Potentials having two orthogonal families of curves as trajectories

We study, using the tool of Joukovsky’s orthogonal coordinates, the determination of the potentials having two families of orthogonal trajectories. We show for compatible cases the existence and the uniqueness, up to a constant factor, of the solution. We note the importance of the ‘isothermal’ nets of curves. We study as an example the net of geometrically similar conic curves and orthogonal trajectories. This revised version was published online in July 2006 with corrections to the Cover Date.     

Invariant distributions and collisionless equilibria

This paper discusses the possibility of constructing time-independent solutions to the collisionless Boltzmann equation which depend on quantities other than global isolating integrals such as energy and angular momentum. The key point is that, at least in principle, a self-consistent equilibrium can be constructed from any set of time-independent phase-space building blocks which, when combined, generate the mass distribution associated with an assumed time-independent potential. This approach provides a way to justify Schwarzschild's method for the numerical construction of self-consistent equilibria with arbitrary time-independent potentials, generalizing thereby an approach developed by Vandervoort for integrable potentials. As a simple illustration, Schwarzschild's method is reformulated to allow for a straightforward computation of equilibria which depend only on one or two global integrals and no other quantities, as is reasonable, for example, for modelling axisymmetric configurations characterized by a non-integrable potential.     

Ordered and chaotic Bohmian trajectories

We discuss the issue of ordered and chaotic trajectories in the Bohmian approach of Quantum Mechanics from points of view relevant to the methods of Celestial Mechanics. The Bohmian approach gives the same results as the orthodox (Copenhagen) approach, but it considers also underlying trajectories guided by the wave. The Bohmian trajectories are rather different from the corresponding classical trajectories. We give examples of a classical chaotic system that is ordered quantum-mechanically and of a classically ordered system that is mostly chaotic quantum mechanically. Then we consider quantum periodic orbits and ordered orbits, that can be represented by formal series of the “third integral” type, and we study their asymptotic properties leading to estimates of exponential stability. Such orbits do not approach the “nodal points” where the wavefunction ψ vanishes. On the other hand, when an orbit comes close to a nodal point, chaos is generated in the neighborhood of a hyperbolic point (called X-point). The generation of chaos is maximum when the X-point is close to the nodal point. Finally we remark that high order periodic orbits may behave as “effectively ordered” or “effectively chaotic” for long times before reaching the period.     

Invariant rotational curves in Sitnikov's Problem

The Sitnikov's Problem is a Restricted Three-Body Problem of Celestial Mechanics depending on a parameter, the eccentricity,e. The Hamiltonian,H(z, v, t, e), does not depend ont ife=0 and we have an integrable system; ife is small the KAM Theory proves the existence of invariant rotational curves, IRC. For larger eccentricities, we show that there exist two complementary sequences of intervals of values ofe that accumulate to the maximum admissible value of the eccentricity, 1, and such that, for one of the sequences IRC around a fixed point persist. Moreover, they shrink to the planez=0 ase tends to 1.     

Invariant geomagnetic co-ordinates for epoch 1977.25

Global maps of invariant geomagnetic co-ordinates calculated for epoch 1977.25 are presented. A more detailed report including computer programs for the rapid calculation of invariant co-ordinates is available.     

GCM simulation of balloon trajectories on Titan

Possible trajectories of passive balloons in Titan's troposphere are simulated with the instantaneous wind field predicted by a GCM (general circulation model). In most areas the basic motion of a balloon is a predominantly eastward or westward drift, depending on altitude, latitude and season of the balloon release point. Some meridional oscillation is always superposed on this basic motion, resulting in a wavy trajectory, with a maximum extent (of 40^°) at high latitudes of the winter hemisphere. As a general rule, the meridional oscillation can be maximised if a balloon is deployed in altitudes and latitudes where the mean zonal wind is eastward and smaller than the phase speed of Saturn's gravitational tide on Titan. A balloon's groundtrack does not repeat as it makes successive circuits around Titan, but rather makes a spiral or braided pattern. The summer pole is rather difficult to access for a balloon not directly introduced there because of small meridional oscillation, while the winter pole can be readily accessed and left several times. A preferred zonal drift direction can be achieved by choosing a proper altitude and hemisphere, but choosing a preferred meridional direction is not possible.     

