Orbital Mechanics

5 Key excerpts on "Orbital Mechanics"

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  eBook - ePub

  Manned Spacecraft Design Principles

  • Pasquale M. Sforza, Pasquale M Sforza(Authors)
  • 2015(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Chapter 5

  Orbital Mechanics

  Abstract

  The equations of orbital motion with particular reference to characteristics of earth orbits are developed and the manner of altering those orbits is discussed. The basic ideas of conservation of energy and angular momentum for closed and open orbits are used to illustrate the maintenance of orbits and the achievement of escape from orbit for interplanetary missions. The ground track of orbits, effects of earth’s rotation and precession, determination of longitude, the spacecraft horizon and effects on communication are analyzed. Interplanetary trajectories are discussed and the orbital transfer process for atmospheric entry is presented.

  Keywords

  Earth orbits; orbital transfer; orbit ground track; interplanetary trajectories

  5.1 Space Mission Geometry

  Orbital Mechanics has its foundation in three important developments which took place over the last half of the seventeenth century:

  • Tyco Brahe’s detailed observations of the motion of the planets.

  • Kepler’s laws describing the nature of the orbits of the planets in a manner consistent with Brahe’s observational data. The first law is that the orbits are ellipses with the sun at the focus, the second is that the radius vector between the sun and any planet sweeps out equal areas in equal times, and the third, that the square of the orbital period of any planet is proportional to the cube of its mean distance from the sun.

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  eBook - ePub

  Spacecraft Systems Engineering

  • Peter Fortescue, Graham Swinerd, John Stark, Peter Fortescue, Graham Swinerd, John Stark(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Chapter 4 Celestial Mechanics

  John P. W. Stark1 , Graham G. Swinerd2 and Peter W. Fortescue2

  1 School of Engineering and Material Science Queen Mary, University of London

  2 Aeronautics and Astronautics, Faculty of Engineering and the Environment, University of Southampton

  4.1 Introduction

  The theory of celestial mechanics underlies all the dynamical aspects of the orbital motion of spacecraft. The central feature is the mutual gravitational force of attraction that acts between any two bodies. This was first described by Newton, and together with his laws of motion (see Chapter 3), it provides us with the theoretical framework for celestial mechanics. The orbits that it forecasts will be relative to an Inertial Frame of Reference (IFR) that is fixed with respect to the stars; the consequential motion relative to the ground will also be covered in Chapter 5.

  The simplified case in which the gravitational force acts between two pointlike objects gives a good approximation to orbital motion for most spacecraft situations. It may easily be shown that if a body has a uniform mass distribution within a spherical surface, then outside it the gravitational force from the body does indeed appear to emanate from a pointlike source. This so-called two-body problem has a solution—a Keplerian orbit.

  Kepler, whose major works were published during the first 20 years of the seventeenth century, consolidated the observations of planetary motion into three simple laws, which are illustrated in Figure 4.1 . These are

  1. the orbit of each planet is an ellipse with the Sun occupying one focus; 2. the line joining the Sun to a planet sweeps out equal areas in equal intervals of time; 3. a planet's orbital period is proportional to the mean distance between Sun and the planet, raised to the power 3/2.

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  eBook - ePub

  Space Flight Dynamics

  • Craig A. Kluever, Peter Belobaba, Jonathan Cooper, Allan Seabridge(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  2 Two‐Body Orbital Mechanics

  2.1 Introduction

  In this chapter, we will develop the fundamental relationships that govern the orbital motion of a satellite relative to a gravitational body. These relationships will be derived from principles that should be already familiar to a reader who has completed a course in university physics or particle dynamics. It should be no surprise that we will use Newton’s laws to develop the basic differential equation relating the satellite’s acceleration to the attracting gravitational force from a celestial body. We will obtain analytical (or closed‐form) solutions through the conservation of energy and angular momentum, which lead to “constants of motion.” By the end of the chapter the reader should be able to analyze a satellite’s orbital motion by considering characteristics such as energy and angular momentum and the associated geometric dimensions that define the size and shape of its orbital path. Understanding the concepts presented in this chapter is paramount to successfully grasping the subsequent chapter topics in orbit determination, orbital maneuvers, and interplanetary trajectories.

