## Polar Radius as a Function of Rotation

In first approximation, one may consider that the polar radii are independent of rotation and use values such as given by Fig. 25.7. In reality the polar radii Rp(a) have a slight dependence on m, which results from the small changes of internal structure brought about by centrifugal force. The rate of change is given by the models of internal structure with rotation. While the equatorial radius strongly inflates, the polar radius decreases by a few percent in general (Fig. 2.7), mostly as a result of a slight decrease of the internal T due to the lower effective gravity. Below 40 M0, the decrease of Rp(m) at the critical velocity amounts to less than 2% [408]. Near 1M0, the decrease of Rp(m) with rotation is larger.

Surprisingly, at 60 M0 there is an increase of the polar radius with growing rotation (Fig. 2.7). This results from the fact that the radiation pressure is relatively important. As the temperature in the polar regions is much higher than at the equator as a result of von Zeipel's theorem (Sect. 4.2.2), the relative increase of the radiation pressure in the outer layers is much higher with a consequent inflation of the polar radius.

One may rather well represent the change of the polar radius as a function of m by a form

where a is a constant for models of a given mass [181]. Figure 2.8 provides information on the changes of polar radii at other Z values. It shows the changes of the polar radius at m = 0.90 for stars of different masses and metallicities. One again notices the general slight decrease of the polar radius at high rotation for most stellar masses, while the most massive stars in particular at the higher metallicities experience an increase.

Fig. 2.7 Variations of the polar radius as a function of the rotation parameter m normalized to the value without rotation for stars of different initial masses at Z = 0.02. From S. Ekstrom et al. [176]

Fig. 2.7 Variations of the polar radius as a function of the rotation parameter m normalized to the value without rotation for stars of different initial masses at Z = 0.02. From S. Ekstrom et al. [176]

Fig. 2.8 Variations of the ratio of the polar radius at a = 0.90 to the value at zero rotation as a function of initial masses for different Z. Courtesy from S. Ekstrom

Fig. 2.8 Variations of the ratio of the polar radius at a = 0.90 to the value at zero rotation as a function of initial masses for different Z. Courtesy from S. Ekstrom

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