Constraining the degree of the dominant mode in QQ Vir

1 MB·English

by  Andrzej Baran

AstrophysicsandSpaceScience
DOI10.1007/s•••••-•••-••••-•
Constraining the degree of the dominant mode in QQ Vir
Andrzej Baran1 • John Telting2 • Roy Østensen3 •
Maciej Winiarski1 • Marek Droz˙dz˙1 •
Dorota Koziel 4 • Mike Reed5 • Raquel Oreiro3 •
Roberto Silvotti6 • Micha l Siwak4 • Uli Heber7 •
0 Peter Papics8
1
0
2

n
a
J
(cid:13)c Springer-Verlag••••
8
2
was to compare the value of the ℓ parameter derived

for the main mode in QQVir to previously published
] Abstract Wepresentearlyresultsoftheapplicationof
R values derived by using different methods.
a method which uses multicolor photometry and spec-
S
troscopy for ℓ discrimination. This method has been
h. successfully applied to the pulsating hot subdwarfBal- Keywords hot subdwarfs; pulsating stars
p
loon090100001. Here we apply the method to QQVir
-
o (PG1325+101). This star was observed spectroscop-
r ically and photometrically in 2008. Details on spec- 1 The method
t
s troscopy can be found in Telting et al. (this volume)
a
[ while photometryandpreliminaryresultsonℓdiscrim- The method we used for mode degree ℓ discrimination
ination are provided here. The main aim of this work has been already described and the results of its ap-
1
plication to the main sequence stars presented in sev-
v
eral papers. This method is very challenging, since it
9 AndrzejBaran
2 requires observations performed in at least three fil-
CracowPedagogical University,IowaStateUniversity
2 ters. Moreover, time-series spectroscopy could also be
5 JohnTelting employed. For this reason, the best objects to apply
. NordicOpticalTelescope,LaPalma,Spain
1 this method to are bright stars with relatively long pe-
0 RoyØstensen riods and relatively high amplitudes, so they are ac-
0 Institute forAstronomy,K.U.Leuven
cessible even with small size telescopes. It is possi-
1
MaciejWiniarski
: ble then to achieve good sampling of the brightness
v CracowPedagogical University changes to derive amplitudes and periods of the de-
i
X MarekDroz˙dz˙ tected modes. Although hot subdwarfs (sdB) are rel-
r CracowPedagogical University atively faint objects, in 2004 independent photometric
a
DorotaKoziel and spectroscopic observations of the brightest pulsat-
JagiellonianUniversity ingsdBstar,Balloon090910001(Bal09),allowedasuc-
MikeReed cessful application of the method (Baran et al. 2008).
MissouriState University For the same star, but with high precision multicolor
RaquelOreiro photometry, Charpinet et al. (2008) also derived good
Institute forAstronomy,K.U.Leuven discriminationforafewmodeswithhighestamplitudes.
RobertoSilvotti Fromthesetwopaperswecandrawoneimportantcon-
INAF–OsservatorioAstronomicodiTorino clusion. The method may give positive discrimination
Micha lSiwak of the ℓ parameter either having high precision three–
filterphotometry,orafew(atleasttwo)oflowprecision
JagiellonianUniversity
photometrysupportedbyspectroscopy. Encouragedby
UliHeber
the successful application of the method for Bal09 we
Dr.Remeis–Sternwarte,Institute forAstronomy,UniversityEr-
langen–Nu¨rnberg decided to try with another pulsating subdwarfB star,
Peter Papics QQVir.
KonkolyObservatory
2
The main goal of performing another observation of
QQVir was to decide which (or if any) value of the
ℓ parameter for the dominant mode, obtained in pre-
vious works, is the correct one. Telting & Østensen
(2004) analyzed spectroscopic data and stated that
600
the main mode is consistent with ℓ=0. Quite inter-
estingly, Charpinet et al. (2006), based on photometry 500 RV
data only, derived ℓ=2 from frequency matching. The
Charpinet et al. (2006) result is even more surprising 400
amsoidteinwdiicthateℓs=u2numsuoadlley. hIigthisinitnrtinerseicstainmgpltihtuendetooftuhsee mag [mma] 300 U
bcoonthstrkaiinndsthoefvdaalutae,opfhtohteommeotdrye adnegdresepefcotrrotshceopdyo,mtio- Delta 200 B
nant mode in QQVir and if possible, either to confirm 100
one of the already obtained values or to exclude both
0 V
of them.
-100
960 970 980 990 1000 1010 1020 1030 1040 1050 1060
Frequency [mHz]
2 Observations
Fig. 1 Light curve of QQVir in three filters. Data in
different filters are colored according to the bandpass and
From the previous observations we knew that the pul- denotedontheleftside. Reddataarefortheradialvelocity
