The discovery of a frequency quintuplet and distorted dipole mode in the rapidly oscillating Ap star HD 6532

Transcription

1 Mon. Not. R. Astron. Soc. 281, (1996) The discovery of a frequency quintuplet and distorted dipole mode in the rapidly oscillating Ap star HD 6532 D. W. Kurtz/ Peter Martinez/* C. Koen 2 and D. J. Sullivan 3 'Department of Astronomy, University of Cape Town, Rondebosch 7700, South Africa 'South African Astronomical Observatory, PO Box 9, Observatory 7935, South Africa 3Department of Physics, Victoria University of Wellington, Wellington, New Zealand Accepted 1996 February 27. Received 1996 Feburary 21; in original form 1995 December 5 1 INTRODUCTION The peculiar magnetic Ap and Bp stars have many fascinating properties: most are spectrum, light and magnetic variables; all show complex peculiarities of atmospheric structure, elemental abundances, and non-uniform surface distribution of elements; and some are high-overtone p-mode pulsators. From the synchronism of the spectrum, light, pulsation and magnetic variations with rotation, it is clear that the magnetic field governs or constrains many of the physical processes in these stars, including convection, turbulence, meridional circulation, elemental diffusion, the orientation of the pulsation axis and selection of the pulsation mode - in both overtone and degree. Thus, determining the surface configuration of the magnetic field is of primary importance for understanding these other physical phenomena. Observed surface magnetic fields in Ap and Bp stars are basically dipolar with the magnetic axis inclined to the * Visiting Astronomer, Cerro Tololo Inter-American Observatory, operated by the Associated Universities for Research in Astronomy, Inc., under contract with the National Science Foundation RAS ABSTRACT We have analysed 77 h of new, multisite observations of HD 6532 obtained in 1994, along with 20 h of observations from 1984 and 90 h from A frequency quintuplet with frequency separations equal to exactly the rotation frequency, and two harmonic frequencies, give a complete solution. The amplitude spectrum of the residuals shows white noise with its highest peaks of amplitude 0.1 mmag. We interpret the frequency quintuplet in terms of a distorted oblique dipole mode, which we model with a spherical harmonic series that is primarily dipolar with small radial and quadrupole contributions. This model does not fully describe the rotational modulation of pulsation amplitude and phase, which leads us to suggest that a completely decentred dipole model needs development. HD 6532 is only the second roap star, after HR 3831, with a polarity-reversing pulsation mode which has been studied in sufficient detail to characterize the geometry of the mode. Key words: stars: individual: HD stars: oscillations - stars: peculiar - stars: rotation - stars: variables: other. rotation axis. Field strengths are typically hundreds of G, frequently a few kg, and occasionally a few tens of kg. These fields are simplistically modelled with oblique centred dipoles. When observations of sufficient precision are available, however, it is always found that one magnetic pole is stronger than the other, and a simple dipole description is not adequate. This more complex field structure can be described by a dipole with its centre linearly displaced along the magnetic axis, or, equivalently, by a spherical harmonic series of oblique axisymmetric centred multipoles. Usually a dipole plus quadrupole is sufficient, but in a few interesting cases a strong octupole is also required, e.g. the Bp Si star HD (Landstreet 1990) and the He-strong B star HD (Thompson & Landstreet 1985). Early photographic magnetic measurements often showed strongly non-sinusoidal rotational variations which could not be completely modelled by a linearly decentred dipole. This led Stift (1975) to develop a generalized decentred dipole model (to describe magnetic fields in Ap stars analogous to those seen in, for example, Uranus and Neptune). A completely decentred dipole field can be equivalently modelled with a centred spherical harmonic series, although, in this case, non-axisymmetric components are needed.

