------------------------------------------------------------------------ From: N. Rodriguez rodrigue@iram.fr To: gcnews@aoc.nrao.edu Subject: submit gcdynamics.tex A&A in press %astro-ph/0806.4252 % % %\documentclass[referee]{aa} % for a referee version %\documentclass[onecolumn]{aa} % for a paper on 1 column %\documentclass[longauth]{aa} % for the long lists of affiliations %\documentclass[rnote]{aa} % for the research notes % \documentclass{aa} % \usepackage{graphicx} % \usepackage{aalongtable} \usepackage[latin1]{inputenc} \usepackage{natbib} \bibpunct{(}{)}{;}{a}{}{,} % to follow the A&A style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\usepackage{txfonts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %---------------------------DEFINITIONS % units definitions \def\kms{~km~s$^{-1}$} \def\kmskpc{~km~s$^{-1}$~kpc$^{-1}$} \def\cmmt{~cm$^{-3}$} \def\cmmd{~cm$^{-2}$} \def\smu{s$^{-1}$} \def\um{$\mu$m} % commands \newcommand{\gsim}{\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}} \newcommand{\lsim}{\raisebox{-.4ex}{$\stackrel{<}{\scriptstyle \sim}$}} \def\ls{\lsim} \def\gs{\gsim} \def\le{$\leq$} \def\ge{$\geq$} \def\deg{^\circ} % \def\lvd{$lv$-diagram} \def\Msun{M$_\odot$} \begin{document} % \title{Gas flow models in the Milky Way embedded bars} % \subtitle{I} \author{N. J. Rodriguez-Fernandez \inst{1} \and F. Combes \inst{2} } \offprints{N. J. Rodriguez-Fernandez} \institute{IRAM, 300 rue de la Piscine, 38406 St. Martin d'Heres, France \\ \email{rodriguez@iram.fr} \and LERMA, Observatoire de Paris, 61 Av de l'Observatoire, 75014 Paris, France } \date{Received September 15, 1996; accepted March 16, 1997} % \abstract{}{}{}{}{} % 5 {} token are mandatory %\authorrunning{Rodriguez-Fernandez } %\titlerunning{Dynamics and molecular chemistry in the Galactic center} \begin{abstract} % context heading (optional) % {} leave it empty if necessary { The gas distribution and dynamics in the inner Galaxy present many unknowns as the origin of the asymmetry of the \lvd ~ of the Central Molecular Zone (CMZ). On the other hand, there are recent evidences in the stellar component of the presence of a nuclear bar that could be slightly lopsided. } % aims heading (mandatory) {Our goal is to characterize the nuclear bar observed in 2MASS maps and to study the gas dynamics in the inner Milky Way taking into account this secondary bar.} % methods heading (mandatory) { We have derived a realistic mass distribution by fitting the 2MASS star counts map of Alard (2001) with a model including three components (disk, bulge and nuclear bar) and we have simulated the gas dynamics, in the deduced gravitational potential, using a sticky-particles code.} % results heading (mandatory) { Our simulations of the gas dynamics reproduce successfully the main characteristics of the Milky Way for a bulge orientation of $20\deg-35\deg$ with respect to the Sun-Galactic Center (GC) line and a pattern speed of 30-40 \kmskpc. In our models the Galactic Molecular Ring (GMR) is not an actual ring but the inner parts of the spiral arms, {while the 3-kpc arm and its far side counterpart are lateral arms that contour the bar.} Our simulations reproduce, for the first time, the parallelogram shape of the \lvd ~ of the CMZ as the gas response to the nuclear bar. This bar should be oriented by an angle of $\sim 60\deg-75\deg$ with respect to the Sun-GC line and its mass amounts to $(2-5.5)\,10^9$ \Msun. We show that the observed asymmetry of the CMZ cannot be due to lopsidedness of the nuclear bar as suggested by the 2MASS maps. } % conclusions heading (optional), leave it empty if necessary {We do not find clear evidences of lopsidedness in the stellar potential. We propose that the observed asymmetry of the central gas layer can be due to the infalling of gas into the CMZ in the {\it l=1.3$\deg$-complex}.} \end{abstract} \keywords{Galaxy: structure -- Galaxy: center -- Galaxy: kinematics and dynamics -- ISM: kinematics and dynamics -- Methods: numerical } \maketitle % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Studying the structure and the dynamics of the Milky Way galaxy is necessary to understand its formation and evolution. For example, the large scale star formation rate, the gas transfer to the nucleus and the fueling of the central super-massive black hole are strongly influenced by the large scale dynamics of the Galaxy. At least three components are needed to explain the rotation curve of the Galaxy: a disc, a bulge and a massive dark halo. In near-infrared images, the bulge is a triaxial structure difficult to distinguish from a bar. The first direct evidence of the barred nature of the Milky Way was found by \cite{Blitz91}. The analysis of COBE/DIRBE data \citep{Dwek95, Binney97, Freudenreich98} supported Blitz \& Spergel results. \cite{Dwek95} found that the peanut-shaped bulge seen in the COBE image is best fitted by {\it boxy} Gaussians functions. Since COBE, the direct evidence of the bar has increased based on photometric surveys and star counts in the near-infrared \citep[see for instance][and references therein]{Lopez-Corredoira00, Lopez-Corredoira05, Babusiaux05}. Although, there is a consensus on the triaxial nature of the bulge, the actual shape and the inclination is still under debate with currently no clear agreement among different authors. The inclination angles found range from $\sim 10\deg-15\deg$ \citep{Binney91, Freudenreich98, Lopez-Corredoira97} to $\sim 30\deg-45\deg$ \citep{Weiner99, Sevenster99, Lopez-Corredoira01} with a number of works giving values in the range $15\deg-30\deg$ \citep{Mulder86, Binney97, Fux99, Englmaier99, Bissantz02, Bissantz03, Babusiaux05, Lopez-Corredoira05}. In addition to the bar-like {\it boxy}-bulge, some recent works have also proposed the existence of another bar that would be longer and thinner than the bulge \citep{Picaud03, Lopez-Corredoira01, Lopez-Corredoira07, Benjamin05}. As the bulge, the near side of this long and thin bar will also be located in the first Galactic quadrant. In contrast, the long bar would be oriented at an angle of $\sim 45\deg$ with respect to the line of sight towards Galactic center. In the present paper we will not discuss this component as the mass should be dominated by the bulge. Hereafter, we will refer to the triaxial boxy bulge as ``large bar" or simply ``bulge". Due to the difficulty to measure the stellar velocities, most of our knowledge of the Galactic dynamics come from spectroscopic observations of the atomic and molecular interstellar medium \citep{Rougoor60, Scoville72, Liszt78, Bally87, Dame01}. The gas exhibit large non-circular motions in the inner Galaxy. Among the possible explanations for these kinematics, de Vaucouleurs (1964) proposed the gas response to a bar potential. This explanation has also been