Operator algebra and the stationary states of stellar magnetic fields

A vector-operator algebra technique for solving magnetic field problems in a toroidal/poloidal representation is illustrated with physical examples. Among the illustrative examples are calculations of necessary and/or sufficient conditions for the existence of stationary magnetic fields in stellar interiors.Paper dedicated to Professor Hannes Alfvén on the occasion of his 70th birthday, 30 May, 1978.This research was supported in part by the National Science Foundation.     

On the origin and structure of stellar magnetic fields

Newly formed stars have magnetic fields provided by the compression of the interstellar field, and contrary to a widely accepted idea these fields are not destroyed by convective motions. For the same reason, the fallacy of ‘turbulent diffusion’, turbulent dynamo action is not possible in any star. Thus all stellar magnetic fields have a common origin, and persist throughout the lifetime of each star, including degenerate phases. This common origin, and a general similarity in stellar evolutionary processes, suggest that the fields may develop similar structural characteristics and MHD effects. This would open new possibilities of coordinating the studies of different types of stars and relating them to solar physics which has tended to become isolated from general stellar physics. As an initial step we consider three features of solar magnetic fields and their MHD effects. First, the solar magnetic field comprises two separate components: a poloidal field and a toroidal field. The former is a dipole field, permeating the entire Sun and closely aligned with the rotational axis; at the surface it is always concealed by much stronger elements of the toroidal field. The latter is probably wound from the former by differential rotation at latitudes below about 35°, where sections emerge through the solar surface and are then carried polewards. The second feature of solar magnetic fields is that all flux is concentrated into flux tubes of strength some kG, isolated within a much larger volume of non-magnetic plasma. The third feature is that the flux tubes are helically twisted into flux ropes (up to ?10²²Mx) and smaller elements ranging down to flux fibres (? 10¹⁸Mx). Some implications of similar features in other stars are discussed.     

The oscillation frequencies of magnetic stellar models

A first-order perturbation theory method developed by Goossens to determine the perturbation to the eigenfrequencies of stellar models caused by the presence of a magnetic field is modified slightly, and applied to models with toroidal and poloidal fields. Some limitations of the analysis are pointed out.     

On the oscillations of a composite stellar model with a weak magnetic field

The unrestricted second-order virial tensor formalism has been used to calculate the characteristic frequencies of linear adiabatic oscillations of a composite stellar model having an isothermal core and a polytropic envelope in presence of a weak poloidal magnetic field. The frequencies of the transverse shear mode and the nonradial pulsation mode for both a radiative and a convective envelope (corresponding to polytropic index 3 and 1.5, respectively) alongwith that of the toroidal mode for the radiative envelope get increased in presence of the magnetic field. However, the frequency of the toroidal mode for the convective envelope registers a decrease in presence of the field. The corrections to the various frequencies decrease with increasing values of the parameter characterizing the lowering of the core temperature in presence of the magnetic field.     

Toroidal versus poloidal magnetic fields in Sun-like stars: a rotation threshold

From a set of stellar spectropolarimetric observations, we report the detection of surface magnetic fields in a sample of four solar-type stars, namely HD 73350, HD 76151, HD 146233 (18 Sco) and HD 190771. Assuming that the observed variability of polarimetric signal is controlled by stellar rotation, we establish the rotation periods of our targets, with values ranging from 8.8 d (for HD 190771) to 22.7 d (for HD 146233). Apart from rotation, fundamental parameters of the selected objects are very close to the Sun's, making this sample a practical basis to investigate the specific impact of rotation on magnetic properties of Sun-like stars.
We reconstruct the large-scale magnetic geometry of the targets as a low-order  (ℓ < 10)  spherical harmonic expansion of the surface magnetic field. From the set of magnetic maps, we draw two main conclusions. (i) The magnetic energy of the large-scale field increases with rotation rate. The increase in chromospheric emission with the mean magnetic field is flatter than observed in the Sun. Since the chromospheric flux is also sensitive to magnetic elements smaller than those contributing to the polarimetric signal, this observation suggests that a larger fraction of the surface magnetic energy is stored in large scales as rotation increases. (ii) Whereas the magnetic field is mostly poloidal for low rotation rates, more rapid rotators host a large-scale toroidal component in their surface field. From our observations, we infer that a rotation period lower than ≈12 d is necessary for the toroidal magnetic energy to dominate over the poloidal component.     

