Shock Cooling Emission from Extended Material Revisited
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20, 2020Typeset using L A TEX twocolumn style in AASTeX62
Shock Cooling Emission from Extended Material Revisited A NTHONY
L. P
IRO , A NNASTASIA H AYNIE ,
2, 1
AND Y UHAN Y AO The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA; [email protected] Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, USA Cahill Center for Astrophysics, California Institute of Technology, MC 249-17, 1200 E California Boulevard, Pasadena, CA 91125, USA
ABSTRACTFollowing shock breakout, the emission from an astrophysical explosion is dominated by the radiation ofshock heated material as it expands and cools, known as shock cooling emission (SCE). The luminosity of SCEis proportional to the initial radius of the emitting material, which makes its measurement useful for investigatingthe progenitors of these explosions. Recent observations have shown some transient events have especiallyprominent SCE, indicating a large radius that is potentially due to low mass extended material. Motivated bythis, we present an updated analytic model for SCE that can be utilized to fit these observations and learn moreabout the origin of these events. This model is compared with numerical simulations to assess its validity andlimitations. We also discuss SNe 2016gkg and 2019dge, two transients with large early luminosity peaks thathave previously been attributed to SCE of extended material. We show that their early power-law evolution andphotometry are well matched by our model, strengthening support for this interpretation.
Keywords: radiative transfer — supernovae: general — supernovae: individual (SN 2016gkg, SN 2019dge) INTRODUCTIONObservations of supernovae (SNe) and other transientsduring the first few days provide valuable information abouttheir progenitors (Piro & Nakar 2013). Current and forth-coming surveys are making this an ideal time to study theseearly properties. The first electromagnetic emission is theshock breakout (SBO) when the shock reaches an opticaldepth of ∼ c / v , where v is the shock speed (Waxman & Katz2017, and references therein). SBO is short lived and highenergy, and thus has only been seen in a handful of cases(e.g., Soderberg et al. 2008; Gezari et al. 2015). In the hoursto days following SBO, the hot shock-heated material ex-pands and cools, giving rise to shock cooling emission (SCE,Nakar & Sari 2010; Piro et al. 2010; Rabinak & Waxman2011). The timescale and temperature of SCE makes it well-suited for ground based optical observatories.As SCE has been detected more regularly, it has often beenfound to be more prominent than expected from typical blueor red supergiant models. This was first noticed for a sub-class of SNe IIb that show double-peaked light curves, suchas SNe 1993J, 2011dh, 2011fu, and 2013df (Wheeler et al.1993; Arcavi et al. 2011; Kumar et al. 2013; Van Dyk et al.2014). It is now generally accepted that their first peaks aredue to SCE from low mass ( ∼ . − . M ⊙ ) extended ( ∼ cm) material (Woosley et al. 1994; Bersten et al. 2012;Nakar & Piro 2014). This unique structure is also reflectedin pre-explosion imaging, which identified the progenitors as yellow supergiants (Aldering et al. 1994; Maund et al. 2011;Van Dyk et al. 2011, 2014).More recently, similar structures involving extended ma-terial have been invoked to explain a wide variety oftransient events (e.g., De et al. 2018b; Taddia et al. 2018;Fremling et al. 2019; Ho et al. 2020; Jacobson-Galán et al.2020; Yao et al. 2020). Constraining the mass and radiusof extended material usually involves comparing the ob-servations to rough analytic scalings or fitting semi-analyticmodels (Nakar & Piro 2014; Piro 2015; Nagy & Vinkó 2016;Sapir & Waxman 2017). The typical approach taken in de-veloping these models is to begin with a density profile