Vol. 000 No.0, 000–000

11institutetext: School of Mathematics and Computer, Tongling University,
Tongling 244061, Anhui, China<EMAIL_ADDRESS>
22institutetext: School of Foundation, Zhejiang Shuren University, Shaoxing
312028, Zhejiang, China
33institutetext: School of Physics and Electronic Engineering, Nanyang Normal
University, Nanyang, Henan 473061, China 44institutetext: School of Physics,
Peking University, Beijing 100871, China<EMAIL_ADDRESS>
55institutetext: Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, China

# Revise thermal winds of remnant neutron stars in gamma-ray bursts

Shuang Du 11 Tingting Lin 22 Shujin Hou 33 Renxin Xu 4455

(Received 2023 xx xx; accepted 20xx xx xx)

###### Abstract

It seems that the wealth of information revealed by the multi-messenger
observations of the binary neutron star (NS) merger event, GW170817/GRB
170817A/kilonova AT2017gfo, places irreconcilable constraints to models of the
prompt emission of this gamma-ray burst (GRB). The observed time delay between
the merger of the two NSs and the trigger of the GRB and the thermal tail of
the prompt emission can hardly be reproduced by these models simultaneously.
We argue that the merger remnant should be an NS (last for, at least, a large
fraction of $1\rm s$), and that the difficulty can be alleviated by the
delayed formation of the accretion disk due to the absorption of high-energy
neutrinos emitted by the NS and the delayed emergence of an effective viscous
in the disk. Further, we extend the consideration of the effect of the energy
deposition of neutrinos emitted from the NS. If the NS is the central object
of a GRB with a distance and duration similar to that of GRB 170817A, thermal
emission of the thermal bubble inflated by the NS after the termination of
accretion may be detectable. If our scenario is verified, it would be of
interest to investigate the cooling of nascent NSs.

###### keywords:

gamma-ray bursts–stars: neutron

##  1 Introduction

The multi-messenger observations of the binary neutron star (NS) merger event,
GW170817/GRB 170817A/kilonova AT2017gfo, provide abundant information for
relevant topics (Abbott et al., 2017a), for example, the origin of short
gamma-ray bursts (sGRBs; e.g., Abbott et al. 2017b; see Margutti & Chornock
2021 for a review). However, the central object left by the binary NS merger
is still uncertain (Abbott et al., 2017c, 2019). Some works have discussed the
connection between the merger remnant and the corresponding kilonova (Margalit
& Metzger, 2017; Yu et al., 2018) and the GRB afterglow (Piro et al., 2019;
Hajela et al., 2022; Ren & Dai, 2022). Likewise, there may be also a certain
connection between the remnant and the GRB prompt emission, for example, Tong
et al. (2018) had considered a magnetar collapse scenario for the prompt
emission (but, this model is not supported by the afterglow observation and
difficult to explain the spectral evolution of the prompt emission).

The burst triggered $\sim 1.74\rm s$ after the merger of the two NSs (Abbott
et al., 2017b). _Fermi_ -GBM observation of GRB 170817A (Goldstein et al.,
2017) shows that the burst has two periods that a main pulse from $t_{\rm
t}-0.320\rm s$ to $t_{\rm t}+0.256\rm s$ and a lower-significance tail from
$t_{\rm t}+0.832\rm s$ to $t_{\rm t}+1.984\rm s$, where $t_{\rm t}$ is the GRB
trigger time. The spectrum of the main pulse shows a non-thermal feature that
can be best fitted by a Comptonized function. However, the spectrum of the
weak tail indicates a black-body radiation with temperature $T_{\rm
th}=10.3\pm 1.5\rm keV$.

The abundant observations give many constraints on GRB models and make these
models faultily (see Margutti & Chornock 2021 for a review). The basic
difficulty is that, if the thermal tail of GRB 170817A is reliable, physical
models are hard to naturally reproduce the observed time delay of $\sim
1.74\rm s$ and the black-body radiation with temperature $T_{\rm th}=10.3\pm
1.5\rm keV$, simultaneously.

