Angular Momentum and the Orbit Formulas
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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 -
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. -
A Physical Constants1
A Physical Constants1 Constants in Mechanics −11 3 −1 −2 Gravitational constant G =6.672 59(85) × 10 m kg s (1996) −11 3 −1 −2 =6.673(10) × 10 m kg s (2002) −2 Gravitational acceleration g =9.806 65 m s 30 Solar mass M =1.988 92(25) × 10 kg 8 Solar equatorial radius R =6.96 × 10 m 24 Earth mass M⊕ =5.973 70(76) × 10 kg 6 Earth equatorial radius R⊕ =6.378 140 × 10 m 22 Moon mass M =7.36 () × 10 kg 6 Moon radius R =1.738 × 10 m 9 Distance of Earth from Sun Rmax =0.152 1 × 10 m 9 Rmin =0.147 1 × 10 m 9 Rmean =0.149 6 × 10 m 8 Distance of Moon from Earth Rmean =0.380 × 10 m 7 Period of Earth w.r.t. Sun T⊕ = 365.25 d = 3.16 × 10 s 6 Period of Moon w.r.t. Earth T =27.3d=2.36 × 10 s 1 From Particle Data Group, American Institute of Physics 2002 http://pdg.lbl.gov/2002/contents http://physics.nist.gov/constants 326 A Physical Constants Constants in Electromagnetism def Velocity of light c =. 299, 792, 458 ms−1 def. −7 −2 Vacuum permeability μ0 =4π × 10 NA =12.566 370 614 ...× 10−7 NA−2 1 × −12 −1 Vacuum dielectric constant ε0 = 2 =8.854 187 817 ... 10 Fm μ0c Elementary charge e =1.602 177 33 (49) × 10−19 C e2 =1.439 × 10−9 eV m = 2.305 × 10−28 Jm 4πε0 Constants in Thermodynamics −23 −1 Boltzmann constant kB =1.380 650 3 (24) × 10 JK =8.617 342 (15) × 105 eV K−1 23 −1 Avogadro number NA =6.022 136 7(36) × 10 mole Derived quantities : Gas constant R = NLkB Particle number N,molenumbernN= NLn Constants in Quantum Mechanics Planck’s constant h =6.626 068 76 (52) × 10−34 Js h = =1.054 571 596 (82) × 10−34 Js 2π =6.582 118 89 (26) × 10−16 eV s 2 4πε0 -
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. -
2. Orbital Mechanics MAE 342 2016
2/12/20 Orbital Mechanics Space System Design, MAE 342, Princeton University Robert Stengel Conic section orbits Equations of motion Momentum and energy Kepler’s Equation Position and velocity in orbit Copyright 2016 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE342.html 1 1 Orbits 101 Satellites Escape and Capture (Comets, Meteorites) 2 2 1 2/12/20 Two-Body Orbits are Conic Sections 3 3 Classical Orbital Elements Dimension and Time a : Semi-major axis e : Eccentricity t p : Time of perigee passage Orientation Ω :Longitude of the Ascending/Descending Node i : Inclination of the Orbital Plane ω: Argument of Perigee 4 4 2 2/12/20 Orientation of an Elliptical Orbit First Point of Aries 5 5 Orbits 102 (2-Body Problem) • e.g., – Sun and Earth or – Earth and Moon or – Earth and Satellite • Circular orbit: radius and velocity are constant • Low Earth orbit: 17,000 mph = 24,000 ft/s = 7.3 km/s • Super-circular velocities – Earth to Moon: 24,550 mph = 36,000 ft/s = 11.1 km/s – Escape: 25,000 mph = 36,600 ft/s = 11.3 km/s • Near escape velocity, small changes have huge influence on apogee 6 6 3 2/12/20 Newton’s 2nd Law § Particle of fixed mass (also called a point mass) acted upon by a force changes velocity with § acceleration proportional to and in direction of force § Inertial reference frame § Ratio of force to acceleration is the mass of the particle: F = m a d dv(t) ⎣⎡mv(t)⎦⎤ = m = ma(t) = F ⎡ ⎤ dt dt vx (t) ⎡ f ⎤ ⎢ ⎥ x ⎡ ⎤ d ⎢ ⎥ fx f ⎢ ⎥ m ⎢ vy (t) ⎥ = ⎢ y ⎥ F = fy = force vector dt -
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. -
Polar Alignment of a Protoplanetary Disc Around an Eccentric Binary
MNRAS 000, 1–18 (2019) Preprint 25 September 2019 Compiled using MNRAS LATEX style file v3.0 Polar alignment of a protoplanetary disc around an eccentric binary III: Effect of disc mass Rebecca G. Martin1 and Stephen H. Lubow2 1Department of Physics and Astronomy, University of Nevada, Las Vegas, 4505 South Maryland Parkway, Las Vegas, NV 89154, USA 2Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA ABSTRACT Martin & Lubow (2017) found that an initially sufficiently misaligned low mass protoplan- etary disc around an eccentric binary undergoes damped nodal oscillations of tilt angle and longitude of ascending node. Dissipation causes evolution towards a stationary state of polar alignment in which the disc lies perpendicular to the binary orbital plane with angular mo- mentum aligned to the eccentricity vector of the binary.We use hydrodynamicsimulations and analytic methods to investigate how the mass of the disc affects this process. The simulations suggest that a disc with nonzero mass settles into a stationary state in the frame of the binary, the generalised polar state, at somewhat lower levels of misalignment with respect to the bi- nary orbital plane, in agreement with the analytic model. Provided that discs settle into this generalised polar state, the observational determination of the misalignment angle and binary properties can be used to determine the mass of a circumbinary disc. We apply this constraint to the circumbinary disc in HD 98800. We obtain analytic criteria for polar alignment of a circumbinary ring with mass that approximately agree with the simulation results. Very broad misaligned discs undergo breaking, but the inner regions at least may still evolve to a polar state. -
