INFLUENCE OF THERMOHALINE CONVECTION ON DIFFUSION-INDUCED IRON ACCUMULATION IN A STARS

In the present computations, important improvements were introduced in the TGEC code concerning the opacities and radiative accelerations. The opacities were computed using the OPCD v3.3 codes and data⁴ (Seaton 2005). It allows us to compute self-consistent Rosseland opacities taking into account the detailed composition of the chemical mixture. The opacities were recalculated at each time step.

To more realistically treat atomic diffusion in the TGEC code, the radiative accelerations on C, N, O, Ca, and Fe were included following the improved version (LeBlanc & Alecian 2004) of the semianalytical prescription proposed by Alecian & LeBlanc (2002). This method allows for very fast computations of radiative accelerations with a reasonable accuracy. The SVP approximation may be implemented in existing codes in a simple way. There are much less data to process than for detailed radiative acceleration calculation, because complete and detailed monochromatic opacities for each ion are not needed. A new grid of SVP parameters, well fitted to the stellar mass range considered in this work, was computed following the procedure described in LeBlanc & Alecian (2004).

The diffusion of H, He and the five metals mentioned above were included in the stellar models. The abundance of the other elements was assumed to be constant throughout the star. The radiative accelerations of these five heavy elements as well as the detailed abundances for all seven elements were computed at each diffusion time step, chosen to be about 50 times smaller than the evolution time step.

For our modeling, the convective regions were instantaneously homogenized. The H i and He ii convective regions were supposed close enough to be connected by overshooting and mixed together. On the other hand, the iron convective region, which may appear in some models in much deeper regions, was supposed disconnected from the surface convective zone.

The models were evolved from pre-main sequence up to hydrogen core exhaustion. They were assumed homogeneous on the pre-main sequence, and atomic diffusion was introduced at the beginning of the main sequence.

We first computed models without thermohaline convection. To avoid the appearance of steep and unrealistic abundance gradients at the transition between radiative and convective regions, we introduced mild mixing at the bottom of each convective zone. This mixing was modeled as a diffusion process through a coefficient D_turb introduced in the chemical transport equation:

$$\begin{equation} \rho \frac{\partial \bar{c_i}}{\partial t}=\frac{1}{r^2}\frac{\partial }{\partial r} \left(r^2 \rho D_{\rm turb} \frac{\partial \bar{c_i}}{\partial t} \right), \end{equation} \tag{ 3 }$$

where $\bar{c_i}$ represents the mean concentration of a species i, ρ is the density, r is the radius, D_turb is given by

$$\begin{equation} D_{\rm turb}=D_{\rm czb} \exp \left(\ln 2 \frac{r-r_{\rm czb}}{\Delta } \right). \end{equation} \tag{ 4 }$$

D_czb and r_czb are respectively the value of D_turb and the value of the radius at the base of the convective zone, Δ is the half width of the mixing region. D_czb and Δ are free parameters, chosen to produce a mild mixing to a small extent. The value of D_czb is taken equal to 10⁵ cm² s⁻¹, and Δ is fixed to 0.5% of the stellar radius below the surface convective region and to 0.05% of the stellar radius below the iron convective zone.

As discussed in the previous sections, thermohaline convection arises in regions where the mean molecular weight increases toward the surface. It leads to mixing on short timescales, compared to those of evolution, and it disappears when the mean molecular weight gradient vanishes. To simulate this phenomenon, we found that the most efficient and simple way was to model the mixing as usual, provided that the chosen diffusion coefficient is large enough to flatten the μ-profile any time a destabilizing μ-gradient is created (at each time step).

The diffusion coefficient proposed by Kippenhahn et al. (1980) is convenient in this framework:

$$\begin{equation} D_{\rm {th}}=\frac{H_p}{\nabla _{\rm {ad}}-\nabla } \frac{16 a c T^3}{c_p \kappa \rho ^2} \left|\frac{d \ln \mu }{dr}\right|, \end{equation} \tag{ 5 }$$

where H_P is the pressure scale height, a is the radiative pressure, c is the speed of light, T is the temperature, $\nabla _{\rm {ad}}$ is the adiabatic gradient, ∇ is the real temperature gradient, c_P is the specific heat at constant pressure, κ is the opacity, and μ is the mean molecular weight of the material. We used this prescription in our computations and checked that the μ-gradient remained always null or negative at each time step in all our models.

