The spiral pattern rotation speed of the Milky Way Galaxy in the Lin-Shu theory of small-amplitude density waves

The lack of a consensus on the spiral pattern rotation speed of the Milky Way Galaxy in the Lin-Shu density wave theory frame proposed in the 1960s shows the urgency of making use of the large volume and high-quality observational data. We study the kinematics of the spiral density wave pattern in the 3 kpc solar vicinity, using main-sequence OB, A, and F stars with Gaia DR3. Small nonaxisymmetric perturbations of the background axially symmetric gravitational potential that generate the spiral tightly wound waves are assumed. Systematic variations of stellar velocities, induced by these perturbations, are examined numerically utilizing a nonlinear least-squares fit. A calculation leads to the pattern rotation speed of ${\mathrm{\Omega }}_{\mathrm{p}}=41\text{–}49$ $\Omega _{\mathrm{{p}}} = 41\text{--} 49$   $\text{km}{\text{s}}^{-1}{\text{kpc}}^{-1}$ $\text{km}\,\text{s}^{-1}\,\text{kpc}^{-1}$ . This ${\mathrm{\Omega }}_{\mathrm{p}}$ $\Omega _{\mathrm{{p}}}$ differs from the angular velocity of mean circular rotation near the Sun of ${\mathrm{\Omega }}_0\approx 27\text{ km}{\text{s}}^{-1}{\text{kpc}}^{-1}$ $\Omega _{0} \approx 27\text{ km}\,\text{s}^{-1}\,\text{kpc}^{-1}$ and sets the corotation circle well inside of the solar circle, at Galactocentric distances $r_{\mathrm{cor}}=4\text{–}6$ $r_{\mathrm{{cor}}} = 4\text{--}6$ kpc (for the solar distance $r_0=8.2$ $r_{0} = 8.2$ kpc). In agreement with the concept of slowly growing waves, the estimated amplitude of the perturbed spiral potential is small, i.e. $\lesssim 7\mathrm{\%}$ $\lesssim 7\%$ of the background potential $|{\mathrm{\Phi }}_0|\sim r_0^2{\mathrm{\Omega }}_0^2$ $|\Phi _{0}| \sim r_{0}^{2} \Omega _{0}^{2}$ . The system’s central bar and density wave driven spirals might be dynamically coupled because the bar rotates almost with the same angular velocity, with the value of the bar rotation speed of ${\mathrm{\Omega }}_{\mathrm{bar}}\sim 40\text{ km}{\text{s}}^{-1}{\text{kpc}}^{-1}$ $\Omega _{\mathrm{{bar}}} \sim 40\text{ km}\,\text{s}^{-1}\,\text{kpc}^{-1}$ drawn from the literature on bar dynamics.