Survival of the Fittest: Numerical Modeling of Supernova 2014C
Felipe Vargas, Fabio De Colle, Daniel Brethauer, Raffaella Margutti, Cristian G. Bernal
DDraft version February 26, 2021
Preprint typeset using L A TEX style emulateapj v. 12/16/11
SURVIVAL OF THE FITTEST: NUMERICAL MODELING OF SUPERNOVA 2014C
Felipe Vargas , Fabio De Colle , Daniel Brethauer , Raffaella Margutti & Cristian G. Bernal Draft version February 26, 2021
ABSTRACTInitially classified as a supernova (SN) type Ib, ∼
100 days after the explosion SN 2014C made atransition to a SN type II, presenting a gradual increase in the H α emission. This has been interpretedas evidence of interaction between the supernova shock wave and a massive shell previously ejectedfrom the progenitor star. In this paper, we present numerical simulations of the propagation of theSN shock through the progenitor star and its wind, as well as the interaction of the SN ejecta withthe massive shell. To determine with high precision the structure and location of the shell, we couplea genetic algorithm to a hydrodynamic and a bremsstrahlung radiation transfer code. We iterativelymodify the density stratification and location of the shell by minimizing the variance between X-rayobservations and synthetic predictions computed from the numerical model. By assuming sphericalsymmetry, we found that the shell has a mass of 2.6 M (cid:12) , extends from 1.6 × cm to 1 . × cm, implying that it was ejected ∼ / ( v w /
100 km s − ) yrs before the SN explosion, and has a densitystratification decaying as ∼ r − . We found that the product of metallicity by the ionization fraction(due to photo-ionization by the post-shock X-ray emission) is ∼ Subject headings: supernovae: individual: SN2014C - stars: mass loss - X-rays: individual: SN 2014C- circumstellar matter - methods: numerical INTRODUCTION
While mass loss is one of the key mechanisms regu-lating the evolution of massive stars, a complete under-standing of it is still missing, specially during the finalphases before the supernova (SN) explosion (e.g., Smith2014). The mass loss history (cid:46) − M w ∼ − − − M (cid:12) yr − . Thewind is often inhomogenous, as proven by radio emission,showing small flux fluctuations of ∼ a few over timescalesof tens-hundreds of days after the explosion (e.g., Bieten-holz & Bartel 2005; Soderberg et al. 2006; Schinzel et al.2009; Wellons et al. 2012; Salas et al. 2013; Bietenholz etal. 2014; Corsi et al. 2014; Palliyaguru et al. 2019).In a few cases, type Ib/c SNe show signs of muchstronger interaction between the ejecta and shells of ma-terial ejected before the explosion. For instance, SN Instituto de Ciencias Nucleares, Universidad NacionalAut´onoma de M´exico, A. P. 70-543 04510 D. F. Mexico Universidade Federal do Rio Grande, Av. Italia km 8 BairroCarreiros, Rio Grande, RS, Brazil Center for Interdisciplinary Exploration and Research in As-trophysics (CIERA) and Department of Physics and Astronomy,Northwestern University, Evanston, IL 60208 ∼ two years before the SN. SN 2001em, initially classi-fied as type Ib SN, presented prominent H α emissionlines at 2.5 yrs. Associated with strong radio and X-rayemission, this was interpreted as evidence of the inter-action between the SN ejecta and a massive ( ∼ (cid:12) )hydrogen-rich shell located at ∼ × cm (Chugai &Chevalier 2006; Chandra et al. 2020). Several other SNeshow similar signs of early interaction with massive shells(see, e.g., Anupama et al. 2005; Moriya & Maeda 2014;Chen et al. 2018; Pooley et al. 2019; Dwarkadas et al.2010; Ben-Ami et al. 2014; Mauerhan et al. 2018; Suzukiet al. 2021). Nevertheless, while in all these cases the in-termediate phases of the transition between type Ib andtype IIn SN were not observed, this transition has beenobserved in detail in the SN 2014C.Discovered by the Lick Observatory Supernova Search (Kim et al. 2014) in the NGC 7331 galaxy at a dis-tance of 14.7 Mpc and initially classified as a type IbSN, SN 2014C made a transition to a type IIn SN about ∼
