Asteroseismology of the Beta Cephei star Nu Eridani: photometric observations and pulsational frequency analysis

0.94 MB·English

by  G. Handler

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2)
Asteroseismology of the β Cephei star ν Eridani:
photometric observations and pulsational frequency
analysis
G. Handler1, R. R. Shobbrook2, M. Jerzykiewicz3, K. Krisciunas4,5,
5
0
T. Tshenye6, E. Rodr´ıguez7, V. Costa7, A.-Y. Zhou8, R. Medupe6,9,
0
2
W. M. Phorah6, R. Garrido7, P. J. Amado7, M. Papar´o10, D. Zsuffa10,

n
a L. Ramokgali6, R. Crowe11, N. Purves11, R. Avila11, R. Knight11,
J

3 E. Brassfield11, P. M. Kilmartin12, P. L. Cottrell12
1

1
1 Institut fu¨r Astronomie, Universit¨at Wien, Tu¨rkenschanzstrasse 17, A-1180 Wien, Austria
v
2 VisitingFellow, Australian National University,Canberra, ACT, Australia
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3 Wroclaw University Observatory,ul. Kopernika 11, 51-622 Wroclaw, Poland
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4 Cerro Tololo Inter-AmericanObservatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile
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5 Las Campanas Observatory, Casilla 601, La Serena, Chile
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6 Theoretical Astrophysics Programme, Universityof the North-West, Private Bag X2046, Mmabatho 2735, South Africa
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7 Institutode Astrofisica de Andalucia, C.S.I.C., Apdo. 3004, 18080 Granada, Spain
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8 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China
0
9 South African Astronomical Observatory, P.O. Box 9, Observatory 7935, South Africa
/
h 10 Konkoly Observatory, Box 67, H-1525 Budapest XII, Hungary
p 11 Department of Physics and Astronomy, University of Hawaii - Hilo, 200 West Kawili Street, Hilo, Hawaii, 96720-4091, USA
- 12 Department of Physics and Astronomy, University of Canterbury, Christchurch, New Zealand
o
r
t
s
Accepted 2003July17.Received2003August13;inoriginalform2003September 10
a
:
v
i ABSTRACT
X
r We undertook a multisite photometric campaign for the β Cephei star ν Eridani.
a
Morethan600hoursofdifferentialphotoelectricuvyV photometrywereobtainedwith
11 telescopes during 148 clear nights.
The frequency analysis of our measurements shows that the variability of ν Eri
can be decomposed into 23 sinusoidal components, eight of which correspond to in-
−1
dependent pulsation frequencies between 5–8 cd . Some of these are arranged in
multiplets, which suggests rotational m-mode splitting of nonradial pulsation modes
as the cause. If so, the rotation period of the star must be between 30 – 60 d.
One of the signals in the light curves of ν Eri has a very low frequency of 0.432
−1
cd . It can be a high-order combination frequency or, more likely, an independent
pulsation mode. In the latter case ν Eri would be both a β Cephei star and a slowly
pulsating B (SPB) star.
The photometric amplitudes of the individualpulsation modes of ν Eriappearto
have increased by about 20 per cent over the last 40 years. So do the amplitudes of
the dominant combination frequencies of the star. Among the latter, we only could
identifysumfrequencieswithcertainty,notdifferencefrequencies,whichsuggeststhat
neitherlight-curvedistortioninitssimplestformnorresonantmodecouplingaretheir
single cause.
Oneofourcomparisonstars,µEridani,turnedouttobevariablewithadominant
timescaleof1.62d.WebelievethatitiseitheranSPBstarjustleavingitsinstability
strip or that its variations are of rotational origin.
Key words: stars: variables: other – stars: early-type – stars: oscillations – stars:
individual: ν Eridani – stars: individual: µ Eridani – techniques: photometric
2 G. Handler et al.
1 INTRODUCTION profilevariationsorphotometriccolouramplituderatiosand
phase shifts. Unfortunately, such methods may not always
Lengthymultisiteobservationsofmultiperiodicallypul-
yield unambiguous results (e.g. see Balona 2000). However,
sating stars have become a standard tool in variable star
Handler et al. (2003) recently showed that mode identifi-
research.Thebenefitofsucheffortsarelong,uninterrupted
cation from photometric colour amplitudes works well for
time series of the variations of the target stars, which are
slowly rotating β Cephei stars and they estimated that a
necessary to resolve complicated pulsational spectra. The
relative accuracy of 3 per cent in theamplitude determina-
more individual variations are present in a given pulsator,
tionsaresufficienttoachieveanunambiguousdetermination
the more one can learn about its interior by modelling the
of ℓ.
observed mode spectra. This technique is called asteroseis-
Consequently, the β Cephei stars are indeed suitable
mology.
