Central Force Field Research Articles

Nuclear Spin Relaxation Driven by Intermolecular Dipolar Interactions: The Role of Solute−Solvent Pair Correlations in the Modeling of Spectral Density Functions

Nuclear spin relaxation provides useful information related to the dynamics of molecular systems. When relaxation is driven by intermolecular dipolar interactions, the relevant spectral density functions (SDFs) also have significant contributions, in principle, from distant spins all over the dynamic range typically probed by NMR experiments such as NOESY. In this work, we investigate the intermolecular dipolar spin relaxation as driven by the relative diffusion of solvent and solute molecules taking place under a central force field, and we examine the relevant implications for (preferential) solvation studies. For this purpose, we evaluate the SDFs by employing a numerical approach based on spatial discretization of the time-propagation equation, and we supply an analytical solution for the simplest case of a steplike mean-field potential. Several situations related to different solute-solvent pair correlation functions are examined in terms of static/dynamic effects and relaxation modes, and some conclusions are drawn about the interpretation of NOE measurements. While we confirm previous results concerning the spoiling effect of long-range spins (Halle, B. J. Chem. Phys. 2003, 119, 12372), we also show that SDFs are sufficiently sensitive to pair correlation functions that useful, yet rather complicated, inferences can be made on the nature of the solvation shell.

Stabilization of the non-trivial relative equilibria of a gyrostat with an elastic element in a circular orbit

The motion of a gyrostat, regarded as a rigid body, in a circular Kepler orbit in a central Newtonian force field is investigated in a limited formulation. A uniformly rotating statically and dynamically balanced flywheel is situated in the rigid body. A uniform elastic element, which, during the motion of the system, is subjected to small deformations, is rigidly connected to the rigid body-gyrostat body. The problem is discretized without truncating the corresponding infinite series, based on a modal analysis or using a certain specified system of functions, for example, of the assumed forms of the oscillations, which depend on the spatial coordinates and which satisfy appropriate boundary-value problems of the linear theory of elasticity. The elastic element is specified in more detail (a rod, plate, etc.), as well as its mass and stiffness characteristics and the form of the fastening, and the choice of the system of functions is determined. Non-trivial relative equilibria of the system (the state of rest with respect to an orbital system of coordinates when the elastic element is deformed) is sought approximately on the basis of a converging iteration method, described previously. It is shown, using Routh's theorem, that by an appropriate choice of the gyrostatic moment and when certain conditions, imposed on the system parameters are satisfied, one can stabilize these equilibria (ensure that they are stable).

Orbits of the n-dimensional Kepler–Coulomb problem and universality of the Kepler laws

In the standard classical mechanics textbooks used at undergraduate and graduate levels, no attention is paid to the dimensional aspects of the Kepler–Coulomb problem. We have shown that the orbits of the n-dimensional classical Kepler–Coulomb problem are the usual conic sections in a fixed two-dimensional subspace and the Kepler laws with their well-known forms are valid independent of dimension. The basic characteristics of motion in a central force field are also established in an arbitrary dimension. The approach followed is easily accessible to late undergraduate and recent graduate students.

Using magnetic fluids to simulate convection in a central force field in the laboratory

Abstract. Large-scale convection in planetary or stellar interiors plays a significant role but it is difficult to reproduce the central force field of those systems in experimental studies. A technique to approximate a central force field through the magnetic field from magnets acting on a magnetic liquid is presented. The thermomagnetic convection in a spherical shell filled with a magnetic liquid is analyzed in the context of a terrestrial laboratory using a 2D Finite Element model. Two configurations of magnetic fields were investigated, one resulting in a radially decreasing force field, and the other in a radially increasing force field. The results suggest that, while the actual force field does not reproduce the central gravity in planetary interiors accurately, it captures the essential qualitative character of the flow unlike other terrestrial experiments, which were either dominated by gravity or by a cylindrical radial force field. It is therefore suggested that such an experiment would provide a useful tool to investigate thermal convection in planetary interiors.

