Very Low-energy Supernovae: Light Curves and Spectra of Shock Breakout

2.1.1. Analytic Representation

At the temperatures, densities, and metallicities explored in this paper, a major contribution to scattering opacity comes from photons colliding with free electrons (Compton scattering), which in these low-energy models can be considered to be in the Thomson limit. This process also contributes to absorption opacity, however, because the collisions are not perfectly elastic and a small amount of energy is exchanged between photon and electron. The energy exchanged per collision is

$\begin{eqnarray}&&{\rm{\Delta }}E=\displaystyle \frac{h\nu }{{m}_{e}{c}^{2}}(4{{kT}}_{g}-h\nu ),\end{eqnarray} \tag{ 1 }$

where T_g is the gas temperature. If the photon energy $h\nu$ is less than the gas/electron energy 4kT_g, then the photons will gain energy (the inverse Compton process); otherwise, they will lose energy to the gas. As the radiation temperature drops, the average photon energy $h\nu$ does as well, and each scattering exchanges less energy. If the only absorption source considered is Compton scattering, then as the breakout energy drops, absorption will become inefficient at equilibrating the radiation. The simulation will therefore show the chromosphere retreating within the star to higher temperatures and densities, increasing the ${T}_{c}$ /${T}_{\mathrm{ef}}$ ratio. This will be discussed further in Section 3.

Bremsstrahlung emission occurs when an electron passes close to another charged particle, generally an atomic nucleus, and the change in its energy causes the emission of a photon. There is an associated inverse bremsstrahlung process wherein a photon strikes an electron moving near a nucleus and causes an increase in its energy. This results in a transfer of energy from radiation to gas and therefore functions as an absorptive opacity. Inverse bremsstrahlung in regions with fully ionized hydrogen and helium follows a Kramer's law opacity and can be calculated analytically as

$\begin{eqnarray}&&{\kappa }_{P}{}_{b}={C}_{b}{T}^{-3.5}{n}_{e}{n}_{i}{l}^{2},\end{eqnarray} \tag{ 2 }$

where C_b is a numerical constant and l is the ionization level of the nucleus.

While Compton scattering depends only on the electron number density n_e, since it considers only photons scattering off electrons, inverse bremsstrahlung depends on both n_e and n_i, as both an electron and an ion are involved in each interaction. For a completely ionized atmosphere consisting of mostly hydrogen and helium, ${n}_{i}\approx {n}_{e}$. This gives ${\kappa }_{P}{}_{b}\propto {n}_{i}^{2}$, and in fully ionized regions $\rho \approx {n}_{i}\mu$, where μ is the mean molecular weight, resulting in a ${\rho }^{2}$ dependency. Inverse bremsstrahlung therefore drops off more rapidly with ρ than Compton scattering, relevant since shock breakout takes place in a region of sharply decreasing density. The steepness of this density profile is a property of the progenitor star and therefore does not differ between standard CCSNe and VLE SNe.

The gas temperature at the shock front, however, does vary significantly between standard CCSNe and VLE SNe. Compton scattering depends linearly on gas temperature, but inverse bremsstrahlung goes as $\propto {T}^{-3.5}$. Thus, a small drop in temperature causes the inverse bremsstrahlung opacity to increase much faster than Compton scattering, fast enough to overcome the suppression from the ${\rho }^{2}$ term, making inverse bremsstrahlung much more significant in models at lower temperatures. In the VLE case temperatures during breakout are on the order of 5 $\times \ {10}^{4}$–3 $\times \ {10}^{5}$ K, and in this regime inverse bremsstrahlung plays a significant role and must be considered in simulation. The different contributions of various opacity processes at different energies are illustrated at low energy in Figure 1 and at a more average energy in Figure 2.

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2.1.2. Implementation

Photoionization is a complex process. Because cross sections for ionization depend strongly on the energy of the incoming photon, precise formulae for this opacity are heavily frequency dependent. However, above 5000 K hydrogen may be assumed to be ionized, and above 4.5 $\times \ {10}^{4}$ K helium may also be assumed to be completely ionized. Therefore, when $T\gt 4.5\times \ {10}^{4}$ K, only metals will contribute to photoionization opacity. In a star of solar metallicity the fraction of the atmosphere that consists of metals is small, but their large cross sections and low ionization energies make their opacities significant. In this regime the photoionization can be treated as a gray opacity with a Kramer's law form:

$\begin{eqnarray}&&{\kappa }_{P}{}_{p}=4.34\times \ {10}^{25}\,Z(1.0+{X}_{{\rm{H}}})\rho {T}^{-3.5},\end{eqnarray} \tag{ 3 }$

where X_H is the hydrogen mass fraction and Z the metal mass fraction (not the proton number). However, this raises the question of how to treat photoionization when $T\lt 4.5\times \ {10}^{4}$ K. Here the Kramer's law opacity breaks down, and truly accurate calculations are quite time intensive. An approximate gray opacity may be found by performing a number of frequency-dependent calculations and taking their Planck mean, but this is again quite time intensive. It is therefore worth considering whether photoionization opacities below the helium ionization limit are relevant; in these models the opacity network ceases to calculate bound–free opacities below this temperature.

Line opacities are not accounted for in these simulations, due to the complexity and computational cost. Incorporating them would require either generating tables for the appropriate temperature and densities or manually implementing a much larger opacity network that would be frequency dependent, negating the work done to place other opacities in simpler gray forms and requiring lengthy multigroup simulations for even the bolometric light curves.

The effects of line broadening due to velocity shear in the region of the photosphere are also not included in these calculations, but fortunately the shocks considered here are low velocity as a rule, and at the explosion energies where electron scattering does not dominate envelope velocities are ∼500 km s⁻¹. Thus, velocity shear effects can be ignored.

To examine this analytic opacity model, tabulated values from the OPAL database were used for comparison. OPAL tables incorporate the effects of lines, as well as many other opacity processes, to a high degree of precision. OPAL can only give total opacity and not separate absorption and scattering opacities, which these simulations require. Still, it is a useful point of comparison. Figure 3 presents opacities as calculated by the analytic formulae discussed here compared to calculations using OPAL for a low-energy explosion near shock breakout. The OPAL opacities are systematically higher; this is expected to be the case as OPAL includes more absorption processes. Thus, the opacities in these simulations should be taken as lower bounds. This implies that the peak luminosity and color temperature results from these simulations are upper bounds.

