# GRB 171205A: Hypernova and Newborn Neutron Star

Yu Wang  ICRA, Dip. di Fisica, Università di Roma “La Sapienza”, Piazzale Aldo
Moro 5, I-00185 Roma, Italy  ICRANet, Piazza della Repubblica 10, I-65122
Pescara, Italy  INAF – Osservatorio Astronomico d’Abruzzo,Via M. Maggini snc,
I-64100, Teramo, Italy  L. M. Becerra  Escuela de Física, Universidad
Industrial de Santander, A.A.678, Bucaramanga, 680002, Colombia  ICRANet,
Piazza della Repubblica 10, I-65122 Pescara, Italy  C. L. Fryer  Center for
Theoretical Astrophysics, Los Alamos National Laboratory, Los Alamos, NM,
87545, USA  Computer, Computational, and Statistical Sciences Division, Los
Alamos National Laboratory, Los Alamos, NM, 87545, USA  The University of
Arizona, Tucson, AZ 85721, USA  Department of Physics and Astronomy, The
University of New Mexico, Albuquerque, NM 87131, USA  The George Washington
University, Washington, DC 20052, USA  J. A. Rueda  ICRA, Dip. di Fisica,
Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, I-00185 Roma, Italy
ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy  ICRANet-Ferrara,
Dip. di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via
Saragat 1, I-44122 Ferrara, Italy  Dip. di Fisica e Scienze della Terra,
Università degli Studi di Ferrara, Via Saragat 1, I-44122 Ferrara, Italy
INAF, Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del Cavaliere
100, 00133 Rome, Italy  R. Ruffini  ICRA, Dip. di Fisica, Università di Roma
“La Sapienza”, Piazzale Aldo Moro 5, I-00185 Roma, Italy  ICRANet, Piazza
della Repubblica 10, I-65122 Pescara, Italy  INAF,Viale del Parco Mellini 84,
00136 Rome, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

GRB 171205A is a low-luminosity, long-duration gamma-ray burst (GRB)
associated with SN 2017iuk, a broad-line type Ic supernova (SN). It is
consistent with being formed in the core-collapse of a widely separated
binary, which we have called the binary-driven hypernova (BdHN) of type III.
The core-collapse of the CO star forms a newborn NS ($\nu$NS) and the SN
explosion. Fallback accretion transfers mass and angular momentum to the
$\nu$NS, here assumed to be born non-rotating. The accretion energy injected
into the expanding stellar layers powers the prompt emission. The
multiwavelength power-law afterglow is explained by the synchrotron radiation
of electrons in the SN ejecta, powered by energy injected by the spinning
$\nu$NS. We calculate the amount of mass and angular momentum gained by the
$\nu$NS, as well as the $\nu$NS rotational evolution. The $\nu$NS spins up to
a period of $47$ ms, then releases its rotational energy powering the
synchrotron emission of the afterglow. The paucity of the $\nu$NS spin
explains the low-luminosity characteristic and that the optical emission of
the SN from the nickel radioactive decay outshines the optical emission from
the synchrotron radiation. From the $\nu$NS evolution, we infer that the SN
explosion had to occur at most $7.36$ h before the GRB trigger. Therefore, for
the first time, the analysis of the GRB data leads to the time of occurrence
of the CO core-collapse leading to the SN explosion and the electromagnetic
emission of the GRB event.

gamma-ray bursts: general – black hole physics – pulsars

##  1 Introduction

The Burst Alert Telescope of the Neil Gehrels Swift Observatory onboard
(Swift-BAT) triggered and located GRB 171205A at $07:20:43$ UT on December
$17$, $2017$. Swift’s X-Ray Telescope (XRT) began to observe $144.7$ s after
the BAT trigger (D’Elia et al., 2017). Soon, Izzo et al. (2017a) found that
the burst was located in a nearby galaxy at redshift $z=0.0368$, which was
later confirmed by the X-shooter telescope of the Very Large Telescope
(VLT/X-shooter) (Izzo et al., 2017b). About $5$ d after, the associated type
Ic supernova (SN) started to emerge and was detected by the $10.4$-m Gran
Telescopio Canarias (GTC) (de Ugarte Postigo et al., 2017) and SMARTS $1.3$-m
telescope (Cobb, 2017).

This source has gained much observational attention since it was the third
nearest GRB at the time of its discovery. D’Elia et al. (2018) performed the
multi-wavelength analysis of GRB 171205A using the data from the Swift and
Konus-Wind satellites, covering from the optical to the sub-MeV energies.
Their cutoff power-law fit gives the peak energy at $\sim 100$ keV and the
isotropic energy in the order of $10^{49}$ erg, which implies this burst is a
low luminosity GRB and is an outlier of the Amati relation. Wang et al. (2018)
reported the spectroscopic observation of the SN associated with the GRB, SN
2017iuk, and of the host galaxy. These observations showed that SN 2017iuk is
a typical type Ic SN that resembles SN 2006aj, and that the host is an early-
type, star-forming galaxy of high mass, low star formation rate, and low solar
metallicity. In this source, for the first time, it was observed the
polarization in the millimeter and radio bands during the afterglow phase,
thanks to the intensive combined use of the Submillimeter Array (SMA), the
Atacama Large Millimeter/submillimeter Array (ALMA) and the Very Large Array
(VLA), which shows a linear polarization $<1\%$ indicative of Faraday
depolarization (Urata et al., 2019; Laskar et al., 2020). The observation
continued for years, the ASKAP, ATCA and $\rm\mu$GMRT radio observations
lasted till $\sim 1000$ d, the radio afterglow decays following a shallow
power-law and no jet break was exhibited (Leung et al., 2021; Maity & Chandra,
2021). Figure 1 shows the multiwavelength light-curve of GRB 171205A.

Figure 1: Luminosity light-curve of Swift-BAT (deep red), Swift-XRT (red),
optical B band from D’Elia et al. (2018) (grey), and radio $1255$ MHz from
Maity & Chandra (2021) (brown), the triangles represent the upper limit. We
also plot the thermal luminosity (yellow). The Swift-XRT data at time
$>8\times 10^{4}$ s is fitted by a power-law of index $1.01\pm 0.06$ and
extrapolated to the earlier and the later time (red solid line, the red shadow
represents the $68\%$ confidence interval). Here $T_{0}=0$ s is the starting
time of the burst, corresponds to $38$ s before the BAT trigger time.

###  1.1 GRB 171205A in the traditional scenario

The origin of low-luminosity GRBs is still an open debate, and some
interpretations include that these are bursts observed off-axis (Waxman, 2004;
Soderberg et al., 2006b, a; Kathirgamaraju et al., 2016; Fraija et al., 2019a;
Izzo et al., 2020), shockwave breakout from the progenitor’s shell (Campana et
al., 2006; Irwin & Chevalier, 2016; Li, 2007; Soderberg et al., 2008; Barniol
Duran et al., 2015; Fraija et al., 2019b), and emission from a jet-heated
cocoon (Nakar, 2015; Kasliwal et al., 2017; Gottlieb et al., 2018). GRB
171205A, as a low luminosity GRB at a low redshift, provides a testing ground
for the theoretical models. Izzo et al. (2019) found thermal X-ray and optical
emissions radiated from material whose velocity evolves from $\sim 0.3~{}c$ to
$0.1~{}c$ in the first $7$ d, and with a chemical composition that differs
from that of SN 2017iuk which has a lower velocity ($<0.1~{}c$) evidenced by
the spectroscopic analysis. They proposed the high-velocity material is a
portion of the accelerated cocoon, which becomes transparent at $\sim 7$ d,
and then the SN dominates the optical emission. Suzuki & Maeda (2022)
performed hydrodynamic simulations of a powerful jet penetrating the
progenitor star and showed that jet-induced chemical mixing can lead to the
observed chemical composition of the high-velocity material. Maity & Chandra
(2021) analyzed GRB 171205A with the shockwave breakout and the canonical off-
axis jet models and show that both are inconsistent with the $1000$ d
observations. Compared to the observation, the shockwave breakout model
predicts a longer duration, a lower peak energy, and requires a higher column
density. Moreover, the radius ($\sim 10^{13}$ cm) derived from the thermal
component is too large for a typical progenitor. For the off-axis model, the
discrepancies arise because the burst does not exhibit expected off-axis
properties like a low peak energy, a luminosity increasing in the afterglow,
and a frequency-independent break in the light-curve (D’Elia et al., 2018).
There are alternative models, e.g., Suzuki et al. (2019) modeled the burst as
mild-relativistic spherical ejecta interacting with an ambient wind-like
medium producing forward and reverse shocks and forming a thin shell. In their
model, the prompt gamma-ray and X-ray emissions are produced when the optical
depth of the shell reaches transparency, and subsequently, the radio and X-ray
emissions are produced in the shock fronts by synchrotron and inverse Compton
processes. They claimed this model can fit the prompt luminosity and duration,
as well as the late-time X-ray, optical, and radio light-curves.

