Celestial Mechanics and Dynamical Astronomy (CELEST MECH DYN ASTR)

Publications in this journal

• Article: Analytical solutions for J(2)-perturbed unbounded equatorial orbits

  V Martinusi, P Gurfil
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  ABSTRACT: While solutions for bounded orbits about oblate spheroidal planets have been presented before, similar solutions for unbounded motion are scarce. This paper develops solutions for unbounded motion in the equatorial plane of an oblate spheroidal planet, while taking into account only the J (2) harmonic in the gravitational potential. Two cases are distinguished: A pseudo-parabolic motion, obtained for zero total specific energy, and a pseudo-hyperbolic motion, characterized by positive total specific energy. The solutions to the equations of motion are expressed using elliptic integrals. The pseudo-parabolic motion unveils a new orbit, termed herein the fish orbit, which has not been observed thus far in the perturbed two-body problem. The pseudo-hyperbolic solutions show that significant differences exist between the Keplerian flyby and the flyby performed under the the J (2) zonal harmonic. Numerical simulations are used to quantify these differences.
  Celestial Mechanics and Dynamical Astronomy 01/2013; 115(1):35-57.
• Article: The impact of Earth’s shadow on the long-term evolution of space debris

  Ch. Hubaux, A. Lemaître
  Celestial Mechanics and Dynamical Astronomy 01/2013;
• Article: Some analytical results about periodic orbits in the restricted three body problem with dissipation

  A. Margheri, R. Ortega, C. Rebelo
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  ABSTRACT: We present some analytical results about the existence of periodic orbits for the planar restricted three body problem with dissipation considered recently by Celletti et al. (CMDA 109, 265, 2011) We show that, under fairly general conditions on the dissipation term, the circular orbits cannot be continued to the dissipative framework. Moreover, we give general conditions for the occurrence or not of a Hopf bifurcation around the libration points L 4 and L 5. Our results are consistent with the numerical findings of Celletti et al.
  Celestial Mechanics and Dynamical Astronomy 07/2012; 113(3):279-290.
• Article: Alessandra Celletti: Stability and Chaos in Celestial Mechanics

  Alice Quillen
  Celestial Mechanics and Dynamical Astronomy 05/2012; 110(4):399-400.
• Article: Families of first kind periodic solutions of the spatial and planar elliptic restricted three-body problem with collision

  Lúcia Brandão Dias, Claudio Vidal
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  ABSTRACT: We consider a restricted three-body problem where the primaries are moving in an elliptic collision orbit and the infinitesimal mass moves in a three dimensional space. This paper is devoted to prove analytically the existence of several families of symmetric periodic solutions as continuation of Keplerian circular orbits. In our approach the perturbing parameter is related with the energy of the primaries. KeywordsRestricted three-body problem-Spatial restricted three-body problem with collision-Averaging method-Periodic solutions-Symmetries-Continuation’s method-Poincaré-Delaunay variables
  Celestial Mechanics and Dynamical Astronomy 05/2012; 106(4):339-370.
• Article: Collinear Central Configuration in Four-Body Problem

  Tiancheng Ouyang, Zhifu Xie
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  ABSTRACT: In the n-body problem a central configuration is formed if the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. We consider the problem: given a collinear configuration of four bodies, under what conditions is it possible to choose positive masses which make it central. We know it is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However for an arbitrary configuration of four bodies, it is not always possible to find positive masses forming a central configuration. In this paper, we establish an expression of four masses depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically we show that there is a compact region in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the compact region, it is always possible to choose positive masses to make the configuration central.
  Celestial Mechanics and Dynamical Astronomy 05/2012; 93(1):147-166.
• Article: Detection by MEGNO of the gravitational resonances between a rotating ellipsoid and a point mass satellite

