Modeling The Most Luminous Supernova Associated with a Gamma-Ray Burst, SN 2011kl
Shan-Qin Wang Zach Cano Ling-Jun Wang Wei-Kang Zheng Zi-Gao Dai Alexei V. Filippenko Liang-Duan Liu
astro-ph.HE






We study the most luminous known supernova (SN) associated with a gamma-ray
burst (GRB), SN 2011kl.
The photospheric velocity of SN 2011kl around peak brightness is
21,000±7,000 km s^-1. Owing to different assumptions related to the
light-curve (LC) evolution (broken or unbroken power-law function) of the
optical afterglow of GRB 111209A, different techniques for the LC
decomposition, and different methods (with or without a near-infrared
contribution), three groups derived three different bolometric LCs for
SN 2011kl. Previous studies have shown that the LCs without an early-time
excess preferred a magnetar model, a magnetar+^56Ni model, or a white
dwarf tidal disruption event model rather than the radioactive heating
model. On the other hand, the LC shows an early-time excess and dip that
cannot be reproduced by the aforementioned models, and hence the
blue-supergiant model was proposed to explain it. 
Here, we reinvestigate the energy sources powering SN 2011kl. We find that
the two LCs without the early-time excess of SN 2011kl can be explained by
the magnetar+^56Ni model, and the LC showing the early excess can be
explained by the magnetar+^56Ni model taking into account the cooling
emission from the shock-heated envelope of the SN progenitor, demonstrating
that this SN might primarily be powered by a nascent magnetar.




1School of Astronomy and Space Science, Nanjing
University, Nanjing 210093, China; dzg@nju.edu.cn
2Key Laboratory of Modern Astronomy and Astrophysics (Nanjing
University), Ministry of Education, China
3Department
of Astronomy, University of California, Berkeley, CA 94720-3411, USA
4Instituto de Astrofísica de Andalucía (IAA-CSIC),
Glorieta de la Astronomía s/n, E-18008, Granada, Spain
5Astroparticle Physics,
Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, China
6Miller Senior Fellow, Miller Institute for Basic
Research in Science, University of California, Berkeley, CA 94720, USA



§ INTRODUCTION




The optical spectra of Type Ic supernovae (SNe Ic) lack hydrogen and helium
absorption lines (see, e.g.,  for reviews). A small
fraction of SNe Ic have spectra with broad absorption troughs indicating
large photospheric velocities and were dubbed “broad-lined SNe Ic
(SNe Ic-BL)" <cit.>. Some SNe Ic-BL have very large kinetic
energies (≳ 10^52 erg; e.g.,
) and were coined
“hypernovae" <cit.>. Hypernovae are usually rather luminous (with
peak luminosities ∼ 10^43 erg s^-1), indicating that they must
have synthesized ∼ 0.5 M_⊙ of ^56Ni if they can be explained
by the radioactive heating model <cit.>.
Moreover, some SNe Ic-BL are associated with long-duration gamma-ray bursts
(GRBs; e.g., ) and have been called
“GRB-SNe" (see , and references therein).
While most GRB-SNe are Type Ic-BL, not all SNe Ic-BL are associated with
GRBs. The most luminous SNe Ic are Type Ic superluminous supernovae (SLSNe;
); their peak luminosities are ≳
7×10^43 erg s^-1 <cit.>, tens of times greater than
those of ordinary SNe.

In this paper, we study a very luminous SN Ic, SN 2011kl, which is
associated with an ultralong (T_90≈ 10^4 s) GRB 111209A
(see, e.g., ) at
redshift z=0.677 <cit.>. By using WMAP ΛCDM cosmology
<cit.> and adopting H_0 =71 km s^-1 Mpc^-1, Ω_M =
0.27, and Ω_Λ= 0.73, <cit.> derived a luminosity
distance (D_L) of 4076.5 Mpc for GRB 111209A/SN 2011kl. With a peak
luminosity ∼ 3.63^+0.17_-0.16× 10^43 erg s^-1
<cit.>[<cit.> found a slightly lower peak
luminosity (∼ 2.8^+1.2_-1.0× 10^43 erg s^-1) since they
ignored the near-infrared contribution.], SN 2011kl is the most luminous
GRB-SN yet detected and is significantly more luminous than all other
GRB-SNe, whose average peak luminosity is 1 × 10^43 erg s^-1
with a standard deviation of 0.4 × 10^43 erg s^-1
<cit.>. The time of peak brightness of SN 2011kl is ∼ 14
days, slightly larger than the average (13.0 days with a standard deviation
of 2.7 days; ) of other GRB-SNe.