2 884 D. W Kurtz et al. More recent photoelectric magnetic measurements are much more nearly sinusoidal in their rotational variation (Borra & Landstreet 1980). Usually a centred dipole plus quadrupole provides an adequate model for them, although, as mentioned above, sometimes a significant octupole component is needed. However, it is clear that when the highest precision measurements are made, non-axisymmetric components are needed (Mathys 1993). Excellent discussions of magnetic field measurements in general, and of the observed magnetic fields in Ap and Bp stars, are given by Mathys (1989) and Landstreet (1992, 1993). The spherical harmonic degrees of the pulsation modes in the rapidly rotating Ap (roap) stars are governed by their magnetic fields, as probably is the selection of their overtones. Most of the identified pulsation modes are oblique dipoles with the pulsation and magnetic axes coinciding (see Martinez 1993, Matthews 1991 and Kurtz 1990a for general reviews of the roap stars). Because the pulsation axis is oblique to the rotation axis, the pulsation mode is seen from varying aspect with the rotation period of the star. This allows the derivation of the rotation period, and puts constraints on the rotational inclination i, and the magnetic obliquity, p. In fact, studies of oblique pulsation modes derive essentially the same information as studies of the effective magnetic field. A reasonable hypothesis is that the pulsation mode and the magnetic field geometries are very similar. This can be tested by further high-precision studies of both, studies which are complementary. Newer spectroscopic magnetic observations, particularly using line profiles (Mathys 1995 and references therein), contain much more information than broad-band photometric observations of the integrated pulsational light variations, but the photometry is orders of magnitude more precise and can, therefore, be used to derive some parameters much more accurately. Although about two-thirds of the magnetic Ap and Bp stars have rotational inclinations, i, and magnetic obliquities, p, such that both magnetic poles are seen over a rotation period (i + p > 90 ), only four of the 28 known roap stars show pulsational polarity reversal. One reason for this low fraction is that some of the roap stars have exceedingly long rotation periods. The case of y Equ is wellknown; its rotation period is of the order of 77 yr (Leroy et al. 1994). From magnetic field studies Mathys (private communication - in preparation) finds three other roap stars to have long rotation periods. They are HD (33 Lib - P> 75 yr), HD (P> 1 yr) and HD (P> 3.2 yr). All four of these long rotation period roap stars should show pulsation amplitude modulation with their rotation periods. No test has yet been made for this. The four roap stars which are known to show polarity reversal are: HD 6532 (Kurtz & Cropper 1987), HD (Kurtz 1990b ), HR 3831 (Kurtz et al. 1994a; Kurtz, Kanaan & Martinez 1993, hereafter KKM) and HD (Matthews, Kurtz & Wehlau 1987). For HD the second pole is seen from very poor aspect (Kurtz, van Wyk & Marang 1990) and the numerous modes have lifetimes of only a few days, making a study of the rotational variation impractical. HD has a very small amplitude and is poorly studied. HD 6532 is the subject of this paper. Only HR 3831 is previously well-studied. It is largely observations of and frequency analyses of HR 3831 which have driven theoretical studies of oblique pulsation. Simple oblique dipole pulsation gives rise to a frequency triplet (Kurtz 1982) split by the rotation frequency with amplitude symmetry about the central frequency. The fact that the rotational sidelobes of the frequency triplet in HR 3831 do not have this amplitude symmetry led Dziembowski & Goode (1985), then Shibahashi (1986, and in Kurtz & Shibahashi 1986) to modify Kurtz's (1982) simple oblique pulsator. When it is assumed that the magnetic perturbation to the pulsation frequency is greater than the Coriolis perturbation, the inequality of the amplitudes of the rotational sidelobes is explained, and the amount of the inequality gives a measure of the integrated global interior magnetic field strength - a value which is not accessible to surface magnetic measurements. Although the theory of Dziembowski & Goode and that of Shibahashi explained and exploited the amplitudes of the frequency triplet, it required that all three frequencies have the same phase at the time of pulsation maximum. That is not observed in HR 3831; the rotational sidelobes are in phase, but the phase of the central frequency differs from them by a non-obvious 2 radians. Further observational analysis by Kurtz, Shibahashi & Goode (1990) showed that more frequencies are present than just the triplet. It was later discovered that these additional frequencies are generated because the pulsation frequency in HR 3831 is cyclically, but not strictly periodically, variable on a time-scale of about 1.6 yr (Kurtz et al. 