\section{Calculations} \label{s.calc}

\subsection{Overview}

\begin{table}
\begin{tabular}{cccc}
$\Mstar\ [\Msun]$ & $\Lstar\ [\Lsun]$ & $\Rpole\ [\Rsun]$ & $\period\ (\dy)$ \\ 
31.73             & 88,300            & 10.23             & 3.0 \\ \hline
$\kappa\ [{\rm cm^{2}\,g^{-1}}]$ & $\ncell$ & $\dApix\ [\Rsun^{2}]$ & \\ 
1                             & 64     & $2.4\times10^{-5}$       & 
\end{tabular}
\caption{Parameters for the O6V+O6V model. $\ncell$ is the RMT
  resolution parameter, and $\dApix$ the pixel area (see Hill \&
  Townsend)} \label{t.model-params}
\end{table}

We apply the numerical method described previously to explore the
phometric and spectroscopic binary of a `matched' massive-star binary
system in which both components have $\Mstar = 31.7\,\Msun$. This mass
corresponds to the O6V spectral type given in the calibration by
\citet[][their table~1]{Martins:2005aa}, and the polar radius and
luminosity of the stars are determined from the same
calibration. Table~\ref{t.model-params} summarizes these and other
parameters adopted for the modeling.

\subsection{Surface Properties} \label{s.calc.props}

Figures~\ref{f.props} illustrates the mesh structure
and surface effective temperature and gravity of one star in the
O6+O6 model, viewed at quadrature from pole-on ($i=0^{\circ}$) and
equator-on ($i=90^{\circ}$) vantages. The center of mass of the system
is at sky coordinates $(x,y) = 0,0$, and the other star is the mirror
image through $x=0$. 

The effective gravity is weakest around the equator, in particular
along the line of centers, due to partial cancellation between
gravitational and centrifugal forces. The effective temperature
follows the gravity due to equation~(\ref{e.grav-dark}), leading to
hotter poles and cooler equatorial regions. However, the external
radiation from the companion counteracts this gravity darkening on the
inward-facing side of the star, raising the effective temperature at
the substellar point $(x,y) = (0.6,0)$ by $\approx 1,500\,\kelv$.

Figure~\ref{f.vel} shows the Cartesian components of the surface flow
velocity for the same model, calculated for a braking time $\tbrake =
10^{4}\,\sec$. With $\corpar \approx 0.5$, the flow follows the
general behavior seen in the top-right panel of Figure~\ref{f.flow}:
near the substellar point it is symmetric about the line of centers,
but further away it is deflected eastward by the Coriolis force. The
maximum flow velocity reached is $\approx 15\,\kms$. We find
empirically that this maximum varies approximately linearly with
$\tbrake$, as would be predicted by equation~(\ref{e.flow-speed}) if
we neglect the square-root term.

\subsection{Line-Profile Variations}

\begin{figure}
\includegraphics[width=\columnwidth]{t_brake_inc_sum_effect}
\caption{The difference (in continuum units) between the depths of the
  blue- and redshifted \hetwo\,4026\,\AA\ line profiles for the O6+O6
  model at quadrature, plotted as a function of braking timescale $\tbrake$ for
  various choices of the inclination. The maximum surface flow
  velocity $v_{\rm max}$ is also shown.}
\label{f.scaling}
\end{figure}

In the absence of surface flows, the isovelocity contours of a
synchronous binary system appear as vertical lines parallel to the
rotation axis. The principal consequence of the flows shown in
Figure~\ref{f.vel} is to deform these contours, making them wider
apart on the star approaching the observer and narrower together on the
star receding from the observer. Figure~\ref{f.spectra} illustrates
this process in the O6+O6 model at four different orbital phases,
for $\tbrake=10^{4}\,\sec$ and an inclination $i=30^{\degr}$. The
$\phase=0.25$ (quadrature) panel in particular illustrates the contour
widening (narrowing) on the approaching (receding) star.

The figure also plots profiles for the \hetwo\,4026\,\AA\ line,
calculated using \glados. When isovelocity contours are wider apart,
the resulting profile is deeper than it would be in the absence of
surface flows (shown in the figure by dashed lines); and vice-versa
when the contours are narrower together. Thus, in spite of the fact
that the stars are identical, the line profiles at quadrature phase
are always deeper for the approaching star, than for the receding
star.

This behavior is strongly reminiscent of the so-called Struve-Sahade
(SS) effect \citep[see][and references therein]{Howarth:1997aa}, where
the line profiles of the secondary component in a massive-star binary
system are enhanced when blueshifted, relative to when redshifted. An
example of the SS effect was reported by \citet{Arias:2002aa} for
HD~165052, an O6.5+O7.5 binary with a $2.955\,\dy$ period. Their
observed \heone\,4471\,\AA\ line profile from the secondary appears to
deepen by $10-20\%$ when the secondary is approaching, compared to
when it is receeding. For comparison, the difference between the blue-
and redshifted profiles in Figure~\ref{f.spectra} is $\approx 0.015$
(in continuum units), corresponding to $\approx 10\%$ of the overall
profile depth. Therefore, it appears quite plausible that the SS
effect is the observational consequence of the radiative surface flows
we consider here.

To explore how the strength of the effect varies,
Figure~\ref{f.scaling} plots the difference between the depths of the
blue- and redshifted \hetwo\,4026\,\AA\ profiles of the O6+O6 model,
as a function of $\tbrake$ for a variety of inclinations. The depth
difference grows stronger as $\tbrake$ increases, because the surface
flows become progressively faster (the maximum velocity $v_{\rm max}$
is also shown in the figure). It also is larger at lower inclinations,
because --- with a smaller projected rotation velocity --- the
isovelocity contours of the stars are more easily distorted by the
surface flows.
