Long-Wavelength, Free-Free Spectral Energy Distributions from Porous Stellar Winds

0.57 MB·

by  Richard Ignace

MNRAS000,1–??(2015) Preprint26January2016 CompiledusingMNRASLATEXstylefilev3.0
Long-Wavelength, Free-Free Spectral Energy Distributions
from Porous Stellar Winds
⋆
R. Ignace
Department of Physics& Astronomy, East Tennessee State University,Johnson City, TN, 37614, USA
6
1 AcceptedXXX.ReceivedYYY;inoriginalformZZZ
0
2
ABSTRACT
n
a
J The influence of macroclumps for free-free spectral energy distributions (SEDs)
of ionized winds is considered. The goal is to emphasize distinctions between micro-
5
clumping and macroclumping effects. Microclumping can alter SED slopes and flux
2
levels if the volume filling factor of the clumps varies with radius;however,the modi-

] ficationsareindependentoftheclumpgeometry.Towhatextentdoesmacroclumping
R alter SED slopes and flux levels? In addressing the question, two specific types of
S macroclump geometries are explored: shell fragments (“pancake”-shaped) and spher-
. ical clumps. Analytic and semi-analytic results are derived in the limiting case that
h
clumpsneverobscureoneanother.Numericalcalculationsbasedonaporosityformal-
p
- ism is used when clumps do overlap.Under the assumptions of a constant expansion,
o isothermal, and fixed ionization wind, the fragment model leads to results that are
r essentially identical to the microclumping result. Mass-loss rate determinations are
t
s not affected by porosity effects for shell fragments. By contrast,spherical clumps can
a
lead to a reduction in long-wavelength fluxes, but the reductions are only significant
[
for extreme volume filling factors.

1
v Key words: infrared:stars–radio:stars–stars:early-type–stars:mass-loss–stars:
5 massive – stars: winds, outflows
1
7
6
0
1 INTRODUCTION explain certain UV-line strengths (e.g., Bouret et al.
.
1 2005; Fullerton et al. 2006; Oskinovaet al. 2007;
0 The issue of structured winds from massive stars stretches
Zsarg´o et al. 2008; Sundqvistet al. 2010), optical re-
6 back several decades, and the prominence of wind“clump-
combination lines (Hillier 1991; L´epine& Moffat 1999),
1 ing” for influencing observables has steadily grown (e.g.,
infrared excesses (e.g., Ignace 2009; Bonanos et al. 2009),
: Hamann et al. 2008). Evidence for clumping comes from
v longer wavelength emissions (e.g., Abbottet al. 1981;
i a variety of indicators. The favored mechanism for ex- Puls et al. 2006; Blomme & Runacres 1997; Nugis et al.
X plaining the winds of massive stars is the line-driven
1998; Gonza´lez & Cant´o 2008), and polarimetry (e.g.,
r wind theory (Lucy& Solomon 1970; Castor et al. 1975;
a Lupie& Nordsieck 1987; Taylor et al. 1991; Brown et al.
Pauldrach et al. 1986). Highly supersonic, massive star
1995; Davies et al. 2007; Li et al. 2009).
winds are driven outwards by the action of the UV-
bright luminosities acting on metal line opacities. But The focus of this contribution is to expand the consid-
the process is intrinsically unstable and naturally leads erationsofhow“macroclumping”or“porosity”caninfluence
to structured flows (Lucy& Solomon 1970; Lucy& White observablesofmassivestarwindsintheinfrared(IR)andra-
1980; Owocki et al. 1988; Macfarlane & Cassinelli 1989; dioregimes.Inrelationtotheeffectofclumpingforinferring
Feldmeier et al. 1997; Dessart & Owocki 2003, 2005). mass-loss rates, M˙, many researchers invoke“microclump-
Observationally, the evidence for structured massive-star ing”.Microclumpingisthelimitinwhichtheradiativetrans-
winds is multi-wavelength in nature,includingX-rays(e.g., feris not impacted bythepresenceof clumping.Formicro-
Cassinelli & Swank 1983; Feldmeier et al. 2003; Naz´e et al. clumping all clumps must be optically thin, and the radia-
2013), the UV band, in the form discrete/narrow ab- tivetransfercalculationsproceedessentiallyasiftheflowis
sorption components (e.g., Prinja & Howarth 1986; “smooth”.Theinfluenceofmicroclumpingemergesinterms
Massa et al. 1995; Prinja et al. 2012), and attempts to ofthevolumefillingfactor ofclumpsfV,oran emissiveen-
hancementfactor(whichis1/f ,e.g.Hamann & Koesterke
V
1998;Ignace et al.2003).Thepresenceofclumpingleadsto
⋆ E-mail:[email protected] anincreaseofthesourcefluxrelativetoasmoothwind,and
(cid:13)c 2015TheAuthors
2 R. Ignace
for microclumping the enhancement is unconnected to the sion of the problem begins with a review in 2 of long-
§
specificsoftheclumpgeometry,suchaswhetherclumpstake wavelength emissions from free-free opacity in a strictly
the form of spheres, filaments, flattened “pancake” struc- smoothwindandinonewithmicroclumping.anexploration
tures, or anyother shape. of porosity effects for the IR/radio band in 4 for the two
§
By contrast, macroclumping deals with the case when different selected geometries. Concluding remarks are given
the radiative transfer is influenced by the geometry of in 5.
