Influence of macroclumping on type II supernova light curves
AAstronomy & Astrophysics manuscript no. ms c (cid:13)
ESO 2019July 5, 2019
Influence of macroclumping on type II supernova light curves
Luc Dessart and Edouard Audit Unidad Mixta Internacional Franco-Chilena de Astronomía (CNRS, UMI 3386), Departamento de Astronomía, Universidad deChile, Camino El Observatorio 1515, Las Condes, Santiago, Chile Maison de la Simulation, CEA, CNRS, Université Paris-Sud, UVSQ, Université Paris-Saclay, 91191, Gif-sur-Yvette, France.Received; accepted
ABSTRACT
Core-collapse supernova (SN) ejecta are probably structured on both small and large scales, with greater deviations from sphericalsymmetry nearer the explosion site. Here, we present 2D and 3D gray radiation-hydrodynamics simulations of type II SN lightcurves from red (RSG) and blue supergiant (BSG) star explosions to investigate the impact on SN observables of inhomogeneities indensity or composition, with a characteristic scale set to a few percent of the local radius. Clumping is found to hasten the release ofstored radiation, boosting the early time luminosity and shortening the photospheric phase. Around the photosphere, radiation leaksbetween the clumps where the photon mean free path is greater. Since radiation is stored uniformly in volume, a greater clumping canincrease this leakage by storing more and more mass into smaller and denser clumps containing less and less radiation energy. Aninhomogeneous medium in which di ff erent regions recombine at di ff erent temperatures can also impact the light curve. Clumping canthus be a source of diversity in SN brightness. Clumping may lead to a systematic underestimate of ejecta masses from light curvemodeling, although a significant o ff set seems to require a large density contrast of a few tens between clumps and interclump medium. Key words. radiative transfer – radiation hydrodynamics – supernovae: general – supernova: individual: SN 1987A
1. Introduction
The ejecta mass of core-collapse supernovae (SNe) is a funda-mental parameter characterizing these events (Heger et al. 2003).This is in part because numerous quantities inferred from obser-vations scale in one way or another with the ejecta mass, such asthe characteristic ejecta expansion rate or the photon di ff usiontime (Arnett 1980). Inferring the ejecta mass is also essential forestimating the progenitor mass, the nucleosynthetic yields, andbuilding a physically consistent picture of core-collapse SN ex-plosions (see e.g. Sukhbold et al. 2016).Paradoxically, estimating the ejecta mass, and by extensionthe progenitor mass, is very challenging. An origin for this dif-ficulty may lie in the fact that the SN material holds little in-ternal energy, which is instead all stored in radiation (producedinitially by the shock but also continuously through radioactivedecay). Mass is only a source of opacity, trapping the radiation,controlling its rate of escape and thus producing the resultingbolometric light curve. However, the shock-deposited energy perunit ejecta mass varies with ejecta depth. For example, the shockpassage through a low-density extended envelope produces anobvious luminous burst in some type IIb SNe, which allows foran estimate of the associated mass (Nomoto et al. 1993; Podsi-adlowski et al. 1993; Woosley et al. 1994; Bersten et al. 2012;Dessart et al. 2018b). In contrast, the shock passage through adense and massive He core may produce a feeble luminosity sig-nature in type II SNe (Dessart & Hillier 2019b), making the in-ference of its mass a very delicate matter. In GRB / SNe, the ejectamasses inferred from observations around the time of bolometricmaximum do not typically agree with those inferred from latetime observations (Maeda et al. 2003; Dessart et al. 2017). Alarge mass must be present at low velocity, but being somewhat“dark”, it is harder to constrain. Hence, mass inferences in dif- ferent SN types, using early or late time constraints, are subjectto complex uncertainties.In SNe powered primarily by Ni decay, ejecta masses areinferred using various incarnations of a di ff usion model, usingassumptions such as homogeneity, fixed opacity, fixed ioniza-tion, and a negligible contribution from shock-deposited energy(see e.g. Arnett 1982; Chatzopoulos et al. 2012). In radiation hy-drodynamics models for these ejecta, spherical symmetry, grayopacity and various levels of mixing are used (see e.g. Berstenet al. 2012). In explosions of red-supergiant (RSG) stars, theejecta mass inferences are generally based on semi-analyticalmodels or radiation hydrodynamics simulations (see e.g. Litvi-nova & Nadezhin 1985, Popov 1993, Utrobin 2007). The ejectais assumed spherically symmetric and smooth, and the gas istreated in local thermodynamic equilibrium (LTE). This impliesthat the gas ionization is set by the Saha equation.In reality, core-collapse SN ejecta are complex environ-ments. The progenitor stars may not explode in a vacuum butinstead in a dense and confined environment (Yaron et al. 2017).The explosion is likely asymmetric on all scales, as evidenced bylight echoes (Rest et al. 2011), nebular-phase spectra (Fransson& Chevalier 1989; Jerkstrand et al. 2012), late-time integral-fieldspectroscopic observations (Kjær et al. 2010), late-time radio ob-servations (Abellán et al. 2017), or spectro-polarimetric observa-tions (Leonard et al. 2006). From the theoretical point of view,these departures from spherical symmetry may arise from a vari-ety of causes, including Rayleigh-Taylor instabilities, post-shockneutrino-driven convection, the standing-accretion-shock insta-bility, or the e ff ect of progenitor rotation (Mueller et al. 1991;Fryxell et al. 1991; Wongwathanarat et al. 2013). These give riseto asymmetries on a wide range of scales (from tens of percent toa few percent of the local radius). Smaller scale structures may Article number, page 1 of 13 a r X i v : . [ a s t r o - ph . S R ] J u l & A proofs: manuscript no. ms also exist but are not resolved by current multi-D hydrodynami-cal simulations.It is therefore of interest to explore the impact of such com-plicated ejecta properties on SN radiation and quantify the im-pact they may have on the inferences we make from more sim-plistic assumptions. One such complication is the 3D inhomo-geneous structure of core-collapse SN ejecta and its impact onSN observables. For example, microclumping has an e ff ect ontype II SN light curves and spectra (Dessart et al. 2018a). Mi-croclumping takes the form of density inhomogeneities that areoptically thin, meaning that their scale is shorter that the typi-cal photon mean free path. By boosting the recombination rate,microclumping hastens the recession of the photospheric layers,increases the radiation leakage from the ejecta, boosts the lumi-nosity, shortens the rise time to maximum in BSG star explo-sions, and leads to a shorter photospheric phase in a type II SN.By reducing the electron density above the photosphere, it alsoleads to a reduction in the H α line strength.The present study investigates the influence of macroclump-ing on type II SN radiation properties. In contrast to micro-clumping, it corresponds to density inhomogeneities that arelarge compared to the photon mean path. Hence, macroclump-ing can influence the transport and escape of radiation from aSN ejecta. Here, we will consider macroclumps with a size of afew percent of the local radius, associated with local variations ineither density or composition. Because it is not at present possi-ble to conduct nonLTE as well as multi-D time-dependent radia-tive transfer, the combined influence of microclumping (whichrequires a solution to the statistical-equilibrium equations) andmacroclumping (which requires multi-D radiative transfer) can-not be assessed. A conclusion of this study is, however, that thetwo e ff ects act in the same direction, and that assuming spher-ical symmetry and a smooth homogeneous medium leads to anunderestimate of the ejecta mass.In the next section, we present our numerical approach. Us-ing gray radiation hydrodynamics in 1D, 2D, and 3D, we ex-plore the influence of macroclumping on the SN radiation duringthe photospheric phase for a RSG explosion model (Section 3).We consider various levels of density contrast (Section 4) andcomposition (Section 5) between the clumps and the interclumpmedium. We also explore the influence of the progenitor radiuswith the case of a BSG star explosion model (Section 6). Finally,we quantify the underestimate in ejecta mass that results fromassuming a smooth ejecta. In Section 8, we present our conclu-sions.
