Libration of the close approaches of Pluto to Neptune
Abstract
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- ... Although the orbits of them cross, they always avoid close – 5 – approach. Neptune-Pluto system are long-term stable (Cohen and Hubbard 1965). In a gas disk, growing protoplanets migrate toward their central stars due to type-I migration. ...Many extrasolar planetary systems containing multiple super-Earths have been discovered. N-body simulations taking into account standard type-I planetary migration suggest that protoplanets are captured into mean-motion resonant orbits near the inner disk edge at which the migration is halted. Previous N-body simulations suggested that orbital stability of the resonant systems depends on number of the captured planets. In the unstable case, through close scattering and merging between planets, non-resonant multiple systems are finally formed. In this paper, we investigate the critical number of the resonantly trapped planets beyond which orbital instability occurs after disk gas depletion. We find that when the total number of planets ($N$) is larger than the critical number ($N_{\rm crit}$), crossing time that is a timescale of initiation of the orbital instability is similar to non-resonant cases, while the orbital instability never occurs within the orbital calculation time ($10^8$ Kepler time) for $N\leq N_{\rm crit}$. Thus, the transition of crossing time across the critical number is drastic. When all the planets are trapped in 7:6 resonance of adjacent pairs, $N_{\rm crit} = 4$. We examine the dependence of the critical number of 4:3, 6:5 and 8:7 resonance by changing the orbital separation in mutual Hill radii and planetary mass. The critical number increases with increasing the orbital separation in mutual Hill radii with fixed planetary mass and increases with increasing planetary mass with fixed the orbital separation in mutual Hill radii. We also calculate the case of a system which is not composed of the same resonance. The sharp transition of the stability can be responsible for the diversity of multiple super-Earths (non-resonant or resonant), that is being revealed by $Kepler$ mission.
- ... The resonant protection provided by the 9:2 resonance is very similar to that provided by the 3:2 and 5:2 mean- motion resonances (Cohen & Hubbard 1965;Gladman et al. 2012). Specifically, libration of the resonant argument Φ 92 = 9λ − 2λ N − 7 around 180 • (Fig. 2) means that when a resonant TNO is at perihelion, Neptune is never near the same mean orbital longitude λ N , preventing close encounters (in this expression, λ is the mean longitude of the particle and = Ω + ω is the longitude of perihelion). ...We report the discovery and orbit of a new dwarf planet candidate, 2015 RR$_{245}$, by the Outer Solar System Origins Survey (OSSOS). 2015 RR$_{245}$'s orbit is eccentric ($e=0.586$), with a semi-major axis near 82 au, yielding a perihelion distance of 34 au. 2015 RR$_{245}$ has $g-r = 0.59 \pm 0.11$ and absolute magnitude $H_{r} = 3.6 \pm 0.1$; for an assumed albedo of $p_V = 12$% the object has a diameter of $\sim670$ km. Based on astrometric measurements from OSSOS and Pan-STARRS1, we find that 2015 RR$_{245}$ is securely trapped in the 9:2 mean-motion resonance with Neptune. It is the first TNO identified in this resonance. On hundred-Myr timescales, particles in 2015 RR$_{245}$-like orbits depart and sometimes return to the resonance, indicating that 2015 RR$_{245}$ likely forms part of the long-lived metastable population of distant TNOs that drift between resonance sticking and actively scattering via gravitational encounters with Neptune. The discovery of a 9:2 TNO stresses the role of resonances in the long-term evolution of objects in the scattering disk, and reinforces the view that distant resonances are heavily populated in the current Solar System. This object further motivates detailed modelling of the transient sticking population.
- ... The first famous numerical computation of planetary orbits was by Eckert et al. (1951) who did a 350-year simulation of the four outer planets (Jupiter, Saturn , Uranus, and Neptune) using a large mainframe computer . This was extended by Cohen and Hubbard (1965) to 120,000 years, and by Cohen et al. (1973) ...We review some of the major achievements of celestial mechanics in the twentieth century, and discuss some unsolved problems that are left to the twenty-first century. The four major research fields in celestial mechanics are treated: dynamical systems, three-body problems, solar system dynamics, and numerical methods. Over the past decades, celestial mechanics has become extremely specialized and categorized along many trajectories, from purely mathematical to quite practical points of view, as well as in what has involved digital computer innovation. These various ends of celestial mechanics sometimes exist in isolation, losing their connections with each other. The purpose of this manuscript is to find clues for rediscovering connections between each of these fields by listing some of major achievements in hoping to find new paths for celestial mechanics in the new century.
