SHOCK BREAKOUT IN DENSE MASS LOSS: LUMINOUS SUPERNOVAE

Supernova shock breakouts from normal massive stars typically give rise to X-ray/ultraviolet bursts with a timescale ≲10³ s (Klein & Chevalier 1978; Falk 1978; Ensman & Burrows 1992; Matzner & McKee 1999; Nakar & Sari 2010). The timescale is generally determined by R_*/c, where R_* is the stellar radius, because the radiative diffusion time at breakout is less than the light travel time across the star. The largest red supergiants have a radius of ∼10¹⁴ cm (Levesque et al. 2005) so that the longest time of the breakout event is about 1 hr. The short timescale of the bursts makes it difficult to detect them, other than with a wide-field X-ray telescope. The breakout detection of SN 2008D with Swift (Soderberg et al. 2008) was fortunate.

The situation changes if there is dense mass loss prior to the supernova that creates an optically thick region. Calculations of this case go back to early computer simulations of supernova light curves. Model 5 of Grassberg et al. (1971) and Model B of Falk & Arnett (1977) include a dense circumstellar shell with radius ∼10¹⁵ cm that determines the properties of the shock breakout. The peak luminosity occurs on a timescale of ∼15 days in the model of Grassberg et al. (1971). Grasberg & Nadyozhin (1987) considered a steady wind of limited duration just before the supernova. Chugai et al. (2004) calculated a model for SN 1994W which allows for dense mass loss around the supernova, leading to a peak luminosity at an age of ∼20 days. A systematic numerical study of such events has recently been carried out by Moriya et al. (2010). Because of the high surrounding density, the supernova can be especially bright and this type of model has been invoked for luminous supernovae. In addition to SN 1994W (Chugai et al. 2004), the application of the dense mass-loss model has been made to SN 2006gy (Smith & McCray 2007), PTF 09uj (Ofek et al. 2010), and SN 2009kf (Moriya et al. 2010) among others.

The aim here is to give an analytical description of the shock breakout process for the case where a radiation-dominated shock propagates into the mass-loss region. Progress on this front has been made by Ofek et al. (2010). The model is presented in Section 2 and compared to observations in Section 3.

We assume that the dense mass loss can be described by a steady wind acting for some time t_ml before the supernova explosion; the extent is R_w = v_wt_ml, where v_w is the wind velocity. The actual case may be more complex, but is not well determined; the wind assumption has been made in other work (Grasberg & Nadyozhin 1987; Ofek et al. 2010; Moriya et al. 2010). If the mass loss is in a steady wind, the density $\rho _w=\dot{M}/4\pi r^2v_w\equiv Dr^{-2}$ can be specified by a density parameter, D_*, scaled to a $\dot{M}=10^{-2} \,M_\odot \ \rm yr^{-1}$ and $v_w=10\,\rm km\, s^{-1}$ wind so that ρ_w = 5.0 × 10¹⁶D_*r⁻² in cgs units. For typical supernova parameters, the explosion drives a radiation-dominated shock through the star. A radiation-dominated shock in a uniform medium has a characteristic thickness described by an optical depth τ_sh ≈ c/v_sh, where v_sh is the shock velocity (Weaver 1976). If the wind optical depth τ_w < c/v_sh, then a radiation-dominated shock does not form in the wind and the radiation diffuses out through an optically thick wind (e.g., Nakar & Sari 2010). In the case that τ_w>c/v_sh, the radiation-dominated shock front propagates into the mass-loss region. This is the case considered here.

After the shock wave passes through the star and the supernova ejecta tend toward free expansion, a shocked layer develops in the wind that is bounded by forward and reverse shock waves at R_fs and R_rs, respectively. The shock waves are initially energy conserving. Once the forward shock front reaches a place where the diffusion time equals the expansion time or, equivalently, τ_w ≈ c/v_sh, radiation diffusion becomes important. We have τ_w = (R⁻¹_fs − R⁻¹_w)κD, so the diffusion condition can be written as R⁻¹_d ≈ R⁻¹_fs − R⁻¹_w, where R_d ≡ κDv_sh/c = 5.7 × 10¹⁴v_sh4kD_* cm, where v_sh4 is the shock velocity in units of 10⁴ km s⁻¹ and k is the opacity κ in units of 0.34 cm² g⁻¹. Two cases are of interest: R_w>R_d so that R_fs ≈ R_d when diffusion becomes important and R_w < R_d so that R_fs ≈ R_w at that time.

