11institutetext: Department of Theoretical Physics and Astrophysics, Faculty
of Science, Masaryk University, Kotlářská 2, Brno, Czech Republic,
11email<EMAIL_ADDRESS>Institute of Theoretical
Physics, Charles University, V Holešovičkách 2, Praha 8, Czech Republic

# Intrinsic polarization of Wolf-Rayet stars due to the rotational modulation
of the stellar wind

S. Abdellaoui 11 J. Krtička 11 P. Kurfürst 2211

###### Abstract

Context. Fast rotating Wolf-Rayet stars are expected to be progenitors of long
duration gamma-ray bursts. However, the observational test of this model is
problematic. Spectral lines of Wolf-Rayet stars originate in expanding stellar
wind, therefore a reliable spectroscopical determination of their rotational
velocities is difficult. Intrinsic polarization of Wolf-Rayet stars due to the
rotational modulation of the stellar wind may provide an indirect way to
determine the rotational velocities of these stars. However, detailed wind
models are required for this purpose.

Aims. We determine the intrinsic polarization of Wolf-Rayet stars from
hydrodynamical wind models as a function of rotational velocity.

Methods. We used 2.5D hydrodynamical simulations to calculate the structure of
rotating winds of Wolf-Rayet stars. The simulations account for the
deformation of the stellar surface due to rotation, gravity darkening, and
nonradial forces. From the derived models, we calculated the intrinsic stellar
polarization. The mass loss rate was scaled to take realistic wind densities
of Wolf-Rayet stars into account.

Results. The hydrodynamical wind models predict a prolate wind structure,
which leads to a relatively low level of polarization. Even relatively large
rotational velocities are allowed by observational constrains. The obtained
wind structure is similiar to that obtained previously for rotating optically
thin winds.

Conclusions. Derived upper limits of rotational velocities of studied Wolf-
Rayet stars are not in conflict with the model of long duration gamma-ray
bursts.

###### Key Words.:

stars: Wolf-Rayet – stars: rotation – polarization – stars: winds, outflows

##  1 Introduction

Classical Wolf-Rayet (WR) stars are massive stars that lost their hydrogen
envelope during their evolution (Conti, 1975; Chiosi et al., 1979; Sander et
al., 2012). The collapse of fast rotating WR stars to a black hole could
generate long duration gamma-ray bursts (Woosley, 1993). In this context, Vink
& de Koter (2005) proposed that fast rotating WR stars are progenitors of a
long duration gamma-ray burst. However, a direct observational test of this
model is difficult due to the problematic determination of rotational
velocities of Wolf-Rayet stars. An estimation of rotational velocities from
polarized light may provide an indirect way to test the nature of gamma-ray
burst progenitors.

Winds of WR stars are ionized, consequently the light scattering on free
electrons is an important source of opacity in the envelopes of these stars.
As a result of light scattering on free electrons, an initially unpolarized
beam coming from the stellar photosphere becomes linearly polarized. After
scattering into the direction making an angle of $\Theta$ with respect to the
incident beam, the ratio of intensities in the directions perpendicular and
parallel to the plane of scattering (given by the directions of incident and
scattered light) is $1:\cos^{2}\Theta$ (Chandrasekhar, 1950). However, in
spherically symmetric envelopes, the contribution of light polarized in
different directions cancels out resulting in zero net polarization.

Most stars are known to be more or less spherical, but the rapid rotation
could break the sphericity and lead to an axisymmetric density structure. If
electron scattering takes place in an axisymmetric envelope, the intrinsic
polarization occurs. From the measurement of Serkowski (1970), few WR stars
have been found to be intrinsically polarized. McLean et al. (1979) discussed
the dependence of the continuum polarization on wavelength in WR stars and
discovered the decrease in polarization across the emission features.

To interpret the observed polarization, Brown & McLean (1977) developed an
analytical method to compute the polarization due to optically thin electron
scattering in axisymmetric envelopes. They assumed that the continuum
polarization mainly depends on the inclination angle toward the line of sight,
the geometry, and the optical depth. Cassinelli et al. (1987) rederived the
polarization expression of Brown & McLean and added the depolarization
correction factor of finite stellar size. Brown et al. (2000) showed that the
depolarization factor has a negligible effect on the value for the
polarization.

In the observational context (Schulte-Ladbeck, 1994), there are several
methods to separate the interstellar polarization and the intrinsic
polarization, each being based on different assumptions. Stevance et al.
(2018) applied Serkowski’s law to fit the interstellar polarization of WR93b
and WR102. They conclude that these two stars are not intrinsically polarized,
hence no line effect could be found, resulting from the dilution of continuum
polarization by unpolarized lineflux. They placed the upper limit on the
rotational velocity, which is lower than the value determined by Sander et al.
(2012) from the shape of emission line profiles, where WR93b and WR102 are
among a small group of WR stars with round emission line profiles, possibly
implying fast rotation. High rotational velocities are required to form a
collapsar, which is likely a source of long duration gamma-ray bursts (Woosley
et al., 1999).

In this work, we discuss the limits of the rotational velocity of WR stars
from polarimetry, in addition to the dependence of polarization on the
inclination angle and rotational velocity. Our study is based on hydrodynamic
stellar wind models of the selected WR stars WR93b and WR102 that include the
effect of rotation. The obtained density was used to calculate the residual
polarization. In Section 2, we describe the method by which the polarization
was calculated. Sections 3-4 provide the discussion of the results with some
comparison to the results of Stevance et al. (2018). In the last section, we
provide the conclusions and some suggestions that might prove fruitful in the
future.

##  2 Computational method

###  2.1 Hydrodynamic model

Stellar wind of hot stars is described by coupling radiative force with
hydrodynamic model equations. The radiative force is based on the Sobolev
approximation for stellar wind (Sobolev, 1960; Castor, 1974; Cranmer & Owocki,
1995). The hydrodynamic model equations describing the wind expressed in
spherical geometry are as follows in the axisymmetric case (Owocki et al.,
1994):

$\displaystyle\frac{\partial\rho}{\partial
t}+\frac{1}{r^{2}}\frac{\partial(r^{2}\rho\varv_{r})}{\partial
r}+\frac{1}{r\sin\theta}\frac{\partial(\rho\varv_{\theta}\sin\theta)}{\partial\theta}$
$\displaystyle=0,$ (1) $\displaystyle\frac{\partial\varv_{r}}{\partial
t}+\varv_{r}\frac{\partial\varv_{r}}{\partial
r}+\frac{\varv_{\theta}}{r}\frac{\partial\varv_{r}}{\partial\theta}-\frac{\varv_{\theta}^{2}+\varv_{\phi}^{2}}{r}$
$\displaystyle=-\frac{1}{\rho}\frac{\partial P}{\partial
r}+g_{r}^{\text{ext}},$ (2)
$\displaystyle\frac{\partial\varv_{\theta}}{\partial
t}+\varv_{r}\frac{\partial\varv_{\theta}}{\partial
r}+\frac{\varv_{\theta}}{r}\frac{\partial\varv_{\theta}}{\partial\theta}-\cot\theta\frac{\varv_{\phi}^{2}}{r}+\frac{\varv_{r}\varv_{\theta}}{r}$
$\displaystyle=-\frac{1}{r\rho}\frac{\partial
P}{\partial\theta}+g_{\theta}^{\text{ext}},$ (3)
$\displaystyle\frac{\partial\varv_{\phi}}{\partial
t}+\varv_{r}\frac{\partial\varv_{\phi}}{\partial
r}+\frac{\varv_{\theta}}{r}\frac{\partial\varv_{\phi}}{\partial\theta}+\cot\theta\frac{\varv_{\theta}\varv_{\phi}}{r}+\frac{\varv_{r}\varv_{\phi}}{r}$
$\displaystyle=g_{\phi}^{\text{ext}},$ (4) $\displaystyle P$
$\displaystyle=c_{\text{s}}^{2}\rho,$ (5)

where $\rho$ is the density of the fluid, $\varv_{r}$, $\varv_{\theta}$, and
$\varv_{\phi}$ are the velocity components in every direction, $c_{\text{s}}$
is the sound speed, and $P$ is the gas pressure. For a star with a radius
$R_{\star}$, mass $M_{\star}$, luminosity $L_{\star}$, and temperature
$T_{\star}$, the external force $\@vec{g}^{\text{ext}}$ is the vector sum of
the line radiative force $\@vec{g}_{\text{line}}$ and the effective gravity
$\@vec{g}_{\text{eff}}$.

