# Afterglow Linear Polarization Signatures from Shallow GRB Jets: Implications
for Energetic GRBs

Gal Birenbaum,1 Ramandeep Gill,2,3 Omer Bromberg,1 Paz Beniamini3,4,5 and
Jonathan Granot3,4,5
1Tel Aviv University, The Raymond and Beverly Sackler School of Physics and
Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
2Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de
México, Antigua Carretera a Pátzcuaro # 8701, Ex-Hda. San José de la
Huerta, Morelia, Michoacán, México C.P. 58089, México
3Astrophysics Research Center of the Open University (ARCO), The Open
University of Israel, P.O. Box 808, Ra’anana 4353701, Israel
4Department of Natural Sciences, The Open University of Israel, P.O. Box 808,
Ra’anana 4353701, Israel
5Department of Physics, The George Washington University, Washington, DC
20052, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Gamma-ray bursts (GRBs) are powered by ultra-relativistic jets. The launching
sites of these jets are surrounded by dense media, which the jets must cross
before they can accelerate and release the high energy emission. Interaction
with the medium leads to the formation of a mildly relativistic sheath around
the jet resulting in an angular structures in the jet’s asymptotic Lorentz
factor and energy per solid angle, which modifies the afterglow emission. We
build a semi-analytical tool to analyze the afterglow light curve and
polarization signatures of jets observed from a wide range of viewing angles,
and focus on ones with slowly declining energy profiles known as shallow jets.
We find overall lower polarization compared to the classical top-hat jet
model. We provide an analytical expression for the peak polarization degree as
a function of the energy profile power-law index, magnetic field configuration
and viewing angle, and show that it occurs near the light curve break time for
all viewers. When applying our tool to GRB 221009A, suspected to originate
from a shallow jet, we find that the suggested jet structures for this event
agree with the upper limits placed on the afterglow polarization in the
optical and X-ray bands. We also find that at early times the polarization
levels may be significantly higher, allowing for a potential distinction
between different jet structure models and possibly constraining the
magnetization in both forward and reverse shocks at that stage.

###### keywords:

gamma-ray burst: general – radiation mechanisms: non-thermal – methods:
numerical – gamma-ray bursts

††pubyear: 2024††pagerange: Afterglow Linear Polarization Signatures from
Shallow GRB Jets: Implications for Energetic GRBs–F

##  1 Introduction

The understanding that gamma-ray bursts (GRBs) originate from narrow, ultra-
relativistic jets is based on three main observational lines of evidence. i)
The high observed power in prompt $\gamma$-rays requires the flow to move with
a high Lorentz factor to reduce the pair-production and Thomson scattering
opacity and allow the $\gamma$-ray photons to escape without substantially
changing the spectrum (Krolik & Pier, 1991; Woods & Loeb, 1995; Lithwick &
Sari, 2001). ii) The high observed fluence, which calls for narrow emission
sites to avoid an energy crisis in emission from a spherical flow (Piran,
1999). iii) Achromatic “jet-breaks” in the afterglow light curves, which occur
when the Lorentz factor of the decelerating flow becomes comparable to inverse
of the jet half-opening angle, making the entire jet visible and leading to a
steepening of the light curve at later times (Rhoads, 1997, 1999; Sari et al.,
1999). Typically, the observed afterglow light curve has a power-law behavior
in frequency ($\nu$) and time ($t_{\rm obs}$), such that
$F_{\nu}\propto\nu^{-\alpha}t_{\rm obs}^{-\beta}$. The simple model of a
sharp-edged jet, with uniform energy and Lorentz factor distributions and no
sideways expansion (known as a "top-hat" jet), produces light curves that
should steepen by $\Delta\beta\gtrsim\frac{1}{2}$ during the jet-break. The
change in temporal index, as well as the duration of the jet-break, depend on
the density profile of the external medium. Such steepenings were seen in
about half of Swift long GRBs (Panaitescu, 2007; Wang et al., 2018) and were
consistent with an averaged jet half opening angle of $\sim 6.5^{\circ}$,
confirming this picture (Racusin et al., 2009).

The traditional view of this sharp-edged, “top hat” jet was challenged with
the landmark discovery of GRB 170817A/GW 170817. Analysis of the afterglow
radio light-curve showed that the jet was observed most likely at a viewing
angle $\sim 3-6$ times larger than the inferred half-opening angle of the
relativistic jet (Mooley et al., 2018; Lazzati et al., 2018; Gill & Granot,
2018; Troja et al., 2019; Gill et al., 2019; Ghirlanda et al., 2019; Mooley et
al., 2022; Govreen-Segal & Nakar, 2023). Such an observation is possible if
the jet’s energy per solid angle and Lorentz factor gradually and continuously
drop with the angle $\theta$ from its symmetry axis outside of some core angle
$\theta_{c}$, with part of the outflowing material directed along the line-of-
sight (LOS). In particular, the jet core may be surrounded by a wide-angle
mildly relativistic outflow component. Such structure was seen in several
numerical simulations of GRB jets breaking out of the confining medium of the
progenitor system111the envelope of a collapsing star in the case of long GRBs
or the sub-relativistic neutron star merger ejecta in the case of short GRBs..
It originates from instability processes on the jet boundary, such as the
Rayleigh–Taylor instability (Matsumoto et al., 2017; Matsumoto & Masada, 2019;
Gottlieb et al., 2021; Abolmasov & Bromberg, 2023), which mix jet with medium
material and typically has a power-law angular profile of energy density and
Lorentz factor. The degree of mixing and amount of energy in the mildly
relativistic component depends on the properties of both the jet and confining
medium, and contains information on the conditions at the jet injection site
(e.g. Gottlieb et al. (2021)).

In several cases of powerful GRBs with long lasting X-ray afterglows and very
high energy emission, the afterglow light curves show a gradual transition in
their temporal decay indexes during the jet-break with late time values of
$1.4\lesssim\beta\lesssim 1.7$, rather than a relatively sharp steepening to a
canonical value of $\beta\sim 2.2$ (O’Connor et al., 2023). Such a behavior
fits a case where the energy in the mildly relativistic component is
comparable to the energy in the relativistic core, leading to a shallow
angular distribution of the energy per solid angle and Lorentz factor in the
jet. An example for such a case is the extremely luminous GRB 221009A, which
showed only a slight steepening in its X-ray afterglow. The multi-wavelength
afterglow light curves were fitted with a narrow cored jet surrounded by a
shallow structure in both the energy per unit solid angle and Lorentz factor
distributions with combined contributions from the reverse and forward shocks
(Gill & Granot, 2023). The fit, however, is not unique and other similar
shallow jet models are able to describe the forward shock afterglow (e.g.
O’Connor et al., 2023). To determine the jet structure, it is important to be
able to distinguish between the different jet models that fit the light curve.

In addition to the observed light curve, upper limits were placed on the
linear polarization of GRB 221009A by Negro et al. (2023) 3.5 days post-burst
at $13.8\%$ and $8.3\%$ in the X-ray and optical bands, respectively. This is
the first time limits on the X-ray afterglow polarization were obtained for
any GRB and in some cases, theory predicts X-ray polarizaton levels would be
higher than optical ones (Granot, 2003; Rossi et al., 2004; Shimoda & Toma,
2021; Birenbaum & Bromberg, 2021)

Unlike the total flux, linear polarization is sensitive to the angular
distribution of energy and emissivity across the jet, making it a promising
way to distinguish between different jet structures. Rossi et al. (2004)
explored the temporal behavior of linear polarization signatures from jets
with uniform (top hat), Gaussian and a power-law angular energy distribution
that follows $\theta^{-2}$ from various viewing angles, discussing their
overall differences. Gill & Granot (2018) fitted different jet structures to
the X-ray and radio afterglow light curves of GW 170817 and modeled the
consequent temporal evolution of the radio afterglow polarization. In a
followup work, Gill & Granot (2020) used the structures they found, combined
with an upper limit on the radio afterglow polarization of GW 170817, and
obtained a limit on the magnetic field structure in the forward shock. This is
also done independently by Corsi et al. (2018); Stringer & Lazzati (2020);
Teboul & Shaviv (2021).

In this work we look at jets with power-law energy and Lorentz factor
distributions in their wings, motivated by results from numerical simulations
(Gottlieb et al., 2021). We explore the evolution of the afterglow light curve
and linear polarization for different energy power-law profiles, magnetic
field configurations in the forward shock and observer viewing angle, focusing
on the difference in the observational signature of jets with shallow and
steep angular profiles. We obtain an analytical expression for the dependence
of the peak polarization degree on the various system parameters. Finally, we
employ our method on the observations of GRB 221009A and fit two different
shallow structures (Gill & Granot, 2023; O’Connor et al., 2023) to the optical
and X-ray light curves. We find the produced polarization curves agree with
the upper limits on the linear polarization in both energy bands (Negro et
al., 2023).

##  2 Methods

To model the polarized afterglow emission from the forward shock we expand the
numerical method developed in Birenbaum & Bromberg (2021). We model a
relativistic shock decelerating in a cold ambient medium with a power-law
rest-mass density profile, $\rho\propto R^{-k}$. The emitting region is
approximated as a 2D surface located just behind the shock radiating
synchrotron emission. To calculate the emission, we divide the surface into
small cells, where in each cell we place a uniform magnetic field with some 3D
orientation. The comoving magnetic field direction,
$\boldsymbol{\rm{\hat{B}}}^{\prime}$, and its corresponding absolute value,
$B^{\prime}$, are randomly drawn from a probability distribution corresponding
to an isotropic field (in 3D) that is stretched in the radial direction by a
factor of $\xi$ (as described in Appendix B). We then calculate the
$I_{\nu},Q_{\nu},U_{\nu}$ Stokes parameters in each cell and sum up the
contributions from all cells to calculate the total emission and polarization.
Below we briefly describe the methods we use for each step of the calculation.
A more detailed description can be found in appendixes A\- C and in Birenbaum
& Bromberg (2021).

###  2.1 Jet structure and forward shock kinematics

Figure 1: Two of the angular energy profiles considered in this work,
featuring a core opening angle of $\theta_{\text{c}}=2^{\circ}$ and power-law
wings with $a=2$. We study the broken power-law (red dash-dotted line) jet in
the context of on-axis structured jets in order to focus on the interplay
between the core and the jet wings and corresponding edge effects. The smooth
power-law jet is investigated with off-axis jets (red solid line).