2. Numerical approach
We have used the Eulerian multi-dimensional radiation-hydrodynamics code heracles (González et al. 2007; Vaytetet al. 2011) to perform 1-D, 2-D, and 3-D simulations of typeII SNe. The code treats the hydrodynamics using a standard sec-ond order Godunov scheme. All simulations employ a gray ra-diation transport solver, which is based here on the M1 momentmodel (Dubroca & Feugeas 1999). As discussed in Gonzálezet al. (2007), the M1 model is well suited for the study of radia-tion transport in a structured medium. It captures well the shad-owing e ff ect of a high density clump as well as the propagationspeed of radiation in a transparent medium. Hence, it handlesadequately the di ff erent transport properties from the optically-thick to the optically-thin layers. Because we start from SN ejecta that are already in homolo-gous expansion, there is no shock on the grid. Dynamical e ff ectsare negligible and the ejecta material evolves ballistically. Thegas and the radiation are in equilibrium at large optical depthand deviate modestly from each other through and above thephotosphere. The need for a multi-group treatment of the radia-tive transfer is therefore not crucial for the computation of thebolometric light curve, so the assumption of gray transport isadequate.We assume a uniform H-rich composition and treat the gasas ideal, with a mean atomic weight µ = γ = /
3. Forsimplicity, we use a simple prescription for the opacity. Since weassume a plasma at the solar composition (thus dominated by Hand He), the opacity at high temperature is well described by theRosseland mean value κ high = g − . At low temperature,we adopt the low value κ low = g − . As in Khatami &Kasen (2018), we find it convenient to use an analytical form forthe temperature dependence of the opacity (we ignore any ex-plicit dependence of the mass absorption coe ffi cient on density,but the inverse mean-free path depends explicitly on ρ ) with κ ( T ) = κ low + (cid:16) κ high − κ low (cid:17) (cid:16) + π arctan (cid:0) T − T ion ∆ T ion (cid:1)(cid:17) , (1)where T is the gas temperature, T ion is a representative recombi-nation temperature for the gas (e.g., which is about 6000 K forH-rich material at representative SN ejecta densities), and ∆ T ion is the range over which the plasma opacity transitions from κ low to κ high as the temperature is raised from below to above T ion (thistransition typically occurs over a narrow temperature range, sowe set ∆ T ion to 200 K). We assume that this gray opacity is splitbetween a scattering component κ sca and an absorption compo-nent κ abs , with an assumed albedo κ sca / ( κ sca + κ abs ) fixed at a valueof 0.9. This is rough but adequate for a type II SN (see, e.g.,Dessart et al. 2015, as well as appendix A for a more extendeddiscussion).Adopting an ideal gas equation of state ignores the impactof changes in excitation and ionization on the pressure, the tem-perature, or the energy of the gas. The thermal energy of the gasis, however, a small fraction of the total ejecta energy, which isdominated by radiation and kinetic energies. This choice allowsa quick determination of the thermodynamic properties analyt-ically, saving time and avoiding numerical issues with interpo-lation between pre-computed table values. Because the focus ofthe present study is to compare between 1D, 2D, and 3D simula-tions with and without macroclumping and for a given SN ejecta,these simplifications are not a concern, provided we use (and wedo) the same choices for all simulations.We use spherical coordinates R , θ (and µ = cos θ ), φ , with n R , n θ , and n φ zones in each direction. The lowest resolution cor-responds to n R = n θ =
48 (in 2D and 3D) and n φ = n R and n θ . A higher resolutionis needed when the simulation starts at a young SN age, hencewhen the ejecta has not yet expanded to a large radius. In 2D and3D, we simulate wedges placed arbitrarily along φ but centeredin latitude along the equatorial plane (i.e., θ = π/ θ and φ extends over 20 ◦ so that a su ffi cient number ofclumps can be used to fill the grid (see Section 2.3). The starting conditions for the radius R , velocity V , density ρ ,and temperature T of the SN ejecta are prescribed analytically. Article number, page 2 of 13uc Dessart and Edouard Audit: Macroclumping and type II SN radiation
The advantage is flexibility. The specification of the density ver-sus velocity follows the approach of Chugai et al. (2007). Theejecta density distribution ρ ( V ) is given by ρ ( V ) = ρ + ( V / V ) k (2)where ρ and V are constrained by the adopted ejecta kineticenergy E kin , the ejecta mass M ej , and the density exponent k through M ej = πρ ( V t ) C m ; E kin = C e C m M ej V , (3)and where C m = π k sin(3 π/ k ) ; C e = π k sin(5 π/ k ) . (4)The grid, which is Eulerian, must cover initially the spacethat the SN ejecta will occupy over the whole simulated evolu-tion. The choice of grid is dictated by the impact it has on theCourant time (especially relevant when doing multi-D) or thenumber of zones needed to resolve the ejecta at all times. Thus,for practical reasons, the simulation is started at a SN age suf-ficiently large that the inner ejecta has expanded to a significantradial scale. Starting at a SN age of days to weeks after explo-sion, we can adopt a minimum ejecta radius between 10 and10 cm. For the outer radius, we must make sure that it is largeenough to encompass all ejecta regions that trap radiation en-ergy during the high brightness phase. For the ejecta propertiesconsidered here, we found that a maximum grid radius R max of4 × cm was suitable. A large fraction of the ejecta leavesthe grid during the simulation but with this choice of R max , theescaping material is always