Poincaré and the Three-Body Problem
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The Solar System's Extended Shelf Life
Vol 459|11 June 2009 NEWS & VIEWS PLANETARY SCIENCE The Solar System’s extended shelf life Gregory Laughlin Simulations show that orbital chaos can lead to collisions between Earth and the inner planets. But Einstein’s tweaks to Newton’s theory of gravity render these ruinous outcomes unlikely in the next few billion years. In the midst of a seemingly endless torrent of charted with confidence tens of millions of ‘disturbing function’ could not be neglected, baleful economic and environmental news, a years into the future. An ironclad evaluation and consideration of these terms revived the dispatch from the field of celestial dynamics of the Solar System’s stability, however, eluded question of orbital stability. In 1889, Henri manages to sound a note of definite cheer. On mathematicians and astronomers for nearly Poincaré demonstrated that even the gravita- page 817 of this issue, Laskar and Gastineau1 three centuries. tional three-body problem cannot be solved report the outcome of a huge array of computer In the seventeenth century, Isaac Newton by analytic integration, thereby eliminating simulations. Their work shows that the orbits was bothered by his inability to fully account any possibility that an analytic solution for of the terrestrial planets — Mercury, Venus, for the observed motions of Jupiter and Saturn. the entire future motion of the eight planets Earth and Mars — have a roughly 99% chance The nonlinearity of the gravitational few-body could be found. Poincaré’s work anticipated the of maintaining their current, well-ordered problem led him to conclude3 that, “to consider now-familiar concept of dynamical chaos and clockwork for the roughly 5 billion years that simultaneously all these causes of motion and the sensitive dependence of nonlinear systems remain before the Sun evolves into a red giant to define these motions by exact laws admit- on initial conditions5. -
Three Body Problem
PHYS 7221 - The Three-Body Problem Special Lecture: Wednesday October 11, 2006, Juhan Frank, LSU 1 The Three-Body Problem in Astronomy The classical Newtonian three-body gravitational problem occurs in Nature exclusively in an as- tronomical context and was the subject of many investigations by the best minds of the 18th and 19th centuries. Interest in this problem has undergone a revival in recent decades when it was real- ized that the evolution and ultimate fate of star clusters and the nuclei of active galaxies depends crucially on the interactions between stellar and black hole binaries and single stars. The general three-body problem remains unsolved today but important advances and insights have been enabled by the advent of modern computational hardware and methods. The long-term stability of the orbits of the Earth and the Moon was one of the early concerns when the age of the Earth was not well-known. Newton solved the two-body problem for the orbit of the Moon around the Earth and considered the e®ects of the Sun on this motion. This is perhaps the earliest appearance of the three-body problem. The ¯rst and simplest periodic exact solution to the three-body problem is the motion on collinear ellipses found by Euler (1767). Also Euler (1772) studied the motion of the Moon assuming that the Earth and the Sun orbited each other on circular orbits and that the Moon was massless. This approach is now known as the restricted three-body problem. At about the same time Lagrange (1772) discovered the equilateral triangle solution described in Goldstein (2002) and Hestenes (1999). -
Orbital Mechanics
Orbital Mechanics These notes provide an alternative and elegant derivation of Kepler's three laws for the motion of two bodies resulting from their gravitational force on each other. Orbit Equation and Kepler I Consider the equation of motion of one of the particles (say, the one with mass m) with respect to the other (with mass M), i.e. the relative motion of m with respect to M: r r = −µ ; (1) r3 with µ given by µ = G(M + m): (2) Let h be the specific angular momentum (i.e. the angular momentum per unit mass) of m, h = r × r:_ (3) The × sign indicates the cross product. Taking the derivative of h with respect to time, t, we can write d (r × r_) = r_ × r_ + r × ¨r dt = 0 + 0 = 0 (4) The first term of the right hand side is zero for obvious reasons; the second term is zero because of Eqn. 1: the vectors r and ¨r are antiparallel. We conclude that h is a constant vector, and its magnitude, h, is constant as well. The vector h is perpendicular to both r and the velocity r_, hence to the plane defined by these two vectors. This plane is the orbital plane. Let us now carry out the cross product of ¨r, given by Eqn. 1, and h, and make use of the vector algebra identity A × (B × C) = (A · C)B − (A · B)C (5) to write µ ¨r × h = − (r · r_)r − r2r_ : (6) r3 { 2 { The r · r_ in this equation can be replaced by rr_ since r · r = r2; and after taking the time derivative of both sides, d d (r · r) = (r2); dt dt 2r · r_ = 2rr;_ r · r_ = rr:_ (7) Substituting Eqn. -
