NEW TRENDS IN ASTRODYNAMICS AND APPLICATIONS III

A New Class of Low Energy Lunar Orbits And Mission Applications
View Description Hide DescriptionA new class of low energy trajectories are discussed which offer substantial reductions in the Delta‐V required to orbit the Moon and perform maneuvers for inclination and altitude changes. For example, inclination changes can be made for a reduction in Delta‐V by a factor of 12 over that of standard methods. Such trajectories about the Moon enable a spacecraft to stay in orbit about for the extended periods of time while performing a wide range of flexible maneuvers for relatively low Delta‐V, thus enhancing data mining at a reduced cost. These lunar orbiting trajectories also provide a way to construct a new type of lunar communication system, which may offer significant advantages to that of placing communication satellites in halo orbits about the Earth‐Moon equilateral Lagrange points. This lunar communication system, together with a special class of low energy lunar transfers from the Earth to the Moon, offers a lunar architecture with a number of advantages over that of other approaches. This architecture may be especially advantageous for the purpose of supporting a human outpost on the Moon.

Transfer to Small Distant Retrograde Orbits
View Description Hide DescriptionThe purpose of this work is to show that using the hyperbolic network around L _{1}, the diversity of transfer options to the Distant Retrograde Orbits (DROs) in the Sun‐Earth Planar Circular Restricted Three Body Problem can be increased. In contrast to previous works, the current paper will concentrate on the range of smallest available DROs. The attractiveness of the transfer options to these orbits will be revealed by including the stable manifolds of the Horizontal Lyapunov Orbits (HLOs) as a part of the transfer path. It will become clear that for a large range of DROs, lower ΔV budgets and/or far lower travel times are achievable.

Update On STEREO And Other Missions Of The JHU/Applied Physics Laboratory
View Description Hide DescriptionThis paper describes how lunar and planetary gravity assists have been used to design trajectories that have enabled challenging missions, currently flying or in development at the Johns Hopkins University’s Applied Physics Laboratory (APL), to explore the Sun, and the planets closest to and farthest from it. This is a continuation of the paper, “Recent Gravity‐Assist Trajectories for Interplanetary and Solar Exploration” presented at the second “New Trends in Astrodynamics and Applications” conference in July 2005. That paper concentrated on MESSENGER, New Horizons, and the early work on STEREO, emphasizing the ground‐breaking orbits that those spacecraft used to accomplish their ambitious goals. This paper gives the current status of those missions, which are now all flying, en route to their targets, especially the newly‐launched STEREO mission.

Rescriptive and Descriptive Gauge Symmetry in Finite‐Dimensional Dynamical Systems
View Description Hide DescriptionGauge theories in physics constitute a fundamental tool for modeling interactions among electromagnetic, weak and strong forces. They have been used in a myriad of fields, ranging from sub‐atomic physics to cosmology. The basic mathematical tool generating the gauge theories is that of symmetry, i.e. a redundancy in the description of the system. Although symmetries have long been recognized as a fundamental tool for solving ordinary differential equations, they have not been formally categorized as gauge theories. In this paper, we show how simple systems described by ordinary differential equations are prone to exhibit gauge symmetry, and discuss a few practical applications of this approach. In particular, we utilize the notion of gauge symmetry to question some common engineering misconceptions of chaotic and stochastic phenomena, and show that seemingly “disordered” (deterministic) or “random” (stochastic) behaviors can be “ordered”. This brings into play the notion of observation; we show that temporal observations may be misleading when used for chaos detection. From a practical standpoint, we use gauge symmetry to considerably mitigate the numerical truncation error of numerical integrations.

Quasi‐Periodic Orbits of the Restricted Three‐Body Problem Made Easy
View Description Hide DescriptionA new fully numerical method is presented which employs multiple Poincaré sections to find quasi‐periodic orbits. The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasi‐periodic orbits and the minimal memory required to store these orbits. This method reduces the calculation of the search for the two‐dimensional invariant torus to a search for the closed orbits, which are the intersection of the invariant torus with the Poincaré sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant torus is reduced to maps between the consecutive Poincaré maps. A Newton iteration scheme makes use of the invariancy of the circles of the maps on these Poincaré sections in order to find the Fourier coefficient that define the circles to any given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasi‐periodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi‐Halo and Lissajous families of the Sun‐Earth Restricted Three‐Body Problem (RTBP) around the L1 and L2 libration points are obtained via this method. Results are compared with the existing literature.

