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A approximate analytic solution to the low thrust constant acceleration energy increase trajectory using a tangential steering program is presented. The solution is based on a two-variable asymptotic expansion of the equations of motion formulated in regularized variables. The high accuracy of the second order solution derived is demonstrated by comparing it with numerically integrated trajectories.
An analytical solution is obtained for the problem of free and forced vibrations of a finite Euler Bernoulli beam with arbitrary (partially fixed) boundary conditions. The effects of linear viscous damping, Winkler foundation, constant axial tension, a concentrated mass, and an arbitrary forcing function are included in the analysis. No restriction is placed on the values of the parameters involved, and the solution presented here contains all cited previous solutions as special cases.
We consider collective emission of a single photon from a cloud of N two-level atoms (one excited, N-1 ground state). For a dense cloud the problem is reduced to finding eigenfunctions and eigenvalues of an integral equation. We discuss an exact analytical solution of this many-atom problem for a spherically symmetric atomic cloud. Some eigenstates decay much faster then the single atom decay rate, while the others undergo very slow decay. We show that virtual processes yield a small effect on the evolution of rapidly decaying states. However, they change the long time dynamics from exponential decay into a power-law behavior which can be observed experimentally. For trapped states virtual processes are much more important yielding additional decay channels which results in a slow decay of the otherwise trapped states. We also show that quantum mechanical treatment of spontaneous emission of weakly excited atomic ensemble is analogous to emission of N classical harmonic oscillators.
We propose an analytical solution of the multi-density Ornstein-Zernike equation supplemented by the associative Percus-Yevick closure relations specifically designed to describe the equilibrium properties of the novel class of patchy colloidal particles represented by the inverse patchy colloids with arbitrary number of patches. Using Baxter’s factorization method, we reduce solution of the problem to the solution of one nonlinear algebraic equation for the fraction of the particles with one non-bonded patch. We present closed-form expressions for the structure (structure factor) and thermodynamic (internal energy) properties of the system in terms of this fraction (and parameters of the model). We perform computer simulation studies and compare theoretical and computer simulation predictions for the pair distribution function, internal energy, and number of single and double bonds formed in the system, for two versions of the model, each with two and three patches. We consider the models with formation of the double bonds blocked by the patch-patch repulsion and the models without patch-patch repulsion. In general very good agreement between theoretical and computer simulation results is observed.
Planetary obliquity determines the meridional distribution of the annual mean insolation. For obliquity exceeding 55°, the weakest insolation occurs at the equator. Stable partial snow and ice cover on such a planet would be in the form of a belt about the equator rather than polar caps. An analytical model of planetary climate is used to investigate the stability of ice caps and ice belts over the widest possible range of parameters. The model is a non-dimensional diffusive Energy Balance Model, representing insolation, heat transport, and ice−albedo feedback on a spherical planet. A complete analytical solution for any obliquity is given and validated against numerical solutions of a seasonal model in the “deep-water” regime of weak seasonal ice line migration. Multiple equilibria and unstable transitions between climate states (ice-free, Snowball, or ice cap/belt) are found over wide swaths of parameter space, including a “Large Ice-Belt Instability” and “Small Ice-Belt Instability” at high obliquity. The Snowball catastrophe is avoided at weak radiative forcing in two different scenarios: weak albedo feedback and inefficient heat transport (favoring stable partial ice cover), or efficient transport at high obliquity (favoring ice-free conditions). From speculative assumptions about distributions of planetary parameters, three-fourths to four-fifths of all planets with stable partial ice cover should be in the form of Earth-like polar caps.
We study theoretically the optical properties of an exciton in a two-dimensional ring threaded by a magnetic flux. We model the quantum ring by a confining potential that can be continuously tuned from strictly one-dimensional to truly two-dimensional with finite radius-to-width ratio. We present an analytic solution of the problem when the electron-hole interaction is short ranged. The oscillatory dependence of the oscillator strength as a function of the magnetic flux is attributed to the Aharonov-Bohm effect. The amplitude of the oscillations changes upon increasing the width of the quantum ring. We find that the Aharonov-Bohm oscillations of the ground state of the exciton decrease with increasing the width, but, remarkably, the amplitude remains finite down to radius-to-width ratios less than unity. We attribute this resilience of the excitonic oscillations to the nonsimple connectedness of our chosen confinement potential with its centrifugal core at the origin.
