# Fallback Accretion Model for the Years-to-Decades X-ray Counterpart to
GW170817

Wataru Ishizaki Kunihito Ioka Center for Gravitational Physics, Yukawa
Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan
Kenta Kiuchi Center for Gravitational Physics, Yukawa Institute for
Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan  Max Planck
Institute for Gravitational Physics (Albert Einstein Institute), Am
Mühlenberg, Potsdam-Golm, 14476, Germany

(Received June 1, 2019; Revised January 10, 2019)

###### Abstract

A new component was reported in the X-ray counterpart to the binary neutron-
star merger and gravitational wave event GW170817, exceeding the afterglow
emission from an off-axis structured jet. The afterglow emission from the
kilonova/macronova ejecta may explain the X-ray excess but exceeds the radio
observations if the spectrum is the same. We propose a fallback accretion
model that a part of ejecta from the neutron star merger falls back and forms
a disk around the central compact object. In the super-Eddington accretion
phase, the X-ray luminosity stays near the Eddington limit of a few solar
masses and the radio is weak, as observed. This will be followed by a power
law decay. The duration of the constant luminosity phase conveys the initial
fallback timescale $t_{0}$ in the past. The current multi-year duration
requires $t_{0}>3$–$30$ sec, suggesting that the disk wind rather than the
dynamical ejecta falls back after the jet launch. Future observations in the
next decades will probe the timescale of $t_{0}\sim 10$–$10^{4}$ sec, around
the time of extended emission in short gamma-ray bursts. The fallback
accretion has not been halted by the $r$-process heating, implying that
fission is weak on the year scale. We predict that the X-ray counterpart will
disappear in a few decades due to the $r$-process halting or the depletion of
fallback matter.

hydrodynamics — accretion, accretion discs — gamma-ray burst: general

††journal: ApJL

##  1 Introduction

Gravitational waves from the first binary neutron star merger event (BNS
merger) GW170817 were observed by the Laser Interferometer Gravitational-Wave
Observatory (LIGO) and the Virgo Consortium (LVC) (Abbott et al., 2017a). This
event was accompanied by a short-duration gamma-ray burst (sGRB) GRB 1780817A
(Abbott et al., 2017b). The superluminal motion of the radio counterpart was
detected, indicating that the relativistic jet was launched from the merged
compact stars (Mooley et al., 2018; Ghirlanda et al., 2019) in the off-axis
direction (e.g., Ioka & Nakamura, 2018, 2019). In addition, a
kilonova/macronova is also associated with GW170817 (Arcavi et al., 2017;
Chornock et al., 2017; Coulter et al., 2017; Cowperthwaite et al., 2017; Drout
et al., 2017; Kasen et al., 2017; Kasliwal et al., 2017; Kilpatrick et al.,
2017; McCully et al., 2017; Nicholl et al., 2017; Shappee et al., 2017; Smartt
et al., 2017; Soares-Santos et al., 2017; Tanaka et al., 2017; Tanvir et al.,
2017). From the modelling of the kilonova/macronova (e.g., Cowperthwaite et
al., 2017; Kasen et al., 2017; Kasliwal et al., 2017; Villar et al., 2017;
Kawaguchi et al., 2018), it is suggested that the merger ejecta consists of at
least two components: the dynamical ejecta and the post-merger ejecta. The
former is released promptly after coalescence in the dynamical timescale
(e.g., Bauswein et al., 2013; Hotokezaka et al., 2013; Radice et al., 2016;
Sekiguchi et al., 2015, 2016; Bovard et al., 2017; Dietrich et al., 2017), and
the latter is released in the secular viscous timescale (e.g., Fernández &
Metzger, 2013; Just et al., 2015; Fernández et al., 2019; Fujibayashi et al.,
2020b).

In GRB 170817A, an off-axis afterglow where an off-axis relativistic jet
interacts with circumstellar medium (CSM), has been observed (Haggard et al.,
2017; Margutti et al., 2018; Troja et al., 2017). The temporal evolution of
the afterglow shows a much slower brightening than seen in top-hat jet
afterglows, which is understood to be an afterglow from a structured jet
(e.g., Lamb et al., 2019; Ghirlanda et al., 2019; Troja et al., 2019;
Beniamini et al., 2020; Takahashi & Ioka, 2020, 2021). The modeling of the
afterglow has been successful, constraining not only the physical quantities
of the jet (e.g., opening angle of the jet, jet luminosity, and structure),
but also the density of CSM (e.g., Margutti et al., 2018; Alexander et al.,
2018; Lamb et al., 2019; Troja et al., 2019; Takahashi & Ioka, 2021).

Recently, the X-ray flux was observed several years after the coalescence
(Hajela et al., 2019, 2020a, 2020b, 2021a; Troja et al., 2020; Balasubramanian
et al., 2021). It has been reported that a 2–3$\sigma$ excess component over
the prediction of the jet afterglow model (Hajela et al., 2019;
Balasubramanian et al., 2021; Hajela et al., 2021a). Hajela et al. (2019)
argued that this is a kilonova/macronova afterglow, which is generated by the
interaction of the merger ejecta with CSM (Nakar & Piran, 2011; Kyutoku et
al., 2014; Takami et al., 2014; Asano & To, 2018; Hotokezaka et al., 2018;
Kathirgamaraju et al., 2019; Nathanail et al., 2021). This model expects an
increase of synchrotron emission also in the radio band. However, no such
radio re-brightening has been observed (Hajela et al., 2019; Balasubramanian
et al., 2021). This may imply a different spectral index.

