\section{Analytic Treatment} \label{s.analytic}

\subsection{Radiation Field} \label{s.analytic.field}

We begin by considering the radiative acceleration of an atmosphere
due to an external source of radiation. The following analysis draws
extensively on the historical literature concerning the reflection
effect, in particular the seminal studies by \citet{Eddington:1926ab}
and \citet{Milne:1926aa}. Given the approximations made in subsequent
sections, it suffices to treat the irradiated atmosphere as both
locally plane parallel and gray. Let $\Itot(\tauv,\vartheta,\varphi)$
denote the frequency-integrated specific intensity of radiation at a
vertical optical depth $\tauv$, traveling in a direction given by the
spherical polar colatitude $\vartheta$ and azimuth $\varphi$ (with the
usual convention that the Cartesian $z$ axis, $\vartheta=0$,
corresponds to the upward vertical direction). This specific intensity
is conveniently expressed as the sum of three terms,
\begin{equation} \label{e.Itot}
\Itot(\tauv,\mu,\varphi) = \Iin(\tauv,\vartheta) + \Iex(\tauv,\vartheta,\varphi) + \Isc(\tauv,\vartheta)
\end{equation}
The $\Iin$ term represents the intrinsic radiation field of the
unirradiated atmosphere, which carries away the luminosity generated
in the stellar core. The external radiation is split into two parts:
the $\Iex$ term represents the external radiation that has yet to
interact with the atmosphere \citep[`\emph{\ldots though attenuated
    [it] maintains its individuality at all
    depths}';][]{Milne:1926aa}; while the $\Isc$ term represents the
radiation which has interacted one or more times with the atmosphere
by scattering or thermalization; we assume that these processes are
isotropic meaning that, like $\Iin$, $\Isc$ is axisymmetric.

Without any loss of generality we assume the external radiation is a
parallel beam with a flux $\Fex$ per unit area normal to itself,
traveling in the direction $\vartheta = \thetaex > \pi/2$ and
$\varphi = 0$. Then, $\Iex$ is given by the simple attenuation
expression
\begin{equation}
\Iex(\tauv,\vartheta,\varphi) = \Fex \,\delta(\thetaex - \vartheta)
\,\delta(\varphi) \,\ee^{-\tauv/\cos\thetaex},
\end{equation}
where $\delta()$ is the Dirac delta function. Combining this with
equation~(\ref{e.Itot}), and taking the first angular moments, yields
expressions for the Cartesian components of the total flux vector
$\vFtot$:
\begin{align}
 \label{e.flux-x}
\Ftotx(\tauv) &= \sin\thetaex\, \Fex\, \ee^{-\tauv/\cos\thetaex} \\
\label{e.flux-y}
\Ftoty(\tauv) &= 0 \\
\label{e.flux-z}
\Ftotz(\tauv) &= \Fin(\tauv) + \cos\thetaex\, \Fex\, \ee^{-\tauv/\cos\thetaex} + \Fsc(\tauv),
\end{align}
where
\begin{equation}
\Fin(\tauv) = 2\pi \int_{0}^{\pi} \Iin(\tauv,\vartheta) \cos\vartheta \, \diff\vartheta
\end{equation}
and similarly for $\Fsc$.

At great depth ($\tauv \gg 1$) the atmosphere should remain unaffected
by the external radiation, and thus $\Ftotz \rightarrow \Fin$
there. Radiative equilibrium\footnote{Since the present work focuses
  on hot, early-type stars, we can reasonably ignore any contributions
  toward thermal equilibrium by sub-surface convection.} requires that
\begin{equation}
\frac{\diff \Ftotz}{\diff \tauv} = 0,
\end{equation}
and furthermore
\begin{equation}
\frac{\diff \Fin}{\diff \tauv} = 0;
\end{equation}
hence, the identity $\Ftotz = \Fin$ must hold at all depths, and not
just for $\tauv \gg 1$. This means that the vertical flux is
\emph{unaffected} by the external radiation, and as a corollary that
\begin{equation} \label{e.flux-balance}
\Fsc(\tauv) = - \cos\thetaex\, \Fex\, \ee^{-\tauv/\cos\thetaex}.
\end{equation}
In the words of \citet{Eddington:1926ab}, `\emph{a star necessarily
  re-emits completely the radiation falling on it}'.

