On the importance of the wind emission to the optical continuum of OB supergiants
aa r X i v : . [ a s t r o - ph ] J a n Astronomy&Astrophysicsmanuscript no. 8991 c (cid:13)
ESO 2018November 5, 2018
On the importance of the wind emission to theoptical continuum of OB supergiants
M. Kraus , J. Kub´at , and J. Krtiˇcka Astronomick´y ´ustav, Akademie vˇed ˇCesk´e republiky, Friˇcova 298, 251 65 Ondˇrejov, Czech Republice-mail: [email protected]; [email protected] ´Ustav teoretick´e fyziky a astrofyziky PˇrF MU, 611 37 Brno, Czech Republice-mail: [email protected] Received; accepted
ABSTRACT
Context.
Thermal wind emission in the form of free-free and free-bound emission is known to show up in the infrared and radio contin-uum of hot and massive stars. For OB supergiants with moderate mass loss rates and a wind velocity distribution with β ≃ . . . . . λ < ∼ . µ m, is expected. Investigations of stellar and wind parameters ofOB supergiants over the last few years suggest, however, that for many objects β is much higher than 1.0, reaching values up to 3.5. Aims.
We investigate the influence of the free-free and free-bound emission on the emerging radiation, especially at optical wave-lengths, from OB supergiants having wind velocity distributions with β ≥ . Methods.
For the case of a spherically symmetric, isothermal wind in local thermodynamical equilibrium (LTE) we calculate the free-free and free-bound processes and the emerging wind and total continuum spectra. We localize the generation region of the opticalwind continuum and especially focus on the influence of a β -type wind velocity distribution with β > Results.
The optical wind continuum is found to be generated within about 2 R ∗ which is exactly the wind region where β stronglyinfluences the density distribution. We find that for β >
1, the continuum of a typical OB supergiant can indeed be contaminatedwith thermal wind emission, even at optical wavelengths . The strong increase in the optical wind emission is dominantly producedby free-bound processes.
Key words.
Stars: early-type – supergiants – Stars: winds, outflows – Stars: mass-loss – circumstellar matter
1. Introduction
It is well established that for massive stars the appearance of(thermal) excess emission at infrared (IR) and radio wavelengthsis caused by free-free and (to a small fraction also by) free-boundemission generated in their winds (see e.g. Panagia & Felli 1975;Olnon 1975). At radio wavelengths, the free-free excess emis-sion is usually used to derive the mass loss rates of hot and mas-sive stars (see e.g. Lamers & Leitherer 1993; Puls et al. 1996).Waters & Lamers (1984) have investigated this excess emis-sion in detail, especially in the near-IR region. These authorsstudied the influence of free-free and free-bound emission to thetotal continuum and pointed already to the importance of thewind velocity (and hence the wind density) distribution that canseverely alter the wind contribution in the IR.However, the investigations of Waters & Lamers (1984) wererestricted to the IR and radio range, i.e. they calculated the windcontribution for λ > ∼ µ m only, while during earlier studiesBrussaard & van de Hulst (1962) had noted that free-bound pro-cesses might become very important in the optical and UV rangefor temperatures typically found in the winds of hot stars and su-pergiants.Whether the free-bound emission in the wind indeed influ-ences the optical continuum, depends severely on the densitydistribution. For line-driven winds, the density follows from theequation of mass continuity, i.e. it is proportional to the massloss rate, and inversely proportional to the wind velocity. A high wind density can therefore be reached by either a high mass lossrate, or a low wind velocity.For massive stars with pronounced high mass loss rates likeWolf-Rayet stars, Luminous Blue Variables, or the group of B[e]stars, it is well known that the wind not only influences, but evendominates, the optical spectrum. In these objects, the wind isusually optically thick even in the visual range and thus com-pletely hides the stellar spectrum. Some recent examples of thethermal wind influence in the form of free-free and free-boundemission at optical wavelengths have been published, e.g. byGuo & Li (2007) for the case of Luminous Blue Variables, andby Kraus et al. (2007) for a Magellanic Cloud B[e] supergiant.The second density triggering parameter is the wind velocitydistribution. For OB-type stars with line-driven winds, the windvelocity is very often approximated by a so-called β -law, where β is in the range of 0 . . . . .