The peak luminosity of SN 2011kl is comparable to that of some
not-quite-superluminous “SLSNe" (e.g.,
PTF11rks, PTF10hgi, iPTF15esb, iPTF16bad, and PS15br) and commensurate with
the SN-SLSN gap transients observed by <cit.>; moreover,
<cit.> pointed out that the spectrum of SN 2011kl is very
different from those of GRB-SNe, but consistent with that of SLSNe. So, the
energy sources powering its LC might be similar to those powering the
SLSN LCs instead of the ordinary SNe Ic. The most popular prevailing models
for explaining SLSNe are the magnetar model
<cit.>
and the ejecta plus circumstellar medium (CSM) interaction model
<cit.>.
[Almost all SLSNe (except maybe pair-instability SLSNe —
e.g., SN 2007bi ;
but see  and ) cannot be explained by the
radioactive heating model <cit.>. Some
SLSNe I exhibiting late-time rebrightening and Hα emission lines
<cit.> might be powered by a hybrid model containing
contributions from magnetars and ejecta-CSM interaction <cit.>.]

There have been several papers modeling the LCs of SN 2011kl. The LC
obtained by <cit.> (hereafter the G15 LC) cannot be explained by
the ^56Ni model but can be by the magnetar model <cit.>
[<cit.> showed that GRB 111209A and SN 2011kl can be
explained purely by the magnetar model, both for the afterglow and
SN-powered phases.], the magnetar+^56Ni model <cit.>,
the white dwarf tidal disruption event (WD TDE) model <cit.>,
and the collapsar model <cit.> involving a stellar-mass
black hole and a fallback accretion disk <cit.>. By
subtracting the contribution from the ultraviolet (UV)/optical/near-infrared
(NIR) afterglow <cit.>
of GRB 111209A, <cit.> obtained a new bolometric LC
(hereafter the K16 LC) and modeled it with the radioactive heating model,
finding 2.27 ± 0.64 M_⊙ of ^56Ni and 6.79_-2.84^+3.67 M_⊙
of ejecta, suggesting that this model is disfavored. The LC derived by
<cit.> (hereafter the I16 LC) has an early-time excess and a dip
[This early-time excess arose from the assumption that a
single power law
could describe the evolution of the optical LC during the afterglow phase. A
broken power law showed no such excess (e.g., ),
so this excess is model-dependent. Moreover, a single power law was used by
<cit.>, and no such excess was observed, making it inconclusive.]
which cannot be reproduced by the models mentioned above, but it can be fit
by the blue supergiant (BSG) model with explosive energy injection
(“the BSG model"; ).
However, it is premature to exclude
the magnetar model in explaining the I16 LC since the cooling emission from
the shock-heated extended envelope <cit.> or the emission from
magnetar shock breakout <cit.> can reproduce the early-time excess.

Thus, the issue of the energy source of SN 2011kl deserves additional
attention. In this paper, we reinvestigate the possible mechanisms powering
the LC of SN 2011kl and discuss their implications. In Section <ref>
, we use the radioactive heating model, the magnetar model, the
magnetar+^56Ni model, and the
magnetar+^56Ni+cooling model to fit the LCs of
SN 2011kl. Our discussion and conclusions are presented in Sections
<ref> and <ref>, respectively.



§ MODELING THE LIGHT CURVES OF SN 2011KL




Extracting the LC of a SN associated with a GRB is difficult since this
process requires a decomposition of the observations to account for the
contributions from the optical afterglow, the SN, and the underlying host
galaxy (e.g., ); moreover, it depends on the assumptions
adopted for modeling the GRB optical afterglow. By assuming that the LC of
the optical afterglow of GRB 111209A can be described by a broken power-law
function and subtracting the contributions from the optical afterglow and
the host galaxy, <cit.> obtained the LC of SN 2011kl (the G15 LC).
The G15 LC did not include the NIR contribution and is a
pseudobolometric LC. By adding the NIR contribution, <cit.>
obtained a genuine bolometric LC (the K16 LC)[The
K16 LC must be brighter than the pseudobolometric LC (the G15 LC)
without the NIR contribution. However, the median values of the luminosities
of the first two points of the K16 LC are dimmer than that of the G15 LC.
This fact is puzzling, perhaps indicating that the uncertainties are
rather large.]. However, based on the assumption that the optical afterglow
of GRB 111209A can also be described by a single (unbroken) power-law
function, <cit.> derived another LC (the I16 LC) for SN 2011kl.
This LC has an early-time excess and a dip. Hence, SN 2011kl has three
different LCs (see Figure <ref>) that must be modeled.

