Astronomy Principles and Practice Fourth Edition.pdf

14.3 General properties of the many-body problem
General perturbation theories 187
If the masses of the planets were vanishingly small compared to the Sun’s mass, then the orbit of any
planet would be unchanging and the six elements would be constant. Indeed, Kepler’s three laws are
the solution to the many-body problem in such a case. But the planetary masses are by no means
negligible and, in the case of comets, near approaches to planets can occur so that, in general, the
problem is much more complicated.
In the past three centuries, it has inspired (and frustrated!) many eminent astronomers and
mathematicians. It is perhaps not obvious that even the three-body problem is of a much higher degree
of complexity than the two-body problem. But if we consider that each body is subject to a complicated
variable gravitational 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 and these were proved quite early on in its history. They were known to Euler (1707–83) but
since then no further overall properties have been discovered or are likely to be.
The statements follow from the only known integrals of the differential equations and refer to the
centre of mass of the system, the total energy of the system and its total angular momentum. Without
saying anything about the trajectories of the individual particles, the following statements can be made:
(a) The centre of mass of the system moves through space with constant velocity, i.e. it moves in a
straight line at a fixed speed.
(b) The total energy of the system (the sum of all the kinetic energies and potential energy) is constant.
Thus, although there is a continual trade-off among the members in kinetic energy and potential
energy, the total energy is unaffected.
(c) The total angular momentum of the system is constant.
In addition to these properties, particular solutions of the three-body problem that exist when
certain relationships hold among the velocities and mutual distances of the particles were found by
Lagrange. He showed that if the three bodies occupy the vertices of an equilateral triangle, their speeds
being equal in magnitude and inclined at the same angle to each mutual radius vector, they will remain
in an equilateral triangle formation, though the triangle will rotate and may change its size. Lagrange
also showed that if the three bodies are placed on a straight line at mutual distances depending upon
the ratios of their masses, they will remain on that line, though it will rotate. Although these equilateral
triangle and collinear solutions of the three-body problem were thought to be of theoretical interest only
at the time of their presentation, it was subsequently discovered that they occur in the Solar System.
Two groups of asteroids, called the Trojans, revolve about the Sun in Jupiter’s orbit, so that their
periods of revolution equal that of Jupiter. In their orbit about the Sun, they oscillate about one or
other of the two points 60 ◦ ahead or behind Jupiter’s heliocentric position. Among Saturn’s moons,
Telesto and Calypso remain 60 ◦ ahead or behind the more massive Tethys while Helene, in Dione’s
orbit, keeps 60 ◦ ahead of Dione.
14.4 General perturbation theories
It has been seen that, because the planets’ mutual attractions are so much smaller than the Sun’s
attraction upon them, the planets’ orbits, to a high degree of approximation, are ellipses about the
Sun. This two-body approximation has been the starting point in many attempts to obtain theories of
the planets’ motions. The two-body orbit of a planet about the Sun is supposed to vary in size, shape
and orientation as if it were a soft plasticine ring moulded by the spectral fingers of the other planets’
gravitational fields.