Maximum Likelihood Estimation of Local Stellar Kinematics
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A&A 551, A9 (2013) Astronomy DOI: 10.1051/0004-6361/201220430 & c ESO 2013 Astrophysics Maximum likelihood estimation of local stellar kinematics T. Aghajani and L. Lindegren Lund Observatory, Lund University, PO Box 43, 22100 Lund, Sweden e-mail: [email protected], [email protected] Received 21 September 2012 / Accepted 28 November 2012 ABSTRACT Context. Kinematical data such as the mean velocities and velocity dispersions of stellar samples are useful tools to study galactic structure and evolution. However, observational data are often incomplete (e.g., lacking the radial component of the motion) and may have significant observational errors. For example, the majority of faint stars observed with Gaia will not have their radial velocities measured. Aims. Our aim is to formulate and test a new maximum likelihood approach to estimating the kinematical parameters for a local stellar sample when only the transverse velocities are known (from parallaxes and proper motions). Methods. Numerical simulations using synthetically generated data as well as real data (based on the Geneva-Copenhagen survey) are used to investigate the statistical properties (bias, precision) of the method, and to compare its performance with the much simpler “projection method” described by Dehnen & Binney (1998, MNRAS, 298, 387). Results. The maximum likelihood method gives more correct estimates of the dispersion when observational errors are important, and guarantees a positive-definite dispersion matrix, which is not always obtained with the projection method. Possible extensions and improvements of the method are discussed. Key words. methods: numerical – methods: analytical – astrometry 1. Introduction dispersion) for a group of stars, when the tangential velocities are known, but not the radial velocities. This method has be- Statistical information about the motions of stars relative to the come popular and can be considered a standard for this purpose. sun may contain important hints concerning the origin and his- We will refer to it as the projection method (PM) throughout tory of the stars themselves (e.g., by identifying kinematic pop- this paper. In addition to the original work by Dehnen & Binney ulations and streams) as well as of the physical properties of (1998), the same method (or variants of it) has been used, e.g., the Galaxy (through dynamical interpretation of the motions). by Mignard (2000), Brosche et al. (2001), van Leeuwen (2007), Ideally this requires that all six components of phase space (po- and Aumer & Binney (2009). sitions and velocities) are known for all the stars in the investi- Despite its wide usage in the literature, the projection method gated sample. This can in principle be achieved through a com- is not founded on any particular estimation principle such as bination of astrometric data (providing positions and distances, maximum likelihood (ML) or Bayesian estimation, but simply from the parallaxes, and tangential velocities from the proper fit the projected first and second moments of the space veloc- motions and parallaxes) and spectroscopic radial velocities. Very ities to the corresponding observed moments of the tangential often, however, such complete data are not available for all the velocities. This works well for large samples, provided that the stars in a sample. For example, the Hipparcos Catalogue (ESA observational uncertainties in the data are negligible compared 1997) gives the required astrometric information for large sam- to the uncertainties due to the sampling, but there is no guar- ples of nearby stars, but not all of them have known radial ve- antee that it is unbiased, or asymptotically efficient as expected locities. Restricting the investigation to stars with measured ra- for an ML estimate. On the contrary, by neglecting the observa- dial velocities could introduce a serious selection bias (Binney tional errors the resulting velocity dispersions are probably over- et al. 1997). The Gaia mission (de Bruijne 2012) will not only estimated. Moreover, for small samples the projection method provide vastly improved astrometric data for much larger stellar may sometimes give unphysical results, in that the estimated dis- samples, but also radial-velocity measurements for stars brighter persion tensor is not positive-definite – implying that the mean than 17 mag; for the fainter stars, however, the phase space square velocity is negative in some directions. Since it is desir- data will still be incomplete. On the other hand, on-going spec- able that the kinematic parameters can be consistently and effi- troscopic surveys such as RAVE (Steinmetz et al. 2006) already ciently estimated also for small samples and in the presence of provide radial velocities for large samples