For the discussion here, we specifically focus on the case of 1.7 M_☉ models. Figures 2–6 compare the results obtained with (solid lines) and without (dashed lines) thermohaline mixing, at various epochs during the H-burning phase (crosses in Figure 1). Figure 2 displays the ratio of the iron mass fraction at various times to that of its initial value. Figure 3 shows the opacity profiles at the same evolution times. Figure 4 presents the differences between the radiative and adiabatic gradients, which allows us to locate the convective regions. Figure 5 displays the mean molecular weight profiles, and Figure 6 displays the mean molecular weight gradient, for the interval 4.5 ⩽ log T ⩽ 6.5.

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The models reach the main sequence with chemically homogeneous envelopes. Then, due to atomic diffusion, helium sinks and creates a stable (negative) μ-gradient below the convective envelope, whereas the radiative acceleration on iron induces an increase of its abundance in the upper stellar layers, and more specifically in the iron peak elements opacity bump centered at log T ≃ 5.2. For the 1.7 M_☉ case, the models with and without thermohaline mixing begin to diverge at about 400 Myr.

At the age of 62 Myr, in both cases the iron accumulation reaches a factor of 7 below the surface, which does not alter the opacity, and a factor 4 in the opacity bump region, which leads locally to a small opacity increase. Consequently, the radiative gradient ($\displaystyle \nabla _{\rm rad}=3/(16 \pi a c G). (P \kappa L_r)/(T^4 M_r)$) increases, but it remains smaller than the adiabatic gradient so that the corresponding region is still convectively stable. At this time (62 Myr) the model only has a single surface convective zone, due to the ionization of hydrogen. As can be seen in Figure 4, from the close-to-zero value of (∇_rad − ∇_ad) around log T ≃ 4.6, the He ii convective zone has just disappeared, due to He settling. At this point, the iron accumulation is not large enough to significantly alter the μ-profile. Here, the effect of the He-induced μ-gradient is indeed much more important than the iron contribution. During stellar evolution, as the atomic diffusion proceeds, the region where the helium-induced stable μ-gradient is the steepest slowly deepens inside the star. As soon as it occurs below the iron opacity bump, the effects of the iron accumulation on the μ-profiles become important.

At 86 Myr, iron accumulation exceeds a factor of 19 and affects the μ-profile. It is responsible for the change in the slope of the μ-profile at log T ≃ 5.3. The iron enrichment leads to a significant opacity increase and therefore to an increase of the radiative gradient which now exceeds the adiabatic one. In both cases, with or without thermohaline mixing, a new very narrow convective region appears around log T ≃ 5.2.

At 299 Myr, this iron convective region becomes significant and leads to chemical homogenization between log T ≃ 5.2 and 5.4 as shown in the μ- and (Fe/Fe₀)-profiles. The homogenization of the iron convective zone leads to an expansion of the iron accumulation region and reduces the iron maximum value from a factor of 19 to a factor of 12.5. From the beginning of the main sequence until 299 Myr, the μ-gradient remains null or negative in most models, except below the surface convective zone where it sometimes takes a slightly positive value, due to iron enrichment in the outer regions.

After 400 Myr, the μ-gradient induced by He settling is then located below the iron accumulation region and the effects of iron accumulation become visible. From now on, models including thermohaline convection below the iron accumulation layers significantly differ from those without thermohaline convection.

During the subsequent stellar evolution, radiative levitation leads to iron accumulation in the iron opacity bump region. In models without thermohaline convection, this iron accumulation drastically increases with time, reaching a factor of 20 at 400 Myr and up to a factor of 95 at 1388 Myr (Figure 2, dashed lines). The consequences on the opacity profiles can be seen in Figure 3, and those on the radiative gradients in Figure 4. It also leads to a new convective zone which persists during most of the main-sequence lifetime.

Iron accumulation in these specific layers leads to a spectacular increase of the μ-values (Figure 5, dashed lines). After 400 Myr, "μ-bumps" develop with a rapid increase slightly above log T ≃ 5.2 and a steep decrease below log T ≃ 5.4, which is highly unstable. The corresponding μ-gradients are shown in Figure 6.

