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.     

Seismology of stellar envelopes: probing the outer layers of a star through the scattering of acoustic waves

The outer layers of Sun-like stars are regions of rapid spatial variation which modulate the p-mode frequencies by partially reflecting the constituent acoustic waves. With the accuracy that has been achieved by current solar observations, and that is expected from imminent stellar observations, this modulation can be observed from the spectra of the low-degree modes. We present a new and simple theoretical calculation to determine the leading terms in an asymptotic expansion of the outer phase of these modes, which is determined by the structure of the surface layers of the star. Our procedure is to compare the stellar envelope with a plane-parallel polytropic envelope, which we regard as a smooth reference background state. Then we can isolate a seismic signature of the acoustic phase and relate it to the stratification of the outer layers of the convection zone. One can thereby constrain theories of convection that are used to construct the convection zones of the Sun and Sun-like stars. The accuracy of the diagnostic is tested in the solar case by comparing the predicted outer phase with an exact numerical calculation.     

The phase function for stellar acoustic oscillations — IV. Solar-like stars

In recent years there has been some progress towards detecting solar-like oscillations in stars. The goal of this challenging project is to analyse frequency spectra similar to that observed for the Sun in integrated light. In this context it is important to investigate what can be learned about the structure and evolution of the stars from such future observations. Here we concentrate on the structure of the upper layers, as reflected in the phase function. We show that it is possible to obtain this function from low-degree p modes, at least for stars on the main sequence. We analyse its dependence on several uncertainties in the structure of the uppermost layers. We also investigate a filtered phase function, which has properties that depend on the layers around the second helium ionization zone.     

Rotation of the solar core from BiSON and LOWL frequency observations

Determination of the rotation of the solar core requires very accurate data on splittings for the low-degree modes which penetrate to the core, as well as for modes of higher degree to suppress the contributions from the rest of the Sun to the splittings of the low-degree modes. Here we combine low-degree data based on 32 months of observations with the BiSON network and data from the LOWL instrument. The data are analysed with a technique that specifically aims at obtaining an inference of rotation that is localized to the core. Our analysis provides what we believe is the most stringent constraint to date on the rotation of the deep solar interior.     

Torsional oscillations of magnetized relativistic stars

Strong magnetic fields in relativistic stars can be a cause of crust fracturing, resulting in the excitation of global torsional oscillations. Such oscillations could become observable in gravitational waves or in high-energy radiation, thus becoming a tool for probing the equation of state of relativistic stars. As the eigenfrequency of torsional oscillation modes is affected by the presence of a strong magnetic field, we study torsional modes in magnetized relativistic stars. We derive the linearized perturbation equations that govern torsional oscillations coupled to the oscillations of a magnetic field, when variations in the metric are neglected (Cowling approximation). The oscillations are described by a single two-dimensional wave equation, which can be solved as a boundary-value problem to obtain eigenfrequencies. We find that, in the non-magnetized case, typical oscillation periods of the fundamental     torsional modes can be nearly a factor of 2 larger for relativistic stars than previously computed in the Newtonian limit. For magnetized stars, we show that the influence of the magnetic field is highly dependent on the assumed magnetic field configuration, and simple estimates obtained previously in the literature cannot be used for identifying normal modes observationally.     

Anomalous variations in low-degree helioseismic mode frequencies

We compare changes in the frequencies of solar acoustic modes with degree between 0 and 2, as derived from Global Oscillation Network Group (GONG), Birmingham Solar Oscillations Network (BiSON) and Michelson Doppler Imager (MDI) spectra obtained between 1995 and 2003. We find that, after the solar-activity dependence has been removed from the frequencies, there remain variations that appear to be significant, and are often well correlated between the different data sets. We consider possible explanations for these fluctuations, and conclude that they are likely to be related to the stochastic excitation of the modes. The existence of such fluctuations has possible relevance to the analysis of other low-degree acoustic mode spectra such as those from solar-type stars.     

An asteroseismic signature of helium ionization

We investigate the influence of the ionization of helium on the low-degree acoustic oscillation frequencies in model solar-type stars. The signature in the oscillation frequencies characterizing the ionization-induced depression of the first adiabatic exponent γ is a superposition of two decaying periodic functions of frequency ν, with 'frequencies' that are approximately twice the acoustic depths of the centres of the He  i and He  ii ionization regions. That variation is probably best exhibited in the second frequency difference  Δ₂ν _{n ,  l} ≡ν _{n −1,  l} − 2ν _{n ,  l} +ν _{n +1,  l}   . We show how an analytic approximation to the variation of γ leads to a simple representation of this oscillatory contribution to Δ₂ν which can be used to characterize the γ variation, our intention being to use it as a seismic diagnostic of the helium abundance of the star. We emphasize that the objective is to characterize γ, not merely to find a formula for Δ₂ν that reproduces the data.     

