4 A kinematical overview of the catalogue

4.1 Galactic distribution

Figure 3 (top) displays the 867 radial velocities versus galactic longitude; most PNe are located in the direction of the galactic Centre and characterized by highly elongated orbits. According to their galactic positions (see middle panel of Fig. 3), most PNe are concentrated towards the galactic disk, and a few PNe may belong to a halo population. Only BoBn 1, a very old and metal-deficient halo star (108.4-76.1, see top of Fig. 3) exhibits a completely atypical motion. Additional measurements of its velocity (the only one available was made in 1977 by Boesharr & Bond) would be desirable in order to definitively confirm this value.

Figure 3 (bottom) displays the distribution in longitude of the 867 PNe according to their velocity uncertainties: about 90$\%$ of the sample have velocity errors better than 20 km$\,$s^-1. The largest uncertainties tend to be found among the bulge PNe.

  

Figure 3: Top: heliocentric radial velocities of the 867 PNe versus their galactic longitude. Middle: galactic positions of the 867 PNe of the present catalogue. Bottom: distribution in longitude of the 867 PNe according to their velocity uncertainties. White represents an error of < 10 km$\,$s-1 (482 PNe), grey 10 $\leq$ error < 20 km$\,$s-1 (281 PNe), and black an error of $\geq$ 20 km$\,$s-1 (104 PNe)

The extrema of the radial velocities towards the galactic Centre have decreased since STPP83 paper: for example, the value of $241.0 \pm 11.0$ km$\,$s^-1 of M1-37 (2.6 - 3.4) in STPP83 now becomes $220.5 \pm 0.9$ km$\,$s^-1, and that of M4-6 (358.6 + 1.8) has gone from $-292.0 \pm 11.0$ km$\,$s^-1 to $-268.1 \pm 5.5$km$\,$s^-1.

4.2 The kinematics of disk PNe

The use of PNe for establishing the Disk rotation curve is hampered by the large uncertainty of the distances. In our study we use the statistical distance scale of Zhang (1995), which is an average of two distance scales: one is based on the correlation between the ionized mass and the radius, the other on the correlation between the radio continuum surface-brightness temperature and the nebular radius. The intrinsic uncertainty in this scale (not counting possible systematic effects) is not known but is likely to be in excess of 30$\%$ (1 sigma) for each object, from comparison with van de Steene & Zijlstra (1994, 1995). These large uncertainties will tend to smooth out structure in the rotation curve and may also introduce systematic effects (Zijlstra & Pottasch 1991).

We selected from our sample 100 PNe located at |l| > 7$^{\circ}$ (in order to avoid contamination by bulge objects) and < 200 pc above the galactic plane (in order to select objects with near-circular orbits). 4 PNe with residual velocities larger than 100 km$\,$s^-1 were considered interlopers and removed from the sample. Figure 4 displays the galactic distribution of the 96 PNe projected onto the galactic plane (triangles) superimposed on all other PNe with estimated distances in the same region. Our sample seems to abruptly end at $R_{\rm gc} = 5$ kpc which could be due to a local spiral arm (see for instance Georgelin & Georgelin 1976, for a description of the galactic spiral structure from HII regions data). Extinction by dusty molecular clouds in such an arm could hide farther PNe from sight.

  

Figure 4: Galactic distribution of the 96 disk PNe projected on the XY plane; the positions of the sun (-8.5,0) and of the galactic center (0,0) are indicated

4.2.1 The local galactic rotation

Consider a star on a purely circular orbit at galactocentric radius R: its heliocentric radial velocity can be expressed by the well-known formula:

$$\begin{eqnarray} V_{\rm r}^{\rm mod} & = & - u_{\odot}\,\cos\,l\,\,\cos\,b - v... ...}\,(R - R_{\odot})^{2})\,\,\sin\,l\,\,\cos\,b \nonumber \\ & + & K\end{eqnarray}$$ (4)

where:

