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Astron. Astrophys. 325, 149-158 (1997)

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2. Effects of a spiral shock on stellar kinematics

The 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 [FORMULA] km s-1, i.e., less than 10% of the circular velocity at the position of the Sun, and the induced radial excursions are therefore small as compared to the radius of the solar circle (also less than 10%), we will use the first order epicyclic approximation to find out the kinematic signature expected in the stars formed by this process. This enables us to obtain a simple formulation, which predicts observable correlations among the constants of the epyciclic orbits, and a criterion of membership in moving groups based on these correlations. In doing so, we will be forced to introduce some simplifications, such as neglecting the streaming motions in the velocity induced by the gravitational potential associated to the density waves; including these perturbations would require to link the local spiral-arm structure to a grand design spiral pattern in our Galaxy. However, such a connection between the solar neighbourhood and the overall spiral structure of the Galaxy has been conflictive ever since the density wave theory was proposed. We will further discuss this point in Sect. 3.3, and will provide an estimate of the errors introduced by our simplifications by integrating stellar orbits, including the effects of the spiral perturbation of the gravitational field.

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 [FORMULA]. The x axis is directed towards the galactic center and the y axis points in the sense of galactic rotation. Orbits deviating little from circularity can be described by the simple epicyclic expressions (e.g. King 1989)

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where A is the corresponding Oort constant, [FORMULA] is the epicyclic frequency, and [FORMULA], [FORMULA], C, [FORMULA] are constants describing the size and position of the orbit and that of the star in it. They can be easily determined from observations by fixing [FORMULA] at the present time.

The spiral pattern and its associated shock rotate with an angular velocity [FORMULA]. 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 i (the pitch angle of the spiral pattern) with the direction of the galactic rotation, that we will assume to be small. As the interstellar gas crosses the shock, its velocity conserves the component parallel to the shock front, while, assuming the shock to be strong and rapidly dissipative, the velocity component perpendicular to it is greatly reduced. The shocked gas thus moves in a direction nearly tangent to the spiral arm; we will henceforth assume that this is also the initial velocity of the stars formed as a consequence of the compression associated to the passage of the gas by the spiral arm. In doing so, we implicitly assume that stars are formed out of clouds resulting from a phase transition in the diffuse interstellar medium or, alternatively, that the drag exerted by the shocked diffuse medium on pre-existing clouds can trigger star formation in them and slow them down to the velocity of the shocked gas in a time which is short as compared to the epicyclic period. This is also the approximation used by Bash & Peters 1976. Yuan & Grosbol 1981 have discussed the case of longer drag timescales, based on the results of Woodward 1976. In that case, the period over which star formation proceeds in the clouds is shorter than the timescale for drag of the clouds by the diffuse medium. If this were so, then the initial velocities of the stars should be of order of the streaming motions induced by shockless spiral arms (as they would form before the kinematic effects of the shock could be transmitted to the star forming clouds), rather than reflecting the post-shock velocity of the diffuse gas. Given that such streaming motions have smaller amplitudes than the deviations from circularity observed in the moving groups discussed here (see Sect. 3.3), we have chosen the first scenario to proceed in our study.

The circular velocity in the reference frame moving with the spiral shock is

[EQUATION]

where we will use for R (the distance to the galactic center) and [FORMULA] the values of 8.5 kpc and 25.9 km s-1 kpc-1, respectively, appropriate for the solar neighbourhood (Kerr & Lynden-Bell 1986). The velocity deviations of the gas from the circular motion in the galactocentric direction and in the direction of galactic rotation are respectively [FORMULA] and [FORMULA]. The components tangential and perpendicular to the shock are:

[EQUATION]

[EQUATION]

For a strong shock to exist, the condition [FORMULA] must be fulfilled, with [FORMULA] 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 [FORMULA] is large. Moreover, the response of the gas to the spiral gravitational potential provides a positive contribution to [FORMULA] from the term [FORMULA] 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 [FORMULA] ; 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, [FORMULA], [FORMULA]. In the reference frame used to define the epicyclic approximation, where [FORMULA], [FORMULA], we obtain

[EQUATION]

[EQUATION]

