A fully general, non-perturbative treatment of impulsive heating

ABSTRACT Impulsive encounters between astrophysical objects are usually treated using the distant tide approximation (DTA) for which the impact parameter, b, is assumed to be significantly larger than the characteristic radii of the subject, rS, and the perturber, rP. The perturber potential is then expanded as a multipole series and truncated at the quadrupole term. When the perturber is more extended than the subject, this standard approach can be extended to the case where rS ≪ b < rP. However, for encounters with b of order rS or smaller, the DTA typically overpredicts the impulse, Δv, and hence the internal energy change of the subject, ΔEint. This is unfortunate, as these close encounters are the most interesting, potentially leading to tidal capture, mass stripping, or tidal disruption. Another drawback of the DTA is that ΔEint is proportional to the moment of inertia, which diverges unless the subject is truncated or has a density profile that falls off faster than r−5. To overcome these shortcomings, this paper presents a fully general, non-perturbative treatment of impulsive encounters which is valid for any impact parameter, and not hampered by divergence issues, thereby negating the necessity to truncate the subject. We present analytical expressions for Δv for a variety of perturber profiles, apply our formalism to both straight-path encounters and eccentric orbits, and discuss the mass-loss due to tidal shocks in gravitational encounters between equal-mass galaxies.

Rotating and non-rotating AdS black holes in $$f(\mathcal{T})$$ gravity non-linear electrodynamics

We derive new exact charged d-dimensional black hole solutions for quadratic teleparallel equivalent gravity, $$f(\mathcal{T})=a_0+a_1\mathcal{T}+a_2\mathcal{T}^2$$f(T)=a0+a1T+a2T2, where $$\mathcal T$$T is the torsion scalar, in the case of non-linear electrodynamics. We give a specific form of electromagnetic function and find out the form of the unknown functions that characterize the vielbeins in presence of the electromagnetic field. It is possible to show that the black holes behave asymptotically as AdS solutions and contain, in addition to the monopole and quadrupole terms, other higher order terms whose source is the non-linear electrodynamics field. We calculate the electromagnetic Maxwell field and show that our d-dimensional black hole solutions coincide with the previous obtained one (Awad et al. in J High Energy Phys 13:1706.01773, 2017). The structure of the solutions show that there is a central singularity that is much mild in comparison with the respective one in general relativity. Finally, the thermodynamical properties of the solutions are investigated by calculating the entropy, the Hawking temperature, the heat capacity, and other physical quantities. The most important result of thermodynamics is that the entropy is not proportional to the area of the black hole. This inanition points out that we must have a constrain on the quadrupole term to get a positive entropy otherwise we get a negative value.

Quantitative Susceptibility Mapping of Magnetic Quadrupole Moments

We modeled the magnetic field up to the quadrupole term to investigate not only the average susceptibility (dipole), but also the susceptibility distribution (quadrupole) contribution. Expanding the magnetic field up to the 2nd order provides the quadrupole (0th: monopole, 1st: dipole). Numerical simulations were performed to investigate the quadrupole contribution with subvoxel nonuniformity. Conventional dipole and our dipole + quadrupole models were compared in the simulation, the phantom and human brain. Furthermore, the quadrupole field was compared with the anisotropic susceptibility field in the dipole tensor model. In a nonuniformity case, numerical simulations showed a nonnegligible quadrupole field contribution. Our study showed a difference between the two methods in the susceptibility map at the edges; both the phantom and human studies showed sharper structural edges with the dipole + quadrupole model. Quadrupole moments showed contrast mainly at the structural boundaries. The quadrupole moment field contribution was smaller but nonnegligible compared to the anisotropic susceptibility contribution. Nonuniform and uniform source distributions can be separately considered by quadrupole expansion, which were mixed together in the dipole model. In the presence of nonuniformity, the susceptibility maps may be different between the two models. For a comprehensive field model, the quadrupole might need to be considered along with susceptibility anisotropy and microstructure effects.

High-frequency effective viscosity of a dilute suspension of particles in Poiseuille flow between parallel walls

It is shown that the formal expression for the effective viscosity of a dilute suspension of arbitrary-shaped particles in Poiseuille flow contains a novel quadrupole term, besides the expected stresslet. This term becomes important for a very confined geometry. For a high-frequency flow field (in the sense used in Feuillebois et al. (J. Fluid Mech., vol. 764, 2015, pp. 133–147), the suspension rheology is Newtonian at first order in volume fraction. The effective viscosity is calculated for suspensions of $N$-bead rods and of prolate spheroids with the same length, volume and aspect ratio (up to 6), entrained by the Poiseuille flow between two infinite parallel flat hard walls. The numerical computations, based on solving the Stokes equations, indicate that the quadrupole term gives a significant positive contribution to the intrinsic viscosity $[{\it\mu}]$ if the distance between the walls is less than ten times the particle width, or less. It is found that the intrinsic viscosity in bounded Poiseuille flow is generally smaller than the corresponding value in unbounded flow, except for extremely narrow gaps when it becomes larger because of lubrication effects. The intrinsic viscosity is at a minimum for a gap between walls of the order of 1.5–2 particle width. For spheroids, the intrinsic viscosity is generally smaller than for chains of beads with the same aspect ratio, but when normalized by its value in the bulk, the results are qualitatively the same. Therefore, a rigid chain of beads can serve as a simple model of an orthotropic particle with a more complicated shape. The important conclusion is that the intrinsic viscosity in shear flow is larger than in the Poiseuille flow between two walls, and the difference is significant even for relatively wide channels, e.g. three times wider than the particle length. For such confined geometries, the hydrodynamic interactions with the walls are significant and should be taken into account.

