Follow-up observations of pulsating subdwarf B stars: Multisite
campaigns on PG 1618+563B and PG 0048+091
M. D. Reed 1, S. J. O’Toole2,3, D.M. Terndrup4, J. R. Eggen1, A.-Y. Zhou1, D. An4, C.-W.
Chen5, W. P. Chen5, H.-C. Lin5, C. Akan6, O. Cakirli6, H. Worters7,8, D. Kilkenny7, M.
Siwak9, S. Zola9,10, Seung-Lee Kim11, G. A. Gelven1, S. L. Harms1, and G. W. Wolf1
1 Department of Physics, Astronomy, & Materials Science, Missouri State University, 901
S. National, Springfield, MO 65897 U.S.A.
MikeReed@missouristate.edu
2 Anglo-Australian Observatory, PO Box 296, Epping NSW 1710, Australia
3 Dr Remeis-Sternwarte, Astronomisches Institut der Universität Erlangen-Nürnberg,
Sternwartstr. 7, Bamberg 96049, Germany
4 The Ohio State University, 140 W. 18th Avenue, Columbus, OH 43210, U.S.A.
5 Graduate Institute of Astronomy, National Central University, Chung-Li, Taiwan
6 Ege University Observatory, 35100 Bornova-Izmir, Turkey
7 South African Astronomical Observatory, PO Box 9, Observatory 7935, South Africa
8 Centre for Astrophysics, University of Central Lancashire, Preston, PR1 2HE, UK
9 Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30-244 Cracow, Poland
10 Mt. Suhora Observatory of the Pedagogical University, ul. Podchora̧zych 2, PL-30-084
Cracow, Poland
11 Korea Astronomy and Space Science Institute, Daejeon, 305-348, South Korea
ABSTRACT
We present follow-up observations of pulsating subdwarf B (sdB) stars as
part of our efforts to resolve the pulsation spectra for use in asteroseismologi-
cal analyses. This paper reports on multisite campaigns of the pulsating sdB
stars PG 1618+563B and PG 0048+091. Data were obtained from observatories
placed around the globe for coverage from all longitudes. For PG 1618+563B,
our five-site campaign uncovered a dichotomy of pulsation states: Early during
the campaign the amplitudes and phases (and perhaps frequencies) were quite
http://arxiv.org/abs/0704.1496v1
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variable while data obtained late in the campaign were able to fully resolve five
stable pulsation frequencies. For PG 0048+091, our five-site campaign uncovered
a plethora of frequencies with short pulsation lifetimes. We find them to have
observed properties consistent with stochastically excited oscillations, an unex-
pected result for subdwarf B stars. We discuss our findings and their impact on
subdwarf B asteroseismology.
Subject headings: variable stars: general — asteroseismology, close binaries
1. Introduction
Subdwarf B (sdB) stars are thought to have masses about 0.5M⊙, with thin (<10
−2M⊙)
hydrogen shells and temperatures from 22 000 to 40 000 K (Saffer et al. 1994). They are
horizontal branch stars that have shed nearly all of their H-rich outer envelopes near the
tip of the red giant branch and as He-flash survivors, it is hoped that asteroseismology can
place constraints on several interesting phenomena. Subdwarf B star pulsations come in two
varieties: short period (90 to 600 seconds; EC 14026 stars after that prototype, officially
V361 Hya stars, or sdBV stars) with amplitudes typically near 1%, and long period (45
minutes to 2 hours; PG 1716 stars after that prototype or LPsdBV stars) with amplitudes
typically <0.1%. For more on pulsating sdB stars, see Kilkenny (2001) and Green et al.
(2003) for observational reviews and Charpinet, Fontaine, & Brassard (2001) for a review of
the proposed pulsation mechanism. For this work, our interest is the sdBV (EC 14026) class
of pulsators.
In order for asteroseismology to discern the internal conditions of variable stars, the
pulsation “mode” must be identified from the temporal spectrum (also called pulsation
spectrum or Fourier transform; FT). The mode is represented mathematically by spherical
harmonics with quantum numbers n (or k), ℓ, and m. For nonradial, multimode pulsators
the periods, frequencies, and/or the spacings between them are most often used to discern the
spherical harmonics (see for example Winget 1991). These known modes are then matched to
models that are additionally constrained by non-asteroseismic observations, typically Teff and
log g from spectroscopy. Within such constraints, the model that most closely reproduces the
observed pulsation periods (or period spacing) for the constrained modes is inferred to be the
correct one. Occasionally such models can be confirmed by independent measurements (Reed
et al. 2004, Reed, Kawaler, & O’Brien 2000, Kawaler 1999), but usually it is impossible to
uniquely identify the spherical harmonics and asteroseismology cannot be applied to obtain
a unique conclusion. Such has been the case for sdBV stars, which seldom show multiplet
structure (i.e., even frequency spacings) that may be used to observationally constrain the
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pulsation modes. However, relatively few sdBV stars have been observed sufficiently to know
the details of their pulsation spectra. The goal of our work is to fully resolve the pulsation
spectrum, search for multiplet structure, and examine the characteristics of the pulsation
frequencies over the course of our observations.
In this paper we report on multi-site follow-up observations of the pulsating sdB stars
PG 1618+563B (hereafter PG 1618B) and PG 0048+091 (hereafter PG 0048) obtained
during 2005. PG 1618B was discovered to be a variable star by Silvotti et al. (2000; hereafter
S00) who detected frequencies of 6.95 and 7.18 mHz (P ≈ 144 and 139 s respectively) from
short data runs (< 2.3 hrs) obtained during seven nights, three of which were separated
by three months. PG 1618 is an optical double consisting of a main sequence F-type star
(component A) with an sdB star (component B) at a separation of 3.7 arcseconds. The
combined brightness is V = 11.8 while the sdB component has V ≈ 13.5. The discovery
data used a combination of photoelectric photometry, which did not resolve the double,
and CCD data which did. The combined flux of the double in the former would have
reduced the pulsation amplitudes. From spectra obtained at Calar Alto, S00 determined
that Teff = 33 900± 1 500K and log g = 5.80 ± 0.2.
PG 0048 was discovered to be a variable star during 10 observing runs varying in length
from 1 to 4 hours obtained in 1997 and 1998 (Koen et al. 2004; hereafter K04). As they used
a variety of instruments spread over the span of a year, K04 were only able to combine two
consecutive runs, from which they detected seven frequencies. However, it is obvious from
their temporal spectrum that pulsation amplitudes and possibly frequencies were changing
(their Fig. 3). K04 contributed this to unresolved frequencies caused by their short duration
data runs. K04 also obtained an optical spectrum and examined 2MASS colors to determine
that PG 0048 has a G0V-G2V companion; though the orbital parameters are unknown and
no Teff or log g estimates were given.
Here we report the results of a new program to resolve the pulsation spectra of these
two stars. Section 2 describes the observations, reductions, and analysis for PG 1618B, and
§3 the same for PG 0048. Section 4 compares the results for both stars and discusses the
implications for asteroseismology.
2. PG 1618B+563B
2.1. Observations
PG 1618B was observed from 5 observatories (Baker, MDM, McDonald, Lulin, and
Suhora) over a 45 day period during spring 2005. Data obtained at MDM (2.4 m) and
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McDonald (2.1 m) observatories used the same Apogee Alta U47+ CCD camera. This camera
is connected via USB2.0 for high-speed readout, and our binned (2×2) images had an average
dead-time of one second. Observations at Baker (0.4 m) and Lulin (1.0 m) observatories
were obtained with Princeton Instruments RS1340 CCD cameras. Data obtained at Baker
Observatory were binned 2× 2 with an average dead-time of one second, while observations
from Lulin Observatory used a 392× 436 subframe at 1 × 1 binning with an average dead-
time of six seconds. The Mt. Suhora Astronomical Observatory (0.6 m) data were obtained
with a photomultiplier tube photometer which has microsecond dead-times. Observations
obtained at McDonald, Baker, and MDM observatories used a red cut-off (BG40) filter, so
the transmission is virtually the same as the blue photoelectric observations from Suhora
observatory. Observations from Lulin Observatory used a Johnson B filter, which slightly
reduced the amount of light collected compared to other observations, but does not impose
any significant phase and/or amplitude changes compared to other observations (Koen 1998;
Zhou et al. 2006). Accurate time was kept using NTP (Baker, McDonald, and MDM
observatories) or GPS receivers (Lulin and Suhora observatories) and corrected to barycentric
time during data reductions.
Standard procedures of image reduction, including bias subtraction, dark current and
flat field correction, were followed using iraf1 packages. Differential magnitudes were ex-
tracted from the calibrated images using momf (Kjeldsen & Frandsen 1992) or occasionally
they were extracted using iraf aperture photometry with extinction and cloud corrections
using the normalized intensities of several field stars, depending on conditions. Photoelec-
tric data reductions proceeded using standard Whole Earth Telescope reduction packages
(Nather et al. 1990). As sdB stars are substantially hotter, and thus bluer, than typical
field stars, differential light curves using an ensemble of comparison stars are not flat due
to differential atmospheric and color extinctions. A low-order polynomial was fit to remove
these trends from the data on a night-by-night basis. Finally, the lightcurves are normalized
by their average flux and centered around zero so the reported differential intensities are
∆I = (I/〈I〉) − 1. Amplitudes are given as milli-modulation amplitudes (mma) with an
amplitude of 10 mma corresponding to an intensity change of 1.0% or 9.2 millimagnitudes.
The companion of PG 1618B adds a complication to the reductions in that data obtained
at McDonald and MDM observatories resolved the optical double, but those from other
observatories did not. Using our data for which the stars are resolved, we determined that
component A contributes 67.3% of the total flux. To correct the unresolved data, we created
iraf is distributed by the National Optical Astronomy Observatories, which are operated by the As-
sociation of Universities for Research in Astronomy, Inc., under cooperative agreement with the National
Science Foundation.
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a fitting function by smoothing the data over many points (around 50 points per box),
multiplying it by 0.673 and subtracting it from the unresolved data. While this process
effectively removes the flux from PG 1618A, it cannot correct for the noise of this component,
which remains behind. As such, the corrected data are noisier, limiting their usefulness.
Multiple-longitude coverage was only obtained during the first week of the campaign.
A total of 73.5 hours of data were collected from three observatories (McDonald, Lulin, and
Suhora) which provided a 47% duty cycle. Subsequent data were obtained only in Missouri
(Baker Observatory) and Arizona (MDM Observatory). These data serve to extend the
timebase of observations (increasing the temporal resolution) and to decrease the noise in the
temporal spectrum. Lightcurves showing the coverage of the first six nights of observations,
as well as a portion of a typical MDM run, are provided in Fig. 1.
2.2. Analysis
Our campaign was quite long (about 45 days) with a concentration of data at the
beginning, but the best data (highest S/N and best conditions) were obtained at the end.
We therefore grouped combinations of nightly runs into the subsets given in Table 2 for
analysis. Table 2 also provides the temporal resolution (calculated as 1/trun where trun is
the length of the observing run) and the 4 σ detection limit (calculated using areas adjacent
to the pulsation but outside of their window functions). The temporal spectra and window
functions of these subsets are plotted in Fig. 2. A window function is a single sine wave of
arbitrary, but constant amplitude sampled at the same times as the data. The central peak
of the window is the input frequency, with other peaks indicating the aliasing pattern of
the data. Each peak of the data spectrum intrinsic to the star will create such an aliasing
pattern. As is evident from Fig. 2, the MDM data was significantly better than the rest, so
we began our analysis with that subset.
Analysis of the MDM data was relatively easy and straightforward. In Fig. 3, the top
panel shows the original FT, while the bottom panel shows the residuals after prewhitening
by the frequencies indicated by arrows. The insets show the window function (top right)
and an expanded view of a 65µHz region around the close doublet. Frequencies, amplitudes
and phases were determined by simultaneously fitting a nonlinear least-squares solution to
the data. Since during the MDM observations the amplitudes were relatively constant, the
solution proceeded as expected and prewhitening effectively removed the peaks and their
aliases. The formal solution and errors for the MDM subset are given in Table 3.
Examination of the other data sets indicates that while PG 1618B was well-behaved
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during the MDM observations, it was not at other times. This was most noticeable in our
examination the data collected during the first week. While the temporal spectrum has the
cleanest window function, no peaks are detected above the 4σ detection limit (1.53 mma)
even though peaks are detected in individual Lulin and McDonald runs (see Fig. 4 which
will be discussed in §2.3). Combining the well-behaved MDM data with any other data set
results in a decrease of amplitudes, indicating that outside of the MDM data, the amplitudes,
phases, or even frequencies are not constant. If the pulsation properties were consistent
throughout the campaign, data collected at smaller telescopes, with low S/N would still
have been useful for reducing the overall noise. Unfortunately, such was not the case so all
we can really conclude is that the MDM data detected all the pulsations that were occurring
(above the detection limit) at that time while the pulsations intrinsic to PG 1618B must
have been more complex at other times. It would be interesting to study the longer-term
variability of PG 1618B, but using only 2 m-class telescopes.
Outside of the combined data sets, there are two frequencies that are detected above the
4σ detection limit during individual runs. The least-squares solutions for these frequencies
are provided at the end of Table 3. The frequency at ≈ 9199µHz was above the noise only
in the March 22 McDonald data, though a peak at the same frequency also appears in the
Suhora data during March 16 and 21. The frequency at ≈ 8179µHz was above the noise only
in the April 30 MDM data though corresponding peaks appear in the McDonald March 18,
Lulin March 18, and Suhora March 21 runs. Since they are detected above the 4σ detection
criteria for those runs, we include them in our discussion that follows.
2.3. Discussion
Silvotti et al. (2000) detected two frequencies in their discovery data while we clearly
resolve four frequencies from our MDM dataset, and two more from individual data runs,
bringing the total to six independent frequencies. We calculate the S00 resolution to be
5.5 µHz and estimate their noise to be about 1 mma though this is misleading in that
because of their short data runs, their window function effectively covers all of the remaining
pulsations. However, for a strict comparison, we can say that our MDM data alone are 3×
better in resolution and have a detection limit twice as good, though in a practical sense our
MDM data are far superior solely based on the duration of our individual data runs. Had S00
observed for longer durations (particularly with their CCD setup), their data would likely
have been similar to the same number of runs from our MDM set. However, it is clearly safe
to say that our MDM data alone are insufficient to describe the complexity of pulsations
occurring within PG 1618B. As such, the remainder of our discussion which is based on the
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MDM data, can only be a minimum of what is really occurring.
2.3.1. Constraints on the pulsation modes
One of our goals is to observationally identify or constrain the pulsation modes of indi-
vidual frequencies. Differing m components of the same degree ℓ have degenerate frequencies
unless perturbed, typically by rotation. If a star is rotating, then each degree will separate
into a multiplet of 2ℓ+ 1 components with spacings nearly that of the rotation frequency of
the star. As such, observations of multiplet structure can constrain the pulsation degree (for
examples, see Winget et al. 1991 for pulsating white dwarfs and Reed et al. 2004 for sdB
stars). For PG 1618B, there are no two frequency spacings that are similar, though there are
not many frequencies to work with. The lack of observable multiplet structure is typical of
sdBV stars but is likely limited to four possibilities: i) Rotation is sufficiently slow that all
m values remain degenerate within the frequency resolution of our data; ii) our line of sight
is along the pulsation axis, with sin i ≈ 0, leaving only the m = 0 mode observable because
of geometric cancellation (Pesnell 1985; Reed, Brondel, & Kawaler 2005); iii) rapid internal
rotation is such that m multiplets are widely spaced and uneven (Kawaler & Hostler 2005);
or iv) at most one pair is part of a multiplet with an unobserved component of the multiplet.
Spectroscopy can only rule out large splittings for possibility (i) as spectroscopic limits
are typically ≈ 10 km/s and from Fig. 1 of S00, PG 1618B appears as a “normal” sdB
devoid of rapid rotation. Possibility (ii) can only be determined for cases in which the sdB
star is part of a close binary such that the rotation and orbital axes can be inferred to be
aligned. Since PG 1618 is only an optical double at wide separation, it does not constraint
the alignment of the surface spherical harmonics. Similarly, possibility (iii) is virtually
impossible to decipher unless the star pulsates in many (tens of) frequencies, and would
still require some interrelation of spacings for modes of the same degree (Kawaler & Hostler
2005). Possibility (iv) also remains an option, though a difficult one to constrain. Higher
resolution (and perhaps longer duration) spectroscopy would help to answer this question,
and multicolor photometry or time-series spectroscopy might also be able to discern the
spherical harmonics (see Koen 1998 and O’Toole et al. 2002 for examples of each).
Another quantity that can be used to constrain the pulsation modes is the frequency
density. Using the assumptions that no two frequencies share the same n and ℓ values (except
possibly the close pair at 6946µHz), and that high-degree ℓ ≥ 3 modes are not observationally
favored because of geometric cancellation (Charpinet et al. 2005; Reed, Brondel, & Kawaler
2005), we can ascertain whether the frequencies are too dense to be accounted for using only
ℓ ≤ 2 modes. From stellar models, a general rule of thumb is to allow three frequencies
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per 1000µHz. We will ignore f1 which is too distant in frequency space and count f5 and
f6 as a single degree ℓ. This leaves four frequencies within 1235 µHz; which can easily be
accounted for using only ℓ ≤ 2 modes. Indeed, even if f5 and f6 do not share their n and ℓ
values, the frequency spectrum can still accommodate all of the detected frequencies without
invoking higher degree modes. Of course this does not mean that they are not ℓ ≥ 3 modes,
only that the pulsation spectrum is not sufficiently dense to require their postulation.