In the models including thermohaline convection induced by these unstable μ-gradients, iron accumulation in the opacity bump region is drastically reduced and never exceeds a factor of 15. It is never completely suppressed however, due to the stabilizing effect of helium settling. The consequences on the opacity profiles, radiative gradients, μ-values, and their gradients can be seen in Figures 3–6 (solid lines). The evolution of the internal stellar structure in the two cases studied here is sensibly different.

In contrast to the models without thermohaline mixing, those which include it do not have persistent iron convective zones, although marginal convection appears and disappears several times during stellar evolution. Actually, in models including thermohaline mixing, the iron abundance in the opacity bump always stays close to the critical value for the onset of convection. When iron accumulation exceeds this critical value, convective mixing occurs and reduces the iron accumulation, making this region stable again. Note that when this occasional convective zone appears, it has a smaller extent than the iron bump itself, so that only part of the iron-enriched zone is homogenized. This is the reason why a small iron peak remains around log T ≃ 5.2 in the 58 Myr, 788 Myr, and 988 Myr models.

The results obtained for 1.5 and 1.9 M_☉ models are qualitatively similar to those of the 1.7 M_☉ models. To illustrate the efficiency of thermohaline convection in these models, we present in Figure 7 the molecular weight and the iron profiles obtained with and without thermohaline mixing for one model for each mass, located at the middle of the main sequence (cf. Figure 1). The radiative acceleration on iron increases with the stellar mass, so that, when no thermohaline convection is taken into account, the iron accumulation is larger in more massive models. The consequent "μ-bump" is also more prominent, as can be seen in Figure 7 (dashed lines). In the results obtained including thermohaline mixing (solid lines), we can see that iron accumulation is again drastically reduced. In the 1.9 M_☉ model, it does not exceed a factor of 10 whereas it goes up to 150 when thermohaline mixing is neglected.

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Figure 8 presents the position of the boundaries of the convective zones inside the models with and without thermohaline mixing for the three masses studied here. The crosses locate the boundaries of the iron convective zone during phases with mixing episodes. When the convective zone settles durably, its boundaries are represented with solid lines.

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For the three stellar masses considered, an iron convective zone resulting from a diffusion-induced iron accumulation rapidly appears after the beginning of the main sequence. In models without thermohaline mixing (Figure 8, upper panels), a first phase occurs during which convective episodes are followed by convectively stable periods. When convection appears in the opacity bump region, the iron peak decreases due to convective mixing. Consequently, the opacity also decreases and the radiative gradient becomes smaller than the adiabatic gradient. The region becomes radiatively stable again. In the 1.5 M_☉ models, this succession of convective and radiative episodes goes on until the end of the main sequence. In the 1.7 and 1.9 M_☉ models, after a few hundred megayears, iron accumulation has become large enough (more than a factor of 15), so that convective mixing cannot reduce the iron abundance enough to stop convection. The iron convective zone then persists during the rest of the main-sequence lifetime.

In models including thermohaline convection, the situation is different. Here again, there is a first phase during which mixing episodes alternate with convectively stable periods. Afterward, the onset of thermohaline mixing strongly modifies the convective (un)stability of the iron opacity bump region. Iron accumulation is reduced and leads, most of the time, to a radiative gradient slightly below the adiabatic gradient. When, during stellar evolution, the accumulation of iron exceeds, somewhere in the opacity bump region, the critical value for the onset of convection, the induced mixing rapidly reduces the iron abundance and the radiative stability is restored. As a result, the iron convective zone does not persist in these models.

