# GRB 190114C: Fireball Energy Budget and Radiative Efficiency Revisited

Liang Li1,2,3 Yu Wang1,2,3
1ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy
2ICRA and Dipartimento di Fisica, Università di Roma “La Sapienza”, Piazzale
Aldo Moro 5, I-00185 Roma, Italy
3INAF – Osservatorio Astronomico d’Abruzzo, Via M. Maggini snc, I-64100,
Teramo, Italy
E-mail<EMAIL_ADDRESS>(LL)E-mail<EMAIL_ADDRESS>(YW)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The jet composition of gamma-ray bursts (GRBs), as well as how efficiently the
jet converts its energy to radiation, are long-standing problems in GRB
physics. Here, we reported a comprehensive temporal and spectral analysis of
the TeV-emitting bright GRB 190114C. Its high fluence ($\sim$
4.436$\times$10-4 erg cm-2) allows us to conduct the time-resolved spectral
analysis in great detail and study their variations down to a very short time-
scale ($\sim$0.1 s) while preserving a high significance. Its prompt emission
consists of three well-separated pulses. The first two main pulses ($P_{1}$
and $P_{2}$) exhibit independently strong thermal components, and 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 interpret the observation. By combining the thermal ($P_{1}$ and $P_{2}$)
and the non-thermal ($P_{3}$) observations based on two different scenarios
(global and pulse properties) and following the method described in Zhang et
al. (2021), we measure the fireball parameters and GRB radiative efficiency
with little uncertainties for this GRB. A relevantly high GRB radiative
efficiency is obtained based on both the global and pulse properties,
suggesting that if GRBs are powered by fireballs, the efficiency can be high
sometimes. More interestingly, though the observed parameters are individually
different (e.g., the amount of mass loading $M$), the radiative efficiency
obtained from $P_{1}$ ($\eta_{\gamma}=36.0\pm 6.5\%$) and $P_{2}$
($\eta_{\gamma}=41.1\pm 1.9\%$) is roughly the same, which implies that the
central engine of the same GRB has some common properties.

###### keywords:

gamma-ray burst: general, gamma-ray burst: individual: 190114C, radiation
mechanisms: non-thermal, radiation mechanisms: thermal

††pubyear: 2023††pagerange: GRB 190114C: Fireball Energy Budget and Radiative
Efficiency Revisited–LABEL:lastpage

##  1 Introduction

On 14 January 2019 at 20:57:02.63 Universal Time (UT) (hereafter $T_{0}$), an
ultra-bright burst, GRB 190114C, was first detected by Gamma-ray Burst Monitor
(GBM) onboard the NASA Fermi Gamma-ray Space Telescope (Hamburg et al., 2019)
and the Neil Gehrels Swift Observatory’s Burst Alert Telescope (BAT) (J.D.
Gropp et al., 2019), and soon after Konus-_Wind_ , _AGILE_ /MCAL, _INTEGRAL_
/SPI-ACS, and the _Insight-HXMT_ /HE were triggered as well. Most
interestingly, for the first time the teraelectronvolt (TeV) emission was
detected by the Major Atmospheric Gamma Imaging Cherenkov (MAGIC) telescopes
from $T_{0}$+57 s until $T_{0}$+15912 s, and its rich multi-wavelength
observations were simultaneously observed from optical (Bolmer & Schady, 2019;
Castro-Tirado et al., 2019; Alexander et al., 2019; Tremou et al., 2019) to
TeV (MAGIC Collaboration et al., 2019a) gamma-ray emissions. The duration
($T_{90}$), the time is taken to accumulate $90\%$ of the burst fluence
starting at the 5% fluence level, reported by Fermi term is $\sim$116 s and
therefore belong to the long-duration burst class. The 1024 ms peak flux and
the fluence during $T_{90}$ duration at 10-1000 keV measured by Fermi-GBM are
246.864$\pm$0.859 photon cm-2 s-1 and (4.436$\pm$0.005)$\times$10-4 erg cm-2,
respectively. The measurement of redshift, $z$=0.424, is announced by Castro-
Tirado et al. (2019). The isotropic energy with a $k$-correction to the rest-
frame (1-104 keV), therefore, is estimated, $E_{\gamma,\rm
iso}$=(2.48$\pm$0.22)$\times$1053 erg. The prompt emission light curve
consists of three well-separated emission pulses. The first emission episode
(i.e., $P_{1}$) starts at $T_{0}$ and lasts for $\sim$2.35 s, the second
emission episode (i.e., $P_{2}$) exhibits multiple peaks and lasts from
$T_{0}$+2.35 s to $T_{0}$+15 s and slightly overlapping with $P_{1}$, and the
significantly fainter third emission episode (i.e., $P_{3}$) extends from
$T_{0}$+15 s to $T_{0}$+25 s. The first two emission episodes have very hard
$\alpha$ indices, the majority of the $\alpha$ indices in $P_{1}$ and $P_{2}$
beyond the line-of-death of synchrotron emission (-2/3, Preece et al., 1998),
while the third emission episode pulse has relatively soft $\alpha$ indices,
indicating a transition from fireball (thermal, $P_{1}$ and $P_{2}$) to
Poynting-flux-dominated (nonthermal, $P_{3}$) outflow. The afterglow emission
measured by Swift-XRT starts at $\sim$ $T_{0}$+68 s. The first GeV photon was
observed by Fermi-LAT at $T_{0}$+2.1 s and the highest-energy photon is a 22.9
GeV event which is observed at $T_{0}$+15 s (Wang et al., 2019).

Thermal photons of the prompt emission generated by the photosphere are
recognized as one of the leading radiative process in GRB physics (e.g.,
Goodman, 1986; Paczynski, 1986; Pe’er et al., 2007). The violation of the
synchrotron emission limit encourages us to search for an additional thermal
component (Li et al., 2022). A highly detailed time-integrated and time-
resolved spectral analysis for GRB 190114C has been presented in a recent
study (Li et al., 2022), its high fluence ($\sim$ 4.436$\times$10-4 erg cm-2)
allows us to conduct the time-resolved spectral analysis in great detail and
to study their variations down to a very short time-scale ($\sim$0.1 s) while
preserving a high significance. By carrying out the detailed spectral analysis
for GRB 190114C, Li et al. (2022) reported that the spectra in both $P_{1}$
and $P_{2}$ exhibit significant deviation from a single Band function (Band et
al., 1993) and can be best fitted using two components: a sub-dominant thermal
(blackbody, hereafter BB) component (e.g., Ryde, 2004; Guiriec et al., 2015;
Li, 2022a), and is accompanied by a dominant non-thermal (Band-like)
component, suggesting the existence of a strong thermal component. Indeed, the
thermal component confidently presents in the time bins from $T_{0}$+0.55 s to
$T_{0}$+1.93 s in $P_{1}$ and from $T_{0}$+2.45 s to $T_{0}$+5.69 s in
$P_{2}$, covering the peaks of the initial two emission pulses, and precisely
corresponds to the episodes that $\alpha$ indices beyond the synchrotron limit
(Li et al., 2022). More interestingly, the two thermal components between the
well-separated thermal pulses ($P_{1}$ and $P_{2}$) evolve independently,
inferred from their observational and physical parameters derived from the
standard fireball model (Pe’er et al., 2007). A large percentage of $\sim$20%
energy of thermal presenting in the gamma-ray prompt emission, making it one
of the most thermal-prominent _Fermi_ GRBs. The strong and independent pulse-
wise thermal components observed in the time-resolved spectral analysis of GRB
190114C make it a good case to study the photosphere properties, allowing 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. These
observational features found in GRB 190114C strongly support the evidence of
shell-like structure during the prompt emission phase and give a good
opportunity to study GRB ejecta composition and GRB radiation efficiency as
well.