100 days after the explosion, showing strong H α emis-sion (Milisavljevic et al. 2015). The modelling of theoptical/UV light curve shows that SN 2014C has a ki-netic energy of 1.75 ± × erg, with an ejectamass of 1.7 ± (cid:12) and a Nickel mass of 0.15 ± (cid:12) (Margutti et al. 2017).SN 2014C has also been extensivel observed with X-rays Telescope (XRT, (Burrows et al. 2005)) on boardthe Neil Gehrels Swift Observatory (XRT) (Gehrels etal. 2004) and the
Chandra X-ray Observatory (CXO) inthe 0.3-10 keV energy band, and by the
Nuclear Spec-troscopic Telescope Array (NuSTAR) from 3 keV to 79keV (Margutti et al. 2017; Brethauer et al. 2020, 2021).Most of the detected X-ray emission is concentrated in a r X i v : . [ a s t r o - ph . H E ] F e b the 1-40 keV energy band, while emission below ∼ × erg s − to 5 × erg s − . Then, it peaked ∼ − ∼ ArcminuteMicrokelvin Imager between 16 and 567 days showedthat the flux increased rapidly at ∼ Very Large array (Margutti et al. 2017). Furthermore, observations doneby using the
Very Long Baseline Interferometry foundthat the shock expansion has already strongly deceler-ated 384 days after the explosion (Bietenholz et al. 2018,2021).Altogether, the evolution of optical, radio and X-rayemission have been interpreted by considering the inter-action of the SN ejecta with a wind with ˙ M = 5 × − M (cid:12) yr − around the progenitor star, and a massive shelllocated at R sh = 5 × cm, with an extension of ap-proximately 0.25 R sh and a density of ∼ cm − (Mil-isavljevic et al. 2015; Anderson et al. 2017; Margutti et al.2017; Bietenholz et al. 2018, 2021). Assuming a velocityof 10-100 km s − for the shell, its position implies thatit was ejected ∼ Fig. 1.—
Schematic evolution of a supernova shock interactingwith an external shell (not-to-scale). (A) the supernova explosionproduces a outwardly propagating shock; (B) the reverse shockbecomes stronger as the shock front collides with the external shell,moving back towards (in mass coordinates) the supernova center. pled to a genetic algorithm (GA). The GA changes theshell density with time by minimizing the difference be-tween the synthetic X-ray emission (computed by post-processing the results of the hydrodynamical simula-tions) and observations of this SN presented in Marguttiet al. (2017); Brethauer et al. (2020, 2021). In this ap-proach, each density ρ ( r i ) (being i = 1, . . . , N ) is con-sidered a parameter of the model.The paper is organized as follows: in Section 2 we de-scribe the hydrodynamics code and the initial conditionsof the simulations, the bremsstrahlung radiation transfercode and the genetic algorithm employed to solve the op-timization problem and to find the density stratificationof the shell. In Section 3 we present the results of ournumerical calculations. In Section 4 we discuss the limitsof the simulations presented, and the implications of ourfindings. Finally, in Section 5 we draw our conclusions. METHODS
Numerical codes and initial conditions
We study the interaction of a SN shock with a massiveshell by running a set of one-dimensional (1D), spheri-cally symmetric simulations with the adaptive mesh re-finement (AMR) code
