Nonlinear pseudo-radial oscillation of a rotating magnetic star

The nonlinear pseudo-radial mode of oscillation of a rotating magnetic star is studied. It is shown that for a general rotational field, the coupling between magnetic field and rotation tends to reduce the average rotational energy parameterT. This result in a lowering of the maximum pulsation amplitudeq _max, which depends on strength of rotation and magnetic field. The configuration tends, therefore, to a new equilibrium state at lower value ofq _max. The analytic solution of the pulsation equation for the case ofy=5/3 in the presence of rotation and magnetic field has also been derived in the Appendix.     

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.     

Surface trapping and leakage of low-frequency g modes in rotating early-type stars – II. Global analysis

A global analysis of the surface trapping of low-frequency non-radial g modes in rotating early-type stars is undertaken within the Cowling, adiabatic and traditional approximations. The dimensionless pulsation equations governing these modes are reviewed, and the boundary conditions necessary for solution of the equations are considered; in particular, an outer mechanical boundary condition, which does not enforce complete wave trapping at the stellar surface, is derived and discussed in detail. The pulsation equations are solved for a 7-M_⊙ model star over a range of rotation rates, using a numerical approach.
The results of the calculations confirm the findings of the preceding paper in the series: modes with eigenfrequencies below a cut-off cannot be fully trapped within the star, and exhibit leakage in the form of outwardly propagating waves at the surface. The damping rates resulting from leakage are calculated for such 'virtual' modes, and found to be appreciably larger than typical growth rates associated with opacity-driven pulsation. Furthermore, it is demonstrated that the surface perturbations generated by virtual modes are significantly changed from those caused by fully trapped modes; the latter result suggests differences in the line-profile variations exhibited by these two types of mode.
The findings are discussed in the context of the 53 Per, SPB and pulsating Be classes of variable star. Whilst wave leakage will probably not occur for overstable g modes in the 53 Per and slowly rotating SPB stars, the adoption of the new outer mechanical boundary condition may still affect the pulsational stability of these systems. Wave leakage for overstable modes remains a possibility in Be stars and the more rapidly rotating SPB stars.     

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.     

Sunspot's electromagnetism with a locally gauge-invariant chern-simons term

We present a (2+1)-dimensional sunspot model whose Lagrange density contains, in addition to the classical Maxwellian term, the so-called Chern-Simons term generalized here in a gauge-invariant way. It is shown that it is namely this term which is responsible for the confinement of the spot's electromagnetic field into a finite-dimensional domain. We further demonstrate that besides the total magnetic flux it is also the spot's electric charge which is non-zero and that both quantities are topologically quantized, i.e. can acquire discrete values only. Finally, a cylindrically symmetric sunspot carryingp magnetic flux quanta,p being a positive integer, is revealed to possess a non-zero total angular momentum, the magnitude of which is proportional top ². The latter fact also implies the stability of rotating sunspots against their fragmentation (splitting).     

On large-scale solar convection

We examine the effects of rotation about a vertical axis on thermal convection with a simple model in which an inviscid, incompressible fluid of zero thermal conductivity and electrical resistivity is contained in a thin annulus of rectangular cross-section. The initial steady state assumed is one of no motion relative to the rotating frame with constant (unstable) vertical temperature gradient and uniform toroidal magnetic field. Small periodic disturbances are then introduced and the linearized perturbation equations solved. We also determine the second-order mean circulations and magnetic fields that are forced by non-zero Reynolds and thermal stresses and magnetic field transports.The solutions have several properties which are relevant to large-scale solar phenomena if giant long-lived convection cells exist on the sun. In particular, the convective cells are tilted in latitude in the same sense as bipolar magnetic regions, and induce vertical magnetic fields with the same tilt. They transport momentum across latitude circles through Reynolds stresses and induced meridional circulations thus setting up a differential rotation. Cells which grow slowly compared to the rotation rate and have comparable dimensions in latitude and longitude transport momentum toward the equator. The cells also form a poloidal magnetic field from initial toroidal field, in a manner similar to that put forth by Parker.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Application of the Method of Matched Asymptotic Expansions to the radial oscillations of a magnetic cylinder

This study deals with the singular character of the perturbation introduced into the eigenvalue problem of the linear and adiabatic oscillations of a gaseous configuration by a magnetic field that is non-zero on the boundary surface of the configuration. This singular character implies that a regular perturbation scheme cannot yield uniformly valid expansion for the eigenfunctions.This investigation considers the application of the Method of Matched Asymptotic Expansions (M.M.A.E.) to the latter singular perturbation problem in order to obtain uniformly valid expansions for the eigenfunctions and first-order expressions for the eigenfrequencies. As an illustrative example, the M.M.A.E. is applied to the eigenvalue problem of the linear, radial, and adiabatic oscillations of a homogeneous cylindrical plasma with a constant longitudinal magnetic field.     

