FAST RADIATION MEDIATED SHOCKS AND SUPERNOVA SHOCK BREAKOUTS

The energy and momentum flux are dominated by the protons' momentum and kinetic energy in the upstream and by the photons in the far downstream. Proton number, momentum, and energy conservation require that

$$\begin{eqnarray} &n_u\beta _s=n_d\beta _d,\cr &p_{\gamma,d}=n_u\beta _s(\beta _s-\beta _d)m_pc^2,\cr &4p_{\gamma,d}\beta _d=n_u\beta _s\big(\beta _s^2-\beta _d^2\big)m_pc^2/2, \end{eqnarray} \tag{ 3 }$$

where the subscript d denotes the downstream values of the parameters and we used the fact that the energy density in photons is e_γ = 3p_γ. Equations (3) imply β_d = β_s/7. Thermal equilibrium implies

$$\begin{equation} e_{\gamma,d}=a_{{\rm BB}}T_d^4. \end{equation} \tag{ 4 }$$

The far downstream temperature is thus

$$\begin{eqnarray} &T_d = \left(\frac{315}{4\pi ^2}\,n_u\,\varepsilon \,\hbar ^3\,c^3\right)^{1/4}\approx \,0.16 (\,\varepsilon _{1}\,n_{u,15})^{1/4} \mbox{ keV}\cr &\approx 0.13 \,n_{u,15}^{1/4}\,\beta _{s,-1}^{1/2}\,\mbox{ keV}, \end{eqnarray} \tag{ 5 }$$

where n_u = 10¹⁵ n_u,15, ε = 0.5 β²_s m_p c² = 10 ε₁MeV, and β_s = 0.1β_s,−1.

Next, we estimate the proton deceleration length scale. Once a proton reaches a point in the shock where the energy is dominated by photons, it experiences an effective force

$$\begin{equation} \beta _s\frac{d\beta }{dx}m_p\, c^2\sim \sigma _T\,\beta _s e_\gamma \sim \sigma _T\, n_u\,\beta _s^3\,m_pc^2, \end{equation} \tag{ 6 }$$

implying a deceleration length of

$$\begin{equation} L_{{\rm dec}}\equiv \beta _s \left(\frac{d\beta }{dx}\right)^{-1}\sim \frac{1}{\sigma _Tn_u\beta _s}. \end{equation} \tag{ 7 }$$

We note that once the photons dominate the pressure, they cannot drift with the protons, as this will imply an energy flux greater than the total energy flux. Thus, the drag estimated in Equation (6) is unavoidable. The energy and momentum of the protons are thus transferred to the photons on a length scale L_dec irrespective of the details of the mechanisms that generate the energy in the radiation field. We note that the transition length, L_dec, is of the order of the distance a photon can diffuse upstream before being advected with the flow. The deceleration of the flow can be solved analytically (e.g., Blandford & Payne 1981, and references therein; see Equation (A6)). For the strong shocks considered here,

$$\begin{equation} x=\frac{1}{21\sigma _Tn_u\beta _s}\ln \left[\frac{(\beta _s-\beta)^7}{(7\beta -\beta _s)\beta _s^6}\right], \end{equation} \tag{ 8 }$$

where x is the distance measured in the shock frame and the point x = 0 corresponds to β/β_d − 1 ≈ 0.25.

The region of the shock profile over which the temperature changes before it reaches T_d can be extended to distances that are much larger than L_dec.

To see this, consider the length scale that is required to generate the density of photons of energy ∼T_d in the downstream, determined by thermal equilibrium, n_γ,eq ≈ p_γ,d/T_d,

$$\begin{equation} L_{T}\sim \beta c\frac{n_{\gamma,{\rm eq}}}{Q_{\gamma,\rm eff}}, \end{equation} \tag{ 9 }$$

where Q_γ,eff is the effective generation rate of photons of energy 3T_d. We use here the term "effective generation rate" due to the following important point. Photons that are produced by the emission mechanism at energies ≪T_d may still be counted as contributing to the production of photons at T_d, since they may be upscattered by inverse-Compton collisions with the hot electrons to energy ∼T_d on a timescale shorter than that of the passage of the flow through the thermalization length, L_T/β_dc. Note that the relevant Compton y parameter,

$$\begin{eqnarray} y&=&\frac{4T}{m_ec^2}\,L_T\,n_d\,\sigma _T\,\beta _d^{-1}=49\frac{4T}{m_ec^2}(L_T\,\beta _s\,n_u\,\sigma _T)\beta _s^{-2}\cr &&\sim 4 (L_T\,\beta _s\,n_u\,\sigma _T)\frac{T}{100 \;{\rm eV}}\,\beta _{s,-1}^{-2}, \end{eqnarray} \tag{ 10 }$$

is much larger than unity for L_Tβ_sn_uσ_T ≫ 1/4(T/100 eV)⁻¹β²_s,−1. Q_γ,eff includes, therefore, all photons produced down to an energy that allows them to be upscattered to T_d. For bremsstrahlung emission, which we assume to be the main source of photons, the number of photons generated diverges logarithmically at low energy, so that Q_γ,eff may be significantly larger than the bremsstrahlung generation rate of photons at T_d.

In order for the photon energy to significantly increase by scattering, it must be scattered ∼m_ec²/(4T) times before getting re-absorbed. This sets a lower limit to the frequency of photons that should be included in Q_γ,eff,

$$\begin{eqnarray} h\nu _{a}&=&m_ec^2\left(\frac{T}{m_e c^2}\right)^{-5/4}\left(\frac{\alpha _e g_{{\rm ff}} (m_ec^2)^{-3}h^3 c^3 n}{32\pi }\right)^{1/2}\cr &&\sim 60g_{{\rm ff}}^{1/2}\left(\frac{T}{100\;{\rm eV}}\right)^{-5/4}n_{15}^{1/2}\;\mbox{eV}, \end{eqnarray} \tag{ 11 }$$

where n = 7n_u = 7 × 10¹⁵n₁₅cm⁻³. The generation rate should include all the photons that are generated with energy above hν_a and are able to be upscattered during the available time L_T/(βc). The bremsstrahlung effective photon generation rate is thus given by

$$\begin{equation} Q_{\gamma,{\rm eff}}=\alpha _e n_pn_e\sigma _T c\sqrt{\frac{m_ec^2}{T}}\Lambda _{{\rm eff}}g_{{\rm eff}}, \end{equation} \tag{ 12 }$$

where g_eff is the Gaunt factor and Λ_eff ∼ log(T/(hν_a)). As long as the Compton y parameter is large, y>log(T/hν_a), all photons produced above ν_a are upscattered to ∼3T energies.

