Dv

where, as we recall, v (1 - x2)1/2 cos p. This solution is actually the vectorial sum of the velocity fields (4.129) and (4.130), with both solutions now being written in the rotating frame of reference (r, x, p). Note that the functions uk and vk are still related to each other by Eq. (4.131) because

Fig. 4.8. Tidally driven currents in a synchronously rotating star. The vertical arrow points toward the companion; the tidal distortion of the model is not depicted. From left to right: Contributions from the P2, P3, and P4 terms in Eq. (4.130), and the sum of these three contributions (when M = M' = 3M0 and e = 0.25; that is, d/R = 2, veq = 290 km s-1, and P = 2nR/veq = 0.31 day). Even for this relatively large value of e, it is the P2 term that dominates in the expansion. The streamlines do not penetrate into the convective core, but there is an accumulation of streamlines in the core boundary layer. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J., 261, 265, 1982.

Fig. 4.8. Tidally driven currents in a synchronously rotating star. The vertical arrow points toward the companion; the tidal distortion of the model is not depicted. From left to right: Contributions from the P2, P3, and P4 terms in Eq. (4.130), and the sum of these three contributions (when M = M' = 3M0 and e = 0.25; that is, d/R = 2, veq = 290 km s-1, and P = 2nR/veq = 0.31 day). Even for this relatively large value of e, it is the P2 term that dominates in the expansion. The streamlines do not penetrate into the convective core, but there is an accumulation of streamlines in the core boundary layer. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J., 261, 265, 1982.

As usual, once the functions u and uk have been obtained, Eq. (4.125) can be solved to O(e3/2) to give a unique solution for the velocity fields u3/2.

Figure 4.8 illustrates the pure tidally driven component of the circulation (see Eq. [4.130]). Following Section 4.3.1, we have considered a 3M0 Cowling point-source model, with electron-scattering opacity and pV = 106prad in the boundary layers (see Eq. [4.62]). Although this large-scale motion is the combination of three terms, it is immediately apparent that the contribution from the P2(v) term dominates over the two others. Their time scale is of the order of the Kelvin-Helmholtz time, tKH, divided by the ratio of the tidal force to gravity at the equator, (M'/M)(R/d)3. These axially symmetric motions are the strict analog of the rotationally driven currents depicted in Figure 4.3.

4.6.2 The reflection effect in close binaries

To the best of my knowledge, Hosokawa (1959) was the first to point out that the mutual heating of the components in a close binary generates large-scale circulatory currents in their superficial layers. To illustrate the problem, we shall calculate the pho-tospheric flow caused by the presence of a permanent "hot spot" on the surface of an early-type star that is a synchronously rotating component of a close binary.

Again consider a rotating frame of reference in which the origin is at the center of the primary (of mass M, radius R, and luminosity L when neglecting the "hot spot"). The x axis points toward the point-mass secondary (of mass M' and luminosity L ), and the z axis is parallel to the overall angular velocity of the primary. In this case, the appropriate expansion parameter, n (say), is the ratio of the fluxes at r = R,

where d is the separation between the two centers of mass.

To discuss the boundary-layer currents caused by the reflection effect in a gray atmosphere, we shall assume that the prescribed irradiating flux F takes the form

d2 Kp where k is the opacity and t is the optical thickness. As usual, v is the cosine of the colatitude from the x axis, and P1(v) = v. Admittedly, this is a crude approximation of the irradiating flux in the surface layers of a star. Yet, Eq. (4.137) adequately models the fact that (a) the epicenter of the permanent "hot spot" is located on the x axis (v = 1) and (b) the irradiating flux is attenuated exponentially with optical depth. By virtue of Eq. (4.137), Eq. (4.5) must be replaced by pTu ■ grad S = div (x grad T) + divF. (4.138)

It is a simple matter to prove that, correct to O(r\), the velocity of the currents can be written in the form

ur = qu(r)P:(v) and uv = qv(r)(1 - v2)—^, (4.139)

dv where

2 pr2 dr

(Compare with Eqs. [4.130]-[4.131].) Retaining turbulent friction in the surface layers, one can also show that the function u satisfies the following equation:

where n is the effective polytropic index. (Compare with Eq. [4.63].)

Following Section 4.3.1, we shall prescribe the usual radiative-zero boundary condition. For electron-scattering opacity, we have k = constant, n = 3, p = pbz4, p = pbz3, T = Tbz, and xV = 10w¡xbz, where z = R — r. Letting next

one can rewrite Eq. (4.141) in the form

where

Equation (4.75), with n = 3, defines the boundary-layer thickness S. (Compare Eq. [4.143] with Eq. [4.76].) Of course, the solutions of Eq. (4.143) must satisfy the boundary conditions (4.77)-(4.79). However, because the motions generated by the reflection effect must vanish at some depth from the surface, condition (4.80) must be replaced by the following condition:

Equations (4.143), (4.145), and (4.77)-(4.79) form the basic equations of the problem.