The effects of viscosity on sound radiation near solid surfaces

Although the acoustic analogy developed by Lighthill, Curle, and Ffowcs Williams and Hawkings for sound generation by unsteady flow past solid surfaces is formally exact, it has become accepted practice in aeroacoustics to use an approximate version in which viscous quadrupoles are neglected. Here we show that, when sound is radiated by non-rigid surfaces, and the smallest dimension is comparable to or less than the viscous penetration depth, neglect of the viscous-quadrupole term can cause large errors in the sound field. In addition, the interpretation of the viscous quadrupoles as contributing only to sound absorption is shown to be inaccurate. Comparisons are made with the scalar wave equation for linear waves in a viscous fluid, which is extended using generalized functions to describe the effects of solid surfaces. Results are also presented for two model problems, one in a half-space and one with simple cylindrical geometry, for which analytical solutions are available.

On the exterior magnetic field and silent sources in magnetoencephalography

Two main results are included in this paper. The first one deals with the leading asymptotic term of the magnetic field outside any conductive medium. In accord with physical reality, it is proved mathematically that the leading approximation is a quadrupole term which means that the conductive brain tissue weakens the intensity of the magnetic field outside the head. The second one concerns the orientation of the silent sources when the geometry of the brain model is not a sphere but an ellipsoid which provides the best possible mathematical approximation of the human brain. It is shown that what characterizes a dipole source as “silent” is not the collinearity of the dipole moment with its position vector, but the fact that the dipole moment lives in the Gaussian image space at the point where the position vector meets the surface of the ellipsoid. The appropriate representation for the spheroidal case is also included.

The Equations of Motion for Binary Systems with Relativistic Quadrupole-Quadrupole Moments Interaction

The study of binary systems is one of the most important problems in astronomy. Especially recently, gravitational wave detection made possible by Laser Interferometer Gravitational wave Observatory - LIGO and VIRGO, LISA opens up a completely new window for the observation of our universe, it becomes one of the most important and forward area in the modern general relativistic astrophysics. Coalescing binary neutron star (NS) systems are believed to be the most important source emitted high-frequency gravitational wave. Therefore the study of NS coalescence is regarded as a major challenge in modern relativistic astrophysics. Indeed, if the two-body problem could be solved with a sufficient accuracy, the wealth of information might be extracted from the waveforms of coalescing binaries. Many early works to derive and investigate the gravitational two-body system with spin and quadrupole moment interaction have been done already. A detail appraisal of their works has been made by Xu, Wu and Schäfer where they derived the first post-Newtonian equations of motion for binary systems with monopole, spin and quadrupole interaction by making use of the scheme developed by Damour, Soffel and Xu (DSX)As we know, in the last stages of coalescence in binary system, the distance between two stars is closer, the tidal force is stronger, so the nonspherical size l is larger, l2/r2 can reach the level of v2/c2, where r is the distance between two bodies. In this case, the relativistic qudrupole-quadrupole term is of 3-PN order, so one can not neglect the 1-PN contribution of the q-q terms when 3-PN equations of motion (for mass-monopole) are considered. In order to fit the requirement of more accurate solution for binary system, the relativistic q-q terms in the post Newtonian equation of motion have been calculated in this paper. Our work is the first to obtain explicit 1-PN equations of motion for binary systems with relativistic quadrupole-quadrupole interaction in terms of only collective coordinates and B-D moments.

New Results on CMB Structure from the Tenerife Experiments

Temperature fluctuations in the CMB (Cosmic Microwave Background) are a key prediction of cosmological models of structure formation in the early Universe. Observations at the Teide Observatory, Tenerife using radiometers operating at 10, 15 and 33 GHz have revealed individual hot and cold features in the microwave sky at high Galactic latitudes. These well-defined features are not atmospheric or Galactic in origin; they represent the first detection of individual primordial fluctuations in the CMB. Their intensity is defined by an intrinsicrmsamplitude of 54−10+14μK for a model with a coherence angle of 4°. The expected quadrupole term for a Harrison-Zel'dovich spectrum isQRMS–PS= 26 ± 6 μK. When our data at Dec=+40° are compared with the COBE DMR two-year data, the presence of individual features is confirmed. New experiments to detect structure on smaller scales are described.

SEMICLASSICAL DESCRIPTION OF SPIN EXCITATIONS OF THE PARTICLE-CORE INTERACTION SYSTEM

A model Hamiltonian describing a nucleon moving in a shifted oscillator well and inter-acting with a harmonic core through a quadrupole-quadrupole term is semiclassically treated. The solutions of the linearised equations of motion are quantised by the Bohr-Sommerfeld procedure. Among the four RPA modes, there is one which corresponds to the harmonic vibration of the spin degree of freedom. This mode is analogous to the spin wave mode describing a system of interacting spins placed in a magnetic field. The state describes a harmonic wobbling motion around a stationary state belonging to the ground rotational band.