Mezcal (De Colle et al. 2012).The code solves the special relativistic hydrodynamicsequations, and has been extensively used to run numer-ical simulations of astrophysical flows (e.g., Gonz´alez-Casanova et al. 2014; De Colle et al. 2014, 2018a,b).umerical modeling of Supernova 2014c 3We follow the propagation of the SN shock front asit moves through a computational grid covering ∼ nineorders of magnitude in space, from ∼ × cm (theouter edge of the iron core) to ∼ × cm. We do soby running two sets of simulations. First, we follow thepropagation of the SN shock front as it moves through theprogenitor star, e.g., from ∼ cm to 10 cm. Then,after remapping the results of the small scale simulationinto a much larger computational box, we follow its prop-agation through the wind of the progenitor star and itsinteraction with the massive shell located at (cid:38) cm.In the small scale simulation, we set the density profileof the progenitor star by using the E25 pre-supernovamodel from Heger et al. (2000). This corresponds to astar which has lost its hydrogen and helium envelope.The resulting Wolf-Rayet star has a mass of 5.25 M (cid:12) and a radius of 3 × cm. The computational gridextends radially from 2 . × cm to 6 . × cm. Weemploy 20 cells at the coarsest level of refinement, with22 levels of refinement, corresponding to a resolution of1 . × cm. The SN energy ( E SN ≈ erg) is imposedby setting the pressure of the two inner cells of the com-putational box as p = E SN (Γ ad − /V , being Γ ad = 4 / V the volume of the two cells.Outside the stellar surface, we take ρ = ˙ M w / (4 πr v w ),being ˙ M w = 5 × − M (cid:12) yr − and v w = 10 cm s − the mass loss rate and velocity of the wind launched bythe Wolf-Rayet star before the collapse. As the velocityof the SN shock front is about two orders of magnitudelarger than the Wolf-Rayet wind (i.e., ∼ cm s − vs. ∼ cm s − ), we assume that the wind mediumis static. The propagation of the SN shock front is fol-lowed as it breaks out of the progenitor star and arrivesto 4 . × cm in 50 seconds.In the large scale simulations, the computational boxgoes from 10 cm to 5 × cm. For r < . × cm we set the density, pressure and velocity by usingthe values determined from the small scale simulation.For larger radii, we take the density stratification as ρ = ˙ M w / (4 πr v w ), with ˙ M w = 5 × − and v w = 10 cm s − as in the small scale simulation. We employ 150cells, with 20 levels of refinement in the AMR grid, cor-responding to a resolution of 2 . × cm. By runningdifferent simulations in which we change the number oflevels of refinement between 14 and 22 levels, we verifythat 20 levels of refinement are enough to achieve con-vergence.Trying to reproduce the observed X-ray emission, wehave first considered a massive, uniform cold shell locatedat the radius R s , with mass M s and thickness ∆ R s . Todetermine the best values for these three parameters, werun a grid of 1815 models by using 11 values of M s (inthe range 1.5 M (cid:12) to 2.5 M (cid:12) ), 15 values of ∆ R s (rangingfrom 7 . × up to 2 . × cm), and 11 values of R s (from 1 . × to 7 . × cm). To compare the re-sults of the numerical simulations with observations, wehave then employed a ray-tracing code (see Section 2.2).Unfortunately, none of these models give an acceptablefit to the observations, implying that the density of themassive shell is not constant.To find the density at several radii is a task that is notpossible to achieve with a grid of numerical models. Forinstance, a grid of ten values for the density at 10 differ- ent radii implies running 10 simulations. Thus, to de-termine the density stratification of the shell, we decidedto solve the “full” optimization problem. This is done bycoupling the Mezcal code with other two codes: a radi-ation transfer code which computes the bremsstrahlungradiation (see Section 2.2), and a genetic algorithm (de-scribed in Section 2.3) that automatically and randomlychanges the density profile inside the shell by minimaz-ing the variance between the synthetic observations com-puted from the numerical model and the X-ray observa-tions.
