# Non-thermal emission from mildly relativistic dynamical ejecta of neutron
star mergers

Gilad Sadeh, Or Guttman, Eli Waxman
Dept. of Particle Phys. & Astrophys., Weizmann Institute of Science, Rehovot
76100, Israel  E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Binary neutron star mergers are expected to produce fast dynamical ejecta,
with mildly relativistic velocities extending to $\beta=v/c>0.6$. We consider
the radio to X-ray synchrotron emission produced by collisionless shocks
driven by such fast ejecta into the interstellar medium. Analytic expressions
are given for spherical ejecta with broken power-law mass (or energy)
distributions, $M(>\gamma\beta)\propto(\gamma\beta)^{-s}$ with $s=s_{\rm KN}$
at $\gamma\beta<\gamma_{0}\beta_{0}$ and $s=s_{\rm ft}$ at
$\gamma\beta>\gamma_{0}\beta_{0}$ (where $\gamma$ is the Lorentz factor). For
parameter values characteristic of merger calculation results- a "shallow"
mass distribution, $1<s_{\rm KN}<3$, for the bulk of the ejecta (at
$\gamma\beta\approx 0.2$), and a steep, $s_{\rm ft}>5$, "fast tail" mass
distribution- our model provides an accurate (to 10’s of percent) description
of the evolution of the flux, including at the phase of deceleration to sub-
relativistic expansion. This is a significant improvement over earlier
results, based on extrapolations of results valid for $\gamma\beta\gg 1$ or
$\ll 1$ to $\gamma\beta\approx 1$, which overestimate the flux by an order of
magnitude for typical parameter values. It will enable a more reliable
inference of ejecta parameters from future measurements of the non-thermal
emission. For the merger event GW170817, the existence of a "fast tail" is
expected to produce detectable radio and X-ray fluxes over a time scale of
$\sim 10^{4}$days.

###### keywords:

gravitational waves – stars: neutron – neutron star mergers

††pubyear: 2022††pagerange: Non-thermal emission from mildly relativistic
dynamical ejecta of neutron star mergers–A

##  1 Introduction

Mergers of binary neutron stars (BNS) are expected to produce ejecta expanding
at mildly relativistic velocities, $\gamma\beta>0.1$ (see Fernández & Metzger,
2016; Radice et al., 2020, for reviews). The amount of ejected mass and its
velocity distribution depend on the parameters of the merging system and on
the equation of state (EoS) of the nuclear matter. Measurements of the
electromagnetic emission generated by the expanding ejecta will enable us to
constrain its properties, and hence the parameters of the merging binary and
of the EoS.

Observations of the optical (UV-IR) electromagnetic counterpart of GW170817
are consistent with an ejecta of $\approx 0.05M_{\odot}$ expanding at
$\beta>0.1$ and undergoing continuous heating by radioactive decays
(Cowperthwaite et al., 2017; Arcavi et al., 2017; Nicholl et al., 2017;
Metzger, 2017). The early evolution of the photospheric radius is consistent
with a power-law mass distribution,
$M(>\gamma\beta)\propto(\gamma\beta)^{-s_{\rm KN}}$ with $s_{\rm KN}\approx
1.6$ at $0.1<\gamma\beta<0.3$ (Waxman et al., 2018). The "shallow" dependence
of mass on velocity at $\gamma\beta<0.3$ is consistent with the results of
numeric calculations, while the amount of mass ejected is larger than
typically obtained in calculations (e.g. Radice et al., 2018; Fernández et
al., 2019; Shibata & Hotokezaka, 2019; Bernuzzi et al., 2020; Hajela et al.,
2022; Nedora et al., 2021b).

Numeric merger calculations find that the velocity distribution of the ejecta
may extend to large velocities, well above the characteristic velocity,
$\beta\sim 0.2$, of the bulk of the ejecta that is responsible for the
generation of the UV-IR "kilonova" (KN) emission (e.g. Bauswein et al., 2013;
Ishii et al., 2018; Radice et al., 2018; Fernández et al., 2019; Shibata &
Hotokezaka, 2019; Ciolfi & Kalinani, 2020; Bernuzzi et al., 2020; Nedora et
al., 2021a; Dean et al., 2021; Hajela et al., 2022). The existence of such a
fast ejecta "tail" has recently been suggested as an explanation of a possible
"re-brightening" of the X-ray emission of the GW170817 electromagnetic
counterpart (Hajela et al., 2022; Nedora et al., 2021a). However, the evidence
for rebrightening is not conclusive (Balasubramanian et al., 2021, 2022; Troja
et al., 2022; O’Connor & Troja, 2022). The mass and velocity profile of the
"fast tail" depend strongly on the parameters of the binary system and on the
EoS. Depending on parameter values, the ejecta mass at $\beta>0.6$ varies, for
example, between $10^{-7}M_{\odot}$ and $10^{-4}M_{\odot}$, and the maximal
velocity varies from $\gamma\beta\approx 0.6$ to $\gamma\beta>3$. This implies
that measurements of the non-thermal emission driven by the fast ejecta may
provide stringent constraints on model parameters.

Two points should, however, be noted here. First, a reliable determination of
constraints would require a significant reduction of the current large
numerical uncertainty in the numerically calculated fast tail parameters.
Second, if BNS mergers are the sources of (short) gamma-ray bursts (see
Mészáros, 2002; Piran, 2004; Nakar, 2007, for reviews), then they are expected
to launch highly relativistic jets. Depending on the jet axis with respect to
our line of sight, and on the ratio of energies carried by the jet and by the
dynamical ejecta, the emission of non-thermal radiation driven by the jet may
dominate that driven by the dynamical ejecta. In the case of GW170817, the
rapid decline of the observed non-thermal emission (Troja et al., 2019; Fong
et al., 2019; Lamb et al., 2019; Hajela et al., 2019; Troja et al., 2020;
Makhathini et al., 2020; Nakar & Piran, 2020) and the super-luminal motion of
the radio centroid suggest that the observed non-thermal emission is driven by
a relativistic jet (the presence of plasma expanding at $\gamma>10$ cannot be
directly inferred from observations, which may be accounted for by a
$\gamma\sim 4$ jet Mooley et al., 2018). In general, due to the lower mass and
higher Lorenz factor of the jet, we may expect the emission to be dominated at
late time by the dynamical ejecta. The interpretation of future observation
will require disentangling the two components.

We consider in this paper the radio to X-ray synchrotron emission produced by
collisionless shocks driven by mildly relativistic ejecta expanding into an
interstellar medium (ISM) with uniform (number) density $n$. Analytic
expressions are derived for the case of spherical ejecta with a broken power-
law dependence of mass on momentum,

$M(>\gamma\beta)=M_{0}\begin{cases}\left(\frac{\gamma\beta}{\gamma_{0}\beta_{0}}\right)^{-s_{\text{ft}}}&\text{for}\quad\gamma_{0}\beta_{0}<\gamma\beta,\\\
\left(\frac{\gamma\beta}{\gamma_{0}\beta_{0}}\right)^{-s_{\text{KN}}}&\text{for}\quad
0.1<\gamma\beta<\gamma_{0}\beta_{0},\end{cases}$ (1)

with parameter values characteristic of the results of numerical calculations-
a "shallow" mass distribution, $1<s_{\rm KN}<3$, for the bulk of the mass (at
$\gamma\beta\approx 0.2$), and a steep, $s_{\rm ft}>5$, fast tail mass
distribution (e.g. Radice et al., 2018; Fernández et al., 2019; Shibata &
Hotokezaka, 2019; Hajela et al., 2022; Nedora et al., 2021b). Analytic
expressions for a broken power-law dependence of ejecta energy on momentum,

$E(>\gamma\beta)=E_{0}\begin{cases}\left(\frac{\gamma\beta}{\gamma_{0}\beta_{0}}\right)^{-\alpha_{\text{ft}}}&\text{for}\quad\gamma_{0}\beta_{0}<\gamma\beta,\\\
\left(\frac{\gamma\beta}{\gamma_{0}\beta_{0}}\right)^{-\alpha_{\text{KN}}}&\text{for}\quad
0.1<\gamma\beta<\gamma_{0}\beta_{0},\end{cases}$ (2)

are also provided, based on the results obtained for a broken power-law
dependence of mass on momentum.

The non-thermal flux is derived assuming that fractions $\varepsilon_{e}$ and
$\varepsilon_{B}$ of the post-shock internal energy density are carried by
non-thermal electrons and magnetic fields respectively, and the electrons are
assumed to follow a power-law energy distribution,
$dn_{e}/d\gamma_{e}\propto\gamma_{e}^{-p}$, where $\gamma_{e}$ is the electron
Lorenz factor (at the plasma rest-frame) and $2\leq p\leq 2.5$. This
phenomenological description of the post-shock plasma conditions is supported
by a wide range of observations and plasma calculations, for both relativistic
and non-relativistic shocks (see Blandford & Eichler, 1987; Waxman, 2006;
Bykov & Treumann, 2011; Sironi et al., 2013, for reviews).

