Non-Local Thermodynamic Equilibrium Radiative Transfer Simulations of Sub-Chandrasekhar-Mass White Dwarf Detonations
Ken J. Shen, Stéphane Blondin, Daniel Kasen, Luc Dessart, Dean M. Townsley, Samuel Boos, D. John Hillier
DDraft version February 17, 2021
Typeset using L A TEX twocolumn style in AASTeX62
Non-Local Thermodynamic Equilibrium Radiative Transfer Simulationsof Sub-Chandrasekhar-Mass White Dwarf Detonations
Ken J. Shen, St´ephane Blondin,
2, 3
Daniel Kasen,
1, 4, 5
Luc Dessart, Dean M. Townsley, Samuel Boos, andD. John Hillier Department of Astronomy and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720, USA Unidad Mixta Internacional Franco-Chilena de Astronom´ıa, CNRS/INSU UMI 3386 and Instituto de Astrof´ısica, Pontificia UniversidadCat´olica de Chile, Santiago, Chile Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France Department of Physics, University of California, Berkeley, CA 94720, USA Lawrence Berkeley National Laboratory, Berkeley, CA, USA Institut d’Astrophysique de Paris, CNRS-Sorbonne Universit´e, 98 bis boulevard Arago, F-75014 Paris, France Department of Physics & Astronomy, University of Alabama, Tuscaloosa, AL, USA Department of Physics and Astronomy & Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (PITT PACC), University ofPittsburgh, Pittsburgh, PA 15260, USA
ABSTRACTType Ia supernovae (SNe Ia) span a range of luminosities and timescales, from rapidly evolving sub-luminous to slowly evolving overluminous subtypes. Previous theoretical work has, for the most part,been unable to match the entire breadth of observed SNe Ia with one progenitor scenario. Here, forthe first time, we apply non-local thermodynamic equilibrium radiative transfer calculations to a rangeof accurate explosion models of sub-Chandrasekhar-mass white dwarf detonations. The resulting pho-tometry and spectra are in excellent agreement with the range of observed non-peculiar SNe Ia through15 d after the time of B -band maximum, yielding one of the first examples of a quantitative match tothe entire Phillips (1993) relation. The intermediate-mass element velocities inferred from theoreticalspectra at maximum light for the more massive white dwarf explosions are higher than those of brightobserved SNe Ia, but these and other discrepancies likely stem from the one-dimensional nature of ourexplosion models and will be improved upon by future non-local thermodynamic equilibrium radiationtransport calculations of multi-dimensional sub-Chandrasekhar-mass white dwarf detonations. INTRODUCTIONType Ia supernovae (SNe Ia) are the explosions of C/Owhite dwarfs (WDs) in interacting stellar systems (seeMaoz et al. 2014 and Jha et al. 2019 for recent reviews).Their standardizable light curves, powered by the ra-dioactive decay of Ni (Pankey 1962; Colgate & McKee1969), can be observed from nearly halfway across theUniverse, enabling their use as cosmological distance in-dicators (Riess et al. 1998; Perlmutter et al. 1999), andtheir thermonuclear ashes contribute significantly to themetal content of galaxies (Timmes et al. 1995). How-ever, despite scrutiny dating back to at least 185 AD anda sharp rise in understanding over the past few decades,fundamental uncertainties remain as to the identity ofthe companion star(s) and the means by which the com-panion ignites the exploding WD.
Corresponding author: Ken J. [email protected]
One such explosion mechanism, the double detonationof a sub-Chandrasekhar-mass WD, in which a heliumshell detonation triggers a carbon core detonation, hasbeen explored as a potential SN Ia explosion model be-ginning four decades ago (e.g., Nomoto 1982; Woosleyet al. 1986). However, in this early work, the heliumshells were relatively massive ( ∼ . M (cid:12) ), leading tolarge amounts of thermonuclear ash on the outside ofthe ejecta, which yielded light curves and spectra thatwere inconsistent with observations (e.g., Nugent et al.1997).More recently, it was realized that both stable and un-stable mass transfer in double WD systems can lead todouble detonations with far smaller helium shells thatyield much better agreement with observations of SNe Ia(Bildsten et al. 2007; Fink et al. 2007; Guillochon et al.2010; Pakmor et al. 2013). Using a more realistic nu-clear reaction network, which includes additional reac-tion pathways that are neglected in standard networkswith a limited number of isotopes, Shen & Moore (2014) a r X i v : . [ a s t r o - ph . H E ] F e b Shen et al. found that helium shell ashes for these lower-mass en-velopes are dominated by Si and Ca and would thusproduce observables that better agree with observations(perhaps even explaining the high-velocity features seenin most SNe Ia). This result has been confirmed byrecent multi-dimensional simulations (Townsley et al.2019; Gronow et al. 2020).These studies, combined with significant issues withother SN Ia scenarios (see Maoz et al. 2014 and Jha et al.2019 and references therein), have made it increasinglylikely that sub-Chandrasekhar-mass double detonationsin double WD systems give rise to a fraction, if not themajority, of non-peculiar SNe Ia, ranging from sublumi-nous to overluminous subtypes. In fact, the strongest ev-idence to date for a successful SN Ia scenario came with
