Optical Color of Type Ib and Ic Supernovae and Implications for Their Progenitors

We construct SN models from helium-rich and helium-poor progenitors for various M_ej, different amounts of ⁵⁶Ni, and different ⁵⁶Ni distributions in the SN ejecta to explore the effects of these factors on the B − V color.

We still do not fully understand the evolutionary paths of massive stars toward SNe Ib or Ic and their mass-loss histories (see Yoon 2017 for a detailed discussion). In this study, we do not aim to investigate pre-SN evolution but instead consider the diverse physical properties of SN Ib/Ic progenitors at the pre-SN stage to investigate their impact on the SN light curves and colors. For this purpose, we adopt different mass-loss rates from helium stars to construct both helium-rich and helium-poor models having various final masses as explained below. All the progenitor models have solar metallicity (i.e., Z = 0.02) and are evolved until the infall velocity of the iron core exceeds 1000 km s⁻¹ using the MESA code (Paxton et al. 2011, 2013,2015, 2018, 2019).

The helium-rich progenitor models contain more than 0.6 M_⊙ of helium (He models) and the helium-poor progenitor models contain less than 0.2 M_⊙ of helium (CO models). Each progenitor model is named to indicate their progenitor type and total mass. For example, He3.1 refers to a helium-rich progenitor with a total mass of 3.1 M_⊙. We use the same notation when referring to the SN model from the respective progenitor model. See Table 2 for details on the progenitor properties.

He3.1 and He3.9 are taken from the binary models Sm11p200d and Sm15p50 of Yoon et al. (2017), respectively. He3.5, He4.2, and He5.6 are obtained by evolving pure helium star models having initial masses of, respectively, 4.0, 5.0, and 7.0 M_⊙ using the Wolf–Rayet mass-loss rate prescription of Nugis & Lamers (2000). He5.3 is obtained by evolving a 9.0 M_⊙ helium star with the Wolf–Rayet mass-loss rate prescription of Yoon (2017). CO3.2, CO3.6, and CO4.2 are obtained by evolving 7.0, 8.0, and 10.0 M_⊙ helium stars, respectively, with the Yoon (2017) mass-loss prescription until core-helium exhaustion, and with an artificially enhanced mass-loss rate (i.e., 500 times the standard Nugis & Lamers 2000 rate) thereafter until core collapse. The CO3.9 and CO5.9 models are constructed in a similar fashion by evolving, respectively, 7.0 and 10.0 M_⊙ helium stars, but the Nugis & Lamers (2000) mass-loss prescription is used during the core-helium burning phase. CO5.3 is calculated with a 11.0 M_⊙ helium star model using the Yoon (2017) mass-loss prescription throughout the whole evolution. Some example MESA in list files for constructing these progenitor models can be found at ZENODO:10.5281/zenodo.7797106.

The final masses span M_tot = 3.1 ⋯ 5.7 M_⊙, and the corresponding ejecta masses span M_ej = 1.7 ⋯ 4.1 M_⊙, with the assumption of this study that the mass cut in the SN explosion is located at the outer boundary of the iron core. This range encompasses a large fraction of the inferred ejecta masses of M_ej ≈ 1.0 ⋯ 5.0 M_⊙ of SNe Ib/Ic (e.g., Drout et al. 2011; Cano 2013; Lyman et al. 2016; Taddia et al. 2018; Zheng et al. 2022). External material having a mass of about 1% of the ejecta mass and an extent of 10¹⁴ cm is attached to each progenitor model to avoid acceleration of the forward shock to a relativistic velocity because relativistic effects cannot be properly handled in the version of the STELLA code that is used for this study (see Yoon et al. 2019 for more discussion on this.). This external matter does not affect the light curve after about one day from the shock breakout, and thus does not affect the focus of our study.

We use the STELLA code that is a one-dimensional multigroup radiative-hydrodynamics code for calculating SN light curves in multiple bands. The SN explosion is simulated by a thermal bomb at the mass cut, and time-dependent radiative transfer equations and hydrodynamics equations are solved simultaneously for approximately 100 frequency bins. For more detailed information, see Blinnikov & Tolstov (2011). See also Yoon et al. (2019) for a recent example of SNe Ib/Ic models calculated with the STELLA code.