The accuracy of the analytic approximations derived in this paper for the non-
thermal flux is verified through comparison to numerical calculations for a
wide range of parameter values. It is significantly improved compared to that
of earlier work, mainly due to an improved treatment of the dynamics of the
ejecta at the relevant velocity range, $0.1<\gamma\beta<3$. The accuracy of
the approximate description of the dynamics is important due to the strong
dependence of the flux on the velocity of the emitting plasma (approx.
$\propto(\gamma\beta)^{4}$, see below). In addition, we provide an approximate
analytic description for both the phase of rising flux, which was addressed in
earlier papers (Nakar & Piran, 2011; Kathirgamaraju et al., 2019), and for the
phase of flux decline, which was not addressed in detail before.

The paper is organized as follows. The analytic expressions providing an
approximate description of the non-thermal flux are derived in § 2. Their
accuracy is estimated by a comparison to the results of numeric relativistic-
hydrodynamic calculations in § 3. Our results are compared to those of earlier
work in § 4. The implications of our results for future observations of the
electromagnetic counterpart of GW170817 are discussed in § 5, and our
conclusions are summarized in § 6.

##  2 Analytic model

As the ejecta expands, a forward shock is driven into the ISM and a reverse
shock is driven into the ejecta (see Fig. 1). The kinetic energy of the ejecta
decreases due to its deceleration by the reverse shock, and the energy is
transferred to the shocked ISM plasma. The shocked ISM mass increases with the
expanding shock radius, while its energy density decreases due to shock
deceleration. The increase of the mass of hot plasma acts to increase the non-
thermal emission, while the deceleration acts to suppress it (due both to the
decreasing energy density and to the decreasing effect of flux enhancement due
to relativistic expansion). For sufficiently steep drop of mass with velocity,
$s\gtrsim 5$ as shown below, the former effect dominates and the non-thermal
flux increases with time. This is due to the fact that the deceleration is
slower for a steeper mass distribution, since the shocked ISM mass that needs
to be accumulated in order to decelerate the shock below a certain velocity
increases faster with decreasing velocity (as the ejecta mass increases faster
with decreasing velocity).

For the steep mass profiles of the fast tail, inferred from numeric
calculations, we thus expect the non-thermal flux to increase with time as
long as the reverse shock propagates through the fast tail of the ejecta. Once
the reverse shock penetrates into the bulk of the ejecta, where the mass
distribution as function of velocity is shallow, the deceleration becomes more
rapid and the flux begins to decrease. At late time, after the reverse shock
crosses the entire ejecta and the expansion velocity becomes sub-relativistic,
the flow approaches the non-relativistic Sedov-von Neumann-Talyor self-similar
flow (Sedov, 1946; Taylor, 1950).

This section is organized as follows. The phase of rising flux,
$t<t_{\text{peak}}$, is discussed in § 2.1, the asymptotic Sedov-von Neumann-
Talyor phase, $t>t_{\text{ST}}$ is discussed in § 2.2, and the intermediate
phase between them is discussed in § 2.3. A summary of the analytic results is
given in § 2.4.

The hydrodynamic profiles are determined by three dimensional parameters,
$\\{M_{0},n,c\\}$, and three dimensionless parameters, $\\{\beta_{0},s_{\rm
ft},s_{\rm KN}\\}$. The dependence of the profiles on the dimensional
parameters follows directly from dimensional considerations. The flow depends
non-trivially only on the dimensionless parameters. The approximations that we
provide are valid over a wide range of values of these parameters, $5<s_{\rm
ft}<12,~{}1<s_{\rm KN}<3,~{}0.3<\beta_{0}<0.9$.

###  2.1 Rising phase

In § 2.1.1 we first derive an approximate description of the dependence of the
velocity of the shocked plasma, $\beta_{\rm SP}$, and of the velocity
$\beta_{\rm ej,RS}$ of the un-shocked plasma at the position reached by the
reverse shock, on radius $R$. We then derive an approximate expression for
$t_{\rm peak}$, the (observed) time at which the flux peaks. In § 2.1.2 we use
these results to derive an approximate description of the non-thermal flux.

####  2.1.1 Dynamics

Figure 1: A schematic illustration showing the four regions of the forward-
reverse shock structure described in § 2.1: (1) un-shocked ISM, (2) shocked
ISM, (3) shocked ejecta, (4) un-shocked ejecta. The pressure in the unshocked
regions is negligible compared to that in the shocked regions, $p_{1},p_{4}\ll
p_{2},p_{3}$. The pressure and velocity are approximated in our analytic
calculations as uniform within the shocked region, and the density is
approximated as uniform within the two regions separated by the contact
discontinuity.

Let us first obtain a qualitative estimate of the relation between the
accumulated shocked ISM mass and the shocked ejecta mass. Consider ejecta of
mass $M_{\rm ej}$ with uniform initial velocity, $\beta_{\rm ej}$. A
significant deceleration of the shell occurs when a significant fraction of
its energy is transferred to the ISM. Assuming that the velocity of the
shocked plasma is not far below the initial shell velocity, i.e. that the
deceleration by the reverse shock is modest as justified below, this implies
significant deceleration when the shocked ISM mass, $M_{\rm ISM}$, reaches
$M_{\rm ISM}\approx M_{\rm ej}/\gamma_{\rm ej}$. This is due to the fact that
the shell’s initial energy is $0.5\beta_{\rm ej}^{2}M_{\rm ej}c^{2}$ for
$\beta_{\rm ej}\ll 1$ and $\gamma_{\rm ej}M_{\rm ej}c^{2}$ for $\gamma_{\rm
ej}\gg 1$, while the shocked ISM energy is $\approx\beta_{\rm ej}^{2}M_{\rm
ISM}c^{2}$ and $\approx\gamma_{\rm ej}^{2}M_{\rm ISM}c^{2}$ respectively. This
result suggests that the position of the reverse shock within the ejecta, i.e.
the initial velocity $\beta_{\rm ej,RS}$ of the ejecta shell that is reached
by the reverse shock, is (implicitly) determined by $M(>\gamma_{\rm
ej,RS}\beta_{\rm ej,RS})=\gamma_{\rm ej,RS}(R)M_{\rm ISM}(R)$, where
$\gamma(R)$ is the Lorenz factor corresponding to $\beta(R)$.

A more accurate approximation, in particular taking into account the
deceleration due to the reverse shock, is obtained as follows. We assume that
at any given $R$ the velocity and pressure profiles within the shocked
ejecta/ISM plasmas (see fig. 1) are uniform, and that the shocked ejecta/ISM
densities are uniform within the regions separated by the contact
discontinuity. Denoting the mass of the ejecta shocked by the reverse shock as
$M_{\rm ej,RS}(R)$, we obtain (using Eq. (1)) the un-shocked ejecta number
density and velocity ahead of the reverse shock, $n_{\rm ej,RS}(R,M_{\rm
ej,RS}(R))$ and $\beta_{\rm ej,RS}(R,M_{\rm ej,RS}(R))$. Given $n_{\rm
ej,RS}(R,M_{\rm ej,RS}(R))$ and the number density of the ISM, $n$, a relation
between $\beta_{\rm ej,RS}$ and the velocity of the shocked plasma,
$\beta_{\rm SP}$, is obtained by requiring the pressures behind the forward
and reverse shocks to be equal, thus determining $\beta_{\rm SP}(R,M_{\rm
ej,RS}(R))$. Finally, $M_{\rm ej,RS}(R)$ is determined by requiring the energy
in the shocked ejecta and ISM plasma to equal the initial kinetic energy
carried by the ejecta mass $M_{\rm ej,RS}(R)$ (the thicknesses of the two
shocked plasma regions, which are required for determining the internal energy
of the shocked plasma, are determined by conservation of mass).

The solution for $\beta_{\rm SP}(R)$ is obtained by numerically solving the
pressure equilibrium and energy conservation equations described in the
preceding paragraph. We use an EOS of the form $p=(\hat{\gamma}-1)e$ with
adiabatic index $\hat{\gamma}(e/n)$ varying from $4/3$ to $5/3$ following the
analysis of Synge (1957) for a plasma of protons and electrons. This is not
accurate for the shocked ejecta plasma, which is composed of a wide range of
nuclei. However, the dependence of the results on the exact form of
$\hat{\gamma}(e/n)$ for the shocked ejecta plasma is weak (as verified by the
numeric calculations). Once the hydrodynamic evolution is determined, we
calculate the non-thermal flux as described in the following subsection.