###  1.2 The BdHN scenario

Therefore, a satisfactory explanation of the multiwavelength data and the
evolution with time of GRB 171205A remains an open issue. In this work, we
analyze this source from the perspective of the binary-driven hypernova (BdHN)
model of long GRBs. The progenitor of the GRB in the BdHN model is a binary
system composed of a carbon-oxygen (CO) star and a neutron star (NS)
companion. Numerical simulations of the sequence of physical processes
occurring in a BdHN has been performed in the last decade and have led to a
detailed picture and interpretation of the GRB observables (see, e.g., Rueda &
Ruffini, 2012; Izzo et al., 2012; Fryer et al., 2014, 2015; Becerra et al.,
2015, 2016; Ruffini et al., 2018b; Becerra et al., 2019). The core-collapse of
the CO star leads to the formation of a newborn NS ($\nu$NS) at its center and
ejects the outer layers of the star in a SN explosion. The ejecta accretes
onto the NS companion and due to matter fallback there is also accretion onto
the $\nu$NS (see, e.g., Becerra et al., 2022; Wang et al., 2022a; Rueda et
al., 2022, and references therein). Both accretion processes are hypercritical
(i.e., highly super-Eddington) in view of the activation of a very efficient
neutrino emission (Becerra et al., 2016, 2018). For orbital periods of a few
minutes, the NS companion reaches the critical mass for gravitational
collapse, leading to a Kerr black hole (BH). These BdHN have been called of
type I (BdHN I). BdHN I explain the energetic GRBs with isotropic energies
$\gtrsim 10^{52}$ erg. The accretion processes are observed as precursors of
the prompt emission (see, e.g., Wang et al., 2019). The gravitomagnetic
interaction of the newborn Kerr BH with the surrounding magnetic field induces
an electric field. For a sufficiently supercritical magnetic field, the
electric field becomes also supercritical leading to an electron-positron
($e^{+}e^{-}$) pair plasma. The self-acceleration of this plasma to Lorentz
factors $\Gamma\sim 100$ and its transparency explain the ultra-relativistic
prompt emission (UPE) phase (see Moradi et al., 2021b for the UPE analysis of
GRB 190114C, and Rastegarnia et al., 2022 for GRB 180720B). The electric field
accelerates electrons to ultra-relativistic energies leading to synchrotron
radiation that explain the observed GeV emission (Ruffini et al., 2019; Rueda
& Ruffini, 2020; Moradi et al., 2021a; Rueda et al., 2022). There is an
additional synchrotron radiation process by relativistic electrons in the
ejecta expanding in the $\nu$NS magnetic field. The $\nu$NS also injects
energy into the ejecta. This synchrotron radiation explains the afterglow
emission in the X-rays, optical, and radio wavelengths (see, e.g., Ruffini et
al., 2018a; Wang et al., 2019; Rueda et al., 2020). Finally, the release of
nickel decay (into cobalt) in the SN ejecta powers the bump observed in the
optical in the late afterglow.

For longer orbital periods, of the order of tens of minutes, the NS companion
does not reach the critical mass, so it remains as a massive, fast rotating
NS. These BdHN have been called of type II (BdHN II). BdHN II explain the less
energetic GRBs with isotropic energies $\lesssim 10^{52}$ erg. The physical
processes and related observables associated with the presence of the BH are
clearly not observed in the BdHN II (e.g., the UPE and the GeV emission). The
synchrotron afterglow in the X-rays, optical, and radio wavelengths, instead,
is present both in BdHN I and II because it is powered by the $\nu$NS and the
SN ejecta (see Wang et al., 2019, 2022a for GRB 180728A and GRB 190829A).

###  1.3 GRB 171205A and the quest for BdHN III

When considering BdHN with longer and longer orbital period, possibly of
hours, the effects associated with the presence of the binary companion become
observationally irrelevant. Therefore, there is no GRB observable that can
discriminate the presence or absence of a binary companion. Under the above
circumstances, we model GRB 171205A neglecting the observational consequences
of a companion NS. We shall call these low-luminous sources with energies
$\lesssim 10^{49}$–$10^{50}$ erg, as BdHNe III.

Table 1 summarizes the sequence of physical phenomena that occur in BdHN I,
II, and III and their corresponding observables in the GRB data. Signatures
from a binary companion appear only in BdHN I and II, while BdHN III shows
only observables associated with the SN and the $\nu$NS.

Table 1: Physical phenomena that occur in BdHN I, II and III, and their associated observables in the GRB data. Physical phenomenon | BdHN | GRB observable
---|---|---
| type |  $\nu$NS-rise | UPE | GeV | SXFs | Afterglow
|  | (soft-hard X-rays) | (MeV) | emission | HXFs | (X/optical/radio)
Early SN emissiona | I, II, III | $\bigotimes$ |  |  |  |
Hypercritical accretion onto $\nu$NSb | I, II, III | $\bigotimes$ |  |  |  |
Hypercritical accretion onto NSb | I, II | $\bigotimes$ |  |  |  |
BH formation from NS collapsec | I |  |  | $\bigotimes$ |  |
Transparency of $e^{+}e^{-}$ (from vacuum | I |  | $\bigotimes$ |  |  |
polarization) with low baryon load regiond |  |  |  |  |  |
Synchrotron radiation inner engine: | I |  |  | $\bigotimes$ |  |
BH + $B$-field+SN ejectae |  |  |  |  |  |
Transparency of $e^{+}e^{-}$ (from vacuum | I |  |  |  | $\bigotimes$ |
polarization) with high baryon loadf |  |  |  |  |  |
Synchrotron emission from SN ejecta with | I, II, III |  |  |  |  | $\bigotimes$
energy injection from $\nu$NSg |  |  |  |  |  |
Pulsar-like emission from $\nu$NSg | I, II, III |  |  |  |  | $\bigotimes$

In Sec. 2, we analyze the Swift observations and fit the time-resolved spectra
using the MCMC method, then we generate the light-curves for the prompt
emission and afterglow, shown in Figs. 1 and 2. The special feature of this
burst is the presence of a thermal component in the early afterglow, where the
temperature drops from about $90$ eV to $70$ eV in the first $300$ s. In Sec.
3, we describe the physical process of this burst, we suggest that this low-
luminosity burst originates from a strong SN (or a hypernova). The fallback
accretion after the SN collapse heats up the SN ejecta, accelerating its
outermost layer to mild-relativistic and the heated ejecta emits thermal
radiation. This process is similar to the cocoon model, but the opening angle
for the energy release of the fallback accretion is much larger than the
traditional jet. This large opening angle is consistent with the absence of
the jet break signal in the afterglow. In the meanwhile, the fallback
accretion spins up the central NS, which in turn injects energy to power the
afterglow by losing its rotational energy. In Sec. 4, we establish the
analytical solutions for the spin-up of the $\nu$NS due to the mass and
angular momentum transfer during the accretion. We derive an analytical
solution for the time required for the spin-up process using an accurate Padè
approximant in the expression of the angular velocity as a function of time
(see Figs. 3 and 4). The spin period of the NS required by the theory can be
obtained from the observation by assuming that the energy of the X-ray
afterglow is mainly contributed by the rotational energy of the $\nu$NS. From
the observation of GRB 171205A, we derive that the $\nu$NS is possibly
accelerated to a spin period of $47$ ms, and $0.026~{}M_{\odot}$ are accreted
by the $\nu$NS via fallback. We show that this process takes $7.36$ h for a
$\nu$NS born with zero spin. In Sec. 5, we present the model of the afterglow
in the X-rays, optical, and radio wavelengths as originating from synchrotron
radiation in the expanding SN ejecta with the energy injection from the
central $47$ ms spinning $\nu$NS pulsar. Section 6 shows the results of the
fit of the X, optical and radio light-curves with the above model (see Fig.
5). The conclusions are given in Sec. 7.

##  2 Spectrum and light-curve

Swift-BAT and Swift-XRT data are retrieved from UKSSDC
111http://www.Swift.ac.uk, the data reduction are performed by Heasoft 6.29
222http://heasarc.gsfc.nasa.gov/lheasoft/, then the exported spectra are
fitted by the Multi-Mission Maximum Likelihood framework (3ML) (Vianello et
al., 2015). In order to produce the luminosity light-curve, the BAT data are
binned following the thresholds that the signal to noise ratio (SNR) is at
least 6 and the maximal bin size is at most $50$ s. Then each binned spectrum
is fitted by a cutoff power-law (CPL) function and is integrated from $15$ keV
to $150$ keV according to the BAT bandwidth to obtain the flux. After having
the fitting parameters, the fluxes and by adopting the FRW cosmology333The
Friedman-Lemaître-Robertson-Walker metric is used for computing the luminosity
distance, Hubble constant $H_{0}=67.4\pm 0.5$ km/s/Mpc, and matter density
$\Omega_{M}=0.315\pm 0.007$ (Planck Collaboration et al., 2018)., the
k-corrected luminosity light-curve is obtained (Bloom et al., 2001). We
generate the light-curve of XRT in the energy range $0.3$–$10$ keV following a
similar procedure, the corresponding binning thresholds change to at least
$200$ counts and $10$ s duration for the windows timing (WT) mode, as well as
at least $100$ counts and $100$ s duration per bin for the photon counting
(PC) mode. All the XRT spectra are fitted by a power-law function444To have
more data points for the light-curve, our binning is more concerned with
sufficiently short time resolution than with exact spectra. Therefore, the
power-law model is used uniformly to fit the spectra, rather than the more
accurate power-law plus blackbody model for which the data of each small bin
cannot constrain all parameters. This introduces an error of less than $5\%$,
which is in an acceptable level. with the photoelectric absorption models of
our Galaxy and the host galaxy. The generated Swift luminosity light-curves
are presented in Fig. 1. We notice that this burst is seen since $\sim 38$ s
before the BAT trigger, hence we set $T_{0}$ as $38$ s before the BAT trigger
time. The XRT light-curve later than $8\times 10^{4}$ s is fitted by a power-
law function using lmfit (Newville et al., 2021), a python package for non-
linear optimization and curve fitting. lmfit implements the Levenberg-
Marquardt method for optimization and is extended by
numdifftool555https://numdifftools.readthedocs.io to estimate the covariance
matrix then to calculate parameter uncertainties. We obtain a power-law index
$1.01\pm 0.06$ with the 1-$\sigma$ uncertainty ($68\%$ confidence level). We
show the power-law fit in Fig. 1 with the 1-$\sigma$ uncertainty region. The
extrapolation of the power-law function coincides with the initial prompt
luminosity.

The $T_{90}$ of the BAT observation lasts $189.19$ s, its time-integrated can
be described by a cutoff power-law model with power-law index $\alpha=1.10\pm
0.35$, while the peak energy cannot be precisely constrained $E_{p}=148.55\pm
121.97$ keV. These parameters are consistent with D’Elia et al. (2018), which
jointly fitted BAT and Konus-Wind data. They obtained
$\alpha=0.85^{+0.54}_{-0.41}$ and $E_{p}=122^{+111}_{-32}$ keV, where the
uncertainty of peak energy has been tightened because Konus-Wind covers higher
energies than BAT. The integrated flux gives $(1.56\pm 0.31)\times 10^{-8}$
erg cm-2 s-1 in the observed $15$–$150$ keV bandwidth, and extrapolated to
$(2.63\pm 0.54)\times 10^{-8}$ erg cm-2 s-1 in $1$–$10^{4}$ keV, which
corresponds to the isotropic energy $E_{\rm iso}=(1.71\pm 0.35)\times 10^{49}$
erg.