The SPB and the β Cephei stars are main-sequence B-type pulsators. The SPB stars oscillate with periods between 0.5 and 5 days, while the β Cephei stars possess 2–8 hr oscillations. The former are interpreted as high-order g-modes, the latter as low order p and/or g-modes. In the early nineties, the new OPAL opacity determinations (Rogers & Iglesias 1992) were allowed to solve the long-standing enigma of B-type pulsators. Rosseland mean opacities computated from these tables were substantially increased in the Z-bump region compared to the previously computed opacities. This opacity enhancement (as large as a factor of 3) provided a boost for the κ-mechanism potentially activated in the metal opacity bump region and could then explain the excitation of pulsations in β Cephei and SPB theoretical standard models (Cox et al. 1992; Kiriakidis et al. 1992; Moskalik & Dziembowski 1992; Dziembowski et al. 1993a). However, as observational capabilities progress, B-type pulsators offer new challenges to the theorists. Some of them are now found in low-metallicity environments (e.g., Kołaczkowski et al. 2006, and references therein). A large number of pulsation modes are indeed detected in B stars, with frequency spectra sometimes revealing new discrepancies between theory and observations. As an example, the two β Cephei stars, ν Eri and 12 Lac (Handler et al. 2006; Jerzykiewicz et al. 2005, and references therein), present frequency spectra which cannot be reproduced with B-type star standard models (e.g., for ν Eri, Ausseloos et al. 2004). Pamyatnykh et al. (2004) showed that an "ad hoc" factor 4 enhancement of the iron group elements in the Z-bump region could help reproducing the observed pulsation spectrum of ν Eri. The combined effects of radiative levitation and gravitational settling provides a natural explanation for this local metal enrichment and could then explain the observation of more than 60 β Cephei stars in low-metallicity environments.

Following the results of Pamyatnykh et al. (2004), Bourge & Alecian (2006) presented preliminary evolutionary models of β Cephei stars evolved from the zero-age main-sequence up to 1 Myr: their evolutionary models did not include consistent diffusion computations but they included, at each time step, a rough estimation of the diffusion-induced iron accumulation in the Z-bump region. After one million years, their inferred iron enrichment is increased by a factor of 12 in the iron opacity bump. The stability analysis of these models showed that such iron enrichment makes the theoretical frequency spectra denser, which could help reducing the discrepancies between asteroseismic observations and theoretical stability analysis.

Following the same idea, Miglio et al. (2007a) carried out a parametric study of the effects of local iron enhancement on the stability of SPB and β Cephei stars. The parametric iron accumulation profile introduced in their models is described by a Gaussian function centered at log T ≃ 5.2, calibrated to be consistent with the iron accumulation profile deduced by Richard et al. (2001) in their A and F star models. As Bourge & Alecian (2006), Miglio et al. (2007a) showed that such accumulation profiles widen the theoretical SPB and β Cephei instability strips and increase the number of excited modes.

The computations by Bourge & Alecian (2006) and Miglio et al. (2007a) reduced the discrepancies between theoretical and observed pulsations. However, they did not discuss the credibility of the iron profiles introduced in their models, nor the stability of the induced μ-gradients.

On the other hand, Miglio et al. (2007b) analyzed the effects of uncertainties in the opacity computations on the excitation of pulsation modes in B-type stars. They compared models computed with OPAL opacities (Iglesias & Rogers 1996) with models computed with the recently updated OP opacity tables (Seaton 2005; Badnell et al. 2005). The OP updated data lead to an enhancement of 18% of the opacity in the Z-bump. Miglio et al. (2007b) showed that these differences considerably affect the theoretical instability strips of β Cephei and SPB stars. Using the OP opacities drastically increases the number of excited modes and significantly extends the theoretical instability strips. These new opacity computations could help solve, at least partly, the discrepancies between theoretical predictions and observations in β Cephei, SPB, and hybrid stars. Up to now, no evolutionary models have been computed for B-type stars including consistent diffusion computations and the new OP opacities. However, in the light of Miglio et al. (2007b) results, we expect the iron accumulation derived by Pamyatnykh et al. (2004) to reconcile observations and theory (by a factor of 4) to be overestimated. In this context, the iron enrichment inferred by Bourge & Alecian (2006; by a factor of 12) for a 1 Myr model should also be too large, which suggests the presence of a "forgotten" process (such as the thermohaline convection) which could reduce the diffusion-induced iron accumulation.

