On the Formation of the W-shaped O ii Lines in Spectra of Type I Superluminous Supernovae

For calculations of synthetic spectra, we utilize a one-dimensional Monte Carlo radiative transfer code (Mazzali & Lucy 1993; Mazzali 2000). Here, we briefly describe the code by focusing on the assumptions relevant to this work. The code calculates the plasma conditions under the modified nebular approximation. The modified nebular approximation takes into account the fact that, in the SN atmosphere, the densities are low, and radiative processes dominate, having the greatest influence on the level of the population (Abbott & Lucy 1985; Mazzali & Lucy 1993). Thus, the code does not assume LTE both for ionization and excitation. The modified nebular approximation is known to work well for spectra of normal SNe (e.g., Pauldrach et al. 1996; Sauer et al. 2006; Tanaka et al. 2008; Hachinger et al. 2012; Teffs et al. 2020). For more details of the code, we refer readers to Mazzali & Lucy (1993) and Mazzali (2000).

The code assumes a sharp photosphere inside the ejecta. At the photosphere, blackbody radiation is emitted. Photon packets replicating flux are then propagated through the SN ejecta outside the photosphere. For the photon propagation in the ejecta, electron scattering and bound–bound absorption are taken into account. The degree of scattering and absorption is determined by the level of the populations and ionization in the ejecta, which depends on density and temperature. A temperature structure in the ejecta outside the photosphere is established by tracing the photon packets, differentiating between a local radiation temperature (T_R) and an electron temperature (T_e). The local electron temperature is crudely assumed to be T_e = 0.9 T_R. The temperatures are estimated by an iterative process as the temperatures give the level of the populations and ionization degrees, which in turn affect the energy flux, and thus, the temperatures.

The level of the population (number density) of the jth ionized element with atomic number i at the lth excitation level n_i,j,l is (at some given radius)

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{i,j,l}}{{n}_{i,j,0}}=W\displaystyle \frac{{g}_{i,j,l}}{{g}_{i,j,0}}{e}^{-{E}_{i,j,l}/{{kT}}_{{\rm{R}}}},\end{eqnarray} \tag{ 1 }$$

where g_i,j,l is the statistical weight, E_i,j,l is the excitation energy from the ground state, k is the Boltzmann constant, and T_R is a radiation temperature. W in Equation (1) is the dilution factor calculated as

$$\begin{eqnarray}&&J={WB}({T}_{{\rm{R}}}),\end{eqnarray} \tag{ 2 }$$

where J is the frequency-integrated mean intensity from the simulation and B(T_R) is the frequency-integrated Planck function with the radiation temperature T_R at some given radius.

Ionization is estimated by the modified nebular approximation (Mazzali & Lucy 1993):

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{i,j+\mathrm{1,0}}{n}_{{\rm{e}}\ }}{{n}_{i,j,0}}=\eta W\sqrt{\displaystyle \frac{{T}_{{\rm{e}}}}{{T}_{{\rm{R}}}}}2\displaystyle \frac{{g}_{i,j+\mathrm{1,0}}}{{g}_{i,j,0}}\displaystyle \frac{{(2\pi {m}_{{\rm{e}}}{{kT}}_{{\rm{R}}})}^{3/2}}{{h}^{3}}{e}^{-{I}_{i,j}/{{kT}}_{{\rm{R}}}},\end{eqnarray} \tag{ 3 }$$

where n_e is the number density of electrons, m_e is the mass of the electron, h is the Planck constant, and I_i,j is the ionization potential of the jth ionized element with the atomic number i. In Equation (3), η is defined as

$$\begin{eqnarray}&&\eta =\delta \zeta +W(1-\zeta ),\end{eqnarray} \tag{ 4 }$$

where δ is a correction factor for the optically thick region at wavelengths shorter than λ₀ = 1050 Å (the Ca ii edge), and ζ is the fraction of recombinations going directly to the ground state. Typically, η ∼ 0.4 for O just outside the photosphere (W ∼ 0.5).

Line scattering by bound–bound transitions from a lower-level l to an upper-level u in homologously expanding ejecta is treated in the Sobolev approximation (Sobolev 1957). A particularly important quality in this context is the Sobolev optical depth

$$\begin{eqnarray}&&\begin{array}{l}{\tau }_{{lu}}=\displaystyle \frac{{hc}}{4\pi }({B}_{{lu}}{n}_{i,j,l}-{B}_{{ul}}{n}_{i,j,u}){t}_{\mathrm{expl}}\\ \,\simeq \displaystyle \frac{\pi {e}^{2}}{{m}_{{\rm{e}}}c}{f}_{{lu}}{n}_{i,j,l}{t}_{\mathrm{expl}}{\lambda }_{{lu}},\end{array}\end{eqnarray} \tag{ 5 }$$

where B_lu and B_ul are Einstein B-coefficients, f_lu is the oscillator strength of the transition, t_expl is the time since explosion, and λ_lu is the wavelength of the transition. The effect of line fluorescence is also taken into account (Mazzali 2000).

