------------------------------------------------------------------------
 hcn_co.tex ApJ 493, in press

% Paglione et al. ``Interpreting the HCN/CO Intensity Ratio in the
% Galactic Center'' 
%
% MACROS for general astrophysics
\def\alwaysmath#1{\ifmmode {#1} 
                  \else {$#1\mkern-5mu$} \fi}
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%--------some general math things-------
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%--------some common symbols ------------
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% ---- Some Chemical Symbols ------
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%end my macros

\documentstyle[12pt,aasms4]{article}

\begin{document}

\title{Interpreting the HCN/CO Intensity Ratio in the Galactic Center}

\author{Timothy A. D. Paglione$^1$, James M. Jackson and Alberto D.
Bolatto$^2$}
\affil{Department of Astronomy, Boston University, 725 Commonwealth
Ave., Boston, MA 02215}
\altaffiltext{1}{Current address: Instituto Nacional de
Astrof\'{\i}sica, \'Optica y Electr\'onica, Apartado Postal 216 y 51,
72000 Puebla, Pue., M\'exico.}
\altaffiltext{2}{also Departamento de Astronom\'{\i}a, Universidad de
la Rep\'ublica, Montevideo, Uruguay}
\centerline{Electronic-mail: paglione@inaoep.mx}

\and
\author{Mark H. Heyer}
\affil{Five College Radio Astronomy Observatory, Lederle Research
Tower, University of Massachusetts, Amherst, MA 01003}

\begin{abstract}

We studied the dense molecular gas in the Galactic center using maps
of HCN and CO \jon\ emission from the central 630 pc of the Milky Way,
and images of HCN \jth\ emission, which requires high densities for
excitation (\nhh $\gsim\,10^6$ \percc), from Sgr~A and Sgr~B.  The
ratio of integrated HCN and CO \jon\ intensities is a sensitive
measure of molecular gas pressure, and the ratio of integrated HCN
\jth\ and \jto{1}{0} intensities uncovers density enhancements in the
maps.  However, the HCN/CO ratio is difficult to model without knowing
the relative HCN and CO abundances.  Further, because of the different
filling factors of HCN and CO emission, models that use homogeneous
clouds may not be accurate for analyzing the HCN/CO ratio.

Most of the mass traced by HCN and CO in the Galactic center is at
high densities (\nhh $\sim10^4$ \percc), roughly an order of magnitude
higher than cloud densities in the Galactic disk.  Most of the dense
gas traced by HCN \jth\ emission is coincident with star forming
regions and cloud interaction zones, and not necessarily emission
peaks.  We smoothed the HCN and CO maps to the typical spatial
resolution of extragalactic observations, and repeated the analysis.
The single large-scale measurement was sensitive to the mass-weighted
average properties of the map.  Therefore, if we can extrapolate this
result to other spirals, studies such as this are sensitive to the
{\em average} gas properties in galactic nuclei, despite poor spatial
resolution.

\end{abstract}

\keywords{ISM: clouds --- ISM: molecules --- Galaxy: center ---
galaxies: ISM}

\section{Introduction}

Recent single dish work has demonstrated the value of HCN as a probe
of the dense gas in galactic nuclei (e.g., Solomon, Downes, \& Radford
1992; Helfer \& Blitz 1993; Jackson et~al. 1995; Paglione, Jackson, \&
Ishizuki 1997a). Because of its high electric dipole moment, HCN
requires large gas densities for collisional excitation.  Thus HCN, a
relatively abundant interstellar molecule, is an excellent probe of 
the dense, star forming gas in galaxies.  It is especially luminous
in starburst galaxies (Solomon et~al. 1992; Helfer \& Blitz 1993;
Jackson et~al. 1996, hereafter Paper~1).  Multiline HCN observations
have shown that, in general, starburst nuclei have both higher average
gas densities, and larger fractions of dense gas by mass, than more
quiescent galaxies like the Milky Way (Paglione et~al. 1997a).

However, the dense gas properties of normal spiral galactic nuclei are
only just now being studied.  The ratio of integrated HCN and CO \jon\
intensities (defined here as ${\cal R}$) has been the most widely used
density probe since these lines are easy to observe and may be
detected in many galaxies (Helfer \& Blitz 1993; Aalto et al. 1995;
Paper~1).  A high HCN/CO ratio indicates large molecular gas densities
because the critical density to excite HCN collisionally is higher
than that of CO.  Unfortunately, because the HCN/CO ratio may be a
strong function of the spatial scale of the observations, the results
are difficult to compare from galaxy to galaxy.  Also, modeling ${\cal
R}$ is problematic due to the unknown abundances of HCN and CO in any
source.

The goals in this paper are to address these two problems by: 1)
studying the dense gas in the center of a normal galaxy at high
sensitivity and spatial resolution, 2) determining the utility of the
HCN/CO ratio as a probe of the gas properties in galaxies, and 3)
quantifying the effects of the usual modeling assumptions on the
inferred cloud properties.  We completed a survey of the HCN and CO
\jon\ emission from the central 630 pc of the Milky Way at 50$''$ (2
pc) sampling (Paper~1).  The Milky Way is a good choice for a
reference quiescent galaxy because it is observable at the highest
spatial resolution and sensitivity of any galaxy.  Also, the star
formation rate and FIR luminosity are much lower in the Galactic
center than in starburst nuclei (Solomon et~al. 1992; Morris 
1993).  However, to compare Galactic and extragalactic data on the
same spatial scales, the Milky Way must be studied over a large
angular area.  To this end, we utilized innovative techniques and
instruments to map large regions of the sky toward the Galactic
center.

We use the ratio of integrated HCN and CO \jon\ intensities ${\cal
R}$, and for comparison, the ratio of integrated HCN \jth\ and
\jto{1}{0} intensities, to estimate the gas properties of the clouds
in the central 630 pc of the Galaxy.  Although the HCN \jth\ line is
more difficult to observe, the HCN \jth/\jon\ ratio is easier to
analyze because the problem of unknown relative abundances is
eliminated.

%A major goal of this study is to compare the emission from
%dense gas in the Milky Way to that from other galaxies.  To make this
%comparison, we convolved the QUARRY map to one ``beam'' roughly 630
%pc in diameter.  This analysis affords us the unique opportunity to
%discern the properties and distribution of the gas in a galactic
%nucleus, information which is unattainable from extragalactic
%observations due to poor resolution.  We apply these findings to
%those of extragalactic molecular line observations (Paglione et~al.
%1996) to find the specific gas components or cloud populations
%traced by single-dish observations, and to ascertain how averaging
%over large areas affects our conclusions.