Gaia ’s second data release, in the form of hypervelocitystars that could only have been realistically producedby the disruption of double WD binaries (Shen et al.2018b). This is a natural outcome of double detona-tions in double degenerate systems undergoing dynam-ically driven mass transfer (the D scenario) for whichthe explosion of the sub-Chandrasekhar-mass primaryWD may occur before the complete tidal disruption ofthe secondary WD. While only three such stars werefound, they may represent the brightest examples of amuch larger underlying population of D survivors inthe Solar neighborhood.These successes have spurred an increasing interest ingenerating detailed observables of sub-Chandrasekhar-mass WD detonations in order to compare to observedSNe Ia (Sim et al. 2010; Blondin et al. 2017, 2018; Shenet al. 2018a; Goldstein & Kasen 2018; Polin et al. 2019;Townsley et al. 2019; Gronow et al. 2020; Kushnir et al.2020). These studies simulate the exploding WD in arange of detail, from one-dimensional models of bareC/O WDs to multi-dimensional calculations with real-istic helium shells. However, all but Blondin et al. (2017,2018) have been performed assuming that energy levelpopulations of ions and, in most cases, ionization statepopulations are given by local thermodynamic equilib-rium (LTE), which becomes an increasingly inaccurateapproximation as SNe evolve. In this work, we presentnon-LTE radiative transfer calculations of a suite of one-dimensional bare C/O WD explosions with accurate nu-cleosynthesis. We find that radiation transport calcula-tions using a combination of non-LTE and more real-istic explosion models yield excellent photometric andspectroscopic agreement with observations, further sup-porting the idea that sub-Chandrasekhar-mass WD det-onations can reproduce the full range of observed SNIa properties. We do find some discrepancies with ob-servations, but these will likely be overshadowed by the changes to predicted observables when calculations withrealistic helium shells are performed in multiple dimen-sions. HYDRODYNAMICAL EXPLOSION MODELSThe starting models for our radiative transfer calcu-lations are one-dimensional solar metallicity WD explo-sions from Shen et al. (2018a). In that study, artificiallybroadened detonations were ignited at the center of theWDs and followed with the reactive hydrodynamic code,
FLASH (Fryxell et al. 2000), augmented with a 41-isotopenuclear network using the
MESA (Paxton et al. 2011) nu-clear reaction module with rates from JINA’s
REACLIB (Cyburt et al. 2010). Tracer particles were then post-processed with a 205-isotope nuclear network, again us-ing
REACLIB and
MESA ’s infrastructure. No mixing pre-scriptions are applied. In this work, we use the 0 . .
90, 1 .
00, and 1 . M (cid:12) simulations from Shen et al.(2018a) with uniform C/O mass ratios of 50 /
50 and30 /
70 for a total of eight models. Their explosion prop-erties are listed in Table 1 in the Appendix. We notethat C/O mass ratios in WDs are likely lower than 50/50(Giammichele et al. 2018 use asteroseismology to infer aC/O ratio as low as 20/80 for one WD), but we includethe 50/50 models for comparison to previous literature.The nucleosynthetic yields from these models are ingood agreement with recent simulations of bare C/OWD detonations that take care to treat the unresolveddetonation realistically such as Miles et al. (2019) andKushnir et al. (2020). However, they disagree signifi-cantly with past studies including Sim et al. (2010) andBlondin et al. (2017), due primarily to the older stud-ies’ less accurate implementation of detonation physicsand/or oversimplification of the nuclear network. Thediscrepancies are particularly large for the lower massexplosions, for which the previous Ni masses are asmuch as a factor of ∼ (cid:38) . M (cid:12) because more mass is processed through nu-clear statistical equilibrium, which is relatively insen-sitive to the choice of nuclear network and the treat-ment of the detonation. The nucleosynthetic differencesare particularly important given the tantalizing corre-spondence to the observed Phillips (1993) relation thatBlondin et al. (2017) found for masses (cid:38) . M (cid:12) usingthe non-LTE radiative transfer code CMFGEN . Since thesehigh-mass models are in agreement with more accurateexplosion calculations, there is hope that non-LTE ra-diative transfer of a wide range of accurate models willyield a better match to the entirety of SNe Ia, from sub-luminous to overluminous subtypes. LIGHT CURVES
Wavelength [˚A] S c a l e dflu x d e n s i t y + o ff s e t LTE time-dep.LTE snapshotNLTE snapshotLTE time-dep.LTEsnapshotNLTE snapshotLTE time-dep.LTEsnapshotNLTE snapshot
Sedona , 1 . M (cid:12) , C / O = 30 / +0 d+15 d+30 d Figure 1.
Scaled and offset
Sedona spectra of a 1 . M (cid:12) ,C / O = 30 /
70 WD detonation +0, +15, and +30 d from B -band maximum. Time-dependent and snapshot LTE spectrausing the resolved line opacity formalism described in Sec.A.2 are shown in orange and green, respectively, and non-LTE snapshot spectra are shown in red. To this end, we process our eight explosion modelswith the radiative transfer code
Sedona and a subset ofthese models with
CMFGEN . We focus on non-LTE cal-culations in this Letter, but in the Appendix, we detailresults using various LTE prescriptions in
Sedona to in-form past and future LTE calculations.3.1.