In terms of group theory—the language of symmetries, the concept of spontaneous symmetry breaking is represented in terms of chains of group-subgroup structures that define the dynamical symmetry of the system under consideration. This framework enables exact analytic solutions of the associated eigenvalue problems. We review two types of applications of dynamical symmetries in contemporary theoretical nuclear structure physics: first for a classification of the many-body systems under consideration, with respect to an important characteristic of their behavior; and second for the creation of exactly solvable algebraic models that describe specific aspects of this behavior. This is illustrated with the boson and fermion realizations of symplectic structures. In the first case with an application of the sp(4, R) classification scheme of even-even nuclei within the major nuclear shells and next with of the sp(4) microscopic model for the description of isovector pairing correlations.
The shape phase transition from spherical to soft deformed at N=88-90 was observed long ago. After the prediction of the X(5) symmetry, for which analytical solution of the nuclear Hamiltonian is given [1], good examples of X(5) nuclei were identified in the N=90 isotones of Nd, Sm, Gd and Dy, in the recent works. The N=90 isotones have almost the similar deformed level structure, forming the isotonic multiplet in Z=50-66, N=82-104 quadrant. This is explained microscopically in terms of the Nilsson level diagram. Using the Dynamic Pairing-Plus-Quadrupole model of Kumar-Baranger, the quadrupole deformation and the occupancies of the neutrons and protons in these nuclei have been calculated, which support the formation of N=88, 90 isotonic multiplets. The existence of F-spin multiplets in Z=66-82, N=82-104 quadrant, identified in earlier works on the Interacting Boson Model, is also explained in our study.
We previously demonstrated an extension of time-domain thermoreflectance (TDTR) which utilizes offset pump and probe laser locations to measure in-plane thermal transport properties of multilayers. However, the technique was limited to systems of transversely isotropic materials studied using axisymmetric laser intensities. Here, we extend the mathematics so that data reduction can be performed on non-transversely isotropic systems. An analytic solution of the diffusion equation for an N-layer system is given, where each layer has a homogenous but otherwise arbitrary thermal conductivity tensor and the illuminating spots have arbitrary intensity profiles. As a demonstration, we use both TDTR and time-resolved magneto-optic Kerr effect measurements to obtain thermal conductivity tensor elements of <110> α-SiO{sub 2}. We show that the out-of-phase beam offset sweep has full-width half-maxima that contains nearly independent sensitivity to the in-plane thermal conductivity corresponding to the scanning direction. Also, we demonstrate a Nb-V alloy as a low thermal conductivity TDTR transducer layer that helps improve the accuracy of in-plane measurements.
We present exact analytical solutions describing spatial bright, trigonometric, and kink-type of self-similar waves, as well as the trains of such waves to the nonlinear Schroedinger equation appropriately modified to model beam propagation in graded-index, nonlinear fiber amplifier with an external source. We show that this model is appropriate for the self-similar propagation in asymmetric twin-core fiber amplifier. The asymmetric twin-core fiber is composed of two adjoining, closely spaced, single mode fibers in which the active one is a graded-index nonlinear fiber and the passive one is a step-index fiber. We obtain these self-similar waves using a Moebius transformation. Our procedure is applicable both for self-focusing and self-defocusing Kerr nonlinearities.
