Time-series Paschen-beta spectroscopy of SU Aurigae

0.61 MB·

by  Ryuichi Kurosawa

Mon.Not.R.Astron.Soc.000,1–13(2004) Printed2February2008 (MNLATEXstylefilev2.2)
Time-series Paschen-β spectroscopy of SU Aurigae
Ryuichi Kurosawa⋆, Tim J. Harries and Neil H. Symington
SchoolofPhysics,UniversityofExeter,StockerRoad,ExeterEX44QL
5
Datestobeinserted
0
0
2
ABSTRACT

n
a We present time-series echelle spectra of the Paβ line of the T Tauri star SU Aur, ob-
J servedoverthreeconsecutivenights.Thelineshowsstrongvariability(∼ 10percent)over

0 the velocity range (100kms−1, 420kms−1) in the red broad absorption component, and
1 weaker variability (∼ 2 per cent) over the velocity range (−200kms−1, 0kms−1) in the
bluewing.Thevariabilityinthevelocityrange(−200kms−1, 0kms−1)iscorrelatedwith
1 thatin(200kms−1, 400kms−1),andthevariabilityinthesevelocityrangesanti-correlates
v
withthatin(0kms−1, 100kms−1).Themeanspectrumfromthesecondnightshowsasug-
5
6 gestionofablue-shiftedabsorptioncomponentatabout−150kms−1,similartothatfound
1 intheHαandHβ lines.Wefindthepositionofthesubpeakintheredabsorptioncomponent
1 changessteadily with time, and its motion modulateson half the rotationalperiod.We also
0 findthatthemodulationofthelineequivalentwidthisassociatedwithahalfandathirdofthe
5 rotationalperiod,whichisconsistentwiththesurfaceDopplerimagesofSU Aur.Radiative
0 transfermodelsofarotationallymodulatedPaβ line,producedintheshock-heatedmagneto-
h/ sphericaccretionflow,arealsopresented.Modelswithamagneticdipoleoffsetreproducethe
p overallcharacteristicsoftheobservedlinevariability,includingthelineequivalentwidthand
- themotionofthesubpeakintheredabsorptiontrough.
o
r Keywords: stars:formation– stars:individual:SU Aur– circumstellarmatter– stars: pre-
t
s main-sequence
a
:
v
i
X
1 INTRODUCTION CollierCameron&Campbell 1993; Shuetal. 1994a). This
r
a picture of the accretion flows is supported by the obser-
T Tauri stars (TTS) are young (<∼ 3 × 106yrs, vation that CTTS have relatively strong (∼ 103G) mag-
Appenzeller&Mundt 1989) low-mass stars, and are thought
netic fields (e.g. Johns-Krull,Valenti,Hatzes,&Kanaan
to be the progenitors of solar-type stars. Classical T Tauri stars
1999; Guenther&Emerson 1996; Symingtonetal. 2004b)
(CTTS) exhibit strong Hα emission, and typically have spectral
and by radiative transfer models which reproduce the
types of F–K. Some of the most active CTTS show emission in
gross characteristics of the observed profiles for some TTS
higher order Balmer lines and metal lines (e.g. CaII, H and K).
(e.g. Muzerolle,Calvet,&Hartmann2001).
They also exhibit an excess amount of continuum flux in the
SU Aurigae is a bright (J ∼ 7, Chakraborty&Ge 2004)
ultraviolet(UV)andinfrared(IR).
young star with a G2 spectral type (Herbig 1960), and has
Line profile studies of CTTS show evidence for both
been classified either as a CTTS (e.g. Giampapaetal. 1993;
outflows and inflows, as seen in the blue-shifted absorption
Bouvieretal.1993)orearly-typeTTS(e.g.Herbstetal.1994).It
features in Hα profiles (e.g. Herbig 1962) and the inverse
isalsothe prototype ‘SU Aur type’ (Herbig&Bell1988). Based
P Cygni (IPC) profiles (e.g. Kenyonetal. 1994; Edwardsetal.
onmodellingofthespectralenergydistribution(SED),SUAuris
1994) respectively. Typical mass-loss rates of CTTS are about
10−9M⊙yr−1 to10−7M⊙yr−1 (e.g.Kuhi1964;Edwardsetal. known tohave an accretion disc (∼ 400au) along withan outer
envelope (e.g. Akesonetal. 2002). Near-infrared coronagraphic
1987; Hartiganetal. 1995), and mass-accretion rates range from
10−9M⊙yr−1to10−7M⊙yr−1(e.g.Kenyon&Hartmann1987; observations of Nakajima&Golimowski (1995) and Gradyetal.
(2001) have shown that the object is associated with a reflection
Bertout,Basri,&Bouvier1988;Gullbringetal.1998).
nebulaandoutflows.Mostrecently,Chakraborty&Ge(2004)have
In the currently favoured model of accretion on to
presented high resolution (0.3 arcsec) J-, H- and K-band adap-
CTTS, the accretion discs are disrupted by the stellar mag-
tiveopticsimagesdisplayingnebulositywithadisc-likestructure
netosphere which channel the gas from the disc on to the
aroundSUAur.
stellar surface (e.g. Uchida&Shibata 1985; Koenigl 1991;
SUAurhasbeenasubjectofanumberofvariabilitystudies,
mainlyintheoptical,e.g. Giampapaetal.(1993),Johns&Basri
⋆ E-mail:[email protected] (1995),Petrovetal.(1996),Oliveiraetal.(2000)andmostrecently
2 R. Kurosawaetal.
Unruhetal.(2004).TheHαline,whichismostcommonlyusedfor trabyusingtheleastsquarefitofthespectrawithacombination
variability studies of TTS, is produced over a large circumstellar oftwoGaussianfunctionsandathirdorderpolynomial.Afterthe
volume, and shows evidence of outflow (winds) as well as mag- preliminary reduction by the pipeline was finished, a wavelength
netosphericinflow.Johns&Basri(1995)studiedthevariabilityof calibration was performed using a combination of OH sky emis-
HαandHβ profilesofSUAur,andfoundthatthemodulationof sionlinesandcomparisonarclines.Finallyweappliedheliocentric
thewindandtheaccretioncomponentsintheprofilesareapprox- velocity corrections and shifted the spectra into therest-frame of
imately180◦ outofphase.Thisledthemtosuggestthatthemag- SUAur, adopting theradial velocity measurement (+16kms−1)
neticfieldaxisisslightlytiltedwithrespecttotherotationaxisof ofHerbig&Bell(1988).
thestar.