Nonlinear Semi‐Analytic Methods for Spacecraft Trajectory Design, Control, and Navigation
View Description Hide DescriptionIn this paper, we present a review of nonlinear semi‐analytic methods for spacecraft trajectory design, control, and navigation. We first discuss previous and recent development of higher‐order relative dynamics theory for a general dynamical system and stress the importance of incorporating the system nonlinearity in the model. For a given reference trajectory, the localized nonlinear dynamics is represented by the higher order state transition tensors, which are computed numerically by integrating the higher order Taylor series terms. We then analytically propagate the a priori Gaussian probability density function via solutions of the Fokker‐Planck equations and compute the associated moments, such as the mean and covariance matrix. Using this analytic realization of the system dynamics and statistics, we derive analytic methods which allows one to propagate, control, and estimate the spacecraft trajectory while incorporating the system nonlinearity. As examples, three‐body problems are considered and the results are compared with first‐order (linear) methods.

Set Oriented Approximation of Invariant Manifolds: Review of Concepts for Astrodynamical Problems
View Description Hide DescriptionDuring the last decade set oriented methods have been developed for the approximation and analysis of complicated dynamical behavior. These techniques do not only allow the computation of invariant sets such as attractors or invariant manifolds. Also statistical quantities of the dynamics such as invariant measures, transition probabilities, or (finite‐time) Lyapunov exponents, can be efficiently approximated. All these techniques have natural applications in the numerical treatment of problems in astrodynamics. In this contribution we will give an overview of the set oriented numerical methods and how they are successfully used for the solution of astrodynamical tasks. For the demonstration of our results we consider the (planar) circular restricted three body problem. In particular, we approximate invariant manifolds of periodic orbits about the L _{1} and L _{2} equilibrium points and show an extension to the application of a continuous control force. Moreover, we demonstrate that expansion rates (finite‐time Lyapunov exponents), which so far have mainly been applied in fluid dynamics, can provide useful information on the qualitative behavior of trajectories in the context of astrodynamics. The set oriented numerical methods and their application to astrodynamical problems discussed in this contribution serve as further important steps towards understanding the pathways of comets or asteroids and the design of energy‐efficient trajectories for spacecraft.

Combined Optimal Low‐Thrust and Stable‐Manifold Trajectories to the Earth‐Moon Halo Orbits
View Description Hide DescriptionIn this paper we incorporate the low‐thrust propulsion in the stable manifold technique to design transfer trajectories to the halo orbits around L _{1} and L _{2} of the Earth‐Moon system. The problem is stated in an optimal control scheme and solved using direct transcription and collocation; the dynamics is discretized over an uniform time grid using a sixth‐order linear multi‐point method. The resulting transfers are made up by a spiral arc that targets a piece of the stable manifold associated to the final orbit. Thanks to the generality of this approach, halo‐to‐Moon transfers are also computed combining unstable manifolds and low‐thrust. Furthermore, complete Earth‐to‐Moon transfers via halos can also be constructed. Results show the feasibility of this kind of transfers requiring moderate propellant mass fractions and feasible times of flight.

Longevity of the Human Spaceflight Program
View Description Hide DescriptionThe longevity of the human spaceflight program is important to our survival prospects. On May 27, 1993 I proposed a method for estimating future longevity, based on past observed longevity using the Copernican Principle: if your observation point is not special the 95% confidence level prediction of future longevity is between (1/39)th and 39 times the past longevity. The prediction for the future longevity of the human spaceflight program (then 32 years old) was greater than 10 months but less than 1248 years. We have already passed the lower limit. This Copernican formula has been tested a number of times, correctly predicting, among other things, future longevities of Broadway plays and musicals, and the Conservative Government in the United Kingdom. Recently, a study of future longevities of the 313 world leaders in power on May 27, 1993 has been completed. Assuming none still in office serve past age 100, the success rate of the 95% Copernican Formula is currently 94.55% with only one case (out of 313) left to be decided. The human spaceflight program has not been around long and so there is the danger its future will not be long enough to allow us to colonize off the earth. Policy implications are discussed. A smart plan would be to try to establish a self‐supporting colony on Mars in the next 45 years. This should not require sending any more tons of material into space in the next 45 years than we have in the last 45 years.

