Be-star rotation: how close to critical?

0.28 MB·English

by  Richard H. D. Townsend

Mon.Not.R.Astron.Soc.000,1–8(2003) Printed2February2008 (MNLATEXstylefilev2.2)
Be-star rotation: how close to critical?
R. H. D. Townsend1,2⋆, S. P. Owocki2,1 & I. D. Howarth1
1 Department of Physics & Astronomy, UniversityCollege London, Gower Street, London WC1E 6BT, UK
2 Bartol Research Institute, Universityof Delaware, Newark, DE 19716, USA
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Received: ....................................Accepted: ....................................
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2

n ABSTRACT
a We argue that, in general, observational studies of Be-star rotation have paid insuf-
J
ficient attention to the effects of equatorial gravity darkening. We present new line-

0 profile calculations that emphasize the insensitivity of line width to rotation for fast
2
rotators.Coupledwithacriticalreviewofobservationalprocedures,thesecalculations

suggestthattheobservationalparameterv sinimaysystematicallyunderestimatethe
2
v true projected equatorialrotationvelocity, ve sini, by some tens of per cent for rapid
rotators.AcrucialimplicationofthisworkisthatBestarsmayberotatingmuchcloser
3
1 totheircriticalvelocitiesthanisgenerallysupposed,bringingarangeofnewprocesses
1 into contention for the elusive physical mechanism responsible for the circumstellar
2 disk thought to be central to the Be phenomenon.
1
3 Keywords: stars:emission-line,Be–stars:rotation–stars:fundamentalparameters
0 – line: profiles – techniques: spectroscopic
/
h
p
-
o
1 INTRODUCTION (Owocki2002).However,ifthisistobeapromisingavenue
r
t of exploration, then rotation must be much closer to crit-
s Be stars are near-main-sequence, B-type stars that have
a ical than is generally supposed at present: at ve/vc ≃ 0.7
beenobservedtoshowBalmer-lineemission.Itisapproach-
: the equatorial gravity is reduced by only about a factor
v ing a century and a half since the discovery of the first Be
i star (Secchi 1867), and in recent decades compelling em- 2, and it requires ve/vc ≃ 0.95 to reduce geff by an order
X of magnitude. Alternatively, to launch material into orbit
pirical evidence has associated the emission-line episodes
ar withtheformationofquasi-Keplerian,equatorialdisks(e.g. b10a0llikstmicsa−ll1y;laatunvec/hvvcel=oci0t.i7esrceoqmuimreesnvsuelroactietiwesitihntehxecseosusnodf
Dachs et al. 1986; Hanuschik 1996). However, there is still
speed(atwhichtheaforementionedweakprocessestypically
no general agreement on the underlying physical mecha-
operate)areachievedonlywhenarotationrateve/vc ≃0.95
nisms responsible for disk formation (Smith et al. 2000).
isreached.Similarargumentsmaybebroughttobearonthe
A phenomenologically crucial characteristic of Be stars
angular momentum budget needed to achieve orbit.
is that they rotate rapidly – more rapidly than any other
In the present paper we review thebases for determin-
class of nondegenerate star. Virtually as soon as this was
ing the true projected equatorial rotation velocity, ve sini,
recognized,itwashypothesizedthattheequatorialrotation
and the relationship to its observational counterpart v sini
velocities, ve, may be sufficiently close to the critical ve- (whereiis theanglebetween thelineofsight and therota-
locity, vc, for material easily to ‘leak’ into a disk (Struve tion axis). Weargue that equatorial gravity darkeningmay
1931; here vc is the velocity at which centrifugal forces plausibly have led to rotational velocities of Be stars be-
balance Newtonian gravity at the equator, so that the ef-
ing systematically underestimated, and that Struve’s origi-
fective gravity, geff, vanishes). Slettebak and others later nalhypothesisof near-critical rotation in Bestarstherefore
scrutinized this hypothesis quantitatively (e.g., Slettebak
has not been ruled out byexisting observational studies.
1949, 1966; Slettebak et al. 1992), establishing the current
canonical view that Be stars’ rotation is actually signifi-
cantly subcritical, with ve/vc ≃ 0.7–0.8 (e.g., Porter 1996;
Chauville et al. 2001). 2 PREAMBLE
Ifrotationdoesplayadirect,causalroleintheBephe-
For a centrally condensed star, the critical velocity of rota-
nomenon,thenitistemptingtospeculatethatitdoessoby
tion is given by
reducing the effective equatorial gravity to an extent that
allows ‘weak’ processes to move material into orbit easily vc = GM∗/Re = 2GM∗/3Rp, (1)
p p
where ‘e’ and ‘p’ subscripts are used throughout to denote
⋆ [email protected] equatorial and polar values, respectively (throughout this
2 R. H. D. Townsend et al.
paperweassumeuniformangularvelocityatthestellarsur-
Table1.FundamentalparametersfortheB-typemodelsconsid-
face). We note that, even if it were possible to measure the eredherein.
equatorial rotation velocity with complete accuracy, these
Sp. M∗ Rp vc logL∗
expressions imply considerable observational uncertaintyin Subtype (M⊙) (R⊙) (kms−1) (dexL⊙)
establishing vc. In particular, it is customary to adopt
massesandradiifromlook-uptables,onthebasisofspectral B0 17.5 7.7 538 4.64
type; and yet not only is the spectral type affected by ro- B0.5 14.6 6.9 519 4.41
B1 12.5 6.3 502 4.21
tation,inanaspect-dependentmanner(e.g.Slettebak et al.