Periodic Orbits in Rotating Second Degree and Order Gravity Fields

Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates.     

Periodic Orbits in Rotating Second Degree and Order Gravity Fields

Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates.     

The effect of a poloidal magnetic field on the radial and non-radial oscillations of a gaseous mass

The oscillations of a polytrope with infinite electric conductivity containing a weak internal poloidal magnetic field which is continuous with an external dipole field are examined with the aid of a variational equation. The corrections to the fundamental characteristic frequencies of the radial and non-radiall=2 pulsation modes are calculated. The magnetic field removes a degeneracy which occurs between these two modes and the resulting frequency splitting is evaluated. The relevance of the results to the known magnetic stars is briefly discussed.     

Stability of accretion discs threaded by a strong magnetic field

We study the stability of poloidal magnetic fields anchored in a thin accretion disc. The two-dimensional hydrodynamics in the disc plane is followed by a grid-based numerical simulation including the vertically integrated magnetic forces. The three-dimensional magnetic field outside the disc is calculated in a potential field approximation from the magnetic flux density distribution in the disc. For uniformly rotating discs we confirm numerically the existence of the interchange instability as predicted by Spruit, Stehle & Papaloizou . In agreement with predictions from the shearing sheet model, discs with Keplerian rotation are found to be stabilized by the shear, as long as the contribution of magnetic forces to support against gravity is small. When this support becomes significant, we find a global instability which transports angular momentum outwardly and allows mass to accrete inwardly. The instability takes the form of a m =1 rotating 'crescent', reminiscent of the purely hydrodynamic non-linear instability previously found in pressure-supported discs. A model where the initial surface mass density Σ( r ) and B _z ( r ) decrease with radius as power laws shows transient mass accretion during about six orbital periods, and settles into a state with surface density and field strength decreasing approximately exponentially with radius. We argue that this instability is likely to be the main angular momentum transport mechanism in discs with a poloidal magnetic field sufficiently strong to suppress magnetic turbulence. It may be especially relevant in jet-producing discs.     

Modeling helioseismic modes in the magnetic atmosphere of the Sun

The pulsation of the solar surface is caused by acoustic waves traveling in the solar interior. Thorough analyses of observational data indicate that these f and p helioseismic oscillation modes are not bounced back completely at the surface but they partially penetrate into the atmosphere. Atmospheric effects and their possible observational application are investigated in one‐dimensional magnetohydrodynamic models. It is found that f and p mode frequencies are shifted of the order of μHz due to the presence of an atmospheric magnetic field. This shift varies with the direction of the wave propagation.Resonant coupling of global helioseismic modes to local Alfvén and slow waves reduce the life time of the global modes. The resulting line width of the frequency line is of the order of nHz, and it also varies with propagation angle. These features enable us to use helioseismic observations in magnetic diagnostics of the lower atmosphere. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The equilibrium and stability of magnetopolytropes

The theory of the oscillations of axisymmetric gaseous configurations with a prevalent magnetic field is presented. The virial tensor method is used to obtain the nine second harmonic modes of oscillations of the system. It is found that out of the nine modes, three are neutral, four are non-radial, and two are coupled. For the Prendergast spherical model it is found that one of the coupled modes is radial and the other non-radial. Both the radial and the non-radial modes obtained in this case agree with the corresponding formulae obtained byChandrasekhar andLimber (1954) andWoltjer (1962).The equilibrium structure of gaseous polytropes with toroidal magnetic fields is also investigated in detail for values of the polytropic indexn=1, 1.5, 2, 3 and 3.5. For this model the components of the moment of intertia and potential energy tensors together with the non-zero components of the supermatrix potential are obtained. The final results in terms of the effect of weak toroidal magnetic fields on the characteristic frequencies of distorted polytropes are presented in the form of tables.     

The subdwarf-O pulsator SDSS J160043.6+074802.9: comments on atmospheric parameters and pulsation amplitude variations

SDSS J160043.6+074802.9 appears to be a binary consisting of a subwarf-O (sdO) star and a late-type Main Sequence companion; the sdO is the only known pulsator in this class of stars and some modes have pulsation amplitude and frequency variations. Surface gravity and helium abundance determinations in the literature do not agree within quoted error limits; these appear to have unidentified systematic errors which, once accounted for, should show that surface gravity and helium abundance determinations to date are not as discordant as is currently supposed. Non-linear pulsation effects are proposed as a possible interpretation of pulsation amplitude and frequency variations observed in some modes.     

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.     

