Op Op

where O() is the order of magnitude symbol.

Let us assume next that in Eq. (4.53) the three terms are of the same order of magnitude. Then, comparing the advection term on the right-hand side to the horizontal dissipation term on the left-hand side, one readily sees that the d dependence can be neglected (i.e., 22 ^ £2) if and only if one has tH ^ tES. Similarly, comparing the advection term on the right-hand side to the vertical dissipation term on the left-hand side, one notices that the r dependence can be neglected (i.e., 22 ^ (2)) if and only if one has tv ^ tES. Thus, if one has simultaneously tH ^ tES and tv ^ tES in a slowly rotating star, the angular velocity is nearly constant throughout the radiative envelope. The latter case is particularly simple because, as we shall see in Section 4.3.1, one can then expand the unknown function 2 in powers of the small parameter e (see Eq. [4.9]). The former case, which is much more involved, will be considered in Section 4.3.2.

As was originally pointed out by Krogdahl (1944), the condition that u vanishes with 2 plus the obvious properties that u must be an even function of e1/2, whereas 2 is to be odd in e1/2, suggest the following choice for the velocities:

2 = 20(1 + ew1 + e 2w2 +----), u = eui + e U2 +----.

Correct to 0(e), it follows at once from Eq. (4.55) that Q2 = Q2 + 0(e2). Thus, to that order of approximation, Eqs. (4.50) and (4.51) do not depend on w1, so that it is possible to calculate u1 from Eqs. (4.3)-(4.6) and (4.50)-(4.51), replacing of course the function Q by the constant Q0. Thence, one calculates the function w1 from Eq. (4.49). Correct to 0(e3/2), this equation becomes

74 dr 1 IXvrr

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