© ESO, 2014

1. Introduction

Blue supergiants (BSG) are key creators of heavy elements, thereby contributing to the composition of the interstellar medium; they are visible out to large distances, and dominate the spectra of star-formation galaxies. Moreover, they heat the interstellar gas and dust, and produce the far-infrared luminosities of galaxies. In particular, B-type supergiants (BSGs) are indissoluble troublemakers in modern astrophysics, as a proper understanding of their evolution is still in its infancy (Langer 2012). Even their fundamental properties are not yet known, and the basic issue as to whether they are core hydrogen (H) burning main-sequence (MS) or core helium (He) burning post-MS stars is still under debate (Vink et al. 2010; Georgy et al. 2013).

Lamers et al. (1995) reported the existence of a drop in terminal wind velocities of BSGs by a factor of two, which is referred to as the “bi-stability jump” (Pauldrach & Puls 1990). According to model predictions of Vink et al. (1999), it is expected that the faster winds of hotter stars switch to slower winds with a ~5 times higher Ṁ below T_eff ~ 22 000 K. Whereas the drop in terminal velocities has been confirmed by observations (Markova & Puls 2008), the predicted increase in Ṁ is still controversial.

While Benaglia et al. (2007) and Markova & Puls (2008) uncovered a local maximum in both radio and Hα mass-loss rates at the location of the bi-stability jump, several works have highlighted significant discrepancies between theoretical and empirical mass-loss rates for B1 and later B-type supergiants (Vink et al. 2000; Trundle et al. 2004; Trundle & Lennon 2005; Crowther et al. 2006; Benaglia et al. 2007; Markova & Puls 2008). Vink et al. (2000) noted that their predictions agreed reasonably well with empirical rates from both radio and Hα emission for late BSGs, but they found huge (order of magnitude) discrepancies when the Hα line was P Cyg shaped or in absorption. Searle et al. (2008) emphasised that empirical models for BSGs likely have an incorrect ionisation structure as they found it challenging to reproduce the optical Hα line simultaneously with key ultraviolet (UV) diagnostics.

To make the picture even more complex, one should be aware of discrepancies between mass-loss rates estimated from Hα, UV, and radio observations for OB stars in general (Massa et al. 2003; Bouret et al. 2005; Puls et al. 2006; Fullerton et al. 2006), which may be due to distance-dependent wind clumping and/or porosity effects (Oskinova et al. 2007; Sundqvist et al. 2010, 2011; Muijres et al. 2011; Šurlan et al. 2012).

In any case, the general trend seems to indicate Ṁ_vink>Ṁ_Hα for B1 and later supergiants, whilst the reverse holds for earlier O supergiants. The key question is whether this discrepancy is the result of incorrect predictions or alternatively that we may not understand the mass-loss indicator Hα well enough. What would one expect to happen when Feiv recombines, and Feiii starts to control the wind driving (Vink et al. 1999)?

Over the last decade, we have made a dedicated effort to improve the physics in the Monte Carlo line driving calculations. We now solve the wind dynamics more locally consistently (Müller & Vink 2008; Muijres et al. 2012), we added Fe to the statistical equilibrium calculations in the isa-wind model atmosphere (de Koter et al. 1993) rather than treating this important line-driving element in a modified nebular approximation, and finally, we have studied the effects of wind clumping and porosity on the line driving (Muijres et al. 2011). After all these improvements in the Monte Carlo line-driving physics, we have to admit that the basic problem of Ṁ_vink>Ṁ_Hα for B1 and later supergiants, is still present, and it is time that we also consider the possibility that it is not the predictions that are at fault, but that we do not understand Hα line formation in BSGs sufficiently well to allow for accurate mass-loss determinations from Hα.

In order to obtain a more complete picture of Hα as a mass-loss diagnostic, we need to know not only the clumping properties of BSGs, but also how sensitive Hα is to the clumping on both sides of the bi-stability jump. Currently, T_eff dependence of the mass-loss rates from Hα of BSGs is still uncertain, and a better knowledge is required, especially as it impacts the question of the evolutionary nature of BSGs.

Despite the fact that the spectral modelling of BSGs is an active area of research (Zorec et al. 2009; Fraser et al. 2010; Castro et al. 2012; Clark et al. 2012; Firnstein & Przybilla 2012), because of a lack of observational data, the nature of BSGs is a long-standing issue in stellar evolution. Hunter et al. (2008) argued on the basis of high-resolution VLT-FLAMES data that the slowly rotating BSGs in the Large and Small Magellanic Clouds (LMC; SMC) are post-MS objects, although their large numbers would remain unexpected. Vink (2008) and Vink et al. (2010) established a general lack of fast rotating BSGs (v sin i > 50 km s^-1) and suggested that these objects could naturally be explained as MS stars, if they lose their angular momentum via an increased mass-loss rate due to the bi-stability jump (the so-called bi-stability braking). They also pointed out that this mechanism would be efficient if the stars spend significant amounts of time on the MS. While the first hypothesis received some support by the apparently brightest supernova (SN) in the telescopic era: SN 1987A, bi-stability braking for stars with initial masses above 40 M_⊙ was confirmed by Markova et al. (2014).