Negating the Yearly Eccentricity Magnitude Variation of Super-Synchronous Disposal Orbits Due to Solar Radiation Pressure
Negating the Yearly Eccentricity Magnitude Variation of Super-synchronous Disposal Orbits due to Solar Radiation Pressure by S. L. Jones B.S., The University of Texas at Austin, 2011 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Aerospace Engineering Sciences 2013 This thesis entitled: Negating the Yearly Eccentricity Magnitude Variation of Super-synchronous Disposal Orbits due to Solar Radiation Pressure written by S. L. Jones has been approved for the Department of Aerospace Engineering Sciences Hanspeter Schaub Dr. Penina Axelrad Dr. Jeffrey Parker Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. Jones, S. L. (M.S., Aerospace Engineering Sciences) Negating the Yearly Eccentricity Magnitude Variation of Super-synchronous Disposal Orbits due to Solar Radiation Pressure Thesis directed by Dr. Hanspeter Schaub As a satellite orbits the Earth, solar radiation pressure slowly warps its orbit by accelerating the satellite on one side of the orbit and decelerating it on the other. Since the Earth is rotating around the Sun, this perturbation cancels after one year for geostationary satellites. However, during the year the eccentricity, and therefore the radius of periapsis, changes. Since disposed geo- stationary satellites are not controlled, traditionally they must be reorbited higher than otherwise to compensate for this change, using more fuel. This issue has increased interest in specialized or- bits that naturally negate this eccentricity perturbation. -
Generalisations of the Laplace-Runge-Lenz Vector
Journal of Nonlinear Mathematical Physics Volume 10, Number 3 (2003), 340–423 Review Article Generalisations of the Laplace–Runge–Lenz Vector 1 2 3 1 4 P G L LEACH † † † and G P FLESSAS † † 1 † GEODYSYC, School of Sciences, University of the Aegean, Karlovassi 83 200, Greece 2 † Department of Mathematics, University of the Aegean, Karlovassi 83 200, Greece 3 † Permanent address: School of Mathematical and Statistical Sciences, University of Natal, Durban 4041, Republic of South Africa 4 † Department of Information and Communication Systems Engineering, University of the Aegean, Karlovassi 83 200, Greece Received October 17, 2002; Revised January 22, 2003; Accepted January 27, 2003 Abstract The characteristic feature of the Kepler Problem is the existence of the so-called Laplace–Runge–Lenz vector which enables a very simple discussion of the properties of the orbit for the problem. It is found that there are many classes of problems, some closely related to the Kepler Problem and others somewhat remote, which share the possession of a conserved vector which plays a significant rˆole in the analysis of these problems. Contents arXiv:math-ph/0403028v1 15 Mar 2004 1 Introduction 341 2 Motions with conservation of the angular momentum vector 346 2.1 The central force problem r¨ + fr =0 .....................346 2.1.1 The conservation of the angular momentum vector, the Kepler problemandKepler’sthreelaws . .346 2.1.2 The model equation r¨ + fr =0.....................350 2.2 Algebraic properties of the first integrals of the Kepler problem . 355 2.3 Extension of the model equation to nonautonomous systems.........356 Copyright c 2003 by P G L Leach and G P Flessas Generalisations of the Laplace–Runge–Lenz Vector 341 3 Conservation of the direction of angular momentum only 360 3.1 Vector conservation laws for the equation of motion r¨ + grˆ + hθˆ = 0 . -
Kepler's Laws of Planetary Motion
Kepler's laws of planetary motion In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion ofplanets around the Sun. 1. The orbit of a planet is an ellipse with the Sun at one of the twofoci . 2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.[1] 3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Most planetary orbits are nearly circular, and careful observation and calculation are required in order to establish that they are not perfectly circular. Calculations of the orbit of Mars[2] indicated an elliptical orbit. From this, Johannes Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. Kepler's work (published between 1609 and 1619) improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets' speeds varied, and using elliptical orbits rather than circular orbits withepicycles .[3] Figure 1: Illustration of Kepler's three laws with two planetary orbits. Isaac Newton showed in 1687 that relationships like Kepler's would apply in the 1. The orbits are ellipses, with focal Solar System to a good approximation, as a consequence of his own laws of motion points F1 and F2 for the first planet and law of universal gravitation. and F1 and F3 for the second planet. The Sun is placed in focal pointF 1. 2. The two shaded sectors A1 and A2 Contents have the same surface area and the time for planet 1 to cover segmentA 1 Comparison to Copernicus is equal to the time to cover segment A .