Focus on Comet Lexell (second part
continued from TS
number 2)
Le Verrier's
computations and the concept of Chaos
by Giovanni Valsecchi*  Copyright Tumbling
Stone 2001
The work of Johann Anders
Lexell on the periodic comet that bears his name was
brilliant and innovative, but did not put an end to the
investigations on the motion of that extraordinary
object. About seventy years later, in the early forties
of the XIX century, Urbain Le Verrier reexamined the
subject from the beginning, asking himself a question
that previous astronomers had essentially avoided.
The question was about the reliability of the orbit
computed for the comet: did the available observations
determine uniquely the orbital elements? (click here to know more about
the orbital elements) At that time the work of Gauss on the recovery of the first asteroid, Ceres (see T.S. number 1: " Ceres: the missing planet?"), was a `fait accompli' and, as a consequence, the quantitative treatment of observational errors in the determination of orbits had become a well posed mathematical problem. While working on comet Lexell, Le Verrier was also working on the determination of the orbit of the planet perturbing Uranus. When that planet was discovered (and named Neptune), at the end of the decade, Le Verrier became a celebrity. 

Let us go back to Lexell's comet.
Le Verrier critically examined again the available observations,
and identified a subset of them that he trusted; he then tried to
compute an accurate orbit for the comet, taking also into account
the gravitational action of the Earth. After many
computations, described in detail in his papers of 1844, 1848 and
1857, he realized that it was not possible to determine a unique
`best' orbit for the comet, since the constraints given by the
observations were insufficient.
However, the situation was not totally
desperate: he was able to express the six orbital elements of
comet Lexell as functions of a single unknown parameter, that he
called mu; he also showed that the observations could be used to
find the permissible range of variation of mu, since outside a
certain range, the path of the comet on the sky would have been
measurably different from the observed one. Said in other words,
Le Verrier had established the direct ancestor of the now
fashionable concept of confidence
region (dict.) for the orbit of a nearEarth object, and in
fact the spirit of his researches is surprisingly modern.
Also Le Verrier, like Lexell a few decades
before, did not content himself with his first important finding,
and went on to explore its consequences. Lexell had shown that,
because of the orbital period of the comet  it was resonant with
that of Jupiter  there had been a first close encounter with the
giant planet before the discovery, in 1767, and that another,
deeper one, was to take place in 1779. Le Verrier computed the
effects of these encounters on the orbit for various values of
mu, so as to obtain a global view of all the possible outcomes 
much like the systematic computations of virtual asteroids (dict.)
carried out nowadays.
He established, among other things, that
the comet could approach Jupiter extremely closely in 1779, as
close as less than three and a half radii of the planet from its
centre; nevertheless, the comet could not become a satellite of
Jupiter, not even temporarily, for any allowed value of mu. The
range of post1779 orbits included even the possibility, for the
comet, to leave the solar system on a hyperbolic orbit.
The reason for this wide range of possible
outcomes was the extreme sensitivity of the subsequent evolution
to the precise value adopted for mu; this sensitivity is a
crucial part of the modern concept of chaos, and in fact Le
Verrier's computations probably represent the first instance of
this concept in scientific literature.
*Giovanni Valsecchi  Istituto di Astrofisica Spaziale  CNR
Who is Le Verrier? Urbain Jean Joseph Le Verrier (18111877) was born at SaintLô, France and at the age of 26, as a teacher of astronomy at the Ecole Polytechnic of Paris, he began an intensive study of the motion of Mercury.
Another important discover made by Le Verrier is a discrepancy in the motion in the perihelion of Mercury which he observed in 1855, when he already was the director of the Paris Observatory. This advance of the perihelion of Mercury was to become in 1915 an important evidence for Einstein's general theory of relativity. Le Verrier, however, attributed this motion to a perturbing body, which he identified as a planet closer to the Sun than Mercury or to a second asteroid belt so close to the Sun as to be invisible, spending much effort searching for asteroids inside the orbit of Mercury in an attempt to prove his theory. 
Reading an orbit: the orbital
elements by Livia Giacomini  Copyright Tumbling Stone 2001 To know the state of
motion of a body, it would be necessary to know the
values of the six parameters corresponding to its
position and its speed. But, during a general motion,
these values change constantly and knowing them at every
moment is a very difficult task. The first 2 orbital elements,
needed to define size and shape of the orbit on this
plane, are the semimajor axis a and the eccentricity
e.