Hot stars with high mass-loss rates are expected to have strong emission lines, predominantly formed by recombination. The Hα line is an excellent tracer for hot-star mass loss, as its detection is possible for large numbers of stars. Moreover, the strong velocity dependence of Hα combined with high-resolution spectroscopy could provide valuable information about velocity fields and density structures in Luminous Blue Variables (LBVs) and SN progenitors (Groh & Vink 2011). That is why understanding the behaviour of the Hα line is crucial to our understanding of mass loss.

In Sect. 2, we describe briefly the used input parameters for our supergiant models. In Sect. 3, we show that the computed line profiles exhibit non-monotonic temperature behaviour, and provide the most likely explanation for it. Section 4 explores the impact of clumping on Hα line equivalent width (EW) and optical depth. In Sect. 5, we discuss the importance of our findings and, finally, in Sect. 6, we summarise our results and discuss possible implications for future work.

2. Method and input model parameters

In order to study Hα line formation, we use the non-LTE radiative transfer code cmfgen (Hillier & Miller 1998; Hillier & Lanz 2001), and we calculate a model grid over a range of temperature and log  g appropriate for BSGs (cf. Table 1). Apart from changes in T_eff and log  g, Hα line formation is sensitive to luminosity, L_⋆, mass, M_⋆, mass-loss rate Ṁ, clumping, and velocity structure. As cmfgen does not solve the momentum equation of the wind, a velocity structure needs to be assumed. For the accelerating part of the wind, we adopt a standard β-type velocity law with β = 1, whilst a hydrostatic solution is adopted for the sub-sonic region.

To understand first how the Hα line profile changes in a qualitative sense, we use models with a simplistic H chemical composition, keeping L_⋆, M_⋆, ν_∞/ ν_esc and Ṁ fixed, whilst we vary T_eff over a range from 30 000 K down to 12 500 K. Following Vink et al. (2001), we adopt the parameters listed in Table 1.

Note that for fixed L_⋆, the changes in T_eff inevitably lead to different R_⋆ and ν_esc values, which may influence the Hα EW. To account for this, we used an optical depth parameter tool that was introduced by Puls et al. (1996); see Sect. 3.1. To keep as many model parameters as possible fixed, we used ν_∞/ ν_esc = 2 – the mean value of ν_∞/ ν_esc ratio at both sides of the bi-stability jump (Lamers et al. 1995; Markova & Puls 2008). It is more natural to keep the ν_∞/ ν_esc ratio constant rather than the value of ν_∞ itself, as the models have different radii.

It is important to keep in mind that due to metal line blocking and a blend with He II the Hα line could be sensitive to changes in He as well as metal abundance. To estimate these effects we also compare how the Hα EW behaves in models with different chemical complexities: (i) pure H; (ii) H + He, with two different He mass fractions (Y_He = 0.25 and 0.6); as well as (iii) metal line-blanketed models.

3. Results

3.1. Hα line profile and equivalent width

Our systematic examination of the Hα line for supergiant models over T_eff range between 30 000 and 12 500 K shows non-monotonic changes in the Hα line profiles with T_eff. As shown in Fig. 1, the Hα line becomes stronger when T_eff drops from 30 000 to 22 500 K, where the line reaches its peak value. Below 22 500 K the line becomes weaker when the effective temperature is further reduced to 12 500 K.

Although we have kept all stellar parameters fixed, the models have different radii and ν_∞. Note that in the hottest models the radii are up to a factor of 6 smaller, whilst the terminal velocities are only up to a factor 2.4 higher. To extract the true temperature effect, the different radii and terminal velocities should be taken into account. One way to do this is through the use of the wind-strength parameter Q concept introduced by Puls et al. (1996). It was demonstrated that for O-type stars recombination lines remain unchanged by specific sets of individual values of Ṁ,v_∞ and R_⋆ as long as the wind-strength parameter Q: (1)is invariant and T_eff remains unchanged. Here Ṁ is calculated in M_⊙/yr, v_∞ in km s^-1 and R_⋆ in R_⊙. As an example, in Fig. 1 we show that the Hα line profile from model C (red dashed line) is basically unaffected when the mass-loss rate is decreased by factor of two and the radius by 2^{2 / 3} (black triangles).

The right-hand panel of Fig. 1 displays how the Hα EW behaves as a function of T_eff for models with similar parameters, but with different individual Ṁ, which scale in such a way that Q is constant (blue squares). The peak is still present and this implies that the Hα behaviour is not due to the changes in v_∞ × R_⋆, i.e., it is a real temperature effect. Note that the peak is slightly shifted towards T_eff ~ 20 000 K.