The presence of a thermal component in the afterglow of GRB 171205A has been
reported in several articles (Campana et al., 2017; D’Elia et al., 2018; Izzo
et al., 2019). Our time-resolved analysis also confirms that the additional
thermal component significantly improves the fit to the low-energy band of the
XRT ($<1$ keV) till $324$ s with a fitting blackbody temperature that drops
from $\sim 90$ eV to $\sim 70$ eV, with an uncertainty of $\sim 10$ eV.
Afterward, the thermal spectrum gradually fades out of the XRT band
($0.3$–$10$ keV) as the temperature decreases. The WT data of XRT is unable to
constrain the temperature at a times later than $\sim 4000$ s, while the
optical telescopes start to capture the thermal component that cools to the
optical band (Izzo et al., 2019).

There is a common time window for BAT and XRT observing the source, from $\sim
151$ s when XRT had slewed to the GRB position, till $\sim 162$ s, the end of
the $T_{90}$ of BAT. The BAT data at the end of the prompt emission is
adequate to constrain the cutoff energy, hence the model of a power-law of
index $\alpha=-2.00\pm 0.17$ plus a blackbody component of $kT=77.53\pm 8.28$
eV is implemented to fit the entire data, as shown in Fig. 2.

The optical and radio light-curves in Fig. 1 are reproduced from D’Elia et al.
(2018) and Maity & Chandra (2021), respectively. The optical luminosity is
unusually bright compared to the X-rays. Izzo et al. (2019) found that the
evolution of the optical spectrum before and after 7 days is dominated by two
blackbodies with different evolution laws. The 1000-day radio light-curve
shows a shallow decay without any jet break signature. We refer to D’Elia et
al. (2018); Izzo et al. (2019); Maity & Chandra (2021) for the detailed
analyses and discussion of the optical and radio data, including the SN
optical observation.

Figure 2: Top: Spectrum of $T_{90}$ observed by BAT, fitted by a cutoff power-
law model with photon index $\alpha=1.10\pm 0.35$ and peak energy
$E_{p}=148.55\pm 121.97$ keV. Bottom: Jointly spectral fitting of BAT and XRT
from $151$ s to $162$ s after the BAT trigger with a composite spectrum of
power-law of index $\alpha=2.00\pm 0.17$ plus a blackbody of temperature
$kT=77.48\pm 7.46$ eV.

##  3 Physical Picture

At a given moment, a type Ic SN occurs from the core-collapse of the CO star,
forming at the same time a $\nu$NS at its center. The fallback accretion spins
up the $\nu$NS (see Sec. 4), while releasing the accretion energy. From
Becerra et al. (2019), the initial accretion rate is up to a few of
$10^{-3}~{}M_{\odot}$ s-1 and lasts tens of seconds, then it drops following a
power-law depending on the SN density profile. Therefore, in the initial phase
of tens of seconds, the total energy generated from the accretion and to be
injected into the stellar shells reaches $\sim 10^{52}$ erg, which is
comparable to the kinetic energy of SN ejecta inferred from the optical
emissions at a later time. Different from the traditional jetted model of
GRBs, this amount of energy is emitted in a large opening angle of probably
tens of degrees, it propagates in a portion of shells and accelerates the
outermost shell to the mild-relativistic velocity. The hydrodynamics can be
referred to the simulation in Ruffini et al. (2018b), where has been simulated
the propagation of GRB injected energy in the expanding stellar shells. The
Lorentz factor of the shockwave is lower than $5$ when it breaks out the
outermost shell at $\sim 10^{12}$ cm. The acceleration of the accretion-
powered blastwave is similar to that proposed for the shock-accelerated GRB
model (Colgate, 1974). In this scenario, a supernova blastwave accelerates as
it propagates down the steep density gradient at the edge of a massive star
(Colgate, 1974; Tan et al., 2001). Although these models can produce highly-
relativistic ejecta in idealized conditions, the bulk of the material reaches
only mildly relativistic velocities. Our model mirrors this evolution,
differing only from this picture because the blastwave is propagating through
an exploding CO star and is not spherical. Our asphericity has many of the
features of the cocoon produced in jet models (see e.g. Mészáros & Rees, 2001;
Ramirez-Ruiz et al., 2002; Zhang et al., 2004; Nakar & Piran, 2017; Eisenberg
et al., 2022; Soker, 2022), that the jet pushes the stellar shells sideways to
form a hot cocoon, a part of the cocoon emerges from the shells and expands
outward with a mild-relativistic velocity. Hence, both our picture and the
cocoon picture involve some heated high-velocity material originated from the
stellar shells expanding and emitting a thermal spectrum. The evolution of
such this blackbody spectrum has been indeed observed by Swift-XRT and several
optical telescopes, and a mass of $1.1\times 10^{-3}M_{\odot}$ moving above
$10^{5}$ km s-1 has been inferred; see Fig. 2 and Izzo et al. (2019). The
difference is that in our picture, we expect a wider opening angle than in a
jet, as we consider this low-luminosity GRB originates from a strong SN or
hypernova in which the central compact object is the $\nu$NS. From the
observations, there is no signature of any jet break in the afterglow till
$\sim 1000$ days (Leung et al., 2021; Maity & Chandra, 2021), hence preferring
a large opening angle description.

At this stage, our system has three energy sources; the accretion, the
spinning $\nu$NS, and the high-velocity material. For the prompt emission,
this low-luminosity GRB deviates from the Amati relation (Amati et al., 2002);
its peak energy ($E_{p}=148.55$ keV, see Fig. 2) is about one order of
magnitude higher than the typical value of a weak GRB with isotropic energy
$\sim 10^{49}$ erg (D’Elia et al., 2018). The deviation indicates this burst
could be an extreme case or is formed by a different mechanism. Izzo et al.
(2019) suggests that the jet deposits the majority energy in the creation of
the cocoon and only a small fraction of energy emitted in gamma-rays. In our
framework, accretion dominates the energy release once the SN explodes, and
the majority of energy is injected into the stellar shells, converting to the
internal and kinetic energy of the SN ejecta, and producing the fast moving
material. The low isotropic energy of the prompt emission can be either
produced by the tail of accretion or by the fast moving material (De Colle et
al., 2018). For the X-ray afterglow, it can be accounted for, at early times,
by the synchrotron emission converted from the kinetic energy of the fast
moving material, and at times after the plateau, by the release of rotational
energy of the $\nu$NS that has been spun up to periods of the order of
milliseconds. We have performed the numerical fitting of the spectrum and
light-curve using this scenario for several GRBs (see, e.g., Ruffini et al.,
2018a; Wang et al., 2019; Rueda et al., 2020). This is also supported by that
the ending time of the plateau coincides with the transparency of the fast
moving material at $\sim 10^{5}$ s. For the optical afterglow, we share the
same opinion with Izzo et al. (2019), that the fast expanding mass dominates
the optical emission before $4$ days, then the dominance is overtaken by
photons diffused out from the massive SN ejecta heated by the nickel
radioactive decay.

The above picture contains many different physical processes, most of which
have been discussed in detail and simulated, as the references mentioned in
the text. However, after the birth of $\nu$NS, the fallback accretion, the
mass change and the spin-up process have been rarely discussed in GRB studies.
Hence, we will focus on modelling the properties of the newborn NS in the next
section.

##  4 Spin-up and fallback accretion onto the $\nu$NS

We turn now to estimate the spin-up and the amount of mass that the $\nu$NS
has accreted to gain enough rotational energy to power the X-ray afterglow
emission, as specified in the BdHN model (see, e.g., Ruffini et al., 2021, for
the analysis of 380 BdHNe).

Assuming the X-ray luminosity as a good proxy of the bolometric luminosity of
the afterglow, we can estimate the change in the $\nu$NS rotational energy
from a time $t_{1}$ to a time $t_{2}>t_{1}$ from the energy balance equation,
i.e.

$\int_{t_{1}}^{t_{2}}\dot{E}_{\rm rot}\,dt=E_{\rm rot}(t_{2})-E_{\rm
rot}(t_{1})\approx-\int_{t_{1}}^{t_{2}}L_{X}dt.$ (1)

After infinite time, the $\nu$NS will have lost all its rotational energy,
therefore when $t_{2}\to\infty$, we have $E_{{\rm rot},\infty}(t_{2})\to 0$.
So, assuming the time $t_{1}$ to be a generic time $t$, and the power-law
luminosity

$L_{X}=A_{X}t^{-\alpha_{X}},$ (2)

we obtain from Eq. (1) that the $\nu$NS angular velocity evolves as

$\Omega(t)\approx\sqrt{\frac{2A_{X}\,t^{1-\alpha_{X}}}{(\alpha_{X}-1)I}},$ (3)

where $I$ is the stellar moment of inertia which we have assumed constant with
time, and can be estimated, for instance, using the EOS-independent
approximate expression (Wei et al., 2019)

$I\approx\left(\frac{G}{c^{2}}\right)^{2}M^{3}\sum_{i=1}^{4}\frac{b_{i}}{(M/M_{\odot})^{i}},$
(4)

where $b_{1}=1.0334$, $b_{2}=30.7271$, $b_{3}=-12.8839$, and $b_{4}=2.8841$.

In the case of GRB 171205A, the X-ray luminosity is fitted by a power-law at
times $t>t_{\rm pl}\approx 8\times 10^{4}$ s, with $A_{X}=(3.165\pm
0.238)\times 10^{47}$ erg s-1, and $\alpha_{X}=1.022\pm 0.055$. Using these
values, we estimate from Eq. (3) that the rotation period of the $\nu$NS at
$t=t_{\rm pl}$ is $P(t_{\rm pl})\approx 51.01$ ms. If we assume that the
$\nu$NS is spinning down from the $\nu$NS-rise, i.e., from $t=t_{\nu\rm
NS}\approx 35$ s, but the emission from it is partially absorbed by the high-
velocity material which is opaque before $\sim 10^{5}$ s, then by
extrapolating from $t=t_{\rm pl}$ backward in time to $t=t_{\nu\rm NS}$, we
infer that at the $\nu$NS-rise time, the $\nu$NS rotation period was
$P_{\nu\rm NS}\equiv P(t_{\nu\rm NS})\approx 46.85$ ms, i.e.,
$\Omega(t_{\nu\rm NS})=134.11$ rad s-1.

