Astron. Astrophys. 350, 434–446 (1999) ASTRONOMYAND

ASTROPHYSICS

On the evolution of moving groups:an application to the Pleiades moving group

R. Asiain, F. Figueras, and J. Torra

Departament d’Astronomia i Meteorologia, Universitat de Barcelona, Avda. Diagonal 647, 08028 Barcelona, Spain

Received 25 May 1999 / Accepted 6 August 1999

Abstract. The disruption of stellar systems, such as open clus-ters or stellar complexes, stands out as one of the most rea-sonable physical processes accounting for the young movinggroups observed in the solar neighbourhood. In the present studywe analyse some of the mechanisms that are important in thekinematic evolution of a group of unbound stars, such as thefocusing phenomenon and its ability to recover the observedmoving group’s velocity dispersions, and the efficiency of discheating and galactic differential rotation in disrupting unboundstellar systems. Our main tools used to perform this analysis areboth the epicycle theory and the integration of the equations ofmotion using a realistic gravitational potential of the Galaxy.

The study of the trajectories followed by stars in each ofthe Pleiades moving group substructures found by Asiain et al.(1999) allows us to determine their stellar spatial and velocitydistribution evolution. The kinematic properties of these sub-structures are compared to those of a simulated stellar complexwhich has evolved under the influence of the galactic gravita-tional potential and the disc heating. We conclude that a constantdiffusion coefficient compatible with the observational heatinglaw is able to explain the velocity and spatial dispersions ofthe Pleiades moving group substructures that are younger than∼ 1.5 · 108 yr.

Key words: stars: kinematics – Galaxy: kinematics and dynam-ics – Galaxy: solar neighbourhood

1. Introduction

A “supercluster” can be defined as a group of stars gravitation-ally unbound that share the same kinematics and may occupyextended regions of the Galaxy. A “moving group” (MG here-after) or “moving cluster”, however, is the part of the superclus-ter that can be observed from the Earth (see, for instance, Eggen1994).

Independent of the physical process that leads to the for-mation of superclusters, there are basically two factors actingagainst their persistence in the general stellar background. First,

Send offprint requests to: R. AsiainCorrespondence to: rasiain@am.ub.es

when a group of unbound stars concentrated in the phase spacehas a certain velocity dispersion, the galactic differential rota-tion tends to spread them out very quickly in the direction of thegalactic rotation (e.g., Woolley 1960). Secondly observationsshow that the velocity dispersion of disc stars is increased withtheir age, which is usually interpreted as the result of a con-tinuous process of gravitational acceleration that may be pro-duced by different agents (e.g., Lacey 1991). This second factor,generally referred to asdisc heating, also disperses stars veryefficiently. It is therefore striking to verify that some of the clas-sical MGs are several108 yr old. Some authors have overcomepart of this problem by assuming that the velocity dispersionof these stellar groups must be very small – e.g. Eggen (1989),Yuan & Waxman (1977), Soderblom & Mayor (1993). Nev-ertheless, some recent studies (Chen et al., 1997; Sabas, 1997;Figueras et al., 1997; Chereul et al., 1998; Asiain et al., 1999)do not support such a small dispersions.

There have been several attempts to explain the origin of su-perclusters. Perhaps one of the most widely accepted explana-tion is the “evaporation” of the outermost stellar component (orcorona) of open clusters (e.g. Efremov 1988). Over time, openclusters are disrupted by the gravitational interaction with mas-sive objects in the Galaxy (such as giant molecular clouds), andas a result the open cluster members fill a long tube in space. Thepart of this tube that we can observe should have a small velocitydispersion – e.g. Weidemann et al. (1992), Eggen (1994). Al-though based in old stellar evolutionary models by Iben (1967),Wielen (1971) found that about 50% of the open clusters disin-tegrate in less than 2·108 years, a result supported by posteriorstudies (Terlevich, 1987; Theuns, 1992). If superclusters comefrom the evaporation of open clusters’s coronae, then Wielen’s(1971) results indicate that most superclusters should be youngor the last tracers of former clusters.

MGs may also be produced by the dissolution of largerstellar agglomerations, such as stellar complexes or fragmentsof old spiral arms (Woolley, 1960; Wielen, 1971; Yuan, 1977;Elmegreen, 1985; Efremov, 1988; Comeron et al., 1997). Thus,a MG could be a mixture of stars coming from different openclusters’ coronae and disrupted associations that share the sameorigin (and motion). This process of MG formation includes thefirst hypothesis mentioned here. Alternatively, Casertano et al.(1993) proposed that MGs are open clusters that evolved under

R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group 435

the gravitational influence of a large local mass that surroundsand traps them. However, it seems difficult to specify the natureof such a large mass. Finally, it has been proposed that super-clusters could actually be made of stars trapped around stableperiodic orbits (e.g. Mullari et al. 1994; Raboud et al. 1998).

In this paper we assume that superclusters occupied a smallvolume in the phase space in their first stages, when they becamegravitationally unbound. From that point on, they evolved underthe influence of the gravitational potential of the Galaxy, as itwould be the case in the classical picture of stellar evaporationfrom open clusters’ coronae. We then analyse different aspectsrelated to the evolution of MGs.

First, in Sect. 2 we use epicycle approximation to study theability of the “focusing phenomenon” (Yuan, 1977) to period-ically group MG stars when no disc heating is considered. InSect. 3 we show how to determine the stellar trajectories froman analytic expression of the galactic potential. The disc heat-ing effect on moving groups’ properties can be considered byintroducing random perturbations to the velocity componentsof stars. These trajectories allow us to study the evolution ofunbound stellar systems independent of the local galactic prop-erties (in contrast with epicycle approximation).

We focus our study on the origin and evolution of thePleiades moving group, since it is the youngest moving groupin the solar neighbourhood and, therefore, we may find it easierto recover its past properties with certain confidence (Sect. 4).We consider the different Pleiades substructures found in Asi-ain et al. (1999, Paper I), and estimate their kinematic age andgalactic position at birth. Finally, in Sect. 5 we simulate a stellarcomplex and determine the trajectory of its stars to study itsdisruption over time. The disc heating effect on the evolution ofthe stellar complex is evaluated.

