# Supernova Polarization Signals From the Interaction with a Dense
Circumstellar Disk

Xudong Wen School of Physics and Astronomy, Beijing Normal University, Beijing
100875, China  Institute for Frontier in Astronomy and Astrophysics, Beijing
Normal University, Beijing 102206, China<EMAIL_ADDRESS>He Gao*
School of Physics and Astronomy, Beijing Normal University, Beijing 100875,
China  Institute for Frontier in Astronomy and Astrophysics, Beijing Normal
University, Beijing 102206, China<EMAIL_ADDRESS>Yi Yang (杨轶)  Physics
Department, Tsinghua University, Beijing, 100084, China  Liangduan Liu
Institute of Astrophysics, Central China Normal University, Wuhan 430079,
People’s Republic of China  Key Laboratory of Quark and Lepton Physics
(Central China Normal University), Ministry of Education, Wuhan 430079,
People’s Republic of China  Shunke Ai Department of Astronomy, School of
Physics and Technology, Wuhan University, Wuhan 430072, China  Zongkai Peng
School of Physics and Astronomy, Beijing Normal University, Beijing 100875,
China  Institute for Frontier in Astronomy and Astrophysics, Beijing Normal
University, Beijing 102206, China

###### Abstract

There is increasing evidence that massive stars may exhibit an enhanced mass
loss shortly before their termination explosion. Some of them also indicate
the enhancement of their circumstellar matter (CSM) is not spherically
symmetric. Supernova (SN) interacting with aspherical CSM could induce special
polarization signals from multiple radiation components that deviate from
spherical symmetry. We investigate the time-evolution of the continuum
polarization induced by the SN ejecta interacting with a disk/torus-like CSM.
Our calculation suggests that the interaction between the SN ejecta and an
immediate disk-like CSM with a thin, homogenous density structure would
produce a high continuum polarization, which may reach a peak level of
$\displaystyle\sim$15%. The interplay between the evolving geometry of the
emitting regions and the time-variant flux ratio between the polar ejecta and
the equatorial CSM interaction may produce a double-peaked feature in the
polarization time sequence. A similar trend of the time evolution of the
polarization is also found for a radially extended CSM disk that exhibits a
wind-like density structure, with an overall relatively lower level of
continuum polarization ($\displaystyle<2.5\%$) during the interaction process.
We also identify a non-uniform temperature distribution along the radial
direction of the CSM disk, which yields a strong wavelength dependence of the
continuum polarization. These signatures provide a unique geometric diagnostic
to explore the interaction process and the associated extreme mass loss of the
progenitors of interacting transients.

Supernovae (1668); Polarimetry(1278)

##  1 Introduction

Modern high-cadence wide-field optical surveys (e.g., Bellm et al., 2019;
Graham et al., 2014) are opening up a new era in our understanding of
supernovae (SNe) and various types of transients by extending the boundaries
of the transient phase space. Lightcurves and follow-up observations spanning
from the first few hours to years or even decades after the transient
explosion enable systematic search and comprehensive characterization of any
signatures of the expanding ejecta interacting with the circumstellar matter
(CSM). CSM in the immediate vicinity of SNe would be ionized by the initial
shock breakout flash and manifest as bright and narrow emission lines of
elements at rather high ionization states. Spectroscopy before such short-
lived ‘flash’ features swept away by the ejecta reveals that Type II and Ib
SNe are commonly surrounded by a shell of matter from enhanced pre-explosion
mass loss Quimby et al. (2007); Gal-Yam et al. (2014); Yaron et al. (2017);
Bruch et al. (2021, 2023). Additionally, core-collapse (CC) SNe may also
embedded in a substantial amount of dense CSM that extends to a rather large
distance from the progenitor. They would sine for years owing to radiation
from the expanding ejecta that interacts with the H-rich CSM, e.g., the Type
IIn SNe. Moreover, superluminous supernovae (SLSNe) (Inserra et al., 2018) and
fast optical transients (Smith et al., 2007; Ho et al., 2019; Ofek et al.,
2010; Moriya et al., 2013) reveal occasional extreme mass loss of massive
stars during the final evolutionary phase. It is currently believed that
stellar instability (Yoon & Cantiello, 2010; Quataert & Shiode, 2012; Fuller,
2017) and binary interaction (Chevalier, 2012; Soker & Kashi, 2013), including
the Roche-lobe overflow (RLOF Mohamed & Podsiadlowski, 2007), which leads to
an elevated mass-loss in a binary system compared to that of the homogeneous
stellar wind, are potential mechanisms causing the formation of CSM. However,
the origin of mass loss of massive stars during the final evolutionary phase
remains unclear, posing one of the biggest challenges in stellar evolution
theory.

Substantial effort has gone into investigating the ejecta-CSM interaction
process under a spherically-symmetric framework (Chevalier, 1982; Chevalier &
Fransson, 1994; Moriya et al., 2013; Dessart et al., 2015; Chevalier &
Fransson, 2017; Liu et al., 2017; Margalit & Ho, 2022), including numerical
and analytical work to predict the lightcurves and spectra from the
interaction process (Chevalier & Irwin, 2011; Ginzburg & Balberg, 2012;
Morozova et al., 2017; Suzuki et al., 2020; Metzger, 2022), as well as
modelling to explain specific events (Ofek et al., 2010; Chatzopoulos et al.,
2013; Moriya et al., 2013). However, these models become inapplicable when the
geometry of the CSM deviates from spherical symmetry, which could be described
more properly with multi-dimensional calculations. In the binary system
scenarios where massive stars are prevalent, there could be equatorially-
confined CSM disks formed by mass transfer and common envelope evolution
(Chevalier, 2012; Pejcha et al., 2016; Metzger & Pejcha, 2017). Such an
intensive CSM enrichment process towards the orbital plane takes place as the
companion fills its Roche lobe. For example, the early-time optical and X-ray
properties and polarization evolution of AT2018cow can be attributed to the
ejecta interacting with CSM confined in a compact and surrounding disk
(Margutti et al., 2019; Maund et al., 2023). The strong early polarization
observed in some Type II SNe originates from the asymmetry of the disk-like
CSM Bilinski et al. (2023). Some of the H-poor superluminous supernovae
(SLSNe) present evidence of both H-rich (Yan et al., 2017) and H-poor CSM
(Pursiainen et al., 2022), the latter may also exhibit a high degree of
asphericity as indicated by its high continuum polarization.