In Section 2, we clarify the conflictions between observations and models and
further show a solution to the difficulty that, if the merger remnant is an
NS, the delay can be partly resulted by the delayed disk formation due to the
absorption of high-energy neutrinos emitted by the NS and delayed emergence of
effective viscous in the accretion disk. In Section 3, we extend the
consideration shown in Section 2 and discuss the electromagnetic signal of the
thermal bubble inflated by the NS. Section 4 is the summary and discussion.

##  2 conflictions and the solution

###  2.1 Conflictions

The prompt emission of GRB 170817A is different from previous GRBs with
thermal components being detected. There is no such an evident interval and
boundary between the power-law and thermal components in pervious ones (as
summarized in Li 2019). As shown in Figure 6 of Goldstein et al. (2017), the
flux of the soft tail is compatible to that of the main pulse in $10-50\rm
keV$. This means that if there is a similar thermal component in the main
pulse, this thermal component should be visible too 111This is an important
aspect that has been overlooked and indicates the separation between the
radiation region of the main gamma-ray pulse and the soft thermal tail.. The
Comptonized component followed by a thermal component challenges the classical
framework of GRBs. (i) According to the widely accepted model that the prompt
emission of a GRB is powered by the energy release in the GRB jet after the
jet turning into optically thin (e.g., Rees & Meszaros 1994; see Zhang 2018
for more details), if the jet is hot enough at the photosphere, the thermal
component should be detected at the first but not following the Comptonized
main pulse (see also in Abbott et al. 2017b); (ii) No matter how the jet is
produced, the jet should be uniform on its component at the line of sight (but
not the stratified jet; Cheng et al. 2021). The production of the jet that a
cold segment followed by a hot segment is confusing. All these facts indicate
that just considering the dissipation of jet energy (e.g., Rees & Meszaros
1994; Zhang & Yan 2011) cannot explain the time delay and thermal tail,
uniformly. To explaining these characteristics, some geometrical effects
should be invoked.

The absence of evident thermal component in the main pulse implies that there
should be a separation between the radiation regions (as well as relevant
parameters) of the two components. Meng et al. (2018) showed that an off-axis
high-latitude-enhanced photosphere emission plus an on-axis photosphere
emission of a structured jet can explain the observed spectrum. However, the
Lorentz factor along the line of sight of the jet is too large, $\sim 20$,
which is deviated from the constraint of the GRB afterglow (e.g., see Figure 7
in Margutti & Chornock 2021), and a significant delay between the merger and
jet launching is required since the photosphere is too close to the central
engine so that the time delay of $1.74\rm s$ cannot be totally attributed to
the time wasting of jet moving. Gottlieb et al. (2018) showed another scenario
that the hard-to-soft spectral evolution can be induced by the transition (a
planar to a spherical phase) of the emission arose from a shock breakout of a
cocoon from the merger’s ejecta. Similarly, this scenario also requires a
delayed jet injection. Therefore, to explain the time delay between GRB
170817A and GW170817 and the hard-to-soft spectral evolution simultaneously,
under both of the structured-jet scenario and jet-cocoon scenario (should be
the only two jet profiles that allowed by the observed properties of GRB
170817A; Abbott et al. 2017b; see Geng et al. 2019 for simulation side), a
delayed jet injection is required. Granot et al. (2017) suggested that the
merger remnant is a short-lived (lifetime $\leq 1\rm s$) hypermassive NS, and
the jet only can be launched after the hypermassive NS collapses into a black
hole. However, this consideration is difficult to be supported by both of
theory (see, Liu et al. 2017 for a review; even the remnant is an NS, the
accretion is still ongoing) and observation (Jordana-Mitjans et al., 2022).