We assume a conical jet propagating in a medium with a power-law density
profile $\rho(R)=m_{\text{p}}n(R)=AR^{-k}$, where $m_{\text{p}}$ is the proton
mass, $n$ is the number density of the ambient medium and
$A=m_{\text{p}}n(R_{0})R_{0}^{k}$ evaluated at a reference radius $R_{0}$. The
propagation leads to the formation of a semi-spherical shock with the same
opening angle as the jet, which emits the afterglow radiation. The jet has an
angular structure consisting of a uniform core surrounded by extended wings
with power-law distributions of kinetic energy and Lorentz factor (Gill &
Granot, 2018). The structure is expressed in terms of the angular profile
function:

$\Theta\equiv\sqrt{1+\left(\frac{\theta}{\theta_{\text{c}}}\right)^{2}},$ (1)

where$\theta$ is the polar angle measured from the jet symmetry axis and
$\theta_{c}$ is the half-opening angle of the jet core. We define the profiles
of the energy per solid angle, $\epsilon=dE/d\Omega$, and the initial
(coasting phase) Lorentz factor, $\Gamma_{0}$, according to

$\epsilon(\theta)=\epsilon_{\text{c}}\Theta^{-a},$ (2)
$\Gamma_{0}\left(\theta\right)=1+\left(\Gamma_{\text{c}}-1\right)\Theta^{-b},$
(3)

where $\epsilon_{c}$ and $\Gamma_{c}$ are the jet core energy per unit solid
angle and the initial Lorentz factor of the shocked radiating fluid behind the
forward shock, respectively. This structure allows for a smooth transition
between the core and wings. We name this structure a “Power-law Jet” (see Fig.
1, solid line) and use it to explore the emission seen by observers located at
angles larger than $\theta_{\text{c}}$ (off-axis observers). Another type of
structure we investigate is the “Broken Power-law Jet” which we explore in the
context of observers located at an observing angle $\theta_{\rm
obs}<\theta_{c}$ (on-axis observers). In such jets, it is easier to control
$\theta_{c}$ when changing the power-law structure of the wings. (see Fig. 1,
dash dotted line). The structure of the “Broken Power-law Jet” is defined as :

$\epsilon(\theta)=\begin{cases}\epsilon_{c}&\theta\leq\theta_{c}\ ,\\\
\epsilon_{c}\left(\frac{\theta}{\theta_{c}}\right)^{-a}&\theta>\theta_{c}\
.\\\ \end{cases}$ (4)

We follow the propagation of the forward shock from the coasting phase through
the relativistic and the non-relativistic deceleration phases. To model the
shock dynamics, we assume that each cell in the shock at an angle $\theta$,
propagates as if it was part of a spherical shock with an energy
$E_{\text{k,iso}}=4\pi\varepsilon(\theta)$ and an initial Lorentz factor
$\Gamma_{0}(\theta)=[1-\beta_{0}^{2}(\theta)]^{-1/2}$ (Gill & Granot, 2018;
Beniamini et al., 2020). The dynamics of each cell is scaled according to the
deceleration radius

$R_{\text{dec}}(\theta)=\left[\frac{\left(3-k\right)E_{\text{k,iso}}(\theta)}{4\pi
Ac^{2}\Gamma_{0}^{2}(\theta)\beta_{0}^{2}(\theta)}\right]^{1/\left(3-k\right)},$
(5)

where $c$ is the speed of light. At this radius the shock begins to decelerate
according to the Blandford-Mckee solution (Blandford & McKee, 1976; Gill &
Granot, 2018), for as long as it remains relativistic. The Lorentz factor
evolution of each cell can be described in terms of the normalized shock
radius at the cell location, $\zeta\equiv R/R_{\text{dec}}$, and the initial
Lorentz factor $\Gamma_{0}$,

$\Gamma(\zeta,\Gamma_{0})=\frac{\Gamma_{0}+1}{2\,\zeta^{3-k}}\left[\sqrt{1+\frac{4\Gamma_{0}}{\Gamma_{0}+1}\zeta^{3-k}+\left(\frac{2\zeta^{3-k}}{\Gamma_{0}+1}\right)^{2}}-1\right]\;.$
(6)

This relation follows the hydrodynamic evolution of each cell from the
coasting phase, through the Blandford-Mckee phase, until it becomes non-
relativistic and converges to the Sedov-Taylor solution (Taylor, 1950; Sedov,
1959).

###  2.2 The emission and the emitting surface

The emission is modeled as slow-cooling synchrotron radiation, originating
from electrons located on a 2D surface at the shock immediate downstream that
has a bulk Lorentz factor222The actual Lorentz factor of the shock is
$\sqrt{2}\,\Gamma$ where $\Gamma$ is the Lorentz factor of its immediate
downstream, however we neglect this factor in our calculation following Gill &
Granot (2018). $\Gamma$. The properties of the emitting electrons, such as the
downstream density $n^{\prime}=4\Gamma n$, are obtained from the shock jump
conditions. We use a Taub-Matthews adiabatic
index333$\hat{\gamma}=\frac{4\Gamma+1}{3\Gamma}$ in the equation of state,
which transitions smoothly from $4/3$ to $5/3$ when $\Gamma\approx 1$. We
further assume that a fraction $\chi_{\text{e}}$ of the electrons are
accelerated to relativistic velocities behind the shock and emit the observed
synchrotron radiation (e.g. Eichler & Waxman, 2005; Ryan et al., 2020). The
emission spectrum in each cell is calculated in the shocked fluid frame, with
critical frequencies and a peak spectral power (per unit frequency and per
unit volume) (Gill & Granot, 2018)

$\nu^{\prime}_{\text{m}}=\frac{3}{\sqrt{2\pi}}\left(\frac{p-2}{p-1}\right)^{2}\frac{q_{e}}{m_{e}^{3}c^{5}}S(\xi)\sqrt{\epsilon_{\text{B}}}\epsilon_{\text{e}}^{2}e^{\prime
5/2}(\chi_{\text{e}}n^{\prime})^{-2}\sin\psi^{\prime}\;,$ (7)
$\nu^{\prime}_{\text{c}}=\frac{27}{32}\sqrt{\frac{2}{\pi}}\frac{q_{e}m_{e}c}{\sigma_{t}^{2}}\left(S(\xi)\right)^{-3}(\epsilon_{\text{B}}e^{\prime})^{-3/2}\left(\frac{\Gamma}{t}\right)^{2}\sin\psi^{\prime}\;,$
(8)
$P^{\prime}_{\nu^{\prime},\text{max}}=0.88\frac{1024}{27}\sqrt{\frac{8}{\pi}}\frac{p-1}{3p-1}\frac{q_{e}^{3}}{m_{e}c^{2}}S(\xi)\sqrt{\epsilon_{\text{B}}e^{\prime}}(\chi_{\text{e}}n^{\prime})\sin\psi^{\prime}\;,$
(9)

where $t$ is the lab frame time, $\sigma_{t}$ is the Thomson cross section,
$m_{e}$ is the electron mass and $q_{e}$ is its charge. In addition, $p$ is
the power-law index of the energy distribution of the radiating electrons with
$N\left(\gamma_{e}\right)\propto\gamma_{e}^{-p}$ for
$\gamma_{e}>\gamma_{\text{m}}$, $\epsilon_{\text{e}}$ and
$\epsilon_{\text{B}}$ are the energy fractions that go to the electrons and
the magnetic field respectively, $S(\xi)$ is the stretching factor of the
magnetic field according to the choice of $\xi$, the magnetic field structure
parameter (Eq. (31)), $e^{\prime}$ is the bulk internal energy density

$e^{\prime}=\left(\Gamma-1\right)n^{\prime}m_{\text{p}}c^{2}\;,$ (10)

and $\psi^{\prime}$ is the pitch angle of the electrons gyrating around the
magnetic field lines. Since the electrons are highly relativistic, their
emission is beamed into a cone of angular size $1/\gamma_{e}$ around their
velocity vectors and the observer sees emission from electrons at a pitch
angle
$\psi^{\prime}=\arccos\left(\boldsymbol{\rm{\hat{B}}}^{\prime}\cdot\boldsymbol{\rm{\hat{n}}}^{\prime}_{\rm
obs}\right)$, where $\boldsymbol{\rm{\hat{n}}}^{\prime}_{\rm obs}$ and
$\boldsymbol{\rm{\hat{B}}}^{\prime}$ are unit vectors pointing towards the
observer and along the direction of the magnetic field in the proper frame
respectively (see Eqs. (25), (35)). The corresponding spectral power per unit
volume in the slow cooling regime, which is typically valid during most of the
afterglow emission, maintains:

$P^{\prime}_{\nu^{\prime}}=P^{\prime}_{\nu^{\prime},\text{max}}\begin{cases}\left(\frac{\nu^{\prime}}{\nu^{\prime}_{\text{m}}}\right)^{1/3}&\nu^{\prime}<\nu^{\prime}_{\text{m}}\\\
\left(\frac{\nu^{\prime}}{\nu^{\prime}_{\text{m}}}\right)^{-(p-1)/2}&\nu^{\prime}_{\text{m}}<\nu^{\prime}<\nu^{\prime}_{\text{c}}\\\
\left(\frac{\nu^{\prime}_{\text{c}}}{\nu^{\prime}_{\text{m}}}\right)^{-(p-1)/2}\left(\frac{\nu^{\prime}}{\nu^{\prime}_{\text{c}}}\right)^{-p/2}&\nu^{\prime}_{\text{m}}<\nu^{\prime}_{\text{c}}<\nu^{\prime}\\\
\end{cases}.$ (11)

From this we calculate the proper isotropic equivalent luminosity by
multiplying the spectral power per unit volume (Eq. 11) by the emitting volume

$L^{\prime}_{\text{iso},\nu^{\prime}}=P^{\prime}_{\nu^{\prime}}\cdot 4\pi
R^{2}\Delta^{\prime},$ (12)

with $\Delta^{\prime}\equiv\frac{R}{4(3-k)\Gamma}$ (Gill & Granot, 2018).

To calculate the total observed emission we sum the contributions from all
cells that have the same photon arrival time to the observer. These cells
constitute a surface known as the equal arrival time surface (EATS; Sari,
1998; Granot et al., 1999). To associate the EATS radius at each angle with an
arrival time to the observer $t_{\rm obs}$, we numerically solve the equation
(see appendix C)

$\frac{t_{\rm
obs}}{(1+z)}-\int^{R}_{0}\left(\frac{1-\beta\tilde{\mu}}{\beta}\right)\frac{dR}{c}=0,$
(13)

where
$\tilde{\mu}=\boldsymbol{\rm{\hat{\beta}}}\cdot\boldsymbol{\rm{\hat{n}}}_{\rm
obs}$. We then employ Eq. (6) to calculate the corresponding $\Gamma(R)$ and
use both values to calculate the emission observed at a time $t_{\rm obs}$.
The overall observed flux is found by integrating over the emission from all
cells on the EATS and accounting for the source redshift, $z$, giving

$F_{\nu}(t_{\rm obs})=\frac{\left(1+z\right)}{16\pi^{2}d_{\text{L}}^{2}}\int
D^{3}(\tilde{\mu})L^{\prime}_{\text{iso},\nu^{\prime}}d\Omega,$ (14)

where $d_{\text{L}}$ is the luminosity distance,
$\tilde{\mu}=\hat{\beta}\cdot\hat{n}_{\rm obs}$ with $\hat{\beta}$ as the unit
vector in the velocity direction of the emitting cell, and
$D(\tilde{\mu})=\left[\Gamma(1-\beta\tilde{\mu})\right]^{-1}=(1+z)\nu/\nu^{\prime}$
is the Doppler factor.