The LCs of SLSNe PTF11rks and PTF10hgi <cit.> are also plotted in
Figure <ref>. By comparing the K16 LC of GRB-SN 2011kl, whose peak
luminosity L_peak,bol is ∼ 3.6 × 10^43 erg s^-1,
to the LCs of SLSNe PTF11rks and PTF10hgi, we find that the peak of the LC
of SN 2011kl is as luminous as that of these two SLSNe. Furthermore, we
point out that other SLSNe also have similar values of L_peak,bol
 — e.g., iPTF15esb (L_peak,bol≈ 4× 10^43
erg s^-1; ), iPTF16bad (L_peak,bol≈
4× 10^43 erg s^-1; ), and PS15br
(L_peak,bol≈ 4.15× 10^43 erg s^-1;
).[SN 2011kl,
along with PTF11rks, PTF10hgi, iPTF15esb, iPTF16bad, and PS15br,
are dimmer than the strict threshold for SLSNe (L_peak>7×
10^43 erg s^-1 and the peak absolute magnitudes
M_peak<-21 mag in any band; ).]

In this section, we use some models to fit these LCs. For the G15 and
K16 LCs, we adopt the Markov Chain Monte Carlo (MCMC) method to minimize
the values of χ^2/dof and get the best-fit parameters and the range.
For the I16 LC, we don't use the MCMC method because its data have no
known error bars.



 §.§ The Radioactive Heating Model


Based on their derived LC (the G15 LC), <cit.> fit the ^56Ni
model and found that the required mass of ^56Ni is ∼ 1 M_⊙ if
the LC was powered by ^56Ni decay. <cit.> suggested that their
bolometric LC (the K16 LC), which included the NIR contribution, needed
2.27 ± 0.64 M_⊙ of ^56Ni to reproduce its luminosity if the LC
was powered by radioactive heating.

Here we employ the semi-analytic ^56Ni model
<cit.> to repeat the fits for the G15 LC,
the K16 LC and I16LC. The free parameters of the ^56Ni model are the
optical opacity κ, the ejecta mass M_ej, the ^56Ni
mass M_Ni, the initial scale velocity of the
ejecta[The
scale velocity of the ejecta v_sc is the velocity of the
surface of the ejecta. The radius of the ejecta can be calculated from R(t)
= R(0)+v_sct <cit.>.] v_sc0, and the
gamma-ray opacity of ^56Ni-cascade-decay photons
κ_γ,Ni.
The value of κ is rather uncertain and has been
assumed to be 0.06 cm^2 g^-1 (e.g., ), 0.07
cm^2 g^-1 (e.g., ), 0.07±0.01
cm^2 g^-1 (e.g., ), 0.08 cm^2 g^-1
(e.g., ), 0.10 cm^2 g^-1 (e.g.,
), and 0.20 cm^2 g^-1 (e.g.,
). Assuming that the dominant opacity source
is Thomson electron scattering and the temperature of the SN ejecta consisting
of carbon and oxygen is not very high (≲ 10,000 K), we adopted 0.07 
cm^2 g^-1 to be the value of κ. The value of κ_γ,Ni
is fixed to be 0.027 cm^2 g^-1 (e.g.,
).
 The photospheric velocity of SN 2011kl inferred from the spectra is ∼
21,000±7,000 km s^-1 <cit.>.
We assume that the moment of explosion (t_expl) for SN
2011kl is equal to that of GRB 111209A.

The LCs reproduced by the radioactive heating models A1, A2, and A3 are
shown in Figure <ref>, and the best-fit parameters are listed
in Table <ref>. While the best-fit model parameters for the
G15 LC are approximately equal to that derived by <cit.>, our
inferred ^56Ni mass (1.42_-0.04^+0.04 M_⊙) for the
K16 LC is smaller than the value derived by <cit.> (2.27±0.64
M_⊙). Moreover, the value of the ejecta mass
(4.57_-1.03^+0.80 M_⊙) is also smaller than that
derived by <cit.> (6.79_-2.84^+3.67 M_⊙).