In the wind, for diffusion over a distance ΔR, the diffusion time is t_d = ΔR²/λc = κρ_wΔR²/c. Taking ΔR ≈ R and noting ρ_w = Dr⁻², we have t_d = κD/c, independent of radius (see also Ofek et al. 2010). If the extent of the optically thick region (to R_ph = min(R_w, κD)) is not much beyond R_d, then κD/c provides an estimate of the diffusion time for the radiation to escape. The time is longer if R_ph>R_d. If the diffusion is assumed to proceed in steps with size ∝R, the timescale is increased by a term that is logarithmic in R. For the approximate results presented here, we neglect the logarithmic dependence and take

$$\begin{equation} t_d=\kappa D/c=6.6 kD_*\ {\rm day}. \end{equation} \tag{ 1 }$$

We now examine a specific model for expansion into the dense wind. After the shock wave has expanded beyond the stellar radius, the ejecta tend toward free expansion (v = r/t) in which there is a steep power-law section on the outside and a flat profile in the inner region (e.g., Matzner & McKee 1999). Here we make the approximation that ρ_in ∝ r⁻¹ at velocities v < v_t and ρ_sn ∝ r⁻⁷ at v>v_t. The outer profile is less steep than the profile inferred for the asymptotic profile expected at high velocities; the wind interaction causes a flatter part of the density distribution to be relevant. We then have v_t = (2E/M_e)^1/2 and ρ_sn = Bt⁻³(r/t)⁻⁷, where B = 4E²/3πM_e (Chevalier & Fransson 1994). The interaction between this power-law density distribution and a surrounding wind, with density ρ_w = Dr⁻², can be described by a self-similar solution with adiabatic index γ = 4/3 (Chevalier 1983), which is appropriate for a radiation-dominated gas. The radius of the contact discontinuity is

$$\begin{equation} R_{\rm cd}=(0.23B/D)^{0.2}t^{0.8}=5.6\times 10^{14}E_{51}^{0.4}M_{e1}^{-0.2}D_*^{-0.2}t_1^{0.8}\ \rm { cm}, \end{equation} \tag{ 2 }$$

where E₅₁ is the explosion energy in units of 10⁵¹ erg, M_e1 is the ejecta mass in units of 10 M_☉, and t₁ is the age in units of 10 days. The highest velocities occur just inside the reverse shock wave, where the velocity is $v_{\rm rs}=0.978R_{\rm cd}/t=6.3\times 10^3E_{51}^{0.4}M_{e1}^{-0.2}D_*^{-0.2}t_1^{-0.2}\,\rm km\, s^{-1}$. The applicability of the solution requires that v_t < v_rs or E^0.5₅₁M^−1.5_e1D_*t₁ < 32.

We initially assume that R_d < R_w. At time t_d,

$$\begin{equation} R_{\rm cd}=R_d=4.0\times 10^{14} k^{0.8}E_{51}^{0.4}M_{e1}^{-0.2}D_*^{0.6} \ {\rm cm}. \end{equation} \tag{ 3 }$$

The amount of energy in the interaction shell above some velocity v₀ is

$$\begin{equation} E_{\rm sh}=\int ^{\infty }_{v_0t}{1\over 2}\rho _{\rm sn}v^2 4\pi r^2 dr={4\over 3}{E^2\over M_ev_0^2}. \end{equation} \tag{ 4 }$$

The internal energy, which is in the form of radiation, is E_rad = 0.32E_sh (Chevalier 1983). Using v₀ = R_cd(t_d)/t_d at time t_d, we have

$$\begin{equation} E_{{\rm rad}}=0.44\times 10^{50}k^{0.4}E_{51}^{1.2}M_{e1}^{-0.6}D_*^{0.8} \ {\rm erg} \end{equation} \tag{ 5 }$$

for the radiative energy in the breakout shell. The timescale for the radiation to escape is t_d, so the luminosity at shock breakout is

$$\begin{equation} L_{b}\approx E_{{\rm rad}}/t_d=7.6\times 10^{43}k^{-0.6}E_{51}^{1.2}M_{e1}^{-0.6}D_*^{-0.2}\ \rm erg\, s^{-1}. \end{equation} \tag{ 6 }$$

