# A Cosmological Fireball with Sixteen-Percent Gamma-Ray Radiative Efficiency

Liang Li ICRANet, Piazza della Repubblica 10, 65122 Pescara, Italy  INAF –
Osservatorio Astronomico d’Abruzzo, Via M. Maggini snc, I-64100, Teramo, Italy
Dip. di Fisica and ICRA, Sapienza Universita di Roma, Piazzale Aldo Moro 5,
I-00185 Rome, Italy  Yu Wang  ICRANet, Piazza della Repubblica 10, 65122
Pescara, Italy  INAF – Osservatorio Astronomico d’Abruzzo, Via M. Maggini snc,
I-64100, Teramo, Italy  Dip. di Fisica and ICRA, Sapienza Universita di Roma,
Piazzale Aldo Moro 5, I-00185 Rome, Italy  Felix Ryde  Department of Physics,
KTH Royal Institute of Technology, and the Oskar Klein Centre for
Cosmoparticle Physics, 10691 Stockholm, Sweden  Asaf Pe’er  Department of
Physics, Bar-Ilan University, Ramat-Gan 52900, Israel  Bing Zhang  Department
of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA
Sylvain Guiriec  Department of Physics, The George Washington University, 725
21st Street NW, Washington, DC 20052, USA  NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA  Alberto J. Castro-Tirado  Instituto de Astrofísica
de Andalucía (IAA-CSIC), PO Box 03004, 18008 Granada, Spain  Departamento de
Ingeniería de Sistemas y Automática, Escuela de Ingenierías, Universidad de
Málaga, Málaga, Spain  D. Alexander Kann  Instituto de Astrofísica de
Andalucía (IAA-CSIC), PO Box 03004, 18008 Granada, Spain  Magnus Axelsson
Department of Astronomy, Stockholm University, SE-106 91 Stockholm, Sweden
Kim Page  School of Physics and Astronomy, University of Leicester, University
Road, Leicester LE1 7RH, UK  Péter Veres  Center for Space Plasma and
Aeronomic Research, University of Alabama in Huntsville, Huntsville,AL,USA
Space Science Department,University of Alabama in Huntsville, Huntsville,
AL,USA  P. N. Bhat  Center for Space Plasma and Aeronomic Research, University
of Alabama in Huntsville, Huntsville,AL,USA Liang Li, Yu Wang, Bing Zhang
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Received ddmmyyyy; Accepted ddmmyyyy)

###### Abstract

Gamma-ray bursts (GRBs) are the most powerful explosions in the universe. How
efficiently the jet converts its energy to radiation is a long-standing
problem and it is poorly constrained. The standard model invokes a
relativistic fireball with a bright photosphere emission component. A
definitive diagnosis of GRB radiation components and measurement of GRB
radiative efficiency require prompt emission and afterglow data with high-
resolution and wide-band coverage in time and energy. Here we report a
comprehensive temporal and spectral analysis of the TeV-emitting bright GRB
190114C. Its fluence is one of the highest of all GRBs detected so far, which
allows us to perform a high-resolution study of the prompt emission spectral
properties and their temporal evolution down to a timescale of about 0.1 s. We
observe that each of the initial pulses has a thermal component contributing
$\sim 20\%$ of the total energy, the corresponding temperature and the
inferred Lorentz factor of the photosphere evolve following broken power-law
shapes. From the observation of the non-thermal spectra and the light-curve,
the onset of afterglow corresponding to the deceleration of the fireball is
considered at $\sim 6$ s. By incorporating the thermal and the non-thermal
observations, as well as the photosphere and the synchrotron radiative
mechanisms, we can directly derive the fireball energy budget with little
dependence on hypothetical parameters and to measure a $\sim 16\%$ radiative
efficiency for this GRB. With the fireball energy budget derived, the
afterglow microphysics parameters can also be constrained directly from the
data.

Gamma-ray bursts (629); Astronomy data analysis (1858)

ddmmyyyy

##  1 Introduction

On 14 January 2019 at 20:57:02.63 Universal Time (UT) (hereafter $T_{0}$), an
ultra-bright burst, GRB 190114C, first triggered the Gamma-ray Burst Monitor
(GBM) onboard the _Fermi_ Gamma-ray Space Telescope(Hamburg et al., 2019) and
the Neil Gehrels Swift Observatory’s (Swift hereafter) Burst Alert Telescope
(BAT)(J.D. Gropp et al., 2019). Soon after, the Large Area Telescope (LAT)
onboard _Fermi_ , Konus-_Wind_ , _INTEGRAL_ /SPI-ACS, _AGILE_ /MCAL, and the
_Insight-HXMT_ /HE were also triggered. Long-lasting and multi-wavelength
afterglow observations were carried out by the Major Atmospheric Gamma Imaging
Cherenkov (MAGIC) telescopes(MAGIC Collaboration et al., 2019b) in the
Teraelectronvolt band and _Swift_ in the X-ray and optical bands, and by
several ground-based optical and radio telescopes, such as GROND(Bolmer &
Schady, 2019), GTC(Castro-Tirado et al., 2019), VLA(Alexander et al., 2019),
MeerKAT(Tremou et al., 2019).

In this paper, we present a comprehensive analysis of GRB 190114C and derive
fireball parameters using rich observation data. The paper is organized as
follows. In Section 2, we outline the main observing properties of the burst
and the purpose of the paper. The methodology is presented in Section 3. The
models that apply to GRB 190114C are presented in Section 4. We presented our
results in Section 5. Our conclusions are summarized in Section 6. Throughout
the paper, the standard $\Lambda$-CDM cosmology with the parameters
$H_{0}=67.4$ ${\rm kms^{-1}}$ ${\rm Mpc^{-1}}$, $\Omega_{M}=0.315$, and
$\Omega_{\Lambda}=0.685$ are adopted (Planck Collaboration et al., 2018).

##  2 Overview

The $T_{90}$ duration111The time taken to accumulate $90\%$ of the burst
fluence starts at the 5% fluence level and ends at the 95% fluence level.
reported by the Fermi term is $\sim$116 s, and therefore GRB 190114C belongs
to the class of long-duration bursts. The 1024 ms peak flux and the fluence
during $T_{90}$ duration at 10-1000 keV measured by Fermi-GBM are
246.9$\pm$0.9 photon cm-2 s-1 and (4.436$\pm$0.005)$\times$10-4 erg cm-2,
respectively. A measured redshift of $z$=0.424 was announced by Castro-Tirado
et al. (2019), therefore, the isotropic-equivalent $\gamma$-ray energy with a
$k$-correction to the rest-frame (1-104 keV) is estimated to be $E_{\gamma,\rm
iso}$=(2.8$\pm$0.3)$\times$1053 erg. Fermi-LAT observed the first GeV photon
at $T_{0}$+2.1 s, and the highest-energy photon is a 22.9 GeV event was
observed at $\sim T_{0}$+15 s (Wang et al., 2019b). The afterglow emission
measured by Swift-XRT begins at $\sim$ $T_{0}$+68 s.

The prompt emission light curve consists of three distinct emission pulses
(Ajello et al., 2020; Wang et al., 2019b) (Figure 1). The first pulse (i.e.,
$P_{1}$) starts at $\sim$ $T_{0}$ and lasts for $\sim$ 2.35 s, the second
pulse (i.e., $P_{2}$) exhibits multiple peaks and lasts from $\sim$
$T_{0}$+2.35 s to $\sim$ $T_{0}$+6 s, and the significantly fainter third
pulse (i.e., $P_{3}$) extends from $\sim$ $T_{0}$+15 s to $\sim$ $T_{0}$+25 s.
A majority of the $\alpha$ indices in $P_{1}$ and $P_{2}$ are beyond the line-
of-death of synchrotron emission (-2/3, Preece et al., 1998), suggesting the
origin of the photosphere emission. Such a feature was observed in GRB 190114C
and it is fully consistent with what the fireball photosphere model predicts.
In the standard GRB fireball “internal-external” shock model(Rees & Meszaros,
1994; Mészáros & Rees, 2000), after a relativistic jet is launched from a
central engine, the energy can be dissipated either by early, short-lived
prompt emissions (generated by photosphere emissions where the jet becomes
transparent or an internal shock via synchrotron emission), occurring at a
close distance from the progenitor and mostly observed in $\gamma$-rays; or
later, long-lasting, multi-wavelength afterglow emissions (generated by an
external shock at a large distance where the GRB jets interact with the
ambient medium), observed in X-ray, optical, and radio wavelengths. Therefore,
a bright thermal component originating from the fireball photosphere and a
nonthermal component presumably originating from internal shocks whose radii
are greater than the photosphere radius would be expected to be observed in
their prompt emission spectra. The emission in $P_{3}$ can be interpreted as a
flare of synchrotron radiation. However, the flux, energy band and power-law
decay index (Figure 2) during the epoch from $\sim$ $T_{0}$+6 s to $\sim$
$T_{0}$+15 s (i.e., $P_{\rm AO}$) between $P_{2}$ and $P_{3}$ are consistent
with external shock emission at $t_{\rm onset}\approx 6$ s defined by the
deceleration of the fireball, where $P_{\rm AO}$ represents the initialization
of the afterglow emission phase generated by the deceleration of the fireball
and $t_{\rm onset}$ represents the onset time of the deceleration of the
fireball. The proposal that $P_{\rm AO}$ is related to afterglow emission is
supported by several independent studies in the literature (e.g., MAGIC
Collaboration et al., 2019b; Ravasio et al., 2019; Ajello et al., 2020; Ursi
et al., 2020). The onset signatures of afterglow emission observed in the MeV
energy range during the prompt emission phase are rare, since they are
typically observed as deceleration bumps in the early afterglow light curve
(e.g., Liang et al., 2010, 2013). The afterglow phase of this GRB has the most
comprehensive observations in terms of spectral coverage (see Figure 3), from
radio to TeV gamma-rays(MAGIC Collaboration et al., 2019b; Ravasio et al.,
2019). This provides a unique opportunity to study the GRB afterglow
properties within the framework of the synchrotron and synchrotron self-
Compton model(MAGIC Collaboration et al., 2019a). Moreover, by combining the
observed properties of the thermal emission in the first two episodes and of
the non-thermal emission in the third episode, GRB 190114C may be the first
case providing us with a unique opportunity to dissect the energy budget of a
GRB fireball.

Another interesting subject related to the GRB prompt emission mechanism,
which describes how efficiently the jet converts its energy to radiation, is
the radiative efficiency of a burst. GRB radiative efficiency may be defined
as (Lloyd-Ronning & Zhang, 2004)

$\displaystyle\eta_{\gamma}$ $\displaystyle\equiv$
$\displaystyle\frac{E_{\gamma}}{E_{\rm
tot}}=\frac{E_{\gamma}}{E_{\gamma}+E_{k}}=\frac{L_{\gamma}}{L_{w,0}},$ (1)

where $E_{\gamma}$, $E_{k}$ and $E_{\rm tot}$ are isotropic-equivalent gamma-
ray energy, afterglow kinetic energy, and total energy, respectively, and
$L_{\gamma}$ and $L_{w,0}$ are the isotropic-equivalent average gamma-ray
luminosity and total wind luminosity at the central engine, respectively. In
order to evaluate the radiative efficiency of a GRB, according to Eq.(1), one
needs to know the isotropic-equivalent gamma-ray energy $E_{\gamma}$ and the
blastwave kinetic energy $E_{\rm k}$. The $E_{\gamma}$ term can be measured
from the spectral parameters. The $E_{\rm k}$ term, on the other hand, is
usually estimated from afterglow data through modeling, but the estimated
values typically have large uncertainties (Zhang et al., 2007). By combining
the prompt emission and early afterglow data, Zhang et al. (2021) proposed a
new method to directly dissect the GRB fireball energy budget into three
components and measure their values. As a result, GRB radiation efficiency can
also be directly calculated with little uncertainty. The method requires a GRB
with a dominant thermal spectral component, a deceleration bump feature in the
early afterglow light curve, and a measured redshift. The measured parameters
include the initial dimensionless specific enthalpy ($\eta$), bulk Lorentz
factors at the photosphere radius ($\Gamma_{\rm ph}$), and before fireball
deceleration ($\Gamma_{0}$), the amount of mass loading ($M$), and the GRB
radiative efficiency ($\eta_{\gamma}$). These measured parameters only weakly
depend on the density $n$ of the interstellar medium when the composition
${\cal Y}$ parameter (typically unity) is specified, where ${\cal Y}$ is the
lepton-to-baryon number ratio, which equals unity for a pure hydrogen fireball
but could be greater (for a pair-loaded fireball) or slightly smaller (for a
neutron-rich fireball without pair loading) than unity.

Once these fireball parameters can be precisely measured, one can also
estimate the blastwave kinetic energy as $E_{\rm K}=\Gamma_{0}Mc^{2}$, as well
as the GRB radiative efficiency $\eta_{\gamma}$. In this paper, using the
photosphere data observed in $P_{1}$ and $P_{2}$, and the early afterglow data
observed in $P_{\rm AO}$ as supported by several independent studies in the
literature, as well as a measured redshift, we apply the method proposed in
Zhang et al. (2021) to directly dissect the GRB fireball energy budget and
therefore to measure GRB radiative efficiency for GRB 190114C.

##  3 Methodology

###  3.1 Data Reduction

We reduced the GBM data using a Python package, namely, _The Multi-Mission
Maximum Likelihood Framework_ (3ML,Vianello et al. 2015). The data we used for
our spectral analysis includes the two most strongly illuminated sodium iodide
(NaI) scintillation detectors (n3, n4) and the most-illuminated bismuth
germanium oxide (BGO) scintillation detector (b0) onboard Fermi-GBM, as well
as the corresponding response files (.rsp2 files are adopted). The detector
selections were made by considering an angle of incidence less than(Goldstein
et al., 2012; Narayana Bhat et al., 2016) $60^{\circ}$ for NaI and the lowest
angle of incidence for BGO. The Time-Tagged Event (TTE) data type is used for
the NaI data (8 keV–1 MeV) and BGO data (200 keV–40 MeV). To avoid the K-edge
at 33.17 keV, the spectral energy range was also cut from 30 to 40 keV. The
background fitting is chosen using two off-source intervals, including the
pre-burst (-20$\sim$-10 s) and post-burst (180$\sim$200 s) epochs, and with
the polynomial order determined (0-4) by applying a likelihood ratio test. The
source interval is selected over the duration (-1$\sim$116 s) reported by the
Fermi-GBM team. The maximum likelihood-based statistics, the so-called Pgstat,
are used, given by a Poisson (observation)-Gaussian (background) profile
likelihood(Cash, 1979).

