Kepler observations of a roAp star: Scuti and Doradus pulsations in Ap stars
Observations of the A5p star KIC 8677585 obtained during the Kepler 10-d commissioning run with 1-min time resolution show that it is a roAp star with several frequencies with periods near 10 min. In addition, a low frequency at 3.142 d is also clearly present. Multiperiodic Dor and Sct pulsations, never before seen in any Ap star, are present in Kepler observations of at least three other Ap stars. Since Dor pulsations are seen in Ap stars, it is likely that the low-frequency in KIC 8677585 is also a Dor pulsation. The simultaneous presence of both Dor and roAp pulsations and the unexpected detection of Sct and Dor pulsations in Ap stars present new opportunities and challenges for the interpretation of these stars.
keywords:Stars: oscillations – stars: variables – stars: individual (KIC 8677585; BD+44 3063) – stars: magnetic.
The Kepler Mission is designed to detect Earth-like planets around solar- type stars (Koch et al., 2010). To achieve that goal, Kepler will continuously monitor the brightness of over 150 000 stars for at least 3.5 yr in a 105 square degree fixed field of view. Photometric results from the 10-d commissioning run show that mag precision in amplitude can be attained for the brighter stars (8–10 mag) for long-cadence (29.4-min) exposures. With this level of precision interesting pulsational behaviour never seen before is being found in many stars (Gilliland et al., 2010). In addition, Kepler has a small allocation for short-cadence (1-min) exposures. In this mode it is possible to detect and study the light variations of short-period pulsating stars such as solar-like pulsators, Sct stars, and rapidly-oscillating Ap stars (roAp stars).
The roAp stars, discovered by Kurtz (1982), are found amongst the coolest subgroup of Ap stars, namely, the SrCrEu group (6 400–10 000 K). About 40 roAp stars are known at present, with temperatures in the range K, and exhibiting either single or multiperiodic pulsations with periods in the range min. These oscillations are interpreted as acoustic modes of low degree and high radial order (typically 15), which are modified at the surface layers by strong, large-scale magnetic fields. In that respect they are quite different from the Sct stars which, having similar mass and effective temperature, tend to have oscillations with radial order not exceeding 4 or 5. The difference in the radial orders of the oscillations found in these two classes of pulsators is thought to result from the difference in the regions where the modes are excited. While in roAp stars the oscillations are believed to be excited in the region of hydrogen ionization (Balmforth et al., 2001; Cunha, 2002; Saio, 2005; Théado et al., 2009), which is partially or fully stabilized against convection by the presence of strong magnetic fields, in Sct stars the excitation takes place in the region of second helium ionization.
Despite the general understanding of the driving of oscillations in roAp stars, there are still a number of puzzling questions to be answered. Whereas models of hot roAp stars show that driving of pulsations is possible when convection is suppressed, this is not the case for cooler roAp stars. For example, Saio et al. (2010) find that all roAp-like pulsations are stable in magnetic models of HD 24712 (HR 1217; DO Eri) in which convection is suppressed. A second puzzling aspect is that apparently non-oscillating Ap stars called “noAp stars” occupy a similar part of the HR Diagram as the roAp stars. At present it is not clear what determines which of these peculiar magnetic stars show high radial overtone p-mode pulsation, and which are apparently non-variable. This may either be the result of a selective driving mechanism or simply an observational bias. It is hoped that the Kepler Mission, with its ability to detect much lower amplitude roAp stars than has previously been possible, will illuminate these problems.
The global magnetic field which is present in all Ap stars plays an essential role in roAp oscillations, influencing their geometry, frequencies, and energy balance. The presence in some roAp stars of frequency multiplets with spacings exactly equal to the angular frequency of rotation led Kurtz (1982) to introduce the oblique pulsator model, according to which the observed pulsations are axisymmetric about the magnetic axis, which is tilted with respect to the rotational axis. This simple picture is challenged by the fact that in some stars the peaks lying symmetrically about the central peak in the multiplet do not have identical amplitudes. This could potentially be explained by the combined effect of rotation and magnetic field on the pulsations, which may break the alignment between the pulsation and magnetic axis (Bigot & Dziembowski, 2002; Gough, 2005). However, that requires the magnetic and centrifugal effects on the oscillation frequencies to be comparable, something that is not expected except, possibly, for particular combinations of magnetic field strength and oscillation frequencies (Cunha, 2007).
