2 The basic equations for the outer stellar layers

The basic equation for the conservation of angular momentum for a stellar layer is (cf. Zahn 1992)

 

$${ \frac{\partial}{\partial t} \left( \rho r^2 \overline{\Omega}\r... ...} \left(\rho \nu r^4 \frac{\partial \overline{\Omega}}{\partial r} \right)\cdot$$ (1)

The quantity $\overline{\Omega}$ is the average angular velocity over the considered isobar. The so-called approximation of shellular rotation is made, it assumes that $\overline{\Omega}$ is essentially a function of r, because the differential rotation on an isobar is small. The physical reason for this assumption rests on the strong horizontal turbulence in a differentially rotating star. Indeed, I am not fully convinced that the currently used coefficient of horizontal turbulence, as proposed by Zahn (1992), is high enough to garantee the homogeneity of the angular velocity on an equipotential. Thus, the basic physics and assumptions underlaying the derivations of this coefficient are being re-examined (Maeder 2002). This appears to result in a new expression for the coefficient of horizontal turbulence, which happens to be at least one order of a magnitude higher. As a consequence, the assumption of shellular rotation as originally proposed by Zahn (1992) will be reinforced.

The above equation expresses that the change of angular momentum of a certain mass element in a star results from the transport by the meridional circulation with a velocity U(r) and from the turbulent diffusion with a coefficient $\nu$. If U(r) and $\nu$ are zero, this equation just says that the specific angular momentum $r^2 \overline{\Omega}$ of a mass element remains constant. The expression for U(r) is that given by Maeder & Zahn (1998) and for $\nu$ by Maeder & Meynet (2001).

If the anisotropic stellar winds remove some matter, for example, at the pole, the lack of mass, say the "hole'' at the pole, will be filled almost instantaneously by some material moving horizontally on the equipotential and bringing its angular momentum. Indeed, the timescale for this compensation is very short. It is the local dynamical timescale, of the orders of hours or days at most in massive stars. Thus, the equation of the surface will always be an equipotential, given for example by the Roche model. However, in the above example the polar mass loss removes less angular momentum than the corresponding amount of mass lost spherically. Let us call $\dot{\mathcal{L}}_{\rm excess}$ the difference of the angular momentum lost by unit of time between an anisotropic stellar wind and a spherical wind for the same amount of mass loss in a star of given angular velocity $\overline{\Omega}$

$$\dot{\mathcal{L}}_{{\rm excess}} (\overline{\Omega}) = \dot{... ...Omega}) - \dot{\mathcal{L}}_{{\rm iso}} (\overline{\Omega}) ,$$ (2)

where $\dot{\mathcal{L}}_{{\rm anis}} (\overline{\Omega})$ and $\dot{\mathcal{L}}_{{\rm iso}} (\overline{\Omega})$ are the rates of losses of the angular momentum by the anisotropic and isotropic winds respectively; these are negative values.

Whether anisotropic mass loss is present or not, the basic Eq. (1) for the conservation of angular momentum remains the same in the stellar interior. The external torque $\dot{\mathcal{L}}_{{\rm excess}} (\overline{\Omega})$due to an anistropic mass loss is only directly acting on the outer surface layer. This means that the modifications of the system of equations only concern the outer boundary conditions. Of course indirectly, the change of boundary conditions can affect the internal distribution of $\overline {\Omega (r)}$, the patterns of meridional circulation, the evolution, etc.

Focusing on the outer boundary conditions, we integrate Eq. (1) over the last thin shell of mass $M_{{\rm shell}}$ centered at a level r just below the stellar surface. We get with account for the anisotropic loss of angular momentum by stellar winds at the surface,

$$\frac{\rm d}{{\rm d}t} \left(M_{{\rm shell}} \overline{\Omega} \;... ...al \overline{\Omega}}{\partial r}+ \frac{3}{2} \dot{\mathcal{L}}_{{\rm anis}} .$$ (3)

We can derive the left hand side term

$$\frac{\rm d}{{\rm d}t} \left(M_{{\rm shell}} \overline{\Omega} \;... ...M_{{\rm shell}} \frac{\rm d}{{\rm d}t} \left( \overline{\Omega} \; r^2\right) ,$$ (4)

where the first term on the right corresponds to the rate of the loss of angular momentum taken by the isotropic mass loss. If we assume that the mass ejected spherically is just embarking its own angular momentum, one has the equality

$$\frac{2}{3} \dot{M}_{{\rm shell}} \; \overline{\Omega} \; r^2 = \dot{\mathcal{L}}_{{\rm iso}} .$$ (5)

Now, if we substract this isotropic component on both sides of Eq. (3), we get

 

$$M_{{\rm shell}} \frac{\rm d}{{\rm d}t} \left( \overline{\Omega} \... ...\overline{\Omega}}{\partial r} + \frac{3}{2} \dot{\mathcal{L}}_{{\rm excess}} .$$ (6)

If U(r) > 0, the circulation removes angular momentum from the outer shell to bring it toward the interior; if $\frac{\partial \overline{\Omega}}{\partial r} > 0$, the diffusion does the same. In addition, there is a contribution $\frac{3}{2} \dot{\mathcal{L}}_{{\rm excess}}$, which may be positive or negative, because the anisotropic mass loss may take away less or more angular momentum than the spherically symmetric case.

