PAIR INSTABILITY SUPERNOVAE: LIGHT CURVES, SPECTRA, AND SHOCK BREAKOUT

The electron–positron PI SN mechanism was originally discussed by Rakavy & Shaviv (1967) and Barkat et al. (1967), and has since been explored with a number of numerical and analytic models (see Heger & Woosley 2002, and references therein). Most recently, Heger & Woosley (2002, hereafter HW02) explored the explosion of non-rotating bare helium cores with a range of masses from 64 to 133 M_☉. A subset of these models, with masses 70–130 M_☉ (in steps of 10 M_☉), will be examined here.

Models of bare helium cores, though computationally expedient, likely do not fully represent the class of PI SNe, as not all stars will have lost their hydrogen envelopes to mass loss or binary mass exchange just prior to exploding. We therefore evolved a new set of models especially for this study of the breakout transients, light curves, and spectra.⁵ They consist of five models each of "hydrogenic" (i.e., possessing hydrogen) stars in the main-sequence mass range 150–250 M_☉, with initial metallicities of 0 and 10⁻⁴ times solar. The surface zoning of these models was chosen much finer than those of HW02 in order to facilitate calculation of the shock breakout emission. In addition, one helium core of mass 100 M_☉ was calculated with three orders of magnitude finer surface zoning (to 10⁻⁸ M_☉) than in HW02. All models were calculated using the Kepler code and the physics discussed in HW02 and Woosley et al. (2002). One difference with the previous studies is that mass loss was included in the hydrogenic stars with non-zero metallicity; however, the mass lost was both small and very uncertain.

Properties of all presupernova stars and their explosions are given in Table 1. The major distinction between the zero metallicity and 10⁻⁴ solar metallicity presupernova stars was that the former died as compact blue supergiants (BSGs), while the latter were red supergiants (RSGs) with radii 10–50 times larger. To some extent, this difference also relies on a particular choice of uncertain parameters—in particular primary nitrogen production and mixing—so not too much weight should be placed on the metallicity of the model. The production of primary nitrogen was, by design, minimal in all models. In the zero metallicity series, especially the higher mass ones, this required reducing semiconvection by a factor of 10 compared with its usual setting in Kepler and turning off overshoot mixing. Unless this was done, only the 150 M_☉ model ended up as a BSG while the other four were red owing to primary nitrogen production.

For the 150, 175, and 200 M_☉ stars with 10⁻⁴ solar metallicity, both semiconvection and overshoot mixing had their nominal settings, but RSGs resulted without primary nitrogen production. The 225 and 250 M_☉ stars with 10⁻⁴ solar metallicity used the reduced mixing—again to avoid large ¹⁴N production—but also ended up as RSGs. Since even a moderate amount of rotation would lead to some mixing between the helium core and hydrogen envelope in the zero metal stars, and since the nominal semiconvection and overshoot parameters also lead to primary nitrogen production, it is likely that a fraction, perhaps all, of the zero metallicity pair instability stars also die as RSGs. This would imply that BSGs are a rare population of supernova progenitors for such massive stars even at ultra-low metallicity. Our goal here though was to prepare a set of blue and RSG progenitors with a range of masses to explore how the observable properties of the explosion may depend on the structure of the hydrogen envelope.

The pair instability is triggered in helium cores above about 40 M_☉ once the temperature in the stellar core exceeds ∼10⁹ K, i.e., after helium burning and during carbon ignition. The creation of e⁺/e⁻ pairs in the core then softens the equation of state to below γ < 4/3 leading to instability and collapse. For helium cores below about 135 M_☉ the collapse is eventually reversed by explosive nuclear burning, first of oxygen and, for more massive cores, silicon. Non-rotating helium cores more massive than 133 M_☉ experience a photodisintegration instability after silicon burning and collapse directly to a black hole (HW02). Between 40 and 60 M_☉, the pair instability in bare helium cores leads to multiple violent mass ejections, but does not disrupt the entire star on the first try (Woosley et al. 2007). Between 60 and 133 M_☉, helium stars collapse to increasing central temperature and density, explode with greater violence, and produce more heavier elements, especially ⁵⁶Ni (see Table 1).