re-lated to the physical conditions of the extended material(e.g., convective, radiative, wind-like) and introduce a shockvelocity profile (Matzner & McKee 1999) to understand theshock energy and ejecta velocity as a function of depth.Here we explore a different approach in which we focuson the velocity profile of extended material once it is in thehomologous phase. At such times, the ejecta naturally has atwo component profile (Chevalier & Soker 1989), consistingof outer material with a strong velocity gradient and innermaterial with a more modest velocity gradient. The advan-tage of this approach over Piro (2015) is that it better matchesthe expected properties of early SCE when the luminosity isgenerated by the outermost material. We discuss useful rulesof thumb for assessing whether SCE from extended materialis appropriate for explaining a particular observation. In Section 2, we present an outline of our analytic modelfor SCE. We assess the validity of the model in Section 3by comparing to numerical calculations. In Section 4, weapply our SCE model to two specific SNe with the purpose ofdemonstrating how the main features of SCE identified hereare exemplified by these explosive events. We conclude inSection 5 with a summary of our results and a discussion offuture work. In Appendix A, we provide further comparisonsbetween our analytic SCE model and numerical calculations. GENERAL FRAMEWORKWe consider extended material with mass M e and radius R e , which is imparted with an energy E e as the shock passesthrough it. The material then expands homologously withradius r = vt , where v is different for each shell of material.In essence, one can think of this as a coordinate system whereeach layer in the exploding ejecta is labeled by its velocity.We use a density structure that is inspired by the work ofChevalier & Soker (1989), in which the extended material isdivided into an outer region with a steep radial dependence, ρ out ( r , t ) = KM e v t t (cid:18) rv t t (cid:19) − n , (1)and an inner density with a shallower radial dependence ρ in ( r , t ) = KM e v t t (cid:18) rv t t (cid:19) − δ . (2)Typical values are n ≈
10 and δ ≈ . n and δ do not drasticallyalter our results as long as n ≫ δ &
1. The parameter K is set by mass conservation, K = ( n − − δ )4 π ( n − δ ) . (3)For typical values of n and δ , K = 0 . v t is the transition velocity between the outer and inner regions.Using energy conservation, this is found to be v t = (cid:20) ( n − − δ )( n − − δ ) (cid:21) / (cid:18) E e M e (cid:19) / , . (4)Although in detail there should be some minimum velocityat the base of the extended material, for this work we assumeit is negligible in comparison to v t .The optical depth as a function of radius and time is τ ( r , t ) = Z ∞ r κρ ( r , t ) dr , (5)where for this work we use a constant electron scatter-ing opacity due to the hot temperatures during SCE. See Rabinak & Waxman (2011) for a treatment that considersmore complicated opacity scalings. The result of this inte-gral for the outer material is τ ( r , t ) = κ KM e ( n − v t t (cid:18) rv t t (cid:19) − n + . (6)Setting the photosphere to be the depth where τ = 2 /
3, thephotospheric radius evolves as r ph ( t ) = ( t ph / t ) / ( n − v t t , t ≤ t ph , (7)where t ph = (cid:20) κ KM e n − v t (cid:21) / , (8)is the time when the photosphere reaches the depth wherethe velocity is v t . After this time, the photospheric radiusshould show a break in its evolution and decline more steeply.To solve for this, we find the τ = 2 / r ph ( t ) = " δ − n − t t − ! + − / ( δ − v t t , t ≥ t ph , (9)In the limit that t ≫ t ph , this can estimated as r ph ( t ) ≈ (cid:18) n − δ − (cid:19) / ( δ − ( t ph / t ) / ( δ − v t t , t ≫ t ph , (10)for the photospheric evolution. Formally, this suggests thatthe photosphere moves quickly inward in radius at thesetimes since δ ≈ .
