Secular Perturbations of Planetary Orbits and their Representation as Series
Abstract
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.
- Mean orbital elements are obtained from their instantaneous, osculating counterparts by removal of the short periodic perturbations. They can be computed by means of different theories, analytical or numerical, depending on the problem and accuracy required. The most advanced contemporary analytical theory (Knežević 1988) accounts only for the perturbing effects due to Jupiter and Saturn, to the first order in their masses and to degree four in eccentricity and inclination. Nevertheless, the mean elements obtained by means of this theory are of satisfactory accuracy for majority of the asteroids in the main belt (Knežević et al. 1988), for the purpose of producing large catalogues of mean and proper elements, to identify asteroid families, to assess their age, to study the dynamical structure of the asteroid belt and chaotic phenomena of diffusion over very long time spans. In the vicinity of the main mean motion resonances, however, especially 2:1 mean motion resonance with Jupiter, these mean elements are of somewhat degraded accuracy.
- 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.
- Though the concept of the Lie transform dates back to more than a century ago, it is only in about the last thirty years that this concept has been introduced into perturbative theories and then applied on a vast scale in various fields of physics. As we recall in the course of this chapter, the field where the concept of the Lie transform was introduced for the first time is celestial mechanics and, incredibly, this concept is the only development of perturbation theory which cannot in some way be made to date back to Poincaré. Equally surprising is that the “old” canonical perturbation theory, in spite of the awkwardness involved by the use of a generating function with “mixed” variables, has ruled up until now, never falling into discredit, not even as a consequence of exaggerations like that of Delaunay, who had to calculate no less than 505 successive canonical transformations. We think that the record for absurdity has been set in plasma physics, where the use was established by somebody of quantizing classical systems, applying quantum perturbation theory (which provides more practical rules) and then letting the Planck constant h → 0 in the result. That was the situation until three decades ago. For these reasons, we thought it right to follow, in our exposition, wherever it has been possible, the chronological order in which the various contributions have appeared, at the end showing how the Lie transform method is substantially the right method for implementing KAM techniques. In this chapter, as in the preceding ones, we have tried to isolate what appeared to us to be the fundamental concepts and to insist on them, instead of dwelling upon the exposition of complicated examples or involved formulae for calculations. For the latter, the reader will find all the necessary information in the bibliographical notes.
- We analyze the dynamics of a driven, damped pendulum as used in mechanical clocks. We derive equations for the amplitude and phase of the oscillation, on time scales longer than the pendulum period. The equations are first order ODEs and permit fast simulations of the joint effects of circular and escapement errors, friction, and other disturbances for long times. The equations contain two averages of the driving torque over a period, so that the results are not very sensitive to the fine structure of the driving. We adopt a constant-torque escapement and study the stationary pendulum rate as a function of driving torque and friction. We also study the reaction of the pendulum to a sudden change in the driving torque, and to stationary noisy driving. The equations for the amplitude and phase are shown to describe the pendulum dynamics quite well on time scales of one period and longer. Our emphasis is on a clear exposition of the physics.
- The purpose of this series of lecture notes is to give an outline of the basic tools required to show the occurrence of chaotic motions in the simplest non--integrable problems in Celestial Mechanics, such as the circular restricted planar 3--body problem. No formal proofs will be given here; they can be found in the references given for each section. Section 1 describes the linear and local theory of ordinary differential equations in the neighbourhood of a fixed point; the problems arising in the embedding of invariant stable and unstable manifolds are also discussed. Section 2 is about periodic orbits; the subjects discussed include variational equations, surfaces of section, the continuation of periodic orbits in the restricted 3--body problem, and bifurcation of hyperbolic periodic orbits from resonant periodic orbits. Section 3 covers fundamental models of resonance, the global behaviour of separatrices, and their intersections; all this allows to give at least an outline of the proof of the fundamental result --presented by Poincaré in his book Les méthodes nouvelles de la mécanique céleste-- by which homoclinic points must necessarily occur in the restricted problem. A short conclusion underscores the fact --shown in a rigorous vay much later-- that this in turn implies that chaos in the strongest possible sense occurs in the restricted problem, and is an essential feature of every non--integrable system, even very simple ones with only two degrees of freedom. I apologize for reporting here my lectures in a very short format, almost without comments in between the formulas and statements of the main results; my understanding of the purpose of this notes is that they should serve as a reminder of the existence of many subjects to be studied, rather than a complete presentation which could not be contained in this format.
