Confirmation of simultaneous p and g mode excitation in HD 8801 and γ Peg from time-resolved multicolour photometry of six candidate 'hybrid' pulsators

We carried out a multicolour time-series photometric study of six stars claimed as 'hybrid' p and g mode pulsators in the literature. γ Peg was confirmed to show short-period oscillations of the β Cep type and simultaneous long-period pulsations typical of Slowly Pulsating B (SPB) stars. From the measured amplitude ratios in the Strömgren uvy passbands, the stronger of the two short period pulsation modes was identified as radial; the second is  ℓ= 1  . Three of the four SPB-type modes are most likely  ℓ= 1  or 2. Comparison with theoretical model calculations suggests that γ Peg is either a  ∼8.5 M_⊙  radial fundamental mode pulsator or a  ∼9.6 M_⊙  first radial overtone pulsator. HD 8801 was corroborated as a 'hybrid'δ Sct/γ Dor star; four pulsation modes of the γ Dor type were detected, and two modes of the δ Sct type were confirmed. Two pulsational signals between the frequency domains of these two known classes of variables were confirmed and another was newly detected. These are either previously unknown types of pulsation or do not originate from HD 8801. The O-type star HD 13745 showed small-amplitude slow variability on a time-scale of 3.2 d. This object may be related to the suspected new type of supergiant SPB stars, but a rotational origin of its light variations cannot be ruled out at this point. 53 Psc is an SPB star for which two pulsation frequencies were determined and identified with low spherical degree. Small-amplitude variability was formally detected for 53 Ari but is suspected not to be intrinsic. The behaviour of ι Her is consistent with non-variability during our observations, and we could not confirm light variations of the comparison star 34 Psc previously suspected. The use of signal-to-noise criteria in the analysis of data sets with strong aliasing is critically discussed.     

On the properties of strange modes

Properties of the so-called strange modes occurring in linear stability calculations of stellar models are discussed. The behaviour of these modes is compared for two different sets of stellar models, for very massive zero-age main-sequence stars and for luminous hydrogen-deficient stars, both with high luminosity-to-mass ratios. We have found that the peculiar behaviour of the frequencies of the strange modes with the change of a control parameter is caused by the pulsation amplitude of a particular eigenmode being strongly confined to the outer part of the envelope, around the density inversion zone. The frequency of a strange mode changes because the depth of the confinement zone changes with the control parameter. Weakly non-adiabatic strange modes tend to be overstable because the amplitude confinement quenches the effect of radiative damping. On the other hand, extremely non-adiabatic strange modes become overstable because the perturbation of radiation force (gradient of radiation pressure) provides a restoring force that can be out of phase with the density perturbation. We discuss this mechanism by using a plane-parallel two-zone model.     

On the dynamics of superfluid neutron star cores

We discuss the nature of the various modes of pulsation of superfluid neutron stars using comparatively simple Newtonian models and the Cowling approximation. The matter in these stars is described in terms of a two-fluid model, where one fluid is the neutron superfluid, which is believed to exist in the core and inner crust of mature neutron stars, and the other fluid represents a conglomerate of all other constituents (crust nuclei, protons, electrons, etc.). In our model, we incorporate the non-dissipative interaction known as the entrainment effect, whereby the momentum of one constituent (e.g. the neutrons) carries along part of the mass of the other constituent. We show that there is no independent set of pulsating g-modes in a non-rotating superfluid neutron star core, even though the linearized superfluid equations contain a well-defined (and real-valued) analogue to the so-called Brunt–Väisälä frequency. Instead, what we find are two sets of spheroidal perturbations whose nature is predominately acoustic. In addition, an analysis of the zero-frequency subspace (i.e. the space of time-independent perturbations) reveals two sets of degenerate spheroidal perturbations, which we interpret to be the missing g-modes, and two sets of toroidal perturbations. We anticipate that the degeneracy of all these zero-frequency modes will be broken by the Coriolis force in the case of rotating stars. To illustrate this we consider the toroidal pulsation modes of a slowly rotating superfluid star. This analysis shows that the superfluid equations support a new class of r-modes, in addition to those familiar from, for example, geophysical fluid dynamics. Finally, the role of the entrainment effect on the superfluid mode frequencies is shown explicitly via solutions to dispersion relations that follow from a 'local' analysis of the linearized superfluid equations.     