###  2.2 A possible solution

Under the classical GRB framework (see Zhang 2018 for more details), the
delayed jet injection indicates the delayed disk formation since the jet is
launched by the central object-disk system. Simulations show that the disk is
formed during several dynamical time scale of the system after the merger
($t_{\rm for}\sim 10{\rm ms}$) regardless of the central object (Shibata &
Hotokezaka, 2019). However, the type of the central object may affect the
formation of the disk. According to the observations of SN1987A (Kunkel et
al., 1987; Hirata et al., 1987; Bionta et al., 1987), the temperature of the
nascent NS is several $\rm MeV$ and the NS can stay very hot for a few seconds
after the birth (see Arnett et al. 1989 for a review). If the central object
of GRB 170817A is an NS, comparing with the black-hole case, there will be an
extra energy shedding from the central engine that a strong neutrino emission
and neutrino-driven wind can emerge from the nascent NS (Duncan et al., 1986;
Qian & Woosley, 1996). The absorption of high-energy neutrinos may turn the
gravity-bound ejecta into free so that the formation of the disk will be put
off until some of the ejecta falls back due to, for example, viscosity and
friction in the ejecta and the cooling of the NS. Therefore, the disk
formation would be determined by the competition between the release of
gravitational potential energy and energy deposition in the gravity-bound
ejecta. The time necessary for the gravity-bound ejecta to absorb enough
energy to overcome the gravity can be estimated as (Perego et al., 2014)

$\displaystyle t_{\rm fre}\sim\frac{e_{\rm gra}}{\dot{e}_{\rm neu}}=0.02{\rm
s}\left(\frac{M_{\rm ns}}{2.5{\rm M_{\odot}}}\right)\left(\frac{R_{\rm
in}}{30\rm km}\right)\left(\frac{L_{\nu_{\rm e}}}{3\times 10^{52}\rm
erg\;s^{-1}}\right)^{-1}\left(\frac{\xi}{1.5}\right)^{-1}\left(\frac{E_{\nu}}{15\rm
MeV}\right)^{-2},$ (1)

where $e_{\rm gra}$ is the specific gravitational energy, $R_{\rm in}$ is the
inner radius of the disk which is assumed to be close to the innermost stable
circular orbit of a Schwarzschild black hole with the same mass as the NS,
$\dot{e}_{\rm neu}$ is the specific heating rate provided by neutrino
absorption at a radial distance $R_{\rm in}$ from the centre, $M_{\rm ns}$ is
the NS mass, $L_{\nu_{\rm e}}$ is the luminosity of electron neutrinos from
the NS, $\xi$ is the factor to describe the aspect ratio of the disk, and
$E_{\nu}$ is the neutrino energy from the NS. Comparing $t_{\rm for}$ and
$t_{\rm fre}$, one can find that they are of the same order, that is, it is
possible to put off the formation of the disk through the absorption of high-
energy neutrinos222If the value of $t_{\rm fre}$ is smaller than that of
$t_{\rm for}$ but larger than the duration of the accretion, there will be a
pause of the accretion, as well as the jet launching, when the accretion last
for $t_{\rm fre}$, and then energy release in the foregoing jet will appear as
a precursor. . The upper limit of the delay of disk formation due to the
absorbing of neutrinos can be estimated as the cooling timescale of the NS,
that is (Perego et al., 2014)

$\displaystyle t_{\rm cool,ns}\sim\frac{3R_{\rm ns}^{2}}{c\lambda_{\rm
N\nu}}\approx 0.7{\rm s}\left(\frac{R_{\rm ns}}{15\rm
km}\right)^{2}\left(\frac{\rho_{\rm ns}}{10^{14}\rm
g\;cm^{-3}}\right)\left(\frac{E_{\nu}}{15\rm MeV}\right)^{2},$ (2)

where $R_{\rm ns}$ is the radius of the NS, $\lambda_{\rm N\nu}$ is the mean
free path of neutrinos in the NS which is given by

$\displaystyle\lambda_{\rm N\nu}\approx 7.44\times 10^{3}{\rm
cm}\left(\frac{\rho_{\rm ns}}{10^{14}\rm
g\;cm^{-3}}\right)^{-1}\left(\frac{E_{\nu}}{15\rm MeV}\right)^{-2},$ (3)

and $\rho_{\rm ns}$ is the matter density of the NS.

