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%% Paper for the Astrophysical Journal

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\usepackage{amsmath}

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%% Units

\newcommand{\Msun}{\ensuremath{\mathrm{M}_{\odot}}}
\newcommand{\Rsun}{\ensuremath{\mathrm{R}_{\odot}}}

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\newcommand{\Nobs}{\ensuremath{N}}

\newcommand{\timei}{\ensuremath{t_{i}}}

\newcommand{\Fap}{\ensuremath{A}}
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\newcommand{\dFi}{\ensuremath{\sigma_{i}}}

\newcommand{\Fstarper}{\ensuremath{\tilde{F}}}
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\newcommand{\etime}{\ensuremath{t}}
\newcommand{\etimei}{\ensuremath{\etime_{i}}}

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\newcommand{\phasei}{\ensuremath{\phase_{i}}}

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\newcommand{\maski}{\ensuremath{\mu_{i}}}

\newcommand{\fback}{\ensuremath{f}}
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\newcommand{\freq}{\ensuremath{\nu}}

\newcommand{\errper}{\ensuremath{\epsilon(\perfunc)}}
\newcommand{\errCV}{\ensuremath{\epsilon(\Fapsyn)}}

\newcommand{\mstar}{\ensuremath{m}}
\newcommand{\mstarper}{\ensuremath{\tilde{m}}}
\newcommand{\mstarres}{\ensuremath{\hat{m}}}

\newcommand{\pgram}{\ensuremath{\mathcal{P}}}
\newcommand{\pgramc}{\ensuremath{\mathcal{P}_{\rm c}}}

\newcommand{\mstarp}{\ensuremath{\mstar_{\rm p}}}
\newcommand{\mstars}{\ensuremath{\mstar_{\rm s}}}
\newcommand{\mstarb}{\ensuremath{\mstar_{\rm b}}}

\newcommand{\Dmstarp}{\ensuremath{\Delta \mstarp}}
\newcommand{\Dmstars}{\ensuremath{\Delta \mstars}}

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\newcommand{\Ocrit}{\ensuremath{\Omega_{\rm c}}}

%% Abbreviations

\newcommand{\soriab}{$\sigma$~Ori~AB}
\newcommand{\sorid}{$\sigma$~Ori~D}
\newcommand{\sorie}{$\sigma$~Ori~E}

\newcommand{\MOST}{\textit{MOST}}

\newcommand{\mesa}{\textsc{mesa}}

%% Main Document

\begin{document}

%% Title

\title{Tick tock: \MOST\ times the magnetospheric clock of $\sigma$ Ori
  E} \shorttitle{Tick tock: \MOST\ times the magnetospheric clock of
  $\sigma$ Ori E}

%% Authors

\author{R. H. D. Townsend\altaffilmark{1}, 
        Th. Rivinius\altaffilmark{2},
        J. F. Rowe\altaffilmark{3},
        A. F. J. Moffat\altaffilmark{4},
        J. M. Matthews\altaffilmark{5},
        D. Bohlender\altaffilmark{6},
        C. Neiner\altaffilmark{7},
        J. H. Telting\altaffilmark{8},
        D. B. Guenther\altaffilmark{9},
        T. Kallinger\altaffilmark{10,5},
        R. Kuschnig\altaffilmark{10,5},
        S. M. Rucinski\altaffilmark{11},
        D. Sasselov\altaffilmark{12},
        W. W. Weiss\altaffilmark{10}}

\email{townsend@astro.wisc.edu}

\altaffiltext{1}{Department of Astronomy, University of Wisconsin-Madison, 2535 Sterling Hall, 475 N. Charter Street, Madison, WI 53706, USA}
\altaffiltext{2}{ESO - European Organisation for Astronomical Research in the Southern Hemisphere, Casilla 19001, Santiago 19, Chile}
\altaffiltext{3}{NASA Ames Research Center, Moffett Field, CA 94035, USA}
\altaffiltext{4}{D\'epartment de physique, Universit\'e de Montr\'eal C.P. 6128, Succursale Centre-Ville, Montr\'eal, QC H3C 3J7, Canada}
\altaffiltext{5}{Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada}
\altaffiltext{6}{National Research Council of Canada, Herzberg Institue of Astrophysics, 5071 West Saanich Road, Victoria, BC, V9E 2E7, Canada}
\altaffiltext{7}{LESIA, UMR 8109 du CNRS, Observatoire de Paris, UPMC, Universit\'e Paris Diderot, 5 place Jules Janssen, 92195, Meudon Cedex, France}
\altaffiltext{8}{Nordic Optical Telescope, Apartado 474, 38700 Santa Cruze de La Palma, Spain}
\altaffiltext{9}{Department of Astronomy and Physics, St. Mary’s University, Halifax, NS B3H 3C3, Canada}
\altaffiltext{10}{University of Vienna, Institute for Astronomy, T\"urkenschanzstrasse 17, A-1180 Vienna, Austria}
\altaffiltext{11}{Dept. of Astronomy and Astrophysics, University of Toronto, 50 St George Street, Toronto, ON M5S 3H4, Canada}
\altaffiltext{12}{Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA}

\shortauthors{Townsend et al.}

%% Abstract

\begin{abstract}
We present results from three weeks' photometric monitoring of the
magnetic helium-strong star $\sigma$~Ori~E using the
\MOST\ microsatellite. The star's light curve is dominated by
rotational modulation arising predominantly from plasma trapped in its
co-rotating, centrifugally supported magnetosphere. The measured
rotation period $P = 1.190847\pm0.000015\,{\rm d}$ is consistent with
the ephemeris published by \citet{Tow2010}, confirming that the star's
rotation is steadily slowing due to magnetic braking. Analysis of the
depths of light minima and the light-curve residuals fails to reveal
any evidence for abrupt centrifugal breakout of plasma from the
magnetosphere; together with a demonstration that the mass contained
in the magnetosphere is significantly less than the theoretical
asymptotic mass, these findings leads us to argue that breakout
episodes do not play a significant role in establishing the star's
magnetospheric mass budget.
\end{abstract}

