Planar Shock Breakout Cooling

• Planar shock breakout cooling is the early radiative phase following a radiation-mediated shock’s emergence from a thin breakout layer, where the optical depth meets τ ~ c/v_shock.
• The phenomenon is characterized by universal analytic solutions using parameters like v0, ρ0, and t0, which underpin the t^(-4/3) decline in the luminosity during the cooling phase.
• Its spectral and light curve evolution, projected from a plane-parallel approximation to spherical geometry, enables probing progenitor properties even when asymmetry or circumstellar effects are present.

Searching arXiv for the cited shock breakout papers to ground the article in the literature. arxiv_search.query({"^^^^^^^^9search_query9 radiation mediated shock breakouts\"^^^^^^^^9 planar shock breakout cooling Sapir Katz Waxman9^^^^^^^^", "^^^^^^^^9start9 ^^^^9search_query9^^^^, "^^^^^^^^9max_results9 ^^^^9all:\9search_query9^^^^}) arxiv_search.query({"^^^^^^^^9search_query9 shock breakout spectra\" OR ^^^^^^^^9all:\9 breakout theory\" OR ^^^^^^^^9all:\9 Breakout in ^^^^9start9^^^^-Dimensional Red Supergiant Envelopes\"", "^^^^^^^^9start9 ^^^^9search_query9^^^^, "^^^^^^^^9max_results9 ^^^^9all:\9search_query9^^^^}) arxiv_search.query({"^^^^^^^^9search_query9 OR id:(&&&^^^^9all:\9^^^^&&&) OR id:(&&&^^^^9 planar shock breakout cooling Sapir Katz Waxman9^^^^&&&) OR id:(&&&^^^^9start9^^^^&&&) OR id:(&&&^^^^9max_results9^^^^&&&)", "^^^^^^^^9start9 ^^^^9search_query9^^^^, "^^^^^^^^9max_results9 ^^^^9all:\9search_query9^^^^}) Planar shock breakout cooling is the early radiative phase that follows the emergence of a non-relativistic radiation-mediated shock from a stellar surface or, more generally, from the edge of an optically thick medium whose breakout layer is geometrically thin compared with its radius. In this regime the local flow is treated as plane-parallel, breakout occurs when the optical depth ahead of the shock satisfies PRESERVED_PLACEHOLDER_^^^^9search_query9^^^^, and the post-breakout bolometric luminosity enters a characteristic cooling decline that is asymptotically PRESERVED_PLACEHOLDER_^^^^9all:\9^^^^ under the standard diffusion, constant-opacity, radiation-dominated assumptions. The modern formulation is anchored by the exact planar breakout solutions of Sapir, Katz, and Waxman and their spherical projection to observable supernova light curves, with subsequent extensions to spectral classification, three-dimensional structure, circumstellar breakout, and opacity-dependent spectral formation (&&&^^^^9all:\9^^^^&&&, &&&^^^^9search_query9^^^^&&&, &&&^^^^9max_results9^^^^&&&, &&&^^^^9 planar shock breakout cooling Sapir Katz Waxman9^^^^&&&).

^^^^9all:\9^^^^. Physical definition and regime of validity

Planar shock breakout cooling refers to the phase in which the emitting layer has expanded by much less than the characteristic radius of the system, so curvature is negligible and each emitting patch can be approximated as a slab. In the standard stellar-surface problem, the preshock density near the surface is taken to follow a power law, PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman9^^^^, the flow is non-relativistic, radiation pressure dominates, transport is handled by diffusion with constant opacity, and breakout is triggered once the remaining optical depth is comparable to the radiation-mediated shock width, PRESERVED_PLACEHOLDER_^^^^9start9^^^^. The approximation is accurate when the scale height of the outer envelope is much smaller than the stellar radius and when the planar breakout duration is much shorter than the time on which spherical expansion becomes dynamically important (&&&^^^^9all:\9^^^^&&&, &&&^^^^9search_query9^^^^&&&).

In the spherical projection used for observable light curves, four approximations are central: spherical symmetry in the global geometry, constant Thomson scattering opacity PRESERVED_PLACEHOLDER_^^^^9max_results9^^^^, a steady-state planar angular intensity at the surface, and a small scale-height condition. The surface angular distribution is taken from the steady-state Thomson-scattering solution of Chandrasekhar and is well approximated by

PRESERVED_PLACEHOLDER_^^^^9search_query9^^^^

with normalization PRESERVED_PLACEHOLDER_^^^^9all:\9^^^^. These assumptions are explicitly intended for progenitors whose outer breakout layer is thin compared with PRESERVED_PLACEHOLDER_^^^^9 OR all:\9^^^^ (&&&^^^^9search_query9^^^^&&&).

