# Reverse shock forming condition for magnetized relativistic outflows:
reconciling theories and simulations

Jing-Ze Ma1 and Bing Zhang2,3
1School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
2Nevada Center for Astrophysics, University of Nevada Las Vegas, Las Vegas, NV
89154, USA
3Department of Physics and Astronomy, University of Nevada Las Vegas, Las
Vegas, NV 89154, USA E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Reverse shock (RS) emission can be used to probe the properties of the
relativistic ejecta, especially the degree of magnetization $\sigma$, in
gamma-ray burst (GRB) afterglows. However, there has been confusion in the
literature regarding the physical condition for the RS formation, and the role
of magnetic fields in the RS dynamics in the Poynting-flux-dominated regime is
not fully understood. Exploiting the shock jump conditions, we characterize
the properties of a magnetized RS. We compare the RS dynamics and forming
conditions from different theories and numerical simulations, and reconcile
the discrepancies among them. The strict RS forming condition is found to be
$\sigma<\sigma_{\mathrm{cr}}=(8/3)\gamma_{4}^{2}(n_{1}/n_{4})$, where $n_{4}$
and $n_{1}$ are the rest-frame number densities of the ejecta and the ambient
medium, respectively, $\gamma_{4}$ is the bulk Lorentz factor, and
$\sigma_{\mathrm{cr}}$ is the critical magnetization. Contrary to previous
claims, we prove that this condition agrees with other theoretical and
simulated results, which can be further applied to the setup and consistency
check of future numerical simulations. Using this condition, we propose a
characteristic radius for RS formation, and categorize the magnetized shell
into three regimes: ‘thick shell’ (relativistic RS), ‘thin shell’ (trans-
relativistic RS), and ‘no RS’ regimes. The critical magnetization
$\sigma_{\mathrm{cr}}$ is generally below unity for thin shells, but can
potentially reaches $\sim 100-1000$ in the ‘thick shell’ regime. Our results
could be applied to the dynamical evolution of Poynting-flux-dominated ejecta,
with potential applications to self-consistent lightcurve modelling of
magnetized relativistic outflows.

###### keywords:

MHD – shock waves – gamma-ray burst: general – ISM: jets and outflows –
galaxies: jets

††pubyear: 2015††pagerange: Reverse shock forming condition for magnetized
relativistic outflows: reconciling theories and simulations–C

##  1 Introduction

Relativistic shock waves often develop when a relativistic ejecta is
decelerated by the ambient medium. Under certain circumstances, the excited
waves would steepen into a reverse shock (RS) propagating into the ejecta as
well as a forward shock (FS) propagating into the medium. Relativistic shocks
are believed to be efficient particle accelerators. The standard FS-RS model
is widely invoked to interpret the nonthermal emission in various
astrophysical systems, such as pulsar wind nebulae (e.g. Kennel & Coroniti,
1984), gamma-ray bursts (GRBs, e.g. Rees & Meszaros, 1992), and fast radio
bursts (FRBs, e.g. Lyubarsky, 2014).

In particular, the RS radiation signature in GRB afterglows can serve as a
tool to probe the properties of the ejecta (for a review of GRBs and the role
of RS, see Kumar & Zhang, 2015). Both the RS and FS emissions may contribute
to the GRB early afterglows. The composition and the bulk Lorentz factor of
the ejecta can be constrained by analyzing the afterglow light curves, e.g.,
from the presence (or absence) of the RS emission and the relative importance
of RS and FS components (Sari & Piran, 1999a, b; Kobayashi, 2000; Kobayashi &
Zhang, 2003; Zhang et al., 2003; Nakar & Piran, 2004; Harrison & Kobayashi,
2013; Japelj et al., 2014; Gao et al., 2015).

One of the ejecta properties reflected by the RS emission is the degree of
magnetization $\sigma$, the ratio of the Poynting flux to the matter energy
flux measured in the lab frame. When the magnetic field is dynamically
unimportant ($\sigma\ll 1$), the RS is classified into two regimes: the ‘thick
shell’ case where a relativistic RS (RRS) decelerates the ejecta, and the
‘thin shell’ case where the RS is Newtonian (Sari & Piran, 1995; Kobayashi,
2000). The relative magnetization of the RS with respect to the FS can be
deduced from the afterglow light curves, thereby indirectly imposing
constraints on the magnetization of the ejecta (Zhang et al., 2003; Harrison &
Kobayashi, 2013; Japelj et al., 2014; Gao et al., 2015; Huang et al., 2016;
Jordana-Mitjans et al., 2020).

However, the physics of the RS remains not well-understood in the
high-$\sigma$ regime. The general picture is as follows: As $\sigma$ increases
from below, the RS emission becomes stronger due to the progressively stronger
synchrotron radiation until reaching a peak at $\sigma\sim 0.1-1$, and then
weakens as the RS becomes Newtonian and suppressed by a dynamically-important
magnetic field (Fan et al., 2004; Zhang & Kobayashi, 2005b; Chen & Liu, 2021).
This is partly supported by the particle acceleration mechanism. For ultra-
relativistic shocks, numerical simulations suggested that particles can only
be accelerated efficiently via the diffusive shock acceleration when
$\sigma\lesssim 10^{-3}$ (Sironi & Spitkovsky, 2009, 2011; Sironi et al.,
2013), but precursor waves generated by the synchrotron maser instability may
operate as an alternative acceleration mechanism up to $\sigma\sim 0.1-1$
(Gallant et al., 1992; Iwamoto et al., 2017, 2018; Plotnikov & Sironi, 2019;
Sironi et al., 2021; Iwamoto et al., 2021). It is less clear whether these
conclusions hold for mildly relativistic shocks (Sironi & Spitkovsky, 2011;
Ligorini et al., 2021), which could be important for a trans-relativistic RS.
The allowed range of magnetization for a detectable RS emission is thus still
poorly constrained, both from theoretical and observational perspectives.

A primary step to understand this problem is to study the $\sigma$-dependent
forming condition of the RS, which is under debate in the literature. In a
one-dimensional (1D) Cartesian geometry, Zhang & Kobayashi (2005b) (hereafter
ZK05) found analytically that the RS cannot form when the magnetization is
higher than a critical value
$\sigma_{\mathrm{cr}}=(8/3)\gamma_{4}^{2}({n_{1}}/{n_{4}})$, where $n_{4}$ and
$n_{1}$ are the rest-frame number densities of the ejecta and the ambient
medium, respectively, and $\gamma_{4}$ is the bulk Lorentz factor
($\sigma_{\mathrm{cr}}\sim 100$ for typical GRB conditions). However, it was
argued that although this criterion is correct in 1D magnetohydrodynamic (MHD)
Riemann problem (as supported by the numerical simulations of Mizuno et al.,
2009), it may not hold in reality due to the multi-dimensional instability of
shocks (Lyutikov, 2010, 2011) or density fluctuations (Granot, 2012).
Alternatively, Giannios et al. (2008) (hereafter GMA08) claimed that the
ejecta may adjust its structure through fast magnetosonic waves before the RS
crosses the ejecta, thus giving a tighter constraint
($\sigma_{\mathrm{cr}}\sim 1$ for typical GRB conditions). This argument was
later criticized by Granot (2012) because it contradicts with the fact that a
fast MHD shock always travels faster than any linear wave in the upstream.
Detailed numerical simulations by Mimica et al. (2009) (hereafter MGA09) found
near-Newtonian RSs, though with low dissipative efficiencies, in the “no RS”
regime claimed by GMA08. However, MGA09 showed that the near-Newtonian RS
emerges from a rarefaction fan instead of the contact discontinuity (CD) due
to the evolution of the flow in spherical geometry, which cannot be captured
by the 1D Cartesian geometry as adopted in ZK05 and Mizuno et al. (2009). The
confusion persists even to this day (Chen & Liu, 2021; Ai & Zhang, 2021), and
a clarification is needed.

In this work, we extend the results of ZK05, and clarify the misunderstandings
in previous works on the MHD RS forming condition. In Section 2, we present
the theoretical basis that determines the properties of a magnetized RS. In
Section 3, we discuss the different RS forming conditions in the literature.
We argue that the ZK05 condition, being consistent with the numerical
simulations by MGA09, is the strict condition. The GMA09 condition, though
conceptually correct, is only an order-of-magnitude estimate of the actual
condition given by ZK05. Using the ZK05 condition, we define a RS forming
radius, and discuss the dynamical evolution of the ejecta in the external
shock stage in Section 4. Conclusions are given in Section 5.

##  2 Properties of the reverse shock

For a general astrophysical scenario where two flows of plasma collide with
each other, the system can be approximately cast into a Riemann problem where
nonlinear waves develop from the initial discontinuity of physical quantities.
A hydrodynamic Riemann problem admits a three-wave solution, i.e., a contact
discontinuity in the middle and two nonlinear waves (shocks or rarefaction
waves). In this hydrodynamic limit, it is possible to determine the wave
pattern (either two shocks, two rarefaction waves, or one shock and one
rarefaction wave) from the initial condition prior to solving the problem
(Rezzolla & Zanotti, 2013). An MHD Riemann problem, on the other hand,
generally admits a seven-wave solution including one contact discontinuity,
two slow waves, two Alfven waves and two fast waves. Each of the fast or slow
waves may either be a shock or a rarefaction wave. This constitutes a much
more complex problem for which there has been no practical method to predict
the wave patterns before obtaining the complete numerical solution via
iterations (Giacomazzo & Rezzolla, 2006). However, a special case of MHD
Riemann problem where the magnetic field is orthogonal to the flow velocity
(i.e., the perpendicular configuration) could form a degeneracy so that slow
waves and Alfven waves vanish (Komissarov, 1999).