2.3.2. Amplitude and phase stability
If pulsating sdB stars are observed over an extended time period, it is common to detect
amplitude variability in many, if not all, of the pulsation frequencies (eg. O’Toole et al. 2002;
Reed et al. 2004; Zhou et al. 2006). Such variability can occasionally be ascribed to beating
between pulsations too closely spaced to be resolved in any subset of the data. However,
variations often appear in clearly resolved pulsation spectra where mode beating cannot be
the cause. For PG 1618B, frequencies f1 and f2 are only detected during a single run each
and frequencies f5 and f6 are too closely spaced to be resolved during individual runs,
leaving only frequencies f3 and f4 available for analysis of amplitude variations.
Figure 4 shows the amplitude and phases of these two frequencies for individual data
runs from McDonald and MDM observatories as well as a single Lulin run (marked by a
triangle); these frequencies were not detected elsewhere. During the MDM observations,
the amplitudes and phases for both frequencies are nearly constant (to within the errors)
except for one low amplitude, but they have a significant variation in the McDonald and
Lulin data. Of particular interest are the phases and amplitudes of f4, especially those
during day three, in which we have both a McDonald and Lulin run that do not overlap in
time. Between these two runs, the amplitude, which had been decreasing during the previous
three days, suddenly increases to begin the same declining pattern again. The phases also
show a bimodal structure early in the campaign with phases near −0.20 and +0.25 with the
first phase jump occurring coincident with the amplitude increase. Except for the lack of
sinusoidal amplitude variation, this has the appearance of unresolved pulsations. However,
if the two unresolved frequencies had intrinsic amplitude variability, then it could reproduce
the observations. However the MDM observations, which are not only steady, but have f4
phases intermediate to the McDonald and Lulin data, do not support this. Clearly, during
the week of MDM observations, PG 1618B had neither amplitude nor phase variations and
since the MDM phases do not coincide with phases from earlier in the campaign, unresolved
pulsations are unlikely. Since the data obtained at MDM and McDonald observatories used
the same acquisition system and time server (NTP), errors in timing also seem unlikely.
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3. PG 0048+091
3.1. Observations
We originally observed PG 0048 as a secondary target during a campaign on KPD 2109+4401
(Zhou et al. 2006). Those data revealed a complex pulsation spectrum which we could not
resolve with such limited sampling and a short time base. As such, PG 0048 was re-observed
as a multisite campaign during Fall 2005. Five observatories participated in the campaign
with the specifics of each run provided in Table 4. Though we were a bit unlucky with
weather, over the course of our 16 night campaign we obtained 167.4 hours of data for a
duty cycle of 44%. Details of the observing instruments and configurations are the same
as for PG 1618B, except for the following: SAAO (1.9 m) used a frame transfer CCD with
millisecond dead-times but only an ≈ 30 × 40 arcsecond field of view, which resulted in no
comparison stars within the CCD field. As such no transparency variations could be cor-
rected and only photometric nights were used. Tubitak Observatory used a Fairchild CCD447
detector; during the first run the images had 1× 1 binning with a dead-time of 102 seconds,
while subsequent runs used 2×2 binning with a dead-time of 29 seconds. Bohyunsan Optical
Astronomy Observatory (BOAO 1.9 m) data were obtained with a SiTe-424 CCD windowed
to 580×445 pixels, binned 2×2 with an average dead time of 14 seconds. MDM and SAAO
used red cut-off filters, making their responses very similar to blue-sensitive photoelectric
observations, while Lulin, BOAO, and Tubitak used no filter making their sampling more
to the red. As pulsations from sdB stars have little amplitude dependence in the visual and
no phase dependence (Koen 1998; Zhou et al. 2006), mixing these data is not seen as a
problem.
The standard procedures of image reduction, including bias subtraction, dark current
and flat field correction, were followed using iraf. Differential magnitudes were extracted
from the calibrated images using momf (Kjeldsen & Frandsen 1992), except for the SAAO
data for which we used aperture photometry because there were no comparison stars. As
described for PG 1618B, we again used low-order polynomials to remove airmass trends
between our blue target star and the redder comparison stars. The lightcurves are normalized
by their average flux and centered around zero, so the reported differential intensities are
δI = (I/〈I〉) − 1. Figure 5 shows the lightcurve of PG 0048 with each panel covering two
days.
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3.2. Analysis
During the campaign, we completed a “quick-look” analysis of data runs as early as
possible to ascertain the data quality and the pulsation characteristics of the star. We
noticed early on that the temporal spectra of PG 0048 changed on a nightly basis with
pulsation frequencies appearing and then disappearing on subsequent nights. Likewise, we
knew that our analysis would be complicated by severe amplitude variations which would
limit the usefulness of prewhitening techniques and could create aliasing. Figure 6 shows the
effects of amplitude variations. The full panels are pulsation spectra for three groups of data:
All of the data; data obtained from September 30 through October 3; and from October 7
through October 11. The right insets are the corresponding window functions plotted on the
same horizontal scale. At such large scales, the windows appear as single peaks and show
that the changes in the FTs are not caused by aliasing. The central insets are individual
data runs within the larger set and show the variability between runs. When sets of data
are combined in which the peak amplitudes are not constant, an FT will show the average
amplitude. For the frequencies that appear in only a few runs, the amplitudes are effectively
quashed in the combined FT. As PG 0048 is the most pulsation variable sdB star currently
known, our immediate goal is to glean as many observables from these data as possible.
While we do provide some interpretation, our aim is to provide sufficient information for
theorists to test their models.
The complexity of the data meant it was necessary to analyze it using multiple tech-
niques: We performed standard Fourier analyses on combined sets of observations to increase
temporal resolution and lower the overall FT noise and analyzed individual runs of the best
quality data. The analysis of individual runs represents a time-modified Fourier analysis,
which is essentially a Gabór transform, except that we replace a Gaussian time discrimina-
tor with the natural beginnings and endings of the individual runs. As the best individual
runs are not continuous with time (and nearly all are from MDM Observatory) the use
of a Gaussian-damped traveling temporal wave discriminator (a standard Gabór transform)
would not enhance the results. The temporal spectra of these runs are shown in Fig. 7. Runs
mdm1005 and mdm1009, though long in duration, have gaps in them because of inclement
weather, whereas the other 12 runs are gap-free. For these 12 runs aliasing in the FT is not
a problem and the only constraints are the width of the peaks, which are determined by run
length, and the noise of the FT, which is a combination of the signal-to-noise of each point
and the number of data points within the run.
Frequencies, amplitudes and phases were determined using two different software pack-
ages, Period04 (Lenz & Breger 2004) and a custom (Whole Earth Telescope) set of non-linear
least squares fitting and prewhitening routines. Each of the three data combinations in Fig. 6,
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the 12 gap-free data runs plotted in Fig. 7, the three data runs obtained during 2004, and
the 10 runs from the discovery data (kindly provided by Chris Koen) were reduced using
both software packages. Overall, more than 35 frequencies were fit during at least one data
run. Table 5 provides information for 28 frequencies which have been detected above the
4σ detection limit. Column 1 lists a frequency designation; column 2 the frequency as fit
to the highest temporal resolution data set in which each frequency is detected with the
formal least-squares errors in column 3. Column 4 provides the standard deviation of the
corresponding frequencies detected in individual runs and column 5 gives the number N of
individual runs in which that frequency was detected (from twelve 2005 runs and three 2004
runs). Tables 6 and 7 provide the corresponding amplitudes as fit for individual runs and var-
ious combinations of data acquired during the 2005 campaign, a re-analysis of the discovery
data, and the 2004 MDM data. The last two rows of these tables provide the 4σ detection
limits and temporal resolutions for the runs. Our determination that these frequencies are
real and intrinsic to the star is based on i) detection by both fitting software packages, and
amplitude(s) higher than the 4σ detection limit with ii) detection during several data runs,
and/or iii) detection at amplitudes too large to be associated with aliasing.
3.3. Discussion
3.3.1. Frequency content
During our 2005 multisite campaign, we detected 24 pulsation frequencies from indi-
vidual data runs plus an additional frequency from the combined data set (which was also
detected in 2004). We recover all seven of the frequencies detected in the discovery data
(K04), but only 14 of the 16 frequencies detected from our 2004 data. As can be seen from
Table 5, PG 0048 shows an atypically large range of frequencies for sdBV-type pulsators,
especially when considering that only one frequency, f23 (11103.3µHz), can be identified as a
linear combination (of f6 and f7). As noted in Table 5, only one frequency is detected in all
of our data (f2: ∼5244.9µHz) while the next most common frequency (f13: ∼7237.0µHz) is
detected in only 11 of the 15 runs. Several frequencies are only detected once or twice (e.g.
f12: ∼7154.3µHz), and so we should test if their amplitudes are sufficient to consider them to
be real. If a particular frequency has amplitudes that are 1σ above the detection limit, then
we might only expect to detect it 68% of the time2. Figure 8 shows the amplitudes and 1σ
errors for 12 different frequencies (four frequencies per panel) and the detection limit (solid
2This is a lower limit since statistically, the pulsation amplitude is equally likely to be higher than 1σ
from the detected level rather then below it.
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line) for individual runs. The (black) circles in the top panel are for f2, which is detected in
every run. However, the two frequencies indicated by (magenta) squares in the middle and
bottom panels are only detected once, even though they are > 1σ above the detection limit.
If their amplitudes were nearly constant (to within their errors), they would be detected at
least 68% of the time. Another way to show this is in panel a of Fig. 9 where the detections
are plotted against their significance. We detect a total of 24 frequencies from 12 individual
runs from our 2005 data. If we detected all 24 frequencies from every run, we would have
made 288 detections, while we only actually made 75. The solid line shows the number of
individual detections cumulative with significance (the number of standard deviations the
detection was above the detection limit). In other words, 49 of our 75 detections were 2σ
or less above the detection limit. The dashed line is the standard Gaussian probability
distribution which shows that at 1σ significance, we should have made at least 196 (68%)
detections. Since 64% of our detections are ≥ 1σ, our 75 detections is well short of what we
should have detected, indicating that the pulsation amplitudes are really falling below the
detection limit. Panel b compares the number of actual detections to the maximum pulsation
amplitude. As expected, there is some correlation as the higher the amplitude, the easier it
is to detect that frequency. Additionally, the highest amplitude (and therefore most easily
detected) frequency is the same during 1997, 1998, and 2004 (f8: ∼5612.2µHz) but is only
detected in ∼ 1/3 of our 2005 data runs during which f2 (∼5244.9µHz) had the highest
amplitudes. This change in pulsation amplitudes will be further discussed in §3.3.3.
3.3.2. Constraints on mode identifications
As in §2.3.1, one of the best ways to relate pulsation frequencies to pulsation modes is
via multiplet structure. With such a rich pulsation spectrum, it seems likely that some of
the frequencies should be related by common frequency splittings. If not, then the pulsation
spectrum is too dense (discussed below) for the frequencies to consist of only low-order ℓ ≤ 2
modes. There are two different spacings that occur many times with splittings near 972
and 41.1 µHz. Table 8 lists these frequencies and the deviation from the average spacing
between them, while Fig. 10 shows them graphically. While the spacings may be important,
it is difficult to attach any physical meaning to them. A spacing of 972 µHz is far too large to
be associated with stellar rotation, as it corresponds to a rotation period of only 17 minutes.
There is currently no high-resolution spectrum of PG 0048, and the star’s main-sequence
companion would complicate any attempts to measure its rotation velocity. However, the
typical v sin i of sdBs is less than ∼ 5 km s−1 (Heber, Reid, & Werner 2000). If we try to
explain this large splitting using asymptotic theory (consecutive n overtones rather than m
multiplets), we would expect successive overtones of the radial index to be roughly evenly
– 13 –
spaced for large n, with one series for each ℓ degree. But the spacings observed in PG 0048 are
irregular, requiring eight degrees (one for every line in Table 8), and 3 ≤ ℓ ≤ 7 modes have
reduced visibility because of geometric cancellation (Charpinet et al. 2005; Reed, Brondel,
& Kawaler 2005). Theory suggests that sdBV stars should pulsate in low overtone modes
and the spacings between successive low overtone modes should differ by several hundreds
of µHz (Charpinet et al. 2002), which again does not match what is observed.
Another asymptotic-like relation is the “Kawaler scheme” which has been recently pre-
sented by Kawaler et al. (2006) and Vuc̆ković et al. (2006). Though it is not compatible with
the low overtone (n) pulsation theory associated with sdB pulsators, an improved frequency
fit can sometimes be obtained using an asymptotic-like formula;
f(i, j) = fo + i× δ + j ×∆
where i has integer values, j is limited to values of −1, 0, and 1, δ is a large frequency spacing
and ∆ a small one. However, for the case of PG 0048, the small spacings are unrelated to the
larger spacings and the larger spacings themselves do not interrelate, but rather appear in
sets with differing spacings between the sets. So this scheme is not applicable for PG 0048.
The smaller, 41.1 µHz spacings are more akin to what we would expect for rotationally
split multiplets, though still large compared to typical measured rotation rates. If the low-
frequency set are all components of a single multiplet, it would require very high degree
(ℓ ≥ 5) pulsations, which are not observationally favored (Charpinet et al. 2005; Reed,
Brondel, Kawaler 2005). The same is true for the high-frequency set if f13 and f16 belong
to the same set. Since they are separated by 7×41.0µHz, it is certainly possible that f12-f13
and f16-f18 are just two pairs, but if they are all combined into a single multiplet, it would
require ℓ ≥ 5 again. Yet these could also be just chance superpositions.
Since PG 0048 pulsates in so many frequencies, it is important to test the significance of
the spacings discussed above. We did this by producing Monte Carlo simulations, randomly
placing 28 frequencies within 6000 microHz of each other, and counting how often we could
detect 14 frequency splittings the same to within about 5%. This criterion would find all but
one of the splittings we actually observe. After analysing over one million simulations, we
detected at least 14 splittings in nearly every case. In other words, the splittings we observe
are not statstically significant.
Another tool we can use to place constraints on mode identifications is the mode density.
As mentioned in §2.3.1, models predict roughly one overtone (n) per degree (ℓ) per 1000 µHz
and though we have detected 28 frequencies, they are spread across nearly 6000 µHz. Since we
do not detect any multiplets that can be unambiguously associated with rotational splitting,
it is likely the pulsations are degenerate in m. The average mode density is 4.8 frequencies
– 14 –
per 1000 µHz, which is too high to accommodate only ℓ ≤ 2, with degenerate m modes
and it gets worse as the frequencies are not quite distributed equally, but rather fall into
loose groups, enhancing the density locally. The regions between 5200 and 6200 and 6600 to
7600 µHz each contains 9 frequencies, and the region from 8800 to 9800 µHz has 5 frequencies.
If all 2ℓ+1 multiplets were filled (9 frequencies per 1000 µHz), the frequency density would
not require any ℓ ≥ 3 modes. However, as we do not detect appropriate multiplets within
these frequency regions, the most likely result is that PG 0048 has too high a frequency
density to exclude ℓ ≥ 3 modes using current models.
3.3.3. Amplitude and phase variability
Nearly all sdBV stars show some amount of amplitude variability. However, no other
sdBV star has shown variability like that detected in PG 0048. An example of this is shown
in Fig. 11, where the same 350 µHz region is shown for the three data sets in Fig. 6. The
bottom two panels are subsets of the top panel, which includes all of the data, and indicate
how strikingly the pulsation spectrum changes with time.
We can apply some constraints to the timescale of amplitude variability. Since pulsa-
tion frequencies can change amplitudes (even to the point of being undetectable) between
individual runs, and in particular between runs from different observatories but for the same
date, the next step was to divide up our longer data runs into halves. The FTs for six such
runs are shown in Fig. 12 with each half containing about four hours of data. Particularly
for runs mdm1007 (which can also be compared to saao1007) and mdm1010, the amplitudes
change by factors of two over times as short as four hours. Low amplitude frequencies can
easily become undetectable within that time. Figure 13 compares the maximum and aver-
age amplitudes detected in individual runs (the black circles and blue squares, respectively)
to detections in groups of data (the lines) from Table 6. If the amplitudes were simply
wandering around between values detected in individual runs, then the amplitude of the
combined data would be an average of these values (the blue squares). However, since the
combined amplitudes are significantly lower then the average amplitudes from individual
runs, something else must be occurring to reduce the amplitudes.
Since changes in phase can impact pulsation amplitudes, we investigated that next.
Figure 14 shows phases for eight of the most-often detected frequencies. f2 is the only
frequency detected in every run, and we include the half-night analysis for it. For the other
seven frequencies, we only determined phases for those runs in which they were detected
above the 4σ limit. While Fig.14 indicates that most phases do not appear constant with
time, most are within 20-30% of a central value.
– 15 –
To isolate and test the impact phase variation creates on data like ours, we analyzed
simulated data with the following properties: The data are represented by a noise-free, single-
frequency sine wave sampled corresponding to our 12 best individual runs with frequency
f2. The amplitudes of each individual run are fixed at the measured values for f2 from
Table 6 and we assume that no phase changes occur during an individual run. While the
properties of f2 may not be the most representative of the variations detected, it is the only
frequency detected every time, and so by using it, we sample the full range of amplitudes.
