3. An application to nearby B, A and F stars
To 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, U, V are the components of the velocity in a non rotating reference frame (while , are defined in a rotating reference frame): U is directed towards the galactic center, and V towards the direction of the galactic rotation. The relation between U, V and , is
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-1 ; the Ursae Majoris moving group (also called the Sirius moving group) near km s-1 ; and the Pleiades moving group near km s-1. For the components of the solar motion, we adopt km s-1.
As shown by Eq. (13), stars from a moving group with a deviation from the circular motion (characterized by the value of C) considerably larger than their average distance to the Sun can be expected to have similar values of . A diagram of vs. C can thus be used too to identify moving groups, in a similar way as is done in a U, vs. V. diagram. The diagram corresponding to our sample of stars appears in Fig. 2.
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 C are similar for these three groups, lying around 600 pc. This result is in an acceptable agreement with the value pc obtained by using the best fitting parameters to the HI kinematics in our Galaxy, obtained by Lin et al. 1969 (, km s-1).
3.1. Ages of the moving grups
Eq. (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 n in Eq. (8). This equation can then be used to derive a kinematic age for the group.
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 n best fitting the actual observations. To estimate the possible ages of moving groups, we use Eq. (8), with the approximation (12) and the condition that, near the Sun, . In this way, we will adopt the following ages:
* Hyades group: yr,
* Ursae Majoris group: yr,
* Pleiades group: yr,
These ages are found assuming in Eq. (8); however, using km s-1 as a representative value of the amplitude of the streaming motions induced by the spiral arms would change the ages by only years.
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 predictions
The 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 x. The lines in Fig. 3 show the average locus expected to be occupied by stars formed in the same passage of a spiral arm, whose vertical coordinate (depending on the adopted age ) has been adjusted according to the position of the actually observed moving groups, as explained in Sect. 3.1. The observational uncertainties may thus move these curves upwards or downwards, while the inflection centered near is a feature of the model, independent of the spiral structure parameters and common to all groups.
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 -x curve; data extending out to a few hundred parsecs away from the galactic center may reveal velocity gradients consistent with this picture. It should be pointed out that, although the presence of a spiral gravitational field is expected to distort the -x relationship, as will be discussed in the next section, the qualitative behavior remains similar, still leaving Ursae Majoris as the best placed group to test our hypothesis. Interestingly, a possible indication of the local maximum in the - x diagram associated to the Ursae Majoris group near the Sun may be present in the results of Méndez et al. 1992, who found a smaller spatial extent for the Ursae Majoris group as compared to the Hyades group: if one bases the criterion for membership allocation on the similarity of velocities, then the Ursae Majoris group should have a small extent in the radial direction, The results of Méndez et al. 1992 may thus be reflecting the abrupt velocity gradient with x expected at the location of the Ursae Majoris group.
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 introduced
As 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:
In these expressions, R, are galactocentric polar coordinates, having the Sun at the present time at , ); m is the number of arms, and is an adimensional factor representing the relative importance of the spiral perturbation as compared to the dominant axisymmetric galactic field.
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-1, , and . The value of estimated by Lin et al. 1969 is , which has been adopted in most subsequent works; however, the streaming motions of young stars and molecular clouds in our Galaxy derived from more recent observations suggest a smaller value of : for a population of objects with a small velocity dispersion , the amplitude of the perturbation in the tangential component of the velocity, , is
Using the values of and found by Clemens 1985 for galactic molecular clouds ( km s-1, km s-1), one finds , a value still sufficient to trigger galactic shocks (Shu, Milione, and Roberts 1973). Streaming motions in other low velocity dispersion populations such as OB stars also show small streaming amplitudes (Burton & Bania 1974, Comerón & Torra 1991), consistent with a value of below 0.05. We have adopted for the calculations presented here.
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. x as given by Eq. (13) may become possible, as the cyclic nature of the perturbing force only produces oscillations around the mean given by the non-armed case.
© European Southern Observatory (ESO) 1997
Online publication: May 5, 1998