A Note on a General Algorithm for Two-Body Expansions

This note describes briefly a technique, easily implemented with most computer algebra systems, for the purpose of computing the two-body expansions (in powers of the eccentricity and in Fourier series of the mean-anomaly) of a large class of functions of the distance, the true anomaly and/or the eccentric anomaly. This revised version was published online in July 2006 with corrections to the Cover Date.     

A note on expansions of functions of velocity in the two-body problem

Fourier expansions of functions of velocity in the two-body problem are obtained in terms of both the true anomaly and the mean anomaly.     

Book Review

The aim of this book is to present techniques for the study of motion of solar system objects in highly eccentric orbits. Instead of using the usual anomalies (mean, true, eccentric), the authors define and use a new kind of anomaly, the elliptic anomaly.In this way, it is possible, in a theory using perturbation series expansions, to make the ratio: (accuracy)/(number of needed terms), higher than in the classical techniques. The book consists of six chapters. The first chapter deals with the elliptic anomaly in the two-body problem and the second chapter presents the general technique to construct first-order perturbation theory in elliptic function expansions. The next three chapters deal with applications of the new technique to artificial satellites and asteroids, in highly eccentric orbits. The last chapter describes the basic algorithms of the theory.The tools developed in the book demand the use of computer algebra, which is implemented by means of Mathematica 3.0.The book is well written and the new technique is clearly presented and related to the existing techniques, making it useful to all those who use analytical or semi-analytical methods for the study of highly eccentric motion. Celestial Mechanics at High Eccentricities, Gordon and Breach Publishers, US$95, GBP 59, EUR 79, ISBN 90-5699-212-0     

Numerical Computation of Hansen-like Expansions

Fourier expansions of elliptic motion functions in multiples of the true, eccentric, elliptic and mean anomalies are computed numerically by means of the fast Fourier transform. Both Hansen-like coefficients and their derivatives with respect to eccentricity of the orbit are considered. General behavior of the coefficients and the efficiency (compactness) of the expansions are investigated for various values of eccentricity of the orbit. This revised version was published online in July 2006 with corrections to the Cover Date.     

Holomorphy of coordinates with respect to eccentricities in the planetary three-body problem

The boundaries of the domains of holomorphy of the coordinates of unperturbed elliptic motion with respect to the eccentricities of planetary orbits are determined for the cases when any of the five anomalies of one of the planets-eccentric, true, tangential, or one of two mutual anomalies suggested by M.F. Subbotin—is used as an independent variable. The resulting equations are a generalization of the known equations for the boundaries of the domains of the holomorphy of coordinates for the cases when the time is the independent variable and determine the bisymmetric ovals, whose size and shape depend on the eccentricities and on the ratio of the planetary mean motions. The largest domains of holomorphy are obtained when the tangential anomaly or one of the Subbotin mutual anomalies is used. A function was found that conformally maps the domain of holomorphy to the unit disk. It was demonstrated that the application of any anomaly of the outer planet as the independent variable can result in a significant shrinking of the domain of the holomorphy of the coordinates of the inner planet, so that the analytic continuation of the initial power series with the center at the origin of the coordinates of a complex plane becomes impossible.     

Position and velocity perturbations in the orbital frame in terms of classical element perturbations

The transformation of classical orbit element perturbations to perturbations in position and velocity in the radial, transverse and normal directions of the orbital frame is developed. The formulation is given for the case of mean anomaly perturbations as well as for eccentric and true anomaly perturbations. Approximate formulas are also developed for the case of nearly circular orbits and compared with those found in the literature.     

An analytical theory for the orbit of nereid

A complete analytical dynamic theory for the motion of Nereid has been constructed, accurate to approximately 0.01 arc second over several hundred years. The solution uses the Lie transform approach advanced by Deprit and is consistent with respect to the magnitudes of the disturbing functions, including all perturbations to an accuracy of 10^–8 relative to the two-body potential (oblateness and third-body). Multiple short-period variables in the third-body perturbations are related via the ratio of their mean motions, reducing the number of independent variables. Extensive use is made of expansions giving trigonometric functions of the true anomaly as analytical Fourier series in the mean anomaly. Initial constants and mass parameters come from the data obtained during the Voyager II encounter with Neptune in 1989.     

