Pulsations in β Pictoris (original) (raw)

Abstract

We report on the discovery of at least 18 pulsation modes in β Pictoris from 697 high-dispersion spectra obtained over a two-week period. These are seen as absorption subfeatures moving across the spectral line profiles, indicating that they are of relatively high spherical harmonic order. They do not resemble the features seen in the Ca H and K lines generally attributed to infalling bodies. We use the correlation function between an unbroadened synthetic spectrum and the observed spectrum to obtain what is essentially a high signal-to-noise ratio representation of the mean line profile. From the correlation intensity as a function of time, we calculated periodograms for every point across the correlation function. In this way, we are able to determine accurate periods for each mode. By phasing the correlation function with the period belonging to a given mode, we can isolate the line profile variations for the mode and deduce the approximate value of the azimuthal order, m. Two of the modes are retrograde, the remainder are prograde and most of them lie in the range 4 ≤|m|≤ 10. Comparison with non-adiabatic models shows that the periods are in the range expected for unstable modes in a star with the mass and temperature appropriate to β Pic. Some constraints can be placed on the effective temperature.

We also report on photometric observations of β Pic obtained during the same run. From our 14 318 photometric observations combined with the discovery data from Koen (2003), we find two definite pulsation modes with frequencies of 47.4 and 38.1 cycle d−1. A third frequency at 47.3 cycle d−1 may also be present. The amplitudes are all smaller than 1.5 mmag.

1 Introduction

The A5V star β Pictoris has long been known to be surrounded by a massive disc of gas and dust seen nearly edge-on (Smith & Terrile 1984). High-resolution spectroscopic observations led to the discovery of strong redshifted absorption features with terminal velocities as high as 300 km s−1 (Lagrange-Henri, Vidal-Madjar & Ferlet 1988) which have been interpreted as a signature of evaporation of infalling km-sized solid bodies in star-grazing orbits. This ‘Falling Evaporating Bodies’ (FEB) scenario appears to reproduce very satisfactorily the observed variable Ca ii and Mg ii features (Beust et al. 1990).

Further observations by Lagrange-Henri et al. (1992) and Ferlet et al. (1993) produced evidence for a clumpy structure of the Ca ii cloud and yielded an estimate of at least 200 infalling events per year. Moreover, there appears to be two different regimes for the variable Ca ii lines: a low-redshift regime (≈20 km s−1 with respect to the star) and a high-redshift regime (up to 90 km s−1 with respect to the star). Variability in the low-redshift regime seems to persist for longer periods (up to 4 d), while the high-redshift variability lasts only for a few hours.

In an even more intensive multisite campaign, Lagrange et al. (1996) confirmed the existence of two velocity regimes and their time-scales. The observations lend further support to the FEB scenario. The high-velocity events appear to be explained by the evaporation of a single body, but the low-velocity events can only be understood in terms of the evaporation of a family of bodies on adjacent orbits.

The variable absorption events discussed above are almost always redshifted with respect to the star. Weak blueshifted components have been detected on a very few occasions: Crawford, Beust & Lagrange (1998) detected a component, comparable in strength with the redshifted components, blueshifted by 14 km s−1 with respect to the star. This can be interpreted in terms of the FEB model by an object on a significantly different orbit from the more common redshifted events. Petterson & Tobin (1999) reported many blueshifted features, sometimes rivaling the strength of the redshifted ones. About 20 per cent of the features appear to be blueshifted, judging from their fig. 56.

The conclusions concerning FEBs are based almost entirely on observations of the Ca H and K lines.

Numerous observations have been devoted to the study of environment of the star. Studies of the star itself have mainly focused on the determination of its classification and age in order to evaluate the evolutionary status of the system. Recent age determinations of the star are 20 ± 10 Myr (Barrado y Navascues et al. 1999), 12+8−4 Myr (Zuckerman et al. 2001) and 11.5 Myr (Ortega et al. 2002). The star itself does not appear to be peculiar in any way. Observations from the Far Ultraviolet Spectroscopic Explorer (FUSE) reveal a broad emission spectrum due to highly ionized C iii and O iv (Deleuil et al. 2001). These spectral features probably originate in a solar-like extended chromosphere or from magnetospheric accretion. This discovery was modelled by Bouret et al. (2002) by a thin region heated up to a few 105 K located close to the stellar photosphere. The discovery of a chromosphere in an A-type star presents a challenge, as chromospheres are not expected in a star with a radiative envelope.

