THE EARLY UV/OPTICAL EMISSION FROM CORE-COLLAPSE SUPERNOVAE

The UV/O emission on a day timescale arises from the outer ≲10⁻² M_☉ shell of the ejecta (Waxman et al. 2007). Neglecting the shell's self-gravity and its thickness (relative to R_*), the pre-explosion density profile within the shell may be approximated by Chandrasekhar (1939):

$$\begin{equation} \rho _0(r_0) = \rho _{1/2} \delta ^n, \end{equation} \tag{ 1 }$$

where δ ≡ (1 − r₀/R_*), r₀ is the radius, and n = 3, 3/2 for radiative and efficiently convective envelopes, respectively. Matzner & McKee (1999) have shown that the velocity of the SN shock within the envelope is well approximated by an interpolation between the Sedov–von Neumann–Taylor and the Gandel'Man–Frank-Kamenetskii–Sakurai self-similar solutions, (Von Neumann 1947; Sedov 1959; Taylor 1950; Gandel'Man & Frank-Kamenetskii 1956; Sakurai 1960):

$$\begin{equation} v_s(r_0) = A_v \left[ \frac{E}{m(r_0)} \right]^{1/2} \left[ \frac{m(r_0)}{\rho r_0^3} \right]^{\beta _1}, \end{equation} \tag{ 2 }$$

where E is the energy deposited in the ejecta, m(r₀) is the ejecta mass enclosed within r₀, A_v ≃ 0.79, and β₁ ≃ 0.19. For r₀ → R_*, Equation (2) reduces to

$$\begin{equation} v_s \simeq A_v \left(\frac{E}{M} \right)^{1/2} \left(\frac{4\pi }{3f_{\rho }} \right)^{\beta _1} \delta ^{-\beta _1 n}, \end{equation} \tag{ 3 }$$

where $f_{\rho } \equiv \rho _{1/2}/ \overline{\rho }_0$, and $\overline{\rho }_0$ is the average ejecta density, $\overline{\rho }_0\equiv 3M/4\pi R^3_*$. Calzavara & Matzner (2004, Appendix A) have derived the values of f_ρ expected for various progenitors (using their notation, f_ρ = (3/4π)ρ₁/ρ_*). For BSGs, they find f_ρ varying nearly linearly with mass, from 0.031 to 0.062 for 8.5 < M/M_☉ < 18.5, and for RSGs, they find 0.079 ≲ f_ρ ≲ 0.13. We show below that the results are not very sensitive to the value of f_ρ.

In what follows, we replace the Lagrangian coordinate r₀ with

$$\begin{equation} \delta _m(\delta) \equiv M^{-1}\int ^{R_*}_{(1-\delta)R_*} \mbox{\it \mbox{d}}r 4 \pi r^2 \rho _0(r) \simeq \frac{3 f_{\rho }}{n+1} \delta ^{n+1}, \end{equation} \tag{ 4 }$$

the fraction of the ejecta mass lying initially above r₀.

As the radiation mediated shock passes through a fluid element lying at r₀, it increases its pressure to

$$\begin{equation} p_0 = \frac{6}{7} \rho _0 v_s^2, \end{equation} \tag{ 5 }$$

and its density to 7ρ₀. (Recall that the post-shock energy density is dominated by radiation.) As the shocked fluid expands, it accelerates, converting its internal energy to kinetic energy. Matzner & McKee (1999) have shown that the final velocity, v_f(r₀), of the fluid initially lying at r₀ is well approximated by v_f(r₀) = f_v(r₀)v_s(r₀), with f_v ≈ 2. The value of f_v depends on the curvature of the shells. The effect of this dependence is considered below.

Equations (3) and (5) hold as long as the shock width is much smaller than the width of the stellar envelope shell lying ahead of the shock. When the shock reaches a radius at which the optical depth of the shock transition layer, τ_s ≅ c/v_s, becomes comparable to the optical depth of the shell lying ahead of the shock, τ₀ ⋍ Mδ_mκ/4πR²_*, the radiation "escapes" ahead of the shock, producing a "shock breakout flash," and the shock can no longer be sustained by radiation. The mass fraction at which breakout takes place is (e.g., Matzner & McKee 1999)

$$\begin{equation} \delta _{m,\rm BO}\backsimeq 2 \times 10^{-5} \frac{f_{\rho }^{-0.07} R_{*,13}^{2.3}}{E_{51}^{0.57} (M/M_{\odot })^{0.57} \kappa ^{1.1}_{0.34} }. \end{equation} \tag{ 6 }$$

For smaller values of δ_m, δ_m < δ_m,BO, the velocity and pressure profiles are shallower than given by Equations (3) and (5).

Assuming adiabatic expansion, neglecting photon diffusion below the photosphere, the pressure and density of the expanding fluid are related by

$$\begin{equation} p(\delta _m,t) = \bigg [ \frac{\rho (\delta _m,t)}{7\rho _0(\delta _m)} \bigg ]^{4/3} p_0(\delta _m). \end{equation} \tag{ 7 }$$

Once a fluid shell expands to a radius significantly larger than R_*, its pressure drops well below p₀ and its velocity approaches the final velocity v_f. At this stage, v_ft ≫ R_*, the shell's radius is approximately given by

$$\begin{equation} r(\delta _m,t) \cong v_f(\delta _m) t, \end{equation} \tag{ 8 }$$

and its density is then given by

$$\begin{equation} \rho = -\frac{M}{4\pi r^2 t } \left(\frac{dv_f}{d\delta _m} \right)^{-1} \cong \frac{n+1}{\beta _1\,n} \, \frac{ M}{4 \pi t^3 v_f^3 } \delta _m. \end{equation} \tag{ 9 }$$

The resulting density profile is steep, dln ρ/dln r = dln ρ/dln v_f = −3 − (n + 1)/β₁n ≈ −10.

For a time- and space-independent opacity κ (which applies, e.g., for opacity dominated by Thomson scattering with constant ionization), the optical depth of the plasma lying above the shell marked by δ_m is

$$\begin{eqnarray} \tau (\delta _m,t) & \equiv & \int _{r(\delta _m,t)}^{\infty } \mbox{\mbox{\it d}}r \kappa \rho (r,t) = \frac{\kappa M }{4\pi } \int _{0}^{\delta _m} \frac{\mbox{\mbox{\it d}}\delta _m^{\prime }}{r^2(\delta _m^{\prime })} \nonumber \\ & = & \frac{1}{1+2 \beta _1 n/(1+n)} \frac{\kappa M \delta _m}{4\pi t^2 v_f^2(\delta _m)}, \end{eqnarray} \tag{ 10 }$$

where the last equality holds when Equation (8) is satisfied. We define the Lagrangian location of the photosphere, δ_m,ph, by τ(δ_m = δ_m,ph, t) = 1. We consider two type of envelopes: radiative envelopes typical to BSGs, for which we take n = 3, and efficiently convective envelopes typical to RSGs, for which we take n = 3/2. (Note that inefficient convection may lead to a more complicated density profile.) For n = 3/2 and n = 3 envelopes, we have

