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

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5. Discussion

The determination presented in this paper yields a significantly different value for the rotation velocity compared with previous studies, notably the ones based on HII regions or HI gas (Fich & Tremaine 1991, Merrifield 1992). Fig. 11 compares the HII rotation results reviewed in Fich et al. (1989) with our results. Beyond [FORMULA], the first show very scattered data around a rising curve apparently incompatible with the flat or decreasing cepheid rotation curve.

[FIGURE] Fig. 11. Comparison of HII region and cepheid rotation curve data. Open circles: HII regions (north), open squares: HII regions (south), filled circles: cepheids. The increase at large radii towards [FORMULA] km s-1 is not observed in the cepheids.

Random or systematic errors on CO and cepheid radial velocities are small and cannot account for the difference. The disagreement can then have two types of causes: either it results from a distance scale difference, or it reveals an intrinsic difference of kinematics between HII regions and cepheids.

5.1. Kinematical difference

A lag on gas rotation of the order of 50 km s-1 for objects as young as classical cepheids -generally younger than a galactic orbital period- would be hard to explain. An Inner Lindblad Resonance of the halo could be invoked as a possible cause of the difference in gas and stellar kinematics, an effect mentioned by Spergel (1993). However this model implies an inversion of the radial component of the stellar velocity field around the resonance as well as a sizeable increase in the velocity dispersion, which are not observed (Lewis & Freeman 1989, Blitz & Spergel 1991).

Another cause could be the presence of large radial motions in the gas, induced for instance by spiral arms. Unlike cepheids, HII regions do show a strong north-south asymmetry in their kinematics (see Fig. 11), that could imply large radial motions. If one assumes that the HII regions follow the rotation curve indicated by the cepheids and that the difference is explained by radial motions alone, then the radial motion needed is 20.4 km s-1 on average. Although this value is rather large and above what is usually expected from spiral arms, it is not implausible.

5.2. Distance scale difference

On a rotation curve plot like Fig. 8, an error on distance moves the points diagonally, since the distance enters the calculation of both R and [FORMULA]. Moreover, an error on the distance modulus µ has an asymmetrical effect on the rotation curve because of the exponential dependence of distance on µ. Therefore a systematic error on µ or a high dispersion on distance determination can artificially cause the rotation curve to appear to be increasing at large R.

The distance shift necessary to bring the HII data of Fich et al. (1989) into agreement with the cepheid data is about 1.2 mag. A distance scale difference of that size is unlikely. On the other hand, the rotation curve of HII regions between R=10 and R=14 kpc has a shape that may awaken some suspicion: its increase is more or less parallel to the line along which a point is moved if its distance is changed while its radial velocity is kept constant. The combination of a distance scale shift and high errors on distances could artificially create such a trend.

We studied this possibility using Monte Carlo simulations: on a synthetic sample orbiting the galactic centre with a constant circular velocity [FORMULA] km s-1, were added a velocity dispersion of 8 km s-1, a dispersion on distances [FORMULA], and a systematic distance shift [FORMULA]. Fig. 12 shows the results of this simulation for three sets of ([FORMULA]). The conclusion is that it is possible to infer an artificially rising rotation curve similar to the one observed in HII regions, from a tracer with in reality a constant rotation velocity, but with parameters like [FORMULA] mag and [FORMULA] mag (Fig. 12 bottom). While high, these values are not unrealistic, given the difficulty of determining HII region distances (Turbide & Moffat 1993). The effect gets even easier to reproduce if the distance scale shift is proportional to R, for instance via metallicity corrections to the theoretical ZAMS.

[FIGURE] Fig. 12. Monte Carlo simulations showing the effect of distance uncertainties on the rotation curve. The filled circles show the "true" synthetic data, with [FORMULA] km s-1, and the open circles the data recovered after adding a distance dispersion [FORMULA] and a systematic shift [FORMULA]. The results are displayed for three sets of parameters ([FORMULA]): 1.0 mag, 0 (a); 0.2 mag, 0.6 mag (b); 0.4 mag, 0.8 mag (c). The plot (c) shows how a rotation curve similar to the one observed in HII regions can result from a lower, flat rotation curve once distance dispersion and shift are added.

