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.     

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.     

Solar-like Oscillations of the Red Giant ɛ Ophiuchi

Solar-like oscillations have recently been observed in the red giant ? Ophiuchi (G9.5III). The large frequency separation is found to be 4.8 μHz, non-radial oscillation mode has been shown to exist. Based on the observed frequency of oscillations and the position of ? Ophiuchi on the Hertzsprung-Russell diagram and by adopting a method of combined calculation of stellar evolution and oscillations, some preliminary constraints on the mass, metal abundance, age and radius of this star have been obtained.     

The plasma radiation of flare kernels

We survey the mathematics of non-linear Hamiltonian oscillations with emphasis being laid on the more recently discovered Kolmogorov instability. In the context of radial adiabatic oscillations of stars this formalism predicts a Kolmogorov instability even at low oscillation energies, provided that sufficiently high linear asymptotic modes have been excited. Numerical analysis confirms the occurrence of this instability. It is found to show up already among the lowest order modes, although high surface amplitudes are then required (¦δr¦/R ～ 0.5 for an unstable fundamental mode - first harmonic coupling). On the basis of numerical evidence we conjecture that in the Kolmogorov unstable regime the enhanced coupling due to internal resonance effects leads to an equipartition of energy over all interacting degrees of freedom. We also indicate that the power spectrum of such oscillations is expected to display two components: A very broad band of overlapping pseudo-linear frequency peaks spread out over the asymptotic range, and a strictly non-linear l/f-noise type component close to the frequency origin. It is finally argued that the Kolmogorov instability is likely to occur among non-linearly coupled non-radial stellar modes at a surface amplitude much lower than in the radial case. This lends support to the view that this instability might be operative among the solar oscillations.     

Self-similar pressure oscillations in neutron star envelopes as probes of neutron star structure

We study eigenmodes of acoustic oscillations of high multipolarity l ∼ 100–1000 and high frequency (∼100 kHz), localized in neutron star envelopes. We show that the oscillation problem is self-similar. Once the oscillation spectrum is calculated for a given equation of state (EOS) in the envelope and given stellar mass M and radius R , it can be rescaled to a star with any M and R (but the same EOS in the envelope). For l ≳ 300, the modes can be subdivided into the outer and inner ones. The outer modes are mainly localized in the outer envelope. The inner modes are mostly localized near the neutron drip point, being associated with the softening of the EOS after the neutron drip. We calculate oscillation spectra for the EOSs of cold-catalyzed and accreted matter and show that the spectra of the inner modes are essentially different. A detection and identification of high-frequency pressure modes would allow one to infer M and R and determine also the EOS in the envelope (accreted or ground state) providing a new, potentially powerful method to explore the main parameters and internal structure of neutron stars.     

Are There Pulsationally Unstable Low-degree p1 Modes in the Sun?

By using a non-local time-dependent theory of stellar convection, the solar non-adiabatic pulsations of the low- and intermediate-degree (l < 25) modes are calculated. The results show that the non-radial p1 modes of l = 1–5 are pulsationally unstable. However, the adjacent g, f, p2-p5 modes and the p1 modes of l > 5 are stable. From the analysis of the diagram of integrated work it is discovered that the excitation of oscillations comes from the radiation zone beneath the convective region. Whether the sun possesses unstable low-degree p1 modes is of signi?cant importance for clarifying the excitation mechanism of solar ?ve-minute oscillations.     

Seismic signatures of strange stars with crust

We study acoustic oscillations (eigenfrequencies, velocity distributions, damping times) of normal crusts of strange stars. These oscillations are very specific because of huge density jump at the interface between the normal crust and the strange matter core. The oscillation problem is shown to be self-similar. For a low (but non-zero) multipolarity l , the fundamental mode (without radial nodes) has a frequency of ∼300 Hz and mostly horizontal oscillation velocity; other pressure modes have frequencies ≳20 kHz and almost radial oscillation velocities. The latter modes are similar to radial oscillations (having approximately the same frequencies and radial velocity profiles). The oscillation spectrum of strange stars with crust differs from the spectrum of neutron stars. If detected, acoustic oscillations would allow one to discriminate between strange stars with crust and neutron stars and constrain the mass and radius of the star.     

