P dm dm

68 Rotating stars and

Many useful properties can be deduced from these equations. For this purpose, let us define the effective gravity

\dm J dz where 1m and 1z are the unit vectors in the m and z direction. Equations (3.16) and (3.17) become

It follows at once that the effective gravity is everywhere orthogonal to the surfaces of constant pressure (i.e., the isobaric surfaces). This is a general property, which is valid no matter whether one has Q = Q(m) or Q = Q(m, z).

Let us now assume the star rotates as a solid body. Equation (3.18) then reduces to g = "grad O, (3.20)

where, except for an additive constant, one has

Under what circumstances can we also derive the effective gravity from a potential in a differentially rotating star? By virtue of Eq. (3.18), this is possible if and only if Q does not depend on z, that is, when the angular velocity is a constant over cylinders centered about the axis of rotation. Then, instead of Eq. (3.21), one has fm n , ,

Various interesting conclusions can be inferred from the existence of such a potential. First, by virtue of Eq. (3.19), one can always write

By definition, for any displacement on a level surface O = constant one has dO = 0. Since Eq. (3.24) shows that dp = 0 on the same surface, it follows at once that the isobaric surfaces coincide with the level surfaces. If so, then, we can write p = p(O) or O = O(p). (3.25)

By virtue of Eq. (3.24), one readily sees that 1 dO(p)

Accordingly, the density is also a constant over an isobaric surface. Thus, the surfaces upon which p, p, and O remain a constant all coincide. As a consequence, when a potential O does exist, the vector g is also normal to the surfaces of constant density (i.e., the isopycnic surfaces).

Reciprocally, let us consider a system for which the surfaces of constant pressure and constant density coincide. If we let

As function of the coordinates, the differential dO is an exact total differential. Accordingly, Eq. (3.20) must hold true, and the vector g may be derived from a potential.

Finally, let us suppose that the effective gravity is everywhere normal to the isopycnic surfaces. By virtue of Eq. (3.23), any displacement over one of these surfaces gives dp = 0, so that the pressure is a constant over an isopycnic surface. The coincidence of the surfaces of constant pressure and constant density is thus established.

If we now collect all the pieces together, it is a simple matter to see that we have proved the equivalence of the following statements:

(a) The angular velocity depends on m only.

(b) The effective gravity can be derivedfrom a potential.

(c) The effective gravity is normal to the isopycnic surfaces.

(d) The isobaric surfaces and the isopycnic surfaces coincide.

Thus, any of these statements implies the three others. By definition, a system for which these statements hold true is called a barotrope.

Following current practice, we shall call a system for which these statements do not hold true a barocline. The major distinction between a barotrope and a barocline lies in their respective stratification. Of particular importance is the fact that the isopycnic surfaces are in general inclined to and cut the isobaric surfaces in a barocline.

Note that slow but inexorable meridional currents do exist in a rotating star. As we shall see in Chapters 4 and 5, however, these currents are so slow that they do not upset the mechanical balance defined by Eqs. (3.16) and (3.17). Hence, they do not modify the basic conclusions reported in this section.

3.3 Some tentative solutions

In Section 3.2.1 we have demonstrated some simple mechanical properties of an axially symmetric star that rotates with some assigned angular velocity. Yet, because we have hitherto circumvented the use of the condition of energy conservation, we do not know whether we can apply these results, without modification, to a radiating star. For example, is there any constraint imposed by the condition of radiative equilibrium on the angular velocity distribution in a barotropic star? Similarly, to what extent is it necessary to modify the conclusions of the Poincare-Wavre theorem when turbulent friction is properly taken into account in a star in strict convective equilibrium? We shall devote this section to the study of these two questions.

3.3.1 The case of radiative equilibrium

Consider a barotropic star in strict radiative equilibrium. By making use of Eq. (3.4), we can write div F = peNuc- (3.29)

Equations (3.19) and (3.20) further imply that

where O is defined in Eq. (3.22). By virtue of the Poincare-Wavre theorem, we immediately deduce that p = p(O) and p = p(O). If the chemical composition is constant (or a function of p and p only), one also has T = T(O). Hence, if we assume that eNuc = eNuc(p, T) and k = k(p, T), both the energy generation rate and the opacity coefficient depend on O only. It follows that

where

4ac T3 dT

Let us consider next the divergence of Eq. (3.32). One obtains

where dn is along the outward normal to a level surface, and a prime denotes a derivative with respect to O. We also have

Clearly, dO/dn is the magnitude of the effective gravity g. Combining Eqs. (3.2) and (3.22), one can write