A recurring distinction in the literature is between the planar phase and the later spherical cooling-envelope phase. In the planar phase, the breakout shell remains the luminosity-determining shell and the luminosity falls steeply; after the ejecta expands by a factor of order unity, the flow becomes spherical, the luminosity shell moves inward in mass coordinate, and the temporal decline is shallower. The transition is therefore dynamical as well as geometric, rather than merely a change of notation (&&&^^^^9all:\9 planar shock breakout cooling Sapir Katz Waxman9^^^^&&&, &&&^^^^9max_results9^^^^&&&).

^^^^9 planar shock breakout cooling Sapir Katz Waxman9^^^^. Universal planar dynamics and bolometric scalings

The exact non-relativistic planar solution is formulated in terms of the breakout shock velocity PRESERVED_PLACEHOLDER_^^^^9 OR all:\9^^^^, the preshock density at breakout $\rho_0$, and the opacity PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9^^^^. The natural scales are

PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9^^^^

and, in the notation of the exact planar problem,

PRESERVED_PLACEHOLDER_^^^^9all:\9 planar shock breakout cooling Sapir Katz Waxman9^^^^

with PRESERVED_PLACEHOLDER_^^^^9all:\9start9^^^^. In these variables the solution is universal for fixed density index PRESERVED_PLACEHOLDER_^^^^9all:\9max_results9^^^^: apart from weak PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9^^^^-dependence, the dimensionless hydrodynamic and radiative profiles do not depend on the absolute physical scale of the progenitor (&&&^^^^9all:\9^^^^&&&).

The universal planar luminosity rises before formal breakout because radiation diffuses ahead of the shock, peaks slightly before the Sakurai breakout time, and then relaxes to a cooling tail. The robust asymptotic result is

PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9^^^^

while the special case PRESERVED_PLACEHOLDER_^^^^9all:\9 OR all:\9^^^^ gives PRESERVED_PLACEHOLDER_^^^^9all:\9 OR all:\9^^^^. The light-curve shape depends only weakly on PRESERVED_PLACEHOLDER_^^^^9all:\99^^^^: for PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman9search_query9^^^^ to PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman9all:\9^^^^, the luminosity differs by less than about PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman9 planar shock breakout cooling Sapir Katz Waxman9^^^^ over the interval containing most of the energy. For the commonly used cases, the peak properties are explicitly tabulated: for PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman9start9^^^^, PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman9max_results9^^^^, PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman9search_query9^^^^, and PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman9all:\9^^^^; for PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman9 OR all:\9^^^^, PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman9 OR all:\9^^^^, PRESERVED_PLACEHOLDER_^^^^9 planar shock breakout cooling Sapir Katz Waxman99^^^^, and PRESERVED_PLACEHOLDER_^^^^9start9search_query9^^^^ (&&&^^^^9all:\9^^^^&&&).

A central exact relation links the escaping flux to the acceleration of the surface: PRESERVED_PLACEHOLDER_^^^^9start9all:\9^^^^ As a result, the asymptotic surface velocity is determined by the radiative fluence. In the spherical projection this gives two especially robust outputs: PRESERVED_PLACEHOLDER_^^^^9start9 planar shock breakout cooling Sapir Katz Waxman9^^^^ both accurate to about PRESERVED_PLACEHOLDER_^^^^9start9start9^^^^ and largely insensitive to the detailed density profile, moderate asymmetry in arrival times, or modest changes in the surface angular intensity (&&&^^^^9all:\9^^^^&&&, &&&^^^^9search_query9^^^^&&&).

^^^^9start9^^^^. Projection to observable light curves

The observable bolometric breakout signal is not the local planar flux itself but the surface emission convolved with both angular intensity and geometric time delay across the stellar disk. In the standard formulation,

PRESERVED_PLACEHOLDER_^^^^9start9max_results9^^^^

where PRESERVED_PLACEHOLDER_^^^^9start9search_query9^^^^, PRESERVED_PLACEHOLDER_^^^^9start9all:\9^^^^ is the limb-darkened intensity profile, and the delay PRESERVED_PLACEHOLDER_^^^^9start9 OR all:\9^^^^ accounts for finite light-travel time across the visible hemisphere. With the universal planar light curve written as PRESERVED_PLACEHOLDER_^^^^9start9 OR all:\9^^^^ and PRESERVED_PLACEHOLDER_^^^^9start99^^^^, the observable bolometric history is determined by PRESERVED_PLACEHOLDER_^^^^9max_results9search_query9^^^^, PRESERVED_PLACEHOLDER_^^^^9max_results9all:\9^^^^, PRESERVED_PLACEHOLDER_^^^^9max_results9 planar shock breakout cooling Sapir Katz Waxman9^^^^, and PRESERVED_PLACEHOLDER_^^^^9max_results9start9^^^^, plus only a weak dependence on PRESERVED_PLACEHOLDER_^^^^9max_results9max_results9^^^^ (&&&^^^^9search_query9^^^^&&&).