optically thin and thus no longer in-fluences the trapping or the di ff usion of SN radiation (some pho-tons interact with optically-thick lines at large velocities, whichwe neglect, but these interactions influence the spectral proper-ties and not the bolometric luminosity).The ejecta is in homologous expansion. Because the Eu-lerian grid extends to a large maximum radius and becausethe presence of a pre-SN wind is ignored, the outer velocitymay become unrealistically large. Hence, the outer velocity isforced to slowly level o ff at V max = − if it exceeds V lim = − , following the expression : V ( R ) = V lim + ( V max − V lim ) (1 − R lim / R ) . if V > V lim , (5)where R lim = V lim t and t is the elapsed time since explosion.This feature is used for simulations of BSG explosions, whichare started at an earlier time (see below). This non-homologousexpansion at large velocity has no impact since it concerns onlythe optically thin regions of the ejecta (which quickly advect outof the grid).In type II SNe, the shock deposited energy plays an essentialpart in the resulting bolometric light curve so the initial temper-ature structure matters. We use the following expression (similarto the opacity formulation above; Eq. 1) T = T low + (cid:16) T high − T low (cid:17) (cid:16) + π arctan (cid:0) R T − R ∆ R T (cid:1)(cid:17) , (6)where T low and T high are the ejecta temperatures far from the ra-dius R T and where ∆ R T controls the scale over which T variesbetween T low and T high . This expression is useful since one canmimic the presence of a temperature jump (e.g., across a recom-bination front) or adjust the temperature gradient as desired. In practice, using previous simulations for type II SNe as a guide(Dessart et al. 2013; Dessart & Hillier 2019a), heracles was runin 1-D and low resolution until we obtained a bolometric lightcurve that approximately resembled that of a standard type II-Plateau or the type II-pec SN 1987A. One can switch betweenthe two light curve morphologies by raising T high (SNe II-P) orlowering T high (SNe II-pec) in the initial model. In addition, all simulations treat the radioactive decay of Niand Co. The code can treat both local and nonlocal energydeposition. However, for the present simulations of type II SNejecta prior to 100 −
200 d, we can assume that the decay poweris deposited locally. This requires following one species acrossthe simulation. The adopted initial profile for Ni is of the form X ( Ni) = X ( Ni ) exp( − Y ) with Y = V − V Ni ∆ V Ni ; V ≥ V Ni (7)and X ( Ni) = X ( Ni ) for V < V Ni . Here, X ( Ni ) is the in-ner ejecta mass fraction of Ni, which is constant until V Ni anddrops exponentially beyond with a characteristic scale ∆ V Ni . Itis set so that the total (spherical equivalent ejecta) mass matchesa desired value (irrespective of clumping; see below).The above expressions can be used to set the boundary con-ditions analytically for V , ρ , and T . For the SN ejecta, we usean inflow inner boundary. V ib is given from the inner boundaryradius R ib at post-explosion time t + t as R ib / ( t + t ) ( t is the SNage at the start of the heracles simulation). For convenience, theinternal energy was set to be constant across the inner boundary.The boundary density is determined by ρ ib = ρ + ( V ib / V ) k . (8)The SN age is incremented at each time step in order to computethe decay power and update the inner boundary condition forthe velocity and density. The SN age is given as the initial SNage plus the elapsed time since the start of the simulation. Inaddition, ρ ib at post-explosion time t + t is directly determinedafter updating ρ at each time step (i.e., set through the constraintthat ρ t is constant in time; this comes from the constraint ofmass conservation, as seen also in Eq. 3). Alternatively, we havealso used a reflecting inner boundary condition (there is then noinflow of material). In this case, the results for the bolometriclight curve are identical (this occurs because the mass injectedon the grid contains a negligible amount of radiation energy).For the outer boundary, we adopt a constant internal energythrough the boundary, a velocity set by homology, and a densityfall-o ff with a power law of exponent six at all times (in practice,it should evolve with time, being nine initially (this is our choicefor the value of k in our simulations; see Table 1) and decreasingas the velocity declines, but this is irrelevant given the super-sonic outflow speed at the outer boundary). For the radiation, weassume a reflecting inner boundary (zero flux) and a free flowouter boundary. This di ff erence arises from the greater cooling from expansion thata ff ects the explosions from more compact stars like BSGs relative toRSGs. This temperature di ff erence means that type II-P SN ejecta holdmore radiative energy at 10 −
20 d after explosion than SNe II-pec, whichbrighten to a delayed maximum because of radioactive decay heating.Consequently, SNe II-P are much more luminous early on than SNeII-pec; see for example Section 5.1 and Figs. 11–12 of Dessart et al.(2011). Article number, page 3 of 13 & A proofs: manuscript no. ms R [10 cm]-15 -14 -13 -12 -11 -10log( ρ /g cm − ) V T − − l og¯ ρ s m . . . ξ . . . . R [10 cm]0 . . . . ¯ X ( N i ) Fig. 1.
Two dimensional density structure (left) for a clumped model set-up over a 90 ◦ wedge (in our heracles simulations, this wedge straddlesthe equatorial plane at θ = π/ θ = π/ θ = π/ − ( V ), the temperature in units of 10 K( T ), the normalized smooth density in the log, the radial variation of clumping ( ξ ), and the normalized Ni distribution ¯ X ( Ni) versus radius R .The model corresponds to the case of a 20 d old type II-P SN ejecta with the following model parameters: M ej =
