Jacobi Integral Research Articles

The motion of a star with variable mass inside a layered inhomogeneous elliptical galaxy

We consider the motion of a variable-mass star inside a layered, inhomogeneous, rotating elliptical variable-mass galaxy. Under certain conditions, the differential equations of motion of a star inside the galaxy admit autonomization. The autonomized differential equations in this problem are similar to those for the motion of a constant-mass star. We have found analogs for the Jacobi integral and the Eddington-Jeans law for the variation of the density of the galaxy in its center and layers.

Multiparticle systems. The algebra of integrals and integrable cases

Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector. A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a particle-interaction potential homogeneous of degree α = −2 are considered. The most general form of the additional integral of motion, which we term the Jacobi integral, is presented for such systems. A new nonlinear algebra of integrals including the Jacobi integral is found. A systematic description is given to a new reduction procedure and possibilities of applying it to dynamics with the aim of lowering the order of Hamiltonian systems. Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations of the Lagrangian identity for systems with a particle-interaction potential homogeneous of degree α = −2 are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane.

Многочастичные системы. Алгебра интегралов и интегрируемые случаи

Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector. A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a particle-interaction potential homogeneous of degree α = −2 are considered. The most general form of the additional integral of motion, which we term the Jacobi integral, is presented for such systems. A new nonlinear algebra of integrals including the Jacobi integral is found. A systematic description is given to a new reduction procedure and possibilities of applying it to dynamics with the aim of lowering the order of Hamiltonian systems. Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations of the Lagrangian identity for systems with a particle-interaction potential homogeneous of degree α = −2 are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane.

Two-Step Simulation for the Formation of Asteroid Itokawa from the Restricted Three-Body Problem to the Multi-Body Problem

This paper studies a formation of Asteroid Itokawa in 2-dimension, dividing the formation process into two steps. The first step is a collapse of parent bodies and the second step is the attachment of small fragments to the early Itokawa that is created by the first step. In the first step, the aggregation of fragments after a head-on impact of two parent bodies is investigated by utilizing the N-body problem with the spring-dumper system. The radius of each fragment and the total mass in the system are set to be 32 meters and 10 times as large as Itokawa, respectively. It is found in the first process that the collision of two parent bodies allows a formation of the early Itokawa that is made by a soft contact of two primary bodies. In the second step, 500 fragments are randomly positioned around the early Itokawa and their motions are numerically analyzed by applying the circular restricted three-body problem. It shows that the population of fragments around the early Itokawa is almost constant regardless of the Jacobi integral, the conserved value in the problem. It implies that fragments may play a role in creating the current shape of Itokawa.

The Construction of Nonspherical Models of Quasi‐Relaxed Stellar Systems

Spherical models of collisionless but quasi-relaxed stellar systems have long been studied as a natural framework for the description of globular clusters. Here we consider the construction of self-consistent models under the same physical conditions, but including explicitly the ingredients that lead to departures from spherical symmetry. In particular, we focus on the effects of the tidal field associated with the hosting galaxy. We then take a stellar system on a circular orbit inside a galaxy represented as a frozen external field. The equilibrium distribution function is obtained from the one describing the spherical case by replacing the energy integral with the relevant Jacobi integral in the presence of the external tidal field. Then the construction of the model requires the investigation of a singular perturbation problem for an elliptic partial differential equation with a free boundary, for which we provide a method of solution to any desired order, with explicit solutions to 2 orders. We outline the relevant parameter space, thus opening the way to a systematic study of the properties of a two-parameter family of physically justified nonspherical models of quasi-relaxed stellar systems. The general method developed here can also be used to construct models for which the nonspherical shape is due to internal rotation. Eventually, the models will be a useful tool to investigate whether the shapes of globular clusters are primarily determined by internal rotation, by external tides, or by pressure anisotropy.

Existence of axially symmetric solutions to the Vlasov-Poisson system depending on Jacobi's integral

We prove the existence of axially symmetric solutions to the Vlasov– Poisson system in a rotating setting for sufficiently small angular velocity. The constructed steady states depend on Jacobi’s integral and the proof relies on an implicit function theorem for operators.

Problem of the motion of a star inside a layered inhomogeneous elliptical galaxy with a variable mass

The problem of the motion of a star inside a layered inhomogeneous rotating elliptical galaxy with a variable mass is considered. We have found an analogue of the Jacobi integral and determined the possible regions of motion. A solution to the equations of perturbed motion has been obtained.

