Spiral arms, bar shape and bulge microlensing in the Milky Way

Abstract A new model for the luminosity distribution in the inner Milky Way is found, using a non-parametric penalized maximum-likelihood algorithm to deproject a dereddened COBE/ DIRBE L-band map of the inner Galaxy. The model is also constrained by the apparent magnitude (line-of-sight) distributions of clump giant stars in certain bulge fields. An important new feature is the inclusion of a spiral arm model in the disc. Spiral arms make the model appear broader on the sky; thus our bar is more elongated than in previous eight-fold symmetric models. They also lead to a smoother disc model interior to the Sun. The bar length is ≈3.5 kpc, and its axis ratios are 1:(0.3–0.4):0.3, independent of whether the spiral arm model is four-armed or two-armed. The larger elongation in the plane makes it possible to reproduce the observed clump giant distributions as well. With only the surface brightness data, a small model degeneracy is found even for fixed orientation of the bar, amounting to about ±0.1 uncertainty in the in-plane axial ratio. Including the clump giant data removes most of this degeneracy and also places additional constraints on the orientation angle of the bar. We estimate 15°≲ϕbar≲30°, with the best models obtained for 20°≲ϕbar≲25°. We use our reference model to predict a microlensing optical depth map towards the bulge, normalizing its mass by the observed terminal velocity curve. For clump giant sources at (l,b)=(3°.9, −3°.8) we find τ−6≡τ/10−6=1.27, within 1.8σ of the new MACHO measurement given by Popowski et al. The value for all sources at (l,b)=(2°.68, −3°.35) is τ−6=1.1, still >3σ away from the published MACHO DIA value. The dispersion of these τ−6 values within our models is ≃10 per cent. Because the distribution of sources is well fitted by the near-infrared model, increasing the predicted optical depths by >20 per cent will be difficult. Thus the high value of the measured clump giant optical depth argues for a near-maximal disc in the Milky Way. Galaxy: centre, Galaxy: structure, galaxies: spiral Introduction Observations of the Milky Way (MW) show significant systematic differences between the near-infrared (NIR) surface brightness of the MW at l<0° and l>0° (e.g. Blitz & Spergel 1991; Weiland et al. 1994; Bissantz et al. 1997). It is widely accepted that these variations reflect the fact that the MW is a barred spiral galaxy. Evidence for a barred component of the luminosity density in the inner MW also comes from starcount observations (e.g. Nikolaev & Weinberg 1997; Stanek et al. 1997; Hammersley et al. 1999; Sevenster 1999; López-Corredoira et al. 2000), from gas dynamics (e.g. Englmaier & Gerhard 1999; Fux 1999; Weiner & Sellwood 1999), and microlensing observations (e.g. Paczynski et al. 1994; Zhao, Rich & Spergel 1996). Further references can be found in Gerhard (2001). The starcount data show significant asymmetries between lines of sight that are symmetrical with respect to the l=0 axis; this is the signature of a bar with its near end at positive Galactic longitudes. Most importantly, starcount data contain information about the distances to the surveyed stars. This is complementary to the all-sky coverage of surface brightness maps, and is valuable for constraining the line-of-sight (LOS) structure of the bulge even if available only for a restricted number of fields. In this paper we will take one step towards combining the information from both kinds of data, and use the clump giant observations of Stanek et al. (1994, 1997) together with the COBE/DIRBE NIR data to determine a model for the luminosity distribution in the inner Galaxy. With this model we can be more confident about the LOS distribution of microlensing sources, and are thus in a much better position to predict microlensing optical depths for comparison with the recent determinations from the MACHO group (Alcock et al. 2000a; Popowski et al. 2000). Most previous models of the inner MW have been parametric, and are thus restricted towards certain classes of densities for the bulge and/or disc. Binney & Gerhard (1996) developed a non-parametric approach to the deprojection of the COBE/DIRBE data based on the Richardson–Lucy algorithm, in which by construction the luminosity models are eight-fold symmetric with respect to the three main planes of the bar/bulge. Models constructed with this approach (Binney, Gerhard & Spergel 1997; Bissantz et al. 1997) give a good fit to the COBE/DIRBE L-band data, but predict less asymmetric LOS distributions towards the fields observed by Stanek et al. (1994) than observed, by more than 0.1 mag. Eight-fold symmetry also excludes modelling the spiral arms of the MW (see, e.g., Englmaier & Gerhard 1999 and Drimmel & Spergel 2001). In the present paper we describe a non-parametric penalized likelihood approach to infer the luminosity density of the inner MW from the COBE/DIRBE data which allows us to include a spiral arm model. This paper is organized as follows. Section 2 describes our new deprojection algorithm. In Section 3 we test the method with known parametric distributions, and analyse the uniqueness of the deprojected bar shape. In Section 4 we present models for the luminosity distribution of the MW which are consistent with both the COBE/DIRBE L-band data and the observed asymmetry in the distribution of clump giant stars, and give constraints on the orientation angle of the Galactic bar. In Section 5 we predict the microlensing optical depths for these models and compare them to recently published results of the MACHO experiment. We close with a summary and conclusions in Section 6. Maximum Likelihood Deprojection Method In this section we describe the technique we have used to construct models for the luminosity distribution of the MW. It is a non-parametric technique that maximizes a likelihood function, which includes penalty terms encouraging smoothness, eight-fold (triaxial) symmetry and a spiral arm component in the model. The minimization procedure is iterative, starting from an initial parametric model. The following subsections describe the initial parametric models (Section 2.1), the algorithm (Section 2.2), the choice of optimal penalty parameters (Section 2.3), and the performance of the algorithm (Section 2.4). The results of using the algorithm to recover known solutions from artificial data are described in Section 3. Parametric models We define parametric models for the luminosity distribution of the MW on a Cartesian grid. The coordinate system has the Galactic Centre at its origin. The axes are parallel to the main axes of the bar. In this coordinate system the position of the Sun is [x=R0 cos(ϕbar), y=R0 sin(ϕbar), Z0], where R0 is the distance of the Sun from the Galactic Centre projected on to the main plane of the Milky Way, Z0 is the position of the Sun above the xy-plane, and ϕbar is the ‘bar angle’, i.e., the angle in the xy-plane between the major axis of the bar and the projected L (...truncated)