For the purpose of deriving analytic expressions for the flux, it is useful to
derive analytic approximations for the dynamics. We give below several useful
approximations, which we found to provide a good description of the dynamics
obtained by solving the equations described above for a wide range of ejecta
parameters, $5<s_{\rm ft}<12,~{}0.2<\gamma_{\rm ej,RS}\beta_{\rm ej,RS}<6$. We
find that $\beta_{\rm ej,RS}(R)$ is well approximated by the implicit relation

$M(>\gamma_{\rm ej,RS}(R)\beta_{\rm ej,RS}(R))=\gamma_{\rm ej,RS}(R)M_{\rm
ISM}(R),$ (3)

that the ratio of the pre- and post-shock velocities is approximately given by

$\frac{\gamma_{\rm ej,RS}(R)\beta_{\rm ej,RS}(R)}{\gamma_{\rm SP}(R)\beta_{\rm
SP}(R)}=1.7,$ (4)

and that $R\approx\beta_{\rm ej,RS}ct_{\rm s}$, where $t_{\rm s}$ is the time
measured at the rest frame of the source ejecting the expanding plasma.

Let us consider next the radius and velocity of the shocked plasma dominating
the emission that contributes to the flux observed by a distant observer at
that time $t$. Photons emitted from the shock front along a line making an
angle $\theta$ with respect to the line of sight are observed at time $t$
related to $R$ by $R=\beta_{\rm ej,RS}ct/(1-\beta_{\rm ej,RS}\cos\theta)$
(where we have used $R\approx\beta_{\rm ej,RS}ct_{\rm s}$ and
$t_{s}=t/(1-\beta_{\rm ej,RS}\cos\theta)$). The emission from a ring of
opening angle $\theta$ and thickness $d\theta$ increases with $\theta$ due to
the increase in ring area, and decreases with $\theta$ due to the decrease in
the Lorenz boost, $1/(\gamma^{2}(1-\beta\cos\theta)^{2})$. We therefore
estimate the angle dominating the flux observed at time $t$ as the angle for
which
$\partial_{\theta}(2\pi\sin\theta/(\gamma^{2}(1-\beta\cos\theta)^{2}))=0$,

$\cos\theta=-\frac{1}{2\beta}+\sqrt{2+\frac{1}{4\beta^{2}}}.$ (5)

We therefore approximate

$R=\frac{\beta_{\rm ej,RS}ct}{\left(1.5-\sqrt{2\beta_{\rm
ej,RS}^{2}+0.25}\right)}.$ (6)

Adopting this approximation we have

$\displaystyle\gamma_{\rm ej,RS}M_{\rm ISM}(t)$
$\displaystyle=\frac{4\pi}{3}nm_{p}c^{3}\frac{\gamma_{\rm ej,RS}\beta_{\rm
ej,RS}^{3}}{\left(1.5-\sqrt{2\beta_{\rm ej,RS}^{2}+0.25}\right)^{3}}t^{3}$ (7)
$\displaystyle\approx\frac{4\pi}{3}nm_{p}c^{3}12(\gamma_{\rm ej,RS}\beta_{\rm
ej,RS})^{5.5}t^{3},$

where the last approximation holds (to $\sim 10$%) for $0.3<\gamma\beta<2.5$.
Using Eq. (3) we finally obtain

$\gamma_{\rm ej,RS}\beta_{\rm
ej,RS}=\left(\frac{t}{t_{R}}\right)^{-\frac{3}{5.5+s}},\quad
t_{R}\equiv\left(\frac{M_{0}(\gamma_{0}\beta_{0})^{s}}{16\pi
nm_{p}c^{3}}\right)^{\frac{1}{3}}.$ (8)

Eqs. (4) and (8) determine the the velocity of the shocked plasma dominating
the emission of flux measured by a distant observer at time $t$.

Using the above approximations, we find that the observed time at which the
flux peaks, i.e. the observed time at which the reverse shock reaches the
shallow $\gamma\beta<\gamma_{0}\beta_{0}$ part of the ejecta, is given by

$\displaystyle t_{\text{peak}}$ $\displaystyle\approx\left(\frac{3M_{0}}{4\pi
nm_{p}\gamma_{0}}\right)^{\frac{1}{3}}\left(\frac{1.5-\sqrt{0.25+2\beta_{0}^{2}}}{c\beta_{0}}\right).$
(9)

####  2.1.2 Non-thermal flux

The solution obtained for $\beta_{\rm SP}(R)$ (and $\beta_{\rm FS}(R)$,
$\beta_{\rm ej,RS}(R)$) by numerically solving the pressure equilibrium and
energy conservation equations described in the preceding subsection, provide,
under the approximation of uniform pressure,velocity, and density within the
shocked plasma region, a full approximate description of the hydrodynamics-
$p(r,t)$, $v(r,t)$, and $n(r,t)$. We use the hydrodynamic profiles to
calculate the specific emissivity at the plasma frame, $j^{\prime}_{\nu}(r,t)$
(following Rybicki & Lightman, 1979), using the phenomenological description
adopted for the magnetic field and for the electron energy distribution-
assuming that the magnetic field and electrons carry fractions
$\varepsilon_{B}$ and $\varepsilon_{e}$ of the internal energy density
respectively, and that the electrons follow a power-law energy distribution.
Using $j_{\nu}$, the specific intensity $I_{\nu}(\hat{\Omega})$ and the
specific flux, $F_{\nu}$, are derived numerically (properly accounting for
relativistic and arrival time effects).

We derive below analytic expressions for $F_{\nu}$, which provide a good (to
10’s of percent) approximation for $F_{\nu}$ obtained numerically as described
above for a wide range of parameters, $5<s_{\rm ft}<12,~{}0.2<\gamma_{\rm
ej,RS}\beta_{\rm ej,RS}<6$. We consider in this derivation only the emission
from the shocked ISM, that dominates over the emission from the shocked ejecta
(as verified by the numeric calculations). The derivation given below provides
the dependence on model parameters. Order unity corrections are applied to the
numeric coefficients obtained in the derivation, in order to improve the
agreement with the numeric results. We do not show in this section comparisons
of the approximations to the numeric results. Instead, we show in § 3 direct
comparison to the full numeric calculation, which includes a full numeric
solution of the hydrodynamics equations (instead of the approximate treatment
used in this subsection).

We first consider the flux produced by a spherical shock driven by a piston of
fixed Lorenz factor $\gamma$, and then derive the flux obtained for the ejecta
under consideration by using the approximate analytic dependence of
$\gamma(t)$ derived in the preceding subsection. Approximating the internal
energy carried by the shocked plasma as $E(R)=\gamma(\gamma-1)M(R)c^{2}$, the
luminosity observed by a distant observer, for the case where electrons lose
all their energy via synchrotron emission on a time scale short compared to
the expansion time, is $L=\varepsilon_{e}dE/dt$. We denote by $\gamma_{c}$ the
(plasma frame) electron Lorenz factor above which the electron energy loss
time is short (compared to the expansion time), and by $\nu_{c}$ the
corresponding (observer frame) synchrotron frequency. Noting that for a power-
law energy distribution of electrons the specific flux at $\nu>\nu_{c}$
follows $F_{\nu}\propto\nu^{-p/2}$, we have for $\nu>\nu_{c}$

$L_{\nu}=2\pi\gamma(\gamma-1)\frac{R^{3}}{t}\varepsilon_{e}nm_{p}c^{2}l_{p}\frac{\nu^{-p/2}}{\nu_{m}^{1-p/2}},$
(10)

where

$l_{p}=\frac{p-2}{1-(\gamma_{\text{max}}/\gamma_{\text{min}})^{2-p}},\quad
l_{p}\xrightarrow[p\to
2]{}\frac{1}{\ln(\gamma_{\text{max}}/\gamma_{\text{min}})},$ (11)

and $\nu_{m}$ is the characteristic observed frequency of the synchrotron
emission of the lowest energy electrons, with $\gamma_{e}=\gamma_{\rm min}$.
The results are not sensitive to the ratio of the maximum to minimum electron
Lorenz factors, $\gamma_{\text{max}}/\gamma_{\text{min}}$, which is expected
to be very large. For our numerical results we use
$\gamma_{\text{max}}/\gamma_{\text{min}}=10^{5}$.

For $\nu<\nu_{c}$, the specific luminosity is suppressed by a factor
$(\nu/\nu_{c})^{1/2}$. The self-absorption frequency, $\nu_{a}$, is expected
to be low, and we therefore consider the emission only at $\nu>\nu_{a}$.