For polar mass loss, the contribution to the rotation of the outer shell is positive, i.e.

$$\frac{3}{2} \dot{\mathcal{L}}_{{\rm excess}} > 0 ,$$ (7)

because $\vert\dot{\mathcal{L}}_{{\rm anis}} (\overline{\Omega})\vert < \vert\dot{\mathcal{L}}_{{\rm iso}} (\overline{\Omega})\vert$ in this case. This means the following: the removal of some mass by polar stellar winds embarks less angular momentum that for an equivalent spherical mass loss. Thus, this is as if we add some angular momentum on the last remaining layer. Conversely, for equatorial mass loss we have

$$\frac{3}{2} \dot{\mathcal{L}}_{{\rm excess}} < 0 .$$ (8)

In this case, the equatorial mass loss pumps more angular momentum than the equivalent spherical mass loss and this accelerates the braking of stellar rotation. In Sect. 4 below, we give the appropriate expression for $\dot{\mathcal{L}}_{\rm excess}$in the general case of anisotropic mass loss.

We insist on the fact that Eq. (6) is not just a mathematical trick to account for the anisotropic mass loss, but it closely corresponds to the physical conditions in the case of anisotropic mass loss. Let us, for example, consider the case of polar enhanced mass loss. As said above, the mass removed by stellar winds at the pole will be almost instantaneously replaced by a flow of material coming from equatorial regions on the equipotential. This material will have more angular momentum than the average material on the equipotential and thus it exerts a positive torque on the surface layer, enhancing its rotation. The strong horizontal turbulence rapidly smears out the effect of this torque on the considered equipotential. We do think that this one dimensional treatment of a physical problem, which by essence is two dimensional, is a correct although simplified representation. The two hypotheses, a) of the instantaneous replacement of the matter on a surface layer and b) that this layer is preserving its equipotential shape, do not seem unreasonable in view of the considered dynamical timescales. Of course, very critical would be the presence of magnetic field which would introduce some coupling of the surface layer both upwards with the ejected matter and downwards with the layers below, as well as horizontally on the equipotential. In this context, let us note that Eq. (6) can also be applied in stars, which are slowed down by magnetic braking. In this case the quantity $\dot{\mathcal{L}}_{\rm excess}$ is the torque exerted by the magnetic field.

There is an additional problem in massive stars. If the mass loss is high, the outer layers are removed before the transport of angular momentum by circulation and turbulent diffusion has the time to bring the angular momentum inward. The condition for the transport of angular momentum to occur is that the characteristic time $t_{\Omega}$ for the readjustement of $\overline {\Omega (r)}$ by circulation and diffusion is shorter than the caracteristic timescale  $t_{\dot{M}}$for mass loss, i.e.

$$t_{\Omega} < t_{\dot{M}} .$$ (9)

For an outer shell of mass thickness $\Delta M$, one has $t_{\dot{M}} \simeq \frac{\Delta M}{ (-\dot{M})}$. Usually in stellar interiors, the meridional circulation is more efficient that turbulent mixing to transport the angular momentum. However, as shown below in Sect. 3.2, the thermal timescale at the surface of massive stars becomes so small that the departure from thermal equilibrium in rotating stars will not contribute to the driving of circulation, thus U(r) = 0 is a good approximation at the stellar surface. Thus, one may write for the characteristic time $t_{\Omega} = \frac{(\Delta r)^2}{\nu \frac{\vert\Delta \Omega\vert}{\Omega}}$. Thus, the condition (9) means that the actual mass loss rate $\dot{M}$ must be smaller in absolute values than a critical mass loss $\dot{M}_{{\rm crit}}$,

$$\dot{M}_{{\rm crit}} = \; - \; \left\vert \frac{\nu}{\Omega... ...\frac{\frac{\Delta \Omega }{\Omega}}{\Delta r}\right\vert\cdot$$ (10)

Thus, if this condition is not realised the external torque $\dot{\mathcal{L}}_{\rm excess}$will have no effect, since the layers will have been lost before they are able to transfer toward the interior the excess or lack of angular momentum which is applied at their surface. From the numerical example discussed in Sect. 4, we may estimate that the order of magnitude of $\dot{M}_{{\rm crit}}$ is of the order of $- 10^{-5}~ M_{\odot}$ yr^-1 which corresponds to an initial mass of about 100 $M_{\odot }$. This means that in the most massive stars, as well as in LBV and WR stars, the mass loss rates are so high that there is little or no transfer of the excess (or lack) of angular momentum into the underlaying layers. Thus, the effects due to the anisotropic mass loss are likely not important for the internal evolution of these extreme stars, provided they have no magnetic field. However, the anisotropies of mass loss will of course shape the ejected nebulae.