Bremsstrahlung Emission Code
We post-process the results of the numerical simula-tions by computing synthetic spectra. The specific fluxis given by F ν = (cid:90) I ν cos θd Ω , (1)where d Ω = 2 π sin θdθ/D , being D = 14 . I ν is the specific intensity.The observed X-rays radiation is due to thermalbremsstrahlung emission caused by the interaction be-tween the SN shock and a massive shell (Margutti et al.2017). To determine the specific intensity, we solve theradiation transfer equation dI ν dτ ν = S ν − I ν , (2)where S ν = j ν /α ν is the source function, τ ν = (cid:82) α ν dl isthe optical depth, and j ν and α ν are the emissivity andthe absorption coefficient respectively. In addition to thebremsstrahlung self-absorption, we also consider photo-electric absorption, which at early times dominates ab-sorption for frequencies (cid:46) Hz, so that α ν = α ν, ff + α ν, bf , j ν = j ν, ff . (3)To compute the bound-free absorption, we use tab-ulated cross-sections for solar metallicity (Morrison &McCammon 1983). For the bremsstrahlung coefficientswe take (e.g., Rybicki & Lightman 1986) j ν = 6 . π × − Z n e n i T − / e − hνkT G (4) α ν = 3 . × T − / Z n e n i ν (cid:16) − e − hνkT (cid:17) G , (5)where Z , n e and n i are the atomic number, electron andion densities respectively, and G is the Gaunt Factor ( ∼ t ff =60( T / K)( n e / cm − ) yrs, so it is much larger thanthe timescales studied here. Genetic Algorithm
Genetic algorithms (GA hereafter, e.g., Rajpaul 2012)are based on the theory of natural selection. GA are
Initial population Hydro codeRadiation codeSurvival Selection Fitness Function (comparison with observations)MutationsCrossoverConvergence StopNoYes
Fig. 2.—
Flowchart showing the genetic algorithm employed todetermine the density stratification of the massive shell interactingwith the SN 2014C. First, we initialize a population of randomdensities at different shell radii. We modify the initial populationby mutations and cross-over (see the text for details). We runhydrodynamical simulations following the interaction of the SNejecta with the shell and we compute synthetic X-ray spectra whichare compared with the observations. We select the “best” elementsof the population using a “fitness function” (the χ test). Theprocess is repeated a thousand times. commonly used optimization methods , employed alsoin astrophysics to solve problems with many degrees offreedom, in which finding the optimal solution would bevery hard otherwise (e.g., Cant´o et al. 2009; De Geyteret al. 2013; Morisset 2016). Nevertheless, this is the firsttime, as far as we know, that optimization methods arecoupled directly to hydrodynamical simulations.As mentioned before, we made a single small scale sim-ulation for the propagation of the shock-wave in the in-terior of the star, and we applied the GA to find thebest density stratification of the shell by running severalsimulations of the interaction of the SN ejecta with themassive shell.Figure 2 shows schematically the GA algorithm im-plemented. A population formed by 10 elements (the“chromosomes” in the GA terminology) is chosen at thebeginning of the iterative process. Each element of thepopulation is defined by setting 11 values of the densityat different radii inside the shell (the “genes”). Nine ofthe densities are defined at a radius given by each of thenine observational epochs available. Additionally, we de-fine two densities, one at a smaller and one at a largerradius. At each step, we create 90 new elements of thepopulation. Half of them are defined by randomly choos-ing two elements of the original population and applyingto them cross-over and mutation (see below), while theother half is initialized by directly copying the densityvalues from one random element of the population andmodifying it only by mutation.In the cross-over process, the two “parents” are mixedby choosing randomly a certain number of densities fromeach element (e.g., the first and second densities from thefirst element, the third density from the second elementand so on). This process is inspired by the genetic mixingpresent in biological evolution. In the mutation processwe modify randomly one density in each element. We doso by setting a Gaussian distribution around the originalvalue of the density ρ , with a width given, in 90% ofthe cases, by ρ /