On nonadiabatic waves in the photospheres of cool stars

The coupled set of equations of hydrodynamics and radiative transfer is derived for small disturbances in a plane, grey atmosphere. Only radiative transfer is taken into account in the energy equation; dynamical effects of radiation are ignored. A mean stationary radiative flux through the photosphere is taken into account. The radiative transfer equation is used by assuming the Eddington approximation, moreover, an exponential height profile of the temperature and an analytical opacity formula are supposed. For this model we obtained an asymptotic solution for plane nonadiabatic acoustic waves and radiation waves. The approach provides a detailed discussion of the interaction of nonadiabatic p‐modes and radiation waves in a realistic model of the photosphere of a solar‐like star.     

On the r-mode spectrum of relativistic stars in the low-frequency approximation

The axial modes for non-barotropic relativistic rotating neutron stars with uniform angular velocity are studied, using the slow-rotation formalism together with the low-frequency approximation, first investigated by Kojima. The time-independent form of the equations leads to a singular eigenvalue problem, which admits a continuous spectrum. We show that for     , it is nevertheless also possible to find discrete mode solutions (the r modes). However, under certain conditions related to the equation of state and the compactness of the stellar model, the eigenfrequency lies inside the continuous band and the associated velocity perturbation is divergent; hence these solutions have to be discarded as being unphysical. We corroborate our results by explicitly integrating the time-dependent equations. For stellar models admitting a physical r-mode solution, it can indeed be excited by arbitrary initial data. For models admitting only an unphysical mode solution, the evolutions do not show any tendency to oscillate with the respective frequency. For higher values of l it seems that in certain cases there are no mode solutions at all.     

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.     

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.     

Variations of the low‐degree solar p‐mode frequencies for 10 years of GOLF observations

We experiment with a method of measuring the frequency of solar p modes, intended to extend the passband for the variations of the frequency spectrum as high as possible. So far this passband is limited to a fraction of μ Hz for the classical analysis based on numerical fits of a theoretical line profile to a power spectrum averaged over periods lasting at least several weeks. This limit for the present analysis can be shifted to the mHz range, corresponding to some of the “5 min” oscillations, but in this range we use a lower resolution which allows us to separate odd and even p modes. We show an example of the results for long term variations and apply this analysis to search for a modulation of the p‐mode frequency spectrum by asymptotic series of solar g modes. A faint signal is found in the analysis of 10 years of GOLF data. This very preliminary result possibly indicates the detection of a small number of g modes of degree l = 1. A tentative determination of an observational value of the parameter P₀ follows. P₀ is the scaling factor of the asymptotic series of g modes and is a key data for solar core physics. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the dipolar f mode of stellar oscillation

The classification of adiabatic modes of non-radial stellar oscillation was established by Cowling in 1941. In addition to acoustic and gravity modes he identified an intermediate mode, which he labelled the f mode, and which in simple stellar models has no radial node. The motion of a dipolar f mode (of spherical-harmonic degree l =1) shifts the centre of mass, and must have zero frequency. On the other hand, if the perturbation to the gravitational potential is neglected (the case considered by Cowling) the f mode has a frequency intermediate between those of the gravity and acoustic modes; this is true of modes of any degree ( l ≥1) . Here we consider the properties of the dipolar f mode, elucidating the origin of these differences through continuous transformations between the various relevant cases; in addition, we discuss the broader issues of the classification of modes of non-radial oscillation.     

Distribution of magnetic flux on the solar surface and low-degree p-modes

The frequencies of solar p-modes are known to change over the solar cycle. There is also recent evidence that the relation between frequency shift of low-degree modes and magnetic flux or other activity indicators differs between the rising and falling phases of the solar cycle, leading to a hysteresis in such diagrams. We consider the influence of the changing large-scale surface distribution of the magnetic flux on low-degree ( l ≤3) p-mode frequencies. To that end, we use time-dependent models of the magnetic flux distribution and study the ensuing frequency shifts of modes with different order and degree as a function of time. The resulting curves are periodic functions (in simple cases just sine curves) shifted in time by different amounts for the different modes. We show how this may easily lead to hysteresis cycles comparable to those observed. Our models suggest that high-latitude fields are necessary to produce a significant difference in hysteresis between odd- and even-degree modes. Only magnetic field distributions within a small parameter range are consistent with the observations by Jiménez-Reyes et al. Observations of p-mode frequency shifts are therefore capable of providing an additional diagnostic of the magnetic field near the solar poles. The magnetic distribution that is consistent with the p-mode observations also appears reasonable compared with direct measurements of the magnetic field.     