B1.5 10.8 5.7 491 4.01
1980), but the physical relationship between spectral type,
B2 9.6 5.4 475 3.85
mass,andradiusisalsomodified(Maeder & Meynet2000).
B2.5 8.6 5.0 468 3.68
Thus systematic errors in ve/vc may easily arise from this B3 7.7 4.7 456 3.52
source alone. B4 6.4 4.2 440 3.24
The primary focus here, however, is on the determina- B5 5.5 3.8 429 3.00
tion of ve sini. Almost all published surveys of rotational B6 4.8 3.5 418 2.78
velocities in Be stars rely on one of two approaches. The B7 4.2 3.2 408 2.56
first is to infer ve sini values from, typically, line width at B8 3.8 3.0 401 2.39
halfdepth,calibratingtheresultsthroughthestandard-star B9 3.4 2.8 393 2.20
systemestablishedbySlettebak et al.(1975)oritsextension
torapidrotators(Slettebak1982).Thesecondistocompare
observationswith‘spun-up’versionsofnarrow-linestars1 or
model-atmospherefluxspectra,typicallyusingaconvolution
approach (cf.Gray1992).Weemphasizethatthelatterap- aretakenfromPorter(1996),withthecriticalvelocityeval-
proach is unphysical: the convolution between intrinsic line uatedusingequation(1).Toobtaintheaccompanyinglumi-
profileandDopplerbroadeningfunctionassumesthetwoto nosity data we use the Warsaw–New Jersey code3 to calcu-
beindependent,whichisclearly notthecasewhenthestel- late evolutionary tracks at the masses tabulated by Porter,
lar surface parameters (i.e., Teff & geff) vary with latitude determiningL∗fromtheepochatwhichtheradiuscoincides
as a result of gravity darkening. with Porter’s values.
Both techniques imply the assumption of a single- Foreachofthethirteenstellarmodels,weconstructtwo
valued, (approximately) linear relationship between line sequences, differing only in thetreatment of thecentrifugal
width and ve sini. Aswe demonstrate below, thetrue situ- force due to rotation. For the ‘complete’ set of models, the
ation is significantly more complicated. In particular, for a effects of this force are fully accounted for: the surface ge-
rigidlyrotatingstarwithradiativeenvelopethatfollows the ometry is that of a gravitational equipotential within the
von Zeipel (1924) form for gravity darkening, Roche approximation, and the surface temperature distri-
bution follows the von Zeipel (1924) gravity-darkening law
Teff ∝ge0ff.25, (2) appropriate to radiative envelopes (cf. equation 2). The lu-
the contribution of the cooler (and hence fainter) equato- minosityL∗ ispreservedunderchangingmodelgeometryby
adjustingthepolareffectivetemperatureinaccordancewith
rialregionstotheobservedspectrummaybereducedtothe
therelation
point where they are effectively invisible. A necessary con-
wseiqtuheonucteporfopsuecrhabtteehnatvioionutroisgtrhavatityv dsianrikevnailnugeswdielltesrymstienmed- Tp = gpL∗ 0.25; (3)
atically underestimate true projected equatorial velocities. (cid:18)σΣ1 (cid:19)
This point is developed in the following sections, through
here, σ is the Stefan-Boltzmann constant, and Σ1 is the
the use of illustrative calculations and by a critical review
surface-area weighted gravity of the rotating star (e.g.,
of observational studies.
Cranmer 1996).
Incontrast,forthe‘uniform’setofmodelsthecentrifu-
gal force is neglected, so that the star remains spherical in
3 MODELS shape,andexhibitsauniformsurfacetemperatureandgrav-
ity; these models mimic the traditional methodology devel-
InTable1weintroduceparametersforthirteenstellarmod- oped by Carroll (1928) and Shajn & Struve (1929), which
els,representativeofthespectraltypes2spanningtheB0-B9 isthebasisfortheconvolution approachdescribed byGray
interval.ThemassandpolarradiusofthemodelsinTable1 (1992)andusedbymanyauthors(e.g.,Howarth et al.1997;
Chauville et al. 2001).
1 Notnecessarilycorrespondingtoslowly-rotatingstars
2 Westressthathere,andthroughout,ouruseoftheterm‘spec-
tral type’ in reference to our models is little more than a no-
tational convenience: although, for instance, a non-rotating star
withthesameparametersasourB0modelwillexhibitaspectrum
loosely resembling a star with MK type B0, it should always be 3 DetailsofthiscodearegivenbyDziembowskietal.(1993)and
rememberedthattheMKscheme(e.g.,Morgan&Keenan1973) Dziembowski&Pamyatnykh (1993), the only significant differ-
is two-dimensional, and variations ina thirdparameter, such as enceinthepresentworkbeingtheadoptionofmore-recentopal
rotation,negate anydirectcorrespondence withphysicalparam- tabulations foropacity (Iglesias&Rogers 1996)and equation of
eters. state(Rogersetal.1996).
Be-star rotation: how close to critical? 3
bySlettebak et al.1992),andalternativeparameterizations
oftheeffectsof rotation (e.g., fixingthepolar temperature,
ratherthanvaryingitinaccordancewithequation3).These
variantsintroduceonlysmallquantitativechangesinthere-
sults.
