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%\documentclass[referee]{aa} % for a referee version
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\documentclass{aa}
\usepackage{graphicx}

\newcommand \htwo  {$\mbox{H}_2$}
%\newcommand \arcmin {\mbox{$^\prime$}}
%\newcommand \ie {{\em i.e.}}
\newcommand \eg {{\em e.g.}}
\newcommand \degree {\mbox{$^\circ$}}
%\newcommand \arcsec {\mbox{$^{\prime\prime}$}}
\newcommand \msun {\mbox{$\mathcal{M}_{\odot}$}}
\newcommand \co  {\mbox{$^{12}\mbox{CO(1-0)}$}}
\newcommand \kms {\mbox{km.s$^{-1}$}}
\newcommand \micron  {\mbox{$\mu$m}}
\newcommand \jyperbeam {\mbox{Jy.beam$^{-1}$}}
\newcommand \mjyperbeam {\mbox{mJy.beam$^{-1}$}}
\newcommand \twelvecoonezero {\mbox{$^{12}$CO(1--0) }}
\newcommand \twelvecotwoone  {\mbox{$^{12}$CO(2--1) }}

%
\begin{document}
%
   \title{Molecular outflow in the composite galaxy NGC 6764}

%   \subtitle{I. Overviewing the $\kappa$-mechanism}

   \author{S. Leon,
          \inst{1,2}
           A. Eckart, \inst{2}
           E. Schinnerer, \inst{3}
           S.-W. Lee \inst{4}
	   M. Krips, \inst{3}
           S. Laine, \inst{5}
	   J. Scharwächter, \inst{3}
                     }

   \offprints{S. Leon, stephane@iaa.es}

   \institute{$^1$ Instituto de Astrof\'{\i}sica de Andaluc\'{\i}a, Granada, Spain \\
	      $^2$ University of Cologne, I. Physikalische Institute, Cologne, Germany\\
              $^3$ Caltech, Los Angeles, USA \\
	      $^4$ University of Toronto, Astronomy Department, Toronto, Canada \\
	      $^5$ Space Telescope Insitute, Baltimore, USA
              }

   \date{Received XX}


   \abstract{
NGC~6764 is a composite galaxy exhibiting central activity (LINER) and recent
massive star formation (Wolf-Rayet galaxies). We present new high-resolution interferometric data 
of the  \twelvecoonezero and \twelvecotwoone emission in the central part of the galaxies together 
with high resolution VLA data of the radiocontinuum emission at 20 and 3.5 cm. The nucleus shows 
a carateristic bubble-like radiocontinuum emission centered on the CO and H$\alpha$ emission and 
slightly offset from the radiocontinuum peak at the center, identified as the AGN. 
The total molecular gas in the center is found to be about 7 $10^8$ \msun, with a CO-to-H$_2$ 
conversion 3 times lower than the standard one. A \twelvecoonezero emission is found to correlate
with the edge of the radiocontinuum emission in the outflow with a projected expansion velocity  
up to 50 \kms relative to the systemic velocity of the galaxy. About $10^7$ \msun of molecular
gas is detected in the outflows, likely ejected by the the stellar winds from the recent starburst.
   \keywords{Galaxy  -- ISM -- Individual: NGC 6764
               }
   }
 \titlerunning{Molecular outflow in NGC~6764}
 \authorrunning{Leon et al.}
   \maketitle
%
%________________________________________________________________

\section{Introduction}
NGC~6764 is a barred spiral classified as a low-ionization nuclear emission-line
region (LINER) from optical spectroscopy (Osterbrock \& Cohen 1982). This nearby
active galaxy (D=32 Mpc for $H_0 = 75 \kms$; 1\arcsec = 160 pc) is as well interesting
for its nuclear starburst which is very young as shown by its Wolf-Rayet (W-R) feature
at 0.47 \micron (HeII) as described in Schinnerer et al. (2000).



\begin{table}
\begin{tabular}{ll}
\hline
Parameter  & Value \\
\hline
\hline
$\alpha_{2000}$   & 19:08:16.494 \\ 
$\delta_{2000}$   & 50:55:59.46  \\
Type$^a$  & SB(s)bc  \\
Distance$^b$ (Mpc)  &  32  \\
m$_B^a$   & 12.6  \\
L$_{B}$ (10$^{10}$ L$_\odot$) &  1.5 \\
Inclination$^c$  & 61.5\degree \\
Major Axis PA$^c$   & 62\degree \\
Bar PA$^c$  & 74.6\degree \\
S$_{12\micron}$ (Jy) &  0.36 \\
S$_{25\micron}$ (Jy) &  1.29 \\
S$_{60\micron}$ (Jy) &  6.33 \\
S$_{100\micron}$ (Jy) & 11.56 \\
L$_{\mbox{FIR}}$ (10$^{10})$L$_\odot$) & 1.1  \\
L$_{\mbox{H$\alpha$}}$ (10$^{40}$ erg.s$^{-1}$)  & 23 \\
M$_{\mbox{HI}}$ (10$^9$\msun)  & 7. \\
S$_{326}^d$ (mJy)   & 309 \\
S$_{365}^a$ (mJy) &  244 \\
S$_{1400}^a$ (mJy) &  133 \\
S$_{4850}^a$ (mJy) &  34 \\ 
S$_{8460}^e$ (mJy) &  31 \\
\hline
\end{tabular}
\begin{tabular}{l}
$^a$ From NED (frequencies in MHz)\\
$^b$ Assuming H$_0=75$ \kms Mpc$^{-1}$.  \\
$^c$ From LEDA.\\
$^d$ From WENSS. \\

\end{tabular}
\caption{NGC~6764 global parameters.}
\label{tab_glob_param}
\end{table}


%__________________________________________________________________

\section{Observations and data reduction}
\subsection{CO data}
The \twelvecoonezero and \twelvecotwoone lines have been observed with the IRAM interferometer 
located on the Plateau de Bure (France) in 1998 (??).
 We calibrated the visibilites using the
 1823+568 and 1828+487 as amplitude and phase references. The PdBI was phase-centered
at the position $\alpha_{2000}=19:08:16.370$ and $\delta_{2000}=50:55:59.58$. The 
velocity resolution was set to 10 \kms for the two CO transitions and the CLEANed maps
have been restored with a synthetised beam of $2.0\arcsec\times 1.6\arcsec$ 
(PA=7\degree ) for the \twelvecoonezero data and $1.25\arcsec\times 1.25\arcsec$ (PA=0\degree)
for the \twelvecotwoone data. The rms noise for the \twelvecoonezero channel maps is 2 mJy.beam$^{-1}$
(58 mK) for the \twelvecoonezero data and 5.5 mJy.beam$^{-1}$ (82.5 mK) for the \twelvecotwoone
data. Hereafter the velocity reference is fixed to be 2400 \kms, 114348.900 MHz rest
frequency for the \twelvecoonezero observation. The \twelvecotwoone observation heve been made at the
same rest velocity.\\
The total integrated \twelvecoonezero flux was measured by Sanders \& Mirabel (1985)
to be 225 Jy.km/s in the 55\arcsec beam of the NRAO-12m telescope.
 Our PdBI \twelvecoonezero observation detect a flux of 112 Jy.km/s, i.e filtering out
about 50 \% of the total flux, assuming all the flux is concentrated in the 
45\arcsec primary beam of the PdBI. It indicates that a large fraction of the molecular 
gas must be present in a diffuse component.
The integrated intensity map for the \twelvecoonezero and
\twelvecotwoone are shown in Fig. \ref{fig_ico_pdbi}. The position of the CO emission
peak has been estimated using the higher spatial resolution of the \twelvecotwoone transition
and the procedure is the same as used for  the optical center (see hereafter). 
The \twelvecotwoone peak position is at $\alpha_{CO}(2000)=19:08:16.402$, 
$\delta_{CO}(2000)=50:56:0.29$.
Assuming that the spiral arms are trailing and with  the receding velocities being 
on the eastern side of the galaxy, the northern side is closer to us.