Regularization and the computation of optimal trajectories

A completely regular form for the differential equations governing the three-dimensional motion of a continuously thrusting space vehicle is obtained by using the Kustaanheimo-Stiefel regularization. The differential equations for the thrusting rocket are transformed using the K-S transformation and an optimal trajectory problem is posed in the transformed space. The canonical equations for the optimal motion in the transformed space are regularized by a suitable change of the independent variable. The transformed equations are regular in the sense that the differential equations do not possess terms with zero divisors when the motion encounters a gravitational force center. The resulting equations possess symmetry in form and the coefficients of the dependent variables are slowly varying quantities for a low-thrust space vehicle.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Estimate of the Transition Value of Librational Invariant Curves

We investigate the break-down threshold of librational invariant curves. As a model problem, we consider a variant of a mapping introduced by M. Hénon, which well describes the dynamics of librational motions surrounding a stable invariant point. We verify in concrete examples the applicability of Greene's method, by computing the instability transition values of a sequence of periodic orbits approaching an invariant curve with fixed noble frequency. However, this method requires the knowledge of the location of the periodic orbits within a very good approximation. This task appears to be difficult to realize for a libration regime, due to the different topology of the phase space. To compute the break-down threshold, we tried an alternative method very easy to implement, based on the computation of the fast Lyapunov indicators and frequency analysis. Such technique does not require the knowledge of the periodic orbits, but again, it appears very difficult to have a precision better than Greene's method for the computation of the critical parameter.     

Chaotic behaviour of trajectories for the asteroidal resonances

A systematic study of the main asteroidal resonances of the third and fourth order is performed using mapping techniques. For each resonance one-parameter family of surfaces of section is presented together with a simple energy graph which helps to understand and predict the changes in the surfaces of section within the family. As the truncated Hamiltonian for the planar, elliptic, restricted three-body problem is used for the mapping, the method is expected to fail for high eccentricities. We compared, therefore, the surfaces of section with trajectories calculated by symplectic integrators of the fourth and six order employing the full Hamiltonian. We found a good agreement for small eccentricities but differences for the higher eccentricities (e 0.3).     

Logarithmic spiral trajectories generated by Solar sails

Analytic solutions to continuous thrust-propelled trajectories are available in a few cases only. An interesting case is offered by the logarithmic spiral, that is, a trajectory characterized by a constant flight path angle and a fixed thrust vector direction in an orbital reference frame. The logarithmic spiral is important from a practical point of view, because it may be passively maintained by a Solar sail-based spacecraft. The aim of this paper is to provide a systematic study concerning the possibility of inserting a Solar sail-based spacecraft into a heliocentric logarithmic spiral trajectory without using any impulsive maneuver. The required conditions to be met by the sail in terms of attitude angle, propulsive performance, parking orbit characteristics, and initial position are thoroughly investigated. The closed-form variations of the osculating orbital parameters are analyzed, and the obtained analytical results are used for investigating the phasing maneuver of a Solar sail along an elliptic heliocentric orbit. In this mission scenario, the phasing orbit is composed of two symmetric logarithmic spiral trajectories connected with a coasting arc.     

Invariant mass approach to the spiral structure of galaxies

Following an earlier suggestion that spiral density waves in galaxies primarily are a response phenomenon to the increase with time of a central quadrupole moment. The object of this note is to describe a simple procedure adequate for estimating the shape of the density response in a stellar disk for given structures of the background and perturbative gravitational fields, respectively, under the premises of first order epicyclic and adiabatic approximation.     

Optimal switching strategy for radially accelerated trajectories

This paper studies the problem of a spacecraft subject to an outward radial thrust, with constant modulus, that may be switched on or off at suitable time intervals. The problem is to find the optimal strategy to guarantee the possibility of transferring the spacecraft from an initial to a final position in a given time interval using the least amount of thrust level. The problem is solved in an optimal framework, using an indirect approach. A number of different mission scenarios are studied in detail: escape missions, flyby missions and rendezvous missions. In the latter case the spacecraft uses a hybrid system comprising an high thrust propulsion system for the final impulsive maneuver. The optimal switching strategy allows one to substantially decrease the thrust level when compared to the continuous case (without thrust modulation).     

The accuracy of proper orbital elements and the properties of asteroid families: Comparison with the linear theory

The accuracy and reliability of the proper orbital elements used to define asteroid families are investigated by simulating numerically the dynamical evolution of families assumed to arise from the “explosion” of a parent object. The orbits of the simulated family asteroids have then been integrated in the frame of the elliptic restricted three-body problem Sun-Jupiter-asteroid, for times of the order of the circulation periods of perihelia and nodes. By filtering out short-periodic perturbations, we have monitored the behavior of the proper eccentricities and inclinations, computed according to the linear secular perturbation theory. Significant long-period variations have been found especially for families having nonnegligible eccentricities and/or inclinations (like the Eos family), and strong disturbances due to the proximity of mean motion commensurabilities with Jupiter have been evidenced (for instance, in the case of the Themis family). These phenomena can cause a significant “noise” on the proper eccentricities and inclinations, probably affecting in some cases the derived family memberships. They can also give rise to a spurious anisotropy in the fragment ejection velocity fields computed from the dispersion in proper elements observed in each family, and this could explain the puzzling anisotropies of this kind actually found in real families by D. Brouwer (1951, Astron. J. 56, 9–32) and by V. Zappalà, P. Farinella, Z. Knežević, and P. Paolicchi (1984), Icarus 59, 261–285).     

