Accretion onto stars with octupole magnetic fields: Matter flow, hot spots and phase shifts

Recent measurements of the surface magnetic fields of classical T Tauri stars (CTTSs) and magnetic cataclysmic variables show that their magnetic fields have a complex structure. Investigation of accretion onto such stars requires global three-dimensional (3D) magnetohydrodynamic (MHD) simulations, where the complexity of simulations strongly increases with each higher-order multipole. Previously, we were able to model disc accretion onto stars with magnetic fields described by a superposition of dipole and quadrupole moments. However, in some stars, like CTTS V2129 Oph and BP Tau, the octupolar component is significant and it was necessary to include the next octupolar component. Here, we show results of global 3D MHD simulations of accretion onto stars with superposition of the dipole and octupole fields, where we vary the ratio between components. Simulations show that if octupolar field strongly dominates at the disc-magnetosphere boundary, then matter flows into the ring-like octupolar poles, forming ring-shape spots at the surface of the star above and below equator. The light-curves are complex and may have two peaks per period. In case where the dipole field dominates, matter accretes in two ordered funnel streams towards poles, however the polar spots are meridionally-elongated due to the action of the octupolar component. In the case when the fields are of similar strengths, both, polar and belt-like spots are present. In many cases the light-curves show the evidence of complex fields, excluding the cases of small inclinations angles, where sinusoidal light-curve is observed and ‘hides’ the information about the field complexity.We also propose new mechanisms of phase shift in stars with complex magnetic fields. We suggest that the phase shifts can be connected with: (1) temporal variation of the star’s intrinsic magnetic field and subsequent redistribution of main magnetic poles; (2) variation of the accretion rate, which causes the disc to interact with the magnetic fields associated with different magnetic moments. We use our model to demonstrate these phase shift mechanisms, and we discuss possible applications of these mechanisms to accreting millisecond pulsars and young stars.     

A non-adiabatic analysis for axisymmetric pulsations of magnetic stars

This paper presents the results of a non-adiabatic analysis for axisymmetric non-radial pulsations including the effect of a dipole magnetic field. Convection is assumed to be suppressed in the stellar envelope, and the diffusion approximation is used to radiative transport. As in a previous adiabatic analysis, the eigenfunctions are expanded in a series of spherical harmonics. The analysis is applied to a  1.9-M_⊙  , main-sequence model  (log  T _eff= 3.913)  . The presence of a magnetic field always stabilizes low-order acoustic modes. All the low-order modes of the model that are excited by the κ-mechanism in the He  ii ionization zone in the absence of a magnetic field are found to be stabilized if the polar strength of the dipole magnetic field is larger than about 1 kG. For high-order p modes, on the other hand, distorted dipole and quadrupole modes excited by the κ-mechanism in the H ionization zone remain overstable, even in the presence of a strong magnetic field. It is found, however, that all the distorted radial high-order modes are stabilized by the effect of the magnetic field. Thus, our non-adiabatic analysis suggests that distorted dipole modes and distorted quadrupole modes are most likely excited in rapidly oscillating Ap stars. The latitudinal amplitude dependence is found to be in reasonable agreement with the observationally determined one for HR 3831. Finally, the expected amplitude of magnetic perturbations at the surface is found to be very small.     