12 M (cid:12) , E kin = . × erg, k = T high = ,
000 K, T low =
100 K, R T = × cm, ∆ R T = × cm, σ cl = . ξ = . V cl = − , ∆ V cl = − , M ( Ni) = .
05 M (cid:12) , V Ni = − , and ∆ V Ni = − . We simulate the e ff ect of macroclumping on SN radiation byadjusting the smooth density profile given by Eq. 2. We first im-pose a radial variation of the magnitude of clumping using thefunction ξ ( V ) = + ( ξ −
1) exp( − Y ) with Y = V − V cl ∆ V cl if V ≥ V cl (9)and ξ = ξ if V < V cl . With this choice, we can impose clump-ing in the inner (lower velocity) ejecta regions while leaving theouter regions untouched (same density as in the smooth casegiven by Eq. 2).In our clumped models, the parameter ξ controls the den-sity contrast between the clump (or interclump) medium withthe corresponding smooth model. When initializing a 2D / ρ inter − cl ( R , µ, φ ) = ρ sm ( R , µ, φ ) ξ ( R ) . (10)This defines the “background" density. We then randomly dis-tribute clumps between R min and R max , µ min and µ max , φ min and φ max . At ( R , µ, φ ), the density associated with a clump at location( R cl , µ cl , φ cl ) is given by ρ cl ( R , µ, φ ) = (1 − ξ ( R )) ξ ( R ) ρ sm ( R , µ, φ ) exp (cid:16) − d σ R (cid:17) , (11) where d cl is the distance between the clump center ( R cl , µ cl , φ cl )and the location ( R , µ, φ ). The characteristic scale of a clump is σ cl R cl to reflect spherical expansion. Our choice also implies thatall clumps have the same spatial extent at a given R .In reality, macroclumps may have a distribution of sizes, per-haps growing continuously from being much smaller than a pho-ton mean free path (microclumps) to being much larger. Unlikefor the treatment of microclumping, which considers that clumpsare surrounded by vacuum, our macroclumps are surrounded byinter-clump material of finite density. Our parameterization isnumerically convenient but others are possible. Because of nu-merical limitations, we adopt relatively large clumps so that ahigh resolution is not needed. We therefore do not consider adistribution of clump sizes, nor consider how the radiation trans-port may be a ff ected as clumps are increased from a microscopicto a macroscopic scale.To initialize a simulation, we keep adding such clumps untilthe cumulative mass of the clumps plus the interclump mediumequals that for the corresponding smooth ejecta model. Equa-tions 10 and 11 indicate that for a clump at R the ratio ρ cl /ρ inter − cl is equal to (1 − ξ ( R )) /ξ ( R ), which is at most (1 − ξ ) /ξ – thisratio is unity at large velocities relative to V cl because we im-pose that clumping eventually dies out as we progress from theinner to the outer ejecta layers. Since the ejecta mass (cid:82) ρ dv isunchanged in the presence of clumping, the clump density goes Article number, page 4 of 13uc Dessart and Edouard Audit: Macroclumping and type II SN radiation as the inverse of the volume filling factor when ρ cl /ρ inter − cl >> Ni. The adopted Ni distribution isuniform in angle but varies with radius. In other words, at a givenradius or velocity, the clump and the interclump media have thesame composition (this holds in all simulations apart from Sec-tion 5). For a given choice of mixing and clumping properties,the mass fraction of Ni is renormalized so that the volume in-tegral (cid:82) dv ρ X ( Ni) is equal to a prescribed value (independentof the adopted clumping). With this procedure, the impact ofmacroclumping on a type II SN light curve can be gauged fora given ejecta mass, kinetic energy, and Ni mass. This treat-ment of clumping leaves the bulk ejecta properties unchanged –it merely redistributes the density over the ejecta volume underspecified geometric constraints.For most of the 2D and 3D simulations presented here, theangular wedge extends over 20 ◦ in θ and φ . The characteristicscale of clumps is typically 0 . R , so about 17 can fit in thelateral direction, and a few hundred clumps are used to fill thegrid. There is thus no need to use a larger angular extent. We focus on ejecta conditions typical of red-supergiant (RSG)and blue-supergiant (BSG) progenitors, producing two sets ofmodels called “2P” and “2pec”. The main characteristic distin-guishing the two sets is the initial temperature. In the 2pec set,the temperature is low initially so that the SN brightness in-creases with time because of the contribution from Ni decay.This case corresponds to events like SN 1987A. In the 2P set, thetemperature is high initially so that the brightness is high earlyon and decreases with time as the ejecta releases its stored ra-diation energy. This corresponds to standard type II-P SNe. Inthis case, the decay of Ni merely lengthens the high brightnessphase.We ran simulations in 1D, 2D, and 3D, with a resolution thatis larger for the 2pec set compared to the 2P set. The clump-ing magnitude is such that ξ varies between 1 (smooth ejecta)and 0.1 (maximum density contrast of 90). The radial varia-tion of clumping varies between models but is such that clump-ing is greater in the inner ejecta and progresses towards unityat the largest velocities (see Eq. 9). Unless otherwise stated,and as explained in Section 2.1, the opacity parameters arethe same in all simulations and such that κ high = g − , κ low = g − , T ion = ∆ T ion =
200 K. For Ni, the adopted mass is 0.078 M (cid:12) for the 2pec set and 0.05 M (cid:12) for the 2P set (with the exception of models discussed in Sec-tion 5). This choice is arbitrary, except for the 2pec models inwhich Ni is essential for producing a high, SN-like, luminos-ity. The same level of Ni mixing is used for all models withina set (i.e., the set 2P or 2pec; see Table 1).It is not clear whether there is a tight correlation between theprofiles of Ni and clumping. Both stem from fluid instabili-ties. The Ni “fingers" may stretch in velocity space further thanthe region of high clumping. Hence, the grid of simulations pre-sented here use two di ff erent distributions for Ni and clumping,and with characteristics that we allow to vary to cover a range ofpossibilities.While 3D simulations have greater consistency, there is agreat benefit in performing 2D simulations. They are computa-tionally cheaper, allowing one to cover a large parameter space,and they also capture the main features of clumped ejecta on the
20 30 40 50 60 70 80 90 100 110Days since explosion41.441.641.842.042.242.442.642.8 l o g ( L b o l / e r g s − ) Fig. 2.
Bolometric light curves for a set of 2P models with the sameejecta properties but assuming spherical symmetry (1D and smooth;dashed line), axial symmetry (2D; clumps have a torus shape), andno symmetry (3D; clumps are spheres). The clumping parameters are ξ = V cl = − , ∆ V cl = − , and σ cl = SN radiation. Computationally expensive 3D simulations onlyprovide a slight quantitative o ff set with respect to 2D counter-parts. Hence, numerous simulations were performed in 2D andonly four in 3D (each costs about 90,000 CPU hours; see Sec-tion 3 and 7).Table 1 presents a summary of the model parameters used forthe grid of models discussed in the following sections. Figure 1gives an illustration for one setup over a 90 ◦ wedge. In that case,the spatial extent of the clumped regions was enlarged to betterreveal the properties of the clumps.
3. Results for a reference case
Figure 2 shows the bolometric light curve for the 2P model sim-ulated in 1D (smooth ejecta density structure) or in 2D and 3D(clumped structure). The adopted clumping is strong at low ve-locity but rises quickly to unity beyond a few thousand km s − .The clumping parameters are ξ = V cl = − , ∆ V cl = − , and σ cl = ff ect the outermost ejectalayers, and because all models have the same Ni mass (i.e., ir-respective of clumping), the model luminosity is the same at thestart of the simulation (when the photosphere is in the smoothouter ejecta layers) and at nebular times (when the total lumi-nosity equals the total decay power, i.e., L bol = L decay ). However,because of the di ff erent ejecta density structures (i.e., smooth orclumped), the rate at which the radiation is released from theejecta di ff ers between models.In the clumped models, the early-time luminosity is greaterthan in the 1D smooth model for up to about 65 d, after which itis below the predictions for the 1D model. The clumped modelsalso transition earlier to the nebular phase. In our setup, the orig-inal ejecta temperature varies with radius but is independent ofangle. The initial temperature structure is independent of clump-ing, and so does the total radiative energy stored in the ejecta.This energy is of the form (cid:82) a R T dv , where a R is the radiationconstant and dv is a volume element. The di ff erent models inFig. 2 therefore radiate roughly the same time-integrated lumi-nosity (modulated by expansion losses), but clumping influences Article number, page 5 of 13 & A proofs: manuscript no. ms
Table 1.
Summary of initial model parameters (see Section 2.4 for discussion).