Workshop on Astronomy and Dynamics at the Occasion of Jacques Laskar’S 60Th Birthday Testing Modified Gravity with Planetary Ephemerides
Workshop on Astronomy and Dynamics at the occasion of Jacques Laskar's 60th birthday Testing Modified Gravity with Planetary Ephemerides Luc Blanchet Gravitation et Cosmologie (GR"CO) Institut d'Astrophysique de Paris 29 avril 2015 Luc Blanchet (IAP) Testing Modified Gravity Colloque Jacques Laskar 1 / 25 Evidence for dark matter in astrophysics 1 Oort [1932] noted that the sum of observed mass in the vicinity of the Sun falls short of explaining the vertical motion of stars in the Milky Way 2 Zwicky [1933] reported that the velocity dispersion of galaxies in galaxy clusters is far too high for these objects to remain bound for a substantial fraction of cosmic time 3 Ostriker & Peebles [1973] showed that to prevent the growth of instabilities in cold self-gravitating disks like spiral galaxies, it is necessary to embed the disk in the quasi-spherical potential of a huge halo of dark matter 4 Bosma [1981] and Rubin [1982] established that the rotation curves of galaxies are approximately flat, contrarily to the Newtonian prediction based on ordinary baryonic matter Luc Blanchet (IAP) Testing Modified Gravity Colloque Jacques Laskar 2 / 25 Evidence for dark matter in astrophysics 1 Oort [1932] noted that the sum of observed mass in the vicinity of the Sun falls short of explaining the vertical motion of stars in the Milky Way 2 Zwicky [1933] reported that the velocity dispersion of galaxies in galaxy clusters is far too high for these objects to remain bound for a substantial fraction of cosmic time 3 Ostriker & Peebles [1973] showed that to -
Laplace-Runge-Lenz Vector
Laplace-Runge-Lenz Vector Alex Alemi June 6, 2009 1 Alex Alemi CDS 205 LRL Vector The central inverse square law force problem is an interesting one in physics. It is interesting not only because of its applicability to a great deal of situations ranging from the orbits of the planets to the spectrum of the hydrogen atom, but also because it exhibits a great deal of symmetry. In fact, in addition to the usual conservations of energy E and angular momentum L, the Kepler problem exhibits a hidden symmetry. There exists an additional conservation law that does not correspond to a cyclic coordinate. This conserved quantity is associated with the so called Laplace-Runge-Lenz (LRL) vector A: A = p × L − mkr^ (LRL Vector) The nature of this hidden symmetry is an interesting one. Below is an attempt to introduce the LRL vector and begin to discuss some of its peculiarities. A Some History The LRL vector has an interesting and unique history. Being a conservation for a general problem, it appears as though it was discovered independently a number of times. In fact, the proper name to attribute to the vector is an open question. The modern popularity of the use of the vector can be traced back to Lenz’s use of the vector to calculate the perturbed energy levels of the Kepler problem using old quantum theory [1]. In his paper, Lenz describes the vector as “little known” and refers to a popular text by Runge on vector analysis. In Runge’s text, he makes no claims of originality [1]. -
Writing the History of Dynamical Systems and Chaos
Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as -
A Long Term Numerical Solution for the Insolation Quantities of the Earth Jacques Laskar, Philippe Robutel, Frédéric Joutel, Mickael Gastineau, A.C.M