We now estimate the mass accreted by the $\nu$NS before the $\nu$NS-rise, so
to spin it up to the above rotation rate. The accretion rate onto the $\nu$NS,
set by the amount of mass from the inner layers of the expanding matter that
fallback onto the $\nu$NS and their infalling speed, proceeds at hypercritical
rates (see, e.g., Fryer et al., 1996). The accretion process makes the $\nu$NS
to increase its mass-energy and rotation rate from the transfer of baryonic
mass and angular momentum. The evolution of the $\nu$NS gravitational mass and
angular momentum can be calculated from (Becerra et al., 2019)

$\displaystyle\dot{M}$ $\displaystyle=\left(\frac{\partial M}{\partial
M_{b}}\right)_{J}\,\dot{M}_{b}+\left(\frac{\partial M}{\partial
J}\right)_{M_{b}}\,\dot{J},$ (5) $\displaystyle\dot{J}$
$\displaystyle=\tau_{\rm acc},$ (6)

where $J=I\Omega$ is the angular momentum, $M$ is the gravitational mass,
$M_{b}$ the baryonic mass, $\dot{M}_{b}$ is the baryonic mass accretion rate,
and $\tau_{\rm acc}$ is the accretion torque.

Equation (5) must be complemented with the expressions of the two partial
derivatives. These relations can be calculated from the fitting formula of the
NS binding energy obtained in Cipolletta et al. (2015)

$\mu_{b}-\mu=\frac{13}{200}\mu^{2}\left(1-\frac{1}{130}j^{1.7}\right),$ (7)

where $j\equiv cJ/(GM_{\odot}^{2})$ is the dimensionless angular momentum and
$\mu=M/M_{\odot}$. From it, we redily obtain

$\displaystyle\left(\frac{\partial\mu}{\partial\mu_{b}}\right)_{j}$
$\displaystyle=\frac{1}{1+\frac{13}{100}\mu\left(1-\frac{1}{130}j^{1.7}\right)},$
(8) $\displaystyle\left(\frac{\partial\mu}{\partial j}\right)_{\mu_{b}}$
$\displaystyle=\frac{\frac{1.7}{2000}\mu^{2}j^{0.7}}{1+\frac{13}{100}\mu\left(1-\frac{1}{130}j^{1.7}\right)}.$
(9)

The numerical simulations of BdHNe performed in Becerra et al. (2019) show
that the material accreted by the $\nu$NS circularizes around it in a sort of
Keplerian disk structure before being accreted. Therefore, we assume that the
accreted matter exerts onto the $\nu$NS the torque

$\tau_{\rm acc}=\chi\,l\,\dot{M}_{b},$ (10)

where $l$ the specific (i.e. per unit mass) angular momentum of the innermost
stable circular orbit around the $\nu$NS, and $\chi\leq 1$ is an efficiency
parameter of angular momentum transfer. For the angular momentum of the last
stable circular orbit, we use the approximate EOS-independent results
presented in Cipolletta et al. (2017)

$l=2\sqrt{3}\frac{GM}{c}\left[1\mp
0.107\left(\frac{j}{M/M_{\odot}}\right)^{0.85}\right].$ (11)

We can obtain an approximate, analytic solution to Eq. (6). For this task, we
use the following analytic formula that fits the numerical results of the
fallback accretion rate calculated in Becerra et al. (2019, 2022)

$\dot{M_{b}}\approx\dot{M}_{0}\left(1+\bar{t}\right)^{-p},$ (12)

where $\dot{M}_{0}=7.2\times 10^{-4}M_{\odot}$ s-1, $t_{\rm acc}=12$ s,
$p=1.3$, and we have introduced the notation $\bar{t}=t/t_{\rm acc}$.

For the involved rotation rates ($j\sim 0.01$), the contribution of the
rotation terms in Eqs. (7) and (11) is negligible, so we can retain only the
first term in those equations. With this assumption, and integrating Eq. (12),
we have

$\displaystyle\mu_{b}$ $\displaystyle=\mu_{b}(t_{0})+\frac{\dot{M}_{0}t_{\rm
acc}}{p-1}\left[1-\left(1+\bar{t}\right)^{1-p}\right],$ (13)
$\displaystyle\mu$
$\displaystyle\approx\frac{100}{13}\left(\sqrt{1+\frac{13}{50}\mu_{b}}-1\right),$
(14) $\displaystyle l$ $\displaystyle\approx
2\sqrt{3}\frac{GM_{\odot}}{c}\mu,$ (15)

where $\mu_{b}(t_{0})\approx\mu_{0}+(13/200)\mu_{0}^{2}$, being
$\mu_{0}=M(t_{0})/M_{\odot}$ the initial $\nu$NS gravitational mass, and we
have inverted Eq. (7) to write the gravitational mass in terms of the baryonic
mass. Equations (13) and (14) implies that in the limit $t\to\infty$ the
baryonic mass and the gravitational mass approaches a maximum value

$\displaystyle\mu_{b,\rm max}$
$\displaystyle=\mu_{b}(t_{0})+\frac{\dot{M}_{0}t_{\rm
acc}}{p-1}=\mu_{b}(t_{0})+0.0288,$ (16) $\displaystyle\mu_{\rm max}$
$\displaystyle=\frac{100}{13}\left(\sqrt{1+\frac{13}{50}\mu_{b,\rm
max}}-1\right).$ (17)

We now approximate the angular momentum derivative as $\dot{J}\approx
I\dot{\Omega}\approx I_{\rm max}\dot{\Omega}$, where $I_{\rm max}=I(\mu_{\rm
max})$, so that Eq. (6) becomes

$\dot{\Omega}\approx\beta\mu(t)(1+\bar{t})^{-p},\quad\beta=\frac{2\sqrt{3}GM_{\odot}^{2}\chi\dot{\mu}_{0}}{cI_{\rm
max}},$ (18)

whose solution can be written as

$\Omega(t)=\Omega(t_{0})+\beta\int_{t_{0}}^{t}\mu(t)(1+\bar{t})^{-p}dt.$ (19)

Making the change of variable $x=(1+\bar{t})^{1-p}$, the integration of Eq.
(19) is straightforward leading to

$\displaystyle\Delta\Omega=\Omega(t)-\Omega(t_{0})$
$\displaystyle=\omega\left\\{x+\frac{2}{3}k\left[\left(1+\frac{13\mu_{b}}{50}\right)^{3/2}-\alpha^{3/2}\right]-1\right\\},$
(20)

where we have defined

$\displaystyle\omega$ $\displaystyle=\frac{100}{13}\frac{\beta\,t_{\rm
acc}}{p-1},\quad\Delta\mu_{b}=\frac{\dot{M}_{0}t_{\rm acc}}{p-1}=0.0288,$ (21)
$\displaystyle k$
$\displaystyle=\frac{50}{13}\frac{1}{\Delta\mu_{b}}=133.547,\quad\alpha=1+\frac{13}{50}\mu_{b,0}$
(22)

and we have set the initial time $t_{0}=0$ since the fallback accretion begins
soon after the SN explosion (see, e.g., Becerra et al., 2019). Figure 3
compares the approximate analytic solution (4) with the solution from the full
numerical integration of Eqs. (5) and (6), in the case of $\mu(t_{0})=1.4$,
$\Omega(t_{0})=0$, and $\chi=0.15$.

Figure 3: Comparison of the approximate solution of the Eqs. (5) and (6) given
by Eq. (4), with the results from the full numerical integration, in the case
$\mu(t_{0})=1.4$, $\Omega(t_{0})=0$, and $\chi=0.15$.

Equation (4) tells us that in the limit $t\to\infty$ ($x\to 0$), the $\nu$NS
reaches asymptotically a maximum angular velocity gain

$\Delta\Omega_{\rm
max}=\omega\left\\{\frac{2}{3}k\left[\left(1+\frac{13\mu_{b,\rm
max}}{50}\right)^{3/2}-\alpha^{3/2}\right]-1\right\\},$ (23)

which as expected is larger for larger values of the angular momentum transfer
efficiency parameter, $\chi$. Since we assume that after the $\nu$NS-rise the
$\nu$NS is spinning down, we seek for solutions with a spinning up phase that
ends with an angular velocity approaching the value that we have inferred at
the $\nu$NS-rise, i.e.

$\Omega_{\rm max}\approx\Omega(t_{\rm\nu NS}),$ (24)

where $\Omega_{\rm max}=\Delta\Omega_{\rm max}+\Omega(t_{0})$. We have used
the approximate symbol in Eq. (24) because by definition the value
$\Omega_{\rm max}$ is reached only asymptotically. For practical purposes, we
seek for solutions in which $\Omega(t_{\rm\nu NS})=0.9\,\Omega_{\rm max}$.
Therefore, given values of $M$ and $\Omega(t_{\rm\nu NS})$, the above
constraint leads to a specific value of $\chi$ that leads to the self-
consistent spin-up phase. For instance, for a $\nu$NS mass $M=1.4M_{\odot}$
and

$\Omega_{\nu\rm NS}\equiv\Omega(t_{\rm\nu NS})=\sqrt{\frac{2A_{X}\,t_{\nu\rm
NS}^{1-\alpha_{X}}}{(\alpha_{X}-1)I}}\approx 134.11\,\,\text{rad s}^{-1},$
(25)

we obtain $\chi=0.182$.