Model R min R max M ej E kin k Age T high T low R T ∆ R T M ( Ni) ∆ V Ni V Ni [10 cm] [M (cid:12) ] [10 erg] [d] [kK] [kK] [10 cm] [M (cid:12) ] [1000 km s − ]2P 0.1 4.0 12 1.2 9 20 50 0.1 0.7 0.3 0.050 1.0 2.02pec 0.1 4.0 13 1.2 9 11 60 0.1 0.3 0.1 0.078 0.5 2.0 V = ¯ F R ( V , θ ) ¯ ρ ( V , θ ) V = V = V =
280 85 90 95 100 θ [deg]0.00.51.0 118.4 d/ V = V = ¯ F R ( V , θ ) ¯ ρ ( V , θ ) V = V = V =
380 85 90 95 100 θ [deg]0.00.51.0 118.4 d/ V = V = ¯ F R ( V , θ ) ¯ ρ ( V , θ ) V = V = V =
480 85 90 95 100 θ [deg]0.00.51.0 118.4 d/ V = Fig. 3.
Evolution of the normalized radial radiative flux ¯ F r ( V , θ ) (solid) and mass density ¯ ρ ( V , θ ) (dashed) at three di ff erent velocities (the label V gives the corresponding ejecta shell velocity in units of 1000 km s − ) for the 2D simulation (model 2P-2D-xi0p1-vcl2e8) shown in Fig. 2.
20 40 60 80 100 120 140Days since explosion1.52.02.5 < R > p h o t [ c m ] Fig. 4.
Evolution of the angle-averaged photospheric radius for the 1D,2D, and 3D simulations shown in Fig. 2. the rate at which the stored radiative energy is released. Becausethe radiative energy is stored uniformly in volume, the segrega-tion of mass into clumps lowers the trapping e ffi ciency of the ejecta. The radiative flux is boosted between the clumps and thestored radiative energy can escape more freely.Figure 3 illustrates this e ff ect for three comoving velocitiesof 2000, 3000, and 4000 km s − for model 2P-2D-xi0p1-vcl2e8.Where the optical depth is large (i.e., at smaller velocities andearlier times), the normalized flux is impacted by the changein photon mean free path caused by clumping. Although theflux is not large at high optical depth, the flux contrast betweenclump and interclump medium is large. As we go to lower opticaldepth, the contrast between clump and interclump medium is un-changed at a given velocity, but the lateral fluctuations in radia-tive flux is weaker. The impact of clumping is most pronouncedin the vicinity of the photosphere. Beyond the photosphere, thephoton mean free path is larger so the material cannot trap ef-ficiently the radiation, whether it is clumped or not. Below thephotosphere, at high optical depth where the photon mean freepath is small, the modulations caused by clumping have a weakinfluence on radiation leakage. But at moderate optical depth,the presence of clumping can allow radiation to leak out frombetween the clumps when the clumps are still optically thick. Inthis region, clumping can foster an earlier escape of radiation.Consequently, the angle-averaged photospheric radius in-creases more slowly (i.e., the photosphere recedes faster in massspace) in the 2D and 3D clumped models compared to thesmooth 1D counterpart (Fig. 4). The greater recession is what Article number, page 6 of 13uc Dessart and Edouard Audit: Macroclumping and type II SN radiation r F r [Normalized] 25.8d 31.6d 37.4d 43.1d48.9d54.7d60.5d66.3d72.1d77.9d83.7d89.4d95.2d101.0d106.8d 112.6d 118.4d 124.2d-7 -5 -3 -1log ( ρ / ρ max ) Fig. 5.
Clock plots for the radial radiative flux (left; the quantity shown is R F r , normalized to its maximum value at each time) and of the massdensity ρ (right; also normalized) as a function of time for the 2D simulation 2P-2D-xi0p1-vcl2e8 shown in Figs. 2–4. Each wedge corresponds toone post-explosion epoch, starting at noon and progressing clockwise, with a time increment of 5.78 d. In each panel, the black line correspondsto the photosphere (the maximum grid radius is 4 × cm). causes the boost to the bolometric light curve. The e ff ect is anal-ogous to that caused by microclumping (Dessart et al. 2018a)but the process is di ff erent. With the microclumping treated in1D nonLTE radiative transfer, the recombination rate at the pho-tosphere is enhanced, which lowers the ionization and helps thephotosphere to recede in mass space. With the macroclump-ing treated in (LTE) multi-D radiative transfer, it is the en-hanced radiative losses that increase the photospheric coolingand causes the faster photospheric recession. In reality, both mi-cro and macroclumping should be present. Because their e ff ectacts in the same sense, the combination of both forms of clump-ing should yield a greater influence on the light curve than whenonly one form of clumping is present.The evolution of the radiative flux and mass density at multi-ple epochs, from the start of the 2D simulation until the nebularphase, is shown in Fig. 5. At the earliest epochs, the photosphere(which is sensitive to the downstream density structure) is es-sentially spherical. The impact of clumping is first born in op-tically thick regions. By ∼
60 d, clumping is present below, at,and beyond the photosphere but its influence has been felt sincethe start. At this epoch, the bolometric luminosity in the 2D and3D simulations drops below the value in the 1D smooth model(Fig. 2). These “clock plots” also show how radiation progressesmore e ffi ciently through the lower density regions between theclumps (see also Fig. 3). The process is time dependent becauseof ejecta expansion and the depth-dependence of clumping, andalso because the radiant energy is typically more abundant in theouter ejecta than in the inner ejecta. Di ff erent clumping proper-ties, combined with di ff erent ejecta properties, would yield dif-ferent behaviors.Figure 2 also includes the bolometric light curve for the 3Dmodel. Interestingly, going from 1D to 2D leads to a greaterchange to the light curve than going from 1D to 3D, even if the e ff ect is qualitatively the same. The di ff erent quantitative behav-ior may result from the greater porosity of the ejecta in 2D sinceclumps are structured as tori, mimicking the e ff ect of alignedclumps in 3D. When clumps are randomly distributed in 3D,they more e ffi ciently cover ejecta-centered sight lines and thusbetter trap the stored radiation. These properties depend on theadopted clumping properties.
