Modeling of aggregation of fractal dust clusters in a laminar protoplanetary disk

The evolutionary hydrodynamic model for the formation and growth of loose dust aggregates in the aerodisperse medium of a laminar disk, which was originally comprised of the gas and solid (sub)micrometer particles, is considered as applied to the problem of the formation of planetesimals in the Solar protoplanetary cloud. The model takes into account the fractal properties of dust clusters. It is shown that the clusters partly merge in the process of cluster-cluster coagulation, giving rise to the formation of large fractal aggregates that are the basic structure-forming elements of loose protoplanetesimals arising as a result of physicochemical and hydrodynamic processes similar to the processes of growth of the fractal clusters. Earlier, the modeling was conventionally performed in an “ordinary” continuous medium without considering the multifractional structure of the dust component of the protoplanetary cloud and the fractal nature of the dust clusters being formed during its evolution. Instead, we propose to consider a complex of loose dust aggregates as a special type of continuous medium, namely, the fractal medium for which there exist points and regions that are not filled with its particles. We suggest performing the hydrodynamic modeling of this medium, which has a noninteger mass dimensionality, in a fractional integral model (its differential form) that takes the fractality into account using fractional integrals whose order is determined by a fractal dimensionality of the disk medium.     

Modification of the jeans instability criterion for fractal-structure astrophysical objects in the framework of nonextensive statistics

Unlike classical studies in which the gravitational instability criterion for astrophysical disks is derived in the framework of traditional kinetics or hydrodynamics, we propose to consider the totality of fluffy dust clusters of various astrophysical objects, in particular, protoplanetary subdisks, as a special type of continuous medium, i.e., fractal medium for which there are points and areas not filled with its components. Within the deformed Tsallis statistics formalism, which is intended to describe the behavior of anomalous systems with strong gravitational interaction and fractal nature of phase space, we derive, on the basis of the modified kinetic equation (with the collision integral in the Bhatnagar-Gross-Krook form), the generalized hydrodynamic Euler equations for a medium with the fractal mass dimension. Considering the linearization of the q-hydrodynamics equations, we investigate the instability of an infinitely homogeneous medium to obtain a simplified version of the modified gravitational instability criterion for an astrophysical disk with fractal structure.     

Stochastic gravitational fluctuations of stellar or galactic systems with a fractal distribution of field sources

The stochastic gravitational fluctuations for a fractal mass distribution are analyzed by means of a functional integral approach. A general method is developed for evaluating the stochastic properties of vectorial additive random fields generated by a variable number of point sources obeying inhomogeneous Poisson statistics. A closed expression for the generating functional of the field is given in terms of the generating functional of the sources. The moments of the resulting vectorial field are finite if the correlation functions of the sources have short tails. In this case all cumulants of the field can be computed exactly: they are averages of the central moments of sources computed in terms of the probability density of the position of a source. The method is applied for analyzing the stochastic gravitational fluctuations generated by a fractal distribution of field sources (stars or galaxies). For a Newtonian force law the correlation functions of the sources are slowly decaying, the cumulants of the stochastic gravitational field are infinite and the probability density of the field intensityF is given by a Lévy fractal stable law with a scaling exponentH depending on the fractal dimensiond _f of the distribution of stars or galaxies:H =d _f /2.     

Evolution of the fractal structure of interstellar clouds in the galaxy

The internal stability of the fractal structure of interstellar clouds in the Galaxy is considered. The conditions under which gravitational interactions between components lead to disruption of the fractal structure are clarified. Model numerical calculations and analytical estimates of the characteristic time of decay of structures are made. It is found that this time is on the same order as the lifetime of interstellar molecular clouds determined from observations. Translated from Astrofizika, Vol. 41, No. 1, pp. 81–90, January-March, 1998.     

