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.     

Migration of Comets

Based on one of the particular cases of twice averaged model Hill problem with the allowance for the oblateness of the central body a quadrature is derived for the determination of the migration time of cometary nuclei from various cometary reservoirs and a (14-th order) algebraic equation for the determination of the initial conditions that allow the escape of the cometary nucleus (which at the initial instant of time moves in an orbit with arbitrary eccentricity (0 < e < 1) and inclination (0° < i < 180°) deeply inside the sphere o f action of the central body) from the sphere of action of the central body or its impact onto the central body. We analyze the shape of the boundaries of the hypothetical cometary reservoirs and the method of searching for regions of high concentration of interstellar particles in the Solar System.     

On the existence of the relative equilibria of a rigid body in the J 2 problem

The motion of a point mass in the J ₂ problem has been generalized to that of a rigid body in a J ₂ gravity field for new high-precision applications in the celestial mechanics and astrodynamics. Unlike the original J ₂ problem, the gravitational orbit-rotation coupling of the rigid body is considered in the generalized problem. The existence and properties of both the classical and non-classical relative equilibria of the rigid body are investigated in more details in the present paper based on our previous results. We nondimensionalize the system by the characteristic time and length to make the study more general. Through the study, it is found that the classical relative equilibria can always exist in the real physical situation. Numerical results suggest that the non-classical relative equilibria only can exist in the case of a negative J ₂, i.e., the central body is elongated; they cannot exist in the case of a positive J ₂ when the central body is oblate. In the case of a negative J ₂, the effect of the orbit-rotation coupling of the rigid body on the existence of the non-classical relative equilibria can be positive or negative, which depends on the values of J ₂ and the angular velocity Ω _e . The bifurcation from the classical relative equilibria, at which the non-classical relative equilibria appear, has been shown with different parameters of the system. Our results here have given more details of the relative equilibria than our previous paper, in which the existence conditions of the relative equilibria are derived and primarily studied. Our results have also extended the previous results on the relative equilibria of a rigid body in a central gravity field by taking into account the oblateness of the central body.     

Liapunov stability of spinning satellites with long flexible appendages

The equations of motion are derived for the case of a spinning satellite which has a central rigid body, and long flexible appendages which are nominally in the spin plane. The major-axis theorem is found to be necessary, but not sufficient for this case, and appropriate sufficiency criteria are derived from a Liapunov function. The results are presented in a form amenable to satellite design.     

Classification of orbits in the plane isosceles three-body problem

The general plane isosceles three-body problem is considered for different ratios of the central body mass to the masses of other bodies. The central body goes through the middle of the segment connecting the other bodies along the perpendicular to this segment. The initial conditions are chosen by two parameters: the virial ratio k and the parameter     , where r˙ is the relative velocity of the 'outer' bodies, and R˙ is the velocity of the 'central' body with respect to the mass centre of the 'outer' bodies. The equations of motion are numerically integrated until one of three times: the time of escape of the central body, its time of ejection with   R >100 d   , or 1000 τ (here d is the mean size, and τ is the mean crossing time of the triple system). The regions corresponding to escapes of the central body after different numbers of triple approaches are found at the plane of parameters   k ∈(0,1)  and   μ ∈(-1,1)  . The regions of stable motions are revealed. The zones of regular and stochastic orbits are outlined. The fraction of stochastic trajectories increases with the central mass. The fraction of stable orbits is highest for equal masses of the bodies.     

