Photospheric Radius Evolution of Homologous Explosions

Physics & Astronomy Faculty Publications Physics and Astronomy

11-21-2018

Liang-Duan Liu Nanjing University; University of Nevada, Las Vegas

Bing Zhang University of Nevada, Las Vegas, [email protected]

Ling-Jun Wang Chinese Academy of Sciences

Zi-Gao Dai Nanjing University

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Repository Citation Liu, L., Zhang, B., Wang, L., Dai, Z. (2018). Photospheric Radius Evolution of Homologous Explosions. Astrophysical Journal Letters, 868(2), 1-7. http://dx.doi.org/10.3847/2041-8213/aaeff6

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This Article has been accepted for inclusion in Physics & Astronomy Faculty Publications by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. The Astrophysical Journal Letters, 868:L24 (7pp), 2018 December 1 https://doi.org/10.3847/2041-8213/aaeff6 © 2018. The American Astronomical Society. All rights reserved.

Photospheric Radius Evolution of Homologous Explosions

Liang-Duan Liu1,2,3, Bing Zhang3 , Ling-Jun Wang4, and Zi-Gao Dai1,2 1 School of Astronomy and Space Science, Nanjing University, Nanjing 210093, Peopleʼs Republic of China; [email protected] 2 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Peopleʼs Republic of China 3 Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA; [email protected] 4 Astroparticle Physics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, Peopleʼs Republic of China Received 2018 September 13; revised 2018 October 25; accepted 2018 November 9; published 2018 November 21

Abstract Recent wide-field surveys discovered new types of peculiar optical transients that showed diverse behaviors of the evolution of photospheric properties. We develop a general theory of homologous explosions with constant opacity, paying special attention to the evolution of the photospheric radius Rph.Wefind that regardless of the density distribution profile, Rph always increases early on and decreases at late times. This result does not depend on the radiation and cooling processes inside the ejecta. The general rising/falling behavior of Rph can be used to quickly diagnose whether the source originates from a supernova-like explosion. The shape of the Rph evolution curve depends on the density profile, so the observations may be used to directly diagnose the density profile as well as the temperature profile of the ejecta. All of the well-monitored supernovae show such a Rph rising/falling behavior, which is consistent with our theory. The recently discovered peculiar transient AT 2018cow showed a continuous decay of Rph, for which a supernova-like explosion origin is disfavored. Our result therefore supports the interpretation of this transient as a tidal disruption event. Key words: opacity – supernovae: general

