Motions in the field of two rotating magnetic dipoles I: Equilibrium points

The equilibrium points of charged particles moving under the Lorentz forces of two parallel or anti-parallel magnetic dipoles located at the two primaries of the Restricted Three-Body Problem are calculated. The magnetic moments of the dipoles are taken perpendicular to the plane of motion of the primaries. The configuration studied simulates the case of close-binary stars with dominant zero-order harmonics in the magnetic field of each star. Depending on the mass parameter and the magnetic moment ratio of the two dipoles, it is found that there exist from one to nine equilibrium points in the plane of motion of the primaries. On the synodical axis of the primaries, depending on the above two ratios, there exist one, two three, or five equilibrium points, while off this axis the number of equilibrium points is zero, two or four.     

Motions in the field of two rotating magnetic dipoles. II. Stability of the equilibrium points

The stability of the equilibrium points found to exist (cf. Goudaset al., 1985, referred to henceforth as Paper I) in the problem of two parallel, or antiparallel, magnetic dipoles that rotate about the centre of mass of their carrier stars, is studied by computing the characteristic roots of their variational equations. The characteristic equation, a biquadratic, solved for many combinations of and showed that all equilibrium points of this problem are unstable.     

Zero-velocity surfaces in the three-dimensional ring problem of N + 1 bodies

The zero-velocity surfaces in the three-dimensional ring problem of N+1 bodies and their parametric evolution is the subject of this paper. These surfaces, which are also known as Hill's or Jacobian surfaces, provide us with valuable information concerning the regions of the permissible particle motion and the existence of equilibrium positions.     

Rigidly rotating modes of the solar magnetic field

Discrete rigidly rotating components (modes) of the large-scale solar magnetic field have been investigated. We have used a specially calculated basic set of functions to resolve the observed magnetic field into discrete components. This adaptive set of functions, as well as the expansion coefficients, have been found by processing a series of digitized synoptic maps of the background magnetic field over a 20-year period. As a result, dependences have been obtained which describe the spatial structure and the temporal evolution of the 27-day and 28-day rigidly rotating modes of the Sun's magnetic field.The spatial structure of the modes has been compared with simulations based on the known flux-transport equation. In the simulations, the rigidly rotating modes were regarded as stationary states of the magnetic field whose rigid rotation and stability were maintained by a balance between the emergence of magnetic flux from stationary sources located at low latitudes and the horizontal transport of flux by turbulent diffusion and poleward directed meridional flow. Under these assumptions, the structure of the modes is determined solely by the horizontal velocity field of the plasma, except for the low-latitude zone where sources of magnetic flux concentrate. We have found a detailed agreement between the simulations and the results of the data analysis, provided that the amplitude of the meridional flow velocity and the diffusion constant are equal to 9.5 m s^–1 and 600 km² s^–1, respectively.The analysis of the expansion coefficients has shown that the rigidly rotating modes undergo rapid step-like variations which occur quasi-periodically with a period of about two years. These variations are caused by separate surges of magnetic flux in the photosphere, so that each new surge gives rise to a rapid replacement of old large-scale magnetic structures by newly arisen ones.     

Main-sequence magnetic CP stars III. Results of magnetic field measurements

We present the third part of the survey of magnetic parameters of main-sequence magnetic CP stars. We analyze the main definitions and terminology, basic data on the magnetic fields of CP stars (catalogs, the history of the stellar magnetism research, the main observational results obtained over 60 years of studies). We describe the modern views on the properties of magnetic CP stars, i.e. their geometric structure, distribution of field strengths, magnetic field and rotation, magnetic field and energy distribution anomalies, and the evolutionary status of magnetic CP stars. We conclude that the observational data mostly support the theory of the relict origin and evolution of magnetic fields of CP stars.     

Formation of a plasmasphere in a rotating magnetic field

Stationary flows of planetary plasmaspheres are analysed within the framework of magnetohydrodynamics. The necessary conditions for existence of a laminar axisymmetric plasma flow around the planet are formulated. A general solution of the problem is obtained for a plasma with the magnetic viscosity greatly exceeding the hydrodynamic one. It has been ascertained that the flow features are strongly dependent on the azimuthal component of the magnetic field. Detailed calculations are performed for a magnetic field similar to a dipole. A strong difference between rather slow-rotating plasmaspheres with plasmapauses and rather fast-rotating continuous plasmaspheres has been revealed. Results of the calculations are useful to explain some properties of the Jovian plasmasphere.     

Equilibrium and stability of the force-free magnetic field III

The condition of stability obtained earlier is generalized for the case of arbitrary force-free fields (f.f.f.). It is shown that the configurations are stable, if the ratio of the current to the field and its gradient are small enough.     