0. Such a β value causes a rather fastwind acceleration at the base of the wind, and the wind reachesits terminal value within a few stellar radii (see e.g. Lamers &Cassinelli 1999). Consequently, the region of very high densitythat might cause enhanced free-bound emission is restricted toan extremely small volume around the stellar surface. We can,therefore, expect that for OB-type stars, which have only moder-ate mass loss rates and whose wind velocity distributions have β values between 0.8 and 1.0, there will be no noticable influenceof the wind on their optical continuum emission.During the last few years, huge e ff ort has been made to de-termine precisely the stellar and wind parameters of OB-type M. Kraus et al.: On the importance of the wind emission to the optical continuum of OB supergiants supergiants in the Galaxy (e.g. Kudritzki et al. 1999; Markovaet al. 2004; Fullerton et al. 2006; Crowther et al. 2006; Puls etal. 2006), in the Magellanic Clouds (see e.g. Evans et al. 2004;Trundle et al. 2004; Trundle & Lennon 2005), and beyond, e.g.,in M 31 (e.g. Bresolin et al. 2002). Interestingly, many OB su-pergiants are found to have rather high β values (up to β = . β -values, resulting in a strong densityincrease close to the stellar surface due to a much slower windacceleration, should have noticable e ff ects on the thermal windemission not even in the near-IR, but extending also to opticalwavelengths. In this paper, we, therefore, aim to investigate anddiscuss in detail the influence of high β values on the wind emis-sion of OB supergiants, especially at optical wavelengths.
2. Description of the model OB supergiant
The calculation of the continuum emission of a typical OB su-pergiant is performed in three steps: (i) first we calculate thestellar emission of the supergiant with no stellar wind, (ii) then,we calculate the emission of the wind with the stellar param-eters as boundary conditions, (iii) and finally, we combine thetwo continuum sources whereby the stellar emission still has topass through the absorbing wind. A justification of the use ofthis so-called core-halo approximation together with a discus-sion of several other assumptions and simplifications are givenin Sect. 4.
To simulate a typical OB supergiant we adopt the followingset of stellar parameters: T e ff =
33 000 K; R ∗ = . R ⊙ ;log L ∗ / L ⊙ = .
5; and log g = .
4. With these parameters, wecompute the stellar continuum emission of a hydrogen plus he-lium atmosphere, given by the Eddington flux, H ν . These calcu-lations are performed with the code of Kub´at (2003, and refer-ences therein), which is suitable for the calculation of non-LTEspherically-symmetric model atmospheres in hydrostatic and ra-diative equilibrium. For simplification, we assume that hydrogen in the wind is fullyionized while further contributions to the electron density dis-tribution from, e.g., helium and the metals, are neglected. Thismeans that for a given mass loss rate the real number density offree electrons is underestimated. This leads to an underestima-tion of the total wind emission generated via free-free and free-bound processes. However, since we are interested only in thee ff ect on the emerging wind emission caused by di ff erent veloc-ity distributions, such a simplification is reasonable. We furtherneglect electron scattering, and describe the wind zone follow-ing Panagia & Felli (1975) with a spherically symmetric sta-tionary model in LTE. We show the frame of reference used forour computations in Fig. 1. The electron number density distri-bution, n e ( r ), then equals the hydrogen number density distribu-tion, n H ( r ), which follows from the equation of mass continuity, n e ( r ) = n H ( r ) = ˙ M πµ m H r ( r ) . (1)This equation relates the density at any location r in the windto the mass loss rate, ˙ M , of the star and the wind velocity, ( r ).The parameters µ and m H are the mean atomic weight, for which ζ rR* to the observer zsy Fig. 1.
