4 Resolution of the condition equations

We determined the kinematic parameters of the galactic model via least squares fit from the equations:

 

$$v_{{\rm r}} \sum\limits_{j=1}^{10} a_j f_j^{{\rm r}}(R,l,b)$$ =

$$v_{{\rm l}} = k \, R \, \mu_{{\rm l}} \, \cos b \sum\limits_{j=1}^{10} a_j f_j^{{\rm l}}(R,l,b)$$ =

$$v_{{\rm b}} = k \, R \, \mu_{{\rm b}} \sum\limits_{j=1}^{10} a_j f_j^{{\rm b}}(R,l,b)$$ = (8)

where $v_{{\rm r}}$ is the radial velocity of the star in km s^-1, k = 4.741 km yr (s pc $\hbox{$^{\prime\prime}$ }$)^-1, R is the heliocentric distance of the star in pc, $\mu_{{\rm l}}$ and $\mu_{{\rm b}}$ are the proper motion in galactic longitude and latitude of the star in $^{\prime\prime}$ yr^-1 and b the galactic latitude. To derive the parameters a_j, an iterative scheme extensively explained in Fernández (1998) was followed.

The weight system was chosen as:

$$p_k = \frac {1} {\sigma_{k,{\rm obs}}^2 + \sigma_{k,{\rm cos}}^2}$$ (9)

where $\sigma_{{\rm obs}}$ are the individual observational errors in each velocity component of the star, calculated by taking into account the correlations between the different variables provided by the Hipparcos Catalogue, and $\sigma_{{\rm cos}}$ is the projection of the cosmic velocity dispersion ellipsoid in the direction of the velocity component considered (see Paper I).

To check the quality of the least squares fits we considered the $\chi ^2$statistics for N - M degrees of freeedom, defined as:

$$\chi^2 = \sum_{k=1}^{N} \frac {\left[ y_k - y(x_k; a_1,...,a... ... \right]^2} {\sigma_{k,{\rm obs}}^2 + \sigma_{k,{\rm obs}}^2}$$ (10)

where x_k are the independent data (sky coordinates and distances), y_k the dependent data (radial and tangential velocity components), N the number of equations and M number of parameters to be fitted. According to Press et al. (1992), if the uncertainties (cosmic dispersion and observational errors) are well-estimated, the value of $\chi ^2$ for a moderately good fit would be $\chi^2 \approx N - M$, with an uncertainty of $\sqrt{2 \; (N - M)}$.

As we did in Paper I, to eliminate the possible outliers present in the sample due to both the existence of high residual velocity stars (Royer 1997) or stars with unknown large observational errors, we rejected those equations with a residual larger than 3 times the root mean square residual of the fit (computed as $\sqrt{[ y_k - y(x_k; a_1,..., a_{10})]^2 / N}$) and recomputed a new set of parameters.