To gain additional insight, the EW was separated into the line and continuum flux (respectively, illustrated in Fig. 2 with red circles and squares). The reason for doing so is that the EW is a measurement for the ratio of the line flux over the continuum flux, and models with similar EW may have completely different line fluxes. As can be seen in Fig. 2, the coolest model has a seven times larger line flux than the line flux in the hottest model, whilst in Fig. 1 it seems that both models have a similar line strength. The similar Hα EW in both models is due to the increase of continuum flux when T_eff is reduced, and the higher line flux in the coolest model. The peak in right-hand panel of Fig. 1 results at 22 500 K (and not at 20 000 K as in Fig. 2) because the ratio between the line and continuum fluxes is the largest for that T_eff. Note that the cooler supergiants models have larger radii, which increases the continuum flux with decreasing T_eff.

The right-hand panel of Fig. 1 also presents behaviour patterns in models with different chemical complexities. The simplest set of models (with only H; i.e., without He) presented with red crosses show practically identical Hα EWs¹ as the EWs from models including both H and He (displayed with grey circles). In other words, He does not seem to influence the Hα line. The reason could be that for this T_eff range Heii is diminished and the blueward Heii blend is not essential, whilst the HeI continuum plays only a minor role at these temperatures. However, the latter holds only when the He abundance is less than 25% by mass. The red asterisks in the right-hand panel demonstrate that the He rich models (comprising 60% He mass fraction) have a factor 2–3 lower Hα EW. This behaviour is in agreement with the results of Dimitrov (1987) who found that a high abundance of He produces stronger absorptions in H lines. Despite the quantitative differences, He is not important for the qualitative behaviour of Hα versus T_eff.

The line EW values from the simplistic H+He models (grey circles) are also compared to those determined from more realistic chemical models (green triangles), which include C, N, O, Si, S, P, and Fe (with atomic data as listed in the table from Appendix B and half-solar metal abundances). While there are quantitative differences due to line blanketing, the qualitative behaviour in Hα EW is similar in both sets of models with different complexities. This implies that the reason for their behaviour is fundamentally driven by the properties of H. Despite their simplification, the models including H+He only (with atomic data as indicated in Table 2), provide an overall picture of the effective temperature dependence of Hα for BSGs. It is, therefore, reasonable to take advantage of these H+He models, using them as a starting point for our investigation.

As an aside, we found that for models with a flatter velocity law (β = 2) or with different mass-loss rates (varying from ~10^-7 to 10^-5M_⊙ yr^-1) the resulting Hα EW behaviour was qualitatively similar to those presented in Fig. 1 with Hα EW showing a peak at T_eff ≃ 22 500 K.

3.2. Two branches of Hα behaviour

The Hα EW changes drastically with T_eff in all sets of models. In general, we find there to be two branches in Figs. 1 and 2: the “hot” and “cool” ones. The hot branch is located between 30 000 to 22 500 K, where the Hα line emission becomes stronger with decreasing T_eff. At the cool branch, from 22 500 to 12 500  K, the behaviour of the line flux changes in the opposite direction. This implies that there is a qualitative change in the behaviour of Hα around 22 500 K.

3.2.1. The “hot” branch

The formation of the Hα line involves transitions between the 3rd and 2nd level of H. Therefore, the emission should be proportional to the total number of H atoms in the 3rd level in the wind above τ_ROSS = 2 / 3 (N₃). As N₃ (as do the number of H atoms in other levels) scales with the fraction of neutral H, we show the H ionisation structure of our models in Fig. 3. It is evident that the wind is mostly ionised in all models and as T_eff drops the wind recombines slightly. The fraction of neutral H increases to almost two percent when T_eff is reduced to 12 500 K (at log τ_ross ~ − 1.5, log  (H / H⁺) ~–1.8). Furthermore, in an absolute sense, the total number of H atoms in the second level (N₂) and N₃ increase, as illustrated in Fig. 4. As a result, the flux in Hα, which is proportional to N₃, should increase to first order as T_eff drops. Therefore, the trend in Hα on the “hot“ side of Figs. 1 and 2 can be understood in terms of a simple recombination effect (see also Puls et al. 1996). Difficulties with such a simple explanation would arise if we were to try to explain the existence of the cool branch in a similar way. Figure 4 illustrates that when T_eff is reduced from 22 500 to 12 500 K, N₃ is still increasing and this should probably produce a stronger Hα line for cooler models. Contrary to expectation, the opposite behaviour of Hα is produced for this branch.

3.2.2. The “cool” branch

On the cool branch, small changes in T_eff lead to qualitatively different Hα line profiles. Most notable is the appearance of a P Cygni profile in Fig. 1 when the effective temperature is reduced from 15 000 to 12 500 K. In fact, these differences between both Hα line profiles contain major clues to the unexpected changes in the line flux of the models on the cool branch. If Hα were a pure recombination line only the third level would be relevant. However, as shown later, Hα increases its optical depth at cooler T_eff. It is thus necessary to assess the source function and the ratio of the number of H atoms per cm³ in the 3rd over the 2nd level (n₃/n₂).