Here, we restrict ourselves to considering a perpendicular RS-FS system with
an ambient medium of a constant number density (e.g., interstellar medium;
ISM). Following the convention of ZK05, we define four regions separated by
three discontinuity surfaces: (1) FS upstream (unshocked), (2) FS downstream
(shocked), (3) RS downstream (shocked), (4) RS upstream (unshocked). Regions 2
and 3 are separated by a contact discontinuity. $Q_{ij}$ represents the
quantity $Q$ in region $i$ measured in the rest frame of $j$.

We define $p_{\mathrm{tot,r}}$ and $p_{\mathrm{tot,f}}$ as the total pressures
(i.e., the sum of magnetic pressure and thermal pressure) in the shocked
regions near the RS and FS, respectively. Following the derivation in ZK05
(for a brief summary, see Appendix A), we find that the shock jump conditions
yield

$\displaystyle
p_{\mathrm{tot,r}}=(\hat{\Gamma}_{3}\gamma_{43}+1)(\gamma_{43}-1)n_{4}m_{\mathrm{p}}c^{2}F_{43}\,,$
(1) $\displaystyle
p_{\mathrm{tot,f}}=(\hat{\Gamma}_{2}\gamma_{21}+1)(\gamma_{21}-1)n_{1}m_{\mathrm{p}}c^{2}F_{21}\,,$
(2)

where $\hat{\Gamma}$ is the adiabatic index given by the equation of state
(EOS), $\gamma$ is the Lorentz factor associated with the dimensionless speed
$\beta$, $n$ is the rest-frame number density, $m_{\mathrm{p}}$ is the proton
mass, $c$ is the speed of light, $F_{21}=F_{21}(\gamma_{21},\sigma_{1})$ and
$F_{43}(\gamma_{43},\sigma_{4})$ are the correction factors due to the
magnetic field as detailed in Appendix A and ZK05. For hydrodynamic shocks, we
have $F_{21}=F_{43}=1$. We assume the pressures near FS and RS balance each
other, i.e., $p_{\mathrm{tot,r}}=p_{\mathrm{tot,f}}$, which is a reasonable
approximation when the shocks initially form111 However, as the shocks
propagate and develop, the two pressures differ, typically within an order of
magnitude from each other (Kobayashi et al., 1999; Beloborodov & Uhm, 2006;
Mimica et al., 2009; Uhm, 2011; Chen & Liu, 2021; Ai & Zhang, 2021; Ai et al.,
2022). . Assuming the pressure balance condition, we find strictly that (see
also equation (32) of ZK05 for a special case)

$\frac{n_{4}F_{43}}{n_{1}F_{21}}=\frac{(\gamma_{21}-1)(\hat{\Gamma}_{2}\gamma_{21}+1)}{(\gamma_{43}-1)(\hat{\Gamma}_{3}\gamma_{43}+1)}\
.$ (3)

Notice that to reach this conclusion, we have already assumed the unshocked
regions (i.e., regions 1 and 4) are cold, such that the gas pressures in these
regions are negligible. We adopt a simple approximation of the adiabatic index

$\hat{\Gamma}_{2}=\frac{4\gamma_{21}+1}{3\gamma_{21}}\,,\
\hat{\Gamma}_{3}=\frac{4\gamma_{43}+1}{3\gamma_{43}}\,,$ (4)

which is suitable for an ideal gas in both non-relativistic and relativistic
regimes (Uhm, 2011). The FS region is nearly unmagnetized
($\sigma_{2},\sigma_{1}\ll 1$), and therefore we get $F_{21}\simeq 1$. The
relation (3) then reduces to (see also equation (4.147) of Zhang, 2018)

$F_{43}\frac{n_{4}}{n_{1}}=\frac{\gamma_{21}^{2}-1}{\gamma_{43}^{2}-1}\ .$ (5)

We treat the shocked region between RS and FS as a blastwave with uniform
Lorentz factor ($\gamma_{2}=\gamma_{3}$), which is the basic assumption of
mechanical models for both hydrodynamical problems (Beloborodov & Uhm, 2006)
and MHD problems (Ai & Zhang, 2021; Ai et al., 2022). This approximation is
confirmed to be reasonable through both hydrodynamical simulations (Kobayashi
et al., 1999) and MHD simulations (Mimica et al., 2009; Ai et al., 2022).
According to the Lorentz transformation
$\gamma_{21}=\gamma_{43}\gamma_{4}(1-\beta_{43}\beta_{4})$ and
$\gamma^{2}-1=\gamma^{2}\beta^{2}$, relation (5) is equivalent to

$\gamma_{41}^{2}\frac{n_{1}}{n_{4}}=\left[\frac{\beta_{43}}{\beta_{21}(1-\beta_{43}\beta_{4})}\right]^{2}F_{43}\,.$
(6)

We assume a relativistic ejecta and an FS that is at least mildly relativistic
with $\beta_{4},\beta_{21}\sim 1$ (typical for the GRB case), and we find

$\gamma_{4}^{2}\frac{n_{1}}{n_{4}}=\left(\frac{\beta_{43}}{1-\beta_{43}}\right)^{2}F_{43}\,.$
(7)

Define a parameter

$\zeta\equiv\gamma_{4}^{2}\frac{n_{1}}{n_{4}}\,,$ (8)

which is only determined by the initial conditions of the unshocked regions.
Noticing that $F_{43}$ is a function of $\sigma$ and $\beta_{43}$ only (see
Appendix A) , we find from equation (7) that $\zeta$ can also be expressed as
$\zeta=\zeta(\sigma,\beta_{43})$. Therefore, when the initial conditions
$(\zeta,\sigma)$ of the unshocked regions are known, the relative Lorentz
factor between the upstream and the downstream of the RS, i.e., $\gamma_{43}$,
is determined straightforwardly from the relation (7). Hereafter, we define

$\gamma_{\mathrm{rel}}^{\mathrm{RS}}\equiv\gamma_{43}\,.$ (9)

Motivated by observations and the discussion in MGA09, we investigate the
dissipative efficiency of the RS. We assume that all the internal energy
behind the shock is carried by the accelerated particles and converted into
radiation, as discussed in Narayan et al. (2011), Komissarov (2012) and Sironi
et al. (2015). This is highly optimized so that the efficiency values obtained
here are the maximum values. The dissipative efficiency of the RS is defined
as the fraction of the total enthalpy that is converted into internal energy
(for slightly different definitions, see Narayan et al., 2011; Komissarov,
2012; Sironi et al., 2015), i.e.,

$\begin{split}f_{\mathrm{sh}}&\equiv\frac{e_{3}}{(n_{3}m_{\mathrm{p}}c^{2}+e_{3}+p_{3})(1+\sigma_{3})}\\\
&=\frac{(\gamma_{43}-1)f_{a,43}}{\left[1+\hat{\Gamma}_{3}(\gamma_{43}-1)f_{a,43}\right](1+\sigma_{3})}\,,\end{split}$
(10)

which is a function of $\gamma_{43}$ and $\sigma$. Here, $f_{a}$ is a
correction factor as detailed in Appendix A and ZK05. As discussed in the last
paragraph, the relative Lorentz factor $\gamma_{43}$ is determined by the
conditions $(\zeta,\sigma)$ of the unshocked regions. Therefore, the
dissipative efficiency can be calculated as long as $(\zeta,\sigma)$ are
known.

##  3 Reverse shock forming condition: Reconciling theories and simulations

Figure 1:  RS properties in the parameter space of
$\zeta=\gamma_{4}^{2}(n_{1}/n_{4})$ and magnetization $\sigma$. We show the
contours of the RS relative Lorentz factor
$\gamma_{\mathrm{rel}}^{\mathrm{RS}}$ (black solid lines labeled on the left
side), and the contours of the RS dissipative efficiency $f_{\mathrm{sh}}$
(colored region with values indicated by the colorbar;
$f_{\mathrm{sh}}=0.5,10^{-1},10^{-2},10^{-3}$ are highlighted in white solid
lines labeled on the top). The quantities
$\gamma_{\mathrm{rel}}^{\mathrm{RS}}$ and $f_{\mathrm{sh}}$ are defined in
equations (9) and (10), respectively. The RS forming conditions given by ZK05
(see Sec. 3.1), Lyutikov (2010, 2011) (see Sec. 3.2), and GMA08 (see Sec. 3.3)
are indicated by the red solid line, the blue solid line, and the orange
dashed line, respectively. Six purple markers in the dotted frame represent
six data points taken from the numerical simulations of MGA09 (see Sec. 3.4).
Specifically, the circles, diamonds, and asterisks represent the parameters
$(\zeta,\sigma)$ when the relative Lorentz factor
$\gamma_{\mathrm{rel}}^{\mathrm{RS}}$ becomes $1.04,1.1,1.25$ according to
MGA09, respectively. The arrow on the last circle indicates that its actual
location on the diagram should be moved upwards to the left due to shell
expansion, making it more consistent with our prediction (as discussed in Sec.
3.4). Below the dark green solid line at the bottom left, the shell spreading
can further complicate the problem (see Sec. 3.3 and 4 for discussions).

In the parameter space of unshocked parameters $(\zeta,\sigma)$ in Fig. 1, we
show the contours of the relative Lorentz factor
$\gamma_{\mathrm{rel}}^{\mathrm{RS}}$ of the RS and the contours of RS
efficiency $f_{\mathrm{sh}}$. They are calculated using equation (7) and
equation (10), respectively. Black solid lines represent the lines of a
constant relative Lorentz factor $\gamma_{\mathrm{rel}}^{\mathrm{RS}}$,
labeled by their values on the left side of the diagram. The
$\gamma_{\mathrm{rel}}^{\mathrm{RS}}=3$ curve is highlighted in bold, as the
approximate boundary between near-Newtonian RS and RRS. The dissipative
efficiencies $f_{\mathrm{sh}}$ are indicated by colors shown in the colorbar,
with $f_{\mathrm{sh}}=0.5,10^{-1},10^{-2},10^{-3}$ highlighted in white solid
lines labeled by their values on the top.