\section{Observations}

The HCN and CO \jon\ maps of the Galactic center used here were
observed between 1993 and 1994 with the 15-beam focal plane array
QUARRY (Erickson et al. 1992) at the 14~m Five College Radio Astronomy
Observatory (FCRAO) in New Salem, MA.  We mapped within the ranges of
Galactic longitude and latitude $-2\pdeg13 \le l \le 2\pdeg13$ and
$-0\pdeg3 \le b \le 0\pdeg2$, which correspond to 630 pc $\times$ 75
pc at the distance of the Galactic center ($D = 8.5$ kpc).  The pixels
are spaced every 50$''$ (2 pc linear scale).  Further details of the
observations are given in Paper~1.

We observed the HCN \jth\ emission ($\nu_0$ = 265.886 GHz) from the
Galactic center remotely with the National Radio Astronomy Observatory
(NRAO\footnote{The National Radio Astronomy Observatory is a facility
of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc.}) 12~m at Kitt Peak, AZ in
1995 April.  We used the facility 260-300 GHz receiver, which had
typical system temperatures of 1300--1600 K.  We used the facility
filterbank spectrometers which consisted of 256 channels of 2~MHz
width.  The main beam efficiency and FWHM were 0.5 and 24$''$.
Calibration was done using the chopper wheel method, and with
observations of W49 and Sgr~B2 in position-switching mode.

For fast mapping, we used the ``on-the-fly'' (OTF) technique (Mangum
1995) to make R.A. $\times$ Decl. = $10' \times 10\pmin5$ maps
centered on the Sgr~A and Sgr~B molecular clouds (R.A. = $17^h 42^m
28^s$, Decl. = $-29\deg 01' 30''$, and R.A. $17^h 44^m 11^s$, Decl. =
$-28\deg 22' 30''$, respectively).  The maps were created by scanning
10$'$ in right ascension at a rate of 10$''$\pers.  The observations
are sampled every 0.1~s as the dish continues to scan.  The position
of the dish is monitored every .01 s to ensure accurate placement of
observed data points in the final map.  The 90 rows in each map are
separated by 7$''$ to achieve an oversampling in declination of
roughly a factor of 3.

The OTFUV and SDGRD routines of the NRAO AIPS reduction package were
used to grid the data into cubes of 43$\times$45 map positions and
256 velocity channels.  The map pixels are separated by 14$''$.  A
parabolic baseline was removed from each gridded spectrum.

\section{Physical Properties of the Dense Gas in the Galactic Center}

\subsection{The HCN/CO Ratio}\label{hcn.co}

To determine the physical conditions of the emitting gas in the
Galactic center, we performed single-component model calculations of
non-LTE HCN and CO excitation assuming that the emission originates
in unresolved, homogeneous, spherical clouds.  A photon escape
probability function was included to account for the radiative
excitation of optically thick lines (see Stutzki \& Winnewisser 1985,
and references therein).

The emission from the first 8 levels of HCN and the first 11 levels
of CO were modeled.  We used the CO collision rates of Flower \&
Launay (1985), and the HCN collision rates of Green \& Thaddeus
(1974).  The HCN/CO intensity ratio was modeled by assuming three
different values for the relative abundances of HCN and CO: [HCN]/[CO]
= $10^{-4}$, \hbox{2.5\ee{-4},} and $10^{-3}$, where [X] is defined as
the abundance of molecule X with respect to molecular hydrogen.  These
ratios cover the range of predicted and observed values for dense
clouds (e.g., Blake et~al. 1987; Bergin, Langer, \& Goldsmith 1995).

\subsubsection{Modeling Procedure}

It is difficult with only two lines to restrict the range of parameter
space to be explored.  Our goal here is to understand how the usual
assumptions for doing so influence the inferred gas properties.
Therefore the temperature and column density are constrained in the
typical way: by assuming that they are directly proportional to the
CO intensity.  The density is determined from ${\cal R}$.  The only
free parameters are the beam filling factor and the abundances of HCN
and CO.  The homogeneous cloud model also implies that the beam
filling factors of HCN and CO are equal (clearly a poor assumption,
but adding more free parameters is unwarranted with only two lines).
In later sections the impact of these assumptions are analyzed.

The CO column density per velocity interval, $N_{CO}/\Delta v$, was
estimated from the CO intensity by assuming
%
\begin{equation}
N_{CO}/\Delta v = \frac{I_{CO} X_{CO} [CO]}{\phi\ \Delta v}\ \ ,
\label{eq.coldv}
\end{equation}

\noindent
where $\phi$ is the area beam filling factor, and $X_{CO}$ is the
conversion factor of CO integrated intensity to \htwo\ column
density.  Although the standard value for $X_{CO}$ is
$\sim\,$3\ee{20} \hbox{\cmtwo/}(\kkms), it was derived from
cold disk clouds (e.g., Scoville et~al. 1987).  We use instead the
smaller value of $X_{CO}=5\ee{19}$ \cmtwo/(\kkms), valid for the
metal-rich, turbulent clouds of the inner Galaxy (Sodroski
et~al. 1995, and references therein).  We use [CO] = 8\ee{-5}
(Frerking, Langer, \& Wilson 1982).  The FWHM velocity width $\Delta
v$ is calculated at each position from the second moment of the CO
spectrum,
%
\begin{equation}
\sigma_v = \left[\frac{\int T (v-\langle v\rangle)^2 dv}
{\int T dv}\right]^{1/2}\ \ ,
\end{equation}

\noindent
and
%
\begin{equation}
\Delta v = 2 \sqrt{\ln 4} \ \sigma_v\ \ .  \label{eq.fwhm}
\end{equation}

\noindent
Equation~\ref{eq.fwhm} corrects the velocity dispersion such that,
for a Gaussian line shape, $\Delta v$ equals the FWHM of the
spectrum.  Nearly all of the dominant emission peaks have roughly
Gaussian profiles at our velocity resolution of 17 \kms.

The kinetic temperature $T_k$ at each position in the HCN/CO ratio
map can be estimated from $T_{CO}$, the peak observed brightness
temperature of CO, since
%
\begin{equation}
T_{CO} = \phi (\tx - T_{bg}) (1-e^{-\tau}) \; ,
\label{eq.tco}
\end{equation}

\noindent
where \tx\ is the excitation temperature, $T_{bg}$ is the 2.7 K
background temperature, and $\tau$ is the CO optical depth.  (All
observed temperatures have been adjusted to the main beam scale.)
In general, $\tx\,\le\,T_k$, and
%
\begin{equation}
T_k \ge\ \frac{T_{CO}}{\phi (1-e^{-\tau})} + T_{bg} \; ,
\label{eq.tk.uneq}
\end{equation}

\noindent
or
%
\begin{equation}
T_k = T_{CO}/f + T_{bg} \; ,
\label{eq.tk}
\end{equation}

\noindent
where we define $f$ as
%
\begin{equation}
0 < f \le\ \phi (1-e^{-\tau}) \le\ 1 \; .
\label{eq.phi}
\end{equation}

