# A detectable ultra-high-energy cosmic ray outburst from GRB 221009A

Hao-Ning He Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210023, China  Astrophysical
Big Bang Laboratory, RIKEN, Wako, Saitama 351-0198, Japan<EMAIL_ADDRESS>B.
Thoedore Zhang Center for Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502,
Japan  Yi-Zhong Fan Key Laboratory of Dark Matter and Space Astronomy, Purple
Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China
School of Astronomy and Space Science, University of Science and Technology of
China, Hefei 230026, China<EMAIL_ADDRESS>
###### Abstract

Gamma-ray bursts (GRBs) have been proposed as one of promising sources of
ultra-high-energy cosmic rays (UHECRs), but observational evidence is still
lacking. The nearby B.O.A.T. (brightest of all time) GRB 221009A, an once-
in-1000-year event, is able to accelerate protons to $\sim 10^{3}$ EeV.
Protons arriving at the Milky Way are dominated by neutron-decay-induced
protons. The inter-galactic magnetic fields would not yield a sizable delay of
the $\geq 10{\rm~{}EeV}$ cosmic rays if its strength is $\lesssim
10^{-13}{\rm~{}G}$, while Galactic magnetic fields would cause a significant
time delay. We predict that, an UHECR burst from GRB 221009A would be
detectable by the Pierre Auger Observatory and the TA$\times$4, within $\sim$
10 years. The detection of such an UHECR outburst will provide the direct
evidence for UHECR acceleration in GRBs.

Gamma-ray bursts (629) — Cosmic rays (329) — Extragalactic magnetic fields
(507) — Cosmic ray sources (328) — High energy astrophysics (739)

††software: Astropy (Astropy Collaboration et al., 2013, 2018), CRPropa 3.2
(Alves Batista et al., 2016), SOPHIA (Mücke et al., 2000a)

##  1 Introduction

The origin of the highest energy cosmic particles is still in debate. GRBs,
the most luminous phenomenon in the universe, are proposed as one of the most
promising sources of UHECRs (Waxman, 1995; Vietri, 1995). However, no observed
UHECRs or TeV-PeV neutrinos have been confirmed to be associated with GRBs
(Icecube Collaboration et al., 2012; Abbasi et al., 2022). One of the
difficulties of identifying UHECRs associated with GRBs is the deflections and
the consequent time delays caused by the magnetic fields. The magnetic
deflection angles and the delay times are dependent on the distance of GRBs
and the properties of the interstellar (both in the host galaxy and the
Galaxy) and inter-galactic magnetic fields. What’s more, UHECRs would lose a
good fraction of energy via interacting with the extragalactic background
light (EBL) and cosmic microwave background (CMB) photons during the
propagation, implying that distant GRBs can hardly contribute to UHECRs. In
addition, the pointing direction of the GRB jet, the capability of
accelerating UHECRs, the total energy deposited into CRs, and the escape rate
of UHECRs from the burst and the host galaxy are also key issues for the
nearby individual GRB (Mészáros, 2002; Kumar & Zhang, 2015).

GRB 221009A, locating at $z=0.151$ (the luminosity distance of 745
Mpc(Malesani et al., 2023)) and releasing an isotropic-equivalent radiation
energy of $E_{\gamma,\rm iso}\simeq 10^{55}{\rm ergs}$ (An et al., 2023), is
the brightest of all time (B.O.A.T.) (Pillera et al., 2022; de Ugarte Postigo
et al., 2022; Castro-Tirado et al., 2022; Malesani et al., 2023; Levan et al.,
2023). Throughout this work, the convention $Q_{\rm x}=Q/10^{x}$ is adopted.
It is also the first GRB with photons of energy extending up to $\sim$10 TeV
(Cao et al., 2023a, b), indicating its potential ability of accelerating
UHECRs (Mirabal, 2023; Alves Batista, 2022; Das & Razzaque, 2023; Zhang et
al., 2023a, b; Isravel et al., 2023). Likely, only 1 of $10,000$ randomly
generated long bursts is as energetic as GRB 221009A, and the rate of GRBs as
nearby and as energetic as GRB 221009A is $\lesssim 1$ per $1000$ years (Burns
et al., 2023; Williams et al., 2023). Therefore, GRB 221009A provides us a
precious opportunity to study UHECRs in our life time.

The energy dissipation radius of this GRB is large, i.e.,
$R=2\Gamma^{2}ct_{\rm v}/(1+z)=10^{15}~{}{\rm
cm}~{}(1+z)^{-1}\Gamma_{2.7}^{2}\frac{t_{\rm v}}{0.082{\rm s}}$, where the
variability timescale is measured as $t_{\rm v}=0.082~{}{\rm s}$ (Liu et al.,
2023), based on the Fermi-GBM data from $210\,{\rm s}$ to $219\,{\rm s}$ after
the Fermi-GBM trigger, and $c$ is the speed of light. The bulk Lorentz factor
$\Gamma$ suggested by LHAASO observations on the afterglow is also large (Cao
et al., 2023a). The large energy dissipation radius and the large bulk Lorentz
factor $\Gamma$ favor a large maximum acceleration energy of protons. If
protons are accelerated at the energy dissipation radius during the prompt
emission phase, the photomeson production interaction and the Bethe-Heitler
pair production process between accelerated protons and prompt photons will
produce neutrons, neutrinos, and electromagnetic particles initiating
electromagnetic cascades (contributing to GeV-TeV photons) (Böttcher & Dermer,
1998; Dermer & Atoyan, 2006; Wang et al., 2018). The non-detection of
neutrinos by the IceCube observatory, observations on GeV photons by Fermi-
LAT, and observations on VHE photons by LHAASO set constraints on the
acceleration and interaction mechanism of protons (Abbasi et al., 2023; Veres
et al., 2022; Bissaldi et al., 2022; Ai & Gao, 2023; Murase et al., 2022; Liu
et al., 2023; Rudolph et al., 2023; Wang et al., 2023). The tightest
constraint is made by (Wang et al., 2023) based on LHAASO observations, and
the baryon loading factor is constrained to be $\xi_{p}\equiv
E_{p}/E_{\gamma,\rm iso}\leq 2$ for a Lorentz factor $\Gamma=500$, suggesting
a possible high isotropic energy deposited in protons $(E_{p})$. LHAASO’s
observations on the afterglow emission suggest that the jet of GRB 221009A is
pointing to Earth. The fact that the jet pointing to Earth allows the
accelerated UHECRs to propagate to the direction of Earth.

In aggregate, the specific features of the B.O.A.T. GRB 221009A, such as the
large energy dissipation radius, the large bulk Lorentz factor, the high
isotropic energy, and the jet pointing to Earth, favor the burst as a possible
UHECR source. To answer the question of whether the UHECR burst from GRB
221009A can be detected in our life time, we study the acceleration and escape
of UHECRs in the burst, and simulate the propagation of UHECRs from the burst.
Since the jet composition of high-luminosity GRBs is likely to be dominated by
protons (Zhang et al., 2018), throughout this work, we assume the composition
of UHECRs from the GRB is pure proton.

##  2 The acceleration and escape of protons

Since most of the radiation energy is released via the prompt emission during
the brightest emission phase from $225.024{\rm~{}s}$ to $233.216{\rm~{}s}$
after the Konus-WIND trigger time in the second pulse (Frederiks et al.,
2023), we assume most of the UHECRs are accelerated during the brightest
emission phase in the second pulse as well. Parameters of the prompt emission
for the brightest phase in the second pulse are listed in Table 1. Protons are
expected to be accelerated at the energy dissipation radius within the dynamic
time, and meantime are undergoing energy loss due to the synchrotron emission,
the hadronic interaction with prompt photons.

