The effect of circumstellar matter on the double-peaked type Ic supernovae and implications for LSQ14efd, iPTF15dtg and SN 2020bvc

We do not calculate the explosive nucleosyntheis, which is not the subject of this work, and instead put a certain amount of radioactive ⁵⁶Ni in the input progenitor models. We assume a nickel distribution which follows a Gaussian profile as in Yoon et al. (2019):

Here, $A$ is the normalization factor, $M_{r}$ the mass coordinate, $M_{\textrm{Fe}}$ the iron core mass, $M_{\textrm{tot}}$ the total mass of the progenitor, and $f_{\textrm{m}}$ the ⁵⁶Ni distribution parameter. For our fiducial models, we fix the total ${}^{56}\textrm{Ni}$ mass to 0.25 $M_{\odot}$, which is the inferred value for LSQ14efd (Barbarino et al., 2017). This amount of ⁵⁶Ni is within the typical range of the ⁵⁶Ni mass distribution of ordinary SNe Ic (Anderson, 2019). We calculate SN models using different values of $f_{\mathrm{m}}$, which determines the degree of ⁵⁶Ni mixing in the SN ejecta: the ⁵⁶Ni distribution becomes flatter for a larger $f_{\mathrm{m}}$ and vice versa as shown in Figure 1.

We assume that the CSM has the density profile of $\rho_{\textrm{CSM}}=\dot{M}/4\pi v_{\textrm{wind}}r^{2}$, with the standard $\beta$-law wind velocity profile: $v_{\textrm{wind}}(r)=v_{0}+(v_{\infty}-v_{0})\left(1-\frac{R_{0}}{r}\right)^{\beta}$. Here $\dot{M}$ is the mass-loss rate of the progenitor, $v_{0}$ is the wind velocity at the progenitor surface, $v_{\infty}$ is the terminal velocity, $R_{0}$ is the radius of the progenitor, and $r$ is the distance from the center of the progenitor. The density profile would follow $r^{-2}$ after the wind is accelerated in a transition layer. According to Fuller & Ro (2018), hydrogen-poor stars are predicted to emit wave-driven outbursts with terminal velocities of a few 100 $\mathrm{km~{}s^{-1}}$. We fix the terminal velocity to $200~{}\mathrm{km~{}s^{-1}}$ in this study. Note that there exists degeneracy between $\dot{M}$ and $v_{\infty}$: a larger $v_{\infty}$ would imply a larger $\dot{M}$ for a given CSM mass, which should be kept in mind when we discuss our result.

For the wind velocity parameter $\beta$, we assume $\beta=3.0$. Although this value can affect the early-time light curves of Type IIP supernovae significantly (Moriya et al., 2018a), our SN Ic models depend on $\beta$ very weakly because of the small radii of the progenitors. The CSM mass can vary by a factor of 5 to 500 for $\beta=1\cdots 5$ for red supergiant SN progenitors (Moriya et al., 2018a) but only by 3% for our models.

Besides, we assume that the chemical composition of the CSM follows that of the outermost part of the progenitor. It would be possible that the CSM is more helium-rich than the progenitor surface. However, our test calculations indicate that the early-time light curves are not meaningfully affected for our considered parameter space even if a helium-rich composition is adopted, although it might be important for the details of early-time spectra.

In the bottom panel of Figure 1, the density profile of the 4P progenitor is presented with the blue dashed line. The density decreases very steeply near the surface of the progenitor. The SN shock is rapidly accelerated in this region, making the time step very small. To avoid this numerical difficulty, we take off $4\times 10^{-5}M_{\odot}$ of the outermost layer of the progenitor when we attach CSM in our models. As discussed below (Section 3.2.1), the choice of the CSM inner boundary is not important for the conclusions of this study.

Note also CSM in our models can also be considered as an extended outward moving envelope created by an energy injection during the final evolutionary stage. The difference between an outward moving envelope and wind matter would be just that an envelope is gravitationally bound to the progenitor while the wind matter is not. As long as the CSM velocity is much lower than the SN shock velocity, wind matter and an extended envelope would not lead to a difference in the resulting light curve as long as the density profile is not very much different (see Section 3.2.1).