If we used a different frequency, we could not know the actual pulsation amplitude during
those runs without detections, and so would only be sampling the higher amplitude data
points. We created simulations with the following phase properties: No change in phase, a
fixed change of ±10% from the previous phase, a fixed change of ±20% from the previous
phase, phases that are randomly set at the beginning of each run, and with the actual phase
values for f2. The results of the simulations are given in Table 9 and the ratios are used in
Fig. 15 where they are compared with our observations. The last line of Table 9 presents
results with unique answers. As expected, there is a correlation between the amplitude and
the amount of phase variation in that increasing changes in phase between individual runs
decreases the measured amplitude of the data set as a whole. More useful are the ratios of
the average amplitude to the average and maximum of the individual amplitudes (〈A〉/〈A〉ind
and 〈A〉/Amax). These ratios can be compared with ratios from all frequencies, as has been
done in Fig. 15. The shaded regions are the 1σ ratios produced from the phase simulations
and the circles represent ratios with G4 data and the squares are for ratios using G1 data.
The frequency ordering is the same as for Fig. 12, and like Fig. 12, the results indicate
that for all but f2 and possibly f6, the amplitudes detected in groups of data are too low
compared to individual amplitudes. Additionally, except for f2, the phases of Fig. 14 (and
their standard deviations given in Table 10) are in discord with the amplitude ratios in that
the amplitudes are too low. There does not seem to be sufficient phase variation to produce
the low amplitudes of the group data. As such, it seems that more extreme circumstances
are required. However, that leads into a more speculative area which we save for §3.3.4.
We conclude our observational portion of this paper with a summary provided in Ta-
ble 10. In this table we have included all measurables (not in previous tables) discussed in
this section and some that will be useful for the next section. Columns 2, 3, and 4 consider
the number of expected detections based on the average significance of the actual detections;
Columns 5 through 8 detail amplitudes detected for individual 2005 observing runs; Columns
9 through 12 provide ratios of individual to group amplitudes; and Column 13 lists the 1σ
deviations of pulsation phases.
– 16 –
3.3.4. A possible cause of the amplitude/phase variability
Can we determine the cause of the erratic behaviour of PG 0048’s frequencies and
amplitudes? We suggest that, with the possible exception of f2, the oscillations may be
stochastically excited. While this is counter to current theory, supporting observations that
we have in hand are 1) amplitudes that vary significantly between every individual run, and
in less than ∼4 hrs; 2) the combined amplitudes are significantly lower than the average value
indicating that phases are not coherent on these timescales; 3) the peaks in the FTs appear
similar to those of known stochastic pulsators (compare Fig. 11 to Fig. 1 of Bedding et al.
(2005) or Fig. 2 of Stello et al. (2006) for stochastic oscillators to those in, for example,
Reed et al (2004) for “normal” sdB stars); and 4) the number of actual frequency detections
compared to the expected number based on significance (the ratio of the two lines in the left
panel of Fig. 9). We note that this is not conclusive evidence, but is suggestive and so we
will pursue a stochastic nature for PG 0048’s pulsations in the remainder of this section.
Recently, Stello et al. (2006) found that short mode lifetimes in red giants can severely
limit the possibility of measuring reliable frequencies. The difficulty arises because the
frequencies can disappear entirely and when they are re-excited (even if this occurs prior to
complete damping), they do not maintain the same phase. The parallel with our analysis
of PG 0048 are clear and this kind of variability has been seen before in sdBVs. In a study
of KPD2109+4401, Zhou et al. (2006) found substantial variation in the amplitudes of two
modes during their 32 night campaign. A brief analysis found that at least one of these
modes, and possibly both of them, satisfied the criterion outlined by Christensen-Dalsgaard
et al. (2001; hereafter JCD01) for stochastically excited pulsations, rather than overstable
driving. The criterion compares the ratio of amplitude scatter to the mean amplitude; for
stochastic pulsations, this ratio should be ≈ 0.52. Stochastic processes in pulsating sdB
stars have also been discussed by Pereira & Lopes (2005) in the context of the complex sdB
pulsator PG1605+072, which is known to have variable amplitudes (O’Toole et al. 2002;
Reed et al. 2007, in press). Using the JCD01 criterion, Pereira & Lopes deduced that none
of the modes of that star were consistent with stochastic excitation. However, O’Toole et al.
(2002) noted amplitude changes between years, while Pereira & Lopes (2005) only studied
7 nights of data, and as such their analysis was likely affected by the short length of their
time series.
A limitation to the JCD01 test is that the damping times of the oscillations should be
longer than the timescale used for determining the amplitudes. Our analysis of ∼ 4 hour
segments of PG 0048 data indicate that amplitude variations are on very short timescales
that are shorter than the observing time for individual runs (see Figs. 7 and 12). We provide
the JCD01 parameter values σA/〈A〉 in Column 7 of Table 10, but we suggest that the JCD01
– 17 –
test is not appropriate for PG 0048. Aside from the JCD01 test, we can attempt to reproduce
some of the observational properties using simple simulations with damped and randomly
re-excited frequencies. The complexity of the actual data is such that we cannot hope to
reproduce it directly, but instead will strive to fit the observations listed at the beginning of
this section. Our simulations follow the simple prescription (equations 2 and 3) of Chaplin
et al. (1997) summarized as follows: The pulsations themselves are described by sine waves
of the form A(t) = A cos((2π · f)(t − φ)), with the amplitude modified in two ways; it is
damped exponentially as A = Ao exp(−te/td) where Ao is the maximum amplitude, te is the
time since the last excitation and td is the damping timescale. The pulsation is re-excited by
setting te = 0 when time t exceeds an excitation timescale (texc). The time before the first
re-excitation is randomly set to some fraction of the excitation timescale and every time the
pulsation is re-excited, the phase is randomly set and the excitation timescale and pulsation
amplitude can vary randomly by up to 20% or their original values. The free parameters of
the simulation are the input amplitude, which is the maximum amplitude attainable and the
excitation and damping timescales (Ao , texc and td, respectively). The simulation includes
frequencies, amplitudes, and phases for up to 100 pulsations with an unlimited number of
data runs (input as run start time, the number of data points, and cycle time). We will
concentrate on matching the MDM runs in Table 6. This is the G1 data set with an average
run length of 9.35 hours (∼ 0.4d), containing ∼ 2800 data points each, an average detection
limit of 0.88 mma, and average ratios AG1/〈A〉 = 0.44± 0.14 and AG1/Amax = 0.37± 0.09.
To match the observational constraint that the pulsation amplitude can reduce by half
in a four hour span, the damping timescale is necessarily less than 5.8 hours. With this
constraint, we produced a grid of simulations with 1 ≤ td ≤ 7 hours in 1 hour increments
and 1 ≤ texc ≤ 33 hours in 2 hour increments. Qualitatively, if no re-excitations occur
during an individual run, the FT is single-peaked whereas multiple re-excitations create a
variety of complex, multi-peaked FTs, depending on how similar the randomized phases were
(the less alike the phases for each re-excitation, the lower the overall amplitude and more
and similar-amplitude peaks appear in the FT). For small texc, the FT becomes increasingly
complex with most simulations resulting in many low-amplitude peaks distributed across a
couple hundred µHz. However, such complex patterns are not consistent with observations
and so we discount small values for texc. Amplitudes in the FT are reduced with large values
of texc and small values of td while the scatter increases with increasing values for both. An
increase in amplitude scatter is necessary to produce the low rate of detections.
As we now have all the pieces in place, we can ask how well the simulations reproduce
the observational constraints we set at the beginning of this section. A selection of the
results are shown in Figs. 16 and 17. Panels a, b, and c of Fig. 16 have td fixed at 5 hours
and vary texc whereas panels d, e, and f fix texc at 19 hours and vary td. Panels a and d
– 18 –
are the results for individual runs and panels b and e are for the combined nine-run data
set. The points represent the average (〈A〉) with 1σ deviations while the lines indicate
maximum (Amax) and minimum amplitudes. Panels c and f show the ratios 〈AG1〉/Amax and
〈AG1〉/〈Aind〉; the average amplitude from the combined data divided by the maximum or
average amplitude from the individual runs. Figure 17 shows the expected rate of detections
calculated in the following manner: The detection limit was calculated using the observed
ratios 〈Amax〉/0.88 = 1.56 and 〈A〉/0.88 = 1.32 for the MDM data with an average detection
limit of 0.88 mma and solving for the new detection limit. The dotted line is the observed
detection rate of 26% and it is interesting that none of simulations are that low if using
the average detected amplitude. While the overall amplitudes can become quite small, the
average follows that, which is used to calculate our detection limit in the top panel. However,
most values of td matched the observed rate near texc = 19-21 hours using Amax. The
simulations that best fit the observational constraints are those which have 4 ≤ td ≤ 6 hrs
and 13 ≤ texc ≤ 21 hrs. The lower values of texc better fit the 〈AG1〉/A ratios while the larger
values are a better match for the detection rate. These relatively simple simulations are able
to fit all of the observed constraints, thus explaining the amplitude variations, their lower
detection limits in groups of data, the relatively low detection rate and the appearance of the
peaks in the FT. What they cannot explain however, is the relative lack of phase variability
in some frequencies (though relatively few were measurable) and why stochastic processes
should occur in the first place.
Stochastic oscillations are usually presumed to be driven by random excitations caused
by convection (see Christensen-Dalsgaard 2004 for a review concerning the Sun). The
He II/He III convection zone in sdB stars was investigated by Charpinet et al. (1996),
who determined that it could not drive pulsations. However, it appears that they did not
investigate this zone for convective motions but rather as a driver for the κ mechanism. So
some ambiguity remains here. There is also convection or semi-convection in the cores of sdB
stars, and it is possible that the eigenfunctions could be sampling this region. Whatever the
case, the extreme amplitude and phase variability of PG 0048 poses a significant challenge
to the iron driving mechanism found by Charpinet et al. (2001) to excite pulsations in sdB
stars. Though it is beyond the scope of this paper to ascertain the cause of the random
amplitude variations, we find that the observed properties are consistent with our simplified
randomly excited simulations and that the amplitude spectrum resembles those of pulsators
that are stochastically driven.
– 19 –
4. Conclusions and Future Work
We have carried out multisite campaigns for two sdB pulsators, PG 1618+563B and
PG 0048+091 and in both cases, our observations were superior to the published discovery
data, yet questions concerning these two stars still remain. Our MDM observations of
PG 1618B (obtained under good conditions) show characteristics typical of about half the
objects in the sdBV (V361Hya) class: A small number of stable (in amplitude) frequencies
with a closely spaced pair. In contrast, the data obtained at McDonald observatory – under
non-photometric conditions – show PG 1618B to be a complex pulsator with four “regions”
of power showing amplitude and phase variability. An ensemble analysis of any combinations
of data other than the MDM set are hindered by poor least-squares fitting and amplitudes
reduced below detectability. Such poor fitting can be caused by unresolved frequencies
with intrinsic amplitude variability (O’Toole et al. 2002) or randomly excited pulsations
(Christensen-Dalsgaard 2004). So despite having expended considerable effort to obtain not
only multisite, but extended time-base observations, we were only reasonably successful at
detecting pulsations from our 2 m telescope data; and these show the star to be two-faced.
With this dichotomy of observational results, PG 1618B remains an interesting target for
more follow-up observations; particularly to examine its long-term frequency stability.
PG 0048 is much more complex than PG 1618B, yet it too has shown somewhat stable
pulsation amplitudes at one epoch (the discovery and 2004 data) and wildly variable ampli-
tudes at another (2005). Though an extremely rich pulsator with at least 28 independent
frequencies, many modes are only excited to amplitudes above the noise occasionally, often
for very short lengths of time. These behaviors are consistent with stochastic pulsations and
we have performed several tests along these lines. We simulated damped and re-excited pul-
sations and found that the observations were best matched with damping timescales between
4 and 6 hours and excitation timescales between 13 and 19 hours. We detected common fre-
quency splittings of 972 and 41 µHz which may be related to multiplet structure, but could
reproduce these using Monte Carlo splittings of random spacings. So while they may be
intrinsic to the pulsation of PG 0048, we cannot be sure. We can be sure that PG 0048’s
rich pulsation spectrum is too dense to be accounted for using only ℓ ≤ 2 modes regardless
of how many m components are present.
The observations presented in this paper provide some very interesting and confusing
results. Pulsations that appear stable during some times and variable at others; attributes
that have also been observed in other sdBV stars as well (KPD 2109+2752, PG 1605+072,
and HS 1824+5745, just to name a few). The pulsations in PG 0048 present observables
that seem best described by randomly excited oscillations which would be in contrast to the
proposed driving mechanism (Charpinet et al. 2001). If validated, it would represent a new
– 20 –
direction in sdB pulsations (and modeling too!). However, a longer time series may be the
only way to clarify the nature of the oscillations in this star. Ideally this would take place
on at least 2m-class telescopes and cover several weeks.
We would like to thank the time allocation committees for generous time allocations,
without which this work would not have been possible; Dave Mills for his time and help
with the Linux camera drivers; Chris Koen for contributing the discovery data and helpful
discussions; and the anonymous Referee for a helpful re-organization of the paper. This
material is based in part upon work supported by the National Science Foundation under
Grant Numbers AST007480. Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s) and do not necessarily reflect the views
of the National Science Foundation. JRE, GAG, and SLH were supported by the Missouri
Space Grant Consortium.
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This preprint was prepared with the AAS LATEX macros v5.2.
– 22 –
Table 1: Observation record for PG 1618B. The first two runs were obtained in 2003, while
the rest were obtained in 2004.
Run Date Start Length Int. Observatory
UT hr:min:sec (Hrs) (s)
suh16mar 17 Mar 00:07:58 3.3 10 Suhora 0.6 m
lul031705 17 Mar 16:46:33 5.6 10 Lulin 1.0 m
McD031805 18 Mar 04:34:00 1.1 5 McDonald 2.1 m
lul031805 18 Mar 15:14:24 6.1 10 Lulin 1.0 m
McD031905 19 Mar 08:40:40 3.1 5 McDonald 2.1 m
lul031905 19 Mar 19:58:41 1.3 15 Lulin 1.0 m
baker032005 20 Mar 04:29:30 5.6 25 Baker 0.4 m
McD032005 20 Mar 06:35:00 5.8 5 McDonald 2.1 m
lul032005 20 Mar 15:17:19 6.0 10 Lulin 1.0 m
suh20mar 20 Mar 18:59:00 8.3 10 Suhora 0.6 m
lul032105 21 Mar 16:44:59 0.7 15 Lulin 1.0 m
suh21mar 21 Mar 18:20:20 8.6 20 Suhora 0.6 m
McD032205 22 Mar 04:33:00 7.8 5 McDonald 2.1 m
suh22mar 22 Mar 18:40:40 2.2 20 Suhora 0.6 m
McD032305 23 Mar 04:12:10 8.0 5 McDonald 2.1 m
mdr299 29 Mar 04:11:15 4.7 15 Baker 0.4 m
mdr301 31 Mar 04:28:10 7.0 15 Baker 0.4 m
mdr302 02 Apr 02:53:10 8.1 10 Baker 0.4 m
bak040305 03 Apr 03:20:46 7.9 15 Baker 0.4 m
bak040405 04 Apr 04:23:10 5.6 15 Baker 0.4 m
bak040505 05 Apr 03:17:10 6.6 10 Baker 0.4 m
bak041405 14 Apr 03:45:00 7.1 10 Baker 0.4 m
bak041505 15 Apr 02:26:30 8.4 10 Baker 0.4 m
bak041605 16 Apr 02:50:30 7.9 10 Baker 0.4 m
bak041705 17 Apr 03:18:50 5.3 15 Baker 0.4 m
bak041805 18 Apr 02:56:00 7.7 15 Baker 0.4 m
mdm042605 26 Apr 04:25:30 7.3 5 MDM 2.4 m
mdm042705 27 Apr 04:16:50 7.6 5 MDM 2.4 m
mdm042805 28 Apr 04:14:00 7.8 3 MDM 2.4 m
mdm042905 29 Apr 04:18:00 1.6 5 MDM 2.4 m
mdm043005 30 Apr 04:04:30 7.8 3 MDM 2.4 m
mdm050105 01 May 04:05:30 7.8 5 MDM 2.4 m
mdm050205 02 May 03:36:40 5.3 5 MDM 2.4 m
– 23 –
Table 2: Subsets of data for PG 1618B. For column 2, observatories are 1) McDonald Obser-
vatory; 2) Suhora Observatory; 3) Lulin Observatory; 4) Baker Observatory; and 5) MDM
Observatory.
Set Observatory(ies) Inclusive Dates Resolution 4σ detection limit
µHz mma
McD 1 18 - 23 March 2.2 1.64
MDM 5 26 Apr - 02 May 1.9 0.55
McD+MDM 1, 5 18 - 02 May 0.3 0.59
Week 1 1, 2, 3 17 - 23 March 1.5 1.53
All 1, 2, 3, 4, 5 17 - 02 May 0.2 0.77
– 24 –
Table 3: Our least-squares fit solution for the pulsation periods, frequencies, and amplitudes
detected in PG 1618B. Formal least-squares errors are given in parentheses. Frequencies
marked with a dagger (†) were only detected during individual runs (one each) while the
remaining frequencies are from the MDM data set.
