The ESA-JAXA BepiColombo mission will provide simultaneous measurements from two spacecraft, offering an unprecedented opportunity to investigate magnetospheric The BepiColombo mission to Mercury Edited by Johannes Benkhoff, Go Murakami and Ayako Matsuoka B A. Milillo

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.

The problem of analytical determination of the stability domain around the points L4 or L5 of the Lagrangian equilateral configuration of the three‐body problem has served in the literature as a basic celestial mechanical model probing the predictive power of the so‐called Nekhoroshev theory of exponential stability in non‐linear Hamiltonian dynamical systems. All analytical investigations in this framework have so far been based on the circular restricted three‐body problem (CRTBP). In this work, we extend the analytical estimates of Nekhoroshev stability in the case of the planar elliptic‐restricted three‐body problem (ERTBP). To this end, we introduce an explicit symplectic mapping model for the planar ERTBP, obtained via Hadjidemetriou's method, which generalizes the family of mappings discussed in earlier papers. The mapping is based on an expansion of the disturbing function up to a sufficiently high order in the eccentricities and the variations of the semimajor axis, and it is given as a series around a period‐one fixed point of the system. Within the domain of the mapping's convergence, we then compute a Birkhoff normal form for 4D symplectic mappings as well as the associated approximate integrals of motion which can be expressed in terms of proper elements. The variations of the integrals predicted by the remainder of the normal form series provide a lower bound for the domain of Nekhoroshev stability for a time at least equal to the age of the Solar System. In the case of Jupiter's Trojans, the domain of the mapping's convergence lies entirely within the region of librational motion, in which the longitude of the perihelion of the asteroid librates around a fixed point value of ϖ. For most asteroids outside this domain macroscopic chaotic diffusion cannot be ruled out. The present analysis provides a physically relevant estimate of the domain of Nekhoroshev stability for proper librations (Dp < 100), and marginal for proper eccentricities (ep < 0.01). The formalism is developed in a general way allowing for applications in both, our Solar System and extrasolar system dynamics.

We investigate the stability of motion close to the Lagrangian equilibrium points L 4 and L 5 in the framework of the spatial, elliptic, restricted threebody problem, subject to the radial component of Poynting-Robertson drag.For this reason we develop a simplified resonant model, that is based on averaging theory, i.e. averaged over the mean anomaly of the perturbing planet. We find temporary stability of particles displaying a tadpole motion in the 1:1 resonance. From the linear stability study of the averaged simplified resonant model, we find that the time of temporary stability is proportional to β a 1 n 1 , where β is the ratio of the solar radiation over the gravitational force, and a 1 , n 1 are the semi-major axis and the mean motion of the perturbing planet, respectively. We extend previous results (Murray (1994)) on the asymmetry of the stability indices of L 4 and L 5 to a more realistic force model.Our analytical results are supported by means of numerical simulations. We implement our study to Jupiter-like perturbing planets, that are also found in extra-solar planetary systems.