2.1.2. Internal Energy and Adiabatic Losses

In specific cases, the adiabatic losses and the internal energy content can be expressed in an analytic form. The following treatment to estimate adiabatic losses and the internal energy content of the blast-wave assumes that, right behind the shock, the freshly shocked electrons instantaneously radiate a fraction *e*rad of their internal energy and then they cool only due to adiabatic losses [55]. By assuming that the accelerated electrons promptly radiate at the shock, and then they evolve adiabatically, one is implicitly considering either a fast cooling regime or quasi-adiabatic regime, in which case the radiative losses do not affect the shell dynamics.

Defining *e*<sup>e</sup> as the fraction of energy *dE*0 *sh* dissipated by the shock and gained by the leptons, the mean random Lorentz factor of post-shock leptons becomes (for a more detailed discussion see Section 2.2):

$$
\gamma\_{\rm acc,\varepsilon} - 1 = (\Gamma - 1)\varepsilon\_{\varepsilon}/\mu\_{\varepsilon} \,. \tag{6}
$$

Here, *µ<sup>e</sup>* = *ρe*/*ρ* is the ratio between the mass density *ρ<sup>e</sup>* of shocked electrons and positrons (simply "electrons" from now on) and the total mass density of the shocked matter *ρ*. In the absence of electron-positron pairs *µ<sup>e</sup>* = *me*/(*m<sup>e</sup>* + *mp*) ' *me*/*mp*.

Leptons then radiate a fraction *e*rad of their internal energy, i.e., the energy lost to radiation is *dE*0 *rad* = −*erade<sup>e</sup> dE*<sup>0</sup> *sh* = −*e dE*<sup>0</sup> *sh*, with *e* ≡ *e*rad*e*<sup>e</sup> being the overall fraction of the shock-dissipated energy that goes into radiation. After radiating a fraction *e*rad of their internal energy, the mean random Lorentz factor of the freshly shocked electrons decreases down to:

$$
\gamma\_{rad,\varepsilon} - 1 = (1 - \epsilon\_{rad})(\gamma\_{acc,\varepsilon} - 1) = (1 - \epsilon\_{rad})(\Gamma - 1)\frac{\epsilon\_{\varepsilon}}{\mu\_{\varepsilon}}.\tag{7}
$$

The assumption of instantaneous radiative losses is verified in the fast cooling regime (*erad* ∼ 1), which is required (but not sufficient) to have *e* ∼ 1 (i.e., a fully radiative blast-wave). In the opposite case *erad* 1, the evolution is nearly adiabatic (*e* 1), regardless of the value of *e*e, and the details of the radiative cooling processes are likely to be unimportant for the shell dynamics. The case with intermediate values of *e*rad and *e* is harder to treat analytically, since the electrons shocked at radius *R* may continue to emit copiously at larger distances as well, affecting the blast-wave dynamics.

A similar treatment can be adopted for protons: if protons gain a fraction *e*<sup>p</sup> of the energy dissipated by the shock (with *e<sup>p</sup>* = 1 − *e<sup>e</sup>* − *eB*), their mean post-shock Lorentz factor will be:

$$
\gamma\_{\rm acc,p} - 1 = (\Gamma - 1) \frac{\epsilon\_p}{\mu\_p} \, , \tag{8}
$$

where *µ<sup>p</sup>* = *ρp*/*ρ* is the ratio between the mass density of shocked protons *ρ<sup>p</sup>* and the total shocked mass density *ρ*. In the standard case, when pairs are absent, *µ<sup>p</sup>* ' 1. Since the proton radiative losses are negligible, the shocked protons will lose their energy only due to adiabatic cooling.