Deviations from spherical symmetry of the CSM can introduce high uncertainty
in the observable properties of interacting transient sources. In particular,
the lightcurve shape and the emission line profile could be highly dependent
on the viewing angle. Major departures from spherical symmetry could also lead
to an inadequate estimate of the bolometric luminosity and the total mass
content of the CSM (see, e.g., Yang et al., 2023). Despite numerical (Nagao,
Maeda & Ouchi, 2020; Uno et al., 2023) and hydrodynamic calculations
(McDowell, Duffell & Kasen, 2018; Kurfürst & Krtička, 2019) were carried out
to characterize the photometric and polarimetry properties of the SN ejecta
interacting with an aspherical CSM, there is a lack of multi-dimensional
exploration using polarization signals, in particular, the polarization
evolution throughout the process that the ejecta engulfing an aspherical CSM
distribution. One of the simplest representations of asphericity yields the
CSM centered on the SN and exhibits a density concentration towards a circular
disk. This configuration possesses a single axial symmetry, which has been
inferred from the well-defined dominant axis on the spectropolarimetric
$\displaystyle Q-U$ plane for various cases. Such a disk-like CSM can be
naturally expected as a consequence of binarity (e.g., Vasylyev et al., 2023,
2024; Mauerhan et al., 2024).

Polarimetry provides a powerful diagnostics of the SN explosion geometry
(Shapiro & Sutherland, 1982; Höflich, 1991; Kasen, 2003; Wang & Wheeler,
2008). As the SN ejecta expands and interacts with the CSM disk, additional
radiation emitted from the interaction zone would be integrated into the
observed bolometric signal of the SN. Wen et al. (2023) proposed that two-
component radiation would produce an observable polarization signal that is
dependent on the viewing angle and the distribution of the emitting zone on
the photosphere. The process of interacting with a CSM disk may induce a
polarized signal similar to that produced by the large-scale inhomogeneous
emission injection through multiple radiation components.

In this paper, we investigate the polarization properties of SN interacting
with a dense disk-like CSM, which could be used as a probe to explore the CSM
geometry. We use a 3-dimensional (3D) Monte Carlo polarization simulation code
(MCPSC) to calculate the lightcurves, spectra, and polarization signals
generated from SN ejecta and CSM interactions, where a dense disk-like CSM is
centered at the progenitor star. We investigate the polarization and its
temporal evolution for different CSM disk structures, namely across a range of
opening angles of the CSM measured from the center of the exploding progenitor
star. We also examine the viewing-angle dependence of various observables.
Finally, we discuss the feasibility of using wavelength-dependent polarization
signals as a new method to study CSM geometry and decipher the SN pre-
explosion mass history.

##  2 Methods

Figure 1: Schematic illustration of the SN ejecta interacting with a dense,
torus-like CSM concentrated to an arbitrarily defined equatorial plane. When
the interaction takes place with a half-tension angle $\displaystyle\theta$,
the global geometry is divided into the interacting CSM disk (region I) and
the expanding ejecta along the poles (region II). And the ejecta freely
expanding towards the regions perpendicular to the plane of the CSM torus. As
the ejecta continues to expand, it will overrun the the CSM disk and may even
engulf the CSM disk.

In this section, we describe the landscape landscape of the ejecta-CSM
interaction, with the latter exhibits a density enhancement towards an
arbitrarily defined plane, namely the equatorial plane. When a massive star
explodes as an SN, a total mass $\displaystyle M_{\rm ej}$ with an explosive
energy $\displaystyle E_{\mathrm{SN}}$ is ejected. We assume that the ejecta
undergoes a homologous expansion with velocity $\displaystyle v_{\rm{SN}}$ and
interact with a CSM disk with inner boundary $\displaystyle
R_{\text{CSM,in}}$. Two shocks are formed in the interaction region: a forward
shock (FS) wave that traverses the CSM, and a reverse shock (RS) wave that
sweeps over the supernova ejecta (Chevalier, 1982; Chevalier & Fransson,
1994). This interaction converts kinetic energy into radiation, injecting
additional energy into the equatorial plane. The emitting zone can therefore
be divided into two parts, namely the interaction (Region I,
$\displaystyle\pi/2-\theta_{\rm csm}<\theta<\pi/2+\theta_{\rm csm}$)), and the
polar (Region II, ($\displaystyle|\pi/2-\theta|>\theta_{\rm csm}$) regions,
where $\displaystyle\theta_{\rm csm}$ denotes the half-opening angle of the
CSM disk measured from the equatorial direction (see, Figure 1). The emission
of the former zone is a combination of the ejecta-CSM interaction and the
radioactive decay of Fe-group elements. The radiation of the latter region
would be entirely dominated by radioactive decay.

In the case of CSM that extended to a large distance from the progenitor star
(i.e., $\displaystyle\gtrsim 10^{15}$ cm), the early phase of the interaction
takes place at the shocked region. The unshocked CSM exposed outside the SN
photosphere thus forms a torus along the equatorial plane. As the ejecta
continues to expand, it will sweep across the entire CSM disk and expand
radially outward, forming a shape characterized by elongated poles and a
shorter equator. The interaction process terminates when the shock wave sweeps
across the outer radius of the CSM $\displaystyle R_{\text{CSM,out}}$.
Eventually, as suggested by certain numerical simulation results, the ejecta
may completely engulf the CSM as suggested by numerical calculations
(McDowell, Duffell & Kasen, 2018). During this process, the relative strengths
of the light emitted from the two regions, as well as the non-spherical
geometry of the CSM and SN ejecta, may give rise to observable polarisation
signals.

In order to describe the temporal evolution of the polarization properties, we
conduct three-dimensional calculation of the polarization properties using the
Monte Carlo Polarization Simulation Code (MCPSC, Wen et al., 2023). The code
computes the flux and the polarization spectra by simulating the random walk
of $\displaystyle N_{\rm{MC}}$ Monte Carlo (MC) photon packets through the
propagating medium. There are $\displaystyle N_{\rm{I}}$ and $\displaystyle
N_{\rm{II}}$ photon packets emitted from the given boundaries in Regions I and
II, respectively, and satisfying $\displaystyle
N_{\rm{MC}}=N_{\rm{I}}+N_{\rm{II}}$. The computational domain is discretized
into a $\displaystyle 100\times 100\times 100$ Cartesian grid to configure the
computing environment.

