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4 The difference of angular momentum losses between anisotropic and isotropic winds

We need now to express the quantity $\dot{\mathcal{L}}_{\rm excess}$appearing in the previous sections. According to the definition, one has

 
$\displaystyle \dot{\mathcal{L}}_{{\rm excess}} = \left[ ~ \dot{I}_{{\rm anis}}(...
...anis}}(\overline{\Omega})}{\dot{I}_{{\rm iso}}(\overline{\Omega})} - 1\right] ,$     (20)

where $\dot{I}_{{\rm iso}}(\overline{\Omega})$ and $\dot{I}_{{\rm anis}}(\overline{\Omega})$ are the time variations (negative values) of the momentum of inertia due to the mass loss by isotropic and anisotropic stellar winds respectively. The momentum of inertia $I_{{\rm iso}}(\overline{\Omega})$ in a shell between the radii r1 and r2 in a star rotating with angular velocity $\overline{\Omega}$ is

\begin{displaymath}I_{{\rm iso}}(\overline{\Omega})= 2\pi \int^\pi_0 \int^{r_2}_...
... \sin^3
\vartheta}{\cos \epsilon} {\rm d} \vartheta {\rm d}r .
\end{displaymath} (21)

For a thin spherical shell of radius R and mass $\Delta M$, $I_{{\rm iso}}$ is just the usual expression

\begin{displaymath}I_{{\rm iso}} = \frac{2}{3} ~ R^2 \Delta M .
\end{displaymath} (22)

In a rotating star, the radial direction does not in general coincide with the normal to the surface (direction of the gravity). The angle $\epsilon$ between the radial direction and the direction of the effective gravity is a function of $\Omega$ and $\vartheta $, it is given by
$\displaystyle \cos \epsilon = - \frac{\vec{g_{\rm eff}} \cdot \vec{r}}{\vert\ve...
...}{27} \omega^2 x(\omega, \vartheta) \sin^2\vartheta \right)
\psi^{-\frac{1}{2}}$     (23)

with
$\displaystyle \psi = \left( -\frac{1}{x^2(\omega, \vartheta)} +
\frac{8}{27} \o...
...{8}{27} \omega^2 x(\omega, \vartheta) \sin\vartheta \cos\vartheta
\right)^{2} .$     (24)

The effective gravity $\vec{g_{\rm eff}}$ includes the effect of the gravitational potential and centrifugal force, but not the effect of radiation pressure (cf. Maeder & Meynet 2000). The rotation parameter $\omega$is the fraction of the angular break-up velocity. The quantity $x(\omega, \vartheta)$ is the ratio of the radius at colatitude $\vartheta $ with respect to the polar radius $R_{{\rm pb}}$ at break-up velocity. (Remark: the exact value of the polar radius $R_{{\rm p}}(\omega)$ also slightly depends on rotation through the structural equations; this effect is accounted for in the numerical models.) The value of $x(\omega, \vartheta)$ is a solution of the equation of the stellar surface in the Roche approximation for given values of $\omega$ and colatitude $\vartheta $, i.e.

\begin{displaymath}\frac{1}{x(\omega, \vartheta)} + \frac{4}{27} \omega^2 x^2
(\...
...^2 \vartheta = \frac{R_{{\rm p}}(\omega)}
{R_{{\rm pb}}} \cdot
\end{displaymath} (25)


 \begin{displaymath}{\rm with} \; x(\omega, \vartheta) =
\frac{r(\overline{\Omeg...
... \omega^2 = \frac{\overline{\Omega}^2
r^3_{\rm eb}}{GM} \cdot
\end{displaymath} (26)

$r_{{\rm eb}}$ is the equatorial radius at break-up. The time variation of the momentum of inertia by a star which loses mass at a rate $\dot{M}$ over its actual surface $\Sigma$ is

 \begin{displaymath}\dot{I}_{{\rm iso}}(\overline{\Omega})= 2\pi \int_{\Sigma}
...
...line{\Omega},\vartheta) \sin^2{\vartheta} \; {\rm d} \dot{M} .
\end{displaymath} (27)

The quantity d$\dot{M}$ is defined by the local mass flux. Usually an isotropic mass flux is considered in the stellar wind theory. However, for the reasons expressed in Sect. 1, the mass loss in a rotating star is anisotropic. Depending on whether we consider the isotropic or anisotropic cases, we have

\begin{displaymath}{\rm d} \dot{M} = \left( \frac{ \Delta \dot{M}(\overline{\Ome...
...)}
{\Delta \sigma } \right)_{{\rm iso/anis}} {\rm d} \sigma ,
\end{displaymath} (28)

where d$\sigma$ refers to the surface element on the rotating star. Thus, the corresponding expression for the time variation of the momentum of inertia is in the isotropic or anisotropic case

\begin{displaymath}\dot{I}_{{\rm iso/anis}}(\overline{\Omega}) = 2 \pi \int^\pi_...
...artheta) \sin^3{\vartheta}}{\cos \epsilon} {\rm d} \vartheta .
\end{displaymath} (29)

The expression for the anisotropic mass flux has been obtained by the application of the stellar wind theory to a rotating star (cf. Maeder & Meynet 2000), by taking into account the variations of $T_{{\rm eff}}$ and effective gravity with latitude and the change of the local Eddington factor. Also, the force multipliers k and $\alpha$ (cf. Castor et al. 1975; Lamers et al. 1995; Puls et al. 1996), which characterize the stellar opacity, may vary over the stellar surface since the temperature and gravity are varying,
 