Using Equations (12) and (9), we obtain

$$\begin{eqnarray} L_T\beta _sn_u\sigma _T&\sim \frac{1}{100\alpha _e\Lambda _{{\rm eff}}g_{{\rm eff}}}\frac{\varepsilon ^2}{\sqrt{m_e c^2 T}m_pc^2}\cr &\approx 16 \varepsilon _{1}^{15/8}n_{15}^{-1/8}\Lambda _{{\rm eff}}^{-1}g_{{\rm eff}}^{-1}. \end{eqnarray} \tag{ 13 }$$

This implies that for large shock velocities,

$$\begin{equation} \beta _s>0.07 n_{15}^{1/30}(\Lambda _{{\rm eff}}g_{{\rm eff}})^{4/15}, \end{equation} \tag{ 14 }$$

the length required to produce the downstream photon density is much larger than the deceleration scale. For lower shock velocities, thermal equilibrium is approximately maintained throughout the shock.

The photons that stop the plasma at the deceleration region must be generated within a distance ∼(n_uβ_sσ_T)⁻¹ of the deceleration region in order to be able to reach it (before being swept downstream by the flow). Consider the following simple model for the first few diffusion lengths downstream of the deceleration region. We assume that the velocity is β_d downstream from some point x = x₀ and is much larger upstream of it. As the e-folding distance of the residual velocity δβ = β − β_d is very short, =β_sn_uσ_T/21, this is a good approximation. We neglect any generation of photons upstream of the transition, $Q_{\gamma,\rm eff}(x<x_0)=0$. This is reasonable as the contribution of photons from a slab, one diffusion length wide, is roughly proportional to β⁻³ and β changes by a factor of 7 along the transition region. We further assume that the generation rate of photons downstream of the transition is roughly constant and equal to $Q_{\gamma,\rm eff}(x=x_0)$. The latter simplification is justified, to an accuracy of a few tens of percent, due to the fact that the temperature changes by order unity across the available (diffusion length) distance and Q_eff ∝ T^0.5. For such a flow, the density of effective photons at x = x₀ is

$$\begin{equation} n_\gamma (x_0)=\frac{1}{\beta _d^2 c^2}Q_{\gamma,\rm eff}\frac{c}{3 n_d \sigma _T}. \end{equation} \tag{ 15 }$$

As the flow at this point is close to the downstream velocity, we have n_γ(x₀)T_s = p_γ,d, where T_s ≡ T(x₀) is the temperature in the immediate downstream. Using Equations (12) and (15), the momentum conservation equation can be written as

$$\begin{equation} \frac{1}{3}\alpha _e\Lambda _{{\rm eff}}g_{{\rm eff}}\sqrt{m_e c^2 T_s}n_d\beta _d^{-2}=\frac{6}{7}n_u\beta _s^2m_pc^2, \end{equation} \tag{ 16 }$$

where Λ_eff = 10Λ_eff,1 ∼ log(T_s/(hν_a)) with hν_a given by Equation (11). Here we assumed that the temperature does not change much within a distance of L_dec downstream of the point x = x₀. The Compton y parameter, relevant for the upscattering of the photons that diffuse up to the deceleration region,

$$\begin{eqnarray} &y=\frac{4T_s}{m_ec^2}\beta _d^{-2}\sim 400 \left(\frac{T_s}{10\mbox{ keV}}\right)\beta _{s,-1}^{-2}, \end{eqnarray} \tag{ 17 }$$

is much larger than 1 for T_s ≫ 30β²_s,−1 eV. We, therefore, assume that all photons above hν_a are upscattered all the way to T_s.

Solving Equation (16) for β_s, we find

$$\begin{eqnarray} \beta _s&=\frac{7}{\sqrt{3}}\left(\frac{1}{2}\alpha _e \Lambda _{{\rm eff}}g_{{\rm eff}}\right)^{1/4}\left(\frac{m_e}{m_p}\right)^{1/4}\left(\frac{T_s}{m_ec^2}\right)^{1/8}\cr &\approx 0.2\Lambda _{{\rm eff},1}^{1/4}\left(\frac{g_{{\rm eff}}}{2}\right)^{1/4}\left(\frac{T_s}{10\mbox{ keV}}\right)^{1/8}. \end{eqnarray} \tag{ 18 }$$

Equation (18) is in good agreement with the results of Weaver (1976) for T_s < 50keV. For β_s>0.07 and n_u = 10¹⁵–10²²cm⁻³, the temperature (velocity) given by Equation (18) for a given velocity (temperature) agrees with the numerical calculations to within a factor of 1.5 (20%; with the gaunt factor calculated as in Weaver 1976). We note that the scaling with parameters in Equation (18) can be found by writing the RMS equations in dimensionless form (see Equations (A11)–(A16)).

Comparing Equation (18) with the equation for the far downstream temperature T_d, Equation (5), we see that the shock temperature is much larger than the downstream temperature for shock temperatures T_s ≳ 1keV corresponding to shock velocities β_s ≳ 0.1 (see also Weaver 1976).