1994a). Once that was recognized, it was possible to remove its effect and analyse many years of data simultaneously to reduce the noise level. Thus it was discovered that the pulsation mode of HR 3831 is a 'distorted dipole' (analogous to the decentred magnetic fields observed in many Ap stars) which gives rise to a frequency septuplet. Using Shibahashi's formulation of the oblique pulsator theory (Kurtz & Shibahashi 1986), Kurtz (1992) obtained a method for deconvolving the frequency septuplet into a spherical harmonic series ofaxisymmetric multipoles. KKM used that method to find that the septuplet can be decomposed into t = 0, 1, 2 and 3 components, with the t = 1 dipole dominant. Shibahashi & Takata (1993) looked for a theoretical justification for this observed septuplet. They found that secondorder terms in the pulsation equations lead to the expectation of t = 1 and 3 components, but not t = 0 and 2. They then made further refinements (Takata & Shibahashi 1994; Shibahashi & Takata 1995), the latter including the effects of a quadrupole component of the magnetic field. That then leads to the expectation of an undecuplet (ll-fold) frequency fine structure, where the lowest two frequencies and highest two frequencies have very low (currently undetectable) predicted amplitudes, hence the observed septuplet. The amplitudes of the observed septuplet are explained within this theory, but not yet all of the phases of the frequency components. That suggests that non-axisymmetric components to the magnetic field and their effects on the pulsation need to be considered - that is, that a completely (but only slightly) decentred magnetic field and pulsation mode are needed. Thus one can see how the observations

3 and theory of both magnetic fields and pulsation leap-frog each other and intertwine in the Ap stars. Observationally it is extremely desirable to have stars in addition to HR 3831 to test oblique pulsator theory. HD 6532 is an obvious candidate. It has an amplitude in Johnson B which exceeds 2 mmag at pulsation maximum. It has a polarity-reversing dipole pulsation which Kurtz & Cropper (1987) found gives rise to a rotationally split frequency triplet. The inclination of the rotation axis is not near to 90 in this star, so one pole is seen from a much more direct aspect than the other, and also for a much larger fraction of the rotation period. That is, one gets a look at the pulsation mode from a large range of aspect over the rotation period. The proximity of the rotation period to 2 d means that frequency analyses of observations obtained from a single site suffer from severe alias problems. These problems have stifled work on this star. Recently Kurtz et al. (1996) used observations of the rotational mean-light variations from 1986 to 1994 to derive an accurate rotation period of Prot = ± d. This new period is the key to our successful frequency analysis in this paper. Kurtz & Cropper (1987) reduced the alias problems in HD 6532 by combining 5 h of observations from Mt. Stromlo (MSSSO) with 85 h from the South African Astronomical Observatory (SAAO). In addition to the dipolar frequency triplet, they found two other frequencies - each separated from the central frequency of the triplet by about 24 ~Hz - which were plausibly from consecutive (n - 1 and n + 1) pulsational overtones. The signal-to-noise (SIN) ratio for those frequencies was poor, however, as it was for the dipole triplet from which the harmonic decomposition must be done. We therefore decided to conduct a multisite observing campaign on HD 6532 in 1994 September for two weeks from the SAAO, Cerro Tololo Inter-American Observatory (CnO) and Mt. John University Observatory (MJUO) to improve the SIN ratio in the frequency analysis while minimizing the alias problems. We found that HD 6532 has a frequency quintuplet, rather than just a triplet. The frequencies of Kurtz & Cropper which were separated by 24 ~Hz from the central frequency of the triplet are aliases of the two new rotational sidelobes. The pulsational variation in HD 6532 is fully represented by this frequency quintuplet plus two first harmonic frequencies. With the new, accurate rotation frequency of Kurtz et al. (1996) we were able to analyse observations spanning 1984 to 1994 simultaneously with significant improvement in the SIN ratio. In Section 2 we present the observations and in Section 3 the frequency analysis. We have used the method of Kurtz (1992) to deconvolve the frequency quintuplet into t = 0, 1 and 2 components, the results of which are shown and interpreted in Section 4. Future work will apply the new theory of Shibahashi & Takata (1995) to both HD 6532 and 3831 to see what details remain to be explained. 