Frame of reference used for the calculations (afterPanagia & Felli 1975). The x -axis is perpendicular to thedrawing-plane.we use a value of 1.4 (i.e. solar composition), and the atomichydrogen mass, respectively.The velocity increase in line-driven winds of hot stars isapproximated with a β -law of the form (see e.g. Lamers &Cassinelli 1999) ( r ) = + ( ∞ − ) (cid:18) − R ∗ r (cid:19) β (2)where R ∗ is the stellar radius, ∞ is wind terminal velocity, and β describes the ’steepness‘ of the velocity increase at the base ofthe wind. The term gives the velocity at the base of the wind,i.e., at r = R ∗ , for which we use the thermal hydrogen velocitygiven by the wind temperature.The wind temperature for hot stars is known to drop quicklywith distance (see e.g., Drew 1989), and a useful description ofthe radial electron temperature distribution as a function of ef-fective temperature and wind velocity has been derived by Bunn& Drew (1992) T e ( r ) = . T e ff − . ( r ) ∞ T e ff . (3)This relation states that in the vicinity of the stellar surface, theelectron temperature is roughly 0 . T e ff , a value that has beenfound and confirmed by other investigators as well (see e.g.Drew 1989; de Koter 1993; Krtiˇcka 2006).For the purpose of our investigation, it is su ffi cient to restrictthe model computations to an isothermal wind, and throughoutthis paper we will use the value of T e = T e ( R ∗ ) =
25 760 K(which is lower than 0 . T e ff ) following from Eq. (3) for our starwith T e ff =
33 000 K. A justification for these assumptions and adiscussion of their reliability are given in Sect. 4.1.The assumption of a constant wind temperature allows for asimplified treatment of the radiation transfer and therefore of theintensity calculation. Consequently, at each impact parameter, ζ (see Fig. 1), the intensity of the wind emission is given by I ν = B ν ( T e ) (cid:16) − e − τ ν ( ζ ) (cid:17) . (4)The optical depth is defined as the line-of-sight integral over theabsorption coe ffi cient of the free-free and free-bound processes, κ ν ( ζ, s ), τ ν ( ζ ) = s max Z s min κ ν ( ζ, s ) ds (5) . Kraus et al.: On the importance of the wind emission to the optical continuum of OB supergiants 3 with the integration limits s min = q R ∗ − ζ for 0 ≤ ζ < R ∗ , (6) s min = − q R − ζ for R ∗ ≤ ζ < R out , (7)and s max = q R − ζ , (8)where R out is the outer edge of the ionized wind. The total ob-servable flux at earth of the wind continuum emission followsfrom the integration of the wind specific radiation intensity (4)over the wind zone projected to the sky F ν, wind = π d ζ max Z B ν ( T ) (cid:16) − e − τ ( ζ ) (cid:17) ζ d ζ (9)where ζ max = R out , and d is the distance to the object.With the wind absorption along each line of sight, i.e., eachimpact parameter ζ , the stellar flux passing through the windzone can be calculated from, F ν, star = π R ∗ d I ν e − τ ν ( ζ = , (10)with the stellar intensity I ν = H ν . For simplicity, we restrictthe calculation of the wind attenuation to the direction ζ = F ν = F ν, star + F ν, wind . (11) The absorption coe ffi cient of the free-free and free-bound pro-cesses, κ ν ( ζ, s ) (in cm − ), is given by (see e.g. Brussaard & vande Hulst 1962) κ ν ( ζ, s ) ≃ . · n e ( ζ, s ) ν T / (cid:16) − e − h ν kT (cid:17) ( g ff ( ν, T ) + f ( ν, T )) (12)where g ff ( ν, T ) is the Gaunt factor for free-free emission and thefunction f ( ν, T ) contains the Gaunt factors for free-bound pro-cesses. To calculate the absorption coe ffi cient, we need to spec-ify the Gaunt factors for both the free-free and the free-boundprocesses.In the literature there exist several approximations for thecalculation of the free-free Gaunt factors in either the short or thelong wavelength regime. To allow for an appropriate transitionfrom one regime to the other (see e.g. Waters & Lamers 1984;Kraus 2000) we use in the long-wavelength region the relation ofAllen (1973), and in the short-wavelength region the expressionof Gronenschild & Mewe (1978). The resulting Gaunt factorsare in good agreement with those calculated from the approxi-mation given by Mihalas (1967), which is based on calculationsof Berger (1956).In the top panel of Fig. 2, we plotted the free-free Gaunt fac-tors calculated over a large range of wavelengths and for dif-ferent electron temperatures. The curves converge to unity forwavelengths shorter than ∼ µ m, while they increase with wave-length in the IR and radio range, where free-free processes areknown to dominate the continuum emission. The free-free Gaunt Fig. 2.