without the comple- non-negligible observational uncertainties, we introduce here a mentary astrometry. Thus we are often faced with the problem new and more rigorous approach, based on the maximum likeli- to estimate kinematical parameters (such as mean velocities and hood method. velocity dispersions) from incomplete phase space data, lacking either the radial or tangential velocity components, or the ac- curate distances needed to derive the tangential velocities from 2. Kinematic parameters and the projection method proper motions. For a homogeneous population of stars the phase space den- Dehnen & Binney (1998) derive a simple and elegant method sity f (r, u, t) describes the density of stars as a function of po- to derive local stellar kinematics (mean velocity and velocity sition (r), velocity (u) and time t.Bythelocal kinematics we Article published by EDP Sciences A9, page 1 of 7 A&A 551, A9 (2013) mean the distribution function here (r = 0) and now (t = 0), that 3.2. Exact expression for the likelihood is f (u) ≡ f (0, u, 0). It is usually assumed that f (u) is a smooth function, and the most common assumption is the Schwarzschild For a problem involving n stars, the parameters of the model are: u approximation, that f ( ) is a three-dimensional Gaussian distri- – u¯ = the mean velocity of the stellar population (a 3-vector, or bution (velocity ellipsoid), or a combination of a few Gaussian 3 × 1matrix); distributions. The velocity ellipsoid is completely described by –D= the dispersion tensor of the stellar population (a sym- u the mean velocity and the dispersion tensor D. metric 3 × 3 matrix; contains 6 non-redundant elements); Throughout this paper we use heliocentric galactic coordi- – p = the true parallaxes of the stars (an n-vector, or n × nates (x,y,z) with +x pointing towards the Galactic Centre, +y 1matrix). in the direction of rotation at the Sun, and +z towards the north Galactic pole. The corresponding heliocentric velocity compo- It is necessary to introduce the true parallaxes pi as formal model nents are denoted (vx,vy,vz)or(u,v,w). parameters, although the strategy is that they will be eliminated For a stellar population, the mean velocity and dispersion on a star-by-star basis leaving us with a problem with only nine tensor are defined as model parameters, namely the (non-redundant) components ofu ¯ and D. We denote by the vector θ the complete set of model u = u ¯ E[ ] , (1) parameters (i.e., the unknowns to be estimated). For n stars the + D = E (u − u¯)(u − u¯)T , (2) total number of model parameters is n 9. The observables are, for each star i = 1 ...n, the observed where E is the expectation operator (population mean). Given components of proper motion,μ ˜l,i andμ ˜b,i, and the observed par- the 3-dimensional velocities ui, i = 1, 2, ..., n for a sample of allaxp ˜i. The total number of observables is 3n. These have ob- stars it is possible to estimate the population mean values by the servational uncertainties that are given by the 3 × 3covariance sample mean values, matrix Ci. Sometimes it is useful to denote by the vector x the complete set of observables (or data). 1 1 Given the observations, the likelihood function L(θ) ≡ u¯ u , D (u − u¯)(u − u¯)T. (3) n i n i i L(¯u, D, p) numerically equals the probability density function i i (pdf) fx(x|θ) of the observables x, given the model parameters θ θ θˆ Note that this estimate of D is always positive definite (except . The objective is to find the ML estimate of , denoted ,i.e., in trivial degenerate cases), but it requires 3-dimensional veloc- the (hopefully unique) set of parameter values that maximizes θ θ = θ ities and the dispersions are likely to be overestimated when the L( ) or, equivalently, the log-likelihood ( ) ln L( ). The total velocity components have significant observational errors. log-likelihood function is given by The tangential velocity of a star can be written u = μ | + − (¯, D, p) ln fμ˜ ,i(˜i pi) ln gi(˜pi pi) , (5) τ = Au, (4) i where gi is the centered normal pdf with standard deviation where A is a projection matrix depending only on the position 1/2 σp,i = [Ci] , i.e., the uncertainty of the parallax pi. It is clear of the star. In the projection method the assumption that the po- 33 ff sitions (and hence the matrices A) are uncorrelated with the ve- that the parameter pi only a ects the ith term in the sum above. locities is invoked to derive a relation between the mean of τ and Therefore, when maximizing with respect to pi we only need to the mean of A, allowing the latter to be solved. In a similar way, consider that one term. For simplicity we drop, for the moment, the elements of D are derived from the relation between the sec- the subscript i so that the term to consider (for one star) can be ond moments of τ and u. For further details we refer to the paper written: by Dehnen & Binney (1998).