Another method for estimating the value of ^56Ni is to use the “Arnett
law" <cit.>, which says that the ^56Ni energy input LC
intersects the peak of the SN LC. According to the Arnett law
and Eq. 19 of <cit.>, the mass of
^56Ni is [Eq. 19
of <cit.> is valid only in the case of full gamma-ray
trapping. For partial gamma-ray trapping (κ_γ,Ni
= 0.027 cm^2 g^-1), however, the inferred ^56Ni mass is equal to
that corresponding to the case of full gamma-ray trapping, because the gamma-ray
leakage doesn't influence the luminosity before and around the LC peak.]

    M_Ni=(L_peak/10^43 erg s^-1)
    [6.45 e^-t_rise/8.8 days
    +1.45 e^-t_rise/111.3 days]^-1 M_⊙,

since the value of t_rise of SN 2011kl is ∼ 14 days and
the peak luminosity is 3.63^+0.17_-0.16× 10^43 erg s^-1
(for the K16 LC) or 2.8^+1.2_-1.0× 10^43 erg s^-1 (for the
G15 LC), the ^56Ni mass is 1.40^+0.07_-0.06 M_⊙ (for the
K16 LC) or 1.08^+0.46_-0.39 M_⊙ (for the G15 LC).



 §.§ The Magnetar Model


<cit.> analyzed the spectrum of SN 2011kl and pointed out that the
spectral features imply that the ^56Ni mass must be significantly
smaller than 1 M_⊙; they excluded the ^56Ni model for explaining
the LC of SN 2011kl. Alternatively, <cit.> employed the magnetar
model developed by <cit.> to fit the G15 LC and obtained a rather
good result. But the K16 LC has not yet been fit with the magnetar model.

Here, we use the semianalytic magnetar model developed by <cit.>
and <cit.> that considers the leakage, photospheric recession,
and acceleration effects to fit the K16 LC, and we repeat the fits for the
G15 LC. The free parameters of the magnetar model are κ,
M_ej,
the magnetic strength of the magnetar B_p, the initial rotation
period of the magnetar P_0, v_sc0, and the gamma-ray
opacity of magnetar photons κ _γ ,mag. The
value of κ _γ ,mag depends mainly on the energy of
the photons (E_photons) emitted from the nascent magnetars,
varying between ∼ 0.01 and 0.2 cm^2 g^-1 for γ-ray
photons (E_γ≳ 10^6 eV)
and between ∼ 0.2 and 10^4 cm^2 g^-1
for X-ray photons (10^2 eV ≲ E_X≲
10^6 eV); see Fig. 8 of <cit.>.

The LCs reproduced by the magnetar models (B1, B2, and B3) are shown in
Figure <ref> and the best-fit parameters are listed in Table
<ref>. We find that the K16 LC can be powered by a magnetar having
B_p = 5.99_-5.55^+1.75×10^14 G and P_0 =
10.83_-6.94^+0.49 ms, while the G15 LC can be powered by a magnetar
with B_p = 9.72_-4.13^+3.23×10^14 G and P_0 =
12.07_-2.52^+1.15 ms (the values of B_p and P_0 derived by
<cit.> are ∼ (6–9) ×10^14 G and ∼ 13.4 ms,
respectively).



 §.§ The Magnetar Plus ^56Ni Model


The contribution of ^56Ni cannot be ignored when modeling some luminous
SNe Ic <cit.>. <cit.> instead used a
hybrid model containing contributions from ^56Ni and a magnetar to fit
three luminous SNe Ic-BL whose peak luminosities are approximately equal to
that of SN 2011kl. <cit.> and <cit.> used the same model
to fit SN 2011kl, suggesting that 0.2 M_⊙ of ^56Ni must be added
so that a better fit can be obtained. Therefore, we used the hybrid model
(the ^56Ni+magnetar model) to fit the K16 LC and the G15 LC. The free
parameters of this model are κ, M_ej, M_Ni,
B_p, P_0, v_sc0, κ_γ,Ni, and
κ_γ,mag.

Since an energetic SN explosion can synthesize ∼ 0.2 M_⊙
of ^56Ni <cit.>, we assumed that the range of ^56Ni
is 0–0.2 M_⊙ (for the G15 and K16 LCs with error bars, which allow us to
use the MCMC method and determine the best-fit parameters) or 0.2 M_⊙
(for the I16 LC without error bars). The LCs reproduced by the ^56
Ni+magnetar model (C1, C2, and C3) are shown in Figure <ref> and
the best-fit parameters are listed in Table <ref>. We find that
this hybrid model can also reproduce good fits, and that almost all
parameters (except for P_0) adopted by the ^56Ni+magnetar model are
the same as that of the magnetar models. Moreover, the masses of
^56Ni determined by the MCMC are 0.11_-0.07^+0.06 M_⊙ (for
the K16 LC) and 0.10_-0.07^+0.06 M_⊙ (for the G16 LC).