Thetopologyofthemagneticfieldisstillindispute.Despite
of the overall success of the dipole field geometry used in ex-
plainingmanyobservedfeaturesofCTTS,Johns-Krull&Gafford 3 RESULTS
(2002) re-examined the validity of this assumption. They found
that observations didnot support the relationshipsbetween mass, A general overview of the dataset is presented in Fig. 1. The
radius, and rotation period that magnetospheric accretion theory mean spectrum shows a classic inverse P Cygni (IPC) profile,
predicts under the assumption of a dipolar field, but found bet- with the blue-wing of the emission component extending to
ter agreement when a non-dipole fieldtopology was used. In ad- −200kms−1andtheabsorptioncomponentwithmaximumdepth
dition, forSU Aur,Unruhetal. (2004)found alackof thecircu- at+100kms−1 andextendingoutto+200kms−1.Thepeakin-
larpolarisationacrosssomestrongphotosphericlines;onceagain tensity of the line, relative to the continuum, is about 1.25. Al-
in conflict with the expectations for a large-scale dipole. Further, though comparable in strength, the Paβ profiles of SU Aur pre-
Oliveiraetal.(2000)foundthetime-laggedbehaviourinthevari- sentedbyFolha&Emerson(2001)showsignificantemissionred-
abilityofsomeopticallines,andproposedthatthiscanbecaused wardsoftheabsorptiondip,andalsohavemaximumabsorptionoc-
bythepresenceofanazimuthalcomponentinthemagnetosphere. curringat0kms−1 i.e.theyappeartobeshiftedby∼ 90kms−1
Near-IR profiles, such as Paβ and Brγ are likely to become bluewardcomparedtoourdataset.
importantprobesofaccretioninCTTSandtheirlower-masscoun- The mean spectra from each night are shown in Fig. 2. It is
terparts,sincetheyappeartosufferfromlesscontaminationbyout- immediatelyapparentthatthebluesideoftheprofileismorestable
flowsthanHα(e.g.Folha1998;Folha&Emerson2001,although than the red, with variability of less than 10 per cent (in contin-
seeWhelanetal.2004)andarelessaffectedbydustobscuration, uumunits)atthelinepeakandblueward.Thereisasuggestionof
enablingthestudyofaccretioninhighlyembedded objects.Here blue-shiftedabsorptionatabout−150kms−1onthesecondnight.
we present time-series Paβ spectra of SU Aur, which we use to Interestingly, theblue-shifted absorption feature at −150kms−1
probetheaccretiongeometry. InSection2wegivedetailsonthe isalsoseeninHα(Giampapaetal.1993)andHβ (Johns&Basri
observations and the data reduction. The results and analysis are 1995)profiles,andthemodulationperiodoftheabsorptionfeature
presentedinSection3.Radiativetransfermodelsarecomputedin isidentified as therotational period of SU Aur (∼ 3 d). The red
anattempttoexplaintheoverallobservedvariabilityfeatureinSec- sideof the mean profilesshows strong variability, extending red-
tion4.TheconclusionsaregiveninSection5. wardtoover+400kms−1.Onthefinalnightthereisemissionon
theredsideoftheabsorptiontrough,andtheprofilemoreclosely
resemblesthatpresentedbyFolha&Emerson(2001).
The greyscale dynamic spectra (Fig. 1) reveal the gradual
2 OBSERVATIONS changes that occur over a timescale of hours. On the first night
thereisafeaturethatacceleratesmonotonicallyredwardacrossthe
A total of 503 Paβ (λ = 1.28181µm) spectra with a signal-to-
noise ratio (S/N) of ∼ 90 per pixel (in the continuum) were ob- absorptiontrough.Onthesecondnightboththeredandbluesides
of theprofilearelower thanthemean. Finally,on thethirdnight
tainedover3nights(2002December1–3)attheUnitedKingdom
weseethegrowthofanemissionfeatureintheabsorptiondip.The
InfraredTelescope(UKIRT)onMaunaKea,Hawaii.Theapproxi-
characteristiclevelofthisred-wingvariabilityisabout10percent.
matetemporalsamplingratewasonespectrumevery3minutes.
The Cooled GratingSpectrometer No. 4 (CGS4), a 1–5 µm
multi-purpose 2-D grating spectrometer with a 256 × 256 InSb
array, with the echelle grating and long camera (300mm), was 3.1 Temporalvariancespectra
used. This provides a velocity resolution (at 1.28181µm) of ∼
16kms−1 using∼ 1pixslitwidthwithouttheNyquiststepping. Inordertoquantifythelevelandsignificanceofthevariabilityseen
Thewavelengthrangeoftheechellespectraisabout3600kms−1 inthePaβprofiles,thetemporalvariancespectrum(TVS)analysis
methodproposedbyFullerton,Gies,&Bolton(1996)wasapplied
providing enough continuum on either side of the line for accu-
to the time series spectra from each night separately, and also to
raterectification.Unfortunatelythereareinsufficientphotospheric
allthreenightscombined. Thismethod statisticallycompares the
featurestoperformaveilingcorrection(butseeSection4.2). Tel-
deviationsinlinefeatureswiththoseoftheadjacentcontinuum.In
luricstandardstarswereobservedthroughouteachnight(every2–3
other words, thelevel of deviation at agivenwavelength iscom-
hours)tocorrectforinstrumentresponseandatmospherictransmis-
putedwithaweightfunctionwhichisinverselyproportionaltothe
sion.
signal-to-noise(S/N)levelofthecontinuum. Byusingthedefini-
ThedataobtainedwithCGS4werereducedusingastandard
tionofFullertonetal.(1996),thej-thvelocitybinoftheTVS,in
procedure(c.f.Puxleyetal.1992):1.biassubtraction,2.flat-field,
thelimitofhighphoton-count,canbewrittenas:
and3.optimalextraction(Horne1986).Allspectrawerereduced
inthesamemanner,usingtheORAC-DRpipelineprogram.Intrin-
N
sicabsorptioncomponents inthestandardstars,usedinstep3to (TVS) = 1 S −S¯ 2 wi (1)
correctforthetelluriclines,weresubtractedfromtheoriginalspec- j N−1 Xi=1(cid:0) ij j(cid:1) Sij
Time-series Paschen-β spectroscopyofSU Aurigae 3
1.3
1.2
x
u
Fl
malized 1.1
or
N
1
0.9
0.8
-400 -200 0 200 400
Velocity (km s-1)
Figure2.Meanspectrafromeachnight:Thefirstnight(solid),thesecond
night(dashed),andthethirdnight(dotted). Whilemostofthevariability
occurs in the red side (50 kms−1, 500 kms−1) of the line, small yet
noticeablechangesinfluxlevelcanbeseenin(−200kms−1,0kms−1).
Thefluxlevelat∼250kms−1isbelowthecontinuumonthefirstandthe
secondnight,butitisinemission(abovethecontinuum)onthethirdnight.
Ahintofablue-shiftedabsorptioncomponentat∼150kms−1isseenon
thesecondnight.
deviations; hence, it provides a more accurate impression of the
deviationfromthevariabilitywithrespecttothecontinuum.
