# Modeling the Optical to Ultraviolet Polarimetric Variability from Thomson
Scattering in Colliding Wind Binaries

Richard Ignace  Department of Physics & Astronomy, East Tennessee State
University, Johnson City, TN 37614, USA  Andrew Fullard  Department of Physics
and Astronomy, Michigan State University, East Lansing, MI 48824, USA  Manisha
Shrestha  Astrophysics Research Institute, Liverpool John Moores University,
Liverpool Science Park, 146 Brownlow Hill,
Liverpool L3 5RF, UK  Yaël Nazé11affiliation: FNRS Senior Research associate
GAPHE, UR STAR, Université de Liège, Allée du 6 Août 19c (B5C), B-4000 Sart
Tilman, Liège, Belgium  Kenneth Gayley  Department of Physics & Astronomy,
University of Iowa, Iowa City, IA 52242, USA  Jennifer L. Hoffman  Department
of Physics & Astronomy, University of Denver, 2112 E. Wesley Ave., Denver, CO
80208, USA  Jamie R. Lomax  Department of Physics, United States Naval
Academy, 572C Holloway Rd., Annapolis, MD 21402, USA  Nicole St-Louis
Département de physique, Université de Montréal, Complexe des Sciences, 1375
Avenue Thérèse-Lavoie-Roux, Montréal (Qc), H2V 0B3, Canada

###### Abstract

Massive star binaries are critical laboratories for measuring masses and
stellar wind mass-loss rates. A major challenge is inferring viewing
inclination and extracting information about the colliding wind interaction
(CWI) region. Polarimetric variability from electron scattering in the highly
ionized winds provides important diagnostic information about system geometry.
We combine for the first time the well-known generalized treatment of Brown et
al. (1978) for variable polarization from binaries with the semi-analytic
solution for the geometry and surface density CWI shock interface between the
winds based on Cantó et al. (1996). Our calculations include some
simplifications in the form of inverse square-law wind densities and the
assumption of axisymmetry, but in so doing arrive at several robust
conclusions. One is that when the winds are nearly equal (e.g., O + O
binaries), the polarization has a relatively mild decline with binary
separation. Another is that despite Thomson scattering being a gray opacity,
the continuum polarization can show chromatic effects at ultraviolet
wavelengths but will be mostly constant at longer wavelengths. Finally, when
one wind dominates the other, as for example in WR + OB binaries, the
polarization is expected to be larger at wavelengths where the OB component is
more luminous, and generally smaller at wavelengths where the WR component is
more luminous. This behavior arises because from the perspective of the WR
star, the distortion of the scattering envelope from spherical is a minor
perturbation situated far from the WR star. By contrast, the polarization
contribution from the OB star is dominated by the geometry of the CWI shock.

Spectropolarimetry — Binary Stars — Stellar winds — Massive stars — Wolf-Rayet
stars — Shocks

##  1 Introduction

Despite comprising the rarest stellar mass group, massive stars ($>$ 8 M⊙) are
the most important originators of elements in the Universe because they
synthesize and distribute heavy elements when they explode as supernovae
(Nomoto et al., 2013). Massive stars also enrich the interstellar medium
during their pre-supernova lifetime via their strong stellar winds. High
levels of mass loss also affects the evolution of massive stars, in particular
the nature of their remnants (Puls et al., 2008; Langer, 2012; Smith, 2014).

Most massive stars spend a large fraction of their lives in binary systems
with other massive stars; approximately 50% are thought to engage in mass
exchange with a close companion (Sana et al., 2012). Interactions between
companions drive the evolutionary paths that can shape both stars’ subsequent
fates (e.g., Langer, 2012; Song et al., 2016).

Colliding-wind binaries can teach us a great deal about the individual stars
and their winds because the geometry of the interaction region is dependent on
the relative mass-loss rates and velocities of the binary components. Early
theories describing these binaries used momentum flux (e.g. Girard & Willson,
1987) and ram-pressure balance (e.g. Kallrath, 1991), or hydrodynamic models
(e.g. Stevens et al., 1992), and even some semi-analytic work (e.g. Pilyugin &
Usov, 2007; Gayley, 2009; Usov, 1992; Cantó et al., 1996). Further work in the
area has focused on hydrodynamic simulations (Parkin & Pittard, 2008; Lamberts
et al., 2011; MacLeod & Loeb, 2020) and predictions of line profiles in both
optical (Luehrs, 1997; Georgiev & Koenigsberger, 2004; Ignace et al., 2009)
and X-rays (Henley et al., 2003; Rauw et al., 2016; Mossoux & Rauw, 2021). The
result of these models, combined with a number of phase-resolved observations,
is that we have generally a good understanding of the expected geometry of
colliding winds (e.g. Rauw et al., 1999; Gosset et al., 2001; Sana et al.,
2004; Gosset et al., 2009; Williams et al., 2009; Kennedy et al., 2010; Fahed
et al., 2011; Nazé et al., 2012; Cazorla et al., 2014; Rauw et al., 2014;
Lomax et al., 2015; Gosset & Nazé, 2016; Nazé et al., 2018; Callingham et al.,
2020; Rodríguez et al., 2020).

Given that massive star winds are strongly ionized, it is natural to consider
Thomson scattering as the dominant scattering opacity in the winds, which in
turn can polarize the observed light. The resulting polarization is sensitive
to the geometry of the scattering regions. The classic Brown et al. (1978,
hereafter BME) model approximates the time-varying continuum polarization
caused by the illumination of circumstellar material in a binary system viewed
at an arbitrary inclination angle. Those authors assume the electron
scattering region is optically thin. Their approach allows for a general
geometry, but for binary stars they consider two point sources of illumination
and a co-rotating scattering envelope.

Brown et al. (1982) extended the BME model to consider elliptical orbits, and
Fox (1994) further extended the formalism to consider finite-size
illuminators. Fox (1994) showed that occultation is only important in very
close binary systems, where the separation of stars is less than 10 times the
radius of the primary. However, none of these enhancements to the theory
specifically addressed the effects of colliding winds in the time-dependent
polarization results. Furthermore, the effects of the wind collision regions
on the wavelength dependence of polarization have not been considered as part
of this theoretical framework. However, such polarized signals associated with
colliding winds have been observed in several systems (St.-Louis et al., 1993;
Lomax et al., 2015). A modelling effort in this domain is therefore critically
important.

Polarization models of stellar wind bow shock structures produced by the
interaction of stellar winds with a local ambient medium shows that
significant polarization can arise from scattering of light in these
structures (Shrestha et al., 2018, 2021). Modeling the polarization signal
caused by colliding wind geometries has been done specifically for the system
V444 Cyg (St.-Louis et al., 1993; Kurosawa et al., 2002), but a general
formalism has not yet been produced. In this paper we derive a consistent
model for the polarization signal produced by wind collision regions in
massive binary systems. In Section 2 we describe our model of polarimetric
variability from optically thin electron scattering in a shock illuminated by
two stars. In Section 3 we derive expressions for the polarization signal
based on the system parameters, and show how these expressions lead to
chromatic and orbital effects in polarization. We summarize our results in
Section 4.

##  2 Polarimetric Variability from Thin Electron Scattering

Our treatment is based on that of BME, who presented a thorough theoretical
construction for thin electron scattering in a generalized envelope with an
arbitrary number of illuminating point sources. In regards to our application
for a binary system, we assume the colliding wind interaction (CWI) is
axisymmetric about the line of centers (LOC) joining the two stars (i.e., we
ignore the Coriolis effect in our example cases, though this has been detected
in at least one colliding-wind binary; Lomax et al. 2015). We further assume
that the separate winds of the two stars are each spherically symmetric up to
the CWI. As a result, in the notation of BME, we have
$\gamma_{1}=\gamma_{2}=\gamma_{3}=\gamma_{4}=0$, and the only factors that are
nonzero are $\gamma_{0}$ and $\tau_{0}$. In our development we will modify
this notation slightly.

BME then considered the more limited scenario of a binary system with a
circular orbit and corotating envelope. Our approach allows for elliptical
orbits (which Brown et al. 1982 later considered), and for the shape and
density of the bowshock, we employ the analytical solution of Cantó et al.
(1996).

The Cantó et al. (1996) solution is predicated on strong radiative cooling.
There are two initial concerns about adopting this model. The first is that
radiative cooling leads to thin shell instabilities (e.g., Lamberts et al.,
2011). The second is that radiative cooling is limited to relatively close
binaries, orbital periods of order a week (e.g., Antokhin et al., 2004). These
issues have significant relevance for predicting X-ray spectra, where the
temperature distirbution along the shock is important; or when simulating
emission line profile shapes, where the detailed vector velocity field is
crucial. However, our case deals with electron scattering and continuum
polarization. Unlike the case of X-ray diagnostics where the distribution of
hot gas is important, we can safely assume the gas is everywhere highly
ionized for computing scattering polarization. Of chief importance to our case
is that the Cantó et al. (1996) derivation is conveniently analytic and driven
by considerations of ram pressure balance which captures much of the key
physics. Most, but not all, of our examples involve either equal winds or
binaries with one dominant wind, and our general conclusions based on the
Cantó et al. (1996) model are fairly robust.

As in the BME formalism, our approach assumes point source illumination. This
is reasonable when the binary separation is not too small (of order the
stellar radii). One distinction, however, is that we do account for finite
stellar size when evaluating volume integrals. This is not incompatible with
BME; we are merely explicit about its inclusion.

A final point about our use of the BME treatment is that due to the
axisymmetry, we employ the notation and approach of BME for a single star and
use superposition in our application to a binary system. In doing so, our
notation departs from BME, although we still employ similar variables. The
following sections introduce our geometry for the systems, our application of
BME to axisymmetric binaries, and semi-analytic solutions for the variable
polarization based on the Cantó et al. (1996) solution for the CWI.

