PROCEEDING S O F SYMPOSI A IN APPLIE D MATHEMATIC S

Volum e IX

Orbi t Theor y Garret t Birkhof f

an d R. E. Lange r

America n Mathematica l Societ y Providence , Rhod e Islan d

http://dx.doi.org/10.1090/psapm/009

PROCEEDINGS OF THE NINTH SYMPOSIUM IN APPLIED MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

Held at New York University APRIL 4-6, 1957

COSPONSORED BY THE OFFICE OF ORDNANCE RESEARCH

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CONTENTS

INTRODUCTION . . . . . . . . v

Orbit Stability in Particle Accelerators . . . . . . 1 By E. D. COURANT

Motion of Cosmic-ray Particles in Galactic Magnetic Fields . . 10 By STANISLAW OLBERT

Stormer Orbits . . . . . . . . . . 1 9 By W. H. BENNETT

General Theory of Oblateness Perturbations . . . . . 2 9 By PAUL HERGET

Fundamental Problems in Predicting Positions of Artificial Satellites. 36 By F. L. WHIPPLE

Cislunar Orbits . . . . . . . . . . 4 8 By K. A. EHRICKE

Satellite Launching Vehicle Trajectories . . . . . 7 5 By J. W, SIRY

Numerical Determination of Precise Orbits . . . . . 1 4 5 By W. J . ECKERT

Comments on General Theories of Planetary Orbits . . . . 1 5 2 By DIRK BROUWER

Orbits in Birkhoff's Central Field 167 By CARLOS GRAEF-FERNÁNDEZ

INDEX 191

INTRODUCTION

Orbit theory, as applied to celestial mechanics and exterior ballistics, is one of the oldest branches of mechanics. The tendency to associate it with seventeenth, eighteenth and nineteenth century mathematics is therefore readily comprehensible. Despite this, however, this branch of science is neither moribund nor consummate. On the contrary, advances in it in recent decades have been significant. I t was to attract the attention of mathematicians to such advances that this Symposium on Orbit Theory was planned.

The timeliness of a refocusing of attention upon celestial mechanics and ballistics, and therefore of the articles which comprise these Proceedings, hardly needs emphasis. That has been supplied by the recent dramatic launching of man-made satellites and the currently accelerated developments of long-range ballistic missiles.

We hope that this newly enhanced practical significance will engender a response from research mathematicians and scientists whereunder some will be inspired to tackle afresh the unsolved problems, old and new, of orbit theory. Some of these problems are difficult; therefore none the less challenging.*

GARRETT BIRKHOFF

RUDOLPH E. LANGER

* We draw attention to the special satellite issue of the Soviet journal Progress of the Physical Sciences, vol. 63 No. la (September 1957). The articles of this have been issued in translation as The Russian literature of satellites, parts I and II , by the International Physical Index, Inc.

INDEX

Acceleration gradual, due to interaction of

particles, 11 gravitational, 81 vector, 90

Admittance, 3 Aerodynamic

force vector, 104 forces, 103 heating, 79

Affine connection, 168 Alternating-gradient focusing, 4 Altitude, constant, 121 Analogue computer, 20 Angle of attack, 85, 104 Antineutrons, 1 Antiprotons, 1 Aphelion circle, 180 Apogee, second-stage, 123 Area, integrals of, 148 Atmospheric

coordinate system, 91 density, 46, 130 drag, 29, 40 ff., 78

Attack, angle of, 85 Aurora polaris, 20 Axis, longitudinal, 100 Azimuth, initial launching, 100

Bands, instability, 6 Barajas, A., 168 Bevatron, 1, 5 Birkhoff, G. D. , 7

central field, 168 theory of gravitation, 167

Brown, E . W., 150 Burnout

first-stage, 78 second-stage, 79 velocity, 80

Canonical transformation, 7 Capture, relativistic, 182 Cascade rectifier, 1 Center

of mass, 85 of pressure, 85

Centers of periodic oscillation, 52 Central field, Birkhoffs, 168 Christofilos, N . C , 4 Circular photon orbit, 174 Cislunar

orbits, 48 ff. space, 48

Cockcroft, D. , 1 Coefficient of pressure thrust Cpp, 82 Collinear mass points, 54 Computer

analogue, 20 IBM 704, 41

Connection, affine, 168 Conservation laws, 170 Constant

altitude, 121 0-program, 121

Constants of integration, 34, 147, 154 orbital, 145

Constraints, 110 engineering, 111

Coordinate system atmospheric, 91 geographic, 93 inertial, 89

Coordinates, origin of, 114 Cosmic radiation, primary, 10 Cosmic-ray particles, propagation of

through the Milky Way, 11 Cosmic-ray primaries, 19 Cosmotron, 1 Courant, E . D. , 4 Cowell-Cromellin method, 42 Critical Jacobian constants for the Earth-