\keywords{stars: individual (\objectname{HD~37479}) --- stars:
  magnetic fields --- stars: rotation --- stars: chemically peculiar
  --- stars: early-type --- circumstellar matter}

%% Introduction

\section{Introduction} \label{sec:intro}

The B2Vpe star \sorie\ (HD~37479) is a magnetic helium-strong star
characterized by variations in many of its observables, including
photometric indices \citep{Hes1977}, H$\alpha$ emission
\citep{Rei2000}, UV wind absorption lines \citep{ShoBro1990}, radio
emission \citep{LeoUma1993}, linear continuum polarization
\citep{KemHer1977} and circular line polarization \citep{Oks2012}. The
$1.19\,{\rm d}$ period of these variations is identified with the
star's rotation period, recently discovered to be gradually
lengthening due to magnetic braking \citep[][hereafter
  T10]{Tow2010}. The variability originates from a combination of
surface abundance inhomogeneities and wind-originated plasma trapped
in a circumstellar, co-rotating, centrifugally supported magnetosphere
\citep[][hereafter T05]{Tow2005}.

In this paper we present data from three weeks' photometric monitoring
of \sorie\ by the \MOST\ microsatellite \citep{Wal2003}, beginning
November 2007. The original motivation behind this observing campaign
was to better characterize the star's light curve, and to search for
any cycle-to-cycle changes arising from the loss of magnetospheric
plasma. The following section briefly reviews the physics underlying
the magnetosphere of \sorie\ and related magnetic massive stars. Then,
\S\ref{sec:obs-reduce} describes the observations and explains the
procedure used to reduce the raw data, and \S\ref{sec:analysis}
analyzes various aspects of the light curve. The findings are
discussed in \S\ref{sec:discuss} and then summarized in
\S\ref{sec:summary}.

%% Background

\section{Background} \label{sec:back}

Magnetospheres around \sorie\ and other He-strong stars of recent
interest (e.g., HR~7355 --- \citealp{Oks2010,Riv2010,Riv2012};
$\delta$~Ori~C --- \citealp{Leo2010}; HR~5907 --- \citealp{Gru2012})
arise from the delicate interplay between magnetic fields, wind
outflow and rapid rotation. Within the narrative of the
\emph{magnetically confined wind shock} (MCWS) paradigm proposed by
\citet{BabMon1997a,BabMon1997b}, radiation-driven wind streams
originating from opposing footpoints of closed magnetic loops undergo
head-on collisions near the loop summits. The wind plasma is
shock-heated to millions of Kelvin and subsequently cools radiatively,
initially by emission at X-ray wavelengths; this explains why a number
of magnetic massive stars, including \sorie, are X-ray sources.

The fate of the cooled post-shock plasma is considered in the
\emph{rigidly-rotating magnetosphere} (RRM) model developed by
\citet{TowOwo2005}. On sufficiently strong field loops the plasma is
forced into co-rotation with the star. Outside the Kepler radius ---
the point where the local azimuthal velocity of the plasma matches the
local orbital velocity --- the centrifugal force exceeds the downward
pull of gravity and can in principle support the plasma in
magnetohydrostatic equilibrium for indefinite periods of time. Based
on an empirical prescription for the wind mass flux at the footpoints
of each closed loop, the RRM model predicts the relative density
distribution of this centrifugally supported plasma throughout the
magnetosphere. In the case of \sorie\ the predicted distribution
consists primarily of a pair of circumstellar clouds situated at the
intersections between magnetic and rotational equators, and these
clouds are able successfully to reproduce key features of the star's
H$\alpha$ and photometric variability (see T05).

With continual wind feeding the total mass of plasma accumulated in a
magnetosphere will grow with time, unless a countervailing mass
leakage mechanism allows some kind of quasi-steady state to be
reached. \citet{TowOwo2005} propose a mechanism involving the
stressing and eventual breaking of magnetic loops by the centrifugal
force, which grows in strength as plasma
accumulates. Magnetohydrodynamic (MHD) simulations by \citet{udD2006}
support this \emph{centrifugal breakout} hypothesis, and moreover
suggest that the reconnection heating arising during breakout episodes
could explain the X-ray flares seen in \sorie\ (over and above its
quiescent MCWS-originated emission) by \citet{GroSch2004} and
\citet{San2004}. Cycle-to-cycle changes in the light curve might be
expected from centrifugal breakout episodes; searching for these
changes requires precise photometry over a long time series, a task
well suited to \MOST.

%% Observations & Data Reduction

\section{Observations and Data Reduction} \label{sec:obs-reduce}

\subsection{Overview} \label{ssec:obs-reduce-over}

\MOST\ observed \sorie\ and four other bright B-type stars (HD~37525;
\sorid; HD~37744; HD~294272) over the interval November 12 -- December
3 2007 with a cadence of around 60\,s. The telescope operated in
direct imaging mode, where targets are placed on the open area of the
science CCD not covered by the Fabry microlens array
\citep[see][]{Row2006a,Row2006b}; this comes at the cost of a degraded
instrumental stability and precision, but is necessary because
\sorie\ is too faint ($V=6.66$) to observe in Fabry mode. Individual
subexposures of 0.530\,s were co-added onboard the satellite prior to
downloading, to avoid saturating the telemetry link
\citep[see][]{Row2008}. The number of subexposures per co-added
exposure was initially set at 31, but was then increased to 61 after
the first 17 hours of the run. 

At the beginning of the run \sorie\ fell outside the \MOST\ continuous
viewing zone (CVZ); therefore, for $\sim 25$ minutes of every
101.413-minute orbit the satellite slewed to observe an alternative
field in the Hyades, resulting in periodic gaps in the data (see
Fig.~\ref{fig:light-curve}). On November 23, \sorie\ entered the CVZ
and \MOST\ switched to observing the star continuously. Around half a
day prior to this switch the onboard computer crashed, leading to a
$\sim 0.25\,{\rm d}$ gap in the data. The orientation of the
spacecraft after the switch initially led to increased solar heating
and a climb in the CCD temperature, accounting for certain features in
the residual light curve (see
\S\ref{ssec:analysis-residual}). Finally, gaps in the data on December
2 and December 3 arose due to science data buffer overruns.

