Deciphering the period spacing pattern in the oscillation spectrum of the SPB star KIC 7760680

0.09 MB·

by  Wojciech Szewczuk

Deciphering the period spacing pattern in the oscillation spectrum of the SPB
star KIC7760680
WojciechSzewczuk1,⋆,JadwigaDaszyn´ska-Daszkiewicz1,andPrzemysławWalczak1
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1AstronomicalInstituteoftheWrocławUniversity,Kopernika11,51–622Wrocław,Poland
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Abstract. We present the analysis of KIC7760680, the rotating Slowly Pulsating B–type star identified in

n the Kepler photometry. The oscillation spectrum of the star exhibits a series of 36 frequencies which are
a quasi-equally spaced in period. We confirm that this series can be associated with prograde dipole modes
J of consecutive radial orders. In our studies, the effects of rotation were included in the MESA equilibrium

5 modelsaswellasinthepuslational calculationsintheframeworkofthetraditionalapproximation. Wefind
thatpulsationalmodelscomputedwiththeOPLIBopacitiesbestreproducetheobservedfrequencyrange. The

] modifiedopacitydatawithanenhancement oftheopacityatlogT = 5.3, 5.46and5.06weretestedaswell.
R
IncreasingtheOPLIBopacitiesbyabout50%atlogT =5.3issufficienttoexcitemodesinthewholerangeof
S
36frequencypeaksofKIC7760680.
.
h
p
-
o 1 Introduction 2 Seismic modelling
r
t
s Knowledge of the geometry of pulsational modes is es-
a
[ sentialinseismicmodelling. InthecaseofKIC7760680,
KIC7760680 was classified as the Slowly Pulsating B- the quasi-equally spaced in period frequency peaks sug-
1
type star by Pápics et al. [1] who found a series of 36 gestthatthesearethemodeswiththesamedegree,ℓ,az-
v
frequencies quasi-equally spaced in period from the Ke- imuthal order, m, and consecutive radial orders, n. Ac-
6
5 plerphotometry. Fromspectroscopicobservationstheau- cordingtoPápicsetal[1], the frequenciesareassociated
2 thors derived the effective temperature, Teff = 11650 ± withprogradedipolemodes(ℓ=1, m=+1).
1 210K, surface gravity, logg = 3.97 ± 0.08, metallic- Inordertocheckwhetherthisstatementisvalidifthe
0 ity, [M/H] = 0.14 ± 0.09 (which correspond to about effectsofrotationareincluded,bothintheequilibriumand
.
1 Z = 0.017) and the minimum value of the rotational ve- pulsationalcalculations,weconstructedagridofrotating
70 locity, Vrotsini = 61.5 ± 5 kms−1. Based on the zero- stellarmodelswithinthe3σerrorboxinTeffandlogg.To
rotation asymptotic theory, Pápics et al. [1] attributed thisendweusedMESAcode[10–12]consideringawide
1
: these frequencies to the prograde dipole gravity modes rangeofthe rotationalvelocity, fromVrot = 62kms−1 to
v of high consecutive radial orders. Recently, Moravveji the upper limit of the validity of the traditional approxi-
i
X et al. [2] made a detailed seismic modelling of the star mation. Astheupperlimitweadopted200kms−1 which
r taking into account effects of rotation in the framework is about a half of the break-up velocity. The step was
a ofthetraditionalapproximation[e.g.,3–9]appliedtothe ∆V = 1 kms−1. We included various rotational mix-
rot
non-rotating MESA [10–12] evolutionary models. The ingmechanisms,i.e.,dynamicalshearinstability,Solberg-
authors found that KIC7760680is a moderatelyrotating Høiland instability, secular shear instability, Eddington-
starwiththerotationalfrequencyamountingto26%ofits Sweet circulation and the Goldreich-Schubert-Fricke in-
Roche break up frequency(0.4805 d−1) and with a mass stability[see 11,13,14].Weassumedtheinitialhydrogen
∼ 3.25M⊙. Furthermore, they constrained the exponen- abundance,X0 = 0.7,metallicityZ = 0.017,thechemical
tiallydecayingconvectivecoreovershootingparameterto mixture of Asplund et al. [15], and the overshootingpa-
fov ≈0.024±0.001.Additionally,theirbestseismicmod- rameterinexponentialdescription, f = 0.02. Threecom-
elsemployedextradiffusivemixingintheradiativeenve- monly used opacity data were adopted: OP [16], OPAL
lope. Here, we repeat the Moravveji et al. [2] approach [17]andOPLIB[18,19].
butourcomputationsincludetheeffectsofrotationalsoin Then,wecalculatednonadiabaticpulsationmodelsin
equilibrium models. In Sect.2 we present results of our theframeworkoftraditionalapproximation(herethestep
seismicmodellingandconclusionsendsthepaper. in V was decreased to 0.1 kms−1) for modes which
rot
seemedtobegoodcandidatestoreproduceobservedseries
of frequencies, i.e., (1,+0), (1,+1), (2,+0) and (2,+1).
⋆e-mail:[email protected] Since for modes other than (1,+1) the best models give
0.50
4
ℓ=1 m=+1
OP AGSS09
X =0.7, Z=0.017
3.5 0
f =0.02
ov
3
0.00
]
m 2.5
p
p

