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.     

On the effect of overshooting as predicted by the modelling of the pre-main-sequence evolution of a 2 M⊙ star

We discuss the effects of convective overshooting in the pre-main-sequence (PMS) evolution of intermediate-mass stars, by analysing in detail the early evolution towards the main sequence of a  2 M_⊙  stellar model. These effects can be extremely important in the end of the PMS, when the abundances in CNO elements approach the equilibrium in the centre. We provide a possible physical explanation on why a moderate amount of overshooting produces, as the star approaches the zero-age main-sequence, an extra loop in the evolutionary tracks on the Hertzsprung–Russell diagram.
An interesting feature is that there is a very well defined amount of overshooting (for a given stellar mass and chemical composition) beyond which a loop is produced. For smaller amounts of overshooting such a loop does not take place and the evolutionary tracks are similar to those found in the literature. The amount of overshooting needed to produce the loop decreases with stellar mass.
We discuss the underlining physical reasons for the behaviour predicted by the evolution models and argue that it provides a crucial observational test for convective overshooting in the core of intermediate-mass stars.     

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.     

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.     

Quasi-periodic oscillations in low-mass X-ray binaries and constraints on the equation of state of neutron star matter

Recently discovered quasi-periodic oscillations in the X-ray brightness of low-mass X-ray binaries are used to derive constraints on the mass of the neutron star component and the equation of state of neutron star matter. The observations are compared with models of rapidly rotating neutron stars which are calculated by means of an exact numerical method in full relativity. For the equations of state we select a broad collection of models representing different assumptions about the many-body structure and the complexity of the composition of superdense matter. The mass constraints differ from their values in the approximate treatment by ∼10 per cent. Under the assumption that the maximum frequency of the quasi-periodic oscillations originates from the innermost stable orbit, the mass of the neutron star is in the range M ∼1.92–2.25 M_⊙. The quasi-periodic oscillation in the Atoll-source 4U 1820−30 in particular is only consistent with equations of state that are rather stiff at high densities, which is explainable, so far, only with pure nucleonic/leptonic composition. This interpretation contradicts the hypothesis that the protoneutron star formed in SN 1987A collapsed to a black hole, since this would demand a maximum neutron star mass below 1.6 M_⊙. The recently suggested identification of quasi-periodic oscillations with frequencies of about 10 Hz with the Lense–Thirring precession of the accretion disc is found to be inconsistent with the models studied in this work, unless it is assumed that the first overtone of the precession is observed.     

On the infant weight loss of low- to intermediate-mass star clusters

Star clusters are born in a highly compact configuration, typically with radii of less than about 1 pc roughly independently of mass. Since the star formation efficiency is less than 50 per cent by observation and because the residual gas is removed from the embedded cluster, the cluster must expand. In the process of doing so it only retains a fraction f _st of its stars. To date there are no observational constraints for f _st, although N -body calculations by Kroupa, Aarseth & Hurley suggest it to be about 20–30 per cent for Orion-type clusters. Here we use the data compiled by Testi et al., Testi, Palla & Natta and Testi, Palla & Natta for clusters around young Ae/Be stars and by de Wit et al. and de Wit et al. around young O stars and the study of de Zeeuw et al. of OB associations and combine these measurements with the expected number of stars in clusters with primary Ae/Be and O stars, respectively, using the empirical correlation between maximal stellar mass and star cluster mass of Weidner & Kroupa. We find that   f _st < 50  per cent with a decrease to higher cluster masses/more massive primaries. The interpretation would be that cluster formation is very disruptive. It appears that clusters with a birth stellar mass in the range  10–10³ M_⊙  keep at most 50 per cent of their stars.     

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.     

Magnetospheric accretion and pre-main-sequence stellar masses

We present a method of determining lower limits on the masses of pre-main-sequence (PMS) stars and so constraining the PMS evolutionary tracks. This method uses the redshifted absorption feature observed in some emission-line profiles of T Tauri stars, indicative of infall. The maximum velocity of the accreting material measures the potential energy at the stellar surface, which, combined with an observational determination of the stellar radius, yields the stellar mass. This estimate is a lower limit owing to uncertainties in the geometry and projection effects. Using available data, we show that the computed lower limits can be larger than the masses derived from PMS evolutionary tracks for M   0.5 M. Our analysis also supports the notion that accretion streams do not impact near the stellar poles but probably hit the stellar surface at moderate latitudes.     

