6 HR diagram, mass-luminosity relation and lifetimes

6.1 A brief review of the effects of rotation on the stellar models

Figure 6: Evolutionary tracks for non-rotating (dotted lines) and rotating (continuous lines) models for a metallicity Z = 0.004. The rotating models have an initial velocity $v_{\rm ini}$ of 300 kms-1. For the rotating $60~M_{\odot}$ model, only part of the evolution is plotted.

Many effects induced by rotation at solar metallicity are also found here at lower metallicity. These effects have been described in detail by Heger et al. (2000), Maeder & Meynet (2000a) and Meynet & Maeder (2000). Thus we will be very brief here. Rotation modifies the evolutionary tracks in the HR diagram through the following main physical effects: rotation lowers the effective gravity $g_{\rm eff}$; it enlarges the convective cores and smoothes the chemical gradients in the radiative zones (the main effect); it enhances the mass loss rates; it produces atmospheric distortions. Since the star has lost its spherical symmetry, the tracks may also change in appearance depending on the angle of view. Here we suppose an average angle of view, as was done in Paper V.

As a consequence of these effects, depending on the degree of mixing (see below), the evolutionary tracks on the MS are extended towards lower effective temperatures (as would do a moderate overshooting) and/or are made overluminous. The evolution towards the red supergiant stage is favoured (see Sect. 7), as well as the evolution into the WR phase. Moreover, as will be discussed below, the MS lifetimes are increased and the surface abundances are modified.

Since there are different rotational velocities, a star of given initial mass and metallicity can follow different tracks corresponding to various initial rotational velocities. To minimize the number of tracks to be computed, we have to choose a value of the initial velocity which produces an average velocity during the MS not too far from the observed value. The problem here is that one does not know what this average rotational velocity is for OB stars at the metallicity of the Small Magellanic Cloud (SMC). In the absence of such an information, we adopt here the same initial rotational velocity as in our grid at Z = 0.020, namely the value $v_{\rm ini} = 300$ kms^-1. Defining a mean equatorial velocity $\overline{v}$ during the MS as in Meynet & Maeder (2000), such a value for $v_{\rm ini}$ corresponds to values of $\overline{v}$ between 220 and 260 kms^-1. These values are close to the average rotational velocities observed for OBV type stars at solar metallicity, which are between 200-250 kms^-1.

6.2 The HR diagram

Figure 6 shows the evolutionary tracks of non-rotating and rotating stellar models for initial masses between 9 and 60$~M_{\odot}$. One sees that the MS width is increased, as would result from a moderate overshoot. Let us recall here that two counteracting effects of rotation affect the extension of the MS band (see Paper V). On one hand, rotational mixing brings fresh H-fuel into the convective core, slowing down its decrease in mass during the MS. This effect produces a more massive He-core at the end of the H-burning phase and this favours the extension of the tracks towards lower effective temperatures. On the other hand, rotational mixing transports helium and other H-burning products (essentially nitrogen) into the radiative envelope. The He-enrichment lowers the opacity. This contributes to the enhancement of the stellar luminosity and favours a blueward track. Clearly here, the first effect dominates over the second one. In this context, it is interesting to recall that, due to the account of horizontal turbulence in the present models, the mixing of the chemical elements by the shear has been effectively reduced in the regions of steep $\mu$-gradient with respect to Paper V (see Sect. 2). This favours, for a given initial rotational velocity, an extension of the MS towards lower effective temperatures. Indeed when rotational mixing is decreased, either as a consequence of the reduction of the initial velocity or by a reduction of the shear diffusion coefficient as in the present models, the time required for helium mixing in the whole radiative envelope is considerably increased, while the time for hydrogen to migrate into the convective core, although it is also increased, remains nevertheless relatively small since hydrogen just needs to diffuse through a small amount of mass to reach the convective core (see Meynet & Maeder 2000). Thus, some increase in the size of the core results, while the effect on the helium abundance in the envelope is not significant. The same kind of effect can be seen in the models by Talon et al. (1997). Of course the $\mu$-gradients are strong near the core and can slow down the diffusion process mentioned above, but on the other hand, the efficiency of the diffusion of hydrogen will also increase with the increasing H-abundance gradient at the border of the core. These are the various reasons why the numerical models show that, for a moderate rotational mixing, the effect of rotation on the convective core mass overcomes the effect of helium diffusion in the envelope.