2. The focusing phenomenon

The epicycle approximation allows us to study, in an analyticalway, the evolution of an unbound system of stars under theinfluence of the galactic potential. Under such approximation,the equations of motion of stars can be expressed as:

ξ′ = ξ′a + ξ′

b cos(κt+ φ)

η′ = η′a − 2Aξ′

at− 2ωξ′b

κsin(κt+ φ)

ζ ′ = ζ ′a cos(νt+ ψ) (1)

in the coordinate system (ξ′, η′, ζ ′) centered at the current po-sition of the Sun.ξ′ points towards the galactic center (GC),η′ is a linear coordinate measured along a circumference of ra-dius R (galactocentric distance of the Sun) and positive in thesense of the galactic rotation (GR), andζ ′ points towards thenorth galactic pole (NGP).κ is the epicyclic frequency, ν isthe vertical frequency, t is the time (=0 at present) andω isthe angular velocity of the Galaxy at the current position of theSun.ξ′

a, ξ′b, η

′a, ζ

′a, φ andψ are integration constants related to

the current position and velocity of a star (ξ′, η′

, ζ ′, ξ′

, η′, ζ ′

)

as:

ξ′a = −2ξ′

ω + η′

2B

ξ′b = −

(2Aξ′

+ η′

2B

)2

+

(ξ′κ

)2

1/2

η′a = η′

− 2ωξ′

κ2

ζ ′2a = ζ ′2

+ζ ′2ν2

φ = arctan

(−2ξ′

Bκ(η′ + 2ξ′A)

)

ψ = arctan

(−ζ ′

νζ ′

), (2)

where the B and A are the Oort’s constants. In Appendix A:we use Eqs. 1 to evaluate the evolution of space and velocitydispersions of a group of stars with low peculiar velocity withrespect to theirRegional Standard of Rest(RSR). Eqs. A2 showthat dispersions in position and velocity components oscillatearound constant values, except for the azimuthal coordinateη′,whose dispersionση′ increases witht. After a few107 yr, thisincrease is dominated by the secular terms for typical positionand velocity dispersions of young stellar groups, and it can beapproximated by the linear relationship

ση′ =AB

(4ω2

σ2ξ′

+ σ2η′

)1/2t . (3)

If we consider a sample of stars withσξ′ ≈ 0 pc andση′ ≈ 0 km s−1, then all dispersions in position and veloc-ity would oscillate around their mean values with an epicycle

frequencyκ. Stars would meet every∆t =2πκ

(at the position

of the Sun∆t ≈ 1.5 · 108 yr), a fact that has been referred toas “focusing phenomenon” (Yuan, 1977). However, when weapply Eq. 3 to a more realistic case where dispersions are closeto those of the open clusters and associations, i.e. a few parsecsin σξ′ and∼ 1–2 km s−1 in ση′ , we obtainση′ ≈ 1 kpc after at ≈ 5–10·108 yr – a period of time shorter than the age of somemoving groups detected in the solar neighbourhood (e.g. Chenet al. 1997; Paper I). Thus, galactic differential rotation disruptsvery efficiently unbound systems of stars. Nonetheless, if thedistribution of stars inη′ is supposed to be gaussian after thedisruption of the stellar system, an important proportion of theoriginal sample will be still concentrated in space after a longperiod of time; for instance, forση′ = 1 kpc we can find still∼ 24% of the initial stars in a region 600 pc long inη′

after5–10·108 yr.

3. Stellar trajectories

The epicycle approximation holds for stars that follow almostcircular orbits around the center of the Galaxy. Although validfor most of our stars, some of them, with high peculiar veloci-ties with respect to theLocal Standard of Rest(LSR), perform

436 R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group

radial galactic excursions of a few kpc in length. The galacticgravitational potential changes significatively at different pointsof their trajectories, and so the first order approximation is nolonger valid. In addition, the vertical motion is poorly describedby a harmonic oscillation as stars gain certain height over thegalactic plane. For a more rigorous analysis, we need to use a re-alistic model of the galactic gravitational potential. Stellar orbitswill be determined from this model by integrating the equationsof motion (Sect. 3.1). Moreover, the addition of a constant scat-tering in this process will allow us to account for the disc heatingeffect on the stellar trajectories (3.2).

3.1. The galactic potential

Expressed in a cartesian coordinate system (ξ, η, ζ) centeredat the position of the Sun1 and rotating at a constant angularvelocityω,

ω =

√√√√ 1R

(∂Φ∂R

)

, (4)

the equations of motion of a star, assuming that the gravitationalpotential of the GalaxyΦG(R, θ, z; t) (in cylindrical coordinatescentered on the GC) is known, are:

ξ = −∂ΦG

∂ξ− ω2

(R − ξ) − 2ωη

η = −∂ΦG

∂η+ ω2

η + 2ωξ

ζ = −∂ΦG

∂ζ. (5)

A fourth order Runge-Kutta integrator allows us to numericallysolve these equations whenΦG is known, obtaining the tra-jectory of the star. To get a realistic estimation of the galacticgravitational potential we decompose it into three parts: the gen-eral axisymmetric potentialΦAS, the spiral armΦSp, and thecentral barΦB perturbations to the first contribution, i.e.

ΦG = ΦAS + ΦSp + ΦB.

We adopt the model developed by Allen & Santillan (1991)for the axisymmetric part of the potential,ΦAS(R, z), becauseof both its mathematical simplicity – which allows us to deter-mine orbits with a very low CPU consumption – and its updatedparameters. The model consists of a spherical central bulge anda disk, both of the Miyamoto-Nagai (1975) form, plus a mas-sive spherical halo. This model is symmetrical with respect toan axis and a plane. The authors adopted the recommendationsof the IAU (Kerr & Lynden-Bell, 1986) for the galactocentricdistance of the Sun (R = 8.5 kpc) and the circular velocity atthe position of the Sun (Θ= 220 km s−1). The Oort constantsderived from this model are well within the currently acceptedvalues.

1 ξ points towards the GC,η towards the sense of the GR, andζtowards the NGP

Spiral arm perturbation to the potential is taken from Linand associates’ theory (e.g. Lin 1971 and references therein),that is:

ΦSp(R, θ; t) = A cos(m(Ωpt− θ) + φ(R)) , (6)

where

A =(Rω)2fr0 tan i

m,

φ(R) = − m

tan iln(R

R

)+ φ0.

A is the amplitude of the potential,fr0 is the ratio between theradial component of the force due to the spiral arms and that dueto the general galactic field.Ωp is the constant angular velocityof the spiral pattern,m is the number of arms,i is the pitchangle,φ is the radial phase of the wave andφ0 is a constantthat fixes the position of the minimum of the spiral potential.According to Yuan (1969), we adoptfr0 = 0.05 (this valuehas been confirmed in a recent study by Fernandez, 1998) andΩp = 13.5 km s−1 kpc−1.

We consider a classical two arm pattern (m = 2)(Yuan, 1969; Vallee, 1995). If we assume that the Sagittariusarm is located at a galactocentric distance RSag = 7.0 kpc,and that the interarm distance in the Sun position is∆R =3.5 kpc as suggested by observations on spiral arm trac-ers (Becker & Fenkart, 1970; Georgelin & Georgelin, 1976;Liszt, 1985; Kurtz et al., 1994), then the pitch angle can be de-termined to be:

tan i =m

2πln(1 +R−1

Sag ∆R). (7)

For m = 2, we obtaini = 7.35 (close to the Yuan’s (1969)valuei = 6.2).

