Numerical simulations of internal shocks in spherical geometry: Hydrodynamics and prompt emission

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Table 2.

List of derived parameters.

Symbol	Definition	Expression
R₀	Collision radius	$\frac{β_{1} β_{4} c t_{off}}{β_{4} - β_{1}}$ $\dfrac {\beta _1\beta _4 c t_\mathrm {off}}{\beta _4-\beta _1}$
χ	Radial width ratio	$\frac{Δ_{0, 1}}{Δ_{0, 4}}$ $\dfrac {\Delta _{0,1}}{\Delta _{0,4}}$
a_u	Proper speed ratio	$\frac{u_{4}}{u_{1}} > 1$ $\dfrac {u_4}{u_1}>1$
f	Proper density ratio	$\frac{n_{4}^{'}}{n_{1}^{'}} = χ \frac{E_{k 0, 4}}{E_{k 0, 1}} \frac{Γ_{1} (Γ_{1} - 1)}{Γ_{4} (Γ_{4} - 1)}$ $\dfrac {n^{\prime}_4}{n^{\prime}_1}=\chi \dfrac {E_{\mathrm {k}0,4}}{E_{\mathrm {k}0,1}}\dfrac {\Gamma _1(\Gamma _1-1)}{\Gamma _4(\Gamma _4-1)}$

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