Lanthanide Features in Near-infrared Spectra of Kilonovae

To investigate which elements can become absorption sources in kilonova photospheric spectra, we systematically calculate the strength of bound–bound transitions for a given density, temperature, and element abundances. The strength of a line is approximated by the Sobolev optical depth (Sobolev 1960) for each bound–bound transition,

$$\begin{eqnarray}\begin{array}{rcl}{\tau }_{l} & = & \displaystyle \frac{\pi {e}^{2}}{{m}_{e}c}{n}_{i,j,k}t{\lambda }_{l}{f}_{l}\\ & = & \displaystyle \frac{\pi {e}^{2}}{{m}_{e}c}{n}_{i,j}t{\lambda }_{l}{f}_{l}\displaystyle \frac{{g}_{k}}{{g}_{0}}{e}^{-\tfrac{{E}_{k}}{{kT}}},\end{array}\end{eqnarray} \tag{ 1 }$$

in homologously expanding ejecta. The Sobolev approximation is valid for matter with a high expansion velocity and a large radial velocity gradient. Here, n_i,j,k is the number density of ions at the lower level of a transition (i-th element, j-th ionization stage, and k-th excited state), f_l and λ_l are the oscillator strength and the transition wavelength, g₀ is the statistical weight at the ground state, and g_k and E_k are the statistical weight and the lower energy level of a bound–bound transition, respectively. As in previous work on kilonovae (e.g., Barnes & Kasen 2013; Tanaka & Hotokezaka 2013), we assume local thermodynamic equilibrium (LTE); we solve the Saha equation to obtain ionization states, and assume Boltzmann distribution for the population of excited levels, which appears in Equation (1) (see Pognan et al. 2022a for non-LTE effects).

For the abundances in the ejected matter from an NS merger, we use the same model as in Domoto et al. (2021) based on a multicomponent free-expansion (mFE) model of Wanajo (2018). Here, we use the Light (L) model as our fiducial model (the left panel of Figure 1), which exhibits a similar abundance pattern to that of metal-poor stars with weak r-process signature (e.g., HD 122563, Honda et al. 2006, the right panel of Figure 1). This is motivated by the fact that the blue emission of GW170817/AT2017gfo a few days after the merger is suggested to have stemmed from the ejecta component dominated by relatively light r-process elements (e.g., Arcavi et al. 2017; Nicholl et al. 2017; Tanaka et al. 2017, 2018). Although the model includes the abundances with the atomic number of Z = 1–110, we use only the abundances at t = 1.5 days with Z = 20–100 in our calculations as shown in the right panel of Figure 1. The mass fractions of elements relevant to this study are summarized in Table 1. Note that the calculated abundances, which are recomputed with an updated nucleosynthesis code (Fujibayashi et al. 2020, 2022), slightly differ (within a factor of 2 at 1.5 days) from those presented in Domoto et al. (2021). We have confirmed that our new nucleosynthetic abundances give almost the same results as those in Domoto et al. (2021).

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For the atomic data, we use a theoretical line list from Tanaka et al. (2020). This line list was constructed for the elements with Z = 30–88 from neutral atoms up to triply ionized ions by systematic atomic structure calculations using the HULLAC code (Bar-Shalom et al. 2001). Since transition data of this line list are not necessarily accurate in terms of wavelengths, we here only study the species of ions to produce strong transitions.

Note that the theoretical line list does not include the data for actinides (Z = 89–100) due to the difficulty of atomic structure calculations (Tanaka et al. 2020). While actinides just work as zero-opacity sources without atomic data, we keep these elements in the list to discuss the spectral features of actinides in Section 4.2.

The strength of bound–bound transitions at t = 1.5 and 3.5 days for the L model is shown in the top panels of Figure 2. We evaluate the Sobolev optical depth for the density of ρ = 10⁻¹⁴ g cm⁻³ and the temperature of T = 5000 K at t = 1.5 days, and ρ = 10⁻¹⁵ g cm⁻³ and T = 3000 K at t = 3.5 days. These are typical values in the line-forming region when we adopt the abundance distribution of the L model (see Figure 6).