discussed in other observational works as those of \cite{Kerr67}, \cite{Bania77} or \cite{Liszt78}. Nevertheless, it is the comparison of the observations with numerical simulations of the gas dynamics what have allowed to better understand the structure of the Galaxy and to derive important dynamical information as the pattern speed of the bar. The first attempt was that of \cite{Mulder86} solving the gas-dynamical equations to get the gas flow in a weak bar for the whole Galaxy. { A major step in our understanding of the gas kinematics in the Galactic Center was the paper by \cite{Binney91}. These authors made a comparison of the CO(1-0) and HI longitude-velocity diagrams (hereafter \lvd) of \cite{Bally87} and \cite{Burton78}, with the $lv$ locus of closed stellar orbits in a bar potential (x1 and x2 orbits). The \lvd \ of the molecular gas in the inner $\sim 4 \deg$ of the Galaxy shows a parallelogram shape that resembles the $lv$ locus of the \textit{cusped} x1 orbit for a bar with inclination of $\sim 16 \deg$. In spite of the globally successful interpretation of the CO and HI \lvd s, the \cite{Binney91} model cannot account for the observed asymmetry of the CO parallelogram, which is lopsided towards positive longitudes while the modeled parallelograms are centered at longitude $0\deg$. In addition the expected upper right vertex of the locus of the \textit{cusped} x1 orbit has a velocity close to 0, while the upper right vertex of the observed CO parallelogram shown in Fig. 2 of \cite{Binney91} has velocity of $\sim 150$ \kms. } The qualitative insight on the gas dynamics that one could obtain by comparing the \lvd~ of the gas with that of a few stellar orbits, should be tested performing simulations of the gas dynamics that take into account the dissipative nature of the gas clouds. This was done by \cite{Jenkins94}, who tried to reproduce the \lvd ~ of the CMZ using a sticky-particles code but with moderate success. { Using a combination of SPH and collisionless particles, \cite{Lee99} were able to reproduce an inner ring with the right inclination in the \lvd \ and also some of the gas clouds with non-circular velocities in the range $l<|6\deg|$.} To our knowledge, the works by \cite{Binney91}, \cite{Jenkins94} and \cite{Lee99} are the only dynamical models that have tried to interpret the dynamics of the complex Galactic center region (the central 500 pc). Other models of the Galactic dynamics have been mainly devoted to larger scales, from the works of \cite{Mulder86} and Wada et al (1994) to the more recent works by \cite{Fux99}, \cite{Englmaier99}, \cite{Weiner99} and \cite{Bissantz03}. The bar pattern rotation speed for most of those models are in the range from 40 to 60 \kmskpc. Regarding the corotation radius, it is estimated to be in the range 3-4 kpc by some works \citep{Englmaier99, Bissantz03, Habing06} and in the range 4-5 kpc by other works \citep{Combes96, Sevenster99, Fux99}. In contrast, \cite{Binney91} placed the corotation at 2.4 kpc and found a pattern speed of 80 \kmskpc. As pointed out by \cite{Combes96}, the discrepancy between Binney et al. estimation and other estimations can be due to the fact that the molecular gas distribution in the Galactic center region used by Binney et al. could be the response to a small nuclear bar instead of the large bar of the Galaxy. Indeed, there are recent evidences of the presence of a distinct structure in the nuclear region of the Galaxy. The star counts map made by \cite{Alard01} with 2MASS data shows an excess of counts in the inner four degrees of the Galaxy that resembles a small nuclear bar. This bar could be lopsided with the barycenter displaced to negative longitudes. \cite{Nishiyama05} have also found evidence of a structure different to the large bar or bulge in the inner degrees of the Galaxy. In the present paper, we want to investigate the gas response to the mass distribution inferred from the 2MASS data. In particular, we model the central mass distribution as a small nuclear bar. We are interested in the role of this nuclear bar to explain the differences among the results of previous works \citep{Binney91, Fux99, Combes96}. We also want to study whether a lopsided nuclear bar could explain the apparent lopsidedness of the molecular gas distribution in the nuclear region, whose complex dynamics are far from being understood. Therefore, we have modeled the 2MASS star counts maps of Alard fitting different parametric functions using three components: disc, bulge/bar and nuclear bar. We use that mass distribution to determine the galactic stellar potential and to model the gas dynamics using a N-body code that takes into account cloud collisions. { Nested bars are common in external galaxies. As much as one third of barred galaxies have a secondary nuclear bar \citep{Laine02}. In some cases the secondary bar is coupled with the primary bar \citep{Shaw93} while in other cases the two bars rotate with different speeds \citep[][]{Wozniak95}. The present paper is the first one dealing with the gas flow in the Milky Way nested bars and we will assume that both bars are dynamically coupled (they rotate with the same speed). As we discuss in Sect. 8, this seems a good assumption since our simulations explain for the first time some characteristics of the inner Galaxy. } The paper is organized as follows: in the next section we summarize our understanding of the distribution and kinematics of the interstellar gas. In Sects. 3 and 4 we present the 2MASS star counts model and the fits to the data. The gas dynamics simulations are described in Sect. 5, and the results are presented in Sects. 6 an 7. We discuss the nature of the nuclear bar in Sect. 8. Our conclusions are summarized in Sect. 9. \begin{figure*}[ht!] \centerline{\includegraphics[angle=-90,width=16cm]{f1_dame.eps}} \caption{Longitude-velocity ($lv$) diagram of the CO(1-0) emission \citep{Dame01}. The solid lines trace the position of some remarkable features as the locus of the spiral arms, the 3-kpc arm and the Connecting Arm. { The black dashed lines indicate the contour of the Galactic Molecular Ring. The solid circles are the terminal velocities measurements of \cite{Fich89} using CO while the triangles are the terminal velocities determined from the HI data of \cite{Burton93}. The boxes mark the position of the Sagittarius, Scutum, Norma and Centaurus tangent points, located respectively at $l \sim 50 \deg ,\ 30 \deg, \, -30\deg$ and $-50\deg$. The lines concerning the Nuclear Disk or Central Molecular Zone are blue but they are better shown in the next figure. } All these lines can be used to compare with the figures showing the simulations results. } \label{fig:dame} \end{figure*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \section{The distribution and