\noindent
Note that $f \approx \phi$, since CO is typically thermalized and
optically thick ($\tx \approx T_k$ and $\tau \gg 1$).  However, the
kinetic temperature and beam filling factor are not directly
calculable in this analysis.  $T_k$ is estimated by choosing $f_1$ (an
initial value for $f$), and setting $\phi=f_1$.  The model then
predicts a CO intensity and optical depth.  By comparing the model and
observed CO intensities, the beam filling factor is estimated from
%
\begin{equation}
\phi = T_{CO}(observed)/T_{CO}(predicted) \; .
\end{equation}

\noindent
If Equation~\ref{eq.phi} is not satisfied, $f$ (and therefore $T_k$)
is adjusted to lower or raise the predicted $T_{CO}$ accordingly
(Equation~\ref{eq.tk}), and the process is repeated.  The procedure
stops when the variations in $T_k$ fall below 5\%\ and
Equation~\ref{eq.phi} holds.  Convergence is typically reached in less
than a few iterations.

%To summarize, at each position the kinetic temperature and the column
%density are estimated from the CO temperature, and the density is
%determined from ${\cal R}$.  The only free parameters are $f_1$ (the
%initial value for the factor relating $T_{CO}$ and $T_k$), the
%abundance ratio [HCN]/[CO], and the quantity $X_{CO} [CO]$.

\subsubsection{Effects of Assumptions and Model}

In this section we discuss how the model assumptions could affect the
inferred gas properties.  This discussion is important since often
extragalactic analyses are also poorly constrained due to a lack of
data.  (They also suffer from poor spatial resolution and sensitivity,
which we address later.)

The inequality in Equation~\ref{eq.phi} implies that $\phi$ may be
poorly estimated.  For high values of $f_1$, requiring that $\phi \ge
f_1$ eliminates any low beam filling factor solutions, thus
underestimating $T_k$ and $N_{CO}/\Delta v$.  Conversely, for low
values of $f_1$, Equation~\ref{eq.phi} can be satisfied with very
small beam filling factors.  Though HCN/CO does not vary much with
$T_k$, at high column densities (where trapping is important), the 
inferred densities may be poorly determined.  The relation between
density and $\phi$ is tested in the following section.

It is also difficult to separate beam filling and optical depth in
Equation~\ref{eq.phi}.  However, $\tau > 1$ almost always, so this
problem should be minor. 

Because the critical density of HCN is a factor of $\sim\,$300
higher than that of CO, the emission from both molecules will not come
from the same regions.  This was shown by modeling the emission from
CO and another high dipole moment molecule, CS (Gierens, Stutzki, \&
Winnewisser 1992).  Therefore, the observed HCN/CO ratio (and thus the
density) is a lower limit to the actual value in the dense,
HCN-emitting regions of interest, and a homogeneous cloud may be a
poor model.  However, as mentioned before, this problem is very
difficult to constrain with only two lines, so it is set aside for
future work.

\subsubsection{Model Results}\label{results}

The results of one model are shown in Figure~\ref{fig.histo} for 
7155 positions in the HCN and CO images of Paper~1.  Here
the fraction of the total mass (estimated from $M_\htwo \propto
I_{CO}$, the observed CO intensity) is displayed as a function of the
model variables, assuming [HCN]/[CO] = 2.5\ee{-4} and $f_1$ = 0.1.
The mass-weighted average values of the parameters in
Figure~\ref{fig.histo} are given in Table~\ref{tab.means}.

A wide range of temperatures and CO column densities are found for the
clouds within the central 630 pc of the Milky Way.  These solutions
are simply scaled by the beam filling factor which is constrained by
the choice of $f_1$.  %Rather high beam filling factors
%($\phi\,\gsim\,$0.2) satisfy the model requirements suggesting that
%the gas features are nearly resolved at 2~pc scales.  
The distribution of mass as a function of $T_k$ is truncated at low
temperatures because map positions without significant emission have
been discarded from the analysis.  Thus ``cold'' regions, which are
also confused within the noise level of our map, have been ignored.
We find $\langle T_k\rangle \approx 75$ K, with roughly 75\%\ of the
mass at temperatures below 100 K.  This result matches that found from
\amm\ measurements of the same clouds (H\"uttemeister et~al. 1993).
Therefore this model solution is used in subsequent analysis and
discussion.

The distribution of mass with density in Figure~\ref{fig.histo} is
dominated by a narrow peak near \nhh = $10^4$ \percc.  To estimate the
dependence of the mean density on the input parameters, $\langle
n\rangle$ is plotted as a function of $f_1$, $\langle\phi\rangle$, and
[HCN]/[CO] in Figure~\ref{fig.n.vs.phi}.  The mean density is directly
proportional to $\langle\phi\rangle$, the average beam filling factor.
This dependence comes about because $N_{CO}/\Delta v$ is inversely
proportional to $\phi$, and at low column densities (optical depths),
trapping is minor, so a higher density is needed to produce a given
HCN/CO ratio.  However, the mean density is more sensitive to the
assumed HCN abundance.  A higher abundance increases the column
density of HCN, thus decreasing, due to radiative trapping, the
density needed to generate a particular HCN/CO ratio.

\subsubsection{Model Biases}\label{biases}

To ensure that the above solutions were not the results of biases in
the method or model, the same procedure was done for a set of 1000 HCN
and CO intensities, generated from a set of random values of \nhh,
$N_{CO}/\Delta v$, $\phi$, and $T_k$.  The only constraint on the
inputs was that the resultant CO intensities had to be above the clip
level used in the Galactic center HCN/CO map.  Three simulations were
run to solve for these gas properties, given the fabricated HCN and CO
intensities.  Biases exist if the original distributions of \nhh,
$N_{CO}/\Delta v$, $\phi$, and $T_k$ are not recovered.  [HCN]/[CO]
was fixed at 2.5\ee{-4}.

Histograms of the input distributions are displayed in
Figure~\ref{fig.rand.in}.  Shown are the average values of the mass
fraction from three simulations, and the 1$\sigma$ dispersions.  The
distribution of mass with density is flat, while the other variables
are affected by the lower limit set on the CO intensity.  Not
surprisingly, most of the mass (intensity) is in optically thick (high
column density), hot gas with high beam filling factors.

Figure~\ref{fig.rand.out} shows the model solutions for \nhh,
$N_{CO}/\Delta v$, $\phi$, and $T_k$ based on the fabricated HCN and
CO intensities (for $f_1 = 0.1$ and [HCN]/[CO] = 2.5\ee{-4}).
Figure~\ref{fig.out.vs.in} compares the output solutions with the
inputs.  Because $\phi$ is not solved for exactly, the kinetic
temperature, column density and beam filling factor show bias.  The
model indicates more mass at low $T_k$, $N_{CO}/\Delta v$ and $\phi$.
The mass distribution with column density also narrows.

