The upper three panels show the dependency of the sheath electron density profiles (N_e(τ)) on the three parameters, ρ_e (panel (a)), d_sheath (b), and d_front (c), while the other two parameters are fixed. The lower panels are the modeled Thomson-scattering brightness profiles I ′ (L) in mean solar brightness (B_⊙), resulting from the given N_e(τ) in the same line-styles in the upper panels. Lines in solid, dotted, dashed, dash-dotted, dash-dot-dotted refers to the cases when ρ_e = (1.6, 3.1, 4.7, 6.2, 7.8) × 10⁴, d_sheath = 1 R_⊙, and d_front = 0.25 R_⊙ in panel (a); ρ_e = 6.2 × 10⁴, d_sheath = (0.6, 1.0, 1.4, 1.8, 8.0) R_⊙, and d_front = 0.25 R_⊙ in panel (b); and ρ_e = 6.2 × 10⁴, d_sheath = 1 R_⊙, and (0.0, 0.1, 0.3, 0.4, 0.5) R_⊙ in panel (c). Because the Gaussian tail is not zero even at large distances, N_e(τ < 0) is not exactly zero, if d_front ≠ 0. By definition, we simply set N_e(τ < 0) = 0. The 3D geometry of the shock used to calculate I′ is shown in Figure 1b.