For a weakly-magnetized ejecta ($\sigma\ll 1$), the strength of the RS is
solely determined by the parameter $\zeta=\gamma_{4}^{2}(n_{1}/n_{4})$. The RS
becomes relativistic at $\zeta\gg 1$ and Newtonian at $\zeta\ll 1$, as
proposed in the early work of Sari & Piran (1995). The RRS is efficient at
dissipating energy, whilst the Newtonian RS is not. The highest efficiency is
reached in the case of a non-magnetized RRS, where we have
$f_{\mathrm{sh}}\simeq 3/4$.

The increase in the magnetization degree $\sigma$ effectively weakens the RS,
as the relative Lorentz factor of the RS can be approximately solved using the
relation $\zeta\sim\gamma_{43}^{4}\beta_{43}^{2}(1+\sigma)$. The dissipative
efficiency is further suppressed in the high-$\sigma$ regime because a large
portion of magnetic energy cannot be converted into internal energy. For any
RS with $\sigma\gtrsim 1$, it dissipates energy at a low efficiency
($f_{\mathrm{sh}}\lesssim 0.01-0.1$). As widely recognized, this suggests that
the RS emission may become too weak to be observed for $\sigma\gtrsim 1$
(Zhang & Kobayashi, 2005b; Giannios et al., 2008; Mimica et al., 2009; Granot,
2012; Zhang, 2018).

To reconcile theories and simulations on RS formation, we integrate the
results from different works in Fig. 1, as discussed below separately.

###  3.1 Treatment of ZK05

A shock exists only if the upstream total pressure exceeds the downstream
total pressure (Lichnerowicz, 1967; Majorana & Anile, 1987; Rezzolla &
Zanotti, 2013). At the critical point where pressure balance is established
between the upstream and downstream, the shock vanishes and the two regions
merge into one continuous flow. When the RS transits to a rarefraction wave,
the total pressure at the RS is simply the the magnetic pressure in the
unshocked region 4, i.e., $p_{\mathrm{tot,r}}\rightarrow 2\sigma
n_{4}m_{\mathrm{p}}c^{2}$. The total pressure at the FS is given by the jump
condition (2) with $\gamma_{21}$ approaching $\gamma_{4}$, i.e.,
$p_{\mathrm{tot,f}}\rightarrow(4/3)(\gamma_{4}\beta_{4})^{2}n_{1}m_{\mathrm{p}}c^{2}$.
The RS forming condition for a relativistic ejecta is thus

$\gamma_{4}^{2}\frac{n_{1}}{n_{4}}>\frac{3}{8}\sigma\,.$ (11)

In the 1D planar symmetry, this condition is supported by the numerical
simulations (Mizuno et al., 2009). This criterion given in ZK05 is shown in
Fig. 1 as a red solid line, below which there is no RS.

###  3.2 Treatment of Lyutikov

Lyutikov (2010, 2011) argued that although the criterion (11) holds in 1D
problems, the RS may not form in multi-dimensions if the CD is subsonic with
respect to the ejecta. He found that this condition can be expressed
asymptotically as

$\gamma_{4}^{2}\frac{n_{1}}{n_{4}}>\begin{cases}3\sigma/8\,,\sigma\ll 1\\\
6\sigma^{3}\,,\sigma\gg 1\end{cases}.$ (12)

A rigorous description of this condition is to have
$\gamma_{\mathrm{rel}}^{\mathrm{RS}}>\gamma_{\mathrm{A}}$, where
$\gamma_{\mathrm{A}}=\sqrt{1+\sigma}$ is the Lorentz factor of the Alfven
speed (the speed of the fast magnetosonic wave for cold flows). In Fig. 1, we
plot the critical condition where
$\gamma_{\mathrm{rel}}^{\mathrm{RS}}=\gamma_{\mathrm{A}}$ as the blue solid
curve, which approaches the asymptotic expression (12) at both ends.

There has been no evidence to support this tighter constraint from multi-
dimensional simulations on relativistic MHD (RMHD) shocks (e.g., Komissarov,
1999; Romero et al., 2005; Giacomazzo & Rezzolla, 2006), and the GRB
afterglow-related simulations were mostly restricted to 1D (e.g., Mimica et
al., 2009; Mizuno et al., 2009). Nevertheless, in Fig. 1 we confirm the
statement of Lyutikov (2010, 2011) that the RS, even if it exists when
$\gamma_{\mathrm{rel}}^{\mathrm{RS}}<\gamma_{\mathrm{A}}$, is ‘weak’, in a
sense that it can not dissipate energy efficiently as a near-Newtonian or
strongly-magnetized shock.

###  3.3 Treatment of GMA08

GMA08 considered a magnetized shell with the width $\Delta$ and the bulk
Lorentz factor $\gamma_{4}$ at radius $r$, all measured in the lab frame222
The shell-spreading effect in spherical geometry is not taken into account in
this section. Therefore, we use the notation $\Delta$ and the initial shell
width $\Delta_{0}$ interchangeably in this section, unless otherwise noted. .
The strength of the RS was conventionally parameterized by

$\xi\equiv\sqrt{l/\Delta}\gamma_{4}^{-4/3}\,,$ (13)

where $l\equiv(3E/4\pi n_{1}m_{\mathrm{p}}c^{2})^{1/3}$ is the Sedov length
associated with the total energy of the ejecta $E=4\pi
r^{2}\Delta\gamma_{4}^{2}n_{4}m_{\mathrm{p}}c^{2}(1+\sigma)$, and $n_{4}$ is
the rest-frame particle number density of the shell at $r$.

GMA08 claimed that the RS only forms when the RS crosses the shell before the
casual contact is established through MHD waves, i.e.,
$r_{\mathrm{c}}>r_{\Delta}$. Here, $r_{\mathrm{c}}$ is the ‘contact’ radius
where MHD waves cross the shell on the expansion timescale $r/\gamma_{4}c$,
and $r_{\Delta}$ is the RS crossing radius where the RS crosses the shell. The
radii are given by (see equation (8) of GMA08 and equation (38) of ZK05)

$\displaystyle
r_{\mathrm{c}}\sim\Delta\gamma_{4}^{2}\left(\sqrt{\frac{1+\sigma}{\sigma}}-1\right)\,,$
(14) $\displaystyle
r_{\Delta}\sim\Delta\gamma_{4}^{2}\zeta^{-1/2}(1+\sigma)^{-1/2}\,.$ (15)

Consequently, the RS forming condition in GMA08 can be cast into

$\gamma_{4}^{2}\frac{n_{1}}{n_{4}}\gtrsim(1+\sigma)^{-1}\left(\sqrt{\frac{1+\sigma}{\sigma}}-1\right)^{-2}$
(16)

when the shell does not spread. Note that here the quantity $\zeta$ is
evaluated at the RS crossing radius $r_{\Delta}$. If the shell-spreading is
taken into account, GMA08 argued that the RS always exists as long as the
contact radius is larger than the shell-spreading radius
$r_{\mathrm{s}}\sim\Delta_{0}\gamma_{4}^{2}$, which is equivalent to
$\sigma\lesssim 1/3$. This criterion is plotted in Fig. 1 as an orange dashed
line, which is given by equation (16) in the $\sigma\gtrsim 1/3$ regime and
goes straight down in the $\sigma\lesssim 1/3$ regime due to the shell-
spreading effect. GMA08 claimed that there is no RS on the right hand side of
this line in the parameter space.

Here, we argue that the ZK05 condition (11) and the GMA08 condition (16) are
of the same physical origin as noticed in Mizuno et al. (2009), with the
former being the strict one and the latter being an order-of-magnitude
estimate. This conclusion is reached based on two observations:

  * $\bullet$

As the RS grows weak and vanishes, the upstream total pressure balances the
downstream total pressure, and at the same time the shock speed decreases and
becomes equal to the magnetosonic wave speed (see, e.g., Section 4.4.3 in
Rezzolla & Zanotti (2013) for relativistic hydrodynamic shocks, Figure 20.5 in
Goedbloed et al. (2010) for Newtonian MHD shocks, Figure 2 in Mizuno et al.
(2009) for RMHD shocks). The former is the theoretical basis for ZK05
condition, and the latter is the starting point for GMA08. Therefore, these
two physical concepts can both serve as the RS forming condition333 Granot
(2012) criticized the GMA08 condition for violating the fact that fast shocks
travel faster than linear waves. Here, we point out that although his
observation is correct, it is important to acknowledge that the fast shock
reduces to a linear wave when the shock is at the critical point to vanish. .

  * $\bullet$

Both criteria show the same scaling with $\sigma$ in Fig. 1. The GMA08
condition (16) gives $\zeta\gtrsim 4\sigma$ when $\sigma\gg 1$, which is
similar to the ZK05 condition (11) but roughly an order of magnitude higher444
This is supported by the same scaling law of critical $\sigma$ with the bulk
Lorentz factor, ejecta width, and total energy in ZK05 and GMA08. Comparing
the scaling law (43) in ZK05 and (11) in GMA08, we find that they express the
same scaling relation, but the numerical factors are different. . We attribute
this discrepancy to the approximation used in GMA08 to estimate the
characteristic radii, which omits the detailed numerical factors. As a proof,
we provide a rigorous derivation of the contact radius $r_{\mathrm{c}}$ and
the RS crossing radius $r_{\Delta}$ in Appendix B, and explain that they can
only become equal to each other when the pressure balance is established at
the ZK05 condition (11).

