Matching the frequency spectrum of pre-main sequence stars by means of standard and rotating models

We applied the aton evolutionary code to the computation of detailed grids of standard (non-rotating) and rotating pre-main sequence (PMS) models and computed their adiabatic oscillation spectra, with the aim of exploring the seismic properties of young stars. As, until now, only a few frequencies have been determined for ∼40 PMS stars, the way of approaching the interpretation of the oscillations is not unique. We adopt a method similar to the matching mode method by Guenther and Brown making use, when necessary, also of our rotating evolutionary code to compute the models for PMS stars. The method is described by a preliminary application to the frequency spectrum of two PMS stars (85 and 278) in the young open cluster NGC 6530. For the Star 85, we confirm with self-consistent rotating models, previous interpretation of the data, attributing three close frequencies to the mode   n = 4, l = 1  and   m = 0  , +1 and −1. For the Star 278, we find a different fit for the frequencies, corresponding to a model within the original error box of the star, and dispute the possibility that this star has a T _eff much cooler that the red boundary of the radial instability strip.     

The β Cephei instability strip

We carried out a series of linear stability analyses of the radial and low-degree non-radial p modes for stellar models with initial masses of     . The stellar models were computed by using convective overshoot distance     , 0.25 and 0.40  H _P. Our numerical results show that the β Cephei instability strip forms a horn-shaped region pointing upwards near the main sequence on the Hertzsprung–Russell diagram (HRD). The lower part of the instability strip for the radial modes join the zero-age main-sequence (ZAMS) at     , while the top of the instability strip extends up to     . The instability strip for the non-radial modes is even wider. The overall instability strip is dominated by the radial and non-radial fundamental modes. The first overtone (the radial-order index     is also pulsationally unstable. We have shown that the β Cephei stability is almost independent of the overshoot parameter d _over used for the stellar models, while it depends critically on the metal abundance. With decreasing metal abundance, the instability region shrinks and eventually disappears for     .     

Unstable non-radial modes in radial pulsators: theory and an example

We study the possibility of the excitation of non-radial oscillations in classical pulsating stars. The stability of an RR Lyrae model is examined through non-adiabatic non-radial calculations. We also explore stability in the presence of non-linear coupling between radial and non-radial modes of nearly identical frequency.   In our model, a large number of unstable low-degree (ℓ = 1,2) modes have frequencies in the vicinity of unstable radial mode frequencies. The growth rates of such modes, however, are considerably smaller than those of the radial modes. We also recover an earlier result that at higher degrees (ℓ = 6–12) there are modes trapped in the envelope with growth rates similar to those of radial modes.   Subsequently, monomode radial pulsation of this model is considered. The destabilizing effect of the 1:1 resonance between the radial mode and nearby non-radial modes of low degrees is studied, with the assumption that the excited radial mode saturates the linear instability of all other modes. The instability depends on the radial mode amplitude, the frequency difference, the damping rate of the non-radial mode, and the strength of the non-linear coupling between the modes considered. At the pulsation amplitudes typical for RR Lyrae stars, the instability of the monomode radial pulsation and the concomitant resonant excitation of some non-radial oscillation modes is found to be very likely.     

Thermal instability in a weakly ionized plasma

We revisit the problem of clump formation due to thermal instabilities in a weakly ionized plasma with the help of a linear perturbation analysis, as discussed by Nejad-Asghar & Ghanbari. In the absence of a magnetic field and ambipolar diffusion the characteristic equation reduces to the thermal instability described by Field. We derive the critical wavelengths, which separate the spatial ranges of stability and instability. Contrary to the original analysis of Nejad-Asghar & Ghanbari, perturbations with a wavelength larger than the critical wavelength destabilize the cloud. Moreover, the instability regime of isentropic perturbations is drastically reduced. Isobaric modes with real values of the critical wavelength appear only if the density dependence of the cooling rate is more pronounced than the temperature dependence. Isentropic modes arise only if the power of the density in the cooling rate is smaller than 1/2, which is not fulfilled for CO cooling. We find that ambipolar diffusion is not a dominating heating process in molecular gas.     