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Initial attempts to simulate shock breakout in SN 1987A produced unexpectedly high radiation color temperatures, particularly at the shock front; gas temperature remained steady, but ${T}_{c}$ increased by an order of magnitude. This discrepancy was ultimately resolved by implementing higher, more accurate absorption opacities. It is worth considering why the simulation results were so sensitive to changes in absorption opacity, as an illustration of the necessity of considering opacity details. Examining the different terms affecting the spectrum in the flux-limited diffusion approximation provides the solution. During breakout, two terms in the radiation-hydro equations dominate the spectrum: an advection term in frequency space and the matter–radiation coupling term. The multigroup treatment of flux-limited diffusion takes the form

$\begin{eqnarray}&&\displaystyle \frac{\partial {E}_{g}}{\partial t}+{\boldsymbol{\nabla }}\cdot \left(\displaystyle \frac{3-{f}_{g}}{2}{E}_{g}{\boldsymbol{v}}\right)-{\boldsymbol{v}}\cdot {\boldsymbol{\nabla }}\left(\displaystyle \frac{1-{f}_{g}}{2}{E}_{g}\right)\end{eqnarray} \tag{ 5 }$

$\begin{eqnarray}&&=\,c{\kappa }_{g}(\alpha {T}^{4}-{E}_{g})+{\boldsymbol{\nabla }}\cdot \left(\displaystyle \frac{c\lambda }{{\chi }_{R}\,}{\boldsymbol{\nabla }}{E}_{g}\right)\end{eqnarray} \tag{ 6 }$

$\begin{eqnarray}&&+{\int }_{g}\displaystyle \frac{\partial }{\partial \nu }\left[\left(\displaystyle \frac{1-f}{2}{\boldsymbol{\nabla }}\cdot {\boldsymbol{v}}+\displaystyle \frac{3f-1}{2}\hat{n}\hat{n}:{\boldsymbol{\nabla }}{\boldsymbol{v}}\right)\nu {E}_{\nu }\right]\end{eqnarray} \tag{ 7 }$

$\begin{eqnarray}&&-\displaystyle \frac{3f-1}{2}{E}_{g}\hat{n}\hat{n}:{\boldsymbol{\nabla }}{\boldsymbol{v}}.\end{eqnarray} \tag{ 8 }$

Consider the optically thick limit where the gas has the most influence on the radiation, in which case ${f}_{g}=1/3$ and all $\hat{n}\hat{n}$ terms (representing free-streaming radiation) disappear:

$\begin{eqnarray}&&\displaystyle \frac{\partial {E}_{g}}{\partial t}+\displaystyle \frac{4}{3}{\boldsymbol{\nabla }}\cdot {E}_{g}{\boldsymbol{v}}-\displaystyle \frac{1}{3}{\boldsymbol{v}}\cdot {\boldsymbol{\nabla }}({E}_{g})=\end{eqnarray} \tag{ 9 }$

$\begin{eqnarray}&&c{\kappa }_{g}(\alpha {T}^{4}-{E}_{g})+{\boldsymbol{\nabla }}\cdot \left(\displaystyle \frac{c}{3{\chi }_{R}\,}{\boldsymbol{\nabla }}{E}_{g}\right)\end{eqnarray} \tag{ 10 }$

$\begin{eqnarray}&&+\displaystyle \frac{1}{3}{\int }_{g}\displaystyle \frac{\partial }{\partial \nu }[({\boldsymbol{\nabla }}\cdot {\boldsymbol{v}})\nu {E}_{\nu }].\end{eqnarray} \tag{ 11 }$

On the left-hand side, the first term in the equation represents the quantity to be solved for, the change in group energy with time. The second and third terms represent energy transfer by bulk fluid motion. The three terms on the right-hand side respectively represent energy exchange via absorption/emission, diffusion of radiation, and energy transfer between frequency groups.

This equation can be interpreted as an advection equation in frequency space (see Zhang et al. 2013, for details). The "sound speed" in this medium includes a term dependent on the divergence of velocity, ${\rm{\nabla }}\cdot {\boldsymbol{v}}$. In practice, this means that as the gas is compressed by the shock, a velocity divergence arises that shifts the spectrum toward the blue. This term can cause the radiation spectral temperature to diverge from the gas temperature even in optically thick regions. In particular, if this term is the only one considered, the spectrum will harden significantly just before breakout as the shock reaches the stellar atmosphere and increases in velocity. The magnitude of this effect is directly linked to the velocity at the shock front—higher velocity leads to greater divergence and more hardening of the spectrum.

Conversely, the absorption–emission source exchange term, which depends on absorptive opacity, will tend to equilibrate the radiation and the matter, so in the case of shock breakout it will cool the radiation. The relative strength of these two terms determines the ultimate color temperature. If the source exchange term dominates, ${T}_{c}$ will follow the gas temperature. If the advection term dominates, however, ${T}_{c}$ can be significantly higher. Therefore, the relative timescales of these terms must be considered in order to ensure that the resulting color temperature has the correct qualitative behavior.

Let the scale length $L=c/{\chi }_{R}\,{\boldsymbol{v}}$, where ${\chi }_{R}$ is the scattering opacity. The terms that will alter the spectrum are the frequency group coupling term $(1/3)({\boldsymbol{\nabla }}\cdot {\boldsymbol{v}})\nu {E}_{g}$, which depends on velocity divergence, and the absorption/emission energy exchange term $c{\kappa }_{P}\,(\alpha {T}^{4}-{E}_{r})$. The timescale of the exchange term is the inverse of $c{\kappa }_{P}$. The timescale of the velocity divergence term is the inverse of ${\boldsymbol{v}}/3L$:

$\begin{eqnarray*}{\tau }_{x}^{-1} & = & c{\kappa }_{P}\,\\ {\tau }_{v}^{-1}=\displaystyle \frac{{\boldsymbol{v}}}{3L} & = & \displaystyle \frac{1}{3}{\boldsymbol{v}}\left(\displaystyle \frac{{\chi }_{R}\,{\boldsymbol{v}}}{c}\right)\\ \displaystyle \frac{{\tau }_{v}}{{\tau }_{x}}=\theta =(c{\kappa }_{P}\,)\left(\displaystyle \frac{3c}{{\chi }_{R}\,{{\boldsymbol{v}}}^{2}}\right) & = & 3{\left(\displaystyle \frac{c}{v}\right)}^{2}\left(\displaystyle \frac{{\kappa }_{P}\,}{{\chi }_{R}\,}\right).\end{eqnarray*}$

If $\theta \gt 1$, ${\tau }_{v}\gt {\tau }_{x}$, the exchange term dominates, and the radiation spectral temperature will follow the gas. If $\theta \lt 1,{\tau }_{v}\lt {\tau }_{x}$, the velocity term dominates, and the radiation spectral temperature will increase above the gas temperature in regions of high velocity divergence, e.g., shock fronts.

