ECCENTRICITY SAMPLES: IMPLICATIONS ON THE POTENTIAL AND THE VELOCITY DISTRIBUTION

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1 Serb. Astron. J. (017), OnLine-First UDC DOI: Original sientifi paper ECCENTRICITY SAMPLES: IMPLICATIONS ON THE POTENTIAL AND THE VELOCITY DISTRIBUTION R. Cubarsi 1,M.Stojanović and S. Ninković 1 Departament de Matemàtiques, Universitat Politènia de Catalunya, Baelona, Spain E mail: rafael.ubarsi@up.edu Astronomial Observatory, Volgina 7, Belgrade 38, Serbia E mail: mstojanovi@aob.rs, sninkovi@aob.rs (Reeived: February 1, 017; Aepted: Mah 1, 017) SUMMARY: Planar and vertial epiyle frequenies and loal angular veloity are related to the derivatives up to the seond order of the loal potential and an be used to test the shape of the potential from stellar dis samples. These samples show a more omplex veloity distribution than halo stars and should provide a more realisti test. We assume an axisymmetri potential allowing a mixture of independent ellipsoidal veloity distributions, of separable or Staekel form in ylindrial or spherial oordinates. We prove that values of loal onstants are not onsistent with a potential separable in addition in ylindrial oordinates and with a spherially symmetri potential. The simplest potential that fits the loal onstants is used to show that the harmonial and non-harmonial terms of the potential are equally important. The same analysis is used to estimate the loal onstants. Two families of nested subsamples seleted for dereasing planar and vertial eentriities are used to borne out the relation between the mean squared planar and vertial eentriities and the veloity dispersions of the subsamples. Aording to the first-order epiyle model, the radial and vertial veloity omponents provide aurate information on the planar and vertial epiyle frequenies. However, it is impossible to aount for the asymmetri drift whih introdues a systemati bias in estimation of the third onstant. Under a more general model, when the asymmetri drift is taken into aount, the rotation veloity dispersions together with their asymmetri drift provide the orret fit for the loal angular veloity. The onsisteny of the results shows that this new method based on the distribution of eentriities is worth using for kinemati stellar samples. Key words. Stars: kinematis Galaxies: kinematis and dynamis Galaxies: statistis Galaxy: solar neighbourhood 1. INTRODUCTION Several kinemati analyses suggest that the Galati thin dis has a non-vanishing vertex deviation, the thik dis has a radial mean motion differing from that of the thin dis, and the halo veloity ellipsoid is likely to be tilted (Pasetto et al. 01a,b, Moni Bidin et al. 01, Casetti-Dinesu et al. 011, Carollo et al. 010, Fuhs et al. 009, Smith et al. 009a,b, Siebert et al. 008). What type of potentials allow to desribe these kinemati features in the solar neighbourhood? There is a large family of potentials onsistent with one ellipsoidal stellar veloity distribution a- 1

2 R. CUBARSI et al. ording to Chandrasekhar s (1960) time dependent systems. For three-dimensional models these potentials were studied by Sala (1990). They inlude Eddington s (1915) and Lynden-Bell (196) stationary potentials. However, for a mixture of stellar populations, by assoiating eah stellar populations with an ellipsoidal veloity distribution, only few possible potentials remain. To determine whih potentials are onsistent with the integrability onditions imposed by a mixture of populations two approahes were made. In a first approah (Cubarsi 014a), axisymmetri potentials satisfying the time-dependent olisionless Boltzmann equation (CBE) allowing a number of independent populations were studied. By independent populations we mean that they may have different mean veloities and arbitrary orientation of veloity ellipsoids. In a Galatoentri ylindri oordinate system (r, θ, z), with θ positive in the diretion of the Galati rotation and z perpendiular to the Galati plane and positive towards the NGP, the potential must have the form U = M (r + z )+ 1 r F (z /r ), (1) where F is an arbitrary funtion of its argument s = z /r (or of the polar angle ϕ =atan s). It is an axisymmetri potential of Stäkel form in ylindrial and spherial oordinates. If the potential is stationary then M is onstant, otherwise it is the only part of the potential that may depend on time-dependent kinemati parameters. For this reason it was alled quasi-stationary potential. Sine Pasetto et al. (01b) and Steinmetz (01) had suggested that the axisymmetry assumption should be relaxed towards a model with pointaxial symmetry in order to aount for suh a kinemati features, in a seond approah (Cubarsi 014b) the point-asymmetri model i.e. rotational symmetry of 180 for the potential and the phase spae density funtions, was studied. The result was that the potential had to be also axisymmetri and, in partiular, spherial, aording to: U = M (r + z )+ N r + z. () However, suh a spherial potential did not allow for either vertex deviation of the population veloity ellipsoids in the Galati plane or tilt of the ellipsoids out of the Galati plane. However, the apparent vertex deviation of dis samples ould also be produed, at least from a stritly mathematial point of view, from a mixture of several populations having different radial and rotation mean veloities, eah one without vertex deviation or even spheroidal. On the other hand, as Smith et al. (009a) suggested, the tilted veloity ellipsoids of the thik dis and halo ould be the result of stellar samples not suffiiently mixed in order to produe well defined veloity ellipsoids or of samples ontaminated by dis stars. Similarly, Evans et al. (016) argue that a spherial potential is onsistent with the veloity distribution of halo stellar samples. But halo stars are not expeted to show the omplexity of dis stellar samples. Therefore, the shape or the symmetry of the potential should be justified from other and better reasons, if possible from dis stellar samples. The main purpose of the urrent study is to test the onsisteny of the potential in the general form of Eq. (1) against the loal veloity distribution, in partiular, the veloity dispersions and the asymmetri drift of the samples. Thus, we shall fous on the main trends of the dis stars whih involve the loal kinemati onstants, namely the planar and vertial epiyle frequenies, and the loal angular veloity. Under the first-order epiyle model, for a given stellar sample the epiyle frequenies are related to the seond veloity entral moments 1 μ rr and μ zz through the mean squared planar and vertial star eentriities. This model, however, is unable to provide a realisti value of the loal angular veloity from a similar relationship between the moment μ θθ and the mean squared planar eentriity. This will be addressed by leaving aside the epiyle model and by taking into aount the asymmetri drift of the samples. In Cubarsi (010) it was proved that, in order to obtain kinematially three-dimensional veloity stellar samples kinematially of the Galati dis, the orbital eentriity behaves as an exellent sampling parameter whih allows to distinguish a number of small-sale features of the veloity distribution. Instead, other sampling parameters suh as the absolute value of the helioentri veloity, metaliity [Fe/H], or olour b y, produe kinematially biased samples and population estimates if they are not omplemented with other sampling riteria suh as the limit of the absolute spae motion. In partiular, by using the stars of the Geneva-Copenhagen Survey II (Nordström et al. 004, Holmberg et al. 007) whih inlude star s eentriities, maximum planar eentriity e =0.3and maximum distane to the Galati plane z max =0.5 kp led to a kinematially representative sample of the thin dis. Therefore, the purpose of the urrent work is to use, along with star s veloities, the eentriity distribution to study the potential and the veloity distribution. The planar and vertial eentriities e and e z, aording to the generalised notion of Ninković (009), are related to the star s orbit and an be omputed by using approximations onerning the gravitational potential (Ninković 011). At a distane r 0 from the Galati entre (GC), z max and the vertial eentriity e z are proportional: z max = r 0 e z. When these data are available, their use as sampling parameters provides veloity samples far more representative of the moving groups they ontain than if the samples had been seleted by the absolute spae motion. In order to draw several nested subsamples of the dis and to ompute the mean squared een- 1 The entral moments are here expressed in the omponent notation, aording to the mean value μ i1 i...i n = u i1 u i...u in with indies in the set {1,, 3}, depending on the peuliar veloity omponent.