 §.§ The magnetar plus ^56Ni plus cooling model


The I16 LC has an early-time excess which cannot be reproduced by the
^56Ni model,
the magnetar model, or the magnetar+^56Ni model (A3, B3, and
C3, respectively; see Figure <ref>). The WD-TDE model proposed by
<cit.> also cannot explain the I16 LC (see Figure 1 of
). <cit.> suggested that the I16 LC can be
explained by the BSG model.

Here, we use another model to explain the I16 LC. We suggest that the
early-time excess might be due to the cooling emission from the shock-heated
extended envelope <cit.>, while the main peak of the LC
might be powered by a magnetar or a magnetar plus a moderate amount of
^56Ni. In this scenario, the progenitor of the SN is surrounded by an
extended, low-mass envelope which is heated by the SN shock and produces a
declining bolometric LC when it cools. At the same time, sources at the
center of the SN release energy and produce a rising LC until it peaks. By
combining the magnetar(+^56Ni) model developed by <cit.> and
<cit.> and the cooling model developed by <cit.>, a LC
with an early-time excess and a dip can be produced. In this model, two
additional parameters must be added: the mass of the extended envelope
(M_env)
and the radius of the extended envelope (R_env).
[The energy of the SN, E_SN, must also be added into the
modeling. However, it can be supposed that
E_SN=E_K=
0.3 M_ejv_sc^2, so this parameter is determined
by M_ej and v_sc, while v_sc is
determined by v_sc0, B_p, and P_0 (the acceleration
effect).]


The LCs reproduced by the cooling model
and the magnetar+^56Ni+cooling model (D1–D3) are shown in Figure
<ref> and the corresponding parameters are listed in Table
<ref>. We find that 
the magnetar+^56Ni+cooling models (D2 and D3) can well fit the entire
I16 LC. In the scenario containing the contribution from the cooling
emission, the progenitor of SN 2011kl was supposed to be surrounded by an
extended envelope whose mass and radius are ∼ 0.63 M_⊙ (or ∼
0.45 M_⊙) and ∼ 51.4 R_⊙ (or ∼ 103 R_⊙),
respectively. Owing to the parameter degeneracy, there must be other choices
of these two parameters.

One advantage of the magnetar-dominated model is that it can explain the
I16 LC if the cooling emission from the extended envelope is added, and it
can explain the K16 LC and the G15 LC if the progenitor was not surrounded
by an extended envelope. Another advantage is that the LC reproduced by the
model with cooling emission can trace the entire LC (including the early
excess, the dip, the peak, and the post-peak) while the LC reproduced by the
BSG model is a smooth, monotonically decreasing LC that is brighter than the
dip and dimmer than the peak. Although the uncertainties of the early-time
data indicate that the LC reproduced by the BSG model might fit the data,
the magnetar+^56Ni+cooling model can give a better match.



 §.§ Summary


In summary, the K16, G15, and I16 LCs cannot be powered solely by the
^56Ni model since (a) ∼ 1.00–1.42 M_⊙
of ^56Ni is inconsistent
with the spectral analysis performed by <cit.>, and (b) the ratio
of ^56Ni masses (∼ 1.0–1.4 M_⊙) to the ejecta mass (∼
2–5 M_⊙) is ∼ 0.3, which is significantly larger than the upper
limit (∼ 0.20; ). Alternatively, the K16 and G15 LCs
can be explained by the magnetar model and the ^56Ni+magnetar model; the
values of M_ej, B_p, and P_0 can be found in
Table <ref>.

Owing to the degeneracy of model parameters, the values of these parameters
have many choices that can also give the best-fit LC. For example,
<cit.> obtained P_0 = 3.5 ms, B_p=1.95×10^15 G,
M_ej = 2.5 M_⊙ if κ = 0.2 cm^2 g^-1, and
M_Ni = 0.2 M_⊙; <cit.> derived P_0 ≈
2 ms, B_p≈ 4×10^14 G, M_ej = 3
M_⊙ if κ = 0.2 cm^2 g^-1,
and M_Ni = 0.2 M_⊙; and
<cit.> derived P_0 = 11.5–13.0 ms, B_p= (1.1–1.3)
×10^15 G, and M_ej = 5.2 M_⊙ if κ =
0.07 cm ^2 g^-1.