§
the clump structures (e.g., Brown et al. 2004b). For ex-
ample, based on the expectations of the line-driving in-
stability operating in massive star winds, Oskinovaet al. 2 SEDS FROM SMOOTH OR EFFECTIVELY
(2004) invoked“shell fragments”, in the shape of flattened SMOOTH WINDS
pancakes, to model a time-averaged spherical distribution
Theobjectiveof thiswork istoexploretheconsequencesof
of wind shocks to explain resolved X-ray emission pro-
macroclumping for IR/radio spectral energy distributions
file shapes observed in massive stars. In a smooth wind,
(SEDs) from ionized stellar winds. To do so, a number of
the line shapes should be significantly asymmetric (e.g.,
simplifications are imposed to aid comparison of cases. For
Ignace2001;Owocki & Cohen 2001;Ignace & Gayley 2002;
example, the model SEDs are restricted to free-free opac-
Leutenegger et al. 2007). Clumping can alter the line pro-
ity. It is possible to include bound-free opacity as well, but
file shape for the emergent X-ray radiation. In particular,
at long wavelengths, its addition will have a similar scaling
it can reduce line asymmetry (e.g., Feldmeier et al. 2003;
as the free-free opacity and will not impact qualitative and
Oskinovaet al. 2004; Owocki & Cohen 2006).
comparative trends described here. A focus on long wave-
InrelationtotheUV,initialresultsfromaFUSEstudy
of the Pv doublet from massive star winds suggested dra- lengthsensuresthatthewindwillbeopticallythick,sothick
that a pseudo-photosphere forms in the wind. In this case
matic reductions in mass-loss rates of O star winds owing
the continuum emission formed in the wind dominates the
to severe levels of clumping (Fullerton et al. 2006). The ar-
highlyabsorbedstellaremissionatthewindbase.Although
gument was that mass-loss rates from line profile fitting of
massive star winds have overall declining temperature and
UV lines is independent of wind clumping for an ion in its
dominant stage. Comparing M˙ values from Pv with those varyingionfractions(e.g.,Drew1989),thewindatlargera-
diustendstobemorenearlyisothermalwithslowlyvarying
derivedfromclumping-dependentdiagnostics(suchasradio
ion fractions. For thisstudyit is convenientto assume that
emissions) provides a measure of the volume filling factor
the wind is isothermal with fixed ionization. Also, the flow
for microclumping. However, the downward revisions were
is assumed to expand radially at constant speed (i.e., the
so severe as to change the expectations of stellar evolu-
tionmodelssubstantially(Hirschi2008).Oneresolution1 to wind terminal speed, v∞).
Central to modeling the SEDs is the frequency-
theproblemwasfoundinaconsiderationofmacroclumping
(Oskinovaet al. 2007; Sˇurlan et al. 2013). dependentoptical depth,τν. The optical depth from a dis-
tantobservertoapointinthewindthatliesalongtheline-
This paper extends consideration of macroclump-
of-sight (los) to thestar centeris given by
ing effects to the IR and radio bands. Others have
also explored macroclumping for radio emissions, such as
∞
Blomme & Runacres (1997) and Gonza´lez & Cant´o (2008). τν = κνρR∗dr˜′. (1)
Theformerconsideredtheimpactofdiscreteshellsandshell
Zr˜
fragmentsonthelongwavelengthemission(butonlyforsin-
where κ is the absorption coefficient and prime indicates
ν
gle shells or fragments). The latter considered an evolving
a“dummy variable”for integration purposes. Tildes signify
shell,withapplicationtowardexplainingvariableradioemis-
lengthsthatarenormalizedtothestellarradius,asforexam-
sions from PCygni. ple ˜l = l/R∗. The one exception will be the inverse radius,
In this paper two particular clump geometries are
u = R∗/r, that will be used in a number of the integral
constrasted: shell fragments and spherical clumps. Spher-
expressions tobe derived.
ical clumps loosely represent the kinds of structures that
Working at relatively long wavelengths for hot-star
can evolve from Rayleigh-Taylor instabilities. Rayleigh-
winds,theRayleigh-Jeans limit is adopted,for which hν
Taylor instabilities lead to filamentary structures that can ≪
kT,and
further devolve into “knots” and roundish clumps (e.g.,
Ellinger et al.2012).Forexample,sucheffectscanhaverel-
Z2 ρ2
evance for the interaction of a wind and the interstellar κ ρ=0.018 i T−3/2g ν−2, (2)
ν µ µ m2 ν
medium (e.g., Mohamed et al. 2012). The shell fragments i e H
have long been a favorite for the highly supersonic massive whereZ isthermsioncharge,µ andµ aremeanmolecular
i i e
star winds that are thought to be filled with shock struc- weights per free ion and per free electron, respectively, ρ is
tures(e.g.,Runacres& Owocki2002),althoughmorerecent themassdensityof thegas, m isthemassof hydrogen,T
H
simulations and observations now favor spherical clumps isthegastemperature,g isthefree-free Gauntfactor, and
ν
(Dessart & Owocki 2003; Leuteneggeret al. 2013). Discus- ν isthefrequencyofobservation.Theadoptedassumptions
imply that Z, µ, and µ are constants in the flow for this
i i e
study.
The radial optical of equation (1)can beexpressed as
1 Macroclumpingisnottheonlypotentialresolution.Thepres-
enceofX-rayemissionscouldinvalidatetheassumptionthatPv
∞
isindeedthedominantionstageforthatelement,asforexample τ =τ (λ) [ρ(r˜′)/ρ ]2dr˜′, (3)
Waldron&Cassinelli(2010)andKrtiˇcka&Kub´at(2012). ν 0 0
Zr˜
MNRAS000,1–??(2015)
Long-Wavelength, Free-Free Spectral Energy Distributions from Porous Stellar Winds 3
where ρ is the density at the base of the wind, and τ = The fluxthen becomes
0 0
κ0(λ)ρ0R∗isthecharacteristicopticaldepthscaleasafunc-
tion ofwavelength,with κ0(λ)=κ(λ)ρ(r˜)/ρ0.Thelatteris F = Lν = 1B (T)r2 g2/3λ−2/3. (8)
given by ν 4πD2 4 ν eff ∝ ν
With g λ0.11 in the radio band (Cox 2000), the radio
ν
∝
τ (λ)=5 105 Zi2 ρ2T−3/2 R∗ g λ2 , (4) SED is a power law with a logarithmic slope exponent of
0 × µiµe 0 0 (cid:18)10R⊙(cid:19) ν cm about−0.6withλ.Observationsofthisvaluegenerally sig-
nal that the wind is isothermal, spherical, and at termi-
where fiducial density and temperature values of ρ0 = nal speed. Slight deviations, especially somewhat steeper
10−13 gcm−3 and T0 =104 K havebeen assumed,thestel- negative slopes, may indicate variations in the tempera-
lar radius has been expressed in terms of ten solar radii, tureorionizationofthewind.FortheRayleigh-Jeanslimit,
and the wavelength is given in centimeters. For a wind of Cassinelli & Hartmann(1977)generalizedtheirresultstore-
completely ionized, pure hydrogen, ρ0 would correspond to lateanobservedSEDslopeintermsofpower-lawexponents
anumberdensityofabout1011cm−3,whichischaracteristic for thedensity and temperaturedistributions.