  A. Compère, A. Lemaître, N. Delsate
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  ABSTRACT: Nowadays the scientific community considers that more than a third of the asteroids are double. The study of the stability of these systems is quite complex, because of their irregular shapes and tumbling rotations, and requires a full body–full body approach. A particular case is analysed here, when the secondary body is sufficiently small and distant from the primary to be considered as a point mass satellite. Gravitational resonances (between the revolution of the satellite and the rotation of the asteroid) of a small body in fast or slow rotation around a rigid ellipsoid are studied. The same model can be used for the motion of a probe around an irregular asteroid. The gravitational potential induced by the primary body is modelled by the MacMillan potential. The stability of the satellite is measured thanks to the MEGNO indicator (Mean Exponential Growth Factor of Nearby Orbits). We present stability maps in the plane (\fracbd, \fraccd){\left(\frac{b}{d}, \frac{c}{d}\right)} where d, b, and c are the three semi-axes of the ellipsoid shaping the asteroid. Special stable conic-like curves are detected on these maps and explained by an analytical model, based on a simplification of the MacMillan potential for some specific resonances (1 : 1 and 2 : 1). The efficiency of the MEGNO to detect stability is confirmed. KeywordsGravitational resonance–MEGNO–Satellite of asteroid–Binary asteroid–1:1 Resonance–Synchronous case
  Celestial Mechanics and Dynamical Astronomy 05/2012; 112(1):75-98.
• Article: Kinematical modeling of the Earth rotation, focusing on the Oppolzer terms in a rigid Earth and the Oppolzer-like terms in an elastic Earth

  Yoshio Kubo
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  ABSTRACT: Under perturbations from outer bodies, the Earth experiences changes of its angular momentum axis, figure axis and rotational axis. In the theory of the rigid Earth, in addition to the precession and nutation of the angular momentum axis given by the Poisson terms, both the figure axis and the rotational axis suffer forced deviation from the angular momentum axis. This deviation is expressed by the so-called Oppolzer terms describing separation of the averaged figure axis, called CIP (Celestial Intermediate Pole) or CEP (Celestial Ephemeris Pole), and the mathematically defined rotational axis, from the angular momentum axis. The CIP is the rotational axis in a frame subject to both precession and nutation, while the mathematical rotational axis is that in the inertial (non-rotating) frame. We investigate, kinematically, the origin of the separation between these two axes—both for the rigid Earth and an elastic Earth. In the case of an elastic Earth perturbed by the same outer bodies, there appear further deviations of the figure and rotational axes from the angular momentum axis. These deviations, though similar to the Oppolzer terms in the rigid Earth, are produced by quite a different physical mechanism. Analysing this mechanism, we derive an expression for the Oppolzer-like terms in an elastic Earth. From this expression we demonstrate that, under a certain approximation (in neglect of the motion of the perturbing outer bodies), the sum of the direct and convective perturbations of the spin axis coincides with the direct perturbation of the figure axis. This equality, which is approximate, gets violated when the motion of the outer bodies is taken into account. KeywordsEarth rotation–Precession and nutation–Rigid Earth–Elastic Earth–Angular momentum axis–Figure axis–Rotational axis–Oppolzer terms–Poisson terms–Convective terms
  Celestial Mechanics and Dynamical Astronomy 05/2012; 110(2):143-168.
• Article: Sequential solution to Kepler’s equation

  Jeremy J. Davis, Daniele Mortari, Christian Bruccoleri
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  ABSTRACT: Seven sequential starter values for solving Kepler’s equation are proposed for fast orbit propagation. The proposed methods have constant complexity (not iterative), do not require pre-computed data, and can be implemented in just a few lines of code. The resulting sequential orbit propagation techniques can be done at different levels of accuracy and speed, depending essentially on the value of orbit eccentricity. Accuracy and algorithmic complexity are evaluated for all the proposed approaches and compared with several existing single-point techniques to solve Kepler’s equation. The new methods obtain improved accuracy at lower computational cost as compared to the best existing methods. KeywordsKepler equation-Orbit propagation
  Celestial Mechanics and Dynamical Astronomy 05/2012; 108(1):59-72.
• Article: A closed-form solution to the minimum Lambert’s problem