We can also obtain a simple analytic estimate of the mass accreted by assuming
that during the spin up phase, the accretion rate, the gravitational mass, and
the moment of inertia are constant and have their maximum values. Under this
assumption, Eqs. (6) and (10) lead to the accreted mass in a time $\Delta t$,

$\Delta\mu_{b}\approx\frac{cI_{\rm max}\Delta\Omega}{2\sqrt{3}\chi
GM_{\odot}^{2}\mu_{\rm max}}.$ (26)

For the above parameters, Eq. (26) gives $\Delta\mu_{b}\approx 0.02570$. This
is very close to the value obtained from the full numerical integration,
$\Delta\mu_{b}=0.02592$, which represents an error of only $0.85\%$. The
accuracy of Eq. (26) resides in the fact that the fallback accretion rate
decreases as a power-law, see Eq. (12), hence most of the baryonic mass is
accreted in the first minutes of the evolution. This explains why the above
value of the accreted mass is close to the maximum accreted mass given by Eq.
(16), i.e., $\Delta\mu_{b,\rm max}=0.0288$.

We turn to obtain an analytic expression of the time interval $\Delta t$
elapsed since the beginning of the fallback accretion, up to the instant when
the $\nu$NS reaches a given angular velocity, or a given angular velocity
gain, $\Delta\Omega$. In principle, we can obtain it by inverting Eq. (4).
However, the equation is highly non-linear, so to obtain a relatively simple
expression for it we use an accurate Padè approximant for the quantity
involving the baryonic mass, i.e.

$\displaystyle\left(1+\frac{13\mu_{b}}{50}\right)^{3/2}$
$\displaystyle=b^{3/2}(\tilde{\alpha}+X)^{3/2}\approx b^{3/2}({\cal
F}+\tilde{\alpha}^{3/2}),$ $\displaystyle{\cal F}$
$\displaystyle=\frac{6\tilde{\alpha}^{3/2}X}{4\tilde{\alpha}-X},$ (27)

where $b=1/k=(13/50)\Delta\mu_{b,\rm max}$, $\tilde{\alpha}=\alpha/b$, and we
have introduced the variable $X=1-x$. For the same example of Fig. 3, we show
in Fig. 4 the excellence performance of the Padè approximant (27), which
approximate the expression with a tiny error of only $10^{-9}$.

Figure 4: Comparison of the Padè approximant given by Eq. (27) with the result
of the full numerical integration.

Using the approximant (27), Eq. (4) becomes a second-order polynomial in the
variable $X$ whose solution is straightforward, leading to the time interval:

$\Delta t=t_{\rm acc}\left[\left(1-X\right)^{\frac{1}{1-p}}-1\right],$ (28)

where

$\displaystyle X$ $\displaystyle=\frac{B+\sqrt{B^{2}+4C}}{2}$ (29)
$\displaystyle B$
$\displaystyle=4\tilde{\alpha}-4\tilde{\alpha}^{3/2}\sqrt{b}-\Delta\Omega/\omega,$
(30) $\displaystyle C$ $\displaystyle=4\tilde{\alpha}\Delta\Omega/\omega.$
(31)

The relevance of the above time interval is that it allows to compute the time
elapsed to reach the angular velocity at the $\nu$NS-rise, $\Omega(t_{\nu\rm
NS})$. Since it is close to the maximum value reachable by the fallback
accretion, that time interval gives an estimate of the time elapsed since the
SN explosion (SN-rise), $t_{\rm SN}$. For the present example, we obtain

$t_{\rm SN}=\Delta t(\Delta\Omega)=\Delta t(\Omega_{\nu\rm NS})\approx
7.36\,\,\text{h},$ (32)

where we have used $\Delta\Omega=\Omega_{\nu\rm NS}-\Omega(t_{0})=134.11$ rad
s-1, as given by Eq. (25). The full numerical integration leads to $7.20$ h,
which implies that Eq. (28) estimates the time interval with an error of only
$2.2\%$.

##  5 Synchrotron and pulsar emission

We turn now to the specific modeling of the multiwavelength afterglow of GRB
171205A. In the present scenario, the non-thermal component of the afterglow
originates from the synchrotron radiation in the SN ejecta. The SN ejecta gets
energy injected from the $\nu$NS fallback accretion and the multipolar
emissions. Numerical calculations of this model applied to the description of
the afterglow of specific GRBs can be found in Ruffini et al. (2018a); Wang et
al. (2019); Rueda et al. (2020). An analytic treatment of the model has been
presented in Rueda (2022), and Wang et al. (2022b) have applied it to model
the afterglow of GRB 180720B. Our afterglow model relies more on the
continuous energy injections than the traditional forward shockwave model that
relies on the kinetic energy of jet. And unlike the traditional model that
only considers the injection of dipole emission as an additional energy source
to explain the short duration plateau (for e.g. internal plateau) (Dai & Lu,
1998a, b; Zhang & Mészáros, 2001; Metzger et al., 2011; Lehner et al., 2012;
Chen et al., 2017; Li et al., 2018; Zhao et al., 2020), our modelling process
takes into account the fallback accretion, the dipole and quadrupole
radiation, such continuous energy injections produce the long-lasting
afterglow. We here follow the latter to estimate for GRB 171205A the emission
generated by the synchrotron mechanism in the X-rays, in the optical, and in
the radio, and the $\nu$NS pulsar emission.

###  5.1 Synchrotron emission by the expanding ejecta

The distribution of radiating electrons per unit energy, $N(E,t)$, is obtained
from the solution of the kinetic equation (Kardashev, 1962)

$\frac{\partial N(E,t)}{\partial t}=-\frac{\partial}{\partial
E}\left[\dot{E}\,N(E,t)\right]+Q(E,t),$ (33)

where $Q(E,t)$ is the number of injected electrons into the ejecta per unit
time $t$, per unit energy $E$, and $\dot{E}$ is the electron energy loss rate.

Following Rueda (2022); Wang et al. (2022b), we adopt the solution to Eq. (33)
for a self-similar uniform expansion

$\displaystyle N(E,t)\approx\begin{cases}\frac{q_{0}}{\beta
B_{*,0}^{2}(\gamma-1)}\hat{t}^{2}E^{-(\gamma+1)},&t<t_{q}\\\
\frac{q_{0}(t_{q}/t_{*})^{k}}{\beta
B_{*,0}^{2}(\gamma-1)}\hat{t}^{2-k}E^{-(\gamma+1)},&t_{q}<t<t_{b},\end{cases}$
(34)

where $E_{b}<E<E_{\rm max}$, being

$E_{b}=\frac{\hat{t}}{{\cal M}t_{*}},\quad t_{b}={\cal M}t_{*}^{2}E_{\rm
max}.$ (35)

The model parameters are defined as follows. The ejecta expands self-similarly
with the radiating layer being $r=R_{*}=R_{*,0}\,\hat{t}$, $\hat{t}\equiv
t/t_{*}$, $t_{*}=R_{*}/v_{*}=R_{*,0}/v_{*,0}$, $v_{*}=R_{*}(t)/t=v_{*,0}$,
$B_{*}(t)=B_{*,0}R_{*,0}/r=B_{*,0}\hat{t}^{-1}$ is the magnetic field strength
at $r=R_{*}$, ${\cal M}\equiv\beta B^{2}_{*,0}/2$,
$\beta=2e^{4}/(3m_{e}^{4}c^{7})$. We assume the injection power-law
distribution $Q(E,t)=Q_{0}(t)E^{-\gamma}$ (Kardashev, 1962; Pacini & Salvati,
1973; Rybicki & Lightman, 1979; Longair, 2011), where $\gamma$ and $E_{\rm
max}$ are parameters to be determined from the observational data, and
$Q_{0}(t)$ can be related to the power released by the $\nu$NS and injected
into the ejecta from $L_{\rm inj}(t)=L_{0}(1+t/t_{q})^{-k}=\int_{0}^{E_{\rm
max}}E\,Q(E,t)dE$, so $Q_{0}(t)=q_{0}\left(1+t/t_{q}\right)^{-k}$, where
$q_{0}\equiv(2-\gamma)L_{0}/E_{\rm max}^{2-\gamma}$.

The bolometric synchrotron radiation power of a single electron is given by
(see, e.g., Longair, 2011)

$P_{\rm syn}(E,t)=\beta B_{*}^{2}(t)E^{2}\approx\frac{\beta}{\alpha}B_{*}\nu,$
(36)

where in the last equality we have used the fact that most of the radiation is
emitted at frequencies near the so-called critical frequency, $\nu_{\rm
crit}=\alpha B_{*}E^{2}$, where $\alpha=3e/(4\pi m_{e}^{3}c^{5})$. By setting
$N(E,t)=\eta\,\hat{t}^{l}E^{-p}$, so with the constants $\eta$ , $l$ and $p$
obtained by comparing this expression with Eq. (34), the synchrotron
luminosity radiated at frequencies from $\nu_{1}$ to $\nu_{2}>\nu_{1}$ can be
written as

$\displaystyle L_{\rm syn}(\nu_{1},\nu_{2};t)$
$\displaystyle=\int_{\nu_{1}}^{\nu_{2}}J_{\rm syn}(\nu,t)d\nu\approx\nu J_{\rm
syn}(\nu,t),$ $\displaystyle\approx\frac{\beta}{2}\alpha^{\frac{p-3}{2}}\eta
B_{*,0}^{\frac{p+1}{2}}\hat{t}^{\frac{2l-n(p+1)}{2}}\nu^{\frac{3-p}{2}}.$ (37)

where $\nu_{1}=\nu$, $\nu_{2}=\nu+\Delta\nu$, being $\Delta\nu$ the bandwidth.
Here, $J_{\rm syn}$ is the spectral density which is given by $J_{\rm
syn}(\nu,t)d\nu\approx P_{\rm syn}(\nu,t)N(E,t)dE$ (see, e.g., Longair, 2011).
In Eq. (5.1), we have made the approximation $\Delta\nu/\nu\ll 1$ because of
the power-law character of the spectral density. Despite the synchrotron
radiation of a single electron is beamed along the velocity of the particle,
we here consider an isotropic distribution of a large number of electrons with
an isotropic distribution of pitch angles, hence leading to an isotropic total
synchrotron luminosity.

