# On the ‘Loose’ Constraint from IceCube Neutrino Non-Detection of GRB 230307A

Xin-Ying Song Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

The recent extremely bright gamma-ray burst (GRB), GRB 230307A from a binary
neutron star merger may offer a good probe for the production of GRB-
neutrinos. Within the constraint from IceCube neutrino non-detection, the
limitations for key physical parameters of this burst are extracted in
different scenarios including the fireball, Poynting-flux-dominated (PFD) and
hybrid jet. Different from the former nearby ‘monsters’ and due to its smaller
isotropic equivalent radiated energy ($E_{\gamma,\rm iso}\sim 4\times 10^{52}$
erg), the constraint seems loose if non-thermal neutrinos produced from
photomeson interactions are the only consideration. However, a quasi-thermal
neutrino emission from hadronuclear processes is constrained in this neutron-
rich post-merger environment, and the upper limit of the allowed nucleon
loading factor is about a few. Based on this, a discussion is presented on the
possible prompt emission mechanism and jet composition for GRB 230307A in the
context of multi-messenger astrophysics.

Gamma-ray bursts(629); Neutrino astronomy (1100)

##  1 Introduction

For decades, gamma-ray bursts (GRBs) are proposed to be ultra-high-energy
cosmic-ray (UHECR) accelerators and power potential sources of astrophysical
high energy neutrinos in the GRB fireball scenario (e.g. Paczynski & Xu, 1994;
Waxman, 1995; Vietri, 1995; Hümmer et al., 2012). There are mainly two
mechanisms for the production of the neutrinos relevant to GRBs: 1) the
photomeson ($p\gamma$) process (e.g. Waxman & Bahcall, 1997), in which the
photons of GRB are guaranteed photons; as predicted in a model-dependent
theory, the prompt emission sites are different, which could affect the
fluence of this kind of non-thermal (NT) neutrinos (e.g. Zhang & Kumar, 2013).
2) Hadronuclear ($pn$, $pp$ and $nn$) interactions (e.g. Bahcall & Mészáros,
2000; Koers & Giannios, 2007; Murase et al., 2013; Bartos et al., 2013) which
produce quasi-thermal (QT) neutrinos during the expanding of hot jet material
of the outflow; neutrons could be decoupled from protons if the bulk Lorentz
Factor ($\Gamma$) is large enough. Thus inelastic collisions could occur and
pions could be produced. The neutrinos with 5-10 GeV could be generated after
pions decays; alternatively, if the coasting occurs earlier than $np$
decoupling, the dissipation of neutrons via internal collisions between the
compound flows may happen, which could also generate QT neutrinos in the sub-
TeV range (e.g. Murase et al., 2013). Note that the neutrino emission could
occur during the prompt phase of GRB as well as the (choked) jet’s propagation
within the macronova/kilonova ejecta or a massive progenitor envelope (e.g.
Mészáros & Waxman, 2001; Murase & Ioka, 2013; Senno et al., 2016; Kimura et
al., 2018), as a precursor for GRBs. However, till now, no evident correlation
between neutrino events and observed GRBs is observed (e.g. Aartsen et al.,
2017; Abbasi et al., 2023a).

It is usually proposed that long GRBs are from the collapses of massive stars,
while short GRBs are produced from mergers of compact stars. However, there
are some exceptions, such as GRB 060614 (e.g. Gehrels et al., 2006) and GRB
211211A (e.g. Yang et al., 2022). Recently, another exception, GRB 230307A,
associated with kilonova emission (Levan et al., 2023b; Fermi GBM Team, 2023;
Xiong et al., 2023; Svinkin et al., 2023) arises. The isotropic-equivalent
radiated energy $E_{\gamma,\rm iso}\sim 4\times 10^{52}$ erg of GRB 230307A is
smaller compared with other nearby bright GRBs (e.g., $E_{\gamma,\rm iso}\sim
10^{54}$ erg for GRB 130427A and $E_{\gamma,\rm iso}\sim 2\times 10^{54}$ erg
for GRB 221009A), thus it could be inferred that the features of the burst can
not be well constrained if the neutrino emission from $p\gamma$ interactions
is the only consideration. However, with the favoured redshift $z\sim 0.065$
(e.g. O’Connor et al., 2023; Levan et al., 2023a), $d^{2}_{L}$ of GRB 230307A
is about 6 times smaller than that of GRB 221009A, and 37 times smaller than
that of GRB 130427A, where $d_{L}$ is the distance of the GRB source.
Therefore it may be still a good probe for the GRB-neutrinos.

It is worth to note that the IceCube Neutrino Observatory provides an upper
limit (U.L.) of muon neutrinos ($\nu_{\mu}$), $1.0$ GeV cm-2 at 90% C.L.
(IceCube Collaboration, 2023), which seems less constraining compared with
$3.9\times 10^{-2}$ GeV cm-2 at 90% CL of GRB 221009A IceCube Collaboration
(2022). Besides, there exists a significant photospheric emission in the
prompt phase in GRB 230307A, which is different from synchrotron radiation-
dominated GRB 221009A whose jet is dominated by the Poynting flux (Yang et
al., 2023). The emission mechanism as well as the jet composition could affect
the production of the neutrinos (Hümmer et al., 2012; Gao et al., 2012), thus
one may wonder the limitations on key features of this burst within this
‘loose’ constraint in different scenarios of emission mechanism and the jet
composition.

The paper is organized as follows: in Section 2, the limitations for key
physical parameters of GRB 230307A are extracted within the constraint of the
neutrino fluence in different GRB-neutrino scenarios; in Section 3, basic
properties of the prompt emission for GRB 230307A are introduced and
discussed; the discussion and summary are given in Section 4.

##  2 The possible neutrino emissions from GRB 230307A

In this section, we consider the possible neutrino emissions via hadronuclear
processes as well as $p\gamma$ interactions. For convenience, we use spectral
parameters of GRB 230307A in $[2.2,11.7]$ s after the trigger time:
$\alpha=-0.8$, $\beta=-8$ and $E_{\rm peak}=1.1$ MeV. The duration is taken to
be 40 s and $E_{\gamma,\rm iso}=4\times 10^{52}$ erg. For comparison, GRB
221009A is also analyzed, with parameters of $\alpha=-1.0$, $\beta=-2.3$,
$E_{\rm peak}=1.0$ MeV, a duration time of 327 s and $E_{\gamma,\rm
iso}=2\times 10^{54}$ erg. The convention $Q=10^{n}Q_{n}$ is adopted for CGS
units hereafter.

###  2.1 photomeson ($p\gamma$) process

The decay chains of inelastic photomeson interactions are

$\displaystyle\rm p+\gamma\rightarrow n+\pi^{0},$ (1)
$\displaystyle\rm\pi^{0}\rightarrow\gamma+\gamma,$ (2)

and

$\displaystyle\rm p+\gamma\rightarrow p+\pi s,$ (3)
$\displaystyle\rm\pi^{+}\rightarrow\mu^{+}+\nu_{\mu},$ (4)
$\displaystyle\rm\pi^{-}\rightarrow\mu^{-}+\overline{\nu}_{\mu},$ (5)
$\displaystyle\rm\mu^{+}\rightarrow e^{+}+\nu_{e}+\overline{\nu}_{\mu},$ (6)
$\displaystyle\rm\mu^{-}\rightarrow e^{-}+\overline{\nu}_{e}+\nu_{\mu}.$ (7)

In the processes shown in (3)-(7), one pion could produce four leptons. Before
deducing formulae, some important definitions are introduced as below:

  * •

the fractions of energy dissipated in protons, random magnetic fields (with a
strength $B$), and radiated as $\gamma$-rays are $\epsilon_{p}$,
$\epsilon_{\rm B}$ and $\epsilon_{\rm\gamma}$. The proton energy is estimated
via $E_{\rm proton}=\epsilon_{p}/\epsilon_{e}E_{\gamma,\rm iso}$, with the CR
loading factor $\xi_{\rm CR}=\epsilon_{p}/\epsilon_{e}$; note that the mark
(′) in the superscript denotes that the quantity is in the comoving frame,
while an overline denotes the rest frame of proton, hereafter; one has
$B^{\prime}=\left[\frac{L_{\rm
tot}(\epsilon_{B}/\epsilon_{e})}{2R^{2}_{d}\Gamma^{2}c}\right]^{1/2}$ in the
comoving frame, where $L_{\rm tot}$ (or $E_{\rm tot}$) is the total luminosity
(or energy);

  * •

the energy distribution of accelerated protons by random magnetic fields is
$\frac{dN_{p}}{dE_{p}}\propto E_{p}^{-p}$, where the index $p=2$ is assumed;
$E_{p,\rm max}$ and $E_{p,\rm min}$ represent the maximum and minimum energies
of accelerated protons in the observer’s rest frame;
$\frac{dN_{p}}{dE_{p}}=\frac{(\epsilon_{p}/\epsilon_{e})E_{\rm tot}}{{\rm
ln}(E_{p,{\rm max}}/E_{p,{\rm min}})}E_{p}^{-p}$, where $E_{\rm
proton}=\epsilon_{p}E_{\rm tot}$; the minimum energy of proton is set to be
$E_{p,\rm min}=\Gamma m_{p}c^{2}$ assuming it is rest in the comoving frame;
the maximum energy $E_{p,\rm max}$ is the energy of protons which are
accelerated by the magnetic field, and estimated by equaling the dynamical
timescale $t_{\rm dyn}^{\prime}\sim R/\Gamma c$ with the accelerating
timescale of protons $t_{\rm acc}^{\prime}\sim E_{p}^{\prime}/(eB^{\prime}c)$;

  * •

$R_{\rm d}$ is the dissipation radius: for photospheric (PH) emissions,
$R_{\rm d,PH}\sim 10^{11}$ cm; for IS mechanism, $R_{\rm d,IS}\sim 10^{12-13}$
cm; for ICMART mechanism, $R_{\rm d,ICMART}\sim 10^{15}$ cm; they could be
estimated from the parameters of the burst;

  * •

the cross section of $p\gamma$ interaction is $\sigma_{p\gamma}=5\times
10^{-28}$ cm-2; $\pi^{+}$ decay time scales is $\tau_{\pi^{+}}=2.8\times
10^{-8}$ s; in the comoving frame, the Lorentz factor of $\pi^{+}$ is denoted
as $\gamma^{\prime}_{\pi^{+}}$. $\varepsilon_{\gamma}$ denotes the specific
energy of a photon; $\gamma_{p}$, $\beta_{p}$ is the Lorentz Factor and
velocity of a specific proton in the comoving spectrum, and so on;

  * •

the width of the Breight-Weigner form of $\Delta(1232)$ resonance is
$\Delta\overline{\varepsilon}_{\rm pk}\sim 0.2$ GeV;
$\overline{\varepsilon}_{\rm pk}\simeq 0.3$ GeV are the photon energy at the
resonance peak in the photomeson process.

