11institutetext: Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA
Leiden, the Netherlands

# Central molecular zones in galaxies:
Multitransition survey of dense gas tracers HCN, HNC, and HCO+

F.P. Israel 11

(Received ????; accepted ????)

New measurements of 46 nearby galaxy centers in up to three transitions of
HCN, HNC, and $\,{\rm HCO^{+}}$ combined with literature surveys establish a
database of 130 galaxies measured in both HCN and HCO+, and 94 galaxies in HNC
as well, allowing a systematic exploration of the relations between normalized
luminosities and line ratios. The almost linear relations between luminosities
are predominantly caused by distance effects and do not reflect galaxy
physical properties. Individual galaxies show significant dispersion in both
their luminosity and line ratio, which will be analyzed in more detail in a
later paper. Very few line ratios correlate either with luminosities or with
other line ratios. Only the normalized transition ladders of HCN and HCO+ and
the $J$=1-0 $\,{\rm{}^{12}CO}$/$\,{\rm{}^{13}CO}$ isotopologue ratio are
positively correlated with CO and far infrared (FIR) luminosity. On average,
HCN and HCO+ have very similar intensities and trace the same gas. In galaxies
dominated by an active nucleus, HCO+ intensities appear to be depressed
relative to HCN intensities. Only a small fraction of CO emission is
associated with gas emitting in HCN and $\,{\rm HCO^{+}}$, yet a significant
fraction of even that gas appears to be translucent molecular gas. In the
observed galaxy centers, the HCN/CO line intensity ratio is not a proxy for
the dense gas fraction, and the FIR/HCN and FIR/CO ratios are not proxies for
the star formation efficiency. A proper understandig of star formation
requires a more appropriate determination of gas mass than provided by the
intensities of individual HCN or CO transitions. The observed molecular line
emission is fully consistent with UV-photon heating boosted by significant
mechanical heating. The molecular gas sampled by HCN and $\,{\rm HCO^{+}}$ has
low kinetic temperatures $T_{\rm kin}\,=\,10-50$ K, low densities
$n_{rmH}\,=\,10^{4}-10^{5}\,\,{\rm cm^{-3}}$, and low optical depths in the
ground-state lines. Most of the gas sampled by CO has densities lower by one
to two orders of magnitude. For a mechanical heating fraction of 0.5, a modest
energy input of only $G\,=\,300$ G0 is required.

###### Key Words.:

Galaxies: galaxies: centers – interstellar medium: molecules – millimeter
lines – observations

††offprints: F.P. Israel

##  1 Introduction

This paper presents multitransition measurements of the HCN, HNC and $\,{\rm
HCO^{+}}$ molecular species tracing the dense molecular gas in the centers of
nearby galaxies. Many late-type galaxies contain massive concentrations of
molecular hydrogen gas close to the nucleus. These concentrations form the
reservoirs that feed mass inflow into supermassive black holes (SMBH), mass
outflow from the nucleus into space, and bursts of circumnuclear star
formation. In a previous paper (Israel 2020), we have explored the physical
characteristics of the inner-disk gas from measurements of the lower $J$
transitions of $\,{\rm{}^{12}CO}$ and $\,{\rm{}^{13}CO}$, augmented by
literature measurements of neutral ($\,{\rm[CI]}$) and ionized
($\,{\rm[CII]}$) atomic carbon.

Table 1: Critical densities

Molecule | Tran- | Frequency |  Eu/ka | $n^{a,b}_{crit}$ | $n^{c}_{eff}$
---|---|---|---|---|---
| sition | (GHz) | (K) |  ($\,{\rm cm^{-3}}$) |  ($\,{\rm cm^{-3}}$)
(1) | (2) | (3) | (4) | (5) | (6)
CO | 1-0 | 115.271 | 5.5 | 4 (2) | …
| 2-1 | 230.538 | 16.6 | 5 (3) | …
| 3-2 | 345.796 | 33.2 | 2 (4) | …
| 4-3 | 461.041 | 55.3 | 2 (5) | …
HCN | 1-0 | 88.632 | 4.3 | 2 (5) | 3 (3)
| 3-2 | 265.886 | 25.5 | 8 (6) | 1 (4)
| 4-3 | 354.506 | 42.5 | 2 (7) | 2 (5)
HCO+ | 1-0 | 89.189 | 4.3 | 3 (4) | 4 (2)
| 3-2 | 267.558 | 25.7 | 1 (6) | 4 (3)
| 4-3 | 356.734 | 42.8 | 3 (6) | 2 (4)
HNC | 1-0 | 90.664 | 4.4 | 1 (5) | 2 (3)
| 3-2 | 271.981 | 26.1 | 4 (6) | 3 (4)
| 4-3 | 362.630 | 43.5 | 9 (6) | 1 (5)
C2H | 1-0 | 87.317 | 4.3 | 2 (5) | 2 (4)
CS | 2-1 | 97.981 | 7.1 | 9 (4) | 9 (3)
| 3-2 | 146.969 | 14.1 | 3 (5) | 3 (4)

Notes: a. Jansen (1995) and Shirley (2015); b. Calculated for $T_{\rm kin}$ =
30 K in the optically thin limit; c. Shirley (2015), with log $N_{ref}$ = 14.0
(HCN, $\,{\rm HCO^{+}}$, and HNC) and log $N_{ref}$ = 13.5 (CS, C2H)