We develop an R-matrix approach to treating collision processes which explicitly takes into account, by means of a simple energy-dependent analytic function, the out-of-phase oscillations of the incident and scattered standing waves in the interior region. Thereby we avoid the use of the Bloch operator. In place of the Bloch operator the incident wave provides the source term in an inhomogeneous equation for the scattered wave. We take those subchannels not treated exactly into account via the optical potential, which is generally non-Hermitian due to dissipation at the boundary. The optical potential is constructed on a real analytic basis using a resolvent that satisfies outgoing-wave boundary conditions. The use of an analytic basis together with the direct determination of the K matrix, rather than the R matrix, at the boundary (this is done by matching the interior wave function to the nearly exact analytic solution beyond the boundary) makes the method particularly well suited to the treatment of ultracold collisions. We have tested our method by applying it to one-photon single-ionization of (He1s{sup 2}) accompanied by excitation to He{sup +}(2s) or He{sup +}(2p) for photon energies above the complete breakup threshold, where the optical potential is non-Hermitian. Excellent agreement with experiment is obtained for the cross sections for photoionization to both He{sup +} (n=1) and to He{sup +} (n=2). The 2s-to-2p branching ratio is strongly influenced by both the optical potential and, at photon energies less than a few tens of eV above the breakup threshold, the nonadiabatic dipole mixing of the 2s and 2p states.
Irradiation-induced Pyrocarbon (PyC) creep affects the stresses of a TRISO-coated particle. An analytical solution with a 4-parameter linear viscoelastic model was used to investigate the effects of PyC creep on the particle stresses. PyC transient and steady-state creep effects on particle stresses were analyzed. PyC creep parameter effects on the particle stresses were studied, including PyC creep coefficient, creep Poisson s ratio in the steady-state component, and Young s modulus in the transient component. The results showed that PyC irradiation-induced creep decreased the particle stresses. Transient creep affected particle stresses only at the irradiation early stage. Lower Young s modulus and higher creep coefficient in the transient component resulted in a larger stresses decrease, and large Poisson s ratio and creep coefficient in the steady-state creep decreased particle stresses. (authors)
Upscaling permeability of grid blocks is crucial for groundwater models. A novel upscaling method for three-dimensional fractured porous rocks is presented. The objective of the study was to compare this method with the commonly used Oda upscaling method and the volume averaging method. First, the multiple boundary method and its computational framework were defined for three-dimensional stochastic fracture networks. Then, the different upscaling methods were compared for a set of rotated fractures, for tortuous fractures, and for two discrete fracture networks. The results computed by the multiple boundary method are comparable with those of the other two methods and fit best the analytical solution for a set of rotated fractures. The errors in flow rate of the equivalent fracture model decrease when using the multiple boundary method. Furthermore, the errors of the equivalent fracture models increase from well-connected fracture networks to poorly connected ones. Finally, the diagonal components of the equivalent permeability tensors tend to follow a normal or log-normal distribution for the well-connected fracture network model with infinite fracture size. By contrast, they exhibit a power-law distribution for the poorly connected fracture network with multiple scale fractures. The study demonstrates the accuracy and the flexibility of the multiple boundary upscaling concept. This makes it attractive for being incorporated into any existing flow-based upscaling procedures, which helps in reducing the uncertainty of groundwater models.
Resonant scattering of fast particles off low frequency plasma waves is a major process determining transport characteristics of energetic particles in the heliosphere and contributing to their acceleration. Usually, only Alfvén waves are considered for this process, although dispersive waves are also present throughout the heliosphere. We investigate resonant interaction of energetic electrons with dispersive, right-handed waves. For the interaction of particles and a single wave a variable transformation into the rest frame of the wave can be performed. Here, well-established analytic models derived in the framework of magnetostatic quasi-linear theory can be used as a reference to validate simulation results. However, this approach fails as soon as several dispersive waves are involved. Based on analytic solutions modeling the scattering amplitude in the magnetostatic limit, we present an approach to modify these equations for use in the plasma frame. Thereby we aim at a description of particle scattering in the presence of several waves. A particle-in-cell code is employed to study wave–particle scattering on a micro-physically correct level and to test the modified model equations. We investigate the interactions of electrons at different energies (from 1 keV to 1 MeV) and right-handed waves with various amplitudes. Differences between model and simulation arise in the case of high amplitudes or several waves. Analyzing the trajectories of single particles we find no microscopic diffusion in the case of a single plasma wave, although a broadening of the particle distribution can be observed.