Des. Period Frequency Amplitude
(s) (µHz) (mma)
f1† 108.7092 (0.0518) 9198.85 (4.38) 1.79 (0.39)
f2† 122.2574 (0.0597) 8179.46 (4.00) 1.04 (0.21)
f3 128.9549 (0.0008) 7754.64 (0.05) 1.71 (0.09)
f4 139.0571 (0.0008) 7191.28 (0.04) 2.04 (0.09)
f5 143.9290 (0.0011) 6947.87 (0.05) 2.22 (0.10)
f6 143.9759 (0.0014) 6945.60 (0.07) 1.64 (0.10)
– 25 –
Table 4: Observations of PG 0048
Run Date Start Length Int. Observatory
UT hr:min:sec (Hrs) (s)
mdr285 10 Oct 04:21:00 6.2 15 MDM 1.3 m
mdr290 12 Oct 03:39:00 7.3 15 MDM 1.3 m
mdr295 14 Oct 03:34:00 7.4 15 MDM 1.3 m
boao 26 Sept. 10:30:20 5.0 10 BOAO 1.9 m
mdm092805 28 Sep 07:32:30 4.5 15 MDM 1.3 m
mdm092905 29 Sep 02:53:00 9.5 15 MDM 1.3 m
mdm093005 30 Sep 02:44:00 9.5 15 MDM 1.3 m
turkSep3sdb 30 Sep 17:37:54 9.3 10 Tubitak 1.5 m
mdm100105 01 Oct 02:46:00 9.4 10 MDM 1.3 m
turk1Octsdb 01 Oct 21:26:58 4.1 10 Tubitak 1.5 m
mdm100205 02 Oct 09:57:30 2.3 15 MDM 1.3 m
turk2Octsdb 02 Oct 20:51:36 4.8 10 Tubitak 1.5 m
mdm100305 03 Oct 02:32:00 9.1 10 MDM 1.3 m
turk3Octsdb 03 Oct 17:41:49 8.5 10 Tubitak 1.5 m
mdm100405 04 Oct 09:24:00 2.2 15 MDM 1.3 m
mdm100505 05 Oct 02:33:00 9.4 12 MDM 1.3 m
mdm100605 06 Oct 02:23:00 9.4 10 MDM 1.3 m
mdm100705 07 Oct 02:15:00 9.6 12 MDM 1.3 m
lul7Oct 07 Oct 18:32:20 1.6 20 Lulin 1.0 m
a024 07 Oct 20:28:14 6.2 10 SAAO 1.9 m
mdm100805 08 Oct 03:17:00 8.5 15 MDM 1.3 m
a039 08 Oct 21:31:48 5.4 10 SAAO 1.9 m
mdm100905 09 Oct 02:17:00 9.3 10 MDM 1.3 m
a057 09 Oct 20:11:53 4.1 10 SAAO 1.9 m
mdm101005 10 Oct 02:06:00 9.6 10 MDM 1.3 m
a077 10 Oct 19:58:49 6.5 10 SAAO 1.9 m
mdm101105 11 Oct 02:00:00 9.6 10 MDM 1.3 m
– 26 –
Table 5. Frequencies detected for differing subsets of PG 0048 data. Column 1 is the
frequency designation; column 2 is the frequency determined from a combined data set;
column 3 is the formal least-squares error from the combined data set; column 4 is the
standard deviation of frequencies detected from individual runs; and column 5 is the
number of individual 2005 and 2004 runs in which that frequency is detected. All
frequencies are given in µHz. († indicates frequencies that were not detected in 2005.)
Des. Freq. σfit σ N
f1 5203.1 - 5.9 2
f2 5244.9 0.16 4.6 15
f3 5287.6 0.03 3.6 8
f4 5356.9 0.05 9.8 8
f5 5407.0 0.07 3.2 1
f6 5465.1 - - 1
f7 5487.2 0.05 8.5 7
f8 5612.2 0.05 4.5 8
f9 5652.9 - - 1†
f10 6609.2 - 9.0 2
f11 6834.3 0.15 9.0 3†
f12 7154.3 0.08 1.4 1
f13 7237.0 0.04 7.0 10
f14 7430.1 - - 1
f15 7501.3 0.07 - 1
f16 7523.9 0.06 15.0 8
f17 7560.0 - - 1
f18 7610.1 0.07 5.6 5
f19 8055.5 0.06 10.1 8
f20 8651.4 0.08 - 1
f21 8820.6 0.08 6.4 4
f22 9352.8 0.07 16.3 1
f23 9385.3 0.28 8.9 1†
f24 9694.6 - - 1
f25 9795.1 0.08 18.8 2
f26 10366.8 0.08 1.4 2
f27 11103.3 - - 1†
f28 11159.8 0.07 - 1
– 27 –
Table 6. Amplitudes detected for differing subsets of PG 0048 data during the 2005 multisite campaign. The last
two rows give the 4σ detection limit and the temporal resolution for each subset. Note that all frequencies are given
in µHz. Runs 1 through 12 are boao, mdm0929, mdm9030, mdm1001, mdm1003, mdm1006, mdm1007, saao1007,
mdm1008, mdm1010, saao1010, and mdm1011 and runs G1 through G4 are MDM only, Oct. 1 - 3, Oct. 7 - 11, and
all the 2005 data, respectively.
Des. 1 2 3 4 5 6 7 8 9 10 11 12 G1 G2 G3 G4
f1 1.11 1.69
f2 2.44 1.03 1.48 1.25 2.26 1.12 0.93 2.34 1.70 2.19 2.42 1.77 1.38 1.07 1.77 1.40
f3 1.21 1.39 1.22 1.01 1.26 1.77 0.60 0.62 0.78 0.83
f4 1.13 1.30 1.24 1.07 1.01 0.59 0.94 0.53 0.54
f5 1.19 0.41 0.52 0.42
f6 0.86 0.59
f7 1.06 1.05 1.02 0.87 1.59 0.56 0.66
f8 0.98 1.12 1.39 1.06 1.02 0.92 0.53
f10 0.94
f12 0.81 0.36
f13 1.36 1.04 0.95 1.68 1.50 1.46 2.13 1.07 0.79 1.08 0.78
f14 0.83
f15 1.52 0.37 0.64 0.79 0.37
f16 0.97 1.36 2.26 1.39 1.78 1.09 0.53 0.86 0.48
f17 1.10 0.50 0.48
f18 1.00 1.09 1.10 0.99 0.48 0.63 0.59 0.36
f19 1.12 0.89 1.08 1.13 1.74 0.43 0.58 0.69 0.47
f20 0.90 0.36 0.44
f21 1.41 0.81 0.94 0.39 0.63
f22 1.10 0.41 0.46
f23 0.44
f24 0.85
f25 1.22 0.88 0.36 0.48 0.35
f26 0.91 1.01 0.36 0.56
f28 1.18 0.40 0.54 0.35
4σ 0.96 0.75 0.84 0.85 0.90 1.02 0.76 1.60 0.84 1.07 1.00 0.91 0.31 0.47 0.43 0.28
1/T 53 29 29 31 29 29 29 46 31 29 39 36 0.9 2.2 2.2 0.8
– 29 –
Table 7. The same as Table 6 for the discovery data (1997 and 1998; courtesy of C. Koen) and 2004 MDM data.
Runs dd1 through dd10 are tex151, tex157, tex177, tex182, tex186, dmk157, dmk163, ck261, ck263, and ck265, runs 1
through 3 are mdr285, mdr290, and mdr295, and runs G1 through G4 are all 1997, October 1998, November 1998,
and 2004 data, respectively.
Des. dd1 dd2 dd3 dd4 dd5 dd6 dd7 dd8 dd9 dd10 1 2 3 G1 G2 G3 G4
f1 2.63
f2 3.18 1.11 1.95 2.01 2.78 1.65
f3 5.86 3.52 2.77 1.76 4.29 2.10 1.54
f4 3.39 2.33 3.40 1.96 1.14 1.90 1.93 1.34 1.57
f5 1.96 0.55
f7 1.96 1.20 2.12 0.93 0.84 1.94 2.16 0.75
f8 4.94* 5.80 3.97 4.39 3.10 3.43 3.57 3.95 4.96 2.85 3.10 3.14 4.05 3.34 2.01 2.97
f9 1.28 0.65
f10 1.18 0.68
f11 1.03 1.16 0.98 1.56 1.00
f13 2.76 1.55 1.56 1.56
f16 4.31* 2.28 1.22 1.26
f17 1.55 2.17 1.69 1.40 1.56
f18 1.25 0.77
f19 2.36 3.29 1.21 1.08 2.03 1.17 1.99 1.06
f20 1.75
f21 0.93 0.74
f22 2.43 1.26
f23 1.04 0.65
f24 0.53
f27 1.00
4σ 5.38 3.54 1.50 1.20 1.70 2.06 1.63 3.80 3.17 2.44 0.75 0.82 0.97 1.14 1.21 1.97 0.50
1/T 194 132 69 96 84 96 84 174 185 111 41 38 37 0.8 5.4 0.9
– 31 –
Table 8. PG 0048 pulsation frequencies split by integer multiples near 972 and 41.1 µHz.
The first column gives the minimum degree ℓ for the multiplet and the order for each row
is from the top of Table 5, or in descending frequencies and the number in parentheses
indicates the integer multiple between itself and the previously listed frequency and the
percent deviation from 972 or 41.1 µHz.
ℓmin Des. Designations for related frequencies
Spacings of 972 µHz
3 f1 f12 (2, 0.2%); f27 (4, 1.4%)
3 f7 f14 (2, 0.2%); f23 (2, 0.4%); f26 (1, 0.8%)
1 f8 f10 (1, 2.4%); f17 (1, 2.4%)
2 f9 f18(2, 0.5%); f20 (1, 6.9%)
2 f11 f21 (2, 2.0%); f25 (1, 0.07%)
1 f3 f13 (2, 0.4%)
2 f6 f22 (4, 0.2%)
Spacings of 41.1 µHz
6 f1 f2 (1, 1.7%); f3 (1, 3.9%); f5 (3, 3.2%); f7 (2, 2.4%); f8 (3, 1.5%); f9 (1, 0.1%)
6 (1, 1) f12 f13 (2, 0.05%); f16† (7, 0.02%); f18 (2, 5.9%)
– 32 –
Table 9. Results of simulations for phase changes using f2 as the template and a
comparison with observations. Column 1 indicates how the phase was changed in the
simulation and columns 2 and 3 provide the average amplitude and standard deviation of
> 100, 000 simulations. Columns 4 and 5 provide ratios comparing amplitudes between
individual runs and that of the combined simulated data set. The bottom row provides the
average amplitude from the individual runs and results from the simulations with constant
phase and fixed amplitudes and phases with those observed for f2.
Sim 〈A〉 σ 〈A〉/〈A〉ind 〈A〉/Amax
10% 1.50 0.03 0.97 0.66
20% 1.35 0.10 0.88 0.60
Random 1.06 0.11 0.69 0.47
〈A〉ind constant phase Aobs & φobs
1.54 1.55 1.49
Table 10. Observed properties of individual frequencies.
Des. #det 〈σd〉 #exp 〈A〉 σA σA/〈A〉 Amax AG4/〈A〉 AG4/Amax AG1/〈A〉 AG1/Amax σφ (%)
f1 2 0.38 3 1.30 0.41 0.32 1.69 - - - - -
f2 12 3.69 12 1.69 0.53 0.31 2.44 0.83 0.57 0.82 0.57 9.6
f3 5 2.63 11 1.32 0.22 0.17 1.77 0.63 0.47 0.45 0.34 -
f4 5 1.59 10 1.14 0.15 0.13 1.30 0.47 0.44 0.52 0.48 8.4
f5 2 0.92 7 1.06 0.19 0.18 1.19 0.40 0.35 0.39 0.34 -
f6 1 0.11 1 0.86 - - 0.86 0.67 0.67 - - -
f7 5 1.16 8 1.09 0.19 0.17 1.59 - - 0.51 0.35 7.9
f8 5 1.50 10 1.11 0.09 0.08 1.39 0.48 0.38 - - 6.3
f9 1 - - - - - - - - - - -
f10 1 1.06 8 0.94 - - 0.94 - - - - -
f11 1 - - - - - - - - - - -
f12 1 0.33 3 0.81 - - 0.81 - - 0.44 0.44 -
f13 7 2.32 11 1.43 0.36 0.25 2.13 0.55 0.37 0.75 0.50 15.8
f14 1 0.41 3 0.83 - - 0.83 - - - - -
f15 1 2.75 11 1.52 - - 1.52 0.24 0.24 0.24 0.24 -
f16 6 2.60 11 1.38 0.44 0.32 2.26 0.35 0.21 0.38 0.23 15.0
f17 1 0.50 4 1.10 - - 1.10 0.44 0.44 0.45 0.45 -
f18 4 0.94 7 1.05 0.18 0.17 1.10 0.43 0.33 0.46 0.44 25.9
f19 5 1.79 10 1.19 0.26 0.22 1.74 0.39 0.27 0.36 0.25 7.3
f20 1 0.32 2 0.90 - - 0.90 - - 0.40 0.40 -
f21 3 1.16 8 1.03 0.26 0.25 1.41 - - 0.38 0.28 -
f22 1 1.32 9 1.10 - - 1.10 - - 0.37 0.37 -
f23 1 - - - - - - - - - - -
f24 1 0.00 1 - - - - - - - - -
f25 2 2.25 11 1.03 0.24 0.23 1.22 0.34 0.29 0.35 0.30 -
f26 3 0.76 6 0.94 0.17 0.18 1.01 - - 0.41 0.36 -
f27 1 - - - - - - - - - - -
f28 1 1.42 9 1.18 - - 1.18 0.30 0.30 0.34 0.34 -
– 34 –
Fig. 1.— Lightcurves for PG 1618B data. The top four panels show the coverage during the
multisite portion of the campaign from March 17 – 24, 2005, while the bottom panel shows
an enlarged section of a run obtained at MDM. Note that the scales are different for the
bottom panel. The line indicates our fit to the data.
– 35 –
Fig. 2.— Temporal spectra of various subsets of PG 1618B data. Insets are the window
functions. Solid (blue) horizontal line is the 4σ detection limit while the dashed (blue) lines
are at 2 mma in all panels.
– 36 –
Fig. 3.— Temporal spectrum of PG 1618B data. Top panel shows the original FT of
the combined MDM data while the bottom panel shows the residuals after prewhitening.
Prewhitened frequencies are indicated by arrows. Insets show an enlarged view of the fre-
quency doublet and the window function.
– 37 –
Fig. 4.— Amplitudes and phases of the two frequencies in PG 1618B that are resolvable
nightly. The data are only those from McDonald and MDM observatories except for a single
Lulin run (marked by a triangle). The dashed lines indicate our fits to the MDM data. Note
that the time axis is discontinuous.
– 38 –
Fig. 5.— Lightcurves for the PG 0048 data. Each panel is two days.
– 39 –
Fig. 6.— Comparison of temporal spectra for varying groups of PG 0048 data. The large
panels show FTs for combinations of data (inclusive dates are labeled). The insets on the
right are the window functions for the combinations of data to scale. The central insets are
slightly enlarged FTs of individual runs obtained congruent with the larger panels and are
labeled. They go from early to late within the campaign.
– 40 –
Fig. 7.— Temporal spectra for individual PG 0048 observing runs.
– 41 –
Fig. 8.— Amplitudes and errors of 12 frequencies detected in PG 0048. Each panel contains
amplitudes for four frequencies (differentiated by point type and color) which may be shifted
by ±0.1 day if they overlap. The solid line is the 4σ detection limit.
– 42 –
Fig. 9.— a) The number of detections of frequencies from individual runs for PG 0048
compared to Gaussian probability. The number of possible detections (dashed line) compared
to the amount of actual detections (solid line) depending on the detection significance. b) A
comparison between the number of detections for individual frequencies and the maximum
amplitude detected.
– 43 –
Fig. 10.— A schematic of the pulsation content of PG 0048 to indicate the nearly evenly
spaced frequencies (provided in Table 8). The top panel shows the spacings that are integer
multiples near 972 µHz and the bottom panel shows those near 41.1 µHz. For clarity, each
set of multiplets are connected via horizontal bars at differing heights and have different
colors (electronic version only, and black is used twice in the top panel).
– 44 –
Fig. 11.— A 350µHz region of PG 0048’s FT covering three modes at ∼7497.4µHz,
∼7524.1µHz and ∼7605.9µHz. The panels correspond to those in Fig. 6. The spectral
windows at the top of each panel cover the same frequency range as the observations, to give
the reader an appreciation of how dense these regions are.
– 45 –
Fig. 12.— Temporal spectra for halves of individual PG 0048 observing runs.
– 46 –
Fig. 13.— Amplitudes of frequencies detected in both individual runs and grouped data
during 2005 for PG 0048. Dots (black) indicate the maximum amplitude, squares (blue) the
average amplitude from individual runs. The lines indicate amplitudes from the following
group datasets (from Table 6): dotted (blue) is G1, short-dashed (green) is G2, long-dashed
(magenta) is G3, solid (black) is G4, and the dot-dashed (black) line is the G4/Amax ratio
(or G1/Amax if no G4 value).
– 47 –
Fig. 14.— Phases of pulsation frequencies. Frequency designations are provided in each
panel with phases determined only for individual runs in which the frequency was detected.
– 48 –
Fig. 15.— Comparison of amplitude ratios from simulations to those observed for PG 0048.
Black circles are for G4 and blue squares are for G1 data. The verticle-lined (top and green)
area is for simulations with 10% phase variations, the +45o-lined (middle and red) area is
for simulations with 20% phase variations, and the −45o-lined (bottom and blue) area is for
simulations with random phases.
– 49 –
Fig. 16.— Simulations of randomly excited and damped pulsations. Panels a, b, and c show
the effect of changing the excitation timescale for a fixed damping timescale of td = 5 hours.
Panel a and b show the results of simulating a single 9.35 hour run and the nine MDM
runs in Table 2, respectively. The points indicate the average amplitude with 1σ error bars,
the solid line is the maximum amplitude from an individual simulation and the dotted line
indicated the minimum amplitude detected from an individual simulation. Panel c shows
the ratios 〈AG1〉/Amax (blue circles) and 〈AG1〉/〈Aind〉 (black triangles). The dashed blue
(dotted black) lines indicate the observed range in PG 0048 for 〈AG1〉/Amax (〈AG1〉/〈Aind〉).