3 ECCENTRICITY SAMPLES: IMPLICATIONS ON THE POTENTIAL AND THE VELOCITY DISTRIBUTION triities and the seond veloity entral moments, we shall use the updated Geneva-Copenhagen Survey III (hereafter GCSIII) atalogue (Holmberg et al. 009). Thefirststepistodesribehowtheloalonstants and the potential are related. A first approah for dis stars involves the use of the epiyle approximation in order to model the distribution of star eentriities. Although this is a lassial astronomial topi (e.g. Binney and Tremaine 008), we shall detail it in order to introdue notation and definitions. In addition, the referred book does not take into aount the eentriity distribution when studying the epiyle approximation. The loal kinemati properties are afterwards referred to the omoving referene frame, for whih we are able to get the kinemati estimates. A seond step onsists in study of how the veloity distribution and eentriities are related. We analyse how these relationships onstrain the ratios of the semiaxes of the veloity ellipsoids. Now, it is neessary to adopt a more general approah than the epiyle model in order to onsider the asymmetri drift of stellar samples. This approah is studied for the general ase where the three mean veloity omponents of a stellar population may differ from the iular veloity orbit. In a third step we obtain the loal Galati onstants from nested stellar samples seleted by planar and vertial eentriities. The planar epiyle frequeny is easily dedued from nested stellar subsamples seleted by planar eentriity. The vertial epiyle frequeny is straightforward obtained from nested stellar subsamples seleted by maximum height. Nested stellar subsamples seleted by planar eentriity should also provide a good estimation of the loal angular veloity. However, this does not our sine the first order epiyle model neglets the asymmetri drift. When this term is taken into aount, the loal angular veloity is orretly estimated as well. This approah leads to a simple way for evaluating the asymmetri drift from the seond veloity entral moments. Finally, we disuss the impliations that the atual values of loal kinematial onstants have on the shape of the potential. By assuming the above family of potentials allowing mixtures of independent ellipsoidal veloity distributions, we prove that the potential annot be separable in addition in ylindrial oordinates and annot be spherially symmetri.. LOCAL KINEMATICAL CONSTANTS The solution of the equations of motion of a point mass partile under a onservative foe field is onstrained by the knowledge about the potential funtion. In the Galaxy, the mutual gravitational interations of the stars determine their orbits. Leaving aside stellar enounters, these interations arise from the smoothed-out stellar distribution of matter, whih is given through a quite unknown gravitational potential. However, some symmetry properties may be generally assumed providing us with a basi set of integrals of motion. The epiyle approximation is a partiular ase of integration of the equations of motion under a minimum set of hypotheses allowing to obtain solutions for nearly iular orbits in the three dimensional spae. Although it is desribed in many standard books on astronomy, in order to fix the notation, we explain the method from srath by determining the solution for iular orbits in Appendix A (Epiyle approximation). Under this approah, the orbit of any star projeted onto the Galati plane desribes an ellipse with origin at a guiding entre or epientre whih moves uniformly in a iular orbit around the GC. The model leads to interesting properties about the star motion, suh as the planar and vertial epiyle frequenies, and the axial ratio of the planar epiyle, whih only depend on loal properties of the potential, as well as properties about the loal stellar veloity distribution and their relation to some eentriity statistis. Linblad s approah onsists in to refer the orbit of a star with position (r, θ, z) near the Galati plane to a referene frame with entre in the position (r,θ, 0) of a star in the Galati plane in iular motion with the same angular momentum integral J. It is the guiding entre C for whih, given the position and veloity of a star, we may obtain r and Ω whih satisfy Eqs. (87) and (90). For the first two oordinates, we may write r = r + ε; θ = θ + δ. (3) The first requirement for the validity of the model is ε r. (4) The seond, sine θ may have an arbitrary origin, by differentiating Eq. (3) we get and we assume ṙ = ε; θ =Ω + δ, (5) δ Ω. (6) The values ε and δ are onstrained sine the angular momentum integral is fixed. Thus, by taking into aount Eq. (79), Eq. (3), and Eq. (5) we have J =(Ω + δ)(r + ε) =Ω r +Ω r ε + r δ +Ω ε +r ε δ + ε δ. Being J =Ω r, by onsidering only the terms up to first order, we get Ω r ε + r δ =0. Therefore, we get the relationship δ = Ω r ε. (7) Notie that, if Ω > 0then δ and ε have opposite signs, as shown in Fig. 1. To obtain the orbit of the star in the iular motion referene frame we write the first equation of motion in Eq. (76) in terms of the effetive potential energy gradient, defined in Eq. (85), so that 3