The rapidly oscillating Ap star HD 99563 and its distorted dipole pulsation mode

We undertook a time-series photometric multisite campaign for the rapidly oscillating Ap (roAp) star HD 99563 and also acquired mean light observations over four seasons. The pulsations of the star, which show flatter light maxima than minima, can be described with a frequency quintuplet centred on 1557.653 μHz and some first harmonics of it. The amplitude of the pulsation is modulated with the rotation period of the star that we determine with 2.91179 ± 0.00007 d from the analysis of the stellar pulsation spectrum and of the mean light data. We break up the distorted oscillation mode into its pure spherical harmonic components and find it is dominated by the ℓ= 1 pulsation, and also has a notable ℓ= 3 contribution, with weak ℓ= 0 and 2 components. The geometrical configuration of the star allows us to see both pulsation poles for about the same amount of time; HD 99563 is only the fourth roAp star for which both pulsation poles are seen and only the third where the distortion of the pulsation modes has been modelled. We point out that HD 99563 is very similar to the well-studied roAp star HR 3831. Finally, we note that the visual companion of HD 99563 is located in the δ Scuti instability strip and may thus show pulsation. We show that if the companion was physical, the roAp star would be a 2.03-M_⊙, object, seen at a rotational inclination of 44°, which then predicts a magnetic obliquity     .     

关于脉冲星非共转模型的研究

本文讨论了共轴、纯转动、force-free的脉冲星系统,在Burman理论基础上建立了一个非共转模型(其中含待定参量,共转情况是它的特例)。利用系统稳定的近似条件——系统能量最小确定特定参量,得到一个与G-J模型类似的共转模型。由于考虑了惯性效应,不存在星风,是稳定态,但没有辐射。在转轴与磁矩有小偏角时,利用非平衡系统更普遍的稳定条件——熵产生最小,确定本模型的参量,得到的辐射能量在量级上与观测值相符合。     

Magnetostatic equilibria for coronal loops on rotating stars

We have investigated magnetostatic equilibria for coronal loops embedded in a potential magnetic field on a rotating star. We find that for any given star, there is a maximum value of the plasma pressure inside a single loop, above which no equilibrium exists. This maximum internal pressure depends on the ratio of the temperatures inside and outside the loop, and on the ratio of the plasma pressure to the magnetic pressure at the base of the external field. Thus, any loop of a large-scale field which is heated or cooled to a different temperature from its immediate surroundings, or which experiences a change in its internal pressure may eventually lose equilbrium. For some values of the base pressure and temperature ratio the relation between summit height and footpoint separation is double-valued. As the summit height of a loop is increased, its footpoint separation increases to a critical value, then decreases to zero at the maximum possible summit height. At the critical footpoint separation the slope of the loop height-footpoint separation relation becomes infinite, and no equilibrium solution exists for greater footpoint separations.We find also that the strength and scale of the field external to the flux tube is the most important factor in determining its maximum height. The effects of varying the stellar rotation rate - and, hence, the variation in pressure with height - are comparatively unimportant, even for very high rotation rates at which the point of balance between gravitational and centrifugal forces lies close to the stellar surface. In this case it is possible to find equilibrium loop solutions whose summits lie outside the centrifugal balance point.We have also investigated the effects of varying the stellar surface gravity. For stellar of fixed mass and rotation rate, the loop dimensions scale approximately linearly with the stellar radius.     

Magnetically distorted polytropes

The oscillations of a gaseous polytrope with a magnetic field having both a toroidal and a poloidal component are examined using the second-order tensor virial equations on the assumption that the magnetic energy is small compared with the gravitational energy. The frequencies of oscillation of the transverse shear, the toroidal and the coupled pulsation modes are tabulated for polytropic indicesn=1, 1.5, 2, 3 and 3.5. It is found that the magnetic field decreases the frequency of oscillation of (i) the transverse shear mode and (ii) the mode which starts as a radial pulsation in the absence of a magnetic field while it increases the frequency of oscillation of (i) the toroidal mode and (ii) the Kelvin mode. In all cases the shift in frequency decreases with increasingn.     

Protodiscs around hot magnetic rotator stars

We develop equations and obtain solutions for the structure and evolution of a protodisc region that is initially formed with no radial motion and super-Keplerian rotation speed when wind material from a hot rotating star is channelled towards its equatorial plane by a dipole-type magnetic field. Its temperature is around 10⁷ K because of shock heating and the inflow of wind material causes its equatorial density to increase with time. The centrifugal force and thermal pressure increase relative to the magnetic force and material escapes at its outer edge. The protodisc region of a uniformly rotating star has almost uniform rotation and will shrink radially unless some instability intervenes. In a star with angular velocity increasing along its surface towards the equator, the angular velocity of the protodisc region decreases radially outwards and magnetorotational instability (MRI) can occur within a few hours or days. Viscosity resulting from MRI will readjust the angular velocity distribution of the protodisc material and may assist in the formation of a quasi-steady disc. Thus, the centrifugal breakout found in numerical simulations for uniformly rotating stars does not imply that quasi-steady discs with slow outflow cannot form around magnetic rotator stars with solar-type differential rotation.