Gas inflow and outflow involve both dense clouds and diffuse intercloud gas
together responsible for the carbon and carbon monoxide emission studied in
that paper. Star formation, in contrast, is exclusively associated with the
dense gas. The determination of the properties of that gas requires
observation of molecules that, unlike CO, need relatively high densities for
excitation at modest temperatures. Molecules such as HCN, $\,{\rm HCO^{+}}$,
HNC, and CS are suitable for this purpose even though they have abundances
much lower than CO and much weaker emission lines. In Table 1 we list, for
these molecules in the relevant transitions, the line frequency, the minimum
temperature for excitation $T_{min}\,=\,E_{upper}$/k, and the temperature-
dependent critical density $n_{crit}$ for an assumed kinetic temperature
$T_{kin}$ = 30 K (cf. Paper I). Critical densities are lower for warmer gas
and higher for cooler gas, roughly by factors of up to two going from 30 K to
either 10 K or 100 K and HCN traces the highest densities. As molecular
emission remains detectable, however, at densities well below $n_{crit}$, we
also list the effective de-excitation density $n_{eff}$. Somewhat arbitrarily,
this is the density that produces a line of intensity 1
$\,{\rm{K\,km\,s^{-1}}}$, for a given temperature and column density (see
Evans 1999; Shirley 2015). The molecules and transitions in Table 1 cover a
wide range of excitation conditions, including densities 102 \- 107 $\,{\rm
cm^{-3}}$ and temperatures 4-55 K. In their study of molecular cloud ensembles
in the inner Galaxy, Evans $\rm et\,al.\ $ (2020) found that the integrated
HCN and $\,{\rm HCO^{+}}$ luminosity can be dominated by emission from even
more tenuous molecular gas with densities as low as 102 $\,{\rm cm^{-3}}$.

After the brightest individual galaxies had been observed in the relatively
accessible $J$=1-0 lines of HCN, $\,{\rm HCO^{+}}$, and HNC, the first surveys
with the $IRAM$ 30m telescope, encompassing some fifteen galaxies, were
published by Nguyen-Q-Rieu $\rm et\,al.\ $ (1992) and Hüttemeister $\rm
et\,al.\ $ (1995). Among early observations carried out between 1993 and 1997,
the large HCN and CO survey of 53 galaxies by Gao $\&$ Solomon (2004a) and the
survey of 37 galaxies in HCN, HNC, $\,{\rm HCO^{+}}$, CO, CN, and CS by Baan
$\rm et\,al.\ $ (2008) stand out. The latter 0combined all observations
available at the time into a heterogeneous database of 117 galaxies with 23
galaxies detected in all three lines, HCN(1-0), HNC(1-0), and $\,{\rm
HCO^{+}}$(1-0). They concluded that the emission from these lines and the far-
infrared (FIR) continuum are tightly correlated, and suggested that the ratio
of the HCN-to-CO luminosity traces the fraction of dense molecular gas. Kohno
$\rm et\,al.\ $ (2001, 2008) and Imanishi $\rm et\,al.\ $ (2007) then
suggested that the ground-state CO-HCN-$\,{\rm HCO^{+}}$ line intensity ratios
from galaxy centers dominated by a burst of star-formation (SB) differ from
those of galaxy centers dominated by nuclear activity (AGN). In their multi-
transition $IRAM$ 30m survey of the HCN and $\,{\rm HCO^{+}}$ emission from a
dozen galaxies, Krips $\rm et\,al.\ $ (2008) found higher average molecular
gas densities in SB galaxies and higher average temperatures and HCN/$\,{\rm
HCO^{+}}$ line ratios in AGN galaxies. An HCN/$\,{\rm HCO^{+}}$ overabundance
was, however, also found in luminous (LIRG) and ultra-luminous (ULIRG)
galaxies representing extreme SBs (Gracía-Carpo $\rm et\,al.\ $ 2008). Bussman
$\rm et\,al.\ $ (2008) conducted a survey of HCN(3-2) emission and found it to
behave differently from HCN(1-0). The major study of dense gas in luminous
galaxies by Baan $\rm et\,al.\ $ (2008) referred to earlier took these results
a step further by constructing diagnostic diagrams to investigate source
differentiation as a function of initial conditions and radiative environment.

More recently, the installation of very sensitive multi-mixer receivers (EMIR)
with very wide back-end coverage (FTS, WILMA) at the $IRAM$ 30m telescope has
enormously expanded the possibilities for extra-galactic molecular line
measurements as illustrated, for instance, by the spectral scans published by
Costagliola $\rm et\,al.\ $ (2011) and Jiang $\rm et\,al.\ $ (2011). Later in
this paper we will refer to their findings as well as those of others obtained
since then. We will do this in the context of analyzing and discussing an
extensive and homogeneous database of HCN, HNC, HCO+, and CO line intensities
in various transitions encompassing the newly obtained observations as well as
directly comparable data from the published literature. The database will be
used to investigate the overall relations between the various lines and
transitions, including verification of previous claims. Detailed modelling of
the results is deferred to a later paper.

This paper is structured as follows. Section 2 presents the new observations.
Section 3 describes the expansion of the database with line intensities from
surveys published by others and the normalization applied to data observed at
different resolutions in order to obtain a homogeneous sample of
intercomparable intensities. Section 4 systematically analyzes the trends and
correlations, either present or absent, between the various observed line
intensities and line ratios. Section 5 discusses the meaning of these
findings. Section 6 summarizes the most important points.

