Adiabatic pulsations and convective instability of rotating gaseous masses

Third order virial equations have been used to investigate the oscillations and the stability of the sequence of differentially rotating, compressible Maclaurin spheroids in the presence of toroidal magnetic fields. It is shown that the neutral point occurring at eccentricitye=0.731 13, which is the analogue of the first point of bifurcation along the Dedekind sequence, remains unaffected by the presence of differential rotation or a toroidal magnetic field. The point of onset of dynamical instability corresponding to the third harmonic deformations does, however, depend upon the magnetic field. It is shifted to values higher thane=0.966 96, the value that obtains in the case of uniform rotation; and a sufficiently large magnetic field can suppress this point. Complete frequency spectra (‘Kelvin’ modes belonging to the harmonicsl=3 and compressible modes belonging tol=1) are obtained in two cases of interest: when the equilibrium state is one of equipartition, and when toroidal magnetic and velocity fields (vanishing at the surface) are present in a configuration rotating with a constant angular velocity.     

The oscillations and the stability of rotating masses with magnetic fields

Using first variations of the integral properties of equilibrium second-order virial relations, the existence of the point of bifurcation of rotating gaseous masses with magnetic fields is substantiated. With the presence of a magnetic field component along the axis of rotation, it is shown that the point of bifurcation, where the Jacobi ellipsoids branch off from the Maclaurin spheroids, is altered, and in fact shifts to higher values of eccentricity compared to the one (namely,e=0.81267) obtained when there is no magnetic field.     

On the adiabatic pulsations of uniformly rotating bodies

The problem of the oscillations and stability of compressible Maclaurin spheroids is reconsidered, on the basis of the third-order virial equations, in an arbitrarily rotating frame of reference. In contrast with the work of Kochhar and Trehan (1974), it is found that the frequencies evaluated in a rotating frame and those evaluated in an inertial frame are related to one another in a very simple way. Numerical calculations made for a wide range of the adiabatic exponent further clarify the effect of compressibility on the natural frequencies.     

The secular instability of slightly viscous magnetic Maclaurin spheroids

We show that the instability induced by viscosity, at the point of bifurcation where the Jacobi ellipsoids branch off from the sequence of Maclaurin spheroids, is not inhibited by the presence of a magnetic field. It has already been shown that a toroidal magnetic field leaves the point of bifurcation unaffected, whereas a magnetic field along the axis of rotation pushes the point of bifurcation to eccentricities higher than the value that obtains in the absence of a magnetic field.     

The oscillations and the stability of rotating masses with magnetic fields

The existence of the point of bifurcation of rotating gaseous masses with toroidal magnetic fields is established from the first variations of the integral properties provided by the second-order virial relations. It is shown that the point of bifurcation, where the Jabcobi ellipsoids branch off from the Maclaurin spheroids, is unaffected by the presence of toroidal magnetic fields.     

Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the <Emphasis Type="Italic">n</Emphasis>-body problem

We use numerical continuation and bifurcation techniques in a boundary value setting to follow Lyapunov families of periodic orbits and subsequently bifurcating families. The Lyapunov families arise from the polygonal equilibrium of n bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, then the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along the Lyapunov families and along bifurcating families, namely for the cases \(n=3\), 4, and 6–9. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. Animations of the families and the choreographies can be seen at the link below.     

Anguilar momentum transport by tidal acoustic wave

We give an analytic expression of the braking torque on a Jacobian ellipsoid rotating steadily in an environmental gas, based on the assumption that the ellipsoid rotates around its shortest principal axis with an angular momentum slightly larger than that at the bifurcation point of the Maclaurin spheroid. This braking torque is effected by the gravitational interaction between the ellipsoid matter and a spiral density configuration in the environmental gas. This spiral configuration, which we call a tidal acoustic wave, is caused by the zone of silence effect in a supersonic flow. With respect to a coordinates system rotating with the ellipsoid, a supersonic region appears outside a certain radius. In this supersonic region, the effect of the non-axisymmetric fluctuation in the ellipsoid potential propagates only along the downstream branches of the Mach waves. This one-sided response of the supersonic part causes the tidal acoustic wave. We restrict ourselves to the equatorial plane, and use an acoustic approximation of the basic equations under the assumption that the self-gravity effect of the environmental gas is negligible in comparison to the main gravity of the ellipsoid. The results are applied to the pre- and post-Main Sequence phases of a rotating star, and relating astrophysical problems are discussed.     