Let us estimate next the frequencies $\nu_{m}$ and $\nu_{c}$. The
characteristic plasma frame frequency of the synchrotron emission of electrons
with Lorenz factor $\gamma_{e}$ and isotropic velocity distribution is
$\gamma_{e}^{2}\sqrt{(\pi u_{B}/2)}e/(m_{e}c)$, where $u_{B}$ is the (plasma
frame) magnetic field energy density. Approximating the post shock energy
density, $2nm_{p}v^{2}$ for non-relativistic and $4\gamma^{2}nm_{p}c^{2}$ for
highly relativistic shocks, by $4\gamma(\gamma-1)nm_{p}c^{2}$, we have
$u_{B}=4\varepsilon_{B}\gamma(\gamma-1)nm_{p}c^{2}$. Defining

$\nu_{s}=\frac{e}{m_{e}}\sqrt{2\pi\varepsilon_{B}nm_{p}},$ (12)

the observer frame synchrotron frequencies are thus

$\nu_{m}=\frac{\gamma_{\rm
min}^{2}\gamma^{-\frac{1}{2}}(\gamma-1)^{\frac{1}{2}}\nu_{s}}{\left(1.5-\sqrt{2\beta^{2}+0.25}\right)},\quad\nu_{c}=\frac{\gamma_{c}^{2}\gamma^{-\frac{1}{2}}(\gamma-1)^{\frac{1}{2}}\nu_{s}}{\left(1.5-\sqrt{2\beta^{2}+0.25}\right)}.$
(13)

$\gamma_{\rm min}$ is obtained by requiring the average energy per electron to
equal a fraction $\varepsilon_{e}$ of the post-shock internal energy per
particle, $(\gamma-1)m_{p}c^{2}$, while $\gamma_{c}$ is obtained by requiring
the (plasma frame) synchrotron loss time,
$m_{e}c^{2}/(\gamma_{e}(4/3)\sigma_{T}cu_{B})$, to be equal to the time
measured at the plasma frame,
$t/\gamma/\left(1.5-\sqrt{2\beta^{2}+0.25}\right)$. This yields

$\displaystyle\gamma_{\rm min}$
$\displaystyle=\frac{l_{p}(\gamma-1)}{p-1}\frac{\varepsilon_{e}m_{p}}{m_{e}},$
(14) $\displaystyle\gamma_{c}$
$\displaystyle=\frac{\left(1.5-\sqrt{2\beta^{2}+0.25}\right)m_{e}}{4(\gamma-1)\varepsilon_{B}\left(\frac{4}{3}\right)\sigma_{T}nm_{p}ct}.$

Eqs. (10-14) provide an approximate description of $L_{\nu}(t)$ for time
independent $\gamma$. In order to obtain an approximate analytic description
of $L_{\nu}$ for the shock driven by the ejecta described by Eq. (1), we
replace the time independent $\beta$ and $R(t)$ in these equations with
$\beta_{\rm SP}(t)$ and $R(t)$ obtained in the preceding sub-section, using
the appropriate time derivative $dE/dt$. We approximate the resulting
functional dependence on $\gamma$ and $\beta$ by power-laws of the form
$\propto(\gamma\beta)^{\lambda}$. This enables us to use Eqs. (4) and (8) to
derive analytic expressions for $\nu_{c}$, $\nu_{m}$ and $L_{\nu}$. For
$\nu_{m}$, for example, we have

$\displaystyle\nu_{m}$
$\displaystyle=\left(\frac{l_{p}(\gamma-1)}{p-1}\frac{\varepsilon_{e}m_{p}}{m_{e}}\right)^{2}\frac{\gamma^{-1/2}(\gamma-1)^{\frac{1}{2}}\nu_{s}}{\left(1.5-\sqrt{2\beta^{2}+0.25}\right)}$
(15)
$\displaystyle\approx\left(\frac{l_{p}\varepsilon_{e}m_{p}}{(p-1)m_{e}}\right)^{2}0.2(\gamma_{\rm
SP}\beta_{\rm SP})^{5.1}\nu_{s},$

where the approximation holds (to 10%) for $0.3<\gamma_{\rm SP}\beta_{\rm
SP}<2.5$. Using this approximation with Eqs. (4) and (8) yields an analytic
expression for $\nu_{m}(t)$. The $(\gamma\beta)^{\lambda}$ power-law
approximations used for $\nu_{c}$ and $L_{\nu}$ are given in App. A, and the
resulting analytic expressions are given in § 2.4.

###  2.2 SvNT phase

At late time, after the reverse shock crosses the entire ejecta and once the
ejecta decelerates to sub-relativistic, $\beta_{\text{s}}\lesssim 0.05$,
velocity, the flow approaches the self-similar behavior described by the SvNT
solution for an explosion with energy given by the initial kinetic energy,
$E$, of the ejecta. At this stage, the shock radius is given by
$R_{\text{ST}}(t)=\zeta\left(\frac{E}{\rho}\right)^{\frac{1}{5}}t^{\frac{2}{5}}$,
where $\zeta$ is an order unity dimensionless parameter depending on the
adiabatic index $\hat{\gamma}$ of the shocked material, $\zeta=1.15$ for
$\hat{\gamma}=\frac{5}{3}$. We therefore estimate the time beyond which the
flow is well approximated by the SvNT solution as

$t_{\text{ST}}\approx\left(0.05\cdot\frac{2.5c}{\zeta}\left(\frac{E}{\rho}\right)^{-\frac{1}{5}}\right)^{-\frac{5}{3}},$
(16)

The synchrotron luminosity produced by the collisionless shock is derived in a
manner similar to that of the preceding sub-section. Assuming that a fraction
$\varepsilon_{e}$ of the internal energy flux through the shock,
$4\pi\frac{4\hat{\gamma}}{(\hat{\gamma}+1)^{2}}\frac{1}{2}\rho
v_{s}^{3}R^{2}$, is carried by electrons, the luminosity at frequencies
$\nu>\nu_{c}$ (where electrons lose all their energy over a time short
compared to the expansion time), is given by

$L_{\nu,\text{ST}}=\frac{4\pi\hat{\gamma}}{(\hat{\gamma}+1)^{2}}\varepsilon_{e}\rho
v_{s}^{3}R^{2}l_{p}\frac{\nu^{-\frac{p}{2}}}{\left(\beta^{2}_{\text{min}}\nu_{m,\text{ST}}\right)^{1-\frac{p}{2}}},$
(17)

where

$\beta_{\text{min}}=\frac{l_{p}}{p-1}\varepsilon_{e}\frac{m_{p}}{m_{e}}\frac{9\beta_{s}^{2}}{32}$
(18)

is the velocity (at the plasma frame) of the lowest energy electrons. Note
that the lowest energy electrons are not relativistic for $\beta_{s}\ll 1$,
and we have therfore used in this calculation an electron energy distribution
that follows a power-law in momentum,
$dn_{e}/d(\gamma\beta)_{e}\propto(\gamma\beta)_{e}^{-p}$ (Krymskii, 1977;
Blandford & Ostriker, 1978; Bell, 1978). The corresponding observer frame
minimum and cooling frequencies are

$\nu_{m,\text{ST}}=\frac{3\beta_{s}}{4\sqrt{2}}\nu_{s},\quad\nu_{c,\text{ST}}=\gamma_{c,\text{ST}}^{2}\frac{3\beta_{s}}{4\sqrt{2}}\nu_{s},$
(19)

where

$\gamma_{c,\text{ST}}=\frac{8m_{e}}{9\beta_{s}^{2}\varepsilon_{B}\left(\frac{4}{3}\right)\sigma_{T}\rho
ct}.$ (20)

The specific flux obtained using Eqs. (17)-(20) is given below by Eqs.
(32-33). The analytic description provides a good approximation to the results
obtained by numerically calculating the synchrotron flux using the analytic
self-similar SvNT hydrodynamic profiles (calculating $j_{\nu}$, $I_{\nu}$ and
$F_{\nu}$ adopting the phenomenological assumptions described above regarding
the magnetic field energy density and the electron energy distribution). This
is demonstrated in figure 2, showing a comparison of the analytic and numeric
results for $\nu_{m,\text{ST}},\nu_{a,\text{ST}}<\nu<\nu_{c,\text{ST}}$.

Figure 2: A comparison of the analytic (solid lines, Eqs. (32-33)) and numeric
(dashed lines) calculations of the synchrotron flux during the phase of non-
relativistic self-similar expansion (where the flow is described by the SvNT
solutions). The parameter values chosen for the calculations are
$E=10^{42}$erg, $\varepsilon_{e}=10^{-1},\varepsilon_{B}=10^{-2}$ and
$n=10^{-2}{\rm cm}^{-3}$. Results are shown for
$\nu_{m,\text{ST}},\nu_{a,\text{ST}}<\nu=3\text{GHz}<\nu_{c,\text{ST}}$ and
observer distance of $D=10$Mpc.

