Helium stars exploding in circumstellar material and the origin of Type Ibn supernovae

We performed 1D multi-group radiation hydrodynamics simulations for a variety of interaction configurations involving an inner shell and an external shell. The inner shell was assumed to result from an explosion, most likely following core collapse. For the outer shell, we considered two possible scenarios. The material was either explosively ejected (as would occur, for example, in a nuclear flash; Woosley et al. 1980; Dessart et al. 2010; Woosley & Heger 2015), or driven out in the form of a super-Eddington wind (for example as a result of excitation by waves born in the core of the progenitor; Quataert & Shiode 2012; Fuller 2017; see also Quataert et al. (2016) and Owocki et al. (2017) for the mechanisms behind such winds). The distinction between these two cases is nontrivial. In SN ejecta, the bulk of the mass lies at low velocity and the velocity is a linear function of radius. In the case of a wind, the mass is more uniformly spread in radius (following a $1/r^{2}$ dependence) and the velocity is constant with radius (for a more detailed discussion of the implication of these two configurations, see discussion in Dessart et al. 2016).

The 1D multi-group radiation hydrodynamics simulations were carried out with HERACLES (González et al., 2007; Vaytet et al., 2011), using the approach discussed in Dessart et al. (2015). An ideal-gas equation of state was used for simplicity, and also because the bulk of the internal energy in explosions and interactions is stored in radiation. For the present simulations on Type Ibn SNe, we considered only hydrogen-free material. The configurations for the inner and outer shells were taken from the helium-star progenitor and explosion models of Dessart et al. (2020b) – these models are very similar to those produced by Woosley (2019) and Ertl et al. (2020). The complex composition of these models was simplified so that HERACLES would only follow helium, oxygen, silicon, and iron (with a suitable renormalization of the sum of mass fractions to unity) in order to capture the basic chemical stratification of these ejecta. This was done for the purpose of using the adequate, composition-dependent, opacities at each location of the grid. In practice, opacities were calculated as a function of composition, density, temperature and energy group (for details, see Dessart et al. 2015). We used eight energy groups positioned at strategic locations to capture the strong variation in absorptive opacity with wavelength. One group covers the Lyman continuum, two groups sample the Balmer continuum, two other groups cover the Paschen continuum, and finally three groups sample the Brackett continuum and beyond. We include radioactive decay power from ${}^{56}\rm{N}$i and ${}^{56}\rm{C}$o and assume that this power is deposited at the site of emission.

For these radiation-hydrodynamics simulations, we were first interested in the bolometric light curve properties, in particular the rise time to peak, the peak luminosity, and the time-integrated bolometric luminosity over the high-luminosity phase. We also wanted to study the properties of the CDS that inevitably forms at the junction of the inner shell and outer shells, composed of swept-up CSM and decelerated ejecta material. The diversity of light curve and dynamical properties was generated by varying the properties of the inner and outer shells. We studied the configuration of a standard-energy core-collapse SN explosion of a Wolf-Rayet star interacting with a dense wind, but also explored other configurations that departed significantly from this.

The default ejecta model for the inner shell is the solar-metallicity model he4 from Dessart et al. (2020b) at 1 d after explosion (this model is a close analog of model he4p0 from Ertl et al. (2020), but differs slightly in ejecta mass and kinetic energy – to distinguish these two models, we use he4 to refer to the model from Dessart et al. 2020b and to he4p0 to refer to the model from Ertl et al. 2020). We adopted the radius, velocity, density, temperature and composition from that model, including ${}^{56}\rm{N}$i. The ejecta is hot and optically thick at that time, with a significant storage of radiation energy so we start with the initial temperature of this ejecta at that time. For the outer shell, we assumed a composition identical to the outermost point in the he4 model (i.e., in the He/N shell), which is essentially pure helium and metals at their solar metallicity value (there is no ${}^{56}\rm{N}$i in the outer shell). For the outer shell, an analytical description of the fluid variables was used for both the ejecta case (for details, see Dessart & Audit 2019), or for the wind case (assuming a wind velocity $V_{\infty}$ and a density $\rho=\dot{M}/4\pi R^{2}V_{\infty}$, where $\dot{M}$ is the wind mass loss rate and $R$ is the radius). The outer shell material was assumed to be cold initially, with a temperature of 2000 K.