1. Introduction 2. A General Theory of Photospheric Radius Evolution The rapid development of several wide-field optical surveys Observationally, the photospheric radius at a particular time ( ( ) 5 4 12 e.g., the intermediate Palomar Transient Factory iPTF , the can be derived by Rph()tLtTt= [ bol ()4ps eff ()] , where the ( ) 6 All-Sky Automated Survey for Supernovae ASASSN , the bolometric luminosity Lbol(t) can be derived from the multi- Panoramic Survey Telescope & Rapid Response System (Pan- color photometry at each epoch t, and the effective temperature ) 7 ( )8) STARRS , and the Dark Energy Survey DES is revolu- Teff(t) can be inferred by fitting the spectra at the same epoch. tionizing the field of time-domain transient astrophysics. In From the theoretical model, the photospheric radius evolution addition to known objects (e.g., supernovae (SNe) and tidal depends on the dynamical evolution and the density profile of ( )) ( disruption events TDEs with extreme properties e.g., the ejecta, but is independent of the cooling and heating ASASSN-15lh, Dong et al. 2016; and iPTF14hls, Arcavi et al. processes.9 In the literature (Arnett 1982), the photospheric ) 2017 , these observations have also discovered several peculiar, radius is often described as rapidly evolving, luminous transients whose nature is not ( properly understood Droutetal.2014;Arcavietal.2016; 2 Whitesides et al. 2017). One example is AT 2018cow, which Rtph ()=- Rt ()l () t,1 ( ) 3 showed a very rapid rise of the lightcurve, and a steady decay of the photospheric radius R (Kuin et al. 2018; Perley et al. 2018; ph for an ejecta with a uniform, time-dependent density ρ(t), Prentice et al. 2018). Such behavior has never been observed where l()tt= 1 rk () is the mean free path of the photons, before in a supernova. Possible interpretations range from special κ fi types of explosions to special types of TDEs, but no definite and is the opacity. In reality, the density pro le of an fi conclusion has been drawn. explosion is not uniform. Different types of density pro les Here we develop a simple theory of the evolution of the will modify Equation (1) to much more complicated forms. photospheric radius, Rph, of a generic explosion, which is In order to simplify the problem to a tractable form, we make homologous (each layer expanding with a constant velocity) several assumptions in the following. First, the supernova but could have arbitrary density profile, heating/cooling ejecta is homologously expanding and spherically symmetric. structure, and hence, arbitrary temperature profile. Assuming Second, Thomson scattering dominates the opacity so that the κ a constant opacity, we derive a generic behavior of the Rph opacity is a constant throughout the evolution. Third, we evolution of such explosions. Section 2 presents the general assume that the emission from the ejecta layers above the theory. Section 3 presents several specific density profile photosphere in the nebula phase does not outshine the emission examples. The results are summarized in Section 4, with some from the photosphere itself. Introducing more complicated discussions on its application to AT 2018cow and other scenarios would introduce more qualitative differences (see transients. 9 This statement is strictly correct for photospheric emission. In practice, 5 R “ ” L T https://www.ptf.caltech.edu/iptf however, ph is determined by the observed bol and eff, which may be 6 dominated by the contribution of emission outside of the photospheric radius http://www.astronomy.ohio-state.edu/~assassin/index.shtml (i.e., the ejecta layers already in the so-called “nebula” phase) during the late 7 https://panstarrs.stsci.edu phase of a supernova explosion. In such cases, the effective Rph derived from 8 https://www.darkenergysurvey.org the data does depend on the heating process in the ejecta.