\subsection{Aperture Photometry} \label{ssec:obs-reduce-photom}

The co-added exposures of \sorie, each a 20 by 20 pixel image, are
reduced using the standard approach of synthetic aperture
photometry. The stellar flux \Fstari\ is calculated as the difference
between the total flux \Fapi\ in a 5-pixel radius circular aperture
centered on the 2-dimensional Gaussian centroid of the image, and the
imputed background flux \Fbacki\ (arising primarily from stray light)
in the same aperture,
\begin{equation} \label{eqn:Fstar}
\Fstari = \Fapi - \Fbacki.
\end{equation}
Here and throughout, the subscript $i=1,\ldots,\Nobs$ is used to index
exposures, with $\Nobs=24\,455$ being the number of co-added exposures
in the full run. The aperture flux is evaluated via
\begin{equation}
\Fapi = \frac{\Cnapi}{\texpi},
\end{equation}
where \Cnapi\ is the measured number of counts in the aperture and
\texpi\ the exposure time. The units of \Fapi\ and other fluxes
defined herein are counts per second.

%%

\subsection{Background Flux Estimation} \label{ssec:obs-reduce-fback-est}

The background flux \Fbacki\ in eqn.~(\ref{eqn:Fstar}) cannot be
measured directly, but is customarily approximated using a `sky' flux
measured in a region surrounding the aperture. However, when applied
to \MOST\ direct imaging data this simple approach gives
unsatisfactory results; although the background and sky fluxes are
strongly correlated \citep[see, e.g.,][their Fig.~3]{Row2006b}, the
relationship between them is not completely linear. Such behavior is
not wholly understood but likely stems from the complex spatial
distribution of stray light falling on the detector, which varies on
the satellite's orbital period \citep{Ree2006}.

Accordingly, we determine the correlation between the background and
sky fluxes empirically using an iterative procedure:
\begin{enumerate}
\renewcommand{\theenumi}{(\roman{enumi})}
%
\item \label{en:back-beg} Begin with the initial estimate
\begin{equation} \nonumber
\Fbacki = \Fskyi,
\end{equation}
where
\begin{equation} \nonumber
\Fskyi = \frac{\nap}{\nsky}\, \frac{\Cskyi}{\texpi}
\end{equation}
is the sky flux rescaled to the aperture size. Here, \Cskyi\ is the
number of counts measured in the sky region (defined by the
non-aperture pixels), and \nsky\ and \nap\ are the number of pixels
composing the sky region and the aperture, respectively.
%
\item \label{en:back-eval-Fstar} Calculate \Fstari\ from \Fapi\ and
  \Fbacki\ via eqn.~(\ref{eqn:Fstar}).
%
\item \label{en:back-split-Fstar} Decompose \Fstari\ into three components,
\begin{equation} \nonumber
\Fstari = \Fstarmean + \Fstarperi + \Fstarresi,
\end{equation}
with \Fstarmean\ being the weighted mean stellar flux, \Fstarperi\ the
periodic light variations due to rotational modulation, and
\Fstarresi\ the residuals. The mean flux is calculated as
\begin{equation} \nonumber
\Fstarmean = \frac{\sum_{i=1}^{\Nobs} \wphoti \Fstari}{\sum_{i=1}^{\Nobs} \wphoti}.
\end{equation}
where
\begin{equation} \nonumber
\wphoti = \frac{1}{\dFi^{2}}
\end{equation}
is the weight of the $i$'th point in terms of its photometric
(Poisson) error \dFi. Likewise, the periodic component is calculated
as
\begin{equation} \nonumber
\Fstarperi = \perfunc(\phasei),
\end{equation}
where 
\begin{equation} \nonumber
\phasei \equiv \freq \etimei
\end{equation}
is the rotation phase of the $i$'th exposure, \etimei\ is the
corresponding mid-exposure time, \freq\ is the rotation frequency of
the star, and the `periodic flux function' $\perfunc(\phase)$ repeats
on the interval $0 \leq \phase \leq 1$. The construction of this
function is discussed in \S\ref{ssec:obs-reduce-perfunc}.
%
\item \label{en:back-prewhiten} Calculate the total aperture flux
  pre-whitened by the mean and periodic signals,
\begin{equation} \nonumber
\Fapwhitei = \Fapi - \Fstarmean - \Fstarperi = \Fstarresi + \Fbacki
\end{equation}
%
\item \label{en:back-update} Update the background flux as
\begin{equation} \nonumber
\Fbacki = \backfunc(\Fskyi),
\end{equation}
where the `background flux function' $\backfunc(\Fsky)$ comes from a
smooth fit to the $(\Fskyi,\Fapwhitei)$ points. The construction of
this function is discussed in \S\ref{ssec:obs-reduce-backfunc}.
%
\item \label{en:back-check} If the maximal relative change in
  \Fbacki\ from the preceding iteration, across all $i$, remains above
  some threshold (taken throughout to be $10^{-3}$), go back to
  step~\ref{en:back-eval-Fstar}; otherwise, the procedure is complete.
\end{enumerate}
The pre-whitening in step~\ref{en:back-prewhiten} is necessary because
the periodic variations \Fstarperi\ are \emph{not} correlated with
\Fskyi, and will distort the fit in step~\ref{en:back-update} unless
removed.