Using equation 1, the temporal deviation spectrum of the
times-seriesspectrafromeachnightwascomputed,andtheresults
were placed in the top panels of Fig. 3. In the same figure, the
spectradividedbythemeanspectrumof eachnight (thequotient
Figure1.Thetime-seriesspectraofPaβ.Themeanspectrum(normalised) spectra) and themean spectrum areshown asa function of time.
ofthe503spectraobtainedduringthethreenight(2002December1–3)is Similarly, (TVS)1/2 computed from all three nights is placed in
shownatthebottom.Inthemiddlepanels,thequotientspectra(dividedby thetopofFig.1.
themeanspectrum)asafunctionoftime(HJD−2452600)areplotted The(TVS)1/2 valuewhichcorresponds tooneper centsta-
asgreyscaleimageswiththecolourscaledfrom1.1(white)to0.9(black). tisticalsignificance(theχ2-probability)isalsoshowninthesame
Thetemporaldeviationspectrum(TVS)1/2fromthe503spectraisshown figure as a reference point. In the absence of real variability, the
onthetoppanel(c.f.Section3.1).Inallplots,thehorizontal axesarein observed (TVS)1/2 values should be below this level for 99 per
velocityspaceinthestellarrestframe.Thegreyscaleimagesclearlyshow cent of time. From the (TVS)1/2 plot in Fig. 1, we confirm
variabilityonbothredandbluesidesofthespectra,butlittlevariabilityis
that the line displays strong variability within the velocity range
seennearthelinecentre.Thedashedlineinthetoppanelshowsthesta-
tisticalsignificanceatthe1percentlevel,anditindicatesthatthevelocity (100kms−1, 420kms−1) in the red wing, and weaker variabil-
range(−200kms−1,420kms−1)hassignificantvariability. itywithinthevelocityrange(−200kms−1, 0kms−1)intheblue
wing.The(TVS)1/2 plotsinFig.3showthatthevelocityranges
inwhichthevariabilityismostactivechangefromnighttonight.
whereS isthesignalofthej-thvelocitybinbelongingtothei-th Using the mass (M∗ = 2.25M⊙) and the radius (R∗ =
ij
spectrumofatime-series,andNisthenumberofobservedspectra. 3.6R⊙)ofSUAurfromCohen&Kuhi(1979),wefinditsescape
S¯ istheweightedmeanspectrumdefinedas velocitytobeVesc = (2GM∗/R∗)1/2 = 490kms−1.Thisvalue
j
isan upper-limit for the velocity of the material as it impacts on
N thestellarsurface,andindeedweseethattheextentoftheredab-
1
S¯j = N wiSij . (2) sorptioninthelineprofileis∼420kms−1,andthatthemaximum
Xi=1 velocityofsignificantvariabilityis∼420kms−1.
Theweightingfactorw forthei-thspectrumisdefinedas
i
1/σ2 3.2 Motionofasubpeakintheabsorptiontrough
ic
w = (3)
i 1 (cid:0)N (1(cid:1)/σ2 ) In the red absorption trough of the mean spectra (see the bottom
N k=1 kc panelsofFig.3),wefindthepresenceofasubpeakatV =∼250,
P
whereσ andσ arethenoiseinthecontinuumforthei-thand ∼ 180 and ∼ 260kms−1 for the first, second and third nights.
ic kc
k-thspectra. Wealsofindthepositionofthesubpeak(indicatedbyarrowsinthe
As mentioned by Fullertonetal. (1996), a more convenient samefigure)changesduringonenight.Itappearsthatamaincause
quantitytobeplottedisthesquarerootofTVS(temporal devia- ofthevariabilityseeninthered-wing(asseeninthegreyscaleim-
tionspectrum)because(TVS)1/2 scaleslinearlywiththespectral agesinthesamefigures)isduetothemotionofthissubpeak,
4 R. Kurosawaetal.
Figure3.Thesummaryofobservationsfromeachnight(thefirstnighttothethirdnightfromlefttorightrespectively).Foreachnight,thenormalisedmean
spectrum(bottom),thequotientspectra(dividedbythemeanspectrum)(middle)withcolourscaledfrom1.08(white)to0.92(black)asafunctionoftime
(HJD−2452600),andthetemporaldeviationspectrumTVS−1/2(top)inunitsof1/100ofthecontinuumflux(%Fc)areshown.Thedashedlineinthetop
plotindicatesthestatisticalsignificancefor1percentlevel.Inthebottomplots,thepositionsofthesubpeakintheredabsorptiontroughsareindicatedwith
arrows,andtheyseemtobemovingfromnighttonight(seeSection3.2).
The positions of the subpeak were estimated for each spec- The two lines (nmax = 2 and nmax = 3) are very similar
trum,byapplyingaquadraticfittothedatapointsaroundthesub- toeach other aswecanseefromthefigureandthevaluesof the
peak. The results are shown in Fig. 4 as a function of time. The Fouriercoefficients.Thecontributionsofthethirdharmonicterms
amplitudeofthechangeinthesubpeakpositionisrelativelylarge (a3 and b3) are very small compared to the first and the second
on the second night compared to those of the first and the third harmonic terms.Totest whether thethirdharmonic termsshould
nights. To demonstrate the motion of the subpeak occurred dur- be included in the fitting functions, we have performed an F-test
ing the second night, we plot the time-averaged (∆t = 70 min) (c.f.Bevington 1969). From the table, the numbers of degree of
spectrainFig.5.Thefigureshowsaweaksubpeak movingfrom freedomareν2 =497andν3 =495fornmax =2andnmax =3
V ∼170kms−1to∼220kms−1duringthecourseofthenight. respectively.Usingthereducedchi-squarevaluesinthetable,we
The subpeak positions were fittedwitha function (f) inthe find Fχ = χ2(nmax =2)−χ2(nmax =3) /χ2ν2 = 2 and
formoftheFourierseries, F = χ2 /χ2(cid:8) = 1. Since F > F, the thir(cid:9)d harmonic terms
ν2 ν3 χ
nmax shouldbeincludedinthefittingalthoughtheircontributionisrela-
a0 2πn 2πn
f(t)= + a cos t +b sin t (4) tivelysmall.
2 Xn=1 n n (cid:16) P (cid:17) n (cid:16) P (cid:17)o From this analysis we found that the motion of the subpeak
in the red absorption trough is possibly associated not only with
wheretandP arethetimeofobservationandtherotationalperiod
of SU Aur respectively, and nmax = 3. The reasons for choos- P, but also with P/2. This is consistent with the the tilted-axis
magnetosphericaccretionmodelmentionedearlier.
ingtheformofthefittingfunctionaboveareasfollows:1.asim-
InspiteoftheexcellentfitsofthedatawithFourierseries,we
plestpossibleformisdesirable,and2.thefirstharmonictermsin
theFourierseriesmight bepresentifthevariationisrelatedwith shouldnotignoreapossibilitythattheratherlargeamplitudeofthe
changeinthesubpeakpositionseenonthesecondnightwascaused
the stellar rotation, 3. the second harmonic terms in the Fourier
byasingleepisodicevent (e.g. byatemporal andlocal enhance-
series might be present if the variation is caused by the tilted-
axis magnetospheric accretion model (e.g. see Shuetal. 1994a; mentoftheaccretionflow).Assumingconservationofenergy,the
Johns&Basri1995),and4.thethirdharmonictermsintheFourier free-falltime(tff)ofanobjectfromtheco-rotationradius(r=Rc)
series might be present because the surface Doppler images of ofthestartothestellarsurface(r=R∗)canbewrittenas
SUAurbyPetrovetal.(1996),Mennessier(1997)andUnruhetal.