• $u_{\odot}$, $v_{\odot}$ and $w_{\odot}$ are the components of the solar motion with respect to the local standard of rest, respectively in the radial, azimuthal and vertical directions ($u_{\odot}$ is taken to be positive toward the galactic center). As our stellar sample is restricted to low vertical heights we will not be able to constrain $w_{\odot}$; instead we fix its value to the standard one (7.3 km$\,$s^-1).
• The second row of the equation expresses the differential galactic rotation to first order; A is the Oort constant.
• In the third row A₂ is the second order coefficient of the derivative of the rotation speed with respect to R. This expansion to second order in $(R - R_{\odot})$ allows us to use the kinematics of stars located further from the Sun.
• The last row gives the expression of the K-term of local galactic expansion.

We fit the above formula by minimizing:

$$\begin{eqnarray} \chi^{2} = \sqrt{\sum_{i=1}^{N} (\frac {v_{\rm r}^{{\rm obs}_{i}} - v_{\rm r}^{{\rm mod}_{i}}} {\Delta \,v_{\rm r}^{i}} )^{2}} \end{eqnarray}$$ (5)

with N the total number of stars in the sample and $\Delta\,v_{\rm r}^{i}$ the uncertainty assigned to the radial velocity of the $i^{\rm th}$ star.

 

Table 3: Kinematical parameters obtained from least-square fitting of Eq. (4)

 

Table 3 presents results obtained from a straightforward least-square fitting of Eq. (4).
We apply our fitting procedure by testing also various cases, as for example the non-inclusion of the A₂ term: it seems necessary to include it in the fitting procedure since without it the fits converge to unphysical values. The value of the A₂ term is largest than for other determinations (see for example Pont et al. 1994, using Cepheids data); this could be explained by the great uncertainties on PNe distances.

$u_{\odot}$ and $v_{\odot}$ appear different from the "standard" values (which are equal to 10.4 and 14.8 km$\,$s^-1 respectively). A high value of $v_{\odot}$ is usually associated with evolved populations: it is related to the asymmetric drift which becomes more important with late-type stars (see Fig. 6 in Jahreiss & Wielen). The mean value of the asymmetric drift of our 96 PNe sample is about 10 km$\,$s^-1 in our model, but with low confidence. The high value of the $u_{\odot}$ parameter may point to the existence of an outward motion of the local standard of rest as already proposed for example by Blitz & Spergel (1991).

The value of the K-term is consistent with zero given the uncertainties inherent in a multi-parameter fit. A non-zero value would be related to imperfections in the data (e.g. a systematic velocity offset or a bias) or imperfections in the rotation curve (e.g. the existence of residual non-axisymmetrical motions). Given the uncertainty, there is no conclusive evidence for a non-zero value.

4.2.2 The disk rotation curve

In order to test the galactic rotation curve as function of galactocentric distance of PNe, we first calculate the PNe galactic-standard-of-rest velocities using the formula:

$$\begin{eqnarray} V_{\rm rot} = \left( \Theta_{\odot} + \frac{V_{\rm lsr}}{\sin\,l\,\,\cos\,b} \right) \,\frac{R_{\rm p}} {R_{\odot}}\end{eqnarray}$$ (6)

where $V_{\rm lsr}$ is calculated from the above galactic rotation model, and $\Theta_{\odot} = R_{\odot}\,(A - B) = 8.7\,(14.4 + 12.4)$ = 233 km s^-1.

In Fig. 5 we display the galactic distribution of the 96 disk PNe and the rotation velocities versus Galactocentric distance. The distance is normalized to the Solar galactocentric distance $R_{\odot}$ = 8.7 kpc. In the bottom panel of the same figure we display the binned galactic rotation curve as provided by our PNe sample. Some authors (Amaral et al. 1996; Maciel & Dutra 1992) have found evidence for large-scale features in their rotation curve, in particular a broad maximum near R = 6 kpc (corresponding here to $R/R_{\odot}$ = 0.75) and a local decrease. Despite the insufficient number of objects and the (likely) smoothing due to the uncertainties in the individual distances, our curve recovers the same behaviour, but with a lower amplitude. Those features have also been detected in CO and HI data (Clemens 1985). There is also a slight indication of lower velocities around 11 kpc.