for asteroseismic studies. If successful, many interesting as-
The most extensive observational efforts for asteroseis-
trophysical results can be expected. For instance, angular
mology have been performed with dedicated telescope net-
momentumtransportinthesestarscanbestudied.Thefre-
works. For instance, the Whole Earth Telescope (Natheret
quenciesofsomepulsationmodesofβ Cepheistarsaresen-
al. 1990) has already observed 40 individual targets, more
sitivetotheamountofconvectivecoreovershooting(Dziem-
thanhalfofwhicharepulsatingwhitedwarfstars,theothers
bowski&Pamyatnykh1991).Deviationsfromtherotational
being pulsating sdB stars, rapidly oscillating Ap stars, cat-
frequencysplittingofnonradialmodemultipletscanbedue
aclysmic variables etc. The Delta Scuti Network (e.g. Zima
to interior magnetic field structure of those stars (Dziem-
et al. 2002) has studied 10 different objects during23 cam-
bowski & Jerzykiewicz 2003). Once the interior structures
paignsandacquiredatotalofmorethan1000hoursofmea-
ofseveralβ Cepheistarsinvariousphasesoftheirevolution
surement for some δ Scutipulsators.
aredetermined,main-sequencestellarevolutioncalculations
The very first coordinated multisite observations were can be calibrated and more accurately extrapolated to the
obtained as early as 1956, on the β Cephei star 12 (DD) supernova stage, which can in turn constrain spectral and
Lacertae (de Jager 1963). This effort even included both chemical evolution theories of galaxies.
spectroscopic and multicolour photometric measurements. Henceitisjustifiedtodevotelargeobservationalefforts
In more recent times, however, β Cephei stars were (aside to β Cephei stars that seem suitable for asteroseismology.
from a large campaign for BW Vul, Sterken et al. 1986) Theselection ofagood candidateisoneofthemost impor-
rarelythetargetsofextendedobservingcampaigns.Therea- tantprerequisitesforsuchastudy.Forthepresentwork,our
son may be the sparse frequency spectra of β Cephei stars choice was ν Eri (HD 29248, V =3.92). Itsmode spectrum
compared to pulsating white dwarfs or δ Scuti stars. For revealshighasteroseismicinterest:fourpulsationfrequencies
many years, the record holder was 12 (DD) Lac with five wereknown,asingletandanequallyspacedtriplet(Kubiak
knownindependentmodesofpulsation(Jerzykiewicz1978), 1980,Cuypers&Goossens1981). Thesinglet hasbeensug-
which was recently superseded by the six modes of V836 gestedtobearadialmode,andthetripletisconsistentwith
Cen (Aertset al. 2003). a dipole (Aerts, Waelkens & de Pauw 1994, Heynderickx,
However, the apparent paucity of frequencies in the Waelkens& Smeyers1994).
modespectra of β Cepheistars may bequestioned.Experi- If this triplet contained at least two rotationally split
ence with δ Scuti stars has shown that the more the detec- m-components of a mode, ν Eri would also be a slow rota-
tion level for periodic light variations is pushed down, the tor,ahypothesissupportedbyitsmeasuredvsini(themost
more pulsation modes are detected (e.g. see the sequence recent determination being 20 km/s, Abt,Levato& Grosso
of papers by Handler et al. 1996, 1997 and 2000). In fact, 2002).Thisisimportantbecausetheadverseeffectsofrota-
mostofthepulsationmodesofthesestarshavelightampli- tional mode coupling (see Pamyatnykh 2003 or Daszyn´ska-
tudes around or below 1 mmag, an amplitude quite easily Daszkiewiczetal.2002)inasubsequenttheoreticalanalysis
detected with the large data sets of 2 to 3 mmag precision would be diminished. Finally, ν Eri is a bright equatorial
differentialphotometryobtainedduringthesecampaigns.As star,soitcanbeobservedfrombothhemisphereswithpho-
the pulsational driving of the β Cephei stars (Moskalik & tometric and high-resolution spectroscopic instruments.
Dziembowski 1992) is based on essentially the same mech-
We therefore organised a multisite campaign for ν Eri,
anism (the κ mechanism) as that of the δ Scuti stars, just
applying both observing methods mentioned above (Han-
operatingonheavierchemicalelements,itcanbesuspected
dler & Aerts 2002). In the following, we will report the re-
thatmanylow-amplitudemodesarealsoexcitedinβCephei
sults from the photometric measurements. The analysis of
stars but have not yet been detected simply because of a
the spectroscopy, pulsational mode identification and seis-
lack of suitable data. Indeed, this idea is supported by re-
mic modelling of the identified oscillations will be the sub-
centhigh-qualityobservations(Stankovetal.2002,Cuypers
ject of future papers.
et al. 2002, Handleret al. 2003).