###  5.2 Newborn NS evolution and pulsar emission

The $\nu$NS is subjected to the angular momentum loss driven by the magnetic
field braking. In the point dipole+quadrupole magnetic field model presented
in Pétri (2015), the total magnetic torque is given by

$\displaystyle\tau_{\rm mag}$ $\displaystyle=\tau_{\rm dip}+\tau_{\rm quad},$
(38) $\displaystyle\tau_{\rm dip}$ $\displaystyle=-\frac{2}{3}\frac{B_{\rm
dip}^{2}R^{6}\Omega^{3}}{c^{3}}\sin^{2}\alpha,$ (39) $\displaystyle\tau_{\rm
quad}$ $\displaystyle=-\frac{32}{135}\frac{B_{\rm
quad}^{2}R^{8}\Omega^{5}}{c^{5}}\sin^{2}\theta_{1}(\cos^{2}\theta_{2}+10\sin^{2}\theta_{2}),$
(40)

where $\alpha$ is the inclination angle of the magnetic dipole moment with
respect to the rotation axis, and the angles $\theta_{1}$ and $\theta_{2}$
specify the geometry of the quadrupole field. The strength of the magnetic
dipole field is $B_{\rm dip}$. The dipole pure axisymmetric mode ($m=0$) is
set by $\alpha=0$, and the pure $m=1$ mode by $\alpha=\pi/2$. The strength of
the quadrupole magnetic field is $B_{\rm quad}$. The quadrupole $m=0$ mode is
set by $\theta_{1}=0$, the $m=1$ mode by $\theta_{1}=\pi/2$ and
$\theta_{2}=0$, while the $m=2$ mode is set by $\theta_{1}=\theta_{2}=\pi/2$.
For the fit of the data, we shall adopt the $m=1$ mode for the dipole while
the quadrupole can range between the $m=1$ and $m=2$ modes. The existence of
multipolar magnetic fields in the newly born NS is supported by some theories
and observations (Tiengo et al., 2013; Mastrano et al., 2013; Rodríguez
Castillo et al., 2016; Pons & Viganò, 2019). Therefore, we can write the total
magnetic torque (38) as

$\tau_{\rm mag}=-\frac{2}{3}\frac{B_{\rm
dip}^{2}R^{6}\Omega^{3}}{c^{3}}\left(1+\xi^{2}\frac{16}{45}\frac{R^{2}\Omega^{2}}{c^{2}}\right),$
(41)

where $\xi$ is the quadrupole to dipole magnetic field strength ratio defined
by

$\xi\equiv\sqrt{\cos^{2}\theta_{2}+10\sin^{2}\theta_{2}}\frac{B_{\rm
quad}}{B_{\rm dip}},$ (42)

and the spindown luminosity as

$L_{\rm sd}=\Omega\,|\tau_{\rm mag}|=\frac{2}{3}\frac{B_{\rm
dip}^{2}R^{6}\Omega^{4}}{c^{3}}\left(1+\xi^{2}\frac{16}{45}\frac{R^{2}\Omega^{2}}{c^{2}}\right).$
(43)

The evolution of the $\nu$NS is obtained from the energy conservation equation

$-(\dot{W}+\dot{T})=L_{\rm tot}=L_{\rm inj}+L_{\rm sd},$ (44)

where $W$ and $T$ are, respectively, the $\nu$NS gravitational and rotational
energy.

##  6 Results

Figure 5: Luminosity of GRB 171205A in the X-rays ($0.3$–$10$ keV), optical (V
band), and radio ($607$ MHZ and $1.255$ GHz) energy bands compared with the
luminosity predicted by the theoretical model. The raising part of the radio
luminosity in the time interval $10^{6}$–$10^{7}$ s is due to synchrotron
self-absorption (see Maity & Chandra, 2021, for details), here unmodeled. The
X-ray data is retrieved from the Swift-XRT repository and analyzed by this
article, the optical and radio data are reproduced from D’Elia et al. (2018)
and Maity & Chandra (2021).

The emission of GRB 171205A comprises thermal and non-thermal components. In
Sec. 1.1, we recalled that Izzo et al. (2019) explains the thermal component
up to $10^{5}$ s in the X-rays and in the optical due to the cooling of fast
moving material. We here address the nature of the non-thermal component once
the material is transparent. Therefore, the present model of synchrotron
radiation described in Sec. 5, aims to explain the data that shows a
decreasing power-law luminosity in the different energy bands with similar
power-law index.

Table 2 summarizes the values of the model parameters that fit the afterglow
of GRB 171205A in the X-rays, optical, and radio energy bands, as shown in
Fig. 5, obtained according to the above guidelines and the fitting procedure
outlined in Appendix A.

Parameter | Value
---|---
$t_{*}$ ($10^{4}$ s) | $2.650\pm 110.276$
$B_{*,0}$ ($10^{5}$ G) | $3.774\pm 157.021$
$\gamma$ | $1.606\pm 0.231$
$E_{\rm max}$ ($10^{4}\ m_{e}c^{2}$) | $3.738$
$k$ | $1.219\pm 0.170$
$L_{0}$ ($10^{47}$ erg s-1) | $1.011\pm 0.801$
$t_{q}$ (s) | $100.00$
$B_{\rm dip}$ ($10^{12}$ G) | $1.000$
$P_{\nu\rm NS}$ (ms) | $46.852\pm 64.910$
Table 2: Value of the synchrotron model parameters that fit the
multiwavelength observational data of GRB 1701205A as shown in Fig. 5.

In the X-rays, the model describes the decreasing power-law behavior at times
$>10^{5}$ s, and in the radio at times $>10^{7}s$. We do not model the rising
part of the radio emission in the time interval $10^{6}$–$10^{7}$ s which is
due to synchrotron self-absorption (see Maity & Chandra, 2021, for details).

The first relevant feature to notice is that the afterglow luminosity fades
with time with an approximate power-law $t^{-1}$. This power-law is shallower
than in GRBs of higher luminosity in which $t^{-1.3}$ (see, e.g., GRB 130427A
or GRB 190114C in Ruffini et al., 2018a; Rueda et al., 2020). The pulsar
emission from magnetic braking predicts a luminosity with a sharper power-law,
in a pure magnetic dipole the luminosity falls as $t^{-2}$, and for a pure
magnetic quadrupole as $t^{-3/2}$ (see equations of Sec. 5.2 and Ruffini et
al., 2018a; Rueda et al., 2020). Therefore, models based on pulsar emission
from magnetic braking alone (even including higher order multipole fields) are
unable to fit the afterglow luminosity of GRB 171205A. This is a first
indication of the necessity of an additional mechanism, in this case the
synchrotron radiation. The second relevant feature is that the afterglow in
the X-rays and in the radio bands show similar power-law index (see the red,
gray and brown data points), as expected from the synchrotron model.

The optical data shows, instead, a flat behavior followed by the bump that
characterizes the peak of the SN emission powered by the decay of nickel in
the ejecta (Arnett, 1996; Izzo et al., 2019). Both the synchrotron radiation
and the SN radioactive decay contribute to the optical emission, but in GRB
171205A the latter dominates over the former. This explains the deviation of
the optical luminosity from the typical power-law behavior of the synchrotron
radiation. This feature is consistent with the BdHN III nature of the source.
In fact, BdHN III are low-luminous sources in which the $\nu$NS is not a very-
fast rotator, so it injects less energy into the ejecta in comparison to BdHNe
I (e.g., GRB 130427A, 180720B or 190114C; see Ruffini et al., 2018a; Rueda et
al., 2020) and BdHNe II (e.g., GRB 190829A; see Wang et al., 2022a).
Therefore, the synchrotron emission is not very luminous and the emergent
optical SN outshines the optical synchrotron luminosity. Interestingly, this
latter feature of the emergent optical SN emission is also fulfilled in the
most general situation of BdHN I and BdHN II (Aimuratov et al., to be
submitted). SN 2017iuk is similar to the SNe associated with high-luminous
GRBs, indicating that the pre-SN progenitor (i.e. the CO star and a NS
companion) leading to the $\nu$NS in its core-collapse event, is similar for
all long GRBs irrespective of their energetics (Aimuratov et al., to be
submitted).

In the X-rays, the synchrotron luminosity fades off after a few $10^{6}$ s,
when $h\nu_{\rm crit}$ falls below keV. At later times, the power-law behavior
continues in the optical and in the radio bands. The pulsar emission is
characterized by a plateau followed by a power-law decay (at times longer than
the characteristic spindown timescale). For a plateau luminosity comparable
(but smaller) to the synchrotron power-law luminosity, the sum of the two
contributions can lead to a luminosity with a less sharp power-law behavior
than the pure synchrotron. The afterglow of GRB 171205A does not show any sign
of change of the power-law of the synchrotron emission (see Fig. 5), so we can
not constrain the magnetic field strength and structure. In Fig. 5, we have
adopted $\approx 47$ ms as initial rotation period of the $\nu$NS and a pure
dipole field ($\xi=0$) of $B_{\rm dip}=10^{12}$ G to guide the eye of the
reader. For magnetic fields $\gtrsim 5\times 10^{13}$ G, the plateau
luminosity of the pulsar emission contributes appreciably to the total X-ray
luminosity affecting the goodness of the fit. Therefore, we can assume the
above estimate as an upper limit to the dipole magnetic field. For the present
synchrotron model parameters, X-ray data after times of a few $10^{6}$ s could
help to constrain the presence of the pulsar emission. A sanity check of the
model is that the energy injected into the ejecta is $\sim 10^{49}$ erg, of
the same order of the rotational energy of the $\nu$NS, for a moment of
inertia of a few $10^{45}$ g cm2.

##  7 Conclusions

In this article, we have interpreted GRB 171205A within the BdHN model of long
GRBs. In particular, because of the low energy release of only a few $10^{49}$
erg, we have classified GRB 171205A as a BdHN III, systems with long orbital
periods, perhaps of the order of hours, in which the NS companion does not
play any role in the cataclysmic event. Most of these binaries are also
expected to be disrupted by the SN explosion (Fryer et al., 2015; Ruffini et
al., 2016, 2018c). Under these circumstances, the GRB event is explained by
the sole activity of the $\nu$NS and its interaction with the SN ejecta.

We have here shown that GRB 171205A is a low-luminous GRB consistent with it
being produced in the core-collapse of a single CO star that forms the $\nu$NS
and the type Ic SN. There are several new results related to the sequence of
physical phenomena occurring in this system and the related GRB observables:

  1. 1.