kinematics of the interstellar gas} In this section we summarize outstanding structures in the longitude-velocity diagram (hereafter $lv$-diagram) of the interstellar gas. The large scale $lv$-diagram derived from observations of the CO(1-0) line by \citep{Dame01} is shown in Fig. \ref{fig:dame} while Fig. \ref{fig:bally} shows the $lv$-diagram of the nuclear region derived also from CO(1-0) observations by \cite{Bally87}. We also discuss the plausible face on view of the Galaxy. The different features discussed here will be used in the following sections to compare with our simulations of the gas dynamics. { \subsection{The terminal velocity curve} The terminal velocity curve (TVC) is not a kinematic feature in itself as many independent features contribute to this curve at different Galactic longitudes. However, we discuss it here since it has been used to constrain several dynamical models of the Milky Way \citep{Binney91,Weiner99, Englmaier99, Bissantz03}. There has been different determinations of the TVC using CO and HI data \citep{Fich89, Burton93, Clemens85, Alvarez90}. The dispersion on the TVC points derived by different authors is typically 10 \kms ~ \citep[see for instance Fig. 6 of][]{Bissantz03}. The TVC measurements by \cite{Fich89} and \cite{Burton93} are shown in Fig. \ref{fig:dame} and in the figures showing the simulations results. In addition, in Fig. \ref{fig:dame} we have indicated with boxes the Sagittarius, Scutum, Norma and Centaurus tangent points, located respectively at $l \sim 50 \deg ,\ 30 \deg, \, -30\deg$ and $-50\deg$ \cite[see for instance][]{Dame01, Englmaier99}. } \subsection{Spiral arms} Many works have studied the spiral structure of the Galaxy using different tracers as HII regions, atomic and molecular gas or photometric observations in different bands from the near to the far infrared. The different models of the spiral arms have two (\cite{Bash81} using HII regions, \cite{Drimmel01} using K-band observations, \cite{Benjamin08} using Spitzer), three \citep[][using HI]{Nakanishi03, Kulkarni82} or four arms (\cite{Georgelin76} using HI regions; \cite{Nakanishi06} using CO or \cite{Drimmel00} using the 240 microns luminosity). Deprojecting a given data set to find out the face on view of the Galaxy is a difficult task in particular for the inner Galaxy. For instance, the kinematical distances based in the rotation curve are not accurate due to the presence of important non-circular motions. Furthermore, after performing the deprojection there is an uncertainty in how different structures are associated to a common arm \citep[see][]{Nakanishi06}. Different authors have even fitted different spirals to the same data. A plausible explanation for the discrepancy among the two arms derived from stellar tracers (K band) and the four arms that are found in most of the interstellar tracers is that the Milky Way has two stellar spiral arms but the gas response is a four-armed spiral \citep{Drimmel00}. In any case, the major arms are logarithmic with pitch angles of $10\deg-15\deg$. In addition, to these grand-design spirals there is a more flocculent structure giving a number of minor arms like the Local Arm, which is a small arm linked to the Perseus spiral arm. In order to compare with our simulations, in Figs. \ref{fig:simu_a70}, \ref{fig:simu_o30} and \ref{fig:simu_a70_sche} we have traced the sketch of the logarithmic spiral arms of \cite{Nakanishi06}, which are very similar to the classical spirals of \cite{Georgelin76}. The locus of different spiral arms in the $lv$-diagram are shown in Fig. \ref{fig:dame}. The Carina arms deserves a special mention since the Carina tangent point is clearly seen at $l \sim -80\deg$. { The determinations of the speed of the spiral pattern give values in the range from 13.5 to 59 \kmskpc \ \citep{Lin69, Fernandez01, Bissantz03, Martos04, Debattista02}. Thus, the spiral pattern speed could be lower that the bar one. } \subsection{The Galactic Molecular ring} The so-called Galactic Molecular Ring (GMR) is one of the most prominent structures in the lv-diagram (Fig. \ref{fig:dame}). The locus of the GMR is more or less a diagonal line that passes trough the origin as a circular ring would do. If the GMR is a real ring, it could be associated to a resonance. \cite{Binney91} proposed that the GMR could be located at the Outer Linblad Resonance. Other models that place the corotation farther out have suggested that the GMR could be better explained by the Ultra Harmonic Resonance \citep{Combes96}. However, the real spatial distribution of the GMR is not known and it could be composed by imbricated spiral arms instead of being an actual ring. Some authors \citep[see for instance][]{Englmaier99} consider that, if formed by several arms, the GMR should be located outside of the corotation since they exhibit almost circular velocities. In their face-on deprojection of the neutral and molecular gas distribution, \cite{Nakanishi06} found that the GMR seems to be the inner part of the Sgr-Carina arm (the far-side of the GMR) and the Scutum-Crux arm (the near-side of the GMR). \subsection{Inner arms: the 3-kpc arm} The 3-kpc arm is clearly seen in the $lv$-diagram of Fig. \ref{fig:dame} with a velocity of -53 km\,s$^{-1}$ at $l=0\deg$. The name of this feature comes from a tangent point at $l=-22\deg$ that corresponds to a distance of $\sim 3$ kpc. This arm is located in between the Sun and the GC since the clouds in the arm absorbs the radiation from the continuum sources in the GC. The 3-kpc arm cannot be explained with a combination of rotation and expansion \citep{Burke64} and has been interpreted as a stationary density wave in a barred potential \citep{Mulder86}. The simulations of \cite{Fux99} suggest that it is a lateral arm that surrounds the bar while, in the simulations by \cite{Englmaier99}, it would be an small arm arising from the extremity of the bar. \cite{Sevenster99} and \cite{Habing06} propose a different explanation. \cite{Sevenster99} found old stars associated with the 3-kpc arm. They interpret the fact that both old stars and gas follow the same trajectories as the probe that the arm would be the locus of closed orbits and not a spiral density wave maximum. They propose that the 3-kpc arm has its origin near one of the two points where the bar meets its corotation radius and that the arm can be a channel to transport gas from the corotation to the GC and to fuel the star formation in cloud complexes like Sgr B2. { The recent discovery of the far-side counterpart of the 3-kpc arm by \cite{Dame08b} will certainly contribute to understand the exact nature of these features (see Sect. 6.3).