The expansion of negative powers of mutual distance between two planets

In this part we obtain the expression for ^–s by the application of Smart's method, which involves Taylor's theorem for functions of several variables. We neglected terms of power higher than the fourth with respect to eccentricities and tangents of inclinations.     

Expansion of the mutual distance between two planets raised to any negative power

An expression for ^–s in terms of the Poincaré variablesL, , H, K, P. Q has been evaluated. The inclinations of the two planets are referred to a common fixed plane. We neglect in the final formula powers higher than the third of the Poincaré variables.     

Expansions of the negative powers of mutual distances between bodies

The negative powers of the mutual distance ^– between two bodies are developed into series converging at any moment but that of collision. On the base of these expansions the series have been constructed representing ^– in the perturbation theory of celestial mechanics. In the general case, including intersecting orbits, the terms are quasi-periodic functions of the time. In the case of non-intersecting orbits the expansion is a double Fourier series in the mean anomalies. All the expansions have a literal form with respect to osculating elements.     

Expansion of the inverse of mutual distance between two bodies raised to any power

We calculate the expression forΔ ^-s in terms of true anomalies and classical orbital elements, referring to a common fixed plane and working up to power four of eccentricities and tangents of inclinations. We obtained two final results: the first wheny ^′>y, the second wheny>y ^′.     

On the expansion of the secular part of the perturbing function of mutual attraction in the satellite systems of planets

We propose a special representation for the secular part of the perturbing function describing the mutual attraction of satellites. In contrast to the known representations, it has a single analytical form for any ratio between the semimajor axes of the perturbed and perturbing satellites. The resulting expression is a partial sum of a power series with respect to the small eccentricities and planet-equatorial inclinations of the satellites’ orbits. This sum includes terms up to and including the fourth degree with respect to these small parameters. The proposed expansion is compared with one of the known expansions for the secular part of the perturbing function.     

Expansion of the planetary mutual distance raised to any negative power by the method of symbolic differential operators

We present a literal approach to evaluate ^–s necessary for the construction of high order planetary theories. This approach is valid to be applied on very large scale digital computers with standard Poisson series programs, for high order and high degree planetary theories. We apply the method of symbolic differential operators for single variable functions, and the binomial theorem expansions, for the evaluation of ^–s . We utilize Laplace coefficients and its derivatives to carry out the development, without dealing with Newcomb operators or Hansen's coefficients.     

Recursive computation of mutual potential between two polyhedra

Recursive computation of mutual potential, force, and torque between two polyhedra is studied. Based on formulations by Werner and Scheeres (Celest Mech Dyn Astron 91:337–349, 2005) and Fahnestock and Scheeres (Celest Mech Dyn Astron 96:317–339, 2006) who applied the Legendre polynomial expansion to gravity interactions and expressed each order term by a shape-dependent part and a shape-independent part, this paper generalizes the computation of each order term, giving recursive relations of the shape-dependent part. To consider the potential, force, and torque, we introduce three tensors. This method is applicable to any multi-body systems. Finally, we implement this recursive computation to simulate the dynamics of a two rigid-body system that consists of two equal-sized parallelepipeds.     

The dynamics of two massive planets on inclined orbits

The significant orbital eccentricities of most giant extrasolar planets may have their origin in the gravitational dynamics of initially unstable multiple planet systems. In this work, we explore the dynamics of two close planets on inclined orbits through both analytical techniques and extensive numerical scattering experiments. We derive a criterion for two equal mass planets on circular inclined orbits to achieve Hill stability, and conclude that significant radial migration and eccentricity pumping of both planets occurs predominantly by 2:1 and 5:3 mean motion resonant interactions. Using Laplace-Lagrange secular theory, we obtain analytical secular solutions for the orbital inclinations and longitudes of ascending nodes, and use those solutions to distinguish between the secular and resonant dynamics which arise in numerical simulations. We also illustrate how encounter maps, typically used to trace the motion of massless particles, may be modified to reproduce the gross instability seen by the numerical integrations. Such a correlation suggests promising future use of such maps to model the dynamics of more coplanar massive planet systems.     