of a wind with a mass-loss rate of M˙ 10−6 M⊙ yr−1 and Alternatively, deviations from the standard value of
terminal speed 103 km s−1, a fairly m∼assive, evolved wind 0.6 may also indicate that the free-free emission forms
from an OB supergiant (e.g., Muijres et al. 2012). For such i−n the wind acceleration zone, whoch is relevant to lower
awind,whichistypicalofthemoreextremewindcaseslike density winds. A rising velocity steepens the density with
WR stars, some OB supergiants, or LBV stars, the optical consequence for the observed SED. A positive slope would
depthat 1 cm is enormous at thelevel of 106. benon-thermal,generallyinterpretedasrelatedtosynchro-
tonemissionandthepresenceofmagnetismintheextended
wind (e.g., Blomme 2011).
2.1 Smooth Winds In this paper the wind is assumed isothermal, with an
ionization structure that is constant throughout, and for
Seminal works for the radio emissions from ionized,
simplicity the sphereical wind will be taken to expand at
massive-star winds include Wright & Barlow (1975) and
constant speed.In thissituation theformal solution for the
Panagia & Felli (1975). Both considered how to relate the
optical depth along any sightline traversing the spherical
radio emission to the wind mass-loss rate, assuming a
wind is given by
smooth, spherical wind. The key results are that for an ob-
server sightline through the spherical wind, the emergent
τ (λ)
intensity is τ(z,p˜)= 0 [θ sin(θ) cos(θ)], (9)
2p˜3 −
where z˜=p˜/tan(θ).
Iν =Bν(T) 1−e−τtot(p˜) , (5) In the limit of large optical depth with τ0 ≫ 1, the
h i “pseudo-photosphere” formed by the wind completely ab-
where isothermality is assumed, τtot is the total optical sorbs the direct emission by the stellar photosphere, and
depthofthewind alonga rayof normalized impact param- thecontinuumemission isconsidered“wind-dominated”.At
eter p˜, and Bν is the Planck function. For a star of radius wavelengths where τ0 1, the wind emission is small, the
R∗ atdistanceDfromEarth,theemergentfluxofradiation absorption of direct st≪arlight is negligible, and the contin-
from thewind is then uum is said to be“star-dominated”. Although model SEDs
will properly account for the starlight component, the dis-
R2 ∞ cussion of fluxes will generally focus on the wind contribu-
Fν =2π D∗2 Bν(T) 1−e−τtot(p˜) p˜dp˜. (6) tion and ignore thethestellar contribution.
Z0 h i Theintegrationforfluxrequiresthetotalopticaldepth,
At long wavelengths where the wind density is an inverse τ for a los of fixed impact parameter p˜. Its value is given
tot
square law, and the wind emission can be considered to by equation (9) for θ =π/2, with τ =πτ /2p˜3. Ignoring
tot 0
dwarf the stellar emission (which is also highly absorbed), the stellar contribution as justified above, the solution for
the optical depthfactor takes on the form τtot =K(λ)p˜−3, thefluxof emission by thewind is
where the stellar, wind, and wavelength parameters of the
oInpatchiitsycaarseectohlelecintteedgirnaltchaenwbaevealnenaglytthi-cdaellpyenevdaelnutaftaecdt.orK. Fν ≈2π DR∗22 Bν(Tw) ∞ 1−e−πτ0/2p˜3 p˜dp˜, (10)
Cassinelli & Hartmann (1977) described how free-free Z0 (cid:16) (cid:17)
Appendix A) details the steps to obtaining an analytic so-
fluxthatformsinthewindcouldbeinterpretedintermsof
lution for thisintegral, with theresult,
a pseudo-photosphere. Their argument was to evaluate an
effective photospheric radius r where τ = 1/3 along the
eff ν
los. From equation (3),one obtains 2 1 πR2 πτ (λ) 2/3
F =F = Γ ∗ B (T ) 0 , (11)
sm ν 3 3 × D2 ν w 2
(cid:18) (cid:19) (cid:20) (cid:21)
reff =[3τ0(λ)]1/3 R∗ ∝gν1/3λ2/3R∗. (7) wsuhlte,reanthdeΓsuibssctrhiept“G“samm”msaig”nfiufinesctitohne. sWmroiogthht &wiBnadrlorew-
Using fiducials of ρ = 10−13 g cm−3, T = 104 K, (1975)showedthatthisexpressioncorrespondstoapseudo-
0 0
and assuming g = 1, the effective radius of the pseudo- photosphere of effective radius where the line-of-sight op-
ν
photospherewill bereff 80R∗ at a wavelength of 1 cm. tical depth achieves τ(λ) = 2π2/81 = 0.244. Hillier et al.
≈
MNRAS000,1–??(2015)
4 R. Ignace
(1983) considered a different measure. They estimated the where τ is the optical depth for a smooth wind of the
sm
optical depth at which a volume integral of the emissivity same M˙, R∗, and wind velocity profile. The expression for
wouldachievetherequiredflux.This“effectivevolume”cor- thefluxofemission,whenthewindisopticallythickfollows
rrespondstoanintegrationofthefree-freeemissivitybeyond from equation (11):
τ(λ)=(2/3π)(3/8)3/2 =0.049.
