SUPERLUMINOUS SUPERNOVAE POWERED BY MAGNETARS: LATE-TIME LIGHT CURVES AND HARD EMISSION LEAKAGE

In recent years, several Wide Field Optical Transient projects, e.g., the Palomar Transient Factory (PTF; Law 2009; Rau 2009), the Panoramic Survey Telescope & Rapid Response System (Pan-STARRS1, PS1; Kaiser 2010; Tonry 2012), the Catalina Real-time Transient Survey (CRTS; Drake et al. 2009), and the La Silla Quest Supernova Survey (LSQ; Baltay 2013), have discovered a class of supernovae (SNe) with peak luminosities 30–100 times more luminous than normal SNe. Accordingly, they were termed "superluminous supernovae" (SLSNe) (Gal-Yam 2012; Quimby 2012) and are distinct from their low-luminosity cousins.

Determining the heating sources for SLSNe and clarifying the explosion mechanisms of their progenitors would be of outstanding importance for the theory of SNe and of stellar evolution. Observational studies of both light curves and spectra of SLSNe have been largely pursued in the past decade (see Gal-Yam 2012, and references therein), and major progress has been made in understanding their nature. However, it is still not completely clear how SLSNe with a variety of characters in their light curves and spectra are powered.

The ⁵⁶Ni cascade decay (radioactivity-powered) model (Colgate & McKee 1969; Colgate et al. 1980; Arnett 1982; Woosley et al. 1989; Valenti et al. 2008) is the most "natural" model accounting for the most SNe and has been strongly supported by a great number of unambiguous observations. However, for very luminous SNe, the masses of ⁵⁶Ni needed are so large that we have to consider the "pair-instability SNe" (PISNe; Barkat et al. 1967; Rakavy & Shaviv 1967; Heger & Woosley 2003; Heger et al. 2003; Chatzopoulos & Wheeler 2012a; Whalen et al. 2013; Dessart et al. 2013; Chatzopoulos et al. 2013a; Kozyreva et al. 2014; Chen et al. 2014a, 2014b, 2014c), which are very rare in the low-redshift universe.⁶ Other challenges posed for the ⁵⁶Ni cascade decay model are that more massive ⁵⁶Ni needs more massive ejecta, resulting in broad theoretical light curves, and that the ⁵⁶Co decay tails have a rather shallow decline rate, both of which are not always consistent with observations (Maeda et al. 2007; Nicholl et al. 2013).

The ejecta–circumstellar medium (CSM) interaction model (Chevalier 1982; Chevalier & Fransson 1994; Chugai & Danziger 1994; Chevalier & Irwin 2011; Chatzopoulos et al. 2012) is also often used to explain luminous and superluminous Type IIn SNe whose optical spectra show narrow and/or intermediate-width Hα emission lines indicative of interactions between the SN ejecta and the CSM that was ejected from the progenitor just prior to the genuine SN explosion. Chatzopoulos et al. (2013b) have applied their semianalytical model (Chatzopoulos et al. 2012) to the observed light curves of a dozen SLSNe and suggested that both hydrogen-deficient (Type I) SLSNe and hydrogen-rich (Type II) SLSNe can be powered by ejecta-CSM interactions.

The rotational energy of a rapidly rotating, strongly magnetized pulsar forming after the core collapse of a massive star could significantly affect the evolution and emission of an outflow surrounding the star. It has been proposed (see Dai & Lu 1998a, 1998b; Zhang & Mészáros 2001; Dai 2004; Yu & Dai 2007; Dall'Osso et al. 2011; Dai & Liu 2012) that the rotational energy of a nascent magnetar can be injected into the relativistic outflow of a gamma-ray burst (GRB) by magnetic dipole radiation and could lead to the shallow decay of a GRB afterglow, which is very consistent with observations by the Swift satellite (Zhang 2007).