For t = 0 andθ = 0, the adopted valueφ0 leads to a mini-mum in the potential at the observed position of the Sagittariusarm (R = RSag = 7.0 kpc, l = 0). φ0 can thus be expressedas:

φ0 = π +m

tan iln(RSag

R

). (8)

For i = 7.35 andm = 2 we obtainφ0 = 0.131 rad.The central bar potential we use here is, for simplicity, a tri-

axial ellipsoid with parameters taken from Palous et al. (1993),i.e.

Φb(R, θ, z; t) = − GMbar(q2bar + x2 +

a2bar

b2bar

y2 +a2

bar

c2bar

z2)1/2 , (9)

wherex = R cos(θ−ΩBt−θ) andy = R sin(θ−ΩBt−θ),with θ = 45 (Whitelock & Catchpole, 1992).abar, bbar andcbar are the three semi-axes of the bar, withqbar its scale length,

and withabar

bbar=

10.42

,abar

cbar=

10.33

, and qbar = 5 kpc.

The adopted total mass of the bar,Mbar, is 109 M, and itsangular velocity,ΩB , is 70 km s−1 kpc−1 (Binney et al., 1991).Although most of the parameters that define the bar are very

R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group 437

uncertain, the effect of the bar on the stellar trajectories becomesimportant only after several galactic rotations, which requires alength of time greater than the age of the stars considered in ourstudy.

3.2. Disc heating in the stellar trajectories

The observational increase in the total stellar velocity dispersion(σ) with time (t), or disc heating, can be approximated by anequation of the form:

σ(t)n = σn + Cvt, (10)

whereσ is the dispersion at birth andCv the “apparent diffusioncoefficient” (Wielen, 1977). The constantsn, σ andCv giveone important information on the physical mechanism respon-sible for the disc heating (Lacey, 1991; Fridman et al., 1994).

Our sample of B and A main sequence type stars, de-scribed in Paper I, allows us to determine an accurate and de-tailed disc heating law for the last109 yr, given the qualityand uniformity of our ages and the size of this sample (2 061stars). A standard nonlinear least-squares method (Levenberg-Marquardt method) has been applied to fit the heating co-efficients to our data (Fig. 1). In this way we obtainσ ≈12 km s−1, n ≈ 5 andCv ≈ 0.01 (km s−1)nyr−1, similarto Lacey’s (1991) results forn = 5, i.e.σ = 15 km s−1 andCv = 0.01 (km s−1)5yr−1. When we fixn to 2, we obtainσ ≈ 15 km s−1andCv ≈ 5.6 · 10−7 (km s−1)2 yr−1, whichis again in agreement with Lacey’s (1991) (σ = 15 km s−1

and Cv = 5 · 10−7 (km s−1)2 yr−1) and Wielen’s (1977)(σ = 10 km s−1 andCv = 6 · 10−7 (km s−1)2 yr−1) fitswith n = 2.

Our accurate observational heating law (circles in Fig. 1) isfar from being smooth. Two special features can be observedon this law: first, the velocity dispersion shows a steep increaseduring the first∼ 4 · 108 yr, probably due to the phase mix-ing of young stars. After this point, this increase becomes lesspronounced. Second, an almost periodic oscillation seems to besuperimposed over the “continuous” heating law. This oscilla-tion (period≈ 3 · 108 yr) could be the signature of an episodicevent (see for instance Binney & Lacey 1988, Sellwood 1999).

Since we do not know the details about the mechanisms thatperturb the stellar orbits and cause the disc heating effect, weassume for simplicity that the accelerating processes acting ona given star can be approximated by a sequence of independentand random perturbations of short duration (Wielen, 1977). Inorder to evaluate the magnitude of these perturbations we gen-erated a sample of 500 stars located around the position of theSun att = 0, and uniformly distributed in a 100 pc/side cube.Initially, these stars are moving with the same velocity as theLSR, with an isotropic dispersion in velocityσ. During theprocess of orbit determination, the velocity of the star is in-stantaneously changed by∆v every∆ yr. This∆v is normallydistributed around zero with a (constant) isotropic dispersionσh in each component. The equations of motion are integratedusing only the axisymmetric part of the galactic potential (ΦAS).This allows us to introduce the heating effect only through the

0 2 4 6 8 10τ [10

8yr]

10

15

20

25

30

σ [k

m s

−1 ]

Fig. 1.Fit of Eq. 10 to our data (see text), using stars closer than 300 pcfrom the Sun, with 100 stars per bin.Solid line: σ, Cv andn arefitted; dashed line: σ andCv are fitted, whereasn is fixed to 2. Theerror bars on points include only statistical uncertainties, calculated asεσ = σ/

√2N , whereN is the number of stars per bin

0 2 4 6 8 10τ [10

8yr]

10

15

20

25

30

σ [k

m s

−1 ]

Fig. 2.Velocity dispersion vs. time, determined by integrating the equa-tions of motion of a simulated sample when an isotropic diffusion isapplied at each time-step.Dashed line: observational fit to disc heatingwith a fixed slopen=2

parameterσh, avoiding possible redundances, such as the scat-tering of disc stars by spiral arms. The best approximation to theobservational heating law is obtained whenσh = 1.45 km s−1

for ∆t = 107 yr, andσ ≈ 15 km s−1. It can be demonstratedthatσh and∆t are related with the “true diffusion coefficient”

D introduced by Wielen (1977) just asD =σ2

h

∆t. On the other

handCv = 2.95 D, as demonstrated by Wielen (1977) using theepicycle theory. We obtain in this way an independent estima-

tion ofCv, i.e.Cv = 2.951.452

107 = 6.2 · 10−7 (km s−1)2 yr−1.

This value is in an excellent agreement with our previous esti-mation. In Fig. 2 the evolution of the total velocity dispersionfor this simulated sample is compared with the fit of Eq. 10 tothe observational data whenn = 2, showing an excellent match.

438 R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group

Table 1. Number of members (N ), velocity components and ages(along with their dispersions) of the Pleiades MGs found in Paper I

Moving N U V W τGroup [km s−1] [km s−1] [km s−1] [108 yr]

B1 34 -4.5(4.7) -20.1(3.3) -5.5(1.9) 0.2(0.1)

B2 75 -10.7(5.3) -18.8(3.7) -5.6(2.2) 0.6(0.2)

B3 50 -16.8(5.1) -21.7(2.7) -5.6(4.6) 3.0(1.2)

B4 53 -8.7(4.8) -26.4(3.3) -8.5(4.7) 1.5(0.5)

This procedure will be used in Sect. 5 to simulate the evolutionof a stellar system under the influence of the disc heating.