We consider two kinetic energies, four ⁵⁶Ni masses, and six ⁵⁶Ni distributions to construct the SN models for a given progenitor model. Kinetic energies of 1B and 2B are obtained by adjusting the explosion energy. We do not calculate explosive nucleosynthesis but instead radioactive ⁵⁶Ni is artificially introduced into the ejecta as in Yoon et al. (2019). The considered ⁵⁶Ni masses are M_Ni = 0.07 M_⊙, 0.14 M_⊙, 0.20 M_⊙, and 0.25 M_⊙. These sets of kinetic energies and ⁵⁶Ni masses cover most of the observed values of ordinary SNe Ib/Ic (e.g., Drout et al. 2011; Cano 2013; Lyman et al. 2016; Taddia et al. 2018; Zheng et al. 2022).

The ⁵⁶Ni mass fraction in the SN ejecta is set to follow either a step distribution or a Gaussian distribution. For a step distribution, we adopt ${X}_{\mathrm{Ni}}({M}_{r})=\tfrac{{M}_{\mathrm{Ni}}}{{f}_{{\rm{m}}}({M}_{\mathrm{tot}}-{M}_{\mathrm{cut}})}$ for M_cut < M_r < f_m(M_tot − M_cut) + M_cut and X_Ni(M_r ) = 0 elsewhere. Here, M_r is the mass coordinate, M_cut is the mass cut which corresponds to the outer boundary of the iron core, M_tot is the total mass of the progenitor, and f_m is a free parameter that controls the shape of the ⁵⁶Ni distribution in the ejecta. For a Gaussian distribution, we adopt ${X}_{\mathrm{Ni}}({M}_{r})=A\exp \left(-{\left[\tfrac{{M}_{r}-{M}_{\mathrm{cut}}}{{f}_{{\rm{m}}}({M}_{\mathrm{tot}}-{M}_{\mathrm{cut}})}\right]}^{2}\right)$ where A is a normalization factor. In the study, f_m = 0.15, 0.5, and 1.0 are considered for both the step and Gaussian ⁵⁶Ni distributions to account for the uncertainty in ⁵⁶Ni mixing. The value of f_m = 0.15 (f_m = 0.5) corresponds to weak (moderate) ⁵⁶Ni mixing. The value of f_m = 1.0 means complete or almost full ⁵⁶Ni mixing for the step and Gaussian distributions, respectively. In Figure 2, we present chemical profiles in the ejecta of the He4.2 and CO4.2 models for various ⁵⁶Ni distributions, as an example. Note that the gamma-ray transfer in STELLA is treated in one-group approximation calibrated against full Monte Carlo transport and, in principle, the deposition of the gamma-ray energy may take place far away from the birthplace of gamma photons in ⁵⁶Ni and ⁵⁶Co decays. In practice this nonlocal heating occurs only at late phases when the mean free path of gamma photons becomes large due to low density. In this study, however, we focus on relatively early phases where the mean free path of gamma photons is short and the heating is effectively local.

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The cumulative distributions of ${(B-V)}_{V\max }$ predicted by the models are presented in Figure 3 for both the step and Gaussian ⁵⁶Ni distributions. Although the SN parameters (i.e., E_K, M_ej, and M_Ni) of our models are chosen to reproduce ordinary SNe Ib/Ic, a precise quantitative comparison of the predicted ${(B-V)}_{V\max }$ with the observation would require a proper consideration of the exact distributions of these parameters within the selected SN Ib/Ic observation sample. However, there exist great uncertainties in the observationally inferred values of these parameters as discussed above and we limit our discussion to a qualitative comparison between the model predictions and the observed values of ${(B-V)}_{V\max }$.

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The He models with the step ⁵⁶Ni distribution have a bluer color (${\overline{(B-V)}}_{V\max }=0.40\mbox{--}0.43$) than the CO models (${\overline{(B-V)}}_{V\max }\sim 0.60$), except for the fully mixed case (f_m = 1.0) that leads to ${\overline{(B-V)}}_{V\max }\simeq 1.0$ for both the He and CO models. The systematic color difference between the He and CO models is ${\rm{\Delta }}{\overline{(B-V)}}_{V\max }\approx$ 0.21 (0.16) for f_m = 0.15 (f_m = 0.5), which is comparable to the observed color difference between the SNe Ib and Ic (i.e., ${\rm{\Delta }}{\overline{(B-V)}}_{V\max }=$ 0.10 and 0.22 for the full and minimally reddened samples, respectively).