The density structure of the ejecta outside the photosphere follows a power law (ρ ∝ r⁻ⁿ , where ρ is density and r is radius) with index n = 7. The radius can be expressed by a velocity v in homologously expanding ejecta (r = vt_expl). In our fiducial model (iPTF 13ajg, see Section 4), the ejecta mass above v = 12,000 km s⁻¹ is M_ej(v ≥ 12,000 km s⁻¹) = 3.7 M_⊙. This ejecta mass is scaled by f_ρ parameter as described below. Note that the total ejecta mass cannot be estimated from our modeling as the radiative transfer is solved only outside of the photosphere. Thus, in the following sections, we give the mass outside of a typical velocity, for which we adopt v = 12,000 km s⁻¹, as a representative value. Abundances in the ejecta are assumed to be homogeneous, and set to be the same as those adopted by Mazzali et al. (2016) for the first spectrum of iPTF 13ajg for all the 66 spectra analyzed in this work: X(He) = 0.10, X(C) = 0.40, X(O) = 0.475, X(Ne) = 0.02, X(Mg) = 5 × 10⁻⁴, X(Si) = 2 × 10⁻³, X(S) = 5 × 10⁻⁵, X(Ca) = 5 × 10⁻⁵, X(Ti) = 2 × 10⁻⁵, X(Fe) = 5 × 10⁻⁴, X(Co) = 3 × 10⁻⁴, and X(Ni) = 1 × 10⁻⁴.

The parameters of the spectral calculations are as follows:

• 1.  
  f_ρ : density scaling factor
• 2.  
  L_bol: bolometric luminosity [erg s⁻¹]
• 3.  
  v_ph: velocity at the photosphere [km s⁻¹]
• 4.  
  t_expl: time since explosion in the rest frame [days]
• 5.  
  b_neb: departure coefficient for the excited states of O ii.

Note that the purpose of this work is to understand the condition for the W-shaped O ii lines to appear as compared with those of normal SNe. Thus, we define the departure coefficient b_neb as a departure from the population expected from the modified nebular approximation (n_{O II} = b_neb × n_{O II, neb}, where n_{O II} and n_{O II, neb} are the number density of O ii in the excited state in ejecta of SLSNe I and that in the modified nebular approximation, respectively).

These parameters in our calculations to model the observed spectra can almost independently be determined from the following physical quantities. We constrain the density scaling parameter f_ρ from the dilution factor so that the dilution factor W at v_ph estimated by Equation (2) is converged to ∼1/2 (i.e., close to the value of the geometric dilution factor) for a given combination of the other parameters. f_ρ is kept the same for the spectral series of each SN. L_bol is constrained from the observed flux (with an accuracy of ∼10%). v_ph is estimated from wavelengths of blueshifted absorption lines. t_expl is then constrained using ${L}_{\mathrm{bol}}=4\pi {({v}_{\mathrm{ph}}{t}_{\mathrm{expl}})}^{2}\sigma {{T}_{\mathrm{eff}}}^{4}$, where σ is the Stefan–Boltzmann constant, and T_eff is the effective temperature. We estimate t_expl for each spectrum so that the entire spectral series for each SN is reproduced by the same explosion date. The estimated explosion date gives a rise time to the peak of the light curve (Table 1). The light curves in Figure 2 are shown as a function of the time since the estimated explosion date, and our estimated rise time is consistent with (or longer than) that estimated from an extrapolation of the light curve. Finally, b_neb is estimated from the depth of the W-shaped O ii lines.

Figure 3 shows an example of a comparison between synthetic spectra and an observed spectrum (the first spectrum of LSQ 14mo). The observed spectrum is reproduced reasonably well with a departure coefficient of b_neb = 1 (no departure from the nebular approximation) at an estimated radiation temperature just outside the photosphere of T_R ∼ 15,000 K. The model with b_neb = 10 produces too deep W-shaped O ii lines. The wavelengths of the W-shaped O ii lines are well matched with a photospheric velocity v_ph = 10,500 km s⁻¹. The UV fluxes are strongly affected by metal absorption, which makes it difficult to estimate temperatures when observed spectral energy distributions are simply fitted by blackbody functions. Models with the best parameters for all the spectra in our sample are shown in Figure 4 and the best parameters are summarized in Table A1 (Appendix 6). The properties of each SN are described below.

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iPTF 13ajg: First, we show the results of modeling for iPTF 13ajg, which we use as our fiducial model, as this object was also modeled in Mazzali et al. (2016). This object was reported in Vreeswijk et al. (2014). The absolute AB magnitude at the u-band peak is ∼ −22.5 mag (Vreeswijk et al. 2014).

For the modeling, we adopt the same parameters as in Mazzali et al. (2016). By definition, the density scaling factor is f_ρ = 1.0, which corresponds to the ejecta mass outside above v = 12,000 km s⁻¹ of M_ej(v ≥ 12,000 km s⁻¹) = 3.7 M_⊙. This object shows the W-shaped O ii lines until t_expl = 54 days. The required departure coefficients to reproduce the depths of the W-shaped O ii lines are b_neb ∼ 50. The depths of the W-shaped O ii lines of this object require one of the largest departure coefficients among all the objects in our sample.