Stringent Criteria for Stable and Unstable Planetary Orbits in Stellar Binary Systems

The existence of planets in stellar binary (and higher order) systems has now been confirmed by many observations. The stability of planetary orbits in these systems has been extensively studied, but no precise stability criteria have so far been introduced. Therefore, there is an urgent need for developing stringent mathematical criteria that allow us to precisely determine whether a planetary orbit in a binary system is stable or unstable. In this Letter, such criteria are defined using the concept of Jacobi's integral and Jacobi's constant. These criteria are used to contest previous results on planetary orbital stability in binary systems.

Semi-Analytical Method for Calculating the Elliptic Restricted Three-Body Problem Monodromy Matrix

R ECENT years have seen a rising interest in launching unmanned spacecraft into libration-point orbits for scientific missions. Past missions such as ISEE-3, SOHO, and Genesis successfully used quasiperiodic orbits about the L1 and L2 collinear libration points of the sun–Earth system. With the embarkation of future NASA and ESA missions such as Darwin, Terrestrial Planet Finder (TPF), and Single Aperture Far Infrared Observatory (SAFIR), to be launched into libration-point orbits, there is an opportunity to design and evaluate novel trajectory planning, simulation, and control schemes. In practical analysis and design of missions around libration points, the circular restricted three-body problem (CR3BP) model is usually adopted [1–4]. Although this model has proven fruitful, it possesses an inherent approximation, assuming that the orbits of the primaries are circular. However, both the motion of the Earth around the sun and the motion of moon around the Earth are eccentric. Incorporating the eccentricity term into the equations of motion renders a more general model, known as the elliptic restricted threebody problem (ER3BP). The ER3BP has significant topological differences compared with the CR3BP. For example, the position of the libration points is not constant, but rather pulsatingwith respect to Earth. Moreover, the Jacobi integral is time(true-anomaly-) dependent. Several works have addressed the problem of finding natural periodic orbits in the planar ER3BP based on specialized regularizations [5,6]. Derivation of such orbits through numerical searches was accomplished in [7,8]. Almost all methods for calculating orbits about the collinear libration points require the monodromy matrix, which is the state-transition matrix evaluated at the orbital period. Moreover, design of stationkeeping maneuvers, whether impulsive [9] or continuous [10], also requires the monodromy matrix. Although the evaluation of the monodromy matrix in the CR3BP is trivial, the problem is more involved in the more general setup of the ER3BP, because the linearized system is nonautonomous. In fact, the linearized system is linear parameter-varying (LPV) and periodic, exhibiting implicit dependence upon time through the true anomaly. It is therefore important to develop reliable algorithms for computing the monodromy matrix. In this work, we develop a new, semianalytical method for calculation of the monodromy matrix. We propose to expand the state-transition matrix into orthogonal Chebyshev polynomials of the first and second kind, shifted to fit the time interval in use. The Chebyshev approximation transforms the nonautonomous differential equations required to calculate the state-transitionmatrix into a set of algebraic equations. This approach can be used as a computationally efficient scheme for computing the state-transition matrix and as a simple check for verifying the accuracy of monodromymatrix calculations using a direct numerical integration.

The Restricted Three-Body Problem: Comments on the Rotational Effect

The synchronization between the orbital motion and axial rotation of the two component stars of a binary system is reviewed. Some previous published papers are mentioned and the general conclusion is outlined: If we shall use a rotating coordinate system synchronous with one of the two stellar axial rotations, it is not possible to obtain a Jacobi integral and the Roche geometry cannot be further analyzed. In addition, a theoretical approach is summarized in order to use the axial rotations of the two component stars, even if the constants of the stellar structure (k2)1, (k2)2must be taken into consideration. So it is found that if the stellar angular velocities are higher than the corresponding Keplerian angular velocity (ω i ≫ ω k , i= \(\overline{1,2}\)), the problem of the rotational effect could be of practical consideration. Finally, a theoretical relationship between the two constants (k2)1and (k2)2of the stellar structure is established.

Energy conservation in the restricted elliptical three-body problem

The energy conservation law for the elliptical three-body problem is derived using an invariant relation corresponding to the Jacobi integral in the circular problem. The minimum-energy surfaces are constructed, which transform in the case of zero eccentricity into zero-velocity surfaces. Some astronomical applications of the results are considered. In particular, it is shown that Roche-lobe overflow displays pulsational behavior in the elliptical three-body problem.