Region of motion in the Sitnikov four-body problem when the fourth mass is finite

This article deals with the region of motion in the Sitnikov four-body problem where three bodies (called primaries) of equal masses fixed at the vertices of an equilateral triangle. Fourth mass which is finite confined to moves only along a line perpendicular to the instantaneous plane of the motions of the primaries. Contrary to the Sitnikov problem with one massless body the primaries are moving in non-Keplerian orbits about their centre of mass. It is investigated that for very small range of energy h the motion is possible only in small region of phase space. Condition of bounded motions has been derived. We have explored the structure of phase space with the help of properly chosen surfaces of section. Poincarè surfaces of section for the energy range ?0.480≤h≤?0.345 have been computed. We have chosen the plane (q ₁,p ₁) as surface of section, with q ₁ is the distance of a primary from the centre of mass. We plot the respective points when the fourth body crosses the plane q ₂=0. For low energy the central fixed point is stable but for higher value of energy splits in to an unstable and two stable fixed points. The central unstable fixed point once again splits for higher energy into a stable and three unstable fixed points. It is found that at h=?0.345 the whole phase space is filled with chaotic orbits.     

Stability of the classical type of relative equilibria of a rigid body in the J 2 problem

The motion of a point mass in the J ₂ problem is generalized to that of a rigid body in a J ₂ gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal harmonic J ₂ and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively. The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system.     

First occurrence of pyrophanite (MnTi03) and baddeleyite (ZrO2) in an ordinary chondrite

Abstract A 220 × 430 μm Mg-Al-chromite fragment in the Raguli H3.8 ordinary chondrite exhibits distinct optical and compositional zoning. The dark central region of the Mg-Al-chromite is enriched in MgO, Al₂O₃ and ZnO and depleted in TiO₂, Cr₂O₃, FeO and MnO relative to the lighter outer region. A subhedral olivine grain attached to radiating euhedral ilmenites is present in the central region of the fragment. One of the ilmenite crystals contains a tiny grain (0.5 × 5.8 μm) of baddeleyite (ZrO₂). Two end-member pyrophanite grains (MnTiO₃) occur in the outer portion of the fragment This is the first occurrence of pyrophanite and baddeleyite in an ordinary chondrite. Although it is possible that the Mg-Al-chromite fragment was derived from an achondritic projectile of unusual composition, we offer two alternative models for its formation. These include (a) a multi-stage process involving nebular melting of a chromian-spinel chondrule that had accreted some olivine-bearing, Mn-rich material followed by metamorphism on the parent body, and (b) formation of the fragment on a metamorphosed parent body from an impact melt of a chromite-rich assemblage containing Mn-rich ilmenite. We favor the latter alternative.     

The general equilibria of a spinning satellite in a circular orbit

Let a rigid satellite move in a circular orbit about a spherically symmetric central body, taking into account only the main term of the gravitational torque. We shall investigate and find all solutions of the following problem: Let the satellite be permitted to spin about an axis that is fixed in the orbit frame; the satellite need not be symmetric, the spin not uniform, and the spin axis not a principal axis of inertia. The complete discussion of this type of spin reveals that the cases found by Lagrange and by Pringle - and the well-known spin about a principal axis of inertia orthogonal to the orbit plane — are essentially the only ones possible; the only further (degenerate) case is uniform spin of a two-dimensional, not necessarily symmetric satellite about certain axes that are orthogonal to the plane containing the body and to the orbit of the satellite around the central body.     

Moment of inertia of a neutron star. I. Relativistic equation for the accumulated moment of inertia

A relativistic, first-order differential equation is derived for the accumulated moment of inertia of a spherically symmetric celestial body. An approximate equation is proposed to describe the contribution of relativistic effects to the moment of inertia of a superdense star. For configurations of an incompressible fluid, this approximation describes the results of the numerical calculations of Chandrasekhar and Miller to within 5% in the entire range of central pressures from 0 to ∞. Translated from Astrofizika, Vol. 40, No. 1, pp. 87–96, January–March, 1997.     

The Parabolic Three-Body Problem

In this communication we present an analytical model for the restricted three-body problem, in the case where the perturber is in a parabolic orbit with respect to the central mass. The equations of motion are derived explicitly using the so-called Global Expansion of the disturbing function, and are valid for any eccentricity of the massless body, as well as in the case where both secondary masses have crossing orbits. Integrating the equations of motion over the complete passage of the perturber through the system, we are then able to construct a first-order algebraic mapping for the change in semimajor axis, eccentricity and inclination of the perturbed body.Comparisons with numerical solutions of the exact equations show that the map yields precise results, as long as the minimum distance between both bodies is not too small. Finally, we discuss several possible applications of this model, including the evolution of asteroidal satellites due to background bodies, and simulations of passing stars on extra-solar planets.     