%%

\subsection{Background Flux Function} \label{ssec:obs-reduce-backfunc}

The background flux function $\backfunc(\Fsky)$ is determined via a
smooth fit to the $(\Fskyi,\Fapwhitei)$ data using the locally
weighted regression algorithm devised by \citet{Cle1979}. In brief,
the function at any \Fsky\ (not necessarily equal to the observed
abscissae \Fskyi) is calculated via a linear least-squares fit to the
data using weights which filter out all but the $r$ points closest to
\Fsky. The weights are usually based on a tri-cubed function
\citep{Cle1979}; however, we choose to introduce additional weighting
based on the photometric errors. The overall weighting of the $i$'th
point in the least-squares fit is thus defined as
\begin{equation} \label{eqn:weights}
\weighti = 
\begin{cases}
\left(1 - \left|\frac{\Fskyi - \Fsky}{h}\right|^{3} \right)^{3} \wphoti & \text{if}\ |\Fskyi - \Fsky| < h, \\
0 & \text{otherwise}.
\end{cases}
\end{equation}
The smoothing length $h$ is set equal to the absolute separation
$|\Fsky_{r} - \Fsky|$ of the $r$'th closest point to \Fsky. A global
smoothing parameter $0 < \fback \leq 1$ is used to set $r$ as the
smallest integer greater than or equal to the product $\fback
\Nobs$. The choice of \fback\ is discussed in
\S\ref{ssec:obs-reduce-params}.

\subsection{Periodic Flux Function} \label{ssec:obs-reduce-perfunc}

To determine the periodic flux function $\perfunc(\phase)$ we
%% initially analyzed the $(\etimei,\Fstari)$ light curve using the
%% multi-harmonic AOV periodogram developed by \citet{Sch1996}. However,
%% as is typical of Fourier-based approaches, reproducing sharp features
%% in the light curve requires large numbers of harmonics ($\gtrsim 50$),
%% which then runs the risk of overfitting smooth regions.
apply the same locally weighted regression algorithm described above
to the $(\phasei,\Fstari-\Fstarmean)$ data.  Weights are calculated
similarly to eqn.~(\ref{eqn:weights}), but periodic boundary
conditions are applied when evaluating the absolute separations
$|\phase_{r} - \phase|$. The parameter $r$ is set using a global
smoothing parameter \fper, defined in the same way as \fback\ above;
the choice of \fper\ is discussed in \S\ref{ssec:obs-reduce-params}.

To allow an independent determination of the star's rotation period
(\S\ref{ssec:analysis-period}), the frequency \freq\ is not specified
\emph{a priori} but rather chosen to minimize the weighted mean square
error
\begin{equation}
\errper = \frac{\sum_{i=1}^{\Nobs} \wphoti [\Fstari - \Fstarmean - \perfunc(\phasei)]^{2}}{\sum_{i=1}^{\Nobs} \wphoti}.
\end{equation}
This period-search approach is similar to the popular phase dispersion
minimization (PDM) algorithm introduced by \citet{Ste1978}.

\subsection{Selection of Smoothing Parameters} \label{ssec:obs-reduce-params}

The fitting procedures described in \S\S\ref{ssec:obs-reduce-backfunc}
and~\ref{ssec:obs-reduce-perfunc} both use the locally weighted
regression algorithm. To select appropriate values for the global
smoothing parameters $(\fback,\fper)$ we use $K$-fold cross validation
(CV), a well-established technique in the statistical community
\citep[see][for a review]{ArlCel2010} which is seeing increasing
application in astrophysical contexts, especially for automated
classification \citep[e.g.,][]{CalFue2004,Fee2005,Blo2011}.

The observational data are randomly partitioned into $K$ approximately
equal-sized sets. For a given $(\fback,\fper)$ the reduction procedure
is applied to the data comprising $K-1$ of the sets taken together
(the `training' set), with the remaining set (the `validation' set)
put aside. The locally weighted regression fits to the background flux
and periodic stellar flux, together with the mean flux, are used to
calculate the synthetic aperture flux
\begin{equation}
\Fapsyn(\Fsky,\phase) = \Fstarmean + \backfunc(\Fsky) + \perfunc(\phase).
\end{equation}
The `predictivity' of the data reduction --- how well it can model the
behavior of the validation set, despite being based on the training
set --- is then quantified via the weighted mean square error
\begin{equation}
\errCV = \frac{\sum_{i=1}^{\Nobs} \maski \wphoti [\Fapi - \Fapsyn(\Fskyi,\phasei)]^{2}}{\sum_{i=1}^{\Nobs} \maski \wphoti}.
\end{equation}
Here, the mask \maski\ is equal to 1 if the $i$'th point is in the
validation set, and 0 otherwise.

This procedure is repeated $K$ times, with each set in turn serving as
the validation set while the remainder are used to form the training
set. The arithmetic mean of the resulting $K$ values of \errCV\ is
adopted as a measure of the predictivity of the data reduction for the
chosen $(\fback,\fper)$ pair. To find the optimal combination of these
parameters, we search over the ranges $-2 \leq \log_{10} \fback \leq
-0.3$ and $-2.5 \leq \log_{10} \fper \leq -1$ for the combination
which minimizes the mean \errCV. For a partition factor $K=10$, the
minimum occurs at $(\fback, \fper) = (0.084, 0.019)$, and we adopt
these parameters throughout (similar values result from choosing $K=5$
%($0.089$, $0.019$)
 or $K=20$).
%($0.074$, $0.018$)

\section{Analysis} \label{sec:analysis}

The ultimate outcome of the data reduction process is presented in
Fig.~\ref{fig:light-curve}, which plots the stellar flux in magnitudes
relative to the mean flux,
\begin{equation}
\mstar = -2.5 \log \left(\frac{\Fstar}{\Fstarmean} \right),
\end{equation}
as a function of time. The following sections explore various aspects
of this light curve.