On axisymmetric hydromagnetic equilibria

The physical characteristics of possible axisymmetric equilibria are examined on the basis of the integrals of hydromagnetic equations. It is shown for nearly spherical configurations that a surface differential rotation is possible only in the absence of a meridional circulation with either purely toroidal or purely poloidal magnetic field. In the presence of a meridional circulation, it is shown that no surface rotation or constant rotation is possible if the magnetic field is purely toroidal, and that no rotation is possible if the magnetic field is purely poloidal. A brief discussion is given on the possible solutions including the case of stellar winds with force-free magnetic fields.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Non-ideal evolution of non-axisymmetric, force-free magnetic fields in a magnetar

Recent numerical magnetohydrodynamic calculations by Braithwaite and collaborators support the 'fossil field' hypothesis regarding the origin of magnetic fields in compact stars and suggest that the resistive evolution of the fossil field can explain the reorganization and decay of magnetar magnetic fields. Here, these findings are modelled analytically by allowing the stellar magnetic field to relax through a quasi-static sequence of non-axisymmetric, force-free states, by analogy with spheromak relaxation experiments, starting from a random field. Under the hypothesis that the force-free modes approach energy equipartition in the absence of resistivity, the output of the numerical calculations is semiquantitatively recovered: the field settles down to a linked poloidal–toroidal configuration, which inflates and becomes more toroidal as time passes. A qualitatively similar (but not identical) end state is reached if the magnetic field evolves by exchanging helicity between small and large scales according to an α-dynamo-like, mean-field mechanism, arising from the fluctuating electromotive force produced by the initial random field. The impossibility of matching a force-free internal field to a potential exterior field is discussed in the magnetar context.     

The effect of rotation on the luminosity of magnetic stars

The effect of rotation and a general magnetic field on the luminosity, radius, and effective temperature of the upper Main-Sequence stars has been investigated using a perturbation analysis. The magnetic field profile prevailing inside the star is assumed to have both poloidal and toroidal components. The case of constant as well as differential rotation is admitted. Model calculations indicate that these stellar parameters modify considerably as a result of coupling between rotation and the magnetic field.     

Stochastic models of stellar winds in T Tauri-stars

We consider the conditions in which emission lines of hydrogen, ionized calcium and ionized magnesium are formed in the spectra of T Tauri-stars. It is well known that there is a problem in interpreting the two-component profiles which are frequently observed in H, Caii K, and Mgii k lines. This problem can be solved if we abandon an assumption which is customarily made in modelling stellar winds: namely, the approximation of a continuous medium. Our calculations indicate that line profiles similar to those which are observed can be obtained by making very simple assumptions about spherical-symmetry and quasistationary gas flow provided that the following conditions are also stratified: (i) the stellar wind consists of separate gas structure (blobs) with a volume-filling factorq of order 0.1; (ii) the blobs decelerate in the outer regions of the wind.The first of these conditions indicates that the outflow of the gas from the surface of a T Tauri-star is extremely inhomogeneous. At any instant, only about 1/10 of the stellar surface acts as a source of stellar wind. Although this conclusion is at first sight very unexpected, it is nevertheless entirely logical if the outflow of gas from the star originates in magnetic activity on the stellar surface.     

Binary stellar winds

Stellar winds from a binary star pair will interact with each other along a contact discontinuity. We discuss qualitatively the geometry of the flow and field resulting from this interaction in the simplest case where the stars and winds are identical. We consider the shape of the critical surface (defined as the surface where the flow speed is equal to the sound speed) as a function of stellar separation and the role of shock waves in the flow field. The effect of stellar spin and magnetic sectors on the field configuration is given. The relative roles of mass loss and magnetic torque in the evolution of orbital parameters are discussed.     

Magnetic arcades in stellar coronae I. Cylindrical geometry

X-ray spectroscopy performed by different astronomical spacecrafts has shown that many active late-type stars possess coronae. For such reason, the magnetic structure of stellar coronae has raised considerable interest and, by analogy with the Sun, it is generally assumed that stellar coronae are structured by magnetic fields having the shape of arcades. Most of those coronal magnetic field configurations assume translational symmetry and are based in planar source surfaces. However, as soon as either the length or the width of such source surfaces become non negligible as compared to the stellar radius, the application of the cylindrical geometry seems to be more appropriate. Then, one way of obtaining coronal magnetic configurations is to deal with source domains extended over a cylindrical surface. In this paper we generate potential coronal arcades based on cylindrical source surfaces with non negligible length or width compared to the stellar radius. The flux function, the magnetic field components, the shape of magnetic field lines and other characteristic magnitudes have been obtained and analyzed for both cases. This revised version was published online in July 2006 with corrections to the Cover Date.     