The pion production rate is deduced to calculate the neutrino spectrum111The
followed deduction could be find in many works, or see a review (e.g. Kimura,
2022) and references therein.. $\sigma_{p\gamma}$ is given as a function of
the photon energy in the proton-rest frame,
$\overline{\varepsilon}_{\gamma}=\gamma_{p}^{\prime}\varepsilon_{\gamma}(1-\beta_{p}^{\prime}\mu)$,
where $\gamma_{p}^{\prime}=\varepsilon_{p}^{\prime}/(m_{p}c^{2})$,
$\beta_{p}^{\prime}$, $\varepsilon_{\gamma}^{\prime}$, $\mu=\cos\theta_{p}$
are the Lorentz factor of protons, the proton velocity, the photon energy, and
the angle between the directions of interacting proton and photon in the
comoving frame, respectively. The energy loss rate of protons by pions
production is written as

$t_{p\gamma}^{-1}=c\int d\Omega\int
d\varepsilon_{\gamma}^{\prime}(1-\beta_{p}^{\prime}\mu)n_{\gamma}(\varepsilon_{\gamma}^{\prime},~{}\Omega)\sigma_{p\gamma}(\overline{\varepsilon}_{\gamma})\kappa_{p\gamma}(\overline{\varepsilon}_{\gamma}),$
(8)

where
$n_{\gamma}(\varepsilon_{\gamma}^{\prime},~{}\Omega)=dN/(d\varepsilon^{\prime}_{\gamma}dVd\Omega)$
and $\kappa_{p\gamma}$ is the inelasticity of the $p\gamma$ interactions. This
rate is approximately equivalent to the pion production rate. Then, to convert
the integration variable from $\mu$ to $\overline{\varepsilon}_{\gamma}$, the
photomeson production rate is represented as

$t_{p\gamma}^{-1}=\frac{c}{2\gamma_{p}^{\prime 2}}\int_{\varepsilon_{\rm
th}}^{\infty}d\overline{\varepsilon}_{\gamma}\sigma_{p\gamma}\kappa_{p\gamma}\overline{\varepsilon}_{\gamma}\int_{\overline{\varepsilon}_{\gamma}/(2\gamma^{\prime}_{p})}^{\infty}\frac{d\varepsilon^{\prime}_{\gamma}}{\varepsilon_{\gamma}^{\prime
2}}n_{\varepsilon^{\prime}_{\gamma}},$ (9)

where we use
$n_{\gamma}(\varepsilon^{\prime}_{\gamma},~{}\Omega)=n_{\varepsilon^{\prime}_{\gamma}}/(4\pi)$,
$n_{\varepsilon^{\prime}_{\gamma}}=dN/(d\varepsilon^{\prime}_{\gamma}dV)$,
$\beta_{p}^{\prime}\sim 1$. Note that the energy threshold $\varepsilon_{\rm
th}$ is determined by the Breit-Weigner mass and width of $\Delta(1232)$
resonance. We have

$\sigma_{p\gamma}\kappa_{p\gamma}\approx\sigma_{\rm pk}\kappa_{\rm
pk}\Delta\overline{\varepsilon}_{\rm
pk}\delta(\overline{\varepsilon}_{\gamma}-\overline{\varepsilon}_{\rm pk}).$
(10)

The pion production efficiency, or the fraction of CR protons producing pions
is given by $f_{p\gamma}={\rm min}(1,t_{p\gamma}^{-1}/t_{\rm dyn}^{-1})$. The
distribution of the target photons in the form of

$\displaystyle n_{\gamma}(\varepsilon_{\gamma})$ $\displaystyle=$
$\displaystyle\frac{dN_{\gamma}(\varepsilon_{\gamma})}{d\varepsilon_{\gamma}}$
(13) $\displaystyle=$ $\displaystyle
n_{\gamma,b}\left\\{\begin{array}[]{cc}\varepsilon_{\gamma,b}^{-\alpha}\varepsilon_{\gamma}^{\alpha},&\varepsilon_{\gamma}<\varepsilon_{\gamma,b}\\\
\varepsilon_{\gamma,b}^{-\beta}\varepsilon_{\gamma}^{\beta},&\varepsilon_{\gamma}\geq\varepsilon_{\gamma,b},\end{array}\right.$

where $\varepsilon_{\rm\gamma,b}=\frac{(\alpha-\beta)E_{\rm peak}}{\alpha+2}$
and $n_{\gamma,b}$ is the specific photon number of
$\varepsilon_{\rm\gamma}=\varepsilon_{\rm\gamma,b}$. About half of the
$p\gamma$ inelastic interactions produce charged pions with the energy of
$\varepsilon_{\pi}\approx 0.2\varepsilon_{p}$. The high-energy pions lose
their energies by either synchrotron radiation or dynamical expansion before
they decay to neutrinos. Considering the average energy fraction in pion
decaying to four leptons, $\varepsilon_{\nu}=1/4\varepsilon_{\pi}$ and half
decays to charged pions in $p\gamma$ inelastic interaction, the differential
total muon neutrino energy is

$E_{\nu}^{2}N_{E_{\nu}}\approx\frac{1}{8}f_{p\gamma}f_{\pi,\rm
sup}{E^{2}_{p}}N_{E_{p}},$ (14)

where $f_{\pi,\rm sup}$ is the pion cooling suppression factor, and
approximated to be

$f_{\pi,\rm sup}\approx 1-\exp(-\frac{t^{\prime}_{\pi,\rm
cl}}{t^{\prime}_{\pi,\rm dec}}),$ (15)

where $t^{\prime}_{\pi,\rm cl}=6\pi
m_{\pi}^{4}c^{3}/(m_{e}^{2}\sigma_{T}B^{\prime 2}\varepsilon^{\prime}_{\pi})$
is the pion cooling time, and $t^{\prime}_{\pi,\rm
dec}=\varepsilon^{\prime}_{\pi}\tau_{\pi}/(m_{\pi}c^{2})$ is the decay time of
charged pions. If
$\varepsilon^{\prime}_{\pi}<\varepsilon^{\prime}_{\rm\pi,cooling}$
($\varepsilon^{\prime}_{\rm\pi,cooling}=\frac{\Gamma}{1+z}\sqrt{\frac{6\pi
m_{\pi}^{5}c^{5}}{m_{e}^{2}\sigma_{T}\tau_{\pi}B^{\prime 2}}}$ is the critical
cooling energy estimated by equating $t^{\prime}_{\pi,\rm syn}$ and
$t^{\prime}_{\pi,\rm dec}$), the cooling effect is not efficient, and
$f_{\pi,\rm sup}\approx 1$, while $f_{\pi,\rm
sup}\approx(\varepsilon^{\prime}_{\pi}/\varepsilon^{\prime}_{\pi,\rm
cooling})^{2}$ for
$\varepsilon^{\prime}_{\pi}>\varepsilon^{\prime}_{\rm\pi,cooling}$.

Considering the cooling process which causes the second break in neutrino
spectrum222The second break is caused by $E_{p,\rm max}$ if the pion cooling
is not significant. and Equations (9$)\mbox{--}$(14), the neutrino spectrum
could be described as a power law with two breaks,

$\displaystyle n_{\nu}(E_{\nu})$ $\displaystyle=$
$\displaystyle\frac{dN_{\nu}(E_{\nu})}{dE_{\nu}}$ (19) $\displaystyle=$
$\displaystyle
n_{\nu,1}\left\\{\begin{array}[]{cc}\varepsilon_{\nu,1}^{\alpha_{\nu}}E_{\nu}^{-\alpha_{\nu}},&E_{\nu}<\varepsilon_{\nu,1},\\\
\varepsilon_{\nu,1}^{\beta_{\nu}}E_{\nu}^{-\beta_{\nu}},&\varepsilon_{\nu,1}\leq
E_{\nu}<\varepsilon_{\nu,2},\\\
\varepsilon_{\nu,1}^{\beta_{\nu}}\varepsilon_{\nu,2}^{\gamma_{\nu}-\beta_{\nu}}E_{\nu}^{-\gamma_{\nu}},&E_{\nu}\geq\varepsilon_{\nu,2},\end{array}\right.$

and one has

$\displaystyle\alpha_{\nu}=p+1+\beta,~{}\beta_{\nu}=p+1+\alpha,~{}\gamma_{\nu}=\beta_{\nu}+2.$
(20)

