\begin{table}
\par
The transport of angular momentum through the stellar interiour is formulated as a diffusive process:
\begin{eqnarray}
\left (\frac{\partial \omega}{\partial t} \right )_m={\frac{1}{i}} \left (\frac{\partial}{\partial
m} \right )_t \left [ \left (4\pi r^2 \rho \right )^2i \nu \left (\frac{\partial \omega}{\partial
m} \right )_t \right ]
-{\frac{2w}{r}} \left (\frac{\partial r}{\partial t} \right )_m{\frac{1}{2}}{\frac{{\rm dln}~i}{{\rm dln}~r}},
\end{eqnarray}
where $\nu$ is the turbulent viscosity and $i$ is the specific angular
momentum of a shell at mass coordinate~$m$.
\par
The specific angular momentum of the accreted matter is determined by integrating the
equation of motion of a test particle in the Roche potential in case the
accretion stream
impacts directly on the secondary star, and is assumed Keplerian otherwise \cite{wellsteinphd}.
Rotationally induced mixing processes and angular momentum transport through
stellar interior are described by \citet{2000ApJ...528..368H}.
Magnetic fields generated due to differential rotation in the stellar interior
\citep{2002AetA...381..923S} are not included here \citep[however, see][]{gammapaper}.
\par
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We calculated the evolution of the binary systems in detail until Case~AB
mass transfer starts. Then we estimated the outcome of this mass transfer
by assuming that it ends when WR star has $\sim$5\% of the hydrogen left at the surface.
For this purpose we calculate the Kelvin-Helmholtz time scale of the primary:
\begin{equation}
{t_{\rm KH}=2\times 10^{\rm 7}{M_{\rm 1}}^{\rm 2}/(L_{\rm 1}R_{\rm l1})~\rm yr}
\end{equation}
where $M_{\rm 1}$, $L_{\rm 1}$ and $R_{\rm l1}$ are mass, luminosity and Roche radius
(in Solar units) of the primary star at the onset of Case~AB mass transfer.
The mass transfer rate is then assumed as:
\begin{equation}\label{mtr}
{\dot M_{\rm tr}=(M_{\rm 1}-M_{\rm WR,in})/{t_{\rm KH}}}
\end{equation}
where $M_{\rm WR,in}$ is the mass of the WR star that has a hydrogen surface abundance of 5\%; all quantities are
taken at the beginning of the mass transfer.
We calculate the change of the orbital period orbit using
constant value of $\beta=0.1$ for non-rotating and $\beta=0.0$ for rotating models
\citep{wellsteinphd}.
Matter that is not retained by the secondary is assumed to leave the system with
a specific angular momentum which corresponds to the secondary's orbital
angular momentum \citep{2001MNRAS.321..327K}.
\par
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Non-rotating models}\label{nonrot}
\par
We concluded in Sect.~\ref{simple} that massive O+O binaries can result in
WR+O systems similar to observed the (HD~186943,
HD~90657 and HD~211853) if accretion efficiency~$\beta$ is low.
Since some O~stars in WR+O binaries have been observed to rotate faster than synchronously,
we concluded that $\beta >0.0$ and assumed a constant value of
$\beta=0.1$ in our detailed evolutionary models.
We already mentioned that the orbital periods of the observed systems are between~6 and 10~days. Since the net effect of Case~A+Case~AB mass transfer is a widening
of the orbit, the initial periods should be shorter than the observed ones, so
we modelled binary systems with initial orbital periods of~3 and 6~days.
\begin{table}%t2
%\centering
\par
\caption{\label{big1}Non-rotating WR+O progenitor models for $\beta=0.1$.
$N$ is the number of the model, $M_{\rm 1,in}$ and $M_{\rm 2,in}$ are initial masses of the primary and
the secondary, $p_{\rm in}$ is the initial orbital period and~$q_{\rm in}$ is the initial mass ratio of
the binary system. $t_{\rm A}$ is time when Case~A mass transfer starts, $\Delta t_{\rm f}$
is the duration of the fast phase of Case~A mass transfer, $\dot M_{\rm tr}^{\rm max}$ is the maximum mass
transfer rate, $\Delta M_{\rm 1,f}$ and $\Delta M_{\rm 2,f}$ are mass loss of the primary and mass gain
of the secondary (respectively) during fast Case~A, $\Delta t_{\rm s}$ is the duration of slow Case~A mass
transfer,
$\Delta M_{\rm 1,s}$ and $\Delta M_{\rm 2,s}$ are mass loss of the primary and mass gain
of the secondary (respectively) during the slow Case~A, $p_{\rm AB}$ is the orbital period at the onset of Case
AB, $\Delta M_{\rm 1,AB}$ is the mass loss of the primary during Case~AB (mass gain of the secondary is
1/10 of this, see Sect.~\ref{code}), $M_{\rm WR,5}$ is the WR mass when the hydrogen surface abundance
is $X_{\rm s}=0.05$, the WR mass at $X_{\rm s}\le0.01$ is given in brackets,
$M_{\rm O}$ is the mass of the corresponding~O~star, $q$ is the mass ratio $M_{\rm WR}/M_{\rm O}$,
and~$p$ is the orbital period of the WR+O system.