Origin of stellar magnetic fields

It is pointed out that the magnetic field of a star is originated from dynamo action associated with the stellar evolution. The magnetic field of a star is related with how much nuclear energy is generated in its phase of evolution. From this we can explain why some stars possess a magnetic field high than that of the Sun. In our case the magnetic field of the star is a by-product of the stars evolution and it has no influence on the internal structure of the star but it does have influence on the flare, chromosphere and coronal activities of the star. Again it is stressed that to confirm the activities of the star, the details of evolution of stars should be calculated according to the photon-neutrino coupling theory.     

The influence of momentum addition on the acceleration of the solar and stellar winds

The influence of the momentum addition, which may be associated with the average or fluctuation transverse component of the magnetic field or others, on the acceleration the solar wind or stellar wind is studied in a local streamtube. The results show that the larger the momentum addition the stronger the acceleration of the wind. For example, if the typical transverse magnetic field is about 0.1 of the longitudinal field, the velocity of the solar wind at 1 AU may be increased by 40%. The coronal hole may be considered as a streamtube, the presence of a high stream from the coronal hole may be explained by the existence of an average or fluctuation transverse magnetic field in the streamtube. A similar conclusion may be applied to the polar region, where the velocity of the solar wind will be larger than elsewhere as if there is a transverse component of magnetic field, as well as to the stellar wind. The influence of other parameters on the acceleration of the solar wind is also discussed. From the viewpoint of the solar wind mechanism, the present paper shows that the momentum addition in the subsonic flow region can increase the velocity of the solar wind at 1 AU.     

Radiation transfer of gyrosynchrotron emission in stellar environments

Non-thermal radio emission has been detected from several kinds of active stars. Polarization and intensity measurements of the quiescent (i.e. non-flaring) emission indicate gyrosynchrotron emission. A three-dimensional magnetic field model for the stellar field is presented and the gyrosynchrotron intensity and polarization emerging from such a model is calculated and compared with observations. Model spectra agree well with observations. Model polarization results indicate that the emission region is more complex than the simple model and may indicate the presence of multipole components in the field, small loops of flux near the stellar surface and significant energy loss in the radiating electron distribution over time.     

The periods of stellar models with linked poloidal-toroidal magnetic fields

First-order perturbation theory results for the changes in pulsation frequencies of a Cowling model star containing a magnetic field with both poloidal and toroidal components are presented. A toroidal field large enough to stabilize the poloidal field may reverse the sign of the frequency change caused by a purely poloidal field for some modes, including the fundamental radial mode.     

Method of ignoring the systematic variations of stellar parallaxes over the celestial sphere in a kinematic analysis of stellar proper motions

A method for determining the velocity field parameters free from the distortions due to the systematic variations of stellar parallaxes over the celestial sphere is proposed. The method is based on the approximation of parallaxes as a function of coordinates on the sphere using spherical harmonics and can be applied in those cases where the solar motion cannot be eliminated from the stellar proper motions. Numerical experiments have shown that our method is able to obtain accurate coordinates of the solar apex and to calculate the kinematic parameters of the Ogorodnikov-Milne model to within three coefficients of the decomposition of parallaxes into first-order spherical harmonics. Examples of applying the method to the stellar proper motions of the Hipparcos catalogue, which admits checking the results using trigonometric parallaxes, are provided. Such a check has been found to yield a positive result only for nearby stars at heliocentric distances that do not exceed 400 pc and for which the parallaxes were determined with a relative error of at least 30%. An interesting feature of this method is the possibility to construct the shape of the figure which is formed by the deviations of the parallaxes from the sphere corresponding to the average parallaxes of the stars under consideration. It should be specially emphasized that all of this is done in the complete absence of information about the stellar parallaxes. The “solar terms” of the stellar proper motions that are formed by the products of the parallaxes by the solar motion components relative to the centroid of stars are the main source of information about the parallaxes here.     

