Multiple resonance in one problem of the stability of the motion of a satellite relative to the center of mass

We investigate the stability of the periodic motion of a satellite, a rigid body, relative to the center of mass in a central Newtonian gravitational field in an elliptical orbit. The orbital eccentricity is assumed to be low. In a circular orbit, this periodic motion transforms into the well-known motion called hyperboloidal precession (the symmetry axis of the satellite occupies a fixed position in the plane perpendicular to the radius vector of the center of mass relative to the attractive center and describes a hyperboloidal surface in absolute space, with the satellite rotating around the symmetry axis at a constant angular velocity). We consider the case where the parameters of the problem are close to their values at which a multiple parametric resonance takes place (the frequencies of the small oscillations of the satellite’s symmetry axis are related by several second-order resonance relations). We have found the instability and stability regions in the first (linear) approximation at low eccentricities.     

Central configurations of four bodies with an axis of symmetry

A complete solution is given for a symmetric case of the problem of the planar central configurations of four bodies, when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry. The positions of the bodies on the axis of symmetry are described by angle coordinates with respect to the outside bodies. The solution is such, that giving the angle coordinates, the masses for which the given configuration is a central configuration, can be computed from simple analytical expressions of the angles. The central configurations can be described as one-parameter families, and these are discussed in detail in one convex and two concave cases. The derived formulae represent exact analytical solutions of the four-body problem.     

Attitude stability of a symmetric satellite at the equilibrium points in the restricted three-body problem

A problem of attitude motion of the smallest body for the restricted three-body problem is analyzed. Axial symmetry is assumed for the body, and attention is focused on the case in which the symmetry axis is normal to the orbit plane. For libration point satellites, results are similar to those for a satellite in orbit about a single body. However, for orbit equilibrium points lying on the line joining the two larger bodies, attitude stability results depart markedly from those for the two-body problem.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

Stability of triangular points in the photogravitational CR3BP with Poynting-Robertson drag and a smaller triaxial primary

The stability of triangular equilibrium points in the framework of the circular restricted three-body problem (CR3BP) is investigated for a test particle of infinitesimal mass in the vicinity of two massive bodies (primaries), when the bigger primary is a source of radiation and the smaller one is a triaxial rigid body with one of the axes as the axis of symmetry and its equatorial plane coinciding with the plane of motion, under the Poynting-Robertson (P-R) drag effect as a result of the radiating primary. It is found that the involved parameters influence the position of triangular points and their linear stability. It is noted that these points are unstable in the presence of Poynting-Robertson drag effect and conditionally stable in the absence of it.     

Are Pulsation and Magnetic Axes Aligned in roAp Stars?

It is commonly assumed that in the rapidly oscillating Ap (roAp) stars the mode axis is aligned or nearly aligned with the magnetic field axis. This would be possible if the field is the only important effect causing departure from spherical symmetry. We show that even though these stars are slow rotators, the centrifugal force cannot be neglected. The consequence is that the modes cannot be in general symmetric about the magnetic field. We argue that such a symmetry is not implied by the observed coincidence between the field and pulsation amplitude maxima.     

Effect of perturbed potentials on the stability of libration points in the restricted problem

The location and the stability in the linear sense of the libration points in the restricted problem have been studied when there are perturbations in the potentials between the bodies. It is seen that if the perturbing functions satisfy certain conditions, there are five libration points, two triangular and three collinear. It is further observed that the collinear points are unstable and for the triangular points, the range of stability increases or decreases depending upon whetherP> or <0 wherep depends upon the perturbing functions. The theory is verified in the following four cases:

1. There are no perturbations in the potentials (classical problem).
2. Only the bigger primary is an oblate spheroid whose axis of symmetry is perpendicular to the plane of relative motion (circular) of the primaries.
3. Both the primaries are oblate spheroids whose axes of symmetry are perpendicular to the plane of relative motion (circular) of the primaries.
4. The primaries are spherical in shape and the bigger is a source of radiation.