4. Influence of some variations in clumpingcharacteristics
Figures 6 and 7 illustrate the impact of clumping properties onthe resulting bolometric light curves of clumped 2D ejecta forthe 2P ejecta conditions (Table 1). For the models in Fig. 6, theadopted clumping properties are ξ = V cl = − , and ∆ V cl ranges between 1000 and 6000 km s − . In Fig. 7, the ejectaproperties are analogous except that V cl = − .For the model with V cl = − and ∆ V cl = − (Fig. 6), the light curve is una ff ected by clumping (it overlapswith that for the 1D smooth model). This arises because clump-ing is confined to the innermost ejecta layers, which containvery little radiative energy at ∼
100 d. Whether this material isclumped or not makes no di ff erence since there is no energy torelease. As ∆ V cl is enhanced, clumping covers a larger range ofthe ejecta so a greater fraction of the volume that stores the ra-diative energy reacts to the change in photon mean-free path. Agreater impact on the light curve is obtained when V cl is raisedfrom zero to 3000 km s − . This velocity threshold correspondsroughly to the edge of the progenitor core (see Fig. 1).In type II SN progenitors, clumping should be stronger inthe inner ejecta, which corresponds to the shocked progenitorHe core. This material, which contains less radiative energy thanthe shocked H-rich envelope, should thus be made even more Article number, page 7 of 13 & A proofs: manuscript no. ms transparent because of clumping, compromising even more theinference of its mass from light curve modeling. Clumping hasa visible e ff ect on the light curve only if it takes place within theH-rich layers of the progenitor where the bulk of the radiativeenergy is stored.Figure 8 shows the impact of the adopted clump size on thebolometric light curve. The larger are the clumps, the stronger isthe impact on the light curve, with a great boost at early times, ashorter optically-thick phase duration (earlier transition to thenebular phase), and greater fluctuations. What drives this be-haviour is that the number of clumps drops as their size is in-creased, facilitating the escape of radiation between fewer andlarger clumps. For smaller clumps, the lines of sight are moreevenly covered by clumps, preventing the escape of radiation.For an infinitely small clump size, the radial optical depth con-verges towards that for the smooth case and the trapping e ffi -ciency of the material is unchanged. This is the case described inDessart et al. (2018a), in which clumping acts primarily throughits influence on the recombination rate.
5. Influence of chemical inhomogeneities
We have also used our clumping-formalism to mimic chemicalsegregation. Isolated regions of space (i.e. clumps) were turnedinto pure helium, while the surrounding material was a mixtureof H and He. In this simulation of chemical segregation, the den-sity was untouched and thus identical to the 1D (smooth) modelcounterpart. Within the simulation, this compositional di ff erencewas conveyed through a distinct recombination temperature, setto 10,000 K for the He-rich material and 6000 K for the rest – theactual material opacity was kept the same as for the solar metal-licity mixture because the reduced electron-scattering contribu-tion in a He-rich plasma is compensated by the greater contri-bution from metal lines. The critical feature of He-rich (H-poor)material is that it recombines at higher temperatures than H-richmaterial.For simplicity, we used the 2P ejecta model with no Ni andthus no decay power. We performed one simulation in 2D, noclumping, and uniform composition (the model name is 2P-2D-smooth). A second simulation was done with the same setup, butin which 30 blobs of pure helium were randomly distributed inboth latitude and radius, between the innermost ejecta layer and5000 km s − . The spatial extent of the He blobs is 2% of the localradius (the blobs are tori in 2D).In the 2D heracles simulation with such He-rich blobs, thelight curve presents two broad bumps and a slightly faster tran-sition to the nebular phase compared to the 2D smooth (homo-geneous) ejecta (Fig. 9). This feature is caused by the faster re-combination in the He-rich blobs, hastening the recession of thephotosphere and the release of stored energy (from within andbelow the blobs). Since the blobs do not change the radiative en-ergy budget of the ejecta, the slightly greater release of energyearly on leads to a faster transition to the nebular phase. The ef-fect found here in 2D with He-rich blobs is similar to the resultsof 1D simulations by Khatami & Kasen (2018) in which the re-combination temperature of the ejecta material is increased.In Nature, type II SN light curves could present fluctuationsassociated with the presence of chemical inhomogeneities. Thisdepends also on the size, number, and composition of such in-homogeneities. Their presence is very likely a result from thechemical mixing caused by Rayleigh-Taylor instabilities andpost-shock neutrino-driven convection (see, e.g., Wongwatha-narat et al. 2013).
6. Results for a more compact, BSG, progenitor
We have also explored the influence of macroclumping in anejecta produced by the explosion of a more compact progeni-tor, namely a BSG star. The initial conditions (especially for thetemperature) for the calculation were adjusted to deliver a bolo-metric light curve similar to that of SN 1987A (Catchpole et al.1987; Hamuy et al. 1988) – see also Section 2.4 and Table 1.For the 2pec set, the simulations are started a little earlier, at11 d, when the ejecta has already started to recombine, but nottoo early so that it has expanded sizably. To resolve this morecompact structure initially, we used a higher resolution, with960 radial zones, and 96 zones in θ — no 3D simulation wasperformed because there are too costly for our computer capa-bilities.Figure 10 shows the bolometric light curve for two clumpedmodels and the 1D-smooth counterpart, all using the 2pec ejectaparameters shown in Table 1. For the clumped models, the pa-rameters are V cl = − and ∆ V cl = − , with ξ = −
15 d beforethe 1D smooth counterpart. The e ff ect of clumping here is analo-gous to the e ff ect of Ni mixing in BSG explosion models (see,e.g., Blinnikov et al. 2000), although for di ff erent reasons. With Ni mixing, power is generated further out in the ejecta, circum-venting the long delay otherwise needed for di ff usion (or for therecession of the photosphere into the layers rich in Ni). In con-trast, with clumping, the recession of the photosphere and thedi ff usion of stored radiation are both hastened.
7. Implications for the inferred type II ejecta masses
All the simulations presented in the preceding sections showthat clumping leads to a shortening of the photospheric phase.A clumped ejecta may thus appear as a smooth ejecta of a lowermass. In this section, we try to quantify this e ff ect by running arestricted set of simulations for 3D clumped ejecta and comparethe resulting light curve to smooth (1D) ejecta having a lowermass.To limit the parameter space, we first performed a set of 2Dclumped simulations based on the 2P ejecta model parameters(Table 1) in which we varied the parameters ξ (values 0.1, 0.25,and 0.5) and V cl (values 1000, 2000, 3000, and 4000 km s − ). Allcombinations of ξ and V cl were done. The results showed thatmodels with ξ = V cl = − are very similar tothe 1D smooth result. Similarly, the results for V cl of 3000 and4000 km s − are very close to each other. Hence, discarding thesesuperfluous choices, we performed 3D simulations for four casesonly, using ξ of 0.1 and 0.25, and V cl of 2000 and 4000 km s − (the 3D model with ξ = V cl = − was alreadypresented in Section 3).As discussed in the preceding sections and now demon-strated in the 3D clumped simulations shown in Fig. 11, decreas-ing ξ or increasing V cl leads to an enhancement of the early-timeluminosity and a shortening of the photospheric phase. Figure 11also shows the light curves for 1D smooth model counterparts inwhich the ejecta mass is reduced. The morphology of the result-ing set of light curves di ff ers from that of the 12 M (cid:12) ejecta simu-lations because the association between density and temperature(i.e., for a given profile T ( R )) is no longer the same. However, Article number, page 8 of 13uc Dessart and Edouard Audit: Macroclumping and type II SN radiation
20 40 60 80 100 120Days since explosion41.441.641.842.042.242.442.642.8 l o g ( L b o l / e r g s − ) V [10 km s − ]0 . . . . . . ξ Fig. 6.
