# The outer solar system for 200 million years

DOI: 10.1086/114149 · Source: NTRS

Abstract

A special-purpose computer is used to integrate the orbits of the outer five planets for more than 100 Myr into the future and more than 100 Myr into the past. The strongest features in the Fourier transforms of the orbital elements of the Jovian planets can be identified with the frequencies predicted by linear secular theory. Many of the weaker features in the Fourier spectra are identified as linear combinations of the basic frequencies. Serious differences are noted between the present measurements and the predictions of Bretagnon (1974). The amplitude of the 3.796 Myr period libration of Pluto's longitude of perihelion is modulated with a period of 34 Myr. Very long periods, on the order of 137 Myr, are also seen. The orbit of Pluto is stable for the duration of the integration; the maximum Liapunov characteristic exponent is less than 10 to the -6.8 power/yr.

- ... This test was done in Wisdom & Holman (1991) (hereafter WH91) using WHJ with the same h and t = 1 Gyr; we repeat this test but with varying initial conditions. We collect sample output every 10,000 yrs as in Applegate et al. (1986). The integration lasted under an hour using unvectorized C-code running on a 2.5 GHz Intel Core i7 processor. ...... We shifted the coordinates into a barycentric frame. The WH91 initial rectangular coordinate ephemerides are written in Applegate et al. (1986); theirˆztheirˆ theirˆz is defined by the total angular momentum vector of the Solar System at some time. Fig. 9shows three of Pluto's Keplerian elements as a function of time. ...Article
- Dec 2016
- MON NOT R ASTRON SOC

The symplectic Wisdom-Holman map revolutionized long-term integrations of planetary systems. There is freedom in such methods of how to split the Hamiltonian and which coordinate system to employ, and several options have been proposed in the literature. These choices lead to different integration errors, which we study analytically and numerically. The Wisdom-Holman method in Jacobi coordinates and the method of Hernandez, H16, compare favorably and avoid problems of some of the other maps, such as incorrect center-of-mass position or truncation errors even in the one-planet case. We use H16 to compute the evolution of Pluto’s orbital elements over 500 million years in a new calculation. - ... – 3 – noted that the precession frequency of Saturn's spin axis has a value close to s 8 −0.692 arcsec yr −1 , where s 8 is the mean nodal regression of Neptune's orbit (or, equivalently, the eighth nodal eigenfrequency of the planetary system; e.g., Applegate et al. 1986; Laskar 1988). Similarly, Ward & Canup (2006) pointed out that the precession frequency of Jupiter's spin axis has a value close to s 7 −2.985 ...... (e.g., Applegate et al. 1986; Laskar 1988), with the linear terms having typically the largest amplitudes A i . As discussed in Section 1, terms with present frequencies s 7 −2.985 ...Article
- May 2015
- Astrophys J

We study the effects of planetary late migration on the gas giants obliquities. We consider the planetary instability models from Nesvorny & Morbidelli (2012), in which the obliquities of Jupiter and Saturn can be excited when the spin-orbit resonances occur. The most notable resonances occur when the $s_7$ and $s_8$ frequencies, changing as a result of planetary migration, become commensurate with the precession frequencies of Jupiter's and Saturn's spin vectors. We show that Jupiter may have obtained its present obliquity by crossing of the $s_8$ resonance. This would set strict constrains on the character of migration during the early stage. Additional effects on Jupiter's obliquity are expected during the last gasp of migration when the $s_7$ resonance was approached. The magnitude of these effects depends on the precise value of the Jupiter's precession constant. Saturn's large obliquity was likely excited by capture into the $s_8$ resonance. This probably happened during the late stage of planetary migration when the evolution of the $s_8$ frequency was very slow, and the conditions for capture into the spin-orbit resonance with $s_8$ were satisfied. However, whether or not Saturn is in the spin-orbit resonance with $s_8$ at the present time is not clear, because the existing observations of Saturn's spin precession and internal structure models have significant uncertainties. - ... It is a brief overview of the works on the study of the Solar system orbital evolution. The secular orbital evolution of 8 planets of the Solar system is numerically studed in work of J. Laskar [24] over 30 million years and it is studed in works of J. Applegate [1] and J. Laskar [25,26] over 200 million years. The chaotic properties of the Solar system, the estimations of the size of the chaotic zones and the chaotic diffusion of the planets are investigated in the set works of J. Laskar [27,28,29]. ...Preprint
- Sep 2018