\begin{table}
\begin{tabular}{l|ll}
Parameter  &  CO(1-0) & CO(2-1) \\
\hline
\hline
Field of View (FWHM)   &  45\arcsec (6.9 kpc)   & 22\arcsec  (3.4 kpc)\\
Bandwidth	       &  270 MHz  		& 473 MHz   \\
Velocity resolution	& 10 \kms		& 10 \kms  \\
Visibility Calibrator  & \multicolumn{2}{c}{1823+568 and  3C380}		\\
Synthetised beam  &	$2\arcsec \times 1.6\arcsec$,7\degree &  $1.25\arcsec \times 1.25\arcsec$ \\
Equivalent $T_b$ for 1 \jyperbeam  & 29 K	& 15 K   \\
rms noise (per channel)     & 2 \mjyperbeam  & 6 \mjyperbeam \\
\hline
\end{tabular}
\caption{PdBI observations}
\end{table}



\begin{figure*}
\resizebox{!}{!}{\includegraphics{./figure/n6764_ico_10_21.eps}}
\caption{Integrated intensity for the \twelvecoonezero transition (a) and for the \twelvecotwoone
transition (b). The contours for (a) are 1,3,..,27 \jyperbeam .\kms and for
(b)  3,6,..,54 \jyperbeam .\kms}
\label{fig_ico_pdbi}
\end{figure*}


\subsection{Optical center}
From the new POSS-II digitalization (REFERENCE), the optical center has been 
estimated on the central part of the red image. 
To determine the center, we have filtered out, using the wavelet transform (e.g
Leon et al. 2000), the large scale emission of the central part. On the peak emission
a Gaussian fit has been performed to measure the position with a sub-pixel precision.
The optical center is found to be $\alpha_{opt}(2000)=19:08:16.494$, 
$\delta_{opt}(2000)=50:55:59.46$. The astrometry precision given by the plate is
better than 0.5\arcsec (McLean 2001).



\subsection{Radio continuum}
%The radio continuum observations have been kindly provided by L. Seppo, previously to
%publication. 
The data were obtained with the Very Large Array (VLA) at both 3.5 and 20 
cm in, respectively, the configurations C and A (see Fig. \ref{fig_vla20}. 
The different configurations allow
to have similar spatial resolutions at the two frequencies: $1.39\arcsec\times 
1.26\arcsec$ (PA=-45\degree ) at 20cm and $2.50\arcsec\times 2.0\arcsec$ 
(PA=15.5\degree). The peak position of the 20cm emission has been computed in
a similar way that the optical center. It is located at $\alpha_{20cm}(2000)=
19:08:16.316$, $\delta_{20cm}(2000)=50:55:59.37$. The accuracy is better than 
0.1\arcsec.

\begin{figure}
\resizebox{8.5cm}{8.5cm}{\includegraphics{./figure/n6764_vla20.eps}}
\caption{Radiocontinuum at 20 cm (VLA) with a synthesized beam of 
$2.50\arcsec\times 2.0\arcsec$. The contour levels are 0.1,0.35,..,3.85,4,5,..,14 
mJy.beam$^{-1}$ for a rms noise of 70 $\mu$Jy.beam$^{-1}$.}
\label{fig_vla20}
\end{figure}

The 20cm flux from the VLA observations is 90 mJy, which is slightly resolved
out compared to the single dish measurement of 133 mJy. The 3.5cm flux is
 31 mJy, which indicated a flattening spectra between 6cm and 3.5cm (see Fig. 
\ref{fig_radioSED_flux}). The thermal contribution  at high frequency is more important
(REFERENCE). So what ??


\begin{figure}
\resizebox{8.5cm}{8.cm}{\includegraphics{./figure/n6764_radioSED.eps}}
\caption{Radiocontinuum flux at different wavelength from single-dish or low
spatial resolution measurements, except for the 8.4 GHz data (this work).}
\label{fig_radioSED}
\end{figure}



\section{The molecular gas}
\subsection{Distribution}
Adopting a conversion factor of X$_{\mbox{CO}} = 3\times10^{20}$cm$^{-2}$
(K.km.s$^{-1}$)$^{-1}$ (e.g. Scoville et al. 1987), the mass of molecular hydrogen
is:

\begin{equation}
M_{H_2} = 1.2\times 10^4 S_{CO} D^2
\end{equation}

where $S_{CO}$ is the CO flux density in Jy.\kms and D the distance in Mpc. The total
flux of our integrated \twelvecoonezero map gives a total molecular hydrogen mass of 2.0 
10$^9$ \msun. The surface mass density is computed by assuming an inclination of the 
nuclear disk similar to the outer disk of 61.5\degree (Eckart et al. 1991). 
%This 
%hypothesis is confirmed  by a gaussian fit of the \twelvecoonezero distribution in the
%nucleus which gives a ratio 2:1 for the ellipsoide, i.e an inclination of 60\degree.
On Fig. \ref{fig_h2dens} the molecular hydrogen surface mass density is shown, 
without including the minor components (mainly He) which would lead to 1.36 times 
the molecular hydrogen surface mass density. It reaches a mass density of up to 1850
\msun .pc$^{-2}$ at a projected distance of 1.25\arcsec ($\sim$ 190 pc) from the 
radiocontinuum peak which will be assumed to be the nucleus of the galaxy (see hereafter).
More than 98 \% of the total molecular gas mass measured by the PdBI is concentred
within the inner 800 pc, and as shown on the Fig. \ref{fig_h2cumu}, as much as  20 \% of
the total mass ($\sim 4 \times 10^8$ \msun) is concentrated in one beamsize 
($\sim$ 310 pc) around the \twelvecoonezero peak. Assuming a z-height of 100 pc, typical height
scale for molecular gas disk in galaxies (REFERENCE), it leads to a lower-limit
of 53 \msun .pc$^3$ for the molecular mass gas density in this region.