- A simple model is presented for the coupled dynamics of the orbit-rotation-climate system of Mars. Changes in the orientation of the spin pole, relative to the orbit pole, influence the spatiotemporal pattern of incident radiation and thus drive climatic mass transport into and out of the polar regions on a variety of timescales. Changes in the mass distribution occur from direct climatic forcing and compensating viscous flow in the interior. The net change in mass distribution influences the rate of spin axis precession and thereby influences obliquity. The rate of secular obliquity drift depends on several poorly known parameters, including the magnitudes and response times of volatile inventories and viscosity structure within Mars. Even relatively modest secular obliquity drift can lead to trapping in nearby resonances. The dissipative nature of the coupled dynamical system makes reconstruction of past evolution much more difficult than for a purely inertial system. The long-term obliquity history of Mars is dominated by climate.
- The long term evolution of the orbits of the asteroids is studied by means of proper elements, which are quasi-integrals of the motion. After a short review of the classical theories for secular perturbations, this paper presents the state of the art for the computation of proper elements. The recent theories have been extended to higher degree in the eccentricities and inclinations, and to the second order in the perturbing masses; they use new iterative algorithms to compute secular perturbations with fixed initial conditions but variable frequencies. This allows to compute proper elements stable over time spans of several million years, within a range of oscillations small enough to allow the identification of asteroid families; the same iterative algorithm can also be used to automatically detect secular resonances, that is to map the dynamical structure of the main asteroid belt. However the proper element theories approximate the true solution of the N-body problem with a conditionally periodic solution of a truncated problem, while the orbits of most asteroids are not conditionally periodic, but chaotic; positive Lyapounov exponents have been detected for a large number of real asteroids. The phenomenon of stable chaos occurs whenever the range of oscillation of the proper elements, as computed by state of the art theories, remains small for time spans of millions of years, while the Lyapounov time (in which the orbits diverge by a factor exp(1) is much shorter, e.g. a few thousand years. This can be explained only by a theory which accounts correctly for the degeneracy of the unperturbed 2-body problem used as a first approximation. The two stages of computation of mean and proper elements are each subject to the phenomana of resonance and chaos; stable chaos occurs when a weak resonance affects the computation of mean elements, but the solution of the secular perturbation equations is regular.
- The problem of calculating the proper elements of asteroids for the purpose of family identification is examined analytically. The derivation of the Lie-series theoretical model of Yuasa (1973) is reviewed; the selection of a coordinate system and of data for the motion of the major planets is explained; and an indirect technique for estimating the accuracy of the elements calculated is described. Results for 158 Koronis, 221 Eos, 24 Themis, and 8 Flora are presented in extensive tables and graphs and discussed in detail. While the elements for Flora are found to be inaccurate, calling into question the membership or even the reality of its family of asteroids, those for Koronis, Eos, and Themis are more accurate than those previously available, although not yet at the 0.001 level required for reliable family classification.
