RELATIVISTIC SHOCK BREAKOUTS—A VARIETY OF GAMMA-RAY FLARES: FROM LOW-LUMINOSITY GAMMA-RAY BURSTS TO TYPE Ia SUPERNOVAE

Consider a relativistic blast wave that propagates near a stellar edge pre-explosion mass density gradient of the form ρ∝zⁿ, where z = (R_* − r), with R_* being the stellar radius and r the distance from the star center. In compact stars, where shocks are more likely to become relativistic, the envelope is radiative and typically n ≈ 3, which is the value that we use throughout the paper. The shock acceleration stops when the width of the shock is comparable to z and the shock breaks out. As we show in Appendix A, this takes place in the shell where the pre-explosion optical depth to the stellar edge is τ ∼ 0.01γ³_s or smaller, where γ_s is the shock Lorentz factor. Namely, breakout takes place at z ≲ 0.01γ³_s/(ρκ_T), where κ_T ≈ 0.2 cm² g⁻¹ is the Thomson cross section per unit of mass.³ This sets the maximal Lorentz factor of the shock and we refer to that shell as the outermost shell. Unlike the Newtonian case, the energy released from a relativistic breakout is not dominated by the outermost shell. Instead it is dominated by the shell in which the pre-explosion optical depth is τ ∼ 1 (i.e., z ∼ 1/(ρκ_T)), which we therefore refer to as the breakout shell and denote its properties by the subscript "₀." In the case of properties that evolve with time, this subscript refers to their value in the breakout shell right after it is crossed by the shock. Given the above assumptions the system is completely defined by the following parameters of the breakout shell.

• 1.  
  γ₀: the Lorentz factor of the shock when it crosses the breakout shell.
• 2.  
  m₀: the mass of the breakout shell.
• 3.  
  d₀: the post-shock width (i.e., after compression) of the breakout shell in the lab frame.
• 4.  
  R_*: the radius of the star.
• 5.  
  n: the power law index describing the pre-explosion density profile, here we use n = 3.

Other characteristics of the breakout shell can be calculated from above. For example, the initial thermal energy of the breakout shell is E₀ ≈ γ²₀m₀c².

The pre-breakout and post-breakout dynamics of a planar relativistic blast wave in a decreasing density profile was investigated by Johnson & McKee (1971), Tan et al. (2001), and Pan & Sari (2006). If the density follows a power law then the evolution is self-similar both before and after the breakout. Before the breakout a relativistic shock accelerates as γ_i∝ρ^−μ, where $\mu =\left(\sqrt{3}-\frac{3}{2}\right) \approx 0.23$ and the subscript "_i" indicates initial values (note that ρ is the pre-shock density). This propagation profile sets the entire pre-explosion hydrodynamic profile. Conceptually, it is useful to treat the whole expanding envelope as a series of successive shells. The properties of any shell deeper than the outermost shell can be characterized by its mass, m.

• 1.  
  $\gamma _i=\gamma _0 (m/m_0)^{-\mu n \over n+1} \propto m^{-0.17}$ is the initial Lorentz factor.
• 2.  
  $d_i \approx z/\gamma _i^2 \propto m^\frac{1+2n\mu }{(n+1)} \approx m^{0.6}$ is the width of a shocked shell at the shock breakout, in the lab frame.
• 3.  
  $d_i^{\prime }=d_i \gamma _i \propto m^\frac{1+n\mu }{(n+1)} \approx m^{0.42}$ is the width of a shocked shell at the shock breakout, in the shell rest frame.
• 4.  
  $E_i \approx \gamma _i^2 mc^2 \propto m^{\frac{1+n(1-2\mu)}{n+1}}\approx m^{0.65}$ is the internal energy of a shell at the shock breakout, in the lab frame.
• 5.  
  $t_i \approx \frac{d_i}{c} \gamma _i^2 \propto m^\frac{1}{n+1} \propto m^{0.25}$ is the lab frame time for the shell expansion, which is also the time between the shock crossing and the breakout.

Following breakout all the shells expand and accelerate. The acceleration is faster than that of a freely expanding relativistic shell as outer shells gain energy from inner ones. In this subsection we follow the acceleration of shells with mass m ≳ m₀. These shells remain opaque during the whole planar phase, also after their pairs annihilate. The final Lorentz factor of less massive shells depends on their evolving opacity and is discussed shortly in the next section.

The acceleration during the planar phase, i.e., before a shell doubles its radius, follows (Johnson & McKee 1971; Pan & Sari 2006):

$$\begin{equation} \gamma = \gamma _i \left(\frac{t}{t_i}\right)^\frac{\sqrt{3}-1}{2}\propto m^{-0.27}t^{0.37}, \end{equation} \tag{ 1 }$$

where t is the lab frame time (not to be confused with observer time) measured since the breakout. If the shell is optically thick at the end of acceleration and acceleration ends during the planar phase then the final Lorentz factor is (Johnson & McKee 1971; Tan et al. 2001; Pan & Sari 2006; Oren & Sari 2009)

$$\begin{equation} \gamma _f=\gamma _i^{1+\sqrt{3}}\propto m^{-0.48}. \end{equation} \tag{ 2 }$$

This relation is general as it does not depend on the exact density profile, as long as there is a large energy reservoir behind the accelerating shell. Equations (1) and (2) imply that the acceleration ends at

$$\begin{equation} t_f = t_i \gamma _i^{3+\sqrt{3}}\propto m^{-0.57}. \end{equation} \tag{ 3 }$$

Thus, more massive shells end their acceleration at earlier times and lower Lorentz factors.

