## 3. An application to nearby B, A and F starsTo study the properties of the spiral structure in the solar
neighbourhood as derived from some moving groups, we used a sample of
3373 B-, A-, and F-type main sequence stars with known proper motions,
radial velocities, and Strömgren photometry. The data are mostly
from the Hipparcos Input Catalogue (Turon et al. 1992), complemented
with Strömgren photometry and new radial velocities from several
sources; see details in Jordi et al. 1996 and Chen et al. 1997. The
existence of moving groups in this sample can be appreciated in the
diagram shown in Fig. 1. In this diagram, where , are the
components of the solar motion with respect to the circular velocity.
Fig. 1 essentially reproduces the features in Fig. 1 of Palous &
Hauck 1986, but with a considerably larger sample of stars. There are
three major clusterings of stars visible in Fig. 1: the Hyades
moving group near km s
As shown by Eq. (13), stars from a moving group with a deviation
from the circular motion (characterized by the value of
The clusters in the diagram translate into
clusters in the diagram. The stars of the
Hyades, Pleiades and Ursae Majoris moving groups now cluster around
, , and
, respectively. Although with a rather large
scatter, due in a large part to observational errors, the average
values of ## 3.1. Ages of the moving grupsEq. (8) predicts a relationship between and the age. In principle, this could be used as a consistency test for our initial hypothesis if the ages of the groups were known with precission. However, some of the simplifications we have made in our treatment, such as the neglection of the gravitational perturbation associated to the density wave, can be expected to influence the real evolution of with time. We will evaluate this influence in Sect. 3.3; however, to do it, and to use Eq. (13) as a first approximation to the recognition of moving groups with our alternative criterion, it is convenient to have at least an estimate of the ages of the moving groups observed in the solar neighbourhood. The age of groups may be in principle better determined when there is an open cluster associated to them, as is the case of the Hyades and the Pleiades. The situation is somewhat worse for the Ursae Majoris group, whose age determination relies on stars selected only by their velocity, regardless of their position in the sky. An independent approach, used by Chen et al. 1997, is based on the joint use of proper motions, radial velocities, and Strömgren photometry to detect moving groups and simultaneously derive the kinematical properties and mean age of each one. Unfortunately, even a short review of the literature shows that
uncertainties in the determination of the age of even thoroughly
studied open clusters are comparable or even larger than the epicyclic
period. Discussions of this problem and its relation to the features
of the adopted stellar models can be found in Bertelli et al. 1992,
and it is illustrated with ages derived for single clusters by
different authors in Janes & Phelps 1994. Therefore, the published
values are of use only to the extent of allowing an estimate of the
number of epicyclic orbits elapsed since the formation of the group,
related to Several kinematic ages are in principle possible for each moving
group. As the differences among them are an entire number of epicyclic
periods ( yr), it is usually possible to choose
only one or two values of * Hyades group: yr, * Ursae Majoris group: yr, * Pleiades group: yr, These ages are found assuming in Eq. (8);
however, using km s It should be noted that these ages are those of the group members which are passing by the solar neighbourhood at present, and are in general different from the average age defined before and used in Eqs. (13). The ages are found from Eq. (13), using the fact that the coefficients , , are linear in , and replacing by its value in the solar vicinity and by . In this way, and assuming as before , we obtain: * Hyades group: yr * Ursae Majoris group: yr * Pleiades group: yr These are the values to be used in Eqs. (13); however, the former set of - values should be used when comparing to independent age estimates of each moving group in the solar neighbourhood. The distinction is not very important for the two older groups, but can be relevant for the Pleiades, as the difference between both values amounts to over 50 % of the present age of the group. On the other hand, it will be shown in Sect. 3.3 that best-fitting ages based only on stars with are not reliable, due to our neglection of the spiral arm gravitational field which can cause group displacements several hundred parsecs in amplitude. The age adopted for the Hyades group is consistent with the several age determinations listed by Janes & Phelps 1994 for both the Hyades and the Praesepe clusters, both having similar kinematics and age and possibly belonging to the same moving group. The average adopted by Janes & Phelps 1994 suggests