In contrast, the output and input densities agree well except at very
high and very low densities.  Above $10^6$ \percc, the HCN/CO ratio
drops with density as HCN becomes thermalized, and there is no unique
solution for density (the model chooses the lower value).  Also, if
the observed ${\cal R}$ is greater than the model at all densities,
the point with the highest HCN/CO ratio is taken.  This density is
near $10^6$ \percc, and explains the extra mass near this value.
Below $10^2$ \percc, both CO and HCN are subthermally excited and the
HCN/CO ratio is not as sensitive to changes in density.  Therefore,
the model reproduces the input densities well for $10^2 \percc < \nhh
< 10^6$ \percc, and the density distribution in
Figure~\ref{fig.histo}, which shows over 30\%\ of the total mass at a
density of $\sim 10^4$ \percc, is probably not an artifact of the
molecular probes, model, or analysis procedure.

HCN and CO intensities were also generated using the Galactic center
solutions as inputs.  Unlike with the randomly generated parameters,
there is extremely good agreement between the input and output in this
exercise.  Apparently more self-consistent or realistic cloud
properties are well reproduced by the model.  Despite the expected
biases, none of the distributions are greatly affected
(Figure~\ref{fig.redo}).  With each iteration, the peaks all increase
slightly, and the dispersions decrease.  The deviations in the means
are well within 1$\sigma$.

\subsubsection{The Dependence of HCN/CO on Gas Properties}
\label{press}

Figure~\ref{fig.press} shows the HCN/CO ratio versus the
estimated physical properties of the gas at each of the 7155 modeled
positions in our Galactic center map.  There is no clear correlation
between $\phi$ and the HCN/CO ratio.  However, ${\cal R}$ rises with
kinetic temperature, and it is tightly correlated with density and
pressure, $P=nT_k$.  Therefore, according to our model, the HCN/CO
ratio may be a sensitive measure of molecular gas pressure over
orders of magnitude in ${\cal R}$ and $P$.  Recall however, that the
derived gas density depends strongly on the assumed [HCN]/[CO]
abundance ratio and $\phi$.

The tightness of the correlation between thermal gas pressure and
${\cal R}$ is curious.  It was expected that ${\cal R} \propto \nhh$,
but the model indicates that ${\cal R}$ is fairly insensitive to, or
drops with $T_k$ (except for very high values of ${\cal R}$, $T_k$ and
\nhh).  However, Figure~\ref{fig.press} shows ${\cal R}$ rising with
$T_k$.  Also, a tight correlation results if the dispersion in column
densities is very small, as is the case.  These points question
whether $T_k$ and $N_{CO}/\Delta v$ are estimated properly.

The small dispersion in column densities (Figures~\ref{fig.histo},
\ref{fig.rand.out} and \ref{fig.out.vs.in}) may result from using a
constant CO-to-\htwo\ conversion factor when it is predicted to change
with cloud conditions (e.g., Sakamoto 1996).  In fact, the dispersion
in column density found from modeling \ihcn\ and \amm\ emission from
these clouds is 2--4 times that derived from a simple conversion of CO
intensity (Paglione et~al. 1997b; H\"uttemeister et~al. 1993).  Also
the CO abundance changes markedly from cloud core to surface (Blake
et~al. 1987).  Therefore, a tight correlation may result because a
constant CO-to-\htwo\ conversion does not properly uncover the
structure in these dense clouds.

That the HCN/CO ratio increases with $T_k$ when the model shows no
such trend is more difficult to understand.  This effect probably
arises from estimating $T_k$ from $T_{CO}$ (Equation~\ref{eq.phi}),
and the physics ignored in the homogeneous cloud model.  The worst
assumption of the model is that $\phi(HCN) = \phi(CO)$.  We define
$\phi$ as ${\cal N} A_c/A_B$, where ${\cal N}$ is the number of clumps
in the beam, and $A_c$ and $A_B$ are the clump and beam radii,
respectively.  It can be shown that $A_c(HCN)$ increases with $T_k$.
According to the model, $T_{HCN}$ rises roughly like $\nhh^{1/2}$.
The density of a typical clump follows a power law with radius of
$\nhh\propto r^{-3/2}$ (Gierens et~al. 1992 and references therein).
Therefore $T_{HCN}$ should decrease with radius as $r^{-3/4}$.
In comparison, $T_{CO}$ is fairly insensitive to \nhh, so it does not
change with $r$ until the clump edge is reached.  This radial
structure was found by Gierens et~al. (1992) by modeling the molecular
emission from a clump.  Therefore, as $T_{CO}$ and $T_{HCN}$ increase
with $T_k$, $A_c(HCN)$ increases, but $A_c(CO)$ does not.

The number of HCN-emitting clumps ${\cal N}$ will also increase with
$T_k$.  Within the beam are clumps of various densities, some of which
are dense enough to have detectable HCN emission.  CO emission is
presumably seen from every clump.  As $T_k$ increases, HCN emission
will now be detectable from less dense clumps, thus increasing ${\cal
N}(HCN)$ while ${\cal N}(CO)$ remains unchanged.

This argument can be tested by estimating $T_k$ through other means.
In accordance with the model, no correlation between $T_k$ and ${\cal
R}$ is found when $T_k$ is derived from \ihcn\ or \amm\ measurements
(Paglione et~al. 1997b; H\"uttemeister et~al. 1993).  Therefore,
though the HCN/CO ratio may be a good measure of gas pressure, the
real scatter is most likely higher than shown in
Figure~\ref{fig.press}.

\subsection{Comparison with Other Galaxies: Modeling the Large-Scale
Emission}

To compare the HCN and CO spectra from the Milky Way to those from
other galaxies, the QUARRY maps were convolved to one ``beam'' with a
FWHM of 630 pc, centered on $l=0\deg$, $b=0\deg$.  This linear scale
corresponds to 26$''$ at a distance of 5 Mpc, comparable to
observations of nearby galaxies with the IRAM 30~m (Rieu et~al. 1992;
Aalto et~al. 1995).  The maps can also be compared to interferometric
data (Paglione, Tosaki, \& Jackson 1995) though few galaxies have been
studied in HCN to date (Helfer \& Blitz 1997; Kohno et~al. 1996;
Brouillet \& Schilke 1993; Jackson et~al. 1993; Downes et~al. 1992).
The convolved HCN and CO spectra (Figure~\ref{fig.spec}) have
integrated intensities ($I_{HCN}$ = 9.8 K \kms\ and $I_{CO}$ = 113.3
\kkms) and peak main beam brightness temperatures ($T_{HCN}$ = 67 mK
and $T_{CO}$ = 534 mK) typical for nearby quiescent galaxies (Helfer
\& Blitz 1993; Aalto et~al. 1995).  In fact, after lowering the
signal-to-noise ratio of the Milky Way spectra, they are qualitatively
indistinguishable from extragalactic observations.