As a comparison to equation (16), GMA08 wrote the condition in terms of $\xi$
as $\xi^{3}\lesssim(1+\sigma)(\sqrt{(1+\sigma)/\sigma}-1)^{2}$. The
dimensionless parameters can thus be converted into each other through a
simple relation at the RS crossing radius $r_{\Delta}$ as

$\zeta\sim\xi^{-3}\,.$ (17)

It should be pointed out here that all the derivations and simulations of
GMA08 and MGA09 (see Sec. 3.4) are based on the assumption that the $\xi$
parameter (converted to $\zeta$ in our context) is evaluated at the RS
crossing radius $r_{\Delta}$. Only under this assumption can the relation
$\zeta\sim\xi^{-3}$ hold. As discussed in more detail in Sec. 4.5, we argue
that the parameters evaluated at $r_{\Delta}$ should not be used to track the
evolution of the shell or to determine the RS forming condition555 For
reference, ZK05 gave the critical $\sigma_{\mathrm{cr}}$ based on the
parameters evaluated at the deceleration radius, which we find is also not a
satisfying answer. . To track the evolution of the shell, we should use the
parameters $\xi$ (or $\zeta$) evaluated at the radius comoving with the shell.
To estimate the critical $\sigma_{\mathrm{cr}}$ at which the RS is formed, we
should use the parameter $\xi$ (or $\zeta$) evaluated at the radius where the
critical RS forming condition is satisfied.

Here, we assume that the shell does not spread. If the shell spreading is
taken into account, we need to distinguish between the initial parameters
$(\zeta_{0},\sigma_{0})$ before shell spreading and the parameters
$(\zeta,\sigma)$ after shell spreading666 It is important to highlight at this
point that the diagram in Fig. 1 only holds when the parameters
$(\zeta,\sigma)$ are taken in the unshocked regions right behind the RS (or
the reverse rarefaction wave when the RS does not form). . GMA08 suggested
that when $\sigma_{0}\lesssim 1/3$, the RS always forms due to the shell-
spreading effect. Although we find that the RS will always form eventually for
$\sigma_{0}\lesssim 1$ if shell spreading is taken into account (see Sec. 4),
we do not find the breaking of causal contact at $\sigma_{0}\lesssim 1/3$ (as
suggested by GMA08) a convincing argument for the formation of RS. This is
discussed more thoroughly in Sec. 4.4.

Motivated by GMA08, we further explore the regime where the shell could spread
before the RS crosses the shell, i.e., $r_{\mathrm{s}}<r_{\Delta}$. We adopt a
more rigorous expression (38) of $r_{\Delta}$, and compute the ratio of the RS
crossing radius $r_{\Delta}$ to the shell-spreading radius $r_{\mathrm{s}}$.
In Fig. 1, we plot the critical condition where $r_{\mathrm{s}}=r_{\Delta}$ as
the dark green solid line. We find that below this line, the shell spreads
before the RS crosses the shell ($r_{\mathrm{s}}<r_{\Delta}$; ‘thin shell’
regime discussed in Sec. 4.2). All the weakly-magnetized thin shells that have
undergone shell spreading are expected to enter this regime and form RSs which
cross the shell as trans-relativistic shocks with
$\gamma_{\mathrm{rel}}^{\mathrm{RS}}\simeq 1.04$ (see a detailed discussion in
Sec. 4). Nevertheless, for a wide range of parameter space above the dark
green solid line and the red solid line, the RS crosses the shell before shell
spreading ($r_{\mathrm{s}}>r_{\Delta}$; ‘thick shell’ regime discussed in Sec.
4.1). We refer to Sec. 4 for a more detailed discussion on shell spreading.

###  3.4 Treatment of MGA09

MGA09 performed high-resolution simulations in 1D spherical geometry, focusing
on the evolution of RS and FS in RMHD. They confirmed the finding of GMA08
that the evolution of the system is characterized by two parameters, namely
$\xi$ and $\sigma$. Using the relation (17) to convert $\xi$ into $\zeta$, we
plot six simulated data points taken from Figure 4 of MGA09, shown as six
purple markers highlighted by a dotted frame in Fig. 1. The models in MGA09
are taken for $\sigma=1,3$ when the relative Lorentz factor
$\gamma_{\mathrm{rel}}^{\mathrm{RS}}$ becomes $1.04$ (circles), $1.1$
(diamonds), and $1.25$ (asterisks), respectively.

Our predictions of the relative Lorentz factors fit perfectly with the
simulated results of MGA09 for $\gamma_{\mathrm{rel}}^{\mathrm{RS}}=1.1,1.25$.
Therefore, we conclude that the theoretical formulation of ZK05 is consistent
with the simulations of MGA09 if the RS is not too weak
($\gamma_{\mathrm{rel}}^{\mathrm{RS}}\gtrsim 1.1$). We confirm the results of
MGA09 that the magnetized weak RS (circles and diamonds) dissipates less than
$\sim 1\%$ of the total energy, indicating a suppressed emission.

However, our theory underestimates the relative Lorentz factor for the
$\zeta\sim 1.1,\sigma=3,\gamma_{\mathrm{rel}}^{\mathrm{RS}}=1.04$ model, which
we predict to be at the verge of vanishing. The discrepancy for near-Newtonian
RS ($\gamma_{\mathrm{rel}}^{\mathrm{RS}}\lesssim 1.1$) may be attributed to
the internal evolution of the ejecta profile in spherical geometry, which is
not captured by our theory. For instance, Figure 1 in MGA09 demonstrated that
the rarefaction wave propagating backwards results in the radial expansion of
the shell. This subsequently leads to decreases in both the number density
$n_{4}$ and the magnetization degree $\sigma$, creating an otherwise
impossible near-Newtonian RS in the rarefaction fan. Therefore, it becomes
important in these cases to define the initial parameters $\zeta_{0}$ and
$\sigma_{0}$ to be distinguished from $\zeta$ and $\sigma$ right in front of
the RS. We expect that the actual location of the near-Newtonian RS (circles)
on Fig. 1 should be moved upwards to the left (indicated by the purple arrow),
where $\sigma<\sigma_{0}$ and $\zeta>\zeta_{0}$ due to the radial expansion of
the shell. This would explain the underpredicted relative Lorentz factor of
the near-Newtonian RS. For stronger RS
($\gamma_{\mathrm{rel}}^{\mathrm{RS}}\gtrsim 1.1$), the RS has either
propagated to the end of the rarefaction fan or formed at the CD, which limits
the effect of shell expansion induced by rarefaction waves.

##  4 Reverse shock forming radius: an order-of-magnitude view of shell
evolution

Figure 2:  Order-of-magnitude classification for the dynamical evolution of
the shell in the parameter space of $\xi=\sqrt{l/\Delta}\gamma_{4}^{-4/3}$ and
magnetization $\sigma$. Yellow region: Generalized thick shell regime. Green
region: Generalized thin shell regime. Red region: Direct deceleration regime
with no RS. White region: Uncertain regime where the shell spreads and
decelerates with no RS. The approximate critical condition
$\xi\sim\sigma^{-1/3}$ for the RS formation is highlighted in red, which
separates the RS forming regime (left to the red line;
$r_{\mathrm{F}}<r_{\Delta}$) and the initially no RS regime (right to the red
line; $r_{\Delta}<r_{\mathrm{F}}$), consistent with Figure 1 in GMA08. In the
red region, the shell is directly decelerated by the FS with no RS or shell
spreading. In the yellow region, the RS is formed, becomes an RRS, and fully
decelerates the shell when it reaches the rear part of the shell at the double
coincidence of $r_{\Delta}\sim r_{\mathrm{dec}}$, which is the conventional
‘thick shell’ scenario in hydrodynamics (Sari & Piran, 1995). In the green
region, shell spreading causes decreases in both $\xi$ and $\sigma$, leading
to the formation of the RS which strengthens into a trans-relativistic shock
at the triple coincidence $r_{\mathrm{N}}\sim r_{\Delta}\sim r_{\mathrm{dec}}$
(the ‘thin shell’ scenario in hydrodynamics; Sari & Piran, 1995). The
evolution of shell parameters is indicated by the dashed arrows, attributed to
the shell spreading in the ‘thin shell’ regime and the deceleration by RRS in
the ‘thick shell’ regime, respectively. In the green region on the right of
the red dashed line, the RS will not form if the subsequent shell spreading is
omitted. However, in the high-$\sigma$ thin shell regime (white region), how
the shell responds to the shell-spreading effect may determine whether it
forms a RS or directly enters the deceleration phase, which is not well
understood. We expect that the shell may spread and then be decelerated by the
FS without forming a RS in the white region.

Using the pressure balance condition for RS formation, we can define a new
characteristic radius, i.e., the RS forming radius $r_{\mathrm{F}}$ where the
RS is formed. This can be done by substituting the number density $n_{4}$ with
the expression for total energy $E=4\pi
r^{2}\Delta\gamma_{4}^{2}n_{4}m_{\mathrm{p}}c^{2}(1+\sigma)$ in ZK05 condition
(11), which gives (see also equation (7) in Mimica et al., 2009, although in a
different context)

$r_{\mathrm{F}}\equiv\left(\frac{3E\sigma}{32\pi
n_{1}m_{\mathrm{p}}c^{2}\gamma_{4}^{4}\Delta(1+\sigma)}\right)^{1/2}\,.$ (18)