The dynamic timescale is $t^{\prime}_{\rm dyn}=R/(c\Gamma)\simeq
67{\rm~{}s}R_{15}\Gamma_{2.7}^{-1}$. The magnetic field at the energy
dissipation radius can be calculated by assuming the energy fraction of the
magnetic field and electrons as $\epsilon_{B}$ and $\epsilon_{e}$, i.e.,
$B^{\prime}=(8\pi\epsilon_{B}L_{\gamma}/(\epsilon_{e}4\pi
R^{2}\Gamma^{2}c))^{1/2}\simeq 5.2\times 10^{3}~{}{\rm
G}~{}\epsilon_{e,-1}^{-\frac{1}{2}}\epsilon_{B,-2}^{\frac{1}{2}}\Gamma_{2.7}^{-1}R_{15}^{-1}L_{\gamma,54}^{\frac{1}{2}}$,
where $L_{\gamma}$ is the calibration luminosity. The acceleration timescale
of protons with energy of $\varepsilon_{p}$ is $t^{\prime}_{\rm
acc}(\varepsilon_{p})=R^{\prime}_{L}(\varepsilon_{p})/c\simeq 4.9~{}{\rm
s}~{}(1+z)\varepsilon_{p,20}\eta_{0}^{-1}\epsilon_{e,-1}^{\frac{1}{2}}\epsilon_{B,-2}^{-\frac{1}{2}}R_{15}L_{\gamma,54}^{-\frac{1}{2}}$.
The synchrotron cooling time scale of protons is $t^{\prime}_{\rm
syn}(\varepsilon_{p})=9m_{p}^{4}\Gamma/(4ce^{4}B^{\prime
2}\varepsilon_{p}(1+z))\simeq 7.3\times 10^{2}~{}{\rm
s}~{}(1+z)^{-1}\varepsilon_{p,20}^{-1}\epsilon_{e,-1}\epsilon_{B,-2}^{-1}\Gamma_{2.7}^{3}R_{15}^{2}L_{\gamma,54}^{-1}$,
where $m_{p}$ is the rest mass of protons.

The number density of the GRB prompt photons is described by
$\frac{dn(\varepsilon_{\gamma})}{d\varepsilon_{\gamma}}=A_{\gamma}\left(\frac{\varepsilon_{\gamma}}{\varepsilon_{\gamma,\rm
b}}\right)^{-q}$ where the normalization factor is $A_{\gamma}\sim
L_{\gamma}/(4\pi cR^{2}2\varepsilon_{\gamma,\rm b}^{2})$, with $q=\alpha$ for
$\varepsilon_{\gamma}<\varepsilon_{\gamma,\rm b}$, $q=\beta$ for
$\varepsilon_{\gamma}>\varepsilon_{\gamma,\rm b}$. Adopting the $\Delta$
resonance approximation, the cross section peaks when the energy of photons is
$\varepsilon_{\gamma,\Delta}\simeq 0.3\rm GeV$ in the proton rest frame. The
energy loss timescale of the photomeson production interaction between protons
and prompt photons is approximated to be (Waxman & Bahcall, 1997; He et al.,
2012)

$\displaystyle t^{\prime-1}_{p\gamma}(\varepsilon_{p})$ $\displaystyle\simeq
5.3\times 10^{-3}{\rm
s^{-1}}(1+z)^{-1}L_{\gamma,54}R_{15}^{-2}\Gamma_{2.7}^{-1}\varepsilon_{\gamma,\rm
b,3MeV}^{-1}$
$\displaystyle\times\left\\{\begin{array}[]{ll}\left(\frac{\varepsilon_{p}}{\varepsilon_{p,\rm
b}}\right)^{\beta-1},\,\,\,\,\varepsilon_{p}<\varepsilon_{p,\rm b}\\\
\left(\frac{\varepsilon_{p}}{\varepsilon_{p,\rm
b}}\right)^{\alpha-1}.\,\,\,\,\varepsilon_{p}>\varepsilon_{p,\rm
b}\end{array}\right.$

where $\varepsilon_{p,\rm
b}\equiv\Gamma^{2}\varepsilon_{\gamma,\Delta}m_{p}c^{2}/(2(1+z)^{2}\varepsilon_{\gamma,\rm
b})\simeq 8.9\times 10^{15}{\rm
eV}\Gamma_{2.7}^{2}(1+z)^{-2}\varepsilon_{\gamma,\rm b,3\rm MeV}^{-1}$ is the
characteristic energy of protons that interacting with photons of energy
$\varepsilon_{\gamma,\rm b}$ via $\Delta$ resonance.

Adopting parameters summarized in Table 1, the acceleration timescale, the
energy loss timescales, and the dynamic time of protons at the energy
dissipation radius are calculated and plotted in Figure 1. The curve of the
energy loss timescale via the photomeson interaction is calculated
numerically, since the analytical approximation shown in Eq. (2) is not
accurate for protons at the high energy end, where the multi-pion channel
plays an important role.

By comparing the dynamic timescale $t^{\prime}_{\rm dyn}$, the proton
synchrotron cooling timescale $t^{\prime}_{\rm syn}$, the photomeson
production timescale $t^{\prime}_{p\gamma}$ with the proton acceleration
timescale $t^{\prime}_{\rm acc}$, i.e., setting ${\rm min}[t^{\prime}_{\rm
dyn},~{}t^{\prime}_{\rm syn}(\varepsilon_{p,\rm
max}),~{}t^{\prime}_{p\gamma}(\varepsilon_{p,\rm max})]=t^{\prime}_{\rm
acc}(\varepsilon_{p,\rm max})$, we can estimate the maximum energy of the
accelerated protons $\varepsilon_{p,\rm max}$. As shown in Figure 1, we have
$t_{\rm syn}(\varepsilon_{p,\rm max})=t^{\prime}_{\rm acc}(\varepsilon_{p,\rm
max})<t^{\prime}_{\rm dyn}<t^{\prime}_{p\gamma}(\varepsilon_{p,\rm max})$, and
the maximum energy of accelerated protons is

$\displaystyle\varepsilon_{p,\rm max}=1.2\times 10^{21}{\rm
eV}(1+z)^{-1}\Gamma_{2.7}^{3/2}\eta_{0}^{1/2}\epsilon_{e,-1}^{1/4}\epsilon_{B,-2}^{-1/4}R_{15}^{1/2}L_{\gamma,54}^{-1/4},$
(4)

with $\eta$ as the acceleration efficiency.

Protons with lower energy will be confined in the acceleration site due to the
strong magnetic field. Those protons with the Larmor radius
$R^{\prime}_{L}(\varepsilon_{p})$ and the cooling scale of the photomeson
interaction $ct^{\prime}_{p\gamma}(\varepsilon_{p})$ larger than the shell
thickness $\Delta r^{\prime}$, would be able to escape directly. The escape
rate of protons with energy of $\varepsilon_{p}$ is estimated as (Baerwald et
al., 2013)

$f_{\rm esc}=\lambda^{\prime}_{\rm mfp}(\varepsilon_{p})/\Delta
r^{\prime}={\rm min}[\Delta
r^{\prime},R^{\prime}_{L}(\varepsilon_{p}),ct^{\prime}_{p\gamma}(\varepsilon_{p})]/\Delta
r^{\prime}$ (5)

where $\lambda^{\prime}_{\rm mfp}(\varepsilon_{p})\equiv{\rm min}[\Delta
r^{\prime},R^{\prime}_{L}(\varepsilon_{p}),ct^{\prime}_{p\gamma}(\varepsilon_{p})]$
is the mean free path of protons with energy of $\varepsilon_{p}$. The shell
thickness is $\Delta r^{\prime}\simeq\Gamma ct_{\rm v}/(1+z)=1.2\times
10^{12}{\rm cm}~{}(1+z)^{-1}\Gamma_{2.7}\frac{t_{\rm v}}{0.082{\rm s}}$, the
Larmor radius of protons is
$R^{\prime}_{L}(\varepsilon_{p})=\varepsilon_{p}(1+z)/(\Gamma\eta
eB^{\prime})=1.3\times 10^{12}{\rm
cm}~{}(1+z)\varepsilon_{p,21}\eta_{0}^{-1}\epsilon_{e,-1}^{\frac{1}{2}}\epsilon_{B,-2}^{-\frac{1}{2}}R_{15}L_{\gamma,54}^{-\frac{1}{2}}$,
and $e$ as the elementary charge. For protons with energy $\varepsilon_{\rm
p}>\varepsilon_{\rm p,esc}=9.6\times 10^{20}{\rm
eV}(1+z)^{-2}\Gamma_{2.7}\eta_{0}\epsilon_{e,-1}^{-\frac{1}{2}}\epsilon_{B,-2}^{\frac{1}{2}}R_{15}^{-1}L_{\gamma,54}^{\frac{1}{2}}\frac{t_{\rm
v}}{0.082{\rm s}}$, there are $\Delta r^{\prime}<R^{\prime}_{\rm L}$ and
$\Delta r^{\prime}<ct^{\prime}_{p\gamma}$. According to Eq. (5), the escape
efficiency is $f_{\rm esc}=1$ for protons with energy $\varepsilon_{\rm
p}>\varepsilon_{\rm p,esc}$ 111To be noted here, we only consider the energy
loss of protons due to interactions with prompt photons but ignore possible
interactions with afterglow photons. .