1. The exact evolution depends sensitivelyon δ though, so it is difficult to predict r ph ( t ) without fittingto simulations. Furthermore, the photospheric evolution canbe more complicated because effects like recombination andradioactive heating begin to be important before t ph .We estimate the initial thermal energy in each shell withvelocity v to be E th , ( v ) ≈ Z ∞ vt π r ρ ( v / dr = π KM e v t n − (cid:16) v t v (cid:17) n − . (11)The factor of 2 scaling for the velocity takes into accountthat the initial velocity imparted by the shock is acceleratedby pressure gradients before reaching homologous expansion(Matzner & McKee 1999). This material adiabatically coolsas it expands with the scaling E th ∝ / r . The outer materialall roughly starts with a similar initial radius of R e , so that E th ( v , t ) ≈ E th , ( v ) (cid:18) R e vt (cid:19) . (12)An observer at time t observes radiation from the diffu-sion depth where 3 τ ≈ c / v . Although τ ≈ c / v is com-monly used as an approximation (e.g., Nakar & Sari 2010),we include the factor of 3 to better match numerical results(Morozova et al. 2016). Using Equation (6), the velocity atthis depth is v d ( t ) = ( t d / t ) / ( n − v t , t ≤ t d , (13)where t d = (cid:20) κ KM e ( n − v t c (cid:21) / , (14)is the time at which the diffusion reaches the depth where thevelocity is v t . The radius of the diffusion depth is r d ( t ) = v d ( t ) t = ( t d / t ) / ( n − v t t , t ≤ t d , (15)At any time t , an observer sees a luminosity L ( t ) ≈ E th , ( v d ( t )) t (cid:18) R e v d ( t ) t (cid:19) . (16)Putting together the above expressions, this is simplified to L ( t ) ≈ π ( n − n − cR e v t κ (cid:16) t d t (cid:17) / ( n − , t ≤ t d . (17)This is just the classic result that the SCE luminosity isproportional to the initial radius of the shock heated mate-rial (e.g., Nakar & Sari 2010; Piro et al. 2010; Piro & Nakar2013), with the addition of a power-law dependence withtime, L ∝ t − / ( n − , dictated by the outer velocity profile.To understand how the luminosity evolves for times after t d , one might try to solve for the velocity of the diffusiondepth just as we did above for the outer material. This wouldresult in v d ( t ) ∝ t / (2 − δ ) , which has a positive exponent be-cause δ ∼ .
1. Such a scaling would seemingly imply thediffusion depth has reversed direction and is now moving intoshallower material. The problem is that this material cannotradiate much more because, by this time, it has already lostmost of its thermal energy. Instead, what happens is that thediffusion depth stays roughly fixed at r d ≈ v t t where it con-tinues to radiate and cool.To understand how the luminosity from the layer at v t changes with time, we solve the differential equation for thethermal energy of a radiation dominated gas subject to radia-tive cooling and adiabatic expansion (e.g., Piro 2015) dE th ( v t , t ) dt = − L − E th ( v t , t ) t , (18)where the radiative luminosity is L = tE th ( v t , t ) / t d . (19)We integrate Equation (18) from t d to t to solve for E th ( v t , t )and then substitute this result into Equation (19) to find L ( t ) = E th ( v t , t d ) t d exp (cid:20) − (cid:18) t t d − (cid:19)(cid:21) , t ≥ t d , (20) where the prefactor is E th ( v t , t d ) t d = π ( n − n − cR e v t κ , (21)which matches onto the luminosity at t ≤ t d given by Equa-tion (17). This result shows that because the luminosity ismostly originating from a single layer at r d ≈ v t t , it falls ex-ponentially (rather than a power law like for the outer mate-rial). This is similar to the conclusion of Piro (2015), whopresents a one-zone treatment of SCE.Following the exponential drop, heating from interior re-gions of the explosion (e.g., radioactive powering) will be-gin competing with the SCE. This will typically occur beforethe time t ph because by the time the photosphere has passedthrough the depth at v t , the extended material will have lostmost of its thermal energy.SCE roughly radiates as a black body (although seeSapir & Waxman 2017 for a discussion of thermalization)and thus the observed temperature is estimated as T BB = L π r σ SB ! / , (22)where σ SB is the Stefan-Boltzmann constant. Just as for theluminosity, the