On the stability of high inclination Trojans

Numerical integrations have been performed for orbits of Venus', Earth's and Mars' Trojan-asteroid test particles in a self consistent model of the solar system to study the stability of inclined Trojan orbits. In the case of Mars low inclination orbits tend to be unstable while the contrary seems to apply to Venus and Earth, although the stability of some very high inclination orbits may not be excluded on the basis of these computations.     

Dynamics of Mars Trojans

In this paper we explore the dynamical stability of the Mars Trojan region applying mainly Laskar's Frequency Map Analysis. This method yields the chaotic diffusion rate of orbits and allows to determine the most stable regions. It also gives the frequencies which are responsible for the instability of orbits. The most stable regions are found for inclinations between about 15° and 30°. For inclinations smaller than 15°, we confirm, by applying a synthetic secular theory, that the secular resonances ν₃, ν₄, ν₁₃, ν₁₄ rapidly excite asteroid orbits within a few Myrs, or even faster. The asteroids are removed from the Trojan region after a close encounter with Mars. For large inclinations, the secular resonance ν₅ clears a small region around 30° while the Kozai resonance rapidly removes bodies for inclinations larger than 35°. The dynamical lifetimes of the three L5 Trojans, (5261) Eureka, 1998 VF31, 2001 DH47, and the only L4 Trojan 1999 UJ7 are determined by numerically integrating clouds of corresponding clones over the age of the Solar System. All four Trojans reside in the most stable region with smallest diffusion coefficients. Their dynamical half-lifetime is of the order of the age of the Solar System. The Yarkovsky force has little effect on the known Trojans but for bodies smaller than about 1-5 m the drag is strong enough to destabilize Trojans on a timescale shorter than 4.5 Gyr.     

Long term wind erosion on Mars

Wind erosion and deposition are powerful agents of surface change on Mars. Erosion is sensitive to the atmospheric density, so feedback between orbit variations and atmospheric density can enhance the sensitivity of erosion rates to orbital parameters. We use statistics derived from a 1 Gyr integration of the spin axis of Mars, coupled with runs of the NASA Ames 3-D general circulation model (GCM) at a variety of orbital conditions and pressures, to explore this feedback. We find that wind erosion in the GCM is associated with two factors: baroclinic zone winds, and strong cross-equator solstice flows. Both of these factors are influenced by topography, producing an asymmetry in the erosion pattern between the north and the south. The erosion model, averaged over 1 Gyr, produces potential erosion rates of in the north and in the south, which increase by an order of magnitude in an early 28 mbar atmosphere. The stability of these potential erosion patterns over geological time indicates that the lowland regions of Mars are continuously eroded, and that wind is capable of revealing ancient sedimentary deposits, or eroding substantial deposits that may have otherwise been preserved.     

Long term evolution of comet Halleys orbit

The aim of the paper is to study the long term evolution of comet Halleys orbit taking into account small errors in the initial conditions. Recent papers deal with mapping methods to model cometary dynamics; (e.g. Petrosky and Broucke, 1987 and Chirikov and Vecheslavov, 1986). They will be discussed critically and compared with our own results. We then tested the model using numerical integration methods. For the moment we limited our calculation to 2.10⁵ years, but a 10⁶ year integration is still in progress. We show the expected dynamical evolution of Hallyes orbit taking into account also smaller and larger errors of the initial conditions (nongravitational effects are only roughly estimated). Finally we discuss alsothe controversal opinions concerning the role of the planets (especially the earth).     