Besides, to start the accretion and launch the jet (e.g., through neutrino
pair annihilation; see Liu et al. 2017 for a review)333The tangible surface of
the NS will prevent Blandford-Znajek process (Blandford & Znajek, 1977) even
if the NS has an ergosphere. However, other collimation mechanisms for
Poynting-flux-dominated jets under magnetar case may exist (e.g., Bucciantini
et al. 2012)., the disk should transfer angular momentum and generate thermal
energy. It is difficult to estimate when the viscous works. If only the main
gamma-ray pulse of GRB 170817A is produced by the jet and the thermal tail
originates from other reasons (e.g., from the cocoon), to match the duration
of the main pulse $t_{\rm 90,m}\approx 0.58\rm s$ (Goldstein et al., 2017),
the viscosity parameter (Shakura & Sunyaev, 1973) can be estimated as

$\displaystyle\alpha\sim t_{\rm 90,m}^{-1}\left(\frac{H}{R_{\rm
out}}\right)^{-2}\Omega_{\rm k}^{-1}\approx 0.02\left(\frac{t_{\rm
90,m}}{0.6\rm
s}\right)^{-1}\left(\frac{H/R}{1/3}\right)^{-2}\left(\frac{M_{\rm ns}}{2.5\rm
M_{\odot}}\right)^{-1/2}\left(\frac{R_{\rm out}}{100\rm km}\right)^{3/2},$ (4)

where $H$ is the disk height, $H/R$ is the aspect ratio of the disk, and
$R_{\rm out}$ is the outmost radius of the disk. Usually, without some
magnetic instabilities, the value of the viscosity parameter cannot be up to
$\sim 0.01-0.1$ (see Kato et al. 2008 for more details). Therefore, if
effective magnetic field cannot emerge in the disk timely, the jet launching
may be also put off. As an example, if the seed of the magnetic field in the
disk is from the NS, only the magnetic field of the NS is generated (e.g., in
$\sim 1\rm s$ after the formation of the proto-NS; Thompson & Duncan 1993),
can the disk begin to amplify this magnetic seed. Note that, a large-scale
magnetic field in the disk should be essential. The magnetic field can prevent
proton deposition in the jet core, so that excessive mass loading due to the
neutrino-driven wind in the jet (Perego et al., 2014) will be avoided.

###  2.3 An interim summary

Based on the assumptions that the thermal gamma-ray tail of GRB 170817A is
real and the structured jet and the jet-cocoon structure are the only two
allowed jet profiles to explain the thermal tail, we get the following claims.
(a) The incompatibility between observations and models of GRB 170817A
indicates the requirement of the delayed jet injection. (b) Although
quantitative estimation is absence due to the unknown cooling and
magnetization of nascent NSs, neutrino deposition and delayed emergence of
effective viscous in the disk should be physical mechanisms that may result a
fraction of the time delay of $\sim 1.74\rm s$ (e.g., $\sim 0.7\rm s$ after
the merger as required in Gottlieb et al. 2018).

The delay may not be the only effect of the energy deposition of neutrinos in
GRBs. As long as the duration of a GRB is short enough and the central object
of the GRB is an NS, when the accretion terminates, the NS could be still hot
enough, so that a clearer fireball (compared with the neutrino-driven wind
launched before the termination of the accretion) will be inflated through
neutrino pair annihilation (Goodman et al., 1987) and even thermal radiation
(Usov, 2001). Although, GRBs should be powered by energy release in jets, it
is meaningful to revise the evolution of the fireball since it is directly
powered by the NS and may reveal the information of the GRB central engine and
test above idea. In the next section, we discuss possible electromagnetic
counterpart of the fireball.

