Fraction Of Available Energy Research Articles

A constraint on pre-Main-Sequence mass loss

The principal energy source of a pre-Main-Sequence (PMS) star is gravitational contraction. Mass ejection, regardless of its actual mechanism, in effect transfers gravitational potential energy from the entire star to the escaping matter. The virial theorem limits the efficiency of such a process. Theoretical PMS mass-radius relationships limit the total loss to a few tenths of the initial mass. The observed luminosities and estimated mass loss rates of T-Tauri stars are used to estimate the fraction of available energy expended on mass ejection. Actual losses appear to be much smaller than the theoretical limit in most cases. Only stars of emission line intensity classes 4 and 5 may be capable of significant mass loss.

Rupture velocity and radiation efficiency

So long as there is some finite specific fracture energy involved in earthquake rupture, the rupture velocity cannot exceed the Rayleigh-wave velocity CR in the direction of mode II extension, or the S -wave velocity β in the direction of mode III extension. The radiation efficiency ηR, which is the fraction of available energy that goes into seismic waves, depends upon the rupture velocity. It is zero if the crack grows quasi-statically, increases with rupture velocity, and tends to unity as the velocity approaches the limiting value appropriate to the mode of extension. A model involving a semi-infinite crack in antiplane shear (mode III) which accelerates rapidly to a velocity v , and runs at this velocity until it is arrested by a barrier of higher fracture energy, yields a formula for estimating the specific fracture energy γ , γ 0 = L T 0 2 μ π β − V β + V where L is the length of extended rupture, T is the traction and μ is the rigidity of the elastic medium. This agrees, within a factor of 2 for 0 < V /β < 0.9, with the formula deduced by Husseini et al . (1975) for a model involving the “seismic gap” mode of arrest, γ 0 = L T 0 2 2 μ π . This formula therefore provides a reasonable estimate of the specific fracture energy involved in an earthquake that is insensitive to the mode of arrest of the rupture.

Cavitation dynamics: II. Free pulsations and models for cavitation bubbles

A mathematical formalism developed for investigating the dynamics of cavitation bubbles has been used to obtain numerical solutions describing the behavior of bubbles of different initial radii that are damped by heat conduction, viscosity, and compressibility. Calculations have been made to determine two measures of damping, the maximum temperatures, the maximum pressures, and the resonance frequencies of bubbles set into pulsations by a pressure pulse. In general, these quantities are controlled by heat conduction and viscosity at small amplitudes and mainly by compressibility at large amplitudes of motion. One measure of damping—the energy dissipation modulus—has a peak at a well-defined maximum radius. This peak serves to define a transition radius; for pulsations with amplitudes greater than this transition radius, the fraction of available energy dissipated in a cycle decreases and hence very large internal energy densities may occur. A second transition occurs at a radius called the critical radius; for pulsations with amplitudes greater than this critical radius, inertial forces control the collapse of a bubble and very large kinetic energy densities may occur in the liquid at the interface. The transition radius and the critical radius, which are functions of the initial size of a cavity, have been used to define a dynamical threshold for cavitation. In a motion where the maximum radius exceeds this proposed threshold, the characteristic phenomena of cavitation may be expected to appear. The dynamical threshold has in turn been used as the basis for models of cavitation bubbles useful in interpreting such phenomena. Subject Classification: 30.70, 30.60.

Constant energy partitioning in the photoionization of NO and O2

Published data on the photoelectron spectra of NO and O2 are analyzed, and the energy distributions are found to have averages proportional to the available energy. Hence, vibration takes a constant fraction of the energy. Average distributions are found as a function of the fraction of available energy.

Monte Carlo calculations of reaction rates and energy distributions among reaction products. IV. F+HF(ν) → HF(ν′)+F and F+DF(ν) → DF(ν′)+F

Rate constants are calculated for the vibrational relaxation HF(ν) and DF(ν) molecules by F atoms with ν = 1, 2, 3, and 6. Three-dimensional classical trajectories of the collision dynamics of these reactions were calculated by means of a modified London-Eyring-Polanyi-Sato (LEPS) potential energy surface used to calculate rate constants for the reactions between H atoms and F2 molecules, and D atoms and F2 molecules. The Monte Carlo procedure is used to start each collision trajectory. By means of this calculation, it is predicted for the reactant HF molecule in the ν = 2, J = 8 state that 13.3% of the mean fraction of available energy will become rotation in the product HF, 80.7% will become vibrational energy in the product HF, and 6% will become relative translational energy of the products. As the vibrational energy of the reactant HF molecule increases the mean fraction of available energy that becomes rotational energy increases slightly. For example, if the reactant molecule HF is in the ν = 6, J = 8 state, it is found that the mean fraction of available energy that will become rotational energy in HF is 15.8%, 75.3% will become vibrational energy in HF, and 8.9% will become translational energy in the product. Two different methods are used to calculate the rate constants kν→ν−1 for the de-excitation of vibrationally excited HF molecules by F atoms. The main mechanism for the de-excitation of vibrationally excited HF or DF molecules by F atoms is vibration-rotation energy transfer. Multiple quantum transitions occur in the de-excitation of higher (ν ≥slant 3) vibrationally excited states of HF molecules by F atoms. Chemical effects provide an important mechanism for the efficient vibrational relaxation of HF and DF molecules by F atoms. Rate constants are provided for many reactions that have not been measured experimentally.

