An inspection of Figure 1.6 shows that the mean projected equatorial velocity of main-sequence stars increases slowly with spectral type, reaching a maximum of about 200 km s-1 in the late B-type stars. Thence, the mean velocity (v sin i) decreases slowly for later spectral types until about F0, where it starts dropping precipitously through the F-star region. As is well known, this rapid transition to very small rotational velocities occurs at approximately the spectral type where subphotospheric convection zones become suddenly much deeper on the main sequence. Accordingly, because Sunlike stars are most likely to develop episodic mass ejections and magnetically channeled stellar winds, it is generally thought that these stars are losing mass - and, hence, angular momentum - as they slowly evolve on the main sequence. Postponing to Chapter 7 the study of these low-mass stars (M < 1.5M0), in this chapter we shall consider stars more massive than the Sun (M > 1.5M0) that are in radiative equilibrium in their surface layers.
In Chapter 4 we have already discussed the large-scale meridional currents and concomitant differential rotation in the radiative envelope of an early-type star, when the departures from spherical symmetry are not too large. Admittedly, the aim of that chapter was to develop a clear understanding of the many hydrodynamical phenomena that arise in a rotating star. In the following sections of this chapter we shall instead examine a selection of practical topics dealing with rotation, meridional circulation, and turbulence in the early-type stars. The chapter is organized as follows. The modifications brought by axial rotation on the overall structure of a main-sequence star are discussed in Section 6.2.1. Section 6.2.2 is devoted to the effects of rotation on the observable parameters, which depend on the inclination of the rotation axis to the line of sight. Section 6.3 presents a detailed study of axial rotation along the upper main sequence. In Section 6.4, which is of direct relevance to the study of chemically peculiar stars, we consider the interaction between microscopic diffusion and rotationally driven motions in a stellar radiative envelope. We conclude the chapter with a brief discussion of the changes in rotation as an early-type star evolves off the main sequence.
The main objective of this section is the construction of reliable numerical models of rotating stars consisting of a convective core, in which hydrogen burning is taking place, and a chemically homogeneous radiative envelope. In fact, very little is known about the interaction between rotation and convection in the core of an early-type star. For mathematical simplicity, it is often assumed that convective cores rotate uniformly; as was correctly pointed out by Tayler (1973), however, there is still considerable uncertainty about this point. The state of motion in the outer envelope of an early-type star has received comparatively much greater attention. Unfortunately, the study of a stellar radiative zone is complicated by the necessity to come to terms with a whole spectrum of eddylike motions that continuously interact with the mean flow, that is, the overall rotation and the slow but inexorable meridional currents. Following Section 3.6, we shall explicitly resolve these large-scale motions, while parameterizing the smaller-scale transient eddies through the use of Reynolds stresses and eddy viscosities. In cylindrical polar coordinates (m, y, z), the mean velocity v becomes v = u + Qm (6.1)
where u is the two-dimensional meridional velocity. Since we are considering an axially symmetric configuration, mass conservation implies that div(p u) = 0, (6.2)
where p is the mean density. Neglecting the acceleration and inertia of the meridional flow, we can rewrite the poloidal part of Eq. (3.123) in the form
pp where p is the pressure, V is the gravitational potential, and Fp (u) is the poloidal part of the turbulent viscous force per unit volume acting on the circulation. Similarly, by use of Eq. (6.1), one can show that the y component of Eq. (3.123) has the form d ~ ~ p —(Qm2) + pu ■ grad(Qm2) = F9(Q), (6.4)
d t where Fy (Q) is the azimuthal component of the turbulent viscous force per unit volume acting on the differential rotation (see Eq. [3.133]). To complete these equations we must add Poisson's equation,
an equation of state,
and the energy equation, p Tu ■ grad S = peNuc — div Ft, (6.7)
where S is the specific entropy and Ft is the total (radiative and convective) flux vector (see Eqs. [5.11]-[5.13]). Remaining symbols have their standard meanings.
The above set of partial differential equations provides seven scalar relations among the seven unknown functions Q, u, p, p, T, and V. Thus, in principle, the internal structure of a rotating star with meridional circulation is entirely determined by these equations, together with some initial conditions and the usual set of boundary conditions
(see Section 2.2.2). The main difficulty of the problem lies in the fact that neither the internal stratification of a rotating star nor the shape of its free surface are known in advance. Another difficulty arises because we know very little about the transport of specific angular momentum, 2, in a stellar interior. In principle, the angular velocity ^ can be calculated from Eq. (6.4), which merely expresses that the advection of specific angular momentum by the meridional currents must balance the effects of turbulent friction acting on the mean azimuthal flow. In practice, because the coefficients of eddy viscosity cannot be calculated from first principles alone, the actual dependence of the angular velocity on the coordinates and time remains quite uncertain. As was pointed out in Section 4.8, the precise determination of the rotation law in a stellar radiative envelope must await the development of numerical models that resolve the transient eddylike motions in sufficient detail to reproduce their transport properties adequately. Parenthetically note that the presence of a weak poloidal magnetic field does not solve the problem either since, as was shown in Section 4.7.2, such a field does not necessarily maintain almost uniform rotation throughout the radiative envelope of an early-type star.
