The Lindblad–Oort idea of a rotation of the Galaxy around the Galactic Centre (GC) proposes that for any type of Galactic object there is at each point in the plane a mean circular, differential motion (Oort 1927a,b). In the conventional lowest order approach to the determination of the rotation parameters of the system, a strictly circular model is adopted (Mihalas & Binney 1998). Small non-axisymmetric spiral-like perturbations of the basic axisymmetric gravitational potential induce non-circular variations of stellar velocities proportional to $\widetilde{v}_r \cos \phi$ and $\widetilde{v}_\varphi \sin \phi$, and we are looking for these systematic motions in our study. Here, $\widetilde{v}_r$ and $\widetilde{v}_\varphi$ are the amplitudes of the radial and tangential wave motions that depend weakly on the Galactocentric distance r, and ϕ is the phase of the wave (Lin et al. 1969; Yuan 1969; Rohlfs 1977; Griv et al. 2020, annexure B therein). The method of analysis used for this purpose follows the development of Crézé & Mennessier (1973), Byl & Ovenden (1978), Mishurov, Pavlovskaya & Suchkov (1979), and Grivnev (1981). Specifically, in addition to the ordinary circular rotation of the system that is axisymmetric in the mean, Galactic objects have small streaming wave motions. The radial and tangential components of the velocity of a star in the plane are represented as
(1)$$\begin{eqnarray*}
V_r = \widetilde{v}_r \cos \phi ,
\end{eqnarray*}$$(2)$$\begin{eqnarray*}
V_\varphi = r \Omega + \widetilde{v}_\varphi \sin \phi ,
\end{eqnarray*}$$where Ω(r) is the angular velocity of the mean motion at the star’s distance r from the GC for the type of object considered, and the amplitudes $\widetilde{v}_r$, $\widetilde{v}_\varphi$ and the phase ϕ are to be found. Distinct populations of Galactic objects will have different average velocities Vφ at r, so the value of Ω should vary from population to population. An observed value of heliocentric line-of-sight velocity of a star $v$los corrected for solar motion towards the apex may be modeled in the following form:
(3)$$\begin{eqnarray*}
v_{\mathrm{los}} &amp;=&amp; \left\lbrace r_0 \left[ \Omega - \Omega _0 \right] \sin \ell - \widetilde{v}_r \cos \phi \cos (\ell +\varphi) \right. \nonumber \\
&amp;&amp;\left. \quad + \widetilde{v}_\varphi \sin \phi \sin (\ell +\varphi) + u_0 \cos \ell - v_0 \sin \ell \right\rbrace \cos b \nonumber \\
&amp;&amp;\quad - w_0 \sin b ,
\end{eqnarray*}$$where ℓ and b are the Galactic coordinates, the angle φ is measured clockwise in the direction of overall rotation from the radius passing through the location of the Sun, Ω0(r0) is the angular velocity of the mean motion at the Sun’s distance r0, ϕ = ϕ0 − m[φ − (1/tan p)ln (r/r0)], ϕ0 is the phase of the wave at the Sun’s location of the different Fourier m-modes, and the constants u0, $v$0, $w$0 are the components of solar peculiar motion relative to the mean linear speed of rotation at the Sun’s distance r0Ω0 (Mihalas & Binney 1998). To reiterate, the terms ∝(Ω − Ω0) and u0, $v$0, $w$0 describe the mean rotation of the system under study and the peculiar motion of the Sun. The deviation of the motion of objects from the circular motion due to a wave perturbation is characterized by terms ${\propto} \widetilde{v}_r \cos \phi$ and $\widetilde{v}_\varphi \sin \phi$.
Citation: Griv et al. 2020
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