The line force $\@vec{g}_{\text{line}}$ at the position $\@vec{r}$ was
obtained by integrating numerically the stellar intensity $I$ over the solid
angle $\Omega_{\star}$ weighted by the velocity gradient (Cranmer & Owocki,
1995)

$\@vec{g}_{\text{line}}(\@vec{r})=\frac{k\sigma_{\text{T}}^{1-\alpha}}{\rho(\@vec{r})^{\alpha}c\varv_{\text{th}}^{\alpha}}\int_{\Omega_{\star}}(\@vec{n}\cdot\mathbf{\nabla}[\@vec{n}\cdot\@vec{\varv}(\@vec{r})])^{\alpha}\@vec{n}\,I(\@vec{n},\@vec{r})\,\text{d}\Omega.$
(6)

Here, $\alpha$ and $k$ are the Castor et al. (1975, hereafter CAK) and Abbott
(1982) parameters for the radiative force, respectively, $\sigma_{\text{T}}$
is the cross section due to Thomson scattering on free electrons,
$\varv_{\text{th}}$ is the thermal velocity, and $c$ is the speed of light. We
note that the line force also depends on the dilution factor due to ionization
which has been removed here as the $\delta$ parameter can be put equal to zero
(see Table 2).

To solve the hydrodynamic model equations (1) – (5), we used the VH-1 (Blondin
et al., 1990) piecewise parabolic method (PPM) code coupled with the
subroutine developed by Owocki et al. (1996) to determine the radiative force.
The well-known PPM algorithm was developed by Colella & Woodward (1984), and
it is the third-order finite difference scheme. The VH-1 code solves the
hydrodynamic equations in 1D, 2D, and 3D planar, cylindrical, and spherical
geometry. For our problem, we used 2.5D spherical geometry, with three
velocity components ($\varv_{r}$, $\varv_{\theta}$, $\varv_{\phi}$), where the
flow is specified along the radial and colatitudinal coordinate components
($r$ and $\theta$). The grid is discretized into $n_{i}=320$ zones in radius,
from stellar surface $r_{\min}=R_{\star}$ to $r_{\max}\approx 10R_{\star}$.
The latitudinal direction is subdivided into $n_{j}=100$ equidistant zones
from the north pole ($\theta=0$) to the south pole ($\theta=\pi$).

The most important factor for hydrodynamic simulation of stellar winds is to
define the lower boundary condition in the radial direction to introduce the
oblate stellar surface due to rotation. Owocki et al. (1994) proposed a
specified stair-casing boundary condition to solve this difficulty We have
chosen the same configuration to formulate our problem. To assure a subsonic
inflow of matter at the base of the wind, we set the base wind density to a
constant value proportional to the mass loss rate $\dot{M}$,

$\rho(r=R_{\star})\propto\frac{\dot{M}}{4\pi R_{\star}^{2}c_{\text{s}}},$ (7)

whereas for the velocity component, we assume a subsonic outflow for polar
velocity and rigid body rotation for the azimuthal velocity. To control the
oblateness of the stellar surface, the radius was set as a function of
colatitude $\theta$ and the ratio of rotational velocity and the critical
velocity $\omega$ (Collins & Harrington, 1966; Owocki et al., 1994)

$\frac{R_{\star}(\theta)}{R_{p}}=\frac{3}{\omega\sin\theta}\cos\left(\frac{\pi+\arccos(\omega\sin\theta)}{3}\right),$
(8)

where $R_{\star}(\theta)$ is the surface radius and $R_{p}$ is the polar
radius. For the upper boundary, we set an outflow condition. Because of the
symmetry, we set a reflecting boundary conditions (default from VH-1 code) in
the latitudinal direction.

###  2.2 CAK parameter optimization

In order to make the numerical simulation of the two WR stars, we have to know
the CAK parameters of the line force. To our knowledge, there has not been a
previous calculation of line force parameters for these two stars. Therefore,
to get the value of $\alpha$ and $k$, we required the terminal velocity and
mass loss rate from observations. We varied the line force parameters to fit
the observational values. We adopted the stellar parameters from Tramper et
al. (2015) and Stevance et al. (2018, see Table 1). The selected values for
CAK parameters are given in Table 2. The critical velocity is
$1734\,\text{km}\,\text{s}^{-1}$ for WR93b and
$2286\,\text{km}\,\text{s}^{-1}$ for WR102 (Stevance et al., 2018).

The mass loss rate corresponding to the selected CAK parameters is $5.8\times
10^{-7}\,M_{\odot}\,\text{yr}^{-1}$ for WR93b and $4.7\times
10^{-7}\,M_{\odot}\,\text{yr}^{-1}$ for WR102, which is lower than what was
derived from observations. Therefore, for the calculation of the polarization,
we multiplied the density by the ratio of the observed mass loss given in
Table 1 to the numerical mass loss from the hydrodynamic simulation. We
multiplied the final density by a factor of $17$ for WR93b and $24$ for WR102.
We could increase the mass loss rate to the observational values (e.g.,
including the $\delta$ parameter), but running the model with rotation leads
to a nonphysical structure of the density and the velocity, as we could get
higher velocity at the polar regions and some numerical instability occurred
inside the computational domain. The obtained terminal velocity is about
$6000\,\text{km}\,\text{s}^{-1}$ for WR93b and around
$8000\,\text{km}\,\text{s}^{-1}$ for WR102.

Table 1: Adopted stellar and wind parameters. WR | $\log(L_{\star}/L_{\odot})$ | $T_{\star}$ | $R_{\star}$ | $M_{\star}$ | $\varv_{\infty}$ | $\log(\dot{M}/1\,M_{\odot}\,\text{yr}^{-1})$
---|---|---|---|---|---|---
|  | (kK) | $(R_{\odot})$ | ($M_{\odot}$) | ($\text{km}\,\text{s}^{-1}$) |
93b | 5.30 | 160 | 0.58 | 7.1 | 5000 | -5
102 | 5.45 | 210 | 0.39 | 7.0 | 5000 | -4.92

We ran the simulations for different rotational velocities. To check the
influence of individual assumptions on the intrinsic polarization, we
calculated several sets of wind models with an increasing level of complexity.
First, we included radial force only leading to a compression effect as
discussed by Bjorkman & Cassinelli (1993). Within these assumptions, an
equatorial density enhancement forms, as found by Owocki et al. (1994).
Second, we added the nonradial forces, which inhibit the formation of the disk
(Owocki et al., 1996). In the most complex model, we also included the gravity
darkening. Figure 1 shows the density contours as a function of the radius and
colatitude for $V_{\text{rot}}=900\,\text{km}\,\text{s}^{-1}$; the results are
similar to those found by Owocki et al. (1996) and show a reduced density at
the equator due to the gravity darkening and the effect of nonradial forces.
Our aim is not to investigate the equatorial disk formation, but rather we
want to make use of the hydrodynamic simulation in order to obtain the
resulting polarization. In the following section, we show how the continuum
polarization can be calculated from the wind density $\rho$ derived from the
hydrodynamic model.

Table 2: Adopted CAK parameters. | $\alpha$ | $k$ | $\delta$
---|---|---|---
WR93b | 0.52 | 0.61 | 0
WR102 | 0.52 | 0.61 | 0
Figure 1: Density (in g$\cdot$cm-3) contours of stellar wind of the star WR93b
versus radius $r$ and colatitude $\theta$, in log scale spaced by 0.8 dex,
denoted contour correspond to $\log(\rho)=-11.2$, for rotation
$V_{\text{rot}}=1100\,\text{km}\,\text{s}^{-1}$, with nonradial forces and
gravity darkening.