###  3.2 Bayesian Spectral Analysis

The spectral parameters are obtained by adopting a fully Bayesian analysis
approach. The main idea is that after the experimental data are obtained,
Bayes’s theorem is applied to infer and update the probability distribution of
a specific set of model parameters. Building up a Bayesian profile model
($M$), and given an observed data set ($D$), the posterior probability
distribution $p(M\mid D)$, according to the Bayes’s theorem, is given by

$p(M\mid D)=\frac{p(D\mid M)p(M)}{p(D)},$ (2)

where, $p(D\mid M$) is the likelihood that combines the model and the observed
data, and expresses the probability to observe (or generate) the dataset $D$
from a given model $M$ with its parameters; $p(M)$ is prior on the model
parameters; and $p(D)$ is called the evidence, which is a constant with the
purpose of normalisation. We utilise the typical spectral parameters from the
$Fermi$-GBM catalogue as the prior distributions:

$\displaystyle\left\\{\begin{array}[]{ll}A_{\rm
Band}\sim\log\mathcal{N}~{}(\mu=0,\sigma=2)&\rm cm^{-2}\ keV^{-1}\ s^{-1}\\\
\alpha_{\rm Band}\sim\mathcal{N}~{}(\mu=-1,\sigma=0.5)\\\ \beta_{\rm
Band}\sim\mathcal{N}~{}(\mu=-2,\sigma=0.5)\\\ E_{\rm
Band}\sim\log\mathcal{N}~{}(\mu=2,\sigma=1)&\rm keV\\\ A_{\rm
CPL}\sim\log\mathcal{N}~{}(\mu=0,\sigma=2)&\rm cm^{-2}\ keV^{-1}\ s^{-1}\\\
\alpha_{\rm CPL}\sim\mathcal{N}~{}(\mu=-1,\sigma=0.5)\\\ E_{\rm
CPL}\sim\log\mathcal{N}~{}(\mu=2,\sigma=1)&\rm keV\\\ A_{\rm
BB}\sim\log\mathcal{N}~{}(\mu=-4,\sigma=2)&\rm cm^{-2}\ keV^{-1}\ s^{-1}\\\
kT_{\rm BB}\sim\log\mathcal{N}~{}(\mu=2,\sigma=1)&\rm keV\\\
\end{array}\right.$ (12)

We employ a Markov Chain Monte Carlo (MCMC) sampling method ($emcee$, Foreman-
Mackey et al. 2013) to sample the posterior. The parameter estimation is
obtained at a maximum a posteriori probability from the Bayesian posterior
density distribution, and its uncertainty (or the credible level) is evaluated
from the Bayesian highest posterior density interval at 1$\sigma$ (68%)
Bayesian credible level.

###  3.3 Model Comparison

The best-fit model is reached by comparing the DIC values of different models
and picking the one with the lowest value. The DIC is defined as
DIC=-2log[$p$(data$\mid\hat{\theta}$)]+2$p_{\rm DIC}$, where $\hat{\theta}$ is
the posterior mean of the parameters, and $p_{\rm DIC}$ is the effective
number of parameters. The preferred model is the one that provides the lowest
DIC score. We report the $\Delta$DIC values by comparing the best model with
other models in Table 1. Log(posterior) is adopted by the method of the
maximum likelihood ratio test, which is treated as a reference of the model
comparison(Vuong, 1989).

###  3.4 BBlocks Methods

We use a method called Bayesian blocks (BBlocks, Scargle et al. 2013) to rebin
the Time Tagged Event (TTE) light curve. Time bins are selected in such a way
as to capture the true variability of the data. Such a calculation requires
each bin to be consistent with a constant Poisson rate. In each bin, it allows
for a variable time width and signal-to-noise (S/N) ratio. As such, we first
apply the BBlocks method with the false alarm probability $p_{0}=0.01$ to the
TTE light curve of the most strongly illuminated GBM detector (n4). The other
detectors (n3 and b0) are then binned into matching slices. We notice that the
BBlocks analysis generates two slices ($0.70\sim 1.58$ s and $1.58\sim 1.71$
s) from $0.70$ s to $1.71$ s. On the other hand, the two slices have a very
high significance (263.97 and 115.59). To study the parameter evolution in
great detail, we, therefore, rebin the time intervals with five narrower
slices of significance $>$ $80$ instead. We also did the same analysis on the
last slice of $P_{2}$ ($5.51\sim 5.69$ s), generating two narrower slices
($5.51\sim 5.65$ s and $5.65\sim 5.69$ s), with significance $>$ $70$ each, to
study the temperature evolution in more detail. We, therefore, obtain 8 slices
for $P^{\rm th}_{1}$ and 16 slices for $P^{\rm th}_{2}$ to study the
photosphere properties.

##  4 Models

###  4.1 Deriving the Photosphere Properties Using the Traditional Method

The traditional method to derive the photosphere properties invokes the
standard fireball model(Mészáros & Rees, 2000; Pe’er, 2015; Zhang, 2018).
Within this framework, the fireball invokes thermally accelerated, matter-
dominated, and finally shocks-decelerated ejecta(Goodman, 1986; Paczynski,
1986).

The thermal emission of GRB 190114C is extremely strong, ranking second in
thermal-to-total flux ratio (21%) among the over 2700 GRBs observed by Fermi-
GBM up to date. The identification of the strong thermal component in GRB
190114C allows us to determine the physical properties of the relativistic
outflow within the framework of the non-dissipative photosphere theory(Pe’er
et al., 2007; Vereshchagin & Aksenov, 2017), which also applies to moderately
dissipated photospheres where the photosphere spectrum is only mildly
modified. The photosphere photons observed at a given time, corresponding to
the one-time bin in our time-resolved analysis, are assumed to be emitted from
an independent thin shell. Therefore, the observed BB temperature $kT_{\rm
obs}$, the BB flux $F_{\rm BB}$, and the total flux $F_{\rm tot}$
(thermal+non-thermal) of a given time bin determine the photosphere properties
of the corresponding shell. The entire duration of photosphere emission is
conjugated by the emissions from a sequence of such shells. One can infer the
bulk Lorenz factor $\Gamma$, and the initial size of the flow $R_{0}$ in each
time bin and their temporal evolutions.

Within the framework of the standard fireball model (Pe’er et al., 2007), for
a given shell, it is generated at an initial radius

$r_{0}(r_{\rm ph}>r_{\rm s})=\frac{4^{3/2}d_{\rm
L}}{(1.48)^{6}\xi^{4}(1+z)^{2}}(\frac{F^{\rm obs}_{\rm BB}}{\mathbb{Y}F^{\rm
obs}})^{3/2}\Re,$ (13)

and self-accelerates to reach a saturated Lorentz factor

$\eta(\equiv\Gamma)(r_{\rm ph}>r_{\rm s})=\left[\xi(1+z)^{2}d_{\rm
L}\left(\frac{\mathbb{Y}F^{\rm obs}\sigma_{\rm T}}{2m_{\rm
p}c^{3}\Re}\right)\right]^{1/4}$ (14)

in the coasting phase. If the photosphere radius is greater than the
saturation radius, it reads

$r_{\rm ph}(>r_{\rm s})=\frac{L_{0}\sigma_{T}}{8\pi m_{\rm
p}c^{3}\Gamma^{3}},$ (15)

where the dimensionless parameter

$\Re=\left(\frac{F_{\rm BB}}{\sigma_{\rm
B}T^{4}}\right)^{1/2}=\xi\frac{(1+z)^{2}}{d_{\rm L}}\frac{r_{\rm ph}}{\Gamma}$
(16)

presents the effective transverse size of the photosphere. The burst
luminosity $L_{0}=4\pi d^{2}_{L}\mathbb{Y}F_{\rm tot}$ is given by the
observation, $\mathbb{Y}$ is the ratio between the total fireball energy and
the energy emitted in gamma-rays. The numerical factor $\xi$ is of the order
of unity that can be obtained from angular integration. The luminosity
distance $d_{\rm L}$ of redshift $z$ is integrated by assuming the standard
Friedmann–Lemaître–Robertson–Walker (FLRW) metric. Other physical constants
are the Thomson cross section $\sigma_{\rm T}$, the proton rest mass $m_{\rm
p}$, the speed of light $c$, and the Stefan-Boltzmann constant $\sigma_{\rm
B}$.

###  4.2 Directly Deriving the Fireball Properties from Observations

GRB 190114C has a redshift measurement. Its prompt emission is thermally sub-
dominated and its lightcurve has a clear early pulse indicating the afterglow
initiation. These three properties make it the first case where one can use
observational properties to directly determine the fireball characteristics
including the dimensionless specific enthalpy at the engine $\eta$, isotropic
equivalent total mass $M$, bulk Lorentz factor at the site of the photosphere
$\Gamma_{\rm ph}$, initial afterglow Lorentz factor before the deceleration
phase $\Gamma_{\rm s}$, the kinetic energy in the fireball $E_{\rm k}$, and
$\gamma$-ray radiative efficiency $\eta_{\gamma}$. The method described below
follows Zhang et al. (2021).

The initial, total energy of a fireball is

$E_{\rm tot}=\eta Mc^{2}.$ (17)

The fireball undergoes rapid acceleration and reaches a Lorentz factor
$\Gamma_{\rm ph}$ at the photosphere. The internal energy released as thermal
emission can be estimated as

$E_{\rm th}=(\eta-\Gamma_{\rm ph})Mc^{2},$ (18)

Afterwards, the fireball moves at an almost constant speed until internal
dissipation at internal shocks occurs at a larger distance. The emitted non-
thermal emission can be estimated as

$E_{\rm nth}=(\Gamma_{\rm ph}-\Gamma_{0})Mc^{2},$ (19)

where $\Gamma_{\rm s}$ is the Lorentz factor after the dissipation and also
the initial Lorentz factor in the afterglow phase.

The Lorentz factor at photosphere radius $\Gamma_{\rm ph}$ can be estimated as
(modified from Pe’er et al. (2007); Bégué & Iyyani (2014), see Zhang et al.
(2021) for details)

$\begin{split}\Gamma_{\rm ph}&=\left[(1+z)^{2}D_{\rm L}\frac{{\cal
Y}\sigma_{\rm T}F^{\rm obs}_{\gamma}}{2m_{p}c^{3}{\cal
R}}\frac{\eta^{3/2}}{\eta-\Gamma_{0}}\right]^{2/9},\\\ {\cal
R}&=\left(\frac{F^{\rm obs}_{\rm BB}}{\sigma_{\rm
B}T^{4}}\right)^{1/2}.\end{split}$ (20)

which involves several direct observables including redshift $z$, total flux
$F^{\rm obs}_{\gamma}$, thermal flux $F^{\rm obs}_{\rm BB}$ and the observed
temperature $T$. Other parameters are the pair multiplicity parameter ${\cal
Y}$ which is commonly taken as $1$, the luminosity distance $D_{\rm L}$
computed from the redshift adopting the FLRW cosmology, and fundamental
constants such as speed of light $c$, proton mass $m_{\rm p}$, Thomson cross
section $\sigma_{\rm T}$, and Stefan-Boltzmann constant $\sigma_{\rm B}$.

The initial Lorentz factor of the afterglow phase $\Gamma_{\rm s}$ can be
derived by equating the kinetic energy to the swept-up ISM mass at the
deceleration time $t_{\rm dec}$, which is an observable indicated by a light-
curve pulse (the third pulse for 190114C). Using Eq.(7.81) of Zhang (2018) and
the above arguments, we derive

$\Gamma_{\rm s}\simeq 170t_{\rm
dec,2}^{-3/8}\left(\frac{1+z}{2}\right)^{3/8}\left(\frac{E_{\rm th,52}+E_{\rm
nth,52}}{n}\right)^{1/8}\left(\frac{\Gamma_{0}}{\eta-\Gamma_{0}}\right)^{1/8}.$
(21)

where $n$ is the ISM density assumed as one particle per cubic centimetre as
usual222Note that we do not discuss the case of a wind medium (Dai & Lu, 1998;
Mészáros et al., 1998; Chevalier & Li, 1999) in our theoretical model (Zhang
et al., 2021). This is because afterglow observations suggest that the
majority of GRBs, especially those with the clear deceleration signature, are
consistent with having a constant density medium (Zhang et al., 2007; Liang et
al., 2010). More importantly, because for a wind medium the fireball dynamics
should be in the “thick shell” regime (Kobayashi & Zhang, 2003; Wu et al.,
2003) while the observations do not require a thick shell dynamics for this
burst, we only consider a constant-density medium in our calculation..

Simultaneously solving Eqs. 18 – 21, we obtain fireball parameters $\eta$,
$\Gamma_{\rm ph}$, $M$ and $\Gamma_{\rm s}$, and in turn, we can calculate the
kinetic energy of the afterglow

$E_{k}=\Gamma_{0}Mc^{2},$ (22)

and the efficiency of the prompt gamma-ray emission

$\eta_{\gamma}=\frac{E_{\rm th}+E_{\rm nth}}{E_{\rm
tot}}=\frac{\eta-\Gamma_{0}}{\eta}.$ (23)

##  5 Results

###  5.1 Multi-wavelength Observations

(1) TeV (MAGIC) Observations: The Major Atmospheric Gamma Imaging Cherenkov
(MAGIC) telescopes observed for the first time very-high-energy gamma-ray ($>$
1 TeV) emission from $T_{0}$+57 s until $T_{0}$+15912 s (MAGIC Collaboration
et al., 2019b), setting the record of one of the highest energy photon
detected from any GRB. Both the TeV lightcurve and spectrum can be well-
described by a power-law model333The convention
$F_{\nu,t}=t^{\hat{\alpha}}\nu^{\hat{\beta}}$ is adopted throughout the
paper., with the temporal decay index $\hat{\alpha}_{\rm
MAGIC}$=-1.60$\pm$0.07 (Figure 3a) and the spectral decay index
$\hat{\beta}_{\rm MAGIC}$=-2.16$\pm$0.30 (Figure 3b). The total TeV-band
(0.3-1 TeV) energy integrated between $T_{0}$+62 s and $T_{0}$+2454 s is
$E^{\rm MAGIC}_{\gamma,\rm iso}$$\sim$ 4.0$\times$1051 erg (MAGIC
Collaboration et al. 2019b).

(2) GeV (Fermi-LAT) Observations: The first GeV photon was observed by Fermi-
LAT at $T_{0}$+2.1 s. The highest-energy photon detected by LAT is a 22.9 GeV
event detected at $T_{0}$+15 s(Kocevski et al., 2019), therefore, the
bandwidth (0.1-10 GeV) was reasonably adopted to measure the total GeV energy
detected by LAT. For comparison, in Table 2 we also report the results based
on the other two bandwidths: (0.1-100 GeV) and (0.1-1 GeV). After that time,
the lightcurve and spectrum as measured by LAT (0.1-10 GeV) from $T_{0}$+55 s
to $T_{0}$+8000 s are well-fitted by a power-law model with temporal decay
index $\hat{\alpha}_{\rm LAT}$=-1.29$\pm$0.01 (Figure 3a) and spectral slope
index $\hat{\beta}_{\rm LAT}$=-2.01$\pm$0.98 (Figure 3b). The total GeV-band
(0.1-10 GeV) energy integrated between $T_{0}$+2.1 s and $T_{0}$+8000 s is
$E^{\rm LAT}_{\gamma,\rm iso}$ =(1.09$\pm$0.24)$\times$1053 erg, which can be
separated into two emission components: the prompt emission ($\leq$ 6 s)
accounts for $E^{\rm LAT}_{\gamma,\rm iso}$=(8.49$\pm$1.80)$\times$1051 erg,
while the afterglow emission ($>$6 s) accounts for $E^{\rm LAT}_{\gamma,\rm
iso}$=(1.01$\pm$0.24)$\times$1053 erg.