2 HIGH-SPEED PHOTOMETRIC OBSERVATIONS OF HD6532 With a rotation period so near to 2 d, cross-talk between the 2-d -1 aliases of the rotational components of the frequency The rapidly oscillating Ap star HD multiplet in HD 6532 requires multisite observations to reduce the amplitudes of those aliases. We therefore organized a two-week observing campaign using the 1-m telescopes of the SAAO in South Africa, cno in Chile and MJUO in New Zealand. All observations were made through Johnson B filters using lo-s integrations. They were corrected first for dead-time losses, sky background and extinction, then co-added to produce 40-s integrations. Finally, low-frequency sky transparency noise (well separated from the pulsation frequencies) was filtered to produce white noise in the amplitude spectrum. This facilitates the comparison of the frequency solution to the light curves by examination, and it gives better least-squares estimates of the errors in amplitude and phase which are not affected by the low-frequency noise. Previously published observations (Kurtz & Kreidl 1985; Kurtz & Cropper 1987) were then re-examined. We rejected a few of the light curves as too noisy; others we filtered further at low frequencies in the same way as the new data. This resulted in a set of observations of duration 20 h on five nights in 1984, 90 h on 23 nights in 1985, and 77 h on 11 nights in Table 1 gives a journal of all the observations used in this paper. Fig. 1 shows the 7.4-h light curve obtained at the SAAO on JD fitted with the frequency solution given in Table 2. From this light curve one can judge the SIN ratio on a good night, and the quality of the fitted solution which is typical for all nights over the entire lo-yr time-span. HD 6532 rotates in d. With polarity reversal this means the amplitude goes from maximum to minimum in a quarter of that time, or 11.7 h. That amplitude modulation is clearly visible over the time span of 7.4 h shown in Fig FREQUENCY AN ALYSIS The light curves of HD 6532 were analysed for their component frequencies using a variety of procedures with crosschecks. Frequencies were first identified in an amplitude spectrum generated using Kurtz's (1985) faster implementation of Deeming's (1975) discrete Fourier transform (DFT). Interpolation in complex space (O'Donoghue 1981; O'Donoghue & Warner 1982) was employed to reduce computing time. Frequencies, amplitudes and phases were optimized and errors estimated using a combination of linear least-squares and non-linear least-squares fitting procedures along with sequential pre-whitening. The linear least-squares procedure optimizes amplitude and phase for all fitted frequencies simultaneously. The results of that are used as initial values for the non-linear least-squares routine which then iteratively optimizes and yields error estimates for frequency, amplitude and phase. The following discussion is easier to follow with the answer in mind. Fig. 2(a) shows a schematic amplitude spectrum of our final solution for HD 6532 from an analysis of all the available data. HD 6532 pulsates in a single distorted dipole mode. The rotational amplitude modulation which results from the changing viewing aspect gives rise to the frequency quintuplet seen in Fig. 2(a). This quintuplet (along with two low-amplitude harmonic frequencies) is a complete description of the pulsation variations of HD 6532 over the entire time-span of the observations. The highest

4 886 D. W Kurtz et al. Table 1. Journal of observations of HD Civil Date HID n.o t a telescope obs 1984 hours mmag November SAAO O.5-m DWK November SAAO 0.5-m DWK November SAAO O.5-m DWK November SAAO 0.5-m DWK November SAAO 0.5-m DWK 1: <1.73> July SAAO O.5-m DWK July SAAO O.5-m DWK July SAAO 0.5-m DWK July SAAO 0.5-m DWK August SAAO 1.00m DWK August SAAO 1.00m DWK August SAAO 1.00m DWK August SAAO 1.0-m DWK September SAAO I.O-m DWK September SAAO 1.0-m DWK 29/30 September SAAO I.O-m DWK 00/01 October SAAO 1.0-m DWK 15/16 October SAAO 1.0-m DWK October SAAO 1.0-m DWK October SAAO 1.0-m DWK October SAAO 1.0-m DWK October SAAO 1.00m DWK October SAAO 1.0-m DWK October SAAO 1.00m DWK October SAAO 1.0-m DWK October SAAO 1.0-m DWK October MSSSO 0.6-m MSC October MSSSO 0.6-m MSC October SAAO 1.0-m DWK 1: <1.22> September SAAO 1.00m CK September SAAO 1.00m CK September MJUO I.O-m DJS September SAAO I.O-m CK September SAAO 1.0-m CK September SAAO 1.0-m DWK September SAAO I.O-m DWK September cno 1.00m PM September CTlO 1.0-m PM September eno 1.0-m PM September CTIO 1.0-m PM September CTlO 1.00m PM September CTIO 1.0-m PM 1: <1.33> 1: all years <1.32> Key to observers: DWK=D. W. Kurtz; DJS=D. J. Sullivan; PM = Peter Martinez; CK=Chris Koen; MSC=M. S. Cropper. Notes to Table 1. The Julian Dates are plus the date shown; n40 is the number of 40-s integrations obtained; u is the standard deviation per observation of the residuals to the fit of the solution given in Table 2; the rows marked with a 'I;' give the total number of nights observed, the number of 40-s observations and number of hours observed; the brackets in the u column indicate that the value is a weighted mean. The 1984 data were first reported in Kurtz & Kreidl (1985); the 1985 data were first reported in Kurtz & Cropper (1987).

