Rotational Splitting Research Articles

Equidistant frequency triplets in pulsating stars: the combination mode hypothesis

Multiplet structures are a common feature in pulsating stars, and can be the consequence of rotational splitting, mode interaction or sinusoidal amplitude variations. In this paper we examine the phenomenon of (nearly) equidistant triplets, which are unlikely to be caused by rotational splitting, in different types of pulsating stars: a Delta Scuti variable (1 Mon), an RR Lyrae variable (RR Lyr) and a short-period Cepheid (V743 Lyr). We examine the hypothesis that one of the modes forming the triplet results from a combination of the other two modes. The analyses were carried out on recent data sets by using multiple-frequency analyses and statistics with the package PERIOD04. In particular, the small departures from equidistance were calculated for the three selected stars. For the Delta Scuti variable 1 Mon, the departure from equidistance is only 0.000079 +- 0.000001 c/d (or 0.91 +- 0.01 nHz). For 1 Mon the Combination Mode Hypothesis with a mode excited by resonance is the most probable explanation. For the star RR Lyr, the hypothesis of resonance through a combination of modes should be considered. The results for the best-studied cepheid with a Blazhko period (V743 Lyr) are inconclusive because of an unfavorable period of 1.49d and insufficient data. Comment: 6 pages, 1 figure, accepted for publication in Astronomy and Astrophysics

The rapidly pulsating subdwarf B star PG 1325+101

In this article we present the results of 215 h of time-series photometry on the rapidly pulsating subdwarf B star PG 1325+101 (T_eff = 35 000 K, log g = 5.8, log N(He)/N(H) = -1.7), obtained during 25 days of observation in Spring 2003 from nine different sites. As in previous observations, the temporal spectrum is dominated by the main peak at 7255.55 ?Hz, with an amplitude of about 2.7% which, however, is dropped to about 1.7% in February 2005. No secondary peaks close to the dominant pulsation mode are clearly detected. In addition, at least fourteen more pulsation frequencies are found: three of them at 7704.92, 9380.17 and 14511.10 ?Hz were already present in the discovery run with small differences in frequency, probably due to 1-day aliasing effects. The peak at 7704.92 ?Hz belongs to a triplet of almost equally spaced frequencies that could be due to rotational splitting and would imply a rotational period of about 1.6 days. Based on the results of this article, a detailed asteroseismic analysis of PG 1325+101 is presented in a separate paper (Charpinet et al. 2006b, A&A, 459, 565, Paper II).

Mössbauer spectra and synchrotron-radiation-based perturbed angular correlations in special cases of rotational dynamics in fluids

The effect of a deterministic behavior of the molecular angular momentum between molecular collisions on the shapes of the resonance absorption spectra and the curves of synchrotron-radiation-based perturbed angular correlations (SRPAC) is considered and taken into account for description of the chaotic reorientation of “molecular” axes in fluids, in contrast to the standard statistical approaches employed previously for the interpretation of data of Mossbauer spectroscopy and recently developed SRPAC techniques. A general formalism is proposed for the calculation of the Mossbauer spectra and SRPAC curves in the presence of time-dependent hyperfine interactions. This formalism is used to describe the fluctuating quadrupole interaction within the framework of two models of the deterministic behavior of the principal axis of the electric field gradient tensor, which either rotates or oscillates (at an arbitrary angle) about some axis. Nontrivial transformation of the Mossbauer spectra and SRPAC curves depending on the parameters in both models is demonstrated, which describes the qualitatively different physical natures of the rotational dynamics and the corresponding resonance effects in fluids. In some limiting cases, simple analytical expressions are obtained that describe the Mossbauer spectra and SRPAC curves in terms of the effective constants of the quadrupole interaction and the rotational splitting of the main spectral lines. The revealed features of the hyperfine structure formation should be taken into account in developing models used for the analysis and interpretation of experimental data.

Frequency, splitting, linewidth and amplitude estimates of low- p modes of Cen A: analysis of Wide-Field Infrared Explorer photometry

We present results of fitting the 50-d time series of photometry of α Cen A taken by the Wide-Field Infrared Explorer (WIRE) satellite in 1999. Both power spectrum and autocovariance function (ACF) fitting techniques were used in an attempt to determine mode frequencies, rotational splittings, lifetimes and amplitudes of low-l p modes. In all, using both techniques, we managed to fit 18 modes (seven l= 0, eight l= 1 and three l= 2) with frequencies determined to within 1–2 μHz. These estimates are shown to be 0.6 ± 0.3 μHz lower, on average, than the frequencies determined from two other more recent studies, which used data gathered about 19 months after the WIRE observations. This could be indicative of an activity cycle, although due to the large uncertainty, more data would be needed to confirm this. Over a range of 1700–2650 μHz, we were also able to use the ACF fitting to determine an average lifetime of 3.9 ± 1.4 d, and an average rotational splitting of 0.54 ± 0.22 μHz, which is the first ever reliable estimate of this parameter. In contrast to the ACF, the power spectrum fitting was shown to return significantly biased results for these parameters.

