbital distribution of the Trojan asteroids does not offer a strong constraint on the dynamical origin of this population.

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“…Many investigations of the elliptic restricted three-body problem have been done, e.g., Rabe (1967) and Lohinger and Dvorak (1993), who considered the stability of the Lagrangian points, which depend on the eccentricity of the orbit and the mass ratio of the primaries (as shown in Fig. 2).…”

Section: Preparations For the Calculationsmentioning

confidence: 99%

Trojans in Habitable Zones

et al. 2005

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With the aid of numerical experiments we examined the dynamical stability of fictitious terrestrial planets in 1:1 mean motion resonance with Jovian-like planets of extrasolar planetary systems. In our stability study of the so-called "Trojan" planets in the habitable zone, we used the restricted three-body problem with different mass ratios of the primary bodies. The application of the three-body problem showed that even massive Trojan planets can be stable in the 1:1 mean motion resonance. From the 117 extrasolar planetary systems only 11 systems were found with one giant planet in the habitable zone. Out of this sample set we chose four planetary systems-HD17051, HD27442, HD28185, and HD108874-for further investigation. To study the orbital behavior of the stable zone in the different systems, we used direct numerical computations (Lie Integration Method) that allowed us to determine the escape times and the maximum eccentricity of the fictitious "Trojan planets." Key Words: Trojan planets-Exoplanets-Habitable zone-Dynamical astronomy. Astrobiology 5, xxx-xxx. 583

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Third-order stability of the long-period Trojan librations

Cited by 25 publications

References 0 publications

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

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

Collisional and dynamical evolution of theL$\mathsf{_{4}}$ Trojan asteroids

Trojans in Habitable Zones

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