The planar approximation remains valid until spherical expansion appreciably reduces the optical depth of the emitting layers. The transition time is estimated as

PRESERVED_PLACEHOLDER_^^^^9max_results9search_query9^^^^

so the exact projected planar light curves are applicable for PRESERVED_PLACEHOLDER_^^^^9max_results9all:\9^^^^. A particularly important regime is

PRESERVED_PLACEHOLDER_^^^^9max_results9 OR all:\9^^^^

which is valid for most progenitors with roughly PRESERVED_PLACEHOLDER_^^^^9max_results9 OR all:\9^^^^. In that limit the intrinsic planar pulse is brief, the observed light curve is dominated by light-travel-time smearing, and analytic expressions simplify substantially (&&&^^^^9search_query9^^^^&&&).

For PRESERVED_PLACEHOLDER_^^^^9max_results99^^^^, relevant to large red supergiants, the late planar tail persists observationally over

PRESERVED_PLACEHOLDER_^^^^9search_query9search_query9^^^^

with

PRESERVED_PLACEHOLDER_^^^^9search_query9all:\9^^^^

In the same regime the luminosity can be inverted to estimate the progenitor radius,

PRESERVED_PLACEHOLDER_^^^^9search_query9 planar shock breakout cooling Sapir Katz Waxman9^^^^

Moderately asymmetric explosions enter through a spread in shock arrival times, PRESERVED_PLACEHOLDER_^^^^9search_query9start9^^^^, which modifies the early observed light curve but does not change PRESERVED_PLACEHOLDER_^^^^9search_query9max_results9^^^^ or PRESERVED_PLACEHOLDER_^^^^9search_query9search_query9^^^^ (&&&^^^^9search_query9^^^^&&&).

^^^^9max_results9^^^^. Thermalization, spectral formation, and the planar spectrum

The bolometric planar solution does not by itself fix the observed temperature or spectral shape. A separate thermalization problem determines whether the radiation remains in equilibrium. In the analytic treatment of Nakar and Sari, the luminosity shell is defined by PRESERVED_PLACEHOLDER_^^^^9search_query9all:\9^^^^, the color shell is where the observed photon energy is set, and the thermal coupling parameter

PRESERVED_PLACEHOLDER_^^^^9search_query9 OR all:\9^^^^

distinguishes equilibrium from non-equilibrium emission. If PRESERVED_PLACEHOLDER_^^^^9search_query9 OR all:\9^^^^, the radiation is thermalized; if PRESERVED_PLACEHOLDER_^^^^9search_query99^^^^, photon production is insufficient and the observed temperature exceeds the blackbody temperature. During the planar phase, when the breakout shell itself controls the luminosity, the bolometric luminosity declines as

PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9^^^^

while the observed temperature can evolve quite differently depending on whether the breakout shell is initially thermalized (&&&^^^^9all:\9 planar shock breakout cooling Sapir Katz Waxman9^^^^&&&).

The spectral classification of the planar phase can be organized by three timescales: the breakout-shell diffusion time

PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9^^^^

the light-crossing time

PRESERVED_PLACEHOLDER_^^^^9all:\9 planar shock breakout cooling Sapir Katz Waxman9^^^^

and the thermal-equilibrium reveal time PRESERVED_PLACEHOLDER_^^^^9all:\9start9^^^^. Because PRESERVED_PLACEHOLDER_^^^^9all:\9max_results9^^^^, there are five allowed orderings and therefore five breakout scenarios. At any given time the spectrum is one of five types: a smeared blackbody, a smeared free-free spectrum, a mixed smeared free-free plus blackbody spectrum, an unsmeared self-absorbed free-free spectrum, or a blackbody. Before PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9^^^^ the spectrum is a Comptonized free-free spectrum; after PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9^^^^ a true blackbody component from thermalized material becomes visible. Once PRESERVED_PLACEHOLDER_^^^^9all:\9 OR all:\9^^^^, the flash is over and the later planar phase follows the familiar cooling scalings PRESERVED_PLACEHOLDER_^^^^9all:\9 OR all:\9^^^^ and PRESERVED_PLACEHOLDER_^^^^9all:\99^^^^ if thermalized layers dominate (&&&^^^^9 planar shock breakout cooling Sapir Katz Waxman9^^^^&&&).

