Evidence for sub-Chandrasekhar-mass progenitors of Type Ia supernovae at the faint end of the width-luminosity relation

TL;DR

This study uses radiative transfer simulations to show that sub-Chandrasekhar-mass white dwarf explosions can explain the rapid light-curve evolution and faintness of certain Type Ia supernovae, aligning with observed relations.

Contribution

It provides the first detailed modeling of sub-Chandrasekhar-mass WD explosions, demonstrating their consistency with faint, fast-declining SNe Ia at the width-luminosity relation's faint end.

Findings

Sub-Chandrasekhar models produce faster light-curve evolution.

These models exhibit larger post-maximum decline rates.

They better match the observed width-luminosity relation at low luminosities.

Abstract

The faster light-curve evolution of low-luminosity Type Ia supernovae (SNe Ia) suggests that they could result from the explosion of white dwarf (WD) progenitors below the Chandrasekhar mass ($M_{\rm Ch}$). Here we present 1D non-LTE time-dependent radiative transfer simulations of pure central detonations of carbon-oxygen WDs with a mass ($M_\rm{tot}$) between 0.88 M$_{\odot}$ and 1.15 M$_{\odot}$, and a $^{56}\rm{Ni}$ yield between 0.08 M$_{\odot}$ and 0.84 M$_{\odot}$. Their lower ejecta density compared to $M_{\rm Ch}$ models results in a more rapid increase of the luminosity at early times and an enhanced $\gamma$-ray escape fraction past maximum light. Consequently, their bolometric light curves display shorter rise times and larger post-maximum decline rates. Moreover, the higher $M(^{56}\rm{Ni})/M_\rm{tot}$ ratio at a given $^{56}\rm{Ni}$ mass enhances the temperature and ionization level in the spectrum-formation region for the less luminous models, giving rise to bluer colours at maximum light and a faster post-maximum evolution of the $B-V$ colour. For sub-$M_{\rm Ch}$ models fainter than $M_B\approx -18.5$ mag at peak, the greater bolometric decline and faster colour evolution lead to a larger $B$-band post-maximum decline rate, $\Delta M_{15}(B)$. In particular, all of our previously-published $M_{\rm Ch}$ models (standard and pulsational delayed detonations) are confined to $\Delta M_{15}(B) < 1.4$ mag, while the sub-$M_{\rm Ch}$ models with $M_\rm{tot}\lesssim 1$ M$_{\odot}$ extend beyond this limit to $\Delta M_{15}(B)\approx 1.65$ mag for a peak $M_B\approx -17$ mag, in better agreement with the observed width-luminosity relation (WLR). Regardless of the precise ignition mechanism, these simulations suggest that fast-declining SNe Ia at the faint end of the WLR could result from the explosion of WDs whose mass is significantly below the Chandrasekhar limit.

Tables

Table 1. Table 1: Basic properties of the SN Ia models considered in this study, with photometric quantities relevant to the width-luminosity relation.