- A new theory for the calculation of proper elements, taking into account terms of degree four in the eccentricities and inclinations, and also terms of order two in the mass of Jupiter, has been derived and programmed in a self contained code. It has many advantages with respect to the previous ones. Being fully analytical, it defines an explicit algorithm applicable to any chosen set of orbits. Unlike first order theories, it takes into account the effect of shallow resonances upon the secular frequencies; this effect is quite substantial, e.g. for Themis. Short periodic effects are corrected for by a rigorous procedure. Unlike linear theories, it accounts for the effects of higher degree terms and can thus be applied to asteroids with low to moderate eccentricity and inclination; secular resonances resulting from the combination of up to four secular frequencies can be accounted for. The new theory is self checking : the proper elements being computed with an iterative algorithm, the behaviour of the iteration can be used to define a quality code. The amount of computation required for a single set of osculating elements, although not negligible, is such that the method can be systematically applied on long lists of osculating orbital elements, taken either from catalogues of observed objects or from the output of orbit computations. As a result, this theory has been used to derive proper elements for 4100 numbered asteroids, and to test the accuracy by means of numerical integrations. These results are discussed both from a quantitative point of view, to derive an a posteriori accuracy of the proper elements sets, and from a qualitative one, by comparison with the higher degree secular resonance theory.
- Planetary and satellite theories have been historically and are presently intimately related to the available computing capabilities, the accuracy of observational data, and the requirements of the astronomical community. Thus, the development of computers made it possible to replace planetary and lunar general theories with numerical integrations, or special perturbation methods. In turn, the availability of inexpensive small computers and high-speed computers with inexpensive memory stimulated the requirement to change from numerical integration back to general theories, or representative ephemerides, where the ephemerides could be calculated for a given date rather than using a table look-up process. In parallel with this progression, the observational accuracy has improved such that general theories cannot presently achieve the accuracy of the observations, and, in turn, it appears that in some cases the models and methods of numerical integration also need to be improved for the accuracies of the observations. Planetary and lunar theories were originally developed to be able to predict phenomena, and provide what are now considered low accuracy ephemerides of the bodies. This proceeded to the requirement for high accuracy ephemerides, and the progression of accuracy improvement has led to the discoveries of the variable rotation of the Earth, several planets, and a satellite. By means of mapping techniques, it is now possible to integrate a model of the motion of the entire solar system back for the history of the solar system. The challenges for the future are: Can general planetary and lunar theories with an acceptable number of terms achieve the accuracies of observations? How can numerical integrations more accurately represent the true motions of the solar system? Can regularly available observations be improved in accuracy? What are the meanings and interpretations of stability and chaos with respect to the motions of the bodies of our solar system? There has been a parallel progress and development of problems in dealing with the motions of artificial satellites. The large number of bodies of various sizes in the limited space around the Earth, subject to the additional forces of drag, radiation pressure, and Earth zonal and tesseral forces, require more accurate theories, improved observational accuracies, and improved prediction capabilities, so that potential collisions may be avoided. This must be accomplished by efficient use of computer capabilities.
- The variations in the obliquity of Mars are considered to be the likely source of major climatic variations on that planet. This paper explores the range of uncertainty in the obliquity history of Mars associated with the present uncertainty in the axial precession rate, applying three different analytic techniques. It is shown that, within the observationally allowed range of axial precession rates, there are some intervals where the obliquity history of Mars is only weakly dependent on the precession rate, and other intervals where the obliquity is very sensitively dependent on the precession rate. A very wide range of obliquity histories are possible, including some which involve resonance passages within the relatively recent past. It is estimated that obliquities as high as 51.4 deg or as low as 0.2 deg may have occurred within the last ten million years.
- Equations of motion for the general four-body problem are derived in terms of Jacobian coordinates. A reduction to the three-body problem yields an approach for studying the perturbations experienced by a binary about whose center of mass a third mass revolves. A stability criterion for the binary and the revolving mass is proposed. This stability criterion can be related to a similar criterion developed by Zare (1977). Known triple star systems, data from the solar system, and numerical experiments conducted for triple mass problems are analyzed through use of the stability criterion. Difficulties in applying the stability criterion to the Saturn-Titan-Hyperion and sun-Neptune-Pluto systems also receive attention.
- 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.
