An Analytic Bolometric Light Curve Model of Interaction-Powered Supernovae and its Application to Type IIn Supernovae
Takashi J. Moriya, Keiichi Maeda, Francesco Taddia, Jesper Sollerman, Sergei I. Blinnikov, Elena I. Sorokina
aa r X i v : . [ a s t r o - ph . H E ] J u l Mon. Not. R. Astron. Soc. , 1–29 (2013) Printed 22 April 2018 (MN L A TEX style file v2.2)
An Analytic Bolometric Light Curve Model ofInteraction-Powered Supernovae and its Application toType IIn Supernovae
Takashi J. Moriya , , ⋆ , Keiichi Maeda , Francesco Taddia , Jesper Sollerman ,Sergei I. Blinnikov , , , and Elena I. Sorokina Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, University of Tokyo,5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan Department of Astronomy, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan Research Center for the Early Universe, Graduate School of Science, University of Tokyo,7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, 10691 Stockholm, Sweden Institute for Theoretical and Experimental Physics, Bolshaya Cheremushkinskaya 25, 117218 Moscow, Russia Novosibirsk State University, Novosibirsk 630090, Russia Sternberg Astronomical Institute, M.V.Lomonosov Moscow State University, Universitetski pr. 13, 119992 Moscow, Russia
Accepted 2013 July 24. Received 2013 July 24; in original form 2013 March 24
ABSTRACT
We present an analytic model for bolometric light curves which are powered bythe interaction between supernova ejecta and a dense circumstellar medium.This model is aimed at modeling Type IIn supernovae to determine the prop-erties of their supernova ejecta and circumstellar medium. Our model is notrestricted to the case of steady mass loss and can be applied broadly. We onlyconsider the case in which the optical depth of the unshocked circumstellarmedium is not high enough to affect the light curves. We derive the luminos-ity evolution based on an analytic solution for the evolution of a dense shellcreated by the interaction. We compare our model bolometric light curves toobserved bolometric light curves of three Type IIn supernovae (2005ip, 2006jd,2010jl) and show that our model can constrain their supernova ejecta and cir-cumstellar medium properties. Our analytic model is supported by numericallight curves from the same initial conditions. c (cid:13) T. J. Moriya et al.
Key words: circumstellar matter — stars: mass-loss — supernovae: general— supernovae: individual: SN 2005ip — supernovae: individual: SN 2006jd —supernovae: individual: SN 2010jl
Massive stars which die as supernovae (SNe) do not end their lives as they were born. Theychange their mass, size, temperature, luminosity and many other properties during theirevolution toward their death. The final fate, or the SN type, of a massive star is determinedby these changes during their evolution (e.g., Heger et al. 2003).One of the most critical factors which dramatically change the properties of a star andlargely affect its final fate is mass loss. Massive stars continue to lose mass until the end oftheir lives because of their huge luminosities. From X-ray and radio observations of youngSNe, it has been possible to estimate the mass-loss rates of SN progenitors shortly before theirexplosions (e.g., Chevalier, Fransson, & Nymark 2006; Chevalier & Fransson 2006; Maeda2013). In most cases, the estimated mass-loss rates are within the range expected from theradiation-driven mass loss (e.g., Owocki, Gayley, & Shaviv 2004).However, there are a number of SNe which seem to be strongly affected by the inter-action with circumstellar media (CSM) whose densities are too high to be explained bythe standard radiation-driven mass loss. Most of them are spectroscopically classified asType IIn because of the narrow hydrogen emission lines seen in their spectra (Schlegel 1990;Filippenko 1997). The mass-loss rates of the progenitors before their explosions are estimatedto be ∼ − − ∼ . M ⊙ yr − (e.g., Taddia et al. 2013; Kiewe et al. 2012; Fox et al. 2011).Because of the high-density CSM, a cool dense shell is suggested to be created betweenSN ejecta and CSM (e.g., Chevalier & Fransson 1994; Chugai et al. 2004). The cool denseshell can create dust grains efficiently and SNe IIn are promising sites for dust formation(e.g., Kozasa et al. 2009). SNe IIn can also be used as a distance ladder thanks to the denseshell (e.g., Blinnikov et al. 2012; Potashov et al. 2013). In addition, some SNe IIn can beobserved at very high redshifts and may provide us with information about star formationand initial-mass functions in the early Universe (e.g., Cooke 2008; Cooke et al. 2009, 2012;Tanaka et al. 2012; Tanaka, Moriya, & Yoshida 2013; Whalen et al. 2013).Despite the many interesting phenomena associated with SNe IIn, the nature of SNe IIn ⋆ [email protected] c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe α emitting regions within their host galaxies also seem consistent withthese relatively low-mass progenitors (Anderson et al. 2012, but see also Crowther 2013).However, the other SN IIn progenitors detected in archival images are very massive starswhich are rather consistent with luminous blue variables (LBVs) (e.g., Gal-Yam & Leonard2009; Mauerhan et al. 2013; Pastorello et al. 2013, see also Smith et al. 2011a,b). LBVs aretheoretically interpreted as an evolutionary stage to a Wolf-Rayet star and they have notbeen considered as a pre-SN phase (e.g., Crowther 2007; Maeder & Meynet 2000, but seeGroh, Meynet, & Ekstr¨om 2013; Moriya, Groh, & Meynet 2013; Langer 2012). In addition,extreme mass-loss mechanisms of such very massive stars are not well-known. There areseveral suggested mechanisms to induce extreme mass loss, like pulsational pair-instability(e.g., Woosley, Blinnikov, & Heger 2007), rotation (e.g., Maeder & Desjacques 2001), poros-ity (e.g., Owocki, Gayley, & Shaviv 2004), gravity-mode oscillations (e.g., Quataert & Shiode2012), or binary interaction (e.g., Chevalier 2012), but we still do not know which mecha-nisms are actually related to SNe IIn (see also Dwarkadas 2011).For a better understanding of SNe IIn, especially their progenitors and extensive mass-loss mechanisms, we need more theoretical investigation of SNe IIn as well as more observa-tional data. In this paper we develop a simple analytic bolometric LC model which can beused to fit SN IIn observations to estimate CSM and SN ejecta properties. We believe thatthe information obtained by applying our LC model to many SNe IIn can lead to a betterunderstanding of SNe IIn.This paper is organized as follows. We present our analytic LC model in Section 2. Wederive the evolution of the shocked shell analytically and use it to obtain the bolometric LCevolution. We apply it to some observational SN IIn bolometric LCs and obtain constraintson the properties of SN ejecta and CSM for these SNe in Section 3. The discussion is givenin Section 4 and we conclude this paper in Section 5. c (cid:13)000