The He models with a Gaussian ⁵⁶Ni distribution also have a bluer color (${\overline{(B-V)}}_{V\max }\approx$ 0.45 and 0.88) than the CO models (${\overline{(B-V)}}_{V\max }\approx$ 0.65 and 1.04) when the ⁵⁶Ni is not fully mixed (f_m = 0.15 and 0.5), and the differences between them are ${\rm{\Delta }}{\overline{(B-V)}}_{V\max }\approx$ 0.20 and 0.16, which are also comparable to the observed color difference. Compared to the models with the step ⁵⁶Ni distribution, the models with a Gaussian ⁵⁶Ni distribution were systematically redder for a given f_m. The difference is most prominent when f_m = 0.5, showing a color difference of ${\rm{\Delta }}{\overline{(B-V)}}_{V\max }\approx$ 0.45 between the step and the Gaussian ⁵⁶Ni distributions. This is because the ⁵⁶Ni abundance in the outer layer of the ejecta is higher in the models with a Gaussian distribution for a given f_m value. See Section 4.3 for a detailed discussion on the effect of mixing. On the other hand, the results with f_m = 1.0 are similar to the case of the step ⁵⁶Ni distribution because ⁵⁶Ni is fully mixed throughout the ejecta for both cases.

SN ejecta with more ⁵⁶Ni for given ejecta mass and explosion energy would have a larger thermal energy due to the extra heating. Figure 4 compares the evolution of the Rosseland mean opacity and gas temperature in the ejecta of the He4.2 model with f_m = 0.5 for two different ⁵⁶Ni masses, M_Ni = 0.07 M_⊙ and 0.25 M_⊙. The ejecta with M_Ni = 0.25 M_⊙ are hotter at every mass zone for a given epoch compared to the case of M_Ni = 0.07 M_⊙. Accordingly, the ejecta are more opaque because of more free electrons in the outer layers of the ejecta, and the photosphere retreats more slowly. The photospheric gas temperature remains in-between $3.9\lt \mathrm{log}T[{\rm{K}}]\lt 4.1$ at t = 8–29 days. When the V-band peak appears at t = 27.5 days, the effective temperature at the Rosseland mean photosphere (T_eff) and the blackbody fit temperature of the SN spectrum (T_bb), which is more relevant to the SN color than the effective temperature, are T_eff = 7820 K and T_bb = 8320 K, respectively. The corresponding B − V is ${(B-V)}_{V\max }=0.22$. On the other hand, the photospheric temperature of the M_Ni = 0.07 M_⊙ model stays in $3.9\lt \mathrm{log}T[{\rm{K}}]\lt 4.1$ at t = 11–19 days, the duration of which is shorter than the M_Ni = 0.25 M_⊙ case. At the V-band peak (t = 19.2 days), the effective and blackbody fit temperatures are T_eff = 7370 K and T_bb = 7240 K, respectively. The corresponding B − V is ${(B-V)}_{V\max }=0.47$. This comparison illustrates that a higher M_Ni to M_ej ratio leads to a bluer optical color at the V-band peak for a given set of initial conditions.

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The dependency of ${(B-V)}_{V\max }$ on M_Ni/M_ej can also be observed in Figure 5. As M_Ni/M_ej becomes larger, ${(B-V)}_{V\max }$ of the models tends to decrease for a given ⁵⁶Ni distribution (note that the y-axis in the figure is flipped). By comparing the model color distributions presented in Figure 3 with the top and middle panels of Figure 5, we can find that the spread of ${(B-V)}_{V\max }$ for a given progenitor model and ⁵⁶Ni distribution mainly originates from the M_Ni/M_ej spread. For example, the ${(B-V)}_{V\max }$ distribution of the CO models with a step ⁵⁶Ni distribution of f_m = 0.5 has the red end of the color distribution (${(B-V)}_{V\max }$ = 1.1) when the models have the smallest M_Ni/M_ej = 0.02 and the blue end of the color distribution (${(B-V)}_{V\max }$ = 0.3) when the models have the largest M_Ni/M_ej = 0.14.

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It seems that the observed SN Ib/Ic sample does not follow the relation between M_Ni/M_ej and ${(B-V)}_{V\max }$ predicted by the STELLA models (see the bottom panel of Figure 5). This might be due to the limited sample size, the lack of ⁵⁶Ni mixing information, or poor estimates of the ejecta/explosion parameters and/or host extinction. Future extensive investigation into SNe Ib/Ic photometry would test the validity of our model prediction.

