DIVERSITY OF LUMINOUS SUPERNOVAE FROM NON-STEADY MASS LOSS

We consider a spherically symmetric dense wind extending from r = R_i to r = R_o, where r is the radius. The density ρ is assumed to follow ρ(r) = Dr^−w with a constant D. Throughout this paper, the opacity κ of the wind is assumed to be constant. Although our model is applicable to any uniform compositions with a constant κ, we mainly consider H-rich winds throughout this paper. We assume that an SN explosion has occurred in the dense wind and that the shock breakout occurs in the wind (e.g., Chevalier & Irwin 2011; see Section 1). The radius of the forward shock at the time of the shock breakout is set to r = xR_o. Note that the entire wind above the progenitor star is optically thick; this supports the assumption that the shock breakout occurs in dense wind. We also introduce a radius y_τR_o where the optical depth evaluated from the wind surface becomes τ, i.e.,

$$\begin{equation} \tau =\int ^{Ro}_{y_\tau R_o}\kappa \rho dr. \end{equation} \tag{ 1 }$$

Shock breakout in a wind had been simply defined to occur when the diffusion timescale of the entire wind is comparable to the shock propagation timescale of the entire wind, i.e.,

$$\begin{equation} \int ^{R_o}_{xR_o}\kappa \rho dr\simeq \frac{c}{v_s} \end{equation} \tag{ 2 }$$

(see, e.g., Weaver 1976; Nakar & Sari 2010, for the details of the shock breakout condition).

However, the wind could contain a large optically thin region even if the entire wind is optically thick. Figure 1 is a simplified illustration of the effect of w in the dense wind. Two dense winds with different w but the same R_o and xR_o are compared in the figure: (a) a wind with a constant density and (b) a wind with a steep density gradient. In both cases, the shock breakout is assumed to occur in the wind at the same radius r = xR_o (x < 1) with the same forward shock velocity v_s; thus, the optical depth between xR_o and R_o is exactly the same in both cases (τ_b). One of the important differences between the two winds is in the radius of the last scattering surface y₁R_o, where τ = 1 (see Figure 2 for the value of y₁). Even if the entire wind is optically thick in both cases, the region where τ becomes larger than 1 (r < y₁R_o) is more concentrated than the central region, and the wind contains a larger optically thin region outside whose size is ΔR = R_o − y₁R_o in the case of large w.

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If the wind contains an extended optically thin region, Equation (2) is no longer an appropriate condition for the shock breakout, because it postulates that the entire wind at r ⩽ R_o is optically thick enough for photons to be diffusive. The shock breakout condition should be evaluated only at the optically thick region where photons are diffusive. Thus, the shock breakout should occur when the diffusion timescale of the optically thick region of the wind is comparable to the shock propagation timescale of the region. Hence, the shock breakout condition should be set as

$$\begin{equation} \tau _x\equiv \int ^{y_1 R_o}_{xR_o}\kappa \rho dr\simeq \frac{c}{v_s}, \end{equation} \tag{ 3 }$$

where c is the speed of light and τ_x = τ_b − 1. In Equation (3), we presume for simplicity that photons diffuse in the region where the optical depth evaluated from the wind surface exceeds 1; i.e., photons are assumed to diffuse at R_i < r < y₁R_o and freely stream at y₁R_o < r < R_o. In this sense, Equation (3) may also be interpreted as Equation (2) combined with a kind of flux-limited diffusion approximation. For the case of the shock breakout at the surface of the wind (x ≃ 1), the conditions for Equations (2) and (3) are similar.

Since the observations of Type II LSNe display a constant shock velocity until some time in the declining phase of their LCs (e.g., Smith et al. 2010), we assume that v_s is constant. This assumption can be wrong if the shocked wind mass becomes comparable to the SN ejecta mass. Equations (1) and (3) lead us to

$$\begin{eqnarray} D&\simeq &\left\lbrace \begin{array}{@{}l@{\quad}l@{}} \dsty \frac{\left(1-w\right)\left(1+c/v_s\right)}{\kappa \left(1-x^{1-w}\right)R_o^{1-w}} & (w\ne 1),\\\ms\ms \dsty\frac{1+c/v_s}{\kappa (-\ln x)} & (w=1). \end{array} \right. \end{eqnarray} \tag{ 4 }$$

In addition, using Equation (4), we can express y_τ as

$$\begin{eqnarray} y_\tau &\simeq &\left\lbrace \begin{array}{@{}l@{\quad}l@{}l} \dsty \left(\frac{c/v_s+\tau x^{1-w}}{c/v_s+\tau }\right)^{\frac{1}{1-w}}&& (w\ne 1),\\\ms\ms \dsty x^{\frac{\tau }{c/v_s+\tau }} && (w=1). \end{array} \right. \end{eqnarray} \tag{ 5 }$$

Figure 2 shows a plot of y₁ as a function of x for several w. Larger w leads to smaller y₁ for a given x, as discussed qualitatively above.