Planar spectral calculations with local Compton equilibrium and bremsstrahlung photon production show that the surface temperature is determined primarily by PRESERVED_PLACEHOLDER_^^^^9 OR all:\9search_query9^^^^ and PRESERVED_PLACEHOLDER_^^^^9 OR all:\9all:\9^^^^ and only weakly by PRESERVED_PLACEHOLDER_^^^^9 OR all:\9 planar shock breakout cooling Sapir Katz Waxman9^^^^. For hydrogen-helium envelopes, the peak surface temperature is fit by

PRESERVED_PLACEHOLDER_^^^^9 OR all:\9start9^^^^

with more complete fits including a weak density dependence. The time-integrated emitted spectrum is a particularly robust prediction: it depends on PRESERVED_PLACEHOLDER_^^^^9 OR all:\9max_results9^^^^ and PRESERVED_PLACEHOLDER_^^^^9 OR all:\9search_query9^^^^ alone, is relatively insensitive to light-travel-time smearing and slight deviations from spherical symmetry, and peaks at

PRESERVED_PLACEHOLDER_^^^^9 OR all:\9all:\9^^^^

in the paper’s notation. This robustness is one reason breakout fluence spectra are often treated as cleaner diagnostics than instantaneous spectra (&&&^^^^9 planar shock breakout cooling Sapir Katz Waxman9start9^^^^&&&).

^^^^9search_query9^^^^. Departures from the idealized planar picture

The idealized planar solution is local. Observable breakout signals can depart from the textbook form because different surface patches need not reach breakout simultaneously. In the analytic spherical projection, a spread in shock arrival times PRESERVED_PLACEHOLDER_^^^^9 OR all:\9 OR all:\9^^^^ changes the early-time shape but leaves PRESERVED_PLACEHOLDER_^^^^9 OR all:\9 OR all:\9^^^^ and PRESERVED_PLACEHOLDER_^^^^9 OR all:\99^^^^ essentially unchanged. This already implies that the early light curve is more sensitive to angular and temporal smearing than the integrated energetics are (&&&^^^^9search_query9^^^^&&&).

Three-dimensional radiation-hydrodynamic calculations of red-supergiant envelopes make this point explicit. In the Athena++ calculations, the outer envelope contains a low-density halo and large-scale density fluctuations, so breakout occurs at lower densities than in one-dimensional models and at different radii at different times. Local diffusion times are PRESERVED_PLACEHOLDER_^^^^9 OR all:\9search_query9^^^^–PRESERVED_PLACEHOLDER_^^^^9 OR all:\9all:\9^^^^ hr, but the dominant global timescale is the shock traversal across the corrugated breakout surface,

PRESERVED_PLACEHOLDER_^^^^9 OR all:\9 planar shock breakout cooling Sapir Katz Waxman9^^^^

with PRESERVED_PLACEHOLDER_^^^^9 OR all:\9start9^^^^ and PRESERVED_PLACEHOLDER_^^^^9 OR all:\9max_results9^^^^–PRESERVED_PLACEHOLDER_^^^^9 OR all:\9search_query9^^^^ hr. Measured breakout durations are PRESERVED_PLACEHOLDER_^^^^9 OR all:\9all:\9^^^^–PRESERVED_PLACEHOLDER_^^^^9 OR all:\9 OR all:\9^^^^ hr in ^^^^9start9^^^^D, versus PRESERVED_PLACEHOLDER_^^^^9 OR all:\9 OR all:\9^^^^–PRESERVED_PLACEHOLDER_^^^^9 OR all:\99^^^^ hr in ^^^^9all:\9^^^^D, and the longer duration lowers the predicted luminosity by a factor of $\rho_0$^^^^9search_query9^^^^–ρ0\rho_0^^^^9all:\9^^^^ to $\rho_0$^^^^9 planar shock breakout cooling Sapir Katz Waxman9^^^^. The post-breakout decline of each local patch remains approximately planar, $\rho_0$^^^^9start9^^^^, but the observed light curve is a superposition of many patches at different stages of breakout and cooling. A common misconception is therefore that the observed rise time directly measures $\rho_0$^^^^9max_results9^^^^; in these ^^^^9start9^^^^D models, it does not (&&&^^^^9start9^^^^&&&).