Critical Trajectories for the Human Settlement of the High Frontier
View Description Hide DescriptionIf preservation and prosperity of humanity on the Earth and human settlement of space are our goals, we should concentrate on a commercial path to get there. Commercial enterprise has a long history of fortuitously aiding scientific progress. We expect radical changes in the cost of earth to orbit transportation, and in the methods and efficacy of deep space transportation, within the next two decades.
A successful space tourism industry, beginning with suborbital tourism, will greatly drive down the cost of access to orbit over the next 15 years. Inexpensive transportation to low Earth orbit is the first requirement for a great future on the High Frontier. Inexpensive means the cost associated with a mature transportation system. A mature system has a cost of three to five times the cost of the propellant. The first cheap, reliable and highly reusable rocket engines are just now appearing in vehicles. With an assured market and high flight rate, the heretofore glacial progress in reducing the cost of space transportation is likely to become rapid. This is the first critical enabling example of synergy between free market economics and scientific and technical progress in space. It will not be the last.
New high power switches and ultracapacitors developed for the automotive market make possible cheap, robust and reliable mass driver engines. In space construction, using masses of nonterrestrial materials make the gravity tractor technique much more capable than previous schemes to maneuver asteroids. Ion propulsion will continue to improve and the first solar sails will be flown. Advanced robotics will allow remarkable improvements in productivity. The computing power available to robots began to follow the exponential Moore’s law less than decade ago. The first commercial autonomous mobile robots appeared in late summer 2006. Humans, however, will be required for the foreseeable future in repair and supervisory roles, particularly in unstructured settings such as asteroid mines.
The evolution from small tourist stations of the next decade to large space hotels will make economical the use of fully closed life‐support systems. These could be considered the first space colonies. Derivatives of these commercial space hotels may form suitable Moon and asteroid mining habitats.
Using nonterrestrial materials is a key to opening the space frontier. Dozens of rendezvous missions to Near Earth Objects will be needed to assay their resources and to plan rational NEO diversion. The development of NEO mining techniques serves two purposes, raw materials supply and planetary defense. We need economical trajectories to and from these bodies. These trajectories must not only be economical in terms of delta V or time, but in dollars, and in the time value of money, factors not generally considered by the OMB.
Satellite solar power stations may be a $500 billion per year market worldwide and cheap nickel steel from asteroids may be an enabler of power satellite construction. One asteroid of the right size and composition in a suitable orbit could open this market. Platinum group metals may be an important export, either as a primary product, or as a byproduct of nickel steel alloy production. Other products, derived from carbon, may also be important. The first economical product from an asteroid mine is likely to be water, for propellant or life‐support and radiation shielding in space hotels.

On the Theory of Bodily Tides
View Description Hide DescriptionDifferent theories of bodily tides assume different forms of dependence of the angular lag δ upon the tidal frequency χ. In the old theory (Gerstenkorn 1955, MacDonald 1964, Kaula 1964) the geometric Iag angle is assumed constant (i.e., δ ∼ χ^{0}), while the new theory (Singer 1968; Mignard 1979, 1980) postulates constancy of the time lag Δt (which is equivalent to saying that δ ∼ χ^{1}).
Each particular functional form of δ(χ) unambiguously determines the form of the frequency dependence of the tidal quality factor, Q(χ), and vice versa. Through the past 20 years, several teams of geophysicists have undertaken a large volume of experimental research of attenuation at low frequencies. This research, carried out both for mineral samples in the lab and for vast terrestrial basins, has led to a complete reconsideration of the shape of Q(χ). While in late 70s – early 80s it was universally accepted that at low frequencies the quality factor scales as inverse frequency, by now it is firmly established that Q ∼ χ^{α}, where the positive fractional power α varies, for different minerals, from 0.2 through 0.4 (leaning toward 0.2 for partial melts) — see the paper by Efroimsky (2006) and references therein. That paper also addresses some technical difficulties emerging in the conventional theory of land tides, and offers a possible way of their circumvention — a new model that is applicable both for high inclinations and high eccentricities (contrary to the Kaula expansion which converges only for i ≠ π/2 and e < 0.6627434). Here we employ this new model to explore the long‐term evolution of Phobos and to provide a more exact estimate for the time it needs to fall on Mars. This work is a pilot paper that anticipates a more comprehensive study in preparation (Efroimsky & Lainey 2007).

Phase space reconstruction in the restricted three‐body problem
View Description Hide DescriptionWe study the motion of the infinitesimal mass in the planar circular restricted three‐body problem. The infinitesimal mass can undergo regular or chaotic motions. It can be captured by one of the primaries, it can make transfers from one primary to another, or it can even escape those captures. We analyze through statistical methods the time‐series given by the time intervals between successive crossings made by the infinitesimal particle to a given Poincaré section. We apply Takens/Yorke embedding theory to reconstruct the phase space from these time series, and we use the correlation dimension of the reconstructed phase space as a tool to distinguish between various types of motions.

