Arbitrary Mass Distribution Research Articles

Quasilinear static solutions in massive gravity

The static vacuum spherically symmetric solutions of massive gravity theories possess two integration constants: the mass $M$ and a scalar charge $S$. The presence of this scalar charge reflects the modification of the gravitational interaction as compared to general relativity. Surprisingly, these solutions have an asymptotic behavior different from that obtained in the linear perturbation theory. The aim of this paper is to understand how these modified spherically symmetric solutions emerge from a quasilinear approximation in order to generalize them to any arbitrary mass distribution. Along with these modified solutions, we found a new class of static solutions having a Yukawa shape.

Construction of Measure by Mass Distribution

In this note we show that a measure can be constructed on an arbitrary set by iterated arbitrary mass distribution over arbitrary subsets of the set.

Figure–figure interaction between bodies having arbitrary shapes and mass distributions: a power series expansion approach

We derive an expression for the mutual gravitational force and torque of two bodies having arbitrary shapes and mass distributions, as an expansion in power series of their products of inertia and of the relative coordinates of their centres of mass. The absolute convergence of all the power series developed is rigorously demonstrated. The absence of transcendental functions makes this formalism suitable for fast numerical applications. The products of inertia used here are directly related to the spherical harmonics coefficients, and we provide a detailed analysis of this relationship.

Phase models of the Milky Way stellar disc

We present a new iterative method for constructing equilibrium phase models of stellar systems. Importantly, this method can provide phase models with arbitrary mass distributions. The method is based on the following principle. Our task is to generate an equilibrium N-body system with a given mass distribution. For this purpose, we let the system reach equilibrium through its dynamical evolution. During this evolution we hold mass distribution in this system. This principle is realized in our method by means of an iterative procedure. We have used our method to construct a phase model of the disc of our Galaxy. In our method, we use the mass distribution in the Galaxy as input data. Here we used two Galactic density models (suggested by Flynn, Sommer-Larsen and Christensen and by Dehnen and Binney). For a fixed-mass model of the Galaxy we can construct a one-parameter family of equilibrium models of the Galactic disc. We can, however, choose a unique model using local kinematic parameters that are known from Hipparcos data. We show that the phase models constructed using our method are close to equilibrium. The problem of uniqueness for our models is discussed, and we discuss some further applications of our method.

Emergent gravity from noncommutative gauge theory

We show that the matrix-model action for noncommutative U(n) gauge theory actually describes SU(n) gauge theory coupled to gravity. This is elaborated in the 4-dimensional case. The SU(n) gauge fields as well as additional scalar fields couple to an effective metric G_{ab}, which is determined by a dynamical Poisson structure. The emergent gravity is intimately related to noncommutativity, encoding those degrees of freedom which are usually interpreted as U(1) gauge fields. This leads to a class of metrics which contains the physical degrees of freedom of gravitational waves, and allows to recover e.g. the Newtonian limit with arbitrary mass distribution. It also suggests a consistent picture of UV/IR mixing in terms of an induced gravity action. This should provide a suitable framework for quantizing gravity.

Numerical implementation of the exact dynamics of free rigid bodies

In this paper the exact analytical solution of the motion of a rigid body with arbitrary mass distribution is derived in the absence of forces or torques. The resulting expressions are cast into a form where the dependence of the motion on initial conditions is explicit and the equations governing the orientation of the body involve only real numbers. Based on these results, an efficient method to calculate the location and orientation of the rigid body at arbitrary times is presented. This implementation can be used to verify the accuracy of numerical integration schemes for rigid bodies, to serve as a building block for event-driven discontinuous molecular dynamics simulations of general rigid bodies, and for constructing symplectic integrators for rigid body dynamics.

Simulation of the full two rigid body problem using polyhedral mutual potential and potential derivatives approach

Herein we investigate the coupled orbital and rotational dynamics of two rigid bodies modelled as polyhedra, under the influence of their mutual gravitational potential. The bodies may possess any arbitrary shape and mass distribution. A method of calculating the mutual potential’s derivatives with respect to relative position and attitude is derived. Relative equations of motion for the two body system are presented and an implementation of the equations of motion with the potential gradients approach is described. Results obtained with this dynamic simulation software package are presented for multiple cases to validate the approach and illustrate its utility. This simulation capability is useful both for addressing questions in dynamical astronomy and for enabling spacecraft missions to binary asteroid systems.