Elastic theory of surface deformation in molecular physisorption

In molecular physisorption, beyond attractive and repulsive forces, a third contribution arises from the elastic deformation of the surface. In the present work the dimple appearing on a deformable surface upon physisorption of a foreign molecule is studied and its effect on the adsorption energy is calculated. The depth and shape of the physisorption dimple are studied on the basis of the theory of elasticity. The problem of the deformation of a semi-infinite elastic medium, subject to central body forces, is solved, with the centre (i.e., the molecule) placed outside the medium. A vectorial Poisson equation for the displacement field is solved with the boundary condition that no surface forces act. For a centrosymmetrical molecule the solution is obtained in two steps, whose results are then superposed: (1) the deformation field in an infinite elastic medium subject to a central body force field, whose centre coincides with the centre of the molecule, is evaluated (the solution generates mismatch of the boundary conditions); (2) a complementary homogeneous problem is solved with boundary conditions compensating the spurious contributions generated in step (1). Specific attention is given to methane adsorption both on a metal surface (Au) and on a molecular crystal surface (solid Ar). The shape function ζ( R) is always positive (indicating a bulging surface) if the distance z of the molecule (i.e., of the carbon atom) from the surface is large, but becomes largely negative for z smaller, showing a definite dimple. The minimum energy (adsorption equilibrium) corresponds to the latter case. The dimple depth turns out to be 10.0 pm in the case of Au, while for Ar we find a depth of 30.5 pm at 80 K and 25.6 pm at 0 K. A discussion of the asymptotic behaviour (proportional, off equilibrium, to R −1) of the deformation at long distances R from the adsorbed molecule is also given.

The Dynamics of a Satellite-Gyrostat with a Single Nonzero Component of the Vector of Gyrostatic Moment

The dynamics of a satellite-gyrostat moving in the central Newtonian force field along a circular orbit is studied. In the particular case when the vector of gyrostatic moment is parallel to one of the satellite’s principal central axes of inertia, all the equilibrium states are determined. For each equilibrium orientation, sufficient conditions of stability are obtained as a result of the analysis of the generalized energy integral, and necessary conditions of stability are determined as a result of analysis of the linearized equations of motion. The evolution of regions of validity for the conditions of stability of equilibrium positions are studied in detail depending on the parameters of the problem. All the bifurcation values of the parameters at which qualitative changes of the regions of stability take place are determined.

The GeoFlow experiment—spherical Rayleigh-Bénard convection under the influence of an artificial central force field

Spherical Rayleigh-Bénard convection under the influence of an artificial central force field produced by the so-called dielectrophoretic effect is studied as a simplified model of the flow in the outer earth core. The fluid motion there is most probably driving the earth's dynamo and the energy source for the earth's magnetic field. Studying convective flows in earth-like geometry could lead to a deeper understanding of the basics of these processes.This research is a preparatory study for the experiments on the International Space Station (ISS). A bifurcation-theoretical approach shows the existence of heteroclinic cycles between spherical modes (l, l + 1) for the non-rotating system. This behavior depends strong on the radius ratio of the spheres and will be hard to detect in the experiment. For slow rotations interactions of the azimuthal modes (m, m + 1) found in numerical simulations for supercritical states are supposed to be experimentally observable.

The (3,4) Mode Interaction in the GeoFlow Framework

AbstractThe problem of convection in a self‐gravitating spherical shell of fluid is commonly encountered in sciences like astrophysics and geophysics (earth's liquid core). The GEOFLOW‐experiment is a project of the European Space Agency in order to perform the spherical Rayleigh‐Bénard convection problem on the International Space Station in a micro‐gravity environment: the central force field is simulated by a dielectrophoretic one. Beyond a critical Rayleigh number Rac, generically an unique spherical ℓ mode becomes unstable and only stationary or travelling waves solutions are expected near the onset. But, for a critical aspect ratio ηc two consecutive modes (ℓ, ℓ + 1) are unstable. The (1,2) and (2,3) interactions have showed a rich bifurcation diagram, in particular, we have found heteroclinic cycles predicted by the theoretical study. Because of the experiment requirements, only the (3,4) one is possible. So, this paper purposes to analyse this bifurcation in non‐rotating case in the GEOFLOWframework using the theory of bifurcation with the spherical symmetry.

