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
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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.
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Reading an orbit: the orbital
elements
by Livia Giacomini - Copyright Tumbling Stone 2001To 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.
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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):
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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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