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A&A manuscript no.
(will be inserted by hand later)
Your thesaurus codes are:
6 (08.03.2; 08.15.1; 08.09.2 (LSS 3184); 08.22.3)
ASTRONOMY
AND
ASTROPHYSICS
April 10, 2000
Physical properties of the pulsating hydrogenídeficient star
LSS 3184 (BX Cir) ?
V. M. Woolf and C. S. Jeffery
Armagh Observatory, College Hill, Armagh, BT61 9DG, Northern Ireland
email: vmw@star.arm.ac.uk, csj@star.arm.ac.uk
Received 16 February 2000; accepted 3 April 2000
Abstract. We report new determinations of the radius
and mass of the pulsating heliumírich, hydrogenídeficient
star LSS 3184 (BX Cir) using measurements of radial veí
locity and angular radius throughout its pulsation cycle.
Measurements of radial velocity, and thus changes in stelí
lar radius (\DeltaR ? ), were made using AngloíAustralian Teleí
scope echelle spectra. Hubble Space Telescope ultraviolet
spectra and groundíbased BV photometry were used to
find temperatures and fluxes throughout the pulsation cyí
cle. The temperatures and fluxes were used to find the
angular radius of the star (ff). The ff, \Deltaff, and \DeltaR values
thus found were used to calculate the mean stellar radius
hR ? i = 2:31 \Sigma 0:10R fi . If we use the previously determined
log g = 3:35 \Sigma 0:1 for LSS 3184 and our radius estimate,
we find its mass to be M ? = 0:42 \Sigma 0:12M fi .
Key words: stars: chemically peculiar -- stars: oscillations
-- stars: variables -- stars: individual: LSS 3184
1. Introduction
Pulsations in stars provide tools for researchers in seví
eral fields of astronomy. They provide standard candles for
measuring Galactic and extragalactic distances and they
provide methods for measuring stellar parameters, even
below the directly observable photosphere. Pulsations in
hydrogenídeficient, heliumírich stars such as extreme heí
lium, R Coronae Borealis, and hydrogenídeficient carbon
stars have not been studied in as much detail as those
in stars with more `normal' chemical abundances. While
this is partly understandable since most pulsating stars
have normal compositions, studying stars with little or no
hydrogen is important to allow tests of pulsation theory.
Send offprint requests to: V. M. Woolf
? Based on observations obtained with the NASA/ESA Hubí
ble Space Telescope, which is operated by STScI for the Assoí
ciation of Universities for Research in Astronomy, Inc., uní
der NASA contract NAS 5í26555. Based on observations obí
tained at the AngloíAustralian Telescope, Coonabarabran,
NSW, Australia.
Extreme helium stars, as their name would imply, have
weak or noníexistent hydrogen absorption lines and very
large helium abundances ( ? ¦ 99 per cent). Two extreme
helium stars, V652 Her and LSS 3184, are known to pulí
sate. Saio (1993) showed that in extreme helium stars with
temperatures around 2 \Theta 10 5 K, like V652 Her, the ßí
mechanism caused by ironígroup (Zíbump) opacity can
excite the observed pulsations. Saio (1994, 1995) predicted
that LSS 3184 should pulsate because of its location in the
Zíbump instability finger. Kilkenny & Koen (1995) discoví
ered that LSS 3184 shows photometric variations with a
period of about 0.107 d. The similarity of V652 Her and
LSS 3184 in temperature, surface gravity, and pulsation
period implies that they are very similar in other physical
parameters.
Kilkenny et al. (1999) have recently reported an obserí
vational analysis of LSS 3184, including a determination of
its photometric period (0.1066 d). In addition, they used
medium resolution (Ö=\DeltaÖ ‹ 4000) spectra to measure
radial velocity variations and to show that the photometí
ric variability is caused by pulsations. Drilling, Jeffery, &
Heber (1998) reported an analysis of LSS 3184 in which
they found T eff = 23 300 \Sigma 700 K, log g = 3:35 \Sigma 0:1 and
nH=nHe ß 0:00015. Kilkenny et al. (1999) report that usí
ing the radius they determined and log g from Drilling
et al. (1998) gives a mass of 0:15M fi , which is much smaller
than 0:7M fi , the mass accepted for V652 Her (LynasíGray
et al. 1984), and small enough to imply that some input
parameter or procedure is in error.