We joined the inner and outer shells at a radius $R_{\rm t}$ of about 10¹⁴ cm (see Fig. 15). This corresponds to an age for the inner shell of about one day after core collapse, while the outer shell was produced months to years before core collapse (depending on its velocity). Different values of $R_{\rm t}$ reflect different physical evolutions for the progenitor star but do not alter much the overall trends discussed here, as long as $R_{\rm t}$ remains below a few 10¹⁵ cm (for larger values of $R_{\rm t}$, the outer shell would be optically-thin, the thermalization would be weaker, the deceleration of the inner shell less efficient etc.)

For the inner shell, the default kinetic energy is $E_{\rm kin}=7\times 10^{50}$ erg, $M_{\rm ej}=$ 1.49 $M_{\odot}$, and $M(^{56}$Ni$)=$ 0.08 $M_{\odot}$. Variants were produced in which we scaled the velocity by a factor $\sqrt{0.1}$ (corresponding to a scaling of $E_{\rm kin}$ by 0.1) and others in which we scaled the density by a factor of 0.1 (corresponding to a scaling of both $E_{\rm kin}$, $M_{\rm ej}$, and $M(^{56}$Ni) by 0.1). These scaled variants are not fully consistent but they are useful to explore the outcome of interactions involving a less energetic or a less massive inner ejecta. For the outer shell, we considered either an ejecta of 1 $M_{\odot}$ with $E_{\rm kin}$ of 10⁴⁷ or 10⁴⁹ erg (models E1 to E6), or winds with $V_{\infty}=$ 1000 km s^-1 and wind mass loss rates of 0.001, 0.01, or 0.1 $M_{\odot}$ yr^-1 (models W1 to W9). Table 1 presents a summary of the HERACLES simulations, both for the initial interaction configurations and for the resulting light curve and CDS properties. We show the initial radial profile of the velocity, density, and temperature for all interaction configurations in Fig 15.

The resulting dynamical and radiative properties reflect how the kinetic energy stored in the outer parts of the inner shell is absorbed by the outer shell. In turn, the optical depth of the outer shell sets the typical diffusion time over which the bulk of this power is radiated (this timescale is also comparable to the rise time to maximum). When the configuration becomes optically thin, the model luminosity is equal to the shock luminosity $L_{\rm shock}=2\pi R^{2}\rho V_{\rm shock}^{3}$ (here, $R$ is the radius of the shock, $\rho$ is the CSM density ahead of the shock, and $V_{\rm shock}$ if the shock velocity) together with the contribution from radioactive decay power.

In the next two sections, we present some results from radiation-hydrodynamics simulations. We do not intend in this work to be quantitative and to propose a specific model for any specific observation. Rather, we explore the systematics for various configurations, and give special attention to the salient features of transient light curves, namely the rise time to peak, the peak luminosity, and the time-integrated bolometric luminosity over the high-luminosity phase. Detailed comparisons of models and observations are deferred to a future study.

The results from the previous section indicate that a suitable configuration for Type Ibn SNe like 2006jc, 2011hw, or 2018bcc may be a 10⁵⁰ erg ejecta with $\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}$ 1 $M_{\odot}$  ramming into a slowly moving 1 $M_{\odot}$ outer shell. At late times, all the mass in the system has piled up into a dense shell moving at a velocity of order 1000 km s^-1. There is little mass either below or above the dense shell, hence most of the absorption and emission arises from the dense shell. In this situation, when focusing on the low-energy radiation, it is no longer necessary to consider the global and complex structure of the interaction, and one can instead focus exclusively on the CDS.