%\begin{figure}
%\resizebox{9cm}{8cm}{\includegraphics{./figure/n6764_h2dens.eps}}
%\caption{Molecular hydrogen  surface mass density in the center of NGC~6764 
%(i=61.5\degree)
%in \msun pc${-2}$. The contours levels are starting at 2-$\sigma$ level (100 \msun 
%pc$^{-2}$) up to the maximum of 2600 \msun pc$^{-2}$.}
%\label{fig_h2dens}
%\end{figure}


\begin{figure}
\resizebox{8.5cm}{8.5cm}{\includegraphics{./figure/n6764_vla20_ico_bar.eps}}
\caption{VLA 20cm emission overlaid by CO(1-0) emission with contour levels from
1 to 27 \jyperbeam\kms by step of 1 \jyperbeam\kms. The position of the major
axis and the bar are shown respectively by the solid and dashed lines.}
\label{fig_vla_ico_bar}
\end{figure}



\begin{figure}
\resizebox{8.5cm}{8.5cm}{\includegraphics{./figure/n6764_halfaLog_co10_color.eps}}
\caption{H$\alpha$ emission in NGC6764 (Log scale) overlaid by the integrated intensity of the
CO(1-0) emission (contour levels 1,3,..,30 \jyperbeam).}
\label{fig_halpha_co10}
\end{figure}


\begin{figure}
\resizebox{9cm}{8cm}{\includegraphics{./figure/n6764_gasmass_cumu.eps}}
\caption{Cumulative distribution of the molecular gas mass on the position
centered on the \twelvecoonezero peak.}
\label{fig_h2cumu}
\end{figure}


Apart from the central concentration the CO distribution (see Fig. \ref{fig_ico_pdbi}) shows an
extension towards the western side  and a spur roughly 
perpendicular to the major axis. Comparing the CO distribution with the bar orientation 
(see Fig. \ref{fig_vla_ico_bar}) reveals that the western feature extending up to 10\arcsec
from the center is aligned with the primary bar. It is commonly accpeted that the gas is 
inflowing
along the leading edge of the bars towards the center since the seminal paper of Athanassoula 
(1992). When the molecular gas distribution  is compared to the H$\alpha$ emission in this 
region, it appears that it is associated with two star forming regions 
($5 10^{39} and 3 10^{39}$ erg/s H$\alpha$ flux  respectively) which are  among the brightest
of the galaxy, excluding the strong H$\alpha$ emission at the tip of the bar (Rozas et al. 1996).
Since the galaxy is very inclined, the location of these regions and the molecular gas associated
 with is more ambiguous. Comparing with hydrodynamical simulations of gas in barred (Piner et al.
1995) the location of these star forming region is consistent with an inner ring (or equivalently
with an inner Lindblad resonance): in this picture the western feature of the molecular gas
is associated with the inflowing gas connecting to the inner ring where star forming regions are
present.\\
The CO-spur visible on the north can  be associated with the radiocontinuum superbubble 
observed at 20cm (see Fig \ref{fig_vla20} \& \ref{fig_vla_ico_bar}). The CO emission is 
correlated
with the norh-eastern  edge of the 20cm radiocontinuum bubble. The CO emission is detected  up 
to 8\arcsec above the disk plane; correcting for the disk inclination (i=61$^{\degree}$) 
it is found
that the molecular gas reaches 1.5 kpc above the plane. Since there is no clear  kinematic 
signature of the outflowing molecular gas, the mass-load by the outflow is estimated from the
CO spatial distribution only: cutting at one beamsize above the CO-peak and integrating to the 
northern feature gives a molecular gas mass in the out-of-plane outflow of 1.7 10$^7$\msun.


\subsection{$^{12}$CO  as molecular gas mass tracer ?}


To estimate the molecular gas mass, we have used the ``standard'' CO-to-H$_2$ conversion factor,
$X^{std}_{CO}= 3 10^{20} cm^{-2}K^{-1} km^{-1}s$ estimated
in the Milky Way galactical disk from the comparison between the $\gamma$ ray emission and the
\twelvecoonezero integrated line intensity (Strong et al. 1988). This standard value takes into 
account the 
Helium content. In the galactical disk this empirical value is found to match quite well the
theoritical one (Radford et al. 1991). Nevertheless there are growing evidence that this
conversion factor is overestimated by a facotr 3-10 in the central part of galaxies, especially
in starbursts. The physical reasons for this discrepancy have been discussed by various authors
already: metallicity gradient (Dahmen et al.1998), broadening of the CO line by bulge  
potential (Stacy et al. 1989, Mauersberger et al. 1996), physical condition of the gas (optical 
depth, temperature, density).
In NGC 253, a nearby starburst, Mauersberger et al. (1996) found a conversion factor $\sim 7$ times
lower than $X^{std}_{CO}$ in the central part of the galaxy, where they argue that the 
central potential well is broadening the CO lines and its emission could be partially optically
thin. This conversion factor has been found to be 3 to 10 times lower in the galactic bulge than in
the disk (Sodroski et al. 1995). Based on line transfer models, measurements of the cold 
dust emission and comparison with the optical extinction A$_v$, this factor is about 3-4 lower in 
M51, NGC891 and NGC3593 (Adler et al. 1992, Garc\'{\i}a-Burillo et al. 1992, Garc\'{\i}a-Burillo 
et al. 2000).
In a sample of ultraluminous infrared galaxies (ULIRGs), Solomon et al. (1997) found that in these
CO-bright object the conversion factor $X^{std}_{CO}$ overestimates by a factor 3 the molecular
gas mass. In the  ULIRG Arp 220, Downes  \& Solomon (1998) found that the ratio is 6 times
lower compared to the Milky Way derivation, and their whole sample gives a mean value of $\sim 5$ 
times lower.\\
The optical extinction A$_V$ can be estimated from the gas column density (Bohlin et al. 1978)
using:
\begin{equation}
\frac{(N_{HI}+2N_{H_2}}{A_V} = 1.87 10^{21} \mbox{cm$^{-2}$.mag$^{-1}$}
\end{equation}
Applying this to the molecular gas density computed with $X^{std}_{CO}$ and neglecting the 
HI contribution in the center (Wilcots et al. 2001), the peak A$_V$ is found to be $\sim$ 125 mag
and extinction of A$_V \sim 20$ mag extending up to 900 pc eastern  from the CO peak.  
This extinction is clearly overestimated through the high column density computed with the standard
conversion factor. Eckart et al. (1991) estimate an extinction A$_V$=3--4 towards the nucleus, which
is not compatible with the exctintion derived from the molecular gas column density, even when the
CO peak is offset from the very center, as we propose hereafter.  
The IRAS FIR flux at 100 \micron can be used to give a lower limit of the extinction A$_B$ (Laureijs
et al. 1987) by:
\begin{equation}
\frac{IRAS(100\micron)}{A_B}=6.3 MJy.sr^{-1}.mag^{-1}
\end{equation}