evolution of T BB will probably become morecomplicated before t ph because of additional heating sources.Additionally, both the photospheric depth and observed tem-perature will be impacted by other physics, such as recombi-nation, not included here. For these reasons, the early scalingfor T BB and the break in r ph at t d are the most robust charac-teristics to compare with observations.Figure 1 summarizes the main conclusions of the abovediscussion schematically. As SCE proceeds, the luminositysteepens from a power law to an exponential once the diffu-sion depth transitions from outer to inner material at time t d .A key point is that as long as the opacity stays roughly con-stant, then the observed photospheric radius keeps the samepower law even as the luminosity dropping faster. This evolu-tion only changes at the later time of t ph ≈ ( c / v ) / t d , whenthe photosphere transitions into the inner material. COMPARISON TO NUMERICAL MODELSWe next turn to simulations to demonstrate how theyroughly follow the properties described above. These cal-culations are similar to the Type IIb SN models of Piro et al.(2017), but we save discussing the full details until Ap-pendix A. The key points to mention here are that the modelsconsist of a 3 . M ⊙ helium core (once the inner 1 . M ⊙ hasbeen excised to represent formation of a neutron star) withhydrogen-rich extended material siting atop. For the particu-lar example here we use M e = 0 . M ⊙ and R e = 125 R ⊙ .These models are exploded with our open-source numer-ical code SNEC (Morozova et al. 2015) with an explosion R ad i u s Time Lu m i no s i t y t d ~ t -4/(n-2) t ph R a d i o a c t i v e P o w e r i n g r ~ t r ~ t δ -1)ph BB T e m pe r a t u r e ~ t -1/2-1/(n-2)+1/(n-1) R a d i o a c t i v e P o w e r i n g r ~ t r ~ v t d t ~ e - (t/t ) ~ e - (t/t ) t -1/2+1/(n-1) Figure 1.
Diagram showing the main phases of the luminosity(turquoise), photospheric radius (red), diffusion radius (purple), andblack body temperature (orange) for SCE. The axis are logarithmicto emphasize the power-law time dependencies. The timescales of t d and t ph correspond to where the luminosity and photospheric radiusshow breaks, respectively. At late times (but likely before t ph ), theSCE luminosity and black body temperature evolution will be over-taken by radioactive heating (dark blue curves) and may not followthese scalings. These results also do not account for recombination,which can alter the depth of r ph and r d and in turn also impact theluminosity and temperature. energy of E SN = 10 erg, but only a fraction of this energymakes its way into the extended material. This is estimatedto be (Nakar & Piro 2014) E e ≈ × E (cid:18) M c M ⊙ (cid:19) − . (cid:18) M e . M ⊙ (cid:19) . erg , (23)where E = E SN / erg and M c is the mass of the heliumcore. For this case, we find E e = 2 . × erg. Substitutingthis energy into Equation (4) results in v t = 2 . × cm s − .In Figure 2, we present a snapshot of the density profilefrom this numerical model at 1 day following explosion (redcurve). The strong density break above ≈ cm corre-sponds to the top of the helium core, above which the ex- Figure 2.
Numerical density profile for a SN model taken 1 day afterexplosion (red curve). The strong break at a radius a little above10 cm roughly divides the interior helium core ( M c = 3 . M ⊙ )from the hydrogen-rich extended material ( M e = 0 . M ⊙ ). Thedashed line shows the analytic density profile using Equations (1)and (2) with v t = 2 . × cm. tended hydrogen-rich material sits. Although the overall den-sity profile can be fairly complicated given the many differentcompositional layers within the star, the extended materialclearly shows the shallow inner and steep outer density pro-file. The dashed line corresponds to the analytic density pro-file using Equations (1) and (2) with v t = 2 . × cm. Thiscomparison shows that the analytic density profile is a rea-sonable, albeit not exact, description of the numerical result.In Figure 3, we compare the observables of the simula-tions to our analytic model. In the top panel, we comparethe SCE luminosity using Equations (17) and (20) where t d = 0 .
98 days. The analytic solutions are plotted out to t ph = 2 .