The observation of an increasing number of anomalous diffusion phenomena motivates the study to reveal the actual reason for such stochastic processes. When it is difficult to get analytical solutions or necessary to track the trajectory of particles, lattice Monte Carlo (LMC) simulation has been shown to be particularly useful. To develop such an LMC simulation algorithm for the Galilei variant anomalous diffusion, we derive explicit solutions for the conditional and unconditional first passage time (FPT) distributions with double absorbing barriers. According to the theory of random walks on lattices and the FPT distributions, we propose an LMC simulation algorithm and prove that such LMC simulation can reproduce both the mean and the mean square displacement exactly in the long-time limit. However, the error introduced in the second moment of the displacement diverges according to a power law as the simulation time progresses. We give an explicit criterion for choosing a small enough lattice step to limit the error within the specified tolerance. We further validate the LMC simulation algorithm and confirm the theoretical error analysis through numerical simulations. The numerical results agree with our theoretical predictions very well.
We propose a formulation of quantum mechanics for a finite level system whose potential function is not realizable and/or analytic solution of the wave equation is not feasible. The system's wavefunction is written as an infinite sum in a complete set of square integrable functions. Interaction in the theory is introduced in function space by a real finite tridiagonal symmetric matrix. The expansion coefficients of the wavefunction satisfy a three-term recursion relation incorporating the parameters of the interaction. Information about the structure and dynamics of the system is contained in the scattering matrix, which is defined in the usual way. The bound state energy spectrum (system's structure) is finite. Apart from the 2M- 1 dimensionless parameters of the interaction matrix, whose rank is M, the theory has one additional scale parameter. In the development, we utilize the kinematic tools of the J-matrix method.
In this paper, we consider two outstanding intertwined problems in modern high-energy astrophysics: (1) the vertical-thermal structure of an optically thick accretion disk heated by the dissipation of magnetohydrodynamic turbulence driven by the magnetorotational instability (MRI), and (2) determining the fraction of the accretion power released in the corona above the disk. For simplicity, we consider a gas-pressure-dominated disk and assume a constant opacity. We argue that the local turbulent dissipation rate due to the disruption of the MRI channel flows by secondary parasitic instabilities should be uniform across most of the disk, almost up to the disk photosphere. We then obtain a self-consistent analytical solution for the vertical thermal structure of the disk, governed by the balance between the heating by MRI turbulence and the cooling by radiative diffusion. Next, we argue that the coronal power fraction is determined by the competition between the Parker instability, viewed as a parasitic instability feeding off of MRI channel flows, and other parasitic instabilities. We show that the Parker instability inevitably becomes important near the disk surface, leading to a certain lower limit on the coronal power. While most of the analysis in this paper focuses on the case of a disk threaded by an externally imposed vertical magnetic field, we also discuss the zero net flux case, in which the magnetic field is produced by the MRI dynamo itself, and show that most of our arguments and conclusions should be valid in this case as well.
We present a new multi-dimensional radiation-hydrodynamics code for massive stellar core-collapse in full general relativity (GR). Employing an M1 analytical closure scheme, we solve spectral neutrino transport of the radiation energy and momentum based on a truncated moment formalism. Regarding neutrino opacities, we take into account a baseline set in state-of-the-art simulations, in which inelastic neutrino–electron scattering, thermal neutrino production via pair annihilation, and nucleon–nucleon bremsstrahlung are included. While the Einstein field equations and the spatial advection terms in the radiation-hydrodynamics equations are evolved explicitly, the source terms due to neutrino–matter interactions and energy shift in the radiation moment equations are integrated implicitly by an iteration method. To verify our code, we first perform a series of standard radiation tests with analytical solutions that include the check of gravitational redshift and Doppler shift. A good agreement in these tests supports the reliability of the GR multi-energy neutrino transport scheme. We then conduct several test simulations of core-collapse, bounce, and shock stall of a 15M{sub ⊙} star in the Cartesian coordinates and make a detailed comparison with published results. Our code performs quite well to reproduce the results of full Boltzmann neutrino transport especially before bounce. In the postbounce phase, our code basically performs well, however, there are several differences that are most likely to come from the insufficient spatial resolution in our current 3D-GR models. For clarifying the resolution dependence and extending the code comparison in the late postbounce phase, we discuss that next-generation Exaflops class supercomputers are needed at least.