###  2.1 The Density Profile of the SN Ejecta

Following Matzner & McKee (1999), we adopt a broken power law to describe the
radial density profile of the SN ejecta, i.e.,

$\rho_{\mathrm{ej}}(v,t)=\left\\{\begin{array}[]{ll}\zeta_{\rm{\rho}}\frac{M_{\mathrm{ej}}}{R_{\mathrm{tr}}^{3}}\left(\frac{r}{R_{\mathrm{tr}}}\right)^{-\delta},&r<R_{\mathrm{tr}}\\\
\zeta_{\rm{\rho}}\frac{M_{\mathrm{ej}}}{R_{\mathrm{tr}}^{3}}\left(\frac{r}{R_{\mathrm{tr}}}\right)^{-n},&r\geq
R_{\mathrm{tr}}\end{array}\right.$ (1)

where the transition radius $\displaystyle R_{\text{tr}}$ of the ejecta is
expressed as

$R_{\mathrm{tr}}=\zeta_{\rm{v}}\left(\frac{E_{\mathrm{SN}}}{M_{\mathrm{ej}}}\right)^{1/2}t.$
(2)

The coefficients can be obtained from the normalized continuity condition,
namely

$\zeta_{\rm{\rho}}=\frac{(n-3)(3-\delta)}{4\pi(n-\delta)},\hskip
10.00002pt\zeta_{\rm
v}=\left[\frac{2(5-\delta)(n-5)}{(n-3)(3-\delta)}\right]^{1/2}.$ (3)

For a red supergiant progenitor, the typical values of the density power
indices are $\displaystyle\delta=1,n=12$ (Matzner & McKee, 1999).

###  2.2 The Density Profile of the CSM

Following the prescription of (Suzuki et al., 2019), we describe the density
profile of a torus-like CSM as’

$\rho_{\text{CSM}}\left(r,\theta\right)=qr^{-s}\exp\bigg{[}-\left(\frac{r}{R_{\text{CSM,out}}}\right)^{p}-\left(\frac{\theta}{\theta_{\rm{CSM}}}\right)^{m}\bigg{]}.$
(4)

For the sake of simplicity, we follow Nagao, Maeda & Ouchi (2020) and neglect
the terms that smooth the edge density involving $\displaystyle
R_{\text{CSM,out}}$ and $\displaystyle\theta_{\rm{csm}}$. The density profile
thus yields $\displaystyle\rho_{\text{CSM}}\left(r\right)=qr^{-s}$ along the
radial direction. The scaling factor $\displaystyle q$ can be determined as
$\displaystyle q=\rho_{\text{CSM,in}}R_{\text{CSM,in}}^{s}$, where
$\displaystyle\rho_{\text{CSM,in}}$ denotes the density at the inner radius
$\displaystyle R_{\text{CSM,in}}$ of the CSM. The power-law index of the CSM
density profile yields $\displaystyle s$, which $\displaystyle s=0$ and
$\displaystyle s=2$ characterize a thin shell and a homogeneous wind,
respectively (Chatzopoulos et al., 2012). The mass of the CSM disk
$\displaystyle{M}_{\rm{disk}}$ can be written as
$\displaystyle{M}_{\rm{disk}}={M}_{\rm{CSM,sphere}}\sin{\theta_{\rm{csm}}}$,
where $\displaystyle{M}_{\rm{CSM,sphere}}$ is the total mass of the spherical
CSM.

###  2.3 Shock dynamics and luminosity input

The dynamical evolution of shock waves along the radial direction of the CSM
disk follows the self-similar solutions found by Chevalier (1982). The radius
of the forward ($\displaystyle R_{\mathrm{sh,FS}}$) and reverse shocks
($\displaystyle R_{\mathrm{sh,RS}}$) can be expressed as a function of time ,
i.e.,

$\displaystyle R_{\mathrm{sh,FS}}=$ $\displaystyle
R_{\text{CSM,in}}+\beta_{\rm
F}\left[\frac{Ag^{n}}{q}\right]^{\frac{1}{n-s}}t^{\frac{n-3}{n-s}}$ (5)
$\displaystyle(t_{i}<t<t_{i}+t_{\mathrm{FS}}),$

and

$\displaystyle R_{\mathrm{sh,RS}}=$ $\displaystyle
R_{\text{CSM,in}}+\beta_{\rm
R}\left[\frac{Ag^{n}}{q}\right]^{\frac{1}{n-s}}t^{\frac{n-3}{n-s}}$ (6)
$\displaystyle(t_{i}<t<t_{i}+t_{\mathrm{RS}}),$

Where $\displaystyle\beta_{\rm{F}}$, $\displaystyle\beta_{\rm{R}}$, and
$\displaystyle A$ are constants determined by the values of n and s, as
detailed in Table 1 of Chevalier (1982). Here $\displaystyle t_{i}$ denotes
the time when the SN eject start to interact with the CSM, $\displaystyle
t_{\mathrm{FS}}$ and $\displaystyle t_{\mathrm{RS}}$ represent the termination
times of the FS sweeping through all the optically thick CSM material and the
RS sweeping through the SN ejecta, respectively.