Gravity Behaves Like That?
View Description Hide DescriptionMany concepts of chaotic action in astrodynamics can be appreciated through simulations with home computers and software. Many astrodynamical cases are illustrated. Although chaos theory is now applied to spaceflight trajectories, this presentation employs only inert bodies with no onboard impulse, e.g., from rockets or outgassing. Other nongravitational effects are also ignored, such as atmosphere drag, solar pressure, and radiation. The ability to simulate gravity behavior, even if not completely rigorous, on small mass‐market computers allows a fuller understanding of the new approach to astrodynamics by home astronomers, scientists outside orbital mechanics, and students in middle and high school. The simulations can also help a lay audience visualize gravity behavior during press conferences, briefings, and public lectures. No review, evaluation, critique of the programs shown in this presentation is intended. The results from these simulations are not valid for — and must not be used for — making earth‐colliding predictions.

A Simple Approximate Analysis of the Linear Stability of Ring Systems
View Description Hide DescriptionWe give a simple linear stability analysis of a system of n equal mass bodies in circular orbit about a single more massive body. A full analysis requires the possibility of perturbing all bodies. If the massive body is sufficiently dominant, then one can ignore perturbations to it. In this paper, we give a linear stability analysis based on perturbations to just one of the small ring bodies. Such an analysis could be justified by assuming that this one body has mass zero. But, we do not make this assumption. Therefore, it is surprising that the result we obtain agrees to within a factor of 2 with the result one obtains by considering perturbations to all ring bodies. We also give a simple back‐of‐the‐envelope computation that shows that our stability mass threshold is consistent with the observed optical density of Saturn’s rings.

Dangers of Truncating the Disturbing Function In Small Body Solar System Dynamics
View Description Hide DescriptionPlanetary and satellite systems are replete with orbital resonant configurations that have been modeled by a variety of techniques. We highlight the dangers of models which attempt to trace numerically or analytically the orbital evolution of any two resonant bodies with a truncated disturbing function. Using a semianalytic model based on a traditional disturbing function expansion about zero eccentricities and inclinations, we pinpoint the nature and number of terms needed to model a typical Jovian asteroid and Kuiper Belt Object, and estimate the eccentricities at which convergence of disturbing function coefficients breaks down. We find the notion of “order” to represent an inappropriate metric for accuracy in the orbital solution for these classes of objects, and deduce that even for dynamical configurations with “massless” objects, including more than a few disturbing function terms is often necessary.

Natural Sources of Antiparticles in the Solar System and the Feasibility of Extraction for High Delta‐V Space Propulsion
View Description Hide DescriptionAntiparticles have a mass‐based energy density nearly 10 orders of magnitude greater than the best chemical propellants. This attribute, particularly with antiprotons, enables exciting new approaches to spacecraft propulsion and design. However, these advantages have not been realized due to the inherent limitations associated with the artificial production and storage of the antiparticles. In comparison, antiparticles are produced and trapped naturally in the space environment due to the interaction of high‐energy galactic cosmic rays (GCR) with residual matter in the interstellar medium and around solar system bodies. We assess the stable and transient antiparticle content of these sources and subsequently consider their capture and application to high delta‐v space propulsion.
The magnetosphere surrounding a planet offers a unique environment for the generation and trapping of antiprotons. Using Earth’s magnetic field as an example, we have considered the various source mechanisms that are applicable to a planetary magnetosphere, the confinement duration versus transport processes, and the antiparticle loss mechanisms. We have estimated the trapped population of antiprotons magnetically confined following production in the atmosphere due to nuclear interactions between high‐energy cosmic rays and constituents of the residual planetary atmosphere. We present results of the estimated particle fluxes due to pair production of antiprotons in the exosphere, the decay of albedo antineutrons generated in the atmosphere, and the focusing the transient GCR antiprotons by the magnetic field of the planet. We discuss relevant scaling parameters and extend the terrestrial results to the Jovian planets and other solar system objects to estimate the total supply of antiprotons surrounding these bodies.
The expediency of utilizing an electromagnetic scoop to extract antiparticles for practical use is subsequently considered. A large scale magnetic field generated by a spacecraft can be used as a funnel to direct charged antiparticles towards a trap. We discuss and explore the fundamental performance limits of such a device and estimate the total antiproton collection rate for a given flux level. Based on predicted fluxes, it is potentially feasible to extract tens to hundreds of micrograms of antiprotons from the natural environment over the course of a year. Near the throat of the collection device, the particle can be transferred to closed field lines where it is stably trapped in the mini‐magnetosphere that is formed in the space surrounding the exterior of the vehicle. Following capture and trapping, the particles can be used as a fuel for propelling spacecraft to high velocities.