1
F = F (15)
2.2 Winds with Microclumping mic f2/3 × sm
V
A feature of the radio-band emission is the opportunity to
which is the well-known result that for a given mass-loss
infer the mass-loss rate of the wind. The scale of the flux
rate,microclumpingenhancesthefluxofemissionbyafactor
is set by (M˙/v∞)2/3. Since thewind terminal speed can be f−2/3.Toignorewindclumpingwouldimplyoverestimating
independentlyconstrained from wind-broadened lines (e.g., V
thewindmass-lossratebyafactorf−1/2,sinceF M˙4/3
iPnrfienrjaweintdal.M˙19v9a1l)u,eso,bwsehrvicehdhraasdiboeeflnuxpeusrscuaendbbeyunsuemdetro- and Fmic∝(M˙2/fV)2/3. V sm ∝
However, there is the possibility that the clumps are
ousresearchers(e.g.,Abbottet al.1986;Bieging et al.1989;
not optically thin, in which case the radiative transport
Leitherer et al.1995).However,evidencefortheinfluenceof
musttakeaccountoftheclumpgeometry(e.g.,Brown et al.
clumping effects became apparent through a variety of di-
2004a), a situation that has been dubbed macroclumping
agnostics, as described in 1. The question arises of how
§ (e.g.,Oskinovaet al.2007).Theopacityatlongwavelengths
clumpingimpacts thelong-wavelength emissions.
scales as λ2. Clump structures that are optically thin at
Manyhaveadoptedamodelof“microclumping”toinfer
better M˙ values from radio studies of dense winds. Micro- shorter wavelengths can become optically thick at longer
ones,thusthedegreetowhichstochasticallystructuredflows
clumpingisthelimitinwhichallindividualclumpstructures
may be treated as microclumping versus macroclumping is
are optically thin. The radiative transport through such a
a λ-dependentconsideration.
flow is the same as for a smooth wind, but with a correc-
tionfactor.Thusmicroclumpingis“effectivelysmooth”inits
treatment of the flux calculation. Microclumping does not
altertheSEDslopeascomparedtoasmoothwindapproach
(e.g., Nugis et al. 1998), but it does alter inferred values of
M˙.
Becausethefree-freeopacityscalesasρ2,clumpinggen- 3 WINDS WITH MACROCLUMPING:
SPECIAL CASE WITH DISCRETE CLUMPS
eratesmoreopticallythinemissionperunitvolumeascom-
paredtoanunclumpedwind.Definingf asthevolumefill-
V The radiative transfer through macroclumps requires spec-
ing factor for the clumping, mass-loss rates are lowered by
ification of the clumps themselves, as for example the in-
themultiplicativefactor,√f (i.e.,forf constantthrough-
V V vestigations of Feldmeier et al. (2003) and Oskinovaet al.
outthewind),ascompared tonoclumping.This isderived
(2004).Ultimately arealization of thestructuredflowmust
next,and will be used as a reference against which to com-
beimposed,andthentheradiativetransportcanbenumer-
pare macroclumping results.
ically evaluated by shooting rays through the medium to
Let ρ be the volume-averaged wind density. Let ρ
h i sm computeemergentintensitiesandthenfluxesforunresolved
betheunclumped,smoothwinddensityforasphericalwind
sources. Oskinovaet al. (2004) use a discrete approach for
with
thesimulation of X-ray observables.
An alternative approach has been described under the
M˙ heading of porosity. Owocki & Cohen (2006) simulate the
ρ = , (12)
sm 4πr2v(r) effects of macroclumping with integral relations based on
probabilistic considerations associated with the radiative
where v(r) is the velocity law of the wind. The density of
transfer. The approach relies on constructing an effective
clumpswith microclumping becomes
opacity for expressing the consequences of macroclumping
(e.g., Sundqvistet al. 2012, 2014). The porosity formalism
1 M˙ is adopted in the radiative transport for calculations of the
ρ = . (13)
mic fV × 4πr2v(r) free-free fluxesfrom winds.
However, before pursuing a solution for the full radia-
Theformofthewindspeedwillbeirrelevant,asthefocusof
tion transport through theclumped outflow, thereis a spe-
applications is on asymptotic results for thick winds where
cialcircumstancethatcan beinstructivetoconsider.Imag-
theobservedfree-freeemissionformsatlargeradii.Atlarge
ine that clumped structures are sparsely distributed such
radius, v(r) v∞. thatfromtheobserverpoint-of-view,noclumpsoverlapeach
≈
When calculating the optical depthintegral, thefactor
otherasprojectedonthesky.Furthersupposethatthesolu-
κ ρ ρ2 f−2. However, the clumps are encountered
ν ∝ mic ∝ V tionforagivenclumpisδFν,i.Thenthefluxfortheensemble
only over a fraction of the integration path length, namely
total becomes
thefraction f .Consequently,for microclumping, theopti-
V
cal depthbecomes
N
F = δF . (16)
ν ν,i
τ =τ /f , (14)
mic sm V i
X
MNRAS000,1–??(2015)
Long-Wavelength, Free-Free Spectral Energy Distributions from Porous Stellar Winds 5
Inthelimit thatclumpsarenumerous,thesumcanberep-
resented in integral form:
F = n (r˜,µ,φ)R3δF (r˜,µ,φ)r˜2dr˜dµdφ, (17)
ν cl ∗ ν
Z
where µ = cos(θ) and n is the number density of clumps
cl
distributed throughoutthewind.
Themodelsunderconsiderationwillbeforlargeoptical
depthsforasymptoticconditionswiththeemission forming
at largeradiuswherethewindisat terminalspeedv∞.For
a uniform distribution,thenumberdensity of clumps is
N˙
n = 0 , (18)
cl 4πr2v∞
whereN˙ istheinjectionrateofclumpsintotheflowatthe
0
wind base.
Note that the complete absence of overlap among the
ensembleisactuallytoostrongarequirement.Whatmatters
isthatthecumulativeopticaldeptharisingfromanyoverlap
be optically thin. This was the limit implicitly used in the
application by Ignace & Churchwell (2004, hereafter “IC”)
to explain radio SEDs from ultracompact Hii regions. This
Figure 1. Geometry for the shell fragment clump. In cross-
special case will be referred to as the“sparse”limit, even
section a shell fragment is a thin line that is normal to a radial
though the overlap of structures that are optically thin is fromthestar.Heretheobserverisalongthez-axisandviewsthe
formally allowed. shellfragmentatanobliqueangleθ.
