Using broadband photometry to examine the nature of Long Secondary Periods in red giants

1.7 MB·English

by  M. Takayama

Mon.Not.R.Astron.Soc.000,1–21(2002) Printed6January2015 (MNLATEXstylefilev2.2)
Using broadband photometry to examine the nature of
Long Secondary Periods in red giants
5 M. Takayama1⋆, P. R. Wood2 and Y. Ita1
1
0 1Astronomical Institute, Graduate School of Science, Tohoku University,Sendai, Miyagi 980-8578, Japan
2 2Australian National University,Research School of Astronomy and Astrophysics, Cotter Road, Weston Creek ACT, 2611 Australia

n
a
J Accepted 1988December 15.Received1988December14;inoriginalform1988October11

5

ABSTRACT
]
R Long-termJHK light curves have recently become available for largenumbers of the
S more luminous stars in the SMC. We have used these JHK light curves, along with
. OGLE V and I light curves, to examine the variability of a sample of luminous red
h
giantsintheSMCwhichshowprominentlongsecondaryperiods(LSPs).Theoriginof
p
the LSPs is currently unknown. In oxygen-richstars,we found that while mostbroad
-
o bandcolours(e.g.V−I)getredderwhenanoxygen-richstardimsduringitsLSPcycle,
r the J-K colour barely changes and sometimes becomes bluer. We interpret the J-K
t
s colour changes as being due to increasing water vapour absorption during declining
a lightcausedbythe developmentalayerofdensecoolgasabovethephotosphere.This
[
result and previous observations which indicate the development of a chromosphere

1 between minimum to maximum light suggestthat the LSP phenomenon is associated
v with the ejection of matter from the stellar photosphere near the beginning of light
8 decline. We explore the possibility that broadband light variations from the optical
4 to the near-IR regions can be explained by either dust absorption by ejected matter
7 or large spots on a rotating stellar surface. However, neither model is capable of
0
explainingtheobservedlightvariationsinavarietyofcolour-magnitudediagrams.We
0
conclude that some other mechanism is responsible for the light variations associated
.
1 with LSPs in red giants.
0
5 Key words: circumstellar matter – infrared: stars.
1
:
v
i
X
1 INTRODUCTION longest periods of all thesequences. SequenceD stars show
r photometricvariationswithperiodsof∼400–1500 daysand
a All luminous red giants appear to be variables and they
theyalsoshowanothervariationswithashorterperiodthat
aregenerally knownasLongPeriod Variables(LPVs).They
usually correspond to the period of sequence B. In these
vary with periods which fall on at least eight period-
stars, the shorter period is called the primary period and
luminosity sequences which have been labeled A′, A, B,
the longer period corresponding to sequence D is known
C′,C,D,Eand F(Wood et al.1999;Soszyn´ski et al. 2004;
as a Long Secondary Period (LSP). LSP stars have been
Ita et al.2004;Taburet al.2010;Soszyn´ski & Wood2013).
found within the LMC, SMC and Galactic populations of
SequencesA′-CandFarethoughttoconsistofstarspulsat-
LPVs.Infact,approximately25-50%ofluminousredgiants
ing in the radial fundamental and overtone modes and the
havebeenfoundtoshowLSPsinMACHOandOGLEdata
non-radialdipoleandquadrupolemodes(Wood et al.1999;
(Wood et al.1999;Soszyn´ski et al.2007;Fraser et al.2008).
Ita et al. 2004; Takayamaet al. 2013; Soszyn´ski & Wood
However,theoriginoftheLSPphenomenonisstillunknown
2013; Stello et al. 2014). These stars have been classified
and many explanations havebeen suggested. Some of these
as Miras, Semi-Regulars(SRs) and OGLE Small Amplitude
are now highlighted.
Red Giants (OSARGs) (Wood et al. 1999; Soszyn´ski et al.
2004;Soszyn´ski et al.2007).Ontheotherhand,sequenceE The most straightforward explanation is pulsation.
isknowntoconsistofclosebinarysystemshowingellipsoidal However, all pulsation explanations have problems. Since,
and eclipsing variability (Wood et al. 1999; Derekas et al. at the given luminosity, the typical LSP is approximately
2006;Soszyn´ski 2007). 4 times longer than the period of a Mira (on sequence C),
The last sequence, called sequence D, exhibits the andsincetheMiras areknowntobepulsatingin theradial
fundamentalmode,anexplanationintermsofnormal-mode
radialpulsationisnotpossible.Non-radialg-modepulsation
⋆ E-mail:[email protected] was proposed as the explanation of the LSP phenomenon
2 M. Takayama, P. R. Wood and Y. Ita
by Wood et al. (1999) since g-mode periods can be longer model is somewhat similar to the rotating spot model in
thantheradialfundamentalmodeperiod.However,because that it involves variable amounts of the visible stellar sur-
red giants have a large convective envelope and only a thin face having a temperature cooler than normal. The photo-
outerradiative layerwhere g-modes could develop,theam- metric variations seen in both the rotating spot model and
plitudeofg-modesshouldbetoosmalltocorrespondtothe thegiant convectivecell model should besimilar.
observed light and radial velocity amplitudes of LSP stars Another potential origin for the LSP is a variable
(Wood et al. 2004). amount of circumstellar dust. Wood et al. (1999) proposed
Someresearchershavealsoarguedforaclosebinaryhy- that the periodic dust formation found in the circumstellar
pothesis(likesequenceE) toexplain theLSPphenomenon. regionsoftheoreticalmodelsofAGBstarsbyWinters et al.