Top:
Gaunt factors for free-free processes. They con-verge to unity in the optical, but increase with wavelength inthe IR and radio range.
Middle:
The function f ( ν, T ) contain-ing the Gaunt factors for the free-bound processes. The valuesof f ( ν, T ) increase steeply in the optical, while for λ > µ mthey drop quickly and become unimportant compared to the free-free Gaunt factors. Bottom:
Absorption coe ffi cient of the free-free and free-bound processes (see Eq. (12)). Clearly visible isthe growing influence of the free-bound processes in the opti-cal range, especially for decreasing temperature. The line stylesindicate the electron temperature.factors thereby depend only weakly on the chosen electron tem-perature.The situation is completely di ff erent for the function f ( ν, T ),which itself is not a Gaunt factor but that contains the Gauntfactors of the free-bound processes. We calculate this functionwith Eq. (25) of Brussaard & van de Hulst (1962). The factors g n entering this equation are thereby the Gaunt factors for thetransitions from the free level E = h ν / n + h ν to the bound level E = h ν / n . These individual Gaunt factors can, in principle,be calculated with the exact formula given by Menzel & Pekeris(1935) or by the approximation provided by Mihalas (1967). Butit turns out, that these Gaunt factors have values between 0.8 and1.1 in the wavelength range of our interest (i.e. for λ > ∼ . µ m)and converge to unity for λ > ∼ µ m (see Fig. 8 of Brussaard& van de Hulst 1962). It is therefore reasonable to use g n ≃ f ( ν, T ) over the same wavelength range and for the M. Kraus et al.: On the importance of the wind emission to the optical continuum of OB supergiants
Fig. 3.
Continuum emission of a typical OB supergiant (solidline), consisting of the stellar atmosphere having passed throughthe absorbing wind (dotted) and the thermal wind emission(dashed). Also included is the emission from a wind with purefree-free processes (long-dashed line) and the emission of thestar without wind (dashed-dotted).same electron temperatures as the free-free Gaunt factors. Theresults are shown in the middle panel of Fig. 2.For λ > ∼ µ m, the values of f ( ν, T ) are ≪
1. The hydrogenfree-bound processes, therefore, play no role in the IR and ra-dio regimes. At optical wavelengths however, i.e., for λ < µ m, f ( ν, T ) starts to grow steeply, with f ( ν, T ) ≫ ∼
30 000 K, i.e., for values typicallyfound in the winds of OB supergiants. This e ff ect has been men-tioned already by Brussaard & van de Hulst (1962) and Kraus(2000) who pointed to the possible importance of the contribu-tion of free-bound processes to the total continuum emission.The resulting absorption coe ffi cient, κ ν , is shown in the bot-tom panel of Fig. 2 where we plot κ ν / n calculated with Eq. (12)for the di ff erent electron temperatures. Besides the well-knownincrease with increasing wavelength, the absorption coe ffi cientalso peaks at short wavelengths due to the growing influence ofthe free-bound processes.
3. Results
We fix the terminal wind velocity at ∞ = − and themass loss rate at ˙ M = × − M ⊙ yr − . In addition, we place theobject to an arbitrary distance of 1 kpc. The resulting continuumemission is first calculated for the case of β = . β = . β lies typically in the range 0 . . . . . β = . λ > ∼ µ m, while the optical spectrum remains uninfluenced. Thestellar continuum only su ff ers in the IR and radio range, wherefor λ > ∼ µ m the stellar emission is completely absorbed by Fig. 4.
Increase of the fluxes generated within wind zones withouter edge R out , with respect to an infinitely large wind zone. Thethermal wind emission in the optical and near-IR is generatedcompletely in the vicinity of the stellar surface, i.e., within 5 R ∗ only. Table 1.