In this paper, we restrict our treatment to cases where the breakout shell ends its acceleration during the planar phase, i.e., t_{f, 0} < t_s where

$$\begin{equation} t_s=\frac{R_*}{c} \end{equation} \tag{ 4 }$$

is the lab frame time of transition between the planar and spherical phases. Since pre-shocked optical depth of the breakout shell is of order unity (i.e., κ_Tρ₀z₀ ≈ 1),

$$\begin{equation} \frac{t_0}{t_s}= \frac{z_0}{R_*} \approx \left(\frac{R_*^2}{\kappa _T M_*}\right)^{\frac{1}{n+1}}=3\times 10^{-3} M_{5}^{-0.25} R_{5}^{0.5}, \end{equation} \tag{ 5 }$$

where M_x and R_x are the progenitor star mass and radius in units of x solar masses and radii, respectively. Thus, there is a critical Lorentz factor, γ_{0, s}, for which t_{f, 0} = t_s:

$$\begin{equation} \gamma _{0,s} = 3.5 M_{5}^{0.05} R_{5}^{-0.1}. \end{equation} \tag{ 6 }$$

The corresponding critical final Lorentz factor of the breakout shell is

$$\begin{equation} \gamma _{f,s} = \gamma _{0,s}^{\sqrt{3}+1} = 30 M_{5}^{0.14} R_{5}^{-0.27}. \end{equation} \tag{ 7 }$$

If γ₀ ⩽ γ_{0, s}, then acceleration ends in the planar phase and the corresponding final Lorentz factor is given by Equation (2). Otherwise it ends during the spherical phase and Equation (2) is not valid anymore, a case that we treat elsewhere.

Acceleration continues beyond the breakout shell, up to the outermost shell, which lies at⁴ z ∼ 0.01γ³_i/(ρκ_T). This shell achieves the maximal initial Lorentz factor

$$\begin{equation} \gamma _{{\rm max},i} \approx 1.7 \gamma _0^{0.65}. \end{equation} \tag{ 8 }$$

Note that our calculation is limited to γ₀ < γ_{0, s} ≲ 4.5, where γ₀ < γ_{max, i} and the description above is consistent.

The radiation in the expanding gas is trapped by pairs and γγ annihilation opacity at the time of the breakout. In the previous section, we have shown that for most progenitors in the regime that we consider, the breakout shell becomes transparent after it ended acceleration and before, or soon after, the transition to the spherical phase, i.e., t_{f, 0} < t_{th, 0} ≲ t_s. Therefore we derive below the observed light curve in that case. Note that the breakout shell is not the only shell that becomes transparent during the planar phase. All the lighter shells (m < m₀) out to the outermost shell become transparent during the planar phase as well. Shells that are deeper (and more massive and slower) than the breakout shell remain opaque during the planar phase and become transparent only during the spherical phase. Figure 1 depicts a schematic sketch of the luminosity and temperature evolution during the breakout and planar phase.

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Since slower moving shells carry significantly more energy, E_i∝γ^−3.75_i, the breakout emission is dominated by the energy confined to the breakout shell at t_{th, 0}, when its local frame temperature is T'_th:

$$\begin{equation} E_{{\rm bo}}= E_0 \frac{\gamma _{f,0}}{\gamma _0} \frac{T^{\prime }_{{\rm th}}}{T^{\prime }_0} \approx \frac{1}{4} m_0 c^2 \gamma _0 \gamma _{f,0} \approx 2 \times 10^{45} R_{5}^2 \gamma _{f,0}^\frac{1+\sqrt{3}}{2} {\rm erg}, \end{equation} \tag{ 14 }$$

where we use τ₀ ≈ κ_Tm₀/(4πR²_*) = 1 to find

$$\begin{equation} m_0 \approx 4 \times 10^{-9} M_\odot R_{5}^2. \end{equation} \tag{ 15 }$$

The observed temperature of this emission is

$$\begin{equation} T_{{\rm bo}}=T^{\prime }_{{\rm th}} \gamma _{f,0} \sim 50 \gamma _{f,0} \, {\rm keV}, \end{equation} \tag{ 16 }$$

and it is smeared by light travel time effects over a duration that is comparable to t^obs_s:

$$\begin{equation} t_{{\rm bo}}^{{\rm obs}} \approx t_s^{{\rm obs}} \approx \frac{R_*}{c \gamma _{f,0}^2} \approx 10 \frac{ R_{5}}{\gamma _{f,0}^2} \, {\rm s}, \end{equation} \tag{ 17 }$$

where t^obs_s is the time of transition from the planar to the spherical phase as seen by the observer. Clearly, Equations (15)–(17) are general and are applicable to any gas density profile. In fact, Equation (14) is also applicable to any sharply decreasing density gradient, even if it is not a power law. The reason is that m₀ does not depend on this profile and so does the relation between γ₀ and γ_{f, 0} (Equation (2)), as long as acceleration ends during the planar phase and there is a large energy reservoir behind the breakout shell. Together, Equations (14)–(17) provide three generic observables that depend on only two parameters γ_{f, 0} and R_*. Thus, the energy, temperature, and timescale over constrain R_* and γ_{f, 0}, where any two observables out of these three are enough to determine R_* and γ_{f, 0}. In case that all three are measured, they must satisfy

$$\begin{equation} t_{{\rm bo}}^{{\rm obs}} \sim 20 \, {\rm s} \left(\frac{E_{{\rm bo}}}{10^{46} \,{\rm erg }}\right)^{1/2} \left(\frac{T_{{\rm bo}}}{50 \, {\rm keV}}\right)^{-\frac{9+\sqrt{3}}{4}}, \end{equation} \tag{ 18 }$$

if the source of the flare is a quasi-spherical⁶ relativistic shock breakout. This relativistic breakout relation can be used to test whether any observed gamma-ray flare is consistent with a relativistic breakout origin.