For the SN 1987A model, $v\sim {\rm{15,000}}$ km s⁻¹ at breakout. If ${\kappa }_{P}$ /${\chi }_{R}$  = 10⁻⁶, $\theta =0.0012;$ if ${\kappa }_{P}$ /${\chi }_{R}$ increases to 10⁻³, then $\theta =1.2$, implying a significant shift in the spectral behavior of the simulation. Figure 4 shows the temperatures of three SN 1987A simulations run up to the moment of breakout in order to test this hypothesis. Since at these temperatures electron scattering dominates both ${\kappa }_{P}$ and ${\chi }_{R}$ , the opacity network was simplified to only calculate this process, which allows easy variation by changing the parameter $\epsilon ={\kappa }_{P}\,/{\chi }_{R}\,$. Solid lines represent gas temperature, and dashed lines represent color temperature. Where the two lines overlap, the radiation and gas are in equilibrium.⁴ These simulations used 64 frequency groups spaced logarithmically in the range 1 $\times \ {10}^{15}$–1 $\times \ {10}^{18}$ Hz.

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The red and green dashed lines, representing color temperature in simulations with $\epsilon ={10}^{-6}$ and 10⁻⁴, respectively, both spike above the gas temperature at the shock front—the region of highest velocity divergence—and then diffuse forward. By contrast, in the $\epsilon ={10}^{-3}$ simulation (blue) the color temperature follows the gas temperature through the hydrodynamic shock. Thus, the anomalously high color temperatures in the initial simulations, which were run with $\epsilon ={10}^{-6}$, are due to velocity effects. Changing this ratio to a more realistic $\epsilon \sim {10}^{-3}$ brought ${T}_{c}$ in line with previous simulations, as will be discussed in Section 4.

Shock front velocities in the red supergiant models, especially the VLE explosions modeled here, are much slower than in SN 1987A. Velocities at breakout range from 80 to 1200 km s⁻¹ for RSG15 and from 150 to 500 km s⁻¹ for RSG25. Even for ${\kappa }_{P}$ /${\chi }_{R}$  = 10⁻⁶ the exchange term dominates in most RSG cases. Velocity effects are therefore unlikely to play a role in VLE SN breakouts.

In cases with high velocity, low metallicity, or both, this effect may still come into play. SNe of more standard energies exploding in very metal-poor blue supergiants, as theorized for some models of first-generation stars, might also have unusually high color temperatures. While a full simulated exploration of this space is beyond the scope of this paper, we can qualitatively predict some behavior. Stars with high envelope metallicities and/or low shock velocities will likely show the same 2–3 ratio of ${T}_{c}$ to ${T}_{\mathrm{ef}}$ as SN 1987A, but stars with very low envelope metallicities, such as Pop III stars, might exhibit spectral temperatures dramatically higher than predicted. Kasen et al. (2011) estimated the shock velocity at breakout in pair SNe in a red supergiant model at $1.5\times \ {10}^{4}$ km s⁻¹, similar to the SN 1987A model discussed in this section; in blue supergiants this number can be substantially higher. At the temperatures of a standard-energy breakout inverse bremsstrahlung is highly suppressed and metals contribute most of the photoionization opacity; in a low-metal star total absorption opacity may therefore be quite low. At high velocities and low absorptive opacity the velocity divergence discussed here can easily dominate the exchange term and make the spectrum much hotter than opacity arguments would predict.

Based on a number of recent surveys (e.g., O'Connor & Ott 2011; Pejcha & Thompson 2015; Sukhbold et al. 2015), the pre-SN stars that most commonly produce black holes in stars that still retain their envelopes had main-sequence masses in the range of 15–35 ${M}_{\odot }$ . Especially prolific in black hole production may be the stars with 18–26 ${M}_{\odot }$ (Horiuchi et al. 2011; Ugliano et al. 2012; Brown & Woosley 2013; Kochanek 2014; Sukhbold & Woosley 2014; Clausen et al. 2015). In order to sample this range, two progenitor stars are used here, both red supergiants, with main-sequence masses 15 ${M}_{\odot }$ (RSG15) and 25 ${M}_{\odot }$ (RSG25). Their pre-SN structures are shown in Figures 8 and 9 and Table 2. Since the radiation produced by breakout depends chiefly on the shock wave energy, which will be varied, and the pre-SN radius, which changes only by about a factor of two in the mass range of interest, these two models should suffice. The models here are taken from Woosley & Heger (2007) and are very similar to those more recently explored by Sukhbold & Woosley (2014) and Sukhbold et al. (2015).

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Table 2.  Pre-SN Star Parameters

Star	Final Mass (${M}_{\odot }\,$)	He Core Mass (${M}_{\odot }\,$)	Radius (cm)
SN 1987A	15	8	3.2 × 1013
RSG15	12.79	4.27	6 × 1013
RSG25	15.84	8.20	1.07 × 1014

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These stars were evolved using the KEPLER code (Weaver et al. 1978; Woosley et al. 2002) from ignition on the main sequence to the time of collapse. An artificial shock of variable energy was initiated by first extracting the iron core and then dumping energy in the bottom 10 zones. Since the calculations here focus on the early light curve and its plateau and do not depend on nucleosynthesis or details of how the explosion was powered, this simple approach should suffice.

It is important to note that the energy deposited, the shock energy at the time of breakout, and the final kinetic energy at infinity of the ejecta are three different quantities. Not all energy deposited becomes kinetic, and the shock energy at breakout is not the same as the total energy, which includes both positive internal and negative gravitational energy. Both the kinetic energy at breakout, which is relevant for breakout, and the final kinetic energy, which is relevant for the later light curve, are given in Table 3.

