# The Peak Frequency and Luminosity of Synchrotron Emitting Shocks: from Non-
Relativistic to Ultra-Relativistic Explosions

Ben Margalit School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA  Eliot Quataert Department of Astrophysical
Sciences, Princeton University, Princeton, NJ 08544, USA

###### Abstract

Synchrotron emission is ubiquitous in explosive astrophysical events—it is a
natural byproduct of shocks formed when matter expelled by the explosion
collides with ambient material. This emission is well-observed in various
classes of transients, and is often interpreted within a canonical
‘equipartition’ framework that allows physical properties of the shock to be
inferred from the frequency and luminosity at which the observed spectral
energy distribution (SED) peaks. This framework has been remarkably successful
in explaining observations of radio supernovae. It has also been used for
trans-relativistic explosions, where the shock velocities approach the speed
of light. However, the conventional framework does not incorporate
relativistic effects. Neither does it account for thermal electrons, which
have been shown to be important for high-velocity shocks. In this paper we
describe a revised framework that accounts for these two effects, and is
applicable to non-relativistic, trans-relativistic, and ultra-relativistic
explosions. We show that accounting for these effects can dramatically change
the inferred parameters of high-velocity shocks, and in particular—that the
shock velocity, ambient density, and total energy are overestimated by the
conventional non-relativistic framework. We delineate the phase-space where
such modifications are important in terms of observationally measurable
parameters. We also find a novel upper limit on the peak synchrotron
luminosity of shock-powered transients, which is remarkably consistent with
existing observations. Finally, we discuss a prediction of the model—that the
SED will qualitatively change as a function of shock velocity—and show that
this is broadly consistent with data for representative events (e.g.,
SN1998bw, AT2018cow, CSS161010, AT2020xnd).

Time domain astronomy (2109); High energy astrophysics (739); Shocks (2086);
Radio transient sources (2008); Supernovae (1668); Gamma-ray bursts (629);

††software: numpy

##  1 Introduction

Transient astrophysical events produce emission that spans the entire
electromagnetic spectrum and can be accompanied by multi-messenger
counterparts in the form of neutrinos, gravitational-waves, and cosmic rays.
These events are often the harbingers of extreme and energetic “explosions”
such as core-collapse and thermonuclear supernovae (SNe), gamma-ray bursts
(GRBs), compact object mergers, or tidal disruption events (TDEs). In these
cases and in many other scenarios, the explosive event expels some material at
relatively high velocities. The ejected material will inevitably collide with
the ambient medium in the vicinity of the source and form a shock. Such shocks
are well known sources of broadband synchrotron emission that can be detected
at different frequencies (most commonly, in the radio).

Well-studied examples include the classes of radio SNe and GRB afterglows
(e.g., Weiler et al., 2002). In both cases, it is usually assumed that
observed emission is produced by a population of non-thermal relativistic
electrons accelerated at the shock front through a first-order Fermi process.
These electrons produce synchrotron emission by gyrating in a magnetic field
that is typically assumed to be amplified self-consistently by the
(collisionless) shock. This model has worked remarkably well in explaining
observations of “normal” radio SNe and GRB afterglows (e.g., Chevalier, 1982;
Weiler et al., 1986; Chevalier, 1998; Sari et al., 1998). It is also broadly
consistent with expectations from theory and kinetic simulations (e.g., Bell,
1978; Blandford & Eichler, 1987; Blandford & Ostriker, 1978; Spitkovsky, 2008;
Sironi & Spitkovsky, 2009, 2011; Sironi et al., 2013; Park et al., 2015).

In recent years, interest in synchrotron-emitting shock-powered transients has
been renewed with the discovery of a growing sample of mildly- and trans-
relativistic explosions including broad-lined Ic SNe (Ic-BL) and the emerging
class of Fast Blue Optical Transients (FBOTs) which are most known by their
prototypical event AT2018cow (Margutti et al., 2019; Ho et al., 2019).
Margalit & Quataert (2021) showed that for such events, observed synchrotron
emission could instead be powered by the population of electrons that are
thermalized in the post-shock region (rather than non-thermal electrons, which
must undergo an additional acceleration process). This thermal-electron model
reproduced well the observational data for several FBOTs (Ho et al., 2022),
and is a natural expectation for shocks whose velocity exceed $\beta_{\rm
sh}c\gtrsim 0.2c$ (Margalit & Quataert, 2021). The contribution of thermal
electrons to observed synchrotron emission has also been discussed in the GRB
literature (Giannios & Spitkovsky, 2009; Warren et al., 2017; Ressler &
Laskar, 2017; Warren et al., 2018; Samuelsson et al., 2020; Warren et al.,
2022). Still, the importance of synchrotron emission from thermal electrons in
shock-powered transients is not yet widely appreciated.

In this paper, we extend our previous work (Margalit & Quataert, 2021) by
incorporating relativistic effects and developing an analytic formalism for
shock-powered synchrotron emission (and absorption) that is correct for
arbitrary shock velocities. This allows us to properly survey the full
parameter-space from non-relativistic radio SNe through trans-relativistic
FBOTs and SNe Ic-BL, to ultra-relativistic GRBs, all within a single
framework. A particular focus of this paper is applying this formalism to
describe the frequency and luminosity at which the spectral energy
distribution (SED) of the resulting synchrotron emission peaks.

In a canonical paper, Chevalier (1998) showed that the radio data of typical
radio SNe is consistent with a synchrotron self-absorption (SSA) model in
which the SED is self-absorbed at low frequencies and the peak of the SED
coincides with the SSA frequency (that is, the frequency at which the SSA
optical depth is $\sim 1$). Furthermore, Chevalier (1998) showed that—in cases
where this model applies—observations of the frequency $\nu_{\rm pk}$ and
specific luminosity $L_{\nu_{\rm pk}}$ at which the SED peaks are sufficient
to fully determine the physical properties of the shock. In particular,
Chevalier (1998) show how observed values of $\nu_{\rm pk}$, $L_{\nu_{\rm
pk}}$, and the time $t$ at which they are observed (measured from the
explosion epoch) can be used to infer the shock radius and post-shock magnetic
field. These can subsequently be used to determine all other physical
variables of the system, including the shock velocity, the density of the
ambient medium, and the explosion energy. The method described by Chevalier
(1998) has been used extensively throughout the literature, and with great
success. However, this formalism is inapplicable when the shock velocities are
high. In particular, the Chevalier (1998) analysis does not include
relativistic corrections which are important when the proper-velocity of the
shock is $(\Gamma\beta)_{\rm sh}\gtrsim 1$. It also neglects the contribution
of thermal electrons that has been shown to be important when $\beta_{\rm
sh}\gtrsim 0.2$ (Margalit & Quataert, 2021). The aim of the present work is to
extend this formalism such that it can be appropriately applied to shocks of
arbitrary velocity.

Our primary result is a set of expressions that can be used to infer the shock
velocity and ambient density from observed values of $\nu_{\rm pk}$,
$L_{\nu_{\rm pk}}$, and $t$. These can subsequently be used to find the shock
radius, post-shock magnetic field, and energy (see §6 and Appendix A), and are
valid for any shock velocity. We begin by presenting our formalism in §2. In
§3 we describe the resulting synchrotron SEDs. The SED will appear
qualitatively different depending on the shock velocity, and is separated into
three regimes: (a) the SED peak is set by SSA and non-thermal (power-law)
electrons dominate emission; (b) SSA governs the SED at peak but thermal
electrons dominate emission at this frequency; (c) emission at the peak of the
SED is optically-thin. These regimes are treated in §4.1, §4.2, and §5,
respectively. We discuss implications of our results and compare to
observations in §6. Finally, we summarize our findings and conclude in §7.

##  2 The Physical Model

We follow the non-relativistic framework of Margalit & Quataert (2021),
extending this model to arbitrary shock velocity. We additionally adopt a
line-of-sight (LOS) approximation in which flux is considered to be dominated
by material along the LOS to the source. For relativistically expanding
sources, emission from different annular regions will reach the observer at
different times. A complete calculation of the observed flux in this case
therefore necessitates full knowledge of the shock expansion history. This has
been performed numerically for the case of GRBs assuming Blandford-McKee
expansion (Granot et al., 1999a, b), or as post-processing of hydrodynamic
simulations (e.g., van Eerten et al., 2012). Wishing to avoid imposing any
specific expansion history (which may differ in different contexts and for
different shock velocities), and in the interest of obtaining analytically-
tractable results—we adopt the LOS approximation. The Doppler factor is
therefore taken to be $\mathcal{D}=\Gamma$, where
$\Gamma=1/\sqrt{1-\beta^{2}}$ is the Lorentz factor of post-shock emitting
matter with velocity $\beta c$. The lab-frame shock radius can be expressed as

$R=\sqrt{1+\left(\Gamma\beta\right)_{\rm sh}^{2}}\left(\Gamma\beta\right)_{\rm
sh}c\bar{t}~{};~{}~{}~{}\bar{t}\equiv\ell_{\rm dec}t$ (1)

where $t$ is the elapsed time measured in the lab (observer) frame and
$\ell_{\rm dec}\gtrsim 1$ is a dimensionless parameter that accounts for
potential deceleration of the shock. Here $\Gamma_{\rm sh}$ is the shock
Lorentz factor and $\beta_{\rm sh}c$ its corresponding velocity, and we have
expressed $R$ in terms of the proper velocity $\left(\Gamma\beta\right)_{\rm
sh}=\sqrt{\Gamma_{\rm sh}^{2}-1}$. The shock velocity is related to the post-
shock velocity via the Blandford-McKee relations (Blandford & McKee 1976;
their Equation 5)111 This relationship is exact for arbitrary velocities and
does not depend on the shock’s evolution history (in particular, it does not
assume that the shock evolves according to the ultra-relativistic Blandford-
McKee self-similar solution).  where we assume that the post-shock adiabatic
index can be approximated by $\hat{\gamma}=(4+\Gamma^{-1})/3$ (see Equation
B7).

Equation (1) reduces to $R=\beta_{\rm sh}c\bar{t}$ in the non-relativistic
regime and to $R=2\Gamma^{2}c\bar{t}$ in the ultra-relativistic limit. For
$\ell_{\rm dec}=1$ it describes a constant velocity shock. A decelerating
shock with the same instantaneous velocity would have a larger radius, and can
be described by taking $\ell_{\rm dec}\gtrsim 1$. For example, $\ell_{\rm
dec}=8$ for an ultra-relativistic shock decelerating according to the
Blandford-McKee solution and viewed along the LOS. (Sari 1997; accounting for
emission off the LOS leads to an effective $\ell_{\rm dec}\sim 3-7$; Sari
1998) For a Newtonian shock that expands as $R\propto t^{m}$ (e.g., Chevalier,
1998) we instead have $\ell_{\rm dec}=1/m$. This implies $1\lesssim\ell_{\rm
dec}\lesssim 3/2$ for the hydrodynamic solution described by Chevalier (1982)
and $\ell_{\rm dec}=5/2$ for the case of a Sedov-Taylor blast-wave propagating
into a constant-density medium.

The approximate adiabatic index $\hat{\gamma}$ adopted above combined with the
exact Blandford-McKee solutions imply that $n^{\prime}=4\Gamma n$ for
arbitrary $\Gamma$, where $n^{\prime}$ ($n$) is the downstream (upstream)
density. The downstream energy-density is similarly given by
$u^{\prime}=4\Gamma(\Gamma-1)n\mu m_{p}c^{2}$ where $\mu$ is the mean
molecular-weight, and we adopt $\mu\simeq 0.62$ appropriate for solar
composition. This can be used to find the downstream electron temperature
$\Theta\equiv kT_{e}^{\prime}/m_{e}c^{2}$ via Equation (2) of Margalit &
Quataert (2021) with $\Theta_{0}=\epsilon_{T}(\mu
m_{p}/\mu_{e}m_{e})(\Gamma-1)$. We assume the mean molecular weight per
electron is $\mu_{e}\simeq 1.18$, as expected for fully-ionized solar
composition material. The equation for $\Theta$ implies that a fraction
$\epsilon_{T}$ of the post-shock energy density is carried by thermal
electrons. We similarly adopt the usual assumption that a post-shock magnetic
field $B$ is amplified such that $B^{2}/8\pi=\epsilon_{B}u^{\prime}$, and that
a fraction $\epsilon_{e}\equiv\delta\epsilon_{T}$ of the post-shock energy
density is carried by non-thermal electrons that are accelerated into a power-
law distribution $dn/d\gamma_{e}\propto\gamma_{e}^{-p}$ in electron Lorentz
factor $\gamma_{e}$ (note that $B$ is the magnetic field in the post-shock
frame; we omit the prime notation for consistency with previous work).

Based on the results of kinetic simulations (e.g., Sironi et al. 2013) we
adopt the following fiducial values for the microphysical parameters described
above: $\epsilon_{T}=0.4$, $\epsilon_{e}=0.01$ ($\delta=0.025$),
$\epsilon_{B}=0.1$, and $p=3$. This implies that 40% of the post-shock energy
is carried by thermal electrons, 10% by magnetic fields, and 1% by non-thermal
(power-law) accelerated electrons. The remaining $\sim$50% of post-shock
energy is carried by thermal and non-thermal (cosmic ray) ions. Typical
kinetic simulations suggest that $\sim$10% of energy goes into cosmic-ray ions
and therefore that thermal ions carry $\sim$40% of the post-shock energy. This
situation implies approximate post-shock equilibration between electrons and
ions such that $T_{e}\approx T_{\rm ion}$. Considering lower values of
$\epsilon_{T}$ would relax this assumption and generally allow for
$T_{e}<T_{\rm ion}$.