PTF 09atu. This object was reported in Quimby et al. (2011b). The absolute AB magnitude at the u-band peak is ∼ −21.5 mag (Quimby et al. 2011b). To reproduce the time series of the spectra, the density scaling factor for this object is found to be f_ρ = 0.5, which corresponds to the ejecta mass outside v = 12,000 km s⁻¹ of M_ej(v ≥ 12,000 km s⁻¹) = 1.8 M_⊙. This object shows the W-shaped O ii lines at all the epochs at which spectra were taken (t_expl = 23–46 days). The required departure coefficients to reproduce the depths of the W-shaped O ii lines are b_neb ∼ 1–50.

PTF 09cnd. This object was also reported in Quimby et al. (2011b). The absolute AB magnitude at the u-band peak is ∼ −22 mag (Quimby et al. 2011b). The spectra taken at MJD 55055 (2009 August 12) and MJD 55068 (2009 August 25) were also modeled in Mazzali et al. (2016). We updated the flux calibration of the spectra using the newly published photometric data (De Cia et al. 2018); the fluxes are found to be higher than those in Mazzali et al. (2016) by a factor of ∼2.5. Thus, the bolometric luminosity of the best model in this work is also higher than that in Mazzali et al. (2016) by a factor of ∼2.5. With the increase of the bolometric luminosity, the time since explosion is longer than that in Mazzali et al. (2016) to have a larger photospheric radius. This updated time is also consistent with the light curve. The density scaling factor for this object is found to be f_ρ = 0.7, which corresponds to the ejecta mass outside v = 12,000 km s⁻¹ of M_ej(v ≥ 12,000 km s⁻¹) = 2.6 M_⊙. Despite the differences in the luminosity and time, the overall properties of the model are consistent with those derived by Mazzali et al. (2016), e.g., the inferred density scaling factor (or mass outside of a certain velocity) for PTF 09cnd is smaller than that for iPTF 13ajg.

This object shows the W-shaped O ii lines until t_expl = 65 days. The required departure coefficients to reproduce the depths of the W-shaped O ii lines are b_neb ∼ 1–10. Note that the absorption line around 2700 Å cannot be reproduced by our models because of the fixed homogeneous abundance. Mazzali et al. (2016) reproduce those lines by Mg ii with the abundance of X(Mg) ∼0.1, but we fixed the abundance to X(Mg) = 5 × 10⁻⁴.

SN 2010gx. This object was reported in Pastorello et al. (2010b), Mahabal et al. (2010), and Quimby et al. (2011b) as PTF10cwr, and in Pastorello et al. (2010a) as SN 2010gx. The absolute AB magnitude at the u-band peak is ∼ −21.7 mag (Quimby et al. 2011b). For this object, the density scaling parameter is found to be f_ρ = 3.0, which corresponds to the ejecta mass outside v = 12,000 km s⁻¹ of M_ej(v ≥ 12,000 km s⁻¹) = 11 M_⊙. This object shows the W-shaped O ii lines until t_expl = 25 days. The required departure coefficients to reproduce the depths of the W-shaped O ii lines are b_neb ∼ 1–3. After t_expl = 25 days, the W-shaped O ii lines disappear. The observed spectra taken after MJD 55323 (t_expl = 63 days) could not be reproduced, possibly because of the fixed abundance and/or the single-power density law and the mass assumption. Thus, we only show models for the observed spectra before MJD 55323.

SN 2011kg. This object was reported in Quimby et al. (2011a). The absolute AB magnitude at the g-band peak is ∼ −20.8 mag (Inserra et al. 2013). For this object, the density scaling parameter is found to be f_ρ = 0.1, which corresponds to the ejecta mass outside v = 12,000 km s⁻¹ of M_ej(v ≥ 12,000 km s⁻¹) = 0.4 M_⊙. This object shows the W-shaped O ii lines until t_expl = 37 days. The required departure coefficients to reproduce the depths of the W-shaped O ii lines are b_neb ∼ 1–10 (small departure). After t_expl = 37 days, the W-shaped O ii lines disappear. The absorption lines around ∼3000 and ∼3500 Å may be due to Fe ii and/or Si ii, which are not reproduced with our fixed abundance. However, this does not affect the departure coefficient required for the O ii lines.

PTF 12dam. This object was reported in Quimby et al. (2012). The absolute AB magnitude at the g-band peak is ∼ −22 mag (Chen et al. 2017). For this object, the density scaling parameter is found to be f_ρ = 0.4, which corresponds to the ejecta mass outside v = 12,000 km s⁻¹ of M_ej(v x02A7E; 12,000 km s⁻¹) = 1.5 M_⊙. This object shows the W-shaped O ii lines until t_expl = 59 days. The required departure coefficients to reproduce the depths of the W-shaped O ii lines are up to b_neb ∼ 50. After t_expl = 59 days, the W-shaped O ii lines disappear.