In this paper we report an analysis of LSS 3184 usí
ing high resolution optical spectra for radial velocity meaí
surement and ultraviolet Hubble Space Telescope (HST)
spectra and groundíbased BV photometry for temperaí
ture, luminosity, and angular radius measurement. The
new data provide better temperature and angular radius
estimates and a much cleaner radial velocity curve for the
star, allowing a more reliable estimate of its radius and
mass than was possible previously.

2 V. M. Woolf & C. S. Jeffery: Physical properties of LSS 3184
Fig. 1. Plot showing velocity and phase coverage for LSS 3184 spectroscopic observations, with error bars shown for velocity.
Note that the velocity here is not corrected for projection effects.
2. Observations and data reduction
2.1. AAT visible spectral observations
Spectra of LSS 3184 were obtained during the nights of
1996 May 18 and 19 using the University College Loní
don Echelle Spectrograph at the 3.9ím AngloíAustralian
Telescope. Exposure times were between 4 and 5 miní
utes. Standard iraf packages were used for bias and
flat field correction, reducing echelle orders to one dií
mensional spectra, and applying the wavelength scale usí
ing thoriumíargon spectra. As the wavelengths covered
by adjacent orders overlapped, the spectra covered the
range 3850--5055 Ú A completely. The spectral resolution
was Ö=\DeltaÖ ‹ 48 000. Velocity corrections for Earth's moí
tion were found for each exposure using rvcorrect and
were applied using dopcor.
2.2. Hubble Space Telescope ultraviolet observations
Ultraviolet spectra of LSS 3184 were obtained using the
Faint Object Spectrograph of the HST. Observations were
made in RAPID mode using the blue detector, the 0: 00 86
square aperture and grating G160L over three orbits on
1997 February 7. Data files reduced using the most recent
calibration files were downloaded from the HST archives
on 1999 October 12. Individual exposures were made apí
proximately 19.25 seconds apart. The usable part of the
spectra covered the range 1150--2500 Ú A. Based on the
ephemeris of Kilkenny et al. (1999), observations durí
ing the three orbits covered the phase (OE) ranges from
0.3154--0.6099, 0.9327--0.2376, and 0.5602--0.8651, where
maximumV magnitude is at OE = 0 and again at OE = 1.
Thus the star was observed by the HST over 85.5 per cent
of its pulsational cycle.
3. Analysis
3.1. Radial velocity determinations
The velocity shifts between spectra were measured using
the cross correlation package fxcor in iraf. For each
measurement the velocities found from 28 of the 35 orders
of the echelle spectra were used to find a weighted average
velocity. Weights for the averaging were the inverse of the
velocity errors reported by fxcor. The unused spectral

V. M. Woolf & C. S. Jeffery: Physical properties of LSS 3184 3
orders either had no strong absorption lines or had strong
interstellar lines which did not allow a reliable stellar veí
locity determination.
The procedure took two iterations. In the first iteration
the best results were obtained by using the sum of all the
second night's LSS 3184 spectra as the cross correlation
template for the first night's spectra and vice versa. The
spectra from both nights were then shifted by the velocií
ties thus found and coíadded to provide the template for
the second iteration. The absorption lines in the second
template were much sharper, as the velocity smearing due
to the stellar pulsations was effectively removed. A third
iteration was performed with the spectra shifted by the
velocities found in the second iteration before coíadding
to make the template, but the velocities from the third
iteration were effectively identical to those from the secí
ond. The velocities from the second iteration, shifted so
that the mean velocity is zero, are shown in Fig. 1. Error
bars show the weighted standard deviations of the velocí
ity from the echelle orders. Gaps are present in the data
where clouds prevented observations. From the the data in
Fig. 1, we find that the peakítoípeak radial velocity varií
ation is 40:0 \Sigma 0:3 km s \Gamma1 , which is larger than 30 km s \Gamma1 ,
the value found by Kilkenny et al. (1999).
Velocity data from the two nights were phased to
the pulsation cycle using the ephemeris of Kilkenny
et al. (1999): T 0 = 2 449 477:4691(\Sigma0:0016), Period =
0:1065784(\Sigma0:0000005)d. The measured radial velocities
were multiplied by the factor \Gamma1:42 to correct for proí
jection effects and give the surface velocity through the
pulsation cycle in the stellar rest frame. The projection
factor was chosen based on preliminary work by Monta~n'es
Rodriguez et al. (2000) (See also Albrow & Cottrell 1994;
Gautschy 1987; and references therein). We will discuss
later the effects of choosing a different projection factor.
