Kirkwood gaps

J. Wisdom. The origin of the Kirkwood gaps A mapping for asteroidal motion near the 3/1 commensurability. Astr. J., 87 577-593, 1982. 

Wisdom, J. The Origin of the Kirkwood Gaps A Mapping for Asteroidal Motion Near the 3/1 Commensurability. Astron. J. 87 (1982) 577-593 Tuckerman, M., Martyna, G. J., Berne, J. Reversible Multiple Time Scale Molecular Dynamics. J. Chem. Phys. 97 (1992) 1990-2001 Tuckerman, M., Berne, J. Vibrational Relaxation in Simple Fluids Comparison of Theory and Simulation. J. Chem. Phys. 98 (1993) 7301-7318 Humphreys, D. D., Friesner, R. A., Berne, B. J. A Multiple-Time Step Molecular Dynamics Algorithm for Macromolecules. J. Chem. Phys. 98 (1994) 6885-6892... 

The main-belt region is not evenly populated with asteroids, and several zones have been found in which virtually no asteroids reside (Fig. 1). The American astronomer Daniel Kirkwood (1814-1895) first noticed these empty regions, or gaps, in 1866. Now called Kirkwood gaps, these asteroid-devoid zones are located near orbits for which the time to complete one circuit around the Sun is a simple fraction (e.g., 1/2, 2/3, 3/4) of Jupiter s orbital period. For example, given that Jupiter orbits the Sun once every 11.86 years, an asteroid belt gap is expected at a distance of 3.3 astronomical units (AU) from the Sun, where any orbiting body would have a period of 5.93 years, one-half that of Jupiter. Such a gap does indeed exist. These Kirkwood gaps are produced by... 

Kirkwood gap Any of several spaces in the distribution of asteroids in the main belt that correspond to locations of orbital resonance with Jupiter. Consequently any asteroids found there have long ago b n perturbed by Jupiter s enormous gravitational influence into more eccentric orbits. The gaps were discovered by the American astronomer Daniel Kirkwood (1814-95). 

Between the main concentration in the Main Belt are relatively empty regions known as Kirkwood gaps. These are regions were an object s orbital period is a simple fraction of that of Jupiter (resonance). 

Gap in eqs A2 and A4 is expressed as volume per molecule. These equations are valid for any -component system. Expressions for the Kirkwood—Buff integrals in ternary mixtures can be obtained from eq A4. ° In particular, one can obtain the following expressions for Gu and G23 for an infinitely dilute solute (component 2)... 

With this notation we can express all the thermodynamic quantities paj>, Va, and Kt in terms of the Kirkwood-Buff integrals, Gap ... 

This cannot be interpreted in terms of PS. Because of the long-range correlations imposed by the closure condition, these Gap are not the Kirkwood-Buff integrals. In the same sense, the relations (8.49), (8.51), and (8.53) hold true because of the closure condition with respect to the individual particles A and C. Clearly, one cannot conclude from (8.49) that the PS of W is zero. The sign of PS of W is determined by the difference GWa and Gwc, provided that GWa and GWc are evaluated in a system open with respect to the three components. 

The next experiment we consider is that studied by Fraenkel (29) and by Kirkwood and Auer (39). In this experiment a fluid is placed between two flat plates perpendicular to the y-direction, one of which is fixed and the other is oscillating in the x-direction with very small amplitude. The gap between the plates is sufficiently small that the velocity profile is almost linear the velocity gradient is varying sinusoidally with time, so that fc(t) = PAe kV , where k° is a complex quantity which specifies the amplitude and phase of the oscillating velocity gradient, and o> is the... 

Relation (3.7.27) is very general. First, it applies to any two-component system at chemical equilibrium, as well as to any classification procedure we have chosen for the two quasi-components. Second, because of the application of the Kirkwood-Buff theory of solutions, we do not have to restrict ourselves to any assumption of additivity on the total potential energy of the system. Furthermore, the quantities Gap appearing... 

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