Most of the helium core explosions studied here were taken from the survey by HW02, but the 100 M_☉ model was recalculated with finer surface zoning. A possible point of confusion is whether shocks ever form in these sorts of helium stars or whether, given their small radius, the surface remains in sonic communication with the center throughout the collapse and initial expansion. We find that for the 100 M_☉ model the outer layers do not participate in the collapse and, at about 2 s after maximum compression, a very strong shock forms initially about 2 M_☉ beneath the surface at a radius of 8 × 10⁹ cm. Due to the large explosion energy and acceleration in the steep density gradient, material in the shock reached a speed of about one-third the speed of light before erupting through the photosphere.

The explosions of the hydrogenic stars had characteristics, including kinetic energy and nucleosynthesis, approximately set by the mass of their helium cores (Table 1). There were, however, some important differences. For the helium stars the mass was set at the beginning of the calculation and held fixed, while in the hydrogenic stars the helium core grew significantly after central hydrogen depletion owing to hydrogen shell burning. In some cases it also shrank due to convective dredge up. At death the helium core thus had a different composition and entropy than a corresponding helium star evolved at constant mass.

An even more significant difference of the hydrogenic stars is that their exploding helium cores encounter the lower density hydrogen envelope and interact with it hydrodynamically. This has several important consequences. First, the expansion of the helium core is slowed, which produces Rayleigh–Taylor instabilities and mixing. Up until this point, the explosion had been determined by simple physics and (assuming no rotation) well represented by a spherical one-dimensional calculation. While the mixing can be calculated in a multi-dimensional code (e.g., Joggerst et al. 2009), it has not been yet and is parameterized here as in Kasen & Woosley (2009). Except for two models that experienced significant fallback (see below), the mixing has no effect on the nucleosynthesis or energetics and does not affect the breakout emission. However, the late-time spectra may be somewhat sensitive to mixing. Figures 1 and 2 plot the final density and mixed compositional structures for a few representative models.

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In some cases, the deceleration of the helium core by the envelope is so severe that the star does not completely explode. This is particularly true for the BSG models. As pointed out by Chevalier (1989) and explored by Zhang et al. (2008) and Joggerst et al. (2010), braking and fallback occur to a greater extent in more compact stars. The helium core encounters its own mass earlier and the reverse shock returns to the center when the density is still high. In our case, this resulted in models B150 and B175 failing to eject anything other than their hydrogen envelope. Of course the story does not end there—these bound helium cores oscillate awhile, settle down, and evolve again. Given their residual helium masses, models B150 and B175 will probably become PI SNe again and disrupt entirely on the second try. They thus represent an extension to higher masses of the pulsational PI SNe studied by Woosley et al. (2007). Depending upon the timing, the second mass ejection may produce an extremely bright supernova as the very energetic second supernova plows into the ejecta of the first. This future evolution is beyond the scope of the present paper, but the evolution and explosion of BSGs with zero-age main sequence masses of 100–200 M_☉ is clearly an area worth further study.

In the hydrogenic models, the expansion of the exploded helium core into the hydrogen envelope drives a radiation dominated shock. When this shock approaches the surface of the star, the postshock radiation can escape in a luminous X-ray/UV burst (Klein & Chevalier 1978; Ensman & Burrows 1992; Matzner & McKee 1999). This event occurs at a distance ΔR from the stellar surface such that the diffusion time from the shock front (t_d ∼ τΔR/c) is comparable to the dynamical time for the shock to travel the same distance (t_e ∼ ΔR/v_s). This implies an optical depth τ ≈ c/v_s at breakout. The shock velocity v_s scales with (E/M)^1/2 and for pair SNe is of order 3 × (10³–10⁴) km s^–1, similar to that of ordinary core-collapse supernovae. Shock breakout thus occurs at optical depths of τ_b ≈ 30–100.

The surface layers of PI SNe progenitors are difficult to model; a proper representation of the atmospheric structure would require a detailed radiation transport within the stellar evolution code and possibly the inclusion of three-dimensional effects. The limitations of the one-dimensional stellar evolution models thus introduce some uncertainty into the breakout predictions. While the current models were much more finely zoned than previous calculations, they still did not fully resolve the optically thin layers of the star. They did, however, determine the slope of the steep density profile at the surface, out to an optical depth of a few. To extend the profile into the optically thin region, we fit a power law to the outermost zones and linearly interpolated and extrapolated the density structure. The re-zoned model had 100 zones within the region τ < τ_b and extended to a minimum τ of 10⁻².

