7 The ratio of blue to red supergiants in the SMC

7.1 Model results

The observed ratio B/R of blue to red supergiants in the SMC cluster NGC 330 lies between 0.5 and 0.8, according to the various sources discussed in Langer & Maeder (1995). Not many new results have been obtained since then. New IR searches have revealed some AGB stars in the SMC (Zilstra et al. 1996) and ISO observations (Kucinskas et al. 2000) have led to the detection of an IR source in NGC 330, which may be a Be supergiant or a post AGB-star, but this does not change the statistics significantly. Notice that the definition of B/R is not always the same, e.g. for Humphreys & McElroy (1984), B means O, B and A-supergiants. Here, we strictly count in the B/R ratio the B star models from the end of the MS to type B9.5 I, which corresponds to $\log T_{{\rm eff}} = 3.99$according to the calibration by Flower (1996). We count as red supergiants all star models below $\log T_{{\rm eff}} = 3.70$ since red supergiants in the SMC are not as red as in the Galaxy (Humphreys 1979). We note that the exact definition of this limit has no influence on the observed or theoretical B/R ratios, since the evolution through types F, G, K is always very fast.

As noted by Langer & Maeder (1995), the current models (without rotation) with Schwarzschild's criterion predict no red supergiants in the SMC (cf. Schaller et al. 1992). This is also seen in Fig. 9 which illustrates for models of $20~M_\odot$ at Z = 0.004 the variations of the $T_{{\rm eff}}$ as a function of the fractional lifetime in the He-burning phase for different rotation velocities. For zero rotation, we see that the star only moves to the red supergiants at the very end of the He-burning phase, so that the B/R ratio, with the definitions given above, is ${\rm B/R} \simeq 47$. For average rotational velocities $\overline{v}$ during the MS, $\overline{v} = 152$, 229 and 311 kms^-1, one has respectively ${\rm B/R} = 1.11$, 0.43 and 0.28. Thus, the B/R ratios are much smaller for higher initial rotation velocities, as rotation favours the formation of red supergiants and reduce the lifetime in the blue. We notice in particular that for $\overline{v} = 200$ kms^-1, we have a B/R ratio of about 0.6 well corresponding to the range of the observed values.

Figure 9: Evolution of the $T_{{\rm eff}}$as a function of the fraction of the lifetime spent in the He-burning phase for $20~M_\odot$ stars with different initial velocities.

Figure 10: Evolution of the $T_{{\rm eff}}$as a function of the fraction of the lifetime spent in the He-burning phase for 15, 20 and $25~M_{\odot}$ stars at Z = 0.004 with $v_{{\rm ini}} = 0$ and 300 kms-1.

The B/R ratios change with the stellar masses. Figure 10 shows for the models of 15, 20 and 25 $~M_{\odot}$ the changes of $T_{{\rm eff}}$as a function of the fractional lifetimes in the He-burning phase for different rotation. For all masses, we notice that the non-rotating stars spend nearly the whole of their He-phase as blue supergiants and almost none as red supergiants. For $v_{\rm ini} = 300$ kms^-1 (which corresponds to about $\overline{v} = 220$ kms^-1), we notice a drastic decrease of the blue phase and a corresponding large increase of the red supergiant phase.

Figure 11 shows the same as Fig. 9 but for the models of 9 and 12 $~M_{\odot}$. These models mark the transition from the behaviour of massive stars, which move at various paces from blue to red, to the intermediate mass stars, which go directly to the red giant branch and then describe blue loops in the HR diagram. At zero rotation, the 15 $~M_{\odot}$ model has the "massive star'' behaviour and the 9 $~M_{\odot}$ model shows a most pronounced "blue loop''. For $v_{{\rm ini}} = 0$ kms^-1, the 12 $~M_{\odot}$model is just in the transition between the behaviours of nearby models of 9 and 15 $~M_{\odot}$. The rotating model at 15 $~M_{\odot}$ is first blue and then goes to the red, while the rotating 9 $~M_{\odot}$ model goes first to the red, then back to the blue and red again. The behaviour of the rotating 12 $~M_{\odot}$ is also intermediate between these two, with the consequence that it always stays more or less in the blue, which is surprising at first sight, but well consistent with the mentioned intermediate behaviour. As seen in Sect. 5, this transition zone with almost entirely blue models extends from about 10.5 to 12.2 $~M_{\odot}$.

