2. Observed vertical motions in the Gould Belt
The kinematic structure of the Gould Belt has been extensively studied for nearly one century now (see Frogel & Stothers 1977for an exhaustive review, and Torra et al. 1999for references to more recent, mostly pre-Hipparcos work). Most of the studies have explored the kinematical peculiarities of this structure on the basis of the velocity components of the stars in the directions of the galactic plane, as well as their gradients in those same directions. These motions reveal a rather complex expanding pattern associated to the Gould Belt. Torra et al. 1999have carried out a comprehensive study of the kinematics of the Gould Belt using Hipparcos data, complemented with radial velocities and Strömgren photometry which enabled them to obtain space velocities and ages for a large sample of O and B stars. Similar analyses have been published by Lindblad et al. 1997and Palou 1998. These studies are also restricted to motions in the directions of the galactic plane only, and I will refer to their results for the forthcoming discussions on this aspect.
Studies on the velocity component perpendicular to the galactic plane have been much less abundant, probably due to the subtler systematic patterns that may be expected and to the little use of radial velocities in this case, as nearly all the stars lie at low galactic latitudes. Fortunately, the proper motions of unprecedented quality provided by Hipparcos make it possible now to carry out meaningful studies based on proper motions of stars with precisely known distances, out to a distance from the Sun comparable to the whole extent of the Gould Belt. To complement the results of the works referred to in the previous paragraph, I have prepared a sample of Hipparcos stars fulfilling the following requirements:
The second constraint implies in practice that nearly all the stars are found within a distance of 400 pc from the Sun, which matches fairly well the size of the Gould Belt. At that distance, the average standard error in the Hipparcos proper motion for the stars in the sample, yr-1, results in a velocity error of only 1.7 km s-1, and is below 1 km s-1 for most of the stars in the sample. Therefore, no constrain has been set on the proper motions at the time of selecting the stars. To obtain space velocities, the sample has been complemented with radial velocities taken from the catalogues of Barbier-Brossat 1989, Andersen & Nordström 1983a, 1983b, 1985, Duflot et al. 1995, and Fehrenbach et al. 1997. It should be noted however that, with most of the stars lying at galactic latitudes , the radial velocity adds only a minor contribution to the velocity component perpendicular to the galactic plane discussed here. For this reason, the 20 stars without published radial velocities in the sample of 323 stars selected from the Hipparcos catalog have been retained, arbitrarily assigning them a zero radial velocity.
As shown by Torra et al. 1999, most of the stars with ages below years in the solar neighbourhood (which approximately includes the sample used here) belong to the Gould Belt on the basis of their spatial distribution. Therefore, kinematical contamination by stars not belonging to that structure is not expected to be significant in the present sample.
Let us assume that the perpendicular component of the velocity, W, has systematic variations along the galactic plane, expressed as , , where x is directed towards the galactic center and y towards the direction of galactic rotation. Such derivatives of W allow the definition of an axis of vertical oscillation of the stars around the galactic plane, , which has the property that W is proportional to the distance of the star to it:
where the z subindex indicates the component along the direction perpendicular to the galactic plane, and is the vector perpendicular to the axis of vertical oscillation joining it with the star. Eq. (1) can be written as
with x, y now being the components of the heliocentric position vector of the star in the directions defined above. is the distance of the nearest point of the axis of vertical oscillation to the Sun, which can be positive or negative depending on whether the x coordinate of that point is positive or negative. The sense of is defined so that, if , then the stars located in the hemisphere, limited by the plane defined by and the z direction, that faces the galactic center are moving towards decreasing z. The components of are thus
An expression like Eq. (2) can be written for each star in the catalog, setting up a system with the unknowns , , and that can be solved by least squares. Before doing that, however, it is necessary to consider the contribution to W due to the peculiar motion of the Sun, which otherwise would be engulfed in the last term on the right hand side of Eq. (2). Following Torra et al. 1999, I use km s-1 as the value to add to the observed W before using it as the input for the left hand side of Eq. (2).
The solution of Eq. (2) for the sample of stars used here reveals indeed a significant systematic pattern in the vertical motions of the stars under study, yielding the following values:
where is defined by
The result obtained for is very sensitive to the adopted value of , as one obtains pc when decreasing by only 1 km s-1. The values obtained for and are nevertheless practically independent of this choice. The gradient in W can be appreciated in Fig. 1, where its average over 100 pc-wide bins is plotted as a function of the distance to the axis of vertical oscillation. To test the real existence of such a gradient and its orientation, the system of Eqs. (2) has been solved repeatedly by adding each time a random component compressed between -10 km s-1 and 10 km s-1 to the W component of each stellar velocity, and by removing from the sample stars with km s-1 or km s-1, so as to ensure that the results obtained are not a consequence of a spurious effect or of stars with a highly deviant behaviour. Results within or near the above intervals have been consistently obtained every time for and . It should be noted that at (which is roughly the case for most of the stars in our sample) Eq. (2) can also be written as , where is the proper motion in galactic latitude, k is a conversion factor, and is the trigonometric parallax. Since can be determined directly from Hipparcos proper motions, the determination of and is practically insensitive to systematic effects introduced by random errors in the parallaxes of the stars in the sample.
The meaning of the results found here will be better interpreted in the framework of the models for the internal kinematics of the Belt, and their discussion is therefore deferred to Sect. 4. For the time being, it should be noted that the orientation of the axis of vertical oscillation is markedly different from the direction of the nodal line where the Gould Belt intersects the galactic plane, which runs approximately along the direction (Comerón et al. 1994, Torra et al. 1999). As will be seen, the offset between both directions has important implications on the internal dynamics of the Gould Belt. On the other hand, it will be shown as well that the rather low value of implies that the Gould Belt plane is near its maximum tilt at present. This feature was also noted by Frogel & Stothers 1977, on the basis of the absence of any significant gradient of the W component in the direction perpendicular to the galactic plane.
© European Southern Observatory (ESO) 1999
Online publication: November 3, 1999