  Martín Avendaño, Daniele Mortari
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  ABSTRACT: A closed form solution to the minimum DVtot2{\Delta V_{\rm tot}^2} Lambert problem between two assigned positions in two distinct orbits is presented. Motivation comes from the need of computing optimal orbit transfer matrices to solve re-configuration problems of satellite constellations and the complexity associated in facing these problems with the minimization of DVtot{\Delta V_{\rm tot}}. Extensive numerical tests show that the difference in fuel consumption between the solutions obtained by minimizing DVtot2{\Delta V_{\rm tot}^2} and DVtot{\Delta V_{\rm tot}} does not exceed 17%. The DVtot2{\Delta V_{\rm tot}^2} solution can be adopted as starting point to find the minimum DVtot{\Delta V_{\rm tot}}. The solving equation for minimum DVtot2{\Delta V_{\rm tot}^2} Lambert problem is a quartic polynomial in term of the angular momentum modulus of the optimal transfer orbit. The root selection is discussed and the singular case, occurring when the initial and final radii are parallel, is analytically solved. A numerical example for the general case (orbit transfer “pork-chop” between two non-coplanar elliptical orbits) and two examples for the singular case (Hohmann and GTO transfers) are provided.
  Celestial Mechanics and Dynamical Astronomy 05/2012; 106(1):25-37.
• Article: The eleventh motion constant of the two-body problem

  Andrew J. Sinclair, John E. Hurtado
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  ABSTRACT: The two-body problem is a twelfth-order time-invariant dynamic system, and therefore has eleven mutually-independent time-independent integrals, here referred to as motion constants. Some of these motion constants are related to the ten mutually-independent algebraic integrals of the n-body problem, whereas some are particular to the two-body problem. The problem can be decomposed into mass-center and relative-motion subsystems, each being sixth-order and each having five mutually-independent motion constants. This paper presents solutions for the eleventh motion constant, which relates the behavior of the two subsystems. The complete set of mutually-independent motion constants describes the shape of the state-space trajectories. The use of the eleventh motion constant is demonstrated in computing a solution to a two-point boundary-value problem. KeywordsTwo-body problem–Motion constants–Boundary-value problems
  Celestial Mechanics and Dynamical Astronomy 05/2012; 110(3):189-198.
• Article: Artificial equilibrium points for a generalized sail in the circular restricted three-body problem

  Generoso Aliasi, Giovanni Mengali, Alessandro A. Quarta
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  ABSTRACT: This paper introduces a new approach to the study of artificial equilibrium points in the circular restricted three-body problem for propulsion systems with continuous and purely radial thrust. The propulsion system is described by means of a general mathematical model that encompasses the behavior of different systems like a solar sail, a magnetic sail and an electric sail. The proposed model is based on the choice of a coefficient related to the propulsion type and a performance parameter that quantifies the system technological complexity. The propulsion system is therefore referred to as generalized sail. The existence of artificial equilibrium points for a generalized sail is investigated. It is shown that three different families of equilibrium points exist, and their characteristic locus is described geometrically by varying the value of the performance parameter. The linear stability of the artificial points is also discussed. KeywordsArtificial Lagrangian equilibrium points–Radial propulsive acceleration–Generalized sail–Displaced points–Stability
  Celestial Mechanics and Dynamical Astronomy 05/2012; 110(4):343-368.
• Article: A numerical study of the stabilization effect of steepness

  Nataša Todorović, Massimiliano Guzzo, Elena Lega, Claude Froeschlé
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  ABSTRACT: We present the results of numerical experiments about the influence of steepness on the resonant structure, stability and diffusion in a 4-dimensional symplectic map. The map is designed so that by changing a parameter, we smoothly switch steepness on and off by the change of the so called steepness coefficients. In both cases we measure the diffusion coefficients of the actions within a resonance. According to Nekhoroshev theorem we find that, in the steep case, the diffusion coefficients are definitely smaller than in the non steep one, thus confirming the threshold effect of the steepness coefficients which comes from the proof of Nekhoroshev theorem. KeywordsNekhoroshev theorem–Steepness–Diffusion–Hamiltonian systems–Symplectic maps
  Celestial Mechanics and Dynamical Astronomy 05/2012; 110(4):389-398.
• Article: Dynamics of a particle under the gravitational potential of a massive annulus: properties and equilibrium description