Such escaped protons will be deflected by magnetic fields in the host galaxy.
The deflection caused by the regular magnetic field in the host galaxy is
dependent on the GRB location in the host galaxy and the pointing direction of
the GRB jet. The Hubble space telescope observations reveal a disk-like host
galaxy with an effective radius of about $R_{\rm e}=2.45\pm 0.20\,\rm kpc$
(Levan et al., 2023). The host galaxy is viewed close to edge on with the
burst about 0.65 kpc offset from the nucleus (Levan et al., 2023). Therefore,
deflections on protons by the regular magnetic field in the host galaxy is
approximated to be (Finley, 2006)

$\theta_{\rm reg,hg}\simeq 4.1^{\circ}\varepsilon_{p,19.5}^{-1}\frac{B_{\rm
reg,hg}}{{1\,\mu\rm G}}\frac{R_{\rm e}}{{\rm 2.45\,\rm kpc}},$ (6)

with $B_{\rm reg,hg}$ is the average strength of the regular magnetic field.
The root mean square (rms) of random deflections due to the turbulent magnetic
field in the host galaxy is approximated as (Lee et al., 1995; Harari et al.,
2002; Finley, 2006)

$\theta_{\rm rms,hg}\simeq 1.5^{\circ}\varepsilon^{-1}_{p,19.5}\frac{B_{\rm
rms,hg}}{{2\mu\rm G}}\left(\frac{\lambda_{B,\rm hg}}{{\rm
50~{}pc}}\right)^{1/2}\left(\frac{L_{\rm hg}}{{10\rm~{}kpc}}\right)^{1/2},$
(7)

where $B_{\rm rms,hg}$ and $\lambda_{B,{\rm hg}}$ are the rms strength and the
coherent length of the turbulent magnetic field strength, and $L_{\rm hg}$ is
the propagation length of protons in the turbulent magnetic field. The rms of
the time delay due to the turbulent magnetic field is approximated as (Alcock
& Hatchett, 1978; Alves Batista, 2022)

$\displaystyle\Delta t_{\rm rms,hg}\simeq\frac{L_{\rm hg}\theta_{\rm
rms,hg}^{2}}{12c}\simeq 1.9\,{\rm
yr}\varepsilon^{-2}_{p,19.5}\left(\frac{B_{\rm rms,hg}}{2~{}\mu\rm
G}\right)^{2}\frac{\lambda_{B,\rm hg}}{\rm 50\,\rm pc}\left(\frac{L_{\rm
hg}}{{10\,\rm kpc}}\right)^{2}.$ (8)

If only considering the deflection by the regular magnetic field and meanwhile
$\theta_{\rm reg,hg}$ is larger than the jet half-open angle $\theta_{\rm j}$,
protons escaping from the host galaxy will not point to the Milky Way (MW).
Taking account of the deflection by the turbulent magnetic field in the host
galaxy and in the extragalactic field, a small fraction of protons might still
have a chance to arrive at the MW. If $\theta_{\rm reg,hg}<\theta_{\rm j}$ and
$\theta_{\rm rms,hg}\gg\theta_{\rm j}$, the flux of protons pointing to Earth
would be reduced significantly. LHAASO’s observation on the afterglow emission
suggests a very narrow jet with a half-open angle $\theta_{\rm j}\simeq
0.6^{\circ}E_{\rm k,55}^{-1/8}n_{0}^{1/8}$ (Cao et al., 2023a), where $E_{\rm
k}$ is the isotropic kinetic energy and $n$ is the circumstance density. If
assuming $E_{\rm k}=3\times 10^{55}{\rm erg}$, $n=1$, $B_{\rm
reg,hg}=1~{}\mu\rm G$, $B_{\rm rms,hg}=2~{}\mu\rm G$, $L_{\rm hg}=10\rm~{}kpc$
and $\lambda_{B,{\rm hg}}=50\rm~{}pc$, for protons with energy
$\varepsilon_{p}\gtrsim 300\,\rm EeV$, there are $\theta_{\rm
reg,hg}<\theta_{\rm j}$ and $\theta_{\rm rms,hg}<\theta_{\rm j}$, indicating
that protons with energy $\gtrsim 300\,\rm EeV$ injected into the inter-
galactic space from the host galaxy are still pointing to Earth. The rms of
the time delay for $\gtrsim 300\,\rm EeV$ protons is about 7 days, which will
not affect the flux of detected protons in an observation time of a few years.
If magnetic fields of the host galaxy are weaker, the threshold energy of
protons pointing to Earth might be lower than 300 EeV. For the sake of
conservation, we take account of protons with energy $\gtrsim 300$ EeV that
directly escaping from the burst, as a component of UHECRs that injected into
the inter-galactic space and propagating to the direction of Earth.

##  3  The production and escape of neutrons

Free neutrons are produced via the photomeson production interaction in the
burst,

$p+\gamma\rightarrow\Delta^{+}\rightarrow\left\\{\begin{array}[]{lc}n+\pi^{+}\\\\[5.69046pt]
p+\pi^{0}.\end{array}\right.$ (9)

and would decay into protons via the $\beta-$decay, i.e.,

$n\rightarrow p+e^{-}+\bar{\nu}_{e},$ (10)

with a lifetime of about $t^{\prime}_{n}\simeq 879.6~{}{\rm s}$ in the rest
frame (Workman et al., 2022).

The timescale for the neutron production is

$\displaystyle t^{\prime}_{p\gamma\rightarrow
n}\simeq\frac{\kappa_{p\gamma}}{\rho\kappa_{n}\Re_{n}}t^{\prime}_{p\gamma}\simeq
4.3t^{\prime}_{p\gamma}.$ (11)

where $\rho=0.23$ is the cross-section ratio, $\kappa_{n}=0.4$ is the fraction
of the proton energy converted into secondary neutrons, and $\Re_{n}=1/2$ is
the neutron multiplicity, for the target photon energy in the proton rest
frame being larger than $2000\,\rm MeV$ (Mücke et al., 2000b; Mannheim et al.,
2000; Dermer et al., 2012). Therefore, a fraction of the proton energy is
converted into the neutron energy. The produced neutrons will lose energy via
the $n\gamma$ interaction, with a cross section similar to that of $p\gamma$
interaction, i.e., $t^{\prime}_{n\gamma}\simeq t^{\prime}_{p\gamma}$. The
optical thickness for neutron escape is
$\tau_{n}\simeq(t^{{}^{\prime}-1}_{n\gamma}+t^{{}^{\prime}-1}_{n})/t^{{}^{\prime}-1}_{\rm
dyn}\simeq t^{\prime}_{\rm dyn}/t^{\prime}_{p\gamma}$(Baerwald et al., 2013).
In the case of GRB 221009A, neutrons can escape from the burst since
$\tau_{n}<1$.