A long term numerical solution for the insolation quantities of the Earth Jacques Laskar, Philippe Robutel, Frédéric Joutel, Mickael Gastineau, A.C.M. Correia, Benjamin Levrard To cite this version: Jacques Laskar, Philippe Robutel, Frédéric Joutel, Mickael Gastineau, A.C.M. Correia, et al.. A long term numerical solution for the insolation quantities of the Earth. 2004. hal-00001603 HAL Id: hal-00001603 https://hal.archives-ouvertes.fr/hal-00001603 Preprint submitted on 23 May 2004 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Astronomy & Astrophysics manuscript no. La˙2004 May 23, 2004 (DOI: will be inserted by hand later) A long term numerical solution for the insolation quantities of the Earth. J. Laskar1, P. Robutel1, F. Joutel1, M. Gastineau1, A.C.M. Correia1,2, and B. Levrard1 1 Astronomie et Syst`emesDynamiques, IMCCE-CNRS UMR8028, 77 Av. Denfert-Rochereau, 75014 Paris, France 2 Departamento de F´ısica da Universidade de Aveiro, Campus Universit´ario de Santiago, 3810-193 Aveiro, Portugal May 23, 2004 Abstract. We present here a new solution for the astronomical computation of the insolation quantities on Earth spanning from -250 Myr to 250 Myr. -
Relativistic Acceleration of Planetary Orbiters
RELATIVISTIC ACCELERATION OF PLANETARY ORBITERS Bernard Godard(1), Frank Budnik(2), Trevor Morley(2), and Alejandro Lopez Lozano(3) (1)Telespazio VEGA Deutschland GmbH, located at ESOC* (2)European Space Agency, located at ESOC* (3)Logica Deutschland GmbH & Co. KG, located at ESOC* *Robert-Bosch-Str. 5, 64293 Darmstadt, Germany, +49 6151 900, < firstname > . < lastname > [. < lastname2 >]@esa.int Abstract: This paper deals with the relativistic contributions to the gravitational acceleration of a planetary orbiter. The formulation for the relativistic corrections is different in the solar-system barycentric relativistic system and in the local planetocentric relativistic system. The ratio of this correction to the total acceleration is usually orders of magnitude larger in the barycentric system than in the planetocentric system. However, an appropriate Lorentz transformation of the total acceleration from one system to the other shows that both systems are equivalent to a very good accuracy. This paper discusses the steps that were taken at ESOC to validate the implementations of the relativistic correction to the gravitational acceleration in the interplanetary orbit determination software. It compares numerically both systems in the cases of a spacecraft in the vicinity of the Earth (Rosetta during its first swing-by) and a Mercury orbiter (BepiColombo). For the Mercury orbiter, the orbit is propagated in both systems and with an appropriate adjustment of the time argument and a Lorentz correction of the position vector, the resulting orbits are made to match very closely. Finally, the effects on the radiometric observables of neglecting the relativistic corrections to the acceleration in each system and of not performing the space-time transformations from the Mercury system to the barycentric system are presented. -
Problem Set #6 Motivation Problem Statement
PROBLEM SET #6 TO: PROF. DAVID MILLER, PROF. JOHN KEESEE, AND MS. MARILYN GOOD FROM: NAMES WITHELD SUBJECT: PROBLEM SET #6 (LIFE SUPPORT, PROPULSION, AND POWER FOR AN EARTH-TO-MARS HUMAN TRANSPORTATION VEHICLE) DATE: 12/3/2003 MOTIVATION Mars is of great scientific interest given the potential evidence of past or present life. Recent evidence indicating the past existence of water deposits underscore its scientific value. Other motivations to go to Mars include studying its climate history through exploration of the polar layers. This information could be correlated with similar data from Antarctica to characterize the evolution of the Solar System and its geological history. Long-term goals might include the colonization of Mars. As the closest planet with a relatively mild environment, there exists a unique opportunity to explore Mars with humans. Although we have used robotic spacecraft successfully in the past to study Mars, humans offer a more efficient and robust exploration capability. However, human spaceflight adds both complexity and mass to the space vehicle and has a significant impact on the mission design. Humans require an advanced environmental control and life support system, and this subsystem has high power requirements thus directly affecting the power subsystem design. A Mars mission capability is likely to be a factor in NASA’s new launch architecture design since the resulting launch architecture will need take into account the estimated spacecraft mass required for such a mission. To design a Mars mission, various propulsion system options must be evaluated and compared for their efficiency and adaptability to the required mission duration. -
The Two-Body Problem