Long term evolution of quasi-circular Trojan orbits

Trojan asteroids undergo very large perturbations because of their resonance with Jupiter. Fortunately the secular evolution of quasi circular orbits remains simple—if we neglect the small short period perturbations. That study is done in the approximation of the three dimensional circular restricted three-body problem, with a small mass ratio μ—that is about 0.001 in the Sun Jupiter case. The Trojan asteroids can be defined as celestial bodies that have a “mean longitude”, M + ω + Ω, always different from that of Jupiter. In the vicinity of any circular Trojan orbit exists a set of “quasi-circular orbits” with the following properties: (A) Orbits of that set remain in that set with an eccentricity that remains of the order of the mass ratio μ. (B) The relative variations of the semi-major axis and the inclination remain of the order of ${\sqrt{\mu}}$ . (C) There exist corresponding “quasi integrals” the main terms of which have long-term relative variations of the order of μ only. For instance the product c(1 – cos i) where c is the modulus of the angular momentum and i the inclination. (D) The large perturbations affect essentially the difference “mean longitude of the Trojan asteroid minus mean longitude of Jupiter”. That difference can have very large perturbations that are characteristics of the “horseshoes orbit”. For small inclinations it is well known that this difference has two stable points near ±60° (Lagange equilibrium points L₄ and L₅) and an unstable point at 180° (L₃). The stable longitude differences are function of the inclination and reach 180° for an inclination of 145°41′. Beyond that inclination only one equilibrium remains: a stable difference at 180°.     

Long term evolution of solar sector structure

The large-scale structure of the solar magnetic field during the past five sunspot cycles (representing by implication a much longer interval of time) has been investigated using the polarity (toward or away from the Sun) of the interplanetary magnetic field as inferred from polar geomagnetic observations. The polarity of the interplanetary magnetic field has previously been shown to be closely related to the polarity (into or out of the Sun) of the large-scale solar magnetic field. It appears that a solar structure with four sectors per rotation persisted through the past five sunspot cycles with a synodic rotation period near 27.0 days, and a small relative westward drift during the first half of each sunspot cycle and a relative eastward drift during the second half of each cycle. Superposed on this four-sector structure there is another structure with inward field polarity, a width in solar longitude of about 100° and a synodic rotation period of about 28 to 29 days. This 28.5 day structure is usually most prominent during a few years near sunspot maximum. Some preliminary comparisons of these observed solar structures with theoretical considerations are given.     

Spectroscopy and photometry of Mars Trojans

Mars is the only terrestrial planet known to have co-orbiting “Trojan” asteroids. We have obtained visible and near-IR reflectance spectra of three of these objects: 5261 Eureka and 1998 VF31 in the L5 region and 1999 UJ7 in the L4 region. We also obtained JHK spectrophotometry and a visible lightcurve for 5261 Eureka. The asteroid 5261 Eureka has a visible spectrum that is classified as Sr in the Bus taxonomy, and has infrared colors consistent with the A-class asteroids. The data for 1998 VF31 have a restricted wavelength range, but are most consistent with the Sr or Sa class, though we note a marginal consistency with the D class. We can rule out a C-class classification. 1999 UJ7 has an X-class or T-class spectrum, which is unlike that of the other two Mars Trojans. The photometric data for Eureka are limited, but we can constrain the period to longer than 5 hours (likely 5.5-6 hours) and lightcurve amplitude of at least 0.15 magnitude at this viewing geometry. The spectral differences among the Mars Trojans suggests that either they did not all form at their present solar distances or that they have not always been at their present sizes.     

Spectroscopic observations of Jupiter Trojans

We present the results of a campaign of spectroscopic observations of Jupiter Trojan asteroids. Thirty-four objects were observed during three runs in July and November 1998, and March 2002 using the Danish 1.54-m telescope at ESO. The covered spectral range was between 5000 and 9000 Å. Our observations include objects belonging both L₄ to L₅ clouds. According to analyses of previous investigations of Trojans, the spectra of different taxonomic classes can be separated on the basis of the slope of the reflectance spectrum. The large majority of the objects of our sample have been found to belong to the D taxonomic class, but we found also objects of P- and C-type. In two cases, we found also evidence of blueish spectral trends. Our data are important, since they allow us to substantially enlarge the whole data set of available Trojan spectra.     

