Trojan Planets Research Articles

Stability of fictitious Trojan planets in extrasolar systems

AbstractOur work deals with the dynamical possibility that in extrasolar planetary systems a terrestrial planet may have stable orbits in a 1:1 mean motion resonance with a Jovian like planet. We studied the motion of fictitious Trojans around the Lagrangian points L4/L5 and checked the stability and/or chaoticity of their motion with the aid of the Lyapunov Indicators and the maximum eccentricity. The computations were carried out using the dynamical model of the elliptic restricted three‐body problem that consists of a central star, a gas giant moving in the habitable zone, and a massless terrestrial planet. We found 3 new systems where the gas giant lies in the habitable zone, namely HD99109, HD101930, and HD33564. Additionally we investigated all known extrasolar planetary systems where the giant planet lies partly or fully in the habitable zone. The results show that the orbits around the Lagrangian points L4/L5 of all investigated systems are stable for long times (107 revolutions). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Secondary resonances of co-orbital motions

The size distribution of the stability region around the Lagrangian point L4 is investigated in the elliptic restricted three-body problem as the function of the mass parameter and the orbital eccentricity of the primaries. It is shown that there are minimum zones in the size distribution of the stability regions, and these zones are connected with the secondary resonances between the frequencies of librational motions around L4. The results can be applied to hypothetical Trojan planets for predicting values of the mass parameter and the eccentricity for which such objects can be expected or their existence is less probable.

Trojan planets in HD 108874?

Aims. It turned out recently that, in addition to a large planet with a semimajor axis a ∼ 1 AU and a low eccentricity (e ∼ 0.07), the extrasolar planetary system HD 108874 harbors another massive planet with 2.43 AU < a < 2.93 AU. The inner planet is orbiting the G5 host star in the habitable zone (=HZ); so that we could established stable regions for Earth-like Trojan planets. Methods. We integrated some 10 5 orbits of fictitious Trojans around the Lagrangian points for up to 107 years and checked the stability of the orbital elements and their chaoticity with the aid of the Fast Lyapunov Indicator. Results. It turns out that this multiplanetary system is the first one where - with the uncertainties in eccentricity and semimajor axes of the outer planet - the existence of Trojan terrestrial planets in stable orbits in the HZ is possible for some combinations of the orbital parameters.

Trojan Pairs in the HD 128311 and HD 82943 Planetary Systems?

Two nearby stars, HD 128311 and HD 82943, are believed to host pairs of Jupiter-like planets involved in a strong first-order 2:1 mean motion resonance (MMR). In this work, we reanalyze available radial velocity (RV) measurements and demonstrate that it is also possible to explain the observed RV variations of the parent stars as being induced by a pair of Trojan planets (i.e., in a 1:1 MMR). We show that these Trojan configurations reside in extended zones of stability in which such systems can easily survive in spite of the large masses of the planets, large eccentricities, and nonzero mutual inclinations of their orbits. We also show that HD 82943 could harbor a previously unknown third planet of ~0.5MJ in ~2 AU orbit.

Trojans in Habitable Zones

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."

Could the 47 Ursae Majoris Planetary System be a Second Solar System? Predicting the Earth‐like Planets

(Abridged)We numerically investigated the dynamical architecture of 47 UMa with the planetary configuration of the best-fit orbital solutions by Fischer et al. We systematically studied the existence of Earth-like planets in the region 0.05 AU $\leq a \leq 2.0$ AU for 47 UMa with numerical simulations, and we also explored the packed planetary geometry and Trojan planets in the system. In the simulations, we found that "hot Earths" at 0.05 AU $\leq a < $ 0.4 AU can dynamically survive at least for 1 Myr. The Earth-like planets can eventually remain in the system for 10 Myr in areas involved in the mean motion resonances (MMR) (e.g., 3:2 MMR) with the inner companion. Moreover, we showed that the 2:1 and 3:1 resonances are on the fringe of stability, while the 5:2 MMR is unstable. Additionally, the 2:1 MMR marks out a remarkable boundary between chaotic and regular motions, inside, most of the orbits can survive, outside, they are mostly lost in the orbital evolution. In a dynamical sense, the most likely candidate for habitable environment is Earth-like planets with orbits in the ranges 0.8 AU $\leq a < 1.0$ AU and 1.0 AU $ < a < 1.30$ AU (except several unstable cases) with relatively low eccentricities. The Trojan planets with low eccentricities and inclinations can secularly last at the triangular equilibrium points of two massive planets. Hence, the 47 UMa planetary system may be a close analog to our solar system.