Following Sari et al. (1998, 1999), the flux density emitted by a “spherical”
relativistic source is $F_{\nu}\sim\pi\Gamma R^{2}I_{\nu}^{\prime}/D_{L}^{2}$,
where $D_{L}$ is the luminosity distance to the source and $I_{\nu}^{\prime}$
is the specific intensity in the source frame. The latter satisfies the
radiative transfer equation
$dI_{\nu}^{\prime}/ds=j_{\nu}^{\prime}-\alpha_{\nu}^{\prime}I_{\nu}^{\prime}$
where $j_{\nu}^{\prime}$ and $\alpha_{\nu}^{\prime}$ are the local emissivity
and absorption coefficients. The flux density estimate above is correct as
long as the the source is azimuthally uniform (“spherical”) within the region
from which (potentially beamed) radiation reaches the observer, i.e. within an
opening angle $\sim 1/\Gamma$ from the observer LOS. Therefore, even a
collimated source can be considered spherical if it is expanding
relativistically and its opening angle is $>1/\Gamma$ (Sari et al., 1999).
Defining the isotropic equivalent luminosity of a source as $L_{\nu}\equiv
4\pi D_{L}^{2}F_{\nu}$, we can write

$L_{\nu}\approx 4\pi^{2}\Gamma R^{2}\frac{\left\langle
j^{\prime}_{\nu}\right\rangle}{\left\langle\alpha^{\prime}_{\nu}\right\rangle}\left(1-e^{-\left\langle\alpha^{\prime}_{\nu}\right\rangle\Delta
R^{\prime}}\right).$ (2)

Here $\Delta R^{\prime}\sim R/\Gamma$ is the width of the emitting region
measured in the source frame (in the observer frame, $\Delta R\sim
R/\Gamma^{2}$), and we define an emission volume-filling factor $f$ following
Chevalier (1998) such that

$\Delta R^{\prime}\equiv 4fR/3\Gamma.$ (3)

From hydrodynamic considerations one expects that $\Delta R^{\prime}\lesssim
R/4\Gamma$. We therefore adopt $f=3/16$ as a fiducial value. The LOS averaged
emissivity and absorption coefficients, $\left\langle
j^{\prime}_{\nu}\right\rangle$,
$\left\langle\alpha^{\prime}_{\nu}\right\rangle$, are described in Margalit &
Quataert (2021) and include the contribution of both thermal and non-thermal
(power-law) electrons. In the non-relativistic limit and with $f=3/4$,
Equation (2) reduces to Equation (21) of Margalit & Quataert (2021). In cases
where only power-law electrons contribute to observed emission, it is also
consistent with the estimates of Chevalier (1998) in the non-relativistic
limit, and with Sari et al. (1998) in the ultra-relativistic optically-thin
limit. As previously mentioned, in the relativistic case one must generally
integrate the specific intensity over a surface from which all emitted photons
arrive at the same time (Granot et al., 1999a, b). In Equation (2) we have
instead adopted the LOS approximation and neglected this aspect for the sake
of analytic tractability. The impact of this assumption will be investigated
in future work.

With the assumptions above, the properties of the shock and its resulting
emission can be fully specified with only two physical variables (along with
several dimensionless microphysical and geometric parameters). In this paper
we choose these physical variables to be the proper-velocity of the shock and
the ambient density. We find it convenient to express the ambient (upstream)
density $n$ at the location of the shock in terms of an effective mass-loss
parameter $\dot{M}/v_{\rm w}$, such that

$\dot{M}/v_{\rm w}\equiv 4\pi\mu m_{p}R^{2}n(R).$ (4)

For reasons that will later become apparent, this parameterization is useful
because the mass-loss parameter depends on the peak frequency and observed
time only through the combination $\nu_{\rm pk}t$. Note that Equation (4) is
merely a parameterization choice which replaces
$n\leftrightarrow\dot{M}/v_{\rm w}$. In particular, our results are not
limited to the restricted case of a steady-state wind medium, and can be
applied to arbitrary density profiles.222 One caveat to this statement is that
emission is assumed to be dominated by electrons that have been swept-up by
the most recent shock-doubling time. This typically requires that most of the
swept-up material must be accumulated at $r\sim R$, and for a power-law
density gradient $n\propto r^{-k}$, implies that $k<3$.  The mass-loss
parameter should therefore be viewed as only a proxy for the ambient density
at the current shock location.

##  3 The Spectral Energy Distribution

Figure 1: Representative spectral energy distributions (SEDs) for the three
regimes considered in this work. The progression from left to right is a
function (primarily) of the shock velocity. Solid and dashed grey curves show
the emissivity contributed by the population of thermal and power-law
electrons, respectively. Solid black curves show the resulting SED. The
frequency scaling of $L_{\nu}$ is shown above each corresponding section of
the SED. (a) For $(\Gamma\beta)_{\rm sh}\approx 0.1$. The peak of the SED is
set by synchrotron self-absorption (SSA) such that $L_{\nu_{\rm pk}}$ is
obtained at the self-absorption frequency ($\nu_{\rm pk}\simeq\nu_{\rm a}$).
At low shock velocities, the emissivity at $\nu_{\rm pk}$ is dominated by
power-law electrons such that $L_{\nu_{\rm pk}}<L_{\rm th}$ (Equation 8). This
regime corresponds to the ‘standard’ model considered in the literature (e.g.,
Chevalier 1998) and is discussed here in §4.1. (b) For $(\Gamma\beta)_{\rm
sh}\approx 0.6$. When the shock velocity is higher (typically mildly or trans-
relativistic) then the SED peak is still governed by SSA, however emission at
frequency $\nu_{\rm pk}$ is now dominated by thermal electrons. This is true
when $L_{\nu_{\rm pk}}>L_{\rm th}$ (Equation 8). This regime implies a
potentially steep spectral slope at frequencies slightly above $\nu_{\rm pk}$.
At higher frequencies power-law electrons again dominate the emissivity and
the spectral slope reverts the standard predictions (see Margalit & Quataert
2021 for more details). (c) For $(\Gamma\beta)_{\rm sh}\approx 3$. For ultra-
relativistic shocks the SSA frequency falls below the peak of the emissivity
($\nu_{\rm a}<\nu_{\Theta}$) and the SED peaks at the latter frequency
($\nu_{\rm pk}\simeq\nu_{\Theta}$). Emission at $\nu_{\rm pk}$ is again
dominated by thermal electrons, and this regime is discussed in §5.

Following Chevalier (1998) we are interested in estimating physical properties
of the shock in terms of the observed frequency $\nu_{\rm pk}$ and luminosity
$L_{\nu_{\rm pk}}$ at which the SED peaks. We assume that electrons are slow
cooling at this frequency and neglect extrinsic effects on the SED such as
free-free absorption. As described in §2, we extend the Chevalier (1998) model
by considering both relativistic effects and the contribution of synchrotron
emission from thermal electrons. The latter was shown to be important for
mildly-relativistic shocks where $\beta_{\rm sh}\gtrsim 0.2$ (Margalit &
Quataert, 2021), while the former is clearly relevant when $(\Gamma\beta)_{\rm
sh}\gtrsim 1$.

The unabsorbed SED is proportional to $j_{\nu}(\nu)$ and is intrinsically
peaked at frequency $\nu_{\Theta}=3\Gamma\Theta^{2}eB/4\pi m_{e}c$ (Margalit &
Quataert, 2021), the frequency at which a typical thermal electron radiates
most of its synchrotron power. The observed SED however will be modified due
to SSA and thus does not track the emissivity at frequencies below the SSA
frequency, $\nu_{\rm a}$. Two categories therefore emerge when considering the
observed peak of the SED. In the first case, the intrinsic peak of the
emissivity at $\nu\simeq\nu_{\Theta}$ is hidden due to SSA, and the observed
peak of the SED occurs instead at the SSA frequency $\nu_{\rm a}$. This
happens when $\nu_{\Theta}<\nu_{\rm a}$, that is—when the SSA optical depth at
frequency $\nu_{\Theta}$ is $\tau_{\Theta}>1$. The second case occurs when
$\tau_{\Theta}<1$ and the SSA frequency falls below the thermal frequency,
$\nu_{\rm a}<\nu_{\Theta}$. In this case the intrinsic peak of the emissivity
is unabsorbed and corresponds to the observed peak of the SED ($\nu_{\rm
pk}\simeq\nu_{\Theta}$). The SSA optical depth is a strong function of shock
velocity (Margalit & Quataert, 2021), and we show in subsequent sections that
the two scenarios described above are delineated by whether the velocity is
smaller or larger than $\left(\Gamma\beta\right)_{\rm sh}\sim 1$. The SSA SED
can further be divided into two subcases depending on whether thermal
electrons or power-law electrons dominate emission at frequency $\nu_{\rm
pk}$. As discussed above, these two cases correspond to shock velocities above
or below $\left(\Gamma\beta\right)_{\rm sh}\sim 0.3$, respectively.

We therefore have three cases to consider overall, delineated by increasing
shock velocity. Each of these cases is characterized by a qualitatively
different SED. Representative SEDs for these three regimes are illustrated in
Figure 1. In the left panel (case a) the shock velocity is low and $\nu_{\rm
pk}\simeq\nu_{\rm a}$ with emission dominated by power-law electrons. This is
the ‘standard’ case considered in the literature, corresponding to the
original Chevalier (1998) analysis. The middle panel (case b) shows the SED
for mildly-relativistic shocks where $\nu_{\rm pk}\simeq\nu_{\rm a}$ but
emission at $\nu_{\rm pk}$ is dominated by thermal electrons. This regime was
first discussed by Margalit & Quataert (2021). Finally, the right panel (case
c) shows the SED at even higher velocities (in particular, when
$\Gamma\beta\gg 1$) when $\nu_{\rm a}<\nu_{\rm pk}\simeq\nu_{\Theta}$. In the
following sections we demonstrate how the shock properties can be inferred
from observations of $\nu_{\rm pk}$ and $L_{\nu_{\rm pk}}$. In doing so, we
separate our discussion based on the three regimes discussed above: §4.1
considers case (a); §4.2 treats case (b); and §5 focuses on case (c).

##  4  Synchrotron Self-Absorbed Peak: $\left(\Gamma\beta\right)_{\rm
sh}\ll$–$\lesssim 1$

###  4.1 Power-law Electrons

Chevalier (1998) first pointed out the importance of SSA in shock-powered
transients, and treated the case of a purely power-law distribution of
emitting electrons. This regime corresponds to the SED illustrated in panel
(a) of Figure 1. We begin by briefly recapping these results for sake of
completeness. All numerical values below adopt a fiducial $p=3$, but do not
depend strongly on the power-law exponent.

In the limit $\left(\Gamma\beta\right)_{\rm sh}\ll 1$ and when power-law
electrons dominate both the emissivity and absorption coefficients, the shock
velocity can be expressed as

$\displaystyle\left(\Gamma\beta\right)_{\rm sh}$ $\displaystyle\simeq
0.44\left(\frac{\epsilon_{e}/\epsilon_{B}}{0.1}\right)^{-1/19}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm Hz}^{-1}}\right)^{9/19}$
$\displaystyle\times\left(\frac{f}{3/16}\right)^{-1/19}\left(\frac{\nu_{\rm
pk}\bar{t}}{5\,{\rm GHz}\times 100\,{\rm d}}\right)^{-1};~{}~{}L_{\nu_{\rm
pk}}<L_{\rm th}.$ (5)

The condition $L_{\nu_{\rm pk}}<L_{\rm th}$ is equivalent to the statement
that power-law electrons dominate emission at frequency $\nu_{\rm
pk}\simeq\nu_{\rm a}$, and is discussed in the following subsection (see
Equation 8). The mass-loss parameter can similarly be written as a function of
the peak luminosity and frequency (e.g., Chevalier & Fransson 2006), giving

$\displaystyle\frac{\dot{M}}{v_{\rm w}}$ $\displaystyle\simeq 1.8\times
10^{-4}\frac{M_{\odot}\,{\rm yr}^{-1}}{10^{3}\,{\rm
km\,s}^{-1}}\,\epsilon_{B,-1}^{-11/19}\epsilon_{e,-2}^{-8/19}$
$\displaystyle\times\left(\frac{f}{3/16}\right)^{-8/19}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm Hz}^{-1}}\right)^{-4/19}$
$\displaystyle\times\left(\frac{\nu_{\rm pk}\bar{t}}{5\,{\rm GHz}\times
100\,{\rm d}}\right)^{2};~{}~{}L_{\nu_{\rm pk}}<L_{\rm th}.$ (6)

Here and in the following we use standard notation whereby
$\epsilon_{B,-1}\equiv\epsilon_{B}/0.1$ and
$\epsilon_{e,-2}\equiv\epsilon_{e}/0.01$.