Long term variations in solar flare activity

A statistical analysis of the contemporary (1954-1975) solar flare particle events has been made for the parametersF (integrated, proton fluence in cm^-2 in an event with kinetic energy above 10 MeV) andR ₀ (the characteristic rigidity). These data are compared with the long-term averaged values determined from stable- and radio-nuclide measurements of lunar samples. The analysis shows that the ancient solar flare proton spectrum was harder (higher R₀ values) compared to that observed in contemporary flares. A similar analysis can not be made for the mean long-term averaged flux (ˉJ, cm^-2 S^-1), since the contemporary averages suffer from an uncertainty due to the statistics of a single event. However, the average flux estimates for time durations 〈T〉 exceeding 10³ yr, are free from such uncertainties. The long-term averaged ˉJ values obtained over different time scales (10⁴ - 10⁶ yr) suggest a possible periodic variation in solar flare activity, with enhanced flux level during the last 10⁵ yr. The available data rule out the occurrence of giant flares, with proton fluence exceeding 10¹⁵ cm^-2 during the last million years.     

On the Instability of Jupiter's Trojans

We have numerically explored the mechanisms that destabilize Jupiter's Trojan orbits outside the stability region defined by Levison et al. (1997, Nature385, 42-44). Different models have been exploited to test various possible sources of instability on timescales on the order of ∼10⁸ years.In the restricted three-body model, only a few Trojan orbits become unstable within 10⁸ years. This intrinsic instability contributes only marginally to the overall instability found by Levison et al.In a model where the orbital parameters of both Jupiter and Saturn are fixed, we have investigated the role of Saturn and its gravitational influence. We find that a large fraction of Trojan orbits become unstable because of the direct nonresonant perturbations by Saturn. By shifting its semimajor axis at constant intervals around its present value we find that the near 5:2 mean motion resonance between the two giant planets (the Great Inequality) is not responsible for the gross instability of Jupiter's Trojans since short-term perturbations by Saturn destabilize Trojans, even when the two planets are far out of the resonance.Secular resonances are an additional source of instability. In the full six-body model with the four major planets included in the numerical integration, we have analyzed the effects of secular resonances with the node of the planets. Trojan asteroids have relevant inclinations, and nodal secular resonances play an important role. When a Trojan orbit becomes unstable, in most cases the libration amplitude of the critical argument of the 1:1 mean motion resonance grows until the asteroid encounters the planet. Libration amplitude, eccentricity, and nodal rate are linked for Trojan orbits by an algebraic relation so that when one of the three parameters is perturbed, the other two are affected as well. There are numerous secular resonances with the nodal rate of Jupiter that fall inside the region of instability and contribute to destabilize Trojans, in particular the ν₁₆. Indeed, in the full model the escape rate over 50 Myr is higher compared to the fixed model.Some secular resonances even cross the stability region delimited by Levison et al. and cause instability. This is the case of the 3:2 and 1:2 nodal resonances with Jupiter. In particular the 1:2 is responsible for the instability of some clones of the L4 Trojan (3540) Protesilaos.     

Trojans in Stable Chaotic Motion

The orbits of 13 Trojan asteroids have been calculated numerically in the model of the outer solar system for a time interval of 100 million years. For these asteroids Milani et al. (1997) determined Lyapunov times less than 100 000 years and introduced the notion "asteroids in stable chaotic motion". We studied the dynamical behavior of these Trojan asteroids (except the asteroid Thersites which escaped after 26 million years) within 11 time intervals - i.e. subintervals of the whole time - by means of: (1) a numerical frequency analysis (2) the root mean square (r.m.s.) of the orbital elements and (3) the proper elements. For each time interval we compared the root mean squares of the orbital elements (a, e and i) with the corresponding proper element. It turned out that the variations of the proper elements e_p in the different time intervals are correlated with the corresponding r.m.s.(e); this is not the case for sin I_p with r.m.s.(i). This revised version was published online in July 2006 with corrections to the Cover Date.     