We also calculate some models with a very small amount of CSM (0.001% of the progenitor mass) with $R=10^{14}~{}\mathrm{cm}$ for comparison. In this case, the CSM would correspond to the wind material from an ordinary line-driven Wolf-Rayet wind having $\dot{M}\sim 10^{-5}~{}M_{\odot}~{}\mathrm{yr^{-1}}$. For convenience, these models are denoted by ‘no-CSM’ as the effect of CSM on the light curve is practically negligible in this case.

We focus on the effects of four parameters for a given progenitor mass: CSM mass ($M_{\textrm{CSM}}$), CSM radius ($R_{\textrm{CSM}}$), nickel distribution ($f_{\textrm{m}}$), and explosion energy ($E_{\textrm{burst}}$). In our fiducial models, we consider $M_{\textrm{CSM}}=0.05\cdots 0.3M_{\odot}$, $R_{\textrm{CSM}}=10^{13}\dots 10^{15}~{}\mathrm{cm}$, $f_{\mathrm{m}}=$ 0.15, 0.3, and 0.6. We consider the explosion energy of $E_{\mathrm{burst}}=(1\cdots 3)\times 10^{51}~{}\mathrm{erg}$. for 4P models and $E_{\mathrm{burst}}=(5\cdots 12)\times 10^{51}~{}\mathrm{erg}$ for 8P models. The ${}^{56}\textrm{Ni}$ mass is set to 0.25 $M_{\odot}$ and 0.40 $M_{\odot}$ for reproducing the main peaks of LSQ14efd/iPTF15dtg and SN 2020bvc, respectively.

For simplicity, each model is referred to as xP_$f_{\textrm{m}}$_$E_{\textrm{burst}}$_ $M_{\textrm{CSM}}$_log$R_{\textrm{CSM}}$ in the figures. For example, 4P_fm0.15 _E2_0.15M_R13 denotes the SN model with the 4P progenitor, $f_{\textrm{m}}$=0.15, $E_{\textrm{burst}}$=2.0B, $M_{\textrm{CSM}}$=0.15$M_{\odot}$, and log$R_{\textrm{CSM}}$ [cm] = 13.

In STELLA, SN shock is initiated by a thermal bomb and begins to propagate outward from the assumed mass cut. The density profile is steeper than $r^{-3}$ at the outermost layers of the progenitor (Figure 1) thus the shock moving forward in these layers is accelerated until it reaches the surface of the progenitor (Nadezhin & Frank-Kamenetskii, 1965; Grasberg, 1981; Blinnikov & Tolstov, 2011). As the forward shock enters CSM, in which the density profile follows $r^{-2}$, it decelerates and an inward-moving reverse shock is created. These shocks convert a significant fraction of SN kinetic energy into internal energy, leading to a bright shock-cooling emission.

As an example, we present bolometric light curves of our 4P models with $f_{\mathrm{m}}=0.15$, $E_{\mathrm{burst}}=2.0$B, and $R_{\mathrm{CSM}}=10^{14}~{}\mathrm{cm}$ for various CSM masses in Figure 2 and the corresponding multi-color light curves (i.e., in $U$, $B$, $V$, $R$ and $I$ bands) in Figure 3.

After the shock breakout, the bolometric luminosity of the no-CSM model drops very rapidly until the post-breakout plateau phase is reached (See Dessart et al., 2011, for a detailed discussion on this short-lived plateau phase of SNe Ib/Ic). By contrast, for the models with CSM, it remains brighter and decreases more slowly for several days (e.g., $\sim$ 10 days for $M_{\mathrm{CSM}}=0.15M_{\odot}$). The main power source during this period is the interaction of the SN shock and the CSM. We refer to this period as the ‘interaction-powered phase (IPP)’ following Moriya et al. (2011). After the IPP, the light curve is dominated by the energy due to the radioactive decay of ${}^{56}\textrm{Ni}$, and we refer to this phase as the ‘${}^{56}\textrm{Ni}$-powered phase (NPP)’.