Besidesthedetectionofmanypulsationmodes,another
2 OBSERVATIONS AND REDUCTIONS
necessaryingredientforasteroseismologyisthecorrectiden-
tification of these modes with their pulsational quantum Ourphotometric observations were carried out with 11 dif-
numbers, k, the radial overtone of the mode, the spheri- ferent telescopes and photometers at 10 observatories on
cal degreeℓ and theazimuthal orderm.Forpulsating stars 5 different continents; they are summarised in Table 1. In
whose frequency spectra do not show any obvious regulari- most cases, single-channel differential photometry was ac-
tiescausedbyrotationally splitmodesorconsecutiveradial quiredthroughtheStr¨omgrenuvyfilters.However,atSierra
overtonestheuseofmodeidentificationmethodsisrequired. Nevada Observatory (OSN) a simultaneous uvby photome-
Thismayforinstancebespectroscopicdiagnosticsfromline ter was used, so we included the b filter as well, and at the
Asteroseismology of ν Eridani: photometry 3
fourobservatorieswherenoStr¨omgrenfilterswereavailable multiperiodic signals including harmonic, combination, and
we used Johnson V. Some measurements through the H equally spaced frequencies. As will be demonstrated later,
β
filters were also obtained at OSN. The total time base line ouranalysis requires some of these features.
spanned by ourmeasurements is 157.9 d. We started by computing the Fourier spectral window
We chose two comparison stars for ν Eri: µ Eri (HD ofthefinallightcurvesineachofthefilters.Itwascalculated
30211, B5IV, V = 4.00) and ξ Eri (HD 27861, A2V, V = as theFourier transform of a single noise-free sinusoid with
5.17).Anothercheckstar,HD29227(B7III,V =6.34)was a frequency of 5.7633 cd−1(the strongest pulsational signal
also monitored at OSN. We note that µ Eri was the single ofν Eri)andanamplitudeof36mmagsampledinthesame
comparisonstarinallpublishedextensivephotometricstud- way as were our measurements. The upper panel of Fig. 2
ies of ν Eri (van Hoof 1961, Kubiak& Seggewiss 1991) and containstheresultforthecombinedyandV data.Anyalias
that its HIPPARCOS photometric data (ESA 1997) imply structures that would potentially mislead us into incorrect
someslowvariability(Koen&Eyer2002).Thestarisalsoa frequencydeterminationsarequitelowin amplitudedueto
spectroscopicbinary(Porb =7.35890d,e=0.26,Hill1969). ourmultisite coverage.
Inthehopethatwecouldalsounderstandthevariabilityof
We proceeded by computing the amplitude spectra of
µEriwithourmultisiteobservations,andhopingtousethat
the data itself (second panel of Fig. 2). The signal desig-
knowledge in re-analyses of the published data of ν Eri, we
natedf dominates,butsomeadditionalstructuresnotcom-
1
retainedµEriasacomparison star.Duringdatareduction,
patiblewith thespectral window sidelobes arealso present.
wetookcarethatthevariationsofµEriwouldnotinfluence
Consequently, we prewhitened this signal by subtracting a
theresults on our primary target.
syntheticsinusoidal lightcurvewithafrequency,amplitude
Data reduction was therefore started by correcting for
and phase that yielded the smallest possible residual vari-
coincidence losses, sky background and extinction. Nightly
ance,andcomputedtheamplitudespectrumoftheresidual
extinction coefficients were determined with the Bouguer
light curve(third panel of Fig. 2).
method from the ξ Eri measurements only; second-order
Thisresultedinthedetectionofasecondsignal(f )and
colourextinctioncoefficientswerealsodetermined.Wethen 2
of another variation at the sum frequency of thetwo previ-
calculated differential magnitudes between the comparison
ously detected. We then prewhitened a three-frequency fit
stars (in the sense µ Eri - ξ Eri). Heliocentrically corrected
from the data using the same optimisation method as be-
versionsofthesetimeserieswereset asideforlateranalysis
foreandfixedthecombinationtermtotheexactsumofthe
of the variability of µ Eri, towhich we will return later.
two independent frequencies. We continued this procedure
Thenightly(µEri-ξ Eri)differentialmagnitudeswere
(furtherpanelsofFig.2)untilnosignificantpeakswereleft
fitted with low-order polynomials (n<4). The residuals of
in the residual amplitude spectrum.
the nondifferential µ Eri magnitudes with respect to that
We consider an independent peak statistically signifi-
fit were combined with the ξ Eri data and were assumed
cant if it exceeds an amplitude signal-to-noise ratio of 4 in
toreflecttheeffectsoftransparencyanddetectorsensitivity
the periodogram; combination signals must satisfy S/N >
changesonly.Consequently,thesecombinedtimeserieswere
3.5 to be regarded as significant (see Breger et al. 1999 for
binned into intervals that would allow good compensation
amorein-depthdiscussionofthiscriterion).Thenoiselevel
fortheabovementionednonintrinsicvariationsinthetarget
wascalculatedastheaverageamplitudeina5cd−1interval
startimeseriesandweresubtractedfromthemeasurements
centred on thefrequency of interest.
ofν Eri.Thebinningminimisestheintroductionofnoisein
thedifferential light curveof thetarget. Werepeatedtheprewhiteningprocedurewiththeuand
The timings for this differential light curve were helio- v data independently and obtained the same frequencies
centrically corrected as the next step. Finally, the photo- within the observational errors. We then determined final
metric zeropoints of the different instruments, that may be values for the detected frequencies by averaging the values
different because of thedifferent colours of ν Eri and ξ Eri, from theindividualfilters,weightedbytheirS/N.Thepul-
were examined at times of overlap with a different site and sational amplitudes were then recomputed with those fre-
adjustedifnecessary.MeasurementsintheStr¨omgrenyand quencies; theresult is listed in Table 2.