Sedona snapshots and the impact of non-LTE
Sedona (Kasen et al. 2006) is a Monte Carlo radia-tive transfer code that has been used in many studies ofastronomical transient phenomena. Most of these stud-ies have assumed that energy level and ion state popu-lations possess their LTE values, but these can also becalculated in non-LTE in
Sedona . Due to computationalconstraints, in this Letter, we apply
Sedona ’s non-LTEcapabilities to “snapshot” calculations, in which pho-tons are propagated through ejecta whose velocity, den-sity, and composition profiles are fixed at a particulartime. This is done iteratively until the output luminos-ity matches that of a time-dependent LTE calculationand radiative equilibrium in each zone is established.Further details are outlined in Section A.3.To confirm that this snapshot procedure has thepotential to match cost-prohibitive time-dependentnon-LTE calculations, we compare spectra of time-dependent and snapshot LTE simulations in Figure1 for the 1 . M (cid:12) iii to Fe ii , less blanketing by Fe ii lines at bluer wave-lengths, and a commensurate decrease in the redistribu-tion of flux at redder wavelengths. This has importantimplications for post-maximum photometry as well asthe Phillips (1993) relation, which we discuss in Section3.3.3.2. Time-dependent non-LTE with
CMFGEN andcomparison to observations
We also perform time-dependent radiative transfercalculations with the non-LTE code
CMFGEN (Hillier &Dessart 2012; Dessart et al. 2014) for a subset of themodels: all four of the 50/50 C/O models and the1 . M (cid:12) CMFGEN , but we note herethe significant fact that the atomic data input for thetwo codes is not identical. In particular, the
CMFGEN runsconsider more levels for Fe ii , Co ii , and Co iii than areincluded in the Sedona calculations. Thus, comparisonsof results from the two codes in this study should beregarded as qualitative.Figure 2 shows light curve comparisons of the
Sedona non-LTE snapshot and the
CMFGEN non-LTE time-dependent calculations for our models with C / O =50 /
50; our C / O = 30 /
70 models are displayed in theAppendix in Figure 6. As with the
Sedona
LTE calcu-lations, there is good agreement at the time of B -bandmaximum in all bands (e.g., the two codes yield maxi-mum B -band magnitudes that differ by 0 . V -band, and R -band light curves continueto match through day +30. Moreover, the agreementfor most bands and models persists through day +15.Given the different methods of solving the radiativetransfer problem in Sedona and
CMFGEN , this agreementgives confidence that the multi-band photometry is be-ing accurately calculated in non-LTE through day +15.As shown in Figures 2 and 6, the agreement throughday +15 also extends to the observed photometry ofthe overluminous SN 1999dq (Stritzinger 2005; Jha
Shen et al. − − − −
14 Bolometric
U B −
20 0 20 − − − − V NLTEC/O = 50/50 −
20 0 20 R . M (cid:12) . M (cid:12) . M (cid:12) . M (cid:12) −
20 0 20 I SedonaCMFGEN . . . . . . Time since B -band maximum [days] . . . . . . A b s o l u t e m ag n i t ud e Figure 2.
Multi-band non-LTE light curves of 50/50 C/O mass ratio WD detonations. Masses are as labeled, with brighterpeak B -band magnitudes as the mass increases. CMFGEN results are represented by solid lines;
Sedona snapshots are demarcatedwith circles. Open diamonds, circles, and squares show observations of SNe 1999dq, 2011fe, and 1999by, respectively. Note thatthe observed “bolometric” light curves only consider UVOIR flux and are thus up to ∼
20% lower than the true bolometricluminosities. et al. 2006; Ganeshalingam et al. 2010) , which is well-matched in most bands by the 1 . M (cid:12) explosions; thenormal SN 2011fe (Munari et al. 2013; Pereira et al.2013; Tsvetkov et al. 2013), which corresponds to the1 . M (cid:12) detonations; and the subluminous SN 1999by(Garnavich et al. 2004; Stritzinger 2005; Ganeshalingamet al. 2010), which agrees with the 0 . M (cid:12) explosions.There are some discrepancies: for example, the R - and I -band light curves for the 1 . M (cid:12) explosion are toodim compared to SN 1999dq, the rise times of the more Most of the observational data used in this work was obtainedthrough https://sne.space (Guillochon et al. 2017). massive models are too short compared to observations,and the colors at very early times for all the models aretoo red. But overall, the agreement through day +15for this wide range of SNe Ia is striking.The
CMFGEN and
Sedona light curves do diverge fromeach other and from observations by day +30 in U -, B -, and I -bands. The disagreement in bluer bands isespecially large for the higher masses. Perhaps coinci-dentally, the photometry from the two codes bracketsthe best-matching observed U -band and B -band lightcurves at day +30, with CMFGEN being too bright and
Sedona being too faint.The root cause of the disagreement between the twocodes is unclear. As previously discussed, the input . . . . ∆ m ( B ) − . − . − . − . − . − . − . P e a k a b s o l u t e B - b a nd m ag n i t ud e . M (cid:12) . M (cid:12) . M (cid:12) . M (cid:12) SedonaCMFGEN
C/O = 50 / / Figure 3.
