Sphere Of Uniform Density Research Articles

An approximate analytical solution for radial evolution of homologous collapse

ABSTRACT The homologous collapse of a sphere of uniform density from a resting state under its self-gravity has been used to model the formation of astronomical objects. It is well known that the evolution of the radius with time cannot be obtained explicitly because of the need to solve a transcendental equation of cycloidal parameter with respect to time. By combining the Padé approximation and the Schröder formula, we construct an approximate analytical solution of radius as a function of time. Our method is a direct method rather than an iteration method and it requires only solving a cubic equation and evaluating three trigonometric functions. Compared with the existing methods, the accuracy and effectiveness of this method are clearly illustrated.

A uniform spherical goat (problem): explicit solution for homologous collapse’s radial evolution in time

ABSTRACT The homologous collapse from rest of a uniform density sphere under its self gravity is a well-known toy model for the formation dynamics of astronomical objects ranging from stars to galaxies. Equally well-known is that the evolution of the radius with time cannot be explicitly obtained because of the transcendental nature of the differential equation solution. Rather, both radius and time are written parametrically in terms of the development angle θ. We here present an explicit integral solution for radius as a function of time, exploiting methods from complex analysis recently applied to the mathematically similar ‘geometric goat problem.’ Our solution can be efficiently evaluated using a Fast Fourier Transform and allows for arbitrary sampling in time, with a simple python implementation that is $\sim \, 100\times$ faster than using numerical root-finding to achieve arbitrary sampling. Our explicit solution is advantageous relative to the usual approach of first generating a uniform grid in θ, since this latter results in a non-uniform radial or time sampling, less useful for applications such as generation of sub-grid physics models. This solution is also of interest because it is the e = 1 case of Kepler’s equation. Our subsequent work extends the method here to produce a general one for solving Kepler’s equation, which turns out to be faster than any other currently extant.

Do simulated molecular clouds look like real ones?

ABSTRACT Simulations of molecular clouds often begin from highly idealized initial conditions, such as a uniform-density sphere with an artificially imposed turbulent velocity field. While the resulting structures may appear qualitatively similar to those detected in continuum and line observations, it is unclear whether they are genuinely representative of real molecular clouds. Recent observational work has discovered a tight, often close-to-linear relationship between the integrated intensity of molecular lines and the total column density of the cloud material. We combine magnetohydrodynamical simulations, time-dependent chemistry, and radiative transfer to produce synthetic molecular line observations of model clouds. We find similarly tight correlations between line intensity and column density to those observed, although the linear behaviour is only seen in isolated (as opposed to colliding) model clouds. This linear relationship is not due to optically thin emission; all lines investigated have high optical depths, and the increase in integrated intensity with column density is due to higher velocity dispersion along the line of sight. Overall, the idealized models commonly used in the literature appear to be reasonably accurate representations of real molecular clouds.

Clarifying some common misconceptions about the expansion of the universe

Modern cosmology is based on a 6-parameter model, called Lambda-CDM, which provides an accurate description of the expansion of the Universe on the basis of measured parameters. However, there are attempts to give ‘phenomenological’ descriptions of the dynamics of the Universe that are inconsistent with this model and sometimes lead to misconceptions. They stem from a Newtonian approach that interprets the expansion as due to an ‘initial kick’ that placed all matter and energy in the Universe on ‘inertial trajectories’. These descriptions do not conform to the relativistic approach. Here we show that the Einstein field equations trace the expansion of cosmic space directly to the increase in the energy content of the observable universe caused by the additional space entering it as a consequence of the forward motion of its horizon. We derive here the connection between variations in the scale factor and the corresponding variations in the energy of the observable universe, in the case of the most general cosmological model. We then obtain explicit closed-form expressions of this relationship both in the case of the Universe of any curvature and containing matter and radiation, and in the case of the de Sitter universe, characterized by the cosmological constant only. Some interesting features are pointed out, both in terms of the energy growth rate of the observable universe in the various cosmological models, and in terms of the dynamics of the scale factor. Finally our demonstration is corroborated by an analogy between the expansion of cosmic space and the gravitational stretching of space at the center of a sphere of uniform density whose radius increases with time. This analogy, which allowed us to obtain the novel closed-form expression of the stretching of space at the centre of such a sphere, confirmed that the paradigm that ‘gravity expands space’ is valid in both systems.