If M_Ni/M_ej was systematically larger in SNe Ib than SNe Ic, it could explain the systematically bluer color of SNe Ib than SNe Ic of our SN sample (the middle panel of Figure 1). In contrast to this expectation, however, it seems that SNe Ic have systematically larger M_Ni/M_ej than SNe Ib in our SN sample (the bottom panel of Figure 1). The same trend is also found in the most recent data set of SNe Ib/Ic provided by Zheng et al. (2022). We therefore conclude that the color difference between SNe Ib and Ic cannot be attributed to different M_Ni to M_ej ratios. However, given the large uncertainty in the estimates of M_Ni and M_ej as discussed in Section 2 (see also Table 4), this conclusion should be only considered tentative.

According to the most popular SN scenario, SN Ib and Ic progenitors are helium rich and helium poor, respectively (e.g., Yoon 2015). This scenario has been supported by many observational and theoretical studies as discussed in Section 1 (e.g., Matheson et al. 2001; Dessart et al. 2012; Hachinger et al. 2012; Liu et al. 2016; Fremling et al. 2018; Yoon et al. 2019; Dessart et al. 2020; Moriya et al. 2020; Teffs et al. 2020; Williamson et al. 2021; Shahbandeh et al. 2022). Recent stellar evolution models also suggest that SN Ib and Ic progenitors would not form a continuous sequence in terms of He amount if mass-loss enhancement during the WC phase of Wolf–Rayet stars, which is implied by recent Wolf–Rayet star observations of the local universe, is assumed (see Yoon 2017 for a detailed discussion and references therein; see also Woosley et al. 2021 and Aguilera-Dena et al. 2023). In this section, we explore the consequence of the dichotomic nature of helium content in SN Ib/Ic color (see also Yoon et al. 2019).

In Figure 6, we present the evolution of the ratio of the free electron to baryon numbers (n_e/n_b) and the gas temperature in the ejecta of the He4.2 and CO4.2 models with E_K = 1.0B, M_Ni = 0.2 M_⊙, and a step ⁵⁶Ni distribution (f_m = 0.5). In the He4.2 model, the ratio n_e/n_b in the outermost layers (i.e., M_r ≳ 4.0 M_⊙) rapidly decreases after t ≃ 4 days as the temperature decreases and the Rosseland mean photosphere rapidly moves inwards accordingly in the mass coordinate. However, in the CO4.2 model, more free electrons are available in the outer layers of the ejecta and the decrease of n_e/n_b is relatively slow compared to the He4.2 model. This makes the photosphere of the CO4.2 model retreat inwards more slowly than in the He4.2 model: M_r,ph[M_⊙] = 4.2 → 3.9 → 2.8 (He4.2) and M_r,ph[M_⊙] = 4.2 → 4.1 → 3.4 (CO4.2) at t = 0 days → 6 days → 28 days, respectively. The photospheric gas temperature at t = 5−28 days is higher in the He4.2 model ($3.9\lt \mathrm{log}T[{\rm{K}}]\lt 4.1$) than in the CO4.2 model ($3.7\lt \mathrm{log}T[{\rm{K}}]\lt 3.9$).

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The V-band peak is reached when t = 25.6 days and t = 21.8 days for the He4.2 and CO4.2 models, respectively, and the corresponding effective and blackbody fit temperatures are T_eff = 7610 K and T_bb = 8234 K for He4.2 and T_eff = 6774 K and T_bb = 7327 K for CO4.2, respectively.

For this reason, the He models are systematically bluer than the CO models for a given set of model parameters except for the (almost) full ⁵⁶Ni-mixing cases (i.e., f_m = 1.0; Figure 3). The effect of ⁵⁶Ni is discussed in the section that follows. As shown in Figure 3, the differences in ${(B-V)}_{V\max }$ between the two progenitor types (i.e., He and CO) in our models are ${\rm{\Delta }}{(B-V)}_{V\max }\approx$ 0.14–0.18 for a given ⁵⁶Ni distribution, which is comparable to the ${\rm{\Delta }}{(B-V)}_{V\max }$ value for our observed sample of SNe Ib/Ic. This is in good agreement with the notion that SNe Ib originate from helium-rich progenitors and SNe Ic from helium-poor progenitors.

Radioactive ⁵⁶Ni can significantly affect the SN color by radioactive heating, ionization induced by the heating, and line blanketing. Although a higher M_Ni/M_ej leads to a bluer color at the optical peak for a given ⁵⁶Ni distribution as discussed in Section 4.1, we find that stronger ⁵⁶Ni mixing can make the color redder at the V-band peak for a given M_Ni/M_ej ratio, as shown in Figure 3: the fully mixed case (f_m = 1.0) results in a much redder color (${\overline{(B-V)}}_{V\max }$ ≈1.0) than the weakly/moderately mixed cases (f_m = 0.15 and 0.5, ${\overline{(B-V)}}_{V\max }$ ≈ 0.4–0.6).