$\varepsilon_{\nu,1}$ could be derived in two different cases according to
$\tau_{p\gamma}$. If $\tau_{p\gamma}<5$, one has

$\varepsilon_{\nu,1}=\varepsilon_{\nu,1}^{0}=6.33\times 10^{5}\rm
GeV(1+z)^{-2}(\Gamma/300)^{2}(\frac{\varepsilon_{\rm\gamma,b}}{1MeV})^{-1}.$
(21)

if $\tau_{p\gamma}$ increases above 5, $f_{\rm p\gamma}$ exceeds 0.5 and
quickly approaches 1, and the neutrino flux no longer significantly increases
with $\tau_{p\gamma}$, and $\varepsilon_{\nu,1}$ has a form of

$\displaystyle\varepsilon_{\nu,1}=\varepsilon_{\nu,1}^{0}\times(\frac{\tau_{p\gamma}^{\rm
peak}}{3})^{1+\beta},$ (22)

where $\tau_{p\gamma}^{\rm peak}$ is the peak $p\gamma$ optical depth (the one
for protons with energy $E_{p}^{\rm peak}$ to interact with the photons with a
peak energy) and described as

$\displaystyle\tau_{p\gamma}^{\rm peak}=8.9L_{{\rm
GRB},52}\left(\frac{\Gamma}{300}\right)^{-2}R_{13}^{-1}\left(\frac{\epsilon_{\gamma,b}}{\rm
MeV}\right)^{-1}.$ (23)

$f_{\rm p\gamma\to N\pi}$ is the fraction of proton energy going into the
$\pi$ production, and one has $f_{\rm p\gamma\to
N\pi}=1-(1-\langle\chi_{p\to\pi}\rangle)^{\tau_{p\gamma}}$, where
$\langle\chi_{p\to\pi}\rangle=0.2$ is the the average fraction of energy
transferred from protons to pions333$\tau_{p\gamma}=3$ is used in Zhang &
Kumar (2013), and $f_{\rm p\gamma\to N\pi}$ is much nearer to 100% with
$\tau_{p\gamma}=5$ shown in Ai & Gao (2023)..

From Equation (21) and (22), $\varepsilon_{\nu,1}$ is denoted with

$\displaystyle\varepsilon_{\nu,1}=\varepsilon_{\nu,1}^{0}\times\text{min}\left\\{1,\left(\frac{\tau_{p\gamma}^{\rm
peak}}{5}\right)^{1+\beta}\right\\}.$ (24)

$\varepsilon_{\nu,2}$ is relevant to the
$\varepsilon^{\prime}_{\rm\pi,cooling}$ as

$\displaystyle\varepsilon_{\nu,2}=\frac{1}{4}D\varepsilon_{\rm\pi,cooling}^{\prime},$
(25)

where $D$ is the Doppler factor and $D\simeq 2\Gamma$ if the observation is
head on. The amplitude $n_{\nu,1}$ could be derived from integration of
$E_{p}\frac{dN_{p}}{dE_{p}}dE_{p}$ from $E_{p,\rm min}$ to $E_{p,\rm max}$ as

$\displaystyle n_{\nu,1}=\frac{1}{8}f_{\rm p}f_{\rm p\gamma\to
N\pi}\frac{\epsilon_{\rm p}/\epsilon_{\rm e}E_{\gamma,\rm
iso}}{\ln(\varepsilon_{\nu,2}/\varepsilon_{\nu,1})}\varepsilon_{\nu,1}^{-2},$
(26)

where $f_{\rm
p}=\frac{\ln(\varepsilon_{\nu,2}/\varepsilon_{\nu,1})}{\ln(E_{\rm
p,max}/E_{\rm p,min})}$ .The observed fluence of the muon neutrinos is

$\displaystyle
E_{\nu}^{2}\phi_{\nu}(E_{\nu})=\frac{E_{\nu}^{2}n_{\nu}(E_{\nu})}{4\pi d_{\rm
L}^{2}}.$ (27)

(a)(b)
(c) $R_{d}=10^{11}$ cm(d) $\Gamma=400$

Figure 1: (a) and (b) the non-thermal neutrino spectra with $\Gamma=400$ for
GRB 230307A and GRB 221009A of different emission mechanisms. For GRB 221009A,
$f_{\rm p}$ is calculated to be about 0.3, 0.5 and 0.3 for PH, ICMART and IS
models respectively; for GRB 230307A, $f_{\rm p}$ is calculated to be about
0.3, 0.7 and 0.5 for PH, ICMART and IS models respectively. (c) and (d) are
allowed ranges of $\Gamma$, $R_{d}$ and $\xi_{\rm CR}$. The solid red line
denotes the U.L. at 90% C.L. of GRB 230307A. The blue dotted line denotes one
order of magnitude lower than U.L. = 1 GeV cm-2. The allowed ranges are below
the corresponding lines, while the excluded ranges are above them, the same
below.

With $\Gamma=400$, $\xi_{\rm CR}=\epsilon_{p}/\epsilon_{\gamma}=3$ and
$\epsilon_{B}=\epsilon_{\gamma}$ for each emission mechanism, the predicted
neutrino spectra from GRB 230307A and 221009A are shown in Figure 1 (a) and
(b). The fluence level of GRB 230307A with assuming a photospheric emission is
the maximum of three mechanisms, which is about two orders of magnitude
smaller than the U.L. of 1 GeV cm-2, and one order of magnitude smaller than
that of GRB 221009A. Furthermore, as shown in Figure 1 (c) and (d), $\xi_{\rm
CR}$ of the excluded region is extremely large (even if with a small
$R_{d}=10^{11}$ cm in Figure 1 (c)), which seems unreasonable for GRBs. The
case is similar and the $\xi_{N}$ is still large ($>10$) if with an U.L of one
order of magnitude lower (0.1 GeV cm-2). Thus, the emission mechanisms as well
as $\xi_{\rm CR}$ are not constrained within the U.L. of the neutrino
production via $p\gamma$ interactions.

###  2.2 hadronuclear process

(a)$\zeta_{n}=5.7$ for GRB 230307A(b)$\Gamma=400$
(d)

Figure 2: (a), (b) are allowed and excluded ranges of $\Gamma$, $\xi_{\rm N}$
and $\zeta_{n}$ in $np$ decoupling mechanism. The blank around lower $\Gamma$
and higher $\xi_{\rm N}$ is the $np$ coupling region where
$\Gamma<\Gamma_{np}$. The red dashed line denotes that of GRB 221009A. The
gray vertical lines denote the $\Gamma=400$ for (a) and $\zeta_{n}=2.3$ and
$5.7$ for (b). (c) The allowed and excluded ranges of $\xi_{\rm N}$ in the
$np$ collision mechanism. The vertical line denotes $\tau_{np}=1$.

####  2.2.1 neutron-proton ($np$) decoupling

In the neutron-rich environment, an admixture of baryons exist in the outflow
besides of photons and $e^{\pm}$. During the expanding of the outflow, the
electrons can exchange energy with photons via Compton scattering and with
protons via Coulomb collisions; however, for the neutrons, only protons have
enough mass to collider with them and cause a collisional $np$ coupling (e.g.
Rossi et al., 2006). This means that, if the neutron-to-proton ratio (denoted
as $\zeta_{n}$) is large so that there are not enough protons, or $\Gamma$ is
large enough (above the the critical value $\Gamma_{np}$), the neutrons can
not be accelerated so fast along with the other outflow component, even if
they have drifted into the outflow in the early stage of acceleration. In this
scenario the $np$ decoupling happens. After decoupling both $n$ and $p$
continue to coast together due to inertia, and their relative velocities could
reach the threshold for inelastic collisions, of which decay chains are

$\displaystyle\rm p+\rm n\rightarrow\rm p+\rm n+\pi^{0},$ (28)
$\displaystyle\rm p+\rm n\rightarrow n+n+\pi^{+},$ (29) $\displaystyle\rm
p+\rm n\rightarrow p+p+\pi^{-}.$ (30)

The charged pions and their secondary particles muons, could produce leptons
via the same decay chains as those in processes (4)-(7). The detected fluence
of muon neutrinos number from $np$ decoupling could be described as,

$\displaystyle\Phi_{N_{\nu_{\mu}}}=\frac{N_{n}N_{\nu_{\mu}}\tau_{np}}{4\pi
d_{L}^{2}},$ (31)

where $N_{n}$ denotes the number of neutrons contained in the outflow,
$N_{\nu_{\mu}}$ denotes the number of muon neutrinos created per inelastic
$np$ interaction with considering the neutrinos flavour mixing, and
$\tau_{np}$ denotes the optical depth for the $np$ collision, which could be
approximated to the inelastic $np$ interaction probability.

In the acceleration phase, the radius ($R$) and $\Gamma(R)$ has a relation of

$\Gamma(R)=\Gamma_{0}\left(\frac{R}{r_{0}}\right)^{m},$ (32)

where $r_{0}$ is the initial radius of acceleration and $\Gamma_{0}$ is the
initial Lorentz factor. $r_{0}$ is taken to be $10^{8}$ cm444$r_{0}=10^{8}$ cm
is the mean value of the initial acceleration radius (Pe’er et al., 2015).;
$m$ ranges from 1/3 to 1. For the rapid acceleration of the fireball, $m=1$,
while for the slow acceleration by the Poynting flux, $m$ could be 1/3. Note
that in a collapsar scenario of a massive star, $2/3<\zeta_{n}\simeq 1$, while
in a NS-NS merger scenario, $\zeta_{n}\geq 1$ (e.g. Bahcall & Mészáros, 2000).
In the neutron-rich environment after the NS-NS merger for GRB 230307A, the
electron (or proton) to nucleon ratio $Y_{e}=\frac{1}{1+\zeta_{n}}\sim 0.15$
for a fast dynamical ejecta and $0.3$ for slow disk-wind ejecta as reported in
Bulla et al. (2023), which corresponds to $2.3<\zeta_{n}<5.7$. The neutrons
could penetrate the outflow, and be contained in the outflow (e.g.
Beloborodov, 2003).