Note that for temperatures T ≳ 1keV the typical photons in the immediate downstream have energies hν ∼ 3T higher than the maximal energies attainable by bulk Compton acceleration in non-relativistic flows, given by Equation (2). Indeed, the typical photon energies, as derived from Equation (18), rise much faster with velocity than hν_max ∝ β²_s, and at T ∼ 10keV, corresponding to β_s ≈ 0.2, are already much higher (hν ∼ 3T ∼ 30keV compared to hν_max ∼ 4keV). At temperatures exceeding T ∼ 50keV, where pair production becomes important, Equation (16) is not applicable anymore. However, the velocity required for bulk Comptonization to be able to accelerate to the typical photon energies at those temperatures, hν ∼ 150keV, approaches c. Thus, for non-relativistic and mildly relativistic shock velocities with β_s ≳ 0.1, bulk photon acceleration cannot reach energies that are considerably higher than the typical "thermally" Comptonized photon energies in the immediate downstream.

We can broadly divide the shock into four separate regions.

• 1.  
  Near upstream: a few diffusion lengths, (β_sσ_Tn_u)⁻¹, upstream of the deceleration region. In this region, characterized by velocities that are close to the upstream velocity, β ≈ β_s, and temperatures, T ≫ T_u, the temperature changes from T_u to ∼T_s. It ends when the fractional velocity decrease becomes significant.
• 2.  
  Deceleration region: a (β_sσ_Tn_u)⁻¹ wide region where the velocity changes from β_u to β_d and the temperature is roughly constant, T ≃ T_s.
• 3.  
  Immediate downstream: roughly a diffusion length, (β_sσ_Tn_u)⁻¹, downstream of the deceleration region. In this region, characterized by velocities close to the downstream velocity, β ≈ β_d, and temperature, T ∼ T_s, the photons that stop the incoming plasma are generated. Upstream of this region β>β_d and the photon generation rate is negligible. Photons that are generated downstream of this region are not able to propagate up to the transition region.
• 4.  
  Intermediate downstream: the region in the downstream where most of the far downstream photons are generated and T changes from T_s to T_d. This region has a width L_T given by Equation (13), much greater than (β_sσ_Tn_u)⁻¹. Thus, diffusion within this region can be neglected. The temperature profile is expected to follow T ∝ x⁻². To see this, note that the photon density at a distance x from the shock is proportional to the integral of the photon generation, n_γ ∝ T^−1/2x. Since the photon pressure equals the downstream pressure, we have n_γ ∝ T⁻¹ and T ∝ x⁻² (this is valid for a constant value of Λ_effg_eff and is somewhat shallower in reality). Using this dependence of the temperature on distance, we find that L_Tβ_sn_uσ_T ∼ Λ_effg_eff|_dT^−1/2_d((Λ_effg_eff)|_sT^−1/2_s)⁻¹, in agreement with Equations (13) and (18).

As an illustration, the profiles of the velocity and temperature of a non-relativistic RMS (β_s = 0.25, n_u = 10¹⁵cm⁻³) are shown in Figure 1. A description of a novel method used to find such solutions is described in the Appendix. As can be seen, the analytic estimates are in good agreement with the numerical calculation.

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The analysis presented above is valid for temperatures up to about T_s ∼ 30keV, where a number of effects that were not taken into account begin to affect the temperature profile. The main process that should be taken into consideration is the creation of electron–positron pairs at T ≳ 50keV. Additional corrections include relativistic corrections to the bremsstrahlung emission rate, inclusion of photon creation by double Compton scattering, Klein–Nishina (K-N) corrections to the Compton cross section, reduction of the Compton y parameter due to the smaller optical depth, τ ∼ β⁻²_d, and the anisotropy due to the high velocities, β_s ∼ 1.

The following important characteristic of the shock structure is likely unchanged: the plasma decelerates due to scattering with photon and pairs that are produced in the first β⁻¹_d optical depths of the downstream (a discussion of the deceleration velocity profile, neglecting K-N corrections, is given in Levinson & Bromberg (2008); note that K-N corrections and significant electron–positron production significantly affect the result; Budnik et al. 2010). As the downstream velocity is β_d < 1/3 for all β_s, the diffusion approximation in the downstream is reasonable and an equation similar to Equation (16), with the relativistic corrections included, can be used to constrain the temperature in the transition region.

Note that Weaver (1976) claimed that once a significant amount of pairs is created, the pairs will "short out" the electrostatic field which is responsible for the deceleration of the protons. He claimed that the electrons and ions will no longer be fully coupled, and suggested that once there are many pairs per proton, there will be a relative drift of order β_s between the leptons and protons, and that the main mechanism for stopping the protons will be Coulomb collisions with the leptons. We find this picture to be unrealistic in real plasmas. A relatively weak magnetic field, which will be freezed into the pair plasma, will suffice to deflect the protons on scales which are much shorter than determined by Coulomb collisions. Several length scales may be important here. The gyroradius of the protons in a magnetic field B = 10²B₂G is

$$\begin{equation} R_L=\gamma \beta \frac{m_pc^2}{eB}\sim 3\times 10^4 \gamma \beta B_2^{-1}\mbox{ cm}, \end{equation} \tag{ 19 }$$

the skin depth of the pair plasma is

$$\begin{equation} l_{{\rm sd}}=\sqrt{\frac{m_ec^2}{4\pi e^2n_l}}\sim 1.7\times 10^{-2}n_{l,15}^{-1/2}\mbox{ cm}, \end{equation} \tag{ 20 }$$

and the Compton mean free path is

$$\begin{equation} l_{T}=\frac{1}{n_l\sigma _T}\sim 1.5\times 10^9 n_{l,15}^{-1}\mbox{ cm}. \end{equation} \tag{ 21 }$$

As long as the Compton mean free path is much larger than the other two scales, it is probably safe to assume that the protons and electrons are highly coupled on the dynamical scales. We thus assume in what follows that collective plasma effects will keep the pair plasma and the protons coupled without the need for an electrostatic field.

As long as the Compton y parameter is much larger than unity, the spectrum will be close to a Wein spectrum. For the entire parameter space considered here, the positron number is in equilibrium. To see this, note that the typical time a positron takes to annihilate is similar to the typical time a photon takes to Compton scatter, thus pairs can be annihilated (or generated) in β_d Compton optical depths compared with the β⁻¹_d available.