Ultra-high-energy neutrons with energy of $\varepsilon_{n}$ will be able to
travel a distance of

$D_{n}=ct^{\prime}_{n}\frac{\varepsilon_{n}(1+z)}{m_{n}}\simeq 88~{}{\rm
kpc}~{}\varepsilon_{n,19}(1+z)$ (12)

before decaying into protons, which is considerably larger than the scale of
most galaxies for $\varepsilon_{n}\gtrsim 10^{19}{\rm eV}$, where $m_{n}$ is
the rest mass of neutrons. Therefore, neutrons with energy larger than 10 EeV,
produced in the photomeson production interactions, are able to escape from
the host galaxy with no deflections or time delay, and then decay into protons
in the inter-galactic space. The neutron-decay-induced protons will be an
important component of UHECRs that injected into the inter-galactic space.

##  4 Spectra of accelerated protons, escaped protons and secondary particles

Protons are assumed to be accelerated following a power-law spectrum with an
exponential cutoff at the highest energy range,

$\varepsilon_{\rm p}^{2}\frac{dN^{\rm acc}_{\rm p}}{d\varepsilon_{\rm
p}}=A_{p}\varepsilon_{\rm p}^{-p+2}{\rm
exp}(-\varepsilon_{p}/\varepsilon_{p,\rm max}),$ (13)

where the normalization coefficient $A_{p}=\xi_{p}E_{\gamma,\rm iso}/{\rm
ln}(\varepsilon_{p,\rm max}/\varepsilon_{p,\rm min})$ for $p=2$,
$\varepsilon_{p,\rm min}$ is assumed to be the rest energy of protons. The
luminosity of the total protons are $L_{p}=\xi_{p}L_{\gamma}=10^{54}{\rm
erg}~{}\xi_{p,0}L_{\gamma,54}$. The adopted values of $\xi_{p}$,
$E_{\gamma,\rm iso}$, $L_{\gamma}$, $\varepsilon_{p,\rm max}$ and
$\varepsilon_{p,\rm min}$ are listed in Table 1.

The spectrum of direct escaping protons can be approximated as

$\varepsilon^{2}_{\rm p}\frac{dN^{\rm esc}_{\rm p}}{d\varepsilon_{\rm
p}}=\varepsilon_{p}^{2}\frac{dN_{p}^{\rm acc}}{d\varepsilon_{p}}f_{\rm
esc}=A_{p}{\rm exp}(-\varepsilon_{p}/\varepsilon_{p,\rm max})f_{\rm esc}.$
(14)

The production of secondaries from the photomeson production process are
calculated via the following formula numerically,

$\frac{dN_{s}}{d\varepsilon_{s}}=\int\frac{d\varepsilon_{p}}{\varepsilon_{p}}\frac{dN_{p}^{\rm
acc}}{d\varepsilon_{p}}\int
d\varepsilon_{\gamma}\frac{dn_{\gamma}}{d\varepsilon_{\gamma}}\mathcal{R}(\varepsilon_{s},\varepsilon_{p}),$
(15)

where $s$ denotes neutrons and neutrinos, and
$\mathcal{R}(\varepsilon_{s},\varepsilon_{p})$ represents the production rate
of secondary particles, including neutrons and neutrinos with energy
$\varepsilon_{s}$, produced from primary protons with energy
$\varepsilon_{p}$. The production rate table
$\mathcal{R}(\varepsilon_{s},\varepsilon_{p})$ can be generated with the Monte
Carlo generator SOPHIA (Mücke et al., 2000a). Parameters adopted in the
calculation are listed in Table 1, following measurements and constraints
based on the multi-messenger observations.

In the calculation, the synchrotron cooling of the secondary pions and muons
(He et al., 2012) have been taken into account. Charged pions produced in the
photomeson interaction will decay into 4 final state leptons, via processes
$\pi^{\pm}\rightarrow\nu_{\mu}(\bar{\nu}_{\mu})\mu^{\pm}\rightarrow\nu_{\mu}(\bar{\nu}_{\mu})e^{+}(e^{-})\nu_{e}(\bar{\nu}_{e})\bar{\nu}_{\mu}(\nu_{\mu})$,
which approximately share the pion energy equally. The all flavor neutrino
spectrum can be calculated as (He et al., 2012)

$\epsilon_{\nu}\frac{dn_{\nu}}{d\epsilon_{\nu}}\mathrm{d}\epsilon_{\nu}=\frac{3\Re(\epsilon_{p})}{4}f_{p\gamma}(\epsilon_{p})\theta_{\nu}(\epsilon_{p})\epsilon_{p}\frac{dn_{p}}{d\epsilon_{p}}\mathrm{d}\epsilon_{p},$
(16)

with the factor
$\theta_{\nu}(\epsilon_{p})=\frac{1}{3}\zeta_{\pi}(\epsilon_{p})+\frac{2}{3}\zeta_{\pi}(\epsilon_{p})\zeta_{\mu}(\epsilon_{p})$
accounting for the cooling of secondary particles, where
$\zeta_{\pi}=1-\exp{(-t_{\pi,\rm syn}/\tau_{\pi})}$ and
$\zeta_{\mu}=1-\exp{(-t_{\mu,\rm syn}/\tau_{\mu})}$ are suppression factors
caused by the synchrotron cooling of pions and muons, and $\Re$ is the ratio
between the amount of the charged pions and the total pions. The synchrotron
cooling timescales of pions and muons can be calculated as $t_{\pi,\rm
syn}=0.041~{}{\rm
s}~{}(1+z)^{-1}\varepsilon_{\pi,18}^{-1}\epsilon_{e,-1}\epsilon_{B,-2}^{-1}\Gamma_{2.7}^{3}R_{15}^{2}L_{\gamma,54}^{-1}$
and $t_{\mu,\rm syn}=0.015~{}{\rm
s}~{}(1+z)^{-1}\varepsilon_{\mu,18}^{-1}\epsilon_{e,-1}\epsilon_{B,-2}^{-1}\Gamma_{2.7}^{3}R_{15}^{2}L_{\gamma,54}^{-1}$,
and the life time of pions and muons can be written as
$\tau_{\pi}=0.37{\rm~{}s}\epsilon_{\pi,18}\Gamma_{2.7}^{-1}(1+z)$ and
$\tau_{\mu}=41.5{\rm~{}s}\epsilon_{\mu,18}\Gamma_{2.7}^{-1}(1+z)$. The energy
of pions and muons in the observer’s frame are approximated to be
$\epsilon_{\pi}\simeq 0.2\epsilon_{\rm p}$ and
$\epsilon_{\mu}=0.15\epsilon_{\rm p}$.

The characteristic cutoff energy of pions and muons due to the synchrotron
cooling, can be calculated via equaling the life time and the synchrotron
cooling timescale, i.e.,

$\varepsilon_{\pi,\rm cut}=1.9\times 10^{17}~{}{\rm
eV}~{}(1+z)^{-1}\epsilon_{e,-1}^{1/2}\epsilon_{B,-2}^{-1/2}\Gamma_{2.7}^{2}R_{15}L_{\gamma,54}^{-1/2}$
(17)

and

$\varepsilon_{\mu,\rm cut}=1.0\times 10^{16}~{}{\rm
eV}~{}(1+z)^{-1}\epsilon_{e,-1}^{1/2}\epsilon_{B,-2}^{-1/2}\Gamma_{2.7}^{2}R_{15}L_{\gamma,54}^{-1/2}.$
(18)

In Figure 2, we plot the all-flavor neutrino fluence. The IceCube found none
track-like events to the direction of the GRB from the fast-response analysis
(FRA; IceCube Collaboration (2022); Abbasi et al. (2023)) in a time window of
$-1$ hour to $+2$ hours from the Fermi-GBM trigger time, assuming a power law
neutrino spectrum with different index p (IceCube Collaboration, 2022). We
plot the IceCube $90\%$ CL upper limit of all flavor neutrinos for index
$p=1.5$ in Figure 2, which shows that our predicted neutrino flux does not
violate the IceCue upper limit, assuming the flavor ratio as $1:1:1$ after
oscillation.