Massive stars which die as supernovae (SNe) do not end their lives as they were born. Theychange their mass, size, temperature, luminosity and many other properties during theirevolution toward their death. The final fate, or the SN type, of a massive star is determinedby these changes during their evolution (e.g., Heger et al. 2003).One of the most critical factors which dramatically change the properties of a star andlargely affect its final fate is mass loss. Massive stars continue to lose mass until the end oftheir lives because of their huge luminosities. From X-ray and radio observations of youngSNe, it has been possible to estimate the mass-loss rates of SN progenitors shortly before theirexplosions (e.g., Chevalier, Fransson, & Nymark 2006; Chevalier & Fransson 2006; Maeda2013). In most cases, the estimated mass-loss rates are within the range expected from theradiation-driven mass loss (e.g., Owocki, Gayley, & Shaviv 2004).However, there are a number of SNe which seem to be strongly affected by the inter-action with circumstellar media (CSM) whose densities are too high to be explained bythe standard radiation-driven mass loss. Most of them are spectroscopically classified asType IIn because of the narrow hydrogen emission lines seen in their spectra (Schlegel 1990;Filippenko 1997). The mass-loss rates of the progenitors before their explosions are estimatedto be ∼ − − ∼ . M ⊙ yr − (e.g., Taddia et al. 2013; Kiewe et al. 2012; Fox et al. 2011).Because of the high-density CSM, a cool dense shell is suggested to be created betweenSN ejecta and CSM (e.g., Chevalier & Fransson 1994; Chugai et al. 2004). The cool denseshell can create dust grains efficiently and SNe IIn are promising sites for dust formation(e.g., Kozasa et al. 2009). SNe IIn can also be used as a distance ladder thanks to the denseshell (e.g., Blinnikov et al. 2012; Potashov et al. 2013). In addition, some SNe IIn can beobserved at very high redshifts and may provide us with information about star formationand initial-mass functions in the early Universe (e.g., Cooke 2008; Cooke et al. 2009, 2012;Tanaka et al. 2012; Tanaka, Moriya, & Yoshida 2013; Whalen et al. 2013).Despite the many interesting phenomena associated with SNe IIn, the nature of SNe IIn ⋆ [email protected] c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe α emitting regions within their host galaxies also seem consistent withthese relatively low-mass progenitors (Anderson et al. 2012, but see also Crowther 2013).However, the other SN IIn progenitors detected in archival images are very massive starswhich are rather consistent with luminous blue variables (LBVs) (e.g., Gal-Yam & Leonard2009; Mauerhan et al. 2013; Pastorello et al. 2013, see also Smith et al. 2011a,b). LBVs aretheoretically interpreted as an evolutionary stage to a Wolf-Rayet star and they have notbeen considered as a pre-SN phase (e.g., Crowther 2007; Maeder & Meynet 2000, but seeGroh, Meynet, & Ekstr¨om 2013; Moriya, Groh, & Meynet 2013; Langer 2012). In addition,extreme mass-loss mechanisms of such very massive stars are not well-known. There areseveral suggested mechanisms to induce extreme mass loss, like pulsational pair-instability(e.g., Woosley, Blinnikov, & Heger 2007), rotation (e.g., Maeder & Desjacques 2001), poros-ity (e.g., Owocki, Gayley, & Shaviv 2004), gravity-mode oscillations (e.g., Quataert & Shiode2012), or binary interaction (e.g., Chevalier 2012), but we still do not know which mecha-nisms are actually related to SNe IIn (see also Dwarkadas 2011).For a better understanding of SNe IIn, especially their progenitors and extensive mass-loss mechanisms, we need more theoretical investigation of SNe IIn as well as more observa-tional data. In this paper we develop a simple analytic bolometric LC model which can beused to fit SN IIn observations to estimate CSM and SN ejecta properties. We believe thatthe information obtained by applying our LC model to many SNe IIn can lead to a betterunderstanding of SNe IIn.This paper is organized as follows. We present our analytic LC model in Section 2. Wederive the evolution of the shocked shell analytically and use it to obtain the bolometric LCevolution. We apply it to some observational SN IIn bolometric LCs and obtain constraintson the properties of SN ejecta and CSM for these SNe in Section 3. The discussion is givenin Section 4 and we conclude this paper in Section 5. c (cid:13)000 , 1–29 T. J. Moriya et al.
In this section, we develop an analytic SN bolometric LC model under the assumption thatits main power source is the kinetic energy of SN ejecta colliding with a dense CSM. Atfirst, we analytically investigate the evolution of the dense shell created by the interactionin Section 2.1. The analytic solution for the evolution of the dense shell before time t t (seebelow) is essentially the same as obtained in previous works (e.g., Chevalier 1982a, 1990;Chevalier & Fransson 1994, 2003) but our solution does not assume steady mass loss (seealso Fransson, Lundqvist, & Chevalier 1996).After deriving the evolution of the dense shell, we provide an analytic expression for bolo-metric LCs. This method was introduced by Chugai & Danziger (1994) (see also Wood-Vasey, Wang, & Aldering2004; Svirski, Nakar, & Sari 2012) to explain the luminoisity due to the interaction but theirmodel assumes a CSM from steady mass loss. We generalize this method for the cases of non-steady mass loss and apply our model to entire bolometric LCs. Chatzopoulos, Wheeler, & Vinko(2012); Chatzopoulos et al. (2013) also follow a similar approach to obtain an analytic LCmodel from the interaction but they consider the case where the unshocked CSM is opti-cally thick. Here, we consider the case in which the unshocked CSM does not affect thebolometric LC so much. Some SNe IIn are suggested to have very optically thick CSMto explain their huge luminosities (e.g., Chevalier & Irwin 2011; Moriya & Tominaga 2012;Moriya et al. 2013b; Ginzburg & Balberg 2012) but they are beyond the scope of this pa-per. High energy photons are expected to be emitted when the CSM is optically thin (e.g.,Chevalier & Fransson 1994) but they are presumed to be absorbed by the dense shell becauseof its high column density and re-emitted as optical photons which are mainly observed (e.g.,Wilms, Allen, & McCray 2000). The inverse Compton scattering and other effects can alsoreduce the energy of the photons (e.g., Chevalier & Irwin 2012). The shocked dense CSM and SN ejecta form a thin dense shell because of the efficientradiative cooling. We assume that the thickness of the shocked shell is much smaller thanits radius and it can be denoted by a radius r sh ( t ). The conservation of momentum requires M sh dv sh dt = 4 πr h ρ ej ( v ej − v sh ) − ρ csm ( v sh − v w ) i , (1) c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe M sh is the total mass of the shocked SN ejecta and CSM, v sh is the velocity of the shell, ρ ej is the SN ejecta density, v ej is the SN ejecta velocity, ρ csm is the CSM density, and v w isthe CSM velocity. We derive the evolution of r sh based on this equation. We do not use theequation for the conservation of energy which is necessary to derive the self-similar solutionincluding the reverse shock and the forward shock (Chevalier 1982b; Nadyozhin 1985). Thisis because of the radiative energy loss from the dense shell. When the radiative cooling isefficient, the shocked region does not extend as wide as the width expected from the self-similar solution due to the loss of the thermal pressure caused by the radiative energy loss.Thus, our approximation to neglect the shell width is presumed to be a good approximationto the shell evolution.We further assume that the CSM density follows ρ csm = Dr − s and that the CSM velocity v w is constant. We adopt a double power-law profile for the density of homologously ( v ej = r/t ) expanding SN ejecta ( ρ ej ∝ r − n outside and ρ ej ∝ r − δ inside) based on numericalsimulations of SN explosions (e.g., Matzner & McKee 1999). With SN kinetic energy E ej and SN ejecta mass M ej , the SN density structure is expressed as ρ ej ( v ej , t ) = π ( n − δ ) [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / t − v − n ej ( v ej > v t ) , π ( n − δ ) [2(5 − δ )( n − E ej ] ( δ − / [(3 − δ )( n − M ej ] ( δ − / t − v − δ ej ( v ej < v t ) , (2)where v t is obtained from the density continuity condition at the interface of the two densitystructures as well as E ej and M ej as follows, v t = " − δ )( n − E ej (3 − δ )( n − M ej . (3)The outer density slope n depends on the SN progenitor and n ≃ n = 6 .