$$\begin{eqnarray} \delta _{m,\rm ph}(t) &= 2.4 \times 10^{-3} f_{\rho }^{-0.12} \frac{ E_{51}^{0.81}}{(M/M_{\odot })^{1.6} \kappa ^{0.81}_{0.34} } t_5^{1.63} \, \left(n=\frac{3}{2}\right),\nonumber\\ \delta _{m,\rm ph}(t) &= 2.6 \times 10^{-3} f_{\rho }^{-0.073} \frac{ E_{51}^{0.78}}{(M/M_{\odot })^{1.6} \kappa ^{0.78}_{0.34} } t_5^{1.56} \, (n=3), \nonumber\\ \end{eqnarray} \tag{ 11 }$$

where E = 10⁵¹E₅₁erg, t = 10⁵t₅s, and κ = 0.34κ_0.34cm² g⁻¹. Here, and in what follows, we use (following Matzner & McKee 1999) β₁ = 0.1909, f_v = 2.1649, andA_v = 0.7921 for n = 3/2 and β₁ = 0.1858, f_v = 2.0351, andA_v = 0.8046 for n = 3. Using Equations (3), (8), and (7), we find that the radius and the effective temperature of the photosphere are given by

$$\begin{eqnarray} r_{\rm{ph}}(t) &= 3.3 \times 10^{14} f_{\rho }^{-0.062} \frac{E_{51}^{0.41} \kappa _{0.34}^{0.093} }{(M/M_{\odot })^{0.31}} t_5^{0.81}\mbox{ cm}\, \left(n=\frac{3}{2}\right),\nonumber\\ r_{\rm{ph}}(t) &= 3.3 \times 10^{14} f_{\rho }^{-0.036} \frac{E_{51}^{0.39} \kappa _{0.34}^{0.11} }{(M/M_{\odot })^{0.28}} t_5^{0.78}\mbox{ cm}\, (n=3),\qquad \end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray} T_{\rm{ph}}(t) &=& 1.6 \, f_{\rho }^{-0.037} \frac{E_{51}^{0.027} R_{*,13}^{1/4} }{(M/M_{\odot })^{0.054} \kappa ^{0.28}_{0.34}} t_5^{-0.45} \mbox{ eV}\, \left(n=\frac{3}{2}\right),\nonumber\\ T_{\rm{ph}}(t) &=&1.6 \, f_{\rho }^{-0.022} \frac{E_{51}^{0.016} R_{*,13}^{1/4} }{(M/M_{\odot })^{0.033} \kappa ^{0.27}_{0.34}} t_5^{-0.47} \mbox{ eV}\, (n=3).\qquad \end{eqnarray} \tag{ 13 }$$

Here, R_* = 10¹³R_*,13cm. The dependence on n and on f_ρ is weak. Note that Equation (13) corrects a typo (in the numerical coefficient) in Equation (19) of Waxman et al. (2007).

As mentioned in the introduction, the photospheric temperature is weakly dependent on E and M and is approximately linear in (R_*/κ)^1/4. The photospheric radius, on the other hand, does not depend on R_*, is weakly dependent on κ and is approximately linear in E^0.4/M^0.3. The luminosity predicted by the simple model described here, L = 4πσr²_phT⁴_ph, is

$$\begin{equation} L = 8.5\times 10^{42} \frac{E^{0.92}_{51} R_{*,13} }{f^{0.27}_{\rho } (M/M_{\odot })^{0.84}\kappa ^{0.92}_{0.34} }t^{-0.16}_5 \mbox{ erg}\, {\rm s}^{-1} \end{equation} \tag{ 14 }$$

for n = 3/2, and

$$\begin{equation} L = 9.9\times 10^{42} \frac{E^{0.85}_{51} R_{*,13} }{f^{0.16}_{\rho } (M/M_{\odot })^{0.69}\kappa ^{0.85}_{0.34} }t^{-0.35}_5 \mbox{ erg}\, {\rm s}^{-1} \end{equation} \tag{ 15 }$$

for n = 3.

Our simple description of r_ph, T_ph, and L, Equations (12)–(15), holds for δ_m,ph > δ_m,BO, i.e., for (comparing Equations (6) and (11))

$$\begin{equation} t>t_{\rm BO}=0.05\times 10^5 \frac{(M/M_{\odot })^{0.6} R_{*,13}^{1.4} }{\kappa ^{0.2}_{0.34}E^{0.9}_{51} }\rm \ s. \end{equation} \tag{ 16 }$$

This requirement also ensures that the ejecta shells have expanded and cooled significantly, and thus reached their terminal velocity v_f. For these times, the approximation of Equation (7) holds, and the radius of each shell is well approximated by r = t v_f.

The value of f_v deviates from 2 for large δ_m, due to the increasing curvature of the shells (e.g., Matzner & McKee 1999). Requiring the deviation not to exceed 30%, in which case the error in Equation (13) is smaller than 15%, implies limiting the analysis to times

$$\begin{equation} t < (f_{\rho }/0.07)^{0.69} (M/M_{\odot }) \kappa _{0.34}^{0.5} E_{51}^{-0.5} \times t_{\rm fv}, \end{equation} \tag{ 17 }$$

where t_fv = 4.5 × 10⁶ sand1.2 × 10⁵ s for n = 3/2 and n = 3 envelopes, respectively. Thus, the curvature effect is negligible on day–week timescale for M ∼ 10 M_☉, and may be significant on day timescale only for low mass ejecta.

Let us next examine the assumption that photon diffusion does not lead to strong deviations from adiabatic expansion below the photosphere. The size of a region around r(δ_m, t) over which the diffusion has a significant effect is $D(\delta _m,t) \simeq \sqrt{ct/3k\rho (\delta _m,t)}$. Thus, the radius r_d = r(δ_m,d, t) above which diffusion affects the flow significantly may be estimated as D(δ_m = δ_m,d, t) = r(δ_m = δ_m,d, t). This gives

$$\begin{eqnarray} r_d(t) &= 3.7 \times 10^{14} f_{\rho }^{-0.069} \frac{E_{51}^{0.45} \kappa _{0.34}^{0.1} }{(M/M_{\odot })^{0.35}} t_5^{0.79}\mbox{ cm}\, \left(n=\frac{3}{2}\right),\nonumber\\ r_d(t) &= 3.8 \times 10^{14} f_{\rho }^{-0.04} \frac{E_{51}^{0.44} \kappa _{0.34}^{0.12} }{(M/M_{\odot })^{0.32}} t_5^{0.75}\mbox{ cm}\, (n=3).\qquad \end{eqnarray} \tag{ 18 }$$

This radius is similar to, and somewhat larger than, the photospheric radius given by Equation (12). The rapid increase of the diffusion time, ∼3κρr²/c, at smaller radii implies that diffusion does not significantly affect the fluid energy density below the photosphere. Next, we note that in regions where the diffusion time is short, the luminosity carried by radiation, L ∝ r²dp/dτ, is expected to be independent of radius. The steep dependence of the density on radius (dln ρ/dln r ∼ −10) then implies that the energy density roughly follows p ∝ τ, i.e., T ∝ τ^1/4. This temperature profile is close to the adiabatic profiles derived in Section 2.1, for which T ∝ τ^0.28andτ^0.27 for n = 3and3/2, respectively. Thus, diffusion does not lead to a significant modification of the pressure and temperature profiles also at radii where the diffusion time is short.