63 days, which roughly matches when the simulationsbegin diverting from the analytics due to radioactive heating.While the luminosity shows a break at t d , the photosphericradius continues as a power law that closely follows the evo-lution predicted by Equation (7) all the way up until t ph . Atthis point, the photosphere even decreases in radius as foundin Equation (10), but the time dependence of r ph for t ≫ t ph is so strongly dependent on δ that the analytics do not havemuch predictive power at these times.Nevertheless, the takeaway from this comparison is thateven if the density profile is only roughly replicated by theanalytic model, the observables are reproduced fairly ro-bustly. This provides some confidence that as these modelsare compared with observations, at least the main physicalparameters will be constrained with some fidelity. COMPARISON TO OBSERVATIONSWe next compare our analytic results to a few observationswhere the presence of extended material and associated SCEhas been claimed. This demonstrates how the observations
Figure 3.
Luminosity (turquoise), photospheric radius (red), andblack body temperature (orange) from the numerical model used forthe profile plotted in Figure 2. The dashed lines show the analyticresults for extended material with M e = 0 . M ⊙ and R e = 125 R ⊙ using n = 10, δ = 1 . v t = 2 . × cm s − . show the main features we identify here, strengthening theinterpretation of SCE.4.1. SN 2019dge
SN 2019dge was a helium-rich supernova with a fast-evolving light curve indicating a low ejecta mass (Yao et al.2020). Its early rise was too rapid to be explained as radioac-tively powered diffusion, and was thus interpreted as beingdue to SCE from extended material.An important difference between Piro (2015) and the up-dated model here is that we now include the power-law evo-lution of L ( t ) for t < t d as well as a more careful treatmentof the scaling of r ph . It is therefore interesting to investigatewhether SN 2019dge shows such evolution since most otherSNe do not have such detailed early observations. In Fig-ure 4, we plot the observed bolometric luminosity and photo-spheric radius using logarithmically spaced coordinates. Pre-senting the data in this way immediately makes the power-law evolution of these quantities clear in a way that may notbe as obvious if the coordinates were plotted linearly.We fit the first six data points of L and r ph using our ana-lytic model and evaluating χ . Motivated by the helium-richcomposition of this event, we set κ = 0 . g − . Contoursof constant log ( χ /χ ) are plotted in Figure 5 as a func- Figure 4.
Observed bolometric luminosity (turquoise) and photo-spheric evolution (red) of SN 2019dge (Yao et al. 2020). Dashedlines are from our analytic model using M e = 0 . M ⊙ , R e = 205 R ⊙ ,and E e = 5 . × erg. Figure 5.
Contours of constant log ( χ / χ ) as a function of R e and v t from fitting the first six data points of SN 2019dge. The redstar marks the location of χ with M e = 0 . M ⊙ , R e = 205 R ⊙ , and v t = 7 . × cm s − . The contours correspond to log ( χ / χ ) =0 .
25 to 1 .
75 spaced by intervals of 0 . tion of R e and v t . The red star marks the location of χ .The best fitting value of M e does not change much, rangingfrom M e ≈ . − . M ⊙ over this parameter space. To betterunderstand how these parameters are set by the observations,note that t d is well constrained by when the luminosity be-gins to drop, which puts constraints on the ratio ( M e / v t ) / .Combining this with the normalization of r ph ( t ), which scalesas M / e v / t for n = 10, allows us to independently constrain M e and v t . Finally, using the value of v t we find, the overallnormalization of L ( t ) constrains R e .The best fitting model is plotted in Figure 4 with param-eters M e = 0 . M ⊙ , R e = 205 R ⊙ , and E e = 5 . × erg(or equivalently v t = 7 . × cm s − ). If we were insteadto use the model from (Piro 2015) to fit this event, the re-sult would be a much smaller radius (by a factor of ∼
5) anda larger explosion energy. This is mostly due to the differ-ences in the scaling of r ph between the two models. In thisupdated work, correctly including the velocity gradient re-sults in larger photospheric velocities. In contrast, since thework of (Piro 2015) would predict smaller photospheric ve-locities, this must be compensated by inferring a larger en-ergy to match the observed colors of SN 2019dge, which inturn implies a smaller radius since there is a degeneracy be-tween E e and R e in the luminosity.Nevertheless, the main point of this comparison is not thespecific parameters but how the data clearly scales as ex-pected from our analytic work. This provides even strongersupport for the SCE interpretation. Looking for power-lawbehavior at early times will be a useful way to identifywhether SCE is being seen in other future events.4.2. SN 2016gkg
We next consider our model in comparison to SN 2016gkg.This was a Type IIb SN that was caught especially early afterexplosion and shows a prominent double-peaked light curve.It has well-sampled multi-band coverage including ultravi-olet wavelengths (Arcavi et al. 2017; Kilpatrick et al. 2017;Tartaglia et al. 2017). Especially unique is the amateur datafound during the rise that greatly restricts the explosion time(Bersten et al. 2018). Although early work constrained theradius of the extended material using a variety of analyticand semi-analytic work (e.g., Rabinak & Waxman 2011;Nakar & Piro 2014; Piro 2015; Sapir & Waxman 2017), agrid of numerical simulations was required to provide a de-tailed fit to the multi-band light curves of SCE (Piro et al.2017).In Figure 6, we plot contours of constant log ( χ /χ )from fitting the first ≈ . Figure 6.
