Initial Rotational Angular Momentum Research Articles

PHYSICAL PROPERTIES OF THE B AND Be STAR POPULATIONS OFhAND χ PERSEI

We present a study of the B and Be star populations of the Double Cluster h and chi Persei. Blue optical spectroscopy is used to first measure projected rotational velocity, V sin i, effective surface temperature, T_eff, and surface gravity, log g, for B-type sample stars, while available Stromgren photometry is used to calculate T_eff and log g for the Be stars showing emission. In our sample of 104 objects for which we measured these stellar parameters, 28 are known or proposed Be stars. Of these Be stars, 22 show evidence of emission at the times of our observations, and furthermore, we find evidence in our data and the literature for at least 8 transient Be stars in the clusters. We find that the Be stars are not rotating near their critical velocity, contrary to the results of studies of similar open clusters. We compare the results of our analysis with other previous studies, and find that the cluster members are more evolved than found by Huang & Gies but still retain much of their initial rotational angular momentum.

Stereodynamics of the F + HD(v = 0, j = 1) reaction: direct vs. resonant mechanisms

The stereodynamics and mechanism of the F + HD(v = 0, j = 1) → HF (DF) + D (H) reactions have been thoroughly analysed at collision energies in the 0-160 meV range. Specifically, this study is focused on (i) the comparison between the stereodynamics of the collisions leading to HF and DF formation, and (ii) the stereodynamical fingerprints of the resonance that occurs at low collision energies in the HF channel and whose manifestation in the total cross section is greatly diminished for initial j > 0. While previous studies were limited to the analysis of integral cross sections (ICS), differential cross sections (DCS) and reaction probabilities, in the present work we have included the analysis of vectorial quantities such as the direction of the initial rotational angular momentum and internuclear axis, and their effect on reactivity. In particular, polarisation parameters (PP) and polarisation dependent differential cross sections (PDDCS), quantities that describe how the intrinsic HD rotational angular momentum and molecular axis polarisations contribute to reaction, are calculated and examined. The evolution of the PPs with the collision energy differs markedly between the two reaction channels. For the DF channel, the PP values are small and change very little in the energy range in which DF formation is appreciable. In contrast, rapid fluctuations in the magnitude and sign of the PPs are observed in the HF channel at low collision energies in and around the resonance. As the collision energy increases, direct (non-resonant) scattering prevails, and the various quantities are reasonably well accounted for by the QCT calculations, as in the case of the DF channel. The intrinsic directional information has been used to access the extent of control that can be achieved through polarisation of the HD molecule prior to collision. It was found that the same extrinsic preparation leads to very different outcomes on the HF channel DCS when the collision energy is close to the resonance. It is also shown that polarisation of the HD internuclear axis along the initial relative velocity enhances the effect of the resonance and allows its clear identification. Finally, the effect of different extrinsic preparations on the angle-velocity DCS is found to be strong, thus allowing considerable control of product angular distributions.

Cooling and fragmentation of gas in rotating protogalaxies

The dynamical, thermal, and chemical evolution of gas in protogalaxies with non-zero angular momentum is considered. It is shown that, in protogalaxies with a total mass (dark and baryonic) of M = 107M⊙ at redshifts z = 12 whose gas has rotational angular momentum (spin parameter λ ≳ 0.005), a disk-like structure forms during the initial collapse of the galaxy, in contrast to non-rotating protogalaxies, whose collapse is spherically symmetric. The existence of initial angular momentumfor gas in protogalaxies increases the cooling time of the gas, delaying the formation of the first stars. Increasing the rotational angular momentum of the gas leads to cooling of the gas to lower temperatures (T < 100 K), at which HD molecules dominate in the cooling, while the total mass of cool gas (T < 1000 K) is decreased. The stability of disk-like structures in the central regions of protogalaxies is analyzed. It is shown that the disk that is formed is gravitationally unstable, and multiple fragmentation at various distances from the center is possible when the initial rotational angular momentum of the protogalaxy is increased. The possible birth of several stars in the first protogalaxies is discussed.

