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\section{}

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Use
\begin{equation}
u^2 {\dd\ln u\over\dd r}=-\cs^2\left({\dd\ln\rho\over\dd r}+{\dd s/\cp\over\dd r}\right)-{GM\over r^2}
\end{equation}
together with $\ln\rho+\ln u+\ln r^2=\const$ to obtain
\begin{equation}
u^2 {\dd\ln u\over\dd r}=\cs^2{\dd\over\dd r}\left(\ln u+\ln r^2+s/\cp\right)-{GM\over r^2}
\end{equation}
\begin{equation}
\left(u^2-\cs^2\right) {\dd\ln u\over\dd r}=\cs^2{\dd\over\dd r}\left(\ln r^2+s/\cp\right)
+{\dd\over\dd r}\left({GM\over r}\right)
\end{equation}
\begin{equation}
\left(u^2-\cs^2\right) {\dd\ln u\over\dd r}=-{\dd\over\dd r}\Phi_{\rm eff}
\end{equation}
where
\begin{equation}
\Phi_{\rm eff}=\Phi+\int_r^\infty \cs^2{\dd\over\dd r'}\left(\ln{r'}^2+s/\cp\right)\,\dd r'
\end{equation}
with $\Phi=-GM/r$.\footnote{
Integration by parts:
\begin{equation}
\Phi_{\rm eff}=\Phi-\cs^2\left(\ln{r}^2+s/\cp\right)
-\int_r^\infty {\dd\cs^2\over\dd r'} \left(\ln{r'}^2+s/\cp\right)\,\dd r',
\end{equation}
but there is no advantage.}

\section{Effective solar wind potential}
\label{CritPoint}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{ppot}
\end{center}\caption[]{
Effective potential $\Phi_{\rm eff}=-\cs^2\ln r^2-GM/r$ (black),
together with the Bernoulli term $-\cs^2\ln r^2$ (red), and
the gravitational potential $-GM/r$ (blue).
}\label{ppot}\end{figure}

The purpose of this appendix is to illustrate the effective
potential experienced by the siphon flow in the isothermal Parker wind.
By comparison, in the case of Roche-lobe overflow, the one-dimensional
Bernoulli equation can be written as
\EQ
\left(u^2-\cs^2\right)\,\frac{\dd}{\dd r}\ln u^2
=-\frac{\dd\Phi_{\rm eff}}{\dd r},
\EN
where $u=\meanU_r$ and $\Phi_{\rm eff}(r)$ is just the usual gravitational
potential in a binary system between the two components.
For the Parker wind, this term takes the form
\EQ
\Phi_{\rm eff}=-\cs^2\ln r^2-GM/r,
\EN
which is plotted in \Fig{ppot} along with its two contributions.
Evidently, $-GM/r$ steadily increases.
Thus, to overcome the gravitational pull, the first term is needed.
It becomes important at large radii and is then negative.
It characterizes the rapid expansion of the cross-sectional area,
$4\pi r^2$, and has then the same effect as the increase of the
cross-sectional area in a Laval nozzle; see \cite{Shore}.

\section{Old text}

Noticeable magnetic energy losses are found only near the rotation axis;
see the blue dotted lines in \Fig{ppmdot_M4096b1_Q001_Om1}.
As a function of radius, similarly to the case of slow rotation,
$\dot{E}_{\rm M}(r)$ has a maximum somewhere in $R<r<\rcrit$, which is
where the Alfv\'en point lies.
Furthermore, Model~B has a much smaller magnetic
energy loss at large radii than Model~A.
Here, we have multiplied $\dot{E}_{\rm M}(r)$ by a factor of $10^4$ to make it
comparable to the values of $\dot{E}_{\rm K}(r)$.

The model shows similarities with earlier simulations of
outflows emanating from stellar accretion disc dynamos \citep{vRBDS03,vRB04},
but there the opening angle was closer to $30\degr$.
In the present simulations, the opening angle is essentially zero.
It corresponds to a cylinder in which most of the magnetic fields are
ejected, although the flow speed is here strongly reduced.

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{ppmdot_M4096b1_Q001_Om1}
\end{center}\caption[]{
Radial dependence of $\dot{M}$ (a) and $\dot{E}_{\rm M}$ (b) for
$0\leq\theta\leq30$ (close to the axis) as blue dotted lines,
$30\leq\theta\leq60$, as black solid lines, and
$60\leq\theta\leq90$ (close to the equator) as red dashed lines
for Model~B.
The radial dependence of $\dot{E}_{\rm K}$ is shown as green
solid lines.
}\label{ppmdot_M4096b1_Q001_Om1}\end{figure}


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%\bibitem[Biskamp \& M\"uller(1999)]{BM99}
%Biskamp, D., \& M\"uller, W.-C.\yprl{1999}{83}{2195}

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