(3) MeV (Fermi-GBM) Observations: The duration of the GRB ($T_{90}$) is about
$116$ s as reported by Fermi-GBM. The 1024 ms peak flux and the fluence at
10-1000 keV measured by Fermi-GBM are 246.9$\pm$0.9 photon cm-2 s-1 and
(4.436$\pm$0.005)$\times$10-4 erg cm-2, respectively. With a known redshift,
$z$=0.4245 $\pm$0.0005(Selsing et al., 2019), and based on the best models for
each emission episode (CPL+BB model for prompt emission and Band model for
afterglow emission), the total $k$-corrected isotropic energy in the rest-
frame 1-104 keV band as derived from Fermi-GBM observations between $T_{0}$+0
s and $T_{0}$+116 s is $E^{\rm GBM}_{\gamma,\rm
iso}$=(2.82${}^{+0.43}_{-0.25}$)$\times$1053 erg(Wang et al., 2019b), and
between $T_{0}$+15 s and $T_{0}$+25 s ($P_{3}$) is $E^{\rm GBM}_{\gamma,\rm
iso}$=$\sim$1.24$\times$1052 erg . The prompt emission ($\leq$ 6 s) accounts
for $E^{\rm GBM}_{\gamma,\rm iso}$=(2.29${}^{+0.10}_{-0.09}$) $\times$ 1053
erg while the afterglow emission ($>$6 s) accounts for $E^{\rm
GBM}_{\gamma,\rm iso}$=(5.33${}^{+4.23}_{-2.34}$) $\times$ 1052 erg. There is
a $\sim$ 3.24 s lag between the GBM emission and the LAT emission.

(4) keV (Swift-XRT) Observations: Following the trigger by Swift-BAT, the
spacecraft slewed immediately to the location of the burst. The X-ray
Telescope (XRT) began observing the afterglow at $T_{0}$+64 s. Pointed
Windowed Timing mode data were collected from $T_{0}$+68 s to $T_{0}$+626 s,
after which the count rate was low enough for Photon Counting mode to be
utilised. The burst was followed for more than 28 days, although the last
detection occurred on $T_{0}$+20 day. The XRT lightcurve showed a typical
power-law behaviour with a power-law index $\hat{\alpha}_{\rm XRT}$=-1.39+0.01
(Figure 3). The isotropic X-ray energy release $E^{\rm XRT}_{\rm X,iso}$
measured by Swift-XRT (0.3-10 keV) from $T_{0}$+68 s to $T_{0}$+13.86 days is
$\sim$1.48$\times$1052 erg.

(5) Optical Observations: Optical data have been gathered from Refs.(MAGIC
Collaboration et al., 2019b; Jordana-Mitjans et al., 2020; Misra et al., 2019;
Melandri et al., 2021), as well as GCN data from Refs. (Bikmaev et al., 2019;
Im et al., 2019a, b; Kim & Im, 2019; Kim et al., 2019; Mazaeva et al., 2019;
Watson et al., 2019a, b). The automatically processed UVOT data are also used.
All afterglow data have been host-subtracted using the host-galaxy values
taken from de Ugarte Postigo et al. (2020), and only late data without any
supernova contribution has been used. Note that Jordana-Mitjans et al. (2020)
found chromatic evolution in their early RINGO3 data. However, this effect is
small, which leads to some additional scatter around the first and second
steep-to-shallow decay transition. After the respective host galaxy magnitude
has been subtracted for each band, all the bands are shifted to the $R_{\rm
c}$ band to produce a composite light curve stretching from 33 s to 56.4 days
after the GRB trigger. The light curve can be described by multiple power-law
decay segments in a steep-normal-shallow-normal-steep arrangement. The first
two segments have slopes $\hat{\alpha}_{\rm opt,1}=-1.628\pm 0.012$ and
$\hat{\alpha}_{\rm opt,2}=-1.035\pm 0.006$, with a break time at
$t_{b,1}=429\pm 61$ s and a sharp transition index with $n=-13.3\pm 2.0$
(Figure 3a), such an early steep-normal transition is consistent with the
superposition of a reverse shock component with a forward shock component.
After a second, sharp break ($n=-8.6\pm 1.9$) at $t_{\rm b,2}=4856\pm 216$ s,
the lightcurve goes over into an even flatter phase, decaying with
$\hat{\alpha}_{opt,3}=-0.512\pm 0.035$. At a break time $t_{b,3}=0.548\pm
0.036$ d (with a smoother transition index $n=4.4\pm 2.1$), the decay becomes
steeper again, reaching a value similar to $\hat{\alpha}_{\rm opt,2}$,
$\hat{\alpha}_{opt,4}=-1.146\pm 0.036$, indicating the shallow decay phase may
be interpreted as an energy injection. We find a final break at $t_{\rm
b,4}=6.33\pm 1.26$ d to an even steeper decay $\hat{\alpha}_{\rm
opt,5}=-1.714\pm 0.041$ ($n=10$ had to be fixed).

This final break may represent a jet break. If so, the post-break slope would
be quite shallow but not unprecedented (see the sample of Zeh et al. 2006 for
comparison). There is no conclusive evidence from the X-ray data for this
break. However, we note the last three _Swift_ data points are decaying more
steeply than before, and the X-ray data only extends to $\approx 14$ d, which
does not allow strong conclusions to be drawn.

(6) Radio Observations: The radio data points are taken from (Laskar et al.,
2019). Radio observations were carried out by the Atacama Large
Millimeter/submillimeter Array (ALMA) at Band 3 with a center frequency of
97.5 GHz spanning the period from $T_{0}$+0.0995 days to $T_{0}$+0.217 days
and lasting for 3 hours, and together with NSF’s Karl G. Jansky Very Large
Array (VLA) observations with a full sequence of observations spanning 5–38
GHz, starting at $T_{0}$+0.197 days and ending at $T_{0}$+0.261 days. As shown
in the left panel of Figure 3, the radio afterglow lightcurve from the ALMA
observation is well-fitted with a power-law model with the temporal decay
index $\hat{\alpha}_{\rm radio}=-0.69\pm 0.02$. The radio observations at
$\lesssim T_{0}$+0.03 days, as well as the optical and millimetre
observations, were interpreted as emissions from the reverse-shocked ejecta in
(Laskar et al., 2019).

###  5.2 Time-integrated and Time-resolved Spectral Analysis

We first performed the time-integrated spectral analysis (treating the entire
$T_{90}$ as one-time bin, i.e., from $T_{0}$ to $T_{0}+116$ s) by using
various GRB spectral models, including power-law (PL), blackbody (BB), cutoff
power law (CPL), Band function(Band et al., 1993), smoothly broken power law
(SBKL), PL+BB(Ryde, 2005), PL+Bandcut, CPL+BB(Li, 2019b), and Band+BB(Guiriec
et al., 2011), respectively. Our refined time-integrated spectral analysis
suggests that the CPL+BB model can best characterize the spectral shape of the
burst. The corresponding corner plot is shown in Figure 4.

GRB spectra are known to evolve over different pulses, or even within a pulse.
The time-integrated spectral analysis, therefore, must be replaced by the
time-resolved spectral analysis to study the GRB radiation mechanism in great
detail. We next performed a time-resolved spectral analysis for the Fermi-GBM
observations. Thanks to its high fluence of (4.436$\pm$0.005)$\times$10-4 erg
cm-2 as the fifth-highest fluence GRB ever observed by Fermi-GBM, we were able
to divide its $T_{90}$ duration (116 s) into $48$ slices, with each time bin
containing enough photons to conduct a high-significance spectral analysis. We
used the typical GRB spectral model, the Band model (Band et al., 1993), to
fit the time-resolved spectra in each slice (see Table 3). We found that the
low-energy photon index $\alpha$ exhibits a wide-spread temporal variability
(-0.14 to -1.99), and the majority of $\alpha$ values in the first two pulses
are harder than the typical value of $\alpha$ defined by the synchrotron line
of death ($\alpha$=-2/3, Preece et al. 1998), suggesting a significant
contribution from thermal emission from the fireball photosphere(Ryde et al.,
2010; Mészáros & Rees, 2000). The majority of the high energy photon index
$\beta$ values are not well-constrained, indicating that the CPL model is
preferred in comparison with the Band model (Table 3). The violation of the
synchrotron limit encourages us to search for an additional thermal component.
To search for the best model to characterise the spectral shape of the burst,
we attempted to fit the time-resolved spectra in each slice with both the CPL
and the CPL+BB models. The DIC of the CPL+BB model is at least 10 and can be
hundreds less than the CPL model, indicating that adding a thermal component
improves the spectral fitting greatly ($\rm\Delta DIC>10$, Acuner et al.
2020). The CPL+BB (CPL: Cutoff Power-law, BB: Blackbody) model(Ryde, 2005;
Battelino et al., 2007) gives a better fit in comparison with the CPL (see
Table 4), Band and other models from $T_{0}$+$0.55$ s to $T_{0}$+$1.93$ s in
$P_{1}$ (includes 8 slices, hereafter $P^{\rm th}_{1}$) and from
$T_{0}$+$2.45$ s to $T_{0}$+$5.69$ s in $P_{2}$ (includes 16 slices, hereafter
$P^{\rm th}_{2}$) based on the deviance information criterion (DIC). $P^{\rm
th}_{1}$ and $P^{\rm th}_{2}$ correspond to the peak flux of the $P_{1}$ and
$P_{2}$, respectively, which precisely correspond to the epochs when the
power-law indices $\alpha$ of the single CPL fits are beyond the limits of the
synchrotron line of death(Preece et al., 1998), i.e. $\alpha>-2/3$, indicating
of the existence of a thermal component(Ryde, 2004; Ghirlanda et al., 2007).
An example of an $\nu F_{\nu}$ spectrum for one time slice ($4.95$ s–$5.45$ s)
with the CPL+BB model giving the best fit is displayed in Figure 5. Both
$P_{1}$ and $P_{2}$ include non-thermal and sub-dominant thermal components.
The thermal components observed in GRB 190114C exhibit pulse-wise temporal
properties, i.e., those in $P_{1}$ and $P_{2}$ evolve independently over their
pulse durations (Figure 6). Such a feature provides a unique opportunity to
study the photosphere properties at distinctly different epochs of central
engine activities.

The time-resolved analysis shows that almost all the low-energy photon index
$\alpha$ values of the CPL-only fits in $P_{\rm AO}$ are much softer than
those in $P_{1}$ and $P_{2}$ (Figure 1), suggesting that the emission has a
different origin. The index $\alpha$ gradually decreases toward $-2$, a
typical value for synchrotron radiation, which indicates that the fireball has
entered the afterglow phase. So we set the beginning of the epoch as the
deceleration time when the mass of the ambient medium collected by the
shockwave is comparable to $1/\Gamma$ of fireball energy(Meszaros & Rees,
1993; Sari & Piran, 1999).

###  5.3 Photosphere Properties

We compared the properties of the thermal components identified in $P_{1}$ and
$P_{2}$. The evolutions of the characteristic temperatures (k$T$) in $P_{1}$
and $P_{2}$ follow distinctly broken power-law decays: a smooth decay of the
temperature followed by a fast drop (see the left panel in Figure 6). The
temporal feature in each pulse is consistent with the typical observations
that showed a temperature evolution with a broken power law in time(Ryde,
2004; Ryde & Pe’er, 2009), but such a feature in two independent pulses within
one burst has never been identified in previous observations. The temporal
behaviours showing different decay indices between two different pulses within
a single GRB suggest that the GRB central engine ejects distinct independent
jet components during its active phase. We note that several GRBs with
statistically significant thermal components have been observed by BATSE,
Konus, Swift, and Fermi before(Ryde et al., 2010; Guiriec et al., 2011, 2013;
Axelsson et al., 2012). However, they are either single-pulse bursts (e.g. GRB
110721A, Axelsson et al. 2012) or highly overlapping multi-pulse bursts (e.g.
GRB 090902B, Ryde et al. 2010), or their thermal emission component is not
strong enough (e.g. GRB 100724B, Guiriec et al. 2011), so that the photosphere
properties could not be studied in detail among distinct pulses. The unique
advantages of GRB 190114C, i.e. its low redshift, high fluence, several well-
separated pulses in one single GRB, and strong thermal component, make such a
study possible.

Within the framework of the standard fireball photosphere model(Pe’er et al.,
2007), we can infer the photosphere characteristics and the ratio of thermal
to non-thermal emission to obtain information on the jet properties, such as
the bulk Lorentz factor $\Gamma$ and the initial size of the jet $r_{0}$.
Figure 6 and Figure 7 show the evolution of the bulk Lorentz factor $\Gamma$
and the parameter $\Re$, respectively; where $\Re$ is the effective transverse
size of the emitting region(Ryde & Pe’er, 2009). They exhibit similar temporal
behaviors in $P_{1}$ and $P_{2}$, i.e., a broken power-law evolution behavior,
with $\Re$ increasing with time and $\Gamma$ decreasing over time. The
comparison of the properties of a global view as well as the best-fitting
results of the relevant parameters is summarized in Table 5.

The derived Lorentz factors and the photosphere radii exhibit systematic
variations, with the Lorentz factor decreasing from $\sim 1000$ to $\sim 200$
(Figure 6), and the photosphere radius varying on the order of $10^{12}$ cm
(Figure 7a). This is likely related to the behaviour of the GRB central
engine. The decay of $\Gamma$ in $P_{1}$ and $P_{2}$ is consistent with the
expectation that faster ejecta from the engine tends to reach the photosphere
earlier than slower ejecta, and the rapid decline at the end of each episode
may be related to the abrupt cessation of the engine activity(Li & Zhang,
2021), with the decay slope defined by the ebbing ejection rate of the central
engine. Since the Lorentz factor range is not very wide, it is expected that
the deceleration of the fireball is essentially prompt without a significant
energy injection phase due to the pile-up of the slow materials. This is
consistent with the power-law decay with time of the multi-wavelength
afterglow emission from the source(MAGIC Collaboration et al., 2019b; Wang et
al., 2019b, a).