§ DISCUSSION






 §.§ What Determines the Peak Luminosities of SNe and SLSNe?


The inferred best-fit values of P_0 and B_p of the magnetar
powering the LCs of SN 2011kl are ∼ 9–14 ms and ∼ (6–14) ×
10^14 G, respectively. The values of P_0 obtained by <cit.>,
<cit.>, and <cit.> are 3.5 ms, 2 ms, and 11.5–13.0 ms
(respectively), while the respective values of B_p derived by these groups
are 1.95×10^15 G, ∼ 4×10^14 G, and (1.1–1.3)
×10^15 G. The spin-down timescale of a magnetar (τ_p) is
determined by the values of P_0 and B_p,
τ_p = 1.3 (B_p/10^14 G)^-2(P_0/10 ms)^2 yr
<cit.>. We note that
other SLSNe Ic whose peak luminosities are larger than ∼ 10^44
erg s^-1 must be powered by magnetars with P_0 ≈ 1–10 ms
and 10^13–10^15 (see, e.g.,

), indicating that the values of P_0 and B_p play a crucial role in
determining the peak luminosities of SLSNe. When P_0 ≳ 10 ms and
B_p is a few 10^14 G, the magnetar-powered SNe are luminous
<cit.>; when P_0 ≲ 10 ms and B_p is a few 10^16 G,
the magnetars can explain the LCs of some SNe Ic-BL
(; see also ), while
<cit.> suggested that most GRB-SNe are powered by radioactive
heating.

<cit.> emphasized the role of P_0 while <cit.>
highlighted the role of B_p. We suggest that the more reasonable scheme
discussing the factors influencing the peak luminosities and the shapes of
magnetar-powered SNe and SLSNe is to take into account both of these
parameters. However, it should be noted that the values of τ_m (or
M_ej) also influence the peak luminosities of magnetar-powered SNe
and SLSNe <cit.>. Hence, the luminosities of SNe and SLSNe are
determined by all of these parameters (P_0, B_p, and τ_m, which is
primarily determined by M_ej) and cannot solely be influenced
by any single one. Similarly, the peak luminosities of ^56Ni-powered SNe
and SLSNe are determined by the values of M_Ni and τ_m.



 §.§ Correlations Between the Opacity and Other Parameters


In Section <ref>, we assumed that the value of κ is
0.07 cm^2 g^-1. As mentioned above, however, the value of κ is
rather uncertain (0.06–0.20 cm^2 g^-1). Therefore, it is necessary to
explore the correlations between the opacity and all other parameters
involved in the modeling.

Assuming κ = 0.10 cm^2 g^-1 and 0.20 cm^2 g^-1,
and using the ^56Ni model, the magnetar model, and the magnetar plus
^56Ni model, we repeated the fits for the K16 LC and the G15 LC,
obtaining the best-fit parameters (see Table <ref>).

By comparing the values of the parameters corresponding to different
values for κ, we found that while the values of the ejecta masses are
significantly influenced by the values of κ and M_ej =
aκ^-1 +b (a and b are constants; see Figure <ref>, and a
similar figure can be found in ), all other
parameters are slightly affected by the values of κ and no
correlation between them and κ has been found.

We did not study the correlations between the opacity and other
parameters for the I16 LC, since its data lack error bars, and the
value of M_ej is proportional to that of κ ^-1 while
the values of all other parameters are completely independent of that of
M_ej.



 §.§ The Initial Kinetic Energy of SN 2011kl


The multidimensional simulations (see , and references
therein) for neutrino-driven SNe suggest that the value of E_K
provided by the neutrinos can be up to ∼ (2.0–2.5) × 10^51
erg. Provided the value of v_sc0 is 14,000 km s^-1 (then
the ejecta mass is lowered to 2.4 M_⊙ for the K16 LC or 1.4 M_⊙
for the I16 LC), the origin of E_K,0 of SN 2011kl
(E_K,0=0.3 M_ejv_sc0^2≈ 2.4×
10^51 erg) can be comfortably explained by the neutrino-driven model
and other processes are not required to provide additional E_K,0.