that yr is the most likely value; these authors take yr as an average for Praesepe, a value closer to the estimate of Eggen 1993 and Chen et al. 1997 for members of the group at large, i.e., not only cluster members. Nevertheless, other studies (Soderblom et al. 1993, Williams et al. 1994, Balachandran 1995) support the coevality of both clusters. We will therefore adopt yr, for the Hyades moving group, although adopting , yr could also have some observational support. On the other hand, we should notice that, in his study of the Hyades moving group members, Eggen 1992a finds indications of several age groups with ages ranging from yr to yr. There are also large discrepancies as to the age of the Ursae Majoris group. A short review of derived ages is given by Boesgaard et al. 1988. The commonly accepted value is around yr (e.g. Soderblom & Mayor 1993a), but Eggen 1992b finds indications of a large age spread and the existence of groups considerably older than that value. Younger ages, more concordant with the above quoted value, are found by Palous & Hauck 1986, as well as Eggen 1983. Chromospheric activities among G and K members of the group (Soderblom & Mayor 1993b) also support an age around yr. We will adopt the above estimate of , yr in this paper. As for the Pleiades, the only value compatible with our hypothesis about the origin of its moving group and its estimated age is , yr, near the independent age estimate of Chen et al. 1997. Although the age commonly accepted for the Pleiades is yr (Lyngå 1987), this value has been a subject of controversy, given the apparent evidence for a significantly older age of the lower mass stars (Herbig 1962) or a large age spread among very low mass members (Steele et al. 1993). These problems have been recently reassessed: Mazzei & Pigatto 1989 have found an age in excess of yr for the more massive stars using improved evolutionary models, a result confirmed by Meynet et al. 1993. On the other hand, Stauffer et al. 1995 do not find indications of a significant non-coevality among very low mass members. Convincing evidence for a Pleiades age exceeding yr has been recently presented by Basri et al. 1996 based on the lithium test applied to some of the lowest mass members. We conclude that current estimates are well consistent with the kinematic age inferred by us assuming the Pleiades moving cluster to have been formed in a spiral shock. ## 3.2. Model predictionsThe fact that the relationship (13) implies that at well defined values of makes our definition of moving groups equivalent in practice to the classical ones, in the sense that membership criteria can be established on the basis of the similarity of the space velocities, whose direction is determined by the angle . As discussed above, this is a consequence of the correlations among epicyclic elements which appear when the samples analyzed are restricted to a distance of few hundred parsecs from the Sun, such as the one used in this paper. However, our model makes some predictions about the spatial variations of some epicyclic elements that may be checked in the future, when precise velocities are available for more distant stars, thus possibly revealing distant members of the moving groups discussed here. Fig. 3 shows the relation (13) for the different moving groups,
indicating the variations to be expected with
As can be seen, an extension of the observations out to a distance
of 500 pc or more should clearly reveal our predicted spatial
gradients in , and consequently whether or not
our hypothesis about the origin of the discussed moving groups is
supported by them. The case of the Ursae Majoris moving group is
especially tantalizing, as it is located near a local maximum in the
- Another consequence of our definition of moving group is that coevality is not strictly required, although in practice the age spread of stars of a given group within a radius of a few hundred parsecs should be generally small and hardly detectable. The age spread for a given group can be estimated from (11b), from which, neglecting the spread in (whose effect is much smaller than that of the spread in ), we can write Using spiral structure parameters from Lin et al. 1969, we obtain in the limits of very young () and very old () groups that and , respectively. Given the large uncertainties reviewed in Sect. 3.1, age gradients in the galactocentric direction may be actually difficult to detect even extending the observational horizon out to 1 kpc from the Sun. ## 3.3. Relation to the spiral structure and validity of the approximations introducedAs was already noted in Sect. 2, the treatment of the orbits of the stars subsequent to their formation in a spiral shock under the epicyclic approximation implies a number of simplifications which can be expected to distort the relation (13). A major neglected effect is due to the gravitational potential associated to the spiral density wave, while only the axisymmetric, slowly varying overall gravitational potential is used for the epicyclic approximation. To estimate the effect of our simplifications, we have numerically integrated the individual orbits of stars, which have been evolved in time under the influence of a radial force (where the subindex "0" denotes values at the Sun's position), corresponding to a flat rotation curve, and perturbed by the force components due to the density wave: with In these expressions, For the estimate of the effects induced by the perturbing
