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Astron. Astrophys. 359, 103-112 (2000)

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4. Results and discussion

4.1. Constraints on the spheroid density law

Fig. 3 gives the value of the likelihood as a function of the flattening, power law index and local normalization. On the left, iso-likelihood contours are drawn for four values of the local normalization (0.5, 0.75, 1.0 and 1.25 [FORMULA]). On the right, we show the likelihood values as a function of the power law index for the best fit value of the flattening.

[FIGURE] Fig. 3. Likelihood as a function of power law index, flattening and local density: On the left: iso-likelihood contours at 1, 2 and 3 sigmas in the plane (power law index, flattening) of the spheroid population. On the right, likelihood as a function of the power law index, for the best flattening. Lines are coded by the local density. Solid line: standard local density as defined in Sect. 2.4; dotted line: 0.75 [FORMULA]; dashed: 0.50 [FORMULA], dotted-dashed: 1.25 [FORMULA]

Comparing the results of different local normalization we conclude that the choice of the local normalization sensitively displaces the best fit power law index and flattening, but their likelihood are not similar. The best fit model is obtained either with a local density of 0.75 [FORMULA], a power law index of 2.44 and an axis ratio of 0.76, or a local density of 0.5 [FORMULA], a power law index of 2.24 and an axis ratio of 0.86. The values obtained with a standard local density [FORMULA] are slightly worse but stay within 1 sigma confidence level. They are a power law index of 2.62 and an axis ratio of 0.70. The best fit local densities 0.5[FORMULA] and 0.75[FORMULA] agrees with the Bahcall & Casertano determination of the local luminosity function as seen in Fig. 1, but conflict with Dahn et al., which favors a local density of 1.25 [FORMULA]. However, in the present study the statistics is dominated by stars with absolute magnitudes in the range 3 to 8, a range poorly represented in the Dahn et al. sample. Only deeper counts could give constraints on the fainter part of the luminosity function.

It is worth having a look at the colour distributions as predicted by the best fit model compared with the observational data. Figs. 4 and 5 show the colour distributions observed (dots) and predicted (heavy solid line) by the best fit model ([FORMULA]=0.76, n=2.44, 0.75[FORMULA]) in the selected magnitude interval in each tested field. Superimposed we show the distribution of the spheroid population alone as predicted by the model (light solid line). We see here that some photometric systems are not closely matched by the model, as seen by slight shifts between model and data in some cases. But the way we have selected spheroid stars in the blue peak of the distribution cannot introduce a bias even in case of colour shifts.

[FIGURE] Fig. 4. Colour distributions of selected samples. Contributions of different magnitude interval have been summed up for clarity. Observations in different fields are shown by dots. The ordinates give the true number of observed stars in each colour bin. Heavy solid lines are the predicted number of stars by the model assuming a power law index of 2.44, a flattening of 0.76 and a local density of 0.75[FORMULA] for the spheroid. Thin lines show the contribution of the spheroid alone. 

[FIGURE] Fig. 5. Same as Fig. 4 for other fields

4.2. Sensitivity to the IMF slope

Whatever the assumed IMF slope in the range 1-2.2, the maximum likelihood is obtained for the same density law parameters. There is a slight likelihood variation related to the choice of the IMF, but it is only due to the deepest magnitude bin towards the pole.

A separate analysis of star counts deeper than 22 towards the pole can help determine precisely the IMF slope. In this magnitude range spheroid stars with absolute magnitude 10-11 contribute substantially to star counts, while their frequency is sensitive to the IMF as can be seen on Fig. 1b. The analysis is slightly different from the determination of the density law. In this range of absolute magnitude the subdwarf sequence turns redwards making the colour index a good luminosity indicator. The V-I distribution is used as an additional constraint. A V-I histogram is built with a bin 0.1 magnitude wide over the range 0 to 3. The density law is adopted from the above analysis, so the free parameters are the halo and thick disc IMF slopes. Since these two populations are quite well separated in the (V,V-I) plane, the two IMF slope estimates are de-correlated. Table 2 gives the resulting slope estimates with their likelihood in the magnitude range 22-24 for the different spheroid density laws determined previously.


Table 2. IMF slopes of the spheroid and thick disc populations determined by the maximum likelihood technique for four models of spheroid density laws

Spheroid models with a local density 0.75 [FORMULA] and 1.0 [FORMULA] give the maximum likelihood, well in agreement with the previous result. However the model with 0.50 [FORMULA] is noticeably worse. Eventually, the resulting IMF slopes do not depend significantly on the assumed density laws and the likelihood is well peaked around the maximum indicating a robust determination. We conclude that the IMF of the halo, in the mass range [0.1, 0.8] is:


while the IMF slope of the thick disc seems to be slightly smaller and similar to the disc's (Haywood, 1994). These values do not account for binarity. Thus the true IMF should slightly steepen. We leave the value uncorrected until more data are available on the binary fraction among low mass spheroid stars.