##  2 Observations

Table 2:  List of observed galaxy centers

NGC | Dist. |  lg$I_{FIR}$ |  lg$L_{FIR}$ |  $I_{\rm CO}$(1-0) | $D_{25}$ |  Typea
---|---|---|---|---|---|---
IC | (Mpc) |  (Wm-2) |  ($L_{\odot}$) |  ($Kkm/s$) | (’) |
(1) | (2) | (3) | (4) | (5) | (6) | (7)
N 253 | 3.4 | -10.42 | 10.13 | 1030 | 25x7.4 | S
N 470 | 31.7 | -12.44 | 10.04 | 28 | 2.8x1.7 | S
N 520 | 30.5 | -11.79 | 10.66 | 113 | 4.5x1.8 | S
N 660 | 12.2 | -11.46 | 10.20 | 154 | 9.1 | S
N 891 | 9.4 | -11.53 | 9.90 | 137 | 14x2.5 | (S)
N 972 | 21.4 | -11.75 | 10.39 | 67 | 3.4x1.7 | S
Maff2 | 3.1 | -11.23 | 9.23 | 220 | 5.8x1.6 | S
N1055 | 13.4 | -11.84 | 9.89 | 77 | 7.6x2.7 | S
N1068b | 15.2 | -11.04 | 10.80 | 168 | 7.1x6.0 | A
N1084 | 18.6 | -12.33 | 9.69 | 30 | 3.3x1.2 | (S)
N1097b | 16.5 | -11.81 | 10.10 | 136 | 9.3x6.6 | A+S
N1365b | 21.5 | -11.36 | 10.78 | 260 | 11x6.2 | A+S
I342 | 3.8 | -11.36 | 9.28 | 161 | 21x21 | S
N1808 | 12.3 | -11.31 | 10.35 | 135 | 6.5x3.9 | A
N2146 | 16.7 | -11.16 | 10.77 | 187 | 6.0x3.4 | S
N2559 | 21.4 | -11.78 | 10.36 | 78 | 4.1x2.1 | …
N2623 | 79.4 | -11.94 | 11.34 | 18 | 2.4x0.7 | L
N2903 | 7.3 | -11.65 | 9.56 | 80 | 13x6.0 | S
N3034 | 5.9 | -10.28 | 10.74 | 680 | 11x4.3 | S
N3079b | 20.7 | -11.60 | 10.51 | 235 | 7.9x1.4 | A
N3310 | 19.2 | -11.79 | 10.26 | 7 | 3.1X2.4 | S
N3627 | 6.5 | -11.61 | 9.50 | 74 | 9.1x4.2 | A+S
N3628 | 8.5 | -11.54 | 9.80 | 203 | 15x3.0 | S
N3690 | 48.5 | -11.32 | 11.53 | 69 | 2.9x2.1 | L
N4030 | 26.4 | -11.95 | 10.37 | 42 | 4.3 | …
N4038 | 23.3 | -11.65 | 10.56 | 47 | 5.2x3.1 | S
N4102 | 17.3 | -11.62 | 10.34 | 75 | 2.8x1.2 | A
N4321 | 14.1 | -11.88 | 9.90 | 82 | 7.4x6.3 | S
N4414 | 9.0 | -11.77 | 9.62 | 51 | 3.6x2.0 | A?
N4527 | 13.5 | -11.79 | 9.95 | 88 | 6.2x2.1 | S?
N4569 | 12.3 | -12.19 | 9.47 | 89 | 9.5x4.4 | A+S
N4666 | 27.5 | -11.74 | 10.62 | 74 | 4.6x1.3 | S
N4826b | 3.8 | -11.66 | 9.34 | 91 | 10x5.4 | A+S
N5033b | 17.2 | -12.00 | 9.95 | 53 | 11x5.0 | A
N5055 | 8.3 | -11.66 | 9.66 | 70 | 13x7.2 | (A)
N5194b | 9.1 | -11.59 | 9.81 | 48 | 11x6.9 | A
N5236 | 4.0 | -11.22 | 9.46 | 195 | 13x12 | S
N5775 | 28.9 | -11.97 | 10.43 | 48 | 4.2x1.0 | …
N6240 | 109 | -11.96 | 11.59 | 70 | 2.1x1.1 | L
N6701 | 59.1 | -12.23 | 10.80 | 45 | 1.5x1.3 | A+S
N6946 | 5.5 | -11.06 | 9.90 | 228 | 11.5x9.8 | S
N6951b | 24.3 | -12.04 | 10.21 | 50 | 3.9x3.2 | A+S
N7469∗ | 67.0 | -11.88 | 11.25 | 55 | 1.5x1.1 | L
N7714 | 38.5 | -12.30 | 10.35 | 4 | 1.9x1.4 | S
N7771 | 58.0 | -11.97 | 11.04 | 100 | 2.5x1.0 | L
Arp220 | 82.9 | -11.31 | 12.04 | 110 | 1.5x1.2 | U

Notes: a. S = Starburst, A = AGN, L = LIRG, U = ULIRG; parentheses indicate
marginal case. b. Seyfert nucleus.

Figure 1: Sample of molecular line profiles of galaxy centers observed with
the $IRAM$ 30m telescope. Galaxy and transition depicted are identified at the
top of each panel. The top row shows $J$ 1-0 observations of (in each panel
from left to right) HNC, HCO+, and HCN observed in 2010. The middle row shows
the same lines and includes the C2H at the very right, observed in 2011. The
bottom row shows $J$=3-2 observations of HCO+ (left) and HCN (right). In all
panels the vertical scale is intensity $T_{\rm A}^{*}$ (K) and the horizontal
scale is velocity V(LSR) in $\,{\rm{km\,s^{-1}}}$.

Figure 2: Sample of $J$=3-2 and $J$=4-3 molecular line profiles of galaxy
centers observed with the 15m $JCMT$. Galaxy and transition are depicted at
the top of each panel. In all panels the vertical scale is intensity $T_{\rm
A}^{*}$ (K) and the horizontal scale is velocity V(LSR) in
$\,{\rm{km\,s^{-1}}}$.

###  2.1 Observing list

We based our selection on the multi-transition $\,{\rm{}^{12}CO}$ and
$\,{\rm{}^{13}CO}$ survey of 126 galaxies by Israel (2020), which originally
consisted of the brightest galaxies in the $IRAS$ infrared sky survey but was
later expanded by adding other galaxies bright in $\,{\rm{}^{12}CO}$.
Brightness is an important criterion because the $\,{\rm{}^{13}CO}$ line, with
its low optical depth, is relatively weak. Because the HCN, HCO+ and HNC lines
are even weaker, the survey presented in this paper is limited to a subset of
the brightest galaxies from the CO survey. Because of their greater distance,
few infrared-luminous galaxies are bright enough to be included in this
limited sample.

Table 2 provides the names of the 46 galaxies observed, their distances, FIR
intensities and luminosities, $\,{\rm{}^{12}CO}$(1-0) intensities in the
$IRAM$ $22"$ beam, and overall angular size (Israel 2020 and references
therein). The list contains a single ULIRG and five LIRGs. The remainder are
lower-luminosity galaxies with a central starburst an AGN, or both.

We made observations in the $J$=1-0, $J$=3-2 and $J$=4-3 transitions of HCN,
$\,{\rm HCO^{+}}$ and HNC with the $IRAM$ 30m and the $JCMT$ 15m telescopes.
The $J$=2-1 transitions are at frequencies with poor atmospheric transmission
and observations were not attempted. The $J$=3-2 transitions were observed
with both $IRAM$ and $JCMT$, allowing determination of beam-dilution effects.
As a byproduct of the early (2008) observations with the $IRAM$ ABCD receivers
we also obtained CS(3-2) observations of thirteen galaxies, whereas the later
$IRAM$ EMIR observations also covered C2H(1-0) emission at the band-edge in
eleven galaxies.