Johnson V filters were treated as equivalent and analysed The residuals from this solution were searched for ad-
together due to the same effective wavelength of these fil- ditional candidate signals that may be intrinsic. We have
ters.Theresultingfinalcombinedtimeserieswassubjected firstinvestigated theresiduals intheindividualfilters, then
to frequency analysis; we show some of our light curves of analysedtheaveragedresidualsinthethreefilters(whereby
ν EriinFig.1.Intheend,wehadmorethan3000measure- theu data were divided by1.5 to scale them to amplitudes
ments in each filter with accuracies of 3.7 (u filter), 3.0 (v and rms scatter similar to that in the other two filters),
filter) and 3.0 mmag (y/V filters) per datapoint available. and finally applied statistical weights according to the rec-
ommendation by Handler (2003). Some interesting features
were found and are listed in Table 3.
3 FREQUENCY ANALYSIS Inthistable,thesignalat0.254cd−1isformallysignifi-
cantinthev filterdata,andnoticeablepeaksarepresentat
3.1 The program star
thesamefrequencyinboththeuandyfilterdata.However,
Ourfrequencyanalysesweremainlyperformedwiththepro- wefindtheevidencefromallthedatasetstakentogethernot
gram PERIOD 98 (Sperl 1998). This package applies single- sufficiently convincing to claim reality for this peak. Simi-
frequencypowerspectrumanalysisandsimultaneousmulti- lar comments apply to the other signals listed in Table 3.
frequency sine-wave fitting. It also includes advanced op- Wewouldhoweverliketopointoutthatthevariationsnear
tions, such as the calculation of optimal light-curve fits for 6–8 cd−1may all be components of multiplets that include
4 G. Handler et al.
Table 1. Log of the photometric measurements of ν Eri. Observatories are ordered according to geographical longitude. Sites that
acquiredV measurementsonlyaremarkedwithasterisks.
Observatory Longitude Latitude Telescope Amountofdata Observer(s)
Nights h
SierraNevadaObservatory −3◦23’ +37◦04’ 0.9m 18 53.59 ER,VC,RG,PJA
CerroTololoInteramericanObservatory −70◦49’ −30◦09’ 0.6m 8 43.19 KK
FairbornObservatory −110◦42’ +31◦23’ 0.75mAPT 24 114.54 −−
LowellObservatory −111◦40’ +35◦12’ 0.5m 10 46.01 MJ
MaunaKeaObservatory∗ −155◦28’ +19◦50’ 0.6m 4 7.78 RC,NP,RA,RK,EB
MountJohnUniversityObservatory +170◦28’ −43◦59’ 0.6m 1 3.83 PMK
SidingSpringObservatory +149◦04’ −31◦16’ 0.6m 31 117.70 RRS
Xing-LongObservatory∗ +117◦35’ +40◦24’ 0.85m 3 15.72 AYZ
SouthAfricanAstronomicalObservatory +20◦49’ −32◦22’ 0.5m 37 151.82 GH,TT,RM,WP,LR
SouthAfricanAstronomicalObservatory +20◦49’ −32◦22’ 0.75m 7 39.31 GH
Piszkesteto¨ Observatory∗ +19◦54’ +47◦55’ 0.5m 5 11.86 MP,DZ
Total 148 605.35
Table 2.Multifrequencysolutionforourtime-resolvedphotom- detected modes. The signals at frequencies higher than 17
etry of ν Eri. Formal error estimates (following Montgomery & cd−1all coincide with combinations of detected modes.
O’Donoghue1999)fortheindependentfrequenciesrangefrom±
0.00001 cd−1 for f1 to ± 0.00035 cd−1 for f8. Formal errors on
the amplitudes are ± 0.2 mmag in u and ± 0.1 mmag in v and 3.2 The comparison stars
y.TheS/Nratioquotedisforthey filterdata.
We still have to analyse the light curves of µ Eri−ξ Eri.
To this end, we computed the amplitude spectrum of these
ID Freq. uAmpl. vAmpl. yAmpl. S/N
(cd−1) (mmag) (mmag) (mmag) dataand showthemintheupperpanelof Fig.3.Onepeak
standsout;prewhiteningitleavesstrongevidenceforfurther
f1 5.76327 73.5 41.0 36.9 137.0
f3 5.62006 34.6 23.9 22.7 83.6 variability of this star (Fig. 3, second panel), but no more
f4 5.63716 32.2 22.4 21.0 77.7 periodicities can be detected.