Model	$M_{\mathrm{tot}}$	$E_{\mathrm{kin}}$	$M{(}^{56}\mathrm{Ni})$	${\text{varv}}_{99}{(}^{56}\mathrm{Ni})$	${\dot{e}}_{\mathrm{decay}}$	$t_{\mathrm{rise}}\left(\mathrm{bol}\right)$	$t_{\mathrm{rise}}\left(B\right)$	$L_{\mathrm{bol}}$	$M_B$	$\Delta M_{15}\left(\mathrm{bol}\right)$	$\Delta M_{15}\left(B\right)$	${\left(B-V\right)}_{+0}$	${\left(B-V\right)}_{+15}$
	(M⊙)	(erg)	(M⊙)	(km s-1)	(erg s-1 g-1)	(day)	(day)	(erg s-1)	(mag)	(mag)	(mag)	(mag)	(mag)
Standard Chandrasekhar-mass delayed-detonation models (DDC series)
DDC0	1.41	1.56 (51)	0.86	1.29 (4)	6.32 (9)	16.7	17.7	1.85 (43)	$-$19.65	0.74	0.74	0.02	0.20
DDC6	1.41	1.50 (51)	0.72	1.20 (4)	5.41 (9)	16.8	18.0	1.57 (43)	$-$19.48	0.72	0.83	0.04	0.36
DDC10	1.41	1.48 (51)	0.62	1.15 (4)	4.63 (9)	17.1	17.7	1.38 (43)	$-$19.33	0.74	0.92	0.04	0.50
DDC15	1.41	1.47 (51)	0.51	1.12 (4)	3.72 (9)	17.6	17.1	1.14 (43)	$-$19.15	0.72	1.03	0.04	0.65
DDC17	1.41	1.41 (51)	0.41	1.08 (4)	2.85 (9)	18.6	17.2	9.10 (42)	$-$18.93	0.67	1.27	0.04	0.90
DDC20	1.41	1.38 (51)	0.30	1.03 (4)	2.12 (9)	18.7	17.2	6.65 (42)	$-$18.44	0.58	1.38	0.28	1.14
DDC22	1.41	1.30 (51)	0.21	9.80 (3)	1.40 (9)	19.6	17.5	4.47 (42)	$-$17.62	0.54	1.22	0.73	1.34
DDC25	1.41	1.18 (51)	0.12	8.56 (3)	7.37 (8)	21.0	19.8	2.62 (42)	$-$16.44	0.62	1.01	1.31	1.50
Pulsating Chandrasekhar-mass delayed-detonation models (PDDEL series)
PDDEL1	1.38	1.35 (51)	0.75	1.33 (4)	5.87 (9)	16.4	18.0	1.62 (43)	$-$19.49	0.76	0.95	0.07	0.51
PDDEL3	1.38	1.33 (51)	0.68	1.30 (4)	5.30 (9)	16.6	18.1	1.48 (43)	$-$19.39	0.76	1.04	0.08	0.62
PDDEL7	1.38	1.32 (51)	0.60	1.29 (4)	4.72 (9)	16.6	17.8	1.32 (43)	$-$19.25	0.72	1.12	0.10	0.77
PDDEL4	1.38	1.30 (51)	0.53	1.27 (4)	4.10 (9)	16.9	17.6	1.16 (43)	$-$19.09	0.69	1.22	0.14	0.90
PDDEL9	1.38	1.26 (51)	0.40	1.23 (4)	3.07 (9)	17.5	16.9	8.81 (42)	$-$18.70	0.63	1.28	0.27	1.10
PDDEL11	1.38	1.23 (51)	0.30	1.18 (4)	2.13 (9)	18.8	16.9	6.42 (42)	$-$18.12	0.60	1.21	0.56	1.27
PDDEL12	1.38	1.21 (51)	0.25	1.15 (4)	1.73 (9)	19.5	17.3	5.43 (42)	$-$17.76	0.60	1.14	0.76	1.35
Sub-Chandrasekhar-mass models (SCH series)
SCH7p0	1.15	1.40 (51)	0.84	1.51 (4)	7.91 (9)	15.6	15.8	1.85 (43)	$-$19.64	0.85	0.79	0.06	0.12
SCH6p5	1.13	1.35 (51)	0.77	1.44 (4)	7.43 (9)	15.8	16.2	1.71 (43)	$-$19.58	0.86	0.81	0.05	0.14
SCH6p0	1.10	1.30 (51)	0.70	1.39 (4)	6.91 (9)	16.0	16.7	1.57 (43)	$-$19.50	0.86	0.83	0.05	0.17
SCH5p5	1.08	1.23 (51)	0.63	1.32 (4)	6.31 (9)	16.3	17.3	1.42 (43)	$-$19.39	0.87	0.90	0.06	0.26
SCH5p0	1.05	1.16 (51)	0.55	1.25 (4)	5.67 (9)	16.4	17.3	1.25 (43)	$-$19.25	0.87	0.97	0.06	0.39
SCH4p5	1.03	1.09 (51)	0.46	1.20 (4)	4.95 (9)	16.6	17.5	1.08 (43)	$-$19.09	0.86	1.10	0.07	0.57
SCH4p0	1.00	1.03 (51)	0.38	1.16 (4)	4.23 (9)	16.5	17.3	9.01 (42)	$-$18.88	0.84	1.24	0.11	0.74
SCH3p5	0.98	9.71 (50)	0.30	1.14 (4)	3.50 (9)	16.4	16.5	7.34 (42)	$-$18.62	0.81	1.38	0.16	0.91
SCH3p0	0.95	9.19 (50)	0.23	1.11 (4)	2.73 (9)	16.5	15.2	5.76 (42)	$-$18.30	0.80	1.47	0.22	1.07
SCH2p5	0.93	8.67 (50)	0.17	1.08 (4)	2.08 (9)	16.4	14.9	4.36 (42)	$-$17.86	0.80	1.60	0.39	1.22
SCH2p0	0.90	8.14 (50)	0.12	1.05 (4)	1.52 (9)	15.9	14.6	3.17 (42)	$-$17.27	0.85	1.64	0.67	1.31
SCH1p5	0.88	7.59 (50)	0.08	1.02 (4)	1.09 (9)	15.4	13.4	2.26 (42)	$-$16.64	0.98	1.52	0.89	1.34

Table 2. Table 2: Summary of the model atoms used in our radiative-transfer calculations (see also Dessart & Hillier, 2010 ) . N f refers to the number of full levels, N s to the number of super levels, and N trans to the corresponding number of bound-bound transitions. The last column refers to the upper level for each ion treated. Note that n 𝑛 n w  2 W refers to a state with principal quantum number n 𝑛 n (all l 𝑙 l states combined into a single state), and spin 2.

