Xi-Can Tang, Zhi Li, Iulia T. Simion, Hao Tian, Zhijian Luo, Shuting Fan, Zi-Qi Li

Aims. We constrain the dynamical perturbations of the spiral arms in the Milky Way disk, based on the non-axisymmetric streaming motions of RGB stars revealed by APOGEE and \textit{Gaia}. Methods. We develop a revised steady-state radial-velocity response model that incorporates both the \(V_{R,\sin}\) and the dynamically important \(V_{R,\cos}\) components for a two-armed logarithmic spiral potential. The model is validated using orbit integrations with \texttt{AGAMA} and Bayesian parameter recovery with \texttt{dynesty}, and is applied to the smoothed two-dimensional radial-velocity field of RGB stars while accounting for Lindblad and corotation resonances. Results. The revised model reproduces the phase and amplitude of the mock radial-velocity field to the \(\sim2\%\) level, substantially improving upon earlier \(V_{R,\sin}\)-only formulations. Applied to the observational data, it yields a robust pitch angle of \(p \simeq 10^\circ\) and a local surface density contrast of \(ξ\simeq 5\)--\(18\%\) at the solar radius. The radial scale length is less well-constrained (\(h_{R,1} \simeq 40\)--\(50\,\mathrm{kpc}\)) due to intrinsic parameter covariance. Resonance effects strongly shape the velocity field, thus affecting the fitting: the radial velocity becomes extremely large near the Lindblad resonances, whereas it vanishes close to the corotation resonance. Conclusions. Our results demonstrate that including both the \(V_{R,\sin}\) and \(V_{R,\cos}\) terms is essential for a physically consistent interpretation of stellar streaming motions induced by a spiral potential. The observed kinematics constrain the spiral pattern speed to \(Ω_{p} \approx 10\)--\(20\,\mathrm{km\,s}^{-1}\mathrm{kpc}^{-1}\).