Long-term integrations and stability of planetary orbits in our Solar system

Itô, Takashi; Tanikawa, Kiyotaka

doi:10.1046/j.1365-8711.2002.05765.x

Cited by 109 publications

(82 citation statements)

References 31 publications

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“…Our conclusion about the stability of the outer Solar System agrees with that of Ito and Tanikawa [12]. They performed two simulations of the Jovian planets and Pluto that spanned 5 × 10 10 years, approximately 10 times the age of the Solar System.…”

Section: Discussionsupporting

confidence: 89%

Long simulations of the Solar System: Brouwer's Law and chaos

Grazier¹,

Newman

Hyman

et al. 2005

ANZIAMJ

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The accuracy of long simulations of the Solar System is limited by the accumulation of round-off error. If the round-off error is systematic, the error in conserved quantities grows as t where t is time, and that in dynamical variables as t 2 . If the round-off error is stochastic, the error grows as t 1/2 and t 3/2 respectively. C1087In a previous study, we showed that it was possible to implement the order thirteen Störmer method so the errors grew stochastically for the two-dimensional Kepler problem. Here we show the implementation gives stochastic error growth on three-dimensional simulations of the Solar System. Our integrations are such that the positions of the major planets are known with an estimated error of no more than 2 • after 10 9 years, a precision unmatched by earlier investigations.Further, our numerical results suggest the outer Solar System is not chaotic as has previously been reported, but rather computational errors in positions grow no faster than t 3/2 , conforming with existing models for stochastic error growth in an otherwise well-behaved system of ordinary differential equations.

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Section: Discussionsupporting

confidence: 89%

Long simulations of the Solar System: Brouwer's Law and chaos

Grazier¹,

Newman

Hyman

et al. 2005

ANZIAMJ

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“…The long term integrations of (Varadi et al2003, Laskar et al, 2004a use precise models but span only a few 100 Myr. On the other hand, the long term integration of Ito and Tanikawa (2002) is performed over a few Gyr but do not include the relativistic contribution. …”

Section: Discussionmentioning

confidence: 99%

Chaotic diffusion in the Solar System

Laskar

2008

Icarus

142

176

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The discovery of the chaotic behavior of the planetary orbits in the Solar System (Laskar, 1989(Laskar, , 1990) was obtained using numerical integration of averaged equations. In (Laskar, 1994), these same equations are integrated over several Gyr and show the evidence of very large possible increase of the eccentricity of Mercury through chaotic diffusion. On the other hand, in the direct numerical integration of (Ito and Tanikawa, 2002) performed without general relativity over ±4 Gyr, the eccentricity of Mercury presented some chaotic diffusion, but with a maximal excursion smaller than about 0.35. In the present work, a statistical analysis is performed over more than 1001 different integrations of the secular equations over 5 Gyr. This allows to obtain for each planet, the probability for the eccentricity to reach large values. In particular, we obtain that the probability of the eccentricity of Mercury to increase beyond 0.6 in 5 Gyr is about 1 to 2 %, which is relatively large. In order to compare with (Ito and Tanikawa, 2002), we have performed the same analysis without general relativity, and obtained even more orbits of large eccentricity for Mercury. In order to clarify these differences, we have performed as well a direct integration of the planetary orbits, without averaging, for a dynamical model that do not include the Moon or general relativity with 10 very close initial conditions over 3 Gyr. The statistics obtained with this reduced set are comparable to the statistics of the secular equations, and in particular we obtain two trajectories for which the eccentricity of Mercury increases beyond 0.8 in less than 1.3 Gyr and 2.8 Gyr respectively. These strong instabilities in the orbital motion of Mecury results from secular resonance beween the perihelion of Jupiter and Mercury that are facilitated by the absence of general relativity. The statistical analysis of the 1001 orbits of the secular equations also provides probability density functions (PDF) for the eccentricity and inclination of the terrestrial planets (Mercury, Venus, the Earth and Mars) that are very well approximated by Rice PDF. This provides a very simple representation of the planetary PDF over 5 Gyr. On this time-scale the evolution of the PDF of the terrestrial planets is found to be similar to the one of a diffusive process. As shown in (Laskar, 1994), the outer planets orbital elements do not present significant diffusion, and the PDFs of their eccentricities and inclinations are well represented by the PDF of quasiperiodic motions with a few periodic terms.

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“…The Solar System is known to be "practically stable", in the sense that none of the 9 planets is likely to suffer mutual collisions, or be ejected from the Solar System, over the next several billion years 3,8,9,10 . The motion of Pluto is chaotic with a Lyapunov time of 10-20 million years 11,12,1 , while the inner Solar System has a Lyapunov time of about 4-5 million years 13, 14, 1 .…”

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confidence: 99%

Is the outer Solar System chaotic?

Hayes¹

2007

Nature Phys

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One-sentence summary: Current observational uncertainty in the positions of the Jovian planets precludes deciding whether or not the outer Solar System is chaotic. 100 word technical summary: The existence of chaos in the system of Jovian planets has been in question for the past 15 years. Various investigators have found Lyapunov times ranging from about 5 millions years upwards to infinity, with no clear reason for the discrepancy. In this paper, we resolve the issue. The position of the outer planets is known to only a few parts in 10 million. We show that, within that observational uncertainty, there exist Lyapunov timescales in the full range listed above. Thus, the "true" Lyapunov timescale of the outer Solar System cannot be resolved using current observations. 100 word summary for general public: The orbits of the inner planets (Mercury, Venus, Earth, and Mars) are practically stable in the sense that none of them will collide or be ejected from the Solar System for the next few billion years. However, their orbits are chaotic in the sense that we cannot predict their angular positions within those stable orbits for more than about 20 million years. The picture is less clear for the outer planets (Jupiter, Saturn, Uranus and Neptune). Again their orbits are practically stable, but it is not known for how long we can accurately predict their positions within those orbits.The existence of chaos among the Jovian planets is a contested issue. There exists apparently unassailable evidence both that the outer Solar System is chaotic 1, 2 , and that it is not 3, 4, 5 . The discrepancy is particularly disturbing given that computed chaos is sometimes due to numerical artifacts 6,7 . In this paper we discount the possibility of numerical artifacts, and demonstrate that the discrepancy seen between various investigators is real. It is caused by observational uncertainty in the orbital positions of the Jovian planets, which is currently a few parts in 10 million. Within that observational uncertainty, there exist clearly chaotic 1

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Long-term integrations and stability of planetary orbits in our Solar system

Cited by 109 publications

References 31 publications

Long simulations of the Solar System: Brouwer's Law and chaos

Long simulations of the Solar System: Brouwer's Law and chaos

Chaotic diffusion in the Solar System

Is the outer Solar System chaotic?

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