##  3 Possible connection between NSs and precursors

###  3.1 Basic schematic diagram

After the onset of the binary NS merger, the dynamically mass ejection begins
to be ejected, and the proto-NS soon forms later, as well as the neutrino-
driven wind from the NS (Duncan et al., 1986; Qian & Woosley, 1996)444Note
that the neutrino-driven wind will also impact the gravity-bound ejecta and
impede the accretion. However, once the system begins to accrete matter, it
means that the release of gravitational potential energy dominates the system
and the energy deposition due to absorbing neutrinos becomes a secondary
effect. . Almost simultaneously (e.g., in $\sim 1\rm s$ after the merger
according to the above discussion), an accretion disk emerges. The disk and
the NS make up the GRB central engine, so that the relativistic jet begins to
be launched. Together with the jet launching, baryonic wind driven by
neutrinos from the NS and disk (Ruffert et al., 1997; Dessart et al., 2009;
Perego et al., 2014) is accompanying. After the termination of the accretion,
a fireball with less baryonic mass loading and totally powered by the NS is
inflated. As the cooling of the NS is very fast (Burrows & Lattimer, 1986),
this fireball should be less energetic than the fireball expected in Paczynski
(1986, 1990). Hereafter, this less energetic and clearer fireball is called as
thermal bubble. Thanks to the jet breaking the ejecta, the interspace left by
the jet will be continually filled up by the matter extended from the thermal
bubble. That is, the thermal bubble will mainly inflates along the direction
of the jet moving, and the inflation along other directions will be suppressed
by this prior distribution of energy and the constraint of surrounding matter.
Correspondingly, two bunches will extend from the thermal bubble (see Fig. 1).
As long as the power of the bunch is large enough, the bunch will rush out
from the ejecta following the jet ultimately. Empirically, according to the
isotropic luminosity of on-axis low-luminosity GRBs (e.g., GRB 060218, Campana
et al. 2006; GRB 100316D, Starling et al. 2011)555The fluxes of the X-ray
afterglows of these GRBs are continuous attenuation which is different from
that of the off-axis GRB (GRB 170817) and consistent with the expectation of
on-axis GRBs. Therefore, we believe these GRBs are viewed on-axis and cannot
be explained via jet-cocoon scenario., the critical isotropic luminosity of
the bunch which can make the bunch break out from the surrounding matter
should be

$\displaystyle L_{\rm c}\sim 10^{46}{\rm erg\;s^{-1}}.$ (5)

Although the cooling of the nascent NS is highly uncertain (Yakovlev &
Pethick, 2004; Page et al., 2006), observation (Arnett et al., 1989) and
simulation (Qian & Woosley, 1996) indicate that the temperature of the nascent
NS should maintain $10^{10}-10^{9}\rm K$ for several seconds. Therefore, as
long as the lifetime of the accretion disk is short enough, the total energy
of the thermal bubble inflated in several seconds may be up to
$10^{48}-10^{49}\rm erg$ as expected in Goodman et al. (1987). So, the
luminosity of the bunch can beyond the critical value. With the cooling of the
NS, the luminosity of the thermal bunch will be larger than the power supplied
by the NS at a certain time. If the bunch gets out of the encirclement after
this time point, the totally energy of the thermal bubble will be reduced, as
well as the energy density and pressure. Therefore, ultimately, the escape
route left by the jet will be closed by the squeezing of the surrounding
matter due to the decrease of the energy density in the thermal bubble, and
then the bunch will be cut off. After the breakout, the bunch will be finally
transparent for the thermal radiation. In the next, we discuss the
detectability of the thermal radiation.

Figure 1: Schematic diagram for the evolution of the thermal bubble.