potential, we have used the best fitting parameters to the 21 cm
observations of HI in our Galaxy derived by Lin et al. 1969. It is a
well known fact that this set of parameters can fit rather
successfully the overall HI structure of our Galaxy, but fails in
explaining the details of the distribution of spiral arm tracers in
the solar neighbourhood (Lin & Yuan 1978; Elmegreen 1985). Lin
& Yuan 1978 consider that several spiral structures coexist in the
Galaxy, and that the description of our Galaxy in terms of a single
spiral pattern is an oversimplification; the latter objection would
therefore apply to the perturbations described by Eqs. (17), and would
leave the problem of integrating the orbits indetermined. However, we
may expect that orbit integration with the mentioned set of parameters
should still provide a valid estimate of the influence of the effects
neglected in our treatment. The integration of the orbits has thus
been performed using km s where Therefore, Using the values of and
found by Clemens 1985 for galactic molecular
clouds ( km s In our numerical integrations, we assume a number of probe stars to be formed at each position of the spiral shock, with initial motions given by the initial conditions (5). The component of the pre-shock streaming motion is set to zero; as already discussed in Sect. 3.1, a non-zero value of can be compensated for by a relatively small change in , and it is not expected to have any noticeable observational consequences. The spiral shock is assumed to be forming stars during an interval of time centered at the time of its passage by the local standard of rest; the interval is determined by the condition that stars formed out of it do not have the possibility of migrating to the solar neighbourhood during the time span of our simulations. The orbit of each individual star is tracked until the present moment, and the epicyclic elements are calculated at the end of the integration. The results are presented in Fig. 4, whose axes are the same as those of Fig. 3. The three values of correspond to the three moving groups discussed in Sect. 3.1. For the sake of completeness, each panel shows three sets of data: the filled circles delineate the loci defined by Eq. (13); the open triangles are the result of numerical integrations without spiral arms (); and the open circles show the result of numerical integrations with spiral arms. The results for clearly indicate the validity of the epicyclic approximation in our treatment. Only probe stars lying within 1 kpc from the Sun at the end of the integrations were considered in Fig. 4. The width of the bands of open triangles and circles are mostly due to the scatter in values of , which was neglected in deriving Eq. (13). Further scatter is expected in real observations as a consequence of the scatter in resulting from the velocity dispersion in the pre-shocked medium.
The use and shortcomings of Eq. (13) can be appreciated from Fig. 4. The qualitative behaviour of the integrated orbits with and without spiral arms is similar, and over a large part of the diagram, the solid line representing Eq. (13) is near the average locus of the open circles representing the stars moving under the spiral arm potential. The position of the inflection near is qualitatively reproduced in the orbits integrated with spiral arms, appearing also like an inflection or like a plateau, although this is generally the region where the difference between the cases with and without arms is larger. Given the amplitude of the displacement between the armed and
armless cases, the fit of the curves derived from Eq. (13) to the data
in Fig. 3 (which depends in the single parameter
) is not meaningful as far as their
intersections with the solar circle () are
concerned; put in other words, the intersection of those curves with
the line does not allow a precise dating of
moving groups, as was noted in Sect. 3.1. However, Eq. (13) is
still useful to estimate the gradients to be expected in the
plane, and the distance to the Sun where a
sample of stars should extend in order to test the real existence of
such gradients and the plausibility of the scenario proposed here.
Moreover, if the observational data are extended out to a distance
from the Sun larger than the size of the radial excursions induced by
the spiral structure, more accurate dating of moving clusters based on
the overall behaviour of vs. © European Southern Observatory (ESO) 1997 Online publication: May 5, 1998 |