Inordertoemploythesparselimit expressions,thena-
ture of the macroclumps must be specified. Here two cases
Feldmeier et al. (2003) described the radiative trans-
are considered: shell fragments and spherical clumps. Both
port of X-ray emissions through shell fragments. Although
willbetakenashavingconstantdensitywithintheirbound-
at IR and radio wavelengths, the opacity is different (ρ2
aries,butwithdensitiesthatdependonlocationinthewind
for free-free opacity versus ρ for photoabsorptiveopacity of
flow.Asphericalclumpoffersthesamefluxnomatterfrom
X-rays),thegeometricalconsiderationsaresimilar.Oneim-
what direction it is viewed, but the intensity distribution
portantdifferenceisthatfortheX-rayproblem,thepancake
across the clump is not uniform (i.e., spherical clumps are
structuresconsist of a hot plasma component adjacent to a
brighter along their diameter than at their limb). By con-
cool one. It is the cool one that does the photoabsorbing;
trast the shell fragments will be treated as circular “pan-
the hot plasma produces optically thin emission. For the
cake”shapes.Theprojected shapeofashellfragment when
free-free situation, the emission and absorption both occur
viewedobliquelyiselliptical.Incontrasttosphericalclumps,
in thesame material, in this case thecooler component2.
the intensity across a shell fragment is uniform (i.e., ignor-
Following Feldmeier et al., the optical depth through a
ingedge effectsgiven that thefragments are assumed to be
shell fragment along a ray that intersects it at an oblique
geometricallythin).Evidentlydifferentclumpstructuresof-
angle θ from thenormal will begiven by
fer different source properties, and these can impact their
ensemble trends.
τ
τ = r , (19)
z µ
3.1 Sparse Flattened Shell Fragments | |
where µ = cosθ, z signifies the observer axis (see Fig. 1),
Shellfragmentclumps,withoutwardnormalsdirectedalong and τ is the optical depth normal to the shell fragment.
r
radials from the star, may represent compressed structures Note that this optical depth is taken as a constant across
from shocked gas. One-dimensional, time-dependent, hy- thefragment.
drodynamic simulations predict highly structured winds in Fortheopticaldepthalongaradialthatpassesthrough
terms of dense shells separated by zones of highly rarified ashellfragment,itistemptingtoassumethatthedensityis
gas. inversesquarebyassumingthatthefragmentevolvethrough
However,the1DmodelsoverpredictthevariabilityofX- theflowatconstantsolidangle,δΩ.However,thisisnotob-
rays from massive stars. Although results for ζ Pup clearly vious.Thefactthatthefragmentrepresentsashockimplies
indicate X-ray variability, the number of discretized, non-
spherical clumps implied seems large (Naz´eet al. 2013).
2 Sincethefree-freeopacitydropsasT−3/2,andtheX-rayemit-
Two-dimensional simulations by Dessart & Owocki (2003,
tingplasmaistwoordersofmagnitudehotterthanthecoolergas,
2005) suggest that a highly structured flow can form. Al-
any hot component existing at large radius as an“inter-clump”
though3Dmodelshavenotbeenpresented,1Dand2Dsim-
component will be optically thin compared to the cool clump
ulations have led to a picture of a wind flow in which the component. Also, the optically thin free-freeemissivityscales as
majorityofthematterexistsinafairlyrandomdistribution density squared. The hot component can reasonably be ignored
of clumps. asacontributor tothelong-wavelength free-freeemission.
MNRAS000,1–??(2015)
6 R. Ignace
that matter may accumulate in it, which would make the Evaluating the integral in µ and manipulating the integra-
density less steep than inverse square (e.g., Gayley 2012). tion constants yields
On the other hand, the lateral width of the fragment can
expandowingtogas pressure,resultingin an evolvingsolid 2q/3 3 δΩR2 N˙
angle for thefragment. Fν = 4q 3Γ 1− 2q τ03/2q×πBν D2∗ v∞/0R∗. (26)
Forsimplicitythedensitywithinafragmentistakento − (cid:18) (cid:19)
decline with the radius of its center as a power law, with Note that with q =2 for an inverse square law density, the
r˜−q. The optical depthis SED would be Fν ∝ gν0.75λ−0.5 ∼ λ−0.42. This is a fairly
shallow SED slope. The well-known SED slope of 0.6 for
−
a smooth wind occurs when q 5/4.
τ =τ (λ)µ−1δr˜−2q, (20) ≈
z 0
where
3.2 Sparse Spherical Clumps
The results presented here for spherical clumps are an ex-
τ0 =κ0ρ0R∗δr˜0, (21)
tension of the method discussed in IC, who considered an
with δr0 the radial width of a fragment at the base of the application of the free-free emission from constant density,
wind. spherical clumps to explain anomalous SED slopes seen in
Theintensityalongaraythatisobliquethroughafrag- some ultra-compact H II regions. Here the approach of IC
ment is given by: is applied to a time-averaged spherical wind. Unlikethe in-
terstellar situation, application to a stellar wind imposes a
I =2πB 1 e−τ0µ−1r˜−2q . (22) characteristic optical depth distribution for an ensemble of
ν ν
− clumps owing to the nature of the clumps partaking in a
Again, thishassumes that theiradial optical depth τ is con- spherical outflow. Consider the SED from a single isolated
r
stant across the face of the shell fragment. The flux is clump.ICshowed(alsoinOsterbrock1989)thatthefluxof
δF =δΩµI (r2/D2). emission from a single clump is given by
ν ν
Givenatime-averagedsphericallysymmetricandsparse
distributionofclumps,andusingequation(17),thesolution R2
forthetotalfluxfrom an ensembleofsuchstructures,allof Fν = D∗2 πBνG(τcl), (27)
thesame solid angle extent,becomes
where τ is the optical depth along the diameter of the
cl
spherical clump which κ ρ constant within its interior, as
ν
F =B δΩ R∗2 N˙0 1 ∞ 1 e−τ0µ−1r˜−2q r˜2dr˜. given by
ν ν D2 v∞/R∗ Z0 Z1 h − i
(23) τ =2κ (ρ ,λ)ρ R , (28)
cl ν cl cl cl
Theratiov∞/R∗isaflowrateofclumpsoverthescaleofthe Note that equation (27) is based on equation (4) from IC,
stellarradius,whichisthescaleofthewinddensity,whereas but corrects for an errant factor of 2 that should not have
N˙0 is the injection rate of clumps into the flow. Hence the been present in IC.
ratioN˙0/(v∞/R∗)representsarelativeclusteringofclumps The function G(τcl) is given by
along a radial.