The gravitational instability in the dust layer of a protoplanetary disk:: axisymmetric solutions for nonuniform dust density distributions in the direction vertical to the midplane

The gravitational instability in the dust layer of a protoplanetary disk with nonuniform dust density distributions in the direction vertical to the midplane is investigated. The linear analysis of the gravitational instability is performed. The following assumptions are used: (1) One fluid model is adopted, that is, difference of velocities between dust and gas are neglected. (2) The gas is incompressible. (3) Models are axisymmetric with respect to the rotation axis of the disk. Numerical results show that the critical density at the midplane is higher than the one for the uniform dust density distribution by Sekiya (1983, Prog. Theor. Phys. 69, 1116-1130). For the Gaussian dust density distribution, the critical density is 1.3 times higher, although we do not consider this dust density distribution to be realistic because of the shear instability in the dust layer. For the dust density distribution with a constant Richardson number, which is considered to be realized due to the shear instability, the critical density is 2.85 times higher and is independent of the value of the Richardson number. Further, if a constant Richardson number could decrease to the order of 0.001, the gravitational instability would be realized even for the dust to gas surface density ratio with the solar abundance. Our results give a new restriction on planetesimal formation by the gravitational instability.     

Approaching a homogeneous galaxy distribution: results from the Stromlo–APM redshift survey

Recent results from a number of redshift surveys suggest that the Universe is well described by an inhomogeneous, fractal distribution on the largest scales probed. This distribution has been found to have fractal dimension, D , approximately equal to 2.1, in contrast to a homogeneous distribution in which the dimension should approach the value 3 as the scale is increased. In this paper we demonstrate that estimates of D , based on the conditional density of galaxies, are prone to bias from several sources. These biases generally result in a smaller measured fractal dimension than the true dimension of the sample. We illustrate this behaviour in application to the Stromlo–APM redshift survey, showing that this data set in fact provides evidence for fractal dimension increasing with survey depth. On the largest scale probed, r ≈60  h ⁻¹ Mpc, we find evidence for a distribution with dimension D =2.76±0.10. A comparison between this sample and mock Stromlo–APM catalogues taken from N -body simulations (which assume a CDM cosmology) reveals a striking similarity in the behaviour of the fractal dimension. Thus we find no evidence for inhomogeneity in excess of that expected from conventional cosmological theory. We consider biases affecting future large surveys and demonstrate, using mock SDSS catalogues, that this survey will be able to measure the fractal dimension on scales at which we expect to see full turn-over to homogeneity, in an accurate and unbiased way.     

Mutual gravitational potential,force, and torque of a homogeneous polyhedron and an extended body: an application to binary asteroids

Binary systems are quite common within the populations of near-Earth asteroids, main-belt asteroids, and Kuiper belt asteroids. The dynamics of binary systems, which can be modeled as the full two-body problem, is a fundamental problem for their evolution and the design of relevant space missions. This paper proposes a new shape-based model for the mutual gravitational potential of binary asteroids, differing from prior approaches such as inertia integrals, spherical harmonics, or symmetric trace-free tensors. One asteroid is modeled as a homogeneous polyhedron, while the other is modeled as an extended rigid body with arbitrary mass distribution. Since the potential of the polyhedron is precisely described in a closed form, the mutual gravitational potential can be formulated as a volume integral over the extended body. By using Taylor expansion, the mutual potential is then derived in terms of inertia integrals of the extended body, derivatives of the polyhedron’s potential, and the relative location and orientation between the two bodies. The gravitational forces and torques acting on the two bodies described in the body-fixed frame of the polyhedron are derived in the form of a second-order expansion. The gravitational model is then used to simulate the evolution of the binary asteroid (66391) 1999 KW4, and compared with previous results in the literature.     

Fractal properties of solar magnetic fields

We study the spatial properties of solar magnetic fields using data from the Solar Vector Magnetograph of the Marshall Space Flight Center (MSFC) (FeI 5250.2 Å) and SOHO/MDI longitudinal magnetic field measurements (Ni 6767.8 Å) (96-min full-disk maps). Our study is focused on two objects: the fractal properties of sunspots and the fractal properties of the spatial magnetic field distribution of active and quiet regions considered as global structures. To study the spatial structure of sunspots, we use a well-known method of determining the fractal dimension based on an analysis of the perimeter—area relation. To analyze the fractal properties of the spatial magnetic field distribution over the solar surface, we use a technique developed by Higuchi. We have revealed the existence of three families of self-similar contours corresponding to the sunspot umbra, penumbra, and adjacent photosphere. The fractal coefficient has maxima near the umbra—penumbra and penumbra—photosphere boundaries. The fractal dependences of the longitudinal and transverse magnetic field distributions are similar, but the fractal numbers themselves for the transverse fields are larger than those for the longitudinal fields approximately by a factor of 1.5. The fractal numbers decrease with increasing mean magnetic field strength, implying that the magnetic field distribution is more regular in active regions.     