###  3.2 The photosphere emission of the bunch

If the bunch can break out of the ejecta, the fundamental problem is when or
where the breakout occurs. We need to know the velocities of the ejecta and
bunch. The expansion time of the ejecta is $\sim R_{\rm ns}/v_{\rm e}$, where
$R_{\rm ns}$ is the radius of the NS, and $v_{\rm e}$ is the velocity of the
ejecta. According to the duration of sGRBs, the lifetime of the accretion disk
should be more likely longer than $R_{\rm ns}/v_{\rm e}$. So, after the
termination of the accretion, the ejecta should be in the self-similar
adiabatic expansion stage with an constant velocity ($\sim 0.1c-0.3c$, where
$c$ is the speed of light). Similarly, the bunch also soon inflates to the
maximum speed since the faster jet has cleared the way for the bunch666If the
jet power is sufficiently larger than the critical value defined by equation
(1), we can expect that the acceleration of the jet is almostly unaffected by
the ejecta. Then the acceleration of the followed bunch is also unaffected
since the jet should be faster.. The saturation radius of the bunch can be
estimated as $R_{\rm b,c}\sim\eta_{\rm b}R_{\rm ns}$ (Piran et al., 1993),
where $\eta_{\rm b}$ is the dimensionless entropy of the bunch (i.e., the
maximum Lorentz factor of the bunch). The Lorentz factor of the jet is usually
$\eta_{\rm j}\sim 10^{2}-10^{3}$. Comparing with the jet, the bunch is less
energetic and clearer. Therefore, the order of magnitude of the maximum
Lorentz factor of the bunch that $\eta_{\rm b}\sim 10^{1}$ is adopted, and
then the saturation radius of the bunch is $R_{\rm b,c}\simeq\eta_{\rm
b}R_{\rm ns}\sim 10^{7}\rm cm$. If the breakout radius of the thermal bunch,
$R_{\rm b,b}$, is large enough that $R_{\rm b,b}\gg R_{\rm b,c}$, the velocity
of the bunch is relativistic in the most of the time before breaking out of
the ejecta. Under this case, the breakout radius can be estimated as

$\displaystyle R_{\rm b,b}$ $\displaystyle\sim$ $\displaystyle\frac{{v}_{\rm
b}v_{\rm e}\Delta t}{{v}_{\rm b}-v_{\rm e}}=3.0\times 10^{9}{\rm
cm}\left(\frac{v_{\rm e}}{0.1c}\right)\left(\frac{t_{\rm a}}{1\rm
s}\right)\left(\frac{{v}_{\rm b}/({v}_{\rm b}-v_{\rm e})}{1}\right),$ (6)

where $\Delta t$ is the time interval between the ejection of the ejecta and
the inflation of the thermal bubble, and ${v}_{\rm b}\approx c$ is the
velocity of the bunch. Comparing $R_{\rm b,c}$ and $R_{\rm b,b}$, one can find
that the estimation is self-consistent.

To see the thermal radiation from the bunch, the bunch must reach its
photosphere radius, $R_{\rm ph,b}$, and there should be $R_{\rm ph,b}>R_{\rm
b,b}$. As discussed above, the saturation radius of the bunch is much smaller
than the breakout radius of the bunch. Therefore, only the case that $R_{\rm
b,b}>R_{\rm b,c}$ is left. That is to say, when the bunch reaches the
photosphere, it has already passed the saturation radius. Under this case, the
photosphere radius is given by (Mészáros & Rees, 2000; Zhang, 2018)

$\displaystyle R_{\rm ph,b}=\frac{\sigma_{\rm T}L_{\rm b}Y}{8\pi\eta_{\rm
b}^{3}m_{\rm p}c^{3}}\simeq 2.2\times 10^{11}{\rm cm}\left(\frac{L_{\rm
b}}{10^{49}\rm erg\;s^{-1}}\right)\left(\frac{\eta_{\rm b}}{30}\right)^{-3}Y,$
(7)

where $\sigma_{\rm T}$ is the electron Thomson cross section, $L_{\rm b}$ is
the isotropic luminosity of the bunch, $Y$ is the pair multiplicity parameter,
and $m_{\rm p}$ is the proton mass. As a comparison, the photosphere radius of
the jet is

$\displaystyle R_{\rm ph,j}=\frac{\sigma_{\rm T}L_{\rm j}Y}{8\pi\eta_{\rm
j}^{3}m_{\rm p}c^{3}}\simeq 2.2\times 10^{10}{\rm cm}\left(\frac{L_{\rm
j}}{10^{51}\rm erg\;s^{-1}}\right)\left(\frac{\eta_{\rm
j}}{300}\right)^{-3}Y,$ (8)

where $L_{\rm j}$ is the jet power, and $\eta_{j}$ is the dimensionless
entropy of the jet. Thus, it is unnecessary to overly worry about the
shielding of the jet to the thermal radiation from the bunch. However, we
should consider the cover of the jet prompt emission to the thermal radiation
from the bunch.