4 R. CUBARSI et al. r = V, (8) and we expand it up to the first order around the guiding entre C V V + V ε. The first term, aording to Eq. (86), is null, and the seond term, aording to Eq. (88), is κ = V. (9) Then, from Eq. (3) the radial omponent of the equations of motion beomes ε = κ ε. (10) Therefore, the ondition of a loal minimum of the effetive potential energy at r, given by Eq. (84), ensures the orbit to be stable. The approximation given by Eq. (10) desribes the radial motion of the star as an harmoni osillator around the guiding entre C with frequeny κ ε = a sin(κt p). (11) The positive value of κ is the planar epiyle frequeny, anda and p are integration onstants. The axial omponent is easily obtained by integrating Eq. (7), δ = Ω r a sin(κt p), (1) δ = Ω a os(κt p), (13) r κ where the additive integration onstant an be assumed as null sine ε 0andδ 0whena 0. In a similar way, the vertial omponent z of the star referred to the guiding entre C in the Galati plane is obtained in a first order approximation by expanding the vertial gradient of the potential as U z U z z, so that we obtain z = b sin(νt q), (14) being ν = U z > 0, (15) where ν is the vertial epiyle frequeny, andb and q integration onstants. Usually, the onstant b is notated as the star s maximum height z max Position referred to the iular orbit A star at a position S, with oordinates (x, y, z) referred to the GC, will be referred to the iular veloity of the guiding entre C in the Galati plane with oordinates (r,θ, 0) whih is moving with iular veloity Θ =Ω r (see Fig. 1). Then, the new Cartesian oordinates (ξ,η,ζ) of the star satisfy: ξ = ε = a sin(κt p), η = r δ = Ω κ a os(κt p), ζ = z = b sin(νt q). (16) We fous on the star s motion projeted onto the Galati plane sine the motion in the diretion ζ is independent of the other oordinates. Vertially, the star simply osillates about the Galati plane. For a simpler notation we use the dimensionless onstant γ = Ω κ, (17) depending on the loal values of the star and the potential. It is not, stritly speaking, a new onstant. Then, the oordinates (ξ,η) desribe the following ellipse entred at C ξ + γ η = a (18) where γ 1 is the ratio of axes.. δ x = Π ξ C GC r δ S (ε, r δ) η Θ = Ω r Fig. 1. In the Galati plane, the star S is referred to the GC with oordinates (x, y) and to the iular veloity of the guiding entre C, with iular veloity Θ =Ω r, with oordinates (ξ,η). Notie that the loal veloities, being referred to C, desribe an ellipse with the same axis ratio than the previous one, ξ = aκ os(κt p) =κγ 1 η, η = Ω a sin(κt p) = Ω ξ, ζ = νb os(νt q). y (19)

5 ECCENTRICITY SAMPLES: IMPLICATIONS ON THE POTENTIAL AND THE VELOCITY DISTRIBUTION Hene, it is fulfilled ξ + γ η = κ a. (0) It is worth notiing that above ellipses are relative to the guiding entre C, whih is speifi of eah star. Finally, the motion of the star referred to the GC is easily obtained in ylindrial oordinates aording to r = r + ε = r + a sin(κt p), θ = θ + δ = θ 0 +Ω t + γ a r os(κt p), z = b sin(νt q). 3. POTENTIAL (1) In the past setion we have shown that the radial and transversal epiyle frequenies are the same and, like the vertial frequeny, they are onstant, not depending on the star but on loal values of the potential. This is an important result obtained in the twenties by Oort and Lindblad. Eah region in a galaxy provides stars with a loal osillation mode. As we have adopted an axisymmetri model, the epiyle frequenies would vary depending on the distane to the GC and to the plane. First we study the impliations on the potential, and afterwards we shall estimate how muh the epiyle frequenies may vary from one point to another. We shall assume a quasi-stationary potential given by Eq. (1), i.e. an axisymmetri potential allowing for finite mixture of ellipsoidal veloity distributions and, in addition, arbitrary population mean veloities in the radial, rotation, and vertial omponents (although in the symmetry plane the vertial mean veloity is null). This potential satisfies the time-dependent CBE for axially symmetri stellar systems. The partiular ase of a spherial potential is the solution of the CBE for stellar system with point-to-point axial symmetry (Cubarsi 014b). Let us remember that a stationary potential does not imply a steady-state stellar system, that is, a stationary veloity distribution. Indeed, a time dependent veloity distribution is needed to allow for a non-vanishing radial mean veloity of the stellar populations. It is worth notiing that, for steady-state stellar systems, the potential allowing the alignment of the stress tensor along an orthogonal oordinate system is of separable or Stäkel form (An and Evans 016, Evans et al. 016). These authors suggest that the atual ase should be very lose to the spherial alignment and they obtain a potential similar to that of Eq. (1) where the first term, instead of the harmoni potential, is an arbitrary funtion of r + z. In our ase, where eah population veloity distribution is ellipsoidal in the peuliar veloities, this term beomes totally determined sine this is the only possibilty allowing differential radial motion of populations. In partiular, to study the spherial ase, Eq. (1) an also be written as where U = M (r + z )+ 1 r + z N(z /r ), () ) 1 F (z /r )=N(z /r ) (1+ z. In addition to the general ase, we shall study two partiular ases. One with onstant F in Eq. (1), orresponding to a potential separable in addition in ylindrial oordinates, and another one with onstant N in Eq. (), orresponding to the spherial potential Separable ylindrial potential We assume the potential of Eq. (1) separable in addition in ylindrial oordinates r U = M (r + z )+ F r. (3) where F is onstant. We define the following linear operator, whih appeared in the ondition of orbital stability of Eq. (88), ating over the potential U ( L r [ ]= + 3 ) [ ]. (4) r It satisfies L r [ 1 r + ] = 0 either with 1, onstants or funtions of z. Furthermore, L r [Mr ]=8M. Then, these potentials provide onstant squared epiyle frequenies κ =8M and, aordingtoeq. (15),ν =M regardless the point in the Galaxy Ṫherefore, on one hand, the existene of bounded orbits requires the fator M to be positive. On the other hand, the onstant M determines both epiyle frequenies. Then, the ratio of the epiyle frequenies is κ ν =, whih is not the atual ase, sine, aording to the ommonly aepted values Ω 7 km s 1 kp 1, κ 37 km s 1 kp 1,andν 70 km s 1 kp 1 (e.g. Binney and Tremaine 008, Table 1.), the vertial frequeny must be higher than the planar (the rotation period is about 0 Myr and the vertial period of osillation is approximately 87 Myr). By Eq. (87), in the Galati plane, for a star in iular motion and angular momentum integral J, the radius r of the iular orbit verifies 4 = F + J M. 5