Dynamics of multibody systems in planar motion in a central gravitational field

Multibody systems in planar motion are modelled as two or more rigid components that are connected and can move relative to each other. The dynamics of such multibody systems in planar motion in a central gravitational force field is analysed. The equations of motion of the system include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom gives rise to complex dynamical systems that are usually not integrable. Relative equilibria, corresponding to circular orbits of the multibody system, are obtained. The free dynamics has a symmetry due to a cyclic coordinate. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. The stability of the relative equilibria is analysed using the Routh stability criterion when it is applicable; an expansion of the Hamiltonian in normal form is used otherwise. We apply the general results to a multibody system consisting of two hinged planar bodies, each modelled as a rigid massless link with a point mass at one end with their other ends connected by a hinge joint. We obtain the relative equilibria of this model, and carry out a stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out for perturbations to these relative equilibria, to confirm their stability properties.

Stable analysis of long duration motions of FE‐discretized structures in central force fields

The computation of structures moving in central force fields generally requires long‐time integration including geometrically nonlinear behavior (large rotations) as such, e.g. satellite structures move for a long time. To achieve a numerically stable computation the energy momentum method which fulfills linear and angular momentum as well as energy conservation within the time step is chosen for the time integration. The focus in the contribution is on Hamiltonian systems. A formulation for the gravitational force in a central force field as external force on a rigid or flexible satellite is given. The presented formulation enables the computation of the exact spatial distribution of the gravitational forces acting on a structure using the FE‐discretization which is necessary to analyze, e.g. the orientation of a satellite in a gravitational field. The fulfillment of the conservation laws within the time step is proved. The necessity for considering the spatial distribution of the gravitational forces is discussed based on numerical examples.

Ultrafast dynamics for electron photodetachment from aqueous hydroxide

Charge-transfer-to-solvent reactions of hydroxide induced by 200 nm monophotonic or 337 and 389 nm biphotonic excitation of this anion in aqueous solution have been studied by means of pump-probe ultrafast laser spectroscopy. Transient absorption kinetics of the hydrated electron, e(aq) (-), have been observed, from a few hundred femtoseconds out to 600 ps, and studied as function of hydroxide concentration and temperature. The geminate decay kinetics are bimodal, with a fast exponential component ( approximately 13 ps) and a slower power "tail" due to the diffusional escape of the electrons. For the biphotonic excitation, the extrapolated fraction of escaped electrons is 1.8 times higher than for the monophotonic 200 nm excitation (31% versus 17.5% at 25 degrees C, respectively), due to the broadening of the electron distribution. The biphotonic electron detachment is very inefficient; the corresponding absorption coefficient at 400 nm is <4 cm TW(-1) M(-1) (assuming unity quantum efficiency for the photodetachment). For [OH(-)] between 10 mM and 10 M, almost no concentration dependence of the time profiles of solvated electron kinetics was observed. At higher temperature, the escape fraction of the electrons increases with a slope of 3x10(-3) K(-1) and the recombination and diffusion-controlled dissociation of the close pairs become faster. Activation energies of 8.3 and 22.3 kJ/mol for these two processes were obtained. The semianalytical theory of Shushin for diffusion controlled reactions in the central force field was used to model the geminate dynamics. The implications of these results for photoionization of water are discussed.

Analytical Mechanics. Foundations of Engineering Mechanics Series

Analytical Mechanics. Foundations of Engineering Mechanics Series

The stability of a family of non-trivial equilibrium orientations to the attracting centre of a gyrostat with an elastic rod

The motion of a system consisting of a gyrostat and an elastic rod in a circular Kepler orbit in a central Newtonian force field is considered in a restricted formulation. The gyrostat is treated as a rigid body in which there is a dynamically rotating flywheel and a statically counterbalanced flywheel. The uniform elastic rod, which is rectilinear in the undeformed state, is rigidly fixed to the gyrostat housing at one end. The axis of the undeformed rod is arbitrarily located in the principal plane of inertia of the gyrostat. The relative displacements of the points of the system as a result of a small deformation of its elastic link are represented in the form of an infinite series of its expansion (without its a priori truncation) in a specified system of functions, which depend on the spatial coordinates, with unknown time-dependent coefficients. The orientation of the system for an attracting centre is defined by indicating the position with respect to the associated system of coordinates of the unit vectors of the normal to the plane of the orbit and the radius vector or transversal of the orbit at the centre of mass of the system. Here, these two unit vectors are located in the principal central plane of inertia of the gyrostat, containing the axis of the undeformed rod. The deformations of the rod, which naturally depend on the orientation and the gyrostatic moment which ensures equilibrium of the chosen orientation (non-trival equilibrium since, in this case, generally speaking, the rod is deformed), and its stability in the Lyapunov sense are determined for the two single parameter families of uniaxial orientations of the system to an attracting centre which have been separated out in this way.