The application of computer algebra system Piranha to the investigation of the planetary problem is described in this work. Piranha is an echeloned Poisson series processor authored by F. Biscani from Max Planck Institute for Astronomy in Heidelberg. Using Piranha the averaged semi-analytical motion theory of four-planetary system is constructed up to the second degree of planetary masses. In this work we use the algorithm of the Hamiltonian expansion into the Poisson series in only orbital elements without other variables. The motion equations are obtained analytically in time-averaged elements by Hori-Deprit method. Piranha showed high-performance of analytical manipulations. Different properties of obtained series are discussed. The numerical integration of the motion equations is performed by Everhart method for the Solar system's giant-planets and some exoplanetary systems. - ... The R3BPQP model (Fig. 6, middle) has been obtained by integrating the complete system, but without taking the direct perturbations of Saturn, Uranus, and Neptune into account on the motion of the satellite. The orbit of the Sun is thus quasi-periodic, and its frequencies can be found for example in Applegate et al. (1986), Carpino et al. (1987, or Robutel & Gabern (2006). Resonances can now be created between the frequencies of the satellites and all the frequencies of the outer Solar System contained in the spectrum of the motion of the Sun. ...Context. The dynamical region of the Jovian irregular satellites presents an interesting web of resonances that are not yet fully understood. Of particular interest is the influence of the resonances on the stochasticity of the individual orbits of the satellites, as well as on the long-term chaotic diffusion of the different families of satellites. Aims: We make a systematic numerical study of the satellite region to determine the important resonances for the dynamics, to search for the chaotic zones, and to determine their influences on the dynamics of the satellites. We also compare these numerical results to previous analytical works. Methods: Using extensive numerical integrations of the satellites along with an indicator of chaos (MEGNO), we show global and detailed views of the retrograde and prograde regions for various dynamical models of increasing complexity and indicate the appearance of the different types of resonances and the implied chaos. Results: Along with secular and mean motion resonances that shape the dynamical regions of the satellites, we report a number of resonances involving the Great Inequality, and which are present in the system thanks to the wide range of the values of frequencies of the pericenter available for the satellites. The chaotic diffusion of the satellites is also studied and shows the long-term stability of the Ananke and Carme families, in contrast to the Pasiphae family. Tables 1 and 2 are available in electronic form at http://www.aanda.org
- Article
- Jun 1993
- J GEOPHYS RES

The climate-system variability on the time scale 10 exp 4 to 10 exp 6 years for the late Pliocene and the Pleistocene, and its possible sources, was explored by comparing three deep-sea-sediment delta-(O-18) records (interpreted to reflect changes in continental ice volume and thus changes in insolation and orbital variability) with the insolation series at 65 deg N. The results were also compared with delta-(O-18) simulations from a version of a Birchfield and Grumbine (1958) ice-sheet and bedrock climate model, in order to determine the importance of internal variability arising from feedback between the ice sheets and bedrock through the late Pliocene and Pleistocene in the simulations and in the climate records. The results of these studies indicate that the ice sheet and bedrock dynamics can account for a substantial part of the suborbital variability during the Pleistocene. The internal variability generated by this nonlinear dynamics enhances and modifies orbital forcing at low and very low frequencies (hundreds to thousands of kiloyears). - Article
- Aug 1995
- ICARUS

The dynamical state of the Pluto-Charon binary is distinctive in several respects including its well-known position in the Neptune 3:2 mean motion resonance, its librating argument of perihelion, and its high heliocentric eccentricity and inclination. Here we present a suite of numerical simulations of bodies in the outer Solar System which demonstrate that Pluto's high-heliocentriceccentricity and high-inclination states can both result directly from objects initially on low-inclination, nearly-circular orbits. This evolution occurs entirely due to gravitational interactions with the giant planets in their present orbits and causes objects to be trapped in both the Neptune 3:2 mean motion resonance and a perihelion libration similar to Pluto's, but with a 3:2 libration amplitude that is much larger than that of the Pluto-Charon binary. Therefore, in order to achieve a complete scenario for the evolution of Pluto into its present dynamical state, it is also necessary for Pluto's 3:2 libration amplitude to be damped by some dissipative event or events. We show that there are several mechanisms that can achieve this dissipation including (i) a single giant impact (which may have formed the binary itself), or (ii) a large number of physical collisions and gravitational interactions with the primordial Kuiper belt population. - Article
- Oct 2001
- Chin Astron Astrophys

This is a discussion and personal views on current important topics and methods in solar system dynamics. The topics include dynamical models, orbit resonance, planetary rings, chaos and secular evolution, motion of near-earth asteroids, the Kuiper belt, gravitational theory of the solar system and other related problems. - ArticleFull-text availableWe review some of the major achievements in celestial mechanics and dynamical astronomy in the 20th century to look for their new directions in the 21st century.
- Article
- Jan 2003

In this article the linear combinations of fundamental frequencies of a simplified Solar System were determined. This model consists of Venus, Earth, Mars, Jupiter and Saturn with their actual orbital elements. The frequencies present in the system were derived on one hand using the results of a numerical integration via an appropriate frequency analysis and on the other hand via the analytical Laplace-Lagrange theory. These frequencies were compared to already existing results; it turned out that this model is a very good representation of the whole Solar System concerning the frequencies involved. The amplitudes of the planets in the two different models were also compared. The periods, the corresponding linear combinations and, amplitudes of the five planets are presented in tables. - Article
- Jan 1988

The long term changes of the orbital elements of the planets are described by secular perturbation theories. After a short historical discussion, the secular perturbation equations are derived by means of the formalism of the Lie series transformations. To solve the classical problem of the long term changes in the major semiaxes second order effects have to be computed. As for the long term changes in the eccentricities and inclinations, they can be computed by means of higher degree theories. However the time span over which the latter apply cannot be increased at will. This because of the divergence of the perturbative series, a fundamental property of a non-integrable system such as the N-body problem. Numerical integrations are therefore an essential tool both to assess the reliability of any analytic theory and to provide data on the fundamental frequencies of the secular system and on the occurrence of secular resonances. Examples are taken from the LONGSTOP integrations of the outer planets for 100 million years. - Article
- Jan 2007
- Int J Leader Educ

