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SN 2009ip: Constraining the latest explosion properties by its late-phase light curve

We constrain the explosion and circumstellar properties at the 2012b event of SN 2009ip based on its late-phase bolometric light curve recently reported. The explosion energy and ejected mass at the 2012b event are estimated as 0.01 Msun and 2e49 erg, respectively. The circumstellar medium is assumed to have two components: an inner shell and an outer wind. The inner shell which is likely created at the 2012a event has 0.2 Msun. The outer wind is created by the wind mass loss before the 2012a mass ejection, and the progenitor is estimated to have had the mass-loss rate about 0.1 Msun/yr with the wind velocity 550 km/s before the 2012a event. The estimated explosion energy and ejected mass indicate that the 2012b event is not caused by a regular supernova.


INTRODUCTION
It is widely believed that the efficient conversion of kinetic energy to radiation results in luminous transients. Type IIn supernovae (SNe), which sometimes even become superluminous (e.g., Gal-Yam 2012), are largely powered by the interaction between SN ejecta and circumstellar media (CSM) (e.g., Moriya et al. 2014). There also exist bright transients called 'SN impostors' which are likely brightened by the collision of the material ejected from the progenitor intermittently (e.g., Van Dyk et al. 2000). SN impostors are not caused by the final SN explosions of the progenitors, but they sometimes become as bright as SNe. Type IIn SNe and SN impostors are related to the unsolved problems in stellar mass loss, and it is important to understand their origins (e.g., Smith 2014;Langer 2012). They are also suggested to be an important high-energy cosmic-ray producers (Murase et al. 2011(Murase et al. , 2014. In addition, they can play a role as a distance ladder ) and can also be an important probe of the early Universe (e.g., Cooke et al. 2009;Tanaka et al. 2012).
SN 2009ip is one of the most studied transients powered by the interaction due to its activeness which kept us surprised since 2009. The progenitor of SN 2009ip got bright and was assigned a SN name in August 2009 (Maza et al. 2009). However, the subsequent observations revealed that it was not a genuine SN and the progenitor remained at the location Foley et al. 2011). The progenitor mass is estimated to be above ∼ 60 M ⊙ , and the observed brightening is considered to be a SN impostor from a luminous blue variable star Foley et al. 2011). The progenitor experienced several rebrightening since 2009 (Pastorello et al. 2013).
The origin of the final luminosity increase observed so far (the 2012b event) has been largely debated. The peak luminosity which is comparable to those of SNe and the broad spectral lines led to the suggestion that the 2012b event is triggered by the final SN explosion of the progenitor (e.g., Mauerhan et al. 2013;Smith et al. 2014;Baklanov et al. 2013;Ouyed et al. 2013). The 2012a event, which is similar to the precursor of SN 2010mc (Ofek et al. 2013b), is linked to the pre-SN mass ejection probably caused by the violent late-phase nuclear burning (e.g., Quataert & Shiode 2012, see also Moriya 2014a). On the other hand, it is also suggested that a SN event may not be required to explain the large luminosity observed in the 2012b event because of the efficient conversion from the kinetic energy to radiation. Thus, the 2012b event may not be caused by the final SN explosion of the progenitor (e.g., Pastorello et al. 2013;Fraser et al. 2013Fraser et al. , 2015Margutti et al. 2014). For example, ; Kashi et al. (2013); Tsebrenko & Soker (2013) related the 2012 events to the merger of massive stars.
The aim of this Letter is to reveal the unresolved origin of the enigmatic 2012b event by modeling its late-phase bolometric light curve (LC) until about 750 days after the 2012b event recently reported by Fraser et al. (2015). We constrain the explosion and circumstellar properties at the 2012b event by using an analytic bolometric LC model developed by Moriya et al. (2013b). We begin this Letter by briefly summarizing our LC model in the next section.

Assumptions
We assume a progenitor system schematically shown in Fig. 2 led to the 2012b and later event of SN 2009ip.  Fraser et al. (2015). The origin of time is set at the beginning of the 2012b event. We also show the results of the LC fitting with the function L = L 1 (t + t 0 ) −1.5 after 150 days since the beginning of the 2012b event.
The event is assumed to be caused by an ejection of the mass M ex with the kinetic energy E ex in a dense CSM. The dense CSM are assumed to have two components: an inner shell and an outer regular wind. The inner shell is presumed to be created by the mass ejection during the 2012a event with the mass-loss rateṀ ej , and it is assumed to have the mass M sh . We note that this kind of shells may also be formed by the confinement of the stellar wind (Mackey et al. 2014). The outer wind structure is assumed to be created with wind mass loss with the mass-loss rateṀ s and velocity v w . Although the observed luminosity fluctuations before the 2012a event indicate that the CSM may not be smooth, we assume that the overall density structure is approximately proportional to r −2 . Based on the observations of SN 2009ip in 2009, we assume that v w = 550 km s −1 Foley et al. 2011). The inner and outer components are separated at R sh .
