The spectrum of gravity modes as a function of the solar structure: Model with a mixed core

The asymptotic properties of the gravity modes of solar models with a mixed core have been investigated. Such models have been constructed by Gavryusev and Gavryuseva (1984) to explain the low value of the observed neutrino flux (Cleveland, Davis, and Rowly, 1984). The strong enhancement of the Brunt-Väissälä frequency in the region of variable chemical composition at the boundary of the mixed core gives rise to a nonequidistant spectrum of gravity mode periods (Figure 1), contrary to the case of standard models. These models do not satisfy the helioseismological constraints given by the p-modes. However, the peculiar behavior of the numerically computed periods of the gravity modes is interesting to analyze, in view of observational detection in solar and stellar spectra.     

Gravity modes and mixed modes as probes of stellar cores in main‐sequence stars: From solar‐like to β Cep stars

We investigate how the frequencies of gravity modes depend on the detailed properties of the chemical composition gradient that develops near the core of main‐sequence stars and, therefore, on the transport processes that are able to modify the μ profile in the central regions. We show that in main‐sequence models, similarly to the case of white dwarfs, the periods of high‐order gravity modes are accurately described by a uniform period spacing superposed to an oscillatory component. The periodicity and amplitude of such a component are related, respectively, to the location and sharpness of the μ gradient. We briefly discuss and interpret, by means of this simple approximation, the effect of turbulent mixing near the core on the periods of both high‐order and low‐order g modes, as well as of modes of mixed pressure‐gravity character. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The asymptotic spectrum of gravity modes as a function of the solar structure: Standard solar model

Some results are given on the properties of the second-order asymptotic expression of the periods of low-degree gravity modes and on their rotational splitting. These could be of some help for the detection of these modes in the signal.     

Stochastic excitation of gravity modes in massive main-sequence stars

We investigate the possibility that gravity modes can be stochastically excited by turbulent convection in massive main-sequence (MS) stars. We build stellar models of MS stars with masses M=10?M _⊙,15?M _⊙, and 20?M _⊙. For each model, we then compute the power supplied to the modes by turbulent eddies in the convective core (CC) and the outer convective zones (OCZ). We found that, for asymptotic gravity modes, the major part of the driving occurs within the outer iron convective zone, while the excitation of low n order modes mainly occurs within the CC. We compute the mode lifetimes and deduce the expected mode amplitudes. We finally discuss the possibility of detecting such stochastically-excited gravity modes with the CoRoT space-based mission.     

The effect of turbulent mixing on the g‐mode spectrum of MS stars

Understanding transport processes inside stars is one of the main goals of asteroseismology. Chemical turbulent mixing can affect the internal distribution of μ near the energy generating core, having an effect on the evolutionary tracks similar to that of overshooting. This mixing leads to a smoother chemical composition profile near the edge of the convective core, which is reflected in the behavior of the buoyancy frequency and, therefore, in the frequencies of gravity modes. We describe the effects of convective overshooting and turbulent mixing on the frequencies of gravity modes in B‐type main sequence stars. In particular, the cases of p‐g mixed modes in β Cep stars and high‐order modes in SPBs are considered. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

160-min oscillations of the Sun as the mean of study of its internal structure

The investigation of the models of the contemporary Sun with a mixed core has shown that the amplitude of some gravity modes of oscillations of the star can be mainly concentrated in the central region. This phenomenon takes place if the node of the amplitude of radial displacement coincides with the boundary of the mixed core. In this case the core can be regarded as a driving generator of the oscillations, determining their period and phase. It is suggested as the explanation of the observational properties of the 160-min oscillation.

     

Trapped gravity waves and the five-minute oscillations of the solar atmosphere

The various modes of hydrodynamic waves are considered for a model of the solar atmosphere which is based on the Bilderberg model and includes the effects of ionization. The atmosphere forms a potential well for internal gravity waves, since the stability is low at the base (near the convection) and low again in the region of partial ionization in the chromosphere. Calculations show that there are two resonant (trapped) modes of internal gravity waves for horizontal wavelengths based on the scale of the granulation. The properties of these modes are in close agreement with the two modes of oscillation observed by Frazier (1968). Trapped acoustic modes are found to have periods too short to account for the observations.Presently Visiting Fellow, Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colo.     

An asymptotic representation of unstable gravity modes in stars

An asymptotic representation is developed for high order unstable gravity modes in a star which are associated with a convective zone enclosed between two radiative layers. In the domain which contains both turning points of the differential equation, the solutions are represented by a single asymptotic expansion in terms of Weber functions.     

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)     

Gravity modes in composite polytropic stars

We study gravity modes of composite polytropic stars which consist of two convectively stable zones separated by a convectively unstable zone. In addition to the unstable gravity modes associated with the intermediate zone, we distinguish two types of stable gravity modes, one type being mainly associated with the core, the other one being mainly associated with the envelope. We find also some accidental resonances between the core and the envelope.     