On the quasi-periodic oscillations in magnetars

We study torsional Alfvén oscillations of magnetars, that is neutron stars with a strong magnetic field. We consider the poloidal and toroidal components of the magnetic field and a wide range of equilibrium stellar models. We use a new coordinate system  ( X , Y )  , where     and     and a ₁ is the radial component of the magnetic field. In this coordinate system, the one+two-dimensional evolution equation describing the quasi-periodic oscillations (QPOs), see Sotani et al., is reduced to a one+one-dimensional equation where the perturbations propagate only along the y -axis. We solve the one+one-dimensional equation for different boundary conditions and the open magnetic field lines, that is magnetic field lines that reach the surface and there match up with the exterior dipole magnetic field as well as closed magnetic lines, i.e. magnetic lines that never reach the stellar surface. For the open field lines, we find two families of QPO frequencies: a family of 'lower' QPO frequencies which is located near the x -axis and a family of 'upper' frequencies located near the y -axis. According to Levin, the fundamental frequencies of these two families can be interpreted as the turning point of the continuous spectrum. We find that the upper frequencies are multiples of the lower ones by a constant equalling  2 n + 1  . For the closed lines, the corresponding factor is   n + 1  . By using these relations, we can explain both the lower and the higher observed frequencies in SGR 1806−20 and SGR 1900+14.     

The magnetosphere of oscillating neutron stars in general relativity

Just as a rotating magnetized neutron star has material pulled away from its surface to populate a magnetosphere, a similar process can occur as a result of neutron-star pulsations rather than rotation. This is of interest in connection with the overall study of neutron star oscillation modes but with a particular focus on the situation for magnetars. Following a previous Newtonian analysis of the production of a force-free magnetosphere in this way Timokhin et al., we present here a corresponding general-relativistic analysis. We give a derivation of the general relativistic Maxwell equations for small-amplitude arbitrary oscillations of a non-rotating neutron star with a generic magnetic field and show that these can be solved analytically under the assumption of low current density in the magnetosphere. We apply our formalism to toroidal oscillations of a neutron star with a dipole magnetic field and find that the low current density approximation is valid for at least half of the oscillation modes, similarly to the Newtonian case. Using an improved formula for the determination of the last closed field line, we calculate the energy losses resulting from toroidal stellar oscillations for all of the modes for which the size of the polar cap is small. We find that general relativistic effects lead to shrinking of the size of the polar cap and an increase in the energy density of the outflowing plasma. These effects act in opposite directions but the net result is that the energy loss from the neutron star is significantly smaller than suggested by the Newtonian treatment.     

The interaction of the stellar wind with an extrasolar planet—3D hybrid and drift-kinetic simulations

To be able to simulate the interaction of extrasolar planets with the stellar wind, a number of planetary parameters are required. Some of these (like planetary mass and radius) can be obtained directly from observational data. Other properties are not known very precisely. For example, up to now, there is no observation providing information on the strength of planetary magnetic moments. However, there is good reason to expect only very small magnetic moments for planets in very close orbits around their stars (like HD 209458 b and OGLE-TR-56 b). Thus, as a first step towards a more complete treatment, it seems reasonable to treat the interaction of the stellar wind with an unmagnetized planet. Calculations were performed for a nonconducting as well as for a weakly conducting planet. The interaction with the stellar wind and the resulting induced magnetosphere was simulated using a three dimensional hybrid code as well as in the drift-kinetic approximation. The effect of a interplanetary magnetic field oriented perpendicular to the incoming stellar wind was included. In the case of a weakly conducting body an asymmetrical Mach cone is formed, whereas for a nonconducting body no Mach cone is observed. These investigations will serve as the first step in the search for particular effects occurring at extrasolar planets, which could possibly lead to observable effects, e.g. radio emission. The results are also relevant for plasma structures near weakly conducting, unmagnetized bodies like the Earth's moon.     

Two-dimensional relativistic stellar wind

The anisotropic structure of the relativistic stellar wind is investigated. Both relativistic fluid velocity and relativistic temperature are taken into account. General analysis is carried out in the curvilinear coordinates and the generalization of the dispersion equation is obtained. The topological structure of the individual field lines is the same as in the spherically-symmetric case, except the fact that the magnetic field dependence on distance cannot be establisheda priori. The interaction between neighbouring field lines brings the dependence on the transverse coordinate, numbering the field lines. This dependence leads to the establishing of a new constraint on the global flow topology. The two-dimensional wind structure is analyzed, with the constraint taken into account, in the large distances limit, using the asymptotic expansion into ther ^–1 power series. In the lowest order approximation the constraint reduces to a new global constant of motion. This constant causes the splitting of the two solution families.