6 R. CUBARSI et al. Sine M > 0, then J > F. This is the ondition for a stable orbit. Thus, a repulsive foe assoiated with the potential term with F>0would allow stable orbits for all the stars. This exludes the existene of iular orbits within a radius lower that r min =[F/M] 1 4, orresponding to the value J =0. Otherwise, an attrative foe assoiated with F < 0 would allow bounded orbits at any distane from the GC but only for stars trespassing a threshold angular veloity with a minimum angular momentum integral Jmin = F. Other orbits beome unstable. The angular veloity satisfies Eq. (90). Then, for the potential in Eq. (1), the squared angular veloity in terms of the radius of any star in a iular orbit on the Galati plane is given by Ω (r) =M F r 4, r4 F/M. (5) By expressing M in terms of the planar epiyle frequeny, for r = r we determine F from the loal angular veloity, i.e. ( ) F = r4 κ 4 Ω (r ). (6) The loal values for these onstants allow estimates κ and Ω (r ) 730 (both values in km s kp ). Then, for suh a potential, F is negative. Therefore, an attrative foe requires κ<ω. 3.. Spherial potential We assume the spherial potential of Eq. () with onstant N, that is, Eq. (). In the symmetry plane, the planar epiyle frequeny is, like in the previous ase, onstant κ =8M. On the other hand, the vertial epiyle frequeny at z =0is ν = U z =M N 4. (7) Then, in the Galati plane both epiyle frequenies are onstrained aording to the relationship ( ) N = r4 κ 4 ν. (8) Hene, N is now related to the loal vertial epiyle frequeny. Sine the atual values in the solar neighbourhood satisfy ν>κ, the above equation would provide a negative value for the onstant N. Similarly, an attrative foe requires κ<ν. However, by omparing Eq. (5) and Eq. (7), sine the spherial and separable ylindrial potentials satisfy F = N and κ =8M, weget ν =Ω (r ). Therefore, for the spherial potential in the Galati plane, the loal vertial epiyle frequeny and the 6 absolute value of the loal angular veloity math at r. There is no alternative parameter to fit the loal angular veloity. This ould neither be the atual ase in the solar neighbourhood General ase For the general ase of Eq. (1), aording to the term having the arbitrary funtion F (s) with s = z /r,weget and L r [ U ]=8M + 1 r 4 ( 8sF (s)+4s F (s) ) (9) U z =M + 1 r 4 (F (s)+4sf (s)). (30) Then, in the Galati plane, s = 0, the epiyle frequenies satisfy κ =8M, In addition, Eq. (5) beomes Ω (r ) κ 4 ν κ 4 = F (0) r 4 = F (0) r 4. (31). (3) For the seond term of potential (3), an attrative foe in terms of r and z is assoiated with loal values F (0) < 0andF (0) > 0. Therefore, it implies Ω >κand ν >κ. Then, suh a potential term inreases the foe produed by the harmoni potential. Therefore, we have three independent parameters related to the three loal onstants. They an be adjusted aording to atual values and provide the loal derivatives of the potential funtion U =Ω r, (,0) U = κ 3Ω, (33) (,0) U z = ν. (,0) 4. KINEMATICS We ompare two lose iular orbits at radii r = r and r = r + ε in the Galati plane. Eah harateristi angular veloity Ω (r) isgivenbyeq. (90). The orresponding angular momentum J (r) of a star in iular orbit at a radius r is equal to r Ω (r). By taking into aount the first equation of

7 ECCENTRICITY SAMPLES: IMPLICATIONS ON THE POTENTIAL AND THE VELOCITY DISTRIBUTION motion in Eq. (76), we may determine how the angular veloity, the iular veloity, and the angular momentum integral vary from one iular orbit to another. For a given potential, the iular veloity Θ at a radius r satisfies Θ (r) r and its derivative is: Θ (r) r Θ (r) = U(r, 0), (34) Θ (r) r = U(r, 0). (35) The above equations allow us to obtain the planar epiyle frequeny at r = r in terms of the loal iular veloity and its derivative, instead of in terms of the potential derivatives, so that κ = Θ (r ) r ( Θ (r ) r + Θ (r) ). (36) Conversely, from Eq. (36) we an estimate the first derivative of the iular veloity at r in terms of the loal epiyle frequeny, whih is easily written by using the angular veloity Ω = Θ/r Θ (r) = κ Ω (r ) Ω (r ). (37) Therefore, in a first order approximation, and by taking into aount the definition of γ in Eq. (17), the iular veloity Θ (r + ε) may be estimated as Θ (r + ε) =Θ (r )+ ( κγ 1 Ω (r ) ) ε. (38) By working in a similar way, we an write the following radial gradients Ω (r) J (r) Notie that the approximation: Ω (r + ε) =Ω (r )+ ( κγ 1 = κγ 1 Ω (r ) r, (39) = κγ 1 r. (40) Ω (r ) ) ε r, (41) being ε r, allows us to assume a nearly onstant angular iular veloity for all iular orbits around the radius r. Then, for suffiiently large values of r,theassumption that the angular veloity Ω (r )isnearly onstant around r is muh more aurate than for Θ (r ) Comoving referene frame Wewishtorefertheveloityofastarpassing through the point S, having oordinates (ξ,η,ζ) with respet to the iular orbit entred in C, to the iular orbit entred in a fixed point S 0,whih is the projetion of S onto the plane z =0,withoordinates (ξ,η,0). Therefore, the angular momentum integral of the orbits passing through C and S are the same, while the one of the iular orbit in S 0,aording to Eq. (40), differs by the amount J(r) ξ. There is no radial motion in both iular orbits so that Π Π (C) = 0 and Π (S 0 ) = 0. Hene, the Galatoentri radial veloity of the star, as well as the radial veloity referred to the iular orbit, is Π= ξ. Thus, by Eqs. (17) and (19), we may write Π(S) Π (S 0 )=κγ 1 η = κa os(κt p). (4) Similarly, there is is no vertial motion in both iular orbits in the Galati plane, so that Z Z (C) = 0 and Z (S 0 ) = 0. Then, the vertial veloity of the star satisfies Z = ζ, whihwewrite as Z(S) Z (S 0 )=νbos(νt q). (43) On the other