Peak B -band magnitude vs. decrease in B -bandmagnitude between peak and 15 d after peak: the Phillips(1993) relation. Solid symbols show the results from the Sedona non-LTE snapshots, and open symbols show the
CMFGEN results. Circles (triangles) represent initial C/O frac-tions of 50/50 (30/70). Masses are as labeled. Gray errorbars are values from a sample of SNe Ia, and black error barsare values for SN 1999by, SN 2011fe, and SN 1999dq. atomic data is not identical for the two codes, renderingquantitative comparisons difficult. Another possibility is
Sedona ’s assumption of radiative equilibrium (i.e., thatheating is equal to cooling in each zone), which is not thecase for
CMFGEN . CMFGEN also includes non-thermal pro-cesses, which are neglected in the
Sedona runs. Theseonly contributed tenths of a magnitude of difference atday +30 for the Chandrasekehar-mass WD models inDessart et al. (2014), but their effects may be strongerfor sub-Chandrasekhar-mass WD explosions. As it isbeyond the current scope, we leave a thorough exami-nation of the causes of differences between the two codesto a future study.3.3.
The Phillips relation
Figure 3 shows the peak B -band magnitude vs.∆ m ( B ), the decline in B -band magnitude 15 d af-ter peak, for our Sedona and
CMFGEN simulations. Grayerror bars are observed values taken from the CfA lightcurve data set (Hicken et al. 2009), and black error barsrepresent SN 1999by (Garnavich et al. 2004), SN 2011fe(Pereira et al. 2013), and SN 1999dq (Ganeshalingamet al. 2010).It has been well-established for decades that SNe Iathat reach brighter peak B -band magnitudes evolve more slowly than dimmer SNe Ia; this is the Phillips(1993) relation. Most studies of Chandrasekhar-massexplosions (but c.f. H¨oflich et al. 2017) have been un-able to reproduce the entirety of the relation due tothe constant ejecta mass in the explosions, which is theprimary variable controlling the rate of the light curveevolution. These studies have found that, while themore luminous end of the relation can be achieved byChandrasekhar-mass explosions, they cannot reproduceobservations with ∆ m ( B ) (cid:38) . B -band maximum: our CMFGEN and
Sedona calculationsbecome discrepant soon after day +15, with neither codeadequately matching the light curves of the entire rangeof observed SNe Ia. However, it is reassuring that, whenthe results of the two codes do match each other, theyalso match observations.Peak magnitudes and values of ∆ m ( B ) have alsobeen found to correlate with host galaxy properties.Shen et al. (2017) combined a binary population synthe-sis calculation with an assumed relationship between ex-ploding WD masses and the resulting values of ∆ m ( B )to derive a quantitative explanation for this observedcorrelation of SN Ia properties with host galaxy charac-teristics (in particular, stellar age). The relationshipwe find in the present work between WD mass and∆ m ( B ) is consistent with that assumed in Shen et al.(2017), and thus their conclusions remain valid: namely,sub-Chandrasekhar-mass WD detonations are a plausi-ble mechanism to explain the correlation of SN Ia prop-erties with stellar age due to the evolution of the primaryWD mass in merging double WD binaries.We also note that the relationship between peak mag-nitude and the rate of decline for the bolometric lightcurve is much weaker. The bolometric decline rateparameter ∆ m (bolometric) is indeed larger for the0 . M (cid:12) models than the 1 . M (cid:12) models, but only by0 . B -band width-luminosity relation is not the photon diffusion time but Shen et al. instead the faster color evolution of the dimmer modelsdue to the earlier onset of Fe iii to Fe ii recombination. SPECTRAScaled and offset
Sedona and
CMFGEN spectra of a sub-set of the models near +0, +15, and +30 d from B -bandmaximum are shown in Figure 4, along with compar-isons to observed spectra. (The 0 . M (cid:12) models are omit-ted for simplicity, and models without CMFGEN spectraare displayed in Fig. 7 in the Appendix.) The near-maximum spectra are all in excellent agreement witheach other, except for one significant flaw: as with pre-viously published LTE spectra (e.g., Sim et al. 2010;Shen et al. 2018a; Polin et al. 2019) and those in thiswork, the intermediate-mass element (IME) absorptionfeatures for the more massive WD models are blueshiftedby several thousand km s − compared to observations;see Table 1 in the Appendix for the inferred velocitiesat maximum light.However, the multi-dimensional study performed byTownsley et al. (2019) yielded maximum-light spectralIME velocities for a 1 . M (cid:12) explosion that are muchcloser to that seen in SN 2011fe along most lines of sight.Gronow et al. (2020) found a similar result for their1 . M (cid:12) explosion, albeit in an angle-averaged sense.It is thus likely that the IME velocity discrepancies area product of the one-dimensional nature of our models,given the multi-dimensional LTE results and the similar-ity between our LTE and non-LTE spectra at maximumlight.At later times, the agreement among CMFGEN , Sedona ,and the observations is less definitive but still persists.E.g., the 0 . M (cid:12) Sedona day +15 and +30 models donot appear to be close fits to SN 1999by, but the 50/50