Generation of a toroidal magnetic field in rotating neutron stars

The electromagnetic properties of neutron stars (pulsars) are studied. It is shown that taking the presence of two angular rotation velocities of the components of neutron stars and the first corrections to the general theory of relativity into account in the equations of hydrodynamic equilibrium for the plasma and in Maxwell’s equations leads to the generation of toroidal magnetic fields in the depths of a neutron star. __________ Translated from Astrofizika, Vol. 49, No. 1, pp. 97–101 (February 2006).     

The magnetic field of a rotating cloud and magneto-rotational explosions

The contraction of a massive rotating plasma cloud with the magnetic field perpendicular to the axis of rotation and extending to infinity is considered. At some stage the contracting cloud reaches a state of dynamic quasi-stationary equilibrium. The change of the magnetic field in the cloud atmosphere before its arrival at the quasi-stationary state (stage I) and also in the process of quasi-equilibrium (stage II) are studied.At stage I an essential change of the external magnetic field geometry occurs, namely the formation of zero (neutral) lines and the transformation of the field into a quasi-radial one. Given certain conditions, the reconnection of the field lines in neutral X-type points may occur with the formation of closed loops. In this case the flux of field lines, which connect the contracting cloud with infinity, decreases asymptotically as (R/R _i)^2/3, whereR/R _i is the ratio of the present radius to initial one.After the cloud arrives at the state of dynamic equilibrium (stage II) a considerable increasing of the magnetic field occurs due to twisting of the field lines by rotation. The field strength increases up to some threshold after which instability suddenly occurs. As a result of cumulation occurring in the zero-line direction, and the subsequent dynamic dissipation, the ejection of relativistic particles and plasma in both directions along the rotational axis takes place. The magnetic field restores itself rapidly due to the continual twisting and this leads to the appearance of repeated explosions.The tension of the magnetic field lines as well as plasma outflow carry away the angular momentum. Its diminution determines the rate of secular gravitational contraction. During the contraction the rotational energy increases, so that recurrent bursts, being of magneto-rotational nature are based, finally, on the gravitational energy reservoir.According to our calculations of the time-interval for repetition of explosions, the energy output and certain other parameters, we are able to explain repeated bursts in the nuclei of galaxies and quasars observed, in particular, in the appearance of radio-variability.Extended version of the paper read at All-Moscow Astrophysical Seminar in Sternberg Astronomical Institute 3rd April 1969.     

Gravitating configurations with magnetic field. III. Dedekind and Riemann ellipsoids

Kiev Polytechnic Institute. Translated from Astrofizika, Vol. 30, No. 1, pp. 208–219, January–February, 1989.     

Motions and magnetic fields in filaments

Based on the developed method of jointly using data on the magnetic fields and brightness of filaments and coronal holes (CHs) at various heights in the solar atmosphere as well as on the velocities in the photosphere, we have obtained the following results:

The upward motion of matter is typical of filament channels in the form of bright stripes that often surround the filaments when observed in the HeI 1083 nm line.

The filament channels observed simultaneously in Hα and HeI 1083 nm differ in size, emission characteristics, and other parameters. We conclude that by simultaneously investigating the filament channels in two spectral ranges, we can make progress in understanding the physics of their formation and evolution.

Most of the filaments observed in the HeI 1083 nm line consist of dark knots with different velocity distributions in them. A possible interpretation of these knots is offered.

The height of the small-scale magnetic field distribution near the individual dark knots of filaments in the solar atmosphere varies between 3000 and 20000 km.

The zero surface separating the large-scale magnetic field structures in the corona and calculated in the potential approximation changes the inclination to the solar surface with height and is displaced in one or two days.

The observed formation of a filament in a CH was accompanied by a significant magnetic field variation in the CH region at heights from 0 to 30000 km up to the change of the predominant field sign over the entire CH area. We assume that this occurs at the stage of CH disappearance.

     

Integral surfaces with space-time coordinates,in the gravitational field of a rotating system

A new integration theory is formulated for dynamical systems with two degrees of freedom, in the gravitational field of a rotating system. Four integrals of motion may be determined from complete solutions of a system of three first-order, partial differential equations in three independent variables. The solutions of this system define two integral surfaces with space-time coordinates. These surfaces represent two independent solutions of a second-order kinematic system to which the original fourth-order system has been reduced. An integral curve may be represented as the locus of intersection points of the integral surfaces. The new theory is the theoretical basis for a method of analytic continuation of periodic orbits of the circular restricted problem.     

Rayleigh-Taylor instability of rotating fluid in the presence of a vertical magnetic field

The Rayleigh-Taylor instability of two rotating superposed fluids in the presence of a vertical magnetic field has been investigated. It is shown thatn ² is purely real, wheren is the growth rate of a perturbation. In the basis of this fact it is shown that a unique dispersion relation exists if the lighter fluid lies beneath the heavier one. However, if the heavier fluid lies beneath the lighter fluid, then no unique dispersion relation exists. The effect of rotation is to slow down the rate at which potentially unstable stratification departs from the equilibrium position.     