Integrals of motion in the elliptic three-dimensional restricted three-body problem

Three integrals of motion have been found in the three-dimensional elliptic restricted three-body problem for small eccentricitye of the relative orbit of the primaries and small distancer and eccentricitye of the orbit of the third body around a primary. The integrals are given in the form of formal series in the mass-ratio , the eccentricitiese, e and the coordinates and velocities. These integrals depend periodically on the time.     

The Limit Problems for the Equation of Oscillations of a Satellite

We consider the ordinary differential equation of the second order, which describes oscillations of a satellite with respect to its mass center moving along an elliptic orbit with eccentricity e. The equation has two parameters: e and μ. It is regular for 0 ≤ e < 1 and singular when e = 1. For 1 we obtain three limit problems. Their bounded solution to the first limit problem form a two-dimensional (2D) continuous invariant set with a periodic structure. Solutions to the second limit problem form 2D and 3D manifolds. The μ-depending families of odd bounded solutions are singled out. One of the families is twisted into a self-similar spiral. To obtain the limit families of the periodic solutions to the original problem match together the odd bounded solutions to the first and the second limit problem. The point of conjunction is described by the third (the basic) limit problem. The limit families are very close to prelimit ones computed in earlier studies. This revised version was published online in July 2006 with corrections to the Cover Date.     

Récherché du terme denième ordre des développements de l'astrodynamique

General expressions for the functionsE=f(M), r/a=f(M) anda/r=f(M) in the two-body problem are obtained as functions of eccentricitye without using Bessel functions.     

The equilibrium and stability of uniformly rotating gaseous systems in hydromagnetics

The theory for investigating the equilibrium and stability of a uniformly rotating gaseous system with a prevalent magnetic field is developed by using the virial tensor approach. Most of the discussion in this paper depends on the assumption that on the surface of the system, the magnetic field is zero. In Appendix A, however, we have considered the case in which the surface magnetic field is non-zero.We have obtained the nine modes of oscillations, grouped into the transverse shear, toroidal and pulsation modes. From this analysis have also found the conditions under which the sequence of a uniformly rotating axially symmetric configuration in the presence of a magnetic field should have a point of bifurcation, that is, a point where objects with genuine triplanar symmetry branch off. This condition is also generalized in the Appendix to include the effects of differential rotation and non-zero surface magnetic field.Applications to the cosmogonic fission problem, the study of the pulsation of rotating magnetic stars and some radio astrophysical problems are briefly discussed.     

Families of symmetric-periodic orbits in the elliptic three-dimensional restricted three-body problem

With an orbit of the three-dimensional circular problem as a starting point, we have calculated families of symmetric-periodic orbits in the three-dimensional elliptic problem with a variation of the mass ratio and the eccentricitye. Afterwards, we have studied their evolution and stability.     

Force-Free Rotator

A simple mechanichal problem of the force-free motion of a relativistic bead inside a rotating pipe is examined. A relevant change of shape of the pipe is considered. The force-free motion of the bead along the rotating pipe is examined as the motion in a differentially rotating medium. The differential rotation proves to be resembling a well known case of the Couette flow without outer cylinder. The determination of vectoral fields (magnetic field) in the rotating charged mediums (plasma) is problematic for the laboratory observer. Therefore the same problem is examined in the framework of general relativity, that is the physical quantities are considered in the inertial frame and non-inertial frame with no rotational in homogeneity. The problematic character of the determination of the physical quantity in the differentially rotating madium relative to the inertial observer is explained. It is shown that at certain large distances from the rotation center the pipe does not rotate. However, at rather small distances its shape takes on the appearance of an expanded spiral-like configuration. A possible relevance of the obtained results to the motion of a relativistic plasma flows in pulsar magnetosphere is pointed out. The areas of Crab pulsar's dipole radiation are estimated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Hip-hop solutions of the 2N-body problem

Hip-hop solutions of the 2N-body problem with equal masses are shown to exist using an analytic continuation argument. These solutions are close to planar regular 2N-gon relative equilibria with small vertical oscillations. For fixed N, an infinity of these solutions are three-dimensional choreographies, with all the bodies moving along the same closed curve in the inertial frame.     