The convolved data can be used solve for the physical conditions of
the molecular gas in the Galactic center over large scales with the
techniques from \S\ref{hcn.co}.  The HCN/CO ratio may be
underestimated because more CO than HCN emission is likely to lie
outside of our rectangular map (Paper~1; Paglione 1996).  However, we
overlook this correction since it is less than a 10\%\ change in
${\cal R}$, and within the observational uncertainties.

Initially $\phi$ is constrained from a simple geometric scaling of the
map solution $\langle\phi\rangle$, using the ratio of the area of the
rectangular map to that of the circular Gaussian beam,
%
\begin{equation}
\phi\, \lsim\, \langle\phi\rangle\,\frac{\Delta l \Delta b}{\pi
(FWHM/2)^2 (\ln 2)^{-1}} \; .
\end{equation}

\noindent
With $\Delta l = 630$ pc, $\Delta b = 75$ pc, and FWHM = 630 pc, we
find $\phi \sim 0.02$.  Therefore, $T_k > 30$ K, and $\log
N(CO)/\Delta v \sim$ 17.1 \coldv.  With these constraints, and given
${\cal R} = 0.08\pm0.02$ and [HCN]/[CO] = 2.5\ee{-8}, then $\log
(n/\percc) \sim 3.8\,\pm\,0.4$.  These solutions, within the
uncertainties, are identical to the mass-weighted mean values found by
modeling each position in the original QUARRY map
(Table~\ref{tab.means}, Figure~\ref{fig.histo}).  Therefore, this
technique is sensitive to the {\em average} properties of the
molecular gas in the central 630 pc of the Milky Way.  If these
results may be generalized, then {\em analyses of the large-scale
molecular emission in galactic nuclei probe the mean properties of
their cloud populations.}  Note that this comparison may only be valid
in the HCN-emitting regions of galaxies.

\subsection{Cloud Stability}\label{tides}

With their densities estimated, we can now assess the stability of the
dense clouds in the Galactic center against tidal disruption.  The
mean density and density dispersion as functions of projected distance
$r$ from Sgr~A$^*$ are compared to the critical density for tidal
stability in Figure~\ref{fig.tides}.  This critical density was based
on a power law mass distribution in the central Galaxy proportional to
$r^{1.2}$, containing a central point mass of 3.6\ee{6} \msun\
(Sanders \& Lowinger 1972).  The cloud density required to withstand
the tidal forces of the Galactic center is proportional to $r^{-1.8}$
(G\"usten \& Downes 1980; Ho et~al. 1985).  Figure~\ref{fig.tides}
shows that within $\sim\,$100 pc (projected distance) of the Galactic
center, much of the mass traced by HCN and CO \jon\ emission at our
resolution is not dense enough to be self-gravitating.  Thus, star
formation in this region is suppressed (cf. Morris 1993) and must
be a result of other processes such as cloud collisions and shock
compression.  In support of this scenario, the star forming regions in
the Sgr~A clouds do in fact seem affected by local shocks (Ho
et~al. 1985; Serabyn, Lacy, \& Achtermann 1992).  They may also be
bound by external pressure (Spergel \& Blitz 1992).  In contrast, much
of the gas at $\sim\,$80 and 250 pc, which corresponds to major cloud
features and massive star forming regions such as Sgr~B, Sgr~C, and
the $l=1\pdeg5$ complex, is stable against tidal disruption.

This analysis is hampered by the fact that the distances of these
clouds from Sgr~A$^*$ are not well known.  However, much evidence
indicates that at least the Sgr~A clouds reside near the Galactic
center (e.g., G\"usten \& Downes 1980).  Therefore the cloud features
in Sgr~A traced by CO and HCN are unstable to tidal disruption and
must be bound by external pressure or other influences.

\subsection{HCN \jth\ Emission from the Milky Way}\label{rhcn}

In the previous analysis, the ratio of the abundances of HCN and CO
at each position was a free parameter, and assumed to be constant with
position.  However, this ratio is known to vary by factors of several
within star forming clouds (e.g., Blake et~al. 1987), and is not well
known across the Galactic center.  Our analysis has also shown that
the abundances have a profound effect on the inferred gas density
(Figure~\ref{fig.n.vs.phi}).  Therefore this variable must be either
constrained or eliminated.  The HCN/CO ratio is also difficult to
model because of the different beam filling factors of CO and HCN, and
that there may not be a unique density solution
(Figures~\ref{fig.out.vs.in} and \ref{fig.rats.vs.n}).

Therefore, though the HCN/CO ratio is easier to observe in many
sources, and \S\ref{hcn.co} illustrates its usefulness, ${\cal R}$ may
not be the best tracer of high density gas in galaxies.  The HCN
\jth/\jon\ intensity ratio is preferable in order to eliminate the
problem of unknown relative abundances.  Figure~\ref{fig.rats.vs.n}
shows ${\cal R}$ and HCN \hbox{\jth/}\jon\ as functions of \nhh\ for
two values of column density.  Owing to the high critical density of
the HCN \jth\ line, HCN \jth/\jon\ rises with density over a very
large range, unlike ${\cal R}$ which peaks near the critical density
of the HCN \jon\ line of $\sim 10^6$ \hbox{\percc.}  Another advantage
is that the HCN \jon\ and \jth\ emission arise in nearly the same
regions because their critical densities are more similar.  In
contrast, the critical density of the HCN \jon\ line is over 300 times
higher than that of CO \hbox{\jon.}  Therefore, though their beam
filling factors are most likely not equal, HCN \hbox{\jth/}\jon\ is
more appropriate than ${\cal R}$ in the context of the homogeneous
cloud model.  HCN \jth\ emission has also been proven to be a useful
probe of dense gas in galaxies (Paglione et~al. 1997a).  Thus using
HCN \hbox{\jth/}\jon\ to estimate the densities of molecular clouds in
the Galactic center employs the same techniques used to study external
galaxies.

\subsubsection{Results}

Due to the high opacity of the atmosphere at 265 GHz, the mapping was
limited to the two regions with the highest HCN luminosities: the
giant molecular cloud complexes Sgr~A and Sgr~B.  Integrated intensity
maps of the HCN \jth\ emission from these clouds are shown in
Figures~\ref{fig.sgramap} and \ref{fig.sgrbmap}.  Many of the features
in these maps are seen in the \jon\ data as well (Paper~1).  Three
peaks are seen in HCN \jth\ integrated intensity toward Sgr~A which
correspond to (in order of increasing longitude) the ``20 \kms''
cloud, the circumnuclear ring (CNR), and the ``50 \kms'' cloud
(G\"usten 1989; Jackson et~al. 1993).  Bright emission is seen at the
position of the Sgr~B2 Main continuum source (Benson \& Johnston
1984), and the peak near the edge of the map ($l=0\pdeg77$,
$b=-0\pdeg06$) is nearly coincident with FIR source 38 from the survey
of Odenwald \& Fazio (1984).  The emission below Sgr~B2 near
$b=-0\pdeg1$ has no obvious counterpart in the infrared or radio.
Both of these peaks are seen in \ihcn\ \jon\ emission (Paglione
et~al. 1997b).

