Volume 547, November 2012
|Number of page(s)||17|
|Section||Galactic structure, stellar clusters and populations|
|Published online||01 November 2012|
Here we outline the procedure to pass from a radial velocity set of data to the true velocity first order moments in the case of an all-sky-survey coverage. This is an application of what is shown in Sect. 3.1 that we used as a reference case. In this case the matrix in Eq. (10)can be computed directly using spherical coordinates . The generic element of the matrix, , (A.1)can be computed as (A.2)where S2 = [0,2π[× [− π/2,π/2[ and the solid angle . Or, element by element: (A.3)From its inverse we can easily obtain the elements of the velocity ellipsoid as: (A.4)In general, all this is valid when the data have a spherically symmetric distribution. With real data, owing to the partial sky coverage, the matrix (A.3)may substantially differ from the symmetric case. To take this into account, we use the matrix (A.3)as a mask, i.e. as a constraint on the relative weight that the generic matrix element has with respect to another element when trying to maximize the parameter q of Eq. (17), i.e. to control the coupling of the off-diagonal blocks of the operator matrix p. The generic matrix element to be determined requires integrals of many complex trigonometric functions, thus implying long tedious calculations. Fortunately, many elements are null by symmetry. For instance, because there exist only three linearly independent isotropic fourth-rank tensors related to the Kronecker delta tensor by ℐijkl = δilδik, and , the generic symmetric fourth-rank isotropic tensor X  can be expressed as a linear combination of these (e.g., classically written with coefficient Xs,Xa,XTr): (A.5)so that only the different terms can be easily singled out from Eq. (A.5). Comparing the shape of the matrix in Eq. (A.3)with the true case of RAVE data from Fig. A.1 shows that in most cases nearly symmetric conditions apply. Indeed, matrices in Eq. (10), , Eq. (11), , and (12), closely resemble the case of spherical symmetry (see, e.g., Fig. A.1 for the matrix of Eq. (10)). For the third moment we have (A.6)to get (A.7)
The inverse matrix of the Eq. (10)for the best fit model. The matrix closely resembles the fully analytical case except for the numerical part. Different shades of colour are applied to visualize the symmetries.
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To disentangle the first and second cumulants of the thick disk alone, we need up to the fourth order cumulants of the mixture. The V-cumulants mixture distribution of third and fourth order are introduced here. These moments exhibit a symmetry along the azimuthal velocity vector of the mixture. Hence they differ from what is laid out in the Appendices B, C, and D of Cubarsi & Alcobé (2004). Also the notation differs in order to be consistent with the notation used in our study. The third cumulants from which we can easily compute the two components of the normalized vector defined in Eq. (17), are (B.1)Once the values of and are computed as an SVD solution of the previous overdetermined system, the remaining cumulants can be calculated via (B.2)The fourth order equation can be computed as (B.3)Once we have the whole set of the cumulants we define the tensor (B.4)where . The constraining equations derived in Cubarsi & Alcobé (2004) can be reduced to the following set of fourteen scalar relations (B.5)that we solve in a least-squares sense with respect to the elements cφφ and . The final step to calculate the desired results for q is to take the derived values for cφφ and and to work out the last constraining equation from the relations (B.6)
By testing the method developed in this paper on a completely synthetic catalogue created with the Padova Galaxy Model (e.g., Vallenari et al. 2004,and references therein) we can test our ability to recover the correct results, to refine the method and to improve its performance.
Moreover we obtain here a remarkable example of convergence of the method on the true RAVE data, where the method is forced to work without the knowledge of the previously determined photometric distances by Zwitter et al. (2010). A fundamental selection criterion in order to achieve this particular result is the cut in the surface gravity of the stars. In order to avoid the contamination by giant stars which can enter our sample because of their intrinsic luminosity despite their distances (see, e.g., Klement et al. 2008, 2011) we plot in Fig. C.1 the distance distribution of the dwarf stars selected with a cut in the surface gravity at log 10g = 3.5. Moreover in the introductory consideration (Sect. 2) we explained how the expansion over a parameter ε of the radial component of the Galactic rotation leads to only a weak influence of the photometric distance errors on the results. This parameter has to be small, of the order of . As evident by plotting the distances for an averaged sample of stars of the mixture ( ≈ 38 000 stars) the distribution shows an ε variation well within the confirming that our selection cuts are able to retain stars with distances within a range of without an a priori knowledge of the distance.
Distance distribution for the selected sample of stars. The number is normalized to the highest value to place the peak position at 1. The x axis shows distances divided by the adopted solar position (to better illustrate the parameter ε for which the adopted approximations hold).
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After that, technically speaking, this result is achieved by simply allowing the variation of the randomly sampled distances of each star not only within their photometrically determined errors, but along all of the lines of sight, working only with directions (l,b) instead of the full parameter space of directions and distances (l,b,d). The results are presented in Table C.1. The results are remarkably similar to the ones presented in Table 1 as expected from the selection criteria adopted in Eq. (5).
Values of the thick disk velocity dispersion tensor without the knowledge of the distances.
© ESO, 2012
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