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Astron. Astrophys. 330, 136-138 (1998)

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2. Results

As explained above we consider K2-K5 III stars. The luminosity function [FORMULA] is defined as usual, i. e. [FORMULA] gives the number of stars per [FORMULA] with an absolute K magnitude in the interval [FORMULA]. It is here approximated by a Dirac's delta function, i.e. [FORMULA] centred at [FORMULA]. Its true dispersion is only 0.6 mag. The effect of a non-vanishing dispersion on absolute magnitude could be important, as the resulting value for the scale length could be affected by some sort of Malmquist bias. We have performed several numerical calculations using a gaussian distribution for the luminosity function, obtaining similar results. The final value of the estimated error is not affected either, as it mainly arises from the fitting process. Even though these kind of effects are difficult to evaluate, the approximation of the luminosity function by a [FORMULA] was found to be a satisfactory assumption. In the fundamental equation of star counts (e. g. Mihalas and Binney, 1981; Gilmore and Reid, 1983; Calbet et al., 1995) the density function is also taken into account. It was assumed that:

[EQUATION]

where [FORMULA] is the density function, i. e. the number density of stars taking its local value as unit, R is the galactocentric distance, and z is the vertical coordinate. We take [FORMULA]. The vertical exponential profile and the value of [FORMULA] were adopted from Wainscoat et al. (1992). We have repeated these calculations for different values of [FORMULA] and we have obtained similar results, i. e. neither the proposed value of h nor its estimated error are affected. Actually h is found to be noticeably independent of [FORMULA]. We therefore think that this choice for [FORMULA] cannot be an important source of errors.

Defining as usual [FORMULA], as the number of stars per squared degree with magnitude between 9 and 10, the fundamental equation of star counts becomes

[EQUATION]

where [FORMULA] is the solid angle, C is a constant and r is the distance from the Sun. This expression is easily transformed into

[EQUATION]

where

[EQUATION]

[FORMULA] is another constant, and

[EQUATION]

As b is taken to be constant, [FORMULA] is a function only of l, and therefore [FORMULA] is a function only of l. Eq. (3) predicts a linear function relation between [FORMULA] and [FORMULA]. The results are plotted in Fig. 1. In this figure, error bars represent Poisson errors. After fitting (using a standard chi-squared method as described in Press et al., 1992), we obtained

[EQUATION]


[FIGURE] Fig. 1. Log of the number of K=9-10 magnitude stars per squared degree as a function of [FORMULA] defined in the text.

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© European Southern Observatory (ESO) 1998

Online publication: January 8, 1998
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