Towards a unified view of inhomogeneous stellar winds in isolated supergiant stars and supergiant high mass X-ray binaries

1.8 MB·

by  Silvia Martínez-Núñez

SpaceScienceReviewsmanuscriptNo.
(willbeinsertedbytheeditor)
Towards a unified view of inhomogeneous stellar winds in
isolated supergiant stars and supergiant high mass X-ray
binaries
SilviaMart´ınez-Nu´n˜ez PeterKretschmar Enrico
· ·
Bozzo LidiaM.Oskinova JoachimPuls Lara
· · ·
Sidoli JonOlofSundqvist PereBlay Maurizio
7 Falanga· FelixFu¨rst Ange·lG´ımenez-G·arc´ıa
1 · · ·
IngoKreykenbohm MatthiasKu¨hnel Andreas
0 · ·
Sander Jose´MiguelTorrejo´n Jo¨rnWilms
2 · ·

r PublishedatJournalofSpaceScienceReviews,Springeronline07March2017
a
M
S.Mart´ınez-Nu´n˜ez
0
Instituto de F´ısica de Cantabria (CSIC-Universidad de Cantabria), E-39005, Santander, Spain E-mail:
1 [email protected]

] P.Kretschmar
E EuropeanSpaceAstronomyCentre(ESA/ESAC),Science Operations Department P.O.Box78,E-28691,
VillanuevadelaCan˜ada,Madrid,SpainE-mail:[email protected]
H
E.Bozzo
.
h ISDC, University of Geneva, Chemin dEcogia 16, Versoix, 1290, Switzerland E-mail: En-
p [email protected]
-
o L.M.Oskinova
Institut fu¨r Physik und Astronomie, Universita¨t Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam,
r
t Germany
s
a J.Puls
[ Universita¨tssternwarte derLudwig-Maximilians-Universita¨t Mu¨nchen,Scheinerstrasse1,81679,Mu¨nchen,
Germany
2
v L.Sidoli
INAF,IstitutodiAstrofisicaSpazialeeFisicaCosmica-Milano,viaE.Bassini15,I-20133Milano,Italy
8
1 J.O.Sundqvist
6 CentrodeAstrobiolog´ıa,CSIC-INTA,Ctra.Torrejo´naAjalvirkm.4,28850Madrid,Spain&Instituutvoor
8 Sterrenkunde,KULeuven,Celestijnenlaan200D,3001Leuven,Belgium
0
P.Blay
1. NordicOpticalTelescope-IAC,P.O.Box474,E-38700,SantaCruzdeLaPalmaSantaCruzdeTenerife,
Spain
0
7 M.Falanga
1 International Space Science Institute (ISSI), Hallerstrasse 6, CH-3012 Bern, Switzerland & International
: SpaceScienceInstituteinBeijing,No.1NanErTiao,ZhongGuanCun,Beijing100190,China
v
i F.Fu¨rst
X Cahill Center forAstronomy and Astrophysics, California Institute ofTechnology, Pasadena, CA 91125,
USA
r
a
AG´ımenez-Garc´ıa
InstitutoUniversitariodeF´ısicaAplicadaalasCienciasylasTecnolog´ıas,UniversityofAlicante,P.O.Box
99,E03080Alicante,Spain
I.Kreykenbohm
2 S.Mart´ınez-Nu´n˜ezetal.
Abstract Massivestars,atleast 10timesmoremassivethantheSun,havetwokeyprop-
∼
ertiesthatmakethemthemaindrivers ofevolution ofstarclusters,galaxies,andtheUni-
verse as a whole. On the one hand, the outer layers of massive stars are so hot that they
producemostoftheionizingultravioletradiationofgalaxies;infact,thefirstmassivestars
helped to re-ionize the Universe after its Dark Ages. Another important property of mas-
sivestarsarethestrongstellarwindsandoutflowstheyproduce.Thismassloss,andfinally
theexplosionofamassivestarasasupernova oragamma-rayburst,provide asignificant
input of mechanical and radiative energy into the interstellar space. These two properties
togethermakemassivestarsoneofthemostimportantcosmicengines:theytriggerthestar
formationandenrichtheinterstellarmediumwithheavyelements,thatultimatelyleadsto
formation of Earth-like rocky planets and the development of complex life. The study of
massivestarwindsisthusatrulymultidisciplinaryfieldandhasawideimpactondifferent
areasofastronomy.
In recent years observational and theoretical evidences have been growing that these
winds are not smooth and homogeneous as previously assumed, but rather populated by
dense “clumps”. The presence of these structures dramatically affects the mass loss rates
derived from the study of stellar winds. Clump properties in isolated stars are nowadays
inferred mostly through indirect methods (i.e., spectroscopic observations of line profiles
invarious wavelength regimes, andtheir analysis basedon tailored, inhomogeneous wind
models).Thelimitedcharacterizationoftheclumpphysicalproperties(mass,size)obtained
so far have led to large uncertainties in the mass loss rates from massive stars. Such un-
certaintieslimitourunderstandingoftheroleofmassivestarwindsingalacticandcosmic
evolution.
