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Fig. 6.1. The central temperature Tc as a function of the central density pc for main-sequence stars. The curve is drawn through the data referring to nonrotating stars (dots). The crosses refer to critically rotating stars. The numerals along the curve define the mass of the models. Source: Sackmann, I. J., Astron. Astrophys., 8, 76, 1970.

must also be smaller at the equator than at the poles. In other words, the nonsphericity effect induces a dependence of effective temperature on latitude, with the polar regions appearing hotter than the equatorial belt.

To illustrate these results, I shall summarize the numerical work of Sackmann (1970), who, by making use of Takeda's double-approximation technique, has constructed a large set of models for main-sequence stars in the mass range O.8-2OM0. Her calculations show that for each mass along the main sequence it is possible to construct a series of uniformly rotating models, with each series terminating with a model for which the effective gravity vanishes at the equator. The maximum luminosity change caused by solid-body rotation is about 7% for high-mass stars and somewhat smaller for low-mass stars with a radiative envelope. (For stars with masses below 1. 5M0, this change becomes much larger, though less certain.) Figure 6.1 demonstrates that a uniformly rotating star of mass M has similar central properties as a nonrotating star with mass M — AM, where AM > 0. We observe that the values of Tc and pc for rotating stars on the verge of equatorial breakup fall exactly along the curve for nonrotating stars, with their positions being somewhat shifted in the direction of the lower masses. Note also that the largest deviation between the values for critically rotating stars and nonrotating stars is as small as 0.001 in log10 Tc and 0.004 in log10 pc! Following Sackmann, one has

Table 6.1. The percentage decrease in mass necessary to make the central pressure of critically rotating models equal to that of a nonrotating model.

M/M0

AM/M (%)

M / M0

A M/ M (%)

0.8

3.0

3

2.2

1.0

4.1

5

2.0

1.4

0.7

7

2.7

1.5

0.0

9

2.3

1.8

1.2

10

2.0

2.0

1.4

20

2.8

Source: Sackmann, I. J., Astron. Astrophys., 8, 76, 1970.

Source: Sackmann, I. J., Astron. Astrophys., 8, 76, 1970.

where e is the pressure-weighted average of the ratio of centrifugal force to gravity over the whole star. Table 6.1 illustrates this mass-lowering effect at breakup rotation along the main sequence. For the sake of completeness, in Figure 6.2 we also depict the critical equatorial velocity vc at the point of equatorial breakup. Note that the velocity vc steadily decreases as one passes down the main sequence from 2OM0 to 1.4M0 and that it rises again as the mass is decreased below 1.4M0.

The above results strongly suggest that solid-body rotation can be considered as a small perturbation superimposed on the structure of a nonrotating star. For differentially

Fig. 6.2. The critical equatorial velocity vc as a function of mass along the main sequence. Source: Sackmann, I. J., Astron. Astrophys., 8, 76, 1970.

rotating configurations, however, the situation is quite different because these systems can store a much higher total angular momentum than a uniformly rotating model with the same ratio of centrifugal force to gravity at the equator (cf. Section 2.8.3). Accordingly, we surmise that sequences of stellar models in nonuniform rotation do not terminate, therefore allowing for much larger observable effects than in a uniformly rotating model on the verge of equatorial breakup. That this is indeed the case was properly demonstrated by Bodenheimer (1971) and Clement (1979).

Several series of differentially rotating models have been constructed, each with fixed mass M and fixed angular momentum distribution Om2, but with increasing values for the total angular momentum J. The rotational characteristics of three 3OM0 models are illustrated in Figure 6.3. Note that considerable polar flattening occurs, with the ratio of equatorial to polar radii ranging up to about 4. Yet, none of these models approaches the limit of zero effective gravity at the equator. Not unexpectedly, in contrast to the case of solid-body rotation, conditions in the central regions now show large changes caused by differential rotation. This is illustrated in Figure 6.4, which shows that the effect of an increase in J is to shift the configuration closely parallel to and downward along the curve corresponding to nonrotating stars. A similar mass-lowering effect was found by Clement, who enlarged Bodenheimer's analysis by constructing sequences of differentially rotating models in the whole mass range 1.5-3OM0.