} \subsection{The connecting arm and Bania's clumps : tracing the dust lanes ? } The ``Connecting Arm" is clearly seen in Fig. \ref{fig:dame} as a feature with high inclination with a velocity of $\sim$ 100 km\,s$^{-1}$ at $l=10\deg$ and more than 250 km\,s$^{-1}$ at $l=3\deg$ . The name ``Connecting Arm" comes from the fact that it seems to be connected to the nuclear ring. This a very puzzling structure that has not been discussed much in the literature \citep[see][for a short summary]{Fux99}. On the one hand, \cite{Burton78} and \cite{Liszt80} suggested that the Connecting arm could be the edge of the HI ring found at a radius of $\sim 1$ kpc (see below). On the other hand, in one of his simulations, \cite{Fux99} obtained a feature in the $lv$-diagram that resembles the Connecting Arm. In that simulation, the Connecting Arm-like feature is the locus of the off-axis shocks or dustlanes (since in the optical images of barred galaxies the off-axis shocks are seen as dust lanes). \cite{Marshall08} also interpret the Connecting Arm as the near side dustlane. However, in other of the simulations discussed by \cite{Fux99} the locus of the dustlane in the $lv$-diagram is a vertical feature extending approximately from a velocity of 0 km\,s$^{-1}$ to 200 km\,s$^{-1}$ at an almost constant longitude (see his figures 15 and 16). This reminds the ``clumps" discussed by \cite{Bania77}. These clumps are outstanding cloud complexes found at $l=5.5\deg$ and $l=3.1\deg$ that exhibit a huge velocity dispersion of $\sim 200$ km\,s$^{-1}$ in a very narrow range of longitude. A portion of the clump located at $l=3.1\deg$ (known as Clump 2) is showed in Fig. \ref{fig:bally}. The Clump 2 has been interpreted as a dustlane by \cite{Stark86}, while in the \cite{Fux99} interpretation, Banias's clumps would be gas clouds that are about to enter the dustlanes shocks, although the actual dustlane would be the Connecting Arm as mentioned above. { This is also the interpretation of \cite{Liszt06, Liszt08}, who have studied in detail the spatial and velocity structure of all these features.} \subsection{The HI ring} The HI emission in the GC region can be modeled as a tilted circular ring with inner radius of $\sim$ 300 pc and outer radius of 1-1.5 kpc \citep{Burton78}. This first model was improved by \cite{Liszt80}, who discussed that an elongated ring could also explain the observations. The spatial distribution of the interstellar gas in the innermost 3 kpc of the Galaxy has recently been reviewed by \cite{Ferriere07}. { The model that they consider more plausible, despite its inherent uncertainties, is that of \cite{Liszt80} since an elongated ring seems to be easier to understand if it is composed by gas clouds moving in elongated x1 orbits. Therefore, they modeled the HI ring as an elliptical ring with semi-major axis of 1.6 kpc, and axis ratio of 3.1, with an inner hole of semi-axis 800 pc $\times$ 258 pc that encloses the Central Molecular Zone (see bellow). } \begin{figure}[th!] \centerline{\includegraphics[width=8cm]{f2_bally.eps}} \caption{{\it Upper panel:} Longitude-velocity ($lv$) diagram of the CO(1-0) emission in the Central Molecular Zone using data of \cite{Bally87} and integrating all negative Galactic latitudes. The solid lines trace the position of some remarkable features as the Clump 2, or structures K and J of \cite{Rodriguez-Fernandez06}. The dashed line indicates the contour of the very crowded central region. {\it Lower panel:} Integrated intensity map.} \label{fig:bally} \end{figure} \subsection{The Central Molecular Zone} The Central Molecular Zone (hereafter CMZ) refers to the central accumulation of gas in the inner hundreds of parsec of the Galaxy. This gas is mainly molecular and extends continuously approximately from $-1.5\deg$ to $2\deg$ (see Fig.~\ref{fig:bally}). The spatial distribution is not symmetric due to the prominent cloud complex located at $l=1.3\deg$ (hereafter {\it l=1.3$\deg$-complex}) that lacks of a negative longitude counterpart. Figure~\ref{fig:bally} also shows the $lv$-diagram of the CMZ. This plot has been obtained integrating the CO(1-0) data of \cite{Bally87} for negative Galactic longitudes to remark a number of kinematical component as the Clump~2 (at $l=3.1\deg$), and the features K and J discussed by \cite{Rodriguez-Fernandez06}. The kinematics of the inner region are very complex with a high fraction of the gas exhibiting non-circular velocities. Figure \ref{fig:bally} shows with a dashed line the contour of the inner CMZ that contains the feature M of \cite{Rodriguez-Fernandez06} and the inner arms found by \cite{Sofue95}. This contour resembles a parallelogram but it is not the parallelogram that inspired the \cite{Binney91} work. The whole \lvd ~ shown in Fig. \ref{fig:bally} is very different to that used by \cite{Binney91}. { These authors integrated the CO data of \cite{Bally87} over positive and negative latitudes and studied the region $2.2\deg >l > -2.2\deg$. In their Fig. 2, the feature K is not clearly detached from the inner $lv$ parallelogram and they considered that it is part of a larger $lv$ parallelogram. However, following the analysis of the \cite{Bally87} data by \cite{Rodriguez-Fernandez06}, we reckon that feature K is a distinct kinematic feature. This conclusion is also supported by the large scale ($13\deg > l > -12\deg$) CO(1-0) survey by \cite{Bitran97}, whose \lvd ~ shows clearly that the feature K extends to $l \sim 3\deg$ where it is connected to the Connecting Arm. Interestingly, when one compares the $lv$ parallelograms of the x1 orbits displayed in Fig.~3 of \cite{Binney91} to the parallelogram shown in our Fig.~\ref{fig:bally}, the upper right vertex are in good agreement, showing velocities of $\sim 0$ \kms. Thus, one of the apparent discrepancies (see Sect. 1) between the \cite{Binney91} model and the CO \lvd ~ disappears. } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Modeling the star counts map} \subsection{The star counts map} We have modeled the 2MASS star counts map of \cite{Alard01} to determine a realistic stellar potential. In this section we summarize the methods and the main conclusions of Alard's work. Using the H and K bands and assuming that we know the intrinsic color of the sources ($H$ and $K$), one can correct for extinction one of the bands as follows: \begin{equation} m_K^{corr}=m_K- [ A_K/(A_H-A_k)][(m_H-m_K)-(H-K)] \end{equation} where $m_{K,H}$ are the observed apparent magnitudes in K and H band (thus suffering extinction), $A_{K,H}$ is the extinction in each band and $m_K^{corr}$ is the extinction corrected magnitude in K band. At the distance of the bulge a cutoff $m_K=9$ (see below) implies that the sources are early M-giants (see Lopez-Corredoira et al. 2001). Assuming an intrinsic color $H-K$ for M giants of $\sim 0.17$ \citep{Binney98, Wainscoat92} and taking into account the relative extinctions $A_k/A_v=0.112$ and $A_h/A_v=0.175$ \citep{Rieke85}, one can write: \begin{equation} m_K^{corr}=m_K-1.77 \, (m_H-m_K)+0.30 \end{equation} \cite{Alard01} defined an extinction corrected magnitude $m_e\equiv m_K^{corr}-0.30$ as \begin{equation} m_e \equiv m_K-1.77 \, (m_H-m_K) \end{equation} and constructed his star counts map using this magnitude with a cutoff of 9 mag. In most of this map, the density profile at constant longitude is exponential. This exponential profile is also present in numerical simulations. \cite{Combes90} showed that a disk with a small bulge in its center can evolve in a peanut shaped bar with a nearly exponential profile perpendicular to the plane of the galaxy. However, \cite{Alard01} noticed that the observed density profile in the inner four degrees of the Galaxy shows an excess of sources in the plane. Subtracting an exponential profile corresponding to disk and bulge, he found that the residual seems to be a small, lopsided, nuclear bar in the galactic center. This structure has not been found in previous studies since the data lacked either the depth or the resolution. \subsection{The model} In this section we describe how we have modeled the 2MASS star counts map. The number of stars in a volume $dV$ located at $x$ with a magnitude in the interval $M$ and $M+dM$ is: \begin{equation} dN=\Phi(M,x)dMdV \end{equation} If the distribution of magnitudes is the same in all the space we can write $\Phi$ as $\Phi(M,x)=\phi(M)\rho(x)$, where $\phi(M)$ is the {\it luminosity function} (the fraction of stars with magnitude $M$) and $\rho(x)$ is the { star density} in the point $x$. Therefore, the total number of stars with a magnitude $M$ lower than a cutoff $M_c$ is: \begin{equation} N=\int_{M3.5 \mathrm{kpc} \\ %\rho(r,z)=\rho(3.5 kpc, z) \frac{r_{kpc}}{3.5} & r<3.5 \mathrm{kpc} \\ %\end{array} %\right. %\end{equation} \subsubsection{Triaxial bulge} From the COBE/DIRBE images we know the bulge of the Milky Way is {\it boxy} \citep{Dwek95}. The 2MASS data confirm this result \citep{Lopez-Corredoira05}. Therefore, we have adopted a {\it boxy} Gaussian (Dwek et al. function G2) to represent the bulge. \begin{equation} \label{eq:g2} \rho^B(x',y',z')=\rho_0 \exp (-0.5r_s^2) \end{equation} \begin{equation} r_s=\left( \left[ \left(\frac{x'}{x_0}\right)^2+\left(\frac{y'}{y_0}\right)^2\right]^2+ \left(\frac{z}{z'_0}\right)^4 \right)^{1/4} \end{equation} The bulge function is expressed in a coordinate system $(x',y',z')$ that follows its symmetry axes. Therefore, a rotation is needed to get the expression in the $(x,y,z)$ coordinate system. \begin{equation} \left\lbrace\begin{array}{l} x'=x \cos\beta^B + y \sin \beta^B\\ y'=-x \sin \beta^B + y \cos \beta^B\\ z'=z\\ \end{array} \right. \end{equation} {Figure~\ref{fig:sisref} shows the $(x,y)$ and $(x',y')$ axis seen from the $z=z'>0$ hemisphere. The figure also shows the definition of the angle $\beta^B$, which is measured counterclockwise from the $x>0$ semi-axis. However, many papers measure the inclination of the bar clockwise from the Sun-GC line to the near side of the bar. Therefore, for an easy comparison with previous results we have defined the angle $\alpha^B$ as shown in the figure.} \begin{figure} \begin{center} \includegraphics[width=6cm]{fig-sisref.eps} \caption{ Sketch of the different coordinates systems and angles used to define the bars in the Galactic plane. The $z$ and $z'$ axis are coincident and the $z>0$ semiaxis is in the northern Galactic hemisphere.} \label{fig:sisref} \end{center} \end{figure} \subsubsection{Triaxial nuclear bar} In contrast to the bulge, the small nuclear bar found by Alard does not seem to be boxy. Therefore we have taken a triaxial Gaussian function (Dwek et al. function G1) to represent this component: \begin{equation} \label{eq:g1} \rho^b(x'',y'',z'')=\rho_0 \exp (-0.5r^2) \end{equation} \begin{equation} r=\left[ \left(\frac{x''-x_1}{x_0}\right)^2+\left(\frac{y''}{y_0}\right)^2+\left(\frac{z''}{z_0}\right)^2 \right]^{1/2} \end{equation} Where $x_1$ allows to fit a lopsided bar. To obtain the bar density in the $(x,y,z)$ coordinate system one should apply the rotation: \begin{equation} \left\lbrace\begin{array}{l} x''=x \cos\beta^b + y \sin \beta^b\\ y''=-x \sin \beta^b + y \cos \beta^b\\ z''=z\\ \end{array} \right. \end{equation} {The definition of $\beta^b$ is exactly analogous to that of $\beta^B$ shown in Fig.~\ref{fig:sisref}. Again, in the following, and for coherence with most of the literature in the subject, we will use an angle $\alpha^b$ measured clockwise from the Sun-GC line to refer to the inclination of the nuclear bar.} \subsubsection{Final star counts equation} Finally, we must also convert $\rho(x,y,z)$ to $\rho'(l,b,s)$ since we want to model the star counts in Galactic coordinates $(l,b)$ and to integrate along the line of sight ($s$). If $(x,y,z)$ is a right-handed reference frame centered in the galactic center, with the plane of the sky in the $xz$-plane with $x>0$ for $l<0$, and $z>0$ towards the Galactic north pole, and assuming that the sun is located in $(x,y,z)=(0,-D,0)$ then: \begin{equation} \label{eq:trans} \begin{array}{l} x=-s \cos(b) \sin(l)\\ y=-D+s\cos(b)cos(l)\\ z=s \sin(b)\\ \end{array} \end{equation} The star counts equation towards $(l_0,b_0)$ is then: \begin{equation} \label{eq:starcounts} \begin{array}{l} N(l_0,b_0,m_c) = \Delta l \, \Delta b \times \\ \times \int_{m_{min}}^{m_{max}} dm \int_{0}^{s_{max}} ds\, \phi'(m,s)\, \rho'(l_0,b_0,s)\, s^2 \cos b \\ \end{array} \end{equation} where $s^2 \cos b$ is the Jacobian of the coordinates transformation (Eq. \ref{eq:trans}). We have assumed that the density is constant in the interval from $l_0$ to $l_0+\Delta l$ and from $b_0$ to $b_0+\Delta b$. We have computed the integral in magnitudes from $m_{min}=-10.0 \ \mathrm{mag}$ to $m_{max}=9.0 \ \mathrm{mag}$ with $\Delta m =0.5 \ \mathrm{mag}$. The integral on the distance along the line of sight $s$ has been computed up to $s_{max}=20\ \mathrm{kpc}$ using a variable step $\Delta s$: \begin{equation} \Delta s = \left\lbrace \begin{array}{ll} 0.002\ \mathrm{kpc} & s, R < 0.100\ \mathrm{kpc} \\ 0.008\ \mathrm{kpc} & s, R < 0.300\ \mathrm{kpc} \\ 0.010\ \mathrm{kpc} & s, R < 0.700\ \mathrm{kpc} \\ 0.050\ \mathrm{kpc} & s, R < 1.500\ \mathrm{kpc} \\ 0.100\ \mathrm{kpc} & s, R < 3.100\ \mathrm{kpc} \\ 0.200\ \mathrm{kpc} & s, R < 6.300\ \mathrm{kpc} \\ 0.300\ \mathrm{kpc} & s, R > 6.300\ \mathrm{kpc} \\ \end{array} \right. \end{equation} %................................................... \section{Fitting the star counts map} We have fitted the star counts map using the model described above. This model has 14 free parameters corresponding to the disc ($\rho_0, \, h_r, \,h_z$), the bulge ($\rho_0, x_0, y_0, z_0, \alpha$) and the nuclear bar ($\rho_0, x_0, y_0, z_0, \alpha, x_1$). The star counts