###  5.4 Application to GRB 190114C with Our New Method

In short, GRB 190114C is unique in terms of the following aspects. (1) It has
three well-separated emission episodes, which can be defined as the first,
second, and third pulses. (2) The emission of the first two main pulses
consists of two strong thermally-subdominated episodes, which independently
exhibit similar temporal properties. (3) The first two pulses (thermal) and
the third pulse (non-thermal) have distinct spectral properties. (4) The
thermal component has a thermal to total flux ratio of 21${}^{+6}_{-4}$%,
which is the second-highest among the GRBs observed with Fermi-GBM so far (the
highest one is observed in GRB 090902B, with thermal flux ratio $\sim$ 70%).
(5) Strong TeV emission was observed, setting the record of one of the highest
photon energy in any GRB(MAGIC Collaboration et al., 2019b). The two well-
separated pulses with independent and analogous thermal component evolution
patterns make this extraordinarily bright GRB a unique event to study the jet
composition and evolution of the photospheric properties in a single GRB. We
note that several interesting cases have been observed in the past. For
example, in some GRBs, a hot fireball jet characterized by a quasi-thermal
Planck-like spectrum was observed (e.g. GRB 090902B, Abdo et al. 2009). In
many other GRBs, a Poynting-flux-dominated outflow characterized by a Band (or
cutoff power-law)-only function may also be observed (e.g., GRB 080916C,Abdo
et al. 2009 and GRB 130427A, Preece et al. 2016). More interestingly, we may
also observe a hybrid jet characterized by either a two-component spectral
scenario (composed of a non-thermal component and a thermal component
simultaneously, e.g., GRB 110721A, Axelsson et al. 2012; Gao & Zhang 2015, or
a transition from fireball to Poynting-flux-dominated outflow within a single
GRB (e.g., GRB 160625B, Ryde et al. 2011; Zhang et al. 2018; Li 2019a).
However, GRB 190114C presented unique information not available before.

The above-mentioned two methods (see Section 5.3) of measuring Lorentz factors
both rely on some unknown parameters. By combining the photosphere data in
$P_{1}$ and $P_{2}$ and the afterglow data in $P_{\rm AO}$, one can
discriminate various energy components in the fireball in an essentially
parameter-independent way(Zhang et al., 2021). A systematic search in
previously detected GRBs did not reveal a single case showing both a
significant photosphere signature and an afterglow deceleration
signature(Zhang et al., 2021). GRB 190114C, therefore, provides the first case
in which the determinateness of fireball parameters(Zhang et al., 2021) can be
carried out. We perform a time-integrated spectral fit to the prompt emission
spectra of $P_{1}$ and $P_{2}$ (0.55 - 1.93 s and 2.45-5.69 s) with the CPL+BB
model and derive the observed properties (including both the thermal and non-
thermal components) of the fireball as shown in Table 6. Following Zhang et
al. (2021) (for details see Section 4.2), we can derive the following physical
parameters of a GRB fireball (Table 6): initial dimensionless specific
enthalpy $\eta=854\pm 38$, bulk Lorentz factor at the photosphere $\Gamma_{\rm
ph}=833\pm 38$, bulk Lorentz factor before deceleration $\Gamma_{0}=719\pm
59$, and fireball isotropic-equivalent mass loading $M_{\rm iso}=(1.7\pm
0.4)\times 10^{-3}M_{\odot}$. This gives a direct measurement of the fireball
radiative efficiency $\eta_{\gamma}=(15.8\pm 5.4)\%$. This measured efficiency
has much smaller uncertainties than the values derived for previous GRBs using
afterglow modeling(Zhang et al., 2007). A high fireball radiative efficiency
has been theorized in the past(Mészáros & Rees, 2000; Kobayashi & Sari, 2001).
Our measured $\eta_{\gamma}\sim 16\%$ suggests that a GRB fireball can indeed
emit both thermal and non-thermal gamma-rays efficiently. We also find that
the derived bulk Lorentz factors measured during the prompt emission phase
($\Gamma=854\pm 38$) are slightly higher than the bulk Lorentz factor measured
at the deceleration radius ($\Gamma_{0}=719\pm 59$). This is fully consistent
with the picture described by the GRB fireball model in which a fraction of
the kinetic energy is dissipated during the prompt emission phase.

To solve the above equations described in Section 4.2, we apply the Monte-
Carlo method to obtain the mean and the uncertainty for the measured values
and the uncertainties of $E_{\rm th,iso},E_{\rm nth,iso},F_{\rm BB}^{\rm
obs},F_{\rm\gamma}^{\rm obs},kT^{\rm obs}$. We sample, for each of them,
$10000$ times following the normal distribution. We set a range of $6-10$ s
for $t_{\rm dec}$ while of $0.5-1.5$ cm-3 for $n$. We obtain the values of
$\eta$, $\Gamma_{\rm ph},\Gamma_{0},M_{\rm iso}$, $E_{\rm k,iso}$, $E_{\rm
tot,iso}$ and $\eta_{\gamma}$, computed from the $10000$ samples by the above
equations and find they can be fitted by skew-normal distributions, see e.g.
in Figure 8, from which the mean and the asymmetrical uncertainties are
derived. The average values based on the two thermal episodes of $P^{\rm
th}_{1}$ (from 0.55 to 1.93 s) and $P^{\rm th}_{2}$ (from 2.45 to 5.69 s) are
given in Table 6. All the measured quantities are presented in the upper
panel, and all the derived parameters are presented in the lower panel.

###  5.5 Further Estimate of the Energy Fractions Assigned to Electrons
($\epsilon_{e}$) and Magnetic ($\epsilon_{B}$) fields

Once $E_{\rm k}$ is precisely obtained from the observational data using our
new methods discussed above, one can estimate the energy fractions assigned to
electrons ($\epsilon_{e}$) and magnetic fields ($\epsilon_{B}$) using
afterglow models(Zhang et al., 2007).

The isotropic blastwave kinetic energy ($E_{\rm K,iso}$) can also be measured
from the afterglow emission (normal decay) using the Swift-XRT data. For a
constant density interstellar medium (ISM), e.g., Schulze et al. (2011), the
characteristic synchrotron frequency and the cooling frequency of minimum-
energy injected electrons, and therefore the peak spectral flux, can be given
by Sari et al. (1998); Yost et al. (2003); Zhang et al. (2007)

$\nu_{\rm m}=3.3\times 10^{12}{\rm
Hz}\left(\frac{p-2}{p-1}\right)^{2}(1+z)^{1/2}\varepsilon_{\rm
B,-2}^{1/2}\varepsilon_{\rm e,-1}^{2}E^{1/2}_{\rm K,iso,52}t^{-3/2}_{\rm d},$
(24) $\nu_{\rm c}=6.3\times 10^{15}{\rm
Hz}(1+z)^{-1/2}(1+Y)^{-2}\varepsilon_{\rm B,-2}^{-3/2}E^{-1/2}_{\rm
K,iso,52}n^{-1}t^{-1/2}_{\rm d},$ (25) $F_{\nu,\rm max}=1.6{\rm
mJy}(1+z)D_{28}^{-2}\varepsilon_{\rm B,-2}^{1/2}E_{\rm K,iso,52}n^{-1},$ (26)

where $p$ is the electron spectral distribution index, $\epsilon_{e}$ and
$\epsilon_{B}$ are the energy fractions assigned to electrons and magnetic
fields, $t_{\rm d}$ is the time in the observer frame in units of days,
$D_{28}=D/10^{28}$, is the luminosity distance in units444The convention
$Q=10^{x}Q_{x}$ is adopted in cgs units for all parameters throughout the
paper. of $10^{28}$ cm, $n$ is the number density in the constant density
ambient medium, and

$Y=\left[-1+(1+4\eta_{1}\eta_{2}\varepsilon_{\rm e}/\varepsilon_{\rm
B})^{1/2}\right]/2,$ (27)

is the Inverse Compton (IC) parameter, where
$\eta_{1}=\rm{min}[1,(\nu_{c}/\nu_{m})^{(2-p)/2}]$,
$\eta_{2}=\rm{min}[1,(\nu_{\rm KN}/\nu_{c})^{(3-p)/2}]$ (for the slow cooling
$\nu_{m}<\nu_{x}<\nu_{c}$ case) is a correction factor introduced by the
Klein-Nishina effect, where $\nu_{\rm KN}$ is the Klein-Nishina frequency

$\begin{split}\nu_{\rm KN}=h^{-1}\Gamma
m_{e}c^{2}\gamma^{-1}_{e,X}(1+z)^{-1}\simeq 2.4\times 10^{15}{\rm
Hz}(1+z)^{-3/4}E^{1/4}_{\rm K,iso,52}\varepsilon_{\rm B,-2}^{1/4}t^{-3/4}_{\rm
d}\nu^{-1/2}_{18},\end{split}$ (28)

where $h$ is Plancks constant and $\gamma_{e,X}$ is the electron Lorentz
factor corresponding to the X-ray band emission.

The spectral regime can be determined by using the closure relation in the
afterglow emission via the observed temporal ($\hat{\alpha}$) and spectral
($\hat{\beta}$) indices. The temporal index $\hat{\alpha}_{\rm XRT}$ is
measured from the Swift-XRT lightcurve (see Figure 3a, and the corresponding
spectral index $\hat{\beta}_{\rm XRT}=-(\Gamma_{\rm XRT}-1)=-0.93\pm 0.10$
($\Gamma_{\rm XRT}$ is the photon spectral index) is available from the Swift
online server (Evans et al., 2007, 2009). Using the temporal and spectral
indices, one can therefore determine that the X-ray emission in GRB 190114C is
in the $\nu_{\rm m}<\nu_{\rm x}<\nu_{\rm c}$ regime. With the spectral regime
known, the electron index $p$ can be derived using the observed temporal
index: $p=(3-4\hat{\alpha}_{\rm XRT})/3=2.85\pm 0.01$.

In the case of $p>2$, and in the $\nu_{\rm m}<\nu_{\rm x}<\nu_{\rm c}$ regime,
one can derive the X-ray band energy flux as

$\begin{split}\nu F_{\nu}(\nu=10^{18}{\rm Hz})=F_{\nu,{\rm max}}(\nu_{\rm
m}/\nu_{\rm x})^{(p-1)/2}\\\ =6.5\times 10^{-13}{\rm
ergs^{-1}cm^{-2}}D_{28}^{-2}(1+z)^{(p+3)/4}\\\ \times
f_{p}\varepsilon_{B,-2}^{(p+1)/4}\varepsilon_{e,-1}^{p-1}E^{(p+3)/4}_{\rm
K,iso,52}n^{1/2}t_{d}^{(3-3p)/4}\nu_{18}^{(3-p)/2}.\end{split}$ (29)

This gives,

$\begin{split}E_{\rm K,iso,52}=\left[\frac{\nu F_{\nu}(\nu=10^{18}{\rm
Hz})}{6.5\times 10^{-13}\rm ergs^{-1}cm^{-2}}\right]^{4/(p+3)}\\\ \times
D_{28}^{8/(p+3)}(1+z)^{-1}t_{d}^{3(p-1)/(p+3)}\\\ \times
f_{p}^{-4/(p+3)}\varepsilon_{B,-2}^{-(p+1)/(p+3)}\varepsilon_{e,-1}^{4(1-p)/(p+3)}\\\
\times n^{-2/(p+3)}\nu_{18}^{2(p-3)/(p+3)},\end{split}$ (30)

where $\nu F_{\nu}$ ($\nu=10^{18})$ Hz is the energy flux at frequency
$10^{18}$ Hz in units of ${\rm erg\ s^{-1}\ cm^{-2}}$, and

$f_{p}=6.73\left(\frac{p-2}{p-1}\right)^{p-1}\left(3.3\times
10^{-6}\right)^{(p-2.3)/2}.$ (31)

is a function of the electron power-law index $p$.

Simultaneously solving Eq. 27 and Eq. 30, with the IC parameter $Y^{\rm
IC}(=E_{\rm GeV}/E_{\rm MeV}$) constrained from the observations in GRB
190114C, e.g. $Y^{\rm IC}=0.75$ (e.g., Wang et al., 2019a), and the using
episodes of $P^{\rm th}_{1}$ and $P^{\rm th}_{2}$, we obtain $\epsilon_{\rm
B}$ and $\epsilon_{\rm e}$ during the time interval of $P_{\rm AO}$,

$\displaystyle\left\\{\begin{array}[]{ll}\epsilon_{\rm e,-1}=1.11\pm 0.01,\\\
\epsilon_{\rm B,-2}=0.05\pm 0.01,\\\ \end{array}\right.$ (34)

Knowing values of $\epsilon_{\rm B}$ and $\epsilon_{\rm e}$, we can also solve
for $\nu_{\rm m}$, $\nu_{\rm c}$, and $\nu_{\rm KN}$,

$\displaystyle\left\\{\begin{array}[]{ll}\nu_{\rm m}=(1.30\pm 0.82)\times
10^{17}\rm Hz,\\\ \nu_{\rm c}=(4.44\pm 0.66)\times 10^{17}\rm Hz,\\\ \nu_{\rm
KN}=(6.55\pm 0.16)\times 10^{17}\rm Hz\\\ \end{array}\right.$ (38)

##  6 Conclusions

In this paper, using the photosphere data observed in $P_{1}$ and $P_{2}$, and
the early afterglow data observed in $P_{\rm AO}$, as well as a measured
redshift, we apply the method proposed in Zhang et al. (2021) to directly
dissect the GRB fireball energy budget and therefore to measure GRB radiative
efficiency for GRB 190114C.

We first performed a detailed time-integrated and time-resolved spectral
analysis for the _Fermi_ -GBM observations by using various GRB spectral
models. Its prompt emission consists of three well-separated pulses. We found
a strong thermal component observed in the first two emission pulses. The
spectra in $P_{1}$ and $P_{2}$ are best fitted by a two-component scenario,
with a non-thermal Band-like component accompanied by a thermal blackbody
component. The thermal component has a thermal to total flux ratio of
21${}^{+6}_{-4}$%. Such a strong thermal component found in the time-resolved
spectral analysis between well-separated pulses in GRB 190114C gives a good
opportunity to study the photospheric properties, and allows us for the first
time to study a fine time-resolved spectral analysis and track the blackbody
evolution among the different pulses in a single GRB. Indeed, we found two
well-separated thermal pulses evolving independently and analogically,
inferred from their observational and physical parameters derived from the
fireball model. Such independent and analogical pulse-wise thermal properties
in GRB 190114C are the first case found in GRB history, which strongly
supports the evidence of a shell-like structure during the prompt emission
phase. We also found that starting from the third pulse ($P_{3}$) and
extending to the entire afterglow, the spectra are all non-thermal, the
synchrotron plus Compton up-scattering model well interprets the observation ,
and consequently the fireball parameters are obtained. More interestingly, the
onset signature of afterglow emission corresponding to the deceleration of the
fireball was observed to be from $T_{0}$+$6$ s to $T_{0}$+$15$ s in $P_{\rm
AO}$ due to the fact that multiple pieces of observational evidence (e.g.,
flux, energy band, and power-law index) are consistent with external shock
emissions. By incorporating the thermal ($P_{1}$ and $P_{2}$) and the non-
thermal ($P_{\rm AO}$) observations, as well as the photosphere and the
synchrotron radiative mechanisms, we directly derived the fireball energy
budget with little dependence on hypothetical parameters (Zhang et al., 2021)
and to measure a $\sim$16% radiative efficiency for this GRB.