Each photon packet is initialized with a specific energy and frequency,
determined by the luminosity and temperature of the emitting region’s
photosphere (Mazzali & Lucy, 1993; Lucy, 1999). In this work, the luminosity
input $\displaystyle L_{\mathrm{Ni},\mathrm{I}/\mathrm{II}}$ in the SN ejecta
from radioactive element decay for regions I and II can be expressed as

$L_{\mathrm{Ni},\mathrm{I}/\mathrm{II}}(t)=M_{\mathrm{Ni},\mathrm{I}/\mathrm{II}}[(\mathrm{\epsilon_{Ni}}-\mathrm{\epsilon_{Co}})e^{-t/\tau_{\mathrm{Ni}}}+\mathrm{\epsilon_{Co}}e^{-t/\tau_{\mathrm{Co}}}],$
(7)

where $\displaystyle\mathrm{\epsilon_{Ni}}=3.9\times
10^{10}\mathrm{erg~{}g^{-1}~{}s^{-1}}$ and
$\displaystyle\mathrm{\epsilon_{Co}}=6.8\times
10^{9}\mathrm{erg~{}g^{-1}~{}s^{-1}}$ are the heating rates of 56Ni and 56Co,
respectively. $\displaystyle\tau_{\mathrm{Ni}}$ = 8.8 days and
$\displaystyle\tau_{\mathrm{Co}}$ = 111.3 days are their decay timescales
(Valenti et al., 2008; Khatami & Kasen, 2019). By assuming 56Ni is evenly
distributed throughout the ejecta, the total ejecta mass $\displaystyle
M_{\mathrm{Ni}}$ that is dispersed in regions I and II give’

$M_{\mathrm{Ni},\mathrm{I}}=M_{\mathrm{Ni}}\sin(\theta_{\mathrm{csm}}),\quad
M_{\mathrm{Ni},\mathrm{II}}=M_{\mathrm{Ni}}-M_{\mathrm{Ni},\mathrm{I}},$ (8)

$\displaystyle M_{\mathrm{Ni}}$ is the mass of all 56Ni contained in the whole
ejecta.

Following Chatzopoulos et al. (2012) and Wang et al. (2019), we take the
luminosity input from the forward ($\displaystyle L_{\rm{FS}}$) and reverse
($\displaystyle L_{\rm{RS}}$) shock waves as

$\displaystyle L_{\rm{FS}}(t)=$
$\displaystyle\frac{2\pi}{(n-s)^{3}}\left(Ag^{n}\right)^{\frac{5-s}{n-s}}q^{\frac{n-5}{n-s}}(n-3)^{2}(n-5)$
(9)
$\displaystyle\beta_{\rm{F}}^{5-s}t^{\frac{2n+6s-ns-15}{n-s}}\quad(t_{i}<t<t_{i}+t_{\mathrm{FS}})$

and

$\displaystyle L_{\rm{RS}}(t)=$ $\displaystyle\frac{2\pi
g^{n}}{\beta_{\rm{R}}^{n-5}}\frac{n-5}{n-3}\left(\frac{3-s}{n-s}\right)^{3}\left(\frac{q}{Ag^{n}}\right)^{\frac{n-5}{n-s}}$
(10) $\displaystyle
t^{\frac{2n+6s-ns-15}{n-s}}\quad(t_{i}<t<t_{i}+t_{\mathrm{RS}}).$

Adopting an approach similar to that of Nagao, Maeda & Ouchi (2020), the total
heating luminosity in the CSM disk produced from the interaction process gives

$L_{\mathrm{CSM,disk}}(t)=[\epsilon_{\rm{FS}}L_{\rm{FS}}(t)+\epsilon_{\rm{RS}}L_{\rm{RS}}(t)]\sin{\theta_{\rm{csm}}},$
(11)

where the $\displaystyle\epsilon_{\rm{FS}}$ and
$\displaystyle\epsilon_{\rm{RS}}$ are the corresponding conversion efficiency
from kinetic energy to radiation for forward and reverse shock waves,
respectively.

###  2.4 The Integrated Emission of the SN

In region $\displaystyle\mathrm{I}$, we consider a hybrid model which the
luminosity input is provided by a combination of the the ejecta-CSM
interaction and the radioactive decay. When calculating the luminosity of the
photosphere embedding the CSM disk, it is necessary to consider the photon
diffusion time from the shock plane to different areas of the photosphere,
leading to variations in local luminosity of the photosphere. Photons in the
CSM disk that are exposed outside the supernova photosphere in region
$\displaystyle\mathrm{II}$ are more likely to escape from the latitudinal
photosphere of the CSM disk. This is because the latitudinal optical depth is
relatively smaller compared to the radial optical depth. We therefore focus on
the luminosity produced from photospheric areas at different radii of the CSM
disk, and neglect the slight effect of the latitudinal optical depth. We trace
the random walk process of $\displaystyle N_{\rm{II}}$ photon packets from
forward shock wave surface to the CSM disk photosphere to determine the
distribution of photon packets on the CSM disk photosphere. In practice, the
CSM torus has been divided into $\displaystyle N_{\rm{l}}$ layers along the
radial direction from $\displaystyle R_{\rm{sh,FS}}$ to $\displaystyle
R_{\rm{ph,CSM}}$. For each layer $\displaystyle i$, we track the number of
photon packets passing through ($\displaystyle N_{\rm{p},i}$) and the number
of photon packets radiating out from that layer ($\displaystyle
N_{\rm{rad},i}$). The luminosity generated by the ejecta-CSM interaction in
the $\displaystyle i$-th layer can be expressed as $\displaystyle
L_{\text{disk}}$ multiplied by the contribution rate $\displaystyle
N_{\rm{rad},i}/N_{\rm{II}}$ for that layer, namely

$\displaystyle
L_{\text{CSM},\text{i}}\left(t\right)=\frac{N_{\rm{rad},i}}{N_{\rm{II}}}\frac{1}{t_{\text{dCSM,i}}}\exp\left[-\frac{t}{t_{\text{dCSM,i}}}\right]$
(12)
$\displaystyle\int_{0}^{t}\exp\left[\frac{t^{\prime}}{t_{\text{dCSM,i}}}\right]L_{\text{CSM,disk}}\left(t^{\prime}\right)dt^{\prime},$

The diffusion timescale $\displaystyle t_{\text{dCSM,i}}$ through the i layers
of the CSM disk can be expressed as a function of the radius ($\displaystyle
R_{\text{disk,i}}$) of the $\displaystyle i$-th layer in the radial direction
and the total disk mass ($\displaystyle M_{\text{disk,i}}$) enclosed within
$\displaystyle R_{\text{disk,i}}$, i.e.,

$t_{\text{dCSM,i}}=\frac{\kappa_{\text{CSM}}M_{\text{disk,i}}}{\beta
cR_{\text{disk,i}}\sin{\theta_{\rm{csm}}}},$ (13)

where $\displaystyle\kappa_{\rm CSM}$ is the mean opacity in CSM.
$\displaystyle\beta=13.8$ is a constant for the density distribution of the
ejecta. Similarly, the luminosity contribution to the $\displaystyle i$-th
layer from the decay of radioactive elements in the ejecta from region
$\displaystyle\mathrm{I}$ ($\displaystyle L_{\rm{rad,I},i}$) can be expressed
as