Adiabatic losses can be computed starting from *dE*0 int = −*p* 0 *dV*0 , where *p* 0 is the pressure in the comoving frame. For *N* particles with Lorentz factor *γ*, the internal energy density is:

$$
\mu\_{\rm int}' = \frac{N(\gamma - 1)m\,c^2}{V'} \,. \tag{9}
$$

The radial change of the Lorentz factor, as a result of expansion losses, is:

$$\left(\frac{d(\gamma - 1)}{dr}\right)\_{ad} = -(\hat{\gamma} - 1)(\gamma - 1)\frac{d\ln V'}{dr}.\tag{10}$$

To estimate the adiabatic losses, let us assume that the shell comoving volume scales as *V* 0 ∝ *R* <sup>3</sup>/Γ, corresponding to a shell thickness in the progenitor frame <sup>∼</sup> *<sup>R</sup>*/<sup>Γ</sup> 2 . This scaling is correct for both relativistic and non-relativistic shocks in the decelerating phase (BM76). For re-accelerating relativistic shocks, Shapiro [59] demonstrated that the thickness of the region containing most of the blast-wave energy is still ∼ *R*/Γ 2 . For the sake of simplicity, changes in the comoving volume due to a time-varying adiabatic index or radiative efficiency are neglected. If the scaling *V* 0 ∝ *R* <sup>3</sup>/Γ is assumed, the equation can be further developed analytically, and reads:

$$\left(\frac{d(\gamma - 1)}{dr}\right)\_{ad} = -(\hat{\gamma} - 1)(\gamma - 1)\left(\frac{3}{r} - \frac{d\ln\Gamma}{dr}\right). \tag{11}$$

The comoving Lorentz factor at radius *R*, for a particle injected with *γ*(*r*) when the shock radius was *r*, will be

$$\gamma\_{\rm ad}(\mathcal{R},r) - 1 = \left(\frac{r}{R}\right)^{3(\hat{\gamma}-1)} \left[\frac{\Gamma(R)}{\Gamma(r)}\right]^{(\hat{\gamma}-1)} (\gamma(r) - 1) \,. \tag{12}$$

where *γ*(*r*) is given by *γrad*,*<sup>e</sup>* (*r*) (Equation (7)) for leptons, and by *γacc*,*p*(*r*) (Equation (8)) for protons.

Considering the proton and lepton energy densities separately, the comoving internal energy at radius *R* will be:

$$E\_{\rm int}^{\prime}(\mathbf{R}) = 4\pi c^2 \int\_0^{\mathbf{R}} dr r^2 \left\{ \rho\_p(r) [\gamma\_{ad,p}(\mathbf{R}, r) - 1] + \rho\_\varepsilon(r) [\gamma\_{ad,\varepsilon}(\mathbf{R}, r) - 1] \right\}.\tag{13}$$

With the help of Equation (12), one can explicitly find *E* 0 *int*(*R*) and insert it in Equation (5).

The other term needed in Equation (5) is *dE*0 *ad*/*dR*. First, we have derived (*dγ*/*dR*)*ad* for a single particle. Now integrating over the total number of particles, again considering protons and leptons separately, one obtains:

$$\frac{d E\_{\rm ad}^{\prime}(R)}{dR} = -4\pi c^2 (\hat{\gamma} - 1) \left( \frac{3}{R} - \frac{d \log \Gamma}{dR} \right) \int\_0^R dr r^2 \left\{ \rho\_p(r) (\gamma\_{\rm ad, p} - 1) + \rho\_\varepsilon(r) (\gamma\_{\rm ad, \varepsilon} - 1) \right\}.\tag{14}$$

In Equations (13) and (14), it is assumed that only the swept-up matter is subject to adiabatic cooling, i.e., that the ejecta particles are cold.