4.2 Results: B2 subtype
WefirstconsiderthesurfacemodelsfortheB2subtype,the
most common in the sample of known Be stars (see, e.g.,
Porter 1996). Fig. 1 shows the characteristic widths of the
Heiλ4471˚Alineasafunctionofve sini,ateachofthefour
sinivaluesconsideredherein,forboththeuniform(∆u)and
complete(∆c)models.Wenotethatthefoursolidlineseach
spanadifferentve sinirangebecausethedomainve =[0,vc]
has been scaled by the four different sini values; a similar
Figure 1.Thecharacteristicwidths,∆,oftheHeiλ4471˚Aline effect is not apparent in the dotted lines, because the ∆u
profile for the B2 subtype, plotted as a function of ve sini for curves of the uniform models are self-similar4, and lie atop
the two models and four separate inclinations considered in the
oneanother. Thelatter result is acorollary of thefact that
text. The width data of the complete model (solid lines) are la-
linewidthsfortheuniformmodelfollowalinearrelationship
beledwiththeirassociatedsinivalues;thoseoftheuniformmodel
(dotted line) alllieonthe same curve. For ve sini.0.05vc,the with ve sini.
intrinsiclinebroadeningandblendedcomponentsdominaterota- Unlike the uniform models, the characteristic width
tionaleffects,andthecharacteristicwidthsshouldbedisregarded. curves for the gravity-darkened ‘complete’ models (which
furnish a better representation of real stars) are not self-
similar, an embodiment of departures from a linear ∆c-
4 SPECTROSCOPIC SIMULATIONS ve sinirelationship.Suchdeparturesarisefromanumberof
effects, including thevariations of both line and continuum
4.1 Method strengthwithlatitude,butthedominantfactorforfastrota-
torsisthedecrease incontinuumemission neartheequator
For both sets of surface models, we synthesize profiles for
resulting from von Zeipel darkening. This leads to a rela-
Hei λ4471 and Mgii λ4481˚A (including doublet and for-
tively small contribution from the fast-rotating equatorial
bidden components), the lines most commonly used by ob-
regions to thespatially integrated spectrum, andin turnto
servers to determine B-star v sini values. The calculations
asystematicreductioninlinewidthsrelativetotheuniform
use the bruce code (Townsend 1997), which is readily
model.Thusifv sinivaluesarecalculatedusingthecalibra-
adapted from its original intended application in modeling
tion found for the uniform model, they may underestimate
nonradial pulsation by simply setting the pulsation ampli-
the true value of ve sini by ∼20%, or more. Moreover, the
tudesto zero.
near-invisibility oftheequatorialregions in near-critical ro-
A large grid of precomputed conventional model spec-
traisrequiredbybruce,todescribetheangularandwave- tatorsmeansthatincreasing ve has almost no effect on line
width for such stars.
length dependences of the photospheric radiative intensity
We illustrate this line-width redundancy in Fig. 2, for
povoeler atnhde erqaunagteoro.fWTeeffc,onlosgtrgucvtasluuechs aengcroiudnutesirnegdtbheetwsyeenn- models with ve =395 and 460kms−1 (0.83 and 0.97 vc, re-
spectively).Clearly,evenwithoureffectivelynoise-freesyn-
specspectral-synthesiscodebyI.HubenyandT.Lanz.For
theticdata,thehelium profilesarealmost indistinguishable
the B0–B5 models, the underlying atmospheric structure is
in terms of their widths; there is no practicable means of
calculated using the non-LTE tlusty code (Hubeny 1988;
ascertaining that the two profiles shown belong to models
Hubeny& Lanz 1995), under the assumption that hydro-
with equatorial velocities that differ by almost 20%. This
gen and helium are the only sources of line opacity; for the
underlines the point that the degeneracy of ∆c as a func-
latersubtypes,atmosphericmodelsaretakenfrom theline-
tion of ve must seriously compromise any attempt at devis-
blanketed LTE grids published by Kurucz(1993).
ingav sini(∆)calibration,andparticularlyasingle-valued,
The rotating-star spectral synthesis is undertaken over
near-linearcalibration,forstarsstronglyaffectedbygravity
arange of equatorial velocities 0<ve <vc,and at fourdif- darkening.
fering inclinations: sini = {0.25,0.50,0.75,1.00}. We char- AlsoshowninFig.2aretheMgiiλ4481˚Aprofileforthe
acterize the line widths of the resulting profiles by using sameve =395and460kms−1models.Theequivalentwidth
the first zero, ξ1, of the Fourier transform of the flux spec- (EW)ofthislineincreaseswithdecreasingtemperature(by
trum (Smith & Gray 1976); specifically, we use ∆ ≡ ξ−1.
1 afactortwofrompoletoequatorinourve =0.95vcmodels),
Thesecharacteristic widths∆ haveanearlylinear relation- unlike the Hei lines. This increase in relative line strength
ship with full-width at half depth, but in any case our re-
partly offsets the decline in continuum flux (which falls by
sults are largely insensitive to theparticular approach used
factor of nearly four, from pole to equator), and as a result
to characterize line width.
Toexaminetherobustnessofourcalculations, wecom-
puted a number of variants on the basic models, including 4 By this, we mean that one curve can be transformed into an-
different choices of stellar parameters (e.g., those published otherbyscalingboth∆u andve sinibythesameamount
4 R. H. D. Townsend et al.
Figure 2. Hei λ4471 and Mgii λ4481˚A line profiles for equator-on ‘complete’ B2 models at ve = 395kms−1 (solid lines, left-hand
scales) and ve = 460kms−1 (dashed lines, right-hand scales). To facilitate the comparison of line widths and shapes, the models are
plottedonslightlydifferentverticalscales,chosensoastonormalizethelinedepths.