4. The Pleiades moving groups

In Paper I we used a method based on non-parametric densityestimators to detect MGs among a sample of 2 061 B and A mainsequence type stars in the four-dimensional space (U,V,W,log(age)). We used HIPPARCOS data as well as radial velocitiesfrom several sources (see Paper I for the details) to determinethe stellar spatial velocities, and Stromgren photometry for thestellar ages. Tables 1 and 2 in Paper I show the main properties ofthe MGs when separated in the (U,V,W,log (age)) and (U,V,log(age)) spaces respectively. Since the W-velocity component isless discriminant than the other three variables, we will focusour study on the MGs in Table 2 of Paper I (Table 1 here).In particular, as already mentioned in the Sect. 1, we will dealwith those MGS whose velocity components resemble those ofPleiades open cluster.

It is interesting to compare these results with those recentlyobtained by Chereul et al. (1998), who also found similar sub-structures in the Pleiades and other MGs by means of a waveletanalysis performed at different scales. The velocity dispersionsof their substructures are quite a bit larger than those found forclassical MGs, in agreement with our results. In their analysis,they did not consider the stellar age as a discriminant variable,which prevents them from detecting those substructures that arestrongly defined in age but not so well defined in the velocitycomponents. Another important difference between both meth-ods is that Chereul et al. (1998) did not determine photometricages for A0 to A3 stars, since no reliable metallicity is avail-able for them. Instead, they computed a “paliative” age thatproduces an artificial peak in age of107 yr, and a lack of otheryoung stars (up to∼ 5 · 108 yr). As mentioned in Asiain et al.(1997), a metallicityZ = 0.02 is representative, in a statisticalsense, of (normal) A type stars. Using this value, we did notobserve any lack of stars in the young part of the age distribu-tion (Paper I). On the other hand, since we do not use F typestars in our study because of the high uncertainties involved inthe process to determine their ages, our data do not allow us toconfirm the existence of the∼ 109 yr old Pleiades substructurefound by Chereul et al. (1998).

Using the numerical integration procedure described inSect. 3.1, and the mean properties (nuclei) of the Pleiades sub-structures found in Paper I (Table 1), we have computed thetrajectories of these substructures from the present up to the

Fig. 3. Trajectories of the MG’s nuclei in Table 1 backwards in time,from present (t = 0) until their mean age. The dashed lines representspiral arms, as defined in Sect. 3.1. The reference system is rotatingwith the same angular velocity as the spiral arms (Ωp)

moment they were born (Fig. 3). This latter age is defined asthe average age of the MGs constituent members. The youngestgroup, i.e. B1, is composed of Scorpio-Centaurus (Sco-Cen)OB association members (Paper I), and it was born in the in-terarm region. In Sect. 4.1 we study the evolution of this group.Since B1 is still too young to be affected by the disc heating ef-fect or phase space mixing, we can determine its kinematic agewith some confidence. The B2 group is considered separately inSect. 4.2. This is also quite a young group and contains accurateinformation on some of the closest associations. The birthplaceof the older groups, i.e. B3 and B4, is close to a minimum of thespiral arm potential, which seems consistent with their beingborn around this structure. Details on their spatial and velocityevolution are given in Sect. 4.3.

In recent studies based on the velocity field of Cepheids,Mishurov et al. (1997) and Mishurov & Zenina (1999) obtaineda set of spiral arm parameters which clearly differ from thoseadopted here (e.g., in Mishurov & Zenina (1999),φ0 = 142

andΩp − ω ≈ 0.5 km s−1 kpc−1). When considering thesenew parameters, all Pleiades MG substructures turn out to bealso placed, at birth, around the spiral arms. In this particularcase, the Pleiades substructures appear, at birth, concentratedin a small area of the Galaxy. We also tested that the mainconclusions of the current paper does not critically change withthe selection of any of this two spiral arm patterns, as expected.

4.1. Scorpio-Centaurus association

Because of the short age of these stars and the quality of our datawe are able to determine quite precisely these stars’ kinematicage, defined as the time at which they were most concentratedin space – assuming that they are gravitationally unbound. Weconsider here only those stars in B1 that are concentrated inspace (see Fig. 7 in Paper I). With the exception of HIP84970 andHIP74449, all of these stars were classified as members of Sco-

R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group 439

−0.3−0.2−0.1 0.0 0.1ξ’ [kpc]

−0.1

0.0

0.1

−0.1

0.0

0.1

−0.1

0.0

0.1

η’ [

kpc]

−0.1

0.0

0.1

−0.1

0.0

0.1

−0.2−0.1 0.0 0.1

ζ’ [

kpc]

0

−4

−8

−12

−16

Fig. 4. Spatial distribution of the 27 stars in B1 sub-group that arespatially concentrated in space as a function of time. Numbers in theupper right corners indicate the time in Myr

Cen association by de Zeeuw et al. (de Zeeuw et al., 1999)2.Their position in the galactic and meridional planes as a functionof time are shown in Fig. 4. It is quite evident that betweenaround 4 and 12 Myr ago these stars were closer to each otherthan they are at present. However, observational errors produceadditional dispersions on velocity and position, and thereforethe maximum spatial concentration we find is shifted to thepresent. Fig. 4 also shows us that the last intersection of theSco-Cen association with the galactic plane was between 8 and16 million years ago.

The evolution of the errors or dispersions in the stellar po-sition and velocity can be calculated more precisely by meansof the epicycle approximation (Eqs. A2). We now consider the

2 Though B1 is considered as a single group because of the fewmembers it contains, most of its stars belong to three neighbouring as-sociations: Upper Scorpius (US), Upper Centaurus Lupus (UCL), andLower Centaurus Crux (LCC) (de Geus et al., 1989; Blaauw, 1991;de Zeeuw et al., 1999).

−0.4 −0.3 −0.2 −0.1 0.0 t [10

8 yr]

0.00

0.05

0.10

0.15

0.20

σ ρ [k

pc]

Fig. 5. Evolution of the observed dispersion (dot-dashed line), meanpropagated error (dashed line) and intrinsic dispersion (solid darkerline) in ρ (distance to the Sun), determined from the most spatiallyconcentrated stars in the B1 group

fact that the observed dispersion in position inside a given MG(σρ,obs, whereρ is the distance from the LSR to a given star)can be decomposed into two parts, i.e.

σ2ρ,obs(t) = σ2

ρ,int(t) + σ2ρ,err(t), (11)

whereσρ,int(t) is the MG intrinsic dispersion at the timet, andσρ,err(t) is the MG mean observational error.

