Mean-motion resonance

Periapses alignment (or anti-alignment) may occur in resonant and non-resonant systems alike. In resonant systems, they are the natural states after the system is trapped into a mean-motion resonance (see Section 6). Conversely, in non-resonant systems they are a consequence of the angular momentum variations during resonance crossings without capture (Ferraz-Mello et al., in preparation). However, and independently of how they reached this condition, an important consequence of... 

The above equations were used to find Apsidal corotation solutions in the case of planets in 2/1 and 3/1 mean-motion resonances. The relationship between eccentricities and mass ratios in some of these solutions is shown in Figure 8. The top panels correspond to symmetric solutions. In the left-hand side panel, the periapses are anti-aligned. This is the... 

The initial distances to the star are just behind the 2/1 resonance a = (i /(12 = 0.612. When the semi-major axis of mi increases, ai increases and the mean-motion resonance (a = 0.63) between mi and m2 is reached. Capture then can take place. The probability of capture depends on the rate of variation of ai - if the rate is high, the orbit crosses the resonance without capture, one phenomenon very well studied in the case of one massless particle. Other factors influencing the probability of capture are the orbital eccentricities - capture is more probable when orbital eccentricities are small (Dermott et al., 1988 Gomes, 1995). In our calculations, initial eccentricities were lower than 0.001 and the physical parameters were adjusted to have slow resonance approximation. Figure 9 shows the evolution of the semi-major axes. 

Beauge, C., Ferraz-Mello, S. and Michtchenko, T.A. (2003), Extrasolar planets in mean-motion resonance Apses alignment and other stationary solutions. Astroph. J. 593, 1124-1133. (astro-ph 0219577). 

Callegari Jr., N., Michtchenko, T. and Ferraz-Mello, S. (2004), Dynamics of two planets in 2 1 mean-motion resonance. Cel. Mech. Dynam. Astron. 89 (in press). Dermott, S., Malhotra, R. and Murray, C.D. (1988), Dynamics of the Uranian and Saturnian satellite systems A chaotic route to melting Miranda. Icarus 76, 295-334. 

Lee, M. H. and Peale, S. J. (2003), Extrasolar planets and mean-motion resonances. In Scientific Frontiers in Research on Extrasolar Planets (D. Deming and S. Seager, eds.) ASP Conf. Series (in press). 

Michtchenko, T. and Ferraz-Mello, S. (2001), Modeling the 5 2 mean-motion resonance in the Jupiter-Saturn planetary system. Icarus 149, 357-374. 

Figure 10.6 Proper orbital elements of numbered asteroids, from the catalog of Knezevid Milani (2003). Also shown are the locations of the main mean-motion and secular resonances.

The vibrational motions of the chemically bound constituents of matter have fre-quencies in the infrared regime. The oscillations induced by certain vibrational modes provide a means for matter to couple with an impinging beam of infrared electromagnetic radiation and to exchange energy with it when the frequencies are in resonance. In the infrared experiment, the intensity of a beam of infrared radiation is measured before (Iq) and after (7) it interacts with the sample as a function of light frequency, w[. A plot of I/Iq versus frequency is the infrared spectrum. The identities, surrounding environments, and concentrations of the chemical bonds that are present can be determined. 

According to Eq. (4-62), when woTo < 1, T, is proportional to 1/Tc, whereas when woTc 1, Ti is proportional to Tc. When Tc = Wo, Tj has its minimum value. Figure 4-7 is a schematic representation of the relationship between T and Tc. The physical meaning of this relationship is that coupling between the spin system and the lattice is most efficient when the resonance frequency and the frequency of molecular motion are equal. Tc can be measured by studying the dependence of Ti on wq (by varying the field strength). For small molecules in solution Tc is commonly 10 to 10 s. 

The use of computer simulations to study internal motions and thermodynamic properties is receiving increased attention. One important use of the method is to provide a more fundamental understanding of the molecular information contained in various kinds of experiments on these complex systems. In the first part of this paper we review recent work in our laboratory concerned with the use of computer simulations for the interpretation of experimental probes of molecular structure and dynamics of proteins and nucleic acids. The interplay between computer simulations and three experimental techniques is emphasized (1) nuclear magnetic resonance relaxation spectroscopy, (2) refinement of macro-molecular x-ray structures, and (3) vibrational spectroscopy. The treatment of solvent effects in biopolymer simulations is a difficult problem. It is not possible to study systematically the effect of solvent conditions, e.g. added salt concentration, on biopolymer properties by means of simulations alone. In the last part of the paper we review a more analytical approach we have developed to study polyelectrolyte properties of solvated biopolymers. The results are compared with computer simulations. 

When the size of metals is comparable or smaller than the electron mean free path, for example in metal nanoparticles, then the motion of electrons becomes limited by the size of the nanoparticle and interactions are expected to be mostly with the surface. This gives rise to surface plasmon resonance effects, in which the optical properties are determined by the collective oscillation of conduction electrons resulting from the interaction with light. Plasmonic metal nanoparticles and nanostructures are known to absorb light strongly, but they typically are not or only weakly luminescent [22-24]. 

In 1934 Ingold defined the chemical term "mesomerism" to mean stable intermediate states explained physically by quantal resonance.41 He stressed a preference for the word "mesomerism" over the term "tautomerism" or "resonance" on the grounds that it did not lead to the false impression that alternate structural formulas are passing into each other very rapidly, like tautomerides. If such rapid motion were going on, he argued, every molecule must spend most or all of its time in transforming itself, and the term molecular "state" simply loses its meaning.42... 

Diffusion is defined as the random translational motion of molecules or ions that is driven by internal thermal energy - the so-called Brownian motion. The mean movement of a water molecule due to diffusion amounts to several tenth of micrometres during 100 ms. Magnetic resonance is capable of monitoring the diffusion processes of molecules and therefore reveals information about microscopic tissue compartments and structural anisotropy. Especially in stroke patients diffusion sensitive imaging has been reported to be a powerful tool for an improved characterization of ischemic tissue. 

The amount of decrease of the resonance width may be simply estimated in the following way 50). Let the motion of the spins be characterized by a time tc, that is t is the average time a spin stays in a definite environment or the correlation time for the motion. This environment will cause a difference 5w in the precessional frequency of the spin which may be positive or negative from some average value to. During the time Tc the spin acquires a phase angle 60 = TcSu in addition to that acquired by the uniform precession at to. If we consider the motion to be a random walk process (51), after n such intervals during a time t the mean square phase acquired will be... 

In practice, this model is oversimplified since the exciting wake shedding is by no means harmonic and is itself coupled with the shape oscillations and since Eq. (7-30) is strictly valid only for small oscillations and stationary fluid particles. However, this simple model provides a conceptual basis to explain certain features of the oscillatory motion. For example, the period of oscillation, after an initial transient (El), becomes quite regular while the amplitude is highly irregular (E3, S4, S5). Beats have also been observed in drop oscillations (D4). If /w and are of equal magnitude, one would expect resonance to occur, and this is one proposed mechanism for breakage of drops and bubbles (Chapter 12). 

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