f2 5.65393 37.9 26.4 25.1 92.6 The single periodicity that may be present in the light
ff67 66..2246420085 32..99 21..59 21..68 96..88 curvesofµErihasafrequencyof0.61638±0.0005cd−1and
f8 7.19994 1.3 0.9 1.1 4.3 uvyamplitudesof10.2,6.4and5.0mmag,respectively.Itis
f5 7.89780 4.3 3.1 3.0 11.7
statistically significant with S/N ratios between 6.3 and4.4
fA 0.43218 5.5 3.2 3.2 7.1
f2+f3 11.27399 2.8 1.7 1.4 6.4 inthedifferentfilters,butitisnotclearifitsfrequencyand
f1+f3 11.38333 11.1 7.9 7.5 34.9 amplitudewereconstantthroughouttheobservingwindow.
f1+f4 11.40043 10.9 7.7 7.1 33.2
f1+f2 11.41720 12.6 9.0 8.4 39.3 Theresidualamplitudespectrumafterprewhiteningthissig-
2f1 11.52654 4.5 3.1 2.9 13.8 nal still shows a very strong 1/f component and indicates
f1+f5 13.66107 1.6 1.2 1.1 6.4 thatthevariabilityofµEriiscomplicated.However,further
f1+f3+f4 17.02049 1.0 0.7 0.7 4.7
f1+f2+f3 17.03726 4.4 3.1 2.6 16.9 analysesofthedifferentialµEri−ξ Erilight curves,suchas
f1+f2+f4 17.05435 1.0 0.8 0.9 6.1 searchingfor variations with nonsinusoidal pulseshapes us-
f1+2f2 17.07113 0.8 0.7 0.5 3.5
ing the residualgram method (Martinez & Koen 1994) or
2f1+f3 17.14660 1.8 1.4 1.2 8.0
2f1+f4 17.16370 1.7 1.2 1.0 6.6 by folding the data with the orbital period, all remained
2f1+f2 17.18047 1.9 1.4 1.3 8.2 inconclusive.
2f1+f2+f3 22.80053 1.5 1.0 0.9 6.9
Ontheotherhand,wenoticedthattheamplitudespec-
trum of the de-trended comparison star magnitude differ-
ences (with a low-order polynomial fitted to each night of
Table 3. Possible further signals. The data set in which they datatoremovetheslowvariabilityofµEri)hasthehighest
attainedhighestS/N areindicated. peak at 10.873 cd−1in all the filters. It is even significant
with a S/N = 5.2 in the v filter data, where it reaches an
ID Frequency S/N amplitudeof 0.6 mmag.
(cd−1) To examine whether this peak is real and if so, from
0.2543 4.0(v) what star it originates, we computed the differential (ν Eri
6.221 3.6(uvy,weighted) −µEri)and(νEri−ξEri)lightcurves.Wethenprewhitened
7.252 3.4(uvy,weighted) the frequency solution from Table 2 from these data and
7.914 3.1(u) computedtheresidualamplitudespectra.Wealsoexamined
f1+2f3 17.0034 3.2(u) the (ξ Eri – HD 29227) data from OSN for this purpose.
f1+2f2+f3 22.6912 3.5(u) Unfortunately,thesetestswerenot fully conclusivebecause
3f1+f2+f3 28.5638 3.1(u) the noise level in these amplitude spectra is higher than
in the ones of the de-trended differential comparison star
magnitudes only. All we can say is that if real, the 10.873
Asteroseismology of ν Eridani: photometry 5
Figure1.Somelightcurvesofν Eri.PlussignsaredataintheStro¨mgrenufilter,filledcirclesareourvmeasurementsandopencircles
representStro¨mgreny andJohnsonV data.Thefulllineisafitcomposedofalltheperiodicitiesdetected inthelightcurves(Table2).
Theamountofdatashownhereisabouthalfthetotal.
cd−1variation is more likely to originate from ξ Eri. In any 3.3 Re-analysis of literature data
case, ourassumption that ξ Eriisphotometrically constant
does not affect ouranalysis of ν Eri.
We have reanalysed the photometric measurements by van
Hoof(1961) andKubiak&Seggewiss (1991). Thefirstdata
set was retrieved from the IAU archives (as deposited by
6 G. Handler et al.
Figure2.Amplitudespectraofν Eri.Theuppermostpanelshowsthespectralwindowofthedata,followedbytheperiodogramofthe
data.Successiveprewhiteningstepsareshowninthefollowingpanels;notetheirdifferentordinatescales.Seetextfordetails.
Cuypers & Goossens 1981) and consists of two seasons of minute bins and by removing poor data, sometimes whole
ultravioletU andoneseasonofyellowY measurements(the nights. The frequency analysis of the resulting U data set
Y bandpassisidenticaltoJohnsonV,seeLyng˚a1959).The revealed the presence of frequencies f to f as well as the
1 4
frequencyanalysiswasperformedontheU dataastheyare sumfrequenciesoff withthef ,f ,f triplet.Theampli-
1 2 3 4
considerably more numerous, whereas we determined only tude spectrum after prewhitening the corresponding multi-
the amplitudes in the Y data with frequencies fixed to the frequencyfitshowsastrongincreaseofnoisethatprecludes
values derived from the U measurements. the detection of further signals. We note that neither the
We homogenised these data by averaging them into 7-
Asteroseismology of ν Eridani: photometry 7
Table 4. Multifrequency solution for the U and Y data of van
Hoof (1961). The identifications of the signals are the same as
inTable 2. Formal errorestimates (Montgomery & O’Donoghue
1999) for the independent frequencies range from ± 0.00001
cd−1 for f1 to ± 0.00017 cd−1 for f6. Formal errors on the U
amplitudesare±0.3mmag.However,therealerrorsarebelieved
tobehigherbecauseofthezeropointadjustmentswemade.The
formaluncertainty ontheY amplitudesis±0.8mmag.
