Low-Thrust Maneuvers for the Efficient Correction of Orbital Elements
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The Utilization of Halo Orbit§ in Advanced Lunar Operation§
NASA TECHNICAL NOTE NASA TN 0-6365 VI *o M *o d a THE UTILIZATION OF HALO ORBIT§ IN ADVANCED LUNAR OPERATION§ by Robert W. Fmqcbar Godddrd Spctce Flight Center P Greenbelt, Md. 20771 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. JULY 1971 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. NASA TN D-6365 5. Report Date Jul;v 1971. 6. Performing Organization Code 8. Performing Organization Report No, G-1025 10. Work Unit No. Goddard Space Flight Center 11. Contract or Grant No. Greenbelt, Maryland 20 771 13. Type of Report and Period Covered 2. Sponsoring Agency Name and Address Technical Note J National Aeronautics and Space Administration Washington, D. C. 20546 14. Sponsoring Agency Code 5. Supplementary Notes 6. Abstract Flight mechanics and control problems associated with the stationing of space- craft in halo orbits about the translunar libration point are discussed in some detail. Practical procedures for the implementation of the control techniques are described, and it is shown that these procedures can be carried out with very small AV costs. The possibility of using a relay satellite in.a halo orbit to obtain a continuous com- munications link between the earth and the far side of the moon is also discussed. Several advantages of this type of lunar far-side data link over more conventional relay-satellite systems are cited. It is shown that, with a halo relay satellite, it would be possible to continuously control an unmanned lunar roving vehicle on the moon's far side. Backside tracking of lunar orbiters could also be realized. -
Appendix a Orbits
Appendix A Orbits As discussed in the Introduction, a good ¯rst approximation for satellite motion is obtained by assuming the spacecraft is a point mass or spherical body moving in the gravitational ¯eld of a spherical planet. This leads to the classical two-body problem. Since we use the term body to refer to a spacecraft of ¯nite size (as in rigid body), it may be more appropriate to call this the two-particle problem, but I will use the term two-body problem in its classical sense. The basic elements of orbital dynamics are captured in Kepler's three laws which he published in the 17th century. His laws were for the orbital motion of the planets about the Sun, but are also applicable to the motion of satellites about planets. The three laws are: 1. The orbit of each planet is an ellipse with the Sun at one focus. 2. The line joining the planet to the Sun sweeps out equal areas in equal times. 3. The square of the period of a planet is proportional to the cube of its mean distance to the sun. The ¯rst law applies to most spacecraft, but it is also possible for spacecraft to travel in parabolic and hyperbolic orbits, in which case the period is in¯nite and the 3rd law does not apply. However, the 2nd law applies to all two-body motion. Newton's 2nd law and his law of universal gravitation provide the tools for generalizing Kepler's laws to non-elliptical orbits, as well as for proving Kepler's laws. -
Astrodynamics
Politecnico di Torino SEEDS SpacE Exploration and Development Systems Astrodynamics II Edition 2006 - 07 - Ver. 2.0.1 Author: Guido Colasurdo Dipartimento di Energetica Teacher: Giulio Avanzini Dipartimento di Ingegneria Aeronautica e Spaziale e-mail: [email protected] Contents 1 Two–Body Orbital Mechanics 1 1.1 BirthofAstrodynamics: Kepler’sLaws. ......... 1 1.2 Newton’sLawsofMotion ............................ ... 2 1.3 Newton’s Law of Universal Gravitation . ......... 3 1.4 The n–BodyProblem ................................. 4 1.5 Equation of Motion in the Two-Body Problem . ....... 5 1.6 PotentialEnergy ................................. ... 6 1.7 ConstantsoftheMotion . .. .. .. .. .. .. .. .. .... 7 1.8 TrajectoryEquation .............................. .... 8 1.9 ConicSections ................................... 