The validity of the above conclusions may be tested by using the self-similar solutions of Chevalier (1992), which describe the diffusion of photons in an expanding envelope with a density following ρ ∝ r^−mt^{m − 3} and initial pressure p ∝ r^−lt^{l − 4}. The evolution of the ejecta density and pressure derived in Section 2.1 follows, for v_ft ≫ R_*, ρ = Br^−mt^{m − 3} with m − 3 = (1 + n)/nβ₁, and p = Ar^−lt^{l − 4} with l = (3γ − 2) + (γ + n)/nβ₁, where γ = 4/3. Applying the solutions of Chevalier (1992) to these profiles, one finds the same r_d as given by Equation (18) and

$$\begin{equation} L_c = 9.6\times 10^{42} \frac{E^{0.91}_{51} R_{*,13} }{f^{0.17}_{\rho } (M/M_{\odot })^{0.74}\kappa ^{0.82}_{0.34} }t^{-0.35}_5 \mbox{ erg}\, {\rm s}^{-1}, \end{equation} \tag{ 19 }$$

for n = 3 and

$$\begin{equation} L_c = 1.0\times 10^{43} \frac{E^{0.96}_{51} R_{*,13} }{f^{0.28}_{\rho } (M/M_{\odot })^{0.87}\kappa ^{0.91}_{0.34} }t^{-0.17}_5 \mbox{ erg}\, {\rm s}^{-1}, \end{equation} \tag{ 20 }$$

for n = 3/2. (The parameter q of the self-similar solutions is q = 0.495and0.683 for n = 3 and n = 3/2, respectively. The density and pressure coefficients are A = 2.95E^3.89R_*f^−0.44M^−2.89,  53.7E^4.95R_*f^−0.89M^−3.95 and B = 100E^3.59f^−0.33M^−2.59, 10³E^4.37f^−0.67M^−3.37for n = 3 and n = 3/2, respectively.)

The parameter dependence of L_c derived using the self-similar diffusion solutions is similar to that obtained by the simple model of Section 2.2, and the normalization of L_c derived using the self-similar diffusion solutions differs from the results of Section 2.2, Equations (15) and (14), by ≈10%. The effective temperatures derived from the diffusion solutions, via L = 4πr²_phσT⁴, differ from those derived in the previous section by 1%–5%.

Thus, the effects of diffusion on the luminosity and on the effective temperature are small, as expected. It is important to emphasize in this context that since the diffusion approximation breaks down near the photosphere, the results obtained using the self-similar diffusion solutions are not necessarily more accurate than those derived (e.g., in the previous section) by neglecting photon diffusion below the photosphere. Improving the accuracy of the simple model requires a transport, rather than a diffusion, description of the photon propagation. The differences between the results obtained neglecting diffusion and including it may be considered as a rough estimate of the inaccuracy of the model.

Finally, the following note is in place here. The results of Chevalier (1992) for L and r_d (Equations (3.19) and (3.20)) are different both in normalization and in scaling from those derived here. This is due to some typographical errors in earlier equations of that paper. When corrected, in Chevalier & Fransson (2008), the results obtained using the diffusion solutions are similar to those obtained in Waxman et al. (2007) and here. The difference in the numerical coefficient of the photospheric temperature, Equation (19) of Waxman et al. (2007) and Equation (5) of Chevalier & Fransson (2008), is mainly due to the typo in Equation (19) of Waxman et al. (2007), which is corrected in Equation (13) above.

In order to obtain a more accurate description of the early UV/O emission, we use the mean opacity provided in the OP project tables (Seaton 2005). We replace Equation (10) with

$$\begin{equation} \tau (\delta _{m},t)= \int _{r(\delta _{m},t)}^{\infty }\mbox{\emph {d}}r\rho \,\kappa [T(\delta _{m},t),\rho (\delta _{m},t)], \end{equation} \tag{ 21 }$$

where κ(T, ρ) is the Rosseland mean of the opacity, and solve τ(δ_m = δ_m,ph, t) = 1 numerically for the location of the photosphere. In order to simplify the comparisons with the suggested analytical models, in the remainder of this section we shall take the ejecta properties in the limit of Equation (8).

3.1.1. H Envelopes

Consider first explosions in H-dominated envelopes. In Figure 1, we compare the temperature of the photosphere calculated using the OP tables with the results given by Equation (13) for κ = 0.34 cm² g^-1, corresponding to fully ionized H. The difference in T_ph obtained by the two methods is smaller than 10% for T_ph > 1 eV. At lower temperatures, the κ = 0.34 cm² g^-1 approximation leads to an underestimate of T_ph, by ≈20% at 0.7 eV. This is due to the reduction in opacity accompanying H recombination. The reduced opacity implies that the photosphere penetrates deeper into the expanding envelope, to a region of higher temperature. The photospheric radius is not significantly affected and is well described by Equation (12).

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3.1.2. He Envelopes

Let us consider other explosions in He-dominated envelopes. In this case, the constant opacity approximation does not provide an accurate approximation for T_ph. We therefore replace Equations (12) and (13) with an approximation which takes into account the reduction of the opacity due to recombination. On the timescale of interest, hour ≲t≲ day, the photospheric temperature is in the energy range of 3 eV ≳T≳ 1 eV. In this temperature range (and for the characteristic densities of the photosphere), the opacity may be crudely approximated by a broken power law:

$$\begin{equation} \kappa = 0.085 \,\mbox{cm}^2 \mbox{g}^{-1} {\left\lbrace {\begin{array}{@{}l@{\quad }l@{}}(T/1.07\, \mbox{ eV})^{0.88} & T > 1.07\, \mbox{ eV}, \\ \ms (T/1.07\, \mbox{ eV})^{10} & T \le 1.07\, \mbox{ eV}. \end{array}}\right.}\, \end{equation} \tag{ 22 }$$

Using this opacity approximation, we find that Equation (13) for the photospheric temperature is modified to

$$\begin{equation} T_{\rm{ph}}(t) = {\left\lbrace {\begin{array}{@{}l@{\quad }l@{}}1.33 \mbox{ eV}f_\rho ^{-0.02}R_{*,12}^{0.20} t_5^{-0.38} & T_{\rm{ph}} \ge 1.07\, \mbox{ eV}, \\ \ms 1.07\, \mbox{ eV}(t/t_b)^{-0.12} & T_{\rm{ph}} < 1.07\, \mbox{ eV}. \end{array}}\right.} \end{equation} \tag{ 23 }$$