\subsection{Periodic Variations} \label{ssec:analysis-periodic}

\begin{figure*}[ht]
\includegraphics{fig_light_curve}
\caption{The stellar flux \mstar\, in magnitudes relative to the mean
  flux, plotted as a function of time. The solid curve overlaying the
  data shows the periodic component \mstarper. The vertical dashed
  lines delineate the day boundaries.} \label{fig:light-curve}
\end{figure*}

The periodic component of the stellar flux, defined in magnitudes as
\begin{equation}
\mstarper = -2.5 \log \left(1 + \frac{\Fstarper}{\Fstarmean} \right),
\end{equation}
is plotted over the light curve in Fig.~\ref{fig:light-curve}. It
clearly highlights the signature twice-per-rotation eclipse-like
dimmings of the star first revealed in full by \citet{Hes1976}. As
discussed by T05 and \citet{Tow2008}, these light minima arise when
the two principal magnetospheric clouds (see \S\ref{sec:back}) transit
across and occult the stellar disk.  In contrast to the narrow,
symmetric primary (deeper) minimum, the secondary minimum is broad,
shallow, and shows a distinct asymmetry. This asymmetry, which can
also be seen in the earlier photometry by \citet{Hes1977} and
\citet{OksTow2007}, suggests that the cloud responsible for the
secondary minimum is itself divided into two smaller subclouds, with
the trailing one containing more mass than the leading one \citep[see
  also][]{SmiBoh2007}.

T05 attribute the brightening seen after the secondary eclipse to
photospheric flux variations arising from the inhomogeneous surface
abundance distribution. We briefly discuss future prospects for modeling
these variations in \S\ref{sec:discuss}.

\subsection{Period Measurement} \label{ssec:analysis-period}

\begin{figure}[t]
\includegraphics{fig_period_dist}
\caption{The probability distribution function of synthetic rotation
  periods obtained from the Monte-Carlo simulation. The shaded
  region is centered on the measured period $P = 1.190847\,{\rm d}$
  (shown as the solid vertical line) and encloses 68.2\,\% of the area
  under the curve. The instantaneous period $P = 1.190851$ calculated
  using the T10 ephemeris is shown as the dotted vertical
  line.} \label{fig:period-dist}
\end{figure}

\begin{table}[ht]
\begin{center}
\begin{tabular}{cccc}
$E$ & \tobs\ (HJD) & \tcalc\ (HJD) & $\tobs-\tcalc$ (d) \\ \hline
9774 & $2454418.0657\pm0.0009$ & 2454418.070 & -0.0044 \\
9775 & $2454419.2553\pm0.0006$ & 2454419.261 & -0.0056 \\
9776 & $2454420.4467\pm0.0008$ & 2454420.452 & -0.0051 \\
9777 & $2454421.6375\pm0.0008$ & 2454421.643 & -0.0051 \\
9778 & $2454422.8282\pm0.0007$ & 2454422.833 & -0.0053 \\
9779 & $2454424.0195\pm0.0009$ & 2454424.024 & -0.0048 \\
9780 & $2454425.2081\pm0.0008$ & 2454425.215 & -0.0071 \\
9781 & $2454426.3984\pm0.0007$ & 2454426.406 & -0.0076 \\
9782 & $2454427.5920\pm0.0006$ & 2454427.597 & -0.0049 \\
9783 & $2454428.7825\pm0.0009$ & 2454428.788 & -0.0052 \\
9784 & $2454429.9728\pm0.0009$ & 2454429.979 & -0.0058 \\
9785 & $2454431.1638\pm0.0010$ & 2454431.169 & -0.0057 \\
9786 & $2454432.3544\pm0.0009$ & 2454432.360 & -0.0059 \\
9787 & $2454433.5453\pm0.0010$ & 2454433.551 & -0.0058 \\
9788 & $2454434.7355\pm0.0010$ & 2454434.742 & -0.0065 \\
9789 & $2454435.9276\pm0.0008$ & 2454435.933 & -0.0053 \\
9790 & $2454437.1165\pm0.0007$ & 2454437.124 & -0.0071 \\
\end{tabular}
\end{center}
\caption{Observed (\tobs) and predicted (\tcalc) times of primary
  light minima, labeled by their cycle number $E$ (see T10), together
  with their differences.} \label{tab:minima}
\end{table}

The rotation frequency found by minimizing \errper\ is $\freq =
0.839739\pm0.000010\,{\rm d^{-1}}$, which corresponds to a rotation
period $P = 1.190847\pm0.000015\,{\rm d}$. The quoted uncertainties
are calculated using bootstrap Monte-Carlo simulations
\citep[e.g.,][]{Pre1992}. Random sampling with replacement is applied
to the $(\phasei,\Fstari-\Fstarmean)$ data, to construct $10^{7}$
synthetic light curves; for each of these \freq\ is determined as
before by minimizing \errper. Fig.~\ref{fig:period-dist} plots the
probability distribution function ${\rm d}p/{\rm d}P$ of the resulting
synthetic rotation periods. The shaded region is constructed to be
symmetric about the measured period and enclose 68.2\%\footnote{Chosen
  as the extent of the 1-$\sigma$ confidence interval for a Gaussian
  distribution; but note that the distribution shown in
  Fig.~\ref{fig:period-dist} is only approximately Gaussian.}  of the
area under the curve. The width of this region is $\pm0.000015\,{\rm
  d}$, which is the basis for the quoted uncertainty in $P$.

The ephemeris constructed by T10, which accounts for the braking of
the star's rotation, predicts an instantaneous period $P = 1.190851
\pm 0.000003\,{\rm d}$ at the time of the \MOST\ observations; as
illustrated in Fig.~\ref{fig:period-dist} this is in excellent
agreement with the period reported here. A further check on the
ephemeris comes from comparing \tobs\ and \tcalc, the observed and
calculated (T10, their eqn.~5) times of primary minima in the light
curve, respectively. Table~\ref{tab:minima} summarizes these timing
data, with the \tobs\ values measured using the parabolic minimum
fitting described by T10. The calculated times clearly lag the
observed times, with the difference $\tobs - \tcalc$ having a mean
$-0.0057\,{\rm d}$ and a scatter wholly consistent with the tabulated
\tobs\ measurement errors. This lag may arise from the uncertainties
in the ephemeris, but it could also be a manifestation of the
wavelength-dependent timing effects discussed in \S3.4 of T10, because
the \MOST\ passband \citep[see][their Fig.~4]{Wal2003} is redder than
the Johnson $U$ passband used in the T10 observations. Either way,
however, the lag amounts to only $\sim 0.005$ of the rotation period;
and therefore this good agreement between observed and calculated
minima times serves as additional, strong confirmation of the T10
ephemeris.