Species	Nf	Ns	Ntrans	Upper Level
C i	26	14	120	2s2p3 3Po
C ii	26	14	98	2s24d 2D5/2
C iii	112	62	906	2s8f 1Fo
O i	101	38	743	2s22p3(4So)6f 3F2
O ii	143	63	1868	2s22p2(3P)5p 2P${}_{3/2}{}^{}{}_{}{}^o$
O iii	86	44	516	2s2p2(4P)3p 3Po
Ne i	139	70	1587	2s22p5(2P${}_{3/2}{}^{}{}_{}{}^o$)6d 2[5/2]${}_3{}^{}{}_{}{}^o$
Ne ii	91	22	1107	2s22p4(3P)4d 2P3/2
Ne iii	56	24	240	2s22p3(3So)4p 5P
Na i	71	22	1614	30w 2W
Mg ii	80	31	1993	30w 2W
Mg iii	99	31	775	2p57s 1Po
Al ii	44	26	171	3s5d 1D2
Al iii	80	31	2011	30w 2W
Si ii	59	31	354	3s27g 2G7/2
Si iii	61	33	312	3s5g 3G5
Si iv	48	37	405	10f 2Fo
S ii	324	56	8464	3s3p3(5So)4p 6P
S iii	98	48	840	3s3p2(2D)3d${}^3$P
S iv	67	27	397	3s3p(3Po)4p 2D5/2
Ar i	110	56	1541	3s23p5(2P${}_{3/2}{}^{}{}_{}{}^o$)7p 2[3/2]2
Ar ii	415	134	20197	3s23p4(3P1)7i 2[6]11/2
Ar iii	346	32	6901	3s23p3(2Do)8s 1Do
Ca ii	77	21	1736	3p630w 2W
Ca iii	40	16	108	3s23p55s 1Po
Ca iv	69	18	335	3s3p5(3Po)3d 4D${}_{1/2}{}^{}{}_{}{}^o$
Sc ii	85	38	979	3p63d4f${}^1$P${}_1{}^{}{}_{}{}^o$
Sc iii	45	25	235	7h 2H${}_{11/2}{}^{}{}_{}{}^o$
Ti ii	152	37	3134	3d2(3F)5p 4D${}_{7/2}{}^{}{}_{}{}^o$
Ti iii	206	33	4735	3d6f 3H${}_6{}^{}{}_{}{}^o$
Cr ii	196	28	4162	3d4(3G)4p x4G${}_{11/2}{}^{}{}_{}{}^o$
Cr iii	145	30	2359	3d3(2D2)4p 3D${}_3{}^{}{}_{}{}^o$
Cr iv	234	29	6354	3d2(3P)5p 4P${}_{5/2}{}^{}{}_{}{}^o$
Mn ii	97	25	236	3d4(5D)4s2 c5D4
Mn iii	175	30	3173	3d4(3G)4p y4H${}_{13/2}{}^{}{}_{}{}^o$
Fe i	136	44	1900	3d6(5D)4s4p x5F${}_3{}^{}{}_{}{}^o$
Fe ii	827	275	44 831	3d5(6S)4p2(3P) 4P1/2
Fe iii	698	83	36 807	3d5(2H)4d1K7
Fe iv	1000	100	72 223	3d4(3G)4f 4P${}_{5/2}{}^{}{}_{}{}^o$
Fe v	191	47	3977	3d3(4F)4d 5F3
Fe vi	433	44	14 103	3p5(2P)3d4(1S) 2P${}_{3/2}{}^{}{}_{}{}^o$
Fe vii	153	29	1753	3p5(2P)3d3(2D) 1P${}_1{}^{}{}_{}{}^o$
Co ii	2747	136	593 559	3d7(2D)6p 3P${}_1{}^{}{}_{}{}^o$
Co iii	3917	123	679 412	3d6(3D)6d 4P3/2
Co iv	314	37	9062	3d5(2P)4p 3P${}_1{}^{}{}_{}{}^o$
Co v	387	32	13 605	3d4(3F)4d 2H9/2
Co vi	323	28	9608	3d3(2D)4d${}^1$S
Co vii	319	31	9096	3p5(2P)4d(3F) 2D${}_{3/2}{}^{}{}_{}{}^o$
Ni ii	1000	59	51812	3d8(3F)7f 4I${}_{9/2}{}^{}{}_{}{}^o$
Ni iii	1000	42	66511	3d7(2D)4d 3S1
Ni iv	254	28	6317	3D6(1G1)4p 2G${}_{7/2}{}^{}{}_{}{}^o$
Ni v	183	46	3065	3d5(2D3)4p 3F${}_3{}^{}{}_{}{}^o$
Ni vi	314	37	9569	3d4(5D)4d 4F9/2
Ni vii	308	37	9225	3d3(2D)4d 3P2
Total	18 710	2534	1 717 141

Full text

††pubyear: 2017††pagerange: Evidence for sub-Chandrasekhar-mass progenitors of Type Ia supernovae at the faint end of the width-luminosity relation–B

1 Introduction

The finding that the peak magnitudes of Type Ia supernovae (SNe Ia) are tightly correlated with their post-maximum decline rate in several optical bands (Pskovskii, 1977; Phillips, 1993) revolutionized the use of these events as extra-galactic distance indicators and formed the cornerstone for the discovery of dark energy (Riess et al., 1998; Perlmutter et al., 1999).

Due to its determining role in observational cosmology, this so-called width-luminosity relation (hereafter WLR when considering the $B$-band) has received extensive theoretical attention. The slower post-maximum decline of SNe Ia with larger peak $M_{B}$ has been thought to arise from an increase in opacity with ${}^{56}\rm{Ni}$ mass, which affects the timescale for the radiation to escape the ejecta (e.g., Hoeflich et al., 1996; Pinto & Eastman, 2001; Mazzali et al., 2001). However, in contrast with observational studies that show a significantly weaker relation between peak magnitude and post-maximum decline rate for the integrated $(U)BVRI$ flux (e.g., Contardo et al., 2000), one would expect this relation to also hold for the bolometric luminosity.

A later study by Kasen & Woosley (2007) showed that the decay heating from ${}^{56}\rm{Ni}$ affects the onset of the iii$\rightarrow$ii recombination of iron-group elements (IGEs) in the ejecta, which causes a faster colour evolution for the least luminous events (in particular the $B-V$ colour), and hence a larger decline rate in $B$ relative to other bands. The WLR is then interpreted purely as a colour effect related to the level of Fe ii/Co ii line blanketing, independent of the evolution of the bolometric luminosity.

The mass of the exploding WD (hereafter $M_{\rm tot}$) is known to affect the bolometric evolution around maximum light (Pinto & Eastman, 2000). However, most theoretical studies of the WLR have either focused exclusively on Chandrasekhar-mass ($M_{\rm Ch}$) progenitors (e.g., Pinto & Eastman, 2001; Mazzali et al., 2001; Höflich et al., 2002; Kasen & Woosley, 2007), or excluded sub-$M_{\rm Ch}$ double-detonation models based on their apparent colour mismatch (see, e.g., Hoeflich et al., 1996). A generic problem of these sub-$M_{\rm Ch}$ models resides in the composition of the detonated He shell, which is needed to trigger a secondary detonation in the carbon-oxygen (C-O) core. The synthesis of ${}^{56}\rm{Ni}$ and other IGEs in this external shell causes discrepant colours and spectra at early times (Nugent et al., 1997; Kromer et al., 2010). Nonetheless, Sim et al. (2010) have shown that pure central detonations in single sub-$M_{\rm Ch}$ WDs without an outer He shell are able to reproduce the overall trend of the observed WLR, although it is still unclear whether sub-$M_{\rm Ch}$ models fare better than the standard $M_{\rm Ch}$ scenario, in particular for low-luminosity SNe Ia. However, observational studies have shown that the narrow light curves of fast-declining SNe Ia likely result from sub-$M_{\rm Ch}$ ejecta, and that the WLR is naturally explained as a relation between ejected mass and ${}^{56}\rm{Ni}$ mass (Scalzo et al., 2014a, b).