Panels d, e, and f correspond to panels a, b, and c except that the excitation timescale is
fixed at texc = 19 hours and the damping timescale is varied.
– 50 –
Fig. 17.— Expected fraction of detected frequencies based on simulations. Top panel used
average amplitudes while the bottom panel used maximum amplitudes from the simulations.
Differing lines represent different values of td given in the legend. Dotted line is the observed
26% detection rate. See the electronic edition of the Journal for a color version of this figure.
	Introduction
	PG 1618B+563B
	Observations
	Analysis
	Discussion
	Constraints on the pulsation modes
	Amplitude and phase stability
	PG 0048+091
	Observations
	Analysis
	Discussion
	Frequency content
	Constraints on mode identifications
	Amplitude and phase variability
	A possible cause of the amplitude/phase variability
	Conclusions and Future Work

  We present follow-up observations of pulsating subdwarf B (sdB) stars as part
of our efforts to resolve the pulsation spectra for use in asteroseismological
analyses. This paper reports on multisite campaigns of the pulsating sdB stars
PG 1618+563B and PG 0048+091. Data were obtained from observatories placed
around the globe for coverage from all longitudes. For PG 1618+563B, our
five-site campaign uncovered a dichotomy of pulsation states: Early during the
campaign the amplitudes and phases (and perhaps frequencies) were quite
variable while data obtained late in the campaign were able to fully resolve
five stable pulsation frequencies. For PG 0048+091, our five-site campaign
uncovered a plethora of frequencies with short pulsation lifetimes. We find
them to have observed properties consistent with stochastically excited
oscillations, an unexpected result for subdwarf B stars. We discuss our
findings and their impact on subdwarf B asteroseismology.

Introduction
Subdwarf B (sdB) stars are thought to have masses about 0.5M⊙, with thin (<10
−2M⊙)
hydrogen shells and temperatures from 22 000 to 40 000 K (Saffer et al. 1994). They are
horizontal branch stars that have shed nearly all of their H-rich outer envelopes near the
tip of the red giant branch and as He-flash survivors, it is hoped that asteroseismology can
place constraints on several interesting phenomena. Subdwarf B star pulsations come in two
varieties: short period (90 to 600 seconds; EC 14026 stars after that prototype, officially
V361 Hya stars, or sdBV stars) with amplitudes typically near 1%, and long period (45
minutes to 2 hours; PG 1716 stars after that prototype or LPsdBV stars) with amplitudes
typically <0.1%. For more on pulsating sdB stars, see Kilkenny (2001) and Green et al.
(2003) for observational reviews and Charpinet, Fontaine, & Brassard (2001) for a review of
the proposed pulsation mechanism. For this work, our interest is the sdBV (EC 14026) class
of pulsators.
In order for asteroseismology to discern the internal conditions of variable stars, the
pulsation “mode” must be identified from the temporal spectrum (also called pulsation
spectrum or Fourier transform; FT). The mode is represented mathematically by spherical
harmonics with quantum numbers n (or k), ℓ, and m. For nonradial, multimode pulsators
the periods, frequencies, and/or the spacings between them are most often used to discern the
spherical harmonics (see for example Winget 1991). These known modes are then matched to
models that are additionally constrained by non-asteroseismic observations, typically Teff and
log g from spectroscopy. Within such constraints, the model that most closely reproduces the
observed pulsation periods (or period spacing) for the constrained modes is inferred to be the
correct one. Occasionally such models can be confirmed by independent measurements (Reed
et al. 2004, Reed, Kawaler, & O’Brien 2000, Kawaler 1999), but usually it is impossible to
uniquely identify the spherical harmonics and asteroseismology cannot be applied to obtain
a unique conclusion. Such has been the case for sdBV stars, which seldom show multiplet
structure (i.e., even frequency spacings) that may be used to observationally constrain the
– 3 –
pulsation modes. However, relatively few sdBV stars have been observed sufficiently to know
the details of their pulsation spectra. The goal of our work is to fully resolve the pulsation
spectrum, search for multiplet structure, and examine the characteristics of the pulsation
frequencies over the course of our observations.
In this paper we report on multi-site follow-up observations of the pulsating sdB stars
PG 1618+563B (hereafter PG 1618B) and PG 0048+091 (hereafter PG 0048) obtained
during 2005. PG 1618B was discovered to be a variable star by Silvotti et al. (2000; hereafter
S00) who detected frequencies of 6.95 and 7.18 mHz (P ≈ 144 and 139 s respectively) from
short data runs (< 2.3 hrs) obtained during seven nights, three of which were separated
by three months. PG 1618 is an optical double consisting of a main sequence F-type star
(component A) with an sdB star (component B) at a separation of 3.7 arcseconds. The
combined brightness is V = 11.8 while the sdB component has V ≈ 13.5. The discovery
data used a combination of photoelectric photometry, which did not resolve the double,
and CCD data which did. The combined flux of the double in the former would have
reduced the pulsation amplitudes. From spectra obtained at Calar Alto, S00 determined
that Teff = 33 900± 1 500K and log g = 5.80 ± 0.2.
PG 0048 was discovered to be a variable star during 10 observing runs varying in length
from 1 to 4 hours obtained in 1997 and 1998 (Koen et al. 2004; hereafter K04). As they used
a variety of instruments spread over the span of a year, K04 were only able to combine two
consecutive runs, from which they detected seven frequencies. However, it is obvious from
their temporal spectrum that pulsation amplitudes and possibly frequencies were changing
(their Fig. 3). K04 contributed this to unresolved frequencies caused by their short duration
data runs. K04 also obtained an optical spectrum and examined 2MASS colors to determine
that PG 0048 has a G0V-G2V companion; though the orbital parameters are unknown and
no Teff or log g estimates were given.
Here we report the results of a new program to resolve the pulsation spectra of these
two stars. Section 2 describes the observations, reductions, and analysis for PG 1618B, and
§3 the same for PG 0048. Section 4 compares the results for both stars and discusses the
implications for asteroseismology.
2. PG 1618B+563B
2.1. Observations
PG 1618B was observed from 5 observatories (Baker, MDM, McDonald, Lulin, and
Suhora) over a 45 day period during spring 2005. Data obtained at MDM (2.4 m) and
– 4 –
McDonald (2.1 m) observatories used the same Apogee Alta U47+ CCD camera. This camera
is connected via USB2.0 for high-speed readout, and our binned (2×2) images had an average
dead-time of one second. Observations at Baker (0.4 m) and Lulin (1.0 m) observatories
were obtained with Princeton Instruments RS1340 CCD cameras. Data obtained at Baker
Observatory were binned 2× 2 with an average dead-time of one second, while observations
from Lulin Observatory used a 392× 436 subframe at 1 × 1 binning with an average dead-
time of six seconds. The Mt. Suhora Astronomical Observatory (0.6 m) data were obtained
with a photomultiplier tube photometer which has microsecond dead-times. Observations
obtained at McDonald, Baker, and MDM observatories used a red cut-off (BG40) filter, so
the transmission is virtually the same as the blue photoelectric observations from Suhora
observatory. Observations from Lulin Observatory used a Johnson B filter, which slightly
reduced the amount of light collected compared to other observations, but does not impose
any significant phase and/or amplitude changes compared to other observations (Koen 1998;
Zhou et al. 2006). Accurate time was kept using NTP (Baker, McDonald, and MDM
observatories) or GPS receivers (Lulin and Suhora observatories) and corrected to barycentric
time during data reductions.
Standard procedures of image reduction, including bias subtraction, dark current and
flat field correction, were followed using iraf1 packages. Differential magnitudes were ex-
tracted from the calibrated images using momf (Kjeldsen & Frandsen 1992) or occasionally
they were extracted using iraf aperture photometry with extinction and cloud corrections
using the normalized intensities of several field stars, depending on conditions. Photoelec-
tric data reductions proceeded using standard Whole Earth Telescope reduction packages
(Nather et al. 1990). As sdB stars are substantially hotter, and thus bluer, than typical
field stars, differential light curves using an ensemble of comparison stars are not flat due
to differential atmospheric and color extinctions. A low-order polynomial was fit to remove
these trends from the data on a night-by-night basis. Finally, the lightcurves are normalized
by their average flux and centered around zero so the reported differential intensities are
∆I = (I/〈I〉) − 1. Amplitudes are given as milli-modulation amplitudes (mma) with an
amplitude of 10 mma corresponding to an intensity change of 1.0% or 9.2 millimagnitudes.
The companion of PG 1618B adds a complication to the reductions in that data obtained
at McDonald and MDM observatories resolved the optical double, but those from other
observatories did not. Using our data for which the stars are resolved, we determined that
component A contributes 67.3% of the total flux. To correct the unresolved data, we created
iraf is distributed by the National Optical Astronomy Observatories, which are operated by the As-
sociation of Universities for Research in Astronomy, Inc., under cooperative agreement with the National
Science Foundation.
– 5 –
a fitting function by smoothing the data over many points (around 50 points per box),
multiplying it by 0.673 and subtracting it from the unresolved data. While this process
effectively removes the flux from PG 1618A, it cannot correct for the noise of this component,
which remains behind. As such, the corrected data are noisier, limiting their usefulness.
Multiple-longitude coverage was only obtained during the first week of the campaign.
A total of 73.5 hours of data were collected from three observatories (McDonald, Lulin, and
Suhora) which provided a 47% duty cycle. Subsequent data were obtained only in Missouri
(Baker Observatory) and Arizona (MDM Observatory). These data serve to extend the
timebase of observations (increasing the temporal resolution) and to decrease the noise in the
temporal spectrum. Lightcurves showing the coverage of the first six nights of observations,
as well as a portion of a typical MDM run, are provided in Fig. 1.
2.2. Analysis
Our campaign was quite long (about 45 days) with a concentration of data at the
beginning, but the best data (highest S/N and best conditions) were obtained at the end.
We therefore grouped combinations of nightly runs into the subsets given in Table 2 for
analysis. Table 2 also provides the temporal resolution (calculated as 1/trun where trun is
the length of the observing run) and the 4 σ detection limit (calculated using areas adjacent
to the pulsation but outside of their window functions). The temporal spectra and window
functions of these subsets are plotted in Fig. 2. A window function is a single sine wave of
arbitrary, but constant amplitude sampled at the same times as the data. The central peak
of the window is the input frequency, with other peaks indicating the aliasing pattern of
the data. Each peak of the data spectrum intrinsic to the star will create such an aliasing
pattern. As is evident from Fig. 2, the MDM data was significantly better than the rest, so
we began our analysis with that subset.
Analysis of the MDM data was relatively easy and straightforward. In Fig. 3, the top
panel shows the original FT, while the bottom panel shows the residuals after prewhitening
by the frequencies indicated by arrows. The insets show the window function (top right)
and an expanded view of a 65µHz region around the close doublet. Frequencies, amplitudes
and phases were determined by simultaneously fitting a nonlinear least-squares solution to
the data. Since during the MDM observations the amplitudes were relatively constant, the
solution proceeded as expected and prewhitening effectively removed the peaks and their
aliases. The formal solution and errors for the MDM subset are given in Table 3.
Examination of the other data sets indicates that while PG 1618B was well-behaved
– 6 –
during the MDM observations, it was not at other times. This was most noticeable in our
examination the data collected during the first week. While the temporal spectrum has the
cleanest window function, no peaks are detected above the 4σ detection limit (1.53 mma)
even though peaks are detected in individual Lulin and McDonald runs (see Fig. 4 which
will be discussed in §2.3). Combining the well-behaved MDM data with any other data set
results in a decrease of amplitudes, indicating that outside of the MDM data, the amplitudes,
phases, or even frequencies are not constant. If the pulsation properties were consistent
throughout the campaign, data collected at smaller telescopes, with low S/N would still
have been useful for reducing the overall noise. Unfortunately, such was not the case so all
we can really conclude is that the MDM data detected all the pulsations that were occurring
(above the detection limit) at that time while the pulsations intrinsic to PG 1618B must
have been more complex at other times. It would be interesting to study the longer-term
variability of PG 1618B, but using only 2 m-class telescopes.
Outside of the combined data sets, there are two frequencies that are detected above the
4σ detection limit during individual runs. The least-squares solutions for these frequencies
are provided at the end of Table 3. The frequency at ≈ 9199µHz was above the noise only
in the March 22 McDonald data, though a peak at the same frequency also appears in the
Suhora data during March 16 and 21. The frequency at ≈ 8179µHz was above the noise only
in the April 30 MDM data though corresponding peaks appear in the McDonald March 18,
Lulin March 18, and Suhora March 21 runs. Since they are detected above the 4σ detection
criteria for those runs, we include them in our discussion that follows.
2.3. Discussion
Silvotti et al. (2000) detected two frequencies in their discovery data while we clearly
resolve four frequencies from our MDM dataset, and two more from individual data runs,
bringing the total to six independent frequencies. We calculate the S00 resolution to be
5.5 µHz and estimate their noise to be about 1 mma though this is misleading in that
because of their short data runs, their window function effectively covers all of the remaining
pulsations. However, for a strict comparison, we can say that our MDM data alone are 3×
better in resolution and have a detection limit twice as good, though in a practical sense our
MDM data are far superior solely based on the duration of our individual data runs. Had S00
observed for longer durations (particularly with their CCD setup), their data would likely
have been similar to the same number of runs from our MDM set. However, it is clearly safe
to say that our MDM data alone are insufficient to describe the complexity of pulsations
occurring within PG 1618B. As such, the remainder of our discussion which is based on the
– 7 –
MDM data, can only be a minimum of what is really occurring.
2.3.1. Constraints on the pulsation modes
One of our goals is to observationally identify or constrain the pulsation modes of indi-
vidual frequencies. Differing m components of the same degree ℓ have degenerate frequencies
unless perturbed, typically by rotation. If a star is rotating, then each degree will separate
into a multiplet of 2ℓ+ 1 components with spacings nearly that of the rotation frequency of
the star. As such, observations of multiplet structure can constrain the pulsation degree (for
examples, see Winget et al. 1991 for pulsating white dwarfs and Reed et al. 2004 for sdB
stars). For PG 1618B, there are no two frequency spacings that are similar, though there are
not many frequencies to work with. The lack of observable multiplet structure is typical of
sdBV stars but is likely limited to four possibilities: i) Rotation is sufficiently slow that all
m values remain degenerate within the frequency resolution of our data; ii) our line of sight
is along the pulsation axis, with sin i ≈ 0, leaving only the m = 0 mode observable because
of geometric cancellation (Pesnell 1985; Reed, Brondel, & Kawaler 2005); iii) rapid internal
rotation is such that m multiplets are widely spaced and uneven (Kawaler & Hostler 2005);
or iv) at most one pair is part of a multiplet with an unobserved component of the multiplet.
Spectroscopy can only rule out large splittings for possibility (i) as spectroscopic limits
are typically ≈ 10 km/s and from Fig. 1 of S00, PG 1618B appears as a “normal” sdB
devoid of rapid rotation. Possibility (ii) can only be determined for cases in which the sdB
star is part of a close binary such that the rotation and orbital axes can be inferred to be
aligned. Since PG 1618 is only an optical double at wide separation, it does not constraint
the alignment of the surface spherical harmonics. Similarly, possibility (iii) is virtually
impossible to decipher unless the star pulsates in many (tens of) frequencies, and would
still require some interrelation of spacings for modes of the same degree (Kawaler & Hostler
2005). Possibility (iv) also remains an option, though a difficult one to constrain. Higher
resolution (and perhaps longer duration) spectroscopy would help to answer this question,
and multicolor photometry or time-series spectroscopy might also be able to discern the
spherical harmonics (see Koen 1998 and O’Toole et al. 2002 for examples of each).
Another quantity that can be used to constrain the pulsation modes is the frequency
density. Using the assumptions that no two frequencies share the same n and ℓ values (except
possibly the close pair at 6946µHz), and that high-degree ℓ ≥ 3 modes are not observationally
favored because of geometric cancellation (Charpinet et al. 2005; Reed, Brondel, & Kawaler
2005), we can ascertain whether the frequencies are too dense to be accounted for using only
ℓ ≤ 2 modes. From stellar models, a general rule of thumb is to allow three frequencies
– 8 –
per 1000µHz. We will ignore f1 which is too distant in frequency space and count f5 and
f6 as a single degree ℓ. This leaves four frequencies within 1235 µHz; which can easily be
accounted for using only ℓ ≤ 2 modes. Indeed, even if f5 and f6 do not share their n and ℓ
values, the frequency spectrum can still accommodate all of the detected frequencies without
invoking higher degree modes. Of course this does not mean that they are not ℓ ≥ 3 modes,
only that the pulsation spectrum is not sufficiently dense to require their postulation.
2.3.2. Amplitude and phase stability
If pulsating sdB stars are observed over an extended time period, it is common to detect
amplitude variability in many, if not all, of the pulsation frequencies (eg. O’Toole et al. 2002;
Reed et al. 2004; Zhou et al. 2006). Such variability can occasionally be ascribed to beating
between pulsations too closely spaced to be resolved in any subset of the data. However,
variations often appear in clearly resolved pulsation spectra where mode beating cannot be
the cause. For PG 1618B, frequencies f1 and f2 are only detected during a single run each
and frequencies f5 and f6 are too closely spaced to be resolved during individual runs,
leaving only frequencies f3 and f4 available for analysis of amplitude variations.