With the advent of high-speed computers in the 1960s, significant advances have been made in the study of the internal structure of rotating stars. However, because the actual distribution of angular momentum within a star is still largely unknown, in all numerical models proposed to date the rotation law is always specified in an ad hoc manner. In this section we shall thus assume that there are no internal motions other than rotation, and we shall merely replace Eq. (6.4) by some prescribed rotation law, either ^ = constant or some function ^ = Q.(m) that satisfies the essential stability condition defined in Eq. (3.98). If so, then, Eq. (6.3) simplifies to the usual condition of mechanical equilibrium for a barotrope,
(see Section 3.2.1). Given these simplifications, one readily sees that the basic equations are quite similar in structure to those for nonrotating stars, except that Eq. (6.5) must be solved in two dimensions with an outer boundary that is itself an unknown. Another difficulty stems from the fact that Eq. (6.8) is incompatible with the energy equation in a circulation-free barotrope (see Section 3.3.1). Accordingly, it is also assumed that, though radiative equilibrium does not hold at every point, it does hold on average (i.e., averaged over each level surface O = constant).
A great number of techniques have been devised to determine the equilibrium structure of rotating polytropes and barotropic stars. To the best of my knowledge, Milne (1923) was the first to construct barotropic models for slowly rotating stars, using a first-order perturbation technique and treating the effects of uniform rotation as a small distortion superimposed on a known spherical model (see Eqs. [4.9]-[4.25]). As was originally shown by Takeda (1934), however, fairly accurate results can be obtained by means of a double-approximation technique. In the central regions, where the rotational distortion is small, a first-order expansion is used. This solution is then matched to a solution in the low-density surface layers, where the gravitational field arises mainly from the matter present in the slightly oblate inner core. Since, in general, the domains of validity of the two approximation regimes overlap, self-consistent solutions may readily be constructed. More recently, Kippenhahn and Thomas (1970) have shown that the use of two zones is unnecessary for the same degree of accuracy can be obtained in choosing an appropriate geometrical representation for the level surfaces. Their technique has been widely used because, without much trouble, rotation can be incorporated into the usual programs of stellar evolution (see, e.g., Section 5.4.1). Unfortunately, although it provides satisfactory results for quasi-spherical models in slow uniform rotation, other methods must preferably be used when the level surfaces greatly deviate from concentric spheres.
Progress in the study of rapidly rotating barotropes has been made by using full numerical solutions of all the relevant structure equations. Notably, Ostriker and Mark (1968) have developed the self-consistent-field method, which was especially designed to relax altogether the restrictive assumption of quasi-sphericity. In this method, Eq. (6.5) is replaced by its integral solution, where the triple integral must be evaluated over the volume V of the configuration. Given an angular momentum distribution, an iterative procedure is established in which an approximate expression for the total potential $ is derived from a trial density distribution Po(m, z). A new density distribution p\(m, z) is then obtained from the equilibrium equations. For convenience, the external boundary condition on the gravitational potential is applied on a sphere exterior to the model. This is the basis of the self-consistent-field method, in which Poisson's equation and the equilibrium equations are solved alternately. This iterative scheme works remarkably well for the more massive stars, but it fails to converge even for a nonrotating main-sequence model if its mass is less than about 9M0 (i.e., if its central mass concentration is sufficiently high). This is the reason why Clement (1978) has presented a two-dimensional, finite-difference technique for solving Poisson's equation simultaneously with the equilibrium equations. The method does not appear to be limited by the large central concentrations that characterize intermediate mass stars and those with high angular momentum. Rapidly rotating main-sequence models in the mass range that is not accessible to the self-consistent-field method have been computed with this two-dimensional numerical technique.
As was originally shown by Milne (1923), uniform rotation has two general effects on the structure of a star. It leads to (i) a global expansion of the star due to the local centrifugal force and (ii) a departure from sphericity due to the nonspherical part of the effective gravity. To be specific, because the centrifugal force takes over from the pressure part of the burden of supporting the weight of the overlying layers in the energy-producing regions, the global-expansion effect causes a reduction in the total luminosity of the star when it is compared to its nonrotating counterpart having the same mass. Moreover, because a uniformly rotating star is slightly oblate, in its equatorial belt part of the mass is supported by the centrifugal force whereas this is not the case in the polar regions. Accordingly, the pressure and hence the net outward flux of energy
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