Computational Astrophysics I: Introduction and basic concepts Helge Todt Astrophysics Institute of Physics and Astronomy University of Potsdam SoSe 2021, 3.6.2021 H. Todt (UP) Computational Astrophysics SoSe 2021, 3.6.2021 1 / 32 The two-body problem H. Todt (UP) Computational Astrophysics SoSe 2021, 3.6.2021 2 / 32 Equations of motionI We remember (?): The Kepler’s laws of planetary motion (1619) 1 Each planet moves in an elliptical orbit where the Sun is at one of the foci of the ellipse. 2 The velocity of a planet increases with decreasing distance to the Sun such, that the planet sweeps out equal areas in equal times. 3 The ratio P2=a3 is the same for all planets orbiting the Sun, where P is the orbital period and a is the semimajor axis of the ellipse. The 1. and 3. Kepler’s law describe the shape of the orbit (Copernicus: circles), but not the time dependence ~r(t). This can in general not be expressed by elementary mathematical functions (see below). Therefore we will try to find a numerical solution. H. Todt (UP) Computational Astrophysics SoSe 2021, 3.6.2021 3 / 32 Equations of motionII Earth-Sun system ! two-body problem ! one-body problem via reduced mass of lighter body (partition of motion): M m m µ = = m (1) m + M M + 1 as mE M is µ ≈ m, i.e. motion is relative to the center of mass ≡ only motion of m. Set point of origin (0; 0) to the source of the force field of M. Moreover: Newton’s 2. -
Orbital Mechanics Course Notes
Orbital Mechanics Course Notes David J. Westpfahl Professor of Astrophysics, New Mexico Institute of Mining and Technology March 31, 2011 2 These are notes for a course in orbital mechanics catalogued as Aerospace Engineering 313 at New Mexico Tech and Aerospace Engineering 362 at New Mexico State University. This course uses the text “Fundamentals of Astrodynamics” by R.R. Bate, D. D. Muller, and J. E. White, published by Dover Publications, New York, copyright 1971. The notes do not follow the book exclusively. Additional material is included when I believe that it is needed for clarity, understanding, historical perspective, or personal whim. We will cover the material recommended by the authors for a one-semester course: all of Chapter 1, sections 2.1 to 2.7 and 2.13 to 2.15 of Chapter 2, all of Chapter 3, sections 4.1 to 4.5 of Chapter 4, and as much of Chapters 6, 7, and 8 as time allows. Purpose The purpose of this course is to provide an introduction to orbital me- chanics. Students who complete the course successfully will be prepared to participate in basic space mission planning. By basic mission planning I mean the planning done with closed-form calculations and a calculator. Stu- dents will have to master additional material on numerical orbit calculation before they will be able to participate in detailed mission planning. There is a lot of unfamiliar material to be mastered in this course. This is one field of human endeavor where engineering meets astronomy and ce- lestial mechanics, two fields not usually included in an engineering curricu- lum. -
Hamiltonian Perturbation Theory (And Transition to Chaos) H 4515 H
Hamiltonian Perturbation Theory (and Transition to Chaos) H 4515 H Hamiltonian Perturbation Theory dle third from a closed interval) designates topological spaces that locally have the structure Rn Cantor dust (and Transition to Chaos) for some n 2 N. Cantor families are parametrized by HENK W. BROER1,HEINZ HANSSMANN2 such Cantor sets. On the real line R one can define 1 Instituut voor Wiskunde en Informatica, Cantor dust of positive measure by excluding around Rijksuniversiteit Groningen, Groningen, each rational number p/q an interval of size The Netherlands 2 2 Mathematisch Instituut, Universiteit Utrecht, ;>0 ;>2 : q Utrecht, The Netherlands Similar Diophantine conditions define Cantor sets Article Outline in Rn. Since these Cantor sets have positive measure their Hausdorff dimension is n.Wheretheunper- Glossary turbed system is stratified according to the co-dimen- Definition of the Subject sion of occurring (bifurcating) tori, this leads to a Can- Introduction tor stratification. One Degree of Freedom Chaos An evolution of a dynamical system is chaotic if its Perturbations of Periodic Orbits future is badly predictable from its past. Examples of Invariant Curves of Planar Diffeomorphisms non-chaotic evolutions are periodic or multi-periodic. KAM Theory: An Overview A system is called chaotic when many of its evolutions Splitting of Separatrices are. One criterion for chaoticity is the fact that one of Transition to Chaos and Turbulence the Lyapunov exponents is positive. Future Directions Diophantine condition, Diophantine frequency vector Bibliography A frequency vector ! 2 Rn is called Diophantine if there are constants >0and>n 1with Glossary jhk;!ij for all k 2 Znnf0g : Bifurcation In parametrized dynamical systems a bifur- jkj cation occurs when a qualitative change is invoked by a change of parameters.