Following ZK05, we can now estimate five characteristic radii relevant for the
external shocks: (1) the shell-spreading radius777 Although the shell
spreading in high-$\sigma$ regime has not been carefully explored, there is
evidence that the shell-spreading radius can still be estimated as
$r_{\mathrm{s}}\sim\gamma_{0}^{2}\Delta_{0}$ in highly-magnetized ejecta. The
early analytical treatment of the hydrodynamics of ejecta suggested that the
shell spreading arises from the spherical geometry during the coasting phase
of the cold ejecta (Meszaros et al., 1993; Piran et al., 1993). If the
coasting stage still exists for highly-magnetized ejecta, we expect that the
expression for the shell-spreading radius would be the same. However, if there
is no coasting phase in high-$\sigma$ regime (see, e.g., Granot et al., 2011;
Granot, 2012; Gao & Zhang, 2015), it is unclear if the shell can spread at
all.  $r_{\mathrm{s}}\sim\gamma_{0}^{2}\Delta_{0}$, (2) the RS forming radius
$r_{\mathrm{F}}\sim
l^{3/2}\Delta^{-1/2}\gamma_{4}^{-2}\sigma^{1/2}(1+\sigma)^{-1/2}$, (3)
$r_{\mathrm{N}}\sim l^{3/2}\Delta^{-1/2}\gamma_{4}^{-2}$ where the RS
transforms from a Newtonian shock to a relativistic shock, (4) the RS crossing
radius $r_{\Delta}\sim\Delta^{1/4}l^{3/4}(1+\sigma)^{-1/2}$ where the RS
reaches the rear part of the shell, and (5) the deceleration radius
$r_{\mathrm{dec}}\sim l\gamma_{4}^{-2/3}(1+\sigma)^{-1/3}$ where the FS
collects $1/\gamma_{4}$ of the shell rest mass888 GMA08 defined the
deceleration radius as $r_{\mathrm{dec}}=l/\gamma_{4}^{2/3}$. However, ZK05
argued that it should be defined using the kinetic energy alone, as the
differences in magnetic energy is negligible. For a discussion of this point,
see Lyutikov (2005) and Zhang & Kobayashi (2005a). At ZK05 deceleration
radius, a magnetized shell may decelerate with a shallow deceleration index
$\gamma_{4}\propto r^{-1/2}$ (for ISM; Zhang et al., 2006; Granot et al.,
2011; Granot, 2012), and later on transforms into a Blandford-Mckee self-
similar deceleration phase at the deceleration radius defined by the total
energy (Blandford & McKee, 1976; Zhang, 2018). Here, we follow the definition
of ZK05. We find these two different definitions lead to similar
classifications of the shell evolution shown in Fig. 2. . They are related to
each other by (see also equation (51) in ZK05 for the first four radii)

$\xi_{0}^{2}r_{\mathrm{s}}=(1+\sigma)^{1/3}r_{\mathrm{dec}}=\xi^{1/2}(1+\sigma)^{1/2}r_{\Delta}=\frac{r_{\mathrm{N}}}{\xi}=\left(\frac{1+\sigma}{\sigma}\right)^{1/2}\frac{r_{\mathrm{F}}}{\xi}\,,$
(19)

where $\xi_{0}$ is the parameter $\xi$ defined by the initial width
$\Delta_{0}$ and the initial Lorentz factor $\gamma_{0}$. Notice that only the
shell-spreading radius $r_{\mathrm{s}}$ is defined by the initial conditions,
whereas the other four radii can in principle change with the shell parameters
as the shell evolves. In the parameter space of $(\xi,\sigma)$, there are
multiple regions representing different physical processes. Two features can
be used to simplify the problem: (1) $r_{\Delta}\lesssim r_{\mathrm{F}}$
($\xi\gtrsim\sigma^{-1/3}$; right hand side of the red line in Fig. 2) gives
the unphysical scenario where the RS crosses the shell even before the RS can
be formed, which in fact represents the regime where RS cannot be formed
initially. In these cases, the characteristic radii
$r_{\mathrm{F}},r_{\mathrm{N}},r_{\Delta}$ associated with the RS lose their
physical meanings, and thus should be left out of our discussion. (2)
$r_{\mathrm{F}}$ is always smaller than $r_{\mathrm{N}}$ but approaches
$r_{\mathrm{N}}$ in the high-$\sigma$ regime.

For the following discussion, we only focus on the external shocks. We start
with a shell propagating through the ISM at radius $r_{0}$ from the central
engine, with all the five characteristic radii well beyond $r_{0}$. We assume
that the shell is in a quasi-coasting stage, where the Lorentz factor
$\gamma_{0}$, shell width $\Delta_{0}$, and magnetization $\sigma_{0}$ remain
approximately constant as the shell moves999 This assumption is valid for
weakly-magnetized shells. For highly-magnetized shells, however, the ejecta
may never reach a steady coasting phase. The shell could directly enter the
deceleration phase after the rapid acceleration if the coasting radius is
larger than the deceleration radius (see, e.g., Granot et al., 2011; Granot,
2012; Gao & Zhang, 2015). Here, we limit our discussion to the cases where the
coasting radius is smaller than all the five characteristic radii. . Under
this assumption, the parameter space in Fig. 2 is generally separated into
three regimes: the ‘thick shell’ regime (Sec. 4.1), the ‘thin shell’ regime
(Sec. 4.2), and the ‘no RS’ regime (Sec. 4.3), as discussed below.

###  4.1 ‘Thick shell’ regime (yellow region)

The RS is formed and becomes relativistic, which then fully decelerates the
shell when it crosses the entire ejecta.

This regime is characterized by
$r_{\mathrm{F}}<r_{\mathrm{N}}<r_{\mathrm{dec}}<r_{\Delta}<r_{\mathrm{s}}$.
The shell coasts freely as the number density drops with radius as
$n_{4}\propto r^{-2}$ until the RS forming condition (11) is met. A Newtonian
RS is first formed at $r\sim r_{\mathrm{F}}$, which does not influence the
dynamics of the shell appreciably ($\xi\simeq\xi_{0},\sigma\simeq\sigma_{0}$).
As the shell propagates further, the number density continues to drop until
the RS becomes relativistic at $r\sim r_{\mathrm{N}}$. At $r\gtrsim
r_{\mathrm{N}}$, the shell begins to be decelerated by the RRS, leading to a
decrease in the Lorentz factor estimated to be $\gamma_{4}\simeq
l^{3/4}\Delta^{-1/4}r^{-1/2}$ (Sari & Piran, 1997; Kobayashi et al., 1999),
although it is less clear whether this scaling holds in the high-$\sigma$
regime. With this expression, we find that $r_{\mathrm{N}}$ in relation (19)
can be replaced by the distance $r$, which gives

$\frac{r}{\xi^{3/2}}=\frac{(1+\sigma)^{1/3}}{\xi^{1/2}}r_{\mathrm{dec}}=(1+\sigma)^{1/2}r_{\Delta}=l^{3/4}\Delta_{0}^{1/4},\
r_{\mathrm{N}}\lesssim r\lesssim r_{\Delta}\,.$ (20)

Therefore, when the shell passes $r_{\mathrm{N}}$, $\xi$ begins to increase as
$\xi\propto r^{2/3}$, accompanied by the increase in $\sigma$ ($\sigma\propto
r,\sigma\ll 1$ or $\sigma\propto r^{1/2},\sigma\gg 1$; see Table 1)101010
Here, we assume the ejecta is decelerated by the RRS, leading to
$\gamma_{4}\propto r^{-1/2}$ even in the high-$\sigma$ regime. Under this
assumption, both the magnetic energy density and the rest mass density
decrease with $r$. However, the rest mass density has a steeper dependence on
$r$, thus decreasing faster than the magnetic energy density. This results in
the increase in $\sigma$. . This evolution is indicated by the arrows pointing
upwards to the right in Fig. 2. Following the arrow, $\xi$ continues to
increase as the shell propagates to larger distances. However, as long as the
parameters stay in the yellow region, we always have
$r<r_{\mathrm{dec}}<r_{\Delta}$. This means that the shell has to continue
travelling until reaching the boundary $\xi\sim(1+\sigma)^{-1/3}$ of the
yellow region, where $r\sim r_{\mathrm{dec}}\sim r_{\Delta}$.

In summary, in this ‘thick shell’ regime, a shell that begins with
$\xi_{0}<(1+\sigma_{0})^{-1/3}$ forms an RS which strengthens into an RRS.
Subsequently, the parameters $(\xi,\sigma)$ start to travel within the regime
as the RRS decelerates the shell, until the shell adjusts itself to
$\xi\sim(1+\sigma)^{-1/3}$ so that the RRS crosses and fully decelerates the
shell at the same time (at the double coincidence of $r_{\mathrm{dec}}\sim
r_{\Delta}$).

###  4.2 ‘Thin shell’ regime (green region)

A Newtonian RS is formed either from the beginning or during shell spreading.
The shell is decelerated by both the RS and FS as the RS becomes trans-
relativistic and crosses the shell at approximately the same time. This regime
can be further divided into two sub-regimes according to whether the shell
spreads before the RS formation, as discussed below.

Above the black line $\xi\sim\sigma^{-1/6}$, $r_{\mathrm{s}}$ is the smallest
radius. The coasting phase persists until the shell begins to spread at $r\sim
r_{\mathrm{s}}$ with $\Delta\sim r/\gamma_{0}^{2}$. As long as the shell
spreading is initiated, $r_{\mathrm{s}}$ in relation (19) can be replaced by
the distance $r$, i.e.,

$\xi^{2}r\\!=\\!(1+\sigma)^{1/3}r_{\mathrm{dec}}\\!=\\!\xi^{1/2}(1+\sigma)^{1/2}\\!r_{\Delta}\\!=\\!\frac{r_{\mathrm{N}}}{\xi}\\!=\\!\left(\frac{1\\!+\\!\sigma}{\sigma}\right)^{1/2}\\!\frac{r_{\mathrm{F}}}{\xi}\\!=\\!l\gamma_{0}^{-2/3}\,.$
(21)

At $r\gtrsim r_{\mathrm{s}}$, both $\xi$ and $\sigma$ decrease as the shell
propagates outwards (indicated by the arrow pointing downwards to the left in
Fig. 2; see Table 1). A Newtonian RS is formed at $\xi\sim\sigma^{-1/6}$,
where $r\sim r_{\mathrm{F}}$. Nevertheless, we always find
$r<r_{\Delta}<r_{\mathrm{dec}}<r_{\mathrm{N}}$ within the green region, which
means that the shell continues to expand following the trajectory indicated by
the arrow until reaching the triple coincidence $r_{\mathrm{N}}\sim
r_{\Delta}\sim r_{\mathrm{dec}}$ at $\xi\sim 1$.

