Orbital Motion © IOP Publishing Ltd 2005 © IOP Publishing Ltd 2005 © IOP Publishing Ltd 2005 © IOP Publishing Ltd 2005 Contents Preface to First Edition xv Preface to Fourth Edition xvii 1 The Restless Universe 1 1.1 Introduction.….….….….….….….….……….….….….….….….….….….….….….…1 1.2 The Solar System.….….….….….….….….…...................….….….….….….…...........1 1.2.1 Kepler’s laws.….….….….….….….… ….….….….….….…......................4 1.2.2 Bode’s law.….….….….….….….….….….….….….….…..........................4 1.2.3 Commensurabilities in mean motion.….….….….…….….….….….….…..5 1.2.4 Comets, the Edgeworth-Kuiper Belt and meteors.….….…….….….….…..7 1.2.5 Conclusions.….….….….….….….…….….….….….….….........................9 1.3 Stellar Motions.….….….….….….….….….….….….….….….….….….….….….…...9 1.3.1 Binary systems.….….….….….….…… ….….….….….….…..................11 1.3.2 Triple and higher systems of stars.….….….….….….….….….….….…...11 1.3.3 Globular clusters.….….….….….….…… ….….….….….….…...............13 1.3.4 Galactic or open clusters.….….….….….…..….….….….….….…...........14 1.4 Clusters of Galaxies.….….….….….….….…..….….….….….….….….….….….…..14 1.5 Conclusion.….….….….….….….….……….….….….….….….….….….….….…....15 Bibliography.….….….….….….….….…..….….….….….….….….….….….….…....15 2 Coordinate and Time-Keeping Systems 16 2.1 Introduction.….….….….….….….….…… ….….….….….….….….….….................16 2.2 Position on the Earth’s Surface.….….….….….….… ….….….….….….…................16 2.3 The Horizontal System.….….….….….….….….….….….….….….….......................18 2.4 The Equatorial System.….….….….….….….….….….….….….….…........................20 2.5 The Ecliptic System.….….….….….….….…..….….….….….….…...........................21 2.6 Elements of the Orbit in Space.….….….….….….…….….….….….….…................ 22 2.7 Rectangular Coordinate Systems.….….….….….…… ….….….….….….…..............24 2.8 Orbital Plane Coordinate Systems.….….….….….……….….….….….….….............24 2.9 Transformation of Systems.….….….….….….….….….….….….….….….….….…. 25 2.9.1 The fundamental formulae of spherical trigonometry.….….….….….…...25 2.9.2 Examples in the transformation of systems.….….….…….….….….….…28 2.10 Galactic Coordinate System.….….….….….….….….….….….….…..........................35 2.11 Time Measurement.….….….….….…… ….….….….….….….….….….….….….….36 2.11.1 Sidereal time.….….….….….….…… ….….….….….….…......................36 2.11.2 Mean solar time.….….….….….….….….….….….….….….....................39 vii © IOP Publishing Ltd 2005 viii 2.11.3 The Julian date.….….….….….….…… ….….….….….….…...................41 2.11.4 Ephemeris Time.….….….….….….……….….….….….….…..................41 Problems.….….….….….….….….….… ….….….….….….….….….….….….….…42 Bibliography.….….….….….….….….…..….….….….….….….….….….….….…...43 3 The Reduction of Observational Data 44 3.1 Introduction.….….….….….….….….…… ….….….….….….…...............................44 3.2 Observational Techniques.….….….….….….…… ….….….….….….…...................44 3.3 Refraction.….….….….….….….….….…….….….….….….…................................. 47 3.4 Precession and Nutation.….….….….….….….… ….….….….….….….....................48 3.5 Aberration.….….….….….….….….….… ….….….….….….….….….….….….…..53 3.6 Proper Motion.….….….….….….….….….….….….….….….…...............................55 3.7 Stellar Parallax.….….….….….….….….….….….….….….….…..............................55 3.8 Geocentric Parallax.….….….….….….….…… ….….….….….….…........................56 3.9 Review of Procedures.….….….….….….….….….….….….….….….........................60 Problems.….….….….….….….….….… ….….….….….….….….….….…...............61 Bibliography.….….….….….….….….…..….….….…….….….….….….…..............61 4 The Two-Body Problem 62 4.1 Introduction.….….….….….….….….……….….….….….….…............................... 62 4.2 Newton’s Laws of Motion.….….….….….….……….….….….….….…................... 62 4.3 Newton’s Law of Gravitation.….….….….….….….….….….….….….…..................63 4.4 The Solution to the Two-Body Problem.….….….….…… ….….….….….….…........64 4.5 The Elliptic Orbit.….….….….….….….….… ….….….….….….…...........................67 4.5.1 Measurement of a planet’s mass.….….….….…..….….….….….….….....69 4.5.2 Velocity in an elliptic orbit.….….….….….….….….….….….….….........70 4.5.3 The angle between velocity and radius vectors.….….…..….….….….…..73 4.5.4 The mean, eccentric and true anomalies.….….….….….….….….….…....74 4.5.5 The solution of Kepler’s equation.….….….….….….….….….….….…....76 4.5.6 The equation of the centre.….….….….….….….….….….….….…...........78 4.5.7 Position of a body in an elliptic orbit.….….….…..….….….….….….…...78 4.6 The Parabolic Orbit.….….….….….….….….….….….….….….….............................80 4.7 The Hyperbolic Orbit.….….….….….….….….….….….….….….…..........................83 4.7.1 Velocity in a hyperbolic orbit.….….….….……….….