This article is an interpretive study of the theory of irreversible and dissipative systems process transformation of Nobel Prize winning physicist Ilya Prigogine and how it relates to the phenomenological study of leadership and organizational change in educational settings. Background analysis on the works of Prigogine is included as a foundation for human inquiry and metaphor generation for open, dissipative systems in educational settings. International case study research by contemporary systems and leadership theorists on dissipative structures theory has also been included to form the interpretive framework for exploring alternative models of leadership theory in far from equilibrium educational settings. Interpretive analysis explores the metaphorical significance, connectedness, and inference of dissipative systems and helps further our knowledge of human-centred transformations in schools and colleges. - Article
- Sep 1987
- Proc Math Phys Eng Sci

There are several physical situations in the solar system where chaotic behavior plays an important role. Saturn's satellite Hyperion is currently tumbling chaotically. Many of the other irregularly-shaped satellites in the solar system had chaotic rotations in the past. There are also examples of chaotic orbital evolution. Meteorites are most probably transported to earth from the asteroid belt by way of a chaotic zone. Chaotic behavior also seems to be an essential ingredient in the explanation of certain nonuniformities in the distribution of asteroids. The long-term motion of Pluto is suspiciously complicated, but objective criteria have not yet indicated that the motion is chaotic. - Article
- Oct 1990
- ICARUS

The orbits of Jupiter and Saturn are represented by nonprecessing constant-eccentricity ellipses. Orbits that survive for longer than 800,000 Jupiter periods are found in the band of long-lived orbits centered at 1.35 and 1.45 Jovian distances. The effects of inaccuracies in the numerical integration on the orbit stability are examined, and it is demonstrated that stable orbits remain stable even when the accuracy is degraded by a factor of about 250,000. It is therefore considered that the long-delayed instability onset is not an artifact of the numerical integration but a consequence of the celestial mechanics. - Article
- May 2002
- ANNU REV EARTH PL SC

Until the early 1990s, numerical simulations of Solar System dynamics were done using accurate but slow integrators. The typical timescales were on the order of a million years, apart from exceptions achieved by considering averaged equations or using specifically designed supercomputers. In the last decade, new numerical integration methods for Solar System dynamics have been introduced. The mixed variable symplectic method (Wisdom & Holman 1991) has permitted the study, in the absence of close encounters, of the evolution of planets and small bodies on timescales comparable to the age of the Solar System. The regularized mixed variable scheme (Levison & Duncan 1994) has allowed the compilation of statistics on the evolution of thousands of near-Earth asteroids and comets, from their source regions to their dynamical elimination. The Symba and the Mercury codes (Duncan et al. 1998, Chambers 1999), which treat close encounters between massive bodies in a symplectic way, have permitted the simulations of planetary accretion and of the early phase of the highly chaotic evolution of the Solar System. This paper reviews the most exciting results obtained with these new integrators. Emphasis is given to the conceptual steps that these works represent in our understanding of Solar System - Article
- May 1989
- ICARUS

Prompted by the recent indication by Wisdom (1987) that orbits may appear stable for very long periods and then suddenly become chaotic, an earlier integration of asteroidlike orbits between Jupiter and Saturn in which most of the asteroids had close encounters with these planets and were removed is extended nearly 2000 times longer. Where two bands, centered on 1.35 and 1.45 Jovian distances, harbored asteroids before, all the asteroids are presently removed. Asteroids are found to be removed by Saturn-orbit crossings 5 times more frequently than by Jupiter-orbit crossing. - Article
- Jan 2010
- PHYS TODAY

The study of planet formation has been revolutionized by recent observational breakthroughs, which have allowed the detection and characterization of extrasolar planets, the imaging of protoplanetary disks, and the discovery of the Solar System's Kuiper Belt. Written for beginning graduate students, this textbook provides a basic understanding of the astrophysical processes that shape the formation of planetary systems. It begins by describing the structure and evolution of protoplanetary disks, moves on to the formation of planetesimals, terrestrial and gas giant planets, and concludes by surveying new theoretical ideas for the early evolution of planetary systems. - Article
- Mar 1988
- ASTRON ASTROPHYS

Based upon the assumption that in the long term Pluto has a chaotic orbit (as opposed to its present stable orbit in resonance with Neptune) it is shown that gross orbital changes under close encounters would be expected on a time-scale of a few times 104yr. Control is not rapidly passed on to the sunward planets: rather, the perihelion distance remains at about 30 AU whilst the aphelion distance is increased to >150 AU, and the inclination is reduced to match the orbital plane of Neptune. A possible origin for Pluto therefore appears to be the inner Oort Cloud. Conversely, if Pluto had formed in its present region at the same time as the rest of the planets then is would have only avoided orbital disruption on account of the stability of its resonances with Neptune. Whatever the origin of Pluto, this stable configuration must have been attained very soon after it entered its present orbit. - ArticleFull-text available
- Feb 1991