To better understand the role of CSM in the IPP, we compare the evolution of the local luminosity ($L_{r}$), density ($\rho_{r}$), temperature ($T_{r}$) and velocity ($v_{r}$) of the 4P model having $M_{\mathrm{CSM}}=0.3M_{\odot}$ to those of the no-CSM 4P model in Figures 4, 5, 6 and 7. In the CSM model, the propagation of the forward and reverse shocks can be traced by the positive and negative luminosity peaks in the bottom panel of Figure 4. No aftereffect of the shocks is seen for the no-CSM model since there is barely an interplay of the shock and the wind matter.

The location of the photosphere is also greatly affected by the presence of CSM. As seen in Figure 5, in the CSM model the photosphere (defined by the Rosseland mean opacity) initially moves upward in the CSM in the mass coordinate until $t\simeq 1.0$ d. The photosphere gradually moves downward thereafter along with the SN ejecta expansion.

The shocked layers are heated up and the outermost layers in the CSM model remain much hotter than in the no-CSM model from $t=0.6$ d when the forward shock breaks out the CSM (Figure 6). These shocked layers in the CSM model have a significantly lower velocity compared to the no-CSM model (Figure 7). Note also that a dense shell is formed at $M_{r}\approx 3.2-3.9M_{\odot}$ as the reverse shock sweeps up this region.

These effects of the shock become more prominent for a larger CSM mass; a larger amount of the kinetic energy is transformed into the internal energy as more layers are shock-heated and decelerated. Furthermore the lower expansion velocity makes the expansion cooling less efficient and the photon diffusion time longer. These factors make the IPP longer and the bolometric and optical luminosities during the IPP brighter for a larger CSM mass as seen in Figures 2 and 8.

The CSM also has a great impact on the SN color. Given that the photosphere during the IPP is hotter for a larger CSM mass, the color of the SN during the IPP becomes bluer as seen in Figure 9 (see the upper left panel).

Yoon et al. (2019) showed that, without CSM, the early-time color evolution of a SN Ib/Ic sensitively depends on the ${}^{56}\textrm{Ni}$ distribution in the SN ejecta. Stronger mixing of ${}^{56}\textrm{Ni}$ into the outermost layers would lead to a bluer color in the earliest days followed by a monotonic reddening during the photospheric phase, while fairly weak mixing leads to three distinct phases of initial reddening, blueward evolution, and reddening again. In addition, a strong ${}^{56}\textrm{Ni}$ mixing tends to suppress the post-breakout emission that would otherwise appear during early times (Dessart et al., 2012; Piro & Nakar, 2013; Yoon et al., 2019). Our CSM models in Figures 9 indicate, however, that a monotonic reddening can also be realised with a sufficient amount of CSM (e.g., for the cases of 4P_fm0.15_E2_0.3M_R14 and 4P_fm0.15_E2_0.15M_R15 in Figure 9), even if the ${}^{56}\textrm{Ni}$ mixing is weak (i.e., $f_{\mathrm{m}}=0.15$).

On the other hand, the heat due to ${}^{56}\textrm{Ni}$ in the inner region diffuses outward while the photosphere moves down toward the ${}^{56}\textrm{Ni}$-heated region as can be seen in Figure 4. Once the luminosity is dominated by this ${}^{56}\textrm{Ni}$ heating, the IPP ends and the NPP begins. The light curve during the NPP is determined by the total ${}^{56}\textrm{Ni}$ mass and its distribution and becomes almost independent of the CSM structure (Figures 2 and 8).

In this section, we apply our results to the SN Ic LSQ14efd, which motivated this work. The comparison of our SN models with the observation is done with eyes. The observed data are taken from Barbarino et al. (2017). Although a quantitative fitting procedure (Morozova et al., 2018; Ergon et al., 2015) is also possible with our grid of models, eye inspection is more than enough because our grid resolution is rather coarse, as can be seen in Figures 8 and 11

For the comparison of the models with the observation, the distance modulus of 37.1 is adopted for LSQ14efd. Given that we have only one data point of the IPP of this SN, the explosion date cannot be easily determined from the observation. In Figures 8 and  11, the explosion date is chosen from the model that can best reproduce the $V$-band light around the NPP peak. The corresponding explosion dates are MJD 56875.5 d and 56874.0 d for 4P and 8P models, respectively. However, when we make eye inspection to find the best fit model, we freely shift the explosion date such that the light curve around the $V$-band NPP peak of each model may match the observation.