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We find that, among all the elements, Y ii, Zr ii, La iii, and Ce iii lines are as strong as the Sr ii triplet lines under the condition at t = 1.5 days. On the other hand, under the condition at t = 3.5 days, Y i, Zr i, and Ba ii lines appear instead of Zr ii, La iii, and Ce iii lines. Although the individual lines of Zr i are not by far the strongest, they can be important absorption sources due to the fact that the multiple lines show comparable strength at a certain wavelength range.

The behavior of the strength of the lines can be understood as the dependence of the Sobolev optical depths on temperature and density (Figure 4 of Domoto et al. 2021). When the density is ρ = 10⁻¹⁴–10⁻¹⁵ g cm⁻³, the strength of Sr ii triplet lines does not show a large difference between T = 3000 and 5000 K. This is determined by the combination of the ionization fraction and the population of the levels at these temperatures. While the lines of singly ionized Sr (Z = 38) are not largely affected in this temperature range, the lines of neutral Y (Z = 39) and Zr (Z = 40) appear at T = 3000 K. This is due to the slightly higher ionization potentials of Y and Zr. On the other hand, the strength of Ce iii lines largely changes in this temperature range reflecting the ionization fraction. Most Ce atoms are singly ionized at T = 3000 K, and thus, the lines of Ce iii become weaker at a lower temperature. The behavior of Ba ii (Z = 56) and La iii (Z = 57) lines can be explained in a similar way to that of Ce iii (Z = 58).

Although our line list includes all elements with Z = 30–88, our results indicate that only a few elements, such as Sr, Y, Zr, Ba, La, and Ce can become strong absorption sources in the spectra. The reason can be interpreted by the atomic properties of these elements. From Equation (1), the necessary conditions for a given line to become strong are that (1) the transition probability (g_k f_l ) is high and (2) the lower energy level of the transition (E_k ) is low (i.e., the level population is high).

The left panel of Figure 3 shows the mean values of log gf for all the lines of singly ionized ions as a function of atomic number. The mean gf-values show the pattern according to the orbital angular momentum l of the valence shell. This is more understandable when we use the complexity of a given ion, defined as (Kasen et al. 2013):

$$\begin{eqnarray}&&C={{\rm{\Pi }}}_{m}\displaystyle \frac{{g}_{m}!}{{n}_{m}!({g}_{m}-{n}_{m})!},\end{eqnarray} \tag{ 2 }$$

where g = 2(2l + 1) is the number of magnetic sublevels in the subshell with orbital angular momentum l, and n_m is the number of electrons in the nl-orbital labeled m. The complexity indicates how dense the energy levels are packed, and takes the maximal value when filling half the closed shell (Figure 1 of Kasen et al. 2013). Figure 4 shows the number of lines and the mean value of log gf for singly ionized ions as a function of complexity. We find that the complexity shows a positive correlation with the number of lines. It is natural because the number of transition combinations increases for larger complexity, i.e., denser energy levels. We also find that the complexity shows a negative correlation with the mean gf-value. This can be understood by the sum rule of oscillator strength; when the ion has a larger number of transitions, the oscillator strength of each line tends to be smaller. Other ionization states are not presented in Figure 4, but show similar trends. Therefore, the ions with relatively low complexity are likely to have transitions with relatively high gf-values.

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The right panel of Figure 3 shows the mean values of the lower energy levels of transitions for all the lines of singly ionized ions as a function of atomic number. The mean lower energy level of transitions tends to be pushed up toward higher energy as atomic number increases in an electron shell series. Other ionization states also show similar behavior. As spin–orbit interaction energy strongly depends on atomic number, energy-level spacing at a certain shell increases with atomic number. Also, electron orbital radii become smaller as atomic number increases, so that electron–electron interaction energies become higher for larger atomic numbers. As a result, the distribution of energy levels becomes wider for larger atomic numbers at a given shell (Tanaka et al. 2020). Therefore, the ions with smaller atomic numbers in a certain period on the periodic table tend to have transitions from low-lying energy levels.