\subsection{Residual Flux} \label{ssec:analysis-residual}

\begin{figure*}[ht]
\includegraphics{fig_residuals}
\caption{The residual stellar flux \mstarres\ in magnitudes relative
  to the mean flux, plotted as a function of time. The solid curve
  beneath (shifted down by $0.025\,{\rm mag}$ for clarity) shows the
  $\pm0.1\,{\rm d}$ boxcar mean curve, with the gray envelope
  illustrating the associated one-standard-deviation bounds. For
  comparison, the dashed curve shows the corresponding smoothed light
  curve of HD~37744.} \label{fig:residuals}
\end{figure*}

\begin{figure*}[ht]
\includegraphics{fig_periodogram}
\caption{The Lomb-Scargle periodogram \pgram\ of the residual flux
  \Fstarres, plotted as a function of frequency \freq. Vertical tick
  marks indicate the $14.2\,{\rm d^{-1}}$ orbital signal and its
  harmonics (thick) and the $1\,{\rm d^{-1}}$ aliases of these peaks
  (thin). The horizontal dashed line indicates the $\pgramc = 13.1$
  threshold corresponding to a false-alarm probability $p(\pgram >
  \pgramc) = 0.05$.} \label{fig:periodogram}
\end{figure*}

Figure~\ref{fig:residuals} plots the residual component of the stellar
flux, defined in magnitudes as
\begin{equation}
\mstarres = -2.5 \log \left(1 + \frac{\Fstarres}{\Fstarmean} \right).
\end{equation}
A $\pm0.1\,{\rm d}$ boxcar mean curve, together with the associated
one-standard-deviation bounds, is plotted below the points to
highlight long-term trends in the data. This smoothed curve clearly
reveals an abrupt dimming by about $0.0035\,{\rm mag}$ near the
mid-point of the observations ($t - 2454416.5 \approx 11$), together
with a reduction in the standard deviation. Also visible in the curve
is a low-level ripple with a frequency $\sim 1-2\,{\rm d^{-1}}$.
However, the corresponding smoothed light curve of HD~37744 (also
shown in the figure) reveals similar behavior in both respects; hence,
neither the dimming nor the ripple can be intrinsic to \sorie. The
effects are likely instrumental in origin; the dimming in particular
is correlated with a sharp $5\,{\rm K}$ increase in the temperature of
the CCD pre-amplifier, due to the increased solar heating which
occurred when \MOST\ switched to continuous observation of
\sorie\ (cf. \S\ref{ssec:obs-reduce-over}).

Apart from these instrumental variations, the smoothed curve in
Fig.~\ref{fig:residuals} is relatively devoid of features. In
particular, there are no obvious flares characterized by a sudden
brightening of the star followed by a slow decline. One interpretation
of this result is that there were no centrifugal breakout episodes
during the \MOST\ run, since any breakout would be accompanied by a
large release of magnetic energy. A caveat, however, is that although
a link between magnetic reconnection and optical flaring has been
established in other types of system (e.g., weak-line T Tauri stars
--- \citealp{Fer2004}; M dwarfs --- \citealp{Ste2006}), the same
cannot be said for the centrifugally supported magnetospheres
considered here. The MHD breakout simulations by \citet{udD2006}
cannot offer much guidance, since they lack the necessary microphysics
to determine the energy yield across the electromagnetic spectrum.

To uncover any low-amplitude periodic signals present in the residual
flux, Fig.~\ref{fig:periodogram} plots the Lomb-Scargle periodogram
$\pgram(\freq)$ of \Fstarres\ for frequencies extending up to
$100\,{\rm d^{-1}}$. Also shown is the critical threshold $\pgramc =
13.1$ corresponding to a false-alarm probability $p(\pgram>\pgramc) =
0.05$ \citep[see][their \S13.8]{Pre1992}. Peaks rising above this
threshold can be seen at $\freq = 14.2\,{\rm d^{-1}}$ and its
harmonics, along with $\pm 1,2,3,\ldots\,{\rm d^{-1}}$ beat
frequencies. Recognizing the $14.2\,{\rm d^{-1}}$ signal as the
orbital frequency of \MOST, these peaks can be attributed to the gaps
in the data for the first half of the observing, and/or imperfect
subtraction of the background flux. The beat frequencies arise because
the Earthshine contribution to the stray light (primarily from the
high-albedo polar regions) are modulated on a 1-day period
\citep{Ree2006}. We note that if we bypass the iterative background
flux estimation procedure described in
\S\ref{ssec:obs-reduce-fback-est}, and instead assume $\Fbacki =
\Fskyi$, then the periodogram peaks discussed here are a factor-10
higher. This serves as \emph{a posteriori} justification for adopting
the iterative procedure, even though it is more complex and
time-consuming.

The periodogram also reveals power at $\sim 1\,{\rm d^{-1}}$ and $\sim
2\,{\rm d^{-1}}$ due to the instrumental ripple, and at low
frequencies ($< 1\,{\rm d^{-1}}$) presumably due to the instrumental
dimming. However, no other peaks rise above the false-alarm threshold
\pgramc, strongly suggesting that \sorie\ exhibits no periodic light
variations other than the rotational modulation.