67 exactly) isthe lowest possible n expected from the self-similar solution of Sakurai (1960) (e.g., Chevalier1990). A value of n ≃
10 is expected for SN Ib/Ic and SN Ia progenitors (Matzner & McKee1999; Kasen 2010) and n ≃
12 is expected for explosions of RSGs (Matzner & McKee 1999).The inner density slope δ is δ ≃ − ρ ej ∝ r − n starts to interact with the CSM. In thisphase, M sh becomes M sh = Z r sh R p πr ρ csm dr + Z v ej , max /tv ej /t πr ρ ej dr, (4)= 4 πD − s r − s sh + t n − ( n − δ )( n − r n − [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / , (5) c (cid:13)000
12 is expected for explosions of RSGs (Matzner & McKee 1999).The inner density slope δ is δ ≃ − ρ ej ∝ r − n starts to interact with the CSM. In thisphase, M sh becomes M sh = Z r sh R p πr ρ csm dr + Z v ej , max /tv ej /t πr ρ ej dr, (4)= 4 πD − s r − s sh + t n − ( n − δ )( n − r n − [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / , (5) c (cid:13)000 , 1–29 T. J. Moriya et al. where R p is the radius of the progenitor, v ej , max is the velocity of the outermost layer ofthe SN ejecta before the interaction. In deriving Equation (5) from Equation (4), we haveassumed r sh ≫ R p , v ej , max ≫ v ej , and s < v ej = r sh /t (homologous expansion of the SN ejecta), theequation for the conservation of momentum becomes " πD − s r − s sh + t n − ( n − δ )( n − r n − [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / d r sh dt =1( n − δ ) [2(5 − δ )( n − E ej ] ( n − / t n − [(3 − δ )( n − M ej ] ( n − / r n − r sh t − dr sh dt ! − πDr − s sh dr sh dt ! . (6)Here, we assume that the CSM velocity is much smaller than the shell velocity ( v sh ≫ v w ).Solving the differential equation, we get a power-law solution r sh ( t ) = " (3 − s )(4 − s )4 πD ( n − n − n − δ ) [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / n − s t n − n − s . (7)Note that r sh obtained with this approach has the same time dependence as the self-similarsolution [ t ( n − / ( n − s ) ] (Chevalier 1982a,b; Chevalier & Fransson 2003; Nadyozhin 1985).Equation (7) holds until the time t t when the interacting region reaches down to theinner ejecta, namely when the v ej entering the shell becomes v t or r sh ( t t ) = v t t t is satisfied,i.e., t t = " (3 − s )(4 − s )4 πD ( n − n − n − δ ) [(3 − δ )( n − M ej ] (5 − s ) / [2(5 − δ )( n − E ej ] (3 − s ) / − s . (8)After t t , the density structure of the SN ejecta entering the shell starts to follow ρ ej ∝ r − δ and the equation of the momentum conservation becomes " πD − s r − s sh + M ej − r − δ sh ( n − δ )(3 − δ ) t − δ [2(5 − δ )( n − E ej ] ( δ − / [(3 − δ )( n − M ej ] ( δ − / d r sh dt =1( n − δ ) [2(5 − δ )( n − E ej ] ( δ − / r − δ sh [(3 − δ )( n − M ej ] ( δ − / t − δ r sh t − dr sh dt ! − πDr − s sh dr sh dt ! . (9)Generally, we cannot solve Equation (9) analytically but the solution of Equation (9) isexpected to asymptotically approach the solution of the differential equations M sh d r sh dt = 4 πr ( − ρ csm v ) , (10) (cid:18) πD − s r − s sh + M ej (cid:19) d r sh dt = − πDr − s sh dr sh dt ! . (11)The asymptotic solution from Equation (11) satisfies the equation4 πD − s r sh ( t ) − s + (3 − s ) M ej r sh ( t ) − (3 − s ) M ej E ej M ej ! t = 0 . (12) c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe r sh ( t = 0) = 0 , (13) dr sh dt ( t = 0) = E ej M ej ! , (14)are applied in deriving Equation (12). As the asymptotic solution is derived by assumingthat most of the SN ejecta is in the dense shell, the dependence of r sh ( t ) on the SN ejectastructure ( n and δ ) disappears. Here, we write down r sh ( t ) derived in the previous section for the special case of the steadymass loss ( s = 2). The CSM density structure becomes˙ M = 4 πr ρ csm v w , (15)where ˙ M is the mass-loss rate and D can be expressed as D = ˙ M πv w . (16)Then, r sh ( t ) before t = t t is r sh ( t ) = " n − n − n − δ ) [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / v w ˙ M n − t n − n − ( t < t t ) , (17)and t t = 2( n − n − n − δ ) [(3 − δ )( n − M ej ] / [2(5 − δ )( n − E ej ] / v w ˙ M . (18)The asymptotic solution after t t becomes r sh ( t ) = v w ˙ M M ej − vuut E ej M ˙ Mv w t . (19)As noted in Section 2.1.1, the asymptotic solution is independent of the SN density structure( n and δ ). We construct an analytic bolometric LC based on r sh ( t ) obtained in the previous section. Weassume that the kinetic energy of the SN ejecta is the dominant source of the SN luminosity.The available kinetic energy is c (cid:13)000
12 is expected for explosions of RSGs (Matzner & McKee 1999).The inner density slope δ is δ ≃ − ρ ej ∝ r − n starts to interact with the CSM. In thisphase, M sh becomes M sh = Z r sh R p πr ρ csm dr + Z v ej , max /tv ej /t πr ρ ej dr, (4)= 4 πD − s r − s sh + t n − ( n − δ )( n − r n − [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / , (5) c (cid:13)000 , 1–29 T. J. Moriya et al. where R p is the radius of the progenitor, v ej , max is the velocity of the outermost layer ofthe SN ejecta before the interaction. In deriving Equation (5) from Equation (4), we haveassumed r sh ≫ R p , v ej , max ≫ v ej , and s < v ej = r sh /t (homologous expansion of the SN ejecta), theequation for the conservation of momentum becomes " πD − s r − s sh + t n − ( n − δ )( n − r n − [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / d r sh dt =1( n − δ ) [2(5 − δ )( n − E ej ] ( n − / t n − [(3 − δ )( n − M ej ] ( n − / r n − r sh t − dr sh dt ! − πDr − s sh dr sh dt ! . (6)Here, we assume that the CSM velocity is much smaller than the shell velocity ( v sh ≫ v w ).Solving the differential equation, we get a power-law solution r sh ( t ) = " (3 − s )(4 − s )4 πD ( n − n − n − δ ) [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / n − s t n − n − s . (7)Note that r sh obtained with this approach has the same time dependence as the self-similarsolution [ t ( n − / ( n − s ) ] (Chevalier 1982a,b; Chevalier & Fransson 2003; Nadyozhin 1985).Equation (7) holds until the time t t when the interacting region reaches down to theinner ejecta, namely when the v ej entering the shell becomes v t or r sh ( t t ) = v t t t is satisfied,i.e., t t = " (3 − s )(4 − s )4 πD ( n − n − n − δ ) [(3 − δ )( n − M ej ] (5 − s ) / [2(5 − δ )( n − E ej ] (3 − s ) / − s . (8)After t t , the density structure of the SN ejecta entering the shell starts to follow ρ ej ∝ r − δ and the equation of the momentum conservation becomes " πD − s r − s sh + M ej − r − δ sh ( n − δ )(3 − δ ) t − δ [2(5 − δ )( n − E ej ] ( δ − / [(3 − δ )( n − M ej ] ( δ − / d r sh dt =1( n − δ ) [2(5 − δ )( n − E ej ] ( δ − / r − δ sh [(3 − δ )( n − M ej ] ( δ − / t − δ r sh t − dr sh dt ! − πDr − s sh dr sh dt ! . (9)Generally, we cannot solve Equation (9) analytically but the solution of Equation (9) isexpected to asymptotically approach the solution of the differential equations M sh d r sh dt = 4 πr ( − ρ csm v ) , (10) (cid:18) πD − s r − s sh + M ej (cid:19) d r sh dt = − πDr − s sh dr sh dt ! . (11)The asymptotic solution from Equation (11) satisfies the equation4 πD − s r sh ( t ) − s + (3 − s ) M ej r sh ( t ) − (3 − s ) M ej E ej M ej ! t = 0 . (12) c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe r sh ( t = 0) = 0 , (13) dr