As can be seen in Figures 3 and 8, non-detection with the upper magnitude limit of $m_{\mathrm{V}}=20.6\pm 0.2$ at 22 d before the $V$-band maximum is reported for this SN (Barbarino et al., 2017). The information about the IPP peak and its rise time is missing and only one data point before the end of the IPP is available in the $V$-band (i.e., $m_{V}=20.03$ at $t=56882.48$ MJD; the first observed point). The local minimum in the $V$-band at the trasition between IPP and NPP ($t=56884.44$ MJD; the second observed point) is $m_{V}=20.15$. The data in the other optical bands are available only after the IPP.

To find the model parameters that can give a consistent fit to LSQ14efd, we take the following steps. First, the amount of ${}^{56}\textrm{Ni}$ is fixed to $0.25~{}M_{\odot}$, which is inferred from the NPP peak brightness (Barbarino et al., 2017), as explained above. Second, we only use the models with $f_{\mathrm{m}}=0.15$ because the color evolution of these models is qualitatively same as that of LSQ14efd where the signature of relatively weak ${}^{56}\textrm{Ni}$ mixing is found (see the discussion in Section 3.2.3).

For the remaining sets, we search for the CSM parameters (i.e., $\log R_{\mathrm{CSM}}$ and $M_{\mathrm{CSM}}$) that can best explain the three data points of the observed IPP (i.e, the non-detection limit, the first observed point, and the second observe point which is the local minimum at the IPP to NPP transition; see Figure 8) as well as $E_{\mathrm{burst}}$ that can give a reasonable $V$-band light curve width compared to LSQ14efd.

We find that no 8P model can satisfy the non-detection limit: given the very high energies (i.e,. $E_{\mathrm{burst}}=$ 6.0B and 8.0B), too bright emission is predicted at the point of non-detection for the models that can match the second and third data points of the IPP (see Figure 11). Note also that the color predicted by the 8P models is much redder than the observation (i.e., by more than 0.6 mag at the NPP peak in terms of $B-V$; Figure 12).

Within the grid of our fiducial 4P models with $f_{\mathrm{m}}=0.15$, we find that the IPP brightness can be best reproduced by the model with $M_{\mathrm{CSM}}=0.15M_{\odot}$, $R_{\mathrm{CSM}}=10^{14}~{}\mathrm{cm}$ and $E_{\mathrm{burst}}=2.0$B. This best fit model is presented in Figure 13 (the green line). The model with $R_{\mathrm{CSM}}=10^{15}$ cm which can well reproduce the first observed point is also shown in the figure for comparison (the dashed line). This model predicts too faint emission at the second point of the observed IPP, and can be ruled out.

As discussed in Section 3.2.2, different combinations of $M_{\mathrm{CSM}}$ and $R_{\mathrm{CSM}}$ can yield a similar IPP peak. To investigate this uncertainty, in Figure 13, we also present two test models for which we adopt $M_{\mathrm{CSM}}=2.0M_{\odot}$ and $R_{\mathrm{CSM}}=10^{12}$ cm (the red line) and $M_{\mathrm{CSM}}=0.5M_{\odot}$ and $R_{\mathrm{CSM}}=10^{13}$ cm (the blue line). These models and our best fit model have a similar IPP peak in $V$-band. However, the evolution after the IPP peak is different for each case. The model with $M_{\mathrm{CSM}}=2.0M_{\odot}$ and $R_{\mathrm{CSM}}=10^{12}$ cm have a very long term effect of CSM and predicts brighter emission at the second and third observed points. The $B-V$ color evolution of this model after the NPP peak is also distinctively different from the observation. The model with $M_{\mathrm{CSM}}=0.5M_{\odot}$ and $R_{\mathrm{CSM}}=10^{13}$ cm has a longer decline rate from the IPP peak than our best fit model and predict too bright emission compared to the observation at the first observed point.

We find that the degeneracy between $M_{\mathrm{CSM}}$ and $R_{\mathrm{CSM}}$ could be more easily broken in the U band as seen in the upper right panel of Figure 13. The compared tree models have a similar IPP peak in the V band but the U band IPP peak is systematically brighter for a larger $R_{\mathrm{CSM}}$, for which the adiabatic cooling is less efficient. Therefore, U-band observations during early times would be most useful in future observational studies on the CSM properties.