Table 2.Velocitydeficiencies δV,asapercentage ofthecritical for the B0 models, but grows to 7.0 for the B9 models. We
velocity vc, for the Hei λ4471 and Mgii λ4481˚A profiles of the note in passing that when bolometric fluxesare considered,
13modelsubtypes;ineachcase,ve/vc=0.95andi=90◦. the flux ratio is fixed at ≈ 14.7 by dint of the fact that
ve/vc =0.95, irrespective ofwhich modelisunderconsider-
Sp. δV/vc (%) ation.
Subtype Heiλ4471˚A Mgiiλ4481˚A
TheoverallincreaseinδV/vctowardlaterspectraltypes
B0 12.1 8.7 ismodulatedbythedependenceoflineEWontemperature.
B0.5 13.7 10.0 As discussed in the preceding section, the EW variation of
B1 15.4 11.0 the Mgii line tends to counteract the continuum gravity
B1.5 17.0 11.0
darkeningeffect.Thisexplainswhythevelocity deficiencies
B2 18.5 11.4 exhibitedbythelinearesmaller thanthoseoftheHeiline,
B2.5 20.0 11.5
B3 21.6 11.9 and also why the growth in δV/vc is seen to reverse itself
temporarily around subtypeB5.
B4 24.3 12.8
B5 24.6 12.0
B6 26.6 12.5
B7 28.9 14.0
B8 30.8 15.4 5 PHOTOMETRIC SIMULATIONS
B9 33.4 17.1
5.1 Method
the Mg line widths show a somewhat greater (though still In the preceding section, we have demonstrated how grav-
small) response to increasing equatorial rotation. ity darkening can make a near-critical star appear, when
observed spectroscopically, to be rotating at an ostensibly
slowerrate.Inthissection,webroadenourinvestigationby
4.3 Results: other subtypes
examiningtheimpactofgravitydarkeningonthephotomet-
Thesurfacemodelsfortheothersubtypesexhibitbehaviour ric signatures of rapidly rotating stars.
similar totheB2models discussed above:with theonset of Forthecompletesurfacemodels(cf.Section3),wesyn-
gravity darkening, the widths of line profiles saturate, and thesizeabsolutemagnitudesintheJohnsonB andV bands.
become insensitive to any further increase in the equato- The calculations use our rotcolor code, which is based
rial velocity. To quantify succinctly the degree of this ef- on thebruce code but oriented toward photometric rather
fect, we measure the line width for complete models at than spectroscopic modeling. rotcolor requires an input
sini = 1, ve/vc = 0.95, and then use the uniform models grid of precomputed photometric fluxes, which we obtain
to infer a v sini valuefor that width. Wedenotethe differ- by combining the four-coefficient limb-darkening data pub-
ence (ve sini−v sini) as a ‘velocity deficiency’, δV, which lished byClaret (2000) with normal-emergent absolute flux
we list in Table 2 for each of the13 subtypesconsidered. data made available to us by Claret (personal communica-
In both the Hei and Mgii lines, there is a systematic tion);bothdataarebasedontheKurucz(1993)atmosphere
increase in the velocity deficiency (expressed as a fraction grids.Toconvertsyntheticfluxesintoabsolutemagnitudes,
of vc) toward later spectral types. For the most part, this we use the observed values for the Sun, MB = 5.48 and
increasearisesfromanenhancementofthecontinuumgrav- MV =4.83 (Allen 1976).
itydarkeningeffect,asmeasuredbytheratiobetweenpolar Ourcalculationsareundertakenforfiverotationveloci-
and equatorial continuum fluxes: in the λλ4471–4481˚A re- tiesve/vc ={0.00,0.24,0.48,0.71,0.95}, andat threediffer-
gion appropriatetotheHeiandMgiilines,thisratio is3.7 inginclinationsi={0◦,45◦,90◦}.Forcomparisonpurposes,
Be-star rotation: how close to critical? 5
dening values correspond closely to the colour-magnitude
slope of themain sequencethroughout theBspectral type.
Therefore,foranensembleofrandomlyorientedBstars,the
effect of gravity darkening due to rotation at fixed ve/vc is
toshiftthemainsequenceupandtotherightinthecolour-
magnitude diagram, but to leave it otherwise undistorted.
Wewill return to this point in Section 6.3.
6 DISCUSSION
The calculations in Section 4 are intended to underline our
central point that line widths are a rather poor indicator
of ve sini at near-critical rotation. Although this basic idea
hasbeenpreviouslynotedintheliterature(Stoeckley1968;
Hardorp & Strittmatter 1968a; Collins & Truax 1995), in
practice it has been almost entirely ignored by observers
Figure 3.Selectedcompletesurfacemodels(cf.Section3)plot-
ted as points in the Johnson B−V colour-magnitude diagram, (cf.Howarth2002).Forexample,inthecourseofwritingup
atfiverotationvelocitiesve/vc={0.00,0.24,0.48,0.71,0.95}and this work we discovered that Stoeckley (1968, his Fig. 4)
threeinclinationsi={0◦,45◦,90◦}.Solidlinesjointogether the givesadiagram thatisverysimilartoourFig.1,whilealso
points from models that differ solely in their rotation velocity; addressing the implications for Be stars. However, the low
dottedlinesindicateZAMS–TAMSevolutionarytracksthatpass citation rate for this paper (at the time of writing, a sin-
through thenon-rotating models, thelatter beinglabeledatthe glecitation5 intheADSdatabase since1975) suggests that
bottom oftheplotwiththeirnominalspectraltypes.
