We present results of a search for identification of modes responsible for the six most significant frequency peaks detected in the rapidly rotating SPB star $mu$ Eridani. All published and some unpublished photometric data are used in our new analys
is. The mode identification is carried out with the method developed by Daszynska-Daszkiewicz et al. employing the phases and amplitudes from multi-band photometric data and relying on the traditional approximation for the treatment of oscillations in rotating stars. Models consistent with the observed mean parameters are considered. For the five frequency peaks, the candidates for the identifications are searched amongst unstable modes. In the case of the third frequency, which is an exact multiple of the orbital frequency, this condition is relaxed. The systematic search is continued up to a harmonic degree $ell =6$. Determination of the angular numbers, $(ell,m)$, is done simultaneously with the rotation rate, $V_{rm rot}$, and the inclination angle, $i$, constrained by the spectroscopic data on the projected rotational velocity, $V_{rm rot}sin i$, which is assumed constant. All the peaks may be accounted for with g-modes of high radial orders and the degrees $ellle 6$. There are differences in some identifications between the models. For the two lowest--amplitude peaks the identifications are not unique. Nonetheless, the equatorial velocity is constrained to a narrow range of (135, 140) km/s. Our work presents the first application of the photometric method of mode identification in the framework of the traditional approximation and we believe that it opens a new promising direction in studies of SPB stars.
Results of mode identification and seismic modelling of the $beta$ Cep/SBP star 12 Lacertae are presented. Using data on the multi-colour photometry and radial velocity variations, we determine or constrain the mode degree, $ell$, for all pulsational
frequencies. Including the effects of rotation, we show that the dominant frequency, $ u_1$, is most likely a pure $ell=1$ mode and the low frequency, $ u_A$, is a dipole retrograde mode. We construct a set of seismic models which fit two pulsational frequencies corresponding to the modes $ell= 0,$ p$_1$ and $ell= 1,$ g$_1$ and reproduce also the complex amplitude of the bolometric flux variations, $f$, for both frequencies simultaneously. Some of these seismic models reproduce also the frequency $ u_A$, as a mode $ell= 1,$ g$_{13}$ or g$_{14}$, and its empirical values of $f$. Moreover, it was possible to find a model fitting the six 12 Lac frequencies (the first five and $ u_A$), only if the rotational splitting was calculated for a velocity of $V_{rm rot}approx 75$ km/s. In the next step, we check the effects of model atmospheres, opacity data, chemical mixture and opacity enhancement. Our results show that the OP tables are preferred and an increase of opacities in the $Z-$bump spoils the concordance of the empirical and theoretical values of $f$.
We review the present-day methods of mode identification applied to main sequence pulsators focusing on those that make use of multicolour photometry and radial velocity data. The effects which may affect diagnostic properties of these observables ar
e discussed. We also raise the problem of identification of high degree modes which can dominate oscillation spectra obtained from space-based projects.
We study all possible sources of inaccuracy in theoretical values of the photometric observables, i.e. amplitude ratios and phase differences, of early B-type main sequence pulsators. Here, we discuss effects of parameters coming from both models of
stellar atmospheres and linear nonadiabatic theory of stellar pulsation. In particular, we evaluate for the first time the effect of the departure from the LTE approximation. The atmospheric input comes from line-blanketed, LTE and NLTE plane-parallel, hydrostatic models. To compute the limb-darkening coefficients for NLTE models, we use the Least-Square Method taking into account the accuracy of the flux conservation. We present effects of NLTE atmospheres, chemical composition and opacities on theoretical values of the photometric observables of early B-type pulsators. To this end, we compute tables with the passband fluxes, flux derivatives over effective temperature and gravity as well as the non-linear limb-darkening coefficients in 12 most often used passbands, i.e. in the Stromgern system, $uvby$, and in the Johnson-Cousins-Glass system, $UBVRIJHK$. We make these tables public available at the Wroc{l}aw HELAS Web page, http://helas.astro.uni.wroc.pl.
We undertake another attempt towards seismic modelling of the most intensive studied main sequence pulsators of the early B spectral type, $ u$ Eridani. Our analysis is extended by a requirement of fitting not only pulsational frequencies but also th
e complex amplitude of the bolometric flux variation, $f$, related to each mode frequency. This approach, called {it complex asteroseismology}, provides a unique test of stellar parameters, atmospheres and opacities. In particular, the concordance of the empirical and theoretical values of $f$ we obtained for the high-order g mode opens a new gate in seismic studies of the main-sequence hybrid pulsators. The most intriguing and challenging result is that whereas an agreement of the theoretical and empirical values of $f$ for the radial mode can be achieved only with the OPAL data, a preference for the OP tables is derived from the analysis of the high-order gravity mode.
We present results of a {bf comprehensive} asteroseismic modelling of the $beta$ Cephei variable $theta$ Ophiuchi. {bf We call these studies {it complex asteroseismology} because our goal is to reproduce both pulsational frequencies as well as corres
ponding values of a complex, nonadiabatic parameter, $f$, defined by the radiative flux perturbation.} To this end, we apply the method of simultaneous determination of the spherical harmonic degree, $ell$, of excited pulsational mode and the corresponding nonadiabatic $f$ parameter from combined multicolour photometry and radial velocity data. Using both the OP and OPAL opacity data, we find a family of seismic models which reproduce the radial and dipole centroid mode frequencies, as well as the $f$ parameter associated with the radial mode. Adding the nonadiabatic parameter to seismic modelling of the B-type main sequence pulsators yields very strong constraints on stellar opacities. In particular, only with one source of opacities it is possible to agree the empirical values of $f$ with their theoretical counterparts. Our results for $theta$ Oph point substantially to preference for the OPAL data.
We study how rotation affects observable amplitudes of high-order g- and mixed r/g-modes and examine prospects for their detection and identification. Our formalism, which is described in some detail, relies on a nonadiabatic generalization of the tr
aditional approximation. Numerical results are presented for a number of unstable modes in a model of SPB star, at rotation rates up to 250 km/s. It is shown that rotation has a large effect on mode visibility in light and in mean radial velocity variations. In most cases, fast rotation impairs mode detectability of g-modes in light variation, as Townsend (2003b) has already noted, but it helps detection in radial velocity variation. The mixed modes, which exist only at sufficiently fast rotation, are also more easily seen in radial velocity. The amplitude ratios and phase differences are strongly dependent on the aspect, the rotational velocity and on the mode. The latter dependence is essential for mode identification.