###  2.3 Polarization model

We intend to model the continuum polarization due to electron scattering, as
we expect that the polarization will vanish in the emission lines.
Polarization from radiation scattering depends on the integral of electron
density and the volume of scattering. The analytic expression of the
polarization was developed by Brown & McLean (1977), which depends on the
inclination angle, optical depth, and geometry factor, as follows:

$P_{R}\approx\bar{\tau}(1-3\gamma)\sin^{2}i,$ (9)

where $\bar{\tau}$ is the averaged Thomson scattering optical depth defined as
the integral of the number density over the radius and the colatitudinal angle
$\theta$ (as shown in Fig. 2, see also Brown et al. 1978),

$\bar{\tau}=\frac{3}{16}\sigma_{\text{T}}\int_{r_{1}}^{r_{2}}\int_{\mu_{1}}^{\mu_{2}}n_{\text{e}}(r,\mu)\,\text{d}r\,\text{d}\mu.$
(10)

The factor $\gamma$ in Eq. (9) is given by

$\gamma=\frac{\int_{r_{1}}^{r_{2}}\int_{\mu_{1}}^{\mu_{2}}n_{\text{e}}(r,\mu)\,\mu^{2}\,\text{d}r\,\text{d}\mu}{\int_{r_{1}}^{r_{2}}\int_{\mu_{1}}^{\mu_{2}}n_{\text{e}}(r,\mu)\,\text{d}r\,\text{d}\mu},$
(11)

where $\mu=\cos\theta$, $i$ is the inclination angle, and $n_{\text{e}}$ is
the electron number density, which depends on the mass density $\rho$;

$n_{e}=\frac{\rho}{\mu_{\text{e}}m_{\text{H}}},$ (12)

where $m_{\text{H}}$ is the hydrogen atom mass and $\mu_{\text{e}}$ is the
mean molecular weight per free electron (cf. also the calculation of
polarization signatures in Kurfürst et al., 2020). To take the depolarization
effect of finite stellar geometry into account, Cassinelli et al. (1987)
introduced the correction factor $D(r)=\sqrt{1-R^{2}_{\star}/r^{2}}$, and the
polarization relation becomes the following:

$P_{R}=\frac{3}{16}\sigma_{\text{T}}\sin^{2}i\int_{r_{1}}^{r_{2}}\int_{\mu_{1}}^{\mu_{2}}n_{e}(r,\mu)(1-3\mu^{2})D(r)\,\text{d}r\,\text{d}\mu.$
(13)

Stellar winds of WR stars are optically thick even in the vicinity of the
sonic point. However, Eq. (13) accounts for just the polarization due to
optically thin scattering. Therefore, we selected the lower limit of
integration in Eq. (13) as $r_{1}=1.2R_{\star}$ to exclude the optically thick
part of the wind. The upper limit corresponds to the outer boundary of
numerical simulations, $r_{2}=10R_{\star}$.

###  2.4 Polarization due to surface temperature variation

The shape of the star changes due to the effect of rotation becomes oblate by
the effect of centrifugal forces, and the emergent flux experiences a
redistribution so that the polar regions become bright, and the equator
becomes dark, what is known as the effect of gravity darkening. These effects
can be specified by the known Roche and von Zeipel models for early type stars
(Harrington & Collins, 1968; Cranmer & Owocki, 1995) and they affect the
radiative intensity in the wind. The anisotropic intensity introduces an
additional source of polarization.

The von Zeipel theorem (Collins, 1963; Harrington & Collins, 1968; Cranmer &
Owocki, 1995) shows that the local flux $F(\theta)$ on the surface of a
rotating star is proportional to local effective gravity $g(\theta)$

$F(\theta)=\sigma_{B}T_{\text{eff}}^{4}(\theta)\propto
g_{\text{eff}}(\theta),$ (14)

where $\sigma_{B}$ is the Boltzmann’s constant, $T_{\text{eff}}(\theta)$ is
the effective temperature depending on colatitude, and the magnitude of
surface gravity is expressed as

$g_{\text{eff}}(\theta)=\frac{GM_{\star}}{R_{p}^{2}}\left[\left(-\frac{1}{x^{2}}+\frac{8}{27}\omega^{2}x\sin^{2}\theta\right)^{2}+\left(\frac{8}{27}\omega^{2}x\sin\theta\cos\theta\right)^{2}\right]^{\frac{1}{2}},$
(15)

where $GM_{\star}/R_{p}^{2}$ is the polar gravity term at the critical
velocity. von Zeipel theorem does not provide the constant of proportionality
and many authors (Collins, 1963; Cranmer & Owocki, 1995) fitted this constant
as a function of $\omega$; we assume this constant to be unity for simplicity.

If we assign the temperature to a point in the wind, the specific intensity
$I$ can be computed using

$I_{0}(\@vec{r},\@vec{n})=B_{\nu}(T_{\text{eff}}(\theta)),$ (16)

where $B_{\nu}$ is the Planck function of a black body.

From the work of Brown & McLean (1977, Eq. (20)), we have the intrinsic
polarization

$P_{R}\approx\frac{I_{1}}{I_{0}}P_{1},$ (17)

where $I_{1}\ll I_{0}$ and $P_{1}$ is the polarization of the scattered
intensity $I_{1}$

$P_{1}=\frac{3\pi\sigma_{T}\sin^{2}i}{16}\frac{\int_{r_{1}}^{r_{2}}\int_{\mu_{1}}^{\mu_{2}}n_{\text{e}}(r)B_{\nu}(T_{\text{eff}}(\theta))[1-3\mu^{2}]\text{d}r\text{d}\mu}{I_{1}}.$
(18)

Replacing $P_{1}$ and $I_{0}$ and by their expressions, we get

$P_{R}=\frac{3\pi\sigma_{T}\sin^{2}i}{16}\frac{\int_{r_{1}}^{r_{2}}\int_{-1}^{1}n_{\text{e}}(r)(1-3\mu^{2})B_{\nu}(T_{\text{eff}}(\theta))\text{d}r\text{d}\mu}{B_{\nu}(T_{\star})}.$
(19)

It is possible to add the depolarization $D(r)$ effect due to finite stellar
size as previously in Eq. (13).

###  2.5 Numerical integration

In order to numerically evaluate the integrals, we have used the trapezoidal
rule as implemented in the Numpy library (Harris et al., 2020). Moreover, we
used Pandas for data analysis (McKinney, 2011) and Matplotlib for plotting
(Hunter, 2007).

Calculations were carried out for various values of rotational velocities. The
results are shown in Figs. 3 – 6, where the polarization $P_{R}$ is plotted as
a function of the inclination angle $i$ for different fractions of the
critical velocity.

Figure 2:  Geometry of the plane of integration in spherical geometry
($r$,$\theta$,$\phi$) adopted from Brown et al. (2000).

##  3 Results

Stevance et al. (2018) applied the simplified model of wind-compressed zones
by Ignace et al. (1996) to investigate the dependence of the linear
polarization on the stellar rotation. In our model, we included gravity
darkening and oblateness of the star due to rotation effects; the model is
based on the work of Cranmer & Owocki (1995) and Owocki et al. (1996).

Figures 3– 6 show an increase in absolute values of the polarization by
increasing the rotational velocity and the inclination angle. For higher
rotational velocities, the departure from spherical symmetry becomes stronger,
which leads to higher linear polarization. The inclination is another crucial
parameter to observe the intrinsic polarization. When the inclination tends to
zero, the system is viewed in the direction from the pole, the distribution of
matter becomes symmetric with respect to the viewing direction, and the
polarization drops to zero. The polarization reaches its maximum when the
system is viewed equator on and when the angle coincides with the scattering
plane. The signature of polarization is given by the geometry. The signature
is positive for oblate disk-like geometry, while it becomes negative for
prolate jet-like geometry.

The upper limit for rotation velocities was set to $19\%$ and $10\%$ (of the
critical velocity) for both stars in the work of Stevance et al. (2018). This
was determined from the observed value of the polarization, which is less than
0.077% for WR93 and less than 0.057% for WR102. For a comparison with their
results, we additionally employed the analytical model by Ignace et al. (1996)
to calculate the density structure of the wind. We calculated two sets of
hydrodynamical models, one where the density was numerically computed
including the radial force only, which corresponds to the analytical model.
The second set of more realistic models includes the gravity darkening and
nonradial components of the radiative force.

Due to a nonphysical structure which appears for low rotational velocities, we
chose to calculate the polarization for the two stars with a higher rotational
velocity than the ones employed by Stevance et al. (2018), as seen in Figs. 3
– 4. Figure 3 shows the intrinsic polarization for WR93b, where the values
obtained from the hydrodynamic simulation with only radial forces of about
$10\%$ are higher than for the analytic model viewed equator on, while for the
model with gravity darkening and nonradial forces, the polarization is $2\%$
in absolute value. Figure 4 illustrates the variation of the polarization with
inclination for WR102 star, the polarization from the hydrodynamic model is
$18\%$, which is higher than from the analytical model ($2.52\%$), while from
nonradial forces and gravity darkening it is $0.61\%$ in absolute value. In
general, the polarization is positive for oblate density distribution and this
shows that the plane of polarization is parallel to the symmetry axis. The
polarization is negative when obtained from the model with gravity darkening,
and this indicates that there is a higher density of material at the polar
region than at the equatorial region, and the plane of polarization is
perpendicular to the symmetry axis.

For the hydrodynamic model with the radial force only, we chose to integrate
from the radius $r_{1}=1.2R_{\star}$; the reason is that the oblateness of the
stellar geometry, due to rotation and lower boundary conditions, rises the
polarization degree and gives overestimated values. Wood et al. (1993)
calculated the polarization for a Be star and they developed a relationship to
get an estimate of the degree of the polarization. Applying equation (39),

$P_{R}=\frac{3\sigma_{\text{T}}}{32\pi}\frac{\dot{M}}{2\pi
m_{p}R_{\star}\varv_{\infty}};$ (20)

from their paper to the stellar parameters, we see that $P_{R}\approx 8.6\%$
for WR93b and $P_{R}\approx 18\%$ for WR102, which shows a good agreement with
the hydrodynamic model.