###  2.1 Geometry and Stellar Properties

In our model, the winds of the two stars and the intervening colliding wind
shock are prescribed using primarily polar coordinates for each star. We
define the primary star to be the one with the stronger wind in terms of
momentum flux, $\dot{M}v_{\infty}$, where $\dot{M}$ is the mass-loss rate and
$v_{\infty}$ is the terminal wind speed. The secondary is then the weaker wind
case in terms of this product. We typically use subscripts “1” and “2” to
identify primary and secondary.

We introduce spherical coordinates with respect to the axis that is the line
of centers between the two stars. Coordinates centered on the primary star
$(r_{1},\theta_{1},\phi_{1})$ are such that $\theta_{1}=0$ in the direction of
the secondary. Likewise, the coordinates linked to the secondary star
$(r_{2},\theta_{2},\phi_{2})$ also have $\theta_{2}=0$ in the direction of the
primary. Frequently, our approach employs the standard cosine notation,
$\mu_{1}=\cos\theta_{1}$ and $\mu_{2}=\cos\theta_{2}$.

The individual winds are taken to be spherical with densities varying with the
inverse square of the distance from the star. Consequently, we are ignoring
the wind acceleration zone that is relevant whenever the bow shock forms close
to either or both stars. We do not include the potential for radiative braking
of the stronger wind if it enters the region of dominance of the companion
stellar flux (Gayley et al., 1997; Lomax et al., 2015), nor the possibility
that the stronger wind might in some cases ram directly into the photosphere
of the secondary star.

With spherical winds and constant speed radial outflow, the density relations
for the primary and secondary winds are:

$\displaystyle\rho_{1}$ $\displaystyle=$
$\displaystyle\frac{\dot{M}_{1}}{4\pi\,r_{1}^{2}\,v_{1}}=\rho_{1,0}\,\left(\frac{R_{1,\ast}}{r_{1}}\right)^{2},$
(1) $\displaystyle\rho_{2}$ $\displaystyle=$
$\displaystyle\frac{\dot{M}_{2}}{4\pi\,r_{2}^{2}\,v_{2}}=\rho_{2,0}\,\left(\frac{R_{2,\ast}}{r_{2}}\right)^{2},~{}{\rm
and}$ (2) $\displaystyle n_{1}$ $\displaystyle=$ $\displaystyle
n_{1,0}\,\left(\frac{R_{1,\ast}}{r_{1}}\right)^{2},$ (3) $\displaystyle n_{2}$
$\displaystyle=$ $\displaystyle
n_{2,0}\,\left(\frac{R_{2,\ast}}{r_{2}}\right)^{2},$ (4)

where $n_{\rm j}$ for $j=1,2$ are electron number densities with corresponding
scale parameter $n_{\rm j,0}=\rho_{0}/\mu_{\rm e}m_{H}$, for $m_{H}$ the mass
of hydrogen and $\mu_{\rm e}$ the mean molecular weight per free electron.
Since we consider terminal speed flow, we have omitted the subscript $\infty$
for the primary and secondary wind speeds in equations (1) and (2).

###  2.2 Bow Shock Model

Figure 1: Top-down illustration of the two stars, primary and secondary in
magenta and the bowshock, in blue, formed by the colliding winds. Labeled
variables are defined in text.

To explore polarimetric variability from colliding wind shocks, we choose the
formulation of Cantó et al. (1996) for their semi-analytic bow shock solution
involving radiative cooling and two stellar spherical winds at terminal speed.
This solution specifically assumes radiative cooling, and the bow shock takes
the form of an interface of infinitesimal thickness between the two otherwise
spherical winds. This interface for the CWI is axisymmetric and characterized
with a surface density.

Here we reproduce in brief the key expressions for the solution of Cantó et
al. (1996), but with a few changes of notation. Cantó et al. (1996) use a “1”
subscript for the primary, but no subscript for the secondary; we use a “2”
subscript for the secondary. Cantó et al. (1996) use $\sigma$ for the surface
mass density, while we use $\Sigma=\sigma/\mu_{\rm e}m_{H}$ for the surface
number density of electrons. In principle $\mu_{\rm e}$ is a function of
coordinate in either wind and in the bow shock itself; however, for
simplicity, we assume $\mu_{\rm e}$ is constant throughout the shock.

The bow shock geometry and its surface density are related to two fundamental
ratios:

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{\dot{M}_{2}v_{2}}{\dot{M}_{1}v_{1}},~{}{\rm and}$ (5)
$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle v_{2}/v_{1}.$ (6)

The first of these, $\beta\leq 1$, is the ratio of wind momentum of the
secondary compared to the primary; the second , $\alpha$, is the ratio of wind
terminal speeds. Figure 1 provides a schematic of the binary system with
intervening CWI region between the two stars. The radial distances of the bow
shock from the stars are denoted as $R_{1,S}$ and $R_{2,S}$, with

$\displaystyle
R_{2,S}=D\,\frac{\sin\theta_{1}}{\sin(\theta_{1}+\theta_{2})},~{}{\rm and}$
(7) $\displaystyle
R_{1,S}=\sqrt{D^{2}+R^{2}_{2,S}-2\,D\,R_{2,S}\,\cos\theta_{1}},$ (8)

where $D$ is the separation between the two stars at any moment. We denote the
standoff radii of the bowshock from each star along the line of centers as
$R_{1,0}$ and $R_{2,0}$, with

$\displaystyle R_{2,0}=\frac{\beta^{1/2}}{1-\beta^{1/2}}\,D,~{}{\rm and}$ (9)
$\displaystyle R_{1,0}=\frac{1}{1-\beta^{1/2}}\,D.$ (10)

While the above relations are analytic formulations in $\theta_{1}$ and
$\theta_{2}$, the relation between the two coordinate angles is implicit,
making the solution overall semi-analytic. The relationship of the angles is
given by,

$\frac{\theta_{1}}{\tan\theta_{1}}=1+\beta\left(\frac{\theta_{2}}{\tan\theta_{2}}-1\right).$
(11)

Note that the asymptotic angles (“opening angles”) for the bow shock are given
by

$\displaystyle\theta_{2,\infty}-\tan\theta_{2,\infty}=\frac{\pi}{1-\beta},~{}{\rm
and}$ (12) $\displaystyle\theta_{1,\infty}=\pi-\theta_{2,\infty}.$ (13)

The case $\beta=1$ corresponds to a planar shock between identical stars and
winds, with $\theta_{1,\infty}=\theta_{2,\infty}=\pi/2$.

The final key ingredient for modeling the polarimetric variability is the
surface number density distribution. Again from Cantó et al. (1996), this is
given by

$\displaystyle\frac{\Sigma}{\Sigma_{0}}$ $\displaystyle=$
$\displaystyle\frac{\sin(\theta_{1}+\theta_{2})}{\sin\theta_{1}\sin\theta_{2}}\times$
$\displaystyle\left[\beta(1-\cos\theta_{2})+\alpha(1-\cos\theta_{1})\right]^{2}\times$
$\displaystyle\left\\{\left[\beta(\theta_{2}-\sin\theta_{2}\cos\theta_{2}+(\theta_{1}-\sin\theta_{1}\sin\theta_{2})\right]^{2}\right.$
$\displaystyle\left.+\left(\beta\sin^{2}\theta_{2}-\sin^{2}\theta_{1}\right)^{2}\right\\}^{-1/2}$

where the scaling constant is

$\Sigma_{0}=\frac{\dot{M}_{2}/\mu_{\rm
e}m_{H}}{2\pi\beta\,D\,v_{2}}=\frac{2R_{2,\ast}}{\beta\,D}\,R_{2,\ast}n_{2,0}=\alpha\,\frac{2R_{1,\ast}}{D}\,R_{1,\ast}n_{1,0}.$
(15)

The last step in the above expression represents the surface number density as
being twice the column depth of the wind of the secondary from the bow shock
to infinity.

In the special case of $\beta=1$ for a planar shock forming from two identical
stars, $R_{1,S}=R_{2,S}=D/2\mu$, where $\mu=\cos\theta$ with
$\theta=\theta_{1}=\theta_{2}$. Combined with $\alpha=1$, the surface density
then simplifies to

$\frac{\Sigma}{\Sigma_{0}}=4\,\frac{\cos\theta\,(1-\cos\theta)^{2}}{\sin\theta(\theta-\sin\theta\cos\theta)}.$
(16)

Having defined the geometry of the CWI interface and its properties, we turn
next to characterizing the electron scattering polarization.

###  2.3 Thin Scattering

As previously noted, we employ the approach of BME in application to the
results of Cantó et al. (1996), which is explicitly axisymmetric. In the
treatment of BME, ignoring absorption and assuming that the total amount of
scattered light is small compared to the specific luminosities of either star,
the polarization is given by

$p_{\rm tot}=\tau\,(1-3\gamma)\,\sin^{2}i=p_{0}\,\sin^{2}i,$ (17)

where $p_{\rm tot}$ is the polarization, $\tau$ is an angle-averaged optical
depth of the envelope, $\gamma$ is called the shape factor, and $i$ is the
viewing inclination relative to the symmetry axis of the envelope. We
introduce $p_{0}$ for conveniently representing the product of optical depth
and envelope shape. The definitions of $\tau$ and $\gamma$ are

$\tau=\frac{3}{16}\sigma_{T}\,\int\int n(r,\mu)\,dr\,d\mu,$ (18)

and

$\gamma=\frac{\int\int n(r,\mu)\,dr\,\mu^{2}\,d\mu}{\int\int
n(r,\mu)\,dr\,d\mu},$ (19)

where $n(r,\mu)$ is the axisymmetric distribution of electrons throughout the
scattering volume. If the scattering region is spherically symmetric,
$\gamma=1/3$, and the polarization is zero. For a wind whose density varies as
the inverse of the squared distance, $\tau=3\tau_{\ast}/8$, with
$\tau_{\ast}=n_{0}\,\sigma_{T}\,R_{\ast}$ being the radial optical depth of
the wind in electron scattering.