In contrast to SPB stars, sdB stars are evolved, compact objects with low-mass cores and outer H-rich envelopes. They are experiencing the core helium-burning phase on the extended horizontal branch. They all present chemical peculiarities attributed to gravitational settling and radiative levitation in the presence of weak stellar winds. Subdwarf B stars host two groups of pulsators: the first one, composed of hot sdB stars, is characterized by rapid oscillations, with periods ranging from 80 to 600 s, caused by low-order, low-degree p-modes. The second group, composed of the coolest sdB stars, oscillates with periods in the 2000–9000 s range, due to high-order, low-degree g-modes. Both types of pulsator are driven by the κ-mechanism acting in the iron peak element opacity bump.

The relation between the envelope metal content and the pulsation driving mechanism was first established by Charpinet et al. (1996) for the short-period B subdwarf pulsators. They showed that a solar metal content in the envelope of their models is unable to excite modes but that models with envelopes containing an overabundance of metals can globally destabilize modes such as observed. They argued that this metal enrichment needs only to occur in the driving region itself and not necessarily in the whole envelope as assumed in their models.

A natural explanation for the required local metal enhancement is the radiative levitation process, which is expected to act efficiently in the radiative envelopes of sdB stars. In these stars, as in A and F stars, the competition between gravitational settling and radiative acceleration is expected to produce local accumulations of heavy elements. Charpinet et al. (1997) presented more sophisticated and realistic models of sdB stars, the so-called "second generation models," including nonuniform iron abundances. These envelope models rely on the assumption that a state of diffusive equilibrium is reached between radiative levitation and gravitational settling operating on iron (disregarding other potentially competing processes), leading to stable nonuniform iron profiles. Charpinet et al. (1997) demonstrated that such local iron accumulations in the Z-bump region may produce efficient excitation for pulsation modes. Similar models were successfully used for detailed asteroseismic studies of several sdB stars: they globally nicely reproduced the global properties of the pulsating sdB stars and were able to closely reproduce their observed periods (Brassard et al. 2001; Charpinet et al. 2001, 2005b, 2006, 2008).

As emphasized by Charpinet et al. (2009), these "second generation models" however suffer from several shortcomings. The theoretical instability strip predicted by these models is larger than the observed one, and the observed period ranges in individual stars are also usually narrower than predicted. These discrepancies are likely due to an overestimate of the iron abundance in the Z-bump region.

In the "second generation models" of Charpinet et al., the opacities are computed using the OPAL opacity tables (Iglesias & Rogers 1996). As discussed previously for SPB and β Cephei stars, the use of the most recent OP opacities (Seaton 2005; Badnell et al. 2005) may also have a direct impact on the properties of the sdB pulsation driving mechanism. Similarly to other types of stars (Jeffery & Saio 2007; Miglio et al. 2007b), using OP opacities instead of OPAL data would increase the opacity in the Z-bump, resulting in a more efficient pulsation driving mechanism.

Here again we find evidence that the iron accumulation needed to account for the observations must be lower than obtained by the atomic diffusion alone. Introducing thermohaline convection could certainly help reducing the discrepancies.

The γ Doradus stars are A–F main-sequence pulsators. Their oscillations are interpreted as high-order g-modes with periods ranging from 0.35 to 3 days. They are driven by a flux blocking mechanism at the base of their convective envelope, as shown by Guzik et al. (2000) with frozen convection models and by Dupret et al. (2004) and Dupret et al. (2005) with time-dependent convection models. It has been demonstrated that the "convection blocking" of radiation can drive high-order g-modes only in stellar models for which the temperature at the bottom of the convective envelope ranges between 2  ×   10⁵ K (which corresponds nearly to the Z-bump region central temperature) and 4.8  ×   10⁵ K. The depth of the convective envelope plays a crucial role in the driving mechanism proposed by Guzik et al. (2000) to explain the γ Doradus pulsations.

As a consequence, the seismic properties of γ Doradus stars are quite sensitive to atomic diffusion including radiative levitation because of the way they influence the convective zones. As emphasized by Montalbán et al. (2007), the diffusion effects on convection are of various kinds. The He settling which leads to H enrichment in the external stellar layers induces an opacity increase which affects the depth of the surface convective zone. On the other hand, as discussed in previous paragraphs, radiative levitation produces reservoirs of iron peak elements which may induce the appearance of an iron convective zone near 2  ×   10⁵ K. Consequently, the driving mechanism of γ Doradus stars is expected to be sensitive to the thermohaline convection which occurs below the heavy-element accumulation zone.