Isotope and mass weighting effects in collinear triatomic reactive collisions

A one-dimensional quantum mechanical reaction coordinate method has been developed by Rankin and Light [J. Chem. Phys. 51, 1701 (1969)] and Miller and Light [J. Chem. Phys. 54, 1635, 1643 (1971)] to study vibrational excitation of product molecules in reactive triatomic collinear collisions. In this paper the method is extended to study the isotopic reactions H+Cl2→HCl+Cl and D+Cl2→DCl+Cl. The observed results showed there is no change in the mean fraction of available energy entering product vibration. This is in agreement with previous experimental and classical trajectory study findings. The method is further used to study a reaction with a much different mass combination Cl+HI→HCl+I. We expect this reaction, from classical trajectory studies, to exhibit primarily ``mixed'' energy release. Also from classical studies, the different mass combinations for the reaction H+Cl2→HCl+Cl and the reaction Cl+HI→HCl+I are expected to lead to significantly different reaction dynamics. The one-dimensional quantum mechanical results are consistent with the classical results.

Monte Carlo calculations of reaction rates and energy distribution among reaction products. II. H+HF(v) → H2 (v′) + F and H+HF(v) → HF(v′) + H

Rate constants are calculated for the reactions of atomic hydrogen with vibrationally excited hydrogen fluoride by analyzing the results of classical trajectories on a semiempirical London-Eyring-Polanyi-Sato (LEPS) potential energy surface. Monte Carlo procedures are used to start each collision trajectory. Reaction rate constants are presented for direct reactions into specific vibrational states of the product H2 and HF molecules. By means of this calculation, it is predicted for the reactant HF molecule in the ν = 3 state that 11.3% of the mean fraction of available energy will become vibrational energy in H2, 1.2% will become rotational energy in H2, and 87.5% will become translational energy in the products. As the vibrational energy of the reactant HF molecule increases, the mean fraction of available energy which becomes vibrational energy of the product H2 molecule increases. For example, if the reactant molecule HF is in the v = 6 state, it is found that the mean fraction of available energy that will become vibrational energy in H2 is 35.1%, 8% will become rotational in H2, and 56.9% will become translational energy in the products. For the reactant HF molecule in the v = 3 state, 45% of the mean fraction of available energy will become vibrational energy in the product HF molecule, 0.4% will become rotational energy in HF, and 54.8% will become translational energy in the products. The de-excitation of HF by H atoms is due to vibration-translation energy transfer. Results of this trajectory calculation, indicate that (1) multiple quantum jumps are significant in the reactions of H atoms with vibrationally excited HF, and (2) chemical effects provide an important mechanism for the efficient relaxation of vibrationally excited HF by H atoms. Both theory and experiment indicate that the rate of deactivation of vibrationally excited HF by H atoms is very fast. Rate constants are provided for many reactions that have not been measured experimentally.

Monte Carlo calculations of reaction rates and energy distributions among reaction products. III. H+F2→ HF+F and D+F2→ DF+F

A three-dimensional classical trajectory calculation is made of the collision dynamics of the exothermic reactions H+F2(v, J)→ HF(v′, J′)+F and D+F2(v, J)→ DF(v′, J′)+F by means of a modified London-Eyring-Polanyi-Sato (LEPS) potential energy surface. Trajectory calculations are used to establish the anti-Morse parameters for a modified LEPS potential-energy surface which produce the best agreement with previous experimental measurements of reaction rates and energy distributions among reaction products. Very good agreement is obtained with the experimental overall rate constant, the experimental prediction that the maximum vibrational level population of HF (v′) is achieved in v′ = 5, and the mean fraction of available energy entering vibration of the newly formed HF bond. The maximum vibrational level population of DF (v′) is achieved in v′ = 8. The mean fractions (f̄v′=Ev′/Etotal and f̄R′=EJ′/Etotal) of the total available energy entering vibration plus rotation are (1) for H+F2, f̄v′+f̄R′=(0.54+0.02)=0.56. (2) for D+F2, f̄v′+f̄R′=(0.56+0.02)=0.58. The relative rate constants for both reactions have a slight temperature dependence. Rate constants are presented for reactions directly into specific vibrational states of HF and DF.

Monte Carlo Calculations of Reaction Rates and Energy Distributions Among Reaction Products. I. F+H2→HF+H

A three-dimensional, classical trajectory calculation is made of the collision dynamics of the reaction F+H2(v, J)→HF(v′, J′)+H by means of the London-Eyring-Polanyi-Sato (LEPS) potential energy surface. Monte Carlo procedures are used to start each collision trajectory. A discussion is presented of the temperature dependence of the relative rates of formation of vibrationally excited hydrogen fluoride. By means of this calculation, it can be predicted that 71% of the mean fraction of available energy will become vibration in HF, 10.5% will become rotation in HF, and 18.5% will become translation in the product. The probability that direct chemical reaction between atomic fluorine and molecular hydrogen will lead to the formation of the product HF molecule in the ground vibrational state v′=0 was found to be zero. The ratio k(v′=3)/k(v′=2) appears to be independent of temperature and has a value of 0.49, which is in excellent agreement with available experimental data. The ratio k(v′=2)/k(v′=1) has a slight temperature dependence, which would indicate that the improved or increased inversion could be achieved at low temperatures. Reaction rate constants are presented for reactions directly into specific vibrational and rotational states of the product HF molecule. Translational energy distributions of the products are compared with available experimental data.