5.3. Effect of changes in the assumptions

In the previous sections, the cepheid distance scale was considered to be exact. We examine in this section how it is affected by changing some assumptions and by systematic biases. [FORMULA] kpc and [FORMULA] km s-1 are assumed. Using (8.5 kpc,220 km s-1) gives very similar results.


PL relation zero-point
The change caused in the derived rotation velocity [FORMULA] by a shift in the zero-point of the distance scale [FORMULA] was estimated with Monte Carlo simulations:

[EQUATION]

i.e. [FORMULA] km s-1 for a typical value of 0.1 mag for [FORMULA] (Feast & Walker 1987).

Unrecognized type II cepheids
The presence of type II cepheids can bias the results (see Sect. 3.6) because their distance is overestimated by using type I PL relations. If [FORMULA] is the rate of type II in the sample, and [FORMULA] their rotation velocity (assumed to be constant), then Monte Carlo simulations indicate:

[EQUATION]

[FORMULA] is not known. For values of 150 km s-1 (thick disc) or 200 km s-1 (old disc, as proposed by Harris & Wallerstein 1984), one gets, with [FORMULA] =0.1, [FORMULA] = -1.4 and +4.6 km s-1 respectively.


Binaries
The presence of an unrecognized companion causes the distance of a cepheid to be miscalculated. The radial velocities are also affected, but in a non-systematic way, only amounting to an increase in the final dispersion. In none of the sample objects was a change in the [FORMULA] velocity detected.

The effect of undetected binaries was simulated by adding synthetic blue main-sequence companions inspired from Evans (1992, 1995). We assumed a companion rate of 20%, with a frequency proportional to [FORMULA], to a maximum of [FORMULA] mag, and then computed the effect on the inferred reddenings and distances.

Distances from PC(B-V) and PL(V) (Sect.  3.1) can be affected. The net effect is [FORMULA] km s-1. Distances from PC(V-I) and PL(I) are much less sensitive, with [FORMULA] km s-1.


Metallicity corrections
Repeating the entire procedure with different values for the assumed metallicity gradient in the disc ([FORMULA]) gives the following relations:


[TABLE]


e.g. changing the gradient from -0.07 dex kpc-1 to -0.03 dex kpc-1 increases [FORMULA] by 2.2/7.0 km s-1.


Position in the instability strip
The photographic detection of faint cepheids near the magnitude limit favours objects on the blue and bright side of the instability strip, a bias pointed out by Feast (1995). In terms of the notation used in the appendix, it means [FORMULA] in Eqs. (3) and (4). We modeled this bias and found that with a width of the instability strip of 0.6 mag in V, it causes an overestimation of distances by 0.03 mag in (V-I, [FORMULA]) and 0.08 mag on average in (B-V, [FORMULA]) (or 1-2 km s-1 on [FORMULA]) if the cepheid detection limit can be modelled as a simple magnitude cutoff. As the second distances are in our sample already larger on average than the first, it is likely that the bias is not significant.

5.4. Non-axisymmetric models of the Galaxy

The cepheid data can be compared with some recent suggestions for non-axisymmetric models of the Galaxy. The model by Kuijken (1994), assuming an m=1 asymmetry, implies a radial motion of the LSR. Blitz & Spergel (1991) propose in their elliptical model a radial motion of the LSR of 14 km s-1 relative to the outer disc, to explain features observed in gas kinematics. The value obtained here, [FORMULA] km s-1 (Sect.  4.2), is not compatible with this model. It can be compared with other observational results, such as Lewis & Freeman (1989) or Metzger & Schechter (1994), that give significant LSR motions in the opposite direction. The fact that the cepheids show an LSR radial motion intermediate between younger and older objects could indicate an age dependence of radial motion in the outer disc.

In fact, there is no strong evidence in the cepheids for any non-axisymmetric component in the rotation. However, as pointed out by Kuijken (1993), the outer disc velocity field is sensitive to any non-axisymmetric component, but is not a good detector of it.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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