$\displaystyle
L_{\rm{rad}\mathrm{I},\text{i}}\left(t\right)=\frac{N_{\rm{rad},i}}{N_{\rm{II}}}\frac{1}{t_{\text{drad,i}}}\exp\left[-\frac{t}{t_{\text{drad,i}}}\right]$
(14)
$\displaystyle\int_{0}^{t}\exp\left[\frac{t^{\prime}}{t_{\text{drad,i}}}\right]L_{\rm{Ni},\mathrm{I}}\left(t^{\prime}\right)dt^{\prime},$

and the diffusion timescale $\displaystyle t_{\text{drad,i}}$ through the i
layers of the CSM disk is

$t_{\text{d,i}}=\frac{\kappa_{\text{CSM}}\left[(M_{\text{disk,i}}/\sin{\theta_{\rm{csm}}})+M_{\text{ej}}\right]}{\beta
cR_{\text{disk,i}}}$ (15)

The energy carried by each photon packet radiated from the $\displaystyle
i$-th layer of the photosphere can be written as

$\epsilon_{\rm{i}}=\frac{\left[L_{\text{CSM,i}}\left(t\right)+L_{\rm{rad}\mathrm{I},\text{i}}\left(t\right)\right]\Delta
t}{N_{\rm{rad,i}}},$ (16)

where $\displaystyle\Delta t$ is the duration of Monte Carlo process. The
temperature radiated from layer i in the CSM disk can be expressed in terms of
the luminosity radiated from layer i and the equivalent radial cross-sectional
area $\displaystyle A_{\rm{disk,i}}$ , i.e.,

$T_{\rm{disk}}=\left[\frac{L_{\text{CSM,i}}\left(t\right)+L_{\rm{rad}\mathrm{I},\text{i}}\left(t\right)}{\sigma_{\rm
SB}A_{\rm{disk,i}}}\right]^{1/4}.$ (17)

When the radius of the supernova photosphere exceeds the outer boundary of the
CSM disk, the luminosity calculation for region $\displaystyle\mathrm{I}$ can
be simplified by assuming that photons within the disk can only escape
radially from the photosphere of the disk.

As for region $\displaystyle\mathrm{II}$, the luminosity is mainly contributed
by the radioactive element decay in the expanding SN ejecta. Based on Arnett
(1982), the luminosity could be estimated to be

$\displaystyle\displaystyle L_{\rm{rad}\mathrm{II}}(t)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{2}{t_{\rm{m}}}e^{-\left(\frac{t^{2}}{t_{\rm{m}}^{2}}+\frac{2R_{0}t}{vt_{\rm{m}}^{2}}\right)}~{}\int_{0}^{t}e^{\left(\frac{t^{\prime
2}}{t_{\rm{m}}^{2}}+\frac{2R_{0}t^{\prime}}{vt_{\rm{m}}^{2}}\right)}$ (18)
$\displaystyle\displaystyle\times\left(\frac{R_{0}}{vt_{\rm{m}}}+\frac{t^{\prime}}{t_{\rm{m}}}\right)L_{\mathrm{Ni},\mathrm{II}}(t^{\prime})dt^{\prime},$

where $\displaystyle R_{0}$ is the initial radius of the progenitor, and we
follow Chatzopoulos et al. (2012) to take the limit $\displaystyle
R_{0}\rightarrow 0$. The diffusion time scale $\displaystyle t_{\rm{m}}$ can
be written as

$\displaystyle\displaystyle t_{\rm{m}}=\left(\frac{2\kappa_{\rm{ej}}M_{\rm
ej}}{\beta vc}\right)^{1/2},$ (19)

where $\displaystyle\kappa_{\rm{ej}}$ is the mean opacity to optical photons
in SN ejecta. The energy carried by each photon packet emitted from the
photosphere in region $\displaystyle\mathrm{II}$ is $\displaystyle
L_{\rm{rad}\mathrm{II}}\Delta t/N_{\rm{\mathrm{II}}}$. The photosphere
temperature is

$T_{\rm disk}=\left(\frac{L_{\rm{rad}\mathrm{II}}}{\sigma_{\rm
SB}A_{\rm{ej}}}\right)^{1/4},$ (20)

where $\displaystyle A_{\rm{ej}}$ is the area of the photosphere in the region
$\displaystyle\mathrm{II}$ supernova ejecta.

###  2.5 The Integrated Polarization of the SN

Each photon packet carries polarization information which can be described by
a useful convention, namely the Stokes vector $\displaystyle S=(I,Q,U,V)$,
where $\displaystyle I$ gives the total intensity, $\displaystyle Q$ and
$\displaystyle U$ describe the linear polarization, and $\displaystyle V$
specifies the state of circular polarization. Considering that circular
polarization has never been observed in SNe, also, in a scattering atmosphere
without a magnetic field, the radiative transfer calculations for circular and
linear polarization can be decoupled (Chandrasekhar, 1960), we therefore
neglect the $\displaystyle V$ component. The polarization degree
($\displaystyle P$) and the position angle ($\displaystyle\chi$) can then be
given in terms of the Stokes parameters, namely

$P=\frac{\sqrt{Q^{2}+U^{2}}}{I},$ (21)
$\chi=\frac{1}{2}\tan^{-1}{\bigg{(}\frac{U}{Q}\bigg{)}}~{}.$ (22)

For the sake of convenience, for the case of the orthogonal components of
$\displaystyle U$ is balanced in the projection plane along the line-of-sight
to the observer, the polarization degree can be simplified to $\displaystyle
P=Q/I$.