the significance of this result for Be-star physics has been
largely overlooked. In this section, therefore, we critically
reviewthehistorical basisfor thecanonical viewthat ve/vc
wealso synthesizecoloursfortheevolutionarytrackscalcu-
does not exceed ∼0.8 for Be stars.
lated in Section 3; each track extends from zero-age main
sequence (ZAMS) through to terminal-age main sequence
(TAMS,definedbythecessationofcorehydrogenburning). 6.1 Line profiles
Thebasicmethodfor determinationsof v siniis, ofcourse,
line-profile measurements. Slettebak’s (1982) observational
5.2 Results
study of rotation in a sample of 163 Be stars has been
InFig. 3we plot theresultsof ourphotometric simulations particularly influential; many authors have relied directly
in a colour-magnitude diagram. The general effect of rota- on his v sini data (e.g., Porter 1996), or have used them
tionistodisplacemodelsawayfromtheirzero-rotationloci, to calibrate (e.g., Balona 1975; Halbedel 1996; Steele et al.
moving them along trajectories in the MV/(B−V) plane 1999; Yudin 2001; Abtet al. 2002) or to validate (e.g.,
that are, to a first approximation, straight lines. The dif- Howarth et al. 1997; Chauville et al. 2001) their own mea-
ferential rate of displacement varies in step with ve, being surements.
largest close tocritical rotation, and so small for theslowly Slettebak(1982)usedvisualinspectionofphotographic
rotating models (ve = 0.24vc) that they can barely be dis- spectratoestimatethelinewidthsinhistargets,calibrating
tinguished from their non-rotating neighbours. hisresultswithstandardsfromSlettebak et al.(1975).This
The most striking aspect of Fig. 3 is the strong incli- approach assumes a single-valued correspondence between
nation (i) dependenceof thetrajectories’ orientation in the linewidthandv sini,andneglectstheredundancybetween
MV/(B−V)plane.Forthei=0◦models,rotationincreases theseparametersthatwehaveemphasized.Slettebakisuni-
the observed brightness, due to beaming of radiation from versally acknowledged as an exceptionally skilled observer,
the hot pole; however, this increase happens without any and we believe that his careful and conservative approach
appreciable variation in colour, leading to trajectories that wouldhaveledhimtoadopttheminimumv sinivaluecon-
are close to vertical. The converse occurs when i = 90◦: sistent with his observations (rather than some arbitrarily
the centrifugal distention of equatorial regions produces a larger value), which, as we have shown, will generally un-
drop in the surface-averaged effective temperature, redden- derestimate ve sini.
ingthecoloursatapproximatelyconstantbrightness,andso Furthermore, the v sini values of the underpinning
resulting in nearly horizontal trajectories. For intermediate Slettebak et al. (1975) standard stars were themselves cali-
i = 45◦ case, a combination of brightening and reddening brated against theoretical profiles from Collins (1974). The
produces diagonal trajectories, coincidentally nearly paral-
lel to those produced by theprocess of stellar evolution.
5 Collins&Truax (1995) cite Stoeckley’s work, and also give a
Theincreaseinbrightnessforthepole-onmodelsisap-
similardiagramtoourFig.1(theirFig.4);butwedisagreewith
proximately independent of subtype, being ∆MV ≈−0m. 65 theirconclusionthatanapproximatetreatmentoflimbdarkening
at ve =0.95vc, while thereddeningof theequator-on mod- isthemajorfactor indeparturesfromtheresultsoftheuniform
els,atthesamerotationrate,showsamodestincreasefrom model. Note that even our simple, uniform models incorporate
∆(B −V) = 0m. 032 at B0 to ∆(B −V) = 0m. 066 at B9. a detailed treatment of non-linear, wavelength-dependent limb
By chance, the ratios between these brightening and red- darkening.
6 R. H. D. Townsend et al.
latter’s pioneering calculations required computations that withanyvalue(orarangeofvalues)&0.8,andparticularly
were, by the standards of the day, sophisticated and de- with near-critical rotation – an obvious limiting value.
manding. Unfortunately, subsequent results, including his Toinvestigatethispossibilityinaheuristicway,wecon-
own (cf. Collins & Truax 1995) and those we present here, siderapopulationof completeB2-starmodels with equato-
shed some doubt on this fundamental calibration. For ex- rial velocities distributed uniformly between ve = 0.93vc
ample,Collins (1974)foundlinebroadeningtoberelatively and ve = 0.97vc, and with rotation axes oriented ran-
unaffected by gravity darkening. Indeed, his paper explic- domly. Analyzing this population with Hei λ4471˚A line
itly notes a “remarkably good linear correlation between profiles derived from our uniform models (thereby effec-
the theoretical value for ve sini and the half-width of the tively mimicking Chauville et al.’s procedure), we obtain a
line”–acorrelation from whichourgravity-darkened,com- meanprojectedequatorialvelocityhv sini/vci=0.635,from
plete models depart, quite significantly, in the crucial limit whichtheChandrasekhar & Mu¨nch(1950)relationssuggest
of large ve sini (Fig. 1). a mean equatorial velocity hve/vci=0.809, with a variance
Collins’ remark was illustrated by calculations of the σ2(ve/vc)=−9×10−3. These inferred values6 are remark-
Heiλ4471˚AprofileforaB0-typestellarmodel(hisFig.5). ablyclosetothoseobtainedbyChauvilleetal.,eventhough
That figure indicates line half-widths at half depth that oursyntheticpopulation rotates at significantly more-rapid
aresystematically broader forcriticalrotatorsthanforsub- rates than the hve/vci = 0.83 advanced by those authors.