According to the normal GRB scenario, the prompt emission is emitted at the
internal shock radius which is given by (Rees & Meszaros, 1994)

$\displaystyle R_{\rm is}\simeq 2\eta_{\rm j}^{2}c\Delta t=6\times 10^{11}{\rm
cm}\left(\frac{\eta_{\rm j}}{100}\right)^{2}\frac{\Delta t}{1\rm ms},$ (9)

where $\Delta t$ is the temporal gap between ejecting the two shells of the
jet. When the tail of the jet reaches the internal shock radius, the bunch has
arrived at

$\displaystyle r\sim\beta_{\rm b}c\cdot\left(\frac{R_{\rm is}}{\beta_{\rm
j}c}\right)=\frac{\beta_{\rm b}}{\beta_{\rm j}}R_{\rm is},$ (10)

where $\beta_{\rm b}=v_{\rm b}/c\approx 1-1/(2\eta_{\rm b}^{2})$, and
$\beta_{\rm j}\approx 1-1/(2\eta_{\rm j}^{2})$. Correspondingly, if the
termination of the prompt emission is set as the tail of the jet passing the
internal shock radius, we have the following three situations.

  * I)

If $r<R_{\rm ph,b}$, the thermal radiation from the bunch will be never
covered by the prompt emission.

  * II)

When $r>R_{\rm ph,b}$, comparing with the prompt emission, the thermal
radiation from the bunch will be emitted earlier at $R_{\rm ph,b}$. At this
time, the tail of the jet has arrived at $\sim R_{\rm ph,b}\beta_{\rm
j}/\beta_{\rm b}$. Therefore, there will be a catch-up problem. In the lab
frame, the jet moves with $\beta_{\rm j}$ and leaves $R_{\rm ph,b}$ first.
After a time interval $\delta t$, the thermal radiation leaves $R_{\rm ph,b}$
and begins to pursue the jet (the jet has arrived at $\sim R_{\rm
ph,b}\beta_{\rm j}/\beta_{\rm b}$ at this time). If the thermal radiation from
the bunch is observed after the termination of the prompt emission, the travel
distance which makes the thermal radiation from the bunch catch up with the
jet should satisfy (i.e., when the thermal photon arrives at the internal
shock radius, it has yet catch up with the jet)

$\displaystyle 2\eta_{\rm j}^{2}c\delta t>R_{\rm is}-R_{\rm ph,b},$ (11)

where

$\displaystyle\delta t$ $\displaystyle\sim$ $\displaystyle\left(\frac{R_{\rm
ph,b}}{\beta_{\rm b}c}\beta_{\rm j}c-R_{\rm ph,b}\right)/(\beta_{\rm j}c)$
(12) $\displaystyle\approx$ $\displaystyle\frac{R_{\rm ph,b}}{\beta_{\rm
j}c}\left(\frac{\beta_{\rm j}}{\beta_{\rm b}}-1\right).$

In this situation, a small value of $R_{\rm is}$ is needed, for example,
$R_{\rm is}<\sim 100R_{\rm ph,b}$ by taking $\eta_{\rm j}=100$ and $\eta_{\rm
b}=10$.

  * III)

Alternatively, when $r>R_{\rm ph,b}$, if the thermal radiation is detected in
front of the prompt emission, there should be

$\displaystyle 2\eta_{\rm j}^{2}c\delta t^{\prime}<R_{\rm is}-R_{\rm ph,b},$
(13)

where

$\displaystyle\delta t^{\prime}\sim\left[R_{\rm ph,b}\left(\frac{\beta_{\rm
j}}{\beta_{\rm b}}-1\right)+\beta_{\rm j}ct_{\rm 90}\right]/(\beta_{\rm j}c).$
(14)

Similarly, we have $R_{\rm is}>\sim 100R_{\rm ph,b}+2\times 10^{4}ct_{90}$
with the same parameters as that of the situation (II).