###  2.3 The observed polarization

Since the emission in each cell originates from a uniform field it is linearly
polarized with a polarization degree

$\Pi_{\max}=\frac{\alpha+1}{\alpha+5/3}\;,$ (15)

where $\alpha$ is the spectral index of the emission at the given frequency,
$P^{\prime}_{\nu^{\prime}}\propto\nu^{\prime-\alpha}$, evaluated according to
Eq. (11). The local direction of the polarization vector in the proper frame
is

$\boldsymbol{\rm{\hat{P}}}^{\prime}=\frac{\boldsymbol{\rm{\hat{n}}}^{\prime}_{\rm
obs}\times\boldsymbol{\rm{\hat{B}}}^{\prime}}{|\boldsymbol{\rm{\hat{n}}}^{\prime}_{\rm
obs}\times\boldsymbol{\rm{\hat{B}}}^{\prime}|}\;.$ (16)

This quantity is transformed to the observer frame and projected onto the
plane of the sky according to the procedure described in appendix D. The
polarization angle in each cell is then calculated from the ratio of the two
orthogonal polarization vector components in the observer frame:

$\phi_{\text{p,0}}=\arctan\left(\frac{P_{\text{y}}}{P_{\text{x}}}\right)\;,$
(17)

where $\hat{x}$ points along the vector connecting the symmetry axis with the
line of sight on the plane of the sky (for a choice of $\phi_{\rm
obs}=0^{\circ}$) and $\hat{y}$ points along the orthogonal direction (see Fig.
10 for a detailed illustration). We then use Eqs. (12), (14) and (17) to
calculate the flux weighted Stokes parameters at each cell and sum them up
over all cells:

$\displaystyle\frac{Q_{\nu}}{I_{\nu}}=\frac{\int\Pi_{\max}\cos(2{\phi_{\text{p,0}}})D^{3}(\tilde{\mu})L^{\prime}_{\text{iso},\nu^{\prime}}d\Omega}{\int
D^{3}(\tilde{\mu})L^{\prime}{\text{iso},\nu^{\prime}}d\Omega}\;,$
$\displaystyle\frac{U_{\nu}}{I_{\nu}}=\frac{\int\Pi_{\max}\sin(2{\phi_{\text{p,0}}})D^{3}(\tilde{\mu})L^{\prime}_{\text{iso},\nu^{\prime}}d\Omega}{\int
D^{3}(\tilde{\theta})L^{\prime}_{\text{iso},\nu^{\prime}}d\Omega}\;.$

Using the integrated Stokes parameters from Eq. LABEL:eq:StokesParameters, we
obtain the total observed polarization degree and polarization angle:

$P_{\rm
tot}=\frac{\sqrt{Q_{\nu}^{2}+U_{\nu}^{2}}}{I_{\nu}}=\frac{Q_{\nu}}{I_{\nu}}\;,$
(19) $\phi_{\text{p}}=\frac{1}{2}\arctan\left(\frac{U_{\nu}}{Q_{\nu}}\right),$
(20)

where due to the axial symmetry of the flow, $U_{\nu}$ vanishes, leaving us
with $P_{\rm tot}=Q_{\nu}/I_{\nu}$ for a choice of $\phi_{\rm obs}=0^{\circ}$.
This axial symmetry limits changes in the polarization angle to rotations by
$90^{\circ}$ only. The sign of the observed polarization degree changes when
such a rotation occurs, as shown in sub-section 3.1.1.

The time variation of the overall polarization degree and the polarization
angle depends on the jet structure and on the magnetic field configuration. It
also depends on $\theta_{\rm obs}$ and on the observed frequency, which
together with the afterglow parameters, determine the relevant power-law
segment in the synchrotron spectrum (Sari et al., 1998; Granot & Sari, 2002).
Eqs. (LABEL:eq:StokesParameters) gives a higher weight to more luminous areas,
causing them to dominate the observed polarization. If the emission is
dominated by a small area over which the local polarization vector is ordered,
the observed polarization would be extremely high (for example, when emission
from structured jets turns core dominated at late times). If the emission
originates from an extended region in the shock where the direction of the
polarization vector varies over a range of $\sim 90^{\circ}$, some
cancellations will arise when integrating the polarization, leading to
moderate to low polarization degree (at early times, the emission from off-
axis structured jets is dominated by their extended wings).

##  3 Results

In this paper we study the effects of the jet structure and magnetic field 3D
orientation on the temporal evolution of polarization seen by viewers from
different viewing angles. We focus the study on shallow jets ($a\leq 2$)
observed in the optical band so that the observed frequency $\nu$ will not
cross any critical frequency444To see how the polarization degree is affected
by spectral breaks, we refer the reader to Birenbaum & Bromberg (2021), Granot
(2003) and Rossi et al. (2004). ($\nu_{\text{m}}<\nu<\nu_{\text{c}}$). The
details of the afterglow model we use are given in Table 1. Below we show
results for jets observed from viewing angles within the jet core, with
normalized viewing angles $q\equiv\theta_{\rm obs}/\theta_{c}<1$ (on-axis
jets) and jets observed from viewing angles outside the core, i.e. with $q>1$
(off-axis jets). All cases considered in this section are set with a flat
distribution of initial LF with $b=0$.

Parameter Space
---
Parameter |  Value
$\theta_{c}$ |  $2^{\circ}$
$\Gamma_{c}$ |  250
a (PL index of energy structure) |  0.5, 1, 2, Top hat jet
b (PL index of LF structure) |  0
$E_{\text{c}}$ [erg] |  $10^{50}$
$n_{\text{ISM}}\ [\textrm{cm}^{-3}]$ |  1
Density profile ($n\propto r^{-k})$ |  $k=0$
$\nu_{\text{obs}}$ [Hz] |  $10^{15}$ (PLS G)
$p$ |  $2.5$
$\epsilon_{\text{e}}$ |  0.1
$\epsilon_{\text{B}}$ |  0.005
$\chi_{\text{e}}$ |  1
$d_{\text{L}}$ [cm] |  $10^{28}$
$z$ |  $0.54$
Table 1: Parameter space and afterglow model for the cases considered in
section 3.

###  3.1 Viewing angle and jet structure

####  3.1.1 On-axis jets ($q<1$)

Figure 2: Observed polarization degree (upper panel) and flux (lower panel) as
function of time for the broken power-law jet model at observer frequency
$\nu=10^{15}$ Hz (PLS G) with a flat distribution of initial LF ($b=0$) and a
random magnetic field structure, confined to the shock plane ($\xi\rightarrow
0$). Jet break times for each model are marked with stars. The various values
of $a$ represent different slopes for the power-law jet wings and the observer
is located at $q=0.7$. The signs of the polarization degree represent opposing
directions of it, with polarization angle difference of $90^{\circ}$.

Fig. 2 shows the temporal evolution of the polarization degree (upper panel)
and the afterglow light curve (lower panel) of jets with different power-law
structures in energy $a=(0.5,1,2)$, compared to a top hat jet, and random
magnetic fields confined to the shock surface ($\xi\rightarrow 0$), observed
at a viewing angle $q=0.7$. Each structure is marked by a different color, as
shown in the figure legend. The polarization degrees of all structures show
two peaks at opposing signs corresponding to polarization angles differing by
$90^{\circ}$. This polarization signature is well known in the context of top
hat jets and is formed due to the fact that the observed emission comes from a
ring-like region with an angular radius of $\frac{1}{\Gamma}$ centered around
the observer LOS, rather then the entire shock surface (see Fig. 6 for an
illustration of this ring).555A similar polarization signature also occurs for
a more uniform (less limb-brightened) afterglow image, as expected in other
PLSs, so it is rather robust. In the case of a random magnetic field, confined
to the surface of the forward shock, the polarization vector is radially
ordered along this ring (on the plane of the sky, relative to the center of
the ring-like image). At early times, the entire ring is visible and the
polarization degree is zero. As the shock decelerates, the angular radius of
the ring grows and eventually, parts of the polarized ring disappear beyond
the jet edge, reducing the cancellations of opposing polarization directions
and giving rise to a global polarization degree (see Fig. 6 of Birenbaum &
Bromberg (2021) or Fig. 3 of Sari (1999) for a visual demonstration of this
effect). The sign of the polarization degree is determined by the polarization
angle where a change in sign means the global polarization vector has rotated
by $90^{\circ}$. This rotation occurs when half of the polarized ring
disappears beyond the jet edge, followed by a rise in polarization in the
opposite direction (Ghisellini & Lazzati, 1999; Sari, 1999; Granot & Konigl,
2003; Nava et al., 2016; Birenbaum & Bromberg, 2021).

When power-law wings are added beyond the uniform core, the effective jet core
opening angle becomes larger and the emitting ring reaches the effective core
edge at a later time, resulting in later appearance of the polarization peaks
and rotation of the polarization angle. The size of the effective jet core
opening angle is dictated by the wing power-law index $a$. In shallower setups
(with smaller $a$ values), this effective angle is larger since the
contribution to the observed flux from regions at wider angles increases. This
effect is shown in Fig. 2 with polarization extrema and jet break times, which
are marked with stars, occurring at later times for shallower jets. The second
polarization peak is associated with the time when the opening angle of the
emitting ring is of the order of the jet core opening angle, i.e. when
$1/\Gamma\sim\theta_{c}$. For top hat jets, this is also the typical time when
the jet break occurs, resulting in a relatively tight correlation between the
two times (Sari, 1999; Granot, 2003; Granot & Konigl, 2003; Rossi et al.,
2004; Birenbaum & Bromberg, 2021). This correlation is weaker in shallow jets
although it still holds up to a factor 3 between the two times (see sub-
section 3.3 for a detailed discussion). The larger effective jet opening angle
also changes the heights of the polarization peaks as well. This can be
explained with the effective normalized observed angle, quantified by the $q$
parameter ($q_{\text{eff}}$) becoming smaller. Smaller values of
$q_{\text{eff}}$ are associated with higher degrees of axial symmetry and thus
lower polarization (Ghisellini & Lazzati, 1999; Sari, 1999; Granot & Konigl,
2003; Rossi et al., 2004; Shimoda & Toma, 2021; Teboul & Shaviv, 2021;
Birenbaum & Bromberg, 2021).