Soszyn´ski (2007) found at least 5% of LSP stars showed (1994) and H¨ofner et al. (1995) could give rise to the light
ellipsoidal or eclipse-like variability. He also noted, as did variation of the LSPs. Evidence for dust around red gi-
Derekaset al.(2006),thatifthesequenceEellipsoidalvari- ants with LSPs was found by Wood & Nicholls (2009) who
able were plotted using the orbital period rather than the showed that LSP stars tend to have a mid-infrared ex-
photometricperiod (half theorbital period) thensequences cess. On near- to mid-infrared color-magnitude diagrams,
DandEoverlappedontheperiod-luminositydiagram.How- the LSP stars lie in a region similar to that occupied by R
ever, the large amplitude of many LSP light curves (up to CoronaeBorealisstarswhichareknowntobesurroundedby
1 magnitude) would require a close pair of eclipsing red gi- patchy circumstellar dust clouds. This result suggests that
ants, very unlikely in itself, and this would lead to alter- the variation of LSP stars might be due to the ejection of
nating deep and shallow minima which are not seen. On dust shells that may be patchy. Similarly, in studying the
the other hand, Wood et al. (1999) found that many light effect of dust ejection on MACHO colours and magnitudes,
curves due to the LSPs show faster decline and slower rise Wood et al. (2004) noted that patchy dust clouds seemed
and they proposed that a binary with a companion sur- to be required in order to reproduce the observed colour-
roundedbyacomet-shapedcloudofgasanddustcouldgive magnitudevariations.
rise tothoselight curvefeatures. Various observational and In this paper, we utilize new long-term near-infrared
numerical results have conflicted with the binary hypothe- JHKs light curves of stars in the SMC obtained with the
sis. Hinkleet al. (2002) observed radial velocity variations SIRIUS camera on the IRSF 1.4 m telescope (Ita et al.
in six local red giants with LSPs and found that the ve- 2015). We combine the JHKs observations with V and
locity curves all had a similar shape. For eccentric binaries I monitoring of SMC stars by the Optical Gravitational
witharandomlongitudeofperiastron,thelightcurveshape LensingExperiment(OGLE) (Soszyn´ski et al.2011).Using
should vary from star to star. The lack of such variability this combined data, we examine light and colour variations
led Hinkleet al. (2002) to reject thebinary explanation for of LSP stars over a wide wavelength range. We then test
LSPs.Wood et al.(2004)andNicholls et al.(2009)obtained models involving dust ejection or variable spots to see if
radial velocity curvesfor stars in theGalaxy and LMC and theycan reproducetheobservedlight andcolourvariations
theyarguedthatthesmalltypicalfullvelocityamplitudeof over the wide wavelength range. For our computations, the
∼3.5 km s−1 meant that all binary companions would need dusty code (Ivezi´cet al. 1999) is used to model the effects
amassnear0.09M⊙.Sincecompanionsofthismassarerare ofejecteddustandthe2012versionoftheWilson-Devinney
around main-sequence stars, it seems unlikely that the 25– code(Wilson & Devinney1971)isusedtomodeltheeffects
50%ofredgiantswhichshowLSPscouldhaveacompanion on the light and colour of a spotted star.
of this mass.
Starspotsduetomagneticactivity on AGBstars have
been proposed by some researchers (e.g. Soker & Clayton
1999) and a rotating spotted star has been suggested as
an explanation of the LSPs. Possible evidence for mag- 2 THE OBSERVATIONAL DATA
netic activity in stars with LSPs is provided by the ex-
2.1 The near-IR data
istence of Hα absorption lines whose strength varies with
the LSP (Wood et al. 2004). However, from observations, A long-term multi-band near-infrared photometric survey
Olivier & Wood (2003) found that the rotation velocity of forvariablestarsintheLargeandSmallMagellanic Clouds
LSPstars istypically lessthan3km/sand theynoted that hasbeencarriedoutattheSouthAfricanAstronomicalOb-
this rotation velocity is too small to explain the periods of servatoryat Sutherland(Ita et al. 2015).TheSIRIUScam-
less than 1000 days which are often observed for the LSP. era attached to the IRSF 1.4 m telescope was used for this
For example, a 3 km/s rotation velocity gives rise to the surveyandmorethan10yearsofobservationsinthenear-IR
period of 2868 days for an AGB star with a ∼170 R⊙. A bands J(1.25µm), H(1.63µm) and Ks(2.14µm) band were
way to overcome this objection would be to have multiple obtained. In this work, we select the SMC stars from the
star spots. In another study,Wood et al. (2004) found that SIRIUSdatabase.Variableandnon-variablestarsinanarea
with a simple blackbody model, it was difficult for a rotat- of1squaredegreeinthecentralpartoftheSMChavebeen
ing spotted star to simultaneously reproduce both the am- observedabout99-126 timesintheperiod 2001–2012 anda
plitudeandcolourvariationobservedintheMACHOobser- total of 340147, 301841 and 215463 sources were identified
vations of stars with LSPs. The use of model atmospheres inJ,H andKs,respectively.Wenotethatthephotometric
rather than blackbodies, and allowing for limb darkening, detection range of the SIRIUS camera is about 8∼18 mag-
could alter this conclusion. nitudes in Ks band. A total of 12008 variable sources of all
The giant convective cell model is a relatively new ex- kindsweredetectedofwhich4533weredetectedinallthree
planation of the LSP phenomenon (Stothers 2010). This wavebands(data release ver130501. Itaet al. (2015)).
