Appendix A: The slowdown of a CB

We review the functional form of the time dependent Lorentz factor $\gamma(t)$, which is explicit and analytical in a fair approximation (DDD 2001).

In minutes of observer's time, CBs reach a roughly constant radius $R_{\rm max}$ and are parsecs away from their progenitor star, a domain where a constant-density ISM may be a reasonable approximation. Relativistic energy-momentum conservation in the progenitor's rest frame results in the equation governing the deceleration of a CB in the ISM:

$$M_{_{\rm CB}}\, {\rm d}\gamma =-\, m_p\,c^2\, n_p\, \pi\, R_{\rm max}^2\, \gamma^2\, c\, {\rm d}t_{\rm SN} .$$ (36)

Interestingly, the above expression is correct both if the incoming ISM protons are isotropically reemitted in the CB rest frame, or if they are ingurgitated by the CB (in the first case, they are reemitted with average energy $m_{p}\,c^2\,\gamma^2$ in the progenitor's frame, in the second, the change in $\gamma$ per added proton is ${\rm d}\gamma=-[m_{p}/M_{_{\rm CB}}]\gamma$).

Use the relation ${\rm d}t_{\rm SN}=\gamma(t)\, {\rm d}t$ between the times measured in the supernova and CB rest frames, divide both sides of the Eq. (36) by $M_{_{\rm CB}}\, \gamma^3$ and integrate to obtain the relation:

$${1\over \gamma^2(t)} - {1\over \gamma_0^2}\simeq {2\, c\, t\over x_\infty} ,$$ (37)

where t is CB time, and:

$$x_\infty\equiv{N_{\rm CB}\over\pi\, R_{\rm max}^2\, n_{p}},$$ (38)

with $N_{\rm CB}\approx M_{_{\rm CB}}/m_{p}$ the CB's baryon number.

It is important to know the number of electrons accumulated by a CB as its Lorentz factor decreases from $\gamma_0$ to $\gamma(t)$ (in the approximation n_e=n_p of a Hydrogenic ISM this number equals that of scattered or incorporated protons). The number rate of accumulation is related to the energy-loss rate of Eq. (36) so that:

$${\rm d}N= - \eta\, {M_{_{\rm CB}}\, {\rm d}\gamma\over m_{p}\... ...^2}= -\eta\, N_{_{\rm CB}}\,{{\rm d}\gamma\over \gamma^2}\cdot$$ (39)

Assuming constant $\eta$, the total number of ISM electrons accumulated at a CB time t is then:

$$N(t)=\eta\, N_{_{\rm CB}}\,\left[{1\over \gamma(t)}-{1\over \gamma_0}\right]\cdot$$ (40)

The time-dependence of $\gamma(t)$ with t the observer's time is more complicated than Eq. (37): the relation between the two times ( ${\rm d}t_{\rm obs}={\rm d}t_{_{\rm CB}}(1+z)/\delta$) introduces a $\gamma$ (or t) dependence via $\delta$. The result for $\gamma(t_{\rm obs})$is (DDD 2001):

 

$$\gamma \gamma(\gamma_0,\theta,x_\infty;t) = {1\over B} \,\left[\theta^2+C\,\theta^4+{1\over C}\right]$$ =

$$\equiv \left[{2\over B^2+2\,\theta^6+B\,\sqrt{B^2+4\,\theta^6}}\right]^{1/3}$$ C

$$\equiv {1\over \gamma_0^3}+{3\,\theta^2\over\gamma_0}+ {6\,c\, t\over (1+z)\, x_\infty}\cdot$$ B (41)

The Lorentz factor of the CB decreases from $\gamma_0$ to $\gamma_0/2$ as the CB travels a distance $x_\infty/\gamma_0$, whose reference value is 1.3 kpc, as in Table 1.