Inequation(23),theintegrationoverµreflectsthefact
2
that the circular shell fragments, when seen oblique to the G(τcl)=1− τ2 1−(1+τcl)e−τcl . (29)
normal, appear as ellipses of uniform brightness I (for a cl
ν (cid:2) (cid:3)
given fragment) and area µδΩR∗2. Note that the optical Atlarge opticaldepthwith τcl ≫1,G≈1.Attheopposite
depth appears to diverge as µ 0, which is unphysical; extreme of τcl 1, G 2τcl/3, which is the area-averaged
→ ≪ ≈
however, in this model such edge-on fragments also have valueacross thesphere.
zero area and so nevercontributeto theflux. The optical depth τcl along the diameter of a clump
The solution for thefluxwould beanalytic if thelower located at radiusr˜can be expressed as
limit for the integration in radius were to extend to zero.
Asinpreviousworks(c.f.,Wright & Barlow1975),changing ρ2 R
τ (r˜)=τ (λ) cl cl , (30)
thelowerlimitfrom1to0isacceptableforlongwavelengths cl 0 ρ2R
whereτ 1. Ageneral solution tointegrals with theform (cid:18) 0 0 (cid:19)
0
≫ where ρ and R are fiducial values. The density within
of equation (23) is 0 0
theclumpis constant,butthequestionof howthatdensity
varieswithradiusinthewindmustbeaddressed.Onemight
∞ 1 e−ax−b r2dr=a3/b −Γ(−3/b) , (24) consider spherical clumps to maintain constant solid angle,
− b
Z0 h i (cid:20) (cid:21) likethefragmentclumpcase.Inordertomaintainspherical
if b>3. Forthe application of interest, thesolution is shape, this would imply that Rcl r, and so ρcl r−3.
∝ ∝
However,thereferencecaseinvolvesadensitythatdropsas
r−2.Imposingρ r−2meansthatsphericalclumpsdonot
Fν =πBν δΩDR2∗2 v∞N˙/0R∗ Z01 (cid:18)τµ0(cid:19)3/2q (cid:20)−Γ(−2q3/2q)(cid:21) µdµ. expaTnodaatllocwonsftocarln∝tgrseoaltiderangegnlee.rality, the form ρcl ∝ r˜−m
(25) is adopted. Mass conservation for a clump requires that
MNRAS000,1–??(2015)
Long-Wavelength, Free-Free Spectral Energy Distributions from Porous Stellar Winds 7
R3ρ = constant, and so R r˜m/3. As a result, the is set at low and modest values of τ . At long wavelengths
cl cl cl ∝ cl
position-dependent optical depth of a spherical clump be- theintegraliswell-approximatedasaconstant(butonethat
comes dependson m),and theSED slope is determined primarily
by the factors outside the integral, which for m = 2 and 3
bracket thecanonical result of 0.6 for microclumping.
τcl =τ0(λ)r˜5m/3. (31) Note that these results a−ssume a universal size for
Note that spherical clumps have been used in considera- clumps at the inner wind radius. It is straightforward to
tionsbyOskinovaet al.(2007)andIgnace et al.(2012).Itis introduce a distribution of clump sizes. Suppose that cl is
P
not uncommonfor aspherical shellto“fragment”intoelon- theprobabilityforaclumptohavearadiusRcl intheinter-
gated features ala a Rayleigh-Taylor instability, and these val of Rmin to Rmax. Then the flux for an ensemble of sizes
in turn can further break up into ball-shaped structures. would proceed as follows.
Dessart & Owocki(2005)observethiseffectintheir2Dsim- The result of equation (34) is now indicated by Fˆν for
ulationsfortime-dependentline-drivenwinds.Theirmodels theSEDcontributionfrom clumpsthathaveasizeRcl.For
show that wind structures develop rings that are round in a range of sizes, thefluxbecomes
cross-section, rather than flat, lending support to the idea
that roughly spherical clumpscould form. Consequently,at Rmax
anygiventime,asnapshotoftheflowwouldpossessacom- Fν = (Rcl)Fˆν(Rcl)dRcl, (38)
P
ponent comprised of spherical clumps. ZRmin
For the case of sparse clumps, the total emergent flux where theprobability distribution is normalized, with
from theensemble is:
Rmax
Fν = πBν RDc22l ∞ ncl(r˜′)G(τcl)4πr˜′2dr˜′ (32) ZRmin P(Rcl)dRcl =1. (39)
Z1 The influence of a range of sizes affects the SED cal-
= πBν RDc22l v∞N˙/0R∗ Z1∞ G[τcl(r˜′)]dr˜′. (33) icnulFat0i,osnoinsmtawlolerwcalyusm. Fpsirstte,ntdhteorebies laenssobverirgahllt.faScetcoornodf,Rasc2-l
At this point it is useful to adopt a change of variable sumingthatallclumpshavethesameinitialdensity,smaller
fromnormalizedradiusxtoopticaldepthτ fortheintegra- clumps tend to be more optically thin, which makes them
cl
tion.Letthelargestvalueofτ beτ foraclumplocated less bright, and further means that they become optically
cl max
at x = 1. Then x = (τ /τ )−3/5m. The integration that thickat longer wavelengths as compared to bigger clumps.
cl max
wasovertheclumpensembleinspatiallocationbecomesone Figure 2 shows SEDs for a clump distribution, cl
over the clump ensemble in optical depth space (similar in R−β, with values of β = 0,1,2, and 3, from top toPbo∝t-
cl
spirit tothe work of IC),with tom.Theexamplesaredesignedtodisplaytherelativeeffect
from having a greater representation of small clumps. The
3 τmax magentaindicatestheasymptoticSEDvalue.Thesemodels
Fν = 5mF0(λ)τmγ−ax1 τc−lγG(τcl)dτcl, (34) are for m=2. An ensemble of clump sizes does not change
Z0 the SED asymptote, only the brightness level and where in
where wavelengththewindemissionbeginstodominatethestellar
component.