ID Freq. U Ampl. Y Ampl. S/N
(cd−1) (mmag) (mmag)
f1 5.76345 52.8 28.9 78.2
f3 5.62018 25.5 20.1 37.7
f4 5.63738 26.5 19.5 39.2
f2 5.65385 27.9 20.8 41.3
f6 6.24417 2.7 2.1 4.0
f7 6.26227 3.7 4.0 5.6
f5 7.89830 2.1 2.0 3.1
f1+f3 11.38363 8.1 6.2 13.0
f1+f4 11.40083 8.1 5.5 13.0
f1+f2 11.41730 9.2 7.0 14.7
2f1 11.52690 2.8 3.8 4.4
Figure 3. Upper panel: amplitude spectrum of (µ Eri-ξ Eri)
in the y filter. Second panel: residual amplitude spectrum after
prewhitening f1. Lower panel: residual amplitude spectrum of
(µEri-ξEri)outtotheNyquistfrequency;nofurthervariations
aredetected.
low 0.432 cd−1variation of ν Eri nor the suspected 0.616
cd−1periodicity of µ Eri could bedetected.
To enable a search for further signals known from our
analysis in those data, we determined the zeropoints of the
residuallightcurvesoftheindividualnightsandsubtracted Figure 4. Schematic amplitude spectrum of ν Eri. Solid lines
them from the data. Fourier analysis of this modified data represent detected modes, whereas the dashed lines indicate the
set allowed thedetectionoffourmoresignals established in positionsofpossiblefurthersignals.
ourmeasurements.Theirfrequenciesandamplitudesrecov-
eredinvanHoof’sU dataarelistedinTable4togetherwith
This figure shows intriguing structures. Besides the
thecorresponding Y amplitudes.
knowntripletoffrequenciesnear5.64cd−1,thereisanother
ThephotometricmeasurementsbyKubiak&Seggewiss
doubletnear6.24cd−1withasuspectedfurthercomponent,
(1991) consist of two runs of 17.1 and 5.1 days time base,
and the two other signals near 7.2 and 7.9 cd−1may also
respectively,separatedby4years.Thef ,f ,f tripletcan
2 3 4
bypartsofmodemultiplets.Webelievethatthesearesigns
therefore not be resolved in thesedata, butf can besepa-
1
of rotational splitting of nonradial pulsation modes. If so,
ratedfrom itanditsuvby amplitudescanbeestimated.We
obtain amplitudes of 55±3 mmag in u, 30±2 mmag in v, the rotation period of ν Eri must be between 30 and 60 d,
28±2 mmag in b, and 27±2 mmag in y for signal f . The dependingonthetypesofmodewesee(pand/orgmodes).
1
0.616 cd−1variation of µ Eri may bepresent in these data. The low-frequency triplet is asymmetric. Dziembowski
& Jerzykiewicz (2003) calculated the asymmetry as A =
obs
f + f − 2f = −7.1 ± 0.3 × 10−4cd−1(using the nam-
2 3 4
ing convention from Table 2) from archival data, whereas
4 DISCUSSION our measurements indicate A =−3.3±0.4×10−4cd−1.
obs
ComparingTables2and4,itappearsthatourvalueforthe
4.1 The β Cephei-type pulsation frequencies
triplet centroid frequency f is less accurate than the for-
4
We have detected eight independent signals in the light malerrorswould suggest -whichisplausiblegiventhatour
curves of ν Eri that are in the typical frequency domain observational time base is only about 2.6 times the inverse
for β Cephei star pulsation, i.e. they are pressure (p) and triplet splitting.
gravity(g)modesoflowradialorder.Weshowtheschematic A comparison of Tables 2 and 4 allows another inter-
amplitudespectrumcomposedofthesemodesandoffurther estingconclusion:thepulsationalamplitudesofallmodesof
suspected signals in Fig. 4. ν Eri seem to have increased between van Hoof’s and our
8 G. Handler et al.
measurements. Allowing for the different wavelength pass- Althoughthisslowvariabilitywasdetectedinthemea-
band of the archival U measurements that also included a surementsinallthreeStr¨omgrenfiltersused,wetookspecial
silvered mirror (Lyng˚a1959) and ourStr¨omgren udata,an careindeterminingwhetherthesevariationsareintrinsicto
increase in the pulsational amplitudes of about 20 per cent the star. Consequently we computed amplitude spectra of
canbeestimated.Itispossiblethatmostofthisincreasehas thefourlargesthomogeneoussubsetsofdata(i.e.thosethat
occurredinthelast15–20yearsastheuamplitudeoff in used the same filters and detectors throughout the whole
1
thedatabyKubiak&Seggewiss (1991) isalsoconsiderably campaignandthatspannedatimebaselongerthan70days)