On the detection of mutual perturbations as proof of planets around PSR1257+12

Unambiguous detection of the consequences of mutual perturbations of the hypothesized planets about the pulsar PSR1257+12 would be unassailable proof of their existence. Nearly all of the residuals in the times of arrival (TOA) of the pulses after subtraction of the TOA predicted from the best fit constant period model are accounted for by including the effects of two orbiting planets with constant orbital parameters. The nature and magnitude of additional residuals in the TOA due to the gravitational interactions between the planets are determined by numerically calculating the TOA residuals for the orbital motion including the perturbations and subtracting the TOA residuals from analytic expressions of the orbital motion with orbital parameters fixed at averaged values. The TOA residual differences so obtained oscillate with periods comparable to the orbital periods with the oscillations varying in amplitude as a function of epoch within any given observational period. The signature of the perturbations is thus a quasiperiodic modulation of the residual differences obtained after removal of the effects of the orbital motion with best fit, constant orbital parameters. The amplitudes of this modulation reach about 10sec for observational periods exceeding 1000 days for the minimum planetary masses with sini = 1, and they increase as 1 / sini for 1 / sini < 5, wherei is the inclination of the orbit plane to that of the sky. Greater accumulated phase differences between the effects of perturbed and unperturbed orbital motions are available in the times of zero values in the observed and predicted TOA residuals and these comprise a second signature of the perturbations. The perturbation signatures should become detectable as the observation interval approaches 1000 days.Paper presented at the Conference onPlanetary Systems: Formation, Evolution, and Detection held 7–10 December, 1992 at CalTech, Pasadena, California, U.S.A.     

On the co-orbital motion of two planets in quasi-circular orbits

We develop an analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. The Hamiltonian, averaged over one of the planetary mean longitudes, is expanded in power series of eccentricities and inclinations. The model, which is valid in the entire co-orbital region, possesses an integrable approximation modeling the planar and quasi-circular motions. First, focusing on the fixed points of this approximation, we highlight relations linking the eigenvectors of the associated linearized differential system and the existence of certain remarkable orbits like the elliptic Eulerian Lagrangian configurations, the anti-Lagrange (Giuppone et al. in MNRAS 407:390–398, 2010) orbits and some second sort orbits discovered by Poincaré. Then, the variational equation is studied in the vicinity of any quasi-circular periodic solution. The fundamental frequencies of the trajectory are deduced and possible occurrence of low order resonances are discussed. Finally, with the help of the construction of a Birkhoff normal form, we prove that the elliptic Lagrangian equilateral configurations and the anti-Lagrange orbits bifurcate from the same fixed point $L_4$ L 4 .     

The masses of the principal planets

A set of masses for the principal planets is derived systematically from all available fundamental and independent determinations. In deriving these values an attempt has been made to treat independently those determinations based on differing observational types or analytical methods.Presented at IAU Colloquium No. 9, The IAU System of Astronomical Constants, Heidelberg, Germany, August 12–14, 1970.     

The formation of the outer planets

One of the outstanding problems in planetary cosmogony is to account for the depletion of hydrogen in the outer planets, Neptune and Uranus. It is suggested that these planets were originally similar to the major planets but that the settling towards the centre of grains, enriched by substances such as methane, ammonia and water because of the low temperatures, released enough energy to cause the evaporation of most of the hydrogen.     

On the minimum distance between two Keplerian orbits with a common focus

The methods used so far for determination of the closest approach between two orbits are discussed, and corrected versions of two of them are presented.     

The formation and habitability of terrestrial planets in the presence of close-in giant planets

‘Hot jupiters,’ giant planets with orbits very close to their parent stars, are thought to form farther away and migrate inward via interactions with a massive gas disk. If a giant planet forms and migrates quickly, the planetesimal population has time to re-generate in the lifetime of the disk and terrestrial planets may form [P.J. Armitage, A reduced efficiency of terrestrial planet formation following giant planet migration, Astrophys. J. 582 (2003) L47-L50]. We present results of simulations of terrestrial planet formation in the presence of hot/warm jupiters, broadly defined as having orbital radii ?0.5 AU. We show that terrestrial planets similar to those in the Solar System can form around stars with hot/warm jupiters, and can have water contents equal to or higher than the Earth's. For small orbital radii of hot jupiters (e.g., 0.15, 0.25 AU) potentially habitable planets can form, but for semi-major axes of 0.5 AU or greater their formation is suppressed. We show that the presence of an outer giant planet such as Jupiter does not enhance the water content of the terrestrial planets, but rather decreases their formation and water delivery timescales. We speculate that asteroid belts may exist interior to the terrestrial planets in systems with close-in giant planets.     