The Giant Radio Array for Neutrino Detection (GRAND, Álvarez-Muñiz et al.
(2020)) is a scheduled detector for high energy particles with energy larger
than $10^{17}\,{\rm eV}$, including neutrinos, UHECRs, and gamma-rays. The
full planned configuration of GRAND will cover an area of $200,000\,{\rm
km^{2}}$. The GRAND sensitivity for transient neutrino flares with zenith
angle of $90^{\circ}$ are plotted in Figure 2. Before taking account of the
synchrotron cooling of pions and muons, the neutrino spectrum peaks around
$10^{17}-10^{18}\,{\rm eV}$, which is just the sensitive band of GRAND.
However, by taking account of the synchrotron cooling suppression, the
neutrino fluence is suppressed in the sensitive band of GRAND. Hence, if
adopting the equilibrium energy of magnetic field $\epsilon_{B}=10^{-2}$,
neutrinos from GRB 221009A-like event will not be able to be detected by
GRAND, as shown in Fig. 2. If a lower equilibrium energy of magnetic field is
assumed, for instance, $\epsilon_{B}=10^{-4}$, the energy of neutrinos can
reach the GRAND’s threshold. Therefore, future observations of GRAND on
neutrinos from other GRBs similar to GRB 221009A would constrain the magnetic
field in the burst, and further constrain the acceleration mechanism of cosmic
rays and the GRB central engine model.

##  5 Propagation in the inter-galactic space and in the Milky Way

Protons injected into the inter-galactic space are contributed by two
components, as mentioned above, one component is protons directly escaping
from the burst, with energy above 300 EeV, and the other is the neutron-decay-
induced protons, with energy above 10 EeV. Assuming that protons accelerated
in the burst follows Eq. (13) with $p=2$, as shown by the dashed-dotted line
in Figure 3, the spectra of the directly escaping protons and the neutron-
decay-induced protons injected into the inter-galactic space would follow the
dark orange dotted line and the purple dotted line in Figure 3, respectively.

In the inter-galactic space, protons are deflected by the intergalactic
magnetic field (IGMF) randomly. The corresponding deflection and time delay of
protons depends on the strength of the IGMF. The rms of the deflection angle
is estimated as (Finley, 2006)

$\theta_{\rm rms,IG}=2.9\times 10^{-6\circ}\varepsilon^{-1}_{p,19.5}B_{\rm
IG,-16}\left(\frac{\lambda_{\rm IG}}{1\,\rm
Mpc}\right)^{1/2}\left(\frac{D}{745\rm\,Mpc}\right)^{1/2},$ (19)

where $B_{\rm IG}$ and $\lambda_{\rm IG}$ are the rms strength and the
coherence length of the IGMF, and $D$ is the distance between the source and
the MW. The time delay due to the IGMF is about (Alcock & Hatchett, 1978;
Alves Batista, 2022)

$\Delta t_{\rm IG}\simeq 16\,{\rm s}\,\varepsilon^{-2}_{p,19.5}B^{2}_{\rm
IG,-16}\frac{\lambda_{\rm IG}}{1\,\rm
Mpc}\left(\frac{D}{745\rm\,Mpc}\right)^{2}.$ (20)

If $B_{\rm IG}\lesssim 10^{-13}{\rm G}$ and $\lambda_{\rm IG}\lesssim 1{\rm
Mpc}$, there are $\theta_{\rm rms,IG}\ll\theta_{\rm j}$ and $\Delta t_{\rm
IG}<5{~{}\rm yr}$ for protons with energy larger than $10$ EeV, therefore, the
fluence of protons arriving at the MW within 5 years would not be suppressed
by the deflection of the IGMF significantly. A $\sim 400$ GeV photon to the
direction of GRB 221009A was detected by Fermi-LAT with a time delay of 0.4
days, suggesting an IGMF strength $B_{\rm IG}\sim 4\times 10^{-17}{\rm G}$
(Xia et al., 2022), which is consistent with constraints made from TeV blazars
(Dermer et al., 2011; Podlesnyi et al., 2022). Adopting $B_{\rm IG}\sim
4\times 10^{-17}{\rm G}$, in the inter-galactic space, protons with energy
larger than 10 EeV are deflected by extremely small angles and short time
delays, which can be ignored for the detection of UHECRs on Earth.

A 3D simulation on protons propagating in the inter-galactic space is operated
via the software CRPropa 3.2 (Alves Batista et al., 2016, 2022), taking
account of interactions with the EBL (Gilmore et al., 2012) and CMB photons,
including photo-pion productions, photo-disintegration, and electron-pair
productions. The direct escape protons and neutron-induced protons are
injected from a source at a comoving distance $647$ Mpc to a spherical
observer with a radius of $R_{\rm obs}=100~{}{\rm kpc}$ 222The input comoving
distance of GRB 221009A is calculated via taking a flat $\Lambda$CDM cosmology
with $H_{0}=67.4~{}\rm kms^{-1}Mpc^{-1}$ and $\Omega_{M}=0.315$(Planck
Collaboration et al., 2020).. In the 3D simulation, we assume a Kolmogorov-
type turbulent magnetic field with the rms strength of $B_{\rm rms}=4\times
10^{-17}{\rm~{}G}$ and the coherence length of $l_{c}=192\rm~{}kpc$. We
consider a grid of $256^{3}$ cells covering a total volume of
$\sim(7.7\rm~{}Mpc)^{3}$. The grid is assumed to be periodically repeated to
cover the whole simulation region. The Cash-Karp method (Cash & Karp, 1990) is
adopted for simulating charged particle propagation in a magnetic field. The
simulation shows that protons with energy larger than $30$ EeV lose more than
$50\%$ energy after propagating a comoving distance of 624.7 Mpc. Therefore,
the fluence of ultra-high-energy protons arriving at the edge of the MW is
significantly suppressed. Most of the UHECRs from GRB 221009A lose their
energy to $\lesssim 60$ EeV after propagating in the inter-galactic space. The
protons arriving at the edge of the MW peak around the energy of $30-40$ EeV
with the spectra shown by solid lines in Figure 3. From Figure 3, one can tell
that, the contribution from neutron-decay-induced protons (the purple solid
line) is dominated over the contribution from the direct escape protons (the
dark orange solid line).

For the propagation of UHECRs in the MW, the energy loss is negligible, and
the GMF model we adopt follows Jansson & Farrar (2012a, b)’s model, including
the random large-scale and turbulent small-scale components. We derive a map
of protons arriving at Earth following the probability distribution of UHECRs
after propagating in the Galactic magnetic field (GMF), via applying Galactic
lenses in CRPropa 3.2 (Alves Batista et al., 2022) on protons that arrive at
the edge of the MW. In Figure 4, we plot the map of 1000 UHECRs observed on
Earth, with the energy spectral shape of UHECRs at the edge of the MW
following the summed spectrum of the two solid lines shown in Figure 3. We
backtrack anti-protons from Earth to the edge of the MW at 20 kpc adopting the
Cash-Karp propagation method (Cash & Karp, 1990) via CRPropa 3.2, to measure
the time delay $\Delta t$ caused by the Jansson & Farrar (2012a, b) GMF model
and the fraction of protons that arriving at Earth within a time delay
$f(\Delta t)$. The time delay $\Delta t$ and deflection angles $\theta$ of
those protons are plotted in Figure 5, which indicates that, as time
increases, protons arriving at Earth have a tendency to distribute around the
direction of the GRB within a larger deflection angle. The solid angle
$\Omega_{\rm max}(\Delta t)$ is calculated corresponding to the circle region
around the GRB with a radius of the maximum deflection angle $\theta_{\rm
max}(\Delta t)$. Values of $f(\Delta t)$ and the expected solid angle
$\Omega_{\rm max}(\Delta t)$ for the time delay $\Delta t$ derived via our
simulations on backtracking $2\times 10^{6}$ anti-protons in the GMF, are
listed in Table 2. In reality, the solid angle could be measured from the
detected excess of UHECRs around the direction of the GRB.