Towards mode selection in δ Scuti stars: regularities in observed and theoretical frequency spectra

Only a fraction of the theoretically predicted non-radial pulsation modes have so far been observed in δ Scuti stars. Nevertheless, the large number of frequencies detected in recent photometric studies of selected δ Scuti stars allow us to look for regularities in the frequency spacing of modes. Mode identifications are used to interpret these results.
Statistical analyses of several δ Scuti stars (FG Vir, 44 Tau, BL Cam and others) show that the photometrically observed frequencies are not distributed at random, but that the excited non-radial modes cluster around the frequencies of the radial modes over many radial orders.
The observed regularities can be partly explained by modes trapped in the stellar envelope. This mode selection mechanism was proposed by Dziembowski & Królikowska and shown to be efficient for  ℓ= 1  modes. New pulsation model calculations confirm the observed regularities.
We present the s – f diagram, which compares the average separation of the radial frequencies ( s ) with the frequency of the lowest frequency unstable radial mode ( f ). This provides an estimate for the  log  g   value of the observed star, if we assume that the centres of the observed frequency clusters correspond to the radial mode frequencies. This assumption is confirmed by examples of well-studied δ Scuti variables in which radial modes were definitely identified.     

Crustal oscillations of slowly rotating relativistic stars

We study low-amplitude crustal oscillations of slowly rotating relativistic stars consisting of a central fluid core and an outer thin solid crust. We estimate the effect of rotation on the torsional toroidal modes and on the interfacial and shear spheroidal modes. The results compared against the Newtonian ones for wide range of neutron star models and equations of state.     

A General Relativistic Calculation of Boundary Layer and Disc Luminosity for Accreting Non-magnetic Neutron Stars in Rapid Rotation

We calculate the disc and boundary layer luminosities for accreting rapidly rotating neutron stars with low magnetic fields in a fully general relativistic manner. Rotation increases the disc luminosity and decreases the boundary layer luminosity. A rapid rotation of the neutron star substantially modifies these quantities as compared with the static limit. For a neutron star rotating close to the centrifugal mass shed limit, the total luminosity has contribution only from the extended disc. For such maximal rotation rates, we find that well before the maximum stable gravitational mass configuration is reached, there exists a limiting central density, for which particles in the innermost stable orbit will be more tightly bound than those at the surface of the neutron star. We also calculate the angular velocity profiles of particles in Keplerian orbits around the rapidly rotating neutron star. The results are illustrated for a representative set of equation of state models of neutron star matter.     

Oscillations of rapidly rotating superfluid stars

Using time evolutions of the relevant linearized equations, we study non-axisymmetric oscillations of rapidly rotating and superfluid neutron stars. We consider perturbations of Newtonian axisymmetric background configurations and account for the presence of superfluid components via the standard two-fluid model. Within the Cowling approximation, we are able to carry out evolutions for uniformly rotating stars up to the mass-shedding limit. This leads to the first detailed analysis of superfluid neutron star oscillations in the fast rotation regime, where the star is significantly deformed by the centrifugal force. For simplicity, we focus on background models where the two fluids (superfluid neutrons and protons) corotate, are in β-equilibrium and co-exist throughout the volume of the star. We construct sequences of rotating stars for two analytical model equations of state. These models represent relatively simple generalizations of single fluid, polytropic stars. We study the effects of entrainment, rotation and symmetry energy on non-radial oscillations of these models. Our results show that entrainment and symmetry energy can have a significant effect on the rotational splitting of non-axisymmetric modes. In particular, the symmetry energy modifies the inertial mode frequencies considerably in the regime of fast rotation.     

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.     