Testing models for Long Secondary Periods 3
2.2 The optical data
0.18
We obtained V and I band time series of SMC red giants 0.16
from the OGLE project (Soszyn´ski et al. 2011). Typically,
0.14
about1000observingpointswereobtainedintheI bandby
OGLE-II+IIIwhile a much smaller number of about 50–70 0.12
e
points were obtained in the V band. d
u 0.1
Soszyn´ski et al. (2011) dividedtheLPVsinto LSPand plit
non-LSP stars according to their positions in the period- m 0.08
a
luminosityrelationsforvariableredgiants.Wefoundnearly K
0.06
700LSPstarsintheareaoftheIRSFinfraredsurvey.Inor-
dertodetermineifanLSPstarisoxygen-richorcarbon-rich, 0.04
weinitiallyadoptedtheclassification methodintroducedby
0.02
Soszyn´ski et al.(2011).Thestarsweexaminedindetailhad
0
their oxygen-rich or carbon-rich status checked against op-
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
tical spectral classifications in theliterature.
I amplitude
Figure1.TherelationbetweentheIandKsbandfullamplitude
2.3 The combined near-IR and optical data of 298 oxygen-rich stars and 88 carbon stars. The stars selected
fordetailedstudyhaveI amplitudegreaterthan0.2magnitudes
By combining the SIRIUS data with the OGLE data, we andKs amplitudegreater then0.04magnitudes.
obtained298oxygen-richLSPstarsand88carbon-richLSP
starsintheregionmonitoredbytheSIRIUScamera.Forall
sample stars, a small number of apparent bad data points
were removed from each time series. Then, by using a first star is detected.
order Fourier fit to the I band time series, we obtained the Column7:Nameofthesurveysubregioninwhichthevari-
period and amplitude corresponding to the largest ampli- able star is detected.
tudemodeofthelight curves.Inall cases, theobtained pe- for more detail, see Ita et al. (2015).
riods and amplitudes were almost identical to those in the
OGLE-III catalog (Soszyn´ski et al. 2011). Using the period
from the fit to the I light curve, a fit was also made to the
Ks light curvein order to derive its amplitude. 3 SEPARATING THE LSP VARIATION FROM
Since our aim is to look for and to model colour and
THE PRIMARY PERIOD VARIATION
magnitudechanges,weneedtheobservedchangestobesig-
nificantly larger than the noise in the data. The limiting Light curves of all LSP stars show variations with the pe-
factorinthisstudyisnoiseinthenear-IRdata.Weconcen- riod corresponding to sequence D and also variations with
trate on the brightest stars with Ks <13 and for them the the primary period, which mainly corresponds to sequence
useful lower limit for clearly apparent Ks band variation B. Therefore, in order to study the LSP variation only, we
is about 0.04 magnitudes. Similarly, we set a useful lower need to separate each light curve into the components due
limitforI bandvariationofabout0.2magnitudes.Figure1 to variation with the LSP and variation with the primary
shows thefull amplitudeof theobserved LSPstars intheI period. The separation method we used is now described.
and Ks bands. This figure shows the selected stars, which Atfirst,weadoptedareferencetimet0 andthendelim-
having full amplitudes larger than 0.04 magnitudes in Ks ited the light curve into time intervals equal to the size of
and 0.2 magnitudes in I. This selection gave us a sample theLSP(P),witht0asthestartofaninterval.Thenwefit-
of 7 oxygen-richstars and 14 carbon stars for analysis. The tedtheobserved datapointsin each intervalindependently
spectraltypesofoursamplestarsobtained fromtheOGLE usingasecondorderFourierseriesF0 withtheperiodequal
photometryagreewiththeclassificationsintheSIMBADas- to that of the LSP,
tronomicaldatabasewherethespectraltypeswereobtained
fsrtoamrs saptemctarxoismcoupmicldigehtteramreinlaisttioednsi.nTThaebplero1p.erties of these F0(t)=A0+X2 a0sin(cid:16)2πit−Pt0(cid:17)+b0cos(cid:16)2πit−Pt0(cid:17).
Near infrared JHKs photometric data as a function of i=1
Julian date (JD) for a sample of 7 oxygen-richstars and 14 The fit was performed on independent intervals of length
carbon stars are published along with this paper. The file P rather than thefull light curveinterval because the light
nameof thetime-series data indicatesstar-name and wave- curvesofLSPstarsvarysubstantiallyfromcycletocycleof
band, like ”OGLE-SMC-LPV-07852.J.dat”. Table 2 shows theLSP.Furthermore,inordertominimizethedependence
time-series for a sample of the LSP star, which consists of ofthefinalfitontheadoptedvalueoft0,thefitwasmadeus-
seven columns as follows. ing4valuesfort0,τ0,τ1,τ2andτ3,witheachofthesevalues
Column 1 : Julius day. increasing by 14P, giving fits F0(t), F1(t), F2(t) and F3(t),
Column 2 : Calibrated (2MASS referenced) magnitude. respectively. In this study, τ0 corresponds to JD2450000. A
Column3:Errorindeferentialimageanalysisinmagnitude. requirementforafittobemadewasthattherebeatleast6
Column 4 : Error in reference magnitude. pointsin each of thefirst and second halvesof thefit inter-
Column 5 : Error in conversion offset in magnitude. val. If such points were not available, no fit was made. The
Column6:Nameofthesurveyregioninwhichthevariable finaladopted fit F(t) at theobserved times is given by
4 M. Takayama, P. R. Wood and Y. Ita
Table 1. Basicparametersoftheprogramstarsatmaximumlight
Name RA Dec V I J H Ks L TBB Teff PLSP Sp.
