Thayer Watkins
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Resonance Phenomena in the Solar System

Resonance Phenomena

Within the asteroid belt there are no asteroids having a period which is one third, one half or two fifths of the period of Jupiter. On the other hand there are 19 which have periods which are close to two thirds that of Jupiter. And there are the Trojan and Apollo asteroids which have the same period as that of Jupiter but move in orbits such that they along with the sun and Jupiter form equilateral triangles.

Another resonance phenomenon is the Cassini Division gap between the rings of Saturn. Saturn has three prominent rings; named A, B, and C; as shown below and several fainter rings not shown.

The colors shown bear no relation to reality but the dimensions are to scale. The Cassini Division is a 1700 mile wide gap between Ring A and Ring B that corresponds to a region in which a particle would have a period of revolution near to one half the period of Saturn's moon Mimas, one third the period of the next moon, Enceladus, or one fourth the period of the next moon, Tethys.

The innermost moon, Janus, is a very small moon discovered long after Mimas, Enceladus, Tethys and Dione and the rest. Janus was, in fact, the tenth moon discovered because it is so small.

The diagram below shows the relationship between the radii of the orbits of the six innermost moons and the rings of Saturn. The diameters of the moons are not shown to scale.

The other moons have radii exceding the scale of the diagram.

Orbital Resonances

A body revolving in an elliptical orbit around a central body can be viewed as oscillating between a maximum distant from the body,apogee, and a minimum distance from the body, perigee. If some disturbing force hits the body every time it reaches its apogee (or perigee) the disturbance will accumulate and the body will be moved to a different orbit. If the disturbing force hits in the same direaction at both apogee and perigee the impact would be twice as great.

For other ratios the effect is not as great because one instance of the disturbing force tends to cancel another instance. For example, if the disturbing force hits three times per cycle, say once at apogee, once before perigee and once after perigee the effect of the hit before the perigee ehances the cycle while the one after perigee tends to suppress the cycle.

If the disturbing force hits three times in two cycles there would be a tendency for two hits to offset each other but not completely. Likewise for five hits in two cycles.

Consider a harmonic oscillator which is subject to a harmonic disturbing force; i.e.,

md²r/dt² = -kr + a*cos(ωt)
r(0)=b, r'(0)=0.

Taking the Laplace transform of this equation gives

m(-r'(0+) - sr(0+) + s²L(s)) = -kL(s) + aωs/(s²+ω²)

where L(s) is the Laplace transform of r(t).

Since r'(0)=0, the above equation reduces to

(s² + k/m)L(s) = sr(0) + (a/m)[ωs/(s² + ω²)]
or, equivalently
L(s) = r(0)[s/(s² + k/m)] + (a/m)(ωs/[(s² + ω²)(s² + ν²)]
or, letting k/m=ν², L(s) = r(0)[s/(s² + ν²)] + (a/m)(ωs/[(s² + ω²)(s² + k/m)]

The term ωs/(s² + ω²) is the Laplace transform of cos(ωt) and ν/(s² + ν²) is the Laplace transform of sin(νt).

Taking the inverse Laplace transform of the above equation yields a solution of the form

r(t) = (r(0)/ν)cos(νt) + F(t)

The first term on the right is merely the solution that would prevail if there were no disturbing force. F(t) is the impact of the disturbing force and has the form

A[cos((ω-ν)t)/(ω-ν) + cos((ω+ν)t)/(ω+ν)]

As the frequency of the disturbing force, ω, approaches the frequency of the oscillation, ν, the disturbance term increases without bound. This is resonance.

It is notable that resonance does not depend upon the amplitude of the disturbing force, only upon frequency matching.

In the above exercise resonance only occurred for ω → ±ν, but if the disturbing term is a composite of harmonics then resonance would occur for each of the harmonics.

(To be continued.)

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