As long as the shocked particles remain relativistic, the equations for the comoving internal energy and for the adiabatic expansion losses assume simpler forms:

$$\begin{split} E\_{\rm int}'(\mathbf{R}) &= 4\pi c^2 \int\_0^R dr \, r^2 \frac{r}{R} \left[ \frac{\Gamma(R)}{\Gamma(r)} \right]^{1/3} \Gamma(r) \left\{ \frac{\epsilon\_p}{\mu\_p} \rho\_p + (1 - \epsilon\_{rad}) \frac{\epsilon\_\varepsilon}{\mu\_\varepsilon} \rho\_\varepsilon \right\} \\ &= 4\pi c^2 \int\_0^R dr \, r^2 \frac{r}{R} \left[ \frac{\Gamma(R)}{\Gamma(r)} \right]^{1/3} \Gamma(r) \, \rho(r) \left( \epsilon\_p + \epsilon\_\varepsilon - \epsilon \right) \\ \frac{dE\_{\rm ad}'(\mathbf{R})}{dR} &= -E\_{\rm int}'(\mathbf{R}) \left( \frac{1}{R} - \frac{1}{3} \frac{d \log \Gamma}{dR} \right) . \end{split} \tag{15}$$

In the absence of significant magnetic field amplification, *e<sup>p</sup>* + *e<sup>e</sup>* ' 1 so that *e<sup>p</sup>* + *e<sup>e</sup>* − *e* ' 1 − *e*, and the radiative processes of the blast-wave are entirely captured by the single efficiency parameter *e*. In the fast cooling regime *erad* ∼ 1 and *e* ' *e<sup>e</sup>* . In this case, the term *e<sup>p</sup>* + *e<sup>e</sup>* − *e* reduces to *e*p, meaning that, regardless of the amount of energy gained by the electrons, in the fast cooling regime the adiabatic losses are dominated by the protons, since the electrons lose all their energy to radiation.

Evaluating these expressions for adiabatic blast-waves in a power-law density profile *ρ* ∝ *R* −*s* , one obtains:

$$E\_{\rm int}'(R) = \frac{9 - 3s}{9 - 2s} \Gamma m c^2 \quad , \quad \frac{dE\_{ad}'(R)}{dR} = -\frac{(9 - s)(3 - s)}{2(9 - 2s)} \frac{\Gamma m c^2}{R} \, , \tag{16}$$

where Γ ∝ *R* <sup>−</sup>(3−*s*)/2 as in the adiabatic BM76 solution has been used.

In the fully radiative regime *e* = 1, which implies *E* 0 *int* = 0 and *dE*<sup>0</sup> *ad* = 0, Equation (5) reduces to:

$$\frac{d\Gamma}{dm} = -\frac{(\Gamma\_{\rm eff} + 1)(\Gamma - 1)}{M\_0 + m} \; , \tag{17}$$

which describes the evolution of a momentum-conserving (rather than pressure-driven) snowplow. Replacing Γ*eff* → Γ, the solution of this equation coincides with the result by BM76.

Since the model is based on the homogeneous shell approximation, the adiabatic solution does not recover the correct normalization of the BM76 solution. In this treatment, the total energy of a relativistic decelerating adiabatic blast wave in a power-law density profile *ρ*(*R*) ∝ *R* −*s* is

$$E\_0 \simeq \Gamma\_{\rm eff} \mathcal{E}'\_{\rm int} \simeq \frac{4}{3} \Gamma E'\_{\rm int} \simeq \frac{12 - 4s}{9 - 2s} \Gamma^2 mc^2 \,, \tag{18}$$

so that the BM76 normalization can be recovered by multiplying the density of external matter in Equations (5), (13) and (14) by the factor (9 − 2*s*)/(17 − 4*s*). To smoothly interpolate between the adiabatic regime and the radiative regime, the following correction factor should be adopted:

$$\mathcal{C}\_{BM76,\varepsilon} \equiv \varepsilon + \frac{9 - 2s}{17 - 4s} (1 - \varepsilon) \ . \tag{19}$$

No analytic model properly captures the transition between an adiabatic relativistic blast-wave and the momentum-conserving snowplow, as *e* increases from zero to unity. The simple interpolation in Equation (19) joins the fully adiabatic BM76 solution with the fully radiative momentum-conserving snowplow.

In summary, Equations (5), (13) and (14), complemented with the correction in Equation (19) (which should by applied to every occurrence of external density and external matter) completely determine the evolution of the shell Lorentz factor Γ as a function of the shock radius *R*.