During t < t^obs_bo the observer receives also photons from shells that are outer of the breakout shell, which have higher observed temperature than T_bo. These photons generate a high-energy power law at frequencies T_bo < ν. Shells with γ_i < T'_i/T'_th ≈ 4 obtain their terminal Lorentz factor before they become optically thin, implying that $\gamma _f = \gamma _i^{\sqrt{3}+1}$ and E∝E_iγ_f/γ_i∝γ^−2.01_i, while $T = T^{\prime }_{{\rm th}} \gamma _f \propto \gamma _i^{\sqrt{3}+1}$. Since the emission from each shell is spread over a duration t_obs∝γ⁻²_f, the luminosity from these outer shells dominates at early time: L∝t^−0.65_obs and T∝t^−0.5_obs. When integrating over the entire breakout flare (t < t^obs_bo), the spectrum exhibits a high-energy power law

$$\begin{equation} \nu F_\nu (\nu >T_{{\rm bo}}) \propto \nu ^{-0.74} \end{equation} \tag{ 19 }$$

up to ν = 0.2γ^1.75₀ MeV, which is at most a few times larger than T_bo.

During t_obs < t^obs_bo(= t^obs_s) the observer also receives photons from the breakout shell itself after it becomes transparent. During this phase adiabatic cooling dictates (e.g., Nakar & Sari 2010), T∝t^−1/3_obs and L∝t^−4/3_obs, implying

$$\begin{equation} \nu F_\nu (\nu <T_{{\rm bo}}) \propto \nu. \end{equation} \tag{ 20 }$$

The range of the low-energy power law is rather limited $T_{{\rm bo}}>\nu >T_{{\rm bo}}(c t_{{\rm th},0}/R_*)^{1/3}= 0.5 T_{{\rm bo}} \gamma _0^{1/\sqrt{3}} M_{5}^{-1/12} R_{5}^{1/6}$. Note that Equation (19) and the spectral range where the high- and low-energy power laws are observed do depend on the specific density profile and are given for n = 3.

At t^obs_s the observed luminosity is still dominated by light emitted when the breakout shell becomes transparent, ∼t_{th, 0}. This emission fades quickly until the emission from the spherical phase becomes dominant. The fading light curve can easily be derived in case γ_{f, 0} ≫ 1, since then the observed luminosity is dominated by emission of the breakout shell at large angles (>1/γ_{f, 0}). The luminosity decays then as L∝t⁻²_obs and the temperature as T∝t⁻¹_obs (Kumar & Panaitescu 2000; Nakar & Piran 2003), until the emission from the spherical phase becomes dominant.

During the spherical phase the radius of the expanding sphere is no longer constant. Instead, the radius of the shells is r∝vt and the luminosity is dominated by photons that are leaking from the shell where the optical depth to the observer is ∼c/v. We refer to this shell as the luminosity shell and denote its properties with the superscript " $\, \widehat{ }$  " (see Nakar & Sari 2010 for a detailed discussion), such that, e.g., the luminosity shell satisfies, by definition, $\widehat{\tau }=c/\widehat{v}$. The evolution during the spherical phase depends on whether the luminosity shell is relativistic or not and on the initial temperature of the luminosity shell. As long as $\widehat{\gamma }_f\gg 1$, the initial temperature of the luminosity shell is ∼200 keV and its dynamics is well approximated by the relativistic self-similar solution. Shells with 0.5 < γ_iβ_i < 1 also have an initial temperature of ∼200 keV but their dynamics is better described by the Newtonian approximation, namely, shock propagation according to the self-similar solution of Sakurai (1960) and no significant acceleration after the shock breakout. The transition time between the relativistic and Newtonian phases is calculated below and it takes place around:

$$\begin{equation} t_{{\rm NW}}^{{\rm obs}} = \frac{R_*}{c} \gamma _{f,0}^{1.05}= t_s^{{\rm obs}} \gamma _{f,0}^{3.05} \end{equation} \tag{ 21 }$$

During the Newtonian phase $t_{{\rm obs}} \propto \widehat{v}^{-4}$ (Nakar & Sari 2010). The end of the phase in which the initial temperature of the luminosity shell is constant takes place when $\widehat{v}\approx 0.5$c, which is at t_obs ≈ 10t^obs_NW. Here we present luminosity and temperature evolution during the spherical phase up to that point ($\widehat{v}\approx 0.5$c). At later times the temperature drops quickly until thermal equilibrium is obtained. Therefore, there is a sharp break in the temperature evolution at 10t_NW. The light curve evolution at t > 10t_NW is covered by Nakar & Sari (2010). Figure 1 depicts a schematic sketch of the luminosity and temperature evolution during the spherical phase.

4.2.1. Relativistic Phase (t^obs_s < t_obs < t^obs_NW)

The relativistic velocity of the shells dictates $\widehat{\tau }\sim 1$ and since τ∝m/r² the mass of the luminosity shell is $\widehat{m}\propto t^{2}$. The photons from the luminosity shell arrive to the observer at $t_{{\rm obs}} \approx t/\widehat{\gamma }_f^2$ and their arrival is distributed over a comparable duration. Therefore the luminosity shell satisfies $\widehat{m}\propto t_{{\rm obs}}^{0.69}$, $\widehat{\gamma }_i\propto t_{{\rm obs}}^{-0.12}$, $\widehat{\gamma }_f\propto t_{{\rm obs}}^{-0.33}$, and $\widehat{E}_i^{\prime } \propto \widehat{m}\widehat{\gamma }_i\propto t_{{\rm obs}}^{0.57}$, while $\widehat{r}\propto t \propto t_{{\rm obs}}\widehat{\gamma }_f^2 \propto t_{{\rm obs}}^{0.34}$ and Equation (10) dictates $\widehat{T}^{\prime } \propto t_{{\rm obs}}^{-0.36}$. The observed luminosity and temperature are

