1. Introduction
The fireball model [
1,
2,
3,
4,
5] has long been accepted to interpret the GRB physical process. The fireball is required to move with a relativistic speed towards us to avoid the “compactness problem” [
6]. The dynamical evolution of a fireball includes three phases: acceleration, coasting, and deceleration. During the acceleration phase, the Lorentz factor increases linearly with the radius
r and becomes a constant after most thermal energy has been converted to kinetic [
7]. After the fireball reaches the maximum Lorentz factor, it enters the “coasting” phase and moves with a constant Lorentz factor until it collects a considerable mass of ambient medium at the deceleration radius
, after which the Lorentz factor decays significantly. The maximum Lorentz factor during the “coasting” phase is also the initial Lorentz factor (
) of the ejecta during the deceleration phase. Panaitescu and Kumar [
8] and Molinari et al. [
9] have shown that
, where
is the Lorentz factor at
.
is a crucial parameter to constrain burst models [
10]; however, this parameter is difficult to measure directly, unlike some other parameters, such as the isotropic energy
and the isotropic luminosity
. Several methods are proposed to infer
, among which the most commonly invoked one is the afterglow onset method. The idea is that the peak time
of the early afterglow light curve is taken as the time when the deceleration phase begins (
). Given that
for a constant density (ISM) medium is most sensitive to
but only weakly depends on other parameters,
is possible to estimate by measuring
and
[
9,
10,
11,
12].
Before the
Swift era, afterglow observations mostly started several hours after the burst trigger and the early optical afterglows were rarely detected. The launch of
Swift has changed the situation. With the prompt slewing capability of the X-ray telescope (XRT [
13]) and a UV-optical telescope (UVOT [
14]), it enabled direct observations of the very early afterglow phase of GRBs and gained abundant early afterglow data.
In this paper, we analyze the prompt and afterglow emission of GRB 181110A, a long burst whose afterglow light curve shows a multi-band early peak.
We perform the spectral fitting for the prompt emission and calculate the peak energy and . We present the optical to X-ray light curves observed by Swift and our fitting results. We use the temporal and spectral properties of the afterglow to infer the jet structure, the circumburst medium profile, and the initial Lorentz factor of GRB 181110A.
We will follow the convention
to describe the temporal and spectral evolution of the afterglow. The concordance cosmology adopted has parameters of
,
and
[
15]. Uncertainties are given at 68% (
) confidence level for one parameter unless stated otherwise.
2. Observations and Data
GRB 181110A was detected by
Swift at 08:43:31 UT on 10 November 2018 [
16]. The BAT light curve showed a multi-peaked structure with a duration of
s [
17]. The
Swift XRT began observing the field 64 s after the BAT trigger. A bright, uncatalogued X-ray source was located with an enhanced position of RA (J2000): 20 h 09 m 16.32 s and Dec (J2000): -36d 53
47.9
with an uncertainty of 1.4 arcsec (at
confidence level) [
18]. The
Swift UVOT began settled observations of the field of GRB 181110A 72 s after the BAT trigger [
19]. In the initial exposures, UVOT detected an optical counterpart consistent with the XRT position (
Figure 1). The redshift of GRB 181110A is
z = 1.505 [
20].
The light curve and spectral data of BAT
1 and XRT were obtained from the
Swift online repository
2 on 9 November 2021. The BAT data we selected (15 keV to 150 keV) are analyzed with the HEASOFT package (version 6.28) and Xspec 12.11.1
3.
3. Analysis
3.1. Afterglow Light Curve Modeling
Figure 2 shows the optical, ultraviolet, and X-ray light curves of the afterglow of GRB 181110A. The data were taken with
Swift XRT and UVOT. The light curves of different energy bands evolve nearly synchronously and show a peak around 1200 s.
To derive the peak time and slopes of rise and decay phase, we use a smoothly broken power-law function to fit the light curves [
21]:
where
t is the time after the trigger,
is the normalization constant,
is the smoothness parameter, and
is the slope of rise or decay phase (
). Following [
9,
22], the light curve reaches the maximum at
We choose the data from 600 s after the trigger to fit since as shown in
Figure 2, the earlier part may be affected by the prompt emission.
We set
as free parameters and use
emcee [
23], the Python ensemble sampling toolkit for affine-invariant MCMC to fit the observed data with Equation (
1). We first fit the data of different bands separately and the results are quite similar; thus, we fit them with a unified model. The peak time (
) of optical and UV light curves is about 1200 s. The best-fitting results are listed in
Table 1 and the corresponding curves are shown in
Figure 3.