- Article
- Feb 1993
- ASTROPHYS J

Results are presented of numerical integrations over billion year time scales of the orbital evolution of more than one thousand test particles on initially low-inclination, low-eccentricity orbits within the proposed Kuiper belt beyond Neptune. Particles which eventually crossed Neptune's orbit often showed long periods (up to several billion years) of relatively low-eccentricity oscillations punctuated by a very rapid jump to Neptune-crossing eccentricity. This flux may be the ultimate source of present-day short-period comets. It is found here that there exists a correlation between Liapunov and crossing times in the Kuiper belt. None of the particles in the study with Liapunov time scales greater than about 1 Myr actually became a Neptune-crosser in 4 Gyr. An intricate structure to the region between 35 and 45 AU is found at the end of the billion year simulation. Implications for the origins of short-period comets and the detectability of objects currently in the Kuiper belt are discussed. - Article
- Apr 1990
- ASTRON ASTROPHYS

An iterative method is developed in the construction of general planetary theories. Due to the speed of computers, the study of long-period variations of planetary orbits can now be done by numerical integration over multimillion-year time spans. The iterative method enables the development of higher-order perturbations with respect to the planetary masses. The advantages and disadvantages presented by both numerical integration methods and by analytical methods are described. The equations of motion of the system of eight planets from Mercury to Neptune are considered along with the relevant forces. The iterative method is then developed from the solution of the Lagrange-Laplace system of equations. With the help of each preceding iteration the complete 48 second terms of the Lagrange equations are then calculated, with the only limitations on precision appearing in the truncation of the Fourier series. Finally, results are presented on the application of the method to the Jupiter-Saturn system. - The long-period behavior of the asteroids in the 3:2 and 2:1 resonances with Jupiter is investigated by integrating numerically the equations of motion of the planar three-body system. The results of the numerical integration procedure were submitted to a low-pass filtering and the decimation parameter was chosen in accordance with the filter. The behaviour of the most important physical quantities which describe the orbital evolution in the resonance were Fourier analysed. The topology of phase space was studied by constructing the surfaces of section. The structure of the phase space in the 3:2 and 2:1 resonances showed the presence of both regular and chaotic motion. Regular motion shows a great variety of orbital behaviour. Chaotic motion occurs because of: (1) the proximity of the resonance border; (2) the presence of secondary resonances (when the one fundamental frequency enters into resonance with the other frequencies and creates a pattern of islands inside the primary resonance); (3) the proximity of the chaotic region emanating from the boundary of corotations. The location of the main secondary resonances was computed on the (a_0_,e_0_)-plane. The results obtained with our numerical technique were compared to similar results previously obtained with either analytical or semi-numerical models by several authors.
- ArticleFull-text available
- Jun 1989

- Encke' s method was generalized to a dynamical system with a linear approximate solution. Its superiority to the classical Encke's method was shown analytically for two kinds of perturbed harmonic oscillators. The method was applied to the orbital and rotational motions of celestial bodies expressed in terms of the orbital elements and new sets of variables. Numerical simulations showed that the new formulations reduce integration errors significantly. When compared to the existing schemes, the gain in precision by the new method reaches 3 to 9 digits while the required computational labor remains almost the same.
- Article
- Feb 1988
- EARTH MOON PLANETS

The perturbation theory of Hori-Lie (Hori, 1966) is used to establish a second order Uranus-Neptune canonical planetary theory. Elliptic expansions are performed taking into account the 0, 1, and 2 powers of the eccentricity-inclination. The present analysis only includes the principal part of the planetary Hamiltonian. The canonical variables of Poincare are adopted to avoid the appearance of small divisors in the partial derivatives of the determining function with respect to the linear variables. Only Uranus-Neptune critical terms are taken as the periodic terms. - Article
- Nov 2005
- J GEOPHYS RES

The rotational state of Venus is presumably determined by a balance between two different solar tidal torques: an atmospheric thermal tide yields a torque which acts in the same direction as the retrograde rotation of Venus and a gravitational solid body tide yields a torque which opposes the current rotation. The strengths of these opposing torques both depend on the eccentricity of the orbit of Venus but with different sensitivities. Thus variations in the orbital eccentricity cause fluctuations in the rotation rate at which the tidal torques balance. If Venus is considered to be a single rigid body, the rotational velocity will change by amounts equivalent to ∼1 km/yr at the equator, over timescales of 106 years. Allowance for differential rotation between the core, mantle, and atmosphere increases the range over which the solid surface rotation rate varies. For plausible values of the strength of viscous coupling at the atmosphere-mantle boundary and the core-mantle boundary the variations in solid surface rotational velocity can be 2–3 times larger than in the rigid coupling case. When these long-period forced variations in rotation rate are considered, the present rotation rate of Venus is seen to be close enough to a resonant spin-orbit interaction with the Earth that it may occasionally pass through that resonance. - Article
- Oct 2007

In this paper we summarise the present knowledge on the formation and evolution of the Moon by critically analysing most credited theories. Information is also provided on present understanding of geology, geochemistry and geophysics of our natural satellite. - Book
- Aug 2018

Cambridge Core - Computational Science and Modelling - The Exoplanet Handbook - by Michael Perryman - Article
- May 2018
- SOLAR SYST RES+

