3 A galactic kinematic model for the solar neighbourhood

3.1 Systematic velocity field in the galactic model

We propose an extension of the bidimensional model previously applied for radial velocities by Comerón & Torra (1991). In our model we considered the systematic velocity components:

 

$$v_{{\rm r}} = v_{{\rm r_1}} + v_{{\rm r_2}} + v_{{\rm r_3}}$$

$$v_{{\rm l}} = v_{{\rm l_1}} + v_{{\rm l_2}} + v_{{\rm l_3}}$$

$$v_{{\rm b}} = v_{{\rm b_1}} + v_{{\rm b_2}} + v_{{\rm b_3}}$$ (3)

where subindex 1 refers to the solar motion contribution, subindex 2 to differential galactic rotation and subindex 3 to spiral arm kinematics, respectively (see Appendix A). Solar motion is expressed through the three components of the Sun's velocity in galactic coordinates ( $U_\odot, V_\odot, W_\odot$). Galactic rotation curve was developed up to second-order approximation, $a_{\rm r}$ and $b_{\rm r}$ being the first- and second-order terms, respectively:

 

$$a_{\rm r} = \left( \frac {\partial \Theta}{\partial \varpi} \right)_\odot = \frac {\Theta(\varpi_\odot)}{\varpi_\odot} - 2 A$$

$$b_{\rm r} = \frac {1}{2} \left( \frac {\partial^2 \Theta}{\partial \varpi^2} \right)_\odot$$ (4)

where we show the relationship between $a_{\rm r}$ and the A Oort constant. B Oort constant can be derived from:

$$B = A - \frac {\Theta(\varpi_\odot)}{\varpi_\odot}\cdot$$ (5)

We considered as free parameters the galactocentric distance of the Sun ( $\varpi_\odot$) and the circular velocity at the Sun's position ( $\Theta (\varpi _\odot )$). Finally, spiral arm kinematics was modelled within the framework of Lin's theory. We considered the number of the spiral arms (m) and their pitch angle (i) as free parameters, and we derived the perturbation velocity amplitudes in the antigalactocentric and tangential directions ( $\Pi _{\rm b}$ and $\Theta _{\rm b}$, respectively; they were considered as constant magnitudes, assuming that their variation with the galactocentric distance is smooth), the phase of the spiral structure at the Sun's position ( $\psi _\odot$) and the parameter $f_\odot$, which takes into account the difference in the velocity dispersion between the solar-type stars and the considered stars. This parameter is defined by the relationship between the spiral perturbation velocity amplitudes for the sample stars ( $\Pi _{\rm b}$, $\Theta _{\rm b}$) and for the Sun ( $\Pi_{{\rm b \odot}}$, $\Theta_{{\rm b \odot}}$):

$$\Pi_{{\rm b \odot}} = \frac {1 - \nu^2 + x_{{\rm stars}}}{1 - \nu^2 + x_\odot} \Pi_{{\rm b}} \equiv f_\odot \Pi_{{\rm b}}$$

$$\Theta_{{\rm b \odot}} = \frac {1 - \nu^2 + x_{{\rm stars}}}{1 - \nu^2 + x_\odot} \Theta_{{\rm b}} \equiv f_\odot \Theta_{{\rm b}}$$ (6)

where $\nu$ is the dimensionless rotation frequency of the spiral structure and x is the stability Toomre's number (Toomre 1969), which depends on the velocity dispersion of the considered stars (see details in Appendix A). The inclusion of $f_\odot$ is new with regard to the model proposed by Comerón & Torra ().

Equation (3) can be expressed as:

 

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

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

$$v_{{\rm b}}(R,l,b) \sum\limits_{j=1}^{10} a_j f_j^{{\rm b}}(R,l,b)$$ = (7)

where the constants a_j contain combinations of the kinematic parameters we wish to determine ($U_\odot$, $V_\odot$, $W_\odot$, $a_{\rm r}$, $b_{\rm r}$, $\psi _\odot$, $\Pi _{\rm b}$, $\Theta _{\rm b}$ and $f_\odot$) and f_jⁱ(R,l,b) are functions of the heliocentric distance and the galactic longitude and latitude (see Eqs. (A.21) and (A.22)).

   
3.2 Free parameters of our galactic model

A 2-armed Galaxy was the first proposed view for our stellar system, mainly derived from HI and HII observations, but also from the spatial distribution of supergiant stars and other bright objects. These classical studies show the existence of at least two arms inside the solar circle (the Sagittarius-Carina or -I arm and the Norma-Scutum or -II arm), one local arm (Orion-Cygnus or 0 arm) and one external arm (Perseus or +I arm). The Orion-Cygnus arm seems to be a local spur (Bok 1958). Lin et al. (1969) proposed a galactic system with 2 main spiral arms, where the Norma-Scutum and the Perseus arms are two segments of the same arm. By taking into account the interarm distance between the Sagittarius-Carina and the Perseus arms, these authors deduced a pitch angle of -6$^\circ$. But, as early as the mid-70s, Georgelin & Georgelin (1976) proposed a 4-armed galactic system with a pitch angle of $-12\hbox{$^\circ$ }$ from a study of the spatial distribution of HII regions. However, Bash (1981) examined this 4-armed model and found that a 2-arm pattern predicts HII regions in the same direction and with the same radial velocities as those used by Georgelin & Georgelin (1976), provided that dispersion velocities were considered.