(mag) (mag) (mag) (mag) (mag) (L⊙) (K) (K) (days) type
OGLE-SMC-LPV-07852 12.402540 -72.97075 16.60 14.34 12.91 12.05 11.73 4617 2803 3434 645 O
OGLE-SMC-LPV-09856 13.186620 -72.95178 16.49 14.37 12.96 12.10 11.86 4408 2928 3558 615 O
OGLE-SMC-LPV-10774 13.530120 -72.47989 – 14.98 13.73 12.81 12.57 2170 2844 3474 396 O
OGLE-SMC-LPV-12332 14.166840 -73.07389 – 14.26 12.95 12.08 11.82 4455 2887 3517 632 O
OGLE-SMC-LPV-14045 14.987085 -72.94936 17.50 15.13 13.72 12.79 12.57 2197 2865 3495 390 O
OGLE-SMC-LPV-14715 15.332205 -72.86369 – 14.27 12.92 12.03 11.75 4544 2815 3445 601 O
OGLE-SMC-LPV-14781 15.361920 -72.85678 16.48 14.62 13.30 12.44 12.23 3245 2991 3620 469 O
OGLE-SMC-LPV-08199 12.555540 -73.18697 17.75 14.90 13.29 12.27 11.64 3745 2203 2792 767 C
OGLE-SMC-LPV-08280 12.583995 -72.80533 16.36 13.78 12.33 11.36 10.93 7922 2490 3106 1008 C
OGLE-SMC-LPV-09341 12.990495 -73.20256 – 13.92 12.48 11.63 11.26 6419 2734 3362 1081 C
OGLE-SMC-LPV-10109 13.281915 -73.12992 16.08 13.84 12.52 11.60 11.20 6411 2592 3215 964 C
OGLE-SMC-LPV-12653 14.317380 -72.80128 – 14.83 13.10 12.03 11.44 4465 2200 2789 1356 C
OGLE-SMC-LPV-13340 14.654580 -72.44625 16.28 14.12 12.59 11.66 11.28 6003 2606 3230 1014 C
OGLE-SMC-LPV-13557 14.755710 -72.46908 16.31 13.91 12.45 11.54 11.13 6876 2592 3215 1032 C
OGLE-SMC-LPV-13748 14.847210 -73.04900 – 14.69 13.09 12.11 11.67 3977 2468 3082 925 C
OGLE-SMC-LPV-13802 14.871165 -72.65750 17.08 14.19 12.69 11.71 11.16 6085 2334 2936 1011 C
OGLE-SMC-LPV-13945 14.946045 -72.71094 – 14.06 12.63 11.73 11.29 5872 2560 3180 927 C
OGLE-SMC-LPV-14084 15.004245 -72.89839 16.30 14.27 12.74 11.83 11.53 5042 2759 3388 799 C
OGLE-SMC-LPV-14159 15.048705 -72.66822 – 14.42 12.96 12.04 11.57 4406 2495 3112 913 C
OGLE-SMC-LPV-14829 15.382875 -73.09919 15.69 13.70 12.37 11.54 11.21 6950 2839 3469 780 C
OGLE-SMC-LPV-14912 15.434040 -73.29506 – 14.49 12.92 11.91 11.31 5135 2244 2837 722 C
Notes: TBB is the blackbody temperature derived from J −Ks while Teff is the effective temperature derived from J −Ks using the
formulainBessell,Wood,&Evans(1983).LuminositiesLwerederivedbyapplyingabolometriccorrectiontoKs (seeSection4.1).
Table 2. The first ten records from the time series data for a variable source in J band (OGLE-SMC-LPV-07852). The seven columns
areexplainedinthetext. Thefullversionofthistableisavailableasonlinematerial.
Column1 Column2 Column3 Column4 Column5 Column6 Column7
(day) (mag) (mag) (mag) (mag)
2452092.635160 12.930 0.003 0.019 0.003 SMC0050-7250 C
2452212.344921 12.959 0.002 0.019 0.003 SMC0050-7250 C
2452213.302422 12.958 0.003 0.019 0.003 SMC0050-7250 C
2452214.397972 12.940 0.002 0.019 0.003 SMC0050-7250 C
2452246.289626 12.991 0.003 0.019 0.003 SMC0050-7250 C
2452285.324709 13.007 0.003 0.019 0.003 SMC0050-7250 C
2452298.305372 12.974 0.003 0.019 0.003 SMC0050-7250 C
2452390.680728 12.931 0.002 0.019 0.003 SMC0050-7250 C
2452411.689245 12.948 0.002 0.019 0.003 SMC0050-7250 C
2452415.634797 12.965 0.003 0.019 0.003 SMC0050-7250 C
1 3 the V band and the Ks band. As noted above, fits are not
F(t)= 4XFj(t). available at all times in a given band.
j=0
When there were fewer than 4 fits, the average was taken 4 MODELING THE LSP MAGNITUDE AND
over the number of fits available. An example of the final COLOUR VARIATIONS
lightcurvefitsforeachbandinarepresentativestarisshown
in Figure2. We will now create models for the magnitude variations in
As one can see from Figure2, observation times of the various bands associated with LSPs in order to see if the
OGLE I band usually differed from those of the OGLE V assumed models are consistent with the observations. The
band and the SIRIUSJ, H and Ks bands, the latter being two models we examine are obscuration by dust and the
taken simultaneously. The I band observations are always variation of spots on a star.
the most frequent. In our analysis, we will examine colours
relative to the I band magnitude. In order to derive the
4.1 The dust model
coloursandI bandmagnitudesatsimultaneousdates,theI
band fits to the LSP variations were used. Magnitudes and OnepossibleexplanationoftheLSPphenomenonisepisodic
colours were derived from the fits at observation times of circumstellardustabsorption.Hereweassumethatthecen-
Testing models for Long Secondary Periods 5
16.8 todustabsorption bythoseejected dustcloudswhich liein
17.0
V 17.2 thelineofsighttothestar.Observationally,theejectionve-
17.4 locityofthecloudsis∼200–250kms−1(Feast1975;Clayton
15.0 1996; Feast 1990). In view of these velocity estimates, and
I 15.2 sincewedonotknowwhatcausesdustshellejectionsinLSP
stars,weconsideredtwovaluesforthewindvelocityv of15
13.7 and220kms−1,andthevelocityisassumedtobeconstant
J 13.8 withradius.Weassumethatdustisejectedforatimeinter-
valequaltohalfthelengthoftheLSPsotheshellthickness
12.8 is vP/2, where P is the length of the LSP. Our models are
H
12.9 thusmadewith two values for theshell thickness.