  Eva Tresaco, Antonio Elipe, Andrés Riaguas
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  ABSTRACT: This paper studies the main features of the dynamics around a massive annular disk. The first part addresses the difficulties finding an appropriated expression of the gravitational potential of a massive disk, which will be used to define the differential equations of motion of our dynamical system. The second part describes the main features of the dynamics with special attention to equilibrium of the system. KeywordsPotential theory–Annular disk–Solid disk–Composition of annuli
  Celestial Mechanics and Dynamical Astronomy 05/2012; 111(4):431-447.
• Article: The IAU 2009 system of astronomical constants: the report of the IAU working group on numerical standards for Fundamental Astronomy

  Brian Luzum, Nicole Capitaine, Agnès Fienga, William Folkner, Toshio Fukushima, James Hilton, Catherine Hohenkerk, George Krasinsky, Gérard Petit, Elena Pitjeva, Michael Soffel, Patrick Wallace
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  ABSTRACT: In the 2006–2009 triennium, the International Astronomical Union (IAU) Working Group on Numerical Standards for Fundamental Astronomy determined a list of Current Best Estimates (CBEs). The IAU 2009 Resolution B2 adopted these CBEs as the IAU (2009) System of Astronomical Constants. Additional work continues to define the process of updating the CBEs and creating a standard electronic document. KeywordsNumerical standards–Fundamental Astronomy–Fundamental constants
  Celestial Mechanics and Dynamical Astronomy 05/2012; 110(4):293-304.
• Article: Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations

  Vladimir Martinuşi, Pini Gurfil
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  ABSTRACT: Finding relative satellite orbits that guarantee long-term bounded relative motion is important for cluster flight, wherein a group of satellites remain within bounded distances while applying very few formationkeeping maneuvers. However, most existing astrodynamical approaches utilize mean orbital elements for detecting bounded relative orbits, and therefore cannot guarantee long-term boundedness under realistic gravitational models. The main purpose of the present paper is to develop analytical methods for designing long-term bounded relative orbits under high-order gravitational perturbations. The key underlying observation is that in the presence of arbitrarily high-order even zonal harmonics perturbations, the dynamics are superintegrable for equatorial orbits. When only J 2 is considered, the current paper offers a closed-form solution for the relative motion in the equatorial plane using elliptic integrals. Moreover, necessary and sufficient periodicity conditions for the relative motion are determined. The proposed methodology for the J 2-perturbed relative motion is then extended to non-equatorial orbits and to the case of any high-order even zonal harmonics (J 2n , n≥1). Numerical simulations show how the suggested methodology can be implemented for designing bounded relative quasiperiodic orbits in the presence of the complete zonal part of the gravitational potential. KeywordsSatellite relative motion–Zonal harmonics–Satellite formation flying–Cluster flight–elliptic integrals
  Celestial Mechanics and Dynamical Astronomy 04/2012; 111(4):387-414.
• Article: Asymptotic solution for the two-body problem with constant tangential thrust acceleration

  Claudio Bombardelli, Giulio Baù, Jesus Peláez
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  ABSTRACT: An analytical solution of the two body problem perturbed by a constant tangential acceleration is derived with the aid of perturbation theory. The solution, which is valid for circular and elliptic orbits with generic eccentricity, describes the instantaneous time variation of all orbital elements. A comparison with high-accuracy numerical results shows that the analytical method can be effectively applied to multiple-revolution low-thrust orbit transfer around planets and in interplanetary space with negligible error. KeywordsTwo body problem–Tangential thrust–Asymptotic expansion–Orbit transfer
  Celestial Mechanics and Dynamical Astronomy 04/2012; 110(3):239-256.