Method one: by combining a sampling parameter related to an isolating integral of the stellar motion, an optimisation of
the mixture approach, and a maximisation of the partition entropy for the constituent populations of the stellar sample.
Method two: by segregating into different kinematical components in terms of the stellar orbital parameters.
Method three: by approaching a maximum entropy velocity distribution to samples selected in terms of stellar eccentricity
layers.
Working samples: HIPPARCOS and Geneva-Copenhagen survey catalog.
Results: kinematical characterisation of large-scale structures, such as thin disc, thick disc and halo, and identification of
small-scale structures, such as moving groups in the solar neighbourhood.
Consequences: confirmation of the Titius-Bode-like law for radial velocity dispersions and explanation of the apparent
vertex deviation of the disc from the swinging of two major kinematic groups around the LSR, by predicting a continuously
changing orientation of the disc pseudo ellipsoid.Postprint (published version

Alcobé López, Santiago

Cubarsí Morera, Rafael

Ninkovic, S.

Vidojevic, S.

English

Method one: by combining a sampling parameter related to an isolating integral of the stellar motion, an optimisation of

the mixture approach, and a maximisation of the partition entropy for the constituent populations of the stellar sample.

Method two: by segregating into different kinematical components in terms of the stellar orbital parameters.

Method three: by approaching a maximum entropy velocity distribution to samples selected in terms of stellar eccentricity

layers.

Working samples: HIPPARCOS and Geneva-Copenhagen survey catalog.

Results: kinematical characterisation of large-scale structures, such as thin disc, thick disc and halo, and identification of

small-scale structures, such as moving groups in the solar neighbourhood.

Consequences: confirmation of the Titius-Bode-like law for radial velocity dispersions and explanation of the apparent

vertex deviation of the disc from the swinging of two major kinematic groups around the LSR, by predicting a continuously

changing orientation of the disc pseudo ellipsoid

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Three kinematical methods to identify local galactic structures