First, the fireball scenario is considered with $m=1$. The $np$ collision time
scale is $t^{\prime}_{np}\approx 1/(n^{\prime}_{p}\sigma_{np}c)$, where the
$\sigma_{np}\approx 3\times{10}^{-26}~{}{\rm cm}^{2}$ is the approximate $np$
cross section, $n^{\prime}_{p}\approx L_{p}/(4\pi R^{2}\Gamma\eta m_{p}c^{3})$
is the proton density, $L_{p}$ is the luminosity of the proton outflow and
$\eta$ is the maximum Lorentz Factor that all the energy is used to accelerate
the protons. By equating $t^{\prime}_{np}$ and the dynamical time scale
$t^{\prime}_{\rm dyn}\approx R/(\Gamma c)$, the the decoupling radius is
estimated to be $r_{\rm dec}\approx 8.7\times{10}^{11}~{}{\rm
cm}~{}L_{p,53}^{1/3}r_{0,11}^{2/3}\Gamma_{0,1}^{-2/3}\eta_{,2.9}^{-1/3}$, at
which the Lorentz factor becomes (Bahcall & Mészáros, 2000; Murase et al.,
2022),

$\Gamma^{0}_{n,\rm dec}\approx
65~{}L_{p,53}^{1/3}r_{0,11}^{-1/3}\Gamma_{0,1}^{1/3}\eta_{2.9}^{-1/3}.$ (33)

Note that $\Gamma$ must be greater than $\Gamma_{np}$ which is

$\Gamma_{np}\approx{\left(\frac{\sigma_{np}L_{p}\Gamma_{0}}{4\pi
m_{p}c^{3}r_{0}}\right)}^{1/4}=477L^{1/4}_{p,53}\Gamma^{1/4}_{0}r^{1/4}_{0,8}.$
(34)

Thus the Lorentz Factor of neutron at the decoupling time is

$\Gamma_{n,\rm dec}=\text{min}(\Gamma^{0}_{n,\rm dec},\Gamma_{np})$ (35)

decoupling radius by definition, and the energy of QT neutrinos is

$E_{\nu}^{\rm QT}\approx 0.1\Gamma_{n,\rm dec}m_{p}c^{2}/(1+z),$ (36)

which predicts $\sim 1-10$ GeV neutrinos with $m_{p}c^{2}=0.938$ GeV and
$\Gamma_{n,\rm dec}$ is mild ($\lesssim 100$). The neutrino energy fluence is
estimated to be

$\displaystyle E^{2}_{\nu_{\mu}}\phi_{\nu_{\mu}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{12}\frac{(1+z)}{4\pi
d_{L}^{2}}\zeta_{n}\left(\frac{\Gamma_{n,\rm dec}}{\Gamma}\right)E_{\rm
proton},$ (37)

where $\phi_{\nu_{\mu}}=\frac{1}{4\pi
d^{2}_{L}}\frac{dN_{\nu_{\mu}}}{dE_{\nu_{\mu}}}$. $E_{\rm
proton}\approx\xi_{N}E_{\gamma}^{\rm iso}$ is the kinetic energy of the proton
outflow and $\xi_{N}$ is the proton or nucleon loading factor. The coefficient
1/12 comes from the product of inelasticity (0.5), the fraction of charged
pions (2/3) and the energy fraction of $\nu_{\mu}$ after the flavour mixing
(1/4).

Figure 2 (a) and (b) shows the allowed ranges of $\xi_{\rm N}$, $\Gamma$ and
$\zeta_{n}$ within the constraint from U.L. of neutrino fluence. With
$\Gamma=400$ and $\zeta_{n}=5.7$, the U.L. of $\xi_{\rm N}$ could reach up to
3. From Figure 2 (b), the U.L. of $\xi_{n}$ decreases with the $\zeta_{n}$.
With one order of magnitude lower than 1 GeV cm-2, the U.L. of
$\xi_{N}\lesssim 1$ as shown in dotted lines in Figure 2 (a) and (b). Compared
with the largest allowed $\xi_{\rm N}\lesssim 0.1$ for GRB 221009A (with
$\zeta_{n}=1$), it seems that a heavier baryon loading could be allowed for
GRB 230307A.

If another case that the Poynting flux is dominated ($\sigma_{0}>10$, where
$\sigma_{0}$ is the Poynting-to-kinetic flux ratio) is considered, the
neutrino fluence could be smaller. This is because $\tau_{np}$ in Equation
(31) has a form of

$\displaystyle\tau_{np}(m)=\int^{\infty}_{\chi_{\pi}}(1-\frac{0.75}{\ln\chi})(\chi^{-1}-\chi^{-3})\chi^{-1/m},$
(38)

where $\chi$ is the ratio of Lorentz Factor of proton to that of neutron and
$\chi_{\pi}\simeq 2.15$ is the threshold for pion production (e.g. Landau &
Lifshitz, 1971; Koers & Giannios, 2007). For the Poynting-flux-dominated (PFD)
case, $m=1/3$ and the saturation radius is much larger, $\tau_{np}\simeq
0.01\zeta_{n}^{1/2}$ is several times smaller that that of fireball
($\tau_{np}\simeq 0.05\mbox{--}0.1$). Thus, the corresponding U.L. of
$\xi_{N}$ is even much larger than that obtained in the fireball scenario,
which seems too large for the assumption that the Poynting flux is dominated.
Therefore, the U.L. can not provide a stringent constraint in the PFD
scenario.

A third scenario of a hybrid jet with $1\lesssim\sigma_{0}<10$, could be
considered as well. If the decoupling occurs during the rapid acceleration,
the case is similar to the fireball scenario; similar to the above discussion,
the U.L. of $\xi_{N}\sim$ a few could be also obtained, which seems consistent
with the hybrid scenario if we assume there exists a Poynting flux of which
the energy is compatible with $E_{\gamma,\rm iso}$. Otherwise, it is similar
to the PFD scenario.

####  2.2.2 collisions in neutron-loaded flows

The $pn$ collisions occur not only in the case of $np$ decoupling before the
coasting regime, but also in the case of internal collisions between the
compound flow if $\Gamma_{np}$ is larger than $\eta$ and the coasting occurs
earlier than the decoupling (e.g. Murase et al., 2013). As discussed in Murase
et al. (2013), the energy of the produced neutrinos is

$E_{\nu}^{\rm qt}\approx 0.1\Gamma\Gamma_{\rm rel}^{\prime}m_{p}c^{2}/(1+z),$
(39)

where $\Gamma_{\rm rel}^{\prime}\sim 2$ is the relative Lorentz factor of the
interacting flow and sub-TeV neutrinos are expected for
$\Gamma\sim{10}^{2}\mbox{--}{10}^{3}$. The neutrino energy fluence is

$\displaystyle E_{\nu}^{2}\phi_{\nu_{\mu}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{12}\frac{(1+z)}{4\pi
d_{L}^{2}}\tau_{pn}\xi_{N}{E}_{\gamma,\rm iso}.$ (40)

Internal collisions between neutron-loaded flows may be relevant for sub-
photospheric dissipation (e.g. Beloborodov, 2010). $\xi_{N}\sim 4-20$ is
predicted in this scenario (e.g. Vurm et al., 2011a). As shown in Figure 2
(c), the U.L. of $\xi_{\rm N}$ could reach up to $\sim 5$ with $\tau_{np}=1$,
which could be consistent with the scenario. The U.L. of $\xi_{N}\sim 0.5$ is
obtained within the smaller constraint of U.L=0.1 GeV cm-2, seems too small.
However, it may be reasonable for a hybrid jet in which there exists an
appreciable Poynting flux, because a considerable part of $E_{\gamma,\rm iso}$
could be contributed by the dissipation of the Poynting flux via e.g. ICMART
mechanism.

###  2.3 Other processes: $pp$ collisions, multi-pions and Kaons production in
$pp$ collisions and $p\gamma$ interactions

Other processes, such as $pp$ collisions, multi-pions and Kaons production in
$pp$ collisions and $p\gamma$ interactions (e.g. Wang & Dai, 2009; Gao et al.,
2012) could also contribute to the neutrino emission. However, compared with
the estimations above, their contributions are not much larger, and do not
affect the conclusion.

###  2.4 Implications

The U.L. (1 GeV cm2) of the neutrino fluence for GRB 230307A seems too loose
to constrain a specific GRB emission mechanism, if the non-thermal neutrino
production via $p\gamma$ interactions is the only consideration. For QT
neutrinos, the U.L. of $\xi_{N}\sim$ a few, which seems higher than that
($\xi_{N}\lesssim 0.1$) of GRB 221009A. If the baryon loading is really
appreciable, the jet is not PFD as that in GRB 221009A. Therefore, it may be
necessary to directly investigate the prompt emission mechanism from spectra
in the prompt phase to see if there is something different.