Frequency analysis of Cepheids in the Large Magellanic Cloud: new types of classical Cepheid pulsators

We have performed a detailed systematic search for multiperiodicity in the Population I Cepheids of the Large Magellanic Cloud. In this process, we have identified for the first time several new types of Cepheid pulsational behaviour. We have found two triple-mode Cepheids pulsating simultaneously in the first three radial overtones. In 9 per cent of the first overtone (FO) Cepheids, we have detected weak but well-resolved secondary periodicities. They appear either very close to the primary pulsation frequency or at a much higher frequency with a characteristic period ratio of 0.60–0.64. In either case, the secondary periodicities must correspond to non-radial modes of oscillation. This result presents a major challenge to the theory of stellar pulsations, which predicts that such modes should not be excited in Cepheid variables. Non-radial modes have also been found in three of the fundamental first overtone (FU/FO) double-mode Cepheids, but no such oscillations have been detected in single-mode Cepheids pulsating in the FU mode.
In 19 per cent of double-mode Cepheids pulsating in the first two radial overtones (FO/SO type), we have detected a Blazhko-type periodic modulation of amplitudes and phases. Both modes are modulated with a common period, which is always longer than 700 d. Variations of the two amplitudes are anticorrelated, and maximum of one amplitude always coincides with minimum of the other. We have compared observations of modulated FO/SO Cepheids with predictions of theoretical models of the Blazhko effect, showing that the currently most popular models cannot account for properties of these stars. We propose that the Blazhko effect in FO/SO Cepheids can be explained by a non-stationary resonant interaction of one of the radial modes with another, perhaps non-radial, mode of oscillations.     

Modelling solar atmospheric gravity oscillation modes

Solar oscillations are investigated in a one‐dimensional hydrodynamic plane‐parallel model with an atmosphere. Besides the acoustic pressure (p) modes, the fundamental (f) and Lamb mode, another set of eigenmodes, a group of atmospheric gravity (g) modes, is found in the low‐frequency region of the spectrum. Their frequencies and spatial behaviour are studied. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-radial oscillations and energy transport in rotating solar (stellar) wind

The propagation of non-radial, small amplitude perturbations superposed on a zero-order, stationary, non-magnetic, polytropic, rotating stellar wind is studied in the limit of the local theory, i.e. for k r 1, k being the module of the wave vector and r the characteristic scale of the zero-order flow. The resulting dispersion equation is of the 3rd order in (complex) frequency and the possible modes correspond to two acoustic type waves, and to a gravity-shear wave with strongly anisotropic propagation properties, due to coupling between the internal gravity waves and shear motion. The gravity-shear mode allows velocity differences in the medium to exist with no corresponding density fluctuations and hence with no shock wave formation. It is suggested that this mode corresponds to some of the fast-slow velocity streams observed in the interplanetary medium and may provide means for wave energy being transported outwards with the zero-order flow, with little dissipation in the inner region of the solar wind.     

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.     

Axisymmetric modes of rotating relativistic stars in the Cowling approximation

Axisymmetric pulsations of rotating neutron stars can be excited in several scenarios, such as core collapse, crust- and core-quakes or binary mergers, and could become detectable in either gravitational waves or high-energy radiation. Here, we present a comprehensive study of all low-order axisymmetric modes of uniformly and rapidly rotating relativistic stars. Initial stationary configurations are appropriately perturbed and are numerically evolved using an axisymmetric, non-linear relativistic hydrodynamics code, assuming time-independence of the gravitational field (Cowling approximation). The simulations are performed using a high-resolution shock-capturing finite-difference scheme accurate enough to maintain the initial rotation law for a large number of rotational periods, even for stars at the mass-shedding limit. Through Fourier transforms of the time evolution of selected fluid variables, we compute the frequencies of quasi-radial and non-radial modes with spherical harmonic indices l =0 , 1, 2 and 3, for a sequence of rotating stars from the non-rotating limit to the mass-shedding limit. The frequencies of the axisymmetric modes are affected significantly by rotation only when the rotation rate exceeds about 50 per cent of the maximum allowed. As expected, at large rotation rates, apparent mode crossings between different modes appear. In addition to the above modes, several axisymmetric inertial modes are also excited in our numerical evolutions.     