LSQ 14mo. This object was reported in Nicholl et al. (2015). The absolute AB magnitude at the g-band peak is ∼ −21.2 mag (Leloudas et al. 2015). The spectra of this object were also modeled in Chen et al. (2017). The first spectrum (at −7 days) is modeled with t_expl = 18 days. However, the model spectrum shows a somewhat higher velocity of the W-shaped O ii lines as compared with the observed spectrum. To have a better match in the velocity, we adopt the photospheric velocity to be smaller than that estimated in Chen et al. (2017) by ∼5000 km s⁻¹. Accordingly, the time since explosion in our model for this spectrum is estimated to be t_expl = 33 days. This revised time is compatible with the light curve, and reproduces the later spectra as well. With this choice, the density scaling parameter for this object is found to be f_ρ = 0.2, which corresponds to the ejecta mass outside v = 12,000 km s⁻¹ of M_ej(v ≥ 12,000 km s⁻¹) = 0.7 M_⊙.

This object shows the W-shaped O ii lines until t_expl = 54 days. The required departure coefficients to reproduce the depths of the W-shaped O ii lines are b_neb ∼ 1 (no departure). After t_expl = 54 days, the W-shaped O ii lines disappear.

SN 2015bn. This object was reported in Nicholl et al. (2016). The absolute AB magnitude at the g-band peak is ∼ −22 mag (Nicholl et al. 2016). For this object, the density scaling parameter is found to be f_ρ = 0.4, which corresponds to the ejecta mass outside v = 12,000 km s⁻¹ of M_ej(v ≥ 12,000 km s⁻¹) = 1.5 M_⊙. This object lacks the W-shaped O ii lines throughout the epochs of available spectra (t_expl = 63–135 days). This is the only object that does not show the W-shaped O ii lines at any epochs in our sample.

In this section, we show the behavior of the temperature and the departure coefficient as a function of time. The left panel of Figure 5 shows radiation temperatures just outside the photospheres in the models with the best parameters as a function of time since explosion. The temperatures decrease with time. It is clear that the only spectra with temperatures higher than T_R ∼ 12,000 K show the W-shaped O ii lines (see also Könyves-Tóth 2022). The dependence of the W-shaped O ii lines on temperature is discussed in more detail in Section 5.1. The W-shaped O ii lines can appear only up to t_expl ∼ 60 days. In fact, more luminous objects tend to show the W-shaped O ii lines somewhat longer (see the points with the black edges in Figure 2). This is because, for more luminous objects, higher temperatures can be achieved for a longer time.

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The right panel of Figure 5 shows the relations between time since the explosion and the departure coefficients. The W-shaped O ii lines in many of the observed spectra in the early phases (t_expl ≲ 50 days) are well reproduced with the departure coefficients of b_neb ∼ 1. This means that the W-shaped O ii lines are formed with little departure from the population in the nebular approximation. Some of the spectra with the W-shaped O ii lines around the phases t_expl ∼ 50–70 days require somewhat larger departure coefficients of b_neb ∼ 10–50. For the observed spectra without the W-shaped O ii lines, we obtained the upper limits of the departure coefficients. When the departure coefficients exceed the upper limits, the synthetic spectra will produce the W-shaped O ii lines. The upper limits of the departure coefficients become larger at later times.

Temperatures and departure coefficients are plotted against one another in Figure 6. There is a tendency for spectra with higher (lower) temperatures to require smaller (larger) departure coefficients. The spectra with the temperatures of T_R ≳ 14,000 K show the W-shaped O ii lines with the departure coefficient of b_neb = 1 (no departure). As the temperatures decrease, larger departure coefficients are requisite for the spectra with the W-shaped O ii lines. The departure coefficient for the W-shaped O ii lines in the spectra of the SLSNe I in our sample is b_neb ∼ 50 at most. This is in contrast to the departure coefficient for the He i lines in spectra of SNe Ib (b ∼ 10⁴–10⁵; Lucy 1991).

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Here, we discuss the dependence of the strength of the W-shaped O ii lines on temperature. Figure 7 shows a series of synthetic spectra with various temperatures. For the calculations of the synthetic spectra, the bolometric luminosity L_bol was parameterized to change the temperature. The model with the radiation temperature just outside the photosphere of T_R ∼ 15,000 K shows the strongest W-shaped O ii lines. Neither the models with temperatures of T_R ≲ 13,000 K nor with T_R ≳ 18,000 K show the W-shaped O ii lines.

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To understand this behavior, we evaluate the strength of the absorption lines from the Sobolev optical depth as in Equation (5) under the nebular approximation (see Hatano et al. 1999 for a similar analysis for normal SNe under LTE). Absorption lines can appear when the Sobolev optical depth is τ_lu ≳1 above the photosphere over a range of velocities. A typical velocity range of the line-forming region is v ∼ v_ph−1.5 v_ph to reproduce the width of the absorption line seen in the O ii lines.

The left panel of Figure 8 shows the fraction of O ii at typical photospheric parameters evaluated with the same abundance adopted for the spectral synthesis calculations in Section 3.2. Near the photospheres, density is typically ρ = 3 × 10⁻¹⁴ g cm⁻³, which is almost constant over time. The right panel of Figure 8 shows the Sobolev optical depth τ_lu of the W-shaped O ii lines at the photosphere obtained from the ionization fraction and Equations (1) and (5). The Sobolev optical depth shown here is that of one of the most prominent O ii lines, λ_lu = 4649 Å. For the line, we apply f_lu = 0.34 (Wiese 1996) and t_expl = 40 days in Equation (5).