Let us recall that some core overshooting was needed in stellar models in order to reproduce the observed MS width (see Maeder & Mermilliod 1981; Maeder & Meynet 1989). Typically, the width of the MS band in $\Delta \log L/L_\odot$ at log $T_{\rm eff} = 4.4$ obtained from the present non-rotating models is 0.85, while a value of about 1.20 is required to reproduce the observation. Such a width can be reproduced with a moderate amount of overshooting (see Schaller et al. 1992).

Rotation as we just saw above produces also a wider MS. Indeed, the width of the MS band in $\Delta \log L/L_\odot$ at log $T_{\rm eff} = 4.4$ for the rotating models presented in Fig. 6 is slightly superior to 1. This comparison shows that if the $v_{\rm ini} = 300$ kms^-1models are well representative of the average case, then rotation alone might account for about half of the MS widening required by the observations. This is only a rough estimate. The MS extension is different for different initial rotational velocities (see Fig. 8), it will also be slightly different depending on the angle of view of the stars. Therefore the evaluation of the contributions of rotation and overshooting to the widening of the MS band requires the computations of numerous models for various initial masses and rotational velocities so that detailed population synthesis models can be performed. However, we can say with some confidence that rotation decreases by about a factor of 2 the amount of overshooting needed to reproduce the observed MS width, as was already proposed by Talon et al. (1997).

One striking difference between non-rotating and rotating models after the MS concerns the fraction of the helium burning phase spent as a red supergiant. This point will be discussed in detail in the next section. Let us also mention here that rotation shortens the blue loops of the 9$~M_{\odot}$ model. This is a consequence of the more massive helium cores existing at the end of the H-burning phase in rotating models, as well as of the addition of helium near the H-burning shell (cf. Sect. 7.2). As in the case of the 9$~M_{\odot}$ model and for the same reason, rotation reduces the extension of the "partial'' blue loop associated with the 12$~M_{\odot}$ model.

For the rotational velocities considered here, no model enters the WR phase during the core H-burning phase. But after the MS phase, the rotating 40$~M_{\odot}$ model enters the WR phase while its non-rotating counterpart does not. This illustrates the fact that rotation decreases the minimum initial mass for a single star to become a WR star (Maeder 1987; Fliegner & Langer 1995; Maeder & Meynet 2000a; Meynet 2000). We shall not develop this point further here since it will be the subject of a forthcoming paper.

Figure 7: Evolutionary tracks for non-rotating stellar models at Z = 0.004 in the transition region between stars with and without blue loops. The initial masses are indicated in solar masses.

6.3 Differences between rotating tracks at Z = 0.004 and 0.020

Since the physics has been improved with respect to Paper V (see Sect. 2), we cannot directly compare the solar tracks of Paper V with the present ones. Therefore we computed one $20~M_\odot$ model at solar metallicity with and without rotation with exactly the same physics as in the present paper. These models are plotted together with the $20~M_\odot$ models at Z= 0.004 in Fig. 8.

From this figure, one sees that rotation at low metallicity has similar effects than at solar metallicity. For instance, the increase due to rotation of the He-core masses at the end of the H-burning phase is similar at Z = 0.020 and Z = 0.004. Typically, for the $20~M_\odot$ models shown in Fig. 8, one has that in the non-rotating models, both at Z = 0.020 and Z = 0.004, the helium cores contain 26% of the total mass at the end of the MS phase. In the rotating models with $v_{\rm ini} = 300$ kms^-1, this mass fraction is enhanced up to values between 30-32%.

