It has been seen in this chapter that special sets of formulae exist for elliptic, parabolic and hyperbolic motion as well as for the three corresponding cases of rectilinear motion. Even in elliptic motion itself
a number of the formulae break down when the eccentricity approaches zero (i.e. when the orbit tends to a circle). In section 4.12 for example, in deriving the orbital elements from a given position and ve- locity, the equation
cannot be used in the circular case to obtain f from a knowledge of r, h andµ. In the circular case e = 0 and there is no perihelion or time of perihelion passage. Even if e is slightly greater than zero, the use of orthodox elliptic formulae would lead to very inaccurate determination of e, w andτ.
In a different context, the same problem exists when the inclination tends to zero; in that case the longitude of the ascending node Ω becomes indeterminate and other formulae must be used to over- come this problem (seesection 4.11, alsosection 8.5).
Various attempts have been made to provide sets of universal or unified formulae that can be used with all kinds of two-body conic-section orbital motion, the distinction between the universal and unified sets being that the former can be applied even if e tends to zero whereas the latter cannot. It is not within the scope of this work to describe these attempts. The student should refer to Herrick (1971, 1972) for a full discussion of universal and unified variables and parameters.
Problems
Take the necessary data from the appendices.
4.1 From equations (4.17) and (4.19) derive the equation
4.2 Halley’s comet moves in an elliptical orbit of eccentricity 0·9673. Compare its velocities, both linear and angular, at perihelion and aphelion.
4.3 Obtain the equation of the centre from the series (4.63) and (4.64), correct to O(e3).
4.4 Find the perihelion distance of that comet which, moving in a parabolic orbit in the plane of the ecliptic, remains the longest time within the Earth’s orbit (assumed circular).
4.5 Prove that the mean anomaly M and the true anomaly f in elliptic motion are related by the equation
Hence deduce that, to O(e2)
4.6 A space vehicle is moving in an elliptical orbit of period T under the attraction of the Sun, mass M. The motors are fired momentarily so that its orbital speed V is suddenly increased by the increment ∆V. Show that the resulting change ∆T in period is given by
4.7 A minor planet is moving in an orbit of eccentricity 0·21634 and period 4·3856 years. Calculate the eccentric anom- aly 1·2841 years after perihelion passage, correct to 1 of arc.
4.8 A rocket leaves the Earth’s atmosphere just before burn-out (thrust terminated), which occurs at a height of 640 km. At this instant its geocentric velocity is 10·4 km s− 1. In what direction must it be travelling to achieve maximum distance from the Earth’s centre? Calculate this distance. If the direction of travel of the rocket at burn-out has made an angle of 88° with the geocentric radius vector of the rocket, calculate the period of the rocket’s orbit.
4.9 When first injected into orbit, artificial Earth satellite Sputnik 16 had a semimajor axis of 1·0478 Earth radii and a period of 90·54 minutes. Calculate the mass of the Earth in units of the Sun’s mass.
4.10 On January 10·0 1963, the heliocentric ecliptic rectangular coordinates of position and velocity of an interplanetary probe were x = 0·68, y = 0·52, z = 0·18 and = −2·2, = 28·1, = 2·6 respectively; the distance being measured in units of the Earth’s semimajor axis, the velocity in km s-1. Find the elements of the Earth’s orbit.
Bibliography
Astrand J J 1890 Huelftasein zur Leichten und Genauen Aufloesung des Keplerischen Problems (Auxiliary Tables for Simple
and Accurate Solution of Kepler’s Problems) (Leipzig: Engelmann)†
Bauschinger J 1901 Tafeln zur Theoretischen Astronomie (Tables on Theoretical Astronomy) (Leipzig: Engelmann)† Cayley A 1861 Mem. R. Astron. Soc. 29 191†
Emslie A G and Walker I W 1979 Cel. Mech. 19 147
Herrick S 1953 Tables for Rocket and Comet Orbits AMS 20 (Washington; National Bureau of Standards) ——— 1971, 1972 Astrodynamics vols 1 and 2 (London: Van Nostrand)
Moran P E 1973 Cel. Mech. 7 122
Moran P E, Roy A E and Black W 1973 Cel. Mech. 8 405
Moulton F R 1914 An Introduction to Celestial Mechanics (New York: Macmillan) Roy A E and Moran P E 1973 Cel. Mech. 7 236
Roy A E, Moran P E and Black W 1972 Cel. Mech. 6 468
Schlesinger F and Udick S 1912 Tables for the True Anomaly in Elliptic Orbits 2 No.17 (Publications of the Allegheny Observatory)†
Sconzo P, Le Shak A R and Tobey R 1965 Astron. J. 70 269
Smart W M and Green R M 1977 Textbook on Spherical Astronomy (London: Cambridge University Press) Steffensen J F 1956 K Danske Vidensk. Selsk. Mat.–Fys. Meddr 30 number 18
——— 1957 K. Danske Vidensk. Selsk. Mat.–Fys. Meddr 31 number 3
Stracke G 1928 Tafeln der Elliptischen Koordinaten C = (r/a) cosν und S = (r/a) sinν fuer Exzentrizitaetswinkel von
0° bis 25° (Tables of the Elliptical Coordinates C = (r/a) cosv and S = (r/a) sinν for Eccentricity Angles from
0° to 25°) (Berlin: Veroeffentlichen des Astronomisches Recheninstituts)† Watson J C 1892 Theoretical Astronomy (Philadelphia: Lippincott)
†If the student has access to a library containing these references, it is instructive to look at them and realize that, how- ever helpful they may have been, the labour expended in using them without modern computer technology must have been prodigious.
