Characteristic radii of a dirty fireball as a function of the dimensionless entropy η. Shown are the Planck radius R_P (red solid), the Wien radius R_W (orange dot–dashed), the Thomson photospheric radius R_ph (purple dashed), and the dissipation radius ﹩{R}_{{\rm{dis}}}\sim 2{{\rm{\Gamma }}}_{{\rm{W}}}^{2}c\delta t﹩ for δt = 0.1 s (gray solid) and 1 s (gray dotted). The fiducial parameters are f_± = 1, ϵ_P = 1, ϵ_W = 0.5, ﹩{{\rm{\Gamma }}}_{{\rm{P}}}={\rm{\max }}(\eta /2,1)﹩, and Γ_W = η. The calculation adopts η_γ = 0.03, a fixed X-ray luminosity L_γ,iso = 10⁵⁰ erg s⁻¹, and a total jet luminosity L = L_γ,iso/η_γ.