Finally, the wind mass M_wind is defined as

$$\begin{equation} M_\mathrm{wind}=\int _{R_i}^{R_o}4\pi r^2\rho dr. \end{equation} \tag{ 6 }$$

Using Equation (4), we obtain

$$\begin{eqnarray} M_\mathrm{wind}\simeq \left\lbrace \begin{array}{@{}l@{\quad}l@{}l} \dsty \frac{4\pi (1-w)(1+c/v_s)\big(R_o^{3-w}-R_i^{3-w}\big)}{\kappa (3-w)(1-x^{1-w})R_o^{1-w}}&& (w\ne 1,3),\\\ms\ms \dsty \frac{2\pi (1+c/v_s)\big(R_o^{2}-R_i^{2}\big)}{\kappa (-\ln x)} && (w=1), \\\ms\ms \dsty \frac{8\pi (1+c/v_s)\ln (R_o/R_i)}{\kappa (x^{-2}-1) R_o^{-2}} && (w=3). \end{array} \right. \end{eqnarray} \tag{ 7 }$$

Here, we estimate the timescale of photon diffusion (t_d), which characterizes the LCs of LSNe, and the timescale of the shock propagation (t_s), which represents the timescale for the forward shock to go through the entire wind. t_d corresponds to the timescale for the LC to reach the peak (e.g., Arnett 1980, 1982). t_d can be expressed as

$$\begin{eqnarray} t_d&\simeq &\frac{\tau _x (R_o-xR_o-\Delta R)}{c}, \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} &=&\frac{\tau _x (y_1R_o-xR_o)}{c},\ \ \ \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} &=&\left\lbrace \begin{array}{@{}l@{\quad}l@{}l} \dsty\frac{R_o}{v_s}\left[\left(\frac{c/v_s+ x^{1-w}}{c/v_s+1}\right)^{\frac{1}{1-w}}-x\right]&& (w\ne 1),\\\ms\ms \dsty\frac{R_o}{v_s}\left(x^{\frac{1}{1+c/v_s}}-x\right) && (w=1). \end{array} \right. \end{eqnarray} \tag{ 10 }$$

t_s is defined as the time required for the forward shock to go through the entire wind, including the optically thin region, after the shock breakout,

$$\begin{equation} t_s=\frac{R_o-xR_o}{v_s}. \end{equation} \tag{ 11 }$$

Hence, we can derive the ratio of the two timescales:

$$\begin{eqnarray} \frac{t_d}{t_s}&\simeq &\left\lbrace \begin{array}{@{}l@{\quad}l@{}l} \dsty\frac{1}{1-x}\left[\left(\frac{c/v_s+ x^{1-w}}{c/v_s+1}\right)^{\frac{1}{1-w}}-x\right]&& (w\ne 1),\\\ms\ms \dsty\frac{1}{1-x}\left(x^{\frac{1}{1+c/v_s}}-x\right) && (w=1). \end{array} \right. \end{eqnarray} \tag{ 12 }$$

Figure 3 shows the ratio as a function of x with several w, in which v_s is set to 10,000 km s⁻¹ corresponding to the observational value of SN 2008es (see Section 3.2). Every line reaches t_d/t_s ≃ c/(v_s + c) = 0.97 at x ≃ 1. This corresponds to the case of the shock breakout at the surface of the wind (e.g., Chevalier & Irwin 2011).

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When the shock breakout occurs inside the wind (x < 1), the ratio t_d/t_s varies depending on the density slope of the wind. For a given x, t_d/t_s gets smaller as the density slope of the wind gets steeper (i.e., larger w). This is because the last scattering surface of the wind is located farther away from the surface as the density slope gets steeper, i.e., y₁ gets smaller as w gets larger for a given x (see Figure 2 for values of y₁). In other words, the optically thin region in the wind is spatially larger for wind with a steeper density gradient. This is true even if the wind radius is the same. We note again that, when the shock breakout occurs in the wind, the forward shock reaches the last scattering surface (r = y₁R_o) with the diffusion timescale t_d, i.e., the shock locates at r = y₁R_o at the LC peak.

In this section, we present consequences of the variation in t_d/t_s that results from the variation in w. We show that the two kinds of Type II LSNe, i.e., Type IIn LSNe (e.g., SN 2006gy; Smith et al. 2010) and Type IIL LSNe (e.g., SN 2008es; Miller et al. 2009; Gezari et al. 2009), are naturally expected from the shock breakout in the dense H-rich wind with a different density slope.

If we assume that both types of LSNe are illuminated by the interaction of the H-rich dense wind and SN ejecta and that the wind is so dense that the shock breakout occurs in the wind, the spectral evolution of LSNe is determined by t_d/t_s.