equation is evaluated for an initial guess of the density function parameters. A $\chi^2$ parameter is computed as the difference of the star counts data points ($N_{data_i}$) and the model ($N_{model_i}$). \begin{equation} \chi^2= \frac{1}{n} \sum_{i=1}^n \frac{(N_{data_i}-N_{model_i})^2}{\sigma^2_i} \end{equation} where $n$ is the number of points to be fitted and $\sigma$ is the error of the data points. We have estimated the $\sigma$ from Fig. 1 of \cite{Alard01} as the dispersion of the star counts with respect to an exponential vertical profile. Afterwards we have used an iterative process to minimize $\chi^2$ using the Levenberg-Marquardt algorithm \citep{Press92}. \subsection{Fitting the disc and the boxy bulge} We have not attempted to fit all the free parameters at once. First, we have fitted a star counts model with the disc and the boxy bulge to a reduced data set consisting of one point every 10 pixels (one data point every $0.5\deg$). In the disc region, we have taken one point every 5 pixels ($0.25\deg$) across the disc. Finally, we have neglected the data points in the inner region ($-3 < l (deg) < 3, \ -2 From Fig. 1 of \cite{Alard01} is clear that the dispersion of the star > counts map in the nuclear region is smaller than in the bulge. We have estimated a $\sigma$ of $5 \%$ for these data points. %\begin{figure} %\centerline{\includegraphics*[bb=304 50 533 363, angle=-90,width=9cm]{f_map_m9_all.eps}} %\caption{Data set used to do fit the disc, the bulge and the nuclear bar} %\label{fig:data2} %\end{figure} We have fitted the data using our star counts model with three components: disc, bulge and nuclear bar. We have taken as starting point the best fits for $\alpha^B$=10$\deg$, 20$\deg$, 35$\deg$, 45$\deg$ and 60$\deg$ derived in the previous section (all the parameters of the disc and the bulge are fixed except the factors $\rho_0$). The best fit results for the parameters of the nuclear bar are summarized in Table \ref{tab:res-Bdb}. The full set of results for the nuclear bar fits are listed in Tables \ref{tab:res-Bdb1} - \ref{tab:res-Bdb4}. The best fits are found with a thick triaxial bar with typical axis ratios of $1:(0.7-0.8):(0.5-0.6)$ and $\alpha^b$ in the interval $60-120$ {\it for all the $\alpha^B$'s}. This implies that the parameters of the nuclear bar can indeed be constrained independently of the bulge and that $\alpha^b \neq \alpha^B$. Globally, the best fit is obtained with $\alpha^B=10\deg$ and a nuclear bar with $\alpha^b=90\deg-105\deg$. some examples of good fits are shown in Fig. \ref{fig:res-Bdb}. \begin{table*}[] \caption{Summary of the parameters of the nuclear bar as derived from the best fits to the 2MASS star counts data. The first row are fits with $x_1$ fixed to 0 and the second row corresponds to fits with free $x_1$ to allow for lopsidedness. The full set of results as a function of the orientation of the bulge $\alpha^B$ are listed in Tables \ref{tab:res-Bdb1} to \ref{tab:res-Bdb4}. } \label{tab:res-Bdb} \begin{center} \begin{tabular}{llllll|l} \hline $x_0^b$ & $y_0^b$ & $z_0^b$ &$\rho_0^b$ & $\alpha^b $ & $x_1$ & $\chi^2$ \\ kpc & kpc & kpc & & deg & kpc & \\ \hline 0.141-0.150& 0.130-0.138& 0.095-0.097 & 0.141-0.150 & 60-120 & 0.0 & 0.357-0.376\\ 0.158-0.166& 0.115-0.133& 0.088-0.094 & 0.151-0.178 & 60-120 & 0.022-0.025& 0.331-0.346 \\ \hline \end{tabular} \end{center} \end{table*} \begin{figure*} \vspace{0cm} \centerline{\includegraphics*[bb= 304 47 530 655 , angle=-90,width=15cm]{f4_fit_a80b15.eps}} \centerline{\includegraphics*[bb= 304 47 530 655 , angle=-90,width=15cm]{f4_fit_a45b30_asi.eps}} \caption{{\it Upper panels:} Thin solid contours represent the star counts map. Thick dashed lines are the best fit obtained with a star counts model with $\alpha^B=10\deg$, $h_r=2.5$ kpc and $\alpha^b=75$ deg. The panel on the right is a zoom of the nuclear region. The points indicate the data used to do the fit. {\it Lower panels:} Same that the upper panels for the best fit with $\alpha^B=45\deg$, and a lopsided bar with $\alpha^b=60\deg$.} \label{fig:res-Bdb} \end{figure*} %\subsection{Fitting disc, bulge and {\it lopsided} nuclear bar} Finally, we have also performed fits with a lopsided nuclear bar ($x_1 \neq 0$). The results are shown in Tables \ref{tab:res-Bdb1} - \ref{tab:res-Bdb4} and they are summarized in Table \ref{tab:res-Bdb}. The best fits are still obtained with a nuclear bar that is almost perpendicular to the Sun-GC line. The symmetry center of the bar would be shifted by 22-25 pc towards the third Galactic quadrant if $\alpha^b=60\deg-90\deg$ while it would be shifted by the same quantity but towards the fourth quadrant if $\alpha^b=90\deg-120\deg$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Simulations of the gas dynamics: methods} We have used the code of \cite{Combes85} to simulate the motion of gas clouds in the potential obtained from the mass distribution inferred with the star counts model. This code has already been used successfully to understand the gas distribution and dynamics of a number of galaxies as %M51 \citep{Garcia-Burillo93}, M100 \citep{Garcia-Burillo98}, %NGC 4321 \citep{Sempere95} or NGC 4736 \citep{Gerin91, Mulder96}. \subsection {Stellar potential, rotation curve and masses} The potential is obtained from the mass distribution derived in the previous section, using standard FFT techniques in a Cartesian grid of $512 \times 512 \times 16$ cells. { The cells are cubic with a linear size of 58.6 pc, therefore the physical size of the grid is $30 \times 30 \times 0.94$ kpc$^3$. This is good compromise to model the large scale dynamics at least to a radius of 10 kpc and to have a good enough spatial resolution to study the Galactic center. As described by \cite{Combes81}, the interaction between periodically reproduced images is avoided following the scheme of \cite{Hohl69}, i.e. the potential computations are done in a grid 8 times bigger, of $1024 \times 1024 \times 32$ cells. The disk mass is truncated at 15kpc in radius, and the mass distribution is assumed spherical outside, so that its influence in the center is cancelled. This is equivalent to assume that a spherical dark halo is dominating the mass outside. This assumption should have a negligible influence on the dynamics of the inner Galaxy, which is the main interest of this paper. } The stellar potential is decomposed in its axisymmetric and non-axisymmetric parts. The axisymmetric potential is calculated in radial bins using the averaged mass in the bin. The non-axisymmetric part of the potential is calculated with a mass distribution corresponding to the difference between the total-mass and the axisymmetric distribution. The stellar number density derived in the previous section should be multiplied by a factor $f$ to obtain the total mass density. This factor accounts for the mass of the stars detected by 2MASS but also for the mass of those stars that are not counted in the 2MASS maps. { In addition, we have to introduce a dark halo to explain the observed velocities at large radii.