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The Sobolev optical depth peaks at T_R ∼ 15,000 K with a value τ_lu ∼ 4. As the temperature decreases from T_R ∼ 15,000 K, the Sobolev optical depth steeply decreases since the number density of the excited state decreases with the Boltzmann factor. On the other hand, as the temperature increases from T_R ∼ 15,000 K, the Sobolev optical depth decreases because O ii is ionized to O iii. Therefore, a temperature of around T_R ∼ 15,000 K makes the Sobolev optical depth peak. Only at T_R ∼ 13,000–18,000 K (T_R ∼ 14,000–16,000 K), the Sobolev optical depth exceeds τ_lu ∼ 1 (τ_lu ∼ 3).

This dependence of line strength on temperature can also be seen in Figure 6. The synthetic spectra with temperatures higher than T_R ∼ 14,000 K show the W-shaped O ii lines with a departure coefficient of b_neb = 1. On the other hand, the synthetic spectra with temperatures lower than T_R ∼ 14,000 K require departure (b_neb > 1) to show the W-shaped O ii lines. Most of the observed spectra with temperatures of T_R ≲ 12,000 K do not show the W-shaped O ii lines.

Although SLSNe I are sometimes classified into two types based on the absence or presence of the W-shaped O ii lines in their spectra (e.g., Könyves-Tóth & Vinkó 2021), our results indicate that the difference between the two types would be only temperature. It is natural that all the spectra of SN 2015bn do not show the W-shaped O ii lines because of their low temperatures.

We now examine the effect of host extinction on the departure coefficients via temperature. Although the temperatures estimated in the model are affected by host extinction, it was not corrected for as mentioned in Section 2. Here, we investigate a possible host extinction of iPTF 13ajg, which requires one of the largest departure coefficients (b_neb ≲ 50) among our sample. The upper limit of the host extinction of iPTF 13ajg is estimated to be E(B − V) < 0.12 from the absence of Na I D lines (Vreeswijk et al. 2014). Thus, we applied E(B − V) = 0.05 to the second spectrum of iPTF 13ajg taken at MJD 56391 (one of the best spectra).

The host-extinction-corrected spectra and the models with the best parameters are shown in Figure 9. For the model of the spectrum with E(B − V) = 0.0 (before the host extinction correction), the best parameters are f_ρ = 1.0, L_bol = 10^44.5 erg s⁻¹, v_ph = 12,250 km s⁻¹, t_expl = 50 days, and b_neb = 30. These parameters give a radiation temperature just outside the photosphere of T_R ∼ 13,000 K. When we assume E(B − V) = 0.05, the intrinsic spectrum becomes brighter and bluer. Then, the best parameters are f_ρ = 1.0, L_bol = 10^44.6 erg s⁻¹, v_ph = 12,250 km s⁻¹, t_expl = 47 days, and b_neb = 1. These parameters give a radiation temperature just outside the photosphere of T_R ∼ 14,500 K. By this somewhat higher temperature, the model spectrum with b_neb = 1 produces a deeper W-shaped O ii absorption feature.

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These results demonstrate that the required departure coefficients are quite sensitive to the host extinction, which is always difficult to estimate. Even a small host extinction correction greatly affects the UV fluxes, increasing the estimated temperature. Thus, the required departure coefficient tends to be smaller when the host extinction is applied.

We here examine nonthermal processes in the ejecta of SLSNe despite our finding that the population of the excited states of O ii does not largely deviate from the population in the nebular approximation. This may yield constraints on the power source of SLSNe I because the population would be influenced by nonthermal excitation/ionization to some extent whenever there are γ-rays from either ⁵⁶Ni or a magnetar in the ejecta. We test this for temperatures in the range of T = 8000–12,000 K, where the nebular approximation does not lead to the formation of the W-shaped O ii lines.

In analogy to He i lines in spectra of SNe Ib, the ionization rate can be constrained by the population of the excited states of O ii. For the appearance of the He i lines, the excited states of He i are populated by high-energy electrons that are produced by Compton scattering of γ-rays from ⁵⁶Ni decay (Lucy 1991; Hachinger et al. 2012). The high-energy electrons ionize He i to He ii with a rate determined by the energy deposition rate of γ-rays from ⁵⁶Ni decay. Then, He ii recombines to the excited states of He i (see Figure 1 of Tarumi et al. 2023). Therefore, the population of the excited states of He i is related to the nonthermal ionization rate. While the presence of He i lines in spectra of SNe Ib gives the nonthermal ionization rate, the absence of the W-shaped O ii lines in spectra of SLSNe I can give an upper limit to the nonthermal ionization rate through the population of the excited states of O ii.

The number density of the excited states of O ii should not exceed a certain value that produces the W-shaped O ii lines. The typical value of the Sobolev optical depth producing absorption lines is of the order of τ_lu ∼ 1. This corresponds to densities of n_i,j,l ≳ 1 cm⁻³ for the W-shaped O ii lines obtained from Equation (5) by adopting λ_lu = 4649 Å, f_lu = 0.34 (Wiese 1996), and t_expl = 40 days. This number density is discussed further below and shown to give an upper limit of the ionization rate. The ionization rate can then be converted to an energy deposition rate through the fraction of the deposition energy spent for ionization (among ionization, excitation, and heating of thermal electrons, see, e.g., Fransson & Chevalier 1989), which is called work per ion pair. Finally, this energy deposition rate can be related to power sources through the γ-ray spectra of power sources (and γ-ray transport).