Equilibrium and stability of rotating masses with a toroidal magnetic field

We investigate the equilibrium, oscillations, and stability of uniformly rotating masses with a toroidal magnetic field, proportional with the distance to te axis of rotation. The equilibrium is an oblate or prolate spheroid according as the rotational energy is greater or smaller than the magnetic energy. The sequence of equilibrium figures exhibits a maximum value for the angular velocity in the oblate case and a maximum for the angular momentum in the prolate case. The dispersion relation is derived using Bryan's modified spheroidal coordinates. One obtains 2(n–m)+4 solutions for the oscillation frequency ifm0 and 1/2n or 1/2(n+1) solutions for ² according asn is even or odd ifm=0. The point where the Jacobi ellipsoids bifurcate from the MacLaurin sequence is unaffected by the magnetic field. However, the points of the onset of dynamical instability corresponding to the second and third harmonics and the point where a pear-shaped sequence bifurcate, depend upon the magnetic field. They are shifted to higher values for the eccentricity and can be suppressed by a sufficiently large magnetic field.     

Particle motions around,between or outside the two dipoles of a magnetic binary system

In this paper we study the simple symmetric motions of a charged particle around, between or outside the two primaries of a magnetic-binary system. We also attempt to reveal the influence of the oblateness of the primaries on the evolution of the families of these motions and on their orbital characteristics.     

Motions and magnetic fields in quiescent prominences

An observed relation between line-of-sight velocities and the longitudinal component of the magnetic field in quiescent prominences is discussed. Weak fields in quiescent prominences are associated with large velocities determined from Doppler shifts of resolved emission knots and Doppler line widths measured in Ca ii K line. It is suggested that the observed irregular motions in prominences are driven by photospheric horizontal convection coupled by the prominence magnetic field. An energy flux of 3 × 10⁵ ergs cm^–2 sec^–1 present in the form of Alfvén waves in quiescent prominences is consistent with the observations.     

Dynamical stability of a self-gravitating rotating cylinder in a helical magnetic field

The effect of a helical magnetic field on the oscillations and the stability of a homogeneous self-gravitating rotating cylinder is investigated. The axial field has a tendency to stabilise long wave numbers and to destabilise small wave numbers so that maximum instability occurs for a finite wave number. If the toroidal and the axial component of the field have the same sign, the instability associated with the toroidal field can be removed by the rotation or by the axial field. Rotational instability is reduced but cannot removed by the field. If the components of the field have the opposite sign, rotational instability is increased. The maximum growth rate of the magnetic instability is reduced by a small axial field and tends to a finite value for large axial fields.     

Waves and instabilities in a differentially rotating disc containing a poloidal magnetic field

The theory of waves and instabilities in a differentially rotating disc containing a poloidal magnetic field is developed within the framework of ideal magnetohydrodynamics. A continuous spectrum, for which the eigenfunctions are localized on individual magnetic surfaces, is identified but is found not to contain any instabilities associated with differential rotation. The normal modes of a weakly magnetized thin disc are studied by extending the asymptotic methods used previously to describe the equilibria. Waves propagate radially in the disc according to a dispersion relation which is determined by solving an eigenvalue problem at each radius. The dispersion relation for a hydrodynamic disc is re-examined and the modes are classified according to their behaviour in the limit of large wavenumber. The addition of a magnetic field introduces new, potentially unstable, modes and also breaks up the dispersion diagram by causing avoided crossings. The stability boundary to the magnetorotational instability in the parameter space of polytropic equilibria is located by solving directly for marginally stable equilibria. For a given vertical magnetic field in the disc, bending of the field lines has a stabilizing effect and it is shown that stable equilibria exist which are capable of launching a predominantly centrifugally driven wind.     

Dynamical and secular instability of a self-gravitating rotating cylinder in a toroidal magnetic field

The oscillations and stability of a homogeneous self-gravitating rotating cylinder in a toroidal magnetic field are investigated. It is assumed that the field is proportional to the distance to the axis of the cylinder. We show the existence of four infinite discreta spectra of magnetic (or rotational) modes. Rotation stabilizes the magneticm=1 instability. The magnetic field decreases the growth rate of rotational instability and reduces the interval of unstable wavenumbers. Ifm=1, instability always occurs with the exception of the equipartition state. Ifm>1, the instability can be suppressed by a sufficiently large magnetic field. Resistivity decreases the growth rate of magnetic instability, but increases the growth rate of rotational instability. For zero wavenumber perturbations secular instability occurs due to the action of resistivity before a neutral point is attained where a second secular instabiliity initiates due to the action of resistivity.