####  3.1.2 Off-axis jets ($q>1$)

Figure 3: Observed polarization degree (upper panels) and flux (lower panels)
at observer frequency $\nu=10^{15}Hz$ (PLS G) with a flat distribution of
initial Lorentz factor ($b=0$) and a random magnetic field structure confined
to the face of the shock ($\xi\rightarrow 0$) for off-axis jets. The direction
of the polarization vector remains constant throughout the temporal evolution.
Left panel: We set the viewing angle to be constant for an off-axis jet with
$q=5$ with varying values of $a$. The polarization degree reduces with $a$ and
its peak coincides with a break in the light curve. Right panel: The structure
of the jet is held constant with $a=1$ and the off-axis viewing angle is
changed. As the observer approaches the jet axis with reducing values of $q$,
the polarization peak becomes lower and occurs at earlier times. Figure 4:
Observed polarization degree (upper panel) and flux (lower panel) at observer
frequency $\nu=10^{15}Hz$ (PLS G) with a flat distribution of initial Lorentz
factor ($b=0$), smooth power law energy profile with $a=1$, observed with
$q=3$. The structure of the magnetic field changes with the value of $\xi$.
The direction of the polarization vector changes by $90^{\circ}$ once the
$\xi$ parameter crosses $1$. The polarization peak happens at about the same
time for all curves and is close to the break time of the light curve (marked
with a star). The light curves of the models corresponding to
$\xi=0.707,1.414$ and $\xi\rightarrow 0$ converge to one another due to
similar mean magnitude magnetic field magnitudes.

In the case of off-axis jets the observer is located at an angle larger than
the core half opening angle. In this case the observed emission is initially
dominated by the lower energy per solid angle wings, since the core emission
is beamed away from the observer due to its high Lorentz factor. As the jet
slows down, the beaming relaxes and regions closer to the core gradually
become visible, leading to a light curve evolution, which is different than
the one in the case of an on-axis jet. While for shallow jets the contribution
of the core to the total flux is sub-dominant, it typically still dominates
the polarized flux, and therefore determines the polarization. To study the
effect of the jet structure on the time evolution of the polarization
signature, we begin by viewing the jet at a viewing angle of $q=5$, and hold
it constant while exploring different power-law profiles of the wings
corresponding to shallow jets and comparing them to a top hat jet (see Fig. 3,
left panel). The configuration of the magnetic field is random on the plane of
the shock ($\xi\rightarrow 0$). Similarly to the on-axis case, the observed
flux comes from a “ring” like shape centered around the observer’s LOS.
However, due to the angular structure in energy density and Lorentz factor,
the flux distribution in the ring is not uniform but concentrated in a region
closer to the jet core. As a result, the observed flux is always polarized in
the direction pointing from the LOS towards the jet core, which translates in
our setup to a positive sign of $Q_{\nu}$ (see Fig. 6 for a simplified
illustration). The peak in the polarization curve occurs when the opening
angle of the emitting ring is of order $\theta_{\rm obs}$ and the core beaming
cones open towards the observer, revealing it to the observer with
$1/\Gamma\sim\theta_{\rm obs}$ (this time is defined similarly as $t_{\rm sh}$
in Beniamini et al. (2022)). At this time, the light from the emitting ring
reaches maximal asymmetry, maximizing the fraction of polarized light. For
this reason, the time of the peak is nearly independent of the energy
profile666for a fixed value of $\epsilon_{c}$ there is only a weak dependence
on $a$ that arises for sufficiently shallow jets (see Beniamini et al. (2022)
for details). and depends on the viewing angle alone. This can also be seen in
the right panel of Fig. 3, where the $q$ parameter is changed while keeping
the jet structure constant. As in the on-axis case, decreasing the power-law
index of the energy structure in the wings reduces the asymmetry in the light
distribution and the value of the peak polarization. In the case of a top hat
jet (blue curve in Fig. 3), only the core is luminous. Within the context of a
structured jet, we can say that light is only coming from parts of the
emission ring that intersect with the core. During the polarization peak time,
the entire core is visible and the emitting region on the ring has an angular
size of $\sim 2\theta_{c}$, which is smaller than the angular size of the ring
$2\theta_{\rm obs}\simeq 10\theta_{c}$ corresponding to the chosen value of
$q=5$ (see Fig. 6 for a visual demonstration of this effect). As a result, the
radially polarized emission coming from this narrow region in the ring
resembles a uniform polarization pattern and the polarization degree reaches a
value close to the threshold value given in Eq. (15). The continual
deceleration of the jet at later times adds light from regions that are
progressively further away from the core, and therefore contribute less light.
This causes the light curve to show a break similar to the jet-break at a time
close to the time of the polarization peak. We find that the two times are
correlated with each other within a factor of $3$ (see upper panel of Fig. 5
with further explanations in section 3.3).

###  3.2 Magnetic field 3D orientation

So far we assumed that the magnetic field configuration is random in the plane
of the shock with a small coherence length. Although this assumption is often
used in describing the magnetic field configuration just behind the shock, it
doesn’t take under consideration effects of turbulence and stretching of the
magnetic field further downstream, which can generate a non negligible radial
component (e.g. Granot & Konigl, 2003; Gill & Granot, 2020). Hereon we will
denote the co-moving field component in the plane of the shock by
$B^{\prime}_{\perp}$ and its radial component (parallel to the shock normal)
by $B^{\prime}_{||}$. A parallel field produces local polarization in an
azimuthal direction around the LOS, where most of the observed light is coming
from a ring with an opening angle $1/\Gamma$ around the LOS (see Nava et al.
(2016) for a visualisation of this effect). This polarization vector
orientation is rotated by exactly $90^{\circ}$ from the radial polarization
produced by a purely perpendicular field (in the shock plane; see Fig. 11.8 of
Granot & Ramirez-Ruiz (2012) or Birenbaum & Bromberg (2021)). Since
polarization is a pseudo-vector, the combination of the two polarization
components lowers the overall polarization.

We parameterize the 3D structure of the magnetic field following the formalism
of Sari (1999) and Gill & Granot (2020), in which an initial field with a
constant strength and an isotropic 3D orientations is stretched by a factor
$\xi$ in the radial (shock normal) direction (for details see Appendix B). In
this formalism, $\xi=0$ corresponds to a field confined to the plane of the
shock and $\xi\rightarrow\infty$ to a purely radial field.

Figure 4 shows the polarization curves and light curves from four structured
jet models, all with a power-law index $a=1$ and $q=3$. The models differ by
their $\xi$ values with $\xi=0,0.707,1.414$ and $\infty$. In the first two
models, where $\xi<1$, the light from the perpendicular magnetic field
component dominates, causing the polarization to have a positive sign as
before. For the two models with $\xi>1$, the polarization is mostly in the
azimuthal direction and thus has a negative sign. It can also be seen that the
maximal polarization is lower in the cases where $\xi$ is close to 1,
corresponding to cases where $B^{\prime}_{||}$ and $B^{\prime}_{\perp}$ are of
the same order. The time evolution of the polarization degree is similar to
what we discussed in section 3.1.2. The absolute value of the polarization
levels in the extreme magnetic field configurations corresponding to
$\xi\rightarrow 0,\infty$ differ from one another. The $\xi\rightarrow\infty$
features a uniform field, directed along the local velocity vector, and this
asymmetry results in higher polarization degrees compared to those of the
$\xi\rightarrow 0$ magnetic field configuration. The involvement of random
component in the latter case leads to a rise in cancellations as contributions
from radiating areas are summed to estimate the total polarization, thus
reducing it.

###  3.3 Behaviour of peak polarization

Figure 5: Upper panel: Ratio between the polarization peak times
$t_{P_{\max}}$ and light curve break times $t_{b}$ as function of normalized
viewing angle $q=\theta_{\rm obs}/\theta_{c}$ for all jet structures
considered in this work. We can see that these two times are within a factor
of three from each other except for the extremely shallow structure of
$a=0.5$, combined with a viewing angle of $q=0.7$ that forms a symmetrical
system around the line of sight and very late and slight jet break. Lower
panel: Maximal polarization degree $P_{\max}$ as function of the normalized
viewing angle $q=\theta_{\rm obs}/\theta_{c}$. For the top-hat jet (blue
$\times$ sign), the peak polarization for different $q$ values is described by
two different functions for $q_{\text{eff}}\leq 1$ (solid lines) and
$q_{\text{eff}}>1$ (dotted lines). For the structured shallow jets, the
maximal polarization degree seems to evolve as $P_{\max}\propto q$ (solid
lines), close to the $q<1$ behaviour for the top-hat jet.

The temporal evolution of the observed polarization degree, demonstrated in
Figs. 2\- 4, shows that the value of the peak polarization degree, $P_{\max}$,
varies with changes in the viewing angle ($q$), the jet wings energy power-law
index $a$, and orientation of magnetic fields behind the shock, $\xi$. The
dependence of $P_{\max}$ on $q$ and $a$ can be explained in terms of a simple
toy model for the polarization degree, inspired by the correlation between the
time of peak polarization, $t_{{}_{P_{\max}}}$ and the jet break time
$t_{\text{b}}$. The upper panel of Fig. 5 demonstrates this correlation in
jets with different structures and viewing angles. It shows that in almost all
cases, the ratio between the two times varies between $1/3$ and $3$. In the
case of a jet with $a=0.5$ and $q=0.7$ we see a deviation from this relation,
likely due to our inability to measure $t_{\text{b}}$ correctly. We evaluate
$t_{\text{b}}$ by fitting two asymptotes to the AG light curve at early and
late times and taking the intersection time between them, as shown in the
lower panel of Fig. 4. The outlier jet structure is extremely shallow and is
observed from a relatively small effective viewing angle. As a result the jet
break is very gradual and late, making it difficult to correctly fit an
asymptote to the early time AG light curve.

Figure 6: A toy model of the observed emission on the plane of the sky of a
top-hat jet during the polarization peak time, assuming that this time
coincides with the jet break time. The left and right panels demonstrate cases
of on-axis and off-axis jets respectively. The black circle represents the
angular size of the jet core ($\theta_{c}$), while the green ring covers the
angular size of the observable region during the time of peak polarization
with respect to an observer located at an angle $\theta_{\rm obs}$ from the
jet axis. The red arches track the most luminous regions during the jet-break
time in both types of jets. The polarization degree is obtained by calculating
the normalized $Q_{\nu}$ parameter of the light coming from the red arch,
which scales as $Q_{\nu}/I_{\nu}\propto\sin(2\chi)/2\chi$. See text for
further details.

To model the polarization degree during the polarization peak time, we built a
simplified toy model for the emission observed on the plane of the sky. We
make the following simplifying assumptions:

  1. 1.

We model a top-hat jet, i.e. the emission comes only from the jet core.

  2. 2.

The peak polarization time coincides with the time of the jet-break, i.e.
$t_{P_{\max}}=t_{\text{b}}$.

  3. 3.

The emission originates from a narrow arch across the core with an angular
radius of $1/\Gamma$ from the LOS.

