The collinear central configuration of n equal masses

Moulton's Theorem says that given an ordering of masses m ₁, ..., m _nthere exists a unique collinear central configuration. The theorem allows us to ask the questions: What is the distribution of n equal masses in the collinear central configuration? What is the behavior of the distribution as n → ∞? These questions are due to R. Moekel (personal conversation). Central configurations are found to be attracting fixed points of a flow — a flow we might call an auxiliary flow (in the text it is denoted F(X)), since it has little to do with the equations of motion. This flow is studied in an effort to characterize the mass distribution. Specifically, for a collinear central configuration of n equal masses, a bound is found for the position of the masses furthest from the center of mass. Also some facts concerning the distribution of the inner masses are discovered.     

Bifurcation of central configuration in the 2N+1 body problem

The bifurcation of central configuration in the Newtonian N-body problem for any odd number N ≥ 7 is shown. We study a special case where 2n particles of mass m on the vertices of two different coplanar and concentric regular n-gons (rosette configuration) and an additional particle of mass m₀ at the center are governed by the gravitational law he 2n+1 body problem. This system is of two degrees of freedom and permits only one mass parameter μ =m ₀/m. This parameter μ controls the bifurcation. If n≥ 3, namely any odd N ≥ 7, then the number of central configurations is three when μ ≥ μ _c , and one when μ ≥ μ _c . By combining the results of the preceding studies and our main theorem, explicit examples of bifurcating central configuration are obtained for N ≤ 13, for any odd N ∈ [15,943], and for any N ≥ 945.     

A non singular solution for spherical configuration with infinite central density

A non-singular exact solution with an infinite central density is obtained for the interior of spherically symmetric and static structures. Both the energy density and the pressure are infinite at the center but we have e ^λ(0)=1 and e ^ν(0)≠0. The solution admits the possibility of receiving signals from the region of infinite pressure.     

Restricted four-body problem. The case of a central configuration: Libration points and their stability

We consider the problem of the motion of a zero-mass body in the vicinity of a system of three gravitating bodies forming a central configuration.We study the case where two gravitating bodies of equal mass lie on the same straight line and rotate around the central body with the same angular velocity. Equations for calculating the equilibrium positions in this system have been derived. The stability of the equilibrium points for a system of three gravitating bodies is investigated. We show that, as in the case of libration points for two bodies, the collinear points are unstable; for the triangular points, there exists a ratio of the mass of the central body to the masses of the extreme bodies, 11.720349, at which stability is observed.     

The stability and uniqueness of coronal loops

A hydrodynamic model of high resolution is used to examine the stability of coronal loops to finite amplitude perturbations. The loop is heated by means of a low-amplitude energy input and its subsequent dynamic relaxation is followed.Firstly, the initial atmosphere is generated by solving the time independent form of the hydrodynamic equations. It is shown that the loop structure depends critically on the balance between the radiative losses and the quiescent heating at the base of the transition zone, i.e. on the concavity of the temperature profile in this region. This result already anticipates the need for high spatial resolution across the model transition zone.The dynamic evolution of the loop is then investigated for two classes of lower boundary conditions. In one case the chromospheric temperature is fixed throughout the simulation; in the other the low chromosphere is represented by a rigid insulating barrier. In both cases the loop is found to be stable: The loop is also unique to the extent that it relaxes to a state which is physically indistinguishable from its initial configuration. It is pointed out however, that a loop whose chromosphere is only marginally stable can evolve dynamically away from the initial static configuration.Finally, the observational consequences of the analysis are discussed. The differential emission measure profile is found to change its form as the loop cools, firstly, through an evaporative phase in which the coronal density increases; secondly, through a quasi-steady relaxation in which the enhanced coronal density gradually drains away to the chromosphere. This behaviour represents a possible observational test of the model.     

Existence and uniqueness of stellar equilibrium models

The stellar equilibrium equations for given surface pressureP _* and temperatureT _*, and in the absence of convection, are translated into a nonlinear integral equation, in which the radiusR of the star enters as an eigenvalue. We show that under broad mathematical assumptions on the constitutive equations (equation of state, opacity and energy generation) a global existence and uniqueness property can be formulated. If a valueP _M is selected, which restricts the allowed pressure and temperature range |P(r)–P _*|+E|T(r)–T _*P _M (E, arbitrary constant of dimensions of a pressure over temperature), thenat least one solutionP(r),T(r) exists in the pressure-temperature range chosen, for anyR<R _E . This solution isunique forR<R _c .R _E andR _c are expressed in terms of the constitutive equations, and of the pressure-temperature range adopted. A physical argument in favour of the stability of this solution is presented.     

On the uniqueness of image reconstruction from the amplitude of radiointerferometric response

The theorem is proved that presents a rigorous foundation of the possibility of unique reconstruction of the radio-image of the sourceI(r) from the modulus of visibility function |V(b|=|Î(r)|. This results gives a rigorous explanation of the efficiency of different numerical methods of the image reconstruction using only amplitude data which are widespread in radio-astronomy.     