The four-planet problem is solved by constructing an averaged semi-analytical theory of secondorder motion by planetary masses. A discussion is given of the results obtained by numerical integration of the averaged equations of motion for the Sun–Jupiter–Saturn–Uranus–Neptune system over a time interval of 10 Gyr. The integration is based on high-order Runge–Kutta and Everhart methods. The motion of the planets is almost periodic in nature. The eccentricities and inclinations of the planetary orbits remain small. Short-period perturbations remain small over the entire interval of integration. Conclusions are drawn about the resonant properties of the motion. Estimates are given for the accuracy of the numerical integration. - Penitentes are snow and ice features formed by erosion that, on Earth, are characterized by bowl-shaped depressions several tens of centimetres across, whose edges grade into spires up to several metres tall. Penitentes have been suggested as an explanation for anomalous radar data on Europa, but until now no penitentes have been identified conclusively on planetary bodies other than Earth. Regular ridges with spacings of 3,000 to 5,000 metres and depths of about 500 metres with morphologies that resemble penitentes have been observed by the New Horizons spacecraft in the Tartarus Dorsa region of Pluto (220°-250° E, 0°-20° N). Here we report simulations, based upon a recent model representing conditions on Pluto, in which deepening penitentes reproduce both the tri-modal (north-south, east-west and northeast-southwest) orientation and the spacing of the ridges of this bladed terrain. At present, these penitentes deepen by approximately one centimetre per orbital cycle and grow only during periods of relatively high atmospheric pressure, suggesting a formation timescale of several tens of millions of years, consistent with crater ages. This timescale implies that the penitentes formed from initial topographic variations of no more than a few tens of metres, consistent with Pluto's youngest terrains.
- Chapter
- Jan 1988

Project LONGSTOP was set up to investigate the long term dynamics of the outer solar system over timescales comparable to its age. This was done by means of numerical integrations on a CRAY-1S computer. Comparison with analytic theories required the use of filtering procedures and Fourier analysis. The 6-body point-mass newtonian problem, plus a gaussian ring model for the effect of the inner planets, turned out to be a good approximation to the real system; general relativity corrections can be easily introduced although they are not yet critical over 100 Myr. Long term variations in shape and orientation of planetary orbits from numerical integrations over 9.3 Myr suggested that analytic theories must be improved in order to be valid for such a timespan. Variations in the major semiaxes of Uranus and Neptune with a 1.119 Myr period have been found in the data; they could be recovered also analytically once the amplifying effect of the 2/1 quasi-resonance in mean motion between Uranus and Neptune was taken into account. The 100 Myr integration LONGSTOP 1B revealed the presence of a very small divisor with 31 Myr period. In relation with this small divisor, and with others which could not be identified with combinations of up to 8 fundamental frequencies, there appeared to be an accumulation of spectral lines of comparable amplitude in some regions of the spectrum. This was not the case when the output of the 9.3 Myr integration LONGSTOP 1A was analyzed; it suggests that 100 Myr might be long enough a timespan already to reveal the presence of non regular regions of motion in the phase space. - Chapter
- Oct 2014

Celestial mechanics is the topic of this chapter. The 2-body solution is given, the restricted 3-and n-body solutions discussed, and the effects of perturbations on the orbital elements are treated in detail. Tidal friction and its effects in the Earth-Moon system, spin-orbit and orbit-orbit resonances are discussed. - Chapter
- Jan 1995

The Jupiter-Saturn 2:5 near-commensurability is analyzed in a fully analytic Hamiltonian planetary theory. Computations for the Sun-Jupiter-Saturn system, extending to the third order of the masses and to the 8th degree in the eccentricities and inclinations, reveal an unexpectedly sensitive dependence of the solution on initial data and its likely nonconvergence. The source of the sensitivity and apparent lack of convergence is this near-commensurability, the so-called great inequality. This indicates that simple averaging, still common in current semi-analytic planetary theories, may not be an adequate technique to obtain information on the long-term dynamics of the Solar System. Preliminary results suggest that these difficulties can be overcome by using resonant normal forms. - Article
- Sep 2015

The migration and encounter histories of the giant planets in our Solar System can be constrained by the obliquities of Jupiter and Saturn. We have performed secular simulations with imposed migration and N-body simulations with planetesimals to study the expected obliquity distribution of migrating planets with initial conditions resembling those of the smooth migration model, the resonant Nice model and two models with five giant planets initially in resonance (one compact and one loose configuration). For smooth migration, the secular spin-orbit resonance mechanism can tilt Saturn's spin axis to the current obliquity if the product of the migration time scale and the orbital inclinations is sufficiently large (exceeding 30 Myr deg). For the resonant Nice model with imposed migration, it is difficult to reproduce today's obliquity values, because the compactness of the initial system raises the frequency that tilts Saturn above the spin precession frequency of Jupiter, causing a Jupiter spin-orbit resonance crossing. Migration time scales sufficiently long to tilt Saturn generally suffice to tilt Jupiter more than is observed. The full N-body simulations tell a somewhat different story, with Jupiter generally being tilted as often as Saturn, but on average having a higher obliquity. The main obstacle is the final orbital spacing of the giant planets, coupled with the tail of Neptune's migration. The resonant Nice case is barely able to simultaneously reproduce the {orbital and spin} properties of the giant planets, with a probability ~0.15%. The loose five planet model is unable to match all our constraints (probability <0.08%). The compact five planet model has the highest chance of matching the orbital and obliquity constraints simultaneously (probability ~0.3%). - Article
- Mar 2015
- ICARUS