Table 3:  Galaxy centers: line intensities $\int(T_{mb}dV)$
($\,{\rm{K\,km\,s^{-1}}}$) for HCN, HCO+, HNC

NGC | HCN |  |  |  |  HCO+ |  |  |  | HNC |
---|---|---|---|---|---|---|---|---|---|---
IC | (1-0) | (3-2) | (3-2) | (4-3) | (1-0) | (3-2) | (3-2) | (4-3) | (1-0) | (3-2)
| $IRAM$ | $IRAM$ | $JCMT$ | $JCMT$ | $IRAM$ | $IRAM$ | $JCMT$ | $JCMT$ | $IRAM$ | $JCMT$
| 27.6” | 9.7” | 19.0” | 13.7” | 27.6” | 9.7” | 19.0” | 13.7” | 27.6” | 19.0”
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
253 |  68.4$\pm 1.4^{a}$ |  120$\pm$12 |  77.2$\pm$14.2 |  61.1$\pm$1.0 |  57.9$\pm 1.2^{a}$ |  112$\pm$25 |  62.4$\pm$1.1 |  66.0$\pm$1.0 |  35.5$\pm 3.2^{a}$ |  34.3$\pm$0.7
470 |  0.75$\pm$0.21 | … | … | … | … | … | … | … | … | …
520 |  2.14$\pm$0.14 | … |  1.85$\pm$0.41 | … |  3.12$\pm$0.13 | … |  2.34$\pm$0.27 | … |  1.31$\pm$0.12 | …
660 |  6.54$\pm$0.65 |  5.34$\pm$0.68 |  4.20$\pm$0.90 |  2.55$\pm$0.50 |  5.13$\pm$0.75 |  3.35$\pm$0.90 |  4.33$\pm$0.29 | … |  2.97$\pm$0.31 |  1.74$\pm$0.32
891 |  1.30$\pm$0.12 |  1.90$\pm$0.55 |  1.66$\pm$0.38 | … |  1.33$\pm$0.19 |  1.29$\pm$0.28 |  1.36$\pm$0.35 | … |  0.44$\pm$0.08 | …
972 |  1.64$\pm$0.18 | … |  1.60$\pm$0.55 | … |  1.91$\pm$0.17 | … |  0.71$\pm$0.35 | … |  0.76$\pm$0.17 | …
1055 |  2.35$\pm$0.15 | … |  1.55$\pm$0.49 | … |  1.40$\pm$0.14 | … |  0.44$\pm$0.15 | … |  0.79$\pm$0.14 | …
Maf2 |  15.4$\pm$0.5 |  16.6$\pm$0.7 |  8.68$\pm$0.50 |  3.79$\pm$0.79 |  9.48$\pm$0.30 |  12.7$\pm$1.4 | … |  3.48$\pm$0.70 |  7.3$\pm 0.7^{d}$ |  4.04$\pm$0.46
1068 |  25.0$\pm 1.2^{b}$ |  19.0$\pm 1.2^{c}$ |  14.3$\pm$0.6 |  10.9$\pm$0.6 |  14.8$\pm 0.8^{b}$ |  7.6$\pm 0.8^{c}$ |  5.43$\pm$0.64 |  4.85$\pm$0.50 |  9.3$\pm 1.2^{b}$ |  4.08$\pm$0.55
1084 |  0.71$\pm$0.10 | … |  1.85$\pm$0.40 | … |  0.62$\pm$0.10 | … |  1.26$\pm$0.30 | … |  0.28$\pm$0.09 |  0.37$\pm$0.17
1097 |  9.31$\pm$0.18 | … |  5.99$\pm$0.81 | … |  6.67$\pm$0.20 | … |  4.52$\pm$0.66 | … |  3.06$\pm$0.19 | …
I342 |  10.8$\pm$0.3 |  8.3$\pm$0.9 |  3.53$\pm$0.35 |  2.7$\pm 0.3^{i}$ |  8.41$\pm$0.24 |  7.49$\pm$0.94 |  5.05$\pm$0.75 |  3.8$\pm 0.3^{i}$ |  4.75$\pm 0.11^{j}$ |  1.47$\pm$0.24
1365 |  14.2$\pm$0.7 |  12.0$\pm$0.4 |  9.20$\pm$0.57 | … |  11.7$\pm$1.4 |  9.80$\pm$0.36 | … | … |  7.24$\pm$0.52 |  3.59$\pm$0.74
1808 |  12.5$\pm$0.35 | … | … | … |  11.7$\pm$0.4 | … | … | … |  5.12$\pm$0.35 | …
2146 |  4.67$\pm$0.36 |  4.4$\pm 0.3^{c}$ |  4.30$\pm$0.49 |  4.49$\pm$0.89 |  6.70$\pm$0.36 |  5.2$\pm 0.6^{c}$ | … | … |  1.86$\pm$0.35 |  2.33$\pm$0.83
2559 |  2.96$\pm$0.13 | … |  2.52$\pm$0.33 |  2.40$\pm$0.48 |  3.17$\pm$0.18 | … | … | … |  1.03$\pm$0.13 |  0.14$\pm$0.13
2623 |  1.77$\pm$0.38 |  2.07$\pm$0.76 |  1.82$\pm$0.34 |  1.29$\pm$0.39 |  1.87$\pm$0.25 |  3.18$\pm$0.48 | … | … | … |  1.83$\pm 0.2^{f}$
2903 |  3.35$\pm$0.12 | … |  1.88$\pm$0.19 |  1.36$\pm$0.23 |  1.81$\pm$0.32 | … |  1.62$\pm$0.32 | … |  1.23$\pm$0.12 |  0.61$\pm$0.48
3034 |  29$\pm$3c |  27$\pm 3^{c}$ |  13.4$\pm$0.7 |  7.51$\pm$0.55 |  40.2$\pm 2.5^{c}$ |  34.0$\pm 2.0^{c}$ |  27.2$\pm$1.1 |  19.3$\pm$1.9 |  12$\pm 1.5^{e}$ |  3.58$\pm$1.52
3079 |  7.57$\pm$0.58 |  15.8$\pm$1.5 |  6.91$\pm$0.46 |  3.44$\pm$0.47 |  6.54$\pm$0.23 |  16.2$\pm$1.8 |  6.14$\pm$1.05 |  3.78$\pm$0.49 |  2.2$\pm 0.14^{j}$ |  3.1$\pm$1.2
3310 |  0.30$\pm$0.08 | … |  1.41$\pm$0.72 | … |  0.67$\pm$0.40 | … |  0.74$\pm$0.28 | … |  0.32$\pm$0.12 | …
