Astron. Astrophys. 334, 210-220 (1998)

2. The spin-up mechanism

2.1. Assumptions

In this paper, we discuss the evolution of the rotation rate of post-main sequence stars, i.e. of stars which possess an active hydrogen burning shell source. This shell source converts hydrogen into helium and thus increases the mass of the helium core with time. However, this takes place on the very long time scale of the thermonuclear evolution of the star, and for the following argument we can consider the shell source as fixed in Lagrangian mass coordinate.

The shell source provides an entropy barrier, which separates the high entropy envelope from the low entropy core, and it marks the location of a strong chemical discontinuity, i.e. a place of a strong mean molecular weight gradient. Both, the entropy and the mean molecular weight gradient, act to strongly suppress any kind of mixing through the hydrogen burning shell source. This concerns chemical mixing as well as the transport of angular momentum (e.g., Zahn 1974; Langer et al. 1983; Meynet & Maeder 1997). Therefore, in the following, we shall regard the angular momentum evolution of the hydrogen-rich envelopes of the stars under consideration as independent of the core evolution.

This picture is somewhat simplified, as due to the inhibition of angular momentum transport through the shell source the post-main sequence core contraction and envelope expansion results in a large gradient in the angular velocity at the location of the shell source, i.e. a large shear which may limit the inhibiting effects of the entropy and mean molecular weight gradients. However, with the current formulation of shear mixing in our stellar evolution code, we find the angular momentum transport to be insignificant (cf. Sect. 3). In any case, the total amount of angular momentum in the helium core is much smaller than that in the hydrogen-rich envelope ( in the model discussed below), so that even if all of that would be transported out into the envelope it would not alter its angular momentum balance much.

As a star evolves into a red supergiant, its envelope structure changes from radiative to convective, starting at the surface. If we assume the envelope to be in solid body rotation initially, it can not remain in this state without any transport of angular momentum, unless

remains constant with time everywhere, i.e. remains constant. Here, m is the Lagrangian mass coordinate, r the radius, the density, and is the specific moment of inertia. For a sphere of radius r the gyration constant is , but even for deformed equipotential surfaces of average radius r one still finds to be of order unity. The assumption that does not depend on r was used to derive Eq. (1). Obviously, the above homology condition does not hold for stars which change their mode of energy transport in the envelope, since the polytropic index n changes between in the radiative envelope and in the convective state.

In the following we assume that convection tends to smooth out angular velocity gradients rather than angular momentum gradients, i.e. that convective regions tend to be rigidly rotating. This is certainly a good approximation at least if the rotational period (, where is the angular velocity) is long in comparison to the convective time scale (, where is the convective velocity and is the pressure scale height), and it may also hold for more rapid rotation if convective blobs can be assumed to scatter elastically (cf. Kumar et al. 1995). Latitudinal variations of the rotational velocity as deduced from helioseismological data for the solar convective envelope (Thompson et al. 1996) are not taken into account in our 1D stellar evolution calculation; however, the latitudinal averaged rotation rate of the solar convection zone deviates by no more than from solid body rotation (cf. e.g. Antia et al. 1997).

Although, for the considerations in Sect. 2.2 we assume rigid rotation to persist in convection zones, the necessary condition to make the spin-up mechanism described here work is only that convection transports angular momentum on a time scale which is short compared to the evolutionary time scale of the star, and that it leads to a characteristic angular momentum distribution in between the cases of constant angular velocity and constant angular momentum. The efficiency of the spin-up is largest for constant angular velocity and drops to zero for the case of constant angular momentum. The mechanism we present here is not restricted to convection, but any transport of angular momentum that acts towards solid body rotation is suitable to accomplish what we describe here.

2.2. The spin-down of convective envelopes

The rotation frequency of a rigidly rotating convective envelope depends on its moment of inertia and its angular momentum. Both are altered by mass loss from this envelope, the former by loss of mass, the latter by the accompanied loss of angular momentum. Additionally, the envelope's moment of inertia is also affected by changes of its density stratification.