The dynamical stability of W Ursae Majoris-type systems

Theoretical study indicates that a contact binary system would merge into a rapidly rotating single star due to tidal instability when the spin angular momentum of the system is more than a third of its orbital angular momentum. Assuming that W Ursae Majoris (W UMa) contact binary systems rigorously comply with the Roche geometry and the dynamical stability limit is at a contact degree of about 70 per cent, we obtain that W UMa systems might suffer Darwin's instability when their mass ratios are in a region of about 0.076–0.078 and merge into the fast-rotating stars. This suggests that the W UMa systems with mass ratio   q ≤ 0.076  cannot be observed. Meanwhile, we find that the observed W UMa systems with a mass ratio of about 0.077, corresponding to a contact degree of about 86 per cent would suffer tidal instability and merge into the single fast-rotating stars. This suggests that the dynamical stability limit for the observed W UMa systems is higher than the theoretical value, implying that the observed systems have probably suffered the loss of angular momentum due to gravitational wave radiation (GR) or magnetic stellar wind (MSW).     

Non-adiabatic oscillations of red giants

Using our non-local time-dependent theory of convection, the linear non-adiabatic oscillations of 10 evolutionary model series with masses of  1–3 M_⊙  are calculated. The results show that there is a red giant instability strip in the lower temperature side of the Hertzsprung–Russell diagram which goes along the sequences of the red giant branch and the asymptotic giant branch. For red giants of lower luminosities, pulsation instability is found at high order overtones; the lower order modes from the fundamental to the second overtone are stable. Towards higher luminosity and lower effective temperature, instability moves to lower order modes, and the amplitude growth rate of oscillations also grows. At the high luminosity end of the strip, the fundamental and the first overtone become unstable, while all the modes above the fourth order become stable. The excitation mechanisms have been studied in detail. It is found that turbulent pressure plays a key role for excitation of red variables. The frozen convection approximation is unavailable for the low temperature stars with extended convective envelopes. In any case, this approximation can explain neither the red edge of the Cepheid instability strip, nor the blue edge of the pulsating red giant instability strip. An analytic expression of a pulsation constant as a function of stellar mass, luminosity and effective temperature is presented from this work.     

Magnetic fields of Ap stars as a result of the Tayler instability

Ap star magnetism is often attributed to fossil magnetic fields which have not changed much since the pre‐main‐sequence epoch of the stars. Stable magnetic field configurations are known which could persist probably for the entire mainsequence life of the star, but they may not show the complexity and diversity exhibited by the Ap stars observed. We suggest that the Ap star magnetism is not a result of stable configurations, but is the result of an instability based on strong toroidal magnetic fields buried in the stars. The highly nonaxisymmetric remainders of the instability are reminiscent of the diversity of fields seen on Ap stars. The strengths of these remnant magnetic fields are actually between a few per cent up to considerable fractions of the internal toroidal field; this means field strengths of the order of kGauss being compatible with what is observed. The magnetic fields emerge at the surface rather quickly; rough estimates deliver time‐scales of the order of a few years. Since rotation stabilizes the instability, normal A stars may still host considerable, invisible toroidal magnetic fields (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Secular instability of g-modes in rotating neutron stars

Gravitational radiation tends to drive gravity modes in rotating neutron stars to become unstable. For an inviscid star, the instability sets in when the rotation frequency is about 0.7 times the corresponding mode frequency of the non-rotating star. Neutron stars with spin frequencies ≳100 Hz are susceptible to this instability, with a growth time of the order of years. However, it is likely that viscous dissipation suppresses the instability except for a narrow range of temperatures around 10⁹ K. We also show that the viscosity-driven instability of g-modes is absent.     

The effect of r-mode instability on the evolution of isolated strange stars

We studied the evolution of isolated strange stars (SSs) synthetically, considering the influence of r -mode instability. Our results show that the cooling of SSs with non-ultrastrong magnetic fields is delayed by heating due to r -mode damping for millions of years, while the spin-down of the stars is dominated by gravitational radiation (GR). Especially for the SSs in a possible existing colour–flavour locked (CFL) phase, the effect of r -mode instability on the evolution of stars becomes extremely important because the viscosity, neutrino emissivity and specific heat involving pairing quarks are blocked. It leads to the cooling of these colour superconducting stars being very slow and the stars can remain at high temperature for millions of years, which differs completely from previous understanding. In this case, an SS in CFL phase can be located at the bottom of its r -mode instability window for a long time, but does not spin-down to a very low frequency for hours.     

