Individual P-mode Research Articles

Frequency dependence of p-mode frequency shifts induced by magnetic activity in Kepler solar-like stars

The variations of the frequencies of the low-degree acoustic oscillations in the Sun induced by magnetic activity show a dependence on radial order. The frequency shifts are observed to increase towards higher-order modes to reach a maximum of about 0.8 μHz over the 11-yr solar cycle. A comparable frequency dependence is also measured in two other main sequence solar-like stars, the F-star HD 49933, and the young 1 Gyr-old solar analog KIC 10644253, although with different amplitudes of the shifts of about 2 μHz and 0.5 μHz, respectively. Our objective here is to extend this analysis to stars with different masses, metallicities, and evolutionary stages. From an initial set of 87 Kepler solar-like oscillating stars with known individual p-mode frequencies, we identify five stars showing frequency shifts that can be considered reliable using selection criteria based on Monte Carlo simulations and on the photospheric magnetic activity proxy Sph. The frequency dependence of the frequency shifts of four of these stars could be measured for the l = 0 and l = 1 modes individually. Given the quality of the data, the results could indicate that a physical source of perturbation different from that in the Sun is dominating in this sample of solar-like stars.

Solar-like oscillations in the G9.5 subgiantβAquilae

An interesting asteroseismic target is the G9.5 IV solar-like star beta Aql. This is an ideal target for asteroseismic investigations, because precise astrometric measurements are available from Hipparcos that greatly help in constraining the theoretical interpretation of the results. The star was observed during six nights in August 2009 by means of the high-resolution \'echelle spectrograph SARG operating with the TNG 3.58 m Italian telescope on the Canary Islands, exploiting the iodine cell technique. We present the result and the detailed analysis of high-precision radial velocity measurements, where the possibility of detecting time individual p-mode frequencies for the first and deriving their corresponding asymptotic values will be discussed. The time-series analysis carried out from \sim 800 collected spectra shows the typical p-mode frequency pattern with a maximum centered at 416 \muHz. In the frequency range 300 - 600 \muHz we identified for the first time six high S/N (\gtrsim 3.5) modes with l = 0,2 and 11 < n < 16 and three possible candidates for mixed modes (l = 1), although the p-mode identification for this type of star appears to be quite difficult owing to a substantial presence of avoided crossings. The large frequency separation and the surface term from the set of identified modes by means of the asymptotic relation were derived for the first time. Their values are \Delta \nu = 29.56 \pm 0.10 \muHz and \epsilon = 1.29 \pm 0.04, consistent with expectations. The most likely value for the small separation is \delta\nu_{02} = 2.55 \pm 0.71 \muHz.

Solar-cycle variations of large frequency separations of acoustic modes: implications for asteroseismology

We have studied solar cycle changes in the large frequency separations that can be observed in Birmingham Solar Oscillations Network (BiSON) data. The large frequency separation is often one of the first outputs from asteroseismic studies because it can help constrain stellar properties like mass and radius. We have used three methods for estimating the large separations: use of individual p-mode frequencies, computation of the autocorrelation of frequency-power spectra, and computation of the power spectrum of the power spectrum. The values of the large separations obtained by the different methods are offset from each other and have differing sensitivities to the realization noise. A simple model was used to predict solar cycle variations in the large separations, indicating that the variations are due to the well-known solar cycle changes to mode frequency. However, this model is only valid over a restricted frequency range. We discuss the implications of these results for asteroseismology.

On the Determination of Michelson Doppler Imager High‐Degree Mode Frequencies

The characteristics of the solar acoustic spectrum are such that mode lifetimes get shorter and spatial leaks get closer in frequency as the degree of a mode increases for a given order. A direct consequence of this property is that individual p-modes are resolved only at low and intermediate degrees and that at high degrees individual modes blend into ridges. Once modes have blended into ridges, the power distribution of the ridge defines the ridge central frequency, and it will mask the true underlying mode frequency. An accurate model of the amplitude of the peaks that contribute to the ridge power distribution is needed to recover the underlying mode frequency from fitting the ridge. We present the results of fitting high-degree power ridges (up to l = 900) computed from several 2-3 month long time series of full-disk observations taken with the Michelson Doppler Imager (MDI) on board the Solar and Heliospheric Observatory between 1996 and 1999. We also present a detailed discussion of the modeling of the ridge power distribution, and the contribution of the various observational and instrumental effects on the spatial leakage, in the context of the MDI instrument. We have constructed a physically motivated model (rather than some ad hoc correction scheme) that we believe results in a methodology that can produce an unbiased determination of high-degree modes once the instrumental characteristics are well understood. Finally, we present preliminary estimates of changes in high-degree mode parameters with epoch and thus solar activity level and discuss their significance. These estimates are preliminary because they rely on a simple—if not simplistic—ridge-to-mode correction scheme to account for errors in the plate scale used for the spherical harmonic decomposition. Such a correction scheme produced residual systematics that, as we show, are not always constant with time. These cannot be properly corrected without reprocessing the data back to the level of the spherical harmonic decomposition.