Table 3.  VLE Breakout Model Results

Model	Star	va (km s−1)	KEbb (erg)	KEfc (erg)	Lp (erg s−1)	Lcrd (erg s−1)	${\rm{\Delta }}t$ e (h)	Max ${T}_{\mathrm{ef}}$ (K)	Max ${T}_{c}$ (K)	${\lambda }_{\max }$ (Å)
A15	RSG15	80	3.86 $\times \ {10}^{46}$	6.58 $\times \ {10}^{46}$	9.50 $\times \ {10}^{39}$	9.57 $\times \ {10}^{39}$	68.4	8.15 $\times \ {10}^{3}$	⋯	⋯
B15	RSG15	150	6.43 $\times \ {10}^{47}$	1.54 $\times \ {10}^{48}$	3.89 $\times \ {10}^{41}$	3.82 $\times \ {10}^{41}$	35.2	2.06 $\times \ {10}^{4}$	⋯	⋯
C15	RSG15	320	4.98 $\times \ {10}^{48}$	1.21 $\times \ {10}^{49}$	8.39 $\times \ {10}^{42}$	8.33 $\times \ {10}^{42}$	8.1	4.44 $\times \ {10}^{4}$	6.40 $\times \ {10}^{4}$	797
D15	RSG15	650	2.13 $\times \ {10}^{49}$	5.04 $\times \ {10}^{49}$	5.43 $\times \ {10}^{43}$	5.38 $\times \ {10}^{43}$	5.3	7.08 $\times \ {10}^{4}$	1.15 $\times \ {10}^{5}$	443
E15	RSG15	1200	5.43 $\times \ {10}^{49}$	1.23 $\times \ {10}^{50}$	2.13 $\times \ {10}^{44}$	2.02 $\times \ {10}^{44}$	3.1	9.97 $\times \ {10}^{4}$	2.24 $\times \ {10}^{5}$	228
F15	RSG15	1920	2.16 $\times \ {10}^{50}$	5.07 $\times \ {10}^{50}$	8.25 $\times \ {10}^{44}$	8.07 $\times \ {10}^{44}$	1.83	1.40 $\times \ {10}^{5}$	2.35 $\times \ {10}^{5}$	217
G15	RSG15	2400	2.54 $\times \ {10}^{50}$	1.20 $\times \ {10}^{51}$	1.68 $\times \ {10}^{45}$	1.66 $\times \ {10}^{45}$	1.42	1.67 $\times \ {10}^{5}$	3.18 $\times \ {10}^{5}$	160
A25	RSG25	150	1.38 $\times \ {10}^{48}$	⋯	9.07 $\times \ {10}^{41}$	9.03 $\times \ {10}^{41}$	67.0	1.57 $\times \ {10}^{4}$	⋯	⋯
B25	RSG25	179	1.53 $\times \ {10}^{48}$	⋯	1.23 $\times \ {10}^{42}$	1.23 $\times \ {10}^{42}$	57.9	1.70 $\times \ {10}^{4}$	⋯	⋯
C25	RSG25	210	2.27 $\times \ {10}^{48}$	6.12 $\times \ {10}^{48}$	2.76 $\times \ {10}^{42}$	2.77 $\times \ {10}^{42}$	37.0	2.08 $\times \ {10}^{4}$	⋯	⋯
D25	RSG25	280	3.52 $\times \ {10}^{48}$	⋯	5.85 $\times \ {10}^{42}$	5.83 $\times \ {10}^{42}$	25.9	2.51 $\times \ {10}^{4}$	⋯	⋯
E25	RSG25	480	1.10 $\times \ {10}^{49}$	⋯	3.67 $\times \ {10}^{43}$	3.61 $\times \ {10}^{43}$	9.3	3.97 $\times \ {10}^{4}$	⋯	⋯

Notes.

^aVelocity at breakout. ^bKinetic energy at breakout. ^cFinal kinetic energy of SN as calculated by KEPLER. ^dCorrected for light-travel time. ^eFWHM of a travel-time-corrected light curve.

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In most core-collapse simulations the properties of the pre-collapse stellar atmosphere are irrelevant, since by the time the explosion is seen the layers that governed breakout have already become transparent. Care must be taken to treat it accurately in shock breakout studies. Additional modifications were therefore made to the progenitor models' atmospheres. These modifications are discussed in Section 5.1.2.

Three calculations were done for each case: a precise breakout simulation using CASTRO (Section 7), a simplified breakout simulation using KEPLER for comparison purposes, and a precise calculation of the later plateau-stage light curve using KEPLER (Section 7.3). KEPLER uses only single-temperature flux-limited radiation transport, but as long as a KEPLER progenitor model is mapped into CASTRO at a time when the material is still very optically thick and the radiation is fully thermalized, no loss of precision occurs. Breakout results from KEPLER give information on the bolometric luminosity and effective emission temperature that is useful to compare with CASTRO and, with adequate zoning, give reasonable estimates. These results are discussed further in Section 7.3. Only the CASTRO multigroup calculation gives information on the color temperature and spectrum.

Because the energy in the shock at breakout depends only on the local velocity structure in the hydrogen envelope, models for the 25 ${M}_{\odot }$ case were generated by simply multiplying the velocities in KEPLER model C25 by a constant factor before inputting them into CASTRO. All RSG15 models were calculated individually using KEPLER.

5.1.1. Choice of Energies

5.1.2. Stellar Atmospheres

The KEPLER code is designed to study the internal structure and nucleosynthesis of stars and SNe and does not treat the outer stellar atmosphere very carefully. However, shock breakout is fundamentally an atmospheric phenomenon, since its properties are governed by the critical thick-to-thin transition region near the photosphere. In the SN 1987A models, this atmosphere is on the order of $1\,\times \,{10}^{12}$ cm thick; in the RSG models it is $\sim 6\,\times \,{10}^{12}$ cm. The effect of variations in the stellar atmosphere was tested by running two different simulations of shock breakout in SN 1987A. The first model had its atmosphere replaced by a power-law fit of the form log${}_{10}\,\rho ={\alpha }_{1}{\mathrm{log}}_{10}\,r+{\beta }_{1},{\mathrm{log}}_{10}\,T={\alpha }_{2}{\mathrm{log}}_{10}\,r+{\beta }_{2}$, where $\alpha ,\beta$ were determined by fitting the original KEPLER data. The fits gave ${\alpha }_{1}=-42.1,{\beta }_{1}=518.0,{\alpha }_{2}=-26.4,{\beta }_{2}=334.5$. The second model then had its atmosphere replaced by the same power-law form using ${\alpha }_{1}/2,{\alpha }_{2}/2$, generating a significantly less steep gradient. The resulting atmospheric gradients are shown in Figure 10. As can be seen in Figure 11, the different atmospheres result in quite different breakout profiles. The two breakouts have the same integrated energy, since that energy is being released from the shock wave itself, but the more slowly varying atmosphere influences the timescale and temperature of the observed transient. A shallower atmospheric gradient creates a wider transition region and thus a breakout that is longer lasting and correspondingly less luminous at peak, while a steeper gradient creates a harder, faster transient.

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The true atmosphere of SN 1987A's progenitor is not expected to vary this much between pre-SN models, but the comparison does demonstrate that differences in the atmospheric gradient can produce corresponding and significant differences in the resulting light curves. During KEPLER's modeling of the propagation of the shock waves from the core to the CASTRO link point, physics changes designed to improve simulation of the shock wave structure can lead to secondary responses in the atmosphere. Since explosions at different energies take different amounts of time to reach the surface, this can also lead to variations between different models in the same pre-SN progenitor. These divergences are clearly unphysical, and modifications to the CASTRO input models are therefore required.