In writing Eqs. (4) in this form, we are assuming a trailing spiral and define the pitch angle i to be positive. Under the conditions of validity of our approximation, the term [FORMULA] in (4a) is much smaller than [FORMULA], so that

[EQUATION]

On the other hand, the term [FORMULA] in (4b) is much smaller than the term [FORMULA] in the right hand side of (4a). By comparison to (4a), Eq. (4b) can then be simply approximated by

[EQUATION]

The Eqs. (4c) and (4d) thus determine the approximate initial velocity at the time [FORMULA] of the passage of the spiral shock by a given position [FORMULA], [FORMULA]. If the term [FORMULA] is still larger than [FORMULA], 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:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

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 i (the pitch angle of the spiral arm). The shocked region then moves independently of the spiral pattern, and the galactic differential rotation tends to align its axis with the direction of the galactic rotation in a characteristic timescale given by shearing due to galactic differential rotation, [FORMULA] years. Eventually, this region may approach the solar neighbourhood, appearing as a ridge of stars perpendicular to the direction of the galactic center. If the distance from the Sun until which we can observe these stars is small enough, the fragment of this ridge accessible to observations will actually correspond to a small portion of the spiral arm which we can characterize by a single value of [FORMULA], [FORMULA] or, by Eqs. (5d), (5e), of [FORMULA], [FORMULA].

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, [FORMULA], [FORMULA]. We can easily relate [FORMULA], [FORMULA], and [FORMULA]: the position of the spiral shock can be represented locally by a straight line moving across our reference system:

[EQUATION]

where a is related to the present position of the spiral arm, and

[EQUATION]

Moreover, by Eqs. (5a), (5b),

[EQUATION]

and relating [FORMULA], [FORMULA] to the positions of the epicyclic orbits guiding centers by means of (5d), (5e),

[EQUATION]

[EQUATION]

whence

[EQUATION]

[EQUATION]

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 [FORMULA] can be estimated from (1a) and (5c), making [FORMULA]:

[EQUATION]

and the ratio of the two terms in the denominator of Eq. (10) is

[EQUATION]

For values of A, [FORMULA] typical of a flat rotation curve and small i, we have that [FORMULA], and therefore we can write (10) as

[EQUATION]

[EQUATION]

Developing the products, and retaining only the first-order terms,

[EQUATION]

[EQUATION]

We note that, from Eq. (6), [FORMULA], where [FORMULA] 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, a can be very large for old moving groups.

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 [FORMULA] with galactocentric distance; to simplify, we will use that, for a flat rotation curve,

[EQUATION]

where [FORMULA] is the value at the position of the Sun. Replacing (11b) in (8), using the approximation (12), and neglecting the term containing [FORMULA], which is in general much smaller than those containing [FORMULA], we finally obtain

[EQUATION]

where

[EQUATION]

[EQUATION]

[EQUATION]

with

[EQUATION]

Eq. (13), plus the condition [FORMULA] 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 [FORMULA], [FORMULA], 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 x is of order of or smaller than C, then [FORMULA] and [FORMULA] are always strongly correlated, regardless of the existence of a relationship like (13). Eq. (13), however, clearly shows the dependence of velocity on position implied by our definition of moving group, taking into account that [FORMULA]. Since, by Eqs. (1c), (1d), [FORMULA] essentially determines the direction of the velocity, our definition implies a gradient in both [FORMULA] and [FORMULA] with the x coordinate, independent of the gradient in V derived from the condition of isoperiodicity (see e.g. Eggen 1992a), noticeable only when x becomes of the same order as C. Moreover, Eq. (13) can have several possible solutions for [FORMULA] in the proximities of [FORMULA] for some values of [FORMULA] ; it is therefore possible in principle that stars of the solar neighbourhood having a common origin in the same spiral arm could appear as kinematically distinct groups. On the other hand, in some regions of the [FORMULA] - x diagram the gradient of velocity with position can be expected to be difficult to appreciate, given the scatter arising from uncertainties in the velocity determinations. High precission spatial velocities extending to more distant stars (up to a few hundred parsecs) should permit the detection of this effect and thus provide a justification of our proposed definition of moving groups on physical grounds.

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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