Most roAp stars do not show the rotational multiplets predicted by the oblique pulsator model either because the rotational period is too long (and the multiplets unresolved) or because they have amplitudes below the threshold of detectability. Many do, however, show multiple frequencies which in some cases can be interpreted in terms of modes of alternating spherical harmonic degree, , and of consecutive radial orders, .
In the case of linear, adiabatic oscillations, in a spherically symmetric star, at high radial order the frequencies of modes of the same degree are repeated at approximately regular intervals (Tassoul, 1980). This interval, defined as the frequency difference between modes of the same but successive radial orders , , is called the large separation, and is a measure of the mean density of the star. Moreover, in the same asymptotic limit of high radial orders, if modes of alternating even and odd are present, then the peaks in the frequency spectrum are separated by , while if only modes of the same are present, the separation between consecutive peaks is .
In some roAp stars a pattern of nearly equally spaced frequencies is indeed observed. If these are interpreted in accordance with the asymptotic relation derived by Tassoul (1980), the average large separation allows the stellar parameters to be constrained. In practice, the presence of a magnetic field breaks the spherical symmetry of the pulsating star, and perturbs the oscillation frequencies away from the values predicted by this asymptotic relation. This has been shown theoretically in a number of studies that considered oscillations in the presence of a large scale magnetic field (Cunha & Gough, 2000; Saio & Gautschy, 2004; Saio, 2005; Cunha, 2006). According to these works, a pattern of nearly equally spaced peaks may still be found in particular sections of the frequency spectrum, but it is also clear that the magnetic field can distort the regular frequency pattern. This problem can be resolved by matching of observed frequencies with frequencies calculated from models which include the effect of a magnetic field (Saio, 2005).
The use of the asymptotic relationship or direct frequency matching requires that we associate the observed frequencies with the correct value of . In roAp stars which show rotational modulation, the value of may be inferred from the number and relative amplitudes of the rotational multiplets. However, application of the method to HD 24712 gives conflicting results (Saio et al., 2010). On the other hand, an alternative mode identification cannot reproduce the observed rotational multiplet structure in this star.
It is quite clear that we are far from a complete understanding of pulsations in roAp stars, even though impressive advances have been made both observationally and theoretically over the last few years. The varied properties of roAp pulsations present a wide range of interesting challenges. Each roAp star seems to have its own peculiar characteristics. There were no previously known roAp stars in the Kepler field of view, though several Ap stars were known prior to the satellite’s launch. It was therefore of great interest to discover that one of these stars pulsates in several frequencies in the roAp star range. This star, KIC 8677585 (BD+44 3063; ILF1+44 20; JD2000 position: 19:06:28, +44:50:33, ) is classified as A5p (Macrae, 1952). In this paper we present a frequency analysis of this new roAp star and investigate its relationship with other roAp and the Sct and Dor variables.
The Kepler data for KIC 8677585 consist of 14 264 nearly continuous photometric data points taken during the time interval (truncated Barycentric Julian date) BJD 54953.53–54963.25 (9.73 d) with a mean sampling time of 59.06 s (the exposure time is 58.85 s). The “uncorrected” data used here consist of aperture photometry in which a few points have been flagged as bad for one reason or another and a few more deviate significantly from the mean. Removing these points leaves 14 222 data points that can be analysed. KIC 8677585 has a very small contamination factor of less than 3 percent, so the probability of any of the pulsation signal measured coming from a nearby star is low.
The only ground-based photometric observations available prior to Kepler launch for KIC 8677585 were five-colour photometry using Sloan filters from the Kepler Input Catalogue (KIC). The KIC also contains a number of derived quantities including the stellar radius, , the effective temperature, , and the surface gravity . There is no guarantee that these values are appropriate, especially for Ap stars. For KIC 8677585 these are R, K and , values that are reasonable for a cool Ap star. From these values a luminosity of can be estimated. Frequently, estimates of fundamental parameters for Ap stars from photometry are poor because of the strong line blanketing caused by overabundances of rare earth elements by orders of magnitude compared to normal stars. Nevertheless, for KIC 8677585 spectroscopic estimates of and are in good agreement with those of the KIC.
2.1 Stellar parameters of KIC 8677585
One high resolution spectrum of the star was obtained using HERMES (High Efficiency and Resolution Mercator Echelle Spectrograph, Raskin & Van Winckel (2008)) installed at 1.2-m Mercator telescope at the Roque de los Muchachos Observatory on La Palma, Canary Islands (http://www.mercator.iac.es). The resolving power of HERMES is about 85 000. The reduction was done with software developed for this instrument.