3+7m
γ = , (35)
5m
and 4 WINDS WITH MACROCLUMPING:
POROSITY APPROACH
R2
F =πB cl. (36) The preceding discussion of thesparse limit shows that the
0 ν D2
geometry of the clumps can affect both the SED slope and
Oneoftheinterestingattributesofequation(34)isthe thefluxlevel. However, onegenerally expectsthat multiple
wavelength dependence of the factors appearing in front of clumpswillliealongagivensightline.Insuchcasesofover-
theintegral.IntheRayleigh-Jeanslimit,theproductofthe lap,theradiativetransferismorecomplicated.Theemission
Planck function and τmax yieldsa power law with by each rearward clump along a los must be attenuated by
alltheclumpsinterveningbetweenitandtheobserver,with
B τγ−1 gγ−1λ−4+2γ. (37) corresponding emission increments for each.
ν max ∝ ν Severalresearchershaveconsideredtheradiativetrans-
The integral itself contributes an additional wavelength de- fer in porous, massive star winds both for diagnos-
pendence via the upper limit τ . Given that m = 2 and tics such as X-ray and UV lines (Feldmeier et al. 2003;
max
m = 3 are the relevant density power-laws of interest, γ Oskinovaet al.2004;Owocki & Cohen2006;Oskinovaet al.
equals 1.7 and 1.6, respectively. The factors outside the in- 2007; Sundqvistet al. 2012, 2014), and in terms of wind-
tegral, using g λ0.11, give corresponding wavelength de- driving physics such as super-Eddington and line-driven
ν
pendencesofλ−0∝.53 form=2andλ−0.72form=3.Atlong flows (Owocki et al. 2004; Sundqvistet al. 2014). Here the
wavelengthssuchthatτ 1,theintegralfactorhasonly useoftheporosityformalismisbroughttobearontheprob-
max
a weak dependenceon λ. Th≫e reason is that thefactor τ−γ lem of continuumfree-free emission.
cl
intheintegranddominatessuchthatmostoftheintegration The porosity approach is a way of expressing discrete
MNRAS000,1–??(2015)
8 R. Ignace
sents the mean free path between clumps along a sightline.
Forthick clumps, theeffectiveoptical depth becomes
τ = dz˜/h˜. (41)
Z
This amounts to the number of mean free paths along the
los. Consequently, even if clumps are quite optically thick,
it may be that τ can be relatively small if the environment
is highly porous (i.e., h˜ 1).
≫
4.1 Porosity with Fragments
Consider again the pancake-shaped shell fragment in pro-
jection. For a geometrically thin clump of not overly large
solid angle, such a fragment appears elliptical in shape and
is uniformly bright across its projected face. The porosity
length for a distribution of fragments is
h˜ = 1 = v∞/R∗ 4π . (42)
nclµAclR∗ (cid:18) N˙0 (cid:19) (cid:18)µδΩ(cid:19)
Theshellfragments haveaporosity lengththatdependson
Figure 2. SEDs for spherical clumps in the sparse limitwith a µ,orlocation aroundthestar.Fortheouterwindthatisat
distribution in clump sizes Rcl. In these four models, β signifies terminalspeed,h˜ doesnotvarywithdistancefromthestar,
power-lawdistributionsforclumpsizes.Thecaseofβ=0isaflat
but does vary with location about the star. The associated
distribution(clumpsofdifferentsizesareallequallylikely,within
volumefilling factor for shell fragments is
aspecifiedrange);increasingβresultsinincreasinglymoresmall
clumpsascomparedtolargeones.Allthemodelshavethesame
numberofclumpsandsameoptical depthparameter. N˙ µδΩ
fV =ncVc =ncAcR∗δ˜l0 = v∞/0R∗ 4π δ˜l0, (43)
(cid:18) (cid:19)
where V is a volume over which the filling factor is deter-
c
mined, and δ˜l is the normalized radial width of the frag-
0
ment. This width could be a function of radius, but will be
sumsoverclumpstructuresfortheradiativetransferalonga
taken as constant in thediscussion that follows.
sightlineintermsofintegralexpressions.Theresultreduces
Theexpressionfortheopticaldepthtopositionu=1/x
tothemicroclumpingcasewhenclumpsareopticallythin.It
along thelos tothestar center (i.e., µ=1) is given by
is when individual clumps become optically thick that the
clump geometry must be considered. The ability to move
from discretesumstointegralexpressionsistheoppositeof τ u 1 e−τcl]
τ = 0 − u2du, (44)
the sparse limit: for the sparse limit, the radiative transfer f τ
0 Z0 (cid:20) cl (cid:21)
isconfinedtoindividualclumps;porosityallowsformultiple
where f =f µ,
clumpsalong a sightline. V 0
Feldmeier et al.(2003)showedthattheeffectiveoptical
depthalong a sightline througha porous wind is given by f = N˙0 δΩ δ˜l , (45)
0 v∞/R∗ 4π 0
(cid:18) (cid:19)
and
τ = n R3 1 e−τcl dA˜ dz˜, (40)
cl ∗ cl
−
Z (cid:20)Z (cid:0) (cid:1) (cid:21) τ = τ0 δ˜l0 u4 . (46)
whereA˜cl isthenormalized projectedarea oftheclumpto- cl (cid:18)f0(cid:19) (cid:18)f0 (cid:19) (cid:18) µ (cid:19)
ward the observer. The inner integration is across the face The bracketed factor in equation (44) is the form of an es-
of the clump; the outer is through the wind. The interpre- cape probability. At low values of τ , the factor reduces to
cl
tation is that the product nclR∗3dA˜cldz˜ is the number of unity, in which case the microclumping limit is recovered.
clumps encountered in a differential unit of volume. The Forlargeτ ,thefactorreducesto1/τ ,sothattheeffective
cl cl
factor (1 exp( τcl)) is the fraction of light absorbed by optical depth when clumps become thick can be much sup-
− −
a clump at this location, with τcl the optical depth of the pressedwithmacroclumpingascomparedtomicrcolumping.
clump along the ray. Clearly, if τcl 1, the effective opti- The left side of Figure 3 shows the optical depth for a
≪
cal depth is just the cumulative optical depth for thelos in sightline along a radial to the star based on equation (44).
theintervaldz.However,whenτcl 1,theeffectiveoptical The figure compares the porosity case against the micro-
≫
depthby clumps. clumpingassumption,ignoringthefact thattheclumpsare
Regarding this last point, the normalized porosity opticallythick.Thevolumefillingfactorsaref =0.01(ma-
0
length is defined as h˜ = h/R∗ = (nclAclR∗)−1 and repre- genta), 0.04, 0.09, 0.16, and 0.25 (black), all with τ0 =300.