The radiative efficiency of a burst is another interesting subject related to
the GRB prompt emission mechanism, which describes how efficiently the jet
converts its energy to radiation. 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 calculate GRB radiative efficiency ($\eta_{\gamma}$) 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
$E_{\gamma}$ can be directly measured using spectral parameters. The $E_{\rm
k}$ term, on the other hand, cannot be directly measured from observations.
The traditional method (e.g., Zhang et al., 2007; Wang et al., 2015) used the
afterglow data through modeling to estimate its value, but the estimated value
typically carries large uncertainties since it depends on many uncertain shock
microphysics parameters, primarily $\epsilon_{e}$ (Freedman & Waxman, 2001),
but also $\epsilon_{\rm B}$ and electron spectral index $p$ (Zhang et al.,
2007; Wang et al., 2015), where $\epsilon_{e}$ and $\epsilon_{B}$ are the
energy fractions assigned to electrons and magnetic fields $\epsilon_{B}$,
respectively. This resulted in high uncertainties in the derived GRB radiative
efficiency, ranging from below 10% to more than 90% (Zhang et al., 2007; Wang
et al., 2015; Li et al., 2018). By combining the prompt emission photosphere
emission data 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
observable 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
density ($\eta$), bulk Lorentz factors at the photosphere radius ($\Gamma_{\rm
ph}$), and before fireball deceleration ($\Gamma_{0}$), the amount of mass
loading ($M$), and 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. 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 a result, GRB
radiative efficiency $\eta_{\gamma}\equiv E_{\gamma}/(E_{\gamma}+E_{\rm K})$
can be also derived (Lloyd-Ronning & Zhang, 2004).

By combining the prompt emission and assuming that the afterglow emission
starts at $\sim$ 6 s as 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), Li et al. (2022) first applied this
new method (Zhang et al., 2021) to GRB 190114C, and directly measured all the
parameters for the first time (see Table 6 in Li et al. 2022). The measured
parameters at different emission sites are consistent with the expectation of
a fireball, and such a feature clearly exhibits a whole picture of self-
consistency from early prompt to late afterglow emission. GRB 190114C,
therefore, exhibits the evolution of a textbook relativistic fireball. In this
paper, we show that $P_{3}$ could also be explained as it originating from the
synchrotron self-Compton pulse from the reverse shock afterglow emission,
indicating the onset signature of afterglow emission. Using the strong thermal
components observed in $P_{1}$ and $P_{2}$, along with the early afterglow
data from $P_{3}$, we can recalculate all parameters. On the other hand, we
may also perform a similar pulse-wise analysis if we take into account the
distinct thermal components that were seen in $P_{1}$ and $P_{2}$ and make the
assumption that the emission before and after $P_{3}$ may represent the
afterglow emission component corresponding to the two initial pulses $P_{1}$
and $P_{2}$.

The paper is organized as follows. The spectral analysis is presented in
Section 2. The methodology apply to GRB 190114C based on their global and
pulse-wise properties are presented in Section 3. The discussions are
presented in Section 4 and our conclusions are summarized in Section 5.
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 Time-integrated and Time-resolved Spectral Analysis

We first perform the time-integrated spectral analysis with the entire
duration ($T_{90}$) of the burst (i.e., from $T_{0}$ \+ 0 s to $T_{0}$ \+ 116
s). The background is fitted with polynomial functions with the determined
polynomial order (0-4) by applying a likelihood ratio test, using the two off-
source time intervals, the intervals pre- (-20 s to -10 s) and post- (180 to
200 s) the burst. We select two brightest NaI detectors (n3, n4) to obtain an
angle of incidence less than 60 degrees (Goldstein et al., 2012), as well as
one BGO detector (b0) with the lowest angle of incidence. The Time-Tagged
Event (TTE) data type is used for NaI data (8 keV - 1 MeV) and BGO data (200
keV - 40 MeV). Following the standard practice (Li et al., 2019; Li, 2019b, a;
Yu et al., 2019; Burgess et al., 2019; Li, 2020; Li & Zhang, 2021; Li et al.,
2021; Li, 2022b, a) provided by _Fermi_ team, the spectral analysis is carried
out by using the Bayesian iterations of Markov Chain Monte Carlo (MCMC), which
are performed by a Python package Multi-Mission Maximum Likelihood Framework
(3ML, Vianello et al. 2015). To search for the best model representing the
spectral shape, we first fit the spectrum by examining various frequently used
spectral models by performing a detailed spectral analysis and model
comparisons. The utilized models include PL, CPL, Band, SBKL, PL+BB, CPL+BB,
PL+Bandcut, and Band+BB, respectively111We use the following abbreviations:
simple power-law (PL), CPL (cutoff power-law), smoothly broken power law
(SBKPL), and BB (blackbody).. Our refined spectral analysis suggests that the
CPL+BB (or the Band+BB) fitting is much better than the others, its AIC and
BIC scores are at least dozens of points lower than other models. This is
quite surprising and it indicates that adding a thermal component largely
improves the spectral fitting.

We further conduct the time-resolved spectral analysis from -1.0 s to 150 s
covering the $T_{90}$ duration. The time bins are selected by using the
Bayesian blocks method (BBlocks, Scargle et al. 2013) to TTE light curve of
the most strongly illuminated GBM detector (n4), other used detectors are
binned in matching time bins. In total, 49 time bins of spectra are obtained.
We first use the typical GRB spectral model the CPL model to fit the data, and
to obtain the temporal evolution of the low energy photon index $\alpha$. The
first result we find is that the temporal evolution of the power-law photon
index $\alpha$ (below the spectral energies peak) obtained from the CPL model
is very unusual. It is very hard in the first 6 s, covering the peaks of the
first two pulses and significantly violating the optically thin synchrotron
limit (Preece et al., 1998), then it crosses the synchrotron limit with a
hard-to-soft trend, several spectra have $\alpha$ close to 0 (see Figure 1).
Eventually, it is softer than the fast cooling limit of -3/2 (Preece et al.,
1998). The break of the fast-cooling limit at a later time infers the GRB has
entered the afterglow phase, and the cutoff energy is found to be $<$100 keV
after 50 s. Observing such a wide span of photon index in a single GRB is
unique, especially since the breaks occur in sequences of time bins and are
confirmed by data of high statistical significance ($S>$20). The second result
we find is that a time-integrated fitting of the whole $T_{90}$ shows the
consistency, $\sim$20% of the Fermi-GBM gamma-ray energy is in thermal, and
the average temperature is greater than 130 keV. This high percentage of
thermal flux reminds the GRB 090902B, which is famous for the intense thermal
emission appearing in the first half of the prompt emission. Such a value is
believed to be one of the highest on record for thermal emission in the hybrid
spectral case (a dominated non-thermal emission component with a sub-dominated
thermal component).

##  3 Directly Deriving the fireball parameters and radiative efficiency

###  3.1 The Method

Zhang et al. (2021) proposed a new method to directly dissect the GRB fireball
energy budget into three components and measure their values. The method
requires a GRB observable 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 density ($\eta$), bulk Lorentz factors at the photosphere radius
($\Gamma_{\rm ph}$), and before fireball deceleration ($\Gamma_{0}$), the
amount of mass loading ($M$), and 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. Following is a brief description of the relevant
calculations.