Since bothσρ,obs andσρ,err can be easily determined at dif-ferent epochs (σρ,obs(t) can be calculated by integrating thestellar orbits until timet, andσρ,err(t) can be propagated frompresent using Eqs. A2), we can also derive the timetmin atwhichσρ,int(t) was minimum. In Fig. 5 we show the evolutionof σρ,obs, σρ,err andσρ,int. We observe a clear minimum inσρ,int aroundtmin ≈ 9 · 106 yr, which corresponds to B1 kine-matic age. At the minimum spatial concentration we find theintrinsic spatial dispersions areσξ′ = 20 pc,ση′ = 22 pc, andσζ′ = 15 pc, smaller than their current values ofσξ′ = 30 pc,ση′ = 33 pc, andσζ′ = 16 pc. Also, their intrinsic velocity disper-sions were smaller attmin, and<∼ 2 km s−1 in each component.

There is a clear discrepancy between the kinematic and the“photometric” age of B1. This could be due to several reasons.First of all, most of the stars in B1 are massive, and their at-mospheres are probably rotating very quickly. Thus, the ob-served photometric colors are affected by this rotation, and soare the derived atmospheric parameters. The photometric ages,determined from these atmospheric parameters and stellar evo-lutionary models (Asiain et al., 1997), are systematically over-estimated because of this effect. In order to evaluate this effect,we have corrected the observed photometric colors for rotationby considering a constant angle of inclination between the lineof sight and the rotation angle (i = 60), and a constant atmo-spheric angular velocity (ω = 0.6 ωc, whereωc is the criticalangular velocity). In this way we obtain a mean age≈ 1.1·107 yr,and even smaller values forω > 0.6 ωc. More details on the

440 R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group

Table 2. Number of members (N ), spatial velocity components andages (along with their dispersions) of the Sco-Cen associations foundby de Zeeuw et al. (1999), determined from those members present inour sample

Assoc. N U V W τ[km s−1] [km s−1] [km s−1] [108 yr]

US 14 -3.5(4.3) -15.7(2.1) -6.4(1.8) 0.4(0.5)

UCL 32 -3.7(7.0) -19.7(4.1) -4.9(2.1) 0.7(1.0)

LCC 19 -6.8(4.7) -18.5(6.5) -6.4(1.7) 0.9(1.2)

TOTAL 65 -4.6(6.1) -18.5(4.9) -5.7(2.1) 0.7(1.0)

Table 3. Same as Table 2, now removing a few stars with peculiarvelocity components and ages

Assoc. N U V W τ[km s−1] [km s−1] [km s−1] [108 yr]

US 12 -3.8(4.0) -15.5(2.1) -6.6(1.8) 0.2(0.1)

UCL 28 -1.7(4.4) -20.4(3.6) -4.4(1.5) 0.4(0.3)

LCC 16 -5.1(2.7) -20.6(4.0) -6.0(1.6) 0.4(0.3)

TOTAL 56 -3.1(4.2) -19.4(4.0) -5.3(1.9) 0.4(0.3)

correction for atmospheric rotation can be found in Figueras& Blasi (1998). Second, before applying our method to detectmoving groups (Paper I) we eliminated all stars with relativeerrors in age bigger than 100%, which slightly bias the age ofyoung groups to larger values.

In a recent study de Zeeuw et al. (1999) carried out a cen-sus of nearby OB associations using HIPPARCOS astrometricmeasurements and a procedure that combines both a convergentpoint method and a method that uses parallaxes in addition to po-sitions and proper motions. Their sample of neighbouring starsis much larger than ours, since radial velocities and Stromgrenphotometry are needed for our analysis. They found 521 stars inthe Sco-Cen association, of which only 65 were in our sample.From these stars we have determined the mean velocity compo-nents of each association, which are in close agreement to thoseof B1 (Table 2). However, their dispersions and ages are quitea bit larger than expected for a typical association. A deeperanalysis revealed to us the presence of some stars in de Zeeuwet al.’s (1999) associations whose peculiar velocities are respon-sible for their large velocity dispersion. Consistently, the agesof these stars are also peculiar (τ ≈ 1–5 · 108 yr). Removingthese stars and a few others with anomalously large ages3, all ofwhich probably belong to the field, we find a more reasonablekinematic properties and ages for these associations (Table 3).

4.2. B2 group

Even though B2 is quite a young group, the propagation ofage uncertainties over time prevents us from determining thegroup’s kinematic age by means of the procedure developed for

3 Eliminated stars: HIP79599 and HIP80126 in US; HIP70690,HIP73147, HIP80208 and HIP83693 in UCL; HIP60379, HIP62703and HIP64933 in LCC

−4 −3 −2 −1 0t [10

8 yr]

0

20

40

60

80

Num

ber

of S

tars

B2B3B4

Fig. 6. Number of MG stars placed around their nucleus (at a distance≤ 300 pc) as a function of time, for MGs B2, B3 and B4

the B1 group. Instead, we have determined the trajectories ofeach star in B2, and that of the MG’s nucleus itself. The spa-tial concentration of these stars is propagated back in time bycounting the number of stars found in 300 pc around the MGnucleus (Fig. 6). We do not find any maximum in spatial concen-tration during the last108 yr. To better understand the structureof this group we plot in Fig. 7 the position of these stars andtheir velocity components on the galactic and meridional plane,referred to the LSR and corrected for galactic differential rota-tion. We observe that it is actually composed of several spatialstellarclumps, each of which has its own kinematic behaviour,although certain degree of mixture is also evident. In particular,one of these clumps (most of the filled circles in Fig. 7) per-fectly overlaps with the Sco-Cen association mentioned above(20 of these stars are classified as members of this associationby de Zeeuw et al. 1999). This clumps’s velocity components,and kinematic age, are also very similar to those found for theB1 group. Once again, the photometric age of these stars isprobably overestimated because of the high rotation velocityof their atmospheres4. A second clump is coincident with theCassiopeia-Taurus (Cas-Tau) association in both position andvelocity spaces (most of the empty circles in Fig. 7). There aretwo dominant streams among these stars. In the heliocentric sys-tem, corrected for galactic differential rotation, their velocitiesare (U,V)'(-10,-18) km s−1 and (-13,-23) km s−1 respectively(they share a common W component∼ −6 km s−1). Their agesare5–6 · 107 yr. These values are in good agreement with thosefound by de Zeeuw et al. (1999) for Cas-Tau association, i.e.(U,V,W) = (-13.24,-19.69,-6.38) km s−1 with an age of5·107 yr(actually, about half of them are classified as Cas-Tau membersby these authors). Finally, we observe a less concentrated group

4 Our MG finding algorithm (Paper I) was able to detect 47 of the 56Sco-Cen association members in our sample. However, the classifica-tion procedure we applied could not group them into a single MG. Theage of these very young stars, affected by the important uncertaintiesalready mentioned, is probably causing this misclassification

R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group 441

Fig. 7.Position of B2 stars at present and their velocities referred to theLSR and corrected for galactic differential rotation. Different symbolsare used for three galactic longitude ranges. The size of the symbols isinversely proportional to the age of the stars they represent

of stars atξ ≈ 0.05 kpc andη ≈ 0.2 kpc, orl ≈ 90 (diamondsin Fig. 7). The kinematic properties and age of this clump arevery similar to those of the Cas-Tau association, whereas itsposition quite well agrees with that of the recently discoveredCepheus OB6 association (Hoogerwerf et al., 1997).