Theoretical Study of the Stereodynamics of CO Collisions with CH3- and CF3-Terminated Alkanethiolate Self-Assembled Monolayers

We present a classical-trajectory study of CO collisions with regular (CH3-terminated) and omega-fluorinated (CF3-terminated) alkanethiol self-assembled monolayers (SAMs) with a focus on analyzing the stereodynamics properties of the collision. The CO molecule is scattered with incident angles of either 30 degrees or 60 degrees with respect to the surface normal and with 60 kJ x mol(-1) collision energy, and we analyze final translational and rotational energy, mechanism of the collisions, and orientation and alignment of the rotational angular momentum. Analysis of the alignment of the final rotational angular momentum in collisions involving initially rotationally cold CO indicates a slight preference for "cartwheel" and "corkscrew" rotational motions. In contrast, collisions of initially excited CO slightly favor "helicopter" motion of the recoiling molecule. Moreover, studies of final orientation reveal that, while cartwheel "topspin" motion is favored for collisions in which initially cold CO becomes rotationally excited, no preferred handedness is observed when CO leaves the surfaces with "helicopter" motion. Analysis of trajectories involving initially rotationally excited CO in which the initial rotational angular momentum is aligned and/or oriented shows a non-negligible effect of the initial rotational motion on the dynamics of energy transfer. For instance, CO approaching the SAMs with helicopter motion retains a larger fraction of its initial rotation than molecules colliding with cartwheel-type motions. Conservation of the alignment and orientation of the initial rotational angular momentum vector is also enhanced with helicopter motion relative to cartwheel or random motions. The calculated trends in the stereodynamic properties for the two SAMs indicate that the CH3-SAM is effectively more corrugated than the CF3-SAM.

Quasiclassical determination of reaction probabilities as a function of the total angular momentum

This article presents a quasiclassical trajectory (QCT) method to determine the reaction probability as a function of the total angular momentum J for any given value of the initial rotational angular momentum j. The proposed method is based on a discrete sampling of the total and orbital angular momenta for each trajectory and on the development of equations that have a clear counterpart in the quantum-mechanical (QM) case. The reliability of the method is illustrated by comparing QCT and time-dependent wave-packet QM results for the H+D(2)(upsilon=0,j=4,10) reaction. The small discrepancies between both sets of calculations, when they exist, indicate some genuine quantum effects. In addition, a procedure to extract the reaction probabilities as a function of J when trajectories are calculated in the usual way using a continuous distribution of impact parameters is also described.

Quasi-classical treatment of the Stereodynamics of chemical reactions: k-r-k′ vector correlation for the Li+HF(v=1,j=1)→LiF+H reaction

We present a classical treatment of the k-r-k′ vector correlation for atom–diatom reactions, involving the directions of the initial and final relative velocities and the internuclear axis of the diatom. The formalism is based on the expansion of the joint probability distributions in multipolar moments, and it is analogous to that developed for the vector correlations implying the initial or final rotational angular momentum. Within the framework of classical mechanics, the present treatment allows the determination of the differential cross section for any preparation of the internuclear axis and for any initial rotational angular momentum of the reactants. This methodology has been applied to the study of the steric effect in the Li+HF(v=1,j=1,m=0)→LiF+H reaction and the theoretical results have been compared via simulation with the recent experimental determination by Loesch and co-workers of laboratory angular distributions (LAB ADs) for several distributions of the internuclear axis of HF. Very good agreement has been found between experimental and simulated LAB AD, allowing an interpretation of the experimental results. Although the integral steric effect is very small, the differential and/or state resolved steric effects are more pronounced. Moreover, it has been found that by varying the preparation of the internuclear axis of the HF molecule, the population of final states of the LiF changes noticeably, which represents a clear case of control of the outcome of a chemical reaction.

Detection of collision-induced rotational alignment in counterpropagating beam scattering

Collision-induced rotational alignment of NH 3 scattered from He is detected in a counterpropagating pulsed molecular beam scattering experiment. Starting from an isotope distribution of the initial rotational angular momentum, the generation of a highly aligned product ensemble due to inelastic scattering is observed. For all probed state-resolved differential cross sections the alignment changes from negative values for backward scattering to positive values at small deflection angles which is in qualitative agreement with predictions from a geometric apse model. The degree of alignment strongly depends on the final rotational state, indicating a correlation with the involved energy transfer.