Fig. 6 illustrates our simple toy model for cases of on-axis (left) and off-
axis (right) observers. The angular size of the jet core is marked by a black
solid circle. The observer is located at an angle $\theta_{\rm obs}$ from the
jet axis (the center of the black circle). The green ring centered around the
observer’s LOS marks the observable region at the time of $t_{\text{b}}$,
assuming that at this time $\Gamma(\theta=0)=\theta_{c}^{-1}$ in the on-axis
jet and $\Gamma(\theta=0)=\theta_{\rm obs}^{-1}$ in the off-axis jet. The red
solid arch marks the region from which most of the observed emission
originates. If the magnetic field is bound to the plane of the shock
($\xi\rightarrow 0$) then the light emitted from the red arch is polarized in
a radial direction relative to the LOS on the plane of the sky. The total
polarization degree in this case can be estimated by calculating the total
$Q_{\nu}$ parameter of the light emitted by the arch, normalized by its total
power, $I_{\nu}$:

$P_{\max}=\frac{Q_{\nu}}{I_{\nu}}=\Pi_{\rm
max}\frac{\int_{-\chi}^{\chi}\cos(2\phi)d\phi}{\int_{-\chi}^{\chi}d\phi}=\Pi_{\rm
max}\frac{\sin(2\chi)}{2\chi}\;,$ (21)

where $\phi$ is an azimuthal angle on the arch and $\chi$ is the angle of the
arch at the edge of the jet core (see Fig. 6). For on-axis jets ($q<1$)
$\cos\chi=q/2$ and for off-axis jets ($q>1$) $\sin\chi=1/2q$, as can be seen
in Fig. 6. Substituting these values in Eq. (21) we get the value of
$P_{\max}$ in the two asymptotic limits:

$P_{\max}\simeq\Pi_{\rm
max}\begin{cases}\frac{q}{\pi}\left(1+\frac{q}{\pi}\right)&q\ll 1\\\
q\sin{1/q}&q\gg 1,\end{cases}$ (22)

Note that $P_{\max}$ passes smoothly through $q=1$. The two asymptotes
intersect at $q=1.875$ at an approximated value of $P_{\max}$ that is $\sim
15\%$ larger than the value obtained from Eq. 21, thus in practice Eq. 22 can
provide a reasonable approximation for $P_{\max}$ at the entire range of
viewing angles. The lower panel of Fig. 5 shows the value of $P_{\max}$ in
jets with different structures observed from various viewing angles. The color
scheme is the same as in the upper panel. The dotted blue line represents a
fit of Eq. 21 to the data of the top-hat jet, normalized by $N=0.8$. The data
of the shallow jets ($a=0.5-2$), is fitted by the $q<1$ asymptote of Eq. 22.
The shallow energy profiles in the wings of these jets increase the effective
angular size of the core, leading to a smaller effective $q$ parameter
$q_{\text{eff}}$ than the naive value of $\theta_{\rm obs}/\theta_{c}$.
Universal definitions of the effective core opening angle, based solely on the
jet energy profile, may not fit all structured jets in general. For example,
the definition given in Govreen-Segal & Nakar (2023) does not apply for
shallow jets with $a<2$. We choose to quantify this $q_{\text{eff}}$ parameter
using the angular flux profile during the jet-break time $t_{\text{b}}$, as
the luminosity of a certain emitting region is the observational factor that
determines whether or not it is a part of the core. We set the effective core
opening angle at $10\%$ of the angular luminosity cumulative distribution
function (CDF) peak. As a result, the semi-linear fit of $P_{\max}(q)$ extends
to larger values of $q$. We plan to explore this definition in the context of
afterglow polarization in future work.

Figure 7: Maximal polarization degree $P_{\max}$ as function of the energy
profile of the jet, quantified as a declining power-law $a$. The maximal
polarization degree seems to evolve as $P_{\max}\propto a^{1.4}$ which holds
over all viewing angles.

Fig. 7 shows the change in $P_{\max}$ when we keep the $q$ parameter constant
and change the value of $a$. Larger values of $a$ imply a steeper energy
profile in the jet wings and a smaller effective core angular size. This leads
to larger effective value of $q$ parameter and higher $P_{\max}$ for the same
viewing angle, consistent with the above results. We find that
$P_{\max}\propto a^{1.4}$ in all values of $q$ tested, implying that $q$ and
$a$ are rather independent parameters. Stating this differently, the effective
angular size of the core depends on the $a$ parameter and is the same for all
observers viewing the jet from different angles. This relation also holds for
both smooth power-law and broken power-law jet structures.

Figure 8: Maximal polarization degree $P_{\max}$ as function of the magnetic
field structure parameter $\xi$, under which synchrotron radiation is emitted.
This relation is plotted for a few geometrical combinations and seems to agree
with a hyperbolic tangent of $\log_{10}\xi$ for all values. We mark in shaded
gray the limits on $\xi$ imposed by the upper limits on the polarization of GW
170817, corresponding to $0.75<\xi<1.15$. In the insert we zoom in on the
region around these limits. All curves change sign at $\xi=1$. Detailed models
can be found in table 2.

The above tests were preformed assuming that the magnetic field is bound to
the plane of the shock ($\xi\rightarrow 0$). In this situation the
polarization degree in the most luminous region is close to $\Pi_{\rm max}$
and points along a the radial direction from the observer. It is important to
test how our results change when the value of $\xi$ varies. Fig. 8 shows the
change in $P_{\max}$ in three types of shallow jets observed at different
viewing angles, as a function of $\xi$. All curves show qualitatively similar
shapes that can be fitted by a hyperbolic tangent function:

$P_{\max}(\xi)=A\tanh(-B\log_{10}\xi+C)-D,$ (23)

where $(A-D)$, $-(A+D)$ are the asymptotic values of $P_{\max}$ at
$\xi\rightarrow 0$ and $\xi\rightarrow\infty$ respectively, $B$ is the width
of the transition region and $C=\tanh^{-1}(D/A)$ ensures that
$P_{\max}(\xi=1)=0$ as expected in a case of a perfectly isotropic magnetic
field. Fittings of the function to the $P_{\max}(\xi)$ in cases with different
$a$ and $q$ are given in table 2.

Notice that Eq. (23) is not symmetric about the $x$ axis, namely
$|P_{\max}(\xi\rightarrow 0)|-|P_{\max}(\xi\rightarrow\infty)|=-2D$ (instead
of $0$ in the symmetric case). The reason for that is that in the case of
$\xi\rightarrow 0$ the polarization is obtained by summing contributions
regions with random field on the plane of the shock, whereas in the case of
$\xi\rightarrow\infty$ the contributing field is ordered (radial), and
produces a higher overall polarization. We also note that the asymmetry level
($D/A$) decreases with larger $a$. The reason is that the observed light in
jets with steeper wings (larger $a$) is coming from narrower regions. When
$\xi\rightarrow 0$ the polarization vector in each emitting zone in the ring
will deviate less from this radial direction, giving rise an overall
$P_{\max}$ that is closer to $P_{\max}(\xi\rightarrow\infty)$.777The asymmetry
level also slightly decreases for large $q$, since a larger fraction of the
emission near the time of peak polarization is coming from angle close to
$1/\Gamma$ from the LOS, where $\hat{n}^{\prime}_{\rm obs}$ is close to the
shock plane leading to a more ordered field in projection to the plane normal
to $\hat{n}^{\prime}_{\rm obs}$, which is closer to the ordered radial field
case. The immediate implication is that there is some dependency between $\xi$
and $a$, however as long as we limit the results to $\xi\lesssim 1$ the
dependency is weak, allowing us to obtain a simple analytic expression for
$P_{\max}$ applicable for shallow jets:

$P_{\max}=qa^{1.4}\left[0.055\tanh\left(-2.3\log_{10}\xi+0.34\right)-0.02\right].$
(24)

Following the approach of Gill & Granot (2020)888also derived independently by
Corsi et al. (2018); Stringer & Lazzati (2020); Teboul & Shaviv (2021)., we
use the limits on the radio polarization degree of GW 170817 with
$P_{tot}<|12\%|$ at 244 days, along with its fitted power-law jet structure
from Gill & Granot (2018), to constrain the typical value of the $\xi$
parameter in terms of our model. We find a nearly isotropic magnetic field
structure agrees with the limits on GW 170817 with $0.75<\xi<1.15$ (plotted
gray area in Fig. 8). Eq. (24) is applicable to this range within the $\xi$
parameter space. Assuming a "single zone" model, in which the same shock-
generated field induces the multi-wavelength synchrotron afterglow forward
shock emission, allows us to generalize the constraints on $\xi$ from the
radio to the optical regime. In the context of shallow jets, we can
interpolate the results of Fig. 4 using these limits and conclude the near
isotropic magnetic field structure imposed by observations, would induce very
modest polarization degrees upon shallow jet models, regardless of the off-
axis viewing angle.

Another parameter dependence we did not explore so far was the propagation of
the relativistic shock into a non-uniform medium with $\rho\propto R^{-k}$
where $k>0$. We plan to thoroughly explore this in future work.

##  4 GRB 221009A - A shallow jet?

Figure 9: Upper panel: Temporal evolution of the polarization degree for two
suggested structures for GRB 221009A: Gill & Granot (2023) in solid lines and
O’Connor et al. (2023) in dash-dotted lines. The curves are computed in two
bands: X-ray (blue) and optical (red) where the upper limits on linear
polarization in both bands is presented with triangles in corresponding colors
(Negro et al., 2023). The structure of the magnetic field is taken to be
random on the plane of the emitting region ($\xi\rightarrow 0$). In dark blue
dash-dotted lines we plot the X-ray polarization curve for the O’Connor et al.
(2023) model with $\xi=0.75$ and find reduced polarization at early times. All
proposed models agree with the upper limits on polarization taken at 3.5 days
in both bands. We mark with stars the polarization level at $\sim 2.5$ hours
post-burst. Lower panel: X-ray (blue) and optical (red ) light curves for two
of the proposed models for the structure of the jet of GRB 221009A: Gill &
Granot (2023) in solid lines and O’Connor et al. (2023) in dash-dotted lines.
Both forward shock models fit the observed data (O’Connor et al., 2023; Gill &
Granot, 2023).

GRB 221009A was a bright long GRB with an isotropic equivalent energy
$E_{\rm\gamma,iso}\approx 10^{55}\;$erg (de Ugarte Postigo et al., 2022;
O’Connor et al., 2023), the largest isotropic-equivalent gamma-ray energy
measured to date (Burns et al., 2023). Detection of TeV photons from the early
AG (Huang et al., 2022) also suggests that the intrinsic energy of the event
is quite high. Its measured low redshift of $z=0.151$ (de Ugarte Postigo et
al., 2022; Malesani et al., 2023), together with the high observed flux
allowed for detailed multi-wavelength afterglow measurements (Williams et al.,
2023; Bright et al., 2023; Laskar et al., 2023; O’Connor et al., 2023) as well
as obtaining upper limits on the X-ray polarization of $13.8\%$ placed $3.5$
days post-burst (Negro et al., 2023).