The second input parameter we require for dusty is
12.6
K 12.7 the temperature Tin at the inner edge of the dust shell.
We assume Tin is the dust condensation temperature. The
2000 2500 3000 3500 4000 4500 dust condensation temperature is commonly assumed to
HJD-2450000 lie in the range of 800–1500 K (e.g. van Loon et al. 2005;
Figure 2.ThecompletelightcurvesofOGLE-SMC-LPV-10774 Cassara` et al. 2013) and in ourmodels we consider Tin val-
(P=396d)inthebandsVIJHK.Therawdatapointsareshown ues of 800 and 1500 K. We note that Tin determines the
assmallfilledgreytriangleswhilethefitvaluesareshownaslarge innerradius of thedust shell.
filledblackcircles. Thethirdinputparameterfordusty isthegrain type.
The assumed chemical composition of the grains is dif-
ferent for models of oxygen-rich stars and carbon stars.
tral star ejects mass for several hundred days in each LSP
van Loon et al.(2005)useddustytoempiricallydetermine
cycle,leadingtotheformation ofthinsphericalmassshells.
possible grain types for oxygen-rich and carbon AGB stars
Dustgrainswhichformintheejectedmatterwillbecarried
intheLMCbyrequiringanacceptablefittotheSEDofthe
along with the expanding mass shells leading to dimming
observedstars. Weusethegrain chemical composition that
of the light from the central star. We assume that the dust
they adopted. For carbon stars, we use dust grains of 50%
shellsareejectedsphericallyalthoughthisisnotanessential
amorphous carbon from Hanner (1988), 40% graphite from
aspect of the dust model for the optical and near-infrared
Draine & Lee (1984) and 10 % SiC from P`egouri`e (1988)
bands which we are examining since thermal emission in
and for oxygen-rich stars, we use the astronomical silicates
these bandsis not strong.
of Draine & Lee (1984).
The amount of dimming of a star due to circumstellar
dust depends on the optical depth through the dust. For a The stellar luminosity L and the effective tempera-
sphericaldustshellwithanouterradiusRout(corresponding ture Teff of our LSP stars are also required as input to
tothestart ofamass ejection episode),an innerradiusRin dusty. These were computed from the J and Ks magni-
(corresponding to the end of a mass ejection episode, or tudesat maximum light, when circumstellar extinction is a
the dust formation radius in a newly-forming inner shell), minimum. The Teff values for the observed stars were es-
a constant expansion velocity v and a mass ejection rate timated from the J-Ks color at maximum light using the
M˙, the optical depth τ through the shell is given by (e.g. formula given in Bessell, Wood, & Evans (1983). We also
λ
Groenewegen & de Jong 1993) used J-Ks to calculate a blackbody temperature TBB for
the central star at maximum. Luminosities were computed
τ ∝ M˙ΨQλ/a( 1 − 1 ), (1) using a bolometric correction BCK to the Ks magnitude.
λ vρg Rin Rout For oxygen-rich stars, we adopted the formula BCK =
0.60+2.56(J−K)−0.67(J−K)2 givenbyKerschbaum et al.
whereΨ is thedusttogas mass ratio, Q is theabsorption
λ
efficiencyofthegrain atwavelength λ,aisthegrain radius (2010) and for carbon stars we adopt the formula BCK =
1.70+1.35(J−K)−0.30(J−K)2 givenbyBessell & Wood
and ρg is thegrain density.
We have used the dusty code (Ivezi´cet al. 1999) to (1984). To estimate of absolute bolometric magnitude, we
used the distance modulus to the SMC of 18.93 given by
examine the effect of a spherical dust shell on the broad
band colours of LSP stars. dusty computes the SED of a Keller & Wood (2006). The derived values for L, TBB and
T are listed in Table 1. When using dusty, we assumed
dustenshroudedLSPstarforonegivenconfigurationofthe eff
thatthecentralstaremitsablackbodyspectrumoftemper-
dust shell. From Equation 1,we can see that theinnermost
dustshell playsthemost important rolefor dust extinction atureTBB.Weareonlyinterestedinchangesinmagnitudes
and colours, rather than the absolute values of the colours
whenmultipledustshellsofsimilaroriginexist.Inourmodel
and magnitudes, and the blackbody assumption will have
we only consider theinnermost dust shell.
Thefirstinputparameterfordusty whichweconsider only an insignificant effect on these changes.
isthedustshellthickness.AGBstarsarewell-known mass- Thefinalinputparametertodustyistheopticaldepth
losing stars in which the terminal velocity of the mass flow τV in the V band. The output from dusty is the SED of
is ∼10–20 km s−1 (Decin et al. 2007; DeBeck et al. 2010; thecentral star extincted by a shell of visible optical depth
Lombaert et al.2013).Ontheotherhand,RCoronaeBore- τV. Unless there is significant emission from the dust shell
alis (RCB) stars are similarly luminous evolved stars which in each band under consideration, for a given value of τV
eject mass semi-periodically in random directions (Clayton theextinction of thecentral star in each band is essentially
1996). The large-amplitude variability of these stars is due independentof the details of theshell radius or thickness.