$$\begin{equation} \widehat{L}=\frac{\widehat{E}}{t_{{\rm obs}}}=\frac{\widehat{\gamma }_f\widehat{E}_i^{\prime } (\widehat{T}^{\prime }/T_i^{\prime })}{t_{{\rm obs}}} = \frac{E_0 c}{R_*} \left(\frac{t_0}{t_s}\right)^{1/3} \gamma _0^{\frac{10\sqrt{3}+6}{3}} \left(\frac{t_{{\rm obs}}}{t_s^{{\rm obs}}}\right)^{-1.12}, \end{equation} \tag{ 22 }$$

$$\begin{equation} \widehat{T}\approx 200 \,{\rm keV } \left(\frac{t_0}{t_s}\right)^{1/3} \gamma _0^\frac{4\sqrt{3}+3}{3} \left(\frac{t_{{\rm obs}}}{t_s^{{\rm obs}}}\right) ^{-0.68}. \end{equation} \tag{ 23 }$$

The relativistic phase ends once $\widehat{\gamma }_i\approx 1$ at a time given in Equation (30).

4.2.2. Newtonian Phase with Constant Initial Temperature ($t_{{\rm \rm NW}}^{{\rm obs}}<t_{{\rm obs}}<{\it 10}\, t_{{\rm \rm NW}}^{{\rm obs}}$)

The luminosity during this phase is independent of the temperature since pairs are long gone. It is therefore similar to the luminosity evolution of the spherical phase following a Newtonian breakout (e.g., Chevalier 1992), where (for n = 3)

$$\begin{equation} \widehat{L}\propto t_{{\rm obs}}^{-0.35}. \end{equation} \tag{ 24 }$$

Therefore a significant flattening of the luminosity evolution is expected at t_NW. From this point the luminosity decay is steady until recombination and/or radioactivity starts playing a role.

The initial volume of the luminosity shell is proportional to its initial width (all shells start at R_*), $\widehat{d}_i \propto t^{0.44}$. The observed temperature evolves as

$$\begin{equation} \widehat{T}\approx T^{\prime }_i \left(\frac{\widehat{v}^3 t^3}{\widehat{d}_i R_*^2}\right)^{-1/3} \propto t_{{\rm obs}}^{-0.6}. \end{equation} \tag{ 25 }$$

This slope is not very different than that of the relativistic phase. Therefore there is no significant feature in the temperature evolution at t_NW. Only at t_obs ∼ 10t_NW the temperature starts dropping rapidly until the luminosity shell gains thermal equilibrium.

There are several mechanisms that can result in a shock breakout from a white dwarf. The most famous is Type Ia SNe, where the whole star explodes. Other theoretically predicted scenarios are Type .Ia SNe (Shen et al. 2010), where helium accreted in an AM CVn is detonated, and AIC to a neutron star (e.g., Hillebrandt et al. 1984; Fryer et al. 1999; Dessart et al. 2006). All these explosions are expected to produce a rather constant ejecta velocity (2E/M_ej)^1/2 ∼ 10, 000 km s⁻¹, with explosion energy that ranges between 10⁵¹ erg in Type Ia SNe, where the whole star is ejected (M_ej ≈ 1.4 M_☉), and 10⁴⁹–10⁵⁰ erg in the other explosions, where only 1%–10% of the stellar mass is ejected. There are also various observed SNe, which are probably generated by a range of white dwarf explosions. Some possible examples are SN 2005E (M_ej ≈ 0.3 M_☉, E ≈ 4 × 10⁵⁰ erg; Perets et al. 2010), SN2010X (M_ej ≈ 0.16 M_☉, E ≈ 1.7 × 10⁵⁰ erg; Kasliwal et al. 2010), and SN 2002bj (M_ej ∼ 0.2 M_☉, E ∼ 10⁵⁰ erg; Poznanski et al. 2010).

Since E/M_ej is similar for all these explosions and the dependence of γ₀β₀ on E (or M_ej) when this ratio is constant is very weak (∝E^0.17), shock breakouts from all these types of white dwarf explosions produce a similar signature. The breakout Lorentz factor is γ₀β₀ ≈ 1–3 for a white dwarf radius 3–10 × 10⁸ cm. The total energy released in the breakout is 10⁴⁰–10⁴² erg and the typical photon energy is ∼MeV. The breakout pulse is spread over ∼1–30 ms, having a peak luminosity of ∼10⁴⁴ erg s⁻¹. These results for the breakout emission are different, especially in the temperature prediction, than those of Piro et al. (2010b) who found a Type Ia breakout signal that peaks in the X-rays. The reason is that they assume thermal equilibrium behind the shock, which is not a good assumption for Type Ia SNe (Nakar & Sari 2010), and that they ignore relativistic effects.

Note that this signature is expected only if the line of sight to the observer is transparent. In case the explosion is triggered by white dwarf mergers, the optical depth of the debris that surrounds the exploding core can be high and the shock breakout may take place at a much larger radius than the white dwarf radius, carrying out much more energy at a much lower velocity. For example, Fryer et al. (2010) find that a Type Ia SN shock breakout in the case of a double degenerate merger takes place at a radius of ∼10¹³–10¹⁴ cm, where the shock is Newtonian. Thus, a breakout emission from a Type Ia SN can potentially differentiate between single and double degenerate progenitor systems.