On the numerical integration of an explicit solution of the homologous collapse’s radial evolution in time

ABSTRACT In a recent paper, Slepian and Philcox derive an explicit solution of the homologous collapse from rest of a uniform density sphere under its self-gravity as a function of time. Their solution is given in terms of two curvilinear integrals along a suitable Jordan contour; in practice, it must be approximated by a quadrature rule. The aim of this paper is to examine how the choice of the contour and the quadrature rule affects the accuracy and the efficiency of this integral solution approximation. More precisely, after a study of the complex roots of a transcendental equation that relates time with the variable, some alternative Jordan contours that turn out to be more convenient are proposed. Then, by using as quadrature rule the composite trapezoidal rule because of its reliability and spectral convergence accuracy, some numerical experiments are presented to show that the combination of contours and quadrature rule allows us to obtain numerical results with high accuracy and low computational cost.

Self-consistent dynamical models with a finite extent – I. The uniform density sphere

ABSTRACT The standard method to generate dynamical models with a finite extent is to apply a truncation in binding energy to the distribution function (DF). This approach has the disadvantages that one cannot choose the density to start with, that the important dynamical quantities cannot be calculated analytically, and that a fraction of the possible bound orbits are excluded a priori. We explore another route and start from a truncation in radius rather than a truncation in binding energy. We focus on the simplest truncated density profile, the uniform density sphere. We explore the most common inversion techniques to generate DFs for the uniform density sphere, corresponding to a large range of possible anisotropy profiles. We find that the uniform density sphere cannot be supported by the standard isotropic, constant anisotropy or Osipkov–Merritt models, as all these models are characterized by negative DFs. We generalize the Cuddeford inversion method to models with a tangential anisotropy and present a one-parameter family of dynamical models for the uniform density sphere. Each member of this family is characterized by an anisotropy profile that smoothly decreases from an arbitrary value β0 ≤ 0 at the centre to completely tangential at the outer radius. All models have a positive DF over the entire phase space, and a non-zero occupancy of all possible bound orbits. This shows that one can generate non-trivial self-consistent dynamical models based on preset density profile with a finite extent.

Multilayer Spherical Stellar Clusters with Uniform Density

Various solutions of the kinetic equation for the equilibrium of a gravitating sphere of uniform density with a quadratic gravitational potential and a linear dependence of gravitational force on radius are examined. New analytic solutions are obtained for a uniform sphere with a hollow spherical volume and central mass inside the sphere. Solutions are also obtained for an arbitrary number of spherical layers with the same density, but with different equilibrium distribution functions.

Electron confinement meet electron delocalization: non-additivity and finite-size effects in the polarizabilities and dispersion coefficients of the fullerenes.

In this work, we used finite-field derivative techniques and density functional theory (DFT) to compute the static isotropic polarizability series (αl with l = 1, 2, 3) for the C60-C84 fullerenes and quantitatively assess the intrinsic non-additivity in these fundamental response properties. By comparing against classical models of the fullerenes as conducting spherical shells (or solid spheres) of uniform electron density, a detailed critical analysis of the derived effective scaling laws (α1 ∼ N1.2, α2 ∼ N2.0, α3 ∼ N2.7) demonstrates that the electronic structure of finite-sized fullerenes-a unique dichotomy of electron confinement and delocalization effects due to their quasi-spherical cage-like structures and encapsulated void spaces-simultaneously limits and enhances their quantum mechanical response to electric field perturbations. Corresponding frequency-dependent polarizabilities were obtained by inputting the αl series into the hollow sphere model (within the modified single frequency approximation), and used to compute the molecular dispersion coefficients (Cn with n = 6, 8, 9, 10) needed to describe the non-trivial van der Waals (vdW) interactions in fullerene-based systems. Using first-order perturbation theory in conjunction with >140 000 DFT calculations, we also computed the non-negligible zero-point vibrational contributions to α1 in C60 and C70, thereby enabling a more accurate and direct comparison between theory and experiment for these quintessential nanostructures.