6 M. Takayama, P. R. Wood and Y. Ita
14.0 OGLE-SMC-LPV-07852 14.8 OGLE-SMC-LPV-08199 14.6 OGLE-SMC-LPV-13748
Tin=800K
Tin=1500K 15.0 14.8
14.2
I I 15.2 I 15.0
14.4
15.4 15.2
14.6 15.6 15.4
13.6 14.0
OGL E-SMC- LPV-09 856 OGLE -SMC-L PV-0828 0 OGLE -SMC-L PV-1380 2
14.2 13.8 14.2
14.0 14.4
I 14.4 I I
14.2 14.6
14.6 14.4 14.8
1143..66 1153..08
OGL E-SMC- LPV-10 774 OGLE -SMC-L PV-0934 1 OGLE -SMC-L PV-1394 5
14.8
13.8 14.0
15.0 14.0 14.2
I I I
14.2 14.4
15.2
14.4 14.6
15.4 1143..66 14.8
OGL E-SMC- LPV-12 332 OGLE -SMC-L PV-1010 9 14.2 OGLE -SMC-L PV-1408 4
14.2 13.8
14.4
14.0
I 14.4 I I 14.6
14.2
14.8
14.6 14.4
15.0
14.6 14.2
15.0 OGL E-SMC- LPV-14 045 14.6 OGLE -SMC-L PV-1265 3 OGLE -SMC-L PV-1415 9
14.4
14.8
15.2
14.6
I I 15.0 I
14.8
15.4
15.2
15.0
15.6 15.4
14.0 13.8 15.2
OGL E-SMC- LPV-14 715 OGLE -SMC-L PV-1334 0 13.6 OGLE -SMC-L PV-1482 9
14.0
14.2 13.8
14.2
I I I 14.0
14.4 14.4
14.2
14.6
14.6
14.4
14.8
OGL E-SMC- LPV-14 781 13.8 OGLE -SMC-L PV-1355 7 14.4 OGLE -SMC-L PV-1491 2
14.4
14.6
14.0
14.6 14.8
I I 14.2 I
15.0
14.8 14.4
15.2
14.6
15.0
1.2 1.3 1.4 1.5 1.6 1.7 1.4 1.6 1.8 2 2.2 1.4 1.6 1.8 2 2.2
I-J I-J I-J
Figure 3. Comparisons ofobservations withmodelsof expansion velocity15km s−1. The7panels ofthe leftcolumnrepresent theI
andI−J variationsfortheoxygen-richstarsandthe14panelsofthemiddleandrightcolumnsareforthecarbonstars.Theredpoints
show the LSP variations obtained from Fourier fits to the observed light curves. The blue lines show least squares fits to these points
(thelineisomittediftherelativeerrorintheslopeismorethan0.2).Theblackcontinuous lineswithopencirclesshowthemagnitude
andcolourvariationsformodelswithaninnerdustshelltemperatureof800Kwhilethetrianglesareformodelswithaninnerdustshell
temperature of 1500K. The top end of each curve corresponds to shell optical depth τV = 0 and the points are spaced at intervals of
0.5inτV =0.Theblack dotted linesshow thelocus ofun-extincted blackbodies ofdifferenttemperatures andthe constant luminosity
giveninTable1foreachstar.Notethatthescaleoftheverticalaxisisthesameforalloxygen-richstarsandforallcarbonstars.
Testing models for Long Secondary Periods 7
14.0 OGLE-SMC-LPV-07852 14.8 OGLE-SMC-LPV-08199 14.6 OGLE-SMC-LPV-13748
Tin=800K
Tin=1500K 15.0 14.8
14.2
I I 15.2 I 15.0
14.4
15.4 15.2
14.6 15.6 15.4
13.6 14.0
OGL E-SMC- LPV-09 856 OGLE -SMC-L PV-0828 0 OGLE -SMC-L PV-1380 2
14.2 13.8 14.2
14.0 14.4
I 14.4 I I
14.2 14.6
14.6 14.4 14.8
1143..66 1153..08
OGL E-SMC- LPV-10 774 OGLE -SMC-L PV-0934 1 OGLE -SMC-L PV-1394 5
14.8
13.8 14.0
15.0 14.0 14.2
I I I
14.2 14.4
15.2
14.4 14.6
15.4 1143..66 14.8
OGL E-SMC- LPV-12 332 OGLE -SMC-L PV-1010 9 14.2 OGLE -SMC-L PV-1408 4
14.2 13.8
14.4
14.0
I 14.4 I I 14.6
14.2
14.8
14.6 14.4
15.0
14.6 14.2
15.0 OGL E-SMC- LPV-14 045 14.6 OGLE -SMC-L PV-1265 3 OGLE -SMC-L PV-1415 9
14.4
14.8
15.2
14.6
I I 15.0 I
14.8
15.4
15.2
15.0
15.6 15.4
14.0 13.8 15.2
OGL E-SMC- LPV-14 715 OGLE -SMC-L PV-1334 0 13.6 OGLE -SMC-L PV-1482 9
14.0
14.2 13.8
14.2
I I I 14.0
14.4 14.4
14.2
14.6
14.6
14.4
14.8
OGL E-SMC- LPV-14 781 13.8 OGLE -SMC-L PV-1355 7 14.4 OGLE -SMC-L PV-1491 2
14.4
14.6
14.0
14.6 14.8
I I 14.2 I
15.0
14.8 14.4
15.2
14.6
15.0
1.2 1.3 1.4 1.5 1.6 1.7 1.4 1.6 1.8 2 2.2 1.4 1.6 1.8 2 2.2
I-J I-J I-J
Figure 4. ThesameasFigure3butformodelswithanexpansionvelocityof220kms−1.