A detailed discussion of the emission of plasmas in pair and Compton equilibrium is given by Svensson (1984), who shows that the main emission processes are bremsstrahlung and double Compton, with a domination of bremsstrahlung at temperatures above T ≈ 60keV.

A precise determination of the relation between T_s and β_u is complicated in the range T_s ≳ 30keV due to the relativistic corrections to the emission rates and due to the order-unity y parameter. Such determination requires a careful analysis of these effects. We avoid such a detailed analysis by limiting our analysis to a demonstration of the validity of the following statements, which set stringent constraints on the relation between T_s and β_d.

• 1.  
  Temperatures T_s>30keV are reached for downstream velocities β_d>0.03, with β_d ≈ 0.03 for T_s ≈ 30keV (as shown in Section 2).
• 2.  
  At β_d = 0.1, the shock transition temperature satisfies 60keV < T_s.
• 3.  
  T_s ≲ 200keV for all the possible downstream velocities, β_d < 1/3.

The validity of statements 2 and 3 is demonstrated in Sections 3.3 and 3.4 below by comparing the photon-to-pair ratio that is determined by the balance of photon production and diffusion, Equation (15), with the ratio that is determined by pair production equilibrium. The complications due to the various corrections are avoided since, under the conditions we consider, the following hold: the number of pairs greatly exceeds the number of protons, n_l ≫ n_p, where n_l = n₊ + n₋ ≈ 2n₋; double Compton emission is negligible; the Compton y parameter is large.

Assuming pair production domination and neglecting double Compton emission, Equation (15) can be written as

$$\begin{equation} \frac{n_\gamma }{n_l}=\frac{1}{3}\alpha _e\Lambda _{{\rm eff}} \bar{g}_{{\rm eff,rel}}(\hat{T})\beta _d^{-2}, \end{equation} \tag{ 22 }$$

where we wrote the free–free emission in the form

$$\begin{equation} Q_{\gamma,{\rm eff}}=\alpha _e \sigma _T c n_l^2 \Lambda _{{\rm eff}}\bar{g}_{{\rm eff,rel}}(\hat{T}). \end{equation} \tag{ 23 }$$

Here, $\hat{T}\equiv T/(m_ec^2)$, and $\bar{g}_{{\rm eff,rel}}$ is the total Gaunt factor (defined by Equation (23)) including all lepton–lepton bremsstrahlung emission. For 10 < Λ_eff < 20 and 60keV < T < m_ec², the approximation

$$\begin{equation} \bar{g}_{{\rm eff,rel}}\approx \Lambda _{{\rm eff}}/2 \end{equation} \tag{ 24 }$$

agrees with the results of Svensson (1984) to an accuracy of better than 25%. Substituting the value of Equation (24) into Equation (22), we find

$$\begin{equation} \frac{n_\gamma }{n_l}\approx 10 \left(\frac{\Lambda _{{\rm eff}}}{10}\right)^2\beta _{d,-1}^{-2}. \end{equation} \tag{ 25 }$$

We next estimate the absorption frequency. The absorption is dominated by e + e− bremsstrahlung and for T ≳ 0.1m_ec² photons require a single scattering to change their energy considerably. The absorption frequency is thus roughly equal to the threshold frequency ν_C=B at which the Compton scattering rate equals the absorption scattering rate and is given by

$$\begin{eqnarray} h\nu _{a}\sim h\nu _{C=B}&=& m_ec^2\left(\frac{T}{m_e c^2}\right)^{-3/4}\left(\frac{\alpha _e g_{{\rm ff}} (m_ec^2)^{-3}h^3 c^3 n_l}{8\pi }\right)^{1/2}\cr &&\sim 0.6 \!\left(\frac{g_{{\rm ll,ff}}}{10}\right)^{1/2}\!\!\left(\frac{T}{50\mbox{ keV}}\right)^{-3/4}\!\!n_{l,18}^{1/2}\;\mbox{eV}, \end{eqnarray} \tag{ 26 }$$

where n_l = 10¹⁸n_l,18. The lepton number density n_l is roughly given by

$$\begin{eqnarray} \hspace{-15pt}n_l&=&\left(\frac{n_l}{n_\gamma }\right)n_\gamma \sim \left(\frac{n_l}{n_\gamma }\right)\frac{m_p\beta _s^2c^2}{3T}n_u \sim 0.6\times 10^{18} \cr && \times\left(\frac{10n_l}{n_\gamma }\right)\left(\frac{T}{50\mbox{ keV}}\right)^{-1}\beta _s^2 n_{15}\mbox{ cm}^{-3}, \end{eqnarray} \tag{ 27 }$$

yielding

$$\begin{equation} \Lambda _a=\ln \left(\frac{T}{h\nu _a}\right)\approx 11-\frac{1}{2} \ln (n_{15}), \end{equation} \tag{ 28 }$$

where we used g_ff,ll ∼ 10, n_l/n_γ ∼ 10, and T ∼ 50keV (the result is not sensitive to the specific values adopted for these parameters).

The Compton y parameter is

$$\begin{equation} y=(4\hat{T}+16\hat{T}^2)\beta _d^{-2}> 40 (T/50\mbox{ keV}) \beta _{d,-1}^{-2}. \end{equation} \tag{ 29 }$$

Equations (28) and (29) imply that for T ∼ 0.1m_ec² the Compton y parameter is large enough, y>Λ_a, to allow all the photons above the absorption frequency, hν_a, to be Comptonized (photons are upscattered by a factor of e^y in energy moving all photons above hν_a to the Comptonized energies $3T=3e^{\Lambda _a}h\nu _a$). This implies, in turn, that photon production at all frequencies above hν_a contributes to the effective generation rate and Λ_eff = Λ_a.