Once the above three cases are satisfied, a black-body radiation with
luminosity (Mészáros & Rees, 2000; Zhang, 2018)

$\displaystyle L_{\rm ph,b}$ $\displaystyle=$ $\displaystyle L_{\rm
b}\left(\frac{R_{\rm ph,b}}{R_{\rm b,c}}\right)^{-2/3}$ (15) $\displaystyle=$
$\displaystyle 3.2\times 10^{46}{\rm erg\;s^{-1}}\left(\frac{L_{\rm
b}}{10^{49}\rm erg\;s^{-1}}\right)^{7/12}\left(\frac{R_{\rm ns}}{1.2\times
10^{6}\rm cm}\right)^{-5/6}\left(\frac{\eta_{\rm
b}}{30}\right)^{8/3}Y^{-2/3},$

temperature

$\displaystyle T_{\rm obs}$ $\displaystyle=$ $\displaystyle\left(\frac{L_{\rm
b}}{4\pi R_{\rm ns}^{2}g_{0}\sigma_{\rm B}}\right)^{-1/4}\left(\frac{R_{\rm
ph,b}}{R_{b,c}}\right)^{-2/3}$ (16) $\displaystyle=$ $\displaystyle 31.3{\rm
keV}\left(\frac{L_{\rm b}}{10^{49}\rm
erg\;s^{-1}}\right)^{-5/12}\left(\frac{R_{\ast}}{1.2\times 10^{6}\rm
cm}\right)^{-5/6}\left(\frac{\eta_{\rm b}}{30}\right)^{11/3}Y^{-2/3}$

and duration being several seconds will be emitted, where $g_{0}$ is the
effective degree of freedom of the equipartition theorem, and $\sigma_{\rm B}$
is the Stefan-Boltzmann constant. Note that,in situation (I) and situation
(II), the prompt emission should be followed by the thermal radiation, and in
situation (III), the thermal radiation should be followed by the prompt
emission. If an on-axis sGRB with the similar distance as that of GRB 170817A
is detected in the future, the discussion shown in this section may can be
tested.

##  4 Summary and discussion

In this paper, we suggest a scenario to eliminate the conflictions between
observations and models of GRB 170817A under the assumption that the remnant
of the binary NS merger is an NS (last for, at least, a large fraction of
$1\rm s$). We argue that a fraction of the time delay of $\sim 1.74\rm s$
between GW170817 and GRB 170817A can be resulted by the absorption of high-
energy neutrinos emitted by the central NS and delayed emergence of effective
viscous in the disk. If our scenario is true, when the central object of an
on-axis GRB with the similar distance and duration as that of GRB 170817A is
an NS, a weak thermal pulse should be detected before or after the main gamma-
ray pulse. Besides, our suggestion is hopefully to be tested when the merger
remnant is a black hole as long as the delayed jet injection is mainly induced
by the delayed disk formation since, when the merger remnant is a black hole,
there should be an evident decrease of the delay. Once the idea presented here
is tested, the time of the delayed jet injection is meaningful to investigate
the cooling of nascent NSs, as well as the constraint of the equation of state
of NSs.

It is worth noting that, although, we consider a bunch with the similar cross
sectional area (CSA) as that of the jet, the CSAs of the two may be different.
As long as the thermal bubble shrinks to satisfy equation (5), the bunch can
break out from the ejecta. It is no reason to demand that the bunch is exactly
shaped by the channel left by the jet. Therefore, the CSA of the bunch may be
larger than the CSA of the jet. This could be another mechanism to form
sheathes of sGRB jets (then the jet-bunch scenario could be a replacement of
the jet-cocoon scenario for explaining the thermal tail of GRB 170817A).

##  5 Acknowledgement

We thank the anonymous referee for very useful comments that have allowed us
to improve our paper. We would like to thank Drs. Liang Li and Chengjun Xia
for useful discussion. This work is supported by the National SKA Program of
China (2020SKA0120100), research projects of Henan science and technology
committee (212300410378) and the NSFC grants (U1938116).

##  6 Data Availability

no new data generated.

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