CMFGEN model at day +14 does adequately reproduceSN 1999by’s day +17 spectrum. Meanwhile, the 1 . M (cid:12) day +15 Sedona and
CMFGEN spectra for both C/O ra-tios provide good matches to SN 2011fe’s spectrum, andat day +30,
Sedona ’s spectra continue to yield a goodfit, while the
CMFGEN spectra appear suitable. A similarpattern arises for the 1 . M (cid:12) spectra and SN 1999dq.We also note that the maximum-light IME velocity dis-crepancies are not a problem at these later times.In summary, it appears that non-LTE radiative trans-fer of one-dimensional sub-Chandrasekhar-mass WDdetonations is able to reproduce observed spectra of theentire range of SNe Ia, from the subluminous SN 1999byto the overluminous SN 1999dq, up to 30 d from the timeof B -band maximum. The most notable discrepancy isin the IME velocities at peak light for the brighterSNe Ia, but at later times, this difference disappears. CONCLUSIONS Here, we summarize the main results of this work: • The predicted photometry from non-LTE radia-tive transfer simulations of sub-Chandrasekhar-mass WD detonations using two different codes(
Sedona and
CMFGEN ) matches through day +15 af-ter B -band maximum for all models and in almostall bands. Moreover, this multi-band photometryagrees relatively well with observations of a widerange of SNe Ia through +15 d. With minor ex-ceptions, the theoretical bolometric, V -band, and R -band light curves match each other and obser-vations through at least +30 d. Some discrepan-cies do exist, but they will likely be overshad-owed by the changes that future multi-dimensionalcalculations will engender. Clumping, which wedo not consider in this work, will also quanti-tatively change the results presented here (Wilket al. 2020). • Non-LTE radiative transfer of sub-Chandrasekhar-mass WD detonations is able to reproduce theentirety of the Phillips (1993) relation, from sub-luminous to overluminous SNe Ia. Since the pho-tometry generated with both
CMFGEN and
Sedona agrees through day +15, this appears to be arobust finding. • The simulated peak light spectra mostly agreewith observations for the whole range of SNe Ia,but the minima of the IME absorption features forthe higher mass explosions are faster than those forthe brighter SNe, whose luminosity they match.Day +15 and +30 spectra are also reproduced(with no velocity shifts). • The predicted observables from our non-LTE sim-ulations agree with those from our LTE calcula-tions at the time of maximum B -band light. Thatis to say, non-LTE effects have little visible im-pact at this time. After this time, non-LTE cal-culations yield hotter, more ionized, and less Fe ii -line-blanketed atmospheres as compared to LTEsimulations, leading to changes in flux in the ultra-violet and near-infrared. Non-LTE and LTE bolo-metric and V -band light curves (and to a lesser ex-tent, R -band light curves) continue to match eachother until at least +30 d from maximum.Our non-LTE radiative transfer calculations show thatsub-Chandrasekhar-mass WD detonations match obser-vations of the entire range of SNe Ia, from subluminousto overluminous, perhaps as well as could be expectedfrom one-dimensional simulations. Radiative transfer Sedona − . − . CMFGEN − . Sedona +14 . . CMFGEN +16 . Sedona +29 . . . M (cid:12) , C / O = 50 / a) Sedona +0 . − . CMFGEN − . Sedona +15 . . CMFGEN +14 . Sedona +30 . . CMFGEN +29 . . M (cid:12) , C / O = 50 / b)3000 5000 7000 900001234567 Sedona +0 . − . CMFGEN +0 . Sedona +15 . . CMFGEN +13 . Sedona +30 . . CMFGEN +28 . . M (cid:12) , C / O = 30 / c) 3000 5000 7000 9000 Sedona +0 . . CMFGEN +0 . Sedona +15 . . CMFGEN +15 . Sedona +30 . . CMFGEN +30 . . M (cid:12) , C / O = 50 / d)0 . . . . . . Wavelength [˚A] . . . . . . S c a l e dflu x d e n s i t y + o ff s e t Figure 4.
Scaled and offset
CMFGEN and
Sedona non-LTE spectra with comparisons to spectra of observed SNe with similarlight curves. Days from the time of B -band maximum are as labeled. studies of multi-dimensional double detonation simula-tions with low-mass helium shells are in their infancyand have only been performed in LTE (Townsley et al.2019; Gronow et al. 2020), but multi-dimensional effectshave already proven important for reducing the discrep-ancy in IME velocities at maximum light. However,given the work presented here, these multi-dimensionalLTE light curves and spectra are likely only accurateuntil maximum light. After this time, the bolometric and V -band light curves will probably agree with future,more physical non-LTE results, but multi-dimensionalnon-LTE simulations will be crucial for accurate predic-tions in other bands and for spectra at later times.We thank the anonymous referee for their commentsand acknowledge helpful discussions with Alison Miller.We are indebted to the other developers of Sedona , withextra thanks to David Khatami, Hannah Klion, and
Shen et al.