Modulation and symmetry changes in stellar dynamos

Stellar dynamos are governed by non-linear partial differential equations (PDEs) which admit solutions with dipole, quadrupole or mixed symmetry (i.e. with different parities). These PDEs possess periodic solutions that describe magnetic cycles, and numerical studies reveal two different types of modulation. For modulations of Type 1 there are parity changes without significant changes of amplitude, while for Type 2 there are amplitude changes without significant changes in parity. In stars like the Sun, cyclic magnetic activity is interrupted by grand minima that correspond to Type 2 modulation. Although the Sun's magnetic field has maintained dipole symmetry for almost 300 yr, there was a significant parity change at the end of the Maunder Minimum. We infer that the solar field may have flipped from dipole to quadrupole polarity (and back) after deep minima in the past and may do so again in the future. Other stars, with different masses or rotation rates, may exhibit cyclic activity with dipole, quadrupole or mixed parity. The origins of such behaviour can be understood by relating the PDE results to solutions of appropriate low-order systems of ordinary differential equations (ODEs). Type 1 modulation is reproduced in a fourth-order system while Type 2 modulation occurs in a third-order system. Here we construct a new sixth-order system that describes both types of modulation and clarifies the interactions between symmetry-breaking and modulation of activity. Solutions of these non-linear ODEs reproduce the qualitative behaviour found for the PDEs, including flipping of polarity after a prolonged grand minimum. Thus we can be confident that these patterns of behaviour are robust, and will apply to stars that are similar to the Sun.     

An equatorially symmetric rotator model for magnetic stars. II. Inhomogeneous element distribution

For the construction of more realistic models of magnetic stars the influence of large inhomogeneities in the distribution of line absorbing elements over the stellar surface on the determination of the mean longitudinal magnetic field component, Beff, is discussed. In this paper we assume the same magnetic field geometry as in paper I, i.e. the superposition of a dipole and an axissymmetrical quadrupole both lying in the equatorial plane with their axes parallel to one another. The models for the inhomogeneties can be classified by the concentration of the line absorbing elements: (1) at the polar caps, (2) in rings surrounding the polar caps, (3) in a belt in the neighbourhood of the “magnetic” latitude δ = 90°. The result of the computations shows that the amount of compensations of both polarities by integration over the visible hemisphere may be remarkably reduced, yielding values of Beff not so far away from the polar field strength as in the homogeneous models. Moreover, the large quadrupole contribution necessary in some cases to represent the observations in the homogeneous models may be considerably smaller with inhomogeneites taken into account. First comparisons between computed and observed values of Beff(t) are made for the stars α²CVn, 53 Cam and HD 215441.     

A search for the age-dependency of Ap star parameters

Some observational data of the sample of the magnetic chemically peculiar stars (MCP stars) are investigated statistically. For the MCP stars of spectral types later than A2 both the frequency distribution and the R ⋅ sin i-values suggest the existence of a linear relation between stellar diameter and rotation period. The MCP stars of spectral types earlier than B9 show an overpopulation of small R ⋅ sin i which may indicate the existence of a second group with smaller radius in this sample. The equatorially symmetric rotator is used as the magnetic model. With respect to its temporal behaviour the effective magnetic field is separated into dipolar and quadrupolar contribution. Both signs of the axisymmetric quadrupole moment appear with equal frequency. The dipole moment which produces the amplitude of the B_eff(t) curve forms for longer periods two groups which are separated by a distinct gap. Both of the groups exhibit magnetic fields which are the stronger the greater the stellar radius is, contrary to what is expected for frozen-in fields. The dominance of magnetic curves without polarity reversal for longer-period stars is in accordance with predictions of the dynamo theory.     

Magnetic Models of HD 115708 and HD 119419

The magnetic fields of the chemically peculiar stars HD 115708 and HD 119419 were modeled using observed curves of variation of the magnetic field with the phase of the rotational period. It turned out that the field of HD 115708 is described, in a first approximation, by a central dipole, while the field of HD 119419 is described by an off-center dipole. The main parameters of the magnetic fields of both stars and maps of the surface field-strength distribution were obtained. The dipole axis of the first star lies in the equatorial plane while that of the second is almost parallel to the axis of rotation.     

Magnetic field models of CP stars. II: HD11503, HD12098, HD12447, HD14437, HD34452, HD40312, HD178892

We model magnetic fields of seven magnetic stars using a program for studying the structure of magnetic fields in CP stars. It appears that five of them clearly manifest the structure of a central dipole, and the remaining two can be explained by a shifted dipole model. Our previous research and the results of this study demonstrate that the dipole orientation inside the stars relative to the rotation axis can vary from 0° to 90°, both for fast and slow rotators. We can not yet solve the question of the existence of a dominant orientation due to lack of statistics. Our modeling results are consistent with those calculated using Preston’s technique in the case of a dipole field configuration.     