sationalperiodofthedominantmodeisrelativelyshort curveand thescale on Y-axisis not relevant for thesedata
( 2min.). Along with a low brightness ( 13.5mag in
∼ ∼
the V filter), data required for the method were not
easy to obtain, particularly with small size telescopes.
For this reason, we performed only one filter photom-
etry on each telescope with the exposure time short
enough to get appropriate sampling but not necessar-
ily long enough to have high precision data. Details
on the observations are presented in Table1, while all
photometrygatheredinthis projectis plottedinFig.1.
This figurealsoincludes spectroscopicdata whichhave
25
been used in this method. Details on spectroscopy can
be found in the paper by Telting et al. (2010) in this
volume. 20
Data in different filters were taken at different tele-
scopes. This is the reason why there are not the same ma] 15
nspuemctbreurmofofntihghetsstaforranadllsfipletcetrrsa.lrWesipthonrseespofectthetoCCthDe Amplitude [m 10
we used, the highest signal has been obtained in the
V filter. These data have been used for precise fre-
quencydeterminationofthedominantmode. Thatwas 5
also the reason why observations in the V filter have
beenperformedforthelongesttimefromthesiteswith
0
6 6.5 7 7.5 8 8.5 9
easy access, like: Suhora, Mercator and Baker. In ad-
Frequency [mHz]
dition to the V filter, data in the U filter were taken Fig. 2 Amplitude spectrum of QQVir in V filter. The
at Loiano and Nordic Optical Telescope (NOT) during dominant mode is clearly seen at ∼7.25mHz. Significant
three nights in total. Fortunately, the dominant mode signalisalsopresentathigherfrequencies. Thisspectrumis
is clearly resolved with only one night of data. Some- constrainedtotheregionwherethedominantsignalappears
time later, it turned out that, quite luckily, almost at
thesametimeasourcampaign,absolutelyindependent
data in B filter were taken during the course of a dif-
ferent project. That enables us to use three filter data.
ModedegreeinQQVir 3
The B filter data were not crucial, but this method of
Table 1 ObservationallogforQQVirdata. Alldatawere
mode discrimination improves with the number of fil- taken in 2008.
ters used.
Date Site Exp [s] Length [h] Filter
Although the observations in different filters were
16 Mar Loiano 20 2.1 U
not conducted simultaneously, we can use published
17 Mar Loiano 20 5.8 U
results to see if the amplitude of the dominant fre-
28 Mar NOT 22 2.5 U
quency changes with time. Results obtained by
01 Mar Loiano 15 4.4 B
Telting & Østensen (2004) and Silvotti et al. (2006)
13 Mar Konkoly 15 4.8 B
have radial velocity and B filter amplitudes consis-
16 Feb Suhora 15 4.2 V
tent with ours. Those observations were performed
26 Feb Suhora 15 0.4 V
independently and at differing times during 2003. We
15 Mar Mercator 20 0.8 V
therefore conclude that the amplitudes are not chang-
21 Mar Mercator 20 1.9 V
ing significantly and remain stable to within the er-
22 Mar Mercator 20 1.1 V
rors. Moreover, results from the multisite campaign
24 Mar Baker 25 8.1 V
of Silvotti et al. (2006) indicate there are no nearby
25 Mar Baker 30 7.5 V
frequencies to the dominant one, even at very low am-
27 Mar Mercator 20 2.6 V
plitudes. Thissuggeststhatourdatadonotsufferfrom
28 Mar Mercator 20 2.6 V
amplitude variability or the effects of nearby frequen-
30 Mar Mercator 20 2.2 V
cies.
31 Mar Mercator 20 7.0 V
01 Apr Mercator 20 6.8 V
03 Apr Mercator 20 5.9 V
3 Fourier analysis
04 Apr Mercator 20 4.4 V
05 Apr Mercator 20 0.8 V
A prewhitening procedure was applied to derive fre-
20 Apr Suhora 15 1.3 V
quencies, amplitudes and phases of the pulsation
24 Apr Suhora 15 4.9 V
modes. First, we assumed that amplitudes can vary
25 Apr Suhora 15 7.3 V
linearly. As it turned out, the change was insignificant
26 Apr Suhora 15 7.3 V
(withintheerrors)andwehavedecidedtoassumecon-
27 Apr Suhora 15 4.3 V
stant amplitude and redid the prewhitening process.
28 Apr Suhora 15 3.9 V
Phases were calculated relative to epoch 2454552.0
30 Apr Suhora 15 1.2 V
which was chosen arbitrarily. Amplitudes and phases
02 May Suhora 15 5.1 V
for the dominant mode at frequency 7.2554776
± 11 May Suhora 15 3.8 V
0.0000004mHz (derived from V filter data) derived
12 May Suhora 15 4.9 V
fromthedatainUBVfilters,aswellasfromtheradial
14 May Suhora 15 2.8 V
velocity curve are presented in Table2. A raw ampli-
16 May Suhora 15 2.3 V
tude spectrum calculated from the data in V filter is