Let us now compare the photon–lepton ratio derived for the balance of photon production and diffusion, to the ratio derived from pair production equilibrium. For large chemical potential, μ_ch ≫ T, pair production equilibrium gives

$$\begin{equation} \frac{n_{l}}{n_\gamma }\Big |_{{\rm eq}}=\int _0^{\infty }dxx^2e^{-\sqrt{x^2+\hat{T}^{-2}}}=\frac{K_2(\hat{T}^{-1})}{\hat{T}^2}, \end{equation} \tag{ 30 }$$

where K₂ is the modified Bessel function of the second kind of order 2. At the temperature range 60keV < T < 90keV, this ratio changes by an order of magnitude: n_γ/n_l|_eq(T = 60keV) ≈ 130 while n_γ/n_l|_eq(T = 90keV) ≈ 13. Comparing these values to the result given by Equation (25), we conclude that at β_d = 0.1 the temperature must be above 60keV.

In the regime 200keV < T < m_ec², the pair production equilibrium is approximately given by

$$\begin{equation} n_\gamma /n_l\approx 0.5 m_ec^2/T. \end{equation} \tag{ 31 }$$

The number of leptons is now given by (see Equation (27))

$$\begin{eqnarray} n_l&\sim \left(\frac{n_l}{n_\gamma }\right)\frac{\Gamma _s^2 m_p c^2}{3T}n_u\cr &\sim 0.6\times 10^{18}\left(\frac{n_l}{n_\gamma }\right)\left(\frac{T}{m_e c^2}\right)^{-1}\Gamma _s^2 n_{15}\mbox{ cm}^{-3}, \end{eqnarray} \tag{ 32 }$$

and the absorption photon energy by

$$\begin{eqnarray} &&h\nu _{a}\sim 0.1 \left(\frac{g_{{\rm ll,ff}}}{10}\right)^{1/2}\left(\frac{T}{m_ec^2}\right)^{-3/4}n_{l,18}^{1/2}\;\mbox{eV}, \end{eqnarray} \tag{ 33 }$$

yielding

$$\begin{equation} \Lambda _a=\ln \left(\frac{T}{h\nu _a}\right)\approx 15-\frac{1}{2} \ln (n_{15})-\ln (\Gamma _s), \end{equation} \tag{ 34 }$$

where we used g_ff,ll ∼ 10, n_l/n_γ ∼ 1, and T ∼ m_ec² (the result is not sensitive to the specific values adopted for these parameters).

We can rewrite Equation (25) for β_d ≈ 1/3 and 60keV < T < m_ec² as

$$\begin{equation} \frac{n_\gamma }{n_l}\approx 2.5 \left(\frac{\Lambda _{{\rm eff}}}{15}\right)^2(3\beta _{d})^{-2}. \end{equation} \tag{ 35 }$$

Comparing Equations (35) and (31), we see that if T ≳ 200keV there would be too many photons generated per lepton. At these high temperatures, the Compton y parameter is large and radiative Compton emission is negligible. At temperatures T ≳ 100keV, K-N corrections to the Compton cross section of photons with energy hν ∼ 3T become significant. The possible effect that K-N corrections have is to lower the temperature due to the larger photon generation depth. This in turn would suppress the K-N corrections. Hence, K-N corrections cannot change the conclusion that T ≲ 200keV. We conclude that for β_d>0.1, 60keV < T ≲ 200keV.

Consider a shock with velocity β_s ≳ 0.1 approaching the photosphere of a spherical mass distribution. The breakout will occur once the optical depth down to the photosphere is τ ∼ β⁻¹ ≲ 10.

The energy carried by the photons in a shell of shocked material with a width of the order of the shock width is roughly given by (e.g., Waxman et al. 2007)

$$\begin{equation} \delta E_{\gamma }= f_E\delta m \beta _s^2c^2/2, \end{equation} \tag{ 36 }$$

where δm is the mass of the shell and f_E is a coefficient that depends on the velocity and geometry. For a shocked shell in the downstream of a non-relativistic planar shock, f_E = 36/49. f_E is of order unity for non-relativistic and mildly relativistic shocks in spherical geometry, as long as the distance to the photosphere is not much larger than the radius. The mass of a shell that has an optical depth τ is roughly given by

$$\begin{equation} \delta m\sim 4\pi R^2\tau \kappa ^{-1}\approx 10^{26} R_{12}^2 (\kappa /\kappa _T)^{-1}\tau _{0.5} \mbox{g}, \end{equation} \tag{ 37 }$$

where κ_T = σ_T/m_p ∼ 0.4cm²g⁻¹, τ = 10^0.5τ_0.5, and R = 10¹²R₁₂cm. Assuming that the energy of shocked plasma with optical depth τ ∼ 3τ_0.5 is emitted, we find that there is a simple relation between the released energy, breakout radius, and shock velocity at breakout:

$$\begin{equation} \delta E \sim 0.5\delta m (\beta _s c)^2\sim 4\times 10^{46} \beta _s^2R_{12}^2 (\kappa /\kappa _T)^{-1}\tau _{0.5}\,\mbox{ erg}. \end{equation} \tag{ 38 }$$

Once a significant fraction of the energy in the shock region is emitted, the steady-state shock solution is no longer applicable. Note that before the solution becomes invalid, a significant fraction of the energy in the shock region must be emitted. We thus expect that photons with energies of the order of 3T_s carrying an energy similar to the energy estimated in Equation (38) are inevitably emitted.

In this subsection, we discuss the dynamics of shock acceleration near the stellar surface, and its evolution within the wind, and derive characteristic breakout velocities. Many of the relations derived can be found in the literature (e.g., Matzner & McKee 1999).