The integrals for $\tau$ and $\gamma$ are defined with respect to the star
center, and the treatment for axisymmetry does not require top-down symmetry.
Thus one can introduce $\tau_{1}$ and $\gamma_{1}$ associated with
polarization from scattering of starlight from the primary, and then
$\tau_{2}$ and $\gamma_{2}$ for scattering of starlight by the secondary.
Being optically thin, the results add linearly as weighted by the wavelength-
dependent luminosities of the two stars:

$p_{0}=\frac{L_{1}(\lambda)\,p_{1}+L_{2}(\lambda)\,p_{2}}{L_{1}(\lambda)+L_{2}(\lambda)}.$
(20)

This result, in our notation, is equivalent to equations (6a) and (7) from BME
for axisymmetry. The dependence on viewing inclination is implicit in equation
(20) via equation (17). As pointed out by BME, we have that $i_{1}=i_{2}=i$ so
that $p_{\rm tot}=p_{0}\sin^{2}i$ as in equation (17) earlier.

Note that for a particular geometry as expressed by the wind and orbital
properties, $p_{1}$ and $p_{2}$ have generally different values but are
defined with respect to the same axis, the LOC between the stars. While these
values are not wavelength-dependent (i.e., chromatic), $p_{0}$ can be
chromatic because the two illuminating sources will generally have different
spectral energy distributions (SEDs).

This last point deserves additional comment. When one star dominates the
brightness of the system in a given wavelength range, the polarization will be
flat and take the polarization value of the dominant star. At wavelengths for
which the emission of both stars follows a Rayleigh-Jeans law, the relative
contribution to the total luminosity of the stars will always be the same,
hence $p_{\rm tot}$ will also be flat. However, for stars of unequal
temperatures, at wavelengths around the Wien peak, the ratio of specific
luminosities will vary, and either $p_{1}$ or $p_{2}$ may dominate, or the
dominant terms may switch. Thus for hot stars, $p_{0}$ will be chromatic at
short wavelengths, despite the fact that electron scattering is gray.

##  3 Model Results

###  3.1 Expressions for the Polarization

Recall that for BME, the polarization depends on source parameters $\tau$ and
$\gamma$, but these in turn depend on angle-averaged column densities of free
electrons. It is convenient to introduce two varieties of angle-averaged
column densities, one that is a zeroth-order moment and one that is a second-
order moment:

$\langle N\rangle=\frac{1}{2}\int\int n(r,\mu)\,dr\,d\mu,$ (21)

and

$\langle\tilde{N}\rangle=\frac{1}{2}\int\int n(r,\mu)\,dr\,\mu^{2}\,d\mu.$
(22)

Then

$\tau=\frac{3}{8}\sigma_{T}\,\langle N\rangle,$ (23)

and

$\gamma=\langle\tilde{N}\rangle/\langle N\rangle,$ (24)

where these parameters would have subscripts 1 or 2 for the primary or
secondary stars, respectively. For example, the primary wind with density
given by equation (3) would have a first-moment column density of $\langle
N_{1}\rangle=n_{1,0}R_{1,\ast}$.

In application to the colliding wind binary, there are 3 key angular regimes
to consider in relation to each of the two stars. These 3 solid angle sectors
contribute to the parameters $\gamma_{1,2}$ and $\tau_{1,2}$. These will be
detailed next in terms of Cases A, B, and C, with reference to Figure 2.

Figure 2 is a copy of Figure 1 in terms of the stars and the CWI (labeled as
“Shock”), now used to emphasize the 3 angular regimes. There is still the LOC
with separation $D$. Using the secondary as example, three rays are labeled
with A2, B2, and C2. Similar rays could be drawn as originating from the
primary, which would then be labeled A1, B1, and C1. Consequently,

$\langle N\rangle=\langle N_{A}\rangle+\langle N_{B}\rangle+\langle
N_{C}\rangle,$ (25)

and

$\langle\tilde{N}\rangle=\langle\tilde{N}_{A}\rangle+\langle\tilde{N}_{B}\rangle+\langle\tilde{N}_{C}\rangle.$
(26)

Figure 2: Similar to Fig. 1, the two stars and colliding wind shock are
illustrated. For calculation of the polarization properties, column density
moments involve three different ray categories, here identified as A2, B2, and
C2 in relation to the secondary star. The dotted line is the line of centers
(LOC) for separation $D$ between the stars. The impact parameter $\varpi_{1}$
is the same as for Fig 1. For Case B2, the contributions associated with I
(path segment), II (point), and III (semi-bound line segment) are identified.

####  3.1.1 Case A: Rays not intersecting the colliding wind shock

For $\theta_{1}>\theta_{1,\infty}$ or $\theta_{2}>\theta_{2,\infty}$, the ray
does not intersect the shock nor the opposite star, but travels strictly
through its own wind. In this case the angle-averaged column densities are
trivial for an inverse square law, with

$\displaystyle\langle N_{A1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{1,0}R_{1,\ast}\,(1+\mu_{1,\infty}),$ (27)
$\displaystyle\langle\tilde{N}_{A1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{6}n_{1,0}R_{1,\ast}\,(1+\mu^{3}_{1,\infty}),$ (28)
$\displaystyle\langle N_{A2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{2,0}R_{2,\ast}\,(1+\mu_{2,\infty}),$ (29)
$\displaystyle\langle\tilde{N}_{A2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{6}n_{2,0}R_{2,\ast}\,(1+\mu^{3}_{2,\infty}).$ (30)

####  3.1.2 Case B: Rays intersecting the shock but not the opposite star

When a ray intersects the shock, there are three distinct contributions to the
relevant angle-averaged column densities: a segment of the star’s own wind,
the surface density at the shock itself, and finally a segment through the
wind of the opposite star. The ray does not intercept the surface of the
opposite star, a case treated in the next section.

We denote these three contributions as I for own wind, II for the shock, and
III for the opposite wind, and use these as subscripts to accompany 1 and 2 to
signify whether the rays originate from the primary or the secondary star.
Note that the angular integrations are affected by the ray missing or
intercepting the opposite star.

The first segment is the column within the star’s own wind from its surface to
the shock. The columns are:

$\displaystyle\langle N_{B1,I}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{1,0}R_{1,\ast}\,\int_{\mu_{1,\infty}}^{\mu_{2,\ast}}\,\left[1-\frac{R_{1,\ast}}{R_{1,S}(\mu)}\right]\,d\mu,$
(31) $\displaystyle\langle N_{B2,I}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{2,0}R_{2,\ast}\,\int_{\mu_{2,\infty}}^{\mu_{1,\ast}}\,\left[1-\frac{R_{2,\ast}}{R_{2,S}(\mu)}\right]\,d\mu,$
(32) $\displaystyle\langle\tilde{N}_{B1,I}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{1,0}R_{1,\ast}\,\int_{\mu_{1,\infty}}^{\mu_{2,\ast}}\,\left[1-\frac{R_{1,\ast}}{R_{1,S}(\mu)}\right]\,\mu^{2}\,d\mu,$
(33) $\displaystyle\langle\tilde{N}_{B2,I}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{2,0}R_{2,\ast}\,\int_{\mu_{2,\infty}}^{\mu_{1,\ast}}\,\left[1-\frac{R_{2,\ast}}{R_{2,S}(\mu)}\right]\,\mu^{2}\,d\mu,$
(34)

where $\mu_{1,\ast}=\cos\theta_{1,\ast}=\sqrt{1-(R_{1,\ast}/D)^{2}}$ and
$\mu_{2,\ast}=\cos\theta_{2,\ast}=\sqrt{1-(R_{2,\ast}/D)^{2}}$. In the
integrand, the fraction being subtracted represents the missing column in the
star’s own wind that would be present if not for the shock.

Figure 3: Similar to Fig. 2, the two stars and colliding wind shock are
illustrated. The cases A2, B2, and C2 in relation to the secondary star are
indicated. The $z$ coordinate is indicated in this version, with $z_{0}$
labeled. For columns evaluated along radials from the secondary, $z$ is
defined in the wind of the primary; and vice versa.

At the shock itself, the surface density contributes to the column along the
ray, owing to the accumulation of material entering the shock and outflowing
along its surface. However, the kinematics are not of relevance here; only the
surface density itself matters. What is relevant is the projection of the
surface with respect to the ray. The contribution to the column depends on a
direction cosine between the radial unit vector and the local normal to the
surface. We represent these as $K_{1}(\mu)$ and $K_{2}(\mu)$. Omitting the
subscripts, $K$ for each star is given by

$K(\mu)=\frac{1}{\sqrt{1+\left(\frac{1}{R_{S}}\frac{dR_{S}}{d\theta}\right)^{2}}}.$
(35)

Then the contribution by the shock to the columns becomes

$\displaystyle\langle N_{B1,II}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{1,0}R_{1,\ast}\,\int_{\mu_{1,\infty}}^{\mu_{2,\ast}}\,K_{1}(\mu)\,\Sigma(\mu)\,d\mu,$
(36) $\displaystyle\langle N_{B2,II}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\int_{\mu_{2,\infty}}^{\mu_{1,\ast}}\,K_{2}(\mu)\,\Sigma(\mu)\,d\mu,$
(37) $\displaystyle\langle\tilde{N}_{B1,II}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\int_{\mu_{1,\infty}}^{\mu_{2,\ast}}\,K_{1}(\mu)\,\Sigma(\mu)\,\mu^{2}\,d\mu,$
(38) $\displaystyle\langle\tilde{N}_{B2,II}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\int_{\mu_{2,\infty}}^{\mu_{1,\ast}}\,K_{2}(\mu)\,\Sigma(\mu)\,\mu^{2}\,d\mu.$
(39)

The third region is in the wind of the opposite star, beyond the shock. The
integration in radius depends on whether the ray strikes the star or not. The
former case is treated in the next section. Here we express the columns along
the ray from the shock that stretches to infinity with impact parameter
$\varpi$. Consider, for example, a ray originating from the secondary at
orientation $\theta_{2}$. The ray intercepts the shock at $R_{2,S}$. Further,
the ray is in the wind of the primary. The impact parameter for that ray is
$\varpi_{1}=D\sin\theta_{2}$.