Literature β values for OB supergiants. References are:M04 = Markova et al. (2004); C06 = Crowther et al. (2006);K99 = Kudritzki et al. (1999); E04 = Evans et al. (2004); T05 = Trundle & Lennon (2005); T04 = Trundle et al. (2004).
Galaxy Sp.Types β Ref.Milky Way O4 – O9.7 0.7 – 1.25 M04Milky Way O9.5 – B3 1.2 – 3.0 C06Milky Way B0 – B3 1.0 – 3.0 K99Magellanic Clouds O8.5 – B0.5 1.0 - 3.5 E04Magellanic Clouds B0.5 – B2.5 1.0 - 3.0 T05Magellanic Clouds B0.5 – B5 1.0 - 3.0 T04 the wind. In Fig. 3, we also included the results for a wind withpure free-free emission. From a comparison of the total windemission with the pure free-free wind emission, it is obvious thatfree-bound processes dominate the wind emission in the opticaland near-IR wavelengths, i.e., for λ < ∼ µ m.To understand where the optical wind continuum is gener-ated, we re-calculate the wind emission for di ff erent values ofthe outer edge of the wind zone. We then compared the resultingemission to the emission from an infinitely large wind, and weplotted the ratios versus frequency in Fig. 4. From this plot, it isevident that the wind emission in the optical and near-IR is gen-erated in the vicinity of the stellar surface, i.e., within 5 R ∗ . Mostof the emission for λ . µ m (about 95 %) is even generatedwithin the innermost 1 . R ∗ . β > . β pa-rameter in the velocity law given by Eq. (2) varies over a muchlarger range. Even values as high as 3.5 are reported. A list of β values of OB supergiants found in the literature is provided inTable 1.The main e ff ect of a higher β value is the less steep increasein the wind velocity with distance from the stellar surface. Thisis shown in Fig. 5 where we plotted the velocity increase within5 R ∗ from the stellar surface in terms of the terminal velocity fordi ff erent values of β . According to the equation of mass conti-nuity (Eq. (1)), the wind density is proportional to ( r ) − . Winds . Kraus et al.: On the importance of the wind emission to the optical continuum of OB supergiants 5 Fig. 5.
Velocity increase in winds with di ff erent β values. Thehigher the β , the more slowly the wind is accelerated and thefarther away from the star it reaches its terminal value. Fig. 6.
Density ratios close to the stellar surface for winds with β > . β = .
0. Due to the slowerwind acceleration for higher β , the wind density remains muchhigher over a larger wind volume, resulting in a relative densityenhancement close to the stellar surface, i.e., within 2 . . . R ∗ .with higher β values consequently have a (much) higher densityin the accelerating wind regions.We calculated the density distributions in winds with in-creasing β and plotted the densities in terms of the density distri-bution for the wind with β = . β > β =
1. The higherthe β , the stronger become these relative density enhancementsdue to (much) slower wind acceleration (see Fig. 5). These rela-tive density enhancements extend over the innermost ∼ . . . R ∗ of the wind, while for larger distances, where the velocity distri-butions for the higher β values also start to approach the terminalvelocity, the densities converge.In Sect. 3.1 we showed that for a wind with β = . < R ∗ . This is exactly therange where β has its strongest influence on the wind density. Ifthe optical continuum for winds with β > . β . We determined the wind volume in which the optical flux isgenerated by comparing the flux generated within an infinitelylarge wind zone with the flux generated within winds with dif-ferent outer edges, R out . The results for winds with di ff erent val- Fig. 7.
Flux ratio F ff − fb ( R ∞ ) / F ff − fb ( R out ) as a function of outerwind edge R out for di ff erent values of β . The calculations shownare for λ = µ m and are also valid for all smaller wavelengths.As reference, we include the ratio for R out = R ∞ (solid line). Forall values of β , the optical wind continuum emission is generatedwell within the plotted wind size of 5 R ∗ . Fig. 8.