sh dt ( t = 0) = E ej M ej ! , (14)are applied in deriving Equation (12). As the asymptotic solution is derived by assumingthat most of the SN ejecta is in the dense shell, the dependence of r sh ( t ) on the SN ejectastructure ( n and δ ) disappears. Here, we write down r sh ( t ) derived in the previous section for the special case of the steadymass loss ( s = 2). The CSM density structure becomes˙ M = 4 πr ρ csm v w , (15)where ˙ M is the mass-loss rate and D can be expressed as D = ˙ M πv w . (16)Then, r sh ( t ) before t = t t is r sh ( t ) = " n − n − n − δ ) [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / v w ˙ M n − t n − n − ( t < t t ) , (17)and t t = 2( n − n − n − δ ) [(3 − δ )( n − M ej ] / [2(5 − δ )( n − E ej ] / v w ˙ M . (18)The asymptotic solution after t t becomes r sh ( t ) = v w ˙ M M ej − vuut E ej M ˙ Mv w t . (19)As noted in Section 2.1.1, the asymptotic solution is independent of the SN density structure( n and δ ). We construct an analytic bolometric LC based on r sh ( t ) obtained in the previous section. Weassume that the kinetic energy of the SN ejecta is the dominant source of the SN luminosity.The available kinetic energy is c (cid:13)000 , 1–29 T. J. Moriya et al. dE kin = 4 πr ρ csm v dr sh , (20)and thus the bolometric luminosity will be L = ǫ dE kin dt = 2 πǫρ csm r v , (21)where ǫ is the conversion efficiency from kinetic energy to radiation. Especially, the bolo-metric luminosity before t t can be expressed as a power-law function L = L t α , (22)where L = ǫ πD ) n − n − s (cid:18) n − n − s (cid:19) " (3 − s )(4 − s )( n − n − n − δ ) [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / − sn − s , (23) α = 6 s −
15 + 2 n − nsn − s . (24)In Figure 1, α is plotted as a function of s for n = 12 , , t t , the asymptotic bolometric LC can be obtained based on the asymptotic radiusevolution from Equation (12). L = 2 πǫρ csm r v (25)= 2 πǫDr − s sh (3 − s ) M ej (cid:16) E ej M ej (cid:17) πDr − s sh + (3 − s ) M ej . (26) In the case of steady mass loss ( s = 2), we can use 4 πD = ˙ M /v w and express L before t t as L = ǫ Mv w v (27)= ǫ ˙ Mv w ! n − n − (cid:18) n − n − (cid:19) " n − n − n − δ ) [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / n − t − n − . (28)This equation is basically the same as obtained in previous studies (e.g., Chugai & Danziger1994; Wood-Vasey, Wang, & Aldering 2004).We can also express the asymptotic bolometric LC after t t using Equation (19). L = ǫ Mv w E ej M ej ! Mv w E ej M ! t − . (29)By defining two parameters a and b as a = ǫ Mv w E ej M ej ! , (30) c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe α n=12n=10n=7 Figure 1. α ( L ∝ t α before t t ) as a function of s for some n . n ≃
12 represents RSG explosions and n ≃
10 is for SNe Ib/Icand SNe Ia progenitors. n ≃ n . b = 2 ˙ Mv w E ej M ! , (31)we can express L in a simple way. Namely, L = 2 − n − n − ab − n − (cid:18) n − n − (cid:19) [2(5 − δ )( n − n − n − [( n − n − n − δ )] n − [(3 − δ )( n − n − n − t − n − , (32)before t t and L = a (1 + bt ) − , (33)long after t t . Here, t t is expressed as t t = 4 [(3 − δ )( n − ( n − n − n − δ ) [(5 − δ )( n − b . (34)The physical parameters of CSM and SN ejecta have the relations E ej = 2 aǫb , (35)˙ Mv w M − ej = 14 ǫb a ! . (36) c (cid:13)000
10 is for SNe Ib/Icand SNe Ia progenitors. n ≃ n . b = 2 ˙ Mv w E ej M ! , (31)we can express L in a simple way. Namely, L = 2 − n − n − ab − n − (cid:18) n − n − (cid:19) [2(5 − δ )( n − n − n − [( n − n − n − δ )] n − [(3 − δ )( n − n − n − t − n − , (32)before t t and L = a (1 + bt ) − , (33)long after t t . Here, t t is expressed as t t = 4 [(3 − δ )( n − ( n − n − n − δ ) [(5 − δ )( n − b . (34)The physical parameters of CSM and SN ejecta have the relations E ej = 2 aǫb , (35)˙ Mv w M − ej = 14 ǫb a ! . (36) c (cid:13)000 , 1–29 T. J. Moriya et al.
We first show examples of procedures to fit our analytic bolometric LC to bolometric LCsconstructed from observations. The actual processes for the comparison depend on the avail-able information from observations but the basic concepts will be essentially the same asthe examples presented here.Our bolometric LC model consists of two components. Before t t , the model LC has apower-law dependence on time ( L = L t α ). Thus, we can first use the function L t α to fitan early LC and obtain L and α . Assuming n , the CSM density slope s can be constrainedjust by α through Equation (24) (Figure 1).If there are spectral observations at these epochs and the shell velocity evolution can beestimated by them, we can use v sh ( t ) = dr sh dt (37)= n − n − s " (3 − s )(4 − s )4 πD ( n − n − n − δ ) [2(5 − δ )( n − E ej ] ( n − / [(3 − δ )( n − M ej ] ( n − / n − s t − − sn − s (38) ≡ v t − − sn − s , (39)to fit the velocity evolution and obtain v . Just from the three values, L , α , and v , we canobtain the CSM density structure for given ǫ and n , D = 12 πǫ (cid:18) n − n − s (cid:19) − s L v s − . (40)This means that we can estimate the mass-loss rate without assuming M ej and E ej . As thetime dependence of v sh is small, the velocity information of just a single epoch can constrain D . So far, M ej and E ej are degenerated and we have to assume either M ej or E ej to constrainthe other parameter.The formulae L = L t α and v sh = v t − (3 − s ) / ( n − s ) can only be applied before t t . Afterobtaining the physical values, we have to check whether t t is larger than the epochs used forthe fitting. If there is an available bolometric LC after t t , we can fit Equation (26) to theLC and obtain further constraints to break the degeneracy between E ej and M ej .We show how this procedure works in the next section by using actual bolometric LCsfrom observations. c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe Table 1.
SN IIn properties estimated by the bolometric LCmodel ( ǫ = 0 . s ( ρ csm ∝ r − s ) h ˙ M i a E ej n = 10 n = 12 ( M ⊙ yr − ) (10 erg)2005ip 2.3 2.4 1 . − . × − b . − . × − b b b ba Average rate derived by the CSM mass within 10 cm andthe CSM wind velocity 100 km s − . b Derived assuming M ej = 10 M ⊙ . Here we compare our analytic bolometric LCs to observed LCs of SNe IIn 2005ip, 2006jd,and 2010jl, and estimate CSM and SN ejecta properties of them. We assume ǫ = 0 . ǫ is affected by SN ejecta mass and CSM mass but it is typically of the order of 0.1(e.g., Moriya et al. 2013b). All the fitting procedures in this section are performed by usingthe least-squares method unless otherwise mentioned. Table 1 is a summary of the SN IInproperties derived in this section. SN 2005ip was intensively observed by Stritzinger et al. (2012) from ultraviolet to near-infrared wavelengths. They derived a bolometric LC that we use for the comparison to ourbolometric LC model. Optical photometric and spectroscopic observations are also reportedby Smith et al. (2009), whereas Fox et al. (2009, 2010, 2011, 2013) summarize the near-infrared observations of SN 2005ip. We assume that the explosion date of SN 2005ip was 9days before its discovery and all the following dates are since the explosion.At first, we fit the obtained bolometric LC up to 220 days by L = L t α and we get L = 1 . × t ! − . erg s − . (41)In Figure 2 we show the result. α = − .