Table 4.4. The functions u andr v in the surface boundary layer.

N

= 5

N

=6

x

u

r v

u

r v

0.0

0.

-3.6299E+5

0.

-5.2076E+4

0.2

1.3084E+2

-3.6179E+5

2.3604E+1

-5.1793E+4

0.4

2.5672E+2

-3.4923E+5

4.5883E+1

-4.9218E+4

0.6

3.6435E+2

-3.1731E+5

6.4100E+1

-4.3798E+4

0.8

4.4216E+2

-2.7198E+5

7.6742E+1

-3.7005E+4

1.0

4.8658E+2

-2.2146E+5

8.3607E+1

-2.9828E+4

1.2

4.9889E+2

-1.7091E+5

8.5073E+1

-2.2837E+4

1.4

4.8297E+2

-1.2372E+5

8.1863E+1

-1.6411E+4

1.6

4.4429E+2

-8.2151E+4

7.4931E+1

-1.0814E+4

1.8

3.8927E+2

-4.7645E+4

6.5365E+1

-6.2057E+3

2.0

3.2453E+2

-2.0864E+4

5.4292E+1

-2.6548E+3

2.2

2.5680E+2

-1.7779E+3

4.2781E+1

-1.4281E+2

2.4

1.9138E+2

1.0236E+4

3.1753E+1

1.4236E+3

2.6

1.3276E+2

1.6263E+4

2.1925E+1

2.1955E+3

2.8

8.3842E+1

1.7674E+4

1.3763E+1

2.3593E+3

3.0

4.5990E+1

1.5942E+4

7.4771E+0

2.1127E+3

3.2

1.9177E+1

1.2467E+4

3.0481E+0

1.6416E+3

3.4

2.2549E+1

8.4233E+3

2.7242E-1

1.1010E+3

3.6

-6.6589E+0

4.6728E+3

-1.1723E+0

6.0390E+2

3.8

-9.7740E+0

1.7310E+3

-1.6591E+0

2.1675E+2

Source: Tassoul, J. L., and Tassoul, M., Astrophys. J., 261, 273, 1982.

Source: Tassoul, J. L., and Tassoul, M., Astrophys. J., 261, 273, 1982.

In Table 4.4 we list the functions u and r v (in cms-1) for a 3M0 Cowling point-source model, with n = 3, k = 0.34 cm2 g-1, and u1 = 6.45 x 107 cm s-1. The values are listed for N = 5 (a1 = 20 and 8/R = 3.6 x 10-3)and N = 6(a1 = 50 and 8/R = 4.6 x 10-3). Figure 4.9 illustrates the function y(x) when a1 = 20. It is apparent from Table 4.4 and Eq. (4.139) that the axially symmetric circulation pattern consists of a main cell (or gyre) within the boundary layer (0 < R — r < 0.01 R) and secondary cells at lower depths (R — r > 0.01 R). Because the flow speed decreases exponentially with optical depth, the dominant mass flow takes place within the outermost external layer of the absorbing star, however. The circulatory currents are symmetrical with respect to the line joining the centers of gravity, with rising motions in the "hot spot" (v = 1) and sinking motions at the antipode (v = —1). There is thus a mean steady current that is flowing away from the "hot spot" on the stellar surface and a mean steady countercurrent that is flowing away from the antipode at a somewhat lower level. The whole flow, in fact, takes place within a very thin superficial shell (0.99 < r/R < 1). Typically, with n = 10-2 and N = 6, Table 4.4 indicates that |ur| < 0.85 cms-1 and |ue| < 520 cms-1. Even though there are still uncertainties about these maxima (again because u and rv are quite sensitive to the values of N), these speed estimates are far removed from the various evaluations based on frictionless solutions that can be found in the literature. All these evaluations are utterly inadequate because they do not satisfy the kinematic boundary condition (4.38).

-

1 1

1 1

-

-

I06y

-

-

n = 3

-

-

X

\ a,= 20

-

Fig. 4.9. Function y(x) in the surface boundary layer, when n = 3 and a1 = 20. Source: Tassoul, J. L., and Tassoul, M., Astrophys. J., 261,273, 1982.

4.7 Meridional circulation in a magnetic star

In Section 4.3.1 we have obtained a self-consistent description of meridional streaming and concomitant differential rotation in the chemically homogeneous envelope of an early-type, nonmagnetic star. Since these matters have been largely clarified by now, here we shall go a step further and discuss the role of a prescribed magnetic field in an early-type star. For the sake of simplicity, we shall assume that the large-scale field is not maintained by a contemporary dynamo operating in the convective core, but rather that it is the slowly decaying relic of the field present in the gas from which the star formed.

4.7.1 The magnetically driven currents

In an inertial frame of reference, the momentum equation for the large-scale flow becomes

Dv = -grad V - 1 grad p + 1 F(v) + —^ curl H x H, (4.146)

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