Unstable non-radial modes in radial pulsators: theory and an example

We study the possibility of the excitation of non-radial oscillations in classical pulsating stars. The stability of an RR Lyrae model is examined through non-adiabatic non-radial calculations. We also explore stability in the presence of non-linear coupling between radial and non-radial modes of nearly identical frequency.   In our model, a large number of unstable low-degree (ℓ = 1,2) modes have frequencies in the vicinity of unstable radial mode frequencies. The growth rates of such modes, however, are considerably smaller than those of the radial modes. We also recover an earlier result that at higher degrees (ℓ = 6–12) there are modes trapped in the envelope with growth rates similar to those of radial modes.   Subsequently, monomode radial pulsation of this model is considered. The destabilizing effect of the 1:1 resonance between the radial mode and nearby non-radial modes of low degrees is studied, with the assumption that the excited radial mode saturates the linear instability of all other modes. The instability depends on the radial mode amplitude, the frequency difference, the damping rate of the non-radial mode, and the strength of the non-linear coupling between the modes considered. At the pulsation amplitudes typical for RR Lyrae stars, the instability of the monomode radial pulsation and the concomitant resonant excitation of some non-radial oscillation modes is found to be very likely.     

Eight new δ Scuti stars

HD 23194, a member of the Pleiades, was found to pulsate with a period of about 30 min. The literature on the star is reviewed, and it is concluded that it may be a marginal Am star in a binary system. HD 95321 is an evolved Am ( ρ Puppis) star with a 5.1-h periodicity. Mode identification of its pulsation, based on multicolour photometry, suggests that the oscillation is probably non-radial with ℓ=2. We also report on the discovery of six other new δ Scuti stars, some of which may be pulsating in gravity modes.     

UBVRIJH photometry of two new luminous δ Scuti stars and the discovery of δ Scuti pulsation in the most evolved Ap star known

Time-series photometry of the Hipparcos variable stars HD 199434 and 21190 is reported. Both stars are pulsators of the δ Scuti type. Reclassifications of the MK types of the stars, based on new spectrograms, are given. HD 21190 is found to be F2III SrEuSi:, making it the most evolved Ap star known. Its Strömgren photometric indices support the peculiar spectral type. It is also one of the most evolved δ Scuti stars known. Its combined Ap– δ Scuti nature makes it an important test of models of pulsation in peculiar stars recently developed by Turcotte et al., although it is more extreme than any model they examined. Physical parameters of both stars are estimated from Strömgren and H β photometry, and Hipparcos absolute magnitudes. We attempt mode identifications based on amplitude ratios and phase differences from our photometry. The dominant pulsation of HD 21190 may be an overtone radial mode. The model fits for HD 199434 are even less satisfactory, but favour an ℓ=2 mode. Given the good quality and wavelength coverage of our data, the poor results from the application of the photometric theory of mode identification may call into question the use of that technique.     

Low-frequency variability and binarity of the δ Scuti star XX Pyx

We analyse 147 h of single-site CCD time series photometry of the multiperiodic low-amplitude δ Scuti star XX Pyx with the aim of investigating variability at low frequencies. Part of the data were obtained in the context of the 1998 multisite campaign on XX Pyx, the results of which were described by Handler et al. We find that periodic low-frequency variations are present in the XX Pyx light curves, and we detect two frequencies at f _A=0.8695 cycle d⁻¹ and f _B=1.7352 cycle d⁻¹ , respectively, with amplitudes of 4.5 and 5.4 mmag. The low-frequency variability is intrinsic to XX Pyx, and cannot be attributed to instrumental or atmospheric effects. The near 2:1 ratio of the frequencies leads us to suggest that XX Pyx is a δ Scuti star in a binary system, with a possible binary period of 27.6 h. This is strongly supported by the detection of radial velocity variations from the re-analysis of echelle spectra obtained by Handler et al. However, in the absence of a spectroscopic period, alternative explanations of the photometric variability involving pulsation are also possible; the variations occur close to possible combination frequencies of the short-period ( δ Scuti) variations, but high Q values of 0.57 and 0.28 d suggest that the variations are not a result of normal (p-)modes. They could possibly be due to g-modes excited to observable amplitudes as a result of resonance effects. Surface features (spots) are unlikely to be the cause of the variations.
We searched for combination frequencies (  f _i − f _j ) , f being the normal δ Scuti frequencies detected by Handler et al., but failed to detect any.     