Trojans and Comets of the Jupiter Group

In a recent paper (Rabe, 1970), I suggested that at least some comets of the Jupiter group may have originated from the relatively dense Trojan clouds that, according to a recent survey by the van Houtens and Gehrels (1970), seem to be associated with the equilateral points L4 and L5 of the Jupiter orbit. This suggestion was inspired by the finding that the periodic comet Slaughter-Burnham has been captured (or perhaps recaptured) into unstable or temporary “Trojan” librations lasting approximately 2500 yr and by the circumstance that the Jacobi “constants” of most Jupiter group comets have values between 3.0 and 2.5, or just in that range which would also be occupied by all known and unknown Trojans associated with Jupiter in stable or unstable librations involving heliocentric eccentricities up to about 0.5 and inclinations as large as 30°. Moreover, it is well known that nearly all comets of the Jupiter group are able to approach Jupiter rather closely, thus providing the possibility for such drastic orbital changes as temporary capture into, or escape from, librational motion of the Trojan type. All, presumably stable, orbits of actual Trojan planets presently known have eccentricities e not exceeding 0.15, but we also know that stable short-period librations with much larger e values do exist in the restricted Sun-Jupiter problem, so that even in the real, nonrestricted situation there should be a possibility for corresponding librations with these more substantial short-period components. These librations may be unstable but may have long lifetimes nevertheless. The case of P/Slaughter-Burnham has proved that such motions are indeed possible, even with an e as large as 0.52.

Third-order stability of the long-period Trojan librations

view Abstract Citations (25) References (1) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Third-order stability of the long-period Trojan librations Rabe, Eugene Abstract Nearly all existing studies of the stability properties of periodic solutions of the restricted problem of three bodies are based on linearized variation equations, so that the results are valid only for infinitesimally small displacements from the given periodic orbit. Since the normally nonperiodic Trojan-type librations in the sun-Jupiter plane, about either one of the two equilateral points of Lagrange, may be treated as pre- dominantly short-period oscillations about a suitable long-period reference solution, the question of the actual dependence of the librational stability on the magnitude of such superposed oscillations is a matter of considerable interest. The second- and third-order terms of the variation equations are therefore included in the present analysis of the stability of the long-period Trojan librations which are available since their earlier determination (Rabe 1961, 1962) by numerical methods. For any such reference libration, the first- order solution from the linearized variation equations involves a purely imaginary characteristic exponent ic, independent of the arbitrary oscillation amplitude. In the second-order increments to this solution, new short-period terms involving 2ic emerge, together with additional terms of the basic long period T, while the "stability constant" c, of order 12, remains unchanged and independent of the constants of integration. At the third-order stage of the solution, however, c becomes a function of the principal amplitude of the short- period oscillations which is nearly proportional to the helicentric eccentricity e of such a nonperiodic Trojan planet (the long-period reference Trojans have nearly circular orbits around the sun). The initial phase of the principal oscillation has a comparatively higher-order effect on c. For sufficiently small e, and for basic long-period libration amplitudes typical of the real Trojans, the oscillating librational motion remains stable, because the properly adjusted c (e) are found to remain real. For e values exceeding a certain critical em depending on the reference libration, however, instability is indicated by the resulting two conjugate complex values of c. This limiting stable eccentricity e is about 0.19 for very small basic librations, but decreases with the increasing size of the reference orbit. When the latter extends over about 430 in longitude, for example, has decreased to roughly 0.08. The stability limits indicated by the present results agree very well with the observed maximum eccentricities and libration amplitudes of the known Trojan planets. Publication: The Astronomical Journal Pub Date: February 1967 DOI: 10.1086/110196 Bibcode: 1967AJ.....72...10R full text sources ADS |

Trojan orbits: I. d'Alembert Series at L4

The analytical continuation of the family of long period orbits at L 4 represents the normal Castesian coordinates of the asteroid as d'Alembert series in function of an orbital parameter. By means of a program that enables a computer to calculate symbolically in the real graded algebra of d'Alembert series, the family of Trojan orbits is computed numerically up to the fourteenth order. Within the range of amplitudes of real Trojan planets, the d'Alembert series are in very close agreement with the numerical results of Prof. E. Rabe. Outside this range, the poor convergence of the Fourier series is found to be imputed, not to a 13:1 commensurability between the Trojan's libration period and Jupiter's orbital period, but to the slow convergence of the d'Alembert series all along the family of long period librations.

Our Astronomical Column

A New Trojan Minor Planet.—Circular 294 of the Astronomisches Recheninstitut contains an announcement by Dr. G. Stracke that he has discovered a seventh Trojan planet. It was found by K. Reinmuth, assistant at K¶nigstuhl Observatory, on Jan. 27, and observed by him again on Feb. 19 and 24; Dr. Stracke deduced the orbit from these three places.and finds the mean daily motion to be 297.852″, almost exactly that of Jupiter. Both i and φ are moderate, being 3° 9′ and 8° 42′ respectively, so the oscillations from the equilateral position will not be so large as in some cases. It makes a fifth member of the group of Trojans the longitude of which is 60° greater than that of Jupiter, while there are only two on the other side of Jupiter.

University and Educational Intelligence

University and Educational Intelligence