S. Alcobe

A Titius-Bode like law for stellar populations in the velocity space

The statistical algorithm MEMPHIS (Cubarsi & Alcobe 2006) was applied to a large sample from the Hipparcos catalogue with the full space motions (Cubarsi & Alcobe 2004), to segregate the kinematic populations of the solar neighbourhood. Four stellar populations were obtained, namely early-thin disk, young-thin disk, the whole thin disk (which contains both previous populations plus the continuum of old thin disk stars), and the thick disk population. Now, we wish to point out two main results from the analysis of such a segregation (Alcobe & Cubarsi 2005). First, the relationship between the maximum stellar velocity of a sample and its average age τ can be approximated by the relation |V|max ∝ τ . Second, the local stellar populations can be described from a Titius-Bode like law (TBLL) for the radial velocity dispersion, σ1 = 6.6 ( 4 3 ) , so that for values n = 2, 3, 5, 8 it determines some average energy levels of discrete populations, while for continuous intervals n 5 and n 7 it describes the velocity-age evolution of thin and thick disk components, as shown in the Table below. Thus, the velocity dispersions of the local kinematic populations seem to follow a geometrical progression, allowing us to do an analogy with the old Titius-Bode distribution for keplerian orbits, although a physical explanation for the later law remains still open (Lynch 2003). Indeed, such a TBLL in the velocity space could be already conjectured from previous published kinematic parameters of the Galactic components (e.g. Alcobe & Cubarsi 2001). As in the keplerian case, it is possible to argue that velocity dispersion values have too much uncertainty, but, even so, it is not possible to ignore anymore such a resemblance. Such results are consistent with Galactic formation models that predict some quasi-continuous stellar populations in the sense that the continuity is constricted by σ1 levels of the TBLL. The physical meaning of the variable n involved in the TBLL may be related with the average epicycle energy ER ∼ σ 1 of the stars representative of the disk heating process. It shows continuity from n = 3 to 5 for the thin disk, and from 7 to 8 for the thick disk, but discreteness from n = 2 to 3 between early-thin and young-thin disk, and from 5 to 8 between thin and thick disk components. For the thin disk, for example, the level n = 5 should represent the saturation point of maximum velocity dispersion, likely corresponding to the limited predicted by the observed wavenumber of spiral structure of the Milky Way, while the discontinuity from n = 5 to 7 indicates an abrupt jump in the average energy, that was produced when the thick disk was formed about 10± 1Gyr ago. n TBLL σ1 τ Population 2 12 12 Early-thin 3 16 16 < 4 Gyr Young-thin 5 28 28 10 Thin disk 8 66 65 14 Thick disk

Mean Velocity of Local Populations: Axiality, Age and Time Dependence

The segregated disk populations obtained from applying the MEMPHIS algorithm (Cubarsi & Alcobe, This symposium) to the Hipparcos catalogue are the basis of the following important kinematical results (Cubarsi & Alcobe 2006). In particular, the thick disk population was extrapolated until reaching an asymptotic ideal axisymmetric population. The obtained no net radial motion point had a radial heliocentric velocity of 20± 1 Km s−1 toward the galactic anticentre, which is the opposite of the solar radial motion. According to the obtained population mean velocities, an intriguing question arose: Why the young thin disk population, the continuum of old thin disk stars, and the thick disk, are moving, on average, in the same direction, as belonging to an unique main stream? On the other hand, the early thin disk population, in the region of the Hyades-Pleiades supercluster, remains as an isolated population, with a nearly null radial mean velocity. The thick disk stars drawn from increasing Hipparcos subsamples showed a decreasing trend of vertex deviation (Cubarsi & Alcobe 2004, Alcobe & Cubarsi 2005), by indicating a clear trend to axial-symmetry. If we assume that the asymptotic thick disk is nearly in steady state, then we can also admit a null radial mean velocity for the oldest thick disk. Thus, in the plane, using galactocentric cylindrical coordinates, with star velocities Π (toward the Galactic anticentre) and Θ (in the rotational direction), the mean velocity of thick disk stars satisfies

Entropy of The Mixture Probability as Indicator of Population Discontinuities: MEMPHIS Algorithm

A statistical population is always associated with the distribution of a random variable. If the distribution is too complex, it is useful to think about a superposition of simpler components. For the stellar velocities, the distribution is generally assumed as a mixture of gaussian distributions, since these are particular solutions of Chandrasekhar equations for statistical equilibrium, easy to interprete dynamically. In any case, the nature of the phase space density function might be explained from the dynamics of large stellar groups, sharing a common potential, through integrals like the energy, the angular momentum, or more general quadratic integrals, rather than from the kinematics of particular groups of stars, such as those producing streaming motions. Another fact to point out is that a well defined population must be adult, that is, well described from some constant and consistent statistics. Otherwise, if a sample estimate varies depending on an external parameter, for example the mean age of a sample, then, either the type of distribution might be exchanged, or the population might be divided into an adequate number of subpopulations. In this context, the MEMPHIS algorithm (Alcobe & Cubarsi 2005) has demonstrated to be useful in order to identify nearby stellar populations, by using the full space motions of a stellar sample obtained from the Hipparcos catalog (Cubarsi & Alcobe 2004). A sampling parameter, the modulus of the velocity |V|max, produces a hierarchical set of nested samples allowing to detect significant population components of their velocity distribution. Although the method was designed to identify normal distributions, it also provides, as a less prior information pattern, a good approach to segregate non-gaussian populations. The entropy variations of the mixture probability allow us to estimate the number of populations without any prior assumption about such a number, according to the parameters of the Table below (sampling parameter, velocity dispersions, means –in Km s−1– and vertex deviation are displayed in the UVW -cartesian heliocentric coordinates system). The main disk structure is supported by two gaussian populations, thin and thick disk. Two subcomponents were found within the thin disk, which have a high deviation from gaussianity in the radial direction, as a consequence of a non-random behaviour (Cubarsi & Alcobe 2006), but not in the other directions. Old disk stars were obtained as a broad wing of the young-thin disk. All together, with the early-thin population, they make up the thin disk component. In total 2+2 populations, since halo stars were not included in the sample. |V|max Population % σU σV σW U0 V0 W0 e 51 Early-thin 37 12± 31 11± 1 7± 1 −21.3± 4.3 −14.7± 0.3 −5.9± 0.2 Young-thin 38 16± 22 14± 1 14± 1 6.3± 4.3 −7.7± 0.3 −6.9± 0.2 209 Thin disk 91 28± 1 16± 2 13± 1 −10.6± 0.4 −14.0± 0.3 −7.2± 0.3 10± 2 Thick disk 09 65± 2 39± 9 41± 2 −14.7± 2.9 −64.8± 2.7 −8.3± 2.7 7± 3