According to the equation E_rot,0≈ (1/2)IΩ_0^2
≈ 2× 10^52(P_0/1 ms)^-2 erg (I
≈ 10^45 g cm^2 is the rotational inertia
of the magnetar), we find that the initial rotational energy of the magnetar
(P_0=11.4–14.2 ms) powering the LCs of SN 2011kl is ∼ 1.5 ×
10^50 erg ( obtained E_rot,0= (1.2–1.6)
×10^50 erg), significantly smaller than their E_K,0.
Even if the initial rotational energy of this magnetar is all converted to
the E_K,0 of the ejecta, the ratio of final E_K to
E_K,0 is ∼ 1.1. Therefore, the acceleration effect is
rather small and can be neglected.



 §.§ The Ejecta Mass and Binarity


Assuming that κ = 0.07 cm^2 g^-1 and according to
the results of our work based on the magnetar models (the ^56Ni
models have been excluded), the inferred ejecta mass of SN 2011kl
is 3.83_-1.46^+2.06 M_⊙ (for the K16 LC), 4.76_-1.78^+3.05 M_⊙ (for the G15 LC), or ∼ 2.12 M_⊙
(for the I16 LC). If κ = 0.10 cm^2 g^-1 or
0.20 cm^2 g^-1 were adopted, the inferred masses would decrease
by a factor of about 2–3, ∼1–2 M_⊙.

By adopting κ = 0.04 cm^2 g^-1 and v (v_sc0) =
20,000 km s^-1, <cit.> concluded that the mass of the ejecta
is 2 M_⊙ if SN 2011kl is powered by a magnetar. Assuming that κ
= 0.07 cm^2 g^-1 and v (v_sc0) = 21,000 km s^-1,
the ejecta mass must be 2 M_⊙×0.04/0.07 ×21,000/20,000
=1.323 M_⊙, significantly smaller than the value derived here (2.12
M_⊙).

This difference is caused by the fact that we adopted the equation
τ_m=(2κM_ej/βv_scc)^1/2 while
<cit.> adopted the equation
τ_m=(3κM_ej/4πv_scc)^1/2.
The ratio of the masses derived from these two
equations is 3β/8 π=3×13.8/8×3.1416=1.647. By
multiplying this factor by 1.323 M_⊙, we obtain 2.18 M_⊙,
consistent with the value derived here (2.12 M_⊙). The ejecta mass
would be 1.45 M_⊙ if κ = 0.1 cm^2 g^-1 and τ_m=
(2κM_ej/βv_scc)^1/2.

<cit.> and <cit.> argued that the ejecta mass of
SN 2011kl inferred from the magnetar model is too low to be produced by a
core-collapse SN whose remnant harbors a magnetar. However, it must be
pointed out that such a small value is not disfavored by the explosion of a
SN Ic and magnetar formation, since the massive hydrogen and helium
envelopes of the progenitors of SNe Ic had been stripped by their companions
or winds, and the mean value of the ejecta mass of SNe Ic is 2.9 M_⊙
with a standard deviation of 2.2 M_⊙ (see Table 6 of ). The inferred ejecta masses of Type Ic-BL SNe 2002ap, 2006aj, 2009bb, and
2010bh are 2.0_-0.7^+0.8 M_⊙, 1.4_-0.2^+0.4 M_⊙,
1.9_-0.5^+0.6 M_⊙, and 0.9_-0.2^+0.2 M_⊙,
respectively (see Table 5 of ). Note, especially, that SN
2006aj with M_ej=1.4_-0.2^+0.4 M_⊙ (or ∼ 2
M_⊙; ) is a SN Ic-BL associated with an X-ray flash (a
“soft GRB") that has long been believed to be powered by a nascent magnetar
<cit.>.

Such small ejecta masses indicate that the progenitor of SN 2011kl might be
located in a binary system and most of its mass had been stripped by its
companion before the explosion. Although the mass-transfer process reduced
the mass of the progenitor, it can enhance the angular momentum and angular
velocity of the progenitor, making it easier for the nascent neutron stars
left by the SN explosions to be millisecond (P_0 ≈ 1–10 ms)
magnetars.



§ CONCLUSIONS




In this paper, we modeled the LCs of SN 2011kl, which is the most luminous (
L_peak = 3.63^+0.17_-0.16× 10^43 erg s^-1)
GRB-SN identified thus far. By using the bolometric LC of SN 2011kl derived
by <cit.> and assuming that this SN is powered by ^56Ni
cascade decay, we find that the required ^56Ni mass is 1.42_-0.04^+0.04 M_⊙, which is smaller than that (2.27±0.64
M_⊙) inferred by <cit.>. The ^56Ni model can be excluded
from explaining the LCs of SN 2011kl, since the spectral features indicate
that the amount of ^56Ni must be significantly smaller than ∼ 1
M_⊙ <cit.>.