1 The Astrophysical Journal Letters, 868:L24 (7pp), 2018 December 1 Liu et al. discussion in Section 4), but the general features discussed in where this Letter may not alter substantially. 1 4 For an energetic explosion such as a supernova, the ejecta IxxdxK º h(),9 () would enter a homologous expansion phase after a few times of ò0 R the expansion timescale Rp v, where p is the radius of the is a dimensionless factor for kinetic energy that is related to the v progenitor and is the mean expansion velocity of the ejecta. assumed density profile. For a homologous expansion of the ejecta with a velocity Combining Equations (6) and (8), the velocity of the gradient, fast ejecta layers propagate in front and slow ejecta ( fi outermost layer of the ejecta which is the maximum velocity layers lag behind. The inner boundary of the ejecta is de ned in the ejecta) is given by by the slowest ejecta, and its radius reads ⎛ ⎞12 E I Rtmin()=+ R min,0 vt min ,2 ( ) 2 K M vmax = ⎜ ⎟ .10() ⎝ Mej IK ⎠ where vmin is the minimum velocity of the ejecta and Rmin,0 is the initial radius of the innermost radius when the explosion For a uniform density distribution, one has vmax = 12 enters the homologous phase. The outer boundary of the ejecta (10EMKej 3) . It is worth noting that vmax is a parameter in is defined by our semi-analytic model, which usually cannot be measured directly. Rt()=+ R0max v t,3 ( ) The total optical depth of the ejecta τtot is where R0 is the initial radius of the outermost radius in the Rt() v tkr= dr.11() homologous phase, and max is the maximum velocity of the tot ò Rtmin() ejecta. The homologous expansion conditions imply Rp < Rmin,0< R 0 and vvmin max. For a constant opacity κ, one has We define a comoving, dimensionless radius x as ⎡ RR- ⎤2 tt()t = (0,12 )⎢ 0min,0⎥ () rR- min tot tot ⎣ ⎦ x º ,4() Rt()- Rmin () t RR- min where ttot (0) is the initial optical depth, i.e., where r is the radius of a particular layer in the ejecta from the center of explosion, and 0x1 is satisfied for all of the tkrtot()0,0,= (RRI 0 ) 0 t ( 13 ) elements within the ejecta. and For a homologous expansion, the density of the ejecta can be written as 1 Ixdxt º h() ò0 ⎡ RR- ⎤3 rrh()rt,,0= ( R )() x⎢ 0min,0⎥ ,5() 0 ⎣ ⎦ is a dimensionless factor for the optical depth that is related to Rt()- Rmin () t the density profile. The total optical depth ttot ()t decreases with -2 where r()R0,0 is the initial density at the outermost radius of the time following t . When ttot = 23the whole ejecta becomes ejecta, h()x is a function to describe the density profile of the ejecta, transparent. We introduce a critical time so that ttot ()tt = 2 3 and for a uniform density distribution, one has h()x = 1.The is satisfied, which reads --3 ⎧ ⎫ [(RR0 min,0 ) ( RtRt ( ) min ( ))] scaling describes the homo- ⎛ RR- ⎞ ⎡ 30t ()⎤12 t = ⎜ 0min,0⎟⎨⎢ tot ⎥ - 1,⎬ () 14 logous expansion of the ejecta. t ⎝ ⎠⎩⎣ ⎦ ⎭ The total ejecta mass can be derived through integrating over vvmax- min 2 the density profile, i.e., after tτ the explosion enters the so-called “nebular” phase, Rt() when the assumption of blackbody emission becomes invalid. 2 Mrrtdrej = ò 4,pr() Based on the Eddington approximation, the relation between Rtmin() the externally observed effective temperature T and the 3 eff = [(4,0,pr RRI0 )]0 M () 6 internal temperature T at an optical depth τ=2/3 is given by (Arnett 1980; Arnett & Fu 1989) where 3 ⎛ 2 ⎞ 1 TT4 =+4 ⎜⎟t .15() 2 eff ⎝ ⎠ IxxdxM º h(),7 () 4 3 ò0 Therefore, the location of the photospheric radius R is at is a dimensionless factor for ejecta mass that is related to the ph t ()Rph = 23,which is defined as assumed density profile. The total kinetic energy with a given density profile can be Rt() 2 ò krdr = .16() derived as Rtph() 3 Rt() ( ) ( ) 1 22 Using Equations 5 and 12 , this condition can be rewritten as EtK ()= ò rp v4 rdr Rtmin() 2 ttot ()t 2 3 2 Itph ()= ,17 ( ) = [(2,0,pr RRvI0 )]0 max K () 8 It 3