We followed the supernova explosion using the Kepler code until the shock front began to approach the stellar surface (τ ≈ 10⁵). The model structure was then mapped into a modified version of the SEDONA transport code (Kasen et al. 2006) which coupled multi-wavelength radiation transport to a staggered mesh one-dimensional spherical Lagrangian hydrodynamics solver. A standard artificial viscosity prescription was included to damp oscillations behind the shock front. We followed the radiation transport using implicit Monte Carlo methods (Fleck & Cummings 1971) coupled to the hydrodynamics in an operator split way. The transport was treated in a mixed-frame formalism, in which photon packets were propagated in the inertial frame, but the opacities and emissivities were computed in the comoving frame, and the proper Lorentz transformations were applied to move between the frames. The radiation energy and momentum deposition terms in the hydro equations were estimated by tallying the energy and direction of packets moving through each zone. Further details of the numerical methods in the context of shock breakout will be given in a separate publication (D. Kasen & S. Woosley 2011, in preparation).

The opacity and emissivity of the models were discretized into 25,000 wavelength bins covering the range λ = 0.1–25000 Å, while the output spectra were binned to a coarser resolution to improve the photon statistics. The dominant opacity in the supernova shock is electron scattering, while free–free is the most important absorptive opacity. Because the current models had zero or very low metallicity, we ignored bound–free and line opacity. At wavelengths near the blackbody peak, the ratio of free–free opacity to total opacity is typically small _a ≲ 10⁻⁴. In the postshock region, Compton upscattering may then become an important means of energy exchange between matter and radiation (Weaver 1976). On average, the fractional change of a photon's energy in a single Compton scattering is _c = 4kT/m_ec², which gives an effective _c = 6 × 10⁻⁵ at T = 10⁵ K. Comptonization will significantly alter the radiation spectrum after scattering through an optical depth τ_c ≳ ^−1/2_c which, for higher temperatures, may be less than the thermalization depth to free–free opacity τ_a = ^−1/2_a. In our calculations, we therefore approximated the Compton effects by taking a fraction _c of the Thomson opacity to be thermalizing. This gray absorptive component was added to the wavelength-dependent free–free opacity. Clearly, a direct treatment of inverse Compton scattering is needed to more accurately predict the spectrum of the breakout burst; nevertheless, our treatment of the full non-gray radiation transport offers some advance over previous numerical calculations.

Once a shock front reaches the steep outer layers of the star, it accelerates down the steep density gradient. For model R250, the shock velocity reaches v_s ≈ 1.5 × 10⁴ km s^–1 around breakout. The temperature of the postshocked gas is determined by the jump conditions aT⁴_s/3 = ρ₀v²_s/(γ + 1), where ρ₀ is the density of the pre-shocked material and γ = 4/3 is the adiabatic index for a radiation dominated gas. Near the surface, the RSG models had densities of ρ₀ ∼ 10⁻¹¹ g cm⁻³ which gave typical postshock temperatures of T_s ≈ 5 × 10⁵ K. The BSG models, being more compact, had higher densities at the surface and hotter temperatures, T_s ≈ 10⁶ K.

Figure 3 plots the calculated bolometric light curves of the breakout transients. The key observable properties are listed in Table 2. These calculations included the proper light travel time effects, which primarily determined the duration of the observed burst, Δt ≈ R₀/c. The RSG models lasted 1–2 hr and reached peak luminosities of L_peak = 10⁴⁵–10⁴⁶ erg s^–1. The BSG models, being more compact, had briefer transients lasting 0.1 hr, with somewhat lower peak luminosities, L_peak ≈ 10⁴⁴ erg s⁻¹. These values can be compared to the breakout in an ordinary Type II plateau supernova (SN IIP) explosion: L_peak = 5 × 10⁴⁴ erg s^–1, ΔT = 0.5 hr. As it turns out, the shock velocities and postshock energy densities of PI SNe are comparable to that of SNe IIP, but the breakout emission can be significantly brighter due to the larger stellar radii.