showninFig.2. Ithasbeenlimited tothe regionwhere
thesignalappeared,whichis6–9mHz. Theamplitude
for the dominant modes detected both in the radial-
spectracalculatedfromotherfilterslookssimilar,how-
velocity and brightness variations. Discrimination of
ever the noise level and resolution is worse as less data
ℓ, does not depend on the observations alone. It also
were obtained. Since the main aim of this work is to
depends on the model parameters, in particular effec-
derive the most likely value of the dominant mode, the
tive temperature, surface gravity, flux and limb dark-
other modes are not included in Table2. Nevertheless,
eningderivativesovereffectivetemperatureandsurface
during the prewhitening process they were taken into
gravity. For QQVir we assumed the effective tempera-
account to check if their removal has any influence on
tureandsurfacegravityderivedbyTelting & Østensen
the frequency and/or amplitude of the mode in ques-
(2004), Teff = 34800 K, logg = 5.81 dex. Then we
tion. If so, they were included in the solution.
needed to calculate flux distributions in the range be-
tween 300 and 700 nm for a grid of models with Teff
ranging from 32500 to 37500 K and logg, from 5.5 to
4 Application of the method
6.0 dex. To do this, we used a grid of models prepared
fortheinterestedrangeinTeff andlogg. Theresulting
Having derived frequencies, amplitudes and phases we
fluxdistributionsweremultipliedbytransmissionfunc-
could apply the method to discriminate ℓ parameters
4
tions of the filters to get integrated fluxes in the three
photometric bands – UBV – that we used.
For the same grid of models, we also calculated the
specific intensities as a function of µ = cosθ, where θ
is theanglebetweenthe line ofsightandnormaltothe
stellarsurfaceinagivenpointofthestellardisc. Using
these intensities, the limb darkening law was derived
fitting the coefficients c and d of a function
I(µ) = I(0)[1 c(1 µ) d(1 √µ)] (1)
− − − −
which was found to reproduce the calculated changes
of I(µ) best. This was done for different combinations
of input parameters of the model, thus allowing the
calculation of the necessary derivatives.
5 Discussion and conclusions
A disadvantageof the method we applied for mode de- 9
gree discrimination is that we cannot derive a value of
ℓinadirectway. We mustuseaniterativeprocessand
7
employ a merit function to discriminate which mode is
the most probable. The minimum of the function (if
it exists) indicates the possible value. In this study we χ2 5
applied this method to discriminate between divergent
solutions for the main frequency of QQVir. Results of 3
our calculation are shown in Fig.3. The results clearly
show that the best fit was obtained for ℓ>0. However, 1
theminimumofthemeritfunctionisnotverydeepand
cannotexclude ℓ=1and ℓ=2. Itcan, however,,exclude 0 1 2 3 4
l
ℓ >2. This does not help to achieve the main goal of
χ2 0.50 0.94 1.72 9.30 89.80
unique mode identification, but is consistent with pre-
vious work. In the case of Bal09, Baran et al. (2008)
excluded any ℓ value if the χ2 function was at least 3 Fig. 3 Results of application of the method for mode de-
greediscrimination. Theplot shows χ2 asafunction ofthe
timeshigherthanthe smallestvalueofthisfunction. If
ℓ parameter. In the bottom of the figure the precise values
we follow this rule we can also exclude the ℓ 2 value.
≥ of χ2 for each ℓ are given
On the other hand, we can use the probability Q that
the merit function for the correct model will exceed a
givenvaluebychance. TheQvaluesfortheℓ=0,1and
2 are 0.7, 0.4, 0.13, respectively. In this case none of
the firstthree ℓ’s canbe excluded. Howeverthe chance
that ℓ=0 is wrong is more than 5 times smaller than
for ℓ=2.
Acknowledgements This work was partially sup-
portedbyPolishMNiSWgrantNo. 554/MOB/2009/0.
RS wishes to thank A.DeBlasi for having contributed
to the Loiano observations. R.O. is supported by the
Research Council of Leuven University, under grant
GOA/2003/04.
ModedegreeinQQVir 5
Table 2 Resultsoftheprewhiteningprocess. Onlyresultsforthemainmodeat7.2554776mHzarepresented. Phasesare
given for epoch 2454552.0
filter Amplitude [mma] error [mma] phase [rad] error [rad]
U 31.89 1.02 3.425 0.048
B 26.15 0.42 3.438 0.049
V 23.73 0.14 3.434 0.007
Radial Velocity [km/s] error [km/s] phase [rad] error [rad]
RV 11.74 0.55 5.633 0.052
6
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