In order to estimate the population of the excited states of O ii under certain ionization and excitation conditions, we solve a simplified rate equation as illustrated in Figure 10. We only consider three levels: the ground state of O ii, the ground state of O iii (35.1 eV higher than the ground state of O ii), and the excited O ii spin-quartet state (23.0 eV higher than the ground state of O ii). We do not consider the case of the excited state of O ii in the spin-doublet state, which is expected to be similar to that in the spin-quartet state. This is because both the excited states would be balanced to be comparably occupied via O iii. This is also seen in the excited state of He i in the spin-singlet state and that in the spin-triplet state achieved by the balance via He ii (Lucy 1991). Also, we separately solve two equations below: an equation for the balance between the ground state of O ii and the ground state of O iii, and an equation for inflow to and outflow from the excited O ii spin-quartet state.

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We first consider the balance between the ground state of O ii and that of O iii by nonthermal processes. The balance between the ionization from O ii to O iii and the recombination from O iii to O ii gives the number density of O iii n_{O III} :

$$\begin{eqnarray}&&\beta {n}_{{\rm{O}}\,{\rm\small{II}}}={n}_{{\rm{O}}\,{\rm\small{III}}}{n}_{{\rm{e}}}{\alpha }_{{\rm{O}}\,{\rm\small{III}}\to {\rm{O}}\,{\rm\small{II}}}(T),\end{eqnarray} \tag{ 6 }$$

where β is the nonthermal ionization rate (per second), n_{O II} is the number density of the ground state of O ii, n_e is the number density of electrons, and α_{O III→O II} (T) is an effective recombination rate from the ground state of O iii to the ground state of O ii. Since we consider only nonthermal ionization here, the ionization rate β is expressed as

$$\begin{eqnarray}&&\beta =\displaystyle \frac{\dot{e}}{w},\end{eqnarray} \tag{ 7 }$$

where $\dot{e}$ is the energy deposition rate per particle given by power sources (erg per second), and w is the energy (erg) required to ionize an ion (work per ion pair). Here, w is usually obtained from the Spencer–Fano equation (Spencer & Fano 1954; Kozma & Fransson 1992).

Treatment of the recombination processes is not trivial. The direct recombination to the ground state is often largely suppressed because it is immediately followed by absorption. In fact, in a typical density we consider, the optical depth of the recombination photons is quite high (τ > 10⁶). In a realistic ejecta including various elements other than O, the recombination photons can also be absorbed by other ions, and thus, some direct recombination can still occur. Another path is recombination through the excited states. However, since we do not solve a full rate equation including the excited states of O ii (see below), it is difficult to accurately evaluate the effective recombination rate through many excited states. Under these circumstances, we approximately adopt the direct recombination rate to the ground state from Nahar (1999) as an effective recombination rate α_{O III→O II} (T) in Equation (6). This corresponds to the limit of the most efficient recombination to O ii. In reality, the recombination rate would be lower than what is adopted here, and the number density of O iii ions would be enhanced (implications of this assumption are discussed below).

Next, we solve the equation for inflow to and outflow from the excited O ii spin-quartet state. This gives the number density of O ii in that state n_{O II, ex}:

$$\begin{eqnarray}&&\begin{array}{l}{n}_{{\rm{O}}\,{\rm\small{III}}}{n}_{{\rm{e}}}{\alpha }_{{\rm{O}}\,{\rm\small{III}}\to {\rm{O}}\,{\rm\small{II}},\mathrm{ex}}(T)\\ =\,{n}_{{\rm{O}}\,{\rm\small{II}},\ \mathrm{ex}}({R}_{\mathrm{PI}}+{n}_{{\rm{e}}}{q}_{{\rm{O}}\,{\rm\small{II}},\mathrm{ex}\to {\rm{O}}\,{\rm\small{II}}}\\ +{n}_{{\rm{e}}}{q}_{{\rm{O}}\,{\rm\small{II}},\mathrm{ex}\to {\rm{O}}\,{\rm\small{III}}}+{\beta }_{\mathrm{esc}}{A}_{\mathrm{ex}\to \mathrm{gs}}),\end{array}\end{eqnarray} \tag{ 8 }$$

where α_{O III→O II,ex}(T) is the recombination rate from the ground state of O iii to the excited O ii spin-quartet state, R_PI is a photoionization rate, q_{O II,ex→O II} is the collisional transition rate from the excited O ii spin-quartet state to the ground state of O ii, q_{O II,ex→O III} is the collisional transition rate from the excited O ii spin-quartet state to the ground state of O iii, β_esc is the Sobolev escape probability, and A_ex→gs is the Einstein coefficient for the transition from the excited O ii spin-quartet state to the ground state of O ii. Note that the last term on the right side of Equation (8) includes both self-absorption and spontaneous emission. The photoionization rate R_PI is computed as