2, and in 10% of the cases by 10 ρ , sothat in a few cases the system explores density values faraway from the initial one (to avoid being trapped by a We have employed GA in this paper but the results obtainedare independent on the particular optimization method chosen. local minimum).The shell densities were then mapped as initial con-ditions into the HD code. Then, after the simulationwas completed, the bremsstrahlung X-ray emission wascomputed by post-processing the results of the HD calcu-lation. A fitness function (a reduced χ test) was appliedto compare the synthetic spectra produced by the modeland the observational data at 9 epochs: 308, 396, 477,606, 857, 1029, 1257, 1574 and 1971 days after explosion.The fittest 10 elements (out of the new 90 and the old 10elements of the population) were saved and used as ini-tial condition for a new step. This process was repeatedfor ∼
100 iterations, for a total number of ∼ simu-lation. Each simulation took ∼
10 minutes so that theentire process could be completed in less than a day.The simulations were done on a cluster of CPUs, byusing the “Message Passing Interface” library. At eachiteration, the master node initialized and synchronisedthe simulations, selected the best elements and managedthe cross-over/mutation processes, while the other nodesrun (in parallel) each of the 90 hydrodynamics simula-tion, compute the X-ray spectra and the fitness function(a χ test). RESULTS
SN shock propagation through the progenitor wind
The propagation of the shock wave through the starhas been extensively studied both for the non-relativistic(e.g., Sakurai 1960; Matzner & McKee 1999) and mildlyrelativistic regime (e.g., Tan et al. 2001; Nakar & Sari2010; Waxman & Katz 2017). As the shock approachesthe surface of the progenitor star and it moves in thestellar envelope, in which the density drops steeper than ρ ∝ r − , the shock velocity increases to mildly relativisticspeeds (see the blue line in the bottom panel of Figure3).As a result, once the SN shock breaks out in ∼ ∼ km s − , while a small fraction of themass (corresponding to a kinetic energy ≈ ergs) ex-pands with larger velocities (up to v sh ∼ . ρ env ∝ ( R/r − k ) predict an ejecta density strat-ification ρ ∝ r − n after the break-out, with n = 7 − n ∼ n ∼ N u m b e r D e n s i t y [ c m ] T e m p e r a t u r e [ K ] Radius [cm] 0.10.20.30.40.5 V e l o c i t y [ v / c ]
50 s0.01 days0.09 days1.08 days12.0 days
Fig. 3.—
Time evolution of the SN shock as it moves throughthe wind of the progenitor star. From top to bottom: density,temperature and velocity profiles of the SN shock as it interactswith the wind of the progenitor star (at r (cid:46) × cm) and theouter shell (at (cid:38) × cm). ejecta through the progenitor wind leads to the forma-tion of a double shock structure, formed by the forwardshock (FS), which accelerates and heats the WR wind,and the reverse shock (RS), which decelerates and heatsthe SN ejecta. Following Chevalier (1982), the evolutionof the forward and reverse shocks is self-similar, with R ∼ t m , where m = ( n − / ( n − n ∼ m ∼ .
5, which is consistent with the evolutionof the shock wave obtained in our simulation (see Fig-ure 3, bottom panel, and Figure 4). The FS velocityis much larger than the wind velocity. Thus, the post-shock temperature achieves values (cid:38) K, while theSN bulk temperature quickly drops by adiabatic expan-sion (heating by Ni and Co decay is not included inthe calculation).In our simulations, the SN shock acceleration stopsonly when the shock arrives to the very edge of the pro-genitor star. This leads to an overestimation of the trueshock velocity, as the shock acceleration should stop oncethe stellar envelope becomes optically thin to the radia-tion coming from the post-shock region. A detailed cal-culation of the shock acceleration and break-out is anopen problem, which requires a radiation hydrodynam-ics code. Once the SN ejecta interacts with the shell (seebelow), the shock velocity quickly drops. Then, the lateevolution of the system will be independent on particularshock velocity obtained, while it? will depend stronglyon the pre-shock structure of the ejecta. S h o c k p o s i t i o n [ c m ] (a)10 Time [days]00.10.20.30.4 S h o c k V e l o c i t y [ v / c ] (b) Fig. 4.—
Top panel : position of the SN shock as function of time.The break seen at ∼
50 days corresponds to the beginning of theinteraction with the shell.
Bottom panel : shock velocity. Duringthe self-similar phase, the shock velocity drops from ∼ .