Figure 4 shows the amplitude and phases of these two frequencies for individual data
runs from McDonald and MDM observatories as well as a single Lulin run (marked by a
triangle); these frequencies were not detected elsewhere. During the MDM observations,
the amplitudes and phases for both frequencies are nearly constant (to within the errors)
except for one low amplitude, but they have a significant variation in the McDonald and
Lulin data. Of particular interest are the phases and amplitudes of f4, especially those
during day three, in which we have both a McDonald and Lulin run that do not overlap in
time. Between these two runs, the amplitude, which had been decreasing during the previous
three days, suddenly increases to begin the same declining pattern again. The phases also
show a bimodal structure early in the campaign with phases near −0.20 and +0.25 with the
first phase jump occurring coincident with the amplitude increase. Except for the lack of
sinusoidal amplitude variation, this has the appearance of unresolved pulsations. However,
if the two unresolved frequencies had intrinsic amplitude variability, then it could reproduce
the observations. However the MDM observations, which are not only steady, but have f4
phases intermediate to the McDonald and Lulin data, do not support this. Clearly, during
the week of MDM observations, PG 1618B had neither amplitude nor phase variations and
since the MDM phases do not coincide with phases from earlier in the campaign, unresolved
pulsations are unlikely. Since the data obtained at MDM and McDonald observatories used
the same acquisition system and time server (NTP), errors in timing also seem unlikely.
– 9 –
3. PG 0048+091
3.1. Observations
We originally observed PG 0048 as a secondary target during a campaign on KPD 2109+4401
(Zhou et al. 2006). Those data revealed a complex pulsation spectrum which we could not
resolve with such limited sampling and a short time base. As such, PG 0048 was re-observed
as a multisite campaign during Fall 2005. Five observatories participated in the campaign
with the specifics of each run provided in Table 4. Though we were a bit unlucky with
weather, over the course of our 16 night campaign we obtained 167.4 hours of data for a
duty cycle of 44%. Details of the observing instruments and configurations are the same
as for PG 1618B, except for the following: SAAO (1.9 m) used a frame transfer CCD with
millisecond dead-times but only an ≈ 30 × 40 arcsecond field of view, which resulted in no
comparison stars within the CCD field. As such no transparency variations could be cor-
rected and only photometric nights were used. Tubitak Observatory used a Fairchild CCD447
detector; during the first run the images had 1× 1 binning with a dead-time of 102 seconds,
while subsequent runs used 2×2 binning with a dead-time of 29 seconds. Bohyunsan Optical
Astronomy Observatory (BOAO 1.9 m) data were obtained with a SiTe-424 CCD windowed
to 580×445 pixels, binned 2×2 with an average dead time of 14 seconds. MDM and SAAO
used red cut-off filters, making their responses very similar to blue-sensitive photoelectric
observations, while Lulin, BOAO, and Tubitak used no filter making their sampling more
to the red. As pulsations from sdB stars have little amplitude dependence in the visual and
no phase dependence (Koen 1998; Zhou et al. 2006), mixing these data is not seen as a
problem.
The standard procedures of image reduction, including bias subtraction, dark current
and flat field correction, were followed using iraf. Differential magnitudes were extracted
from the calibrated images using momf (Kjeldsen & Frandsen 1992), except for the SAAO
data for which we used aperture photometry because there were no comparison stars. As
described for PG 1618B, we again used low-order polynomials to remove airmass trends
between our blue target star and the redder comparison stars. The lightcurves are normalized
by their average flux and centered around zero, so the reported differential intensities are
δI = (I/〈I〉) − 1. Figure 5 shows the lightcurve of PG 0048 with each panel covering two
days.
– 10 –
3.2. Analysis
During the campaign, we completed a “quick-look” analysis of data runs as early as
possible to ascertain the data quality and the pulsation characteristics of the star. We
noticed early on that the temporal spectra of PG 0048 changed on a nightly basis with
pulsation frequencies appearing and then disappearing on subsequent nights. Likewise, we
knew that our analysis would be complicated by severe amplitude variations which would
limit the usefulness of prewhitening techniques and could create aliasing. Figure 6 shows the
effects of amplitude variations. The full panels are pulsation spectra for three groups of data:
All of the data; data obtained from September 30 through October 3; and from October 7
through October 11. The right insets are the corresponding window functions plotted on the
same horizontal scale. At such large scales, the windows appear as single peaks and show
that the changes in the FTs are not caused by aliasing. The central insets are individual
data runs within the larger set and show the variability between runs. When sets of data
are combined in which the peak amplitudes are not constant, an FT will show the average
amplitude. For the frequencies that appear in only a few runs, the amplitudes are effectively
quashed in the combined FT. As PG 0048 is the most pulsation variable sdB star currently
known, our immediate goal is to glean as many observables from these data as possible.
While we do provide some interpretation, our aim is to provide sufficient information for
theorists to test their models.
The complexity of the data meant it was necessary to analyze it using multiple tech-
niques: We performed standard Fourier analyses on combined sets of observations to increase
temporal resolution and lower the overall FT noise and analyzed individual runs of the best
quality data. The analysis of individual runs represents a time-modified Fourier analysis,
which is essentially a Gabór transform, except that we replace a Gaussian time discrimina-
tor with the natural beginnings and endings of the individual runs. As the best individual
runs are not continuous with time (and nearly all are from MDM Observatory) the use
of a Gaussian-damped traveling temporal wave discriminator (a standard Gabór transform)
would not enhance the results. The temporal spectra of these runs are shown in Fig. 7. Runs
mdm1005 and mdm1009, though long in duration, have gaps in them because of inclement
weather, whereas the other 12 runs are gap-free. For these 12 runs aliasing in the FT is not
a problem and the only constraints are the width of the peaks, which are determined by run
length, and the noise of the FT, which is a combination of the signal-to-noise of each point
and the number of data points within the run.
Frequencies, amplitudes and phases were determined using two different software pack-
ages, Period04 (Lenz & Breger 2004) and a custom (Whole Earth Telescope) set of non-linear
least squares fitting and prewhitening routines. Each of the three data combinations in Fig. 6,
– 11 –
the 12 gap-free data runs plotted in Fig. 7, the three data runs obtained during 2004, and
the 10 runs from the discovery data (kindly provided by Chris Koen) were reduced using
both software packages. Overall, more than 35 frequencies were fit during at least one data
run. Table 5 provides information for 28 frequencies which have been detected above the
4σ detection limit. Column 1 lists a frequency designation; column 2 the frequency as fit
to the highest temporal resolution data set in which each frequency is detected with the
formal least-squares errors in column 3. Column 4 provides the standard deviation of the
corresponding frequencies detected in individual runs and column 5 gives the number N of
individual runs in which that frequency was detected (from twelve 2005 runs and three 2004
runs). Tables 6 and 7 provide the corresponding amplitudes as fit for individual runs and var-
ious combinations of data acquired during the 2005 campaign, a re-analysis of the discovery
data, and the 2004 MDM data. The last two rows of these tables provide the 4σ detection
limits and temporal resolutions for the runs. Our determination that these frequencies are
real and intrinsic to the star is based on i) detection by both fitting software packages, and
amplitude(s) higher than the 4σ detection limit with ii) detection during several data runs,
and/or iii) detection at amplitudes too large to be associated with aliasing.
3.3. Discussion
3.3.1. Frequency content
During our 2005 multisite campaign, we detected 24 pulsation frequencies from indi-
vidual data runs plus an additional frequency from the combined data set (which was also
detected in 2004). We recover all seven of the frequencies detected in the discovery data
(K04), but only 14 of the 16 frequencies detected from our 2004 data. As can be seen from
Table 5, PG 0048 shows an atypically large range of frequencies for sdBV-type pulsators,
especially when considering that only one frequency, f23 (11103.3µHz), can be identified as a
linear combination (of f6 and f7). As noted in Table 5, only one frequency is detected in all
of our data (f2: ∼5244.9µHz) while the next most common frequency (f13: ∼7237.0µHz) is
detected in only 11 of the 15 runs. Several frequencies are only detected once or twice (e.g.
f12: ∼7154.3µHz), and so we should test if their amplitudes are sufficient to consider them to
be real. If a particular frequency has amplitudes that are 1σ above the detection limit, then
we might only expect to detect it 68% of the time2. Figure 8 shows the amplitudes and 1σ
errors for 12 different frequencies (four frequencies per panel) and the detection limit (solid
2This is a lower limit since statistically, the pulsation amplitude is equally likely to be higher than 1σ
from the detected level rather then below it.
– 12 –
line) for individual runs. The (black) circles in the top panel are for f2, which is detected in
every run. However, the two frequencies indicated by (magenta) squares in the middle and
bottom panels are only detected once, even though they are > 1σ above the detection limit.
If their amplitudes were nearly constant (to within their errors), they would be detected at
least 68% of the time. Another way to show this is in panel a of Fig. 9 where the detections
are plotted against their significance. We detect a total of 24 frequencies from 12 individual
runs from our 2005 data. If we detected all 24 frequencies from every run, we would have
made 288 detections, while we only actually made 75. The solid line shows the number of
individual detections cumulative with significance (the number of standard deviations the
detection was above the detection limit). In other words, 49 of our 75 detections were 2σ
or less above the detection limit. The dashed line is the standard Gaussian probability
distribution which shows that at 1σ significance, we should have made at least 196 (68%)
detections. Since 64% of our detections are ≥ 1σ, our 75 detections is well short of what we
should have detected, indicating that the pulsation amplitudes are really falling below the
detection limit. Panel b compares the number of actual detections to the maximum pulsation
amplitude. As expected, there is some correlation as the higher the amplitude, the easier it
is to detect that frequency. Additionally, the highest amplitude (and therefore most easily
detected) frequency is the same during 1997, 1998, and 2004 (f8: ∼5612.2µHz) but is only
detected in ∼ 1/3 of our 2005 data runs during which f2 (∼5244.9µHz) had the highest
amplitudes. This change in pulsation amplitudes will be further discussed in §3.3.3.
3.3.2. Constraints on mode identifications
As in §2.3.1, one of the best ways to relate pulsation frequencies to pulsation modes is
via multiplet structure. With such a rich pulsation spectrum, it seems likely that some of
the frequencies should be related by common frequency splittings. If not, then the pulsation
spectrum is too dense (discussed below) for the frequencies to consist of only low-order ℓ ≤ 2
modes. There are two different spacings that occur many times with splittings near 972
and 41.1 µHz. Table 8 lists these frequencies and the deviation from the average spacing
between them, while Fig. 10 shows them graphically. While the spacings may be important,
it is difficult to attach any physical meaning to them. A spacing of 972 µHz is far too large to
be associated with stellar rotation, as it corresponds to a rotation period of only 17 minutes.
There is currently no high-resolution spectrum of PG 0048, and the star’s main-sequence
companion would complicate any attempts to measure its rotation velocity. However, the
typical v sin i of sdBs is less than ∼ 5 km s−1 (Heber, Reid, & Werner 2000). If we try to
explain this large splitting using asymptotic theory (consecutive n overtones rather than m
multiplets), we would expect successive overtones of the radial index to be roughly evenly
– 13 –
spaced for large n, with one series for each ℓ degree. But the spacings observed in PG 0048 are
irregular, requiring eight degrees (one for every line in Table 8), and 3 ≤ ℓ ≤ 7 modes have
reduced visibility because of geometric cancellation (Charpinet et al. 2005; Reed, Brondel,
& Kawaler 2005). Theory suggests that sdBV stars should pulsate in low overtone modes
and the spacings between successive low overtone modes should differ by several hundreds
of µHz (Charpinet et al. 2002), which again does not match what is observed.
Another asymptotic-like relation is the “Kawaler scheme” which has been recently pre-
sented by Kawaler et al. (2006) and Vuc̆ković et al. (2006). Though it is not compatible with
the low overtone (n) pulsation theory associated with sdB pulsators, an improved frequency
fit can sometimes be obtained using an asymptotic-like formula;
f(i, j) = fo + i× δ + j ×∆
where i has integer values, j is limited to values of −1, 0, and 1, δ is a large frequency spacing
and ∆ a small one. However, for the case of PG 0048, the small spacings are unrelated to the
larger spacings and the larger spacings themselves do not interrelate, but rather appear in
sets with differing spacings between the sets. So this scheme is not applicable for PG 0048.
The smaller, 41.1 µHz spacings are more akin to what we would expect for rotationally
split multiplets, though still large compared to typical measured rotation rates. If the low-
frequency set are all components of a single multiplet, it would require very high degree
(ℓ ≥ 5) pulsations, which are not observationally favored (Charpinet et al. 2005; Reed,
Brondel, Kawaler 2005). The same is true for the high-frequency set if f13 and f16 belong
to the same set. Since they are separated by 7×41.0µHz, it is certainly possible that f12-f13
and f16-f18 are just two pairs, but if they are all combined into a single multiplet, it would
require ℓ ≥ 5 again. Yet these could also be just chance superpositions.
Since PG 0048 pulsates in so many frequencies, it is important to test the significance of
the spacings discussed above. We did this by producing Monte Carlo simulations, randomly
placing 28 frequencies within 6000 microHz of each other, and counting how often we could
detect 14 frequency splittings the same to within about 5%. This criterion would find all but
one of the splittings we actually observe. After analysing over one million simulations, we
detected at least 14 splittings in nearly every case. In other words, the splittings we observe
are not statstically significant.
Another tool we can use to place constraints on mode identifications is the mode density.
As mentioned in §2.3.1, models predict roughly one overtone (n) per degree (ℓ) per 1000 µHz
and though we have detected 28 frequencies, they are spread across nearly 6000 µHz. Since we
do not detect any multiplets that can be unambiguously associated with rotational splitting,
it is likely the pulsations are degenerate in m. The average mode density is 4.8 frequencies
– 14 –
per 1000 µHz, which is too high to accommodate only ℓ ≤ 2, with degenerate m modes
and it gets worse as the frequencies are not quite distributed equally, but rather fall into
loose groups, enhancing the density locally. The regions between 5200 and 6200 and 6600 to
7600 µHz each contains 9 frequencies, and the region from 8800 to 9800 µHz has 5 frequencies.
If all 2ℓ+1 multiplets were filled (9 frequencies per 1000 µHz), the frequency density would
not require any ℓ ≥ 3 modes. However, as we do not detect appropriate multiplets within
these frequency regions, the most likely result is that PG 0048 has too high a frequency
density to exclude ℓ ≥ 3 modes using current models.
3.3.3. Amplitude and phase variability
Nearly all sdBV stars show some amount of amplitude variability. However, no other
sdBV star has shown variability like that detected in PG 0048. An example of this is shown
in Fig. 11, where the same 350 µHz region is shown for the three data sets in Fig. 6. The
bottom two panels are subsets of the top panel, which includes all of the data, and indicate
how strikingly the pulsation spectrum changes with time.
We can apply some constraints to the timescale of amplitude variability. Since pulsa-
tion frequencies can change amplitudes (even to the point of being undetectable) between
individual runs, and in particular between runs from different observatories but for the same
date, the next step was to divide up our longer data runs into halves. The FTs for six such
runs are shown in Fig. 12 with each half containing about four hours of data. Particularly
for runs mdm1007 (which can also be compared to saao1007) and mdm1010, the amplitudes
change by factors of two over times as short as four hours. Low amplitude frequencies can
easily become undetectable within that time. Figure 13 compares the maximum and aver-
age amplitudes detected in individual runs (the black circles and blue squares, respectively)
to detections in groups of data (the lines) from Table 6. If the amplitudes were simply
wandering around between values detected in individual runs, then the amplitude of the
combined data would be an average of these values (the blue squares). However, since the
combined amplitudes are significantly lower then the average amplitudes from individual
runs, something else must be occurring to reduce the amplitudes.
Since changes in phase can impact pulsation amplitudes, we investigated that next.
Figure 14 shows phases for eight of the most-often detected frequencies. f2 is the only
frequency detected in every run, and we include the half-night analysis for it. For the other
seven frequencies, we only determined phases for those runs in which they were detected
above the 4σ limit. While Fig.14 indicates that most phases do not appear constant with
time, most are within 20-30% of a central value.
– 15 –
To isolate and test the impact phase variation creates on data like ours, we analyzed
simulated data with the following properties: The data are represented by a noise-free, single-
frequency sine wave sampled corresponding to our 12 best individual runs with frequency
f2. The amplitudes of each individual run are fixed at the measured values for f2 from
Table 6 and we assume that no phase changes occur during an individual run. While the
properties of f2 may not be the most representative of the variations detected, it is the only
frequency detected every time, and so by using it, we sample the full range of amplitudes.
If we used a different frequency, we could not know the actual pulsation amplitude during
those runs without detections, and so would only be sampling the higher amplitude data
points. We created simulations with the following phase properties: No change in phase, a
fixed change of ±10% from the previous phase, a fixed change of ±20% from the previous
phase, phases that are randomly set at the beginning of each run, and with the actual phase
values for f2. The results of the simulations are given in Table 9 and the ratios are used in
Fig. 15 where they are compared with our observations. The last line of Table 9 presents
results with unique answers. As expected, there is a correlation between the amplitude and
the amount of phase variation in that increasing changes in phase between individual runs
decreases the measured amplitude of the data set as a whole. More useful are the ratios of
the average amplitude to the average and maximum of the individual amplitudes (〈A〉/〈A〉ind
and 〈A〉/Amax). These ratios can be compared with ratios from all frequencies, as has been
done in Fig. 15. The shaded regions are the 1σ ratios produced from the phase simulations
and the circles represent ratios with G4 data and the squares are for ratios using G1 data.
The frequency ordering is the same as for Fig. 12, and like Fig. 12, the results indicate
that for all but f2 and possibly f6, the amplitudes detected in groups of data are too low
compared to individual amplitudes. Additionally, except for f2, the phases of Fig. 14 (and
their standard deviations given in Table 10) are in discord with the amplitude ratios in that
the amplitudes are too low. There does not seem to be sufficient phase variation to produce
the low amplitudes of the group data. As such, it seems that more extreme circumstances
are required. However, that leads into a more speculative area which we save for §3.3.4.