In this Letter, we present an alternative interpretation for the X-ray excess
in GW170817 (see Figure 1). As shown by many theoretical studies, a part of
the ejecta from the BNS merger is inevitably gravitationally bounded,
regardless of whether they are dynamical or post-merger ejecta. Since the mass
$M$ of the central object of GW170817 is estimated as
$M=2.74_{-0.01}^{+0.04}~{}M_{\odot}$ (Abbott et al., 2017a), its Eddington
luminosity is as follows:

$L_{\mathrm{edd}}=\frac{4\pi GMm_{\mathrm{p}}c}{\sigma_{\mathrm{T}}}\sim
3.4\times
10^{38}~{}\mathrm{erg}~{}\mathrm{s}^{-1}\left(\frac{M}{2.7M_{\odot}}\right),$
(1)

where $m_{p}$ is the mass of the proton and $\sigma_{\rm T}$ is the Thomson
scattering cross section. This is consistent with the X-ray excess luminosity
within a factor, implying that the X-ray excess could come from an accretion
disk near the central compact object. However, it is not clear whether the
fallback accretion in the BNS merger can maintain a super-Eddington accretion
for a few years or not111In fact, Margutti et al. (2018) applied the fallback
accretion model to the early X-ray emission from the jet afterglow and
concluded that the emission from the accretion flow never dominates the X-ray
emission from GW170817. However, the parameter they adopted corresponds to the
fallback of the dynamical ejecta (the mass acretion rate,
$\dot{M}(t=1~{}\sec)\sim 10^{-3}M_{\odot}~{}{\rm s}^{-1}$), which is not
appropriate for considering light curves on an annual scale, as we will see
later.. Thus, in this Letter, by constructing a X-ray light curve model from
fallback accretion, we investigate whether the re-brightening is possible or
not. We also constrain the $r$-process heating rate from the X-ray light curve
by applying the semi-analytic model of the mass accretion inhibited by
radioactive heating due to the $r$-process elements developed in Ishizaki et
al. (2021).

##  2 Fallback rate of the ejecta

Figure 1:  A schematic picture of our model. The post-merger ejecta (shown in
blue) is blown out after the dynamical ejecta (shown in red) which released
immediately after the compact binary coalescence. The post-merger ejecta
inside a radius $r_{\rm esc}$ is still gravitationally bounded and will
fallback into the central compact object. Since the fallback ejecta has finite
angular-momentum, an accretion disk (shown in green) is formed around the
central object. Photons from the accretion disk are mainly observed as X-rays
(shown in orange).

In this section, we estimate the fallback accretion rate for each of the two
types of ejecta: the dynamical ejecta and the post-merger ejecta. The
accretion rate of dynamical ejecta, which is released promptly in the
dynamical timescale after the merger when neutron stars are destroyed by tidal
forces or the impact of the collision, can be modeled based on the numerical
simulations. On the other hand, the fallback rate of the post-merger ejecta,
which is the subsequent disk wind launched mainly by viscous heating in the
accretion disk in the secular timescale, is theoretically unclear, because the
numerical relativity simulations of BNS mergers have not been calculated long
enough to capture this fallback accretion (see Fujibayashi et al., 2020b, for
the best effort). Therefore, for the post-merger ejecta, we estimate the mass
accretion rate by constructing a simple model.

The fallback accretion rate decays in proportion to $t^{-5/3}$ after some
initial time (Rees, 1988; Michel, 1988). Thus we write the mass accretion rate
as the following simple power-law form,

$\dot{M}(t)=\frac{2}{3}\frac{M_{\rm
tot,fb}}{t_{0}}\left(\frac{t}{t_{0}}\right)^{-5/3},$ (2)

where $t_{0}$ is the initial fallback time when the fallback accretion starts,
and $M_{\rm tot,fb}$ is the total fallback mass of the ejecta, which satisfies

$M_{\rm tot,fb}=\int_{t_{0}}^{\infty}\dot{M}(t)dt.$ (3)

In Ishizaki et al. (2021), the accretion rate is calculated based on the
profile of the dynamical ejecta derived from the numerical relativity
simulation in Kiuchi et al. (2017). The fallback accretion rate of the
dynamical ejecta with zero temperature calculated using the same method as in
Ishizaki et al. (2021) is fitted by Equation (2) with $M_{\rm
tot,fb}=4.0\times 10^{-3}M_{\odot}$ and $t_{0}=4~{}{\rm msec}$. We find that
the simple power-law model describes well the expected accretion rate after
$t>1~{}\sec$.

As mentioned earlier, the fallback accretion rate for the post-merger ejecta
is theoretically unclear. The state-of-the-art study by Fujibayashi et al.
(2020b) showed that the post-merger ejecta mass is about
$0.05$–$0.2~{}M_{\odot}$ and its velocity is about $v=0.01c$–$0.1c$. Kawaguchi
et al. (2020) calculated the long term evolution of the post-merger ejecta
using the result of Fujibayashi et al. (2020b) as initial conditions. While
the disk outflow is blowing ($t\lesssim 8$ sec), the resultant angle-averaged
mass density profile of the ejecta is approximately proportional to the power-
law of a radial distance $r^{-2.4}$ for the ejecta with velocity less than
about $0.04c$ (Kawaguchi et al. (2020), private communication, see also
Fujibayashi et al. (2020b)). Hence, we assume that the mass density
distribution of the post-merger ejecta is proportional to $r^{-2.4}$.