     

Stationary motions in a restricted problem of three rigid bodies

The present paper is a continuation of papers by Shinkaric (1972), Vidyakin (1976), Vidyakin (1977), and Duboshin (1978), in which the existence of particular solutions, analogues to the classic solutions of Lagrange and Euler in the circular restricted problem of three points were proved. These solutions are stationary motions in which the centres of mass of the bodies of the definite structures always form either an equilateral triangle (Lagrangian solutions) or always remain on a straight line (Eulerian solutions) The orientation of the bodies depends on the structure of the bodies. In this paper the usage of the small-parameter method proved that in the general case the centre of mass of an axisymmetric body of infinitesimal mass does not belong to the orbital plane of the attracting bodies and is not situated in the libration points, corresponding to the classical case. Its deviation from them is proportional to the small parameter. The body turns uniformly around the axis of symmetry. In this paper a new type of stationary motion is found, in which the axis of symmetry makes an angle, proportional to the small parameter, with the plane created by the radius-vector and by the normal to the orbital plane of the attracting bodies. The earlier solutions-Shinkaric (1971) and Vidyakin (1976)-are also elaborated, and stability of the stationary motions is discussed.     

Exact analytic solutions for the rotation of an axially symmetric rigid body subjected to a constant torque

New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion time length and rotation amplitude. The present paper adds further elements to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

Formation of collimated beams

A mechanism is proposed for the formation of collinated beams in radio galaxies. Collimated flows are considered to be non-thermally driven by high energy particles and magneto-hydrodynamic (MHD) waves. The galactic nucleus is regarded as being surrounded by a cool gas. The cool gas accretes onto the nucleus, and then the high energy particles are completely locked to the MHD waves. When a quasi-radial magnetic field is embedded in the accretion flow, the resulting MHD wave packets are collimated into the direction of the symmetry axis of the galactic nuclear disc. The fluid around the nucleus is considered to be accelerated and heated by these MHD waves. The fluid beam is ejected along the symmetry axis.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

A magnetohydrodynamical theorem for steady axisymmetric systems with implications to ring galaxies

A magnetohydrodynamical theorem can be deduced for steady axisymmetric magnetized systems in situations which are usually accepted as representative in interstellar plasmas. It shows that for particular conditions detailed below, magnetic fields have either the direction of the symmetry axis of the system or have field lines encircling this axis of symmetry, but intermediate directions are excluded. Ring galaxies are probably of the second type, which would be important in explaining ring patterns. Our galaxy itself would be of this second type slightly separated from steady state, and the magnetic field radial distribution compatible with the gas content is estimated. The implication of this theorem in actual galaxies is difficult to analyze and gives rise to rather speculative discussions.     

Response of Rigid Rotators to Gravitational Radiation

The response of an axially symmetric rigid rotator to incident gravitational radiation is discussed for particular states of free rotator motion using generalized EULERian equations, and assuming wavelengths large compared with the rotator dimensions. First, if coincident initially, the rotation and the symmetry axes slightly differ after exposed to a radiation flux which has suitable polarization and propagates perpendicular to the rotation axis. Secondly, the angular velocity of a rotation perpendicular to the symmetry axis is changed in a wave field propagating in the direction of the rotation axis (BRAGINSKI-rotator). — For highly monochromatic resonance radiation with wave frequencies equal to the rotation frequency (in the first case) or twice the rotation frequency (second case), the response is sufficiently large to have some interest for future experiments.     

Axially symmetric explosion in a spheroid

The method of Laumbach and Probstein is applied to a point explosion in a spheroid with exponential density distribution. It is shown that the shock wave propagates strongly along the direction of symmetry axis and the envelope of the shock front elongates to the same direction. The rate of elongation of the shock envelope increases with the eccentricity of the spheroid and finally the blowout of the shock wave along the polar axis occurs when the eccentricity exceeds some critical value. It is suggested that such a blowout of the shock may be connected with some galactic outburst having axial symmetry.     

Radiation Effect on a Particle’s Periodic Orbits in a Regular Polygon Configuration of N Bodies

The photo-gravitational problems of two or more bodies have attracted much attention during the last decades. In this paper, radiation is considered as an additional factor influencing the particle motion in a regular polygon formation of N big bodies where the ν = Ν ? 1 primary bodies have equal masses and are located at the vertices of a regular polygon and the Nth primary has different mass and is located at the mass center of the system. We assume that some or all the primary bodies are radiation sources and we numerically explore various cases where symmetry of the resultant force field with respect to the same axis is preserved. For the purposes of our investigation we adopt Radzievski’s theory and assumptions. The material gathered helps us to estimate the radiation effect on the evolution of periodic orbits and their characteristics, such as their periods and their stability. Figures and diagrams illustrate these alterations and document our conclusions.     