The magnetar-powered scenario has also been introduced to explain the light curves of some SNe or SLSNe. Maeda et al. (2007) suggested that a magnetar with surface magnetic field B ∼ 10^14–15 G and initial spin period P₀ ∼ 10 ms can explain the second peak and late-time decline rate of the light curve of SN 2005bf; Kasen & Bildsten (2010) and Woosley (2010) calculated the theoretical light curves powered by millisecond magnetars; Inserra et al. (2013) employed the semianalytical magnetar-powered model to fit the light curves of six SLSNe Ic (PTF10hgi, SN 2011ke, PTF11rks, SN 2011kf, SN 2012il, and SN 2010gx) with a variety of feasible physical values for magnetars, while the ⁵⁶Ni-powered model needs an unrealistic amount of ⁵⁶Ni to explain these SLSNe and therefore can be excluded. Hence, they concluded that the magnetar-powered model can provides a viable explanation for all of the SLSNe Ic. The successful fit for another SLSN Ic, PTF12dam (Nicholl et al. 2013), with the magnetar-powered model supported the conclusion given by Inserra et al. (2013). This so-called magnetar-powered model has attracted more and more attentions because it can explain many bright and/or sophisticated light curves of a variety of SNe, while the traditional radioactivity-powered model cannot.

Recently, Nicholl et al. (2014) fitted the light curves of three SLSNe Ic (LSQ12dlf, SSS120810, SN 2013dg) and one SLSN IIn (CSS121015) using the ⁵⁶Ni cascade decay model, magnetar-powered model, and ejecta-CSM interaction model, respectively. Their results excluded the ⁵⁶Ni cascade decay model and found that the late-time light curves reproduced by the magnetar-powered model are brighter than the observations, posing a challenge for the magnetar-powered model. Hence, they suggested either that the magnetar-powered model fails to explain the light curves of those SLSNe, or that there exists X-ray leakage in ejecta of these SLSNe.⁷ They have alternatively fitted the data of these SLSNe using the ejecta-CSM interaction model and obtained rather good agreement. Comparing these results, they proposed that these two models are valid in explaining SLSNe, without particularly favoring either. This conclusion is different not only from Inserra et al. (2013) and Nicholl et al. (2013), who favor the magnetar-powered model, but also from Chatzopoulos et al. (2013b) and Benetti et al. (2014), who favor the interaction model. Therefore, the energy generation mechanisms responsible for SLSNe Ic are still under debate.

Our major goals in this paper are to check the possibility that the magnetar-powered model can be responsible for some SLSNe Ic or IIn and to confirm quantitatively the leakage effect in the expanding ejecta. For these purposes, we try to incorporate the leakage effect into the original magnetar-powered model, supposing that there exists gamma-ray and X-ray leakage in SNe ejecta after energy was injected to the ejecta.

This paper is organized as follows: in Section 2 we revise the magnetar-powered model by introducing the leakage factor into the original model. We fit the light curves of SN 2010gx, CSS121015, SN 2013dg, LSQ12dlf, and SSS120810 using our revised magnetar-powered model in Section 3. Finally, discussions and conclusions are presented in Section 4.

We first derive the semianalytical equation of SN light curves powered by magnetars in this section and revise it by introducing the leakage effect.

Using Equations (30)–(33), (36), and (47)–(48) derived by Arnett (1982) and neglecting the $(E_{th,0}/t_{0})e^{(t/\tau _{m})^2+(R_{0}/v\tau _{m})(t/\tau _{m})}$ term (where t₀ is the diffusion timescale, E_{th, 0} is the initial total thermal energy and proportional to $R_{0}^3$ and $T_{00}^4$, R₀ ≪ 10¹⁴ cm is the initial radius of the progenitor, and T₀₀ is the initial temperature of the center of the ejecta), we get the photospheric luminosity of SNe powered by a variety of energy sources (radioactive ⁵⁶Ni decay, magnetar spin-down, etc.),

$$\begin{eqnarray} L(t) =&&\frac{2}{\tau _{m}}e^{-\left(\frac{t^{2}}{\tau _{m}^{2}}+\frac{2R_{0}t}{v\tau _{m}^{2}}\right)} \int _0^t e^{\left(\frac{t^{\prime 2}}{\tau _{m}^{2}}+\frac{2R_{0}t^{\prime }}{v\tau _{m}^{2}}\right)} \nonumber \\ &&\times \left(\frac{R_{0}}{v\tau _{m}}+\frac{t^{\prime }}{\tau _{m}}\right)L_{\rm inp}(t^{\prime })dt^{\prime }, \end{eqnarray} \tag{ 1 }$$