For a spherical wind with inverse square density, the integration for the
column along such a chord is analytic:

$\int_{-\infty}^{z_{0}}n_{0}\,\frac{R_{\ast}^{2}}{r^{2}}\,dz=n_{0}\,R_{\ast}\,\frac{R_{\ast}}{\varpi(\mu)}\,\int_{0}^{\theta_{0}}\,d\theta=n_{0}\,R_{\ast}\,\frac{R_{\ast}}{\varpi(\mu)}\,\theta_{0}(\mu),$
(40)

where the subscripts 1 and 2 have been suppressed for this general result.
Take again the example of a ray from the secondary. Then $z_{0}$ refers to the
$z$-coordinate in the wind of the primary corresponding to $R_{1,S}$ and
$\varpi_{1}$. Figure 3 shows the location of $z_{0}$ in relation to the system
components. With $\theta_{0}$ defined with respect to the stellar axis, one
can show that $\theta_{0}+\theta_{1}+\theta_{2}=\pi$.

The next step is to integrate in $\mu$. Again using the secondary as an
example, this integration will be of the form
$\theta_{0}(\mu_{2})d\mu_{2}/\varpi_{1}\sim\theta_{0}(\theta_{2})d\theta_{2}$.
The end result from the columns for section III is:

$\displaystyle\langle N_{B1,III}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{2,0}R_{2,\ast}\,\frac{R_{2,\ast}}{D}$
$\displaystyle\times\int_{\theta_{1,\infty}}^{\theta_{2,\ast}}\,(\pi-\theta_{1}-\theta_{2})\,d\theta_{1},$
$\displaystyle\langle N_{B2,III}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{1,0}R_{1,\ast}\,\frac{R_{1,\ast}}{D}$
$\displaystyle\times\int_{\theta_{2,\infty}}^{\theta_{1,\ast}}\,(\pi-\theta_{1}-\theta_{2})\,d\theta_{2},$
$\displaystyle\langle\tilde{N}_{B1,III}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{2,0}R_{2,\ast}\,\frac{R_{2,\ast}}{D}$
$\displaystyle\times\int_{\theta_{1,\infty}}^{\theta_{2,\ast}}\,(\pi-\theta_{1}-\theta_{2})\,\cos^{2}\theta_{1}\,d\theta_{1},$
$\displaystyle\langle\tilde{N}_{B2,III}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{1,0}R_{1,\ast}\,\frac{R_{1,\ast}}{D}$
$\displaystyle\times\int_{\theta_{2,\infty}}^{\theta_{1,\ast}}\,(\pi-\theta_{1}-\theta_{2})\,\cos^{2}\theta_{2}\,d\theta_{2}.$

where $\sin\theta_{1,\ast}=R_{1,\ast}/D$ and
$\sin\theta_{2,\ast}=R_{2,\ast}/D$. When the ray intercepts the opposite star,
the lower limit to the integral in equation (40) is no longer 0 in $d\theta$.
Thus the integrands for the angular integrations above are different over the
solid angle extent of the opposite star.

For Case B, one adds the contributions from the different segments to obtain

$\displaystyle\langle N_{B1}\rangle$ $\displaystyle=$ $\displaystyle\langle
N_{B1,I}\rangle+\langle N_{B1,II}\rangle+\langle N_{B1,III}\rangle,$ (45)
$\displaystyle\langle N_{B2}\rangle$ $\displaystyle=$ $\displaystyle\langle
N_{B2,I}\rangle+\langle N_{B2,II}\rangle+\langle N_{B2,III}\rangle,$ (46)
$\displaystyle\langle\tilde{N}_{B1}\rangle$ $\displaystyle=$
$\displaystyle\langle\tilde{N}_{B1,I}\rangle+\langle\tilde{N}_{B1,II}\rangle+\langle\tilde{N}_{B1,III}\rangle,$
(47) $\displaystyle\langle\tilde{N}_{B2}\rangle$ $\displaystyle=$
$\displaystyle\langle\tilde{N}_{B2,I}\rangle+\langle\tilde{N}_{B2,II}\rangle+\langle\tilde{N}_{B2,III}\rangle.$
(48)

Figure 4: Similar to Fig. 1, but zoomed to emphasize angular quantities used for case C calculations. The primary star is shown at top with its extent a dashed curve. The shock is a dotted curve. For the secondary star at bottom, only its center is indicated.  Figure 5: For the case of two equal stars and a planar shock, $\tau=\tau_{1}=\tau_{2}$ and $\gamma=\gamma_{1}=\gamma_{2}$. The figure shows the variations of these parameters along with $p_{\rm tot}$ as a function of binary separation, $D/R_{\ast}$, where $R_{\ast}=R_{1}=R_{2}$. Example lines of $D^{-1}$ variation are shown for reference; the parameters determining the colliding wind polarization decline much more slowly than $D^{-1}$. Note that since $p_{\rm tot}<0$, its negative is plotted for convenience. Table 1: Stellar and Wind Properties for Parameter Study Displayed in Figure 6 Panel | $R_{2,\ast}$ | $R_{1,\ast}$ | $\dot{M}_{2}$ | $\dot{M}_{1}$ | $v_{2}$ | $v_{1}$ | $D$
---|---|---|---|---|---|---|---
| $(R_{\odot})$ | $(R_{\odot})$ | $(10^{-7}M_{\odot}~{}{\rm yr}^{-1})$ | $(10^{-7}M_{\odot}~{}{\rm yr}^{-1})$ | (103 km s-1) | (103 km s-1) | $(R_{\odot})$
Top | 10 | 10 | 65 | 100 | 2 | 2 | 100
Top | 10 | 10 | 70 | 100 | 2 | 2 | 100
Top | 10 | 10 | 75 | 100 | 2 | 2 | 100
Top | 10 | 10 | 80 | 100 | 2 | 2 | 100
Top | 10 | 10 | 85 | 100 | 2 | 2 | 100
Top | 10 | 10 | 90 | 100 | 2 | 2 | 100
Top | 10 | 10 | 95 | 100 | 2 | 2 | 100
Top | 10 | 10 | 100 | 100 | 2 | 2 | 100
Mid | 10 | 10 | 100 | 100 | 1.3 | 2 | 100
Mid | 10 | 10 | 100 | 100 | 1.4 | 2 | 100
Mid | 10 | 10 | 100 | 100 | 1.5 | 2 | 100
Mid | 10 | 10 | 100 | 100 | 1.6 | 2 | 100
Mid | 10 | 10 | 100 | 100 | 1.7 | 2 | 100
Mid | 10 | 10 | 100 | 100 | 1.8 | 2 | 100
Mid | 10 | 10 | 100 | 100 | 1.9 | 2 | 100
Mid | 10 | 10 | 100 | 100 | 2.0 | 2 | 100
Bot | 3 | 10 | 100 | 100 | 2 | 2 | 100
Bot | 4 | 10 | 100 | 100 | 2 | 2 | 100
Bot | 5 | 10 | 100 | 100 | 2 | 2 | 100
Bot | 6 | 10 | 100 | 100 | 2 | 2 | 100
Bot | 7 | 10 | 100 | 100 | 2 | 2 | 100
Bot | 8 | 10 | 100 | 100 | 2 | 2 | 100
Bot | 9 | 10 | 100 | 100 | 2 | 2 | 100
Bot | 10 | 10 | 100 | 100 | 2 | 2 | 100

####  3.1.3 Case C: Rays that intercept the opposite star

In the final scenario, case C, the ray intersects the opposite star. This
modifies the upper limit for the integral associated with segment III.
Segments I and II from case B are the same for case C; only segment III
differs. To be explicit, for segment C,I for a star’s own wind up to the
shock, one still uses equations (31)–(34) for B,I but with different limits to
the integrals. The lower limit is $\mu_{\ast}$ for star 1 or 2 as appropriate,
and the upper limit is $+1$ for all the integrals. Similarly, the contribution
C,II at the shock uses equations (36)–(39) for B,II also with $\mu_{\ast}$
(again, as appropriate for 1 or 2) for the lower limit and $+1$ for the upper
limit. It is only C,III that requires reconsideration, as follows.

One way of expressing case C,III is that when $\varpi<R_{\ast}$, or
alternatively when $\theta<\theta_{\ast}$, for primary or secondary as the
case may be, the radial integral along the chord is still given by an angle,
but this angle is not $\theta_{0}$. Instead we introduce new angles $\epsilon$
and $\delta$, with

$\theta_{1}+\theta_{2}+\epsilon+\delta=\pi/2.$ (49)

Figure 4 shows the locations of $\epsilon$ and $\delta$ with respect to the
system. We have that $\varpi_{1}=D\sin\theta_{2}=R_{1,\ast}\cos\epsilon_{1}$,
so that $\epsilon_{1}$ is defined in terms of $\theta_{2}$ for a ray
originating from the secondary and intercepting the primary. When the case is
reversed, all the subscripts are reversed. This allows us to find $\delta$
from the definition above, and the radial integration along the ray segment is
$\delta$.