35c to ∼ . ∼ . SN shock interaction with the massive shell At ∼
50 days after the explosion, the shell begins inter-acting with the massive shell (see Figure 4). Radio andoptical observations showed that the interaction started ∼
100 days after the explosion (Milisavljevic et al. 2015;Anderson et al. 2017). Then, our simulations overesti-mate the average shock velocity by a factor of ∼
2. Alower shock velocity can be due, as mentioned above, tothe loss of thermal energy during the shock breakout or,alternatively, to a less steep density profile in the outerlayers of the progenitor stars.The interaction of the ejecta with the shell is shownin Figures 4 and 5. The shell presents large densityfluctuations (see the upper panel of figure 5). Then,the shock propagation leads to the formation of strongreverse shocks which interaction produces the complexshock structure observed in Figure 5. Once interactingwith the shell, the shock velocity quickly drops to ∼ − , then maintain an approximately constant ve-locity (see Figure 4). Small fluctuations in the shockvelocity are present at ∼ days (see the bottom panelof Figure 3), with increases (drops) in velocity by ∼ − corresponding to drops (increases) in the density.Initially, the reverse shock is stronger then the for-ward shock. Thus, the shocked ejecta is hotter thanthe shocked wind (Figure 5, blue and green lines inthe middle panel). As the ejecta crosses the reverseshock and approaches the reverse shock velocity, the RStemperature drops becoming smaller than the FS tem-perature. Thus, the bremsstrahlung specific emission(larger at smaller temperatures) is initially dominatedby the shocked wind, and later mostly originated intothe shocked ejecta.A fit to the density profile gives ρ ∝ r − . ± . (al-though there are large fluctuations), consistently withthe constant shock velocity seen in Figure 5 (as E ∼ M v ∼ R ρv implies that the velocity is constant aslong as ρ ∝ R − and the shock is adiabatic). The shellmass is 2.6 M (cid:12) . This value is consistent with the 3.0M ± (cid:12) (VLB model) determined by Brethauer et From the Hydrogen line, a velocity of ∼ half of this valueis inferred (Milisavljevic et al. 2015). This is consistent with thehydrogen lines being produced in the post shock clumpy mediumor by recombination of the upstream medium. N u m b e r D e n s i t y [ c m ] (a) Day 308Day 477Day 857Day 1257Day 1971ICNo Data T e m p e r a t u r e [ K ] (b)10 Radius [2×10 cm] 0.0000.0050.0100.0150.0200.0250.0300.035 V e l o c i t y [ v / c ] (c) Fig. 5.—
The same as Figure 3, but for larger radii, showing indetail the interaction between the SN ejecta and the shell. Thelines corresponds to epoch in which there are X-ray observationsavailable. “IC” represents the initial shell density profile. In addi-tion to nine densities corresponding to the epochs with X-ray dataavailable, the amount of unshocked neutral mass is estimated byconsidering bound-free X-ray absorption. As the absorption doesnot depend on the density stratification but only on the amount ofmass crossed by the X-rays, we show this region as uniform in thefigure (dashed line in the top panel). al. (2021) which also assumes spherical symmetry andwithin the range of typical shell masses observed in typeIIn SNe (0.1-10 M (cid:12) , see, e.g., Smith 2017; Branch &Wheeler 2017).To determine the structure of the shell, we left as a freeparameter in the GA algorithm the amount of neutralhydrogen still to be crossed by the ejecta (inferred fromthe bound-free absorption). We get M = 0 .
38 M (cid:12) solarmasses for this component at t = 1971 days, which isrepresented by a constant density dashed line in the toppanel of Figure 5. We notice that the exact structure ofthis region can not be determined. Nevertheless, it willextend to r ∼ . × cm if the shell density continuesdropping as r − , in which case (moving at a constantspeed as discussed above) the SN shock will break out ofit ∼ . Radiation
To compare the model with observations, we com-pute the X-ray bremsstrahlung emission coming from theshocked material. We assume that the shocked material is completely ionized (so that it does not contribute tothe bound-free opacity) and that the unshocked shell isneutral. Extending the radio synchrotron emission toX-rays by assuming F ν ∝ ν − ( p − / ∝ ν − with p ∼ ∼ (cid:46) × Hz than observed.Solid lines, which fits better the observational data, arecomputed by assuming that the neutral medium has halfof solar metallicity. DISCUSSION
The optimization method employed in this work allowsus to determine the detailed structure of the shell. Asdescribed in the previous section, the shell has a mass of2.6 M (cid:12) (2.2 M (cid:12) of shocked and 0.4 M (cid:12) of still unshockedgas), a density stratification ρ ∝ r − and extends from2 × cm to 2 × cm. While uncommon in typeIb SNe, SN IIn show evidence of strong interaction, withshell masses of 0 . −
10 M (cid:12) (see, e.g. Smith 2017; Branch& Wheeler 2017, and references therein). Also, manytype IIn SNe show an X-ray emission inconsistent witha density profile ρ ∝ r − , implying a steeper densitystratification (Dwarkadas & Gruszko 2012). While shar-ing many characteristics with SN IIn, in the case of theSN 2014C the shell is located at a much larger distance,implying that it was ejected ∼ / ( v w /
100 km s − ) yrsbefore the SN explosion.Harris & Nugent (2020) presented a series of numer-ical simulations of a SN ejecta interacting with a walllocated at r ∼ × cm, and with a CSM windat larger radii. They deduce a small mass for the wall( ∼ . − .