where τ_m is the effective light-curve timescale and L_inp(t) is the generalized power source. Using Equation (10) in Chatzopoulos et al. (2012), τ_m can be written as

$$\begin{eqnarray} \tau _{m} &=&\left(\frac{10\kappa M_{\rm ej}}{3\beta vc}\right)^{1/2} \nonumber \\ &\simeq & \left(\frac{10\kappa M_{\rm ej}}{3\times 13.8\times vc}\right)^{1/2} \nonumber \\ &=& 0.5\left(\frac{\kappa M_{\rm ej}}{vc}\right)^{1/2} , \end{eqnarray} \tag{ 2 }$$

where κ, M_ej, v, and c are the Thomson electron scattering opacity, the ejecta mass, the expansion velocity of the ejecta, and the speed of light in a vacuum, respectively. β ≃ 13.8 is a constant that accounts for the density distribution of the ejecta.

Although the rotational energy dissipation mechanism of the magnetars is ambiguous, we can expect that most of the rotational energy released is likely to be Poynting-flux-dominated wind (Metzger et al. 2014). When the high-energy leptons and/or baryons in the magnetar wind hit the SN ejecta, X-ray and gamma-ray photons can be generated and propagate in the ejecta. Therefore, it is reasonable to assume that most of the spin-down energy can be converted to X-ray and gamma-ray photons.

In the magnetar-powered model, adopting the ansatz that all the spin-down energy released can be converted to the heating energy of surrounding SN ejecta, the input power for the ejecta is equal to the magnetic dipole radiation power,

$$\begin{equation} L_{\rm inp({\rm mag})}(t) = \frac{E_{p}}{\tau _{p}} \frac{1}{(1+t/\tau _{p})^{2}}, \end{equation} \tag{ 3 }$$

where $E_{p}\,{\simeq}\ ({1}/{2})I_{\rm NS}\Omega _{\rm NS}^2$ is the rotational energy of a magnetar and $I_{\rm NS} =(2/5)M_{\rm NS}R_{\rm NS}^{2}$ is the moment of inertia of a magnetar whose canonical value can be set to 10⁴⁵ g cm² (Kasen & Bildsten 2010). Thus, E_p ≃ 2 × 10⁵² erg(M_NS/1.4 M_☉) (P_NS/1 ms)⁻² (R_NS/10 km)², and $\tau _{p} = {6I_{\rm NS} c^3}/ {B^2R_{\rm NS}^6\Omega ^2} = 1.3 B_{14}^{-2}P_{10}^2\, {\rm yr}$ is the spin-down timescale of the magnetar (Kasen & Bildsten 2010), in which the magnetar dissipates most of its rotational energy into the expanding ejecta. Here M_NS, P_NS, and R_NS are the mass, rotational period, and radius of the magnetar, respectively. The parameters have been scaled to typical values B₁₄ = B/10¹⁴ G and P₁₀ = P/10 ms (Kasen & Bildsten 2010).

Most previous research that fit observed data with the magnetar-powered model neglected hard (gamma-ray and X-ray) photon leakage at all epochs,⁸ whereas in reality a portion of hard photons can escape from the ejecta while other photons can be down-Comptonized to UV–optical–IR photons and diffuse to the photosphere, especially after the transition from the photospheric phase to the nebular phase.

Observations for SN 1987A (Sunyaev et al. 1987; Matz et al. 1988) and SN 2014J (Churazov et al. 2014; Diehl et al. 2014) have revealed that a large amount of gamma rays and X-rays escape from their ejecta at later times. Recently, Kotera et al. (2013) have explored the effect of mildly magnetized millisecond pulsars (B = 10¹³ G) on SN remnants and demonstrated that the hard emissions could generate bright (10⁴³–10⁴⁴ erg s⁻¹) TeV gamma-ray emissions and a milder X-ray peak (10⁴⁰–10⁴² erg s⁻¹) could appear several hundred days after the core-collapse supernova (CCSN) explosion.