At the outer layers of stars, where the enclosed mass, M, is approximately constant, M = M_* = const, and the composition is approximately uniform, the density distribution is a power law in the distance from the edge of the star,

$$\begin{equation} \rho =\rho _1 \delta ^{n}, \end{equation} \tag{ 39 }$$

where R_* is the stellar radius and

$$\begin{equation} \delta =(R_*-r)/r. \end{equation} \tag{ 40 }$$

This relation is exact if we assume M = M_* and a polytropic equation of state $p=k\rho ^{\gamma _n}$, in which case n = (γ_n − 1)⁻¹ and

$$\begin{equation} \rho _1=\left(\frac{\gamma _n-1}{k\gamma }\frac{{\it GM}}{R_*}\right)^{n}. \end{equation} \tag{ 41 }$$

For an efficiently convective envelope, the entropy is constant. Assuming an adiabatic index γ_n = 5/3, we have n = 3/2. For a radiative envelope, with radius-independent luminosity L = L_* ≪ L_Edd and mass M = M_*, we have p_γ = (L_*/L_Edd)p, where L_Edd = 4πGMc/κ is the Eddington luminosity limit, resulting in a polytropic equation of state p ∝ ρ^4/3, or n = 3. The optical depth from the edge of the star is given by

$$\begin{equation} \tau =\int _r^{R_*}\rho \kappa dr\propto \delta ^{n+1}. \end{equation} \tag{ 42 }$$

The density at the outer edge is given by

$$\begin{equation} \rho =(\rho _1)^{\frac{1}{n+1}}\left(\frac{(n+1)\tau }{\kappa R_*}\right)^{n/(n+1)}. \end{equation} \tag{ 43 }$$

Denoting

$$\begin{equation} \rho _1=f_\rho \frac{M_{{\rm ej}}}{\frac{4\pi }{3}R_*^3}, \end{equation} \tag{ 44 }$$

where M_ej is the mass of the ejected envelope, we find

$$\begin{equation} \rho \approx 2.0\times 10^{-9}f_{\rho }^{1/4}(M_{{\rm ej}}/M_\odot)^{1/4}(\kappa /\kappa _T)^{-3/4}R_{12}^{-3/2}\tau _{0.5}^{3/4}\;\mbox{g}\mbox{ cm}^{-3} \end{equation} \tag{ 45 }$$

for n = 3, and

$$\begin{equation} \rho \approx 2.8\times 10^{-10}f_{\rho }^{2/5}(M_{{\rm ej}}/M_\odot)^{2/5}(\kappa /\kappa _T)^{-3/5}R_{13}^{-9/5}\tau _{0.5}^{3/5}\;\mbox{g}\mbox{ cm}^{-3} \end{equation} \tag{ 46 }$$

for n = 3/2.

As the shock wave approaches the stellar surface, it accelerates down the density gradient. In planar geometry, with density vanishing at x = x₀ following ρ ∝ ((x₀ − x)/x₀)ⁿ = δⁿ, the flow approaches a self-similar behavior as the shock velocity diverges (Gandel'Man & Frank-Kamenetskii 1956; Sakurai 1960). In this limit

$$\begin{equation} v_s\propto \delta ^{-\beta _1n}\propto \tau ^{-\lambda _{v}}, \end{equation} \tag{ 47 }$$

where λ_v = β₁n/(n + 1) and β₁ ≃ 0.19 for γ = 4/3 and n = 3, 3/2. The planar self-similar solution provides a good approximation for the shock behavior near the surface of the star, where the thickness of the layer lying ahead of the shock is small compared to the radius, i.e., for δ ≪ 1 (e.g., Matzner & McKee 1999). Denoting the mass M_* − M(r) of the shell lying ahead of radius r, at δ < (R_* − r)/R_*, by m = M_* − M(r), and noting that near the surface the optical depth τ is τ ∝ m, Equation (47) may be written as

$$\begin{equation} v_s\propto m^{-\lambda _v}, \end{equation} \tag{ 48 }$$

with λ_v = β₁n/(n + 1). The kinetic energy carried by the material having velocity greater than some v_s scales with the velocity as

$$\begin{equation} E(>v_s)\propto v_s^{-\delta _{v}}, \end{equation} \tag{ 49 }$$

where

$$\begin{equation} \delta _{v}=\frac{1+\frac{1}{n}-2\beta _1}{\beta _1}\approx 5, 7 \end{equation} \tag{ 50 }$$

for n = 3 and n = 3/2, respectively.

The velocity of the shock at an optical depth τ is larger than the typical bulk envelope velocity, $v_{\rm ej,b}\approx \sqrt{E_{\rm ej}/M_{\rm ej}}$, where E_ej is the energy deposited in the envelope, by a factor

$$\begin{equation} A_{v_s}\approx \left(\frac{M_{{\rm Ses}}\kappa }{4\pi R_*^2}\tau \right)^{\lambda _{v}}, \end{equation} \tag{ 51 }$$

where M_Ses is the mass of the region in which the planar solution is applicable. For n = 3 we have λ_v ≈ 0.14, while for n = 3/2 we have λ_v ≈ 0.11. The velocity amplification factor for n = 3 is

$$\begin{equation} A_{v_s}\approx 11 (M_{{\rm Ses}}/M_\odot)^{0.14}(\kappa /\kappa _T)^{0.14}R_{12}^{-0.28}\tau _{0.5}^{-0.14}, \end{equation} \tag{ 52 }$$

and for n = 3/2 it is

$$\begin{equation} A_{v_s}\approx 4 (M_{{\rm Ses}}/M_\odot)^{0.11}(\kappa /\kappa _T)^{0.11}R_{13}^{-0.23}\tau _{0.5}^{-0.11}. \end{equation} \tag{ 53 }$$

The density and velocity profiles of the outer envelope are illustrated in Figure 2.