In the triangle region bound by $\xi\sim 1$ and $\xi\sim\sigma^{-1/6}$,
$r_{\mathrm{F}}$ is the smallest radius. If the shell starts the coasting
phase here, it first forms a Newtonian RS at $r\sim r_{\mathrm{F}}$, and then
begins to spread after passing $r_{\mathrm{s}}$. Following the shell-spreading
trajectory, the RS inevitably strengthens into a trans-relativistic shock
while crossing and decelerating the shell at $\xi\sim 1$.

A common feature of this ‘thin shell’ regime is that the shell-spreading
effect ultimately leads to a trans-relativistic RS crossing and decelerating
the entire shell at approximately the same time (the triple coincidence
$r_{\mathrm{N}}\sim r_{\Delta}\sim r_{\mathrm{dec}}$ at $\xi\sim 1$). The
differences between the sub-regimes are also evident: If the shell begins
coasting in the triangle region ($1\lesssim\xi_{0}\lesssim\sigma_{0}^{-1/6}$),
an RS is first formed before the shell spreads. If the initial shell width is
thinner ($\xi_{0}\gtrsim\sigma_{0}^{-1/6}$), the shell spreading leads to RS
formation. Above the red dashed line ($\xi_{0}\gtrsim\sigma_{0}^{-1/3}$), the
RS is not even formed if shell spreading is omitted. One plausible example is
the numerical simulation by MGA09 with $\xi_{0}=0.5,1.1$ and $\sigma_{0}=1$.
Although their simulated results lie at the intersection of the regimes and
thus are difficult to be categorized via order-of-magnitude arguments, the
results support the claim that shell spreading (induced by rarefaction waves
in MGA09) could lead to the formation of an otherwise impossible RS.

###  4.3 ‘No RS’ regime (red and white regions)

The shell is directly decelerated by the FS without forming an RS, either from
the beginning (red region) or after shell spreading (white region). The radii
associated with the RS are not relevant, as the RS is not formed
($r_{\Delta}<r_{\mathrm{F}}$). This regime can be further divided into two
sub-regimes according to whether the shell spreads before deceleration, as
discussed below.

In the red region ($\sigma^{-1/3}\lesssim\xi\lesssim\sigma^{1/6}$), the
deceleration radius becomes the smallest radius. Consequently, as the shell
propagates, it continues coasting until reaching the deceleration radius
$r_{\mathrm{dec}}=r_{\mathrm{dec,0}}\sim
l\gamma_{0}^{-2/3}(1+\sigma_{0})^{-1/3}$, and begins to be decelerated by the
FS following the Blandford-Mckee self-similar solution (Blandford & McKee,
1976).

In the white region ($\sigma^{1/6}\lesssim\xi\lesssim\sigma$), the shell-
spreading radius is the smallest radius. Consequently, the coasting phase
subsequently leads to shell spreading at $r\gtrsim r_{\mathrm{s}}$. The
relative magnitudes of the five radii are related to each other through $\xi$
and $\sigma$ according to relation (21). As the shell propagates to larger
distances, both $\xi$ and $\sigma$ drop as $\xi\propto r^{-1/2},\sigma\propto
r^{-1/2}$ (indicated by the arrow in the white region following
$\xi\propto\sigma$; see Table 1). Therefore, in the white region, the shell
spreading does not lead to the formation of an RS as in the ‘thin shell’
regime, but instead leads to deceleration. The shell keeps expanding following
the trajectory indicated by the arrow until $r\sim r_{\mathrm{dec}}$ at
$\xi\sim\sigma^{1/6}$, where the shell is decelerated by the FS.

However, the boundary between the white region and the ‘thin shell’ regime is
solely determined by the estimate that a highly-magnetized shell spreads
following $\xi\propto\sigma$. This has not been discussed in the literature
yet, and detailed numerical simulations are needed to determine the boundary
between the ‘no RS’ regime and the ‘thin shell’ regime. For this reason, we
label the white region as the ‘no RS?’ regime to acknowledge the ambiguity at
the boundary.

###  4.4 Comparison with previous works

Here, we point out that Fig. 2 is essentially the combination of Figure 4 in
ZK05 and Figure 1 in GMA08, modified due to our current understanding of RS
formation. Figure 4 in ZK05 is the upside-down version of this diagram. As
done in our Fig. 2, both ZK05 and GMA08 correctly categorized the parameter
space into three regimes, and interpreted the ‘thick shell’ (region I in ZK05
and yellow region in Fig. 2) and ‘thin shell’ (region III in ZK05 and green
region in Fig. 2) regimes similar to our discussion. However, there are two
major improvements in our Fig. 2 compared to ZK05 and GMA08:

  * $\bullet$

The existence of a ‘no RS’ regime is clarified. ZK05 suggested that RS may
still form and even lead to broad emission peak in their region II (the red
region in Fig. 2) based on the observation that $r_{\Delta}$ is the smallest
radius. We find that this is not the case. This discrepancy arises because
ZK05 did not include the RS forming radius in their calculations. Although the
RS crossing radius $r_{\Delta}$ is the smallest radius in the red region, the
RS does not even form since $r_{\Delta}<r_{\mathrm{F}}$. This further suggests
the importance of including the RS forming radius $r_{\mathrm{F}}$ in the
discussion of magnetized external shocks. This point is corrected in Figure 1
of GMA08, where they showed the RS forming regions similar to our diagram.

  * $\bullet$

The boundary between the ‘no RS’ regime and the ‘thin shell’ regime is
updated, though some uncertainties remain. In ZK05, this boundary between
their region II and III was found to be $\xi\sim\sigma^{1/3}$, where
$r_{\Delta}\sim r_{\mathrm{s}}$. However, we have shown that the RS does not
form in these regions, which makes $r_{\Delta}$ irrelevant to the evolution of
the shell. In these regions, we should only discuss whether the shell spreads
first or decelerates first. GMA08 proposed that the shell spreading only
operates at low $\sigma$, which leads to an extended ‘no RS’ regime as well as
a reduced ‘thin shell’ regime compared to our Fig. 2. They seem to indicate
that when $\sigma\gtrsim 1/3$, the shell establishes causal contact through
fast magnetosonic waves and limits shell spreading. We do not find this
argument convincing, as there is no evidence that the variations in flow
velocity can be suppressed by causal contact, although the causal contact may
help maintain the coherence of magnetic field (e.g., Lyutikov & Blandford,
2003). In our Fig. 2, this boundary is put at $\xi\sim\sigma$, which is our
estimate of whether the shell spreading leads to direct deceleration or RS
formation. This is closely related to the shell-spreading process in the
high-$\sigma$ regime, which has not been probed in previous works. The
location of this boundary still merits further investigations, in particular
through numerical simulations111111 The numerical simulations conducted by
MGA09 did not explore this shell-spreading regime of the parameter space.
Furthermore, we argue that the simulated results of MGA09 shown in Fig. 1
cannot be plotted in Fig. 2 for direct comparison (as was done in MGA09). This
is because the parameters chosen in MGA09 for their Figure 4 are at the
intersection of different regimes ($\xi_{0},\sigma_{0}\sim 1$), which may not
be correctly categorized by an order-of-magnitude argument. They also adopted
an artificially defined $\xi_{\mathrm{eq}}$ from a long-lasting ejecta to
mimic the $\xi$ parameter evaluated at RS crossing for a finite-width shell,
which may not be applicable for a direct comparison with Fig. 2. .

###  4.5 Estimating the critical degree of magnetization

GRB parameters have wide ranges, e.g, initial Lorentz factor $\gamma_{0}\sim
10^{2}-10^{3}$, initial shell width $\Delta_{0}\sim 10^{7}-10^{13}$ cm, total
energy $E\sim 10^{50}-10^{54}$ erg, and medium rest-frame number density
$n_{1}\sim 10^{-1}-10^{3}\ \mathrm{cm^{-3}}$ (Japelj et al., 2014; Gao et al.,
2015; Kumar & Zhang, 2015; Zhang, 2018). Medium estimate gives

$\xi_{0}\simeq
1.3\times\gamma_{0,2.5}^{-4/3}\Delta_{0,11}^{-1/2}E_{52}^{1/6}n_{1,0}^{-1/6}\,,$
(22)

where $Q_{n}\equiv Q/10^{n}$ is adopted in cgs units hereafter. Note that
$\xi_{0}$ is more sensitive to our choices of $\gamma_{0}$ and $\Delta_{0}$,
and both ‘thin shell’ ($\xi_{0}\gtrsim 1$) and ‘thick shell’ ($\xi_{0}\lesssim
1$) are plausible scenarios.

As shown in Fig. 2, for the ejecta to eventually form an RS, the initial
degree of magnetization should satisfy

$\sigma_{0}\lesssim\sigma_{\mathrm{0,cr}}\simeq\max(\xi_{0}^{-3},\xi_{0})\,,$
(23)

which is sensitive to $\gamma_{0}$ and $\Delta_{0}$. Here,
$\sigma_{\mathrm{0,cr}}$ is the critical value of $\sigma_{\mathrm{0}}$ below
which an RS can eventually form. The most stringent constraint is achieved for
$\xi_{0}\sim 1$. Therefore, we find that all the ejecta with
$\sigma_{0}\lesssim 1$ should be able to eventually form an RS. This is
approximately the estimate given by GMA08, and is consistent with the argument
that RS will always exist in the hydrodynamic cases. Nevertheless, in the
extreme cases where the shell is ultra-thin ($\xi_{0}\sim 100$) or ultra-thick
($\xi_{0}\sim 0.1$), even an initially highly-magnetized ejecta with
$\sigma_{0}\sim 10^{3}$ can eventually form an RS.