The optical extinction A$_B$  is then found to be about 5.5 mag. This is a lower limit 
assuming that the 100 \micron flux is spread over the whole beam of IRAS (4\arcmin).
Taking a typical ratio of $A_B/A_V$=1.3, obtained by modelling the dust and stellar content
in spiral galaxies (Xilouris et al. 1999), then we get $A_V > 4$ mag in NGC6764.
The dust mass computed from
the IRAS fluxes at 60 and 100 \micron gives a total mass of M$_d$=$1.4\times10^6\msun$ with a mean 
temperature of 36.5 K. It is known from FIR observations with ISO that the IRAS data are missing
a cold component, T$_d < 20$ K,  from the dust emission (Alton et al. 1998, Trewhella et al. 2000) 
emitting beyond 150 \micron. This component can be 2-10 times more massive than 
the warm dust mass M$_d$. A similar result has been found in elliptical galaxies, observing
the $\lambda$1250 \micron bolometric emission (wiklind \& Henkel 1995).
In the outer Galaxy, Sodroski et al (1994) find a gas-to-dust ratio about 200, using the 
COBE data reaching larger wavelength than IRAS.
In NGC~6764
the gas-to-mass ratio, using the standard CO-to-H$_2$ conversion factor leads to $M_{gas}/M_{dust} 
\sim 1400$. This factor is by far larger than the typical galactical value, indicating an 
overestimation of the molecular gas mass from the \twelvecoonezero luminosity.\\
In order to reach a visual extinction  A$_V$ in accordance with the previous modelling 
(Eckart et al.  1991, Schinnerer et al. 2000), and to get a more realistic gas-to-dust ratio, we
 choose to take a conversion factor X$\sim X^{std}_{CO}/3$.  This  scaling match the one 
found in previous study for the center of (starburst) galaxies and it has been chosen to fit
  the gas-to-dust mass modeled by Xilouris et al.(1999).
Hereafter the molecular gas mass will be given with this conversion factor, but the reader should 
keep in mind that this calibration is likely correct in a range of a factor 2. 

\subsection{Kinematics}

{\bf m=1 mode, PV diagram, molecular outflow PV-features; streaming motions; connection
with the primary bar.}
 
The Figure \ref{fig_co_mom1} shows the beam-smeared map of the intensity-weighted \twelvecoonezero 
velocity, derived from the first moment of the \twelvecoonezero data cube overlaying the 20cm
radiocontinuum emission.The observations
are consistent with a global rotation model, dynamically centered on the 20cm peak, combined with
streaming motions. The strongest deviation from the rotation model is along the molecular outflow
originating from the \twelvecoonezero peak. The \twelvecoonezero integrated intensity peak does
not correlate with the dynamical center, indicating a possible strong mode m=1 in the very center.

The Figure \ref{fig_co_mom2} shows the beam-smeared map of the intensity-weighted \twelvecoonezero 
velocity dispersion  in the center, derived from the second moment  of the \twelvecoonezero data 
cube overlaid by the \twelvecoonezero integrated intensity. In the center, the velocity dispersion
$\sigma_V$ reaches value up to 200 \kms. Such high velocity dispersion are not owing to intrinsic
velocity dispersion in GMCs (few tens of \kms) but is nearly completely formed of the beam-smeared
velocity gradient along the rotation curve. The maximum velocity dispersion is thus located on
the dynamical center, which corresponds to the 20 cm peaks at the coordinates $\alpha_{20cm}$(J2000)
= 19:08:16.316  and $\delta_{20cm}$(J2000)= 50:55:59.37. An area of high-velocity dispersion which is
not completely caused by the gradient velocity is visible at the north of the \twelvecoonezero
peak, where the velocity dispersion $\sigma_V$ is about 150 \kms. 

\begin{figure}
\resizebox{9cm}{8cm}{\includegraphics{./figure/n6764_vla20_co10mom1.eps}}
\caption{Isovelocity field contours overlaying the 20cm radiocontinuum emission.
Thick lines are negative velocities by 10 \kms step. The reference velocity is 
2400 \kms 
(Heliocentric).}
\label{fig_co_mom1}
\end{figure}

\begin{figure}
\resizebox{9cm}{8cm}{\includegraphics{./figure/n6764_co10disp_Ico10.eps}}
\caption{Velocity dispersion from the \twelvecoonezero observations overlaid by the
\twelvecoonezero integrated intensity with contours levels 2,4,..30 \jyperbeam\kms}
\label{fig_co_mom2}
\end{figure}

It is tempting to associate this
high velocity dispersion with the starburst region and stellar winds at the origin of the molecular
outflow. To estimate the amount of velocity dispersion created by the starburst region,  
we compute the odd-decomposition of $\sigma_v^2$ relative to the 20cm radiocontinuum peak, namely
\begin{equation}
\sigma_{V,odd}^{2}(\mbox{\bf x})=\frac{\sigma_V^2(\mbox{\bf x})-\sigma_V^2(\mbox{\bf -x})}{2}
\end{equation}

Then, the excess of $\sigma_V$ on one side relative to the other can be computed as $\sigma_V(+)=
\sqrt{2\times \sigma_{V,odd}^{2}}$ , which is shown for a cut along the major axis on the figure 
\ref{fig_co_disp_excess}. Two bumps marking a local excess of velocity dispersion are present: the
peak at 2\arcsec (84 \kms) is associated with the \twelvecoonezero integrated intensity connected to
the central starburst at about 200-300 pc from the nucleus; the peak at 4--6\arcsec (78 \kms), 
$sim$ 800 pc from the nucleus, is likely connected to streaming motion (inflow, outflow) of
the gas in the bar connecting to the central part  and, as well the smaller starforming region 
visible in H$\alpha$ (see figure XX). Since the rotation gradient is less pronounced at this 
radius, the actual value is more accurate regarding the beam-smearing.
Using the corrected correction factor, an upper limit for the  kinematic energy $E_{kin}^{SB}$ 
deposited by the starburst in the molecular gas can be estimated to $E_{kin}^{SB}=\frac{1}{2} M_{H_2}
<\sigma_V^2(+)>$, i.e. $9.9\times10^{54}$ ergs, with $<\sigma_V^2(+)>$=55 \kms and $M_{H_2}=3.3
\times 10^8$\msun ($\sim$ 50\% of the total molecular gas mass, see figure \ref{fig_h2cumu}).


\begin{figure}
\resizebox{9cm}{8cm}{\includegraphics{./figure/n6764_disp_oddpart.eps}}
\caption{The asymmetry of the \twelvecoonezero  velocity dispersion $\sigma_V(+)$ is 
computed along the major axis (see text for explanation). The two bumps are connected to the
starburst (84 \kms) and to the bar reconnection zone (78 \kms)}
\label{fig_co_disp_excess}
\end{figure}