We note that the numerical prefactors in Equations (4.1,4.1) differ slightly
from the corresponding expressions in Chevalier (1998). Specifically, the
shock velocity implied by Equation (4.1) is $\sim$30% larger than would be
inferred from Equation (13) in Chevalier (1998). This is due to two factors.
First, Chevalier (1998) expresses results in terms of a “peak”
luminosity/frequency that is defined by the intersection of the optically-thin
and optically-thick power-law limits, instead of using the true peak values as
implied by Equation (2). This method overestimates the peak luminosity and
underestimates peak frequency.333 More precisely, the “peak” frequency
$\nu_{*}$ as defined in Chevalier (1998) is related to the true peak frequency
$\nu_{\rm pk}$ via $\nu_{*}=\nu_{\rm pk}\tau_{\rm pk}^{2/(p+4)}$. The peak
luminosity is similarly given by $L_{\nu_{*}}=L_{\nu_{\rm pk}}\tau_{\rm
pk}^{5/(p+4)}/(1-e^{-\tau_{\rm pk}})$. The optical depth at peak $\tau_{\rm
pk}$ can be found by numerically solving the equation $e^{-\tau_{\rm
pk}}\left[(p+4)\tau_{\rm pk}+5\right]=5$ (e.g., Ghisellini 2013 § A.5). For
$p=3$, $\tau_{\rm pk}\simeq 0.639$ and therefore $\nu_{*}\simeq 0.88\nu_{\rm
pk}$ and $L_{\nu_{*}}\simeq 1.54L_{\nu_{\rm pk}}$.  Second, a small
multiplicative difference between the two prescriptions arises due to
different treatments of the electron pitch angle. Chevalier (1998) sets
$\sin\theta=1$ for simplicity whereas we assume a more realistic scenario
where emitting electrons are randomly oriented (see Margalit & Quataert 2021).
Combining these effects, one can show that the above expressions are
consistent with the Chevalier (1998) results if one replaces $L_{\nu_{\rm
pk}}\to 2.02L_{\nu_{\rm pk}}$ and $\nu_{\rm pk}\to 1.01\nu_{\rm pk}$ in the
equations of Chevalier (1998).

###  4.2 Thermal Electrons

We now turn to treating the second regime where thermal electrons dominate the
emissivity and absorption coefficients at frequency $\nu_{\rm pk}$. This
corresponds to the SED pictured in panel (b) of Figure 1.

In this scenario, Equation (2) can be reduced to an expression for the peak
luminosity

$\displaystyle L_{\nu_{\rm pk}}\approx$
$\displaystyle\frac{3\pi^{2}\mu}{2\mu_{e}}m_{p}c^{2}\left(1-e^{-\tau_{\rm
pk}}\right)\epsilon_{T}\left(\nu_{\rm pk}t\right)^{2}$
$\displaystyle\times\frac{\Gamma}{\Gamma+1}\left[1+\left(\Gamma\beta\right)_{\rm
sh}^{2}\right]\left(\Gamma\beta\right)_{\rm sh}^{4},$ (7)

where $\tau_{\rm pk}\sim 1$ is the SSA optical depth at the peak frequency
$\nu_{\rm pk}\simeq\nu_{\rm a}$. Thermal electrons only dominate when the
shock velocity, and therefore (according to the above equation) the peak
luminosity, are high. Specifically, one can use Equation (4.2) to find that
thermal electrons dominate when $L_{\nu_{\rm pk}}>L_{\rm th}$, where

$\displaystyle L_{\rm th}$ $\displaystyle\approx 1.7\times 10^{28}\,{\rm
erg\,s}^{-1}\,{\rm Hz}^{-1}\,\epsilon_{B,-1}^{-4/15}\epsilon_{e,-2}^{8/15}$
(8)
$\displaystyle\times\left(\frac{\epsilon_{T}}{0.4}\right)^{-7/5}\left(\frac{f}{3/16}\right)^{4/15}\left(\frac{\nu_{\rm
pk}\bar{t}}{5\,{\rm GHz}\times 100\,{\rm d}}\right)^{34/15}.$

This limit is obtained by requiring that $\nu_{\rm pk}=\nu_{j}$, where
$\nu_{j}$ is the frequency below which the emissivity is dominated by thermal
electrons (Margalit & Quataert, 2021). The dependence on $\epsilon_{e}$ here
relies on the approximate relation
$\nu_{j}/\nu_{\Theta}\propto(\epsilon_{e}/\epsilon_{T})^{-0.25}$ (Margalit &
Quataert, 2021), and the fact that $L_{\rm
th}\propto\epsilon_{e}^{4/15}(\nu_{j}/\nu_{\Theta})^{-2(p+5)/15}$.

In the limit $\left(\Gamma\beta\right)_{\rm sh}\ll 1$, Equation (4.2) implies
that the shock velocity is

$\displaystyle\left(\Gamma\beta\right)_{\rm sh}$ $\displaystyle\approx
0.4\left(\frac{\epsilon_{T}}{0.4}\right)^{-1/4}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm Hz}^{-1}}\right)^{1/4}$
$\displaystyle\times\left(\frac{\nu_{\rm pk}\bar{t}}{5\,{\rm GHz}\times
100\,{\rm d}}\right)^{-1/2};~{}~{}L_{\nu_{\rm pk}}>L_{\rm th}$ (9)

when thermal electrons dominate, an approximation that is typically accurate
to within $\lesssim$15% for $\left(\Gamma\beta\right)_{\rm sh}\leq 1$. Note
that, in contrast with the power-law electron case, here the shock velocity
does not depend on $\epsilon_{B}$.

The mass-loss rate can similarly be expressed as a function of the peak
luminosity and $\nu_{\rm pk}t$. In general, finding $\dot{M}/v_{\rm w}$ when
thermal electrons dominate requires numerically solving a transcendental
equation. However in the limit $\left(\Gamma\beta\right)_{\rm sh}\ll 1$, the
following analytic approximation holds to within a factor of a few,

$\displaystyle\frac{\dot{M}}{v_{\rm w}}$ $\displaystyle\sim 4\times
10^{-5}\frac{M_{\odot}\,{\rm yr}^{-1}}{10^{3}\,{\rm
km\,s}^{-1}}\,\epsilon_{B,-1}^{-2/3}\left(\frac{\epsilon_{T}}{0.4}\right)^{-11/12}$
$\displaystyle\times\left(\frac{f}{3/16}\right)^{-1/3}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm Hz}^{-1}}\right)^{-3/4}$
$\displaystyle\times\left(\frac{\nu_{\rm pk}\bar{t}}{5\,{\rm GHz}\times
100\,{\rm d}}\right)^{19/6};~{}~{}L_{\nu_{\rm pk}}>L_{\rm th}.$ (10)

The above expression makes use of the approximate relation $\nu_{\rm
pk}/\nu_{\Theta}\propto\tau_{\Theta}^{0.2}$ from Margalit & Quataert (2021),
where $\tau_{\Theta}$ is the SSA optical depth at frequency $\nu_{\Theta}$.

We can additionally use this approximate relation along with Equation (22) of
Margalit & Quataert (2021). to express the spectral slope of the SED at
$\nu\gg\nu_{\rm pk}$ as

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\alpha(\nu)\equiv\left(\frac{d\ln
L_{\nu}}{d\ln\nu}\right)\approx\frac{5}{6}-\tilde{\alpha}(\nu)~{};$ (11)
$\displaystyle\tilde{\alpha}(\nu)$ $\displaystyle\sim
3.2\left(\frac{\nu}{\nu_{\rm
pk}}\right)^{1/3}\epsilon_{B,-1}^{-1/18}\left(\frac{\epsilon_{T}}{0.4}\right)^{-13/72}\left(\frac{f}{3/16}\right)^{1/18}$
$\displaystyle\times\left(\frac{L_{\nu_{\rm pk}}}{10^{29}\,{\rm
erg\,s}^{-1}\,{\rm Hz}^{-1}}\right)^{-5/24}\left(\frac{\nu_{\rm
pk}\bar{t}}{5\,{\rm GHz}\times 100\,{\rm d}}\right)^{17/36}.$

In principle, this provides a closure relation that can be tested against
observed data. In practice however, this expression is only valid within a
limited range of parameters where $\nu_{\Theta},\nu_{\rm pk}\ll\nu\ll\nu_{j}$.
These conditions are only satisfied if both $1\ll 4\tilde{\alpha}(\nu)^{3}\ll
380\left(\delta/0.025\right)^{-1/4}$ and $\nu\gg\nu_{\rm pk}$. For the
fiducial parameters above, these conditions are not met (the first condition
is analogous to the requirement $L_{\rm th}\ll L_{\nu_{\rm pk}}\ll L_{\rm
crit}$; Figure 2 shows that this is rarely satisfied).

###  4.3 The Phase-Space

Figure 2: Phase-space of shock-powered synchrotron transients delineated in
terms of the frequency $\nu_{\rm pk}$ and specific luminosity $L_{\nu_{\rm
pk}}$ at which the spectral energy distribution (SED) peaks at time $t$: here,
showing the regime where synchrotron self-absorption (SSA) sets these peak
properties (such that $\nu_{\rm pk}\simeq\nu_{\rm a}$). Dashed black lines
show contours of constant shock four-velocity $\left(\Gamma\beta\right)_{\rm
sh}$, and dashed purple curves show contours along which the mass-loss
parameter $\dot{M}/v_{\rm w}$ is constant (this is related to the density
ahead of the shock through Equations 1,4). Below the solid grey curve,
$L_{\nu_{\rm pk}}<L_{\rm th}$ (Equation 8) and power-law electrons dominate
the emissivity at frequency $\nu_{\rm pk}$ (§4.1). In this case, and when
$\left(\Gamma\beta\right)_{\rm sh}\ll 1$, the results are consistent with the
original analysis of Chevalier (1998), and the SED takes the form shown in
Figure 1 panel (a). Alternatively, when $L_{\nu_{\rm pk}}\gtrsim L_{\rm th}$
then thermal post-shock electrons dominate the synchrotron emissivity at
frequency $\nu_{\rm pk}$, and must be taken into account (§4.2; Margalit &
Quataert 2021). The velocity and mass-loss contours within this region differ
from the power-law-electron results and imply lower densities and shock
velocities than would have otherwise been inferred from the standard Chevalier
(1998) relations. The radio SED within this region is also qualitatively
different and follows that shown in Figure 1 panel (b). The red curve labeled
$L_{\rm crit}$ is an upper limit on the peak luminosity that we discuss in §5
(see also Equation C7). This curve corresponds to a SSA optical depth of
$\tau_{\Theta}\sim 1$ at frequency $\nu_{\Theta}$, where the unabsorbed SED
peaks, and is satisfied for shocks with $\left(\Gamma\beta\right)_{\rm
sh}=\left(\Gamma\beta\right)_{\rm sh,crit}\sim 1$. The SED of higher velocity
shocks is optically-thin at peak, with $\nu_{\rm pk}\simeq\nu_{\Theta}$. This
situation is described by Figure 3. Here we have adopted fiducial values:
$\epsilon_{B}=0.1$, $\epsilon_{T}=0.4$, $\epsilon_{e}=0.01$, $p=3$, $f=3/16$,
and $\ell_{\rm dec}=1$.

Figure 2 shows the phase-space of shock-powered SSA transients as a function
of the frequency and specific luminosity at which the SED peaks at time $t$
(all measured in the observer frame). It accounts for relativistic effects as
well as the contribution of thermal electrons—elements that are crucially
important when considering high-velocity shocks. A plot of this kind was first
introduced by Chevalier (1998) for non-relativistic shocks (without thermal
electrons), and is widely used in analyses of radio transients. Its power
derives from the fact that it allows one to infer and compare physical
properties of different sources as a function of purely observational
parameters. In particular, the shock velocity $\left(\Gamma\beta\right)_{\rm
sh}$ can be inferred from Figure 2 using the dashed-black contours, while the
upstream density (parameterized via the mass-loss parameter $\dot{M}/v_{\rm
w}$; Equation 4) can be found using the dashed-purple contours.

At low velocities, relativistic corrections and the effect of thermal
electrons are negligible, and Figure 2 is consistent with the standard
Chevalier (1998) plot (although with minor quantitative corrections; see 4.1).
However events that fall above $L_{\nu_{\rm pk}}>L_{\rm th}$ (thick grey
curve; Equation 8) have their peak properties dominated by synchrotron
emission from the population of thermal electrons in the post-shock region.
This modifies the Chevalier (1998) relationships and is most readily seen as
kinks in the (purple) density curves, and an increased spacing of the shock-
velocity contours (black curves).

Observed peak properties of various transients are also shown in Figure 2. Red
stars denote FBOTs, circles denote different types of SNe, and squares show
jetted TDEs. A subset of these events are identified by name, adjacent to each
marker. A growing number of events are found within the mildly-relativistic
region where $L_{\nu_{\rm pk}}>L_{\rm th}$ and where our current relativistic
model must be considered. In particular, we find that using the non-
relativistic Chevalier (1998) relations over-estimates both the velocity and
density of such events (compare Equations 4.2,4.2 with 4.1,4.1), and we
discuss some implications of this fact in §6.

The solid red curve in Figure 2 shows an upper limit on the peak luminosity
that we derive in the following section. It is interesting to note that,
despite spanning many orders of magnitude in both $L_{\nu_{\rm pk}}$ and
$\nu_{\rm pk}t$, not a single event exceeds the theoretical upper limit.