Moon system, 60 Cunningham, L. E. , 38 Curvature, upward, 122

Density, atmospheric, 46, 130 Displacement, orbit, 5 Disturbing function, 157 Drag, 29, 40 ff., 78, 81

coefficient CD(M), 82 force vector, 103

192 INDEX

Earth-Moon system, 60, 68 Earth's rotational velocity, 87 Einstein's equivalence principle, 167 Elementary particles, 1 Elements, 146 Elliptic motion, 152 Ellipticity of the spheroid, 93 Energy

integral of, 148 satellite of a given, 132

Equatorial radius, 93 Equidistant mass points, 53 Equilibrium orbit, 2 Equivalence principle, Einstein's, 167 Errors, determination of in the system of

star positions, 149 Expansion in harmonic series, 146 Extrapolation, polynomial, 132 Extreme value of a function of n variables,

113

Fermi, Enrico, 11, 15 Firing time, 89 First-stage burnout, 78 Fixed weight, 107 Flight, vacuum ,110 Focusing

alternating-gradient, 4 strong, 4

Force function of Earth's gravitational field, 30

Galactic frame of reference, 12 magnetic fields, 10 ff.

General perturbations, method of, 146 Generalized potential well, 175 Geocentric

latitude <j>c, 92, 96 longitude A, 92

Geographic coordinate system, 93 latitude <f>g, 93, 96

Geophysical Year, International, 36, 75 Gradual acceleration process, due to inter-

action of cosmic particles, 11 Gravitation, BirkhofT's theory of, 167 Gravitational

acceleration, 81 field, force function of Earth's, 30 force, 102 noise, 45

Gravitational potential tensor, 168

Gravity turn, 84

Hamiltonian function, 7, 43 Hansen's method, 30 ff., 161 Harmonic series, expansion in, 146 Heating, aerodynamic, 79 Height, 93 Hill's

boundary curves, 65 equation, 4

Hyperplane, (n—1)-dimensional, 113 Hypersurfaces, (n— 1)-dimensional, 113

IGY satellites, lifetime of, 75 Impulse, specific, 81 Inertial coordinate system, 89 Initial

conditions, 105 launching azimuth, 100 perigee height, 75, (minimum) 126 tilt angle, 84, 112, 127

Instability bands, 6 Integral

of energy, 148 vis viva, 51

Integrals of areas, 148 Integration

determination of constants of, 147 numerical, 146

Interstellar plasma, 12 International Geophysical Year, 36, 75 Isochrones, 138

Jacobian constant, 65 critical, 60

Jacobian integral, 65 Jupiter, orbit of, 147

Latitude geocentric, 92, 96 geographic, 93, 96

Launch point, 97 Launching azimuth, initial, 100 Libration points, 52, 69 Lifetime

orbital, 138 satellite, .122, 130

Lift coefficient CLa(M), 85 force, 84, 103

INDEX 193

Light polarization of from distant stars, 11 rays, bending of, 167

Livingston, M. S., 4 Local vertical, 93 Longitude, geocentric, 92 Longitudinal axis, 100 Lunar tables (E. W. Brown), 150

McMillan, E. M., 2 Mach number, 82 Magnetic fields, motion of with respect to

galactic frame of reference, 12 Mars, 149 Mass, center of, 85 Mass points, collinear, 54 Mass ratio, 80

propellant, 83 propellant-tank, 108

Masses of the planets, determination of, 149

Maximum projection velocity, 107 turning rate, 112

Matrix, symplectic, 6 Mercury, advance of the perihelion of, 167 Meridian plane, 96 Milky Way, propagation of cosmic-ray

particles through, 11 Minimum initial perigee height, 126 Minitrack, 36 Minor planets, 150 Moon, orbit of, 150 MOONWATCH, 36 Moser, J., 7 Multi-stage rocket, 106

NORC, 149 Neptune, orbit of, 147 Nonlinearities, 7 Numerical integration, 146

Oblateness, effect of Earth's, 29 Observations, 147 One-dimensional satellite rocket launching

vehicle trajectories, 80 Optical tracking program, 36 Optimization analyses

orbit criteria for, 122 satellite rocket launching vehicle stage

weight, 106 ff. trajectory shape, 110

Optimum trajectory, 121

Orbit criteria for optimization analyses, 122 displacement, 5 equilibrium, 2 of the moon, 150 of the photon, 172

Orbital constants, 145 lifetime criterion, 138 speed, 75 stability, 1 ff., 187

Orbits about barycenter of finite masses, 70 about libration points, 62, 69 circular, 173 cislunar, 48 ff. of planets, 147, 151 ff. periodic, 55 Stormer, 19 ff.