Figure \ref{fig_co10_chanmap} shows the \twelvecoonezero channel map at 10 \kms velocity resolution.
\twelvecoonezero emission is detected in the individual channels at 2.5$\sigma$ level from the 
velocity 2210 to 2600 \kms (LSR). Two \twelvecoonezero peaks are developping in the very center
at the negative and positive velocities. From 0 to 110 \kms, extended weak feature are showing
up. One is visible from the \twelvecoonezero peak oriented NS from the velocities 10 to 130 \kms, 
the other is more extended ($\sim$ 10\arcsec) from the velocities 10 to 100 \kms. The first streaming
motion can be associated with the molecular outflow, visible on the \twelvecoonezero  integrated
intensity, flaring up to 6\arcsec ($\sim$ 1 kpc)  (see figure \ref{fig_vla_ico_bar}). The redshifted
velocities are compatible with the inclination of the galaxy and z-motion. The other feature is 
compatible with in-the-disk-plane motions (spiral arm or bar reconnection features). {\bf ?????}



\begin{figure*}
\resizebox{17.5cm}{15cm}{\includegraphics{./figure/n6764_co10_chanmap_2.eps}}
\caption{Channel maps for the \twelvecoonezero emission. The rms noise corresponds to 
2 mJy.beam$^{-1}$. The contours are 5,15,25,..,95 mJy.beam$^{-1}$. The 
synthesized beam is $2\arcsec\times 1.6\arcsec$. The rest velocity is 2400 \kms.
The cross is located at the phase center, i.e. $\alpha_{2000}=$19:08:16.370  
and $\delta_{2000}$=50:55:59.58.}
\label{fig_co10_chanmap}
\end{figure*}


%\begin{figure*}
%\resizebox{17.5cm}{15cm}{\includegraphics{./figure/n6764_co21_channelmapR.eps}}
%\caption{Channel maps for the \twelvecotwoone emission. The rms noise corresponds to 
%2 mJy.beam$^{-1}$. The contours are -20,0,..,320 mJy.beam$^{-1}$. The 
%synthesized beam is $1.25\arcsec\times 1.25\arcsec$. The rest velocity is 2400 \kms.}
%\label{fig_co21_chanmap}
%\end{figure*}

The PV diagram of the \twelvecoonezero and \twelvecotwoone are shown on the figure \ref{fig_co_pvdiag}
along and perpendicular to the major axis (PA=62\degree) and at a PA=90\degree to compare with the
\twelvecoonezero observations of the Nobeyama Array (ckart et al. 1996). 


\begin{figure*}
\begin{center}
\resizebox{14cm}{18cm}{\includegraphics{./figure/n6764_co_PV.eps}}
\caption{Position-velocity diagram of the \twelvecoonezero (left) and \twelvecotwoone (right)
emission in the center of NGC~6764. The offset 0 is taken at the CO peak.
The major axis PA is 62\degree, and the PA of 90\degree has been added to compare
with Fig. 1 of Eckart et al. (1996). The units are in \jyperbeam}
\label{fig_co_pvdiag}
\end{center}
\end{figure*}


\subsection{CO line ratio}
The line ratio for the integrated intensity between the \twelvecotwoone and \twelvecoonezero
transitions,  
$\Re = \frac{I_{\twelvecoonezero}}{I_{\twelvecotwoone}}$, is computed by convolving
the \twelvecotwoone integrated map to the spatial resolution of the \twelvecoonezero data and is 
shown on  the Fig.  \ref{fig_co_line_ratio}. The maximum $\Re$ is found to be 
0.84 combining all the channels. This ratio is in aggreement with the estimation
made by Eckart et al. (1991) from single dish observations. The
global line ratio measured there is interpreted as arising from moderately  warm,
dense and optically thick gas. Their conclusion remain globally  valid in the 
light of our new observations. Nevertheless new informations brought by the 
high spatial resolution have to be emphasized.


\begin{figure}
\resizebox{9cm}{8cm}{\includegraphics{./figure/n6764_coratio_total_grey.eps}}
\caption{\twelvecotwoone to \twelvecoonezero line ratio integrated on the whole range of
channels overlaid by the \twelvecoonezero contour levels (1,3,5,..,30 \jyperbeam\kms).}
\label{fig_co_line_ratio}
\end{figure}


\begin{figure}
\resizebox{8cm}{8cm}{\includegraphics{./figure/n6764_coratioMax_vel.eps}}
\caption{\twelvecotwoone to \twelvecoonezero maximum line ratio $\Re_{max}$ 
computed in the center of NGC~6764 for each \twelvecoonezero channel (10 \kms width).}
\label{fig_coratiomax_vel}
\end{figure}


The maximum line ratio $\Re_{max}$ for each \twelvecoonezero and \twelvecotwoone channel (10 \kms 
width)
is shown on the  Fig. \ref{fig_coratiomax_vel}. Clearly a bimodal feature is present with a peak
on the blue and red side. The blue side with a peak at -140 \kms is associated with the molecular
gas closer to the very center, as shown on the CO line ratio channel maps on the Fig. 
\ref{fig_co_line_ratio_chan}. The red side with a peak at 140 \kms reaches a value maximum of
about 1.2. This high  CO line ratio is associated with the bulk of the molecular gas as it
will be shown from the CO line ratio channel maps.\\



\begin{figure*}
\begin{center}
\resizebox{16cm}{20cm}{\includegraphics{./figure/n6764_coratio_chan.eps}}
\end{center}
\caption{\twelvecotwoone to \twelvecoonezero line ratio for different velocities, 
indicated on each panel, 
with a channel width of 10 \kms. The first contour is for 0.4 and by step of 
0.1.}
\label{fig_co_line_ratio_chan}
\end{figure*}


The \twelvecotwoone/\twelvecoonezero line ratio has been computed for each channel 
(10 \kms width). The lack  of the 
short spacing data for the two transitions can affect the true value by missing the extended 
molecular gas emission (Weiß et al. 2001). Nevertheless this line ratio is an indicator of the 
most extreme conditions of the moelcular gas (hot kinetic temperature, low density). The figure
\ref{fig_co_line_ratio_chan} shows this CO line ratio from the channel -180 \kms up to 200 \kms with