smaller than in our data. inallthreefilters.Wefoundapeakat0.432cd−1presentin
all these amplitude spectra, and we are therefore sure it is
not an artifact of the observing or reduction procedures; it
4.2 The combination signals is dueto intrinsic variations of ν Eri.
Unfortunately, there is some doubt as to whether this
The light curves of ν Eri are not perfectly sinusoidal
is an independent frequency or not. A possible high-order
(cf. Fig.1); therefore combination signals result from our combination frequency (3f −3f ) would be located only
1 3
method of frequency determination. It is not well known 0.0026 cd−1away from f , which is much larger than the
A
whatthephysicalcauseofcombinationfrequenciesis.Some formalerrorestimate of±0.0001 cd−1forthefrequencyun-
ofthemostprominenthypothesesincludesimplelight-curve certainty of f , butsmaller than the0.0063 cd−1frequency
A
distortions due to the pulsations propagating in a nonlin-
resolutionofourdata.Itwouldbesurprisingifsuchahigh-
early responding medium or independent pulsation modes
order difference frequency were present in our data when
excited by resonances.
there is no evidence for lower-order ones. We thus suspect
Incaseofthelight-curvedistortionhypothesis,theam-
thatf isanindependentfrequency,butonlynewmeasure-
A
plitudes and phases of the combination frequencies can be
mentsincreasing thetime baseof our dataset will allow us
predicted. They also contain some information about the
to answer this question unambiguously.
medium that distorts the light curves (see Wu 2001 and
If f were an independent periodicity, what would be
A
references therein). In the simplest case, the amplitudes of
thephysicalreason for suchavariability?Asargued above,
the combination signals scale directly with the product of
the rotation period of ν Eri must be between 30 and 60 d;
the amplitudes of the parent modes (e.g. see Garrido &
theobserved2.3-dperiodcanthereforenotbeconnectedto
Rodr´ıguez 1996).
rotation. ν Eri is also not known to be a binary; extensive
As we do not have a pulsational mode identification
radial velocity studies are available and no variability ex-
available at this point, we cannot examine the two above-
cept that due to the short-period pulsations has ever been
mentioned hypotheses quantitatively, but some statements
reported.Inaddition,thewavelengthdependenceoftheam-
canstillbemade.Forinstance,simplelightcurvedistortion plitude of the 0.432 cd−1signal excludes a pure geometric
should producecombination sum and differencefrequencies
origin of this variability.
of about the same amplitude. However, difference frequen-
Hence, the slow variations are probably due to high-
cies are completely absent in our multifrequency solution
order g-mode pulsations of the star, which is also consis-
(Table 2) although the strongest of these signals should be
tentwith thecolouramplitudes. Wenotethat Jerzykiewicz
easily detectable in our data.
(1993) also detected a low-frequency variation in his light
The amplitude variations we reported before can also
curves of the β Cephei star 16 (EN) Lac, which he inter-
be examined. We note that the amplitude of the sum fre-
pretedtobeeitherduetoapairofspots-orduetog-mode
quenciesoff with thef ,f ,f triplet increased byabout
1 2 3 4 pulsation.
20percent,whichisthesameastheincreaseoftheindivid-
ualamplitudes of theparent modes, butless than expected
for the simple light curvedistortion hypothesis. 4.4 The variability of µ Eri
The absence of the difference frequencies also poses a
In Sect. 3.1 we reported that one of our comparison
problem for the resonant mode coupling hypothesis.As the
stars, µ Eri, shows rather complex light variations with
stellar eigenmode spectrum is much denser at low frequen-
a time scale of about 1.6 days. To examine their origin,
cies than at high frequencies due to the presence of many
we first determined its position in the HR diagram. As
gravity modes, one would expect to see many more differ-
a start, we retrieved the standard Str¨omgren and Geneva
ence frequencies in case of mode coupling, which is not the
colours of µ Eri from the Lausanne-Geneva data base
case.
(http://obswww.unige.ch/gcpd/gcpd.html).
Thestar’sGenevacoloursthenimplyT =15670±100
eff
KaccordingtothecalibrationsbyKu¨nzlietal.(1997).The
4.3 The low-frequency variation
star’s HIPPARCOS parallax (ESA 1997), combined with
Wefoundasignalof0.43218cd−1inthelightcurvesofν Eri a reddening correction of A = 0.06 determined from its
V
and evidence for other periodic low-frequency signals. Such Str¨omgren colours and Crawford’s (1978) calibration, bolo-
variability is an order of magnitude slower than β Cephei- metriccorrections byFlower (1996) andDrilling &Landolt
type pulsation. These variations are not due to the slowly (2000) resultinM =−3.6±0.4.WecanthusplaceµEri
bol
variablecomparisonstarµEri:theyoccuratfrequenciesdif- in the theoretical HRdiagram (Fig. 5).