(2004)weaklysuggestthepresenceofcoolspotsonthesurfaceat tff =− 1 R∗ Rc −1 −1/2dr (5)
threedifferentlongitudeswhichareapproximatelyequallyspaced. Vesc(Rc)ZRc (cid:16) r (cid:17)
withTnhmeaxfit=wit1hannmdanxm=ax3=is2shfoowrncoinmFpiagr.is4ona.loInngawllicthastehse, tfihtes whereVesc(Rc) = (2GM∗/Rc)1/2,andR∗ andM∗arethestel-
lar radius and mass respectively. By evaluating the integral, one
fixedperiodof2.7d(Unruhetal.2004)wasadopted.SeeTable1
obtains
forthesummaryofthefittingparameters.Thefigureclearlyshows
thhaantdt,htehefitfiwtsiwthitnhmnamxa=x =12faailnsdtonmreapxre=se3ntatrheesdigantaifi.cOannttlhyeboetthteerr. tff = VescR(cRc)nπ2 +q1/2(1−q)1/2−arcsin(cid:0)q1/2(cid:1)o (6)
Time-series Paschen-β spectroscopyofSU Aurigae 5
0.5
300
-1m s)250 0.4
k
ns (
o
ositi 0.3
P
k 200
a
Sub-Pe ed Flux 0.2
z
ali
150 m
or
N
0.1
9.5 10 10.5 11 11.5 12 12.5
HJD - 2452600 (days)
0
Figure4.Thepositionsofthesubpeak(circles)intheredwingofthePaβ
profilefromthethreenightsareplottedasafunctionoftime.Thedatawere
fittedwiththeFourierseries(equation4)withthetermsuptonmax = 1
(dash-dotline),nmax = 2(dashedline)andnmax = 3(solidline).The -0.1
corresponding fittingcoefficients aresummarisedinTable1.Itisclearly
shownthatthelinewithnmax =1doesnotfitthedataverywell,butthe
50 100 150 200 250 300 350
lineswithnmax =2andnmax =3fitthedataequallywell.Thecontri-
butionfromthethirdharmonictermsintheFourierseriesisverysmall.The Velocity (km/s)
motionofthesubpeakintheredabsorptiontroughisassociatednotonly
withtherotationalperiodofSUAur(P),butalsowithhalfoftheperiod
Figure5.Thetime-seriesspectraofthePaβfromthesecondnight(solid)
(P/2).Atypicalsizeoftheuncertaintyinthepeakpositionsisindicatedon
andthequadraticfitsforthesubpeakpositions(dashed).Eachspectrumis
thelower-leftcorner.
theaverageof27consecutivespectra(∆t=70min)inchronologicalorder
fromthetoptobottom.Eachspectrumisverticallydisplacedby-0.1from
whereq = R∗/Rc.Usingthemass(M∗ = 2.25M⊙)andthera- trheed-pwrienvgioaupspesparescttroumbefmorovcilnargitfyr.oTmhVep∼osi1ti7o0nkomfas−w1eatok∼sub2p2e0akkmins−th1e
dius(R∗ = 3.6R⊙)ofSUAurfromCohen&Kuhi(1979)along
duringthecourseofthesecondnight.Thepositionsofthesubpeakwere
with the co-rotation radius (Rc = 3R∗) in equation 6, we find estimatedusingtheoriginalspectra(beforebeingaveragedover27spectra),
tff = 10.5h.Thistime-scaleiscomparabletotheobservingtime byapplyingaquadraticfittothedatapointsaroundthesubpeak(Fig.4).
spanofagivennight,sotheassociationofthesubpeakmotionon
Thearrowsindicatethemeasuredpositionsofthesubpeak.
thesecond night withasingleepisodicevent cannot beruledout
by timescale arguments alone. However, the systematic trends in
theaccelerationofthefeatureonthefirstandthirdnightsstrongly The map is useful to visually identify velocity bin regions
suggeststhatthefeatureisproducedbyrotationalmodulation.To thatcorrelateoranti-correlatewitheachother(e.g.Johns&Basri
excludethepossibility,theobjectmustbeobservedforafewrota- 1995; Oliveiraetal. 2000). The figure shows the profile vari-
tionalperiods. ability in the velocity range (−200kms−1,0kms−1) weakly
correlates with that for (200kms−1,400kms−1). On the other
hand, they seem to weakly anti-correlate with the variability in
3.3 Auto-correlationmap (0kms−1,100kms−1).Thepatternseeninthemapisverysimi-
lartothatseenintheHβauto-correlationfunctionofOliveiraetal.
Theauto-correlationmapforthePaβspectra(withall503spectra)
(2000).Theyalsofoundthreesimilarwavelengthrangesinwhich
was calculated toexamine ifvariability at a given velocity bin is
thefluxlevelscorrelateandanti-correlatewithoneanother.
correlatedwiththoseatadifferentpartofspectra.Theresultwas
The auto-correlation maps with time lags of P/2, P/3 and
placed in Fig.6 asa greyscale image. Thecorrelation coefficient
P/4 where P is the rotational period were also computed (with
value (C ) of the i-th row and the j-th column in the map was
ij P =2.7dfromUnruhetal.2004).Becauseofthepoortimecov-
computedbyusing:
erages(3nightsatasinglelocation)ofthedata,wewereunableto
N S −S¯ S −S¯ drawanysignificantconclusionfromthosemaps,andtheyarenot
Cij = Pm=1(cid:0)Nmi S i(cid:1)−(cid:0)S¯mj2 j(cid:1) (7) shownhere.
m=1 mi i
P (cid:0) (cid:1)
whereS isthesignalofthen-thvelocitybininthem-thspec-
mn 3.4 Lineequivalentwidth
trumof thetime-series,andN isthetotalnumber of spectra.S¯
n
isthemeansignalofthen-thvelocitybininthetime-series.The Thelineequivalentwidths(EWs)ofPaβprofiles(showninFig.1)
mapissymmetricaboutthediagonal(i = j pixels).Therangeof are computed by averaging/rebinning over 10 profiles obtained
thecorrelationvalues variesfrom−1to1whichcorrespond toa consecutively (approximately over 30 minutes), in order to de-
stronganti-correlationandastrongcorrelationrespectively. crease the size of variance due to the relatively low average S/N
6 R. Kurosawaetal.
Table1.Thesummaryofthefittingparametersinequation4forthepositionsofthesubpeakshowninFig.4.Nandχ2are
ν
thenumberofdatapointsandthereducedchi-squarerespectively.
nmax N χ2ν a0 a1 b1 a2 b2 a3 b3
1 503 3.5 224. −35.7 −13.2 ... ... ... ...
