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Astron. Astrophys. 358, 299-309 (2000)

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3. Ballistic orbits for the case of an explosive event

We consider the motions of a group of test particles, which have been ejected simultaneously a time [FORMULA] ago from a point E located in the galactic plane in the direction of galactic longitude [FORMULA], at a distance [FORMULA] from the Sun. We assume that: i) an energetic explosive event within a dense and massive molecular cloud produced an expanding shell; ii) the braking forces due to the accretion of gas acted drastically in the very initial stages of the event disrupting the cloud and reducing the expansion velocity of a given test particle to an equivalent initial velocity [FORMULA]; iii) the distances r of the test particles are always small as compared with the distances to the galactic center C; iv) the z-motions can be decoupled from the motions parallel to the galactic plane. These assumptions appear to be adequate for studying the gas at [FORMULA] since mostly, we avoid the motions near to the galactic plane.

The motions are refered to a system of coordinates [FORMULA], with its origin at E, which is rotating about C with the angular velocity [FORMULA] = 25 km [FORMULA] kp[FORMULA]. The [FORMULA]-axis points in the direction [FORMULA], the [FORMULA]-axis in the direction [FORMULA], and the z-axis in the direction [FORMULA]. Under the assumptions stated above the equations of motion as functions of the time t are well-known (e.g. Olano 1982). If the explosive event is characterized by an isotropic ejection of particles with velocity [FORMULA] at [FORMULA], the initial conditions will be

[EQUATION]

[EQUATION]

where [FORMULA] (azimuth, measured clockwise from the [FORMULA]-axis) and [FORMULA] (altitude, measured northwards from the galactic plane) refer to the direction of ejection.

In this Section we consider that the components [FORMULA], [FORMULA] and [FORMULA] of any interaction force between the ejected test particle and the surrounding medium can be neglected (ballistic orbits). Therefore, Chandrasekhar's (1942) analytical solution can be adopted for the [FORMULA]- and [FORMULA]-motions, whereas the z-motions can be approximated by small oscillations about the galactic plane with a period T. The latter could become an oversimplification, since the proportionality between the galactic gravitational force per unit mass [FORMULA] and the altitude z breaks down at about 350 pc above the plane (cf. Dickey 1993). Nevertheless, small oscillations appear as an acceptable insight into most of the z-motions occurring in our model. We refer to Olano (1982) for further details. In order to allow a comparison with the observations of the CNM, the computed results are referred to an inertial system [FORMULA] which we identify with the LSR. If the position and the velocity of the observer are [FORMULA], and [FORMULA] respectively, and the observer measures [FORMULA], and V, we have:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The computations of r can be checked independently of the 21-cm observations in all those cases where identifications of the clouds are possible and the distances are known. Furthermore, in our computations we considered also an initial altitude [FORMULA] of E above the galactic plane (cf. Paper I), a velocity [FORMULA] of the observer, as well as an altitude [FORMULA] of the Sun above the galactic plane.

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000
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