Invariant imbedding and Chandrasekhar's planetary problem of radiative transfer

In connection with Chandrasekhar's planetary problem of radiative transfer the total scattering and the diffuse transmission functions have been discussed by several authors (cf. Chandrasekhar, 1950; van de Hulst, 1948; Sobolev, 1948; Bellman,et al., 1967; Kagiwada and Kalaba, 1971). With the aid of the Bellman-Krein formula for the resolvent kernel of the auxiliary equation governing the source function, we show how the invariant imbedding equations governing the diffuse scattering and transmission functions can readily be obtained. So far as we know, the Cauchy system of the functional equations for the scattering and transmission functions is new and is well-suited for the numerical computation.Supported by the National Science Foundation under Grant No. GP 29049, and by the Atomic Energy Commission under Grant No. AT (40-3)-113 Project 19.     

The Role of Hyperbolic Invariant Sets in Stickiness Effects

With the standard map model, we study the stickiness effect of invariant tori, particularly the role of hyperbolic sets in this effect. The diffusion of orbits originated from the neighborhoods of hyperbolic points, periodic islands and torus is studied. We find that they possess similar diffusion rules, but the diffusion of orbits originated from the neighborhood of a torus is faster than that originated near a hyperbolic set. The numerical results show that an orbit in the neighborhood of a torus spends most of time around hyperbolic invariant sets. We also calculate the areas of islands with different periods. The decay of areas with the periods obeys a power law, and the absolute values of the exponents increase monotonously with the perturbation parameter. According to the results obtained, we conclude that the stickiness effect of tori is caused mainly by the hyperbolic invariant sets near the tori, and the diffusion speed becomes larger when orbits diffuse away from the torus.     

Properties of collision trajectories of asteroids with the earth

Close encounters of asteroids with the Earth can lead to a transition to resonant orbits, close encounters, and collisions in the future. The structure of the sets of possible collisions of asteroids with the Earth is similar to a fractal structure thanks to resonant returns of asteroids. Areas of semi-major axes and positions and sizes of gaps leading to collisions with the Earth for Apophis asteroids 2007 VK184 and 2011 AG5 are given.     

J2 Invariant Relative Orbits for Spacecraft Formations

An analytic method is presented to establish J ₂ invariant relative orbits. Working with mean orbit elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits. By having both orbits drift at equal angular rates on the average, they will not separate over time due to the J₂ influence. Two first order conditions are established between the differences in momenta elements (semi-major axis, eccentricity and inclination angle) that guarantee that the drift rates of two neighboring orbits are equal on the average. Differences in the longitude of the ascending node, argument of perigee and initial mean anomaly can be set at will, as long as they are setup in mean element space. For near polar orbits, enforcing both momenta element constraints may result in impractically large relative orbits. It this case it is shown that dropping the equal ascending node rate requirement still avoids considerable relative orbit drift and provides substantial fuel savings.This revised version was published online in October 2005 with corrections to the Cover Date.     

Invariant manifolds in the measure-preserving mappings with three-dimensions

In this paper, we study the following three-dimensional mappings $$T:\left\{ \begin{gathered} x_{n + 1} = x_n + y_n + B sin z_n , \hfill \\ y_{n + 1} = y_n + A sin x_{n + 1} , \hfill \\ z_{n + 1} = z_n + C sin y_{n + 1} + D, \hfill \\ \end{gathered} \right.\left( {\bmod 2\Pi } \right)$$ where A, B, C, D are parameters. When D>BC and 2π/D is an irrational number, we find numerically-two-dimensional and one-dimensional invariant manifolds, but when DBC and 2π/D is a rational number we find numerically one-dimensional manifolds and the fixed points for some cycles.     

Comparison of low-energy lunar transfer trajectories to invariant manifolds

In this study, transfer trajectories from the Earth to the Moon that encounter the Moon at various flight path angles are examined, and lunar approach trajectories are compared to the invariant manifolds of selected unstable orbits in the circular restricted three-body problem. Previous work focused on lunar impact and landing trajectories encountering the Moon normal to the surface, and this research extends the problem with different flight path angles in three dimensions. The lunar landing geometry for a range of Jacobi constants is computed, and approaches to the Moon via invariant manifolds from unstable orbits are analyzed for different energy levels.