A complementary ^^^^9start9^^^^D core-collapse simulation of a $\rho_0$^^^^9search_query9^^^^ Type II-P progenitor finds shock breakout at $\rho_0$^^^^9all:\9^^^^ s in the southern hemisphere, $\rho_0$^^^^9 OR all:\9^^^^ s in the equatorial direction, and $\rho_0$^^^^9 OR all:\9^^^^ s in the northern hemisphere, so the angular spread in breakout time is almost a full day. The paper explicitly states that this would smear out the initial breakout flash, but it also explicitly states that the shock-breakout light curves themselves are not calculated before homology is reached. The early post-breakout evolution is instead treated hydrodynamically, with the thermal energy assumed to decline as $\rho_0$9 in nearly adiabatic homologous expansion. This sharpens the distinction between hydrodynamic breakout structure and the dedicated planar radiative-cooling calculation (&&&^^^^9 planar shock breakout cooling Sapir Katz Waxman9all:\9^^^^&&&).

Circumstellar material can change the problem qualitatively. If the optical depth of the CSM satisfies PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9search_query9^^^^, breakout occurs in the CSM rather than at the stellar surface, at a larger radius and with durations that can extend to days. In a dense shell with a sharp outer edge, the early evolution is still planar because the shocked layer is thin, but radiative losses during the planar phase become more important than in the stellar-envelope case. The analytic CSM solution retains the classic PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9all:\9^^^^ limit when the shell is optically thick, but if PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9 planar shock breakout cooling Sapir Katz Waxman9^^^^ is only marginally above PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9start9^^^^ the decline steepens and the total luminosity can show an intermediate-time behavior roughly PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9max_results9^^^^ because multiple diffusion eigenmodes contribute (&&&^^^^9max_results9^^^^&&&, &&&^^^^9 planar shock breakout cooling Sapir Katz Waxman9 OR all:\9^^^^&&&).

^^^^9all:\9^^^^. Opacity revisions, observational diagnostics, and interpretive use

A major revision of the classical spectral picture comes from frequency-dependent opacity with bound-free and bound-bound contributions from heavy elements. In fast Newtonian planar breakouts with PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9search_query9^^^^, adding TOPS opacity rather than assuming a fully ionized free-free medium increases photon production, helps the radiation maintain LTE to higher velocities, and can reduce the emission temperature by half and even an order of magnitude in the planar shock-breakout problem. In the representative case PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9all:\9^^^^ and PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9 OR all:\9^^^^, the characteristic breakout temperature is around PRESERVED_PLACEHOLDER_^^^^9all:\9search_query9 OR all:\9^^^^ eV in the free-free-only case and around PRESERVED_PLACEHOLDER_^^^^9all:\9search_query99^^^^ eV with TOPS opacity. For red-supergiant envelope breakout, the paper argues that the SED will very likely remain in LTE without stellar wind, implying much weaker X-ray emission than earlier simplified predictions (&&&^^^^9 planar shock breakout cooling Sapir Katz Waxman99^^^^&&&).

Observationally, planar shock breakout cooling is valuable because the early signal depends mainly on a small set of breakout parameters. The rise time of the bolometric luminosity measures PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9search_query9^^^^, the total duration measures PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9all:\9^^^^, the appearance of a blackbody component constrains PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9 planar shock breakout cooling Sapir Katz Waxman9^^^^, the total radiated energy constrains PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9start9^^^^, and the low-frequency turnover probes the self-absorption frequency. In scenarios with PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9max_results9^^^^, measuring the rise time, pulse duration, and total radiated energy allows direct estimates of PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9search_query9^^^^, PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9all:\9^^^^, and PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9 OR all:\9^^^^. UV coverage is especially important because that is often where the transition from free-free to thermal emission is most diagnostic; facilities specifically highlighted for this purpose are ULTRASAT, UVEX, and Einstein Probe (&&&^^^^9 planar shock breakout cooling Sapir Katz Waxman9^^^^&&&).

The practical astrophysical role of planar shock breakout cooling is therefore twofold. First, it provides the local building block for realistic breakout calculations in stars, winds, and compact dense shells. Second, it supplies diagnostic relations that are unusually insensitive to uncertain details: the bolometric light curve is weakly dependent on the density profile, the integrated breakout energy and maximum ejecta velocity are robust, and the late planar PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9 OR all:\9^^^^ decline can constrain radius when geometric smearing and strong asphericity are under control. At the same time, the literature shows that these same diagnostics can be degraded or reinterpreted by asymmetry, three-dimensional envelope structure, dense CSM, and realistic high-ionization opacity. Planar shock breakout cooling is thus best understood not as a complete description of early supernova light, but as the universal local regime from which more global and more realistic breakout phenomenology is constructed (&&&^^^^9all:\9^^^^&&&, &&&^^^^9search_query9^^^^&&&, &&&^^^^9start9^^^^&&&, &&&^^^^9 planar shock breakout cooling Sapir Katz Waxman99^^^^&&&).