With the fireball parameters that have been determined, the isotropic kinetic
energy(Panaitescu & Kumar, 2001) of the fireball at the afterglow phase is
measured as $E_{\rm k,iso}=(1.6\pm 0.7)\times 10^{54}$ erg. This allows us to
make use of this prompt-emission-measured $E_{\rm k,iso}$ in the afterglow
model to constrain shock microphysics parameters. Using broad-band afterglow
data, we can derive an electron injection power law index $p\simeq 2.85$ and
the inverse Compton parameter $Y^{\rm IC}\sim 0.75$. This leads to the
determination of the two equipartition parameters of electrons and magnetic
fields: $\epsilon_{e}=(1.11\pm 0.01)\times 10^{-1}$ and $\epsilon_{\rm
B}=(0.5\pm 0.1)\times 10^{-3}$. These parameters are usually poorly
constrained in other GRBs unless there is complete multi-wavelength afterglow
data(Panaitescu & Kumar, 2002). We are able to measure these values more
precisely, and they are also broadly consistent with the afterglow modeling of
the event(MAGIC Collaboration et al., 2019b).

We thank the anonymous referee for his/her valuable comments and suggestions.
We also thank Damien Bégué, Hüsne Dereli-Bégué, Michael S. Briggs, Xue-Feng
Wu, Zi-Gao Dai, Ye-Fei Yuan, Yi-Fu Cai, En-Wei Liang, Remo Ruffini, and
ICRANet members for many helpful discussions on GRB physics and phenomena. In
particular, LL would like to dedicate this piece to the memory of Dr. Magnus
Axelsson, a close colleague who passed away recently and was one of its main
contributors. AJC-T acknowledges financial support from the State Agency for
Research of the Spanish MCIU through the “Center of Excellence Severo Ochoa”
award to the Instituto de Astrofísica de Andalucía (SEV-2017-0709). DAK
acknowledges support from Spanish National Research Project
RTI2018-098104-J-I00 (GRBPhot). We also acknowledge the use of public data
from the Fermi Science Support Center (FSSC) and the UK Swift Science Data
Center.

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$t_{1}\sim t_{2}$ |  $\Delta$DIC(1) |  $\Delta$DIC(2) |  $\Delta$DIC(3) |  $\Delta$DIC(4) |  $\Delta$DIC(5) |  $\Delta$DIC(6) |  $\Delta$DIC(7)
---|---|---|---|---|---|---|---
(s) | (CPL+BB-PL) | (CPL+BB-BB) | (CPL+BB-CPL) | (CPL+BB-Band) | (CPL+BB-SBKPL) | (CPL+BB)-(PL+BB) | (CPLBB)-(PL+Bandcut)
0$\sim$116 | -3523 | -19565 | -266 | -262 | -34 | -457 | -68

Table 1: Comparison of $\Delta$DIC between the best model (CPL+BB) and other
various models (PL, BB, CPL, Band, SBKPL, PL+BB, PL+Bandcut) in GRB 190114C,
which is based on the time-integrated spectral analysis.

Satellite-Instrument |  $T_{0}$+[$t_{\rm start}$,$t_{\rm stop}$] | Observed Bandwidth | Isotropic Energy | Model | Reference
---|---|---|---|---|---
| (s) |  | (erg) | (For energy) |
MAGICa |  $T_{0}$+[62, 2454] | 0.3$\sim$1 TeV |  $\sim$4.0$\times$1051 | SPL | MAGIC Collaboration et al. (2019b)
Fermi-LATb |  $T_{0}$+[2.1, 8000] | 0.1$\sim$10 GeV | (1.09$\pm$0.24)$\times$1053 | SPL | this paper
Fermi-LATb |  $T_{0}$+[2.1, 6] | 0.1$\sim$10 GeV | (8.49$\pm$1.80)$\times$1051 | SPL | this paper
Fermi-LATb |  $T_{0}$+[6, 8000] | 0.1$\sim$10 GeV | (1.01$\pm$0.24)$\times$1053 | SPL | this paper
Fermi-GBMc |  $T_{0}$+[0, 116] | 0.001$\sim$10 MeV | (2.82${}^{+0.43}_{-0.25}$)$\times$1053 | (CPL+BB)/Band | this paper
Fermi-GBMc |  $T_{0}$+[0, 6] | … | (2.29${}^{+0.10}_{-0.09}$)$\times$1053 | CPL+BB | this paper
Fermi-GBMc |  $T_{0}$+[6, 116] | … | (5.33${}^{+4.23}_{-2.34}$)$\times$1052 | Band | this paper
Fermi-GBMc |  $T_{0}$+[0, 6] | … | (3.69${}^{+0.78}_{-0.67}$)$\times$1052 | CPL+BB | this paper
Fermi-GBMc |  $T_{0}$+[0, 6] | … | (1.92${}^{+0.12}_{-0.11}$)$\times$1053 | CPL+BB | this paper
Swift-XRTd |  $T_{0}$+[68, 1197626] | 0.3$\sim$10 KeV |  $\sim$1.48$\times$1052 | SPL | this paper

Table 2: Various isotropic energy releases were observed by different
satellite instruments at different wavelengths and different time intervals.
Notes. a Time-integrated-isotropic-equivalent energy releases observed by
MAGIC from $T_{0}$+62 to $T_{0}$+2454 s as reported in MAGIC Collaboration et
al. (2019b). b Time-integrated-isotropic-equivalent energy releases by the
Fermi-LAT observation with a (0.1-10GeV) bandwidth, as well as separated into
the prompt and afterglow emission as defined in the Methods. c Time-
integrated-isotropic-equivalent energy releases by the Fermi-GBM observation
using the best models; as well as those separated into the prompt and
afterglow emission, and into the thermal and non-thermal energy releases
during the prompt emission phase.d The total-isotropic-equivalent energy
release observed by the Swift-XRT.