5 HD6532 JD B 40 The rapidly oscillating Ap star HD I+- MMAG HOURS Figure 1. The light curve of HD 6532 obtained on 1995 September 18/19 at SAAOo Each panel is 2 h long; the light curve is continuous for 7.4 h and is read like lines of print. The oscillation period is 6.9 min. The fitted curve uses the parameters of the final frequency solution given in Table 2. The rotational amplitude modulation is apparent. Table 2. Frequency solution for HD frequency ~Hz v -2v rot ± v -Vrot ± v ± v + Vrot ± v + 2v rot ± v ± v + Vrot ± amplitude mmag phase radians 0.17± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.22 (J = mrnag 1:0 = HID peaks in the amplitude spectrum of the residuals after this quintuplet is fitted to the data are 0.10 mmag. Ideally, a frequency analysis should be governed only by the assumptions and limitations of the analysis technique. When model-dependent theoretical expectations do not influence the determination of the frequencies, the testing of theory with those frequencies is most fruitful. Because of aliases, and because of the long time-span from 1985 to 1994

6 888 D. W. Kurtz et al ~V -Vrot a ~V ~V +vrot ~V +2vrot o 00 ~~~----~~~--~~-L-L~~~~--~~~--~~~ AMPLITUDE mmag FREQUENCY mhz b Figure 2. Amplitude spectra for HD (a) A schematic amplitude spectrum illustrating our frequency solution. There is only one pulsation mode - a distorted oblique dipole mode; it has a frequency of mhz which is the central frequency, v, of the five shown. All of the frequencies in this schematic spectrum are separated by exactly the rotation frequency. The ensemble, which we refer to as the quintuplet, describes the pulsation amplitude and phase modulation as a function of rotational phase. (b) The amplitude spectrum of the data. The highest peak is v - v mt By examining the solution of panel (a), the frequencies v and v + v mt can also be seen. Panel (b) is plotted at a resolution of 2 x 10-7 mhz, i.e. there are 10 6 frequency steps. Hence the individual peaks are much narrower than the width of the finest lines in the plot. We used very high-resolution amplitude spectra to inspect the details of this figure for our analysis, but do not display those many detailed diagrams. (c) The amplitude spectrum of the residuals after v - V mu v and v + V mt have been pre-whitened from the data. The frequencies v - 2v mt and v + 2v mt can be selected by reference to panel (a). The two peaks to the left of v - 2v rot are its -1-d- 1 and - 2-d -I aliases which disappear in (d) which is the amplitude spectrum of the residuals after the frequency quintuplet has been prewhitened. The ordinate scale is identical for all four panels of this figure.

7 when no observations were obtained, we cannot meet this ideal in this paper. We rely on expectations from the oblique pulsator model to choose among statistically equivalent aliases to arrive at our solution. Fig. 3 shows the spectral window of the new 1994 data. We were not fortunate with the weather during this multisite campaign and obtained only a 22 per cent duty cycle. Hence the 1-d- 1 and 2-d- 1 aliases are significant. Because the rotation period of HD 6532 is so close to 2 d, this means that alternating members of the quintuplet have cross-talk between their spectral windows. Independent analyses of the 1994 and 1985 data both yield a frequency triplet which is the central three frequencies of Fig. 2(a), but spectral window cross-talk perturbs the outer two frequencies of the triplet in such a way that, although further frequencies are obviously present, the values of those frequencies depend on the subset of the data which is analysed. This problem was overcome by analysing all of the data listed in Table 1 simultaneously. There is then substantially redundant coverage of all rotational phases and the timespan is so long that the V rot rotational sidelobe frequencies are resolved from the 2 d -I aliases. A new problem then arises, however - there are many statistically equivalent 1 cycle per 9 yr (1/9 yr-i) aliases caused by the gap from 1985 to 1994 when no observations were made. We managed this new problem by assuming that HD 6532 is an oblique pulsator which demands frequencies separated by exactly the rotation frequency. This constraint on the frequency solution is plausible from previous work on the star, and from other roap stars - especially HR3831. It does mean, however, that the elegant frequency solution we find in this paper is only consistent with, and is not an independent confirmation of, the oblique pulsator model AMPLITUDE mmag 0.60 The rapidly oscillating Ap star HD We analysed the 1985 and 1994 data independently, then used the results of those analyses to guide an analysis of the entire JD , , data set. Within the frequency errors, the analyses of both the 1985 and 1994 data gave the same highest peak, so we selected amongst the many 1/9 yr-i aliases in the data the highest peak, v - Vrot> closest to the centroid of the 1985 and 1994 peaks. Fig. 2(b) shows the relevant section of the amplitude spectrum. The spectral window is similar to that in Fig. 3 in terms of the amplitudes of the daily aliases, but, of course, contains many more-finely-spaced aliases. The rotation frequency in HD 6532 is known to such high precision (Kurtz et al. 1996) that once the frequency of highest amplitude, v - V rot' is selected and pre-whitened, the correct alias for the second frequency, v, is unambiguously specified by the (assumed) oblique pulsator model requirement that the frequency separation must be equal to the rotation frequency. After pre-whitening simultaneously by v - Vrot and v, the third frequency, v + Vrot> is determined in a similar manner. That reproduces the previously known frequency triplet. The amplitude spectrum of the residuals after prewhitening by the triplet is shown in Fig. 2( c), where the noted frequencies are v - 2v rot and v + 2v rol" Hence we find that the rotational amplitude modulation is described by a frequency quintuplet. Pre-whitening by that quintuplet gives the amplitude spectrum of the residuals shown in Fig. 2( d) where there are no significant peaks. The discovery of the quintuplet solves the problem of the two unexplained frequencies of Kurtz & Cropper which were separated from v by 2411Hz. They are the - 1-d -\ alias of v-2v rot and the +1_d- 1 alias of v+2v rot We find that when the quintuplet is fitted to Kurtz & Cropper's data, the HD6532 JD SPECTRAL WINDOW FREQUENCY mhz Figure 3. The spectral window for the 1994 data. The multisite campaign was intended to eliminate the 2-d -I aliases. With a 22 per cent duty cycle they were reduced, but not eliminated. The highest peak is scaled to the amplitude of the highest peak in the 1994 data.