Figure 4 reveals that N₃ is constantly increasing when T_eff is reduced, however, N₂ increases more steeply. The (n₃/n₂) ratio has been plotted versus Rosseland optical depth in Fig. 5. First of all, the (n₃/n₂) ratio is close to unity for the models at the “hot” branch. However, for the models on the cool branch, n₂ becomes significantly larger than n₃, particularly in the outer wind. Although N₁,N₂,N₃ are all increasing as T_eff drops, the ratio(n₃/n₂) is always decreasing, and the second level becomes more populated than the third level on the cool branch. This leads to a “dip” in (n₃/n₂) in the outer wind of the cooler models. The “dip” is quite pronounced, especially for the coolest model, where n₂ is ten times higher than n₃. The increased n₂ in the outer wind produces absorptions for Hα photons emitted close to the photosphere. This naturally decreases the Hα flux. Hence the question about the decreasing EW over the cool branch, could be referred to as an issue regarding the behaviour of the (n₃/n₂)ratio.

Therefore, the next question is, why does n₂ become significantly larger than n₃? Is the “dip” in (n₃/n₂) ratio is predominantly caused by an increase of n₂ or by a decrease of n₃? In order to address these questions, we take a detailed look into the behaviour of the continua and their effects on individual levels.

3.3. Two possible explanations for the existence of the “cool” branch

3.3.1. A decrease of n₃?

In Fig. 6 we show spectral energy distributions (SEDs) at the surface. The Lyman continuum flux is greatly reduced from 30 000 to 12 500 K. Despite that, nothing dramatically happens with the Lyman continuum around 22 500 K. The changes in the continuum flux are rather gradual. To understand the behaviour of the Lyman continuum in Fig. 7, we compare the continuum optical depth at λ ~ 900 Å (black solid line) to changes in the H ionisation structure (red dashed line with ordinate in red colour placed on the right side). It is evident that the Lyman continuum becomes optically thick at distances closer than d_τ ~ 0.06, 0.25, 0.6 and 7 [R_⊙] from the photosphere (roughly τ_ross = 2 / 3), respectively for the models with T_eff = 30 000, 22 500, 17 500, and 12 500 K. The fraction of neutral H at those distances is between ~10^-4.9 and 10^-4.3. The comparison between the Lyman continuum optical depth and H ionisation structure shows that as soon as neutral H atoms exceed critical values, the Lyman continuum becomes optically thick. Moreover, the steep increase of neutral H close to the star leads to a significant increase of the optical depth of the Lyman continuum. Consequently, a large fraction of Lyman ionising photons are blocked, and the Lyman continuum is no longer the main source of ionisation.

Najarro et al. (1997) studied the appearance of H and He lines in the wind of the LBV P Cygni, and they found that H recombination, crucial for the Ly continuum and the Lyα optical depth, block the ionising Lyman flux. In their models, the increased n₂ is due to the high optical depth of Lyα, giving rise to strong absorption, similar to those produced in our coolest B supergiant model.

As n₁,n₂ and n₃ are controlled from different continuum ranges, knowledge about the behaviour of Balmer and Paschen fluxes (not only of the Lyman continuum) is required to understand their behaviour. In Fig. 6 it is shown that the Balmer and Paschen continuum fluxes decrease when T_eff is reduced. To understand this, we plot the wind optical depth in the Balmer and Paschen continua in Fig. 8 (right panel). Since the cross-section for the photo-ionisation of a H atom in quantum state n by a photon of wavelength λ is: (2)the opacity would be proportional to λ³. Therefore, we have chosen to plot the wind optical depth in the Balmer and Paschen continua at wavelengths close to their corresponding jumps (λ ~ 3500 and 8100 Å, respectively).

Figure 8 shows that the wind optical depth in the Balmer continuum remains fairly constant for all models. Note that when T_eff is reduced below 22 500 K, the wind optical depth in the Paschen continuum is steeply decreased. Models that are cooler than 17 500 K even have τ_Paschen<τ_Balmer. This should produce larger changes in the Paschen flux than in the Balmer flux. To quantify the reaction of the continuum fluxes Fig. 9 shows the number of photons – N_λ, versus T_eff for the Lyman, Balmer, and Paschen continua respectively. The number of photons is given by: (3)where λ_c is the wavelength boundary of corresponding continuum series; R_⋆ is the stellar radius. The computed numbers of photons in the Lyman and Balmer continua are in reasonable agreement (by a factor of ~4) with those from previous studies (Thompson 1984; Diaz-Miller et al. 1998) if the different radii are taken into account. Balmer and Paschen fluxes are able to ionise H atoms respectively from level 2 and 3 and regulate those levels. It is evident from Fig. 9 (left panel) that these fluxes are about 4 orders of magnitude larger than the Lyman flux at the cool branch. This is a consequence of high wind optical depth in the Lyman continuum (reported in Fig. 8) and the optically thinner wind in the Balmer and Paschen continua. Hence, the Balmer and Paschen fluxes are the main sources of ionisation of H atoms in second and third levels over the cool branch. Figure 9 indicates that the total number of photons able to ionise atoms in level 2 is nearly the same for all models. This would provide nearly the same number of H atoms ionised from the second level over both branches. Therefore, we do not expect dramatic changes in n₂ due to the Balmer continuum. By contrast, the “Paschen photons” are gradually increasing in number as T_eff is reduced. This is seen in the right panel of Fig. 9, where we zoomed in around N_λ for Balmer and Paschen continua. For cooler models, the increasing flux in the Paschen continuum may thus depopulate more H atoms in the third level.