Thus, B2 seems to be the superposition of several OB as-sociations from the Gould Belt, which are mixing with eachother in the process of disintegration. These stars were classi-fied as belonging to the same MG in Paper I since they roughlyshare the same kinematics and age, a consequence of their beingformed from the same material.

4.3. Older Pleiades subgroups

Groups B3 and B4 in Table 1 are considerably older than B1and B2, and therefore only few details on the conditions inwhich they were formed can be recovered. In particular, theuncertainties inη′ due to observational error increase almostmonotonically with time. Though this increase can be closely

Fig. 8.Stellar trajectories of the B3 members during the last 3.5·108 yr.The position of the B3 nucleus att = 0, -1.5 and -3108 yr (in units of108 yr) is indicated with an open circle. The reference system is thesame as in Fig. 3

approximated by Eq. 3 for an axysimmetric galactic potential,when considering the termsΦSp and ΦB this approximationbreaks (it only works during the first≈ 108 yr). To estimatethe effect of typical errors in both the position (≈ 10–15 pc)and velocity (≈ 2–3 km s−1) components of moving groupson η′ as a function of time, we have simulated a stellar groupwhose dispersions in the phase space equals those errors, thendetermined their orbits. We obtain a dispersion inη′ due to theseerrors of≈ 500 pc after∼ 3 ·108 yr. Moreover, we cannot knowthe way in which individual orbits have been perturbed due tothe disc heating effect, producing additional spatial and velocitydispersions.

As mentioned above, these older groups seem to have beenborn in the vicinity of the spiral arms. The position at birth ofgroups B3 and B4 are especially interesting: on the one hand, thetrajectories followed by these groups show a maximum galac-tocentric distance at the moment they were born, which cor-responds to a minimum kinematic energy (Figs. 3 and 8); onthe other hand, by determining the trajectories of the B3 mem-bers we observe two focusing points close to the B3 and B4birthplaces (Fig. 8). An identical result is obtained when usingB4 stars. Following Yuan (1977), these points could be inter-preted as the birthplace of B3 and B4. For spatially concentratedgroups of stars with small velocity dispersions epicycle theorypredicts (Sect. 2) the focusing phenomenon will be produced

every∆t =2πκ

. Since the Pleiades groups oscillate around a

guiding center placed atR ≈ 8.0–8.2 kpc, the correspondingepicycle frequency isκ ≈ 38–40 km s−1 kpc−1(determinedfrom ΦAS, Sect. 3.1), with∆t ≈ 1.5–1.6·108 yr. This period∆t is compatible with the ages of B3 and B4.

By following the same procedure used for the B2 group westudy the spatial concentration of the older groups (Fig. 6). ForB4 there is a maximum concentration att ≈ −1.5 · 108, whichcorresponds to this MG’s age, whereas we observe two peaksin concentration for B3 att ≈ −1.5 and−3 · 108 respectively,

442 R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group

the first one corresponding to the last focusing event, and thesecond one corresponding to the average age of its members.

5. Evolution of a stellar complex

As defined by Efremov (1988), Stellar Complexes (SC) are“groupings of stars hundreds of parsecs in size and up to 108 yr inage, apparently uniting stars born in the same gaseous complex”.Associations and open clusters are the brighter and denser partsof these huge gaseous complexes. In other words, and accord-ing to Elmegreen & Elmegreen (1983), the fundamental unit ofstar formation is an HI-supercloud of∼ 107 M which, dur-ing fragmentation into∼ 105-106 M giant molecular clouds,gives birth to a SC. From these giant molecular clouds clustersand associations are formed. Examples of SCs are the GouldBelt in our Galaxy, 30 Doradus in the LMC, and NGC 206 inAndromeda.

In this section we explore the possibility that such objects arethe progenitors of MGs. Since both galactic differential rotationand disc heating effect tend to disrupt any unbound system ofstars on a short timescale, the large number of stars born insidea SC – and therefore roughly sharing the same kinematics andages – may ensure a high spatial concentration for a longer time,especially at the focusing points.

In order to analyse the possible link between SCs and MGs,we have generated a SC as a set of different unbound systemsborn at different epochs. Its main characteristics are taken fromthe literature and are described in what follows. Although thereare no preferences on the galactic plane region where SCs areformed (they can even be found in the interarm region, as sug-gested by the observations of OB Associations in Andromeda,e.g. Magnier et al. 1993), we select the position and velocity ofthe B4 nucleus at birth (t = −1.5 · 108 yr), since in this waywe can compare the results here obtained with observationson this real group. According to several authors’ estimations(Efremov, 1988; Elmegreen, 1985; Efremov & Chernin, 1994),SCs are several hundreds of parsecs in size. The original shape ofour simulated complex has been designed as an ellipsoid whoseequatorial plane is 250 pc in radius and lies on the galactic plane,and its vertical semi-axis is 70 pc. Six associations are born in-side it at different epochs, randomly distributed inside this vol-ume. Since the dispersion in age can be large when consideringthe SC as a whole (Efremov, 1988), we impose the conditionthat one associations borns every 107 yr. Hence, the first one isborn att = −1.5 · 108 yr, the second one at−1.4 · 108 yr, andso on. The mean velocities of these associations at birth followa gaussian distribution around the B4 nucleus velocity, with an(isotropic) dispersion similar to the velocity dispersion insidea molecular cloud, i.e.≈ 5/

√3 km s−1 in each component

(Stark & Brand, 1989). Each of these associations is defined asfollows: they contain 500 stars, which are randomly distributedinside a sphere of 15 pc in diameter, an intermediate value be-tween the smallest associations (e.g. the Trapezium cluster) andthe largest ones (Blaauw, 1991; Brown, 1996); and their internalvelocity dispersion is 2 km s−1 in each velocity component, asobserved in most of the closer OB Associations (Blaauw, 1991)

and in molecular clouds (Scoville, 1990); their internal disper-sion in ages is 107 yr (Efremov, 1988). We also assume that allthe simulated stars survive up to the time (t) when we study thesystem.

The orbit of each simulated star from the moment it was bornup to the present (t = 0) is determined considering the galacticgravitational potential (ΦAS + ΦSp + ΦB), without taking intoaccount the disc heating effect. Their current spatial distribu-tion is shown in Fig. 9, where different symbols are used for thestellar “groups” coming from each simulated association. Ac-cording to our simulation, these groups are still concentrated inspace, although their distribution is, at present, far from beingisotropic due to the differential rotation. Each of them occupiesan ellipsoidal volume≈ 150–250 pc wide, 500 pc long, and70 pc height, and are perfectly separated in space (four of themcan be observed at different positions inside a 300 pc radiussphere around the Sun). The dispersion in each velocity com-ponent of any of the groups is still≈ 2 km s−1, as expectedfrom Eqs. A2. Although the mean velocity of these stars is verysimilar to that of B4, the dispersions in both velocity and spaceare too small when compared to dispersions in detected MGsthat are a few 108 yr old (Table 1). Thus, the spiral arms andcentral bar perturbations to the galactic gravitational potentialcannot account for the observational MGs’ velocity dispersions.

