6 HR diagram, mass-luminosity relations and lifetimes

   

Table 2: Properties of the stellar models at the end of the H-burning phase, at the end of the He-burning phase and at the end of the C-burning phase or during the thermal pulse-AGB phase. The masses are in solar mass, the velocities, in km s^-1 and the lifetimes, in million years. The abundances are in mass fraction. The abundance ratios are normalized to their initial values, which are, in mass fraction, (N/C) $_{\rm ini}= 0.309$ and (N/O) $_{\rm ini}= 0.035$.

			End of H-burning					End of He-burning					End of C-burning
M	$v_{\rm ini}$	$\overline v$	$t_{{\rm H}}$	v	$Y_{{\rm s}}$	N/C	N/O	$t_{{\rm He}}$	v	$Y_{{\rm s}}$	N/C	N/O	$M_{\rm fin}$	v	$Y_{{\rm s}}$	N/C	N/O
60	0	0	3.883	0	0.23	1.00	1.00	0.332	0	0.23	1.00	1.00	59.57	0	0.23	1.00	1.00
	300	327	3.513	545	0.35	64.2	25.8	0.345	2	0.47	126	49.8	50.42	5	0.51	145	63.0
40	0	0	4.974	0	0.23	1.00	1.00	0.419	0	0.23	1.00	1.00	39.86	0	0.23	1.00	1.00
	300	289	4.279	355	0.25	80.8	14.3	0.436	21	0.28	123	19.1	38.61	4	0.30	139	22.4
20	0	0	8.773	0	0.23	1.00	1.00	0.886	0	0.23	1.00	1.00	19.97	0	0.23	1.00	1.00
	200	157	7.445	139	0.23	12.8	4.89	0.978	8	0.23	21.3	6.55	19.97	1	0.26	45.2	11.6
	300	240	7.624	228	0.23	28.6	7.20	0.902	81	0.24	54.2	10.4	19.97	75	0.24	54.8	10.6
	400	325	7.737	338	0.23	69.1	9.23	0.932	155	0.27	150	16.6	19.90	208	0.27	153	17.3
15	0	0	12.12	0	0.23	1.00	1.00	1.473	0	0.23	1.00	1.00	14.99	0	0.30	42.3	19.5
	300	234	10.65	212	0.23	21.9	6.74	1.281	111	0.24	53.6	11.3	14.98	2	0.29	126	23.2
9	0	0	25.12	0	0.23	1.00	1.00	3.285	0	0.23	1.00	1.00	8.998a	0	0.24	34.5	15.5
	200	151	22.03	128	0.23	5.45	3.16	3.414	72	0.23	24.6	8.00	8.997	1	0.26	79.3	18.0
	300	230	22.50	212	0.23	79.5	10.5	3.040	73	0.23	107	12.0	8.997a	2	0.27	234	23.4
	400	307	22.77	294	0.23	365	13.7	3.612	86	0.24	477	17.4	8.997a	2	0.27	579	25.7
													AGB phase
													$M_{\rm fin}$	$Y_{{\rm s}}$	C	N	O
7	0	0	38.28	0	0.23	1.00	1.00	5.917	0	0.23	1.00	1.00	6.999	0.24	2.5E-7	2.4E-6	4.9E-6
	300	229	34.66	209	0.23	78.6	9.67	5.576	74	0.24	160	12.9	6.998	0.36	1.7E-3	7.0E-4	6.6E-4
5	0	0	70.71	0	0.23	1.00	1.00	14.38	0	0.23	1.00	1.00	5.000	0.24	2.5E-7	2.2E-6	5.0E-6
	300	226	65.69	198	0.23	16.3	5.32	13.61	94	0.24	68.2	12.7	4.996	0.34	7.5E-4	1.0E-3	3.9E-4
3	0	0	200.6	0	0.23	1.00	1.00	50.22	0	0.23	1.00	1.00	2.990	0.27	2.9E-7	1.7E-6	5.5E-6
	300	229	208.0	228	0.23	26.9	6.97	52.52	29	0.26	177	15.8	2.910	0.29	1.3E-5	7.4E-4	2.0E-4
2	0	0	637.5	0	0.23	1.00	1.00	109.3	0	0.24	4.47	2.59	2.000	0.25	3.5E-7	1.0E-6	6.3E-6
	300	251	688.3	318	0.23	3.27	2.29	107.8	9	0.28	257	10.8	1.863	0.29	6.2E-8	6.0E-6	5.7E-6
a Models at the beginning of the C-burning phase.