In order to estimate the fallback mass for the post-merger ejecta, we
calculate the boundary radius between the gravitationally bounded and
unbounded parts. At this radius $r_{\rm esc}$, the kinetic and gravitational
energies are balanced. If the ejecta at $r_{\rm esc}$ moves under the energy
conservation law, the radius $r_{\rm esc}$ evolves in time according to the
following differential equation:

$\frac{1}{2}\left(\frac{dr_{\rm esc}}{dt}\right)^{2}-\frac{GM}{r_{\rm
esc}}=0.$ (4)

Integrating this over time and evaluating it at $t=t_{0}$, we obtain

$r_{\mathrm{esc}}=\left(\frac{9}{2}GMt_{0}^{2}\right)^{1/3},$ (5)

where we assume that $r_{\rm esc}$ at $t_{0}$ is sufficiently large relative
to the initial position. At $t=t_{0}$, the mass inside $r=r_{\rm esc}$ is the
total mass to fall back $M_{\rm tot,fb}$. Given the mass density distribution
$r^{-2.4}$, the ratio of the fallback mass to the total mass $M_{\rm ej}$ of
the post-merger ejecta, $f_{\rm fb}$, can be calculated as

$f_{\rm fb}=\frac{M_{\rm tot,fb}}{M_{\rm ej}}=\frac{\int_{0}^{r_{\rm
esc}}\rho_{\rm esc}\left(r/r_{\rm esc}\right)^{-2.4}4\pi
r^{2}dr}{\int_{0}^{v_{\rm ej}t_{\rm 0}}\rho_{\rm esc}\left(r/r_{\rm
esc}\right)^{-2.4}4\pi r^{2}dr}=\left(\frac{r_{\rm esc}}{v_{\rm ej}t_{\rm
0}}\right)^{0.6},$ (6)

where $\rho_{\rm esc}$ is the mass density at $r=r_{\rm esc}$ and $v_{\rm ej}$
is the velocity of the post-merger ejecta. Note that the mass $(1-f_{\rm
fb})M_{\rm ej}$ that does not fall back radiates the kilonova/macronova. The
numerical evaluation of the quantities in the post-merger ejecta is as
follows:

$r_{\rm esc}\sim 3.43\times 10^{9}~{}{\rm cm}~{}\left(\frac{t_{0}}{5~{}{\rm
sec}}\right)^{2/3}~{}\left(\frac{M}{2.7~{}M_{\odot}}\right)^{1/3},$ (7)
$f_{\rm fb}\sim 0.72~{}\left(\frac{t_{0}}{5~{}{\rm
sec}}\right)^{-0.2}~{}\left(\frac{M}{2.7~{}M_{\odot}}\right)^{0.2}~{}\left(\frac{v_{\rm
ej}}{0.04c}\right)^{-0.6},$ (8) $M_{\rm tot,fb}\sim 5.72\times
10^{-2}~{}M_{\odot}~{}\left(\frac{t_{0}}{5~{}{\rm
sec}}\right)^{-0.2}~{}\left(\frac{M}{2.7~{}M_{\odot}}\right)^{0.2}~{}\left(\frac{M_{\rm
ej}}{0.08~{}M_{\odot}}\right)~{}\left(\frac{v_{\rm ej}}{0.04c}\right)^{-0.6}.$
(9)

The time $t_{0}$ at which the fallback starts in the post-merger ejecta is for
example the time when the outflow from the central object stops blowing and
the fallback cannot be supported by the ram pressure of the outflow. However,
because the theoretical understanding of the fallback accretion of the post-
merger ejecta is not enough as mentioned above, the exact value of $t_{0}$ is
not known. Therefore we treat $t_{0}$ as a free parameter in this study. For
the mass $M_{\rm ej}$ and velocity $v_{\rm ej}$ of the ejecta, we adopt the
values that are consistent with the kilonova/macronova light curve used in
Kawaguchi et al. (2020), who properly treat the radiation transfer across the
different ejecta components.

##  3 X-ray light curve of the fallback accretion in BNS merger

The ejecta that falls back onto the central object forms an accretion disk,
and emits mainly X-rays as will be discussed later. Here, we assume that the
central object collapses to a black hole (BH) with mass $M=2.7M_{\odot}$ at
least at the time $t\sim 1$ yr. These X-rays are only observable if they are
not absorbed by the kilonova/macronova ejecta in the line of sight (Kisaka et
al., 2016; Murase et al., 2018; Margutti et al., 2018). According to Murase et
al. (2018), the time required for the dynamical ejecta to become optically
thin is estimated as follows:

$t_{\rm thin}=\left(\frac{3\hat{K}_{\rm X}M_{\rm ej}}{4\pi v_{\rm
ej}^{2}}\right)^{1/2}\sim
0.65{\rm~{}yr}\left(\frac{\hat{K}_{\mathrm{X}}}{100\mathrm{~{}cm}^{2}\mathrm{~{}g}^{-1}}\right)^{1/2}\left(\frac{M_{\rm
ej}}{0.08~{}M_{\odot}}\right)^{1/2}\left(\frac{v_{\rm ej}}{0.1c}\right)^{-1},$
(10)