Results drastically change when considering the effect ofconstant heating on each individual star’s trajectory, as de-scribed in Sect. 3.2. To determine these trajectories we use onlythe axisymmetric part of the galactic potential, since the asym-metric perturbations to this potential are already included in thetreatment of the heating. The distribution of the simulated starsat present is shown in Fig. 10. The members of the differentgroups are now completely mixed with each other. The totalSC forms a long structure∼ 3 kpc long and∼ 1 kpc wide.Since in this simulation the Sun has been considered to be inthe very center of the SC att = 0, the members of this complexcan be found up to some 500 pc from us in the radial direc-tion, and much further in the tangential direction. By enlarginga square region (600 pc/side) around the Sun we observe inFig. 10 that, although the members of two groups dominate inthe solar neighbourhood, all the other evolved associations arealso represented. Again, the mean velocity components of B4are recovered when considering only the (636) stars closer than300 pc from the Sun. However, as expected, the dispersionsin velocities are now higher than in the former simulation – thetotal velocity dispersion is≈ 8–10 km s−1, depending on the re-gion of the evolved SC. Hence, a constant diffusion of the stellarvelocities can perfectly account for B4 velocity dispersions.

In order to study the properties of younger MGs, we anal-yse the general properties of the complex at its earlier stages,only 5 · 107 yr after the first association was born. At that mo-ment (t = −1 · 108 yr) some stars in the youngest simulatedgroups are not yet born, while some others are as young as B1and B2 members. The associations are still very concentratedin the phase space, though some merging can be observed, ashappens with B2 group. Velocity dispersions inside each asso-ciation clearly depend upon their ages, as expected. Thus, in the

R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group 443

−1.5 −0.5 0.5 1.5Y [kpc]

7.0

7.5

8.0

8.5

9.0

9.5

10.0

X

[kpc

]

GC

GR

a)

7.0 7.5 8.0 8.5 9.0 9.5 10.0X [kpc]

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

Z [

kpc]

GC

NP b)

Fig. 9a and b.Position on the galactica and meridionalb planes of the simulated stellar complex att = 0 (see text) when no disc heating effectis considered. Three square regions (600 pc/side) on the galactic plane have been enlarged to better distinguish the details. A different symbolis used for each group of stars coming from the same association. GC: galactic center; NP: north galactic pole; GR: galactic rotation

−1.5 −0.5 0.5 1.5Y [kpc]

7.0

7.5

8.0

8.5

9.0

9.5

10.0

X

[kpc

]

GC

GR

a)

7.0 7.5 8.0 8.5 9.0 9.5 10.0X [kpc]

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

Z [

kpc]

GC

NPb)

Fig. 10a and b.Same as Fig. 9 but considering now the effect of the disc heating on the trajectory of the stars

444 R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group

U-component the velocity dispersion varies from≈ 2.7 km s−1

for youngest groups to≈ 4.1 km s−1 for the oldest (in the othercomponents the dispersions are smaller).

Finally, it is interesting to take a look to the general prop-erties of a simulated SC at a largert. The mean properties ofthe simulated SC are now taken from the B3 nucleus at birth(t = −3 · 108 yr). At present (t = 0 yr), assuming a constantheating ofσh = 1.45 km s−1 every∆t = 107 yr, the 3000simulated stars occupy a huge curved region, about 2 kpc wideand almost 10 kpc long. In the most dense parts of this region,∼ 200 stars can be found in a 300 pc radius sphere, whose to-tal velocity dispersions are very high (12–14 km s−1). Starswith such large dispersions would be completely merged withfield stars, so they could not be detected as MG members. Amuch smaller diffusion coefficient is needed to recover the B3’svelocity dispersions (σh ≈ 0.7 km s−1).

Thus, the disgregation of SCs by means of a constant heatingmechanism is not able to account for the whole set of PleiadesMG substructures. A diffusion coefficient that depends on thestellar peculiar velocity and/or on the time (e.g. episodic diffu-sion) probably might explain the observed properties of thesegroups. Moreover, the presence of some open clusters in theoriginal SC could keep the stars together in phase space duringlonger periods. Some young open clusters, i.e. IC 2391,αPersei,Pleiades, etc, share roughly the same kinematics as the groupsin Table 1, which favours this hypothesis.

6. Conclusions

In this paper we studied the disruption of unbound systems ofstars as a mechanism to understand the origin and evolution ofmoving groups. The epicycle theory was used to find an analyticexpression for the time dependence of dispersion in both thestellar position and velocity coordinates. We have obtained inthis way a simple expression for the secular increase of thedispersion in the azimuthal galactic coordinate over time as afunction of the initial conditions. This increase in dispersion is,in fact, a direct consequence of the galactic differential rotation.

In order to overtake the constraints of the epicycle theory,we concentrated our analysis on the determination of the stellartrajectories using numerical integration of the equations of mo-tion, which provided us with an independent and more accurateestimation of the evolution of unbound systems. To perform thisintegration we used an analytic and axisymmetric galactic po-tential, along with spiral arm and central bar perturbations. Thislatter procedure allowed us to include random perturbations thatmimic the disc heating effect on stellar trajectories.

The trajectories followed by the members of the Pleiadesmoving group substructures found in Paper I were used to com-pare the kinematic and photometric ages of these structures, andto establish their position at birth. The youngest group, B1 (theSco-Cen association), was found to be most spatially concen-trated some9 · 106 yr ago, a value considerably smaller thanits photometric age. This is probably due to the effect of thehigh stellar atmospheric rotation of early type stars on the ob-served photometric colors. Groups B4 and B3 were born at a

time from the present equal to one and two times the epicycle pe-riod respectively, which means that they are spatially focused atpresent (probably that is the reason why we can observe them).At birth, these two groups were found to be close to the spiralarm structure. Concerning B2, a detailed analysis revealed to usthat it is actually composed of several associations which aredisintegrating at present. As well, its averaged photometric ageis probably overstimated, as in the case of B1.