$t_{\rm start}$$\sim$$t_{\rm stop}$ | $S$ | $K$ | $\alpha$ | $E_{\rm c}$ | $F$ | $K$ | $\alpha$ | $\beta$ | $E_{\rm p}$ | $F$ |  $\Delta$DIC | $p_{\rm DIC,CPL}$ | $p_{\rm DIC,Band}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14)
$P_{1}$ |  |  |  |  |  |  |  |  |  |  |  |  |
-0.067$\sim$0.029 | 8.40 | 0.37${}^{+0.07}_{-0.07}$$\times$10-1 | -0.98${}^{+0.17}_{-0.17}$ | 891${}^{+616}_{-519}$ | 4.68${}^{+4.85}_{-2.45}$$\times$10-6 | 0.37${}^{+0.08}_{-0.08}$$\times$10-1 | -0.98${}^{+0.19}_{-0.19}$ | -6.17${}^{+2.72}_{-2.71}$ | 921${}^{+614}_{-501}$ | 4.42${}^{+4.94}_{-1.99}$$\times$10-6 | 0.0 | 0.5 | 0.3
0.029$\sim$0.141 | 21.45 | 134.00${}^{+38.90}_{-38.70}$$\times$10-1 | -1.11${}^{+0.08}_{-0.08}$ | 1170${}^{+517}_{-479}$ | 10.37${}^{+10.65}_{-4.94}$$\times$10-6 | 0.71${}^{+0.05}_{-0.05}$$\times$10-1 | -1.15${}^{+0.06}_{-0.07}$ | -5.12${}^{+2.74}_{-3.12}$ | 1211${}^{+386}_{-370}$ | 12.53${}^{+3.27}_{-3.19}$$\times$10-6 | 81.7 | 0.6 | 1.4
0.141$\sim$0.294 | 39.83 | 1.41${}^{+0.08}_{-0.08}$$\times$10-1 | -1.11${}^{+0.05}_{-0.05}$ | 944${}^{+230}_{-232}$ | 16.99${}^{+5.07}_{-3.33}$$\times$10-6 | 1.41${}^{+0.08}_{-0.08}$$\times$10-1 | -1.11${}^{+0.05}_{-0.05}$ | -6.34${}^{+2.50}_{-2.54}$ | 824${}^{+165}_{-165}$ | 17.33${}^{+3.78}_{-2.84}$$\times$10-6 | 0.3 | 2.6 | 2.8
0.294$\sim$0.415 | 50.05 | 2.78${}^{+0.17}_{-0.17}$$\times$10-1 | -0.90${}^{+0.05}_{-0.05}$ | 451${}^{+65}_{-66}$ | 21.90${}^{+4.86}_{-3.63}$$\times$10-6 | 2.78${}^{+0.18}_{-0.18}$$\times$10-1 | -0.90${}^{+0.05}_{-0.05}$ | -5.92${}^{+2.68}_{-2.72}$ | 492${}^{+55}_{-56}$ | 22.58${}^{+3.64}_{-3.11}$$\times$10-6 | 0.0 | 2.8 | 2.9
0.415$\sim$0.546 | 73.35 | 4.42${}^{+0.18}_{-0.18}$$\times$10-1 | -0.78${}^{+0.04}_{-0.04}$ | 452${}^{+42}_{-42}$ | 40.15${}^{+5.55}_{-5.07}$$\times$10-6 | 4.44${}^{+0.18}_{-0.18}$$\times$10-1 | -0.78${}^{+0.04}_{-0.04}$ | -6.21${}^{+2.53}_{-2.57}$ | 545${}^{+39}_{-40}$ | 41.10${}^{+4.81}_{-3.84}$$\times$10-6 | -0.0 | 2.9 | 3.0
0.546$\sim$0.701 | 96.49 | 6.47${}^{+0.22}_{-0.22}$$\times$10-1 | -0.66${}^{+0.03}_{-0.03}$ | 354${}^{+23}_{-23}$ | 49.72${}^{+5.44}_{-5.02}$$\times$10-6 | 6.63${}^{+0.26}_{-0.26}$$\times$10-1 | -0.65${}^{+0.04}_{-0.03}$ | -4.05${}^{+1.24}_{-1.89}$ | 454${}^{+26}_{-25}$ | 54.13${}^{+6.04}_{-5.79}$$\times$10-6 | -6.0 | 2.9 | 0.8
0.701$\sim$1.579 | 263.62 | 7.71${}^{+0.08}_{-0.08}$$\times$10-1 | -0.68${}^{+0.01}_{-0.01}$ | 451${}^{+10}_{-11}$ | 80.76${}^{+3.02}_{-2.94}$$\times$10-6 | 7.72${}^{+0.08}_{-0.08}$$\times$10-1 | -0.68${}^{+0.01}_{-0.01}$ | -6.43${}^{+1.96}_{-2.31}$ | 594${}^{+10}_{-10}$ | 81.07${}^{+2.16}_{-1.95}$$\times$10-6 | -1.1 | 3.0 | 2.6
1.579$\sim$1.713 | 117.21 | 8.32${}^{+0.16}_{-0.16}$$\times$10-1 | -0.47${}^{+0.02}_{-0.02}$ | 515${}^{+22}_{-22}$ | 147.00${}^{+12.37}_{-11.77}$$\times$10-6 | 8.32${}^{+0.16}_{-0.17}$$\times$10-1 | -0.46${}^{+0.02}_{-0.02}$ | -6.73${}^{+2.01}_{-2.14}$ | 788${}^{+25}_{-25}$ | 146.90${}^{+9.38}_{-8.28}$$\times$10-6 | -0.1 | 3.0 | 3.0
1.713$\sim$1.805 | 88.47 | 8.93${}^{+0.34}_{-0.34}$$\times$10-1 | -0.56${}^{+0.04}_{-0.04}$ | 319${}^{+21}_{-21}$ | 67.00${}^{+8.01}_{-6.55}$$\times$10-6 | 9.12${}^{+0.41}_{-0.42}$$\times$10-1 | -0.54${}^{+0.04}_{-0.04}$ | -5.38${}^{+2.41}_{-2.98}$ | 447${}^{+26}_{-27}$ | 70.25${}^{+10.60}_{-7.62}$$\times$10-6 | -1.9 | 2.9 | 1.9
1.805$\sim$1.933 | 80.82 | 6.01${}^{+0.33}_{-0.33}$$\times$10-1 | -0.84${}^{+0.04}_{-0.04}$ | 275${}^{+26}_{-26}$ | 28.49${}^{+3.60}_{-3.22}$$\times$10-6 | 9.62${}^{+1.71}_{-1.71}$$\times$10-1 | -0.57${}^{+0.11}_{-0.11}$ | -2.15${}^{+0.10}_{-0.09}$ | 199${}^{+27}_{-27}$ | 42.18${}^{+14.55}_{-9.59}$$\times$10-6 | -20.9 | 2.9 | 0.5
1.933$\sim$2.137 | 72.22 | 3.10${}^{+0.16}_{-0.16}$$\times$10-1 | -1.01${}^{+0.04}_{-0.04}$ | 392${}^{+46}_{-46}$ | 18.98${}^{+2.76}_{-2.28}$$\times$10-6 | 3.11${}^{+0.16}_{-0.16}$$\times$10-1 | -1.01${}^{+0.04}_{-0.04}$ | -6.43${}^{+2.35}_{-2.36}$ | 381${}^{+32}_{-32}$ | 19.17${}^{+2.09}_{-1.81}$$\times$10-6 | 0.4 | 2.9 | 3.1
2.137$\sim$2.406 | 63.87 | 2.48${}^{+0.16}_{-0.16}$$\times$10-1 | -1.00${}^{+0.05}_{-0.04}$ | 301${}^{+36}_{-36}$ | 11.75${}^{+1.70}_{-1.52}$$\times$10-6 | 2.49${}^{+0.17}_{-0.17}$$\times$10-1 | -1.00${}^{+0.05}_{-0.05}$ | -6.41${}^{+2.39}_{-2.41}$ | 297${}^{+25}_{-26}$ | 11.82${}^{+1.54}_{-1.24}$$\times$10-6 | 0.5 | 2.8 | 3.1
2.406$\sim$2.452 | 37.17 | 2.83${}^{+0.19}_{-0.19}$$\times$10-1 | -1.01${}^{+0.06}_{-0.06}$ | 1040${}^{+285}_{-291}$ | 42.69${}^{+18.09}_{-10.96}$$\times$10-6 | 2.90${}^{+0.21}_{-0.21}$$\times$10-1 | -1.00${}^{+0.07}_{-0.07}$ | -4.30${}^{+2.00}_{-3.02}$ | 916${}^{+225}_{-210}$ | 45.91${}^{+15.24}_{-10.74}$$\times$10-6 | -3.8 | 2.3 | 1.0
$P_{2}$ |  |  |  |  |  |  |  |  |  |  |  |  |
2.452$\sim$2.642 | 152.49 | 9.41${}^{+0.14}_{-0.14}$$\times$10-1 | -0.35${}^{+0.02}_{-0.02}$ | 464${}^{+15}_{-15}$ | 169.90${}^{+11.71}_{-10.59}$$\times$10-6 | 9.43${}^{+0.14}_{-0.14}$$\times$10-1 | -0.35${}^{+0.02}_{-0.02}$ | -6.95${}^{+1.98}_{-2.08}$ | 763${}^{+17}_{-18}$ | 170.30${}^{+8.70}_{-6.92}$$\times$10-6 | -0.2 | 3.0 | 2.9
2.642$\sim$2.882 | 135.79 | 6.40${}^{+0.10}_{-0.10}$$\times$10-1 | -0.51${}^{+0.02}_{-0.02}$ | 559${}^{+22}_{-22}$ | 117.20${}^{+8.53}_{-7.44}$$\times$10-6 | 6.41${}^{+0.10}_{-0.10}$$\times$10-1 | -0.51${}^{+0.02}_{-0.02}$ | -5.55${}^{+1.68}_{-2.30}$ | 827${}^{+24}_{-23}$ | 118.90${}^{+6.59}_{-5.62}$$\times$10-6 | -2.4 | 3.0 | 2.4
2.882$\sim$3.088 | 102.21 | 4.47${}^{+0.09}_{-0.09}$$\times$10-1 | -0.53${}^{+0.02}_{-0.02}$ | 666${}^{+33}_{-33}$ | 102.80${}^{+9.27}_{-8.80}$$\times$10-6 | 4.47${}^{+0.09}_{-0.09}$$\times$10-1 | -0.53${}^{+0.02}_{-0.02}$ | -5.93${}^{+1.99}_{-2.47}$ | 974${}^{+37}_{-37}$ | 103.50${}^{+6.94}_{-5.88}$$\times$10-6 | -1.2 | 3.0 | 2.5
3.088$\sim$3.208 | 92.58 | 4.99${}^{+0.11}_{-0.11}$$\times$10-1 | -0.45${}^{+0.03}_{-0.03}$ | 814${}^{+50}_{-50}$ | 180.70${}^{+22.92}_{-18.31}$$\times$10-6 | 5.16${}^{+0.12}_{-0.12}$$\times$10-1 | -0.39${}^{+0.03}_{-0.03}$ | -2.85${}^{+0.13}_{-0.13}$ | 1099${}^{+49}_{-49}$ | 192.00${}^{+16.16}_{-15.55}$$\times$10-6 | -53.8 | 3.0 | 4.0
3.208$\sim$3.605 | 147.68 | 4.58${}^{+0.06}_{-0.06}$$\times$10-1 | -0.35${}^{+0.02}_{-0.02}$ | 619${}^{+20}_{-20}$ | 131.60${}^{+10.15}_{-8.08}$$\times$10-6 | 4.64${}^{+0.06}_{-0.07}$$\times$10-1 | -0.33${}^{+0.02}_{-0.02}$ | -2.88${}^{+0.09}_{-0.09}$ | 968${}^{+24}_{-24}$ | 149.00${}^{+7.72}_{-7.26}$$\times$10-6 | -93.1 | 3.0 | 4.0
3.605$\sim$3.739 | 80.43 | 4.00${}^{+0.10}_{-0.10}$$\times$10-1 | -0.32${}^{+0.04}_{-0.04}$ | 594${}^{+36}_{-36}$ | 115.80${}^{+15.62}_{-14.13}$$\times$10-6 | 4.04${}^{+0.10}_{-0.10}$$\times$10-1 | -0.30${}^{+0.04}_{-0.04}$ | -3.06${}^{+0.21}_{-0.20}$ | 966${}^{+42}_{-42}$ | 129.30${}^{+12.41}_{-11.48}$$\times$10-6 | -19.7 | 2.9 | 3.9
3.739$\sim$3.959 | 140.01 | 6.34${}^{+0.10}_{-0.10}$$\times$10-1 | -0.20${}^{+0.02}_{-0.02}$ | 533${}^{+19}_{-19}$ | 193.50${}^{+15.29}_{-14.73}$$\times$10-6 | 6.55${}^{+0.11}_{-0.11}$$\times$10-1 | -0.14${}^{+0.03}_{-0.03}$ | -2.71${}^{+0.07}_{-0.07}$ | 873${}^{+23}_{-23}$ | 224.70${}^{+13.46}_{-13.54}$$\times$10-6 | -170.4 | 3.0 | 4.0
3.959$\sim$4.096 | 129.78 | 9.68${}^{+0.18}_{-0.18}$$\times$10-1 | -0.19${}^{+0.03}_{-0.03}$ | 399${}^{+14}_{-14}$ | 175.70${}^{+14.96}_{-12.87}$$\times$10-6 | 9.75${}^{+0.19}_{-0.19}$$\times$10-1 | -0.19${}^{+0.03}_{-0.03}$ | -3.65${}^{+0.35}_{-0.23}$ | 709${}^{+19}_{-19}$ | 187.10${}^{+12.17}_{-10.94}$$\times$10-6 | -11.0 | 3.0 | 3.8
4.096$\sim$4.442 | 171.47 | 8.48${}^{+0.13}_{-0.13}$$\times$10-1 | -0.44${}^{+0.02}_{-0.02}$ | 365${}^{+11}_{-11}$ | 90.91${}^{+4.75}_{-5.11}$$\times$10-6 | 8.55${}^{+0.15}_{-0.15}$$\times$10-1 | -0.44${}^{+0.02}_{-0.02}$ | -3.60${}^{+0.39}_{-0.21}$ | 560${}^{+13}_{-13}$ | 97.41${}^{+4.96}_{-4.85}$$\times$10-6 | -9.2 | 3.0 | 3.5
4.442$\sim$4.509 | 67.63 | 7.06${}^{+0.35}_{-0.34}$$\times$10-1 | -0.73${}^{+0.04}_{-0.04}$ | 350${}^{+32}_{-33}$ | 49.84${}^{+7.13}_{-6.25}$$\times$10-6 | 7.07${}^{+0.35}_{-0.36}$$\times$10-1 | -0.73${}^{+0.05}_{-0.05}$ | -5.87${}^{+2.69}_{-2.78}$ | 440${}^{+30}_{-30}$ | 51.28${}^{+6.70}_{-5.57}$$\times$10-6 | -0.4 | 2.9 | 2.8
4.509$\sim$4.770 | 142.78 | 7.16${}^{+0.12}_{-0.12}$$\times$10-1 | -0.65${}^{+0.02}_{-0.02}$ | 493${}^{+20}_{-20}$ | 88.02${}^{+5.56}_{-5.44}$$\times$10-6 | 7.16${}^{+0.12}_{-0.12}$$\times$10-1 | -0.65${}^{+0.02}_{-0.02}$ | -6.78${}^{+2.03}_{-2.13}$ | 666${}^{+19}_{-19}$ | 88.41${}^{+4.12}_{-3.73}$$\times$10-6 | 0.0 | 3.0 | 3.1
4.770$\sim$4.950 | 134.52 | 9.70${}^{+0.21}_{-0.21}$$\times$10-1 | -0.45${}^{+0.02}_{-0.02}$ | 351${}^{+14}_{-14}$ | 96.19${}^{+6.57}_{-6.45}$$\times$10-6 | 9.70${}^{+0.20}_{-0.20}$$\times$10-1 | -0.45${}^{+0.02}_{-0.02}$ | -7.33${}^{+1.86}_{-1.84}$ | 542${}^{+14}_{-14}$ | 96.30${}^{+4.52}_{-4.29}$$\times$10-6 | 0.6 | 3.0 | 3.2
4.950$\sim$5.451 | 184.67 | 6.94${}^{+0.10}_{-0.10}$$\times$10-1 | -0.62${}^{+0.02}_{-0.02}$ | 422${}^{+13}_{-13}$ | 72.22${}^{+3.53}_{-3.51}$$\times$10-6 | 6.94${}^{+0.10}_{-0.10}$$\times$10-1 | -0.62${}^{+0.01}_{-0.01}$ | -6.97${}^{+1.97}_{-2.05}$ | 582${}^{+13}_{-12}$ | 72.22${}^{+2.46}_{-2.36}$$\times$10-6 | 0.1 | 3.0 | 3.1
5.451$\sim$5.514 | 77.55 | 10.10${}^{+0.41}_{-0.41}$$\times$10-1 | -0.42${}^{+0.04}_{-0.04}$ | 302${}^{+20}_{-20}$ | 83.08${}^{+10.90}_{-9.34}$$\times$10-6 | 10.12${}^{+0.40}_{-0.40}$$\times$10-1 | -0.41${}^{+0.04}_{-0.04}$ | -6.81${}^{+2.16}_{-2.15}$ | 475${}^{+21}_{-21}$ | 83.12${}^{+6.69}_{-6.74}$$\times$10-6 | 0.5 | 2.9 | 3.2
5.514$\sim$5.689 | 109.97 | 10.10${}^{+0.43}_{-0.43}$$\times$10-1 | -0.55${}^{+0.03}_{-0.03}$ | 193${}^{+10}_{-10}$ | 37.07${}^{+3.57}_{-3.20}$$\times$10-6 | 10.49${}^{+0.58}_{-0.57}$$\times$10-1 | -0.53${}^{+0.04}_{-0.04}$ | -4.13${}^{+1.15}_{-1.47}$ | 270${}^{+12}_{-12}$ | 39.58${}^{+4.51}_{-4.22}$$\times$10-6 | -5.6 | 2.9 | 1.1
5.689$\sim$5.808 | 69.83 | 6.33${}^{+0.59}_{-0.59}$$\times$10-1 | -0.89${}^{+0.06}_{-0.06}$ | 171${}^{+20}_{-20}$ | 17.29${}^{+3.08}_{-2.54}$$\times$10-6 | 5.15${}^{+0.21}_{-0.21}$$\times$10-1 | -1.00${}^{+0.04}_{-0.04}$ | -5.10${}^{+2.35}_{-3.08}$ | 226${}^{+5}_{-5}$ | 19.52${}^{+2.96}_{-1.50}$$\times$10-6 | 2.1 | 2.6 | 0.9
5.808$\sim$6.000 | 60.35 | 2.46${}^{+0.32}_{-0.32}$$\times$10-1 | -1.35${}^{+0.07}_{-0.08}$ | 202${}^{+41}_{-41}$ | 8.22${}^{+1.64}_{-1.45}$$\times$10-6 | 4.53${}^{+1.21}_{-1.15}$$\times$10-1 | -1.06${}^{+0.13}_{-0.13}$ | -2.20${}^{+0.11}_{-0.11}$ | 83${}^{+11}_{-11}$ | 11.53${}^{+5.05}_{-3.72}$$\times$10-6 | -18.9 | 1.6 | -0.6
$P_{AO}$ |  |  |  |  |  |  |  |  |  |  |  |  |
6.000$\sim$6.436 | 62.10 | 1.02${}^{+0.10}_{-0.11}$$\times$10-1 | -1.63${}^{+0.06}_{-0.06}$ | 440${}^{+133}_{-138}$ | 5.80${}^{+1.17}_{-0.88}$$\times$10-6 | 1.09${}^{+0.14}_{-0.16}$$\times$10-1 | -1.60${}^{+0.07}_{-0.08}$ | -5.13${}^{+2.87}_{-3.31}$ | 139${}^{+29}_{-32}$ | 5.98${}^{+2.08}_{-1.32}$$\times$10-6 | -2.0 | 1.0 | 0.1
6.436$\sim$6.867 | 50.89 | 0.71${}^{+0.07}_{-0.07}$$\times$10-1 | -1.73${}^{+0.05}_{-0.05}$ | 537${}^{+201}_{-182}$ | 4.66${}^{+0.85}_{-0.66}$$\times$10-6 | 0.88${}^{+0.14}_{-0.20}$$\times$10-1 | -1.64${}^{+0.10}_{-0.11}$ | -3.58${}^{+1.50}_{-2.81}$ | 113${}^{+33}_{-40}$ | 5.21${}^{+2.31}_{-1.58}$$\times$10-6 | -19.6 | 1.4 | -15.2
6.867$\sim$8.221 | 64.64 | 0.44${}^{+0.01}_{-0.01}$$\times$10-1 | -1.77${}^{+0.02}_{-0.02}$ | 892${}^{+84}_{-86}$ | 3.55${}^{+0.14}_{-0.15}$$\times$10-6 | 0.43${}^{+0.02}_{-0.05}$$\times$10-1 | -1.81${}^{+-0.00}_{-0.05}$ | -4.96${}^{+2.68}_{-2.72}$ | 630${}^{+275}_{-265}$ | 4.75${}^{+0.43}_{-0.81}$$\times$10-6 | -31.8 | 2.2 | -23.1
8.221$\sim$9.567 | 49.76 | 0.32${}^{+0.01}_{-0.01}$$\times$10-1 | -1.78${}^{+0.03}_{-0.03}$ | 830${}^{+132}_{-140}$ | 2.59${}^{+0.14}_{-0.17}$$\times$10-6 | 0.29${}^{+0.01}_{-0.01}$$\times$10-1 | -1.85${}^{+0.02}_{-0.03}$ | -5.02${}^{+2.64}_{-2.64}$ | 647${}^{+264}_{-275}$ | 3.50${}^{+0.22}_{-0.37}$$\times$10-6 | -6.2 | 2.2 | 1.5
9.567$\sim$12.400 | 54.86 | 0.22${}^{+0.01}_{-0.01}$$\times$10-1 | -1.86${}^{+0.03}_{-0.03}$ | 706${}^{+193}_{-192}$ | 1.83${}^{+0.17}_{-0.16}$$\times$10-6 | 0.21${}^{+0.02}_{-0.02}$$\times$10-1 | -1.91${}^{+0.06}_{-0.05}$ | -5.15${}^{+2.52}_{-2.54}$ | 305${}^{+362}_{-221}$ | 2.33${}^{+0.43}_{-0.52}$$\times$10-6 | -7.6 | 2.1 | -6.5
12.400$\sim$15.547 | 47.53 | 0.17${}^{+0.01}_{-0.01}$$\times$10-1 | -1.93${}^{+0.03}_{-0.03}$ | 699${}^{+207}_{-205}$ | 1.47${}^{+0.14}_{-0.14}$$\times$10-6 | 0.15${}^{+0.00}_{-0.00}$$\times$10-1 | -1.99${}^{+0.01}_{-0.01}$ | -5.54${}^{+2.37}_{-2.31}$ | 512${}^{+321}_{-312}$ | 2.11${}^{+0.06}_{-0.09}$$\times$10-6 | 0.3 | 2.0 | 1.0
$P_{3}$ |  |  |  |  |  |  |  |  |  |  |  |  |
15.547$\sim$15.872 | 38.28 | 0.74${}^{+0.07}_{-0.07}$$\times$10-1 | -1.49${}^{+0.06}_{-0.06}$ | 552${}^{+199}_{-182}$ | 4.59${}^{+1.03}_{-0.93}$$\times$10-6 | 0.76${}^{+0.09}_{-0.10}$$\times$10-1 | -1.47${}^{+0.08}_{-0.08}$ | -5.07${}^{+2.60}_{-2.62}$ | 278${}^{+76}_{-90}$ | 4.77${}^{+1.65}_{-0.98}$$\times$10-6 | -1.8 | 1.7 | 0.0
15.872$\sim$16.173 | 49.52 | 1.06${}^{+0.09}_{-0.09}$$\times$10-1 | -1.49${}^{+0.05}_{-0.06}$ | 649${}^{+197}_{-214}$ | 6.94${}^{+1.74}_{-1.13}$$\times$10-6 | 1.14${}^{+0.08}_{-0.08}$$\times$10-1 | -1.45${}^{+0.05}_{-0.05}$ | -6.18${}^{+2.61}_{-2.61}$ | 248${}^{+32}_{-31}$ | 6.59${}^{+0.74}_{-0.76}$$\times$10-6 | 0.5 | 1.7 | 2.4
16.173$\sim$16.927 | 98.83 | 1.99${}^{+0.14}_{-0.14}$$\times$10-1 | -1.39${}^{+0.04}_{-0.04}$ | 198${}^{+23}_{-23}$ | 6.71${}^{+0.78}_{-0.64}$$\times$10-6 | 2.94${}^{+0.38}_{-0.38}$$\times$10-1 | -1.20${}^{+0.07}_{-0.07}$ | -2.36${}^{+0.09}_{-0.08}$ | 88${}^{+7}_{-7}$ | 8.23${}^{+1.70}_{-1.38}$$\times$10-6 | -18.7 | 2.6 | 3.1
16.927$\sim$17.324 | 59.55 | 1.19${}^{+0.14}_{-0.14}$$\times$10-1 | -1.59${}^{+0.07}_{-0.06}$ | 245${}^{+53}_{-56}$ | 5.11${}^{+0.96}_{-0.77}$$\times$10-6 | 1.23${}^{+0.16}_{-0.16}$$\times$10-1 | -1.57${}^{+0.07}_{-0.07}$ | -5.79${}^{+2.79}_{-2.80}$ | 96${}^{+10}_{-10}$ | 5.23${}^{+1.19}_{-0.93}$$\times$10-6 | 0.7 | 1.7 | 2.5
17.324$\sim$17.719 | 51.00 | 0.86${}^{+0.12}_{-0.12}$$\times$10-1 | -1.69${}^{+0.07}_{-0.07}$ | 283${}^{+83}_{-87}$ | 4.31${}^{+0.94}_{-0.74}$$\times$10-6 | 1.00${}^{+0.17}_{-0.16}$$\times$10-1 | -1.62${}^{+0.08}_{-0.08}$ | -3.45${}^{+1.21}_{-2.00}$ | 70${}^{+11}_{-10}$ | 4.57${}^{+1.54}_{-1.06}$$\times$10-6 | -5.3 | -0.1 | -1.2
17.719$\sim$20.397 | 95.90 | 0.64${}^{+0.04}_{-0.04}$$\times$10-1 | -1.72${}^{+0.04}_{-0.03}$ | 210${}^{+26}_{-27}$ | 3.04${}^{+0.29}_{-0.28}$$\times$10-6 | 0.77${}^{+0.10}_{-0.10}$$\times$10-1 | -1.63${}^{+0.06}_{-0.06}$ | -2.86${}^{+0.48}_{-0.09}$ | 53${}^{+4}_{-4}$ | 3.27${}^{+0.77}_{-0.59}$$\times$10-6 | -14.2 | 2.5 | -3.3
20.397$\sim$21.699 | 50.58 | 0.34${}^{+0.03}_{-0.03}$$\times$10-1 | -1.95${}^{+0.04}_{-0.04}$ | 197${}^{+36}_{-37}$ | 2.32${}^{+0.29}_{-0.26}$$\times$10-6 | 0.64${}^{+0.13}_{-0.12}$$\times$10-1 | -1.65${}^{+0.08}_{-0.08}$ | -3.21${}^{+0.67}_{-0.11}$ | 31${}^{+1}_{-1}$ | 2.11${}^{+0.68}_{-0.55}$$\times$10-6 | 6.0 | 1.7 | -2.4
21.699$\sim$23.330 | 37.70 | 0.21${}^{+0.02}_{-0.02}$$\times$10-1 | -1.96${}^{+0.04}_{-0.04}$ | 240${}^{+51}_{-53}$ | 1.49${}^{+0.18}_{-0.16}$$\times$10-6 | 0.27${}^{+0.06}_{-0.06}$$\times$10-1 | -1.85${}^{+0.09}_{-0.08}$ | -4.14${}^{+1.83}_{-2.89}$ | 17${}^{+5}_{-5}$ | 1.55${}^{+0.71}_{-0.62}$$\times$10-6 | -8.1 | 1.7 | -7.0
23.330$\sim$26.530 | 39.21 | 0.14${}^{+0.02}_{-0.02}$$\times$10-1 | -1.91${}^{+0.06}_{-0.06}$ | 347${}^{+122}_{-113}$ | 1.04${}^{+0.18}_{-0.14}$$\times$10-6 | 0.19${}^{+0.05}_{-0.05}$$\times$10-1 | -1.82${}^{+0.12}_{-0.10}$ | -4.78${}^{+2.52}_{-2.73}$ | 31${}^{+8}_{-8}$ | 1.04${}^{+0.52}_{-0.39}$$\times$10-6 | -6.9 | 1.2 | -5.5
26.530$\sim$33.075 | 37.67 | 0.10${}^{+0.01}_{-0.01}$$\times$10-1 | -1.93${}^{+0.05}_{-0.05}$ | 387${}^{+123}_{-116}$ | 0.75${}^{+0.10}_{-0.09}$$\times$10-6 | 0.11${}^{+0.01}_{-0.02}$$\times$10-1 | -1.88${}^{+0.06}_{-0.07}$ | -5.01${}^{+2.57}_{-2.69}$ | 30${}^{+8}_{-9}$ | 0.75${}^{+0.28}_{-0.21}$$\times$10-6 | 0.4 | 1.5 | 1.5
33.075$\sim$47.327 | 35.24 | 0.07${}^{+0.00}_{-0.00}$$\times$10-1 | -1.84${}^{+0.03}_{-0.03}$ | 493${}^{+81}_{-83}$ | 0.53${}^{+0.04}_{-0.04}$$\times$10-6 | 0.07${}^{+0.01}_{-0.01}$$\times$10-1 | -1.87${}^{+0.08}_{-0.10}$ | -4.59${}^{+2.50}_{-2.80}$ | 74${}^{+20}_{-27}$ | 0.62${}^{+0.28}_{-0.19}$$\times$10-6 | -11.8 | 2.2 | -7.3
47.327$\sim$73.490 | 28.11 | 0.05${}^{+0.01}_{-0.01}$$\times$10-1 | -1.83${}^{+0.06}_{-0.06}$ | 583${}^{+268}_{-250}$ | 0.35${}^{+0.06}_{-0.06}$$\times$10-6 | 0.06${}^{+0.01}_{-0.01}$$\times$10-1 | -1.71${}^{+0.08}_{-0.07}$ | -4.03${}^{+1.90}_{-2.91}$ | 67${}^{+15}_{-14}$ | 0.34${}^{+0.12}_{-0.09}$$\times$10-6 | -2.7 | 1.0 | -0.7