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If there is an optically thick wind surrounding the star, breakout occurs within the wind. Assuming a wind with a constant mass-loss rate $\dot{M}$ and velocity v_w, $\rho _w=\dot{M}/4\pi v_{w}R^{2}$, the optical depth between R and infinity is related to the mass m_w(<R) by τ(>R) = m_w(<R)κ/4πR² (for R ≫ R_*), and breakout is expected to occur at a radius

$$\begin{eqnarray} &&R_{{\rm br}}=\frac{\dot{M}}{v_w}\frac{\kappa }{4\pi \tau } \sim 0.6\times 10^{11}\left(\frac{\dot{M}_{-5}}{v_{w,8}}\right)(\kappa /\kappa _T)\tau _{0.5}^{-1}\mbox{ cm}.\nonumber\\ \end{eqnarray} \tag{ 54 }$$

At early time, the velocity v_s,w of the shock driven into the wind at radius R is approximately given by the velocity of the fastest shell of the ejecta. Since the final velocity reached by a fluid element shocked at v_s, following its acceleration due to the post-shock adiabatic expansion, is v_f ≃ 2v_s (Matzner & McKee 1999), the velocity of the shock driven into the wind is initially $\approx 2 A_{v_s}v_{{\rm ej}}$. At later time, when the accumulated wind mass begins to decelerate the ejecta, the shock velocity is approximately obtained by equating the mass of the shocked wind, m_w(<R), with the mass of the part of the ejecta that was accelerated to velocity v>v_s,w. Using Equation (48) we have $m_w(<R)\approx M_{\rm ej}(v_{s,w}/2v_{\rm ej})^{-\lambda _v}$ or $v_{s,w}/v_{\rm ej}\approx 2(M_{\rm ej}/m_w(<R))^{\lambda _v}$. This gives

$$\begin{equation} \frac{v_{s,w}}{v_{\rm ej}}=\min \left[2A_{v_s}, 30\left(\frac{M_{\rm ej}}{M_\odot }\right)^{1/7}\left(\frac{\dot{M}_{-5}}{v_{w,8}}\right)^{-1/7}R_{12}^{-1/7} \right] \end{equation} \tag{ 55 }$$

for BSG (n = 3) and

$$\begin{equation} \hspace{-15pt}\frac{v_{s,w}}{v_{\rm ej}}=\min \left[2A_{v_s}, 13\left(\frac{M_{\rm ej}}{M_\odot }\right)^{1/9}\left(\frac{\dot{M}_{-5}}{v_{w,8}}\right)^{-1/9}R_{13}^{-1/7} \right] \end{equation} \tag{ 56 }$$

for RSG (n = 3/2).

The assumption that the shock velocity in the wind is equal to the velocity of the shocked ejecta is correct as long as the density encountered by the reverse shock in the ejecta is much larger than the density encountered by the forward shock in the wind.

Assuming typical bulk envelope velocities of $v_{{\rm ej,b}}\approx \sqrt{E_{\rm ej}/M_{\rm ej}}\sim (3\hbox{--}10)\times 10^8\mbox{ cm}\;{\rm s}^{-1}$, or β_ej,b ∼ 0.01–0.03, the typical shock velocities at the last few optical depths of BSGs and W-R stars may reach values of β_s ≳ 0.2, for which temperatures exceeding T ∼ 10keV may be reached in the shock transition region. The shock velocities in the wind may be up to twice higher than the maximal shock velocity achieved within the star.

Let us consider the radiation that may be emitted during the breakout of a shock from the stellar surface. We consider the emission during a time interval R_*/c following the time at which the shock reached the surface (as long as the star is not resolved, any emission will be spread in time over this timescale). On similar timescales, the outer part of the star expands significantly, over a distance β_fR_* ∼ R_*, where β_f ≈ 2β_s is the final velocity reached by a fluid element shocked at β_s (following its acceleration due to the post-shock adiabatic expansion, e.g., Matzner & McKee 1999). On the same timescale, R_*/c, photons that originated at an optical depth τ_esc ∼ δ^−1/2(τ = 1) are capable of escaping. Using Equation (42), we have

$$\begin{equation} \delta (\tau =1)\sim \left(\frac{M_{{\rm ej}}\kappa }{4\pi R_*^2}\right)^{-\frac{1}{n+1}}. \end{equation} \tag{ 57 }$$

This implies

$$\begin{equation} \tau _{{\rm esc}}\sim 10(M_{{\rm ej}}/M_\odot)^{1/8}(\kappa /\kappa _T)^{1/8}R_{12}^{-1/4} \end{equation} \tag{ 58 }$$

for BSG typical parameters, and

$$\begin{equation} \tau _{{\rm esc}}\sim 15(M_{{\rm ej}}/M_\odot)^{1/5}(\kappa /\kappa _T)^{1/5}R_{13}^{-2/5} \end{equation} \tag{ 59 }$$

for RSG parameters. For 1 ≳ β_s ≳ 0.1, these values are not much larger than the shock widths, β⁻¹_s, and a non-negligible fraction, β⁻¹_s/τ_esc, of the photons in these regions is likely to escape over ∼R_*/c. The amount of energy emitted during R_*/c is thus not much larger than the estimate given in Equation (38).

Note that the radiation observed at a given time is the sum of radiation emitted from different positions of the star at different stages of the breakout (up to differences of R_*/c). Photons that originated from shocked material in deeper layers, τ ≫ β⁻¹_s, will have characteristic energies which are lower than the photon energies at the transition region, hν ∼ 3T_s, due to photon production and possible adiabatic cooling. Therefore, we expect to see a spectrum which is a sum of Wein-like spectra with temperatures reaching the tens of keV shock temperatures T_s and starting from lower energies.

Let us consider next breakouts that take place within an optically thick wind. In this case, the optical depth of the shocked wind material, which is compressed to a thin shell, is comparable to the optical depth of the wind ahead of it. Thus, we expect all photons contained within this region to escape on a timescale similar to R/v_s, the timescale on which the optical depth changes significantly. Again, the spectrum would be composed of Wein-like spectra with temperatures reaching T_s.

Finally, the following point should be stressed. As the shock approaches an optical depth of 1/β from the surface, the shock structure deviates from the steady-state solution due to the escape of energy from the system and due to the steep density gradient. Note that at this stage the shock propagates within a density profile which varies on a length scale comparable to the shock thickness. As for the effect of energy escape, a significant fraction of the energy in the shock transition region must be emitted before the structure is changed and thus the steady-state estimate for the photon energies will correctly describe the photons carrying most of the escaping energy. It is reasonable to expect that the effect of the density gradient on the temperature will be of order unity for fast shocks, as the pre-shock density across a shock width changes only by a factor of few—by ∼1/β in the case of stellar surface (see Equation (45)) and by ∼1 in the case of a wind. An exact calculation of the emerging spectrum requires a time-dependent calculation and is beyond the scope of this paper.