hand, for the veloity omponent of the star along Galati roation, bearing in mind Eq. (3) and the equivalene ξ = ε, wehave Θ(S) = (r + ξ)(ω (C) + δ) =Ω (C) r + r δ +Ω (C) ξ + ξ δ. We only onsider up to the first order terms and take into aount Eq. (7). Thus Θ(S) = Ω (C) r Ω (C) ξ =Θ (C) Ω (C) ξ. (44) Also, aording to Eq. (38), the iular veloity at S 0 may be approximated from the iular orbit at Cas Θ (S 0 )=Θ (C) + ( κγ 1 Ω (C) ) ξ. (45) Therefore, by subtration of Eq. (45) from Eq. (44) and using Eqs. (16), we get Θ(S) Θ (S 0 )= κγ 1 ξ = κγ 1 a sin(κt p). (46) Finally, by substitution of ξ and η from Eqs. (4) and (44) into Eq. (18), we get [Π Π (S 0 )] + γ [Θ Θ (S 0 )] = κ a. (47) This is the equation for a family of ellipses entred at the planar iular veloity in S 0, whose semiaxes are in inverse proportion with regard to the ellipses of Eq. (18) and Eq. (0). While the latter ellipses are entred at a different guiding entre C for eah star, the family of Eq. (47) has a ommon entre at S 0 for all the stars passing through S and, in general, due to the axial symmetry and to the symmetry plane, for all the stars with the same oordinates (r, θ). In 7

8 R. CUBARSI et al. addition, we may assume that κ and γ are similar at the points C, S 0, and S. Notie that the values κ and γ in Eq. (17) depend on Ω, whih is nearly onstant around C as dedued from Eq. (41). However, the epiyle model introdues a larger error in assuming a similar value for Θ at C, S 0,andS. Usually the loal parameters κ and γ are expressed in terms of the Oort onstants, A and B, of the field of differential motions at the point S 0 in the Galati plane, as desribed in Appendix B (Oort onstants). 5. VELOCITY STATISTICS Forafixedtimet, we wish to alulate some veloity statistis for a loal sample omposed of stars around the point S in order to estimate the loal values κ, ν, andγ. In this proess the variables involved are those of Eqs. (4), (46) and (43), namely, the veloity omponents (Π, Θ, Z), the star amplitudes a and b, and the phases p and q. In partiular, the values a and b are related to the planar and vertial eentriities. The planar eentriity is a dimensionless measure of deviation from iular motion in the Galati plane, defined as e = r a r p, (48) r 0 where r a and r p are the extremal distanes to the rotation axis, and r 0 = r a + r p, (49) is their arithmeti mean. We assume r 0 = r.therefore, the amplitude a of Eq. (16) orresponds to r a r p, and the planar eentriity is e = a r 0. (50) On the other hand, the vertial eentriity is defined as e z = 1 z a + z p (51) r 0 where z a and z p are the amplitudes of the distanes to the Galati plane. For stars about the Galati plane, the amplitude b z max in Eq. (14) is the same as za + zp. Then, the vertial eentriity satisfies e z = z max. (5) r 0 The veloity variables have a trivariate distribution that, for large stellar samples, may be managed as a Gaussian population mixture. A priori it is not easy to selet a stellar sample ontaining a single population sine the sampling parameters always produe trunated or overlapped veloity distributions. The eentriity e and the planar amplitude 8 a = er 0 are random variables that, for large dis samples, have an approximate lognormal distribution (Cubarsi 010). The distane r 0 to the GC may be assumed the same for all stars of a loal sample. The maximum height z max behaves like the planar amplitude. Sine the vertial and planar motions were solved separately, they involve independent random variables. The planar and vertial phases p, q, with no prior information about them, taken for granted that the stars are well mixed, may be reasonably assumed as uniformly distributed in [ π, π]. Furthermore, phases and eentriities are independent variables, sine they are the two integration onstants of the seond-order differential equation governing the star s motion. In Cubarsi (010) the eentriity was proven a very useful sampling parameter in order to isolate partiular kinemati behaviours of thin dis subsamples. Both planar and vertial amplitudes have to be ombined in order to exlude thik dis or halo stars. Several nested stellar samples seleted as inreasing the star eentriity did provide a very detailed representation of the small and large sale kinemati struture of loal stars assoiated with the main moving groups. It ould be said that the eentriity is a highly respetful sampling parameter of kinematially homogeneous stars. Now, we will deepen in the kinemati information provided by the eentriities Stars in nearly iular orbits For fixed time and position, the harateristi iular motion values κ, ν, γ are onstant. We alulate the expeted values E( ) of the stars in a sample Σ for the above mentioned random variables. The sample Σ will be hosen to have maximum eentriity e 0 =max (e), e Σ and maximum height z 0 = max z max Σ (z max). Moreover, this will be done for a series of subsamples in order to study the main trends and stability of the estimates. For a sample Σ(e 0 ) seleted by planar eentriity, the mean values E(e) ande(e ) are inreasing funtions of e 0. Similarly, for of a sample Σ(z 0 ) seleted by the star height, the mean values E(z max )ande(z max) are inreasing funtions of z 0, as shown in Fig. 4. By taking expeted values in the Eqs. (4) and (46), we get Π Π (S 0 )=κ a os(κt p), Θ Θ (S 0 )= κγ 1 a sin(κt p). (53) If we write the mean veloity omponents at S as Π 0 = Π and Θ 0 = Θ, sineπ (S 0 )=0and os(κt p) = sin(κt p) =0, (54)