Taking the radiation energy E_rad and the shocked volume of 4π(R³_fs − R³_rs)/3 at time t_d, where R_fs and R_rs are the forward and reverse shock radii, respectively, the mean radiation energy density gives a radiation temperature

$$\begin{equation} T_{{\rm rad}}=6.4\times 10^4 k^{-0.25}t_{d1}^{-0.25}=7.1\times 10^4 k^{-0.5}D_*^{-0.25}\ {\rm K}, \end{equation} \tag{ 7 }$$

where we have used R_fs/R_cd = 1.208 and R_rs/R_cd = 0.978 (Chevalier 1983), and t_d1 is t_d in units of 10 days. Another estimate of T_rad can be obtained from the forward shock condition on the pressure: aT⁴/3 = (6/7)ρ_wv²_fs for a γ = 4/3 gas, where a is the radiation constant (see also Ofek et al. 2010). In this case, the temperature coefficient in Equation (7) is 7% higher; the difference is due to the fact that there is a drop in the pressure behind the forward shock front in the shocked region. The results assume a smooth wind with an r⁻² density profile, but the temperature of the breakout shell is not strongly dependent on this assumption. More generally, if ΔR is the size of the breakout region, the optical depth through the region is ρκΔR ≈ c/v_sh and the shock crosses the shell in the diffusion time, or ΔR ≈ v_sht_d. The resulting density ρ = c/κv²_sht_d, when combined with the shock jump condition for the radiation pressure, gives a temperature comparable to that in Equation (7).

These results can be compared to the numerical results of Moriya et al. (2010). Taking their model s13w2r20m2e3 with E₅₁ = 3, M_e1 = 1.3, and D_* = 1, the breakout radius in our model is R_d = 5.9 × 10¹⁴ cm, which is inside the outer radius of 2 × 10¹⁵ cm in the numerical model. Substituting the parameters into Equation (5), the radiated energy is E_rad = 1.4 × 10⁵⁰ erg, which is close to the 2.0 × 10⁵⁰ erg found in the numerical model (Moriya et al. 2010). The numerical result may be larger, in part, because the shock wave generates power in the more extended circumstellar medium. We also examined the scaling of E_rad with the parameters and found reasonable agreement with the numerical results. The scaling depends on the supernova density gradient, which is only approximately treated here.

Estimates of the observable color temperature of the radiation require considerations of whether radiation equilibrium is attained in the emitting region. Following Nakar & Sari (2010), we define a thermal coupling coefficient $\eta =n_{{\rm BB}}/(t_d\dot{n}_{\rm ph,ff}(T_{{\rm BB}}))$, where n_BB ≈ aT⁴_BB/3k_BT_BB is the photon number density in thermal equilibrium, k_B is Boltzmann's constant, and $\dot{n}_{\rm ph,ff}(T_{{\rm BB}})=3.5\times 10^{36}\rho ^2T^{-1/2}$ s⁻¹ cm⁻³ is the production rate of photons by the free–free process. If sufficient photons are produced to maintain the blackbody number density, or η ≲ 1, thermal equilibrium is achieved. Using Equations (1) and (7) for t_d and T_BB, and ρ = 7ρ_w(R_fs) (taking into account the factor 7 compression in the shock wave), we estimate η for the breakout shell:

$$\begin{equation} \eta _0\approx 0.4 k^{0.45}E_{51}^{1.6}M_{e1}^{-0.8}D_*^{-1.48}. \end{equation} \tag{ 8 }$$

For the standard parameters, the breakout shell is marginally in thermal equilibrium. As the radiation propagates into the unshocked mass-loss region, the lower density results in a deviation from thermal equilibrium. The frequency dependence of the opacity can play a role (Moriya et al. 2010) and we do not treat the details of spectrum production here.

The loss of radiative energy from the shocked region results in the formation of a dense shell at radius R, as seen in numerical simulations (Grassberg et al. 1971; Falk & Arnett 1977). The expansion of the shell into additional mass loss produces continuing power for the supernova, L = 2πR²ρ_w(v_fs − v_w)³, where the wind velocity v_w may be affected by preshock radiative acceleration. The simulations of Moriya et al. (2010) show some evidence for acceleration, but it is only significant near the breakout radius because of the r⁻² dependence of the radiative flux, and we neglect it here. The expansion of R can be described by the thin shell approximation (Chevalier 1982), yielding R = 0.94R_cd. The resulting power is

$$\begin{equation} L=7.1\times 10^{43}E_{51}^{1.2}M_{e1}^{-0.6}D_*^{0.4}t_1^{-0.6}\ \rm erg\, s^{-1}. \end{equation} \tag{ 9 }$$

The magnitude of the luminosity is similar to that produced by the initial breakout radiation, as is seen in numerical simulations (Grasberg & Nadyozhin 1987; Moriya et al. 2010). The luminosity lasts until the shock wave at R reaches the edge of the dense wind, R_w, at t_w = 1.1E^−0.5₅₁M^0.25_e1D^0.25_*R^1.25_w16 yr, where R_w16 is in units of 10¹⁶ cm. Figure 1(a) illustrates the luminosity evolution with the late flattening from the shell interaction.