Bolometric light curves (left) for a set of 2D clumped models for the 2P case in which the radial clumping profile ξ ( V ) is modified. Thethin black line corresponds to the instantaneous decay power. For the clumping parameters, the present set uses ξ = V cl = − , and ∆ V cl ranges between 1000 and 6000 km s − (right). The dashed line corresponds to the 1D smooth model counterpart.
20 40 60 80 100 120Days since explosion41.441.641.842.042.242.442.642.8 l o g ( L b o l / e r g s − ) V [10 km s − ]0 . . . . . . ξ Fig. 7.
Same as Fig. 6, but now using V cl = −
20 40 60 80 100 120Days since explosion41.441.641.842.042.242.442.642.8 l o g ( L b o l / e r g s − ) Fig. 8.
Comparison of bolometric light curves for 2D 2P ejecta modelswith ξ = V cl = − , and with di ff erent choices for theclump lateral size σ cl , which covers from 0.005 to 0.04 times the localradius (see label as well as Eq. 11 for details). the overall behavior is similar in the sense that the lower ejectamasses yields larger luminosities early on (i.e. during the first
20 40 60 80Days since explosion41.441.641.842.042.242.442.642.8 l o g ( L b o l / e r g s − ) Fig. 9.
Bolometric light curves for two 2P ejecta models simulated in2D, one being smooth and of uniform (solar-metallicity) composition,the other including 30 pure-He blobs (with a characteristic size set to0 . R ) randomly distributed up to a velocity of 5000 km s − . Radioac-tive decay power is ignored in this set of simulations. two third of the photospheric phase, but with the exception ofthe first week) and an earlier transition to the nebular phase since Article number, page 9 of 13 & A proofs: manuscript no. ms
10 20 30 40 50 60 70 80 90 100 110 120Days since explosion41.241.441.641.842.0 l o g ( L b o l / e r g s − ) Fig. 10.
Same as Fig. 2, but now for the 2pec models. For the clumpedmodels, the parameters are V cl = − and ∆ V cl = − ,with ξ = Ni mass of 0.078 M (cid:12) . the ejecta contains the same amount of trapped radiation energyinitially.If we use the duration of the photospheric phase as a proxyfor ejecta mass, our 3D clumped models correspond to variousreductions in ejecta mass compared to a 1D smooth ejecta modelwith identical parameters (as given Table 1). Model 2P-3D-xi0p25-vcl2e8 matches the 1D smooth model that is 0.5 M (cid:12) lessmassive. For models 2P-3D-xi0p25-vcl4e8 and 2P-3D-xi0p1-vcl2e8, this reduction is 1 − . (cid:12) . For model 2P-3D-xi0p1-vcl4e8, the reduction is about 3.5 M (cid:12) .This exploration is somewhat artificial since one cannot inprinciple change the ejecta mass without changing the otherquantities. But it suggests that clumping, if strong and extended,can lead to a significant underestimate of the ejecta mass. Themagnitude of the e ff ect on the light curve depends on the levelof clumping in the ejecta regions that contain a sizable amountof trapped radiation. Hence, it depends both on the density struc-ture (average density and clumping profile) and on the tempera-ture structure (how the radiation energy is distributed in velocityor mass space).
8. Conclusions
We have presented a set of gray radiation-hydrodynamics simu-lations in 1D, 2D, and 3D for type II SN ejecta from BSG andRSG progenitors. The simulations are limited to the phase ofhomologous expansion, starting at 10 −
20 d after explosion. Forsimplicity, the initial ejecta conditions are set analytically us-ing guidance from more sophisticated radiative transfer simula-tions of type II SNe. Macroclumping is introduced in the form ofradially and laterally confined high-density regions (tori in 2D,spheroids in 3D), with an extent set to some fraction (typically ∼ ff ect can be strong because at agiven radius or velocity, the clumps and the interclump mediumhave the same temperature in optically-thick regions, given bywhat was produced by the shock and subsequently degradedby expansion. By segregating more and more mass into dense clumps, a greater amount of stored radiation becomes trappedwithin a lower density medium (the interclump medium), so thatits escape is facilitated. The general e ff ect is thus to boost theearly time luminosity and shorten the photospheric phase. InBSG explosions, macroscopic clumping also leads to a shorterrise time to bolometric maximum.The exact impact of clumping on the SN radiation dependson numerous aspects. The e ff ect of clumping increases as thesize of the clumps increases, which also tends to introducelarge amplitude fluctuations in the light curve. For small enoughclumps, the medium acts as if it was smooth (if we neglect the in-fluence on the recombination rate and the ionization; see Dessartet al. 2018a). Although potentially stronger, clumping at low ve-locity has little impact on the light curve during the photosphericphase because the inner ejecta layers contain only a small frac-tion of the total radiation budget. The larger volume occupiedby the ejecta regions at larger velocities stores more radiationenergy, but clumping is expected to be weaker there. Hence, itis the clumping at intermediate velocities of a few 1000 km s − that probably has the strongest impact on type II SN light curves.In our set of simulations, the strongest impact was obtained forcases in which clumping corresponded do a maximum densitycontrast of a few tens out to about 4000 km s − . In the case ofstrong clumping, the 3D clumped model showed a light curveanalogous to that of a smooth ejecta model with a 30% lowerejecta mass. For lower values of clumping, the o ff set in ejectamass may be only 10 or 20%.Clumping may also appear in the form of composition inho-mogeneities rather than density variations. An interesting e ff ectis the case of He-rich clumps since their recombination temper-ature is larger than for a mixture of H and He. In 2D heracles simulation with such He-rich blobs, the light curve presents low-frequency variations compared to the smooth (homogeneous)ejecta counterpart, as well as a slightly earlier transition to thenebular phase. In reality, there may be simultaneously densityvariations and chemical inhomogeneities, with distinct proper-ties at di ff erent depths, yielding a complicated e ff ect on the SNlight curve.Clumping may also be associated with a microscopic e ff ect,not included in the present simulations, but discussed in Dessartet al. (2018a). With clumping, the recombination rate (whichscales with the square of the local gas density) is boosted sothat dense clumps will recombine on a shorter time scale thanthe surrounding lower density medium. In practice, a clumpedmedium will be a complicated mixture of regions with di ff erentdensity, temperature, ionization (as well as composition if chem-ical segregation is taken into account). But clumping shouldlead to a faster recession of the photosphere through the lower-density interclump medium as well as the lower-temperaturelower-ionization higher-density clumps. Clumping is probablypresent on a variety of scales in SN ejecta, but on all scales,clumping tends to facilitate the release of stored energy. Quan-titatively, the simulations in this study suggest that large-scaleclumping may not significantly impact type II SN light curvesbecause this requires density contrasts of a few tens betweenclump and interclump medium. A greater impact on SN observ-ables may arise from a microphysical e ff ect of clumping, througha boost of recombination rates, which can occur for density con-trasts of a few rather than a few tens (Dessart et al. 2018a).Observationally, micro- and macroclumping may be at theorigin of some of the diversity of type II SNe, including visualdecline rates and photospheric phase duration (see e.g. Andersonet al. 2014), colors (see e.g. de Jaeger et al. 2018), and spectralpeculiarities (see e.g. Dessart & Hillier 2019a for the case of Ba ii Article number, page 10 of 13uc Dessart and Edouard Audit: Macroclumping and type II SN radiation
20 30 40 50 60 70 80 90 100 110Days since explosion41.441.641.842.042.242.442.642.8 l o g ( L b o l / e r g s − ) Fig. 11.
