tokens: that the mass point C is under the action of an arbitrary additional force F (per unit mass) that is not gravitational and acts on the gas component of the supercloud. In our case, F will represent an hydrodynamic force due to the friction between the gas supercloud and the surrounding interstellar medium or due to the pass of a shock wave that affects the gas supercloud. Then, the motion equations in the reference system ( x, y ) are given by Figure 1. Illustration of the stationary coordinate system ( x, y ) centred at the Galactic centre and the rotating coordinate system (η, ξ) of origin O moving on a circular orbit of radius R 0 . The figure plane represents the Galactic plane and the rotation sense is clockwise. The angle θ, between the y -axis and ξ-axis, increases in the sense of the Galactic rotation. where F x and F y are the components of F and subscript c refers to the coordinates of the point C. Because the force K ( R c ) is equal to the centripetal force at R c , we will write K ( R c ) = R c ω( R c ) 2 , where ω( R c ) refers to the Galactic angular velocity at R c and for short will be written as ω c . Since we are interested in referring the motions to the rotating system, we should convert equations ( 1 ) into ones depending on (η, ξ). The formulae that govern this transformation are which are obtained from simple geometrical considerations of Fig. 1 . Besides, we should have into account that F x = F ξ sin θ + F η cos θ and F y = F ξ cos θ − F η sin θ, and that θ is a function of time t given by θ = θ 0 + ω 0 t . We adopt θ 0 = 0, i.e. the y -axis and ξ-axis agree at t = 0. Differentiating twice equations ( 2 ) with respect to the time t, we obtain where and . From the second members of equations ( 1 ), with the corresponding replacements, we get that where and . By equating equations ( 3 ) to the corresponding equations ( 4 ), we have The system of equations ( 5 ) is equivalent to the system a 1 = c 1 and a 2 = c 2 . Thus, the exact epicyclic equations of motion for the mass point C are If we adopt a flat rotation curve for the Galaxy, with a rotation velocity V 0 = 220 km s −1 , in approximate agreement with the measurements in a large range of Galactocentric distances (e.g. Sofue, Honma Omodaka 2009 ; Bovy et al. 2012 ), and . Since (see Fig. 1 ), we have For the study of the orbit of the local supercloud, the origin of the system (η, ξ) should be located in the solar neighbourhood at the Galactocentric distance of the Sun (i.e. R 0 ≈ 8 kpc). Replacing equation ( 7 ) into equations ( 6 ), this equation system can be solved numerically. The initial conditions needed to solve the differential equation system ( 6 ), i.e. η c (0), ξ c (0), , , will be calculated from the LSR velocity and position of the supercloud's gravitational centre at t = 0. With this purpose, we use the relations ( 2 ) to obtain η c = x c cos θ − y c sin θ and ξ c = x c sin θ + y c cos θ − R 0 . Hence, and . Since θ = ω 0 t = 0 at t = 0, the former
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