} The total density used is then: \begin{equation} \label{eq:rhot} \rho^T=f \ ( \rho^D + \rho^B + \rho^b ) + \rho^H \end{equation} where the halo density is defined by a Plummer law: \begin{equation} \label{eq:halo} \rho^H(r)=\frac{\rho_0^H}{[1+(r/r_c)^2]^{5/2}} \end{equation} { To determine the scaling factors $f$ and $\rho_0^H$ we have compared the terminal velocities and the rotation curve of the models with those derived from spectroscopic observations \citep{Clemens85, Fich89, Burton93}. To obtain terminal velocities of the right magnitude (see Figs. \ref{fig:simu_a70} and \ref{fig:simu_o30}) and a flat rotation curve at large radii (Fig. \ref{fig:rotcur1})}, one should apply a scaling factor $f=3.9 \ 10^8$ and a halo with $ \rho_0^H= 1.12\,10^5$ and $r_c=15.0 \, \mathrm{kpc}$. With these factors and the parameters given in Tables \ref{tab:res-Bdb1} - \ref{tab:res-Bdb4}, the function $\rho^T$ gives the total mass density in units of $10^3$~\Msun/kpc$^3$. Integrating in the total grid size, the typical masses of the different components are: $M^H=7\,10^{10}$~M$_\odot$, $M^D=2.9\,10^{10}$~M$_\odot$, $M^B=1.9\,10^{10}$~M$_\odot$ and $M^b=2.0\,10^{9}$~M$_\odot$, which are in good agreement with previous determinations \citep[see for instance][]{Weiner99, Fux99}. Figure \ref{fig:rotcur1} shows an example of rotation curve. The modeled curve compares well with the rotation curve derived from observations of the interstellar gas for radius larger than 2 kpc. The agreement seems worse for small radii. However the comparison in this region is not straightforward. On the one hand, the measured curve has been derived using the tangent point method assuming that the clouds move in circular orbits, which is not true in the innermost region. In addition, this curve depends on the position of the observer in the Galaxy. On the other hand, the modeled rotation curve has been derived with the axisymmetrical potential and with an azimuthal average of the non-axisymmetrical potential. This rotation curve is a kind of average curve independent of the position of the observer. %\begin{figure} %\centerline{\includegraphics[angle=-90,width=7cm]{f_rotcur_0.eps}} %\caption{Rotation curve. $\rho^T=f ( \rho^D + \rho^B + \rho^b )$ } %\label{fig:rotcur} %\end{figure} \begin{figure} \centerline{\includegraphics[angle=-90,width=8.5cm]{f5a2.eps}} \caption{Rotation curve obtained with the halo and the scaling discussed in Sect. 5.1 for a bulge and nuclear bar orientation of $\alpha^B=20\deg$ and $\alpha^b=75\deg$, respectively. The different lines show the contribution of the different mass components to the total rotation curve. The red points indicate the rotation curve measured from observational data. The mass of the halo , disc, and bulge are $M^H=7\,10^{10}$~M$_\odot$, $M^D=2.9\,10^{10}$~M$_\odot$, $M^B=1.9\,10^{10}$~M$_\odot$, respectively. The mass of the nuclear bar is $M^b=2.0\,10^{9}$~M$_\odot$.} \label{fig:rotcur1} \end{figure} Figure \ref{fig:omega} gives the rotation velocity $\Omega$ as a function of galactocentric radius. It also shows the $\Omega + \kappa/m$ curves (where $\kappa$ is the epicyclic frequency). For a given bar pattern speed $\Omega_p$, these curves give the location of the corotation ($\Omega=\Omega_p$), the Outer Lindblad resonance ($\Omega+\kappa/2=\Omega_p$), the Inner Lindblad resonances ($\Omega-\kappa/2=\Omega_p$) and the Ultra Harmonic Resonance ($\Omega-\kappa/4=\Omega_p$). \begin{figure} \centerline{\includegraphics*[ angle=-90, width=8cm]{f6_omega.eps}} \caption{Angular frequencies as a function of radius in the epicyclic approximation for the potential discussed in Sect. 5.1} \label{fig:omega} \end{figure} \subsection{Gas cloud dynamics} The gas clouds are modeled as particles that are initially launched in the axisymmetric potential with a velocity corresponding to the circular velocity of the potential at the radius of the particle. The initial radial distribution is an exponential disk with radial scale of 2 kpc. %Its distribution perpendicular to the plane is Gaussian. The non-axisymmetric part of the potential is then introduced gradually multiplying by the factor $1-\exp[-t/\tau]^2$ with a time delay of $\tau=$100 Myr. The angular speed of the non-axisymmetrical pattern, $\Omega_p$, is constant along the simulation. We do not use a mass spectrum for the clouds and we do not simulate star formation. %All the clouds have a constant mass of 3.1\,$10^4$ M$_\odot$. The simulations contain typically $10^6$ particles. The effects of the gas self-gravitation are neglected. The clouds move as test particles in the stellar potential except when they collide inelastically. The forces due to the stellar potential are evaluated every 1 Myr and the particles are moved according to them. The particles are followed in three dimensions but the collisions are computed in a two-dimensional grid with 480 $\times$ 480 cells. Clouds in the same grid cell can collide. The relative velocity of two collisions partners loses 60$\%$ of its absolute value in the collision. A detailed discussion of the collisional schema can be found in \cite{Combes85} and \cite{Casoli82}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Simulations results and discussion: large scale structure} Figures \ref{fig:simu_a70} and \ref{fig:simu_o30} show face-on views and \lvd \ for a grid of models with $\alpha^B$ from $10 \deg$ to $45 \deg$ and pattern speeds from 20 to 50 \kmskpc. For an easy comparison with the observations, we have overlaid in the \lvd s the TVC points, the boxes indicating the tanget points and the solid and dashed lines shown in Fig. \ref{fig:dame}. We have also plotted the four logarithmic spirals of \cite{Nakanishi06} in the face-on views. The models shown in Figs. \ref{fig:simu_a70} and \ref{fig:simu_o30} exhibit a number of common characteristics. The galaxies have two major spiral arms but they also exhibit a second pair of less prominent spiral arms. In the inner $\sim 4$ kpc, there are arms that contour the bar forming a kind of oval (hereafter lateral arms). The density of gas inside this oval decreases down to a radius of $\sim 1$ kpc, where there is a ring. This ring is connected to the lateral arms by the off-axis arms (or dustlanes). In the inner hundreds of parsecs there is a nuclear condensation of gas. { The Milky Way galaxy is a very complex system and up to now there is not a single simulation of the gas dynamics that can explain all the observed features, nor quantitatively nor qualitatively. \cite{Englmaier99} have compared their models to the