Equatorially trapped Rossby waves in radiative stars

Observations by recent space missions reported the detection of Rossby waves (r-modes) in light curves of many stars (mostly A, B, and F spectral types) with outer radiative envelope. This article aims to study the theoretical dynamics of Rossby-type waves in such stars. Hydrodynamic equations in a rotating frame were split into horizontal and vertical parts connected by a separation constant (or an equivalent depth). Vertical equations were solved analytically for a linear temperature profile and the equivalent depth was derived through free surface boundary condition. It is found that the vertical modes are concentrated in the near-surface layer with a thickness of several tens of surface density scale height. Then with the equivalent width, horizontal structure equations were solved, and the corresponding dispersion relation for Rossby, Rossby-gravity, and inertia-gravity waves was obtained. The solutions were found to be confined around the equator, leading to the equatorially trapped waves. It was shown that the wave frequency depends on the vertical temperature gradient as well as on stellar rotation. Therefore, observations of wave frequency in light curves of stars with known parameters (radius, surface gravity, rotation period) could be used to estimate the temperature gradient in stellar outer layers. Consequently, the Rossby mode may be considered as an additional tool in asteroseismology apart from acoustic and gravity modes.     

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     .     

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.     

Solar five-minute oscillations of low,intermediate, and high degree

The overstability of acoustic modes trapped inside the Sun is studied with mechanical and thermal effects of turbulence included in an approximate manner through the eddy diffusivities. Many of the acoustic modes are found to be overstable with the most rapidly growing modes occupying a region centred around 3.3 mHz and spread over a wide range of length-scales. The numerical results turn out to be in reasonable accord with the observed power-spectrum of the five-minute oscillations of arbitrary degree. It is demonstrated that these oscillations are most likely to be driven by a simultaneous operation of the -mechanism and the convective Cowling mechanism, the dominant contribution to the generation of self-excited acoustic waves arising from the turbulent diffusion.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

Discovery and asteroseismological analysis of a new pulsating DB white dwarf star, PG 2246+121

We report 36.6 h of time-resolved CCD photometry of the DB white dwarf star PG 2246+121 and the discovery that it is a new pulsating variable. Analysis of our compact single-site data set allowed the detection of three mode multiplets, two triplets at 256 and 329 s, respectively, and one doublet at 286 s. The frequency splitting within those structures is exactly the same within the length and accuracy of our data set.
We argue that these multiplets are the result of non-radial g-mode pulsations, most probably of spherical degree ℓ=1, which then yields a formal stellar rotation period of 2.00±0.12 d. We suggest that the excited modes are three consecutive radial overtones of order 3–7, most likely k =4,5,6. This discovery's impact on the understanding of pulsating DB white dwarfs is discussed.     

How to drive roAp stars

Rapidly oscillating Ap stars constitute a unique class of pulsators with which to study non-radial oscillations under some — even for stars — unusual physical conditions. These stars are chemically peculiar, they have strong magnetic fields and they often pulsate in several high-order acoustic modes simultaneously. We discuss here an excitation mechanism for short-period oscillation modes based on the classical κ mechanism. We particularly stress the conditions that must be fulfilled for successful driving. Specifically, we discuss the roles of the chemical peculiarity and strong magnetic field on the oscillation modes and what separates these pulsators from δ Scuti and Am-type stars.     

Pulsational properties of V2109 Cyg

In this study V2109 Cyg (a pulsating δ Scuti star) has been modelled. In treating the oscillation equations perturbation in gravitational potential energy has been taken into account. Both radial and nonradial oscillations are treated with adiabatic approximation. The so called radial fundamental frequency (5.3745 c/d) and the nonradial frequency (5.8332 c/d) were obtained within a satisfactory precision. It was found that the Cowling approximation introduced more error as one went from low overtones to high overtones in radial oscillations. A similar trend was observed in nonradial case with low values of l. By keeping the effective temperatures almost the same as with V2109 Cyg two more models with different masses have also been calculated to see the effect of inclusion of perturbation in gravitational potential energy on oscillation frequencies in different masses. Conclusion arrived is that one must be careful to employ the Cowling approximation especially for high nonradial oscillation frequencies. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semiclassical approximation for low-degree stellar p modes – I. The classical eigenfrequency equation

A new eigenfrequency equation for low-degree solar-like oscillations in stars is developed, based on the assumption of purely classical propagation in the stellar interior of acoustic waves modified by buoyancy and gravity . Compared with high-frequency asymptotic analysis, the eigenfrequency equation has a new functional form, with expansion in powers of ℓ(ℓ+1) instead of 1/ ω . Basic observable quantities, the 'large' and 'small' frequency separations , are interpreted as the dependence on frequency and refraction angle of a classical action integral for waves propagating close to the stellar diameter. The new eigenfrequency equation gives a significant improvement in accuracy over previous analyses when tested with solar p modes, suggesting this as an alternative and more powerful tool for applications in stellar seismology.