The effects of rotation at Z=0.020have already been discussed in Talon et al. (1997), Denissenkov et al. (1999), Heger et al. (2000a), Heger & Langer (2000), Meynet & Maeder (2000). At the metallicity of the Small Magellanic Cloud, the effects of rotation have been discussed by Maeder & Meynet (2001). Let us very briefly recall the most important effects of rotation:

• The Main Sequence (MS) evolutionary tracks with rotation are extended toward lower effective temperatures and reach higher luminosities at the end of the MS phase than their non-rotating counterparts. In that respect rotation acts as a moderate overshoot.

• The MS lifetimes increase with rotation, typically by about 10% for a 20 $M_\odot$ model with an average rotational velocity on the MS corresponding to the observed ones.

• For a given value of the initial mass and metallicity, the evolutionary tracks may be different, due to different initial rotational velocities. Even for a given initial rotation, the tracks may appear different depending on the angle of view, since the polar regions of a rotating star are in general hotter than the equatorial ones (cf. Maeder & Peytremann 1970). This induces some scatter in the position of the end of the MS phase in the HR diagram.

• The theoretical period-luminosity relation for the Cepheids is changed by rotation. A Cepheid, at a given position in the HR diagram, if originating from a rotating progenitor, will pulsate with a longer period than a Cepheid having a non-rotating progenitor.

• The evolution toward the red supergiant phase is favoured by rotation. Rotation also facilitates the formation of Wolf-Rayet stars.

• The surface abundances are modified.

We shall see that the models at the very low metallicity Z = 10^-5present some striking differences with respect to what happens at higher metallicities: in particular, rotation implies smaller main sequence lifetimes and rotation does not favour the evolution toward the red supergiant stage (at least for the range of initial velocities explored here).

6.1 The HR diagram

For most of the stellar models, we computed the rotating tracks for an initial velocity $v_{\rm ini}$ = 300 km s^-1. This value of $v_{\rm ini}$ corresponds to a mean velocity $\overline v$ during the MS between 226 and 240 km s^-1 for initial masses below 20 $M_\odot$ (see Table 2). These values are close to the mean rotational velocities observed for OBV type stars at solar metallicity, which are between 200-250 km s^-1. For more massive stellar models at Z= 10^-5the average velocities are higher. This results essentially from the larger outward transport of angular momentum by circulation in more massive stars (see Sect. 4 and Maeder & Meynet 2001).

Figure 10 shows the evolutionary tracks of non-rotating and rotating stellar models for initial masses between 2 and 60 $M_\odot$. The effective temperatures plotted correspond to an average orientation angle (see also Paper I). At the beginning of the evolution on the ZAMS, rotational mixing has no impact on the structure, since the star is homogeneous. At this stage, only the hydrostatic effects of rotation are present, i.e. the effects due to the centrifugal acceleration term in the hydrostatic equilibrium equation. As is well known, these effects shift the ZAMS position toward lower values of L and $T_{{\rm eff}}$ (see e.g. Paper I). From Fig. 10, one sees that the less massive the star, the greater the shift. This results from the facts that, for a given $v_{\rm ini}$, the lower the initial mass, the greater the ratio of the centrifugal force to the gravity. Indeed this ratio, equal to ${v_{\rm ini}^2\over R} { R^2 \over GM}$, varies as about $1/M^{\alpha}$ with $\alpha$ equal to about 0.4.