The dynamics of inclined Neptune Trojans

We investigated the stable area for fictive Trojan asteroids around Neptune’s Lagrangean equilibrium points with respect to their semimajor axis and inclination. To get a first impression of the stability region we derived a symplectic mapping for the circular and the elliptic planar restricted three body problem. The dynamical model for the numerical integrations was the outer Solar system with the Sun and the planets Jupiter, Saturn, Uranus and Neptune. To understand the dynamics of the region around L ₄ and L ₅ for the Neptune Trojans we also used eight different dynamical models (from the elliptic problem to the full outer Solar system model with all giant planets) and compared the results with respect to the largeness and shape of the stable region. Their dependence on the initial inclinations (0° < i < 70°) of the Trojans’ orbits could be established for all the eight models and showed the primary influence of Uranus. In addition we could show that an asymmetry of the regions around L ₄ and L ₅ is just an artifact of the different initial conditions.     

The fate of primordial lunar Trojans

Multiple large impact basins on the lunar nearside formed in a relatively-short interval around 3.8-3.9 Gyr ago, in what is known as the Lunar Cataclysm (LC; also known as Late Heavy Bombardment). It is widely thought that this impact bombardment has affected the whole Solar System or at least all the inner planets. But with non-lunar evidence for the cataclysm being relatively weak, a geocentric cause of the Lunar Cataclysm cannot yet be completely ruled out [Ryder, G., 1990. Eos 71, 313, 322-323]. In principle, late destabilization of an additional Earth satellite could result in its tidal disruption during a close lunar encounter (cf. [Asphaug, E., Agnor, C.B., Williams, Q., 2006. Nature 439, 155-160]). If the lost satellite had D>500 km, the resulting debris can form multiple impact basins in a relatively short time, possibly explaining the LC. Canup et al. [Canup, R.M., Levison, H.F., Stewart, G.R., 1999. Astron. J. 117, 603-620] have shown that any additional satellites of Earth formed together with (and external to) the Moon would be unable to survive the rapid initial tidally-driven expansion of lunar orbit. Here we explore the fate of objects trapped in the lunar Trojan points, and find that small lunar Trojans can survive the Moon's orbital evolution until they and the Moon reach 38 Earth radii, at which point they are destabilized by a strong solar resonance. However, the dynamics of Trojans containing enough mass to cause the LC (diameters >150 km) is more complex; we find that such objects do not survive the passage through a weaker solar resonance at 27 Earth radii. This distance was very likely reached by the Moon long before the LC, which seems to rule out the disruption of lunar Trojans as a cause of the LC.     

Long term observations of the Hα chromospheric network

Frequent filtergrams of the quiet sun at H + 0.65 Å were taken from above the Arctic Circle during a period of 62 hr. Features observed in individual filtergrams or movies are described with the dynamic changes they undergo. Thirty filtergrams taken at 2-hr intervals are presented and the development of a typical supergranule is shown in some detail. A study of supergranules shows that they lose their identity in about 21 hr on the average.     

Long term dynamical evolution and classification of classical TNOs

Classical trans-Neptunian objects (TNOs) are believed to represent the most dynamically pristine population in the trans-Neptunian belt (TNB) offering unprecedented clues about the formation of our Solar System. The long term dynamical evolution of classical TNOs was investigated using extensive simulations. We followed the evolution of more than 17000 particles with a wide range of initial conditions taking into account the perturbations from the four giant planets for 4 Gyr. The evolution of objects in the classical region is dependent on both their inclination and semimajor axes, with the inner (a<45 AU) and outer regions (a>45 AU) evolving differently. The reason is the influence of overlapping secular resonances with Uranus and Neptune (40–42 AU) and the 5:3 (a∼ ∼42.3 AU), 7:4 (a∼ ∼43.7 AU), 9:5 (a∼ ∼44.5 AU) and 11:6 (a∼ ∼ 45.0 AU) mean motion resonances strongly sculpting the inner region, while in the outer region only the 2:1 mean motion resonance (a∼ ∼47.7 AU) causes important perturbations. In particular, we found: (a) A substantial erosion of low-i bodies (i<10°) in the inner region caused by the secular resonances, except those objects that remained protected inside mean motion resonances which survived for billion of years; (b) An optimal stable region located at 45 AU<a<47 AU, q>40 AU and i>5° free of major perturbations; (c) Better defined boundaries for the classical region: 42–47.5 AU (q>38 AU) for cold classical TNOs and 40–47.5 AU (q>35 AU) for hot ones, with i=4.5° as the best threshold to distinguish between both populations; (d) The high inclination TNOs seen in the 40–42 AU region reflect their initial conditions. Therefore they should be classified as hot classical TNOs. Lastly, we report a good match between our results and observations, indicating that the former can provide explanations and predictions for the orbital structure in the classical region.     

