## 2. Effects of a spiral shock on stellar kinematicsThe possibility that the perturbation on the galactic potential which induces the spiral structure could trigger star formation via a large scale spiral shock in the interstellar medium was already studied by Roberts 1969 and Shu et al. 1972; see also the reviews of Wielen 1974, Rohlfs 1977, Toomre 1977, and Chapter 3 of Bertin & Lin 1995 for an overview of the process of shock-induced star formation in spiral arms. The actual existence of such a shock has been controversial, as it depends on the dominant phase of the galactic interstellar medium. As noted by Combes & Gerin 1985, a medium dominated in volume by coronal gas and containing most of the denser phases in the form of discrete molecular clouds would react very differently to a spiral perturbation as would do a more homogeneous medium dominated in volume by warm atomic hydrogen. Also, Elmegreen 1992 has discussed other ways in which spiral arms can trigger cloud collapse and star formation without the need for a spiral shock. However, the existence of a large scale spiral shock has been assumed in other galaxies in order to explain the azimuthal separation of tracers of different stages of star formation (see Bertin & Lin 1995 and references therein), radial trends in the efficiency of star formation (Roberts, Roberts, & Shu 1975; Cepa & Beckman 1990; Puerari & Dottori 1996), and color gradients across the spiral arms (Yuan & Grosbol 1981). In our Galaxy, the kinematic properties associated to a spiral shock producing a phase transition in the diffuse interstellar medium have been applied to explain the overall dynamics of molecular clouds (Bash & Peters 1976; Bash et al. 1977) and the vertex deviation of O and B stars in the solar neighbourhood (Hilton & Bash 1982). The kinematical perturbations induced by spiral arms in the solar neighbourhood have been applied to address problems such as the vertex deviation (Yuan 1971), the local values of the Oort constants (Lin et al. 1978; Lindblad 1980) and the heating of the galactic disk (Binney & Lacey 1988). Our main hypothesis throughout this paper is that the main moving
groups presently observed in the solar neighbourhood were formed in
such shocks, and that the jump in the velocity of the gas crossing the
shocks is the main responsible of their deviation from the circular
motion. As such deviations are observed to be of order
km s Let us consider a rotating reference frame whose center,
momentaneously occupied by the Sun, rotates in a circular orbit around
the galactic center with a velocity . The
where The spiral pattern and its associated shock rotate with an angular velocity . As long as the Sun is far from the corotation circle, the spiral shock can be pictured as a ridge rapidly crossing this reference system at periodical intervals, leaving star formation behind it. The velocity vector of the gas entering the shock, together with the shock jump conditions, determine the initial conditions in the motions of the stars. The spiral shock forms an angle The circular velocity in the reference frame moving with the spiral shock is where we will use for For a strong shock to exist, the condition must be fulfilled, with being the effective sound speed in the interstellar gas. In a tightly wound spiral, this condition can be fulfilled far from the corotation circle, where is large. Moreover, the response of the gas to the spiral gravitational potential provides a positive contribution to from the term at the position of the shock fronts (Shu et al. 1972). On the other hand, the velocity dispersion in the unshocked gas adds random terms in (3a) and (3b), expected to be of order of ; therefore, this condition also ensures that the velocity dispersion of the gas does not introduce any essential changes in our treatment. Such velocity dispersion in the gas entering the shock, implying a range of values in the initial values for the jump conditions, may be expected to be reflected in the internal velocity dispersion of the moving groups formed out of it. After the shock, , . In the reference frame used to define the epicyclic approximation, where , , we obtain In writing Eqs. (4) in this form, we are assuming a trailing spiral
and define the pitch angle On the other hand, the term in (4b) is much smaller than the term in the right hand side of (4a). By comparison to (4a), Eq. (4b) can then be simply approximated by The Eqs. (4c) and (4d) thus determine the approximate initial velocity at the time of the passage of the spiral shock by a given position , . If the term is still larger than , the impulse is mostly in the radial direction, and the resulting expressions are essentially the same as if we had assumed a purely circular motion for the gas at the start. Eqs. (4c) and (4d), together with Eqs. (1), give the following relations for the epicyclic elements: Let us consider now the evolution of a star forming region along a
spiral shock. Initially, the region is elongated along the spiral arm
and forming with the direction of the galactic rotation an angle
The solar neighbourhood can be expected to contain some such fragments of ancient star forming ridges, left behind by the spiral arm as it moved across the local region of the galactic disk. Each such fragment will be characterized by the position of the guiding center of the epicyclic orbits of the stars belonging to it, , . We can easily relate , , and : the position of the spiral shock can be represented locally by a straight line moving across our reference system: where Moreover, by Eqs. (5a), (5b), and relating , to the positions of the epicyclic orbits guiding centers by means of (5d), (5e), whence This expression can be somewhat simplified if the Sun is far from the corotation and if the observational horizon is restricted to no more than a few hundred parsecs; in that case, the maximum value of can be estimated from (1a) and (5c), making : and the ratio of the two terms in the denominator of Eq. (10) is For values of Developing the products, and retaining only the first-order terms, We note that, from Eq. (6), , where
is the average age of the group defined as the
time when the star-forming spiral shock passed by the local standard
of rest; therefore, We can obtain a relationship among the epicyclic elements of stars with their origin in the same spiral arm by using Eq. (8). In doing so, we should include the dependence of with galactocentric distance; to simplify, we will use that, for a flat rotation curve, where is the value at the position of the Sun. Replacing (11b) in (8), using the approximation (12), and neglecting the term containing , which is in general much smaller than those containing , we finally obtain where with Eq. (13), plus the condition constant, may thus be used to provide an alternative definition of moving group, by which member stars are identified by having been formed in the same passage of a spiral arm by the solar neighbourhood. Notice that this definition, unlike classical ones, does not imply that the members of a moving group have isoperiodic orbits around the galactic center, that their space velocity vectors are similar, or that the stars are coeval. However, the restriction in heliocentric distance of the samples of stars whose motions we can analyze implies strong correlations among their orbital elements, resulting in kinematical similarities among the stars belonging to a group and observed in the solar neighbourhood. On the other hand, Eq. (10), together with the restriction in accessible values of , , and correlations among these and other orbital elements limits the range of ages among the stars belonging to the so-defined moving groups. It is in practice convenient to use an alternative form of Eq.
(13), due to the fact that if © European Southern Observatory (ESO) 1997 Online publication: May 5, 1998 |