To simplify further the approach, we may no longer distinguish between inner-shell and outer-shell material within the CDS. Weeks after the bolometric maximum, fluid instabilities will probably have had sufficient time to produce a mixed composition of both shells.⁴⁴4A foretaste of such instabilities in interacting SNe can be gleaned from the study of Blondin et al. (1996). More work, in 3D, including radiation transport, with realistic progenitors, and in the parameter space relevant for Type Ibn SNe, is needed to determine the level of mixing and clumping in the CDS.. Furthermore, the pre-SN ejecta occurs just months to years before core collapse, so one can assume that the inner shell and the outer shell material involved in the interaction arise from the envelope above the iron core in the progenitor. For our radiative-transfer calculations, we can therefore use a helium-star explosion model, and assume that at late times, this ejecta is a good representation of the combined inner and outer shells ejected in two consecutive events. In this work, we thus use the helium-star explosion models originally presented by Ertl et al. (2020). We enforce a chemical mixing by running four times a boxcar width of $0.06M_{\rm ej}$ through the ejecta. In our calculations, we consider solar-metallicity progenitor models with an initial helium-star mass between 2.9 and 12 $M_{\odot}$ (for details on these models, see Dessart et al. 2021). Following our assumption for the late-time conditions, we put the total mass of each model into the dense shell. The density structure adopted follows a Gaussian with a center at 2000 km s^-1 and a standard deviation of 70 km s^-1. With this adopted structure, we rescale the density to match the ejecta mass of each helium-star explosion model – the density in the dense shell is 100 to 1000 times greater than in the inner ejecta of the same helium-star explosion model but expanding freely without interaction. The original composition is then re-interpolated in mass space onto that new mass distribution.

We vary the age of the configuration by adopting different radii for the CDS, here chosen to be at 2, 3, or $6\times 10^{15}$ cm – an example for the model he4p0 is shown in Fig. 4. For reference, a shell moving at 500 km s^-1 for two years would reach a radius of $3.15\times 10^{15}$ cm, which gives an idea of how far the 2004 pre-SN outburst of SN 2006jc may have travelled (Pastorello et al., 2007). Because the spectrum formation occurs locally within the CDS, homologous expansion may be assumed. We no longer capture the non-monotonic velocity structure like in Dessart et al. (2015), but this allows us to run CMFGEN in standard, steady-state, line-blanketed mode, with a proper account of the effect of lines on the temperature and ionization structure within the CDS.

Finally, we inject power to mimic the contributions from the interaction and radioactive decay. We choose a Gaussian power profile whose volume integral yields a prescribed total power – we explore with values of 2 to $10\times 10^{42}$ erg s^-1, which are typical of the high-luminosity phase of Type Ibn SNe like 2006jc. For the decay power, we adopt the same initial ${}^{56}\rm{N}$i mass as given in each helium-star explosion model of Dessart et al. (2021), and we assume that at the time of the computation, the ${}^{56}\rm{N}$i has entirely decayed into a mixture composed of 50% ${}^{56}\rm{C}$o and 50% ${}^{56}\rm{F}$e, as expected for a post-explosion epoch of about three months, roughly representative of the nebular epochs that we aim to study here.⁵⁵5The time when the number of ${}^{56}\rm{C}$o and ${}^{56}\rm{F}$e nuclei are equal is given by $\tau_{\rm Co}\log(2\tau_{\rm Co}/(\tau_{\rm Co}-\tau_{\rm Ni}))$, where $\tau_{\rm Co}$ and $\tau_{\rm Ni}$ are the lifetimes of ${}^{56}\rm{C}$o and ${}^{56}\rm{N}$i nuclei. This time corresponds to 86.4 d. When comparing to specific observations, using the inferred post-explosion time (and the adequate mixture of ${}^{56}\rm{C}$o and ${}^{56}\rm{F}$e – only traces of ${}^{56}\rm{N}$i would remain) would have been superior but it is not essential for this first exploration on the basic properties of Type Ibn SNe. It would also imply rerunning a model for each observation, while there still remains much ambiguity on the actual ${}^{56}\rm{N}$i mass produced in Type Ibn SNe in the first place.