As was the case at higher metallicities, the MS width is increased by rotation. Rotational mixing brings fresh H-fuel into the convective core, slowing down its decrease in mass during the MS. A more massive He-core is produced at the end of the H-burning phase, which favours the extension of the tracks toward lower effective temperatures. Rotational mixing also transports helium and other H-burning products (essentially nitrogen) into the radiative envelope. The He-enrichment lowers the opacity. This contributes to the more rapid increase of the stellar luminosity during the MS phase and limits the redwards motion in the HR diagram.

The widening of the MS produced by rotation mimics the effect of an overshoot beyond the convective core (see Talon et al. 1997, Paper VII). Since the observed width of the MS has often been taken to parameterize the size of the convective core, the above argument shows that rotating models tend to decrease the amplitude of the overshoot necessary to reproduce the observed MS width.

Figure 11 shows the evolutionary tracks of 20 $M_\odot$ models for different initial velocities and metallicities during the H-burning phase. One sees that the extension of the MS due to rotation decreases when the metallicity decreases. This results from the smaller increase of the He-core due to rotation at low Z. Typically, at the end of the MS at Z = 10^-5, the helium core mass in the rotating 20 $M_\odot$ model ( $v_{\rm ini}$ = 300 km s^-1) is greater by 11% with respect to its value in the non-rotating model. The corresponding increase at Z=0.004is 23%. The reason for this difference is the following one. When rotation brings fresh hydrogen fuel, in the core, it brings also carbon and oxygen which act as catalysts in the CNO burning. These catalyst elements are of course in much lower abundances at Z=10^-5 than at Z = 0.004 and thus the core enhancement is less pronounced.

Figure 11: 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 = 10-5.

At Z = 0.004, the rotating star models with initial masses between 9 and $\sim$25 $M_\odot$, are evolving, after the MS phase, much more rapidly toward the red supergiant stage (RSG) than the non-rotating models, in agreement with the observed number ratio of blue to red supergiants in the Small Magellanic Cloud cluster NGC 330 (Maeder & Meynet 2001). As was extensively discussed, the balance between the blue and the red is very sensitive to many effects. At the very low metallicity considered here, rotation does not succeed in producing red supergiants, at least for the range of initial rotational velocities explored. The dominant reason appears to be the smaller He-cores produced at lower Z and the fact that the growth of this core due to rotation is smaller than at higher Z. In terms of the discussion by Maeder & Meynet (2001), this reduces the central potential enough to keep a blue location during the whole He-burning phase, and the amount of helium diffused in the region of the shell is unable to compensate for the smaller core. Only, during the very last stages in the C-burning phase, when the core heavily contracts, does the central potential grow enough to produce a red supergiant.

The rotating models do not present any well developed blue loop, except in the case of the 2 $M_\odot$ model. In that respect the situation is similar to the case of the non-rotating models (see Sect. 5 above). Interestingly, we note that the rotating models, with initial mass below 7 $M_\odot$, evolve redwards during the AGB-phase. This a consequence of the third dredge-up, which brings at the surface carbon and oxygen synthesized in the He-burning shell, as well as primary nitrogen built up in the H-burning shell (see Sects. 7 and 8). The important enhancements of these elements at the surface make the star to behave as a more metal rich star and thus push it to a redder location in the HR diagram.

In the present grid no model enters the Wolf-Rayet phase. At the end of the C-burning phase, the mass fraction of hydrogen at the surface of the rotating 60 $M_\odot$ model is still important ($\sim$0.48), although much lower than at the surface of the corresponding non-rotating model ($\sim$0.77). It is likely that more massive or faster rotating star models may enter the Wolf-Rayet phase before central He-exhaustion.

6.2 Masses and mass-luminosity relations

Table 2 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 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 8. The Cols. 9 to 13 present some properties of the models at the end of the core He-burning phase; $t_{{\rm He}}$ is the He-burning lifetime. Some characteristics of the last computed models are given in Cols. 14 to 18; $M_{\rm fin}$ is the final stellar mass. For stars with initial mass superior or equal to 9 $M_\odot$, the final stage corresponds to the end of the C-burning phase. For the lower initial mass stars, it corresponds to the beginning of the Thermal Pulse AGB (TP-AGB) phase. Typically the rotating 3 $M_\odot$ model was computed until the fifth thermal pulse. For the intermediate mass stars, the mass fractions of carbon (C), oxygen (O) and nitrogen (N) at the surface of the stars are given.