Figure 3: Percentage polarization of WR93b from the model with radial force
only (hydro with radial), nonradial forces, and gravity darkening (hydro with
full model), compared to the model of the Ignace et al. (1996) wind compressed
disk (WCD) for $V_{\text{rot}}=900\,\text{km}\,\text{s}^{-1}$. Figure 4:  As
in Fig. 3, but for WR102.

Figures 5–6 show the variations of the polarization for the stars WR93 and
WR102, respectively, and for different rotational velocities by adding
nonradial forces and the gravity darkening. For the reason stated above, we
chose to integrate from $r_{1}=1.2R_{\star}$. The polarization is negative,
which shows a prolate structure of the geometry, and for low rotational
velocities, the intrinsic polarization tends to $-0.95\%$ for WR93b and to
$-0.17\%$ for WR102. Increasing the rotational velocity decreases the net
polarization; for a rotation about $60\%$ of the critical velocity, we get
$P_{R}=-2.4\%$ and $P_{R}=-1.32\%$ for WR93b and WR102, respectively.

To obtain the upper limits for rotational velocity, we carried out a
logarithmic least square fitting for the model with nonradial forces and
gravity darkening. Table 3 shows a comparison of upper limits of rotational
velocity, at the inclination angle $90^{\circ}$ (edge-on), from our model and
the model of Stevance et al. (2018).

Calculating the angular momentum $j$ from the obtained rotational velocity,
and comparing this with the threshold $j\geq 3\times
10^{16}\,\text{cm}^{2}\,\text{s}^{-1}$ of the collapsar derived by MacFadyen &
Woosley (1999) in Table 3, the lower limits of angular momentum exceed the
threshold of MacFadyen & Woosley (1999). Therefore, the studied WR stars may
be progenitors of a long gamma-ray burst. This agrees with the results of
Gräfener et al. (2012b), who determined the rotational velocities from
photometry, and with the original results of Stevance et al. (2018).

Table 3: Comparison between the fitted maximum observationally allowed rotational velocity (in $\text{km}\,\text{s}^{-1}$) in this model and the model of Stevance et al. (2018). The last column gives the maximum specific angular momentum determined from the maximum allowed rotational velocity. WR |  $P_{R}$ (%, observation) |  $V_{\text{rot}}$ (Stevance’s model) |  $V_{\text{rot}}$ (Current model) | $\log(j/1\,\text{cm}^{2}\,\text{s}^{-1})$
---|---|---|---|---
93b | 0.077 | 324 | 277 | 17.88
102 | 0.057 | 234 | 444 | 17.85
Figure 5: Polarization of WR93b as a function of the inclination for different
rotational velocities (denoted in the graph in units of
$\text{km}\,\text{s}^{-1}$) determined from the models including nonradial
forces and gravity darkening.  Figure 6: As in Fig. 5, but for WR102.

In hydrodynamic simulations, we have considered an isothermal wind and assumed
that variations of the effective temperature all over the stellar surface did
not contribute to the polarization. In order to see the effect due to the
change in the temperature, we computed the intrinsic polarization from Eq.
(19), where the temperature changes due to the effective gravity (gravity
darkening) and the number density is assumed to be radial dependent
($n_{\text{e}}(r)\propto\dot{M}/r^{2}v(r)$). For WR93b star with rotation
$\omega=20\%$ of the critical velocity, the obtained polarization was found to
be $0.0027\%$, which is smaller than the upper limit obtained from the
observation (Stevance et al., 2018), and this means that this effect is not
sufficient to break the symmetry of the star. On the other hand, for the
critical rotation, the polarization can reach up to $14.52\%$. For WR102 with
rotation $\omega=20\%$, the polarization is similar to the previous star and
has a value of $0.006\%$, and the rotation at critical velocity leads to a
polarization with the value of $32.5\%$.

##  4 Discussion

We have determined the polarization due to electron scattering from
hydrodynamical models of optically thin winds. The analysis involves several
simplifications, including the neglect of the optically thick part of the wind
on the density distribution and polarization which are perhaps the most
significant ones. Therefore, in this section, we discuss the effect of the
optical depth on the density distribution and polarization, as WR stars are
known to have optical thick winds.

###  4.1 Density distribution

The adopted density distribution was derived from hydrodynamical models
appropriate for winds which are optically thin in continuum. However, winds of
WR stars may be optically thick and the radius of the hydrostatic core could
be significantly lower than the radius of the unity mean optical depth
(Gräfener et al., 2012a; Poniatowski et al., 2021). If we consider the effect
of gravity darkening, the density has a prolate distribution; however, in the
optically thick regime, the radiative diffusion effects may further complicate
the structure of the flow (Gayley, 2004). Using the model outlined in Owocki
et al. (1998), and Dwarkadas & Owocki (2002), which was adopted to optically
thick winds, the mass flux is expressed as

$\dot{m}(\theta)=\dot{m}(\theta=0)(1-\omega^{2}\sin^{2}\theta),$ (21)

where $\dot{m}(0)$ is the mass flux at the pole. The mass conservation implies
that in addition to the mass flux, the wind speed would be modified by the
rotational effect. The wind terminal speed is expressed as a function of
colatitude,

$\varv_{\infty}(\theta)=\varv_{\infty}(\theta=0)\sqrt{1-\omega^{2}\sin^{2}\theta},$
(22)

where $\varv_{\infty}(0)$ is the terminal speed at the pole. For
$\omega=20\%$, the polarization is $-0.48\%$ and $-1.03\%$ for WR93b and for
WR102, respectively; also, at the critical rotation, the polarization is
$-20.5\%$ for WR93b and $-43\%$ for WR102. Comparing this with the
hydrodynamic model including nonradial forces and gravity darkening, the
polarization values showed comparable results, where, for example,
$\omega=30\%$, $P_{R}=-1.1\%$, and $P_{R}=-2.3\%$ and from hydrodynamic with
nonradial forces and gravity darkening $P_{R}=-0.95$ and $P_{R}=-0.11$ for
WR93b and for WR102, respectively. We note that in the hydrodynamic model, we
integrated from $r=1.2R_{\star}$ as mentioned earlier. This indicates that the
effect of modifying the wind density due to the optical thickness of WR star
winds does not significantly change the main results of our paper.

###  4.2 Optical depth

The polarization does not only depend on the inclination or rotational
velocity, but it also depends on the opacity of the medium. In our model, we
assumed the optical thin regime, but using Monte Carlo radiative transfer,
Carlos-Leblanc et al. (2019) show that the polarization increases by
increasing the optical depth ($\tau>1$). Due to their high mass loss rate, WR
stars have an optical thick wind even to the electron scattering. To
investigate the effect of the optical depth, using a more detailed treatment
of the polarized radiative transfer, for instance with the Monte Carlo method
of the radiative transfer, is needed. A work is under way using the latter
method in couple with the hydrodynamic model, but we note that our models
account for the strongest effect which is due to the last scattering in an
optically thin envelope.

We assumed that the stars produce unpolarized light, which propagates into the
wind where each photon scatters once at most. This is a simplification,
especially for the winds of WR stars with a high density. However, multiple
scattering leads to depolarization; therefore, we selected integration radii
outside the optically thick envelope.

###  4.3 Magnetic field effect and the ionization structure

In this work, we did not consider the magnetic field. As a result of the
magnetic field, the wind flows along the field lines, the mass-flux depends on
the magnetic field tilt, and the circumstellar environment deviates from
spherical symmetry (Ud-Doula et al., 2008; Küker, 2017). This should also
result in intrinsic polarization. Also, we neglected the variations in
electron density due to the change of ionization, which is a reasonable
approximation when the surface abundances are dominated by helium and for
relatively high effective temperatures.

To maintain the acceleration around the sonic point, high effective
temperatures are needed (Gräfener & Hamann, 2005),which allow for strong
dependence of the iron peak element opacity on the temperature in WR stars. As
showed by Gräfener & Hamann (2005), the ”Iron Bump” opacity may lead to an
optically thick wind. Another factor that could impact the wind optical depth
is the combination between the Eddington limit and the limit of rotation. As
discussed by Maeder & Meynet (2000), and Gräfener (2021) and known as the
$\Gamma\Omega$ limit, for high $\Gamma$, just moderate rotation is sufficient
to increase the mass loss rates and as consequence the wind becomes optically
thick. These effects can further modify our results.