Solar [CLC][ITAL]p[/ITAL][/CLC]-Modes Modulated by Interior [CLC][ITAL]g[/ITAL][/CLC]-Modes

The process by which frequency modulations of solar p-modes with shorter periods (~5 minutes) occur because of the presence of solar interior g-modes with longer periods (greater than a few hours approximately) is studied. The key results of a model analysis are given. In addition to the effects of stochastic excitation, this modulation by g-modes would give rise to fine spectral structures in the form of frequency sideband peaks closely packed around individual p-modes (with narrow frequency intervals on the order of g-mode frequencies) rather than of separate g-mode spectral power peaks in the low-frequency bands. For those p-modes of low degree that penetrate deep into the solar interior, this effect of frequency modulation by interior g-modes (if they exist) should be more pronounced. Observationally, individual p-modes usually contain numerous fine peaks that have to be somehow modeled (or smoothed) in order to estimate p-mode frequencies, amplitudes, and line widths, etc. By carefully weeding out noises of various origins in high-resolution p-mode frequency spectra, one might be able to extract the desired signals of g-modes trapped in the solar interior that have so far evaded the scrutiny of advanced helioseismological experiments.

Line Profiles and Rotational Splitting of Individual P-Modes

We used GONG time series for ℓ = 0,1,2,3 from June 10, 1995 to January 7, 1997, 578 days in total. The duty cycle varies significantly during this period of time but on average it is sufficiently high, and the remaining gaps are irregular. Because of this we do not see daily side lobes in the spectra obtained.

A Comment on the Relationship between the Modal and Time‐Distance Formulations of Local Helioseismology

The relationship between the time-distance and modal-decomposition approaches to solar active region seismology is clarified through the consideration of the oscillations of a plane-parallel, isentropic polytrope. It is demonstrated by direct construction that a wave packet formed through the superposition of neighboring p-modes interferes constructively along a ray bundle that follows the appropriate WKBJ ray path obtained by using the eikonal approximation. Because the actual power envelope of the solar 5 minute oscillations restricts the excited p-modes to rather low radial orders, the ray bundles are diffuse and sample portions of the solar envelope that are some ≈ 10-30 Mm distant from the nominal WKBJ ray path. This behavior is consistent with the fact that the eikonal approximation becomes valid only in the limiting case of large radial orders (n 1). The p-mode wave packets that are isolated by employing the time-distance methods must therefore be described either as a superposition of individual p-modes (a wave packet), or as a sum of ray paths (a ray bundle), depending upon which representation proves to be optimal for the given circumstances.

Excitation of solar p-modes

We investigate the rates at which energy is supplied to individual p-modes as a function of their frequencies v and angular degrees l. The observationally determined rates are compared with those calculated on the hypothesis that the modes are stochastically excited by turbulent convection. The observationally determined excitation rate is assumed to be equal to the product of the mode's energy E and its (radian) line width Г. We obtain E from the mode's mean square surface velocity with the aid of its velocity eigenfunction. We assume that Г measures the mode's energy decay rate, even though quasi-elastic scattering may dominate true absorption. At fixed l, EГ rises as v^7 at low v, reaches a peak at v ≈ 3.5 mHz, and then declines as v^(-4•4) at higher v. At fixed v, EГ exhibits a slow decline with increasing l. To calculate energy input rates, P_ α, we rely on the mixing-length model of turbulent convection. We find entropy fluctuations to be about an order of magnitude more effective than the Reynolds stress in exciting p-modes. The calculated P_ α mimic the v^7 dependence of EГ at low v and the v^(-4•4) dependence at high v. The break of 11.4 powers in the v-dependence of EГ across its peak is attributed to a combination of (1) the reflection of high-frequency acoustic waves just below the photosphere where the scale height drops precipitously and (2) the absence of energy-bearing eddies with short enough correlation times to excite high-frequency modes. Two parameters associated with the eddy correlation time are required to match the location and shape of the break. The appropriate values of these parameters, while not unnatural, are poorly constrained by theory. The calculated P_ α can also be made to fit the magnitude of EГ with a reasonable value for the eddy aspect ratio. Our results suggest a possible explanation for the decline of mode energy with increasing l at fixed v. Entropy fluctuations couple to changes in volume associated with the oscillation mode. These decrease with decreasing n at fixed v, becoming almost zero for the ƒ-mode.