536 corresponds to s = 2 . s = 2 . n = 10and n = 12, respectively. Thus, the CSM around the progenitor of SN 2005ip likely hadslightly steeper density structure than the expected density structure from steady mass loss.The deviation from the steady mass loss of SN IIn progenitors is also suggested from X-rayobservations (Dwarkadas & Gruszko 2012).One interesting feature in Figure 2 is the similarity of the analytic LC from the SNejecta-CSM interaction to the available energy from the radioactive decay of 0 . M ⊙ Ni c (cid:13) , 1–29 T. J. Moriya et al. b o l o m e t r i c l u m i n o s i t y ( e r g s − ) SN 2005ip bolometric LC (Stritzinger et al. 2012)best fit analytic interaction model0.18 M ⊙ Ni- Co- Fe decay (full trapping) b o l o m e t r i c l u m i n o s i t y ( e r g s − ) SN 2005ip bolometric LC (Stritzinger et al. 2012)best fit analytic interaction model0.18 M ⊙ Ni- Co- Fe decay (full trapping)
Figure 2.
Bolometric LC of SN 2005ip (Stritzinger et al. 2012) and some LC models. The solid line is the best fit to L = L t α up to 220 days. The dot-dashed line is the available energy from the radioactive decay of 0 . M ⊙ Ni. The luminosity inputsfrom the two power souces resemble each other up to about 100 days and the later LC is required to distinguish between them. before around 100 days. The available energy from the radioactive decay Ni → Co → Feis (Nadyozhin 1994) h . × e − t/ (8 . + 1 . × e − t/ (111 . i M Ni M ⊙ erg s − , (42)where M Ni is the initial Ni mass. We cannot distinguish between the two power sourcesonly from the bolometric LC before about 100 days. The two energy sources can only bedistinguished by LCs at later epochs. The similarity, especially at around 50 days, is becauseof the decay time of Co. At around 50 days, the radioactive energy from Co is dominantand the available energy from the decay follows ∝ e − t/ (111 . . The values and the declinerates (the first derivatives) of the functions following ∝ e − t/ (111 . and ∝ t − m ( m is aconstant) can get similar at t = 111 . m days. Looking at Figure 1, m ≃ . s ≃ t ≃
50 days. For a LC from the interactionto have a similar decline rate to that from the Co radioactive decay after ≃
100 days, m should be close to unity and the CSM density slope should be steep ( s ≃ ≃ ,
500 km s − (Stritzinger et al. 2012). Then, based on Equation (40), we get ρ csm ( r ) = . × − (cid:16) r cm (cid:17) − . g cm − ( n = 10) , . × − (cid:16) r cm (cid:17) − . g cm − ( n = 12) . (43)The Thomson scattering optical depth τ sh of the solar-metallicity unshocked CSM when theshell is at the radius 10 cm, above which the shell is located at the epochs we fit the LC,is c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe τ sh = .
22 ( n = 10) , .
25 ( n = 12) , (44)and our assumption that the unshocked CSM does not affect the LC at the epochs we usefor the fitting is justified. We estimate an average mass-loss rate h ˙ M i by using the CSMmass within 10 cm. Assuming v w = 100 km s − , the CSM mass within 10 cm is lost fromthe progenitor in 32 years before the explosion. The average mass-loss rate in this period is h ˙ M i = . × − M ⊙ yr − ( n = 10) , . × − M ⊙ yr − ( n = 12) . (45)The bolometric luminosity of SN 2005ip after 300 days is almost constant ( ≃ . × erg s − ). The asymptotic solution (Equation 26) can have a constant luminosity ata certain condition. For example, the asymptotic solution for s = 2 (Equation 33) canbe constant if bt ≪
1. However, for the case of SN 2005ip, we could not find a constantasymptotic solution which is consistent with the early LC before 300 days. The constantluminosity may be due to, e.g., another CSM component or light echos.To constrain the SN properties, we assume M ej = 10 M ⊙ . Then, from L above, weobtain E ej = . × erg s − ( n = 10) , . × erg s − ( n = 12) . (46) t t becomes t t = . × days ( n = 10) , . × days ( n = 12) , (47)so the epochs we used for the fitting ( t <
220 days) are justified.The average mass-loss rate we obtained (10 − M ⊙ yr − ) is consistent with the rateestimated by Fox et al. (2011) (1 . × − M ⊙ yr − ) but larger than the rate suggestedby Smith et al. (2009) (2 × − M ⊙ yr − ). Based on these mass-loss rates, Smith et al.(2009) conclude that the progenitor of SN 2005ip is a massive RSG like VY CMa (e.g.,Smith, Hinkle, & Ryde 2009), while Fox et al. (2011) prefer a more massive progenitor likea LBV. Our results seem to support the latter scenario but depend on the value of ǫ assumedin deriving D so we cannot constrain the progenitor strongly (see Section 4.1). In principle,we may be able to distinguish between the two progenitors with n , but our results are foundnot to depend much on n . Binary evolution may also be related to the dense CSM (e.g.,Chevalier 2012). c (cid:13) , 1–29 T. J. Moriya et al.3.2.2 SN 2006jd
SN 2006jd was also observed in a wide spectral range by Stritzinger et al. (2012) and theyobtained a bolometric LC. We use their bolometric LC for our modeling. We assume thatthe date of the explosion is 9 days before its discovery and the following dates are sincethe explosion. Chandra et al. (2012a) estimate CSM properties of SN 2006jd based on theX-ray and radio observations after about 400 days since explosion. They conclude that theCSM density profile is rather flat ( s ≃ . − .
6) and the CSM density is ∼ cm − at ∼ × cm. Fox et al. (2011, 2013) estimate the mass-loss rate based on near-infraredobservations (2 . × − M ⊙ yr − ).By fitting the LC before 250 days with L = L t α , we obtain L = 3 . × t ! − . erg s − . (48)From α = − . s = 1 . s = 1 . n = 10 and n = 12, respectively. The shell velocity of SN 2006jd around 100 days since theexplosion is likely ≃ ,
000 km s − (Stritzinger et al. 2012). Then, based on Equation (40),we get ρ csm ( r ) = . × − (cid:16) r cm (cid:17) − . g cm − ( n = 10) , . × − (cid:16) r cm (cid:17) − . g cm − ( n = 12) . (49)The Thomson scattering optical depth τ sh of the solar-metallicity unshocked CSM when theshell is at the radius 10 cm is τ sh = .
22 ( n = 10) , .
26 ( n = 12) , (50)and our model is self-consistent. We estimate an average mass-loss rate by using the CSMmass within 10 cm and v w = 100 km s − as we did for SN 2005ip in the previous section.The average mass-loss rate is h ˙ M i = . × − M ⊙ yr − ( n = 10) , . × − M ⊙ yr − ( n = 12) . (51)The estimated average mass-loss rate is consistent with the rate derived by Fox et al. (2011)from dust emission (2 . × − M ⊙ yr − ). Interestingly, the mass-loss rate is very close tothose of SN 2005ip estimated in the previous section, although the density slopes are quitedifferent ( s = 2 . − . s = 1 . − . c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe b o l o m e t r i c l u m i n o s i t y ( e r g s − ) SN 2006jd bolometric LC (Stritzinger et al. 2012)best fit analytic interaction model b o l o m e t r i c l u m i n o s i t y ( e r g s − ) SN 2006jd bolometric LC (Stritzinger et al. 2012)best fit analytic interaction model
Figure 3.