\subsubsection{Modeling}

The HCN \jth\ data were convolved to the 58$''$ beam size and 17
\hbox{\kms} velocity resolution of the \jon\ observations, and
regridded to match the QUARRY map.  The HCN \jth/\jon\ ratio is
calculated from the ratio of the main beam integrated intensities at
each position in the HCN \jon\ map.  This ratio was modeled just as in
\S\ref{hcn.co}.  The kinetic temperature and column density are
initially constrained by the CO intensity at each position, but now
$\Delta v$ is determined from the second moment of the HCN \jth\
spectra, $\phi$ is estimated from the HCN \jon\ brightness temperature
and optical depth, and \nhh\ is estimated from HCN \jth/\jon.

\subsubsection{Model Results}

The model results for Sgr~A and Sgr~B are listed in
Table~\ref{tab.means.ab}, and shown in Figures~\ref{fig.histoa} and
\ref{fig.histob} (for $f_1 = 0.1$).  The distributions of $T_k$ and
HCN column density per velocity interval for the two complexes are
quite different.  The gas in Sgr~B appears warmer and has a narrower
range of column densities.  The larger dispersion in $\Delta v (HCN)$
for Sgr~A may account for its broader distribution of estimated column
densities.  In general, the distributions for Sgr~A are broader than
those of Sgr~B.  This difference, plus the larger mass fraction of low
\nhh\ and $\phi$ solutions in Sgr~A, support the interpretation that
the clouds there may be perturbed (\S\ref{tides}).

It is interesting to note that the highest densities derived for the
Sgr~A clouds are not located at the 50 \kms\ cloud, the peak of the
HCN \hbox{\jth} emission, but at the circumnuclear ring around
Sgr~A$^*$ and the 20 \kms\ cloud (Figure~\ref{fig.sgramap}).  In fact,
the \hii\ regions in Sgr~A reside in relatively low density areas
perhaps because they have cleared out some of the gas.  However, the
CNR and the shock-compressed region of the 20 \kms\ cloud (Ho et
al. 1985) are density peaks.  Therefore this technique successfully
identifies density enhancements not obvious as brightness peaks.  That
we are able to discern these features despite our limited resolution
shows the reliability of HCN \hbox{\jth/}\jon\ as a measure of
molecular gas densities in GMCs.

\subsubsection{Model Biases}

To determine the biases in the model and procedure, 100 HCN \jth, HCN
\hbox{\jon,} and CO \jon\ intensities were generated from random
inputs as in \S\ref{biases}.  The output parameters are plotted
against the inputs in Figure~\ref{fig.out.vs.in32} for $f_1 = 0.1$
and [HCN]/[CO] = 2.5\ee{-4}.  The biases from this analysis are
similar to those from the HCN/CO ratio, though nearly all input
densities are recovered using HCN \jth/\jon.

\subsection{Comparing HCN \jth/\jon\ with ${\cal R}$}
\label{hcnco.hcn32}

Slightly higher average densities are found from the HCN \jth/\jon\
ratio than from the HCN/CO analysis (Figure~\ref{fig.hcnco.hcn32}).
The logarithm of the CO and HCN column densities differ by $3.3\pm
0.3$, nearly the chosen abundance ratio of log [HCN]/[CO] = 3.6.  (The
extra factor of 0.3 results because $\Delta v_{CO} \sim 2\times\Delta
v_{HCN}$).  Thus we find self-consistency in the model despite the
known biases.

In Figure~\ref{fig.rats} the HCN \jth/\jon\ ratio is shown versus
${\cal R}$.  These ratios were modeled to find a physical basis for
the differences in the derived densities.  The homogeneous cloud
model cannot simultaneously account for the two ratios.  For a given
value of HCN \jth/\jon, the model predicts a higher HCN/CO ratio.
This results again because $\phi(CO) > \phi(HCN)$.  Therefore the
observed ${\cal R}$ is a lower limit to the actual ratio, and the real
cloud is not homogeneous.  This error is most pronounced at the cloud
edges where the HCN emission fades yet the densities are sufficient to
excite the CO line.

\section{Summary}

We made wide-field maps of the emission from dense molecular gas in
the Galactic center.  The HCN/CO ratio ${\cal R}$ and the HCN
\jth/\jon\ ratio were used to estimate the physical properties of the
mapped regions.

When convolved to the spatial resolution of extragalactic
measurements, the HCN and CO spectra from the Milky Way closely
resemble those observed from other galaxies.  The density derived
from the convolved spectra of the Milky Way was identical to the mean
density found by modeling individual map positions.  This result
implies that similar measurements of other galaxies are most likely
sensitive to the {\em average} properties of their molecular cloud
populations, despite poor spatial resolution.

Most of the mass in the Galactic center traced by HCN and CO \jon\
is at rather high densities (\nhh $\sim\,10^4$ \percc), roughly an
order of magnitude higher than typical values in the disk.  The gas
features are nearly resolved on parsec scales.  The density structure
found from modeling the HCN \jth/\jon\ ratio corresponds to known
sites of star formation, interaction zones, and the circumnuclear ring
around Sgr~A$^*$.

Overall, the HCN/CO ratio is a useful probe of the molecular gas
properties in the Galactic center and, most likely, other galaxies.
It is apparently quite sensitive to pressure, $P=nT_k$, over a wide
range of values.  However, the derived properties depend on the
initial assumptions.  With only HCN and CO \jon\ data, the beam
filling factor is poorly constrained and the resulting column
densities and temperatures are biased.  As a result, the derived 
densities depend on $\phi$ as well, but are affected most by the
chosen [HCN]/[CO] abundance ratio.  Also, because the beam filling
factors of HCN and CO are so different, models of homogeneous clouds
should be used with caution when analyzing the HCN/CO ratio.  On the
other hand, the HCN \jth/\jon\ ratio can be modeled without knowing
the abundance of HCN, and with less uncertainty in relative beam
filling.

\acknowledgements

This work was funded in part by NSF grant AST-9318849.