Resonance Overlap Is Responsible for Ejecting Planets in Binary Systems

A planet orbiting around a star in a binary system can be ejected if it lies too far from its host star. We find that instability boundaries first obtained in numerical studies can be explained by overlap between sub-resonances within mean-motion resonances (mostly of the j:1 type). Strong secular forcing from the companion displaces the centroids of different sub-resonances, producing large regions of resonance overlap. Planets lying within these overlapping regions experience chaotic diffusion, which in most cases leads to their eventual ejection. The overlap region extends to shorter-period orbits as either the companion's mass or its eccentricity increase. Our analytical calculations reproduce the instability boundaries observed in numerical studies and yield the following two additional results. Firstly, the instability boundary as a function of eccentricity is jagged; thus, the widest stable orbit could be reduced from previously quoted values by as much as 20%. Secondly, very high order resonances (e.g., 50:3) do not significantly modify the instability boundary despite the fact that these weak resonances can produce slow chaotic diffusion which may be missed by finite-duration numerical integrations. We present some numerical evidence for the first result. More extensive experiments are called for to confirm these conclusions. For the special case of circular binaries, we find that the Hill criterion (based on the critical Jacobi integral) yields an instability boundary that is very similar to that obtained by resonance overlap arguments, making the former both a necessary and a sufficient condition for planet instability.

Symmetry breaking: A heuristic approach to chaotic scattering in many dimensions

As the theory of chaotic scattering in high-dimensional systems is poorly developed, it is very difficult to determine initial conditions for which interesting scattering events, such as long delay times, occur. We propose to use symmetry breaking as a way to gain the insight necessary to determine low-dimensional subspaces of initial conditions in which we can find such events easily. We study numerically the planar scattering off a disk moving on an elliptic Kepler orbit, as a simplified model of the elliptic restricted three-body problem. When the motion of the disk is circular, the system has an integral of motion, the Jacobi integral, which is no longer conserved for nonvanishing eccentricity. In the latter case, the system has an effective five-dimensional phase space and is therefore not amenable for study with the usual methods. Using the symmetric problem as a starting point we define an appropriate two-dimensional subspace of initial conditions by fixing some coordinates. This subspace proves to be useful to define scattering experiments where the rich and nontrivial dynamics of the problem is illustrated. We consider in particular trajectories which take very long before escaping or are trapped by consecutive collisions with the disk.

A variational proof of existence of transit orbits in the restricted three-body problem

Because of the Jacobi integral, solutions of the planar, circular restricted three-body problem are confined to certain subsets of the plane called Hill's regions. For certain values of the integral, one component of the Hill's region consists of disk-like regions around of the two primary masses, connected by a tunnel near the collinear Lagrange point, L 2. A ‘transit orbit’ is a solution which crosses the tunnel, in a sense which can be made precise using Conley's isolating block construction. For values of the Jacobi integral sufficiently close to its value at L 2, Conley found transit orbits by linearizing near the equilibrium point. The goal of this paper is to develop a method for proving existence of transit orbits for values of the Jacobi constant far from equilibrium. The method is based on the Maupertuis variational principle but isolating blocks turn out to play an important role.

A Basic Power Decomposition in Lagrangian Mechanics

A Basic Power Decomposition in Lagrangian Mechanics

Spatial Structure of Regular and Chaotic Orbits in Self-Consistent Models of Galactic Satellites

In several previous papers we had investigated the orbits of the stars that make up galactic satellites, finding that many of them were chaotic. Most of the models studied in those works were not self-consistent, the single exception being the Heggie and Ramamani (1995) models; nevertheless, these ones are built from a distribution function that depends on the energy (actually, the Jacobi integral) only, what makes them rather special. Here we built up two self-consistent models of galactic satellites, freezed theirs potential in order to have smooth and stationary fields, and investigated the spatial structure of orbits whose initial positions and velocities were those of the bodies in the self-consistent models. We distinguished between partially chaotic (only one non-zero Lyapunov exponent) and fully chaotic (two non-zero Lyapunov exponents) orbits and showed that, as could be expected from the fact that the former obey an additional local isolating integral, besides the global Jacobi integral, they have different spatial distributions. Moreover, since Lyapunov exponents are computed over finite time intervals, their values reflect the properties of the part of the chaotic sea they are navigating during those intervals and, as a result, when the chaotic orbits are separated in groups of low- and high-valued exponents, significant differences can also be recognized between their spatial distributions. The structure of the satellites can, therefore, be understood as a superposition of several separate subsystems, with different degrees of concentration and trixiality, that can be recognized from the analysis of the Lyapunov exponents of their orbits.