Supergiant high mass X-ray binaries (SgXBs) are among the brightest X-ray sources
inthe sky. A largenumber of them consist ofa neutron staraccreting from thewind of a
massivecompanionandproducingapowerfulX-raysource.Thecharacteristicsofthestellar
wind together with the complex interactions between the compact object and the donor
stardeterminetheobserved X-rayoutput fromallthesesystems.Consequently, theuseof
SgXBsforstudies ofmassivestarsisonly possiblewhenthephysicsof thestellarwinds,
thecompactobjects,andaccretionmechanismsarecombinedtogetherandconfrontedwith
observations.
Thisdetailedreviewsummarisesthecurrentknowledgeonthetheoryandobservations
ofwindsfrommassivestars,aswellasonobservationsandaccretionprocessesinwind-fed
highmassX-raybinaries.Theaimistocombineinthenearfutureallavailabletheoretical
Dr.KarlRemeis-Observatory&ECAP,Universita¨tErlangen-Nu¨rnberg,Sternwartstr.7,D-96049Bamberg,
Germany
M.Ku¨hnel
Dr.KarlRemeis-Observatory&ECAP,Universita¨tErlangen-Nu¨rnberg,Sternwartstr.7,D-96049Bamberg,
Germany
A.Sander
Institut fu¨r Physik und Astronomie, Universita¨t Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam,
Germany
J.M.Torrejo´n
InstitutoUniversitariodeF´ısicaAplicadaalasCienciasylasTecnolog´ıas,UniversityofAlicante,P.O.Box
99,E03080Alicante,Spain
J.Wilms
Dr.KarlRemeis-Observatory&ECAP,Universita¨tErlangen-Nu¨rnberg,Sternwartstr.7,D-96049Bamberg,
Germany
UnifiedviewofofinhomogeneousstellarwindsinsupergiantstarsandHXMB 3
diagnostics and observational measurements to achieve a unified picture of massive star
windsinisolatedobjectsandinbinarysystems.
Keywords Massivestars stellaroutflows X-raybinary wind-fedsystems accretion
· · · ·
processes SgXBs SFXTs
· ·
1 Introduction
Massive stars (M >10M ) play an important role in the evolution of star clusters and
initial
galaxies.Massivestar∼sgene⊙rateionizingultravioletradiation,andheatthedust.Thewinds
of massive stars, and their final explosions as supernovae or gamma-ray bursts provide a
significant input of energy and chemically enriched matter into the interstellar medium
(Kudritzki, 2002). Massive stars are among the most important drivers of cosmic evolu-
tion, they regulatestarformation and, together withlow-mass stars,enrich the interstellar
mediumwithheavyelements.AmongthebrightX-raysourcesintheskyasignificantnum-
berconsistsofacompactobjectaccretingfromthewindofsuchmassivestars.Thesewinds
arefast(withtypicalterminalvelocitiesupto2500kms 1),dense(withmass-lossratesup
−
toM˙ >10 5 10 7M yr 1),anddrivenbylinescatteringofthestar’sintensecontinuum
− − −
radiat∼ion field−. Exampl⊙es of asystemcomprising amassivestarand acompact object are
CygX-1/HDE226868,thefirstdetectedstellar-massblackhole,andVelaX-1,theprototype
ofwindaccretingneutronstarX-raybinaries.Bothinisolatedmassivestarsandinbinary
systemswithaccretingcompact objects,thebasicpictureofthewindformation andwind
accretionprocess has beenestablishedfordecades. However,new findings concerning in-
homogeneitiesinthemassivestarwindsandtheunexpectedlypronouncedX-rayvariability
insomewind-fedbinariesquestionedourpreviousunderstandingofthesesystems.
Thefirstquantitativedescription ofline-driven stellarwindswasprovided inthesem-
inal paper by Castoretal. (1975), which assumed a stationary, homogeneous, and spheri-
callysymmetricoutflow. Laterworks (e.g.,Owockietal.,1988;Feldmeieretal.,1997b,a;
DessartandOwocki, 2005) showed that theline-driven hot starwinds areinfact unstable
1 to velocity perturbations (the so-called ”line-driven instability”, hereafter LDI), leading
tohigh-speed rarefactions thatsteepen intostrong shocks, whereby most material iscom-
pressed into spatially narrow ’clumps’ (or shells in 1-D simulations) separated by large
regionsofmuchlowerdensities.Thepresenceofclumpsinthewindsofmassivestarsissup-
portedbynumerousobservationalevidencesinmanydifferentwavebands(seeHamannetal.,
2008;Sundqvistetal.,2011, forcomprehensive overviews). Innumerical simulations, the
LDI is observed to generate strong wind shocks, which provide a possible explanation
(Feldmeieretal.,1997b,a)forthesoftX-rayemissionobservedfrom”normal”(putatively
single,non-magnetic) OB-stars,aswellasfortheirlackofsignificanttime-variability(see
Naze´etal., 2013, for arecent review). Clumps affect several stellarwind diagnostics in a
non-trivialway,anddiscussionsareon-goingtoinferthephysicalpropertiesofthesestruc-
turesfromtheresultsofthemostrecentobservationalcampaigns.