2 The Astrophysical Journal Letters, 868:L24 (7pp), 2018 December 1 Liu et al. where Equation (25), we have 1 Itph ()º h ( xdx ),18 ( ) 4It ò ttr = .26() xtph() 3h()()xxtph ph and xRRRR=-()() - is a dimensionless para- ph ph min min fi meter of the photosphere radius. As the expansion proceeds, We can then nd out the time tph,max when the photospheric ( ) xph decreases with time, which means that the photospheric radius reaches the maximum by substitution Equation 26 into radius recedes in the comoving coordinate of the ejecta. When Equation (12). Therefore, according to our general theory, we reach the tt= t, one has xph = 0, i.e., the photospheric radius reaches the innermost radius of the ejecta, and the photons produced following conclusion: in an ejecta undergoing homologous expansion, for an arbitrary density distribution profile, the anywhere in the ejecta can escape directly without being photospheric radius Rph always displays an initially rising scattered. phase and a later declining phase. The result does not depend The photospheric radius is on the radiation and cooling process inside of the ejecta. Rtph()=- [ RtRtxtRt () min ()] ph () + min ().19 ( ) 3. Examples The evolution of the photospheric radius depends on the R competition between the expansion and the recession of x in In this section, we consider the ph evolution in several ph fi fi ( the comoving coordinate of the ejecta. examples with different speci c density pro les a spherical ) The time derivative of the photospheric radius reads symmetry is assumed throughout , and show the differences among these examples. dRph =-()vvxmax min ph dt 3.1. CASE I: Uniform Density Profile dxph fi +-[()Rt Rmin ()] t +vmin.20 ( ) If the density pro le of the ejecta is uniform, one has dt h()x = 1. Substituting it into Equation (17), one can obtain ( ) It is worth noting that dRdtis not the so-called xph=-123t tot. This is equivalent to Equation 1 . As the ph R(t) photospheric velocity v as measured by observers based on ejecta expands homologously, linearly increases with time, ph λ∝t3 spectral information, which is the instantaneous velocity of the while the mean free path of the photons evolves as . ( ) fi τ = layer of ejecta that reaches the photosphere radius. In our According to Equation 26 ,we nd that tr 2 corresponding calculation, we have assumed that the ejecta is homologously to the maximum photospheric radius. The evolution of the expanding, which means the local velocity v is proportional to photospheric radius and velocity with an uniform density is shown in Figure 1 (red dashed lines). the radius r. Therefore, the photospheric velocity vph is given by To calculate Rph, we need to solve Equation (17) and then vph RRph- min apply Equation (19). For vvmin max, given a certain density = .21() fi vmax RR- min pro le h()x , there are four main free parameters that may significantly affect the results: the ejecta mass Mej, the initial Comparing vph with the observational photospheric velocity radius of the outer layer of the ejecta R0, the initial kinetic evolution obtained from absorption spectral features could help energy E , and the opacity κ. In the following, we investigate us to constrain the velocity profile of the explosion ejecta. K ( ) how different parameters affect the result for the uniform Taking the time derivative of Equation 17 , one has density case. The fiducial parameters are chosen as (plotted ⎡ ⎤ ⎛ ⎞ with red dashed line in Figure 2): MM= 3.0 ; E =´2 dxph d 1 d 2 I ej  K ⎢ h()xdx⎥ = ⎜ t ⎟.22 ( ) 51 R = 13 κ= 2 −1 ò ⎝ ⎠ 10 erg, 0 10 cm, and 0.1 cm g . dt dx ⎣ x ph ⎦ dt 3 t ph tot We first investigate the effect of ejecta mass. Three values M = M We can then obtain the time derivative of xph as are adopted: ej 1, 3, 8 e. The results are shown in panel (a) of Figure 2. One can see that the ejecta mass has dxph 4Ivt max- v min 1 fi fl =- .23() signi cant in uence on the photospheric radius evolution. As dt 3 Rt()- Rmin () tht ( x ph ) tot () t Mej increases, the maximum Rph is larger and the time it takes to reach the maximum is longer. Substituting it into Equation (20), we get Next, we investigate the effect of kinetic energy by adopting 50 51 51 ⎡ ⎤ EK =´510,210,410 ´ ´ erg. As shown in panel (b) dRph 4It E fl R =-()vvx⎢ - ⎥.24() of Figure 2, K mainly in uences the time when ph reaches the max min⎣ ph ⎦ dt 3ht()()xtph tot maximum, but has little influence on the peak value of Rph. Because we fixed the ejecta mass as MMej = 3.0 , a higher The location of the maximum photospheric radius is found kinetic energy corresponds to a larger velocity scale, resulting in by setting dRdtph = 0 in Equation (24), giving a faster evolution of Rph. ( ) R 4I The panel c of Figure 2 shows that initial radius 0 has a xt()-=t 025() ph negligible effect on the evolution of Rph. We adopt three 3ht()()xtph tot 12 13 14 values, i.e., R0=10 ,10,10 cm, and find that Rph Let us define a “transitional” optical depth τtr by essentially does not change. This is because during the dRdtph = 0, i.e., when the total optical depth τtot equals ttr, evolution, we are mostly investigating the epochs when the photospheric radius reaches its maximum. Using vt R0, so that the initial conditions do not matter much.