The absorption of p-modes by sunspots - Variations with degree and order

A spherical harmonic decomposition of the p-modes into inward and outward propagating waves is employed to investigate the absorption of solar p-modes by an isolated sunspot. The absorption coefficient (averaged over frequency and azimuthal order) is found to increase with increasing horizontal wavenumber k over the range 0-0.8/Mm. For larger horizontal wavenumbers, in the range 0.8-1.5/Mm, the absorption coefficient decreases with increasing k. The absorption along each individual p-mode ridge tends to peak at an intermediate value of the spherical harmonic degree in the range 200-400. The highest absorption is found along the p(1) ridge, and the absorption decreases with increasing radial order.

Excitation Mechanisms of Oscillations in Stars

AbstractExcitation mechanisms of oscillations in stars are discussed, in particular on the problems of the Beta Cephei pulsations and of the solar oscillations. A long-standing mystery about the excitation mechanism of the Beta Cephei pulsations seems to have finally been solved, which is due to availability of the new “OPAL” opacities. The very mechanism is the classical к-mechanism, however, due to enhanced heavy element opacities at temperature around 2 × 105K. By using the new opacities, three independent groups have found pulsational instabilities for the fundamental radial mode and a few low-degree non-radial modes in Beta Cephei models. Stochastic excitation of solar oscillations by turbulent convection is then discussed. It is shown that the noise generation by quadrupole radiation concentrated at the top of the convection zone may well explain the frequency dependence of the observed solar acoustic power pumped into individual p-modes.

Sunspot Seismology: The Interaction of Solar p-Modes with a Sunspot

AbstractProgress in sunspot seismology is briefly reviewed. New observations of the absorption of p-modes by a sunspot (Bogdan, Brown, Lites, and Thomas 1993) show that the absorption coefficient (averaged over frequency and azimuthal order) increases with horizontal wavenumber to a maximum near k = 0.9 Mm−1 and then decreases monotonically to zero at about k = 1.5 Mm−1. The absorption along individual p-mode ridges shows a sinusoidal-like modulation, which may be an important feature for comparison with theory. Results for the quiet Sun show no net absorption, but do show apparent emission along p1 over the same range of k where the sunspot shows its greatest absorption. This behavior, and the overall decrease in absorption at higher wavenumbers, can be explained as an effect of the decreasing lifetime of p-modes with increasing wavenumber k (increasing mode degree l).

Wave generation by turbulent convection

We consider wave generation by turbulent convection in a plane parallel, stratified atmosphere that sits in a gravitational field, g. The atmosphere consists of two semi-infinite layers, the lower adiabatic and polytropic and the upper isothermal. The adiabatic layer supports a convective energy flux given by mixing length theory; F_c ~ pv^3_H, where p is mass density and v_H is the velocity of the energy bearing turbulent eddies. Acoustic waves with ω > ω_(ɑc) and gravity waves with ω < 2k_h H_iωb propagate in the isothermal layer whose acoustic cutoff frequency, ω_(ac), and Brunt-Vaisala frequency, ω_b, satisfy ω^2_(ɑc) = yg/4H_i and ω^2_b = (y-1)g/yH_i, where y and H_i denote the adiabatic index and scale height. The atmosphere traps acoustic waves in upper part of the adiabatic layer (p-modes) and gravity waves on the interface between the adiabatic and isothermal layers (f-modes). These modes obey the dispersion relation ω^2≈2/m gk_h(n + m/2), for ω < ω_(ɑc). Here, m is the polytropic index, k_h is the magnitude of the horizontal wave vector, and n is the number of nodes in the radial displacement eigenfunction; n = 0 for f-modes. Wave generation is concentrated at the top of the convection zone since the turbulent Mach number, M = v_H/c, peaks there; we assume M_t « 1. The dimensionless efficiency, η, for the conversion of the energy carried by convection into wave energy is calculated to be η~M_t^(5/12) for p-modes,f-modes, and propagating acoustic waves, and η~M, for propagating gravity waves. Most of the energy going into p-modes, f-modes, and propagating acoustic waves is emitted by inertial range eddies of size h ~ M_t^(3/2)H_t, at ω ~ ω_(ɑc) and k_h ~ 1/H_t. The energy emission into propagating gravity waves is dominated by energy bearing eddies of size ~ H_t and is concentrated at ω ~ v_t/H_t ~ M_t ω_(ɑc) and k_h ~ 1/H_t. We find the power input to individual p-modes, E_p, to vary as ω<^(2m^2+7m-3)/(m+3) at frequencies ω « v_t/H_t. Libbrecht has shown that the amplitudes and linewidths of the solar p-modes imply E_p ∝ ω^8 for ω « 2 x 10^(-2) s^(-1). The theoretical exponent matches the observational one for m ≈ 4, a value obtained from the density profile in the upper part of the solar convection zone. This agreement supports the hypothesis that the solar p-modes are stochastically excited by turbulent convection.