g
o
l 2 η
[

A

g
o
l 1.5
-0.50
1
observations
χ2=5348, V =68.1 km s-1, M=3.34M , log T =4.0873
0.5 rot ⊙ eff
χ2=5357, V =68.3 km s-1, M=3.34M , log T =4.0868
rot ⊙ eff
χ2=5564, V =78.6 km s-1, M=3.46M , log T =4.0811
rot ⊙ eff
0 -1.00
0.7 0.8 0.9 1 1.1 1.2
ν [d-1]
Figure 1. The best seismic models calculated with OP opacities. The observed frequencies and their amplitudes are plotted with
verticallines(leftY-axis). Colourcirclesmarkedtheoreticalfrequenciesandtheirinstabilityparameter,η,canbeereedontheright
Y-axis.
worsequalityofthefit, χ2 (definedasinMoravvejietal. plotted. As one can see only 3–4 modes are marginally
[2]), by the order of magnitude, we confirm that the fre- stable, η ≈ 0. Parametersof these modelsas wellas ob-
quenciesintheseriescanbeassociatedwiththeprograde tainedwiththeOPALopacitiesaresummarisedinTable1.
dipolemodes.
InFig.1, the theoreticalfrequenciesof(1,+1)modes
for the best three OP models are compared with the ob-
served ones. The parameters of these models are sum-
marisedinTable1. Withoutadoubt,thequalityoffitting Then,werecalculatedthebestOPLIBmodelwithdif-
needsimprovement. Forthebestmodels,weobtainedthe ferentvaluesofthemetallicityandovershootingparameter
valuesofχ2oftheorderof5500whichisthevaluealmost as well as with the modifiedOPLIB opacityprofile. The
three times higherthan the one obtainedby Moravvejiet modificationsoftheOPLIBdataconsistedofanenhance-
al. [2]. Thisisbecauseinourcalculationswefixedmetal- mentoftheopacityatthedepthscorrespondingtotemper-
licity and overshooting parameter compared to the cited atureslogT =5.3,5.46andlogT =5.06.Theincreaseof
authors. Moreover, Moravveji et al. fitted coefficient of theopacityatlogT = 5.06mimicsthenewopacitybump
the extra diffusive mixing in the radiatively stable enve- identifiedinKuruczmodelsbyCugier[20]. Themodified
lopewhereasweemployedrotationalinducedmixingwith models are shown in Fig.3. The increase of the opacity
fixedcoefficients.InFig.1ontherightY-axisthereisalso at logT = 5.3 by about50% gives theoreticalinstability
shown the instability parameter η (η ≤ 0 – stable mode, inthewholerangeoftheobservedfrequencies.Moreover,
η > 0 – unstable mode). As one can see modeswith the foreachobservedfrequencywefoundacorrespondingun-
lowestfrequenciesin ourbestmodelsare predictedtobe stable mode. Increasingopacityat logT = 5.46helps in
stable. Thisisobviouslycontrarytotheobservationsand excitinglowfrequencymodeswhereasinstabilityofhigh
needstobeimproved. frequenciesis unchanged. On the other hand, addingthe
Thequalityofthefitisquantitativelythesameforeach Kuruczbump results in reducingthe pulsationalinstabil-
considered opacity source but for models with OPLIB ity, both, at low and high frequencies. Larger value of
opacities the theoretical instability domain is in an ex- themetallicitygivestheinstabilitydomainshiftedtowards
cellent agreementwith the observedfrequencies. This is higher frequencies whereas more effective overshooting
shown in Fig.2, in which three best OPLIB models are fromtheconvectivecoreactsintheoppositedirection.
4
ℓ=1 m=+1
OPLIB AGSS09
X =0.7, Z=0.017 0.20
3.5 0
f =0.02
ov
3
0.00
]
m 2.5
p
p