Consequently, it is natural that Sr, Y, Zr, Ba, La, and Ce show strong lines. According to the two properties mentioned above, the ions on the left side of the periodic table are anticipated to show strong lines. In fact, all of the elements showing strong lines belong to groups 2 to 4 on the periodic table; in other words, they have a relatively small number of valence electrons (low complexity) and relatively low-lying energy levels.

Among these ions, La iii and Ce iii show strong lines at the NIR wavelengths (Figure 2). This can be understood from the properties of lanthanide elements. While lanthanide elements are characterized by having the 4f-electrons, the configurations of low-lying energy levels for lanthanides also involve the outer 5d and 6s shells. This means that the energy scales of 4f, 5d, and 6s orbitals for lanthanides are similar; in other words, the energy differences between these orbitals are small. In fact, the strong transitions of La iii and Ce iii at the NIR wavelengths involve an electron jump between 4f and 5d orbitals (see Tables 3 and 4). Thus, it is natural that La iii and Ce iii lines tend to appear in the NIR region.

We find that Y, Zr, Ba, La, and Ce, as well as Sr, can show strong lines under the physical conditions of NS merger ejecta. To enable us to inspect absorption lines produced by these elements in kilonova photospheric spectra, we calibrate the energy levels and resulting transition wavelengths of theoretical atomic data with experimental data. The details for calibration procedures are given in Appendix A.

To use the calibrated lines in radiative transfer simulations (Section 3), we need their transition probabilities. After calibrating the energy levels and wavelengths, the transition probabilities of the calibrated lines are taken from the VALD database (Piskunov et al. 1995; Kupka et al. 1999; Ryabchikova et al. 2015) or Kurucz's atomic data (Kurucz 2018) if the lines are listed, otherwise theoretical values from Tanaka et al. (2020) are adopted. The general validity of the theoretical transition probabilities is discussed in Appendix A. The transitions with the theoretical transition probabilities are summarized in Tables 3–4 (Appendix A).

Finally, we construct a hybrid line list by combining the VALD database for Z = 20–29 and the results of atomic calculations from Tanaka et al. (2020) for Z = 30–88. Among the data of Z = 30–88, strong transitions of Sr ii, Y i, Y ii, Zr i, Zr ii, Ba ii, La iii, and Ce iii are replaced with those calibrated with experimental data. The information for the hybrid line list is summarized in Table 2. The bottom panels of Figure 2 show the strength of bound–bound transitions at t = 1.5 and 3.5 days calculated with the new hybrid line list. In these panels, the wavelengths of strong transitions are accurate as they are calibrated with experimental data. We find that La iii and Ce iii lines become strong absorption sources at the NIR wavelengths at t = 1.5 days. For Y i, Y ii, Zr i, Zr ii, and Ba ii, most of the wavelengths of relatively strong lines are found to be placed in the optical region (λ < 10000 Å). Thus, their lines are unlikely to produce absorption features in the NIR region, although they may be important absorption sources at the optical wavelengths (see Section 3).

We adopt the theoretical transition probabilities when they are not available in the VALD database or Kurucz's line list. To validate the accuracy of theoretical gf-values for NIR lines, we use the relative intensities of Ce iii lines. Johansson & Litzén (1972) measured the emission lines of Ce iii at the NIR wavelengths in a laboratory and showed the relative intensities of measured lines. Assuming LTE for ionization and excitation, the intensity of an emission line can be calculated as

$$\begin{eqnarray}\begin{array}{rcl}I & = & b\ {g}_{u}A\ {e}^{-\tfrac{{E}_{u}}{{kT}}}\\ & = & b\displaystyle \frac{8\pi {e}^{2}}{{m}_{e}c{\lambda }_{l}^{2}}{g}_{l}{f}_{l}{e}^{-\tfrac{{E}_{u}}{{kT}}},\end{array}\end{eqnarray} \tag{ 3 }$$

where A, g_u , and E_u are Einstein's A coefficient, the statistical weight, and the energy level of the upper level for a transition, respectively, and b is a constant depending on the ion species. Since plasma in experiments is typically in LTE due to the high density of ions (Kielkopf 1971), we can use this formula to evaluate the gf-values of Ce iii lines.