- The secular system of the 8 main planets of the solar system has been computed up to order 2 and degree 5. It was numerically integrated over 30 million years and Fourier analysed to obtain a solution similar to the results of analytical theories.
- In Poincarés “Méthodes Nouvelles de la Mécanique Celeste”, pp. 127–135, there is given a formal solution of the problem of three bodies in motion under their mutual gravitational attractions, in the case where one of the bodies (the “primary”) has mass appreciably larger than the others, and where the motion of the others relative to the primary is nearly circular, and nearly coplanar. We may call this the “planetary case” of the general gravitational problem of three bodies. The solution is derived using Von Zeipel type transformations, in a number of stages, making use of infinite series in powers of the ratios of the smaller masses to that of the primary, and, in the treatment of the secular variations, powers of quantities of the order of the orbital eccentricities and orbital inclinations. In the present treatment, the simple extension to n planets is made, and the transformations employing power series are introduced making use of the Lie series method introduced by Hori (Pub. Astron. Soc., Japan, Vol. 18, pp. 287-295, 1966) which gives explicit expressions for the transformed variables in terms of the untransformed, and vice-versa. The short-period terms are removed by a single transformation at the outset, so that the elements of the matrix defining the linear transformation employed in the secular variation theory are functions of the constant transformed major axes, and so do not themselves possess short-period terms as those in Poincare’s solution do.
- After a presentation of Lyapunov characteristic exponents (LCE) we recall their basic properties and numerical methods of computation. We review some numerical computations which are concerned with LCEs mainly those concerning the dimensions of invariant manifolds and chaotic attractors.
- A theory of general perturbations based on Lie series is presented together with its predecessor, the von Zeipel method. Some drawbacks of the von Zeipel method are shown together with the motivation for an improved theory by a pedagogical purpose. It is shown that perturbation theories based on canonical transformations and averaging principles yield the same results (through the second order) irrespective of the implicit or explicit type of the transformations. The motion of a near earth satellite in an orbit with small eccentricity and inclination, and that with small angular momentum are presented as two examples of the use of the theory.
- Results are reported from long-term numerical integrations of the motion of the outer planets, undertaken as part of the Longstop project (Nobili, 1987). The theoretical basis and numerical implementation of the computations are reviewed, and the results are presented graphically and analyzed in detail. In the 9.5-Myr computations of Longstop 1A, the dynamical structure of the outer solar system is shown to be significantly affected by the secular frequency g5-g7 (1.119 Myr), involving the pericenters of Jupiter and Uranus. The accumulation of spectral lines observed in the long-period spectrum of the outer solar system is tentatively attributed to a secular small divisor of period about 31 Myr, found in the 100-Myr integration Longstop 1B.
- For the ‘planetary case’ of the gravitationaln-body problem in three dimensions, a sequence of Lie series contact transformations is used to construct asymptotic series representations for the canonical parameters of the instantaneous orbits in a Jacobi formulation. The series contain only periodic terms, the frequencies being linear combinations of those of the planetary orbits and those of the secular variations of the apses and nodes, and the series are in powers of the masses of the planets in terms of that of the primary, and of a quantity of the order of the excursions of the eccentricities and inclinations of the orbits. The treatment avoids singularities for circular and coplanar orbits. It follows that the major axes are given by series of periodic terms only, to all orders in the planetary masses.
- Modern theories of dynamical systems have very clearly demonstrated the unexpected fact that systems governed by the equations of Newtonian dynamics do not necessarily exhibit the 'predictability' property. Indeed, very recent researches have shown that in wide classes of very simple systems satisfying those equations predictability is impossible beyond a certain definite time horizon.
- 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
- The 'LONGSTOP' project, which investigates solar system dynamical stability over timescales that are comparable to its lifetime through numerical integrations, has concentrated on obtaining a compression of the output which will be suitable for comparison with analytical theories. Attention is presently given to output decimation and digital filtering, which facilitate the study of long term changes in the orbital elements as deduced from the output of a numerical integration, and to Fourier analysis techniques, by means of which prominent lines in the dynamical spectrum can be identified when theoretical constraints on the allowed combination frequencies are taken into account. In this way, a synthetic secular perturbation theory is built from the numerical experiments for comparison with available analytical theories.
