Laplace resonance

How can a phenomenon of orbital dynamics and celestial mechanics, such as the Laplace resonance, result in distinctive tectonic patterns on a planetary body The linking process is tides. The orbital resonance forces orbital eccentricities. Thus, as a satellite orbits Jupiter, even if its... 

The tides also have important effects on the dynamics (both rotation and orbit) of Europa. Because the orbit is eccentric, tidal torques tend to drive spin to a state that is slightly faster than synchronous (Greenberg and Weidenschilling 1984). This effect in turn can increase the tidal stress on the surface substantially, although it acts on a much longer timescale ( -MO5 years) than the 3 1/2 day orbital period. The tides also effect the long-term evolution of the orbits of the satellites, so that the Laplace resonance itself probably changes over MO7 years or more. 

The mechanism by which tides create the cycloidal crack patterns was discovered by B.R. Tufts and G.V. Hoppa (Hoppa et al. 1999c). Due to the orbital eccentricity, which is pumped and maintained by the Laplace resonance, the tidal stress on Europas ice crust changes periodically with each orbit. We call this variation the diurnal tide, because the 85 hr orbital period is approximately the length of a Europan day, and is comparable to the length of a day on Earth. 

In that way, a crack propagating across Europa can record the stress, which originates with the action of the Laplace resonance. 

Strike-slip displacement, in which one crustal plate shears past another, is relatively rare in the solar system except in a few places. On Earth, with its global tectonics, strike-slip between plates is well known. The San Andreas fault in California is a famous example. Similar strike-slip is ubiquitous on Europa. Faults there include some longer than the San Andreas, with Strike-slip displacement of tens of km. Strike-slip displacement on Europa is probably driven by diurnal tides, and thus by the Laplace resonance. Over the course of a day, the tidal stress follows a sequence that can open a crack, shear it, then close it. Then, stress that would reverse the shear is resisted while the crack is closed. This process repeats on a daily basis. In this process, which is analogous to walking, one plate of crust can shear past another, with Strike-slip displacement visible along the boundary. This process has been described in detail by Hoppa et al. (1999b). 

The description of imaging experiments in reciprocal space is not restricted to k space, the Fourier conjugate space of physical space. The modification of the spin density by other parameters like resonance frequencies, coupling constants, relaxation times, etc., can be treated in a similar fashion [Miil4]. For the frequency-dependent spin density, the Fourier transformation with respect to 2 is already explicitly included in (5.4.7). Introduction of a Ti-dependent density would require the inclusion of another integration over T2 in (5.4.7) and lead to a Laplace transformation (cf. Section 4.4.1). 

The system evolves with the innermost orbit receding from the central body (because of the non-conservative forces acting on mi) up to the moment where the system is captured into a resonance, a2 is almost constant. When the 2/f-resonance is reached, the system is trapped by the resonance. As known since Laplace, after the capture, mi continuously transfers one fraction of the energy that it is getting from the non-conservative source to m2, so that 02 also increases. One may note from Figure 9 that, after the capture into the resonance, ai increases at a smaller pace than before the capture. The increase of the semi-major axes is such that the ratio ai/a2 remains constant. 

Gravitational N-body dynamical systems usually have only the current positions and velocities of the bodies as the observational evidence of their past history. The traces of their past behavior exist only as conceptual, theoretically computed orbits. However, the Laplace orbital resonance among the galilean satellites actually leaves a visible record of its behavior in a surprising and seemingly unlikely form Linear traces in distinctive geometric patterns clearly visible on the surface of the satellite Europa are a direct recording of the effect of the orbital resonance. 

Figure 1. A Voyager image of a far southern region on Europa, showing many distinctive cycloidal ridges, which mark the paths of cracks in Europa s crust. These linear trajectories are the result of the Laplace orbital resonance among the Galilean satellites. Three of the Cycloids have IAU assigned names, as shown. Cycloids are chains of arcs, each typically 100 km long, often with a dozen or so arcs in each chain.

In the seventeenth century, modern experimental acoustics originated when the Italian mathematician Galileo explained resonance as well as musical consonance and dissonance, and theoretical acoustics got its start with Sir Isaac Newton s derivation of an expression for the velocity of sound. Although this yielded a value considerably lower than the experimental result, a more rigorous derivation by Pier re-Simon Laplace in 1816 obtained an equation yielding values in complete agreement with experimental results. 

Nanobubbles located on the surface of a shear-wave resonator behave peculiarly. First, they may look like rigid objects due to Laplace pressure. They may decrease the frequency and increase the bandwidth and thereby seemingly contradict the simple-minded models of slip. Also, they act as a source of acoustic streaming. Since acoustic streaming increases mass transport close to the interface, the bubbles thereby speed up their own dissolution. 

Y. Song, Laplace Inversion for NMR Data Analysis, in Magnetic Resonance in Food Science, ed. J. van Du mhoven, P. S. Belton, G. A. Webb and H. van As,... 

The linearized and Laplace-transformed equations of the models described above are used to evaluate the various system transfer functions as functions of the Laplace variables s = cr + jco, where a is the real part and co is the imaginary part of the complex variable s. a refers to the damping constant (or damped exponential frequency) and co refers to the resonant oscillation frequency of the system. 

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