Here, we want to discuss two processes which can change the rotation frequency of a rigidly rotating convective envelope without employing global contraction or expansion of the star, but rather by mass outflow through its upper or lower boundary. We will show that the first case leads to a spin-down, while the latter spins the envelope up.

The spin-down of mass losing rigidly rotating envelopes can be understood by breaking up the continuous mass and angular momentum loss into three discrete steps (see the left hand side of Fig. 1, cf. also Langer 1998), neglecting secular changes of the stellar structure. Starting from a rigidly rotating envelope extending from to (A1), the outer part of the convective envelope located between and is removed by stellar mass loss within the time interval (A1 A2). In the second step, for which we assume local angular momentum conservation, the envelope re-expands to roughly its original size (A2 A3). This results in a slow-down of the surface rotation and, as the layers deep inside the envelope expand less, differential rotation below (A3). In the third step, rigid rotation is re-established by an upward transport of angular momentum (A3 A4). As this implies an averaging of the angular velocity, it is clear from Fig 1 (A4) that now the whole convective envelope rotates slower than at the beginning of step 1. Obviously, the redistribution of angular momentum towards the stellar surface leads to an increase of the angular momentum loss rate.

Fig. 1. Mass loss from a rigidly rotating stellar envelope from the surface (case A: left panels) and through its lower boundary (case B: right panels). For illustration, the continuous process is split up into three steps. First (panels ), mass is removed from the envelope, second, the envelope restores its original radial extent through expansion (panels ), and third (panels ), the angular momentum is redistributed such that rigid rotation is restored. It leads to spin-down (spin-up) and decrease (increase) of the specific angular momentum for the case of mass loss trough the upper (lower) boundary of the convective envelope. Thin lines show the state of the preceding step.

The efficiency of the angular momentum loss induced by mass loss from the surface of a rigidly rotating envelope, i.e. the amount of angular momentum lost per unit mass lost relative to average specific angular momentum of the envelope, is given by

where and are total mass and total angular momentum of the envelope, and are stellar mass and angular momentum loss rate, and are the specific angular momentum and moment of inertia at the surface, respectively, and is the stellar radius. The mean value of a quantity x over the envelope is defined by

where M is the mass of the star. The larger the more efficient is the loss of angular momentum per unit mass lost. A value of corresponds to a decrease of the mean specific angular momentum of the envelope, to an increase. For the case of mass loss from the surface of a rigidly rotating stellar envelope is always greater than 1.

The density stratification in the envelope determines how much angular momentum is stored in the layers close to the surface relative to the total angular momentum of the envelope. An envelope structure which holds most of the mass close to its bottom favors high angular momentum loss rates, since this decreases . In red supergiants, on the other hand, is rather high, since those stars store a large fraction of their mass in layers far from the stellar center, thus reducing .

Fig. 2. Moment of inertia per radius as a function of radius (in units of the stellar radius) for four different stellar models. The solid line shows a ZAMS model, the dotted line a red supergiant model with an almost fully convective hydrogen-rich envelope, the dashed line a red supergiant model just before the blue loop, and the dash-dotted line a blue supergiant during the blue loop. All models are taken from the sequence described in detail below.

This is demonstrated in Fig. 2, where we plotted the moment of inertia per unit radius, for spherical symmetry, as a function of the fractional radius for different types of stars. For a red supergiant model with extended convective envelope we find the major contribution to the total moment of inertia I of the star at radii around (), whereas for the zero-age main sequence (ZAMS) model matter at dominates the moment of inertia of the star (). As main sequence stars can be approximated by rigid rotators (cf. Zahn 1994), the whole star takes the rôle of the stellar envelope as far as our definition of is concerned. Therefore, the considered main sequence star loses its angular momentum four times more efficient than the red supergiant model plotted in Fig. 2.

Two other cases are shown in Fig. 2, i.e., a red supergiant model shortly before its transition into the blue supergiant stage, where I is dominated by the density inversion at the upper edge of the convective region, making the angular momentum loss from the envelope of this star quite inefficient (), and a blue supergiant model during central helium burning in which the moment of inertia is concentrated even more towards the center of the star () than for the ZAMS model. Thus, if the envelopes of blue supergiants stay close to solid body rotation - which is indicated by our time-dependent calculations - they experience a more efficient spin-down than main sequence stars.