critical rotators at the same ve sini. We find it difficult to Of course, this simulation is not particularly realistic, but
deviseanyphysicalexplanationforsuchbehaviour,andsus- itnonethelessclearlyestablishesthatstatisticalstudiesthat
pectthatitmayindicatesomeproceduralorcodingerrorin donotadequatelytreattheeffectsofgravitydarkeningmust
hisstudy.Indeed,laterworkby,e.g.,Collins & Truax(1995) inevitably lead to systematic underestimates of ve sini.
givesfindingsbroadlyconsistentwithourown,andalsocon-
tradicts the linear correlation found by Collins (1974). In
any event, our Table 2 indicates that the B0 subtype ex- 6.3 Photometry
hibits the smallest velocity deficiencies of any of the stellar
The calculations in Section 5 indicate that near-critical
models that we have considered. Therefore, notwithstand-
ing any putative problems with Collins’ early calculations, rotation (by which we mean ve/vc ≃ 0.95) displaces
stars toward brighter magnitudes or redder colours, de-
weexpectthedataplottedinhisFig.5toprovidethepoor-
pending on the inclination. This phenomenon, first dis-
estillustration ofthedegreetowhichgravitydarkeningcan
covered by Roxburgh& Strittmatter (1965), has been in-
influenceline broadening.
vestigated and confirmed by a number of authors (e.g.,
Fromthepointofviewofbothobservationalprocedure
Collins 1966; Faulkneret al. 1968; Hardorp & Strittmatter
and underlying calibration, we are therefore led to the con-
1968b; Maeder & Peytremann 1970). Our qualitative find-
jecture that the system of Be-star velocities underpinned
ings are very similar to these studies, although we obtain
by thework of Slettebak et al. (1975) and Slettebak (1982)
may systematically underestimate ve sini values, particu- s(m∆a(Blle−r cVh)an≃ge0sm. 0in6)btrhigahntnfoeussnd(∆hMistVori≃cal−ly0m.(f6o)rarenadsocnoslowuer
larlyattheupperendoftherotationscale.Thisconjectureis
discuss below).
givencredencebytherecentinvestigationbyChauville et al.
Evidently, if Be stars are indeed near-critical rotators,
(2001),whoobtainedv sinivaluesfor116Bestarsbyleast-
squaresfitsofsyntheticspectratoobservedHeiλ4471˚Aline they should naturally occupy a brighter/redder position in
thecolour-magnitude diagram than normalB-typestars. A
profiles. As for our uniform models, Chauville et al.’s syn-
searchthroughtherelevantliteraturerevealssubstantiveev-
thetic profiles were calculated under the assumption of no
idenceforsuchananomalousposition:Zorec & Briot(1991)
rotational distortion or gravity darkening, and they there-
find their sample of Be stars to be over-luminous with re-
forecertainlyunderestimateve sini.However,theyobtained
spect to main sequence stars, and of the fourteen previous
v sini values in excellent overall agreement with those of
studies that they cite, eleven disclose similar behaviour.
Slettebak (1982), encouraging the view that there is also
Intheerawhentheobservationalconsequencesofgrav-
a systematic error in the latter’s results (and hence in all
ity darkening were first being investigated, it was natural
subsequentwork built on them).
to attribute this observed over-luminosity to rapid rotation
(e.g.,Stoeckley1968).However,manyofthestudiesfromthe
6.2 Statistics 1960’speriod(includingthemajorityofthoseweciteabove)
were based on interior calculations by Roxburgh et al.
Chauville et al. (2001) measured a mean projected equa-
(1965),whichwerefoundbySanderson et al.(1970)tocon-
torial velocity hv sini/vci = 0.65 for their sample of Be tain a serious error. Correction of this error resulted in a
stars. Using the Chandrasekhar & Mu¨nch (1950) relations
reduction ofthebrightness/reddening changecaused byro-
between projected and intrinsic rotation velocities, they in-
tation(towardsvaluescommensuratewiththosewefoundin
ferred from their data a mean equatorial velocity hve/vci= Section 5), to such an extent that rotation alone was hard-
0.83±0.03. This extremely small apparent intrinsic scatter
pushedto explain theluminosity anomaly.
led them to conclude that “all studied Be stars rotate at
EversincethediscoveryoftheerrorbyRoxburgh et al.
nearly thesame ratio [of] ve/vc”.
Weknowof nophysicalreason why all Bestars should
rotateatthesameve/vc ≃0.8,butexaminationofourFig.1 6 Note that the variance for our synthetic population is less
shows that at and above ve/vc ≃ 0.8 there is effectively no than zero due to the marginal breakdown of the assumption
change in line width with increasing ve. It is therefore rea- by Chandrasekhar &Mu¨nch (1950) that the v sini distribution
sonable tosuppose that theobservations may beconsistent arisesfromthespatialprojectionofanintrinsicdistribution.