Fraction of stellar radiation escaping from the wind zonefor di ff erent values of β . While for β = . λ < µ m passes unabsorbed through the wind,the situation is di ff erent for higher β values for which even theoptical wind continuum su ff ers from wind absorption along itsway through the wind. The higher β the more stellar emission isabsorbed also from the near- and mid-IR stellar spectrum.ues of β are shown as a function of the outer edge, R out , in Fig. 7.These results are evaluated for λ = µ m and are also valid forall smaller wavelengths (see Fig. 4). Obviously, for winds with β > .
0, more than 90% of the optical continuum is generatedwithin 1 . R ∗ , and more than 98% are produced within 2 R ∗ .Before we turn to the calculation of the total continuumemission from winds with β > .
0, we first investigate the in-fluence of β on the stellar emission. Since the wind absorptioncoe ffi cient, κ ff − fb , is proportional to n (see Eq. (12)), an increas-ing density leads to a (strongly) increasing optical depth. Thestellar flux will therefore be more strongly absorbed in the windwith a high β compared to a wind with β =
1. This is illustratedin Fig. 8. Of course, the strongest depression of the stellar emis-sion occurs at IR wavelengths ( λ > µ m); with increasing β alsosome part of the stellar flux at optical wavelengths, especially inthe red part of the spectrum, is also absorbed.But while the increasing absorption coe ffi cient results in adecrease of the stellar flux, it leads at the same time to a strongincrease of the free-free and free-bound emission in the optical M. Kraus et al.: On the importance of the wind emission to the optical continuum of OB supergiants
Fig. 9.
Comparison of total continuum spectra of the star pluswind system (solid lines) for di ff erent values of β . As in Fig. 3,a wind with pure free-free emission is included in each panel(long-dashed line) to emphasize the growing importance of thefree-bound contributions for λ ≤ µ m. With increasing β thefree-free and free-bound contributions (dashed) increase espe-cially in the near-IR and optical spectrum. At the same time, thestellar continuum (dotted) su ff ers from the increasing absorptiv-ity of the wind zone. For comparison, the pure stellar continuumis included (dashed-dotted).and IR part of the spectrum. This can be seen upon inspection ofFig. 9 where we plotted the total continuum emission for windswith di ff erent β values. The optical wind continuum increaseswith β leading to an enhanced total continuum in the IR andeven at optical wavelengths. This increase, especially at opticalwavelengths, is more evident in Fig. 10. There we plot the fluxratio between the total continuum emission of the star plus windsystem with β > β = β > ∼ .
0, the totalcontinuum exceeds the one for β = β . Whilefor β = . Fig. 10.
Continuum flux ratio of the star plus wind system fordi ff erent values of β with respect to β = .
0. With increasing β the increase in optical flux due to the increasing importance ofthe free-bound emission from the wind becomes visible. Fig. 11.
Ratio of the wind emission with respect to the total con-tinuum emission for di ff erent values of β . While for β = . β , reaching values upto 30% in the red part of the optical spectrum for β > . >
10 % for β = . β > ∼ . even at optical wave-lengths the wind plays a non-negligible role for the continuumemission of OB supergiants having wind velocity distributionswith β > .
4. Discussion
For our calculations we made use of several assumptions andsimplifications like a constant wind temperature, the core-haloapproximation, LTE even for the bound-free processes, and theneglect of electron scattering. These are severe restrictions tokeep the model as simple as possible. Here, we want to discussthe influence and possible consequences of these assumptionsand simplifications on the model results. In addition, we wantto address shortly the topic of wind clumping and its expectedinfluence on our results.