The columns now become

$\displaystyle\langle N_{C1,III}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{2,0}R_{2,\ast}\,\frac{R_{2,\ast}}{D}\int_{0}^{\theta_{2,\ast}}\,\delta(\theta_{1})\,d\theta_{1},$
(50) $\displaystyle\langle N_{C2,III}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{1,0}R_{1,\ast}\,\frac{R_{1,\ast}}{D}\int_{0}^{\theta_{1,\ast}}\,\delta(\theta_{2})\,d\theta_{2},$
(51) $\displaystyle\langle\tilde{N}_{C1,III}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{2,0}R_{2,\ast}\,\frac{R_{2,\ast}}{D}\int_{0}^{\theta_{2,\ast}}\,\delta(\theta_{1})\,\cos^{2}\theta_{1}\,d\theta_{1},$
(52) $\displaystyle\langle\tilde{N}_{C2,III}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{1,0}R_{1,\ast}\,\frac{R_{1,\ast}}{D}\int_{0}^{\theta_{1,\ast}}\,\delta(\theta_{2})\,\cos^{2}\theta_{2}\,d\theta_{2}.$
(53)

###  3.2 Special Case of a Planar Shock

For a binary consisting of two identical stars, $\alpha=\beta=1$, and the
resulting CWI is a planar shock located midway between the stars at $D/2$ from
either one. Appendix Special Case of $\alpha=\beta=1$ details the
simplifications that result for this scenario, in particular an analytic
expression for $R_{S,1}=R_{S,2}$ in terms of $\mu$, $K_{1}=K_{2}$, and the
surface density distribution.

We introduce the simplifying notation $\tau=\tau_{1}=\tau_{2}$ and
$\gamma=\gamma_{1}=\gamma_{2}$, and display in Figure 5 how these properties
vary with separation between the stars. Note also that $p_{1}=p_{2}\equiv
p_{0}$, and because the two stars have $L_{1}=L_{2}$, $p_{0}$ is a constant at
all wavelengths. For this figure we assume a stellar radius of $10R_{\ast}$,
wind speed $2000$ km s-1, and mass loss rates $10^{-5}M_{\odot}$ yr-1. The
polarization amplitude scales with $\dot{M}/v$ for the case of equal stars.

By our convention for Figure 1, both $(1-3\gamma)$ and $p_{0}$ are negative,
so are shown in Figure 5 as multiplied by $-1$ for convenience. Since many,
but not all, terms associated with the calculation scale as $D^{-1}$, dotted
magenta curves are shown with that scaling for comparison. The outcome is that
the polarization amplitude $p_{0}$ does indeed decline with binary separation,
but much less steeply than $D^{-1}$; its behavior is closer to linear for the
chosen parameters and distances shown.

###  3.3 Parameter Study

We conducted a parameter study of relatively similar winds for primary and
secondary components. Results are displayed in Figure 6, with model parameters
identified in Table 1. The figure has 3 panels – top, middle (“mid”), and
bottom (“bot”) – with model parameters similarly grouped in Table 1. The top
is for variation of the ratio of mass-loss rates (which turns out to be
$\beta$ because $\alpha$ is fixed); the middle is for variation of the ratio
of the wind terminal speeds (which is $\alpha=v_{2}/v_{1}$, but also
$\beta=\alpha$ for fixed mass-loss rates); and bottom is for variation of
radii. Note that for the top and middle panels, the far right side corresponds
to equal winds and a planar shock. For the bottom panel, $\beta=1$ and the
shock is always planar.

In the top panel, the ratio of mass-loss rates varies from 0.65 up to 1.0. The
four curves are for $\tau_{1}$ in red, $\tau_{2}$ in purple, $1-3\gamma_{1}$
in blue, and $1-3\gamma_{2}$ in green. While the primary wind has the higher
optical depth, the deviation of the envelope from spherical increases faster
for the secondary (green) than the primary (blue) as $\beta$ declines. This is
an important feature of the discussion in Section 3.6, where small $\beta$
values are emphasized as being typical of WR+OB binary systems.

For the middle panel, $\beta=\alpha$, yet the behavior is reversed. Lowering
the wind speed of the secondary actually elevates the density scale for its
wind.

Finally for the bottom panel, $\beta=\alpha=1$ is fixed, and the wind shock is
planar. Despite the geometry being invariant, the polarization depends on the
radii of the two stars. This arises because the polarization properties scale
with column density, which are generally inverse to radius. For example, with
a secondary smaller than the primary yet having the same mass-loss rate and
wind speed, the column density is higher in the secondary wind, so $\tau_{2}$
(purple) increases with decreasing $R_{2}$.

Figure 6: Variation of optical depth $\tau_{1}$ (red) and $\tau_{2}$ (purple),
shape factors $(1-3\gamma_{1})$ (blue), and $(1-3\gamma_{2})$ (green) for the
parameter study with model properties detailed in Table 1; solid points are
optical depths and open squares are for the shape factors involving $\gamma$.
The top panel shows how the polarization varies with the ratio of mass-loss
rates for the stars; middle is for the ratio of wind speeds,
$v_{2}/v_{1}=\alpha$; bottom is for the ratio of stellar radii. For each panel
(“top”, “mid”, “bot”), the 8 points on the curves correspond to model
calculations for the 8 parameter combinations shown in Table 1.

###  3.4 Chromatic Effects

To illustrate chromatic effects, we fixed a particular set of binary
parameters while allowing the temperatures of the two stars to vary. The fixed
properties lead to $p_{1}=0.009$ and $p_{2}=0.192$. In this example, the
binary components are separated by $40~{}R_{\odot}$. The secondary is twice as
large ($10R_{\odot}$) as the primary ($5R_{\odot}$); the two winds have equal
speeds (2000 km s-1); and the mass-loss rate for the primary is $3.3\times$
larger than the secondary ($\dot{M}_{2}=3\times 10^{-6}\,M_{\odot}$ yr-1).
Additionally, we took $\sin i=1$; the effect of viewing inclination is to
scale the curves by $\sin^{2}i$ at all wavelengths.

To illustrate changes in the spectropolarimetric continuum shape, we treated
the two stars as simple Planckian sources with effective temperatures $T_{1}$
and $T_{2}$. We fixed the temperature of the secondary at $T_{2}=25,000$ K. We
varied the temperature of the primary from $16,000$ K to $40,000$ K in $3,000$
K intervals, and display the results in Figure 7, where polarization is shown
as positive. These variations may not be consistent with actual combinations
of parameters for real stars. The point of the exercise is to highlight the
fact that when the stars have different spectral energy distributions, the
continuum polarization is not generally constant with wavelength, even though
electron scattering is gray. Only when the two stars have equal temperatures
is the continuum polarization truly flat at all wavelengths.

Note especially that the polarization signal changes strongly from the FUV
through the optical to 1 micron. For massive stars with typical temperatures
well in excess of $10,000$ K, both the primary and secondary spectra are in
the Rayleigh-Jeans tail in the optical, and so the continuum will always be
flat or nearly flat in that waveband. It is only in the UV that the
polarization deviates significantly from constant. For the selected
parameters, the polarization actually drops toward the UV when the primary is
hotter (i.e., more luminous), owing to the fact that $p_{1}\ll p_{2}$. By
contrast, when the secondary is hotter (i.e., more luminous), the polarization
increases significantly. Ultimately, for any combination of binary parameters,
when the more luminous star in the UV also has the higher polarimetric
component (i.e., $p_{1}$ or $p_{2}$), the polarization is enhanced in the UV
relative to the optical; when the more luminous UV source has the lower
polarimetric component, the polarization will drop toward the UV.

The behavior in Figure 7 is included in the formalism of BME, but is specific
to binaries with two hot stars. Other categories of binaries can certainly
show rather different behavior. For example, consider symbiotic stars, which
involve a hot white dwarf and a cool giant star (e.g., Muerset et al., 1991).
In such a case, the polarigenic opacity may be more complex than simple
electron scattering (i.e., not simply gray opacity), and may not involve a
colliding wind but perhaps instead accretion onto a disk. Nonetheless, if the
scattering opacity is dominated by Thomson scattering, the combination of an
optically-bright component with a UV-bright component would yield a
wavelength-dependent polarization that would reveal a telltale gradual
variation from the FUV to the IR.

In any situation where intrinsic polarization from Thomson scattering is
observable over a wide spectral domain (separable from interstellar
polarization either by its binary variation in the time domain, or line
effects in the stellar winds, or modeling the wavelength dependence of the
interstellar polarization), these results show that the residual wavelength
dependence of the intrinsic polarization offers a unique and important
diagnostic. Via the following analysis, we obtain complementary leverage in
our understanding of both the different polarizations produced by the two
stellar light sources, and also the spectral shape of the continua of both
stars. This stems from the grayness of Thomson scattering, which implies that
the sole source of wavelength dependence in the intrinsic polarization derives
from the contrasting brightnesses of the two stars. Hence if ${\cal
L}_{21}(\lambda)$ is the wavelength-dependent ratio of the secondary
brightness to the primary, then the total polarization $q(\lambda)$ presents
as a brightness-weighted average of the wavelength independent polarization
induced by the primary light source, $q_{1}$, and that induced by the
secondary, $q_{2}$, according to

$q(\lambda)=\frac{q_{1}\ +\ q_{2}{\cal L}_{21}(\lambda)}{1\ +\ {\cal
L}_{21}(\lambda)}\ .$ (54)

If the spectral shape contrast ${\cal L}_{21}(\lambda)$ is regarded as known
by the stellar spectral types, then observing $q$ at two different wavelengths
that sample suitably different values of ${\cal L}_{21}(\lambda)$ allows the
above equation to separate the $q_{1}$ and $q_{2}$ contributions. This
separation of the polarizations caused by the two different light sources
allows a unique probe of the geometry of the wind collision zone.