##  6 Results and Prospects

The expected counts of protons with the energy of $>10$ EeV from the GRB
detected by October of the year $2022+\Delta t$ can be calculated as

$N_{\rm s}(\Delta t)=163f(\Delta t)\xi_{p,0}\frac{A_{\rm
eff}}{310\rm~{}km^{2}}.$ (21)

where $A_{\rm eff}$ is the annual exposure of the detector.

We assume the background spectrum is the same as the spectrum of the isotropic
UHECRs detected by the PAO. The expected UHECR background counts within a
solid angle $\Omega_{\rm max}(\Delta t)$ around the GRB integrated over time
by October of the year $2022+\Delta t$ can be estimated via 333Here we
calculate the integration adopting half-a-year bins.

$N_{\rm b}(\Delta t)=A_{\rm eff}\int^{\Delta t}_{0}\int^{\infty}_{10\rm
EeV}J_{\rm CR}(\varepsilon_{\rm CR})\Omega_{\rm max}(dt)d\varepsilon_{\rm
CR}dt,$ (22)

where $J_{\rm CR}(\varepsilon_{\rm CR})$ follows the spectrum of isotropic
UHECRs detected by the PAO, as described by Eq. (8) in Aab et al. (2020), in
units of $\rm km^{-2}sr^{-1}yr^{-1}EeV^{-1}$.

The statistical significance of the UHECR excess can be calculated via the Li
$\&$ Ma method (Li & Ma, 1983; Abbasi et al., 2014)

$S_{\rm LM}=\sqrt{2}\left[N_{\rm on}{\rm ln}\left(\frac{2N_{\rm on}}{N_{\rm
on}+N_{\rm off}}\right)+N_{\rm off}{\rm ln}\left(\frac{2N_{\rm off}}{N_{\rm
on}+N_{\rm off}}\right)\right]^{1/2}$ (23)

where $N_{\rm on}=N_{\rm s}+N_{\rm b}$ and $N_{\rm off}=N_{\rm b}$.

The Pierre Auger Observatory (PAO), Telescope Array (TA), and TA$\times$4 are
the largest UHECR detectors in the world. The exposure of the cosmic ray
detector depends on the declination of the arriving UHECRs. According to the
directional exposure for surface detectors of the PAO and TA shown in the left
panel of Figure 1 in Biteau et al. (2019), the averaged annual exposure of the
PAO and TA for the GRB’s declination $19.7^{\circ}$ is adopted as $\sim
310{\rm~{}km^{2}}$ and $\sim 170{\rm~{}km^{2}}$, with compiled dataset of
events with energies $E>8.9~{}{\rm EeV}$ and $E>10~{}{\rm EeV}$, respectively.
The exposure of fully completed TA$\times$4 is approximately 4 times the TA
exposure, while so far about half of the full TA$\times$4 is completed (Kido,
2023). Counts of GRB UHECRs that will be detected by the PAO and the full
TA$\times$4, for the baryon loading factor $\xi_{p}=1$, are plotted in Figure
6. The significance of the PAO and TA$\times$4 detection peaks around
2030-2035, reaches $4.3~{}\sigma$ and $6.4~{}\sigma$, respectively, and then
decreases since the solid angle of the arriving UHECRs increases, more
background is included, as the time increases. By October 2027, after 5-year
observations, the PAO and the full TA$\times 4$ can detect $\sim 17$ and $\sim
36$ GRB UHECRs (plus $\sim 5$ and $\sim 11$ background UHECRs) with energy
larger than 10 EeV and the maximum deflection angle $\theta_{\rm max}$ of
$\sim 5^{\circ}$ around the direction of the GRB. The corresponding
significance of the excess is estimated to be $\sim 3.4~{}\sigma$ and
$5.0~{}\sigma$, respectively. If combining the PAO and the fully completed
TA$\times$4 data, the combined significance would become higher, which would
exceed 5 $\sigma$ by 2027.

The annual exposure of GRAND for UHECRs from GRB 221009A is about 10 times the
PAO according to Figure 10 in the GRAND white paper (Álvarez-Muñiz et al.,
2020). If the construction of GRAND can be completed before 2035, GRAND can
detect $\sim 148$ GRB UHECRs plus $\sim 347$ background UHECRs with the energy
of $>10\rm\,EeV$ around the GRB, with 8-year operation from the beginning of
2035 to the end of 2042, for $\xi_{p}=1$, with a significance of the excess as
$\sim 5.1~{}\sigma$. The projected significance of the UHECR excess measurable
for these three detectors is shown in Figure 6, and the prospects are indeed
promising.

In He et al. (2016), a Monte Carlo Bayesian method is proposed to study the
correlation between UHECRs and source candidates, via studying the spatial
distribution of UHECRs with different energies. In the future, we can take
into account not only the spatial distribution of UHECRs depending on the
energy but also the temporal information of the arriving UHECRs, to further
check the correlation between the detected UHECRs and the GRB at high
statistics.

In this work, we adopt a high bulk Lorentz factor $\sim 500$, as suggested by
LHAASO observations on the afterglow. With a typical variability timescale of
$\sim 0.082$ s, as observed by Fermi-GBM, we then take an energy dissipation
radius $R\sim 10^{15}{\rm cm}$, where protons are accelerated. A large energy
dissipation radius and a high bulk Lorentz factor of the ejecta are helpful in
enhancing the maximum energy of the accelerated protons, as indicated by Eq.
(4), and further leads to a high flux of observed UHECRs. Interestingly, a
large energy dissipation radius of $\sim 10^{15}$ cm may challenge the
standard fireball internal shock model unless just a small fraction of
electrons have been accelerated otherwise the synchrotron radiation of the
shocked electrons will not be able to peak at MeV energies. Instead, a large
energy dissipation radius of $10^{15}-10^{16}$ cm is natural for a Poynting-
flux-dominated GRB outflow (e.g., Fan et al., 2005; Zhang & Yan, 2011; Kumar &
Zhang, 2015). Indeed, recently, Zhang et al. (2023) suggested that the ejecta
of GRB 221009A was composed of a narrow ($\sim 0.6^{\circ}$ half opening
angle) Poynting-flux-dominated outflow composition and a broader matter-
dominated jet wing. In the future, a successful detection of the UHECR
outburst from GRB 221009A will not only directly establish the GRB-UHECR
connection, but also provide us a valuable chance to measure or stringently
constrain the IGMF and the GMF, and key parameters of the GRB physics, such as
the energy dissipation radius, the bulk Lorentz factor and the baryon loading
factor of the jet, and further reveal the radiation mechanism and the central
engine of the GRB.

Table 1:  Adopted Parameters of GRB 221009A. Descriptions | Symbols | Values
---|---|---
Redshift | $z$ | 0.151
Dissipation radius | $R$ | $10^{15}{\rm cm}$
Low energy photon index 444Parameters for the brightest emission from $T_{0}+225.024{\rm s}$ to $T_{0}+233.216{\rm s}$ in the second pulse are listed (Frederiks et al., 2023), with $T_{0}=47821.648{\rm s}$ UT (13:17:01.648) on 2022 October 9 as the Konus-WIND trigger time. | $\alpha$ | 0.76
High-energy photon indexa | $\beta$ | 2.13
Minimum energy of photon spectruma | $\varepsilon_{\gamma,\rm min}$ | $20\,\rm keV$
Maximum energy of photon spectruma | $\varepsilon_{\gamma,\rm max}$ | $10\,\rm MeV$
Peak energy of photon spectruma | $\varepsilon_{\gamma,{\rm b}}$ | $3\,\rm MeV$
Calibration luminosity555The luminosity at 20 keV–10 MeV. | $L_{\gamma}$ | $1\times 10^{54}\,\rm erg\,s^{-1}$
Bulk Lorentz factor | $\Gamma$ | 500
Baryon loading factor | $\xi_{\rm p}$ | 1
Proton index | $p$ | 2
Minimum energy of proton spectrum | $\varepsilon_{\rm p,\rm min}$ | $9.8\times 10^{10}\,\rm eV$
Maximum energy of proton spectrum | $\varepsilon_{\rm p,max}$ | $1.2\times 10^{21}\,\rm eV$
Fraction of magnetic field energy | $\epsilon_{B}$ | 0.01
Fraction of electron energy | $\epsilon_{e}$ | 0.1
Total radiation energy | $E_{\gamma,\rm iso}$ | $10^{55}\,\rm erg$

