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Resonance - Part 2 of 5

Asteroids can even be in resonance with one another. Opik (1970)
examined the effect of Ceres on Pallas, which have nearly identical mean
distances from the Sun, mean angular motions and periods of revolution.
This 1/1 commensurability results in conjunctions approximately every
2,640 years (the last one being about 313 years ago). Opik concludes
that the two asteroids are little affected by resonance, having retained
their orbital characteristics since their formation in the same "ring of
diffuse matter." The stability of a resonance is determined by its phase
angle. If it librates about some angle, usually 0 or 180°, the resulting
resonances will be very stable. Such is the case for 4/3, 3/2, and 2/1.
At 1/1, the phase angle librates about 60°, a case that will be
considered in Chapter 15.
As an example, consider the asteroid 1362 Griqua at the 2/1 resonance.
In 1970, Brian Marsden did a study of Griqua's orbit. Fig. 5-8 shows the
oscillations of the phase angle for a period of 3,000 years. Griqua
librates about 0° with an amplitude of 100° to 120° in a period of close
to 400 years. The numbers at the maxima and minima of the curve show the
least distances from Jupiter in AU. These take place when the libration
is near its extremes. Marsden concluded that Griqua's nodal period is
34,000 years. Liu and Innanen (1985) have found that the resonant
phenomenon becomes more pronounced as the orbital eccentricity of the
asteroid increases. In addition, the variations of the elements a, e and
i are larger in the regions of the resonant zone which lie just inside
or outside the boundary of the libration region.
While the mathematical theory of resonances is beyond the scope of this
book, it is important to understand the basic principles and their
shortcomings. Most models have been based on the three-body problem
(Sun-Jupiter-asteroid), which is a reasonable approximation since the
disturbing influence of Jupiter far exceeds that of the other planets
(Giffen, 1973).
In the study of planetary motions, the theory of secular perturbations
can be used over intervals of 10^5 years. This theory fails in the case
of commensurable motion, however, due to the presence of the infamous
small divisors in the series expansions of the perturbations. The
alternative of numerically integrating the equations of motion directly
also fails for long time intervals due to cumulative round-off errors.
Thus, almost every theoretical and numerical study of the asteroid
resonances rely on the so-called averaging principle. It was first
introduced by Lagrange and Laplace, and was explicitly stated by Gauss,
who replaced each planet by a ring of mass whose density at each
longitude was inversely proportional to the planet's velocity at that
longitude.
Following an extension of this idea by Poincaré in 1902, J. Schubart
developed a model in the 1960's for the investigation of commensurable
motion at a resonance. This averaging principle remains an intuitive
assertion which is, strictly speaking, untrue. According to V.I. Arnold,
"This principle is neither a theorem, an axiom, nor a definition, but a
physical proposition. Such assertions are often fruitful sources of
mathematical theorems."
The averaging principle is used simply because it is too expensive to
study unaveraged equations of motion. The averaging process eliminates
short-period variations, making the equations tractable. With this
caveat in mind, we can now look at the four theories put forward to
explain the Kirkwood gaps.

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