A smooth curve (high order polynomial) was fit to the
velocity data (Fig. 2). The velocity values on this curve
were used to calculate change in stellar radius (\DeltaR ? ) and
surface acceleration. The phase bin centers for acceleraí
tion and \DeltaR ? are shifted by half a bin with respect to
the velocity bins, i.e. the end of an acceleration bin is the
center of the next velocity bin and the end of a velocity
bin is the center of the next \DeltaR ? bin.
3.2. Temperature determination
To find temperature variations through the pulsation cyí
cle, synthetic spectra were fit to the groundíbased BV phoí
tometry of Kilkenny et al. (1999) and our HST ultraviolet
spectrophotometry. Before the fitting, the spectra were
binned in wavelength. No information is lost through the
binning since the fits are to the shape of the spectrum, not
to individual lines. The spectra are noisy at short waveí
lengths, reflecting a drop in detector sensitivity. To avoid
possible problems caused by this increased noise, only the
part of the spectra with Ö Ö 1270 Ú A was used.
Fig. 2. Fit (solid line) to velocity data corrected for projection
effects (points). The data have been folded over by 0.2 cycles
in Figs. 2, 4, and 5.
The pulsation period was divided into phase bins and
the spectra and photometry within each bin were averí
aged. No significant difference in the results was found
with the period divided into 15, 25, 50, or 99 bins. We
will report the results found for 25 bins.
A grid of lineíblanketed model atmospheres was calí
culated under the assumption of planeíparallel geometry,
hydrostatic equilibrium and local thermodynamic equilibí
rium using the code sterne described by Jeffery & Heber
(1992) and by Drilling et al. (1998). Following the latí
ter, we assumed a composition for LSS 3184 given by
nH = 0; nHe = 0:99 and nC = 0:003, where n represents
fractional abundance by number, and all other elements
were assumed to have solarílike relative abundances. The
grid extended between 15000 and 30000 K, log g = 3:00
and 4.00 (cgs units). As will be shown later, changing the
log g used in the model atmospheres makes only a minor
difference in the temperatures and angular radii derived.
In the fitting procedure T eff , angular radius (ff), and
EB\GammaV were allowed to vary and the downhill simplex proí
gram amoeba (Press et al. 1992) was used to find the
minimum ¼ 2 difference between the synthetic spectrum
and the observed spectral and photometric data at each
phase (Jeffery et al. 2000). An example of the fit is shown
in Fig. 3.
When the extinction EB\GammaV was allowed to vary we
found the mean to be EB\GammaV = 0:239 \Sigma 0:008. Because the
extinction is not expected to vary with pulsation phase,
we chose to use EB\GammaV = 0:24 throughout the cycle and
did a second iteration allowing only temperature and ff to
vary.
3.3. Radius determination
In determining the radius of LSS 3184, we make two así
sumptions. First, we assume that the temperature (and
thus ff) and radial velocity (and thus \DeltaR ? ) were measured

4 V. M. Woolf & C. S. Jeffery: Physical properties of LSS 3184
Fig. 3. Best fit synthetic spectrum (dashed curve) plotted with
ultraviolet spectrum (histogram) and BV photometry values.
This is for the phase bin centered at 0.580.
at approximately the same layer in the stellar atmosphere.
Second, because we measure ff perpendicular to the line
of sight and \DeltaR ? parallel to the line of sight, we are así
suming that the pulsation is spherically symmetric. Thus
we get \DeltaR ? =R ? = \Deltaff=ff. With these assumptions we can
use a modified Baade's method (Baade 1926) to find R ? ,
using our previously determined values of ff and \DeltaR ? .
There are two ways we can do this. In the first, we
choose two phase bins, determine \Deltaff=ff and \DeltaR ? between
them and find R ? = \DeltaR ? ff=\Deltaff, with ff and R ? defined
at one of the chosen phase bins. This method has the
disadvantage of using data from only two phase bins and
thus ignoring additional information available from the
rest of the pulsation cycle. In the second method we plot
\Deltaff=ff versus \DeltaR ? . The slope of a linear fit to the data
points is then 1=R ? . We use the second method.