As mentioned, the problem is complicated by the fact that we have no direct knowledge of the angular momentum distribution within a star. Fortunately, the Bodenheimer-Clement calculations indicate that, given a mass M and a total angular momentum J, the changes in central temperature and density and in total luminosity are not strongly dependent on the interior angular velocity gradient. In view of the rather arbitrary nature of the assumed rotation laws, this is a most useful result.

In summary, uniform rotation has a mass-lowering effect on the internal structure of a main-sequence star, which gives a rotating model some of the characteristics of a nonrotating model of lower mass. Thus, uniform rotation leads to lower interior temperatures, lower luminosities, and either higher or lower interior densities depending on whether the star's mass is greater or smaller than about 1.5M0, which is the point where main-sequence stars change from convective cores to convective envelopes. Detailed calculations strongly suggest that this mass-lowering effect is generally valid since it applies to solid-body rotation as well as to various degrees of differential rotation. This is consistent with the view that rotating stars on the upper main sequence have less massive convective cores and, therefore, shorter lifetimes than their nonrotating counterparts.*

* Recall that all barotropic models presented in this section have rotation laws that satisfy the constraint imposed by dynamical stability with respect to axisymmetric motions; that is, their specific angular momentum Om2 increases outward so that their angular velocity falls off more slowly than m-2, where m is the distance from the rotation axis (see Eq. [3.98]). More recently, Clement (1994) has probed the limiting case Om2 = constant, which corresponds to a marginally stable configuration. Accurate two-dimensional models have been computed, assuming that one has O a m-2 outside the cylinder containing the convective core and a solid-body rotation inside that cylinder. Calculations show that these extreme models have more massive convective cores than their nonrotating or rigidly rotating counterparts, at least for stars with masses below 12M0. In more massive configurations, however, the convective cores always decrease in mass fraction for any distribution ofspecific angular momentum.

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

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Fig. 6.3. Detailed structure of three models for 3OM0. Re is the total equatorial radius and ve is the equatorial velocity. The shaded area indicates the convective core. The upper portions show isopycnic contours enclosing mass fractions 0.2,0.4,0.6,0.8,0.95,0.999, and 1.0. The lower portions give the ratio of the angular velocity ^ to the central value , the fraction mm of the total mass interior to the corresponding cylindrical surface about the rotation axis, and the ratio of the circular velocity v to the surface value ve. The boundary of the convective core is indicated by an asterisk. Source: Bodenheimer, P.,Astrophys. J., 167, 153, 1971.

6.2.2 Effects of rotation on the observable parameters

The most conspicuous effect of rotation is to distort a star into an oblate configuration. This is well illustrated in Figure 6.3, although it is not known whether such high degrees of differential rotation are present in real stars. Yet, it is these departures from sphericity and the luminosity changes that are of paramount importance for the observable effects of rotation on the radiation emanating from a star. As we recall from Section 3.3.1, a barotropic model with a radiative envelope has an emergent flux \F| that varies in proportion to the surface effective gravity g (see Eq. [3.41]). Since this quantity is smaller at the equator than at the poles, both the local effective temperature and surface brightness are, therefore, lower at the equator than at the poles. This implies in turn that the various magnitudes and color indices of a rotating star will be functions of the aspect angle i between the line of sight and the rotation axis.

The theoretical problem divides naturally into three parts: (1) Building an interior model so that the effective temperature and gravity become known as functions of latitude on its free surface, (2) computing the energy spectrum of radiation as a function of aspect angle when a suitably realistic model atmosphere is fitted at each point of the free surface, and (3) integrating the emergent flux to obtain the usual photometric parameters for

Fig. 6.4. Sequences of rotating models with increasing angular momentum J (solid curves) in the (log pc — log Tc )-plane. Numbers on curves give the decimal logarithm of J in cgs units. Source: Bodenheimer, P., Astrophys. J., 167, 153, 1971.

each aspect angle. Figure 6.5 illustrates the results obtained by Maeder and Peytremann (1970), who have computed the energy spectrum of radiation for uniformly rotating stars of5M0, 2M0, and 1.4M0. Each rotational track represents configurations ranging from the nonrotating model to the uniformly rotating model for which Q/Qc = 0.99, where is the angular velocity at breakup rotation. For each mass, different values of the inclination i have been considered, with the aspect angle increasing from i = 0° ("pole-on" stars) to i = 90° ("equator-on" stars). For the 2M0 models, the percentage of stars under the random-orientation hypothesis is also indicated. (This is of course valid for all masses.) We observe that a pole-on star appears brighter than a nonrotating star of the same mass, but has almost the same color. This is so because one is directly facing the brighter polar regions as well as a larger projected area resulting from the star's oblateness. Figure 6.5 also shows that an equator-on star appears fainter and considerably redder than a nonrotating star of the same mass. The reason lies in the fact that limb darkening reduces the brightness of the polar regions while gravity darkening makes the equatorial belt cooler.

