A two-level perturbation method for connecting unstable periodic orbits with low fuel cost and short time of flight: application to a lunar observation mission

The proposed method connects two unstable periodic orbits by employing trajectories of their associated invariant manifolds that are perturbed in two levels. A first level of velocity perturbations is applied on the trajectories of the discretized manifolds at the points where they approach the nominal unstable periodic orbit in order to accelerate them. A second level of structured velocity perturbations is applied to trajectories that have already been subjected to first level perturbations in order to approximately meet the necessary conditions for a low \(\varDelta \text {V}\) transfer. Due to this two-level perturbation approach, the number of the trajectories obtained is significantly larger compared with approaches that employ traditional invariant manifolds. For this reason, the problem of connecting two unstable periodic orbits through perturbed trajectories of their manifolds is transformed into an equivalent discrete optimization problem that is solved with a very low computational complexity algorithm that is proposed in this paper. Finally, the method is applied to a lunar observation mission of practical interest and is found to perform considerably better in terms of \(\varDelta \text {V}\) cost and time of flight when compared with previous techniques applied to the same project.     

Improved grid search method: an efficient tool for global computation of periodic orbits

We present an improved grid search method for the global computation of periodic orbits in model problems of Dynamics, and the classification of these orbits into families. The method concerns symmetric periodic orbits in problems of two degrees of freedom with a conserved quantity, and is applied here to problems of Celestial Mechanics. It consists of two main phases; a global sampling technique in a two-dimensional space of initial conditions and a data processing procedure for the classification (clustering) of the periodic orbits into families characterized by continuous evolution of the orbital parameters of member orbits. The method is tested by using it to recompute known results. It is then applied with advantage to the determination of the branch families of the family f of retrograde satellites in Hill’s Lunar problem, and to the determination of irregular families of periodic orbits in a perturbed Hill problem, a species of families which are difficult to find by continuation methods.      

A grid search for families of periodic orbits in the restricted problem of three bodies

A method is described for the numerical determination of families of periodic orbits in the planar restricted problem of three bodies. The families are sought in their representation as curves in a two-dimensional space of parameters. A grid search is applied to the study of the evolution of satellite motion when the mass parameter is varied. Only that part of the space of parameters is investigated for which one of them, the relative energy constant, takes values larger than that corresponding to the inner Lagrangian pointL ₂. Critical values of the mass parameter are determined for which new families of simple or double periodic orbits appear inside the closed ovals of zero velocity.     

Broad search for unstable resonant orbits in the planar circular restricted three-body problem

Unstable resonant orbits in the circular restricted three-body problem have increasingly been used for trajectory design using optimization and invariant manifold techniques. In this study, several methods for computing these unstable resonant orbits are explored including grid searches, flyby maps, and continuation. Families of orbits are computed focusing on orbits with multiple loops near the secondary in the Jupiter–Europa system, and their characteristics are explored. Different parameters such as period and stability are examined for each set of resonant orbits, and the continuation of several specific orbits is explored in more detail.     

The use of invariant manifolds for transfers between unstable periodic orbits of different energies

Techniques from dynamical systems theory have been applied to the construction of transfers between unstable periodic orbits that have different energies. Invariant manifolds, trajectories that asymptotically depart or approach unstable periodic orbits, are used to connect the initial and final orbits. The transfer asymptotically departs the initial orbit on a trajectory contained within the initial orbit’s unstable manifold and later asymptotically approaches the final orbit on a trajectory contained within the stable manifold of the final orbit. The manifold trajectories are connected by the execution of impulsive maneuvers. Two-body parameters dictate the selection of the individual manifold trajectories used to construct efficient transfers. A bounding sphere centered on the secondary, with a radius less than the sphere of influence of the secondary, is used to study the manifold trajectories. A two-body parameter, κ, is computed within the bounding sphere, where the gravitational effects of the secondary dominate. The parameter κ is defined as the sum of two quantities: the difference in the normalized angular momentum vectors and eccentricity vectors between a point on the unstable manifold and a point on the stable manifold. It is numerically demonstrated that as the κ parameter decreases, the total cost to complete the transfer decreases. Preliminary results indicate that this method of constructing transfers produces a significant cost savings over methods that do not employ the use of invariant manifolds.     

A new tool for the study of periodic orbits

We present the generalization of the method of rational Fourier-series approximations for nonlinear ordinary differential equations in galactic dynamics. In previous papers we presented the formulae which describe simple families of periodic orbits with very good accuracy. In this Letter we generalize our formulae to describe periodic orbits of any resonancem:n.     

A new family of periodic orbits for the restricted problem

A new family of periodic orbits of the three-dimensional restricted three-body problem which continue off from a consecutive collision orbit are numerically studied. Their behavior for varying energy is unexpected. In particular, associated with our system is a countable set of resonant energy values and each time the energy passes through one of them the periodic orbit forms a loop by self-intersection. Any number of loops can form by this process and the resulting orbits take on an interesting appearance.     

A search for collision orbits in the free-fall three-body problem I. Numerical procedure

A numerical procedure is devised to find binary collision orbits in the free-fall three-body problem. Applying this procedure, families of binary collision orbits are found and a sequence of triple collision orbits are positioned. A property of sets of binary collision orbits which is convenient to search triple collision orbits is found. Important numerical results are formulated and summarized in the final section.     