\subsection{Radiative Acceleration} \label{s.analytic.accel}

The foregoing analysis lays the groundwork for calculating the
radiative acceleration $\vgrad$. For an isotropic opacity
$\kappa$\footnote{Anisotropic opacities typically result from the
  interaction between velocity gradients and line opacity sources, and
  can give rise to line-driven winds from irradiated stars
  \citep[e.g.,][]{Gayley:1999aa}. The present analysis, however, is
  restricted to a consideration of \emph{continuum} radiative
  driving.}, this acceleration is
\begin{equation}
\vgrad = \frac{\kappa \vFtot}{c}.
\end{equation}
Combining this with equations~(\ref{e.flux-x}--\ref{e.flux-z}), the
non-vanishing Cartesian components of $\vgrad$ are
\begin{align}
\label{e.grad-x}
\gradx(\tauv) &= \frac{\kappa \Fex}{c} \sin\thetaex\, \ee^{-\tauv/\cos\thetaex}, \\
\label{e.grad-z}
\gradz(\tauv) &= \frac{\kappa \Fin}{c} + \frac{\kappa \Fex}{c} \cos\thetaex\, \ee^{-\tauv/\cos\thetaex} + \frac{\kappa \Fsc}{c}.
\end{align}
But by equation~(\ref{e.flux-balance}), the second a third terms in the
expression for $\gradz$ must be equal and opposite; therefore,
\begin{equation}
\gradz(\tau) = \frac{\kappa \Fin}{c},
\end{equation}
and, like the flux, the vertical radiative acceleration in the
atmosphere is unaffected by the external radiation.

In their treatment \citet{Drechsel:1995aa} neglect the third term in
equation~(\ref{e.grad-z}), even though it is this term which exactly
cancels the second. This casts serious doubt on their claim that
significant modifications to stellar surface geometry might arise from
irradiation by a binary companion. In fact, the consequences of such
irradiation are to be found in the expression~(\ref{e.grad-x}) for
$\gradx$. Since this component depends on $\tauv$, it is follows that
$\nabla \times \vgrad \neq 0$. This in turn implies that it is
impossible to balance the radiative force with any combination of
gravitational, centrifugal or gas-pressure forces, all of which may be
written as gradients of scalar functions. Thus, radiatively driven
transverse flows are inevitable, just as a heavy wind must induce
surface currents on a tranquil lake.

\subsection{The Transverse Velocity} \label{s.analytic.vel}

To constrain the surface flow pattern arising from the transverse
radiative accelerations discussed above, we begin by considering the
hydrodynamic momentum conservation equation, expressed in the frame of
reference co-rotating with the star \citep[e.g.,][]{Tassoul:2000aa},
\begin{equation} \label{e.hydro-momentum}
\frac{\partial \vvel}{\partial t} + (\vvel \cdot \nabla) \vvel + 2
\vOmega \times \vvel = - \frac{1}{\rho} \nabla p + \vgeff + \vgrad +
\vgdrag
\end{equation}
Here, $\vgeff$ is the effective gravitational acceleration, combining
contributions from Newtonian gravity and the centrifugal force;
$\vgrad$ is the net radiative acceleration given by the expressions in
the preceding section; and $\vgdrag$ is a generic drag term introduced
for reasons we discuss below. For simplicity, it is assumed throughout
that the rotation angular velocity vector $\vOmega$ is constant, with
magnitude $\Omega$, in both time and space.

Under the conditions of steady-state flow, the time derivative term in
the above equation vanishes, leading to a vector condition
\begin{equation} \label{e.hydro-momentum-steady}
(\vvel \cdot \nabla) \vvel + 2
\vOmega \times \vvel = - \frac{1}{\rho} \nabla p + \vgeff + \vgrad + \vgdrag
\end{equation}
prescribing force balance between the inertia and Coriolis terms on
the left-hand side, and the effective gravity, pressure gradient,
radiative and drag terms on the right-hand side. We do not explicitly
concern ourselves with force balance in the vertical direction, since
we assume that the atmospheric stratification remains largely
unaffected by the external irradiation from the companion star, and by
the surface flows established by this irradiation. This assumption is
based largely on the observation that, as demonstrated in the
preceding section, the external irradiation has no effect on the
vertical component of $\vgrad$. Accordingly, the analysis now focuses
exclusively on the transverse parts of
equation~(\ref{e.hydro-momentum-steady}).