First evidence of inertial modes in γ Doradus stars: The core rotation revealed

The advent of space photometry with CoRoT and Kepler has allowed for the gathering of exquisite and extensive time series for a wealth of main-sequence stars, including γ Doradus stars, whose detailed seismology was not achievable from the ground. γ Doradus stars present an incredibly rich pulsation spectra, with gravito-inertial modes, in some cases supplemented with δ Scuti-like pressure modes – for the hybrid stars – and, in many cases, with Rossby modes. The present paper aims to show that in addition to these modes which have been established in the radiative envelope, pure inertial modes that are trapped in the convective core can be detected in Kepler observations of γ Doradus stars thanks to their resonance with the gravito-inertial modes. We started by using a simplified model of perturbations in a full sphere of uniform density. Under these conditions, the spectrum of pure inertial modes is known from analytical solutions of the so-called Poincaré equation. We then computed coupling factors, which helped select the pure inertial modes which interact best with the surrounding dipolar gravito-inertial modes. Using complete calculations of gravito-inertial modes in realistic models of γ Doradus stars, we are able to show that the pure inertial and gravito-inertial resonances appear as “dips” in the gravito-inertial mode period spacing series at spin parameters that are close to those predicted by the simple model. We find the first evidence of such dips in the Kepler γ Doradus star KIC 5608334. Finally, using complete calculations in isolated convective cores, we find that the spin parameters of the pure inertial and gravito-inertial resonances are also sensitive to the density stratification of the convective core. In conclusion, we have discovered that certain dips in gravito-inertial mode period spacings that have been observed in some Kepler stars are, in fact, signatures of resonances with pure-inertial modes that are trapped in the convective core. This holds the promise that it would be possible to finally access the central conditions, namely, the rotation and density stratification, of intermediate-mass stars in the main sequence.

Klein–Gordon equation in curved space-time

We report the methods and results of a computational physics project on the solution of the relativistic Klein–Gordon equation for a light particle gravitationally bound to a heavy central mass. The gravitational interaction is prescribed by the metric of a spherically symmetric space-time. Metrics are considered for an impenetrable sphere, a soft sphere of uniform density, and a soft sphere with a linear transition from constant to zero density; in each case the radius of the central mass is chosen to be sufficient to avoid any event horizon. The solutions are obtained numerically and compared with nonrelativistic Coulomb-type solutions, both directly and in perturbation theory, to study the general-relativistic corrections to the quantum solutions for a 1/r potential. The density profile with a linear transition is chosen to avoid singularities in the wave equation that can be caused by a discontinuous derivative of the density. This project should be of interest to instructors and students of computational physics at the graduate and advanced undergraduate levels.

A Hybrid Quantum-Classical Model of Electrostatics in Multiply Charged Quantum Dots

We present a general model for describing the properties of excess electrons in multiply charged quantum dots (QDs). Key factors governing Fermi-level energies and electron density distributions are investigated by treating carrier densities, charge compensation, and various material and dielectric medium properties as independently tunable parameters. Electronic interactions are described using a mean-field electrostatic potential calculable through Gauss's Law by treating the quantum dot as a sphere of uniform charge density. This classical approximation modifies the Particle in a Sphere'' Schrodinger equation for a square well potential and reproduces the broken degeneracy and Fermi-level energies expected from experiment and first-principles methods. Several important implications emerge from this model: (i) Excess electron density drifts substantially towards the QD surfaces with high electron densities and large radii, and when solvated by a high dielectric medium. (ii) The maximum density of condu...