On the evolution of giant protoplanets forming in circumbinary discs

We present the results of hydrodynamic simulations of Jovian mass protoplanets that form in circumbinary discs. The simulations follow the orbital evolution of the binary plus protoplanet system acting under their mutual gravitational forces, and forces exerted by the viscous circumbinary disc. The evolution involves the clearing of the inner circumbinary disc initially, so that the binary plus protoplanet system orbits within a low density cavity. Continued interaction between disc and protoplanet causes inward migration of the planet towards the inner binary. Subsequent evolution can take three distinct paths: (i) the protoplanet enters the 4 : 1 mean motion resonance with the binary, but is gravitationally scattered through a close encounter with the secondary star; (ii) the protoplanet enters the 4 : 1 mean motion resonance, the resonance breaks, and the planet remains in a stable orbit just outside the resonance; (iii) when the binary has initial eccentricity   e _bin≥ 0.2  , the disc becomes eccentric, leading to a stalling of the planet migration, and the formation of a stable circumbinary planet.
These results have implications for a number of issues in the study of extrasolar planets. The ejection of protoplanets in close binary systems provides a source of 'free-floating planets', which have been discovered recently. The formation of a large, tidally truncated cavity may provide an observational signature of circumbinary planets during formation. The existence of protoplanets orbiting stably just outside a mean motion resonance (4 : 1) in the simulations indicate that such sites may harbour planets in binary star systems, and these could potentially be observed. Finally, the formation of stable circumbinary planets in eccentric binary systems indicates that circumbinary planets may not be uncommon.     

Automated,closed form integration of formulas in elliptic motion

In some perturbation theories it is possible to avoid expansion of the perturbations in powers of the eccentricity, obtaining results in closed form by using the true or eccentric anomaly instead of the mean anomaly. This paper describes an algorithm (which has been programmed for the 6600 computer using the formula manipulation system TRIGMAN) for automatically performing the integrals which arise in these theories.     

Chaotic zone boundary for low free eccentricity particles near an eccentric planet

We consider particles with low free or proper eccentricity that are orbiting near planets on eccentric orbits. Through collisionless particle integration, we numerically find the location of the boundary of the chaotic zone in the planet's corotation region. We find that the distance in semimajor axis between the planet and boundary depends on the planet mass to the 2/7 power and is independent of the planet eccentricity, at least for planet eccentricities below 0.3. Our integrations reveal a similarity between the dynamics of particles at zero eccentricity near a planet in a circular orbit and with zero free eccentricity particles near an eccentric planet. The 2/7th law has been previously explained by estimating the semimajor at which the first-order mean motion resonances are large enough to overlap. Orbital dynamics near an eccentric planet could differ due to first-order corotation resonances that have strength proportional to the planet's eccentricity. However, we find that the corotation resonance width at low free eccentricity is small; also the first-order resonance width at zero free eccentricity is the same as that for a zero-eccentricity particle near a planet in a circular orbit. This accounts for insensitivity of the chaotic zone width to planet eccentricity. Particles at zero free eccentricity near an eccentric planet have similar dynamics to those at zero eccentricity near a planet in a circular orbit.     

Improvement of the position of planet X based on the motion of nearly parabolic comets

Based on the motion of nearly parabolic comets, we have improved the position of planet X in its orbit obtained by Batygin and Brown (2016). By assuming that some of the comets discovered to date could have close encounters with this planet, we have determined the comets with a small minimum orbit intersection distance with the planet. Five comets having hyperbolic orbits before their entry into the inner Solar system have been separated out from the general list. By assuming that at least one of them had a close encounter with the planet, we have determined the planet’s possible position. The planet’s probable ephemeris positions at the present epoch have been obtained by assuming the planet to have prograde and retrograde motions. In the case of a prograde motion, the planet is currently at a distance Δ whose value belongs to the interval Δ ∈ (1110, 1120) AU and has a right ascension α and declination δ within the intervals α ∈ (83?, 90?) and δ ∈ (8?, 10?); the true anomaly υ belongs to the interval υ ∈ (176?, 184?). In the case of a retrograde motion: α ∈ (48?, 58?), δ ∈ (?12?, ?6?), Δ ∈ (790, 910) AU, and υ ∈ (212?, 223?). It should be noted that in the case of a retrograde motion of the planet, its ephemeris position obtained from the motion of comets agrees with the planet’s position obtained byHolman and Payne (2016) from highly accurate Cassini observations and is consistent with the results of Fienga et al. (2016).     