Discussions of the details of the gamma-ray and X-ray emission leakage are beyond the scope of this paper. We then introduce the leakage factor $ke^{-\tau _{\gamma }}$ and the trapping factor $(1-ke^{-\tau _{\gamma }})$ in the magnetar-powered model to represent gamma-ray/X-ray leakage and trapping rate, where k ≃ 1 is a dimensionless parameter.⁹ τ_γ is the optical depth of the ejecta to gamma rays and can be written as τ_γ = At⁻² (Clocchiatti & Wheeler 1997; Chatzopoulos et al. 2009, 2012). The larger τ_γ is, the larger the trapping rate is and the smaller the leakage rate is.

The density profile of the ejecta can be represented by a power law, ρ(r, t)∝[r(t)]^−η. Provided that the SN ejecta has a uniform density distribution (η = 0, ρ=ρ(t), M_ej = (4/3)πρR³) and the expansion velocity is constant, we have E_k = (3/10)M_ejv² and A can be written as ¹⁰

$$\begin{eqnarray} A&=&\frac{9\kappa _{\gamma } M_{\rm ej}^2}{40\pi E_{\rm k}} \nonumber \\ &=&\frac{3\kappa _{\gamma } M_{\rm ej}}{4\pi v^2 }\nonumber \\ &=&4.748 \times 10^{13}\frac{\kappa _{\gamma,0.1}M_{\rm ej,\odot }}{v_{9}^2} {\rm s}^2, \end{eqnarray} \tag{ 4 }$$

where v₉ = v/10⁹ cm s⁻¹, M_{ej, ☉} = M_ej/M_☉, and κ_{γ, 0.1} = κ_γ/0.1 cm² g⁻¹, where κ_γ is the effective gamma ray opacity. The typical value of A for normal CCSNe is between 10¹³ s² and 10¹⁵ s².

Using Equations (1), (3), and (4) and letting k = 1, R₀ → 0, the photospheric luminosity function in the magnetar-powered model with leakage can be written as

$$\begin{eqnarray} L_{\rm mag}(t)=&&\frac{2E_{p}}{\tau _{p}\tau _{m}} e^{-\left(\frac{t}{\tau _{m}}\right)^{2}} (1-e^{-At^{-2}})\nonumber \\ &&\times \int _{0}^{t}\frac{1}{(1+t^{\prime }/\tau _{p})^{2}} e^{\left(\frac{t^{\prime }}{\tau _{m}}\right)^{2}} \frac{t^{\prime }}{\tau _{m}}dt^{\prime }. \end{eqnarray} \tag{ 5 }$$

We refer to the model corresponding to Equation (5) as the "revised magnetar-powered model ." When A → ∞, the leakage can be neglected and the model can be called the "original magnetar-powered model."

Therefore, the free parameters of our model are M_ej, v and κ (which determine τ_m; Equation (2)), B and P (which determine E_p and τ_p), and A. The velocity v, if measured, can be fixed at its measured value and cannot be regarded as a free parameter.

If we assume η = 0, A can be calculated from M_ej, v, and κ_γ using Equation (4). Furthermore, as has been shown by (Kotera et al. 2013), for gamma rays κ_γ ≃ κ. However, as the density profile and κ_γ are quite uncertain,¹¹ the value of A may be quite different from that calculated by Equation (4). We therefore also investigate the effect of varying A to represent cases where η > 0 and/or κ_γ ≠ κ.

In this section we try to fit the observed data of SN 2010gx, CSS121015, SN 2013dg, LSQ12dlf, and SSS120810 using our revised magnetar-powered model. Our fits follow the research performed by Inserra et al. (2013) and Nicholl et al. (2014), whose fit parameters for these SLSNe are listed in Table 1 and are all denoted as "Model A." The common feature of the theoretical light curves reproduced by "Model A" is that the early-time theoretical light curves were well fitted (except for LSQ12dlf), while the late-time theoretical light curves are brighter than the observed luminosities, i.e., the original magnetar-powered model fails to fit their late-time light curves. An excess in the theoretical light curve reproduced for SN 2013dg is not obvious but may still exist (see discussion in Section 3.3).

We now employ our revised magnetar-powered model to fit the observed data and upper limits of aforementioned SLSNe and explore whether the late-time excess can be eliminated or decreased.