Colloidal droplets of the concentrated phase in aqueous methanol solutions of a hydrophilically modified poly(dimethylsiloxane)

When water is added to dilute MeOH solutions of a hydrophilically modified poly(dimethylsiloxane) (HPM-PDMS), molecularly dispersed HPM-PDMS chains associate to form colloidal droplets of the concentrated phase, which are charged spheres of uniform density with the radius of gyration less than 100 nm. Those droplets are remarkably different from those in solutions of HPM-PDMS directly dissolved in MeOH–water mixtures with0.5 and including 0.1 m NaAc, which may contain the coexisting dilute phase.

Are turbulent spheres suitable initial conditions for star-forming clouds?

Abstract To date, most numerical simulations of molecular clouds, and star formation within them, assume a uniform density sphere or box with an imposed turbulent velocity field. In this work, we select molecular clouds from galactic scale simulations as initial conditions, increase their resolution, and re-simulate them using the smoothed particle hydrodynamics code gadget2. Our approach provides clouds with morphologies, internal structures and kinematics that constitute more consistent and realistic initial conditions for simulations of star formation. We perform comparisons between molecular clouds derived from a galactic simulation, and spheres of turbulent gas of similar dimensions, mass and velocity dispersion. We focus on properties of the clouds such as their density, velocity structure and star formation rate. We find that the inherited velocity structure of the galactic clouds has a significant impact on the star formation rate and evolution of the cloud. Our results indicate that, although we can follow the time evolution of star formation in any simulated cloud, capturing the entire history is difficult as we ignore any star formation that might have occurred before initialization. Overall, the turbulent spheres do not match the complexity of the galactic clouds.

Incompressible Einstein–Maxwell fluids with specified electric fields

The Einstein–Maxwell equations describing static charged spheres with uniform density and variable electric field intensity are studied. The special case of constant electric field is also studied. The evolution of the model is governed by a hypergeometric differential equation which has a general solution in terms of special functions. Several classes of exact solutions are identified which may be considered as charged generalizations of the incompressible Schwarzschild interior model. An analysis of the physical features is undertaken for the uniform case. It is demonstrated that uniform density spheres with constant electric field intensity are not realizable with isotropic pressures. This highlights the necessity of studying the criteria for physical admissability of gravitating spheres in general relativity which are solutions to the Einstein–Maxwell equations.

ON THE INTERNAL DYNAMICS OF STARLESS CORES: STABILITY OF STARLESS CORES WITH INTERNAL MOTIONS AND COLLAPSE DYNAMICS

In order to understand the collapse dynamics of observed low-mass starless cores, we revise the conventional stability condition of hydrostatic Bonnor-Ebert spheres to take internal motions into account. Because observed starless cores resemble Bonnor-Ebert density structures, the stability and dynamics of the starless cores are frequently analyzed by comparing to the conventional stability condition of a hydrostatic Bonnor-Ebert sphere. However, starless cores are not hydrostatic but have observed internal motions. In this study, we take gaseous spheres with a homologous internal velocity field and derive stability conditions of the spheres utilizing a virial analysis. We propose two limiting models of spontaneous gravitational collapse: the collapse of critical Bonnor-Ebert spheres and uniform density spheres. The collapse of these two limiting models are intended to provide the lower and the upper limits, respectively, of the infall speeds for a given density structure. The results of our study suggest that the stability condition sensitively depends on internal motions. A homologous inward motion with a transonic speed can reduce the critical size compared to the static Bonnor-Ebert sphere by more than a factor of two. As an application of the two limiting models of spontaneous gravitational collapse, we compare the density structures and infall speeds of the observed starless cores L63, L1544, L1689B, and L694-2 to the two limiting models. L1689B and L694-2 seem to have been perturbed to result in faster infall motions than for spontaneous gravitational collapse.