A comparison between the intensities of Ce iii lines calculated with the theoretical transition probabilities and those measured by experiments (Johansson & Litzén 1972) is shown in Figure 5. We adopt the temperature of T = 12000 K, which is a typical plasma temperature in experiments (Kielkopf 1971). Here a normalization factor b is set so that the calculated values are close to the experimental values. We find that the calculated and experimentally measured intensities are in good agreement except for a few lines. Since in the situation we consider (E_u ≲ 2 eV), the intensities are mainly determined by transition probabilities, the trend suggests that our theoretical gf-values of Ce iii lines are reasonable. We note that, although the gf-values of a few lines should be higher than our estimates (blue circles in Figure 5), they are relatively weak and do not affect our conclusions. Nevertheless, to determine the exact values of transition probabilities for these lines, more experimental and observational calibrations are necessary for the NIR region.

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In this section, we calculate realistic synthetic spectra of kilonovae by using the new hybrid line list. We use a wavelength-dependent radiative transfer simulation code (Tanaka & Hotokezaka 2013; Tanaka et al. 2014, 2017, 2018; Kawaguchi et al. 2018, 2020). The photon transfer is calculated by the Monte Carlo method. To compute the opacity for bound–bound transitions, we adopt the expansion opacity (Karp et al. 1977) and use the formula from Eastman & Pinto (1993):

$$\begin{eqnarray}&&{\kappa }_{\exp }(\lambda )=\displaystyle \frac{1}{{ct}\rho }\sum _{l}\displaystyle \frac{{\lambda }_{l}}{{\rm{\Delta }}\lambda }(1-{e}^{-{\tau }_{l}}),\end{eqnarray} \tag{ 4 }$$

where τ_l is the Sobolev optical depth (Equation (1)). In the equation, the summation is taken over all transitions within a wavelength bin Δλ (see below). The Sobolev optical depth is evaluated by assuming LTE for ionization and excitation as in Section 2.1.

For the atomic data, we use the new hybrid line list constructed in Section 2.3. The hybrid line list still includes weak transitions whose wavelengths are not necessarily accurate. To avoid the substantial effects of these lines on spectra, we adopt a wide wavelength grid for the opacity calculation with the atomic data from theoretical calculations (i.e., lines for Z = 30–88). The wavelength grid is typically set to Δλ = 10 Å (Tanaka & Hotokezaka 2013), but here a 20 times wider grid is adopted for the theoretical line list. This smears out the individual effect of each line on the bound–bound opacity. We also performed the same opacity calculations with the typical fine wavelength grid, and confirmed that the resultant total opacity is almost unchanged. For the accurate transitions (i.e., lines for Z = 20–29 and calibrated lines), we adopt Δλ = 10 Å. By combining the opacity calculated with the theoretical atomic data and the strong transitions calculated with the accurate data, we are able to discuss whole spectral features, i.e., an overall shape, absorption lines, and their time evolution.

In the radiative transfer code, the temperature in each cell is determined by the photon flux (Lucy 2003; Tanaka & Hotokezaka 2013). The photon intensity is evaluated as

$$\begin{eqnarray}&&{J}_{\nu }d\nu =\displaystyle \frac{1}{4\pi {\rm{\Delta }}{tV}}\sum _{d\nu }\epsilon {ds},\end{eqnarray} \tag{ 5 }$$

where is the comoving-frame energy of a photon packet. The temperature is estimated by assuming that the wavelength-integrated intensity 〈J〉 = ∫J_ν d ν follows the Stefan–Boltzmann law, i.e.,

$$\begin{eqnarray}&&\langle J\rangle =\displaystyle \frac{\sigma }{\pi }{T}_{R}^{4}.\end{eqnarray} \tag{ 6 }$$

The kinetic temperature of electrons T_e is assumed to be the same as the radiation temperature T_R , i.e., T = T_e = T_R under LTE.