We conclude our observational portion of this paper with a summary provided in Ta-
ble 10. In this table we have included all measurables (not in previous tables) discussed in
this section and some that will be useful for the next section. Columns 2, 3, and 4 consider
the number of expected detections based on the average significance of the actual detections;
Columns 5 through 8 detail amplitudes detected for individual 2005 observing runs; Columns
9 through 12 provide ratios of individual to group amplitudes; and Column 13 lists the 1σ
deviations of pulsation phases.
– 16 –
3.3.4. A possible cause of the amplitude/phase variability
Can we determine the cause of the erratic behaviour of PG 0048’s frequencies and
amplitudes? We suggest that, with the possible exception of f2, the oscillations may be
stochastically excited. While this is counter to current theory, supporting observations that
we have in hand are 1) amplitudes that vary significantly between every individual run, and
in less than ∼4 hrs; 2) the combined amplitudes are significantly lower than the average value
indicating that phases are not coherent on these timescales; 3) the peaks in the FTs appear
similar to those of known stochastic pulsators (compare Fig. 11 to Fig. 1 of Bedding et al.
(2005) or Fig. 2 of Stello et al. (2006) for stochastic oscillators to those in, for example,
Reed et al (2004) for “normal” sdB stars); and 4) the number of actual frequency detections
compared to the expected number based on significance (the ratio of the two lines in the left
panel of Fig. 9). We note that this is not conclusive evidence, but is suggestive and so we
will pursue a stochastic nature for PG 0048’s pulsations in the remainder of this section.
Recently, Stello et al. (2006) found that short mode lifetimes in red giants can severely
limit the possibility of measuring reliable frequencies. The difficulty arises because the
frequencies can disappear entirely and when they are re-excited (even if this occurs prior to
complete damping), they do not maintain the same phase. The parallel with our analysis
of PG 0048 are clear and this kind of variability has been seen before in sdBVs. In a study
of KPD2109+4401, Zhou et al. (2006) found substantial variation in the amplitudes of two
modes during their 32 night campaign. A brief analysis found that at least one of these
modes, and possibly both of them, satisfied the criterion outlined by Christensen-Dalsgaard
et al. (2001; hereafter JCD01) for stochastically excited pulsations, rather than overstable
driving. The criterion compares the ratio of amplitude scatter to the mean amplitude; for
stochastic pulsations, this ratio should be ≈ 0.52. Stochastic processes in pulsating sdB
stars have also been discussed by Pereira & Lopes (2005) in the context of the complex sdB
pulsator PG1605+072, which is known to have variable amplitudes (O’Toole et al. 2002;
Reed et al. 2007, in press). Using the JCD01 criterion, Pereira & Lopes deduced that none
of the modes of that star were consistent with stochastic excitation. However, O’Toole et al.
(2002) noted amplitude changes between years, while Pereira & Lopes (2005) only studied
7 nights of data, and as such their analysis was likely affected by the short length of their
time series.
A limitation to the JCD01 test is that the damping times of the oscillations should be
longer than the timescale used for determining the amplitudes. Our analysis of ∼ 4 hour
segments of PG 0048 data indicate that amplitude variations are on very short timescales
that are shorter than the observing time for individual runs (see Figs. 7 and 12). We provide
the JCD01 parameter values σA/〈A〉 in Column 7 of Table 10, but we suggest that the JCD01
– 17 –
test is not appropriate for PG 0048. Aside from the JCD01 test, we can attempt to reproduce
some of the observational properties using simple simulations with damped and randomly
re-excited frequencies. The complexity of the actual data is such that we cannot hope to
reproduce it directly, but instead will strive to fit the observations listed at the beginning of
this section. Our simulations follow the simple prescription (equations 2 and 3) of Chaplin
et al. (1997) summarized as follows: The pulsations themselves are described by sine waves
of the form A(t) = A cos((2π · f)(t − φ)), with the amplitude modified in two ways; it is
damped exponentially as A = Ao exp(−te/td) where Ao is the maximum amplitude, te is the
time since the last excitation and td is the damping timescale. The pulsation is re-excited by
setting te = 0 when time t exceeds an excitation timescale (texc). The time before the first
re-excitation is randomly set to some fraction of the excitation timescale and every time the
pulsation is re-excited, the phase is randomly set and the excitation timescale and pulsation
amplitude can vary randomly by up to 20% or their original values. The free parameters of
the simulation are the input amplitude, which is the maximum amplitude attainable and the
excitation and damping timescales (Ao , texc and td, respectively). The simulation includes
frequencies, amplitudes, and phases for up to 100 pulsations with an unlimited number of
data runs (input as run start time, the number of data points, and cycle time). We will
concentrate on matching the MDM runs in Table 6. This is the G1 data set with an average
run length of 9.35 hours (∼ 0.4d), containing ∼ 2800 data points each, an average detection
limit of 0.88 mma, and average ratios AG1/〈A〉 = 0.44± 0.14 and AG1/Amax = 0.37± 0.09.
To match the observational constraint that the pulsation amplitude can reduce by half
in a four hour span, the damping timescale is necessarily less than 5.8 hours. With this
constraint, we produced a grid of simulations with 1 ≤ td ≤ 7 hours in 1 hour increments
and 1 ≤ texc ≤ 33 hours in 2 hour increments. Qualitatively, if no re-excitations occur
during an individual run, the FT is single-peaked whereas multiple re-excitations create a
variety of complex, multi-peaked FTs, depending on how similar the randomized phases were
(the less alike the phases for each re-excitation, the lower the overall amplitude and more
and similar-amplitude peaks appear in the FT). For small texc, the FT becomes increasingly
complex with most simulations resulting in many low-amplitude peaks distributed across a
couple hundred µHz. However, such complex patterns are not consistent with observations
and so we discount small values for texc. Amplitudes in the FT are reduced with large values
of texc and small values of td while the scatter increases with increasing values for both. An
increase in amplitude scatter is necessary to produce the low rate of detections.
As we now have all the pieces in place, we can ask how well the simulations reproduce
the observational constraints we set at the beginning of this section. A selection of the
results are shown in Figs. 16 and 17. Panels a, b, and c of Fig. 16 have td fixed at 5 hours
and vary texc whereas panels d, e, and f fix texc at 19 hours and vary td. Panels a and d
– 18 –
are the results for individual runs and panels b and e are for the combined nine-run data
set. The points represent the average (〈A〉) with 1σ deviations while the lines indicate
maximum (Amax) and minimum amplitudes. Panels c and f show the ratios 〈AG1〉/Amax and
〈AG1〉/〈Aind〉; the average amplitude from the combined data divided by the maximum or
average amplitude from the individual runs. Figure 17 shows the expected rate of detections
calculated in the following manner: The detection limit was calculated using the observed
ratios 〈Amax〉/0.88 = 1.56 and 〈A〉/0.88 = 1.32 for the MDM data with an average detection
limit of 0.88 mma and solving for the new detection limit. The dotted line is the observed
detection rate of 26% and it is interesting that none of simulations are that low if using
the average detected amplitude. While the overall amplitudes can become quite small, the
average follows that, which is used to calculate our detection limit in the top panel. However,
most values of td matched the observed rate near texc = 19-21 hours using Amax. The
simulations that best fit the observational constraints are those which have 4 ≤ td ≤ 6 hrs
and 13 ≤ texc ≤ 21 hrs. The lower values of texc better fit the 〈AG1〉/A ratios while the larger
values are a better match for the detection rate. These relatively simple simulations are able
to fit all of the observed constraints, thus explaining the amplitude variations, their lower
detection limits in groups of data, the relatively low detection rate and the appearance of the
peaks in the FT. What they cannot explain however, is the relative lack of phase variability
in some frequencies (though relatively few were measurable) and why stochastic processes
should occur in the first place.
Stochastic oscillations are usually presumed to be driven by random excitations caused
by convection (see Christensen-Dalsgaard 2004 for a review concerning the Sun). The
He II/He III convection zone in sdB stars was investigated by Charpinet et al. (1996),
who determined that it could not drive pulsations. However, it appears that they did not
investigate this zone for convective motions but rather as a driver for the κ mechanism. So
some ambiguity remains here. There is also convection or semi-convection in the cores of sdB
stars, and it is possible that the eigenfunctions could be sampling this region. Whatever the
case, the extreme amplitude and phase variability of PG 0048 poses a significant challenge
to the iron driving mechanism found by Charpinet et al. (2001) to excite pulsations in sdB
stars. Though it is beyond the scope of this paper to ascertain the cause of the random
amplitude variations, we find that the observed properties are consistent with our simplified
randomly excited simulations and that the amplitude spectrum resembles those of pulsators
that are stochastically driven.
– 19 –
4. Conclusions and Future Work
We have carried out multisite campaigns for two sdB pulsators, PG 1618+563B and
PG 0048+091 and in both cases, our observations were superior to the published discovery
data, yet questions concerning these two stars still remain. Our MDM observations of
PG 1618B (obtained under good conditions) show characteristics typical of about half the
objects in the sdBV (V361Hya) class: A small number of stable (in amplitude) frequencies
with a closely spaced pair. In contrast, the data obtained at McDonald observatory – under
non-photometric conditions – show PG 1618B to be a complex pulsator with four “regions”
of power showing amplitude and phase variability. An ensemble analysis of any combinations
of data other than the MDM set are hindered by poor least-squares fitting and amplitudes
reduced below detectability. Such poor fitting can be caused by unresolved frequencies
with intrinsic amplitude variability (O’Toole et al. 2002) or randomly excited pulsations
(Christensen-Dalsgaard 2004). So despite having expended considerable effort to obtain not
only multisite, but extended time-base observations, we were only reasonably successful at
detecting pulsations from our 2 m telescope data; and these show the star to be two-faced.
With this dichotomy of observational results, PG 1618B remains an interesting target for
more follow-up observations; particularly to examine its long-term frequency stability.
PG 0048 is much more complex than PG 1618B, yet it too has shown somewhat stable
pulsation amplitudes at one epoch (the discovery and 2004 data) and wildly variable ampli-
tudes at another (2005). Though an extremely rich pulsator with at least 28 independent
frequencies, many modes are only excited to amplitudes above the noise occasionally, often
for very short lengths of time. These behaviors are consistent with stochastic pulsations and
we have performed several tests along these lines. We simulated damped and re-excited pul-
sations and found that the observations were best matched with damping timescales between
4 and 6 hours and excitation timescales between 13 and 19 hours. We detected common fre-
quency splittings of 972 and 41 µHz which may be related to multiplet structure, but could
reproduce these using Monte Carlo splittings of random spacings. So while they may be
intrinsic to the pulsation of PG 0048, we cannot be sure. We can be sure that PG 0048’s
rich pulsation spectrum is too dense to be accounted for using only ℓ ≤ 2 modes regardless
of how many m components are present.
The observations presented in this paper provide some very interesting and confusing
results. Pulsations that appear stable during some times and variable at others; attributes
that have also been observed in other sdBV stars as well (KPD 2109+2752, PG 1605+072,
and HS 1824+5745, just to name a few). The pulsations in PG 0048 present observables
that seem best described by randomly excited oscillations which would be in contrast to the
proposed driving mechanism (Charpinet et al. 2001). If validated, it would represent a new
– 20 –
direction in sdB pulsations (and modeling too!). However, a longer time series may be the
only way to clarify the nature of the oscillations in this star. Ideally this would take place
on at least 2m-class telescopes and cover several weeks.
We would like to thank the time allocation committees for generous time allocations,
without which this work would not have been possible; Dave Mills for his time and help
with the Linux camera drivers; Chris Koen for contributing the discovery data and helpful
discussions; and the anonymous Referee for a helpful re-organization of the paper. This
material is based in part upon work supported by the National Science Foundation under
Grant Numbers AST007480. Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s) and do not necessarily reflect the views
of the National Science Foundation. JRE, GAG, and SLH were supported by the Missouri
Space Grant Consortium.
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This preprint was prepared with the AAS LATEX macros v5.2.
– 22 –
Table 1: Observation record for PG 1618B. The first two runs were obtained in 2003, while
the rest were obtained in 2004.
Run Date Start Length Int. Observatory
UT hr:min:sec (Hrs) (s)
suh16mar 17 Mar 00:07:58 3.3 10 Suhora 0.6 m
lul031705 17 Mar 16:46:33 5.6 10 Lulin 1.0 m
McD031805 18 Mar 04:34:00 1.1 5 McDonald 2.1 m
lul031805 18 Mar 15:14:24 6.1 10 Lulin 1.0 m
McD031905 19 Mar 08:40:40 3.1 5 McDonald 2.1 m
lul031905 19 Mar 19:58:41 1.3 15 Lulin 1.0 m
baker032005 20 Mar 04:29:30 5.6 25 Baker 0.4 m
McD032005 20 Mar 06:35:00 5.8 5 McDonald 2.1 m
lul032005 20 Mar 15:17:19 6.0 10 Lulin 1.0 m
suh20mar 20 Mar 18:59:00 8.3 10 Suhora 0.6 m
lul032105 21 Mar 16:44:59 0.7 15 Lulin 1.0 m
suh21mar 21 Mar 18:20:20 8.6 20 Suhora 0.6 m
McD032205 22 Mar 04:33:00 7.8 5 McDonald 2.1 m
suh22mar 22 Mar 18:40:40 2.2 20 Suhora 0.6 m
McD032305 23 Mar 04:12:10 8.0 5 McDonald 2.1 m
mdr299 29 Mar 04:11:15 4.7 15 Baker 0.4 m
mdr301 31 Mar 04:28:10 7.0 15 Baker 0.4 m
mdr302 02 Apr 02:53:10 8.1 10 Baker 0.4 m
bak040305 03 Apr 03:20:46 7.9 15 Baker 0.4 m
bak040405 04 Apr 04:23:10 5.6 15 Baker 0.4 m
bak040505 05 Apr 03:17:10 6.6 10 Baker 0.4 m
bak041405 14 Apr 03:45:00 7.1 10 Baker 0.4 m
bak041505 15 Apr 02:26:30 8.4 10 Baker 0.4 m
bak041605 16 Apr 02:50:30 7.9 10 Baker 0.4 m
bak041705 17 Apr 03:18:50 5.3 15 Baker 0.4 m
bak041805 18 Apr 02:56:00 7.7 15 Baker 0.4 m
mdm042605 26 Apr 04:25:30 7.3 5 MDM 2.4 m
mdm042705 27 Apr 04:16:50 7.6 5 MDM 2.4 m
mdm042805 28 Apr 04:14:00 7.8 3 MDM 2.4 m
mdm042905 29 Apr 04:18:00 1.6 5 MDM 2.4 m
mdm043005 30 Apr 04:04:30 7.8 3 MDM 2.4 m
mdm050105 01 May 04:05:30 7.8 5 MDM 2.4 m
mdm050205 02 May 03:36:40 5.3 5 MDM 2.4 m
– 23 –
Table 2: Subsets of data for PG 1618B. For column 2, observatories are 1) McDonald Obser-
vatory; 2) Suhora Observatory; 3) Lulin Observatory; 4) Baker Observatory; and 5) MDM
Observatory.
Set Observatory(ies) Inclusive Dates Resolution 4σ detection limit
µHz mma
McD 1 18 - 23 March 2.2 1.64
MDM 5 26 Apr - 02 May 1.9 0.55
McD+MDM 1, 5 18 - 02 May 0.3 0.59
Week 1 1, 2, 3 17 - 23 March 1.5 1.53
All 1, 2, 3, 4, 5 17 - 02 May 0.2 0.77
– 24 –
Table 3: Our least-squares fit solution for the pulsation periods, frequencies, and amplitudes
detected in PG 1618B. Formal least-squares errors are given in parentheses. Frequencies
marked with a dagger (†) were only detected during individual runs (one each) while the
remaining frequencies are from the MDM data set.