On the moving subfeatures in the Be star ζ Oph

We analyse a series of line profile observations of the He  i 6678 line in ζ Oph. A period analysis on these data using the mode and moments of the line profile confirms the two previously known periods. We describe a new method of mode identification for pulsating stars in which the calculated profiles are directly fitted to observed profiles. The method yields the full set of pulsational parameters including the spherical harmonic degree, ℓ, and azimuthal number, m . Application of the method to these data confirms the mode identifications previously suggested for the two periodicities. We find that the derived pulsational parameters are physically realistic and conclude that non-radial pulsation is the most likely explanation for the travelling subfeatures. However, a unique mode identification is still not possible – several non-sectorial modes fit the data as well as the usually adopted sectorial identifications. The predicted photometric amplitudes are in good accord with upper limits derived from photometric observations. We conclude that ζ Oph is a star in the β Cep instability strip in which two modes of high degree (probably ℓ=4 and ℓ=8) are excited. We present an interpretation of these findings in which the cause of the low-order line profile and light variations in periodic Be stars is corotating photospheric clouds, while the travelling subfeatures are incidental to the Be phenomenon and are a result of non-radial pulsation.     

On the incidence of magnetic fields in slowly pulsating B, β Cephei and B-type emission-line stars

We have obtained 40 high-resolution circular spectropolarimetric measurements of 12 slowly pulsating B (SPB) stars, eight β Cephei stars and two Be stars with the Echelle Spectropolarimetric Device for the Observation of Stars at CFHT (ESPaDOnS) and Narval spectropolarimeters. The aim of these observations is to evaluate recent claims of a high incidence of magnetic field detections in stars of these types obtained using low-resolution spectropolarimetry by Hubrig et al. The precision achieved is generally comparable to or superior to that obtained by Hubrig et al., although our new observations are distinguished by their resolution of metallic and He line profiles, and their consequent sensitivity to magnetic fields of zero net longitudinal component. In the SPB stars, we confirm the detection of magnetic field in one star (16 Peg), but find no evidence of the presence of fields in the remaining 11. In the β Cep stars, we detect a field in  ξ¹  CMa, but not in any of the remaining seven stars. Finally, neither of the two B-type emission-line stars shows any evidence of magnetic field. Based on our results, we conclude that fields are not common in SPB, β Cep and B-type emission-line stars, consistent with the general rarity of fields in the broader population of main sequence B-type stars. A relatively small, systematic underestimation of the error bars associated with the UV Focal Reducer and Low Dispersion Spectrograph for the Very Large Telescope (FORS1) longitudinal field measurements of Hubrig et al. could in large part explain the discrepancy between their results and those presented here.     

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.     

Oscillations of rapidly rotating stratified neutron stars

We use time evolutions of the linear perturbation equations to study the oscillations of rapidly rotating neutrons stars. Our models account for the buoyancy due to composition gradients and we study, for the first time, the nature of the resultant g modes in a fast spinning star. We provide detailed comparisons of non-stratified and stratified models. This leads to an improved understanding of the relationship between the inertial modes of a non-stratified star and the g modes of a stratified system. In particular, we demonstrate that each g mode becomes rotation dominated, i.e. approaches a particular inertial mode, as the rotation rate of the star is increased. We also discuss issues relating to the gravitational wave driven instability of the various classes of oscillation modes.     

Time-series spectroscopy of the rapidly oscillating Ap star HR 3831

We present time-series spectroscopy of the rapidly oscillating Ap (roAp) star HR 3831. This star has a dominant pulsation period of 11.7 min and a rotation period of 2.85 d. We have analysed 1400 intermediate-resolution spectra of the wavelength region 6100–7100 Å obtained over one week, using techniques similar to those we applied to another roAp star, α  Cir.
We confirm that the H α velocity amplitude of HR 3831 is modulated with rotation phase. Such a modulation was predicted by the oblique pulsator model, and rules out the spotted pulsator model. However, further analysis of H α and other lines reveals rotational modulations that cannot easily be explained using the oblique pulsator model. In particular, the phase of the pulsation as measured by the width of the H α line varies with height in the line.
The variation of the H α bisector shows a very similar pattern to that observed in α Cir, which we have previously attributed to a radial node in the stellar atmosphere. However, the striking similarities between the two stars, despite the much shorter period of α Cir (6.8 min), argues against this interpretation unless the structure of the atmosphere is somewhat different between the two stars. Alternatively, the bisector variation is a signature of the degree ℓ of the mode and not the overtone value n .
High-resolution studies of the metal lines in roAp stars are needed to understand fully the form of the pulsation in the atmosphere.     