COLLAPSE AND FRAGMENTATION OF MAGNETIC MOLECULAR CLOUD CORES WITH THE ENZO AMR MHD CODE. I. UNIFORM DENSITY SPHERES

Magnetic fields are important contributers to the dynamics of collapsing molecular cloud cores, and can have a major effect on whether collapse results in a single protostar or fragmentation into a binary or multiple protostar system. New models are presented of the collapse of magnetic cloud cores using the adaptive mesh refinement (AMR) code Enzo2.0. The code was used to calculate the ideal magnetohydrodynamics (MHD) of initially spherical, uniform density and rotation clouds with density perturbations, i.e., the Boss and Bodenheimer (1979) standard isothermal test case for three dimensional (3D) hydrodynamics (HD) codes. After first verifying that Enzo reproduces the binary fragmentation expected for the non-magnetic test case, a large set of models was computed with varied initial magnetic field strengths and directions with respect to the cloud core axis of rotation (parallel or perpendicular), density perturbation amplitudes, and equations of state. Three significantly different outcomes resulted: (1) contraction without sustained collapse, forming a denser cloud core, (2) collapse to form a single protostar with significant spiral arms, and (3) collapse and fragmentation into binary or multiple protostar systems, with multiple spiral arms. Comparisons are also made with previous MHD calculations of similar clouds with barotropic equations of state. These results for the collapse of initially uniform density spheres illustrate the central importance of both magnetic field direction and field strength for determining the outcome of dynamic protostellar collapse.

Zeta potential of colloidal particle in solvent primitive model electrolyte solution: a density functional theory study

A systematic study of zeta potential for a spherical double layer (SDL) around a colloidal particle in electrolyte solutions, is performed using density functional theory and Monte Carlo simulation. The usual recipe under the solvent primitive model is employed to model the system, where macroion, counterions, and coions are represented by charged hard spheres of uniform charge density and the presence of solvent is taken into account by modelling it as neutral hard spheres. All the components of the system are embedded in a dielectric continuum in order to consider the electrostatic effect of the solvent. The density functional theory employs a suitable weighted density approximation to calculate the hard-sphere contribution, whereas the residual electrostatic interactions are calculated as a small perturbation around the uniform fluid. The zeta potential profiles of a SDL in the presence of a number of electrolytes have been calculated and are found to be considerably influenced in the presence of solvent with an increase in the concentration of the electrolyte. The theory successfully predicts the maxima and sign reversal of the zeta potential profiles at high macroion surface charge density and in the presence of multivalent counterions, as obtained from the Monte Carlo simulation.

Hydrodynamic forces on steady and oscillating porous particles

AbstractWe derive new analytical results for the hydrodynamic force exerted on a sinusoidally oscillating porous shell and a sphere of uniform density in the Stokes limit. The coupling between the spherical particle and the solvent is done using the Debye–Bueche–Brinkman (DBB) model, i.e. by a frictional force proportional to the local velocity difference between the permeable particle and the solvent. We compare our analytical results and existing dynamic theories to lattice–Boltzmann simulations of the full Navier–Stokes equations for the oscillating porous particle. We find our analytical results to agree with simulations over a broad range of porosities and frequencies.

Accurate van der Waals coefficients from density functional theory

The van der Waals interaction is a weak, long-range correlation, arising from quantum electronic charge fluctuations. This interaction affects many properties of materials. A simple and yet accurate estimate of this effect will facilitate computer simulation of complex molecular materials and drug design. Here we develop a fast approach for accurate evaluation of dynamic multipole polarizabilities and van der Waals (vdW) coefficients of all orders from the electron density and static multipole polarizabilities of each atom or other spherical object, without empirical fitting. Our dynamic polarizabilities (dipole, quadrupole, octupole, etc.) are exact in the zero- and high-frequency limits, and exact at all frequencies for a metallic sphere of uniform density. Our theory predicts dynamic multipole polarizabilities in excellent agreement with more expensive many-body methods, and yields therefrom vdW coefficientsC6,C8,C10for atom pairs with a mean absolute relative error of only 3%.

Uniform density static fluid sphere in Einstein-Gauss-Bonnet gravity and its universality

In Newtonian theory, gravity inside a constant density static sphere is independent of spacetime dimension. Interestingly this general result is also carried over to Einsteinian as well as higher order Einstein-Gauss-Bonnet (Lovelock) gravity notwithstanding their nonlinearity. We prove that the necessary and sufficient condition for universality of Schwarzschild interior solution describing a uniform density sphere for all $n\geq4$ is that its density is constant.