An alternative to a simple power-law extrapolation of the density structure in the KEPLER pre-SN star's outer zones is to use a model stellar atmosphere. For the 15 ${M}_{\odot }$ red supergiant models, realistic model atmospheres were available from the MARCS database (Gustafsson et al. 2008). The MARCS atmospheres were calculated using a specialized code designed to help observers fit spectral data for red supergiants. The atmospheres include LTE and non-LTE effects not simulated in either KEPLER or CASTRO, as well as the effects of line blanketing. The 15 ${M}_{\odot }$ progenitor has ${T}_{\mathrm{ef}}$  ∼ 3550 K, solar metallicity, and log surface gravity of −0.32. The two MARCS atmospheres closest to the 15 ${M}_{\odot }$ progenitor were selected based on ${T}_{\mathrm{ef}}$, metallicity, and log specific surface gravity and then interpolated to fit the KEPLER progenitor's properties. The MARCS atmosphere data extend approximately $5\times \ {10}^{12}$ cm below and $1\times \ {10}^{12}$ cm above the photosphere. In order to ensure a smooth transition, a power law is fit to the combined MARCS atmosphere and KEPLER model and used to replace all RSG15 atmospheres. This replacement provided a much more accurate atmospheric gradient. The final progenitor atmospheres are shown in Figure 12.

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The atmosphere of a 25 ${M}_{\odot }$ red supergiant is more complex and has been less well studied in simulation. This progenitor's envelope is very extended ($\sim {10}^{14}$ cm) and loosely bound. Once the star reaches this stage of its life, it will have likely lost part or all of the envelope to winds or other instabilities; uncertainties in mass loss make it difficult to reliably estimate the atmosphere's final structure. The section of parameter space explored by MARCS did not extend near enough to our progenitor star to reliably extrapolate the data. Thus, for RSG25 a power law of the same form as the 87A fit described above was fit to the existing KEPLER model and used to replace the progenitor atmospheres.

The CASTRO simulations used an 18-species network, terminating at nickel, with two auxiliary variables, electron fraction Y_e and mean molecular weight $1/\mu$. Nuclear reactions were turned off for the present study, and all species were passively advected. In the regions of interest the composition was dominated by hydrogen and helium at relatively low temperatures, so an ideal gas law plus radiation provided an accurate equation of state. KEPLER has its own general equation of state that is good under all conditions except extremely high density. There was good agreement between the KEPLER pressure, internal energy, temperature, and density and the equivalent quantities in CASTRO after the remap.

As discussed in Section 2, shock breakout is a transition of the shock wave from an optically thick to optically thin region, and an accurate treatment of opacity is key to obtaining realistic results. The physics of shock breakout occur at and behind the shock front, where the gas temperature is in the range of 4.5 $\times \ {10}^{4}$–1 $\times \ {10}^{6}$ K when breakout occurs, depending on the model's energy. At these temperatures and the relevant mass fractions the primary species of interest are H and He. A Saha solver is used to calculate the ionization fraction of H and He. The electron abundance is then used to calculate both a scattering opacity ${\chi }_{R}$ and a small contribution to absorptive opacity ${\kappa }_{P}{}_{c}$, computed by assuming a fixed ratio $\epsilon ={\kappa }_{P}{}_{c}/{\chi }_{R}$. Both Thomson scattering and its contribution to absorptive opacity are gray, i.e., insensitive to frequency. Inverse bremsstrahlung absorption is calculated according to the equations discussed in Section 2.1. Photoionization absorption contributed by metals is calculated according to the equations discussed in Section 2.2. Line opacities, bremsstrahlung processes below 4.5 $\times \ {10}^{4}$ K, and photoionization processes below 4.5 $\times \ {10}^{4}$ K are not accounted for, as the former requires much more detailed calculations (Section 2.3) and the gray opacity laws for the latter two processes break down once their assumptions of complete H and He ionization are significantly violated. For details on both the theory and implementation of these opacities, see Section 2.

CASTRO is an Eulerian code and thus requires a certain amount of mass in every cell on the grid, or severe instabilities will result. A very thin ambient medium is therefore placed around the progenitor star to keep it stable while the processes of interest run. In the case of shock breakout this ambient medium must be made optically thin to ensure that its presence does not distort the resulting light curves. The ambient medium is generally made as cold and thin as possible while still maintaining numerical stability, to avoid affecting simulation results. If the opacity does not have the correct low-temperature behavior, this medium can become opaque and influence the results in an unphysical way. The H–He Saha solver becomes (and stays) correctly transparent in the ambient medium. Photoionization opacity is neglected (see previous section). If it were not, this medium might have some slight absorptive opacity. The medium used in this study had a density of order 10⁻¹² g cm⁻³. Most of the shock breakout transients themselves, and thus our simulations, are completed by the time the hydrodynamic shock itself reaches the ambient medium, and if the medium is appropriately transparent, it will not interact with the radiation either. Our assumed density for the medium therefore does not affect the breakout transient hydrodynamically in our simulations.

In reality stars near the ends of their lives—especially red supergiants—are prone to shedding large and uncertain amounts of mass. Narrow-line SNe II demonstrate clear signs of CSM interactions. Modeling the full scope of breakout–CSM interactions, however, is beyond the scope of this paper.

Due to their large radii, breakout transients in red supergiants have a much longer timescale than SN 1987A, further lengthened by the low energies considered. Typical durations of the breakout peak are hours to days. For some of the lower energies, the SN fails almost completely. For others the envelope is ejected, but not the helium core. The wide range of kinetic energies examined results in a diverse set of peak luminosities. Perhaps unfortunate for their detection, duration and peak brightness are inversely related—that is, the brighter the breakout, the shorter it will likely be.

In RSG15, the final kinetic energies for the VLE models ranged from $6.6\times \ {10}^{46}$ erg to $1.23\times \ {10}^{50}$ erg as listed in Table 3. Their peak luminosities ranged from 9.57 $\times \ {10}^{39}$ to 2.02 $\times \ {10}^{44}$ erg. The two standard-energy models, F15 (0.5 B) and G15 (1.2 B), had peak luminosities of 8.07 $\times \ {10}^{44}$ and 1.66 $\times \ {10}^{45}$, respectively. Bolometric light curves for RSG15 are shown in Figure 13. FWHM durations range from 3 to 35 hr, with an outlying 70 hr for the lowest-energy model, A15. Though mass loss leaves RSG15 and RSG25 with similar pre-SN masses (12.79 ${M}_{\odot }$ versus 15.84 ${M}_{\odot }$ ), the radius of RSG25 is nearly double the radius of RSG15. Breakouts with similar energies are thus expected to have longer light curves and lower peak energies in RSG25 versus RSG15. Bolometric light curves for RSG25 (Figure 14) show peak luminosities around 10⁴² erg s⁻¹ and durations in the 25–70 hr range, excepting E25, whose kinetic energy at breakout was significantly higher.