For the ejecta density structure, we assume a single power law (ρ ∝ r⁻³) for the velocity range of ejecta v = 0.05–0.3c (e.g., Metzger et al. 2010). The total ejecta mass is set to be M_ej = 0.03 M_⊙, which is suggested to explain the observed luminosity of AT2017gfo (e.g., Tanaka et al. 2017; Kawaguchi et al. 2018). For the abundance distribution, we use the same model (L model) as described in Section 2.1. The heating rate of radioactive nuclei as a function of time is consistently computed for this model. The thermalization efficiency of γ-rays and radioactive particles follows the analytic formula given by Barnes et al. (2016). The resulting temperature structure of the ejecta is shown in Figure 6.

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Figure 7 shows the synthetic spectrum at t = 1.5 days after the merger. To show the contribution of different elements, we also plot the Sobolev optical depths in the ejecta at v = 0.16 c. The wavelengths of lines are blueshifted according to v = 0.16 c. Note that we plot only spectroscopically accurate lines that contribute to absorption features.

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The spectrum shows two strong absorption features at λ ∼ 8000 Å (red arrow) and λ ∼ 6500 Å (blue arrow). These are mainly caused by Sr ii and Ca ii as found in Domoto et al. (2021), although some Y ii (dark-blue) and Zr ii (green) lines also slightly affect the feature. In addition, the spectrum shows wide absorption features around λ ∼ 12,000 and 14,000 Å (pink and orange arrows). These are produced by La iii and Ce iii lines, respectively. The presence of these absorption features is reasonable, because Ca ii, Sr ii, La iii, and Ce iii are found to be strong absorption sources in the one-zone analysis (Section 2). The central wavelengths of absorption lines for La iii and Ce iii show that the photospheric velocity at the NIR region is v ∼ 0.16c, while that for Ca ii and Sr ii is v ∼ 0.2c. This indicates that the line-forming regions for different wavelength ranges do not coincide owing to the wavelength-dependent opacity.

Figure 8 shows a comparison between our results and the observed spectra of AT2017gfo at t = 1.5, 2.5, and 3.5 days after the merger (Pian et al. 2017; Smartt et al. 2017; see Gillanders et al. 2022 for the latest calibration). We here focus only on the spectral features in the NIR region, because the absorption features by Ca and Sr in the observed spectra of AT2017gfo and their implication for the ejecta condition have already been discussed in Domoto et al. (2021). We find that the overall slopes of synthetic spectra in the NIR region are quite similar to the observed ones. The Doppler shift of absorption lines becomes smaller with time, because the density of the ejecta becomes lower and the photosphere moves inward (in mass coordinate). Interestingly, the positions of absorption features at the NIR wavelengths in our results are consistent with those seen in AT2017gfo, especially at t ≥ 2.5 days. Although this model motivated by the observed luminosity of AT2017gfo is quite simple, the NIR features appear to agree with the observed ones without an adjustment of, e.g., density distribution. This implies that the absorption features at the NIR wavelengths in the spectra of AT2017gfo may be caused by the La iii and Ce iii lines.

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It should be noted that the assumption of LTE may not be valid in a low-density region. In the results here, neutral atoms, especially for Y and Zr, which appear in the outer ejecta, are the dominant opacity sources at t ≥ 2.5 days at the optical wavelengths (Tanaka et al. 2020; Kawaguchi et al. 2021; Gillanders et al. 2022). On the other hand, recent work on the nebula phase of kilonovae suggests that ionization fractions as well as the temperature structure of ejecta can be deviated from those expected in LTE with time, i.e., as the ejecta density decreases (Hotokezaka et al. 2021; Pognan et al. 2022b). These non-LTE effects may change the emergent spectra a few days after the merger, mainly at the optical wavelengths, where many strong lines of neutral atoms exist (Kawaguchi et al. 2021). Nevertheless, since the photosphere for the NIR region is located at the inner ejecta where the density is enough high, non-LTE effects are expected to be subdominant (Pognan et al. 2022a).

Our results show that kilonova photospheric spectra exhibit absorption features of La iii and Ce iii in the NIR region, which are in fact similar to those seen in the spectra of AT2017gfo. In this subsection, we examine a possible range of these lanthanide mass fractions in the ejecta of AT2017gfo by using the NIR features.