8 M. Takayama, P. R. Wood and Y. Ita
4.2 Comparison of the dust model with is no way that such variations can be the result of variable
observations amountsof dust extinction only.
The most probable explanation for the variation of
4.2.1 The variation of I with I−J
J −Ks in O-rich LSP stars is strong H2O absorption in
In general, the I light curves are the best sampled of the Ks band. Significant broad H2O band absorption is found
light curves in the different filters, the J, H and K light between about 1.4µm and 1.9µm on the edges of the H
curves are the next best sampled and the V light curves and Ks filter bandpasses. In non-variable stars, this ab-
are mostly only sparsely sampled. Thus when comparing sorption is seen from about M6 III to later spectral type
observations against models in magnitude-colour plots, we in M giants (e.g. Tsuji 2000; Rayneret al. 2009). Strong
useI asthemagnitudeandacolourinvolvingI andanother water absorption in the H and K bands is also com-
magnitude. The other magnitude should be one of J, H or monly seen in large amplitude pulsating red giants (Mira
K becauseofthehigherfrequencyofobservation compared variables) (e.g. Johnson & M´endez 1970). The reason for
to V. Ideally, I − K would be best because of the large this absorption is the existence of a relatively dense shell
wavelength range spanned by this colour, but as we will with a temperature of about 1800K elevated above the
show below, theK bandin O-rich starsis contaminated by photosphere by shock waves associated with the pulsation
variableH2Oabsorption, as is theH band,so weuseI−J (Ireland,Scholz, & Wood 2011). Using the models atmo-
as the colour in our first comparison of observations and sphere results of Houdashelt et al. (2000), we find that the
models. V − I colours of the stars we are examining here yield
In Figures3 and 4 we show the I magnitude plotted Teff ∼3400–3600K,correspondingtospectraltypesM3–M5.
against theI−J color for the observed samples of oxygen- Due to these relatively early M spectral types, the results
rich and carbon stars. Figure3 displays models with a shell noted above mean that strong water absorption in the Ks
expansion velocity of 15 km s−1 (thin shell) while Figure4 band can not occur in a hydrostatic atmosphere. The ele-
displays models with a shell expansion velocity of 220 km vation of a shell of mass above the photosphere is required
s−1 (thick shell). Note that it is not the absolute values of to produce strong water absorption, as in the case of Mira
points in the diagrams that is important, it is the slope of variables. We suggest that this shell elevation, which may
thelinerepresentingthevariationofI withI−J thatmat- also lead to dust formation, occurs at the beginning of the
ters.IftheLSPvariationsarecausedbyvariableamountsof luminositydeclinesassociatedwiththeLSPs,leadingtothe
circumstellar dust alone, then the slope will depend almost formationofincreasingH2OabsorptionintheKs bandand
entirely on the dust properties and not on the spectrum of blueing of theJ−Ks colour.
thecentralstar. Thecentralstar, which inthesemodels we This result is a second piece of observational evidence
assumetoemitasablackbodywiththeLandTBB givenin for the existence of a disturbance above the photosphere
Table 1, determines the position of the model in Figures3 caused by the LSP. Wood et al. (2004) found variable Hα
and 4 when thereis nocircumstellar extinction. absorptioninstarswithLSPswhichtheyattributetoachro-
Firstly,wemakesomecommentsonthemodels.Inboth mosphereofvariablestrength,withthestrongestHαabsorp-
Figure3andFigure4,thetwomodellinesdifferslightlydue tion near maximum light. Combined with theresults above
tore-emissionfromthedustshellatJ andtoalesserextent for the H2O absorption layer, we have a picture whereby
I.ComparingFigure3withFigure4,correspondingtomod- matterisejected abovethephotospherebeginningnear the
elswith different dustshellthickness,wesee thatthereisa luminositydeclineoftheLSPformingarelativelydenseshell
slightly faster increase in I−J with I for the thinner shell of temperature ∼1800K. As the LSP cycle progresses, this
ofFigure3,especiallyforthehotterinnerdustshelltemper- shell appears to be slowly heated by non-thermal processes
ature of 1500K. The reason for this is that for the thinner to∼8000Kgivingrisetoachromospherewhichisthendis-
dust shell there is more emission at J. Also, for a given op- placed bythe nextejected shell containing H2O.
ticaldepthτV tothecentreofthediskoftheobservedstar, Turning to the carbon stars, we see that they do get
thetotalextinction averagedoverthestellar diskisslightly redderastheygetfainterinI butinallcasestheyclearlydo
greater for thethinnershell in Figure3 than for thethicker notreddenasmuchaspredictedbythemodels.Tobringthe
shellinFigure4(thelinesinFigure3arelongerthaninFig- observedvariationinJ−KswithdeclineinI intoagreement
ure4). This is because the extinction near the limb of the withthemodels,whilekeepingthevariationofI−J withI
diskissmaller relativetotheextinctionatthecentreofthe inagreementwith themodels,would requireastrangedust
disk for a thickershell. opacitywithalargebumpintheKs band.Wedonotknow
The observed variation of I with I−J associated with of any dust composition that could produce such a bump
theLSP(redpoints)isgenerallyconsistentwiththeslopeof (see Section 4.2.4 for a discussion of the effect of different
one of themodel lines in Figures3 or 4. This indicates that grain opacities).
