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 ago from a point E located in the galactic plane in the direction of galactic longitude , at a distance 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 ; 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 since mostly, we avoid the motions near to the galactic plane.
The motions are refered to a system of coordinates , with its origin at E, which is rotating about C with the angular velocity = 25 km kp. The -axis points in the direction , the -axis in the direction , and the z-axis in the direction . 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 at , the initial conditions will be
where (azimuth, measured clockwise from the -axis) and (altitude, measured northwards from the galactic plane) refer to the direction of ejection.
In this Section we consider that the components , and 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 - and -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 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 which we identify with the LSR. If the position and the velocity of the observer are , and respectively, and the observer measures , and V, we have:
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 of E above the galactic plane (cf. Paper I), a velocity of the observer, as well as an altitude of the Sun above the galactic plane.
© European Southern Observatory (ESO) 2000
Online publication: June 26, 2000