It is evident from the $\tauv$ dependence of $\gradx$
(equation~\ref{e.grad-x}) that a different transverse flow
equilibrium will be reached at each depth in the atmosphere, typically
with the larger flow velocities at shallower depths where the
radiative acceleration is stronger. Indeed, this follows naturally
from the non-zero curl of the radiative force, discussed previously.
Rather than attempting to simulate this complex shear flow, which
would no doubt involve the treatment of intricate phenomena such as
Kelvin-Helmholtz instabilities and turbulence, we choose instead a
simple model where the transverse velocity throughout the visible
layers of the atmosphere is characterized by a single vector
$\vvmean(\theta,\phi)$ tangential to the stellar surface (here,
$\theta$ and $\phi$ are now the polar and azimuthal angular
coordinates in the global spherical polar coordinate system of the
star). This vector is defined by averaging the flow velocity over the
optical depth interval $0 \leq \tauv \leq 2/3$,
\begin{equation} \label{e.vvmean}
\vvmean(\theta,\phi) = \frac{3}{2} \int_{0}^{2/3}
  \vvel(\tauv,\theta,\phi) \,\diff \tauv.
\end{equation}
If the opacity $\kappa$ is assumed to be depth-independent, then
$\vvmean$ represents the mean momentum per unit mass throughout the
visible layers.

A similar averaging can be applied to the transverse parts of
equation~(\ref{e.hydro-momentum-steady}), to obtain expressions for
the components of $\vvmean$. In attempting to simplify these
expressions, our initial efforts focused on achieving force balance
solely between the transverse radiative acceleration and the flow
inertia $(\vvmean \cdot \nabla) \vvmean$, with the assumption that the
other terms would be unimportant. However, this resulted in highly
supersonic velocities, that are neither observed nor physically
plausible on the surface of stars, underlining the fact that for
more-realistic subsonic flows, inertia is wholly insufficient at
balancing radiative forces. In fact, for subsonic flows, it seems more
reasonable to assume instead that flow inertia itself is negligible
(we review this assumption below in
Section~\ref{s.analytic.props}). Furthermore, recognizing that
acoustic waves will tend to smooth out any pressure fluctuations, the
pressure gradient term in equation~(\ref{e.hydro-momentum-steady}) can
also be expected to be unimportant.

This then leaves the remaining Coriolis, radiative and drag forces to
determine the transverse velocity. First considering the case of a
spherical stellar surface (the generalization to arbitrary surface
geometries is deferred until Section~\ref{s.analytic.nonsph}),
$\vvmean$ is conveniently represented in terms of its components
$(\vmeantheta,\vmeanphi)$ in the $(\theta,\phi)$ directions. For
transverse force force balance, with inertia and pressure forces
neglected, these components must satisfy the equations
\begin{equation} \label{e.force-balance}
\begin{aligned}
-2 \Omega \cos\theta \vmeanphi &= \gradmeantheta + \gdragmeantheta, \\
2 \Omega \cos\theta \vmeantheta &= \gradmeanphi + \gdragmeanphi.
\end{aligned}
\end{equation}
The terms appearing on the right-hand sides are the appropriate
vertical averages of the components of $\vgrad$ and
$\vgdrag$. Assuming that $\kappa$ is depth-independent, the results of
the preceding section can be used to derive the averaged components of
$\vgrad$ as
\begin{equation} \label{e.flow-grad}
\begin{aligned}
\gradmeantheta &= \frac{\kappa \atten(\muex) \Fextheta}{c}, \\
\gradmeanphi &= \frac{\kappa \atten(\muex) \Fexphi}{c}.
\end{aligned}
\end{equation}
Here $(\Fexr,\Fextheta,\Fexphi)$ are the $(r,\theta,\phi)$ components
of the external flux vector $\vFex$, with
\begin{equation}
\muex = -\Fexr/|\vFex|,
\end{equation}
while the attenuation function $\atten(\muex)$ is introduced as
\begin{equation}
\atten(\mu) = \frac{3}{2} \int_{0}^{2/3} \ee^{-\tau/\mu} \,\diff\tau = \frac{3}{2} \mu \left(1 -
  \ee^{-2/3\mu}\right).
\end{equation}
Strictly speaking, these expressions are only applicable when the
external radiation is a parallel beam; however, we assume for
simplicity we can also apply them to a divergent beam.

To complete the derivation, we must consider the drag term
$\vgdrag$. This term is included to allow force balance around the
stellar rotational equator, because there the Coriolis force has no
transverse component.  Physically, the drag term represents viscous
coupling with the deeper, near-stationary parts of the star, that
tends to decelerate the moving visible layers. The simplest treatment
of the drag is to express it as a deceleration proportional to
$\vvmean$,
\begin{equation} \label{e.flow-gdrag}
\vgdrag = -\frac{\vvmean}{\tbrake}.
\end{equation}
Here $\tbrake$ is a characteristic braking timescale, a free parameter
of our model.