Table 2:  The time delay $\Delta t$, the fraction of arriving UHECRs $f(\Delta t)$, the solid angle $\Omega_{\rm max}(\Delta t)$, and the significance detected by the PAO and TA$\times$4 by the October of certain years, assuming the baryon loading factor $\xi_{\rm p}=1$.  Time |  $\Delta t$[yr] | $f(\Delta t)$ |  $\Omega_{\rm max}(\Delta t)$[sr] | PAO Significance[$\sigma$] | TA$\times$4 Significance[$\sigma$]
---|---|---|---|---|---
2025.10 | 3 | 3.5$\%$ | 0.013 | 1.9 | 2.8
2027.10 | 5 | 10.2$\%$ | 0.020 | 3.4 | 5.0
2029.10 | 7 | 15.5$\%$ | 0.027 | 4.0 | 5.8
2031.10 | 9 | 19.4$\%$ | 0.034 | 4.2 | 6.2
2033.10 | 11 | 22.7$\%$ | 0.040 | 4.3 | 6.3
2035.10 | 13 | 25.7$\%$ | 0.046 | 4.3 | 6.4
2037.10 | 15 | 29.8$\%$ | 0.059 | 4.1 | 6.1

Figure 1:  Various timescales as functions of the proton energy. All the
timescales are measured in the comoving frame of the GRB outflow. The adopted
parameters are listed in the Table 1. Figure 2:  The fluence of all flavor
neutrinos from the source, adopting parameters as in Table 1. The dark green
dotted line is the neutrino spectrum without considering the synchrotron
cooling of pions and muons, and the dark green solid line is the one
considering the synchrotron cooling effect. The black solid line is the
sensitivity of GRAND detecting neutrinos with a zenith angle of
$\theta=90^{\circ}$ assuming null background. The cyan solid line is the upper
limit flux of all-flavor neutrinos in the IceCube’s 3 hours FRA by assuming
the spectral index as $p=1.5$.  Figure 3: The spectra of protons involved in
this work. The dash-dotted line, dotted lines, and solid lines represent
protons accelerated in the burst, injected to the inter-galactic space, and
arriving at the edge of the MW, respectively.  Figure 4: The expected arrival
distribution of 1000 UHECRs from GRB 221009A in the Galactic coordination, in
the Hammer projection. Colors represent energies of arriving UHECRs. The red
star denotes the coordinate of GRB 221009A.  Figure 5: The expected time delay
and deflection angles of the arriving UHECRs from GRB 221009A. The deflection
and the time delay are mainly contributed by the Galactic magnetic fields.
Colors represent energies of arriving UHECRs.  Figure 6: The prospect of
detecting the UHECR outburst from GRB 221009A. The expected integral counts
detected by the PAO and the full TA$\times$4, as a function of the time since
the trigger time of GRB 221009A, are shown in solid lines. The teal lines
denote the significance of the UHECR excess, detected by the PAO, the full
TA$\times$4 and GRAND, respectively. The X-axis starts from the year of 2025.

This work is supported by Project for Young Scientists in Basic Research of
Chinese Academy of Sciences (No. YSBR-061), and by NSFC under the grants of
No. 12173091, No. 11921003, No.12321003 and No. 12333006.