Furthermore, to the extent that $q_{1}$ and $q_{2}$ are expected to be
wavelength independent, a self-consistency check on the assumed ${\cal
L}_{21}(\lambda)$ becomes possible by inverting the above equation into

${\cal L}_{21}(\lambda)=\frac{q_{1}\ -\ q(\lambda)}{q(\lambda)\ -\ q_{2}}\ .$
(55)

To whatever extent this inferred brightness contrast deviates from its assumed
value, we have the opportunity to update it to recover consistency with the
polarized spectrum $q(\lambda)$. For example, a $(q_{1},q_{2})$ pair can be
inferred from wavelength pairs generated by fixing a wavelength at the UV end
of the observed $q(\lambda)$ and sweeping the second wavelength over the full
observed range. If the assumed ${\cal L}_{21}(\lambda)$ contains errors, that
would generate a curve in $(q_{1},q_{2})$ space rather than a single
consistent point. Then by fixing the second wavelength at its longest value
and sweeping the first wavelength back toward the UV, the curve is closed back
to its starting $(q_{1},q_{2})$ point. The resulting closed curve then gives
an estimate of the preferred $(q_{1},q_{2})$ value near the center of this
curve, and that preferred $(q_{1},q_{2})$ pair then allows ${\cal
L}_{21}(\lambda)$ to be self-consistently updated via the above equation. The
wavelength independence of the intrinsic polarization contributions $q_{1}$
and $q_{2}$, assuming they are dominated by Thomson scattering in the
colliding winds, then provides an improved estimate of ${\cal
L}_{21}(\lambda)$ and an independent check of our understanding of binary
spectral types. Also, when both $q_{1}$ and $q_{2}$ are appreciable,
independent knowledge of both allows an important probe of the colliding wind
geometry, since the two stars illuminate that geometry differently.

On the other hand, in situations where one contribution dominates, say
$q_{1}$, as may be the case in WR/O binaries discussed below, the wavelength
dependence of the intrinsic $q(\lambda)$ directly inherits the wavelength
dependence of ${\cal L}_{21}(\lambda)$ via $q(\lambda)=q_{1}/\left(1+{\cal
L}_{21}(\lambda)\right)$. Hence in this case we have an even more direct
handle on the correct brightness contrast between the two stars over the full
wavelength regime of the observed polarization. The wider that wavelength
regime accessed by our technology, the more powerful is this constraint,
underscoring the value of extending our polarization capabilities into the FUV
range for understanding binaries containing hot stars.

Figure 7: Variation of the polarized continuum with wavelength, here shown
from the FUV to 1 micron. The stars are treated as Planckian. The temperature
of the secondary is fixed at 25,000 K, and the temperature of the primary
varies from 16,000 K to 40,000 K in 3,000 K increments. The particulars of the
stellar and wind parameters for this illustrative case are described in Sect.
3.4. For the selected parameters, the limiting polarizations $p_{1}$ and
$p_{2}$ are indicated with horizontal red and blue dotted lines, respectively.

###  3.5 Orbital Effects

Figure 8: Comparison of the percent polarization from a face-on binary system
with colliding winds (CWs) against that from a binary with non-interacting
winds (NIWs; § 3.5). The points represent separations of $25R_{\odot}$ to
$115R_{\odot}$ in $5R\odot$ increments, with monotonic sequence as indicated
by “Closer” and “Farther”. Wind collision significantly increases the level of
polarization of CWs over NIWs, but less so as the binary separation increases
to large values. The green line represents the analytic derivation from App
Special Case of $\alpha=\beta=1$ for wide-binary separations.

The polarimetric properties of the colliding wind system depend on the binary
separation, $D$. When the orbit is circular (i.e., $D$ is constant), the
values $p_{1}$ and $p_{2}$ are constant as well. For an eccentric orbit with
eccentricity $e$ and semi-major axis $a$, the binary separation varies as

$D(\varphi)=a\,\frac{1-e^{2}}{1+e\cos\varphi},$ (56)

where $\varphi$ is the orbital azimuth, defined so that $\varphi=0$
corresponds to periastron. Thus $p_{1}$ and $p_{2}$ are functions of orbital
phase through the variation of $D(\varphi)$.

The polarization also varies throughout the orbital motion because the
inclination, $i$, of the LOC between the stars changes relative to the
observer’s line of sight. Let $i_{\rm orb}$ be the viewing inclination of the
orbital plane, so that $i_{\rm orb}=0^{\circ}$ is a top-down view of the orbit
and $i_{\rm orb}=90^{\circ}$ is an edge-on view. Despite a fixed orientation
of the orbital plane, $i_{\rm orb}$, our construction for calculating
polarization depends on the system axis defined by the LOC between the stars,
and this rotates in the fixed orbital plane to produce variability.

The time-variable polarization is given by

$\displaystyle q$ $\displaystyle=$ $\displaystyle p_{0}(D)\,\sin^{2}i(t)\,\cos
2\psi(t),~{}{\rm and}$ (57) $\displaystyle u$ $\displaystyle=$ $\displaystyle
p_{0}(D)\,\sin^{2}i(t)\,\sin 2\psi(t),$ (58)

where $D=D(t)$ for eccentric orbits, $t$ depends on orbital phase through the
azimuth ($\varphi$ of the LOC), and the polarization position angle $\psi$
relates to the orbital azimuth and the fixed viewing inclination of the
orbital plane via

$\tan\psi=-\cos i_{\rm orb}/\tan\varphi.$ (59)

The inclination, $i$, for the LOC to the viewer’s line of sight is given by

$\cos i=\sin i_{\rm orb}\,\cos\varphi.$ (60)

In order to understand the relevance of CWIs for the polarization level, we
introduce the idea of “non-interacting winds” (NIWs). The concept of a NIW
provides a reference against which to compare the physical case of CWI
polarization arising from the shock. What CWI ultimately represents is a
redistribution of matter from the two stellar winds via the wind collision.
There is polarization without a CWI, because each of the two stars shines on
the wind of the other, even if both winds remain spherical. The CWI represents
another contribution by breaking spherical symmetry. For wide binary
separations ($D\gg R_{\ast}$), we expect the two cases to become proportional,
since the column depths of the various regions will scale as $D^{-1}$ (see
App. Special Case of $\alpha=\beta=1$).

Figure 9: (Top) Example $q-u$ loops for a binary consisting of identical
stars, for which $p_{1}=p_{2}\equiv p_{0}$. Eccentricities $e$ and
inclinations $i_{\rm orb}$ are indicated for line type and color. There are
two loops per orbit, but with eccentric orbits, the pair separate into
different sizes except for $e=0$. (Bottom) Same 6 models as top, now displayed
as lightcurves in $q$ (red), $u$ (blue), and total polarization $p_{\rm tot}$
(black). Periastron passage is at phase 0.0, and apastron at phase 0.5. The
line types still relate to $e$ as in the top panel. The case $i_{\rm
orb}=45^{\circ}$ has the higher polarization in $p_{\rm tot}$ and larger
amplitude variation in $q$ and $u$.

We thus define a NIW simply as a superposition of the two separate binary
winds as if no collision takes place and neither wind impacts the opposite
star. This means each wind is spherically symmetric about its own star, which
contributes no polarization. Instead, polarization arises only from scattering
of starlight from the secondary by the wind of the primary, and vice versa. We
do, however, account for occultation of the wind behind each respective star
in calculating the polarization.

Our results are shown in Figure 8, with the polarization amplitude in percent
for the NIW along the horizontal, and for the CWI along the vertical. For this
example we assume two equal winds and thus a planar shock for the CWI case.
The points represent stellar separations of $D=25\,R_{\odot}$ to
$115\,R_{\odot}$ in $5\,R_{\odot}$ increments, with closer and farther
separations labeled for the sequence. The values are for a face-on binary,
oriented so that $u=0$ and $q<0$, and $p=|q|$ for this plot. The dotted
diagonal indicates where the polarizations would be equal; the solid green
line is the asymptotic relation derived in Appendix Special Case of
$\alpha=\beta=1$.

For the CWI case, we fitted a linear regression to $p_{0}(D)$ in the case of a
binary with two identical stars (i.e., planar CWI shock) to obtain
$p_{0}=1.328-0.0111D$, normalized so that $p_{0}=1\%$ at $D=30~{}R_{\odot}$.
For the NIWs, we used the same stellar wind and star properties as for the
wind collision case. The polarization from a CWI is larger than for a NIW,
often by a significant factor. At larger separations, the trend is for the
points to approach the diagonal line, signifying that the wind collision is
becoming irrelevant.

We recognize that we are using a model for a bow shock with radiative cooling,
and that at large separations, the cooling will be adiabatic. Even so, with
identical stars and winds, the shock will still be planar, and the qualitative
conclusion remains valid even if the quantitative values are inaccurate.

Figure 9 displays a suite of polarimetric variations for CWIs with different
values of $e$ and $i_{\rm orb}$. We show model variable polarization curves
for inclinations and eccentricities, as labeled. The top panel displays the
resulting $q-u$ loops; the bottom panel shows polarized light curves as a
function of orbital phase. At $i_{\rm orb}=90^{\circ}$ (not shown), all curves
in Figure 9 would become horizontal lines with only $q$ variation but no $u$
variation. Note also that $p_{\rm tot}=\sqrt{q^{2}+u^{2}}$.