Compositional evidence for Titan's stratospheric tilt

Five years of Cassini CIRS infrared spectra have been used to determine the tilt of Titan's stratospheric symmetry axis with respect to the solid body rotation axis. Measurements of HCN abundance centred around 5 mbar (125 km altitude) at equatorial latitudes show the symmetry axis is tilted by 4.0±1.5^° in a direction 70±40^°W of the sub-solar point. This value is consistent with tilts determined from temperature and haze measurements by Achterberg et al. (2008a) and Roman et al. (2009). The consistency of results from three independent methods suggests that Titan's entire stratosphere is tilted and provides a powerful constraint on the underlying atmospheric dynamics.     

Polar orbits around binary stars

Oks proposes the existence of a new class of stable planetary orbits around binary stars, in the shape of a helix on a conical surface whose axis of symmetry coincides with the interstellar axis, and rotates with the same orbital frequency as the binary pair. We show that this claim relies on the inappropriate use of an effective potential that is only applicable when the stars are held motionless. We also present numerical evidence that the only planetary orbits whose planes are initially orthogonal to the interstellar axis that remain stable on the time scale of the stellar orbit are ordinary polar orbits around one of the stars, and that the perturbations due to the binary companion do not rotate the plane of the orbit to maintain a fixed relationship with the axis.     

The masses in a symmetric solution of the four body problem

It is proved that if a non-collinear motion of the four body problem has a symmetry axis (or plane), then the center of mass lies on this axis (plane) and the symmetric masses are equal. We also remark that this result is true for the generalized attraction law given by the inverse (α+1)-power of the distance, with α > 0.     

Effect of perturbed potentials on the non-linear stability of libration pointL 4 in the restricted problem

The non-linear stability of the libration pointL ₄ in the restricted problem has been studied when there are perturbations in the potentials between the bodies. It is seen that the pointL ₄ is stable for all mass ratios in the range of linear stability except for three mass ratios depending upon the perturbing functions. The theory is applied to the following four cases:

Substructure of the core of the perseus supercluster of galaxies

The hierarchical substructure of the core of the Perseus cluster of galaxies (A426, A262, A347, N507, and N383) is investigated using the method of S-tree diagrams. In the main system M (100 galaxies), two substructures AM (35 galaxies) and BM (13 galaxies) are determined. The group AM can be identified with the cluster A262 and the group N507; BM can be identified with the group N383; A426 and A347 are in the main system. The axis of symmetry of the system M passes through the Seyfert galaxy N1275 and is parallel to the axis of symmetry of the group AM, which is the core of increased density of the cluster of galaxies. Radial segregation of galaxies by morphological types is observed in the systems M and AM. The distribution of galaxies by position angles is uniform. Translated from Astrofizika, Vol. 41, No. 1, pp. 65–72, January-March, 1998.     

On the restricted three body problem with oblate primaries

We present a study of the Lagrangian triangular equilibria in the planar restricted three body problem where the primaries are oblate homogeneous spheroids steadily rotating around their axis of symmetry and whose equatorial planes coincide throughout their motion.     

CO observations of Southern molecular clouds. Outflows from young stellar objects GRV 8 and GRV 16

¹²CO (1-0) observations of two Southern dark clouds (globules) associated with cometary nebulae GRV 8 (a biconical nebula) and GRV 16 (a conelike nebula) are presented. GRV 8 shows an outflow from the central part of the nebula (where in 2MASS images a star is located, which is perhaps responsible for this outflow); however, both lobes of the outflow are redshifted with a velocity of +1.95 km/s with respect to the molecular cloud. The two opposite redshifted lobes are a rather rare phenomenon that could be explained by the presence of a double star instead of a single one as the engine responsible for the outflow. The two lobes are almost parallel to the axis of symmetry of the biconical nebula. In the case of the conelike nebula GRV 16 we observe a bipolar outflow, where the eastern blueshifted lobe has a velocity of –4 km/s with respect to the molecular cloud, and the western redshifted one has a velocity +2.5 km/s. The outflow has a direction almost coinciding with the axis of symmetry of the conelike nebula. The star associated with the conelike nebula is responsible for this outflow.Published in Astrofizika, Vol. 48, No. 1, pp. 101–112 (February 2005).