Models of rapidly rotating neutron stars: remnants of accretion-induced collapse

Equilibrium models of differentially rotating nascent neutron stars are constructed, which represent the result of the accretion-induced collapse of rapidly rotating white dwarfs. The models are built in a two-step procedure: (1) a rapidly rotating pre-collapse white dwarf model is constructed; (2) a stationary axisymmetric neutron star having the same total mass and angular momentum distribution as the white dwarf is constructed. The resulting collapsed objects consist of a high-density central core of size roughly 20 km, surrounded by a massive accretion torus extending over 1000 km from the rotation axis. The ratio of the rotational kinetic energy to the gravitational potential energy of these neutron stars ranges from 0.13 to 0.26, suggesting that some of these objects may have a non-axisymmetric dynamical instability that could emit a significant amount of gravitational radiation.     

The pulsating DA white dwarf star EC 14012−1446: results from four epochs of time-resolved photometry

The pulsating DA white dwarfs are the coolest degenerate stars that undergo self-driven oscillations. Understanding their interior structure will help us to understand the previous evolution of the star. To this end, we report the analysis of more than 200 h of time-resolved CCD photometry of the pulsating DA white dwarf star EC 14012−1446 acquired during four observing epochs in three different years, including a coordinated three-site campaign. A total of 19 independent frequencies in the star's light variations together with 148 combination signals up to fifth order could be detected. We are unable to obtain the period spacing of the normal modes and therefore a mass estimate of the star, but we infer a fairly short rotation period of  0.61 ±0.03 d  , assuming the rotationally split modes are  ℓ= 1  . The pulsation modes of the star undergo amplitude and frequency variations, in the sense that modes with higher radial overtone show more pronounced variability and that amplitude changes are always accompanied by frequency variations. Most of the second-order combination frequencies detected have amplitudes that are a function of their parent mode amplitudes, but we found a few cases of possible resonantly excited modes. We point out the complications in the analysis and interpretation of data sets of pulsating white dwarfs that are affected by combination frequencies of the form   f _A + f _B − f _C   intruding into the frequency range of the independent modes.     

Surface magnetic fields on two accreting T Tauri stars: CV Cha and CR Cha

We have produced brightness and magnetic field maps of the surfaces of CV Cha and CR Cha: two actively accreting G- and K-type T Tauri stars in the Chamaeleon I star-forming cloud with ages of 3–5 Myr. Our magnetic field maps show evidence for strong, complex multipolar fields similar to those obtained for young rapidly rotating main-sequence stars. Brightness maps indicate the presence of dark polar caps and low-latitude spots – these brightness maps are very similar to those obtained for other pre-main-sequence and rapidly rotating main-sequence stars.
Only two other classical T Tauri stars have been studied using similar techniques so far: V2129 Oph and BP Tau. CV Cha and CR Cha show magnetic field patterns that are significantly more complex than those recovered for BP Tau, a fully convective T Tauri star.
We discuss possible reasons for this difference and suggest that the complexity of the stellar magnetic field is related to the convection zone; with more complex fields being found in T Tauri stars with radiative cores (V2129 Oph, CV Cha and CR Cha). However, it is clearly necessary to conduct magnetic field studies of T Tauri star systems, exploring a wide range of stellar parameters in order to establish how they affect magnetic field generation, and thus how these magnetic fields are likely to affect the evolution of T Tauri star systems as they approach the main sequence.     

The rapidly pulsating sdO star, SDSS J160043.6+074802.9

A spectroscopic analysis of Sloan Digital Sky Survey (SDSS) J160043.6+074802.9, a binary system containing a pulsating subdwarf-O (sdO) star with a late-type companion, yields   T _eff= 70 000 ± 5000 K  and  log  g = 5.25 ± 0.30  , together with a most likely type of K3 V for the secondary star. We compare our results with atmospheric parameters derived by Fontaine et al. and in the context of existing evolution models for sdO stars. New and more extensive photometry is also presented which recovers most, but not all, frequencies found in an earlier paper. Therefore, it seems probable that some pulsation modes have variable amplitudes. A non-adiabatic pulsation analysis of uniform metallicity sdO models show those having  log  g > 5.3  to be more likely to be unstable and capable of driving pulsation in the observed frequency range.     

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.     

The distance to the Orion Nebula cluster

The distance to the Orion Nebula cluster (ONC) is estimated using the rotational properties of its low-mass pre-main-sequence (PMS) stars. Rotation periods, projected equatorial velocities and distance-dependent radius estimates are used to form an observational sin  i distribution (where i is the axial inclination), which is modelled to obtain the distance estimate. A distance of  440 ± 34 pc  is found from a sample of 74 PMS stars with spectral types between G6 and M2, but this falls to  392 ± 32 pc  when PMS stars with accretion discs are excluded on the basis of their near-infrared excess. Since the radii of accreting stars are more uncertain and probably systematically underestimated, then this closer distance is preferred. The quoted uncertainties include statistical errors and uncertainties due to a number of systematic effects including binarity and inclination bias. This method is geometric and independent of stellar evolution models, though does rely on the assumption of random axial orientations and the Cohen & Kuhi effective temperature scale for PMS stars. The new distance is consistent with, although lower and more precise, than most previous ONC distance estimates. A closer ONC distance implies smaller luminosities and an increased age based on the positions of PMS stars in the Hertzsprung–Russell diagram.     