Stability of magnetic flux tubes in close binary stars

Photometric and Doppler imaging observations of active binaries indicate the existence of starspots at preferred longitudes (position angles with respect to the companion star). We investigate the stability of magnetic flux tubes in the convection zone of close, fast‐rotating binary stars and explore whether the observed preferred longitudes could be caused by tidal forces and the deformation of the active star. We assume a synchronized binary system with spin axes perpendicular to the orbital plane and a rotation period of a few days. The tidal force and the deviation from spherical structure are considered in lowest‐order perturbation theory. The magnetic field is in the form of toroidal magnetic flux rings, which are stored in mechanical equilibrium within the stably stratified overshoot region beneath the convection zone until the field has grown sufficiently strong for the undulatory instability to initiate the formation of rising loops. Frequencies and geometry of stable as well as growth rates of unstable eigenmodes are determined by linear stability analysis. Particular consideration is given to the question whether the effects of tidal forces and perturbations of the stellar structure can force a rising flux loop to enter the convection zone at specific longitudes.     

Excitation and visibility of slow modes in rotating B-type stars

We use the traditional approximation to describe oscillations with frequencies comparable to the angular rotation rate. Validity of this approximation in application to main-sequence B stars is discussed. Numerical results regarding mode stability and visibility are presented for a model of the Be star HD 163868. For this object, Walker et al. detected a record number of mode frequencies using data from the small space telescope MOST . Our interpretation of these data differs from that of Walker et al. In particular, we interpret peaks in the lowest frequency range as retrograde g modes. We find instability in a large number of modes that remain undetectable because of unfavourable aspect and/or effect of cancellation. There is no clear preference to excitation of prograde modes.     

Kelvin–Helmholtz instability and circulation transfer at an isotropic–anisotropic superfluid interface in a neutron star

A recent laboratory experiment suggests that a Kelvin–Helmholtz (KH) instability at the interface between two superfluids – one rotating and anisotropic, the other stationary and isotropic – may trigger sudden spin-up of the stationary superfluid. This result suggests that a KH instability at the crust–core (  ¹ S₀–³ P₂  –superfluid) boundary of a neutron star may provide a trigger mechanism for pulsar glitches. We calculate the dispersion relation of the KH instability involving two different superfluids including the normal fluid components and their effects on stability, particularly entropy transport. We show that an entropy difference between the core and crust superfluids reduces the threshold differential shear velocity and threshold crust–core density ratio. We evaluate the wavelength of maximum growth of the instability for neutron star parameters and find the resultant circulation transfer to be within the range observed in pulsar glitches.     

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.     

A new oscillating model suggestion for FG Vir

Evolutionary stellar models of FG Vir have been developed theoretically and are compared with earlier observational results. Using the models, we performed calculations to obtain radial and non-radial adiabatic oscillation frequencies. The results show that, if the observational splitting was considered and the observational mode identifications were followed, 1.85M _⊙ star models with the rotational velocities in the range from 32 to 66 kms⁻¹ seem to be representative models of FG Vir.     

A numerical study of the r-mode instability of rapidly rotating nascent neutron stars

The first results of numerical analysis of classical r-modes of rapidly rotating compressible stellar models are reported. The full set of linear perturbation equations of rotating stars in Newtonian gravity is solved numerically without the slow rotation approximation. A critical curve of gravitational wave emission induced instability, which restricts the rotational frequencies of hot young neutron stars, is obtained. Taking the standard cooling mechanisms of neutron stars into account, we also show the 'evolutionary curves' along which neutron stars are supposed to evolve as cooling and spinning down proceed. Rotational frequencies of 1.4-M_⊙ stars suffering from this instability decrease to around 100 Hz when the standard cooling mechanism of neutron stars is employed. This result confirms the results of other authors, who adopted the slow rotation approximation.     