To understand the key question whether the “dip” in (n₃/n₂) results solely from an increase of level 2, or from a decrease of level 3 as well, we show in Fig. 10 how their number densities (and the number density of levels 4 and 10, n₄ and n₁₀) change with T_eff. It is evident from the plot that when T_eff is reduced, n₃ behaves in a fashion more similar to the number densities of the higher levels, e.g., n₄. This is expected if the levels are mainly fed by recombinations. Therefore, a decrease of level 3 does not occur and consequently the (n₃/n₂) ratio is not affected by changes in n₃. It seems that the changes in the Paschen continuum with T_eff are not large enough to cause a decrease of n₃. As a result, n₂ should play a major role in the (n₃/n₂) ratio, causing the “dip.”

3.3.2. An increase of n₂?

Due to the enormous optical depth in the Ly continuum (reported in Fig. 8), the ionising flux is blocked for the “cool” models and n₁ can no longer be affected by the Lyman continuum. As a result, level 2 becomes the effective ground state, exhibiting a depopulation close to the photosphere, and an overpopulation in the outer wind in model C (T_eff = 12 500 K) as shown in Fig. 11. This is similar to the findings of Puls et al. (1998) who concluded that Hα appears like a P-Cygni line for A supergiants. It should be noted that the produced peak in the departures from LTE for the second level (b₂) in model C occupies the same position as the “dip” in the (n₃/n₂) ratio. This coincidence indicates that the “dip” is mainly produced by an increase of n₂.

The difference in behaviour of level 2 compared to higher levels can clearly be seen in Fig. 10, where the number densities of level 3 (blue triangles), 4 (squares) and 10 (green solid line) are similar for all T_eff. For the models on the “hot” branch even n₂ behaves in an identical manner to the number densities of the higher levels. This behaviour is to be expected when the level populations are solely fed by recombinations, and when they should scale as ρ² ~ r^-4 (via the continuity equation). The population of the ground state is, however, affected by ionisations as well, and it is thus inversely proportional to the dilution factor of the radiation field. The latter effect makes it increase as distance-squared, and in the outer wind the final dependence is 1/r², as shown in Fig. 10. The black asterisks, connected by a dashed line, represent the number density of the ground state (n₁), which is decreasing as 1/r² (illustrated as a blue dashed line). When T_eff is reduced to 12 500 K, n₂ diverges from higher-lying energy levels and behaves similar to the ground level (decreasing as 1/r²). At the same time, the Hα line profile forms an absorption component.

3.4. Lyα and the second level

In order to understand why level 2 diverges from the higher levels when T_eff drops, we ran additional models in which we artificially removed the Lyα transition by reducing its oscillator strength by a huge factor (10⁴). The reaction of Hα is displayed in the upper panels of Fig. 12. It highlights the key role of Lyα. The effects on model M (T_eff=22 500  K) are striking: Hα now switches from a pure emission line into a P Cygni line, and the line flux is decreased significantly. Furthermore, for the cooler model the removal of Lyα leads to a deeper absorption component. In the middle panels in Fig. 12, we present produced changes in n₂ (dashed) and n₃ (solid) due to the absence of Lyα – as a function of τ_ross. The middle panels demonstrate that the Lyα removal leads to a tremendous increase of n₂ (in comparison to the initial model) in the outer wind, leading to a stronger absorption component in model C, and the appearance of a P Cygni profile in model M, where it is noted that the lack of Lyα leads to significant changes in the third level as well: n₃ is surprisingly reduced, and as a result the Hα flux decreases.

To understand the changes in the level populations due to the removal of Lyα, we show its net radiative rate in the lower panels of Fig. 12. This quantity is defined as the difference between the number of radiative transitions from the upper (second) level to the lower (ground) state and the number of radiative transitions from the ground level to the second level. As a result, the net radiative rate is positive when there is a net decay of electrons from the upper level, and it is negative when there is a net excitation of electrons. It is seen that as soon as n_2noLyα and n_3noLyα diverge from their initial ones, at log (τ_ross) < ~ − 1.8 (in model C) and log (τ_ross) < ~ − 1.5 (in model M), the total Lyα radiative rate in the initial model (black solid line) is positive². Therefore, the line effectively acts as a “drain” for the second level. When we remove it, the decay of electrons from the second level is suppressed, and n₂ is tremendously increased, as shown in the middle panels.