##  5  Optically-Thin Peak: $\left(\Gamma\beta\right)_{\rm sh}\gg$–$\gtrsim 1$

In the previous section we treated the case where the SED peak is set by SSA.
However, for sufficiently high-velocity shocks the peak is instead determined
by the intrinsic peak of the emissivity and $\nu_{\rm pk}\simeq\nu_{\Theta}$.
This occurs once the SSA optical depth at frequency $\nu_{\Theta}$ drops below
unity such that $\nu_{\rm a}<\nu_{\Theta}$ and $L_{\nu_{\rm
a}}<L_{\nu_{\Theta}}\simeq L_{\nu_{\rm pk}}$. The characteristic SED in this
case is illustrated in Figure 1 panel (c). In this scenario the peak
luminosity is described by optically thin emission from thermal electrons
whose characteristic Lorentz factor is $\gamma_{e}\sim 2\Theta$, and can be
written from Equation (2) as

$L_{\nu_{\rm pk}}\overset{\left(\Gamma\beta\right)_{\rm sh}\gg
1}{\approx}\frac{4\pi^{3}m_{e}c^{2}f}{3\epsilon_{B}\epsilon_{T}^{6}}\left(\frac{2\sqrt{3}\mu_{e}m_{e}}{\mu
m_{p}}\right)^{7}\frac{\left(\nu_{\rm
pk}\bar{t}\right)^{3}}{\left(\Gamma\beta\right)_{\rm sh}^{4}}.$ (12)

This equation is valid in the relativistic limit
$\left(\Gamma\beta\right)_{\rm sh}\gg 1$, and can be used to obtain an
explicit expression for the proper-velocity of the shock

$\displaystyle\left(\Gamma\beta\right)_{\rm sh}$
$\displaystyle\overset{\left(\Gamma\beta\right)_{\rm sh}\gg
1}{\approx}3.1\,\epsilon_{B,-1}^{-1/4}\left(\frac{\epsilon_{T}}{0.4}\right)^{-3/2}\left(\frac{f}{3/16}\right)^{1/4}$
(13) $\displaystyle\times\left(\frac{L_{\nu_{\rm pk}}}{10^{29}\,{\rm
erg\,s}^{-1}\,{\rm Hz}^{-1}}\right)^{-1/4}\left(\frac{\nu_{\rm
pk}\bar{t}}{5\,{\rm GHz}\times 100\,{\rm d}}\right)^{3/4}$

in the optically-thin regime. Using the fact that $\nu_{\rm
pk}\simeq\nu_{\Theta}$ and the definition of $\nu_{\Theta}$ allows us to
similarly express the mass-loss parameter in this regime as

$\displaystyle\frac{\dot{M}}{v_{\rm w}}$
$\displaystyle\overset{\left(\Gamma\beta\right)_{\rm sh}\gg
1}{\approx}1.2\times 10^{-7}\frac{M_{\odot}\,{\rm yr}^{-1}}{10^{3}\,{\rm
km\,s}^{-1}}\,\left(\frac{\epsilon_{T}}{0.4}\right)^{2}\left(\frac{f}{3/16}\right)^{-1}$
$\displaystyle\times\left(\frac{L_{\nu_{\rm pk}}}{10^{29}\,{\rm
erg\,s}^{-1}\,{\rm Hz}^{-1}}\right)\left(\frac{\nu_{\rm pk}\bar{t}}{5\,{\rm
GHz}\times 100\,{\rm d}}\right)^{-1}.$ (14)

These expressions are relevant for ultra-relativistic shocks such as in GRBs.
Emission by thermal electrons in GRBs has been investigated by several studies
in recent years (Giannios & Spitkovsky, 2009; Warren et al., 2017; Ressler &
Laskar, 2017; Warren et al., 2018; Samuelsson et al., 2020; Warren et al.,
2022). Our present results add to this growing literature by providing
analytic expressions that allow one to infer the shock properties from the
(thermal-electron-dominated) SED peak, if a clear peak in the SED is detected.
This regime may also be relevant in other astrophysical contexts as well. For
example the ‘synchrotron-maser’ model for fast radio bursts (FRBs) invokes
magnetized ultra-relativistic shocks as a mechanism for powering FRBs. A
prediction of this model is that FRBs will be accompanied by high-energy
synchrotron emission from thermal electrons behind the shock (Lyubarsky, 2014;
Beloborodov, 2017; Metzger et al., 2019; Beloborodov, 2020; Margalit et al.,
2020b, a). Our present analysis would be suitable for describing such
emission, albeit only if emitting electrons are slow-cooling (in the FRB
context it is more likely that emission is instead fast-cooling; e.g., Metzger
et al. 2019). To date, only a single FRB has an observed high-frequency
counterpart that may be attributed to this model (the Galactic FRB 20200428;
Margalit et al. 2020a). However future Galactic or local-Universe FRBs may
increase the number of such detections.

Finally, we note the strong sensitivity on the composition of the ambient
medium in the ultra-relativistic regime, namely $L_{\nu_{\rm
pk}}\propto(\mu_{e}/\mu)^{7}$. For our assumed solar composition this amounts
to a factor of $(\mu_{e}/\mu)^{7}\approx 90$. Throughout much of the
literature the composition is assumed to be pure H and the mean molecular
weights are implicitly taken to be $\mu=\mu_{e}=1$. Equation (12) demonstrates
that, at least for peak emission in the ultra-relativistic limit, even a
modest change to this assumption can cause a dramatic change to the luminosity
(though note that this is degenerate with $\epsilon_{T}$, which may itself
depend on composition).

###  5.1 A Limit on $L_{\nu_{\rm pk}}$

An important consequence arises from Equations (4.2,12): while $L_{\nu_{\rm
pk}}$ increases as a function of $\left(\Gamma\beta\right)_{\rm sh}$ when the
shock is sub-relativistic (Equation 4.2), it decreases with velocity when the
shock is ultra-relativistic (Equation 12). This implies that, for fixed
$\nu_{\rm pk}t$, there is a maximum attainable peak luminosity such that
$L_{\nu_{\rm pk}}\leq L_{\rm crit}$. From the above argument it is also clear
that this maximum is realized when $\left(\Gamma\beta\right)_{\rm sh}\sim 1$.
More precisely, there is a critical shock velocity
$\left(\Gamma\beta\right)_{\rm sh,crit}$ at which $L_{\nu_{\rm pk}}=L_{\rm
crit}$. Shocks with $\left(\Gamma\beta\right)_{\rm
sh}>\left(\Gamma\beta\right)_{\rm sh,crit}$ are characterized by an optically-
thin peak (case C in Figure 1), while lower velocity shocks are SSA at peak
(cases A and B in Figure 1). Expressions for $\left(\Gamma\beta\right)_{\rm
sh,crit}$ are derived in Appendix C (see Equations C4,C5).

A crude estimate of the maximum peak luminosity $L_{\rm crit}$ that is
acceptable for the fiducial parameters considered here is

$\displaystyle L_{\rm crit}$ $\displaystyle\sim 10^{31}\,{\rm
erg\,s}^{-1}\,{\rm
Hz}^{-1}\,\epsilon_{B,-1}^{-4/15}\left(\frac{\epsilon_{T}}{0.4}\right)^{-13/15}$
$\displaystyle\times\left(\frac{f}{3/16}\right)^{4/15}\left(\frac{\nu_{\rm
pk}\bar{t}}{5\,{\rm GHz}\times 100\,{\rm d}}\right)^{34/15},$ (15)

however this expression can be wildly inaccurate if $\epsilon_{B}\ll 0.1$,
$\epsilon_{T}\ll 0.4$, or for extreme ranges of $\nu_{\rm pk}\bar{t}$ (even
for our fiducial parameters, it is only accurate to within a factor of a few).
In Appendix C and Equations (C6,C7) we derive more accurate expressions for
$L_{\rm crit}$ that should generally be used instead of Equation (5.1).

Figure 3:  Same as Figure 2, but for the case where the SED peak is governed
by optically-thin emission rather than SSA (that is, when $\nu_{\rm
a}<\nu_{\Theta}\simeq\nu_{\rm pk}$; see Figure 1 panel c). This regime
corresponds to $(\Gamma\beta)_{\rm sh}>(\Gamma\beta)_{\rm sh,crit}\sim 1$ and
is relevant for relativistic shocks. See §5 for further details.

Figure 3 shows the parameter-space of relativistic shocks, where the SED peak
is set by optically-thin synchrotron emission from thermal electrons. Similar
to Figure (2), black dashed curves show contours of constant shock four-
velocity while purple dashed curves show contours of constant mass-loss
parameter. In this optically-thin regime the shock velocity increases as one
moves rightwards on the diagram (towards large $\nu_{\rm pk}t$). This is the
direct opposite of the situation in the SSA (sub-relativistic) regime shown in
Figure 2. The limiting peak luminosity $L_{\rm crit}$ can be understood by
comparing these two figures. Consider a non-relativistic shock with
$(\Gamma\beta)_{\rm sh}\ll 1$, as described by Figure 2. Now consider the
change in $\nu_{\rm pk}$ and $L_{\nu_{\rm pk}}$ as one increases the shock
velocity while keeping the mass-loss rate $\dot{M}/v_{\rm w}$ fixed. This is
akin to moving upwards along a dashed-purple contour—towards higher
$L_{\nu_{\rm pk}}$. Eventually, the peak luminosity hits the critical $L_{\rm
crit}$ curve along which $\nu_{\rm pk}\approx\nu_{\rm a}\approx\nu_{\Theta}$.
As one increases the shock velocity further, $\nu_{\rm a}<\nu_{\rm
pk}\approx\nu_{\Theta}$ and the SED peak becomes governed by optically-thin
emission. This situation is described by Figure 3. As can be seen, curves of
constant mass-loss-rate undergo an inflection with respect to Figure 2 such
that—following the same mass-loss contour and increasing the velocity now
causes one to move rightwards in the diagram, and away from the limiting
$L_{\rm crit}$ curve. The limit $L_{\rm crit}$ can therefore be understood as
a consequence of the fact that, in the optically-thin regime, the peak
frequency $\nu_{\rm pk}$ increases with shock velocity much more rapidly than
the peak luminosity does.

We conclude this section with an important note about the limiting luminosity.
The luminosity $L_{\rm crit}$ is defined as the maximum peak luminosity
attainable given observed values of $\nu_{\rm pk}t$. That is—$L_{\rm crit}$ is
in general an upper limit on the specific luminosity at frequency $\nu_{\rm
pk}$, where the SED peaks. When $\nu_{\Theta}<\nu_{\rm a}\simeq\nu_{\rm pk}$
and SSA governs the peak flux (typically for sub-relativistic velocities) then
$L_{\rm crit}$ is also a hard limit on the specific luminosity at any
frequency (not just at $\nu_{\rm pk}$). This is because the SED at low
frequencies $\nu<\nu_{\rm pk}\simeq\nu_{\rm a}$ is steep
($L_{\nu}\propto\nu^{2}$ or $\nu^{5/2}$ compared with $L_{\rm
crit}\propto\nu^{34/15}$). However when $\nu_{\rm
a}<\nu_{\Theta}\simeq\nu_{\rm pk}$ (optically-thin peak) then $L_{\rm crit}$
is an upper limit only on the luminosity at the peak of the SED (at frequency
$\nu_{\rm pk})$. The specific luminosity at low frequencies $\nu\ll\nu_{\rm
pk}$ can exceed $L_{\rm crit}$ because the SED is relatively shallow between
$\nu_{\rm pk}$ and the SSA frequency $\nu_{\rm a}$ ($L_{\nu}\propto\nu^{1/3}$;
shallower than $L_{\rm crit}\propto\nu^{34/15}$).

##  6 Disucssion and Implications

A clear prediction of our model is that the radio SED of trans-relativistic
and ultra-relativistic transients, where thermal electrons are important,
should appear qualitatively different than the SED of non-relativistic shocks
(Figure 1). Testing this hypothesis requires contemporaneous broadband
observations with good frequency coverage. Ho et al. (2022) showed that the
SEDs of the FBOTs AT2020xnd, CSS161010, and AT2018cow could be well-fit by a
thermal electron model (e.g., see their Figure 11). Particularly noteworthy in
this context is CSS161010 (Coppejans et al., 2020). It has one of the highest
$L_{\nu_{\rm pk}}/L_{\rm th}$ ratios of any transient plotted in Figure 2,
which implies that it should be deep within the thermal-electron-dominated
regime. This event is also particularly well sampled with multi-frequency
observations. As shown in Ho et al. (2022), CSS161010 is remarkably well-fit
by the thermal electron model, exhibiting the hallmark $L_{\nu}\propto\nu^{2}$
SSA slope expected in this regime (at least within a spherically-symmetric
one-zone model) along with an apparently steepening optically-thin slope that
is characteristic of the non-power-law electron distribution.

To further test this model, we reinterpret data for the relativistic SN1998bw
(Kulkarni et al., 1998) using the model described in this paper. SN1998bw has
one of the highest $L_{\nu_{\rm pk}}/L_{\rm th}$ ratios of any transient in
our sample (see Figure 2), and is therefore a compelling test case for the
thermal electron scenario. Figure 4 shows multi-epoch fits to the radio SEDs
of SN1998bw over the first $40$ days of its evolution. The inset shows the
inferred shock velocity as a function of time, which is $(\Gamma\beta)_{\rm
sh}\approx 1$ and decreasing with time, and the mass-loss rate is
$\dot{M}/v_{\rm w}\sim 5\times 10^{-7}\,M_{\odot}\,{\rm yr}^{-1}/1000\,{\rm
km\,s}^{-1}$. In this regime, thermal electrons dominate the observed
emission. One can see that the thermal electron spectra provide an excellent
fit to the data, further supporting this model. In fact, in reviewing the
literature on SN1998bw we became aware that such a model had already been
proposed for this event due to the inconsistency of the observed SED with
standard power-law model predictions (Waxman & Loeb, 1999). Our renewed
analysis lends further support to this idea. Note that Waxman & Loeb (1999)
adopted a mono-energetic electron distribution function for simplicity, while
our treatment accounts for emission arising from a full thermal (Maxwell-
Jüttner) distribution. The thermal distribution produces an SED that decays
significantly less steeply at high frequencies (as
$\propto\nu^{5/6}e^{-\frac{3}{2}\left(2\nu/\nu_{\Theta}\right)^{1/3}}$,
instead of $\propto\nu^{1/2}e^{-\nu/\nu_{\Theta}}$) and is therefore more
broadband than that of a mono-energetic electron distribution.