The rotational and translational dynamics of molecular hydrogen physisorbed in activated carbon: A direct probe of microporosity and hydrogen storage performance

We have measured Incoherent Inelastic Neutron Scattering (IINS) spectra of H 2 physisorbed in high purity chemically activated carbon (AC) at different surface coverage and at temperatures near the triple point of bulk hydrogen. Our experimental results and DFT calculations show that at low surface coverage, due to the very low corrugation of the adsorption potential, and in the absence of H 2–H 2 lateral interactions, the adsorbed molecules are practically free to translate in the 2D plane parallel to the surface. Model calculations show that a complete mixing between the sub-states of the J = 1 manifold occurs on the free surface. The J = 0-to-1 rotational transition should split if the H 2 molecule is adsorbed in a slit type pore. Rotational splitting of up to 13 meV is found in the narrowest pores of around 6 Å investigated. The calculated isosteric heat of adsorption for molecules adsorbed on the free surface, at different sites and molecule orientations, range between −39 and −42 meV/H 2 at 77 K. In the optimum size slit pores, these numbers double up. Micropore volume of 0.34–0.45 ml/g carbon, and an upper limit of 4 wt% hydrogen storage is anticipated for the investigated material.

Asteroseismological Studies of Long‐Period Variable Subdwarf B Stars. I. A Multisite Campaign on PG 1627+017

We present the results of an extensive multisite campaign on the long-period variable subdwarf B star PG 1627+017. We gathered 300 hr of useful R-band and ~50 hr of simultaneous U/R differential photometry. From the R-band data we were able to extract 23 periodicities in the 4500-9000 s range with amplitudes between 0.05% and 0.5% of the star's mean brightness. The oscillations with the highest amplitudes cluster between 6300 and 7050 s and are thought to exhibit frequency splitting due to binary-synchronous stellar rotation. Interestingly, we find the observed period distribution to be extremely nonuniform, with dense frequency multiplets occurring in several narrow band passes. In order to compare the observed period spectrum to theoretical predictions, we constructed a set of newly updated and improved subdwarf B star models. We find that by invoking degree indices of l = 2, 3, and 4, nonadiabatic calculations can qualitatively reproduce the range of periodicities measured for PG 1627+017 if its atmospheric parameters are pushed to the lower end of their spectroscopic temperature uncertainties. However, the exploitation of rotational splitting and the U/R photometry, as well as the mean spacing between periodicities, indicate that at least the four highest amplitude peaks probably correspond to modes with l = 1. While this points to deficiencies in our models at the nonadiabatic level, the resulting constraints on mode identification are invaluable to first attempts at asteroseismology. Indeed, we identify only a few families of models that can closely reproduce the main periodicities observed in terms of dipole modes. This leaves us hopeful that, given a larger number of partially identified observed frequencies, asteroseismology may be achieved for long-period variable subdwarf B stars.

Solar FLAG★ hare and hounds: on the extraction of rotational p-mode splittings from seismic, Sun-as-a-star data

We report on results from the first solar Fitting at Low-Angular degree Group (solar FLAG) hare-and-hounds exercise. The group is concerned with the development of methods for extracting the parameters of low-l solar p-mode data ('peak bagging'), collected by Sun-as-a-star observations. Accurate and precise estimation of the fundamental parameters of the p modes is a vital pre-requisite of all subsequent studies. Nine members of the FLAG (the 'hounds') fitted an artificial 3456-d data set. The data set was made by the 'hare' (WJC) to simulate full-disc Doppler velocity observations of the Sun. The rotational frequency splittings of the l = 1, 2 and 3 modes were the first parameter estimates chosen for scrutiny. Significant differences were uncovered at l = 2 and 3 between the fitted splittings of the hounds. Evidence is presented that suggests this unwanted bias had its origins in several effects. The most important came from the different way in which the hounds modelled the visibility ratio of the different rotationally split components. Our results suggest that accurate modelling of the ratios is vital to avoid the introduction of significant bias in the estimated splittings. This is of importance not only for studies of the Sun, but also of the solar analogues that will be targets for asteroseismic campaigns.

