1 Introduction

Young stars have been traditionally used as probes of the galactic structure in the solar neighbourhood. Their luminosity makes them visible at large distances from the Sun, and their age is not great compared to the dynamical evolution timescales of our galaxy.

These stars show kinematic characteristics that cannot be explained by solar motion and differential galactic rotation alone. Small perturbations in the galactic gravitational potential induce the formation of density waves that can explain part of the special kinematic features of young stars in the solar neighbourhood and also the existence of spiral arms in our galaxy. The first complete mathematical formulation of this theory was done by Lin and his associates (Lin & Shu 1964; Lin et al. 1969).

Lin's theory has several free parameters that can be derived from observations. Two of the main parameters are the number of spiral arms (m) and their pitch angle (i). Although the original theory proposed a 2-armed spiral structure with $i = -6\hbox{$^\circ$ }$, as early as the mid-70s Georgelin & Georgelin (1976) found 4 spiral arms with $i = -12\hbox{$^\circ$ }$ from a study of the spatial distribution of HII regions. This controversy is still not resolved: in a review Vallée (1995) concluded that the most suitable value is m = 4, whereas Drimmel (2000) found that emission profiles of the galactic plane in the K band - which traces stellar emission - are consistent with a 2-armed pattern, whereas the 240 $\mu$m emission from dust is compatible with a 4-armed structure. In a recent paper, Lépine et al. () described the spiral structure of our Galaxy in terms of a superposition of 2- and 4-armed wave harmonics, studying the kinematics of a sample of Cepheids stars and the l-v diagrams of HII regions.

The angular rotation velocity of the spiral pattern $\Omega _{{\rm p}}$ is another parameter of the galactic spiral structure. It determines the rotation velocity of the spiral structure as a rigid body. The classical value proposed by Lin et al. (1969) is $\Omega_{\rm p} \approx$ 13.5 km s^-1 kpc^-1. The angular rotation velocity in the solar neighbourhood due to differential galactic rotation is $\Omega_\odot \approx$ 26 km s^-1 kpc^-1 (Kerr & Lynden-Bell ). Thus, the value of $\Omega _{{\rm p}}$ implies that the so-called corotation circle (the galactocentric radius where $\Omega_{{\rm p}} = \Omega$) is in the outer region of our galaxy ( $\varpi_{{\rm cor}} \approx 15{-}20$ kpc, depending on the galactic rotation curve assumed). Nevertheless, several authors found higher values of $\Omega _{{\rm p}}$, about 17-29 km s^-1 kpc^-1 (Marochnik et al. ; Crézé & Mennessier 1973; Byl & Ovenden 1978; Avedisova 1989; Amaral & Lépine 1997; Mishurov et al. 1997; Mishurov & Zenina 1999; Lépine et al. 2001). These values place the Sun near the corotation circle, in a region where the difference between the galactic rotation velocity and the rotation of the spiral arms is small. This fact has very important consequences for the star formation rate in the solar neighbourhood, since the compression of the interstellar medium due to shock fronts induced by density waves could be chiefly responsible for this process (Roberts 1970).

Other parameters of Lin's theory are the amplitudes of induced perturbation in the velocity (in the antigalactocentric and the galactic rotation directions) of the stars and gas, and the phase of the spiral structure at the Sun's position. The interarm distance and the phase of the spiral structure can be determined from optical and radio observations (Burton 1971; Bok & Bok 1974; Schmidt-Kaler 1975; Elmegreen 1985). The interarm distance gives us a relation between the number of arms and the pitch angle.

In this paper we obtain the galactic kinematic parameters from two samples of Hipparcos stars described in Sect. 2: one that contains O- and B-type stars, and another one composed of Cepheid variable stars. In Sect. 3 we propose a model of our galaxy, a generalization of that previously used by Comerón & Torra (1991). The authors only applied their model to radial velocities. The accurate astrometry of the Hipparcos satellite offers a good opportunity to also use the proper motion data. The resolution of the condition equations, based on a weighted least squares fit, is explained in Sect. 4. An extensive set of simulations is performed in Sect. 5 in order to assess the capabilities of the method, that is, to analyze the influence of the observational errors and biases in the kinematic parameters. We finally present our results and discussion in Sects. 6 and 7, respectively.