8 1.10 Relating Energy and Semi-major Axis . ........ 9 2 Two-Dimensional Analysis of Motion 11 2.1 ReferenceFrames................................. 11 2.2 Velocity and acceleration components . ......... 12 2.3 First-Order Scalar Equations of Motion . ......... 12 2.4 PerifocalReferenceFrame . ...... 13 2.5 FlightPathAngle ................................. 14 2.6 EllipticalOrbits................................ ..... 15 2.6.1 Geometry of an Elliptical Orbit . ..... 15 2.6.2 Period of an Elliptical Orbit . ..... 16 2.7 Time–of–Flight on the Elliptical Orbit . .......... 16 2.8 Extensiontohyperbolaandparabola. ........ 18 2.9 Circular and Escape Velocity, Hyperbolic Excess Speed . .............. 18 2.10 CosmicVelocities -
Maneuver Detection of Space Object for Space Surveillance
MANEUVER DETECTION OF SPACE OBJECT FOR SPACE SURVEILLANCE Jian Huang*, Weidong Hu, Lefeng Zhang $756WDWH.H\/DE1DWLRQDO8QLYHUVLW\RI'HIHQVH7HFKQRORJ\&KDQJVKD+XQDQ3URYLQFH&KLQD (PDLOKXDQJMLDQ#QXGWHGXFQ ABSTRACT on the technique of a moving window curve fit. Holzinger & Scheeres presented an object correlation Maneuver detection is a very important task for and maneuver detection method using optimal control maintaining the catalog of the orbital objects and space performance metrics [5]. Kelecy & Jah focused on the situational awareness. This paper mainly focuses on the detection and reconstruction of single low thrust in- typical maneuver scenario where space objects only track maneuvers by using the orbit determination perform tangential orbital maneuvers during a relative strategies based on the batch least-squares and long gap. Particularly, only two classical and extended Kalman filter (EKF) [6]. However, these commonly applied orbital transition manners are methods are highly relevant to the presupposed considered, i.e. the twice tangential maneuvers at the maneuvering mode and a relative short observation gap. apogee and the perigee or one tangential maneuver at In this paper, to address the real observed data, we only an arbitrary time. Based on this, we preliminarily consider the common maneuvering modes and estimate the maneuvering mode and parameters by observation scenarios in practical. For an orbital analyzing the change of semi-major axis and maneuver, minimization of fuel consumption is eccentricity. Furthermore, if only one tangential essential because the weight of a payload that can be maneuver happened, we can formulate the estimation carried to the desired orbit depends on this problem of maneuvering parameters as a non-linear minimization. -
Reduction of Saturn Orbit Insertion Impulse Using Deep-Space Low Thrust
Reduction of Saturn Orbit Insertion Impulse using Deep-Space Low Thrust Elena Fantino ∗† Khalifa University of Science and Technology, Abu Dhabi, United Arab Emirates. Roberto Flores‡ International Center for Numerical Methods in Engineering, 08034 Barcelona, Spain. Jesús Peláez§ and Virginia Raposo-Pulido¶ Technical University of Madrid, 28040 Madrid, Spain. Orbit insertion at Saturn requires a large impulsive manoeuver due to the velocity difference between the spacecraft and the planet. This paper presents a strategy to reduce dramatically the hyperbolic excess speed at Saturn by means of deep-space electric propulsion. The inter- planetary trajectory includes a gravity assist at Jupiter, combined with low-thrust maneuvers. The thrust arc from Earth to Jupiter lowers the launch energy requirement, while an ad hoc steering law applied after the Jupiter flyby reduces the hyperbolic excess speed upon arrival at Saturn. This lowers the orbit insertion impulse to the point where capture is possible even with a gravity assist with Titan. The control-law algorithm, the benefits to the mass budget and the main technological aspects are presented and discussed. The simple steering law is compared with a trajectory optimizer to evaluate the quality of the results and possibilities for improvement. I. Introduction he giant planets have a special place in our quest for learning about the origins of our planetary system and Tour search for life, and robotic missions are essential tools for this scientific goal. Missions to the outer planets arXiv:2001.04357v1 [astro-ph.EP] 8 Jan 2020 have been prioritized by NASA and ESA, and this has resulted in important space projects for the exploration of the Jupiter system (NASA’s Europa Clipper [1] and ESA’s Jupiter Icy Moons Explorer [2]), and studies are underway to launch a follow-up of Cassini/Huygens called Titan Saturn System Mission (TSSM) [3], a joint ESA-NASA project. -