From the magnetically null Fe i line at 5434.523 Å we measured the projected rotational velocity to be km s. Lines of Nd iii, Pr iii and Eu ii are very strong, as is typical for the roAp stars. The equivalent width of the interstellar Na i D1 line (39 mÅ) and an empirical calibration by Munari & Zwitter (1997) shows that the colour excess for the star is less than .
The Balmer lines profiles are good indicators of effective temperature for roAp stars. We have compared observed and synthetic profiles of the H and H lines. The synthetic calculations of Balmer line profiles were done using the SYNTH code by Piskunov (1992) with model atmospheres from the NEMO grid (Heiter et al., 2002). By fitting the Balmer profiles we derived K. The Balmer line profiles are not very sensitive to at this temperature. We estimated from the ionisation equilibrium of Fe i and Fe ii lines and Cr i and Cr ii lines, taking into account the indication from photometry.
2.2 Magnetic field
The mean magnetic field modulus can be detected in Ap stars from high resolution spectra using spectral lines with resolved Zeeman components. The line of Fe ii at 6149.24 Å is commonly used for this in cool Ap stars. In KIC 8677585 it shows partial Zeeman splitting. To obtain the mean magnetic field modulus we calculated synthetic spectra with the SYNTHMAG code of Piskunov (1999) for a range of abundances and magnetic field strengths. The spectral line list was taken from the Vienna Atomic Line Database (VALD, Kupka et al. 1999), which includes lines of rare earth elements from the DREAM database (Biémont et al., 1999). The synthetic spectra were then compared with the observations for the best match which yields an estimate of the magnetic field modulus of kG from line of Fe ii 6149.24 Å. We note that this line in KIC 8677585 is blended in the blue wing, probably with the line of Sm ii 6149.063 Å.
3 Position in the HR Diagram
In Fig. 1 we show the location of KIC 8677585 in the theoretical HR diagram and compare it with other roAp stars and with noAp stars for which temperatures and luminosities are available in the literature. We have used the spectroscopic temperature and the which is the luminosity from the KIC parameters and assuming a standard deviation of 0.5 mag for the absolute magnitude.
KIC 8677585 appears to be on the zero-age main sequence with an effective temperature close to the mean value for the roAp stars. Further observations are desirable to refine the effective temperature and luminosity.
4 Frequency analysis
A periodogram of the light curve shows that KIC 8677585 varies in two quite distinct frequency regions: a low-frequency and a high-frequency domain. In the low-frequency domain the only periodicity is d ( Hz) with an amplitude of mag. The high-frequency domain contains at least seven components in the range 1450– 1680 Hz. Outside these two regions, the background noise level in the periodogram is approximately 5 mag. The significant frequencies are listed in Table 1. In the table we display the false alarm probability (FAP) (Scargle, 1982), showing that there is some doubt about the reality of and especially . It should be noted that that the FAP is only to be taken as a guideline. Different approaches to calculate it yield different probabilities, which need to be interpreted in the context of their underlying definitions and assumptions. We consider to be almost certainly of instrumental origin since Kepler data have not yet been corrected for long-term drift.
The periodograms in the two regions are shown in Fig. 2. In this figure the region around twice the frequency of the main peaks is plotted to show the probable detection of the first harmonic of and . Harmonics are quite common in the light curves of roAp stars, but it seems extraordinary that they should be present in the extremely low amplitude variations seen here.
Inspection of Fig. 2 shows a certain regularity in the frequency spacing of the peaks. One can quantify this by calculating the autocorrelation function (ACF) of the spectral significance (Reegen, 2007) within a certain frequency range. Fig. 3 shows the absolute value of the ACF as a function of frequency spacing for the frequency range 1390–1850 Hz. There is no change when restricting the frequency range even further, but the ACF obviously gets worse as the range is expanded. From the figure, the highest peak occurs at a frequency lag of 38.1 Hz with smaller peaks at 72.3, 73.9, 16.2 and 54.2 Hz. From these values the large separation most likely is in the ranges [36–39] Hz or [72–76] Hz.
5 Modelling of KIC 8677585
Despite the limitations posed by the lack of reliable stellar parameters, the frequencies found in the high-frequency domain do provide sufficient information for a first modelling of this star.
As mentioned in the introduction, among other effects, the magnetic field modifies the oscillation frequencies. A typical oscillation spectrum of a multiperiodic roAp star may still show a pattern of frequencies that resembles that found in a non-magnetic star, but often some of the frequencies are displaced in relation to the non-magnetic positions. Also, the large separations are usually slightly enlarged (by a few Hz) in relation to their non-magnetic counterparts.
