3. Data sets and fitting methods
Obtaining good constraints on the spheroid density law requires a good photometric accuracy. This generally depends on using CCD detectors on large telescopes, on fields as wide as possible to cover large samples, and a large range of galactocentric distances. This can be obtained with a number of data sets at various galactic latitudes and longitudes. We have collected such data sets from the literature. Most have been made for extragalactic purposes.
3.1. Available data
The main data characteristics are summarized in Table 1. The photometric systems are close to the Johnson-Cousins system. Spheroid star selection was based on their magnitude and colour (either B-V, V-R, or V-I, depending on the available observations), in order to avoid presence of contamination by other populations. Aiming at model independent results, the model was used essentially to select colour and magnitude ranges and fields where the contamination by thick disc stars remains negligible under any reasonable thick disc hypothesis. For this reason all data brighter than magnitude 20 at intermediate and low latitudes were excluded. A small number of disc white dwarfs is also present in the selection but the proportion is at most a few percent and has no consequence on the result.
Our survey program include at the moment two fields, one towards the north galactic pole, another at intermediate latitude (l=150,b=60). The NGP field is the deepest up to now: it is complete and free from galaxy contamination up to magnitude 24. A full description of these data sets will be given in a forthcoming paper.
The other selected data sets are the six fields of the DMS survey (Hall et al., 1996; Osmer et al., 1998) observed in V and R bands at medium latitude, 4 fields from the Canada-France Redshift survey (CFRS, (Lilly et al., 1995; Le Fevre et al., 1995; Hammer et al., 1995)) dedicated to galaxy counts, two fields from the Koo & Kron investigation for quasars (Koo & Kron, 1982; Koo et al., 1986), another field from Reid & Majewski near the north galactic pole (Reid & Majewski, 1993).
The absolute visual magnitude of halo stars in the selected samples ranges between 3 and 8, except our north galactic pole field which reaches MV 11. All these fields taken together cover a large part of the (R,z) plane, as can be seen in Fig. 2 where the distributions in R and z of 90% of halo stars in each field of view are drawn.
3.2. Analysis method
Population synthesis simulations have been computed in every observed field using photometric errors as close as possible to the true observational errors, generally with photometric errors growing as a function of the magnitude and assumed to be Gaussian. Monte Carlo simulations are done in a solid angle much larger than the data in order to minimize the Poisson noise.
Then we compare the number of stars produced by the model with the observations in the selected region of the plane (magnitude, colour) and we compute the likelihood of the observed data to be a realization of the model (following the method described in Bienaymé et al., 1987a, appendix C). The likelihood has been computed for a set of models, varying the power law index between 2.0 and 3.5, the flattening between 0.3 and 1.0, the local density between 0.5 and 1.25 times the standard value as defined in Sect. 2.4, and the spheroid IMF slope from 1.0 to 2.2.
The confidence limits of estimated parameters are determined by the likelihood level which can be reached by pure random change of the sample: a series of simulated random samples are produced using the set of model parameters. The rms dispersion of the likelihood about the mean of this series gives an estimate of the likelihood fluctuations due to the random noise. It is then used to compute the confidence limit. Resulting errors are not strictly speaking standard errors, they give only an order of magnitude.
© European Southern Observatory (ESO) 2000
Online publication: June 30, 2000