2 503 2.8 288. −46.2 36.6 −36.7 29.9 ... ...
3 503 2.8 234. −42.3 33.9 −39.4 28.2 −9.14 −1.06
but not on the third night. With much larger time steps (∼ 1d),
Johns&Basri(1995)foundasimilaranti-correlationbehaviourof
theHαEWsmeasuredusingthevelocitybinsV ∼−150kms−1
andthosemeasuredusingV ∼100kms−1.Thisisconsistentwith
theauto-correlationmap(Fig.6)shownearlier.
ThedatapointswerefittedwiththeFourierseries(equation4)
for nmax = 2 and nmax = 3 cases separately. As done for the
fittingofthethesubpeakpositionsinSection3.2,theperiodwas
constrainedtobe2.7d(Unruhetal.2004)inthefittingprocedure.
TheresultsarealsoshowninFig.7,andthecorrespondingFourier
coefficientsaresummarisedinTable2alongwiththereducedchi-
square(χ2)values.
ν
Forallcases(total,redandblueEWs),thelineswithnmax =
3arebetterrepresentationsofthedatapoints,aswecanseefrom
the figure and the χ2 values in the table. To test the validity of
ν
the statement above, we performed an F-test. For the total EWs
fits, the numbers of degrees of freedom are ν2 = 42 and ν3 =
40 for nmax = 2 and nmax = 3 (Table 2) respectively. Using
the values of the reduced chi-square values in the table, we find
Fχ = χ2(nmax =2)−χ2(nmax =3) /χ2ν2 = 37 and F =
χ2 /χ2(cid:8)=8.1;hence,F >F.Fromth(cid:9)istest,wefoundthatthe
ν2 ν3 χ
inclusionofthethirdharmonictermsarearealimprovementinthe
fittingandtheyshouldbeincluded.Similarconclusionswerefound
Figure6.Auto-correlationmapofthePaβspectra(3nights)withnotime fortheblueandtheredEWvariabilitycurves.
delay.Thestrongerthecorrelation,thebrighterthepixelsinthemap.Sim- Interestingly,thecontributionofthethirdharmonictermsare
ilarly, thethestronger theanti-correlation, thedarker thepixels are.The
asimportantasthefirstandthesecondharmonictermsasonecan
fluxchangesseenbetween(−200kms−1,0kms−1)and(200kms−1,
400kms−1)arecorrelated witheachother.Ontheotherhand,theflux seefromthevaluesof theFourier coefficientsinTable2.Thisis
consistentwiththeideathatvariationiscausedbythecombination
changesseenbetweenthosewavelengthrangesweaklyanti-correlatewith
thefluxchangesseenin(0kms−1,100kms−1). ofthetilted-axismagnetospheric accretionmodel (forthesecond
harmonicterms)andthepresenceofthecoolspotsofthesurfaceat
threedifferentlongitudeswhichareapproximatelyequallyspaced
(∼ 90) intheoriginal profiles.Threedifferent ranges of velocity fromthesurfaceDopplerimages(forthethirdharmonicterms),as
brieflymentionedinSection3.2.
binswereusedforcomputingtheEWs.ThetotalEWswerecom-
putedusingthevelocityrange−500kms−1 <V <500kms−1. Since thedata sampling span of one night isvery similar to
P/3,itisverydifficulttodistinguishtheP/3componentfoundin
The EWs of the red wing (the red EWs) were computed using
0kms−1 < V < 500kms−1, andthose of thebluewing(the the Fourier analysis from a spurious detection. To overcome this
blue EWs) with −500 kms−1 < V < 0 kms−1. The results shortcoming,asimilarobservationmustbeperformedatmultiple
sites(forcontinuous phase coverage) for at least afew rotational
areshown inFig. 7. During the firstnight, thetotal EW changes
between−0.1and−0.8A˚.Itchangesbetween+0.1and−0.4A˚ periods.Also,authorswouldliketoremindreadersthatthesurface
Doppler images of SU Aur obtained by Petrovetal. (1996) and
onthesecondnight,andduringthethirdnightitchangesbetween
−0.7 and −1.9A˚. The negative sign in the EW values indicates Mennessier (1997) suffer from the same data sampling problem
as ours. The surface Doppler images constructed by Unruhetal.
that the lineisinemission. Fromnight to night, theaverage EW
changesfrom−0.4to−0.2A˚,andthento−1.5A˚. (2004),ontheotherhand,arebasedonthemulti-siteobservations
whichhasafairlycontinuous phase coverage and for afew peri-
As we can see from this figure and also from the temporal
ods;however,theyhassignificantproblemswithnon-periodicline
variance spectrainFig.1,theamplitudeof thevariabilityforthe
variability.
redwing(ortheredEW)ismuchlargerthanthatforthebluewing;
hence,thebasicbehaviourofthetotalEWvariabilityfollowsthat
oftheredwing(theredEW).
According to the figure, the variability of the total EW and
4 MODELS
that of the red EW appear to be correlated with each other. On
other hand, the variability of the total EW and that of the blue These data, along with other spectroscopic time-series observa-
wingappeartobeanti-correlatedforthefirstandthesecondnight, tions, provide some strong constraints on the possible magneto-
Time-series Paschen-β spectroscopyofSU Aurigae 7
Table2. Thesummaryofthefittingparametersinequation4fortheequivalentwidth(EW)datashowninFig.7.N and
χ2 arethenumberofdatapointsandthereducedchi-squarerespectively.Thevelocitybinsusedforthetotal,red,andblue
ν
EWsare−500kms−1 < V < 500kms−1,0kms−1 < V < 500kms−1,and−500kms−1 < V < 0kms−1
respectively.
nmax N χ2ν a0 a1 b1 a2 b2 a3 b3
Total 2 48 3.1 −1.20 0.39 −0.06 −0.24 0.41 ... ...
Total 3 48 0.4 −0.36 0.59 0.43 −0.52 0.20 0.18 −0.72
Red 2 48 7.2 −1.21 −0.21 0.06 0.29 −0.25 ... ...
Red 3 48 0.8 −1.17 −0.22 −0.09 0.26 −0.11 −0.39 0.31
Blue 2 48 0.7 2.38 −0.17 −0.20 0.01 −0.23 ... ...
Blue 3 48 0.2 1.55 −0.37 −0.34 0.25 −0.09 0.21 0.39
star.ThischaracteristicisalsorecoveredinsurfaceDopplerimages
n = 2 byPetrovetal.(1996),Mennessier(1997)andUnruhetal.(2004),
max
1 andisindicativeofthepresenceofcoolspotsonthesurfaceatthree
differentlongitudeswhichareapproximatelyequallyseparated.