9 ECCENTRICITY SAMPLES: IMPLICATIONS ON THE POTENTIAL AND THE VELOCITY DISTRIBUTION then, the epiyle model provides a loal entroid at S with veloities Π 0 =Π (S 0 )=0, Θ 0 =Θ (S 0 ). (55) Therefore, suh a first-order epiyle model annot give an aount for the asymmetri drift i.e. the differene between the loal iular veloity at S 0 and the loal mean rotation veloity at S of the stellar population omposing the sample Σ, Δ Δ θ =Θ (S 0 ) Θ 0 (S 0 ). (56) This is one of the main limitations of the epiyle model. We now ompute the veloity varianes i.e. the diagonal seond entral moments, from Eqs. (4) and (46). Sine Π (S 0 )andθ (S 0 )areonstants,we have μ rr V [Π] = V [Π Π (S 0 )], (57) μ θθ V [Θ] = V [Θ Θ (S 0 )]. Bearing in mind Eq. (54), as well as the independene of the variables a and p, we then obtain μ rr = [κa os(κt p)] κa os(κt p) and, similarly = κ a os (κt p) μ θθ = [κγ 1 a sin(κt p)] κγ 1 a sin(κt p) = κ γ a sin (κt p). It is easy to see that the uniform distribution of p leads to os (κt p) = sin (κt p) = 1. (58) Thus, the relationships between the seond veloity entral moments and the mean value of the squared planar amplitude of a sample Σ are given by μ rr = 1 κ a, μ θθ = 1 κ γ a. Let us remark that the expeted value a = r 0 e (59) is not proportional to the square of the average eentriity of the sample a as it ould be misinterpreted from Binney and Tremaine (008, p170, equations 3.97 and 3.99). Sine they do not take into aount the eentriity distribution i.e. they do not onsider the eentriity as a random variable and use X for a and X for a, whih is not orret. By adding the expressions of Eq. (59), we get μ rr + γ μ θθ = κ a. (60) Therefore, aording to the epiyle model, the ratio of varianes is μ rr = γ, (61) μ θθ whih is a value depending on the loal properties of the stellar system, but not depending on the average values of the eentriities of the sample. However, this is not exat, as we shall see in the numerial appliation. If the model aounts for the asymmetri drift, we will get a more aurate relationship as explained in the next Setion. For the vertial motion, a similar reasoning for the epiyle approximation leads to Z 0 = Z (S 0 )=0. Then, sine μ zz V [Z] =V [Z Z (S 0 )], we get from Eq. (43) the relationship between the seond entral moment μ zz and the mean value of the squared vertial amplitude: μ zz = 1 ν z max. (6) Thus, the moments of suh a sample satisfy μ rr = κ a μ zz ν zmax. (63) Therefore, they do not keep a onstant ratio, like that of Eq. (61). The ratio is proportional to the ratio of the mean squared amplitudes of the sample. 5.. General ase In models that are more omplex than the epiyle approximation, possible drifts along the other diretions, similarly to the asymmetri drift for the rotation omponent of Eq. (56), are written as Δ r =Π (S 0 ) Π 0 (S 0 ), (64) Δ z = Z (S 0 ) Z 0 (S 0 ), although Π (S 0 )=Z (S 0 ) = 0, we prefer to maintain suh a notation. Nevertheless, for samples near the Galati plane, these quantities generally vanish. It is straightforward to estimate the ratio of the semiaxes of the veloity ellipsoid for a model without negleting the asymmetri drift and the mean radial veloity. We rewrite Eqs. (4) and (46) as (Π Π 0 )+(Π 0 Π (S 0 )) = κa os(κt p), (Θ Θ 0 )+(Θ 0 Θ (S 0 )) = κγ 1 a sin(κt p). (65) By squaring those expressions and alulating their expeted value, we get: μ rr +(Π 0 Π (S 0 )) = 1 κ a, μ θθ +(Θ 0 Θ (S 0 )) = 1 κ γ a. (66) 9

10 R. CUBARSI et al. Adding the foregoing equations and reordering terms, we get a general expression for Eq. (60): μ rr + γ μ θθ = κ a (Π 0 Π ) γ (Θ 0 Θ ). (67) For the ratio, we get μ rr μ θθ = 1 κ a (Π 0 Π (S 0 )) 1 κ γ a (Θ 0 Θ (S 0 )). (68) This equation generalises that of Ninković (199), where a vanishing radial mean veloity was assumed. Its right-hand side term depends on the average values of the sample and, in partiular, would require a higher-order model to express the veloity differenes in terms of the averaged amplitudes. This is left for a future work. In partiular, by using Eqs. (56) and (64), the expressions in Eq. (66) beome μ rr +Δ r = 1 κ a, μ θθ +Δ θ = 1 κ γ a, (69) so that the onstant γ aounts for something slightly different than the ratio of axes, μ rr +Δ r μ θθ +Δ = γ. (70) θ This expression is more realisti than Eq. (61). Sine the asymmetri drift Δ θ annot be measured in terms of averaged eentriities aording the epiyle model, it must be estimated in an alternative way. For the vertial motion, for stars with orbits far from iular, we an do a similar reasoning. Then, the orresponding equations an be written by using Eq. (64) as follows. For the vertial veloity distribution we have μ zz +Δ z = 1 ν z max, (71) and, by ombining the vertial and radial distributions, we have μ rr +Δ r μ zz +Δ z = κ ν a z max. (7) Therefore, we find that the ratio in Eq. (70) should be maintained for different stellar subsamples, sine it does not depend on the distribution of their eentriities. Instead, the ratio in Eq. (7) depends on the distribution of the planar and vertial eentriities, hene it is not onstant for different stellar subsamples. In partiular, for samples with Δ r =Δ z =0,theratioμ rr /μ zz is not neessarily the same. 6. APPROACH AND RESULTS The urrent analysis an also be used to dedue the loal kinematial onstants from subsamplesofthethindisdrawnfromthegcsiiiatalogue, seleted with planar eentriities 0 e < 0.30 and z max 0.5 kp (Cubarsi 010). We will not repeat here the disussion about the sampling parameters, although we do point out that other sampling riteria like metaliity or Strömgren photometry suh as b y olour were not able to desribe the small-sale struture of the veloity distribution as eentriities did (ibid. Figures 7 and 8). The eentriities in the atalogue were realulated in Stojanović (015) and ompared with those obtained by omputing the star orbits by following the approah of Vidojević and Ninković (009), resulting in a totally similar distribution. For veloities, we now use a helioentri oordinate system, with the radial helioentri veloity omponent U positive towards the GC, the helioentri veloity omponent V positive in the diretion of the Galati rotation, and the veloity omponent Z perpendiular to the Galati plane and positive towards the NGP. The veloities (Π, Θ, Z) of previous