\subsection{Magnetospheric Evolution} \label{ssec:analysis-minima}

\begin{figure}[ht]
\includegraphics{fig_minima}
\caption{Depths of primary (\Dmstarp; filled symbols) and secondary
  (\Dmstars; open symbols) light minima, plotted in magnitudes as a
  function of time.  The \Dmstars\ data have been shifted vertically
  to facilitate comparison between the two; the missing data point
  comes from the corresponding gap in the light curve on November 22
  (cf.~Fig.~\ref{fig:light-curve}).} \label{fig:minima}
\end{figure}

To examine any changes in the magnetospheric mass during the observing
run (as expected, for instance, due to centrifugal breakout episodes),
Fig.~\ref{fig:minima} plots the relative depths of the primary and
secondary minima as a function of time. These are defined as
\begin{equation} \label{eqn:minima}
\Dmstarp = \mstarp - \mstarb, \qquad \Dmstars = \mstars - \mstarb,
\end{equation}
respectively, where \mstarp\ and \mstars\ are the stellar flux in
magnitudes at the primary and secondary light minima, determined from
the parabolic fitting discussed in Section~\ref{ssec:analysis-period};
while \mstarb\ is the mean flux between each pair of minima, averaged
over the phase interval 0.15 -- 0.25. Although the figure reveals
changes in \Dmstarp\ and \Dmstars\ at a level $\sim 0.002\,{\rm mag}$
which exceeds the error bars, there are no abrupt and ongoing
reductions in the depths which might signify the loss of material from
the magnetospheric clouds. Indeed, the changes appear more consistent
with the instrumental variations mentioned in the previous section.

%%

\section{Discussion} \label{sec:discuss}

\begin{table}
\begin{tabular}{cccccc}
$M_{\ast}$ (\Msun) & $R_{\ast}$ (\Rsun) & $\Omega$ ( \Ocrit) & $B_{\ast}$ (kG) & $\beta$ ($\degr$) & $\epsilon_{\ast}$ \\ \hline
8.30 & 3.77 & 0.454 & 11.0 & 55 & $10^{-3}$
\end{tabular}
\caption{Parameters adopted in calculating the inferred magnetosphere
  mass \Mmag\ and asymptotic magnetosphere mass \Masymp\ of
  \sorie.} \label{tab:params}
\end{table}

An important outcome from the \MOST\ observations is the measurement
of the rotation period of \sorie\ with a precision sufficient to test
the T10 ephemeris. The good agreement found between observed and
calculated periods, and likewise for the timing of the light minima,
validates the ephemeris and moreover confirms that the magnetic
braking of \sorie\ is steady rather than episodic. This is in contrast
to the `glitchy' spindown seen in some Ap stars \citep[e.g., CU Vir;
  see][and references therein]{Mik2011}, but agrees with the MHD
simulations by \citet{udD2009} which suggest that angular momentum
loss from massive-star magnetospheres is a smooth process.

Another noteworthy result is the failure to find any evidence for
centrifugal breakout episodes, either in the form of optical flares in
the residual flux (\S\ref{ssec:analysis-residual}) or as systematic
reductions in the depths of the light minima
(\S\ref{ssec:analysis-minima}). This could be due simply to unlucky
scheduling of the \MOST\ run, coupled with the fact that the breakout
recurrence timescale is poorly constrained (as it depends on the
unknown wind mass-loss rate). However, there are a number of
independent arguments which favor the alternative conclusion that
centrifugal breakout simply \emph{does not occur} in \sorie, at least
at a level where it has any impact on the magnetospheric mass budget.

Foremost amongst these is the discrepancy between the magnetospheric
mass \Mmag\ inferred from the observations together with the RRM
model, and the asymptotic magnetosphere mass \Masymp\ predicted by the
breakout analysis of \citet[][their Appendix A2]{TowOwo2005}; if
centrifugal breakout plays a role in governing the magnetospheric mass
budget, then these two values should be
comparable. Table~\ref{tab:params} lists the stellar and magnetosphere
parameters adopted here to evaluate \Mmag\ and \Masymp; the field
strength $B_{\ast}$, magnetic obliquity $\beta$ and magnetosphere
scale-height parameter $\epsilon_{\ast}$ are taken from T05, while the
other parameters are derived in Appendix~\ref{app:params}. Applying
the light-curve synthesis procedure described by T05, the RRM model
requires $\rhomax \kappa R \approx 7$ to reproduce the observed depth
$\Dmstarp \approx 0.065$ of the primarily light minima
(cf.~Fig.~\ref{fig:minima}), with \rhomax\ being the maximum mass
density in the magnetosphere, $\kappa$ the flux-mean opacity in the
\MOST\ passband, and $R$ the stellar radius. A lower limit on the
opacity is given by the electron scattering value, $\kappa_{\rm es} =
0.34\,{\rm cm^{2}\,g^{-1}}$ for a fully ionized solar-abundance
composition. With $R = 3.77\,\Rsun$ from Table~\ref{tab:params}, we
therefore obtain an upper limit $\rhomax \lesssim 8 \times
10^{-11}\,{\rm g\,cm^{-3}}$ on the maximum density\footnote{This is a
  very conservative estimate; using the Inglis-Teller effect, both
  \citet{GroHun1982} and \citet{SmiBoh2007} derive a bound $n_{\rm e}
  \lesssim 2 \times 10^{12}\,{\rm cm^{-3}}$ on the electron number
  density, which corresponds to the tighter upper limit $\rho \lesssim
  4 \times 10^{-12}\,{\rm g\,cm^{-3}}$.}. Integrating over the RRM
density distribution leads to a corresponding upper mass limit $\Mmag
\lesssim 2 \times 10^{-10}\,\Msun$. This is almost two orders of
magnitude smaller than the asymptotic mass $\Masymp = 1.2 \times
10^{-8}\,\Msun$ predicted by the breakout analysis, indicating that
the magnetosphere is well short of the level required for significant
breakout episodes to occur.

Further corroborating arguments are presented in a forthcoming paper
(Townsend et al., in preparation), which demonstrates that the low-mass
companion discovered by \citet{Bou2009} is responsible for 70\% of the
X-ray flux from the \sorie\ system. It seems likely that the X-ray
flares believed originally to arise during centrifugal breakout (see
\S\ref{sec:back}) instead come from the magnetic activity of the
companion.