Additional independent observational evidence of clumped stellar winds comes from
supergiant high mass X-ray binaries (SgXBs), i.e. those systems in which a compact ob-
ject(ablackholeoraneutronstar)orbitsasupergiantO-Bstar.Sakoetal.(2003)wasthe
first to review spectroscopic results obtained by X-ray observatories for several wind-fed
SgXBs.Theyconcludedthattheobservedspectraandtimevariabilityoftheseobjectscould
bebestexplainedbyassumingthataccretionontothecompactobjectistakingplacefrom
1 AlreadyLucyandSolomon(1970)pointedoutthatradiativeline-drivingissubjecttoastronginstability.
4 S.Mart´ınez-Nu´n˜ezetal.
ahighly structured stellarwindwherecool denseclumps areembedded inararefiedpho-
toionizedgas.SimilarstudieswerelatercarriedoutonanumberofbrightSgXBs,including
4U1700 37 (vanderMeeretal., 2005), VelaX-1 (Kreykenbohmetal., 2008; Fu¨rstetal.,
−
2010; Mart´ınez-Nu´n˜ezetal., 2014), Cyg X-1 (Misˇkovicˇova´ etal., 2011), and GX 301 2
−
(Fu¨rstetal., 2011). Although the presence of structured clumped winds in SgXBs seems
thus well established, there is astillconsiderable uncertainty inthephysical properties of
thoseclumpsandthemechanismsbywhichthestructuredwindisabletofeedthecompact
object. Particularly puzzling is the pronounced X-ray variability (a factor of 100-1000
∼
higher than in classical SgXBs) of the supergiant fast X-ray transients (SFXTs) sources.
Thisvariabilityisunlikelytobeonlyduetothepresenceofmassivestructuresinthewind
ofthesupergiantstarsandrequiresad-hocassumptionsontheon-goingaccretionprocesses.
The layout of the review is as follows: Section 2 first introduces the basic physics of
line-driven winds in detail, from the pioneering ’CAK model’ to modern simulations in-
cludingsmallandlargescalestructuresinthewind.Thesectioncontinueswiththetheory
ofaccretionofthesewindsontocompactobjectsandespeciallyneutronstars,treatingalso
differentaccretionregimesandtheinhibitionofaccretion.Section3discussesthedetermi-
nation of stellar and wind parameters by quantitative spectroscopy in the optical and UV
regime,includingtheeffectsofwindclumpingonthemasslossdiagnostics.Measurements
anddiagnosticsintheX-rayregimearediscussedinSection4,togetherwithcaveatswhen
applyingthese.Thissectionalsosummarisesthecurrentknowledgeonboththe”classical”
SgXBsandtheSFXTs.Finally,Section5summarisesthemaincurrentlyopenquestionson
stellarwindsandwindpropertiesofmassivestars.
2 Basicphysics
Wesummariseinthenexttwosectionsthebasicphysicsoftheline-drivenwindsinmassive
starsandaccretionprocessesinwind-fedbinaries.
2.1 Basicsofline-drivenwinds
Adecisiveproperty ofhot, massivestarsistheirstellarwind, withtypical mass-lossrates
(forsolarmetallicity),M˙ 10 7...10 5 M /yr,andterminalvelocities,υ ,ranging from
− − ∞
200... 3,500kms 1.The≈originofthesew⊙indsisattributedtoradiativeline-driving,i.e.,
−
stellar continuum photons are scattered in a multitude of spectral lines and transfer their
momentumtothewind.Sincethisprocessrequiresalargenumberofphotons (i.e.,ahigh
luminosity),suchwindsoccurinthehotteststars,likeO-typestarsofallluminosityclasses,
butalsoincoolerBA-supergiants,becauseoftheirlargerradii.Efficientline-drivingfurther
requires a large number of spectral lines close to the flux-maximum and a high interac-
tion probability (i.e., a significant optical depth). Since most spectral lines originate from
various metals, a strong dependence of M˙ on metallicity is thus to be expected, and such
line-drivenwindsshouldonlyplayaminorrole(ifatall)intheearlyUniverse.2Thetheory
of line-driven winds has been pioneered byLucyandSolomon (1970) andparticularly by
Castoretal.(1975,henceforth ’CAK’),withessentialimprovements regarding aquantita-
tivedescriptionandapplicationprovidedbyFriendandAbbott(1986)andPauldrachetal.