8 890 D. W. Kurtz et al. amplitude spectrum of the residuals is flat with no indication of any other frequencies. The presence of harmonics to the fundamental frequencies in the amplitude spectra of many roap stars shows that their light variations are non-linear. In HR 3831, for example, there is a low-frequency septuplet, a first-harmonic quintuplet, a second-harmonic triplet and a thirdharmonic singlet (KKM 1993). Kurtz & Cropper (1987) found a single-harmonic frequency in HD 6532 at 2v. We also find this frequency. After pre-whitening by 2v we find another significant peak at 2v + V rot ' Pre-whitening by both these peaks leaves only noise in the amplitude spectrum with the highest peaks at 0.06 mmag. If there is a third frequency at 2v - V rot' it has an amplitude too low to be detected. Hence our final frequency solution given in Table 2 is a quintuplet plus two frequencies which may be part of a triplet at the first harmonic. The errors have been estimated by fitting the quintuplet to the data by linear least-squares and then by non-linear least-squares to optimize frequency, amplitude and phase. The errors in frequency are appropriate if the correct first alias was selected. Given the large number of nearly equivalent alias choices, guessing the right one is improbable. Therefore, there may be a substantially larger systematic error in the frequencies. Importantly, however, the quoted errors are appropriate for the relative differences in the frequencies. This is supported by the nonlinear least-squares fit of the quintuplet which adjusts the frequencies for an optimum least-squares fit. Those optimum frequencies differ from the values in Table 2 (which require that they be separated by exactly V rot) by less than 3/T in all cases. 4 DISCUSSION OF THE DISTORTED DIPOLE The frequency quintuplet and harmonics are a complete solution to the light variations of HD 6532 to the precision of the data. Assuming the oblique pulsator model, HD 6532 has only one pulsation mode with a frequency of v= j.lhz, which is the central frequency of the quintuplet. The other four frequencies of the quintuplet describe the amplitude and phase modulation as a function of the rotation of the star. As such, they constrain the geometry of the mode. A useful way to visualize this is to look at the plot of the pulsation phase and amplitude as a function of rotation shown in Fig. 4. The rotational inclination and magnetic obliquity are such that one pulsation pole (pole 1) is seen for two-thirds of the rotational cycle and the other (pole 2) for one-third. Pulsation maximum occurs when pole 1 is seen from its most favourable aspect. That time has been calculated by requiring that the phases given in Table 2 for the first rotational side lobes be equal, i.e. cp(v-vrot)=cp(v+vrot). Within 1/T this time is equal to the time when the rotational light variations are at an extremum (maximum in U and B; minimum in V, R and I: Kurtz et al. 1996). This is consistent with the oblique pulsator model and implies that the magnetic axis and pulsation axis do coincide, and that the abundance patches which give rise to the rotational light variations are centred on the magnetic poles. The pulsation phases are approximately constant with a 1t-radian phase flip at the time of quadrature when one pulsation pole disappears and the other appears over the horizon, as is expected for oblique pulsation. The theoretically fitted lines in Fig. 4 were calculated using the method of Kurtz (1992) which assumes the oblique pulsator model formulation of Shibahashi (1986; Kurtz & Shibahashi 1986). Those calculated fits assumed that the pulsation mode can be represented as a spherical harmonic series of t = 0, 1 and 2 components. Table 3 gives the contributions of each of those components. The dipole obviously dominates. The quadrupole is needed to explain the ± 2v rot sidelobes of the quintuplet; it contributes a small amount to the ± V rot sidelobes, too. There is also a small t = 0 contribution to the central frequency. This is needed in any solution because the central frequency does not have exactly the same phase as the ± V rot sidelobes at the time of pulsation maximum. We picked