TodatetheobservationalphenomenaassociatedwithSUAur
Å) 0 havebeeninterpretedusingcartoon-likemodelsofthecircumstel-
W ( lar geometry and dynamics. Although useful, it is not clear that
E these necessarily simplistic descriptions provide a reasonable ex-
-1 planation of the changes in line-profile shape that are observed.
However, it isalsotruethat thenear-star geometry issufficiently
complicatedthat adetailedfittotheobservations isnot currently
-2 tractable,andwillprobablyrequireacombinationofsimultaneous
time-series observations (spectroscopy, circular polarimetry and
n = 3 photometry) spanning a wide range of wavelength. In this sec-
max
tion we adopt a modelling approach that, while falling short of
1
a formal fit, provides a greater quantitative insight into the line
profile variability than a cartoon description. Our intention is to
Å) 0 develop radiative-transfer models based on the simple magneto-
W ( spheric geometries that have been proposed in previous studies
(Johns&Basri1995;Petrovetal.1996).Wewillbeabletodeter-
E
minewhether or not thesegeometriesarecapableof reproducing
-1
thegrosscharacteristicsofthelineprofilevariability,andtherefore
makeabetterassessmentoftheapplicabilityofthemodels.
-2
10 11 12 4.1 Themodellingcode
HJD - 2452600.0 (day) The three-dimensional Monte Carlo radiative trans-
fer code TORUS (Harries 2000; Kurosawaetal. 2004;
Symington,Harries,&Kurosawa 2004a) is used to compute
Figure 7. The equivalent widths (EW) of Paβ using the velocity bins
−500kms−1 < V < 500kms−1 (circles), −500kms−1 < V < the Paβ line profiles as a function of time (rotational phase).
0kms−1(diamonds)and0kms−1 < V < 500kms−1(squares)are First, the model computes the non-LTE populations of 14-level
plottedasafunctiontime.Thetoppanelshowsthedatafitsusingequation4 Hydrogen atoms in the magnetosphere which is funnelling the
withnmax = 2,andthebottompanelshowsthosewithnmax = 3.The gas through the magnetic field lines from the inner edge of
fitsforthetotalEW(−500kms−1 <V < 500kms−1),theblueEW the accretion disc. Second, the model computes the observed
(−500kms−1 < V < 0kms−1)andtheredEW(0kms−1 < V < profile of Paβ as a function of rotational phase. In our models,
500kms−1)areshowninsolid,dashedanddash-dotlinesrespectively. the Sobolev approximation (c.f. Mihalas 1978) is used in the
Therotationalperiod,2.7d(Unruhetal.2004),waskeptconstantinthe framework of core-plus-halo Monte Carlo radiative transfer
fittingprocedure.Table2summarisesthefittingparameters.
method, in which the photosphere is treated separately from the
outer atmosphere (e.g. accretion streams) of a star. Readers are
referred to Symingtonetal. (2004a) for a detailed description
sphericgeometryofSUAur.Specifically,inSection3.2,wehave of the spectroscopic model of hydrogen lines arising from the
shownthatthemotionofthesubpeakintheredabsorptiontroughis accretionstreamsinthemagnetosphere. Alloftheaccretionflow
relatednotonlytotherotationalperiodofthestar,butalsotohalfof is assumed to be constant in our models i.e. there is no periodic
therotationalperiod.Thisphenomenonisconsistentwiththetilted- pulsationofdensityenhancementetc.
magnetic axis models (e.g. see Shuetal. 1994a; Johns&Basri No rotational velocity component is included in our models
1995)asmentionedearlier.Furthermore,theanalysisinSection3.4 asitwasneglectedbyHartmannetal.(1994)andSymingtonetal.
suggests that the variability of the Paβ equivalent width is asso- (2004a). The effect of the rotation on Paβ is expected to be
ciated with a half and and a third of the rotational period of the small according to the calculations of Muzerolleetal. (2001, see
8 R. Kurosawaetal.
their fig. 8). A slightly larger amount of rotational broadening z Rotational axis
To observer
than that of Muzerolleetal. (2001) is expected for SUAur since
vsini ∼ 60kms−1 (Johns-Krull1996;Unruhetal.2004)while Magnetic axis
Muzerolleetal.(2001)usedvsini ≈ 10kms−1 intheircalcula- i
θ
tion.Interestingly,Oliveiraetal.(2000)foundthetime-laggedbe- m
haviourinthevariabilityofsomeopticallines,andproposedthat
thiscanbecausedbythepresenceoftheazimuthalcomponentin p
themagnetospherewhichmightbecausedbytheinteractionofthe r
rotating magnetosphere with the circumstellar disc. Although the θ
effectoftherotationonthePaβlinemaybeimportantforaquanti-
o
tativemeasurement,itshouldnotaffecttheconclusiondrawnfrom x
thequalitative analysispresented in thissection. Theexact effect
oftherotationonthelineformationandvariabilitywillbeinvesti-
gatedmorecarefullyinafuturepaper.
R
4.2 Modelconfigurations Magnetic field line m
A star with radius R∗ and mass M∗ is placed at the origin (O)
of a Cartesiancoordinate system (x,y,z)as shown inFig. 8. The
rotational axis of the star is set to be identical to the z-axis, and
thesenseofrotationiscounter-clockwisewhenthestarisviewed Figure 8.Basic model configuration. Astar radius R∗ andmass M∗ is
pole-on.They-axisisperpendicularandintothepage.Anobserver placedattheorigin(O)ofaCartesiancoordinatesystem(x,y,z).Therota-
is placed on the z-x plane with an inclination angle i measured tionalaxisofthestarisidenticaltothez-axis,andthesenseofrotationis
counter-clockwisewhenthestarisviewedpole-on(viewedfrom+zdirec-
fromthez-axis.Thestructureofthemagnetosphereisthesameas
tion).They-axisisperpendicularandintothepage.Anobserverisplaced
thatofHartmannetal.(1994)i.e.theshapeofthedipole(poloidal)
onthe z-xplane with an inclination angle imeasured from z-axis. The
magneticfieldlinesisdescribedby
magneticaxisistiltedbyasmallangleθmwithrespecttotherotational
r=Rm sin2θ (8) arx=is.RTmhessihna2peθoafstihneed.gip.oHlear(tpmoalonindaelt)aml.a(g1n9e9t4ic).fieldlinesisdescribedby
wherer,Rmandθaretheradialdistancefromthecentreofthestar
toapoint(p)alongafieldline,thedistancetothepointwherethe
fieldlineintersectswiththeequatorial plane, andthepolar angle
measured from the magnetic axis (normally z-axis) respectively. i = 60o
Themagneticaxiscanbeinclinedbyasmallangleθmwithrespect i = 70o
therotationalaxis.Weconsiderthefollowingthreemodels: i = 80o
1.4 i = 85o
ModelA A star is surrounded by the axi-symmetric magnetic i = 90o
field(describedbyequation8)withconstantaccretion
x
flow along themagnetic fieldlines, but the magnetic u
axistiltedby10◦withrespecttotherotationalaxis. d Fl
e
ModelB A star is surrounded by the axi-symmetric magnetic aliz 1.2
field(describedbyequation8)exceptthreethin(10◦) m
or
gaps,wherethereisnomagneticfieldandgas,arelon- N
gitudinally placed with equal separations (i.e. 120◦).