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Table 2. Shock Breakout Transients

Name	Lpeak	Δtha	Teff	Tcol	λpc	Epd	Etote	$E_{> L_\alpha }$f
	(erg s–1)		(105)	(105)b
R250	9.6 × 1045	5858	1.3	3.5	82.5	0.15	5.9 × 1049	2.2 × 1047
R200	2.6 × 1045	6422	1.0	2.2	130.5	0.09	1.8 × 1049	1.6 × 1047
R150	1.2 × 1045	7051	0.9	1.7	168.5	0.07	9.3 × 1048	1.3 × 1047
B250	1.4 × 1045	966	3.3	6.3	45.5	0.27	1.3 × 1048	6.5 × 1044
B200	6.7 × 1044	714	3.9	7.7	37.5	0.33	5.8 × 1047	1.9 × 1044

Notes. ^aDuration of burst, full width at half-maximum, in seconds. ^bColor temperature. ^cSpectral wavelength peak, in Å. ^dSpectral energy peak, in keV. ^eTotal energy emitted in burst, in erg. ^fTotal energy emitted in burst at wavelengths greater than Lyα, in erg.

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Figure 4 shows synthetic spectra of the bursts, averaged over the peak of the breakout light curve. The RSG spectra peak at wavelengths λ_p = 80–170 Å (∼0.07–0.15 keV), while the BSG spectra peak near 37–45 Å (∼0.3 keV). These peak wavelengths are comparable to that of ordinary Type IIP breakout (λ_p ≈ 100 Å). The spectra are reasonably approximated by a blackbody, but with excess emission at both high and low frequencies. This is because the time-averaged spectra represent a convolution of several blackbodies at different temperatures.

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Because the opacity in the atmosphere is strongly scattering dominated, radiation is thermalized at optical depths τ ≈ ^−1/2_a well below the photosphere. The emergent spectrum is thus characterized by the higher temperature of these deeper layers. Defining the color temperature T_col of the mean spectrum as the temperature of a blackbody peaking at the same wavelength, we find that the observed T_col is a typically a factor 2–3 times higher than the effective temperature at the photosphere (Table 2). A similar effect was noted by Ensman & Burrows (1992) for models of SN 1987A. In addition, recent analytic work by Katz et al. (2010) has shown that when the shock velocity is relatively high, ≳ 0.1c (and when the thermalization by Compton scattering is treated properly) non-equilibrium effects may lead to a significantly harder non-thermal component in the spectrum. In the BSG models, the shock velocities do in fact reach ∼0.1c and such a non-thermal component could modify the predicted spectra shown here. The RSG models, on the other hand, have lower shock velocities and are likely not strongly affected.

Following shock breakout, radiation continues to diffuse out of the expanding, cooling ejecta. This leads to a longer lasting emission that may be visible at optical wavelengths for several weeks. Such an initial thermal component to the light curves is discussed in the following section. The extreme luminosity of the breakout bursts themselves, and their short durations, makes them appealing transient to search for in high-redshift surveys. We consider their detectability in Section 4.

Following shock breakout, the luminosity of PI SNe can be powered by three different sources: (1) the diffusion of thermal energy deposited by the shock, (2) the energy from the radioactive decay of synthesized ⁵⁶Ni, or (3) the interaction of the ejecta with a dense surrounding medium. The internal energy suffers adiabatic losses on the expansion timescale t_ex = R₀/v, so source (1) will be most significant for stars with large initial radius R₀. Interaction has not been included in the models discussed here, but as already mentioned, can be very significant for stars which undergo pulsations before exploding completely.

We calculated light curves of the explosion models using the SEDONA code (Kasen et al. 2006). The initial density, composition, and temperature structures were taken from the Kepler calculations extended into the nearly homologous expansion phase (∼10 days after explosion). The energy deposition from ⁵⁶Ni decay was followed using a multi-wavelength transport scheme treating the emission, propagation, and absorption of gamma rays. For the optical radiation transport, the opacities used included electron-scattering, bound–free, free–free, and the aggregate effect of millions of Doppler broadened line transitions treated in the expansion opacity formalism (Eastman & Pinto 1993). Atomic level populations were calculated assuming local thermodynamic equilibrium, typically a reasonable approximation for supernovae in the photospheric phase.