Modification of the Jeans and Toomre instability criteria for astrophysical fractal objects within nonextensive statistics

Within the formalism of Tsallis nonextensive statistics designed to describe the behavior of anomalous systems, systems with a strong gravitational interaction between their individual parts and the fractal nature of phase space, we have obtained linearized equations for the oscillations of a rigidly rotating disk by taking into account dissipative effects and give a derivation of the dispersion equation in the WKB approximation. Based on the previously derived modified Navier—Stokes hydrodynamic equations (the so-called equations of q-hydrodynamics), we have analyzed the axisymmetric oscillations of an astrophysical, differentially rotating gas—dust cosmic object and obtained modified Jeans and Toomre gravitational instability criteria for disks with a fractal phase-space structure.     

Capillary action in a crack on the surface of asteroids with an application to 433 Eros

Some asteroids contain water ice, and a space mission landing on an asteroid may take liquid to the surface of the asteroid. Gas pressure is very weak on the surface of asteroids. Here we consider the capillary action in a crack on the surface of irregular asteroids. The crack is modeled as a capillary which has a fixed radius. An asteroid's irregular gravitational potential influences the height of the liquid in the capillary. The height of the liquid in the capillary on the surface of such asteroids is derived from the asteroid's irregular gravitational potential. Capillary mechanisms are expected to produce an inhomogeneaous distribution of emergent liquid on the surface. This result is applied to asteroid 433 Eros, which has an irregular, elongated, and concave shape. Two cases are considered: (1) we calculate the height of the liquid in the capillary when the direction of the capillary is perpendicular to the local surface of the asteroid; (2) we calculate the height of the liquid in the capillary when the direction of the capillary is parallel to the vector from the center of mass to the surface position. The projected height in the capillary on the local surface of the asteroid seems to depend on the assumed direction of the capillary.     

Stochastic-thermodynamic modeling of turbulent chaos with nonzero memory using fractional Fokker—Planck—Kolmogorov equations

A stochastic-thermodynamic approach to the derivation of the generalized fractional Fokker—Planck—Kolmogorov (FFPK) equations is considered. The equations describe turbulent transfer processes in a subsystem of turbulent chaos on the basis of fractional dynamics, which takes into account the structure and metric of fractal time. The actual turbulent motion of a fluid is known to be intermittent, since it demonstrates the properties that are intermediate between the properties of regular and chaotic motions. On the other hand, the process of the flow turbulization may be non-Markovian because of the multidimensional spatiotemporal correlations of pulsating parameters; in a physical language, this means that the process has a memory. The introduction of fractional time derivatives into the FFPK kinetic equations, used to find the probability distribution functions for different statistical characteristics of structured turbulence, makes it possible to use an unified mathematical formalism in considering the effects of memory, nonlocality, and time intermittence, with which we usually associate the presence of turbulent bursts against the background of less intense low-frequency oscillations in the background turbulence. This study is aimed at creating representative models of space and natural media. It is a development of the synergetic approach to the modeling of structured turbulence in astrogeophysical systems, which has been developed by the author in a series of papers (Kolesnichenko, 2002–2005).     

Polarization of the CMB Photons Crossing the Galaxy Clusters

In this paper, I investigate a local effect of polarization of the Cosmic Microwave Background (CMB) in clusters of galaxies, induced by the Thomson scattering of an anisotropic radiation. A local anisotropy of the CMB is produced by some scattering and gravitational effects, as, for instance, the Sunyaev Zel‘dovich effect, the Doppler shift due to the cluster motion and the gravitational lensing. The resulting anisotropy ΔI/I depends on the physical properties of the clusters, in particular on their emissivity in the X band on their size, on their gravitational potential and on the peculiar conditions characterizing the gas they contain. By solving the Boltzmann radiative transfer equation in presence of such anisotropies I calculate the average polarization at the centre of some clusters, namelyA2218, A576 and A2163, whose properties are quite well known. I prove that the gravitational effects due to the contraction or to the expansion have some importance, particularly for high density structures; moreover, the peculiar motion of the cluster, considered as a gravitational lens, influences the propagation of the CMB photons by introducing a particular angular dependence in the gravitational anisotropy and in the scattering integrals. Thus, the gravitational and the scattering effects overally produce an appreciable local average polarization of the CMB, may be observable through a careful polarization measurements towards the centres of the galaxy clusters. This revised version was published online in July 2006 with corrections to the Cover Date.     