MNRAS000,1–??(2015)
Long-Wavelength, Free-Free Spectral Energy Distributions from Porous Stellar Winds 9
Figure3. Left:Theline-of-sight(los)opticaldepthasafunctionofnormalizedinverseradiusuascomparedwithmicroclumpingfora
rangeofvolumefillingfactors.Upperpanelisforthefullrangeofufrom0to1;lowerisforsmallu(i.e.,r≫R∗)tohighlighttheregion
where microclumping and porosity diverge. The volume filling factors are for f0 = 0.01 (magenta), 0.04, 0.09, 0.16, and 0.25 (black).
Right:Associatedcontoursforopticaldepthunityinthez−pplane.Thecolorschemeandlinetypesarethesameasintheleftpanel.
clumping and porosity with shell fragments only begins to
developnearopticalunity;departuresbetweenthetwothen
increase toward higher optical depth.
The right side of Figure 3 displays a cross-section of
the axisymmetric optical-depth unity contours in the z p
−
plane. The colors and line types correspond to those of the
left side of the figure. The central hashed region is for a
smooth, unclumped wind with the same value of τ . Note
0
that the contours are fairly closely matched, save for the
“dimple”that results for the shell fragments. That feature
ariseswhereshellfragmentsareseenmorenearlyedge-onso
that theporosity length is large.
Figure4showsmodelSEDsforbothmicroclumpingand
forporositywithshellfragments.Theopticaldepthparam-
eterτ =300isthesameasinFigure3,butagreaterrange
0
of volume filling factors are used to spread out the transi-
tioninwavelengthfromstar-dominantedtowind-dominated
continua. For Figure 4, the filling factor constant ranges
from f =100 (lowest curve) to 10−4 (highest curve). Note
0
Figure4. SEDcalculationsthatcomparemicroclumpingmodels thatthetransition from star-dominatedtowind-dominated
withporositymodelsforshellfragments.Allcurvesareforτ0 = would sample the inner, accelerating portion of the wind,
300, with f0 = 0.0001 (top), 0.001, 0.01, 0.1, and 1.0 (bottom). which is not currently included the model. Such effects are
The two sets of curves are so close as to be indistinguishable in ignored to emphasize comparisons between microclumping
thisplot. and macroclumping effects.
Solid lines are for the shell fragments; dashed lines are for The two sets of SEDs are so close as to be indistin-
microclumpingusingthesameparameters.Theradialwidth guishable in the figure. That microclumping and porosity
ofashellfragmentischosentobeδ˜l =f .Theupperpanel with shell fragments yield essentially identical SEDs aligns
0 0
showscurvesforthefullrangeofu;lowerisablow-uparound well with the fact that the two models have such close los
theregionofopticaldepthunity.Differencesbetweenmicro- optical depthsfor theoptically thin portion of thewind.
MNRAS000,1–??(2015)
10 R. Ignace
Figure5. ThemodelsherearelikethoseinFig.3,butforporositywithsphericalclumps.Themodelshavethesamevalueofτ0=300,
fV ahs the same range of values as did f0, with the same corresponding color designations. Major differences as compared to shell
fragments are (a) the overall lower optical depths achieved at the star, (b) the fact that departures between microclumping and the
porosity start at lower optical depths of only a few tenths, and (c) the crossing of curves, both in the los optical depths (left) and in
somecasesthecontour curves(right).
4.2 Porosity with Spherical Clumps where
The nature of spherical clumps is different from the case of
2
shellfragments.Forspheresthevolumefillingfactorisgiven G(τ )=1 1 e−τcl τ e−τcl . (50)
by cl − τcl − − cl
(cid:2) (cid:3)
The optical depth integral can be more conveniently ex-
pressed as
1 N˙
f =n (r)V = 0 R˜3, (47)
V cl cl 3 v∞/R∗ 0
τ 1 θ 3G(τ )
τ = 0 cl sin2θdθ, (51)
where n r−2, V r2, R˜ is the normalized radius of a f p˜3 2τ
cl ∝ cl ∝ 0 V Z0 (cid:20) cl (cid:21)
clumpasthebaseofthewind.Whereasshellfragmentshave
where as before, tanθ = p˜/z˜. The ratio 3G/2τ acts like
avolumefillingfactorthatdoesnotdependsonradiusfrom c
an escape probability for the clump. When τ 1, the
thestar,butdoesdependon location aboutthestar,owing cl ≪
ratio3G/2τ reducestounity,asexpectedforopticallythin
to their flattened shapes, the f for spheres is the same at cl
V
clumps. When τ 1, the ratio becomes 1/(2τ /3). The
all locations. Therelated porosity length scales as: cl ≫ c
factor of 2/3 arises from area-averaging the optical depth
across theface of thespherical clump.
h˜ = 1 =4 v∞/R∗ 1 r˜2/3 = 4 R˜0 u−2/3. becoAmleosng the los to the star, the optical depth expression
nclAclR∗ (cid:18) N˙ (cid:19) (cid:18)R˜02(cid:19) 3 f0
(48)
τ u 3G(τ )
τ(u)= 0 cl u2du, (52)
whichhastheinversescalingwithradiusastheclumpsolid fV Z0 (cid:20) 2τcl (cid:21)
angle.Consequently,theporositylengthbecomeslargerwith
Theoptical depthacross thediameter of theclump is
increasing distance from the star.
For spheres the lines of sight intersect the spherical
clumpsalong differentchords. Notingthat τcl is theoptical τ = τ0 2R˜cl u10/3. (53)
depthalongthediameterof asphericalclump,AppendixB cl (cid:18)fV (cid:19) (cid:18) fV (cid:19)
detailstheanalyticderivationfortheinnerintegralforequa-
The unusual power-law exponent with radius derives from
tion (40). Theresult, simply stated here, is that
howthesphericalclumpevolvesinsizethroughoutthewind,
as explained next.
To illustrate the effects of porosity with spherical
τ = nclAclR∗G(τc)dz, (49) clumps,Figure5showsthelosopticaldepthintheporosity
Z
MNRAS000,1–??(2015)