##  3 Basic spectral properties of GRB 230307A

GBM NaI detector (NaI 10), and GBM BGO (BGO 1) detector recorded the brightest
flux, however, dead time (DT) and pile-up instrumental (PUI) effects occur in
[$T_{0}+2.5$, $T_{0}+11.0$] s (GCN 33551, Dalessi & Fermi GBM Team, 2023), as
shown in the red shadow in the light curve in Figure 3 (a). Fortunately, data
from GECAM-B are not distorted by PUI effect. Therefore, data analysis is
performed with data from the brightest detector (GRD 4) of GECAM-B from
[$T_{0}+2.2$, $T_{0}+11.7$] s, while in other time bins, data from the two
brightest detectors (NaI 10 and BGO 1) of GBM are used. A polynomial is
applied to fit all the energy channels and then interpolated into the signal
interval to yield the background photon count estimate for GRB data. The
Markov Chain Monte Carlo (MCMC) fitting is performed to find the parameters
with the maximum Poisson likelihood.

From Figure 3 (a), the values of $\alpha\gtrsim-2/3$ (synchrotron death line,
Preece et al., 1998) concentrate in the first 7 s, which implies that the
emission at the beginning may have a photospheric origin. The fit results are
shown in Table 1 in Appendix A. After about 7 s, $\alpha$ gradually decreases
to about -1.6, which means that the non-thermal emission may become dominant
gradually afterwards. The typical QT spectra in the first few seconds are
shown in Figure 3 (c)-(e), which is narrow with the maximum of
$\alpha\sim-0.3$ and more like a broadened Planck spectrum after subtracting
the non-thermal emission denoted by a power-law (PL) function. The
photospheric emission is a natural consequence of the fireball (e.g., Cavallo
& Rees, 1978; Goodman, 1986; Paczynski, 1986). The Planck spectrum related to
the photospheric emission could be broadened by geometrical broadening (Pe’er,
2008; Lundman et al., 2013), as well as by dissipations below the photosphere
(e.g., Rees & Mészáros, 2005), such as magnetic reconnection (e.g., Giannios &
Spruit, 2005), and hadronic collision shocks (e.g., Vurm et al., 2011b).

From Figure 3 (f)-(h) and the decreasing $\alpha$ with time, it is found that
the so-called QT spectral shape becomes broader. This may be caused by the
enhanced dissipation below the photosphere with time. In summary, an
significant existence of the photospheric emission is observed in the prompt
phase of GRB 230307A, which is greatly different from that in GRB 221009A.

(a)(b)
(c)[-0.7,1.2] s (d)[-1.2,1.7] s (e)[-1.7,2.2] s
(f)[11.7,12.2] s (g)[13.7, 14.2] s (h)[35.7, 39.7] s

Figure 3: (a) The light curves (logarithm scale) of GBM na detector and
$\alpha$, $E_{\rm p}$ values, with a sub-plot (b) for the first 1 s. Red
circles and blue triangles denote $\alpha$ values from the fit results with
BAND and CPL functions (the same below), respectively. Green diamonds denote
the $E_{\rm p}$ values from the fit results. The sub-plot shows $\alpha$ and
$E_{\rm p}$ values of the first 1 s. (c)-(h) Time resolved spectra with
modeling with multi-blackbody (mBB) or mBB+PL functions.

##  4 Discussion and Summary

I. Constraints from the possible neutrino emission on GRB 230307A

$E_{\rm iso,\gamma}$ for GRB 230307A is not large enough, compared with the
nearby extremely bright GRB 130427A and 221009A. The U.L. seems so loose that
the emission mechanisms for this burst are not well constrained, with assuming
that the neutrinos production from $p\gamma$ interactions is the only
consideration. The case is similar even if the U.L. is one order smaller. It
is proposed that the QT neutrinos from hadronuclear processes may contribute
to the neutrino emission. The U.L. of allowed nucleon loading factor is about
a few in this neutron-rich post-merger environment of GRB 230307A, while it
could be $\lesssim 1$ if an U.L. of one order of magnitude smaller is assumed.

II. The nucleon loading and emission mechanism in GRB 230307A

The neutrino emission is relevant to the emission mechanism and the jet
composition for GRBs. Three different jet properties are considered in the
estimation for the QT neutrino fluence. The U.L. of the nucleon loading
$\xi_{\rm N}\sim$ a few for GRB 230307A, while the U.L. of $\xi_{\rm
N}\lesssim 0.1$ for GRB 221009A, which may imply that a heavier nucleon
loading is allowed in GRB 230307A than that in GRB 221009A.

For GRB 221009A, the U.L. of allowed $\xi_{N}\lesssim 0.1$ is very small,
which is consistent very well with its jet composition and emission properties
inferred from the spectral modeling (Yang et al., 2023): a PFD jet (with a
synchrotron-dominated prompt emission). The prompt emission of GRB 230307A
seems quite different from that of GRB 221009A. A significant photospheric
emission dominates the beginning of the prompt phase; the spectrum becomes
broader and broader with time, which could be interpreted by enhanced sub-
photosphere dissipations via magnetic reconnection, or hadronic collisions.
The dissipation via hadronic collisions could occur in a fireball or a hybrid
jet, while the magnetic dissipation may happen in a hybrid or PFD jet. If the
jet is PFD, it is inferred that there must be a thermalization below the
photosphere, to interpret the observed photospheric emission.

GRB 211211A is another nearby GRB ($z=0.076$) and has a same origin as GRB
230307A (Yang et al., 2022). There is no report about the relevant neutrino
emission, however, $\sigma_{0}$ tells some information about the nucleon
loading. $\sigma_{0}\sim 10\mbox{--}100$ is estimated for GRB 211211A (Chang
et al., 2023). This means that the jet for GRB211211A is PFD and nucleon
loading is small. Besides, there exist neither significant photospheric
emissions, nor evident spectral evolution (from photospheric to non-thermal)
in the prompt phase. It may be inferred that the jet composition of GRB
230307A is somewhat different from those of GRB 211211A and GRB 221009A, and a
heavier baryon loading may account for it.

III. Assuming a more stringent constraint from neutrinos detection

A data set of high-energy track events are used in the analysis by IceCube to
extract the U.L. of neutrino fluence, and the sensitivity can vary greatly
over different declinations ($\delta$) (Abbasi et al., 2021, 2023b). GRB
230307A is located at the southern hemisphere with $\delta=-75.4^{\circ}$,
while $\delta=+19.8^{\circ}$ for GRB 221009A. The U.L. for non-detection in
GRB 230307A might be about one order of magnitude lower if it was in northern
hemisphere with a reduced background. Thus, with assuming that it has the same
sensitivity as that in the northern hemisphere, 0.1 GeV cm-2 as a U.L. is also
used in the analysis as shown in Section 2. However, this smaller U.L. still
can not provide a stringent constraint on GRB emission mechanisms via
$p\gamma$ interactions, as shown in Figure 2. For the thermal neutrino
emission, a smaller U.L. is obtained, however, it is still much larger than
that in GRB 221009A or GRB 211211A. The conclusion is similar as that with 1
GeV cm-2 as a U.L..

The authors thank supports from the National Program on Key Research and
Development Project (2021YFA0718500). This work is partially supported by
International Partnership Program of Chinese Academy of Sciences (Grant
No.113111KYSB20190020). The author is very grateful to the suggestions from
Jessie Thwaites and the IceCube Collaboration. Dr Xin-Ying Song thanks public
GRB data of Fermi/GBM and GECAM-B data group. I am very grateful for the
comments and suggestions from the anonymous referees. I thank Dr. Rui Qiao for
his suggestions on the response matrix generating for GECAM-B data. I thank Dr
Michael S. Briggers and Dr Michelle Hui for their suggestions on GBM data. I
thank Dr. Xi-Lu Wang and Prof. Shuang-Nan Zhang for their suggestions.

##  Appendix A The fit results of the prompt emission of GRB 230307A

The fit results modeling with mBB or mBB+PL models and empirical fucntions are
listed in Table. 1.