4. Results and discussion
The velocities, effective temperatures, and angular radii
measured through the pulsation cycle of LSS 3184 are
listed in Table 1. The velocities are not corrected for proí
jection effects and are reported for the center of the phase
bins. Temperatures and angular radii are reported for the
beginning of the phase bins. Data are missing where HST
observations did not fall in the affected phase bins. Typical
uncertainties are indicated.
The surface acceleration, surface velocity, and change
in radius determined for LSS 3184 using the AAT spectra
are shown in Fig. 4. Note that the surface velocity reported
here is the velocity measured from the spectra multiplied
by \Gamma1:42 to correct for projection effects and make posií
tive velocity be away from the star's center. The change
in radius has been set so that it is zero at photometric
maximum, OE = 0.
Fig. 5 shows the integrated flux between 1270 and
2508 Ú A and the temperature and angular radius deterí
Table 1. Velocity, temperature, and angular raí
dius through the pulsation cycle of LSS 3184. Veí
locity is not corrected for projection effects.
OE bin V a
T eff
b
ff b
range (km s \Gamma1 ) (K) (10 \Gamma11 arcsec)
\Sigma0:40 \Sigma90: \Sigma0:005
0.00--0.04 18.58 22500 1.799
0.04--0.08 14.80 22480 1.799
0.08--0.12 3.39 22480 1.794
0.12--0.16 í9.42 22450 1.790
0.16--0.20 í16.00 22350 1.793
0.20--0.24 í18.31 22230 1.796
0.24--0.28 í19.18 22040 1.807
0.28--0.32 í18.56
0.32--0.36 í17.53 21700 1.831
0.36--0.40 í15.04 21500 1.846
0.40--0.44 í12.94 21340 1.857
0.44--0.48 í10.36 21180 1.867
0.48--0.52 í7.59 21100 1.868
0.52--0.56 í5.38 21000 1.874
0.56--0.60 í2.82 20940 1.877
0.60--0.64 í0.19 20930 1.876
0.64--0.68 2.36 20900 1.877
0.68--0.72 5.39 20900 1.877
0.72--0.76 7.47 20990 1.870
0.76--0.80 11.07 21100 1.864
0.80--0.84 13.85 21330 1.849
0.84--0.88 16.91 21540 1.840
0.88--0.92 18.83 21750 1.832
0.92--0.96 19.86
0.96--1.00 20.81 22410 1.802
a center of phase bin
b beginning of phase bin
mined by fitting synthetic spectra to HST UV spectra
and groundíbased BV photometry through the pulsation
cycle.
\Deltaff=ff 0 is plotted against \DeltaR ? in Fig. 6. The `0' subí
script indicates that the reference bin is at OE = 0. As
is seen in Figures 4 and 5, the shapes of the ff and \DeltaR ?
curves are not identical, which is the cause of the noní
linear, looped shape of the \Deltaff=ff versus \DeltaR ? curve. If
the ff and \DeltaR ? curves are normalized and placed on the
same plot (Fig. 7), the differences in the curves are more
noticeable. Figure 12 from Kilkenny et al. (1999) shows a
similar loop on the right hand side of the angular radius
versus stellar linear radius plot. Our work cannot be used
as an independent confirmation of the nonílinear angular
versus stellar radius curve, however, as we have used the
same BV photometry as Kilkenny et al.
The nonílinear shape makes determining the radius
more difficult. It raises questions about the assumption
made in using Baade's method to find R ? , that \DeltaR ?
and \Deltaff are measurements of the same quantity, with
\Deltaff decreased by a factor proportional to the distance to
LSS 3184. One possible explanation of the discrepancy is

V. M. Woolf & C. S. Jeffery: Physical properties of LSS 3184 5
Fig. 4. Surface acceleration, surface velocity, and change in
radius of LSS 3184 through its pulsation cycle.
that we are measuring \DeltaR ? and ff at different layers in
the atmosphere. \DeltaR ? is measured using optical spectra,
while ff is measured using ultraviolet spectra and optical
photometry. It is possible that the layer where the optical
lines are formed expands and contracts a bit differently
than the layer where the ultraviolet continuum is formed.
It is also possible that the star and/or its pulsations are
not spherically symmetric, as might occur if the star were
flattened by rotation, so that the measured \Deltaff, which
is perpendicular to the line of sight, and \DeltaR ? , which is
parallel to the line of sight, act differently. Further, the
atmosphere is not static. The atmosphere's temperature
is constantly changing, and the pulsating layers undergo
a substantial compression at minimum radius, as shown
by the spike in the acceleration in Fig 4, which may cause
nonadiabatic effects. So the temperature may be acting
differently in the expanding part of the phase than in the
contracting part.