Table 3: Time-resolved spectral fit results of GRB 190114C. (1) The start and
stop times (in units of s) of the BBlocks time bins; (2) the significance $S$;
(3-6) the best-fit parameters for the CPL model; (7-11) the best-fit
parameters for the Band model; (12) the difference between the Deviance
Information Criterion (DIC) for the CPL and the Band model,
$\Delta$DIC=DICBand-DICCPL; (13-14) the effective number of parameters
($p_{\rm DIC}$) for the CPL and Band model, respectively.

| $t_{\rm start}\sim t_{\rm stop}$ | $S$ | Model |  $\Delta$DIC | Temperature | Thermal Flux | Total Flux | Ratio
---|---|---|---|---|---|---|---|---
| (s) |  |  | DIC(CPL+BB)-(CPL) | (keV) | (erg cm-2 s-1) | (erg cm-2 s${}^{-1})$ |
| 0.55$\sim$0.70 | 96.49 | CPL+BB | -177 | 351${}^{+99}_{-100}$ | 0.59${}^{+1.82}_{-0.45}$$\times$10-5 | 0.57${}^{+0.19}_{-0.11}$$\times$10-4 | 0.10${}^{+0.32}_{-0.08}$
| 0.70$\sim$0.98 | 153.65 | CPL+BB | -285 | 283${}^{+59}_{-55}$ | 0.58${}^{+1.27}_{-0.42}$$\times$10-5 | 0.72${}^{+0.15}_{-0.11}$$\times$10-4 | 0.08${}^{+0.18}_{-0.06}$
| 0.98$\sim$1.45 | 196.39 | CPL+BB | -43 | 186${}^{+40}_{-34}$ | 0.91${}^{+1.72}_{-0.80}$$\times$10-5 | 0.86${}^{+0.22}_{-0.24}$$\times$10-4 | 0.10${}^{+0.20}_{-0.10}$
$P^{\rm th}_{1}$ | 1.45$\sim$1.58 | 105.17 | CPL+BB | -148 | 163${}^{+20}_{-19}$ | 2.42${}^{+2.10}_{-1.13}$$\times$10-5 | 1.05${}^{+0.03}_{-0.20}$$\times$10-4 | 0.23${}^{+0.20}_{-0.11}$
| 1.58$\sim$1.64 | 80.78 | CPL+BB | -29 | 151${}^{+11}_{-11}$ | 5.26${}^{+2.68}_{-1.69}$$\times$10-5 | 1.38${}^{+0.33}_{-0.28}$$\times$10-4 | 0.36${}^{+0.21}_{-0.14}$
| 1.64$\sim$1.71 | 83.47 | CPL+BB | -21 | 136${}^{+28}_{-24}$ | 2.06${}^{+4.18}_{-1.63}$$\times$10-5 | 1.49${}^{+0.68}_{-0.41}$$\times$10-4 | 0.14${}^{+0.29}_{-0.12}$
| 1.71$\sim$1.80 | 88.47 | CPL+BB | -114 | 73${}^{+13}_{-18}$ | 0.09${}^{+0.20}_{-0.07}$$\times$10-5 | 0.71${}^{+0.17}_{-0.14}$$\times$10-4 | 0.01${}^{+0.03}_{-0.01}$
| 1.80$\sim$1.93 | 80.82 | CPL+BB | -139 | 30${}^{+4}_{-5}$ | 0.07${}^{+0.05}_{-0.04}$$\times$10-5 | 0.32${}^{+0.02}_{-0.02}$$\times$10-4 | 0.02${}^{+0.02}_{-0.01}$
| 2.45$\sim$2.64 | 152.16 | CPL+BB | -21 | 174${}^{+13}_{-13}$ | 3.88${}^{+1.89}_{-1.40}$$\times$10-5 | 1.76${}^{+0.26}_{-0.24}$$\times$10-4 | 0.22${}^{+0.11}_{-0.09}$
| 2.64$\sim$2.88 | 136.19 | CPL+BB | -622 | 197${}^{+17}_{-16}$ | 4.07${}^{+1.91}_{-1.34}$$\times$10-5 | 1.17${}^{+0.28}_{-0.28}$$\times$10-4 | 0.35${}^{+0.18}_{-0.14}$
| 2.88$\sim$3.09 | 103.16 | CPL+BB | -12 | 188${}^{+22}_{-22}$ | 2.36${}^{+1.83}_{-1.11}$$\times$10-5 | 1.01${}^{+0.38}_{-0.29}$$\times$10-4 | 0.23${}^{+0.20}_{-0.13}$
| 3.09$\sim$3.21 | 92.86 | CPL+BB | -20 | 162${}^{+12}_{-8}$ | 4.93${}^{+1.71}_{-1.58}$$\times$10-5 | 1.95${}^{+0.85}_{-0.57}$$\times$10-4 | 0.25${}^{+0.14}_{-0.11}$
| 3.21$\sim$3.60 | 146.18 | CPL+BB | -37 | 149${}^{+7}_{-8}$ | 3.16${}^{+1.03}_{-0.79}$$\times$10-5 | 1.36${}^{+0.47}_{-0.31}$$\times$10-4 | 0.23${}^{+0.11}_{-0.08}$
| 3.60$\sim$3.74 | 82.69 | CPL+BB | -20 | 151${}^{+11}_{-11}$ | 3.36${}^{+1.65}_{-1.28}$$\times$10-5 | 1.23${}^{+0.84}_{-0.49}$$\times$10-4 | 0.27${}^{+0.23}_{-0.15}$
| 3.74$\sim$3.96 | 140.29 | CPL+BB | -63 | 140${}^{+5}_{-5}$ | 7.11${}^{+1.13}_{-1.01}$$\times$10-5 | 2.23${}^{+0.81}_{-0.55}$$\times$10-4 | 0.32${}^{+0.13}_{-0.09}$
$P^{\rm th}_{2}$ | 3.96$\sim$4.10 | 130.66 | CPL+BB | -44 | 108${}^{+6}_{-7}$ | 3.52${}^{+1.85}_{-1.83}$$\times$10-5 | 1.79${}^{+1.71}_{-0.79}$$\times$10-4 | 0.20${}^{+0.22}_{-0.13}$
| 4.10$\sim$4.44 | 170.04 | CPL+BB | -922 | 115${}^{+7}_{-7}$ | 1.89${}^{+0.80}_{-0.59}$$\times$10-5 | 0.94${}^{+0.13}_{-0.11}$$\times$10-4 | 0.20${}^{+0.09}_{-0.07}$
| 4.44$\sim$4.51 | 69.23 | CPL+BB | -207 | 97${}^{+15}_{-13}$ | 1.22${}^{+1.57}_{-0.74}$$\times$10-5 | 0.55${}^{+0.23}_{-0.13}$$\times$10-4 | 0.22${}^{+0.30}_{-0.15}$
| 4.51$\sim$4.77 | 142.52 | CPL+BB | -133 | 111${}^{+5}_{-5}$ | 3.45${}^{+1.02}_{-0.72}$$\times$10-5 | 0.85${}^{+0.22}_{-0.16}$$\times$10-4 | 0.41${}^{+0.16}_{-0.12}$
| 4.77$\sim$4.95 | 134.52 | CPL+BB | -54 | 90${}^{+4}_{-4}$ | 3.21${}^{+0.89}_{-0.76}$$\times$10-5 | 0.97${}^{+0.33}_{-0.23}$$\times$10-4 | 0.33${}^{+0.15}_{-0.11}$
| 4.95$\sim$5.45 | 184.46 | CPL+BB | -176 | 91${}^{+3}_{-3}$ | 2.45${}^{+0.45}_{-0.39}$$\times$10-5 | 0.70${}^{+0.14}_{-0.10}$$\times$10-4 | 0.35${}^{+0.09}_{-0.07}$
| 5.45$\sim$5.51 | 76.02 | CPL+BB | -93 | 82${}^{+5}_{-5}$ | 3.18${}^{+1.30}_{-1.22}$$\times$10-5 | 0.80${}^{+0.95}_{-0.32}$$\times$10-4 | 0.40${}^{+0.49}_{-0.22}$
| 5.51$\sim$5.65 | 100.84 | CPL+BB | -27 | 53${}^{+4}_{-4}$ | 1.10${}^{+0.51}_{-0.43}$$\times$10-5 | 0.44${}^{+0.11}_{-0.16}$$\times$10-4 | 0.25${}^{+0.13}_{-0.12}$
| 5.65$\sim$5.69 | 48.93 | CPL+BB | -26 | 32${}^{+2}_{-2}$ | 1.07${}^{+0.33}_{-0.26}$$\times$10-5 | 0.35${}^{+0.07}_{-0.06}$$\times$10-4 | 0.30${}^{+0.11}_{-0.09}$
$T_{90}$ | 0.00$\sim$116.00 | 190.61 | CPL+BB | -266 | 132${}^{+4}_{-4}$ | 0.12${}^{+0.04}_{-0.03}$$\times$10-5 | 0.06${}^{+0.01}_{-0.01}$$\times$10-4 | 0.21${}^{+0.07}_{-0.05}$