###  2.3 Intermediate phase

We consider here the phase during which the reverse shock propagates through
the shallower, $\beta<\beta_{0}$ part of the ejecta. Using the semi-analytic
calculation of the dynamics described in § 2.1.1, we find that $\gamma_{\rm
ej,RS}\beta_{\rm ej,RS}$ is well described for $1<s_{\rm KN}<3$ and
$0.1<\gamma_{\rm ej,RS}\beta_{\rm ej,RS}<0.7$ by

$\gamma_{\rm ej,RS}\beta_{\rm ej,RS}\propto t^{-3/(4.7+s_{\rm KN})}.$ (21)

The synchrotron flux is then derived in a manner similar to that of § 2.1.2.
The $(\gamma\beta)^{\lambda}$ power-law approximations used for $\nu_{c}$ and
$L_{\nu}$ in this regime are given in App. A, and the resulting analytic
expressions are given in § 2.4.

###  2.4 Summary of the analytic model results

We summarize below the analytic results derived in the preceding sub-sections
for the specific flux at the three time regimes, $t<t_{\rm peak}$, $t>t_{\rm
ST}$ and $t_{\rm peak}<t<t_{\rm ST}$, and provide at the end an interpolation
formula that provides a good approximation for the specific flux at all times.
The numerical values of the parameters are normalized to those expected to be
relevant for NS merger ejecta.

Analytic expressions for the synchrotron emission produced by an ejecta with a
power-law dependence of ejecta energy on momentum, Eq. (2), may be obatined
from the results given in this sub-section by the substitutions

$\displaystyle M_{0}$
$\displaystyle=1.5\frac{E_{0}}{(\gamma_{0}\beta_{0})^{2}c^{2}},$ (22)
$\displaystyle s_{\rm ft}$ $\displaystyle=\alpha_{\rm ft}+2,\quad
s_{\text{KN}}=\alpha_{\text{KN}}+1.5.$

The three characteristic time scales are given by Eqs. (8), (9) and (16)

$\displaystyle t_{R}$
$\displaystyle=51\left(\frac{M_{R,-6}}{n_{-2}}\right)^{\frac{1}{3}}\text{days},$
(23) $\displaystyle t_{\text{peak}}$
$\displaystyle=550g(\beta_{0})\left(\frac{M_{0,-4}}{n_{-2}}\right)^{\frac{1}{3}}\text{days},\quad
g(\beta_{0})=\frac{1.5-\sqrt{0.25+2\beta_{0}^{2}}}{\gamma_{0}^{\frac{1}{3}}\beta_{0}},$
$\displaystyle t_{\text{ST}}$ $\displaystyle=2.9\times
10^{4}\left(\frac{E_{50}}{n_{-2}}\right)^{\frac{1}{3}}\text{days},$

where

$M_{R}\equiv M(\gamma\beta>1)=M_{0}(\gamma_{0}\beta_{0})^{s_{\rm ft}},$ (24)

and we use $n=10^{-2}n_{-2}{\rm cm}^{-3}$, $E=10^{50}E_{50}$erg,
$M_{R}=10^{-6}M_{R,-6}M_{\odot}$ and $M_{0}=10^{-4}M_{0,-4}M_{\odot}$. The
function $g(\beta_{0})$ is plotted in Fig. 3. For an ejecta with a power-law
dependence of ejecta energy on momentum, Eq. (2),

$E_{R}\equiv
E(\gamma\beta>1)=E_{0}(\gamma_{0}\beta_{0})^{\alpha}=\frac{M_{R}c^{2}}{1.5}.$
(25)

Figure 3: The functions $g(\beta_{0})$ and $g_{w,q}(\beta_{0},p,s)$ (Eqs. (23)
and (31)) for $s=7$, $p=2.2$.

The specific flux at $t<t_{\text{peak}}$ is given by Eqs. (10-14)

$\displaystyle F_{\nu,\rm ft}$
$\displaystyle=D_{26.5}^{-2}\varepsilon_{e,-1}^{p-1}f_{\rm ft}(p)$ (26)
$\displaystyle\times\begin{cases}7.7\varepsilon_{B,-2}^{\frac{p-2}{4}}n_{-2}^{\frac{3p-2}{4}}M_{R,-6}^{\frac{2}{3}}\nu_{9.5}^{-\frac{p}{2}}\left(\frac{t}{t_{R}}\right)^{w_{\rm
ft}}m\text{Jy}&\text{for}\quad\nu>\nu_{c},\\\
0.1\varepsilon_{B,-2}^{\frac{p+1}{4}}n_{-2}^{\frac{p+1}{4}}M_{R,-6}\nu_{9.5}^{\frac{1-p}{2}}\left(\frac{t}{t_{R}}\right)^{q_{\rm
ft}}\mu\text{Jy}&\text{for}\quad\nu<\nu_{c},\end{cases}$

with $\varepsilon_{e}=10^{-1}\varepsilon_{e,-1}$,
$\varepsilon_{B}=10^{-2}\varepsilon_{B,-2}$,

$w_{\rm ft}=\frac{5-7.5p+2s_{\rm ft}}{5.5+s_{\rm ft}},\quad q_{\rm
ft}=\frac{4.5-7.5p+3s_{\rm ft}}{5.5+s_{\rm ft}},$ (27)

and

$f_{\rm ft}(p)=960\times 8.8^{-p}(2.5-0.7p)(p-1)^{2-p}(l_{p})^{p-1}.$ (28)

$f_{\rm ft}(p)$ deviates from unity by less than $30\%$ for $2<p<2.5$ (the
origin of $8.8^{-p}(2.5-0.7p)$ factor is in the normalization chosen for the
parameter values and the power-law approximation (App. A)).

The cooling frequency and $\gamma_{\rm SP}\beta_{\rm SP}(t)$ at
$t<t_{\text{peak}}$ are given by Eqs. (4), (8) and (13)

$\displaystyle\nu_{c,\rm ft}$ $\displaystyle=1.9\times
10^{10}\varepsilon_{B,-2}^{-\frac{3}{2}}n_{-2}^{-\frac{5}{6}}M_{R,-6}^{-\frac{2}{3}}\left(\frac{t}{t_{R}}\right)^{\frac{0.7-2s_{\rm
ft}}{5.5+s_{\rm ft}}}\text{GHz},$ (29) $\displaystyle\gamma_{\rm SP}\beta_{\rm
SP}$ $\displaystyle=0.6\left(\frac{t}{t_{R}}\right)^{-\frac{3}{5.5+s}}.$

Notice that in the limit $s_{\rm ft}\rightarrow\infty$, in which the mass
increases very rapidly as the velocity is reduced, we have $\gamma\beta\propto
t^{0}$ and $L_{\nu}\propto t^{3(2)}$ below (above) $\nu_{c}$, corresponding to
no deceleration.

The peak flux, $F_{\nu,\rm ft}(t=t_{\text{peak}})$, is given by

$\displaystyle F_{\nu,\rm peak}$
$\displaystyle=D_{26.5}^{-2}\varepsilon_{e,-1}^{p-1}f_{\rm ft}(p)$ (30)
$\displaystyle\times\begin{cases}170\varepsilon_{B,-2}^{\frac{p-2}{4}}n_{-2}^{\frac{3p-2}{4}}M_{0,-4}^{\frac{2}{3}}\nu_{9.5}^{-\frac{p}{2}}g_{w}m\text{Jy}&\text{for}\quad\nu>\nu_{c},\\\
10\varepsilon_{B,-2}^{\frac{p+1}{4}}n_{-2}^{\frac{p+1}{4}}M_{0,-4}\nu_{9.5}^{\frac{1-p}{2}}g_{q}\mu\text{Jy}&\text{for}\quad\nu<\nu_{c},\end{cases}$

where

$\displaystyle g_{w}$ $\displaystyle=2.3^{w_{\rm
ft}}(\gamma_{0}\beta_{0})^{\frac{s_{\rm ft}}{3}(2-w_{\rm
ft})}(g(\beta_{0}))^{w_{\rm ft}}\approx 1.1(\gamma_{0}\beta_{0})^{2.3p-1},$
(31) $\displaystyle g_{q}$ $\displaystyle=2.3^{q_{\rm
ft}}(\gamma_{0}\beta_{0})^{s_{\rm ft}\left(1-\frac{q_{\rm
ft}}{3}\right)}(g(\beta_{0}))^{q_{\rm ft}}\approx
1.2(\gamma_{0}\beta_{0})^{2.2p-0.5}.$

The functions $g_{w,q}(\beta_{0},p=2.2,s_{\rm ft}=7)$ are plotted in Fig. 3.