Mode identification in the δ Scuti Star 1 Mon

We present new multicolour photometry and simultaneous high-dispersion spectroscopy for the δ Scuti star 1 Mon. The two main periodicities in the star are still present, but the third known period is not directly detected in the new radial velocities or light variations. However, it is detected in the periodogram of the second moment of the line profile variations. We use the cross-correlation function as an approximation for the line profile variations. By computing theoretical profiles for a given mode and comparing them with phased cross-correlation profiles, we are able to determine a goodness-of-fit criterion and estimate the most probable spherical harmonic degree,     the azimuthal order, m , of the pulsation and also the angle of inclination. We then compare the relative amplitudes and phases of the photometric variations in five wavebands and obtain the best estimates of     for the two visible periodicities. We confirm the earlier determinations that the main periodicity is a radial mode and that the other periodicity is probably         We show that the line profile variations and light variations give consistent results. We point out the importance of a long wavelength range when using the photometric mode identification technique. Finally, we attempt to match the two periods with unstable modes from linear, non-adiabatic calculations. We are able to show that the principal period is well matched by either the fundamental or first overtone radial mode, but could not find a satisfactory fit to the     mode. We discuss implications for mode identification of δ Scuti stars based on what we have learned from this star.     

Confirmation of simultaneous p and g mode excitation in HD 8801 and γ Peg from time-resolved multicolour photometry of six candidate 'hybrid' pulsators

We carried out a multicolour time-series photometric study of six stars claimed as 'hybrid' p and g mode pulsators in the literature. γ Peg was confirmed to show short-period oscillations of the β Cep type and simultaneous long-period pulsations typical of Slowly Pulsating B (SPB) stars. From the measured amplitude ratios in the Strömgren uvy passbands, the stronger of the two short period pulsation modes was identified as radial; the second is  ℓ= 1  . Three of the four SPB-type modes are most likely  ℓ= 1  or 2. Comparison with theoretical model calculations suggests that γ Peg is either a  ∼8.5 M_⊙  radial fundamental mode pulsator or a  ∼9.6 M_⊙  first radial overtone pulsator. HD 8801 was corroborated as a 'hybrid'δ Sct/γ Dor star; four pulsation modes of the γ Dor type were detected, and two modes of the δ Sct type were confirmed. Two pulsational signals between the frequency domains of these two known classes of variables were confirmed and another was newly detected. These are either previously unknown types of pulsation or do not originate from HD 8801. The O-type star HD 13745 showed small-amplitude slow variability on a time-scale of 3.2 d. This object may be related to the suspected new type of supergiant SPB stars, but a rotational origin of its light variations cannot be ruled out at this point. 53 Psc is an SPB star for which two pulsation frequencies were determined and identified with low spherical degree. Small-amplitude variability was formally detected for 53 Ari but is suspected not to be intrinsic. The behaviour of ι Her is consistent with non-variability during our observations, and we could not confirm light variations of the comparison star 34 Psc previously suspected. The use of signal-to-noise criteria in the analysis of data sets with strong aliasing is critically discussed.     

On the frequency and amplitude variations of the δ Scuti star CD−24 7599 (=XX Pyx)

We present 132 h of new time-series photometric observations of the δ Scuti star CD−24 7599 acquired during 86 nights from 1993 to 1996 to study its frequency and amplitude variations. By using all published observations we demonstrate that the three dominating pulsation modes of the star can change their photometric amplitudes within one month at certain times, while the amplitudes can remain constant within the measurement errors at other times. CD−24 7599 also exhibits frequency variations, which do not show any correspondence between the different modes.   The typical time-scale for the amplitude variations is found to be several hundred days, which is of the same order of magnitude as the inverse linear growth rates of a selected model. We find no evidence for periodic amplitude modulation of two of the investigated modes ( f ₂ and f ₃), but f ₁ may exhibit periodic modulation. The latter result could be spurious and requires confirmation. The observed frequency variations may either be continuous or reflect sudden frequency jumps. No evidence for cyclical period changes is obtained.   We exclude precession of the pulsation axis and oblique pulsation for the amplitude variations. Beating of closely spaced frequencies cannot explain the amplitude modulations of two of the modes, while it is possible for the third. Evolutionary effects, binarity, magnetic field changes or avoided crossings cannot be made responsible for the observed period changes. Only resonance between different modes may be able to explain the observations. However, at this stage a quantitative comparison is not possible. More observations, especially data leading to a definite mode identification and further measurements of the temporal behaviour of the amplitudes and frequencies of CD−24 7599, are required.     