the central velocity being the systemic velocity of NGC~6764 (2400 \kms). From -180 to -40 \kms
the peak of the CO line ratio is correlated with the nucleus (the radiocontinuum peak is 
indicated 
by  a cross). The peak at -140 \kms seen in the figure \ref{n6764_coratioMax_vel.eps} is likely 
to be
the edge effect seen on the CO line ratio at this velocity. The ``true'' maximum is reached at 
-130
\kms with a value of 0.9. at the exact position of the radiocontinuum peak (within a pixel). 
The western feature of CO emission has a mean CO line ratio of 0.3, which is likely 
underestimated
because of the filtering-out of the extended emission. From -140 to the highest velocity 
channels,
the peak of the CO line ratio is displacing to the eastern side of the nucleus as the CO emission
in a whole, but the peak the CO line ratio is not correlated with the CO emission peak, both 
ingrated and channel by channel. There is an offset of 2 arcsec to the south-eastern fo the CO
line ratio peak. A pattern is present on the positive velocity channel with the isocontour of
CO line ratio aligned along  a PA of 160\degree. On the figure \ref{fig_coratio_halpha_120}, 
the CO
line ratio at the channel 120 \kms is overlaid by the H$\alpha$ emission. The H$\alpha$ emission
peaks with the starburst emission, which is slightly offset from the center (see below) and and 
extended plume is visible on the south-eastern part, extending down to connect with the external
superbubble. The extended H$alpha$ emission is created by superwinds from O new-born stars.
This high-velocity superwinds (1000--2000 \kms, Chevalier et al. 1985) will excite the 
low density molecular gas  on its path to give a higher CO line ratio. This anisotropy of the 
CO line ratio relative to the plane of the galaxy (about 2\arcsec$\sim$ 300 pc  below the plane)
together with the extended H$alpha$ emission may indicate that the star forming region is 
slightly offset towards the south and the lower z-column density of molecular gas to the south is
creating this anisotropie of the H$alpha$ emission, together with the CO line ratio. 
The radiocontinuum at 20 cm show as well this feature and present a total intensity stronger
in the southern part than in the norther part (XX vs. YY).
This particular orientation is clearly visible as well on the CO integrated line ratio on the
figure \ref{fig_co_line_ratio_chan}


\begin{figure}
\resizebox{8cm}{8cm}{\includegraphics{./figure/n6764_coratio_halpha_color.eps}}
\caption{\twelvecotwoone/\twelvecoonezero line ratio at the channel 120 \kms (10 \kms width) 
overlaid by the H$alpha$ isocontours.}
\label{fig_coratio_halpha_120}
\end{figure}




% Ask Kotaro Kohno for the HCN/CO ratio they computed (Kohno et al. 2001). They
% refer to NGC~6764 as a composite galaxy with AGN/starburst composite
% 


\begin{table*}
\begin{tabular}{lllllll}
\hline
  Region  &  $\frac{CO(2-1)}{CO(1-0)}_{\mbox{Tot$^a$}}$ & 
$\frac{CO(2-1)}{CO(1-0)}_{\mbox{Low$^a$}}$ &
$\frac{CO(2-1)}{CO(1-0)}_{\mbox{High$^a$}}$  &  $\sigma_{Max}$  
& $\sigma_{V_{1-0}}$  & $\sigma_{V_{2-1}}$ \\
& &  & &  (\msun .pc$^{-2}$)  & (\kms) & (\kms) \\
\hline
\hline
\twelvecoonezero peak          &  0.82 &  0.93  & 0.85   & 1860        & 190  & 184  \\
Nucleus$^b$            &  0.76 &  1.12  &  0.63  & 1000        & 185  & 164 \\
Nucleus 2\arcsec N     &  0.72 &  1.47   &  -  & 670            & 150  & 220 \\
Outflow$^c$ 4\arcsec N &   -   & $\sim$ 0.25   &  -  & $\sim$ 140  & 220  & -   \\
Arm/bar 10\arcsec W    &   $\sim$ 0.25  & -   &  -  & $\sim$ 210  & 48   & -   \\
\hline
%\multicolumn{4}{1}{$^a$ \twelvecoonezero kinematics.}
\multicolumn{7}{l}{$^a$ CO line ratio relative to the total, low and high velocity channel
map, see text.} \\
\multicolumn{7}{l}{$^b$ Radiocontinuum peak (20cm), see text.} \\
\multicolumn{7}{l}{$^c$ Offset position relative to \twelvecoonezero peak.}\\
\end{tabular}
\end{table*}



\section{Star Formation}
From the 60 and 100 \micron IRAS flux, the total FIR luminosity can be estimated. 
Following Condon (1992), star formation rate for the massive star can be computed with
$SFR(M>\msun) = 9.1 10^{-11}(L_{FIR}/L_\odot)$, which gives a SFR of 1 \msun/yr. 
According to Xu et al. (1998) the IRAS S$_{25\micron}$ flux is entirely due to the AGN, when
40 \% of the  S$_{12\micron}$ flux is due to the NSB. Indeed, according the classification
made by Xu et al. (2001) from the IRAS flux, NGC 6764 is classified as an AGN and marginally
as a starburst. 
The H$\alpha$ line is mainly emitted in the HII regions where the new-born massive stars are
ionizing the gas. The total H$\alpha$ flux can be used as a tracer of the star formation
as computed by Kennicutt (1988) where: \\
\begin{equation}
\mbox{L(H}\alpha) = \frac{1.26\times10^{41}}{\mbox{erg/s}} \frac{\mbox{SFR}}{\mbox{\msun/yr}}
\end{equation}
For the NSB, the H$\alpha$ flux is estimated to be 1.3 10$^{41}$ erg/s (see Table 
\ref{tab_fluxes hereafter}, which gives a SFR of 1 \msun/yr, consistent with the FIR.
luminosity.
These two derivations of the SFR are concerning the massive stars only, taking into account
all the stellar range and given the optical extinction, Eckart et al. (1991) derived a value 
of 4 \msun/yr for the NSB. They found a supernova rate of $1.2 10^{-2}$/yr. The  supernovae
of type II
produce a radio continuum non-thermal emission L$_{NT}$, which can be estimated  
(Condon \& Yin 1990, Hill et al. 2001):
\begin{equation}
L_{NT} =  1.3\times 10^{23} \nu^{-\alpha}N_{SN}    
\end{equation}
where $L_{NT}$ is the non-thermal radio luminosity  in $W.Hz^{-1}$ at the frequency $\nu$, 
in GHz, for the non-thermal spectral index $\alpha$, with a production rate of $N_{SN}$ per 
year. For the NSB, the drawback is the determination of the spectral index from the 
non-thermal contribution. Since it  should rapidly steepens (Colina et al.XXX??), the 
mean spectral index in the NSB is likely to give an upper limit for the non-thermal 
contribution. The mean value in the inner 6\arcsec is $-0.51\pm0.22$, which leads to the 
following range for the non-thermal radio fluxes (see Fig. \ref{fig_nonthermalflux}): 
14 to 16 mJy at 20 cm for a spectral index between -0.3 and -0.7, 21 to 39 mJy at 6 cm and 
24 to 56 mJy at 3.5 cm. The uncertainty increase with the frequency, but it appears that the 
main contribution of the non-thermal flux for the SNII is at 3.5 cm. The total flux measured 
at this frequency is 31 mJy, with a spectra flattening at high frequency 
(Fig. \ref{fig_radioSED}). 
When the flux at low frequency is mainly produced 
by the AGN, the 3.5 cm contribution from the NSB is more important and is creating this 
flattening spectra. By fitting the radiocontinuum between 326 MHz and 4.8 MHz where the 
non-thermal contribution is dominating, the spectral index is found to be $\alpha=-0.46$ and 
extrapolating the non-thermal contribution up to 8.5 GHz, we found a contribution of 6 mJy which 
is much below the measured value of 31 mJy, indicating a large contribution of the thermal gas
produced by the NSB.