The chemical atmospheric composition of the giant planets

For a long time it was believed that the atmospheres of the giant planets, dominated by molecular hydrogen and helium, were similar in composition to the primordial nebula from which they formed. However, this image has strongly evolved over the past twenty years, due to new developments of ground-based infrared spectroscopy, coupled with the success of the Voyager space mission.Significant differences were measured in the abundances of helium, deuterium and carbon of the four giant planets. The variations in the C/H and D/H ratios have given support to the "nucleation" formation scenario, in which the four giant planets first accreted a nucleus of about ten terrestrial masses, big enough to bind gravitationally the surrounding gaseous nebula; the helium depletion in Saturn has been interpreted as a differentiation effect in Saturn's interior; the apparent helium excess in Neptune, coupled with the recent unexpected detection of CO and HCN in this planet, might imply the presence of molecular nitrogen. In the case of Jupiter and Saturn, disequilibrium species have been detected (CO, PH₃, GeH₄, AsH₃), which are tracers of vertical dynamical motions.In the future, significant progress in our knowledge of the Jovian composition, including the noble gases, should be obtained with the mass spectrometer of the Galileo probe. The ISO mission is expected to provide new far-infrared spectroscopic data which should lead to the detection of new minor species and a better determination of the D/H ratio.     

The interiors of the giant planets — 1983

The last few years brought progress in our understanding of the interiors of the giant planets especially of the two larger ones which have been visited by Pioneer and Voyager spacecraft. An analysis of the formation of the giant planets also heped to clarify certain important common features. The presently available model of Jupiter is still based on certain somewhat bothersome approximations but it appears to satisfy the main observational constraints. Saturn's interior is much better understood than it was previously although the quantitative aspects of the role of the miscibility gap in the hydrogen-helium system have not yet been entirely resolved. Much attention has been directed at the interiors of Uranus and Neptune and the outstanding question appears to be the location and the amount of ices and methane present in their outer layers. Both the two-and the three-layer models are moderately successful. Serious difficulties arise from the considerable uncertainties concerning the rotational periods of both planets. Also the estimates of the internal heat fluxes and of the magnetic fields of both planets are not sufficiently certain. It is hoped that the forthcoming flyby of these two planets by a Voyager spacecraft will provide important new data for a future study of their interiors.     

The mutual potential and gravitational torques of two bodies to fourth order

The mutual gravitational potential of two bodies of arbitrary shape is expressed to fourth order in an extension of MacCullagh's Formula for a single body. The expressions for the gravitational torques acting on each body are derived in a form convenient for use in the differential equations describing the rotational dynamics.     

Computing the critical points of the distance function between two Keplerian orbits via rigorous global optimization

A novel method to compute all critical points of the distance function between two Keplerian orbits (either bounded or unbounded) with a common focus is presented. The problem is attacked as a global optimization problem, solved by a rigorous global optimizer based on Taylor models. Thus, thigh enclosures of the stationary points are obtained. The embedded capability of the method of delivering high-order Taylor expansions is then used to analyze how uncertain orbital parameters affect the position of the stationary points and the associated distance values. Sample orbital sets and Apophis asteroid are used as test cases.     

An intriguing correlation between the masses and periods of the transiting planets

We point out an intriguing relation between the masses of the transiting planets and their orbital periods. For the six currently known transiting planets, the data are consistent with a decreasing linear relation. The other known short-period planets, discovered through radial-velocity techniques, seem to agree with this relation. We briefly speculate about a tentative physical model to explain such a dependence.