Table 1: The time-resolved results with mBB, mBB+PL and BAND functions. Time bins | Favoured Model | $m$ | $kT_{\rm min}$ | $kT_{\rm max}$ | $F_{\rm T}$ | $\alpha$ | $F_{\rm NT}$ | BIC | $\frac{\chi^{2}}{ndof}$
---|---|---|---|---|---|---|---|---|---
(s) |  |  | (keV) | (keV) | (10-6 erg cm-2 s-1) |  | (10-6 erg cm-2 s-1) |  |
[-0.01, 0.39] |  | -0.35${}^{+0.07}_{-0.09}$ | 5.9${}^{+0.7}_{-0.4}$ | 105.2${}^{+7.0}_{-5.2}$ | 30.3${}^{+0.5}_{-0.4}$ |  |  | 249.0 | $\frac{239.7}{207}$
| mBB+PL | -0.44${}^{+0.09}_{-0.09}$ | 8.5${}^{+1.2}_{-1.2}$ | 108.5${}^{+50.6}_{-20.0}$ | 27.2${}^{+1.8}_{-1.2}$ | -2.09${}^{+0.05}_{-0.01}$ | 7.0${}^{+7.2}_{-2.5}$ | 235.6 | $\frac{221.6}{205}$
[0.39, 0.7] | BAND | $\alpha$ | $\beta$ | $E_{\rm p}$ |  |  |  |  |
|  | $-1.61^{+0.24}_{-0.12}$ | $-6.88^{+3.66}_{-3.49}$ | $101.3^{+17.0}_{-12.7}$ |  |  | $5.89^{+0.89}_{-0.61}$ | 122.3 | $\frac{113.6}{147}$
[0.7, 1.2] | mBB | 0.30${}^{+0.02}_{-0.03}$ | 1.9${}^{+1.0}_{-0.6}$ | 259.9${}^{+5.2}_{-4.6}$ | 106.6${}^{+1.1}_{-1.5}$ |  |  | 250.2 | $\frac{240.6}{236}$
|  | 0.24${}^{+0.03}_{-0.03}$ | 13.4${}^{+1.6}_{-1.6}$ | 268.5${}^{+8.6}_{-7.3}$ | 103.0${}^{+1.3}_{-1.6}$ | -1.97${}^{+0.03}_{-0.02}$ | 7.5${}^{+1.1}_{-1.1}$ | 254.9 | $\frac{240.6}{234}$
[1.2, 1.7] | mBB | 0.35${}^{+0.01}_{-0.01}$ | 1.8${}^{+0.9}_{-0.5}$ | 418.3${}^{+6.0}_{-5.6}$ | 232.3${}^{+2.7}_{-1.6}$ |  |  | 222.2 | $\frac{212.7}{236}$
|  | 0.38${}^{+0.03}_{-0.02}$ | 8.5${}^{+2.1}_{-1.4}$ | 413.2${}^{+6.7}_{-6.2}$ | 229.4${}^{+3.7}_{-3.1}$ | -1.95${}^{+0.05}_{-0.03}$ | 6.0${}^{+1.0}_{-1.1}$ | 228.9 | $\frac{214.6}{234}$
[1.7, 2.2] |  | 0.53${}^{+0.01}_{-0.01}$ | 1.6${}^{+0.6}_{-0.4}$ | 462.9${}^{+5.5}_{-4.7}$ | 382.0${}^{+4.3}_{-2.5}$ |  |  | 333.6 | $\frac{324.1}{236}$
| mBB+PL | 0.64${}^{+0.03}_{-0.03}$ | 9.0${}^{+4.1}_{-3.2}$ | 445.3${}^{+6.7}_{-20.0}$ | 374.7${}^{+3.6}_{-3.6}$ | -1.95${}^{+0.03}_{-0.05}$ | 8.4${}^{+2.0}_{-2.5}$ | 302.1 | $\frac{287.8}{234}$
[2.2, 4.6] |  | 0.18${}^{+0.02}_{-0.02}$ | 10.0${}^{+7.3}_{-0.7}$ | 439.1${}^{+127.0}_{-15.7}$ | 270.3${}^{+115.9}_{-6.3}$ |  |  | 269.1 | $\frac{259.7}{220}$
| mBB+PL | 0.21${}^{+0.03}_{-0.03}$ | 11.0${}^{+2.2}_{-1.8}$ | 431.4${}^{+20.0}_{-15.0}$ | 263.1${}^{+7.5}_{-7.5}$ | -1.98${}^{+0.20}_{-0.11}$ | 6.9${}^{+10.0}_{-5.5}$ | 268.7 | $\frac{254.6}{218}$
[4.6, 7.0] |  | 0.14${}^{+0.03}_{-0.04}$ | 10.0${}^{+4.8}_{-0.6}$ | 492.0${}^{+33.0}_{-16.9}$ | 394.6${}^{+15.6}_{-8.6}$ |  |  | 283.5 | $\frac{274.1}{220}$
| mBB+PL | 0.16${}^{+0.10}_{-0.04}$ | 12.5${}^{+1.5}_{-1.5}$ | 493.4${}^{+126.0}_{-21.0}$ | 385.9${}^{+13.6}_{-17.8}$ | -1.98${}^{+0.37}_{-0.05}$ | 15.8${}^{+13.0}_{-10.4}$ | 282.9 | $\frac{268.8}{218}$
[7.0, 9.3] |  | -0.03${}^{+0.03}_{-0.19}$ | 10.5${}^{+3.3}_{-0.6}$ | 507.3${}^{+110.0}_{-22.1}$ | 271.0${}^{+73.0}_{-6.1}$ |  |  | 311.6 | $\frac{302.2}{220}$
| mBB+PL | -0.02${}^{+0.03}_{-0.04}$ | 11.4${}^{+1.9}_{-0.9}$ | 506.4${}^{+40.0}_{-20.0}$ | 265.4${}^{+5.9}_{-5.9}$ | -1.98${}^{+0.20}_{-0.40}$ | 8.0${}^{+8.5}_{-7.7}$ | 310.6 | $\frac{296.5}{218}$
[9.3, 11.7] |  | -0.23${}^{+0.03}_{-0.21}$ | 10.7${}^{+1.5}_{-0.5}$ | 491.8${}^{+110.0}_{-30.3}$ | 177.0${}^{+60.1}_{-3.1}$ |  |  | 270.7 | $\frac{261.3}{220}$
| mBB+PL | -0.22${}^{+0.03}_{-0.04}$ | 11.5${}^{+4.1}_{-0.8}$ | 490.7${}^{+80.0}_{-52.1}$ | 171.5${}^{+11.1}_{-8.4}$ | -1.99${}^{+0.20}_{-0.32}$ | 9.4${}^{+10.0}_{-7.6}$ | 267.7 | $\frac{253.6}{218}$
[2.2, 11.7] |  | 0.03${}^{+0.01}_{-0.01}$ | 10.1${}^{+0.4}_{-0.3}$ | 504.4${}^{+13.6}_{-10.5}$ | 288.2${}^{+4.2}_{-3.8}$ |  |  | 340.0 | $\frac{330.6}{220}$
| mBB+PL | 0.04${}^{+0.02}_{-0.05}$ | 10.9${}^{+25.1}_{-0.8}$ | 502.9${}^{+12.0}_{-10.0}$ | 281.9${}^{+4.1}_{-4.1}$ | -2.03${}^{+0.43}_{-0.15}$ | 12.8${}^{+10.0}_{-10.0}$ | 319.3 | $\frac{305.2}{218}$
[11.7, 12.2] |  | -0.18${}^{+0.02}_{-0.02}$ | 6.5${}^{+0.4}_{-0.4}$ | 454.2${}^{+12.9}_{-12.1}$ | 151.8${}^{+2.0}_{-2.0}$ |  |  | 298.2 | $\frac{288.6}{236}$
| mBB+PL | -0.19${}^{+0.01}_{-0.03}$ | 9.8${}^{+0.7}_{-0.7}$ | 461.5${}^{+10.0}_{-10.0}$ | 146.2${}^{+1.9}_{-1.9}$ | -1.98${}^{+0.02}_{-0.02}$ | 9.3${}^{+1.5}_{-1.7}$ | 291.8 | $\frac{277.5}{234}$