Figure 4: Multi-epoch fit to the radio SEDs of SN1998bw over the first 40 days
(data from Kulkarni et al. 1998), assuming fiducial microphysical parameters
(e.g., as in Figure 2). The best fit parameters imply that thermal electrons
dominate emission in the observed frequency bands. It can be seen that the
thermal synchrotron model provides excellent fits to the SEDs of 1998bw. The
inset shows the inferred shock velocity that results from each epoch’s SED
fit. This is in line with the shock velocity shown in Figure 2.

We have also briefly tested this model against the SEDs of other high-velocity
events shown in Figure 2. The jetted TDEs Swift J164449.3+573451 (J1644; Bloom
et al. 2011; Zauderer et al. 2011; Berger et al. 2012; Eftekhari et al. 2018,
and references therein) and AT2022cmc (Andreoni et al., 2022; Pasham et al.,
2023) can be well fit by a thermal-electron SED, and we will explore the
implications of this in future work. We have also tested this model against
data for several “relativistic” SNe, and find that in many cases the thermal-
electron model can provide adequate fits. In a few cases, however, it seems
that the thermal SED does not describe the data as well as a power-law
electron model (one example is SN2009bb; Soderberg et al. 2010). One
possibility is that the microphysical parameters in such events differ from
our fiducial values (e.g., $\epsilon_{B}\ll 0.1$), such that $L_{\rm
th}\gtrsim L_{\nu_{\rm pk}}$. For SN2009bb this would require that
$\epsilon_{e,-2}^{-8/15}\epsilon_{B,-1}^{4/15}(\epsilon_{T}/0.4)^{7/5}\ell_{\rm
dec}^{-34/15}\lesssim 0.03$. In general, the radio data for most SNe is
sparse, so it is difficult to characterize the SED with great detail. As a
byproduct, it is also common in many cases that the physical properties of the
shock are determined by the peak of the light-curve in a given band, rather
than the peak of the SED. Although the two may coincide in certain cases, this
need not be the case more generally. If the density profile does not follow a
simple power-law, or if there is energy injection into the shock, the light-
curve can have peculiar and/or non-monotonic behavior, and it’s peak need not
coincide with the SSA frequency passing through the band. We warn against
using events without a peak frequency that is well-determined from the SED in
such analyses, and exclude events of this kind from our sample.

###  6.1 Energy

A primary impetus for this work is the increasing rate at which trans-
relativistic transients are discovered. These transients are typically modeled
using the canonical non-relativistic framework of Chevalier (1998). However,
as shown in this work, both relativistic corrections and the contribution of
thermal electrons are important when modeling trans-relativistic explosions. A
major implication of our present work is that properly taking these effects
into account changes the inferred physical parameters for these events.

In particular, both thermal electrons and relativistic corrections act to
decrease the inferred shock velocity and mass-loss rate. A direct consequence
is that the associated shock energy is reduced when including these effects.
This can be seen by expressing the post-shock energy
$U=VB^{2}/8\pi\epsilon_{B}$ as a function of observed peak properties. Taking
$V=4\pi fR^{3}/3$ and $B\approx 3\epsilon_{B}^{1/2}(\dot{M}/v_{\rm
w})^{1/2}/2\bar{t}$ (Equation A2) we find using Equations (4.1,4.1,4.2,4.2)
that

$\displaystyle U$ $\displaystyle\approx 1.5\times 10^{50}\,{\rm
erg}\,\epsilon_{B,-1}^{-8/19}\epsilon_{e,-2}^{-11/19}\left(\frac{f}{3/16}\right)^{8/19}$
(16) $\displaystyle\times\left(\frac{L_{\nu_{\rm pk}}}{10^{29}\,{\rm
erg\,s}^{-1}\,{\rm Hz}^{-1}}\right)^{23/19}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{-1};~{}~{}L_{\nu_{\rm pk}}<L_{\rm th}$

when power-law electrons dominate, while

$\displaystyle U$ $\displaystyle\sim 3\times 10^{49}\,{\rm
erg}\,\epsilon_{B,-1}^{-2/3}\left(\frac{\epsilon_{T}}{0.4}\right)^{-5/3}\left(\frac{f}{3/16}\right)^{2/3}$
(17) $\displaystyle\times\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{5/3}\left(\frac{\bar{t}}{100\,{\rm
d}}\right)^{8/3};~{}~{}L_{\nu_{\rm pk}}>L_{\rm th}$

when thermal electrons are responsible for the observed emission at frequency
$\nu_{\rm pk}$. In the regime where it is appropriate ($L_{\nu_{\rm
pk}}>L_{\rm th}$), Equation (17) gives a lower value of the post-shock energy
than would have otherwise been inferred using the standard Chevalier (1998)
model (Equation 16). In particular, the ratio of these inferred energies can
be expressed as $U[\text{thermal }e^{-}]/U[\text{power-law
}e^{-}]\sim\left(L_{\nu_{\rm pk}}/L_{\rm th}\right)^{-23/19}$, with only very
weak dependence on other parameters.444 $U[\text{thermal
}e^{-}]/U[\text{power-law }e^{-}]\propto\left(\epsilon_{B}/f\nu_{\rm
pk}\bar{t}\right)^{0.08}\epsilon_{T}^{0.03}\epsilon_{e}^{-0.07}$ This
illustrates how the inferred energy drops when one accounts for thermal
electrons in the regime $L_{\nu_{\rm pk}}\gg L_{\rm th}$ where they are
important.

Figure 5: Post-shock energy versus shock velocity inferred from peak
properties of synchrotron-powered transients. Colored symbols show results
derived using the model described in this paper (different symbols and colors
correspond to different classes of events). Connected grey points show the
corresponding properties that would have been inferred using the commonly-
adopted Chevalier (1998) model, which does not incorporate relativistic
corrections or thermal electrons. These inferences overestimate the shock
velocity and total energy, an effect that becomes pronounced for high-velocity
explosions where $L_{\nu_{\rm pk}}>L_{\rm th}$ (to the right of the leftmost
vertical black curve; see also Figure 2 and Equation 8). The fact that the
energetic constraints on some events can be reduced by more than an order of
magnitude underpins the importance of considering relativistic effects when
studying trans-relativistic transients. As one moves from low to high
velocities the resulting SED changes in qualitative ways that provide a
consistency check on the increasing importance of thermal electrons at high
shock speeds. The regimes (a), (b), and (c), denoted here correspond to the
three different SEDs illustrated in Figure 1. The same fiducial microphysical
and geometric parameters as in Figure 2 are adopted here.

Figure 5 illustrates this point. Colored symbols show the inferred energy and
shock velocity for different transients as implied by our model. Red stars
show the currently known sample of FBOTs, while differently colored open
circles show various SNe. Each colored point is connected to a corresponding
smaller grey symbol that shows the properties that would have been inferred
for the same event if relativistic corrections and thermal electrons would
have been neglected (in some cases the points overlap such that the grey
symbols are hidden behind the colored ones). Events that fall to the left of
this plot (lower velocities) satisfy $L_{\nu_{\rm pk}}<L_{\rm th}$ such that
their observed emission is dominated by power-law electrons and their SED will
be similar to that shown in Figure 1 panel (a). Conversely, events that fall
to the right of the leftmost vertical dashed-black curve have their peak
emission dominated by thermal electrons ($L_{\nu_{\rm pk}}>L_{\rm th}$).555
The exact velocity at which $L_{\nu_{\rm pk}}=L_{\rm th}$ depends on $\nu_{\rm
pk}t$ (albeit weakly; Equation 8) and will therefore vary for different
events, see Figure 2. In Figure 5 we plot this transition for a representative
$\nu_{\rm pk}\bar{t}=5\,{\rm GHz}\times 100\,{\rm day}$.  Figure 5 shows that
while the non-relativistic power-law-electron framework of Chevalier (1998)
works well for events where the shock velocity is $\lesssim 0.2c$, it is
inaccurate for trans-relativistic events with higher velocity. In particular,
the shock velocity and energy are overestimated by this model (compare grey
points with their associated colored symbols), the latter by as much as two
orders of magnitude. This discrepancy becomes more acute as one moves towards
higher shock velocities and further away from the $L_{\nu_{\rm pk}}=L_{\rm
th}$ line. This highlights the importance of considering relativistic
corrections and thermal electrons when modeling such events.

We note that the discussion above applies to the physically-motivated scenario
where $\epsilon_{e}\ll\epsilon_{T}\lesssim 1$ (that is, $\delta\ll 1$). A
minimum on the explosion energy is often estimated by adopting ‘equipartition’
microphysical parameters instead,
$\epsilon_{e}\sim\epsilon_{B}\sim\mathcal{O}(1)$. One can immediately see from
Equation (16) that such values of $\epsilon_{e}$ and $\epsilon_{B}$ indeed
minimize the inferred energy. In this scenario $\epsilon_{e}\sim\epsilon_{T}$
(i.e., $\delta\sim 1$), so that $L_{\rm th}$ is very large and there is little
to no phase space in which thermal electrons dominate peak emission. That is,
if power-law electrons are assumed to be in equipartition with magnetic fields
and thermal electrons + ions, then $L_{\rm th}\sim L_{\rm crit}$ such that
$L_{\nu_{\rm pk}}$ is almost always smaller than $L_{\rm th}$, and therefore
Equation (16)—and not (17)—is appropriate for estimating the energy. In other
words, in the equipartition scenario thermal electrons are almost never
relevant, even if they are present. This implies that the standard minimum-
energy estimates, which neglect thermal electrons entirely, are still valid
lower-limits on $U$. The reduction in inferred energies when including thermal
electrons that is discussed in previous paragraphs applies to the physically
motivated case where $\epsilon_{e}\ll 1$ rather than equipartition. This would
alleviate energetic constraints for more reasonable expected values of
$\epsilon_{e}$ and $\epsilon_{B}$. We also note that an equipartition argument
holds true also in the case where thermal electrons are included and dominant
over power-law electrons ($\epsilon_{e}\ll\epsilon_{T}$, $L_{\nu_{\rm
pk}}>L_{\rm th}$). This can be derived from Equation (17), which again shows
that $U$ is minimized when $\epsilon_{T}\sim\epsilon_{B}\sim\mathcal{O}(1)$.

Although it is reasonable to expect that $\epsilon_{e}\ll\epsilon_{T}$, the
efficiency of electron-ion equilibration is not fully understood and therefore
the extent to which $\epsilon_{T}\sim 1$ is somewhat uncertain. Particle-in-
cell simulations tend to support the idea that plasma processes can mediate
efficient energy transfer between ions and electrons (Coulomb collisions are
typically ineffective at this), however this may not necessarily lead to full
equilibration (where the temperature of ions and electrons are the same), and
the details are still debated (e.g., Sironi et al., 2013; Tran & Sironi, 2020;
Raymond et al., 2023; Groselj et al., 2024).666Note that a strict lower limit
$\epsilon_{T}>\mu_{e}m_{e}/\mu m_{p}\simeq 10^{-3}$ arises from considering
the case where electrons and ions do not exchange energy at all.

Finally, we discuss the results of Barniol Duran et al. (2013) in light of our
present work. Barniol Duran et al. (2013) generalized Chevalier (1998) and
similar equipartition methods (e.g., Scott & Readhead, 1977) to the case of
ultra-relativistic sources. In many ways this is similar to the objective of
our present work. However, there are significant differences between the two
analyses. First, Barniol Duran et al. (2013) assume a purely power-law
electron distribution and do not incorporate the effects of thermal electrons.
As we have shown here, thermal electrons are critically important in ultra-
relativistic shocks as they—and not the power-law electrons—determine the
properties at the peak of the SED (see Figure 1 panel c). Second, Barniol
Duran et al. (2013) express their results in terms of a dimensionless
parameter $\eta\equiv\min(\nu_{m}/\nu_{\rm a},1)$ that is not specified in
that work. Here $\nu_{m}$ is the characteristic (minimum) frequency of power-
law electrons, and is analogous to $\nu_{\Theta}$ in the thermal electron
scenario. Using the results of Barniol Duran et al. (2013) with $\eta$ treated
as a constant $\sim\mathcal{O}(1)$ is therefore equivalent to the assumption
that the peak of the SED occurs at $\nu_{\rm pk}\sim\nu_{\rm a}$. We have
shown here instead that emission at the peak of the SED is typically
optically-thin for ultra-relativistic shocks. To the extent that this
conclusion applies also in the purely power-law electron scenario, then one
would generally expect $\nu_{\rm a}\ll\nu_{\rm pk}\simeq\nu_{\rm m}$, or
$\eta\gg 1$. The fact that $\eta\nsim 1$, and that the value of $\eta$ is not
explicitly determined (it will, among other things, depend on $\Gamma$)
therefore makes it difficult to compare the Barniol Duran et al. (2013)
expressions with our present results.

##  7 Summary

In this paper we have extended the seminal work of Chevalier (1998), which
considered SSA in non-relativistic shocks, to apply to shocks of arbitrary
velocity. For the first time, we account for both relativistic corrections and
the impact of post-shock thermal electrons with closed-form analytic
expressions. These effects become significant when the shock proper-velocity
exceeds $(\Gamma\beta)_{\rm sh}\gtrsim 0.2$ (Margalit & Quataert, 2021), and
are particularly important when considering trans-relativistic explosions. We
also treat the case where the SSA frequency falls below the intrinsic peak of
the emissivity, and the observed peak emission is governed by optically-thin
emission from thermal electrons. This regime occurs when $(\Gamma\beta)_{\rm
sh}>(\Gamma\beta)_{\rm sh,crit}\sim 1$ (Equations C4,C5), and is relevant for
relativistic shocks.