3.2. Spectral Analysis
3.2.1. Prompt Emission Spectral Analysis
We perform both time-integrated and time-resolved spectral analyses within the
time interval from
− 39.21 s to
+ 99.16 s. The time-resolved spectral analysis is performed by slicing the
interval into time bins with the Bayesian block [
24]. The cutoff power-law function (CPL) and the single power-law (PL) function are adopted to fit the data. The CPL function is described as
where
A is the normalization coefficient,
is the low-energy photon spectral index, and
is the break energy of the photon spectrum, and it is related to the peak energy (
) of the
spectrum by
. The single power-law (PL) function is
where
A is the normalization coefficient and
is the single power-law photon index.
The time-integrated BAT spectrum ( s to s) is well fitted by CPL function (), with a photon index and keV. We calculate the isotropic energy erg.
For the time-resolved spectrum, we first fit the spectrum using the CPL function. Given the narrow energy band of BAT, we use the PL function to fit for the time bins in which
cannot be reliably constrained. Our result indicates the temporal evolution of the spectra, which is shown in
Table 2. The energy spectrum index of GRB 181110A is found to be soft on the whole, and the spectral evolution is observed, as shown in
Figure 4;
shows a hard-to-soft pattern at first and then shows an intensity-tracking pattern.
3.2.2. Afterglow SED Fitting
In
Figure 5, we build the spectral energy distribution (SED) at 1400 s after the trigger. This epoch is chosen because multiband data are available. The SED has been corrected for extinction in the Milky Way
[
25]) and X-ray absorption (
[
26]). In addition, to account for host galaxy dust extinction, we fit the afterglow SED with the Small Magellan Cloud (SMC) template extinction law [
27,
28,
29] and derive a small visual extinction (
), taking the Lyman alpha absorption at
into account. Overall, the SED is consistent with a simple power-law extending from optical to X-ray band and the spectral slope is
, which is in line with the average spectral index for XRT PC mode data (
) retrieved from the online repository
4.
3.3. Afterglowpy Modeling
To further investigate the properties of the relativistic jet, we numerically model the afterglow light curves (from 5000 s after the trigger) using
afterglowpy [
31]. Three structures for the jet’s energy profile are considered: the top-hat, the Gaussian, and the power-law jets [
32,
33,
34,
35,
36]. The physical parameters for the top-hat model are the viewing angle (
), the jet core opening angle (
), the isotropic energy (
), the circumburst medium density (
), and the fraction of shock energy that is transferred to electron and the magnetic field, respectively (
and
). The Gaussian and power-law models have the truncation angle (
) as an additional free parameter, and the power-law model has the power-law index (
b) as another additional parameter. Then, with
emcee, we show the results of physical parameters in
Table 3. The posterior distribution of physical parameters of top-hat jet for GRB 181110A is presented in
Figure 6.
4. Discussion
4.1. Constraint on the Medium Profile and Jet Structure
In the standard afterglow model, the interaction between the relativistic fireball and the circumburst medium leads to external shocks and produces the multiband afterglow emission. The temporal and spectral behavior of multiband afterglow can be used to diagnose the profile of the circumburst medium. For GRB 181110A, we firstly consider the ISM case. When
, the fireball has not been decelerated significantly, and the flux of the forward shock emission can be described by the scaling law [
37]:
for
, where
and
are the typical synchrotron frequency and the cooling frequency of electrons, respectively. The best-fitting parameters listed in
Table 1 are therefore consistent with the temporal behavior above. When
, the temporal decay index (
) in
Table 1 and the spectral index (
) derived from our analysis are consistent with the closure relation
in
case [
37]. The electron energy spectral index is estimated to be
with the relation
, while in the wind scenario, the rise of the forward shock emission for
cannot be steeper than
[
38]. Hence, a homogeneous medium model is favored for GRB 181110A.
Note that the electron energy spectral index
is closer to the results of top-hat and Gaussian jet in
Table 3. In addition, for the Gaussian model, the constrained jet profile is similar to that of the top-hat model due to
[
31]. The similar results between these two models in
Table 3 also favor a uniform jet over a structured one.
Using the parameters in
Table 3, we further estimate the typical frequencies of electrons and find
Hz and
Hz, which is consistent with our analysis above (
).