3627 |  2.79$\pm$0.14 | … |  2.23$\pm$0.40 |  1.64$\pm$0.16 |  2.93$\pm$0.15 |  2.1$\pm 0.2^{j}$ | … | … |  0.78$\pm$0.19 |  0.58$\pm$0.20
3628 |  5.35$\pm$0.36 |  3.25$\pm$0.73 |  3.42$\pm$0.20 |  2.01$\pm$0.17 |  5.60$\pm$0.23 |  4.67$\pm$0.93 | … | … |  3.4$\pm 0.3^{d}$ | …
3690 |  2.51$\pm$0.30 |  1.58$\pm 0.38^{g}$ |  0.99$\pm$0.21 |  0.44$\pm$0.19 |  3.62$\pm$0.18 |  4.72$\pm 0.44^{g}$ | … | … |  0.90$\pm$0.19 | …
4030 |  3.45$\pm$0.14 |  2.00$\pm$0.50 |  1.37$\pm$0.45 |  1.57$\pm$0.17 |  1.76$\pm$0.17 |  0.90$\pm$0.26 | … | … |  1.80$\pm$0.17 | …
4038 |  2.57$\pm$0.10 |  0.91$\pm$0.19 |  0.95$\pm$0.21 |  1.10$\pm$0.71 |  3.53$\pm$0.10 |  1.53$\pm$0.33 |  1.44$\pm$0.28 | … |  1.13$\pm$0.07 | …
4102 |  3.75$\pm$0.32 |  3.64$\pm$0.68 |  2.01$\pm$0.62 | … |  2.28$\pm$0.51 |  2.02$\pm$0.61 | … |  2.74$\pm$0.17 |  2.81$\pm$0.23 | …
4321 |  4.93$\pm$0.28 |  3.09$\pm$0.26 |  2.23$\pm$0.60 |  1.75$\pm$0.46 |  3.19$\pm$0.12 |  2.12$\pm$0.39 | … | … |  1.64$\pm$0.08 | …
4414 |  3.76$\pm$0.39 | … |  1.05$\pm$0.39 |  0.92$\pm$0.17 |  2.68$\pm$0.30 |  1.34$\pm$0.41 | … | … |  1.03$\pm$0.15 | …
4527 |  3.41$\pm$0.24 |  1.90$\pm$0.21 |  1.33$\pm$0.44 | … |  3.07$\pm$0.23 |  1.87$\pm$0.20 | … | … |  1.54$\pm$0.28 | …
4569 |  3.21$\pm$0.15 |  2.35$\pm$0.38 |  0.76$\pm$0.23 |  0.69$\pm$0.25 |  2.50$\pm$0.14 |  2.08$\pm$0.33 | … | … |  0.73$\pm$0.15 | …
4666 |  2.19$\pm$0.07 |  1.82$\pm$0.46 |  0.60$\pm$0.38 |  1.36$\pm$0.11 |  1.42$\pm$0.10 |  0.63$\pm$0.24 | … | … |  1.36$\pm$0.14 | …
4826 |  7.19$\pm$0.39 |  1.77$\pm$0.36 |  1.94$\pm$0.70 |  1.22$\pm$0.32 |  4.46$\pm$0.10 |  1.33$\pm$0.21 | … | … |  3.03$\pm$0.15 |  1.32$\pm$0.31
5033 |  1.28$\pm$0.13 | … |  1.26$\pm$0.51 | … |  0.95$\pm$0.15 | … |  0.62$\pm$0.26 | … |  1.07$\pm$0.17 | …
5055 |  2.59$\pm$0.44 |  1.85$\pm$0.48 |  0.90$\pm$0.60 |  0.61$\pm$0.22 |  1.43$\pm$0.28 | … | … | … |  0.71$\pm$0.15 | …
5194 |  7.15$\pm$0.29 |  2.31$\pm$0.24 |  1.38$\pm$0.70 |  1.36$\pm$0.43 |  3.36$\pm$0.15 |  1.43$\pm$0.29 | … | … |  2.74$\pm$0.14 |  0.48$\pm$0.29
5236 |  8.54$\pm$0.26 | … |  3.86$\pm$0.30 |  1.76$\pm$0.20 |  10.4$\pm$0.4 | … | … |  2.52$\pm$0.36 |  4.3$\pm 0.3^{d}$ |  1.30$\pm$0.28
5775 |  1.39$\pm$0.36 |  0.85$\pm$0.19 | … | * |  1.30$\pm$0.39 |  0.46$\pm$0.31 | … | … |  0.27$\pm$0.17 | …
6240 |  3.80$\pm$0.20 |  7.8$\pm 0.31^{h}$ |  4.70$\pm$0.29 |  4.61$\pm$1.42 |  6.37$\pm$0.20 |  7.1$\pm 1.9^{h}$ |  1.79$\pm$1.15 | … |  0.78$\pm$0.15 | …
6701 |  2.32$\pm$0.06 |  2.55$\pm$0.25 |  1.09$\pm$0.20 |  1.85$\pm$0.46 |  2.37$\pm$0.07 |  2.41$\pm$0.24 |  3.43$\pm$0.75 | … |  0.96$\pm$0.09 |  0.50$\pm$0.50
6946 |  9.9$\pm 0.1^{c}$ |  9.2$\pm 0.6^{c}$ |  3.85$\pm$0.36 |  1.53$\pm$0.22 |  8.5$\pm 0.1^{c}$ |  8.6$\pm 0.6^{c}$ |  3.19$\pm$0.38 | … |  4.0$\pm 0.4^{d}$ |  1.98$\pm$0.54
6951 |  2.68$\pm$0.13 |  1.99$\pm$0.38 |  1.13$\pm$0.39 | … |  1.95$\pm$0.08 |  2.24$\pm$0.41 |  1.30$\pm$0.44 | … |  1.50$\pm$0.09 |  0.67$\pm$0.20
7469 |  2.56$\pm$0.13 |  2.76$\pm 0.34^{g}$ |  2.01$\pm$0.51 |  0.73$\pm$0.25 |  2.68$\pm$0.12 |  2.08$\pm 0.42^{g}$ |  1.51$\pm$0.19 | … |  1.32$\pm$0.13 |  1.09$\pm$0.44
7714 |  0.56$\pm$0.18 | … | … | … |  0.18$\pm$0.17 | … | … | … | … | …
7771 |  5.35$\pm$0.60 |  3.33$\pm 0.58^{g}$ |  1.19$\pm$0.23 | … |  5.59$\pm$0.26 |  1.98$\pm 0.37^{g}$ |  2.48$\pm$0.56 | … |  2.31$\pm$0.15 |  1.21$\pm$0.43
A220 |  8.16$\pm$0.17g |  18.0$\pm 0.5^{g}$ |  12.5$\pm$0.6 |  3.89$\pm$0.59 |  3.77$\pm 0.21^{g}$ |  4.57$\pm 0.23^{g}$ | … |  2.96$\pm$0.49 |  7.8$\pm 0.8^{g}$ | …