Our main findings can be summarized as follows:

  * •

The physical properties of the shock can be determined from measurements of
the peak frequency $\nu_{\rm pk}$ and spectral luminosity $L_{\nu_{\rm pk}}$
at which the synchrotron SED peaks. Technically, this relation is double-
valued: observed $\left\\{\nu_{\rm pk},L_{\nu_{\rm pk}}\right\\}$ can be
interpreted as either a SSA shock with $(\Gamma\beta)_{\rm
sh}<(\Gamma\beta)_{\rm sh,crit}$, or a higher-velocity shock with
$(\Gamma\beta)_{\rm sh}>(\Gamma\beta)_{\rm sh,crit}$ characterized by an
optically-thin peak. In practice however, it should be easy to distinguish
between the two scenarios given the different SED morphology expected in
either case (Figure 1).

  * •

For shocks governed by SSA: thermal electrons dominate the emissivity and
absorption at frequency $\nu_{\rm pk}$ when $L_{\nu_{\rm pk}}>L_{\rm th}$
(Equation 8; solid grey curve in Figure 2). For luminous events that fall
within this regime, the standard Chevalier (1998) relations (4.1,4.1) are no
longer applicable. Instead, the physical properties of the shock should be
estimated from Equations (4.2,4.2; see also Appendix A) or using the numerical
procedure outlined in Appendix B (implemented in code available at
https://github.com/bmargalit/thermal-synchrotron-v2).

  * •

Accounting for thermal electrons in the SSA regime reduces the velocities and
mass-loss rates with respect to what would have otherwise been inferred using
the standard Chevalier (1998) formalism. This leads to a significant reduction
in the implied energy of trans-relativistic explosions, which may have
previously been overestimated by as much as two orders of magnitude (Figure 5;
Equations 16,17).

  * •

A prediction of our model is that the radio SED will change qualitatively as a
function of shock velocity. With increasing shock velocity, emission at
$\nu_{\rm pk}$ transitions from being synchrotron self-absorbed and governed
by power-law electrons, to a SSA peak dominated by thermal electrons, and
eventually to an optically-thin peak (Figure 1). This prediction can be tested
with broadband observations of high-velocity events. As an example, we have
modeled observations of SN1998bw and shown that the thermal electron model
provides an excellent fit to the SEDs at $t\leq 40\,{\rm days}$ (Figure 4).
The SEDs of several high-velocity FBOTs have also been shown to be well-
described by the thermal electron model (Ho et al. 2022; their Figure 11),
lending further support to this scenario.

  * •

There is a maximum allowed peak luminosity $L_{\nu_{\rm pk}}$, such that
$L_{\nu_{\rm pk}}\leq L_{\rm crit}$ (Equation C7; see also Equation 5.1 and
Figure 2). This is a consequence of the transition between optically-thin and
optically-thick emission at the peak of the SED, and is a generic feature of
synchrotron shock models.777 The expression for $L_{\rm crit}$ is determined
by properties of the thermal electron population (§5; Appendix C). However, a
qualitatively similar upper limit would apply also in the (unphysical) case
where thermal electrons are neglected. This gives $L_{\rm crit}\sim 2\times
10^{33}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}\,\epsilon_{e,-2}^{-2}\epsilon_{B,-1}^{-23/55}(\nu_{\rm
pk}\bar{t}/5{\rm GHz}\times 100{\rm d})^{133/55}$.  While the specific
luminosity at low frequencies $\nu\ll\nu_{\rm pk}$ can exceed this limit (that
is, $L_{\nu}>L_{\rm crit}$ is possible) when peak emission is optically-thin,
the peak luminosity should always satisfy $L_{\nu_{\rm pk}}\leq L_{\rm crit}$.
It is interesting to note from Figure 2 that no observed event falls above
this limit (red curve), despite clear observational biases towards detecting
more luminous events. We argue that the limit $L_{\rm crit}$ that we have
derived here provides a natural explanation to this fact.

We have worked here under the assumption that electrons that dominate emission
at frequency $\nu_{\rm pk}$ are slow cooling (that is, they do not lose
appreciable energy throughout the duration of the observed emission). In
Appendix D we derive the necessary conditions required for this assumption to
be valid. The result shows that the slow-cooling assumption is typically valid
in the radio band, but does not always hold at higher frequencies. In cases
where electrons are fast cooling, our present results cannot be used (the
shock velocity inferred from the expressions in this paper would underestimate
the true velocity). The upper limit $L_{\nu_{\rm pk}}<L_{\rm crit}$ is likely
to remain valid however, since the luminosity is lowered when electrons are
fast-cooling.

We also neglect external propagation effects such as free-free absorption. If
free-free absorption is important, it may modify the SED peak, in which case
our formalism cannot be used. This is likely only important for events where
the inferred ambient densities are very large. Such events generally fall
towards the bottom-right corner of Figure 2. Explosions with large
velocities—which are of primary interest here—are less-likely to be impacted
by free-free absorption.

For the sake of clarity, we have also omitted the dependence on cosmological
redshift $z$ from our expressions, in essence assuming that $z\approx 0$.
Incorporating cosmological corrections is straightforward however. One need
only replace $t\to(1+z)t$ and $\nu_{\rm pk}\to\nu_{\rm pk}/(1+z)$ in any
equation, and recall that $L_{\nu_{\rm pk}}=4\pi D_{\rm lum}^{2}F_{\nu_{\rm
pk}}/(1+z)$ (e.g., Hogg, 1999), so that

$L_{\nu_{\rm pk}}\approx\frac{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}{1+z}\left(\frac{F_{\nu_{\rm pk}}}{\rm mJy}\right)\left(\frac{D_{\rm
lum}}{300\,{\rm Mpc}}\right)^{2}.$ (18)

Here $D_{\rm lum}$ is the luminosity distance and $F_{\nu_{\rm pk}}$ the
measured peak flux density. Conveniently, many parameters are only dependent
on time and frequency through the combination $\nu_{\rm pk}t$ which is
invariant, and can therefore be calculated in any frame of reference.

Finally, we note that good spectral coverage is critical to properly
diagnosing shocks within this framework. Our method relies on a reliable
estimate of the frequency and luminosity at the peak of the SED, which can
only be determined accurately with a well-sampled SED. In lieu of a well-
sampled SED, the Chevalier (1998) methodology has often been used by
substituting $\nu_{\rm pk}\leftrightarrow\nu$ and $t\leftrightarrow t_{\rm
pk}$. That is, the method is sometimes used with an estimate of the time
$t_{\rm pk}$ at which a single-band light-curve at some frequency $\nu$ peaks
(as opposed to using the frequency $\nu_{\rm pk}$ at which the SED peaks).
This approach is only reasonable if the peak of the light-curve is set by the
time at which $\nu_{\rm pk}$ passes through the observing band. This may
indeed be true in certain cases, but is not generally guaranteed. The light-
curve is influenced by the uncertain ambient density profile and shock
expansion/deceleration history, neither of which need be well-behaved. Using
the SED peak rather than the light-curve peak sidesteps these uncertainties,
and is a more robust way of using the framework. This is true both in our
present work, and for the original analysis of Chevalier (1998).

The code developed in this analysis is publicly available at
https://github.com/bmargalit/thermal-synchrotron-v2, and may be particularly
useful when the analytic approximations derived in the main text are not
sufficiently accurate. This code can be imported as a python module and used
to infer physical properties of observed transients, for example by fitting
model-generated flux densities to real data. It may also be useful for
generating plots similar to Figures (2,5).

B.M. thanks Nayana A. J. for sharing a compilation of $\nu_{\rm pk}$ and
$L_{\nu_{\rm pk}}$ for SNe which provided a basis for the sample of events
shown in Figures 2,5. This research benefited from interactions that were
funded by the Gordon and Betty Moore Foundation through Grant GBMF5076.
(Harris et al., 2020), scipy (Virtanen et al., 2020), matplotlib (Hunter,
2007).

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##  Appendix A Expressions for Additional Physical Variables

In the following, we use our main results to explicitly write solutions to
other physical variables that describe the shock and that are of general
interest. We begin by treating the case where SSA sets the peak of the SED,
and find analytic solutions in the approximate limit where
$\left(\Gamma\beta\right)_{\rm sh}\ll 1$. We then similarly treat the
optically-thin case which is appropriate in the ultra-relativistic limit,
$\left(\Gamma\beta\right)_{\rm sh}\gg 1$.

###  A.1 SSA Peak

When the peak of the SED is set by SSA (cases ‘a’ and ‘b’ in Figure 1) we can
use the results of §4. In the limit $\left(\Gamma\beta\right)_{\rm sh}\ll 1$,
the shock velocity is given by Equations (4.1,4.2) depending on whether power-
law or thermal electrons dominate emission at frequency $\nu_{\rm pk}$. Using
these expressions along with Equation (1), the shock velocity can be written
as

$R\approx\begin{dcases}1.1\times 10^{17}\,{\rm
cm}\,\left(\frac{\epsilon_{e}/\epsilon_{B}}{0.1}\right)^{-1/19}\left(\frac{f}{3/16}\right)^{-1/19}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}\right)^{9/19}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{-1}&;~{}L_{\nu_{\rm pk}}<L_{\rm th}\\\ 10^{17}\,{\rm
cm}\,\left(\frac{\epsilon_{T}}{0.4}\right)^{-1/4}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}\right)^{1/4}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{-1/2}\left(\frac{\bar{t}}{100\,{\rm
d}}\right)^{1/2}&;~{}L_{\nu_{\rm pk}}>L_{\rm th}\end{dcases}$ (A1)

where the top ($L_{\nu_{\rm pk}}<L_{\rm th}$) case corresponds to the power-
law-dominated regime (Chevalier, 1998), and the bottom case ($L_{\nu_{\rm
pk}}>L_{\rm th}$) corresponds to the thermal-electron-dominated scenario
(Margalit & Quataert, 2021). As discussed in the main text, $L_{\rm th}$ is
the spectral luminosity that separates these two regimes (Equation 8; see also
Figure 2)

Next we find the post-shock magnetic field, which is related to the mass-loss
rate via

$B=\frac{c}{R}\sqrt{8\epsilon_{B}(\dot{M}/v_{\rm w})\Gamma(\Gamma-1)}$ (A2)

In the limit $\left(\Gamma\beta\right)_{\rm sh}\ll 1$ this reduces to
$B\approx 3\epsilon_{B}^{1/2}(\dot{M}/v_{\rm w})^{1/2}/2\bar{t}$. Using this
result along with the mass-loss rate as given by Equations (4.1,4.2) we find
the post-shock magnetic field

$B\approx\begin{dcases}0.58\,{\rm
G}\,\left(\frac{\epsilon_{e}/\epsilon_{B}}{0.1}\right)^{-4/19}\left(\frac{f}{3/16}\right)^{-4/19}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}\right)^{-2/19}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)&;~{}L_{\nu_{\rm pk}}<L_{\rm th}\\\ 0.3\,{\rm
G}\,\epsilon_{B,-1}^{1/6}\left(\frac{\epsilon_{T}}{0.4}\right)^{-11/24}\left(\frac{f}{3/16}\right)^{-1/6}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}\right)^{-3/8}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{19/12}\left(\frac{\bar{t}}{100\,{\rm
d}}\right)^{7/12}&;~{}L_{\nu_{\rm pk}}>L_{\rm th}\end{dcases}$ (A3)

in the power-law and thermal electron cases, respectively. The upstream
density can similarly be derived using the mass-loss rate (Equations 4.1,4.2)
along with the radius (Equation A1) and Equation (4), giving

$n\approx\begin{dcases}680\,{\rm
cm}^{-3}\,\epsilon_{B,-1}^{-13/19}\epsilon_{e,-2}^{-6/19}\left(\frac{f}{3/16}\right)^{-6/19}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}\right)^{-22/19}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{4}\left(\frac{\bar{t}}{100\,{\rm d}}\right)^{2}&;~{}L_{\nu_{\rm
pk}}<L_{\rm th}\\\ 200\,{\rm
cm}^{-3}\,\epsilon_{B,-1}^{1/6}\left(\frac{\epsilon_{T}}{0.4}\right)^{-5/12}\left(\frac{f}{3/16}\right)^{-1/3}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}\right)^{-5/4}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{25/6}\left(\frac{\bar{t}}{100\,{\rm
d}}\right)^{13/6}&;~{}L_{\nu_{\rm pk}}>L_{\rm th}\end{dcases}.$ (A4)

Finally, for the sake of completeness, we repeat the post-shock energy as
given by Equations (17,16),

$U\approx\begin{dcases}1.5\times 10^{50}\,{\rm
erg}\,\epsilon_{B,-1}^{-8/19}\epsilon_{e,-2}^{-11/19}\left(\frac{f}{3/16}\right)^{8/19}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}\right)^{23/19}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{-1}&;~{}L_{\nu_{\rm pk}}<L_{\rm th}\\\ 3\times 10^{49}\,{\rm
erg}\,\epsilon_{B,-1}^{-2/3}\left(\frac{\epsilon_{T}}{0.4}\right)^{-5/3}\left(\frac{f}{3/16}\right)^{2/3}\left(\frac{\nu_{\rm
pk}}{5\,{\rm GHz}}\right)^{5/3}\left(\frac{\bar{t}}{100\,{\rm
d}}\right)^{8/3}&;~{}L_{\nu_{\rm pk}}>L_{\rm th}\end{dcases}.$ (A5)

Note that the bottom cases in Equations (A3,A4,A5) depend on the mass-loss
rate in the thermal-electron regime. The analytic expression for
$\dot{M}/v_{\rm w}$ in this case (Equation 4.2) is an approximate relationship
that is only accurate to within a factor of a few. The magnetic field,
density, and energy estimates are therefore also subject to these errors in
the thermal electron scenario. These are a consequence of the transcendental
nature of the equations that arise in the thermal electron case, which have no
closed-form analytic solution. For analytic tractability we have adopted in
the above equations an approximate solution to this transcendental equation
(in particular, the equation relating the SSA frequency $\nu_{\rm a}$ to the
optical depth at $\nu_{\Theta}$, $\tau_{\Theta}$) as described in Margalit &
Quataert (2021). The code we provide can obtain exact solutions to these
variables (under the assumptions of the model, see §2).