Nathan Roth. K.J.S., D.M.T., and S.B. received sup-port for this work from NASA through the AstrophysicsTheory Program (NNX17AG28G). D.K. is supported inpart by the U.S. Department of Energy, Office of Sci-ence, Office of Nuclear Physics, under contract num-ber DE-AC02-05CH11231 and DE-SC0004658, and bya SciDAC award DE-SC0018297. D.J.H. acknowledgessupport for the development of
CMFGEN from STScI the-ory grant HST-AR-12640.001-A and NASA theory grantNNX14AB41G. This research was supported in part bythe Gordon and Betty Moore Foundation through grantGBMF5076, by a grant from the Simons Foundation(622817DK), and by the Exascale Computing Project(17-SC-20-SC), a collaborative effort of the U.S. Depart-ment of Energy Office of Science and the National Nu- clear Security Administration. This research used theSavio computational cluster resource provided by theBerkeley Research Computing program at the Univer-sity of California, Berkeley (supported by the UC Berke-ley Chancellor, Vice Chancellor for Research, and ChiefInformation Officer). This research also used resourcesof the National Energy Research Scientific ComputingCenter (NERSC), a U.S. Department of Energy Officeof Science User Facility located at Lawrence BerkeleyNational Laboratory, operated under Contract No. DE-AC02-05CH11231.
Software:
CMFGEN (Hillier & Dessart 2012),
FLASH (Fryxelletal.2000), matplotlib (Hunter2007),
MESA (Pax-ton et al. 2011),
Sedona (Kasen et al. 2006)APPENDIX A. RADIATION TRANSPORT WITH
Sedona
A.1.
Expansion opacity formalism
For the majority of previous studies using
Sedona , local thermodynamic equilibrium (LTE) is assumed to hold forthe matter, so that ionization fractions and level populations are given by Saha-Boltzmann distributions. Furthermore,most of these studies have treated line opacities in the “expansion opacity” approximation (Karp et al. 1977), in whichthe contributions from bound-bound transitions are grouped together into discretized frequency bins.In the
Sedona implementation of expansion opacity, the source function for bound-bound transitions can be expressedin terms of the mean intensity integrated over the line profile, ¯ J ν , and the Planck function, B ν ( T ), as S ν = (1 − (cid:15) ) ¯ J ν + (cid:15)B ν ( T ) , (A1)where (cid:15) is the absorption probability, and the gas temperature is T . In principle, (cid:15) is different for every transition ofeach element, but it is typically a single user-specified constant in Sedona ; we adopt this simplification here, but c.f.Goldstein & Kasen (2018) where it is set separately for low-mass and high-mass elements.In our previous study (Shen et al. 2018a), (cid:15) was set equal to 1. Figure 5 shows the effects of varying (cid:15) on LTElight curves of our sub-Chandrasekhar-mass WD detonations. Dotted, dashed-dotted, and dashed lines representsimulations with (cid:15) = 0 .
5, 0 .
8, and 1 .
0, respectively. (Solid lines and circles represent other treatments of bound-boundtransitions and will be discussed in Secs. A.2 and A.3.) Bolometric light curves are largely insensitive to the choiceof (cid:15) for the duration of the simulations, and the same is true for the light curves of each individual band except I -band through day +15 from B -band maximum. After day +15, decreasing (cid:15) results in bluer light curves, with acolor difference that increases with time. The differences are relatively minimal in the V - and R -bands, but they aresignificant for the other bands, reaching 0 . B -band for the 0 . M (cid:12) model at +30 d.A.2. Resolved line opacity formalism
Line opacities can also be implemented in
Sedona in a less approximate way by assuming the absorption and emissionprofiles are specified functions that are then directly mapped to an opacity grid. For simplicity, we assume the twoprofiles are equivalent to each other. In LTE, this implies the source function is the Planck distribution, so the effectiveabsorption probability (cid:15) = 1.In this work, we choose the line profile to be a Voigt profile. If the only line broadening is thermal, resolving thelines would require an extremely finely spaced frequency grid. We thus specify an artificial broadening velocity, set inthis work to be 100 km s − . We use a logarithmic frequency spacing of dν/ν = 3 × − , yielding ∼
10 frequency binsper line. We verify that our results are converged with a broadening velocity as high as 200 km s − , but we use a lower velocity of 100 km s − for added certainty. − − − −
14 Bolometric
U B −
20 0 20 − − − − V Sedona
C/O = 30/70 −
20 0 20 R . M (cid:12) . M (cid:12) . M (cid:12) . M (cid:12) −
20 0 20 I (cid:15) = 0 . (cid:15) = 0 . (cid:15) = 1 . . . . . . . Time since B -band maximum [days] . . . . . . A b s o l u t e m ag n i t ud e Figure 5.
Multi-band
Sedona light curves of C / O = 30 /
70 WD detonations with various treatments of bound-bound opacity.Masses are as labeled, with brighter peak B -band magnitudes as the mass increases. Dotted, dashed-dotted, and dashed linesshow results using the LTE expansion opacity formalism (discussed in Sec. A.1) with (cid:15) = 0 .
5, 0 .
8, and 1 .
0, respectively. Solidlines represent calculations using resolved lines in LTE (Sec. A.2), and non-LTE snapshots (Sec. A.3) are demarcated withcircles.
