          Astron. Astrophys. 321, 465-476 (1997)

## 3. Hydrostatic effects of rotation

By hydrostatic effects of rotation we mean here the influence on the stellar structure of the centrifugal force, letting aside the effects produced by the transport mechanisms induced by rotation. Such hydrostatic effects have been already studied by numerous authors in the past (Faulkner et al. 1968; Kippenhahn et al. 1970; Kippenhahn & Thomas 1970; Sackmann 1970; Endal & Sofia 1976; Kippenhahn 1977). However it appears interesting to briefly consider them in this paper in order to estimate, in the frame of the "shellular" stellar models, the importance of these hydrostatic effects with respect to the transport processes, and to up-date with modern physical ingredients the results obtained by previous works.

### 3.1. Effects of rotation on the ZAMS

As starting models, we considered uniformly rotating models on the zero age main sequence (ZAMS). Some characteristics of these models are given in Table 1 as a function of the initial mass and angular velocity: in column 2 are listed the ratios of the angular velocity to the critical velocity ( corresponds to the breaking equatorial velocity at the surface on the ZAMS), the equatorial velocities and the positions in the theoretical HR diagram are indicated in columns 3 to 5, the oblatenesses (polar radius over the equatorial radius ) and the ratios between the equatorial radius obtained with rotation and that obtained without rotation ( ) are given in columns 6 and 7 respectively. Table 1. Effect of rotation on the ZAMS

The effective temperatures listed in Table 1 have a different meaning than usual in the sense that they represent some kind of a mean effective temperature of the star. Let us recall that at the surface of a rotating star, the flux (and therefore the effective temperature) is proportional to the local effective gravity (theorem of Von Zeipel 1924). To illustrate this point we have plotted on Fig. 1 the variations of the local effective temperature with the colatitude at the surface of rotating 40 ZAMS models, using the relation given by Maeder (1971). At 90% of the critical angular velocity, the polar regions have enhanced by 1.8 with respect to temperatures of the equatorial zones. In order to associate only one representative value of the effective temperature to each of our stellar models, we have used the relation L  where L is the luminosity, S the rotationally deformed stellar surface, the Stefan-Boltzmann constant and the effective temperature (see also Appendix A.3). Fig. 1. Local effective temperatures as a function of the colatitude at the surface of 40 ZAMS models rotating at different velocities.

From Table 1, one can see that the effects of rotation on the hydrostatic structure are modest. Fig. 2 presents the ZAMS we have obtained in the theoretical HR diagram. As was found by Kippenhahn & Thomas (1970), rotation provokes a decrease of the luminosity and of the effective temperature. These decreases are quite modest (2% for , between 5 and 11% for ) and do not much depend on the initial mass. Thus the increase of the rotation rate has qualitatively a similar effect on the location of the ZAMS as an increase of the initial metallicity. The ZAMS for non rotating models with twice the solar metallicity ( ) lies at a position which would roughly corresponds to a value of equal to 0.7. The ZAMS internal structure of the star is little affected by rotation. The temperature profiles inside the three 40 ZAMS models presented in Table 1 differ by less than 5%. Fig. 2. Upper ZAMS in the theoretical HR diagram for different rotation rates.

### 3.2. Hydrostatic effects of rotation on the main sequence

From the ZAMS presented above, models of 40 and 60 were evolved during the main sequence (MS) with account of only the hydrostatic effects and local conservation of the angular momentum. Some characteristics of the models are indicated in Table 2: the duration of the main sequence is indicated in column 3, the position in the HR diagram at the end of the H-burning phase is given in columns 4 and 5, the central temperature and density at this stage are given in columns 6 and 7 respectively. As for the ZAMS models, the hydrostatic effects alone have a modest impact on the model outputs: when increases from 0 to 0.9, the lifetimes of the considered models are increased by 1 to 2%. The lifetime enhancement is a consequence of the decrease of the central temperature in rotating models due to the decreased effective gravity. The evolutionary tracks for the models with are compared with the non rotating models in Fig. 3. The tracks are shifted towards lower luminosities. One can say roughly that a massive star having an initial rotational velocity equal to 90% of the critical velocity will have a behaviour similar to a non rotating star with an initial mass smaller by 0.5 to 2.5 solar masses. Table 2. Effect of rotation on the MS Fig. 3. Evolutionary tracks for 40 and 60 models for and .

Due to the envelope expansion which occurs during the main sequence, the surface equatorial velocity for the 40 model decreases by more than a factor three during this phase. The profile of the angular velocity at the end of the main sequence is presented in Fig. 4. The star has a radiative core at this stage with a steep angular velocity gradient. In the central regions, as a result of the core contraction, the angular velocity is enhanced by about a factor 30 with respect to the surface angular velocity. Fig. 4. Profile as a function of the langrangian mass coordinate of the angular velocity for the 40 model at the end of the main sequence. The arrow on the left of the figure indicates the initial value of the angular velocity ( ).    © European Southern Observatory (ESO) 1997

Online publication: June 30, 1998 