Be-star rotation: how close to critical? 7
(1965), there has been a shift towards interpreting the The most direct method for examining rotational dis-
anomalyasaconsequenceofthecircumstellaremission(CE) tortion is through direct imaging. Once the aspect ratio
producedbythedisksofBestars(e.g.,Fabregat et al.1996). x ≡ Re/Rp of the centrifugally-distorted star is known, its
Thishypothesiscertainly seemspromising, butit hasnever equatorial velocity can be calculated in the Roche approxi-
beenestablishedwhetherCEcanexplaintheobservationsin mation from the expression
toto; as Zorec & Briot (1997) point out, it may be the case
thatmultiplemechanisms (includingrapid rotation) are re- ve =x 3(x−1), (4)
quired to explain the anomalous position of Be stars in the vc r x3
colour-magnitude diagram. Indeed,it is still not clear what where we note that x = 1.5 corresponds to ve/vc = 1.
theeffectsareoftheintrinsic‘rotation reddening’discussed Recently,Domiciano deSouza et al.(2003)appliedthisap-
by Maeder (1975), which arise from the distention of the proach to the Be star α Eri; their VLTI observations in-
envelopearoundtheequatorialregionsofarapidlyrotating dicate an aspect ratio entirely consistent with near-critical
star(suchthatthelocalatmosphericstructuredepartsfrom rotation. Clearly, thepower ofthisapproach lies in itsabil-
that predicted byplane-parallel models). itytoavoidtheneedforcomplexmodelling ofstellaratmo-
spheres or interiors. Unfortunately, its applicability is lim-
ited to those stars which are resolvable via interferometry.
6.4 Future Approaches
HavingpresentedevidencethatcurrentmeasurementsofBe
7 CONCLUSION
starrotationareunderestimates,letusbrieflyconsiderwhat
approaches might be deployed in the future to address this We have argued that the available observations, and their
problem. To reiterate, the essence of the problem is this: interpretation,leaveopenthepossibilitythatBestarsrotate
when a star rotates at a significant fraction of its critical atorneartotheircriticalvelocities,contrarytotheprevious
velocity vc, the insensitivity of line widths toward any fur- consensus.InquestioningthecanonicalviewthatBe-starro-
ther increase in the rotation rate means that these widths tation is substantially sub-critical, we have been motivated
are poor indicators of the star’s true projected equatorial by the issue of the fundamental physical processes respon-
velocity ve sini. How might this degeneracy belifted? sible for the Be phenomenon. A range of phenomenological
One possible approach lies in leveraging the combined models have been developed to account for various aspects
temperature/wavelengthdependenceofthecontinuumflux. of Be-star behaviour,some quitedetailed and sophisticated
From far ultraviolet (FUV) observations of Be stars, Heap (e.g.,Okazaki1991),butthemechanismsunderlyingthefor-
(1976) discovered the line broadening at these wavelengths mation of circumstellar disksremain elusive. This islargely
to be significantly less than at visible wavelengths. As due to the substantial energetic requirements for levitating
pointed out by Hutchings (1976), this effect can be under- material in a strong gravitational field.
stood as a result of the different surface brightness distri- Near-critical rotation alleviates this problem to the
butions at visible and FUV wavelengths, caused by gravity pointwhereknownprocesses,suchaspulsation orgaspres-
darkening.Hutchings& Stoeckley(1977)subsequentlysug- sure,mayreadily providesufficient energyand angularmo-
gestedthattheeffectmightbeusedtoobtainreliablevalues mentum.Ourcritiquearguesthatsuchnear-criticalrotation
forve/vc andi.Suchanapproachcertainlylookspromising, has not beenruledoutbyexistingstudies,betheyspectro-
butsuffersfromthedrawbackofrequiringsimultaneousvis- scopic or photometric. Indeed, these studies furnish neces-
ible, FUVand (if possible) infrared observations, which are sary evidencethattheBestarsare rotatingclosetocritical;
often difficult to secure. we hope that this paper will provide a motivation towards
An alternative technique for lifting the degeneracy lies obtaining, perhaps using one of the approaches we suggest
in exploiting the differing temperature sensitivities of dif- in Section 6.4, sufficient evidence in support of the same
ferent spectral lines. As we demonstrated in Sections 4.2– conclusion.
4.3, the inverse temperature/EW dependence exhibited by
theMgii λ4481˚A linepartly counteracts theeffect of grav-
ity darkening in the continuum, and lessens the impact of
ACKNOWLEDGMENTS
the degeneracy. In the case of an individual line such as
this, the effect will probably be too small to be of practi- We thank Dietrich Baade and Thomas Rivinius for their
cal use7, but the effect could be exploited by simultaneous helpfulsuggestionstowardimprovingthispaper.RHDTac-
modelling of multiple lines. In principle, the different tem- knowledges PPARC support. SPO received support from a
perature/EWdependenceofeachlinemeansthatagoodfit PPARC fellowship for sabbatical research in the UK, and
for all lines will occur only at a single point in parameter fromNSFgrantAST00-97983totheUniversityofDelaware.
space(thatis,asinglecombinationofve,i,M∗etc.).Inreal- IDHis a Jolligoode fellow.
ity,ofcourse,noiseandotherfactorswilllimittheprecision
of this approach; nevertheless, the approach certainly ap-
pears promising, and hasalready been applied with success REFERENCES
to rapidly-rotating O stars (see Howarth & Smith 2001).