The specific flux at $t>t_{\text{ST}}$ is given by Eqs. (16-20)

$\displaystyle F_{\nu,\text{ST}}$
$\displaystyle=D_{26.5}^{-2}\varepsilon_{e,-1}^{p-1}f_{\rm ST}(p)$ (32)
$\displaystyle\times\begin{cases}1\varepsilon_{B,-2}^{\frac{p-2}{4}}n_{-2}^{\frac{3p-2}{4}}E_{50}^{\frac{2}{3}}\nu_{9.5}^{-\frac{p}{2}}\left(\frac{t}{t_{\text{ST}}}\right)^{w_{\text{ST}}}m\text{Jy}&\text{for}\quad\nu>\nu_{c},\\\
0.1\varepsilon_{B,-2}^{\frac{p+1}{4}}n_{-2}^{\frac{p+1}{4}}E_{50}\nu_{9.5}^{\frac{1-p}{2}}\left(\frac{t}{t_{\text{ST}}}\right)^{q_{\text{ST}}}\mu\text{Jy}&\text{for}\quad\nu<\nu_{c},\end{cases}$

with

$w_{\text{ST}}=\frac{20-15p}{10},\quad q_{\text{ST}}=\frac{21-15p}{10},$ (33)

and

$f_{\rm ST}(p)=1.4\times 10^{10}\left(2\times
10^{4}\right)^{-p}(p-1)^{2-p}(l_{p})^{p-1}.$ (34)

$f_{\rm ST}(p)$ is bounded by $0.75<f_{\rm ST}(p)<3$ for $2<p<2.5$ (The origin
of the $\left(2\times 10^{4}\right)^{-p}$ factor is the normalization chosen
for the values of the parameters).

The cooling frequency at $t>t_{\text{ST}}$ is given by Eqs. (19-20)

$\displaystyle\nu_{c,\text{ST}}$ $\displaystyle=3.7\times
10^{8}\varepsilon_{B,-2}^{-\frac{3}{2}}E_{50}^{-\frac{2}{3}}n_{-2}^{-\frac{5}{6}}\left(\frac{t}{t_{\text{ST}}}\right)^{-\frac{1}{5}}\text{GHz},$
(35)

For $t_{\text{peak}}<t<t_{\text{ST}}$ the specific flux is given by

$F_{\nu,\text{KN}}=F_{\nu,\rm
peak}\times\begin{cases}\left(\frac{t}{t_{\text{peak}}}\right)^{w_{\rm
KN}}\text{for}\quad\nu>\nu_{c},\\\
\left(\frac{t}{t_{\text{peak}}}\right)^{q_{\rm
KN}}\text{for}\quad\nu<\nu_{c},\end{cases}$ (36)

with

$w_{\rm KN}=\frac{7.4-7.5p+2s_{\rm KN}}{4.7+s_{\rm KN}},\quad q_{\rm
KN}=\frac{7.5-7.5p+3s_{\rm KN}}{4.7+s_{\rm KN}}.$ (37)

The cooling frequency is given by

$\nu_{c,\text{KN}}=\nu_{c,\text{ft}}(t_{\rm peak})\left(\frac{t}{t_{\rm
peak}}\right)^{\frac{0.5-2s_{\rm ft}}{4.7+s_{\rm ft}}}.$ (38)

The $\nu<\nu_{c}$ specific flux is well described at all time by the
interpolation

$F_{\nu}=\left(0.5F_{\nu,\text{peak}}^{-5}\left(\left(\frac{t}{t_{\text{peak}}}\right)^{-5q_{\rm
ft}}+\left(\frac{t}{t_{\text{peak}}}\right)^{-5q_{\text{KN}}}\right)+F_{\nu,\text{ST}}^{-5}\right)^{-0.2}.$
(39)

Here, $F_{\nu,\text{peak}}$ is given by Eq. (30), $t_{\text{peak}}$ is given
by Eq. (23), $q_{\rm ft}$ is given by Eq. (27), $q_{\text{KN}}$ is given by
Eq. (37) and $F_{\nu,\text{ST}}$ is given by Eq. (32). The specific flux at
$\nu>\nu_{c}$ is well described by a similar expression, replacing $q_{X}$
with $w_{X}$ given by Eqs. (27), (37) and (33).

##  3 Comparison to numeric calculations

We compare in this section the approximate analytic description of the non-
thermal specific flux derived in the preceding section with the results
obtained using numerical solutions of the relativistic hydrodynamics
equations. The emission of radiation was calculated, given a numeric solution
for the flow fields, in a manner similar to that described at the opening of §
2.1.2. The EOS used in the numeric calculations is same as that used for the
approximate hydrodynamic solutions described in § 2.1.1, $p=(\hat{\gamma}-1)e$
with adiabatic index $\hat{\gamma}(e/n)$ varying from $4/3$ to $5/3$ following
the analysis of Synge (1957). As explained at the opening of § 2, the
dependence of the dynamics on the dimensional parameters $\\{M_{0},n,c\\}$
follows directly from dimensional considerations, and the flow depends non-
trivially only on the dimensionless parameters, $\\{\beta_{0},s_{\rm
ft},s_{\rm KN}\\}$. We have explored a wide range of these parameters
$5<s_{\rm ft}<12,~{}1<s_{\rm KN}<3,~{}0.3<\beta_{0}<0.9$, and found good
agreement with our analytic approximation. We give below a few examples.

  1. 1.

In Fig. 4 we compare the numeric solution for the forward shock velocity and
radius with the results of the semi-analytic solution for $t<t_{\rm peak}$
described in § 2.1.1, and with the SvNT solution valid at late times. The good
agreement supports the validity of both the numeric code and the semi-analytic
solution of the dynamics.

  2. 2.

In Fig. 5 we compare the specific flux obtained numerically with that given
analytically in § 2.4, Eq. (39). The agreement is good for a wide range of
relevant values of $\\{\beta_{0},s_{\rm ft},s_{\rm KN}\\}$.

  3. 3.

In Fig. 6 we compare $F_{\nu,\text{peak}}$ obtained numerically with the
analytic expression of Eq. (30) for a range of $\gamma_{0}\beta_{0}$ and
$s_{\rm ft}$ values. The numeric and analytic results are in close agreement.

  4. 4.

In Fig. 7 we compare the numeric logarithmic slope of the temporal dependence
of the specific flux at $t<t_{\rm peak}$, $q_{\rm ft}=d\log F_{\nu}(t)/d\log
t$, with the analytic expression of Eq. (27) for a range of $p$ and $s_{\rm
ft}$ values, with a fixed $\beta_{0}$. The numeric and analytic results are in
close agreement.

  5. 5.

In Fig. 8 we compare $t_{\text{peak}}$ obtained numerically with the analytic
expression of Eq. (23) for a range of $\gamma_{0}\beta_{0}$ and $s_{\rm ft}$
values. The numeric and analytic results are in close agreement.

A comparison of the analytic results with numeric results for a power-law
dependence of ejecta energy on momentum, Eq. (2), is given in § 4.

Figure 4: The forward shock $\gamma\beta$ (upper panel) and radius (lower
panel) as a function of $t_{\rm s}$, the time measured at the rest frame of
the source ejecting the expanding plasma. The results of the numeric solution
of the hydrodynamics (red) are compared with the semi-analytic solution
described in § 2.1.1 (yellow), valid at early time (up to the time at which
the reverse shock crosses the fast tail, $s_{\text{ft}}>5$, $\sim 150$days),
and with the SvNT self-similar solution (red) valid at late times. The
parameter values used for this calculation are $n=3\times
10^{-2}\text{cm}^{-3},M_{0}=2\times 10^{-6}M_{\odot},s_{\rm
ft}=7,s_{\text{KN}}=1.5$ and $\gamma_{0}=1.35$.

Figure 5: A comparison of the specific flux given analytically in § 2.4, Eq.
(39) (solid lines) and the specific flux obtained numerically (dashed lines),
for different values of the dimensionless parameters $\\{s_{\rm ft},s_{\rm
KN},\gamma_{0},p\\}$ and
$\varepsilon_{e}=10^{-1},\varepsilon_{B}=10^{-2},M_{0}=1.2\times
10^{-5}M_{\odot}/5.2\times 10^{-5}M_{\odot}/7.6\times
10^{-7}M_{\odot}/1.1\times 10^{-5}M_{\odot}$ (for $s_{\text{ft}}=6/7/8/9$) and
$n=3\times 10^{-2}\text{cm}^{-3}$. Results are shown for
$\nu_{m},\nu_{a}<\nu=3\text{GHz}<\nu_{c}$ and observer distance of $D=100$Mpc.
Figure 6: A comparison of the peak flux, $F_{\nu,\rm peak}$, given by the
analytic model, Eq. (30) (dashed lines), and that derived from numeric
calculations (’+’ and ’x’ marks), as a function of $p$ and of
$\gamma_{0}\beta_{0}$, for $M_{0}=10^{-4}M_{\odot}$ and $n=7\times
10^{-2}\text{cm}^{-3}$. Results are shown for
$\nu_{m},\nu_{a}<\nu=3\text{GHz}<\nu_{c}$ and observer distance of $D=100$Mpc.
Figure 7: A comparison of the power-law index $q_{\rm ft}$ describing the
$F_{\nu}\propto t^{q_{\rm ft}}$ time dependence of the specific flux before
the peak, given by the analytic model, Eq. (27) (dashed lines), and that
derived from numeric calculations (’+’ and ’x’ marks), as a function of $p$
and of $s_{\rm ft}$, for $M_{R}=10^{-6}M_{\odot},n=3\times
10^{-3}\text{cm}^{-3}$ and $\gamma_{0}=1.35$. Figure 8: A comparison of the
peak time, $t_{\text{peak}}$, given by the analytic model, Eq. (23) (solid
line), and that obtained by numeric calculations (’+’ marks) for a range of
$\gamma_{0}\beta_{0}$ values, for $M_{0}=10^{-4}M_{\odot}$ and $n=7\times
10^{-3}$.