Mode trapping in sdO models

Mode trapping of high-radial order gravity modes was found for a particular sdO model. The trapping is caused by the change in composition from the helium radiative shell to the hydrogen burning shell. A non-adiabatic effect of this trapping is the higher tendency to instability of the trapped modes. Low- to intermediate-radial order pressure modes (in sdO models they correspond to mixed modes with most nodes in the P-mode region) are found to be trapped by the chemical transition from the carbon-oxygen core to the He burning shell. As the trapping is produced in the deep interior of the star, where energy interchange is negligible at the p-mode frequencies, it has no particular effect on the driving.     

Tidally induced surface displacements, external potential variations, and gravity variations on Mars

We have used and extended Roosbeek’s tidal potential for Mars to calculate tidal displacements, gravity variations, and external gravitational potential variations. The tides on Mars are caused by the Sun, and to a lesser degree by the natural satellites Phobos (8%, relative to the Sun) and Deimos (0.08%, relative to the Sun). To determine the reaction of Mars to the tidal forcing, the Love numbers h, l, and k and the gravimetric factor δ were calculated for interior models of Mars with different state, density, and radius of the core and for models which include mantle anelasticity. The latitude dependence and frequency dependence of the Love numbers have been taken explicitly into account. The Love numbers are about three times smaller than those for the Earth and are very sensitive to core changes; e.g., a difference of about 30% is found between a model with a liquid core and an otherwise similar model with a solid core. Tidal displacements on Mars are much smaller than on Earth due to the smaller tidal potential, but also due to the smaller reaction of Mars (smaller Love numbers). For both the tidal diplacement and the tidal external potential perturbations, the tidal signal is at the limit of detection and is too small to permit properties of Mars’s interior to be inferred. On the other hand, the Phobos tidally induced gravity changes, which are subdiurnal with typical periods shorter than 12 h, can be measured very precisely by the very broad band seismometer with thermal control of the seismological experiment SEIS of the upcoming NetLander mission. It is shown that the Phobos-induced gravity tides could be used to study the Martian core.     

Radiative damping of trapped gravity waves in the solar atmosphere

The radiative damping of trapped gravity waves in an optically thin atmosphere is studied for a stratified Boussinesq fluid. The character of the atmospheric eigenmodes depends on the distribution of the Brunt-Väisälä frequency N and the radiative relaxation time . The calculations for simple layer models show that if N is large over some finite fraction of the trapping region, then modes of long lifetime can exist. In order to suppress gravity waves entirely, it is necessary that N < 1 over the entire trapping region. Qualitative application of the results to the solar atmosphere leads to the conclusion that gravity wave eigenmodes of the solar atmosphere, although damped, are by no means eliminated by radiative effects.     

Effect of Gravity on the Fast Modes of a Horizontal Coronal Slab

Theoretical studies of the normal modes of a coronal slab often neglect gravity, as in Edwin and Roberts (Solar Phys. 71, 239, 1982). Here we study analytically the effect of gravity acting on a horizontal slab as a first step away from a homogeneous medium. Because of the inclusion of gravity, the symmetry of a homogeneous slab is lost, so the normal modes cannot be classified into kink and sausage modes. The presence of gravity also modifies the oscillatory frequencies of the slab, as well as the lower cutoff frequency, resulting in the possible transition between surface and body modes. For general coronal parameters, the dimensionless gravity term turns out to be small, so these effects are also small. A.J. Díaz’s current affiliation: Universitat de les Illes Balears, Palma, E-07122, Spain.     

The EC 14026 stars – VII. PG 1605+072, a star with many pulsation modes

We report the discovery of large-amplitude (∼0.25 mag) pulsations in the bright ( V =12.8) sdB star, PG 1605+072. The dominant period is 480 s, but more than 20 periods were present on at least three separate occasions. Frequency analysis of the complete data set yields more than 30 periods. A few of these are harmonics or linear combinations of the strongest modes. Excluding the latter, the periods span a range of almost 400 s, which contrasts with the typical range <20 s for most other EC 14026 stars.
Analysis of multicolour photometry limited any cool companion to being a main-sequence star of type M0 or later. Balmer line profile fitting yielded an effective temperature of 32 100±1000 K and a log g of 5.25±0.10, significantly smaller than in the other stars of the EC 14026 class, and possibly indicative of a more evolved state. The lower gravity is probably responsible for the fact that the pulsation periods and amplitudes are respectively much longer and larger than in other stars of the class. This star is an obvious target for asteroseismological investigation using a multilongitude photometric campaign.     