Here, R_* = 10¹²R_*,12 cm, and t_b is the time at which T_ph = 1.07 eV, and we have neglected the dependence on E and M, which is very weak. The photospheric radius, which is less sensitive to the opacity modification, is approximately given by

$$\begin{equation} r_{\rm{ph}}(t) = 2.8\times 10^{14} f_\rho ^{-0.038} E_{51}^{0.39} (M/M_{\odot })^{-0.28} t_5^{0.75} \mbox{ cm}. \end{equation} \tag{ 24 }$$

Here, we have neglected the dependence on R_*, which is weak. For T_ph > 1.07 eV, the bolometric luminosity is given by

$$\begin{equation} L = 3.3\times 10^{42} \frac{E^{0.84}_{51} R_{*,13}^{0.85} }{f^{0.15}_{\rho } (M/M_{\odot })^{0.67} }t^{-0.03}_5 \mbox{ erg}\, {\rm s}^{-1}. \end{equation} \tag{ 25 }$$

The deviation of f_v from the assumed value of 2 for large δ_m (discussed in Section 2.2) has only a negligible effect on Equations (23) and (25).

In Figure 2, we compare the approximation of Equation (23) for T_ph with a numerical calculation using the OP opacity tables. The approximation of Equation (23) holds to better than 8% down to T_ph ≃ 1 eV. The temperature does not decrease significantly below ≃1 eV due to the rapid decrease in opacity below this temperature, which is caused by the nearly complete recombination.

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The following comment is in place here. The strong reduction in opacity due to He recombination implies that the photosphere reaches deeper into the envelope, to larger values of δ_m. The plus signs in Figure 2 denote the time at which δ_m,ph = 0.1. For such a large mass fraction, the initial density profile is no longer described by Equation (1) and the evolution of the ejecta is no longer given by equations of Section 2.1. This further complicates the model for the emission on these timescales (see Section 6.1 for further discussion).

3.1.3. He–C/O Envelopes

Let us consider next envelopes composed of a mixture of He and C/O. At the relevant temperature and density ranges, the C/O opacity is dominated by Thomson scattering of free electrons provided by these atoms, and is not very sensitive to the C:O ratio. Denoting by 1 − Z the He mass fraction, the C/O contribution to the opacity may be crudely approximated, within the relevant temperature and density ranges, by

$$\begin{equation} \kappa = 0.043\,{Z}\,\mbox{cm}^2 \mbox{g}^{-1} (T/1\mbox{ eV})^{1.27}. \end{equation} \tag{ 26 }$$

This approximation holds for a 1:1 C:O ratio. However, since the opacity is not strongly dependent on this ratio, T_ph obtained using Equation (26) (Equation (27)) holds for a wide range of C:O ratios (see discussion at the end of this subsection). At the regime where the opacity is dominated by C/O, Equation (13) is modified to

$$\begin{equation} T_{\rm{ph}}(t) = 1.5 \mbox{ eV}f_\rho ^{-0.017} {Z}^{-0.2} R_{*,12}^{0.19} t_5^{-0.35}. \end{equation} \tag{ 27 }$$

In the absence of He, i.e., for Z = 1, T_ph is simply given by Equation (27). For a mixture of He–C/O, Z < 1, T_ph may be obtained as follows. At high temperatures, where He is still ionized, the He and C/O opacities are not very different and T_ph obtained for a He envelope, Equation (23), is similar to that obtained for a C/O envelope, Equation (27). At such temperatures, we may use Equation (23) for an envelope containing mostly He, and Equation (27) with Z = 1 for an envelope containing mostly C/O (a more accurate description of the Z-dependence may be straightforwardly obtained by an interpolation between the two equations). At lower temperatures, the He recombines and the opacity is dominated by C/O. At these temperatures, T_ph is given by Equation (27) with the appropriate value of Z. The transition temperature is given by

$$\begin{equation} T_{\rm He\hbox{--}C/O} = 1\, Z^{0.1} \mbox{ eV}. \end{equation} \tag{ 28 }$$

The photospheric radius, which is less sensitive to the opacity variations, is well approximated by Equation (24). At the stage where the opacity is dominated by C/O, the bolometric luminosity is given by

$$\begin{equation} L = 4.7\times 10^{42} \frac{E^{0.83}_{51} R_{*,13}^{0.8} }{f^{0.14}_{\rho } Z^{0.63}(M/M_{\odot })^{0.67} }t^{-0.07}_5 \mbox{ erg}\, {\rm s}^{-1}. \end{equation} \tag{ 29 }$$

The deviation of f_v from the assumed value of 2 for large δ_m (discussed in Section 2.2) does not significantly affect the results of Equations (27) and (28), but may significantly affect the result given in Equation (29). Equations (17), (26), and (27) indicate that for an explosion with M = 10 M_☉ and E₅₁ = 1, the luminosity will be reduced by a factor of ∼2 (compared to the predictions of Equation (29)) when T_ph ≈ 1eV. This further complicates the model for the emission on these timescales. See Section 6.1 for further discussion.

In Figures 3 and 4, we compare the analytic approximation for T_ph derived above to the results of numerical calculations using the OP opacity tables. For the C/O envelopes (Figure 3), the approximation of Equation (27) holds to better than 6% down to T_ph ≃ 0.5 eV. For the Z = 0.3 mixed He–C/O envelopes (Figure 4), the approximations obtained by using Equations (23) and (27) with a transition temperature given by Equation (28) hold to better than 10% down to T_ph ≃ 0.8 eV. Using similar comparisons for different compositions, we find that similar accuracies are obtained over the range 0.7>Z > 0.3, and for increasing or decreasing the C:O ratio by an order of magnitude.

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We have shown in Section 2.3 that photon diffusion is not expected to significantly affect the luminosity. Such diffusion may, however, modify the spectrum of the emitted radiation. We discuss below in some detail the expected modification of the spectrum.

For the purpose of this discussion, it is useful to define the "thermalization depth," r_ther, and the "diffusion depth," r_diff. r_ther(t) < r_ph(t) is defined as the radius at which photons that reach r_ph(t) at t "thermalize," i.e., the radius from which photons may reach the photosphere without being absorbed on the way. This radius may be estimated as the radius for which τ_sctτ_abs ≈ 1 (Mihalas & Mihalas 1984), where τ_sct and τ_abs are the optical depths for scattering and absorption provided by plasma lying at r > r_ther(t). r_ther is thus approximately given by

$$\begin{equation} 3 (r_{\rm{ther}}-r_{\rm{ph}})^2 \kappa _{\rm{sct}}(r_{\rm{{ther}}}) \kappa _{\rm{abs}}(r_{\rm{{ther}}}) \rho ^2(r_{\rm{{ther}}}) = 1, \end{equation} \tag{ 30 }$$

where κ_sct and κ_abs are the scattering and absorption opacities, respectively. (Typically, the opacity is dominated by electron scattering.) r_diff is defined as the radius (below the photosphere) from which photons may escape (i.e., reach the photosphere) over a dynamical time (i.e., over t, the timescale for significant expansion). We approximate r_diff by

$$\begin{equation} r_{\rm{ph}} = r_{\rm{diff}}+\sqrt{c \, t/3 \kappa _{\rm{{sct}}}(r_{\rm{diff}}) \rho (r_{\rm{diff}})}, \end{equation} \tag{ 31 }$$

where c is the speed of light.