The initial, total energy of a fireball is

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

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

$E_{\rm th}=(\eta-\Gamma_{\rm ph})Mc^{2},$ (3)

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

$E_{\rm nth}=(\Gamma_{\rm ph}-\Gamma_{0})Mc^{2},$ (4)

where $\Gamma_{0}$ 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)

$\begin{split}\Gamma_{\rm ph}&=\left[(1+z)^{2}D_{\rm L}\frac{{\cal
Y}\sigma_{\rm T}F^{\rm obs}_{\gamma}}{2m_{p}c^{3}{\cal
R}}\frac{\eta^{3/2}}{\eta-\Gamma_{0}}\right]^{2/9},\\\ {\cal
R}&=\left(\frac{F^{\rm obs}_{\rm BB}}{\sigma_{\rm
B}T^{4}}\right)^{1/2}.\end{split}$ (5)

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 $\Gamma_{0}$ 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

$\begin{split}\Gamma_{0}\simeq 0.9^{3/8}\left(\frac{3E_{\rm
k}(1+z)^{3}}{2\Pi\hat{\gamma}nm_{\rm p}c^{5}t^{3}_{\rm dec}}\right)^{1/8}\\\
\simeq 170t_{\rm
dec,2}^{-3/8}\left(\frac{1+z}{2}\right)^{3/8}\left(\frac{E_{\gamma,52}}{n}\right)^{1/8}\left(\frac{\Gamma_{0}}{\eta-\Gamma_{0}}\right)^{1/8}.\end{split}$
(6)

where $\hat{\gamma}$(=4/3) is the numerical coefficient, $n$ is the ISM
density assumed as one particle per cubic centimetre as usual, $t_{\rm dec}$
is the peak time of the third pulse since we assume that this point has
already in the deceleration time, and $E_{\rm k}$ is the isotropic blastwave
kinetic energy $E_{\rm K,iso}$ as we have calculated above, and
$E_{\gamma,52}=E_{\rm th,52}+E_{\rm nth,52}$.

Simultaneously solving Eqs.(3–6), we obtain the fireball parameters $\eta$,
$\Gamma_{\rm ph}$, $M$ and $\Gamma_{0}$ (see Table GRB 190114C: Fireball
Energy Budget and Radiative Efficiency Revisited), and in turn, we can
calculate the kinetic energy of the afterglow

$E_{k}=\Gamma_{0}Mc^{2},$ (7)

and the efficiency of the prompt gamma-ray emission (see Table GRB 190114C:
Fireball Energy Budget and Radiative Efficiency Revisited)

$\eta_{\gamma}=\frac{E_{\rm th}+E_{\rm nth}}{E_{\rm
tot}}=\frac{\eta-\Gamma_{0}}{\eta}.$ (8)

###  3.2 Global properties

####  3.2.1 The Deceleration of Pulse $P_{3}$: Synchrotron Self-Compton Pulse
from the Reverse Shock Afterglow Emission

The shell-merged fireball continues to move approximately at the Lorentz
factor $\Gamma_{0}$ and then decelerates when the swept ISM mass equals to its
kinetic energy. This moment symbolizes the starting of the afterglow phase,
accompanied by a bulging pulse on the light curve and the softening of the
spectral index. This apparently coincides with the third weal pulse $P_{3}$ of
which the time-resolved spectral indices (see Figure 1) ($\alpha\sim$ -1.5 to
-2) are much softer than those in the first two pulses ($P_{1}$ and $P_{2}$),
but are consistent with the typical values of synchrotron radiation of
afterglow origin.

Considering that the observational properties in $P_{3}$ are likely to be
consistent with the theoretical model of synchrotron self-Compton emission
from the reverse shock (e.g., Mészáros & Rees, 1997; Sari & Piran, 1999;
Kobayashi & Zhang, 2003; Wei, 2003a), we try to check the interpretation that
the emission of the third $P_{3}$ pulse (during the time intervals between 15
s and 25 s) as from the optical photons of the reverse shock up-scattered to
the X- and $\gamma$-rays.

Before performing the calculation of some feature quantities, we first
determine that the reverse shock of this burst can be simplified to the thick
shell, as $\Delta$ is about two times of $l/2\Gamma_{0}^{8/3}$, where
$\Delta=cT_{\rm prompt}$ is the shell width approximately from the product of
the speed of light and duration of prompt emission, and $l=(3E_{\rm k}/4\pi
nm_{\rm p}c^{2})^{1/3}$ is the Sedov length. The duration of $P_{3}$ ($\sim
10$ s) is close to the prompt duration ($\sim 6-10$ s), which confirms the
thick shell approximation. We also need to notice that $\Delta$ is not much
larger than $l/2\Gamma_{0}^{8/3}$, this is in line with that $P_{3}$ is a
faint pulse, its energy released is $E_{\gamma,\rm iso}\approx 1.20\times
10^{52}$ erg, taking up a very small portion ($1.5\%$) of the kinetic energy
$E_{\rm k,iso}$ = (7.8$\pm$0.6) $\times$ 1053 erg, which enables us to
simplify the deceleration Lorentz factor $\Gamma_{\rm d}\simeq\Gamma_{0}$.

The ratio of electron and magnetic field equipartition parameters
$\epsilon_{\rm e}/\epsilon_{\rm B}=132\gg 1$ indicates an SSC-dominated
regime, where $\epsilon_{\rm B}$ and $\epsilon_{\rm e}$ are the fractions of
the shock energy transferred to the magnetic field and electrons,
respectively. For a reverse shock, with $\Gamma_{\rm d}>500$, the first-order
SSC normally up-scatters the photons to keV/MeV, and the second-order IC
boosts photons again to the GeV range. Observationally, the luminosity of
photons above GeV is below the keV/MeV luminosity, suggesting that the second-
order SSC is suppressed by the Klein-Nishina effect. Hence, we may consider
only the first-order IC and derive the ratio of Compton scattering parameter
(Sari & Esin, 2001) $x=(\epsilon_{\rm e}/\epsilon_{\rm B})^{1/2}=12$.

Reverse shock has been widely studies, for similar previous cases and the
related theories, we refer to Wang et al. (2001a, b); Wei (2003b); Kobayashi
et al. (2007); Urata et al. (2014) and the reference therein. Our applied
model generally follows Wang et al. (2005), in which the reverse-shock
synchrotron self-Compton emission is modelled and applied, and equations are
mainly adopted from Wei (2003b); Wang et al. (2005).

A contact discontinuity separates the reverse and forward shocks, maintaining
equal pressure on both sides. Hence, the typical frequencies and the flux from
the synchrotron emission of the reverse shock are connected to the
corresponding values of the forward shock. Assuming the magnetic fields are
the same in both shocked, we have

$\begin{split}\nu_{\rm m,r}&\simeq\Gamma_{\rm d}^{-2}\nu_{\rm m,f}\\\
&=3.89\times 10^{15}\left(1+z\right)^{1/2}(\frac{p-2}{p-1})^{2}\Gamma_{\rm
d,3}^{2}\epsilon_{\rm B,-2}^{1/2}\epsilon_{\rm e,-1}^{2}n_{0}^{1/2}\;\;\;{\rm
Hz}\\\ &=0.62\;\;\;{\rm eV}\end{split}$ (9)

$\begin{split}\nu_{\rm c,r}&=\nu_{\rm c,f}\\\ &=7.94\times
10^{16}\left(1+z\right)^{-1/2}\epsilon_{\rm
B,-2}^{-3/2}E_{k,52}^{-1/2}n_{0}^{-1}t_{\rm dec,2}^{-1/2}\;\;\;{\rm Hz}\\\
&=2.9\;\;\;{\rm keV}\end{split}$ (10)

where f and r denote forward and reverse respectively. The cooling frequency
is much larger than the minimal frequency ($\nu_{\rm c,r}\gg\nu_{\rm m,r}$).
These frequencies are up-scattered to