However, it is encouraging that the slopes of the exí
panding (the upper half of the loop in Fig. 6) and coní
tracting (lower half of the loop) parts of the \Deltaff=ff versus
Fig. 5. Flux hF1270\Gamma2508 i, effective temperature, and angular
radius of LSS 3184 through its pulsation cycle. The curves
simply connect the data points and are not fits to the data.
Fig. 6. \Deltaff=ff 0 versus \DeltaR ? through the pulsation cycle of
LSS 3184. Points are connected in order of phase with OE = 0
at the origin. The dashed line is a linear least squares fit to all
data points. Error bars for \DeltaR ? are smaller than the symbols.
Numbers next to symbols indicate the corresponding pulsation
phase.

6 V. M. Woolf & C. S. Jeffery: Physical properties of LSS 3184
Fig. 7. ff (solid curve) and \DeltaR (dashed curve) versus pulsation
phase.
\DeltaR ? curve are not too different. \Deltaff and \DeltaR ? are still
correlated.
If we find the slope of the curve using a least squares
fit to all of the data points (dashed line in Fig. 6) then
the radius at photometric maximum (OE = 0), the inverse
of the slope, is 2:27 \Sigma 0:10R fi , where the uncertainty is
derived from the standard error in the least squares fit
to the slope. The average \DeltaR ? over the pulsation cycle,
with \DeltaR ? j 0 at OE = 0, is 28190 km, or 0:04R fi . So if
we use all the \Deltaff=ff 0 versus \DeltaR ? data points, we find the
mean R ? to be hR ? i = 2:31R fi . This is larger than the
1:35R fi mean radius found by Kilkenny et al. (1999) for
LSS 3184 and is closer to the 1:91R fi mean radius found
for V652 Her by LynasíGray et al. (1984).
We tested the effects of using only a portion of the
\Deltaff=ff versus \DeltaR ? curve to determine R ? , though we have
no reason to reject any particular data points. If we use
the points on the upper part of the curve, with phase
0:16 ß OE ß 0:56, then we find hR ? i = 2:16 \Sigma 0:07R fi . If
we use the points on the lower part of the curve, ignoring
the points to the left of the change in slope at \DeltaR ? ‹
10 000 km, so that 0:60 ß OE ß 0:96, we find hR ? i = 1:75 \Sigma
0:08R fi . If we use only the points to the right of \DeltaR ? =
0 km then we find hR ? i = 1:97 \Sigma 0:10R fi . In all cases,
even the extreme one where we reject all points but those
on the lower part of the curve, our hR ? i is larger than
R ? = 1:35 \Sigma 0:15R fi , the value found by Kilkenny et al.
(1999).
Drilling et al. (1998) estimated log g = 3:35 \Sigma 0:1 for
LSS 3184. If we use this with our hR ? i = 2:31R fi in the
formula g = GMR \Gamma2 , we find M ? = 0:42 \Sigma 0:12M fi .
This is larger than 0:15M fi , the mass found by Kilkenny
et al. (1999) for LSS 3184 and closer to 0:7 +0:4
\Gamma0:3 , the estií
mated mass of V652 Her (LynasíGray et al. 1984). Our
larger estimate results mainly from the larger peakítoí
peak range of radial velocities we measured. It is likely
that the smaller velocity amplitude measured by Kilkenny
et al. resulted from a combination of the lower spectral resí
olution and the lower signal to noise in the spectra they
used. This meant that even the best cross correlation temí
plate still contained some velocity broadening, thus dilutí
ing the velocity amplitude.