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Three kinematical methods to identify local Galactic structures1R. Cubarsi1, S. Alcobe´1, S. Vidojevic´2,3,4, S. Ninkovic´4,51Dept. Matema`tica Aplicada IV, Universitat Polite`cnica de Catalunya, E08034-Barcelona, Catalonia, Spain. 2Faculty of Mathematics, University ofBelgrade, Studentski trg 16, 11000 Beograd, Serbia. 3Observatoire de Paris, Section de Meudon, 5 Place Jules Janssen, 92195 Meudon, France. 4 InstituteIsaac Newton of Chile, Yugoslavia Branch. 5 Astronomical Observatory, Volgina 7, 11060 Beograd 38, Serbia.AbstractMethod one: by combining a sampling parameter related to an isolating integral of the stellar motion, an optimisation ofthe mixture approach, and a maximisation of the partition entropy for the constituent populations of the stellar sample.Method two: by segregating into different kinematical components in terms of the stellar orbital parameters.Method three: by approaching a maximum entropy velocity distribution to samples selected in terms of stellar eccentricitylayers.Working samples: HIPPARCOS and Geneva-Copenhagen survey catalog.Results: kinematical characterisation of large-scale structures, such as thin disc, thick disc and halo, and identification ofsmall-scale structures, such as moving groups in the solar neighbourhood.Consequences: confirmation of the Titius-Bode-like law for radial velocity dispersions and explanation of the apparentvertex deviation of the disc from the swinging of two major kinematic groups around the LSR, by predicting a continuouslychanging orientation of the disc pseudo ellipsoid.Method I: MEMPHIS algorithmA sampling parameter P is defined to introduce a hierarchy into the sample (Alcobe´ & Cubarsi 2005 and Cubarsi et al. 2010),so that a set of nested subsamples is recursively drawn from the total sample. Some properties, which are associated withisolating integrals of the star motion, such as the absolute values of the velocity component perpendicular to the Galacticplane, P = |W |, of the rotational velocity, P = |V |, or of the total velocity P = |(U,V,W)| , are used as sampling parameters todiscriminate between populations. A bimodal pattern of two Gaussian distributions is recursively applied to identify differentkinematic behaviours within each subsample. In each case, the optimal sampling parameter provides the least informativesubsample associated with the more representative mixture parameters, as well as the least error in the mixture approach.For samples drawn from the HIPPARCOS catalogue (ESA 1997) and the Geneva-Copenhagen survey (GCS) of the Solarneighbourhood (Nordtro¨m et al. 2004), the best segregation of thin and thick discs is obtained by using the sampling parameterP = |(U,V,W)|, while the halo is detached from the total disc from the sampling parameter P = |W |.Sample #S Pop. P σU σV σW U V W ε [◦]HIP 12,516 t 91.5% |(U,V,W)| =230 28.2 ± 0.2 16.9 ± 0.2 12.5 ± 0.1 −11.2 ± 0.3 −14.7 ± 0.2 −7.1 ± 0.1 13 ± 11,003 T 7.3% 69.3 ± 1.3 37.9 ± 0.9 42.9 ± 0.9 −7.1 ± 2.2 −60.8 ± 1.2 −9.7 ± 1.4 4 ± 2159 H 1.2% |W | =180 179.6 ± 7.8 89.0 ± 6.3 90.8 ± 6.0 −0.61 ± 14.2 −234.9 ± 7.0 −10.8 ± 7.2 −6 ± 3GCS 12,415 t 93.8% |(U,V,W)| =230 29.7 ± 0.2 17.8 ± 0.2 13.9 ± 0.1 −10.4 ± 0.3 −15.2 ± 0.2 −7.1 ± 0.1 11 ± 1763 T 5.7% 65.7 ± 1.5 36.7 ± 1.1 41.5 ± 0.9 −3.4 ± 2.4 −59.3 ± 1.3 −7.1 ± 1.5 5 ± 262 H 0.5% |W | =170 178.6 ± 12.2 113.7 ± 15.0 110.5 ± 14.2 1.6 ± 22.7 −230.8 ± 14.4 −16.4 ± 14.0 −5 ± 10Table 1: Optimal mixture parameters for the HIPPARCOS (HIP) and GCS samples. The displayed quantities are: size ofthe optimal sample, sampling parameter, segregated population (t=thin disc, T=thick disc, H=halo) and mixture proportion,velocity dispersions, mean velocities (both in km s−1), and vertex deviation.Method II: Galactocentric orbitsThe orbital parameters are used to classify stars into Galactic subsystems such as the thin disc, the thick disc, and the halo(Vidojevic´ & Ninkovic´ 2008, 2009; Cubarsi et al. 2010). For the GCS catalogue, we assume that any star crossing theboundary of 70 kpc adopted in the model of the Galaxy is a halo star. There are 13 such stars in the sample, of which 5have angular momentum that differs from that of the Galactic rotation. Also, 26 other sample stars with a Galactocentric