β Pictoris falls in the region of the Hertzsprung–Russell (HR) diagram populated by δ Scuti stars. It seems reasonable to suppose that it might show detectable periodic light and radial velocity variations characteristic of δ Scuti stars. Lanz, Heap & Hubeny (1995) derived the following parameters: _T_eff= 8200 ± 150 K, log _g_= 4.25 ± 0.1, _L_bol= 11.3 ± 3.5 L⊙ and _M_= 1.8 ± 0.1 M⊙. Models of stars with these parameters are linearly unstable with frequencies in the range 50 < ν < 60 cycle d−1. One may thus expect to see light and line profile variations with a period between 25 and 30 min. Recently, Koen (2003) (hereafter K2003) reported the discovery of three pulsation modes in the light curve of β Pic in this period range.

On the basis of this discovery, we decided to investigate the pulsations of β Pic by means of simultaneous photometric and high-dispersion spectroscopic observations. In this paper we present the results of this investigation.

2 Observations and Reduction Procedure

Spectroscopic observations at SAAO were obtained using the GIRAFFE echelle spectrograph fibre-fed attached to the Cassegrain focus of the 1.9-m telescope. The GIRAFFE spectrograph has a resolving power of about 32 000. The 1024 × 1024 TEK CCD chip gives a dispersion of 0.06–0.09 Å per pixel. A Th–Ar arc lamp was used for wavelength calibration with arc spectra taken at regular intervals to calibrate possible drifts. Flat-fielding was accomplished by illuminating the camera with uniform light using a tungsten filament lamp and a diffusing screen. The blaze correction was determined by measuring the response across each order when the fibre was illuminated by a tungsten lamp. The wavelength range was 4400–6680 Å spread over 45 orders. Exposure times were kept at 5 min for a signal-to-noise (S/N) ratio of about 150–250. A total of 697 spectra of β Pic was obtained on 13 nights over a two-week period in 2003 January 14–27 (Table 1).

Table 1.

Observing log for the spectroscopy of β Pic. The first column is the starting Julian day with respect to JD 2452200. The second column is the duration, ΔT, in hours of observing and this is followed by the number of observations, N.

In order to study the line profile variations, the continuum needs to be determined. Placing the continuum by hand is not an easy process. We preferred to rectify the spectra with the aid of a synthetic spectrum calculated using the spectrum code (Gray & Corbally 1994) using a Kurucz solar-abundance model atmosphere appropriate to a A5 V star (_T_eff= 8250 K, log _g_= 5.00) and broadened with v sin _i_= 130 km s−1.

All photometric measurements logged in Table 2 were acquired with the Modular Photometer attached to the SAAO 0.5-m telescope. A neutral density filter was used to attenuate the light sufficiently so as not to overload the photoelectric photometer. As explained in K2003, continuous monitoring of the star, rather than the traditional differential photometry, is deemed more efficient for the relatively rapid variations and very low amplitudes. Therefore, all observations for a given night were made using single filter. After some experimentation, the R band was selected as optimal: the amplitudes were found to be comparable to those measured in the bluer wavebands, while contamination by atmospheric transparency changes is considerably smaller. Integration times of 10 s were used throughout.

Table 2.

Observing log for photometry of β Pic. The first column is the filter through which the star was observed. The second column is the starting Julian day with respect to JD 2452200. The third column is the duration, Δ_T_, in hours of observing and this is followed by the number of observations, N.

We obtained 14 318 photometric observations, over 42.6 h of monitoring. These data are easily combined with the 7730 measurements, spread over 26.7 h, accumulated by K2003: his observations spanned 8 nights, and the gap between his last run and the first run in Table 2 is 10 d.

3 The Periods — Photometry

Analysis of our data shows that two significant periodicities are present in the light curve. Although the star was observed through different filters, we simply combined all the data irrespective of the filter used. There will probably be very small phase differences between the variation through different filters and there will also be slight amplitude variations from filter to filter, but this will not affect the frequencies. The periodogram of the combined data in Table 2 (Fig. 1) reveals the highest peak to be at ν1= 49.459 cycle d−1, but the 1-d aliasing is strong so that values of ν1 = 46.459, 47.459, 48.459, 49.459 or 50.459 cycle d−1 are all equally probable. Pre-whitening by any of these possible frequencies leads to a second frequency at ν2= 38.057 or 39.057 cycle d−1. No further significant frequencies are found.

Figure 1.