Bolometric LC of SN 2006jd (Stritzinger et al. 2012) and the best fit L = L t α model up to 250 days. The secondrise starting around 400 days cannot be explained by our model and may be due to, e.g., another CSM component. another CSM component. Since we can only fit the early phases, we cannot constrain M ej and E ej independently. Here, we assume M ej = 10 M ⊙ to estimate E ej . The estimated E ej is E ej = . × erg ( n = 10) , . × erg ( n = 12) . (52)Note again that we assume ǫ = 0 . t t obtained by these valuesare t t = . × days ( n = 10) , . × days ( n = 12) . (53)The epochs we used to fit L = L t α ( t <
250 days) are justified for the n = 10 case. Forthe n = 12 case, t t is smaller than 250 days. However, there are only two observational datapoints beyond 180 days and we find that the results of fitting by using t <
180 days arealmost the same as the results we obtained with t <
250 days.The CSM properties we derived are consistent with s ≃ . − . ∼ cm − at ∼ × cm as obtained by Chandra et al. (2012a) from X-ray and radioobservations. However, the X-ray and radio observations were performed after the epochswhen the bolometric LC starts to rise (after about 400 days since the explosion). Our modelis not applicable at these epochs as is discussed above and this correspondence can be acoincidence. SN 2010jl has been extensively observed in a wide range of wavelengths (Smith et al. 2011b,2012; Stoll et al. 2011; Andrews et al. 2011; Chandra et al. 2012b; Fox et al. 2013; Maeda et al. c (cid:13) , 1–29 T. J. Moriya et al. V -band LC peak reported by Stoll et al. (2011). We apply ourspherically symmetric bolometric LC model but the observation of polarization indicates anasymmetric nature of the CSM around SN 2010jl (Patat et al. 2011).At first, we use L = L t α to fit the bolometric LC and get L = 2 . × t ! − . erg s − (54) α = − .
486 corresponds to s = 2 . s = 2 . n = 10 and n = 12, respectively.However, t t become t t = . n = 10) , . n = 12) , (55)for E ej = 10 erg or t t =
57 days ( n = 10) ,
23 days ( n = 12) , (56)for E ej = 2 . × erg with the obtained L . This means that the L = L t α formula weused for the fitting is not applicable for most of the epochs we used for the fitting. Thus, weneed to use the asymptotic formula (Equation 26) to fit the LC.In Figure 4, we show some asymptotic LC models from Equation (26). We have searchedfor a good fit by changing s , D , and E ej . We assume M ej = 10 M ⊙ . The best model we foundis shown in Figure 4 and it has ρ csm ( r ) = 2 . × − (cid:18) r cm (cid:19) − . g cm − , (57)and E ej = 2 . × erg . (58)The Thomson scattering optical depth of the solar-metallicity unshocked CSM when theshell is at 10 cm is τ sh = 7 . τ sh becomes ∼ ∼ × cm and the shell is above ∼ × cm at the epochs we apply our model (after about 30 days since the explosion).The average mass-loss rate estimated by the CSM mass within 10 cm for v w = 100 km s − is h ˙ M i = 0 . M ⊙ yr − . (59) c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe b o l o m e t r i c l u m i n o s i t y ( e r g s − ) SN 2010jl bolometric LC (Zhang et al. 2012)best fit L =L t α models =2.2 asymptotic interaction models =2 asymptotic interaction model3.4 M ⊙ Ni- Co- Fe decay (full trapping)
Figure 4.
Bolometric LC of SN 2010jl (Zhang et al. 2012) and some model fits to it. The dashed line represents the best fitfor L = L t α . However, t t expected from the result of the fit is too small to apply this model to the entire LC. Thus, we needto apply the asymptotic LC formula which is applicable after t t . We get a good fit with s = 2 . s = 2 model. The dot-dashed line is the radioactive decay energy available from 3 . M ⊙ Ni. Theradioactive decay model is suggested by Zhang et al. (2012) to explain the early bolometric LC but our interaction model canexplain the entire LC with a single component.
The estimated rate is consistent with those obtained based on the infrared emission (Maeda et al.2013; Fox et al. 2013). t t = 29 days for n = 10 and δ = 1 and t t = 15 days for n = 12 and δ = 1. Thus the usage of the asymptotic formula is justified.Since s obtained above is close to the case of the steady mass loss ( s = 2), we also tryto fit the bolometric LC by the asymptotic formula L = a (1 + bt ) − / for s = 2 (Equation33). We obtain a = 4 . × erg s − and b = 6 . × − s − with t t = 22 days ( n = 10and δ = 1) or t t = 13 days ( n = 12 and δ = 1). By using a , b , and ǫ = 0 .
1, we get E ej = 1 . × erg , (60)from Equation (35). Assuming M ej = 10 M ⊙ and v w = 100 km s − , we obtain˙ M = 0 . M ⊙ yr − , (61)from Equation (36). The rate is similar to the average rate from the s = 2 . s = 2 . s = 2 models, we find that the s = 2 model has flatterLC than the s = 2 . s smaller, the model LC gets flatter and it getsharder to explain the bolometric LC of SN 2010jl. Thus, we presume that the CSM around c (cid:13)000
1, we get E ej = 1 . × erg , (60)from Equation (35). Assuming M ej = 10 M ⊙ and v w = 100 km s − , we obtain˙ M = 0 . M ⊙ yr − , (61)from Equation (36). The rate is similar to the average rate from the s = 2 . s = 2 . s = 2 models, we find that the s = 2 model has flatterLC than the s = 2 . s smaller, the model LC gets flatter and it getsharder to explain the bolometric LC of SN 2010jl. Thus, we presume that the CSM around c (cid:13)000 , 1–29 T. J. Moriya et al. the progenitor of SN 2010jl may be a bit steeper than the CSM expected from steadymass loss. This conclusion contradicts that obtained by Chandra et al. (2012b) from X-ray observations. Chandra et al. (2012b) suggest s = 1 . r sh ∝ t ( n − / ( n − s ) which is notlikely applicable at the epochs when they obtained X-ray data ( ≃
60 days and ≃
360 dayssince the explosion). This is because of the small t t mainly due to the high CSM density asis shown above.So far, we fit the entire bolometric LC up to about 200 days by a single component.On the contrary, Zhang et al. (2012) suggested a two-componet model for the bolometricLC. They suggested that the LC before around 100 days is mainly powered by 3 . M ⊙ of Ni whose available radioactive energy is shown in Figure 4. They suggested that theSN ejecta-CSM interaction started playing a role at later epochs by using a model LC ofthe interaction developed by Wood-Vasey, Wang, & Aldering (2004). However, the required Ni mass is very large (3 . M ⊙ ) and this amount of Ni is rather difficult to produce in acore-collapse SN explosion (e.g., Umeda & Nomoto 2008). In addition, no signatures of Feelements are observed in the late phase spectra of SN 2010jl which are expected if there islarge amount of Ni production (e.g., Dessart et al. 2013). As noted in Section 3.2.1, thebolometric LC powered by the interaction resembles to the LC powered by the radioactivedecay of Ni at early epochs and we need to use additional late-phase LCs to distinguishbetween the two power sources. We have shown here that we need only one component fromthe interaction model to explain the whole LC of SN 2010jl.