2 Contrastedtothealmostmetallicity-independent,porosity-moderatedcontinuum-drivenwindshypoth-
esizedbyOwockietal.(2004).
UnifiedviewofofinhomogeneousstellarwindsinsupergiantstarsandHXMB 5
(1986). Line-driven winds have beenreviewed by KudritzkiandPuls(2000)and morere-
centlybyPulsetal.(2008).Inthefollowing,wewillbrieflyconsidersomerelevantaspects,
mostlyintermsofthe’standardmodel’andthetheorydevelopedbyCAK.
2.1.1 TheCAKmodelandbeyond
Fromstudyingthetemporalvariabilityoftypicalwind-features(UVP-Cygniprofiles,H ,
α
HeII4686, see Sect. 3) and from analysing these lines, it turned out that the global quan-
tities describing the outflow (M˙,υ ) typically show only little variations. This and other
∞
evidencemotivatesthedefinitionofastationary,sphericallysymmetric,andhomogeneous
standardmodel.Effectsfromrotationandmagneticfieldsarebrieflyoutlinedattheendof
thissection,anddeviationsfromahomogeneousstructurearediscussedinSects.2.1.2and
2.1.3.
Insuchastandardmodel,themass-lossrateM˙ =4πr2ρ(r)υ(r)remainsconstantover
thewind,andtheequationofmotionisgovernedbypressuretermsandexternalforces,in
our case the inward gravitational pull and an outward directed radiative acceleration. For
simplicity, the Thomson-acceleration due to electron scattering will be included as a cor-
rectiontogravity3intermsoftheconventionalEddington-Gamma,Γ =g /g ∝
Edd Thomson grav
L/M,andtheremainingcontinuumaccelerationcanbeneglectedinmosthotstarwinds.
Thus, the ‘only’ difficulty regards calculating the radiative line force. Basically, this
forcecanbederivedfromthemomentumtransferoccurringduringtheabsorptionand(re-
)emissionof(mostly) stellarphotons, whereon average theemissionprocess cancels out
becauseofitsfore-aft-symmetry. Sincemostphotons areabsorbed inmetallines,themo-
mentum needs to be redistributed to the bulk plasma (H and He), by means of Coulomb
collisions(SpringmannandPauldrach,1992)4.
Incertainfrequencyintervals,theline-densitycanbesohighthatphotonsontheirway
out of the wind are not only scattered in one line before they escape, but also in a sec-
ond one, a third one, etc., until they ultimately find their way out. This process is called
multi-line-scattering,andleadstoacertaincomplexityinanalyticalcalculationsoftheline
force.5 Forsimplicity,weassumeinsteadthateachlinecanbetreatedseparately,i.e.,that
stellarphotonscaninteractwithonlyonelineandthenleavethewind,irrespectiveofline-
density.AsshownbyPuls(1987),thisisnottoobadanapproximation forOBA-stars.6 In
thiscase,thetotallineforcecanbecalculatedbysumminguptheindividualcontributions
fromallparticipatinglines,expressedintermsofilluminatingintensityandlineopacity.In
rapidlyexpandingatmospheres,thisexpressioncanbesimplifiedbymeansoftheso-called
Sobolevapproximation(Sobolev,1960).Ifatfirstweonlyconsiderradiallystreamingpho-
tons (relaxedlateron), theradiativeline-acceleration forlineiatresttransition-frequency
ν resultsin
0,i
grad,i= Lνciν20,i4dπυr/2dρrh1−exp(cid:0)−kLd,iυse/ρdυrth(cid:1)i= Lνciν20,i4dπυr/2dρrh1−e−τSobi (1)
3 Bothaccelerationsdependonr 2,atleastinahomogeneousmedium.
−
4 Asignificantdriftbetweenmetallicionsandthebulkplasmaorevenacompletedecouplingofcertain
ionsmightbecomepossibleinwindsoflowmetallicityand/orlowdensity,e.g.,Babel(1995);Krticˇkaetal.
(2003);Krticˇka(2006);OwockiandPuls(2002).
5 For details, see, e.g., FriendandCastor (1983); Puls (1987); LucyandAbbott (1993); Gayleyetal.
(1995).
6 InthedensewindsofWolf-Rayetstars(seebelow),multi-linescatteringneedstobeaccountedfor.
6 S.Mart´ınez-Nu´n˜ezetal.
withL thespectralluminosity,s themass-absorptioncoefficientforThomsonscattering,
νi e
andυ thethermalvelocityforarepresentativeion.k istheso-calledline-strength,corre-
th L,i
spondingtotheratiobetweenfrequency-integratedline-opacityχ¯ andThomson-scattering
i
opacitys overatypicalline-width∆ν ,
e Dop,i
χ¯
kL,i= s ρ∆νi . (2)
e Dop,i
Forthedominatingresonancelinesfrommajorions,k isroughlyconstantoverthewind.