- The short-periodic perturbations of asteroid elements are calculated by means of the fourth-degree analytic theory of Yuasa (1973) and, for comparison, by means of a combination of numerical integration and filtering techniques. The accuracy and reliability of the obtained mean elements are analysed from the point of view of their applicability as parameters for asteroid classification into families. Special attention has been paid to the influence of near resonances and to the problem of the deterioration of the accuracy for high eccentricity and high inclination objects. It has been found that in most cases a simple and fast analytic procedure can provide data accurate enough for a reliable classification.
- 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.
- Within the LONGSTOP (Long-term Gravitational Stability Test of the Outer Planets) research project, numerical integrations of the orbits of the outer planets show, for the first time, dynamical features in the behavior of the semimajor axes of these planets over time scales of the order of millions of years. The most interesting one is an oscillation, in antiphase, of the semimajor axes of Uranus and Neptune revealing an almost exact exchange of energy of the two planets with one another. The period of the oscillation is 1,119,416 years - the same as the period of the exchange in angular momentum between Jupiter and Uranus.
- Long term numerical integrations of planetary orbits designed to study the stability of the Solar System over timescales comparable to its age have become very promising thanks to the availability of very powerful computers and to a substantial improvement in methods of investigating the stability of hierarchical dynamical systems. The stability of such numerical integrations relies on the ability to control all possible sources of error. Among the errors caused by the inadequacy of the physical model are those due to the fact that Newton's theory of gravitation is used instead of general relativity. It is shown that the secular advance of perihelia predicted by general relativity can be simulated exactly by a 1/r-squared perturbing potential with almost negligible additional cost in computer time.
- One of the oldest problems of celestial mechanics is that of the long-term behaviour of the semimajor axes a of the planetary orbits. Analytical theories1,2 predict periodic variations in a, some of which may have very long periods, but these terms have never been computed. We have now performed a 9.3-Myr numerical integration of the orbits of the outer planets, using a pure newtonian point mass model. An accurate integrator and an effective low-pass filtering of the output allow us to detect high-order variations in the energies, and hence also in a, with periods ranging from tens of thousand to millions of years. The most interesting feature is an energy exchange between Uranus and Neptune with a period of 1,119,000 years, the same as the period of the libration between the perihelia of Jupiter and Uranus3,4. The mechanism involves Jupiter and also Saturn; moreover, their energy shows puzzling longer-term trends. The energy of Pluto changes mostly with periods close to that of the 3:2 libration in mean motion with Neptune. Its spectrum in this region shows a very complicated structure; however, we have found no indication of chaotic behaviour.
- 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 analytical stability criterion applicable to coplanar hierarchical three-body systems described in the first paper of this series, Walkeret al. (1980), is modified to give an exact representation ofHill-type stability in all such cases. The dependence of the stability on all orbital parameters (in the coplanar case) is taken into account. The criterion for stability is now dependant upon the participating masses, the elements of the initial osculating Keplerian orbits of the system (viz. the orbits ofm 2 aboutm 1 andm 3 about the mass-centre of the (m 1,m 2) system) and the positions within these orbits. The behaviour of the stability of such systems is demonstrated (both analytically and numerically) with respect to certain of the parameters involved to consider effects not dealt with in the above-mentioned paper. In particular two interesting real cases of triple systems in the Solar System are discussed, namely Sun-Jupiter-Saturn and Earth-Moon-Sun. The results of the present paper are compared with those of past authors who considered the same systems. Finally some general features arising out of our analysis are discussed.
- 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.