A laser-induced fluorescence study of product rotational state distributions in the charge transfer reaction: Ar+(2P3/2)+N2→Ar+N+2 (X) at 0.28 and 0.40 eV

The nascent rotational state distributions of N+2 produced in the charge transfer reaction of Ar+ (2 P3/2 ) with N2 at 0.28 and 0.40 eV are remeasured by laser-induced fluorescence. A supersonic expansion is used to reduce the initial rotational angular momentum of the N2 . The N+2 product rotational distributions, in both v″=0 and v″=1, have low and high energy components. For ease of reference, we describe each distribution as a summation of two Boltzmann distributions. At a relative collision energy of 0.28 eV, the Boltzmann temperatures are 100±20 K and 745±120 K for N+2 (v″=0) and 80±10 K and 680±30 K for N+2 (v″=1). Adiabatic potential energy curves for the lowest vibronic states are calculated and a simple curve hopping model is presented. Applying this model to the production of N+2 (v″=1), for example, those reactants that charge transfer on the outgoing leg of a reactive trajectory interact with a deep potential well in the entrance channel for collinear geometry. We postulate that rotationally excited products result. In comparison, reactants that charge transfer on the ingoing leg (or in perpendicular geometry) do not sample the collinear potential well and the resulting products are less rotationally excited.

Theoretical investigation of the vibrational relaxation process Ar + OH(ν = 1) → Ar + OH(ν = 0)

For the system Ar + OH electronic adiabatic as well as non-adiabatic vibrational relaxation mechanisms have been studied on the basis of the classical trajectory and the surface-hopping trajectory method, respectively, using simple model potentials for the two lowest electronic states. The collision processes are discussed in detail, especially regarding the influence of the initial rotational angular momentum of the OH molecule which greatly enhances the vibrational deactivation cross sections. Furthermore, thermal rate constants are calculated and compared to recent results from transition state theory; considerable disagreement is found for both mechanisms.

Sum rules for the rotational structure in the molecular transition spectrum

This paper is concerned with the spectra, such as the oscillator strength distribution for transitions of linear, symmetric-top or asymmetric-top molecules, induced by any external perturbation. The sum over a rotational structure in the spectrum for a fixed initial rotational state Gamma is expressible in a simple form independent of Gamma . This sum rule is a general form of a theorem proved recently by the present author for the mean-energy-loss cross section. The theorem is a geometrical statement, and is valid regardless of the physical mechanism of the transition. A proof different from that given previously shows that the geometrical meaning of the theorem is the second cosine formula J'2=J2+Jt2+2J.Jt for the addition of the initial rotational angular momentum J and the angular momentum J'. As examples, sum rules for the dipole and generalised oscillator strengths for both discrete and continuum transitions are discussed. Also, sum rules for the rate constant and for the integral and differential cross sections for the mean energy loss hold for the collision of a molecule with any particle having or not having an internal structure, if the adiabatic-rotation approximation is valid.

Vibrational relaxation and the statistics of triplet states and matrix elements in methylglyoxal

Quantum beat spectroscopy is presently the only direct probe of intramolecular couplings and spacings between individual highly exicted states of polyatomic molecules with state densities of the order of 1000 states per wave number. We have applied this method to methylglyoxal in an attempt to discover if there are preferred pathways of intramolecular energy flow and if it is possible, in this and similar molecules, to localize energy in particular bonds. The method has enough resolution to allow observation of hyperfine splittings of excited triplet states as well as states which are different vibrationally excited triplet states. Consequently, the vibrational coupling and spacing trends are slightly obscured. However, our data indicate that there are no selection rules for intersystem crossing other than rigorous conservation laws and a requirement for conserving nuclear spin states. The dependence on the initial rotational angular momentum N is evidently not important at all at low N but might be useful at high N.