Vcomponent of opposite sign to that of the Galactic rotation are classified as halo stars. Stars of the Galactic disc are expectedto move around the Galactic centre in nearly planar orbits so their vertical eccentricities should not be significant. For thisreason, 25 stars in the sample with ev > 0.4 are classified as halo stars. In the case of nearly planar orbits, stars of the disc are1DYNAMICS AND EVOLUTION OF DISC GALAXIES, 27th Annual Pushchino Conference (May 31 - June 04, 2010). Sternberg AstronomicalInstitute (Moscow State University) and Pushchino Radio Astronomy Observatory, Moscow, Russia.1not expected to have a very high interval of R so a limit to the planar eccentricity should also exist. So 10 sample stars withep > 0.8 are classified halo stars. There is one sample star that, despite belonging to the halo according to any criterion givenabove, is assigned to the halo because its highest Z amplitude takes place almost in the middle of the interval in R. Therefore,we finally identify 75 (about 0.5%) halo stars.Sample #S Pop. σU σV σW U V W ε [◦]GCS 12,566 t 95.0% 30.2 ± 0.4 19.0 ± 0.3 13.5 ± 0.2 −9.6 ± 0.6 −16.1 ± 0.4 −7.1 ± 0.3 10 ± 1599 T 4.5% 67.8+4.1−3.6 40.9+2.5−2.2 46.2+2.8−2.5 −16.4 ± 5.5 −52.4 ± 3.3 −7.5 ± 3.7 1 ± 213,165 D 99.5% 32.9 ± 0.4 21.9 ± 0.3 16.5 ± 0.2 −9.9 ± 0.6 −17.4 ± 0.4 −7.1 ± 0.3 10 ± 175 H 0.5% 165+32−23 125+24−17 110+21−15 −6.9 ± 2.1 −201 ± 24 −14.4 ± 3.3 −9 ± 7Table 2: Segregation of populations for GCS sample from the Galactic orbits of the stars.The remaining stars belong to the Galactic disc. The variations in the shape and size of Galactocentric orbits for stars ofthe Galactic disc are correlated with their eccentricities, and for sufficiently high values of both ep and ev, the sides of theorbital trapezia become curvilinear. We identified an approximate border where curving of the sides of orbital trapezia begins.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800.050.10.150.20.250.30.350.4epevFigure 1: Vertical eccentricity versus planar eccentricityfor total disc stars. The broken line (red) separates starsof the thin disc (green) from those of the thick disc (blue).We present this border as a broken line in the ev versus ep plot forthe disc stars. The equations of the straight lines containing thesegments areep ∈ [0, 0.2] : ev = −0.25 ep + 0.15,ep ∈ [0.2, 0.5] : ev = −0.33 ep + 0.17. (1)The vertex occurs at (0.2, 0.1). The points lying inside this bro-ken line, 12,566 (95%), represent the stars of the thin disc, thoselying outside it, 599 (4.5%), the thick-disc ones. The elements ofthe velocity ellipsoid, the mean heliocentric velocity components,dispersions, and the vertex deviation with their uncertainties forthe halo, thick disc and thin disc are given in Table 2. In thecase of the mean values and dispersions, the uncertainties corre-spond to the 95% confidence intervals, whereas the uncertainty inthe vertex deviation is determined following the formula for errorpropagation.Titius-Bode-like law (TBLL)This empirical law (Alcobe´ & Cubarsi 2005) may be related with the average epicycle energy ER ∼ σ2U of the stars rep-resentative of the disk heating process. For each population, the radial velocity dispersion continuously increases with thesampling parameter P = |(U,V,W)|, up to reach some steady values, which are collected by the TBLL. Now, by adding thehalo component, we may certainly certify this relationship, and associate the discrete local stellar populations with the radialdispersions expressed by the equationσU(n) = 6.6(43)3n+2; n = 0, 1, 2, 3. (2)n σU -TBLL σU -MEMPHIS stellar pop.0 6.6 - spherical pop. at birth1 12 12 early-type stars2 28 28 thin disk3 66 65 thick disk4 156 156 haloMethod III: Maximum entropy approachIf an extended set of moments is available, this method provides a linear algorithm leading to a fast and suitable estimation ofthe velocity distribution (Cubarsi 2010). It can be used to model multimodal distributions that cannot be described throughGaussian mixtures. The maximum entropy function has the formf (V) = eP(V), (3)where P(V) is a power series of the velocity components containing as many terms as the number of moment constraints.The boundary conditions, which are usually assumed for the solutions of the stellar hydrodynamic equations, along