Themagneticaxisisalignedwiththerotationalaxis.
1
ModelC AcombinationofModelAandModelB.Themagne-
tospherehasthree10◦ gaps,andthemagneticaxisis
tiltedby10◦withrespecttotherotationalaxis.
Inallthreemodels,wehaveadoptedthefollowingstellarpa-
-400 -200 0 200 400
rameters:themassM∗ = 2.25M⊙ andtheradiusR∗ = 3.6R⊙
(Cohen&Kuhi 1979). The magnetospheric radius (Rm) is as- Velocity (km/s)
sumedtobecomparabletothesizeoftheco-rotationradius(Rc)of
thestar(e.g.Pringle&Rees1972;Ghosh&Lamb1979;Shuetal.
Figure9.ThePaβlineprofilescomputedatfivedifferentinclinations(i=
1994b;Romanovaetal.2002).Rc ≈3.0R∗forR∗andM∗values 90◦,85◦,70◦,60◦)using the samemagnetosphere as inModel A, but
givenabove.Therangeofthemagnetosphericradiusischosento withouttiltingthemagneticaxisi.e.θm =0.Thesubpeakintheredwing
beRm =2.5−3.5R∗,whichisinproportiontothesmall/narrow appears in emission only at the high inclination (i >∼ 85◦). At lower
modelofMuzerolleetal.(2001).Thetemperaturestructurealong inclinations(i<∼70◦),thefluxlevelofthebluesideoftheprofileisvery
themagneticfieldwasadoptedfromMuzerolleetal.(2001) with sensitivetochangeininclination. Ontheotherhand,athigherinclination
themaximumtemperature(Tmax)of8000K(seetheirfig.2).The angles (i >∼ 70◦),theyareinsensitive tothechange intheinclination
angle.
stellar continuum of the core star is described by a model atmo-
sphereofKurucz(1979)withTeff =5750Kandlogg=4(cgs).
Asimplegeometricallythinandopticallythickaccretiondisc
isplacedjustoutsideoftheouteredgeofthemagnetosphere. All
Time-series Paschen-β spectroscopyofSU Aurigae 9
shapeand relativeintensityof theobservations (Fig.1),although
Table3.Modelparameters
themodellineprofilesarerathermoretriangular.Littledifference
Model θm #ofgaps M˙ is seen in the shape of the mean spectra from three models. The
(◦) 10−7M⊙yr−1 amnodde∼ls−ov1e0r0esktimmast−e1t)hecolemvpelasreodftthoetbhleueo-bwseinrvga(tbioentw. Teehne∼dis−cr2e0p0-
(cid:0) (cid:1)
A 10 0 1.15 ancycouldbecausedbye.g.thewronginclinationangle,thewrong
B 0 3 1.25 geometryofthemagneticfieldlineusedinthemodel,andthefact
C 10 3 1.15 thatthemodeldoesnotincludetherotationalvelocitycomponentin
theaccretionstreams.Ifalower(e.g.60◦)inclinationangleisused,
theprofilewillbecomelesstriangular.Althoughalowerinclination
the photons which encounter the discare absorbed. Withthisas-
modelwillgiveusabetterfittotheobservedmeanprofile,thevari-
sumptionofthedisc,aphotonemittedfromthebottomhalfofthe
abilityintheredabsorptiontroughwillbemuchsmallerthanthat
magnetosphere can beocculted by thedisc, but aphoton emitted
oftheobservation,andthevariabilityinthebluesidewillbemuch
fromthetophalfcannotbeoccultedbythedisc.Thisisreasonable
larger than that of the observation. The mean spectra from Mod-
forcomputingPaβprofilesexceptforcaseswithveryhighinclina- elsAandCshowthesubpeakintheredabsorptiontroughsimilar
tionangles.Ifamorerealisticdisc(e.g.aflareddisc)wasusedin
totheoneinseenintheobservation;howeverthepositionsofthe
themodel,wewouldexpectthereductionofthephotosphericcon- subpeakinbothmodelsare∼100kms−1smallerthanthatofthe
tinuum(e.g.Chiang&Goldreich1999)aswellasthelineemission
observedspectra.
duetoobscurationbytheouterpartofthedisc.Predictingtheexact
InModelA,themagneticaxisisperpendiculartotheobserver
effectsonthelineprofileshapescausedbythistypeofobscuration
when phase = 0. Around this phase, the subpeak in the red ab-
requiresmodellingofaself-consistentaccretiondisc,whichisbe-
sorptiontroughbecomemostprominent(seenasbrighterpixelsin
yondthescopeofthispaper.
thegreyscaleimage).Thesubpeakarisesfromasimplegeometri-
Sincewedonothavetheestimateoftheamountoftheveiling
caleffect:athighinclinationtheobserver’sline-of-sight(LOS)to-
around Paβ line, the flux contribution from the disc itself is not
wardsthehotspotspassesthroughbothlowvelocitymaterialnear
consideredinourcalculations.Arecentmeasurementoftheveiling
thedisc,andhighvelocitymaterialnearthestellarsurface,leading
at2.2µmforSUAuris0.6±0.3(Muzerolleetal.2003),butthe
to two absorption components on the red side of the line profile.
valueshouldbesignificantlysmallerat1.3µm(Paβ).Iftheveiling
Therelativepositionandstrengthofthesecomponentschangesas
correctionwastakenintoaccount,thelinestrengthofthemodels
themagnetospherebecomesmoreface-on.Thelevelofthesubpeak
presentedinthispaperwouldbeweaker.Themass-accretionrates
shouldbecomelowerasthephasechangesfrom0to0.5because
(M˙) for the models were chosen so that the mean observed line
ofthisgeometrical/projectedvelocityeffect.Thegreyscaleimage
strengthisapproximatelyreproduced.
of Model A shows this effect. Overall line variabilityseen in the
The inclination is set to i = 80◦ which is higher than the
redabsorptiontroughiswellreproducedbyModelsAandC.Since
previouslyassumedvalues(e.g.Muzerolleetal.2003;Unruhetal.
thisvariabilityisprimarilyageometricallyeffect,itisquiteinsensi-
2004).Fig.9showsthePaβlineprofilescomputedatdifferentin-
tivetotheadoptedtemperaturestructure,andwefindqualitatively
clinationanglesusingthesamemagnetosphericaccretionstructure
similar behaviour for models with 6000K < Tmax < 10000K
asinModelA,butwithnodipoleoffseti.e.θm = 0.Atlowerin- (althoughnaturallythemodelsdifferindetail).