References to intensities taken from the literature: a. Aladro $\rm et\,al.\ $
(2015); b. Average from Krips $\rm et\,al.\ $ (2008), Costagliola $\rm
et\,al.\ $ (2011), and Aladro $\rm et\,al.\ $ (2015); c. Krips $\rm et\,al.\ $
(2008); d. Hüttemeister $\rm et\,al.\ $ (1995); e. Average from Hüttemeister
$\rm et\,al.\ $ (1995) and Aladro $\rm et\,al.\ $ (2015); f. Pérez-Beaupuits
$\rm et\,al.\ $ (2007); g. Graciá-Carpio $\rm et\,al.\ $ 2008: h. Li $\rm
et\,al.\ $ (2018); i. Tan $\rm et\,al.\ $ (2018); j. Li et al. (2021)

Table 4:  Galaxy centers: CS, C2H line intensities $\int(T_{mb}dV)$

NGC | CS(3-2) |  C2H(1-0)
---|---|---
IC | $16.8"$ | $27.6"$
| $\,{\rm{K\,km\,s^{-1}}}$ | $\,{\rm{K\,km\,s^{-1}}}$
(1) | (2) | (3)
N 253 | … |  34.64$\pm$0.45a
N 660 |  1.70$\pm$0.22 |  2.77$\pm$0.28
N 891 |  0.41$\pm$0.07 |  0.57$\pm$0.24
Maff2 |  4.41$\pm$0.41 | …
N1055 |  0.40$\pm$0.14 | …
N1068 | … |  8.84$\pm$2.20ab
N1365 |  2.80$\pm$0.35 | …
..I342 |  4.03$\pm$0.20 | …
N1808 | … |  5.43$\pm$1.09
N2146 |  1.22$\pm$0.23c |  3.85$\pm$0.79
N2623 |  0.90$\pm$0.11 | …
N2903 |  0.58$\pm$0.08 | …
N3034 |  12.1$\pm$0.5c |  20.61$\pm$0.23a
N3079 |  4.98$\pm$0.29 |  2.90$\pm$0.30bd
N3628 |  3.77$\pm$0.23 | …
N3690 |  0.68$\pm$0.09c |  1.20$\pm$0.18e
N4102 |  1.09$\pm$0.20 | …
N5055 |  2.89$\pm$0.42 | …
N5194 |  1.27$\pm$0.07c |  1.26$\pm$0.07a
N5236 |  3.36$\pm$0.22 |  4.01$\pm$0.08a
N5775 |  0.27$\pm$0.13c |  1.90$\pm$0.70
N6240 |  0.60$\pm$0.04c |  1.19$\pm$0.20be
N7469 | … |  1.71$\pm$0.29ab
N7771 | … |  1.10$\pm$0.18b

References to measurements from the literature: a. Aladro $\rm et\,al.\ $
(2015); b. Costagliola $\rm et\,al.\ $ (2011); d. Li $\rm et\,al.\ $ (2019);
e. Jiang $\rm et\,al.\ $ (2011).

###  2.2 $IRAM$ 30m observations

We used the $IRAM$ 30 m telescope on Pico de Veleta (Granada, Spain) 111$IRAM$
is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). The
observations enabling the research in this paper have received funding from
the European Commission Seventh Framework Programme (FP/2007-2013) under grant
agreement No 283393 (RadioNet3) in three observing runs (April 2008, November
2010, and February 2011) in beam-switching mode with a throw of 4$\arcmin$. In
the 2008 run, we observed 13 galaxies with the IRAM 3-mm and 1.3-mm SIS (ABCD)
receivers coupled to 4-MHz back-ends, one line per spectrum. In the 2010 and
2011 runs we used the EMIR receiver to observe both the J=1-0 transitions of
HCN, HCO+, and HNC, and the $J$=3-2 transitions of HCN and HCO+ in single
spectra and also reobserved the weaker galaxies from the 2008 run. A sample of
the $IRAM$ observations is shown in Fig. 1. Individual spectra were reduced
with the CLASS package; line parameters were determined by fitting gaussians
after baseline subtraction. Intensities were converted to main-beam brightness
temperatures using main beam efficiencies $\eta_{\rm mb}$ of 0.82 at 89 GHz,
0.74 at 146 GHz, and 0.52 at 255 GHz.