- ... A notable example in the trans-Neptunian belt is Pluto, which has q ~ 29.7 AU (i.e., " inside " the orbit of Neptune at 30.1 AU). However, thanks to the resonant protection mechanism, Pluto is always far away from the giant planet when near perihelion (Cohen & Hubbard 1965). The Kozai mechanism is also responsible for keeping Pluto far from the plane of the solar system during its perihelion approach, contributing to its long-term stability against the giant planet's gravitational influence (Williams & Benson 1971). ...Trans-Neptunian objects (TNOs) are icy/rocky bodies that move beyond the orbit of Neptune in a region known as the trans-Neptunian belt (or Edgeworth-Kuiper belt). In contrast to the predictions of accretion models that feature protoplanetary disk planetesimals evolving on dynamically cold orbits (with both very small eccentricities, e, and inclinations, i), in reality TNOs exhibit surprisingly wide ranges of orbital eccentricities and inclinations. Several theoretical models have addressed the origin and orbital evolution of the main dynamical classes of TNOs, but none have successfully reproduced them all. In addition, none have explained several objects on peculiar orbits, or provided insightful predictions, without which a model cannot be tested properly against observations. Based on extensive simulations of planetesimal disks with the presence of the four giant planets and huge numbers of modeled planetesimals, I explore in detail the dynamics of the TNOs, in particular their (un)stable regions over timescales comparable to the age of the solar system, and the role of resonances across the entire trans-Neptunian region. I also propose that, along with the orbital history of the giant planets, the orbital evolution of primordial embryos (massive planetesimals comparable to Mars-Earth masses) can explain the fine orbital structure of the trans-Neptunian belt, the orbits of Jovian and Neptunian Trojans, and possibly the current orbits of the giant planets. Those primordial embryos were ultimately scattered by the giant planets, a process that stirred both the orbits of the giant planets and the primordial planetesimal disk to the levels observed at 40-50 AU. In particular, the main constraints provided by the trans-Neptunian belt are optimally satisfied if at least one such primordial embryo (planetoid) survived in the outskirts of the solar system.
- ... In spite of heroic efforts by such mathematicians as Poincaré and Jürgen Moser, the resonances remain a mystery. It came as a surprise to the theoreticians when a numerical analysis [36] recently revealed that the orbits of Neptune and Pluto are locked in a stable resonance. Undoubtedly there exist many more missed opportunities to create new branches of pure mathematics out of old problems of applied science. ...
- ... Although the existence of objects some 60° from Neptune and for which the assumption of direct, circular orbits placed them only slightly beyond Neptune might have hinted that they were Neptune " Trojans, " librating in 1:1 orbital resonance with Neptune, it seemed at least as likely that they were instead relatively near the perihelion points of orbits in the 2:3 resonance, which has a much larger phase space. After all, Pluto itself librates in 2:3 resonance with Neptune, a possibility apparently not even suggested until it was firmly established in 1964 (Cohen and Hubbard, 1965). In May 1994, the availability of followup observations of 1993 SC finally provided an opportunity for the publication () of the result that the assumption that this object was near perihelion and in the Neptune 2:3 resonance (mean distance 39 AU) could ensure that it would always be more than 14 AU from Neptune. ...We review the history of the prediction of, and searches for, a population of comets and transneptunian planetesimals. Starting with initial speculations before and after the discovery of Pluto, we examine various predictions by Edgeworth, Kuiper, and others on the existence of such a population and review the increasingly sophisticated theoretical efforts that eventually showed that the number of short-period comets requires that an ecliptic transneptunian population exists. We then recount various search programs that culminated in the discovery of the first few transneptunian objects and led to the realization that this region is dynamically much more complicated than first suspected and has important links both to Centaurs and the dense inner core of the Oort cloud.
- ... Compared with the period of ω, the libration period of θ is much shorter. In the case of Pluto, the period of θ is about 19670 yr (Cohen & Hubbard 1965) and the libration period of ω is about 3.955 × 10 6 yr (Williams & Benson 1971 ). Accordingly , θ in the averaged Hamiltonian is replaced by its averaged value. ...The Kozai resonance of Kuiper Belt objects is explored using a model of a circular-restricted three-body problem. We use an analytical approach to find the topological structure of the plane of the eccentricity and the argument of perihelion. We find that objects inside the 2:3 and 1:2 resonances can be inside the Kozai resonance, with their arguments of perihelion ω librating around 90° or 270°. This is consistent with the fact that Pluto is inside both the 2:3 resonance and the Kozai resonance. Furthermore, objects outside the mean-motion resonances are also found to be inside the Kozai resonance. We discover that there are stable equilibrium points of ω at 90° and 270°, but not at 0° and 180° as was shown in the work of Thomas & Morbidelli. To verify our results, numerical experiments are carried out with various longitudes of the node and mean anomalies. In these experiments, some test particles are found to be in the Kozai resonance around 90° for one billion years, and the libration amplitudes of their arguments of perihelion are small. This is in agreement with our analytical results. No particles are found to stay inside the Kozai resonance around ω= 0° or 180°, although some particles exist temporarily in the Kozai resonance around ω= 0° or 180°. We conclude that the Kozai resonance around ω= 90° or 270° exists both inside and outside the mean-motion resonance for Kuiper Belt objects.
- The size distribution in the Kuiper Belt records physical processes operating during the formation and subsequent evolution of the solar system. This paper reports a study of the apparent magnitude distribution of faint objects in the Kuiper Belt, obtained via deep imaging on the Canada-France-Hawaii Telescope and the ESO Very Large Telescope UT1. We find that the entire range of observed objects (magnitudes mR ~ 20–27) is well represented by an unbroken power law, with the number of objects per square degree brighter than magnitude R being of the form Σ(mR < R) = 10, with α = 0.69 and R0 = 23.5. This luminosity function's slope implies a steep size distribution in the observed range, which should "roll over" to a shallower "collisional" slope once observations extend to even fainter magnitudes and thus sample bodies whose collisional ages become less than the age of the solar system. Our observations indicate the roll over is for diameters of less than 50 km, in agreement with collisional models. Modeling our survey gives a belt mass between 30 and 50 AU of order 0.1 M⊕, relatively insensitive to the roll over diameter as long as the latter is 1 km. We report the discovery of several objects outside of 48 AU and discuss the evidence for a sharp outer edge to the trans-Neptunian distribution.
