Optimized Nekhoroshev stability estimates for the Trojan asteroids with a symplectic mapping model of co-orbital motion

We present new important results about the intermediate-type Seyfert galaxy Mrk 315, recently observed through optical imaging and integral-field spectroscopy. Broad-band images were used to study the morphology of the host galaxy, narrow-band Hα images to trace the star-forming regions, and middle-band [O  iii ] images to evidence the distribution of the highly ionized gas. Some extended emission regions were isolated and their physical properties studied by means of flux-calibrated spectra. High-resolution spectroscopy was used to separate different kinematic components in the velocity fields of gas and stars. Some peculiar features characterize this apparently undisturbed and moderately isolated active galaxy. Such features, already investigated by other authors, are re-analysed and discussed in the light of these new observations. The most relevant results we obtained are: the multitiers structure of the disc; the presence of a quasi-ring of regions with star formation much higher than previous claims; a secondary nucleus confirmed by a stellar component kinematically decoupled by the main galaxy; a new hypothesis about the controversial nature of the long filament, initially described as hook shaped, and more likely made of two independent filaments caused by interaction events between the main galaxy and two dwarf companions.     

High order transfer maps for perturbed Keplerian motion

The paper presents a new semi-analytical technique for the propagation of near-Earth satellite motion. The approach uses differential algebra techniques to compute the high order expansion of the solution of the system’s ordinary differential equation for one orbital revolution, referred to as the transfer map. Once computed, a single high order transfer map (HOTM) can be reused to map an initial condition, or a set of initial conditions, forward in time for many revolutions. The only limiting factor is that the mapped objects must stay close to the reference orbit such that they remain within the region of validity of the HOTM. The performance of the method is assessed through a set of test cases in which both autonomous and non-autonomous perturbations are considered, including the case of continuously propelled trajectories.     

High order normal form construction near the elliptic orbit of the Sitnikov problem

We consider the Sitnikov problem; from the equations of motion we derive the approximate Hamiltonian flow. Then, we introduce suitable action–angle variables in order to construct a high order normal form of the Hamiltonian. We introduce Birkhoff Cartesian coordinates near the elliptic orbit and we analyze the behavior of the remainder of the normal form. Finally, we derive a kind of local stability estimate in the vicinity of the periodic orbit for exponentially long times using the normal form up to 40th order in Cartesian coordinates.     

Exchange orbits: an interesting case of co-orbital motion

In this investigation we treat a special configuration of two celestial bodies in 1:1 mean motion resonance namely the so-called exchange orbits. There exist—at least—theoretically—two different types: the exchange-a orbits and the exchange-e orbits. The first one is the following: two celestial bodies are in orbit around a central body with almost the same semi-major axes on circular orbits. Because of the relatively small differences in semi-major axes they meet from time to time and exchange their semi-major axes. The inner one then moves outside the other planet and vice versa. The second configuration one is the following: two planets are moving on nearly the same orbit with respect to the semi-major axes, one on a circular orbit and the other one on an eccentric one. During their dynamical evolution they change the characteristics of the orbit, the circular one becomes an elliptic one whereas the elliptic one changes its shape to a circle. This ‘game’ repeats periodically. In this new study we extend the numerical computations for both of these exchange orbits to the three dimensional case and in another extension treat also the problem when these orbits are perturbed from a fourth body. Our results in form of graphs show quite well that for a large variety of initial conditions both configurations are stable and stay in these exchange orbits.     

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 .     

On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited

In the framework of the planar and circular restricted three-body problem, we consider an asteroid that orbits the Sun in quasi-satellite motion with a planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital resonance with the planet, characterized by a nonzero eccentricity and a resonant angle that librates around zero. Likewise, in the rotating frame with the planet, it describes the same trajectory as the one of a retrograde satellite even though the planet acts as a perturbator. In the last few years, the discoveries of asteroids in this type of motion made the term “quasi-satellite” more and more present in the literature. However, some authors rather use the term “retrograde satellite” when referring to this kind of motion in the studies of the restricted problem in the rotating frame. In this paper, we intend to clarify the terminology to use, in order to bridge the gap between the perturbative co-orbital point of view and the more general approach in the rotating frame. Through a numerical exploration of the co-orbital phase space, we describe the quasi-satellite domain and highlight that it is not reachable by low eccentricities by averaging process. We will show that the quasi-satellite domain is effectively included in the domain of the retrograde satellites and neatly defined in terms of frequencies. Eventually, we highlight a remarkable high eccentric quasi-satellite orbit corresponding to a frozen ellipse in the heliocentric frame. We extend this result to the eccentric case (planet on an eccentric motion) and show that two families of frozen ellipses originate from this remarkable orbit.     

Delta-v requirements for earth co-orbital rendezvous missions

Earth co-orbital asteroids present advantages as potential targets for future asteroid rendezvous missions. Their prolonged proximity to Earth facilitates communication, while their Earth-like orbits mean a steady flux of solar power and no significant periodic heating and cooling of the spacecraft throughout the course of the mission. Theoretical studies show that low-inclination co-orbital orbits are more stable than high-inclination orbits. As inclination is the most significant indicator of low delta-v rendezvous orbits, there is the potential for a large population of easily accessible asteroids, with favorable engineering requirements. This study first looks at phase-independent rendezvous orbits to a large number of objects, then looks in more detail at the phase-dependent orbits to the most favorable objects. While rendezvous orbits to co-orbital objects do not have a low delta-v necessarily, some objects present energy requirements significantly less than previous rendezvous missions. Currently we find no ideal co-orbital asteroids for rendezvous missions, although theoretical Earth Trojans present very low-energy requirements for rendezvous.     