L,i
Aline-strengthofunitythusreferstoaweaklineofcontinuumelectron-scatteringstrength,
whereasstronglinescanhavek 106orevenmore.
L,i
≈
ThemostintriguingquantityappearinginEq.1istheradialvelocitygradient,whichre-
sultsfromtheDoppler-effectexperiencedbytheabsorbingmatterinanexpandingmedium.
AsobviousfromEq.1,theradiativeaccelerationfromopticallythinlines(withlineoptical
depth in Sobolev approximation τSob <1) is proportional to kL,i and does not depend on
velocityand density, whilstforoptically thicklines(τ >1)g becomes independent
Sob rad,i
ofline-strength(saturation),butnowdependson(dυ/dr)/ρ.
The basic trick of CAK was to write the total line acceleration, i.e., the sum over all
contributinglinesi,asanintegraloveraline-strengthdistribution
gtot =∑g g (k ,ν)dN(k ,ν) (3)
rad rad,i→ZZ rad L L
i
wherethisdistributiondependsonline-strengthandfrequency.Fromsomepreliminaryem-
piricalargumentswhichhavebeenconfirmedmeanwhile(e.g.,Pulsetal.2000andFig.1),
CAKassumedapower-lawdistributionw.r.t.k andafrequentialdistribution∝1/ν,
L
dN(kL,ν)=−N0kαL−2dkLdν/ν. (4)
Inthiscase,integralsinEq.3canbesolvedanalytically,andoneobtains
L s υ N Γ(α) dυ/dr α s L
gtot = e th 0 := e k kα=g (r)Γ k kα, (5)
rad 4πr2c2 1 α s υ ρ 4πr2c CAK 1 grav Edd CAK 1
(cid:16) e th (cid:17)
−
whereΓ(α)istheGamma-function,
υ N Γ(α) dυ/dr
k = th 0 , k = . (6)
CAK 1
c 1 α s ρυ
e th
−
k (on the order of 0.1 for O-stars and early B-stars) is one of the so-called force-
CAK
multiplier parameters, and k the line-strength where the exponent in Eq. 1, the optical
1
depth in Sobolev approximation, becomes unity.7 αis the 2nd force-multiplier parameter
( 0.6...0.7 for O-star winds, see Fig. 1, left panel), either corresponding to the slope of
≈
theline-strengthdistributionfunction,Eq.4,oralternativelyinterpretedastheratioofline-
accelerationfromopticallythicklinestothetotallineacceleration.
Afteraccountingfornon-radialphotonsandionizationeffects(notdiscussedhere),we
caninsertthetotallineaccelerationintothetime-independent equationofmotion.There-
sultingnon-lineardifferentialequationcanbesolvedeithernumerically(e.g.,Pauldrachetal.
1986; FriendandAbbott 1986) or, applying certain simplifications, alsoanalytically (e.g.,
7 k1correspondstot−1inthenotationofCAK.
UnifiedviewofofinhomogeneousstellarwindsinsupergiantstarsandHXMB 7
Fig.1 Left:Frequencyintegratedline-strengthdistributionfunctionforanO-typewind(Teff=40kK,solar
abundance),andcorrespondingpower-lawfit(slope=α 2).Right:ObservedWLRforGalacticO-stars
−
(asterisks, plus-signs and rectangles for luminosity classes I, III and V objects, respectively). Data from
Pulsetal.(1996).Dashed:linear regressiontol.c.Iobjects; dotted: linear regression toluminosity class.
IIIandVobjects.Analytical considerations (seetext)andtheoretical models(fromVinketal.2000,solid
red)donotpredictadependence onluminosityclass.The‘observed’difference ispresumablyduetothe
neglectofwind-inhomogeneities(clumping)inthemass-lossanalysis(seealsoRepolustetal.(2004)).Note
thattheobservedwind-momentadeviatetowardslowvaluesbelowlogL/L <5.2(‘weakwindproblem’,
cf.Marcolinoetal.2009andreferencestherein,andHuenemoerderetal.20⊙12forapotentialexplanation).
BluedotsindicatetheoreticalmodelsfromPauldrachetal.(2003)calculatedforfivestarsfromtheobserved
sample,resultinginsimilarwind-momentaaspredictedbyVinketal.(2000).