When the effect of the magnetic field is not taken into account in the models, matching of observed and calculated frequency differences are certainly to be preferred to matching of individual frequencies themselves, particularly when a regular pattern is found. With this in mind, we will use the effective temperature and derived from the spectrum of the star, i.e, K and , and the two possible intervals for the observed large separation, namely [36–39] Hz and [72–76] Hz, to constrain the models.
We considered a grid of possible equilibrium models for the star, obtained using the ASTEC stellar evolution code (Christensen-Dalsgaard, 2008b). We calculated evolutionary sequences of models for masses in the range 1.3–2.1 M, in steps of 0.2 M, setting the initial relative abundance of metals at the surface to and the helium abundance to . Moreover, we considered no core overshooting and a fixed value for the mixing length parameter of . The intention was not to do a detailed modelling of the star, varying all possible parameters in a fine grid. That we hope to perform later, when significantly longer time-series of Kepler data for this star become available. For the moment we are mostly interested is finding how the choice of the interval for the large separation influences the basic properties of the star.
In Fig. 4 we show the evolutionary sequences considered here in the and diagrams. The position of KIC 8677585, along with the 1 error box is also shown in the former. For models within the error box, we then computed the oscillation frequencies using the ADIPLS code (Christensen-Dalsgaard, 2008a) and, from these, derived the corresponding large separations.
Fig. 5 shows the diagram along with the models whose large separations were found to be within either of the two intervals considered.
In Fig. 4 the position of these models in the HR diagram is shown, along with corresponding evolutionary tracks.
|Model 1||Model 2|
The oscillation frequencies of the two best models of our grid are shown in the upper and middle panels of Fig. 6. A schematic amplitude spectrum of KIC 8677585 in the high-frequency domain, with an indication of the frequency separations between different peaks, is shown for comparison in the lowest panel. The best models correspond to those that minimize a function taking into account both the non-seismic data and the large separation. For the latter we considered the two intervals determined from the observations, hence the existence of two best models. The properties of these two models are presented in Table 2.
From the figure one could naively say that models with the smaller option for the large separation reproduce best the seismic data. However, that is a dangerous statement to make for a number of reasons: First, the comparison of individual frequencies, between models and observations, is known to be risky due to possible systematic effects that may result from the incorrect modelling of the surface layers of stars; secondly, no detailed exploitation of the parameter space was considered in the modelling, i.e, the step considered in stellar mass was large and no variation of chemical composition or convective overshoot was considered in the present grid; thirdly, the effect of the magnetic field on the oscillations was not taken into account.
An example of the importance of taking these effect into account is shown in the middle panel of Fig. 6 where the dashed lines show the model frequencies obtained including the effect of a dipole magnetic field with a polar strength of kG. Those frequencies have been obtained by the method of non-adiabatic analysis described in Saio (2005), where the oscillations are assumed to be axisymmetric to the magnetic axis. The latitudinal dependence of an axisymmetric oscillation mode is represented by a sum of components proportional to spherical harmonics . The latitudinal degree of each mode is identified by which is the value for the component having a largest kinetic energy. The parameters of unperturbed model (shown in Fig. 7) are similar to those of model 2 in Table 2, but a slightly larger mass of is adopted because a composition of is used.
Although the large separation for a given latitudinal degree hardly changes, the relative frequencies between different degrees are affected by the presence of a magnetic field. This comes from the fact that modes of different degree are affected differently by the magnetic field, as seen in Fig. 7 where oscillation frequencies are shown as a function of . In this model, calculated frequencies agree with observed ones at kG and kG as indicated by vertical lines in Fig. 7. Model frequencies are similar at the two different values of because the magnetic effect changes cyclically as first found by Cunha & Gough (2000). The former value of is consistent with the mean modulus kG from the spectroscopic analysis (§ 2.2). We note that, in addition to the frequencies listed in Table 1, the periodogram (Fig. 2) shows minor peaks at mHz and mHz (and possibly at mHz) which correspond to frequencies of modes. A future long-term observation will clarify the reality of these frequencies. Then, we will be able to better constrain the stellar parameters, and to examine the theory for the oscillations of magnetic stars.