For r_diff < r_ther, photons of the characteristic energy 3T(r_ther, t)>3T_ph will reach the photosphere, while for r_ther < r_diff photons of the characteristic energy 3T(r_diff, t)>3T_ph will reach the photosphere. Thus, the spectrum will be modified from a blackbody at T_ph and its color temperature will be T_col > T_ph. We approximate in what follows T_col = T(r_ther) for r_diff < r_ther and T_col = T(r_diff) for r_diff > r_ther.

The lower panels of Figures 1–4 present the ratio T_col/T_ph for the various envelope compositions considered. For this calculation, we have assumed that the scattering opacity is dominated by Thomson scattering of free electrons, used the electron density (as the function of density and temperature) provided by the OP tables for determining κ_sct, and estimated κ_abs = κ − κ_sct. (Recall that κ is the Rosseland mean of the opacity.) It would have been more accurate to use an average of the absorptive opacities over the relevant wave bands, which are not provided by the OP table. However, since the dependence of the color temperature on the absorptive opacity is weak, T_col ∝ κ^(−1/8)_abs, the corrections are not expected to be large. The figures imply that over the relevant timescale, t ≲ 1 day,

$$\begin{equation} f_T \equiv T_{\rm{{col}}}/T_{\rm{{ph}}} \approx 1.2. \end{equation} \tag{ 32 }$$

Using Equation (32) with Equations (13), (23), and (27) for the photospheric (effective) temperature, the progenitor radius may be approximately inferred from the color temperature by

$$\begin{equation} R_{*} \approx 0.70\times 10^{12} \left[ \frac{T_{\rm{col}}}{(f_T/1.2) \mbox{ eV}} \right]^{4}t_5^{1.9} f_{\rho }^{0.1}\,\rm cm \end{equation} \tag{ 33 }$$

for H envelopes,

$$\begin{equation} R_{*} \approx 1.2\times 10^{11} \left[ \frac{T_{\rm{col}}}{(f_T/1.2) \mbox{ eV}} \right]^{4.9} t_5^{1.9} f_{\rho }^{0.1}\, \mbox{ cm} \end{equation} \tag{ 34 }$$

for He envelopes with T > 1.07 eV, and

$$\begin{equation} R_{*} \approx 0.58 \times 10^{11} \left[ \frac{T_{\rm{col}}}{(f_T/1.2) \mbox{ eV}} \right]^{5.3} t_5^{1.9} f_{\rho }^{0.1} Z \,\mbox{ cm} \end{equation} \tag{ 35 }$$

for He–C/O envelopes when the C/O opacity dominates. (The transition temperature is given in Equation (28).)

We have neglected in our analysis the effective broadening of atomic lines due to the velocity gradients in the outflow. Line broadening may have a significant effect on the opacity and on the observed emission (Karp et al. 1977; Wagoner et al. 1991; Eastman & Pinto 1993), as well as on the dynamics (e.g., in the case of stellar winds, see Friend & Castor 1983). We give below a crude estimate of the line-broadening effects for the problem of interest here. A detailed analysis of line broadening, which requires detailed numerical calculations (see, e.g., Section 6.9 of Castor 2004), is beyond the scope of this manuscript.

The analysis of Wagoner et al. (1991) shows that the effective line opacity introduced by the velocity gradients may significantly affect the Rosseland mean opacity (at the relevant densities and velocity gradients, see their Figures 5 and 6) at temperatures where recombination leads to a large reduction of the Thomson electron scattering opacity. The main contribution to this "expansion opacity" is from resonant line scattering of Fe group elements. As explained in Section 3.1, the main effect that recombination introduces to our analysis is the penetration of the photosphere to shells of high enough temperatures where significant ionization is maintained. Since the opacity enhancement due to velocity gradient effects does not prevent the strong reduction of the opacity due to recombination, it will not prevent the penetration of the photosphere to a region of significant ionization. Nevertheless, the enhanced line opacity may introduce an order unity increase of the opacity at temperatures close to the recombination temperature at short, λ < 0.25 μ, wavelengths (see Figures 6 and 7 of Wagoner et al. 1991). A detailed analysis of this effect is beyond the scope of this manuscript.

We should note, nevertheless, that the effective line opacity enhancement due to velocity gradients may be smaller in our case compared to the estimates of earlier analyses. To show this, let us examine the following heuristic derivation of the effective broadening of atomic lines. Consider a photon that travels outward/inward in the region of relatively low optical depth. Due to the velocity gradient of the expanding ejecta, the photon frequency as measured in the plasma rest frame is shifted as it propagates by dν/ν = −dv/c = −(∂v/∂r)dr/c = −(∂v/∂r)dt, where dv = (∂v/∂r)dr is the velocity difference across dr, and we neglect the plasma speed with respect to that of the photon in setting dr = cdt. Assuming that a photon is absorbed/scattered as its frequency is shifted across that of a line, then the probability for scattering/absorption is dP = dν(dN/dν), where dN/dν is the line "density" per unit frequency. The resulting photon mean free path is therefore l⁻¹ ∼ dP/cdt ∼ c⁻¹|∂v/∂r|ν(dN/dν). For v = r/t, which is valid at late time, we have

$$\begin{equation} l_\nu ^{-1}\sim \frac{|\partial v/\partial r|}{c}\nu (dN/d\nu)=(ct)^{-1}\nu (dN/d\nu) \end{equation} \tag{ 36 }$$

(compare, e.g., to Equation (3.10) of Wagoner et al. 1991). The photon is absorbed/scattered provided the line optical depth is large enough. Neglecting the natural width of the lines, we may replace the line opacity with κ_ν = κ_lν₀δ(ν − ν₀). Denoting ν' = (1 − dv/c)ν, we obtain

$$\begin{eqnarray} \tau _l\sim \int cdt^{\prime } \rho \kappa _{\nu ^{\prime }}&\approx & \int \frac{d\nu ^{\prime }}{\nu }\frac{c}{\partial v/\partial r}\rho \kappa _{\nu ^{\prime }} \nonumber \\ &\approx & \frac{c}{\partial v/\partial r}\rho \kappa _l=ct\rho \kappa _l. \end{eqnarray} \tag{ 37 }$$

Thus, the line should be "counted" in determining dN/dν, if ctρκ_l ≫ 1 (compare, e.g., to Equations (3.10) and (2.7) of Wagoner et al. (1991), and Equation (9) of Friend & Castor 1983).