$\displaystyle\nu_{\rm m,r}^{\rm IC}$ $\displaystyle\simeq$
$\displaystyle\gamma_{m}^{2}\nu_{m,r}=13~{}\mbox{keV},$ (11)
$\displaystyle\nu_{\rm c,r}^{\rm IC}$ $\displaystyle\simeq$
$\displaystyle\gamma_{c}^{2}\nu_{c,r}=277~{}\mbox{MeV},$ (12)

where

$\displaystyle\gamma_{m}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{p-2}{p-1}\right)\epsilon_{e}\left(\frac{m_{p}}{m_{e}}\right)=145,$
(13) $\displaystyle\gamma_{c}$ $\displaystyle\simeq$
$\displaystyle\gamma_{m}\left(\frac{\nu_{c}}{\nu_{m}}\right)^{1/2}=9843$ (14)

Here $m_{\rm p}$ and $m_{\rm e}$ are the masses of the proton and the
electron, respectively. The flux between $\nu_{\rm m}^{\rm IC}$ and $\nu_{\rm
c}^{\rm IC}$ drops as $\sim t^{-(3p+1)/3}$, where $p$ is the power-law index
of electron distribution, from the Fermi-GBM observations, the flux drops as
$\sim t^{-3.3}$, so $p=2.9$ is derived and applied to obtain the above
$\nu_{\rm m}$ and $\nu_{\rm c}$ values. The flux density at the $\nu_{\rm m}$
is given by

$\begin{split}f_{\rm\nu_{m},r}&\simeq\Gamma_{d}f_{\rm\nu_{m},f}\\\
&=11\left(1+z\right)\Gamma_{d,3}\epsilon_{\rm
B,-2}^{1/2}E_{k,52}n_{0}^{-1}D_{\rm L,28}^{-2}\;\;\;{\rm Jy}\\\
&=185\;\;\;{\rm Jy}\end{split}$ (15)

where $D_{\rm L}$ is the luminosity distance of the source. It corresponds to
the SSC flux

$\begin{split}\nu_{\rm m,r}^{\rm IC}f_{\nu_{\rm m,r}^{\rm IC}}\simeq x\nu_{\rm
m,r}f_{\nu_{\rm m,r}}=5.7\times 10^{-6}\;\;\;{\rm
erg~{}cm^{-2}~{}s^{-1}}\end{split}$ (16)

The above results suggest that the modeled SSC emission from the reverse shock
in the keV range and the flux in the order of $10^{-6}$ erg cm-2 s-1 is
consistent with observation of $P_{3}$.

The optical afterglow data also offer support to this interpretation. Even
though there is no early optical data before $\sim 33$ s, the optical
lightcurve fitting requires a steeper slope ($\sim-1.76$) early on before
transitioning to a normal slope ($\sim-1$), consistent with the superposition
of a reverse shock component with the forward shock component222Throughout the
paper, the convention $F_{t,\nu}=t^{-\alpha}\nu^{-\beta}$ is applied, where
$\alpha$ and $\beta$ are the temporal and spectral indices, respectively..

In Figure 2, we show the log ($F$)-log($t-t_{0}$) plots with a temporal fit to
the decay phase of the third peak using the single power-law model (typical
model for afterglow emission), setting $t_{0}$ at the trigger time
(log($t-t_{\rm trigger}$), which is physically motivated to study external-
shock-powered light curves (Kobayashi & Zhang, 2007). We get $F\propto
t^{-3.32\pm 0.49}$ s. The obtained decay index ($-3.32\pm 0.49$) in this way
is much steeper than the typical values observed from a forward shock emission
in afterglow, but is in good agreement with a reverse shock afterglow emission
predicted by the model discussed above. Therefore, $P_{3}$ could be originated
from a reverse shock.

In addition, the above computation takes $n=1$, and the adopted parameters
such as $\epsilon_{\rm e}$, $\epsilon_{\rm B}$ and $E_{\rm k}$ are all derived
from $n=1$. Hence, the resulting consistency with the observation of $P_{3}$
proves that $n$ is indeed $\sim$ 1.

####  3.2.2 Results

Following the physical picture that the central engine releases two sequences
of energy forming $P_{1}$ and $P_{2}$, which merge and enter the afterglow
phase generating $P_{3}$, we can directly derive the fireball parameters and
GRB radiative efficiency for GRB 190114C following the method described in
Zhang et al. (2021). We first perform a spectral fit using the CPL+BB model
(see Section 2) by treating both thermal components observed in $P_{1}$ and
$P_{2}$ as a whole. The observed parameters obtained from the spectral
analysis, therefore, include the isotropic equivalent thermal energy $E_{\rm
th,iso}$ and isotropic equivalent non-thermal energy $E_{\rm nth,iso}$, the
thermal $F^{\rm obs}_{\rm BB}$ and total $F^{\rm obs}_{\gamma}$ energy flux,
and the average temperature $kT^{\rm obs}$ (see Table GRB 190114C: Fireball
Energy Budget and Radiative Efficiency Revisited). The deceleration time
$t_{\rm dec}$ ($t_{\rm p}$) is measured from the third pulse assuming it
originates from the synchrotron Self-Compton pulse from the reverse shock
afterglow emission (see Section 3.2.1). The peak time ($t_{\rm p}$) is
determined using the fast-rising and exponential decay (FRED) model (Kocevski
et al., 2003) to fit the 1024 ms counts lightcurve of the third pulse (see
Table GRB 190114C: Fireball Energy Budget and Radiative Efficiency Revisited).
By combining the thermal component in the first two pulses with a non-thermal
component in the third pulse (see the left panel of Figure 1) and following
the method proposed in Zhang et al. (2021), one can directly determine the
fireball characteristics. The measured parameters include the initial
dimensionless specific enthalpy density ($\eta$), bulk Lorentz factors at the
photosphere radius ($\Gamma_{\rm ph}$) and before fireball deceleration
($\Gamma_{0}$), the amount of mass loading ($M$), the kinetic energy in the
fireball $E_{\rm K}$, and GRB radiative efficiency ($\eta_{\gamma}$).

In this scenario, the prompt emission of GRB 190114C lasts from $0$ s to $15$
s, and thermal emission is prominent during $0.55$ s-$1.93$ s and $2.45$
s-$5.69$ s. With the observed properties (see the upper panel of Table GRB
190114C: Fireball Energy Budget and Radiative Efficiency Revisited), and
substituting all these values in Equations (2-8), we obtain the fireball
characteristics (see the middle panel of Table GRB 190114C: Fireball Energy
Budget and Radiative Efficiency Revisited), where $M_{\odot}$ is the mass of
the sun ($1.9891\times 10^{33}\rm~{}g$). The derived fireball parameters
consist of the dimensionless specific enthalpy density at the engine $\eta$,
the bulk Lorentz factor at the site of the photosphere $\Gamma_{\rm ph}$, the
initial afterglow Lorentz factor before the deceleration phase $\Gamma_{0}$,
the isotropic equivalent total mass $M_{\rm iso}$, the kinetic energy in the
fireball $E_{\rm k,iso}$, and the $\gamma$-ray radiative efficiency
$\eta_{\gamma}$, as well as the energy fractions assigned to electrons
($\epsilon_{e}$) and magnetic ($\epsilon_{\rm B}$) fields, the characteristic
synchrotron frequency ($\nu_{\rm m}$) and the cooling frequency ($\nu_{\rm
c}$) of minimum-energy injected electrons, and the Klein-Nishina frequency
($\nu_{\rm KN}$). The measured quantities from observations and the derived
fireball parameters using our new methods with assuming ${\cal Y}=1$ and
$n=1~{}cm^{-3}$. We note that (1) we assume that at the time of $P_{3}$ the
swept mass from the environment is much less than the mass ejected from the
central engine, this assumption is validated by the following numerical
result. The mass ejected from the black hole ($M$= 8.6$\times$10-4$M_{\sun}$)
is much more massive than the mass swept from the ISM within the first 20 s
($\sim 10^{-4}M_{\sun}$), which verifies the feasibility of ignoring the swept
mass during the above modeling. (2) A relevantly high GRB radiative efficiency
is obtained, $\eta_{\gamma}=(28.3\pm 1.4)\%$, suggesting that if GRBs are
powered by fireballs, the efficiency can be high sometimes.