It was long been suggested that a strong outburst of ultraviolet or X-ray radiation may be observable when the RMS reaches the edge of the star (e.g., Colgate 1974b; Falk 1978; Klein & Chevalier 1978; Ensman & Burrows 1992; Matzner & McKee 1999; Blinnikov et al. 2000). These outbursts carry important information about the progenitor star (Arnett 1977) including a direct measure of the star's radius. During the past two years, the wide-field X-ray detectors on board the Swift satellite detected luminous X-ray outbursts preceding an SN explosion in two cases, SN2006aj and SN2008D (Campana et al. 2006; Soderberg et al. 2008). Analysis of the later optical SN emission revealed that both were of type Ib/c, probably produced by compact (W-R) progenitor stars (Pian et al. 2006; Mazzali et al. 2006, 2008; Modjaz et al. 2006; Maeda et al. 2007; Malesani et al. 2009; Tanaka et al. 2009). While some authors argue that the X-ray outbursts are produced by shock breakouts (Campana et al. 2006; Waxman et al. 2007; Soderberg et al. 2008), others argue that the spectral properties of the X-ray bursts rule out a breakout interpretation, and imply the existence of relativistic energetic jets penetrating through the stellar mantle/envelope (Soderberg et al. 2006; Fan et al. 2006; Ghisellini et al. 2007; Li 2007; Mazzali et al. 2008; Li 2008; some argue that the breakout interpretation holds for SN2008D, but not for SN2006aj; Chevalier & Fransson 2008).

The main challenge raised for the breakout interpretation (e.g., Mazzali et al. 2008) was that in both cases, SN2006aj and SN2008D, the observed X-ray flash had a non-thermal spectrum extending to ≳10keV, in contrast with the sub-keV thermal temperatures expected (Matzner & McKee 1999; note that while thermal components with low temperatures or fluxes cannot be ruled out, e.g., Mazzali et al. 2008; Modjaz et al. 2009, such thermal components carry only a small fraction of the observed flux). However, the analysis presented here implies that fast, β_s>0.2, breakouts may be expected for compact, BSG or W-R progenitors, and that for such fast breakouts the X-ray outburst spectrum may naturally extend to tens or hundreds of keV.

Let us discuss in some more detail the possible interpretation of the X-ray outbursts associated with SN2006aj and SN2008D as breakouts. In the case of SN2008D, both the X-ray outburst and early UV/O emission that followed it are consistent with a breakout interpretation (see Soderberg et al. 2008, for detailed discussion). The X-ray burst energy and duration are consistent with the relation of Equation (38), and the UV/O emission is consistent with that expected from the post-shock expansion of the envelope. Moreover, the non-thermal X-ray and radio emission that follow the X-ray outburst are consistent with those expected from a shock driven into a wind by the fast shell that was accelerated by the RMS, for wind density and shell velocity, v/c ∼ 1/4, and energy, E ∼ 10⁴⁷ erg, which are inferred from the X-ray outburst. The fast velocity inferred for the late radio and X-ray emission is consistent with that required to account for the non-thermal spectrum.

The case of SN2006aj is more complicated. The X-ray burst energy and temperature are consistent with a mildly relativistic, v/c ∼ 0.8, shock breakout from a wind surrounding the star, the UV/O emission is broadly consistent with that expected from the expanding envelope, and the non-thermal X-ray emission that follows the X-ray outburst is consistent with that expected from the shock driven into the wind by the fast shell that was accelerated by the RMS (Campana et al. 2006; Waxman et al. 2007). However, the duration of the X-ray outburst is larger than expected, the UV/O emission is not as well fit by the model as in the case of 2008D, and the non-thermal radio emission is higher than expected from the wind–shell interaction. Several authors (Campana et al. 2006; Waxman et al. 2007) have suggested that the deviations from the simple breakout model are due to a highly non-spherical explosion. Some support for the non-spherical nature of the explosion was later obtained by the polarization measurements of Gorosabel et al. (2006; see, however, Mazzali et al. 2007). Other authors (Soderberg et al. 2006; Ghisellini et al. 2007; Fan et al. 2006; Li et al. 2007) have argued that a relativistic jet is required to account for the observations. We believe that an analysis of the modifications to the simple spherical model, introduced by a highly non-spherical breakout, is required to make progress toward discriminating between the two scenarios.

There is an additional important point that should be clarified in the context of SN2006aj. Under the shock breakout hypothesis, the energy and velocity of the accelerated shell that is responsible for the X-ray emission are v/c ∼ 0.8 and E ∼ 10^49.5 erg. This is similar to the parameters of the fast expanding shell inferred to be ejected by SN1998bw (associated with GRB080425), based on long-term radio and X-ray emission, which are interpreted as due to interaction with a low-density wind (Kulkarni et al. 1998; Waxman & Loeb 1999; Li & Chevalier 1999; Waxman 2004a, 2004b). Long-term radio observations strongly disfavor the existence of an energetic, 10⁵¹ erg, relativistic jet associated with SN1998bw (Soderberg et al. 2004). The similarity of SN1998bw & SN2006aj may therefore suggest that such a jet is not present also in the case of SN2006aj. However, the large energy deposited by the explosion in a mildly relativistic shell, v/c ∼ 0.8, is a challenge in itself: the acceleration of the SN shock near the edge of the star is typically expected to deposit only ∼10⁴⁶ erg in such fast shells (Tan et al. 2001), as can be inferred, e.g., from Equation (49), which implies $E(>v_s)/E_{\rm ej}\sim\break (v_s/v_{\rm ej,b})^{-\delta _{v}}$.