###  A.2 Optically-Thin Peak

In the ultra-relativistic $\left(\Gamma\beta\right)_{\rm sh}\gg 1$ limit the
peak of the SED is governed by optically-thin synchrotron emission dominated
by thermal electrons and the results of §5 are applicable. Using these results
and Equation (1) we can express the shock radius as

$R\approx 2.4\times 10^{18}\,{\rm
cm}\,\epsilon_{B,-1}^{-1/2}\left(\frac{\epsilon_{T}}{0.4}\right)^{-3}\left(\frac{f}{3/16}\right)^{1/2}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}\right)^{-1/2}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{3/2}\left(\frac{\bar{t}}{100\,{\rm d}}\right)^{5/2}.$ (A6)

The post-shock magnetic field in the ultra-relativistic limit reduces to
$B\approx 2\epsilon_{B}^{1/2}(\dot{M}/v_{\rm w})^{1/2}/(\Gamma\beta)_{\rm
sh}\bar{t}$ which, combined with Equations (13,5) gives

$B\approx 6.5\,{\rm
mG}\,\epsilon_{B,-1}^{3/4}\left(\frac{\epsilon_{T}}{0.4}\right)^{5/2}\left(\frac{f}{3/16}\right)^{-3/4}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}\right)^{3/4}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{-5/4}\left(\frac{\bar{t}}{100\,{\rm d}}\right)^{-9/4}.$ (A7)

The upstream density is then (Equations 4,5)

$n\approx 9.5\times 10^{-4}\,{\rm
cm}^{-3}\,\epsilon_{B,-1}\left(\frac{\epsilon_{T}}{0.4}\right)^{8}\left(\frac{f}{3/16}\right)^{-2}\left(\frac{L_{\nu_{\rm
pk}}}{10^{29}\,{\rm erg\,s}^{-1}\,{\rm
Hz}^{-1}}\right)^{2}\left(\frac{\nu_{\rm pk}}{5\,{\rm
GHz}}\right)^{-4}\left(\frac{\bar{t}}{100\,{\rm d}}\right)^{-6}.$ (A8)

Finally, the post-shock energy $U=fB^{2}R^{3}/6\epsilon_{B}$ can be expressed
as

$U\approx 1.9\times 10^{50}\,{\rm
erg}\,\epsilon_{B,-1}^{-1}\left(\frac{\epsilon_{T}}{0.4}\right)^{-4}\left(\frac{f}{3/16}\right)\left(\frac{\nu_{\rm
pk}}{5\,{\rm GHz}}\right)^{2}\left(\frac{\bar{t}}{100\,{\rm d}}\right)^{3},$
(A9)

which does not depend on the peak luminosity.

##  Appendix B General Solution

In the main text and the preceding Appendix we have derived analytic solutions
that are appropriate in different regimes. These solutions are, in certain
cases, only approximate, and are never exact at the transition between
different regimes (e.g., near $L_{\nu_{\rm pk}}\sim L_{\rm th}$, or
$L_{\nu_{\rm pk}}\sim L_{\rm crit}$). Here we outline a general methodology
for solving for $(\Gamma\beta)_{\rm sh}$, $\dot{M}/v_{\rm w}$ given $\nu_{\rm
pk}$, $L_{\nu_{\rm pk}}$, that is accurate for any set of parameters, but
which generally requires a numerical procedure.

Following Equation (2), the specific luminosity at frequency $\nu$ (not
necessarily the peak frequency) can be written in a general form as

$\displaystyle L_{\nu}\left[\nu~{};~{}(\Gamma\beta)_{\rm sh},\dot{M}/v_{\rm
w}\right]=4\pi^{2}\Gamma R^{2}\Delta R^{\prime}\langle
j_{\nu}^{\prime}\rangle\frac{1-e^{-\tau(\nu)}}{\tau(\nu)}=\tilde{L}\left[xI^{\prime}(x)+\frac{8\pi}{\sqrt{3}}C_{j}(p)\delta\frac{g(\Theta)}{f(\Theta)}x^{-\frac{p-1}{2}}\right]\frac{1-e^{-\tau(x)}}{\tau(x)},$
(B1)

where

$\tau(x)\equiv\tau_{\Theta}\left[\frac{I^{\prime}(x)}{x}+\frac{3^{3/2}}{\pi}C_{\alpha}(p)\delta\frac{g(\Theta)}{f(\Theta)}x^{-\frac{p+4}{2}}\right],$
(B2)

is the optical depth at frequency $\nu$, $\delta=\epsilon_{e}/\epsilon_{T}$,

$x\equiv\frac{\nu}{\nu_{\Theta}}=\frac{\sqrt{2}\pi
m_{e}c}{3e\epsilon_{B}^{1/2}}\left(\frac{\dot{M}}{v_{\rm
w}}\right)^{-1/2}\frac{\Gamma_{\rm sh}^{2}\beta_{\rm
sh}}{\Gamma^{3/2}(\Gamma-1)^{1/2}\Theta^{2}}\nu\bar{t},$ (B3)

is a normalized frequency, and

$\tilde{L}\equiv\frac{4\sqrt{2}e^{3}\mu_{e}\epsilon_{B}^{1/2}f}{\sqrt{3}\mu
m_{p}m_{e}c}\left(\frac{\dot{M}}{v_{\rm
w}}\right)^{3/2}f(\Theta)\Gamma^{3/2}(\Gamma-1)^{1/2},$ (B4)
$\tau_{\Theta}\equiv\frac{\sqrt{2}e\mu_{e}f}{3^{5/2}\mu
m_{p}c\epsilon_{B}^{1/2}}\left(\frac{\dot{M}}{v_{\rm
w}}\right)^{1/2}\frac{f(\Theta)}{\Theta^{5}}\Gamma^{-1/2}(\Gamma-1)^{-1/2}$
(B5)

are functions of the shock velocity and effective mass-loss rate. In the
above, $f(\Theta)$ and $g(\Theta)$ (not to be confused with the volume filling
factor $f$) are functions of the normalized electron temperature and are given
by Equations (5,8) of Margalit & Quataert (2021).888 Note that there was a
typo in the expression for $g(\Theta)$ presented in Equation (8) of Margalit &
Quataert (2021). The correct expression is
$g(\Theta)=\frac{(p-1)\gamma_{m}}{(p-1)(\gamma_{m}-1)-p+2}\left(\frac{\gamma_{m}}{3\Theta}\right)^{p-1}$.
The electron temperature is itself a function of the shock velocity, and is
determined by Equation (2) of Margalit & Quataert (2021) with
$\Theta_{0}=\epsilon_{T}(\mu m_{p}/\mu_{e}m_{e})(\Gamma-1)$. In the limit
$\Theta\gtrsim 1$ which is relevant for high-velocity shocks,
$f(\Theta)\approx g(\Theta)\approx 1$. The coefficients $C_{j}(p)$ and
$C_{\alpha}(p)$ are given by Equations (15,17) of Margalit & Quataert (2021).
For $p=3$, these attain the values $C_{j}=2/3$ and $C_{\alpha}\simeq 28.6$.
Finally, the function $I^{\prime}(x)$ is given by Equation (13) of Margalit &
Quataert (2021), and is derived in Mahadevan et al. (1996).

At the peak of the SED, $\left.dL_{\nu}/d\nu\right|_{\nu_{\rm pk}}=0$ (by
definition). Given observed values of $\nu_{\rm pk}$ and $L_{\nu_{\rm pk}}$,
we therefore obtain a set of two equations for two variables,
$(\Gamma\beta)_{\rm sh}$, $\dot{M}/v_{\rm w}$. They are

$L_{\nu}\left[\nu_{\rm pk}~{};~{}(\Gamma\beta)_{\rm sh},\dot{M}/v_{\rm
w}\right]\overset{!}{=}L_{\nu_{\rm
pk}}~{},~{}\text{and}~{}~{}\left(\frac{dL_{\nu}}{dx}\right)\left[\nu_{\rm
pk}~{};~{}(\Gamma\beta)_{\rm sh},\dot{M}/v_{\rm w}\right]\overset{!}{=}0.$
(B6)

In general, these equations must be solved using numerical root-finding
procedures. Note that we have treated the luminosity as a function of the
proper-velocity of the shock, $(\Gamma\beta)_{\rm sh}$, whereas several of the
expressions above depend instead on the Lorentz factor of post-shock material,
$\Gamma$. The two are related to one-another via Equation (5) of Blandford &
McKee (1976). With the effective adiabatic index
$\hat{\gamma}=(4+\Gamma^{-1})/3$, this can be rewritten as

$(\Gamma\beta)^{2}=\frac{1}{4}\left[(\Gamma\beta)_{\rm
sh}^{2}-2+\sqrt{(\Gamma\beta)_{\rm sh}^{4}+5(\Gamma\beta)_{\rm
sh}^{2}+4}\right],$ (B7)

and recall that $\Gamma=\sqrt{1+(\Gamma\beta)^{2}}$. Note that
$(\Gamma\beta)^{2}=\frac{9}{16}(\Gamma\beta)_{\rm sh}^{2}$ in the non-
relativistic regime, while $(\Gamma\beta)^{2}=\frac{1}{2}(\Gamma\beta)_{\rm
sh}^{2}$ in the ultra-relativistic case. This implies that
$(\Gamma\beta)^{2}\approx 0.5(\Gamma\beta)_{\rm sh}^{2}$ is a reasonable
approximation for any velocity.

##  Appendix C Critical Luminosity

In the following we derive more accurate expressions for the limiting peak
luminosity, $L_{\rm crit}$. At the critical luminosity, emission (and
absorption) at frequency $\nu_{\rm pk}$ are dominated by thermal electrons.
This implies that the terms proportional to $C_{j}$ or $C_{\alpha}$ in
Equations (B1,B2)—which represent the contributions of power-law electrons—can
be neglected. The condition $\left.dL_{\nu}/d\nu\right|_{\nu_{\rm pk}}=0$
(Equation B6) then introduces an equation relating the peak frequency $x_{\rm
pk}$ to the optical depth at this frequency, $\tau_{\rm pk}\equiv\tau(x_{\rm
pk})$, which can be written as

$\frac{\tau_{\rm pk}}{e^{\tau_{\rm pk}}-1}=1+\left.\frac{1+\left({d\ln
I^{\prime}}/{d\ln x}\right)}{1-\left({d\ln I^{\prime}}/{d\ln
x}\right)}\right|_{x_{\rm pk}}.$ (C1)

This equation must be solved numerically. Once solved however, one can express
the peak luminosity (Equation B1) as a function of a single dimensionless
variable, $\tilde{\tau}_{\rm pk}\equiv x_{\rm pk}\tau_{\Theta}=\tau_{\rm
pk}(x_{\rm pk})x_{\rm pk}^{2}/I^{\prime}(x_{\rm pk})$ (Equation B2),

$L_{\nu_{\rm pk}}(\tilde{\tau}_{\rm
pk})=\frac{2^{59/8}\pi^{17/8}}{3^{9/16}}m_{e}c^{2}\left(\frac{\epsilon_{T}f}{\epsilon_{B}}\right)^{1/8}\left(\nu_{\rm
pk}\bar{t}\right)^{17/8}\tilde{\tau}_{\rm pk}^{-1/8}\left(1-e^{-\tau_{\rm
pk}}\right)\left(\frac{4-1/\Gamma^{2}}{8+1/\Gamma^{2}}\beta\right)^{17/8}\Gamma^{3}.$
(C2)

Note that in the exponent, $\tau_{\rm pk}=\tau_{\rm pk}(\tilde{\tau}_{\rm
pk})$ through Equation (C1). The Lorentz factor of post-shock material
$\Gamma=\Gamma(\tilde{\tau}_{\rm pk})$ is also a function of
$\tilde{\tau}_{\rm pk}$ through the following relation

$\left(\frac{4-1/\Gamma^{2}}{8+1/\Gamma^{2}}\sqrt{1-1/\Gamma^{2}}\right)^{-1/8}\left(\Gamma-1\right)=\frac{\mu_{e}m_{e}}{\mu
m_{p}}\left(\frac{3^{7/2}8\pi f\nu_{\rm
pk}\bar{t}}{\epsilon_{B}\epsilon_{T}^{7}\tilde{\tau}_{\rm pk}}\right)^{1/8}.$
(C3)

In both Equations (C2,C3) we have implicitly assumed that $\Theta\gtrsim 1$.