- We propose a canonical transformation reducing the averaged planar planetary problem near resonance to a one degree of freedom problem when the perturbation is truncated at the first order in the eccentricities. This reducing transformation leads to a very simple explanation of the puzzling behaviour of the Apocentric Librators, a class of asteroids identified by Franklinet al. (1975). An exploration of the phase space of the average problem with the use of the mapping technique shows that the alternation of two libration mechanism is a common feature for initial conditions near, but not inside, the deep resonance region.
- Generalized Jacobian coordinates can be used to decompose anN-body dynamical system intoN-1 2-body systems coupled by perturbations. Hierarchical stability is defined as the property of preserving the hierarchical arrangement of these 2-body subsystems in such a way that orbit crossing is avoided. ForN=3 hierarchical stability can be ensured for an arbitrary span of time depending on the integralz=c 2 h (angular momentum squared times energy): if it is smaller than a critical value, defined by theL 2 collinear equilibrium configuration, then the three possible hierarchical arrangements correspond to three disconnected subsets of the invariant manifold in the phase space (and in the configuration space as well; see Milani and Nobili, 1983a). The same definitions can be extended, with the Jacobian formalism, to an arbitrary hierarchical arrangement ofN[(z)\tilde]23\tilde z_{23} and [(z)\tilde]34\tilde z_{34} , so that both the subsystems are initially hierarchically stable. Then the hierarchical arrangement of the 4 bodies cannot be broken until eitherz 23 orz 34 is changed by an amount [(z)\tilde]ij - zij ( 0 )\tilde z_{ij} - z_{ij} \left( 0 \right) ; that is the whole system is hierarchically stable for a time spain not shorter than the minimum between Dt23 = ( [(z)\tilde]23 - z23 ( 0 ) ) \mathord/ \vphantom ( [(z)\tilde]23 - z23 ( 0 ) ) [(z)\dot]23 [(z)\dot]23 \Delta t_{23} = {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} {\dot z_{23} }}} \right. \kern-\nulldelimiterspace} {\dot z_{23} }} and Dt34 = ( [(z)\tilde]34 - z34 ( 0 ) ) \mathord/ \vphantom ( [(z)\tilde]34 - z34 ( 0 ) ) [(z)\dot]34 [(z)\dot]34 \Delta t_{34} = {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} {\dot z_{34} }}} \right. \kern-\nulldelimiterspace} {\dot z_{34} }} . To estimate how long is this stability time, two main steps are required. First the perturbing potentials have to be developed in series; the relevant small parameters are some combinations of mass ratios and length ratios, the[(z)\dot]ij\dot z_{ij} (we limit ourselves to the planar case). To assess the long term behaviour of the system, we can neglect the short-periodic perturbations and discuss only the long-periodic and the secular perturbations. By using a Poisson bracket formalism, a generalization of Lagrange theorem for semimajor axes and a generalization of the classical first order theories for eccentricities and pericenters, we prove that thez ij do not undergo any secular perturbation, because of the interaction with the other subsystem, at the first order in the ik . After the long-periodic perturbations have been accounted for, and apart from the small divisors problems that could arise both from ordinary and secular resonances, only the second order terms have to be considered in the computation of t 23, t 34. A full second order perturbative theory is beyond the scope of this paper; however an order-of-magnitude lower estimate of the t ij can be obtained with the very pessimistic assumption that essentially all the second order terms affect in a secular way thez ij . The same method could be applied also toN5 body systems. Since almost everyN-body system existing in nature is strongly hierarchical, the product of two ij is very small for almost all the real astronomical problems. As an example, the hierarchical stability of the 4-body system Sun, Mercury, Venus, and Jupiter is investigated; this system turns out to be stable for at least 110 million years. Although this hierarchical stability time is 10 times less than the real age of the Solar System, taking into account that many pessimistic assumptions have been done we can conclude that the stability of the Solar System is no more a forbidden problem for Celestial Mechanics.