where $\hat{K}_{\mathrm{X}}$ is the typical opacity for a 10 keV X-ray photon.
For lower energy X-ray photons, $\hat{K}_{\mathrm{X}}$ is much larger, and the
expected bound-free opacity of neutral or singly ionized heavy $r$-process
nuclei is about $\hat{K}_{\mathrm{X}}\sim
10^{3}\mathrm{~{}cm}^{2}\mathrm{~{}g}^{-1}$ (e.g., Metzger, 2017; Margutti et
al., 2018). In Equation (10), the value is evaluated for the post-merger
ejecta, because the dynamical ejecta has less mass and higher velocity than
the post-merger ejecta. For the ejecta mass, we adopt $M_{\rm
ej}=0.08~{}M_{\odot}$, which is the mass of the wind component based on the
Kawaguchi et al. (2020). This value is consistent with the observation-based
phenomenological model of kilonova/macronova (e.g., Chornock et al., 2017;
Cowperthwaite et al., 2017; Kasliwal et al., 2017; Villar et al., 2017), since
a large fraction of the mass $M_{\rm rj}$ fallbacks and does not contribute to
the kilonova/macronova (see Equation (6) and (8)). For the ejecta velocity, we
adopt $v_{\rm ej}=0.1c$ not $0.04c$ to evaluate Equation (10). The geometric
configuration of GW170817 is such that the line of sight direction is close to
the axial direction (i.e., the viewing angle $\theta_{\rm v}$ is about
$\theta_{\rm v}\sim 25$ deg (Mooley et al., 2018)). According to Kawaguchi et
al. (2020), ejecta on the polar direction are about twice as fast as those in
the equatorial plane, so that we evaluate them with the parameter $v_{\rm
ej}=0.1c$. On the other hand, if the viscous parameter
$\alpha\sim\mathcal{O}(0.01)$, since the post-merger ejecta is expected to
have a larger electron fraction (see Fujibayashi et al., 2020b) than that of
the dynamical ejecta, heavy $r$-process nuclei would be less formed and then
the value of $\hat{K}_{\mathrm{X}}$ would be smaller (For details, see the
mass energy-transfer coefficient data provided by
NIST222https://www.nist.gov/pml/x-ray-mass-attenuation-coefficients). Then,
after a year, the kilonova/macronova ejecta is expected to become optically
thin to X-rays.

The fallback ejecta converts its gravitational energy into radiation by
viscous heating in the accretion disk. Initially the mass accretion rate is
much larger than the Eddington accretion rate. In this super-Eddington
accretion phase, the luminosity is roughly limited by the Eddington luminosity
$L_{\rm edd}$. Many theoretical calculations of the radiative transport in
super-Eddington accretion show that the isotropic bolometric luminosity seen
by slightly off-axis observers (i.e., for GW170817, $\theta_{\rm v}\sim 25$
deg) is about twice that of $L_{\rm edd}$ (e.g., Ohsuga et al., 2005; Sądowski
& Narayan, 2016; Ogawa et al., 2017; Abarca et al., 2018; Kitaki et al.,
2021). Hence, we estimate the bolometric luminosity in the super-Eddington
phase as follows:

$L_{\rm bol}=f_{\rm edd}L_{\rm edd},$ (11)

where $f_{\rm edd}$ is a constant and we adopt $f_{\rm edd}=2$. As the mass
accretion rate becomes small, the radiative transfer effect becomes less
effective and the radiative efficiency increases. For the mass accretion rate
smaller than the Eddington accretion rate, the bolometric luminosity can be
estimated as follows:

$L_{\rm bol}=\eta\dot{M}c^{2}$ (12)

where $\eta$ is the radiative efficiency and we adopt $\eta=0.1$ (e.g., Ohsuga
et al., 2005). Combining Equations (11) and (12), we estimate the luminosity
with the following simple formula:

$L_{\rm bol}=\min\left(\eta\dot{M}c^{2},f_{\rm edd}L_{\rm edd}\right).$ (13)

Note that the isotropic luminosity already accounts for the beaming effect.

In order to convert the bolometric luminosity $L_{\rm bol}$ to the observed
X-ray luminosity $L_{\rm X}$ in Chandra band ($0.3$–$10$ keV), first of all,
we make a rough estimate of the spectrum of the disk emission. By using
equation (11), we can estimate the effective temperature at the innermost
stable circular orbit $R=\xi R_{\rm s}=3\xi_{3}R_{\rm s}$ as

$T_{\rm eff,max}=\left[\frac{f_{\rm edd}L_{\rm edd}}{4\pi\left(\xi R_{\rm
s}\right)^{2}\sigma}\right]^{1/4}\sim 1.50~{}{\rm
keV}~{}\xi_{3}^{-1/2}~{}f_{{\rm
edd},2}^{1/4}~{}\left(\frac{M}{2.7~{}M_{\odot}}\right)^{-1/4},$ (14)

where $f_{\rm edd,2}=f_{\rm edd}/2$, $R_{\rm s}$ is the Schwarzschild radius
of the central object, and $\sigma$ is the Stefan–Boltzmann constant. For the
disk in the super-Eddington phase, the effective temperature depends on the
radius as $T_{\rm eff}\propto R^{-1/2}$ (Watarai et al., 2000). If we assume
that the emission at each radius is the blackbody radiation at the effective
temperature $T_{\rm eff}$, the emission spectrum $L_{\nu}$ of the disk can be
estimated as follows:

$L_{\nu}=2\int_{\xi R_{\rm s}}^{R_{\rm max}}2\pi RB_{\nu}\left(T_{\rm
eff}(R)\right)dR\propto R^{2}T_{\rm eff}^{3}\propto\nu^{-1},$ (15)

where $R_{\rm max}$ is the outer radius of the disk, $B_{\nu}$ is the
blackbody spectrum, and, here we used the fact that the peak of blackbody
radiation is proportional to $T_{\rm eff}^{3}$ and Wien’s displacement law
$\nu\propto T_{\rm eff}$. Hence, the conversion factor can be calculated as
follows:

$f_{\rm X}=\frac{\int_{0.3~{}{\rm keV}}^{10~{}{\rm keV}}L_{\nu}d\nu}{L_{\rm
bol}}\sim\frac{\log\left[\min({T_{\rm eff,max},10~{}{\rm
keV}})\right]-\log\left[\max({T_{\rm eff,min},0.3~{}{\rm
keV}})\right]}{\log(T_{\rm eff,max})-\log(T_{\rm eff,min})},$ (16)

where $T_{\rm eff,min}=T_{\rm eff,max}(R_{\rm max}/(\xi R_{\rm s}))^{-1/2}$ is
the effective temperature at the outer radius. Evaluating the value of $f_{\rm
X}$ for $R_{\rm max}\lesssim 5000~{}{\rm km}$ gives $f_{\rm X}\sim 1$. More
precisely, in the super-Eddington accretion phase, the system is inevitably
optically thick against absorption and scattering, so that we estimate $f_{\rm
X}$ based on numerical calculations that incorporate the effect of radiative
transfer. According to the Monte-Carlo simulation based on hydrodynamic
simulations of the super-Eddington flow, the spectrum with $10M_{\odot}$ BH as
a central source has a peak at a few keV, not depending largely on the
accretion rate, and has approximately $f_{\rm X}\sim 0.5$–$0.8$ (e.g.,
Kawashima et al., 2012; Schnittman et al., 2013; Narayan et al., 2017; Kitaki
et al., 2017; Kawanaka & Mineshige, 2020). The spectral energy distributions
(SEDs) obtained by these simulations show a flat frequency dependence centered
at a few keV, corresponding to $\Gamma\sim 2$. Hajela et al. (2019) reported a
photon index of $\Gamma={1.23}_{-1.03}^{+1.05}$ for the X-ray emission two
years after the merger, which is consistent with the X-ray from super-
Eddington flow within the uncertainty. Kitaki et al. (2017) investigated the
dependence of the spectrum on the BH mass with $10$–$10^{4}~{}M_{\odot}$, and
found that the larger the BH mass, the broader the spectrum centered at a few
keV. This implies that the smaller the BH mass, the larger $f_{\rm X}$ tends
to be. Therefore, extrapolating the result of Kitaki et al. (2017) to the case
for the BH mass $2.7M_{\odot}$, we adopt $f_{\rm X}=0.7$. Furthermore, we
should also take into account the X-ray absorption by ejecta. Since the mean
density of ejecta evolves in time as $\bar{\rho}\propto t^{-3}$, the temporal
evolution of the optical depth $\tau$ can be evaluated as $\tau=\hat{K}_{\rm
X}\bar{\rho}ct\propto t^{-2}$. At $t=t_{\rm thin}$, the ejecta becomes
optically thin (i.e. $\tau\sim 1$), so that the observed X-ray luminosity is
written as

$L_{\rm X}=f_{\rm X}e^{-\left(t/t_{\rm thin}\right)^{-2}}L_{\rm bol}.$ (17)

Figure 2:  The red thick curve shows the X-ray flux from the fallback
accretion of the post-merger ejecta (disk wind), and the light red thick curve
represents the X-ray flux before the absorption (approximately the Eddington
flux). The thin red dashed-and-dotted line is the power-law of time
($t^{-5/3}$), corresponding to the X-ray emission simply proportional to the
mass accretion rate (i.e., ignoring the radiative transfer and the
absorption). The black dashed thick line shows the time before which the
ejecta is optically thick to the X-rays (see Equation (10)). The parameters
used in the calculation of the post-merger ejecta are the total mass of the
ejecta, $M_{\rm ej}=0.08M_{\odot}$, the velocity of the ejecta, $v_{\rm
ej}=0.04c$, the mass of the central object, $M=2.7M_{\odot}$, and the initial
fallback time, $t_{0}=5$ sec. The red long dashed line represents the case for
$t_{0}=100~{}\sec$ with the other parameters being the same. The break of the
thick curve corresponds to the transition time from the super-Eddington regime
to the sub-Eddington regime (see Equation (19)). The green points are the
results of the Chandra observations performed by Hajela et al. (2019, 2020a,
2020b); Troja et al. (2020). The thin black curve is the model curve for the
jet afterglow by Hajela et al. (2019). The thin dotted curves are the model
curves of the X-ray emission from the kilonova/macronova afterglow by Hajela
et al. (2019), and the difference in color indicates the difference in density
distribution of the fast tail.

In Figure 2, the X-ray luminosity associated with the fallback accretion of
the post-merger ejecta in Equation (13) is shown by the red curve. The dashed
line indicates the time before which the kilonova/macronova ejecta is
optically thick and the X-rays are absorbed. The X-ray light curve shows a
constant luminosity up to a certain time $t_{\rm edd}$, after which it
declines in proportion to $t^{-5/3}$. The time $t_{\rm edd}$ corresponds to
the time when Equations (11) and (12) are equal. For the dynamical ejecta, we
have

$t_{\rm edd}\sim 16~{}{\rm days}~{}\eta_{0.1}^{3/5}~{}f_{{\rm
edd},2}^{-3/5}~{}\left(\frac{t_{0}}{4~{}{\rm
msec}}\right)^{2/5}~{}\left(\frac{M}{2.7~{}M_{\odot}}\right)^{-3/5}~{}\left(\frac{M_{\rm
tot,fb}}{4\times 10^{-3}~{}M_{\odot}}\right)^{3/5}.$ (18)

For the post-merger ejecta, we calculate using Equations (2) and (9) as
follows:

$t_{\rm edd}\sim 1395~{}{\rm days}~{}\eta_{0.1}^{3/5}~{}f_{{\rm
edd},2}^{-3/5}~{}\left(\frac{t_{0}}{5~{}{\rm
sec}}\right)^{0.28}~{}\left(\frac{M}{2.7~{}M_{\odot}}\right)^{-0.48}~{}\left(\frac{M_{\rm
ej}}{0.08~{}M_{\odot}}\right)^{3/5}~{}\left(\frac{v_{\rm
ej}}{0.04c}\right)^{-0.36}.$ (19)