Pulsation in the pre-main-sequence Herbig Ae star HD 142666

Asteroseismology of pre-main-sequence δ Scuti stars has the potential not only to provide unprecedented constraints on models of these stars, but also to allow for the possibility of detecting evolutionary period changes, thus providing a direct measure of the pre-main-sequence evolutionary time-scale. In the last two years, the published number of such stars known has doubled from four to eight. Searches are now being conducted amongst the Herbig Ae stars, which are considered to be excellent candidates. We announce the discovery of δ Scuti pulsation in one Herbig Ae star, HD 142666, which lies within Marconi & Palla's theoretically predicted instability strip for pre-main-sequence stars, making this the ninth known pre-main-sequence δ Scuti star. We also demonstrate a lack of δ Scuti pulsation in another such star, HD 142527.     

Discovery of the 'missing' mode in HR 1217 by the Whole Earth Telescope

HR 1217 is a prototypical rapidly oscillating Ap star that has presented a test to the theory of non-radial stellar pulsation. Prior observations showed a clear pattern of five modes with alternating frequency spacings of 33.3 and 34.6 μHz, with a sixth mode at a problematic spacing of 50.0 μHz (which equals  1.5×33.3 μHz)  to the high-frequency side. Asymptotic pulsation theory allowed for a frequency spacing of 34 μHz, but Hipparcos observations rule out such a spacing. Theoretical calculations of magnetoacoustic modes in Ap stars by Cunha predicted that there should be a previously undetected mode 34 μHz higher than the main group, with a smaller spacing between it and the highest one. In this Letter, we present preliminary results from a multisite photometric campaign on the rapidly oscillating Ap star HR 1217 using the 'Whole Earth Telescope'. While a complete analysis of the data will appear in a later paper, one outstanding result from this run is the discovery of a newly detected frequency in the pulsation spectrum of this star, at the frequency predicted by Cunha.     

29 frequencies for the δ Scuti variable BI CMi: the 1997–2000 multisite campaigns

A multisite campaign of BI CMi was carried out with excellent frequency resolution and high photometric accuracy from 1997 to 2000, including two long observing seasons. 29 pulsation frequencies could be extracted from the 1024 h (177 nights) of photometry used. The detected frequencies include 20 pulsation modes in the main pulsation frequency range from 4.8 to 13.0 cycle d⁻¹ (55 to 150 μHz), eight linear combinations of these frequencies, and a very low frequency at 1.66 cycle d⁻¹. Since the value of the low frequency at 1.66 cycle d⁻¹ cannot be identified with a linear combination of other frequencies, g-mode pulsation is suspected, but rotational modulation of abundance spots cannot be ruled out. BI CMi, which is situated near the cool edge of the classical instability strip, may be both a δ Scuti and a γ Doradus star. Another outstanding property of BI CMi is the presence of a number of close frequency pairs in the power spectrum with separations as small as 0.01 cycle d⁻¹.
A rotational velocity of     was determined from a high-dispersion spectrum. From phase differences, the dominant modes can be identified with ℓ values from 0 to 2. The spectral type and evolutionary status of BI CMi are examined.     

Photometric properties of the δ Scuti star BR Cancri

We present the results of a three-year Johnson V and Strömgren uvby H β photometric study of the δ Scuti star BR Cancri (BR Cnc). Our data sets consist of 1293 discrete differential magnitudes in Johnson V and yellow y filters, 883 in Strömgren v and 239 in ub filters. The Fourier analysis of the data suggests four pulsation frequencies for the variable: f ₁=24.978, f ₂=11.358, f ₃=11.808 and f ₄=27.914 cycle d⁻¹. During the three observing years, the main frequency f ₁ kept its V ( y ) amplitude constant at about 6 mmag but its v amplitude seems to be changing. Amplitude variations for all the three other frequencies are also claimed. The pulsation modes of the frequencies are discussed based on the colour data. Using uvbyβ data and calibrations in the literature, we derive the physical parameters for BR Cnc.     