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Models B15, C15, E15, and G15 form a sequence where each model's final kinetic energy increases by approximately an order of magnitude, spanning ${10}^{48}\mbox{--}{10}^{51}$ erg. Their peak luminosities span five orders of magnitude (${10}^{41}\mbox{--}{10}^{45}$ erg s⁻¹), with the largest relative increases occurring between B15 and C15 and between C15 and E15, about a factor of 20. The duration of B15 (1.54 $\times \ {10}^{48}$) is about 35 hr, while the durations of C15 (1.12 $\times \ {10}^{49}$ erg), E15 (1.23 $\times \ {10}^{50}$ erg), and G15 (1.2 $\times \ {10}^{51}$ erg) are 8.1, 3.1, and 1.4 hr, respectively, meaning that a significant shift in breakout duration occurs between $\sim {10}^{48}$ and 10⁴⁹ erg of final kinetic energy.

Models D25 and C15 have similar kinetic energies at breakout (3.52 $\times \ {10}^{48}$ versus 4.98 $\times \ {10}^{48}$ erg); their durations, however, are significantly different. C15 has a duration of 8.1 hr versus 25.9 hr for D25. Models E25 and D15 have similar energies (1.10 $\times \ {10}^{49}$ versus 2.13 $\times \ {10}^{49}$ erg) and similar durations of 9.3 and 5.3 hr. Again, a strong shift in behavior is observed with energy, in this case the influence of progenitor structure on breakout. The much larger radius of RSG25 means that a shock will spend much more time traveling through the envelope than it would in RSG15. This longer travel time and increased interaction with the envelope may amplify the effect of progenitor structure on the resulting breakout. Shock waves energetic enough to traverse the envelope relatively quickly may have less time to interact and therefore show less variation with progenitor structure. Kinetic energy therefore emerges again as the most important parameter governing breakout behavior. This agrees with the analytic equations derived by Piro (2013), discussed further in Section 7.4, which give a dependence $L\propto {E}_{48}^{1.36}$.

Both RSG15 and RSG25 show minor anomalies in their light curves post-peak. RSG15's shock breakouts show a curious blip in their in decline rate post-peak—initially declining quite steeply, but briefly changing to a shallower slope. Four of RSG15's breakouts display this anomaly, and it is correspondingly extended in the longer-duration light curves. RSG25 shows a similar variation post-peak, although in this case it shows multiple peaks/changes in the rate of decline. None of these small features represent significant variations relative to the light curve of each breakout, but the fact that they appear in all light curves of a single progenitor merits further investigation. It is unclear what physical behavior causes these anomalies, but their appearance seems to be related to the formation of a high-density spike at the photosphere as the hydrodynamic shock begins to expand the envelope into the ambient medium. This spike is likely an artifact of 1D simulation. As noted, none of these features are significant relative to the overall light curves, so they can be neglected.

Only a small fraction of the envelope achieves escape speed in model A15, as shown in Figure 15, and the dynamics of this model are not as well determined as the others.

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CASTRO's multigroup transport was used to generate the spectra of RSG15 breakouts with various shock energies. These are shown in Figure 16. All have a dilute blackbody form. Models A15 and B15 were excluded from the multigroup simulation because their large optical depth at breakout posed computational difficulty and the opacity was not realistic anyway (see below). All multigroup models were calculated using 64 logarithmically spaced groups. Model C15 was run with a frequency range $1.5\times \ {10}^{14}\mbox{--}5\times \ {10}^{16}$ Hz. Models D15 and E15 were run with the range 5.0 $\times \ {10}^{14}$–1.0 $\times \ {10}^{17}$ Hz. Models F15 and G15 were run with the range 1.5 $\times \ {10}^{14}$–1.0 $\times \ {10}^{17}$ Hz.

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Near peak luminosity in models C15–E15 ${T}_{c}$ ranges from $6.4\times \ {10}^{4}$ to $3.18\times \ {10}^{5}$ K, moving the blackbody emission peak down from the soft X-ray bands into the hard UV. Previous works have found a color temperature higher than the effective temperature by a factor of roughly 2–3 (Klein & Chevalier 1978; Ensman & Burrows 1992; Tolstov et al. 2013). Ratios of ${T}_{c}$ to T_ef in models C15–G15 in Table 3 range from 1.4 to 2.2, similar to the values suggested by Rabinak & Waxman (2011).

However, as noted in Section 3, these previous works considered only high-energy explosions where electron scattering grossly dominated the opacity. In the low-energy breakouts here, a larger absorptive component helps to thermalize the diffusing photons. Given the small fraction of absorptive opacity needed for thermalization (Section 3) and the opacities shown in Figures 1–3, we expect the near equality of color and effective temperature at peak for models A15, B15, and C15. Higher energies, approaching those of SN 1987A, should be dominated by electron scattering with ${T}_{c}$ close to the maximum values given in Table 3 (models F15 and G15). Models D15 and E15 should have ${T}_{c}$ somewhere between the electron scattering value and T_ef. This is actually good news for the low-energy cases since it implies that breakout there can be calculated without recourse to multigroup transport, i.e., by codes using a single temperature.

Further work on these very low-energy cases would benefit from using an atomic opacity database such as OPAL, but the implementation of this poses a difficulty. OPAL tables give the total opacity and do not distinguish the absorptive and scattering components, as is needed for CASTRO. In the past EB92 overcame a similar limitation by assuming scattering opacity to come entirely from Compton scattering, computing this value, and then subtracting it from the OPAL value to get an absorptive opacity. In cases where the scattering opacity is dominant, this technique can lead to errors as one large number is being subtracted from another similar number, giving an uncertain small difference.

As noted in Section 5.1, the model atmospheres were replaced in the move from KEPLER to CASTRO. Figure 17 shows bolometric luminosity of two breakouts calculated in CASTRO, one using the MARCS atmosphere (blue) and one using a fitted version of the original KEPLER atmosphere (green). The differences are slight but present. KEPLER itself calculates a bolometric light curve that differs from CASTRO, but the divergence appears to be resolution related, as finer zoning in KEPLER significantly reduced the difference. Full results are shown in Table 4. KEPLER slightly underestimates the peak luminosity of the CASTRO results and falls within a factor of 1–3 of the analytic predictions. KEPLER can therefore produce a useful estimate of the bolometric light curve, even with its simpler radiation transport. Of the two codes, CASTRO is the only one that can compute a color temperature, so that result cannot be compared directly. However, if the theoretical work considered in Section 2 can be used to estimate the location of the chromosphere, a color temperature may still be calculated from KEPLER results.