In Figure 7, we show the Rosseland mean opacity and temperature evolution in the ejecta of the He4.2 model for f_m = 0.15 (weak ⁵⁶Ni mixing) and 1.0 (full ⁵⁶Ni mixing) with a step ⁵⁶Ni distribution. It is observed that in the case of f_m = 0.15, the ⁵⁶Ni heating of the outer layers is somewhat delayed, while the ejecta with f_m = 1.0 monotonically cools down given that ⁵⁶Ni is evenly distributed throughout the ejecta. Compared to the f_m = 1.0 case, the Rosseland mean photosphere in the f_m = 0.15 model retreats more rapidly but stays in the hot ⁵⁶Ni-heated region ($\mathrm{log}T[{\rm{K}}]\gt 3.9$ at t = 11–30 days) during which the V-band peak appears at t = 26 days. The Rosseland mean photosphere in the f_m = 1.0 model recedes inwards more slowly and reaches the V-band peak earlier (i.e., at t = 22.1 days). This is because of more free electrons in the outer layers due to ⁵⁶Ni heating and line blanketing due to the presence of ⁵⁶Ni. This leads to lower effective and blackbody fit temperatures for stronger ⁵⁶Ni mixing at V-band peak: T_eff = 7528 K and T_bb = 83420 K for f_m = 0.15 and T_eff = 6760 K and T_bb = 6540 K for f_m = 1.0.

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The effect of line blanketing can also be observed by comparing the evolution of T_eff and T_bb as in Figure 8. In the case of f_m = 0.15, the Rosseland mean photosphere remains in ⁵⁶Ni-free layers and T_bb is systematically higher than the effective temperature at the Rosseland mean photosphere during the epochs around the V-band peak. This is because electron scattering causes the thermalization depth to be located below the Rosseland mean photosphere. The monochromatically scattered photons keep T_bb corresponding to those deep layers. Their number density is reduced by the dilution effect and the result is a bluer spectrum than the blackbody spectrum corresponding to T_eff. By contrast, T_bb is lower than T_eff in the f_m = 1.0 case because line blanketing due to ⁵⁶Ni obscures photons having short wavelengths and this is mainly responsible for the redder SN color with a stronger ⁵⁶Ni mixing.

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We find that the colors of the Gaussian ⁵⁶Ni distribution models with weak and moderate ⁵⁶Ni mixing (f_m = 0.15 and 0.5) are systematically redder than the corresponding step ⁵⁶Ni distribution models (see Figure 3 and Table 3). This result can also be explained by the effects of ⁵⁶Ni on the SN spectral energy distribution. As seen in Figure 2, the ⁵⁶Ni distribution following the Gaussian function extends to the outermost layers even for the case of f_m = 0.15 and 0.5, while with the step function, ⁵⁶Ni is present only in the innermost confined region unless ⁵⁶Ni is fully mixed. Therefore, in the models with a Gaussian ⁵⁶Ni distribution, ⁵⁶Ni is always present at the Rosseland mean photosphere and the resulting line blanketing makes the SN color redder than the corresponding step distribution models.

We conclude that different degrees of ⁵⁶Ni mixing would make the ${(B-V)}_{V\max }$ color different even when their progenitor types are the same in terms of helium content. However, it is not likely that the color difference between SNe Ib and Ic in our observed sample is solely due to this mixing effect.

Yoon et al. (2019) argue that the nonmonotonic and monotonic color evolutions observed in SNe Ib and some SNe Ic during early times implies relatively weak and very strong ⁵⁶Ni mixing in SN Ib and Ic ejecta, respectively. However, stronger ⁵⁶Ni mixing into the helium-rich layer implies the formation of strong He i lines during the photospheric phase (Dessart et al. 2012; Hachinger et al. 2012; Dessart et al. 2020; Teffs et al. 2020; Williamson et al. 2021). This means that if the redder color of SNe Ic compared to SNe Ib were mainly due to stronger ⁵⁶Ni mixing, the SN Ic progenitors must be helium poor. As shown in Section 4.2, on the other hand, helium deficiency would also lead to a redder color compared to the helium-rich case for a given ⁵⁶Ni distribution. Therefore, it is possible that, in reality, the redder color of SNe Ic compared to SNe Ib is related to both helium deficiency and stronger ⁵⁶Ni mixing.