Determination and survey of periodic Trojan orbits in the restricted problem of three bodies

Rabe, Eugene

doi:10.1086/108451

Cited by 116 publications

(36 citation statements)

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“…A comparison with results obtained by numerical integration of the equations of motion indicates that the solution can predict the position of a close-earth satellite with an accuracy of better than 1 cm over a period of 1 month. For this check, a special-purpose Taylor-type integrator is adopted, in which the positions and velocities are expanded into a power series of time and the coefficients of the series are determined by recurrence formulas [Rabe, 1961;Deprit and Zahar, 1966].…”

Section: Current Statusmentioning

confidence: 99%

Applications of Geodesy to Geodynamics an International Symposium

Mueller

1977

Bull. Geodesique

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Section: Current Statusmentioning

confidence: 99%

Applications of Geodesy to Geodynamics an International Symposium

Mueller

1977

Bull. Geodesique

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“…Following Steffensen and Rabe [24,21], recursive computation of Taylor coefficients is used to compute sums, differences, products and powers of power series, etc. (see [3,16], and [9, pp.…”

Section: Introductionmentioning

confidence: 99%

On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

Nguyen‐Ba¹,

Giordano²,

Nguyen-Thu³

et al. 2017

J. of Mod. Meth. in Numer. Math.

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The contractivity-preserving 2-and 3-step predictor-corrector series methods for ODEs (T. Nguyen-Ba, A. Alzahrani, T. Giordano and R. Vaillancourt, On contractivity-preserving 2-and 3-step predictor-corrector series for ODEs, J. Mod. Methods Numer. Math. 8:1-2 (2017), pp. 17-39. doi:10.20454/jmmnm.2017.1130) are expanded into new optimal, contractivity-preserving (CP), d-derivative, k-step, predictor-corrector, Hermite-Birkhoff-Obrechkoff series methods, denoted by HBO(d, k, p), k = 4, 5, 6, 7, with nonnegative coefficients for solving nonstiff first-order initial value problems y = f (t, y), y(t 0 ) = y 0 . The main reason for considering this class of formulae is to obtain a set of methods which have larger regions of stability and generally higher upper bound p u of order p of HBO(d, k, p) for a given d. Their stability regions have generally a good shape and grow generally with decreasing p − d. A selected CP HBO method: 6-derivative 4-step HBO of order 14, denoted by HBO(6,4,14) which has maximum order 14 based on the CP conditions compares satisfactorily with Adams-Cowell of order 13 in PECE mode, denoted by AC(13), in solving standard N-body problems over an interval of 1000 periods on the basis of the relative error of energy as a function of the CPU time. HBO(6,4,14) also compares well with AC(13) in solving standard N-body problems on the basis of the growth of relative positional error, relative energy error and 10000 periods of integration. The coefficients of HBO(6,4,14) are listed in the appendix.

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“…In general, horseshoe orbits are not considered to be long-term stable (Dermott & Murray 1981a;Murray & Dermott 1999). Horseshoe orbits were originally predicted by Brown (1911) and Darwin (1912) and further studied later by, for example, Thuring (1959), Rabe (1961), Giacaglia (1970), Weissman & Wetherill (1974) and Garfinkel (1977), but were largely considered theoretical curiosities until the Saturnian moons Janus and Epimetheus were identified as horseshoe librators (Smith et al 1980;Synnott et al 1981;Dermott & Murray 1981b). The existence of minor planets moving on horseshoe orbits around the major planets was first postulated by Milani et al (1989) and Michel et al (1996) on theoretical grounds.…”

Section: Introductionmentioning

confidence: 99%

Crantor, a short-lived horseshoe companion to Uranus

Marcos

2013

A&A

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Context. Stable co-orbital motion with Uranus is vulnerable to planetary migration, but temporary co-orbitals may exist today. So far, only two candidates have been suggested, both moving on horseshoe orbits: 83982 Crantor (2002 GO 9 ) and 2000 SN 331 . Aims. (83982) Crantor is currently classified in the group of the Centaurs by the MPC although the value of its orbital period is close to that of Uranus. Here we revisit the topic of the possible 1:1 commensurability of (83982) Crantor with Uranus, explore its dynamical past, and look into its medium-term stability and future orbital evolution. Methods. Our analysis is based on the results of N-body calculations that use the most updated ephemerides and include perturbations by the eight major planets, the Moon, the barycenter of the Pluto-Charon system, and the three largest asteroids. Results. (83982) Crantor currently moves inside Uranus' co-orbital region on a complex horseshoe orbit. The motion of this object is primarily driven by the influence of the Sun and Uranus, although Saturn plays a significant role in destabilizing its orbit. The precession of the nodes of (83982) Crantor, which is accelerated by Saturn, controls its evolution and short-term stability. Although this object follows a temporary horseshoe orbit, more stable trajectories are possible and we present 2010 EU 65 as a long-term horseshoe librator candidate in urgent need of follow-up observations. Available data indicate that the candidate 2000 SN 331 is not a Uranus' co-orbital. Conclusions. Our calculations confirm that (83982) Crantor is currently trapped in the 1:1 commensurability with Uranus but it is unlikely to be a primordial 1:1 librator. Although this object follows a chaotic, short-lived horseshoe orbit, longer term horseshoe stability appears to be possible. We also confirm that high-order resonances with Saturn play a major role in destabilizing the orbits of Uranus co-orbitals.

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Determination and survey of periodic Trojan orbits in the restricted problem of three bodies

Cited by 116 publications

References 0 publications

Applications of Geodesy to Geodynamics an International Symposium

Applications of Geodesy to Geodynamics an International Symposium

On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

Crantor, a short-lived horseshoe companion to Uranus

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