The mean anomaly in elliptic motion as random variable

In this paper we deal with the different means for some characteristic quantities and functions in elliptic motion. We then give some probability interpretations of the mean anomaly and discuss the free terms of certain expansions in the Fourier series related to the mean anomaly. Finally we give a therem illustrating the mathematical meaning of results obtained by these interpretations.     

Expansions of elliptic motion based on elliptic function theory

New expansions of elliptic motion based on considering the eccentricitye as the modulusk of elliptic functions and introducing the new anomalyw (a sort of elliptic anomaly) defined byw=u/2K–/2,g=amu–/2 (g being the eccentric anomaly) are compared with the classic (e, M), (e, v) and (e, g) expansions in multiples of mean, true and eccentric anomalies, respectively. These (q,w) expansions turn out to be in general more compact than the classical ones. The coefficients of the (e,v) and (e,g) expansions are expressed as the hypergeometric series, which may be reduced to the hypergeometric polynomials. The coefficients of the (q,w) expansions may be presented in closed (rational function) form with respect toq, k, k=(1–k ²)^1/2,K andE, q being the Jacobi nome relatedk whileK andE are the complete elliptic integrals of the first and second kind respectively. Recurrence relations to compute these coefficients have been derived.on leave from Institute of Applied Astronomy, St.-Petersburg 197042, Russia     

HD 12661行星系统构形的形成与稳定性

最近的多普勒观测表明恒星HD 12661周围存在两颗中等偏心率轨道上运行的行星,内行星的最小质量为2．3木星质量,轨道周期为263．6天;外行星的最小质量为1．57木星质量,轨道周期为1444．5天．该系统的稳定性要求两颗行星处在平运动轨道共振．用数值方法研究了该系统形成初期在恒星气体盘作用下的轨道迁移与稳定性,计算了行星在迁移中被平运动共振俘获的概率．发现这两颗行星目前很可能正处在11:2平运动共振边缘,且运动是混沌的,从而澄清了关于系统目前构形的不同说法,并且很可能在系统形成后行星迁移到目前构形时,气体盘几乎消失了．     

The formation of an eccentric gap in a gas disc by a planet in an eccentric orbit

We investigate the effect of a planet on an eccentric orbit on a two-dimensional low-mass gaseous disc. At a planet eccentricity above the planet's Hill radius divided by its semimajor axis, we find that the disc morphology differs from that exhibited by a disc containing a planet in a circular orbit. An eccentric gap is created with eccentricity that can exceed the planet's eccentricity and precesses with respect to the planet's orbit. We find that a more massive planet is required to open a gap when the planet is on an eccentric orbit. We attribute this behaviour to spiral density waves excited at corotation resonances by the eccentric planet. These act to increase the disc's eccentricity and exert a torque opposite in sign to that exerted by the Lindblad resonances. The reduced torque makes it more difficult for waves driven by the planet to overcome viscous inflow in the disc.     

Recurrence formulae for the Hansen's developments

In this paper we derive some recurrence formulae which can be used to calculate the Fourier expansions of the functions (r/a) ⁿ cosmv and (r/a) ⁿ sinmv in terms of the eccentric anomalyE or the mean anomalyM. We also establish a recurrence process for computing the series expansions for alln andm when the expansions of two basic series are known. These basic series were given in explicit form in the classical literature. The recurrence formulae are linear in the functions involved and thus make very simple the computation of the series.This work was supported by NASA contract No. NASr 54(06).—The paper was presented at the AIAA/AAS meeting, Princeton University, August 1969.     