While the WLR imposes very strict constraints on SN Ia models, its reproduction by no means guarantees a high fidelity to observed events (see, e.g., Blondin et al., 2011). Moreover, the emphasis on the importance of the ionization balance in SN Ia ejecta to explain the WLR highlights the need for an accurate determination of the atomic level populations via a solution to the statistical equilibrium equations, i.e., a fully non-local-thermodynamic equilibrium (non-LTE) solution to the radiative transfer problem, often approximated in previous studies. Here we present 1D non-LTE time-dependent radiative transfer simulations of sub-$M_{\rm Ch}$ models for SNe Ia (Section 2), performed using the cmfgen radiative-transfer code (Hillier & Dessart, 2012).

To better illustrate the impact of a low ejecta mass on the radiative display, we confront these sub-$M_{\rm Ch}$ models to our previously-published $M_{\rm Ch}$ models, which include “standard” $M_{\rm Ch}$ delayed-detonations (DDC series; Blondin et al. 2013) and pulsational $M_{\rm Ch}$ delayed-detonations (PDDEL series; Dessart et al. 2014b). Despite the one-dimensional treatment of the explosion, these $M_{\rm Ch}$ models were found to capture the most salient features of more elaborate multi-dimensional simulations, in particular the asymptotic kinetic energy and chemical stratification (see, e.g., Seitenzahl et al., 2013). More importantly, they provided an excellent match to the observed properties of both normal and low-luminosity SNe Ia at maximum light (Blondin et al., 2013), and to the spectral evolution of normal events out to $\lesssim 100$ d past explosion (SN 2002bo, Blondin et al. 2015; SN 2005cf, Dessart et al. 2014c; SN 2011fe, Dessart et al. 2014b).

We start by analyzing the bolometric evolution of the sub-$M_{\rm Ch}$ models, highlighting the differences in rise time and post-maximum decline compared to $M_{\rm Ch}$ models at a given peak luminosity, i.e., similar ${}^{56}\rm{Ni}$ mass but different WD mass, $M_{\mathrm{tot}}$ (Section 3). We then discuss the impact of a sub-$M_{\rm Ch}$ ejecta on the maximum-light $B-V$ colour (Section 4), and on its post-maximum evolution for low-luminosity SNe Ia (Section 5). Last, we assess the merits of sub-$M_{\rm Ch}$ progenitors in reproducing the observed WLR all the way to the faint end (Section 6). We discuss possible progenitor scenarios leading to the detonation of a sub-$M_{\rm Ch}$ WD in Section 7, followed by our conclusions.

2 Sub-Chandrasekhar-mass models

The sub-$M_{\rm Ch}$ models studied here correspond to pure central detonations of WDs with a mass ranging between 0.88 $\rm{M}_{\odot}$ and 1.15 $\rm{M}_{\odot}$ (Table 1). As such they are similar to the sub-$M_{\rm Ch}$ models of Sim et al. (2010), albeit with a different numerical treatment and radiative-transfer post-processing (see below). The initial WD is assumed to be in hydrostatic equilibrium, and is composed of equal amounts of 12C and 16O by mass, with traces of 22Ne and solar composition for all other isotopes. Importantly, we do not consider the presence of an external He shell, required in the double-detonation scenario to trigger a detonation in the C-O core. Despite the artificial nature of this setup, the models are nonetheless intrinsically coherent and enable us to qualitatively assess the impact of a lower ejecta mass on the radiative display. The 1D hydrodynamical treatment of the explosion phase is analogous to that used in our previous SN Ia studies based on $M_{\rm Ch}$ delayed-detonation models (see Blondin et al., 2013, their section 2). The calculation is carried out until the ejecta mass shells reach a ballistic regime (homologous expansion), less than $\sim 20$ s past explosion. We smooth sharp variations in the composition and density profiles of the hydrodynamical input through convolution with a Gaussian kernel of width $\sigma=800$ $\rm{km\,s}^{-1}$ (see also Blondin et al., 2015).

One important difference is the absence of an initial deflagration phase, required to pre-expand the WD in $M_{\rm Ch}$ models and hence synthesize intermediate-mass elements during the secondary detonation phase. In sub-$M_{\rm Ch}$ models the burning can proceed directly in the form of a detonation wave, since the lower WD density ensures the burning will be incomplete in the outer ejecta layers with no prior pre-expansion. As a result, the initial WD structure sets the density at which the C-O fuel is burned, which is lower for lower WD masses. In particular, the amount of ${}^{56}\rm{Ni}$ synthesized in the explosion is directly proportional to the WD mass, as is the asymptotic kinetic energy (see Table 1). In our sub-$M_{\rm Ch}$ model series (SCH), the initial range of WD masses (0.88–1.15 $\rm{M}_{\odot}$) translates into a ${}^{56}\rm{Ni}$ yield between 0.08 $\rm{M}_{\odot}$ and 0.84 $\rm{M}_{\odot}$.

The lower WD density also limits the production of neutron-rich stable isotopes of IGEs (e.g., 54Fe, 58Ni) in the inner ejecta, while these are more abundantly produced during the deflagration phase of $M_{\rm Ch}$ models. In the outer ejecta layers, however, the burning proceeds at similar densities in both the sub-$M_{\rm Ch}$ and $M_{\rm Ch}$ models considered here, resulting in similar abundance profiles. At a given ${}^{56}\rm{Ni}$ yield, the sub-$M_{\rm Ch}$ models can thus be considered as low-mass analogs of the $M_{\rm Ch}$ models when discussing their radiative properties at early times, including the maximum-light phase discussed in the present paper.