The resulting model light curves (Figure 5) span a wide range of luminosities and durations. The more massive explosions are bright for over 300 days, with luminosities exceeding 10⁴⁴ erg s⁻¹. The long duration reflects the timescale for photons to diffuse through the optically thick supernova ejecta. In a homologously expanding medium, the effective diffusion timescales as t_d ∼ κ^1/2M^3/4_ejE^−1/4, where κ is the effective mean opacity (Arnett 1980). Model He130, for example, is nearly 100 times as massive and energetic as an SN Ia, and thus has a light curve ∼10 times as broad. The slow release of energy moderates the emergent luminosity, so that despite making over 60 times as much ⁵⁶Ni as a typical SNe Ia, model He130 is only ∼10 times brighter than one at peak.

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The morphology of the model light curves depends on the envelope of the progenitor star. The RSG models resemble SNe IIP light curves, though with luminosities and durations ∼3 times greater. The initial luminosity is powered by the thermal energy in the extended hydrogen envelope. Over time, as the ejecta cool, a hydrogen recombination front propagates inward in mass coordinates, which increases the ejecta transparency due to the elimination of electron scattering opacity. This effect regulates the release of the thermal energy (Grasberg & Nadezhin 1976; Popov 1993; Kasen & Woosley 2009). In the inner regions, heating from radioactive decay delays recombination and causes the light curve to rise to a peak at around 200–300 days. Eventually, the transparency wave reaches the base of the hydrogen envelope, after which recombination proceeds much more rapidly through the heavier element core. After a rapid release of the remaining internal energy, the light curve drops off sharply and follows directly the radioactive energy deposition rate.

The light curves of the BSG models have a dimmer initial thermal component, due to the relatively smaller radii of the progenitors. In this respect they resemble scaled up versions of SN 1987A. In models B200, B225, and B250, the light curves rise to a bright ⁵⁶Ni powered peak at ∼300 days after explosion. The less massive events (B150 and B175), which failed to eject any ⁵⁶Ni, only show the initial thermal light curve component lasting ∼150 days, and more closely resemble typical SNe IIP.

The helium core PI SNe models, being the most compact progenitors, lack a conspicuous thermal light curve component altogether. The lower total mass leads to relatively briefer light curves, peaking around 150 days after explosion. Model He130 (∼40 M_☉ of ⁵⁶Ni) reaches an exceptional peak brightness of 2 × 10⁴⁴ erg s^–1at around 180 days. Model He70, on the other hand, proves that despite being massive and energetic, not all PI SNe are bright. This explosion produced only 0.02 M_☉ of ⁵⁶Ni and the light curve reached only 3 × 10⁴¹ erg s⁻¹, rather sub-luminous for a supernova.

Figure 6 compares the synthetic R-band light curves of representative PI SN models with observations of a typical SN Ia (SN 2001el; Krisciunas et al. 2003), a typical SN IIP (SN 1999em; Leonard et al. 2002), and SN 2006gy, one of the most the luminous core-collapse SNe discovered (Smith et al. 2007). The later has been suggested to be a PI SN; however, the predicted model light curve durations are seen to be too long even for this event by a factor of several. Another possibility is that SN 2006gy was an example of a pulsational PI SN (Woosley et al. 2007).

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Figure 7 shows the multi-color optical and near-infrared light curves for the models. At bluer wavelengths, the light curves generally peak earlier and decline more rapidly after peak. This reflects the progressive shift of the spectral energy distribution to the red over time. This shift is not only due to the decrease in effective temperature, but also due to the increase in line opacity from iron group elements of lower ionization stages (Fe ii and Co ii), which blankets the bluer wavelengths.