- The paper develops a method for calculating the perturbing effects of Pluto on the elements of the giant planets: perturbations induced by Pluto are expanded in Chebyshev series. These representations are added to the elements of the giant planets calculated by a planetary theory without Pluto and thus give the value of the elements in the presence of Pluto. Two objectives are accomplished with this method: (1) the form of perturbations of the giant planets by Pluto can be calculated for a 500-year period, and (2) the theory without Pluto is improved.
- We have investigated the possibility of main belt asteroids being trapped in the 1:1 resonance with other asteroids. Using contemporary mass estimates for the large asteroids (1) Ceres, (2) Pallas and (4) Vesta, it is found that four asteroids are temporarily trapped in such resonances with Ceres and Vesta during numerical integrations spanning 2 x 106 yr. In particular, the asteroid (1372) Haremari is likely to be co-orbiting with Ceres at the present time. The corresponding libration period is found to be <~ 105 yr. The implications of our results for the general dynamical evolution of minor bodies and these asteroids in particular is discussed.
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- The pumping-up of inclinations and eccentricities caused by sweeping secular resonances in the asteroid belt is studied through numerical orbital integration and linear analysis. The present asteroids have large mean eccentricities and inclinations that cannot be explained by planetary perturbations alone. Sweeping of secular resonances associated with the depletion of the primitive solar nebula has been proposed for the origin of asteroids' orbits (e.g., by Ward, Colombo, and Franklin in 1976). We performed three-dimensional orbital integrations of asteroids under gravitational forces by Jupiter, Saturn, and the solar nebula. The asteroids' motions are also affected by hydrodynamic gas drag. We consider three types of nebula depletion models: (1) the uniform depletion model, in which the nebula density decreases exponentially with time and uniformly throughout the nebula; (2) the inside-out depletion model, in which nebula gas is depleted from the inside region; (3) the gap-opening model, in which a gap centered at Jupiter gradually expands. Previous studies have concentrated on the two-dimensional uniform depletion model. Our simulation shows that inclinations of asteroids are not pumped up enough in the first model to account for the observed magnitude.Moreover, most asteroids spiral into the Sun by gas drag if the depletion timescale is longer than 105 yr because pumped-up eccentricity induces strong gas drag. On the other hand, in the second and third models, inclinations are pumped up. Our linear analysis shows that the nonuniform depletion model is essential for the secular resonances pumping up inclination to sweep in the asteroid belt. In the case of the inside-out depletion model, both eccentricity and inclination are pumped up enough to be consistent with the observed magnitude in the entire asteroid belt, if the nebula depletion timescale (which is the time required for the nebula edge to migrate by the distance of 5 AU) is longer than 3 × 105 yr. Furthermore, since secular resonances pass after nebula gas has already been depleted in the passing region, gas drag does not damp the pumped-up eccentricities and inclinations and the semimajor axes of the asteroids. Therefore, the resultant eccentricity and inclination in the inside-out depletion model are consistent with observed ones for the nebula depletion time inferred from observed T Tauri stars (106-107 yr). Our gap-opening model shows similar results, although eccentricity and inclination are slightly smaller in the outer asteroid belt region. The essentially required condition to pump up both eccentricity and inclination highly enough in the asteroid region is that the nebula edge moves outward from 5 to 10 AU. The gap formation caused by Jupiter's tidal perturbations may be similar to our inside-out or gap-opening models, inside 10 AU. Therefore, it would be responsible for the high eccentricities and inclinations in the asteroid belt.
- Three resonances, the 3 : 2 mean motion resonance, the Kozai resonance, and the 1 : 1 superresonance, are known to govern the stability of the motion of Pluto concurrently. In this work, we report an extensive numerical exploration of the 1 : 1 superresonance region in element space and the effects of the giant planets on this resonance. It is shown that the 1 : 1 superresonance region is narrower than those for the other two resonances and that it is a second-order resonance. We use the orbits of known Plutinos to investigate our result. None of these Plutinos is in the 1 : 1 superresonance. We also find that the Jovian planets, especially Jupiter, play important roles in concert in the superresonance.
- The purpose of this paper is to derive a symmetric parasitic-free G-symplectic general linear method of order 4 and apply it to problems in celestial mechanics. The numerical method is constructed by satisfying the G-symplectic conditions for general linear methods, together with relevant order conditions while making sure that there is zero parasitism. The internal stages are designed to be diagonally implicit to make the method more efficient. Being multivalue in nature, a starting method is required for the implementation of general linear method and this is also calculated using rooted trees. The general linear method is applied to many body problems and acceptable error in energy and global error are observed.
- It is shown that in addition to four outer planets (Jupiter to Neptune) Pluto should be also taken into account in studies of the orbital dynamics in the trans-Neptunian region. Pluto's effect is particularly large on the orbits in the 2:3 Neptune mean motion resonance. The trajectories found stable over the age of the solar system when only the gravitational effect of four outer planets is considered are often destabilized there in the effect of close Pluto approaches. We estimate that many dynamically primordial bodies moving initially with low to moderate amplitudes in the 2:3 Neptune resonance (semimajor axis 39.45 AU) have been removed from their respective, otherwise stable, locations, when their resonant amplitudes increased in the course of close encounters with Pluto. At large libration amplitude, the orbits became exposed to chaotic changes, and objects were ejected from the 2:3 resonance to Neptune-crossing trajectories. The process of the resonant amplitude excitation was especially efficient for orbits with moderate and large inclinations (i > 8°), where more than 50% of the population has been removed in 4 × 109 yr. We estimate that the remaining part of the primordial resonant population at these inclinations should have had its resonant amplitude excited to about 80°. The effect of Pluto on low-inclination orbits is smaller. We have examined the distribution of 33 objects observed on the 2:3 resonant orbits (Plutinos) and found that there could actually exist indications of the above mechanism. The resonant amplitudes of Plutinos are unusually high for 0.15 < e < 0.3 when compared with randomly generated distribution, and, also, there is only one object (1997 QJ4) on an orbit similar to that of Pluto. In fact, a certain gap may be noticed in the distribution of Plutinos at Pluto's inclination and eccentricity, which, if confirmed by future observations, may be the consequence of Pluto's sweeping effect.