We assume that the inner shell is responsible for the rise and decline observed until ∼ 100 days since the 2012b event ('early phase', Section 2.2). Then, a shock containing both M ex and M sh is assumed to propagate in the outer wind. The shock is assumed to be thin due to the efficient radiative cooling. The continuous interaction between the shock and the outer wind is assumed to be responsible for the later ( 200 days) LC reported by Fraser et al. (2015) ('late phase', Section 2.3).
Finally, we note that the spherical symmetry is assumed in our analytic model. Some observations of the 2012b event are suggested to indicate that the CSM is aspherical (e.g., Levesque et al. 2014;Mauerhan et al. 2014;Graham et al. 2014). However, the mass-loss rates obtained by assuming the spherical symmetry are still likely to be good estimates for the actual mass-loss rates in the aspherical systems (Moriya et al. 2014). In addition, the conversion efficiency discussed below also partly contains the effect of the asphericity (Moriya et al. 2014).

Early phase (diffusion phase)
We assume that the early phase is caused by the diffusion in the inner shell (see also Margutti et al. 2014;Ofek et al. 2013a). If we assume that the shell has an aver- age density ρ, the diffusion time t diff in the shell, which corresponds to the rise time of the LC, is expressed as where κ is opacity and c is the speed of light. We assume κ = 0.34 cm 2 g −1 in this study. Using the diffusion time, M sh can be estimated as As the rise time and the e-folding time of the later luminosity decline are both about 14 days (e.g., Pastorello et al. 2013;Moriya et al. 2014), a shock breakout is likely occurred in the shell (Margutti et al. 2014;Ofek et al. 2013a). This indicates that the entire shell is shocked at the LC peak. Thus, we presume that the blackbody radius at the LC peak of the 2012b event corresponds to R sh and we set R sh ≃ 10 15 cm (Margutti et al. 2014). Assuming t diff ≃ 14 days and R sh ≃ 10 15 cm, we obtain M sh ≃ 0.22 M ⊙ . The estimated mass is consistent with those obtained in the previous studies (∼ 0.1 M ⊙ , e.g., Fraser et al. 2013;Margutti et al. 2014). If the shell is ejected during the 2012a event which lasted for ∼ 40 days, the mass-loss rate during the 2012a event becomeṡ The explosion properties (E ex and M ex ) can be related to the total radiation energy emitted during the 2012b event. The conservation of momentum and energy results in (e.g., Moriya et al. 2013a) The total energy during the 2012b event is estimated to be E rad ≃ 2 × 10 49 erg (Fraser et al. 2013;Margutti et al. 2014).
2.3. Late phase (momentum-driven phase) Fraser et al. (2015) recently reported the late phase LCs until about 750 days after the 2012b event and constructed a bolometric LC (Fig. 1). The late bolometric LC is shown to evolve with a power law L = L 1 t −α , where t is time since the explosion. Fraser et al. (2015) fitted the power-law function by setting t = 0 at the beginning of the 2012a event and obtained α = 1.74 as the best fit parameter. However, we here assume that the inner explosion causing the 2012b event occurred after the 2012a event. If we set t = 0 at MJD = 56193 when the 2012b event began, we obtain α = 1.44 as the best fit parameter.
What is surprising is that α is near 1.5 and it is significantly larger than 1.0. If the interaction between SN ejecta and the dense wind is still ongoing, α is expected to be significantly below 1.0 when the wind is almost steady because of the continuous momentum injection from the SN ejecta (e.g., Moriya et al. 2013b;Ofek et al. 2014). For example, in the case of Type IIn SN 2010jl, the bolometric LC follows a power law with α < 1 even at around 600 days after the explosion (e.g., Maeda et al. 2013), although there exist some arguments for the LC interpretation (Fransson et al. 2014).