Figure 5 shows the resulting CMFGEN spectrum for a CDS composed of the entire he4p0 ejecta material (see composition and density profiles in Fig. 4), centered around a radius of $3\times 10^{15}$ cm, with a velocity of 2000 km s^-1, and subject to a total power of $2\times 10^{42}$ erg s^-1. This model exhibits a very peculiar spectrum by SN standards, but quite typical of Type Ibn SNe several weeks after bolometric maximum. With its strong Fe ii emission between 3000 and 5500 Å  it is in fact reminiscent of the predicted nebular-phase models obtained for helium-star explosions in the mass range 2.6$-$4.0 $M_{\odot}$ (see Dessart et al. 2021). The bottom panel of Fig. 5 shows that the model spectrum is nearly entirely an Fe ii spectrum (with some contribution from Fe i), although the Fe abundance is only 1.6% of the total mass. The spectrum also exhibits a myriad of He i lines at 3889, 3965, 4026, 4471, 5016, 5876, 6678, 7065, and 7281 (note that the feature around 3900 Å is He i 3889 Å and not Ca ii H&K), something that is not expected in nebular phase spectra of Type Ib SNe. In the red part of the optical range, the model exhibits lines of O  i (multiplets around 8446 Å), O  ii (multiplets around 7320 Å), Si ii (around 5041–5056, 5958–5979 and 6347-6371 Å) and Mg ii  (at 4481, 7896, 8234, 9218, and 9244 Å).

Although a gaussian smoothing has been applied to match the typical resolution of Type Ibn SN spectra, the He i line profiles predicted by the model exhibit a red deficit (Fig. 6) – the blue emission is typically three times stronger than the red emission. This asymmetry is caused by optical depth effects. This feature may not persist in a 3D model of the interaction since the CDS would be strongly clumped, with both radial and lateral compression, thereby reducing the effective radial optical depth of the region. The blue-red asymmetry is greater for He i lines in the blue part of the spectrum (e.g., He i 3889 Å) and weaker for those in the red part of the spectrum (e.g., He i 5875 Å). This difference arises from the fact that lines are not only affected by the continuum optical depth, which is essentially gray, but also by other lines, especially in the blue part of the optical where the forest of Fe ii  lines have a strong blanketing effect. He i lines that form further out in the ejecta are less affected (see Fig. 7). This result is reminiscent of the differential optical depth effects affecting H$\beta$ (which overlaps with the Fe ii line forest) and H$\alpha$ (which sits in an opacity hole) in models of SNe Ia that contain in their innermost ejecta layers stripped material from a companion (Dessart et al., 2020a).

The model predicts a number of lines from O i, Mg ii, and Ca ii. All these metals have super-solar abundance in the mixed composition of model he4p0. O and Mg are overabundant in the ONeMg shell of the progenitor, while Ca is overabundant in the Si/S and Fe/He shells, both explosively produced. Hence, the abundance of these metals and the strengths of these O i, Mg ii, and Ca ii lines are strongly model dependent (see Sect. 4.3). All these lines are permitted transitions and are expected given the high density of the emitting region. This is a critical difference with standard Type Ibc SNe unaffected by interaction.