##  5 Conclusions

In this work, we used the outputs of the density distribution from the
hydrodynamic model to calculate the intrinsic polarization for WR93 and WR102
for different rotational velocities. Oblate density distribution leads to a
positive polarization, while prolate distribution leads to a negative
polarization, and this agrees with the model predicted by Brown & McLean
(1977). The intrinsic polarization strongly depends on the rotational
velocities and the inclination.

Based on our assumption, where we have scaled the density to take realistic
values of mass-loss rates of studied stars into account, we show that fast
rotation leads to a prolate density distribution around WR stars. This
produces just a weak polarization signature, which is smaller than the upper
limit of polarization from observations for low rotational velocities.
Therefore, the limits of rotational velocities of WR stars from polarization
do not contradict models of a long duration gamma-ray burst, according to
which the fast rotating WR stars are progenitors of gamma ray bursts.

###### Acknowledgements.

The authors thank Profs. R. Ignace and S. Owocki for discussion. The access to
computing and storage facilities owned by parties and projects contributing to
the National Grid Infrastructure MetaCentrum, provided under the program
“Projects of Large Infrastructure for Research, Development, and Innovations”
(LM2010005) is appreciated. This research was supported by the grant
Primus/SCI/17.

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# COVID-19 and Science Communication: The Recording and Reporting of Disease
Mortality

Ognjen Arandjelović  University of St Andrews
St Andrews KY16 9SX
United Kingdom
Tel: +44(0)1334 46 28 24
E-mail<EMAIL_ADDRESS>
Web: http://oa7.host.cs.st-andrews.ac.uk/

The author declares no conflict of interest. The author is the sole
contributor to all aspects of the present work, including research, analysis,
and manuscript preparation. There is no funding source to declare. No ethical
approval was needed.

###### Abstract

The ongoing COVID-19 pandemic has brought science to the fore of the public
discourse and considering the complexity of the issues involved, with it also
the challenge of effective and informative science communication. A
particularly contentious topic, in that it is both highly emotional in and of
itself, as well as in that it sits at the nexus of the decision-making process
regarding the handling of the pandemic, which has effected lockdowns, social
behaviour measures, business closures, and others, concerns the recording and
the reporting of the disease mortality. To clarify a point which has caused
much controversy and anger in the public debate, the first part of the present
article discusses the very fundamentals underlying the issue of causative
attribution with regards to mortality, lays out the foundations of the
statistical means of mortality estimation, and concretizes these by analysing
the recording and reporting practices adopted in England, and their widespread
misrepresentations. The second part of the article is empirical in nature. I
present data and an analysis of how COVID-19 mortality has been reported in
the mainstream media in the UK and the USA, including a comparative analysis
both across the two countries as well as across different media outlets. The
findings clearly demonstrate a uniform and worrying lack of understanding of
the relevant technical subject matter by the media in both countries. Of
particular interest is the finding that with a remarkable regularity
($\rho>0.998$) the greater the number of articles a media outlet published on
COVID-19 mortality, the greater the _proportion_ of its articles
misrepresented the disease mortality figures.

###### keywords:

COVID-19, epidemic, methodology, decision-making, public policy.

##  1 Introduction

The crucial role that science plays in our everyday lives is hardly something
that needs to be emphasised even to the general public [1, 2]. Moreover, in
the economically developed world, science is for the most part seen as a
positive actor [3]: it is science that has helped us avoid or overcome
previously widespread diseases and illnesses; it is science that has
facilitated our being able to communicate across vast distances using video
and audio; it is science that has made travel fast, efficient, and accessible,
allowing many to explore relatively cheaply distant parts of the globe; it is
science that has made access to large swaths of knowledge freely and readily
accessible to most; and so on. Unsurprisingly, polling consistently shows that
scientists too are seen in a positive light [4]. The negative aspects of
science, which the public does recognize (correctly or incorrectly), are
largely well confined to specific realms: the pace of lifestyle change [1],
applications seen as ‘playing God’ or ‘playing with nature’ (e.g. genetic
modification, creation of new life forms) [5, 6], or rogue actors’ misuse
thereof [2].

However, over the last two years, that is since the emergence of COVID-19, the
place that science plays in our lives appears to have changed substantially.
From the largely benevolent supporting actor working in the background,
supporting, facilitating, and enhancing various everyday pursuits we
undertake, science has come to the fore and is being used to justify – for
better or worse, I state this in a value free sense – in modern times
virtually, if not literally unprecedented restrictions on people’s freedoms in
countries with historically liberal values. Science is used to justify the
prohibition to leave one’s residence [7], to legislate compulsory for the
cessation of normal business operations [8], to impose bans on socializing
with others [7], etc. When science is placed at the crux of decision-making
that effects such severe harmful effects (I am referring to the aforementioned
restrictions themselves only, which are undoubtedly harmful, rather than the
net effect thereof which may very well be beneficial), it is unsurprising that
the public starts to take interest in the relevant science, and seeks to
understand and scrutinize it [9]. Yet, this endeavour is fraught with
difficulty. Firstly, considering the breadth and the depth of competence
required to understand the relevant processes and phenomena to an extent
whereby this understanding (and thereby I am not referring merely to the
knowledge of procedural or factual matters, that is veritism [10], but actual
_understanding_ [11]) is sufficient to facilitate a meaningful critical
assessment of experts’ views, the notion that this competence could be
attained in a short period of time by the general public is rather absurdly
naïve. Secondly, and perhaps this is where the greatest danger lies, it is
this lack of understanding of science and the scientific method, that makes
the lay public unaware of the limitations of its knowledge, making ill-founded
arguments appear convincing and credible, and decreasing trust in the rational
scientific authority [12, 13]. In this context, effective science
communication is crucial, whether it comes from politicians, scientists
themselves, or the media. The overarching message of the present paper is that
science communication in these delicate and febrile times have been found
wanting. Inept and misleading communication, often stemming from a lack of
understanding of the subject matter itself, has resulted in undue (in the
specific context considered herein, which is not to dismiss other, possibly
correct criticisms [14, 15]) public scepticism towards scientific advice, a
reluctance to adopt and follow guidance, increased discontent [16], etc.

Herein I focus on the specific issue of communication regarding COVID-19
mortality, a particularly emotive topic at the nexus of the decision-making
processes which led to the great number of the aforementioned restrictive and
far-reaching measures aimed at dealing with the pandemic, and pivotal in
shaping the public’s attitude and behaviour. I start by discussing the very
fundamentals underlying the issue of causative attribution with regards to
mortality, that is what it means that a person ‘has died of’ something, in
Section 2. Having clarified this notion which has caused much controversy, to
say nothing of anger on all sides of the debate played out in public, and in
particular having explained why the phrasing is epistemologically
inappropriate when applied on a personal level, I lay out the foundations to
the statistical means of mortality estimation on the cohort, or population,
level in Section 2.1. In Section 2.2 I concretize these statistical approaches
by analysing the COVID-19 recording and reporting practices adopted in
England, and explain why certain measures were adopted and how they were
misrepresented first and foremost by the media, but also by some scientists
when communicating with the public. The second part of the article, namely
Section 3, is empirical in nature. Specifically, I present data and analyse
how COVID-19 mortality has been reported in the mainstream media in the UK and
the USA, including a comparative analysis both across the two countries as
well as across different media outlets. The findings of the analysis clearly
demonstrate a uniformly and worrying lack of understanding of the relevant
technical subject matter by the media in both countries. Finally, Section 4
presents a summary of the key points of the article and its conclusions.

##  2 On the quantification of mortality rates

It is unsurprising that the mortality of the novel coronavirus SARS-CoV-2 –
that is the disease it causes, COVID-19 – has quickly become the primary
measure of the virus’s direct impact. Mortality is a simple measure in that it
is underlain by a binary outcome (death or survival) and it is easily
understood by the general public. It is also a highly emotive one.

To a non-specialist, the mortality of a disease also appears as being easily
measurable: it is a simple process involving little more than the counting of
deaths deemed to have resulted from an infection with SARS-CoV-2. However,
underneath this seemingly straightforward task lie a number of nuances. The
primary one of these is presented by the question of when a death can be
attributed to the virus. Even to laypersons who may not be able to express the
reasons behind their reckoning, it is readily evident that this question is
different than, say, that of asking when somebody has died of a gunshot wound,
e.g. as used for the reporting of firearm murders. The key difference between
the two stems from the former being a distal and the latter a proximal factor.
In seeking a link between distal causes and the corresponding outcomes of
interest, the analysis of causality is complicated by the complexity effected
by numerous intervening and confounding factors. Thus, to give a simple
example, while it may be a relatively straightforward matter to establish
respiratory failure as the proximal cause of death, it is far less clear when
COVID-19 can be linked to it as the distal cause, and to what extent,
especially if the patient has pre-existing conditions.