[12.2, 12.7] |  | -0.14${}^{+0.02}_{-0.02}$ | 6.1${}^{+0.4}_{-0.4}$ | 486.2${}^{+12.1}_{-11.9}$ | 160.0${}^{+2.0}_{-2.0}$ |  |  | 258.3 | $\frac{248.8}{236}$
| mBB+PL | -0.15${}^{+0.02}_{-0.02}$ | 9.7${}^{+0.7}_{-0.7}$ | 491.5${}^{+13.2}_{-12.6}$ | 154.6${}^{+2.4}_{-2.4}$ | -1.97${}^{+0.01}_{-0.01}$ | 9.3${}^{+1.7}_{-2.0}$ | 256.2 | $\frac{241.9}{234}$
[12.7, 13.2] |  | -0.17${}^{+0.02}_{-0.02}$ | 6.5${}^{+0.3}_{-0.3}$ | 409.2${}^{+9.9}_{-9.6}$ | 178.4${}^{+2.0}_{-2.3}$ |  |  | 305.4 | $\frac{295.8}{236}$
| mBB+PL | -0.20${}^{+0.02}_{-0.02}$ | 10.2${}^{+0.5}_{-0.9}$ | 419.8${}^{+10.6}_{-10.5}$ | 171.7${}^{+2.6}_{-2.6}$ | -1.98${}^{+0.02}_{-0.01}$ | 11.9${}^{+1.8}_{-1.7}$ | 290.8 | $\frac{276.5}{234}$
[13.2, 13.7] |  | -0.23${}^{+0.01}_{-0.01}$ | 5.7${}^{+0.3}_{-0.3}$ | 469.9${}^{+13.6}_{-12.7}$ | 148.9${}^{+2.0}_{-1.9}$ |  |  | 299.0 | $\frac{289.5}{236}$
| mBB+PL | -0.23${}^{+0.02}_{-0.01}$ | 7.8${}^{+0.9}_{-0.9}$ | 470.1${}^{+14.9}_{-16.2}$ | 143.3${}^{+2.9}_{-2.7}$ | -1.97${}^{+0.02}_{-0.02}$ | 8.6${}^{+2.1}_{-2.3}$ | 298.8 | $\frac{284.6}{234}$
[13.7, 14.2] |  | -0.30${}^{+0.02}_{-0.02}$ | 5.8${}^{+0.3}_{-0.4}$ | 433.1${}^{+19.5}_{-13.9}$ | 106.3${}^{+1.2}_{-2.3}$ |  |  | 238.8 | $\frac{229.3}{236}$
| mBB+PL | -0.31${}^{+0.02}_{-0.02}$ | 9.8${}^{+0.8}_{-0.8}$ | 441.0${}^{+16.8}_{-16.0}$ | 99.3${}^{+1.3}_{-1.3}$ | -1.98${}^{+0.04}_{-0.01}$ | 12.2${}^{+1.5}_{-1.6}$ | 215.5 | $\frac{201.2}{234}$
[14.2, 14.7] |  | -0.34${}^{+0.02}_{-0.01}$ | 5.5${}^{+0.3}_{-0.2}$ | 442.3${}^{+17.6}_{-16.6}$ | 105.4${}^{+2.0}_{-2.0}$ |  |  | 260.7 | $\frac{251.2}{236}$
| mBB+PL | -0.35${}^{+0.03}_{-0.02}$ | 9.0${}^{+0.9}_{-0.7}$ | 449.7${}^{+23.8}_{-15.8}$ | 97.9${}^{+2.7}_{-1.4}$ | -1.99${}^{+0.02}_{-0.01}$ | 13.1${}^{+1.7}_{-1.7}$ | 240.3 | $\frac{226.0}{234}$
[14.7, 15.2] |  | -0.36${}^{+0.02}_{-0.02}$ | 5.6${}^{+0.3}_{-0.3}$ | 415.5${}^{+15.8}_{-15.1}$ | 103.5${}^{+1.2}_{-1.2}$ |  |  | 240.7 | $\frac{231.2}{234}$
| mBB+PL | -0.36${}^{+0.02}_{-0.03}$ | 7.0${}^{+0.8}_{-0.7}$ | 417.2${}^{+17.1}_{-16.0}$ | 99.6${}^{+1.3}_{-2.7}$ | -2.03${}^{+0.03}_{-0.02}$ | 7.0${}^{+2.6}_{-2.4}$ | 239.4 | $\frac{225.2}{232}$
[15.2, 15.7] |  | -0.34${}^{+0.02}_{-0.02}$ | 5.0${}^{+0.2}_{-0.5}$ | 352.2${}^{+14.6}_{-14.6}$ | 82.1${}^{+1.4}_{-1.4}$ |  |  | 271.0 | $\frac{261.5}{234}$
| mBB+PL | -0.34${}^{+0.02}_{-0.03}$ | 7.7${}^{+0.4}_{-0.8}$ | 353.9${}^{+15.1}_{-15.4}$ | 76.1${}^{+1.9}_{-1.4}$ | -1.98${}^{+0.02}_{-0.01}$ | 10.4${}^{+1.8}_{-2.1}$ | 262.0 | $\frac{247.7}{232}$
| BAND | $\alpha$ | $\beta$ | $E_{\rm peak}$ |  |  |  |  |
[-0.01, 0.39] | BAND | -0.77${}^{+0.02}_{-0.02}$ | -7.43${}^{+2.80}_{-0.47}$ | 180.7${}^{+4.3}_{-4.4}$ |  |  | 31.3${}^{+0.6}_{-0.3}$ | 243.30 | $\frac{234.0}{207}$
[0.39, 0.7] | BAND | -1.56${}^{+0.07}_{-0.11}$ | -6.88${}^{+3.66}_{-3.49}$ | 98.3${}^{+17.0}_{-26.5}$ |  |  | 5.8${}^{+0.9}_{-0.6}$ | 126.59 | $\frac{117.9}{147}$
[0.7, 1.2] | BAND | -0.52${}^{+0.03}_{-0.00}$ | -8.98${}^{+3.62}_{-1.50}$ | 599.9${}^{+10.3}_{-2.4}$ |  |  | 109.0${}^{+0.10}_{-0.90}$ | 248.78 | $\frac{239.3}{236}$
[1.2, 1.7] | BAND | -0.49${}^{+0.01}_{-0.03}$ | -11.94${}^{+2.00}_{-3.58}$ | 968.9${}^{+8.5}_{-7.2}$ |  |  | 233.2${}^{+1.8}_{-2.4}$ | 265.97 | $\frac{256.5}{236}$
[1.7, 2.2] | BAND | -0.31${}^{+0.01}_{-0.03}$ | -9.40${}^{+2.90}_{-9.87}$ | 1117.4${}^{+17.1}_{-10.0}$ |  |  | 382.5${}^{+6.2}_{-3.6}$ | 417.51 | $\frac{408.0}{236}$
[2.2, 4.575] | BAND | -0.61${}^{+0.02}_{-0.02}$ | -7.28${}^{+3.13}_{-3.00}$ | 998.4${}^{+41.7}_{-34.0}$ |  |  | 278.3${}^{+5.1}_{-6.6}$ | 280.20 | $\frac{270.8}{220}$
[4.575, 6.95] | BAND | -0.67${}^{+0.01}_{-0.02}$ | -7.36${}^{+10.34}_{-3.00}$ | 1115.0${}^{+26.1}_{-26.1}$ |  |  | 412.1${}^{+4.6}_{-8.2}$ | 297.46 | $\frac{288.1}{220}$
[6.95, 9.325] | BAND | -0.80${}^{+0.01}_{-0.02}$ | -7.21${}^{+3.10}_{-3.00}$ | 1037.8${}^{+44.5}_{-29.7}$ |  |  | 279.9${}^{+9.8}_{-7.5}$ | 330.89 | $\frac{321.5}{220}$
[9.325, 11.7] | BAND | -0.94${}^{+0.02}_{-0.02}$ | -6.70${}^{+0.13}_{-0.13}$ | 838.3${}^{+38.0}_{-44.0}$ |  |  | 178.92${}^{+5.3}_{-10.0}$ | 295.42 | $\frac{286.0}{220}$