- We report here initial results of the Deep Ecliptic Survey, an ongoing new search for Kuiper belt objects (KBOs) and Centaurs using the 8K × 8K Mosaic CCD array on the 4 m Mayall Telescope at Kitt Peak National Observatory. Within the interval covered in this paper, useful observations were obtained during seven nights in 1998 October and November, 1999 April, and 2000 February. We used a novel technique to efficiently find and determine positions of moving objects. Sixty-nine KBOs and Centaurs with apparent magnitudes between 20.6 and approximately the 24th magnitude were discovered. Nine or 10 of the newly discovered KBOs appear to be in the 3 : 2 mean motion resonance with Neptune, and four appear to be scattered-disk objects. Three objects were found that may be in the 4 : 3 resonance. Sixty-two of the objects reported here have been observed on at least one additional night and have received designations. Our own follow-up astrometry was done primarily with the WIYN 3.5 m telescope in queue-scheduled mode and with the Steward Observatory 90 inch (2.3 m) telescope. Others, using a variety of telescopes, recovered a significant number of our objects. Although not a primary objective of the survey, positions of all main-belt asteroids, Trojan asteroids, and nearby fast-moving asteroids seen in our data also have been determined, and most have been reported to the Minor Planet Center. Through simulations and analysis of the existing KBO database, we have investigated the uncertainty to be expected in various KBO orbital parameters as a function of the extent of the astrometric coverage. The results indicate that the heliocentric distance of an object and the inclination of its orbit can be narrowly constrained with observations from a single apparition. Accurate determination of semimajor axis and eccentricity, on the other hand, requires astrometric data extending over additional apparitions. Based on the observed distribution of orbital inclinations in our sample, we have estimated the true distribution of orbital inclinations in the Kuiper belt and find it to be similar to that of the short-period comets. This result is consistent with the commonly held belief that the Kuiper belt is the source region of the short-period comets.
- In this dissertation, the surfaces of Pluto and Triton were investigated by means of spectroscopy over the visible and near infrared wavelength range from 0.5 to 2.5 mum. In this spectral region, the spectra of both bodies are dominated by solar continuum light reflected from their surfaces, interrupted by absorption bands due to the ices of methane, nitrogen, carbon monoxide, and carbon dioxide, in decreasing order of spectral significance. New spectroscopic observations of Pluto and Triton were made at the University of Arizona's 1.54 meter telescope on Mt. Bigelow and 2.3 meter telescope on Kitt Peak. These data were combined with earlier, unpublished observations and with data from the literature. In particular, the Pluto+Charon spectra span a dozen years from 1983 to 1994, the full 360^circ range of subsolar longitudes, and subsolar latitudes from -10 to +13^circ. Laboratory studies were undertaken to measure the spectral properties of nitrogen and methane ices at temperatures comparable to the surface temperatures of Pluto and Triton. These data were used as inputs to radiative transfer models in order to interpret the telescope spectra. The abundances of various ice species on Pluto and Triton were estimated under a wide range of model assumptions. The relative strengths of different methane absorption bands were examined, revealing an enhancement of weak bands relative to stronger bands. Various multiple scattering effects which might be responsible for the observed enhancement were simulated. The spectrum of Pluto's satellite Charon was shown to be consistent with a composition of dirty water ice. Finally, variations in the spectrum of Pluto with rotational phase were examined and used to constrain the geographic distributions of ices on Pluto's surface.
- The differential equations of planetary theory are solved analytically to first order for the two-dimensional case, using only Jacobian elliptic functions and the elliptic integrals of the first and second kind. This choice of functions leads to several new features potentially of importance for planetary theory. The first of these is that the solutions do not require the expansion of the reciprocal of the distance between two planets, even for those variables which depend on two angular arguments. A second result is that the solution is free from small divisors with the exception of two special resonances. In fact, not only are the solutions for resonant orbits free from small divisors, the perturbations for all variables are expressible in closed form. A subset of the resonant orbits maintains this form and in addition has the remarkable feature that the first order perturbations are purely periodic; they contain no secular terms. A solution for the 13 resonance case is given as an example.
- The present condition of planetary theory, i.e., the computation and comprehension of the motion of the planets, is reviewed. Account is taken of the new and anticipated demands of observational accuracy, new mathematical methods in perturbation theory, and the use of the high-speed computer to perform algebraic as well as numerical work.
- A modified periodic orbit of the third kind is introduced that is closely related to periodic orbits of the third kind as defined by Poincar. It is shown that Pluto librates about the periodic orbit with apparent stability. This further explains the librational motion of the resonant argument of Pluto and the avoidance of a Pluto-Neptune close approach as found by Cohen and Hubbard and the long-term motion of Pluto and the librational motion of the perihelion as found by Williams and Benson. With libration about a periodic orbit, the numerical solution of Williams and Benson can be extrapolated to longer times in the past and future.
- 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.
- The known history of the solar system is discussed, also the types of dynamical problems exhibited by members of the solar system and the solutions suggested for a number of such problems. The recent work of Walker, Emslie and Roy, on Empirical Stability Criteria in Many Body Problems is also mentioned.