Based on the same parameters as Inserra et al. (2013) and Nicholl et al. (2014) (except for SN 2013dg and LSQ12dlf), we incorporated the leakage effect into the model, i.e., the values of A are not set to infinity; according to Equation (4), we calculate the values of A for these five SLSNe (see Table 1). The light curves reproduced by both the original and revised models were shown in Figures 1–5.

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3.1. SN 2010gx

3.2. CSS121015

3.3. SN 2013dg

3.4. LSQ12dlf

3.5. SSS120810

In the past five years, many authors have demonstrated that many SLSNe Ic and II/IIn cannot be powered by radioactivity solely. Alternatively, light curves reproduced by the magnetar spin-down model and the ejecta-CSM interaction model can both well mimic the observed data of these SLSNe, indicating that most SLSNe might be powered by newly born magnetars or ejecta-CSM interactions.

Chatzopoulos et al. (2013b) argued that almost all SLSNe can be powered by ejecta-CSM interactions. Furthermore, Nicholl et al. (2014) have already demonstrated that CSS121015, LSQ12dlf, SSS120810, and SN 2013dg, which are difficult to explain by the original magnetar-powered model owing to their late-time excess in the theoretical light curves, can be explained by the ejecta-CSM interaction model, giving us a hint that some SLSNe Ic might be powered by hydrogen- and helium-poor CSM interactions.

However, no spectrum of an SLSN Ic so far has shown narrow emission lines indicative of the interaction of the ejecta with the CSM. This fact seems to suggest that the ejecta-CSM interaction model is unlikely to be responsible for SLSNe Ic,¹² although Chatzopoulos et al. (2013b) argued that the absence of this character does not necessarily imply that the CSM interaction model is invalid for SLSNe Ic.

Besides the observational constraint aforementioned, the difficulties encountered by the interaction model for SLSNe Ic must also be discussed from theoretical perspectives. The first difficulty is that the formation mechanisms of massive (M_CSM ≳ 2 M_☉) H- and He-poor CSM are ambiguous. One of the main mechanisms for explaining unusually high mass-loss rate is the so-called pulsational pair instability (PPI ; Heger et al. 2003; Woosley et al. 2007; Pastorello et al. 2008; Chugai 2009; Chatzopoulos & Wheeler 2012b), which involves a very massive (100 M_☉ ≲ M_ZAMS ≲ 140 M_☉) star that can experience a series of eruptions caused by e⁺e⁻ pair generation and consequent $\nu \bar{\nu }$ pair creation, but the star cannot be destroyed completely until it explodes as a genuine CCSN. The progenitors of SNe/SLSNe Ic are linked to the stars without H and He envelopes that are so compact that their outer materials are difficult to eject by PPI or other possible mechanisms. Combining the fact that no SLSNe presented emission lines, we argue that the CSM interaction model has difficulty accounting for at least some of SLSNe Ic. Another assumption adopted in the CSM interaction model employed by Chatzopoulos et al. (2012), Chatzopoulos et al. (2013b), and Nicholl et al. (2014) is that the photospheres of SLSNe are stationary, whereas in reality the photospheres must expand owing to the law of conservation of momentum.

The magnetar-powered model does not encounter the difficulties mentioned above. Kouveliotou et al. (1998) have demonstrated that magnetars with surface magnetic fields ∼10^14–15 G may constitute 10% of nascent neutron stars (see also Kouveliotou et al. 1994; van Paradijs 1995; Lyne et al. 1998); therefore, Woosley (2010) believed that the birth of rapidly rotating magnetars ("millisecond magnetars") is commonplace.¹³ Furthermore, the stationary photosphere assumption is not adopted in the magnetar-powered model. Successful fits to the observed light curves of PTF10hgi, SN 2011kf, SN 2012il, and PTF12dam using the magnetar-powered model (Inserra et al. 2013; Nicholl et al. 2013) have confirmed the validity of the original magnetar-powered model in explaining the luminosity evolution of the SLSNe Ic.

Yet the original magnetar-powered model is not a complete model since it neglects the hard emission leakage. In reality, although the ejecta of SNe are opaque for gamma-ray and X-ray photons released from the magnetars at early times and these hard emission leakages are not significant, when the fast-expanding ejecta becomes more and more transparent for radiation at advanced epochs, the hard photons would pretty easily escape from the ejecta and cannot be neglected. So it is not surprising that the main problem encountered by the original magnetar-powered model is the late-time excess in the theoretical light curves of some SLSNe (e.g., SN 2010gx (Inserra et al. 2013), CSS121015, LSQ12dlf, SSS120810, and SN 2013dg; (Nicholl et al. 2014)).