In other words, by artificially removing Lyα, we can simulate the appearance of P Cygni profiles for hotter models, showing that the Lyα line is key to the Hα behaviour. We also note that neither He nor Fe are directly required for achieving this, i.e., it is a pure H effect (as was shown in Fig. 1).

Figure 13 illustrates how the Lyα optical depth changes. The location where most of the Hα photons originates form (Appendix A) is shown with thicker line sections. It is evident that, at the hot branch, Lyα is optically thick in the inner Hα forming region and becomes optically thin in the outer region. Furthermore, the Lyα optical depth at the start of the Hα line-formation region is similar for hot models. However, when T_eff drops below 22 500 K, the Lyα optical depth at the start of the Hα line-formation region steeply increases, which continues over the cool branch where Lyα is always optically thick throughout the entire Hα line-forming region. This means that most photons from the photosphere at the Lyα wavelength do not manage to escape; i.e., they are being scattered or absorbed. As a result, the decay from the second level is effectively shut off. This can be seen in the lower panels of Fig. 12. In the region where n₂ and n₃ are affected by Lyα (i.e., for log (τ_ross) < ~ −1.8 for the cool model and log (τ_ross) < ~ −1.5 for the hotter model), the net radiative rate is 2 orders of magnitude lower for the cooler model, i.e., the second level is depopulated less efficiently by 2 → 1 transitions.

In short, the departure of n₂ from the higher level occupation numbers (when T_eff drops from 22 500 K downwards) is related to the Lyα optical depth. When T_eff is reduced below 22 500 K, the Lyα optical depth increases steeply, and level 2 is less efficiently depleted through decay. Unlike the second level, level 3 behaves like higher levels, thus the “dip” in (n₃/n₂) is the result of an increased level 2 population.

4. The effect of clumping

Currently, cmfgen takes optically thin (micro) clumping into account, i.e., the clumps are assumed to have a dimension smaller than the photon mean free path. The density ρ within clumps is assumed to be enhanced by a clumping factor D compared to the wind mean density . This factor can also be understood in terms of volume filling factor f = D^-1, assuming that the inter-clump medium is void. Mass-loss diagnostic techniques that are linearly dependent on density are insensitive to micro-clumping, whilst recombination lines (sensitive to ρ²) tend to overestimate the mass-loss rate of a clumped wind by a factor of .

However, if the clumps are optically thick (macro-clumping), photons may leak through gaps between clumps, and the mean opacity is affected by the distribution, the geometrical size, and the shape of the clumps. In this case the micro-clumping approach is no longer justified.

4.1. The Hα line in a micro-clumping approach

The Hα line emission is a ρ² dependent process and is therefore sensitive to micro-clumping. In order to investigate the potential role of micro-clumping in our model sequences we calculated additional models, identical to the simplistic H+He models but with f_∞ = 0.1, described by the following exponential law: (4)where ν_cl is the velocity at which clumping starts. We have chosen the clumping to start at ν_cl = 20 km s^-1, just above the sonic point.

The Hα line profiles are presented in the left panel of Fig. 14, where it is shown that clumping enables the asymmetry in Hα to appear at hotter T_eff. When clumping was neglected, the increased level 2 produced asymmetric line profile at 15 000 K and 12 500 K (cf. Fig. 1), whilst in clumped models with f_∞ = 0.1, the asymmetry is already present at 20 000 K. The reason for this shift towards higher temperatures is caused by the increased mean density of the micro-clumped winds (in comparison to the mean density of smooth winds), which in turn increases the Lyα optical depth. Consequently, n₂ increases at hotter temperatures. The next question is whether micro-clumping may have an effect in terms of the cool versus hot branch sequences?

Therefore, in the right panel of Fig. 14, we compare how the Hα line EW behaves as a function of T_eff for both clumped and unclumped models. It is found that micro-clumping changes the EW dramatically in the hotter models. Also, at the bi-stability jump location (~22 500 K) micro-clumping has a dramatic impact on the Hα EW. However, micro-clumping progressively plays a lesser role towards the cooler edge of the Bsg regime. The reason for this is that the second level now behaves as a quasi-ground state, i.e., it scales linearly with density ρ, and remains rather unaffected, whilst the more drastic effects for hotter models are the result of the ρ² scaling. Nevertheless, clumping transforms Hα from a pure emission line into a P-Cygni line at 15 000 K. Although micro-clumping is of quantitative relevance in BSGs (especially around the bi-stability jump), the existence of an Hα EW peak remains present in clumped model sequences.

