3 Stationary solutions

3.1 Zahn's solution

This solution applies to the outer boundary of the radiative zone in solar type stars, which have an external convective zone. An asymptotic regime, where the wind properties vary only slowly, on a time long with respect to the characteristic time of meridional circulation is considered by Zahn (1992). He also assumes, which is well verified, that the momentum of inertia of the boundary layer is negligible, so that the flux of angular momentum is nearly constant with depth. Thus, the time derivative in Eq. (6) is zero. In addition, Zahn assumes that the angular momentum transported by the turbulent viscosity is negligible with respect to the advection, an assumption which is also verified in the interior. Thus, one has

$$\frac{\partial\Omega}{\partial r} = 0.$$ (11)

From Eq. (6) one is left with

$$U(r) = \frac{15}{8 \pi} \frac{ \dot{\mathcal{L}}_{{\rm excess}}} {\rho r^4 \overline{\Omega}} \cdot$$ (12)

The two Eqs. (11) and (12) form, according to Zahn, the outer boundary conditions of the internal radiative zone in models of solar type stars. We note that if $\dot{\mathcal{L}}_{{\rm excess}}= 0$, then U(r) = 0 and with $\frac{\partial\Omega}{\partial r} = 0$, one has again the two boundary conditions usually employed in stellar models, (cf. Talon et al. 1997; Meynet & Maeder 2000).

If $\dot{\mathcal{L}}_{{\rm excess}} < 0$, as it would be produced for example by the magnetic braking in solar type stars, then one has U(r) < 0 near the outer edge of the radiative zone. This means that the lack of angular momentum at the edge will be compensated by a meridional circulation flow descending along the polar axis and ascending at the equator. It may be surprising to have a non zero U(r) at the outer boundary. But as pointed by Zahn (1992), there is no problem since in solar type stars the outer convective zone allows the return of mass and its conservation.

3.2 Boundary conditions for upper Main Sequence stars

Future works will tell us whether the above boundary conditions are well appropriate for solar type stars. In the case of Main Sequence (MS) stars with outer radiative layers, the above boundary conditions do not apply for two reasons. Firstly, with a non zero value of U(r) the return of mass is not insured; one cannot consider that the mass loss insures the mass continuity, because stellar winds in massive stars are present independently of the sign of U(r). Secondly, we may not consider that $\frac{\partial\Omega}{\partial r} = 0$, since the positive or negative torque exerted by $\dot{\mathcal{L}}_{\rm excess}$ in the case of anisotropic stellar winds will create a gradient of $\Omega$ in the outer layers. Thus, more appropriate boundary conditions have to be found.

The content of angular momentum of the outer shell is relatively very small, or in other words there is no possibility of stockage in the outer shell. Thus a situation of equilibrium may be reached between the gain (or loss) $\dot{\mathcal{L}}_{\rm excess}$ of angular momentum due to the anisotropic loss and the transport due to advection or diffusion toward (or from) the deeper layers. Thus, we consider that in the external shell the time derivative in the first member of Eq. (6) is zero and we have

 

$$\frac{3}{8 \pi} \dot{\mathcal{L}}_{{\rm excess}} = + \frac{1}{5} ... ...\Omega} U(r) + \rho \nu r^4 \frac{\partial \overline{\Omega}}{\partial r} \cdot$$ (13)

In the deep interior, it is well known that meridional circulation is the most efficient process for transporting the angular momentum. However, this is not necessarily the case very close to the stellar surface. Indeed, meridional circulation results from a departure from thermal equilibrium on an isobar, which drives some motions. However, close to the stellar surface, the timescale for thermal adjustment is

$$t_{{\rm therm}} \sim \frac{(\Delta r)^2}{K} \sim \frac{3 \kappa \rho^2 c_{{\rm P}} (\Delta r)^2}{4 a c T^3} ,$$ (14)

where K is the thermal diffusivity. The timescale $t_{{\rm therm}}$may become smaller than the local dynamical time $t_{{\rm dyn}} \sim \frac{1}{\sqrt{G \rho}}$. Let us examine the ratio of these two timescales

$$\frac{t_{{\rm therm}}}{t_{{\rm dyn}}} \sim \left(\frac{3 \ka... ...4 a c } \right) \frac{\rho^{\frac{5}{2}}}{T^3} (\Delta r) ^2 .$$ (15)

To know how the pressure P is varying close to the surface, we divide the equation of hydrostatic equilibrium by the equation of radiative equilibrium. This leads to

$$\frac{{\rm d}P}{{\rm d}T} = \frac{16 \pi a c G M T^3}{3 \kappa L} \cdot$$ (16)

For electron scattering opacity, this gives $P \sim T^4$. One also has $\beta P = \frac{k}{\mu m_{{\rm H}}} \rho T$, where $\beta$ is the ratio of the gas pressure to the total pressure P. Thus $T^3 \sim \frac{\rho}{\beta}$ and

$$\frac{t_{{\rm therm}}}{t_{{\rm dyn}}} \sim \beta \rho ^{\frac{3}{2}} .$$ (17)

At the stellar surface, $\rho$ and $\beta$ go to zero, which shows that the thermal time scale becomes shorter than the dynamical timescale. Thus, near the stellar surface a departure of thermal equilibrium will be restored very quickly by radiative transfer, without requiring the driving of circulation currents like the meridional circulation. Therefore, one of the boundary conditions at the surface radius R is just

 

U(R) = 0 , (18)

and thus from Eq. (13), one has

$$\frac{\partial \Omega}{\partial r} = \; \frac{3}{8 \pi} \frac{\dot{\mathcal{L}}_{{\rm excess}}}{\rho \nu r^4} \cdot$$ (19)

Equations (18) and (19) form the appropriate outer boundary conditions for stationary situations in stars with radiative layers near the surface. One sees that if $\dot{\mathcal{L}}_{{\rm excess}} > 0$(polar ejection), one has $\Omega$ growing with r at the edge of the star. This is what is expected, since then some rotation is added at the boundary and is thus able to diffuse toward the interior. This is consistent with the fact that at the surface the main transport of angular momentum is by diffusion, since as we have seen U(R)= 0. Conversely, for a dominant equatorial mass loss or for magnetic braking, we shall have $\frac{\partial \Omega}{\partial r} < 0$ and the diffusion of angular momentum toward the exterior will be favoured. Consistently, if $\dot{\mathcal{L}}_{{\rm excess}}= 0$we have again the usual conditions U(R) =0 and  $\frac{\partial\Omega}{\partial r} = 0$.

As shown by Eqs. (9) and (10), the mass loss rates must not be too large in order that the outer layers are not removed before they have had the time to transport the angular momentum. This requirement is even much more severe for an equilibrium stage to be established and we may estimate that mass loss rates smaller by about 2 orders of a magnitude with respect to $\dot{M}_{{\rm crit}}$ are necessary for this. In view of the estimate given after Eq. (10), this means that the mass loss rates must likely be smaller than $\sim$10 $^{-7}~M_{\odot}$ yr^-1, which corresponds to stellar mass lower than about 15 $M_{\odot }$. These are rough estimates and future grids of models may further precise these values.