Due to the distribution of the initial velocities and of the orientations of the angles of view, rotation induces some scatter of the luminosities and effective temperatures at the end of the MS phase (see Paper V). One observes from Fig. 8 that, at low metallicity, for a given initial velocity, the extension towards lower effective temperatures due to rotation is slightly reduced (compare the tracks for $v_{\rm ini} = 300$ kms^-1). Thus, at low Z, our models show that, if the initial rotation velocities are distributed in the same manner as at solar metallicity, the scatter of the effective temperatures and luminosities at the end of the MS will be reduced.

Figure 8: Evolutionary tracks for rotating $20~M_\odot$ models with different initial velocities and various initial metallicities. The initial velocities $\upsilon _{\rm ini}$ are indicated. See Table 1 for more details on the models at Z = 0.004.

6.4 Masses and mass-luminosity relations

When rotation increases, the actual masses at the end of both the MS and the He-burning phases become smaller (cf. Table 1). Typically the quantity of mass lost by stellar winds during the MS is enhanced by 45-75% in rotating models with $v_{\rm ini} = 300$ kms^-1. Similar enhancements were found at solar metallicity (see Paper V). The increase is due mainly to the direct effect of rotation on the mass loss rates (cf. Eq. (5)). The higher luminosities reached by the rotating models and their longer MS lifetimes also contribute somewhat to produce smaller final masses.

Rotation makes the star overluminous for their actual masses. Typically for $v_{\rm ini} = 300$ kms^-1, the luminosity vs. mass ratios at the end of the MS are increased by 15-22%. It is interesting to mention here that even if the rotating and non-rotating tracks in the $\log L/L_\odot$ vs. $\log T_{\rm eff}$ plane are very similar, they may present large differences in the $\log g_{\rm eff}$ vs. $\log T_{\rm eff}$ plane where $g_{\rm eff}$is estimated at an average orientation angle as in Paper V. The large difference in log $g_{\rm eff}$ for similar masses, luminosities and effective temperatures comes from the fact that the effective gravity of the rotating model differs from the gravity of the non-rotating model by an amount equal to the centrifugal acceleration. Typically, the two 40$~M_{\odot}$ tracks plotted in Fig. 6 which, at $\log T_{\rm eff} = 4.4$, differ by only $\sim$0.04 dex in $\log L/L_\odot$, show important differences in the $\log g_{\rm eff}$ vs. $\log T_{\rm eff}$ plane. For instance, the rotating 40$~M_{\odot}$ track overlaps the non-rotating $60~M_{\odot}$ model in this plane. Therefore, one could expect that the attribution of a mass to an observed star position in the $\log g_{\rm eff}$ vs. $\log T_{\rm eff}$ plane is very rotation dependent. The use of non rotating tracks would overestimate the mass (in the example above by 50%), and this might be a cause of the well known problem of the mass discrepancy (see e.g. Herrero et al. 2000). Let us note that in practice the effective gravity and the other physical quantities are derived from the spectral lines, which shapes and equivalent widths are also affected by rotation.

6.5 Lifetimes

 

 

Table 1: Properties of the stellar models at the end of the H-burning phase, at $\log T_{{\rm eff}} = 4.0$(see text) and at the end of the He-burning phase. The masses are in solar mass, the velocities in kms^-1, the lifetimes in million years and the helium abundance in mass fraction. The abundance ratios are normalized to their initial value.