This significant challenge confronted by the original magnetar-powered model motivates us to explore possible revisions. We therefore consider the leakage effect and find that the late-time light curves derived from our revised magnetar-powered model are in good agreement with observed data and upper limits, indicating not only that the magnetars can be responsible for some SLSNe but also that the gamma-ray and X-ray leakages are inevitable when these photons were down-Comptonized to UV–optical–near-IR (NIR) photons and diffused out of the ejecta.

Accordingly, we argue that it is not necessary to apply the ejecta-CSM interaction model in explaining the observed light curves for some SLSNe Ic (e.g., SN 2010gx, SSS120810, LSQ12dlf, and SN 2013dg). Furthermore, we find that SLSN IIn CSS121015, which can be explained by ejecta-CSM interaction, also can be explained by our revised magnetar-powered model, indicating that H-rich SLSNe powered by magnetars may exist in the universe. Hence, we can expect that most SLSNe Ic can be powered by nascent millisecond magnetars and a (minor) portion of SLSNe II can be powered by nascent millisecond magnetars or in rare cases by recycling millisecond magnetars (Barkov & Komissarov 2011).

The fact that the late-time excess can be eliminated or decreased by hard emission leakage indicates that it is nontrivial to introduce the leakage factor in the magnetar-powered model. The values of A adopted in our sample are ≃ 10¹⁴–10¹⁵ s². When t² ≳ A, i.e., t ≳ 10⁷ − 3 × 10⁷ s ≃ 120–350 days, the light curves reproduced by the magnetar-powered models with and without leakage can be easily distinguished from each other; see Figures 1–5. For the same reason, the leakage effect does not significantly influence the early-time (≲ 100–300 days) light curves of these SLSNe.

When we take the limit A → ∞, the leakage in the ejecta can be neglected, i.e., all gamma-ray and X-ray photons should be trapped and thermalized to UV–optical–NIR photons. The limit A → ∞ might be a good approximation for some other SLSNe-Ic, e.g., PTF10hgi, SN 2011kf, and SN 2012il, but we have no idea whether PTF11rks needs to be explained by the revised magnetar-powered model, as a result of the lack of later-time (≳ 100 days after explosion) data and strict upper limits (see Figure 12 in Inserra et al. (2013)). The possibility that the late-time leakage can be compensated by ⁵⁶Co decay tails cannot be excluded, and it might be worthwhile to verify this possibility in future modelings. Another possibility that accounts for full trapping is that the gamma-ray and X-ray photons are so soft that κ_γ (κ_γ/X) is very large and the values of A far exceed 10¹⁵ s².

Whether the leakage of hard emissions can play a key role in later phases is of great interest and of great importance in future studies. A larger sample of SLSNe with high-quality observed data, especially recorded after the SLSNe transitioned to the nebular phases (≳ 60–100 days after the explosions), is required to determine the precise values of fading rate at nebular phases and therefore precise values of the parameter A in Equation (5).

Finally, as addressed by Dessart et al. (2012, 2013), theoretical light curves of SLSNe alone are inadequate to unveil the nature of SLSNe. To completely clarify the energy-generating mechanisms, both theoretical light curves and spectra for every individual event need to be examined carefully. Further detailed modelings are also needed to determine precisely the parameters of putative nascent magnetars.

We thank an anonymous referee for helpful comments and suggestions that have allowed us to improve this manuscript. This work is supported by the National Basic Research Program ("973" Program) of China (grant Nos. 2014CB845800 and 2013CB834900) and the National Natural Science Foundation of China (grant Nos. 11033002 and 11322328). X.F.W. was also partially supported by the One-Hundred-Talent Program, the Youth Innovation Promotion Association, and the Strategic Priority Research Program "The Emergence of Cosmological Structures" (grant No. XDB09000000) of the Chinese Academy of Sciences, and the Natural Science Foundation of Jiangsu Province.