Contours of constant log ( χ / χ ) as a function of R e and v t from fitting the SCE of SN 2016gkg. The red star marks thelocation of χ with M e = 0 . M ⊙ , R e = 115 R ⊙ , and v t = 2 . × cm s − . Contours are spaced the same as Figure 5. are able to consider a much wider range of models over ashorter period of time. More specifically, Piro et al. (2017)considered 2 ,
400 models with varying M e , R e , and E e , whilewe consider 250 ,
000 models in a fraction of the time. Ourbest fit model is slightly higher energy (and smaller radius)than Piro et al. (2017), and this was partially due to our abil-ity to consider a wider range of possible parameters here.In Figure 7, we plot the best fitting model in comparisonto the multi-band photometry. The model presented here isespecially good for fitting the early rise in comparison to themodel by Piro (2015), which has trouble fitting the earliest,bluest data as shown in Arcavi et al. (2017). DISCUSSION AND CONCLUSIONIn this work, we have presented an updated analytic modelfor SCE from extended material. Our approach is differentfrom previous work in that we consider the ejecta once it hasalready reached the homologous phase to derive our results.This has the advantage of resolving the early, power-law evo-lution for the luminosity that is not addressed by Piro (2015)as well as improving our treatment of the photospheric evolu-tion. This allows us to better match the earliest, bluest phasesof SCE. We provide comparisons to numerical models andobservations of SCE to demonstrate the advantage of this ap-proach. The luminosity and photospheric radius evolution ofSN 2019dge exhibit the scalings expected from our model,strengthening the support for a SCE interpretation.
Figure 7.
Colored points are photometry from SN 2016gkg(Piro et al. 2017), focusing on the first peak due to SCE. Dashedlines are our analytic model using M e = 0 . M ⊙ , R e = 115 R ⊙ and E e = 9 . × erg. The analytic model we present can be used to fit observa-tions with the parameters M e , R e , and v t (or alternatively E e ).In principle, one can also fit for the outer density steepness n , but we show that the solutions are fairly insensitive to theexact value of n as long as n ∼
10. The main features of SCEof extended material are summarized as follows. • The density is described as a steep outer profile and ashallow inner profile with a break at a transition veloc-ity v t . • The luminosity scales as L ∼ t − / ( n − up to time t d when the diffusion wave reaches the depth where thevelocity is v t . • The diffusion depth is then roughly fixed at the depthof v t . This causes the luminosity to drop exponentiallyfrom time t d up to a time ∼ t ph = ( c / v t ) / t d . • The photospheric radius scales as r ph ∼ t − / ( n − andremains as roughly this power law even past time t d .Unfortunately, few observations have sufficient early datalike SN 2019dge to resolve such scalings. For exam-ple, iPTF14gqr (De et al. 2018a) and iPTF16hgs (De et al.2018b) could in principle be ideal candidates for applyingthis theory, but they simply have insufficient early data toconclusively resolve if our predicted power laws are occur-ring. Our work provides strong motivation for high cadencesat the earliest times to better test whether SCE from extendedmaterial is taking place. Multi-band coverage is also key sothat a bolometric light curve can be reliably constructed.In the future, there are a number of improvements that canbe made to this this work. Here we fit the SCE component,but the second radioactively-powered peak should be fit si-multaneously. Such a fit should also attempt to consistentlyresolve E e versus E SN . Here we relate these quantities usingEquation (23), which is from Nakar & Piro (2014), but thisrelation would benefit from calibration to numerical models.Other details that can be improved are the relation between n and the initial density profile of the extended material, aswell as the treatment of the opacity. We consider it a strengthof the model here that the results are fairly robust to un-certainties in n , nevertheless, as more information is knownabout the progenitor (for example, from pre-explosion imag-ing and numerical stellar model) it may be useful to considera specific n value other than 10. For the opacity, we havefocused on a constant value motivated by the high temper-atures during SCE, but especially for helium-rich extendedmaterial, recombination may play a role which could makethe outermost layers more transparent to electron scattering.We thank Drew Clausen for generating the 15 M ⊙ modeland associated helium core with MESA that was used for thenumerical calculations. A.L.P. acknowledges financial sup-port for this research from a Scialog award made by theResearch Corporation for Science Advancement. A.H. ac-knowledges support from the USC-Carnegie fellowship. Y.Y.thanks the Heising-Simons Foundation for financial support.APPENDIX A. FURTHER COMPARISONS WITH NUMERICAL MODELSTo test our analytic models, we run numerical SN simulations of helium cores surrounded by hydrogen-rich extended material.A similar model to these ones was the focus of Section 3, but here we present additional models and discuss the numericalmethods in more detail.These calculations are similar to the work of Piro et al. (2017). We start with a helium core that was generated from a 15 M ⊙ zero-age main-sequence star using the 1D stellar evolution code MESA (Paxton et al. 2013). Using the overshooting and mixingparameters recommended by Sukhbold & Woosley (2014), the star is evolved until a large entropy jump between the core andenvelope was established. The convective envelope is removed to mimic mass loss during a common envelope phase. The
Figure A.1.
Comparison of the density profiles of three additional numerical models taken at 1 day following explosion. These were chosento sample a range of M e and R e values, These specifically correspond to M e = 0 . M ⊙ , R e = 100 R ⊙ , v t = 2 . × cm s − (left panel), M e = 0 . M ⊙ , R e = 200 R ⊙ , v t = 1 . × cm s − (center panel), and M e = 0 . M ⊙ , R e = 125 R ⊙ , v t = 1 . × cm s − (right panel). Thesevalues of v t are derived by combining Equations (4) and (23) with E SN = 10 erg. In each case, the dashed lines correspond to the analyticdensity model using Equations (1) and (2). Figure A.2.
Numerical results for the luminosity (turquoise), photospheric radius (red), and temperature (orange) for the same models asFigure A.1. Dashed lines are the analytic models using the same values of M e , R e , and v t . resulting helium core has a mass of ≈ . M ⊙ . Above the helium core we stitch a low mass hydrogen envelope with a ρ ∝ r − / density profile to represent the extended material. This specific scaling is meant to mimic the expected profile for a convectivelayer, but as shown in (Piro et al. 2017), SCE is fairly insensitive to this exact choice as long as M e and R e are the same.These models are then exploded with our open-source numerical code SNEC (Morozova et al. 2015). We assume that the inner1 . M ⊙ of the models form a neutron star and excise this region before the explosion. A Ni mass of 0 . M ⊙ is placed at the inneredge of the ejecta, with the exact value not being critically important because we only compare the SCE phase of the simulationswith our analytic model. We use a “thermal bomb mechanism” for the explosion, where a luminosity is provided to the inner0 . M ⊙ of the model for a duration of 0 .