The key insight stemming from the above is that indeed, it is fundamentally
impossible to claim with certainty that any _particular_ death was caused by
COVID-19. Rather, the approach has to be on _cohort_ (or population) based
analysis. The basic idea is reasonably simple: in order to assess how a
particular factor of interest affects survival, a comparison is made between
cohorts which differ in the aforementioned factor but which are otherwise
statistically matched in the potentially relevant characteristics. Indeed, the
entire field of study usually termed ‘survival analysis’ [17], widely used in
biomedical sciences and engineering amongst others, is focused on the
development of techniques that can be utilized for such analysis and which are
suitable to different scenarios (for example, for problem settings when not
all data is observable, or for different types of factor of interest such as
discrete or continuous, etc.). In the specific case that we are considering
here, the situation is rather straightforward in principle, in that the factor
of interest is also binary, namely an individual in a cohort either has had or
has not had a positive SARS-CoV-2 diagnosis in the past.

While an in-depth overview of survival analysis is out of scope of the present
article, for completeness and clarity it is useful to illustrate just some of
the more common methods used to this end in the literature and practice. An
understanding of these will help set the ground for the topics discussed
thereafter, namely how SARS-CoV-2 deaths should be recorded and how they
should be reported, and why the two are different.

###  2.1 Survival analysis

####  2.1.1 Kaplan–Meier estimation

The frequently used Kaplan–Meier estimator [18, 19, 20] is a non-parametric
estimator of the _survival function_ within a cohort. The survival function
$s(t)$ captures the probability of an individual’s death being no earlier than
$t$:

$\displaystyle s(t)=Prob(\tau>t),$ (1)

where $\tau$ is the time of death of an individual in the cohort, treated as
an outcome of a random variable. In the simplest setting, the challenge is
thus of estimating $s(t)$ given the set $\left\\{\tau_{j}\right\\}_{j=1\ldots
n}$ where $\tau_{j}$ is the time of death of the $j$-th individual in a cohort
numbering $n$. It is assumed that the outcomes corresponding to different
individuals are independent from one another, and identically distributed.
However, usually not all $\tau_{j}$ are available because the time of analysis
precedes the death of at least some individuals in a cohort. In our specific
example, many of the individuals whose data is analysed, whether previously
infected with SARS-CoV-2, will not die for many years in future. To account
for this, rather than assuming that all $\tau_{j}$ are known, the estimate is
sought from a set of pairs $\left\\{(\hat{\tau}_{j},c_{j})\right\\}_{j=1\ldots
n}$ where $c_{j}$ are the censoring times. The censoring time $c_{j}$ is the
latest time for which the survival or non-survival of the $j$-th individual is
known; thus, $\hat{\tau}_{j}$ is meaningful only if $\hat{\tau}_{j}\leq c_{j}$
(e.g. the time of death of those individuals who have not died by the time of
analysis is not known). It is important to emphasise the assumption that
censoring is non-informative, i.e. that censoring statistics in both cohorts
are identical.

It is then a straightforward matter to derive the following estimate for
$s(t)$ (for further technical detail and a step by step derivation see e.g.
the work of Goel et al. [21]):

$\displaystyle\hat{s}(t)=\prod_{i;t_{i}<t}\left(1-\frac{d_{i}}{n_{i}}\right)$
(2)

with $t_{i}$ a time when at least one death occurred, $d_{i}$ the number of
deaths that happened at time $t_{i}$, and $n_{i}$ the individuals known to
have survived up to time $t_{i}$. The plot in Figure 1 shows a typical example
of two survival functions obtained in this manner (the specific example is
from a study of immunological features in muscle-invasive bladder cancer).
Both estimates, the red and the blue one, start at 1, as all participants in
the study are initially alive (and hence, by design, the probability of being
alive is 1). Thereafter, the faster decline of the red curve as compared with
the blue one, captures a more rapid death rate in the cohort corresponding to
the former, i.e. a lower probability of survival past a certain point in the
future.

Figure 1: Example of the Kaplan-Meier derived estimate of two survival
functions (red and blue lines). The corresponding shared areas indicate the
standard deviations of the estimates across time.

####  2.1.2 Cox’s regression

Another widely used technique for survival analysis is Cox’s regression (also
often referred to as Cox’s proportional hazards model), which adopts a
somewhat different, semi-parametric approach from that of the Kaplan-Meier
estimator in several important ways. Firstly, unlike in the case of the
latter, no explicit stratification of the entire patient cohort is performed.
Rather, the same effect is achieved statistically. Secondly, the approach is
multivariate rather than univariate in nature, which makes the method more
appropriate for many real-world analyses when it is not possible to perform
randomization or to ensure the satisfaction of other criteria required for
Kaplan-Meier analysis.

Cox’s regression for survival analysis in its general form is a method for
investigating the effect of several variables on the time of death. Central to
it is the concept of the _hazard function_ , $h(t)$, also called the hazard
rate, which is the instantaneous death rate in a cohort (often, it is
incorrectly described as the probability of death at a certain time [22, 23]).
It is modelled as a product of the baseline hazard function, $h_{0}(t)$ and
the exponential of a linear combination covariates (that is, factors of
interest), $\left\\{x_{i}\right\\}_{i=1\ldots p}$:

$\displaystyle h(t)$
$\displaystyle=h_{o}(t)\times\exp{\left\\{w_{1}x_{1}+w_{2}x_{2}+\ldots
w_{p}x_{p}\right\\}}$ (3) $\displaystyle=h_{o}(t)\times\exp{\sum_{i=1\ldots
p}w_{i}x_{i}}$ (4)

The coefficients $\left\\{w_{i}\right\\}_{i=1\ldots p}$ can be interpreted as
quantifying the effect of the corresponding covariates and can be inferred
from data by means of partial maximum likelihood estimation over all observed
deaths. The resulting _hazard ratios_ $\left\\{\exp{w_{i}}\right\\}_{i=1\ldots
p}$ provide a simple way of interpreting the findings, with values around 1
indicating a lack of effect of the factor, and those greater or lesser than 1
respectively, increased and decreased associated hazard of death. A factor in
this analysis can be, for example, the presence of a specific disease (e.g.
COVID-19), a particular demographic characteristic, a comorbidity of interest,
etc. Thus, Cox’s analysis allows us to interrogate the data as regards the
effect of specifically, say, a past positive test for COVID-19, on mortality,
adjusted for other factors which too may affect it.

Note that both methods described, namely both Cox’s regression and
Kaplan–Meier estimation, are statistical and therefore phenomenological in
nature, as opposed to mechanistic – neither approach models the underlying
processes that effect the connection of observable predictor data (e.g. the
presence of a historical positive test for COVID-19, or one’s sex, age, etc.)
with the outcome of interest (time of death).

###  2.2 The recording and reporting of deaths

An understanding of the technical basics of survival analysis covered in the
previous section erects a solid basis for the consideration of how the
recording and the reporting of deaths due to a distal cause should be
performed. Indeed, the latter have attracted considerable attention and
criticism.

The recording of SARS-CoV-2 deaths during the ongoing pandemic has varied
across different jurisdictions in a multitude of ways. My focus here is not on
the many practical aspects of this process (e.g. how deaths in different
settings such as hospitals, homes, and care homes are aggregated, etc.), as
important as these are, but on its fundamental, methodological underpinnings
which are unaffected by geographical, social, and similar factors. In this
regard, we find rather more uniformity so I will use England as a
representative example.

####  2.2.1 England

Up to August 2020, for England the COVID-19 Data Dashboard reported all deaths
in people who have had a prior laboratory confirmed positive SARS-CoV-2 test.
Thereafter, two further indicators were included, namely the numbers of deaths
of individuals ($\alpha$) who have had their first positive test within 28
days of dying, and ($\beta$) either who have had their first positive test
within 60 days of dying _or_ who have had COVID-19 on the death certificate
(as recorded by a registered medical practitioner).

The reasoning underlying these choices and the evolution of the reporting
system is straightforward to understand. The original intention was to ensure
a high degree of confidence in there having been a SARS-CoV-2 infection in a
deceased person; hence the requirement of a laboratory confirmed positive
SARS-CoV-2 test. The subsequent expansion of the reporting parameters can be
seen as an attempt to account for those deceased individuals, possibly many of
them, who have been infected but were never actually tested for the virus. It
is difficult to argue that these reporting choices are anything other but
sensible. It is in the use and the interpretation of them that the nuance
lies, as I discuss shortly.