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In the case of shock breakout from a red supergiant, the shock front takes ∼1 day to traverse the star and the time for shock breakout is ∼10³ s (Klein & Chevalier 1978). The shock breakout timescale is much less than the time since explosion. In the dense mass-loss case considered here, the time for the shock front to move to the breakout region is ∼R_d/v_sh, which is also the timescale for the breakout event. This property of the luminosity evolution can be seen in simulations of such events (Grassberg et al. 1971; Falk & Arnett 1977; Chugai et al. 2004; Moriya et al. 2010). The rise to maximum light can have complications due to variations in the gas opacity. At the high circumstellar densities considered here, it is likely that the gas is initially neutral and that most of the opacity is due to dust in the presupernova environment. The radiation-dominated shock wave from the supernova has a precursor in the mass-loss region that is expected to heat the circumstellar dust. As the temperature rises through 1000–2000 K, the dust evaporates, giving a decrease in the opacity and the photospheric radius drops to where there is a sharp gradient in the opacity as the gas becomes ionized. In Type IIP supernovae, this property of the opacity causes a recombination wave to back into the expanding envelope with a constant photospheric temperature T ∼ 5000–6000 K. Here, the process is inverted and the photosphere is expected to follow the ionization wave moving out through the dense circumstellar gas. This phase of approximately constant temperature can be seen in simulations (Grassberg et al. 1971). Once the circumstellar mass is ionized, the photospheric expansion slows and the temperature rises. The light curve rises fairly sharply due to the temperature rise until the maximum temperature, and luminosity, is reached.

We now consider the case that R_w < R_d. The shock breakout process begins at t_b ≈ R_w/v_sh. The rise time for the light curve is t_r ≈ δR/v_sh, where δR is the distance in from R_w across which the diffusion time equals the shock crossing time. We thus have (δR)²κDR⁻²_w/c = δR/v_sh, leading to t_r ≈ (R_w/v_sh)(R_w/R_d). In this case, the rise time can be considerably shorter than the time for initial heating of the envelope (Figure 1(b)). Since δR/R_w < 1, the region involved in shock breakout is not radially extended, so the pressure of the escaping radiation can accelerate the gas out to R_w (Ensman 1994). High gas velocities are expected around the time of maximum luminosity. A dense shell forms when the radiation can escape, but does not produce continuing high luminosity because the dense gas does not extend far beyond the breakout point and it has been radiatively accelerated.

For our specific supernova model, the shock breakout begins when the forward shock (R_fs) reaches R_w, which occurs at t_w = 16R^1.25_w15E^−0.5₅₁M^0.25_e1D^0.25_* days, where R_w15 is R_w in units of 10¹⁵ cm. The free expansion velocity at the reverse shock is $v_0=5.7\times 10^3 R_{w15}^{-0.25}E_{51}^{0.5}M_{e1}^{-0.25}D_*^{-0.25}\,\rm km\, s^{-1}$, which leads to a radiated energy of $E_{{\rm rad}}=0.65\times 10^{50} R_{w15}^{0.5}\linebreak [2] E_{51}M_{e1}^{-0.5}D_*^{0.5}$ erg. The rise time for the light curve is $t_r=t_wR_w/R_d=41 k^{-0.8} R_{w15}^{2.25}E_{51}^{-0.9}\linebreak [2] M_{e1}^{0.45}D_*^{-0.35}$ days, so that $L\approx E_{{\rm rad}}/t_r=1.8\times 10^{43}k^{0.8}R_{w15}^{-1.75}E_{51}^{1.9}M_{e1}^{-0.95}D_*^{0.85}\rm erg\, s^{-1}$. The temperature in the shocked shell is like that given in the first part of Equation (7) except that t_d is replaced by t_w. The temperature is lowered in the escape out to the photosphere if there is sufficient photon production in this region (Nakar & Sari 2010).