Alternatively, we used the magnetar model and the magnetar+^56Ni model
to fit the K16 LC and the G15 LC, finding that both of the models can well
explain these LCs. It is noteworthy that <cit.> argue that the
fact that the location of GRB 111209A-SN 2011kl is near the nucleus of the
host galaxy favors the WD-TDE model. However, the position of the SN does
not necessarily mean that this event must be a TDE rather than a SN. The
magnetar model is still possible, explaining the K16 LC and the G15 LC,
while the TDE model is also a plausible one for the G15 LC and the K16 LC.

Whereas the magnetar model and the magnetar+^56Ni model can account for
the K16 LC and the G15 LC, they cannot explain the I16 LC because these
models cannot produce the early-time excess and the dip, although the
presence of the early-time excess is entirely model-dependent (i.e., the
parameters of the optical-afterglow component in the LC decomposition
technique). To solve this problem, we added the contribution of the cooling
emission from a low-mass (M_env≈ 0.5 M_⊙), extended
(R_env≈ 50–100 R_⊙) SN progenitor envelope, and
found that the magnetar+cooling model and the magnetar+^56Ni+cooling
model can account for the I16 LC. In these models, cooling emission from the
shock-heated envelope powers the early-time excess, while the magnetar or
magnetar+^56Ni power the main peak of the LC, and the sum of these two
produces the dip.[It should
be mentioned that the model in which the progenitors have a
nonstandard structure <cit.> has been employed to explain
some SLSNe whose bolometric LCs exhibit an early-time excess
<cit.>.] Hence, we conclude that the BSG model is not
unique for explaining the I16 LC even if the LC of SN 2011kl has an
early-time excess, since a progenitor surrounded by a low-mass, extended
envelope can also power a LC with an early excess.

It seems that the magnetar+^56Ni+cooling and the magnetar+^56Ni
models are more reasonable than the magnetar and magnetar+cooling models for
the LCs of SN 2011kl, since core-collapse SNe must synthesize a moderate
amount of ^56Ni. However, discriminating between the LCs reproduced by
the models with and without ^56Ni is very difficult because the
contribution of a moderate amount of ^56Ni (∼ 0.1-0.2 M_⊙) is
significantly smaller than that of the magnetar.

Provided that the initial velocity of the ejecta of SN 2011kl is ∼
14,000 km s^-1 (the lower limit of the ejecta velocity; see
), the inferred values of the initial kinetic energy of
this SN is E_K,0≈ 2.0 × 10^51 erg , indicating
that the neutrino-driven mechanism <cit.> is able to provide the
E_K,0 for this SN. But larger velocities require other
mechanisms to provide additional E_K,0.

Furthermore, we used a MCMC method for the G15 and the K16 LCs to
constrain the range of the model parameters (we did not perform MCMC for
the I16 LC owing to the absence of the error bars); see
Table <ref>.
By adopting different values of κ, we found that while
the inferred mass is significantly influenced by the values
of κ (M_ej = aκ^-1 +b (a and b are
constants), all other parameters are only slightly affected by the values
of κ and no correlation between them and κ has been found.

According to these results, we suggest that the magnetar and the
magnetar+^56Ni models,
with or without the cooling effect, can reproduce the LCs
of SN 2011kl.[We caution that the question
of how the magnetar powers an ultralong GRB is
still unsolved. <cit.> suggested that a magnetar with P_0 ≈
2 ms and B_p≈ 4×10^14 G can power the ultralong
GRB 111209A and SN 2011kl, but <cit.> demonstrated that it is
difficult for a magnetar to produce an ultralong GRB. This issue is beyond
the scope of this paper and needs additional research. Moreover, we do not
attribute the optical afterglow of GRB 111209A to magnetar spin-down
emission.] In other words, SN 2011kl might be primarily powered by a nascent
magnetar.

We thank I. Arcavi for helpful discussions of the cooling
model. This work is supported by the National Basic Research Program
(“973" Program) of China (grant 2014CB845800) and the
National Natural Science Foundation of China (grant 11573014). L.J.W. is
supported by the National Program on Key Research and Development Project of
China (grant 2016YFA0400801). A.V.F.'s supernova group is supported by
the Christopher R. Redlich Fund, the TABASGO Foundation, and the Miller
Institute for Basic Research in Science (U.C. Berkeley).



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