# 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 - e2). 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.

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, satellites 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. 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 Jupiter's 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.

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