How do these theoretical results compare with the available observational data for normal main-sequence stars? By comparing their uniformly rotating models with various observed quantities, Maeder and Peytremann (1970) found that there was agreement with observation for stars earlier than about spectral type A7 but that later types showed effects at least two times larger than predicted by solid-body rotation. If so, then, what rotation law do upper-main-sequence stars actually follow? The problem has been considered by Smith (1971), who made a statistical study of the data available for rotating stars in the Praesepe and Hyades clusters. In agreement with other works, it is found that these stars seem not to be rotating uniformly. Unfortunately, a

Fig. 6.5. Color-magnitude diagram with rotational tracks for 5M0, 2M0, and 1.4Me, and various angles i. The termination point are for Q,/Q,c = 0.99. Source (revised): Maeder, A., and Peytremann, E., Astron. Astrophys., 7, 120, 1970. (Courtesy of Dr. A. Maeder.)

detailed study of the errors involved also shows the uncertainties to be such that the observations cannot be said to support any particular law of nonuniform rotation. More recently, Collins and Smith (1985) have made use of detailed stellar atmosphere models to compute the photometric effects of differential as well as rigid rotation in the A-type stars. Their analysis confirms the known qualitative result that differential rotation produces a larger scatter in the color-magnitude diagram than does uniform rotation. As was shown by these authors, however, photometry alone can only put rather weak constraints on the angular momentum distribution of the upper-main-sequence stars. This precludes any more definite conclusion about the nature of the rotation law in these stars.

Let us next consider the modifications brought by rotation on the age estimates of open star clusters. As we know, the age of a cluster is obtained from its color-magnitude diagram by fitting the observed sequence in the turnoff region with isochronous lines derived from nonrotating stellar models. The effects of rotation on age estimates are essentially of two kinds: (i) aspect effects on the color and magnitude of each star belonging to the cluster and (ii) structural effects on the models that are used to draw the theoretical isochronic lines. Both effects have been considered by Maeder (1971) under the assumption of uniform rotation on and above the main sequence. His analysis indicates that the structural effects of uniform rotation on age estimates are negligible in comparison with the aspect effects. However, because the displacement of a rotating star to the right of the main sequence can mimic the displacement due to evolution, neglecting the aspect effects leads to an overestimate in age that may reach up to 70% for clusters with the most rapidly rotating stars. In fact, Maeder has estimated that the age overestimates caused by the neglect of rotation reach about 60-70% for a Persei and the Pleiades. By contrast, the ages of the older clusters undergo very little changes, approximately 10-20%, because the stars in the turnoff region are less massive and so are rotating more slowly.

It is evident that neither theoretical considerations nor observations of the continuum can give a clear expectation for the actual rotation law in the upper-main-sequence stars. To what extent can the study of spectral lines yield useful information about the degree of surface differential rotation in these stars? The major effects of axial rotation on spectral lines is to broaden them, with no change in equivalent width; the amount of broadening depends upon the degree of axial rotation and the aspect angle i . In principle, the extent of surface differential rotation and macroturbulence in a star can be determined from the departures of observed line profiles and concomitant Fourier transforms from their standard theoretical counterparts. Attempts to extract this information from line profiles have been made by Stoeckley and Buscombe (1987) and in the Fourier domain by Gray (1977). Although these and related studies have not yet yielded any definite information on the surface velocity field of a star, Gray's results strongly suggest that differential rotation does not exist or is small in early-type stars. More recently, Collins and Truax (1995) have investigated the extent to which the actual velocity field of these stars can be determined by the information contained within a spectral line profile or its Fourier transform. It is found that one may use the classical model of a rotating star to determine projected rotational speeds as long as one does not expect accuracies greater than 10% under ideal conditions, with significantly larger errors for stars exhibiting extreme rotation. Accordingly, the use of the classical model as a probe of surface differential rotation and macroturbulence in a star remains problematic at best.

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