Figure 11: Evolution of the $T_{{\rm eff}}$as a function of the fraction of the lifetime spent in the He-burning phase for 9 and 12$~M_{\odot}$ stars at Z = 0.004 with $v_{{\rm ini}} = 0$and 300 kms-1.

The rotating 9 $~M_{\odot}$ model has a blue loop smaller than for zero rotation; it only extends to the A-type rather than to the B-type range. At a given L and $T_{{\rm eff}}$, the average density in a rotating model is much smaller than in the non-rotating one, so that the period will be longer. The result is such that the application of the standard period-luminosity relation will lead for a given observed period to a too high luminosity, if the star was a fast rotator on the MS. A more complete study of the effects of rotation on Cepheids will be made in a further work.

Table 2 shows the B/R ratios for the various relevant masses for the models with zero-rotation and $v_{\rm ini} = 300$ kms^-1. Apart from the transition model of 12 $~M_{\odot}$, which stays almost entirely in the blue as discussed above, we notice that the B/R ratios decrease very much with rotation, being in the range 0.1 to 0.4 for $v_{\rm ini} = 300$ kms^-1. As noted for the $20~M_\odot$ model, an average velocity of about 200 kms^-1 corresponds to a B/R ratio of 0.6. The order of magnitude obtained is satisfactory, however, future comparisons in clusters will need detailed convolution over the IMF and the distribution of rotational velocities in clusters studied at various metallicities. This is beyond the scope of this paper and we now examine the effects in the internal physics which determine the B/R ratio.

7.2 Stellar physics and the B/R ratio

There are several studies on the blue-red motions of stars in the HR diagram, for example by Lauterborn et al. (1971), Stothers & Chin (1979), Maeder (1981), Maeder & Meynet (1989) and recently by Sugimoto & Fujimoto (2000). Sugimoto and Fujimoto identify several parameters at the base of the envelope $W, \Theta, V$ and N, which play a role in the redwards evolution. Apart from N, which is the polytropic index, we may note that the other parameters are all some function of the local gravitational potential. V is the ratio of the gravitational potential to the thermal energy as in Schwarzschild's textbook (1958). The parameter W is given by W= V/U, where U is the ratio of the local density to the average internal density. The parameter $\Theta$ is given by $\Theta = \ln (\frac{P}{\rho})_{{\rm c}} - \ln (\frac{P}{\rho})_{{\rm env}}$, where the index "c'' refers to the center and "env'' to the base of the envelope. We can easily check that $\Theta$is also related to the potential at the center and at the base of the envelope, as well as to the local polytropic index.

 

 

Table 2: Values of the B/R ratios for models with zero rotation and for models with $v_{\rm ini} = 300$ kms^-1. B means strictly B-type supergiants and R means K-and M-type supergiants (see text).

$M_{\rm ini}$	B/R	B/R
	$v_{{\rm ini}} = 0$	$v_{\rm ini} = 300$
25	63	0.30
20	47	0.43
15	5.0	0.24
12	20.6	85
9	2.7	0.10

We may thus wonder whether most of the effects determining blue vs. red motions in the HR diagram cannot be understood, at least qualitatively, in terms of mainly the gravitational potential of the core. It is very desirable to try to establish some relatively simple scheme for understanding the results of numerical computations. The role of the core gravitational potential for the inflation or deflation of the stellar radius has been emphasized by Lauterborn et al. (1971) in the case of the occurrence of blue-loops for intermediate mass stars (see also Maeder & Meynet 1989). We shall examine here whether we may extend the very useful "rules'' derived by Lauterborn et al. (1971) to the case of massive stars in rotation as studied here. We call $\Phi_{{\rm c}}$the potential of the He-core, which due to a mass-radius relation for the core behaves as $\Phi_{{\rm c}} \sim M_{{\rm c}}^{0.4}$, where $M_{{\rm c}}$is the core mass. The blue-red motions in the HR diagram mainly depend on the comparison of $\Phi_{{\rm c}}$with some critical potential $\Phi_{{\rm crit}}(M)$, which grows with the stellar mass. One has

$$\Phi_{{\rm c}} > \Phi_{{\rm crit}}(M) \quad \quad \quad {\rm Hayashi \quad line}$$ (8)

$$\Phi_{{\rm c}} < \Phi_{{\rm crit}}(M) \quad \quad \quad {\rm blue \quad location}.$$ (9)