The models are computed with a stellar wind mass loss of Hamann/6, except~$^{\rm *}$~Hamann/3, $^{\rm **}$~Hamann/2.\protect\\
${c}$~indicates a contact phase that occurs
for low masses due to a mass ratio too far from unity, for high masses
due to the secondary expansion during slow phase of Case~A.}
\begin{tabular}{lcccccccccccccccccc}
\hline
\hline
$N$ & $M_{\rm 1,in}$ & $M_{\rm 2,in}$ & $p_{\rm in}$ & $q_{\rm in}$ & $t_{\rm A}$ & $\Delta t_{\rm f}$ &
$\dot M_{\rm tr}^{\rm max}$ & $\Delta M_{\rm 1,f}$ & $\Delta M_{\rm 2,f}$ & $\Delta t_{\rm s}$ &
$\Delta M_{\rm 1,s}$ & $\Delta M_{\rm 2,s}$ & $p_{\rm AB}$ &
$\Delta M_{\rm 1,AB}$ & $M_{\rm WR,5}(1)$ & $M_{\rm O}$ & $q$ & $p$\\
\\
\hline
$ $ & $~{M}_{\odot}$ & $~{M}_{\odot}$ & $\rm d$ & & $10^{\rm 6}~ \rm yr$ & $10^{\rm 4}~ \rm yr$ &
$~{M}_{\odot}/\rm yr$ &
$~{M}_{\odot}$ & $~{M}_{\odot}$ & $10^{\rm 6} ~\rm yr$ &
$~{M}_{\odot}$ & $~{M}_{\odot}$ & $\rm d$ &
$~{M}_{\odot}$ & $~{M}_{\odot}$ & $~{M}_{\odot}$ & & $\rm d$\\
\hline
\\
N1 & $41$ & $20$ & $3$ & $2.05$ & $2.8$ & $c$ & $-$ & $-$ & $-$ &
$-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\\
\\
N2 & $41$ & $20$ & $6$ & $2.05$ & $3.6$ & $3.9$ & $5.4$ & $18.82$ & $1.85$ &
$0.39$ & $0.97$ & $0.06$ & $5.9$ & $7.1$ & $11.8(11.2)$ & $22.5$ & $0.52$ & $12.6$ \\
\\
N3 & $41$ & $20.5$ & $3$ & $2.00$ & $2.8$ & $2.2$ & $18.0$ & $21.13$ & $2.11$ &
$-$ & $-$ & $-$ & $2.85$ & $8.17$ & $7.7(7.2)$ & $23.2$ & $0.33$ & $12.5$\\
\\
N4 & $41$ & $24$ & $3$ & $1.71$ & $2.8$ & $3.1$ & $3.6$ & $15.18$ & $1.51$ &
$1.51$ & $5.13$ & $0.20$ & $3.87$ & $9.05$ & $10.1(9.3)$ & $26.4$ & $0.38$ & $13.5$\\
\\
N5 & $41$ & $24$ & $6$ & $1.71$ & $3.6$ & $4.3$ & $3.2$ & $17.31$ & $1.72$ &
$0.42$ & $1.88$ & $0.09$ & $8.92$ & $7.53$ & $12.1(11.4)$ & $26.3$ & $0.46$ & $21.5$\\
\\
N6 & $41$ & $27$ & $3$ & $1.52$ & $2.75$ & $5.8$ & $1.9$ & $13.82$ & $1.37$ &
$1.51$ & $5.86$ & $0.17$ & $4.38$ & $9.59$ & $10.3(9.8)$ & $29.1$ & $0.35$ & $16.6$\\
\\
N7 & $41$ & $30$ & $3$ & $1.37$ & $2.7$ & $6.7$ & $1.1$ & $12.60$ & $1.24$ &
$1.51$ & $6.72$ & $0.08$ & $5.20$ & $9.76$ & $10.5(10.0)$ & $31.8$ & $0.33$ & $20.8$\\
\\
N8 & $45$ & $27$ & $3$ & $1.67$ & $2.5$ & $3.6$ & $3.3$ & $15.41$ & $1.53$ &
$1.57$ & $7.48$ & $0.25$ & $3.88$ & $8.81$ & $11.5(10.7)$ & $29.4$ & $0.39$ & $12.0$\\
\\
N9 & $56$ & $33$ & $3$ & $1.70$ & $1.9$ & $5.0$ & $4.1$ & $17.2$ & $1.70$ &
$1.86$ & $15.66$ & $0.44$ & $4.07$ & $7.14$ & $13.6(12.7)$ & $35.4$ & $0.38$ & $9.8$\\
\\
N10 & $56$ & $33$ & $6$ & $1.70$ & $2.8$ & $5.8$ & $3.1$ & $19.35$ & $1.9$ &
$0.60$ & $4.77$ & $0.02$ & $7.77$ & $9.18$ & $18.6(17.5)$ & $35.1$ & $0.53$ & $15.2$\\
\\
N11$^{\rm *}$ & $56$ & $33$ & $6$ & $1.70$ & $2.8$ & $5.8$ & $3.1$ & $19.35$ & $1.9$ &
$0.46$ & $3.63$ & $0.06$ & $7.91$ & $7.15$ & $18.6(17.2)$ & $34.9$ & $0.53$ & $13.8$\\
\\
N12$^{\rm **}$ & $56$ & $33$ & $6$ & $1.70$ & $2.8$ & $5.8$ & $3.1$ & $19.35$ & $1.9$ &
$0.43$ & $3.43$ & $0.07$ & $8.89$ & $3.5$ & $18.3(16.4)$ & $34.5$ & $0.53$ & $12.1$\\
\\
N13 & $65$ & $37$ & $3$ & $1.76$ & $1.6$ & $3.2$ & $4.7$ & $18.81$ & $1.87$ &
$c$ & $-$ & $-$ & $-$ & $-$ & $16.2(14.8)$ & $-$ & $-$ & $-$\\
\\
N14 & $75$ & $45$ & $3$ & $1.67$ & $1.3$ & $4.2$ & $3.1$ & $18.57$ & $1.79$ &
$c$ & $-$ & $-$ & $-$ & $-$ & $18.5(16.9)$ & $-$ & $-$ & $-$\\
\\
\hline
\end{tabular}
\normalsize
\end{table}