Numerical determination of stability regions for orbital motion in uniformly rotating second degree and order gravity fields

The stability of orbital motion about a uniformly rotating arbitrary second degree and order gravity field is investigated. A normalized form of the equations of motion are derived and analyzed. A numerical stability criteria is proposed and used to evaluate the stability of initially near-circular orbits in the equatorial plane of the body. Regions of stable and unstable motion are clearly delineated, and are seen to be strongly related to resonances between the mean motion and the body rotation rate.     

Comparison between stability limits for satellite motion

Numerical and analytical comparisons are made between three methods of obtaining stability information on satellite motion using the model of the restricted problem of three bodies. Kuiper's (1961) and Szebehely's (1978) approximate results are compared with computer solutions obtained by successive iterations. The three methods show close agreement regarding the maximum values of the orbital radii for stability. The lowest result and therefore the most conservative estimate is obtained by the simplest formula, _max=(/81)^1/3 where is the ratio of the satellite's orbital radius to the distance between the primaries with massesm ₁>m ₂ and is the mass-ratio given bym ₂/(m ₁+m ₂).     

Application of higher order variations of motion

The theoretical considerations developed in a previous paper (Kazantzis, 1973) are applied with the aid of the CDC-3300 computer and examples of new families of doubly symmetric solutions of the restricted three-dimensional problem obtained thereof are presented. Rough estimates of initial conditions of these families have been obtained by applying an extension of the scanning technique developed by Hénon (1966).     

The constrained normal form algorithm

This paper describes an algorithm which brings a regularizable polynomial perturbation of a three degree of freedom Kepler problem into a normal form which Poisson commutes with the Kepler Hamiltonian. We illustrate the alogrithm with an example: the quadratic Zeeman effect. For other applications of this algorithm see [1],[4], and [5]. The authors have written a program in MAPLE which implements the constrained normal form.     

Polar motion estimates from linear combinations of independent series

This paper describes methods for combining independent observations of polar motion into a single time series, and applies these methods to satellite laser ranging, Doppler, and astrometric polar motion data taken over the period 1981--1983. The polar motion data are treated as complex valued time series, yielding one estimate of the noise level for each series, rather than a separate estimate for X and Y components. Estimates of the noise level which take advantage of the existence of simultaneous observations are preferred. Linear combination of the data may be performed in both the time and the frequency domain, and examples of each are present. The results of analyzing three years of observations show little difference between the Allan and conventional variance. Present day Doppler determinations do not show marked improvement compared to classical astrometric results. Thus, the astrometric observations are still useful.     

High order symplectic integrators for perturbed Hamiltonian systems

A family of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H=A+B was given in (McLachlan, 1995). We give here a constructive proof that for all integer p, such integrator exists, with only positive steps, and with a remainder of order O(^p + ²²), where is the stepsize of the integrator. Moreover, we compute the analytical expressions of the leading terms of the remainders at all orders. We show also that for a large class of systems, a corrector step can be performed such that the remainder becomes O(^p +⁴²). The performances of these integrators are compared for the simple pendulum and the planetary three-body problem of Sun–Jupiter–Saturn.     

Reduced normal form as minimal models for nonlinear radial pulsations

We prove that the time behaviour of the currently adopted deterministicradial stellar pulsation equations is always equivalent, in a sense to be defined, to the time behaviour of a low dimensional parametric differential equation (Reduced Normal Form Family). The algebraic construction of this equation is explicitly given. All simple models so far discussed in the literature are members of this family.     

The stability of the triangular Lagrangian points for commensurability of order two

The two variable expansion method is used to show that the triangular equilibrium points in the planar restricted problem of three bodies are unstable at µ=µ₂=0.024293.... It is also shown that even though the equilibrium points are stable for µµ₂ finite motions in the neighborhood of the equilibrium points may be unstable.     

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.     

On a new form for the differential equations of relative motion of the three-body problem

A new method for the development of the disturbing function of the three-body problem is outlined in this paper. A special process is devised to get the distance between two planetsP ₁ andP ₂ in terms of their heliocentric distances. It is then shown that the differential equations of relative motion of this problem can be brought in an homogeneous set of differential equations.     

Nonlinear stability of Halley cometosheath with transverse plasma motion

Weakly nonlinear MHD stability of the Halley cometosheath determined by the balance between the outward ion-neutral drag force and the inward Lorentz force is investigated including the transverse plasma motion as observed in the flanks with the help of the method of multiple scales. The eigenvalues and the eigenfunctions are obtained for the linear problem and the time evolution of the amplitude is obtained using the solvability condition for the solution of the second order problem. The diamagnetic cavity boundary and the adjacent layer of about 100 km thickness is found unstable for the travelling waves of certain wave numbers. Halley ionopause has been observed to have strong ripples with a wavelength of several hundred kilometers. It is found that nonlinear effects have stabilizing effect.