- In this paper a proof is given of Kolmogorov’s theorem on the existence of invariant tori in nearly integrable Hamiltonian systems. The scheme of proof is that of Kolmogorov, the only difference being in the way canonical transformations near the identity are defined. Precisely, use is made of the Lie method, which avoids any inversion and thus any use of the implicit-function theorem. This technical fact eliminates a spurious ingredient and simplifies the establishment of a central estimate. Nel presente lavoro si dimostra il teorema di Kolmogorov sull’esistenza di tori invarianti in sistemi Hamiltoniani quasi integrabili. Si usa lo schema di dimostrazione di Kolmogorov, con la sola variante del modo in cui si definiscono le trasformazioni canoniche prossime all’identità. Si usa infatti il metodo di Lie, che elimina la necessità d’inversioni e quindi dell’impiego del teorema delle funzioni implicite. Questo fatto tecnico evita un ingrediente spurio e semplifica il modo in cui si ottiene una delle stime principali. В этой работе предлагается доказательство теоремы Колмогорова о существовании инвариантных торов в квази-интегрируемых Гамильтоновых системах. Используется схема доказательства, предложенная Колмогоровым, единственное отличие состоит в способе, которым определяются канонические преобразования. В этой работе используется метод Ли, которыи исключает необходимость инверсии и, следовательно, использование теоремы для неявной функции. Этот технический прием исключает ложный ингрдеиент и упрощает получение главной оценки.
- The accuracy and reliability of the proper orbital elements used to define asteroid families are investigated by simulating numerically the dynamical evolution of families assumed to arise from the “explosion” of a parent object. The orbits of the simulated family asteroids have then been integrated in the frame of the elliptic restricted three-body problem Sun-Jupiter-asteroid, for times of the order of the circulation periods of perihelia and nodes. By filtering out short-periodic perturbations, we have monitored the behavior of the proper eccentricities and inclinations, computed according to the linear secular perturbation theory. Significant long-period variations have been found especially for families having nonnegligible eccentricities and/or inclinations (like the Eos family), and strong disturbances due to the proximity of mean motion commensurabilities with Jupiter have been evidenced (for instance, in the case of the Themis family). These phenomena can cause a significant “noise” on the proper eccentricities and inclinations, probably affecting in some cases the derived family memberships. They can also give rise to a spurious anisotropy in the fragment ejection velocity fields computed from the dispersion in proper elements observed in each family, and this could explain the puzzling anisotropies of this kind actually found in real families by D. Brouwer (1951, Astron. J.56, 9–32) and by V. Zappalà, P. Farinella, Z. Knežević, and P. Paolicchi (1984), Icarus59, 261–285).
- 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.
- The General Uranus Satellite Theory GUST (Laskar, 1986) is used for the construction of an analytical ephemeris for the Uranian satellites. The theory is fitted against earth-based observations from 1911 to 1986, and all radio and optical data obtained during Voyager encounter with Uranus. Earth-based observations alone allow the determination of masses which are within 15 percent of the values determined by the Uranus flyby. The analysis of all the observations confirm the values of the masses obtained during the encounter (Stone and Miner, 1986) and give a complete set of dynamical parameters for the analytical theory. An analytical ephemeris, GUST86, with an estimated precision of about 100 km with respect to Uranus is obtained.
- We have designed and built the Orrery, a special computer for high-speed high-precision orbital mechanics computations. On the problems the Orrery was designed to solve, it achieves approximately 10 Mflops in about 1 ft3of space while consuming 150 W of power. The specialized parallelarchitecture of the Orrery, which is well matched to orbital mechanics problems, is the key to obtaining such high performance. In this paper we discuss the design, construction, and programming of the Orrery. Copyright © 1985 by The Institute of Electrical and Electronics Engineers, Inc.
- 1889, Annales de l’Obs
- M D Eginitis
- The recently recognized failure of predictability in Newtonian dynamics
- J Lighthill
- Memoir on the secular variation of the elements of the orbits of the eight principal planets
- J N Stockwell