Unpolarized photon packets are emitted from the photosphere with random
propagation directions and propagate through the medium. Following the
prescriptions in previous studies (e.g., Code & Whitney, 1995; Mazzali & Lucy,
1993; Lucy, 1999; Kasen, 2003; Whitney, 2011; Bulla et al., 2015), photon
packets with zero polarization were prepared at the emitting layer, whose
initial traveling directions were set randomly. For a photon packet traveling
through the medium, a certain probability of electron scattering is assigned.
After the scattering occurs, the renewed polarization state and the
propagation direction of the photon packet will then be set as the initial
conditions. We remark that our calculations focus on the continuum
polarization that arises from the deviation from spherical symmetry of an
electron-scattering photosphere. We do not attempt to model the polarization
induced by particular line opacities. The continuum polarization in the
optical band is independent of wavelength as Thomson scattering is
approximately a grey process at this frequency (Bulla, 2017). We adopt a
conventional Thomson electron scattering opacity value of
$\displaystyle\kappa=$0.34 for both $\displaystyle\kappa_{\rm CSM}$ and
$\displaystyle\kappa_{\rm{ej}}$ (Nagao, Maeda & Ouchi, 2020). When the photon
packet reaches the outer boundary of the computational domain, the propagation
process ends, and the information of the photon packet at that time is
recorded. The observer frame was set at 100 Mpc from the SN center. The final
collection of the photon packets is carried out within each
$\displaystyle\frac{1}{16}\times 180^{\circ}$ latitudinal bins.

Figure 2:  An example of the interaction between the expanding SN ejecta and a
shell of CSM torus ($\displaystyle s=0$). All phases are given in the rest
frame and relative to the first light of the SN. As summarized by the
schematics in the top row, panels (a)-(d) show the evolution of the outer
radius of the ejecta (black solid line), the radius of the photosphere (red
solid line), and the radius of the forward shock wave (yellow solid line) over
time. Panel (e) presents the corresponding $\displaystyle ugriz$ spectral flux
density per unit wavelength ($\displaystyle F_{\lambda}$) of the SN as seen
from aspect angles of $\displaystyle\theta_{\rm los}=$90, 45, and 0 degrees as
indicated by the legend. Panel (f) displays the calculated time-evolution of
the intensity-normalized polarization degree $\displaystyle Q/I$ for different
viewing angles as marked by the legend. The color-shaded areas indicate
different stages of the interaction as characterized by the relative sizes
among the photospheric radius in the ejecta and the CSM ($\displaystyle
R_{\rm{ph,ej}}$,$\displaystyle R_{\rm{ph,CSM}}$), the outer radius of the
ejecta and the CSM ($\displaystyle R_{\rm{out,ej}}$,$\displaystyle
R_{\rm{out,CSM}}$), and the forward shock radius ($\displaystyle
R_{\rm{sh,FS}}$). Panels (h) and (i) in the bottom row illustrate the
wavelength dependence of the observed $\displaystyle F_{\lambda}$ and
$\displaystyle Q/I$, respectively. Different viewing angles are indicated by
the legend.

##  3 Result

Figure 3: Temperature structure as a function of radial distance of the CSM
torus that lies above the photosphere at 1, 4, and 6 days after the first
light, as shown by the red-, green-, and blue-dashed curves, respectively. The
red, green, and blue histograms present the corresponding probability density
distribution of the photons along the radial direction, normalized to the
total number of photons. The unit of the right-hand ordinate is photons per
meter (photons/m). The probability density measured at each bin, scaled by a
factor of ’1e-14’, can be read off the right-hand ordinates. The three
diamonds represent the temperatures of the supernova ejecta photosphere in
region II. The yellow-shaded area represents the radial extension of the
adopted configuration of the torus-like CSM. The red- and the blue-shaded
areas indicate the ranges within the inner and beyond the outer boundaries of
the CSM torus, respectively.  Figure 4: An example of the interaction between
the expanding SN ejecta and a CSM torus with a wind-like ($\displaystyle s=2$)
density structure. All phases are given in the rest frame and relative to the
first light of the SN. The left column shows, from top to bottom, the SN
lightcurves ($\displaystyle F_{\lambda}$) for
$\displaystyle\theta_{\rm{csm}}=5^{\circ}$,$\displaystyle 15^{\circ}$, and
$\displaystyle 30^{\circ}$, respectively. The right subpanels in each row
display the corresponding $\displaystyle Q/I$. Viewing angles
$\displaystyle\theta_{\rm los}$ are indicated by the legend in each subpanel.
Figure 5: Similar to Figure. 4 but for
$\displaystyle\theta_{\rm{csm}}=15^{\circ}$ and a lower kinetic-luminosity
conversion efficiency (i.e.,
$\displaystyle\epsilon_{\rm{FS}}=\epsilon_{\rm{RS}}=0.1$).

Our calculations suggest two main factors that affect the polarization signal
when the SN ejecta interacts with a torus-like CSM. The first one is the extra
energy injected into the CSM torus by the FS and the RS waves during the
interaction phase, which introduces a significant difference in the emitted
radiations between regions I and II. The second one gives the deviation of the
initially spherical SN ejecta caused by its interaction with the aspherical,
torus-like CSM. Parameters that characterize these two factors include the
time-evolution of the luminosity difference between the SN ejecta and the CSM
torus, $\displaystyle\theta_{\rm{csm}}$, the viewing angle
($\displaystyle\theta_{\rm los}$), and the time scale of the ejecta overriding
the torus-like CSM.

###  3.1 Ejecta interacting with immediate CSM Torus

When the SN ejecta interacts with an ambient thin torus of CSM, whose radial
density profile can be described by a uniform shell, namely $\displaystyle
s=0$ in Equation 4, our calculations suggest that the interaction can produce
a rather high level of continuum polarization. In Figure 2 we present a case
that adopts $\displaystyle s=0$, $\displaystyle R_{\text{CSM,in}}=5\times
10^{13}$ cm, $\displaystyle M_{\rm ej}=10M_{\odot}$, $\displaystyle M_{\rm
CSM}=0.1M_{\odot}$, $\displaystyle E_{\rm{SN}}=10^{51}$ erg, $\displaystyle
v_{\rm{SN}}=10^{4}\text{ km s}^{-1}$,
$\displaystyle\rho_{\text{CSM,in}}=5\times 10^{-13}\text{g cm}^{-3}$,
$\displaystyle\theta_{\rm{csm}}=15^{\circ}$,
$\displaystyle\epsilon_{\rm{FS}}=\epsilon_{\rm{RS}}=0.8$ and $\displaystyle
M_{\rm Ni}=0.032M_{\odot}$. As illustrated in the top row of Figure 2, in
Region I (equatorial), the FS propagates through the optically thick CSM in
the eight days (Figures 2a–2c). According to our calculations, within the
first $\displaystyle\sim$8 days, the emission measured for region I is
$\displaystyle\sim$3 order of magnitude stronger compared to that of region
II, thus being the dominant radiation source during such an early interaction
phase.