A New Code for Nonradial Stellar Pulsations and its Application to Low-Mass, Helium White Dwarfs

We present a finite difference code intended for computing linear, adiabatic, non radial pulsations of spherical stars. This code is based on a slight modification of the general Newton-Raphson technique in order to handle the relaxation of the eigenvalue(square of the eigenfrequency) of the modes and their corresponding eigenfunctions. This code has been tested computing the pulsation spectra of polytropic spheres finding a good agreement with previous work. Then, we have coupled this code to our evolutionary code and applied it to the computation of the pulsation spectrum of a low mass, pure-helium white dwarf of 0.3 M _⊙ for a wide range of effective temperatures. In making this calculation we have taken an evolutionary time step short enough such that eigenmodes corresponding to a given model are used as initial approximation to those of the next one. Specifically, we have computed periods, period spacing, eigenfunctions, weight functions, kinetic energies and variational periods for a wide range of modes. To our notice this is the first effort in studying the pulsation properties of helium white dwarfs. The solution we have found working with these realistic white dwarf models are in good accord with the predictions of the asymptotic theory of Tassoul (1980) for high order modes. This indicates that the code presented here is able to work adequately also with realistic stellar models. This revised version was published online in July 2006 with corrections to the Cover Date.     

The quest for the solar g modes

Solar gravity modes (or g modes)—oscillations of the solar interior on which buoyancy acts as the restoring force—have the potential to provide unprecedented inference on the structure and dynamics of the solar core, inference that is not possible with the well-observed acoustic modes (or p modes). The relative high amplitude of the g-mode eigenfunctions in the core and the evanesence of the modes in the convection zone make the modes particularly sensitive to the physical and dynamical conditions in the core. Owing to the existence of the convection zone, the g modes have very low amplitudes at photospheric levels, which makes the modes extremely hard to detect. In this article, we review the current state of play regarding attempts to detect g modes. We review the theory of g modes, including theoretical estimation of the g-mode frequencies, amplitudes and damping rates. Then we go on to discuss the techniques that have been used to try to detect g modes. We review results in the literature, and finish by looking to the future, and the potential advances that can be made—from both data and data-analysis perspectives—to give unambiguous detections of individual g modes. The review ends by concluding that, at the time of writing, there is indeed a consensus amongst the authors that there is currently no undisputed detection of solar g modes.     

Velocity oscillations in solar plage regions

The properties of wave propagation in a perfectly electrically conducting, plane-stratified, inviscid, compressible atmosphere premeated by a horizontal magnetic field which varies with height are investigated. It is shown that a diagnostic diagram can be constructed through a generalization of the propagation equation to account for the presence of a magnetic field.The effect of the magnetic field on the oscillations in solar plages around the temperature minimum is studied and compared with the non-magnetic case based on the Bilderberg Continuum Atmosphere. Due to the joint action of ionization and the magnetic field, a potential well for internal gravity waves is formed. The properties of the trapped waves are in qualitative agreement with the observations.The Brunt-Väisälä frequencies in the presence (N) and absence (N ₀) of the magnetic field around the temperature minimum are shown to obey the relation N/N ₀ (1 + ²)^–1/2, where is the ratio between the Alfvén and the sonic speeds. The modified Brunt-Väisälä frequency (N) is decreased as the strength of the magnetic field increases. The magnetic effect makes the potential well shallower, thus shrinking the domain for the trapped gravity modes.     

Statified model of a neutron star

Evolutionary calculations based on realistic equations of state indicate the stratified nature of the distribution of hadron matter in the interiors of neutron stars. In the proposed model, the stratified structure of a neutron star is treated as a rigid inert core surrounded by a dynamical layer. The physical basis for the model is the concept of the stellar matter of the peripheral envelope as an elastic Fermi continuum, the motions of which are described by the equations of nuclear elastodynamics, proposed in the macroscopic theory of collective processes in laboratory nuclear physics. It is shown that the vibrational dynamics of a neutron star is characterized by two branches of gravitational—elastic, spheroidal (s-mode) and torsional (t-mode) nonradial eigenvibrations. Estimates obtained for the periods of global, gravitational nonradial modes suggest that variations in the intensity of micropulses observed in the millisecond range of the spectra of C-pulsars may be ascribed to these vibrations. The proposed two-component model of a neutron star enables one to consider a glitch in a pulsar’s radio emission as a starquake due to the passage of the companion through periastron of the binary system. Translated from Astrofizika, Vol. 42, No. 2, pp. 235–252, April–June, 1999     

Discovery of A New Class of Pulsating Stars: Gravity-mode Pulsators among Subdwarf B Stars

During the course of an ongoing CCD monitoring program to investigate low-level light variations in subdwarf B (sdB) stars, weserendipitously discovered a new class of multimode pulsators withperiods of the order of an hour. These periods are a factor of tenlonger than those of previously known multimode sdB pulsators (EC14026 stars), implying the new pulsations are due to gravity modes rather than pressure modes. The iron opacity instability that drives the short period EC 14026 stars is effective in hot sdB's. Thelong period pulsators are found only among cooler sdB stars, wherethey are surprisingly common. The mechanism responsible for excitingthe deeper g-modes in cool sdB's is currently unknown, but thetemperature and gravity range in which these stars occur must be animportant clue. We present the first observational results for thisnew class of pulsating sdB stars, and discuss some possible implications.