ferentfromthedominantvariationofthelatterstarandwe Interestingly, µ Eri seems to be an object just at or
havetakencarethatitsvariabilitydoesnotaffect ouranal- shortly after the end of its main sequence life. It is located
ysisofν Eriinourreductionprocedures.Theyaretherefore within the instability strip of the slowly pulsating B (SPB)
present in our light curvesof ν Eri. stars, and the time scale (cf. De Cat & Aerts 2002, Pamy-
Asteroseismology of ν Eridani: photometry 9
such a pulsator. The frequency analysis of these measure-
ments revealed the presence of eight independent pulsation
modes,whichisthemosteverdetectedforsuchastar.These
are normal β Cephei-typevariations (p and g modes of low
radialorder),butoneadditionalsignalmaybeahigh-order
gravity-mode pulsation. ν Eri could therefore be not only
a β Cephei star, but also an SPB star. It would thus the
second example of a star exhibiting two different types of
pulsation.Thehigh-orderg-modesofthefirstsuchstar,HD
209295, arehoweverbelievedtobetriggered bytidaleffects
(Handleretal.2002).Aswehavenoevidenceforbinarityof
ν Eri,itmaythereforebethefirststarinwhichtwotypesof
pulsationwithtimescalesmorethananorderofmagnitude
different are intrinsically excited.
TheβCephei-typepulsationfrequenciesshowsomereg-
ular structures, i.e. some are contained in multiplets that
may be due to nonradial m-mode splitting. If so, the rota-
Figure 5. Theposition of µEriinthe theoretical HR diagram.
tion period of ν Eri is between 30 – 60 d,dependingon the
Somestellarmodelevolutionarytrackslabelledwiththeirmasses
type of modes in these multiplets. These multiplets are a
(full lines) for vsiniZAMS = 200km/s are included. The theo- strong constraint for pulsational mode identification, which
retical borders of the β Cephei and SPB star instability strips
will be the subject of a future paper. In any case, the low-
(Pamyatnykh 1999, thick dashed and dotted lines) are included
forcomparison.Theevolutionarytracksaresomewhatshiftedto order p and g mode spectrum of ν Eri as reported here is
theZAMS(fulldiagonalline)andtotheinstabilitystripbound- suitable for detailed seismic modelling.
ariesbecausethelatterdonotincluderotational effects.
atnykh2002)andcomplexityofitsvariabilityaswellasthe ACKNOWLEDGEMENTS
colourdependenceofitsamplitudesonwavelength arecon-
This work has been supported by the Austrian Fonds zur
sistent with it being an SPB star. µ Eri may have many
F¨orderung der wissenschaftlichen Forschung under grant
pulsation modes excited which are so closely spaced in fre-
R12-N02. MJ’s participation in the campaign was partly
quency that we cannot resolve them with the time base of
supported by KBN grant 5P03D01420. MJ would also like
our measurements.
to acknowledge a generous allotment of telescope time and
However, pulsation is not the only possible physical
thehospitality of Lowell Observatory.
cause of the light variations of µ Eri. For the 1.622 d time
We thank the following students of the University of
scale of the variability of the star to be due to rotational
Hawaii fortheirassistance duringsomeof theobservations:
effects, an equatorial rotational velocity of 193 km/s is re-
DanBolton,AlexandreBouquin,JoshBryant,ThelmaBur-
quired. The published estimates of the projected rotational
gos, Thomas Chun, Alexis Giannoulis, Marcus Lambert,
velocity vsini of µ Eri range from 150 km/s (Abt et al.
AmandaLeonard,AlexMacIver,DaniellePalmese, Jennale
2000) to 190 km/s (Bernacca & Perinotto 1970), which is
Peacock, David Plant, Ben Pollard, Sunny Stewart, Dylan
consistent with that constraint. A double-wave light varia-
Terry and Brian Thomas.
tionwithtwicethatperiodwouldhoweverbeinconflictwith
GHwishestoexpresshisthankstoLouBoydandPeter
the measured vsini. The complexity of the light variations
Reegen for their efforts in maintaining and controlling the
of µ Eri argues against a rotational origin.
FairbornAPTs,toMarcinKubiakforsupplyinghisarchival
The same argument can be used against an interpreta-
measurementsofν Eri,toJan Cuypersfor helpfulinforma-
tion of µ Eri’s variability in terms of binarity. In addition
tion on the passbands of the archival data, and to Conny
(as argued above for the long period detected for ν Eri),
AertsandAnamarija Stankovfor commentson a draft ver-
the wavelength dependence of the colour amplitudes in our
sion of this paper.
measurements suggests that a pure geometric origin of this
variability is unlikely.Most importantly,however, the7.3-d
spectroscopic binary period of µ Eri is quite different from ThispaperhasbeentypesetfromaTEX/LATEXfileprepared
bythe author.
the observed time scale of its light variability or from an
integral multipleof it.
Hence, we cannot unambiguously determine the cause
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