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w ₀)², wheren is the satellite's orbital mean motion andw ₀ its initial rotational angular speed.     

The transfer of Lyman continuum radiation in chromospheric flares

We study the time evolution of a layer of the middle or lower chromosphere being heated by a stream of energetic particles during a solar flare. The region, which is not in LTE, is allowed to cool by the transfer of Lyman continuum radiation, with collisional as well as radiative processes being considered. The resulting time dependence of the electron density and the effective thickness of the layer are in good agreement with values derived from observations. We assume the supply of energetic particles to be cut off when the central electron density of our model layer reaches the peak value of n _e = 4.4 × 10¹³ cm^–3 derived from observations of an importance 3 flare. Depending on the total hydrogen density assumed, the central electron temperature reaches a value ranging from 8000 to 10000 K. These quantities decrease by 20% during the following minute and at a slower rate thereafter.     

Supermassive black-hole binary systems in active galactic nuclei

An analysis of periodic components of flux variability was carried out based on the long-term monitoring of the nuclei of active galaxies 3C454.3, 1633+382, and 3C120, performed in the Crimean Astrophysical Observatory from 1985 to 2008 at 22.2 and 36.8 GHz. Long-period components of the variability (12–14 yrs) were detected and interpreted in terms of the precessional motion of the central body in binary systems. Short-period components (1.5–3 yrs) related to the orbital periods for the motions of the central supermassive black holes were also detected. We concluded that the brightest active galaxies observed as nonstationary sources in a wide range of wavelengths are binary systems of supermassive black holes at the stage of evolution close to coalescence. For the supposed binary black-hole systems, the masses of the central objects and their companions, the orbital radii of the companions, and the coalescence times were determined. The ratios of the masses in the binary systems in all cases proved to be less than ten, pointing to a strong gravitational effect of the companion on the central black hole. The velocities of the central body motion proved to be high, approximately 1000 km/s. This fact should be accounted for in the calculations of the rate of accretion onto the central body. The orbital radii of the companions fall into a narrow range between 4 × 10¹⁶ cm and 6 × 10¹⁶ cm, demonstrating a strong dependence of the masses of the binary systems on the orbital sizes and the energy loss for the gravitational radiation. Within the orbit of the companion during its motion through the accretion disk, a high temperature of surrounding gas is achieved. The high density of the medium, 10⁹–10¹⁰ cm^?3, combined with the magnetic field and shock waves propagating in the accretion disk, develop the conditions for powerful energy release in the directed jets.     

Collinear Central Configuration in Four-Body Problem

In the n-body problem a central configuration is formed if the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. We consider the problem: given a collinear configuration of four bodies, under what conditions is it possible to choose positive masses which make it central. We know it is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However for an arbitrary configuration of four bodies, it is not always possible to find positive masses forming a central configuration. In this paper, we establish an expression of four masses depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically we show that there is a compact region in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the compact region, it is always possible to choose positive masses to make the configuration central.     