Universal closed-form of lagrangian multipliers for coast-arcs of optimum space trajectories

An universal closed-form solution of Lagrangian multipliers for the coast-arcs of optimum space trajectories in a Newtonian central force field is obtained by means of properties of generalized canonical systems and Sundman transformation. This closed-form solution, valid for all conics, is given as a function of a generalized anomaly.

Modern Approach to Classical Mechanics

: The Foundations of Mechanics One-Dimensional Motion of a Particle Encountering Peculiar Motion in Two Dimensions Motion in a Central Force Field The Gravitational Interaction of Two Bodies Collisions of Particles. Scattering Changing the Frame of Reference Lagrangian Mechanics Conservation Laws and Symmetries in Many Particle Systems The Rigid Body Small Oscillations Hamilton's Canonical Formulation of Mechanics Hamilton-Jacobi Theory From Integrable to Non-Integrable Systems.

The planar n-body problem: regular polygon solutions

In this paper, the planar n-body problem under central force fields is considered. By introducing appropriate matrices and under determinated central force fields, we prove that n masses ( n⩾4) at the vertices of a regular polygon, rotating about their common center of mass with an appropriate angular velocity, describe a periodic solution of the n-body problem if and only if the masses are equal.

The geoflow-experiment on ISS (Part II): Numerical simulation

Convection in spherical shells under the influence of a radial force field is an important problem in classical convection theory. It is difficult to reproduce in a terrestrial laboratory though, because there gravity is everywhere downwards rather than radially inwards. It turns out that one can produce a radial force field, by applying a high voltage difference between the inner and outer spheres. The combination of the electric field and the temperature-dependence of the fluid's dielectric coefficient then produces an r −5 central force field. Of course, in a terrestrial laboratory one still has the external gravity as well. In order to eliminate this effect and produce a purely radial force field, an experiment is planned on the International Space Station. In this paper we will present the results of numerical simulations intended to assist in the design of this experiment, for example in choosing the optimal radius ratios. We solve for the onset of convection in such an r −5 force field, as a function of the radius ratio (varying between 0.3 and 0.6), the Prandtl number (varying between 1 and 100), and the Taylor number (measuring an overall rotation of the whole shell, which will also be possible in the experiment).

The geoflow-experiment on ISS (part I): Experimental preparation and design of laboratory testing hardware

Thermal convection in a spherical shell represents an important model in fluid dynamics and geophysics. Investigations on thermal convective instabilities occuring in the spherical gap flow under terrestrial conditions are of basic importance especially for the understanding of symmetry-breaking bifurcations during the transition to chaos. Microgravity,experiments on thermal convection with a simulated central force field are important for the understanding of large scale geophysical motions as the convective transport phenomena in the Earth's liquid outer core. This report summarizes the concurrent experimental (part I), numerical (part II) and theoretical (part III) studies for the preparation of an International Space Station (ISS) experiment inside the Fluid Science Laboratory (FSL). This special experimental device with respect to geophysical simulations is called GEOFLOW. A central symmetric force field similar to the gravity field acting on planets can be produced using the effect of the dielectrophoretic force field by applying a high voltage potential difference to the inner and outer sphere. Flow visualization, Wollaston Shearing Interferometry and Laser Doppler Velocimetry will be used to determine the expected flow pattern during the space experiments.

Planar motion and stability for a rigid body with a beam in a field of central gravitational force

Planar motion for a rigid body with an elastic beam in a field of central gravitational force was investigated, and both of the orbital motion and attitude motion were under consideration. The equations of motion of the system were derived by the variational principle, and on view point of generalized Hamiltonian dynamics, the sufficient conditions for the stability of one class of relative equilibria were given by the energy-momentum method.

Planar motion and stability for a rigid body with a beam in a field of central gravitational force

Planar motion and stability for a rigid body with a beam in a field of central gravitational force