31 M (cid:12) ), in apparent contradiction with pre-vious estimations (e.g., Margutti et al. 2017). Actually,the density of the CSM wind is assumed to be a factor4-7 smaller than the density of the wall at r . The CSMwind, then, corresponds to a mass loss ˙ M w ∼ − M (cid:12) yr − (assuming v w = 10 cm s − ). Our results (which,we stress, have been obtained by a “blind” fit, i.e. with-out assumptions on the final shell structure) clarify thisinconsistency, showing that the wall and the CSM windare both part of the same extended structure (see figure5), with the densest shell material located at ∼ × (see Figure 5) in agreement with Harris & Nugent 2020,and the outer shell density dropping quickly with radius.Thus, we conclude that the same event is responsible forthe ejection of the full massive shell.The parameters determined for the SN 2014C are re-markably similar to those inferred for the SN 2001em.The X-ray, radio and H α emission from SN 2001em havebeen interpreted as evidence of interaction with a 3 M (cid:12) hydrogen-rich shell (Chugai & Chevalier 2006). VLBIobservations showed that the shell is located at 7 × This assumption is justified by the large post-shock tempera-ture and the presence of strong photo-ionizing X-ray and UV emis-sion. umerical modeling of Supernova 2014c 7 Frequency [Hz] 10 D e n s i t y F l u x [ m J y ] Day 308(×1)Day 396(×3)Day 477(×10)Day 606(×30)Day 857(×100)Day 1029(×300)Day 1257(×1000)Day 1571(×3000)Day 1971(×10000)
Fig. 6.—
Comparison between X-ray observations and syntheticobservations computed for the best model. Full lines representssolar metallicity while dashed lines are for half solar metallicity.We do not include the shaded grey area in the fit, as it is dominatedby Fe emission lines. Curves at different times are rescaled so theydo not overlap with each other. cm, and expanding with a velocity of 5800 ± km s − (Bietenholz & Bartel 2007), which is consistent with the7500 km s − inferred here.The best fit to the data is achieved by consideringa bound-free cross-section, corresponding to half solarmetallicity. This low metallicity is in contradiction withobservations of the Fe line. The prominent Fe emissionline at 6.7-6.9 keV is consistent with a metallicity largerby a factor of ∼ ∝ n H σ ν , being n H the density of neutral hydrogen and σ ν the bound-free cross-section (which depends on metallicity). In thiscase, we expect the mass of the shell to be larger than thevalue obtained in this paper, although by a small factoras the bremsstrahlung emission is ∝ n e . A detailed cal-culation of the ionization of the shell is left for a futurework.Different origins for the shell have been considered.The possibility of the shell being due to a massivewind ejection (e.g., de Jager et al. 1988; Leitherer 2010;Kuriyama & Shigeyama 2020) is unlikely, as it would cor-respond to an extremely large mass loss rate of ∼ − M (cid:12) yr − ( v w /100 km s − ). Other possibilities include asudden outburst some time before collapse, which wouldremove the most external layer of the star where almostall hydrogen is found (Smith & Arnett 2014), or binarysystem interactions in which the envelope of the mostmassive star has been stripped away (Sun et al. 2020).A better understanding of the origin of this ejections canbe achieved only by detailed theoretical models coupledto a larger sample of observed interacting supernovae.The density fluctuations have a periodicity of ∼ × cm. A similar periodicity has been observed in theradio emission from SN 1979C (Weiler et al. 1991) andhave been interpreted as evidence of a binary system inwhich the orbital motion modulates the wind density (seeYalinewich & Portegies Zwart 2019) which interacts withthe stellar outburst. If this is also the case of SN 2014C, itwould imply that the binary system is very detached (asthe binary period is (cid:29) × /v w ∼