Since previous long-term insolation modeling in the early 1990s, new atmospheric pressure data, increased computational power, and the upcoming flyby of the Pluto system by NASA's New Horizons spacecraft have generated new motivation and increased capabilities for the study of Pluto's complex long-term (million-years) insolation history. The two primary topics of interest in studying Pluto's insolation history are the variations in insolation patterns when integrated over different intervals and the evolution of diurnal insolation patterns over the last several decades. We find latitudinal dichotomies when comparing average insolation over timescales of days, decades, centuries, and millennia, where all timescales we consider are short relative to the predicted timescales for Pluto's chaotic orbit. Depending on the timescales of volatile migration, some consequences of these insolation patterns may be manifested in the surface features revealed by New Horizons. We find the Maximum Diurnal Insolation (MDI) at any latitude is driven most strongly when Pluto's obliquity creates a long arctic summer (or "midnight sun") beginning just after perihelion. Pluto's atmospheric pressure, as measured through stellar occultation observations during the past three decades, shows a circumstantial correlation with this midnight sun scenario as quantified by the MDI parameter. - We present N-body simulations of resonant planets with inclined orbits that show chaotically evolving eccentricities and inclinations that can persist for at least 10 Gyr. A wide range of behavior is possible, from fast, low amplitude variations to systems in which eccentricities reach 0.9999 and inclinations 179.9 degrees. While the orbital elements evolve chaotically, at least one resonant argument always librates. We show that the HD 73526, HD 45364 and HD 60532 systems may be in chaotically-evolving resonances. Chaotic evolution is apparent in the 2:1, 3:1 and 3:2 resonances, and for planetary masses from lunar- to Jupiter-mass. In some cases, orbital disruption occurs after several Gyr, implying the mechanism is not rigorously stable, just long-lived relative to the main sequence lifetimes of solar-type stars. Planet-planet scattering appears to yield planets in inclined resonances that evolve chaotically in about 0.5% of cases. These results suggest that 1) approximate methods for identifying unstable orbital architectures may have limited applicability, 2) the observed close-in exoplanets may be produced during the high eccentricity phases induced by inclined resonances, 3) those exoplanets' orbital planes may be misaligned with the host star's spin axis, 4) systems with resonances may be systematically younger than those without, 5) the distribution of period ratios of adjacent planets detected via transit may be skewed due to inclined resonances, and 6) potentially habitable planets in resonances may have dramatically different climatic evolution than the Earth. The GAIA spacecraft is capable of discovering giant planets in these types of orbits.
- Article
- Nov 2014
- Astrophys J

On timescales that greatly exceed an orbital period, typical planetary orbits evolve in a stochastic yet stable fashion. On even longer timescales, however, planetary orbits can spontaneously transition from bounded to unbound chaotic states. Large-scale instabilities associated with such behavior appear to play a dominant role in shaping the architectures of planetary systems, including our own. Here we show how such transitions are possible, focusing on the specific case of the long-term evolution of Mercury. We develop a simple analytical model for Mercury's dynamics and elucidate the origins of its short term stochastic behavior as well as of its sudden progression to unbounded chaos. Our model allows us to estimate the timescale on which this transition is likely to be triggered, i.e. the dynamical lifetime of the Solar System as we know it. The formulated theory is consistent with the results of numerical simulations and is broadly applicable to extrasolar planetary systems dominated by secular interactions. These results constitute a significant advancement in our understanding of the processes responsible for sculpting of the dynamical structures of generic planetary systems. - Article
- May 2007
- MEM AM MATH SOC

KAM theory is a powerful tool apt to prove perpetual stability in Hamiltonian systems, which are a perturbation of integrable ones. The smallness requirements for its applicability are well known to be extremely stringent. A long standing problem, in this context, is the application of KAM theory to “physical systems” for “observable” values of perturbation parameters. We consider the restricted circular planar three-body problem (RCP3BP), i.e., the problem of studying the planar motions of a small body subject to gravitational attraction of two primary bodies revolving on circular Keplerian orbits (which are assumed not to be influenced by the small body). When the mass ratio of the two primary bodies is small, the RCP3BP is described by a nearly-integrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with non-small eccentricities, the system is well described by Delaunay variables. The Sun-Jupiter observed motion is nearly circular, and an asteroid of the asteroidal belt may be assumed not to influence the Sun-Jupiter motion. The Jupiter-Sun mass ratio is slightly less than 1/1000. We consider the motion of the asteroid 12 Victoria taking into account only the Sun-Jupiter gravitational attraction regarding such a system as a prototype of RCP3BP. For values of mass ratios up to 1/1000, we prove the existence of two-dimensional KAM tori on a fixed three-dimensional energy level corresponding to the observed energy of Sun-Jupiter-Victoria system. Such tori trap the evolution of phase points “close” to the observed physical data of Sun-Jupiter-Victoria system. As a consequence, in the RCP3BP description, the motion of Victoria is proven to be forever close to an elliptical motion. The proof is based on: 1) a new iso-energetic KAM theory; 2) an algorithm for computing iso-energetic, approximate Lindstedt series; 3) a computer-aided application of 1) and 2) to the Sun-Jupiter-Victoria system. The paper is self-contained but does not include the (∼ 12000 lines) computer programs, which may be obtained by sending an e-mail to one of the authors. - Article
- Jan 1992
- ICARUS