The results for our C / O = 30 /
70 models using this resolved line opacity formalism assuming LTE ionization stateand level populations are shown in Figure 5 as solid lines. The bolometric light curves are consistent with the expansionopacity simulations described in Section A.1, and the multi-band light curves are mostly consistent with each otheruntil maximum light, after which the resolved line opacity simulations remain somewhat close to the expansion opacitysimulations with (cid:15) = 1. However, in U -band, the resolved line opacity simulations are consistently dimmer than all ofthe expansion opacity simulations. A.3. Non-LTE snapshots
Sedona also allows for the inclusion of non-LTE effects in the resolved line opacity formalism by relaxing the LTEassumption that ionization state and level populations are set by Saha-Boltzmann equilibrium. Instead, we implement0
Shen et al. non-LTE by assuming that statistical equilibrium holds, so that the time derivative of every level population is zero: dn i,j,k dt = n i,j +1 , R c → k + (cid:88) k (cid:48) n i,j,k (cid:48) R k (cid:48) → k − n i,j,k R k → c − n i,j,k (cid:88) k (cid:48) R k → k (cid:48) = 0 , (A2)where the population of level k of species i in ionization state j is n i,j,k , R k → k (cid:48) is the transition rate from level k to k (cid:48) , R c → k is the rate of transitions from the continuum ground state of ionization state j + 1, and R k → c is the transitionrate from level k to the continuum ground state of ionization state j + 1. In Sedona , this constraint is cast as a matrixinversion problem ( d(cid:126)n i /dt = M (cid:126)n i = 0) for each element that is treated in non-LTE. In this work, we calculate theprimary spectral-feature-forming elements Si, S, Ca, Fe, Co, and Ni in non-LTE and assume the rest of the elementsare in LTE to reduce the computational load.While the capability to perform time-dependent non-LTE calculations exists in Sedona , such simulations currentlytake too much computational time to be feasible for production runs. Efforts are underway to accelerate thesecalculations, but for this study, we instead perform non-LTE snapshot calculations, in which a prescribed luminosity ispropagated through the composition and density structure of a model at a given time to generate a spectrum, similarto snapshot procedures in other work (e.g., Nugent et al. 1997; Kerzendorf & Sim 2014). We note that the times ofthe non-LTE results are given with respect to the time of B -band maximum as calculated using LTE. In principle, thistime may be different for a time-dependent non-LTE calculation; however, as we describe below, predicted observablesfrom LTE and non-LTE calculations are quite similar through the time of peak light, so this is likely a negligibledifference.To ensure that such snapshot calculations can adequately reproduce time-dependent results, we first perform snapshotcalculations in LTE using the resolved line opacity formalism and compare them to the spectra produced by the time-dependent resolved line opacity LTE calculations described in Section A.2; we show the results of this comparison inFigure 1. We use the composition and density structures at +0, +15, and +30 d from the B -band maxima for all eightof our models and allow the luminosity from the radioactive decay at those times to propagate outwards. However,the outgoing luminosity in the time-dependent calculation is not equal to the instantaneous radioactive decay powerbecause of the finite diffusion speed of the photons as well as energy lost to expansion. Thus, we also supplementthe radioactive luminosity with additional luminosity emanating from the center of the ejecta, iterating until the totaloutgoing luminosity matches the value from the time-dependent calculation. The converged value of the necessaryadditional luminosity is relatively insensitive to the opacity structure of the ejecta, so we only need to perform thisseries of iterations once for each model epoch.The initial temperature structure from the time-dependent calculation is used to calculate opacities, and photonpackets are propagated outwards until all have left the grid. The temperature structure is then recalculated in orderto yield radiative equilibrium (i.e., heating equal to cooling) in all zones. We perform this iteration procedure 10 timesfor our LTE snapshots. While 10 iterations is sufficient for convergence for all of the LTE and roughly half of thenon-LTE snapshots, some of the non-LTE snapshots do not reach convergence after 10 iterations. For these models,we continue the iteration procedure until none of their broad band magnitudes changes by more than 0 . CMFGEN , but at present, such calculations are too costly with
Sedona .The resulting multi-band photometry at day +0, +15, and +30 from B -band maxima for our C / O = 30 / Sedona non-LTE snapshots is shown as circles in Figure 5. Lines show the results of time-dependent LTE calculations usinga variety of implementations, as described in Sections A.1 and A.2. By construction, the bolometric light curvesare well-matched to the time-dependent resolved line opacity LTE calculations at each epoch, and, as expected fromthe spectral comparison, the photometry at peak light in every band is also very similar. However, the reduction inFe ii line blanketing and commensurate reduction in Ca ii emission after peak when implementing non-LTE leads todiscrepancies compared to the resolved line opacity LTE calculations (Kasen 2006), with the largest differences in the U - B -, and I -bands: 15 d after peak, the non-LTE results are as much as 2 mag brighter (dimmer) in U -band ( I -band)than the resolved line opacity LTE results. Meanwhile, the V -band magnitudes are relatively unchanged between1LTE and non-LTE because the iron line flux redistribution mostly occurs outside of the V -band, leaving it largelyuntouched.It is clear from this comparison that LTE radiative transfer calculations of sub-Chandrasekhar-mass SNe Ia in mostbands and under a variety of approximations are reliable until peak but not afterwards. After B -band maximum,bolometric and, to a lesser extent, V -band light curves are relatively immune from the effects of non-LTE, but allother bands are subject to corrections as large as 2 mag just 15 d from peak. B. DESCRIPTION OF
CMFGEN
AND COMPARISON OF INPUT DATA WITH
SedonaCMFGEN is a non-LTE radiative transfer code that solves the time-dependent radiative transfer equations and thetime-dependent kinetic equation for a homologous flow, including treatments of non-local energy deposition and non-thermal processes. No expansion-opacity formalism is used as all bound-bound transitions are explicitly included inthe transfer equation. To reduce the number of bound-bound levels, super-levels are utilized. In this approach, whichcan be easily modified, levels in the same ion with similar properties are assumed to depart from LTE by the samefactor.We use the same setup as in Blondin et al. (2017, 2018), except for two notable differences: (a) we use a largermodel atom for Fe ii . We consider the first 2698 full levels, which are grouped into 228 super levels for the purposeof solving the time-dependent rate equations, whereas our previous setup consisted of 827 full levels grouped into 275super levels; (b) the radioactive decay energy deposition function is determined assuming a purely absorptive greyopacity with κ γ = 0 . Y e cm g − , as opposed to computed via a Monte Carlo code.The linelist used in the Sedona calculations is a superset of the linelist used in
CMFGEN , with three exceptions: the
Sedona linelist only includes the first 1000 levels of Fe ii , Co ii , and Co iii , whereas CMFGEN considers the first 2698,2747, and 3917 levels, respectively. This is the primary reason that comparisons of results from the two codes in thisstudy cannot be regarded as quantitative; a proper comparison, which will be performed in the future, requires theinput data to be as identical as possible, especially for these important ions. C. SUPPLEMENTAL FIGURES AND DATAREFERENCES
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Table 1.