This result is the first direct measurement of the mass function of field star spheroid with a good statistics, thanks to the wide field of the CCD mosaic. Several previous determinations used kinematically selected samples (see Sect. 2.4 for references) or deep fields. But the latter were limited to narrow fields: the first attempt by Richer & Fahlman (1992) lead to a very steep IMF slope of [FORMULA] which had given hope for a dark matter halo of brown dwarfs. Later results have given shallower slopes but the uncertainties were not significantly decreased.

Gould et al. (1998) analyzed a sample of 166 stars in 53 HST WFPC fields, making difficult the de-correlation between structural parameters of the spheroid and its mass function. They found a luminosity function down by a factor two from the present one and deduced an IMF slope of [FORMULA] (in our notation). Their result relies upon the assumption that the spheroid has a mean metallicity of -1.0, which looks too high considering most direct measurement of its abundances. This high metallicity induces an overestimate of the luminosity at a given colour, hence of the distance, as well as an overestimate of the mass relatively to a smaller assumed metallicity.

4.3. Variations of density law with galactocentric position

If we independently check the results obtained in inner fields and in outer fields, we are able to search for solutions with varying power law index and flattening over the galactic radius. Contrarily to Preston et al. (1991) we find no evidence for varying power law index or flattening. However, a round spheroid is ruled out by the inner field data as well as by fields at low latitudes. Thus our results are compatible with a true power law and a constant flattening all along the tested galactic radius.

When comparing data sets from different sources in close galactic fields, discrepancies appear which are larger than expected on the basis of pure random noise. This may be due either to data incompleteness, or to systematic errors in the photometry (including mismatch of the standard photometric system), or to true inhomogeneities in the spheroid distribution. Currently available data are not sufficient to discriminate between these different causes. Homogeneous wide field surveys will be necessary to clarify these aspects. The scope of the current investigation is for this reason limited to large scale average characteristics.

4.4. Contamination by other populations

The blue peak at these magnitudes may be contaminated by disc white dwarfs or by thick disc main sequence stars. The former are very few compared to the density of the halo. The contamination by disc white dwarfs, as determined by the model, is at most 5% in the magnitude range 22-24.

The contamination from the thick disc has been estimated using our best fit thick disc model as adjusted on medium deep star counts. The contamination can reach about 30% in the magnitude range 18-20 but becomes negligible at magnitude larger than 20 as seen in Fig. 6. Hence, we do not take into account magnitudes lower than 20 in our study.

[FIGURE] Fig. 6. Colour distributions towards the galactic pole in three magnitude intervals. Dots: observational counts; heavy solid lines: model distribution assuming a standard thick disc, light lines: model assuming a doubled thick disc density. The contamination of the blue peak by the thick disc can reach 30% at magnitudes 18-20 but becomes negligible at fainter magnitudes.

Had the thick disc contamination been underestimated, then the contribution assigned to the halo in the blue peak would be too large, resulting in a possible distortion of the density law. In order to evaluate how this would affect our conclusions, we have investigated different thick disc models which could fake the halo contribution to the blue peak. Attempts were limited to realistic thick discs roughly fitting the red peak. We have selected two extreme thick disc parameters for which the contamination to the blue peak becomes significant. A thick disc with a local density of 3.9%, a scale height of 1150 pc and an IMF slope of 1. (referred to as model B). A thick disc with a scale height of 2 kpc, a local density of 0.5% and an IMF slope of 1.75. (model C). With such thick disc models, the process of adjusting the spheroid density law parameters end up to tiny local density of about 25% of the standard value and very small power law index of the order of 1.5. Surprisingly, the fit is good on star counts up to magnitude 22, showing that a large range of parameters can reproduce a wide set of star counts. However at magnitude 22-24, model B and C are unable to reproduce the counts, as exemplified in Fig. 7, where at the top star counts at the pole in the magnitude range 22-24 are overestimated in the blue peak, and in the range 20-22 (bottom) the fit is still acceptable.

[FIGURE] Fig. 7. Observed colour distributions towards the north galactic pole (magnitude range 22-24) and SA68 Koo & Kron data (magnitude range 20-22) compared with model predictions with two thick disc models. On the left side, model B (local density of 3.9% of the thin disc, a scale height of 1150 pc and an IMF slope of [FORMULA]), On the right side, model C (local density of 0.5%, scale height 2 kpc, IMF slope 1.75). Thin lines are the contribution of the adjusted spheroid (see text).

So, the degeneracy between thick disc models and halo parameters holds only if star counts are not deep enough. Keeping reasonable values for the thick disc parameters leads to a small contamination with no risk of underestimation of the halo density. If the thick disc contribution is higher than expected from standard models then we would overestimate the local halo density and power law index, strengthening our conclusion towards a flat spheroid with a small power law index.

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© European Southern Observatory (ESO) 2000

Online publication: June 30, 2000