$$\begin{eqnarray}&&{R}_{\mathrm{PI}}={\int }_{h\nu =12.1\ \mathrm{eV}}^{\infty }{\sigma }_{\mathrm{PI}}(\nu )\displaystyle \frac{{{WB}}_{\nu }(T)}{h\nu }d\nu ,\end{eqnarray} \tag{ 9 }$$

where σ_PI(ν) is a photoionization cross section. The transition rate by electron collisions q_if is computed as

$$\begin{eqnarray}\begin{array}{l}{q}_{i\to f}=\left\{\begin{array}{ll}\displaystyle \frac{8.63\times {10}^{-6}}{{g}_{i}\sqrt{T}}{\gamma }_{{if}}{e}^{-}({E}_{f}-{E}_{i})/{kT} & ({E}_{i}\lt {E}_{f})\\ \displaystyle \frac{8.63\times {10}^{-6}}{{g}_{i}\sqrt{T}}{\gamma }_{{if}} & ({E}_{i}\gt {E}_{f})\end{array}\right.\end{array}\end{eqnarray} \tag{ 10 }$$

(Eissner et al. 1969), where g_i is the statistical weight of an initial state, γ_if is an effective collision strength from an initial state to a final state, and E_i and E_f are an energy level of an initial state and a final state, respectively. The escape probability β_esc is computed as

$$\begin{eqnarray}&&{\beta }_{\mathrm{esc}}=\displaystyle \frac{1-{e}^{-{\tau }_{{lu}}}}{{\tau }_{{lu}}}\end{eqnarray} \tag{ 11 }$$

(Castor 1970) with the Sobolev optical depth τ_lu in Equation (5).

The procedure above gives the number density of the excited states of O ii n_{O II, ex} as a function of the ionization rate (Figure 11). Here, we assumed that all the ejecta consist of O, and that O ii is dominant, as shown in the left panel of Figure 8. This provides a condition of n_e = n_{O II} + 2n_{O III} . Also, we adopted the following parameters: ρ = 3 × 10⁻¹⁴ g cm⁻³ as a typical density near the photosphere, α_{O III→O II,ex}(T) from Nahar (1999), σ_PI = 10⁻¹⁷ cm² (Nahar 1999), g_i = 4, γ_if = 1, E_i = 23.0 eV, E_f = 0.0 eV for the collisional de-excitation, E_f = 35.1 eV for the collisional ionization, A_ex→gs = 9.8 × 10⁸ s⁻¹ (Nahar 2010), and W = 1/2 for τ_lu , f_lu = 4.3 × 10⁻² (Wiese 1996), n_i,j,l = n_{O II} , λ_lu = 539Å, and t_expl = 40 days. Note that the photoionization cross section is assumed to be constant because the ionization cross section has many resonances just above the ionization energy. As the peak of the blackbody radiation with a temperature range of T = 8000–12,000 K is located at much lower energy than the ionization threshold, the assumption of the constant cross section is effectively applied just above the ionization threshold.

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Figure 11 shows the number densities for the excited O ii spin-quartet near the photosphere as a function of the ionization rate. The number density of the excited O ii spin-quartet exceeds the critical density n_{O II,ex} ∼ 1 cm⁻³ (the Sobolev optical depth τ_lu ∼ 1) when the ionization rate is β ∼ 3 × 10⁻⁶ s⁻¹ at T = 10,000 K. At T = 8000 and 2,000 K, the number density for the excited state of O ii exceeds the critical density when the ionization rate is β ∼ 10⁻⁶ s⁻¹ and β ∼ 3 × 10⁻⁵ s⁻¹, respectively.

We note that the estimated number densities for the excited state of O ii are lower-limit estimates because we assume blackbody radiation even at short wavelengths. In actual SNe, photons at short wavelengths are suppressed because of line blanketing. Smaller photoionization rates should increase the estimated number densities of the excited stated of O ii. To show this effect, we compare the number densities for the excited O ii spin-quartet with photoionization rate to those without photoionization since the photoionization rate is uncertain because of line blanketing. The results at all the temperatures of T = 8000, 10,000, and 12,000 K without photoionization are almost the same as the result at T = 8000 K with photoionization. At the temperatures of T = 10,000 and 12,000 K, the effect of the photoionization is the strongest to de-populate the excited state of O ii in comparison to all the other processes (collisional de-excitation, collisional ionization, and spontaneous emission with self-absorption). In light of line blanketing, the result with a radiation temperature of T = 8000 K may best represent the actual condition in the ejecta of SLSNe.

The ionization rate can be translated to the energy deposition rate if the work per ion pair in Equation (7) is given. The work per ion pair becomes larger when the electron fraction is larger since a larger fraction of the deposition energy is spent for heating of thermal electrons, not for ionization (Kozma & Fransson 1992). In electron-rich environments where the ionization of metals produces many electrons, a typical value of the work per ion pair (calculated for SNe Ia) is w ∼ 30 I, where I is an ionization potential for an ion (Axelrod 1980). Hereafter, we assume a work per ion pair of w = 30 I for ionization of O ii to O iii in SLSNe I. The deposition rates then corresponding to an ionization rate β that makes n_{O II,ex} ≳ 1 cm⁻³ are $\dot{e}\sim 2\times {10}^{-15}$, $\dot{e}\sim 5\times {10}^{-15}$, and $\dot{e}\sim 5\times {10}^{-14}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$ at T = 8000, 10,000, and 12,000 K, respectively.