###  3.6 Special Case of WR+OB Binaries

Among the more extreme massive star colliding wind systems ($\beta\ll 1$) are
the ones involving an evolved WR star with an OB companion. While the wind
speeds of the two stars can be comparable in this case, the mass-loss rate of
the WR wind will be one to several orders of magnitude larger than for the OB
component. As a result, the CWI shock is significantly displaced from the WR
star and considerably closer to the OB star; it also significantly confines
the spatial scope of the OB wind. On the other hand, the WR and OB components
may or may not have comparable luminosities. In terms of a UV study, the
situation can be ideal for extracting information about the orbital parameters
and properties of the CWI region from both temporal and chromatic effects.

The scenario of $\beta\ll 1$ offers some simplifications for the problem of
the polarization. Foremost is that the “primary” (defined as above as the WR
wind with higher mass loss, not necessarily the more luminous component) is
relatively far from the CWI shock. Consequently, one expects the angle-
averaged column densities over the WR wind component to approach zero. The
associated column densities for the secondary wind evaluated at the primary
will be small. It may seem that $p_{1}=p_{WR}$ would be dominated by the CWI
shock, but this may not mean that the WR component dominates the polarization,
since the CWI is relatively far removed, and thus only acts as a perturbation
on the otherwise spherical wind of the WR star. The result for the O star,
$p_{2}=p_{O}$, is less clear. Its wind has lower column density than the WR,
but the distorted envelope is closer to the O star at low $\beta$. Also, the
CWI wraps around the O star, leading to polarimetric cancellation. We use the
Cantó et al. (1996) formalism to evaluate the possibilities.

Figure 10: Top-down view of a WR+OB binary with density contours superposed.
The WR star is the smaller component at right; at left is the OB star. The
magenta curve highlights the CWI shock. Two representative contours indicate
relative densities, normalized to unity at the surface of the WR star. The
contours are displayed in roughly 0.5 dex intervals. In this example, the
binary separation is $60~{}R_{\odot}$, the OB star has a radius of
$12~{}R_{\odot}$, and the WR star has a radius of $1~{}R_{\odot}$.

To illustrate some of these features, Figure 10 shows a density contour plot
in the orbital plane for a WR+OB binary. The WR star is the smaller star at
right; the secondary is taken to be an O star. The dotted curves are density
contours at approximately 0.5 dex intervals, with two labeled for
illustration, normalized to unit density at the surface of the WR star.

To explore the polarization expected from WR+OB colliding winds, we adopted
the following stellar and orbital parameters. For the WR as primary and an O
star as secondary, we assumed $M_{1}=M_{2}=30~{}M_{\odot}$; $R_{1}=R_{\odot}$
and $R_{2}=12~{}R_{\odot}$, $v_{1}=v_{2}$ so that $\alpha=1$; and we
considered orbital scenarios ranging from a short-period orbit of $P_{S}=7$ d
to a medium-period orbit of $P_{M}=30$ d (a typical range for colliding winds
in circular orbits; see, e.g., Fahed & Moffat, 2012; Zhekov, 2012). Given the
masses, these two orbits correspond to semi-major axes of
$a_{S}=60~{}R_{\odot}$ and $a_{M}=160~{}R_{\odot}$.

As a fiducial, we also adopted $\dot{M}_{1}=10~{}\dot{M}_{2}$ so that
$\beta=0.1$. At this value of $\beta$, the relative standoff distance for the
bow shock is $R_{S,2}/D=0.24$, which is a fixed ratio regardless of binary
eccentricity, given that we assumed the winds are at terminal speed. For the
short-period binary, we assumed a circular orbit, hence
$R_{S,2}=14.4~{}R_{\odot}$. For a typical O star wind, this would be well
within the zone of wind acceleration, where effects such as radiative braking
could be significant (as seen in the WR+O binary V444 Cyg; Lomax et al.,
2015). For the sake of illustration, we ignored such effects.

We calculated $p_{WR}$ and $p_{O}$ for three scenarios, with a summary of
results displayed in Figure 11. The first case is $\beta=0.1$ with orbital
separation ranging from $60~{}R_{\odot}$ to $160~{}R_{\odot}$ for “slow” winds
of 1000 km s-1 for both stars. The second case is for fast winds at 3000 km
s-1, with all other parameters fixed. The third scenario corresponds to an
intermediate wind speed of 2000 km s-1 at a fixed separation of
$D=160~{}R_{\odot}$, but with $\beta$ ranging between 1/15 and 1/5.

The upper panel of Figure 11 summarizes the comparison between slow and fast
winds. Note that polarization is negative for our convention. In this panel,
the red lines represent $p_{O}$ and the blue lines represent $p_{WR}$. The
solid circles are for the slow wind cases, and the open circles are for the
fast wind cases. The results are plotted against $D^{-1}$, normalized as
indicated. The net result is that the polarization is overall larger for a
slower wind, since the density is larger. We find that $\tau_{WR}$ is roughly
constant as $D$ changes, indicating that its value is dominated by the
relatively extended spherical wind of the WR star, since the CWI is far
removed. Because the CWI is relatively farther from the WR star with
increasing $D$, $(1-3\gamma_{WR})$ becomes smaller with $D$. Consequently,
$p_{WR}$ decreases with increasing $D$. The behavior for the O star is that
the polarization is dominated by the CWI. The surface density of the CWI shock
for the Cantó et al. (1996) solution scales as $D^{-1}$ overall. This is
evidenced by the fact that both blue curves appear quite linear in the plot.

For the lower panel of Figure 11, we display the results differently, as
$\beta$ is allowed to vary between 1/15 and 1/5; with smaller $\beta$, the CWI
is closer to the O star component. Consider first the dashed and dotted curves
in black, for $\tau_{WR}$ and $\tau_{O}$, respectively. As $\beta$ becomes
smaller, $\tau_{WR}$ is larger, approaching the limit of the strictly
spherical wind value. The value of $\tau_{O}$ is much lower, and is plotted as
scaled up by $10\times$.

The blue curve in this lower panel represents the ratio of $p_{WR}/p_{O}$. Its
behavior indicates that from geometrical considerations, the contribution to
the polarization from the O star wind is much greater than for the WR wind,
even more so as $\beta$ becomes smaller. Even though the WR wind has a much
higher optical depth scale, the distortion of the scattering envelope from
spherical is quite minor from the perspective of the WR star. This is made
clear by the red curve, where “shape” is the ratio
$(1-3\gamma_{WR})/(1-3\gamma_{O})$, and scaled up by $100\times$. From the
perspective of the O component, the scattering envelope is highly distorted.

In combination, these results suggest that at wavelengths where the O star is
more luminous, the polarization will overall be larger (biased toward $p_{O}$)
than at wavelengths where the WR star is more luminous (polarization biased
toward $p_{WR}$). Our treatment does have limitations, the most important
being that we ignore the wind acceleration region, and that we treat the WR
wind as optically thin to electron scattering. Radiative transfer models have
shown that multiple scattering in bow shock structures can increase the degree
of polarization as well as changing the polarization behavior with inclination
angle (Shrestha et al., 2018). However, in this case it is clear that the WR
wind is already a minor contributor to the polarization when $\beta\ll 1$, and
a more full treatment of multiple scattering at the inner WR wind is not
expected to impact that conclusion. For the rare case of WR-WR binaries,
multiple scattering could be significant and future modelling will need to
take it into consideration. Inclusion of the wind acceleration region and
associated density distribution, along with radiative inhibition, could
certainly change the detailed outcomes of the models presented here.
Additionally, WR+O binaries can generally be expected to show chromatic
behavior over a broader waveband than indicated in Section 3.4. Free-free
opacity is important in the winds of WR stars at all wavelengths; hence the WR
stellar spectrum is never Rayleigh-Taylor even though the OB SED can be (e.g.,
Hillier, 1987). Nonetheless, the present treatment indicates that $p_{O}\gg
p_{WR}$, another qualitative result that is unlikely to change despite our
more simplistic assumptions.

Figure 11: Model results for a WR+O star binary (§ 3.6). The upper panel shows
results for a fixed value of $\beta=0.1$ for the slow and fast wind cases,
plotted against binary separation as $D^{-1}$. Blue lines represent $p_{O}$
and red lines represent $p_{WR}$; solid circles are for slow models and open
circles for fast wind models. The lower panel shows results for
$D=160~{}R_{\odot}$ with $\beta$ varying between 1/15 and 1/5. Black curves
represent the WR and O optical depths, as labeled. The blue line in this panel
represents the ratio of $p_{WR}/p_{O}$. The red line represents the “shape,”
defined as the ratio $(1-3\gamma_{O})/(1-3\gamma_{WR})$ and scaled up by
$100\times$.

###  3.7 Comment on the Overall Scale of the Polarization

As mentioned above for identical stars, the polarization is proportional to
${\dot{M}}/v$. This extends more generally to unequal stars, where the two
different mass loss rates receive different weightings, as a change in both
mass-loss rates by any given factor produces a change in polarization by that
same factor. This implies that a good model of the polarizing geometry allows
the scale of the observed polarization to yield a constraint on the stellar
mass-loss rates that is independent, and hence complementary, of all other
methods for such determinations. Furthermore, to the extent that the
polarization is due to optically thin scattering, the connection between
polarization and mass-loss rate is independent of local clumping in the winds.
Hence contrasting the mass-loss rates inferred from the overall scale of the
polarization to those obtained from conventional methods that are sensitive to
clumping (often termed “density-squared” type mass-loss rate diagnostics such
as radio free-free emission or H $\alpha$ emission) provide an independent
measure of the degree of clumping. Constraining wind clumping is an important
goal for understanding the basic dynamics of radiatively driven winds.

##  4 Summary

This study has made use of the theory of BME for optically thin electron
scattering polarization for a massive colliding-wind binary. The main novelty
has been to explore the analytic solution for the CWI shock structure in terms
of shape and density from Cantó et al. (1996), who assume radiative cooling to
derive a thin shell result. We assume axisymmetry throughout and explore
polarization characteristics and contributions from the two separate
components. Our results range from the limiting case of equal star scenarios
with $\beta=1$ and a planar shock interface (more appropriate to O+O binaries)
to small $\beta$ scenarios (more appropriate to WR+O binaries).