a rotational inclination of i = 50 as reasonable, since neither i nor f3 can be near 90, or the poles would be seen for equal fractions of the rotational period. With i = 50, we used the oblique pulsator model requirement that A~)l +A~\ A~l) tani tan f3 (Kurtz & Shibahashi 1986) to find f3 = 59 ; other i and f3 which satisfy the given constraint and are not near 90 are also allowed. The fitted curve underestimates the amplitude when pole 2 is visible. This can be seen in the bottom of Fig. 4 between rotational phases 0.3 and 0.7. This effect is not an artefact of negative amplitudes not being plotted in the bottom panel. While this does happen at quadrature (rotational phases 0.33 and 0.67), at the time of secondary pulsation amplitude maximum (rotational phase 0.5) the pulsation phase is constant, so the pulsation amplitude points are not biased to higher amplitude. The fitted solution is dominated by the higher SIN data from the rotational phases when pole 1 dominates, and that predicts a pulsation amplitude when pole 2 dominates which is too small. The implication is that the strengths of the two poles are not equal, just as is often found in magnetic studies. Future work will refine this discussion using the theory of Shibahashi & Takata (1995), which includes the effect of a quadrupole component (i.e. approximately a linearly offset dipole) to the magnetic field. That should account for one pole being stronger than the other. It is possible that the strengthening of pole 2 may also flatten the pulsation phase versus rotation phase theoretical fit in the upper panel. The phases of the quintuplet, especially those of v - 2v rot and v + 2v rot, suggest that the mode is not completely axisymmetric, i.e. that a completely decentred dipole solution is needed. Testing of this suggestion will also have to await further theoretical developments. A measure of the integrated internal magnetic field strength, K(ma g ), can be made from the amplitude asymmetry of the v - V rot and v + V rot components of the quintuplet. Consideration of the magnetic and Coriolis perturbations to the pulsation frequency give (see Kurtz & Shibahashi 1986)

9 U) c co 1.0 rl U co c H QUINTUPLET The rapidly oscillating Ap star HD w (f) -1.0 «I (J) co 4.5 E E W :::l f- 2.5 H J ::::E « a ROTATION PHASE Figure 4. This diagram shows the variation in pulsation phase (upper panel) and amplitude (lower panel) as a function of rotation. Rotational phase zero corresponds to pulsation amplitude maximum. Two rotation periods are shown for convenience. Each point in the diagram has been generated by fitting the central frequency of the quintuplet to five cycles ( min) of the light curves by linear leastsquares fitting. The theoretical line has been calculated by the method of spherical harmonic decomposition (Kurtz 1992). Table 3. Components of the spherical harmonic series description of the pulsation mode in HD 6532; i=50, /3=59. l A(e) -2 A (i) -I 4t) A (i) +1 A(t) +2 <p K(mag) = _ 5Cn,fQ(A~\ +A~)I) 3(A~)I-A~)I), where C n ( = from models of Takata & Shibahashi (1995), Q'is the rotation frequency (in radian S-I) and the amplitudes are taken from the dipole components given in Table 3 with the amplitude errors of Table 2. This gives K(mag) (HD 6532) = 1.0 ± Hz. A similar calculation for HR3831 (with amplitudes from KKM and C n,f=0.008) gives K(ma g ) (HR 3831) = 3.5 ± Hz. Presuming that the surface field is proportional to K(ma g ) leads to the expectation that the measured magnetic field in HD 6532 should be about one-third ofthat in HR Available observations of the longitudinal fields are consistent with this; measurements of the quadratic fields are probably inconsistent with it. The mean longitudinal field for HR 3831 varies from to G (Mathys 1994). A single observation of HD 6532 yielded ± 273 G (Mathys & Hubrig, in preparation), which is effectively a null result. Measurements of the quadratic field strengths give 21.9 ± 4.4 kg for HD 6532 (Mathys & Hubrig, in preparation) and 11.4 ± 0.4 kg for HR 3831 (Mathys 1995). Thus, the few available magnetic observations do not confirm the interpretation of K(ma g ) as a measure of the integrated internal field strength.