Finally, we demonstrate that the energy deposition rate can be related to power sources if the γ-ray spectrum emitted by the power sources and subsequent γ-ray transport are known. Here, to give crude constraints on the mass of ⁵⁶Ni in SLSNe I, we hypothetically assume that ⁵⁶Ni is the power source for SLSNe I. For simplicity, ⁵⁶Ni is assumed to uniformly supply the deposition energy to the ejecta (full mixing). The optical depth of γ-rays is τ_γ ≳ 1 in photospheric phases. Thus, for fully mixed ⁵⁶Ni distribution, the energy deposition rate given by ⁵⁶Ni decay can be approximately written as

$$\begin{eqnarray}&&\dot{e}\simeq \displaystyle \frac{{L}_{\mathrm{decay}}}{N},\end{eqnarray} \tag{ 12 }$$

where L_decay is the decay luminosity of ⁵⁶Ni (erg per second) and N is the number of atoms in the ejecta. The decay luminosity is given as (e.g., Nadyozhin 1994)

$$\begin{eqnarray}&&\begin{array}{l}{L}_{\mathrm{decay}}({t}_{\mathrm{expl}})=(6.45\ {e}^{-\tfrac{{t}_{\mathrm{expl}}}{8.8\ {\rm{d}}}}+1.45\ {e}^{-\tfrac{{t}_{\mathrm{expl}}}{111.3\ {\rm{d}}}})\\ \,\times {10}^{43}\ \mathrm{erg}\,{{\rm{s}}}^{-1}\left(\displaystyle \frac{{M}_{\mathrm{Ni}}}{{M}_{\odot }}\right),\end{array}\end{eqnarray} \tag{ 13 }$$

where M_Ni is the mass of ⁵⁶Ni. Here, we apply t_expl = 40 days. The number of atoms is N_A = M_ej/〈A〉m_p, where M_ej is an ejecta mass, 〈A〉 is the average mass number in the ejecta (here, 〈A〉 = 16 for pure O), and m_p is the mass of a proton. The ejecta mass is fixed to a typical value for SLSNe I, M_ej ∼ 6 M_⊙ (Blanchard et al. 2020). Then, the deposition rate needed for appropriate ionization rates can be converted to a mass ratio of ⁵⁶Ni versus the ejecta M_Ni/M_ej. This ratio is shown on the upper x-axis in Figure 11. The mass ratio corresponding to the ionization rate that makes n_{O II, ex} ≳ 1 cm⁻³ is M_Ni/M_ej ∼ 0.05 at T = 10,000 K. This is translated to ⁵⁶Ni mass of M_Ni ∼ 0.3 M_⊙ for M_ej = 6 M_⊙. If SLSNe I are entirely powered by ⁵⁶Ni, the required ⁵⁶Ni mass is ≳3 M_⊙, but it would lead to too strong a nonthermal ionization.

We emphasize that our estimates above involve a number of assumptions and uncertainties, and the upper limit of the ⁵⁶Ni mass is still quite uncertain. For example, the recombination rate adopted in the ionization balance (i.e., the direct recombination rate to the ground state) is expected to be lower because of immediate reabsorption. A lower recombination rate would increase the number density of O iii ions and enhance the population of the excited state of O ii, giving stronger W-shaped O ii lines. Then, the upper limit of ⁵⁶Ni would become smaller (or tighter). On the other hand, there may also be the effects making the upper limit of the ⁵⁶Ni mass higher. For example, we only evaluate the Sobolev optical depth just outside the photosphere. But, in reality, the line is formed over a velocity range of up to v ∼ 1.5 v_ph (Section 5.1). To have τ_lu > 1 for a wider velocity region, more ⁵⁶Ni would probably be required, making the upper limit of the ⁵⁶Ni mass higher. Also, we assumed full mixing to translate the ionization rate to the ⁵⁶Ni mass. Under less mixing, the energy of nonthermal electrons is spent inside the photosphere, which decreases the nonthermal ionization rate outside the photosphere. Thus, with less mixing, the upper limit of the ⁵⁶Ni mass would also become higher (i.e., less stringent).

There are also other assumptions/uncertainties: the simplified rate equation, the uncertain photoionization rate because of the line blanketing, and the uncertain work per ion pair in the ejecta of SLSNe I. Nevertheless, our work demonstrates that spectroscopic properties can be, in principle, used to give constraints on the mass of ⁵⁶Ni in SLSNe I, which roughly corresponds to the mass ratio M_Ni/M_ej of the order of 0.1. In addition, in the cases where SLSNe I are powered by magnetars, spectroscopic properties can also be used to give constraints on spectra of γ-rays from magnetars via ionization rates if combined with detailed transfer calculations (Murase et al. 2021; Vurm & Metzger 2021).