Equations (8) and (9) essentially apply to a steep hydrogen-profile around the core. If this profile is mild, one should account for additional terms (Lauterborn et al. 1971). The main effect can be represented by a parameter h increasing with the amount $\Delta M_{{\rm He}}$ of helium in the transition region above the core (formally, a simplified expression for h in the cases considered by Lauterborn et al. (1971) is given by $\log h = 8.5 X_{{\rm d}} M_ {{\rm d}}/ M$, where $X_{{\rm d}}$ is the amplitude of the change of the H-mass fraction over the smooth transition zone, $M_ {{\rm d}}$ is the width in mass of the transition zone and M the total stellar mass). The value of h also depends on the distribution of this amount of helium $\Delta M_{{\rm He}}$, h is larger if the amount of helium is close to the shell H-burning source. However, if a sufficient amount of helium is brought far from the shell source, in the outer envelope, h may even decrease as suggested by Lauterborn et al. (1971). This also has some consequences, as discussed below. With these remarks, the two above equations become for mild H-profiles

$$h \; \Phi_{{\rm c}} > \Phi_{{\rm crit}}(M) \quad \quad \quad {\rm Hayashi \quad line}$$ (10)

$$h \; \Phi_{{\rm c}} < \Phi_{{\rm crit}}(M) \quad \quad \quad {\rm blue \quad location}.$$ (11)

We shall now try to see whether we may describe correctly with relations (10) and (11) the different physical effects influencing the blue-red motions of massive stars:

Figure 12: Comparison of the internal distribution of helium in two models of $20~M_\odot$ at the middle of the He-burning phase. The dashed-dot line concerns the models with zero rotation and the continuous line represents the case with $v_{\rm ini} = 300$ kms-1.

Mass loss: Mass loss decreases the total stellar mass and thus $\Phi_{{\rm crit}}(M)$, which favours a motion towards the Hayashi line. $\Phi_{{\rm c}}$ is not very much changed, since the size of the final He-core is not very different. However, there is more helium near the H-burning shell, which increases the parameter h and also favours the formation of red supergiants.

This description is fully consistent with the well known fact that, due to mass loss, the intermediate convective zone is much less important (cf. Stothers & Chin 1979; Maeder 1981). A convective zone imposes a polytropic index $N \simeq 1.5$, which implies only a weak density gradient, making the stellar radius smaller and thus keeping the star in the blue. Thus, the physical connexion we have with the interpretation in terms of $\Phi_{{\rm c}}$ is the following one. The larger He-burning core with respect to the actual stellar mass together with the higher He-content in the H-shell region (higher h) lead to a less efficient H-burning shell, thus there is no large intermediate convective zone and this absence permits a red location of the star in the HR diagram.

Figure 13: Comparison of the internal values of $\nabla _{{\rm ad}}$ and of $\nabla _{{\rm rad}}$ in models of $20~M_\odot$ at the very beginning of the He-burning phase ( $Y_{{\rm c}} = 0.993$). The dashed-dot lines concern the models with zero rotation and the continuous lines represent the case with $v_{\rm ini} = 300$ kms-1.

Figure 14: Comparison of the internal values of $\nabla _{{\rm ad}}$ and of $\nabla _{{\rm rad}}$ in models of $20~M_\odot$ at the middle of the He-burning phase (same model as in Fig.  12). Same remarks as in previous figure.

Overshooting: The overshooting does not change $\Phi_{{\rm crit}}(M)$, but $\Phi_{{\rm c}}$is increased, with no major change of the H-profile and thus of h. Clearly, overshooting is thus favouring a redwards motion to the Hayashi line, with the formation of red supergiants. As for mass loss, the larger core contributes to reduce the intermediate convective zone, which leads to the formation of red supergiants.

Lower metallicity Z: A lower Z decreases the mean molecular weight (essentially because the He-content is also lower, see Sect. 2.3). This decreases the internal temperature and the luminosity during the MS phase, leading to slightly smaller convective cores, as shown by numerical models (cf. Meynet et al. 1994). This produces smaller $\Phi_{{\rm c}}$ which favours a blue location, as is observed.

We also note that a lower Z means a slightly higher electron scattering opacity (due to the higher H-content), which would favour larger cores, however this effect appears as a minor one in the models.