Kudritzkietal.1989;Owockietal.2004),andthemass-lossrateresultsastheeigenvalue
oftheproblem.Overall,weobtainthefollowingscalingrelations:
M˙ ∝ k L 1/α′ M (1 Γ ) 1−1/α′ ∝ k1/α′ L ΓEdd 1/α′−1
(cid:16) CAKL (cid:17) (cid:16)M − Edd (cid:17) CAKL (cid:16)1 ΓEdd(cid:17)
⊙ ⊙ ⊙ −
R β
υ(r)=υ∞ 1 ∗
(cid:16) − r (cid:17)
2.25α 2GM(1 Γ ) 1 2.25α
υ∞ − Edd 2= υesc (7)
≈ 1 α(cid:16) R (cid:17) 1 α
− ∗ −
Here,α =α δ,whereδ( 0.1forO-stars)isAbbott’s(1982)ionizationparameter,and
′
− ≈
υ isthephotosphericescapevelocity,correctedfortheradiativeaccelerationbyThomson
esc
scattering.Theexponentβisontheorderof0.8forO-dwarfs,andontheorderof1.3... 2
forBA-supergiants.ToovercomesomeinherentproblemswiththisinitialCAKformulation
(e.g.,theartificialdependenceonafiducialthermalspeed,seeabove),Gayleyetal.(1995)
usedasomewhatdifferentdefinitionfortheline-strength,aswellasaline-statisticswithan
exponentialcut-off,andreformulatedthestandardCAKapproach.Whilethetwoformula-
tions give identical results, thenew one provides a somewhat modified expression for the
mass-lossrate,whichisfrequentlyusednowadays:
M˙ ∝ L 1/α′ 1 M (1 Γ ) 1−1/α′ ∝ L Q¯ΓEdd 1/α′−1
(cid:16)L (cid:17) (cid:16)Q¯ M − Edd (cid:17) L (cid:16)1 ΓEdd(cid:17)
⊙ ⊙ ⊙ −
Forδ=0, the relation between Q¯ ( 2000 for O-stars) and k is given by Q¯1/α 1 =
CAK −
≈
1/α
c/υ (1 α)k .
th CAK
Fi(cid:0)nall−y,byusin(cid:1)gthescalingrelationsforM˙ andυ∞ (Eqs.7),andapproximatingα′
≈
2/3,oneobtainstheso-calledwind-momentumluminosityrelation-WLR-(Kudritzkietal.,
8 S.Mart´ınez-Nu´n˜ezetal.
1995),
R 1 1 L
logD =log M˙υ 2 log +offset(Z,spectraltype), (8)
mom ∞
(cid:16) (cid:0)R⊙(cid:1) (cid:17)≈α′ (cid:16)L⊙(cid:17)
which relates the modified wind-momentum rate D with the stellar luminosity alone.
mom
ThedependenceonMandΓ (difficultto‘measure’)vanishessincetheproductof(M(1
Edd
ΓEdd))1−1/α′ and(υescR1/2)becomesnegligibleaslongasα′ iscloseto2/3.Theoffset i−n
Eq.8dependsonmetallicityandspectraltype,mostlybecausetheeffectivelinenumberand
thusk (orQ¯)dependonthesequantities,viadifferentopacitiesandcontributingions.
CAK
Originally,ithadbeensuggestedtouseacarefullycalibratedWLRasanindependent
tooltomeasureextragalacticdistances,fromthespectroscopicanalysisofextragalacticA-
supergiants and their winds, and by solving for the stellar radius via Eq. 8. Meanwhile,
however, the WLR is mostly used to test the validity of the line-driven wind theory itself
(e.g.,Fig.1,rightpanel).
Varioustheoreticalmodelshavebeencomputedduringrecentdecades,basedonamore
orlessexact calculations ofthe line-force (i.e.,discarding thestatisticalapproach and ac-
counting for non Local Thermodynamic Equilibrium (hereafter, non-LTE) effects). Most
prominent arethemodelsbyVinketal.(2000,2001),relyingonaMonteCarloapproach,
themodelsbyPauldrach(1987)andPauldrachetal.(1994,2001)(‘WM-Basic’),calculat-
ingtheline-forceinaSobolev-approach,andthemodelsbyKrticˇkaandKuba´t(2000,2001,
2004),whichincludeamore-componentdescription(metalionsplusH/He).Allthesemod-
elsagree intheir quantitative predictions (e.g., Fig. 1,right panel), inparticular regarding
themetallicitydependenceofthemass-lossrate,M˙ ∝Z0.6...0.7.8
Themostimpressiveobservationalconfirmationofthetheoreticalconceptofline-driven
winds and their metallicity dependence has been provided by Mokiemetal. (2007), com-
piling observed stellar- and wind-parameters from Galactic, LMC and SMC O-stars, and
analysingthecorrespondingWLRs.Accountingforwind-inhomogeneities(seeSect.2.1.2)
inanapproximateway,theyderiveM˙ ∝Z0.72 0.15,inverygoodagreementwiththeoretical
±
predictions.