Overall, there are numerous free parameters for the model, from the binary
separation to the wind properties. Even when most star and wind properties are
held fixed, raising and lowering the terminal wind speeds at fixed ratio
$\alpha$ still affects the polarization characteristics, since slower winds
are more dense and faster ones are less dense.

Our three main results are as follows:

  1. 1.

From a detailed consideration of the contributions to the column density
moments, there are various terms that scale with $D^{-1}$ for the binary
separation. However, for the equal wind scenario, the scale of polarization
declines far less steeply than $D^{-1}$, so that even relatively wide binaries
may display a significant polarimetric amplitude, with a telltale orbitally
varying phase angle. (See Fig. 5.)

  2. 2.

Chromatic effects can become quite significant toward UV wavelengths. When
dealing with massive stars, all of which are “hot” at $>10,000$ K, the optical
emission is mostly or even very closely following the Rayleigh-Jeans law. The
consequence is that for optical and longer wavelengths, the continuum
polarization is flat. That polarization can still vary with orbital phase, but
there are no chromatic effects. However, at UV wavelengths for stars with
different temperatures, the continuum polarization will generally deviate from
flat (unless one star dominates the luminosity at all wavelengths). The
wavelength-dependent polarization provides additional diagnostic leverage for
extracting information about the winds and CWI shock (Fig. 7). This motivates
UV polarimetric observations of colliding wind binaries, such as would be
provided by the proposed Polstar satellite (Scowen et al., 2021).

  3. 3.

Orbital effects produce distinguishable shapes in the $q$–$u$ plane. The
shapes are mainly elliptical, as pointed out by BME already. Importantly, we
used the context of orbital effects to explore the influence of the CWI shock,
and its boundary separating the two stellar winds, on the amplitude of
polarization. For this purpose we introduced the “non-interacting winds”
(NIWs) construct. This assumes an (unphysical) superposition of the respective
two winds, with polarization arising solely from each star shining on the
spherical wind of the other. In this way the scenario for NIWs and CWIs can be
compared on the scale of the same mass fluxes. For the case of equal winds and
a planar shock, inclusion of the CWI increases the polarization by factors of
several, until the separation of the two stars becomes large compared to the
stellar radii. As expected, the CWI and NIW polarizations become equal, since
the CWI is far removed from either star and thus adds only a small column
density compared to the spherical winds.

  4. 4.

When $\beta\ll 1$, as for example in the case of WR+OB binaries, we find the
interesting result that the polarization for the OB component is much higher
than for the WR component. Whether the observed polarization is dominated by
the WR or the OB star will depend on the weighting by the specific
luminosities. However, it is clear that at wavelengths where the WR star is
more luminous, the polarization will be lower as set by $p_{WR}$, and where it
is less luminous, the polarization will be higher as set by $p_{O}$ (or
$p_{B}$, as the case may be; Fig. 11.)

In closing it is worth noting that the individual stars in a massive colliding
wind system may themselves be sources of polarization, which may be steady or
variable. For example, around 10% of massive stars are known to be magnetic
(Wade et al., 2016), and it is possible (although very rare) for massive star
binaries to have a component that possesses a significant magnetic field
(e.g., Plaskett’s Star, Grunhut et al., 2013). Munoz et al. (2022) has
recently explored the effects of variable linear polarization from electron
scattering for rotating magnetospheres. While this could complicate efforts to
isolate the variable polarization from the CWI, the polarization from
individual stars will be modulated on a rotation period whereas the colliding
wind polarization is modulated on the orbital period. Unless the binaries are
very close, these periods are unlikely to be the same.

WR stars in particular are known to be sources of polarization. However,
typical polarization behavior from individual WR stars appears stochastic
(e.g., St.-Louis et al., 1987; Drissen et al., 1987). The behavior is likely
associated with the wind flow time, $R_{\ast}/v_{\infty}$, that is much
shorter than binary orbital periods. In addition to being stochastic in
nature, the effect could be averaged out to emphasize the smoother variable
polarization from the CWI on the longer period of the orbit. In addition to
variable polarization, some WR stars may have long-term stable polarizations
(e.g., Harries et al., 1998). Abdellaoui et al. (2022) have explored the
polarization that could result for axisymmetric rotationally distorted winds
of WR stars. However, such polarization would be constant. The effect would be
to contribute to a constant offset to the system polarization, similar to the
effect of interstellar polarization. Variable polarization would arise
entirely from the CWI over the timescale of the orbital period.

RI acknowledges funding support for this research from grants by the National
Science Foundation (NSF), AST-2009412 and AST-1747658. YN acknowledges support
from the Fonds National de la Recherche Scientifique (Belgium), the European
Space Agency (ESA) and the Belgian Federal Science Policy Office (BELSPO) in
the framework of the PRODEX Programme. JLH is grateful for NSF funding under
award AST-1816944, and acknowledges that the University of Denver occupies
land within the traditional territories of the Arapaho, Cheyenne, and Ute
peoples. NSL wishes to thank the National Sciences and Engineering Council
(NSERC) for financial support.

## Special Case of $\alpha=\beta=1$

When $\beta=1$ and $\alpha=1$, with stars of identical stellar and wind
parameters, the CWI is planar, and the solution for the shock properties
simplifies considerably. First, we introduce $\theta=\theta_{1}=\theta_{2}$ as
the angle from either star to a point on the planar shock. The distance of the
shock from either star becomes

$R_{S}=D/2\mu.$ (1)

The projection factors become $K=K_{1}=K_{2}=\mu$. Simplification of the
surface density was noted already in eq. (16). Contributions to the
polarization from the CWI component depends on the following integrals (see
eqs. [36]-[39]):

$\int_{0}^{1}\Sigma(\mu)\,K(\mu)\,d\mu=4\Sigma_{0}\,\int_{0}^{1}\frac{\mu^{2}(1-\mu)^{2}}{\sin\theta(\theta-\mu\sin\theta)}\,d\mu\approx
0.56\,\Sigma_{0},$ (2)

and

$\int_{0}^{1}\Sigma(\mu)\,K(\mu)\,\mu^{2}\,d\mu=4\Sigma_{0}\,\int_{0}^{1}\frac{\mu^{4}(1-\mu)^{2}}{\sin\theta(\theta-\mu\sin\theta)}\,d\mu\approx
0.32\,\Sigma_{0}.$ (3)

Using these two results, we can analytically derive the polarization for the
scenario of two equal stars that are widely separated, with $D\gg R_{\ast}$.
At wide separation we can ignore the finite size of each star (i.e.,
$\theta_{\ast}\rightarrow 0$), which amounts to not having to consider case C
(c.f., § 3.1.3). Additionally, the region of case A for each star (c.f., §
3.1.1) is hemispherical and consequently makes no contribution to the net
polarization. All that remains are contributions I., II., and III. for case B
(c.f., § 3.1.2), where the limits of the angular integrations are 0 to $\pi/2$
in $\theta$ or $+1$ to 0 in $\mu$. The total polarization becomes:

$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{3}{8}\,\sigma_{T}\,\left(\bar{N}-3\tilde{N}\right)$ (4)
$\displaystyle=$
$\displaystyle\frac{3}{8}\,\sigma_{T}\,\left[\bar{N}_{B,I}+\bar{N}_{B,II}+\bar{N}_{B,III}-3\tilde{N}_{B,I}-3\tilde{N}_{B,II}-3\tilde{N}_{B,III}\right]$
(5) $\displaystyle=$
$\displaystyle\frac{3}{8}\,\sigma_{T}\,n_{0}\,R_{\ast}\,\left(\frac{R_{\ast}}{D}\right)\,\left[-1.00+0.56+2.47+075-0.96-6.30\right]$
(6) $\displaystyle=$
$\displaystyle-1.68\,\sigma_{T}\,n_{0}\,R_{\ast}\,\left(\frac{R_{\ast}}{D}\right),$
(7)

where each number in the square brackets of eq. (6) corresponds to each term
in the preceding line of eq. (5). Note that if grouped by region, each of I.,
II., and III. would separately yield net negative polarizations.

For comparison the polarization for the non-interacting wind (NIW) case
introduced in § 3.5 can also be evaluated analytically for $D\gg R_{\ast}$.
The two relevant angle-averaged column densities are

$\bar{N}=n_{0}\,R_{\ast}\,\left(\frac{R_{\ast}}{D}\right)\,\int_{0}^{\pi/2}\pi\,d\theta=\frac{\pi^{2}}{2}\,n_{0}\,R_{\ast}\,\left(\frac{R_{\ast}}{D}\right),$
(8)

and

$\tilde{N}=n_{0}\,R_{\ast}\,\left(\frac{R_{\ast}}{D}\right)\,\int_{0}^{\pi/2}\pi\,\cos^{2}\theta\,d\theta=\frac{\pi^{2}}{4}\,n_{0}\,R_{\ast}\,\left(\frac{R_{\ast}}{D}\right).$
(9)

For an NIW with wide binary separation, the polarization is

$p_{NIW}=\frac{3}{8}\,\sigma_{T}\,n_{0}\,R_{\ast}\,\left(\frac{R_{\ast}}{D}\right)\,\left(\frac{\pi^{2}}{2}-3\frac{\pi^{2}}{4}\right)=-0.93\,\sigma_{T}\,n_{0}\,R_{\ast}\,\left(\frac{R_{\ast}}{D}\right).$
(10)

The ratio of the coefficients from eqs. (7) and (10) is 1.8, which is the
solid green line appearing in Fig. 8.

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