Abt H.A., Levato H., Grosso M., 2002, ApJ, 573, 359
Allen C. W., 1976, Astrophysical Quantities, 3rd edn.
7 We also note on more general grounds that the Mgii λ4481˚A Athlone,London
line is a poor choice for ve sini determination, due to blending Balona L. A., 1975, MemRAS,78, 51
withtheadjacentHeiλ4471˚Aline. Carroll J. A., 1928, MNRAS,88, 548
8 R. H. D. Townsend et al.
Chandrasekhar S., Mu¨nch G., 1950, ApJ, 111, 142 SlettebakA.,CollinsG.W.,ParkinsonT.D.,BoyceP.B.,
Chauville J., ZorecJ., Ballereau D.,Morrell N.,Cidale L., WhiteN. M., 1975, ApJS,29, 137
Garcia A., 2001, A&A,378, 861 SlettebakA.,CollinsG.W.,TruaxR.,1992,ApJS,81,335
Claret A., 2000, A&A,363, 1081 SlettebakA.,KuzmaT.J.,CollinsG.W.,1980,ApJ,242,
Collins G. W., 1966, ApJ,146, 914 171
Collins G. W., 1974, ApJ,191, 157 Smith M. A.,Gray D. F., 1976, PASP,88, 809
Collins G. W., Truax R. J., 1995, ApJ, 439, 860 SmithM.A.,HenrichsH.F.,FabregatJ.,2000,ASPConf.
Cranmer S.R., 1996, PhD thesis, University of Delaware Ser.214:IAUColloq.175:TheBePhenomenoninEarly-
Dachs J., Hanuschik R., Kaiser D., Rohe D., 1986, A&A, TypeStars. ASP,San Francisco
159, 276 Steele I. A., Negueruela I., Clark J. S., 1999, A&AS, 137,
Domiciano de Souza A., Kervella P., Jankov S., Abe L., 147
VakiliF., diFolco E., Paresce F., 2003, A&A,407, L47 Stoeckley T. R., 1968, MNRAS,140, 141
DziembowskiW.A.,MoskalikP.,PamyatnykhA.A.,1993, StruveO., 1931, ApJ, 73, 94
MNRAS,265, 588 Townsend R.H. D., 1997, MNRAS,284, 839
von Zeipel H., 1924, MNRAS,84, 665
Dziembowski W. A., Pamyatnykh A. A., 1993, MNRAS,
YudinR. V., 2001, A&A,368, 912
262, 204
Zorec J., Briot D., 1991, A&A,245, 150
Fabregat J., Torrejon J. M., Reig P., Bernabeu G., Bus-
Zorec J., Briot D., 1997, A&A,318, 443
quetsJ., Marco A.,Reglero V., 1996, A&AS,119, 271
FaulknerJ.,RoxburghI.W.,StrittmatterP.A.,1968,ApJ,
151, 203
Gray D. F., 1992, The observation and analysis of stellar
photospheres. Cambridge University Press
Halbedel E. M., 1996, PASP,108, 833
Hanuschik R.W., 1996, A&A,308, 170
Hardorp J., Strittmatter P. A., 1968a, ApJ,153, 465
Hardorp J., Strittmatter P. A., 1968b, ApJ,151, 1057
Heap S. R., 1976, in IAU Symp. 70: Be and Shell Stars
Reidel, Dordrecht,p. 165
Howarth I. D., 2002, in Maeder A., Eenens P., eds, ASP
Conf. Ser.: IAUSymp:Stellar Rotation p. in press
Howarth I. D., Siebert K. W., Hussain G. A. J., Prinja
R.K., 1997, MNRAS,284, 265
Howarth I. D., Smith K.C., 2001, MNRAS,327, 353
HubenyI., 1988, Comp. Phys. Comm., 52, 103
HubenyI., Lanz T., 1995, ApJ, 439, 875
HutchingsJ. B., 1976, PASP,88, 5
HutchingsJ. B., Stoeckley T. R., 1977, PASP,89, 19
Iglesias C. A., Rogers F. J., 1996, ApJ,464, 943
Kurucz R., 1993, CD-ROM No. 16. Smithsonian Astro-
physical Observatory,Washington D.C.
Maeder A., 1975, A&A,42, 471
Maeder A., Meynet G., 2000, ARA&A,38, 143
Maeder A., Peytremann E., 1970, A&A,7, 120
Morgan W. W., Keenan P.C., 1973, ARA&A,11, 29
Okazaki A.T., 1991, PASJ, 43, 75
Owocki S. P., 2002, in Maeder A., Eenens P., eds, ASP
Conf. Ser.: IAUSymp.:Stellar Rotation p.in press
Porter J. M., 1996, MNRAS,280, L31
RogersF.J.,SwensonF.J.,IglesiasC.A.,1996,ApJ,456,
902
RoxburghI.W.,GriffithJ.S.,SweetP.A.,1965,Zeitschrift
furAstrophysics, 61, 203
Roxburgh I. W., Strittmatter P. A., 1965, Zeitschrift fu¨r
Astrophysics, 63, 15
Sanderson A.D., Smith R. C., HazlehurtsJ., 1970, ApJL,
159, 69
Secchi A., 1867, Astronomische Nachrichten,68, 63
Shajn G., StruveO., 1929, MNRAS,89, 222
Slettebak A.,1949, ApJ, 110, 498
Slettebak A.,1966, ApJ, 145, 126
Slettebak A.,1982, ApJS,50, 55