The 2 isolated points located at $R_{\rm gc} = 12.05$ and 12.65 kpc may give the appearance of an increasing outer rotation curve, but their presence cannot be in any way conclusive.

  

Figure 5: Top: galactic distribution of the 96 disk PNe at $\vert z \vert$ < 200 pc. Middle: galactic longitude versus rotation velocity. Bottom: bin and linear fit of the middle graph. The data are averaged in 0.1 bins. The error bars include only the standard error of the mean in each bin

4.3 The rotation of the bulge

Various tracer populations have been used to constrain the structure and kinematics of the galactic bulge (Frogel et al. 1990; Minniti et al. 1992; Whitelock & Catchpole 1992; Beaulieu 1996). These studies gave rather similar results (see also a brief review in Menzies 1990): the rotation curve increases quasi-linearly with l, with a mean slope of about 10-15 km$\,$s$^{-1}\,l^{-1}$ but possibly somewhat steeper near the center. Velocity dispersions are typically 70-120 km$\,$s^-1. A general trend is shown for the metal-rich populations to rotate a little faster than metal-poor ones; the velocity dispersion of the stars tends to decrease away from the Galactic Center.

KFL88 analysed a sample of 147 PNe ranged between $\vert l\vert$ < 10 and $\vert b\vert$ < 5.5 and found indications for the rotation of the bulge: they fitted a linear equation in ($l,V\rm _c$) and found the relation $V_{\rm c} = (12.0 \pm 1.9) l -(13.6\pm8.6)\,$km$\,$s^-1. They point out that due to projection effects the observed slope should be considered as lower limit.

To compare with Kinman et al. (1988), we relax our selection criteria for bulge membership: $\vert l \vert <$ 10.0 degree and $\vert b \vert <$ 7.0 degree; rejecting PNe with optical angular diameter > 20 arcsec and/or radio flux $F_{\rm 6\, cm}$ > 100 mJy (e.g. Acker & Pottasch 1989a). This sample contains 279 PNe. Figure 6 displays the galactic distribution of this PNe sample; the incompleteness is evident in the galactic plane (top panel). The middle panel shows the galactic-standard-of-rest velocities versus the longitude, and the bottom panel shows the same data in one-degree bins, with error bars representing only standard errors of the mean. The linear fit shown in the bottom of Fig. 6 has a slope of (9.9 $\pm$ 1.3) km$\,$s^-1. The zero longitude offset is -6.7 km$\,$s^-1. All these results are in good agreement with KFL88.

  

Figure 6: Top: galactic distribution of 279 bulge PNe. The effects of the absorption in the galactic plane are evident. Middle: galactic longitude versus galactocentric velocity. Bottom: bin and linear fit of the middle graph. The data are averaged in bins of one degree of longitude. The error bars include only the standard error of the mean in each bin

  

Table 5: bulge rotation fitted by a+bl km$\,$s^-1

Ref. 1: Minniti 1996; 2: Izumiura et al. 1995; 3: Catchpole 1990; 4: Menzies 1990.

Table 5 compares values for the bulge rotation derived from different samples. The first three lines show the results from the restricted criterion for bulge membership, using different limits for the Galactocentric distances and only velocities better than 20 km$\,$s^-1. The fourth line shows the result from the relaxed criterion mentioned above. In all cases there is good evidence for the rotation of the bulge. The gradient b may increase very close to the Centre but the uncertainties are much larger for this smaller samples. Interestingly, the relaxed sample gives essentially the same result. The offset at zero longitude (a) is within the uncertainties zero.

The last rows of Table 5 shows for comparison previous determinations. The values are in general consistent, although the location of the tracer populations are not identical. The planetary nebulae probably provide the best coverage close to the Centre, whereas the AGB stars give better coverage at high latitudes.