\begin{figure}
\resizebox{8cm}{7cm}{\includegraphics{./figure/n6764_radio_nothermal_snr.eps}}
\caption{Non-thermal flux, in mJy,  at 3 different frequencies for a range of non-thermal 
spectral  index between -1 and 0. The flux is computed for a supernovae  production rate of
$1.2\times10^{-2}$ per year (Eckart et al. 1991) in the center of NGC~6764.}
\label{fig_radioflux_snr}
\end{figure}




\section{The outflow/shell}
In the Table \ref{tab_shell} we indicate the basic morphological parameters for the 
shell from the  radiocontinuum  20cm observation. 




\begin{table}
\begin{tabular}{l|ll}
 Parameter &  Observation & Physical Unit \\
\hline
h	&  7\arcsec  & 1080 pc \\
R$_{base}$ & 3.5\arcsec & 540 pc \\
\end{tabular}
\label{tab_shell}
\caption{Morphological parameters for the bubble derived from the 20cm radiocontiuum map.}
\end{table}



{\bf Energy buget (kinematic+gravitational] from the outflow components: 
molecular outflow, hot gas. Energy released by the SB.}


\subsection{Radiocontinuum}

To estimate the distribution of the radiocontinuum emission at 20cm in the outflow/bubble, the 
central component has been subtracted by fitting a gaussian on the central flux above a threshold
of 4 \mjyperbeam. The 20cm-subtracted map is then rotated of 17\degree to put the outflow
principal axis  along the x,y axis. The total flux remaining on the center-subtracted map is 
63 mJy, about 60 \% of the VLA observation. In table \ref{tab_vla20_sub} the 20cm measurements
for the center-subtracter emission in the frame of the outflow are presented: the offset of the 
outflow relative to the central radiocontinuum and relative to the plane are clear.

\begin{table}
\begin{tabular}{l|ll}
Outflow  &  East    & West  \\
 \hline
North & 14 mJy   &  7 mJy  \\
South & 24 mJy  &  13 mJy  \\
\end{tabular}
\caption{The 20cm radiocontinuum emission in the outflow frame (rotation of 17 \degree) after
subtracting the central component associated with the AGN (63 mJy). The center is fixed to 
the central radiocontinuum and the flux is given in mJy.}
\label{tab_vla20_sub}
\end{table}



\begin{figure}
\resizebox{8.5cm}{8cm}{\includegraphics{./figure/n6764_indexspec_color.eps}}
\caption{Spectra index $\alpha$ (S$_\nu \propto \nu^\alpha$) derived from the 20
and 3.5cm radiocontinuum.}
\label{fig_indexspec}
\end{figure}


\subsection{H$\alpha$ emission}
H$\alpha$ emission has been observed in NGC~6764 and provided to us kindly by 
D. Frayer.
Figure \ref{fig_halpha_dss} shows the H$\alpha$ emission in the center of 
the galaxy.

\begin{figure}
\resizebox{8.5cm}{8.5cm}{\includegraphics{./figure/n6764_dss2_halpha_insert.eps}}
\caption{B image (DSS2) of NGC~6764 overlaid by the H$\alpha$ uncalibrated flux in 
the center of NGC~6764, the contours are spaced by 0.15dex step starting at 
0.5 $\sigma$ level.}
\label{fig_halpha_dss}
\end{figure}

\begin{figure}
\resizebox{8.5cm}{8.5cm}{\includegraphics{./figure/n6764_halpha_Ico21.eps}}
\caption{H$\alpha$ emission overlaid by the \twelvecotwoone emission.}
\label{fig_halpha_co21}
\end{figure}

\begin{figure}
\resizebox{9cm}{8.5cm}{\includegraphics{./figure/n6764_Index_halpha.eps}}
\caption{Spectral index $\alpha$ between the 20cm and 3.5cm radiocontinuum emission 
overlaid with the H$\alpha$ emission.}
\label{fig_halpha_vla35}
\end{figure}



\section{Discussion}
\subsection{The nuclear starburst/AGN}

NGC~6764 is harboring a nuclear starburst (NSB): Schinnerer et al. (2000) present
a detailed study of the starburst properties. It is actually shown that the 
galaxy has experienced 2 recent starbursts 15-30 and 3-5 Myr ago. The presence of 
bubles seen in radiocontinuum and H$\alpha$ is the indication of a very recent
starburst indeed. Registering the molecular gas distribution relative
to the other tracers (see Table \ref{tab_astrom_peak}) show that the bulk of the
molecular gas content is offset by 1--1.5\arcsec from the radiocontinuum peak but
is correlated with the $H\alpha$ peak. Moreover the bubble morphology is centered
on the \twelvecoonezero peak rather than on the radiocontinuum peak (see Figures XXXX).
The recent starburst associated should originate from the high-density molecular
gas concentration when the radiocontinuum peak traces the galaxy's nucleus.
The NSB is then located 185 pc away from the nucleus. NGC~6764 is disclosing a very
similar NSB to NGC~2782: as discussed by Jogee et al. (1998, 1999) in the latter
case, the morphology of the bubble is associated to the very recent SB.\\
\\

Without any high-resolution spectroscopy in the very center it
is very problematic to derive the mass of the SBH present to trigger the AGN. Nevertheless we 
can put some upper limit on its mass from the CO kinematics. The CO PV diagram (see Fig. 
\ref{fig_co_pvdiag}) along the minor axis shows mainly the unresolved rotational motion around the 
nucleus. In a beamsize of 2\arcsec (320 pc at the NGC6764 distance) the maximum unresolved 
rotational velocity $V^{unr}_{rot} $ corrected for the inclination can estimated to:

\begin{equation}
V^{unr}_{rot} = \frac{G M_{tot}}{R}
\end{equation}
where$ M_{tot}$ is the total enclosed mass in the radius R. In the unresolved beam size of the
PdBI, the total mass enclosed is in a radius of 150 pc is 1.5 $10^9 \msun$.