Galatoentri oordinate system have the radial veloity omponent Π positive towards the Galati antientre and the other oordinates with similar orientation as the helioentri system. For several subsamples Σ(e 0,z 0 )ofmaximum eentriity e 0 and maximum vertial amplitude z 0, the mean veloities, the entral veloity moments, and the expeted values E(e), E(e ), E(z max ), and E(z max) were omputed. They are shown in Table 1. Our purpose is to use the ommon trends of these nested subsamples to dedue the loal onstants. Instead, we do not intend to disuss whether a partiular subsample is more kinematially representative of the loal neighbourhood than the others Nested subsamples by eentriity For fixed z 0 =0.5, we selet a number of samples Σ(e 0 ) for dereasing values of e 0. The lower the limit e 0, the lower the moments μ rr μ 00 and μ θθ μ 00. This is the expeted behaviour aordingtoeq.(59). The subsamples Σ(e 0 ) are used to estimate the fators 1 = 1 κ r0 and = 1 κ γ r0 in Eq. (59). They allow to ompute the loal values for κ and γ for eah subsample listed in Table 1. While the values for κ derived from the sample moments μ 00 are very stable, the values for Ω obtained through the fit of μ 00 vary within about 30%, showing a dereasing trend as the eentriity dereases. These values are fitted all together by using the least squares. We express the moments in the Greek indies notation, i.e., by making expliit the veloity powers, aording to μ αβγ = u α 1 uβ uγ 3, depending on the peuliar veloity omponents. 10

11 ECCENTRICITY SAMPLES: IMPLICATIONS ON THE POTENTIAL AND THE VELOCITY DISTRIBUTION Table 1. The olumns in grey indiate the sampling parameters. Number of stars N, limiting eentriity e0, limiting height z0 (kp), several average values, mean helioentri veloity omponents (km s 1 ), and seond entral moments (km s ) with their sampling varianes are listed for eah sample. For the samples seleted by eentriity, the resulting planar epiyle frequeny (km s 1 kp 1 ) it is assumed =8.5 kp andthe ratio γ of μ 00 and μ00 are also listed. For samples seleted by the maximum height, the resulting vertial epiyle frequeny and the ratio of both frequenies are displayed. N e0 E(e) E(e ) z0 E(zmax) E(z max) U0 V0 W0 μ00 μ110 μ00 μ101 μ011 μ00 κ γ ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± N e0 E(e) E(e ) z0 E(zmax) E(z max ) U0 V0 W0 μ00 μ110 μ00 μ101 μ011 μ00 ν ν/κ ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

12 R. CUBARSI et al. The orresponding regression lines are shown in the upper and middle panel of Fig.. The fit for μ 00 (red dashed line) is very aurate and yields κ =41.1 ± 0. kms 1 kp 1. However, the regression line for μ 00 is not good enough. In the middle panel, the grey dashed line also fits the three samples with the lowest eentriity e 0 =0.01, 0.03, 0.05, that show a more similar trend. Note that the weight on the overall resulting slope of the samples with lower eentriities is lower, sine the least squares fit must ontain the origin. Therefore, the resulting Ω from the fit is not obtained as an average of values obtained from single samples. This has another onsequene: although the sampling errors of moments of samples with lower eentriities and fewer stars are higher, there is a disrepany in the slope of the two lines. As explained in the previous setion, the missed term of the asymmetri drift is responsible for this less aurate fitting. Aording to Strömgren s law, the asymmetri drift is lower for populations with lower moment μ rr. Then, the better estimation of Ω aording to Eq. (61) is obtained from the lower eentriity samples that have fewer stars and greater unertainty in their estimates. This means that the seond term in Eq. (59) should be interpreted as 1 κ γ μ θθ = lim a 0 a, (73) provided the sample has enough stars for the moment to be signifiant. Otherwise, the asymmetri drift must be taken into aount. In suh a ase, we label the peuliar veloity of the Sun as (u,v,w )=(Π Π, Θ Θ,Z Z ) and the helioentri mean veloity of the sample as (U 0,V 0,W 0 )=(Π 0 Π, Θ 0 Θ,Z 0 Z ). Then, the asymmetri drift may be expressed as Δ=Θ Θ 0 = v V 0. For these samples we estimate the asymmetri drift aording to Strömgren s lassial law Δ= v V 0 = kμ rr, with k onstant. In the top panel in Fig. 3 it is adjusted in the form V 0 = v kμ rr. Fig.. Diagonal seond entral moments (km s ) (blue diamonds) in terms of the average squared eentriities of the samples. The upper and middle panel are for samples seleted by planar eentriity, the bottom panel is for samples seleted by maximum height. Regression lines are plotted as red dashed lines. For the middle panel, the grey dashed line fits the three samples with lower e 0. (See the eletroni edition of the Journal for a olor version of this figure). 1 This provides values v =5.5 ± 0.31 km s 1 and k =0.035 ± km 1 s. We also hek the new Strömgren s non-linear relation (Golubov et al. 013) Δ = μ rr + η(μ rr μ zz ) μ θθ V0 v +k μ rr, Θ Θ with η = 1 whih takes into aount the ratio between the axes of the veloity ellipsoid. The first term on the right-hand side depends on the sample, while the seond one is onstant. In this ase, the resulting peuliar rotation veloity of the Sun is slightly lower, although within the error margin of the above value. Therefore, for these samples it provides a similar estimation. For both ases, the asymmetrial drift is depited in the middle panel in Fig. 3. The estimation of the loal angular veloity obtained for the three samples with lower eentriities,sotosayaordingtoeq. (73),isΩ =8.4±0.4 km s 1 kp 1 orresponding to the grey straight line in the bottom panel in Fig. 3. We assumed r =8.5 kp as we did for the values of Table 1. Notie that the value Ω obtained from these three small samples by the least squares estimation foing to inteept by zero is muh better than the average of the angular veloity of the samples.