Periodograms of the light variation of β Pic for 2003 January. The top panel is a periodogram of the raw data, the bottom panel is the periodogram after removing the frequency at ν1= 49.459 cycle d−1. This shows another periodicity at ν2= 38.057 cycle d−1. The frequency is in cycle d−1, the amplitude is in mmag.

We then combined the photometric observations logged in Table 2 with those of K2003, also ignoring the fact that different filters were used on different nights. The aliasing problem is much reduced in the combined data and we are able to identify a possible third frequency. Results are shown in Fig. 2 and Table 3.

Figure 2.

Periodograms of the light variation of β Pic for the all the photometric data, i.e. including those from K2003. The top panel is a periodogram of the raw data, the middle panel is the periodogram after removing the frequency at ν1= 47.4355 cycle d−1. The bottom panel shows the periodicity after removing this frequency and ν2= 38.0593 cycle d−1. The bottom panel shows a possible third periodicity at ν3= 47.2823 cycle d−1. The frequency is in cycle d−1, the amplitude is in mmag.

Table 3.

Periodicities extracted from the combined photometry of Table 2 and K2003. The first column is the identification number, n, assigned to the periodicity. Its frequency, ν_n_, is in cycle d−1. The amplitude, in mmag, is given by A n and the phase (in radians) relative to HJD 2452600.000, is given in the last column. The second line refers to the radial velocities obtained by cross-correlation with a mean spectrum of the star itself. The amplitude is in m s−1.

Comparison of the three frequencies in Table 3 with those in K2003 is, at first sight, rather disappointing: K2003 extracted 47.055, 38.081 and 52.724 d−1 from his four _B_-band runs. However, close scrutiny of his fig. 2 shows that the second highest periodogram peak is at 47.436 d−1, at an amplitude only 0.09 mmags lower than the primary peak at 47.055 d−1: the first two frequencies found here therefore do, in fact, agree well with those in K2003. As far as the the third frequency is concerned, comparison of the third panel in Fig. 2 with the corresponding panel in fig. 2 of K2003 shows that the shapes of the residual periodograms are entirely different. This can probably be ascribed to the different wavebands involved: predominantly R in our data, and exclusively B in K2003.

4 The Periods — Spectroscopy

In order to study the average properties of the line spectrum, it is convenient to use the correlation function. The correlation function is obtained by cross-correlating the observed spectrum with a synthetic spectrum which closely resembles the spectrum of the star. It can be shown that the correlation function is, in effect, a weighted average of the line spectrum and has, of course, a much higher S/N ratio than an individual line profile. We calculated the correlation profiles by cross-correlating each spectrum of β Pic with an unbroadened synthetic spectrum of solar composition with _T_eff= 8 250 K, log _g_= 5.00. This temperature and gravity was estimated from the spectral classification (A5V), but the method is not sensitive to the precise values. From the mean correlation profile, we estimated the projected rotational velocity by a non-linear least-squares fit to a series of models of a rotating star. The star was assumed spherical with a linear limb-darkening coefficient of _u_= 0.5 but and no gravity darkening. We used the mean intrinsic line profile calculated from the synthetic spectrum. We find v sin _i_= 124 ± 3 km s−1 and a mean radial velocity of 21.3 ± 0.7 km s−1 (Fig. 3).

Figure 3.

Mean line profile in β Pic obtained by co-adding all correlation functions. The thick line is a fit using v sin _i_= 124 km s−1 and velocity 21.3 km s−1.

In Fig. 4 we show the grey-scale residuals of each spectrum when divided by the mean profile for the first two nights. It is evident that there are subfeatures moving across the line profile from blue to red. This is a characteristic of non-radial pulsation of high degree.

Figure 4.

Stacked grey-scale plots of the residuals from the mean correlation function for the first two nights: left-hand panel — JD 2452654, right-hand panel — JD 2452655. Note the absorption features moving from blue to red — a characteristic of non-radial pulsations of high degree.

To determine periodicities associated with such profile variations, we could measure the ‘radial velocity’ of each observation, defined as the velocity shift of the point of maximum line absorption. This is not a satisfactory method, as the ‘radial velocity’ jumps discontinuously as one absorption subfeature becomes marginally deeper than another. One could try to smear out the effects of the moving subfeatures by using a broadened template rather than one with zero rotational velocity. The correlation profiles obtained in this way no longer show the moving absorption subfeatures. Fig. 5 shows the radial velocity obtained by this method as a function of time. As can be seen, the third night appears to be somewhat discrepant.