The analyses of Wagoner et al. (1991) and of Karp et al. (1977) are "local." That is, it is assumed there that as the photon's frequency is shifted by an amount comparable to the strong line separation, the parameters of the plasma within which it propagates do not change (the Karp et al. 1977, assumptions are in fact more restrictive). We have made the same assumption in deriving the final result of Equation (37). However, the validity of this assumption is not obvious in our case. For the self-similar ejecta profiles described in Section 2.1, ρ and T are steeply falling functions of v_f, ρ is roughly proportional to v⁻¹⁰_f and T is roughly proportional to v⁻³_f. Thus, as the photon moves outward and its frequency is shifted (in the plasma frame) by ∼v/c (i.e., by dr/r ∼ 1), ρ and T drop by factors of 10³ and 10¹, respectively. This implies that τ_l may be significantly smaller than given by Equation (36).

Following the observations of SN 1987A, many numerical calculations modeling its light curve have been carried out (see, e.g., Hauschildt & Ensman 1994). The latest and most comprehensive of these calculations was carried out by Blinnikov et al. (2000), and it provides UBV light curves from the time of breakout to several months following the explosion. The Blinnikov et al. (2000) radiation-hydrodynamics calculation, which includes a detailed treatment of the opacities and a multi-group transport approximation for the propagation of radiation, should, to our understanding, capture all the relevant physics.

In Figures 5 and 6, we compare the results of our simple model to those of the detailed numerical calculations of Blinnikov et al. (2000). We use the same progenitor parameters as those used in Blinnikov et al. (2000): a BSG (H envelope with n = 3, f_ρ = 1) of radius R_* = 3.37 × 10¹²cm, ejecta mass M = 14.67 M_☉, and explosion energies E = 1.03and1.34 × 10⁵¹erg for models 14E1 and 14E1.3, respectively (and composition as in the outer part of the progenitor given in Figure 2 of Blinnikov et al. 2000). Figure 5 compares the numerical velocity profile with the Matzner & McKee (1999) approximation we use in our model, given by Equation (3). The two agree to better than 10% over the relevant envelope mass fraction. In Figure 6, we compare the numerical early UBV light curves of Blinnikov et al. (2000) with the ones calculated in our model, using Equation (21) with the OP opacity (Seaton 2005). As can be seen in the figure, our analytic model gives fluxes which are larger by a factor of ∼2 than those of the numerical calculation. This difference may be due to differences between the OP opacities and those used by Blinnikov et al. (2000). The opacity given in Blinnikov et al. (1998) for ρ = 10⁻¹³ gcm⁻³, T = 15,000° K, and solar metallicity is larger than the OP opacity by roughly a factor of two. If a similar difference exists for the modified metallicity used in the SN1987A calculations, it would explain the luminosity discrepancy since the luminosity is roughly inversely proportional to κ, see Equation (15).

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The GALEX far-UV and near-UV measurements of the type IIp SNLS−04D2dc are shown in Figure 7. Given the relatively low signal-to-noise ratio, we do not attempt here to constrain the progenitor parameters using a detailed analysis of the UV emission (as we do for SN 2008D in Section 6). Rather, we show that the observed UV flux is consistent with that expected from an expanding shock heated envelope of an RSG progenitor, and compare our simple model predictions to those obtained using detailed numerical calculations. For the latter purpose, we use the numerical calculations described in Gezari et al. (2008).

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Gezari et al. (2008) performed detailed numerical calculations, aimed at reproducing the early UV/O emission from the SNLS−04D2dc, the progenitor of which is most likely an RSG. Their calculation was performed in two stages. At the first stage, a hydrodynamic calculation of the explosion was performed using the one-temperature Lagrangian radiation-hydrodynamics code KEPLER (Weaver et al. 1978). At the second stage, the emission of radiation at time t was calculated by solving, using the multi-group radiation transport code CMFGEN (Dessart & Hillier 2005), the steady state radiation field for the hydrodynamic profiles obtained at time t, keeping the temperature profile fixed for τ>20 and allowing it to self-consistently change at smaller optical depths.

In Figure 8, we compare the Gezari et al. (2008) calculations with the results obtained by our model, using Equation (21) with the OP opacity (Seaton 2005). The progenitor parameters we use are: an RSG (H envelope with n = 3/2 and f_ρ = 25) of radius R_* = 865 R_☉, explosion energy E₅₁ = 1.44, and the ejecta mass of M = 8.9 M_☉. The stellar radius is similar to that used by Gezari et al. (2008), and f_ρ = 25 was chosen (based on a private communication with S. Gezari and L. Dessart, 2008) to provide an approximate description of the outer RSG envelope profile used in their calculation. The explosion energy and ejecta mass used in our model are 20% larger and smaller, respectively, than those used by Gezari et al. (2008), i.e., the E/M ratio is 40% larger in our calculation. This value was chosen to reproduce the observed luminosity. The light curves calculated by Gezari et al. (2008) were shifted by ∼−1.5 mag, i.e., the calculated luminosity was increased by a factor of ∼4, to fit the observations. Since the luminosity is approximately proportional to E/M (see Equation (14)), this implies that for a given E/M ratio our calculation predicts a luminosity that is larger than that of Gezari et al. (2008) by a factor of ∼3 (we have verified this by comparing the results of Gezari et al. 2008 to our model results for the same E/M ratio).

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The following point should be noted here. Due to the large radius of the progenitor, the photosphere lies within the "breakout shell," i.e., δ_m,ph. < δ_m,BO (see Equation (6)), up to t = t_BO ≃ 1.5 d (see Equation (16)). Our simple model is not valid at t < t_BO. However, we expect it to provide a reasonable approximation for the photospheric temperature and radius also at t < t_BO, for the following reason. As long as the photosphere lies at δ_m,ph. < δ_m,BO, the diffusion time at the photosphere is short compared to t. In this case, we expect the spatial dependence of the radiation pressure to approximately follow p ∝ τ (see Section 2.3), in which case the photospheric temperature is given by aT⁴_ph = 3p(τ)/τ. Since, as explained in Section 2.3, the adiabatic pressure profile, p ∝ τ^1.1, is similar to that obtained for short diffusion time, p ∝ τ, and since the adiabatic pressure profile is valid at all times for δ_m ≫ δ_m,BO, we expect Equation (13) to provide a good approximation for T_ph also at t < t_BO. The temporal dependence of r_ph is expected to be somewhat stronger, at t < t_BO, than r_ph ∝ t^0.8 predicted by the simple adiabatic model, since the velocity profile at δ_m,ph. < δ_m,BO is shallower than predicted by Equation (3). The largest deviation from the r_ph ∝ t^0.8 behavior would be obtained assuming uniform velocity at δ_m,ph. < δ_m,BO, which would yield r_ph ∝ t.