On the stationary configuration of the heliospheric sheet

The expansion of solar coronal plasma is considered for the model described in Koutchmy et al. (1999). In addition to a spherical solar surface, the initial configuration represents a heliospheric sheet of dense plasma in the dipole equatorial plane. The heliospheric-sheet current decreases with distance as 1/r ², with its sign being opposite to the sign of the initial-dipole current. The latter follows from the fact that the plasma sheet is denser than the surrounding corona and that the equilibrium condition for the sheet in the gravitational and magnetic fields is satisfied. The field lines of this configuration are nearly straight. We have obtained a general solution of the steady-state MHD equations, which depends not only on distance r but also on latitude θ. Applicability of the solution to interpreting observational data, in particular, those obtained from the Ulysses spacecraft, is discussed.     

A model of the heliospheric magnetic field configuration

A three-dimensional model of the magnetic field configuration in the heliosphere is constructed by assuming that the interplanetary magnetic field consists of four components, (i) the solar dipole, (ii) a large number of small spherical dipoles located along an equatorial circle just inside the Sun (representing the magnetic field line arcade), (iii) the field of the poloidal current system generated by the solar unipolar induction and (iv) the field of an extensive current disc around the Sun lying in the ecliptic plane. The magnetic field intensity at a distance of 1 A.U. (about 20 R⊙ above the ecliptic plane) is normalized to fit the observed spiral configuration.     

On the ballistic configuration during interferometric imaging

Types of ballistic configuration of the satellite formation for Earth remote sensing for interferometric imaging using a synthetic aperture radar antenna are considered. Possible configurations for multiposition satellite formations are provided.     

The configuration of the loop-like structure of the solar atmosphere

An integral, governing steady flows in an isolated thin magnetic flux tube in the hydrostatic plane-stratified atmosphere, has been obtained. The integral, that we named as the shape integral, is expressed as (1 − M_A²)B cos θ = const. Here M_A² is the Alfven Mach number, B is the magnetic field strength and θ is the flux tube inclination to the horizontal. The shape integral should hold for most loop models because it represents just the momentum balance laws and has no relation to any energy balance mode. Its application to the isothermal and static cases is discussed and illustrated.     

Application of the atomic configuration probability to the equilibrium calculation

This is an extension study of the grand canonical approach of the equation of state to the equilibrium calculation for radiative opacities. We discuss the construction of atomic configurations from the ion stages and excitation states of atoms. The atomic configuration is essentially important to the grand canonical approach, and also useful to the radiative opacity calculation. We present the complete formalism for the calculation of atomic configuration probabilities and population fractions. We show that the grand canonical approach provides a shortcut in the computation of thermodynamic equilibrium. We mainly discuss its use in two cases for the equilibrium calculation.     

On the uniqueness of the determination of the coronal potential magnetic field from line-of-sight boundary conditions

We consider a simple model in which the coronal magnetic field B is assumed to be potential in the region between the solar surface _o and an exterior source-surface ₁ of arbitrary shape. We prove that the boundary value problem that determines B from the value B _lof its component on ₀ along either (orthoradial direction) or (fixed direction) has at most one solution. On the other hand, we show that a solution can exist only if B _lsatisfies some solubility conditions.     

Magnetic field configuration of the heliosphere in interstellar space

A model of the heliospheric magnetic field configuration is constructed by including a uniform interstellar magnetic field to the model proposed by Akasofu et al. (1980).     

Large-scale configuration of the magnetic field in spiral galaxies

Global configuration of magnetic field in several spiral galaxies were determined by analysing characteristic variation of Faraday rotation within the galaxy disks. The majority has an open spiral, bisymmetric field configuration, while some (10–20%) have a ring field.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

Influence of fluctuating field on the configuration of force-free field

The consistent equations for random force-free magnetic fields are obtained from the fluctuation and average equations of both momentum and induction ones based on the statistical approach. Some features of average force-free fields are discussed. This idea may be applied to explain many configurations of magnetic field in the astronomical environment.     

The effects of a minimal length on the Kerr metric

The effects of a minimal length on the Kerr metric are studied within the pseudo-complex General Relativity (pcGR), which has a minimal length parameter and also depends on a $r$-dependent metric, allowing for the accumulation of dark energy around a star. The relevant parameters are the rotational Kerr parameter $a$, the mass of a black hole, and a parameter measuring the amount of dark energy accumulated. It is found that the metric is modified by a factor, depending on $r$, resulting in a maximal acceleration. This factor shows several singularities. For small black holes, the corresponding effective potentials exhibit potential barriers, avoiding the increase of the black hole's mass. It is found that the effects of a minimal length are only of importance for very small mass black holes and vanish for macroscopic black holes.     

A remarkable configuration of sector boundaries

A study of all the observed and well-defined sector boundaries from January 1957 to February 1975, published by Svalgaard (1974, 1975a, b), indicated that sector boundary key-dates, transformed into Bartels' days, have a significant preference to occur on certain days of the solar rotation. The eclectic distribution of these sector boundaries give some Bartels' days that are empty of cases, while on other days there is a significant excess over the average. Using this effect, we can predict, in high levels of significance, the possible occurrence of a (+,–) or (–,+) boundary within particular days of the solar rotation.