Despite the plethora of observations, no classical jet break was detected in
the afterglow light curve throughout the first 100 days after the GRB.
Combined with the record-breaking high fluence and $E_{\rm\gamma,iso}$, the
lack of a jet break poses a possible energy crisis for GRB progenitor models
(O’Connor et al., 2023). One appealing solution is that the relativistic jet
core is surrounded by energetic wings with a shallow angular energy profile.
This assumption allows to obtain good fits to the afterglow light curves using
a reasonable intrinsic GRB energy. However, degeneracies in the classical
afterglow model imply that different jet parameters can fit the observed
multi-wavelength light curve (e.g. Gill & Granot, 2023; O’Connor et al.,
2023). Gill & Granot (2023) featured a flat cored jet with
$\theta_{c}=1.2^{\circ}$ surrounded by wings with shallow power-law profiles
in energy per solid angle and Lorentz factor, that scale as $\theta^{-0.8}$
and $\theta^{-0.3}$, respectively. The viewing angle corresponds to $q=0.95$
and the surrounding medium has a wind-like power-law profile with $k=2$.
O’Connor et al. (2023) fitted a broken power-law jet with no core where both
the inner and outer segments have shallow angular power-law profiles in the
energy per solid angle, proportional to $\theta^{-0.75}$ and $\theta^{-1.15}$,
respectively. The opening angle of the inner power-law segment is $3^{\circ}$,
the outer segment extends up to $23^{\circ}$ and the jet spreads into a
uniform density medium with $k=0$. We refer the reader to appendix 3 for a
full description of the afterglow model parameters. Both models are able to
explain the broad-band spectrum with a combined contribution from the reverse
and forward shocks, where the reverse shock dominates the radio emission
during the first 100 days of observations. According to these models, the
optical emission is reverse shock dominated up to $1$ day and dominated by the
forward shock at later times, while the X-ray emission is forward shock
dominated throughout the entire evolution.

We reproduce the light curve fits with our numerical tool, according to the
jet structures suggested by these models and present them in the lower panel
of Fig. 9. The solid lines present the Gill & Granot (2023) model and the
dash-dotted lines show the O’Connor et al. (2023) fit to the data. The empty
circles represent the data points where the blue and red colors show X-ray and
optical data and fits respectively. Since our model evaluates emission from
the forward shock, we present the optical fits only for times > $1$ day. The
upper panel of Fig. 9 shows the associated polarization curves with the same
color mapping as in the bottom panel, assuming a random magnetic field
confined to the plane of the shock ($\xi\rightarrow 0$). Additionally, we plot
the X-ray polarization curve of O’Connor et al. (2023) with a more realistic
value of $\xi=0.75$ in dark blue dash-dotted lines. The Gill & Granot (2023)
model produces polarization levels $\rightarrow 0$ for $\xi=0.75$. The open
triangles mark the upper limits placed at $3.5$ days post-burst by Negro et
al. (2023) in X-ray and optical. It can be seen that all models predict
polarization degrees well below the observed limits even for the most
optimistic magnetic field configuration. With these current polarization
measurements, we have no way of discriminating between the two suggested
models. While the polarization degree is very low at $3.5$ days for both
models, the X-ray polarization differs greatly at early times.

At $\sim 2.5$ hours post-burst the difference between the polarization degrees
predicted by the two models is $8\%$ in the most optimistic configuration and
$2.3\%$, if using a more realistic $\xi=0.75$ value (this time is marked with
stars in the upper panel of Fig. 9). The difference in the behaviour of the
polarization curves lies in the different angular energy profiles used at the
jet cores. The model by Gill & Granot (2023) features a flat core profile with
a narrow opening angle, similar to the profiles tested in this work, with the
viewer located close to the core edge. The polarization curve in this case has
a peak at a time close to the jet break time, when
$\Gamma\simeq\theta_{c}^{-1}$, seen at about 1 day in the X-ray light curve.
Note that the jet in this case expands into a wind-like medium with $k=2$,
thus the polarization peak is smeared over more than a decade in time. In the
model by O’Connor et al. (2023), the inner power-law segment features a
distribution that peaks on the jet symmetry axis, while the viewing angle is
very small. In this case the polarization peaks when the region near the jet
symmetry axis is revealed as its Lorentz factor drops to $\Gamma=\theta_{\rm
obs}^{-1}>\theta_{c}^{-1}$, and the slight jet break seen in the X-ray light
curve is attributed to the broken power-law structure (O’Connor et al., 2023).
Here, the jet expands into a uniform medium ($k=0$), resulting in a narrower
peak. The parameters used by the two models are presented in Appendix 3. Had
X-ray polarization been measured at earlier times, we may have been able to
differentiate between the models via polarization modeling. In addition to the
demonstrated theoretical benefits of a faster IXPE response time, Swift-XRT
observations of GRB afterglow populations suggest that IXPE requires
repointing within 6 hours of the trigger and exposure times on the order of
several hundred kilo-seconds to reliably detect X-ray polarization below
$10\%$ for at least one GRB annually (Negro & Burns, 2024)999Link to the
poster..

##  5 Discussion

Traditional GRB models, assuming a top hat jet with uniform energy and Lorenz
factor distributions, predict polarization degrees $\gtrsim 10\%$ at the time
of the jet-break when observing the jet on axis ($0.5\lesssim q=\theta_{\rm
obs}/\theta_{c}<1$; for smaller $q$ the polarization is lower) and even higher
levels when observing off-axis with $q>1$ ( e.g. Sari, 1999; Ghisellini &
Lazzati, 1999; Granot, 2003; Rossi et al., 2004; Nava et al., 2016; Birenbaum
& Bromberg, 2021). However, the rapid decline in the observed flux with
viewing angle in off-axis jets makes the later case harder to observe
(however, in some cases the off-axis jet emission can exceed that of an on-
axis one, as shown in Granot et al. (2002)). Meanwhile, most observed
polarization levels remain at a level of $10\%$ or below (e.g. Covino et al.
(2004); Steele et al. (2009); Uehara et al. (2012); Wiersema et al. (2014)),
with a few outliers suspected to be associated with a reverse shock
contribution, e.g. Mundell et al. (2013). Adding a jet structure to GRB
afterglow models reduces the overall polarization and can relax the tension
between theoretical models and observations.

We find that the peak polarization degree inversely depends on the amount of
energy in the extended jet wings. We also found that the light contribution
from the wings doesn’t change the relation between the times of the jet-break
and peak polarization that was shown to exist in top hat jets (Sari, 1999;
Granot, 2003; Rossi et al., 2004; Birenbaum & Bromberg, 2021), it does however
extend the break duration and reduce the change in the light curve slope it
induces. In the extreme case of shallow jets, the slope difference may be too
small to be detected, resulting in a lack of an observed canonical jet-break,
together with extremely low polarization levels (e.g. O’Connor et al., 2023).
The jet structure also delays the occurrence time of jet-breaks seen by on-
axis observers. This occurs as a result of an increase in the effective core
angle due to emission from the jet wings. For off-axis observers, the
energetic wings increase the visibility of the jet at early times while
slightly changing the light curve break time. This delay in the jet break
time, following a decrease in the value of $a$ below a critical value
corresponding to the transition to shallow jets, is also predicted from
analytical arguments (see relations for $t_{\text{sh}}$ in Beniamini et al.
(2022)).

A second factor that can reduce the polarization levels is the existence of a
magnetic field component aligned with the shock normal in addition to the
field on the shock plane, parameterized by $\xi$. For example, polarization
measurements in the radio afterglow of GW 170817, along with its light curve,
indicate that the magnetic field is nearly isotropic with $0.75<\xi<1.15$
(adapted from the analysis of Gill & Granot (2020)).

More generally, the observed polarization is affected by three main
parameters: the jet structure, the magnetic field configuration and viewing
angle. To map this apparent degeneracy we obtained an analytical expression
connecting these three parameters to the peak polarization level, showing that
$P_{\max}\propto qa^{1.4}F(\xi)$, where $q=\theta_{\rm obs}/\theta_{c}$ is the
normalized viewing angle, $a$ is the power-law slope of the jet wing energy
profile, and $F(\xi)$ is a shifted hyperbolic tangent (Eq. 24). Light curve
observations, together with polarization measurements during the jet-break
time, can help alleviate some of these degeneracies.

Polarization measurements at early times can provide an additional constraint
on the jet structure. This can be especially important in case of shallow jets
where a jet break may not be detected and polarization levels are expected to
be low, as we show for GRB 221009A. In this case, we showed that two different
models of shallow jets, producing the same afterglow light curve, can have a
significantly different level of polarization at early times (see Fig. 9). Our
polarization analysis focused on the X-ray band, since it is not expected to
be contaminated by light from the reverse shock and thus should reflect the
magnetic field configuration generated by the forward shock alone. If future
early time X-ray polarization measurements are accompanied by detection of
optical or radio polarization as well, the combination of the two could be
used to constrain the magnetization of the reverse shock, which probes the
relic magnetic field in the outflow.

Low magnetization may be a common feature of shallow jets in general. We look
into a sample of five other luminous events that share common features with
GRB 221009A, such as long lasting X-ray afterglows, lack of a canonical jet
break in their light curves, and very high energy emission, assembled by
O’Connor et al. (2023). In some of these events there are early time
detections of optical and radio polarization, suspected to come from a reverse
shock, that show low levels of polarization $<8\%$ (van der Horst et al.,
2014; Laskar et al., 2019; Jordana-Mitjans et al., 2020; Dichiara et al.,
2022; Arimoto et al., 2024). Numerical simulations show that hydrodynamic jets
develop energetic wings with shallow energy profiles as a consequence of their
passage through the confining medium, as opposed to their weakly magnetized
counterparts which form steeper profiles (Gottlieb et al., 2020; Gottlieb et
al., 2021; Beniamini et al., 2022). The polarization from the reverse shock in
such jets is expected to be low due to random orientation of magnetic fields
behind it, making hydrodynamic jets suitable candidates for the observed
shallow GRB jets.

Out of these five events, only two have measured polarization during the
forward shock afterglow phase, showing levels < 3% in the optical band (Itoh
et al., 2013; van der Horst et al., 2014; Arimoto et al., 2024), consistent
with the expected polarization from shallow jets. As stated above, early time
measurements of polarization in X-ray, optical and radio bands can be used to
constrain the magnetization in the ejecta shortly after it breaks out of the
progenitor star. They may also help confirm the connection between the jet
structure and the magnetization of the outflow and possibly differentiate
between suggested models.

##  6 Conclusions

In this work we explore the effect of a power-law angular structure on GRB
afterglow light curves and observed polarization. We find that increasing the
amount of energy in the jet wings reduces the observed polarization and
produces more moderate breaks in the afterglow light curves for both on-axis
and off-axis jets. When the energy in the jet wings becomes comparable to the
energy in the jet core, the jet-break becomes extremely subtle and may not be
observed. Such a distinctive feature is seen in energetic GRBs with long
lasting X-ray afterglows and very high energy emission and may point to the
involvement of an energetic extended structure. Analysis of our results
allowed us to formulate an analytical relation between the system parameters
and the peak polarization degree, supported by a physical toy model of the
system. Finally, we employ our semi-analytical tool on the light curve fits
for GRB 221009A and find an agreement with the upper limits placed on its
afterglow polarization in both optical and X-ray. While this affirms the
shallow jet models suggested for this event, it does not distinguish between
them. Nevertheless, the polarization levels of both models considered for GRB
221009A differ greatly at early times. Possible IXPE detection could have
allowed the desired distinction between them. The expected afterglow
polarization degrees from shallow GRB jets are low, and if these are a common
feature of energetic GRBs, future instrumentation should feature better
sensitivity, along with faster response times.

