Shock strength, pitch angles, and azimuthally-averaged aspect ratios for three representative models. Optical depth and cooling timescale increase from left to right. Top: shocks are stronger for more massive planets but their dependence on disk parameters is not clear. To filter out unphysical results, we mask points that lie within the gap region (|r − r_p |≤ 2.5 R_Hill) or the corotating region (see Eq. (17)). Middle: pitch angles are roughly the same regardless of planet mass for the optically thin case but deviate with increasing optical depth. Black lines denote the expected values using Eq. (19) for h = h_p (dashed) or Eq. (20) for h = h_pr^2∕7 (dotted). Bottom: a power law fit of aspect ratio is only possible for optically thin case or low mass planets but fidelity for such a fit breaks down even for 10-Earth-mass planets in an optically thick disk. Black dashed lines refer to initial aspect ratio profiles.