4.2. Determination of
According to the fits, the early afterglow peaks at about 1193 s after the trigger, thus
, which agrees with the ISM “thin shell” case [
39]. In the thin shell case, the fireball decelerates at
, where
is the timescale over which the ISM mass collected by fireball is
of the ejecta mass, i.e.,
where
is the ratio between the isotropic gamma-ray energy and the isotropic blast wave kinetic energy,
is the particle number density,
is the proton mass, and
is the fireball Lorentz factor at
, which is approximately half of
[
40]. Therefore, we can estimate
by
The notation denotes in cgs for and , . With the parameters obtained in the previous section, we can obtain , , and then we derive .
The value of
confirms the highly relativistic nature of GRB fireballs and is within the typical range
(see Figure 11 in [
41]). With
, we can also derive the deceleration radius
.
4.3. GRB 181110A in a Statistical Context
By analyzing a sample of GRBs with afterglow onset feature, Liang et al. [
12,
42,
43] found strong correlations among the timescales of the onset “bump” and correlations among
,
and
, where
is the peak energy in the cosmological rest frame. Following [
12], we take the full width at half-maximum (FWHM) of our fitting light curve as the characteristic width (
w) of the peak. The rising and decaying timescales (
and
) are measured at FWHM. Their correlations are as follows [
12,
42]:
We examine whether GRB 181110A satisfies these empirical relations and then we compare GRB 181110A with other bursts in the Amati relation [
44,
45,
46]:
and
As shown in
Figure 7 and
Figure 8, the
and
of this burst lie within the LGRB distribution of Amati relation. The correlations among the timescales of the onset “bump” suggest that a wider onset “bump” tends to peak at a later time, which is consistent with
and
w we derived for GRB 181110A. GRB 181110A also shares the same
and
empirical relations with other typical GRBs, which read [
43]:
These consistencies suggest that GRB 181110A belongs to typical long GRBs with an afterglow onset feature which can be interpreted by a standard external forward shock model.
5. Summary
The early afterglow light curve of GRB 181110A shows a peak at about 1200 s, thus it is possible to estimate the fireball initial Lorentz factor by taking the peak time as deceleration time .
Firstly we perform spectral analyses for the prompt emission of GRB 181110A and the time-integrated BAT spectrum ( s to s) is well fitted by the CPL function with and keV, while we fit the time-resolved spectrum with both CPL and PL function. We derive erg. We also find that GRB 181110A has a soft spectrum and a spectral evolution feature; the evolution shows a hard-to-soft pattern at first and then an intensity-tracking pattern.
We fit the afterglow light curves with a smoothly broken power-law model and find that the light curves of all bands show the same temporal behavior. For GRB 181110A, the sharp rise of the very early afterglow light curve has ruled out a wind-like circumburst medium. Our joint analysis of multiband afterglow indicates that the standard external forward shock model with the ISM scenario is favored, and the cooling frequency is likely above X-ray band.
We also use afterglowpy to model the afterglow light curves numerically and find that the jet of GRB 181110A tends to be uniform rather than structured.
We investigate GRB 181110A in a statistical context and find that it locates within the sample of LGRBs with good afterglow onset features. The and we derived agree with the Amati relation for LGRBs. With the peak time of light curve and parameters given by afterglowpy modeling, the initial Lorentz factor is measured to be , which is consistent with the typical values of GRBs . Overall, for those GRBs with multiband data observed, it is possible to infer the structures of their jets. GRB 181110A is a typical long GRB that shows a clear afterglow onset feature; with more GRBs such as this being detected, our knowledge of the GRB fireball and circumburst environment may be extended.
Author Contributions
Conceptualization, D.W. and Z.J.; methodology, S.H., X.L. and L.J.; software, S.H., X.L. and L.J.; formal analysis, S.H.; investigation, S.H.; data curation, S.H. and L.J.; writing—original draft preparation, S.H.; writing—review and editing, S.H., Y.W. and D.W.; visualization, S.H., X.L. and L.J.; supervision, D.W.; project administration, D.W., Z.J. and H.H.; funding acquisition, D.W. All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported by NSFC (No. 12073080, 11933010, 11921003) and by the Chinese Academy of Sciences via the Key Research Program of Frontier Sciences (No. QYZDJ-SSW-SYS024).
Data Availability Statement
Conflicts of Interest
The authors declare no conflict of interest.
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