Although all the frequencies shown in Fig. 7 are below the critical acoustic frequency, no high-frequency modes in this model are excited. (We note that for low-mass models () high-frequency modes in the observed frequency range are excited by the kappa-mechanism in the hydrogen ionization zone.) Since the stability of high-order p modes seems sensitive to the treatment of the optically thin layers, we need further investigation on the excitation of the high-order p modes.
The frequency of the pulsations seen in KIC 8677585 are typical of roAp stars, but the amplitudes are an order of magnitude below any that have been detected from ground-based observations. The outstanding feature of this star, however, is the presence of the low frequency at d which has never been seen in roAp stars. A simple calculation shows that the critical rotational frequency for roAp stars is typically in the range 2 - 5 d. If is due to rotation, the star must be rotating at close to the critical rate. All known roAp stars rotate slowly, and the measured km s for this star effectively rules out the possibility that could be the rotation frequency. A secondary body orbiting just above the photosphere could have an orbital frequency with this value. However, the light curve at is a pure sinusoid, not an eclipse, but it could conceivably be a tidal distortion.
The limiting factor for observations of faint sources is set by source confusion, rather than the photometric accuracy computed for isolated sources. An estimate of the crowding metric calculated from the point spread function of surrounding objects is provided for most sources in the KIC. For KIC 8677585 the contamination coefficient is 0.029. It is therefore not impossible that may be due to the small contribution from a (faint) neighbouring star. To eliminate this possibility requires extensive photometry with a very small aperture with a precision comparable to that of Kepler. For to arise in the much fainter contaminating star would require that this be the only frequency of variation of that star, which again is unlikely. A spectrum of the speculative faint neighbour is required for better understanding of the probability.
|8677585||ILF1+44 20||A5p||10.27||roAp/ Dor|
|8750029||ILF1+44 257||A5p?||9.66||Sct/ Dor|
|9020199||HD 182895||F0p||8.86||Sct/ Dor|
In general, low-frequency pulsations in Ap stars are thought to be unlikely. The settling and diffusion which give rise to the chemical inhomogeneities in these stars drain out He from the He ii driving zone, rendering them pulsationally stable. In addition, damping of pulsations is expected in these stars due to magnetic slow wave leakage if the magnetic field strength is greater than about 1 kG. Prior to the Kepler observations, the only Ap star known to pulsate in low frequencies is HD 21190 (F2III SrEuSi). In this star there is at least one mode with a frequency of 6.68 d which is probably a radial mode (Koen et al., 2001). The star is multiperiodic, but other frequencies could not be extracted from the photometric data. High-time resolution spectroscopic observations of HD 21190 show moving bumps in the cores of spectral lines, indicating the presence of high-degree nonradial pulsations (González et al., 2008). HD 21190 is the most evolved Ap star known; its unique stage of evolution may offer clues as to why it pulsates.
Kepler observations of other Ap stars indicate that pulsations of low radial order may be more common than previously thought. Table 3 lists information for stars in the Kepler field of view which have been classified as Ap. However, these classifications have never been confirmed and they may turn out to be in error. Ap and Am classifications are easily confused at classification dispersions. In at least three of these stars Sct pulsations are clearly present. In two of the stars, multiperiodic Dor pulsations are also visible (Fig. 8).
The clear presence of Dor pulsations in two presumed Ap stars is most surprising. The multiperiodicity renders any explanation in terms of orbital or tidal interaction invalid. This has a bearing on the mode in KIC 8677585. Clearly if the presumed Ap stars with low-frequency modes are indeed Ap stars, it means that the low-frequency mode in KIC 8677585 is not unique to the star and requires an explanation in the context of Ap stars as a whole rather than specific to this star.
As already mentioned, models suggest that pulsations in Ap stars are not excited because of draining of He from the driving zone and damping by magnetic slow wave leakage. The presence of a low frequency in KIC 8677585 and other Ap stars contradicts these ideas. Confirmation of the classifications of the stars in Table 3 are, however, required to cement this conclusion.
The authors wish to thank the Kepler team for their generousity in allowing the data to be released to the Kepler Asteroseismic Science Consortium (KASC) ahead of public release and for their outstanding efforts which have made these results possible. Funding for the Kepler mission is provided by NASA’s Science Mission Directorate. We particularly thank Ron Gilliland and Hans Kjeldsen for their tireless work on behalf of KASC.
LAB wishes to thank the South African Astronomical Observatory for financial support.
This work was partially supported by the project PTDC/CTE-AST/098754/2008 and the grant SFRH / BD / 41213 / 2007 funded by FCT/MCTES, Portugal. MC is supported by a Ciência 2007 contract, funded by FCT/MCTES(Portugal) and POPH/FSE (EC).