There are several sources of uncertainty in our mass
determination. Uncertainty in the log g used is a major
contributor. We note that it is impossible to determine
the pulsation phase of LSS 3184 when the spectrum used
in the analysis of Drilling et al. (1998) was taken in 1985,
as ×
P is unknown. They estimated the temperature of
LSS 3184 at 23 300 \Sigma 700 K. Our data yield a similar value,
hT eff i = 23 230 K, if we use EB\GammaV = 0:27, the extinction
they used. However, because temperature, line strengths,
and other parameters presumably varied throughout the
1íhour exposure for their spectrum, it is possible that the
errors in their temperature and gravity determinations are
larger than the formal errors quoted. If we assume that
LSS 3184 has the same log g as V652 Her, log g = 3:68
(Jeffery et al. 1999), instead of log g = 3:35, then we find
that it has M ? = 0:92M fi
As mentioned earlier, the model atmospheres used
to calculate temperature and angular radius from HST
UV spectra and groundíbased BV photometry assumed
log g = 3:50. The assumed gravity enters into the stellar
parameter calculations in the stellar atmospheres used and
in calculating the mass from the radius. Changing log g in
the model atmospheres by 0.50 dex (to 3.00 or 4.00), but
using log g = 3:35 to calculate the mass as before, changes
the hT eff i, hffi, and hR ? i derived by about 1 per cent, and
thus has only a small effect (¦ 3 per cent) on the mass
derived. The mass finally derived is proportional to g.
Uncertainty in the extinction also adds uncertainty to
the stellar parameters calculated. Using EB\GammaV = 0:25 iní
stead of 0.24 increases the derived hT eff i by 510 K, or
about 2 per cent, but changes hffi by only 0.2 per cent,
and thus has a very minor effect on the derived stellar
radius and mass.
The derived radius varies proportionally with the proí
jection factor used to transform the measured stellar raí
dial velocities into surface velocities. For example, using a
projection factor of 1.31, as LynasíGray et al. (1984) used
in their analysis of V652 Her, instead of 1.42 would give
hR ? i = 2:12R fi instead of hR ? i = 2:31R fi . The smaller raí
dius would give M ? = 0:37M fi instead of M ? = 0:42M fi .
5. Conclusions
In this paper we report new determinations of the radius
(hR ? i = 2:31\Sigma0:10R fi ) and mass (M ? = 0:42\Sigma0:12M fi ) of
the pulsating hydrogenídeficient star LSS 3184. The radial
velocity data and temperature measurements are more reí
liable than those used for previous radius and mass deterí
minations for this star, so our estimates are likely to be
closer to the star's actual parameters. Further improveí
ments can be accomplished by improving the log g and

V. M. Woolf & C. S. Jeffery: Physical properties of LSS 3184 7
chemical composition estimates for LSS 3184. In addition,
there needs to be further study of the difference between
the shapes of the angular radius and physical radius curves
based on temperature and flux measurements, and radial
velocity measurements, respectively. It is important to deí
termine if the difference in shapes is a sign that some of
our data or a step in our analysis is flawed, or if there is
something happening in the star itself that causes the two
measures of radius to behave differently with pulsational
phase.
Acknowledgements. We thank Dr. D. Kilkenny for providing
the differential photometric data used in the temperature analí
ysis. We thank the referee, Dr. D. Kurtz, for his helpful comí
ments. We acknowledge financial support from the former
Department of Education of Northern Ireland and the UK
PPARC (grant Ref PPA/G/S/1998/00019).
References
Albrow M. D., Cottrell P. L., 1994, MNRAS 267, 548
Baade W., 1926, Astr. Nachr. 228, 359
Drilling J. S., Jeffery C. S., Heber U., 1998, A&A 329, 1019
Gautschy A., 1987, Vistas in Astron. 30, 197
Jeffery C. S., Heber U., 1992, A&A 260, 133
Jeffery C. S., Hill P. W., Heber U., 1999 A&A 346, 491
Jeffery C. S., Starling R. L. C, Hill P. W, Pollacco D., 2000,
MNRAS submitted
Kilkenny D., Koen C., 1995, MNRAS 275, 327
Kilkenny D., Koen C., Jeffery C. S., Hill N. C., O'Donoghue
D., 1999, MNRAS 310, 1119
LynasíGray A. E., Schèonberner D., Hill P. W., Heber U., 1984,
MNRAS 209, 387
Monta~n'es Rodriguez P., Aznar Cuadrado R., Jeffery C. S., Polí
lacco D., 2000, in Szabados L., Kurtz D. (eds.), ASP Conf.
Ser., The Impact of LargeíScale Surveys on Pulsating Star
Research (IAU Coll. 176), in press
Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P.,
1992, Numerical Recipes in FORTRAN, 2nd edn. Camí
bridge Univ. Press, Cambridge
Saio H., 1993, MNRAS 260, 465
Saio H., 1994, in Jeffery C. S., ed., CCP7 Newsletter No. 21,
Univ. St Andrews, p. 50
Saio H., 1995, MNRAS 277, 1393