Towards gravitational wave asteroseismology

We present new results for pulsating neutron stars. We have calculated the eigenfrequencies of the modes that one would expect to be the most important gravitational wave sources: the fundamental fluid f mode, the first pressure p mode and the first gravitational wave w mode, for twelve realistic equations of state. From these numerical data we have inferred a set of 'empirical relations' between the mode frequencies and the parameters of the star (the radius R and the mass M ). Some of these relations prove to be surprisingly robust, and we show how they can be used to extract the details of the star from observed modes. The results indicate that, should the various pulsation modes be detected by the new generation of gravitational wave detectors that come online in a few years, the mass and the radius of neutron stars can be deduced with errors no larger than a few per cent.     

A search for luminous Be stars

As Be stars are restricted to luminosity classes III‐V, but early B‐type stars are believed to evolve into supergiants, it is to be expected that the Be phenomenon disappears at some point in the evolution of a moderately massive star, before it reaches the supergiant phase. As a first stage in an attempt to determine the physical reasons of this cessation, a search of the literature has provided a number of candidates to be Be stars with luminosity classes Ib or II. Spectroscopy has been obtained for candidates in a number of open clusters and associations, as well as several other bright stars in those clusters. Among the objects observed, HD 207329 is the best candidate to be a high‐luminosity Be star, as it appears like a fast‐rotating supergiant with double‐peaked emission lines. The lines of HD 229059, in Berkeley 87, also appear morphologically similar to those of Be stars, but there are reasons to suspect that this object is an interacting binary. At slightly lower luminosities, LS I +56°92 (B4 II) and HD 333452 (O9 II), also appear as intrinsically luminous Be stars. Two Be stars in NGC 6913, HD 229221 and HD 229239, appear to have rather higher intrinsic magnitudes than their spectral type (B0.2 III in both cases) would indicate, being as luminous as luminosity class II objects in the same cluster. HD 344863, in NGC 6823, is also a rather early Be star of moderately high luminosity. The search shows that, though high‐luminosity Be stars do exist, they are scarce and, perhaps surprisingly, tend to have early spectral types. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pulsation in the pre-main-sequence Herbig Ae star HD 142666

Asteroseismology of pre-main-sequence δ Scuti stars has the potential not only to provide unprecedented constraints on models of these stars, but also to allow for the possibility of detecting evolutionary period changes, thus providing a direct measure of the pre-main-sequence evolutionary time-scale. In the last two years, the published number of such stars known has doubled from four to eight. Searches are now being conducted amongst the Herbig Ae stars, which are considered to be excellent candidates. We announce the discovery of δ Scuti pulsation in one Herbig Ae star, HD 142666, which lies within Marconi & Palla's theoretically predicted instability strip for pre-main-sequence stars, making this the ninth known pre-main-sequence δ Scuti star. We also demonstrate a lack of δ Scuti pulsation in another such star, HD 142527.     

Be phenomenon in open clusters: results from a survey of emission-line stars in young open clusters