If t_d/t_s ≃ 1, the shock wave reaches the surface of the dense wind soon after the LC has reached the peak with the timescale t_d. Since the entire wind is shocked just after the LC peak, no signature in spectra from the wind is observable after the LC peak. On the other hand, if t_d/t_s < 1, the shock wave continues to propagate in the optically thin region of the wind even after the LC peak. As there remain unshocked materials in the wind even after the LC peak, we expect to see narrow P-Cygni profiles from the unshocked wind even after the LC peak.

To sum up, Type IIL LSNe can come from the dense wind with t_d/t_s ≃ 1, while Type IIn LSNe can result from the dense wind with t_d/t_s < 1. The ratio of the two timescales is determined by w.

The narrow Lorentzian line profiles which are suggested to be caused by the dense wind (e.g., Chugai 2001; Dessart et al. 2009) can appear before the LC peak depending on the optical depth of the wind. If we apply the model of Chugai (2001), the ratio U of the unscattered Hα line flux to the total Hα line flux of the Lorentzian profile is

$$\begin{equation} U\simeq \frac{1-e^{-\tau }}{\tau }. \end{equation} \tag{ 13 }$$

For the case of flat density slopes, τ remains too high until the forward shock reaches the surface (Figure 1), and U is expected to be very small for a long time before the LC peak. On the other hand, if the density decline is steep, the optical depth decreases gradually with time and the suitable optical depth for the appearance of Lorentzian profiles is realized for a long while. Therefore, Lorentzian lines are expected to be observed well before the LC peak for Type IIn LSNe.

The LC evolution of Type II LSNe is also consistent with our models. In our models for both Type IIn and Type IIL LSNe, the forward shock stays in the dense wind until the LC peak. As the wind with τ > 1 is shocked with the timescale of t_d, the dense wind adiabatically cools down after the LC peak. Thus, the LCs of Type II LSNe are supposed to follow the shell-shocked diffusion model presented by Smith & McCray (2007). The shell-shocked diffusion model is based on the adiabatic cooling of the shocked dense wind, which is basically the same as the LC model suggested for Type II SNe by Arnett (1980), and the model has already been shown to be consistent with the declining phase of the LC of SN 2006gy (Smith & McCray 2007). However, it should be noted that the model is too simple, and many effects which cannot be treated by the formulation of Arnett (1980) are ignored in the model. For example, the model assumes a constant opacity and ignores the presence of a recombination wave, which is supposed to be created in the diffusing shocked shell. Thus, we cannot confirm that our model is consistent with the LCs of Type II LSNe by comparison with the shell-shocked diffusion model alone, and numerical LC modeling is required to determine whether our models are consistent with Type II LSN LCs.

SN 2006gy is extensively studied by, e.g., Ofek et al. (2007), Smith et al. (2007, 2008, 2010), Smith & McCray (2007), Woosley et al. (2007), Agnoletto et al. (2009), Kawabata et al. (2009), and Miller et al. (2010). It is classified as Type IIn, and the luminosity reaches ∼ − 22 mag in the R band (Smith et al. 2007). The detailed spectral evolution is summarized in Smith et al. (2010). The narrow P-Cygni Hα lines with the absorption minimum of ≃ 100 km s⁻¹ are considered to come from the wind surrounding the progenitor of SN 2006gy. As SN 2006gy shows narrow P-Cygni profiles after the maximum luminosity, an unshocked wind is supposed to remain after the maximum. Thus, models with t_d/t_s < 1 and y₁ < 1 are preferred. Based on the observations of Smith et al. (2010), we adopt the following parameters:

$$\begin{eqnarray} v_s&\simeq &5200 \,\mathrm{km\, s^{-1}}, \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} t_d&\simeq &70 \,\mathrm{days}. \quad\ \ \ \end{eqnarray} \tag{ 15 }$$

v_s is constrained by the evolution of the blackbody radius, and t_d is obtained from the rising time of the LC. As the narrow Hα P-Cygni profile is detected at 179 days⁷ and disappears at 209 days (Smith et al. 2010), we presume that the forward shock has gone through the entire wind between 179 days and 209 days. We simply take the central date (194 days) as the time when the forward shock has gone through the entire wind, i.e., t_s ≃ 194 days. With t_d, t_s, and v_s, we can estimate x and R_o for a given w from Equations (10) and (11).