3 The Astrophysical Journal Letters, 868:L24 (7pp), 2018 December 1 Liu et al.

Figure 1. Photospheric radius (left panel) and velocity (right panel) evolution with the different density proﬁles for the following choice of parameters: Mej = 3.0M ; 51 13 2 −1 EK =´210erg, R0 = 10 cm, and κ=0.1 cm g .

Finally, we consider the effect of opacity. In the panel (d) of The dimensionless geometric factor for the kinetic energy of Figure 2, three values of opacity are chosen as κ=0.1, 0.2, the ejecta is (Vinkó et al. 2004) 0.4 cm2 g−1. We can see that a higher opacity leads to a higher 5 maximum Rph and a longer time to reach it. x0 1 n 5 R IK = + ()xx00- .30 () In all of these cases, the shape of the ph evolution curves 5 - d 5 - n remain the same, which only depends on the density profile function η(x). The total optical depth of the outer region ejecta reads ⎡ RR- ⎤2 3.2. CASE II: Broken Power-law Density Profile tt()t = (0,31 )⎢ 0min,0⎥ () tot,out tot,out ⎣ ⎦ Rt()- Rmin () t We next relax the assumption of uniform density distribution. The first case we study is a broken power-law density where the initial optical depth of the outer region is profile, with a flatter profile in the inner region and a steeper n profile in the outer part of the ejecta (e.g., Chevalier 1982; xx0 - 0 tkrtot,out()0,0= (RR 0 ) 0 .32 ( ) Matzner & McKee 1999; Kasen & Bildsten 2010; Moriya et al. n - 1 2013), If ttot,out ()02> 3, Rph is located in the outer region at early ⎧ fi ()xx-d 0, x x epochs. We de ne a timescale tt,out when the outer part region h x = ⎨ 00 () -n ()27 becomes transparent (ttot,out ()t = 23), i.e., ⎩()xx00 x x 1, ⎛ ⎞⎧⎡ ⎤12 ⎫ x RR0min,0- 30ttot,out () where 0 is the dimensionless transition radius from the inner t = ⎜ ⎟⎨⎢ ⎥ - 1.⎬ () 33 t,out ⎝ ⎠ ⎣ ⎦ region to the outer region. It is only for n>5 and δ<3 that vvmax- min ⎩ 2 ⎭ the conditions of finite energy and mass can be satisfied. Such a tt< R profile is often adopted in modeling SNe. The outer density At t,out, ph is in the outer region, which is equivalent to xx> index n depends on the progenitor of the SN. For SN Ib/Ic and ph 0, so that n; ( SN Ia progenitors, one has 10 Matzner & McKee 1999; 1 ) -n Kasen & Bildsten 2010; Moriya et al. 2013 . For explosions of Ixxdxph = ò ()0 red supergiants (RSGs), one has n;12 (Matzner & McKee x ph 1999; Moriya et al. 2013). The slope of the inner region of x n x n = 0 ()1 -»x 10--n x 1 n.34 () the ejecta satisfies δ;0−1. In our calculation, we adopt 1 - n ph n - 1 ph d ==0,n 10 as fiducial values. The dimensionless geometric factor for the ejecta mass due We then obtain fi to the assumed density pro le distribution is a broken power ⎡ ⎤ 1 law, i.e., (Vinkó et al. 2004) 2 1-n xtph()= x 0⎢ ⎥ .35() ⎣ 3ttot,out ()t ⎦ 1 3 1 n 3 IxM = 0 + ()xx00- .28 () 3 - d 3 - n At tt> t,out, the outer region becomes transparent ()t » 23. The photospheric radius R would enter the The mass ratio between the inner and outer regions is tot,out ph inner part region of the ejecta. The total optical depth of the ⎛ 3 ⎞ inner region reads 3 - n x0 M = ⎜ ⎟.29() 3 - d ⎝ xxn - 3 ⎠ ⎡ ⎤2 00 x0 RR0min,0- tkrtot,in()tRR= ( 0,0 ) 0 ⎢ ⎥ .36() - d ⎣ Rt- R t⎦ For x0=0.1, one has M = 2.33. 1 ()min ()

4 The Astrophysical Journal Letters, 868:L24 (7pp), 2018 December 1 Liu et al.

Figure 2. Effects of the varying ejecta mass Mej (panel a), kinetic energy EK (panel b), initial radius R0 (panel c), and opacity (panel d) of the ejecta. The uniform density 51 13 2 −1 proﬁle η(x)=1isadopted.Theﬁducial parameters are (plotted with a red dashed line): Mej = 3.0M ; EK =´210erg, R0=10 cm, and κ=0.1 cm g .