Thebi-stabilityjump Oneofthestillunsolvedproblemsregardingline-drivenwindsisthe
reality of the so-called bi-stability jump9, which should affect the mass-loss rates of B-
supergiants(importantinthecontextofSgXBs).Asitturnsout(Pulsetal.,2000;Vinketal.,
2001),themass-lossratesofradiationdrivenwindsaremostlydeterminedbyiron(group)
lines.BelowT 23kK, theionization ofFe(inthelowerwind)switchesabruptly from
eff
≈
FeIVtoFeIII,whichhasmuchmorelinesclosetofluxmaximum.Consequently,themass-
loss rateis predicted toincrease for suchcooler stars by roughly afactor of fiveor more,
whilstυ shoulddecreasebyafactoroftwo(Vinketal.,2001).Thus,thewind-momentum
∞
ratesforB-supergiants(likeinVelaX-1)arepredictedtobelargerthanofO-starsofsimilar
luminosity.
Though agradual decreaseinυ (moreprecisely, intheratioυ /υ )overtherange
∞ ∞ esc
T = 23kK to 18kK has been confirmed in many studies (e.g., Groenewegenetal. 1989;
eff
Crowtheretal.2006;MarkovaandPuls2008),thisisnottrueforthepredictedincreasein
M˙.DetailedinvestigationsofB-supergiantsbyCrowtheretal.(2006)andMarkovaandPuls
(2008)donotshowsuchabehaviour,butratherindicatethattheirmass-lossratesarelower
(orsimilar)tothosefrom O-starsatthesameluminosity. Sincethetheoretical predictions
8 Themetallicitydependenceofυ∞isratherweak,υ∞∝Z0.06...0.13(Leithereretal.,1992;Krticˇka,2006).
9 introducedbyPauldrachandPuls(1990)toexplainthebi-stablebehaviourofwindmodelsforPCygni.
UnifiedviewofofinhomogeneousstellarwindsinsupergiantstarsandHXMB 9
arequiterobust,whereastheformationoftheprimemass-lossindicator,H ,isquitecom-
α
plex in the B-supergiant range (Petrovetal., 2014), further investigations are required to
solvethislong-standingissue.Onemightnote,however,thatallpresentevolutionarycodes
for massive stars incorporate the theoretical mass-loss predictions, and that the predicted
bi-stability jump has alarge effect on the evolution and rotation of B-supergiants and be-
yond(Vinketal.,2010;Markovaetal.,2014).IfthejumpinM˙ werenotpresent,significant
changesinsuchevolutionaryphasesaretobeexpected.
Mass loss from Wolf-Rayet stars From early on, the mass-loss properties of Wolf-Rayet
starsposedamajorproblemfortheoreticalexplanations,sincetheyareconsiderablylarger
compared to O-stars of similar luminosity. Though LucyandAbbott (1993) showed that
line-overlapeffects,coupledwithasignificantlystratifiedionizationbalance,canhelpalot
to increase the mass-loss, it were Gra¨fenerandHamann (2005, 2006, 2007) who showed
thattherearetwoingredientsthatmightproducetheobservedlargemass-lossratesinpar-
allelwithhighterminalvelocities.First,ahighEddington-Γ isnecessarytoprovidealow
effectivegravityandtoenableadeeplyingsonicpoint athightemperatures.Then,ahigh
mass-loss rate leading to an optically thick wind can be initiated either by the ‘hot’ Fe-
opacitybump(around 160kK,forthecaseofWCsandWNEs)orthecoolerone(around
40to70kK,forthecaseofWNLs).10 Alternativewindmodels havebeenconstructed by
Vinketal. (2011), who argue that forΓ >0.7 the winds (more precisely, the pseudo-
Edd
continuum) become optically thickalready at thesonicpoint, which should enableahigh
M˙.Nevertheless,therearestillanumberofdetailstobeworkedoutbeforethesewindsare
completelyunderstood.
Impact of (fast)rotation When stars rotaterapidly, their photospheres become oblate, the
effective temperature decreases from pole towards equator (‘gravity darkening’), and the
wind is predicted tobecome prolate in most cases(because of the larger illuminating po-
lar fluxes), with a fast and dense polar outflow, and a slow and thinner equatorial one
(CranmerandOwocki,1995).11 Whilststellaroblateness andgravity darkening havebeen
confirmed(atleastthebasiceffects)bymeansofinterferometry(DomicianodeSouzaetal.,
2003;Monnieretal.,2007), thepredictions on thewind-structure ofrapidly rotating stars
havenotbeenverifiedbyobservationssofar(Pulsetal.2010andreferencestherein):first,
only few stars in such phases are known (but they exist, e.g., the most extremely rotating
massive stardetected by Duftonetal. 2011 rotates very close to critical), and second, the
toolstoanalysetheatmospheresandwinds(multi-Dmodels!)ofsuchstarsarerare,ifthey
existatall.Notethatevenformoderaterotationthewindispredictedtobecomeasymmetric
(thoughtoalesserextent),andtheformationofimportantmass-lossdiagnosticssuchasH
α
becomesaffected(e.g.,PetrenzandPuls1996).