The maximum peak luminosity can then be calculated from Equation (C2) by
finding $L_{\rm crit}=\max_{\tilde{\tau}_{\rm pk}}\left(L_{\nu_{\rm
pk}}\left[\tilde{\tau}_{\rm pk}\right]\right)$. When $\Gamma\gg 1$ then
$L_{\nu_{\rm pk}}\propto\tilde{\tau}_{\rm pk}^{-1/2}(1-e^{-\tau_{\rm pk}})$,
which peaks at $\tilde{\tau}_{\rm pk}\simeq 0.1712$ (corresponding to $x_{\rm
pk}\simeq 1.506$, and $\tau_{\rm pk}\simeq 0.0587$). Alternatively, when
$\Gamma-1\ll 1$, then $L_{\nu_{\rm pk}}\propto\tilde{\tau}_{\rm
pk}^{-4/15}(1-e^{-\tau_{\rm pk}})$. This peaks at $\tilde{\tau}_{\rm pk}\simeq
2.645$ (and a corresponding $x_{\rm pk}\simeq 2.581$, $\tau_{\rm pk}\simeq
0.169$). With the aid of these numerical coefficients, analytic solutions may
be obtained in the non-relativistic and ultra-relativistic regimes. In
particular, one can calculate $\Gamma$ at the critical luminosity using
Equation (C3). The shock four-velocity can then be calculated using Equation
(B7). Using this method, we find that the proper-velocity of the shock along
the critical curve $L_{\nu_{\rm pk}}=L_{\rm crit}$ is

$\left(\Gamma\beta\right)_{\rm
sh,crit}=\begin{dcases}\left(\Gamma\beta\right)_{\rm sh,NR}\equiv
1.417\epsilon_{B,-1}^{-1/15}\left(\frac{\epsilon_{T}}{0.4}\right)^{-7/15}\left(\frac{f}{3/16}\right)^{1/15}\left(\frac{\nu_{\rm
pk}\bar{t}}{5\,{\rm GHz}\times 100\,{\rm
d}}\right)^{1/15}&,~{}\left(\Gamma\beta\right)_{\rm sh,crit}\ll 1\\\
\left(\Gamma\beta\right)_{\rm sh,UR}\equiv
1.174\epsilon_{B,-1}^{-1/8}\left(\frac{\epsilon_{T}}{0.4}\right)^{-7/8}\left(\frac{f}{3/16}\right)^{1/8}\left(\frac{\nu_{\rm
pk}\bar{t}}{5\,{\rm GHz}\times 100\,{\rm
d}}\right)^{1/8}&,~{}\left(\Gamma\beta\right)_{\rm sh,crit}\gg 1\end{dcases}$
(C4)

in the non-relativistic $\left(\Gamma\beta\right)_{\rm sh,crit}\ll 1$ and
ultra-relativistic $\left(\Gamma\beta\right)_{\rm sh,crit}\gg 1$ limits,
respectively. Clearly, neither limit is strictly appropriate for the typical
parameters adopted above. This is not surprising since we expect the
transition from a SSA to optically-thin peak to occur around
$\left(\Gamma\beta\right)_{\rm sh}\sim 1$. In such cases one must generally
use a numerical procedure to maximize $L_{\nu_{\rm pk}}(\tilde{\tau}_{\rm
pk})$ (Equation C2) and obtain the shock velocity using Equation (C3; in
general, also using a numerical root-solving scheme). Alternatively, we find
that the solution can be well-approximated by the following interpolating
function,

$\left(\Gamma\beta\right)_{\rm
sh,crit}\approx\left[\left(\Gamma\beta\right)_{\rm
sh,NR}^{n}+\left(\Gamma\beta\right)_{\rm
sh,UR}^{n}\right]^{1/n}~{}~{},~{}~{}n=2$ (C5)

with $\left(\Gamma\beta\right)_{\rm sh,NR}$, $\left(\Gamma\beta\right)_{\rm
sh,UR}$ defined in Equation (C4) above. This approximation has a maximum error
of $<1\%$. The critical luminosity can similarly be found using the same
procedure. In the non-relativistic and ultra-relativistic limits analytic
solutions are possible, and we find that

$L_{\rm crit}=\begin{dcases}L_{\rm crit,NR}\equiv 2.73\times 10^{30}\,{\rm
erg\,s}^{-1}\,{\rm
Hz}\,\epsilon_{B,-1}^{-4/15}\left(\frac{\epsilon_{T}}{0.4}\right)^{-13/15}\left(\frac{f}{3/16}\right)^{4/15}\left(\frac{\nu_{\rm
pk}\bar{t}}{5\,{\rm GHz}\times 100\,{\rm
d}}\right)^{34/15}&,~{}\left(\Gamma\beta\right)_{\rm sh,crit}\ll 1\\\ L_{\rm
crit,UR}\equiv 1.68\times 10^{30}\,{\rm erg\,s}^{-1}\,{\rm
Hz}\,\epsilon_{B,-1}^{-1/2}\left(\frac{\epsilon_{T}}{0.4}\right)^{-5/2}\left(\frac{f}{3/16}\right)^{1/2}\left(\frac{\nu_{\rm
pk}\bar{t}}{5\,{\rm GHz}\times 100\,{\rm
d}}\right)^{5/2}&,~{}\left(\Gamma\beta\right)_{\rm sh,crit}\gg 1\end{dcases}.$
(C6)

Again, it is typically the case that $\left(\Gamma\beta\right)_{\rm
sh,crit}\sim 1$ and neither of these expressions is strictly appropriate. We
therefore employ an interpolating function of a similar form to (C5) and find
that

$L_{\rm crit}\approx\left[L_{\rm crit,NR}^{1/n}+L_{\rm
crit,UR}^{1/n}\right]^{n}~{}~{},~{}~{}n=2$ (C7)

provides an excellent approximation to the critical luminosity for any
combination of parameters (the variables $L_{\rm crit,NR}$, $L_{\rm crit,UR}$
are defined in Equation C6 above). This interpolating function has a maximum
error of $<5\%$. Equation (C7) should be used instead of the crude
approximation given by Equation (5.1), which is only accurate at the order of
magnitude level, and for the range of microphysical parameters and $\nu_{\rm
pk}\bar{t}$ considered in Figure 2 (in particular, if $\epsilon_{T}\ll 1$ or
$\epsilon_{B}\ll 0.1$, Equation 5.1 may be inaccurate by orders of magnitude).

##  Appendix D Fast Cooling

Throughout this paper we have made the simplifying assumption that electrons
are slow cooling, that is—that the timescale over which electrons loose their
energy (e.g., via synchrotron radiative losses) is long compared to other
timescales in the problem. This is typically true at $\sim$GHz frequencies, as
we show below, but may be incorrect for particularly luminous events and peak
frequencies in the sub-mm band or higher. In this Appendix we therefore derive
approximate conditions that delineate the regime where electrons cool quickly
through synchrotron radiation and where synchrotron losses are negligible and
our slow-cooling assumption holds (barring other cooling mechanisms such as
inverse-Compton scattering or free-free emission). Note that we here adopt the
terms ‘slow-cooling’ and ‘fast-cooling’ in referring to the cooling timescale
of electrons that produce the observed emission at frequency $\nu_{\rm pk}$,
as this is the frequency of interest in this paper.

Because the synchrotron power scales as $\propto\gamma^{2}$, only electrons
whose Lorentz factor exceeds

$\gamma_{\rm cool}=\frac{6\pi m_{e}c}{\sigma_{T}B^{2}\Gamma t_{e}}$ (D1)

are able to cool appreciably over a timescale $<t_{e}$. Here $t_{e}\sim t$ is
the time it takes injected electrons to cross the emission region, i.e.
$t_{e}\approx\Delta R/\beta c$ where $\beta$ is the post-shock velocity (here
assumed to be constant throughout the post-shock region). Using our definition
for $\Delta R$ (Equation 3) we find that

$t_{e}=\frac{4f}{3}\left(\frac{\Gamma_{\rm
sh}}{\Gamma}\right)^{2}\left(\frac{\beta_{\rm
sh}}{\beta}\right)\bar{t}\approx\begin{cases}\frac{1}{3}\bar{t}\left(\frac{f}{3/16}\right)&,\left(\Gamma\beta\right)_{\rm
sh}\ll 1\\\
\frac{1}{2}\bar{t}\left(\frac{f}{3/16}\right)&,\left(\Gamma\beta\right)_{\rm
sh}\gg 1\end{cases}.$ (D2)

Using Equations (D1,D2) we can calculate the corresponding cooling frequency
$\nu_{\rm cool}$ above which synchrotron cooling effects will become
important. The requirement that electrons dominating emission at $\nu_{\rm
pk}$ be slow cooling is equivalent to the condition $\gamma_{e}(\nu_{\rm
pk})<\gamma_{\rm cool}$, where $\gamma_{e}(\nu_{\rm pk})$ is the Lorentz
factor of electrons that dominate emission at frequency $\nu_{\rm pk}$. For
power-law electrons $\gamma_{e}(\nu)=\sqrt{4\pi m_{e}c\nu/3eB}$. The
requirement that electrons be slow cooling at peak therefore translates into
the requirement

$\text{slow cooling:}~{}L_{\nu_{\rm pk}}>1.4\times 10^{21}\,{\rm
erg\,s}^{-1}\,{\rm
Hz}^{-1}\,\left(\frac{\epsilon_{e}/\epsilon_{B}}{0.1}\right)^{-2}\left(\frac{f}{3/16}\right)^{13/3}\left(\frac{\nu_{\rm
pk}}{5\,{\rm GHz}}\right)^{38/3}\left(\frac{\bar{t}}{100\,{\rm
d}}\right)^{19/3};~{}L_{\nu_{\rm pk}}<L_{\rm th}$ (D3)

where we have implicitly assumed $\left(\Gamma\beta\right)_{\rm sh}\ll 1$
above. When thermal electrons dominate peak emission and $\nu_{\rm
pk}\gg\nu_{\Theta}$ (as appropriate “deep” within case b where
$\nu_{\Theta}\ll\nu_{\rm pk}\simeq\nu_{\rm a}$; that is, when $L_{\rm
th}<L_{\nu_{\rm pk}}\ll L_{\rm crit}$) then emission at $\nu_{\rm pk}$ is
dominated by the high-frequency “tail” of comparatively lower-energy electrons
whose Lorentz factor scales as $\nu^{1/3}$, specifically
$\gamma_{e}(\nu)/\Theta=(2\nu/\nu_{\Theta})^{1/3}$ (Margalit & Quataert,
2021). In this case, the condition that peak emission be in the slow-cooling
regime translates into the condition

$\text{slow cooling:}~{}L_{\nu_{\rm pk}}>2\times 10^{25}\,{\rm
erg\,s}^{-1}\,{\rm
Hz}^{-1}\,\epsilon_{B,-1}^{20/33}\left(\frac{\epsilon_{T}}{0.4}\right)^{-43/33}\left(\frac{f}{3/16}\right)^{52/33}\left(\frac{\nu_{\rm
pk}}{5\,{\rm GHz}}\right)^{190/33}\left(\frac{\bar{t}}{100\,{\rm
d}}\right)^{118/33};~{}L_{\nu_{\rm pk}}>L_{\rm th}$ (D4)

and again this approximation assumes that $\left(\Gamma\beta\right)_{\rm
sh}\ll 1$. At high luminosities, as one approaches $L_{\rm crit}$ this
assumption, along with the assumption that $\nu_{\rm pk}\gg\nu_{\Theta}$ break
down and the equation above is not accurate (even at lower velocities, this is
only accurate to within a factor of a few because of the approximate relations
used in its derivation; see §4.2).

Equation (D4) is less restrictive than Equation (D3). That is, some events
that would have been in the fast-cooling regime (at frequency $\nu_{\rm pk}$)
had they been interpreted using the standard power-law model, may in fact
reside well within the slow-cooling regime if interpreted as emission from
thermal electrons. This is primarily due to the fact that the Lorentz factors
of electrons contributing to emission at frequency $\nu_{\rm pk}$ is lower in
the thermal-electron scenario (because $\nu\propto\gamma_{e}^{3}$ in this
scenario rather than $\nu\propto\gamma_{e}^{2}$; see Margalit & Quataert
2021). Of course, this is only viable for events where $L_{\nu_{\rm
pk}}\gtrsim L_{\rm th}$ where the contribution of thermal electrons indeed
dominates.

Finally, when peak emission is optically thin and thermal electrons dominate
(case c), the emission at $\nu_{\rm pk}$ is dominated by electrons with
Lorentz factors $\gamma_{e}\approx 2\Theta$.999 Specifically, the emissivity
at frequency $\nu_{\rm pk}$ is proportional to $\propto\int
z^{2}e^{-z}F(x_{\rm pk}/z^{2})dz$ where $x_{\rm pk}\equiv\nu_{\rm
pk}/\nu_{\Theta}\simeq 1.268$ and $z\equiv\gamma_{e}/\Theta$. The integrand of
this function peaks at $z\simeq 2.04$, and is therefore dominated by electrons
whose Lorentz factor is $\gamma_{e}\approx 2\Theta$. 50% of the total emission
at frequency $\nu_{\rm pk}$ is contributed by electrons with Lorentz factors
between $z=1.4$ to $z=3$ (90% is contributed by electrons between $z=0.84$ to
$z=5.2$).  Demanding that $\gamma_{e}<\gamma_{\rm cool}$, and using the
results for the ultra-relativistic $\left(\Gamma\beta\right)_{\rm sh}\gg 1$
limit where this case is appropriate, we find that the peak is in the slow-
cooling regime if

$\text{slow cooling:}~{}L_{\nu_{\rm pk}}<1.8\times 10^{32}\,{\rm
erg\,s}^{-1}\,{\rm
Hz}^{-1}\,\epsilon_{B,-1}^{-1}\left(\frac{\epsilon_{T}}{0.4}\right)^{-3}\left(\frac{\nu_{\rm
pk}}{5\,{\rm GHz}}\right)\left(\frac{\bar{t}}{100\,{\rm
d}}\right)^{2},~{}\text{optically-thin}.$ (D5)

Note that in the relativistic optically-thin scenario, the slow-cooling regime
applies to lower-luminosity shocks. In the radio, optically-thin shocks are
almost always slow cooling. However this may not necessarily be the case if
one considers peak emission at higher frequencies, e.g. in the optical, X-ray,
or gamma-ray bands.