The fallback accretion is initially of a few of $10^{-3}M_{\odot}$ s-1 and
lasts tens of seconds (Becerra et al., 2019, 2022). The accretion energy is
$\sim 10^{52}$ erg, comparable to the kinetic energy of the SN ejecta. This
energy is injected into the ejecta, propagates, and accelerates the outermost
shell to the observed mild-relativistic velocity. The hydrodynamics is similar
to the case of the expanding SN ejecta with the GRB energy injection presented
in Ruffini et al. (2018b). The Lorentz factor of the shockwave is $\lesssim 5$
when it gets transparency at $\sim 10^{12}$ cm, and emit a thermal spectrum.
This scenario explains the prompt emission of GRB 171205A. This is also
similar to the cocoon scenario advanced for this source in Izzo et al. (2019).
Both pictures predict the heating of stellar shells (in our case by the
physical process of the fallback accretion originating from the SN explosion
and in the other by postulation a unspecified jet) that get boosted to high-
velocity and emit a thermal spectrum. The associated blackbody emission has
been indeed observed in GRB 171205A, and it has been inferred that $\approx
10^{-3}M_{\odot}$ of material expand at velocities above $10^{5}$ km s-1 (see
Izzo et al., 2019 and Fig. 2). The main difference between the two models is
that in our picture there is not jet. This solutions seems favoured since the
associated jet break expected in the afterglow of jetted GRB models is not
observed in the data up to the last observations at $\sim 1000$ days (Leung et
al., 2021; Maity & Chandra, 2021).

  2. 2.

Regarding the afterglow emission, we have first inferred from an energy
conservation argument, that the $\nu$NS should have started to lose its
rotational energy at $t=35$ s after the GRB trigger, i.e., from what we call
the $\nu$NS-rise, with a rotation period of $47$ ms.

  3. 3.

We have shown that the afterglow of GRB 171205A can not be explained by the
sole pulsar emission of the $\nu$NS by magnetic braking, even including higher
multipole fields (e.g., quadrupole).

  4. 4.

The multiwavelength afterglow is explained by synchrotron radiation emitted by
electrons in the expanding SN, which is further powered by energy injected by
the $\nu$NS. We have calculated the synchrotron luminosity in the X-rays, the
optical and radio wavelengths with an analytic treatment of the above physical
situation. We have shown that the X-rays and the radio luminosities follow the
expectation from the synchrotron model. The raising part of the radio
luminosity in the time interval $10^{6}$–$10^{7}$ s is due to synchrotron
self-absorption (see Maity & Chandra, 2021, for details). The observed optical
luminosity shows a flat behavior followed by the bump of the optical SN
powered by the energy release in the ejecta of the radioactive decay of nickel
into cobalt. We have shown that the synchrotron luminosity in those optical
wavelengths lies below the luminosity of the emergent SN optical emission.
This implies that the observed optical emission contains the contribution of
both the synchrotron radiation and the optical SN.

  5. 5.

Another remarkable fact to be highlighted is that SN 2017iuk, a SN associated
with the low-luminous GRB 171205A, a BdHN III, shows similar properties (e.g.,
peak luminosity and peak time) to the SNe associated with high-luminous GRBs
(BdHN I and II). This suggests that the pre-SN progenitor (i.e., the CO star)
is similar for all long GRBs, irrespective of their energetics (Aimuratov et
al., to be submitted).

  6. 6.

There is a corollary of the above result. In low-luminous GRBs, i.e., in BdHN
III like GRB 171205A, the relatively slow rotation ($47$ ms period) of the
$\nu$NS implies the less energy injected into the ejecta, hence the low
energetics of the associated synchrotron emission. Under these circumstances,
the optical emission of the SN powered by the nickel radioactive decay is able
to outshine the optical synchrotron luminosity.

  7. 7.

We calculated the evolution of the $\nu$NS mass and angular momentum (assumed
to be initially zero) during the fallback accretion process leading to its
spinning up to the $47$ ms rotation period. From this evolution, we have
inferred that the SN explosion occurred at most $7.36$ h before the GRB
trigger time. This sets an estimate of the time delay between the SN explosion
and the electromagnetic emission of the GRB event, assuming a $\nu$NS born
with zero spin.

We thank the Referee for comments and suggestions that helped us in the
presentation of the article. L.M.B. is supported by the Vicerrectoría de
Investigación y Extensión - Universidad Industrial de Santander Postdoctoral
Fellowship Program No. 2022000293.

##  Appendix A Fitting procedure

In this appendix, we describe how we set the value of the model parameters
from the physical scenario and specific observables, including the attached
uncertainties. The parameters to be specified are the index of the electron’s
energy injection, $\gamma$, the parameters defining the injected power $k$,
$L_{0}$, and $t_{q}$, the maximum energy of the electrons, $E_{\rm max}$, the
self-similar expansion timescale, $t_{*}$, the magnetic field at the initial
time of reference of the expansion, $B_{*.0}$, the $\nu$NS dipole magnetic
field strength $B_{\rm dip}$. In Sec. 4, we have already fixed the $\nu$NS
rotation period, $P_{\nu\rm NS}$. Our aim is to estimate the uncertainty in
each parameter from the propagation of the 1-$\sigma$ uncertainty of the
power-law fit of the X-ray and radio luminosity.

Equation (5.1) shows that the signature of the present synchrotron model is
the power-law luminosity in the different bands with (ideally) the same power-
law index. Therefore, we constrain the synchrotron model parameters using the
observational data showing the above property. Figure 5 shows that the X-ray
($0.3$–$10$ keV) luminosity data behaves as a power-law in the time interval
$t\approx(0.87$–$5)\times 10^{6}$ s, and the radio ($1255$ MHz) data in the
time interval $t\approx(2.92$–$8)\times 10^{7}$ s. The two luminosities are
fitted by

$L_{X}=A_{X}\,t^{-\alpha_{X}},\qquad L_{r}=A_{r}\,t^{-\alpha_{r}},$ (A1)

where $A_{X}=(3.165\pm 0.238)\times 10^{47}$ erg s-1, $\alpha_{X}=1.022\pm
0.055$, $A_{r}=(4.290\pm 0.178)\times 10^{42}$ erg s-1, and
$\alpha_{r}=0.616\pm 0.081$. The uncertainties at 1-$\sigma$ level. To
estimate the uncertainties in the value of the model parameters, derived from
the above fit, we follow the standard theory of error propagation. For
instance, given quantity $f$ that is a function of the independent variables
$a_{i}$, i.e., $f(a_{1},a_{2},...,a_{n})$, its uncertainty can be estimated as
(see, e.g., Ku, 1966)

$\delta f=\sum_{i=1}^{n}\sqrt{\left|\frac{\partial f}{\partial
a_{i}}\right|^{2}(\delta a_{i})^{2}}\approx\sum_{i=1}^{n}\left|\frac{\partial
f}{\partial a_{i}}\right|\delta a_{i}.$ (A2)

Thus, the uncertainties of the luminosities given by the power-law fits (A1),
at a time $t$, can be estimated by

$\delta L_{i}\approx\left|\frac{\partial L_{i}}{\partial A_{i}}\right|\delta
A_{i}+\left|\frac{\partial
L_{i}}{\partial\alpha_{i}}\right|\delta\alpha_{i}=\frac{\delta
A_{i}}{A_{i,c}}+|-\ln t|\,\delta\alpha_{i},\qquad i=X,r,$ (A3)

where $A_{X,c}=3.165\times 10^{47}$ erg s-1, $A_{r,c}=4.290\times 10^{42}$ erg
s-1, $\delta A_{X}=0.238\times 10^{47}$ erg s-1, $\alpha_{X,c}=1.022$,
$\alpha_{r,c}=0.616$, $\delta A_{r}=0.178\times 10^{42}$ erg s-1,
$\delta\alpha_{X}=0.055$, and $\delta\alpha_{r}=0.081$.

We turn to the self-similar expansion timescale, $t_{*}$, for which we must
set values for $R_{*,0}$ and $v_{*,0}$. For $v_{*,0}$, we chose a fiducial
value according to numerical simulations of the SN explosion (see, e.g.,
Becerra et al., 2016, 2019), so we set $v_{*,0}=10^{8}$ cm s-1 and there is no
propagated uncertainty to calculate for. With the above, we set the expansion
timescale and its attached uncertainty

$t_{*}=\frac{R_{*,0}}{v_{*,0}},\qquad\delta t_{*}=\frac{\delta
R_{*,0}}{v_{*,0}}.$ (A4)

According to our working assumption of uniform expansion, the inner radius and
its uncertainty are

$R_{*,0}=v_{*,0}\,t_{\rm SN},\qquad\delta R_{*,0}=v_{*,0}\,\delta t_{\rm SN},$
(A5)

where $t_{\rm SN}$ is the time since SN given by Eqs. (28) and (32), and
$\delta t_{\rm SN}$ its uncertainty. For the above expansion velocity
$v_{*,0}$, and the time since SN estimated in Sec. 4, $t_{\rm SN}\approx
2.650\times 10^{4}$ s $\approx 7.36$ h, we have $R_{*,0}\approx 2.65\times
10^{12}$ cm and $t_{*}=t_{\rm SN}=2.65\times 10^{4}$ s. The uncertainty
attached to the time $t_{\rm SN}$ can be estimated as

$\delta t_{\rm SN}=\left|\frac{\partial t_{\rm SN}}{\partial\Omega_{\nu\rm
NS}}\right|\delta\Omega_{\nu\rm NS},\qquad\frac{\partial t_{\rm
SN}}{\partial\Omega_{\nu\rm
NS}}=\frac{1}{\omega}\frac{4\tilde{\alpha}-X}{2X-B}\frac{t_{\rm
acc}}{1-p}\left[(1-X)^{\frac{p}{1-p}}\right],$ (A6)

where $X$, and $B$ are given by Eqs. (29)–(30), evaluated at the time
$t=t_{\nu\rm NS}$ s, so $\Delta\Omega=\Omega_{\nu\rm NS}$. The uncertainty in
estimating $\Omega_{\nu\rm NS}$ from Eq. (25) is given by

