A branching process $Z$ is said to be non conservative if it hits $\infty$ in a finite time with
positive probability. It is well known that this happens if and only if the branching mechanism $\varphi$
of $Z$ satisfies $\int_{0+}d\lambda/|\varphi(\lambda)|<\infty$. We construct on the same
probability space a family of conservative continuous state branching processes $Z^{(\varepsilon)}$,
$\varepsilon\ge0$, each process $Z^{(\varepsilon)}$ having
$\varphi^{(\varepsilon)}(\lambda)=\varphi(\lambda+\varepsilon)-\varphi(\varepsilon)$ as branching
mechanism, and such that the family
$Z^{(\varepsilon)}$, $\varepsilon\ge0$ converges a.s. to $Z$, as $\varepsilon\rightarrow0$. Then we
study the speed of convergence of $Z^{(\varepsilon)}$, when $\varepsilon\rightarrow0$, referred
to here as the explosion speed. More specifically, we characterize the functions $f$ with
$\lim_{\varepsilon\rightarrow0} f(\varepsilon)=\infty$ and such that the first passage times
$\sigma_\varepsilon=\inf\{t:Z^{(\varepsilon)}_t\ge f(\varepsilon)\}$ converge toward the explosion time
of $Z$. Necessary and sufficient conditions are obtained for the weak convergence and convergence
in $L^1$. Then we give a sufficient condition for the almost sure convergence.
This is a joint work with my PhD student Cl\'ement Lamoureux.