          Astron. Astrophys. 318, 416-428 (1997)

## 4. Galactic rotation

### 4.1. Axisymmetric solution

With the distances computed in the previous section, and the velocities of Sect. 2 (column 1 of Table 1), the rotation curve followed by the sample can be determined.

Assuming a circular rotation around the Galactic Centre, the angular velocity is: where is the sun reflex motion.

The rotation velocity is: where R is the distance of the object to the Galactic Centre, derived from d is the distance, µ the distance modulus. (distance of the Sun to the Galactic Centre), (rotation velocity of the LSR) and (solar motion relative to the LSR) must be assumed.
We use two sets of ( , ):
- ( 8.5 kpc, 220 km s-1) recommended in the review by Kerr & Lynden-Bell (1986)
- ( 8 kpc, 200 km s-1) indicated by some more recent studies (Merrifield 1992, Pont et al. 1994)
and =(9.3, 11.2, 7.0) km s-1 determined from cepheids in Pont et al. (1994).

The resulting rotation and angular velocity curves are displayed in Fig. 8 and 9. The objects labeled as type II suspects are drawn as open circles. Only objects farther than from the anticentre ( ) are plotted. Fig. 8. Rotation curve of the outer disc from cepheids, with kpc and km s-1 (top), or kpc and km s-1 (bottom). "Type II suspects" (see Sect. 3.6) are plotted as open circles. The rotation curve obtained in Pont et al. (1994) is drawn up to 10.5 kpc. The dashed lines are constant rotation curves at 192.9 km s-1 and 167.2 km s-1 respectively. Fig. 9. Angular velocity curve. Symbols as in Fig. 8. kpc, km s-1 assumed. The advantage of this representation is that the distance errors affects only the horizontal axis.

The sample traces clearly the rotation curve between R=10 and R=15 kpc. It outlines a practically flat rotation curve, slightly below . Several simple shapes were tried in fitting an analytical rotation curve to the data. A slightly decreasing curve was found. Fitting a linear rotation curve gives: ( kpc, km s-1).
The slope is not significantly different from zero, and was used in the following discussion. Table 2 shows the values of obtained, with or without the type II suspects of Sect  3.6. The fit, of course, is not made in the (R, ) space, but in the ( ) space. Table 2. Constant rotation velocity between 10.5 and 15 kpc fitted to the data, with and without the type II suspects, for two sets of galactic parameters.

The residuals around a flat rotation curve are shown in Fig. 10 (case kpc, km s-1). The final dispersion is 10.51 km s-1 (9.47 without type II suspects). This value gives an upper limit to the velocity dispersion of cepheids in the outer disc. It is of the order of the local radial dispersion for cepheids, 10.4 km s-1 (Pont et al. 1994), and seems to exclude the presence in the sample of a significant number of type II cepheids, since these are expected to have a much larger velocity dispersion, typical of older objects, km s-1. Fig. 10. Observed radial velocities versus velocities predicted from a flat rotation curve with Vrot=192.9 km s-1 ( kpc and km s-1). The solid line is . The two dotted lines indicate km s-1.

### 4.2. Non-axisymmetric components

A non-axisymmetric component in the rotation of the outer disc can be detected either by a north-south asymmetry in the sample, or, using the control sample near the anticentre, by a radial motion not accounted for by circular rotation.

The fitting of the rotation curve was repeated using only the northern and southern stars separately. The results are and resp., with no significant difference. Unfortunately, the number of points in the north is not large enough to provide a very strong constraint on the asymmetry.

With the complete sample, including anticentre objects ( , 11 objects), two types of non-axisymmetric components were fitted for:
-a radial motion of the LSR relative to the outer disc (towards the anticentre) -a uniform expansion of the disc where .

The results are:  These values show that non-axisymmetric components are weak or absent. Such radial motions of large amplitude - up to 15 km/s - as have been proposed in some models or suggested by some observations (see for instance Spergel 1993), are not compatible with the cepheid results.    © European Southern Observatory (ESO) 1997

Online publication: July 8, 1998 