In principle, the degeneracy between $M_{\mathrm{ej}}$ and $E_{\mathrm{K}}$ which exists when inferring SN parameters with a NPP light curve can be broken by comparing the photospheric velocity and the model prediction. As seen in Figure 13, the photospheric velocity evolution of our best fit model is consistent with the observed $v_{\mathrm{SiII}}$. Barbarino et al. (2017) use $v_{\mathrm{FeII}}$, which is higher than $v_{\mathrm{SiII}}$, to infer SN parameters and as a result obtain higher $M_{\mathrm{ej}}$ and $E_{\mathrm{K}}$ (i.e., $6.3~{}\mathrm{M_{\odot}}$ and 5.6B) than our fiducial values. However, as discussed above, such a high SN energy of 5.6B is not favored when the observed IPP is compared with the models. In our models, the photosphere is defined by the Rosseland mean opacity and it would be an interesting subject of future work to investigate which absorption line better traces the Rosseland mean photosphere.

We conclude that the early-time light curve of LSQ14efd is consistent with the SN model prediction with a massive CSM of about $M_{\mathrm{CSM}}\approx 0.15M_{\odot}$ extending up to about $R_{\mathrm{CSM}}\approx 10^{14}$ cm. This corresponds to a mass loss rate of $\dot{M}\approx 1.0M_{\odot}~{}\mathrm{yr^{-1}}$ during $\sim 0.2$ yr before the SN explosion if we assume the terminal wind velocity as $200~{}\mathrm{km~{}s^{-1}}$. We discuss its implications for the mass loss mechanism below in Section 4.3.

We also compare our models with two other double peaked SNe Ic SN iPTF15dtg and SN 2020bvc. iPTF15dtg is a peculiar SN Ic which is suspected to be powered by a magnetar (Taddia et al., 2019) and its optical light curves around the main peak cannot be easily explained by our grid of models. Given that our main interest is to infer the properties of CSM, here we do not attempt to make a model that can reproduce the NPP (see instead Taddia et al., 2016, who inferred SN parameters from the light curve around the main peak). As discussed above, the IPP light curve is largely determined by $M_{\mathrm{CSM}}$, $R_{\mathrm{CSM}}$ and $E_{\mathrm{burst}}$, which can be fairly well determined independently of the detailed properties of the NPP if the early time data of the IPP are good enough.

Comparison of our model grid with the IPP light curve of this SN indicates that the early-time light curve (0 $\sim$ 15 d after the explosion) of iPTF15dtg is fairly consistent with two of our 4P models: 4P_fm0.4_E3_0.05M_R14 and 4P_fm0.4_E3_0.15M_R13, as seen in Figure 14. Here, the last non-detection date is chosen for the explosion date since we cannot fit the main peak with our grid of models. This implies that the progenitor of iPTF15dtg had CSM with $R_{\mathrm{CSM}}=10^{13}-10^{14}\mathrm{cm}$ and $M_{\mathrm{CSM}}=0.05-0.15M_{\odot}$.

For SN 2020bvc, we already presented a result of our model comparison with the observation in another paper  (Rho et al., 2020). For this particular SN, we use the 8P progenitor and assume that ⁵⁶Ni is uniformly distributed in the inner 90% of the SN ejecta. The ${}^{56}\textrm{Ni}$ mass required to explain the NPP peak is found to be $0.4~{}M_{\odot}$. As in the case of LSQ14efd, the explosion date is determined by matching the observed $V$-band light curve around the main peak with the models. For the properties of these models made for the comparison with SN 2020bvc, see Table 3. Within our grid, we find that the overall light curve properties including the IPP of SN 2020bvc are most consistent with the following two sets of parameters: $M_{\mathrm{CSM}}=0.1M_{\odot}$, $R_{\mathrm{CSM}}=10^{14}~{}\mathrm{cm}$ & $E_{\mathrm{burst}}=12$ B, and $M_{\mathrm{CSM}}=0.2M_{\odot}$, $R_{\mathrm{CSM}}=10^{13}~{}\mathrm{cm}$ & $E_{\mathrm{burst}}=15$ B. The early-time color of this SN is very blue as predicted by the model, which is due to the SN interaction with CSM.