Rotation: The effects of rotation are numerous and subtle, and the balance between them is delicate. We notice the following effects:

1.

A first simple effect is that rotation enhances the mass loss rates as described by Eq. (5) and this contributes to favour the formation of red supergiants. However, this effect although significant does not seem to be the dominant one;

2.

The mixing in the MS phase leads to a slight extension of the core, which favours a redwards motion during the He-burning phase. This is just like overshooting, thus it may be very difficult, except perhaps by asteroseismological studies of stars with different rotations, to distinguish between the core extension by overshooting or by rotation. An example of this effect can be seen in Fig. 12, which shows models of a $20~M_\odot$ star in the middle of the He-burning phase. The core in the non rotating case is 2.7$~M_{\odot}$, while it is 3.6$~M_{\odot}$ in the rotating case, i.e. 1/3 larger in mass. Although this increase of the core is very significant (in particular for nucleosynthesis), its effect on the blue-red motions appears less important than the effect we now discuss;

3.

A mild mixing just outside the core as produced by rotation during the MS phase clearly increases the amount of helium near and above the H-shell. This is well seen in Fig. 12, (in the two models shown, the H-shell is just on the right side of the big He-peak). According to the definition of h given above, this results in a larger value of h, which leads Eq. (10) to be satisfied and this favours a red location in the HR diagram. The analysis of the sequence of the models before that illustrated in Fig. 12 shows that the different H-profile of the rotating star is essentially the consequence of the rotational mixing during the MS phase. Thus, the rotational mixing during the MS is the key effect for the formation of red supergiants at low Z.
It is satisfactory to see that rules expressed by Eqs. (8) to (11) are fully consistent with what we can deduce by studying the consequences of a change of mean molecular weight. The larger He-core in the rotating models means that a larger fraction of the total luminosity is made in the core (0.42 instead of 0.31 for the models of Fig. 12). This means that the H-burning shell in the rotating model produces a smaller fraction of the total luminosity and this contributes to reduce the importance of the convective zone above the H-burning shell. Simultaneously, the higher He-content near the shell in the rotating case leads to a decrease in the opacity, and this also contributes to reduce the importance of the convective zone associated with the H-burning shell. Figures 13 and 14 show the internal values of $\nabla _{{\rm ad}}$ and $\nabla _{{\rm rad}}$ in $20~M_\odot$ stars at the beginning of the He-burning phase and at the middle of this phase respectively. We notice the smaller convective zone associated with the H-burning shell in the rotating model in Fig. 13 and its earlier disappearance in Fig. 14. As long as it exists, the convective zone maintains a small polytropic index over the concerned region, and the larger this region is, the smaller the radius. Only when the intermediate convective zone disappears, the star reaches the red supergiant location in the HR diagram. We see that the earlier disappearance of this intermediate convective zone in rotating stars favours their earlier evolution towards red supergiants, consistent with the result of Eqs. (10) and (11);

4.

We may wonder what are the respective effects of mixing on the MS and during the He-burning phase. What we have shown above is just the consequence of the mixing during the MS phase. Some tests have been made and show the following. If we arbitrarily stop the mixing during the He-burning phase, this makes little difference. However, if on the contrary we arbitrarily enhance the mixing during the He-burning phase, this maintains the star in a blue location in the HR diagram. This result is in agreement with the effect mentioned above and discussed by Lauterborn et al. (1971). The parameter h normally grows with $\Delta M_{{\rm He}}$, however it may reach a maximum and then decrease, if helium is brought far enough from the H-shell. Thus, strong mixing in the He-burning phase by spreading helium throughout the star may make h decrease, which according to Eq. (11) leads to a blue location in the HR diagram.

Why does this occur physically? Mixing of helium also implies mixing of hydrogen. For moderate mixing, the H-burning shell becomes more active, due to the increase of its H-content and T. This activates the intermediate convective zone associated with the H-shell. Additional He at the surface also decreases the opacity and favours bluer stars. For more extreme cases of mixing both in the MS and subsequent phases, one would move to the case of almost homogeneous stars, which occurs for fast rotators such as the 60 $~M_{\odot}$ model at Z = 0.02 with a $v_{\rm ini}$ of $\sim$400 kms^-1(cf. Meynet & Maeder 2000). In this case, a bluewards evolution in the HR diagram occurs during the MS phase, as shown by Schwarzschild (1958).