####  2.2.2 Survival analysis…again

Although as I noted earlier, the techniques outlined in Section 2.1 provide a
good basis for contextualizing the recording of SARS-CoV-2 related deaths in
England, it is important to observe that neither Kaplan-Meier estimation nor
Cox’s regression can be applied in the context just described out of the
proverbial can – further thought is needed to adapt the methods to the problem
at hand. In particular, note that as described both approaches are prospective
in nature, in that observation begins at a set time for a known cohort. In
contrast, in the problem setting of interest the question is retrospectively
posed. This challenge is not new and can be addressed in a principled and
robust manner by extending the original statistical models. The relevant
technical details are involved and not necessary to go into detail herein –
the interested reader would be well advised to consult the work of Prentice
and Breslow [24] or Copas et al. [25] for example.

Nevertheless, there is an important a difficulty of a practical nature which
emerges in the context of COVID-19 and which is often one that one has to
contend with in many other epidemiological settings. Specifically, the
selection of non-diseased individuals is far from straightforward. The reason
for this lies in the potentially asymptomatic presentation of the disease. As
a consequence, there is a possibility of the non-diseased cohort actually
containing individuals who have had COVID-19 at some point, but were unaware
of it. An important yet subtle observation that is key to make here is that
this data contamination does not act merely so as to reduce the accuracy of
analysis or reduce the uncertainty of the conclusions. Rather, there is a
_systematic_ bias which is introduced. In particular, observe that because it
is the asymptomatic individuals, i.e. those with the least disease severity
and thus the most optimistic prognosis, who are removed from the COVID-19
positive cohort, the overall prognosis of the nominal COVID-19 positive cohort
is made to appear worse than it would have been had the asymptomatic cases
been included. As a corollary, any analysis applied is likely to produce an
overestimate, rather than an underestimate or an unbiased estimate, of
COVID-19 mortality. Including a model of the source of bias in the overall
statistical model is difficult because the key variables underlying the
phenomenon are latent by their very nature.

####  2.2.3 Piecing it all together

My closing remark in Section 2.2.1, namely that the COVID-19 reporting choices
in England are eminently sensible may have resulted in some readers raising
their brow. Indeed, when the reporting protocol was first published, the
sciolist mainstream (as well as non-mainstream) media was quick to point out
that somebody who died after being hit by a bus and who has had a recent
positive COVID-19 test, would be included in the reported numbers. This has
been widely repeated and a few examples serve well to illustrate the gist of
the argument. Thus, the Daily Mail, the highest circulation daily newspaper in
the UK following the Sun, reported [26]:

> “…if, for example, somebody tested positive in April but recovered and was
> then hit by a bus in July, they would still be counted as a Covid-19
> victim.”

Rowland Manthorpe, an editor at Wired magazine who has written for the
Guardian, Observer, Sunday Telegraph, Spectator, etc., speaking for Sky News
[27] echoed the thoughts:

> “Essentially, there is no way to recover, statistically. So, if I tested
> positive for COVID-19 today and then I got hit by a bus tomorrow, then
> COVID-19 would be listed as my cause of death.”

The reiteration was not limited to media personalities. For example, in an
article provocatively entitled “Are official figures overstating England’s
Covid-19 death toll?” the Guardian [28] reported that an unnamed Department of
Health and Social Care source summed up the process as:

> “You could have been tested positive in February, have no symptoms, then be
> hit by a bus in July and you’d be recorded as a Covid death.”

At first sight, these criticisms do not seem entirely unreasonable. Why
_would_ a person who was killed by being hit by a bus be counted as a COVID-19
death? The Guardian, to its credit, did report an apparent defence by a
‘source at Public Health England’, who was quoted as saying that:

> “…such a scenario would ‘technically’ be counted as a coronavirus death,
> ‘though the numbers where that situation would apply are likely to be very
> small’.”

This is a rather feeble response; and a misleading one too. The implication is
that those deaths indeed _should not_ have been recorded but that their
infrequency renders the matter of little practical significance. That is
incorrect. If these deaths are indeed entirely confounding, statistical
analyses such as those outlined earlier, would have found them to be such – in
these cases, they would indeed present as noise in the data and be practically
insignificant. However, there is another possibility, which is particularly
important when dealing with novel and poorly understood diseases. Imagine if
statistical analysis _did_ reveal that previously COVID-19 positive people die
in greater numbers by being hit by vehicles than their disease free
counterparts; in other words, that there was statistical significance to this
observation. This would have suggested possibly new knowledge about the virus
and its effects. For example, it could have indicated that the virus has long-
lasting neurological effects which would affect one’s ability to respond in
traffic. Indeed, now we _do_ know that SARS-CoV-2 is a neurotropic virus with
a whole host of neuropathological effects including dizziness, decreased
alertness, headaches, seizures, nausea, cognitive impairment, encephalopathy,
encephalitis, meningitis, anosmia, etc. [29]. Yet, an explanation of this kind
was woefully missing from the mainstream coverage. Also, still working within
the premises of this hypothetical scenario, even before any mechanistic
understanding is developed, the mere new knowledge that in some way or
another, past COVID-19 positivity predicts pedestrian deaths in traffic, as
any knowledge, can only be advantageous. For example, it could lead to timely
advice to the public to take additional care in appropriate situations.

To summarize, the recording of COVID-19 related deaths should indeed include
all deaths, whatever their proximal cause may be or appear to be, of
individuals tested positive any time in the past and these numbers should be
reported to the relevant bodies. Thus, if anything, the criteria for the
recording of COVID-19 related deaths were insufficiently rather than
excessively inclusive. However, the _reporting_ , i.e. the communication of
COVID-19 mortality to the general public should based on robust statistical
analyses on the cohort level. It should be a cause of profound concern that
even such publications as Scientific American failed in observing what is
little more than rudimentary rigour as regards this [30], incorrectly
claiming:

> “Nearly 800,000 people are known to have died of COVID-19. [all emphasis
> added]”

##  3 Analysis of mainstream media

Having equipped ourselves with the understanding of how data on disease
effected mortality ought to be collected and recorded, and interpreted and
reported, we are now in good position to turn our attention to the analysis of
the reporting of COVID-19 mortality by the mainstream media. I do so in the
present section. Its structure follows the usual pattern – I start by
describing my data collection protocol in Section 3.1, then report my analysis
of this data in Section 3.2, and finally discuss the findings in Section 3.3.

###  3.1 Data collection

Data was collected from the web sites of the top circulation national daily
newspapers in the UK and the USA, as well as the web sites of the two leading
UK television broadcasters, namely the BBC (British Broadcasting Corporation)
and ITV (Independent Television; legally Channel 3). Sister publications, e.g.
the Sun and the Sun on Sunday or the Daily Mail and Mail On Sunday, were
considered jointly under the name of the main brand. The collection of all
data was conducted on the 23rd of August 2020. In particular, I recorded the
number of articles on each web site containing the exact phrases ‘deaths with
COVID’, ‘deaths from COVID’, ‘deaths of COVID’ (n.b. the search was not case
sensitive). A summary of the raw data can be found in Tables 1 and 2.

| Brand name | ‘with’ | ‘from’ | ‘of’ | ‘from’+‘of’
---|---|---|---|---|---
Newspapers | The Metro | 96 | 431 | 2820 | 3251
The Sun | 59 | 416 | 288 | 704
Daily Mail | 463 | 9120 | 16000 | 25120
The Times | 36 | 147 | 275 | 422
The Mirror | 3360 | 10700 | 41400 | 52100
The Telegraph | 222 | 627 | 788 | 1415
TV | BBC | 4210 | 5850 | 4200 | 10050
ITV | 1650 | 5860 | 3440 | 9300
Table 1: Summary of UK data.  | Brand name | ‘with’ | ‘from’ | ‘of’ | ‘from’+‘of’
---|---|---|---|---|---
Newspapers | USA Today | 64 | 657 | 861 | 1518
WSJ | 1 | 250 | 3780 | 4030
NY Times | 4 | 5250 | 16600 | 21850
New York Post | 1 | 403 | 612 | 1015
LA Times | 1 | 1540 | 9720 | 11260
Washington Post | 91 | 2670 | 2000 | 4670
Star Tribune | 246 | 2380 | 2480 | 4860
Newsday | 0 | 263 | 1140 | 1403
Chicago Tribune | 9 | 2400 | 7780 | 10180
The Boston Globe | 7 | 229 | 872 | 1101
Table 2: Summary of USA data.