Our result indicates that the inferred CSM properties of the double peaked SNe Ic considered in this study (LSQ14efd, iPTF15dtg and SN 2020bvc) are intriguingly similar: $M_{\mathrm{CSM}}\approx 0.1-0.2M_{\odot}$ and $R_{\mathrm{CSM}}\approx 10^{13}-10^{14}~{}\mathrm{cm}$.

The implied mass loss rate is $\dot{M}\approx 0.6-13.0$ $(v_{\mathrm{w}}/200~{}\mathrm{km~{}s^{-1}})~{}M_{\odot}~{}\mathrm{yr^{-1}}$. The mass eruption should have occurred within about 0.2 yr from the explosion if we adopt $v_{\mathrm{w}}=200~{}\mathrm{km~{}s^{-1}}$.

This high mass loss rate inferred in our study cannot be explained by the conventional line-driven wind of Wolf-Rayet stars. Fuller & Ro (2018) explore the possibility of pre-supernova outbursts via wave heating during core neon and oxygen burning in a 5 $M_{\odot}$ hydrogen-free helium star. The predicted mass loss rate is $\sim$0.01$M_{\odot}~{}\mathrm{yr^{-1}}$ with a wind terminal velocity of a $450~{}\mathrm{km~{}s^{-1}}$. This is smaller by two or three orders of magnitude than our inferred value. Note also that helium-poor SN Ic progenitor stars are supposed to be more compact by several factors than helium-rich SN Ib progenitors which are considered by Fuller et al.  (e.g., Yoon et al., 2019). Therefore, it seems that the inferred CSM property of the double peaked SNe Ic cannot be easily explained by the wave heating model.

Aguilera-Dena et al. (2018) find that the surface layers of helium stars could be spun-up to the critical rotation during the final evolutionary stages if the helium stars retained sufficiently high angular momenta. The predicted mass loss rate is about $\dot{M}\lesssim 0.01M_{\odot}~{}\mathrm{yr^{-1}}$ during the last years of the evolution, which is also much lower than our inferred mass loss rate.

Woosley (2019) find that very massive CSM of $0.02\cdots 0.74~{}M_{\odot}$ can be created by silicon flash for helium stars having initial helium star masses of 2.5 - 3.2 $M_{\odot}$. However, all these He star models with silicon flash have a fairly massive helium envelope ($>0.7M_{\odot}$) and the resulting SN would be a SN Ib rather than SN Ic. In addition, the silicon flash only occurs for a relatively small progenitor mass (i.e, $M_{\mathrm{final}}\lesssim 2.6M_{\odot}$), which could not easily explain the light curves of the double peaked SNe Ic of our sample.

One scenario that could explain the CSM of the double peaked SNe Ic of our sample would be the possibility that the mass loss from the progenitor was induced by the combined effects of wave heating and rotation. If the progenitor were rotating at the critical rotation during the core neon and oxygen burning stages, the binding energy of the outermost layers would be lower than the non-rotating case and the mass loss due to the wave heating would be easier.

This scenario is in line with the fact that iPTF15dtg and SN 2020bvc belong to peculiar SNe Ic (ie., magnetar-powered/broad-lined SNe Ic), for which a rapid rotating progenitor is often invoked. This might also imply that the LSQ14efd progenitor was a rapid rotator even though the inferred properties of LSQ14efd do not look peculiar except for the IPP signature. It is possible that LSQ14efd was in part powered by a mangetar, in which case the actual ${}^{56}\textrm{Ni}$ would be smaller than the inferred value of $0.25~{}M_{\odot}$.

Another possibility is that the progenitors did not really undergo a mass eruption, but simply an expansion of the outer layers due to an energy injection during the pre-SN stage. Althernatively, the IPP observed in our sample might not be related to CSM, but to interactions with a companion star (Kasen, 2010) or to high velocity ${}^{56}\textrm{Ni}$ due to an asymmetric explosion (e.g., Folatelli et al., 2006; Bersten et al., 2013). An elaboration of these scenarios would be a subject of future work.