In the first few days after the SN explosion, due to the radial optical depth
in the CSM disk being significantly greater than unity, most of the photons
generated by the FS are emitted from the surface of the CSM disk near the
shock front along the latitudinal direction. As the FS propagates outward, the
diffusion time of photons in the radial direction of the CSM disk gradually
decreases, leading to an increase in the photospheric luminosity at larger
radial within the CSM disk. The polarization level also gradually increases
due to the brightening of the highly aspherical photosphere. The high-
luminosity photosphere of the CSM disk, exposed beyond the SN photosphere, can
generate significant polarization signals (up to $\displaystyle\sim 15\%$)
when both the latitudinal and radial surfaces of the CSM disk play prominent
roles, as observed on day 4. As the SN ejecta continues to expand, the overall
shape of the photosphere gradually becomes spherical, leading to a
progressively decreasing continuum polarization. At around day 8 when the
photons diffuse out the radial direction of the torus CSM, the lightcurve
reaches its peak at $\displaystyle\theta_{\rm los}=90^{\circ}$ (with
$\displaystyle\theta_{\rm los}=0^{\circ}$ directed towards the +z axis). The
emission excess towards the equatorial plane would lead to rise of the
continuum polarization, which becomes most prominent if seen from an equator-
on perspective. As the viewing perspective moves toward a pole-on scenario,
photons emitted from region II become progressively obscured by region I
(Figure 2c). resulting in a monotonically decrease of the total luminosity and
a flip in the observed polarization position angle. As the interaction process
ends, the luminosity of region I gradually dies out and the steadily-expanding
ejecta in region II would take over. The overall shape of the emitting zone
will then gradually transfer from an oblate ($\displaystyle\lesssim$day 8) to
a prolate ellipsoid because the ejecta goes faster at the poles. Such a
geometric alternation would manifest as a flip of the sign of the
polarization, namely from negative to positive as measured by $\displaystyle
Q/I$ (Figure 2f).

Another remarkable signature during the ejecta interacts with the CSM torus
process is that the polarization exhibits a wavelength dependence (see, Figure
2i). Typically, polarization aroused from Thomson scattering of free electrons
with a globally aspherical distribution is considered wavelength independent.
Our calculation suggests that the polarization of the ejecta-CSM interaction
exhibits a strong wavelength dependence, which is mainly due to the
interaction process induces a significant temperature gradient along the
radial direction of the disk disk. We illustrate this effect by calculating
the radial structure of the temperature in Figure 3. Soon after the SN
explosion, the majority of high-frequency photons are radiated from the
latitudinal surface of the disk away from $\displaystyle R_{\rm{ph,CSM}}$,
while a small fraction of low-frequency photons are radiated from the radial
surface of the disk close to $\displaystyle R_{\rm{ph,CSM}}$. As the shock
front continues to propagate outward, the temperature difference of the CSM
disk in the radial direction decreases and photons can gradually be radiated
from the CSM in all directions, resulting in a wavelength-dependence that is
no longer significant.

We note that different temperatures are also found in regions I
($\displaystyle\approx$3200 K at day 4) and II ($\displaystyle\approx$21000 K
at day 4). However, the significant luminosity disparity between the two
regions suggests that the low-frequency, low-energy photons emitted from
region I have only a relatively weak effect on the wavelength dependence of
the polarization, which can be considered negligible. When the ejecta
photosphere covers the latitudinal surface of region I, photons from the
radial surface of the CSM disk and photons from the low-temperature region I
create a small polarization difference at the red and blue ends (approximately
$\displaystyle 0.2\%$ on day 8).

###  3.2 SN Ejecta Interacting with a Detached CSM Torus

We also considered the case of a radially extended torus of CSM located
relatively far from the progenitor star of the supernova. Following a similar
procedure as detailed in Section 3.1, we found that the presence of a detached
CSM disk would produce overall lower levels of the continuum polarization that
have similar trends to those of an immediate thin shell of the CSM. As
illustrated in Figure 4, we present our calculation adopting the following
parameters: $\displaystyle s=2$, $\displaystyle R_{\text{CSM,in}}=1\times
10^{15}$ cm, $\displaystyle M_{\rm ej}=10M_{\odot}$, $\displaystyle M_{\rm
CSM}=1M_{\odot}$, $\displaystyle E_{\rm{SN}}=10^{51}$ erg, $\displaystyle
v_{\rm{SN}}=10^{4}\text{ km s}^{-1}$,
$\displaystyle\rho_{\text{CSM,in}}=5\times 10^{-14}\text{g cm}^{-3}$,
$\displaystyle\epsilon_{\rm{FS}}=\epsilon_{\rm{RS}}=0.8$ and $\displaystyle
M_{\rm Ni}=0.032M_{\odot}$. In this example, region I undergoes a prolonged
interaction process. As the luminosity from radioactive elements increases,
the difference between the luminosities of Regions I and II decreases
monotonically until day 74, leading to a reduced level of the continuum
polarization. The lightcurve enters a plateau after the peak until
$\displaystyle R_{\rm{sh,FS}}=R_{\rm{ph,CSM}}$, corresponding to day 74 in
Figure 5. After that, the lightcurve is mostly powered by the emission from
the ejecta seen from region II.

We remark that since $\displaystyle R_{\rm{ph,ej}}$ remains smaller than
$\displaystyle R_{\rm{ph,CSM}}$ in the CSM disk along the radial direction
before the shock breakout occurs, there is no scenario where the photosphere
of Region II obscures region I. Therefore, as the FS continuously propagates
towards the disk’s edge, the polarization degree gradually increases, until
reaching its peak when the FS reaches the $\displaystyle R_{\rm{ph,CSM}}$. As
the luminosity of region I decreases after the shock breakout, the ejecta
becomes more like a prolate ellipsoid because the ejecta expands faster at the
poles. Such an oblate-to-prolate transformation of the overall geometry of the
emitting zone thus leads to a sign flip of the integrated polarization (see
the right column of Figure 4 after $\displaystyle\approx$day 75). We also
calculated the time-evolution of the continuum polarization for the cases of
the CSM torus with a range of opening angles. As
$\displaystyle\theta_{\rm{csm}}$ increases, the observed polarization degree
gradually decreases. For instance, when
$\displaystyle\theta_{\rm{csm}}=30^{\circ}$, the polarization level
significantly drops to below 1% (see the bottom row of Figure 4).