A flare of 10⁴¹ erg in MeV γ-rays can easily be detected if it takes place in the Milky Way and possibly also in the Magellanic Clouds. Such detection will provide a constraint on the exact explosion timing as well as on the radius of the exploding white dwarf and explosion energy. It may also shed light on the explosion mechanism. This detection channel is extinction free and may be the main detection channel in case the burst takes place in an obscured Galactic environment. Given the estimated Milky Way rate of Type Ia SNe (∼0.01 yr⁻¹), the chance to detect such a burst in the coming decade is low, but not negligible. The rate of events such as SN 2010X is very hard to constrain observationally, based on current optical surveys. In addition, there are no observational constraints on the rate of events with energy lower than 10⁵⁰ erg, since their optical signatures, where these events are searched for, are too weak and evolve too fast for detection. Similarly there are no tight theoretical constraints on the rates of Type .Ia SNe and AICs. Since the gamma-ray shock breakout signatures of all these events are similar and detectable for any galactic event, it is worth looking for them in the current data of satellites such as BATSE, Konus-Wind, Swift, and Fermi. In fact, Cline et al. (2005) find that the sub-sample of several dozen BATSE and Konus-Wind GRBs, those with a duration shorter than 0.1 s, shows a significant anisotropy with concentration of events in the galactic plane (but not toward the Galactic center). This sample of events may contain some already observed shock breakouts from white dwarfs.

Broad-line Type Ibc SNe are thought to be explosions of Wolf–Rayet progenitors, and in some cases are very energetic (E ≫ 10⁵¹ erg). Some Broad-line Type Ibc SNe are associated with GRBs (these are discussed later), but also those that are not associated with GRBs may produce a detectable γ-ray signal. For example SN 2002ap has a total explosion energy of ∼4 × 10⁵¹ erg and an ejected mass of ∼2.5 M_☉ (Maurer & Mazzali 2010), implying a mildly relativistic breakout with an energy output of ∼5 × 10⁴⁴R^0.7₅ erg within ∼10R₅ s. At a distance of about 7 Mpc, the observed fluence on Earth is ∼10⁻⁷R^0.7₅ erg cm⁻². This fluence is detectable by sensitive gamma-ray detectors even for R_* = R_☉ (although no large field-of-view detector with high sensitivity was operational in 2002). Note, however, that if a dense wind surrounds the progenitor, the flare properties may be very different. The total emitted energy is higher, while the typical photon energy may fall either within or below the observing band of soft γ-ray detectors.

Several types of extremely energetic SNe were detected in recent years. SN 2007bi has shown a typical velocity of 12, 000 km s⁻¹ and >50 M_☉ of ejecta implying E ≳ 10⁵³ erg (Gal-Yam et al. 2009). Most of the optical luminosity observed in this SN is generated by radioactive decay. The radius of the progenitor is unknown, but since there is no trace of either hydrogen or helium its radius is most likely in the range, R_* ∼ 10¹¹–10¹² cm. These values imply γ₀β₀ ≈ 1. Therefore, the breakout is expected to produce a ∼100 keV flash that lasts seconds to tens of seconds and carries an energy of 10⁴⁴–10⁴⁶ erg.

Quimby et al. (2011) report on another type of extremely energetic SNe. They find a typical velocity of 14, 000 km s⁻¹ and >5 M_☉ of ejecta implying E ≳ 10⁵² erg. The observed optical energy in these explosions is not generated by radioactive decay and the possible sources are interaction of the ejecta with matter that moves more slowly or a central engine. In the former case, all the energy is given to the ejecta during the explosion and a bright shock breakout is expected. These explosions also do not show any trace of hydrogen and helium implying that their progenitor is also rather compact. Assuming again R_* ∼ 10¹¹–10¹² cm the signature of the shock breakout from these events is similar to that expected from SN 2007bi.

An energy output of 10⁴⁴–10⁴⁶ in soft gamma rays within a minute is detectable out to a distance of 3–30 Mpc. Therefore the detection of breakouts from such events is likely only if their rate is larger than 10⁴ Gpc⁻³ yr⁻¹. The rate of extremely energetic SNe is unknown, but it is most likely much lower than that since otherwise they should have been detected in optical searches in large numbers. We therefore do not expect current gamma-ray detectors to detect any of these SN types, although nature may surprise us.

The engine of long GRBs is thought to be produced during the collapse of a massive stellar core and thus to be launching relativistic jets into a stellar envelope. Bromberg et al. (2011b) show that in (at least some) low-luminosity GRBs, it is highly unlikely that the jets successfully punch through the star and emerge in order to produce the observed γ-ray emission. In this case, what can be the source of the observed gamma rays?

There are several common features observed in all low-luminosity GRBs. All show 10⁴⁸–10⁵⁰ erg of gamma rays or hard X-rays that are emitted in single pulsed light curves showing spectra that become softer with time (see Kaneko et al. 2007 and references therein). They also show radio afterglows that indicate on a similar energy in mildly relativistic ejecta (Kulkarni et al. 1998; Soderberg et al. 2004, 2006). This energy is only a small fraction of the total energy observed in the associated SNe (10⁵²–10⁵³ erg). The smooth light curves and the energy coupled to a mildly relativistic ejecta motivated several authors to suggest that the origin of this emission is shock breakouts (Kulkarni et al. 1998; Tan et al. 2001; Campana et al. 2006; Waxman et al. 2007; Wang et al. 2007). Later, Katz et al. (2010) showed that the deviation from thermal equilibrium can explain the observed γ-rays, compared with the expected X-rays when equilibrium is wrongly assumed. However, the question of whether shock breakouts are indeed the source of low-luminosity GRBs remained open, mostly since there was no quantitative model that could answer whether shock breakouts can indeed explain even the most basic observables, such as energy, temperature, and timescales, of these bursts. Below, we use our model to answer this question. In Section 4, we have shown that these three observables, E_bo, T_bo, and t^obs_bo, overconstrain the physical parameters since they depend in the case of a quasi-spherical relativistic breakout only on two parameters, γ_{f, 0} and the breakout radius. Thus, in addition to finding the value of these two physical parameters, the three observables must satisfy the relativistic breakout relation (Equation (18)). This can be used as a test that any flare that originates from quasi-spherical relativistic breakout must pass. This result is generic as it is independent of the pre-breakout density profile (as long as it is steeply decreasing). Below we apply this test to the four observed low-luminosity GRBs.