###  2.3 $JCMT$ 15 m observations

The observations with the 15 m James Clerk Maxwell Telescope ($JCMT$) on Mauna
Kea (Hawaii) 222Between 1987 and 2015, the James Clerk Maxwell Telescope
(JCMT) was operated by the Joint Astronomy Centre on behalf of the Particle
Physics and Astronomy Research Council of the United Kingdom, the Netherlands
Organization for Scientific Research (until 2013), and the National Research
Council of Canada. were obtained at various periods between 2010 and 2013,
with a beam-switch throw of 3 $\arcmin$, and mostly in queue or backup service
mode. The $J$=3-2 measurements were done with the A3 receiver, and the $J$=4-3
measurements with the central pixel of the HARP array receiver. We used the
FITS protocol to transport the data from SPECX into CLASS. The effectively
available band-pass was limited to 1050 $\,{\rm{km\,s^{-1}}}$ in the $J$=3-2
transition and 800 $\,{\rm{km\,s^{-1}}}$ in the $J$=4-3 transition. Where weak
lines covered much of these ranges, we used the high S/N CO profiles from
Israel (2020) as a template to separate the emission from the baseline. We
determined integrated intensities by fitting gaussians to the observed
profiles after linear baseline subtraction. Because of the limited free
baseline, several of the integrated $JCMT$ fluxes have larger uncertainties
than those determined with the $IRAM$ telescope. A sample of the $JCMT$
observations is shown in Fig. 2. Antenna temperatures were converted to main-
beam brightness temperatures with efficiencies $\eta_{\rm mb}(265)$ = 0.69 and
$\eta_{\rm mb}(355)$ = 0.63.

Table 5: Galaxy centers: molecular line intensities $\int(T_{mb}dV)$
($\,{\rm{K\,km\,s^{-1}}}$) normalized to 22” beams (see text)

NGC | HCN | HNC |  HCO+
---|---|---|---
| (1-0) |  (2-1)a | (3-2) | (4-3) | (1-0) | (3-2) | (1-0) | (3-2) | (4-3)
(1) | (2) | (3) | (4) | (5) | (6 | (7) | (8) | (9) | (10))
N0253 | 87.60 | … | 70.12 | 44.78 | 45.44 | 31.15 | 74.11 | 54.93 | 43.71
N0520 | 2.85 | … | 1.7 | … | 1.74 | … | 4.15 | 2.20 | …
N0660 | 9.29 | 6.4 | 3.99 | 2.15 | 4.22 | 1.35 | 7.29 | 4.58 | …
N0891 | 1.96 | … | 1.61 | … | 0.66 | … | 2.01 | 1.4 | …
N0972 | 1.97 | … | 1.5 | … | 0.9 | … | 2.29 | 0.5 | …
N1055 | 2.82 | … | 1.4 | … | 0.95 | … | 1.68 | 0.4 | …
MAF2 | 23.25 | 12.0 | 7.54 | 2.40 | 11.02 | 3.51 | 14.31 | 6.10 | 2.19
N1068 | 32.75 | 14.9 | 13.44 | 8.92 | 12.18 | 3.84 | 19.39 | 5.05 | 3.85
N1084 | 0.85 | … | 1.7 | … | 0.3 | … | 0.74 | 1.2 | …
N1097 | 11.17 | … | 5.50 | … | 3.67 | … | 8.00 | 4.20 | …
IC342 | 14.69 | … | 2.93 | 1.48 | 6.46 | 1.22 | 11.44 | 4.60 | 2.88
N1365 | 18.60 | … | 8.68 | … | 9.48 | 3.39 | 15.33 | 7.40 | …
N2146 | 6.12 | 4.3 | 4.28 | 4.42 | 2.44 | 2.32 | 8.78 | 4.70 | …
N2559 | 3.55 | … | 2.30 | … | 1.24 | … | 3.80 | … | …
N2623 | 2.46 | … | 1.77 | 1.2 | … | 1.78 | 2.60 | 4.00 | …
N2903 | 4.19 | … | 1.70 | … | 1.54 | … | 2.26 | 1.50 | …
N3034 | 35.67 | 14.2 | 11.50 | 4.58 | 14.76 | 3.1 | 49.45 | 25.90 | 16.49
N3079 | 9.46 | … | 5.77 | 1.92 | 2.7 | 2.6 | 8.18 | 4.97 | 1.91
N3310 | 0.38 | … | 1.3 | … | 0.4 | … | 0.84 | 0.7 | …
N3627 | 2.79 | … | 1.90 | … | 0.8 | 0.5 | 2.93 | … | …
N3628 | 7.60 | … | 3.46 | 2.08 | 4.83 | … | 7.95 | 4.9 | …
N3690 | 3.29 | … | 0.89 | 0.3 | 1.18 | … | 4.74 | 3.00 | …
N4030 | 4.42 | … | 1.26 | 1.20 | 2.30 | … | 2.25 | 0.6 | …
N4038 | 3.37 | … | 0.96 | 1.1 | 1.48 | … | 4.62 | 1.42 | …
N4102 | 4.80 | … | 1.77 | … | 3.60 | … | 2.92 | 1.1 | …
N4321 | 6.16 | … | 2.08 | 1.4 | 2.05 | … | 3.99 | 1.5 | 1.7
N4414 | 4.63 | … | 1.0 | … | 1.27 | … | 3.30 | … | …
N4527 | 4.09 | … | 1.23 | … | 1.85 | … | 3.68 | 1.30 | …
N4569 | 4.85 | 1.0 | 0.59 | 0.3 | 1.10 | … | 3.78 | 0.70 | …
N4666 | 2.98 | … | 0.47 | 0.62 | 1.85 | … | 1.93 | 0.2 | …
N4826 | 8.99 | 2.5 | 1.98 | 1.3 | 3.79 | 1.3 | 5.58 | 0.7 | …
N5033 | 1.28 | … | 1.2 | … | 1.07 | … | 0.95 | 1.90 | …
N5055 | 3.19 | … | 0.77 | 0.4 | 0.9 | … | 1.76 | … | …
N5194 | 9.51 | 1.4 | 1.23 | 1.0 | 3.64 | 0.4 | 4.47 | 0.90 | …
N5236 | 10.50 | … | 3.50 | … | 5.29 | … | 12.79 | … | 2.52
N5775 | 1.64 | … | … | … | 0.3 | 1.3 | 1.53 | … | …
N6240 | 5.62 | 5.6 | 4.21 | 3.2 | 1.15 | … | 9.43 | 5.25 | …
N6701 | 2.32 | … | 0.91 | 1.0 | 0.96 | … | 2.37 | 3.70 | …
N6946 | 12.97 | 5.3 | 3.18 | 0.83 | 5.24 | 1.6 | 11.13 | 2.57 | …
N6951 | 3.56 | 3.1 | 1.00 | … | 2.00 | 0.6 | 2.59 | 1.16 | …
N7469 | 3.28 | … | 1.88 | 0.6 | 1.69 | 1.0 | 3.43 | 1.41 | …
N7771 | 8.08 | … | 0.95 | … | 3.49 | 1.0 | 8.44 | 2.61 | …
A220 | 9.79 | … | 11.54 | 3.01 | 9.36 | … | 4.52 | 3.20 | …