Substituting equations~(\ref{e.flow-grad}) and~(\ref{e.flow-gdrag})
into the right-hand sides of~(\ref{e.force-balance}), and solving the
resulting linear system, the components of averaged flow velocity in
the visible layers are found as
\begin{equation} \label{e.flow-comp}
\begin{aligned}
\vmeantheta &= \frac{\tbrake \kappa \atten(\muex)}{c(1 + \corpar^{2}
  \cos^{2}\theta)} 
\left(
\Fextheta + \corpar \cos\theta \Fexphi
\right), \\
\vmeanphi &= \frac{\tbrake \kappa \atten(\muex)}{c(1 + \corpar^{2}
  \cos^{2}\theta)} 
\left(
\Fexphi - \corpar \cos\theta \Fextheta
\right),
\end{aligned}
\end{equation}
where now the Coriolis parameter
\begin{equation}
\corpar = 2 \Omega \tbrake
\end{equation}
quantifies how far the star rotates during a braking time, and
characterizes whether Coriolis or drag effects will dominate the flow.

\subsection{General properties of the flow} \label{s.analytic.props}

\begin{figure*}
\includegraphics{figure_flow}
\caption{Flow patterns on the surface of a spherical star irradiated
  by a parallel beam, for four values of the Coriolis parameter
  $\corpar$. The horizontal lines in the lower panels indicate the
  location of the critical boundaries $\cos\theta = \pm 1/\corpar$
  that separate Coriolis- and drag-dominated flows.}
\label{f.flow}
\end{figure*}

From the above equation~(\ref{e.flow-comp}) for the components of
the averaged transverse velocity $\vvmean$, the flow speed at any
point is derived as
\begin{equation} \label{e.flow-speed}
|\vvmean| = \frac{\tbrake \kappa \atten(\muex)}{c} 
\sqrt{\frac{\Fextheta^{2} + \Fexphi^{2}}{1 + \corpar^{2} \cos^{2}\theta}}.
\end{equation}
This expression illustrates the necessity of including a drag term in
the force balance condition; in the limit $1/\tbrake \rightarrow 0$ of
no drag, the flow speed would become infinite around the equator.

The flow direction, measured clockwise from the easterly direction,
can likewise be found as
\begin{equation} \label{e.flow-direc}
\tan \fdirec \equiv \frac{\vmeantheta}{\vmeanphi} = \frac{\Fextheta
  + \corpar \cos\theta \Fexphi}{\Fexphi - \corpar \cos\theta \Fextheta}.
\end{equation}
From this equation, we identify two regimes. In the drag-dominated
limit $|\corpar\cos\theta| \ll 1$, the flow direction is given by
\begin{equation}
\tan \fdirec = \frac{\Fextheta}{\Fexphi},
\end{equation}
which is parallel to the transverse component of the external
flux. Conversely, in the opposite Coriolis-dominated limit
$|\corpar\cos\theta| \gg 1$, the direction is
\begin{equation}
\tan \fdirec = 
\begin{cases}
\frac{\Fexphi}{-\Fextheta} & \mu > 0, \\
\frac{-\Fexphi}{\Fextheta} & \mu < 0,
\end{cases}
\end{equation}
which is now perpendicular to the transverse component of the external
flux. If $\corpar > 1$, then a pair of critical boundaries $\cos\theta
= \pm 1/\corpar$ can be defined on the stellar surface, that segregate
the Coriolis-dominated perpendicular flows, in the polar regions, from
the drag-dominated parallel flows around the equator.

