2 Physics of the models

 

 

Table 1: Initial abundances in mass fraction.

Element	Initial abundance
H	0.76996750
3He	0.00002438
4He	0.22999812
12C	7.5500e-7
13C	0.1000e-7
14N	2.3358e-7
15N	0.0092e-7
16O	67.100e-7
17O	0.0300e-7
18O	0.1500e-7
20Ne	7.8368e-7
22Ne	0.6306e-7
24Mg	3.2474e-7
25Mg	0.4268e-7
26Mg	0.4897e-7

The initial composition is given in Table 1. The composition is enhanced in $\alpha$-elements. As in Paper VII, the opacities are from Iglesias & Rogers (1996), complemented at low temperatures with the molecular opacities of Alexander (http://web.physics.twsu.edu/alex/wwwdra.htm). The nuclear reaction rates are also the same as in Paper VII and are based on the new NACRE data basis (Angulo et al. 1999).

The physics of the present models at Z=10^-5 is the same as for models at Z=0.004 (Maeder & Meynet 2001). For rotation, the hydrostatic effects and the surface distortion are included (Meynet & Maeder 1997), so that the $T_{{\rm eff}}$given here corresponds to an average orientation angle. The diffusion by shears, which is the main effect for the mixing of chemical elements, is included (Maeder 1997), with account of the effects of the horizontal turbulence, which reduces the shear effects in regions of steep $\mu$-gradients and reinforces it in regions of low $\mu$-gradients (Maeder & Meynet 2001).

Meridional circulation is the main effect for the internal transport of angular momentum. We use here the expression by Maeder & Zahn (1998) for the vertical component U(r) of the meridional circulation. It is interesting to represent graphically this circulation. Figure 1 illustrates the patterns of the meridional circulation in a 20 $M_\odot$ star at Z=0.020and initial rotation velocity $v_{\rm ini}=300$ km s^-1on the ZAMS. The figure is symmetrical with respect to the rotation axis, as well with respect to the equatorial plane. The small inner sphere is the edge of the central convective core. The inner tube, in the upper hemisphere, represents an interior cell of the meridional circulation. There is an ensemble of such concentric tubes with different meridional velocities. The motions occur in a meridian plane (i.e. turning around the tube). In the upper hemisphere and around the inner tube, the fluid elements go upward on the inner side of the tube and descend toward the equator on the outer side of the tube (U(r) is positive). The external tube represents an outer circulation cell, due to the Gratton-Opik term, which is important in the outer stellar layers. This term leads to a negative U(r), which means that, in the upper hemisphere, the fluid goes up on the outer side of the tube and down along the inner side. There also, this tube is one among an ensemble of stream lines turning in the meridian plane.

The mass loss rates are based on the same references as in the paper for the Z=0.004 models (Maeder & Meynet 2001) and in particular on the data by Kudritzki & Puls (2000) for the OB stars. Of course, the strong reduction of the mass loss rates with metallicity for stars below 60 $M_\odot$ makes the mass loss rather unimportant for the metallicity Z=10^-5 considered here, as illustrated by Table 1 which shows the values of the final masses. We account for the effects of rotation on the mass loss rates, according to the standard stellar wind theory applied to a rotating star (Maeder & Meynet 2000). The net result is that the very massive stars with initial $M \geq 60~M_{\odot}$may still experience significant mass loss as shown in Table 1, if they rotate very fast.

Star models with initial masses superior or equal to 9 $M_\odot$ were computed up to the end of the carbon-burning phase. Star models with masses between 2 and 7 $M_\odot$ were evolved beyond the end of the He-burning phase through a few thermal pulses during the AGB phase.

Figure 2: Evolution of the angular velocity $\Omega$ as a function of the distance to the center in a 15 $M_\odot$ star with $v_{\rm ini}$ = 300 km s-1 and Z = 10-5. $X_{\rm c}$ is the hydrogen mass fraction at the center. The dotted line shows the profile when the He-core contracts at the end of the H-burning phase.

Figure 3: Variation of the angular velocity $\Omega$ as a function of the distance to the center in 3 and 9 $M_\odot$ star models with $v_{\rm ini}$ = 300 km s-1 at Z = 0.020 and Z = 10-5. The mass fraction of hydrogen at the centre $X_{\rm c} \simeq 0.40$.