a: PV map of SO in Oph IRS 63 along the major axis of the SO ring marked in Fig. 1. The dashed and dotted black lines represent the fits to the velocity distribution of V_ϕ with power-law indices of –1 (the free-fall case with non-zero angular momentum) and –0.5 (the Keplerian case), respectively (see Sect. 4.2.2 for details), b: modeled velocity of an inner disk-like envelope (in a simplified 2D version) with its projection parameters (θ_view and ϕ_view) identical to those of Oph IRS 63. The radial (υ_r) and azimuthal (V_ϕ) velocities are given by the formulas in the legend, following Eqs. (25) and (24), which describe the possible motion of disk-like inner envelopes (see Sect. 4.2.2 for details). Note that the model is scale-free, and we have adjusted the velocity range and central velocity for better agreement with the observations (panel a of Fig. 2).