The long-term evolution is computed with the 1D, time-dependent, non-LTE radiative-transfer code cmfgen (Hillier & Dessart, 2012; Dessart et al., 2014c), which includes the treatment of non-local energy deposition and non-thermal processes (e.g., Dessart et al., 2014a). Apart from the two-step {}^{56}\rm{Ni}$$\rightarrow$${}^{56}\rm{Co}$$\rightarrow$${}^{56}\rm{Fe} decay chain, we treat eight additional two-step decay chains associated with 37K, 44Ti, 48Cr, 49Cr, 51Mn, 52Fe, 55Co, 57Ni, and a further six one-step decay chains associated with 41Ar, 42K, 43K, 43Sc, 47Sc, 61Co (see Dessart et al., 2014c). All atom/ion level populations are determined explicitly through a solution of the time-dependent statistical equilibrium equations, which are coupled to the energy and radiative-transfer equations. We consider the following ions: C i–iii, O i–iii, Ne i–iii, Na i, Mg ii–iii, Al ii–iii, Si ii–iv, S ii–iv, Ar i–iii, Ca ii–iv, Sc ii–iii, Ti ii–iii, Cr ii–iv, Mn ii–iii, Fe i–vii, Co ii–vii, and Ni ii–vii. The number of levels considered for each ion is given in Appendix A, Table 2. We assume a constant effective Doppler width (including both thermal and turbulent velocities) of 50 $\rm{km\,s}^{-1}$ for our line-absorption profiles. While this value largely overestimates the true effective Doppler width (likely a few $\rm{km\,s}^{-1}$ at most) and hence the ejecta opacity, we have run extensive tests imposing an effective Doppler width of 10 $\rm{km\,s}^{-1}$ and found a negligible impact (at the few per cent level) on the predicted rise times and post-maximum decline rates. The output of such calculations are light curves and spectra that can be directly confronted to SN Ia observations. Our assumption of spherical symmetry allows for the most elaborate radiative-transfer treatment to date of such sub-$M_{\rm Ch}$ models.

In what follows we focus on the properties most relevant to the WLR, while the full SCH model series will be presented in more detail in a forthcoming paper.

3 Bolometric rise and post-maximum decline

Pinto & Eastman (2000) have emphasized the impact of the total mass111In this paper we only consider explosions that completely unbind the WD, hence the ejecta mass is equal to the WD mass. on the bolometric light curves of SNe Ia, through its effect on the ejecta density and $\gamma$-ray escape. A decrease in $M_{\mathrm{tot}}$ results in a lower column depth which favours a more rapid release of radiation at early times and an enhanced $\gamma$-ray escape fraction past maximum light, leading to a shorter rise time and a faster post-maximum decline. This is indeed the case for the sub-$M_{\rm Ch}$ models presented here, which display systematically shorter bolometric rise times than the $M_{\rm Ch}$ models for a given ${}^{56}\rm{Ni}$ mass (Fig. 1, top panel).

Due to the shorter rise time, the instantaneous rate of energy deposition by radioactive decays ($L_{\rm decay}$) at maximum light is larger (as is the peak luminosity, $L_{\rm bol}$), resulting in a more efficient heating of the ejecta compared to a $M_{\rm Ch}$ model with the same ${}^{56}\rm{Ni}$ mass. The lower $M_{\mathrm{tot}}$ exacerbates this effect, as the specific heating rate at maximum light (\equiv L_{\rm decay}/$$M_{\mathrm{tot}}, noted $\dot{e}_{\rm decay}$ in Table 1) is even larger. This latter parameter is key in explaining the bluer maximum-light colours of the sub-$M_{\rm Ch}$ models (see Section 4).

Moreover, all of the sub-$M_{\rm Ch}$ models have a bolometric decline rate $\Delta M_{15}(\rm{bol})\geq 0.80$ mag (and up to 0.98 mag), where the $M_{\rm Ch}$ models are confined to smaller values (0.54–0.76 mag; see Fig. 1, bottom panel). This larger bolometric decline for the sub-$M_{\rm Ch}$ models will naturally affect all photometric bands, including the $B$-band on which the WLR is based (Section 6).

That said, both the $M_{\rm Ch}$ and sub-$M_{\rm Ch}$ models share a weak dependence of their post-maximum bolometric decline rate on the ${}^{56}\rm{Ni}$ mass: the variation in $\Delta M_{15}(\rm{bol})$ is less than $\sim 0.2$ mag across a factor $\sim$7–10 variation in ${}^{56}\rm{Ni}$ mass. When excluding the faintest sub-$M_{\rm Ch}$ model (SCH1p5), the difference in $\Delta M_{15}(\rm{bol})$ drops to a mere 0.07 mag for the SCH series. This contrasts with the far stronger dependence of the post-maximum $B$-band decline rate $\Delta M_{15}(B)$ on ${}^{56}\rm{Ni}$ mass (and hence peak luminosity; see Table 1), and lends support to the interpretation of the WLR as a colour effect as opposed to an increase of the diffusion time with ${}^{56}\rm{Ni}$ mass (see Kasen & Woosley, 2007).

In fact, the bolometric evolution more naturally follows an opposite trend to the ($B$-band) WLR, the more luminous models declining more rapidly (this is especially true in the ${}^{56}\rm{Ni}$ mass range 0.2–0.6 $\rm{M}_{\odot}$; see Fig. 1). The bolometric decline rate is related to the magnitude and rate of change of the $\gamma$-ray escape fraction past maximum light, which is larger for higher ${}^{56}\rm{Ni}$ mass (see Stritzinger et al., 2006). This is in part due to the larger outward extent of the ${}^{56}\rm{Ni}$ distribution (see the $\varv_{99}(^{56}\rm{Ni})$ column in Table 1) which favours the earlier and more rapid escape of $\gamma$-rays, thereby limiting the amount of decay energy actually deposited in the ejecta (see also Pinto & Eastman, 2001). This effect is modulated by the release of stored radiation past maximum light, which leads to a secondary bolometric maximum whose timing can affect the value of $\Delta M_{15}(\rm{bol})$.

4 Maximum-light colours

Type Ia supernovae obey a strong relation between their peak $M_{B}$ and their $B-V$ colour at maximum light, brighter events displaying bluer colours (e.g., Tripp, 1998). This relation simply reflects the larger magnitude of decay heating from ${}^{56}\rm{Ni}$ for more luminous events, and holds for both $M_{\rm Ch}$ and sub-$M_{\rm Ch}$ models (Fig. 2).