###  3.3 Pulse-wise Properties

####  3.3.1 Independent Thermal Pulses: the First Pulse $P_{1}$ and the Second
Pulse $P_{2}$

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 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).

The observational and derived photosphere parameters (see Section A0.1 for
details), including the blackbody temperature (k$T$), the effective radius
parameter ($\Re$), the bulk Lorentz factor ($\Gamma$), the characteristic
radii (photospheric radius $R_{\rm ph}$, the nozzle radius $R_{\rm 0}$, and
saturation radius $R_{s}$), in fact, all evolve with time. We study the
thermal components in the two well-separated pulses ($P_{1}$ and $P_{2}$) by
comparing their observed properties. An interesting finding in Li et al.
(2022) is that both the observational and the derived photosphere properties
support the evidence that $P_{1}$ and $P_{2}$ exhibit independently pulse-wise
features. k$T$, $\Re$, $\Gamma$, $R_{\rm ph}$, $R_{\rm 0}$, and $R_{s}$,
obtained from the thermal components in $P_{1}$ and $P_{2}$, evolve
independently (see Li et al. 2022 for detail), suggesting the existence of two
distinct thermal pulses. For instance, in both thermal components in $P_{1}$
and $P_{2}$, the blackbody temperature k$T$ first decays with a power law
before sharply plunging in the end. The photospheric radius $R_{\rm ph}$, and
the radius at which the jet is launched to $R_{0}$ are estimated, and the
saturation radius $R_{\rm s}$ consequently is given by $R_{\rm s}\equiv\Gamma
R_{\rm 0}$. All these characteristic radii ($R_{\rm ph}$, $R_{0}$ and $R_{s}$)
increase with time. The results are quite interesting since such distinct
pulse-wise temporal features have never been clearly seen before. Ryde (2004)
and Ryde (2005) studied the properties of thermal components with a large
BATSE sample and found that k$T$ is approximately constant early on, it decays
as a power-law later on, while $\Re$ exhibits a power-law over the whole
pulse. Such general behavior is universal even for bursts with complex and
heavily overlapping light curves. Thermal flux ratio ($F_{\rm BB}/F_{\rm
tot}$), on the other hand, has no such clear trend. In $P_{1}$, it has a
relatively low value ($\sim 10\%$), but increases to a relatively high value
($\sim 20\%$) quickly. However, in $P_{2}$, it is significantly stronger and
keeps almost a constant value ($\sim 20\%$) during the entire thermal
duration. Additional details regarding the observable and photospheric
properties of GRB 190114C are presented in Li et al. (2022).

####  3.3.2 Possible Afterglow Signifies from $P_{1}$ and $P_{2}$

GRB 190114C consists of two individual jets as indicated by the independent
evolution of thermal emission in $P_{1}$ and $P_{2}$. Correspondingly, we
expect the deceleration time of the first jet $t_{\rm dec}(P_{1})$ to be
earlier than the deceleration time of the second jet $t_{\rm dec}(P_{2})$.
Observationally speaking, at the time when the early light curve reaches a
shallow decay phase of power-law decay index $\sim$ -1 to -1.5 and the
spectral index approximates $\sim-2$, the jet is significantly decelerated.
GRB 190114C exhibits two-time segments showing such behavior, the first one
($S_{1}$) appears after the drop of $P_{2}$ and the second one ($S_{2}$)
connects to the end of $P_{3}$. From fitting the light curve and spectra,
shown in Figure 1 and Figure 2, $S_{1}$ decays following a power-law $\sim-2$
and the spectral indices within its duration vary from $\sim-1.6$ to
$\sim-1.9$, $S_{2}$ starts at $\sim 22$ s and continues for the later time,
the spectral indices are all $\sim-1.9$. The proposal that $S_{1}$ and $S_{2}$
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). We, therefore, propose that the
emissions of these two segments are produced by the two decelerated jets
respectively. We notice that $S_{1}$ has a steeper light-curve decay and a
slightly harder spectral shape than the typical value mentioned above. This
could be due to the influence of the tail of $P_{2}$. A simple extrapolation
of $S_{1}$ to the time of $S_{2}$ shows the flux of $S_{2}$ is $\sim 3-5$
times higher than the extrapolated flux of $S_{1}$. This is consistent with
the fact that the energy emitted in $P_{2}$ is more than $3$ times larger than
the energy emitted in $P_{1}$.

####  3.3.3 Results

Following the physical picture in which the central engine releases two
sequences of energy forming $P_{1}$ and $P_{2}$ as suggested in Section 3.3,
and possibly two independent afterglow signatures ($S_{1}$ and $S_{2}$)
corresponding to $P_{1}$ and $P_{2}$ as suggested in Section 3.3.2, we can
recalculate all the parameters based on their pulse properties (see the right
panel of Figure 1). Based on our refined spectral analysis for each thermal
pulse (see Section 2), we obtain all the relevant observed parameters (see the
upper panel of Table GRB 190114C: Fireball Energy Budget and Radiative
Efficiency Revisited), including the isotropic total ($E_{\gamma_{1}}$ and
$E_{\gamma_{2}}$), thermal ($E_{\rm th_{1}}$ and $E_{\rm th_{2}}$), and
nonthermal ($E_{\rm nth_{1}}$ and $E_{\rm nth_{2}}$) energy, the total
($F^{\rm obs}_{\gamma_{1}}$ and $F^{\rm obs}_{\gamma_{2}}$), and thermal
($F^{\rm obs}_{\rm BB_{1}}$ and total $F^{\rm obs}_{\rm BB_{2}}$) energy flux,
and blackbody component temperature (k$T_{1}$ and k$T_{2}$). By using these
pulse-based observed parameters and individually applying the procedure
proposed in Zhang et al. (2021) for each pulse, we can obtain all the fireball
parameters for both $P_{1}$ and $P_{2}$. The measured parameters, including
the initial dimensionless specific enthalpy density ($\eta_{1}$ and
$\eta_{2}$), bulk Lorentz factors at the photosphere radius ($\Gamma_{\rm
ph,1}$ and $\Gamma_{\rm ph,2}$) and before fireball deceleration
($\Gamma_{0,1}$ and $\Gamma_{0,2}$), the amount of mass loading ($M_{1}$ and
$M_{2}$), and GRB radiative efficiency ($\eta_{\gamma,1}$ and
$\eta_{\gamma,2}$), are presented in the lower panel of Table GRB 190114C:
Fireball Energy Budget and Radiative Efficiency Revisited.