The present model has a total Rosseland-mean optical depth of 1.42 and a total electron-scattering optical depth of 0.85. A higher optical depth would have been achieved with a higher ejecta ionization: here helium (the most abundant element in this model with 0.92 $M_{\odot}$) is partially ionized, O (the second most abundant element in this model with 0.31 $M_{\odot}$) is once ionized, Mg is a mix of Mg⁺ and Mg²⁺, Ca is mostly Ca²⁺, and Fe is present in part as Fe⁺ but mostly Fe²⁺. These ionizations explain the weak strength of O i and Ca ii lines. The high ejecta density also inhibits the formation of [O i] $\lambda\lambda$ $6300,\,6364$ and [Ca ii] $\lambda\lambda$ $7291,\,7323$ that are normally seen in nebular phase spectra of Type Ibc SNe. However, the epochs and the power differ. Standard nebular-phase Type Ibc SNe are usually studied at late times of at least 150 d past explosion, when the absorbed power, from radioactive decay, is much smaller.

In the preceding section, the CDS was filled with the ejecta from the helium-star explosion model he4p0, assuming that at the late times considered here, the ejecta and CSM had “reunited” following the interaction. We now use the same approach and investigate the spectral properties obtained for interactions involving ejecta and CSM from a wide range of helium-star progenitor masses.

Figure 8 shows the UV and optical spectra for a CDS filled with the ejecta from the helium-star explosion models he2p9 to he12p0. The CDS radius is $3\times 10^{15}$ cm, its mean velocity is 2000 km s^-1, and the deposited power is $2\times 10^{42}$ erg s^-1. As the initial helium-star mass is increased from 2.9 to 12.0 $M_{\odot}$, the fractional helium mass $M$(He)/$M_{\rm ej}$ drops from 83% to 4% and the total mass (i.e., the CDS mass) increases from 0.93 to 5.32 $M_{\odot}$.

Although only the yields and the total mass change between models shown in Fig. 8, the spectra show a strong evolution. For the less massive and more helium-rich models, the spectrum is much bluer, with more flux emerging in the UV range, and most of the emission lines are associated with He i. Where the bulk of the shock power is deposited, the CDS is hot ($\sim$ 10000 K) and helium is nearly once ionized. This high ionization arises in part from the dominance of helium and the associated underabundance of metals, which would otherwise drive the gas to lower temperatures and lower ionization. As we progress towards a higher initial helium-star mass, the spectral energy distribution shifts to the red, He i lines weaken, and lines from metals (e.g., O i, Mg ii, or Ca ii) appear. This results from the combined effect of a lower fractional helium abundance and a lower ionization. In this sequence, we see that progenitors with an initial helium-star mass greater than 6 $M_{\odot}$ have weak He i lines that are essentially impossible to discern. This suggests that Type Ibn SNe require a high fractional helium abundance of about 50%, but not too high otherwise the spectrum is too blue (see Sect. 5), therefore favoring a low-mass massive star progenitor in a binary and having lost its hydrogen-rich envelope through Roche-lobe overflow.

The evolution of the strength of the He i 5875 Å line (given as a percentage of the optical flux) is illustrated for a larger grid of models in Fig. 9. Here, we considered a CDS radius of 2, 3, or $6\times 10^{15}$ cm, and deposited powers of 2 and $4\times 10^{42}$ erg s^-1 (assuming a CDS radius of $2\times 10^{15}$ cm). We see that only the composition and mass of the lighter models are compatible with a strong He i 5875 Å. This supports further the notion that Type Ibn SNe are connected with weak eruptions and explosions taking place in low-mass helium stars (but massive enough to undergo core collapse).

For a smaller CDS radius, the CDS density goes up at fixed mass and the ionization goes down. A greater power yields a higher ionization and a bluer optical spectrum. The rise in temperature can also boost the continuum flux, strong in the blue, and make the CDS completely optically thick in the continuum. Conversely, for larger radii, the optical depth is lower and the spectrum is more nebular (i.e., lines sit on a weaker continuum). Clumping can reduce the ionization and influence the spectral signatures at any epoch but it does not change the optical depth. Exploring this parameter space in detail is left to a future study, but one can envision that much diversity can arise from these different interaction configurations.