MG has received financial support through D. Guenther’s NSERC grant.
- pagerange: Kepler observations of a roAp star: Scuti and Doradus pulsations in Ap stars–References
- pubyear: 2010
- Balmforth N. J., Cunha M. S., Dolez N., Gough D. O., Vauclair S., 2001, MNRAS, 323, 362
- Biémont E., Palmeri P., Quinet P., 1999, Ap&SS, 269, 635
- Bigot L., Dziembowski W. A., 2002, A&A, 391, 235
- Christensen-Dalsgaard J., 2008a, Ap&SS, 316, 113
- —, 2008b, Ap&SS, 316, 13
- Cunha M. S., 2002, MNRAS, 333, 47
- —, 2006, MNRAS, 365, 153
- —, 2007, Communications in Asteroseismology, 150, 48
- Cunha M. S., Gough D., 2000, MNRAS, 319, 1020
- Gilliland R. L., Brown T. M., Christensen-Dalsgaard J., Kjeldsen H., Aerts C., Appourchaux T., Basu S., Bedding T. R., Chaplin W. J., Cunha M. S., De Cat P., De Ridder J., Guzik J. A., Handler G., Kawaler S., Kiss L., Kolenberg K., Kurtz D. W., Metcalfe T. S., Monteiro M. J. P. F. G., Szabó R., Arentoft T., Balona L., Debosscher J., Elsworth Y. P., Quirion P., Stello D., Suárez J. C., Borucki W. J., Jenkins J. M., Koch D., Kondo Y., Latham D. W., Rowe J. F., Steffen J. H., 2010, PASP, 122, 131
- González J. F., Hubrig S., Kurtz D. W., Elkin V., Savanov I., 2008, MNRAS, 384, 1140
- Gough D., 2005, The Roger Tayler Memorial Lectures 1999-2003: Special issue of Astronomy and Geophysics, 16
- Heiter U., Kupka F., van’t Veer-Menneret C., Barban C., Weiss W. W., Goupil M., Schmidt W., Katz D., Garrido R., 2002, A&A, 392, 619
- Koch D. G., Borucki W. J., Basri G., Batalha N. M., Brown T. M., Caldwell D., Christensen-Dalsgaard J., Cochran W. D., DeVore E., Dunham E. W., Gautier III T. N., Geary J. C., Gilliland R. L., Gould A., Jenkins J., Kondo Y., Latham D. W., Lissauer J. J., Marcy G., Monet D., Sasselov D., Boss A., Brownlee D., Caldwell J., Dupree A. K., Howell S. B., Kjeldsen H., Meibom S., Morrison D., Owen T., Reitsema H., Tarter J., Bryson S. T., Dotson J. L., Gazis P., Haas M. R., Kolodziejczak J., Rowe J. F., Van Cleve J. E., Allen C., Chandrasekaran H., Clarke B. D., Li J., Quintana E. V., Tenenbaum P., Twicken J. D., Wu H., 2010, ArXiv e-prints
- Koen C., Kurtz D. W., Gray R. O., Kilkenny D., Handler G., Van Wyk F., Marang F., Winkler H., 2001, MNRAS, 326, 387
- Kupka F., Piskunov N., Ryabchikova T. A., Stempels H. C., Weiss W. W., 1999, A&AS, 138, 119
- Kurtz D. W., 1982, MNRAS, 200, 807
- Macrae D. A., 1952, ApJ, 116, 592
- Munari U., Zwitter T., 1997, A&A, 318, 269
- Piskunov N., 1999, in Astrophysics and Space Science Library, Vol. 243, Polarization, K. N. Nagendra & J. O. Stenflo, ed., pp. 515–525
- Piskunov N. E., 1992, in Stellar Magnetism, Y. V. Glagolevskij & I. I. Romanyuk, ed., p. 92
- Raskin G., Van Winckel H., 2008, in Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, Vol. 7014, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series
- Reegen P., 2007, A&A, 467, 1353
- Saio H., 2005, MNRAS, 360, 1022
- Saio H., Gautschy A., 2004, MNRAS, 350, 485
- Saio H., Ryabchikova T., Sachkov M., 2010, MNRAS, 403, 1729
- Scargle J. D., 1982, ApJ, 263, 835
- Tassoul M., 1980, ApJS, 43, 469
- Théado S., Dupret M., Noels A., Ferguson J. W., 2009, A&A, 493, 159