- For the orbits with low to moderate inclination and eccentricity, in the asteroid main belt, the analytically computed proper elements are accurate to a level very close to the best result achievable by any analytical theory. This fundamental limitation results from the infinite web of resonances and because of the occurrence of chaotic motions. Still, there are some regions of the belt in which these proper elements are of degraded accuracy, thus preventing a reliable definition of asteroid families and detailed studies of the dynamical structure. We have used a different method to compute asteroid proper elements, following the approach introduced in the LONGSTOP project to describe the secular dynamics of the major outer planets. By applying purely numerical techniques, we produced so-called ‘synthetic’ proper elements for a catalog of 10,256 asteroids with osculating semimajor axes between 2.5 and 4.0 AU. The procedure consisted of simultaneous integration of asteroid and planetary orbits for 2 Myr, with online filtering of the short-periodic perturbations. The output of the integration was spectrally resolved, and the principal harmonics (proper values) extracted from the time series. For each asteroid we have also tested the accuracy and stability in time of the proper elements, and estimated the maximum Lyapunov Characteristic Exponent to monitor the chaotic behaviors. This provided information on the reliability of the data for each orbit, in particular allowing to select 1,852 cases for an extended integration (10 Myr) of the orbits showing instability. The results indicate that for more than half of the cases the proper elements have a time stability improved by more than a factor 3 with respect to the elements computed by the previous analytical theory. But of course there are also unstable cases for which the proper elements are less accurate and reliable, the extreme examples being 23 orbits exhibiting hyperbolic escape from the solar system. This form of escape from the asteroid belt could be responsible for a significant mass loss over the age of the solar system.
- A substantial fraction of the Edgeworth-Kuiper belt objects are presently known to move in resonance with Neptune (the principal commensurabilities are 1/2, 3/5, 2/3, and 3/4). We have found that many of the distant (with orbital semimajor axes a > 50 AU) trans-Neptunian objects (TNOs) also execute resonant motions. Our investigation is based on symplectic integrations of the equations of motion for all multiple-opposition TNOs with a > 50 AU with allowance made for the uncertainties in their initial orbits. Librations near such commensurabilities with Neptune as 4/9, 3/7, 5/12, 2/5, 3/8, 4/27, and others have been found. The largest number of distant TNOs move near the 2/5 resonance with Neptune: 12 objects librate with a probability higher than 0.75. The multiplicity of objects moving in 2/5 resonance and the longterm stability of their librations suggest that this group of resonant objects was formed at early formation stages of the Solar system. For most of the other resonant objects, the librations are temporary. We also show the importance of asymmetric resonances in the large changes in TNO perihelion distances. Key wordsSolar system–trans-Neptunian objects–resonances–orbital evolution
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- The results of the LONGSTOP 1B numerical integration of the orbits of the outer planets from Jupiter to Pluto, spanning 100 Myr forward and 50 Myr backward from the present, are reported. The results are given in terms of fundamental secular frequencies and long-term spectral lines in the major semiaxes, eccentricities, and inclinations of the planetary orbits, with their amplitudes and phases. The results indicate that the outer solar system, over 100 Myr, shows dynamical features that are typical of nonregular regions of motion in the phase space of nonintegrable dynamic systems.
- 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
- Pluto's motion is chaotic in the sense that the maximum Lyapunov exponent is positive and the Lyapunov time (the inverse of the Lyapunov exponent) is about 20 million years (Myr). We have carried out the numerical integration of Pluto over the age of the solar system (5.7 billion years towards the past and 5.5 billion years towards the future), which is about 280 times of the Lyapunov time. Our integration does not show any indication of gross instability in the motion of Pluto. The time evolution of Keplerian elements of a nearby trajectory of Pluto at first grow linearly with the time and then start to increase exponentially. These exponential divergences stop at about 420 Myr and saturate. The exponential divergences are suppressed by the following three resonances that Pluto has: (1) Pluto is in the 3:2 mean motion resonance with Neptune and the libration period of the critical argument is about 20000 years.(2) The argument of perihelion librates around 90 degrees and its period is 3.8 Myr.(3) The motion of the Pluto's orbital plane referred to the Neptune's orbital plane is synchronized with the libration of the argument of perihelion (a secondary resonance). The libration period associated with the second resonance is 34.5 Myr.We briefly discuss the motions of Kuiper belt objects in a 3:2 mean motion resonance with Neptune and several possible scenarios how Pluto evolves to the present stable state.
- Recently, Sheppard et al. (2016) presented the discovery of 7 new trans-Neptunian objects with perihelia beyond 40 AU with moderate eccentricities and semimajor axes over 50 AU. Like the handful of previously known bodies on similar orbits, these objects' semimajor axes are just beyond the Kuiper belt edge and clustered around mean motion resonances (MMRs) with Neptune. The objects likely obtained their observed orbits while trapped in a MMR, where the Kozai-Lidov mechanism can raise their perihelia. This mechanism generates a high-perihelion population and also weakens Neptune's dynamical influence over these objects. Here we numerically model the production of this population under a variety of different migration scenarios for Neptune, varying both migration speed and migration smoothness. We find that high-perihelion objects near Neptunian MMRs constrain the nature of Neptune's migration. In particular, the population near the 3:1 MMR (near 62 AU) is especially useful due to its large population and short dynamical evolution timescale. If Neptune reaches its modern orbit after just ~100 Myrs or less of total migration time, we predict that ~90% of the high-perihelion objects near the 3:1 MMR will all have semimajor axes within 1 AU of each other, residing very near the modern resonance's center. On the other hand, if Neptune takes ~300 Myrs of total time to migrate to its final orbit, we expect ~50% of this population to be in dynamically fossilized orbits slightly closer (>~1 AU) to the Sun than the modern resonance location. We highlight 2015 KH162 as a likely member of this fossilized 3:1 population. Under any plausible migration scenario, the vast majority of high-perihelion objects in resonances more distant than the 4:1 MMR (near 76 AU) reach their orbits well after Neptune stops migrating and represent a recently generated, dynamically active population.
