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 $\int_{0+}dλ/|\varphi(λ)|<\infty$. We construct on the same probability space a family of conservative continuous state branching processes $Z^{(\varepsilon)}$, $\varepsilon\ge0$, each process $Z^{(\varepsilon)}$ having $\varphi^{(\varepsilon)}(λ)=\varphi(λ+\varepsilon)-\varphi(\varepsilon)$ 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 $σ_\varepsilon=\inf\{t:Z^{(\varepsilon)}_t\ge f(\varepsilon)\}$ 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.
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