Rotation, by enhancing the luminosity and lowering the effective gravity, increases the mass loss rates (Maeder & Meynet 2000). As a consequence, the final masses of the rotating models are smaller. At the metallicity Z = 10^-5, except for the 60 $M_\odot$ model, the effects of rotation on the final stellar masses are very weak (see Table 2).

In general, rotation makes the star overluminous for their actual masses. Typically for $v_{\rm ini}=300$ km s^-1, the luminosity vs. mass (L/M) ratios at the end of the MS are increased by 10-14% for stars in the mass range from 3 to 40 $M_\odot$. This results essentially from the He diffusion in the radiative envelope which lowers the opacity and makes the star overluminous. In the 60 $M_\odot$ model, mixing is particularly efficient and the increase of the L/M ratio amounts to 23%. For the 2 $M_\odot$ model, the L/M ratio is decreased in the rotating model, by 6-7%. In this last case, the convective core during the H-burning phase disappears very early, when the mass fraction of hydrogen in the center is still high ( $X_{\rm c} = 0.52$ in the $v_{\rm ini}=300$ km s^-1 model). This puts farther away from the surface the region where helium is produced and thus slows down the helium diffusion in the outer envelope.

The increase of L/M, due to rotation, at Z =10^-5 are in general inferior to those obtained at Z = 0.004, which are between 15-22% (Maeder & Meynet 2001). This is mainly a consequence of the following fact: at Z=10^-5, the increase of the H-burning convective core due to rotation is inferior to that obtained at Z = 0.004 for the same value of $v_{\rm ini}$.

6.3 Lifetimes

Generally we can say that the MS lifetime duration is affected by rotation at least through three effects:

• 1) Rotation increases the quantity of hydrogen burnt in the core. This increases the MS lifetime.

• 2) The hydrostatic effects of rotation make a star of a given initial mass to behave as a non-rotating star of a smaller initial mass. This tends to increase the MS lifetime.

• 3) Rotation increases the helium abundance in the outer radiative envelope. This tends to make the star overluminous with respect to its non-rotating counterpart and thus to reduce the MS lifetime.

When the metallicity decreases, the effect number 3 tends to become the most important one.

This can be seen from a detailed comparison of the tracks in Fig. 11. Indeed the evolutionary tracks for our rotating 20 $M_\odot$ models ( $v_{\rm ini}$ = 300 km s^-1) become overluminous with respect to the non-rotating tracks at an earlier stage for lower metallicities. This mainly results from the fact that when the metallicity decreases, rotational mixing is more efficient (see Sect. 3). Also, at lower Z, the stars are more compact and therefore the timescale for mixing, which is proportional to the square of the radius, decreases. This favours the helium diffusion in the outer envelope. The diffusion of hydrogen into the core, which would increase the MS lifetime, is not really affected, because hydrogen just needs to migrate over the convective core boundary to be engulfed into the core. One notes also that for given values of the equatorial velocity and of the initial mass, the ratio  $\Omega /\Omega _{\rm c}$ of the angular velocity to the break-up velocity decreases whith the metallicity. Thus the hydrostatic effects, which usually make the star fainter, are in general smaller at lower Z.

As a consequence of the above effects, at Z=10^-5, the MS lifetimes are decreased by about 4-14% for the mass range between 3 and 60 $M_\odot$ when $v_{\rm ini}$increases from 0 to 300 km s^-1 (cf. Table 2).

We notice that the rotating 2 $M_\odot$ model has a longer MS phase than its non-rotating counterparts, in contrast with what happens for higher initial mass stars. This is because, when the initial mass decreases, the hydrostatic effects become more and more important (see Fig. 10).

For what concerns the effects of rotation on the He-burning lifetime, let us simply say that when $v_{\rm ini}$ increases from 0 to 300 km s^-1, the changes in the He-burning lifetimes are inferior to 10%.