  2.2 Two‐Body Problem

  At any given instant, the gravitational forces from celestial bodies such as the Earth, sun, moon, and the planets simultaneously influence the motion of a space vehicle. The magnitude of the gravitational force of any celestial body acting on a satellite with mass m can be computed using Newton’s law of universal gravitation

  (2.1)

  where M is the mass of the celestial body (Earth, sun, moon, etc.), G is the universal constant of gravitation, and r is the separation distance between the gravitational body and the satellite. It is not difficult to see that Eq. (2.1) is an inverse‐square gravity law. The gravitational force acts along the line connecting the centers of the two masses. Figure 2.1 illustrates Newton’s gravitational law with a two‐body system comprising the Earth and a satellite. The Earth attracts the satellite with gravitational force vector F21 and the satellite attracts Earth with force F12 . The reader should note that Eq. (2.1)

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  eBook - PDF

  Satellite Geodesy

  Foundations, Methods, and Applications

  • Günter Seeber(Author)
  • 2008(Publication Date)
  • De Gruyter

    (Publisher)

  motions of three and more celestial bodies under the influence of their mutual gravitation. These problems have no general solution. Orbit perturbations [3.2], orbit determination [3.3] and ephemeris computation are also treated in celestial mechanics. Orbit determination refers to orbital parameters derived from observations [3.3.1]. Ephemeris computation refers to geocentric or topocentric positions of celestial bodies or artificial satellites that are derived from orbital elements (e.g. [3.3.3], [7.1.5]). Modern celestial mechanics has its origin in the year 1687 with the publication of Isaac Newton’s Principia (Philosophiae naturalis principia mathematica). Herein the law of gravitation and the laws of motion are described for the first time. In the subsequent 300 years there were no major revolutions in celestial mechanics. Only the launch of the first artificial satellite and the development of powerful computers gave an impetus for new ideas. Besides the classical observation of directions, the measurements of ranges and range-rates can now be made. Also, the influences of Earth’s anomalous gravitational field and non-gravitational forces have to be modeled in addition to the classical perturbations caused by the Sun, the Moon and the planets. Through the development of high speed computers large amounts of data can be processed, and numerical integration methods can be used. Comprehensive textbooks are available for a detailed study of problems and meth-ods in celestial mechanics, such as Stumpff (1959/1965/1974), Brouwer, Clemence (1961), Kovalevsky (1971); Kovalevsky et al. (1989), Schneider (1981, 1993), Taff (1985), Vinti (1998). Easily readable introductions with special regard to satellite and rocket orbits are Escobal (1965), Bate et al. (1971), Roy (1978), Chobotov (1991), Logsdon (1998), and Montenbruck, Gill (2000). Suitable references with particular emphasis on GPS orbits are Rothacher (1992),Yunck (1996), and Beutler et al.

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  eBook - PDF

  Statistical Orbit Determination

  • Bob Schutz, Byron Tapley, George H. Born(Authors)
  • 2004(Publication Date)
  • Academic Press

    (Publisher)

  Chapter 1 Orbit Determination Concepts 1.1 INTRODUCTION The treatment presented here will cover the fundamentals of satellite orbit determination and its evolution over the past four decades. By satellite orbit de-termination we mean the process by which we obtain knowledge of the satellite’s motion relative to the center of mass of the Earth in a specified coordinate system. Orbit determination for celestial bodies has been a general concern of astronomers and mathematicians since the beginning of civilization and indeed has attracted some of the best analytical minds to develop the basis for much of the fundamen-tal mathematics in use today. The classical orbit determination problem is characterized by the assumption that the bodies move under the influence of a central (or point mass) force. In this treatise, we focus on the problem of determining the orbits of noncelestial satellites. That is, we focus on the orbits of objects placed into orbit by humans. These objects differ from most natural objects in that, due to their size, mass, and orbit charateristics, the nongravitational forces are of significant importance. Further, most satellites orbit near to the surface and for objects close to a central body, the gravitational forces depart from a central force in a significant way. By the state of a dynamical system, we mean the set of parameters required to predict the future motion of the system. For the satellite orbit determination problem the minimal set of parameters will be the position and velocity vectors at some given epoch. In subsequent discussions, this minimal set will be expanded to include dynamic and measurement model parameters, which may be needed to improve the prediction accuracy. This general state vector at a time, t , will be denoted as X ( t ) . The general orbit determination problem can then be posed as follows. 1

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