			End of H-burning						$\log T_{{\rm eff}} = 4.0$				End of He-burning
M	$v_{\rm ini}$	$\overline{v}$	$t_{{\rm H}}$	M	v	$Y_{{\rm s}}$	N/C	N/O	v	$Y_{{\rm s}}$	N/C	N/O	$t_{{\rm He}}$	M	v	$Y_{{\rm s}}$	N/C	N/O
60	0	0	3.951	57.709	0	0.24	1.00	1.00	0	0.57	127	267	0.345	41.733	0	0.60	185	198
	300	259	4.232	56.415	307	0.29	11.3	9.50
40	0	0	4.924	39.066	0	0.24	1.00	1.00	0	0.24	1.00	1.00	0.460	30.222	0	0.49	145	66.0
	300	257	5.312	38.650	332	0.25	7.26	4.75	81	0.60	113	68.2	0.496	20.481	21	0.73	203	595
25	0	0	7.196	24.689	0	0.24	1.00	1.00	0	0.24	1.00	1.00	0.806	24.322	0	0.24	1.00	1.00
	300	239	7.809	24.557	274	0.24	4.61	3.25	56	0.25	5.42	3.75	0.752	20.015	0.6	0.39	29.5	18.0
20	0	0	8.736	19.833	0	0.24	1.00	1.00	0	0.24	1.00	1.00	1.007	19.690	0	0.24	1.65	1.25
	200	152	9.533	19.777	146	0.24	2.94	2.25	33	0.24	3.52	2.50	0.963	18.376	1.2	0.29	13.2	8.00
	300	229	9.700	19.750	244	0.24	4.94	3.25	53	0.24	5.58	3.50	0.960	18.039	0.8	0.31	17.0	9.50
	400	311	9.940	19.683	429	0.24	6.39	3.75	65	0.25	6.84	4.00	0.952	17.758	1.0	0.33	20.4	11.0
15	0	0	12.158	14.910	0	0.24	1.00	1.00	0	0.24	1.00	1.00	1.515	14.686	0	0.25	5.00	3.00
	300	225	13.641	14.854	226	0.24	6.16	3.50	40	0.24	6.68	3.75	1.389	14.124	1.2	0.29	18.0	8.75
12	0	0	16.560	11.969	0	0.24	1.00	1.00	0	0.24	1.00	1.00	2.810	11.902	0	0.24	2.84	1.75
	300	225	18.568	11.950	228	0.24	5.06	3.00	74	0.28	16.5	7.75	1.978	11.807	1.9	0.29	18.6	8.50
9	0	0	25.911	8.996	0	0.24	1.00	1.00	0	0.24	3.19	2.00	4.543	8.966	0	0.24	3.19	1.75
	300	222	29.349	8.993	225	0.24	4.23	2.50	127	0.25	9.81	5.00	4.927	8.542	2.2	0.25	10.1	5.25

Table 1 presents some properties of the models. Columns 1 and 2 give the initial mass and the initial velocity $v_{\rm ini}$ respectively. The mean equatorial rotational velocity $\overline{v}$ during the MS phase is indicated in Col. 3. The H-burning lifetimes $t_{{\rm H}}$, the masses M, the equatorial velocities v, the helium surface abundance $Y_{{\rm s}}$ and the surface ratios N/C and N/O at the end of the H-burning phase and normalized to their initial values are given in Cols. 4 to 9. The Cols. 10 to 13 present some properties of the models when $\log T_{{\rm eff}} = 4.0$ during the crossing of the Hertzsprung-Russel diagram, or when the star enters into the WR phase (for the rotating $40~M_\odot$ models and the non-rotating $60~M_{\odot}$ model), or at the bluest point on the blue loop (for the models with $M\le 12~M_\odot$). The Cols. 14 to 19 present some characteristics of the stellar models at the end of the He-burning phase; $t_{{\rm He}}$ is the He-burning lifetime.

From Table 1 one sees that for Z=0.004 the MS lifetimes are increased by about 7-13% for the mass range between 9 and 60$~M_{\odot}$ when $v_{\rm ini}$ increases from 0 to $\sim$300 kms^-1. In general, the corresponding changes in the He-burning lifetimes are inferior to 10%. As was the case at solar metallicity, the ratios $t_{{\rm He}}/t_{{\rm H}}$ of the He- to H-burning lifetimes are only slightly decreased by rotation and remain around 9-17%. At solar metallicity, the changes of the lifetimes due to rotation are quite similar. The rotating $20~M_\odot$ model with $v_{\rm ini} = 300$ kms^-1, at solar metallicity, has a MS lifetime increased by 14% with respect to the non-rotating model. At Z = 0.004, the corresponding increase amounts to 11%, which is not significantly different.