Effect of Planetary Eccentricity on Ballistic Capture in the Solar System

Ballistic capture of spacecraft and celestial bodies by planets of the solar system is studied considering the elliptic restricted three body model. A preferential region, due to the eccentricity of the planet and the Sun-gravity-gradient effect is found for the capture phenomenon. An analytical formula is derived which determines the limiting value of the satellite capture eccentricity e_c as a function of the pericenter distance x_p and planet’s true anomaly. The analytic values e_c are tested by a numerical propagator, which makes use of planetary ephemeris, and only a small difference with respect to numerical integration is found. It turns out that lower values of e_c occur when the planet anomaly is close to zero; that is, capture is easier when the planet is at its perihelion. This fact is confirmed by the capture of celestial bodies. It is shown that Jupiter comets are generally captured when Jupiter is in its perihelion region. Ballistic capture is also important in interplanetary missions. The propellant saved using the minimum ballistic capture eccentricity is evaluated for different missions and compared with respect to the case in which the insertion orbit is a parabola: a significant saving can be accomplished.     

Some investigations into the atmospheric drag problem

This paper calls into question the validity of the well-known formulae for the perturbations in the Keplerian elements, over one revolution of an orbit, for the motion of a drag-perturbed artificial satellite. These formulae are derived from Gauss's form of the planetary equations, by averaging over a single revolution of the orbit, and using the eccentric anomaly as the independent variable.It is shown that for light balloon-type satellites in near-circular orbits neither the eccentric anomaly nor the true longitude is a suitable choice of independent variable for the averaging procedure. Under these circumstances, it would seem that simple formulae for the variations in the elements cannot be derived from Gauss's equations.     

Fourier analysis of the light curves of eclipsing variables,XIX

The aim of the present paper has been to generalize the methods previously developed for analysis of the light changes of eclipsing binary systems in the frequency-domain to cases in which the components of such systems revolve in eccentric orbits. It is shown that these methods can indeed be generalized to systems with eccentric orbits provided that the light momentsA _2m deduced from such eclipses are suitably re-defined in terms of the true, rather than mean, anomaly in the relative orbit; and that due attention is paid to the unit of length in terms of which the fractional radii of the two stars are expressed. When this is done the Fourier methods continue to be applicable to all types of eclipses exhibited by eccentric binary systems — whether these are occultations or transits; total, annular or partial.An application of these methods to practical cases has been postponed for a subsequent communication.     

Modeling of major martian magnetic anomalies: Further evidence for polar reorientations during the Noachian

Maps of the vector components of the Mars crustal magnetic field are constructed at the mapping altitude (360 to 410 km) using a selected set of data obtained with the Mars Global Surveyor magnetometer during 2780 orbits of the planet in 1999. Forward modeling calculations are then applied to six relatively strong and isolated, dominantly dipolar, magnetic anomalies for the primary purpose of estimating bulk directions of magnetization. Assuming that the magnetizing field was a (dipolar) core dynamo field centered in the planet, paleomagnetic pole positions are calculated for the six primary source bodies together with that for a seventh anomaly analyzed earlier. In agreement with several previous studies, it is found that six of the seven pole positions are clustered in what is now the northern lowlands in a region centered northwest of Olympus Mons (mean pole position: 34°±10° N, 202°±58° E). Assuming that the dynamo dipole moment vector was approximately parallel to the rotation axis, the modeling results therefore suggest a major reorientation of Mars relative to its rotation axis after magnetization was acquired. Such a reorientation may have been stimulated by internal mass redistributions associated with the formation of the northern lowlands and Tharsis, for example. A comparison of the mean paleo (magnetic) equator to the global distribution of crustal fields shows that magnetic anomalies tend to occur at low paleolatitudes. The same appears to be true for the Noachian-aged valley networks, which exhibit a broad spatial correlation with the magnetic anomalies. A possible interpretation is that the formation of magnetic anomalies and the valley networks was favored in the tropics where melting of water ice and snow was a stronger source of both surface valley erosion and groundwater recharge during the earliest history of the planet. This would be consistent with models in which hydrothermal alteration of crustal rocks played a role in producing the unusually strong martian magnetic anomalies.