When tt> t,out, the dimensionless photospheric radius can Similar to the above analysis, we can obtain the dimension- be obtained by less photospheric radius as ⎡ ⎤ 23- t ()t ⎡ -a ⎤ tot,out 1 21()- e -a xtph()=- x 0⎢1 ⎥.37() xt()=- ln ⎢ + e ⎥,41() ⎣ ⎦ ph ⎣ ⎦ ttot,in ()t a 3ttot ()t R v The ph and ph evolution are shown in Figure 1 with black where the total optical depth is solid lines. We find that in this situation the Rph evolution curve shares similar qualitative behaviors with the uniform density 1 - e-a ⎡ RR- ⎤2 one. In particular, it shares the same decline rate after the peak. tkr()tRR= (,0 ) ⎢ 0min,0⎥ .42() tot 0 0 ⎣ ⎦ For the particular parameter set that we have adopted, after a Rt()- Rmin () t tt,out = 52.8 days, xph occurs in the inner region, which has a slope δ=0 corresponding to a constant density profile. The Rph evolution for this case is shown in Figure 1 as the green solid curve. Because the density gradient dh dx is larger 3.3. CASE III: Exponential Density Profile than that of the uniform density profile, the Rph decline rate is much slower. We now consider the density profile in the form of 3.4. CASE IV: Density Increases with Radius h()xax=-exp ( ) , ( 38 ) In the three cases mentioned above, the density profile of the a a= where is a small positive value, with 1.72 representing ejecta is either a constant or decreasing with the radius. It is the Pacyzński RSG envelope (Arnett 1980). interesting to investigate the opposite case, i.e., the density In this case, the dimensionless geometric factors for the increases with radius so that there is a positive density gradient, ejecta mass and the kinetic energy are as follows: even though it is difficult to realize such a density profile in SN explosions. 222-++()aae2 -a We assume the density profile as a power law, i.e., I = ,39() M 3 a h()xx= m,43 ( ) and where the power-law index m is assumed to be a positive value 24+-{ 24 -aaae [ 24 + ( 4 + )]} -a to allow density increasing with radius. The uniform density IK = .40() a5 profile corresponds to m=0.

5 The Astrophysical Journal Letters, 868:L24 (7pp), 2018 December 1 Liu et al.

Figure 3. Photospheric radius evolution of various optical transients: Type I superluminous SN PTF 12dam, Type IIP SN 2015ba, Type Ib SN iPTF13bvn, the kilonova AT 2017gfo associated with GW170817/GRB 170817A, as well as the peculiar event AT 2018cow that is likely not from an explosion.

In this situation, the photospheric radius reads Our treatment assumed a constant opacity. In general, the

1 opacity is a function of density, temperature, and composition ⎛ 2 ⎞ m+1 of the ejecta. It is essentially a constant when Thomson Rt()=- Rt ()⎜1 ⎟ ,44() ph scattering dominates the opacity. If the local temperature of the ⎝ 3ttot ()t ⎠ ejecta drops below the recombination temperature Trec, the where ejecta is mostly neutral, in which case the opacity is almost zero. Taking the recombination effect into account, the ejecta ⎡ ⎤2 kr()RR00,0 RR0min,0- becomes transparent in a shorter timescale (Arnett & Fu 1989). t ()t = ⎢ ⎥ .45() tot ⎣ ⎦ 1 + m Rt()- Rmin () t Considering the effect of the recombination would introduce additional complications of Rph evolution, which is not The time derivative of the photosphere is investigated in this Letter. So far we have ignored emission from the outer layers in the ⎡ ⎤ 1 dRph 2 m+1 nebular phase, so the above theory is applied to the case when =-()vv⎢1 - ⎥ max min ⎣ ⎦ the emission from the nebular phase does not outshine the dt 3ttot ()t ⎡ ⎤ -m emission from the photosphere. At late epochs of an explosion, 4()vv- 1 2 m+1 - max min ⎢1 - ⎥ .46() such an assumption is no longer valid. On the other hand, the ⎣ ⎦ 31(+ mt )tttot ()3 tot () t observed spectrum would deviate from blackbody because the emission is optically thin. For an ideal observational campaign We find that when tttot== tr 2331()()mm + +, the with wide-frequency-band observations, such a phase can be in photospheric radius reaches its peak. principle identified. In practice, photometric observations in We adopt m=2, the Rph and vph evolution are shown in several different colors may not be able to tell the difference, so Figure 1 as blue solid lines. Compared with the three cases that an “effective” R˜ph ()t is derived based on the observed mentioned in previous subsections, the photospheric radius in Lbol(t) and Teff(t), which include the contributions from both the this case decreases very rapidly after the peak due to the rapid true photosphere and gas above. This is not the true decrease of density as the photosphere recedes in the ejecta. photospheric radius, which decays slower than the true Rph(t). This explains the shallow Rph decay in many transients as 4. Conclusions and Discussion revealed by observations (e.g., Nicholl et al. 2016; Dastidar ) We have investigated a general model of homologous et al. 2018 . expansion with an arbitrary density distribution profile and the Some SNe, especially SNe IIn, show evidence of interaction ( ) evolution of the photospheric radius. We discover a generic between the SN ejecta and the circumstellar medium CSM around the progenitor. In this case, because the velocity of the behavior, i.e., Rph always rises at early epochs and falls at late epochs. As shown in Figure 1, different density profiles affect outer layers of the SN ejecta is much higher than the velocity of the CSM, one may assume that the ejecta interacts with a the shape of the Rph evolution curves, especially the rate of decline after the peak. However, the general qualitative relatively stationary CSM. The photospheric radius is located in behavior remains the same. Investigating how various the CSM rather than in the SN ejecta. Photons diffuse through fi ( parameters might affect the Rph evolution curve (Figure 2), an optically thick CSM with a xed photosphere Chatzopoulos we find that the initial radius has a negligible effect, while et al. 2012, 2013). Therefore, in this situation, one has Tt()µ fl 14 R ejecta mass, kinetic energy, and opacity all in uence the Lbol ()t . The difference in the ph evolution behaviors between maximum Rph and the time to reach the peak. the interacting model and the homologous explosion model