Nonetheless, one would expect the greater specific heating rate ($\dot{e}_{\rm decay}$; see previous section) for the sub-$M_{\rm Ch}$ models at a given ${}^{56}\rm{Ni}$ mass to result in even bluer colours. As seen in Fig. 2, this effect is present and more pronounced at the faint end (up to $\sim 0.6$ mag bluer $B-V$ colour at a given ${}^{56}\rm{Ni}$ mass), where the relative difference in $\dot{e}_{\rm decay}$ is largest (see Table 1). In contrast, all models brighter than $M_{B}=-19$ mag at peak have very similar $B-V$ colours at maximum ($\lesssim 0.03$ mag standard deviation).

The sub-$M_{\rm Ch}$ models follow the observed relation all the way to the faint end (peak $M_{B}\gtrsim-17$ mag), while the $M_{\rm Ch}$ models are systematically too red at a given peak $M_{B}$. This is also the case for the least luminous sub-$M_{\rm Ch}$ model (SCH1p5), in which the ${}^{56}\rm{Ni}$ mass is too low ($<0.08$ $\rm{M}_{\odot}$, where all the other models have $\gtrsim 0.12$ $\rm{M}_{\odot}$ of ${}^{56}\rm{Ni}$; see Table 1) to efficiently heat the spectrum-formation layers at maximum light. In fact, the only model to have an even redder $B-V$ colour at maximum is the $M_{\rm Ch}$ model DDC25, which has the lowest specific heating rate of all ($\dot{e}_{\rm decay}<10^{9}$ erg s*-1* g*-1*).

5 The post-maximum colour evolution of low-luminosity

SNe Ia

We focus on three models with similar ${}^{56}\rm{Ni}$ mass (within $\pm 0.02$ $\rm{M}_{\odot}$ of 0.23 $\rm{M}_{\odot}$) to better illustrate the impact of $M_{\mathrm{tot}}$ on the post-maximum colour evolution of low-luminosity SNe Ia: the standard $M_{\rm Ch}$ delayed-detonation model DDC22, the pulsational $M_{\rm Ch}$ delayed-detonation model PDDEL12, and the sub-$M_{\rm Ch}$ model SCH3p0, corresponding to the pure central detonation of a 0.95 $\rm{M}_{\odot}$ C-O WD. Their $B$-band light curves (normalized to the same peak magnitude) and $B-V$ colour evolution is shown in Fig. 3. One clearly notes the faster evolution of the $B$-band light curve for the sub-$M_{\rm Ch}$ model (in part due to the faster bolometric evolution; see Fig. 3 inset), with a $\sim 2$ d shorter rise time and a $>0.3$ mag larger $\Delta M_{15}(B)$ [see Table 1], accompanied by a faster evolution of the $B-V$ colour around maximum light. This results in a gradual decrease in the difference in $B-V$ colour compared to the $M_{\rm Ch}$ models, from $>0.5$ mag at $B$-band maximum to $<0.3$ mag 15 days later.

The greater specific heating rate of the sub-$M_{\rm Ch}$ model ($\dot{e}_{\rm decay}$; see Section 3) results in a relatively modest increase of the temperature in the spectrum-formation region at maximum light compared to the $M_{\rm Ch}$ models (on the order of 10–15%). Combined with the lower ejecta density, this slightly larger temperature is nonetheless sufficient to induce a change in the mean ionization state of the gas, enhancing the iii/ii ionization ratio of IGEs (Sc, Ti, Cr, Fe, and Co) that would otherwise effectively block the flux in the $B$-band and emit in redder bands in their once-ionized state (especially Fe ii; see Appendix B).

The bluer $B-V$ colour of the sub-$M_{\rm Ch}$ model thus results both from an overall shift of the SED to the blue (akin to a purely thermal effect) and a modulation of line-blanketing from changes in ionization. Both effects are clearly visible in the maximum-light spectra shown in Fig. 4. By 15 days past $B$-band maximum, the temperature in the spectrum-formation region of the sub-$M_{\rm Ch}$ model has decreased sufficiently to enhance the absorption by singly-ionized IGEs to a level comparable to that in the $M_{\rm Ch}$ models. The differences in $B$-band flux as well as in the overall SED shape are greatly reduced at this phase, although the sub-$M_{\rm Ch}$ model remains slightly bluer, thereby limiting the $B$-band post-maximum decline rate.

6 The width-luminosity relation

The WLR for the $M_{\rm Ch}$ and sub-$M_{\rm Ch}$ models is shown in Fig. 5. At the bright end (peak $M_{B}\lesssim-18.5$ mag), both $M_{\rm Ch}$ and sub-$M_{\rm Ch}$ models follow the observed WLR (grey data points in Fig. 5), albeit with slightly steeper slopes (similar for the DDC and SCH series, and somewhat shallower for the PDD series). At the faint end (peak $M_{B}\gtrsim-18.5$ mag), the $M_{\rm Ch}$ models display a turnover to an anti WLR, reaching a maximum $\Delta M_{15}(B)$ value of $\sim 1.4$ mag (DDC series) and $\sim 1.3$ mag (PDDEL series). These models correspond to the same ones that have a too red $B-V$ colour at maximum light in Fig. 2 (highlighted with red circles in both figures), due to the less efficient heating of the spectrum-formation region from the lower $\dot{e}_{\rm decay}$ (Section 4). Combined with a modest post-maximum evolution of their $B-V$ colour, the resulting $B$-band decline rate is then too low for the $M_{\rm Ch}$ models to reproduce the faint end of the WLR despite their low luminosity.

In the sub-$M_{\rm Ch}$ models, the more rapid colour evolution around maximum light combined with a larger bolometric decline rate result in a higher $\Delta M_{15}(B)$ for the same ${}^{56}\rm{Ni}$ mass. Consequently, only the sub-$M_{\rm Ch}$ models extend beyond $\Delta M_{15}(B)=1.4$ mag. This confirms the WD mass as an essential parameter in producing rapidly-declining SNe Ia at the faint end of the WLR. In particular, the sub-$M_{\rm Ch}$ model SCH2p0 ($M_{\rm tot}=0.90$ $\rm{M}_{\odot}$) has a $\sim 0.6$ mag larger $\Delta M_{15}(B)$ value than the $M_{\rm Ch}$ model with the same ${}^{56}\rm{Ni}$ mass, DDC25 ($M(^{56}\rm{Ni})=0.12$ $\rm{M}_{\odot}$ for both).