Rotation speed and stellar axis inclination from p modes: how CoRoT would see other suns

In the context of future space-based asteroseismic missions, we have studied the problem of extracting the rotation speed and the rotation-axis inclination of solar-like stars from the expected data. We have focused on slow rotators (at most twice solar rotation speed), first, because they constitute the most difficult case and, secondly, because some of the Convection Rotation and planetary Transits (CoRoT) main targets are expected to have slow rotation rates. Our study of the likelihood function has shown a correlation between the estimates of inclination of the rotation axis i and the rotational splitting δν of the star. By using the parameters, i and δv* = δν sini, we propose and discuss new fitting strategies. Monte Carlo simulations have shown that we can extract a mean splitting and the rotation-axis inclination down to solar rotation rates. However, at the solar rotation rate we are not able to correctly recover the angle i, although we are still able to measure a correct δν* with a dispersion less than 40 nHz.

High-Resolution Photoassociation Spectroscopy of Ultracold Ytterbium Atoms by Using the Intercombination Transition

We observed high-resolution photoassociation spectra of laser-cooled ytterbium (Yb) atoms in the spin-forbidden 1S0 - 3P1 intercombination line. The rovibrational levels in the 0u+ state were measured for red detunings of the photoassociation laser ranging from 2.9 MHz to 1.97 GHz with respect to the atomic resonance. The rotational splitting of the vibrational levels near the dissociation limit were fully resolved due to the sub-MHz linewidth of the spectra in contrast to previous measurements using the spin-allowed singlet transition. In addition, from a comparison between the spectra of 174Yb and those of 176Yb, a d-wave shape resonance for 174Yb is strongly suggested.

Whole Earth Telescope Observations of the Pulsating Subdwarf B Star PG 0014+067

PG 0014+067 is one of the most promising pulsating subdwarf B stars for seismic analysis, as it has a rich pulsation spectrum. The richness of its pulsations, however, poses a fundamental challenge to understanding the pulsations of these stars, as the mode density is too complex to be explained only with radial and nonradial low degree (l < 3) p-modes without rotational splittings. One proposed solution, for the case of PG 0014+067 in particular, assigns some modes with high degree (l=3). On the other hand, theoretical models of sdB stars suggest that they may retain rapidly rotating cores, and so the high mode density may result from the presence of a few rotationally-split triplet (l=1), quintuplet (l=2) modes, along with radial (l=0) p-modes. To examine alternative theoretical models for these stars, we need better frequency resolution and denser longitude coverage. Therefore, we observed this star with the Whole Earth Telescope for two weeks in October 2004. In this paper we report the results of Whole Earth Telescope observations of the pulsating subdwarf B star PG 0014+067. We find that the frequencies seen in PG 0014+067 do not appear to fit any theoretical model currently available; however, we find a simple empirical relation that is able to match all of the well-determined frequencies in this star.

Modeling β Virginis using seismological data

This paper presents the modeling of the F9 V star β Virginis based on seismological measurements. Using the Geneva evolution code including rotation and atomic diffusion, we find that two distinct solutions reproduce all existing asteroseismic and non-asteroseismic observational constraints well: a main-sequence model with a mass of 1.28 ± and an age ± 0.20 Gyr, or a model in the post-main sequence phase of evolution with a lower mass of 1.21 ± and an age ± 0.30 Gyr. The small spacings and the ratio r02 between small and large spacings are sensitive to the differences in the structure of the central layers between these two solutions and are also sensitive to the structural changes due to the rotational mixing. They can therefore be used to unambiguously determine the evolutionary state of β Vir and to study the effects of rotation on the inner structure of the star. Unfortunately, existing asteroseismic data do not enable such precise determination. We also show that the scatter in frequencies introduced by the rotational splittings can account for the larger dispersion of the observed large spacings for the non-radial modes than for the radial modes.

