NGp Qm m d m

This is the condition of strict radiative equilibrium in a barotropic star.

The case of a uniformly rotating barotrope is particularly straightforward, and it was originally discussed by von Zeipel (1924). Equation (3.37) then reduces to f'm)g2 + f m)(4nGp - 2^2) = pfNuc. (3.38)

As we know, g is not constant over a level surface in a rotating body, because the distance from one level to the next one is not the same for every point on it. Accordingly, since p, eNuc, and Q are all constant on level surfaces, the coefficient of g in Eq. (3.38) must vanish separately. We have, therefore, f'(O) = 0 or f(O) = constant. (3.39)

Hence, Eq. (3.38) assumes the form

This is known as von Zeipel's law o/gravity darkening. Obviously, condition (3.40) is never fulfilled in an actual star. It follows at once that rigid rotation is impossible for a barotrope in static radiative equilibrium.

Let us consider next the general conservative law Q = Q(m). In this case, as was pointed out by Rosseland (1926) and Vogt (1935), it is intuitively evident that the law Q = Q(m) is incompatible with condition (3.37). Indeed, while Q will be constant over cylinders centered about the rotation axis, g will be constant over certain oblate surfaces. Therefore, by virtue of Eq. (3.37), conditions (3.39) and (3.41) still pertain, but we must impose the additional condition

After integrating, one obtains

where c1 and c2 denote two arbitrary constants. If c2 = 0, we simply recover the case of a uniformly rotating barotrope. Similarly, if c2 = 0, the rotational law (3.43) becomes singular on the rotation axis; it must be disregarded because it also leads to an impossible constraint on eNuc (i.e., condition [3.40] with Q2 being replaced by c1). This argument shows that a differentially rotating barotrope cannot remain in static radiative equilibrium.

It is not the usual energy generation rates that prevent the rotation laws Q = constant or Q = Q(m) from being realized, but rather the condition of strict radiative equilibrium. Indeed, in the limit Q = 0, g is a constant over each spherical surface O = V = constant, and there is no requirement that some terms in Eq. (3.37) should vanish independently of the remaining terms. Therefore, this equation must be regarded as an indication that for nonspherical stars at least one of the assumptions leading to conditions (3.40) and (3.42) must be relaxed.

This problem can be solved in two different ways: Either we assume strict radiative equilibrium while allowing Q to depend on both m and z or we assume that strict radiative equilibrium breaks down in a rotationally distorted star. The latter solution leads to the formation of a large-scale meridional flow (and concomitant differential rotation) in the radiative zone of a rotating star. The former solution is mainly of academic interest, however, because there is no obvious reason why the angular velocity would adjust itself so as to prevent meridional currents. These matters will be discussed further in Chapter 4.

3.3.2 The case of convective equilibrium

In a region of efficient convection, the energy transport keeps the actual temperature gradient closely equal to the adiabatic lapse rate. Let us assume that such a region has reached a state of mechanical equilibrium, with no large-scale motions in meridian planes passing through the rotation axis. Since the fluid is essentially barotropic, it follows at once that

over the whole region where strict convective equilibrium prevails.

The actual form of the rotation law depends on the azimuthal forces. By assumption, there are no meridional motions; and there is no large-scale magnetic field. Accordingly, the only remaining force is the p component of turbulent friction acting on the rotational motion. An explicit expression for this force will be presented in Section 3.6. Anticipating these results and introducing spherical polar coordinates (r, 9, p), one finds that

1 d ( 4dQ 3 \ 1 d ( 3 dQN ir fvr4 — + Vr3 Q + —^ — fH sin3 9 — = 0, (3.45) r2 dr V dr ) sin3 9 d9 \ d9 )

where fV and fiH are the vertical and horizontal coefficients of eddy viscosity, and XV is a parameter representing the influence of global rotation on the anisotropic convective elements (see Eq. [3.133]).

Equation (3.45) must be solved with appropriate boundary conditions. Because eddy viscosity is always much larger in a convective zone than in the surrounding regions, we shall merely prescribe that the tangential viscous stresses vanish at the boundaries of the convective zone (see Eq. [2.21]). For a slowly rotating solar-type star, these conditions become dQ

dr where Ri and Ro are the inner and outer radii of its (almost) spherical convective layer (see Eq. [3.131]).

For the sake of simplicity, let us assume that the parameter X V and the eddy viscosities are constant. Following Kippenhahn (1963), one can show that Eqs. (3.45) and (3.46) can be satisfied only if the angular velocity is constant on spheres, with the rotation law

where Q0 is a constant and a — Xv /fv. Because the parameter Xv does not in general vanish in a convective zone, it prevents rigid rotation from being a solution of Eq. (3.45). Accordingly, conditions (3.44) and (3.47) cannot be satisfied simultaneously in a con-vective region, so that a pure rotation cannot be a solution of the problem. This result confirms Biermann's (1958) original finding that large-scale meridional currents are always present in a region of efficient convection.

This conclusion is very similar to the result obtained in Section 3.3.1. However, in contrast to the case of a radiative zone, it is the necessity to conserve linear momentum, rather than energy, that drives the meridional flow in a connective zone. A detailed discussion of these large-scale currents will be made in Section 5.2.

3.4 The dynamical instabilities

Consider an axially symmetric star and assume that it rotates with some prescribed angular velocity Π= Q.(m, z). By making use of Eqs. (3.16) and (3.17), we can rewrite these conditions of mechanical equilibrium in the compact form

1 j2

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