## References

  * Aab et al. (2020) Aab, A., Abreu, P., Aglietta, M., et al. 2020, Phys. Rev. D, 102, 062005, doi: 10.1103/PhysRevD.102.062005
  * Abbasi et al. (2022) Abbasi, R., Ackermann, M., Adams, J., et al. 2022, ApJ, 939, 116, doi: 10.3847/1538-4357/ac9785
  * Abbasi et al. (2023) —. 2023, ApJ, 946, L26, doi: 10.3847/2041-8213/acc077
  * Abbasi et al. (2014) Abbasi, R. U., Abe, M., Abu-Zayyad, T., et al. 2014, ApJ, 790, L21, doi: 10.1088/2041-8205/790/2/L21
  * Ai & Gao (2023) Ai, S., & Gao, H. 2023, ApJ, 944, 115, doi: 10.3847/1538-4357/acb3bf
  * Alcock & Hatchett (1978) Alcock, C., & Hatchett, S. 1978, ApJ, 222, 456, doi: 10.1086/156159
  * Álvarez-Muñiz et al. (2020) Álvarez-Muñiz, J., Alves Batista, R., Balagopal V., A., et al. 2020, Science China Physics, Mechanics, and Astronomy, 63, 219501, doi: 10.1007/s11433-018-9385-7
  * Alves Batista (2022) Alves Batista, R. 2022, arXiv e-prints, arXiv:2210.12855, doi: 10.48550/arXiv.2210.12855
  * Alves Batista et al. (2016) Alves Batista, R., Dundovic, A., Erdmann, M., et al. 2016, J. Cosmology Astropart. Phys, 2016, 038, doi: 10.1088/1475-7516/2016/05/038
  * Alves Batista et al. (2022) Alves Batista, R., Becker Tjus, J., Dörner, J., et al. 2022, J. Cosmology Astropart. Phys, 2022, 035, doi: 10.1088/1475-7516/2022/09/035
  * An et al. (2023) An, Z.-H., Antier, S., Bi, X.-Z., et al. 2023, arXiv e-prints, arXiv:2303.01203, doi: 10.48550/arXiv.2303.01203
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Baerwald et al. (2013) Baerwald, P., Bustamante, M., & Winter, W. 2013, ApJ, 768, 186, doi: 10.1088/0004-637X/768/2/186
  * Bissaldi et al. (2022) Bissaldi, E., Omodei, N., Kerr, M., & Fermi-LAT Team. 2022, GRB Coordinates Network, 32637, 1
  * Biteau et al. (2019) Biteau, J., Bister, T., Caccianiga, L., et al. 2019, in European Physical Journal Web of Conferences, Vol. 210, European Physical Journal Web of Conferences, 01005, doi: 10.1051/epjconf/201921001005
  * Böttcher & Dermer (1998) Böttcher, M., & Dermer, C. D. 1998, ApJ, 499, L131, doi: 10.1086/311366
  * Burns et al. (2023) Burns, E., Svinkin, D., Fenimore, E., et al. 2023, ApJ, 946, L31, doi: 10.3847/2041-8213/acc39c
  * Cao et al. (2023a) Cao, Z., et al. 2023a, Science, 380, adg9328, doi: 10.1126/science.adg9328
  * Cao et al. (2023b) —. 2023b.  https://arxiv.org/abs/2310.08845
  * Cash & Karp (1990) Cash, J. R., & Karp, A. H. 1990, ACM Transactions on Mathematical Software, 16, 201–222, doi: 10.1145/79505.79507
  * Castro-Tirado et al. (2022) Castro-Tirado, A. J., Sanchez-Ramirez, R., Hu, Y. D., et al. 2022, GRB Coordinates Network, 32686, 1
  * Das & Razzaque (2023) Das, S., & Razzaque, S. 2023, A&A, 670, L12, doi: 10.1051/0004-6361/202245377
  * de Ugarte Postigo et al. (2022) de Ugarte Postigo, A., Izzo, L., Pugliese, G., et al. 2022, GRB Coordinates Network, 32648, 1
  * Dermer & Atoyan (2006) Dermer, C. D., & Atoyan, A. 2006, New Journal of Physics, 8, 122, doi: 10.1088/1367-2630/8/7/122
  * Dermer et al. (2011) Dermer, C. D., Cavadini, M., Razzaque, S., et al. 2011, ApJ, 733, L21, doi: 10.1088/2041-8205/733/2/L21
  * Dermer et al. (2012) Dermer, C. D., Murase, K., & Takami, H. 2012, ApJ, 755, 147, doi: 10.1088/0004-637X/755/2/147
  * Fan et al. (2005) Fan, Y. Z., Zhang, B., & Proga, D. 2005, ApJ, 635, L129, doi: 10.1086/499489
  * Finley (2006) Finley, C. B. 2006, PhD thesis, Columbia University, New York
  * Frederiks et al. (2023) Frederiks, D., Svinkin, D., Lysenko, A. L., et al. 2023, ApJ, 949, L7, doi: 10.3847/2041-8213/acd1eb
  * Gilmore et al. (2012) Gilmore, R. C., Somerville, R. S., Primack, J. R., & Domínguez, A. 2012, MNRAS, 422, 3189, doi: 10.1111/j.1365-2966.2012.20841.x
  * Harari et al. (2002) Harari, D., Mollerach, S., Roulet, E., & Sánchez, F. 2002, Journal of High Energy Physics, 2002, 045, doi: 10.1088/1126-6708/2002/03/045
  * He et al. (2016) He, H.-N., Kusenko, A., Nagataki, S., et al. 2016, Phys. Rev. D, 93, 043011, doi: 10.1103/PhysRevD.93.043011
  * He et al. (2012) He, H.-N., Liu, R.-Y., Wang, X.-Y., et al. 2012, ApJ, 752, 29, doi: 10.1088/0004-637X/752/1/29
  * IceCube Collaboration (2022) IceCube Collaboration. 2022, GRB Coordinates Network, 32665, 1
  * Icecube Collaboration et al. (2012) Icecube Collaboration, Abbasi, R., Abdou, Y., et al. 2012, Nature, 484, 351, doi: 10.1038/nature11068
  * Isravel et al. (2023) Isravel, H., Begue, D., & Pe’er, A. 2023, Astrophys. J., 956, 12, doi: 10.3847/1538-4357/acefcd
  * Jansson & Farrar (2012a) Jansson, R., & Farrar, G. R. 2012a, ApJ, 757, 14, doi: 10.1088/0004-637X/757/1/14
  * Jansson & Farrar (2012b) —. 2012b, ApJ, 761, L11, doi: 10.1088/2041-8205/761/1/L11
  * Kido (2023) Kido, E. 2023, in European Physical Journal Web of Conferences, Vol. 283, European Physical Journal Web of Conferences, 06003, doi: 10.1051/epjconf/202328306003
  * Kumar & Zhang (2015) Kumar, P., & Zhang, B. 2015, Phys. Rep., 561, 1, doi: 10.1016/j.physrep.2014.09.008
  * Lee et al. (1995) Lee, S., Olinto, A. V., & Sigl, G. 1995, ApJ, 455, L21, doi: 10.1086/309812
  * Levan et al. (2023) Levan, A. J., Lamb, G. P., Schneider, B., et al. 2023, ApJ, 946, L28, doi: 10.3847/2041-8213/acc2c1
  * Li & Ma (1983) Li, T. P., & Ma, Y. Q. 1983, ApJ, 272, 317, doi: 10.1086/161295
  * Liu et al. (2023) Liu, R.-Y., Zhang, H.-M., & Wang, X.-Y. 2023, ApJ, 943, L2, doi: 10.3847/2041-8213/acaf5e
  * Malesani et al. (2023) Malesani, D. B., Levan, A. J., Izzo, L., et al. 2023, arXiv e-prints, arXiv:2302.07891, doi: 10.48550/arXiv.2302.07891
  * Mannheim et al. (2000) Mannheim, K., Protheroe, R. J., & Rachen, J. P. 2000, Phys. Rev. D, 63, 023003, doi: 10.1103/PhysRevD.63.023003
  * Mészáros (2002) Mészáros, P. 2002, ARA&A, 40, 137, doi: 10.1146/annurev.astro.40.060401.093821
  * Mirabal (2023) Mirabal, N. 2023, MNRAS, 519, L85, doi: 10.1093/mnrasl/slac157
  * Mücke et al. (2000a) Mücke, A., Engel, R., Rachen, J. P., Protheroe, R. J., & Stanev, T. 2000a, Computer Physics Communications, 124, 290, doi: 10.1016/S0010-4655(99)00446-4
  * Mücke et al. (2000b) Mücke, A., Rachen, J. P., Engel, R., Protheroe, R. J., & Stanev, T. 2000b, Nuclear Physics B Proceedings Supplements, 80, 08/10, doi: 10.48550/arXiv.astro-ph/9905153
  * Murase et al. (2022) Murase, K., Mukhopadhyay, M., Kheirandish, A., Kimura, S. S., & Fang, K. 2022, ApJ, 941, L10, doi: 10.3847/2041-8213/aca3ae
  * Pillera et al. (2022) Pillera, R., Bissaldi, E., Omodei, N., La Mura, G., & Longo, F. 2022, The Astronomer’s Telegram, 15656, 1
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6, doi: 10.1051/0004-6361/201833910
  * Podlesnyi et al. (2022) Podlesnyi, E. I., Dzhatdoev, T. A., & Galkin, V. I. 2022, MNRAS, 516, 5379, doi: 10.1093/mnras/stac2509
  * Rudolph et al. (2023) Rudolph, A., Petropoulou, M., Winter, W., & Bošnjak, Ž. 2023, ApJ, 944, L34, doi: 10.3847/2041-8213/acb6d7
  * Veres et al. (2022) Veres, P., Meegan, C., & Fermi GBM Team. 2022, GRB Coordinates Network, 33125, 1
  * Vietri (1995) Vietri, M. 1995, ApJ, 453, 883, doi: 10.1086/176448
  * Wang et al. (2018) Wang, K., Liu, R.-Y., Dai, Z.-G., & Asano, K. 2018, ApJ, 857, 24, doi: 10.3847/1538-4357/aab667
  * Wang et al. (2023) Wang, K., Ma, Z.-P., Liu, R.-Y., et al. 2023, arXiv e-prints, arXiv:2302.11111, doi: 10.48550/arXiv.2302.11111
  * Waxman (1995) Waxman, E. 1995, Phys. Rev. Lett., 75, 386, doi: 10.1103/PhysRevLett.75.386
  * Waxman & Bahcall (1997) Waxman, E., & Bahcall, J. 1997, Phys. Rev. Lett., 78, 2292, doi: 10.1103/PhysRevLett.78.2292
  * Williams et al. (2023) Williams, M. A., Kennea, J. A., Dichiara, S., et al. 2023, ApJ, 946, L24, doi: 10.3847/2041-8213/acbcd1
  * Workman et al. (2022) Workman, R. L., Burkert, V. D., Crede, V., et al. 2022, Progress of Theoretical and Experimental Physics, 2022, 083C01, doi: 10.1093/ptep/ptac097
  * Xia et al. (2022) Xia, Z.-Q., Wang, Y., Yuan, Q., & Fan, Y.-Z. 2022, arXiv e-prints, arXiv:2210.13052, doi: 10.48550/arXiv.2210.13052
  * Zhang et al. (2023) Zhang, B., Wang, X.-Y., & Zheng, J.-H. 2023, arXiv e-prints, arXiv:2311.14180, doi: 10.48550/arXiv.2311.14180
  * Zhang & Yan (2011) Zhang, B., & Yan, H. 2011, ApJ, 726, 90, doi: 10.1088/0004-637X/726/2/90
  * Zhang et al. (2023a) Zhang, B. T., Murase, K., Ioka, K., et al. 2023a, Astrophys. J. Lett., 947, L14, doi: 10.3847/2041-8213/acc79f
  * Zhang et al. (2023b) Zhang, B. T., Murase, K., Ioka, K., & Zhang, B. 2023b.  https://arxiv.org/abs/2311.13671
  * Zhang et al. (2018) Zhang, B. T., Murase, K., Kimura, S. S., Horiuchi, S., & Mészáros, P. 2018, Phys. Rev. D, 97, 083010, doi: 10.1103/PhysRevD.97.083010