The full equations for the three-body problem (Sun-Jupiter-asteroid) were integrated in order to study the time variations of eccentricities of fictitious asteroids initially located near the 5:2 Jovian commensurability. The runs covered a time span not less than 5000 Jupiter's revolutions around the Sun for the planar model, 10,000 Jupiter's revolutions for the initial inclinations between 5 and 20 degrees, and 100,000 Jupiter's revolutions for the initial inclinations equaled to 40 degrees. Regions of the initial values of semimajor axes and eccentricities for which at some starting orbital orientations and initial positions in orbits, the fictitious asteroids were Mars- and Earth-crossers were studied. We found that, for initial eccentricities less than 0.2 and initial inclinations less than 20 degrees, these ranges were almost the same. The range in which asteroids are Mars-crossers is close to that free of real asteroids. Close encounters of asteroids with Mars and Earth might be one of the causes of the origin of the 5:2 Kirkwood gap. Usually the fictitious asteroids were Mars- and Earth-crossers for certain types of relationships between variations in asteroidal eccentricity and the difference of longitudes of perihelion of an asteroid and Jupiter as well as for the transitions between these types. Relationships between the periods of asteroidal variations in eccentricity, inclination, argument of perihelion, and ascending node were obtained for some fictitious asteroids with small initial eccentricities and inclinations. - Article
- Jan 2008

The Hipparcos satellite, developed and launched by the European Space Agency (ESA) in 1989, was the first space mission dedicated to astrometry – the accurate measurement of positions, distances, and proper motions of stars. Amongst the key achievements of its measurements are refining the cosmic distance scale, characterizing the large-scale kinematic motions in the Solar neighborhood, providing precise luminosities for stellar modelling, and confirming Einstein’s prediction of the effect of gravity on starlight. This authoritative account of the Hipparcos contributions over the last decade is an outstanding reference for astronomers, astrophysicists and cosmologists. It reviews the applications of the data in different areas, describing the subject and the state of the art before Hipparcos, and summarizing all major contributions to the topic made by Hipparcos. It contains a detailed overview of the Hipparcos and Tycho Catalogues, their annexes and their updates. Each chapter ends with comprehensive references to relevant literature. - Article
- May 2000
- SOLAR SYST RES+

Limits and characteristic periods of variations in orbital elements of planets were studied by numerical integration of equations of motion. Interrelations between the characteristic periods of variations in orbital elements of some planets were found. - Article
- May 1988
- ASTRON ASTROPHYS

The author has computed the differential system giving the secular evolution of the 8 main planets of the solar system up to the order 2 with respect to the masses and degree 5 in eccentricity and inclination including lunar and relativistic contributions. This secular system is numerically integrated over 30 million years. A modified Fourier analysis is performed to obtain a solution for the secular evolution of the orbits on a quasi-periodical form. Comparison with Bretagnon's ephemeris VSOP82 allows to derive uncertainties for the determination of the main frequencies of the secular system. Comparisons are made with the results of long term numerical integrations of Applegate et al. (1986) and Carpino et al. (1986) and with the analytical theory of Bretagnon (1974, 1984). The solutions of the outer solar system appear to be more stable than the solutions of the inner solar system.

- Article
- Jan 1984
- CELEST MECH DYN ASTR

Five outer planets are numerically integrated over five million years in the Newtonian frame. The argument of Pluto's perihelion librates about 90 degrees with an amplitude of about 23 degrees. The period of the libration depends on the mass of Pluto: 4.0×106 years forM pluto=2.78×10−6M sun and 3.8×106 years forM pluto=7.69×10−9M sun, which is the newly determined mass. The motion of Neptune's perihelion is more sensitive to the mass of Pluto. ForM pluto=7.69×10−9M sun, the perihelion of Neptune does circulate counter-clockwise and forM pluto=2.78×10−6M sun, it does not circulate and the Neptune's eccentricity does not have a minimum. With the initial conditions which do not lie in the resonance region between Neptune and Pluto, a close approach between them takes place frequently and the orbit of Pluto becomes unstable and irregular. - Article
- Dec 1963
- RUSS MATH SURV+

CONTENTSIntroduction § 1. Results § 2. Preliminary results from mechanics § 3. Preliminary results from mathematics § 4. The simplest problem of stability § 5. Contents of the paperChapter I. Theory of perturbations § 1. Integrable and non-integrable problems of dynamics § 2. The classical theory of perturbations § 3. Small denominators § 4. Newton's method § 5. Proper degeneracy § 6. Remark 1 § 7. Remark 2 § 8. Application to the problem of proper degeneracy § 9. Limiting degeneracy. Birkhoff's transformation § 10. Stability of positions of equilibrium of Hamiltonian systemsChapter II. Adiabatic invariants § 1. The concept of an adiabatic invariant § 2. Perpetual adiabatic invariance of action with a slow periodic variation of the Hamiltonian § 3. Adiabatic invariants of conservative systems § 4. Magnetic traps § 5. The many-dimensional caseChapter III. The stability of planetary motions § 1. Picture of the motion § 2. Jacobi, Delaunay and Poincaré variables §3. Birkhoff's transformation § 4. Calculation of the asymptotic behaviour of the coefficients in the expansion of \Bar{\Bar F}_1 § 5. The many-body problemChapter IV. The fundamental theorem § 1. Fundamental theorem § 2. Inductive theorem § 3. Inductive lemma § 4. Fundamental lemma § 5. Lemma on averaging over rapid variables § 6. Proof of the fundamental lemma § 7. Proof of the inductive lemma § 8. Proof of the inductive theorem § 9. Lemma on the non-degeneracy of diffeomorphisms § 10. Averaging over rapid variables § 11. Polar coordinates § 12. The applicability of the inductive theorem § 13. Passage to the limit § 14. Proof of the fundamental theoremChapter V. Technical lemmas § 1. Domains of type D § 2. Arithmetic lemmas § 3. Analytic lemmas § 4. Geometric lemmas § 5. Convergence lemmas § 6. NotationChapter VI. Appendix § 1. Integrable systems § 2. Unsolved problems § 3. Neighbourhood of an invariant manifold §4. Intermixing § 5. Smoothing techniquesReferences - A numerical solution is determined for the obliquity of the ecliptic and the longitude of the perihelion referred to the mean equinox of 1950.0. A previous solution which includes terms to the second order with respect to the perturbing masses and to the third degree with respect to the planetary eccentricities and inclinations is used for the eccentricity, the longitude of the perihelion measured from the equinox of 1850.0, the inclination of the ecliptic on the plane of reference, and the longitude of the ascending node. A series of analytical expansions which includes terms to the second degree with respect to earth's eccentricity is employed for the obliquity and the precession in longitude. The long-term variation of the ecliptic elements is analyzed for the periods extending to 5 million years before and 1 million years after the present (1950.0).
- Article
- Mar 1964