The large α near 1.5 in the LC after about 200 days since the 2012b explosion indicates that the shock proceeding in the wind is already in the momentum-driven phase at about 200 days. The momentum-driven phase (also called 'snow-plow phase') is the phase when the momentum injection to the shock has already terminated and the shock moves only with the momentum injected to the shock previously (see, e.g., Svirski et al. 2012;Moriya 2014b). Moriya et al. (2013b) obtained the luminosity evolution during the momentum-driven phase in the steady wind as where ǫ is the conversion efficiency from the kinetic energy to radiation, and E m and M m is the energy and mass released inside the wind, respectively. In the system we are interested in (Fig. 2), the shock propagates in the outer wind component after it has passed through the inner shell. Thus, we can set from the conservation of mass, and from the conservation of momentum. The luminosity evolution in the momentum-driven phase (Eq. 4) can be separated into two parts. At first, when 2(Ṁ s /v w )(2E m /M 3 m ) 1/2 t 1, the luminosity is constant. This is because the wind mass swept by the shock is much smaller than the initial injected mass (M m ) and the shock freely expands with a constant velocity. Then, when starts to hold after the shock has swept a large amount of the wind (see Moriya 2014b for detailed discussion on this condition), the luminosity evolves as L = L 1 t −1.5 , where Using Eq. (8), the condition (7) can be used to constrain M m , i.e., The late-phase bolometric LC of SN 2009ip indicates that the condition (9) is satisfied at least about 200 days after the explosion. In Fig. 1, we show the results of fitting of the function L = L 1 (t + t 0 ) −1.5 to the bolometric LC after 150 days since the beginning of the 2012b event with several different explosion times (t 0 ) relative to the beginning of the 2012b event (MJD = 56193). Although α = 1.5 is fixed in the fitting, decent fits to the bolometric LC are obtained. The best L 1 is found as 1.15 × 10 51 (t 0 = 0 days), 1.18 × 10 51 (t 0 = 5 days), 1.22 × 10 51 (t 0 = 10 days), 1.29 × 10 51 (t 0 = 20 days), and 1.42 × 10 51 (t 0 = 40 days) in the cgs unit. Since L 1 does not depend strongly on t 0 , we use L 1 = 1.15 × 10 51 cgs (t 0 = 0 days) as a representative value in the following discussion.
Finally, the mass ejected at the inner explosion (M ex ) can be expressed as

EXPLOSION AND CIRCUMSTELLAR PROPERTIES AT THE 2012b EVENT OF SN 2009ip
We now look into the explosion and circumstellar properties of the final explosive event observed in SN 2009ip so far. We have already constrained the inner shell mass in the previous section (M sh ≃ 0.22 M ⊙ ). The wind velocity is fixed to v w = 550 km s −1 . We also set ǫ = 0.3, which is typically found in Type IIn SN studies (e.g., van Marle et al. 2010;Fransson et al. 2014). The conversion efficiency is related to the physical properties of radiative shocks, and it is not likely to be altered by the origins of the explosions inside. Thus, we use a typical value found in the SN studies here. The conversion efficiency can be reduced by, e.g., multi-dimensional motions and asphericity (e.g., Moriya et al. 2013a). The observational information we have is E rad ≃ 2 × 10 49 erg and L 1 ≃ 1.15 × 10 51 cgs. We first assume several mass-loss rates for the outer wind (Ṁ s = 10 −1 , 10 −2 , 10 −3 M ⊙ yr −1 ) and give constraints on the other parameters for the assumed mass-loss rates.
The explosion properties at the 2012b event can be easily constrained with the formulae derived in Section 2. First, the ejected mass M ex can be constrained with Eq. (10). Then, the explosion energy E ex can be estimated with Eq. (3) assuming the obtained M ex and M sh . Finally, we need to check if the condition (9) holds for the estimated parameters for consistency. Table 1 summarizes the estimated parameters for SN 2009ip. We can first find that M ex is much smaller than M sh . This means that most of the kinetic energy released at the 2012b event is converted to radiation energy (Eq. 3). In other words, Eq. (3) indicates that E rad ≃ E ex , and only small amount of the released ki- netic energy is available for the late phase. The remaining kinetic energy (E m ) is only below 10% of E ex (Table 1). The total amount of energy radiated after 100 days obtained by assuming L = L 1 t −1.5 is 7.8 × 10 47 erg, and only the 10 −1 M ⊙ yr −1 model is consistent with the total radiated energy. The fact that the late-phase bolometric LC roughly follows ∝ t −1.5 gives the constraint (9). Assuming t ≃ 200 days in Eq. (9), we obtain the following constraint: Because M sh is estimated to be 0.22 M ⊙ and M sh ≫ M ex , this constraint also indicatesṀ s is around 10 −1 M ⊙ yr −1 . The estimated mass-loss rate is consistent with those estimated by the Hα luminosity (Fraser et al. 2013;Ofek et al. 2013a), while it is lower than that estimated by Ofek et al. (2013a) based on the multiwavelength observations. Combining above all, we suggest M ex ≃ 0.011 M ⊙ and E ex ≃ 2.1 × 10 49 erg as the explosion properties at the 2012b event.