terminal velocity curve, they have also compared the position the arms tangent points in the CO \lvd , at $|l|=30\deg$ and $50 \deg$. The two \lvd s presented by \cite{Fux99} has been selected by the global resemblance to the \lvd s obtained from CO and HI data. They reproduce qualitatively the 3-kpc arm and the Connecting Arm. \cite{Weiner99} constrained their models by comparing with the HI terminal velocity curve. Finally, \cite{Bissantz03} have used mainly the terminal velocity curve and some voids in the observed \lvd \ to compare with their simulations. To select the best models in the present work we have compared with: \textit{i)} the terminal velocity curve, \textit{ii)} the inclination of the GMR in the \lvd , \textit{iii)} the existence or not of the 3kpc arm and when it exists, its inclination and absolute velocity \textit{iv)} the arms tangent points \textit{v)} the shape and size of the HI 1kpc ring. } \subsection{Pattern speed and orientation of the bulge} { Figure \ref{fig:simu_a70} shows face-on views and lv-diagrams for a grid of models with $\alpha^B=20\deg$ and pattern speeds from 20 to 50 \kmskpc. The model with $\Omega_p=20$ \kmskpc \ reproduces some spiral tangent points and has an inner lateral arm with the inclination of the 3-kpc arm, however it does not reproduce the terminal velocity curve nor the velocity of the 3-kpc arm, for instance. On the other extremity, the model with $\Omega_p=50$ \kmskpc \ does not reproduce satisfactorily neither the terminal velocity curve nor the GMR nor the 3-kpc arm. In contrast, the models with $\Omega_p=30$ and $40$ \kmskpc \ give a very good overall representation of the Galaxy. Both reproduce satisfactorily the spiral tangent points at $|l|=30, 50 \deg$ within $\pm 5 \deg$ and $\pm 10$ \kms, approximately. In addition, the model with $\Omega_p=40$ \kmskpc \ reproduces very well the Carina arm and its tangent point at $\approx -80 \deg $. However, the locus of the inner spiral arms and the 3-kpc arm in the modeled \lvd~ are too steep in comparison with the observed \lvd. In contrast, the GMR and the the 3-kpc arm is very well reproduced from $|l|=90\deg$ to $15\deg$ by the model with $\Omega_p=30$ \kmskpc. In particular the 3-kpc arm is reproduced with both the good inclination and the velocity within $\sim 10$ \kms. In addition, the terminal velocity curve for $l>5 \deg$ is very well reproduced. Given the difficulty of the task, it will not be realistic to give a very narrow interval for the bar pattern speed, from our best models \textit{we conclude that the most probable value for $\Omega_p$ should be in the range $30-40$ \kmskpc}. However, since this work is devoted to the inner Galaxy structure and kinematics in the following we will mainly discuss models with $\Omega_p = 30$ \kmskpc \ since they explain the terminal velocity curve, the 3-kpc arm and the GMR better than the model with $\Omega_p = 40$ \kmskpc. } Figure \ref{fig:simu_o30} shows face-on views and \lvd s for a grid of models with constant $\Omega=30$ \kmskpc \ and $\alpha^B$'s of 45$\deg$ , 35$\deg$ , 20$\deg$ and 10$\deg$ ($\alpha^b$ is constant to 90$\deg$ for all the models). For $\alpha^B=10\deg$, the velocity along the line of sight is too low to explain the high non-circular velocities of the 3 kpc arm ($-53$ \kms \ at $l=0\deg$) and the observed arm is not reproduced in the $lv$ diagram. In contrast, the 3-kpc arm and the inclination of the GMR is well reproduced for $\alpha^B$ from 20$\deg$ to 45$\deg$. { Nevertheless, the model with $\alpha^B=45\deg$ does not reproduce correctly the positive velocity terminal curve. In addition, there is no ring at 1 kpc in the face on view. Once again, we do not pretend to give a very narrow interval of angles for the inclination angle of the bar since the analysis, as in other published works, is still rather qualitative. We reckon that \textit{the inclination of the bar should be in the range from 20$\deg$ to 35$\deg$}, with some preference for 20$\deg$ since the quantitative agreement of the 3-kpc arm and the terminal velocity curve is better. Therefore, the model with $\Omega_p = 30$ \kmskpc \ and $\alpha^B=20\deg$ will be considered as our \textit{standard} model in the rest of the paper. \subsubsection{Comparison with previous works} The inclination of the bar derived from our models is in good agreement with previous determinations \citep[][see also Sect. 1]{Binney97, Fux99, Lopez-Corredoira05}. Regarding the bar pattern speed, our simulations give results that are lower than those of \cite{Englmaier99} or \cite{Bissantz03}, which suggest $\Omega_p=50-60$ \kmskpc, but in good agreement with the results of \cite{Weiner99}, $\Omega_p=42$ \kmskpc, or \cite{Fux99}. In his self-consistent gas and stars simulations, Fux found that the pattern speed evolves from 50 to 30 \kmskpc . Taking into account our potential, a pattern speed of 30 or 40 \kmskpc \ gives a corotation radius of 7 and 5 kpc, respectively (see Fig. \ref{fig:omega}). Therefore, the corotation radius is well beyond the bar extremity, which is the characteristic of a \textit{slow} bar \citep[see for instance][]{Sellwood88}. To explain the fact that the dustlanes in the inner spiral arms of some galaxies are located in the concave section of the arms, and assuming that the bars are fast (i.e., they extend to the corotation), \cite{Sellwood88} proposed the existence of different speeds for the bar and the spiral patterns. Indeed, for the Milky Way some works have derived low rotations speeds of 13-20 \kmskpc \ for the spiral pattern \citep{Lin69, Morgan90, Amaral97}. Gas flow models with independent speeds for the bar and the spiral patterns have been presented by \cite{Bissantz03}, who discussed that different speeds for the spiral and the bar patterns could explain some regions devoid of gas in the \lvd . However, the overall agreement of their \lvd s with the observations is not better than other models like that of \cite{Fux99} or our models, which are able to reproduce \textit{quantitatively} features as the 3-kpc arm. %Although a discussion of an independent pattern speed for the spirals is beyond the scope of this paper, It is interesting to remark that our simulations favor a relatively slow bar and not a fast bar as assumed by \cite{Sellwood88}. Furthermore, the pattern speed of the relatively slow bars models \citep[30-40 \kmskpc,][this work]{Weiner99, Fux99} is comparable to several determinations of the spiral pattern speed that give values of 30-35 \kmskpc \ \citep{Mishurov99,Fernandez01, Lepine01}. %The determination of the spiral pattern speed by \cite{Debattista02} using observations of OH/IR stars gives even higher values of 59 \kmskpc. On the other hand, \cite{Ibata95} have measured a rotation speed of $25\pm 4$~\kmskpc \ for the bulge in the $0.7