As can be seen from Equation (18), the X-rays from the fallback of the
dynamical ejecta is absorbed by the ejecta on a timescale earlier than a year,
and this cannot be the main component explaining the X-ray excess. On the
other hand, the X-rays associated with the fallback of the post-merger ejecta
can explain the X-ray excess. Figure 3 shows the relationship between the
initial fallback time $t_{0}$ and the mass of the kilonova/macronova ejecta
$(1-f_{\rm fb})M_{\rm ej}$, for a given $t_{\rm edd}$. The longer $t_{\rm
edd}$, the more kilonova/macronova ejecta mass is required for the same
$t_{0}$. Note that for $v_{\rm ej}=0.04~{}c$, at $t_{0}\lesssim 5\sec$,
$f_{\rm fb}$ is roughly about unity, so that the mass of the
kilonova/macronova ejecta is small, but the total ejecta mass itself is very
large.

Our model assumes that the mass accretion rate evolves in time as $t^{-5/3}$,
but there can be variations in the actual temporal evolution (e.g., Cannizzo
et al., 1990; Ogilvie, 1999; Kumar et al., 2008; Metzger et al., 2008). For
example, for the radiatively inefficient disk, the mass accretion rate
decreases in proportion to $t^{-4/3}$. Generally, the temporal evolution of
the light curve diminishing depends on the mass accretion rate and the mass of
the accretion disk. Here, for simplicity, we assume and continue to use
equations (2) and (13) in this letter. Furthermore, equation (19) suggests
that the system continues to be in a super-Eddington phase until now, which
implies that the outflow is ejected from the radiatively inefficient disk. The
effect of the outflow on the radiative transfer has already been incorporated
in the evaluation of $f_{\rm edd}$ and $f_{\rm X}$ based on the results of
numerical simulations. Some of the outflow can interact with the
kilonova/macronova ejecta to form shock and release their kinetic energy.
However, the outflow that is fast enough to catch up with the
kilonova/macronova ejecta is limited to those that escape from the fairly
inner edge of the disk.

Figure 3:  Contours of the timescale of the super-Eddington phase $t_{\rm
edd}$ in the plane of the initial fallback time $t_{0}$ and the mass of the
kilonova/macronova ejecta $(1-f_{\rm ej})M_{\rm ej}$. The red and blue curves
show the cases of $t_{\rm edd}=3~{}{\rm yr}$ and $t_{\rm edd}=10~{}{\rm yr}$,
respectively. The thick and dashed curves show the cases of $v_{\rm
ej}=0.04~{}c$ and $v_{\rm ej}=0.2~{}c$, respectively. The other parameters
used in the calculation are the mass of the central object
$M=2.7~{}M_{\odot}$, the radiation efficiency $\eta=0.1$, and the correction
factor in super-Eddington regime $f_{\rm edd}=2$. The reason why the curves
with $v_{\rm ej}=0.04~{}c$ turn around $t_{0}\lesssim 5~{}\sec$ is because
$f_{\rm fb}\sim 1$. The red shaded region represents the range of the
kilonova/macronova ejecta mass obtained by several observation-based
phenomenological modeling (e.g., Chornock et al., 2017; Cowperthwaite et al.,
2017; Kasliwal et al., 2017; Villar et al., 2017; Kawaguchi et al., 2020).
Note, however, that there is a range of $0.04c$–$0.3c$ in the ejecta velocity
in these models.

##  4 Implication to r-process heating

Ishizaki et al. (2021) showed that the mass accretion in fallback is
suppressed by the heating due to the decay of $r$-process elements. According
to Ishizaki et al. (2021), the mass accretion rate decreases to 10% compared
to the usual power-law behavior $t^{-5/3}$ (see Equation (2)) on a timescale
called the halting time $t_{\rm halt}$, which is determined as the solution to
the following equation,

$t_{\rm halt}=K\left[\frac{\left(GM\right)^{2}}{\dot{q}(t_{\rm
halt})^{3}}\right]^{1/5},$ (20)

where $K\sim 2.6$ is a constant obtained from numerical calculations (see
Ishizaki et al., 2021, for details), and $\dot{q}(t)$ is the radioactive
heating rate per unit mass. With the typical heating rate $\dot{q}\sim
10^{10}~{}{\rm erg}~{}{\rm g}^{-1}~{}{\rm s}^{-1}(t/1~{}{\rm day})^{-1.3}$,
which is suggested by the nucleosynthesis calculations and the
kilonova/macronova modelings (e.g., Wanajo et al., 2014; Tanaka et al., 2017),
the halting time can be estimated as

$t_{\rm halt}\sim 2.4\times 10^{5}~{}{\rm
sec}\left(\frac{M}{2.7~{}M_{\odot}}\right)^{1.82}\left(\frac{\dot{q}(t=1~{}{\rm
day})}{10^{10}~{}{\rm erg}~{}{\rm g}^{-1}~{}{\rm s}^{-1}}\right)^{-2.72}.$
(21)

The halting time is sensitive to the uncertainty of the radioactive heating
rate. As reported by Barnes et al. (2020), due to the uncertainty of nuclear
physics, there is one order magnitude uncertainty of the heating rate at
$\mathcal{O}(1)$ day. The corresponding uncertainty of the halting time is a
factor of $\sim 30$ up and down together (see Equation (21)). Furthermore,
after $\mathcal{O}(10)$ days of the coalescence, the gamma-rays emitted by
radioactive decay can escape from the ejecta without ejecta heating. For
example, for the heating rate calculated by Wanajo et al. (2014), the
effective heating rate drops to about half after $\mathcal{O}(10)$ days. Note
that the fraction of gamma-rays in radioactive heating is also affected by the
uncertainty of nuclear physics. Combining the effects of the insufficient
thermalization and the uncertainty of the heating rate (assuming $0.5$ digit
up and down even at $\mathcal{O}(10)$ years), the uncertainty of the halting
time ranges about $10^{4}$–$10^{8}$ sec (see Ishizaki et al., 2021).