We have fixed ǫ = 0 . M ej = 10 M ⊙ in deriving some SN ejecta and CSM properties inthe previous section. Here, we discuss how sensitive the derived properties are to the assumedvalues of ǫ and M ej . In addition, it is practically difficult determine v observationally fromspectra of SNe IIn because the origins of the spectral features are not understood well. Wealso discuss the effect of the uncertainty in v .At first, we assume that v is well-determined and see the effect of ǫ . The estimated CSMdensity structures or the estimated mass-loss rates depend on ǫ for a given v (Equation40). The average mass-loss rates have a relation h ˙ M i ∝ ǫ − . For example, we obtained c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe h ˙ M i = 1 . × − M ⊙ yr − for SN 2005ip ( n = 10) in the previous section by assuming ǫ = 0 . h ˙ M i = 2 . × − M ⊙ yr − if we assume ǫ = 0 .
5. This uncertainty makes it difficult to distinguish the RSG progenitor and the LBVprogenitor. The assumption of M ej = 10 M ⊙ is used to estimate E ej . E ej depends on both M ej and ǫ as E ej ∝ ǫ − n − M n − n − ej . For the typical values of n , we obtain E ej ∝ ǫ − . M . ( n = 10) or E ej ∝ ǫ − . M . ( n = 12). In either case, E ej is mostly determined by theassumed M ej . For the SN 2005ip model ( n = 10) in the previous section ( E ej = 1 . × erg for ǫ = 0 . M ej = 10 M ⊙ ), E ej can be changed to, e.g., 7 . × erg ( ǫ = 0 . M ej = 5 M ⊙ ), 8 . × erg ( ǫ = 0 . M ej = 10 M ⊙ ), or 5 . × erg ( ǫ = 0 . M ej = 5 M ⊙ ). Thus, the estimated E ej are not much affected by the assumed parameters.We have assumed that v can be determined by spectral observations. However, spectraof SNe IIn have complicated features with several components and it is not obvious whichspectral component originates from the dense shell and can be used to estimate v . Thanksto the formation of the dense shell due to the radiative cooling, the shell velocity is one ofthe fastest components in the system (see also Section 4.4). Thus, we have used the fastestvelocity component in the spectra to estimate v in the previous section (17 ,
500 km s − at 100 days for SN 2005ip). Indeed, h ˙ M i ∝ v s − ( h ˙ M i ∝ v − for s ≃
2) and the estimatedmass-loss rates are rather sensitive to the assumed v . Keeping ǫ = 0 . M ej = 10 M ⊙ , weobtain h ˙ M i = 1 . × − M ⊙ yr − and E ej = 1 . × erg for v sh (100 days) = 15 ,
000 km s − , h ˙ M i = 5 . × − M ⊙ yr − and E ej = 5 . × erg for v sh (100 days) = 10 ,
000 km s − , and h ˙ M i = 3 . × − M ⊙ yr − and E ej = 2 . × erg for v sh (100 days) = 5 ,
000 km s − . Aswe use the highest velocity component to estimate v in the previous section, the estimatedmass-loss rates were rather conservative. Note again that the dense shell velocity shouldbe one of the fastest components in the system we model (see also Section 4.4) and thusadopting the higher observed velocity is preferred.Finally, v w has been assumed to be 100 km s − , which is a typical LBV wind velocity(e.g., Leitherer 1997). SN IIn spectra often show a 100 km s − P-Cygni profile. The mass-loss rates estimated are proportional to v w . If SNe IIn are from RSGs, the wind velocity canbe lower (e.g., Mauron & Josselin 2011) and the mass-loss rates estimated will be decreasedbecause of the lower wind velocities. c (cid:13) , 1–29 T. J. Moriya et al.
In deriving the evolution of the shocked-shell radius r sh ( t ), we have assumed that s issmaller than 3. This condition is also required to derive a physical self-similar solution (e.g.,Nadyozhin 1985). The allowed range of α for s < α > − α → − s → α is a monotonically-decreasing function at n >
5. Thus, if we obtain α < − L = L t α , this is beyond the applicability of our model and we need to consider other waysto explain the LC.First, we need to check t t . If t t is smaller than the time used for the fitting, we need touse the asymptotic formula for the fitting. The asymptotic formula can have a rapid declinein the bolometric LC depending on parameters.Another possibility is a CSM with s >
3. Most of the mass in CSM with s > s > M csm ≡ Z r sh R p πr ρ csm dr (62)= 4 πDs − (cid:16) R − sp − r − s sh (cid:17) (63) ≃ πDs − R − sp = constant ( r sh ≫ R p ) . (64)Thus, most of the CSM is shocked soon after the explosion. If the CSM density is relativelylow, the LCs will decline quickly soon after the explosion when most of the CSM componentis swept up. If the shocked shell becomes optically thick, LCs may resemble the ’shell-shockeddiffusion’ model LC suggested by Smith & McCray (2007) as a model for superluminous SNebased on the formalisms by Arnett (1980) (but see also Moriya et al. 2013a). This is a LCmodel for the declining part of the bolometric LC after the shock wave passes through adense CSM. According to this model, the declining part of the bolometric LCs follows L = L exp " − tτ diff t τ exp ! , (65)where t is the time since the maximum luminosity, τ diff is the characteristic diffusion timescalein the shocked shell and τ exp is the expansion timescale of the shocked shell.Bolometric LCs can also follow Equation (65) even if s <
3. This is the case when thehigh-density CSM is small in radius and the entire high-density CSM is shocked soon afterthe beginning of the interaction. Then, there is no continuous interaction and the bolometricLC should decline quickly, possibly following the shell-shocked diffusion model. However, in c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe Ni, magnetars (e.g., Maeda et al. 2007; Kasen & Bildsten 2010; Woosley 2010),or fallback (Dexter & Kasen 2012).
The bolometric LC model presented in this paper does not have a rising part at the begin-ning. There are several mechanisms to make the initial luminosity increase in LCs which arenot taken into account in our model.We have assumed that the radiation emitted from the dense shell is not affected by theunshocked CSM. However, especially at the early phases just after the explosion, the CSMsurrounding the dense shell can be optically thick and the radiation from the shell can bescattered within the CSM. In this case, the diffusion timescale in the optically thick regiondetermines the evolution of the initial luminosity increase and subsequent decline. Our modelshould only be applied to the epochs when the CSM surrounding the dense shell becomesoptically thin and should not be applied at the epochs when the luminosity increases or justafter the luminosity peaks. When the CSM is optically thick, some signatures can be seenin spectra as well (e.g., Chugai 2001).If the CSM is optically thin, the timescale of the initial luminosity increase is expectedto be very small. Two mechanisms can affect the initial luminosity increase. One is the shockbreakout at the surface of the progenitor and the other is the on-set of the SN ejecta-CSMinteraction. Both are presumed to have a short timescale. If the dense part of the CSM andthe progenitor are detached, we may see two luminosity peaks in the early phases: one fromthe shock breakout and the other from the on-set of the interaction. c (cid:13)000
The bolometric LC model presented in this paper does not have a rising part at the begin-ning. There are several mechanisms to make the initial luminosity increase in LCs which arenot taken into account in our model.We have assumed that the radiation emitted from the dense shell is not affected by theunshocked CSM. However, especially at the early phases just after the explosion, the CSMsurrounding the dense shell can be optically thick and the radiation from the shell can bescattered within the CSM. In this case, the diffusion timescale in the optically thick regiondetermines the evolution of the initial luminosity increase and subsequent decline. Our modelshould only be applied to the epochs when the CSM surrounding the dense shell becomesoptically thin and should not be applied at the epochs when the luminosity increases or justafter the luminosity peaks. When the CSM is optically thick, some signatures can be seenin spectra as well (e.g., Chugai 2001).If the CSM is optically thin, the timescale of the initial luminosity increase is expectedto be very small. Two mechanisms can affect the initial luminosity increase. One is the shockbreakout at the surface of the progenitor and the other is the on-set of the SN ejecta-CSMinteraction. Both are presumed to have a short timescale. If the dense part of the CSM andthe progenitor are detached, we may see two luminosity peaks in the early phases: one fromthe shock breakout and the other from the on-set of the interaction. c (cid:13)000 , 1–29 T. J. Moriya et al. b o l o m e t r i c l u m i n o s i t y ( e r g s − ) SN 2005ip bolometric LC (Stritzinger et al. 2012)analytic interaction model (s =2.3)numerical bolometric LC b o l o m e t r i c l u m i n o s i t y ( e r g s − ) SN 2006jd bolometric LC (Stritzinger et al. 2012)analytic interaction model (s =1.4)numerical bolometric LC b o l o m e t r i c l u m i n o s i t y ( e r g s − ) SN 2010jl bolometric LC (Zhang et al. 2012)analytic interaction model (s =2.2)numerical bolometric LC
Figure 5.