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Table 4.  Comparison to Other Models

Model	KEfa (erg)	Lpeak (erg s−1)	${L}_{{\rm{K}}}$ b (erg s−1)	Pred. L (erg s−1)	Max ${T}_{\mathrm{ef}}$ (K)	Tef,kc(K)	Pred. ${T}_{\mathrm{ef}}$ (K)
A15	6.58 $\times \ {10}^{46}$	9.50 $\times \ {10}^{39}$	5.09 $\times \ {10}^{39}$	4.25 $\times \ {10}^{39}$	8.15 $\times \ {10}^{3}$	6.93 $\times \ {10}^{3}$	6.29 $\times \ {10}^{3}$
B15	1.54 $\times \ {10}^{48}$	3.89 $\times \ {10}^{41}$	3.94 $\times \ {10}^{41}$	3.10 $\times \ {10}^{41}$	2.06 $\times \ {10}^{4}$	1.99 $\times \ {10}^{4}$	1.84 $\times \ {10}^{4}$
C15	1.21 $\times \ {10}^{49}$	8.31 $\times \ {10}^{42}$	6.32 $\times \ {10}^{42}$	5.11 $\times \ {10}^{42}$	4.44 $\times \ {10}^{4}$	4.02 $\times \ {10}^{4}$	3.71 $\times \ {10}^{4}$
D15	5.04 $\times \ {10}^{49}$	5.43 $\times \ {10}^{43}$	3.95 $\times \ {10}^{43}$	3.56 $\times \ {10}^{43}$	7.08 $\times \ {10}^{4}$	6.36 $\times \ {10}^{4}$	6.02 $\times \ {10}^{4}$
E15	1.23 $\times \ {10}^{50}$	2.13 $\times \ {10}^{44}$	1.17 $\times \ {10}^{44}$	1.20 $\times \ {10}^{44}$	9.97 $\times \ {10}^{4}$	8.34 $\times \ {10}^{4}$	8.15 $\times \ {10}^{4}$
F15	5.07 $\times \ {10}^{50}$	8.25 $\times \ {10}^{44}$	6.17 $\times \ {10}^{44}$	8.06 $\times \ {10}^{44}$	1.40 $\times \ {10}^{5}$	1.30 $\times \ {10}^{5}$	1.31 $\times \ {10}^{5}$
G15	1.20 $\times \ {10}^{51}$	1.68 $\times \ {10}^{45}$	1.76 $\times \ {10}^{45}$	2.65 $\times \ {10}^{45}$	1.67 $\times \ {10}^{5}$	1.69 $\times \ {10}^{5}$	1.77 $\times \ {10}^{5}$

Notes.

^aFinal kinetic energy as measured in KEPLER. ^bPeak luminosity of a light curve in KEPLER ^cPeak effective temperature in KEPLER.

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Analytic predictions for shock breakout in "normal" SNe IIp already exist in the literature (Matzner & McKee 1999; Katz et al. 2010; Tominaga et al. 2011). Piro (2013) extended these formulae to consider the specific case of low-energy SNe. These formulae predict the bolometric luminosity and timescale of breakouts based on the properties of the progenitor star and the explosion and can be tested against our numerical results. Piro (2013) gives

$\begin{eqnarray*}{L}_{\mathrm{bo}} & = & 1.4\times \ {10}^{41}\displaystyle \frac{{E}_{48}^{1.36}}{{\kappa }_{0.34}^{0.29}{M}_{10}^{0.65}{R}_{1000}^{0.42}}{\left(\displaystyle \frac{{\rho }_{1}}{{\rho }_{* }}\right)}^{0.194}\mathrm{erg}{{\rm{s}}}^{-1}\\ {T}_{\mathrm{obs}} & = & 1.4\times \ {10}^{4}\displaystyle \frac{{E}_{48}^{0.34}}{{\kappa }_{0.34}^{0.068}{M}_{10}^{0.16}{R}_{1000}^{0.61}}{\left(\displaystyle \frac{{\rho }_{1}}{{\rho }_{* }}\right)}^{0.049}{\rm{K}},\end{eqnarray*}$

where ${E}_{48}={E}_{\mathrm{kin}}/{10}^{48}$, ${M}_{10}={M}_{\mathrm{ej}}/10\,{M}_{\odot }$, ${R}_{1000}\,={R}_{* }/1000\,{R}_{\odot },$ and ${\kappa }_{0.34}=\kappa /0.34$. Stellar radii are given in Table 2. It is assumed in both cases that the ejecta mass M_ej is equal to the size of the hydrogen envelope, that the adiabatic index $\gamma =5/3$, and that the factor $({\rho }_{1}/{\rho }_{* })\sim 1$. The results are shown in Table 4.

There is some ambiguity in the equations as to when the quantity E₄₈ should be measured; the kinetic energy at breakout differs from the ejecta's final kinetic energy at infinity. For RSG15, the peak luminosity predictions fell much closer to the KEPLER and CASTRO results when E₄₈ was assumed to be kinetic energy at infinity as measured in KEPLER. The analytic predictions and the KEPLER results are very close, and both slightly underestimate the CASTRO luminosities. RSG25 shows much greater variance; the analytic predictions underestimate the numerical results by a factor of 4–10, with the inaccuracies increasing with kinetic energy. The analytic formulae therefore give reasonable if not precise estimates for a breakout's brightness.

The practical upshot of the opacity effects discussed in Section 3 is that the color temperatures of these faint breakouts are significantly cooler than those of more energetic events, both because of their lower shock energies and because of the convergence of ${T}_{c}$ and ${T}_{\mathrm{ef}}$ . Although the bolometric luminosity of these events is much lower than normal CCSNe, more of the energy will be emitted at low frequencies, and in an IR band the low-energy breakout may actually appear brighter than its more bolometrically energetic counterpart. The reported detections of shock breakout by the Kepler satellite (Garnavich et al. 2016), which observes primarily in the optical and near-infrared, emphasize the advantage of this cooler spectrum.