In this section, we present a criterion that may help distinguish Type Ibn SNe occurring in lower-mass massive stars in binaries from those that may result from pair-instability pulsations in a super massive star. The latter offer very attractive properties for Type Ibn SNe (Yoshida et al., 2016; Woosley, 2017), although the helium content might be too low to produce strong He i lines – they may produce a Type Icn rather than a Type Ibn SN.

Two important differences may separate pulsational-pair instability in very massive stars from instabilities (such as wave excitation or a nuclear flash) in lower mass massive stars. First, the pulsational-pair instability leads to thermonuclear burning at relatively low densities where no ${}^{56}\rm{N}$i is produced. Hence, at late times, these pulsations eject material that contains no ${}^{56}\rm{C}$o nor any ${}^{56}\rm{F}$e from the decay of ${}^{56}\rm{N}$i since the ${}^{56}\rm{N}$i mass is initially zero. Secondly, although strongly sensitive to the uncertain physics of wind mass loss, a low metallicity seems required for massive stars to encounter the regime of pair production and pulsation-induced ejections. The iron content of these ejecta may therefore be very low, perhaps a tenth of that expected at solar metallicity.

Figure 10 illustrates the spectral predictions for the default model he4p0 together with model counterparts in which the ${}^{56}\rm{N}$i-rich shell produced in the original explosion model was excluded (this model thus contains no ${}^{56}\rm{F}$e nor ${}^{56}\rm{C}$o from ${}^{56}\rm{N}$i decay) and another model in which, in addition, we scaled down the abundances of all metals heavier than Ar. We see that the largest impact is obtained in the model with a subsolar primordial metallicity, with an obvious shift of the spectral energy distribution to the blue, a reduced Fe ii emission in the optical range from 4000 to 5500 Å, and a strengthening of He i lines. In other words, the observation of a Type Ibn SN with strong Fe ii emission is evidence for a solar-metallicity event, potentially in tension with a pulsational-pair instability SN.

Determining precisely the iron abundance in the ejecta is however difficult because of the unknown level of mixing within the CDS. If some ${}^{56}\rm{N}$i was produced during the explosion, it could also be strongly mixed within the CDS and raise the iron abundance throughout, the more so for a large initial ${}^{56}\rm{N}$i mass and a small ejecta mass. We study this possibility using model he4p0 by enhancing the mixing so that the composition becomes uniform. In model he4p0, this raises the iron abundance to 0.016 throughout the CDS, hence about ten times the solar value. The impact on the spectrum is significant, with stronger line blanketing in the blue and a boost to the Fe ii emission in the range from 4000 to 5500 Å, as well as in the range 6000 to 6800 Å (Fig. 11). This hypothesis is an upper limit on the possible mixing in Type Ibn SNe and may never be realized in nature.

Figure 12 presents a comparison of our CDS models, based on the composition and mass of model he4p0, with the observations of SN 2006jc at 14.0 d (left panel) and 54.6 d (right panel) after the time of maximum. In the context of interaction, the shock decelerates as material piles up in the CDS (see, for example, the illustration in Fig. 16 for model E5). So, as the shock power drops in time (the more so for a stronger deceleration), the CDS mass grows, and the CDS velocity drops (see also discussion for a large set of simulations for Type IIn SNe in Dessart et al. 2015, 2016). Because the CDS moves, the CDS radius increases in time. So, when studying the properties of Type Ibn SNe are different epochs, it is natural to invoke different powers and CDS properties. The values used here for SN 2006jc may not be strictly correct since they are taken from the grid of models presented in Sect. 4, which were performed with no specific SN in mind.