with anintegral property involving the velocity moments (Cubarsi 2007), lead to a Gramian system of equations to determine thepolynomial coefficients of the phase density function.2ecc=0.01 ecc=0.02ecc=0.03NGC 1901Coma Berenices Pleiadesecc=0.05MiddlebranchComa BerenicesNGC 1901Pleiadesecc=0.10HyadesPleiadesSiriusecc=0.15Hyades+PleiadesSiriusecc=0.20 ecc=0.30Figure 2: Series of contour plots and distributions on the UV plane for GCS subsamples selected from |zmax| < 0.5 kpc andeccentricities up to 0.01, 0.02, 0.03, 0.05, 0.1, 0.15, 0.2 and 0.3. The origin is at the Solar velocity.3To describe the small-scale disc structure, several truncated distributions have been analysed in terms of metallicity, colour,maximum distance to the Galactic plane, and eccentricity. In particular, the eccentricity, which is directly related to the isolat-ing integrals of the star motion, is more discriminating than the absolute velocity for selecting subsamples. A representativethin disc, containing 90% of the whole sample, is selected from maximum eccentricity 0.3 and |zmax| ≤ 0.5 kpc. Its centralmoments are similar to the ones obtained for the thin disc of Tables 1 and 2. Within the thin disc, the eccentricity behaves asan excellent sampling parameter that distinguishes between different eccentricity layers allowing the subjacent structures ofFigure 2 to be visualised.Orientation of the disc velocity pseudo-ellipsoidFor stars with a similar period of oscillation around the LSR in the radial direction (under the epicyclic approximation),several simulations allow us to confirm that a two-peaked distribution of radial velocities, such as for eccentricity 0.01, is dueto a lognormal distribution of the eccentricities. For a mixture of stars with different periods and a lognormal distributionof the velocity amplitude of the stellar orbits, the bimodal shape is maintained. However, if the number of stars with nearlyvanishing amplitude increases, then the radial velocity distribution becomes unimodal, similar to the total thin disc samplewith e = 0.3. The bimodal behaviour of the central disc, associated with the previous major subsystems, may then beexplained from two different phenomena. On one hand, a perturbation similar to a pressure wave, acting in part along theradial direction, induces an oscillation of the radial velocity around the LSR, with heliocentric velocity (-9.72, -11.45, -6.65)km s−1, obtained from a sample with eccentricities e ≤ 0.15. On the other hand, both kinematical major groups, whichactually are placed at the solar position, are in opposite oscillation states. In addition both groups have a difference of about20 km s−1 in rotation mean velocity, so that one group of stars actually surpasses the other group. Therefore, the apparentvertex deviation of the thin disc may stem from the swinging of those major kinematic groups. A scenario of a continuouslychanging orientation of the disc pseudo ellipsoid is then possible.PK40 K20 0 20 40Q170180190200210220230240UK60 K40 K20 0 20 40VK50K40K30K20K1001020Figure 3: (Left) Velocity ellipsoids depicted from the sample with eccentricities e ≤ 0.15, in blue. They are centred inGalactocentric velocities Π0 = 15,Θ0 = 220, and Π0 = −15,Θ0 = 200, with the LSR placed in the middle of them. In red,the thin disc isocontours, from the sample with eccentricities e ≤ 0.3, have positive vertex deviation and are generated fromthe inner structure. The green dashed partial ellipsoids represent a situation with the opposite radial motions. (Right) Contourplots in the UV plane (heliocentric velocities) for the samples with maximum eccentricity e = 0.15 (blue) and e = 0.3 (red).ReferencesAlcobe´ S., Cubarsi R. 2005, A&A, 442, 929Cubarsi R. 2007, MNRAS 207, 380Cubarsi R., Alcobe´ S., Vidojevic´ S., Ninkovic´ S. 2010, A&A, 510, A102Cubarsi R. 2010, A&A, 510, A103ESA 1997, The Hipparcos Catalogue. ESA SP-1200Nordstro¨m B., Mayor M., Andersen J., Holmberg J., Pont F., Jørgensen B.R., Olsen E.H., Udry S., Mowlavi N. 2004, A&A,418, 989Vidojevic´ S., Ninkovic´ S. 2008, SerAJ, 177, 39Vidojevic´ S., Ninkovic´ S. 2009, AN, 330, 464

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Three kinematical methods to identify local galactic structures

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