For our model calculations of stellar wind emission (seeSect. 2.2) we made two severe assumptions concerning the tem- . Kraus et al.: On the importance of the wind emission to the optical continuum of OB supergiants 7 perature: (i) we assumed the wind to be isothermal, and (ii) weused a rather high temperature, i.e. the temperature at r = R ∗ ,as the global electron temperature in the wind. Here, we want todiscuss why these two assumptions are reasonable.Our choice of the electron temperature at r = R ∗ means thatwe are calculating a lower limit of the wind absorption coe ffi -cient, κ ν , because it is proportional to T − / and the Gaunt factorsincrease with decreasing temperature (see Fig. 2). Consequently,according to the Eqs. (4) and (9), the emission caused by free-free and free-bound processes, equally increases with decreasingelectron temperature. Our results calculated for the high electrontemperature thus underestimate the real wind continuum emis-sion. The error in wind emission can be estimated by lookingat the real expected temperature distribution within the regionwhere the optical wind continuum is generated.During our investigations we found that for each value of β ,95 % of the wind continuum at optical wavelengths is generatedin the vicinity of the stellar surface, i.e. within 1 . . . . . R ∗ .This is shown in the top panel of Fig. 12. The solid horizontalline in this plot indicates the 95 % level of the emission, and thevertical lines indicate, for each β , the distance at which this levelis reached.To see how,in a more realistic wind model, the temperaturewould have changed over the wind region in which the opticalcontinuum emission is generated, we finally calculate the tem-perature distribution according to Eq. (3), with the velocity dis-tribution as defined by Eq. (2). The resulting temperature distri-butions for di ff erent values of β are shown in the bottom panel ofFig. 12. An increase in β results in a lower wind velocity at thesame distance from the star and, after Eq. (3), in a higher windtemperature. Therefore, the higher the β , the hotter the wind re-mains at the same location.For winds with β ≥ .
0, the drop in wind temperature isfound to be less than 5 % within the free-free and free-boundemission generation zone. Such a small decrease in temper-ature results in only a tiny and therefore negligible increasein wind emission. The assumption of an isothermal wind with T e = T e ( R ∗ ) is therefore well justified.The situation is di ff erent for winds with β = .
0. Here,the drop in wind temperature is found to be on the order of20 %, which corresponds to a decrease in electron temperatureby about 5 000 K for our chosen model star. In such a case, thetemperature distribution in the wind is not negligible. The de-crease in electron temperature results in an increase of the windabsorption coe ffi cient of about a factor of two (see Fig. 2), andconsequently to a noticeably enhanced wind emission. However,even with an enhanced wind emission, the stellar spectrum stillclearly dominates the total optical continuum. For OB super-giants with β = .
0, the assumption of an isothermal wind with T e = T e ( R ∗ ) is therefore still an acceptable approximation, aslong as the mass loss rate of the star is not extremely high, as inthe case of Luminous Blue Variables or B[e] supergiants. The construction of our star plus wind model is often referredto as the core-halo approximation, i.e., it is assumed that thestellar atmosphere is in hydrostatic equilibrium, and the wind istreated seperately with a density distribution following from themass continuity equation. With such a treatment the transitionbetween the atmosphere and the wind is not properly accountedfor. However, the intention of our research was to investigate theinfluence of one single parameter, namely the steepness of the
Fig. 12.
Top:
As Fig. 7 but for the innermost 2 R ∗ , only. The solidhorizontal line indicates the 95 % level, and the vertical linesthe distances where, for winds of di ff erent β values, this level isreached. Bottom:
Electron temperature distribution within 2 R ∗ for di ff erent values of β , normalized to the maximum electrontemperature. For β > .