Complete closed-form solutions of the Stark problem

Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direction. For both the two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions.     

Evolution of the cometary cloud under the action of stellar perturbations

The probability of variation of the integrals of the orbit as a result of an encounter was found for a two dimensional system. A method of solution of the Kolmogorov-Feller's equation is obtained using this probability function as a kernel, and it allows us to obtain the distribution of the integrals of the orbit as a function of time. The method is applied to the investigation of the evolution of orbits in the outer cometary cloud under the action of galactic stars. We consider the variations of orbits as a purely discontinuous random process, so we take into account not only distant but also close interactions.     

Application of the Gauss' Method to the Stellar Three Body Problem

In this paper, we deal with the stellar three body problem, that is one star is far away from the other two stars. The outer orbit is assumed to be Keplerian. To analyze the effect of the distant star on the orbit of the close stars, we use the Gauss method; this method consist in replacing the gravitational attraction of the third star by the gravitational attraction of an infinitesimal non-homogeneous elliptic ring. We obtain the force vector for the Gauss method in terms of elliptic integrals. Finally we compare the results obtained by this model with the classical third body model. This revised version was published online in August 2006 with corrections to the Cover Date.     

Convection in the mantle and its effect upon the earth's gravitational field

The failure of an equilibrium model to provide an adequate representation of the Earth's external gravitational field suggests that one should consider a more general hydrodynamical model for the interior of the terrestrial globe, and the most probable cause of motion, which may significantly effect the distribution of density inside the Earth, is convection throughout the mantle. In the present paper we investigate the effects of convection in the mantle on the gravitational field of the Earth and calculate the velocity of convection necessary to account for the observed characteristics of the external gravitational field.     

Mutual gravitational potential and torque of solid bodies via inertia integrals

The mutual gravitational potential and the mutual gravitational torque of two bodies of arbitrary shape are expanded to the fourth order. The derivations are based on Cartesian coordinates, inertia integrals with relation to the principal reference frames of each body, and the relative rotation matrix. The current formulation is convenient to utilize in high precision problems in rotational dynamics.     

Streaming Instability in the Gas-Dust Medium of the Protoplanetary Disc and the Formation of Fractal Dust Clusters

The sequence of evolution of the protoplanetary gas-and-dust disk around the parent star includes, according to modern concepts, its compression in the central plane and decay into separate dust condensations (clusters) due to the occurrence of various types of instabilities. The interaction of dust clusters of a fractal structure during their collisions is considered as a key mechanism for the formation and growth of primary solids, which serve as the basis for the subsequent formation of planetesimals and embryos of planets. Among the mechanisms contributing to the formation of planetesimals, an important place belongs, along with gravitational instability, hydrodynamic instabilities, in particular, the socalled streaming instability of the two-phase gas-dust layer due to its ability to concentrate dispersed particles in dense clots. In contrast to a number of existing models of streaming instability, in which dust particles are considered structurally compact and monodisperse, this paper proposes a more realistic model of polydisperse particles of fractal nature, forming dust clusters as a result of coagulation. The instability of the dust layer in the central plane of the protoplanetary disk under linear axisymmetric perturbations of its parameters is considered. A preliminary conclusion can be drawn that the proposed model of dust fractal aggregates of different scales increases the efficiency of linear growth of hydrodynamic instabilities, including the streaming instabilities associated with the difference between the velocities of the dust and gas phases.

     

由SLOAN观测数据测试大尺度结构的均匀性

宇宙学的基本假设之一是宇宙在大尺度上均匀各向同性.为了验证星系分布在大尺度上的均匀性,分别计算观测样本和观测空间几何体的分形维数,得到SDSS-DR4中星系分布的分形维数.观测空间几何体的分形维数用随机样本来确定.样本中的星系红移z的范围为0.01-0.26.当尺度持续增加至几十个Mpc时,星系分布的分形维数一致地趋向于3.所有的样本均显示了明显的转变尺度,当尺度大于此转变尺度时,星系分布的分形维数D<,G>～3,星系的分布转变为均匀分布.结果支持了宇宙学的基本原理关于宇宙大尺度均匀的假设.样本的转变尺度随着样本的光度增强而变大,说明小尺度上星系的分布不是简单的分形分布,而是多维分形分布.高光度星系的转变尺度非常大,直到100h-1Mpc左右才变得均匀.