Same as Fig. 2, but now comparing the results for the 12 M (cid:12) ejecta model 2P-1D-smooth (thick dashed) with those from the 3D counter-parts in which the clumping parameters ξ and V cl are varied (thick colored lines). We overlay the results from additional 1D smooth models inwhich the ejecta mass is progressively decreased by steps of 0.4 M (cid:12) from 12 M (cid:12) down to 7.6 M (cid:12) (thin black lines), which produces a continuoussequence of events with a shorter photospheric-phase duration than model 2P-1D-smooth. [See Section 7 for discussion.] lines in SNe II-pec). Treating both micro- and macroclumpingin a given type II SN simulation is challenging since it requiresboth nonLTE, time-dependence, and multi-D radiation transport,something that is not currently doable. Both clumps and inter-clump medium need to be explicitly modeled since these regionsof di ff erent density (both at a given radius and at di ff erent depths)will have di ff erent temperatures and ionization levels (even forthe same composition), hence di ff erent opacity and emissivity.In the future, we will investigate the e ff ect of clumping inType Ibc SNe. These ejecta are distinct from type II SNe sincethe radiated energy arises more strongly from the continuous de-cay of unstable isotopes rather than from the release of initiallystored shock-deposited energy. Clumping may nonetheless facil-itate radiation escape and impact our inference of SN Ibc ejectamasses. Acknowledgements.
LD thanks ESO-Vitacura for their hospitality. This workwas granted access to the HPC resources of CINES under the allocation 2018 –A0050410554 made by GENCI. We thank John Hillier for fruitful discussions.
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Fig. A.1.
Variation of the albedo in a type II SN at early and late timeduring the photospheric phase. In this study, we use of value of 0.9,which is comparable to what holds in the photospheric regions wheretakes place the bulk of the radiative di ff usion that influences the lightcurve. Appendix A: Influence of the albedo
In the present study, all simulations are performed using analbedo (defined as the ratio of scattering opacity to total opac-ity) of 0.9 throughout the ejecta. This is taken as a representativevalue for type II SN ejecta, and this approximation seems suit-able given that we already make approximations in our modelingby assuming a grey opacity and neglecting nonLTE and time-dependent e ff ects. Nonetheless, to support this choice we showin Fig. A.1 how the albedo (we use here the ratio of the elec-tron scattering opacity to the Rosseland-mean opacity) varieswith Rosseland-mean optical depth at two epochs (early is be-fore recombination, at 16 d past explosion, and late is during therecombination phase, at 70 d after explosion) in a Type II SNejecta simulated with cmfgen (see, for example, Dessart et al.2013). The conditions at these two epochs and for similar SNmodels bracket the range of ejecta conditions simulated herewith heracles . In the outer regions, the albedo is high due toionization freeze out and the low density (the Rosseland meanopacity does not make much sense physically because condi-tions are nonLTE). At great depth, the albedo is low at late timesbecause of the large contribution from line opacity, while at earlytimes, the conditions are strongly ionized and electron-scatteringdominates. In the photospheric regions, the albedo is around 0.8,which is quite close to our choice of 0.9.We have tested the influence of the albedo in our simula-tions, although retaining for simplicity a uniform value for thewhole ejecta (allowing for a meaningful depth dependent albedoas in Fig. A.1 would require a nonLTE treatment). The left panelof Figure A.2 shows the bolometric light curves for a set of12 M (cid:12)
2P smooth 1D ejecta. In these simulations, the value ofthe adopted albedo was varied to cover from an absorption dom-inated plasma (albedo of 0.1) to a strongly scattering-dominatedplasma (albedo of 0.999). The results for an albedo of 0.1 and0.9 are essentially identical, which, together with the propertiesshown in Fig. A.1, suggests the results presented in this paper aresound. Interestingly, as the albedo is ‘unphysically’ increasedto a value of 0.99 and 0.999, the bolometric light curve starsdipping below the other curves at about 50 d, while the photo- spheric phase is correspondingly extended (each ejecta has thesame amount of stored radiation). Our interpretation is that asthe scattering opacity is increased, the gas becomes less and lesscoupled to the radiation, and its emissivity drops, inhibiting itscooling. This weak coupling makes the gas temperature ‘drift’from the radiation temperature. We find that at 70 d after ex-plosion, the model with an albedo of 0.999 has a twice highergas temperature through most of the ejecta relative to the casewith an albedo of 0.1. This higher temperature implies a muchhigher opacity (see Eq. 1) and di ff usion time, which explain thefainter luminosity and the longer phostospheric phase. A similarbehavior was seen by Kasen & Woosley (2009) in their simula-tion of Type II SN ejecta using an artificially enhanced electron-scattering opacity. They attributed it to the larger opacity andthus larger optical depth of the ejecta. In our simulation, theopacity is unchanged but the e ff ective opacity is increased be-cause of the shift to a higher gas temperature. This e ff ect proba-bly occurs too in the simulation of Kasen & Woosley (2009).We have conducted the same experiment but this time in 2Dusing a clumped ejecta (right panel of Figure A.2). We find thatthe albedo has the same e ff ect as in the 1D simulations. This ef-fect is negligible for an albedo below 0.9, and all 2D clumpedejecta yield a similar light curve. The e ff ect is strong for analbedo greater than 0.9, and dominates over the influence ofclumping. For a strongly scattering dominated plasma (whichis quite unphysical; see Fig. A.1), the photosphere is pushed farout in the ejecta, in layers where the adopted clumping is weak.Hence, the e ff ect of clumping is dwarfed for a high albedo. Article number, page 12 of 13uc Dessart and Edouard Audit: Macroclumping and type II SN radiation
20 40 60 80 100 120 140Days since explosion41.441.641.842.042.242.442.642.8 l o g ( L b o l / e r g s − )
20 40 60 80 100 120Days since explosion41.441.641.842.042.242.442.642.8 l o g ( L b o l / e r g s − ) Fig. A.2.
Left: Bolometric light curves for 12 M (cid:12)
2P smooth 1D ejecta but adopting a fixed albedo of 0.1 (absorption dominates the opacity), 0.9,0.99, and 0.999 (scattering vastly dominates the opacity). Right: Same as left, but now for a 2P-2D clumped ejecta with ξ = V cl = − and ∆ V cl = −1