A greater power and a smaller radius is used for the first epoch, leading to a bluer optical spectrum, a stronger continuum flux, and weaker lines. The model reproduces the overall shape and a number of observed spectral signatures are also predicted by the model. The overall shape of the spectrum is due to Fe  ii emission already at that time, although most of the Fe ii emission occurs below 4000 Å – we are mostly seeing the tail of the emission in the optical. The main He i lines predicted by the model and in general also observed are at 3889, 3965, 4026, 4471, 5016, 5876, 6678, 7065, and 7281 Å. The He i 3889 Å line is not observed while the model predicts it. This may be due to the blanketing effect of Fe ii on this He i line, which is weak in the model because He i 3889 Å is predicted to form in a large part in the outer regions of the emitting region. In reality, chemical mixing may cause a more intertwined emission of He i and Fe ii lines, causing a greater blanketing of He i lines by Fe ii lines.

The strongest predicted O  i lines are multiplets around 8446 Å, and there are also O  ii multiplets around 7320 Å. Two sets of strong Si ii lines are present at 5041–5056, 5958–5979, and 6347-6371 Å. Strong Mg ii  lines are predicted at 4481, 7896, 8234, 9218, and 9244 Å, but these are observed with a much lower strength. Overall, the model underestimates the width of line profiles (this does not seem to be a resolution issue since the observed spectral lines in this early-time spectrum are indeed broader than at later times, with the exception of the Ca ii near-infrared triplet), probably because the spectrum forms over a range of velocities and densities rather than in a narrow CDS as assumed. This assumption holds best at late times when ejecta and CSM have all been swept up into a dense confined region.

The right panel of Fig. 12 shows a comparison of SN 2006jc at 54.6 d after maximum with the CDS now at a larger radius of 6 rather than $2\times 10^{15}$ cm and powered at a rate of 2 rather than $6\times 10^{42}$ erg s^-1. These parameters are those used in the model discussed in Sect. 4.2. The model compares satisfactorily in terms of Fe ii emission in the blue, the presence of numerous He i lines, but the model overestimates the strength of Mg ii lines and underestimates the strength of Ca ii lines. This discrepancy may arise from an inadequate density structure (for example due the presence of clumping, which is neglected here). The CDS is also expected to have a very complicated structure, probably with multiple emission regions having a range of density compressions, ionization, temperature, turbulent velocity etc. Obtaining a better model will require more work, and in particular a better representation of the CDS and the interaction region based on realistic 3D radiation-hydrodynamics simulations (which are currently lacking).

Figure 13 compares the observations of SN 2011hw at 46 d after the inferred time of explosion with the model for a CDS composition and mass given by model he4p0, a radius of $2\times 10^{15}$ cm, a velocity of 2000 km s^-1, and powered at a rate of $2\times 10^{42}$ erg s^-1. The observations are corrected for reddening and for redshift adopting $E(B-V)=$ 0.115 mag and $z=$ 0.023; the adopted time of explosion is $MJD=$ 55870 (Pastorello et al., 2015a).

The agreement is better than for SN 2006jc, in particular for metal lines such as those of Mg ii or Ca ii. Compared to the 54.6 d spectrum of SN 2006jc, the 46.0 d spectrum of SN 2011hw shows stronger Mg ii lines and weaker Ca ii lines. The observed flux in the blue is well matched by the Fe ii emission in the model.

Figure 14 compares the observations of SN 2018bcc on the 14th of May 2018, 27.4 d after the estimated time of explosion at $MJD=$ 58225.5 (Karamehmetoglu et al., 2021), with the model spectrum that adopts a CDS mass and composition based on model he4p0, placed at a radius of $3\times 10^{15}$ cm, moving at a velocity of 2000 km s^-1 and powered at a rate of $2\times 10^{42}$ erg s^-1. The comparison is a close analog to that shown for SN 2011hw but serves to show that despite the diversity of Type Ibn SNe, there is also a strong similarity in spectral properties, in particular a few weeks after the light curve maximum.

Although the features are not always well matched in strength, most of the observed features are predicted in the model. Just like for SNe 2006jc and 2011hw, the Mg ii lines are overestimated and the Ca ii lines are too weak, suggesting an ionization offset in the spectrum formation region.