Asteroseismological constraints on the structure of the ZZ Ceti star HL Tau 76

This paper reports the results derived from an asteroseismological study of the cool ZZ Ceti star HL Tau 76. A grid of models has been computed in a parameter space covering the range of log g and Teff, formerly determined by spectroscopy, and a large range of hydrogen mass fraction. The adiabatic non-radial oscillations for all the models have been computed for the modes of degrees and . An algorithm based on a test was applied to evaluate the quality of the fit between observed and theoretical periods. This method resulted in selecting a best fitting model for which the average relative matching of the periods is 0.7%. Then, a detailed comparison between the observed and the computed periods for the and modes of the best fitting model was achieved in order to identify as many observed modes as possible. To perform this identification we used the calculated periods for which we applied the rotational splitting as deduced from the observations. Through this process we identify the 36 independent modes observed in HL Tau 76. The best fitting model for HL Tau 76 is well constrained due to the large number of oscillations detected in this ZZ Ceti star. The main stellar parameters of HL Tau 76 derived from this analysis are: the total mass , the hydrogen mass fraction qH, estimated as thick as . The helium mass fraction consistent with qH must be . The method is not sensitive to Teff variations in the narrow domain of temperature derived from spectroscopy for HL Tau 76. The best adjustment is found however for K. The other derived stellar parameters are the luminosity () and the radius (). We note some trends in the fit of the observed periods with the computed ones which suggest that the rotational splitting could be non-uniform and that the large amplitude modes might contain information on the convection-driven excitation mechanism.

Probing the internal magnetic field of slowly pulsating B-stars through g modes

modes often occur in multiplets with closely spaced periods (with a typical separation of 1%). In some cases this separation can clearly not be due to rotational splitting, which would yield much larger spacings, as was pointed out by De Cat and Aerts (2002). In this letter we propose that such frequency splitt ings are due to the presence of a magnetic field. If this hypothesis is correct, then the splitting of frequencies can be used to e stimate the field strength in the interior of SPB stars.

Multicolour high-speed photometry of the subdwarf B star PG 0014+067 with ULTRACAM★

The non-radially pulsating subdwarf B star PG 0014+067 has previously been presented as a classic case for asteroseismological study, having a moderately rich mode spectrum uncomplicated by severe rotational splitting. Notwithstanding the excellence of previous work, empirical evidence for mode identification is needed to test the modelling results. Consequently high-speed multicolour photometry was obtained over six nights in 2004 August using the high-speed multichannel photometer ULTRACAM on the 4.2-m William Herschel Telescope with a sampling interval of 5 s. To ameliorate the window function, additional single-channel photometry was obtained on five nights in the same time envelope using the 2-m Faulkes Telescope North with a sampling interval of 23 s. 19 individual frequencies have been identified in a combined `white light' data set. Amplitudes have been measured in filtered light (u', g' and r') for 13 of these. Three groups of closely spaced frequencies have spacings of 3, 13 and 2 muHz. Although on the cusp of the frequency resolution, there is evidence that the rotational period should be nearer to 4 d rather than the 1.35 d reported previously, if we assume that these close frequencies belong to a multiplet. It has not proved possible to identify modes unambiguously using the amplitude ratio method because of the errors on the amplitudes, but we do exclude that the two dominant modes have l= 3 or 4; they must be either l= 0, 1, 2.

Multicolour photometry of Balloon 090100001: linking the two classes of pulsating hot subdwarfs

We present results of the multicolour UBVR photometry of the high-amplitude EC14026-type star, Balloon 090100001. The data span over a month and consist of more than a hundred hours of observations. Fourier analysis of these data led us to the detection of at least 30 modes of pulsation of which 22 are independent. The frequencies of 13 detected modes group in three narrow ranges, around 2.8, 3.8 and 4.7 mHz, where the radial fundamental mode, the first and second overtones are likely to occur. Surprisingly, we also detect 9 independent modes in the low-frequency domain, between 0.15 and 0.4 mHz. These modes are typical for pulsations found in PG1716+426-type stars, discovered recently among cool B-type subdwarfs. The modes found in these stars are attributed to the high-order g modes. As both kinds of pulsations are observed in Balloon 090100001, it represents a link between the two classes of pulsating hot subdwarfs. At present, it is probably the most suitable target for testing evolutionary scenarios and internal constitution models of these stars by means of asteroseismology. Three of the modes we discovered form an equidistant frequency triplet which can be explained by invoking rotational splitting of an $\ell$ = 1 mode. The splitting amounts to about 1.58 $\mu$Hz, leading to a rotation period of 7.1 $\pm$ 0.1 days.

Mössbauer Spectra of the Quadrupole Hyperfine Structure in the Presence of the Rotation of the Principal Axis of the Electric Field Gradient Tensor

It is found that the rotation of the principal axis of the electric field gradient tensor about a certain axis radically changes the shape of the Mossbauer spectra of the quadrupole hyperfine structure. This change may be a manifestation of a qualitatively different physical origin of rotational dynamics in liquids. In particular, the absorption spectra of 57Fe nuclei may contain the triplet, quartet, and octet of lines instead of the standard quadrupole doublet. Simple analytical expressions derived for a number of limiting cases describe the spectra in terms of the effective quadrupole coupling constant and the rotational splitting constant for the main lines. The observed features of the formation of quadrupole hyperfine structure spectra must be taken into account in analysis of experimental Mossbauer spectra in liquids.