Table. 1— Continued.  Time bins Favoured Model $\alpha$ $\beta$ $E_{\rm peak}$
$F_{\rm NT}$ BIC $\frac{\chi^{2}}{ndof}$ (s) (keV) (10-6 erg cm-2 s-1) [11.7,
12.2] BAND -0.92${}^{+0.03}_{-0.01}$ -8.12${}^{+3.82}_{-5.67}$
872.7${}^{+76.9}_{-0.0}$ 154.3${}^{+2.7}_{-0.0}$ 331.96 $\frac{322.4}{236}$
[12.2, 12.7] BAND -0.90${}^{+0.05}_{-0.02}$ -9.61${}^{+2.32}_{-7.22}$
939.0${}^{+10.6}_{-7.1}$ 159.9${}^{+2.0}_{-2.5}$ 278.60 $\frac{269.1}{236}$
[12.7, 13.2] BAND -0.90${}^{+0.01}_{-0.01}$ -10.87${}^{+1.07}_{-8.34}$
756.7${}^{+6.3}_{-11.6}$ 180.7${}^{+3.3}_{-4.1}$ 315.01 $\frac{305.5}{236}$
[13.2, 13.7] BAND -0.99${}^{+0.01}_{-0.01}$ -9.08${}^{+3.72}_{-6.08}$
835.7${}^{+10.0}_{-10.0}$ 149.2${}^{+1.1}_{-2.3}$ 314.46 $\frac{304.9}{236}$
[13.7, 14.2] BAND -1.05${}^{+0.01}_{-0.01}$ -10.71${}^{+1.23}_{-8.51}$
742.5${}^{+5.2}_{-7.9}$ 107.8${}^{+1.4}_{-1.4}$ 291.45 $\frac{281.9}{236}$
[14.2, 14.7] BAND -1.09${}^{+0.01}_{-0.02}$ -8.61${}^{+3.14}_{-6.39}$
722.0${}^{+9.0}_{-5.0}$ 106.9${}^{+1.2}_{-1.2}$ 254.52 $\frac{245.0}{236}$
[14.7, 15.2] BAND -1.08${}^{+0.01}_{-0.03}$ -8.74${}^{+3.20}_{-2.46}$
657.0${}^{+12.5}_{-10.4}$ 104.2${}^{+1.2}_{-1.2}$ 284.62 $\frac{275.1}{236}$
[15.2, 15.7] BAND -1.09${}^{+0.01}_{-0.02}$ -8.55${}^{+9.56}_{-0.00}$
586.1${}^{+17.0}_{-17.0}$ 84.1${}^{+1.4}_{-0.7}$ 284.59 $\frac{275.1}{236}$
[15.7, 16.2] BAND -1.12${}^{+0.01}_{-0.02}$ -8.61${}^{+2.00}_{-2.71}$
793.3${}^{+28.9}_{-19.3}$ 89.4${}^{+1.8}_{-1.2}$ 281.45 $\frac{271.9}{236}$
[16.2, 16.7] BAND -1.07${}^{+0.02}_{-0.03}$ -8.40${}^{+2.73}_{-2.98}$
685.5${}^{+20.0}_{-93.3}$ 81.3${}^{+1.5}_{-1.0}$ 267.13 $\frac{257.6}{236}$
[16.7, 17.2] BAND -1.17${}^{+0.01}_{-0.02}$ -8.44${}^{+2.37}_{-7.47}$
481.6${}^{+20.0}_{-20.0}$ 49.7${}^{+0.6}_{-0.6}$ 266.30 $\frac{256.8}{236}$
[17.2, 17.7] BAND -1.16${}^{+0.06}_{-0.01}$ -8.27${}^{+3.86}_{-2.00}$
511.5${}^{+22.1}_{-14.7}$ 48.8${}^{+1.2}_{-0.6}$ 199.33 $\frac{189.8}{236}$
[17.7, 18.2] BAND -1.29${}^{+0.05}_{-0.03}$ -7.58${}^{+0.12}_{-2.10}$
197.6${}^{+28.3}_{-11.6}$ 13.1${}^{+0.3}_{-0.5}$ 243.51 $\frac{234.0}{236}$
[18.2, 18.7] BAND -1.55${}^{+0.06}_{-0.03}$ -7.06${}^{+3.06}_{-3.00}$
219.6${}^{+32.4}_{-39.3}$ 10.9${}^{+0.6}_{-0.8}$ 184.64 $\frac{175.1}{236}$
[18.7, 19.2] BAND -1.30${}^{+0.03}_{-0.03}$ -7.75${}^{+3.05}_{-2.90}$
380.0${}^{+20.0}_{-20.0}$ 30.6${}^{+0.6}_{-0.6}$ 204.36 $\frac{194.8}{236}$
[19.2, 19.7] BAND -1.23${}^{+0.02}_{-0.02}$ -8.40${}^{+2.63}_{-2.00}$
504.9${}^{+20.0}_{-26.1}$ 56.8${}^{+0.9}_{-0.9}$ 264.82 $\frac{255.3}{236}$
[19.7, 20.2] BAND -1.22${}^{+0.01}_{-0.01}$ -8.70${}^{+0.87}_{-8.67}$
715.6${}^{+12.5}_{-10.4}$ 78.7${}^{+1.7}_{-1.1}$ 308.10 $\frac{298.6}{236}$
[20.2, 20.7] BAND -1.16${}^{+0.03}_{-0.01}$ -8.48${}^{+9.84}_{-3.20}$
730.0${}^{+17.2}_{-9.1}$ 70.3${}^{+1.4}_{-1.4}$ 292.01 $\frac{282.5}{236}$
[20.7, 21.2] BAND -1.12${}^{+0.02}_{-0.00}$ -8.38${}^{+9.56}_{-0.25}$
526.2${}^{+25.1}_{-18.8}$ 54.6${}^{+1.1}_{-1.1}$ 280.33 $\frac{270.8}{236}$
[21.2, 21.7] BAND -1.12${}^{+0.03}_{-0.00}$ -8.31${}^{+5.21}_{-4.59}$
593.3${}^{+20.0}_{-20.0}$ 71.8${}^{+1.1}_{-1.1}$ 267.21 $\frac{257.7}{236}$
[21.7, 22.2] BAND -1.19${}^{+0.02}_{-0.02}$ -8.51${}^{+2.36}_{-7.45}$
561.1${}^{+24.0}_{-24.0}$ 51.5${}^{+0.9}_{-0.9}$ 239.74 $\frac{230.2}{236}$
[22.2, 22.7] BAND -1.21${}^{+0.00}_{-0.03}$ -8.44${}^{+3.12}_{-6.50}$
545.2${}^{+20.0}_{-20.0}$ 45.6${}^{+0.7}_{-0.7}$ 240.01 $\frac{230.5}{236}$
[22.7, 23.2] BAND -1.15${}^{+0.00}_{-0.03}$ -7.95${}^{+2.86}_{-2.80}$
405.4${}^{+20.0}_{-20.0}$ 41.1${}^{+0.7}_{-0.7}$ 221.29 $\frac{211.8}{236}$
[23.2, 23.7] BAND -1.18${}^{+0.00}_{-0.03}$ -7.73${}^{+3.87}_{-3.03}$
292.0${}^{+10.0}_{-10.0}$ 28.9${}^{+0.7}_{-0.7}$ 226.04 $\frac{216.5}{236}$
[23.7, 24.2] BAND -1.26${}^{+0.02}_{-0.02}$ -8.09${}^{+9.75}_{-0.00}$
406.7${}^{+16.0}_{-21.4}$ 40.2${}^{+0.9}_{-0.9}$ 270.98 $\frac{261.5}{236}$
[24.2, 24.7] BAND -1.29${}^{+0.01}_{-0.01}$ -8.31${}^{+2.72}_{-3.93}$
505.9${}^{+23.4}_{-31.2}$ 44.5${}^{+0.5}_{-1.1}$ 256.69 $\frac{247.2}{236}$
[24.7, 25.2] BAND -1.25${}^{+0.02}_{-0.01}$ -8.00${}^{+2.00}_{-2.00}$
526.2${}^{+21.2}_{-28.2}$ 44.5${}^{+1.2}_{-1.0}$ 230.45 $\frac{220.9}{236}$
[25.2, 25.7] BAND -1.31${}^{+0.03}_{-0.02}$ -8.25${}^{+3.36}_{-6.46}$
340.8${}^{+16.0}_{-16.0}$ 41.8${}^{+0.7}_{-0.7}$ 266.04 $\frac{256.5}{236}$
[25.7, 26.2] BAND -1.37${}^{+0.02}_{-0.03}$ -7.97${}^{+9.83}_{-0.00}$
271.0${}^{+12.3}_{-9.3}$ 31.9${}^{+0.7}_{-0.4}$ 228.73 $\frac{219.2}{236}$
[26.2, 26.7] BAND -1.48${}^{+0.02}_{-0.02}$ -7.20${}^{+4.24}_{-2.10}$
192.4${}^{+15.1}_{-10.8}$ 24.1${}^{+0.5}_{-0.5}$ 195.13 $\frac{185.6}{236}$
[26.7, 27.2] BAND -1.61${}^{+0.03}_{-0.03}$ -7.11${}^{+3.20}_{-3.30}$
240.7${}^{+66.7}_{-32.7}$ 25.4${}^{+1.0}_{-1.0}$ 259.15 $\frac{249.6}{236}$
[27.2, 27.7] BAND -1.54${}^{+0.03}_{-0.01}$ -7.69${}^{+2.10}_{-3.60}$
404.0${}^{+8.1}_{-13.2}$ 26.4${}^{+0.9}_{-0.6}$ 240.60 $\frac{231.1}{236}$
[27.7, 28.2] BAND -1.44${}^{+0.01}_{-0.03}$ -8.28${}^{+0.37}_{-9.50}$
453.9${}^{+10.6}_{-4.1}$ 37.3${}^{+0.6}_{-1.2}$ 203.75 $\frac{194.2}{236}$
[28.2, 28.7] BAND -1.41${}^{+0.03}_{-0.01}$ -7.76${}^{+1.20}_{-2.08}$
528.9${}^{+21.1}_{-10.4}$ 40.7${}^{+0.9}_{-0.9}$ 233.97 $\frac{224.4}{236}$
[28.7, 29.2] BAND -1.44${}^{+0.00}_{-0.02}$ -7.38${}^{+0.80}_{-3.20}$
516.8${}^{+45.7}_{-45.7}$ 24.2${}^{+1.1}_{-0.7}$ 232.61 $\frac{223.1}{236}$
[29.2, 29.7] BAND -1.45${}^{+0.02}_{-0.02}$ -7.71${}^{+0.70}_{-1.20}$
394.2${}^{+0.0}_{-58.4}$ 19.0${}^{+0.6}_{-0.8}$ 215.80 $\frac{206.3}{236}$
[29.7, 30.2] BAND -1.43${}^{+0.01}_{-0.03}$ -8.26${}^{+9.87}_{-0.00}$
510.1${}^{+43.1}_{-43.1}$ 24.9${}^{+1.0}_{-0.8}$ 220.93 $\frac{211.4}{236}$
[30.2, 30.7] BAND -1.33${}^{+0.03}_{-0.03}$ -7.16${}^{+2.10}_{-1.20}$
266.1${}^{+6.0}_{-2.0}$ 13.2${}^{+0.4}_{-0.6}$ 222.34 $\frac{212.8}{236}$
[30.7, 31.2] BAND -1.38${}^{+0.03}_{-0.03}$ -7.22${}^{+3.00}_{-2.00}$
211.0${}^{+7.2}_{-3.2}$ 11.6${}^{+0.4}_{-0.4}$ 196.71 $\frac{187.2}{236}$
[31.2, 31.7] BAND -1.40${}^{+0.06}_{-0.06}$ -7.18${}^{+3.00}_{-3.12}$
149.2${}^{+12.0}_{-19.7}$ 9.4${}^{+0.4}_{-0.4}$ 177.98 $\frac{168.5}{236}$
[31.7, 35.7] BAND -1.55${}^{+0.02}_{-0.02}$ -7.98${}^{+5.42}_{-1.69}$
254.5${}^{+9.0}_{-9.0}$ 14.9${}^{+0.2}_{-0.2}$ 300.80 $\frac{291.3}{236}$
[35.7, 39.7] BAND -1.51${}^{+0.02}_{-0.00}$ -7.89${}^{+9.76}_{-0.00}$
192.8${}^{+8.2}_{-10.1}$ 10.4${}^{+0.1}_{-0.1}$ 255.69 $\frac{246.2}{236}$
[39.7, 43.7] BAND -1.76${}^{+0.02}_{-0.02}$ -7.10${}^{+3.00}_{-3.40}$
125.3${}^{+10.0}_{-10.0}$ 7.5${}^{+1.1}_{-1.0}$ 185.46 $\frac{175.9}{236}$
[43.7, 47.7] BAND -1.67${}^{+0.02}_{-0.02}$ -7.49${}^{+3.00}_{-3.00}$
162.8${}^{+9.1}_{-13.8}$ 6.7${}^{+0.1}_{-0.5}$ 176.36 $\frac{166.8}{236}$
[47.7, 51.7] BAND -1.54${}^{+0.02}_{-0.02}$ -7.19${}^{+2.40}_{-1.22}$
142.7${}^{+11.1}_{-4.9}$ 4.8${}^{+0.0}_{-0.2}$ 186.30 $\frac{176.8}{236}$
[51.7, 61.7] BAND -1.67${}^{+0.03}_{-0.03}$ -7.64${}^{+3.99}_{-5.74}$
88.1${}^{+12.1}_{-6.5}$ 3.5${}^{+0.2}_{-0.2}$ 163.82 $\frac{154.3}{236}$
[61.7, 71.7] BAND -1.70${}^{+0.03}_{-0.05}$ -7.22${}^{+3.00}_{-0.12}$
98.3${}^{+8.0}_{-7.0}$ 2.4${}^{+0.0}_{-0.0}$ 168.22 $\frac{158.7}{236}$ [71.7,
81.7] BAND -1.77${}^{+0.20}_{-0.10}$ -6.92${}^{+3.00}_{-0.12}$
43.2${}^{+10.0}_{-12.0}$ 1.3${}^{+0.5}_{-0.2}$ 143.93 $\frac{134.4}{236}$
[81.7, 91.7] BAND -1.79${}^{+0.19}_{-0.10}$ -7.22${}^{+5.49}_{-2.70}$
46.5${}^{+10.0}_{-10.0}$ 1.3${}^{+0.1}_{-0.4}$ 157.30 $\frac{147.8}{236}$

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