Figure 4:  The blue line shows the condition above which halting occurs, and
if the heating rate is higher than this line, the mass accretion rate
decreases rapidly (i.e., the halting). The red line is a simple power-law
approximation of the radioactive heating rate, as suggested by the modeling of
the kilonova/macronova of GW170817. The black dot with an error bar is the
radioactive heating rate at $t=43$ day as obtained by Kasliwal et al. (2019)
using the mid-infrared observations by Spitzer. In order for the halting not
to have occurred, at the current time $t_{\rm now}=3$ yr (solid black line),
the current heating rate must be less than $1.25\times 10^{5}~{}{\rm
erg}~{}{\rm g}^{-1}~{}{\rm s}^{-1}$ (dashed black line).

After the halting time $t_{\rm halt}$, the mass accretion rate decreases more
rapidly than $t^{-5/3}$. If the heating rate can be written as a simple power-
law form $\dot{q}\propto t^{-p}$, the temporal evolution of the mass accretion
rate after the halting time can be written as

$\frac{\dot{M}}{\dot{M}_{0}}=\left(\frac{t}{t_{\rm
halt}}\right)^{(3p-15)/10}\exp\left[-f_{\rm
PL}\frac{5}{5-3p}\frac{3\sqrt{2}}{\beta_{0}^{3}K^{1/2}}\frac{f_{1}}{f_{0}}\left(1-\left(\frac{t_{\rm
halt}}{t}\right)^{\frac{5-3p}{10}}\right)\right],$ (22)

where $\dot{M}_{0}$ is the mass accretion rate at $t=t_{\rm halt}$ and is
about 10% of the value expected from the usual power-law behavior at $t=t_{\rm
halt}$, and $\beta_{0}\sim 0.43$, $f_{1}/f_{0}\sim 0.5$, $f_{\rm PL}\sim 0.24$
are the parameters of the semi-analytical model of Ishizaki et al. (2021)
calibrated using numerical simulations for a GW170817-like event. According to
this formula, at $t\sim 3$–$4t_{\rm halt}$, the mass accretion rate decreases
rapidly to about 1% for $p=1.1$–$1.5$. Therefore we predict that the X-ray
light curve fades very steeply at least in a few decades, regardless of the
duration of the super-Eddington phase.

Figure 4 shows the heating rate and the conditions above which the accretion
is halted. The red line shows the simple power-law heating $\dot{q}\sim
10^{10}~{}{\rm erg}~{}{\rm g}^{-1}~{}{\rm s}^{-1}(t/1~{}{\rm day})^{-1.3}$ and
the red shaded area shows the uncertainty of the heating rate of the upper and
lower $0.5$ digits (Barnes et al., 2020). Note that the uncertainty in the
heating rate is shown for the $\mathcal{O}(10)$ day time scale in (Barnes et
al., 2020), and that the range of uncertainties would be much broader for
later times ($t\gtrsim 10~{}{\rm day}$) (e.g., large uncertainties in the
contribution of fission, which is the predominant process in the late stage,
and compositional uncertainties, see also Barnes et al. (2020)). Furthermore,
the effective heating rate applied to ejecta after $\mathcal{O}(10)$ days is
several times smaller due to the decreasing thermalization efficiency of gamma
rays (Hotokezaka et al., 2016; Barnes et al., 2016; Waxman et al., 2018; Kasen
& Barnes, 2019; Hotokezaka & Nakar, 2020) (in contrast to the fact that
charged particles such as beta decay still contribute as a heating source)
(see discussion in Ishizaki et al. (2021)). The black dot with an error bar is
the heating rate at $t=43$ day obtained by Kasliwal et al. (2019) using the
mid-infrared observations by Spitzer, which are consistent with the heating
rates shown by the red lines. Note that this data point is the intrinsic
radioactive heating rate, which is corrected for the effects of thermalization
efficiency. This heating rate is estimated from data obtained in a relatively
narrow band, and does not include the spectral information of the emission
lines expected from kilonova/macronova emission during the nebular phase, so
that there is likely to be a large systematic error. Since the current X-ray
light curve suggests the super-Eddington phase, the halting time has not
likely been reached yet. This implies a heating rate profile where the heating
rate at $\mathcal{O}(1)$ day is small in the range of the shaded region, and
the heating of the ejecta by gamma rays becomes inefficient around
$\mathcal{O}(10)$ days. Furthermore, it is suggested that the heating rate at
a few years is $\dot{q}\lesssim 1.25\times 10^{5}~{}{\rm erg}~{}{\rm
g}^{-1}~{}{\rm s}^{-1}$, which is about $10$ times below the power-law
extrapolation of the heating rate necessary for the kilonova/macronova
emission. Note that this suppression factor $\sim 10$ is explained by the
uncertainties of the heating rate and the gamma-ray escape as mentioned above.
According to Barnes et al. (2020), when the fission contributes effectively,
the heating rate becomes relatively large after $\mathcal{O}(10)$ days, so
that our result supports a model in which the fission is not significant.

