The dependence of the rate of change of blast wave radius with respect to observer time, |$\mathrm{ d}R/\mathrm{ d}t$|⁠, on Lorentz factor |$\Gamma$| and viewing angle |$\theta$|⁠, for ejecta approaching the observer. The coloured lines show the full equation (11), compared with the on-axis limit from equation (6) shown with a dot-dashed line. The pink and blue shaded regions show approximate initial Lorentz factor ranges for XRBs and GRBs. At |$\theta =0^\circ$|⁠, the solution agrees with the |$2 \Gamma ^2$| asymptote for high |$\Gamma$|⁠. At small angles to the line of sight, the curves also agree with this curve for intermediate |$\Gamma$| before flattening off as the |$\cos \theta$| term becomes important.