The four well-studied low-luminosity GRBs are divided into two pairs with very different properties. GRBs 060218 and 100316D are soft and very long. Their rather soft spectrum is best fitted at early time with a cutoff around a peak energy of ∼40 keV (Kaneko et al. 2007; Starling et al. 2011) and both emitted a total energy of ∼5 × 10⁴⁹ erg. Plugging these values into Equation (18) we find that if these bursts are flares from quasi-spherical breakouts then they should have durations of ∼1500 s, which is indeed comparable to the observed durations of these two bursts. The breakout radius in this model is ∼5 × 10¹³ cm and γ_{f, 0} ≈ 1 implying γ₀β₀ ≈ 0.3–1. In the case of GRB 060218, radio emission suggests breakout velocities on the upper side of this range (Soderberg et al. 2006). The large radius inferred by the long duration implies that the breakout is most likely not from the edge of the progenitor star, but from an opaque material thrown by the progenitor to large radii (possibly a wind) prior to the explosion (Campana et al. 2006). The fact that a spherical model provides a good explanation implies that the suggested a-sphericity (Waxman et al. 2007), if exists, does not play a major role in determining the burst duration. Note though that it may play an important role in shaping the breakout spectrum above and below T_bo and the entire light curve following the breakout flare.

The other two low-luminosity GRBs, 980425 and 031203, are relatively hard, T ≳ 150 keV, and not particularly long, ≈30 s. Their hard spectrum made it difficult to explain these events as shock breakouts, before it was realized that the radiation will be away from thermal equilibrium.⁹ Plugging 10⁴⁸ erg and 150 keV, the total energy and typical frequency of GRB 980425, respectively, into Equation (18) results in a breakout duration of ∼10 s, comparable to the observed one. The breakout radius in that case is ∼6 × 10¹² cm and the final Lorentz factor is¹⁰ γ_{f, 0} ≈ 3. The associated SN of GRB 980425, SN 1998bw, was very energetic, exhibiting M_ej ≈ 15 M_☉ and a kinetic energy of ≈0.5 × 10⁵³ erg (Iwamoto et al. 1998). Interestingly, setting R_* ∼ 6 × 10¹² and M_ej = 15M_☉ in Equation (26) we find that an explosion energy of E ≈ 3 × 10⁵³ produces a breakout with the observed properties of GRB 980425. This energy is slightly larger than the one seen in the SN 1998bw, but it is close enough to suggest that the energy source of the SN explosion is also the one of the shock breakout. The mild discrepancy between the two energies may be a result of the actual stellar density profile (compared to the power law with n = 3) or of a deviation of the explosion from sphericity. Using the energy and typical frequency of GRB 031203, 5 × 10⁴⁹ erg, and >200 keV, Equation (18) predicts a duration ≲ 35 s. The observed duration, ∼30 s, is consistent with this value and indicates that the actual temperature is not much higher than 200 keV. The breakout radius in that case is ∼2 × 10¹³ cm and the final Lorentz factor is γ_{f, 0} ≈ 5. The SN associated with GRB 031203, SN 2003lw, is similar in ejected mass and slightly more energetic than SN 1998bw (Mazzali et al. 2006). Setting R_* ∼ 2 × 10¹³ and M_ej = 15 M_☉ in Equation (26) we find that an explosion energy of E ≈ 10⁵⁴ produces a breakout with the observed properties of GRB 031203. Again this value is larger by about an order of magnitude compared with the one observed in the associated SN.

The breakout radii that we find for GRBs 980425 and 031203 are larger than those of typical Wolf–Rayet stars, the probable progenitors of GRB–SNe. This implies that either the breakout is not from the edge of the progenitor star (e.g., from a wind) or that low-luminosity GRB progenitors are larger than commonly thought (which may help in choking the relativistic jet; Bromberg et al. 2011d). Finally, a prediction of relativistic shock breakouts is a significant X-ray emission that can be comparable in energy to that of the breakout γ-ray flare. The X-rays are emitted during the spherical phase, over a timescale that is significantly longer than that of the breakout emission. GRB 031203 has shown a bright signal of dust scattered X-rays, which indicates ∼2 keV emission with energy comparable to that of gamma rays, during the first 1000 s after the burst (Vaughan et al. 2004; Feng & Fox 2010). These values fit the predictions of a breakout model. For example, if the breakout is from a stellar surface at R_* = 2 × 10¹³ and M_ej = 15 M_☉, then Equations (30)–(34) predict a release of ∼10⁵⁰ erg at ∼10 keV within the first hour after the burst.

To conclude, we find that the energy, temperature, and timescales of all four low-luminosity GRBs are explained very well by shock breakout emission and that all of them satisfy the relativistic breakout relation between E_bo, T_bo, and t^obs_bo (Equation (18)). These observables are largely independent of the pre-shocked density profile and are therefore applicable also for a breakout from a shell of mass ejected by the progenitor prior to the explosion. Breakout emission can also explain many other observed properties, such as the smooth light curve, the afterglow radio emission, the small fraction of the explosion energy carried by the prompt high-energy emission, and the delayed bright X-ray emission seen in GRB 031203. We therefore find it to be very likely that indeed all low-luminosity GRBs are relativistic shock breakouts.

GRB 101225A is a peculiar explosion showing ∼an hour long smooth burst of gamma rays and X-rays followed by a peculiar IR-UV afterglow, which is consistent with a blackbody emission. Its origin is unclear. Thöne et al. (2011) argue for a cosmological origin and estimate its redshift to be ≈0.3, while Levan & Tanvir (2011) argue that its origin is local. Here we do not attempt to determine which, if any, of the two views is correct. Instead, given the similarity of the high-energy emission of GRB 101225A to some low-luminosity GRBs, we ask whether the shock breakout can be the source of this emission, assuming that the burst is cosmological and that the redshift estimate of Thöne et al. (2011) is correct.