However, since for blue supergiants the total moment of inertia I is smaller than for their progenitor main sequence stars, they may get closer to critical rotation (cf. Sect. 3.1) if they keep their angular momentum. This may be in particular the case for metal poor massive stars. Those have much smaller mass loss and therefore angular momentum loss rates, and they can evolve from the main sequence directly into a long-lasting blue supergiant stage without an intermediate red supergiant phase (cf. e.g. Schaller et al. 1992); we found such stars to obtain critical rotation as blue supergiants in preliminary evolutionary calculations.

2.3. The spin-up of convective envelopes

The spin-up of convective envelopes which decrease in mass due to an outflow through their lower boundary is understood in a similar way to the mass loss process discussed above. Again we want to split up this process into three discrete steps (see sequence B on the right hand side of Fig. 1). Starting from a rigidly rotating envelope extending from to (B1), the inner part of the convective envelope, located between and , becomes radiative within some time (B2). Then the envelope re-expands to its original size. As local angular momentum conservation is assumed in this step, it results in a spin-up of most of the convective envelope, which is weaker for the layers farther out. The outcome is a differentially rotating envelope which spins fastest at its bottom (B3). Since the envelope has to go back to rigid rotation, angular momentum is transported upward in the next step until this is achieved (B4). We end up with higher specific angular momentum in the whole convective envelope compared to the initial configuration.

As can be seen in Fig. 5 below, during the evolution of a red supergiant towards the blue part of the HR diagram the radial extent of the convective envelope remains about fixed while mass shells drop out of the convective region. If one imagines the convective envelope to consist of moving mass elements or blobs, the spin-up process can thus also be understood as follows. A blob, starting somewhere in the convective region will, as it approaches the lower edge of the convective envelope, decrease its specific moment of inertia and therefore has to lose angular momentum in order to remain in solid body rotation with the whole convective region. Angular momentum has to be transferred to rising blobs such that also they remain in solid body rotation. Mass elements leaving the convective envelope thus only remove small amounts of angular momentum from the convective region. Therefore, the average specific angular momentum of the remaining convective envelope will increase and thus it spins up.

Replacing , and in Eq. (2) by , and , the specific angular momentum, the specific moment of inertia and the radius at the lower edge of the convective envelope, respectively, one can define an efficiency of angular momentum loss through the lower boundary, . We find , especially for those cases where is small, as e.g. for the red supergiant envelopes under consideration here.

The total change of specific angular momentum of the envelope by mass loss through the upper and lower boundary can then be written as

where is the rate at which mass leaves the envelope through its lower boundary. For our sample calculation it is , of order unity, and during the red supergiant stage preceding the evolution off the Hayashi line, so that almost increases as decreases,

reflecting the fact that angular momentum does not get lost efficiently from the convective envelope through its upper nor its lower boundary. The approximation is hardly affected by the variation of the lower bondary radius seen in Fig. 5. Thus, the convective envelope loses most of its mass but keeps a major part of the angular momentum.

When the convective envelope gets depleted in mass and the stellar radius decreases considerably, the global contraction of the stellar envelope results in an additional contribution to its spin-up. Still, mass elements drop out of the convective region (cf. Fig. 5), but now the contraction leads to an increase of the rotation velocity, and the star can reach rotation velocities of the order of the break-up velocity (see below). However, the contraction does not contribute to the increase of the specific angular momentum at the surface.

Finally, we want to note that if the whole star would remain rigidly rotating, e.g., due to the action of magnetic fields inside the star or by more efficient shear instabilities, its spin-up would occur very similar to the case described here. In that case, also mass shells from the core would transfer part of their angular momentum to layers above, which would make the spin-up somewhat more efficient. However, since in a red supergiant the mass elements lose the major part of their angular momentum to the convective envelope before they leave it, and the amount of angular momentum contained in the core is small anyway (cf. Sect. 2.1), the additional spin-up will be small (cf. Fig. 7 before blue loop).

© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998

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