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Astron. Astrophys. 345, 787-812 (1999)

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7. Geometrical constraints on the bar parameters

Fig. 22 shows how the [FORMULA] diagram of model l10´t2540 changes when the viewing point of the observer is modified. Reducing the bar inclination angle [FORMULA] shrinks the structures longitudinally and amplifies the velocities. The observed knee of the 135-km [FORMULA] arm near [FORMULA] 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 [FORMULA] lowers the forbidden velocities, moves the terminal velocity peaks further away from [FORMULA] 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 [FORMULA] moves the loop associated to the [FORMULA] prolongation of the molecular ring structure out to the real tangent point of the Carina arm near [FORMULA]. 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.

[FIGURE] Fig. 22. Dependence of the [FORMULA] diagrams on the location of the observer relative to the bar in model l10´t2540. The upper diagrams illustrate the effects of changing the bar inclination angle and the lower ones the effects of distance. [FORMULA] represents the galactocentric distance of the observer in initial units, which is 9 in the adopted optimum model displayed in Fig. 15.

Constraining the bar parameters by adjusting gas dynamical models to the observed CO and HI [FORMULA] 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 [FORMULA] and [FORMULA] where the 3-kpc and the 135-km s-1 arms intersect the axis shocks and to adjust in real space a major axis through the Galactic centre crossing the line of sights associated to these two directions at galactocentric distances in the ratio [FORMULA]. Simple trigonometrical considerations in the first and fourth Galactic quadrants respectively yield: -27


and isolating [FORMULA] and [FORMULA] between these two equations:


where q remains as a parametrisation of the asymmetry level between the lateral arms, which is not a priori known. In the ideal bisymmetric case [FORMULA], but in reality [FORMULA] according to Sect. 6.2. These formulae rest on the implicit assumption that the intersections of the lateral arms with the axis shocks and the Galactic centre are collinear. If this is wrong, then [FORMULA] will represent the angle between the line joining these intersections and the direction [FORMULA].

[FIGURE] Fig. 23. Basic geometry for the localisation of the intersections between the lateral arms (solid lines) and the axis shocks (long dashed lines). The nuclear ring (short dashed line) has been enlarged and the location of the Sun is indicated by the [FORMULA] symbol.

Extrapolating the connecting arm down to the 3-kpc arm in the [FORMULA] observations leads to [FORMULA], where a knee of the 3-kpc arm can be barely detected, whereas the 135-km s-1 arm meets the opposite axis shock at [FORMULA] (Fig. 1). The resulting constraints are put together in Fig. 24. Clearly, the distance [FORMULA] increases for smaller values of [FORMULA], but no precise value can be given so far for the inclination angle. However, the CO data in Fig. 1 betray a second fainter far-side lateral arm which extends down to longitude [FORMULA] with lower forbidden velocities than the 135-km s-1 arm, as well as a quasi-vertical feature at about the same longitude probably corresponding to gas from the same arm plunging towards the nuclear ring/disc after apocentre, on orbits parallel to the main axis shock. If this is correct, this arm must be much more symmetrical to the 3-kpc arm and the formulae (28) and (29) can be applied to these two arms using [FORMULA]. The results are [FORMULA] and [FORMULA] kpc, and by the way [FORMULA], i.e. [FORMULA] kpc, for the 135-km s-1 arm.

[FIGURE] Fig. 24. Graphical representation of the geometrical constraints on the bar parameters. The solid curves are derived from Eq. (26) for different values of [FORMULA], and the grey surfaces from Eq. (27) for different values of q, setting [FORMULA] and assuming [FORMULA] to show how uncertainties in [FORMULA] propagate to q.

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 [FORMULA], and offset dustlanes in such galaxies do not extend beyond the bar ends, i.e. [FORMULA] (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 [FORMULA] orbits, [FORMULA]. Adopting [FORMULA] as a good compromise, our value of [FORMULA] would imply a corotation radius of [FORMULA] 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 [FORMULA] 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 [FORMULA], 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-1 arm leads to [FORMULA] for the 3-kpc arm, suggesting that this arm should meet the connecting arm close to the bar major axis and close to the apocentre of the cusped [FORMULA] orbit. This is also confirmed by the fact that there is no obvious velocity gap in the observed [FORMULA] diagrams at the longitude where the 3-kpc arm passes into the connecting arm. Our models suggest that the tangent points of the two molecular ring branches in the first Galactic quadrant are within corotation (see Fig. 21), contrary to Englmaier & Gerhard's (1999) deduction, and therefore we rather advocate [FORMULA] kpc.

[FIGURE] Fig. 25. Location of the intersections between the lateral arms and the axis shocks relative to the bar in simulation l10. Left top: distance of these intersections relative to the offcentred density centre and normalised by the corotation radius (based on the Lagrangian points [FORMULA] and [FORMULA]). Left bottom: phase of the intersections relative to the major axis of the bar, defined positive when the formers lead the latter; the dotted line indicates the locus where both are exactly in phase. Right: correlation between the ordinates of the two previous plots. The solid and dashed lines refer to arm intersections on the same side of the bar, and the bold lines to intersections involving the most contrasted lateral arms.

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 [FORMULA] 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. [FORMULA], and support a corotation radius between 4 and 6 kpc (for [FORMULA] kpc). These authors consider an angle below [FORMULA] unlikely, but it should be noted that gas flow fluctuations like those in our self-consistent simulations can increase the [FORMULA] 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 [FORMULA] orbit, then the gas at the intersection of the two arms should corotate with the bar figure. Evaluating the radial velocity [FORMULA] of this gas relative to the LSR yields its rotation velocity with respect to the Galactic centre:


and resorting to Eq. (26):


where [FORMULA] is the circular velocity of the LSR and [FORMULA]. The bar inclination angle and [FORMULA] have vanished in this last formula and thus the method is independent of them. The value of [FORMULA] is hard to determine in the CO and HI [FORMULA] plots because of overlayed emission from the disc and in particular from the molecular ring, but a reasonable range is [FORMULA] km s-1. With [FORMULA] kpc, [FORMULA] km s-1 and [FORMULA] as before, one gets [FORMULA] km s- 1 kpc-1, in fair agreement with the 50 km s-1 kpc-1 of our gas simulations (Fig. 10) when rescaled to [FORMULA] kpc. Changing the value of [FORMULA] or [FORMULA] by [FORMULA] and 10 km s-1 modifies the result only by [FORMULA]% and 3% respectively. However, [FORMULA] may be affected by a radial motion of the LSR and/or oscillations of the bar density centre. Longitude-velocity maps of very dense gas (like CS) could help to localise more precisely the transition from the 3-kpc arm to the connecting arm. Note that the method does not apply to the very asymmetric model l10´t2540, as can be checked from Fig. 22, because the gas at the transition is not at rest in the frame of the bar.

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999