{\bf Thermal contribution in the radiocontinuum, see 
Lee et al. (2002); Star formation efficiency, scaling the ISO flux in the Starburst
to the total Lfir. Discuss the Xco conversion factor for SB. Discuss the high CO ratio
line with the Halpha feature in the SE direction of the SB.}




%
%NSB properties:

\begin{table}
\begin{tabular}{lllll}
\hline
  & H$\alpha$ &  20cm & 3.5cm & M(H$_2$) \\
   & (10$^{39}$ erg.s$^{-1}$) &  (mJy) & (mJy) & (\msun)\\
\hline
\hline
  NSB  &     $\sim 130^a$ &  &   & $4.3\times10^8$ \\
  Nucleus &   & $90^b$  & $28^b$ & \\
  Outflow &   $\sim 45$   &  45 &  12  & $1.7\times10^7$\\
\hline
\multicolumn{5}{l}{$^a$ Includes as well the Nucleus contribution.}\\
\multicolumn{5}{l}{$^b$ Includes as well the NSB contribution.}
\end{tabular}
\caption{Bubble/Nucleus contributions for H$\alpha$ and 20cm radiocontinuum by
fitting a 2D gaussian on the central contribution. The molecular gas mass for the
NSB is the mass emcompassed by one CO(1-0) beam at the maximum (310$\times$250 pc).}
\label{tab_fluxes}
\end{table}

\begin{table}
\begin{tabular}{llll}
\hline
Tracer &  $\alpha_{2000}$ & $\delta_{2000}$  & D$_{20}$ \\
&    &    &   (\arcsec)  \\
\hline
\hline
20cm    & 19:08:16.316 & 50:55:59.37 &  \\
\twelvecoonezero$_{dyn}$ & 19:08:16.351 &  50:55:59.74  & 0.5   \\
\twelvecotwoone & 19:08:16.402 & 50:56:0.29  & 1.2  \\
H$\alpha$   & 19:08:16.440   &   50:55:59.87 &    1.3    \\
H$\alpha_{off}$ & 19:08:16.362 &  50:55:59.68  &  0.5     \\
Nucleus B-band &  19:08 :16.379 & 50:55:59.82 &	  0.6	  \\
Nucleus i-band &  19:08:16.374 &  50:55:59.69  &  0.6	  \\
\hline
\end{tabular}
\caption{Position (J2000.0)  of the peak emission in radiocontinuum
(20cm), CO emission and H$\alpha$ emission (off-filter and emission line). The
distance to the radiocontinuum peak is given in arcsec.}
\label{tab_astrom_peak}
\end{table}

% The typical error measure doing the astrometry on optical image from two DSS2 images 
% (Red and Blue) give about 1''. 


\subsection{GMC destruction in the very center}
What is the tidal heating of molecular clouds due to a central mass concentration ?
The GMCs (few 10$^6$ \msun) are the location of massive star formation, like in 
NGC~6764. The tidal heating will increase the internal velocity $\sigma_{GMC}$ and
speed-up the destruction of the such GMCs, inhibitating the star formation. Let
us consider a central mass concentration M$_{\otimes}$, we call  hereafter SMBH,
 and a GMC of mass M$_{GMC}$ located at a distance R from this SMBH:
The tidal energy U$_t$ felt by the GMCs is then:

\begin{equation}
U_t(r) \approx  - \frac{GM_{\otimes}}{4R^3}r^2
\end{equation}

where r is the radius inside the GMC. The angular dependency of the tidal energy
in $\cos 2\theta$ is not taken into account. Taking a mass density 
profile $\rho(r) \propto r^{-2}$ for the GMCs (e.g. Williams et al. 1995), 
the total tidal energy U$_t$ of the GMC is given by :

\begin{equation}
U_t = - K \int_0^{r_t} 4 \pi U_t(r) \rho (r) r^2 dr \\
K=\frac{M_{GMC}}{4\pi r_t}
\end{equation}

where $r_t$ is the cutoff radius of the GMC. So the total tidal energy can be 
approximated to :
\begin{equation}
U_t = - \frac{G M_\otimes M_{GMC}}{12 R^3}r_t^2
\label{equa_energy_tidal}
\end{equation}

The internal energy of the GMC in virial equilibrium can be written as:

\begin{equation}
U_{GMC} \approx -\frac{1}{2}M_{GMC} \sigma^2
\label{equa_energy_gmc}
\end{equation}

From the equation \ref{equa_energy_tidal} and \ref{equa_energy_gmc}, the distance R
to the nucleus/SMBH where the tidal energy is comparable to the internal energy can
be approximated to :

\begin{equation}
U_{GMC} \approx U_t \Rightarrow R = \left ( \frac{GM_\otimes r_t^2}{6 \sigma^2} 
\right )^\frac{1}{3}
\end{equation}

For a  massive GMC of 60 pc of diameter and 10$^6$ \msun, the velocity 
dispersion is about 11 \kms (Sanders et al. 1985). Taking a $10^9$ \msun for 
the central SMBH, when Eckart et al. (1990) give an upper limit of $\sim 3\times
10^9$ \msun for the central mass in NGC~6764 within 5\arcsec, leads to the 
following radius tidal-dependency estimation for R:

\begin{equation}
R = 175 \mbox{pc} \left (\frac{M_\otimes}{10^9\msun} \right )^{\frac{1}{3}} \left ( \frac{r_t}{
\mbox{30 pc}}\right)^\frac{2}{3} \left ( \frac{\sigma}{\mbox{11 \kms}} 
 \right )^{-\frac{2}{3}} 
\end{equation}

The computation gives a value of about 200 pc for a central mass of 3
\begin{figure}
\resizebox{9cm}{8.5cm}{\includegraphics{./figure/tidal_trunc_gmc_bis.eps}}
\caption{Distance R (pc) to the nucleus of  mass M (\msun) where the tidal energy
of a GMC, of cutoff radius r$_t$ (pc),  is comparable to its internal 
energy (see text). The velocity dispersion is estimated from Sanders et al.
(1985) to be $\sigma = 1.35 r_t^{0.62} \kms$.}
\label{fig_tidal_trunc}
\end{figure}

See Taniguchi \& Wada (1996) for discussion. They have the opposite view...but not Kenney et al.
(1992). 

{\bf Shear effects close to the center; Safronov-Toomre Q value in the center: instabilities 
tracer ?; m=1 mode efficient way to to initiate the strong starburst. Compare the molecular
gas density and SF with other SB/AGN (M82, N7419, N3079(Sofue et al. 2000: UHD,
ultra-high-density molecular disk in this gal., 10E5 Msun/pc2!!)) \\}

Timescale for a cloud to be torn off by the Coriolis force and differential 
rotation is $\tau_D \sim dV/dR-\omega \approx 10^6$ yrs (To be confirmed). To compare
with the Jeans instability timescale (see Sofue et al. 2000 as well).


\section{Conclusions}

{\bf Offset molecular gas concentration; high density H2; molecular gas outflow;
(central outflow??); Energy budget; Radiocontinuum morphology; thermal/non thermal
contribution; SFE; Bar connection; m=1 mode-SB interplay; etc}


\begin{acknowledgements}
This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG)
via grant SFB 494, by special funding from the Science Ministry of the Land
Nordrhein-Westfalen.
\end{acknowledgements}

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