6 The Astrophysical Journal Letters, 868:L24 (7pp), 2018 December 1 Liu et al. discussed here can be used to diagnose the physical origin of an observed photospheric temperature as a function of time, observed SN event. when coupled with the inferred density profile as well as the R In Figure 3,wecollectasampleofexplosionswhose ph adiabatic evolution law in Equation (47), can be used to / evolution is well observed. One can see that the general rising directly diagnose the temperature structure of the ejecta. R falling behavior of ph predicted in our theory is found in Direct confrontations of our theory with detailed observa- different types of SNe, including superluminous supernova PTF tional data of diverse explosion events will be carried out in 12dam (Vreeswijk et al. 2017), Type IIP supernova SN 2015ba future work. with a long plateau (Dastidar et al. 2018),andTypeIb iPTF13bvn (Fremling et al. 2016). The photospheric radius of the “kilonova” transient AT 2017gfo associated with GW170817 We thank Simon Prentice, Daniel Perley for helpful information, Shanqin Wang, Shaoze Li, Anthony L. Piro for also exhibited such a rising/falling behavior (Drout et al. 2017). discussion, and the referee for helpful suggestions. This Letter The widths of the R peaks depend on the physical parameters ph was supported by the National Basic Research Program (“973” of the explosions (e.g., M , E ,andκ), but the general evolution ej K Program) of China (grant No. 2014CB845800) and the National behavior is similar. Natural Science Foundation of China (grant Nos. 11573014 and The special event AT 2018cow shows a peculiar behavior of 11833003). This Letter was also supported by the National steady decline of R as a function of time (Kuin et al. 2018; ph Program on Key Research and Development Project of China Perley et al. 2018); see Figure 3. According to the theory (grant Nos. 2017YFA0402600 and 2016YFA0400801). L.D.L. discussed in this Letter, this behavior means that it is essentially is supported by a scholarship from the China Scholarship impossible for AT 2018cow to be a supernova. Observation- ( ) ally, R decays from the very beginning, and no rising R was Council No. 201706190127 to conduct research at the ph ph University of Nevada, Las Vegas (UNLV). detected. In order to interpret the source as a SN, the ejecta mass should be very small, e.g., ∼0.05Me, in order to make a ( very rapid rise to satisfy the observational constraint Prentice ORCID iDs et al. 2018). For such a small mass, the ejecta would become // / transparent in a very short period of time, e.g., tτ=3.2 days Bing Zhang https: orcid.org 0000-0002-9725-2524 for uniform density distribution. 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