Emission-line stars in young open clusters are identified to study their properties, as a function of age, spectral type and evolutionary state. 207 open star clusters were observed using the slitless spectroscopy method and 157 emission stars were identified in 42 clusters. We have found 54 new emission-line stars in 24 open clusters, out of which 19 clusters are found to house emission stars for the first time. About 20 per cent clusters harbour emission stars. The fraction of clusters housing emission stars is maximum in both the 0–10 and 20–30 Myr age bin (∼40 per cent each). Most of the emission stars in our survey belong to Classical Be class (∼92 per cent) while a few are Herbig Be stars (∼6 per cent) and Herbig Ae stars (∼2 per cent). The youngest clusters to have Classical Be stars are IC 1590, NGC 637 and 1624 (all 4 Myr old) while NGC 6756 (125–150 Myr) is the oldest cluster to have Classical Be stars. The Classical Be stars are located all along the main sequence (MS) in the optical colour–magnitude diagrams (CMDs) of clusters of all ages, which indicates that the Be phenomenon is unlikely due to core contraction near the turn-off. The distribution of Classical Be stars as a function of spectral type shows peaks at B1–B2 and B6–B7 spectral types. The Be star fraction [N(Be)/N(B+Be)] is found to be less than 10 per cent for most of the clusters and NGC 2345 is found to have the largest fraction (∼26 per cent). Our results indicate there could be two mechanisms responsible for the Classical Be phenomenon. Some are born Classical Be stars (fast rotators), as indicated by their presence in clusters younger than 10 Myr. Some stars evolve to Classical Be stars, within the MS lifetime, as indicated by the enhancement in the fraction of clusters with Classical Be stars in the 20–30 Myr age bin.     

High-frequency modes in solar-like stars

p-mode oscillations in solar-like stars are excited by the outer convection zone in these stars and reflected close to the surface. The p modes are trapped inside an acoustic cavity, but the modes only stay trapped up to a given frequency [known as the acoustic cut-off frequency  (ν_ac)  ] as modes with larger frequencies are generally not reflected at the surface. This means that modes with frequency larger than the acoustic cut-off frequency must be travelling waves. The high-frequency modes may provide information about the physics in the outer layers of the stars and the excitation source and are therefore highly interesting as it is the estimation of these two phenomena that cause some of the largest uncertainties when calculating stellar oscillations.
High-frequency modes have been detected in the Sun, in β Hydri and in α Cen A and α Cen B by smoothing the so-called echelle diagram and the large frequency separation as a function of frequency has been estimated. The large frequency separation has been compared with a simple model of the acoustic cavity which suggests that the reflectivity of the photosphere is larger at high frequency than predicted by standard models of the solar atmosphere and that the depth of the excitation source is larger than what has been estimated by other models and might depend on the order n and degree l of the modes.     

Doppler imaging of BD+22°4409 (LO Peg) using least-squares deconvolution

We present maximum-entropy reconstructions of the rapidly rotating dwarf single star BD+22°4409 (LO Peg) from observations at the William Herschel Telescope in 1993 August. Since this star is too faint to use the conventional single- or three-line Doppler imaging methods, we make use of the novel method of least-squares deconvolution, which utilizes the large number of photospheric lines in an echelle spectrum to produce a single high signal-to-noise ratio profile.
The star-spot distributions from the image reconstructions show cool features at both high and low latitudes, in contradiction to recent theoretical predictions of the dynamo behaviour in rapidly rotating stars. Cross-correlation of the images from consecutive nights shows a good correlation from the small-scale structures, but no evidence of surface differential rotation. From the cross-correlation of the high-latitude spot we are able to reject the period of 9.22 h of Jeffries et al. in favour of their preferred period of 10.17 h, confirming the result of Robb & Cardinal.     

Non-linear hydrodynamical evolution of rotating relativistic stars: numerical methods and code tests

We present numerical hydrodynamical evolutions of rapidly rotating relativistic stars, using an axisymmetric, non-linear relativistic hydrodynamics code. We use four different high-resolution shock-capturing (HRSC) finite-difference schemes (based on approximate Riemann solvers) and compare their accuracy in preserving uniformly rotating stationary initial configurations in long-term evolutions. Among these four schemes, we find that the third-order piecewise parabolic method scheme is superior in maintaining the initial rotation law in long-term evolutions, especially near the surface of the star. It is further shown that HRSC schemes are suitable for the evolution of perturbed neutron stars and for the accurate identification (via Fourier transforms) of normal modes of oscillation. This is demonstrated for radial and quadrupolar pulsations in the non-rotating limit, where we find good agreement with frequencies obtained with a linear perturbation code. The code can be used for studying small-amplitude or non-linear pulsations of differentially rotating neutron stars, while our present results serve as testbed computations for three-dimensional general-relativistic evolution codes.