Chromospheric activity in the late A- and early F-type stars of open clusters – II. Pleiades and Alpha Persei

We report observations of the He  i λ 5876 (D3) line in the late A- and early F-type stars in the Pleiades and Alpha Persei star clusters used to determine chromospheric activity levels. This represents the first sample of young stars in this temperature range with chromospheric activity measurements. We find the same average activity level in the young early F stars as in Hyades-age stars and field stars. In addition, the young star sample shows the same large star-to-star variation in activity as seen in the older stars. Thus, as a whole, chromospheric activity in this photospheric temperature range remains the same over nearly a factor of 100 in stellar age (50 Myr to 3 Gyr), in striking contrast to the behaviour of later-type stars. In the five late A stars we find three certain detections of D3 and one likely detection. This includes the bluest star yet observed with a chromospheric D3 line, Pleiades star HII 1362 at ( B − V )₀=0.22, making it one of the earliest stars with an observed chromosphere. The late A stars have D3 equivalent widths comparable to the weakest early F stars. However, when comparing D3 measurements in the young late A stars with older late A stars, we find evidence for a slight decrease in activity with age based on the large number of non-detections in the older stars. We find an apparently linear relationship between the activity upper limit and B − V over our entire range of B − V . Extrapolated blueward, this relationship predicts that the chromospheric D3 line would disappear for all stars at B − V ≈0.13.     

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.     

Predictions of the extent of self-enrichment in oxygen of giant metal-poor H ii regions

In general, H  ii regions do not show clear signs of self-enrichment in products from massive stars  ( M ≥ 8 M_⊙)  . In order to explore why I modelled the contamination with Wolf–Rayet star ejecta of metal-poor  ( Z = 0.001)  H  ii regions, ionized either by a  10⁶ M_⊙  cluster of coeval stars (cluster 1) or by a cluster resulting from continuous star formation at a rate of  1 M_⊙ yr⁻¹  (cluster 2). The clusters have   Z = 0.001  and a Salpeter initial mass function from 0.1 to  120 M_⊙  . Independent one-dimensional constant density simulations of the emission-line spectra of unenriched H  ii regions were computed at the discrete ages 1, 2, 3, 4 and 5 Myr, with the photoionization code cloudy , using as input, radiative and mechanical stellar feedbacks predicted by the evolutionary synthesis code starburst99 . Each H  ii region was placed at the outer radius of the adiabatically expanding superbubble of Mac Low & McCray. For models with thermal and ionization balance time-scales of less than 1 Myr, and with oxygen emission-line ratios in agreement with observations, the volume of the superbubble and the H  ii region was uniformly and instantaneously polluted with stellar ejecta predicted by starburst99 . I obtained a maximum oxygen abundance enhancement of 0.025 dex, with cluster 1, at 4 Myr. It would be unobservable.     

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.     

The evolution of a rapidly accreting helium white dwarf to become a low-luminosity helium star

We have examined the evolution of merged low-mass double white dwarfs which become low-luminosity (or high-gravity) extreme helium stars. We have approximated the merging process by the rapid accretion of matter, consisting mostly of helium, on to a helium white dwarf. After a certain mass is accumulated, a helium shell flash occurs, the radius and luminosity increase and the star becomes a yellow giant. Mass accretion is stopped artificially when the total mass reaches a pre-determined value. As the helium-burning shell moves inwards with repeating shell flashes, the effective temperature gradually increases as the star evolves towards the helium main sequence. When the mass interior to the helium‐burning shell is approximately 0.25 M_⊙, the star enters a regime where it is pulsationally unstable. We have obtained radial pulsation periods for these models.
These models have properties very similar to those of the pulsating helium star V652 Her. We have compared the rate of period change of the theoretical models with that observed in V652 Her, as well as with its position on the Hertzsprung–Russell diagram. We conclude that the merger between two helium white dwarfs can produce a star with properties remarkably similar to those observed in at least one extreme helium star, and is a viable model for their evolutionary origin. Such helium stars will evolve to become hot subdwarfs close to the helium main sequence. We also discuss the number of low-luminosity helium stars in the Galaxy expected for our evolution scenario.