We note, however, that the sub-$M_{\rm Ch}$ models also transition to an anti-WLR, albeit at much fainter magnitudes (peak $M_{B}\gtrsim-17$ mag), and that they fail to reach the highest observed $\Delta M_{15}(B)$ values. We speculate that small variations in the M(^{56}\rm{Ni})/$$M_{\mathrm{tot}} ratio or in the ${}^{56}\rm{Ni}$ distribution within the ejecta (due to explosion asymmetries or large-scale mixing) could enhance the rate of $\gamma$-ray escape and lead to a better agreement with the data in this respect. We will discuss this in more detail in a companion paper in which we compare the SCH2p0 model to the low-luminosity SN 1999by.

The sub-$M_{\rm Ch}$ models of Sim et al. (2010) yield systematically larger $\Delta M_{15}(B)$ values compared to our SCH models at a given peak $M_{B}$, by up to $\sim 0.6$ mag for their two most luminous models and $\sim 0.3$ mag for their two least luminous models (Fig. 5). Their most luminous model ($M(^{56}\rm{Ni})/M_{\rm tot}=0.81/1.15$ $\rm{M}_{\odot}$, similar to our SCH7p0 model) declines too slowly for its luminosity, with $\Delta M_{15}(B)\approx 1.3$ mag for a peak $M_{B}\approx-20$, where SNe Ia of comparable luminosity (and our SCH7p0 model) have $\Delta M_{15}(B)\approx 0.8$ mag. Their least luminous model, however, is in better agreement with the observed WLR, with $\Delta M_{15}(B)\approx 1.8$ mag where our SCH1p5 model only has $\Delta M_{15}(B)\approx 1.5$ mag for the same peak $M_{B}\approx-16.6$. Differences in the nucleosynthetic post-processing of the explosion models and in the radiative-transfer treatment are the likely causes of these offsets (Sim, priv. comm.).

As was the case for the bolometric light curves (Section 3), the faster $B$-band post-maximum decline of the sub-$M_{\rm Ch}$ models is associated with a shorter $B$-band rise time, as observed for low-luminosity SNe Ia (Ganeshalingam et al., 2011). To match the current practice of observers, the time reference is set to maximum light. Studies of the WLR thus ignore the scatter in rise times for SNe Ia with different peak $M_{B}$. While determining the precise time of explosion is observationally complicated, the recent examples of SN 2011fe (Nugent et al., 2011), SN 2013dy (Zheng et al., 2013), and three SN Ia candidates discovered during the Kepler mission (Olling et al., 2015) illustrate the potential of high-cadence searches to constrain the rise time. Future studies of the WLR will then need to consider the pre-maximum rise as well as the post-maximum decline when distinguishing between fast and slow light-curve evolution.

7 Discussion and Conclusions

The predicted observational signatures of pure central detonations in sub-$M_{\rm Ch}$ WDs corroborate a number of distinctive properties of low-luminosity SNe Ia. At a given ${}^{56}\rm{Ni}$ mass, the lower ejecta mass compared to $M_{\rm Ch}$ models results in a faster bolometric evolution around maximum light, which affects all photometric bands. Furthermore, the larger M(^{56}\rm{Ni})/$$M_{\mathrm{tot}} ratio leads to a bluer $B-V$ colour at maximum light and a higher post-maximum $B$-band decline rate, $\Delta M_{15}(B)$, in better agreement with the observed WLR. Conversely, the lower specific heating rate in the $M_{\rm Ch}$ models considered here results in a $B-V$ colour at maximum that is too red compared to observations (through excess absorption by singly-ionized IGEs and weaker emission in the blue), and hence a more modest post-maximum colour evolution that imposes an upper limit on the $\Delta M_{15}(B)$ value.

Even then, the sub-$M_{\rm Ch}$ models fail to match the fastest decline rates observed for low-luminosity, 91bg-like SNe Ia. Variations in the M(^{56}\rm{Ni})/$$M_{\mathrm{tot}} ratio or in the ${}^{56}\rm{Ni}$ distribution within the ejecta could enhance the rate of $\gamma$-ray escape and result in a more efficient cooling of the spectrum-formation region at these phases. Sim et al. (2010) also find that a modest change in the initial WD composition (replacing 7.5% by mass of 12C with 22Ne) has a non-negligible impact on the resulting $\Delta M_{15}(B)$ value. We note that a similar M(^{56}\rm{Ni})/$$M_{\mathrm{tot}} ratio as our low-luminosity sub-$M_{\rm Ch}$ models can in principle be achieved in the context of $M_{\rm Ch}$ models if the explosion is too weak to completely unbind the WD, leaving behind a bound remnant (see Fink et al., 2014).

The sub-$M_{\rm Ch}$ (and $M_{\rm Ch}$) models studied here remain somewhat artificial for genuine quantitative predictions to be made, yet the level of sophistication enabled by the 1D treatment of the radiative transfer enables us to qualitatively assess the impact of a lower ejecta mass on the ionization balance for a given ${}^{56}\rm{Ni}$ mass, which is crucial in explaining the observed WLR. Ultimately, our results will need to be confirmed by more realistic, multi-dimensional models, although radiative-transfer simulations with the same level of sophistication as ours are currently not feasible in 3D (see, e.g., Sim et al., 2013).

But do such low-mass explosions occur in Nature? Detonations of sub-$M_{\rm Ch}$ WDs require an external trigger, usually via thermal instabilities in a layer of accreted He (double detonation mechanism; see, e.g., Woosley & Weaver 1994). This layer can detonate at a lower mass than previously thought (Bildsten et al., 2007), but the resulting IGEs synthesized at high velocities cause discrepant colours and spectra at early times (Kromer et al., 2010). More recent calculations by Shen & Moore (2014), however, show that a detonation in a 0.005 $\rm{M}_{\odot}$ He shell on a 1.0 $\rm{M}_{\odot}$ WD only produces 28Si and 4He, and that significant 44Ti/48Cr production only occurs for shells more massive than $\sim 0.01$ $\rm{M}_{\odot}$. Double detonations thus remain an attractive scenario for fast-declining SNe Ia for which empirical determinations of the ejecta mass indicate sub-$M_{\rm Ch}$ progenitors (Scalzo et al., 2014a, b).