10 892 D. W. Kurtz et al. 5 FREQUENCY VARIABILITY All of the roap stars in which long-term studies have been made show frequency variability. The best studied case is HR 3831 (Kurtz et al. 1994a) which varies cyclically, but not strictly periodically, on a time-scale of about 1.6 yr. Kurtz et al. (1994b) found frequency variability in Q( Cir on a timescale of hundreds of days. Other work indicates frequency variability in HD ,137949, and HR We are, therefore, surprised that all of the data listed in Table 1 can be fitted with the solution given in Table 2. Examination of O-C diagrams shows only scatter about zero within the expected errors. It appears that HD 6532 does not have frequency variability. There are some notes of caution to this conclusion, however. There is a single, 4-h light curve obtained on JD (Kurtz & Kreid11985) which we did not use in the frequency analysis of this paper precisely because it cannot be fitted with the solution in Table 2. In an O-C diagram its phase differs from zero by about 1t radians. We do not know whether that is an indication of frequency variability, or incorrect data reduction of the light curve, or a timing error. We consider the latter two problems unlikely since we test carefully to avoid them, but with only one discrepant light curve we can delve no further into this problem. Only long-term monitoring can solve it. ACKNOWLEDGMENTS We would. like to thank Andrew Stewart for providing assistance at Mt. John University Observatory on several nights. We thank Gautier Mathys for permission to quote his results prior to publication, and for helpful comments. DJS thanks the University of Canterbury for the allocation of Mt. John observing time, and the VUW Internal Grants Committee for financial assistance. DWK and PM thank the FRD for financial support. PM also thanks the University Research Committee of the University of Cape Town for financial support. REFERENCES Borra E. F., Landstreet J. D., 1980, ApJS, 42, 421 Deeming T. J., 1975, Ap&SS, 36, 137 Dziembowski W., Goode P. R., 1985, ApJ, 296, L27 Kurtz D. W., 1982, MNRAS, 200, 807 Kurtz D. W., 1985, MNRAS, 213, 773 Kurtz D. W., 1990a, ARA&A, 28, 607 Kurtz D. W., 199Ob, MNRAS, 242, 489 Kurtz D. W., 1992, MNRAS, 259, 701 Kurtz D. W., Cropper M. S., 1987, MNRAS, 228, 125 Kurtz D. W., Kreidel T. J., 1985, MNRAS, 216, 987 Kurtz D. W., Shibahashi H., 1986, MNRAS, 223, 557 Kurtz D. W., Shibahashi H., Goode P. R., 1990, MNRAS, 247, 558 Kurtz D. W., van Wyk F., Marang F., 1990, MNRAS, 243, 289 Kurtz D. W., Kanaan A., Martinez P., 1993, MNRAS, 260, 343 (KKM) Kurtz D. W., Martinez P., van Wyk F., Marang F., Roberts G., 1994a, MNRAS, 268, 641 Kurtz D. W., Sullivan D. J., Martinez P., Tripe P., 1994b, MNRAS, 270,674 Kurtz D. W., Marang F., van Wyk F., Roberts G., 1996, MNRAS, 280, 1 Landstreet J. D., 1990, ApJ, 352, L5 Landstreet J. D., 1992, A&AR, 4, 35 Landstreet J. D., 1993, in Dworetsky M. M., Castelli F., Faraggiania R., eds, ASP Conf. Ser. 44, Peculiar versus normal phenomena in A-type and related stars. Astron. Soc. Pac., San Francisco, p. 218 Leroy J. L., Bagnulo S., Landolfi M., Landi Degl'Innocenti E., 1994, A&A, 284, 174 Martinz P., 1993, PhD thesis, Univ. Cape Town, South Africa Mathys G., 1989, Fundam. Cosmic Phys., 13, 143 Mathys G., 1993, in Dworetsky M. M., Castelli F., Faraggiania R., eds, ASP Conf. Ser. 44, Peculiar versus normal phenomena in A-type and related stars. Astron. Soc. Pac., San Francisco, p.232 Mathys G., 1994, A&AS, 108, 547 Mathys G., 1995, A&A, 293, 746 Matthews J., 1991, PASP, 103,5 Matthews J., Kurtz D. W., Wehlau W., 1987, ApJ, 313, 782 O'Donoghue D., 1981, PhD thesis, Univ. Cape Town, South Africa O'Donoghue D., Warner B., 1982, MNRAS, 200, 563 Shibahashi H., 1986; in Osaki Y., ed., Hydrodynamic and Magnetohydrodynamic Problems in the Sun and Stars. Univ. Tokyo, Tokyo, p. 195 Shibahashi H., Takata M., "1993, PASJ, 45, 617 Shibahashi H., Takata M., 1995, PASJ, 47, 219 Shift M. J., 1975, MNRAS, 172, 133 Takata M., Shibahashi H., 1995, PASJ, 47, 219 Thompson I. B., Landstreet J. D., 1985, ApJ, 289, L9