We test whether its observed properties satisfy Equation (18), which with a total energy of ∼10⁵¹ erg and gamma-ray spectrum that peaks at ∼40 keV predicts a duration of ∼10⁴ s. This value is consistent with the constraints on the actual duration of the flare (Thöne et al. 2011). The resulting breakout radius is ∼3 × 10¹⁴ cm. Interestingly Thöne et al. (2011) present a model that attempts to explain the IR-UV afterglow by an expanding opaque shell with velocities at least as high as β ≈ 1/3 and an initial radius of 2 × 10¹⁴ cm. A breakout of a shock that accelerates such a shell produces a high-energy signal that is similar to the observed one. We therefore conclude that if Thöne et al. (2011) redshift estimate is correct, then a relativistic shock breakout probably provides the simplest explanation for the high-energy emission.

In the collapsar model of long GRBs, the relativistic jets pierce through the envelope and emerge in order to produce the observed gamma-ray emission. During their propagation the jets drive mildly relativistic bow shocks into the envelope and inflate a cocoon that collimates the jets to an even narrower angle than their initial launching opening angle (e.g., MacFadyen & Woosley 1999; Matzner 2003; Morsony et al. 2007; Mizuta & Aloy 2009). Recently, Bromberg et al. (2011c) have derived an analytic description of this interaction while the jet is propagating deep in the envelope, before it encounters the sharp density gradient near the stellar edge.

The term shock breakouts from GRBs is used in the literature to describe several different phenomena. Here, similarly to the rest of the paper, we focus on the emission that is escaping from the breakout layer after the forward shock, which is driven by the jet head and the cocoon into the stellar envelope, breaks out. The structure of the shock that is driven into the stellar envelope is highly non-spherical, but at the time of the breakout the size of causally connected regions is small and a local spherical approximation is appropriate. Thus, the theory discussed above, together with the model presented in Bromberg et al. (2011c) and the results seen in numerical simulations (e.g., Mizuta & Aloy 2009), can be used to obtain a rough idea of the expected breakout signal.

We consider a typical GRB jet with a half-opening angle θ₀ = 0.1 rad and a total luminosity of 10⁵⁰ erg s⁻¹. The jet is launched continuously into a 5 M_☉ progenitor with a radius of 5 R_☉. At the point that the jet reaches the steep density drop (i.e., r ∼ R_*/2) its velocity approaches the speed of light and it is narrowly collimated to an angle of ∼θ²₀ (Bromberg et al. 2011c). The opening angle of the cocoon at this point is ≈θ₀. From that point the jet head and the cocoon shock start accelerating toward the stellar edge and to expand sideways. Numerical simulations show that the head is expanding sideways, becoming comparable in size to the cocoon, which does not spread significantly (e.g., Figure 17 in Mizuta & Aloy 2009). Once the fractional distance of the shock to the edge, z/R_*, is comparable to the shock opening angle, in our case ∼θ₀ = 0.1, the shock loses causal connection with the sides and it can be approximated by the spherical solution of an accelerating blast wave (Johnson & McKee 1971). Taking γ_sβ_s(z = 0.1 R_*) ≈ 1 and using the relation γ_sβ_s∝z^−0.69 with Equation (5), we obtain γ₀ ≈ 10. This value is larger than γ_{0, s} so the breakout shell accelerates also after the transition to the spherical phase. We do not provide here the solution to such a case, but the initial energy in the breakout shell is ∼10⁴⁸(R_*/5 R_☉)² erg and its energy increases during acceleration by at most an order of magnitude. Assuming that there is no significant wind surrounding the progenitor, this energy is released by a short, ∼ ms, pulse of >MeV photons.

The main distinguishable feature of the breakout pulse is its harder spectrum and low fluence compared with the total burst emission. The pulse luminosity depends strongly on the progenitor size. If the progenitor is rather large, R_* ≳ 5 R_☉, then the breakout can be detected by Swift and Fermi up to a redshift ∼0.1. If it is much smaller, then the breakout is too faint for detection. Even in case the breakout is detectable, it may be difficult to separate it from the GRB itself. Nevertheless, it is worth looking for an initial hard spike in nearby GRBs. Interestingly, there is a sub-sample of BATSE GRBs that show initial short hard spikes followed by a softer burst (Norris & Bonnell 2006).

Here we derive an analytic description of the transition layer of relativistic radiation-mediated shocks, based on the results of Budnik et al. (2010). We use mostly similar notations to those of Budnik et al. (2010) and carry out the calculation in the shock frame, where the flow is assumed to be in a steady state. We approximate the gas radiation in the transition region as two counter flows. One is a flow of photons with energy ∼m_ec² that are generated in the immediate downstream and flow toward the upstream at a drift velocity ∼c. The other is a gas flow that is moving toward the downstream. This is a flow of the fluid that is shocked, which is moving with a Lorentz factor Γ_u in the far upstream (this is the shock Lorentz factor as seen in the upstream frame) and is slowed down in the transition region to β = 1/3 (i.e., Γ ≈ 1). The downstream moving fluid is composed of protons with a proper density n and pairs¹¹ with a proper density n_±. We are interested in the shape of the transition, i.e., Γ and the pair fraction as a function of optical depth and physical length. Our final goal is to find the shock width in the upstream frame in units of the pre-shock (i.e., with no pairs) upstream Thomson optical depth.