Likewise, He-ignited violent mergers (Pakmor et al., 2013), where the dynamical detonation of 0.01$\rm{M}_{\odot}$ of He is sufficient to trigger a detonation in the C-O core, also synthesize very little IGEs in the outer layers ($\sim 10^{-8}$ $\rm{M}_{\odot}$). In particular, Pakmor et al. (2013) argue that low-luminosity SNe Ia result from the violent merger of a C-O+He WD system, which is less massive (and hence results in a more rapid light-curve evolution) than double C-O WD mergers. The predicted rate of such systems can in principle account for the observed SN Ia rate (Ruiter et al., 2011), and their long delay times would corroborate the association of low-luminosity SNe Ia with older stellar populations (e.g., Howell, 2001).

Another possibility, suggested by van Kerkwijk et al. (2010), involves the merger of two low-mass WDs whose combined mass is less than the Chandrasekhar limit (a $0.6+0.6$ $\rm{M}_{\odot}$ merger in their paper). The explosion does not occur during the merger event, but at a later time via compressional heating through accretion of the thick disk from the merger remnant (see Lorén-Aguilar et al., 2009). However, this scenario is unlikely to hold for a combined merger mass $\lesssim 1.2$ $\rm{M}_{\odot}$, given the high temperature and density conditions required to ignite carbon (van Kerkwijk, priv. comm.), and hence would fail to reproduce the fast-declining SNe Ia at the faint end of the WLR, which we argue result from ejecta masses $\lesssim 1$ $\rm{M}_{\odot}$. In this respect, head-on collisions of two 0.5 $\rm{M}_{\odot}$ WDs represent an interesting alternative: Kushnir et al. (2013) find that such a model yields 0.11 $\rm{M}_{\odot}$ of ${}^{56}\rm{Ni}$, comparable to our SCH2p0 model. However, most studies of this progenitor channel conclude that WD collisions can account for at most a few per cent of the observed SN Ia rate (see, e.g., Papish & Perets, 2016).

Regardless of the precise ignition mechanism, the results presented in this paper strongly suggest that explosions of sub-$M_{\rm Ch}$ WDs are a viable scenario for fast-declining SNe Ia at the faint end of the WLR, whose colours cannot be reproduced by $M_{\rm Ch}$ models. In several upcoming companion papers we will present an in-depth study of the sub-$M_{\rm Ch}$ model SCH2p0 compared to the low-luminosity SN 1999by, and explore more generally the feasibility of such sub-$M_{\rm Ch}$ models to reproduce the observed properties of more luminous events, to address the question of multiple progenitor channels for Type Ia supernovae.

Acknowledgements

SB acknowledges useful discussions with Inma Domínguez, Stuart Sim and Marten van Kerkwijk. Part of this work was realized during a one-month visit of SB to ESO as part of the ESO Scientific Visitor Programme. LD and SB acknowledge financial support from the Programme National de Physique Stellaire (PNPS) of CNRS/INSU, France. DJH acknowledges support from STScI theory grant HST-AR-12640.01, and NASA theory grant NNX14AB41G. This work was granted access to the HPC resources of CINES under the allocation c2014046608 made by GENCI (Grand Equipement National de Calcul Intensif). This work also utilized computing resources of the mesocentre SIGAMM, hosted by the Observatoire de la Côte d’Azur, Nice, France.

Appendix A Model atoms

Table 2 gives the number of levels (both super-levels and full levels; see Hillier & Miller 1998 and Dessart & Hillier 2010 for details) for the model atoms used in the radiative-transfer calculations presented in this paper.

Oscillator strengths for CNO elements were originally taken from Nussbaumer & Storey (1983, 1984). These authors also provide transition probabilities to states in the ion continuum. The largest source of oscillator data is from Kurucz (2009)222Data are available online at http://kurucz.harvard.edu; its principal advantage over many other sources (e.g., Opacity Project) is that LS coupling is not assumed. More recently, non-LS oscillator strengths have become available through the Iron Project (Hummer et al., 1993), and work done by the atomic-data group at Ohio State University (Nahar, 2010). Other important sources of radiative data for Fe include Becker & Butler (1992, 1995a, 1995b), Nahar (1995). Energy levels have generally been obtained from the National Institute of Standards and Technology. Collisional data is sparse, particularly for states far from the ground state. The principal source for collisional data among low lying states for a variety of species is the tabulation by Mendoza (1983); other sources include Berrington et al. (1985), Lennon et al. (1985), Lennon & Burke (1994), Shine & Linsky (1974), Tayal (1997a, b), Zhang & Pradhan (1995b; 1995a; 1997). Photoionization data is taken from the Opacity Project (Seaton, 1987) and the Iron Project (Hummer et al., 1993). Unfortunately Ni and Co photoionization data is generally unavailable, and we have utilized crude approximations. Charge exchange cross-sections are from the tabulation by Kingdon & Ferland (1996).

Appendix B Contribution of individual ions to the total optical

flux

Figure 6 reveals the contribution of individual ions (bottom panels) to the full optical synthetic spectra of the standard $M_{\rm Ch}$ delayed-detonation model DDC22, the pulsational $M_{\rm Ch}$ delayed-detonation model PDDEL12, and the sub-$M_{\rm Ch}$ model SCH3p0 ($M_{\rm tot}=0.95\,\rm{M}_{\odot}$), all of which have a similar ${}^{56}\rm{Ni}$ mass of $0.23\pm 0.02$ $\rm{M}_{\odot}$, at $B$-band maximum light and 15 days later. Only ions that impact the flux at the $>10$ per cent level at either phase are shown.