There are several remarks and results in order. (1) $E_{\gamma_{i}}$ is the
radiated energy in $\gamma$-ray for each pulse, consisting of two parts: the
MeV emission observed from _Fermi_ /GBM ($E^{\rm GBM}_{\gamma,\rm iso}$) and
the GeV emission observed from _Fermi_ /LAT ($E^{\rm LAT}_{\gamma,\rm iso}$).
(2) The amount of mass loading $M$ ($\sim$8.6 $\times$ 10-4 $M_{\odot}$)
derived from Section 3.2 is much more massive than the mass of the sum of the
two thermal pulses [($M_{1}$ \+ $M_{2}$) $\sim$4.5 $\times$ 10-4 $M_{\odot}$]
presented in this Section. This is because the time intervals of thermal
components in $P_{1}$ (from $t_{0}$+0.55 s to $t_{0}$+1.93 s) and $P_{2}$
(from $t_{0}$+2.45 s to $t_{0}$+5.69 s) that we used for the pulse properties
are less than the entire time interval (from $t_{0}$+0 s to $t_{0}$+15 s) of
prompt emission as we used for the global properties. (3) The derived $\eta$,
$\Gamma_{\rm ph}$, and $\Gamma_{0}$ values in $P_{1}$ are systematically
greater than those in $P_{2}$, and derived $M$, $E_{\rm K}$, and $E_{\rm tot}$
values in $P_{1}$ are systematically less than those in $P_{2}$, presenting a
self-consistent picture in time and space as expected by the standard fireball
model. (4) A relevantly high radiative efficiency is obtained for both $P_{1}$
[$\eta_{\gamma}=(36.0\pm 6.5)\%$] and $P_{2}$ [$\eta_{\gamma}=(41.1\pm
1.9)\%$], suggesting that if GRBs are powered by fireballs, the efficiency can
be high sometimes. More interestingly, GRB radiative efficiency obtained from
$P_{1}$ and $P_{2}$ is roughly the same, which supports that the central
engine of the same GRB has some common properties.

##  4 Discussion

GRB 190114C is important for the photosphere models, it is among the highest
value in all GRBs, that thermal contributes $\sim 20\%$ of the prompt
emission.

Why GRB 190114C is important for the photosphere model? If only talking about
the thermal significance, GRB 090902B is the most thermal dominant one up to
date, $\sim 70\%$ of its emission is thermal. GRB 190114C has a relatively
lower, but still among the highest value in all GRBs, that the thermal
contributes $\sim 20\%$ of the prompt emission. The superiority of 190114C is
attributed to the clear separation of its two prompt thermal pulses, while the
pulses in GRB 090902B and all others are highly overlapped. Therefore, if the
purpose is to investigate the activity of the central engine and the dynamics
of the photospheres’ evolution, GRB 190114C acts as the best example.

Why the high significance of thermal emission coincides with the high flux,
and with the violation of the synchrotron limit? The high total flux
corresponds to more photons and electrons, bringing higher opacity,
consequently higher photosphere radius, and more significant thermal emission.
The intense thermal Planck spectrum adheres to the power-law-like spectrum of
the non-thermal emission, forming a bump-like structure, which lifts up the
low-energy spectral index if fitting the spectrum by a power-law-like function
alone assuming only the synchrotron model. Indeed, the low-energy spectral
index is brought down when adding an additional thermal component to the
fitting.

Why the thermal percentage is almost invariable? Between $20\%$ and $40\%$
except the very beginning, regardless of the temperature varies from $50$ keV
or $200$ keV. One possible reason is the Compton up-scattering moderates the
ratio. This GRB occurs in one of the densest GRB environments, which has a lot
of hydrogen atoms, such an environment brings high opacity for photons, and
helps convert a high percentage of outflow energy to baryon kinetic energy. As
a result, thermal equilibrium maintains at a high radius, which is preferred
for producing observable thermal flux. The GeV originates from the photons in
the outflow up-scatter with the dense electrons of the environment, as
proposed by Beloborodov et al. (2014). The thermal photons may contribute to
the most of GeV photons. This expectation of GeV origin is supported by the
data, that prompt MeV and GeV to have a strong correlation, see Figure S8 in
Wang et al. (2019), and thermal intensity influences the MeV and GeV spectral
index, see Figure 6 and Figure S8 in Wang et al. (2019).

The initial radius $R_{0}$ changes significantly, it increases two orders of
magnitude. First, different from the Lorentz factor and the photospheric
radius, the evaluation of $R_{0}$ requires tracing the system at the
observational time back to the initial state, this procedure brings a lot of
uncertainty, especially influenced by the composition evolution and the
violation of the non-dissipative assumption. Second, all the formulae are
established for a single shell, without considering the energy exchange
between different shells, for instance, the photosphere emission of a later
shell injects to the shell ahead. Third, the effect of equal arriving time
surfaces. The initial time bin when the photosphere emission starts may escape
from the deviation brought by the second and third reasoning, the $R_{0}$
derived from it carries more confident information of black hole than the
others, and its value of $\sim 10^{6}$ cm is consistent with the Schwarzschild
radius of a typical GRB black hole of a few solar masses. $R_{0}$ derived from
the later slices may be considered as the effective width of the shell slices
(or width of photosphere), reasons as follows.

For the second possibility that the energy exchange between shells, the width
of the photosphere induced by the variation of the Lorentz factor is the
characteristic width in which the energy exchanges, so we can consider it as
the width of a slice of shell

$\Delta R_{\rm ph}=R_{\rm ph}\Delta\beta\simeq\frac{R_{\rm
ph}\Delta\Gamma}{\Gamma^{3}},$ (17)

the maximum value is taken at $\Delta\Gamma=\Gamma$, then we have

$\Delta R_{\rm ph}=10^{8}{\rm cm}~{}R_{\rm ph,12}~{}\Gamma^{-2}_{2}$ (18)

The obtained value is in the same order of $R_{0}$, $\Delta R_{\rm ph}$ is
$\sim 8$ times smaller than $R_{0}$, see Figure 3. This factor of $8$ can be
eliminated by changing $Y$, the total energy versus the observed energy, and
it is true that this GRB contains more energy than the isotropic energy in the
prompt emission since having strong GeV and TeV. $\Delta R_{\rm
ph}/R_{0}\propto Y^{5/4}$, so $Y$ needs to increase $5$ times (MAGIC
Collaboration et al. (2019b) computed the kinetic energy $\simeq 3\times
10^{55}$ erg, so $Y$ can be much bigger $\sim 100$, we can even have
$\Delta\Gamma=\Gamma^{1/2}$, which is reasonable. For the fittings $\Delta
R_{\rm ph}\propto\Gamma^{-3.57\pm 0.14}$ and $R_{0}\propto\Gamma^{-2.97\pm
0.45}$ (if $\Delta\Gamma=\Gamma^{1/2}$, the coincidence of power-law indices
is better), similar evolutionary behavior.

Another possible explanation, especially for the first time bin, is due to the
observational effect. The function of $R_{0}$ can be written as

$R_{0}\propto\frac{1}{T_{\rm obs}^{2}}\frac{F_{\rm BB}^{2}}{F^{3/2}},$ (19)

If we forget the meaning of $R_{0}$, just look at the right term, which is
small when the thermal flux is low and temperature is high. From the
observation, we know to have a confident thermal signature, the thermal
component needs to be luminous and occupy some fraction of the total flux,
otherwise, it is hidden in the data even if it exits. Also, we know only when
the total flux is high, the system may able to produce such an observable
thermal flux. Considering that the black hole suddenly ejects a lot of energy,
which meets the conditions discussed above, this is the first time that there
is a visible photosphere, and it is the only photosphere at that moment.
Therefore, it is quite possible to observe the first thermal emission at a low
photosphere radius with high temperature and low thermal ratio. As we can see
in Figure A6 in Wang et al. (2019), the photosphere radius of the first point
is the lowest and it is away from others. If we exclude the first two points
of $R_{0}$, others are mostly in the same order of magnitude. If we cut the
first and last points, the radii of Figure A6 in Wang et al. (2019) show no
clear trend of decreasing or increasing.

The fast drop in temperature of the last slices. This is a signal telling that
the non-dissipative assumption is not valid. The thermal energy in a shell is
not only dissipating, but it also receives energy injection from the shell
behind it. The last timebin presents a shell without energy injection, its
temperature decreases faster even than the exponential cut, this decreasing
behavior is much steeper than the non-dissipative prediction.