The author has applied the ideas of Poincaré to a study of the long-period effects in nearly commensurable cases of the restricted three-body problem. After isolating the secular and critical terms of the disturbing function by a numerical averaging process performed on an IBM 7094 computer, he found stability for all real asteroid orbits corresponding to the cases treated here. Periods of the variations are given for some forms of planar motion. The theory indicates planar and nonplanar periodic solutions. No evidence is found for a disintegration of the Trojan and Hilda groups of asteroids. This work supports Brouwer's explanation of differences between the Hecuba and Hilda commensurabilities. - Article
- Mar 1985
- CELEST MECH DYN ASTR

Hierarchical stability of the outer Solar System is monitored through its 3-body subsystems by using numerically computed ephemerides for 5106 yr. It is found that the stability parameters of Sun-Jupiter-Saturn and Sun-Uranus-Neptune oscillate in anti-phase in 1.1106 yr. The mechanism responsible for this locking is a secular resonance between Uranus' perihelion and Jupiter's aphelion: the difference between the two librates within 70 with the same period of 1.1106 yr. - Article
- Jul 1981
- ICARUS

The surfaces for the three strongest secular resonances have been located as a function of proper semimajor axis, eccentricity, and inclination for semimajor axes between 1.25 and 3.5 AU. The results are presented graphically. The ν5 resonance only occurs at high inclinations (≳23°). The ν6 resonance passes through both the main belt and Mars-crossing space. The ν16 resonance starts near the inner edge of the belt and, at low inclinations at least, folds around a portion of the Mars-crossing space until it runs nearly parallel with the Earth-crossing boundary. - Article
- Jun 1978
- ASTRON J

The semianalytical approach to long-term solutions of resonant systems with three degrees of freedom, proposed by Giacaglia in 1965, is used to study the long-term motion of Pluto. The study takes into account the effects of Jupiter, Saturn and Uranus on the motion of Pluto. Modified periodic orbits of the third kind constitute the solutions; Pluto is found to librate about one of these periodic solutions. The long-term eccentricity, inclination, perihelion and librational amplitude of the planet are discussed. - Article
- Nov 1983
- ICARUS

The sudden eccentricity increases discovered by Wisdom (1982) are reproduced in numerical integrations of the planar ecliptic restricted three-body problem, verifying that this phenomenon is real. Mapping derivations are qualitatively reviewed and the maximum Liapunov characteristic exponent and its importance for determining the character of a trajectory are explained. The results of a number of calculations of this exponent using the differential equations for the unaveraged three-body problem are shown and compared to equivalent calculations using a mapping. In all cases the two approaches agree whether the orbits are chaotic or quasiperiodic. The mappings are used to trace out the chaotic zone near the 3/1 commensurability, both in the planar-ecliptic problem and in the three-dimensional elliptic problem. The outer boundary of the chaotic zone coincides with the boundary of the 3/1 Kirkwood gap in the actual distribution of asteroids within the errors of the asteroid orbital elements. - Article
- Apr 1982
- ASTRON J

A mapping of the phase space onto itself with the same low-order resonance structure as the 3/1 commensurability in the planar-elliptic restricted three-body problem is obtained. This mapping is about 1,000 times faster than the usual method of numerically integrating the averaged equations of motion. It exhibits some surprising behavior that might provide a key to the origin of the Kirkwood gaps. It is noted that a test asteroid placed in the gap may evolve for a million years with low eccentricity (less than 0.05) and then suddenly jump to large eccentricity (greater than 0.3), becoming a Mars crosser. The removal of the asteroid by a close encounter with Mars would then be possible. As a first test of this hypothesis, a distribution of 300 test asteroids in the area of the 3/1 commensurability was evolved for two million years. When the Mars crossers are removed, the distribution of initial conditions reveals a gap at the location of the 3/1 Kirkwood gap. - Pluto orbit integration over 4.5 million years by variation of parameters technique and confirming Neptune-Pluto orbital resonances