Astronomy Principles and Practice Fourth Edition.pdf

Chapter 14
Celestial mechanics: the many-body problem
14.1 Introduction
Newton formulated and solved the two-body gravitational problem where two massive particles move
in orbits under their mutual gravitational attraction. He applied the formulas he obtained to such
problems as the orbit of a planet moving round the Sun or the orbit of the Moon about the Earth. His
success was due to two fortunate circumstances. One is the distribution of masses in the Solar System.
All planetary masses are small with respect to the Sun’s mass just as the masses of satellites are small
compared to those of their primaries. Because of this, the mutual gravitational attractions of planets
are small in relation to the Sun’s force of attraction on each planet. The second is that the diameters
of planets are small compared with their distances from each other and from the Sun. To a very good
approximation, therefore, the nature of the orbit of a planet about the Sun is a two-body problem, the
bodies being a planet and the Sun.
Newton realized, of course, that the other planets’ attractions had to be taken into account in
describing precisely what orbit a planet would travel in. The problem was, in fact, a many-body
problem and Newton was the first to formulate it. In its form where the objects involved are pointmasses
(that is they have mass but no volume—a physical impossibility but a remarkably useful
concept!), the many-body problem 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, provide a means by which their positions and velocities can be calculated for any
time, past or future.
The problem is more complicated when bodies cannot be taken to be point-masses so that their
shapes and internal constitutions have to be taken into account as in the Earth–Moon–Sun problem.
In order to discuss the methods that have been invented in celestial mechanics to deal with his
problem, we consider first what is meant by the elements of an orbit.
14.2 The elements of an orbit
In figure 14.1, the orbit of a planet P about the Sun S is shown. The orbit is an ellipse, for at the
moment we consider the Sun to be the only mass attracting the planet.
Let S be the direction of the First Point of Aries as seen from the Sun, and let the fixed reference
plane be the plane of the ecliptic.
Then the planet’s orbital plane will intersect the plane of the ecliptic in some line NN 1 , called the
line of nodes. If motion in the orbit is described in the direction shown by the arrowhead, N is called
the ascending node; N 1 is the descending node.
The angle SN is the longitude of the ascending node, usually denoted by the symbol .
The angle between the orbital plane and the plane of the ecliptic is called the inclination, i.
185