In some recent papers several authors have also called this classical view in question. Vallée (1995) reviewed the subject of the determination of the pitch angle and the number of spiral arms and concluded that the Galaxy has a pitch angle of $i = -12 \pm 1\hbox{$^\circ$ }$ and that, taking into account the observed interarm distance, it would be a system of 4 spiral arms. This is also the opinion expressed by Amaral & Lépine (1997), who, fitting the galactic rotation curve to a mass model of the Galaxy, found an autoconsistent solution with a system of 2 + 4 spiral arms (2 arms for 2.8 $< \varpi <$ 13 kpc and 4 arms for $6 < \varpi < 11$ kpc, with the Sun placed at $\varpi_\odot =7.9$ kpc) and a pitch angle of $i = -14\hbox{$^\circ$ }$. Englmaier & Gerhard (1999) used the COBE NIR luminosity distribution and connected it with the kinematic observations of HI and molecular gas in l-v diagrams. They found a 4-armed spiral pattern between the corotation of the galactic bar and the solar circle. Drimmel (2000) found that the galactic plane emission in the K band is consistent with a 2-armed structure, whereas the 240 $\mu$m emission from dust is compatible with a 4-armed pattern. In a recent work, Lépine et al. (2001) analyzed the kinematics of a sample of Cepheid stars and found the best fit for a model with a superposition of 2+4 spiral arms. Contrary to the model by Amaral & Lépine (), Lépine et al. allowed the phase of both spiral patterns to be independent, deriving pitch angles of approximately $-6\hbox{$^\circ$ }$ and $-12\hbox{$^\circ$ }$ for m=2 and m=4, respectively. They argued that this spiral pattern is in good agreement with the l-v diagrams obtained from observational HII data, though they admit that pure 2-armed model produces similar results. In the visible spiral structure of the Galaxy derived by Lépine et al. (see their Fig. 3) the Orion-Cygnus or local arm is seen as a major structure with a small pitch angle. In contrast, Olano (2001) proposed that the local arm is an elongated structure of only 4 kpc of length and a pitch angle of about $-40\hbox{$^\circ$ }$ (in better agreement with the observational determinations of the inclination of the local arm found in the literature) formed from a supercloud about 100 Myr ago, when it entered into a major spiral arm.

In the case of the galactocentric distance of the Sun and the circular velocity at the Sun's position there are also some inconsistencies among the different values found in the literature. In 1986, the IAU adopted the values $\varpi_\odot =$ 8.5 kpc and $\Theta(\varpi_\odot) =220$ km s^-1 (Kerr & Lynden-Bell 1986). Recently, several authors have found values of nearly 7.5 kpc for $\varpi_\odot$ (Racine & Harris 1989; Maciel 1993). A complete review was done by Reid (1993), who concluded that the most suitable value seems to be $\varpi_\odot = 8.0 \pm 0.5$ kpc. In kinematic studies there are serious discrepancies between different authors. Metzger et al. (1998) found $\varpi_\odot = 7.7 \pm 0.3$ kpc and $\Theta(\varpi_\odot) = 237 \pm 12$ km s^-1, whereas Feast et al. (1998) found $\varpi_\odot = 8.5 \pm 0.3$ kpc (both from radial velocities of Cepheid stars). Glushkova et al. () found $\varpi_\odot = 7.3 \pm 0.3$ kpc from a combined sample including open clusters, red supergiants and Cepheids. Olling & Merrifield (1998) calculated mass models for the Galaxy and concluded that a consistent picture only emerges when considering $\varpi_\odot = 7.1 \pm 0.4$ kpc and $\Theta(\varpi_\odot) = 184 \pm 8$ km s^-1. This value for the galactocentric distance of the Sun is in very good agreement with the only direct distance determination ( $\varpi_\odot = 7.2 \pm 0.7$ kpc), which was made employing proper motions of H₂O masers (Reid 1993).

In the view of all that, we decided to derive the kinematic parameters of our model from different combinations of the free parameters involved. On the one hand, concerning spiral structure, two models of the Galaxy were considered: a first model with m = 2 and $i = -6\hbox{$^\circ$ }$, and a second one with m = 4 and $i = -14\hbox{$^\circ$ }$. Both models are consistent with an interarm distance of about 2.5-3 kpc, depending on the adopted value for the distance from the Sun to the galactic center. On the other hand, concerning the galactocentric distance of the Sun and the circular velocity at the Sun's position, two different cases were also taken into account: a first one with $\varpi_\odot = 8.5$ kpc and $\Theta(\varpi_\odot) =220$ km s^-1, and a second one with $\varpi_\odot = 7.1$ kpc and $\Theta(\varpi_\odot) = 184$ km s^-1. In both cases, the angular rotation velocity at the Sun's position is $\Omega_\odot = 25.9$ km s^-1 kpc^-1.