Magnetic field model for slowly rotating CP-stars. γEqu= HD201601

A magnetic field model is constructed for the extremely slow rotator γEqu based on measurements of its magnetic field over many years and using the “magnetic charge” method. An analysis of γEqu and of all the data accumulated up to the present on the magnetic field parameters of chemically peculiar stars leads to some interesting conclusions, of which the main ones are: the fact that the axis of rotation and the dipole axis are not parallel in γEqu and the other slowly rotating magnetic stars which we have studied previously is one of the signs that the braking of CP stars does not involve the participation of the magnetic field as they evolve “to the main sequence.” The axes of the magnetic field dipole in slow rotators are oriented arbitrarily with respect to their axes of rotation. The substantial photometric activity of these CP stars also argues against these axes being close. The well-known absence of sufficiently strong magnetic fields in the Ae/Be Herbig stars also presents difficulties for the hypothesis of “magnetic braking” in the “pre-main sequence” stages of evolution. The inverse relation between the average surface magnetic field Bs and the rotation period P is yet another fact in conflict with the idea that the magnetic field is involved in the braking of CP stars. We believe that angular momentum loss involving the magnetic field can hardly have taken place during evolution immediately prior “to the main sequence,” rather the slow rotation of CP stars most likely originates from protostellar clouds with low angular momentum. Some of the slowly rotating stars have a central dipole magnetic field configuration, while others have a displaced dipole configuration, where the displacement can be toward the positive or the negative magnetic pole. __________ Translated from Astrofizika, Vol. 49, No. 2, pp. 251–262 (May 2006).     

The Permanent and Inductive Magnetic Moments of Ganymede

Data acquired by the Galileo magnetometer on five passes by Ganymede have been used to characterize Ganymede's internal magnetic moments. Three of the five passes were useful for determination of the internal moments through quadrupole order. Models representing the internal field as the sum of dipole and quadrupole terms or as the sum of a permanent dipole field upon which is superimposed an induced magnetic dipole driven by the time varying component of the externally imposed magnetic field of Jupiter's magnetosphere give equally satisfactory fits to the data. The permanent dipole moment has an equatorial field magnitude 719 nT. It is tilted by 176° from the spin axis with the pole in the southern hemisphere rotated by 24° from the Jupiter-facing meridian plane toward the trailing hemisphere. The data are consistent with an inductive response of a good electrical conductor of radius approximately 1 Ganymede radius. Although the data do not enable us to establish the presence of an inductive response beyond doubt, we favor the inductive response model because it gives a good fit to the data using only four parameters to describe the internal sources of fields, whereas the equally good dipole plus quadrupole fit requires eight parameters. An inductive response is consistent with a buried conducting shell, probably liquid water with dissolved electrolytes, somewhere in the first few hundred km below Ganymede's surface. The depth at which the ocean is buried beneath the surface is somewhat uncertain, but our favored model suggests a depth of the order of 150 km. As both temperature and pressure increase with depth and the melting temperature of pure ice decreases to a minimum at ∼170 km depth, it seems possible that near this location, a layer of water would be sandwiched between layers of ice.     

Model of the magnetic field in the inner heliosphere with regard to radial field strength leveling in the solar corona

On the basis of experimental results from the Ulysses spacecraft, a model is proposed for calculating the magnetic field in the corona and the heliosphere in the potential approximation, which is a modification of the potential-field source-surface model. In addition to the photospheric surface and the source surface, a new demarcated spherical surface (leveling surface) is introduced in this model. The magnitude of the radial component of the magnetic field on this surface is assumed to be constant, and its sign abruptly changes from one hemisphere to another. General analytical formulas are given to calculate the potential and field for this model. Calculations are described in detail using the dipole and quadrupole harmonics as examples. Expressions are obtained for the surface currents. The results of visualization of the magnetic field for an axial dipole are discussed.     