The estimated E ex and M ex withṀ s = 10 −1 M ⊙ yr −1 are also consistent with the interpretation that the shock breakout occurred in the inner shell at the 2012b event (Section 2.2). The explosion velocity v ex ≡ 2E ex /M ex is shown in Table 1. The explosion velocity indicates that the shock breakout occurs where the optical depth is ∼ c/v ex ≃ 30 (e.g., Weaver 1976). If we use ρ ≃ 10 −13 g cm −3 estimated by Eq. (1), the shock breakout occurs at ≃ 2 × 10 14 cm. This radius is near ≃ 4 − 5 × 10 14 cm, which is the smallest photospheric radius observed at the beginning of the 2012b event (Margutti et al. 2014).
The estimated average ejecta velocity of v ex ≃ 10 4 km s −1 is also consistent with the broad spectra observed in the 2012b event of SN 2009ip indicating the ejecta velocity of ≃ 10 4 km s −1 (e.g., Fraser et al. 2013). Pastorello et al. (2013) reported similarly broad spectral lines in SN 2009ip before the 2012 events, and this kind of broad lines are known to be associated with non-SN events.
The estimated explosion energy and mass at the 2012b event are not those of regular SNe which typically have ∼ 10 51 erg and ∼ 1−10 M ⊙ . In particular, the progenitor of SN 2009ip is likely heavier than 60 M ⊙ Foley et al. 2011), and the estimated small ejecta mass is inconsistent with its successful SN explosion. Even if the ejecta has ∼ 10 51 erg and only the ejecta within a certain direction interacted with the dense CSM to emit only 1% of the kinetic energy (∼ 10 49 erg, e.g., Smith et al. 2014), the estimated ejecta mass is much smaller than 1% of the progenitor mass. Thus, the explosion occurred inside at the 2012b event is not likely related to a regular SN. However, some SNe may have properties similar to those estimated here, and we cannot conclude for sure if the core collapse of the progenitor occurred or not. For example, a SN with large fallback can have a small amount of ejecta with a small kinetic energy (e.g., Moriya et al. 2010). Low energy and small mass ejection is also predicted to be caused by failed SNe (Nadezhin 1980;Lovegrove & Woosley 2013). Massive star mergers may also have a similar fast mass ejection (e.g., ).

CONCLUSIONS
We have estimated the explosion and circumstellar properties at the 2012b event of SN 2009ip based on the late-phase bolometric LC recently reported by Fraser et al. (2015). The bolometric LC roughly follows L ∝ t −1.5 at least from about 200 days after the 2012b event ( Fig. 1), and it indicates that the shock is already at the momentum-driven phase without any momentuminjection from inside at that time. Thus, we use an analytic bolometric LC model for the momentum-driven phase to estimate the explosion and circumstellar properties. We assume that an explosion occurred inside the circumstellar medium with two components: an inner shell and an outer wind (Fig. 2). The inner shell is supposed to be created during the 2012a event of SN 2009ip, while the outer wind is made by the wind mass loss of the progenitor prior to the 2012a event.
Combining all the bolometric LC information available after the 2012b event, we suggest that an explosion with the energy 2.1 × 10 49 erg and the mass 0.011 M ⊙ occurred at the 2012b event of SN 2009ip. The ejecta first collided to the inner shell whose mass is estimated to be 0.22 M ⊙ . The collision between the ejecta and the inner shell is responsible for the early LC during the 2012b event which is dominated by the photon diffusion after the shock breakout in the shell. The thin shock made by the efficient cooling, which contains the mass of both the ejecta and the inner shell, continues to travel in the outer wind, powering the late-phase LC of SN 2009ip. The total radiated energy and the fact that the bolometric LC is already at the momentum-driven phase with L ∝ t −1.5 at about 200 days after the explosion indicate that the mass-loss rate of the progenitor prior to the 2012a event is about 0.1 M ⊙ yr −1 with the wind velocity 550 km s −1 .
The estimated explosion properties are not those of regular SNe. Thus, the explosion at the 2012b event is not related to a regular SN. It is likely to be a non-SN explosive event or a peculiar SN like those accompanied with large fallback or caused by failed SN explosions.
I would like to thank the referee for the constructive comments and Morgan Fraser for sending me the electric data of the bolometric LC of SN 2009ip. The author is supported by Japan Society for the Promotion of Science Postdoctoral Fellowships for Research Abroad (26·51).