##  4 Comparison to earlier work

We compare in this section our results with analytic results obtained in
earlier papers. Nakar & Piran (2011) have considered a sub-relativistic shell
of uniform $\beta$, and provided estimates of the peak time $t_{\rm peak}$,
which is valid for $\beta\ll 1$, and for the peak flux, which is based on
extrapolation of expressions valid for highly relativistic ejecta.
Kathirgamaraju et al. (2019) (hereafter KGB19) expanded the analysis of Nakar
& Piran (2011), considering a shell with a single power-law energy
distribution, $E(>\gamma\beta)=E_{R}(\gamma\beta)^{-\alpha}$ for
$\beta>\beta_{0}$. They have used the peak flux derived in Nakar & Piran
(2011), included an analysis of the dynamics to derive an expression for
$t_{\rm peak}$ that takes into account relativistic effects and is valid for
$\beta_{0}<0.5$, and used the results of Barniol Duran et al. (2015), that
were derived for highly relativistic ejecta, for the power-law index $q$
describing the $F_{\nu}\propto t^{q}$ time dependence of the specific flux at
$t<t_{\rm peak}$.

Let us first note that the results of KGB19 are not self-consistent. At time
$t<t_{\rm peak}$, when the reverse shock propagates through the
$\beta>\beta_{0}$ ejecta, the emission should depend on the shell energy,
$E_{0}=E(>\gamma_{0}\beta_{0})$, and break/cutoff velocity,
$\gamma_{0}\beta_{0}$, only through the combination
$E_{0}(\gamma_{0}\beta_{0})^{\alpha}=E_{R}$, since $E_{R}$ completely
determines the dynamics at this stage- the shock propagates through the
$E(>\gamma\beta)=E_{R}(\gamma\beta)^{-\alpha}=E_{0}(\gamma_{0}\beta_{0}/\gamma\beta)^{\alpha}$
ejecta and the existence of a break/cutoff at $\gamma_{0}\beta_{0}$ does not
affect the flow up to $t=t_{\rm peak}$. See for example Eqs. (26) and (30)-
the specific flux at $t<t_{\rm peak}$ depends only on $M_{R}$, corresponding
to $E_{R}$, while the peak flux depends on $M_{0}$, corresponding to $E_{0}$.
The expression that KGB19 provide for $F_{\nu}(t)$ depends on $E_{0}$ and
$\gamma_{0}\beta_{0}$ separately, i.e. not only through the combination
$E_{0}(\gamma_{0}\beta_{0})^{\alpha}=E_{R}$. This implies that it would
predict different fluxes at $t<t_{\rm peak}$ for ejecta with the same $E_{R}$,
for which the predicted fluxes should be identical.

In Fig. 9 we compare the specific flux given by our analytic model, using Eq.
(22), for an ejecta with a broken power-law dependence of ejecta energy on
momentum, Eq. (2), with the specific flux obtained by our numeric calculations
and that given by KGB19. Our analytic expressions provide a good approximation
for the numerical results for different values of $\alpha_{\rm ft}$ ($5,7$)
and $\beta_{0}$ ($0.3,0.65$). The analytic expression of KGB19, based on Nakar
& Piran (2011), overestimates the specific flux by typically an order of
magnitude, and significantly underestimate the peak time for $\beta=0.65$.

Figure 9: A comparison of the specific flux given analytically in § 2.4, Eqs.
(22) and (39) (solid lines) with the specific flux obtained numerically
(dashed lines) and that given analytically in KGB19, for ejecta profiles given
by Eq. (2) with different values of the dimensionless parameters
$\\{\alpha_{\rm KN},\alpha_{\rm ft},\beta_{0},p\\}$ and
$\varepsilon_{e}=10^{-1},\varepsilon_{B}=10^{-2},E_{0}=10^{50}\text{erg}/10^{49}\text{erg}$
(upper/lower panel) and $n=3\times 10^{-2}\text{cm}^{-3}$. Results are shown
for $\nu_{m}/\nu_{a}<\nu=3\text{GHz}<\nu_{c}$ and observer distance of
$D=100$Mpc.

The ratio of the peak flux given by Eqs. (30-31) and (22) and that given by
KGB19 is (pre-factor given for $p=2.2$)

$\frac{F_{\nu,\text{peak}}^{\text{KGB19}}}{F_{\nu,\text{peak}}}\approx
6\gamma_{0}^{2.5-2.2p}\beta_{0}^{0.3p-1}.$ (40)

Fig. 10 shows the ratio between the logarithmic slope $q_{\rm ft}=d\log
F_{\nu}(t)/d\log t$ of the temporal dependence of the specific flux at
$t<t_{\rm peak}$ obtained by KGB19,
$q_{\text{ft}}^{\text{KGB19}}=(3\alpha-6(p-1))/(8+\alpha_{\rm ft})$, and our
result, given by Eqs. (27) and (22), for different values of $\alpha_{\rm ft}$
and $p$. Fig. 11 shows the ratio between the peak time, $t_{\rm peak}$,
obtained by KGB19,
$t_{\text{peak}}^{\text{KGB19}}=3.3\left(\frac{E_{0,51}}{n_{-2}}\right)^{\frac{1}{3}}\beta_{0}^{-\frac{2}{3}}\left(\frac{2+\alpha_{\rm
ft}}{\beta_{0}(5+\alpha_{\rm ft})}-1\right)\text{years}$, and our result,
given by Eqs. (23) and (22). As expected, the KGB19 expression provides a good
approximation for $t_{\rm peak}$ for $\beta_{0}<0.5$.

Figure 10: The ratio between the logarithmic slope $q_{\rm ft}=d\log
F_{\nu}(t)/d\log t$ of the temporal dependence of the specific flux at
$t<t_{\rm peak}$ obtained by KGB19 and our result, given by Eqs. (22) and
(27), for different values of $\alpha_{\rm ft}$ and $p$. Figure 11: The ratio
between the peak time, $t_{\rm peak}$, obtained by KGB19, and our result,
given by Eqs. (22) and (23).

##  5 GW170817 KN afterglow

The radio and X-ray observations of the electromagnetic counterpart of
GW170817 are shown in figure 12. The measurements show a growing tension with
the predictions of expanding jet models, but clear evidence for re-brightening
associated with a fast tail is not (yet) present (Hajela et al., 2022; Nedora
et al., 2021a; Balasubramanian et al., 2021, 2022; Troja et al., 2022;
O’Connor & Troja, 2022).

As noted in the introduction, the early evolution of the photospheric radius
is consistent with a power-law mass distribution,
$M(>\gamma\beta)\propto(\gamma\beta)^{-s_{\rm KN}}$ with $s_{\rm KN}\approx
1.6$ at $0.1<\gamma\beta<0.3$ and a total mass of $\approx 0.05M_{\odot}$
(Waxman et al., 2018). We consider the non-thermal emission that would be
produced by a "fast-tail" extension of this power low to higher velocities,
with $s_{\rm ft}$ in the range of 6-8 for $\beta>\beta_{0}=0.3$
($M_{0}=8\times 10^{-3}M_{\odot}$). Figure 12 shows the non-thermal specific
flux predicted by our model for typical values expected for collisionless
shocks, $\varepsilon_{e}=10^{-1},\varepsilon_{B}=0.5\times 10^{-2},p=2.15$,
and conservatively low density, $n=10^{-3}{\rm cm^{-3}}$. The existence of a
fast tail is expected to produce detectable radio and X-ray fluxes over a time
scale of $\sim 10^{4}$days.

Figure 12: The radio and X-ray observations (symbols) of the electromagnetic
counterpart of GW170817 (Troja et al., 2021, 2022; O’Connor & Troja, 2022;
Balasubramanian et al., 2021, 2022), and the radio flux (solid lines)
predicted to be produced by a "fast-tail" extension,
$M(>\gamma\beta)\propto(\gamma\beta)^{-s_{\rm ft}}$ at $\beta>\beta_{0}=0.3$,
of the mass distribution inferred from observations at lower velocity,
$M(>\gamma\beta)\propto(\gamma\beta)^{-1.6}$ at $0.1<\gamma\beta<0.3$ and a
total mass of $0.05M_{\odot}$ (Waxman et al., 2018). X-ray data are scaled by
a factor $3\times 10^{8}$ (corresponding to electron spectral index $p=2.15$).
Model results are shown for $\varepsilon_{e}=10^{-1},\varepsilon_{B}=0.5\times
10^{-2}$ and $n=10^{-3}{\rm cm^{-3}}$.