Absorption intensities of multipole-field-induced double transitions involving orthohydrogen pairs in a solid parahydrogen crystal

Orthohydrogen molecules exhibit quantum diffusion through a solid parahydrogen lattice. At very low orthohydrogen concentrations $(\ensuremath{\sim}1%)$, stable and isolated ortho-${\mathrm{H}}_{2}$ pairs are formed in the para-${\mathrm{H}}_{2}$ crystal. In this paper, theoretical expressions are derived for the integrated absorption coefficients of various multipole-field-induced zero-phonon double transitions in an isolated pair of ortho-${\mathrm{H}}_{2}$ molecules embedded in para-${\mathrm{H}}_{2}$ crystal. Although, the double transitions involving ortho-${\mathrm{H}}_{2}$ pairs consist entirely of fine structures (FS), the intensity formulas derived here are for the line as a whole, except for ${Q}_{{v}_{1}}(1)+{Q}_{{v}_{k}}(1)$ transition in which the individual components are separately treated. Since the $J=1$ rotational state pair splittings of various ground state FS components are well known, experimentalists can compare the observed intensities to the theoretical calculations for the line as a whole (given in units of per molecular-pair) by reducing the observed intensities of each FS components to per molecular pair and summing them together. The formulas given in this paper can be applied to double transitions involving a pair of $J=1$ dopant molecules (with zero permanent dipole moment) embedded in solid parahydrogen (or orthodeuterium) matrices. Furthermore, it is shown that the intensity formulas given for an isolated pair, after generalization, provide good estimates of the intensities of double transitions in a nearly pure ortho-${\mathrm{H}}_{2}$ crystal.

Measuring Stellar Differential rotation with asteroseismology

The variation of rotation with latitude is poorly known on stars other than the Sun. Several indirect techniques, photometric and spectroscopic, have been used to search for departure from rigid rotation for sufficiently fast rotators. Here we investigate the possibility of measuring stellar differential rotation for solar-type stars through asteroseismology. Rotationally split frequencies of global oscillation provide information about rotation at different latitudes depending on the azimuthal order, m, of the mode of pulsation. We present a method to estimate differential rotation based on the realization that the m =± 1a ndm =± 2 components of quadrupole oscillations can be observed simultaneously in asteroseismology. Rotational frequency splittings can be inverted to provide an estimate of the difference in stellar angular velocity between the equator and 45 ◦ latitude. The precision of the method, assessed through Monte Carlo simulations, depends on the value of the mean rotation and on the inclination angle between the rotation axis and the line of sight.

About the rotation of the solar radiative interior

In the modern era of helioseismology we have a wealth of high-quality data available, e.g., more than 6 years of data collected by the various instruments on board the SOHO mission, and an even more extensive ground-based set of observations covering a full solar cycle. Thanks to this effort a detailed picture of the internal rotation of the Sun has been constructed. In this paper we present some of the actions that should be done to improve our knowledge of the inner rotation profile discussed during the workshop organized at Saclay on June 2003 on this topic. In particular we will concentrate on the extraction of the rotational frequency splittings of low- and medium-degree modes and their influence on the rotation of deeper layers. Furthermore, for the first time a full set of individual |m|-component rotational splittings is computed for modes l≤4 and 1<ν<2 mHz, opening new studies on the latitudinal dependence of the rotation rate in the radiative interior. It will also be shown that these splittings have the footprints of the differential rotation of the convective zone which can be extremely useful to study the differential rotation of other stars where only these low-degree modes will be available.

Asteroseismology of the β Cephei star ν Eridani: interpretation and applications of the oscillation spectrum

The oscillation spectrum of v Eri is the richest known for any variable of the β Cephei type. We interpret the spectrum in terms of normal mode excitation and construct seismic models of the star. The frequency data combined with data on mean colours set the upper limit on the extent of overshooting from the convective core. We use data on rotational splitting of two dipole (l = 1) modes (g 1 and p 1 ) to infer properties of the internal rotation rate. Adopting a plausible hypothesis of nearly uniform rotation in the envelope and increasing rotation rate in the μ-gradient zone, we find that the mean rotation rate in this zone is about three times faster than in the envelope. In our standard model only the modes in the middle part of the oscillation spectrum are unstable. To account for excitation of a possible high-order g mode at v = 0.43 cd -1 as well as p modes at v > 6 cd -1 we have to invoke an overabundance of Fe in the driving zone.