Explosion and non-LTE radiative transfer properties of the models used in this workC / O = 50 /
50 C / O = 30 / M (cid:12) ] 0.85 0.90 1.00 1.10 0.85 0.90 1.00 1.10 Ni mass [ M (cid:12) ] 0.13 0.28 0.56 0.80 0.08 0.23 0.52 0.78Kinetic energy [10 erg] 0.87 1.00 1.22 1.38 0.65 0.82 1.08 1.26 Sedona
Days to B -band max. 15.7 16.6 16.6 16.0 15.2 17.2 17.3 16.6Si ii λ km s − ] a b − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . U -band mag. at +0 d − . − . − . − . − . − . − . − . U -band mag. at +15 d − . − . − . − . − . − . − . − . U -band mag. at +30 d − . − . − . − . − . − . − . − . B -band mag. at +0 d − . − . − . − . − . − . − . − . B -band mag. at +15 d − . − . − . − . − . − . − . − . B -band mag. at +30 d − . − . − . − . − . − . − . − . V -band mag. at +0 d − . − . − . − . − . − . − . − . V -band mag. at +15 d − . − . − . − . − . − . − . − . V -band mag. at +30 d − . − . − . − . − . − . − . − . R -band mag. at +0 d − . − . − . − . − . − . − . − . R -band mag. at +15 d − . − . − . − . − . − . − . − . R -band mag. at +30 d − . − . − . − . − . − . − . − . I -band mag. at +0 d − . − . − . − . − . − . − . − . I -band mag. at +15 d − . − . − . − . − . − . − . − . I -band mag. at +30 d − . − . − . − . − . − . − . − . CMFGEN
Days to B -band max. 14.8 16.3 16.3 15.3 · · · · · · · · · Si ii λ km s − ] a · · · · · · · · · Bolometric mag. at +0 d b − . − . − . − . · · · · · · − . · · · Bolometric mag. at +15 d − . − . − . − . · · · · · · − . · · · Bolometric mag. at +30 d · · · − . − . − . · · · · · · − . · · · U -band mag. at +0 d − . − . − . − . · · · · · · − . · · · U -band mag. at +15 d − . − . − . − . · · · · · · − . · · · U -band mag. at +30 d · · · − . − . − . · · · · · · − . · · · B -band mag. at +0 d − . − . − . − . · · · · · · − . · · · B -band mag. at +15 d − . − . − . − . · · · · · · − . · · · B -band mag. at +30 d · · · − . − . − . · · · · · · − . · · · V -band mag. at +0 d − . − . − . − . · · · · · · − . · · · V -band mag. at +15 d − . − . − . − . · · · · · · − . · · · V -band mag. at +30 d · · · − . − . − . · · · · · · − . · · · R -band mag. at +0 d − . − . − . − . · · · · · · − . · · · R -band mag. at +15 d − . − . − . − . · · · · · · − . · · · R -band mag. at +30 d · · · − . − . − . · · · · · · − . · · · I -band mag. at +0 d − . − . − . − . · · · · · · − . · · · I -band mag. at +15 d − . − . − . − . · · · · · · − . · · · I -band mag. at +30 d · · · − . − . − . · · · · · · − . · · · a Velocities are inferred from the wavelength of the absorption minimum. b Times are with respect to the time of B -band maximum. − − − −
14 Bolometric
U B −
20 0 20 − − − − V NLTEC/O = 30/70 −
20 0 20 R . M (cid:12) . M (cid:12) . M (cid:12) . M (cid:12) −
20 0 20 I SedonaCMFGEN . . . . . . Time since B -band maximum [days] . . . . . . A b s o l u t e m ag n i t ud e Figure 6.
Same as Fig. 2, but for C/O mass ratios of 30/70. Shen et al.
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Same as Fig. 4, but for the two models without
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