The factor of ∼2 discrepancy between the luminosity predicted by our model and that obtained by Gezari et al. (2008) is due to the fact that our model predicts a somewhat, ∼40%, larger velocity for the fast outer shells, and hence a larger photospheric radius. In a subsequent publication (I. Rabinak et al. 2011, in preparation), we examine the accuracy of the approximate ejecta density and velocity profiles described in Section 2.1 for a wide range of progenitor models.

The derivations of r_ph and T_ph in Sections 2.2 and 3 are based on the self-similar model of Section 2.1 for the density and pressure profiles of the ejecta. This model, in which p(v_f) and ρ(v_f) are both roughly proportional to v⁻¹⁰_f, is valid in the limit δ_m → 0 (v_f → ∞). In order to extend the model to larger values of δ_m, we adopt the "harmonic-mean" model suggested by Matzner & McKee (1999). In this model, the density and pressure profiles are obtained by interpolating between the small δ_m (large v_f) self-similar power-law dependence of p(v_f) and ρ(v_f), and the power-law dependence p(v_f) ∝ v^α_f, ρ(v_f) ∝ v^β_f with α ≈ 2 and β ≈ −1, obtained in the approximate analysis of Chevalier & Soker (1989) for the lower velocity ejecta. This power-law dependence is obtained by assuming that the shock propagating within the ejecta may be approximately described, for large δ_m, by the Primakoff self-similar solution (a particular, analytic, case of the Sedov–von Neumann–Taylor solutions for shock propagation into ρ ∝ r^−ω density profiles, obtained for ω = 17/7, e.g., Gaffet 1984; Bernstein & Book 1980), and by an approximate (self-similar) description of the post-breakout acceleration of the shocked plasma. The two power-law solutions describing the large and small v_f behavior of the density are matched in the Matzner & McKee (1999) "harmonic-mean" model at ρ = ρ_break and v_f = v_ρbreak, and the pressure profiles at p = p_break and v_f = v_pbreak (see Equations (46) and (47) of Matzner & McKee 1999). ρ_break and v_ρbreak are determined by requiring the ejecta mass and (kinetic) energy to be equal to M and E, respectively. p_break and v_pbreak are determined in Matzner & McKee (1999) by examining numerical simulation results. We find that their parameter choice of p_break and v_pbreak leads to an overestimate of the temperature, compared to that of the self-similar δ_m → 0 solution, by ∼15% at δ_m = 10⁻³. We therefore modify the value of p_break to obtain the correct self-similar behavior at δ_m → 0.

The "harmonic-mean" density and temperature profiles obtained as described in the preceding paragraph are compared in Figure 9 with those of the δ_m → 0 self-similar solution. A significant deviation from the self-similar profiles is obtained for δ_m ≳ 0.1. As discussed in Section 6.2, this deviation affects the model predictions for t ≳ 2 d. The accuracy of the "harmonic-mean" model was examined in Matzner & McKee (1999) by comparing it to the results of numerical calculations of the explosions of various RSG and BSG progenitors. Since it was found that this analytic model provides a good approximation for the envelope's profiles for different initial density structures of the progenitors, we expect the harmonic-mean model to provide a good approximation also for the SN 2008D envelope profiles. However, additional work, which is beyond the scope of this paper, is required in order to obtain a quantitative estimate of the accuracy of the approximation for δ_m > 0.1.

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Spectroscopic observations have constrained the fraction of hydrogen in the ejecta to ≲5 × 10⁻⁴ M_☉ (Tanaka et al. 2009). We therefore consider below He and He–C/O envelopes. As explained in Section 4, the relative extinction may be inferred from the light curves at different frequencies. However, for the clarity of the presentation, we first analyze the data using two relative extinction curves that differ significantly in their short wavelength behavior, a Milky Way extinction curve with R_v = 3.1 (hereafter MW) and a Small Magellanic Cloud extinction curve (hereafter SMC; Cardelli et al. 1989), and only later show how the extinction curve may be directly inferred from the data.

Figure 10 presents a comparison of the color temperature T_col and bolometric luminosity L inferred from the data with those obtained in our model for different progenitor and explosion parameters. We use the observations of Swift/UVTO (V, B, U, UVW1, UVM2, UVW2), Palomar (g, r, i, z) (both taken from Soderberg et al. 2008), and FLWO (B, V, r, i) (Modjaz et al. 2009). Since observations by different telescopes were carried out at different times, we interpolate the observations in different bands to times close to the observation times of the Swift/UVTO and the Palomar telescopes. In order to derive T_col(t) and L(t) from the observations, the measured fluxes should be corrected for extinction. We carry out this correction by (1) assuming a specific extinction curve (MW or SMC) and (2) determining the absolute value of the extinction by requiring T_col inferred from the observations to agree with the model prediction at t ∼ 2 days. This implies that for each set of model parameters (E, M, R_*, envelope composition), a different value of the absolute extinction, E(B − V) is chosen.

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For all the models shown in the figure, we have used E₅₁ = 6 and M/M_☉ = 7, as suggested by Mazzali et al. (2008) from the spectral analysis of the observations at maximum light. As shown in Sections 2 and 3, E and M (mainly the ratio E/M) determine the normalization of the model luminosity, see Equations (25) and (29), but do not affect the time dependence of the luminosity, and T_col is nearly independent of E and M, see Equations (23) and (27). Since the conclusions drawn from the comparisons in fig. (10) are based on the time dependence of T_col and L, their validity is independent of the exact values chosen for E and M.

Figure 10 shows T_col and L for models with three progenitor radii, R_* = 1, 3, 10 × 10¹¹ cm, and two compositions, He-dominated and mixed He–C/O composition with a He mass fraction 1 − Z = 0.3. We show both the simple analytic approximations for L and T_col given in Section 3.1 (dashed curves), which are based on the self-similar ejecta profiles of Section 2.1, and L and T_col obtained solving Equations (21), (30), and (31) with the OP opacity tables for the modified ejecta profiles described in Section 6.1. Since models with larger initial radii predict higher T_col, the absolute extinction inferred for models with larger radii is larger, E(B − V) = 0.625, 0.7, and0.8 for R_* = 1, 3, and10 × 10¹¹ cm, respectively. Once the absolute extinction is determined, from the comparison of observed and predicted T_col at t ∼ 2 days, T_col(t) and L(t) are inferred from the observations using the two relative extinction curves (MW and SMC). In determining the observed T_col and L, the photometric measurements are converted into monochromatic fluxes at the effective wavelength of the broadband (BB) filters, and a BB temperature is determined by a least-squares fit to these fluxes. The resulting T_col and L are shown in Figure 10. The error bars represent the uncertainties obtained in the least-squares fits.