##  6 Summary

We have derived analytic expressions for the non-thermal synchrotron emission
from spherical ejecta with broken power-law mass or energy distributions given
by Eqs. (1) or (2). The analytic model equations are given in § 2.4. The
analytic model results were compared in § 3 and in § 4 to the results of 1D
numeric calculations for a wide range of ejecta parameter values
characteristic of merger calculation results- a "shallow" mass (energy)
distribution, $1<s_{\rm KN}<3$, for the bulk of the mass (at
$\gamma\beta\approx 0.2$), and a steep, $s_{\rm ft}>5$, "fast tail" mass (or
energy) distribution. The analytic model provides an accurate (to 10’s of
percent) description of the evolution of the specific flux obtained in the
numeric calculations, including at the phase of deceleration to sub-
relativistic expansion- see figures 5-9. This is a significant improvement
over earlier results, based on extrapolations of results valid for
$\gamma\beta\gg 1$ and $\ll 1$ to $\gamma\beta\approx 1$, which overestimate
the specific flux by an order of magnitude for typical parameter values and do
not describe the emission beyond the time of peak flux (see discussion in § 4
and figures 9-11).

The key results are the time and magnitude of the peak flux (at
$\nu<\nu_{c}$), given by

$\displaystyle t_{\text{peak}}$
$\displaystyle=550g(\beta_{0})\left(\frac{M_{0,-4}}{n_{-2}}\right)^{\frac{1}{3}}\text{days},$
(41) $\displaystyle F_{\nu,\rm peak}$
$\displaystyle=10D_{26.5}^{-2}\varepsilon_{e,-1}^{p-1}f_{\rm
ft}(p)\varepsilon_{B,-2}^{\frac{p+1}{4}}n_{-2}^{\frac{p+1}{4}}M_{0,-4}\nu_{9.5}^{\frac{1-p}{2}}g_{q}\mu\text{Jy},$
$\displaystyle q_{\rm ft}$ $\displaystyle=\frac{4.5-7.5p+3s_{\rm
ft}}{5.5+s_{\rm ft}},\quad q_{\rm KN}=\frac{7.5-7.5p+3s_{\rm KN}}{4.7+s_{\rm
KN}},$

where $q_{\rm ft/\rm KN}$ is the index describing the $F_{\nu}\propto
t^{q_{\rm ft/\rm KN}}$ time dependence of the specific flux before/after the
peak (for $6<s_{\rm ft}<8$, $1<s_{\rm KN}<2.5$ and $2<p<2.2$ we find
$0.5<q_{\rm ft}<1$ and $-1<q_{\rm KN}<0$), and

$\displaystyle g(\beta_{0})$
$\displaystyle=\frac{1.5-\sqrt{0.25+2\beta_{0}^{2}}}{\gamma_{0}^{\frac{1}{3}}\beta_{0}},$
(42) $\displaystyle f_{\rm ft}(p)$ $\displaystyle=960\times
8.8^{-p}(2.5-0.7p)(p-1)^{2-p}(l_{p})^{p-1}\approx 1,$ $\displaystyle g_{q}$
$\displaystyle=2.3^{q_{\rm ft}}(\gamma_{0}\beta_{0})^{s_{\rm
ft}\left(1-\frac{q_{\rm ft}}{3}\right)}(g(\beta_{0}))^{q_{\rm ft}}\approx
1.2(\gamma_{0}\beta_{0})^{2.2p-0.5}.$

Here, $\gamma_{0}\beta_{0}$ is the break between the fast part of the ejecta
($s_{\text{ft}}>5$) and the slower part of the ejecta ($1<s_{\text{KN}}<3$),
see Eq. (1), $M_{0}=10^{-4}M_{0,-4}M_{\odot}$ is the (total) mass of the fast
tail, $n=10^{-2}n_{-2}{\rm cm}^{-3}$ is the ISM density,
$\varepsilon_{e}=10^{-1}\varepsilon_{e,-1}$ and
$\varepsilon_{B}=10^{-2}\varepsilon_{B,-2}$ are the fractions of internal
energy carried by electrons and magnetic fields, and $p$ is the electrons
power-law index, $dn_{e}/d\gamma_{e}\propto\gamma_{e}^{-p}$.

The flux peaks when the reverse shock crosses the steep fast tail,
$\gamma\beta>\gamma_{0}\beta_{0}$, part of the ejecta into the shallower part
at lower velocity. It may be detected in the radio and X-ray bands, e.g. by
the Very Large Array radio telescope (with a limiting sensitivity of $\sim
5\mu$Jy for $1-10$GHz frequencies) and by the Chandra X-ray observatory (with
a limiting sensitivity of $\sim 10^{-4}\mu$Jy for $0.3-10$keV photons for
$\sim 10$ks exposure time).

Our analytic model will enable a reliable direct inference of the values and
uncertainties of model parameters from observations. While numeric
calculations may provide higher accuracy predictions, the analytic model has a
great advantage when inferring model parameters from observations, since it
does not require detailed resource consuming calculations for each point in
the model’s multi-parameter space.

The late time non-thermal emission of GW170817 was considered in $\S$5. Figure
12 shows the radio and X-ray flux that would be produced by a fast tail,
extending the observationally inferred
$M(>\gamma\beta)\propto(\gamma\beta)^{-s_{\rm KN}}$ with $s_{\rm KN}\approx
1.6$ for $\beta<0.3$ (Waxman et al., 2018), to higher velocity as
$M(>\gamma\beta)\propto(\gamma\beta)^{-s_{\rm ft}}$ with a range of $s_{\rm
ft}$ values. Using typical values for $\varepsilon_{e,B}$ and $p$, and
adopting a conservatively low density, $n=10^{-3}{\rm cm^{-3}}$, we find that
the existence of a fast tail is expected to produce a detectable (few $\mu$Jy)
radio and ($10^{-15}{\rm erg/cm^{2}s}$) X-ray fluxes over a time scale of
$\sim 10^{4}$days.

## Acknowledgements

We thank Jonathan Morag for insightful discussions during the conduction of
this work. EW’s research is partially supported by ISF, GIF and IMOS grants.

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##  Appendix A Power-law approximations

The $(\gamma\beta)^{\lambda}$ power-law approximations used for the derivation
of the non-thermal specific flux produced by the fast tail in Eqs. (27-29) are
(accurate to 10% for $0.3<\gamma\beta<2.5$)

$\nu_{c}\propto\frac{\left(1.5-\sqrt{2\beta^{2}+0.25}\right)}{\gamma^{\frac{1}{2}}(\gamma-1)^{\frac{3}{2}}}\approx(\gamma\beta)^{-3.9},$
(43)
$L_{\nu}(\nu>\nu_{c})\propto\frac{(\gamma-1)^{(p+2)/2}\gamma^{\frac{3-p}{2}}\beta^{3}}{\left(1.5-\sqrt{2\beta^{2}+0.25}\right)^{\frac{p+4}{2}}}\approx
7.7(1.25-0.35p)(\gamma\beta)^{2.5p+2},$ (44)
$L_{\nu}(\nu<\nu_{c})\propto\frac{(\gamma-1)^{(p+3.5)/2}\gamma^{\frac{3.5-p}{2}}\beta^{3}}{\left(1.5-\sqrt{2\beta^{2}+0.25}\right)^{\frac{p+5}{2}}}\approx
7.7(1.25-0.35p)(\gamma\beta)^{2.5p+4}.$ (45)

The $(\gamma\beta)^{\lambda}$ power-law approximations used for the derivation
of the non-thermal specific flux produced by the slower part of the ejecta in
Eqs. (36-38) are (accurate to 10% for $0.1<\gamma\beta<0.7$)

$\nu_{c}\propto\frac{\left(1.5-\sqrt{2\beta^{2}+0.25}\right)}{\gamma^{\frac{1}{2}}(\gamma-1)^{\frac{3}{2}}}\propto(\gamma\beta)^{-3.3},$
(46)
$L_{\nu}(\nu>\nu_{c})\propto\frac{(\gamma-1)^{(p+2)/2}\gamma^{\frac{3-p}{2}}\beta^{3}}{\left(1.5-\sqrt{2\beta^{2}+0.25}\right)^{\frac{p+4}{2}}}\propto(\gamma\beta)^{2.5p+0.6},$
(47)
$L_{\nu}(\nu<\nu_{c})\propto\frac{(\gamma-1)^{(p+3.5)/2}\gamma^{\frac{3.5-p}{2}}\beta^{3}}{\left(1.5-\sqrt{2\beta^{2}+0.25}\right)^{\frac{p+5}{2}}}\propto(\gamma\beta)^{2.5p+2.2}.$
(48)