the variation of I and J due to an LSP could be explained We note that ejected clouds covering only part of the
by variable dust absorption alone. stellarsurfaceratherthanthecompletesurfaceasforspher-
ical shells can not explain theobserved variation in J−Ks
4.2.2 The variation of I with J−Ks wJi−thKdseccolilnoiunrginI.aFcoloruadllmboudtevleirnycrleaarsgeesosplitgihctallydfeapsttehrs,wtihthe
Figures5 and 6 show the variation of I with J −Ks. As declineinI thanforasphericalshellmodel,whereastheob-
one can easily see, the observational variations due to the servations require a slower increase in J −Ks colour. Also,
LSPsareinconsistentwiththemodelvariationsinallcases. given that the variation of the observed I −J colour with
Mostimportantly,fortheO-richstars,theobservedJ−Ks decline in I agrees with the spherical shell models, a cloud
colourtendstogetbluer as the star gets fainter inI.There modelwillnotagreewiththeobservations(theV −I colour
Testing models for Long Secondary Periods 9
14.0 OGLE-SMC-LPV-07852 14.8 OGLE-SMC-LPV-08199 14.6 OGLE-SMC-LPV-13748
Tin=800K
Tin=1500K 15.0 14.8
14.2
I I 15.2 I 15.0
14.4
15.4 15.2
14.6 15.6 15.4
13.6 14.0
O GLE-SMC -LPV-09856 OGL E-SMC-L PV-082 80 OGL E-SMC-L PV-138 02
14.2 13.8 14.2
14.0 14.4
I 14.4 I I
14.2 14.6
14.6 14.4 14.8
1143..66 1153..08
O GLE-SMC -LPV-10774 OGL E-SMC-L PV-093 41 OGL E-SMC-L PV-139 45
14.8
13.8 14.0
15.0 14.0 14.2
I I I
14.2 14.4
15.2
14.4 14.6
15.4 1143..66 14.8
O GLE-SMC -LPV-12332 OGL E-SMC-L PV-101 09 14.2 OGL E-SMC-L PV-140 84
14.2 13.8
14.4
14.0
I 14.4 I I 14.6
14.2
14.8
14.6 14.4
15.0
14.6 14.2
15.0 O GLE-SMC -LPV-14045 14.6 OGL E-SMC-L PV-126 53 OGL E-SMC-L PV-141 59
14.4
14.8
15.2
14.6
I I 15.0 I
14.8
15.4
15.2
15.0
15.6 15.4
14.0 13.8 15.2
O GLE-SMC -LPV-14715 OGL E-SMC-L PV-133 40 13.6 OGL E-SMC-L PV-148 29
14.0
14.2 13.8
14.2
I I I 14.0
14.4 14.4
14.2
14.6
14.6
14.4
14.8
O GLE-SMC -LPV-14781 13.8 OGL E-SMC-L PV-135 57 14.4 OGL E-SMC-L PV-149 12
14.4
14.6
14.0
14.6 14.8
I I 14.2 I
15.0
14.8 14.4
15.2
14.6
15.0
1 1.1 1.2 1.3 1.2 1.4 1.6 1.8 2 1.2 1.4 1.6 1.8 2
J-K J-K J-K
Figure 5.ThesameasFigure3butusingtheJ−KscolourratherthanI−J.
10 M. Takayama, P. R. Wood and Y. Ita
14.0 OGLE-SMC-LPV-07852 14.8 OGLE-SMC-LPV-08199 14.6 OGLE-SMC-LPV-13748
Tin=800K
Tin=1500K 15.0 14.8
14.2
I I 15.2 I 15.0
14.4
15.4 15.2
14.6 15.6 15.4
13.6 14.0
O GLE-SMC -LPV-09856 OGL E-SMC-L PV-082 80 OGL E-SMC-L PV-138 02
14.2 13.8 14.2
14.0 14.4
I 14.4 I I
14.2 14.6
14.6 14.4 14.8
1143..66 1153..08
O GLE-SMC -LPV-10774 OGL E-SMC-L PV-093 41 OGL E-SMC-L PV-139 45
14.8
13.8 14.0
15.0 14.0 14.2
I I I
14.2 14.4
15.2
14.4 14.6
15.4 1143..66 14.8
O GLE-SMC -LPV-12332 OGL E-SMC-L PV-101 09 14.2 OGL E-SMC-L PV-140 84
14.2 13.8
14.4
14.0
I 14.4 I I 14.6
14.2
14.8
14.6 14.4
15.0
14.6 14.2
15.0 O GLE-SMC -LPV-14045 14.6 OGL E-SMC-L PV-126 53 OGL E-SMC-L PV-141 59
14.4
14.8
15.2
14.6
I I 15.0 I
14.8
15.4
15.2
15.0
15.6 15.4
14.0 13.8 15.2
O GLE-SMC -LPV-14715 OGL E-SMC-L PV-133 40 13.6 OGL E-SMC-L PV-148 29
14.0
14.2 13.8
14.2
I I I 14.0
14.4 14.4
14.2
14.6
14.6
14.4
14.8
O GLE-SMC -LPV-14781 13.8 OGL E-SMC-L PV-135 57 14.4 OGL E-SMC-L PV-149 12
14.4
14.6
14.0
14.6 14.8
I I 14.2 I
15.0
14.8 14.4
15.2
14.6
15.0
1 1.1 1.2 1.3 1.2 1.4 1.6 1.8 2 1.2 1.4 1.6 1.8 2
J-K J-K J-K
Figure 6.ThesameasFigure4butusingtheJ−KscolourratherthanI−J.