g
o
l 2 η
[

A
-0.20
g
o
l 1.5
1
observations -0.40
χ2=5558, V =84.3 km s-1, M=3.52M , log T =4.0772
0.5 rot ⊙ eff
χ2=5622, V =84.6 km s-1, M=3.53M , log T =4.0775
rot ⊙ eff
χ2=5778, V =84.1 km s-1, M=3.52M , log T =4.0778
rot ⊙ eff
0
0.7 0.8 0.9 1 1.1 1.2
ν [d-1]
Figure2.ThesameasinFig.1butOPLIBmodelsareshown.
Table1.Theparametersofthebestseismicmodels.Column1
tive temperature for the same mass. Rotational induced
–opacitydata(#1–OP,#2–OPAL,#3–OPLIB);Column2–
mixing can contribute to the differencesas well. We got
mass(inthesolarmassunits)Column3–logarithmofthe
a little worse quality of the fit, χ2 ∼ 5500, compared to
effectivetemperature;Column4–logarithmoftheluminosity
the cited authors, χ2 ∼ 2000 but in our calculations we
(inthesolarluminosityunits);Column5–centralhydrogen
abundance;Column6–rotationalvelocity(inkms−1);Column have fixed the metallicity and the parameter of the core
7–measureofthefittingquality. overshooting,whichMoravvejietal. [2] setasfree. Itis
prominentthatinstabilitydomainofthe(1,+1)modesob-
op M logT log L X V χ2 tained with the OPLIB data (see Fig.2) nearly cover the
eff L⊙ c rot range of the observed 36 frequency peaks. A slight in-
#1 3.340 4.0873 2.222 0.42 68.1 5348
crease of the opacity at logT = 5.3 gives an excellent
#1 3.340 4.0868 2.224 0.42 68.3 5357
agreement. On the other hand, adding the Kurucz bump
#1 3.460 4.0811 2.327 0.35 78.6 5564
atlogT = 5.06spoilstheagreement. Finally,itshouldbe
#2 3.510 4.0807 2.375 0.31 83.3 5786
addedthatseriesoffrequenciesequidistantinperiodmay
#2 3.460 4.0834 2.336 0.34 78.8 5851
be accidental as it can be reproduced by modes of vari-
#2 3.510 4.0801 2.376 0.31 83.5 5998
ous angular numbers(ℓ, m; see e.g. [21]). Therefore, it
#3 3.520 4.0772 2.371 0.30 84.3 5558
is desirable to confirm the results of mode identification
#3 3.525 4.0775 2.373 0.30 84.6 5622
usinganothermethod,e.g., usingmulticolourtime-series
#3 3.520 4.0778 2.370 0.31 84.1 5778
photometry.
3 Conclusions Acknowledgements
This work was financially supported by the Polish National Science
Our calculations confirmed Pápics et al. [1] finding that Centre grant 2015/17/B/ST9/02082. Calculations have been carried
36 quasi-equidistant in period frequencies observed in out using resources provided by Wrocław Centre for Networking and
Supercomputing(http://wcss.pl),grantno.265.
KIC7760680are most probablyassociated with the con-
secutiveprogradedipolemodes. However,ourbestmod-
els have slightly higher masses (from 3.340M⊙ to 3.525 References
M⊙ dependingon theopacitydata) comparedto 3.25M⊙
obtained by Moravveji et al. [2]. The reason is that we [1] P.I. Pápics, A. Tkachenko, C. Aerts, T. Van Reeth,
usedrotatingevolutionarymodelswhichgivelowereffec- K.DeSmedt,M.Hillen,R.Østensen,E.Moravveji,
4 V =84.3 km s-1, M=3.52M , log T =4.0772
rot ⊙ eff
X =0.7, Z=0.017, f =0.02
0 ov
ℓ=1 m=+1 0.20
3.5
3
0.00
]
m 2.5
p
p

g
o
l 2 η
[

A
-0.20
g
o
l 1.5
observations
OPLIB AGSS09
1
OPLIB AGSS09 T53 +50%
OPLIB AGSS09 T546 +50% -0.40
OPLIB AGSS09 T506 +50%
0.5
OPLIB AGSS09 f=0.03
OPLIB AGSS09 Z=0.02
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0.7 0.8 0.9 1 1.1 1.2
ν [d-1]
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(bluecircles)andatlogT =5.06(blackcircles)aswellincreasedovershootingparameter(magentatriangles)andincreasedmetallicity
(cyantriangles).OriginalOPLIBmodelismarkedbyredcircles.ThecoordinatesmeaningisthesameasinFig1.
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