Stability of planar oscillations of a satellite in an elliptic orbit

A problem of stability of odd 2-periodic oscillations of a satellite in the plane of an elliptic orbit of arbitrary eccentricity is considered. The motion is supposed to be only under the influence of gravitational torques.Stability of plane oscillations was investigated earlier (Zlatoustovet al., 1964) in linear approximation. In the present paper a problem of stability is solved in the non-linear mode. Terms up to the forth order inclusive are taken into consideration in expansion of Hamiltonian in a series.It is shown that necessary conditions of stability obtained in linear approximation coincide with sufficient conditions for almost all values of parameters ande (inertial characteristics of the satellite and eccentricity of the orbit). Exceptions represent either values of the parameters ,e when a problem of stability cannot be solved in a strict manner by non-linear approximation under consideration, or values of the parameters which correspond to resonances of the third and fourth orders. At the resonance of the third order oscillations are unstable, but at the resonance of the fourth order both unstability and stability of the satellite's oscillations take place depending on the values of the parameters ,e.     

Inertial Alfvén wave induced turbulent spectra in aurora

In the present paper, we investigate the localization of weak inertial Alfvén wave (IAW) in the presence of finite amplitude magnetosonic fluctuations in low β plasmas (β?m _e /m _i ). When IAW is perturbed by these fluctuations, localized structures of IAW magnetic field intensity are formed. We have developed a semi analytical model based on paraxial approximation to study this interaction. Numerical method has also been used to analyse the localized structures and magnetic fluctuation spectrum of IAW. From the obtained results, we find that the magnetic turbulent spectrum upto k _x λ _e ≈3 fits power law spectrum with an index consistent with the Kolmogorov $k_{x}^{ - 5/3}$ law, here λ _e is the electron inertial length. Furthermore, at shorter wavelengths the spectrum steepens to about $k_{x}^{ - 3.8}$ . Energy transfer from larger lengthscales to smaller lengthscales through this mechanism may be responsible for the observed parallel electron heating in auroral region. Results obtained from the simulation are consistent with the observations recorded from various spacecrafts like FAST, Hawkeye and Hoes 2.     

Oscillations of a self-gravitating fluid spheroid with prevalent magnetic field

The Kelvin modes of oscillation of a selfgravitating, homogeneous fluid spheroid in hydrostatic equilibrium with a poloidal magnetic field inside and a dipole type field outside, are studied, using a variational principle. On the assumption that the eccentricitye of the spheroid is small, the frequencies of oscillation are calculated to the first order ine ².     

Determination of the electromagnetic field produced by a magnetic oblique-rotator

The inertial effect on the structure of the magnetosphere of a rotating star is investigated, in the corotation approximation for a surrounding quasi-neutral plasma. The equation of motion reduces to a usual static balance equation between the electromagnetic and the centrifugal forces, in the rotating frame. However the MHD condition, which can be regarded as a special form of the generalized Ohm's law, is modified by the inclusion of inertial effect, with a violation of the frozen-in condition in case of a general (i.e., not restricted to corotation) plasma motion. The inertial effect on the electromagnetic field is summarized in a partial scalar potential named the non-Backus potential, which is proportional to the centrifugal potential in the corotation approximation.An approximate solution of this corotation problem is given, in which another characteristic radiusr _M appears besides the light radiusr _L . This radius defines a distance beyond which the inertial effect becomes dominant over the electromagnetic one, and is useful in estimating the magnitude of the terminal velocity of a centrifugal wind. A few examples of the modification of dipole magnetic field due to the inertial effect are visualized. In an oblique-rotation case, it can be seen that such a warp of the neutral sheet (the surface ofB _r =0) is reproduced as observed in the Jovian magnetosphere.     

Energy oscillations and a possible route to chaos in a modified Riga dynamo

Starting from the present version of the Riga dynamo experiment with its rotating magnetic eigenfield dominated by a single frequency we ask for those modifications of this set‐up that would allow for a non‐trivial magnetic field behaviour in the saturation regime. Assuming an increased ratio of azimuthal to axial flow velocity, we obtain energy oscillations with a frequency below the eigenfrequency of the magnetic field. These new oscillations are identified as magneto‐inertial waves that result from a slight imbalance of Lorentz and inertial forces. Increasing the azimuthal velocity further, or increasing the total magnetic Reynolds number, we find transitions to a chaotic behaviour of the dynamo (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)