These findings are significant because they challenge the prevailing
narrative for mass leakage from centrifugally supported
magnetospheres. It is natural to now ask what other leakage
mechanism(s) might be at work to balance the continual feeding of
plasma from the wind, as evidenced by the star's rotationally
modulated UV absorption lines \citep{ShoBro1990}. \citet{HavGoe1984}
explore cross-field diffusive processes such as ambipolar diffusion,
but find them far too slow to be effective; revisiting their
calculations with updated stellar parameters does not change this
conclusion. A related question concerns the process(es) responsible
for magnetospheric features not predicted by the RRM model --- for
instance, the subclouds seen in the secondary light minima
(cf. Fig.~\ref{fig:light-curve}). Corresponding departures from the
RRM model have been noted in a number of other He-strong stars
harboring magnetospheres (in particular, those mentioned in
\S\ref{sec:back}). Are these a consequence of the as-yet-unidentified
mass-leakage mechanism, or instead due to departures from a
pure-dipole field topology? In the case of \sorie, recent
spectropolarimetric measurements by \citet{Oks2012} indeed reveal
departures from dipolarity, although these are not consistent with the
decentered dipole invoked by T05 to explain the overall difference in
the depths of the primary and secondary light minima. Clearly, there
remains much work to be done in understanding the effects of mass
redistribution, mass leakage and field topology in governing the
distribution and overall amount of plasma in these stars'
magnetospheres.

Looking now to the future, a logical next step is to decompose the
\MOST\ light curve into magnetospheric and photospheric
components. This will allow quantitative testing of the assumption
(see \S\ref{ssec:analysis-periodic}) that the brightening after the
secondary light minimum is due to the star's inhomogeneous surface
abundance distribution. For other He-strong stars
\citet{Krt2007,Krt2011} have successfully used surface abundance maps
derived from Doppler imaging to reproduce the stars' light
variations. With abundance maps now becoming available for
\sorie\ \citep[see][]{Oks2012}, it should be possible to apply the
same technique to disentangle the magnetospheric and photospheric
components of the light curve. The magnetospheric component can then
be compared against the light-curve morphologies predicted by the RRM
model \citep[see][]{Tow2008}; with a good match between theory and
observation, further constraints can be placed on parameters such as
the stellar inclination, magnetic obliquity and magnetospheric mass.

\section{Summary} \label{sec:summary}

We have undertaken a careful reduction and analysis of the
\MOST\ observations of \sorie. A key result is that the \MOST\ data
provide independent confirmation (\S\ref{ssec:analysis-period}) of the
ephemeris published by \citet{Tow2010}, confirming that magnetic
braking is occurring in \sorie\ and that this braking is a steady
process.

An equally important result is the absence of any evidence for
centrifugal breakout episodes, either in the residual flux
(\S\ref{ssec:analysis-residual}) or in the depths of the primary light
minima (\S\ref{ssec:analysis-minima}). Moreover, the observationally
inferred magnetospheric mass (\S\ref{sec:discuss}) is around two
orders of magnitude below the asymptotic mass predicted by the
breakout analysis of \citet{TowOwo2005}. These findings lead us to
rule out centrifugal breakout as a mechanism for magnetospheric mass
leakage in \sorie.

%% Acknowledgements

\acknowledgements RHDT acknowledges support from NSF awards
AST-0908688 and AST-0904607, and NASA award NNX12AC72G. AFJM, DBG, JMM
and SMR are grateful for financial support from NSERC (Canada).

\appendix

\section{Fundamental Parameters of \sorie} \label{app:params}

\citet{GroHun1982} determine an effective temperature $\Teff =
22\,500\,{\rm K}$ for \sorie\ by fitting the spectral energy
distribution from UV through to IR. They likewise derive a surface
gravity $\logg = 3.85\,{\rm dex}$ from modeling H and He equivalent
widths. A subsequent more-detailed analysis of Balmer-line wings led
\citet{Hun1989} to revise this value slightly upwards, to $\logg =
3.95\,{\rm dex}$. As discussed by these latter authors, the \Teff\ and
\logg\ together imply that \sorie\ is more distant ($\sim 650\,{\rm
  pc}$) than the $\sigma$~Ori cluster ($\sim 450\,{\rm pc}$), and is
moreover a factor $\sim 10$ older than the cluster. These findings,
however, stand contrary to a number of observational results
indicating that \sorie\ is a bona fide member of the cluster rather
than a background star. The reddening of \sorie\ is the same as
\soriab\ \citep{She2008}, and likewise for interstellar polarization
\citep{KemHer1977}. The radial velocity and proper motion of
\sorie\ are indistinguishable from those of the cluster
\citep{Cab2007}. Finally, the spindown measurements by T10 indicate
that the star is young, with an age $\sim 1.1\,{\rm Myr}$ consistent
with lower-end age estimates for the cluster.

The problem with the \citet{Hun1989} analysis likely resides in the
surface gravity determination. Emission from magnetospheric plasma
fills in the wings of Balmer lines; if not properly corrected this
makes the lines appear less broad, and the gravity consequently
smaller, than is actually the case. Given this complication it seems
better to avoid the gravity measurement altogether, and derive stellar
parameters using a different approach. Accordingly, assuming
\sorie\ is a cluster member, a radius $R = 3.77\,\Rsun$ follows from
the angular diameter $\theta = 0.079\,{\rm mas}$ \citep{GroHun1982}
and the cluster distance $d = 444\,{\rm pc}$ derived for solar
metallicity by \citet{She2008}.

To obtain the corresponding mass, we calculate a sequence of
solar-metallicity evolutionary tracks with masses $M = 7, 7.1, 7.2,
\ldots, 9.9, 10\,\Msun$ using the MESA stellar evolution code
\citep{Pax2011}. For simplicity the calculations neglect the effects
of rotation. The $M=8.3\,\Msun$ track passes closest to $\Teff =
22\,500\,{\rm K}$, $R = 3.77\,\Rsun$ point, and we adopt this as the
stellar mass. With the observed rotation period
(\S\ref{ssec:analysis-period}) the dimensionless angular velocity is
$\omega = \Omega/\Ocrit = 0.454$, where $\Omega \equiv 2\pi\freq$ and
$\Ocrit = \sqrt{8 G M/27 R^{3}}$ is the critical angular velocity.

%% Bibliography

\bibliographystyle{apj}
\bibliography{sigorie-most}

\end{document}