How to drive roAp stars

Rapidly oscillating Ap stars constitute a unique class of pulsators with which to study non-radial oscillations under some — even for stars — unusual physical conditions. These stars are chemically peculiar, they have strong magnetic fields and they often pulsate in several high-order acoustic modes simultaneously. We discuss here an excitation mechanism for short-period oscillation modes based on the classical κ mechanism. We particularly stress the conditions that must be fulfilled for successful driving. Specifically, we discuss the roles of the chemical peculiarity and strong magnetic field on the oscillation modes and what separates these pulsators from δ Scuti and Am-type stars.     

The pulsating DA white dwarf star EC 14012−1446: results from four epochs of time-resolved photometry

The pulsating DA white dwarfs are the coolest degenerate stars that undergo self-driven oscillations. Understanding their interior structure will help us to understand the previous evolution of the star. To this end, we report the analysis of more than 200 h of time-resolved CCD photometry of the pulsating DA white dwarf star EC 14012−1446 acquired during four observing epochs in three different years, including a coordinated three-site campaign. A total of 19 independent frequencies in the star's light variations together with 148 combination signals up to fifth order could be detected. We are unable to obtain the period spacing of the normal modes and therefore a mass estimate of the star, but we infer a fairly short rotation period of  0.61 ±0.03 d  , assuming the rotationally split modes are  ℓ= 1  . The pulsation modes of the star undergo amplitude and frequency variations, in the sense that modes with higher radial overtone show more pronounced variability and that amplitude changes are always accompanied by frequency variations. Most of the second-order combination frequencies detected have amplitudes that are a function of their parent mode amplitudes, but we found a few cases of possible resonantly excited modes. We point out the complications in the analysis and interpretation of data sets of pulsating white dwarfs that are affected by combination frequencies of the form   f _A + f _B − f _C   intruding into the frequency range of the independent modes.     

An attempt to identify the pulsation modes in the δ Scuti star QQ Tel

We present simultaneous UBVRI photometry and high-dispersion spectroscopy of the δ Scuti star QQ Tel. At least seven periodicities are detected in the light curve, but there are likely to be many more. The line profile variations suggest that some of the observed frequencies may be due to modes of moderately high degree  (ℓ≳4)  .     

The rapidly oscillating Ap star HD 99563 and its distorted dipole pulsation mode

We undertook a time-series photometric multisite campaign for the rapidly oscillating Ap (roAp) star HD 99563 and also acquired mean light observations over four seasons. The pulsations of the star, which show flatter light maxima than minima, can be described with a frequency quintuplet centred on 1557.653 μHz and some first harmonics of it. The amplitude of the pulsation is modulated with the rotation period of the star that we determine with 2.91179 ± 0.00007 d from the analysis of the stellar pulsation spectrum and of the mean light data. We break up the distorted oscillation mode into its pure spherical harmonic components and find it is dominated by the ℓ= 1 pulsation, and also has a notable ℓ= 3 contribution, with weak ℓ= 0 and 2 components. The geometrical configuration of the star allows us to see both pulsation poles for about the same amount of time; HD 99563 is only the fourth roAp star for which both pulsation poles are seen and only the third where the distortion of the pulsation modes has been modelled. We point out that HD 99563 is very similar to the well-studied roAp star HR 3831. Finally, we note that the visual companion of HD 99563 is located in the δ Scuti instability strip and may thus show pulsation. We show that if the companion was physical, the roAp star would be a 2.03-M_⊙, object, seen at a rotational inclination of 44°, which then predicts a magnetic obliquity     .     