## Acknowledgements

We thank Brendan O’Connor, Geoffrey Ryan and Michela Negro for helpful
discussions. G.B and O.B. are supported by ISF grants 1657/18 and 2067/22, a
BSF grant 2018312, and an NSF-BSF grant 2020747. R.G. is supported by
PAPIIT-2023 (IA105823) grant. P.B. supported by a grant (no. 2020747) from the
United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel
and by a grant (no. 1649/23) from the Israel Science Foundation.

## Data Availability

The data underlying this paper will be shared on reasonable request to the
corresponding author.

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##  Appendix A Modeling the forward shock and the magnetic field structure
behind it

Figure 10: An illustration of the multiple coordinate systems used to evaluate
polarization in structured GRB afterglows. (a) The global Cartesian
coordinates define the symmetry axis of the jet as $\hat{z}$, the $\hat{x}$
axis connects the LOS (teal) and the symmetry axis for a choice of $\phi_{\rm
obs}=0^{\circ}$ and the $\hat{y}$ axis is perpendicular to both, forming a
left-handed coordinate system. (b) The local spherical coordinate system is
denoted at each angular section $(\theta,\phi)$, where the radial direction
(blue) is parallel to the velocity direction and the
$(\hat{\theta},\hat{\phi})$ directions (magenta and red respectively)
correspond to those denoted by the global coordinate system. We use this
system, for example, to denote the magnetic field direction at each cell (see
Eq. 25). The angle $\theta_{\text{B}}$ (brown) is measured from the local
radial direction and the angle $\phi_{\text{B}}$ (green) is measured from the
local $\hat{\phi}$ direction.

The emitting region, generated by the structured relativistic shock that plows
into the ISM, is modeled by a polar grid which divides it into small angular
sections $\left(\theta,\phi\right)$. These coordinates are centered around the
jet axis. The polar angle $\theta$ is measured from the jet symmetry axis and
the coordinate $\phi$ acts as an azimuthal angle. Our code uses two coordinate
systems which are used to calculate physical qualities in the local and global
frames. Local quantities on our 2D emitting region, which is meant to emulate
a semi-spherical shock, are best described by the spherical coordinates
$(\boldsymbol{\rm{\hat{r}}},\boldsymbol{\rm{\hat{\theta}}},\boldsymbol{\rm{\hat{\phi}}})$
(see panel (b) in Fig. 10 for a detailed illustration). The radial direction
is perpendicular to the face of the shock and parallel to the direction of the
velocity at each cell. The $\boldsymbol{\rm{\hat{\theta}}}$ angular direction
is measured from the radial direction and the $\boldsymbol{\rm{\hat{\phi}}}$
direction is denoted in the plane of the emitting surface. We demonstrate the
use of the local spherical coordinate system by defining the direction of the
magnetic field in the rest frame (inset (b) in Fig. 10). Our model denotes the
magnetic field as uniform in each cell with the following direction:

$\boldsymbol{\rm{\hat{B}}}^{\prime}=\cos\theta_{\text{B}}\boldsymbol{\rm{\hat{r}}}+\sin\theta_{\text{B}}\sin\phi_{\text{B}}\boldsymbol{\rm{\hat{\theta}}}+\sin\theta_{\text{B}}\cos\phi_{\text{B}}\boldsymbol{\rm{\hat{\phi}}},$
(25)

where the angle $\theta_{\text{B}}$ is measured from the local radial
direction and determines whether the magnetic field is confined to the plane
of the shock or not. When $\theta_{\text{B}}=90^{\circ}$, the magnetic field
is solely in the plane of the emitting surface
$\in\left[\boldsymbol{\rm{\hat{\theta}}}-\boldsymbol{\rm{\hat{\phi}}}\right]$.
The angle $\phi_{\text{B}}$ determines the orientation of the magnetic field
on the plane of the emitting surface and it is measured from the
$\boldsymbol{\rm{\hat{\phi}}}$ axis. A random magnetic field, with a small
coherence scale, confined to the plane of the shock, would be denoted by
randomizing the angle $\phi_{\text{B}}$ and setting
$\theta_{\text{B}}=90^{\circ}$. This kind of structure can be explained by
generation of the field in the shock by plasma instabilities such as the two-
stream Weibel instability (Medvedev & Loeb, 1999; Achterberg & Wiersma, 2007).
In the case where $\theta_{\text{B}}=0^{\circ}$, the magnetic field is in the
$\boldsymbol{\rm{\hat{r}}}$ direction and is considered radial. However, these
two cases are extreme and the true structure of the magnetic field is probably
a mixture of these two components. Since the magnetic field is uniform in each
cell, the synchrotron radiation from each cell is maximally polarized.

Global quantities would be defined in terms of a Cartesian coordinate system
and would be useful in determining quantities in the observer frame such as
the direction to the observer (line of sight LOS). We assume the observer
plane, from which emission reaches the observer, is perpendicular to the
symmetry axis of the jet $\hat{z}$ which means it lies in the $x-y$ plane (see
panel (a) in Fig. 10). To later on evaluate the local emission quantities and
polarization, we would also need to translate the global coordinates to the
spherical local ones. The direction to the observer is a constant one and thus
is defined by a constant vector in the global frame. For
$\boldsymbol{\rm{\hat{n}}}_{\rm
obs}=n_{x}\boldsymbol{\rm{\hat{x}}}+n_{y}\boldsymbol{\rm{\hat{y}}}+n_{z}\boldsymbol{\rm{\hat{z}}}$
and an observer located at the spherical angles $\left[\theta_{\rm
obs},\phi_{\rm obs}\right]$, we can solve the following set of equations

$\begin{cases}n_{\text{x}}=\sin\theta_{\rm obs}\cos\phi_{\rm
obs}=n_{\text{r}}\sin\theta\cos\phi+n_{\theta}\cos\theta\cos\phi-
n_{\phi}\sin\phi\\\ n_{\text{y}}=\sin\theta_{\rm obs}\sin\phi_{\rm
obs}=n_{\text{r}}\sin\theta\sin\phi+n_{\theta}\cos\theta\sin\phi+n_{\phi}\cos\phi\\\
n_{\text{z}}=\cos\theta_{\rm obs}=n_{\text{r}}\cos\theta-
n_{\theta}\sin\theta\end{cases},$ (26)

where $\boldsymbol{\rm{\hat{n}}}_{\rm
obs}=n_{\text{r}}\boldsymbol{\rm{\hat{r}}}+n_{\theta}\boldsymbol{\rm{\hat{\theta}}}+n_{\phi}\boldsymbol{\rm{\hat{\phi}}}$
is in terms of the local spherical coordinates. These turn out to be

$\begin{cases}n_{\text{r}}=\sin{\theta_{\rm obs}}\cos{\phi_{\rm
obs}}\sin{\theta}\cos{\phi}+\sin{\theta_{\rm obs}}\sin{\phi_{\rm
obs}}\sin{\theta}\sin{\phi}+\\\ +\cos{\theta_{\rm obs}}\cos{\theta}\\\
n_{\theta}=\sin{\theta_{\rm obs}}\cos{\phi_{\rm
obs}}\cos{\theta}\cos{\phi}+\sin{\theta_{\rm obs}}\sin{\phi_{\rm
obs}}\cos{\theta}\sin{\phi}-\\\ -\cos{\theta_{\rm obs}}\sin{\theta}\\\
n_{\phi}=\sin{\theta_{\rm obs}}\sin{\phi_{\rm obs}}\cos{\phi}-\sin{\theta_{\rm
obs}}\cos{\phi_{\rm obs}}\sin{\phi}\end{cases}.$ (27)

##  Appendix B Magnetic field structure parameterization

The downstream region, well behind the shock wave, can also contribute to the
observed emission and may experience a stretching of the magnetic field
component parallel to the bulk velocity direction, also denoted as the radial
direction $\hat{\text{r}}$ (Granot, 2003; Gill & Granot, 2020). To account for
this effect we adopt the formalism by Gill & Granot (2020), following Sari
(1999), assuming a spherical isotropic field $\bar{B}^{\prime}$ just behind
the shock, which is first stretched only in the radial direction by a factor
$\xi$ (while the field components in the plane normal to the radial direction
remain unchanged), and then a global renormalization by a factor of
$\sqrt{3/(2+\xi^{2})}$ is applies to all field components in order for the
field r.m.s to remain the same as for the original isotropic field,
$\bar{B}^{\prime}$, namely:

$\begin{split}&\boldsymbol{\rm{\hat{B}}}^{\prime}=B^{\prime}\left(\cos\theta_{B}\boldsymbol{\rm{\hat{r}}}+\sin\theta_{B}\sin\phi_{B}\boldsymbol{\rm{\hat{\theta}}}+\sin\theta_{B}\cos\phi_{B}\boldsymbol{\rm{\hat{\phi}}}\right)\\\
&=\bar{B}^{\prime}\sqrt{\frac{3}{2+\xi^{2}}}\left(\xi\cos\bar{\theta}_{B}\boldsymbol{\rm{\hat{r}}}+\sin\bar{\theta}_{B}\sin\bar{\phi}_{B}\boldsymbol{\rm{\hat{\theta}}}+\sin\bar{\theta}_{B}\cos\bar{\phi}_{B}\boldsymbol{\rm{\hat{\phi}}}\right)\end{split}$
(28)

where $\theta_{B}$ ($\bar{\theta}_{B}$) is the angle of the post-stretched
(pre-stretched) proper field from the shock normal (as illustrated in figure
10). The direction of the pre-stretched field is obtained from a uniformly
distributed probability density per solid angle, namely the angular parameters
$(\bar{\mu}_{B}=\cos\bar{\theta}_{B},\bar{\phi}_{B})$, are taken from uniform
probability distributions ($-1\leq\bar{\mu}_{B}\leq 1$,
$0\leq\bar{\phi}_{B}\leq 2\pi$) and then used to obtain the corresponding
angular direction of the post-stretched field,

$\mu_{B}=\cos\theta_{B}=\frac{\xi\bar{\mu}_{B}}{\sqrt{1+\bar{\mu}^{2}_{B}(\xi^{2}-1)}}\
,\quad\quad\phi_{B}=\bar{\phi}_{B}\ .$ (29)

This procedure is equivalent to obtaining $\mu_{B}$ from an angular
probability distribution function101010See Gill & Granot (2020), Appendix A
for a detailed discussion.