Table 4: Spectral parameters of the slices having a thermal component in GRB
190114C. The spectra are best fitted by a two-component scenario, with a
thermal BB component accompanied by a non-thermal CPL component. The table
lists the start and stops times of the BBlocks slices, the significance, the
best-fitted model, the $\Delta$DIC between CPL+BB and CPL models, the
temperature, the thermal and total flux, and the ratio of thermal flux. Flux
is defined in the energy band of $1$ keV to $10$ MeV. For the slices of $\sim
3$ s to $\sim 4$ s, Band+BB offers very close goodness of fitting as CPL+BB,
for the global consistency, and considering the time-integrated spectrum is
best fitted by CPL+BB, here we perform all the thermal analysis using CPL+BB.

Table 5: Thermal-pulse properties of GRB 190114C

| $P^{\rm th}_{1}$ | $P^{\rm th}_{2}$
---|---|---
| (From $t_{\rm obs}=$0.55 s to 1.93 s) | (From $t_{\rm obs}=$2.45 s to 5.69 s)
Observed properties |  |
Duration | 1.38 s | 3.24 s
Spectral cut-off energy [$E_{\rm c}$] | 337${}^{+27}_{-27}$(keV) | 605${}^{+17}_{-17}$(keV)
Temperature [k$T$] | 267${}^{+22}_{-18}$(keV) | 145${}^{+3}_{-3}$(keV)
Thermal energy flux [$F_{\rm BB}$] | (1.51${}^{+0.97}_{-0.68}$)$\times$10-5(erg cm-2 s-1) | (2.32${}^{+0.35}_{-0.30}$)$\times$10-5(erg cm-2 s-1)
Total energy flux [$F_{\rm tot}$] | (8.65${}^{+1.64}_{-1.34}$)$\times$10-5(erg cm-2 s-1) | (1.07${}^{+0.05}_{-0.05}$)$\times$10-4(erg cm-2 s-1)
Flux ratio [$F_{\rm BB}/F_{\rm tot}$] | 0.17${}^{+0.12}_{-0.08}$ | 0.22${}^{+0.03}_{-0.03}$
Thermal fluence [$S_{\rm BB}$] | (2.09${}^{+1.35}_{-0.94}$)$\times$10-5(erg cm-2) | (7.51${}^{+1.13}_{-0.97}$)$\times$10-5(erg cm-2)
Total fluence [$S_{\rm tot}$] | (1.20${}^{+0.23}_{-0.19}$)$\times$10-4(erg cm-2) | (3.46${}^{+0.17}_{-0.17}$)$\times$10-4(erg cm-2)
Isotropic thermal luminosity [$L_{\rm BB,\gamma,iso}$] | (1.04${}^{+0.67}_{-0.47}$)$\times$1052(erg s-1) | (1.60${}^{+0.24}_{-0.21}$)$\times$1052(erg s-1)
Isotropic total luminosity [$L_{\rm\gamma,iso}$] | (5.95${}^{+1.13}_{-0.92}$)$\times$1052(erg s-1) | (7.36${}^{+0.34}_{-0.34}$)$\times$1052(erg s-1)
Isotropic thermal energy [$E_{\rm BB,\gamma,iso}$] | (1.01${}^{+0.65}_{-0.46}$)$\times$1052(erg) | (3.64${}^{+0.55}_{-0.47}$)$\times$1052(erg)
Isotropic total energy [$E_{\rm\gamma,iso}$] | (5.81${}^{+1.10}_{-0.91}$)$\times$1052(erg) | (1.68${}^{+0.08}_{-0.08}$)$\times$1053(erg)
Photospheric properties |  |
Nozzle radius [$r_{\rm 0}$] | (8.55$\pm$2.8)$\times$106(cm) | (5.00$\pm$0.48)$\times$107(cm)
Saturation radius [$r_{\rm s}$] | (4.31$\pm$1.53)$\times$109(cm) | (1.96$\pm$0.19)$\times$1010(cm)
Photospheric radius [$r_{\rm ph}$] | (5.33$\pm$0.47)$\times$1011(cm) | (1.41$\pm$0.04)$\times$1012(cm)
Parameter evolution |  |
Temperature [k$T$(t)] | $\propto t^{-0.93\pm 0.04}$ | $\propto t^{-1.32\pm 0.09}$
Effective transverse size [$\Re$(t)] | $\propto t^{3.12\pm 0.49}$ | $\propto t^{2.37\pm 0.32}$
Bulk Lorentz factor [$\Gamma$(t)] | $\propto t^{-0.48\pm 0.05}$ | $\propto t^{-0.81\pm 0.08}$
Nozzle radius $R_{\rm 0}$ (cm) | $\propto t^{4.69\pm 3.89}$ | $\propto t^{4.10\pm 0.86}$
Saturation radius $R_{\rm s}$ (cm) | $\propto t^{4.47\pm 4.20}$ | $\propto t^{2.10\pm 0.96}$
photospheric radius $R_{\rm ph}$ (cm) | $\propto t^{1.76\pm 0.60}$ | $\propto t^{1.09\pm 0.37}$

Table 6: Global properties of GRB 190114C

Measured Parameters |
---|---
Isotropic equivalent thermal energy [$E_{\rm th,iso}$] | (3.69${}^{+0.78}_{-0.67}$)$\times$1052 erg
Isotropic equivalent non-thermal energy [$E_{\rm nth,iso}$] | [(1.92${}^{+0.12}_{-0.11}$)$\times$1053(GBM)+8.49${}^{+1.80}_{-1.80}$$\times$1051(LAT)] erg
Thermal energy flux [$F^{\rm obs}_{\rm BB}$] | (1.27${}^{+0.27}_{-0.23}$) $\times$ 10-5 erg cm-2s-1
Total energy flux [$F^{\rm obs}_{\gamma}$] | [(7.91${}^{+0.33}_{-0.32}$)$\times$10-5(GBM)+3.41${}^{+0.69}_{-0.69}$$\times$10-6(LAT)] erg cm${}^{-2}s^{-1}$
Deceleration time [$t_{\rm dec}$] | 6-10 s
Temperature [$kT^{\rm obs}$] | 163$\pm$6 keV
Redshift [$z$] | 0.4254$\pm$0.0005
Derived Parameters |
Dimensionless specific enthalpy [$\eta$] | $854\pm 38$
Bulk Lorentz factor at $r_{\rm ph}$ [$\Gamma_{\rm ph}$] | $833\pm 38$
Initial Lorentz factor [$\Gamma_{\rm s}$] | $719\pm 59$
Isotropic equivalent total mass [$M_{\rm iso}$] | $(1.7\pm 0.4)\times 10^{-3}~{}M_{\odot}$
Isotropic kinetic energy [$E_{\rm k,iso}$] |  $(1.6\pm 0.7)\times 10^{54}$ erg
Isotropic total energy [$E_{\rm tot,iso}$] |  $(1.8\pm 0.7)\times 10^{54}$ erg
$\gamma$-ray radiative efficiency [$\eta_{\gamma}$] | $15.8\pm 5.4~{}\%$
Further Derived Parameters |
Energy fractions assigned to electric fields [$\epsilon_{e,-1}$] | 1.11$\pm$0.01
Energy fractions assigned to magnetic fields [$\epsilon_{\rm B,-2}$] | 0.05$\pm$0.01
Characteristic synchrotron frequency [$\nu_{\rm m}$] | (1.30$\pm$0.82)$\times$1017 Hz
Cooling frequency [$\nu_{\rm c}$] | (4.44$\pm$0.66)$\times$ 1017 Hz
Klein-Nishina frequency [$\nu_{\rm KN}$] | (6.55$\pm$0.16)$\times$1017 Hz

Figure 1: Count lightcurve of Fermi-GBM during the time span of $0-30$ s.
Differently shaded regions marked with different colors denote the two
independent thermally-sub-dominated episodes: $P^{\rm th}_{1}$ (pink) and
$P^{\rm th}_{2}$ (blue), the afterglow emission episode $P_{\rm AO}$ (yellow),
and the $\gamma$-ray flare emission $P_{3}$ (grey). Left panel: Two horizontal
dashed lines represent the limiting values of $\alpha$=-2/3 and $\alpha$=-3/2
for electrons in the synchrotron slow- and fast-cooling regimes, respectively.
The data points connected by solid lines in orange colour represent the
temporal evolution of the low-energy photon index $\alpha$ of the CPL-only
fits. Right panel: the data points connected by solid lines in orange colour
represent the temporal evolution of the $E_{\rm p}$ of the Band-only fits
while those in red colour indicate the temporal evolution of $E_{\rm c}$ of
the CPL-only fits.

Figure 2: The GBM light curve (black dotted line) with the best fits to the
decay phase of $P_{1}$ (cyan line), $P_{2}$ (blue magenta line), $P_{3}$
(orange line), and the afterglow emission $A_{1}$ (violet line) and $A_{2}$
(yellow line) using the single power-law model. Note that $A_{1}$ and $A_{2}$
correspond to the afterglow emission generated by $P_{1}$ and $P_{2}$,
respectively. The onset of the afterglow emission $t_{\rm Afterglow}$ (the
vertical green line) is used to estimate $\Gamma$ in equation (21). The decay
index of the afterglow emission from $P_{1}$ is $\hat{\alpha}(A_{1})=-1.93\pm
0.09$, which is significantly steeper than a typical value for afterglow
emission measured from other GRBs, this is because part of the energy flux in
this segment is clearly contributed from $P_{2}$, whereas the decay index of
the afterglow emission from $P_{2}$ is $\hat{\alpha}(A_{2})=-1.09\pm 0.04$,
which is in good agreement with typical values found for afterglow emission.

Figure 3: Left panel: multi-wavelength light curve. The data points indicated
by violet, yellow, black, grey, blue magenta, orange, and cyan represent the
MAGIC, Fermi-LAT, Fermi-GBM, Swift-BAT, Swift-XRT, the optical ($R_{\rm
c}$-band), and the radio (at 97.5 GHz) observations, respectively. The solid
lines are the best power-law fitting to the data. Note that: (1) The LAT data
are separated into two parts at $\sim$6 s, and here we only fit the second
(afterglow) part ($>$6 s). (2) The optical $R_{\rm c}$-band has been corrected
for Galactic and host extinction, and the contribution from the host galaxy
has also been subtracted. This light curve has been created by shifting data
from different bands to the $R$ band. Right panel: multi-wavelength spectrum,
covering the energy in MeV (grey), GeV (blue), and TeV (jacinth) emission,
which is simultaneously observed from $T_{0}$+68 s to $T_{0}$+110 s by Fermi-
GBM, Fermi-LAT, and MAGIC, respectively.

Figure 4: Bayesian Monte Carlo Time-integrated spectral fits for Fermi-GBM
data from $T_{0}+0$ s to $T_{0}+116$ s ($T_{90}$ duration). We apply $20$
chains, each chain iterates $10^{4}$ times and burns the first $10^{3}$ times.
The parameters are normalisation (Norm CPL), cut-off energy and power-law
index of the cut-off power-law model, as well as normalisation (Norm BB) and
temperature ($kT$) of the BB model.

Figure 5: Spectrum from $4.95$ s to $5.45$ s. The spectrum includes data from
_Fermi_ -GBM (2 NaI and 1 BGO detector). The fitting is presented by a solid
line, including the components of a Planck blackbody function indicated by a
dashed line and a cutoff power law indicated by a dotted line.

Figure 6: Temporal evolution of the temperature k$T$ (left panel), and bulk
Lorentz factor $\Gamma$ (right panel). The data points indicated by pink and
cyan colours represent the two different pulses. Solid lines are the best
power-law fits to the data for $P_{1}$ and $P_{2}$ excluding several points
during the drop, and shaded areas are their 2-$\sigma$ (95% confidence
interval) regions. The derived time-resolved evolution of $\Gamma$ is based on
the photosphere properties under the framework of the traditional method(Pe’er
et al., 2007).

Figure 7: Pulse-wise properties of GRB 190114C. (a) Temporal evolution of the
photospheric radius $r_{\rm ph}$ (violet), saturation radius $r_{\rm s}$
(orange), and nozzle radius $r_{\rm 0}$ (cyan). (b) Temporal evolution of the
parameter $\Re$. (c) Temporal evolution of the BB energy flux ($F_{\rm BB}$)
and total energy flux ($F_{\rm tot}$). (d) The total energy flux ($F_{\rm
tot}$) versus the BB energy flux ($F_{\rm BB}$). Same color notation as in
Figure 6.

Figure 8: Histogram of $\eta_{\gamma}$ from 10000 Monte Carlo samplings, each
bin corresponds to the $\eta_{\gamma}$ interval of $0.003$. The histogram is
fitted by a skew-normal distribution function.