Elliptical Orbits
1 Ellipse-geometry 1.1 Parameterization • Functional characterization:(a: semi major axis, b ≤ a: semi minor axis) x2 y 2 b p + = 1 ⇐⇒ y(x) = · ± a2 − x2 (1) a b a • Parameterization in cartesian coordinates, which follows directly from Eq. (1): x a · cos t = with 0 ≤ t < 2π (2) y b · sin t – The origin (0, 0) is the center of the ellipse and the auxilliary circle with radius a. √ – The focal points are located at (±a · e, 0) with the eccentricity e = a2 − b2/a. • Parameterization in polar coordinates:(p: parameter, 0 ≤ < 1: eccentricity) p r(ϕ) = (3) 1 + e cos ϕ – The origin (0, 0) is the right focal point of the ellipse. – The major axis is given by 2a = r(0) − r(π), thus a = p/(1 − e2), the center is therefore at − pe/(1 − e2), 0. – ϕ = 0 corresponds to the periapsis (the point closest to the focal point; which is also called perigee/perihelion/periastron in case of an orbit around the Earth/sun/star). The relation between t and ϕ of the parameterizations in Eqs. (2) and (3) is the following: t r1 − e ϕ tan = · tan (4) 2 1 + e 2 1.2 Area of an elliptic sector As an ellipse is a circle with radius a scaled by a factor b/a in y-direction (Eq. 1), the area of an elliptic sector PFS (Fig. ??) is just this fraction of the area PFQ in the auxiliary circle. b t 2 1 APFS = · · πa − · ae · a sin t a 2π 2 (5) 1 = (t − e sin t) · a b 2 The area of the full ellipse (t = 2π) is then, of course, Aellipse = π a b (6) Figure 1: Ellipse and auxilliary circle. -
Habitability of Planets on Eccentric Orbits: Limits of the Mean Flux Approximation
A&A 591, A106 (2016) Astronomy DOI: 10.1051/0004-6361/201628073 & c ESO 2016 Astrophysics Habitability of planets on eccentric orbits: Limits of the mean flux approximation Emeline Bolmont1, Anne-Sophie Libert1, Jeremy Leconte2; 3; 4, and Franck Selsis5; 6 1 NaXys, Department of Mathematics, University of Namur, 8 Rempart de la Vierge, 5000 Namur, Belgium e-mail: [email protected] 2 Canadian Institute for Theoretical Astrophysics, 60st St George Street, University of Toronto, Toronto, ON, M5S3H8, Canada 3 Banting Fellow 4 Center for Planetary Sciences, Department of Physical & Environmental Sciences, University of Toronto Scarborough, Toronto, ON, M1C 1A4, Canada 5 Univ. Bordeaux, LAB, UMR 5804, 33270 Floirac, France 6 CNRS, LAB, UMR 5804, 33270 Floirac, France Received 4 January 2016 / Accepted 28 April 2016 ABSTRACT Unlike the Earth, which has a small orbital eccentricity, some exoplanets discovered in the insolation habitable zone (HZ) have high orbital eccentricities (e.g., up to an eccentricity of ∼0.97 for HD 20782 b). This raises the question of whether these planets have surface conditions favorable to liquid water. In order to assess the habitability of an eccentric planet, the mean flux approximation is often used. It states that a planet on an eccentric orbit is called habitable if it receives on average a flux compatible with the presence of surface liquid water. However, because the planets experience important insolation variations over one orbit and even spend some time outside the HZ for high eccentricities, the question of their habitability might not be as straightforward. We performed a set of simulations using the global climate model LMDZ to explore the limits of the mean flux approximation when varying the luminosity of the host star and the eccentricity of the planet. -
Analysis of Transfer Maneuvers from Initial