\clearpage

\begin{thebibliography}{}

\bibitem[]{} Aalto, S., Booth, R. S., Black, J. H., Johansson,
L. E. B. 1995, \aap, 300, 369

\bibitem[]{} Benson, J. M. \& Johnston, K. J. 1984, \apj, 277, 181 

\bibitem[]{} Bergin, E. A., Langer, W. D., \& Goldsmith, P. F. 1995,
\apj, 441, 222 

\bibitem[]{} Blake, G. A., Sutton, E. C., Masson, C. R., Phillips,
T. G. 1987, \apj, 315, 621

\bibitem[]{} Brouillet, N., \& Schilke, P. 1993, \aap, 277, 381

\bibitem[]{} Downes, D., Radford, S.J.E., Guilloteau, S., Gu\'elin, M.,
Greve, A., \& Morris, D. 1992, \aap, 262, 424

\bibitem[]{} Erickson, N. R., Goldsmith, P. F., Novak, G., Grosslein,
R. M., Viscuso, P. J., Erickson, R. B., and Predmore, C. R. 1992, IEEE
Transactions in Microwave Theory and Techniques, 40, 1

\bibitem[]{} Flower, D. R., \& Launay, J. M. 1985, \mnras, 214, 271

\bibitem[]{} Frerking, M. A., Langer, W. D., \& Wilson, R. W. 1982,
\apj, 262, 590 

\bibitem[]{} Green, S., \& Thaddeus, P. 1974, \apj, 191, 653

\bibitem[]{} G\"usten, R. 1989, in IAU Symp. 136, The Center of the
Galaxy, ed. M. Morris (Dordrecht:Kluwer), 89

\bibitem[]{} G\"usten, R. \& Downes, D. 1980, \aap, 87, 6

\bibitem[]{} Helfer, T. T. \& Blitz, L. 1993, \apj, 419, 86

\bibitem[]{} Helfer, T. T. \& Blitz, L. 1997, \apj, 478, 162

\bibitem[]{} Ho, P. T. P., Jackson, J. M., Barrett, A. H., Armstrong,
J. T. 1985, \apj, 288, 575

\bibitem[]{} H\"uttemeister, S., Wilson, T. L., Bania, T. M.,
Martin-Pintado, J. 1993, \aap, 280, 255

\bibitem[]{} Jackson, J. M., Geis, N., Genzel, R., Harris, A. I.,
Madden, S., Poglitsch, A., Stacey, G. J., \& Townes, C. H. 1993,
\apj, 402, 173

\bibitem[]{} Jackson, J. M., Heyer, M., Paglione, T. A. D., \&
Bolatto, A. D. 1996, \apj, 456, L91 (Paper~1)

\bibitem[]{} Jackson, J. M., Paglione, T. A. D., Carlstrom, J. E., \&
Rieu, N.-Q. 1995, \apj, 438, 695

\bibitem[]{} Jackson, J. M., Paglione, T. A. D., Ishizuki, S., \&
Rieu, N.-Q. 1993, \apj, 418, L13
 
\bibitem[]{} Kohno, K., Kawabe, R., Tosaki, T., \& Okumura, S. 1996,
\apj, 461, L29
 
\bibitem[]{} Mangum, J. 1995, private communication

\bibitem[]{} Morris, M. 1993, \apj, 408, 469

\bibitem[]{} Odenwald, S. F., \& Fazio, G. G. 1984, \apj, 283, 601

\bibitem[]{} Paglione, T. A. D. 1996, Ph.D. Thesis, Boston University

\bibitem[]{} Paglione, T. A. D., Jackson, J. M., \& Ishizuki, S.
1997a, \apj, 484, 656

\bibitem[]{} Paglione, T. A. D., Tosaki, T., \& Jackson, J. M. 1995,
\apj, 454, L117
 
\bibitem[]{} Paglione, T. A. D., Yam, O., Heyer, M. H., \& Jackson,
J. M. 1997b, in preparation

\bibitem[]{} Rieu, N.-Q., Jackson, J. M., Henkel, C., Truong-Bach, \& 
Mauersberger, R. 1992, \apj, 399, 521

\bibitem[]{} Sakamoto, S. 1996, \apj, 462, 215

\bibitem[]{} Sanders, R. H.,  \& Lowinger, T. 1972, \aj, 77, 292

\bibitem[]{} Scoville, N. Z., Yun, M. S., Clemens, D. P., Sanders, D.
B., \& Waller, W. H. 1987, \apjs, 63, 821

\bibitem[]{} Serabyn, E., Lacy, J. H., \& Achtermann, J. M. 1992,
\apj, 395, 166

\bibitem[]{} Spergel, D. N. \& Blitz, L. 1992, \nat, 357, 665

\bibitem[]{} Sodroski, T. J. et~al. 1995, \apj, 452, 262

\bibitem[]{} Solomon, P. M., Downes, D., \& Radford, S. J. E. 1992,
\apjl, 387, L55 

\bibitem[]{} Stutzki, J., \& Winnewisser, G. 1985, \aap, 144, 13 

\end{thebibliography}

\clearpage

\figcaption[../../aips/histo.ps]
{Fraction of mass (estimated from $M \propto I_{CO}$) at a
particular kinetic temperature {\em (upper left)}, CO column density
per velocity interval {\em (lower left)}, \htwo\ density {\em (upper
right)}, and beam filling factor {\em (lower right)}.  Here $f_1=0.1$
and [HCN]/[CO] = 2.5\ee{-4}.
\label{fig.histo}}

\figcaption[../../aips/n.vs.phi.ps]
{{\em (Left)} Dependence of the mass-weighted mean density on the mean
beam filling factor {\em (dots, solid line)} and $f_1$ {\em (open
circles, dotted line)} for [HCN]/[CO] = 2.5\ee{-4}.  {\em (Right)}
Dependence of $\langle\log n\rangle$ on the abundance ratio [HCN]/[CO]
for $f_1=0.5$.
\label{fig.n.vs.phi}}

\figcaption[../../aips/rand.in.ps]
{Random input parameters displayed as in
Figure~\protect\ref{fig.histo}.  The histograms are the average of
three sets of random $T_k$, $N_{CO}/\Delta v$, \nhh, and $\phi$
inputs, and the errorbars show the dispersion.
\label{fig.rand.in}}

\figcaption[../../aips/rand.out.ps]
{Output of model using the random inputs of
Figure~\protect\ref{fig.rand.in}.  Here $f_1 = 0.1$ and
[HCN]/[CO] = 2.5\ee{-4}.
\label{fig.rand.out}}

\figcaption[../../aips/out.vs.in.ps]
{Model solutions plotted against random inputs.  A slope of unity is
indicated.
\label{fig.out.vs.in}}

\figcaption[../../aips/redo.ps]
{Solutions for the mass distributions in the Galactic center after
one, two, and three iterations of the model ({\em histogram, dots,}
and {\em crosses,} respectively).  Here $f_1 = 0.1$ and [HCN]/[CO] =
2.5\ee{-4}.
\label{fig.redo}}