A simple estimate of the brightness of breakout transients in any given band can be found by assuming that the spectrum is a blackbody at ${T}_{c}$ and calculating the fraction of blackbody energy inside that bandpass. More accurate calculations would be done using a measured filter curve. The Kepler satellite has an IR bandpass of 0.4–0.9 μm. Figure 20 shows the results of calculating the fraction of blackbody energy emitted within that bandpass, assuming that ${T}_{c}$ is equal to the maximum ${T}_{\mathrm{ef}}$ in low-energy events. Color temperature is unlikely to go lower than this value, meaning that these curves represent upper bounds. The dynamic range of peak luminosities is significantly compressed, as the brighter breakouts also have higher color temperatures. However, the duration of the transient is unaffected, and thus breakout events of similar luminosity can still be distinguished in energy by measuring the duration. The peak luminosities for these filtered curves range from 4 $\times \ {10}^{41}$ to 4 $\times \ {10}^{39}$ erg s⁻¹. These are still dim events, but not out of reach of current and future surveys; notably, they are significantly brighter for longer than a standard-energy breakout in these bands.

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Figure 21 shows simulated KEPLER light curves for the same 0.4–0.9 μm range. The KEPLER code cannot directly compute a color temperature, but based on analytic and numerical results, we can make some assumptions about its value. The black curve shows the predicted light curve in the 0.4–0.9 μm range of an explosion with final KE 2.4 B, calculated by applying formulae from Nakar & Sari (2010) to locate the chromosphere and using the gas temperature at that layer as a color temperature. The red curve shows a similar prediction for a VLE SN at 1.2 $\times \ {10}^{49}$ erg, corresponding to model C15. At this energy color temperature is predicted to converge with effective temperature and a light curve can be made with a 1-T code that assumes ${T}_{c}$  = ${T}_{\mathrm{ef}}$. The low-energy SN, though nearly a thousand times less energetic, peaks at a higher luminosity and stays bright for longer in the IR. Again, these are still dim events, but not out of reach.

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The cooler temperatures of RSG breakouts make studying VLE SNe at larger distances a more attractive prospect, as redshifting will move the light into optical and IR windows. But the majority of the SN's energy is still emitted at wavelengths shorter than the Lyα cutoff ($\gt 95$% at a color temperature $1\times {10}^{5}$ K), and if an extragalactic SN is assumed to be embedded in a UV-absorbing interstellar medium within its host galaxy, this energy may be absorbed at the source. Since these transients are already faint, nearby (z ≪ 1) events are the primary target, and the majority of absorption will occur either at the source or in the Galaxy. The UV attenuation will therefore depend on the viewing angle through the Galaxy and the unknown circumstellar medium at the source. Nearby SNe will still offer a better target.

Kochanek et al. (2008) proposed to begin a novel search for completely failed CCSNe by looking not for the presence but for the absence of sources. The "Survey About Nothing" monitors 1 $\times \ {10}^{6}$ red supergiants with the Large Binocular Telescope looking for the abrupt disappearance of any of these stars. In addition to potentially capturing a core-collapse failure, this survey could also detect VLE SNe coming from one of these sources. The SNe themselves would be visible as a sudden brightening of the "star" for of order a year, followed by a gradual but complete disappearance. After 4 yr of observations, Gerke et al. (2015) reviewed the survey data, searching for both complete failures and the neutrino-mediated transients created by the Nadyozhin–Lovegrove effect. Four candidates were initially recovered, but follow-up observations ruled out three sources as they later reappeared. The final candidate event satisfied the criteria for a very low-energy SN and continued to be observed. Recently Adams et al. (2016) reported three more years of observation on this candidate, showing that the source had dimmed significantly below the progenitor luminosity. Modeling of possible dust effects compared to optical and IR source data suggests that the observed event was terminal and that the transient's cool, dim properties cannot be explained by dust. This event may therefore be the first observed example of the Nadyozhin–Lovegrove effect and an excellent example of a real VLE SN. Unfortunately, this survey's cadence is not frequent enough to catch a breakout event.

Reynolds et al. (2015) conducted a search through HST archival data looking for collapse events that were not flagged by survey selection rules at the time and recovered one candidate in the range of 25–35 ${M}_{\odot }$ that may have undergone an optically dark collapse.

Several candidate shock breakout events have been published in the literature, but most are high-energy events suspected to come from compact progenitors. Soft X-ray breakout bursts are easier to detect because of the large number of existing space-based X-ray transient satellites designed specifically for the wide-field coverage and rapid slew time needed to capture breakout. In 2008 Soderberg et al. (2008) serendipitously captured an X-ray transient when an SN went off during a Swift observation of its host galaxy. Soderberg et al. (2008) attribute this event to a Type Ib/c CCSN breaking out from a dense stellar wind surrounding its progenitor, a scenario consistent with the high mass loss rates of Type Ib/c progenitors near the end of their lives. Unfortunately, this rapid high-energy event, while of great interest on its own and as a proof of concept for shock breakout observations, bears little relation to the transients explored in this work.

Closer to the VLE SN regime, UV observations using the GALEX satellite in 2008 detected two CCSNe very close to the time of explosion (Gezari et al. 2008): one with fading and one with rapidly rising UV emission, suggesting that the latter had been caught during its breakout phase. KEPLER hydrodynamic models combined with the CMFGEN radiation transport code were used to model the observed UV light curve of these events as breakout in a 15 ${M}_{\odot }$ red supergiant exploding with a final kinetic energy 1.2 B. The calculated effective temperature was similar to the high-energy RSG15 models presented here. The authors noted an effect that will also be important in the case of VLE SNe, namely, that as the bolometric light curve fades the spectral temperature declines, bringing more luminosity into the UV (or optical) observing window. The net effect is an apparent plateau phase that is actually the result of competing processes.

8.3.1. Kepler Satellite Observations

The key to capturing these breakouts is high survey cadence, preferably hourly. Even a daily measurement can miss the more energetic breakouts entirely. Follow-up will be simpler than for compact star breakouts since observers will have a response window measured in hours rather than seconds. Spectroscopic follow-up is strongly recommended to measure ${T}_{c}$ . The spectrum of a red supergiant breakout will be dominated by the blackbody continuum and hydrogen–helium lines. It may also show absorption features from the surrounding nebula. Unlike the later SN light curve, the breakout will show little to no nickel or iron emission. The light curve will transition to photometric and spectroscopic behavior typical of a normal (albeit dim) SN II. Some planned missions could take good observations of breakout transients. The WFIRST mission would provide wide-field observations in the near-IR, and the ULTRASAT program currently in design would launch a rapid-cadence UV satellite that would be ideal for detecting shock breakouts (Sagiv et al. 2014). Among ground-based observatories, Pan-STARRS, PTF/ZTF, LCOGT, and eventually LSST could all make useful observations, although none of these networks are a perfect fit.