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Simulating colour variations in pulsating sdB stars
B.Ramachandran and C.S. Jeery
Armagh Observatory, College Hill, Armagh BT61 9DG, N. Ireland
brc@arm.ac.uk, csj@arm.ac.uk
Abstract
We combine a grid of high-resolution theoretical in-
tensity spectra with simulated non-radially pulsating
stellar surfaces for a variety of pulsation modes. These
are used to simulate the light curves in a number of
photometric pass bands. We attempt to provide di-
agnostics for identifying the modes in pulsating sdB
stars KPD2109+4401 and HS0039+4302.
Introduction
Mode identication in nonradially pulsating stars is a
major issue to be settled. A number of workers (see
cf. Stamford & Watson 1981; Watson 1988) have at-
tempted to solve this problem through analysing pho-
tometric colour variations. In recent years some of the
subdwarf B stars have been discovered to pulsate with
many frequencies. They oer an oppurtunity to per-
form asteroseismic analysis provided their modes are
identied. The usual method for mode identication
is to obtain the theoretical amplitude ratios and com-
pare these with observations. The theoretical meth-
ods mainly involve modelling the ux variations on
the surface to obtain the corresponding disk integrated
quantities in various photometric pass bands. But this
also involves the introduction of nonadiabatic ad hoc
parameters (Watson 1988) which is not always desir-
able. Here we try to provide diagnostics to identify the
modes of oscillations in these stars by numerically sim-
ulating the colour variations through theoretically gen-
erated disk integrated spectra thereby avoiding photo-
metric modelling.
We take two subdwarf B stars KPD 2109+4401 and
HS0039+4302. KPD2109+4401 was dicovered inde-
pendently by Billeres et al.(1998) and Koen(1998).
KPD2109+4401 exhibits about seven periods closely
spaced between 180s to 200s. It was also a target of
high speed spectroscopy(Jeery & Pollacco 2000) and
they found velocity amplitudes of just over 2km/s for
two modes of the star. HS0039+4302 was discovered to
pulsate by Ostensen et al.(2001). It pulsates in about
four frequencies with the periods between 180s to 195s.
Intensity Spectra
Theoretical intensity spectra were calculated for a
range of model atmospheres chosen to match approx-
imately the observed properties of KPD 2109+4401
and HS0039+4302. The model atmospheres were cal-
culated using sterne (Jeery et al. 2003) on a grid
T eff = 28 000(1 000)36 000K, log g = 5:5(0:1)6:0, a
He/H ratio of 0.1 by number and [Fe/H] = 1. The
intensity spectra were computed using spectrum v3 (
Jeery 2003), in the wavelength range 2900 { 7500  A,
using 6 angles  = 0:(0:2)1:0. Typically 25000 metal
lines and all H and He lines would be included in the
spectral synthesis. (In fact, the high-order members
of the Balmer series are not currently treated in the
code; we omitted the wavelength range 3650 { 3800
Angstrom from the nal analysis).
Simulation of colours
bruce (Townsend 1997) was used to simulate the
surface of non-radially pulsating stars; of particular
importance for us are the temperature and gravity dis-
tributions across the visible hemisphere. For the char-
acteristic models we adopted a polar T eff = 32 000K
and log g = 5:8 appropriate for comparison with
KPD2901+4401 (Heber 2000) and HS0039+4302
(Edelmann et al. 2003), polar inclination 60 ф and
30 ф , equatorial rotation velocity 10 km/s. Simulations
were carried out for pulsation models with (l; m) =
(0,0),(1,0),(1,1),(2,0),(2,1),(2,2),(3,0),(3,1),(3,2),(3,3).
The amplitude for each pulsation was dened to be 10
km/s in velocity. These surface models were convolved
with the intensity spectra using kylie (Townsend 1997)
in order to obtain disk-integrated uxes as a function
of time for a given pulsation mode. We converted
these spectra into the Sloan digital sky survey photo-
metric system(SDSS) (Fukugita 1996) by convolving
the uxes with lter functions for u 0
,g 0
,r 0
lters (Fig
1). We then obtained simulated light curves for each
of the lters. Figure 2 shows the simulated light curve
for a l=1,m=0 mode for KPD2109+4401 at polar
inclination angle of 60 ф .
Fig 1. Theoretical flux spectrum for T eff = 32 000K,
log g = 5:8 and [Fe/H] = 1 in then wavelength region
2900 -- 7500 љ A. The SDSS filter response function for
u 0
,g 0
,r 0
are superimposed.
Fig 2. The light curve for l=1,m=0 mode for
KPD2109+4401 at polar inclination angle of 60 ф . The
amplitude of u 0
filter is the greatest followed by g 0
and
r 0 and there is no phase difference
Fig 3. Photometric amplitude differences in SDSS fil­
ters for 30 ф and 60 ф
Analysis
The total amplitude depends on the amplitude of the
oscillation and not on the mode. we use dierence in
amplitudes a x 0 a u 0 where a x 0 indicates amplitudes of
green and red colours and a u 0 is the amplitude of the
u 0 lter. From the light curves,we calculated the pho-
tometric amplitudes for each lter for two polar incli-
nation angles at 30 ф and 60 ф (Fig 3) and obtained the
amplitude dierences a g 0 a u 0 and a r 0 a u 0 (see ta-
ble 1). As expected, Figure 3 demonstrates the largest
dierence between a u 0 and a g 0 for low order (l = 0,1)
modes; that the a g 0 a u 0 variation is generally smaller
than a r 0 a u 0 . The inclination angle seems to be an
important factor in discriminating between modes of
the same spherical degree. We note that there are no
phase dierences between the various pass bands (see
Figure 1).
Table 1. Amplitude dierences for various modes
a g 0 a u 0 and a r 0 a u 0 . (a g 0 a u 0  g 0 =u 0 ) and
(a r 0 a u 0  r 0 =u 0 ).
i = 60 ф i = 30 ф
l m g 0 =u 0 r 0 =u 0 g 0 =u 0 r 0 =u 0
0 0 -0.0154 -0.0332 -0.0154 -0.0193
1 0 -0.0053 -0.0660 -0.0092 -0.0115
1 1 -0.0093 -0.0115 -0.0053 -0.0066
2 0 -0.0006 -0.0008 -0.0029 -0.0037
2 1 -0.0036 -0.0052 -0.0040 -0.0051
2 2 -0.0035 -0.0042 -0.0011 -0.0014
3 0 -0.0004 -0.0006 -0.0003 -0.0004
3 1 -0.0001 -0.0000 -0.0008 -0.0012
3 2 -0.0007 -0.0010 -0.0004 -0.0006
3 3 -0.0004 -0.0000 -0.0000 -0.0001
Conclusion
We have combined a theoretical grid of spectra with
that of simulated non-radially pulsating stellar sur-
faces appropriate to two subdwarf B stars KPD
2109+4401 and HS0039+4302. We numerically sim-
ulated light curves which were obtained after suitably
convolving the theoretical uxes with the lter func-
tions in Sloan digital sky survey system. We nd that
there are no phase dierences between the pass bands.
We also nd that the inclination angle between the po-
lar axis and line of sight is an important diagnostic for
identifying modes as it can be a discriminating factor
between modes of the same spherical degree.
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