clinations(i<∼70◦),thefluxlevelofthebluesideoftheprofile
ThegreyscaleimageofModelBshowtheperiodicvariation
isverysensitivetochangesininclination.Thiswillcausetoomuch
(with1/3ofaphase)ofthefluxlevelsintheredabsorption.The
variabilityinthebluewingifθm = 10◦ (asinModelA);henceit fluxlevelintheredabsorptionincreasesasalongitudinalgapap-
willbeinconsistentwiththeobservation(c.f.Figs.1and3).Onthe
proachesthelineofsightoftheobserversincelinephotonssuffer
otherhand,athigherinclinationangles(i>∼70◦),theyareinsen-
less absorption. Finally, the greyscale of Model C shows a com-
sitivetothechangeintheinclinationangle.Interestingly,wefound
plicated variation pattern, but it is easy to see that the pattern is
thatthehigherinclinationangle(i>∼85◦ inFig.9)isneededto
essentiallyacombinationofthevariabilitypatternsfromModelsA
havethesubpeak intheredabsorptiontroughinemission(above
andB.
continuum)asseeninthemeanspectraofPaβfromthethirdnight
The temporal deviation spectra of Models A and C re-
oftheobservation(Fig.3).
produce approximately the same range of the line variability
In Models B and C, the locations of the gaps are at the az- (−400kms−1, 500kms−1)asseenintheobservations(Fig.1).
imuthangles105◦,225◦and345◦(measuredfrom+x-axisbefore Model B does not fit the (TVS)−1/2 of the absorption at all. In
tilting)attimet = 0orequivalentlyphase = 0.Forthemodels the(TVS)−1/2 ofModelsAandC,ratherlargediscrepanciesare
withthetiltedmagneticaxis(ModelsAandC),theinclinationof
seenintheamountofthevariabilityaroundthelinecentre(slightly
themagneticaxiswithrespecttotheobserverchangesfrom90◦to
ontheblueside). Whilethepeak ofthevariabilityintheredab-
70◦,thento90◦astherotationalphasechangesfrom0to0.5,then sorptionoccursat∼ 150kms−1 forModelAandC,itoccursat
to1.0.ThemodelparametersaresummarisedinTable3. ∼275kms−1intheobservation.
Apossiblewaytoshiftthepeakofthe(TVS)−1/2inthered
wing(equivalentlythepositionofthesubpeakinthemeanspectra)
4.3 Modelspectra
toahigher velocitybinistohavealargermagnetospheric radius
WehavecomputedthePaβspectraat50differentrotationalphases, (Rm),inwhichtheaccretiongashashigherinfallingvelocity.Al-
andplacedtheresultsinFig.10.Thefigureshowsthemeanspec- though thismeans themagnetic accretion streamshavetoextend
traofawholephase,thedeviationsfromthemeanspectra(quotient beyond the co-rotation radius of the star, this would be still con-
spectra)asafunctionofphaseinthegreyscaleimage,andthetem- sistentwiththemagneto-hydrodynamicmodelofRomanovaetal.
poraldeviationspectra(TVS)−1/2fromthemodels. (2002)whofoundatorque-lessaccretionispossiblewhenRm ≈
Themeanspectraofthemodelsbroadlyreproducetheoverall 1.5RcwhereRcistheco-rotationradiusofastar.
10 R. Kurosawaetal.
Figure10.ThesummaryofthePaβspectracomputedforModelA(left),ModelB(centre)andModelC(right).Foreachmodel,spectrawerecomputedat
50differentrotationalphases.Inthebottompanels,themeanspectraofallrotationalphasesareshown.Inthemiddlepanels,thequotientspectra(divided
bythemeanspectrum)areshownasgreyscaleimageswithincreasingrotationalphasesinverticaldirection.Thegreyscaleimageisscaledfrom1.1(white)
to0.9(black).Thetemporaldeviationspectra(TVS)−1/2areshowninthetoppanels.Whencomputingthetemporaldeviationspectra,thesignal-to-noise
ratioof90(incontinuum)wasusedtomatchthatoftheobservation(Section2).SeeTable3forthemodelparametersadopted.
4.4 Modelauto-correlationmaps
Theauto-correlationmapsofPaβline(withnotimelag)werecom-
putedusingthespectraofModelsA,BandC,andtheresultswere
placed in Fig. 11 along with the auto-correlation map shown in
Fig.6foracomparison.
In the map for Model A, the changes in the flux
levels in the velocity range (−200kms−1, 0kms−1) and
(0kms−1, 400kms−1) are strongly correlated with themselves
while they areanti-correlated witheach other. In other words, as
thefluxlevelin(−200kms−1,0kms−1)increasesthefluxlevel
in(0kms−1,400kms−1)decreases,andviceversa.
The blocks of correlation and anti-correlation (seen
as black and white squares in the map of Model A) are
less pronounced for Model B. The third quadrant block
(−300kms−1, 0kms−1)×(−300kms−1, 0kms−1) of
Model B is more complicated than that of Model A because the
accretionstreamslocatedonthefar(back)sideofthestar(asseen
bytheobserver)willbeseenthroughthegapasitapproachesthe
near (front) side of the star. The map for Model C is essentially
thesameasthatofModelAsincetheoverallvariabilitycausedby
thepresenceofthegapsisrelativelysmallcomparedtothatbythe Figure 11. Auto-correlation maps of Paβ from the observation (upper–
precessionofthemagneticaxis. left) and from the radiative transfer models: Models A (upper–right), B
(lower–left) andC(lower–right). Thevelocity bins ofthecontinuum are
The main differences between the auto-correlation map of
excluded from the model auto-correlation maps, and their values are set
the observation and the models are: 1. the absence of the anti-
correlation between the velocity ranges (0kms−1, 100kms−1) troan0ge(sse(e0nkamsgsr−ay1,o1u0t0erkfmrams−es1)).Nanoda(n2t0i-0cokrmrelast−io1n,4b0e0twkemens−th1e)vieslosceietyn
and(200kms−1,400kms−1)inthemodelsand2.ForModelsA
inthemodelsunliketheobservation.Thepositionofthetwodarkstripes
and C, the two dark stripes (anti-correlation bands) crossing per- (anti-correlationbands)crossingperpendiculartoeachotherisontheblue
pendicular to each other are about two times wider than that of side(−125kms−1and−125kms−1)forModelsAandC,butitison
theobservation,andtheyappearastwolargedarkblocksandtwo theredside(100kms−1and100kms−1),intheobservation.
smallerblocks.Thetotalphasecoverageoftheobservationisap-
proximately1/3ofthewholephase;therefore,thismayalsocon-
tributetothedifferenceseeninthemapsfromtheobservationand
themodels.