Mineralogy of a refractory inclusion in the Allende (C3V) meteorite

Abstract— A spherical, 220-μm diameter, spinel-hibonite-perovskite inclusion from the Allende C3V meteorite contains a central hibonite cluster with an angular boundary. This central hibonite is enclosed within spinel that is zoned from Mg-rich at the hibonite boundary to more Fe-rich at the inclusion boundary. This spinel zone includes lath-shaped hibonites usually oriented subradial to the central hibonites. Two textural types of perovskites are present as exsolution from the central hibonite and as equidimensional grains within both the central hibonite and spinel. These second perovskites have exsolution lamellae of Al₂O₃. Within the central hibonite and adjacent to some equidimensional perovskites, a fine porous phase interpreted as alteration has a composition of nearly pure Al₂O₃ with minor amounts of Na and Si. This is possibly either an intergrowth of corundum and nepheline or a modified Al₂O₃, β-alumina. The central hibonites and equidimensional perovskites are considered relict grains on which the spinel-hibonite layer crystallized. The relict material had undergone slow cooling in a previous event to produce exsolution of original high-temperature compositions. Later alteration caused breakdown of hibonite to give an Al₂O₃-rich phase. This inclusion represents a composite body which formed in a Ca-Al-rich environment.     

Chaotic gravitational zones around a regularly shaped complex rotating body

In preparation for the Rosetta mission, the location and widths of gravitational resonances surrounding a regularly shaped and possibly complex rotating body are mapped following the second fundamental model of resonance. It is found that for uniaxial rotation of the central body, the surrounding resonances are widest for prograde orbits. If the figure axis is tilted with respect to the spin axis of the central body, an additional number of wide resonances appear with a preference for prograde and inclined orbits, and the occurrence of initial conditions which lie in the globally connected chaotic web is significantly increased. For larger rotational excitations, it is seen how these new additional resonances overlap internally at low eccentricity for very large semi-major axes. However, with exceptions for some excited short-axis rotational modes of the central body, it is argued that most resonances vanish for retrograde orbits lying in the plane normal to the body spin, and that resonant or non-resonant stability therefore can be expected for a wide range of mean orbit eccentricities.     

O-rich SR and Mira variables in DENIS

Previous VLA observations of the triple radio continuum source in Serpens showed that it has very unusual and extraordinary characteristics. While this source is associated with a star forming region, its outer components exhibit a combination of thermal and nonthermal spectra and large proper motions, Furthermore, the NW lobe has knotty and extended emission connecting the central source with the bright outer knot. Here, we present results of new VLA radio continuum high-angular resolution observations of this Radio Jet. Combining these observations with those obtained previously, we find that: a) one of the knots along the main body of the radio jet (knot G) exhibits proper motions similar to those observed in the outer NW and SE components, and the time variable knot A; b) the outer knots are moving away from the central source in slightly different directions; and c) the orientation of the central source seems to change with time. These results are consistent with a central precessing source that undergoes periodic ejection of material.     

Light Scattering and Extinction in M 82

Published low-resolution measurements of colour and polarisation over the face of M82 are discussed to separate the contribution of starlight and scattered light. We show that in all places of the middle and outer halo the scattered light comes predominantly from a central source of very high ultraviolett excess, the contribution of the disc is negligble there. The projected distributions of H_α-light and scattered continuum are of considerable similarity. Major extinction occurs in the southern half of the main body and of the inner halo; the northern half of the bright body, and the northern halo, are free of extinction, excluding some regions near the minor axis. The light of the central source is reddened only before it is scattered in the halo. The variation of the true degree of polarisation (after correction for starlight) is interpreted in terms of the variation of the mean scattering angle. From this, conclusions can be drawn concerning the location of the dust and the geometry of the illumination. The high brightness of the scattered light near the minor axis is caused axis is caused by a bright illuminating beam there, strengthened in some places by comparatively low scattering angles (45°) and a higher (projected) density of the scattering material. The stellar populations seen in M82 are different in the northern and in the southern halfs of the galaxy. The main body and the region of the northern “halo” consist of an old population of normal metal content (pop. I); the colours of the southern parts – which are partly considerably influenced by extinction – can be due to either metal poor F-stars (pop. II) or to young B-stars. To solve the latter ambiguity and at the same time the question in what direction the plane of the galaxy is tilted, good spectra of the faint southern parts of M82 outside the minor axis are needed.     

A note on Hansen's coefficients in satellite theory

Recurrence relations are derived for the Eccentricity FunctionsG andH and their derivatives, as they appear in the evaluation of geopotential and third body perturbations of an artificial satellite.