$\delta\Omega_{\nu\rm NS}=\left|\frac{\partial\Omega_{\nu\rm NS}}{\partial
A_{X}}\right|\delta A_{X}+\left|\frac{\partial\Omega_{\nu\rm
NS}}{\partial\alpha_{X}}\right|\delta\alpha_{X},\quad\frac{\partial\Omega_{\nu\rm
NS}}{\partial A_{X}}=\frac{\Omega_{\nu\rm
NS}}{2A_{X}},\quad\frac{\partial\Omega_{\nu\rm
NS}}{\partial\alpha_{X}}=-\frac{\Omega_{\nu\rm
NS}}{2}\left(\frac{1}{\alpha_{X}-1}+\ln t_{\nu\rm NS}\right).$ (A7)

For the present parameters, i.e, $t_{\nu\rm NS}=35$ s and $\Omega_{\nu\rm
NS}=134.11$ rad s-1, we obtain from Eq. (A7), $\delta\Omega_{\nu\rm NS}\approx
185.795$ rad s-1, so an uncertainty in the rotation period, $\delta P_{\nu\rm
NS}=|\partial P_{\nu\rm NS}/\partial\Omega_{\nu\rm NS}|\delta\Omega_{\nu\rm
NS}=2\pi\delta\Omega_{\nu\rm NS}/\Omega^{2}_{\nu\rm NS}\approx 64.910$ ms.
Using the above in Eq. (A6), we obtain $\delta t_{\rm SN}\approx 110.276\times
10^{4}$ s $\approx 306.220$ h. Thus, we get from Eq. (A5), $\delta
R_{*,0}\approx 110.276\times 10^{12}$ cm, and from Eq. (A4), $\delta
t_{*}=\delta t_{\rm SN}$.

At large distances from the $\nu$NS, we expect the toroidal component of the
magnetic field to dominate, which decays with distance as $r^{-1}$ (see, e.g.,
Goldreich & Julian, 1969). Assuming a toroidal field of the same order as the
poloidal field near the $\nu$NS surface, its value at the radius $r=R_{*,0}$
is

$B_{*,0}\approx B_{\rm dip}\frac{R}{R_{*,0}}=B_{\rm
dip}\frac{R}{v_{*,0}t_{*}},$ (A8)

where $B_{\rm dip}$ is the strength of the dipole magnetic field and $R$ is
the fiducial $\nu$NS radius. We shall see in Sec. 7 that the data does not
constraint the dipole field but only set an approximate upper limit of $B_{\rm
dip,max}\approx 5\times 10^{13}$ G. Therefore, we shall adopt a fiducial,
conservative magnetic field value $B_{\rm dip}=10^{12}$ G. By using a fiducial
$\nu$NS radius $R=10^{6}$ cm, and the value of $R_{*,0}$ given by Eq. (A5), we
obtain $B_{*,0}\approx 3.774\times 10^{5}$ G. With the choice (A8), the
attached uncertainty is given by

$\delta B_{*,0}=B_{*,0}\frac{\delta R_{*,0}}{R_{*,0}}=B_{*,0}\frac{\delta
t_{*}}{t_{*}}=B_{*,0}\frac{\delta t_{\rm SN}}{t_{\rm SN}},$ (A9)

which leads to $\delta B_{*,0}\approx 157.021\times 10^{5}$ G.

We now set the index $\gamma$. From Eq. (5.1), we infer that the ratio of the
synchrotron luminosity at two frequencies, $\nu_{1}$ and $\nu_{2}$, is given
by $L_{\rm syn}(\nu_{1})/L_{\rm
syn}(\nu_{2})=(\nu_{1}/\nu_{2})^{\frac{3-p}{2}}$, where $p=\gamma+1$.
Therefore, we can constrain the value of the index $\gamma$ using the data in
the X-rays and in the radio as

$\gamma=2\left[1-\frac{\ln(L_{X}/L_{r})}{\ln(\nu_{X}/\nu_{r})}\right],$ (A10)

where $L_{X}$ and $L_{r}$ are given in Eqs. (A1). Since the fitted power-laws
are not equal, the value of $\gamma$ inferred from Eq. (A10) depends on the
time at which we calculate the ratio of the luminosities. Therefore, we adopt
for $\gamma$ the value given by the mean $\langle\gamma\rangle=\Delta
t^{-1}\int\gamma dt$. We obtained $\langle\gamma\rangle\approx 1.6060$, where
we have used $\Delta t\approx 8\times 10^{7}$ s, $\nu_{X}=10$ keV/$h\approx
2.423\times 10^{18}$ Hz and $\nu_{r}=1255$ MHz. From Eq. (A10), the
uncertainty in the choice of $\gamma$ can be estimated as

$\delta\gamma=\left|\frac{\partial\gamma}{\partial L_{X}}\right|\delta
L_{X}+\left|\frac{\partial\gamma}{\partial L_{r}}\right|\delta
L_{r}=\frac{2}{\ln(\nu_{X}/\nu_{r})}\left(\frac{\delta
L_{X}}{L_{X}}+\frac{\delta L_{r}}{L_{r}}\right),$ (A11)

whose mean for the above parameters is $\langle\delta\gamma\rangle=0.231$.

The synchrotron emission peaks around the critical frequency

$\nu_{\rm crit}=\alpha B_{*}E^{2}=\alpha B_{*,0}E^{2}\frac{t_{*}}{t},$ (A12)

and then cutoffs exponentially, where $E$ is the electron energy. Since the
critical frequency decreases with time, there is a hard-to-soft evolution of
the cutoff and the X-rays data give the strongest constraint. The electrons of
maximum energy, $E_{\rm max}$, produce the maximum critical frequency,
$\nu_{\rm crit,max}$. By requiring that $\nu_{\rm crit,max}=\nu_{X}$ at a
cutoff time $t_{\rm cut,X}>t_{f,X}$, where $t_{f,X}\approx 3.5\times 10^{6}$ s
is the time of the last observational X-ray data, we obtain that the maximum
electron energy must at least have the value

$E_{\rm max}=\sqrt{\frac{\nu_{X}\,t_{\rm
cut,X}}{\alpha\,t_{*}\,B_{*,0}}}=\sqrt{\frac{\nu_{X}\,v_{*,0}\,t_{\rm
cut,X}}{\alpha\,B_{\rm dip}R}},$ (A13)

where in the last equality we have used Eq. (A8). The cutoff time must allow
the power-law luminosity to extend at least up to $t_{f,X}$. Thus, we chose
$t_{\rm cut,X}$ such that the exponential cutoff at the time $t=t_{f,X}$ has
reduced the power-law X-ray luminosity to one part in a thousand. With this
condition, we find $t_{\rm cut,X}\approx 2.418\times 10^{7}$ s, so $E_{\rm
max}=3.738\times 10^{4}m_{e}c^{2}$. Equation (A13) tells us that $E_{\rm
max}$, chosen in this way, depends only on fiducial values that we have set
for $v_{*,0}$, $B_{\rm dip}$, $R$ and $t_{\rm cut,X}$, so we can not estimate
an attached uncertainty to it.

Having set all the above parameters, it remains to set the parameters of the
injected power, $L_{0}$, $k$ and $t_{q}$. The synchrotron luminosity increases
at times $t<t_{q}$ (see Sec. 5.1) and decreases at times $t>t_{q}$. The X-rays
luminosity shows always a decreasing behavior, so we set $t_{q}=100$ s, which
roughly corresponds to the initial time of the X-ray data. For the parameters
$L_{0}$ and $k$, we equate the model synchrotron luminosity (5.1) in the case
of X-rays with the power-law luminosity (A1). From this equality, we obtain

$\displaystyle k$ $\displaystyle=\frac{2+2\alpha_{X}-\gamma}{2},$ (A14)
$\displaystyle L_{0}$
$\displaystyle=2\frac{A_{X}}{t_{*}^{\alpha_{X}}}\frac{\gamma-1}{2-\gamma}\left(\frac{t_{*}}{t_{q}}\right)^{k}\left(\frac{t_{\rm
cut,X}}{t_{*}}\right)^{\frac{2-\gamma}{2}}=2A_{X}\frac{\gamma-1}{2-\gamma}t_{q}^{-\alpha_{X}}\left(\frac{t_{\rm
cut,X}}{t_{q}}\right)^{\frac{2-\gamma}{2}},$ (A15)

where we have used Eq. (A13). For the present parameters, we obtain $k=1.219$
and $L_{0}=1.011\times 10^{47}$ erg s-1. Therefore, we can estimate the error
of the above quantities as

$\displaystyle\delta k$ $\displaystyle=\left|\frac{\partial
k}{\partial\alpha_{X}}\right|\delta\alpha_{X}+\left|\frac{\partial
k}{\partial\gamma}\right|\delta\gamma=\delta\alpha_{X}+\frac{1}{2}\delta\gamma,$
(A16) $\displaystyle\delta L_{0}$ $\displaystyle=\left|\frac{\partial
L_{0}}{\partial A_{X}}\right|\delta A_{X}+\left|\frac{\partial
L_{0}}{\partial\alpha_{X}}\right|\delta\alpha_{X}+\left|\frac{\partial
L_{0}}{\partial\gamma}\right|\delta\gamma=L_{0}\left[\frac{\delta
A_{X}}{A_{X}}+\ln
t_{q}\,\delta\alpha_{X}+\left|\frac{1}{(\gamma-1)(2-\gamma)}-\frac{1}{2}\ln\left(\frac{t_{\rm
cut,X}}{t_{q}}\right)\right|\delta\gamma\right],$ (A17)

which read $\delta k=0.170$ and $\delta L_{0}\approx 0.792L_{0}\approx
0.801\times 10^{47}$ erg s-1.

The large uncertainty in the estimate of $t_{*}$ and $B_{*,0}$ is a
consequence of the propagation of the uncertainty of $R_{*,0}$, which arises
from the uncertainty in the estimate of the SN time, $t_{\rm SN}$, because it
is a sensitive function of $\Omega_{\nu\rm NS}$.

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