\figcaption[../../aips/press.ps]
{The HCN/CO ratio versus $T_k$ {\em (upper left)}, $\phi$ {\em (lower
left)}, \nhh\ {\em (upper right)}, and pressure, $P=nT_k$ {\em (lower
right)}, for each modeled position in the Galactic center.  Here
$f_1=0.1$ and [HCN]/[CO] = 2.5\ee{-4}.
\label{fig.press}}

\figcaption[../Figs/sgr-spec.ps]
{Spectra of the HCN {\em (top left)} and CO {\em (bottom
left)} emission from the Milky Way convolved to 630 pc resolution.  
The right frames are the corresponding spectra with noise added to
simulate the signal-to-noise of typical extragalactic data.
\label{fig.spec}}

\figcaption[../../aips/tides.ps]
{Mean density and density dispersion as functions of projected
distance from Sgr~A$^*$.  The data are binned roughly every 8 pc.
The curve is the density required for stability against tidal forces
(Ho et~al. 1985).
\label{fig.tides}}

\figcaption[../Figs/rats.vs.n.ps]
{HCN \jth/\jon\ {\em (circles)} and ${\cal R}$ {\em (squares)} as
functions of density for $\log N_{CO}/\Delta v = 16$ \coldv\ {\em
(solid lines)} and 17.5 \coldv\ {\em (dashed lines)}, given $T_k$ =
100 K.  For HCN column densities, multiply $N_{CO}$ by [HCN]/[CO] =
2.5\ee{-4}.
\label{fig.rats.vs.n}}

\figcaption[../../aips/sgra-fig.ps]
{{\em (Top)}  Map of the integrated HCN \jth\ emission from Sgr~A.
Contours are 200, 300, 400, and 500 \kkms.  The dotted line outlines
the mapped region.  {\em (Bottom)}  Map of density in Sgr~A with radio
continuum sources ({\em triangles,} Ho et al. 1985).  Contours are log
(\nhh/\percc) = 4.0, 4.5, and 5.0.
\label{fig.sgramap}}

\figcaption[../Figs/sgrb-hcn32.ps]
{Map of the HCN \jth\ emission from Sgr~B.  Contours are 160, 220,
280, and 340 \kkms.  The dotted line outlines the mapped region.  The
tilted box indicates the location and positional errors of FIR source
38 from the survey of Odenwald \& Fazio (1984).  The white cross marks
the location of Sgr~B2 Main (Benson \& Johnston 1984).
\label{fig.sgrbmap}}

\figcaption[../../aips/histoa.ps]
{Fraction of mass as plotted in Figure~\protect\ref{fig.histo} for
Sgr~A from the analysis of HCN \jth/\hbox{\jon.}  Here $f_1=0.1$.
\label{fig.histoa}}

\figcaption[../../aips/histob.ps]
{Same as Figure~\protect\ref{fig.histoa} for Sgr~B.
\label{fig.histob}}

\figcaption[../../aips/out.vs.in32.ps]
{Model solutions plotted against random inputs using HCN
\hbox{\jth/}\jon.  Here $f_1=0.1$.  A slope of unity is indicated.
\label{fig.out.vs.in32}}

\figcaption[../../aips/hcnco.hcn32.ps]
{Comparison between solutions from analyzing HCN \jth/\jon\ and
${\cal R}$.  A slope of unity is indicated.  For column density, an
offset of log [HCN]/[CO] = 3.6 was included. 
\label{fig.hcnco.hcn32}}

\figcaption[../Figs/rats.ps]
{Comparison between HCN \jth/\jon\ and ${\cal R}$.
\label{fig.rats}}

\clearpage

\begin{center}
\begin{deluxetable}{ccr@{$\,\pm\,$}lr@{$\,\pm\,$}lr@{$\,\pm\,$}lr@{$\,\pm\,$}l}
\tablewidth{0pt}
\tablecaption{Model Results: Mean Values and 1$\sigma$ Dispersions.
\label{tab.means}}
\tablehead{
$f_1$ & [HCN]/[CO]& \multicolumn{2}{c}{$\langle T_k \rangle$} &
\multicolumn{2}{c}
{$\left\langle\log\left(\frac{N_{HCN}}{\Delta v}\right)\right\rangle$}
& \multicolumn{2}{c}{$\langle\log n\rangle$} & 
\multicolumn{2}{c}{$\langle\phi\rangle$} \\
 && \multicolumn{2}{c}{(K)} & \multicolumn{2}{c}{(\coldv)} & 
\multicolumn{2}{c}{(\percc)} &\multicolumn{2}{c}{}
}
\startdata
1.0 & 2.5\ee{-4} & 11&5 & 16.3&0.2 & 4.4&0.4 & 0.97&0.03 \nl
0.5 & 1.0\ee{-4} & 16&8 & 16.5&0.3 & 4.7&0.4 & 0.65&0.04 \nl
0.5 & 2.5\ee{-4} & 16&8 & 16.5&0.3 & 4.2&0.4 & 0.65&0.04 \nl
0.5 & 1.0\ee{-3} & 17&8 & 16.5&0.3 & 3.4&0.4 & 0.72&0.07 \nl
0.2 & 2.5\ee{-4} & 38&19& 16.9&0.3 & 3.9&0.5 & 0.30&0.04 \nl
0.1 & 2.5\ee{-4} & 75&37& 17.2&0.2 & 3.7&0.5 & 0.17&0.03 \nl
\enddata
\end{deluxetable}
\end{center}

\begin{center}
\begin{deluxetable}{lccr@{$\,\pm\,$}lr@{$\,\pm\,$}lr@{$\,\pm\,$}lr@{$\,\pm\,$}l}
\tablewidth{0pt}
\tablecaption{Model Results for Sgr~A and Sgr~B.
\label{tab.means.ab}}
\tablehead{
Cloud &
$f_1$ & [HCN]/[CO]& \multicolumn{2}{c}{$\langle T_k \rangle$} &
\multicolumn{2}{c}
{$\left\langle\log\left(\frac{N_{HCN}}{\Delta v}\right)\right\rangle$}
& \multicolumn{2}{c}{$\langle\log n\rangle$} & 
\multicolumn{2}{c}{$\langle\phi\rangle$} \\
 &&& \multicolumn{2}{c}{(K)} & \multicolumn{2}{c}{(\coldv)} & 
\multicolumn{2}{c}{(\percc)} &\multicolumn{2}{c}{}
}
\startdata
Sgr A & 0.1 & 2.5\ee{-4} & 97&26 & 14.1&0.4 & 4.5&0.4 & 0.3 & 0.2 \nl
Sgr B & 0.1 & 2.5\ee{-4} &138&24 & 14.1&0.1 & 4.5&0.4 & 0.2 & 0.1 \nl
\enddata
\end{deluxetable}
\end{center}

\end{document}