4.2. The Hα optical depth in a micro-clumping approach

In Fig. 15, we compare how the Hα optical depth changes with T_eff for homogeneous (left) and clumped (right) models. The line-forming region is indicated by thick lines. White squares illustrate at which point 50% of the line EW is already formed (see Appendix A). According to the left plot most of the Hα photons emerge from regions in which the line is optically thin. Interestingly, when T_eff drops from 30 000 to 22 500 K, Hα becomes optically thinner. Below 22 500 K, the line changes its behaviour and becomes optically thicker with decreasing T_eff.

Figure 15 illustrates that the introduction of micro-clumping would increase the Hα optical depth. It seems that at the bi-stability jump (~22 500 K) clumping has the largest impact on the Hα optical depth: it increases by an order of magnitude at the location where 50% of the line EW is formed (indicated by white squares in Fig. 15). At this point, all homogeneous models are optically thin (left panel), while the clumped models are predominantly optically thick in Hα (right panel). It appears that the character of Hα changes when T_eff drops: from an optically thin to an optically thick line, although, we kept the mass-loss rate and ν_∞/ν_esc  constant in all models. Moreover, it is well known from the observations that ν_∞/ν_esc  drops from 2.6 to 1.3 across the bi-stability jump (Lamers et al. 1995; Markova & Puls 2008). This should produce even sharper differences in the Hα line on both sides of the bi-stability jump, as the higher velocity ratio on the hot side is expected to decrease the mean density and thus Hα line optical depth, whilst the lower velocity ratio on the cool side would favour higher optical depth of Hα line. It is worth mentioning that the inclusion of clumping in the models is equivalent to an increase of the mass-loss rate by factor of (in comparison to a homogeneous model)³. Because of this, Fig. 15 also illustrates how the Hα line optical depth would change if the mass-loss rate were increased by factor of .

All models discussed here have Ṁ = 2.33 × 10^-6M_⊙ yr^-1, which is nearly, three times larger than the predicted mass-loss rate for BSGs around the bi-stability jump. Thus, our homogeneous models are similar to models with mass-loss rates of roughly three times lower, but with clumped winds (with f_∞ = 0.1). Hence, in the context of the predictions, we expect an increase of the Hα line optical depth around the bi-stability jump analogous to what is shown in Fig. 15 (i.e., increased by an order of magnitude or more if the mass-loss rate is increased by a factor 5; Vink et al. 1999).

4.3. Impact of macro-clumping on Hα

Our finding implies that although changes in the Hα optical depth are largest at the cool branch, the Hα line EW is less sensitive to micro-clumping in the coolest models. The reasons for that are as follows:

• the Hα optical depth increases when T_eff is reduced;

• the Lyα optical depth also increases;

• the second level is prevented to recombine to the ground state.

As a consequence, the second level behaves as an effective ground state. Thus, on the cool branch the Hα line behaves as an optically thick resonance line and is therefore more sensitive to macro-clumping.

We thus conclude that macro-clumping could play a major role in Hα line formation on the cool side of the bi-stability jump. This could significantly affect empirically derived mass-loss rates. Šurlan et al. (2013) showed that for O-type supergiants macro-clumping may resolve the discrepancy between empirical mass-loss rates derived from Hα and Pv diagnostics (see also Sundqvist et al. 2011).

5. Discussion

In this theoretical study, we have made progress in our understanding of the Hα line over the hot and cool branches around the bi-stability jump. We have not yet discussed whether our results have a direct bearing on the reported discrepancies of empirical late Bsg mass-loss rates from model atmosphere analyses and Monte Carlo iron-line driving calculations of Vink et al. (2000). In other words, the problem of the general trend that Ṁ_vink>Ṁ_Hα for B1 and later supergiants.

Previous investigators, in particular Trundle et al. (2004); Trundle & Lennon (2005) and Crowther et al. (2006) argued that wind clumping would make the discrepancy between Ṁ_Hα and Ṁ_Vink worse, but this conjecture relies on three assumptions: (i) that the character of the wind clumps (e.g., optically thin versus optically thick) would remain the same at the bi-stability jump; (ii) that the amount of wind clumping would remain the same; and (iii) that the diagnostic effects of micro-clumping on the Hα line would be constant with T_eff. How likely is it that the wind structure remains the same when the physics changes at the bi-stability jump?

In the previous sub-section, we noted that our results are in contradiction to those assumptions. The first assumption is also in contradiction to the results of Prinja & Massa (2010), who concluded on the basis of Siiv lines that the BSGs winds contain optically thick clumps, and Oskinova et al. (2007) have shown for O star winds that if clumps become optically thick, and the wind becomes porous, clumping might in fact underestimate empirical mass-loss rates.

Our results show that Hα may become optically thick below the bi-stability jump, whilst conversely it may remain optically thin for hotter objects. This might imply that previous empirical mass-loss rates below the bi-stability jump may have been underestimated, whilst those from hotter stars could be correct or slightly overestimated as a result of micro-clumping. It might also be relevant that the modelled Hα lines in the work of Trundle et al. (2004) and Trundle & Lennon (2005) do not reproduce the observed Hα line shapes. This suggests that the underlying model used in these analyses might not be correct. Moreover, the “derived” values for β are much higher than predicted, and may be an artefact of an inappropriate modelling procedure in case macro-clumping would be relevant. It would thus be worthwhile for future investigations if the effects seen in Figs. 14 and 15 could indeed explain the reported mass-loss discrepancies.