The dipole-quadrupole cycle of the background solar magnetic field

The evolution of the background magnetic field with the solar cycle has been studied using the dipole-quadrupole magnetic energy behaviour in a cycle. The combined energy of the axisymmetric dipole, non-axisymmetric quadrupole, and equatorial dipole is relatively lowly variable over the solar cycle. The dipole field changed sign when the quadrupole field was near a maximum, andvice versa. A conceptual picture involving four meridional magnetic polarity sectors proposed to explain these features may be in agreement with equatorial coronal hole observations. The rate of sector rotation is estimated to be 8 heliographic degrees per year faster than the Carrington rotation (P = 27.23^d synodic). Polarity boundaries of sectors located 180° apart show meridional migrations in one direction, while the boundaries of the other two sectors move in the opposite direction. A simple model of how the magnetic field energy varies, subject to specifying reasonable initial photospheric magnetic and velocity field patterns, follows the observed evolution of the dipole and quadrupole field energies quite nicely.     

Relativistic models of magnetars: structure and deformations

We find numerical solutions of the coupled system of Einstein–Maxwell equations with a linear approach, in which the magnetic field acts as a perturbation of a spherical neutron star. In our study, magnetic fields having both poloidal and toroidal components are considered, and higher order multipoles are also included. We evaluate the deformations induced by different field configurations, paying special attention to those for which the star has a prolate shape. We also explore the dependence of the stellar deformation on the particular choice of the equation of state and on the mass of the star. Our results show that, for neutron stars with mass   M = 1.4 M_⊙  and surface magnetic fields of the order of 10¹⁵ G, a quadrupole ellipticity of the order of 10⁻⁶ to 10⁻⁵ should be expected. Low-mass neutron stars are in principle subject to larger deformations (quadrupole ellipticities up to 10⁻³ in the most extreme case). The effect of quadrupolar magnetic fields is comparable to that of dipolar components. A magnetic field permeating the whole star is normally needed to obtain negative quadrupole ellipticities, while fields confined to the crust typically produce positive quadrupole ellipticities.     

Magnetic field structures in chemically peculiar stars

We report the results of magnetic field modelling of around 50 CP stars, performed using the “magnetic charges” technique. The modelling shows that the sample reveals four main types of magnetic configurations: 1) a central dipole, 2) a dipole, shifted along the axis, 3) a dipole, shifted across the axis, and 4) complex structures. The vast majority of stars has the field structure of a dipole, shifted from the center of the star. This shift can have any direction, both along and across the axis. A small percentage of stars possess field structures, formed by two or more dipoles.     

Pulsar magnetosphere

Pulsars are presently believed to be rotating neutron stars with large frozen-in magnetic fields normally assumed to be dipole fields. It has been shown that such a star must possess a magnetosphere if it rotates sufficiently rapidly. By assuming that the magnetic field is dipolar, and unaffected by the trapped particles in the magnetosphere, and that the field dipole axis is parallel to the rotation axis, Goldreich and Julian determined many of the properties of the magnetosphere. In this paper is given a self-consistent model of the closed field lines of a pulsar magnetosphere. Using this model, it is shown that, close to the star, the above assumptions of Goldreich and Julian are justified. Their results are extended to the oblique rotator as well as to stars with magnetic multipoles of arbitrary order and arbitrary orientation.Supported in part by the U.S. Atomic Energy Commission under Grant 2171T.     

Nonradial oscillations of low-mass stars in the presence of a magnetic field

Nonradial oscillations of a partially degenerate standard model, approximating a class of low-mass stars, have been studied in the presence of a weak poloidal magnetic field. The magnetic field in the interior of the configuration is taken to be continuous across the equilibrium surface and is matched with an external dipole field. Using a variational formulation, corrections to the oscillation frequencies of the Kelvin mode have been found for different values of the central degeneracy. It has been noted that the effect of the magnetic field is to increase the frequency of nonradial (l=2) mode of pulsation.     