To investigate the effect of the La amount on the spectra, we perform the same simulations as in Section 3 but by varying the mass fraction of La. The resultant spectra at t = 2.5 days after the merger are shown in the left panel of Figure 9. We find that the strength of absorption due to the La iii lines at λ ∼ 12500 Å changes with the mass fraction of La. On the other hand, no matter how the mass fraction changes, the overall spectral shapes hardly change. Because La lines have little effect on the total opacity, the NIR opacity is almost unchanged. Thus, the strong lines of La iii keep producing strong absorption as long as enough La is present. According to the tests shown in the left panel of Figure 9, we estimate that the mass fraction of La is higher than 1/30 times that of the L model, i.e., X(La) > 2 × 10⁻⁶, which is required to identify the visible absorption feature at λ ∼ 12500 Å in the spectra of AT2017gfo.

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By contrast, the situation is different in the case of Ce. To investigate the effect of the Ce amount on the spectra, we perform the same calculations for Ce as done for La above. The resultant spectra at t = 2.5 days after the merger are shown in the right panel of Figure 9. We find that the absorption feature at λ ∼ 14000 Å diminishes as the mass fraction of Ce is substantially reduced (blueish curves). Also, the absorption feature disappears as well, even when the mass fraction of Ce is substantially increased (pink curve). Because Ce lines appreciably contribute to the total opacity, the higher Ce mass fraction results in a higher total opacity. As a result, the photosphere shifts outward compared to that in the L model. This makes the photospheric temperature lower, and thus, the Ce iii lines disappear.

The amount of Ce affects not only absorption features but also overall spectral shapes. The spectra become redder and bluer when X(Ce) is increased and reduced, respectively. This is because Ce has high opacity in the NIR region and is the most dominant opacity source at the NIR wavelengths in this model. As the contribution of other heavy elements to the total opacity is subdominant, even if the mass fractions of all the elements with a mass number larger than 100 are varied by a factor of 10, the results are almost the same as those in the right panel of Figure 9. Note that the element species that dominate the opacity depend on the ejecta conditions (such as density, temperature, and epoch), but generally lanthanide elements with a small atomic number (e.g., Ce and Nd) tend to have a larger contribution as they have more transitions from low-lying energy levels (Even et al. 2020; Tanaka et al. 2020).

We roughly estimate the mass fraction of Ce presented in the ejecta of AT2017gfo from our calculations. It is difficult to determine the exact amount of lanthanides from absorption features, because Ce has complex effects on spectral formation, as discussed above. Nevertheless, our demonstration suggests that a certain amount of Ce must have been present in order to explain the absorption features as well as the NIR fluxes. However, a too large amount of Ce diminishes the absorption features. As a result, the mass fraction of Ce is estimated to be between 1/3 and 30 times that of the L model to account for the absorption feature, i.e., X(Ce) ∼ (1–100) × 10⁻⁵. This corresponds to the lanthanide mass fraction of ∼(2–200) ×10⁻⁴ if assuming the solar abundance pattern of r-process elements. While the lanthanide mass fraction estimated here is consistent with or somewhat higher than the values previously suggested (e.g., McCully et al. 2017; Nicholl et al. 2017; Gillanders et al. 2022), we emphasize that this is the first constraint on the lanthanide abundances using the absorption features in the spectra of AT2017gfo.

It should be noted that the results presented here are calculated with a single structure of ejecta (Section 3). The effects of ejecta properties, e.g., mass and velocity, on the spectra should be systematically examined. Furthermore, the spectra are calculated by assuming a simple one-dimensional morphology of ejecta with homogeneous abundance distribution. It is important to employ more realistic models to elucidate the effects of multidimensional ejecta structures. We leave such exploration to future work.

We have shown that the ions that tend to produce absorption features in kilonova photospheric spectra can be explained by atomic properties (Section 2.2). According to the required properties, one can notice that not only lanthanides but also actinide elements can possibly contribute to the spectral features. However, while we include actinide elements in our abundance input up to Z = 100, actinide elements are not included in the line list due to the difficulty of atomic structure calculations for actinides (Tanaka et al. 2020). On the other hand, some experimental data are available for Th (Z = 90). In the following, we discuss the effects of Th absorption features based on experimental data.