###  3.2 Data analysis

I started by looking at the proportion of articles by the publication which
used the incorrect ‘deaths of COVID’ or ‘deaths from COVID’ phrasing. For the
UK based publishers, averaged across publishers, that is, looking at:

$\displaystyle\frac{1}{n_{publishers}}\sum_{i=1}^{n_{publishers}}\frac{n_{i}^{incorrect}}{n_{i}^{correct}+n_{i}^{incorrect}},$
(5)

where $n_{publishers}$ is the number of different publishers, and
$n_{i}^{correct}$ and $n_{i}^{incorrect}$ the number of publisher $i$’s
articles with the correct (‘with COVID’) incorrect (‘from COVID’ or ‘of
COVID’) phrasings, 89.4% of articles contained an incorrect formulation, with
the corresponding standard deviation being 9.0%. In the USA, the corresponding
average was found to be 98.8%, with the standard deviation of 1.8%. Taking
into account the different numbers of articles which contained any of the
search terms, that is:

$\displaystyle\frac{\sum_{i=1}^{n_{publishers}}n_{i}^{incorrect}}{\sum_{i=1}^{n_{publishers}}\left(n_{i}^{correct}+n_{i}^{incorrect}\right)},$
(6)

the overall proportion of incorrectly phrased articles in the UK was 91.0% and
in the USA 99.3%.

Using the thee one-sample Kolmogorov-Smirnov test [31], the distributions of
per-publisher proportions of incorrectly phrased articles were confirmed to be
log-normal, both for the UK and the USA, at the confidence levels $p=0.0197$
and $p=0.0078$ respectively. The two-sample Kolmogorov-Smirnov test [32]
further confirmed that the UK and the USA distributions different
significantly, that is the null hypothesis of the two sample sets coming from
the same log-normal distribution was rejected at the confidence level
$p=0.0058$.

Lastly, the plots in Figure 2 and Figure 2 show the relationships between the
number of incorrectly phrased articles and the total number of articles
reporting COVID related deaths for the UK and the USA respectively. The
corresponding Pearson’s $\rho$ was found to be $0.998$ for the UK and $1.000$
for the USA.

Figure 2: Relationship between the number of incorrectly phrased articles and
the total number of articles reporting COVID related deaths for the (a) UK and
(b) the USA. A highly linear behaviour is readily observed, with the
corresponding Pearson’s $\rho$ equal to $0.998$ and $1.000$ respectively.

###  3.3 Discussion

To begin with a few brief but important comments on the data collection,
firstly it should be noted that for the same of uniformity and like-for-like
comparison, all data was collected on the same day, namely the 23rd of August
2020, and it includes all historical articles, i.e. all articles published on
or before that date. Although the exact date in question was selected at
random, it was deliberately chosen not to be too early in the pandemic, as a
reasonable argument could have been made that journalists, largely unequipped
with the kind of expertize needed to understand the communicate the highly
technical and to everyday reporting novel kind of information, needed a period
of adjustment and learning. This possible objection is fully addressed by the
choice of the 23rd of August 2020, which is some 9 months following the first
identification of COVID-19 in December 2019 [33] and nearly 6 months following
the declaration of a pandemic by The World Health Organization (WHO) on the
11th of March 2020 [34].

The most immediately apparent finding of my analysis in the previous section
is that of the strikingly high proportion of articles which incorrectly
described COVID-19 deaths as being ‘from’ or ‘of’ COVID-19. Perhaps even more
remarkably, this was found to be the case across all analysed media outlets
and both in the UK and the USA. Interestingly though, although in both
countries all but a few articles used the incorrect phrasings, the
transgression in communication was significantly worse in the USA (as I also
confirmed statistically in the previous section) – in the UK approximately 1
in 11 articles did use the correct ‘with’ phrasing, whereas in the USA it was
fewer than 1 in 100. Indeed, USA’s Newsday did not have a single correctly
worded article despite reporting on COVID-19 mortality in 1403 articles, and
WSJ, New York Post, and LA Times had only a single correctly worded article
each, despite respectively 4030, 21850, and 11260 articles on the topic. The
only significant outlier, in a positive sense yet with a still exceedingly
high proportion of incorrectly phrased articles of approximately 70.5%, is the
BBC; the BBC was approximately twice as likely to use the correct phrasing in
reporting COVID-19 mortality than the next best outlet and more than 3 times
as likely as the analysed UK media outlets on average. None of the USA outlets
stand out from the rest, even the most accurate one, namely USA Today, using
the correct wording in only approximately 4% of its articles.

Perhaps the most concerning finding of my analysis concerns the relationship
between the proportion of incorrect reporting by a media outlet and the
outlet’s volume of COVID-19 mortality reporting, summarized by the plots in
Figure 2 and Figure 2. As stated in the previous section, we find that with a
remarkable regularity (Pearson’s $\rho$ for the UK and the USA outlets being
$0.998$ and $1.000$ respectively) the greater the number of articles an outlet
published on COVID-19 mortality, the greater the _proportion_ (n.b. not the
absolute number, which would be expected) of articles which used an incorrect
phrasing with respect to it. There are different reasons which could explain
this. For example, it is possible that outlets whose journalists’ collective
values are more caution driven have as a consequence of that published more on
the topic. This would make the transgression – for a transgression has
certainly taken place in the sense that an incorrect claim was made – entirely
of an intellectual, rather than ethical nature. On the other hand, it is
impossible to dismiss the possibility of a more sinister cause, such as
increased fear oriented reporting being driven by commercial or other
interests. Numerous other, more complex reasons are possible too. Considering
that the available data do not allow us to favour one hypothesis over another,
it would be inappropriate to speculate on the topic; nevertheless, it is
important to note the trend and highlight it as an important one for future
research. Lastly, note that my analysis found no relationship between the
circulation of a newspaper and the corresponding COVID-19 mortality reporting
phrasing accuracy.

##  4 Summary and conclusions

The coronavirus disease 2019 pandemic has placed the science in the spotlight.
However, science does not exist as an immaterial Platonic form – research is
performed by humans, science based decisions are made by humans, and science
is communicated by humans. Human fallibility cannot be taken out of science.
Scientists, decision-makers, and communicated make not only genuine errors,
but also have egos, compete for jobs and prestige, exhibit biases, hold
political and broader philosophical beliefs, etc., all of which affect how
science is materialized and used. A failure to recognize faults in these
processes when they occur can only serve to undermine the general public trust
in science and its application [35, 14].

In this article I focused on a specific and highly relevant aspect of science
communication in the context of the COVID-19 pandemic, namely that of the
disease mortality reporting. Considering the amount of confusion and anger
across the spectrum that the issue has caused, I began with a discussion of
what it means to say that a person has died _from_ a disease. In particular, I
explained that while this may be an acceptable phrasing in some circumstances,
strictly speaking it is not a meaningful one and as such should generally be
avoided. Indeed, this nuance is crucial in situations when there is a
significant interaction between different causal factors, as is the case of
COVID-19 mortality. Hence, I clarified how mortality ought to be assessed,
namely on a cohort (population) basis, and summarized the technical
fundamentals which underpin the necessary analysis used to arrive at such
estimates. The developed insight was concretized with an analysis of the
controversy causing COVID-19 mortality recording and reporting process in
England. I showed how the decisions made were widely misunderstood and
incorrectly interpreted in the media, and demystified the decisions underlying
the process.

The second part of the article turned its attention to the quantitative
analysis of COVID-19 mortality reporting by the mainstream media, namely by
the top circulation daily newspapers in the UK and the USA, as well as the
UK’s two leading TV broadcasters, the BBC and the ITV. The results are
striking: in both countries the vast majority of articles incorrectly reported
COVID-19 deaths, erroneously attributing a causative link between COVID-19 and
the death of any person with a past positive COVID-19 test. This effect was
observed with a remarkable uniformity over the different outlets considered,
with the transgression was significantly worse in the USA than the UK
(approximately 89% vs 99% respectively).

The effects of poor science communication of the kind considered in this paper
must not be underestimated. Not only does incorrect information provide a
faulty basis for individual decision-making, but can also penetrate the
highest levels of legislative and executive branches of government. Indeed, I
leave the reader with three poignant examples. The first of these is from the
address to the nation in February 2021 by the president of the USA, Joe Biden
[36]:

> “Today we mark a truly grim, heartbreaking milestone – 500,071 dead.”,

the second one from the UK’s Prime Minister Boris Johnson statement on
coronavirus delivered on the 26th of January 2021 [37]:

> “I am sorry to have to tell you that today the number of deaths recorded
> from Covid in the UK has surpassed 100,000… [all emphasis added]”,

and the last one from the speech by Keir Starmer, then Leader of the Labour
Party, at the Labour Party Conference 2021 [38]:

> “We have now lost 133,000 people to Covid. [all emphasis added]”

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