To illustrate the foregoing discussion, we consider the simple case of
a spherical star illuminated by a parallel beam coming from the
$(\theta,\phi)=(\pi/2,0)$ direction. For such a configuration, the
external flux has components
\begin{equation}
\begin{aligned}
\Fexr &= -\Fex \sin\theta \cos\phi \\
\Fextheta &= -\Fex \cos\theta \cos\phi \\
\Fexphi &= \Fex \sin\phi
\end{aligned}
\end{equation}
and the components of the transverse velocity, over the irradiated
hemisphere ($|\phi| < \pi/2$) follow from equation~(\ref{e.flow-comp}) as
\begin{equation}
\begin{aligned}
\vmeantheta &= \frac{\tbrake \kappa \atten(\muex) \Fex}{c(1 + \corpar^{2} \cos^{2}\theta)}
\left(
-\cos\theta \cos\phi + \corpar \cos\theta \sin\phi
\right), \\
\vmeanphi &= \frac{\tbrake \kappa \atten(\muex) \Fex}{c(1 + \corpar^{2} \cos^{2}\theta)}
\left(
\sin\phi + \corpar \cos^{2}\theta \cos\phi
\right),
\end{aligned}
\end{equation}
where now
\begin{equation}
\muex = \sin\theta \cos\phi.
\end{equation}
Figure~\ref{f.flow} shows the flow patterns predicted by these
expressions, for four differing values of the parameter $\corpar$. The
$\corpar = 0.1$ and $\corpar = 10.0$ panels are particularly
instructive, since they reveal the geometry in the drag- and
Coriolis-dominated limits. In the former case, the flow is everywhere
directed away from the center of the stellar disk, corresponding to
the point where the radiative acceleration is normal to the
surface. In the Coriolis-dominated case, however, the pattern is
rather more complex. The speed is highest around the equator, due to
the denominator in equation~(\ref{e.flow-speed}), and in each
hemisphere takes the form of a large whorl that diverts
initially-westward flowing material back around toward the easterly
direction. In \emph{both} cases, the flow speed becomes small at the
stellar limb; this is because the external radiation is almost tangent
to the surface there, and therefore barely penetrates into the
atmosphere.

The flow patterns illustrated in Figure~\ref{f.flow} indicate that
$\vvmean$ is strongly divergent in the transverse direction, and
therefore --- in the steady state --- the flow would appear not to
conserve mass. However, we anticipate that a system of return currents
will quickly establish themselves beneath the visible layers. These
currents, linked to the surface flows by an upwelling at the point
where the radiative acceleration is normal to the surface (i.e., the
disk center in the figure), and by downwellings at the terminator of
the irradiated region, will ensure that mass is conserved globally.

To bring the present section to a close, we briefly review the
approximation that the inertia term $\vInertia \equiv (\vvmean \cdot
\nabla) \vvmean$ can be neglected when determining the transverse flow
velocity. This term scales as $\sim |\vvmean|^{2}/\Rstar$, with
\Rstar\ the stellar radius, whereas the acceleration in the
drag-dominated limit ($\corpar \la 1$) scales as $\sim
|\vvmean|/\tbrake$ (cf. equation.~\ref{e.flow-gdrag}). Thus,
neglecting $\vInertia$ is reasonable when the braking length $\lbrake
\equiv \tbrake |\vvmean|$ is small compared with the stellar
radius. With flow speeds being limited to the photospheric sound speed
(for a hot star, $a \sim 25\,\kms$), and anticipated breaking
timescales extending up to $\sim 10^{4}\,{\rm s}$ (see
Section~\ref{s.calc.props}), an upper limit on $\lbrake$ is found as
$\sim 2.5 \times 10^{8}\,{\rm m} \sim 0.36\,\Rsun$. This is much
smaller than typical massive-star radii, and it follows that ignoring
the inertia term in the present context should not lead to any serious
errors. Indeed, for the flows plotted in Figure~\ref{f.flow}, we find
in all cases that the magnitude of $\vInertia$ does not exceed $10\%$
of the other terms appearing in the force-balance
equation~(\ref{e.hydro-momentum-steady}).

\subsection{Formulation for non-spherical stars} \label{s.analytic.nonsph}

In the foregoing analysis, we develop expressions for the flow
velocity $\vvmean$ under the assumption that the surface the star
remains spherical. In a binary system, this is clearly not the case.
However, it is straightforward to demonstrate that the transverse flow
velocity can also be expressed as
\begin{equation} \label{e.flow-nonsph}
\vvmean = \frac{\tbrake \kappa \atten(\muex)}{c(1 +
  \muOmega^{2}\corpar^{2})} \left[ \left(\venorm \times \vFex\right) + \muOmega \corpar
  \vFex \right] \times \venorm,
\end{equation}
where $\venorm$ is the local unit vector normal to the surface, and
\begin{equation}
\muex = - \frac{\vFex \cdot \venorm}{\Fex}, \qquad
\muOmega = \frac{\vOmega \cdot \venorm}{\Omega},
\end{equation}
are the normalized projections of the external flux vector and
rotation angular frequency vector onto this normal. This simple form
for $\vvmean$ is valid for \emph{arbitrary} surface geometries, and
its independence from any specific coordinate system makes it far more
practical than component equations such as~(\ref{e.flow-comp}).