The shock is generated by the interaction of the two counter-streaming flows. Every collision between an upstream moving photon and a downstream moving fluid lepton transfers energy and momentum to the photon and adds the upstream moving photon to the downstream moving fluid. The temperature of the pairs is ∼Γm_ec², as seen in the fluid rest frame. In a steady state the flux of photons that are moving toward the upstream, n_{γ → us}, is equal to the number flux of pairs that are moving toward the downstream (assuming that the latter dominate over the proton number flux). As long as Γ ≫ 1 this steady state equality implies n_{γ → us} ≈ Γn_±. Now, since an upstream moving photon that is scattered is joining the downstream moving fluid, d(Γn_±) = −n_{γ → us}dτ_s, where τ_s is the optical depth for photons moving toward the upstream, defined to be increasing in the upstream direction.¹² Defining x_± ≡ n_±/n we obtain

$$\begin{equation} \frac{dx_{\pm }}{d\tau _s}=-x_{\pm }. \end{equation} \tag{ A1 }$$

Number and energy flux conservation implies (assuming Γ ≫ 1)

$$\begin{equation} n_u \Gamma _u=n \Gamma \end{equation} \tag{ A2 }$$

$$\begin{equation} \Gamma _u^2 n_u mp c^2=\Gamma ^2 n c^2 (m_p+4x_{\pm }\Gamma m_e). \end{equation} \tag{ A3 }$$

In the energy flux equation, we assume that the energy flux of the photon field that flows toward the upstream is negligible, which is true as long as Γ² ≫ 1. We also assume x_± ≫ 1. These equations imply

$$\begin{equation} x_{\pm }=\frac{m_p}{4m_e} \frac{\Gamma _u-\Gamma }{\Gamma ^2}, \end{equation} \tag{ A4 }$$

and therefore

$$\begin{equation} \frac{d\Gamma }{d\tau _s}=\frac{d\Gamma }{dx_{\pm }}\frac{dx_{\pm }}{d\tau _s}=\Gamma \frac{\Gamma _u-\Gamma }{2\Gamma _u-\Gamma }. \end{equation} \tag{ A5 }$$

Define τ_* as the Thomson optical depth of a photon moving toward the upstream then τ_* ≈ τ_s(σ_T/σ_KN(Γ²)), where σ_T is the Thomson cross section and σ_KN(Γ²) is the Klein–Nishina cross section of electron at rest and photon with energy Γ²m_ec². Thus,

$$\begin{equation} d\tau _*=\frac{\sigma _T}{\Gamma \sigma _{{\rm KN}}(\Gamma ^2)} \frac{2\Gamma _u-\Gamma }{\Gamma _u-\Gamma } d\Gamma. \end{equation} \tag{ A6 }$$

Integrating this equation starting at the end of the transition layer, i.e., Γ(τ_s = 0) = 1 results in the structure of the transition layer. Comparison of this structure to the numerical results of Budnik et al. (2010) is presented in Figure 3.

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In order to find the width of the shock in the upstream frame we use the relation

$$\begin{equation} d\tau _*=\Gamma _un_u x_{\pm }\sigma _T dz^{\prime }, \end{equation} \tag{ A7 }$$

where z' is the length coordinate (positive toward the upstream) in the shock frame. Therefore

$$\begin{equation} n_u \sigma _T dz^{\prime }=\frac{4 m_e}{m_p}\frac{2\Gamma _u-\Gamma }{(\Gamma _u-\Gamma)^2\Gamma _u} \frac{\sigma _T}{\sigma _{{\rm KN}}(\Gamma ^2)} \Gamma d\Gamma. \end{equation} \tag{ A8 }$$

Integrating dΓ from the shock toward the upstream, the value of Γ/Γ_u (for values larger than 1/Γ_u) has a universal profile (independent of Γ_u) as a function of n_uσ_Tz'(1 + 2ln [Γ²_u])/Γ²_u. This profile is presented in Figure 4. It implies that the shock width in units of pair unloaded Thomson optical depth in the shock frame is roughly,¹³ ∝Γ²_u. The value of the proportionality coefficient is somewhat arbitrary and it depends on the value of Γ/Γ_u that defines the "shock width." We chose Γ/Γ_u = 0.9, obtaining n_uσ_Tz' ≈ 0.01Γ²_u (the value of the logarithmic factor is 5 − 10 for Γ_u = 2 − 10). Choosing a different value of Γ/Γ_u, to define the shock width, changes the coefficient, which in turn has a very weak effect on Equation (8). Now, since the length scale in the upstream frame, z, is shorter by a factor Γ_u than in the shock frame (i.e., z' = Γ_uz), and defining τ_u = n_uσ_Tz, we approximate the shock width in units of Thomson optical depth of the pre-shock upstream (i.e., with no pairs) as

$$\begin{equation} \Delta \tau _u \sim 0.01 \Gamma _u. \end{equation} \tag{ A9 }$$

Note that our analytic description is applicable to shocks where Γ²_u ≫ 1 and is therefore not directly applicable to mildly relativistic shocks. We do expect, however, that pair creation in these shocks, which increases the optical depth per proton in the immediate downstream by a factor of the order of m_p/m_e, will result in Δτ_u ≪ 1 and that Equation (A9) can be extrapolated to the mildly relativistic regime.

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When deriving Equation (A9) we assumed a steady state. A relativistic shock with Lorentz factor Γ_u and a width Δz (as seen in the upstream frame) contains gas collected over a length of ΔzΓ²_u. Since near the edge of the star the density varies over a scale of the order of distance to the edge, z, the assumption of a steady state is valid as long as z > ΔzΓ²_u or τ > Δτ_uΓ²_u ∼ 0.01Γ³_u, where τ is the pre-shock Thomson optical depth to the stellar edge. As the shock propagates farther, the assumption of steady state breaks down and its structure must be modified. The shock may still be confined during this phase, which we do not attempt to address here. Our result shows that the shock must be confined to the star at least up to optical depth as low as τ ∼ 0.01Γ³_u.