Finally, following Nagao, Maeda & Ouchi (2020), we examined the interaction
process with a lower kinetic energy conversion efficiency, namely
$\displaystyle\epsilon_{\rm{FS}}=\epsilon_{\rm{FS}}=0.1$ under the same
configuration as the radially-extended envelope. The result for a half opening
angle of $\displaystyle\theta_{\rm{csm}}=15^{\circ}$ is shown in Figure 5.
During the interaction process until day 74, the luminosities of regions I and
II are only differed by an order of magnitude, resulting in a polarization
degree of less than 1%. Towards the end of the interaction process, the fast
expanding ejecta near the poles would produce a prolate photosphere, thus
producing a flip of the sign of the continuum polarization (see the lower
panel of Figure 5).

##  4 Conclusion and Discussion

In this letter, we investigate the temporal evolution of the polarization of
SN interacting with various configurations of dense, torus-like CSM. Our
calculations suggest that the integrated continuum polarization would result
from an interplay between the overall geometry and the non-uniformly
distributed emitting regions, both are time-variant. When the SN ejecta
interact with the CSM disk, the CSM disk exposed outside the ejecta
photosphere leads to an overall oblate ellipsoidal shape of the emitting zone.
Additionally, the increased difference of the emitted flux between the CSM
torus (region I) and the polar regions (region II) within the first 4 days
after the explosion would produce a further elevated level of the continuum
polarization (Figures 1a and 1f). This effect is particularly pronounced in
the case of a shell-like CSM located in close vicinity of the progenitor star.
The estimated level of the peak polarization can be high as
$\displaystyle\approx$15% (Figure 2f), which may account for the observed
record-breaking high polarization a few days after the transient explosion,
e.g., the $\displaystyle\sim$7% polarization in blue wavelength range within
$\displaystyle\sim$5 days of the explosion of AT 2018cow (Maund et al., 2023).
As the ejecta expands faster in polar regions (II), it will progressively
cover the emitting regions along the CSM torus (Figure 2b). As a consequence,
photons produced by the ejecta-CSM interaction will predominantly diffuse out
along the radial direction of the CSM torus. At this point, the continuum
polarization would result in an interplay between the geometry of the emitting
zone and the luminosity difference between the pole ejecta (region II) and the
radial surface of the CSM torus. Especially in cases where the radial
luminosity of the CSM is more dominant, a secondary peak of the polarization
might be observed (see, e.g., Figure 2f at $\displaystyle\approx$day 8). A
reversal of the sign of the polarization may occur after the secondary
polarization peak, followed by a steady increase in the polarization level.
The latter can be understood as the ejecta’s expansion towards the poles,
which would cause an increased deviation from spherical symmetry. Some
dynamical simulations show that as the ejecta expands, it will gradually
engulf the CSM disk (McDowell, Duffell & Kasen, 2018). The polarization at
this stage may be determined by a rather complicated process that carves the
post engulfment geometry, which is beyond the scope of this letter.

We found that when the SN ejecta interacts with a CSM torus, the non-uniform
temperature distribution between the polar and equatorial regions would
produce a wavelength-dependent polarization. This can be tested by multi-band
imaging polarimetry or spectropolarimetry. In previous studies, wavelength-
dependent polarization has typically been attributed to wavelength-dependent
line opacity (Howell et al., 2001; Maund et al., 2013; Inserra et al., 2016;
Yang et al., 2020). In the first few days after the SN explosion, as a
consequence of the ejecta-CSM interaction, the temperature would exhibits a
dramatic decrease towards larger radius of the CSM torus (Figure 3). Photons
emitted from the hotter inner regions of the disk (region I) dominate the
polarization at shorter wavelengths, while a small fraction of photons emitted
from the cooler outer regions manipulate the polarization at longer
wavelengths. The latter is also affected by the photons emitted from the
relatively cooler region II. However, such radiation from the polar ejecta
remains approximately more than one order of magnitude lower compared to that
from the outer part of the region I.

Despite our calculations predict the temporal evolution of the polarization
for both a thin shell and a radially extended torus of CSM, we remark that the
detailed process of interaction can be affected by various other factors. For
instance, multiple components of CSM may form a rather complicated density
structure and thus may manifest as time-variant polarization that differs from
our calculations (Liu et al., 2017). The early-time evolution of the broad-
band polarization measured for the fast blue optical transient AT 2018cow
exhibits two peaks at $\displaystyle\approx$days 5.7 and 13, reaching
$\displaystyle p\sim 7\%$ in $\displaystyle r$ and $\displaystyle p\sim 2\%$
in $\displaystyle b$, respectively (Maund et al., 2023). Such behavior is
broadly consistent with that predicted for the SN ejecta interacting with a
thin shell of CSM torus. For the CSM confined within a rather small distance
to the progenitor star, the corresponding timescale of the interaction will
also be compressed. For instance, an AT2018cow-like event with a compact
$\displaystyle s=0$ CSM disk may require polarization diagnostics to be
carried out within $\displaystyle\sim$day after the explosion. Additionally,
hours-cadence sampling would be necessary to resolve the first and the
secondary peaks in their polarization time evolution. Future deployment of
high-precision polarimeters with rapid-response capability will enable unique
measurements of the geometry of the ejecta-CSM interaction process over a
broad range of timescales. The polarization time sequence for various types of
transients will set constraints on the evolution and final mass loss of their
progenitors.

We thank Bing Zhang, Lifan Wang, Maokai Hu and Justyn R Maund for their
valuable discussions, and the anonymous reviewers for their helpful comments
that improved the manuscript. Xudong expresses his heartfelt gratitude to his
grandmother in heaven for her love and encouragement during her lifetime. This
work is supported by the National Natural Science Foundation of China
(Projects 12373040,12021003), the National SKA Program of China
(2022SKA0130100) and the Fundamental Research Funds for the Central
Universities.

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