6. Conclusions

The behaviour of the Hα line over T_eff range between 30 000 and 12 500 K might be characterised by the management between two processes. Whilst the “rise” is the result of simple recombination (n₃^↑), the “fall” is due to the intricate behaviour of the second level (n₂^↑). As T_eff drops below 22 500 K, the existence of a cool branch may be summarised as follows:

• The high Lyman continuum optical depth makes ionisation fromthe first level unlikely.

• Lyα becomes optical thick.

• The drain from the second level is suppressed.

• At the coolest model, level 2 diverges from higher levels, and it operates like a ground state.

• Hα changes its character and behaves like a scattering line with a P-Cygni profile.

During the transition from a recombination to a scattering line, the EW decreases because recombination lines have a larger (and basically unlimited EW, if the mass-loss rate is increased), whilst a scattering line is confined in its EW as it is dominated by the velocity field. Thus, the EW has to decrease when the line starts to change its character, i.e., over the cool branch.

The qualitatively similar Hα behaviour, including just H, and H+He only models, and metal-blanketed models suggests that the Hα behaviour is not related to He or metal properties. Intriguingly, we also found that the effect of clumping on Hα is largest at T_eff around the bi-stability jump and this is not related to the iron ionisation. Furthermore, if the mass-loss rates are increased at the bi-stability jump (as predicted), then the effect (together with clumping) on Hα optical depth would be even stronger. Therefore it is expected that Hα changes its character from an optically thin to an optically thick line and the micro-clumping approximation may no longer be justified.

Although all codes include the physics explained in this work (except for macro-clumping), it is interesting that independent of model complexity, the Hα EW peaks at the location of the bi-stability jump for all our models. This might have consequences for both the physics of the bi-stability mechanism, as well as the derived mass-loss rates from Hα line profiles, as objects located below the Hα EW peak are predicted to be weaker for a similar mass-loss rate, i.e., higher empirical mass-loss rates are required to reproduce a given Hα EW if the star is located at a T_eff below the peak. Whether this deeper understanding of Hα EW over the bi-stability regime would indeed lead to a resolution of the BSG problem remains to be shown with detailed comparisons of our models and observed Hα profiles.

This may also be relevant for magnetic field confinement by magnetic spots that could be induced by the sub-surface convection zone in BSGs (Cantiello et al. 2009), as Shultz et al. (2013) noted that a pair of spots could have remained undetected by current field searches if the higher theoretical mass-loss rates are employed, but that even the largest spots are ruled out if the lower Hα rates are correct.

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Acknowledgments

We would like to thank Dr. Joachim Puls and an anonymous referee for stimulating discussions and detailed comments on an earlier draft of the manuscript. We acknowledge financial support from the Northern Ireland Department of Culture, Arts, and Leisure (DCAL) and the United Kingdom (UK) Science and Technologies Facilities Council (STFC). We are grateful to Dr. John Hillier for providing the cmfgen code to the astronomical community.

References

Appendix A: Where in the wind do Hα photons originate from?

In order to understand the Hα line-formation we show in Fig. A.1 a typical distribution of the emergent intensity I(p), which is scaled by the impact parameter p (see Dessart & Hillier 2005, for details). The top panel of the figure represents the synthetic line profile, which, for each wavelength corresponds to the integral over all p of the scaled intensity I(p) presented at the lower panel. Hence, we are able to identify the contribution at each p to the total line flux.

In other words, the figure provides information where in the wind most of the emergent Hα photons originate from. This knowledge enables us to display the “evolution” of the Hα line (EW) if larger p values are added (or removed) in Fig. A.2. Note that in Fig. A.1 the absorption component emanates for p/R_⋆ < = ~1 (front of the stellar disc), which is in agreement with the conventional mechanism for P Cygni line formation.

In Fig. A.2, we present how the Hα line EW changes when the emergent flux at larger p/R_⋆ (corresponding to x = r/R_⋆ or τ_ross) is added. Note that the Hα EW is nearly constant for log  (τ_ross) > ~ −1.5: in that range the line is in absorption, mainly produced by the wind in front of the stellar disc (p/R_⋆ < ~ 1), i.e., the Hα photons originate at larger distances. From this figure, we define the Hα line-formation region as the region in which Hα changes its EW from 10% to 90%. Although it is by no means conclusive that the line forms in this region, most of the Hα photons (in the observer’s frame) are emitted from this part of the wind, and the behaviour of the line should depend on the local conditions in that region. Therefore, we investigated the Hα related quantities ((n₃/n₂) ratio, τ_Lyα,τ_Hα) at this side of the wind in different models.

Appendix B: Model atom (sophisticated models)

All Tables

All Figures