Energy eigenvalues for the systems with position-dependent effective mass

Based on the analytical transfer matrix method (ATMM), the universal dispersion equation determining the energy spectrum for an arbitrary potential function experienced by a particle with arbitrary position-dependent effective mass distribution is obtained. As an illustrative example, for a given mass distribution m(x) and a potential function V(x), our predicted results are in complete agreement with the exact ones. The energy spectrum of a harmonic oscillator with nonconstant mass is obtained. Since the exact solutions to Schrodinger equation with nonconstant effective mass are a valuable means for checking and improving models and numerical methods in the study of electronic properties, the ATMM will find wide applications in material science and condensed matter.

Focusing of Radiation of Extensive Nonstationary Source by Means of Gravitational Lens with Arbitrary Mass Distribution

Focusing of Radiation of Extensive Nonstationary Source by Means of Gravitational Lens with Arbitrary Mass Distribution

Torsional Vibration of Non-Uniform Shafts Carrying an Arbitrary Number of Rigid Disks

A new exact approach for the analysis of torsional vibration of a non-uniform shaft carrying an arbitrary number of rigid disks is presented. The exact solutions for non-uniform shafts with arbitrary distribution of torsional stiffness or mass polar moment of inertia are obtained for several important cases. The fundamental solutions developed in this paper lead to a single frequency equation for a non-uniform shaft with classical or non-classical boundary conditions and carrying an arbitrary number of rigid disks.

Planetesimal Disk Evolution Driven by Planetesimal-Planetesimal Gravitational Scattering

We investigate the process of inhomogeneous planetesimal disk evolution caused by planetesimal-planetesimal gravitational scattering. We develop a rather general approach based on kinetic theory that self-consistently describes the evolution in time and space of both the disk's surface density and its kinematic properties—dispersions of eccentricity and inclination. The gravitational scattering of planetesimals is assumed to be in the dispersion-dominated regime, which considerably simplifies the analytical treatment. The resultant equations are of the advection-diffusion type. Distance-dependent scattering coefficients entering these equations are calculated analytically under the assumption of two-body scattering to leading order in the Coulomb logarithm. They are essentially nonlocal in nature. Our approach allows one to explore the dynamics of nonuniform planetesimal disks with arbitrary mass and random-velocity distributions. It can also naturally include other physical mechanisms that are important for the evolution of such disks—gas drag, migration, and so on.

Exact solutions for free vibration of shear-type structures with arbitrary distribution of mass or stiffness.

In this paper, shear-type structures such as frame buildings, etc., are treated as nonuniform shear beams (one-dimensional systems) in free-vibration analysis. The expression for describing the distribution of shear stiffness of a shear beam is arbitrary, and the distribution of mass is expressed as a functional relation with the distribution of shear stiffness, and vice versa. Using appropriate functional transformation, the governing differential equations for free vibration of nonuniform shear beams are reduced to Bessel's equations or ordinary differential equations with constant coefficients for several functional relations. Thus, classes of exact solutions for free vibrations of the shear beam with arbitrary distribution of stiffness or mass are obtained. The effect of taper on natural frequencies of nonuniform beams is investigated. Numerical examples show that the calculated natural frequencies and mode shapes of shear-type structures are in good agreement with the field measured data and those determined by the finite-element method and Ritz method.

Gravity probability tomography: a new tool for buried mass distribution imaging

Following the probability tomography principles previously introduced to image the sources of electric and electromagnetic anomalies, we demonstrate that a similar approach can be used to analyse gravity data. First, we give a coherent derivation of the Bouguer anomaly concept as a Newtonian‐type integral for an arbitrary mass distribution buried below a non‐flat topography. A discretized solution of this integral is then derived as a sum of elementary contributions, which are cross‐correlated with the gravity data function in the expression for the total power associated with the Bouguer anomaly. To image the mass distribution underground we introduce a mass contrast occurrence probability function using the cross‐correlation product of the observed Bouguer anomaly and the synthetic field due to an elementary mass contrast source. The tomographic procedure consists of scanning the subsurface with the elementary source and calculating the occurrence probability function at the nodes of a regular grid. The complete set of grid values is used to highlight the zones of highest probability of mass contrast concentrations. Some synthetic and field examples demonstrate the reliability and resolution of the new gravity tomographic approach.

A new exact approach for determining natural frequencies and mode shapes of non-uniform shear beams with arbitrary distribution of mass or stiffness

A new exact approach which combines the basic solutions with unit matrix property and recurrence formula for determining natural frequencies and mode shapes of non-uniform shear beams is presented in this paper. The function for describing the distribution of mass is arbitrary, and the distribution of shear stiffness is expressed as a functional relation with the mass distribution and vice versa. The governing equation for free vibration of a non-uniform shear beam is reduced to a differential equation of the second-order without the first-order derivative by means of functional transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for six cases. The exact solutions of mode shape functions are thus found. The basic solutions, which have a unit matrix property, are derived and used to obtain the frequency equations and mode shapes of multi-step shear beams with varying cross-sections. Numerical examples show that the calculated natural frequencies and mode shapes of two symmetric buildings are very close to the corresponding field measured data, suggesting that the proposed methods are applicable to engineering application and practice.

