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Resonance - Part 1 of 5

CUNNINGHAM C.J. (1988) Introduction to Asteroids, pp. 41-46: Resonances

The term resonance originated in acoustics, where the audible resonances
of musical instruments have been known since antiquity. A note struck on
a piano can set a violin string vibrating, for example.
The music of the spheres is analogous to this. With Jupiter as the
celestial piano, and the asteroids as violin strings, a complex series
of resonances have been established. These relationships were first
noted by Daniel Kirkwood. "The first statement Kirkwood made of the
asteroid gaps, apparently, and also of the gaps in the rings of Saturn,
was before the Buffalo meeting of the American Association for the
Advancement of Science, in August 1866. " (Hogg, 1950). Based on 87
asteroid orbits, his results were published briefly in 1867 and more
fully in 1868, when he listed 97 asteroids ranging from Flora at 2.20 AU
to Sylvia at 3.49 AU.
A histogram of the asteroids (Fig. 5-7) clearly shows several gaps.
Kirkwood calculated the orbital periods of these gaps and designated
them t. In accordance with Kepler's Third Law, he found they were
related to the orbital period of Jupiter, T, by the relationship t/T =
a/(Ae^3/2) where a is the gap and A the distance of Jupiter in A.U.
Kirkwood found for t/T the following data: A period equal to 1/2 that of
Jupiter is 3.2776 AU and that 4/9 = 3.0299 AU; 3/7 = 2.9574 AU; 2/5 =
2.8245 AU; 1/3 = 2.5012 AU; and 2/7 = 2.2569 AU.
To account for these intervals, Kirkwood formulated a gravitational
hypothesis. "A planetary particle at the distance 2.5 - in the interval
between Thetis and Hestia - would make precisely three revolutions while
Jupiter completes one (the 1/3 resonance); coming always into
conjunction with that planet in the same parts of its path. Consequently
its orbit would become more and more eccentric until the particle would
unite with others, either interior or exterior, thus forming the nucleus
of an asteroid. Even should the disturbed body not come in contact with
other matter, the action of Jupiter would ultimately change its mean
distance, and thus destroy the commensurability of the periodic times.
In either case, the primitive orbit of the particle would be left
destitute of matter."
This means that in the first gap the asteroids have a period one-half
that of Jupiter, in the second gap four-ninths, and so on. As shown in
Fig. 5-7, eight major resonances are now recognized. Asteroids were
first found at resonant positions in the following years: 1875: Hilda at
3/2 - 1888: Thule at 4/3 - 1908: Achilles at 1/1 - 1918: Alinda at 3/1 -
1918: Griqua at 2/1. Three of these resonances are not gaps at all, but
concentrations of asteroids: the Hilda group at 3/2, Thule at 4/3 and
the Trojans at 1/1. Percival Lowell caught the essence of it best when
he stated "If the asteroids were numerous enough we should actually
behold in the sky a replica of Saturn's rings, altered only by the
perspective of our different point of view." (Lowell, 1917).
The early interest in resonances stemmed from the ability to deduce
fairly accurate masses of the objects involved. In the late 19th
century, for example, Simon Newcomb used perturbations in the orbit of
the asteroid Polyhymnia to find the mass of Jupiter to be 1/1047.35 that
of the sun. This is nearly identical to the modern value of 1/1047.355.
One modern goal of theorists is to formulate a theory of the origin of
the resonances, and why some generate gaps and others concentrations of
asteroids. Milani et al. (1985) have found "a difference in the local
topology between the 2/1 Hecuba gap and the 3/2 Hilda group based purely
on gravity." While much research remains to be done, it appears there is
a "protection mechanism in the asteroid motion" that permits clustering
at some resonances and gaps at others.
Resonances such as 2/1 are called commensurabilities, wherein Jupiter's
and the asteroids' orbital periods have a ratio of small whole numbers.
In such a case, the asteroid and Jupiter repeat their relative positions
(eg conjunctions) at certain longitudes ...
While most common commensurabilities depend on the asteroid's orbital
eccentricity, inclination-type resonances are also possible, although
they tend to be much weaker. Mathematically, this is due to the fact
that terms occur at one lower power of the eccentricity than of the
inclination. This power is the difference P - Q, and is called the
degree (denoted by q). Its relative strength at different resonances is
shown along the top of Fig. 5-7.

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