5 Test of robustness

The spiral arm potential is expected to contribute between 5-10% to the whole galactic gravitational field in the solar neighbourhood. This small contribution, and the large observational errors and constraints present on the spatial and kinematic parameters of distant stars, make it very difficult to quantify the kinematic perturbations induced by the spiral potential. Then, the results found in the literature have been characterised by significant uncertainties and discrepancies. A good example of this is the contradictory results obtained for the phase of the spiral structure at the Sun's position (see Sect. 7.2): between arms or near an arm? Interesting questions to answer after the release of the Hipparcos data are:

• How does the unprecedented astrometric precision provided by Hipparcos help to diminish such uncertainties? Are they small enough to measure the spiral arm kinematic parameters to any useful degree of accuracy?

• Can we quantify the biases induced by our observational errors and constraints?

• How can the correlations among variables present in the condition equations affect the kinematic parameters?

With regard to our stellar data, as seen in Sect. 2, both O and B star and Cepheid samples suffer from different observational limitations. Although the O and B star sample is large in number, it is limited in distance to no more than 1.5-2 kpc from the Sun (see Fig. 1 and Table 1). In contrast, the Cepheid sample reaches distances of up to about 4 kpc, but the number of stars with reliable data still remains very small (Fig. 2 and Table 1).

 

 

Table 1: Number of stars with distance and proper motions (in brackets those stars with also radial velocity) in several distance intervals. Distances for Cepheids computed from Luri's (2000) PL relation.

sample of O and B stars
0.1 < R < 2 kpc	0.6 < R < 2 kpc
3418 (1903)	448 (307)
sample of Cepheid stars
0.1 < R < 4 kpc	0.6 < R < 2 kpc	0.6 < R < 4 kpc
119 (111)	103 (95)	164 (145)

Numerical simulations were performed to answer the above questions, that is, to evaluate and quantify all the uncertainties and biases involved in our resolution process. In Appendix B we present the detailed procedure followed to build simulated samples as similar as possible to the real data for both O and B stars and Cepheids. This Appendix also contains an exhaustive analysis of the full set of cases which were simulated. Next we present the main conclusions arising from this work, which substantially contribute to the analysis of the real data:

• As a general trend, owing to the correlations between variables, the biases and uncertainties in some kinematic parameters substantially vary when changing the real values of $\psi _\odot$ and $\Omega _{{\rm p}}$;

• The first- and second-order terms of the galactic rotation curve are accurately derived, with an uncertainty of less than 1.3 km s^-1 kpc^-1 (km s^-1 kpc^-2) in all cases. Whereas a negative bias for both parameters is detected for O and B stars (up to -0.7 km s^-1 kpc^-1 for $a_{\rm r}$ and -1 km s^-1 kpc^-2for $b_{\rm r}$), it is negligible for Cepheids;

• For the spiral structure parameters, we obtain better results from O and B stars than from Cepheids. This is due to the larger number of O and B stars. For both samples we found a clear dependence of some parameters with the assumed values for $\psi _\odot$ and $\Omega _{{\rm p}}$. We can expect a bias in $\psi _\odot$ up to $\pm 20\hbox{$^\circ$ }$, and uncertainties of 10-20 $\hbox{$^\circ$ }$ for O and B stars and 30-60 $\hbox{$^\circ$ }$ for Cepheids. For $\Pi _{\rm b}$ and $\Theta _{\rm b}$ we found biases up to $\pm 2$ km s^-1 and uncertainties of 1-2 km s^-1. We did not find an important bias for $\Omega _{{\rm p}}$ neither for O and B stars nor Cepheids, though its standard deviation is large (4-10 km s^-1 kpc^-1 for O and B stars and 10-20 km s^-1 kpc^-1 for Cepheids). These ranges of values have allowed us to check how compatible are the results independently obtained for O and B stars and Cepheids;

• If we choose an incorrect set of free parameters (m, i, $\varpi_\odot$, $\Theta (\varpi _\odot )$), results change, but not to a large extent. This change is similar to the dispersion obtained when solving the condition equations from the different simulated samples (see Appendix B for details). Therefore, it will be difficult to decide between 2- and 4-armed models.

From these results, in Appendix B we conclude that, with the present available observational data, we are able to determine the kinematic parameters of the galactic model proposed in this paper, though we are not able to decide between a 2- or 4-armed Galaxy.