- (1) Introduction and Survey. The method for studying the structure and evolution of the solar system is discussed. It is pointed out that theories that account for the origin of planets alone are basically insufficient. Instead one ought to aim for a general theory for the formation of secondary bodies around a central body, applicable both to planet and satellite formation. A satisfactory theory should not start from assumed properties of the primitive Sun, which is a very speculative subject, but should be based on an analysis of present conditions and a successive reconstruction of the past states. (2) Orbits of Planets and Satellites. As a foundation for the subsequent analysis, the relevant properties of planets and satellites are presented. (3) The Small Bodies. The motion of small bodies is influenced by non-gravitational forces. Collisions (viscosity) are of special importance for the evolution of the orbits. It is pointed out that the focusing property of a gravitational field (which has usually been neglected) leads to the formation of jet streams. The importance of this concept for the understanding of the comet-meteoroid relations and the structure of the asteroidal belt is shown. (4) Resonance Structure. A survey is given of the resonances in the solar system and their possible explanation. It is concluded that in many cases the resonances must already be produced at the times when the bodies formed. It is shown that resonance effects put narrow limits on the postaccretional changes of orbits. (5) Spin and Tides. Tidal effects on planetary spins and satellite orbits are discussed. It is very doubtful if any satellite except the Moon and possibly Triton has had its orbit changed appreciably by tidal effects. The isochronism of planetary and asteroidal spins is discussed, as well as its bearing on the accretional process. (6) Post-accretional Changes in the Solar System. The stability of the solar system and upper limits for changes in orbital and spin data are examined. It is concluded that much of the present dynamic structure has direct relevance to the primordial processes.
- Up to now nearly 1000 small objects have been found in the solar system beyond Neptune. They are called Kuiper Belt Objects (KBOs) or Edgeworth-Kuiper Belt Objects. A group of KBOs named plutinos are in the 3:2 mean motion resonance with Neptune, and Pluto is one of them. Since the discovery of the first KBO in 1992, several small-scale surveys have been carried out and a few large KBOs are discovered. By numerical calculations we have found two regions in the space of orbital elements, where, just as Pluto, objects are trapped in three resonances. These resonances protect Pluto and plutinos from the strong perturbation of Neptune. Furthermore, objects in these two regions will avoid close encounters with Pluto as well. Consequently, in these two regions astronomers might discover some plutinos with larger masses.
- Comets have three known reservoirs: the roughly spherical Oort Cloud (for long-period comets), the flattened Kuiper Belt (for ecliptic comets), and, surprisingly, the asteroid belt (for main-belt comets). Comets in the Oort Cloud were thought to have formed in the region of the giant planets and then placed in quasi-stable orbits at distances of thousands or tens of thousands of AU through the gravitational effects of the planets and the Galaxy. The planets were long assumed to have formed in place. However, the giant planets may have undergone two episodes of migration. The first would have taken place in the first few million years of the Solar System, during or shortly after the formation of the giant planets, when gas was still present in the protoplanetary disk around the Sun. The Grand Tack (Walsh et al. in Nature 475:206–209, 2011) models how this stage of migration could explain the low mass of Mars and deplete, then repopulate the asteroid belt, with outer-belt asteroids originating between, and outside of, the orbits of the giant planets. The second stage of migration would have occurred later (possibly hundreds of millions of years later) due to interactions with a remnant disk of planetesimals, i.e., a massive ancestor of the Kuiper Belt. Safronov (Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, 1969) and Fernández and Ip (Icarus 58:109–120, 1984) proposed that the giant planets would have migrated as they interacted with leftover planetesimals; Jupiter would have moved slightly inward, while Saturn and (especially) Uranus and Neptune would have moved outward from the Sun. Malhotra (Nature 365:819–821, 1993) showed that Pluto’s orbit in the 3:2 resonance with Neptune was a natural outcome if Neptune captured Pluto into resonance while it migrated outward. Building on this work, Tsiganis et al. (Nature 435:459–461, 2005) proposed the Nice model, in which the giant planets formed closer together than they are now, and underwent a dynamical instability that led to a flood of comets and asteroids throughout the Solar System (Gomes et al. in Nature 435:466–469, 2005b). In this scenario, it is somewhat a matter of luck whether an icy planetesimal ends up in the Kuiper Belt or Oort Cloud (Brasser and Morbidelli in Icarus 225:40–49, 2013), as a Trojan asteroid (Morbidelli et al. in Nature 435:462–465, 2005; Nesvorný and Vokrouhlický in Astron. J. 137:5003–5011, 2009; Nesvorný et al. in Astrophys. J. 768:45, 2013), or as a distant “irregular” satellite of a giant planet (Nesvorný et al. in Astron. J. 133:1962–1976, 2007). Comets could even have been captured into the asteroid belt (Levison et al. in Nature 460:364–366, 2009). The remarkable finding of two “inner Oort Cloud” bodies, Sedna and 2012 \(\mbox{VP}_{113}\), with perihelion distances of 76 and 81 AU, respectively (Brown et al. in Astrophys. J. 617:645–649, 2004; Trujillo and Sheppard in Nature 507:471–474, 2014), along with the discovery of other likely inner Oort Cloud bodies (Chen et al. in Astrophys. J. Lett. 775:8, 2013; Brasser and Schwamb in Mon. Not. R. Astron. Soc. 446:3788–3796, 2015), suggests that the Sun formed in a denser environment, i.e., in a star cluster (Brasser et al. in Icarus 184:59–82, 2006, 191:413–433, 2007, 217:1–19, 2012b; Kaib and Quinn in Icarus 197:221–238, 2008). The Sun may have orbited closer or further from the center of the Galaxy than it does now, with implications for the structure of the Oort Cloud (Kaib et al. in Icarus 215:491–507, 2011). We focus on the formation of cometary nuclei; the orbital properties of the cometary reservoirs; physical properties of comets; planetary migration; the formation of the Oort Cloud in various environments; the formation and evolution of the Kuiper Belt and Scattered Disk; and the populations and size distributions of the cometary reservoirs. We close with a brief discussion of cometary analogs around other stars and a summary.