The discontinuity of temperature and Lorentz factor between two pulses. If two
pulses are separated, we expect the second one to be almost the repetition of
the first one. If they are from a continuous process, we expect the
continuities to appear in the evolution. Although their power-law trends are
similar, the value of the parameters is neither repetitive nor continuous.
Comparing the last points in the first pulse and the first points in the
second pulse, there is a difference but slightly, temperature increases,
Lorentz factor increases, and photosphere radius decreases in the first points
of the second one. It is more likely that two pulses are originally
independent, but still, the first one has an influence on the second one, or
the second one merges with the first cooled one.

##  5 Conclusions

In this paper, by using the uniqueness of observations of GRB 190114C and
applying the new method proposed in Zhang et al. (2021), we revisited
photosphere properties and GRB radiative efficiency as first studied in Li et
al. (2022) based on an assumption that the afterglow emission starts at $\sim$
6 s. The method described in Zhang et al. (2021) requires a GRB with a
dominant thermal spectral component, a deceleration bump feature in the early
afterglow lightcurve, and a measured redshift. One can directly dissect the
GRB fireball energy budget into three components and measure their values. The
observed parameters from prompt emission spectral analysis include the
isotropic equivalent thermal energy $E_{\rm th,iso}$ and the isotropic
equivalent non-thermal energy $E_{\rm nth,iso}$, the thermal $F^{\rm obs}_{\rm
BB}$ and total $F^{\rm obs}_{\gamma}$ energy flux, and the average temperature
$kT^{\rm obs}$; and from the earlier afterglow data include the deceleration
time $t_{\rm dec}$ ($t_{\rm p}$). As a result, the measured parameters include
the initial dimensionless specific enthalpy density ($\eta$), bulk Lorentz
factors at the photosphere radius ($\Gamma_{\rm ph}$), and before fireball
deceleration ($\Gamma_{0}$), the amount of mass loading ($M$), the kinetic
energy in the fireball $E_{\rm K}$, and GRB radiative efficiency
($\eta_{\gamma}$).

We considered two different physical pictures to revisit these properties. (1)
Considering that the third pulse $P_{3}$ exhibits distinctly different
temporal and spectral properties compared with the first two pulses ($P_{1}$
and $P_{2}$), it appears that $P_{3}$ is similar to the typical properties
observed in afterglow emissions. We, therefore, argued that $P_{3}$ may
originate from the reverse shock afterglow emission by giving several pieces
of additional evidence. By combining the prompt emission (by treating both
thermal components observed in $P_{1}$ and $P_{2}$ as a whole) and the earlier
afterglow emission ($P_{3}$), we derived all the parameters, and measured
radiative efficiency for GRB 190114C as $\eta_{\gamma}$ = (28.3$\pm$1.4)%. A
relevantly high GRB radiative efficiency is obtained, suggesting that if GRBs
are powered by fireballs, the efficiency can be high sometimes. (2) Following
the physical picture in which the central engine releases two sequences of
energy forming $P_{1}$ and $P_{2}$, and two possible afterglow signatures
($S_{1}$ and $S_{2}$) from $P_{1}$ and $P_{2}$, we then recalculated all the
parameters based on their pulse-wise properties. Two interesting results are
found based on their pulse properties. First, the derived $\eta$, $\Gamma_{\rm
ph}$, and $\Gamma_{0}$ values in $P_{1}$ are systematically greater than those
in $P_{2}$, and derived $M$, $E_{\rm K}$, and $E_{\rm tot}$ values in $P_{1}$
are systematically less than those in $P_{2}$, presenting a self-consistent
picture in time and space as expected by the standard fireball model. Second,
a relevantly high GRB radiative efficiency is obtained for both $P_{1}$
[$\eta_{\gamma}=(36.0\pm 6.5)\%$] and $P_{2}$ [$\eta_{\gamma}=(41.1\pm
1.9)\%$]. More interestingly, though the observed parameters are individually
different (e.g., the amount of mass loading $M$), GRB radiative efficiency
obtained from $P_{1}$ and $P_{2}$ is roughly the same, which implies that the
central engine of the same GRB has some common properties.

Acknowledgements. We thank the anonymous referee for his/her valuable comments
and suggestions. We also thank the ICRANet members for their useful
discussions. We especially thank Bing Zhang for many useful discussions that
greatly improved the paper. This research made use of the High Energy
Astrophysics Science Archive Research Center (HEASARC) Online Service at the
NASA/Goddard Space Flight Center (GSFC).

Data availability. The data underlying this article will be shared on
reasonable request to the corresponding author.

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Table 1 Global properties of GRB 190114C
Measured Parameters Isotropic equivalent thermal energy [$E_{\rm th,iso}$]
(6.5$\pm$0.5) $\times$ 1052 erg Isotropic equivalent non-thermal energy
[$E_{\rm nth,iso}$] (2.4$\pm$0.1) $\times$ 1053 erg Thermal energy flux
[$F^{\rm obs}_{\rm BB}$] (1.9$\pm$0.2) $\times$ 10-5 erg cm-2s-1 Total energy
flux [$F^{\rm obs}_{\gamma}$] (1.01$\pm$0.03) $\times$ 10-4 erg
cm${}^{-2}s^{-1}$ Deceleration time [$t_{\rm dec}$] 16.4$\pm$ 0.1 s
Temperature [$kT^{\rm obs}$] 144$\pm$2 keV Redshift [$z$] 0.4254$\pm$0.0005
Derived Parameters Dimensionless specific enthalpy density [$\eta$] 708$\pm$8
Bulk Lorentz factor at $r_{\rm ph}$ [$\Gamma_{\rm ph}$] 666$\pm$6 Initial
Lorentz factor [$\Gamma_{0}$] 507$\pm$5 Isotropic equivalent total mass
[$M_{\rm iso}$] (8.6$\pm$0.6)$\times$ 10-4 $M_{\odot}$ Isotropic kinetic
energy [$E_{\rm K,iso}$] (7.8$\pm$0.6) $\times$ 1053 erg Isotropic total
energy [$E_{\rm tot,iso}$] (1.1$\pm$0.1)$\times$1054 erg GRB radiative
efficiency [$\eta_{\gamma}$] (28.3$\pm$1.4) % Further Derived Parameters
Energy fractions assigned to electric fields [$\epsilon_{e,-1}$] 1.36$\pm$0.03
Energy fractions assigned to magnetic fields [$\epsilon_{\rm B,-2}$]
0.09$\pm$0.01 Characteristic synchrotron frequency [$\nu_{\rm m}$]
(1.9$\pm$0.2) $\times$ 1017 Hz Cooling frequency [$\nu_{\rm c}$] (2.6$\pm$0.5)
$\times$ 1017 Hz Klein-Nishina frequency [$\nu_{\rm KN}$]
(6.4$\pm$0.2)$\times$1017 Hz