Sounding stellar cycles with Kepler – I. Strategy for selecting targets

The long-term monitoring and high photometric precision of the Kepler satellite will provide a unique opportunity to sound the stellar cycles of many solar-type stars using asteroseismology. This can be achieved by studying periodic changes in the amplitudes and frequencies of the oscillation modes observed in these stars. By comparing these measurements with conventional ground-based chromospheric activity indices, we can improve our understanding of the relationship between chromospheric changes and those taking place deep in the interior throughout the stellar activity cycle. In addition, asteroseismic measurements of the convection zone depth and differential rotation may help us determine whether stellar cycles are driven at the top or at the base of the convection zone. In this paper, we analyse the precision that will be possible using Kepler to measure stellar cycles, convection zone depths and differential rotation. Based on this analysis, we describe a strategy for selecting specific targets to be observed by the Kepler Asteroseismic Investigation for the full length of the mission, to optimize their suitability for probing stellar cycles in a wide variety of solar-type stars.     

Diagnostics of the Internal Structure of Stars using the Differential Response Technique

We address the problem of the diagnosing the deep interior structure of stars using acoustic p-modes, and investigate the diagnostic capabilities of two complementary approaches both based on the differential response technique (Vorontsov, 1998): (a) direct calibration using a grid of evolutionary stellar models, and (b) linear and non-linear (with consecutive linearisations) inversion of low-degree frequencies. We apply this analysis to the frequencies of a model of an old 0.8M star, and to the solar frequencies obtained from BiSON measurements, using a 2-D grid of reference models of different mass and age. We explore the convergence and stability of the asteroseismic inversion, performed with the adaptive regularisation technique of Strakhov and Vorontsov (2001).     

On mode conversion and wave reflection in magnetic Ap stars

We investigate the effect of a strong large-scale magnetic field on the reflection of high-frequency acoustic modes in rapidly oscillating Ap stars. To that end, we consider a toy model composed of an isothermal atmosphere matched on to a polytropic interior and determine the numerical solution to the set of ideal magnetohydrodynamic equations in a local plane-parallel approximation with constant gravity. Using the numerical solution in combination with approximate analytical solutions that are valid in the limits where the magnetic and acoustic components are decoupled, we calculate the relative fraction of energy flux that is carried away in each oscillation cycle by running acoustic waves in the atmosphere and running magnetic waves in the interior. For oscillation frequencies above the acoustic cut-off, we show that most energy losses associated with the presence of running waves occur in regions where the magnetic field is close to vertical. Moreover, by considering the depth dependence of the energy associated with the magnetic component of the wave in the atmosphere we show that a fraction of the wave energy is kept in the oscillation every cycle. For frequencies above the acoustic cut-off frequency, such energy is concentrated in regions where the magnetic field is significantly inclined in relation to the local vertical. Even though our calculations were aimed at studying oscillations with frequencies above the acoustic cut-off frequency, based on our results we discuss what results may be expected for oscillations of lower frequency.     

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 pulsational behaviour of the rapidly oscillating Ap star HD 122970 during two photometric multisite campaigns

We undertook two time-series photometric multisite campaigns for the rapidly oscillating Ap star HD 122970. The first one, conducted in 1998, resulted in 119 h of data and in the detection of three pulsation frequencies. The presence of possible further modes which held the promise of deriving a mode identification motivated a second worldwide campaign in the year 2001. This second campaign resulted in 203 h of measurement, but did not reveal further modes. Rather, one of the previously detected signals disappeared. The two modes common to both data sets have different spherical degree. They also showed slight frequency modulation, and one of them varied in amplitude as well. Possible causes of the latter behaviour include intrinsic instability of the pulsation spectrum or precession of the pulsational axis and orbital motion in a binary system. Frequency analysis of the Hipparcos observations of the star did not allow us to determine the stellar rotation period. The amplitude and phase behaviour of the two modes of HD 122970 in the Strömgren uvby bands is quite similar to that observed for other roAp stars.