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.

A portrait of Le Verrier

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 near-Earth 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 post-1779 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 (1811-1877) was born at Saint-Lô, 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.

By 1845 Le Verrier became interested in the motion of Uranus which did not have the orbit scientists predicted with their mathematical calculations. Le Verrier suggested the presence of another planet beyond Uranus whose gravitational pull could explain the unusual motion of Uranus. Johann Gottried Galle, an astronomer of the Berlin Observatory, was able to observe the planet using Le Verrier's calculations. However, Le Verrier wasn't declared the unique discoverer of Neptune: Couch Adams, an English mathematician, had accomplished the same feat months before Le Verrier's calculations were completed .
In 1854, Le Verrier became director of the Observatory of Paris which, at the time, was in decay. He continued working at the Observatory for most of his life where his drive for efficiency was to made him very unpopular until he was reinstated as director in 1873.

A picture of Neptune taken by Voyager 2 in 1989.
(Credit: Calvin J. Hamilton)

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.
In the reality, for what concerns constant, stable orbits, the motion of a body can be completely described knowing the values of 6 constant parameters, called orbital elements that characterize the orbit. In this hyothesis, knowing the position and the speed of the object at every instant becomes useless.
First of all, let's consider the orbit of an object of the Solar System (a planet or a minor body). This orbit can be an ellipse (as the first of Kepler's laws states), or an opened orbit such as parabola or hyperbola. Anyhow, this orbit will always be a conic (dict.) and it will always lie on a plane called the orbital plane. The intersection between this plane and a reference plane (which can be chosen as the plane where the Earth's orbit lies) is called nodal line. This nodal line passes through the ascending and descending node.

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.

These two parameters can be defined on the orbital plane; for an
elliptical orbit:
- 2a is the length of the major axis (M in the image on the left);
- e, the eccentricity, gives an indication of how much the ellipse is elongated: it is 0 for a circle, and tends to 1 for more and more elongated orbits. It can be calculated with the formula (where m and M are defined in the image):
Three more parameters are needed to describe where is the orbit, and how it is oriented:
- i the inclination of the orbital plane, the angle that this plane forms with a reference plane;
- the longitude of node, the angle between a reference direction and the nodal line, that goes from the Sun to the ascending node;
- the argument of perihelion, the angle from the nodal line to the line joining the Sun and the perihelion, the point on the orbit closest to the Sun.

Finally, the position on the orbit of the body of interest can be specified by giving the sixth parameter:
- T, the time of passage at the perihelion.



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