Origin of coordinates, 114 Oscillation, centers of periodic, 52 Osculating elements, 155

Particle accelerator, 1 Particles

cosmic-ray, 11 elementary, 1 strange, 1

Paths, 168 Perfect trailing trajectory, 84 Performance parameter w, rocket engine

125 Perigee height

initial, 75 minimum initial, 126

Perihelion circle, 180 of Mercury, advance of, 167

Period, propulsion, 80 Periodic orbits, 55

about libration points, 62, 69 Perturbation theory, 39, 155 Perturbations, 29 ff.

general, 43, 146 special, 43, 146

Phase stability, 2 Phasotron, 1, 5 Photograph stations, 39 Photographic tracking program, 37 Photon, 172

orbit of, 172, 174 Planar three-body problem, 48 Plane, meridian, 96

194 INDEX

Planets determination of masses of, 149 minor, 150

Plasma, interstellar, 12 Pluto, orbit of, 147 Point

launch, 97 sub-vehicle, 93 surface, 94

Points, sky, 93 Polaris, aurora, 20 Polarization of light from distant stars, 11 Polynomial extrapolation, 132 Position vector, 90 Potential tensor, gravitational, 168 Potential well, generalized, 175 Pressure, center of, 85 Pressure thrust, coefficient of, 82 Primary cosmic radiation, 10 Probability

distributions, 126 of success, 122, 127

Projection altitude hp, 122 angle, 122, 139 velocity, 76, 107 velocity vector, 88

Propellant mass ratio, 83 usable, 108 weight, 108

Propellant-tank mass ratio, 108 Propulsion period, 80 Proton synchrotrons, 1

Radiation, primary cosmic, 10 Radio tracking program, 36 Rectifier, cascade, .1 Reflection

type A, 15 type B, 15

Relative wind, 99 Relativistic

capture, 182 effects, 167

Relativity, special, 167 Resonances, 5, 7 Restricted

four-body problem, 74 three-body system, 65

Rocket engine performance parameter w, 125

Rotational velocity, Earth's, 87

Satellite launching vehicle, 75 ff. lifetime, 122, 130 projection velocity vector, 88 tracking, 36 ff. trajectories in q-s space, 130

Satellites artificial, 36 ff. IGY, 75

Saturn, orbit of, 147 Second-stage

apogee, 123 burnout, 79

Secular variations, 163 Sigurgeirsson, T., 3 Six-body problem, 147 Sky points, 93 Small divisors, 159 Snyder, H. S., 4 Solar

protons, 19 system, 145, 183 torque, 72

Special perturbations, method of, 146 Specific impulse, 81 Spheroid, equatorial radius of, 93 SSEC, 147 Stability

of orbits about libration points, 69 orbital, 1 ff., 187 phase, 2

Star positions, determination of errors in system of, 149

Strange particles, 1 Störmer, Carl, 19 Störmer orbits, 19 ff. Störmertron, 20 Strong focusing, 4 Sturrock, P. A., 7 Sub-vehicle point, 93 Success, probability of, 127 Sun-Jupiter system, 70 Surface point, 94 Symplectic matrix, 6 Synchrotrons, proton, 1

Tangency, points of, 128, 139 Tank weight, 108 Tensor, gravitational potential, 168 0 -program, 100 Third stage, 79 Three-body problem, planar, 48

INDEX 195

Three-dimensional satellite rocket launching vehicle trajectories, 89

Three-stage rocket, 77, 110 Thrust

direction, 89 force, 80, 105

Tilt angle, initial, 84, 112, 127 Total variable weight, 108 Tracking, artificial satellite, 36 ff. Tracking programs, 36 ff. Trailing trajectory, perfect, 84 Trajectory

optimum, 121 perfect trailing, 84 shape optimization analyses, 110 zero-lift, 83, 110, 121

Trajectories in q—s space, satellite, 130 1-dimensional satellite rocket launching

vehicle, 80 2-dimensional satellite rocket launching

vehicle, 83 3-dimensional satellite rocket launching

vehicle, 89 Transformation, canonical, 7 Trojans, 70 Turning rate, maximum, 112 Two-dimensional satellite rocket launch­

ing vehicle trajectories, 83 Type A reflection, 15 Type B reflection, 15

UNIVAC, 9 Upward curvature, 122

Uranus, orbit of, 147 Usable propellant, 108

Vacuum flight, 110 Vanguard

project, 36 satellite launching vehicle, 75, 110 satellites, 29 three-stage rocket, 77

Variable weight, 108 total, 108

Variation of arbitrary constants, 154 Variations, secular, 163 Veksler, V., 2 Velocity

angle parameter yp, 123 burnout, 80 Earth's rotational, 87 magnitude parameter vp, 122, 123 projection, 76

Velocity vector, 90 satellite projection, 88

Vertical, local, 93 Vis viva integral, 51

Walton, E. T. S., 1 Weight

propellant, 108 tank, 108 total variable, 108

Wind, relative, 99

Zero-lift trajectory, 83, 110, 121 Zero-rate, 121