On the chaotic rotation of planetary satellites: The Lyapunov exponents and the energy

The chaotic behavior in the rotational motion of planetary satellites is studied. A satellite is modelled as a triaxial rigid body. For a set of twelve real satellites, as well as for sets of model satellites, the full spectra of the Lyapunov characteristic exponents (LCEs) of the chaotic spatial rotation are computed numerically. The applicability of the “separatrix map approach” (Shevchenko 2000, 2002) for analytical estimation of the maximum LCEs of the rotation is studied. This approach is shown to be in a particularly good correspondence with the results of our numerical integrations in the case of a prolate axisymmetric satellite moving in an elliptic orbit. The correspondence is good in a broad range of values of the inertial parameters and the orbital eccentricity. The dependence of the LCEs on the energy (the Jacobi integral) of the system for a triaxial satellite in a circular orbit is investigated in numerical experiments. It is found that the dependence of the maximum LCEs on the energy is linear at relatively small values of the energy, with one and the same slope (but various shifts) for the majority of our sets of the values of parameters. What is more, in the case of a prolate axisymmetric satellite, the dependence seems to be universal (the same) over the studied range of the energy over a broad range of values of the inertial parameters. Upper bounds on the values of the maximum LCEs are inferred. The “energetic approach” provides a complementary method for analytical estimation of the LCEs: evidence is given that it is useful when the energy is high.

Assessment of two methods for gravity field recovery from GOCE GPS-SST orbit solutions

Abstract. In the course of the GOCE satellite mission, the high-low Satellite to Satellite Tracking (SST) observations have to be processed for the determination of the long wavelength part of the Earth’s gravity field. This paper deals with the formulation of the high-low SST observation equations, as well as the methods for gravity field recovery from orbit information. For this purpose, two approaches, i.e. the numerical integration of orbit perturbations, and the evaluation of the energy equation based on the Jacobi integral, are presented and discussed. Special concern is given to the numerical properties of the corresponding normal equations. In a closed-loop simulation, which is based on a realistic orbit GOCE configuration, these methods are compared and assessed. However, here we process a simplified case assuming that non-conservative forces can be perfectly modelled. Assuming presently achievable accuracies of the Precise Orbit Determination (POD), it turns out that the numerical integration approach is still superior, but the energy integral approach may be an interesting alternative processing strategy in the near future.Key words. High-low SST – gravity field – GOCE – variational equations – least squares adjustment

Spacecraft Motion About Slowly Rotating Asteroids

Spacecraft motion about an arbitrary second-degree and second-order gravity eeld is investigated. We assume thatthegravityeeldisinuniformrotation aboutanaxisofitsmaximummomentofinertiaandthatitrotatesslowly as compared to the spacecraft orbit period. We derive the averaged Lagrange planetary equations for this system and use them, in conjunction with the Jacobi integral, to give a complete description of orbital motion. We show that, under the averaging assumptions, the problem is completely integrable and can be reduced to quadratures. For the case of no rotation, these quadratures can be expressed in terms of elliptic functions and integrals. For this problem, the orbit plane will experience nutation in addition to the precession that is found for orbital motion about an oblate body. It is possible for the orbit plane to be trapped in a 1:1 resonance with the rotating body, the plane essentially being dragged by the rotating asteroid. As the asteroid rotation rate is increased, this resonant motion disappears for rotation rates greater than a speciec value. Finally, we validate our analysis with numerical integrations of cases of interest, showing that the averaging assumptions apply and give a correct prediction of motion in this system. These results are applicable to understanding spacecraft and particle motion about slowly rotating asteroids. I. Introduction T HIS paper studies the orbital motion of a spacecraft about slowly rotating asteroids. The asteroid gravity e eld is represented by the second-degree and second-order gravity coefecients C20 and C22. The asteroid itself is assumed to be uniformly rotating about the axis of its major moment of inertia. When the asteroid is slowly rotating, we can apply averaging techniques to the Lagrange planetary equations to derive a simplie edset of equations that describe a spacecraft’s orbit. Also, because this is a uniformly rotatinggravitye eld,the Jacobiintegralthatistraditionally used for restricted three-body problems is deened for this problem. 1 Using the Jacobi integral in conjunction with the averaged equations, we can show that the problem can be reduced to quadratures, that is, the problem is integrable. This is an extension of the results that show that the averaged problem is integrable in terms of elliptic, hyperbolic, and trigonometric functions ifthe central body does not rotate. 2 ByproperinterpretationoftheJacobiintegral,wecanderive

Symbolic–numeric investigations for stability analysis of Lagrange systems

An approach for symbolic–numeric stability analysis of equilibrium positions of a satellite system with given gyrostatic and aerodynamic torques and given principal central moments of inertia is presented. The satellite system is described by Lagrange differential equations. The equations of motion form a closed system, for which the Jacobi Integral is valid. It is shown that 24 isolated equilibrium positions exist when the modulus of the gyrostatic torque vector and the modulus of the aerodynamic torque vector are sufficiently small. The stability of the equilibrium positions are analyzed numerically.