- 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.
- The dynamics of the high-inclination Plutinos is systematically studied. We first present the peculiar features of the 2:3 Neptune mean motion resonance (NMMR) for inclined orbits, especially for the correlation of resonant amplitude Aσ with inclination i. Using the numerical integrations for the age of the Solar system, the dynamical structure of the 2:3 NMMR is mapped out on the plane of semimajor axis versus i for different eccentricities. We have shown that i of stable resonant orbits could be as high as 90°; and the stable region is roughly surrounded by the contours of Aσ = 120°. These new findings allow us to further explore the 2:3 NMMR capture and retention of planetesimals with initial inclinations i0 ≤ 90° in the frame of the planet migration model. We find that the outward transportation of Plutinos is possible for any inclined or even perpendicular orbits. The role of i0 in the formation of Plutinos during Neptune's migration is highlighted and interesting results are obtained: (1) the capture efficiency of the 2:3 NMMR decreases drastically first with the increase of i0, but it then raises instead when i0 exceeds ∼50°; (2) the magnitude of i-variation is limited to less than 5° for any i0, and moreover, for Plutinos with i ≳ 48°, their i are forced to decrease throughout the outward migration; (3) Plutinos with i ≳ 48° are certainly outside the Kozai mechanism, since an inclination increase is prohibited by the migrating 2:3 NMMR; (4) the 7:11 inclination-type NMMR could be responsible for nearly circular Plutinos, and a minimum i0 ∼ 15° is required to intrigue this mechanism.
- A new subject of the solar system dynamics, the orbit dynamics of the Kuiper Belt Objects, is reviewed in this paper. Early studies were connected with the origin of short-period comets. After the first Kuiper Belt Object (KBO) was found, attentions are turned on the phase space structure of the resonant KBOs. Morbidelli and Malhotra adopted different models to study the sizes of the resonance regions, espically the 3:2 mean motion resonance which Pluto is in. For its orbital characters, Pluto should be called a large KBO. There are other two resonances in its motion, the Kozai resonance and 1:1 super resonance. It is because of these resonances that Pluto keeps its orbital stability. Observations show that lots of KBOs are in mean motion resonances with Neptune. In some early theories, these resonances were thought to be caused by catastrophic events such as collision, which can not explain the resonances well for their small probability. Malhotra proposed that Pluto was swept into the 3:2 resonance by planets migration, which happened very easily in the early stage of the solar system. This theory can also explain the resonance formation of KBO successfully.
- Symplectic methods are so far the best numerical methods for qualitative exploration in solar system dynamics. They maintain the symplectic structure and key properties of Hamiltonian systems and do not bring in any artificial dissipation, making possible long-term numerical integrations with a large step size. The symplectic method that has been widely adopted in references on qualitative studies of solar system dynamics is the method worked out by Wisdom and Holman (SYA). It is built in the Jacobian coordinate system and takes an approximation of the Hamiltonian. The Wisdom and Holman's method for an exact Hamiltonian is abbreviated as SYP. Actually a symplectic integrator can be built in the barycentric coordinate system (SYS), which separates the Hamiltonian into two parts, the potential energy and the kinetic energy. Here we propose a quasi-symplectic method SYQ in the barycentric coordinate system. An extensive comparative study of these four types of methods is given, especially on their computation efficiency and error accumulation. This research draws the following conclusion. Considering that symplectic integrators are mainly used in exploring the qualitative evolution of dynamical systems and a high precision is not required, SYS should not be recommended in solar system dynamics for its low efficiency. During a 108 years integration, SYP methods cause almost the same errors on the positions of the planets but they take about 40% more computing time. We thus believe that SYP cannot compete with SYA or SYQ, but it is hard to tell SYA or SYQ is better. Our research has also shown that resonances play a role in keeping the orbit configuration of a planetary system during long-term numerical integrations.
- 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.