A search for very high energy bursts

Of great importance in distinguishing between models for gamma-ray bursts (GRBs) is the experimental determination of the highest energy gamma rays associated with bursts. The EGRET detection of a 15 GeV gamma ray indicates that the spectra of at least some bursts extend well beyond the several MeV limit of the BATSE detectors (Hurleyet al., 1994). The low expected flux means that the collecting area of the present generation of satellite-based detectors is too small to detect gamma rays much above this energy efficiently, and such searches are currently undertaken with ground based detectors. In this paper searches made for very high energy GRBs with a southern hemisphere air shower particle array are described.     

A general algorithm for the solution of Kepler's equation for elliptic orbits

An efficient algorithm is presented for the solution of Kepler's equationf(E)=E–M–e sinE=0, wheree is the eccentricity,M the mean anomaly andE the eccentric anomaly. This algorithm is based on simple initial approximations that are cubics inM, and an iterative scheme that is a slight generalization of the Newton-Raphson method. Extensive testing of this algorithm has been performed on the UNIVAC 1108 computer. Solutions for 20 000 pairs of values ofe andM show that for single precision (10^–8) 42.0% of the cases require one iteration, 57.8% two and 0.2% three. For double precision (10^–18) one additional iteration is required. Single- and double-precision FORTRAN subroutines are available from the author.     

A novel algorithm for generating libration point orbits about the collinear points

This paper presents a numerical algorithm that can generate long-term libration points orbits (LPOs) and the transfer orbits from the parking orbits to the LPOs in the circular-restricted three-body problem (CR3BP) and the full solar system model without initial guesses. The families of the quasi-periodic LPOs in the CR3BP can also be constructed with this algorithm. By using the dynamical behavior of LPO, the transfer orbit from the parking orbit to the LPO is generated using a bisection method. At the same time, a short segment of the target LPO connected with the transfer orbit is obtained, then the short segment of LPO is extended by correcting the state towards its adjacent point on the stable manifold of the target LPO with differential evolution algorithm. By implementing the correction strategy repeatedly, the LPOs can be extended to any length as needed. Moreover, combining with the continuation procedure, this algorithm can be used to generate the families of the quasi-periodic LPOs in the CR3BP.     

A search for ultra-high energy counterparts to gamma-ray bursts

A small air shower array operating over many years has been used to search for ultra-high energy (UHE) gamma radiation ( 50 TeV) associated with gamma-ray bursts (GRBs) detected by the BATSE instrument on the Compton Gamma-Ray Observatory (CGRO). Upper limits for a one minute interval after each burst are presented for seven GRBs located with zenith angles < 20°. A 4.3 excess over background was observed between 10 and 20 minutes following the onset of a GRB on 11 May 1991. The confidence level that this is due to a real effect and not a background fluctuation is 99.8%. If this effect is real then cosmological models are excluded for this burst because of absorption of UHE gamma rays by the intergalactic radiation fields.     

A search for 5 min periodic structure in solar 2 cm emission

Two hundred and eighty-five hours of solar data obtained from the University of Iowa 2 cm radiometer during 1968–1969 were analyzed for evidence of periodic structure related to the 5 min periodic chromospheric oscillations detected in optical line emissions. A power spectral analysis of the data failed to show any statistically significant (> 96 % confidence) periodic activity in the frequency range 1–15 mHz (periods of 1–16 min) for data organized according to solar activity in H, soft solar X-rays (2–12 ), and several microwave frequencies (3–15 GHz).A small shift in power from low to higher frequencies in the power spectrum of the 2 cm data was found to be correlated with H and X-ray activity. This power shift is attributed to a relative increase in chromospheric turbulence at altitudes common to H, X-ray, and 2 cm emission.Consistent statistical analyses of previous works reporting evidence for oscillations at microwave and extreme-ultraviolet frequencies indicate that confidence in these previous results is marginal.A model for chromospheric oscillation bursts in quiescent supergranules is incorporated into a statistical analysis of the power spectrum detectability of the oscillations to understand the negative results obtained.     

A search for high energy gamma-rays from solar active regions

The Elliot model for solar flares predicts weak -ray emission from the flare region prior to large flares. A search has been made for such -radiation of energy > 50 MeV. The experiment was performed using balloon-borne detectors flown from an equatorial station during the 1967/1968 solar maximum. A number of small flares were observed, but no associated -rays were detected. A limit of 2.3 × 10⁴ photons/cm² s was placed on the emission from an importance 1N flare. The general lack of major solar activity during the period of the balloon flights precluded a test for the Elliot model.     

A search for the origin of very low electron temperatures

VLET's (Very Low Electron Temperature) are regions in the solar wind (lasting 12–30 h) in which the electron temperature is abnormally low. Because it is generally believed that the thermal conductivity parallel to the interplanetary magnetic field lines is high for electrons, i.e. the contact with the Sun should be ideal, VLET's are surprising observations. In this work a statistical analysis of many of these events is made with respect to the dependence of this phenomenon on interplanetary plasma and field parameters. In contrast with earlier work it was found that VLET's exist not only after a shock front. The statistical analysis showed further that a VLET is always associated with a VLPT (Very Low Proton Temperature) event and that, whilst on average the temperature of the electrons and protons outside the low temperature regions is about the same, the mean proton temperature inside is three times lower than that of the electrons.A particular model for VLET's is investigated in detail; the closed magnetic loop or “blob” model. By assuming: (a) pure adiabatic expansion in a radially streaming solar wind without pressure equalization across the “blob” boundary, and (b) rapid pressure equalization across the boundary, an attempt was made to quantitatively investigate the feasibility of the “blob” model in the light of plasma and field data. It can be shown that the closed magnetic loop model is unlikely to be the major cause of VLET's.