$f_{\mu}(\mu_{B},\xi)=\frac{\frac{1}{2}\xi^{2}}{\left[\xi^{2}(1-\mu_{B}^{2})+\mu_{B}^{2}\right]^{3/2}}\;.$
(30)

The generated values of $(\theta_{B},\phi_{B})$ are then inserted into Eq.
(28) to obtain the post-stretched field strength, which in contrast to the
original isotropic field depends on $\mu_{B}=\cos\theta_{B}$ or
$\bar{\mu}_{B}=\cos\bar{\theta}_{B}$,

$\displaystyle\quad
S(\mu_{B},\xi)=\frac{B^{\prime}}{\bar{B}^{\prime}}=\sqrt{\frac{3\xi^{2}/(2+\xi^{2})}{\xi^{2}(1-\mu^{2}_{B})+\mu^{2}_{B}}}\;,$
$\displaystyle\quad
S(\bar{\mu}_{B},\xi)=\frac{B^{\prime}}{\bar{B}^{\prime}}=\sqrt{\frac{1+\bar{\mu}^{2}_{B}(\xi^{2}\\!-\\!1)}{\frac{1}{3}(2+\xi^{2})}}\;.$
(31)

In particular, $S(\bar{\mu}_{B},1)=1$ (an isotropic field) as required,
$S(\bar{\mu}_{B},0)=\sqrt{\frac{3}{2}}\sin\bar{\theta}_{B}$ (a field confined
to the plane of the shock), and
$S(\bar{\mu}_{B},\xi\to\infty)=\sqrt{3}\bar{\mu}_{B}$ (a purely radial field).
Moreover, the r.m.s field indeed remains unchanged,

$\left\langle
S^{2}\right\rangle=\frac{1}{2}\int_{-1}^{1}d\bar{\mu}_{B}\,S^{2}(\bar{\mu}_{B},\xi)=1\;.$
(32)

Figure 11 demonstrates our adopted local magnetic field structure for
different values of $\xi$.

Figure 11: An illustration of our adopted local magnetic field structure for
five different values of $\xi$.

Our post-stretched magnetic field configuration in the case of $\xi\rightarrow
0$ is similar in terms of the field direction to that of previous works, which
assumed a random magnetic field purely in the shock plane. This can be seen by
substituting $\xi\rightarrow 0$ in Eq. (28), or equivalently by noting that
the probability function $f_{\mu}(\mu_{B},\xi)$ is narrowly peaked at
$\mu_{B}=0$ with a FWHM of the order $\Delta\mu_{B}\sim 2\xi$. However, our
model is distinct from such works (in particular from Granot & Konigl 2003 who
introduced the parameter $b$ with $b=0$ used for this case), in terms of the
field strength. While most previous works assumed a constant field strength,
we effectively assume a random field strength with a cumulative probability
distribution of $P(>\\!S)=1-P(<\\!S)=\sqrt{1-\frac{2}{3}S^{2}}$ where the
normalized post-stretch magnetic field strength,
$S=B^{\prime}/\bar{B}^{\prime}$, assumes values in the range $0\leq
S\leq\sqrt{3/2}$. This difference can lead to some modest quantitative
differences in the emission properties, while the qualitative emission and
polarization properties remain very similar.

It is also worth noting the main difference between our model and that of Gill
& Granot (2020), which also uses the parameter $\xi$ following Sari (1999),
namely the integration region. While we integrate over the 2D contribution to
the observed emission from the immediate downstream region behind the shock
wave, Gill & Granot (2020) integrate over the entire volume behind the shock.
Moreover, they assume some value $\xi_{f}$ of the parameter $\xi$ just behind
the shock, and follow its evolution behind the shock locally following the
Blandford & McKee (1976) solution (assuming flux freezing). Since the radial
stretching of each fluid element is larger than its tangential stretching,
$\xi$ grows with the distance behind the shock. For this reason, the effective
value of $\xi$ when accounting for the emission from the whole shocked region,
$\xi_{\rm eff}$, is somewhat larger than its value just behind the shock,
$\xi_{f}$ (see Fig. 4 of Gill & Granot 2020). Even though these are different
assumptions, the polarization curves of both models are relatively close for
$\xi\rightarrow 0$, as can be seen in Fig. 5 of Gill & Granot (2020). Similar
final results for $\xi>1$ values would correspond to differing $\xi_{f}$
parameters in terms of Gill & Granot (2020) and this current work. A good
example for this difference can be given by a choice of $\xi=1$, which
produces a completely random magnetic field, leading to a zero polarization
degree, in terms of our model. On the other hand, the Gill & Granot (2020)
model manages to produce polarization for $\xi_{f}=1$, due to additional
stretching of the magnetic field in regions well behind the shock, which leads
to $\xi_{\rm eff}>\xi_{f}=1$.

##  Appendix C Transforming between the observer frame and the rest frame

Now that we have the direction to the observer in the observer frame expressed
in terms of the local spherical coordinates (see eq. 27), all that is left to
do is transform it to the rest frame. This is done in the local frame as
photons from each cell are boosted differently according to their angular
inclination from the line of sight (LOS), energy and LF. The transformation is
done via rotation on the LOS $\boldsymbol{\rm{\hat{n}}}_{\rm obs}$ by an angle

$\eta=\cos^{-1}\left(\frac{\tilde{\mu}-\beta}{1-\beta\tilde{\mu}}\right)-\cos^{-1}\left(\tilde{\mu}\right),$
(33)

which is done about the axis

$\boldsymbol{\rm{\hat{\eta}}}=\boldsymbol{\rm{\hat{\beta}}}\times\boldsymbol{\rm{\hat{n}}}_{\rm
obs},$ (34)

and
$\tilde{\mu}=\boldsymbol{\rm{\hat{\beta}}}\cdot\boldsymbol{\rm{\hat{n}}}_{\rm
obs}$. This rotation is then applied on the direction of the photon using a
rotation matrix (Birenbaum & Bromberg, 2021):

$\boldsymbol{\rm{\hat{n}}}^{\prime}_{\rm
obs}=\mathbf{R}(\eta,\boldsymbol{\rm{\hat{\eta}}})\boldsymbol{\rm{\hat{n}}}_{\rm
obs}.$ (35)

This method will also allow us to transform quantities from the rest frame to
the observer frame by using the inverse of the rotation matrix
$\mathbf{R}(\eta,\boldsymbol{\rm{\hat{\eta}}})$.

##  Appendix D The polarization vector in the observer frame

The observer plane, upon which we evaluate polarization, is perpendicular to
the symmetry axis of the jet $\boldsymbol{\rm{\hat{z}}}$ which means it lies
in the $\boldsymbol{\rm{\hat{x}}}-\boldsymbol{\rm{\hat{y}}}$ plane. Therefore,
the correct and general way to represent the local polarization vector in the
observer frame would be in terms of the global
$\boldsymbol{\rm{\hat{x}}}-\boldsymbol{\rm{\hat{y}}}$ plane. We start by
calculating the direction of the polarization vector in each cell. This is
done in the rest frame in terms of the local coordinates by using the
expressions in eqs. 25 and 35:

$\boldsymbol{\rm{\hat{P}}}^{\prime}=\boldsymbol{\rm{\hat{n}}}^{\prime}_{\rm
obs}\times\boldsymbol{\rm{\hat{b}}}^{\prime}\equiv
P^{\prime}_{\text{r}}\boldsymbol{\rm{\hat{r}}}+P^{\prime}_{\theta}\boldsymbol{\rm{\hat{\theta}}}+P^{\prime}_{\phi}\boldsymbol{\rm{\hat{\phi}}}.$
(36)

We would then transform the rest frame polarization vector to the observer
frame using the inverse of the rotation matrix presented in Eq. (35). This
expression is then translated to the global coordinate system, similar to the
way done in eqs. 26 and 27, which gives an observer frame polarization unit
vector of
$\boldsymbol{\rm{\hat{P}}}=P_{\text{x}}\boldsymbol{\rm{\hat{x}}}+P_{\text{y}}\boldsymbol{\rm{\hat{y}}}+P_{\text{z}}\boldsymbol{\rm{\hat{z}}}$.
Since the observed polarization vector lies in the
$\boldsymbol{\rm{\hat{x}}}-\boldsymbol{\rm{\hat{y}}}$ plane, we will then use
these vector components to evaluate the observed local polarization angle
$\phi_{p,0}$ (see Eq. 17).

##  Appendix E Functional dependence of peak polarization on $\xi$

The analytical expression that relates the peak polarization degree to the
magnetic field structure parameter $\xi$ has a hyperbolic tangent functional
form (Eq. 23). In table 2 we present the functional fits plotted in Fig. 8 for
different parameter combinations.

$P_{\max}(\xi)$
---
Parameters |  Functional dependence
$a=2$, $q=5$ |  $0.53\tanh\left(-2.2\log_{10}\xi+0.15\right)-0.086$
$a=1$, $q=3$ |  $0.185\tanh\left(-2.2\log_{10}\xi+0.3\right)-0.055$
$a=1$, $q=1$ |  $0.055\tanh\left(-2.3\log_{10}\xi+0.34\right)-0.02$
$a=0.5$, $q=3$ |  $0.07\tanh\left(-2.2\log_{10}\xi+0.3\right)-0.02$
Table 2: Analytical functional dependence of the Polarization peak height for
the different models considered in Fig. 8.

##  Appendix F Model Parameters for GRB 221009A

To reproduce to light curve fits of the two geometrical models considered in
this work for the structure of GRB 221009A, we use their suggested afterglow
parameters. The Gill & Granot (2023) model is reproduced as is with our model.
To replicate the light curve fit using the O’Connor et al. (2023) jet
structure, we modify their suggested afterglow model in order to accommodate
differences in our afterglow model implementations. We do so by setting
$E_{\text{k,iso}}=10^{55}$ ergs on the line of sight and normalizing the
energy distribution accordingly. The modified afterglow parameters required to
reproduce the fit are $\xi_{\text{e}}=0.0046$, $\epsilon_{\text{e}}=0.0126$
and $\epsilon_{\text{b}}=8.5\cdot 10^{-6}$. We summarize the afterglow
parameters of the two models in table 3.

Afterglow model parameters GRB 221009A
---
Parameter |  Gill & Granot (2023) |  O’Connor et al. (2023)
$\theta_{c}$ |  $1.2^{\circ}$ |  $3^{\circ}$
$\theta_{\rm obs}$ |  $1.14^{\circ}$ |  $0.57^{\circ}$
$\Gamma_{c}$ |  300 |  500
a (PL index of energy structure) |  0.8 |  0.75, 1.15
b (PL index of LF structure) |  0.3 |  0
$E_{\text{c}}$ [erg] |  $2\cdot 10^{55}$ |  $10^{55}$ on the LOS
$n_{\text{ISM}}$ [cm-3] |  0.1 |  1
Density profile ($n\propto r^{-k})$ |  k=2 |  k=0
$p$ |  $2.4$ |  $2.25$
$\epsilon_{\text{e}}$ |  0.01 |  0.0126
$\epsilon_{\text{B}}$ |  $10^{-4}$ |  $8.5\cdot 10^{-6}$
$\chi_{\text{e}}$ |  0.01 |  0.0046
$A_{*}$ |  0.33 [cm-1] |  1 [cm-3]
Table 3: Afterglow parameters for the Gill & Granot (2023) model and modified
parameters for the O’Connor et al. (2023) model.