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2. The Motion of Comets

        Comets necessarily obey the same physical laws as every
other object. They move according to the basic laws of motion and
of universal gravitation discovered by Newton in the 17th century
(ignoring very small relativistic corrections). If one considers
only two bodies -- either the Sun and a planet, or the Sun and a
comet -- the smaller body appears to follow an elliptical path or
orbit about the Sun, which is at one focus of the ellipse. The
geometrical constants which fully define the shape of the ellipse
are the semimajor axis a and the eccentricity e (see Figure 2).
The semiminor axis b is related to those two quantities by the
equation b = a(1-e^2). The focus is located a distance ae from the
center of the ellipse. Three further constants are required if one
wishes to describe the orientation of the ellipse in space
relative to some coordinate system, and a fourth quantity is
required if one wishes to define the location of a body in that
elliptical orbit.

In Figure 2B several ellipses are drawn, all having the same
semimajor axis but different eccentricities. Eccentricity is a
mathematical measure of departure from circularity. A circle has
zero eccentricity, and most of the planets have orbits which are
nearly circles. Only Pluto and Mercury have eccentricities
exceeding 0.1. Comets, however, have very large eccentricities,
often approaching one, the value for a parabola. Such highly
eccentric orbits are just as possible as circular orbits, as far
as the laws of motion are concerned.

The solar system consists of the Sun, nine planets, numerous
satellites and asteroids, comets, and various small debris. At any
given time the motion of any solar system body is affected by the
gravitational pulls of all of the others. The Sun's pull is the
largest by far, unless one body approaches very closely to
another, so orbit calculations usually are carried out as two-body
calculations (the body in question and the Sun) with small
perturbations (small added effects due to the pull of other
bodies). In 1705 Halley noted in his original paper predicting the
return of "his" comet that Jupiter undoubtedly had serious effects
on the comet's motion, and he presumed Jupiter to be the cause of
changes in the period (the time required for one complete
revolution about the Sun) of the comet. (Comet Halley's period is
usually stated to be 76 years, but in fact it has varied between
74.4 and 79.2 years during the past 2,000 years.) In that same
paper Halley also became the first to note the very real
possibility of the collision of comets with planets, but stated
that he would leave the consequences of such a "contact" or
"shock" to be discussed "by the Studious of Physical Matters."

In the case of Shoemaker-Levy 9 we have the perfect example both
of large perturbations and their possible "consequences." The
comet was fragmented and perturbed into an orbit where the pieces
will hit Jupiter one period later. In general one must note that
Jupiter's gravity (or that of other planets) is perfectly capable
of changing the energy of a comet's orbit sufficiently to throw it
clear out of the solar system (to give it escape velocity from the
solar system) and has done so on numerous occasions. See Figure 3.
This is exactly the same physical effect that permits using
planets to change the orbital energy of a spacecraft in so-called
"gravity-assist maneuvers" such as were used by the Voyager
spacecraft to visit all the outer planets except Pluto.

One of Newton's laws of motion states that for every action there
is an equal and opposite reaction. Comets expel dust and gas,
usually from localized regions, on the sunward side of the
nucleus. This action causes a reaction by the cometary nucleus,
slightly speeding it up or slowing it down. Such effects are
called "non-gravitational forces" and are simply rocket effects,
as if someone had set up one or more rocket motors on the nucleus.
In general both the size and shape of a comet's orbit are changed
by the non-gravitational forces -- not by much but by enough to
totally confound all of the celestial mechanics experts of the
19th and early 20th centuries. Comet Halley arrived at its point
closest to the Sun (perihelion) in 1910 more than three days late,
according to the best predictions. Only after F. L. Whipple
published his icy conglomerate model of a degassing nucleus in
1950 did it all begin to make sense. The predictions for the time
of perihelion passage of Comet Halley in 1986, which took into
account a crude model for the reaction forces, were off by less
than five hours.

Much of modern physics is expressed in terms of conservation laws,
laws about quantities which do not change for a given system.
Conservation of energy is one of these laws, and it says that
energy may change form, but it cannot be created or destroyed.
Thus the energy of motion (kinetic energy) of Shoemaker-Levy 9
will be changed largely to thermal energy when the comet is halted
by JupiterUs atmosphere and destroyed in the process. When one
body moves about another in the vacuum of space, the total energy
(kinetic energy plus potential energy) is conserved.

Another quantity that is conserved is called angular momentum. In
the first paragraph of this section, it was stated that the
geometric constants of an ellipse are its semimajor axis and
eccentricity. The dynamical constants of a body moving about
another are energy and angular momentum. The total (binding)
energy is inversely proportional to the semimajor axis. If the
energy goes to zero, the semimajor axis becomes infinite and the
body escapes. The angular momentum is proportional both to the
eccentricity and the energy in a more complicated way, but, for a
given energy, the larger the angular momentum the more elongated
the orbit.

The laws of motion do not require that bodies move in circles (or
even ellipses for that matter), but if they have some binding
energy, they must move in ellipses (not counting perturbations by
other bodies), and it is then the angular momentum which
determines how elongated is the ellipse. Comets simply are bodies
which in general have more angular momentum per unit mass than do
planets and therefore move in more elongated orbits. Sometimes the
orbits are so elongated that, because we can observe only a small
part of them, they cannot be distinguished from a parabola, which
is an orbit with an eccentricity of exactly one. In very general
terms, one can say that the energy determines the size of the
orbit and the angular momentum the shape.




Acknowledgments:
        This booklet is the product of many scientists, all of
whom have cooperated enthusiastically to bring their best
information about this exciting event to a wider audience. They
have contributed paragraphs, words, diagrams, slides, and
preprints as well as their critiques to this document, which
attempts to present an event that no one is quite sure how to
describe. Sincere thanks go to Mike A'Hearn, Paul Chodas, Gil
Clark, Janet Edberg, Steve Edberg, Jim Friedson, Mo Geller, Martha
Hanner, Cliff Heindl, David Levy, Mordecai-Mark Mac Low, Al
Metzger, Marcia Neugebauer, Glenn Orton, Elizabeth Roettger, Jim
Scotti, David Seal, Zdenek Sekanina, Anita Sohus, Harold Weaver,
Paul Weissman, Bob West, and Don Yeomans -- and to those who might
have been omitted. The choice of material and the faults and flaws
in the document obviously remain the responsibility of the author
alone.

The writing and production of this material was made possible
through the support of Jurgen Rahe and Joe Boyce, Code SL, NASA,
and of Dan McCleese, Jet Propulsion Laboratory (JPL). For help in
the layout and production of this booklet, on a very tight
schedule, additional thanks go to the Design Services Group of the
JPL Documentation Section.

All comments should be addressed to the author:

Ray L. Newburn, Jr.
Jet Propulsion Laboratory, MS 169-237
4800 Oak Grove Dr.
Pasadena, CA 91109-8099