Estimating stem and root-anchorage flexibility in trees

This paper describes a nondestructive method for distinguishing root flexibility from stem flexibility in living trees. It is used here for Sitka spruce (Picea sitchensis (Bong.) Carrière.), but is applicable to any species where the main stem is normally straight and near-vertical. Well-known engineering equations permit the calculation of deflected shape for a vertical cantilever with arbitrary distribution of mass and bending stiffness, when infjected to a lateral force. The equations are used to calculate stem deflections of four Sitka spruce trees for which the stem and branch mass distribution and stem taper have been measured. Free parameters in the mathematical model are a nominal value of Young's Modulus E (assumed uniform and isotropic over the cross section and height of the tree stem) and a root-anchorage stiffness k. The former allows the stem to curve, whereas the latter represents the flexibility of the roots and allows the stem to tilt elastically at ground level. For each of the four trees, the calculated deflection curve is compared with actual deflections measured when the living tree is pulled by a rope at a specified point. By adjusting both E and k, iteratively, a best fit solution is obtained. This provides a simple and effective way to determine both stem stiffness and root hinge stiffness from a single experiment on a living tree.

TRANSVERSE VIBRATION OF MULTI MASS LOADED VARIABLE SECTION BEAMS UNDER RANDOM EXCITATION

Dynamic analysis of non-uniform section beams with general end conditions having arbitrary mass and stiffness distributions and proportional viscous damping distribution with several arbitrarily located point mass loadings has been presented for non-stationary random input excitation. Modal analysis has been employed to obtain expressions for second order response statistics after decoupling the system equations. The natural frequencies and mode shape functions for such beams are obtained in closed form as an intermediate step in the outlined approach. Theoretical results for natural frequencies have been compared with experimental evaluations and with data available in the literature. The complete approach is illustrated with an example of a cantilever beam subjected to support excitation.

Hanging shapes of nonuniform cables

Using a variational approach, I derive the equation governing the equilibrium shape of an inelastic cable with an arbitrary mass distribution hanging freely under uniform gravity except for being supported at both ends. I then develop a parametric solution to this shape equation, giving the coordinates of points on the curve assumed by the cable in terms of the distance along the cable. Using numerical integration to determine values, I plot several examples of the cable’s shape for linearly and exponentially increasing density. I also consider the inverse problem of determining the mass distribution which gives rise to a specified shape. I compare the distributions yielding a semicircle, a parabola, and a catenary.

On the tidal function between two real bodies

The general expression of the tidal function between two real bodies of arbitrary mass distribution has been derived. The case has been regarded when the distances between the bodies are comparable to their sizes. In this case, the effect of the body figure is substantial.The relations obtained are utilized for the tidal influence of Jupiter on the surface of Io. It has been shown that Jupiter's shape (with J2 = 0.015) causes a deformation in the sub-Jupiter point of about 15m. This is approximately 10% of the total tidal deformation of Io caused by Jupiter's gravitation.

Finite‐element analysis of electron waveguide discontinuities with arbitrary potential and effective mass distributions

AbstractFor the two‐dimensional discontinuity problems in an electron waveguide with arbitrary potential distributions and arbitrary effective mass distributions, an analysis method based on the finite‐element method is proposed and formulated. Specifically, the method is applied to a rectangular bend in an electron waveguide so that its validity and effectiveness are presented. Also, a formulation is carried out for a more practical electron wave circuit in which the potential outside the waveguide is finite. The formulation is applied to a rib‐type discontinuity in an electron waveguide and the transmission characteristics are analyzed. The results are compared with those for the case in which the potential outside the waveguide is infinite.

Acceleration sensitivity of crystal resonators affected by the mass and location of electrodes

Predominant thickness-shear frequencies and modes of a crystal plate with electrodes of arbitrary shape and mass distribution are obtained by a finite-element method, based on Mindlin's first-order equations with platings. These frequencies and modes are used in a perturbation method for computing the acceleration sensitivity of crystal resonators with electrodes. Computations are made for a square AT-cut quartz plate that is supported by a four-point mount and coated with identical square and uniform electrodes on the upper and lower faces of the plate. To study the effect of uneven distribution of electrode mass, acceleration sensitivities are calculated when a small mass is added at various locations near the edges of the square electrodes. It is found that the percent increase of the acceleration sensitivity of the resonator with a small added mass to that of the resonator without added mass ranges from 3.8% to 541.7%, depending on the location of the small mass placed at the edges of the electrodes.