$\displaystyle \left(\frac{\Delta\dot{M}(\overline{\Omega},\vartheta)}{\Delta \s...
...ta]^{\frac{1}{\alpha}}}
{(1 - \Gamma_{\Omega}(\vartheta))^{\frac{1}{\alpha}-1}}$      
$\displaystyle \quad {{\rm with}} \quad
A = \left(k\alpha\right)^{\frac{1}{\alpha}} \left(
\frac{1-\alpha}{\alpha}\right)^{\frac{1-\alpha}{\alpha}} \cdot$     (30)

The Eddington factor $\Gamma_{\Omega}(\vartheta)$in a rotating star must be defined in an appropriate way, i.e. as the ratio of the local flux to the limiting local flux. In this way, we have (cf. Maeder & Meynet 2000)
$\displaystyle \Gamma_{\Omega}(\vartheta) =
\frac{F(\vartheta)}{F_{{\rm lim}}(\v...
...eta)]}{4 \pi
cGM \left( 1 - \frac{\Omega^2}{2 \pi G \rho_{\rm {m}}} \right) } ,$     (31)

where $\zeta(\vartheta)$ expresses the deviation from von Zeipel theorem due to differential rotation. This factor is in general small and here we neglect it. The mass $M_{*}=
M (1 - \frac{\Omega^2}{2 \pi G \rho_{{\rm m}}})$ is the reduced mass, which takes into account the reduction of the gravitational potential by rotation. The ratio $ \frac{\dot{I}_{{\rm anis}} (\overline{\Omega})}
{\dot{I}_{{\rm iso}} (\overline{\Omega})}$ in Eq. (20) becomes

 \begin{displaymath}\frac{\dot{I}_{{\rm anis}} (\overline{\Omega})}{\dot{I}_{{\rm...
...eta)}{\cos \epsilon}
\sin^3 \vartheta {\rm d} \vartheta}\cdot
\end{displaymath} (32)

Thus, we have established the various equations necessary to calculate Eq. (20).

We need however make some further comments on the normalisation constants intervening in the expressions for the mass flux. Let us first consider the isotropic case. With the above notations, the total mass loss rate can be written

 \begin{displaymath}\dot{M}_{{\rm iso}}(\overline{\Omega})=
{\left(\frac{\Delta ...
...vartheta)}
{\cos \epsilon} \sin \vartheta {\rm d} \vartheta} .
\end{displaymath} (33)

For $\dot{M}_{{\rm iso}} (0)$ at zero rotation, we take an expression of the mass loss rates derived observationally (cf. Kudritzki & Puls 2000; Vinck et al. 2000,2001). To account for the fact that the observed relation is also based on rotating stars, we multiply the empirical values by a factor 0.8 (cf. Maeder & Meynet 2000) to get the average mass loss rates at zero rotation. Now, we need to obtain the global mass loss rate $\dot{M}_{{\rm iso}}(\overline{\Omega})$ for the rotation velocity considered. For that we use Eq. (4.29) by Maeder & Meynet (2000), which expresses the average global increase of the mass loss rates due to rotation

 \begin{displaymath}\frac{\dot{M} (\overline{\Omega})} {\dot{M} (0)} =
\frac{\lef...
... \rho_{\rm {m}}}-\Gamma \right]
^{\frac{1}{\alpha} - 1}}
\cdot
\end{displaymath} (34)

If $\Omega = 0 $, this ratio is of course equal to 1. The ratio $\frac{\Omega^2}{2 \pi G \rho_{{\rm m}}} \simeq
\frac{4}{9} \frac{v^2}{v_{\rm crit}^2} $ with a very good approximation. Here $v_{\rm crit}$ is the usual expression of the critical velocity. Since now $\dot{M}_{{\rm iso}}(\overline{\Omega})$ is known quantitatively, we may easily get from Eq. (33) the local mass flux $\left(\frac{\Delta \dot{M}(\overline{\Omega})}{\Delta \sigma} \right)_{{\rm iso}}$, necessary to calculate Eq. (32).

There is one more normalisation to do. For the anisotropic mass loss rate, we have

\begin{displaymath}\dot{M}_{{\rm anis}}(\overline{\Omega})=
2 \pi \int^\pi_0
\...
...,\vartheta)}{\cos \epsilon} \sin \vartheta {\rm d} \vartheta .
\end{displaymath} (35)

Indeed, for the anisotropic mass flux we can write in a compact form $\left(\frac{\Delta \dot{M}
(\overline{\Omega},\vartheta)}{\Delta \sigma} \right)_{{\rm anis}} = B f(\overline{\Omega},\vartheta)$, where we put in the term B all the coefficients which do not depend explicitely on $\vartheta $. Thus, we have

 \begin{displaymath}\dot{M}_{{\rm anis}}(\overline{\Omega})=
2 \pi B \int^\pi_0 ...
...a) r^2
\sin \vartheta}{\cos \epsilon} \; {\rm d} \vartheta .
\end{displaymath} (36)

Now, we impose

\begin{displaymath}\dot{M}_{{\rm anis}}(\overline{\Omega}) = \dot{M}_{{\rm iso}}(\overline{\Omega})
\end{displaymath} (37)

and in this way we can fix the value of B in Eq. (36) in a manner which is consistent with our definition of $\dot{\mathcal{L}}_{\rm excess}$ in Eq. (2). In the numerical calculations we have checked that with these prescriptions the star is losing exactly the same amount of mass in the corresponding isotropic and anisotropic cases.

Now, with these various equations, we can express $\dot{\mathcal{L}}_{\rm excess}$ in Eq. (20), which is necessary for the time dependent outer boundary condition (Eq. (19)).


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