Revista Mexicana de Astronom´ıa y Astrof´ısica, 52, 283–295 (2016) ANALYSIS OF TRANSFER MANEUVERS FROM INITIAL CIRCULAR ORBIT TO A FINAL CIRCULAR OR ELLIPTIC ORBIT M. A. Sharaf1 and A. S. Saad2,3 Received March 14 2016; accepted May 16 2016 RESUMEN Presentamos un an´alisis de las maniobras de transferencia de una ´orbita circu- lar a una ´orbita final el´ıptica o circular. Desarrollamos este an´alisis para estudiar el problema de las transferencias impulsivas en las misiones espaciales. Consideramos maniobras planas usando ecuaciones reci´enobtenidas; con estas ecuaciones se com- paran las maniobras circulares y el´ıpticas. Esta comparaci´on es importante para el dise˜no´optimo de misiones espaciales, y permite obtener mapeos ´utiles para mostrar cu´al es la mejor maniobra. Al respecto, desarrollamos este tipo de comparaciones mediante 10 resultados, y los mostramos gr´aficamente. ABSTRACT In the present paper an analysis of the transfer maneuvers from initial circu- lar orbit to a final circular or elliptic orbit was developed to study the problem of impulsive transfers for space missions. It considers planar maneuvers using newly derived equations. With these equations, comparisons of circular and elliptic ma- neuvers are made. This comparison is important for the mission designers to obtain useful mappings showing where one maneuver is better than the other. In this as- pect, we developed this comparison throughout ten results, together with some graphs to show their meaning. Key Words: celestial mechanics — methods: analytical — space vehicles 1. INTRODUCTION The large amount of information from interplanetary missions, such as Pioneer (Dyer et al. -
1961Apj. . .133. .6575 PHYSICAL and ORBITAL BEHAVIOR OF
.6575 PHYSICAL AND ORBITAL BEHAVIOR OF COMETS* .133. Roy E. Squires University of California, Davis, California, and the Aerojet-General Corporation, Sacramento, California 1961ApJ. AND DaVid B. Beard University of California, Davis, California Received August 11, I960; revised October 14, 1960 ABSTRACT As a comet approaches perihelion and the surface facing the sun is warmed, there is evaporation of the surface material in the solar direction The effect of this unidirectional rapid mass loss on the orbits of the parabolic comets has been investigated, and the resulting corrections to the observed aphelion distances, a, and eccentricities have been calculated. Comet surface temperature has been calculated as a function of solar distance, and estimates have been made of the masses and radii of cometary nuclei. It was found that parabolic comets with perihelia of 1 a u. may experience an apparent shift in 1/a of as much as +0.0005 au-1 and that the apparent shift in 1/a is positive for every comet, thereby increas- ing the apparent orbital eccentricity over the true eccentricity in all cases. For a comet with a perihelion of 1 a u , the correction to the observed eccentricity may be as much as —0.0005. The comet surface tem- perature at 1 a.u. is roughly 180° K, and representative values for the masses and radii of cometary nuclei are 6 X 10" gm and 300 meters. I. INTRODUCTION Since most comets are observed to move in nearly parabolic orbits, questions concern- ing their origin and whether or not they are permanent members of the solar system are not clearly resolved. -
Perturbation Theory in Celestial Mechanics
Perturbation Theory in Celestial Mechanics Alessandra Celletti Dipartimento di Matematica Universit`adi Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) ([email protected]) December 8, 2007 Contents 1 Glossary 2 2 Definition 2 3 Introduction 2 4 Classical perturbation theory 4 4.1 The classical theory . 4 4.2 The precession of the perihelion of Mercury . 6 4.2.1 Delaunay action–angle variables . 6 4.2.2 The restricted, planar, circular, three–body problem . 7 4.2.3 Expansion of the perturbing function . 7 4.2.4 Computation of the precession of the perihelion . 8 5 Resonant perturbation theory 9 5.1 The resonant theory . 9 5.2 Three–body resonance . 10 5.3 Degenerate perturbation theory . 