13 ECCENTRICITY SAMPLES: IMPLICATIONS ON THE POTENTIAL AND THE VELOCITY DISTRIBUTION Δ = γ μ rr μ θθ, (74) provided the loal onstants are known. 6.. Nested subsamples by maximum height Nowwefixthevaluefore 0 and selet a set of nested subsamples Σ(z 0 ) with dereasing z 0. For these samples, the lower z 0,thelowerμ zz μ 00. This was predited by Eq. (6). However, the ratio μ 00 /μ 00 remains nearly onstant for all the subsamples. For instane, if the subsample seleted by e 0 =0.05 and z 0 =0.5 kp in Table 1 is now limited to z 0 =0.3, the moments μ 00 and μ 00 do not hange but we get the moment μ 00 =81.15 ± 1.98 km s whih is 63% lower. This fat should be taken into aount in seleting samples to study the veloity distribution. Nevertheless, the loal values derived from both subsamples i.e. the epiyle frequenies and the angular veloity, are signifiantly maintained. In addition, from Table 1 we see that the mean value E(e) also remains nearly onstant. The least squares fitting of Eq. (6) (Fig., bottom panel) provides the vertial epiyle frequeny with good auray, ν =84.0 ± 0.4 kms 1 kp 1. Fig. 3. (Top) Strömgren s law relating the seond veloity moments (km s ) to the helioentri mean veloities (km s 1 ) of the subsamples. (Middle) Asymmetri drift (km s 1 ) aording to Strömgren s lassial law (blue diamonds) and the new relation (red iles). (Bottom) Fitting the seond expression of Eq. (69). The grey dashed line is the fit of the three samples with lower e 0, and the red one is the least squares approximation for all the samples. (See the eletroni edition of the Journal for a olor version of this figure). By taking into aount the asymmetri drift i.e. from the seond expression of Eq. (69), we get Ω =9. ± 0.4 kms 1 kp 1. It orresponds to the red line fit in the bottom panel in Fig. 3. If the loal radius is taken as r =8kp,wegetΩ =7.4 ± 0.4 km s 1 kp 1, whih is totally onsistent with the usually assumed value. Therefore, the asymmetri drift of the stars in the sample an be estimated as Fig. 4. Average values of e and z max for samples limited by eentriity e 0 or maximum height z 0. (See the eletroni edition of the Journal for a olor version of this figure). 13

14 R. CUBARSI et al. Indeed, the planar and the vertial amplitudes are not totally independent. In terms of the mean square eentriity and maximum height, this behaviour is shown in Fig. 4 for the whole GCSIII sample with 0 e<0.30, but without limiting z max (13000 stars in total). The first graph shows the average E(zmax) (in red) in terms of the limiting eentriity e 0. It is a monotonous inreasing funtion for e , growing simultaneously with E(e ) (in blue). This fat should be taken into aount speially when working with samples of lower planar eentriity, sine a kinematially representative sample should not ontain many stars with high values of z max. However, the seond graph shows that E(e ) is independent from z 0 (blue line). Therefore, for a sample Σ(e 0 ) the approximate value z 0 for its maximum height an be determined in the following way. The first graph provides the average E(zmax ) in terms of the limit eentriity e 0. Then, the seond graph for the average E(zmax) in terms of z 0, read bakwards, gives the approximate limit z 0. For samples obtained from the limit z 0 the proedure annot be applied, sine E(e )intermsof z 0 is almost onstant. For instane, a subsample seleted by e 0 = 0.1, aording to the first graph of Fig. 4 has E(zmax) The seond graph tell us that this value orresponds to a sample limited by z 0 1 whih means that we should not inlude stars with z max > 1. For a limiting value e 0 =0.03, whih is the lower limit of this definite trend shown by the first graph, we estimate E(zmax) between 0.04 and 0.5. This average value read in the seond graph tells us that in the sample we should not inlude stars with z max higher than 0.5 or0.6. This was a reasoning based on experimental fats that should be further investigated i.e. how the limits e 0 and z 0 are onstrained. However, sine the epiyle model addresses the vertial and planar motions in an independent way, it does not provide a diret solution The simplest potential Aording to the atual values of the loal onstants, the potential an be none of the two partiular ases of Eq. () and Eq. (3), neither separable in ylindrial oordinates nor spherial. The former ase provides a ratio between the epiyle frequenies κ/ν = and the latter leads to ν =Ω,both unrealisti. We then study the general ase of Eq. (1). In the Galati plane, suh a potential beomes very simple. It has a term proportional to r and another proportional to r. To disuss the values of the loal onstants we propose a simple ase example onsisting in a modifiation of the spherial potential, that is, with U = M (r + z )+ F (s) =N (1 + Qs) 1, N r + Qz, (75) in Eq. (1) with N,Q are onstant. It ould be said that this is the simplest quasi-stationary potential allowing to estimate the three onstants in an independent way. We prefer to interpret it as a slight deviation of the spherial potential instead of using the nearly equivalent potential (for small values of s) with F (s) =N (1 Qs) i.e. a first degree polynomial taken from the power series of F (s). Then, we have F (0) = N and F (0) = NQ from where we determine the onstants M, N, andthedimensionless onstant Q involved in the potential as M = 1 8 κ, ( ) N = r4 κ 4 Ω (r ), Q = 4ν κ 4Ω (r ) κ. Thus, Q = 0 orresponds to the ylindrial potential and Q = 1 to the spherial potential. U r Fig. 5. Loal behaviour of the potential of Eq. (75) (red ontinuous line) ompared with the harmoni potential (green dashed line) with N = 0, and with the potential with M =0(blue dashed line) without the harmoni term. (See the eletroni edition of the Journal for a olor version of this figure). For the loal onstants, this yields Q 11.7, whih means that the term 1 r F ( z N r )= r +Qz will have elliptial isoontours ( r Q ) + z =onstwith Q = 3.4. We also ompare in Fig. 5 the loal behaviour of suh a potential (in red) at r in the symmetry plane with that of a potential with N =0 i.e. the harmoni potential (in green), and with a potential with M = 0 (in blue) i.e. only with the term proportional to r. It is lear that both terms have a signifiant ontribution to the shape of the loal potential. 14