If we adopt the model with w = 2, for example, x and R_o are estimated to be 0.0095 and 8.8 × 10¹⁵ cm, respectively. In this case, the shock breakout occurs at xR_o ≃ 3.2 × 10¹⁴ cm, and the last scattering surface is y₁R_o ≃ 3.2 × 10¹⁵ cm. The total wind mass is M_wind ≃ 0.81 M_☉ (cf. R_i ≪ R_o) and is much smaller than the value estimated from the shell-shocked diffusion LC model (∼10 M_☉; Smith & McCray 2007). However, the shell-shocked diffusion model is too simple a model, and we cannot exclude this model just because of the inconsistency with it, as noted in the previous section. Alternatively, if we adopt a steeper density gradient w = 5, x and R_o are estimated to be 0.17 and 1.05 × 10¹⁶ cm, respectively; thus, xR_o ≃ 1.8 × 10¹⁵ cm and y₁R_o ≃ 4.9 × 10¹⁵ cm. y₁R_o is consistent with the blackbody radius at the LC peak estimated from the observations (6 × 10¹⁵ cm). If we assume that R_i ≃ 10¹⁵ cm, the mass contained in the optically thick region (R_i < r < y₁R_o) is 22 M_☉ in our w = 5 model. In this case, the mass of the entire wind becomes M_wind ≃ 23 M_☉. The left panel of Figure 4 shows the optical depth and the enclosed mass distributions. The existence of the unshocked wind may also account for the weakness of the X-ray emission of SN 2006gy (e.g., Woosley et al. 2007).

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The spectral evolution of SN 2006gy is also consistent with our model. Lorentzian H Balmer lines seen in the spectra of SN 2006gy are presumed to be caused by the optically thick wind (e.g., Chugai 2001; Dessart et al. 2009). For example, for the case w = 5, τ becomes ≃ 10 at around 3 × 10¹⁵ cm (Figure 4). This is consistent with τ ≃ 15 at 36 days, which is estimated from the observed ratio U derived from Hα (Smith et al. 2010). In our model, the Lorentzian line profiles are expected to be observed before the forward shock wave goes through the optically thick region of the wind, i.e., before t_d; thus, the Lorentzian spectra should disappear after the LC peak. This is also consistent with the spectra of SN 2006gy (Smith et al. 2010).

Narrow Hα P-Cygni profiles can be created at the optically thin wind above the last scattering surface of the continuum photons (y₁R_o < r < R_o) because of the larger line opacities. Whether or not the narrow Hα P-Cygni profiles can be formed also depends on the ionization level of the wind; thus, the spectral modeling must be performed to determine whether the narrow Hα profiles are actually synthesized in the unshocked wind in our model.

Here, we apply our model to SN 2008es, which is one of the best observed Type IIL LSNe (Miller et al. 2009; Gezari et al. 2009). Although SN 2008es does not show any features of the wind in the spectra, we assume that it is also illuminated by the interaction, based on their brightness, rapid decline of the LC, and blue spectra. In our model, the lack of wind features after the LC peak can be explained by the small difference in t_d and t_s, because the entire wind is shocked by the forward shock just after the LC peak. In other words, y₁ should be close to 1. This can be achieved by the wind slope with w ≲ 1 for the case of x < 1 (Figure 2).

The following parameters are estimated from the observations:

$$\begin{eqnarray} v_s&\simeq &{\rm 10{,}000} \,\mathrm{km\, s^{-1}}, \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} t_d\simeq t_s&\simeq &23 \,\mathrm{days}. \ \ \end{eqnarray} \tag{ 17 }$$

If we adopt the model of w = 0, y₁ is close to 1 and R_o ≃ v_st_s ≃ y₁R_o ≃ 2 × 10¹⁵ cm. y₁R_o is consistent with the blackbody radius at the LC peak estimated from the observations (3 × 10¹⁵ cm). Assuming x = 0.1 and R_i ≪ R_o, the wind mass is M_wind ≃ 0.85 M_☉. M_wind does not vary so much on x unless it is close to 1. The model with w = 1 also gives y₁ ≃ 1 with M_wind ≃ 0.50 M_☉. The right panel of Figure 4 shows the optical depth and mass distributions. Miller et al. (2009) estimated M_wind ≃ 5 M_☉ based on the shell-shocked diffusion LC model of Smith & McCray (2007). Gezari et al. (2009) obtained M_wind ≃ 0.2 M_☉ based on the peak luminosity. M_wind measurements of both the w = 0 and w = 1 models are almost consistent with those estimates.

Lack of Lorentzian H Balmer lines before the LC peak is another important difference between Type IIL LSN 2008es and Type IIn LSN 2006gy. In the case of the flat density slope, the wind optical depth remains very large until the forward shock reaches the wind surface (Figure 4). Hence, narrow H lines, even if they are emitted from the dense region of the wind, are scattered by the dense wind with a large optical depth for a long while. Then, U will be very small until the forward shock reaches the wind surface, and the Lorentzian H lines will be very weak. Thus, it is likely that the Lorentzian H lines are missed. Detailed modeling is required to see the actual spectral evolution expected from the density profile of our model.