Formation of the planets by accretion of planetesimals: Some statistical problems

We consider some consequences of the hypothesis that planets are formed by the accretion or accumulation of roughly lunar-sized planetesimals. Stochastic coalescence theory predicts that the frequency distribution of masses of planetisimals is approximately an inverse power law with index α, 0 < α < 1 (usually α = 1 2 or α = 2 3 , according to the model chosen). The average mass of the planetary nucleus (largest planetesimal) is about 0.60 ( for α = 1 2 ) or 0.42 ( for α = 2 3 ) the mass of whole planet, and the average mass of the second largest planetesimal (largest planetesimal colliding with the planetary nucleus) is about 0.20 ( for α = 1 2 ) or 0.17 ( for α = 2 3 ) the mass of the whole planet. Some other results about the distribution of relative sizes of the largest planetesimals are also derived. The distribution of the random change in rotational angular momentum of the planet as a result of the accretion of planetesimals with random masses, velocities, and directions is also derived. This distribution is used to show that the discrepancy between our estimate of the relative mass of the second largest planetesimal (0.17–0.20) and Safronov's estimate (10 −3 to 10 −2) is due to the relatively high collision speeds his model requires. The estimated rms collision speeds for the present Solar System planets are derived and turn out to be surprisingly small—from 0.07 km/sec for Mars to 2.5 km/sec for Jupiter. If these speeds are unacceptably small, one must conclude that most of the initial rotational angular momentum imparted by collisions has somehow been lost or dissipated.

Tidal friction

The perturbing potential due to the tidal interaction between the earth, sun, and moon is obtained without restrictive assumptions on the internal constitution of the earth. The derived potential is used in Gauss's equations for the time rate of change of the eccentricity, semimajor axis, and inclination of the moon's orbit. The tidal forces perturb the elements describing the earth's motion about the sun by a negligibly small amount. The variation in the earth's angular momentum due to the solar and lunar tides is described by Euler's equations. It is assumed that the moment of inertia of the earth corresponds to that appropriate for a rotating fluid with the density distribution of the present earth.The rate of change of the moon's orbital elements is determined by the phase lag in the elastic component of the tidal bulge raised by the sun and moon. Astronomical data give a current value of the lag of the lunar tidal bulge of 2.16°. The present phase lag corresponds to a rate of increase of the semimajor axis of 3.2 cm yr−1, a rate of decrease of the inclination of the moon's orbital plane of 9.35 × 10−12 rad yr−1, and a rate of increase in the eccentricity of 1.2 × 10−10 yr−1. The obliquity of the earth's equator to the ecliptic and the period of rotation are increasing at present, owing to the combined effects of the lunar and solar tides. The effects of the solar tides are small at present but become important in the future.Numerical integrations of the coupled Gauss‐Euler equations are used to describe both the past history and the future evolution of the earth‐moon system. If the moon traveled a circular orbit, a backward tracing of the history shows the moon approaching the earth, reaching a minimum distance of 2.72 present earth radii. During the time of closest approach, the inclination and obliquity change rapidly, with the moon's orbital plane passing over the pole and the moon's motion becoming retrograde. If the orbit is eccentric, the eccentricity increases rapidly at the time of close approach. The perigee height remains nearly constant, while the apogee increases without bounds over a time of about 1000 years. If the current phase lag remains constant, the time of closest approach is only 1.78 × 109 years ago. Thus, the current phase lag is not consistent with the hypothesis that the earth‐moon system has existed throughout geologic time.The rotational parameters of Mars, Venus, and Mercury are discussed in terms of the dynamical theory. The distribution of rotational angular momentum of the solar system is described, and it is proposed that the major planets and Mars have lost only a very small proportion of their initial rotational angular momentum. The observed dependence of rotational angular momentum on planetary mass yields an estimate of the initial rotational period of the earth of between 9 and 13 hours.The mechanisms by which the earth's rotational energy can be dissipated are reviewed. It is argued that the phase lag in the tides may have remained constant over geologic time or may have been somewhat larger. This conclusion raises the problem of the age of the earth‐moon system.Theories of the origin of the moon are discussed against the requirements imposed by dynamical considerations. It is concluded that the theories of the fission of the earth to form the moon must be discarded, because the moon, once placed in the equatorial orbit, would remain there. Dynamics places severe constraints on the initial conditions for either a capture origin or the development of a binary system. The theory of several initial moons is briefly described, as are the climatological and tectonic problems raised by the consideration of the past history of the earth‐moon system.