Impact of magnetic fields Recent spectropolarimetric surveys (mostly performed by the
international Magnetism in Massive Stars, MiMeS, collaboration, e.g., Wadeetal. 2012,
andworkdonebyS.Hubrigandcollaborators, e.g.,Hubrigetal.2013)haverevealedthat
roughly10%ofallmassivestarshavealarge-scale,organizedmagneticfieldintheirouter
stellarlayers(theincidenceofinternalfieldsmightbehigher), ontheorderofacoupleof
hundredtoseveralthousandGauss.Theoriginofthesefieldsisstillunknown,thoughmost
10 TheimportanceoftheseopacitybumpswaspointedoutalreadybyNugisandLamers(2002).
11 Alltheseeffectsbecomesignificantiftherotationalspeedexceedsroughly70%ofthecriticalone.
10 S.Mart´ınez-Nu´n˜ezetal.
evidence points toquite stablefossil fields formed sometimes during earlyphases of stel-
larformation(Alecianetal.,2013).Theinteractionofthesefieldswiththestellarwindhas
been theoretically investigated by ud-Doula, Owocki and co-workers in a series of publi-
cations (summarised in ud-Doula 2013), and two different scenarios have been identified,
dependingonrotationalspeedandfieldstrength.Fornottoofastrotation(whentheAlve´n
radiusissmallerthantheKeplerianco-rotationradius),amagneticallyconfinedwindispre-
dicted,inwhichthegravitational pull onthetrapped windplasmacreateslargeregions of
infallingmaterial,whereasforfastrotationandstrongconfinement12 oneobtainsarigidly
rotatingmagnetosphere(Alve´nradiuslargerthanKeplerianradius(theseradiiaredefined
inSect.2.2.2),inwhichthecentrifugalforcepreventsthetrappedmaterialfromfallingback
tothestellarsurface.Bothscenariosareconsistentwithobservationalfindings(Petitetal.,
2013),andarenowadayscalleddynamicalandcentrifugalmagnetospheres.Thesetwopop-
ulationscanbedifferentiatedbytheirdistinctH emission:slowlyrotatingO-typestarswith
α
narrow,strongemissionconsistentwithadynamicalmagnetosphere,andmorerapidlyrotat-
ingB-typestarswithbroader,oftendouble-peaked, emissionassociatedwithacentrifugal
magnetosphere. First attempts (Sundqvistetal., 2012a) to simulate the H emission from
α
thedynamical magnetospheres ofprototypical O-stars(denotedbythespectraltypequali-
fiers‘f?p’)havebeenquitesuccessful,thoughttheseinitialinvestigationscertainlyneedto
berepeatedwithinamulti-DNLTEapproach.
2.1.2 Smallscalestructures
Althoughthestandardtheoryofline-drivenwindsoutlinedintheprevioussectionassumes
a stable, time-independent and homogeneous wind, it is since long known that the radi-
ation line-force in fact is subjected to a very strong, intrinsic instability (Milne, 1926;
LucyandSolomon,1970).Belowwereviewthetheoreticalbackgroundforthisfundamen-
tal instability, whereas the corresponding observational background regarding small-scale
windstructureisgiveninSect.3.4.
Linear perturbation theory Following OwockiandRybicki (1984), let us assume a small
velocityperturbationoftheconventionalsinusoidalformδυ=δυei(kx wt),wherethewave
0 −
numberkistheinverseoftheperturbationwavelengthandthecircularfrequencywmaybe
complex(toaccountforpotentialexponentialgrowthordampingoftheinitialperturbation).
Forasphericallysymmetricwind,inaframeco-movingwiththeunderlyingmeanflow,and
neglectinggaspressureterms,thiscircularfrequencyisgivenby
w=iδg /δυ, (9)
rad
whereδg =g g istheresponseoftheunperturbedlineforceg tothevelocity
rad rad rad,0 rad,0
−
perturbation.Foralinethatisopticallythickinthemeanflow,OwockiandRybicki(1984)
showedtheratioδg /δυcanbeexpressedbya“bridginglaw”
rad
δg L
rad Ωik Sob , (10)
δυ ≈ 1+ikLSob
where Ω υ/L is the growth rate of the perturbation, and L =υ /(dυ/dr) the
0 Sob Sob th 0
radialSob≈olevlength13 oftheunperturbedflow,movingwithυ inthestellarframe.Thus,
0
12 Forexample,alargeratiobetweenmagneticandwindenergy.
13 theradialextentofthatzonewherephotonscanbeabsorbedbyaspecificline.