0, the inner parts of the winds remainmuch hotter and almost at a constant value compared to the windwith β = . β , keeping the stellar and the remaining windparameters fixed. Deviations in density distribution of the starplus wind system introduced by the use of the core-halo approx-imation will consequently appear in all our discussed modelsand will influence all our results in the same way. But they willnot significantly alter our conclusions, which are mainly basedon the comparison of the spectra (i.e. the ratios) for stars withdi ff erent β values. In our calculations, we assumed the wind to be in LTE. Thisis a justified assumption for the free-free processes since theyare collisional. For the bound-free processes the situation isless clear. Here, non-LTE e ff ects might play an important role.According to our non-LTE model atmosphere calculations (e.g.Kub´at 2003), departure coe ffi cients can be (much) larger thanunity in the outer parts of the stellar atmosphere (especially forthe ground level) and, consequently, they lead to a higher bound-free opacity.We tested the influence of non-LTE e ff ects on our results ina qualitative way by increasing artificially the bound-free opac-ity. This results in an increase in wind optical depth and, ac-cording to Eq. (10), to an increased attenuation of the stellaremission, while the wind emission remains largely una ff ected.Consequently, the flux of the total continuum emission decreaseswith increasing bound-free opacity, i.e., increasing influence ofnon-LTE e ff ects. At the same time, the importance of the wind M. Kraus et al.: On the importance of the wind emission to the optical continuum of OB supergiants emission with respect to the total continuum increases . The ne-glect of non-LTE e ff ects thus results in a lower limit of the windcontribution.A similar conclusion can be drawn from the neglect of elec-tron scattering. Since electron scattering attenuates the stellarlight passing through the wind zone, it equally acts as an addi-tional opacity source. The consequences for the total continuumemission are therefore the same: a decrease in total emission(over the wavelength ranges that are dominated by the stellaremission), and simultaneously an increase of the wind emissionwith respect to the total continuum.For our model star plus wind system, we checked the elec-tron scattering optical depth in radial direction. We find that forthe chosen stellar and wind parameters and the di ff erent β values,the electron scattering optical depth is always smaller than unity.However, compared to the free-bound opacity, it is not negligi-ble. This means, that we calculated indeed upper limits for thestellar emission resulting in lower limits for the wind emissionwith respect to the total continuum. A proper treatment of non-LTE e ff ects and electron scattering can therefore only confirmour results: the importance of the wind contribution to the totaloptical continuum emission. In recent years, observational and theoretical investigations pro-vided evidence of wind clumping (see e.g. Hillier 2005). One ofthe most striking results found from detailed spectroscopic anal-yses (e.g. Crowther et al. 2002; Hillier et al. 2003; Bouret et al.2003, 2005) was that if a wind is clumped, the mass loss ratesinferred from spectroscopy might be lower on average by a fac-tor of 3. In addition, Hillier et al. (2003) found that clumpingstarts close to the photosphere, i.e., within the region in whichthe influence of β is strongest.The wind emission calculated in our study is not only deter-mined by the β parameter of the velocity distribution, but also bythe mass loss rate of the star that was kept constant during ouranalysis. Many of the authors who derived the high β values (seeTable 1) also claim that their mass loss rates are derived underthe assumption of unclumped winds and might well be a factorof 3 lower. Therefore, the question arises how our results mightchange if we account for wind clumping.This paper is not aimed to study wind clumping in detail,instead we refer to the recent paper by Kraus et al. (2008) whoinvestigated in more detail the influence of wind clumping onthe optical continuum emission of OB supergiants. Their resultscan be summarized as follows: wind clumping, introduced intothe calculations, e.g., by the filling factor approach of Hiller etal. (2003), results in a slight decrease in wind emission at opticalwavelengths. This decrease, however, was found to be less thanthe increase of the wind emission due to a high β value com-pared to a wind with β = .
0. We can, therefore, conclude thatin clumped winds with high β values the e ff ects discussed in thispaper are still present, but probably slightly weakened.
5. Conclusions
We investigated the influence of the thermal wind emission pro-duced by free-free and free-bound processes to the total contin-uum of normal OB supergiants. While for winds with a veloc-ity distribution following a β -law with β typically in the rangeof 0.8 to 1.0, no influence of the wind at optical wavelengthsis expected, the situation can be di ff erent when β exceeds 1.0. High β values reaching even 3.5 have recently been found formany OB supergiants. Our investigations therefore concentratedon such high beta values and their influence on the wind contin-uum emission especially at optical wavelengths.We found that the wind emission in the optical is generatedwithin 2 R ∗ , only. This region is exactly the region where β hasits highest influence on the wind density structure. Since withincreasing β the wind is accelerated much more slowly than for β = .
0, the density close to the stellar surface is strongly en-hanced, leading to an enhanced production of especially free-bound emission at optical wavelengths. At the same time thestellar emission, which passes through the wind on its way tothe observer, is absorbed. These e ff ects increase with increasing β . The total continuum of OB supergiants, for which β valueshigher than 1.0 are found, can thus contain non-negligible con-tributions of wind emission even at optical wavelengths . Acknowledgements.
We thank the referee, Ian Howarth, for his comments.M.K. acknowledges financial support from GA AV ˇCR under grant numberKJB300030701.
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