## 7. Geometrical constraints on the bar parametersFig. 22 shows how the diagram of model l10´t2540 changes when the viewing point of the observer is modified. Reducing the bar inclination angle shrinks the structures longitudinally and amplifies the velocities. The observed knee of the 135-km arm near is better reproduced, but the transition of the 3-kpc arm to the connecting arm happens at too negative velocity and the connecting arm becomes too steep. Increasing the angle lowers the forbidden velocities, moves the terminal velocity peaks further away from and shifts the connecting arm closer to the northern tangent points of the two molecular ring branches. Increasing the distance of the observer obviously produces a longitudinal contraction, but without modifying the velocity of the structures near the centre. The diagram with the observer at moves the loop associated to the prolongation of the molecular ring structure out to the real tangent point of the Carina arm near . These properties cannot be used to infer a robust inclination angle of the bar because they are based on one specific model and the gas flow is strongly time-dependent.
Constraining the bar parameters by adjusting gas dynamical models
to the observed CO and HI diagrams is
a very delicate task and may lead to unreliable results if the models
are not sufficiently realistic. However, with our interpretation of
the dominant features in these diagrams, it is possible to provide
geometrical constraints on the bar inclination angle and extension
which do not depend on the details of the models. The principle of the
method, illustrated in Fig. 23, is to determine in the CO and HI data
the longitudes and
where the 3-kpc and the
135-km s and isolating and between these two equations: where
Extrapolating the connecting arm down to the 3-kpc arm in the
observations leads to
, where a knee of the 3-kpc arm can
be barely detected, whereas the 135-km s
It remains to see how the arm intersections discussed here are related to the true bar parameters. Numerical simulations and analyses of observations in early-type barred galaxies indicate that the ratio between the bar semi-major axis and the corotation radius amounts to , and offset dustlanes in such galaxies do not extend beyond the bar ends, i.e. (e.g. the review of Elmegreen 1996). Also, bisymmetric hydro simulations in rotating barred potentials with straight offset dustlanes generally have lateral arms intersecting the axis shocks very close to their outer ends and at most a few degrees ahead of the bar major axis. In the standard model 001 of Athanassoula (1992), which has an inner Lindblad resonance but no looped orbits, . Adopting as a good compromise, our value of would imply a corotation radius of kpc for the Milky Way. The situation in our non-symmetrised and time-dependent
simulations, shown in Fig. 25, is much more complicated however. As
mentioned in Sect. 5.2, the radius of the intersections between the
lateral arms and the axis shocks increases with time. At the formation
of a lateral arm, the associated intersection leads the bar major
axis, and as the arm moves outwards, it crosses this axis and becomes
trailing. When both are exactly in phase, the intersection radius
relative to corotation is 0.75 on the average, compatible with the
value adopted above, and compares very well to the apocentre radius of
the cusped orbit (see Fig. 21).
Further out, the lateral arms rapidly dissolve in the spiral arms
emanating from the very end of the axis shocks. The minimum value of
the ratio , which could depend on
the size of the nuclear ring in the models, is about 0.4. Taking this
value as a lower limit for the 135-km s
Our value for the bar inclination angle agrees very well with many other determinations (e.g. Stanek et al. 1997; Nikolaev & Weinberg 1997), and in particular with the recent result of Englmaier & Gerhard (1999), who derived the gas flow in the potential of the deprojected COBE/DIRBE L-band luminosity distribution (Binney et al. 1997). The bisymmetric hydro simulations done by Weiner & Sellwood (1999) and matched to the observed HI terminal velocity curves favour slightly larger values of this angle, i.e. , and support a corotation radius between 4 and 6 kpc (for kpc). These authors consider an angle below unlikely, but it should be noted that gas flow fluctuations like those in our self-consistent simulations can increase the area covered by forbidden velocities and hence substantially bias their method. It is also possible from our data interpretation to give a crude estimate of the bar pattern speed. If the 3-kpc arm indeed encounters the connecting arm near the apocentre of the cusped orbit, then the gas at the intersection of the two arms should corotate with the bar figure. Evaluating the radial velocity of this gas relative to the LSR yields its rotation velocity with respect to the Galactic centre: and resorting to Eq. (26): where is the circular velocity
of the LSR and . The bar inclination
angle and have vanished in this
last formula and thus the method is independent of them. The value of
is hard to determine in the CO and
HI plots because of overlayed
emission from the disc and in particular from the molecular ring, but
a reasonable range is
km s © European Southern Observatory (ESO) 1999 Online publication: April 28, 1999 |