Th iii is one of the possible candidates for the ions that can contribute to the spectral features. The atomic structure of Th iii is analogous to that of Ce iii, which has two electrons in the outermost shell involving with f-shell. Fortunately, the energy levels of Th iii are well established by experiments. For optical lines, not only transition wavelengths but also transition probabilities are available in the VALD database (Biémont et al. 2002). Moreover, the transition wavelengths in the NIR region are measured by experiments (Engleman 2003). However, there is no available data on transition probability for the NIR lines.

To test the possibility of identifying Th iii lines in kilonova photospheric spectra, we estimate the transition probabilities of the NIR Th iii lines by using the measured intensities. Since the relative intensities of the measured NIR lines (Engleman 2003) are listed in the NIST database (Kramida et al. 2021) in a consistent way with those of optical lines, we can directly compare the measured and calculated intensities for the lines over the whole wavelength range. Here, although the same estimate of gf-values can be, in principle, applicable to other ionization stages of Th, e.g., Th ii, only those of Th iii are tested. This is because lines of lower-ionized ions for Th are not expected to show strong transitions compared to those of Th iii, such that Ce iii show strong lines but Ce ii does not (Section 2.2).

First, we calculate the line intensities for the optical lines for Th iii with known gf-values taken from the VALD database using Equation (3). A comparison between the intensities calculated with the VALD gf-values and those measured by experiments for Th iii (Kramida et al. 2021) is shown in Figure 10. We find that the calculated and experimentally measured intensities reasonably agree with each other when we adopt the temperature of T = 5000 K. Therefore, we estimate the transition probabilities of NIR lines for Th iii by using the measured intensities (Engleman 2003; Kramida et al. 2021) and this temperature. Estimated gf-values are summarized in Table 5 (Appendix B).

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Then, we calculate the strength of bound–bound transitions for Th iii lines at t = 1.5 days for the L model, as in Section 2. The top panel of Figure 11 is the same as the bottom left panel of Figure 2 but with the Th iii lines. We find that the strength of the Th iii lines at the NIR wavelengths can be comparable to that of Ce iii lines. This is due to the same reason as discussed in Section 2.2: relatively high transition probabilities and low energy levels of transitions.

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We perform the same radiative transfer simulations as in Section 3 by including the Th iii lines. The synthetic spectra with the Th iii lines are shown in the bottom panel of Figure 11 (blue curves). We find that the fluxes for λ ≥ 20000 Å are slightly pushed up compared to those of the results without the Th iii lines (light-blue curves, see Figure 8). However, the spectral features are not substantially different from those not including the Th iii lines, although a wide and marginal absorption feature can be seen around λ ∼ 18000 Å at t = 1.5 days. This implies that it is difficult to confirm the presence of Th from spectral features.

The difference between the absorption features for Th iii and Ce iii can be explained by the complex atomic structure of actinides. As shown in the top panel of Figure 11, although the Th iii lines in the NIR region are relatively strong, many transitions exhibit similar Sobolev optical depths and no significant line like that of Ce iii at λ ∼ 16000 Å exists. This is due to the fact that Th iii has denser low-lying energy levels involved in 5f-shell compared to those of Ce iii involved in 4f-shell. Silva et al. (2022) recently showed that the opacity of another actinide U iii (Z = 92) is about an order of magnitude larger than that of Nd iii for the same reason. Note that an actinide, Ac (Z = 89, as ²²⁷Ac with the half-life of 21.77 yr), can also exist with a similar amount to those of Th and U in the ejecta of kilonovae. Ac iii may have a similar atomic structure to that of La iii. However, Ac iii is poorly understood both in theory and experiments, and it is not clear if Ac iii can produce prominent features like La iii. Thus, although actinide ions can be important opacity sources in the NIR region, it may not be easy to identify the presence of actinides from the spectral features.