Rotators model

If the rotational motion of the molecules is assumed to be entirely unhindered (e.g., by any environment or by collisions with other molecules), it is appropriate to express the time dependence of each of the dipole time correlation functions listed above in terms of a "free rotation" model. For example, when dealing with diatomic molecules, the electronic-vibrational-rotational C(t) appropriate to a specific electronic-vibrational transition becomes ... 

The functions <([) Eg iiy(Re) Eg ii,f(Re,t) (l)j> describe the time evolution of the dipole-related vector (the electronic transition dipole in this case) for the rotational state J. In a "free-rotation" model, this function is taken to be of the form ... 

When applied to the rotation, vibration-rotation, or eleetronie-vibration-rotation eases within the "unhindered" rotation model treated earlier, the Fourier transform involves integrals of the form ... 

Ref 91. Discounted cash-flow models account for use of capital, working capital, income taxes, time value of money, and operating expenses. Real after-tax return assumed to be 12.0%. Short-rotation model used for sycamore and poplar. Herbaceous model used for other species. Costs ia 1990 dollars. Dry tons. 

Alternately, a benign shock compression model has been invoked to explain the polarizations. An elastic dipole-rotation model was first proposed... 

It was noticed that the longer C-H distances are of little importance in the curve for the free-rotation model the curve calculated by ignoring all C-H terms except those for the bond distance and the next larger distance (which is unchanged by rotation) is qualitatively indistinguishable from the free-rotation curve, and the s values for the peaks of the two curves differ on the average by only 0.02. In our treatment of propane and isobutane we have made use of this simplification. 

Abstract. We present initial results from our study of mixing in M Supergiants. C, N and O abundances are measured in five stars. N/C and N/O ratios indicate extensive mixing in excess of the standard models and in support of the rotational models. 

The rotation models predict significant effects on the properties and the evolution of the massive stars. They alter the ratio of red to blue supergiants and hence the nature of SNII progenitors they affect the properties, formation and evolution of Wolf-Rayet stars they result in the enrichment of He and C in the ISM while the abundance of O decreases they produce higher He and a-element yields from SNII via larger He cores. Many of these effects are metallicity dependent. With such far ranging impact, the effects of rotation and mass loss on the evolution of massive stars should be thoroughly understood. 

Fig. 1. N/C and N/O ratios are shown as a function of luminosity relative to the initial solar values. The lower hatched line in each plot is the standard model prediction and the upper hatched line is the predicted value for an initial rotational velocity of 300 km s 1 [6]. Our measurements show that the low ratios seen in aOri are not commonly seen in supergiants. Instead the ratios indicate extensive mixing as predicted by the rotation models.

Fig. 1. Projected rotational velocity in the Hyades and lithium versus effective temperature in 3 open clusters of the same age (Hyades, ComaB, Praesepe). The open symbols are the observed data (Kraft 1965, Stauffer et al. 1987, Mermilliod 1992, Burkhart Coupry 1998, 2000, Boesgaard 1987). The black symbols are the predictions of the rotating models at the age of the Hyades (compilation of Talon Charbonnel 1998 and Charbonnel Talon 1999)...

Fig. 2. Li abundance in main-sequence and turnoff stars in the open cluster IC 4651 aged of 1.5-1.7 Gyr vs Teff from photometry (open points and triangles for actual determination and upper limits respectively). Black points show rotating model predictions (CT99 and Palacios et al. 2003) at 1.5 Gyr for an initial rotation velocity of 110 km.sec-1 and for various stellar masses. For 1.5Mq models for an initial rotation of 50 and 150 km.sec-1 are also shown (black triangle and square). This provides an estimate of the expected dispersion for stars inside the Li dip. See Pasquini et al. (2004) for more details...

In our most favorable case (model C), the maximum value of the diffusion coefficient in the outer part of the HBS (where abundances of Li, C and N present large variations) is 105 cm2.s 1 (Fig. la), far from the 4 108 cm2.s 1 value that seems to be necessary to reproduce the observations according to [3]. As a result none of our rotating models can reproduce the observed patterns emphasized by [4] (Fig. lb). 

Rotation increases the MS lifetime with respect to non-rotating models (up to about 40 %). 

Rotation strongly affects the lifetimes as blue and red supergiants (RSG). In particular in the SMC, the high observed number of RSG can only be accounted for by rotating models [8]. 

For Wolf-Rayet stars of type WN, rotation makes smoother changes of abundances, due to internal mixing. For WN stars, the transition phase, with still H present, becomes longer due to rotation and this increases the late WN phase (WNL) where H is usually present. The CNO abundances at the end of the WN phase are the same for rotating and non-rotating models, because they are model independent and determined just by CNO nuclear equilibrium. Indeed, CNO abundances in WN stars provide a unique test of the physics of the CNO cycle. 

Non-rotating models at very low metallicity are believed to present the following differences with respect to normal metal rich stars ... 

Numerical calculations inspired in the ZK method for stars have also been applied for the description of the dynamics of model dendrimers. La Ferla [232] used a freely-rotating model, including a topology-dependence stiffness parameter and preaveraged HI. With this model, he obtained a complex analytical expression for the mean size. Cai and Chen [233] used a Gaussian model without HI and performed a detailed analysis of the relaxation motions. They investigated the diffusion of the center of mass, the relaxation of the center of mass position relative to the core monomer, and also the rotational and internal modes. 

In support of this model, it is noted that LnFj - which has F ions on both octahedral and tetrahedral sites of a face-centred-cubic Ln -ion array - becomes a fast F -ion conductor below its melting point without any change in the cation array (O Keeffe and Hyde, 1975). This observation shows that some low-energy excitation other than the displacement of F ions into octahedral sites is operative, as is postulated with the cluster-rotation model for PbF2. 

Flexible Rotation Model" as an Approach for TGF- Receptor Rearrangements... 

In this section, three models will be presented that don t force the reorganization of the solid bed and use screw rotation physics. These screw rotation models cause a significant portion of the energy dissipation to occur in the melt film between the solid bed and screw root. These models are for a conventional transition section, for a barrier melting section, and for a special case referred to as one-dlmenslonal melting. 

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