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The brightest helium core models such as He130 display a pronounced secondary maximum in the infrared light curves. This feature is similar to what is observed in SNe Ia and the physical explanation is essentially the same (Kasen 2006). When the temperature in the ejecta drops below ∼7000 K, doubly ionized iron group elements begin to recombine. As the infrared line emissivity is much greater for singly ionized species (Fe ii and Co ii), the flux can be more efficiently redistributed from bluer to redder wavelengths, leading to an increase in the infrared luminosity. The secondary maximum is therefore more prominent in models with large abundances of iron group elements. Detection of a secondary maximum in an observed supernova would provide strong evidence that the explosion did indeed synthesize large amounts of ⁵⁶Ni. The lack of a secondary maximum does not necessarily rule out the presence of substantial ⁵⁶Ni, as strong radial mixing of the nickel can sometimes smear out the two bumps (Kasen 2006).

The spectra of the PI SN models (Figures 8 and 9) resemble those of ordinary SNe, with P-Cygni line profiles superimposed on a pseudo-blackbody continuum. Because of the low abundance of metals in unburned ejecta, many of the familiar line features are weak or missing in the early-time spectra. For example, at maximum light models R250 and B250 show only features from the hydrogen Balmer lines and calcium in their spectrum. However, at later times, when the photosphere has receded into layers of burned material, other line features appear.

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The maximum light spectrum of the bare helium core model He100 is dominated by lines from freshly synthesized intermediate-mass elements and resembles a Type Ic SN with lines due to Mg i, Mg ii, Si ii, Ca ii, and O i. After peak, the spectrum shows more features from the iron group elements in the ⁵⁶Ni-rich core. Helium lines are not present at any epoch, as the envelope temperatures are too low to thermally excite the lower atomic levels of the optical transitions. However, if ⁵⁶Ni is mixed out into the helium-rich layers, non-thermal excitation by radioactive decay products could generate significant helium line opacity (Lucy 1991).

Although PI SNe are highly energetic events, their large ejected masses imply only moderate characteristic velocities: v ∼ (2E_K/M_ej)^1/2 ∼ 5000 km s^–1, about half that typical of SNe Ia and several times less than the broad lined Type Ic SNe that have been associated with gamma-ray bursts (Galama et al. 1998). Figure 10 shows the time evolution of the velocity measured at the electron scattering photosphere. In the BSG and helium core models, the photosphere initially resides in the outer, high velocity layers of ejecta, but recedes quickly as these layers recombine and become transparent. Eventually, the photosphere settles in the inner regions of burned material, where radioactive energy deposition maintains the ionization state. Interestingly, the photospheric velocity in model B250 in fact increases for some time period as the radioactive energy diffuses outward and reionizes the hydrogen envelope. In contrast, the decline in photospheric velocity in the RSG models is gradual until the recombination front reaches the base of the hydrogen envelope, after which it drops off sharply.

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Certain features in the spectra of PI SNe could, if observed, offer direct confirmation that the progenitor star was of low metallicity. The metallicity has two principle spectroscopic effects (Figure 11). First, for metallicities Z ⩾ 10⁻⁴, some absorption lines from intermediate-mass elements are noticeable, for example, those of Ca ii H&K (near 3800 Å) and the IR triplet (near 8300 Å). Second, higher metallicity leads to a significant reduction in the ultraviolet (UV) flux, due to the iron group line blanketing at bluer wavelengths. Similar metallicity effects have been noted in the spectral modeling of SNe IIP (Baron et al. 2003; Dessart & Hillier 2005). Spectroscopic or rest-frame UV observations of PI SNe may, together with modeling, constrain the metallicity of the progenitor star. This assumes, of course, that the supernova has been observed early enough that we are seeing layers of ejecta unaffected by the explosive nucleosynthesis.

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To draw some comparison to observations, Figure 12 shows the near maximum light spectrum of one model, He100, compared to the most promising candidate to date for an observed pair supernova, SN 2007bi. Gal-Yam et al. (2009) have previously shown that the light curve of this supernova is a good match to model He100. On the whole, the correspondence of the spectra is also rather good, especially considering that the model has not been tuned to match the data. Most of the major line features are reproduced, in particular the unusually prominent Mg i and Mg ii lines. The ejecta velocities (as evidenced by the blueshift of the absorption features) are also roughly in agreement. The model fails to reproduce the forbidden Ca ii line emission at 7300 Å, but this feature results from non-equilibrium effects which are not included in our calculations.

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