Figure 5.

Radial velocities obtained by cross-correlation with the mean spectrum of the star itself plotted as a function of Julian date.

Because there does not seem to be a periodicity associated with the discrepant night, we assume that the discrepancy may be attributed to partial obscuration of the disc in the line of sight. Because the star is rapidly rotating, a small obscuration of the limb would modify the line profile to mimic a small velocity shift. As a consequence, we removed the mean velocity separately for each night and calculated the resulting periodogram (Fig. 6). The peak at 3.99 cycle d−1 is almost certainly an artefact and can be ignored. There are two significant periodicities at 39.050 and 47.438 cycle d−1. The peak at 38.050 cycle d−1 is very nearly the same height as its alias at 39.050 cycle d−1. These are clearly the same periods as found in the photometry. The amplitudes and phases are shown in Table 2.

Figure 6.

Periodogram of the data shown in Fig. 5, but with the nightly means removed.

In an attempt to identify the modes associated with these two frequencies, we constructed stacked difference spectra phased with the particular frequency. In the case of ν1= 47.438 cycle d−1 we can detect a very low amplitude wave running from red to blue. We estimate _m_=+1 or _m_=+2 (retrograde wave). The amplitude for ν2 is too low for such an analysis.

A better way of determining periodicities of stars showing pulsation of high ℓ is to calculate the periodogram at many points across the line profile. The result, when plotted as a grey-scale intensity plot of frequency versus position across the correlation profile (where high intensity indicates large amplitude), shows many lines of high intensity across the profile. This indicates that several periodicities are present. If we sum the amplitudes within a range of 50 km s−1 centred on the correlation profile, we can display this as a simple periodogram (Fig. 7). By successively prewhitening the intensity across the correlation function with a chosen frequency and re-calculating the periodogram, we can extract the most significant periodicities (Table 4). Note that in deriving these frequencies, we simply assumed that the highest peak corresponds to the true frequency. This may not be the case, but apart from obtaining further data it is not possible to resolve this problem. Thus an ambiguity of 1 cycle d−1 may be present in the frequencies shown in Table 4. In general, the peaks are significantly higher than the background noise level (about 0.03 per cent). All frequencies shown in Table 4 satisfy Scargle's (1982) false alarm criterion to better than 95 per cent.

Figure 7.

Periodograms of the intensity variation of the central part of the correlation function. The frequency is in cycle d−1 and the amplitude is in percentage of the mean intensity.

Table 4.

Periodicities extracted from the line profile variations. The first column is the identification number, n, assigned to the periodicity. Its frequency, ν_n_, is in cycle d−1. The amplitude, in terms of percentage variation, is given by A n. The fourth column is an estimate of the azimuthal spherical harmonic number. A negative value of m represents a prograde mode. The last column gives the frequencies in the corotating frame.

Figs 8–11 shows grey-scale plots of the intensity residual from the mean profile phased with a particular frequency. By counting the number of subfeatures travelling across the profile in these figures, we can estimate the azimuthal spherical harmonic order, m. Results are shown in Table 4. We note that for modes of high degree it is impossible to determine the spherical harmonic degree, ℓ, as well as m, with any degree of certainty. The most rigorous method is by direct profile fitting. This method may work if the amplitude is large, the S/N ratio is high and ℓ is small. Application to modes of high degree inevitably suffer from considerable uncertainties in m (see, for example Balona & Kambe 1999). In β Pic the amplitudes are very small and m is large, so there is little point in a sophisticated analysis.

Figure 8.

Grey-scale phase plots of residuals from the mean correlation profile phased with the following frequencies (in cycle d−1). Top left — 24.72; top right — 35.06; bottom left — 41.66; bottom right — 43.76.

Figure 9.

Grey-scale phase plots of residuals from the mean correlation profile phased with the following frequencies (in cycle d−1). Top left — 45.44; top right — 46.26; bottom left — 48.94; bottom right — 50.64.

Figure 10.

Grey-scale phase plots of residuals from the mean correlation profile phased with the following frequencies (in cycle d−1). Top left — 52.92; top right — 53.24; bottom left — 56.92; bottom right — 58.38.

Figure 11.

Grey-scale phase plots of residuals from the mean correlation profile phased with the following frequencies (in cycle d−1). Top left — 61.18; top right — 68.38; bottom left — 68.54; bottom right — 71.06.