Closing Remarks: A Charter for Interstellar Studies
View Description Hide DescriptionMultidisciplinary by necessity, interstellar studies attack a seemingly intractable problem from numerous angles, many of which were on display at the Princeton conference. The past year has seen work that firms up our knowledge of the nearest interstellar target, the triple star system Alpha Centauri. Learning that the Centauri A and B stars may well support terrestrial style worlds, we push ahead into an era of breakthrough investigations in exoplanet detection and imaging. The treatment of the Centauri developments illustrates the changes that online capabilities bring to research, offering new venues and providing for vigorous debate via preprint archives, weblogs and self‐archiving by researchers. These tools also hone our skills at presenting the public case for interstellar research to a lay audience often indifferent to both space‐related activities and the possibility of long‐term planning. We must continue to press the case for vigorous exploration and use these new online capabilities to re‐energize an all too jaded public.

Status of Solar Sail Propulsion: Moving Toward an Interstellar Probe
View Description Hide DescriptionNASA’s In‐Space Propulsion Technology Program has developed the first‐generation of solar sail propulsion systems sufficient to accomplish inner solar system science and exploration missions. These first‐generation solar sails, when operational, will range in size from 40 meters to well over 100 meters in diameter and have an areal density of less than 13 grams‐per‐square meter. A rigorous, multiyear technology development effort culminated last year in the testing of two different 20‐meter solar sail systems under thermal vacuum conditions. This effort provided a number of significant insights into the optimal design and expected performance of solar sails as well as an understanding of the methods and costs of building and using them. In a separate effort, solar sail orbital analysis tools for mission design were developed and tested. Laboratory simulations of the effects of long‐term space radiation exposure were also conducted on two candidate solar sail materials. Detailed radiation and charging environments were defined for mission trajectories outside the protection of the earth’s magnetosphere, in the solar wind environment. These were used in other analytical tools to prove the adequacy of sail design features for accommodating the harsh space environment.
Preceding, and in conjunction with these technology efforts, NASA sponsored several mission application studies for solar sails, including one that would use an evolved sail capability to support humanity’s first mission into nearby interstellar space. The proposed mission is called the Interstellar Probe. The Interstellar Probe might be accomplished in several ways. A 200‐meter sail, with an areal density approaching 1 gram‐per‐square meter, could accelerate a robotic probe to the very edge of the solar system in just under 20 years from launch. A sail using the technology just demonstrated could make the same mission, but take significantly longer. Conventional chemical propulsion systems would require even longer flight times. Spinner sails of the type being explored by the Japanese may also be a good option, but the level of maturity in that technology is not clear. While the technology to support a 200‐meter, ultralightweight sail mission is not yet in hand, the recent NASA investments in solar sail technology are an essential first step toward making it a reality.
This paper will describe the status of solar sail propulsion within NASA, near‐term solar sail mission applications, and the plan to advance the technology to the point where the Interstellar Probe mission can be flown.

Sun’s Focus Comes First, Interstellar Target Comes Second
View Description Hide DescriptionThe nearest stellar system, Alpha Centauri, is located about 4.40 light‐years away. This amounts to 278,261 AU. But at only 550 AU, or, more generally, at only about 1,000 AU, the focus of the gravity lens of the Sun is found, which is then 278 times closer than our nearest interstellar target. In other words, assuming equal engineering problems, the trip to the Sun’s focus takes 278 times less than the trip to the nearest stellar target. This makes the Sun’s gravity focus a reasonable target for our probes to reach within this century. But there is more. Before we send any probe towards even the nearest stellar system, we’ll be in the need to have a good radio map of that stellar system, as well as of everything else that maybe on the way. Thus, we need a huge radio magnification of these objects, and nothing is better than the huge magnification provided by the gravitational lens of the Sun. In conclusion, sending a first probe to 1,000 AU in the direction opposite to the target stellar system clearly must be done before any interstellar flight to that stellar system is even designed. To the stars, by steps. Ad astra, incrementis!