Des. Period Frequency Amplitude
(s) (µHz) (mma)
f1† 108.7092 (0.0518) 9198.85 (4.38) 1.79 (0.39)
f2† 122.2574 (0.0597) 8179.46 (4.00) 1.04 (0.21)
f3 128.9549 (0.0008) 7754.64 (0.05) 1.71 (0.09)
f4 139.0571 (0.0008) 7191.28 (0.04) 2.04 (0.09)
f5 143.9290 (0.0011) 6947.87 (0.05) 2.22 (0.10)
f6 143.9759 (0.0014) 6945.60 (0.07) 1.64 (0.10)
– 25 –
Table 4: Observations of PG 0048
Run Date Start Length Int. Observatory
UT hr:min:sec (Hrs) (s)
mdr285 10 Oct 04:21:00 6.2 15 MDM 1.3 m
mdr290 12 Oct 03:39:00 7.3 15 MDM 1.3 m
mdr295 14 Oct 03:34:00 7.4 15 MDM 1.3 m
boao 26 Sept. 10:30:20 5.0 10 BOAO 1.9 m
mdm092805 28 Sep 07:32:30 4.5 15 MDM 1.3 m
mdm092905 29 Sep 02:53:00 9.5 15 MDM 1.3 m
mdm093005 30 Sep 02:44:00 9.5 15 MDM 1.3 m
turkSep3sdb 30 Sep 17:37:54 9.3 10 Tubitak 1.5 m
mdm100105 01 Oct 02:46:00 9.4 10 MDM 1.3 m
turk1Octsdb 01 Oct 21:26:58 4.1 10 Tubitak 1.5 m
mdm100205 02 Oct 09:57:30 2.3 15 MDM 1.3 m
turk2Octsdb 02 Oct 20:51:36 4.8 10 Tubitak 1.5 m
mdm100305 03 Oct 02:32:00 9.1 10 MDM 1.3 m
turk3Octsdb 03 Oct 17:41:49 8.5 10 Tubitak 1.5 m
mdm100405 04 Oct 09:24:00 2.2 15 MDM 1.3 m
mdm100505 05 Oct 02:33:00 9.4 12 MDM 1.3 m
mdm100605 06 Oct 02:23:00 9.4 10 MDM 1.3 m
mdm100705 07 Oct 02:15:00 9.6 12 MDM 1.3 m
lul7Oct 07 Oct 18:32:20 1.6 20 Lulin 1.0 m
a024 07 Oct 20:28:14 6.2 10 SAAO 1.9 m
mdm100805 08 Oct 03:17:00 8.5 15 MDM 1.3 m
a039 08 Oct 21:31:48 5.4 10 SAAO 1.9 m
mdm100905 09 Oct 02:17:00 9.3 10 MDM 1.3 m
a057 09 Oct 20:11:53 4.1 10 SAAO 1.9 m
mdm101005 10 Oct 02:06:00 9.6 10 MDM 1.3 m
a077 10 Oct 19:58:49 6.5 10 SAAO 1.9 m
mdm101105 11 Oct 02:00:00 9.6 10 MDM 1.3 m
– 26 –
Table 5. Frequencies detected for differing subsets of PG 0048 data. Column 1 is the
frequency designation; column 2 is the frequency determined from a combined data set;
column 3 is the formal least-squares error from the combined data set; column 4 is the
standard deviation of frequencies detected from individual runs; and column 5 is the
number of individual 2005 and 2004 runs in which that frequency is detected. All
frequencies are given in µHz. († indicates frequencies that were not detected in 2005.)
Des. Freq. σfit σ N
f1 5203.1 - 5.9 2
f2 5244.9 0.16 4.6 15
f3 5287.6 0.03 3.6 8
f4 5356.9 0.05 9.8 8
f5 5407.0 0.07 3.2 1
f6 5465.1 - - 1
f7 5487.2 0.05 8.5 7
f8 5612.2 0.05 4.5 8
f9 5652.9 - - 1†
f10 6609.2 - 9.0 2
f11 6834.3 0.15 9.0 3†
f12 7154.3 0.08 1.4 1
f13 7237.0 0.04 7.0 10
f14 7430.1 - - 1
f15 7501.3 0.07 - 1
f16 7523.9 0.06 15.0 8
f17 7560.0 - - 1
f18 7610.1 0.07 5.6 5
f19 8055.5 0.06 10.1 8
f20 8651.4 0.08 - 1
f21 8820.6 0.08 6.4 4
f22 9352.8 0.07 16.3 1
f23 9385.3 0.28 8.9 1†
f24 9694.6 - - 1
f25 9795.1 0.08 18.8 2
f26 10366.8 0.08 1.4 2
f27 11103.3 - - 1†
f28 11159.8 0.07 - 1
– 27 –
Table 6. Amplitudes detected for differing subsets of PG 0048 data during the 2005 multisite campaign. The last
two rows give the 4σ detection limit and the temporal resolution for each subset. Note that all frequencies are given
in µHz. Runs 1 through 12 are boao, mdm0929, mdm9030, mdm1001, mdm1003, mdm1006, mdm1007, saao1007,
mdm1008, mdm1010, saao1010, and mdm1011 and runs G1 through G4 are MDM only, Oct. 1 - 3, Oct. 7 - 11, and
all the 2005 data, respectively.
Des. 1 2 3 4 5 6 7 8 9 10 11 12 G1 G2 G3 G4
f1 1.11 1.69
f2 2.44 1.03 1.48 1.25 2.26 1.12 0.93 2.34 1.70 2.19 2.42 1.77 1.38 1.07 1.77 1.40
f3 1.21 1.39 1.22 1.01 1.26 1.77 0.60 0.62 0.78 0.83
f4 1.13 1.30 1.24 1.07 1.01 0.59 0.94 0.53 0.54
f5 1.19 0.41 0.52 0.42
f6 0.86 0.59
f7 1.06 1.05 1.02 0.87 1.59 0.56 0.66
f8 0.98 1.12 1.39 1.06 1.02 0.92 0.53
f10 0.94
f12 0.81 0.36
f13 1.36 1.04 0.95 1.68 1.50 1.46 2.13 1.07 0.79 1.08 0.78
f14 0.83
f15 1.52 0.37 0.64 0.79 0.37
f16 0.97 1.36 2.26 1.39 1.78 1.09 0.53 0.86 0.48
f17 1.10 0.50 0.48
f18 1.00 1.09 1.10 0.99 0.48 0.63 0.59 0.36
f19 1.12 0.89 1.08 1.13 1.74 0.43 0.58 0.69 0.47
f20 0.90 0.36 0.44
f21 1.41 0.81 0.94 0.39 0.63
f22 1.10 0.41 0.46
f23 0.44
f24 0.85
f25 1.22 0.88 0.36 0.48 0.35
f26 0.91 1.01 0.36 0.56
f28 1.18 0.40 0.54 0.35
4σ 0.96 0.75 0.84 0.85 0.90 1.02 0.76 1.60 0.84 1.07 1.00 0.91 0.31 0.47 0.43 0.28
1/T 53 29 29 31 29 29 29 46 31 29 39 36 0.9 2.2 2.2 0.8
– 29 –
Table 7. The same as Table 6 for the discovery data (1997 and 1998; courtesy of C. Koen) and 2004 MDM data.
Runs dd1 through dd10 are tex151, tex157, tex177, tex182, tex186, dmk157, dmk163, ck261, ck263, and ck265, runs 1
through 3 are mdr285, mdr290, and mdr295, and runs G1 through G4 are all 1997, October 1998, November 1998,
and 2004 data, respectively.
Des. dd1 dd2 dd3 dd4 dd5 dd6 dd7 dd8 dd9 dd10 1 2 3 G1 G2 G3 G4
f1 2.63
f2 3.18 1.11 1.95 2.01 2.78 1.65
f3 5.86 3.52 2.77 1.76 4.29 2.10 1.54
f4 3.39 2.33 3.40 1.96 1.14 1.90 1.93 1.34 1.57
f5 1.96 0.55
f7 1.96 1.20 2.12 0.93 0.84 1.94 2.16 0.75
f8 4.94* 5.80 3.97 4.39 3.10 3.43 3.57 3.95 4.96 2.85 3.10 3.14 4.05 3.34 2.01 2.97
f9 1.28 0.65
f10 1.18 0.68
f11 1.03 1.16 0.98 1.56 1.00
f13 2.76 1.55 1.56 1.56
f16 4.31* 2.28 1.22 1.26
f17 1.55 2.17 1.69 1.40 1.56
f18 1.25 0.77
f19 2.36 3.29 1.21 1.08 2.03 1.17 1.99 1.06
f20 1.75
f21 0.93 0.74
f22 2.43 1.26
f23 1.04 0.65
f24 0.53
f27 1.00
4σ 5.38 3.54 1.50 1.20 1.70 2.06 1.63 3.80 3.17 2.44 0.75 0.82 0.97 1.14 1.21 1.97 0.50
1/T 194 132 69 96 84 96 84 174 185 111 41 38 37 0.8 5.4 0.9
– 31 –
Table 8. PG 0048 pulsation frequencies split by integer multiples near 972 and 41.1 µHz.
The first column gives the minimum degree ℓ for the multiplet and the order for each row
is from the top of Table 5, or in descending frequencies and the number in parentheses
indicates the integer multiple between itself and the previously listed frequency and the
percent deviation from 972 or 41.1 µHz.
ℓmin Des. Designations for related frequencies
Spacings of 972 µHz
3 f1 f12 (2, 0.2%); f27 (4, 1.4%)
3 f7 f14 (2, 0.2%); f23 (2, 0.4%); f26 (1, 0.8%)
1 f8 f10 (1, 2.4%); f17 (1, 2.4%)
2 f9 f18(2, 0.5%); f20 (1, 6.9%)
2 f11 f21 (2, 2.0%); f25 (1, 0.07%)
1 f3 f13 (2, 0.4%)
2 f6 f22 (4, 0.2%)
Spacings of 41.1 µHz
6 f1 f2 (1, 1.7%); f3 (1, 3.9%); f5 (3, 3.2%); f7 (2, 2.4%); f8 (3, 1.5%); f9 (1, 0.1%)
6 (1, 1) f12 f13 (2, 0.05%); f16† (7, 0.02%); f18 (2, 5.9%)
– 32 –
Table 9. Results of simulations for phase changes using f2 as the template and a
comparison with observations. Column 1 indicates how the phase was changed in the
simulation and columns 2 and 3 provide the average amplitude and standard deviation of
> 100, 000 simulations. Columns 4 and 5 provide ratios comparing amplitudes between
individual runs and that of the combined simulated data set. The bottom row provides the
average amplitude from the individual runs and results from the simulations with constant
phase and fixed amplitudes and phases with those observed for f2.
Sim 〈A〉 σ 〈A〉/〈A〉ind 〈A〉/Amax
10% 1.50 0.03 0.97 0.66
20% 1.35 0.10 0.88 0.60
Random 1.06 0.11 0.69 0.47
〈A〉ind constant phase Aobs & φobs
1.54 1.55 1.49
Table 10. Observed properties of individual frequencies.
Des. #det 〈σd〉 #exp 〈A〉 σA σA/〈A〉 Amax AG4/〈A〉 AG4/Amax AG1/〈A〉 AG1/Amax σφ (%)
f1 2 0.38 3 1.30 0.41 0.32 1.69 - - - - -
f2 12 3.69 12 1.69 0.53 0.31 2.44 0.83 0.57 0.82 0.57 9.6
f3 5 2.63 11 1.32 0.22 0.17 1.77 0.63 0.47 0.45 0.34 -
f4 5 1.59 10 1.14 0.15 0.13 1.30 0.47 0.44 0.52 0.48 8.4
f5 2 0.92 7 1.06 0.19 0.18 1.19 0.40 0.35 0.39 0.34 -
f6 1 0.11 1 0.86 - - 0.86 0.67 0.67 - - -
f7 5 1.16 8 1.09 0.19 0.17 1.59 - - 0.51 0.35 7.9
f8 5 1.50 10 1.11 0.09 0.08 1.39 0.48 0.38 - - 6.3
f9 1 - - - - - - - - - - -
f10 1 1.06 8 0.94 - - 0.94 - - - - -
f11 1 - - - - - - - - - - -
f12 1 0.33 3 0.81 - - 0.81 - - 0.44 0.44 -
f13 7 2.32 11 1.43 0.36 0.25 2.13 0.55 0.37 0.75 0.50 15.8
f14 1 0.41 3 0.83 - - 0.83 - - - - -
f15 1 2.75 11 1.52 - - 1.52 0.24 0.24 0.24 0.24 -
f16 6 2.60 11 1.38 0.44 0.32 2.26 0.35 0.21 0.38 0.23 15.0
f17 1 0.50 4 1.10 - - 1.10 0.44 0.44 0.45 0.45 -
f18 4 0.94 7 1.05 0.18 0.17 1.10 0.43 0.33 0.46 0.44 25.9
f19 5 1.79 10 1.19 0.26 0.22 1.74 0.39 0.27 0.36 0.25 7.3
f20 1 0.32 2 0.90 - - 0.90 - - 0.40 0.40 -
f21 3 1.16 8 1.03 0.26 0.25 1.41 - - 0.38 0.28 -
f22 1 1.32 9 1.10 - - 1.10 - - 0.37 0.37 -
f23 1 - - - - - - - - - - -
f24 1 0.00 1 - - - - - - - - -
f25 2 2.25 11 1.03 0.24 0.23 1.22 0.34 0.29 0.35 0.30 -
f26 3 0.76 6 0.94 0.17 0.18 1.01 - - 0.41 0.36 -
f27 1 - - - - - - - - - - -
f28 1 1.42 9 1.18 - - 1.18 0.30 0.30 0.34 0.34 -
– 34 –
Fig. 1.— Lightcurves for PG 1618B data. The top four panels show the coverage during the
multisite portion of the campaign from March 17 – 24, 2005, while the bottom panel shows
an enlarged section of a run obtained at MDM. Note that the scales are different for the
bottom panel. The line indicates our fit to the data.
– 35 –
Fig. 2.— Temporal spectra of various subsets of PG 1618B data. Insets are the window
functions. Solid (blue) horizontal line is the 4σ detection limit while the dashed (blue) lines
are at 2 mma in all panels.
– 36 –
Fig. 3.— Temporal spectrum of PG 1618B data. Top panel shows the original FT of
the combined MDM data while the bottom panel shows the residuals after prewhitening.
Prewhitened frequencies are indicated by arrows. Insets show an enlarged view of the fre-
quency doublet and the window function.
– 37 –
Fig. 4.— Amplitudes and phases of the two frequencies in PG 1618B that are resolvable
nightly. The data are only those from McDonald and MDM observatories except for a single
Lulin run (marked by a triangle). The dashed lines indicate our fits to the MDM data. Note
that the time axis is discontinuous.
– 38 –
Fig. 5.— Lightcurves for the PG 0048 data. Each panel is two days.
– 39 –
Fig. 6.— Comparison of temporal spectra for varying groups of PG 0048 data. The large
panels show FTs for combinations of data (inclusive dates are labeled). The insets on the
right are the window functions for the combinations of data to scale. The central insets are
slightly enlarged FTs of individual runs obtained congruent with the larger panels and are
labeled. They go from early to late within the campaign.
– 40 –
Fig. 7.— Temporal spectra for individual PG 0048 observing runs.
– 41 –
Fig. 8.— Amplitudes and errors of 12 frequencies detected in PG 0048. Each panel contains
amplitudes for four frequencies (differentiated by point type and color) which may be shifted
by ±0.1 day if they overlap. The solid line is the 4σ detection limit.
– 42 –
Fig. 9.— a) The number of detections of frequencies from individual runs for PG 0048
compared to Gaussian probability. The number of possible detections (dashed line) compared
to the amount of actual detections (solid line) depending on the detection significance. b) A
comparison between the number of detections for individual frequencies and the maximum
amplitude detected.
– 43 –
Fig. 10.— A schematic of the pulsation content of PG 0048 to indicate the nearly evenly
spaced frequencies (provided in Table 8). The top panel shows the spacings that are integer
multiples near 972 µHz and the bottom panel shows those near 41.1 µHz. For clarity, each
set of multiplets are connected via horizontal bars at differing heights and have different
colors (electronic version only, and black is used twice in the top panel).
– 44 –
Fig. 11.— A 350µHz region of PG 0048’s FT covering three modes at ∼7497.4µHz,
∼7524.1µHz and ∼7605.9µHz. The panels correspond to those in Fig. 6. The spectral
windows at the top of each panel cover the same frequency range as the observations, to give
the reader an appreciation of how dense these regions are.
– 45 –
Fig. 12.— Temporal spectra for halves of individual PG 0048 observing runs.
– 46 –
Fig. 13.— Amplitudes of frequencies detected in both individual runs and grouped data
during 2005 for PG 0048. Dots (black) indicate the maximum amplitude, squares (blue) the
average amplitude from individual runs. The lines indicate amplitudes from the following
group datasets (from Table 6): dotted (blue) is G1, short-dashed (green) is G2, long-dashed
(magenta) is G3, solid (black) is G4, and the dot-dashed (black) line is the G4/Amax ratio
(or G1/Amax if no G4 value).
– 47 –
Fig. 14.— Phases of pulsation frequencies. Frequency designations are provided in each
panel with phases determined only for individual runs in which the frequency was detected.
– 48 –
Fig. 15.— Comparison of amplitude ratios from simulations to those observed for PG 0048.
Black circles are for G4 and blue squares are for G1 data. The verticle-lined (top and green)
area is for simulations with 10% phase variations, the +45o-lined (middle and red) area is
for simulations with 20% phase variations, and the −45o-lined (bottom and blue) area is for
simulations with random phases.
– 49 –
Fig. 16.— Simulations of randomly excited and damped pulsations. Panels a, b, and c show
the effect of changing the excitation timescale for a fixed damping timescale of td = 5 hours.
Panel a and b show the results of simulating a single 9.35 hour run and the nine MDM
runs in Table 2, respectively. The points indicate the average amplitude with 1σ error bars,
the solid line is the maximum amplitude from an individual simulation and the dotted line
indicated the minimum amplitude detected from an individual simulation. Panel c shows
the ratios 〈AG1〉/Amax (blue circles) and 〈AG1〉/〈Aind〉 (black triangles). The dashed blue
(dotted black) lines indicate the observed range in PG 0048 for 〈AG1〉/Amax (〈AG1〉/〈Aind〉).
Panels d, e, and f correspond to panels a, b, and c except that the excitation timescale is
fixed at texc = 19 hours and the damping timescale is varied.
– 50 –
Fig. 17.— Expected fraction of detected frequencies based on simulations. Top panel used
average amplitudes while the bottom panel used maximum amplitudes from the simulations.
Differing lines represent different values of td given in the legend. Dotted line is the observed
26% detection rate. See the electronic edition of the Journal for a color version of this figure.
	Introduction
	PG 1618B+563B
	Observations
	Analysis
	Discussion
	Constraints on the pulsation modes
	Amplitude and phase stability
	PG 0048+091
	Observations
	Analysis
	Discussion
	Frequency content
	Constraints on mode identifications
	Amplitude and phase variability
	A possible cause of the amplitude/phase variability
	Conclusions and Future Work