However, this is not to be confused with the magnetization of the shell when
the RS forms, which we denote as $\sigma_{\mathrm{F}}$. The RS forming
condition gives
$\sigma_{\mathrm{F}}<\sigma_{\mathrm{F,cr}}=(8/3)\zeta_{\mathrm{F}}$, where
$\zeta_{\mathrm{F}}$ is the value of $\zeta$ parameter when the RS is formed.
Normally, it is difficult to estimate $\zeta_{\mathrm{F}}$, mainly because the
number density $n_{4}$ at $r_{\mathrm{F}}$ depends on the dynamical history of
the ejecta. Here, we resort to Fig. 2 to estimate $\sigma_{\mathrm{F}}$. Based
on scaling laws (see Table 1), we find

$\sigma_{\mathrm{F}}\simeq\left\\{\begin{split}&\sigma_{0},\
\xi_{0}\lesssim\min(\sigma_{0}^{-1/6},\sigma_{0}^{-1/3})\\\
&\sigma_{0}^{3/4}\xi_{0}^{-3/2},\ \xi_{0}^{-6}\lesssim\sigma_{0}\lesssim 1\\\
&\sigma_{0}^{3/2}\xi_{0}^{-3/2},\
1\lesssim\sigma_{0}\lesssim\xi_{0}\,\end{split}\right.\ .$ (24)

In the ‘thick shell’ regime, $\sigma_{\mathrm{F}}$ has essentially the same
value as $\sigma_{0}$, and thus is less constrained. In principle, it could
reach $\sim 10^{3}$ for ultra-thick shells, which is even greater than the
estimate given by ZK05. However, in the ‘thin shell’ regime, only if the shell
enters the triangle region ($1\lesssim\xi\lesssim\sigma^{-1/6}$) can an RS be
formed, which means that $\sigma_{\mathrm{F}}\lesssim 1$. Therefore, for the
‘thin shell’ scenario, the degree of magnetization $\sigma$ typically drops to
well below unity when the RS is formed (Granot et al., 2011; Granot, 2012; Gao
& Zhang, 2015), which then decreases further when the RS crosses and
decelerates the shell as a trans-relativistic RS.

##  5 Conclusions

In this paper, we characterize the RS properties of magnetized relativistic
outflows, and clear the confusion about the RS forming condition. To this end,
we extend the theoretical formulation of ZK05, and compare our theoretical
predictions of RS properties with the simulated results of MGA09. We revisit
the RS forming conditions given by ZK05, GMA08, and Lyutikov (2010, 2011), and
reconcile the differences between them. Using the ZK05 condition to define a
RS forming radius, we discuss the evolution of magnetized shell at the
external shock stage.

Our conclusions are summarized as follows:

  * $\bullet$

For the properties of the RS, we confirm previous arguments that a
dynamically-important magnetic field ($\sigma\gtrsim 1$) suppresses the
strength of RS (Zhang & Kobayashi, 2005b; Giannios et al., 2008; Mimica et
al., 2009; Granot, 2012; Zhang, 2018), forcing it to dissipate energy at
efficiency less than $\sim 0.01-0.1$.

  * $\bullet$

For the RS forming condition, we argue that the ZK05 condition
$\sigma<(8/3)\gamma_{4}^{2}(n_{1}/n_{4})$ is the correct condition, which can
be exploited for the setup and consistency check of relevant numerical
simulations (e.g., Ai & Zhang, 2021; Ai et al., 2022). We find the simulated
results of MGA09 are more consistent with the ZK05 predictions rather than the
GMA08 condition. It is further shown that the GMA08 condition, though correct
in concept, mistakenly gives a tighter constraint due to their procedure of
order-of-magnitude estimate. If formulated more rigorously, we show in
Appendix B that the physical motivation of GMA08 leads to the ZK05 condition.

  * $\bullet$

The evolution of a magnetized ejecta at the external-shock stage can be
classified into three regimes: (1) ‘thick shell’ regime ($\xi_{0}\lesssim
1,\sigma_{0}\lesssim\xi_{0}^{-3}$) where the RS is formed and becomes
relativistic, subsequently decelerating the shell while crossing it. (2) ‘thin
shell’ regime ($\xi_{0}\gtrsim 1,\sigma_{0}\lesssim\xi_{0}$) where a Newtonian
RS is formed either after shell spreading or from the beginning, which then
crosses and decelerates the shell as a trans-relativistic shock. (3) ‘no RS’
regime ($\sigma_{0}^{-1/3}\lesssim\xi_{0}\lesssim\sigma_{0}$) where the shell
is decelerated by the FS, either after shell spreading or from the beginning,
without forming an RS. Here, $(\xi_{0},\sigma_{0})$ refer to the values of
ejecta parameters $(\xi,\sigma)$ at coasting phase. Therefore, all the ejecta
with $\sigma_{0}\lesssim 1$ are capable of forming RSs, either initially or
aided by shell spreading. The boundary between the ‘thin shell’ regime and the
‘no RS’ regime at high-$\sigma$ is still ambiguous, limited by our poor
understanding of shell spreading in a highly magnetized ejecta.

  * $\bullet$

For a typical GRB problem, we can define the critical degree of magnetization
$\sigma_{\mathrm{F,cr}}$ below which the RS is immediately formed. We find
that this value is highly sensitive to the bulk Lorentz factor $\gamma_{0}$
and the shell width $\Delta_{0}$. For the RS formed in the ‘thin shell’
scenario, the critical $\sigma_{\mathrm{F,cr}}$ is generally below unity,
similar to the value proposed in GMA08. However, for the ‘thick shell’ cases,
the critical $\sigma_{\mathrm{F,cr}}$ is less constrained, which can
potentially reach $100-1000$ for ultra-thick shells if there is a pre-existing
coasting phase (similar to the value proposed in ZK05).

Here, we restrict our theoretical formulation to 1D outflows, with only some
order-of-magnitude discussions on shell spreading. Detailed numerical
simulations are needed to probe the shell-spreading effect in magnetized
outflows (as suggested in GMA08 and MGA09), as well as other instabilities
potentially arising from multi-dimensional treatments (as suggested in
Lyutikov, 2010, 2011).

In addition, our discussions in Sec. 4 assume that there is always a coasting
phase before shell spreading or external shocks. This may not be true in the
high-$\sigma$ regime (see, e.g. Granot et al., 2011; Granot, 2012; Gao &
Zhang, 2015). Whether the shell can reach a steady coasting phase depends on
the acceleration mechanism, as well as the comparison between the coasting
radius and other characteristic radii. In the ‘thin shell’ regime, the
impulsive acceleration may accelerate the Poynting-flux-dominated ejecta
beyond the fast magnetosonic point, leading to a prolonged, slower
acceleration phase followed by deceleration (Granot et al., 2011; Granot,
2012). For thick shells that resemble long-lasting magnetized jets, efficient
acceleration beyond the fast magnetosonic point is not trivial (Beskin et al.,
1998; Tchekhovskoy et al., 2008, 2009), which may involve magnetic dissipation
(Lyubarsky & Kirk, 2001; Drenkhahn, 2002; Drenkhahn & Spruit, 2002; Kirk &
Skjæraasen, 2003) or external confinement (Komissarov et al., 2009; Lyubarsky,
2009). Unconfined steady-state jets in ideal MHD could still be accelerated
efficiently if (a) the ‘rarefaction acceleration’ operates at the onset of
deconfinement (Komissarov et al., 2010; Tchekhovskoy et al., 2010), or (b) the
‘causality surface’ (where the flow loses lateral causal contact with the jet
axis) is well beyond the fast magnetosonic point (where the causal contact is
broken along the stream line) (Tchekhovskoy et al., 2009). However, the former
mechanism may only be important shortly after the jet emerges from the stellar
envelope (Komissarov et al., 2010; Gao & Zhang, 2015), and the latter effect
depends on the magnetic field configuration (Beskin & Nokhrina, 2006;
Tchekhovskoy et al., 2008; Pu & Takahashi, 2020; Chen & Zhang, 2021). These
suggest that a coasting phase may be more plausible for thick shells.
Nevertheless, if analyzed more properly (e.g., by considering all the
evolutionary stages of the ejecta), a strongly-magnetized shell could be
directly decelerated or form an RS during the acceleration of the ejecta,
which deserves further investigation. The results shown here could be applied
to such analyses, with further implications for the self-consistent lightcurve
modelling of magnetized relativistic outflows.

## Acknowledgements

We thank the anonymous referee for helpful comments and Shunke Ai for helpful
discussion on mechanical models. J.-Z. Ma acknowledges the support from Tsien
Excellence in Engineering Program (TEEP) in Tsinghua University.

## Data Availability

The code developed to perform the calculation in this paper is available upon
request.