Note that the the two frequencies found in the photometry, 49.459 and 38.057 cycle d−1, were not detected by this method. The reason appears to be that the line profile variations caused by low-degree modes are not only of low amplitude, but do not appear as moving bumps. The variation is instead spread across the whole line profile, so that the intensity at any particular wavelength is of much smaller amplitude.

5 Balmer Lines

The Hα line is badly contaminated by telluric lines and cannot be used for a sensitive period search. It does not appear to vary significantly in intensity. By fitting a quadratic to the points around maximum absorption we were able to see some variation in the radial velocities with a frequency of ν= 0.05 cycle d−1 or its aliases at 0.95 and 1.05 cycle d−1, which does appear to be significant. The variations due to pulsation are not seen.

The Hβ line is clean and the radial velocities can be derived with some accuracy (Fig. 12). There are no significant velocity or intensity variations. As in Hα, there are no visible signals due to the pulsation.

Figure 12.

Radial velocities of Hβ (top) and Hα (bottom).

6 Pulsation Models

In order to compare these observations with predictions, we generated evolutionary models with masses of 1.7, 1.8 and 1.9 M⊙ with solar composition and an equatorial rotational velocity of 124 km s−1 using the New Jersey–Warsaw code. These models were then examined for pulsational instability using the nadrot code for linear, non-adiabatic pulsation. Modes with spherical harmonic degree ℓ= 0–14 were examined. Instability was found in the radial modes up to radial order _n_= 7. Instability was found for most non-radial modes in the frequency range occupied by the unstable radial modes.

In Fig. 13 we show the frequency of all unstable modes, in a non-rotating star, as a function of effective temperature. Also shown are the minimum and maximum frequencies observed in β Pic. From this figure, we can conclude that an effective temperature in the range 3.89 < log _T_eff < 3.91 gives best agreement with the observations. Of course, for a proper comparison, the rotational frequency perturbations ought to be considered. The method assumes that all calculated unstable modes are excited to measurable amplitude (which is known not to be true in general). This introduces additional uncertainties in the temperature constraints thus derived.

Figure 13.

Frequencies of unstable modes, ν, in cycle d−1 as a function of effective temperature in models of 1.7, 1.8 and 1.9 M⊙.

We attempted to place some constraints on the age of the star using the pulsation frequencies and models. The calculated frequencies move rapidly to lower values as the star evolves from the main sequence. From this, it appears that β Pic is probably younger than 500 ×106 yr, which unfortunately is not a very useful constraint.

7 Conclusions

We have shown that β Pic belongs to a group of δ Scuti stars in which modes of high degree are excited. Two modes of low degree are visible in the light curve, but at least 16 other modes are present with spherical harmonic degree in the rage 4 < ℓ < 10. While the presence of pulsations is probably unrelated to the line profile variations seen in the Ca ii lines, it is interesting to note that these have until now been unreported, in spite of highly intensive observations of the star. The reason why the profile variations reported here have not been seen in the Ca ii profiles can most probably be attributed to the fact that the calcium lines are formed very high in the atmosphere and are opaque to the non-radial pulsations in the photosphere.

The presence of pulsations may have some bearing on the non-thermal ionization discovered by FUSE. Although the star has a radiative photosphere, it is possible that pulsational energy could be converted to thermal energy, perhaps through the agency of a magnetic field. In this way one could explain the formation of a chromosphere. If this explanation is correct, one should be able to detect similar pulsations in other A-type stars showing high-excitation lines, such as in the rapidly-rotating A-type dwarfs discovered by Abt, Tan & Zhou (1997). They found that approximately one-quarter of the rapidly rotating A-type dwarfs have hot inner discs that appear in the near-UV lines of Ti ii and sometimes in other lines. The presence of Ti ii implies a high temperature of the inner disc. The fact that the lines are only seen in rapid rotators probably means that the inclination of the rotation axis to the line of sight must be high, implying that the hot circumstellar material is in the form of a disc rather than a spherical shell. The disc lines appear and disappear on time-scales of decades. The narrow width of the Ti ii lines constrains the distances of the discs to about seven stellar radii. The detection of non-radial pulsations of high degree in these stars could provide an important clue to the mechanism producing these high temperatures.

Acknowledgment

We would like to thank Dr A. Pamyatnykh for the evolutionary code and Dr W.A. Dziembowski for the nadrot pulsation code.

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Author notes

†

SAAO summer school student.

© 2003 RAS