The Many-Body Problem
5.1 Introduction
The many-body problem was first formulated precisely by Newton. In its form where the objects in- volved are point masses it may be stated as follows: Given at any time the positions and velocities of three or more massive particles moving under their mutual gravitational forces, the masses also being known, calculate their positions and velocities for any other time.
The problem is more complicated when the bodies’ shapes and internal constitutions have to be taken into account as in the Earth−Moon−Sun problem. The point-mass many-body problem has in- spired (and frustrated!) many eminent astronomers and mathematicians in the last three centuries. It is perhaps not obvious that even the three-body problem is of a much higher degree of complexity than the two-body problem. If we consider, however, that each body is subject to a complicated variable grav- itational field due to its attraction by the other two such that close encounters with either may be brought about, the result of each near-collision being an entirely new type of orbit, we see that it would require a general formula of unimaginable complexity to describe all the consequences of all such encounters. In point of fact, several general and useful statements may be made concerning the many-body problem, such statements being embodied in the ten known integrals of the motion. These integrals were known to Euler; since then no further integrals have been discovered or are likely to be. In addi- tion, particular solutions of the three-body problem were found by Lagrange which are of interest in astrodynamics as well as in astronomy. These solutions exist when certain relationships hold among the initial conditions.
Further progress has been mainly in studying special problems where approximations of various kinds may be utilized. For example, in the circular-restricted three-body problem, two massive parti- cles move in undisturbed circular orbits about their common centre of mass while they attract a parti- cle of mass so small that it cannot appreciably affect their circular orbits. It is possible to draw certain conclusions about the resulting orbit of the particle of infinitesimal mass and to establish the existence of families of periodic orbits of this test particle. Many of Poincar ’s epoch-making researches were devoted to this problem; one of immediate interest when we consider that the Earth, the Moon and a space vehicle in Earth−Moon space constitute an approximate example of this three-body case.
It has also been seen that the planets move in almost perfectly elliptical orbits about the Sun, since the mutual attraction between the planets is so much smaller than the Sun’s attraction upon them. This two-body approximation has been the starting point in many attempts to obtain theories of the planets’ motions. In the two-body solution (termed the reference orbit) the elements are constant; if they are now supposed to vary because of the mutual gravitational attractions of the planets, their differential equa- tions may be set up and solved. The resulting expressions for the elements (in general long sums of sines, cosines and secular terms) can be used to obtain a more accurate approximation still. In practice this method is rapidly convergent though laborious, it being only rarely necessary to go beyond the third approximation. Such analytical expressions, valid for a given period of time, are called general
perturbations. They enable some deductions to be made regarding the past and future states of the plan-
etary system though it must be emphasized that no results valid for an arbitrarily long time may be ob- tained in this way. The method of general perturbations has also been applied to satellite systems, to asteroids disturbed by Jupiter, and to the orbits of artificial satellites. It is in fact a powerful tool in as- trodynamics since the analytical expressions clearly exhibit the various forces at work (for example, the oblateness effect of the Earth on a satellite).
A different approach to the many-body problem is that of using special perturbations, a tool which most workers in celestial mechanics before the days of high-speed computers shrank away from, since it involved the step-by-step numerical integration of the differential equations of motion from the ini- tial epoch to the epoch at which the bodies’positions were desired. Its great advantage, however, is that it is applicable to any orbit involving any number of bodies, and nowadays special perturbations are ap- plied to all sorts of astrodynamical problems, especially since many of these problems fall into regions in which special perturbation theories are absent. One such case is that of a lunar circumnavigation, where the orbit of the vehicle in the Earth−Moon field can be adequately treated only by special per- turbations. The main disadvantage of this method is that it rarely leads to any general formulae; in ad- dition, though they may be of no interest to the worker, the body’s positions at all intermediate steps must be computed in order to arrive at the final configuration.
Perturbations may also be divided into two further classes; periodic and secular. Any disturbance of the reference orbit that is repeated with a given period of revolution is termed a periodic perturba- tion and is usually the result of recurrent similar configurations of the bodies involved. Since these are unlikely to occur exactly, such a periodic perturbation (a short-period one) is often bound up with cyclic behaviour of a much longer period so that one speaks of a long-period perturbation.
A secular perturbation causes a change proportional to the time; for example, the advance of peri- helion or the retrogression of the ascending node of a planetary orbit. In many cases it is difficult to dis- tinguish between very long-period perturbations and secular perturbations if the time over which observations have been made is short compared with the suspected long period.
Finally, we should note that a distinction should be made in the n-body problem between the few- body and the many-body problem. In the Solar System we are concerned with the few-body problem where orbits have to be calculated precisely and too few bodies are involved to enable statistical or hy- drodynamical approaches to be tried. In a stellar system we have a many-body problem, allowing us to utilize such methods. A description of them is however retained until a later chapter.