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##  Appendix A Jump conditions of magnetized relativistic shocks

Following the symbol convention of ZK05, the upstream region (unshocked) is
denoted as region 1 and the downstream region (shocked) is denoted as region
2, separated by the shock (denoted as ‘s’). $Q_{ij}$ ($Q_{a,ij}$) represents
the quantity $Q$ ($Q_{a}$) in region $i$ in the rest frame of $j$ ($i=1,2$ and
$j=1,2,s$). Assuming a cold upstream, the shock jump conditions can be written
in the most general forms as (see ZK05)

$\displaystyle\frac{e_{2}}{n_{2}m_{\mathrm{p}}c^{2}}=(\gamma_{21}-1)f_{a,21}\,,$
(25)
$\displaystyle\frac{n_{2}}{n_{1}}=\frac{\hat{\Gamma}_{2}\gamma_{21}+1}{\hat{\Gamma}_{2}-1}f_{b,21}\,,$
(26) $\displaystyle\frac{p_{\mathrm{tot},2}}{p_{2}}=f_{c,21}\,,$ (27)

supplemented by the EOS with $p_{2}=(\hat{\Gamma}_{2}-1)e_{2}$. Here,
$f_{a,21},f_{b,21},f_{c,21}$ are the correction factors contributed by the
magnetic nature of the shock. They are given by

$\displaystyle f_{a,21}=1-\frac{\gamma_{21}+1}{2u_{1s}u_{2s}}\sigma_{1}\,,$
(28) $\displaystyle
f_{b,21}=\frac{\hat{\Gamma}_{2}-1}{\hat{\Gamma}_{2}\gamma_{21}+1}\gamma_{21}\left(1+\frac{\beta_{21}}{\beta_{2s}}\right)\,,$
(29) $\displaystyle
f_{c,21}=1+\frac{u_{1s}}{2(\hat{\Gamma}_{2}-1)u_{2s}}\left(\frac{e_{2}}{n_{2}m_{\mathrm{p}}c^{2}}\right)^{-1}\sigma_{1}\,.$
(30)

The quantity $\beta_{2s}=\beta_{2s}(\sigma,\gamma_{21},\hat{\Gamma}_{2})$ is
found by solving a three order equation (see a detailed derivation in Ma &
Zhang, 2022)

$\begin{split}&\beta_{2s}(\beta_{2s}+\beta_{21})\left[\beta_{2s}-(\hat{\Gamma}_{2}-1)\frac{\beta_{21}}{1+\sqrt{1-\beta_{21}^{2}}}\right]\\\
&=\left[\beta_{2s}+\left(1-\frac{\hat{\Gamma}_{2}}{2}\right)\beta_{21}\right](1-\beta_{2s}^{2})\sigma\,,\end{split}$
(31)

and $\beta_{1s}$ is found through the Lorentz transformation
$\gamma_{1s}=\gamma_{21}\gamma_{2s}(1+\beta_{2s}\beta_{21})$. Combine
equations (25)-(27) with the EOS, and we get the downstream total pressure

$p_{\mathrm{tot},2}=(\hat{\Gamma}_{2}\gamma_{21}+1)(\gamma_{21}-1)n_{1}m_{\mathrm{p}}c^{2}F_{21}\,,$
(32)

where $F=f_{a}f_{b}f_{c}$.

##  Appendix B A rigorous derivation of the contact radius and the RS crossing
radius

Figure 3:  Ratio of the RS crossing radius $r_{\Delta}$ to the contact radius
$r_{\mathrm{c}}$ in the parameter space. The line styles are associated with
the RS relative Lorentz factor and different treatments of RS forming
conditions, as in Fig. 1. The RS crossing radius coincides with the contact
radius at the ZK05 condition (red solid line) rather than the GMA08 condition
(purple dashed line).

Following GMA08, the contact radius $r_{\mathrm{c}}$ is defined as the radius
where the fast magnetosonic wave crosses the shell at the expansion timescale,
i.e.,

$r_{\mathrm{c}}\equiv\frac{\Delta}{\beta_{4}-\beta_{\mathrm{w}}}\,,$ (33)

where $\beta_{\mathrm{w}}$ is the dimensionless wave speed in lab frame given
by the Lorentz transformation
$\beta_{\mathrm{w}}=(\beta_{4}-\beta_{\mathrm{A}})/(1-\beta_{4}\beta_{\mathrm{A}})$.
Here, $\beta_{\mathrm{A}}=\sqrt{\sigma/(1+\sigma)}$ is the dimensionless
Alfven speed. The contact radius can thus be written rigorously as

$r_{\mathrm{c}}=\Delta\gamma_{4}^{2}\left(\sqrt{\frac{1+\sigma}{\sigma}}-\beta_{4}\right)\,,$
(34)

For typical GRB cases with relativistic ejecta, it reduces to the expression
(14)

$r_{\mathrm{c}}\simeq\Delta\gamma_{4}^{2}\left(\sqrt{\frac{1+\sigma}{\sigma}}-1\right)\,.$
(35)

Following Sari & Piran (1995) and ZK05, the RS crossing radius is defined as
the radius where the RS crosses the shell (for a rigorous derivation see,
e.g., Chapter 8.7.1 of Zhang, 2018), i.e.,

$r_{\Delta}\equiv\frac{\Delta}{\beta_{4}-\beta_{3}}\left(1-\frac{\gamma_{4}n_{4}}{\gamma_{3}n_{3}}\right)\,,$
(36)

where the last factor is taken into account due to the compressed shell after
RS crossing. Substituting the shock jump condition (26)(29) and Lorentz
transformation $\gamma_{43}=\gamma_{4}\gamma_{3}(1-\beta_{4}\beta_{3})$ into
equation (36), we find

$\begin{split}r_{\Delta}&=\frac{\Delta}{\beta_{4}-\beta_{3}}\left(1-\frac{\beta_{4s}(1-\beta_{3}^{2})}{(1-\beta_{4}\beta_{3})(\beta_{4s}+\beta_{43})}\right)\\\
&=\Delta\,\frac{\beta_{4s}\beta_{3}(\beta_{3}-\beta_{4})+(1-\beta_{4}\beta_{3})\beta_{43}}{(\beta_{4}-\beta_{3})(1-\beta_{4}\beta_{3})(\beta_{4s}+\beta_{43})}\\\
&=\Delta\frac{1-\beta_{4s}\beta_{3}}{(1-\beta_{4}\beta_{3})(\beta_{43}+\beta_{4s})}\,.\end{split}$
(37)

As the RS grows weak and approaches the point of vanishing, we get
$\beta_{4s}\rightarrow\beta_{\mathrm{A}},\beta_{4s}\rightarrow\beta_{4},\beta_{43}\rightarrow
0$. Consequently, at the verge of RS formation, the RS crossing radius
approaches the contact radius, i.e., $r_{\Delta}\rightarrow r_{\mathrm{c}}$.
For a system with an FS that is at least mildly relativistic, we have
$\beta_{3}\simeq\beta_{4}\sim 1$. We thus obtain

$r_{\Delta}\simeq\Delta\gamma_{4}^{2}\frac{1-\beta_{4s}}{\beta_{43}+\beta_{4s}}\,,$
(38)

which gives

$\frac{r_{\Delta}}{r_{\mathrm{c}}}\simeq\frac{1-\beta_{4s}}{(\beta_{43}+\beta_{4s})\left(\sqrt{(1+\sigma)/\sigma}-1\right)}\,.$
(39)

With the magnetization degree $\sigma$ fixed, $\beta_{4s}$ drops as
$\beta_{43}$ decreases. The ratio of the two radii thus monotonically
increases as the RS grows weak until the two radii become equal to each other
at the pressure balance condition (11) suggested by ZK05.

In Fig. 3, we present the ratio (39) of two radii in the parameter space, with
GMA08 condition plotted as the purple dashed line and ZK05 condition as the
red solid line. As the RS becomes weak, the RS crossing radius increases until
it coincides with the contact radius at the ZK05 condition rather than the
GMA08 condition. Here, we performed a more careful derivation for the
characteristic radii to show that the physical motivation of GMA08 condition
is fully consistent with ZK05. However, due to the order-of-magnitude
estimates in GMA08, the condition given in GMA08 is mistakenly tighter than
the actual RS forming condition given in ZK05.

##  Appendix C Scaling laws for the dynamical evolution of a magnetized shell

Table 1:  Scaling laws for the dynamical evolution of a magnetized shell.

General assumptionsa : $E=4\pi
m_{\mathrm{p}}c^{2}\gamma_{4}^{2}n_{4}r^{2}\Delta(1+\sigma)=\mathrm{const},\sigma\propto
n_{4}r^{2}$
---
Evolutionary stage | Feature | $\log\xi/\log r$ | $\log n_{4}/\log r$ | $\log\sigma/\log r$ | $\log\xi/\log\sigma$
Coasting | $\gamma_{4}\simeq\mathrm{const},\Delta\simeq\mathrm{const}$ | $0$ | $-2$ | $0$ | $-$
Deceleration by RRS, $r_{\mathrm{N}}\lesssim r\lesssim r_{\Delta}$ | $\sigma\ll 1$ | $\gamma_{4}\propto r^{-1/2},\Delta\simeq\mathrm{const}$b | $2/3$ | $-1$ | $1$ | $2/3$
$\sigma\gg 1$ | $-3/2$ |  $1/2$d | $4/3$
Shell spreading, $r_{\mathrm{s}}\lesssim r\lesssim r_{\mathrm{dec}}$ | $\sigma\ll 1$ | $\Delta\propto r,\gamma_{4}\simeq\mathrm{const}$c | $-1/2$ | $-3$ |  $-1$d | $1/2$
$\sigma\gg 1$ | $-5/2$ | $-1/2$ | $1$

  * a

Mimica et al. (2009).

  * b

Kobayashi et al. (1999).

  * c

Meszaros et al. (1993); Piran et al. (1993).

  * d

Granot (2012).

In Table 1, we present the scaling laws of shell parameters as functions of
radius in different evolutionary stages. It should be pointed out here that
the scaling laws in high-$\sigma$ regime are uncertain. This is mainly because
the evolution of $\gamma_{4}$ and $\Delta$ as the shell travels to larger
distances are less clear for $\sigma\gg 1$. In particular, a coasting phase
may not exist in high-$\sigma$ regime (Granot et al., 2011; Granot, 2012; Gao
& Zhang, 2015), which would make some of our discussions irrelevant to the
real scenario. Here, we assume that the total energy of the ejecta is
conserved during its evolution, and $\sigma\propto n_{4}r^{2}$ as pointed out
by Mimica et al. (2009). We extrapolate the scaling laws of $\gamma_{4}(r)$
and $\Delta(r)$ for $\sigma\ll 1$ to high-$\sigma$ regime. Although some
results given here seem to agree with previous analytical estimates for
magnetized shells (Granot, 2012), more rigorous analyses are needed.