Examining Figure 10, we infer a small progenitor radius, R_* ≈ 10¹¹ cm. Progenitors with larger radii require larger extinction to account for the observed flux distribution at t = 2 days, which in turn imply that the extinction corrected L(t) decreases with time at t < 2 days, in contrast with the roughly time-independent L predicted by the models (see also Equations (25) and (29)), and that the extinction-corrected T_col decreases faster than predicted by the models for t < 2 days. This is due to the fact that at earlier times the flux is dominated by shorter wavelength bands, for which the extinction correction is larger. Comparing model predictions and observations at t > 2 days, we find that a mixed He–C/O composition is preferred over an He-dominated one. The Z = 0.7 model presented provides a good fit to the data. We find that Z ∼ few tens of percent is required to fit the observations. Note, however, that the light curve at t > 2 days depends on the non-self-similar part of the density and pressure profiles of the ejecta, for which we have used the "harmonic-mean" approximation described in Section 6.1. As mentioned in Section 6.1, additional work, which is beyond the scope of this paper, is required in order to obtain a quantitative estimate of the accuracy of this approximation. We cannot rule out, therefore, the possibility that the observations may be explained with an He-dominated contribution and a density profile at large δ_m, that deviated from that given by the "harmonic-mean" approximation.

As explained in Section 4, R_* and the relative extinction between different wavelengths may be determined from the O/UV light curves by requiring that the light curves observed at different wavelengths should all be given, after scaling according to Equations (40) and (45), by a single function, given by Equation (42). In Figure 11, we compare the measured specific intensities, corrected for extinction and scaled according to Equations (40) and (45), with the model prediction, Equation (42). For the scaling, we have used {r_ph(t), T_col(t), T_ph(t)} obtained for the mixed He–C/O composition models presented in Figure 10 with R_* = 10¹¹ cm and R_* = 3 × 10¹¹ cm. The extinction τ_λ was obtained by requiring the scaled intensity to best fit that predicted by Equation (42) (taking into account all data points at t < 4 days). The resulting extinction curves are shown in Figure (12). As can been seen in the figure, the R_* = 10¹¹ cm model provides a much better description of the data than the R_* = 3 × 10¹¹ cm model. The extinction curve is more compatible with an MW extinction than with SMC extinction. It differs from the MW curve at short wavelengths, showing no prominent graphite bump. This, as well as the values obtained for A_V and E(B − V), A_V = 2.39, and E(B − V) = 0.63, is consistent with the extinction inferred in A. M. Soderberg et al. (2008, private communication).

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The following point should be explained here. We have obtained the absolute values of the extinction by adopting some values for E and M. It is important to note, that, as explained in detail in Section 4, the relative extinction is independent of E and M, and may be inferred without making assumptions regarding their values. The model predicted temperature, T_ph, and T_col/T_ph are almost independent of E and M, and depend only on R_* (and on the composition). E and M determine the normalization of r_ph, r_ph ∝ E^0.4/M^0.3 (see Equation (24)), but do not affect its time dependence. Thus, modifying E and M changes the scaled fluxes of Equation (41) by some multiplicative factor, which is wavelength independent. Thus, the ratios of the scaled fluxes at different wavelengths are independent of E and M, and so are the inferred relative extinctions. Using the relative extinction inferred from the data, and adopting some relation between the relative and absolute extinctions, which determines the absolution extinction (τ_λ), we may therefore constrain the E^0.4/M^0.3 ratio by comparing the predicted and measured (absolute) flux at some frequency. Using the inferred E(B − V) = 0.63 implies A_V = 2.39 for an MW extinction, for which we infer E₅₁/(M/M_☉) = 0.75.

Based on our analysis of the early UV/O emission of SN 2008D, we infer a small progenitor radius, R_* ≈ 10¹¹ cm, an E₅₁/(M/M_☉) ≈ 0.8, a preference for a mixed He–C/O composition (with C/O mass fraction of tens of percent), E(B − V) = 0.63 and an extinction curve given by Figure 12. We compare below our conclusions to those of earlier analyses.

Explosion models for SN 2008D were consider by Mazzali et al. (2008) and by Tanaka et al. (2009). For E and M, Mazzali et al. (2008) infer E₅₁ ∼ 6 and M ∼ 7 M_☉, i.e., E₅₁/(M/M_☉) ∼ 0.85 while Tanaka et al. (2009) infer E₅₁ = 6 ± 2.5 and M = 5.3 ± 1 M_☉, and E₅₁/(M/M_☉) in the range 0.8 < E₅₁/(M/M_☉) < 1.3. These values are consistent with our inferred value of E₅₁/(M/M_☉) ≈ 0.8. The progenitor radius inferred from the (Tanaka et al. 2009) analysis is 0.9 ≲ R_*/10¹¹ cm ≲ 1.5, also consistent with our inferred R_* ≈ 10¹¹ cm.

In Figure 13, we compare the photospheric velocity obtained in our model with those obtained by Modjaz et al. (2009) and Tanaka et al. (2009) analyzing SN 2008D spectra. As can be seen in the figure, our model predictions are in good agreement with the results inferred from the spectral analyses. We also note that Mazzali et al. (2008) infer, based on spectral analysis, that the mass of the ejecta shell moving at >0.1c is ∼0.03 M_☉, consistent with our model prediction of ∼0.02 M_☉ at this velocity.

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Next, let us discuss our conclusion that the He envelope contains a significant (tens of %) C/O fraction. This conclusion is consistent with the results of Mazzali et al. (2008), who find a large fraction, ∼10% of C at the fast v > 25,000 km s⁻¹ He shells, rising to ∼30% at the v≃ 20,000 km s⁻¹ shells (P. Mazzali 2010, private communication). Both Mazzali et al. (2008) and Tanaka et al. (2009) find a low, ∼0.01, mass fraction of O at the fast shells.

Finally, we comment on the analysis of Chevalier & Fransson (2008), who find a large progenitor radius, R_* ≈ 10¹² cm, based on both the X-ray and the UV/O emission. Chevalier & Fransson (2008) obtain R_* ≈ 10¹² cm by interpreting the X-ray emission as thermal, L_X = 4πR²_*σT⁴_X, and adopting T_X = 0.36 keV. Their motivation for a thermal interpretation of the X-ray emission, despite the fact that the spectrum is non-thermal and extends beyond 10 keV, was that the non-thermal spectrum is difficult to explain theoretically. As explained in the Introduction, the non-thermal spectrum is a natural consequence of the physics of fast radiation mediated shocks (Katz et al. 2010). Thus, there is no motivation, and it is inappropriate, to infer R_* assuming thermal X-ray emission. Moreover, it should be kept in mind that the X-ray breakout may take place within the wind surrounding the progenitor, at a radius significantly larger than R_* (e.g., Waxman et al. 2007). The discrepancy between their large radius and our small radius inferred from the UV/O data is due to the fact that we take into account the modification of the opacity due to He recombination and the difference between color and effective temperatures. Since R_* ∝ κ, neglecting the reduction of opacity due to HE recombination at T = 1 eV (t ∼ 1 d) leads to a significant overestimate of the radius (compare Equations (33) and (34)), and neglecting the difference between color and effective temperatures leads to an additional overestimate of the radius, R_* ∝ T⁴_col = (T_col/T_ph)⁴T⁴_ph with (T_col/T_ph)⁴ ≈ 2 (see Equation (34)).