Photometric studies of three multiperiodic β Cephei stars: β CMa, 15 CMa and KZ Mus

We have carried out single and multisite photometry of the three β Cephei stars β and 15 CMa as well as KZ Mus. For the two stars in CMa, we obtained 270 h of measurement in the Strömgren uvy and Johnson V filters, while 150 h of time-resolved Strömgren uvy photometry was acquired for KZ Mus. All three stars are multiperiodic variables, with three (β CMa) and four (15 CMa, KZ Mus) independent pulsation modes. Two of the mode frequencies of 15 CMa are new discoveries and one of the known modes showed amplitude variations over the last 33 yr. Taken together, this fully explains the diverse behaviour of the star reported in the literature.
Mode identification by means of the amplitude ratios in the different passbands suggests one radial mode for each star. In addition, β CMa has a dominant  ℓ= 2  mode while its third mode is non-radial with unknown ℓ. The non-radial modes of 15 CMa, which are  ℓ≤ 3  , form an almost equally split triplet that, if physical, would imply that we see the star under an inclination angle larger than 55°. The strongest non-radial mode of KZ Mus is  ℓ= 2  , followed by the radial mode and a dipole mode. Its weakest known mode is non-radial with unknown ℓ, confirming previous mode identifications for the pulsations of the star.
The phased light curve for the strongest mode of 15 CMa has a descending branch steeper than the rising branch. A stillstand phenomenon during the rise to maximum light is indicated. Given the low photometric amplitude of this non-radial mode this is at first sight surprising, but it can be explained by the aspect angle of the mode.     

On the moving subfeatures in the Be star ζ Oph

We analyse a series of line profile observations of the He  i 6678 line in ζ Oph. A period analysis on these data using the mode and moments of the line profile confirms the two previously known periods. We describe a new method of mode identification for pulsating stars in which the calculated profiles are directly fitted to observed profiles. The method yields the full set of pulsational parameters including the spherical harmonic degree, ℓ, and azimuthal number, m . Application of the method to these data confirms the mode identifications previously suggested for the two periodicities. We find that the derived pulsational parameters are physically realistic and conclude that non-radial pulsation is the most likely explanation for the travelling subfeatures. However, a unique mode identification is still not possible – several non-sectorial modes fit the data as well as the usually adopted sectorial identifications. The predicted photometric amplitudes are in good accord with upper limits derived from photometric observations. We conclude that ζ Oph is a star in the β Cep instability strip in which two modes of high degree (probably ℓ=4 and ℓ=8) are excited. We present an interpretation of these findings in which the cause of the low-order line profile and light variations in periodic Be stars is corotating photospheric clouds, while the travelling subfeatures are incidental to the Be phenomenon and are a result of non-radial pulsation.     

Abundance analysis of the γ Doradus–δ Scuti hybrid metallic line (Am) star HD 8801

Low frequency oscillation, typical for γ Doradus g‐mode type stellar core sensitive pulsation, as well as higher frequency δ Scuti type pulsation typical for p ‐modes, sensitive to the envelope, make HD 8801 a remarkable hybrid pulsator with the potential to probe a stellar structure over a wide range of radius. In addition HD 8801 is a rare pulsating metallic line (Am) star. We determined the astro‐physical fundamental parameters to locate HD 8801 in the H‐R diagram. We analyzed the element abundances, paying close attention to the errors involved, and confirm the nature of HD 8801 as a metallic line (Am) star. We also determined an upper limit on the magnetic field strength. Our abundance analysis is based on classical techniques, but uses for the final step a model atmosphere calculated with the abundances determined by us. We also discuss spectropolarimetric observations obtained for HD 8801. This object is remarkable in several respects. It is a nonmagnetic metallic line (Am) star, pulsating simultaneously in p‐ and g‐modes, but also shows oscillations with periods in between these two domains, whose excitation requires explanation. Overall, the pulsational incidence in unevolved classical Am stars is believed to be quite low; HD 8801 does not conform to this picture. Finally, about 75 % of Am stars are located in short‐period binaries, but there is no evidence that HD 8801 has a companion. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An asteroseismic study of the β Cephei star θ Ophiuchi: photometric results

We have carried out a three-site photometric campaign for the β Cephei star θ Oph from 2003 April to August. 245 h of differential photoelectric u v y photometry were obtained during 77 clear nights. The frequency analysis of our measurements has resulted in the detection of seven pulsation modes within a narrow frequency interval between 7.116 and 7.973 c d⁻¹. No combination or harmonic frequencies have been found. We have performed a mode identification of the individual pulsations from our colour photometry that shows the presence of one radial mode, one rotationally split  ℓ= 1  triplet and possibly three components of a rotationally split  ℓ= 2  quintuplet. We discuss the implications of our findings and point out the similarity of the pulsation spectrum of θ Oph to that of another β Cephei star, V836 Cen.