- The following 'irregular' features of the solar system are explained in terms of capture events following planetary collisions in a system of noncoplanar eccentric orbits: (1) the terrestrial planets; (2) the origin and capture of the moon; (3) asteroids, meteorites, and comets; (4) the origin and orbit of Pluto and the retrograde motion of Triton; and (5) the outer satellites of Jupiter. The probability of a planetary collision in the postulated orbital system and the products of such a collision are examined in detail. It is shown that a resisting medium resulting from a capture event would serve to round off the planetary orbits in a short time as compared with the age of the solar system and would also give rise to differential rotations of the lines of apses of the early orbits, leading to a high probability of collisions or close interactions between planets. Interactions are analyzed which could result in either substantial orbital modifications or a direct planetary collision. It is suggested that a planetary collision in the asteroid region could lead to the formation of the earth-moon system and that Triton could have been sufficiently perturbed during a Pluto-Neptune encounter to reverse its orbital motion
- Up to now nearly 1000 small objects have been found in the solar system beyond Neptune. They are called Kuiper Belt Objects (KBOs) or Edgeworth-Kuiper Belt Objects. A group of KBOs named plutinos are in the 3:2 mean motion resonance with Neptune, and Pluto is one of them. Since the discovery of the first KBO in 1992, several small-scale surveys have been carried out and a few large KBOs are discovered. By numerical calculations we have found two regions in the space of orbital elements, where, just as Pluto, objects are trapped in three resonances. These resonances protect Pluto and plutinos from the strong perturbation of Neptune. Furthermore, objects in these two regions will avoid close encounters with Pluto as well. Consequently, in these two regions astronomers might discover some plutinos with larger masses.
- It is not particularly surprising that the properties of asteroids should change between the inner and outer parts of the main asteroid belt between Mars and Jupiter. After all, this belt marks the transition between the rocky planets of the inner Solar System and the gas giants of the outer Solar System. What is perhaps more unexpected is that a similar transition should occur in the Kuiper belt — an even greater ring of icy bodies orbiting beyond Neptune. Three years ago, Tegler and Romanishin1 reported the results of a photometric survey of these distant bodies, showing that they fall into two distinct groups based on their colour, but the reason for the two colours was a mystery. On page 979 of this issue2, Tegler and Romanishin present a follow-up survey that demonstrates a possible link between the colours of these objects and their distances from the Sun.
- We 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.
- Our planetary system is dynamically stable for the lifetime of the solar system according to the long-term numerical integrations of planetary orbits (Ito & Tanikawa, 2002). We discuss various forms of subsystems of a stable planetary system which may contribute to maintain the system stability. It is well known that resonances play an important role in such a long time scale. We stress that contrary to the restricted problem such as the stability of asteroids and comets, multi-planet subsystems may have variety of mechanisms for keeping stability.
- The present consideration of the conditions of formation for the Pluto-Charon system in the protoplanetary disk gives attention to a high angular momentum that would render this binary system rotationally unstable in the case of coalescence into a single body. The accumulation of such great angular momentum can be accounted for by the process of planetesimals' trapping in heliocentric orbit. The results of numerical simulations indicate that a significant portion of the Pluto-sized bodies achieve rotational stability.
- It is pointed out that equations of motion (i.e., the continuity equations and the Euler-Bernoulli equation for the motion of a system of particles in gravitational interaction) are best and most advantageously formulated in terms of a Schrödinger-type equation in which the h3 characterizing the uncertainty principle is replaced by a macroscopic phase-space volume which characterizes the coarse-grainedness of the mass distribution in phase space. This method is applied to the study of a gravitational system of the type of our solar system in its recent development, as the present system is assumed to be preceded by a system of a large number of smaller particles in gravitational interaction. The distribution of orbital elements of the planets in the solar system, and also of the elements of the satellites of the major planets, shows distinct regularities. One of them, the commensurability of the periods of revolution, has been repeatedly discussed. Another, the small eccentricities and inclinations, has found a simple explanation in the Kant-du Ligondès hypothesis of the development of the solar system. Still another one, concerning the distribution of the semimajor axes (or periods), has found a numerological formulation in the Titius-Bode "law," and its explanation has been attempted in terms of a distribution of vortex rings in the early stage of the Kant-du Ligondès model. We are here proposing a simpler analysis of the regularities in this distribution of the semimajor axes. It is suggested that statistically there are changes toward preferential orbital elements due to the gravitational interaction in such a system. Instead of discussing the issue in terms of perturbation expansions of individual orbits, however, it may be more advantageous to pursue the analysis in terms of the statistical distribution of integrals of the equations of motion, in particular the Jacobi integral. These distributions of the integrals (orbital elements) are most elegantly formulated in terms of distributions of the stationary-state wave functions which correspond to those integrals. The calculations yield, for the planets and for the satellites, sequences of orbital elements which coincide strikingly well with the observed elements.
- In this discussion of Bode's law Dermott considers whether the law could have arisen by chance, and how the near resonances between a number of orbits in the Solar System may have developed.
- Summary This document is part of Subvolume A ‘Methods, Constants, Solar System’ of Volume 2 ‘Astronomy and Astrophysics’ of Landolt-Börnstein - Group VI Astronomy and Astrophysics.
- Three resonances, the 3:2 exterior mean motion resonance with Neptune, Kozai resonance and 1:1 super resonance, are known to govern concurrently the stability of the motion of Pluto. We explore numerically the resonance zones in which the three resonance coexist. There might exist plutinos with relatively large masses in these zones. Considering that Pluto's perturbation is important to the long-term evolution of plutinos, the resonance zone is mainly explored in the mirror region of Pluto, which is a mirror image of the region around Pluto with respect to the invariant plane of the solar system. We find six resonance zones in the mirror region. The orbit elements at the centers of the six zones and the corresponding heliocentric distances, longitudes and latitudes on July 1, 2003 have been computed and listed for the reference of observation.
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- D Brouwer
Eckert, W. J., Brouwer, D., and Clemence, G. M., 1951, Astron. Papers Amer. Ephemeris Nautical Almanac Vol. XII.