11 5.4 The precession of the equinoxes . 12 6 Invariant tori 14 6.1 Invariant KAM surfaces . 14 6.2 Rotational tori for the spin–orbit problem . 15 6.3 Librational tori for the spin–orbit problem . 16 6.4 Rotational tori for the restricted three–body problem . 17 6.5 Planetary problem . 18 7 Periodic orbits 18 7.1 Construction of periodic orbits . 18 7.2 The libration in longitude of the Moon . 20 1 8 Future directions 20 9 Bibliography 21 9.1 Books and Reviews . 21 9.2 Primary Literature . 22 1 Glossary KAM theory: it provides the persistence of quasi–periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i.e. a non– degeneracy of the integrable system and a diophantine condition of the frequency of motion. -
Orbital Stability Zones About Asteroids
ICARUS 92, 118-131 (1991) Orbital Stability Zones about Asteroids DOUGLAS P. HAMILTON AND JOSEPH A. BURNS Cornell University, Ithaca, New York, 14853-6801 Received October 5, 1990; revised March 1, 1991 exploration program should be remedied when the Galileo We have numerically investigated a three-body problem con- spacecraft, launched in October 1989, encounters the as- sisting of the Sun, an asteroid, and an infinitesimal particle initially teroid 951 Gaspra on October 29, 1991. It is expected placed about the asteroid. We assume that the asteroid has the that the asteroid's surroundings will be almost devoid of following properties: a circular heliocentric orbit at R -- 2.55 AU, material and therefore benign since, in the analogous case an asteroid/Sun mass ratio of/~ -- 5 x 10-12, and a spherical of the satellites of the giant planets, significant debris has shape with radius R A -- 100 km; these values are close to those of never been detected. The analogy is imperfect, however, the minor planet 29 Amphitrite. In order to describe the zone in and so the possibility that interplanetary debris may be which circum-asteroidal debris could be stably trapped, we pay particular attention to the orbits of particles that are on the verge enhanced in the gravitational well of the asteroid must be of escape. We consider particles to be stable or trapped if they considered. If this is the case, the danger of collision with remain in the asteroid's vicinity for at least 5 asteroid orbits about orbiting debris may increase as the asteroid is approached. -
Orbital Inclination and Eccentricity Oscillations in Our Solar
Long term orbital inclination and eccentricity oscillations of the planets in our solar system Abstract The orbits of the planets in our solar system are not in the same plane, therefore natural torques stemming from Newton’s gravitational forces exist to pull them all back to the same plane. This causes the inclinations of the planet orbits to oscillate with potentially long periods and very small damping, because the friction in space is very small. Orbital inclination changes are known for some planets in terms of current rates of change, but the oscillation periods are not well published. They can however be predicted with proper dynamic simulations of the solar system. A three-dimensional dynamic simulation was developed for our solar system capable of handling 12 objects, where all objects affect all other objects. Each object was considered to be a point mass which proved to be an adequate approximation for this study. Initial orbital radii, eccentricities and speeds were set according to known values. The validity of the simulation was demonstrated in terms of short term characteristics such as sidereal periods of planets as well as long term characteristics such as the orbital inclination and eccentricity oscillation periods of Jupiter and Saturn. A significantly more accurate result, than given on approximate analytical grounds in a well-known solar system dynamics textbook, was found for the latter period. Empirical formulas were developed from the simulation results for both these periods for three-object solar type systems. They are very accurate for the Sun, Jupiter and Saturn as well as for some other comparable systems.