##  5 Discussion and Conclusion

We have found that the year-scale X-ray excess in GW170817 can be explained by
the X-ray from the fallback accretion of the post-merger ejecta (most likely
disk wind). The observed X-ray flux is consistent with that in the super-
Eddington regime. In this model, the X-ray flux will remain constant or
diminish, but never increase, in contrast to the kilonova/macronova afterglow.
In general, although it depends on the temperature distribution of the
accretion disk, the emission spectrum from the accretion flow is much harder
than the broad non-thermal spectrum of the kilonova/macronova afterglow (see
Narayan et al. (2017); Kitaki et al. (2017); Kawanaka & Mineshige (2020) for
accretion flow, and see Hajela et al. (2019); Balasubramanian et al. (2021)
for the kilonova/macronova afterglow). This is consistent with the fact that
the radio emission of GRB 170817A does not show any re-brightening
(Balasubramanian et al., 2021). In addition, by assuming that the accretion is
still in the super-Eddington phase, we have obtained a constraint on the
current radioactive heating rate. If the heating continues as suggested by the
modeling of kilonova/macronova emission, the mass accretion rate will
completely fade due to the halting process shown in Ishizaki et al. (2021), so
that the super-Eddington regime can not maintain. This suggests that the
thermalization of the gamma-ray due to the radioactive decay of $r$-process
elements becomes insufficient after $\mathcal{O}(10)$ day. Furthermore, our
model favors models in which the heating rate is relatively small after
$\mathcal{O}(10)$ days. According to Barnes et al. (2020), in such models, the
fission is not significant after $\mathcal{O}(10)$ days. Future X-ray
observations will further limit the uncertainty of nuclear physics.

As seen in Equation (19) and Figure 2, if we adopt the observation- and
simulation-based value of parameters (i.e, $M_{\rm ej}=0.08M_{\odot}$,
$v=0.04c$), the time $t_{0}$ when the fallback accretion starts is about $5$
sec or more. According to Fujibayashi et al. (2020b), for the BH torus system
with the viscosity parameter $\alpha=0.05$, the duration of the viscously
driven wind is about $1$ sec. This is characterized by the viscous timescale,
which varies inversely with the viscosity parameter $\alpha$. If $t_{0}=5$ sec
corresponds to this viscous timescale, our result implies $\alpha\sim 0.01$.
Furthermore, Fujibayashi et al. (2020a) showed that for an accretion disk with
$\alpha=0.04$, the disk material with a mass of about $0.1M_{\odot}$ at the
inner edge of the disk accretes to the central object in about $1$ sec after
coalescence. Since this also corresponds to the viscous timescale, a similar
value $\alpha\sim 0.01$ makes the accretion last for $4$–$5$ sec. This is
consistent with the activity time of the jet driving the sGRB (e.g., Hamidani
et al., 2020).

Assuming that the super-Eddington phase is still ongoing, we can estimate the
initial fallback timescale $t_{0}$ by using Equation (19), namely

$t_{0}\sim 3.9~{}\sec~{}\eta_{0.1}^{-2.14}~{}f_{{\rm
edd},2}^{2.14}~{}\left(\frac{t_{\rm edd}}{1300~{}{\rm
day}}\right)^{3.57}~{}\left(\frac{M}{2.7~{}M_{\odot}}\right)^{1.71}~{}\left(\frac{M_{\rm
ej}}{0.08~{}M_{\odot}}\right)^{-2.14}~{}\left(\frac{v_{\rm
ej}}{0.04c}\right)^{1.29}.$ (23)

If we adopt phenomenological ejecta parameters $M_{\rm ej}\sim 0.08M_{\odot}$
and $v_{\rm ej}\sim 0.1$–$0.2c$ which are directly derived from the
kilonova/macronova light curve (e.g., Chornock et al., 2017; Cowperthwaite et
al., 2017; Kasliwal et al., 2017; Villar et al., 2017) rather than simulations
that take ejecta morphology into account, such as in Kawaguchi et al. (2020),
the initial fallback time $t_{0}$ is required to be greater than about $30$
sec. Furthermore, as can be seen from Equation (23), future X-ray observations
will give us the opportunity to explore the properties of the accretion system
on longer time scales (e.g., $t_{0}\sim 100$–$1000$ sec in $10$ years of
observation). It is interesting to note that the initial fallback timescale of
$t_{0}\sim 30$–$10^{4}$ sec approximately coincides to the typical timescale
of extended emission seen in sGRB (Rowlinson et al., 2013; Kisaka et al.,
2017; Kagawa et al., 2019). More work is needed to find the connection of the
post-merger ejecta in the BNS merger and sGRB (e.g., Rosswog, 2007; Lee &
Ramirez-Ruiz, 2007; Rossi & Begelman, 2009; Metzger et al., 2010; Kisaka &
Ioka, 2015; Kisaka et al., 2017; Desai et al., 2019).

Note added: As this Letter was being completed, we learned of an independent
study by Hajela et al. (2021b), which comes to a similar theoretical
conclusion. The excess also becomes $\sim 3.5$–$4.3\sigma$.

We thank Kyohei Kawaguchi and Sho Fujibayashi for providing us the data of the
profile of the post-merger ejecta. We also thank Kazuya Takahashi, Hamid
Hamidani, Tomoki Wada, Koutarou Kyutoku, Masaru Shibata, and Takashi Nakamura
for valuable comments. We thank the participants and the organizers of the
workshops with the identification number YITP-T-19-04, YKIS2019 and
YITP-T-20-19 for their generous support and helpful comments. This work is
supported by Grants-in-Aid for Scientific Research No. 18H01213 (KK),
20H01901, 20H01904, 20H00158, 18H01213, 18H01215, 17H06357, 17H06362, 17H06131
(KI) from the Ministry of Education, Culture, Sports, Science and Technology
(MEXT) of Japan.

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