10 yrs ( v w / ) − ).The companion star would then be not responsible forthe loss of the envelope of the primary star. An alterna-tive explanation is that the density fluctuations seen inthe GA fit are the direct result of a modulation in theoutburst from the progenitor star.Finally, we notice that the simulations presented hereassume that the shell is spherically symmetric. Thisis consistent with VLBI observations (Bietenholz et al.2018). Small scales inhomogeneities are expected. Thedensity profile shown in Figure 5 show large scale den-sity fluctuations. Furthermore, it is likely that themedium is, at some scale, clumpy. If the shell is notperfectly homogeneous before interacting with the SNejecta, the interaction will amplify the inhomogeneities,leading to a multi-phase medium with denser/colder re-gions in pressure equilibrium with more tenuous/hotterregions, which is consistent with the strong Fe emissionline observed at ∼ ∝ Z n ,inhomogeneities in the shell and mixing with the highermetallicity ejecta lead to a larger emissivity, implyingthat the mass of the shell should be taken as an upperlimit. CONCLUSIONS
In this paper, we presented hydrodynamical simula-tions of the strongly interacting SN 2014C. First, we fol-low the propagation of a SN shock through the progenitorstar. Then, by using as input the outcome of the smallscale simulation (i.e., density, pressure and velocity pro-files), we run a large set of simulations. As described insection 2.3, we initialize the shell with a uniform den-sity n shell = 10 cm − . We follow the propagation of theSN shock as it interacts with the wind launched by theprogenitor Wolf-Rayet star and with the massive shell.We compute the bremsstrahlung emission using the al-gorithm described in Section 2.2, and compare the resultswith observations. At each step, we run a large numberof simulations changing the shell density profile. As aresult, we determine the shell structure and metallicity.In particular, we get a mass of 2 . (cid:12) for the shell anda density profile ρ ∝ r − . We also found that the shell isvery extended, with a size (cid:38) cm. If the shell strati-fication continues with the same slope, the SN shock willbreak out of it nearly 8 yrs after the explosion, i.e. during2022.Radio and X-ray emission allows us to understand themass loss history of core-collapse SNe progenitor on time-scales which are impossible to study by direct observa-tions. As we have shown in this paper, optimizationmethods can be used, coupled with hydrodynamical sim-ulations, to model the density stratification of the envi- ronment once data at several epochs are available, asin the case of SN 2014C. The X-ray emission tracks theforward and reverse shock emission, depending on thedensity of the environment and the ejecta velocity. TheH α emission tracks the shocked shell and the unshockedmedium fotoionized by the X-ray and UV radiation. Alltogether, a detailed fit of the different components canhelp us to get a better understanding of this system.Then, coupled with detailed modeling of the radio emis-sion, this analysis can allow us to determine the micro-physical parameters as a function of time (which are usu-ally degenerate with the density of the environment andejecta velocity), giving us direct information on the par-ticle acceleration process. In this paper, we describe thistechnique by analyzing the X-ray bremsstrahlung emis-sion. The extension to radio and optical emission will beconsidered in a future study.We thank Luc Binette, Cesar Fern´andez Ram´ırez,Leonardo Ferreira and Claudio Toledo Roy for usefuldiscussions. FV and FDC acknowledge support fromthe UNAM-PAPIIT grant AG100820. We acknowl-edge the computing time granted by DGTIC UNAM(project LANCAD-UNAM-DGTIC-281). R.M. acknowl-edges support by the National Science Foundation underAward No. AST-1909796 and AST-1944985. 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