Low T/|W| dynamical instability in differentially rotating stars: diagnosis with canonical angular momentum

We study the nature of non-axisymmetric dynamical instabilities in differentially rotating stars with both linear eigenmode analysis and hydrodynamic simulations in Newtonian gravity. We especially investigate the following three types of instability; the one-armed spiral instability, the low   T /| W |  bar instability, and the high   T /| W |  bar instability, where T is the rotational kinetic energy and W is the gravitational potential energy. The nature of the dynamical instabilities is clarified by using a canonical angular momentum as a diagnostic. We find that the one-armed spiral and the low   T /| W |  bar instabilities occur around the corotation radius, and they grow through the inflow of canonical angular momentum around the corotation radius. The result is a clear contrast to that of a classical dynamical bar instability in high   T /| W |  . We also discuss the feature of gravitational waves generated from these three types of instability.     

Dynamo-generated magnetic fields at the surface of a massive star

Spruit has shown that an astrophysical dynamo can operate in the non-convective material of a differentially rotating star as a result of a particular instability in the magnetic field (the Tayler instability). By assuming that the dynamo operates in a state of marginal instability, Spruit has obtained formulae which predict the equilibrium strengths of azimuthal and radial field components in terms of local physical quantities. Here, we apply Spruit's formulae to our previously published models of rotating massive stars in order to estimate Tayler dynamo field strengths. There are no free parameters in Spruit's formulae. In our models of 10- and  50-M_⊙  stars on the zero-age main sequence, we find internal azimuthal fields of up to 1 MG, and internal radial components of a few kG. Evolved models contain weaker fields. In order to obtain estimates of the field strength at the stellar surface, we examine the conditions under which the Tayler dynamo fields are subject to magnetic buoyancy. We find that conditions for Tayler instability overlap with those for buoyancy at intermediate to high magnetic latitudes. This suggests that fields emerge at the surface of a massive star between magnetic latitudes of about 45° and the poles. We attempt to estimate the strength of the field which emerges at the surface of a massive star. Although these estimates are very rough, we find that the surface field strengths overlap with values which have been reported recently for line-of-sight fields in several O and B stars.     

Instability of spherical accretion — I. Shock-free Bondi accretion

We examine the spatial stability of spherical adiabatic Bondi accretion on to a point gravitating mass against external perturbations. Both transonic critical and subsonic subcritical accretion are shown to be stable against purely radial acoustic, vortex or entropy perturbations. In the case of non-radial perturbations the amplitude of the perturbations grows without limit with smaller radii. Instability manifests itself only if the size of the accreting body is much less than the Bondi radius so that the inflow is highly supersonic or highly subsonic at the surface of the accretor in the case of critical or subcritical accretion respectively. These asymptotics hold and consequently the instability may develop for adiabatic index of accreting gas γ < 5/3. We suggest that this instability may lead to an essential thermalization of accreting flow thus, particularly, solving the problem of otherwise inefficient energy release in spherical accretion on to a black hole.     

Stability of toroidal magnetic fields in the solar tachocline and beneath

Stability of toroidal magnetic field in a stellar radiation zone is considered for the cases of uniform and differential rotation. In the rigidly rotating radiative core shortly below the tachocline, the critical magnetic field for instability is about 600 G. The unstable disturbances for slightly supercritical fields have short radial scales ∼1 Mm. Radial mixing produced by the instability is estimated to conclude that the internal field of the sun can exceed the critical value of 600 G only marginally. Otherwise, the mixing is too strong and not compatible with the observed lithium abundance. Analysis of joint instability of differential rotation and toroidal field leads to the conclusion that axisymmetric models of the laminar solar tachocline are stable to nonaxisymmetric disturbances. The question of whether sun-like stars can posses tachoclines is addressed with positive answer for stars with rotation periods shorter than about two months. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The local axisymmetric instability criterion in a thin, rotating, multicomponent disc

Purely gravitational perturbations are considered in a thin rotating disc composed of gas and several stellar components. The dispersion relation for the axisymmetric density waves propagating through the disc is found and the criterion for the local axisymmetric stability of the whole system is formulated. In the appropriate limit of two-component gas we confirm the findings of Jog & Solomon and extend consideration to the case when one component is collisionless. Gravitational stability of the Galactic disc in the solar neighbourhood based on the multicomponent instability condition is explored using recent measurements of the stellar composition and kinematics in the local Galactic disc obtained by the Hipparcos satellite.