The radiated energy from SN 2006gy from the initial optical rise to around the peak is ∼1 × 10⁵¹ erg (Ofek et al. 2007; Smith et al. 2007), which makes it a good candidate for the physical situation described here (with wind optical depth τ_w>c/v_sh). In this case, the observed rise time gives an estimate of t_d, the diffusion time. The observations indicate a rise time of 60 days. Using Equation (1), the indicated wind density is D_* ≈ 10. The observed peak luminosity has sensitivity to the supernova energy. At maximum light, the observed luminosity was 4 × 10⁴⁴ erg s⁻¹ (Smith et al. 2010), which implies E₅₁ ≈ 3 for the other standard parameters. With these parameters, Equation (3) gives R_d = 2.5 × 10¹⁵M^−0.2_e1 cm while the dense medium may extend to ∼1 × 10¹⁶ cm (Smith et al. 2010), so that R_w>R_d. We attribute the flattening of the observed light curve (Smith et al. 2010) to the continuing interaction in the extended region. With R_w = 10¹⁶ cm, the implied circumstellar mass is ∼30 M_☉. These parameters are close to those found by Smith & McCray (2007) and Smith et al. (2010), which is expected because the basic physical picture is similar in the two cases. Depending on the ejecta mass, the condition that v_t < v_rs may be violated, but we do not expect the results to be strongly affected.

Smith et al. (2007, 2010) estimated an explosion date of 2006 August 20 for SN 2006gy; this time is just before the beginning of a rise in optical luminosity from 0.01L_max to L_max, where L_max is the maximum optical luminosity. In the shock breakout view, the sharp rise of optical radiation occurs after a time R_d/v_sh during which the shock is traveling in the optically thick region (Figure 1). The explosion date is thus ∼60 days earlier than the estimate of Smith et al. (2007, 2010), and mean velocities of uniformly expanding ejecta are lower than the estimates of Smith et al.

Another aspect of the rise to maximum in shock breakout is that the increasing temperature as the shock breaks out is an important component of the rising luminosity. In the case of SN 2006gy, Smith et al. (2010) find a temperature T ≈ 11, 000 K on days 65 and 71 (from 2006 August 20; their Figure 4), which is close to the time of optical maximum light. On day 36, Smith et al. (2010) estimate T ≈ 9500 K, using the same extinction value as that used for other epochs (their Figure 4). However, they find that with an assumed larger extinction, a photosphere with T ≈ 15, 000 K provides a better fit to the observed spectrum. The higher temperature would bring the temperature evolution more in line with that observed in other supernovae, i.e., a decreasing temperature with time, and is advocated by Smith et al. (2010). However, in the shock breakout view, the increasing temperature is expected and there is no need for a time-dependent extinction.

In our model, the dense shell and shock wave are deep within the circumstellar envelope, outside of which is the last equilibrium shell where the spectrum is formed. The radius of this shell is not the blackbody radius R_bb because of the nonequilibrium conditions in the medium. Outside of the shell is an electron-scattering region of moderate optical depth (between c/v_sh and 1) where the peaked Hα profile with a broad base can be formed (Chugai 2001; Smith et al. 2010); the broadening is due to scattering in the thermal gas as opposed to the Doppler effect.

An example of a different type of luminous supernova is SN 2010gx and related objects (Pastorello et al. 2010; Quimby et al. 2009). In this case, there is a sharp drop from peak luminosity without a flattening, broad lines of O ii are present at maximum light, and the narrow H or He lines often observed in Type IIn supernovae are not present. The evidence points to an explosion in a more compact circumstellar medium for this class of objects. The rise time, ∼50 days, and peak luminosity, $(3\hbox{--}4)\times 10^{44}\,\rm erg\, s^{-1}$, are comparable to the case of SN 2006gy, while the temperature is higher, 15,000 K. The rise time suggests that R_w is not much less than R_d, so R_w ≈ R_d. The density is comparable to SN 2006gy. The higher temperature in this case can be attributed to the lower opacity due to the lack of H and He, and the smaller extent of the opaque circumstellar medium.

Although the case for dense mass-loss years before the supernova appears good, the cause of the mass loss is not known. Woosley et al. (2007) suggested that SN 2006gy was the result of pair instability eruption before the supernova. Since the radiated energy produced in this case is ∼10⁵⁰ erg and the radiated energy in observed luminous supernovae is ≳10⁵¹ erg, we have not specifically treated that case here, although if the energy were larger similar physical arguments would presumably apply. The dense mass loss is typically attributed to luminous blue variable eruptions (Smith & McCray 2007), although such eruptions are not understood in the context of stellar evolution. An eruption was observed two years before the explosion of SN 2006jc, which was an H-poor, Type Ib supernova (Pastorello et al. 2007; Foley et al. 2007). The mass loss in this case was too weak to produce the high density phenomena discussed here, but it does show the possibility of a presupernova outburst, even in the H-poor case.

We thank Claes Fransson for discussions. This research was supported in part by NSF grant AST-0807727.

Note added in proof. A model similar to that developed here was presented by Balberg & Loeb (2011) and applied to the Type Ib SN 2008D.