A search of outer Trojans on ASTROVIRTEL images

This paper discusses the main results of a search of outer Trojans on ESO/ESA ASTROVIRTEL images, and the usefulness and the limitations of such an archive for the discovery and dynamical classification of minor bodies. The work was performed in the frame of a program having the aim to detect bodies orbiting around the Lagrangian points of the outer planets. A large number of images taken from 1999 to 2001 with the wide field imager (WFI) camera of the 2.2 m telescope of ESO at La Silla for entirely different scientific reasons, but happening to be in the wanted Lagrangian positions, was retrieved with a dedicated search engine. The moving objects present on these images were detected and examined with specific software tools that permitted one to find out about 1500 minor bodies (mostly previously unknown). More than 5000 positions were sent to the Minor Planet Center, and the search was given the designation I03. The total sky area covered by I03 is of approximately 50 square degrees, and in some areas the limiting magnitudes were fainter than the 24th in the R band. A preliminary classification of the kinematics of the I03 discoveries (using a dedicated code, named AMIGO apparent motion interpretation of generic orbits) has produced several Trojan candidates of Jupiter and of the outer planets. Furthermore, pre-discovery positions have been provided for one NEO, three Jupiter Trojans and three TNOs.     

Chaotic Diffusion And Effective Stability of Jupiter Trojans

It has recently been shown that Jupiter Trojans may exhibit chaotic behavior, a fact that has put in question their presumed long term stability. Previous numerical results suggest a slow dispersion of the Trojan swarms, but the extent of the ‘effective’ stability region in orbital elements space is still an open problem. In this paper, we tackle this problem by means of extensive numerical integrations. First, a set of 3,200 fictitious objects and 667 numbered Trojans is integrated for 4 Myrs and their Lyapunov time, T_L, is estimated. The ones following chaotic orbits are then integrated for 1 Gyr, or until they escape from the Trojan region. The results of these experiments are presented in the form of maps of T_Land the escape time, T_E, in the space of proper elements. An effective stability region for 1 Gyr is defined on these maps, in which chaotic orbits also exist. The distribution of the numbered Trojans follows closely the T_E=1 Gyr level curve, with 86% of the bodies lying inside and 14% outside the stability region. This result is confirmed by a 4.5 Gyr integration of the 246 chaotic numbered Trojans, which showed that 17% of the numbered Trojans are unstable over the age of the solar system. We show that the size distributions of the stable and unstable populations are nearly identical. Thus, the existence of unstable bodies should not be the result of a size-dependent transport mechanism but, rather, the result of chaotic diffusion. Finally, in the large chaotic region that surrounds the stability zone, a statistical correlation between T_LandT_E is found.     

Long term storage of relativistic particles in the solar corona

A model is presented which shows that large numbers of energetic electrons (0.3-> 10 MeV) and protons (1–30 MeV) can be stored in the solar corona at altitudes around 3 × 10⁵ km for periods in excess of 5 days. Specific reference is made to the time period July 6–16 1968 as an excellent example of energetic solar particle storage. Time histories of interplanetary charged particle intensities observed by the IMP-4 and Pioneer 8 satellites are used to substantiate this contention. Detailed reference is also made to solar X-ray, optical and radio data obtained during the period in question, in addition to interplanetary magnetometer data. This model provides a unique solution to many hitherto unexplained solar particle events, and can also account for the lack of prompt particle emission from certain large solar flares recorded in the past.