Bimodal mass and angular momentum losses by magnetically controlled winds during stellar evolution

In recent times evidence for bimodal distributions of stars in the H–R diagram has reached a striking evidence. These bimodal distributions seem to be correlated with a bimodal distribution of masses and angular velocities. The approach we propose to explain the observed bimodality suggests that this latter is due to a bimodal mass loss by magnetically controlled stellar winds during stellar evolution, owing to different magnetic field configurations. It is assumed a mechanism analogous to that which produces solar wind, with magnetic field generated by dynamo working in the convection zone. Different field geometries (dipole cr quadrupole), which depend on the mode the dynamo operates, can produce different but discrete mass losses during stellar evolution, thus producing bimodal distributions of masses and angular velocities.     

中子星的热演化和磁衰减

本文研究了中子星的热演化、自转演化和磁场演化的相互影响．考虑了一个自洽模型：中子星因磁偶极辐射而自转减慢，在内部产生某些加热过程，中子星磁场通过壳层的欧姆耗散来衰减．结果表明，磁场衰减提高了加热过程的重要性；相反，加热效应减慢了磁衰减．因此可以得出，中子星的热、自转和磁场也许不是独立演化的．不仅如此，这些演化与初始条件有关，因此，人们也许可以从射电和Ｘ射线观测对脉冲星年龄、初始磁场和周期给出某些限制．     

Features of multi-dipole magnetic field structures in CP stars

Ten of the sixty investigated magnetic stars have two- or three-dipole structures. From the viewpoint of the relic hypothesis a wide variety of magnetic field structures and strengths allows to assume that in the initial phases of formation of magnetic stars, their fields were even more entangled and heterogeneous than now. This may be due to the complex structure of protostellar clouds, the consequence of non-stationary processes during the collapse, and, probably, the result of subsequent accretion interactions. The expected variation of the large-scale structure with age is lost at the background of a wide variety of structures, depending on the initial conditions. Complex structures occur both in the stars at ZAMS, and in the stars leaving the Main Sequence. As a result of quadratic dependence of the magnetic structure lifetime on their characteristic dimensions, large-scale configurations can exist for times comparable to the lifetime of stellar magnetic field, i.e. τ ≥ 10⁹ yrs. One of the common properties of multi-dipole stars is that the centers of the dipoles are predominantly located in the equatorial plane of rotation. In the majority of studied objects magnetic dipoles (i.e. the regions with the maximum field) are shifted from the center of the star by the distance greater than the radius of the convective core (approximately 0.1R*). This may indicate that the poloidal field is not compatible with the convective core and is not generated therein. Large distances between the monopoles, comparable to the radii of the stars are typical. This may be a sign indicating that inside the stars the field structure is slightly different from the dipole, what implies that the dipole is not a mathematical point, but rather some highly magnetized volume inside the star, comparable to a magnetized rod.     

Magnetic field models of CP stars HD18296, HD19832, HD22470, HD24712

We model the magnetic fields of four magnetic stars using published longitudinal (Be) field measurements. The structure of the magnetic field of each of the four stars is close to that of the central dipole. Unfortunately, the number of measurements for each star is insufficient for accurate finding of the field parameters, and therefore we find no dipole shift exceeding its error Δa ≈ 0.1, expressed as a fraction of the stellar radius. Our data support the opinion that the results of modeling depend most strongly on the adopted inclination of the star’s rotation axis i.     

Current Flows in Pulsar Magnetospheres

The global structure of current flows in pulsar magnetosphere is investigated, with rough calculations of the circuit elements. It is emphasized that the potential of the critical field lines (the field lines that intersect the null surface at the light cylinder radius) should be the same as that of interstellar medium, and that pulsars whose rotation axes and magnetic dipole axes are parallel should be positively charged, in order to close the pulsar's current flows. The statistical relation between the radio luminosity and pulsar's electric charge (or the spindown power) may hint that the millisecond pulsars could be low-mass bare strange stars.