We considered the evolution of a stellar complex under theinfluence of the galactic gravitational potential as a mechanismto account for the main physical properties of moving groups.The high velocity dispersions of some of the Pleiades movinggroup substructures detected among B and A type stars couldbe recovered when the effect of the disc heating on the indi-vidual stellar trajectories was considered. At the same time, thedisc heating can account for the mixing of the stellar complexassociations, although they are still clearly separated in phasespace during the first tens of million years in the complexes’ life(as observed for the groups B1 and B2). After only1.5 · 108 yrfrom the birth of the stellar complex, the complex occupies anellipsoidal area of∼ 3×1.5 kpc2, with its longest axis orientedin the direction of galactic rotation. If the Sun were presentlylocated in the very center of this disrupted stellar complex, wewould be able to find the complex’s components up to very largedistances on the galactic plane.

Thus, a constant heating mechanism (compatible with theobservational heating law) acting on the stars of a stellar com-plex can explain the velocity dispersions obtained for thosePleiades moving group substructures that are younger than∼ 1.5 · 108. The properties of the older Pleiades substructurescould probably be recovered by considering a diffusion coef-ficient depending on the velocity of the stars, and maybe onthe time, and/or the inclusion of open clusters in the simulatedstellar complex.

Acknowledgements.We would like to thank Dr. Ana E. Gomez andDr. J. Palous for many stimulating discussions and suggestions. We arealso indebted to F. Blasi, who estimated the effect of the atmosphericrotation on the computed age of our youngest stars. This work wassupported by CICYT under contract ESP97-1803, and by the PICSprogramme (CIRIT).

Appendix A: propagation of dispersionswith time in the epicycle theory

Let us consider the vectorx = (ξ′,η′,ζ ′,ξ′,η′,ζ ′) containing theposition and velocity of a star at a given timet. We are interestedin determining the propagation of the observational dispersionsover time, in the epicycle approximation frame (Eqs. 1, and theirderivatives). If we define the vectorx = (ξ′

,η′,ζ ′

,ξ′,η′

,ζ ′)

which contains the initial position and velocity of our star, thenthe initial dispersions can be propagated with time as:

σ2xi

(t) =6∑

j=1

(∂xi(t)∂x

j

)2

t

σ2x

j(A1)

whereσxi

is the observed dispersion in the variablexi, andσxi(t) the dispersion att. The dependence of variablesx on the

R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group 445

Table A1. Elements of the matrixB ≡(∂xi

∂cj

)(see text)

cj xi

ξ′ η′ ζ′ ξ′ η′ ζ′

ξ′a 1 cos(φ+ κt) 0 0 −ξ′

b sin(κt+ φ) 0

ξ′b -2 A t −2ω

κsin(κt+ φ) 1 0 −2ωξ′

b

κcos(κt+ φ) 0

η′a 0 0 0 cos(νt+ ψ) 0 −ζ′

a sin(νt+ ψ)ζ′ 0 −κ sin(κt+ φ) 0 0 −ξ′

bκ cos(κt+ φ) 0φ -2 A −2ω cos(κt+ φ) 0 0 2ωξ′

b sin(κt+ φ) 0ψ 0 0 0 −ν sin(νt+ ψ) 0 −νζ′

a cos(νt+ ψ)

Table A2. Elements of the matrixF ≡(∂ci

∂xj

)(see text)

xj ci

ξ′a ξ′

b η′a ζ′ φ ψ

ξ′ −ω

B0 0 0 − 1

2B0

η′

A cos(φ)B

0 0 − sin(φ)κ

cos(φ)2B

0

ζ′ 0 1 0 −2ω

κ2 0 0

ξ′ 0 0 cos(ψ) 0 0 − sin(ψ)

ν

η′ −A sin(φ)

ξ′bB

0 0 −cos(φ)ξ′

bκ− sin(φ)

2ξ′bB

0

ζ′ 0 0 − sin(ψ)

ζ′a

0 0 −cos(ψ)νζ′

a

Table A3. Elements of the matrixA ≡(∂xi

∂xj

)(see text)

xj xi

ξ′ η′ ζ′ ξ′ η′ ζ′

ξ′

A cos(κt) − ωB

0 0sin(κt)κ

cos(κt) − 12B

0

η′

2ωAB

[t− sin(κt)

κ

]1 0

2ωκ2 [cos(κt) − 1]

1B

[At− ω

κsin(κt)

]0

ζ′ 0 0 cos(νt) 0 0

sin(νt)ν

ξ′

AκB

sin(κt) 0 0 cos(κt)κ

2Bsin(κt) 0

η′

2ωAB

[1 − cos(κt)] 0 02ωκ

sin(κt)1B

[A − ω cos(κt)] 0

ζ′ 0 0 ν sin(νt) 0 0 cos(νt)

initial valuesx is given by Eqs. 2. If the coefficients of epicycleequations are included in a vectorc = (ξ′

a,ξ′b,η′

a,ζ ′,φ,ψ) then the

matrixA = (aij) ≡

(∂xi

∂xj

)in Eq. A1 can be determined as the

matrix productA = BF , where the matrixB = (bij) ≡(∂xi

∂cj

)

andF = (f ij) ≡

(∂ci∂x

j

)(i, j = 1, . . . , 6). By partially dif-

ferentiating Eqs. 1 we can determine the matrixB (Table A1),whereas matrixF is determined by just differentiating Eqs. 2(Table A2). Now, from matricesB andF we can determine the

matrix A, given in Table A3. From matrixA, and by means ofthe relationship A1, the propagated dispersions can be expressedas:

σ2ξ′ =

[A cos(κt) − ω

B

]2σ2

ξ′+[sin(κt)κ

]2σ2

ξ′+

+[cos(κt) − 1

2B

]2σ2

η′

σ2η′ =

[2ωA

B

(t− sin(κt)

κ

)]2σ2

ξ′+ σ2

η′+

446 R. Asiain et al.: On the evolution of moving groups:an application to the Pleiades moving group

+[2ωκ2 (cos(κt) − 1)

]2σ2

ξ′+

+[

1B

(At− ω

κsin(κt)

)]2σ2

η′

σ2ζ′ = cos2(νt)σ2

ζ′+[sin(νt)ν

]2σ2

ζ′

σ2ξ′ =

[AκB

sin(κt)]2σ2

ξ′+ cos2(κt)σ2

ξ′+

+[ κ2B

sin(κt)]2σ2

η′

σ2η′ =

[2Aω

B(1 − cos(κt))

]2σ2

ξ′+[2ωκ

sin(κt)]2σ2

ξ′+

+[

1B

(A − ω cos(κt))]2σ2

η′

σ2ζ′ = [ν sin(νt)]2 σ2

ζ′+ cos2(νt)σ2

ζ′. (A2)

From these equations we observe that all the dispersions invariablesx oscillate around a constant value, exceptσ2

η′ , whoseaveraged value increases with time. For small values oftEqs. A2can be approximated by:

σξ′ ≈ σξ′σ2

η′ ≈ σ2η′

+ t2 σ2η′

σζ′ ≈ σζ′σξ′ ≈ σξ′ση′ ≈ ση′σζ′ ≈ σζ′

.

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