….….….….…........84 4.7.2 Position in the hyperbolic orbit.….….….….…..….….….….….….….......85 4.8 The Rectilinear Orbit.….….….….….….….…..….….….….….….….........................87 4.9 Barycentric Orbits.….….….….….….….……….….….….….….…........................... 89 4.10 Classification of Orbits with Respect to the Energy Constant.….……….….….….… 90 4.11 The Orbit in Space.….….….….….….….……….….….….….….….......................... 91 4.12 The f and g Series.….….….….….….….…..….….….….….….…..............................95 4.13 The Use of Recurrence Relations.….….….….….……….….….….….….…............. 97 4.14 Universal Variables.….….….….….….….……….….….….….….…........................ 98 Problems.….….….….….….….….….… ….….….….….….….….….….….….….…99 Bibliography.….….….….….….….….…..….….….….….….…...............................100 5 The Many-Body Problem 101 5.1 Introduction.…….….….….….….….….….….….….….….….….….….….….….....101 © IOP Publishing Ltd 2005 ix 5.2 The Equations of Motion in the Many-Body Problem.….….……….….….….….…102 5.3 The Ten Known Integrals and Their Meanings.….….….……….….….….….….….103 5.4 The Force Function.….….….….….….….…..….….….….….….…..........................105 5.5 The Virial Theorem.….….….….….….….…..….….….….….….…..........................108 5.6 Sundman’s Inequality.….….….….….….….…..….….….….….….….......................108 5.7 The Mirror Theorem.….….….….….….….…..….….….….….….…........................111 5.8 Reassessment of the Many-Body Problem.….….….….…..….….….….….….….....112 5.9 Lagrange’s Solutions of the Three-Body Problem.….….….…..….….….….….…....112 5.10 General Remarks on the Lagrange Solutions.….….….….….….….….….….….…...117 5.11 The Circular Restricted Three-Body Problem.….….….….… ….….….….….….…..118 5.11.1 Jacobi’s integral.….….….….….….……….….….….….….….................118 5.11.2 Tisserand’s criterion.….….….….….….….….….….….….….….............121 5.11.3 Surfaces of zero velocity.….….….….….…..….….….….….….…..........122 5.11.4 The stability of the libration points.….….….….…….….….….….….…126 5.11.5 Periodic orbits.….….….….….….….…….….….….….….….................130 5.11.6 The search for symmetric periodic orbits.….….….….….….….….….…132 5.11.7 Examples of some families of periodic orbits.….….…..….….….….…..134 5.11.8 Stability of periodic orbits.….….….….….….….….….….….….…........136 5.11.9 The surface of section.….….….….….……….….….….….….…............138 5.11.10 The stability matrix.….….….….….….….….….….….….….…..............139 5.12 The General Three-Body Problem.….….….….….…..….….….….….….….............140 5.12.1 The case C < 0.….….….….….….……….…..….….….….…..................141 5.12.2 The case for C = 0.….….….….….….….….….….….….….…................142 5.12.3 Jacobian coordinates.….….….….….….…….….….….….….….............143 5.13 Jacobian Coordinates for the Many-body Problem.….….….….….….….….….…...144 5.13.1 The equations of motion of the simple n-body HDS.….……….….….….145 5.13.2 The equations of motion of the general n-body HDS.….…......................147 5.13.3 An unambiguous nomenclature for a general HDS.….….…....................151 5.14 The Hierarchical Three-body Stability Criterion.….….….……….….….….….…....151 Problems.….….….….….….….….….…….….….….….….…...................................152 Bibliography.….….….….….….….….…..….….….….….….…................................152 6. The Caledonian Symmetric N-body Problem 154 6.1 Introduction.….….….….….….….….……….….….….….….…...............................154 6.2 The Equations of Motions.….….….….….….……….….….….….….…...................154 6.3 Sundman’s Inequality.….….….….….….….…..….….….….….….….......................157 6.4 Boundaries of Real and Imaginary Motion.….….….….…..….….….….….….….....162 6.5 The Caledonian Symmetric Model for n = 1.….….….….…….….….….….….…....164 6.6 The Caledonian Symmetric Model for n = 2.….….….….…….….….….….….…....168 6.6.1 The Szebehely Ladder and Szebehely’s Constant.….….…......................173 6.6.2 Regions of real motion in the ρ1, ρ2, ρ12 space.….….….….….….….…174 6.6.3 Climbing the rungs of Szebehely’s Ladder.….….….…….….….….…....177 6.6.4 The case when E0 < 0.….….….….….…….….….….….….…................182 6.6.5 Unequal masses µ1 ≠ µ2 in the n = 2 case.….….……….….….….….….182 6.6.6 Szebehely’s Constant.….….….….….….…….….….….….….…............183 6.6.7 Loks and Sergysels study of the general four-body problem.…................184 6.7 The Caledonian Symmetric Model for n = 3.….….….….…......................................185 © IOP Publishing Ltd 2005 x 6.8 The Caledonian Symmetric TV-Body Model for odd TV.….….……….….….….….191 Bibliography.….….….….….….….….…..….….….….….….…................................193 7. General Perturbations 194 7.1 The Nature of the Problem.….….….….….….…..….….….….….….…....................194 7.2 The Equations of Relative Motion.….….….….….…..….….….….….….….............195