Table 2 Pulse-wise properties of GRB 190114C
$P^{\rm th}_{1}$ $P^{\rm th}_{2}$ (From $t_{0}$+0.55 s to $t_{0}$+1.93 s)
(From $t_{0}$+2.45 s to $t_{0}$+5.69 s) Measured Parameters Isotropic
equivalent thermal energy [$E_{\rm th,iso}$]
(1.0${}^{+0.7}_{-0.5}$)$\times$1052 erg (3.6${}^{+0.6}_{-0.5}$)$\times$1052
erg Isotropic equivalent non-thermal energy [$E_{\rm nth,iso}$]
(4.8${}^{+1.3}_{-1.0}$)$\times$1052 erg (1.3${}^{+0.1}_{-0.1}$)$\times$1053
erg Thermal energy flux [$F^{\rm obs}_{\rm BB}$]
(1.5${}^{+0.1}_{-0.7}$)$\times$10-5 erg cm-2 s-1
(2.3${}^{+0.4}_{-0.3}$)$\times$10-5 erg cm-2 s-1 Total energy flux [$F^{\rm
obs}_{\gamma}$] (8.7${}^{+1.6}_{-1.3}$)$\times$10-5 erg cm-2 s-1
(1.1${}^{+0.1}_{-0.1}$)$\times$10-4 erg cm-2 s-1 Deceleration time [$t_{\rm
dec}$] $\sim$6 s $\sim$25 s Temperature [$kT^{\rm obs}$] 267${}^{+22}_{-18}$
keV 145${}^{+3}_{-3}$ keV Redshift [$z$] 0.4254$\pm$0.0005 0.4254$\pm$0.0005
Derived Parameters Dimensionless specific enthalpy density [$\eta$] $898\pm
61$ $661\pm 14$ Bulk Lorentz factor at $r_{\rm ph}$ [$\Gamma_{\rm ph}$]
$842\pm 36$ $602\pm 9$ Initial Lorentz factor [$\Gamma_{0}$] $575\pm 42$
$389\pm 5$ Isotropic equivalent total mass [$M_{\rm iso}$] $(1.0\pm 0.3)\times
10^{-4}~{}M_{\odot}$ $(3.5\pm 0.3)\times 10^{-4}~{}M_{\odot}$ Isotropic
kinetic energy [$E_{\rm k,iso}$] $(1.0\pm 0.4)\times 10^{53}$ erg $(2.4\pm
0.3)\times 10^{53}$ erg Isotropic total energy [$E_{\rm tot,iso}$] $(1.6\pm
0.5)\times 10^{53}$ erg $(4.1\pm 0.7)\times 10^{53}$ erg $\gamma$-ray
radiative efficiency [$\eta_{\gamma}$] $36.0\pm 6.5~{}\%$ $41.1\pm 1.9~{}\%$

Figure 1: Count lightcurve of Fermi-GBM during the time span of $0-30$ s. 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. Left panel: For the case based on global properties (see
Section 3.2). The shaded regions marked with different colours denote the
different episodes: the entire duration of $P_{1}$ and $P_{2}$ (grey) and
whole duration of $P_{3}$ (yellow). Right panel: For the case based on pulse-
wise properties (see Section 3.3). The shaded regions marked with different
colours denote the different episodes: the entire duration of $P^{\rm th}_{1}$
(pink) and ${}^{\rm th}P_{2}$ (blue) and whole duration of $P_{3}$ (yellow).

Figure 2: The GBM energy lightcurve (black) with the best fitting to $P_{3}$
using the FRED model (purple), and with the best fitting to the decay phase of
$P_{1}$ (cyan), $P_{2}$ (cobalt-blue), and $P_{3}$ (violet) using the power-
law model. The value of $t_{\rm p}$ (the vertical solid line) is used to
estimate $\Gamma$ in equation (6) while the value of
$\hat{\alpha}(P_{3})=-3.32\pm 0.49$ denotes the temporal index of the reverse
shock from the afterglow emission. The decay index of the afterglow emission
from $P_{1}$ is $\hat{\alpha}(S_{1})=-1.93\pm 0.09$, 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 $\hat{\alpha}(S_{2})=-1.09\pm 0.04$, which is in good agreement with
typical values found for afterglow emission.

Figure 3: The temporal evolution of the nozzle radius $R_{0}$ and the
photosphere width $\Delta R_{\rm ph}$. The photosphere width is $\sim 8$ times
smaller than $R_{0}$, but follows the similar trend of evolution. This factor
of $8$ can be eliminated by increasing 5 times of $Y$, which is reasonable.

In this appendix, we present the deriving the photosphere properties using the
traditional method (Section A0.1), and energy flux (Section A1).

###  A0.1 Deriving the Photosphere Properties Using the Traditional Method

The photosphere photons observed at a given time, corresponding to one time
bin in our time-resolved analysis, are assumed to be emitted from an
independent thin shell, which is given by considering the fireball optical
depth falling to a unity. 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 properties of a
corresponding shell. The entire duration of photosphere emission is conjugated
by the emissions from a sequence of such shells.

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

$r_{0}(r_{\rm ph}>r_{\rm s})=\frac{4^{3/2}d_{\rm
L}}{(1.48)^{6}\xi^{4}(1+z)^{2}}(\frac{F^{\rm obs}_{\rm BB}}{\mathbb{Y}F^{\rm
obs}})^{3/2}\Re,$ (A20)

and self-accelerates to reach a saturated Lorentz factor

$\eta(\equiv\Gamma)(r_{\rm ph}>r_{\rm s})=\left[\xi(1+z)^{2}d_{\rm
L}\left(\frac{\mathbb{Y}F^{\rm obs}\sigma_{\rm T}}{2m_{\rm
p}c^{3}\Re}\right)\right]^{1/4}$ (A21)

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

$r_{\rm ph}(>r_{\rm s})=\frac{L_{0}\sigma_{T}}{8\pi m_{\rm
p}c^{3}\Gamma^{3}},$ (A22)

where the dimensionless parameter

$\Re=\left(\frac{F_{\rm BB}}{\sigma_{\rm
B}T^{4}}\right)^{1/2}=\xi\frac{(1+z)^{2}}{d_{\rm L}}\frac{r_{\rm ph}}{\Gamma}$
(A23)

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 $\sigma_{\rm
B}$.

##  Appendix A1 Energy flux

Since $\nu F_{\nu}$ is not a directly observed quantity, one needs to convert
it to the observed flux by integrating the observed frequencies over the
entire X-ray energy band. Consider that the flux density of the afterglow
emission follows a power-law scaling,

$F_{\nu}\propto\nu^{-\beta}=A\nu^{-\beta},$ (A24)

where $A$ is a constant coefficient, and

$\nu F_{\nu}=A\nu^{1-\beta}.$ (A25)

The energy flux $F$ (erg cm-2 s-1) can be integrated by flux density $F_{\nu}$
(erg cm-2 s-1 Hz-1) over the energy range ($(\nu_{1},\nu_{2})=(0.3-10)$ keV,
with 1 keV$=2.4\times 10^{17}$ Hz) observed by Swift-XRT can be derived as

$\displaystyle F$ $\displaystyle=$ $\displaystyle\int_{\nu 1}^{\nu
2}F_{\nu}d\nu$ (A26) $\displaystyle=$ $\displaystyle\int_{\nu 1}^{\nu
2}A\nu^{-\beta}d\nu$ $\displaystyle=$ $\displaystyle\nu
F_{\nu}\frac{[\left(\frac{\nu_{2}}{\nu}\right)^{1-\beta}-\left(\frac{\nu_{1}}{\nu}\right)^{1-\beta}]}{(1-\beta)}.$

One therefore has

$\begin{split}F=\int_{\nu 1}^{\nu 2}F_{\nu}d\nu\\\ =\int_{\nu 1}^{\nu
2}A\nu^{-\beta}d\nu\\\
=\frac{1}{1-\beta}A\nu^{1-\beta}|^{\nu_{2}}_{\nu_{1}}\\\
=\frac{1}{1-\beta}A(\nu^{1-\beta}_{2}-\nu^{1-\beta}_{1})\\\
=A\nu^{1-\beta}\frac{[\left(\frac{\nu_{2}}{\nu}\right)^{1-\beta}-\left(\frac{\nu_{1}}{\nu}\right)^{1-\beta}]}{(1-\beta)}\\\
=\nu
F_{\nu}\frac{[\left(\frac{\nu_{2}}{\nu}\right)^{1-\beta}-\left(\frac{\nu_{1}}{\nu}\right)^{1-\beta}]}{(1-\beta)},\end{split}$
(A27) $\nu
F_{\nu}=F\times\frac{1-\beta}{[(\frac{\nu_{2}}{\nu})^{1-\beta}-(\frac{\nu_{1}}{\nu})^{1-\beta}]}.$
(A28)