Comparisons between the numerical bolometric LCs and the analytic bolometric LCs presented in Section 3.2 basedon which the initial conditions for the numerical bolometric LC computations are constructed.c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe cm) 20 19 18 17 16 15 14 l o g d e n s i t y ( g c m − ) v e l o c i t y ( , k m s − ) densityvelocity ⊙ )−20−19−18−17−16−15−14 l o g d e n s i t y ( g c m − ) v e l o c i t y ( , k m s − ) densityvelocity Figure 6.
Density and velocity structures of the numerical model for SN 2005ip at 100 days.
To show the reliability of our analytic LC model, we also performed numerical LC calcula-tions using a one-dimensional radiation hydrodynamics code
STELLA (e.g., Blinnikov & Bartunov1993; Blinnikov et al. 2000, 2006). We show some comparisons in this paper and more de-tailed comparisons will be presented elsewhere. We set the initial conditions following thephysical parameters obtained in Section 3.2. The density structure of the homologouslyexpanding SN ejecta has two power-law components as is assumed in the analytic model.The SN ejecta and CSM are initially connected at 10 cm. The CSM outer radius of all themodels is set to 10 cm. The parameter B q which controls the conversion efficiency from thekinetic energy to radiation in the code (see Moriya et al. 2013b for the details) is adjustedto make ǫ ≃ .
1. Both SN ejecta and CSM in the calculations have solar composition.Figure 5 presents the results of our LC calculations. We performed the LC calculationsof three models in Section 3.2, namely, SN 2005ip ( s = 2 . n = 10, δ = 1, E ej = 1 . × erg, and M ej = 10 M ⊙ ), SN 2006jd ( s = 1 . n = 10, δ = 1, E ej = 1 . × erg, and M ej = 10 M ⊙ ), and SN 2010jl ( s = 2 . n = 10, δ = 1, E ej = 2 . × erg, and M ej = 10 M ⊙ ).The overall features of the analytic LCs are well reproduced by the numerical LCs and theanalytic model presented in this paper is shown to provide a good prediction to the numericalresults.In Figure 6, we show the density and velocity structures of the numerical SN 2005ipmodel at 100 days in radius and mass coordinates. We can see that the dense shell is formedbetween the SN ejecta and the dense CSM and the shell width is much smaller than theshell radius because of the radiative cooling. The plot in the mass coordinate indicates that c (cid:13) , 1–29 T. J. Moriya et al. most of the shocked SN ejecta and CSM is in this thin shell. Thus the assumption in ouranalytic model that the shocked region can be expressed by using a single r sh is verified.This means that the forward and reverse shocks are glued to the cool dense shell and thevelocities of them are not different from each other so much at these early epochs becauseof the radiative cooling. Note that the shell is one of the fastest velocity components at thisepoch and the shell velocity is consistent with v adopted ( v sh ≃ ,
500 km s − at 100 days).We can also see that the density structure ahead of the shell is modified slightly because ofthe precursor. As our LC model takes only the sum of the available energy into account, the LC we obtainfrom the model is bolometric and we have applied our analytic bolometric LC model tobolometric LCs constructed from observations. Here we try to fit the L = L t α formula tooptical and near-infrared LCs of SN 2005ip and SN 2006jd obtained by Stritzinger et al.(2012). We focus on the parameter α which is directly affected by the CSM density slope s for a given n .Figure 7 and Table 2 show the results of the LC fits. As we can see, α obtained withdifferent photometric bands have different values. This means that we need to construct abolometric LC from observations to obtain accurate information. This can be clearly seenin Figure 7 of Stritzinger et al. (2012). The spectra evolve significantly with time and nosingle band can represent the entire evolution of the bolometric LC. We thus clearly needto construct a bolometric LC to apply our model to obtain CSM and SN properties of SNeIIn. We have developed an analytic bolometric LC model for SNe powered by the interactionbetween SN ejecta and dense CSM. This model is suitable for modeling SNe IIn. We haveanalytically derived the evolution of the shocked dense shell created by the interaction. Weobtain the bolometric LC evolution from the derived dense shell evolution. Our model is notrestricted to the CSM from steady mass loss.We have applied our bolometric LC model to three SNe IIn whose bolometric LCs havebeen constructed from observations, i.e., SN 2005ip, SN 2006jd, and SN 2010jl. The results c (cid:13) , 1–29 n Analytic Bolometric LC Model for Interaction-Powered SNe Table 2.
List of α from optical andnear-infrared LCs.Band α SN 2005ip SN 2006jdbolometric -0.536 -0.0708 u -1.01 -0.300 B -0.923 -0.374 g -0.934 -0.387 V -0.995 -0.451 r -0.854 -0.557 i -1.00 -0.592 Y -0.706 -0.414 J -0.630 -0.137 H -0.171 0.0950 show that their CSM density slopes are close to what is expected from the steady massloss ( s = 2 where ρ csm ∝ r − s ) but slightly deviate from it ( s ≃ . − . s ≃ . − . s ≃ . h ˙ M i = 0 . − . M ⊙ yr − , SN 2006jd: h ˙ M i =0 . − . M ⊙ yr − and SN 2010jl: h ˙ M i = 0 . M ⊙ yr − ). We could not constrainSN ejecta properties strongly but E ej of all three SNe likely exceeded 10 erg if we assumethat M ej = 10 M ⊙ and that the conversion efficiency from kinetic energy to radiation is 10%( ǫ = 0 . Ni can be similar to each other up to about 100 days since the explosion. We need tohave LCs also at later phases to distinguish between the two luminosity sources from LCsalone.Our bolometric LC model can only be applied for s <
3. For s >
3, we suggest that theshell-shocked diffusion model proposed by Smith & McCray (2007) (see also Moriya et al.2013a) may be applied for some cases.We have also compared our analytic LCs to synthetic ones calculated with a one-dimensional radiation hydrodynamics code
STELLA . Our analytic LCs are well-reproducedby the numerical modeling.We have applied our model to only three observed SNe IIn. We suggest to systematicallystudy SN ejecta and CSM properties of SNe IIn by applying our LC model to many otherSNe IIn. Such a systematic study will lead to a comprehensive understanding of SNe IIn,i.e., their progenitors and the mass-loss mechanisms related to them. c (cid:13) , 1–29 T. J. Moriya et al. un s c a l e d m a g n i t u d e bol.uBgVriYJH un s c a l e d m a g n i t u d e bol.uBgVriYJH Figure 7.
Multi-color LCs of SN 2005ip and SN 2006jd and the results of the fit to L ∝ t α . α obtained by multi-color LCsare not consistent with α obtained from the bolometric LC. We need a bolometric LC to infer CSM and SN ejecta propertiesfrom LCs properly. ACKNOWLEDGMENTS
We thank the anonymous referee for the comments which improved this paper. T.J.M. andK.M. thank the staff at Stockholm University for their hospitality during their stay as short-term visitors when this project was initiated. T.J.M. is supported by the Japan Society forthe Promotion of Science Research Fellowship for Young Scientists (23 · REFERENCES
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