Note: a Based on IRAM measurements by Krips $\rm et\,al.\ $ (2008)

Figure 3:  Top row: galaxy center luminosity-luminosity relations in identical
$22"$ beams. Leftmost panels: $J$=1-0 HCN and $\,{\rm HCO^{+}}$ versus
$\,{\rm{}^{12}CO}$ with constant ratios $0.10$ and $0.03$ marked by solid
lines. Rightmost panels: HNC(1-0) versus $\,{\rm{}^{12}CO}$ with constant
ratios $0.050$ and $0.008$, and central $\,{\rm{}^{12}CO}$ versus global far-
infrared (FIR) continuum with constant ratios $2.5\times 10^{-2}$ and
$3.2\times 10^{-3}$ K km/s pc2/LO marked. Bottom row, leftmost panel: central
HCN(1-0) versus global FIR continuum with constant ratios of $2\times 10^{-3}$
and $1.25\times 10^{-4}$ marked, very similar to the HCN/CO (top left) and
HCN/FIR (top right) diagrams. Center left panel: relation between HCN
luminosity ($L$(HCN)$\propto\,$I(HCN)$\times$D2) and distance ($D^{2}$).
Center right panel: relation between luminosity $L$(HCN) and intensity
$I$(HCN); vertical line marks constant surface brightness of 1 K km/s.
Rightmost panel: relation between $I$(HCN) and distance ($D^{2}$), with a
horizontal line of resolved surface brightness of 0.5 K km/s and a diagonal
line of unresolved point-like emission (cf. Table 1). In these and following
diagrams, filled circles denote the new data from this paper, open circles
literature data on similar galaxies, crosses galaxies with an identified AGN
and triangles literature survey data on (U)LIRGs.
The results of least-squares regression fits, of the form log(y) = a log(x) +
b, to the data in this figure and the following figures are given in Table 8.
Table 6: Galaxy centers: molecular line ratios

NGC |  HCO+/HCN | HNC/HCN | HCN/CO | HCN | HNC |  HCO+
---|---|---|---|---|---|---
| 1-0 |  2-1a | 3-2 | 3-2 | 4-3 | 1-0 | 3-2 | 1-0/1-0 |  2-1/1-0b | 3-2/1-0 | 4-3/1-0 | 3-2/1-0 | 3-2/1-0 | 4-3/1-0
| 27.6” | 34.7” | 9.7” | 19.0” | 13.7” | 27.6” | 19” | 22” | 22” | 22” | 22” | 22” | 22” | 22”
(1) | (2) | (3) | (4) | (5) | (6 | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14) | 15
253 | 0.85 | … | 0.93 | 0.81 |  1.21c | 0.52 | 0.45 | 0.085 | … | 0.80 | 0.51 | 0.69 | 0.74 | 0.59
520 | 1.46 | … | … | 1.27 | … | 0.61 | … | 0.025 | … | 0.60 | 0.53 | … | … | …
660 | 0.78 | … | 0.63 | 1.03 |  1.04d | 0.45 | 0.34 | 0.060 | 0.69 | 0.43 | 0.23 | 0.32 | 0.63 | …
891 | 1.02 | … | 0.68 | 0.82 | … | 0.34 | … | 0.014 | … | 0.82 | … | … | 0.69 | …
972 | 1.17 | … | … | 0.4: | … | 0.46 | … | 0.029 | … | 0.76 | … | … | 0.18 | …
1055 | 0.60 | … | … | 0.3: | … | 0.34 | … | 0.037 | … | 0.50 | … | … | 0.24 | …
Maf2 | 0.62 | … | 0.77 | … | 0.92 | 0.47 | 0.47 | 0.106 | 0.52 | 0.32 | 0.10 | 0.32 | 0.43 | …
1068 * | 0.59 | 0.74 | 0.40 | 0.38 |  0.46cd | 0.40 | 0.29 | 0.195 | 0.46 | 0.41 | 0.27 | 0.32 | 0.26 | 0.20
1084 | 0.87 | … | … | 0.68 | … | 0.39 | 0.2: | 0.028 | … | … | … | … | … | …
1097 * | 0.72 | … | … | 0.75 | … | 0.33 | … | 0.082 | … | 0.49 | … | … | 0.53 | …
I342 | 0.78 | … | 0.90 | 1.43 |  1.41c | 0.59 | 0.42 | 0.091 | 0.20 | 0.10 | 0.40 | 0.14 | 0.25 | …
1365 * | 0.82 | 0.74 | 0.81 | … |  1.11cd | 0.51 | 0.39 | 0.072 | … | 0.47 | … | 0.36 | 0.48 | …
1808 * | 0.94 | 1.04 | … | … |  0.96cd | 0.41 | … | … | … | … | … | … | … | …
2146 | 1.44 | … | 1.18 | … | … | 0.40 | 0.5: | 0.033 | 0.70 | 0.70 | 0.72 | … | 0.54 | …
2559 | 1.07 | … | … | … | … | 0.35 | … | 0.046 | … | 0.65 | … | … | … | …
2623 | 1.06 | … | 1.54 | … | … | … | 0.99 | 0.137 | … | 0.72 | 0.48 | … | 1.5 | …
2903 | 0.54 | … | … | 0.86 | … | 0.37 | … | 0.052 | … | 0.41 | … | … | 0.66 | …
3034 | 1.39 | … | 1.26 | 2.03 |  2.72c | 0.41 | 0.27 | 0.052 | 0.40 | 0.32 | 0.13 | 0.21 | 0.52 | 0.33
3079 * | 0.86 | … | 1.03 | 0.89 | 1.10 | 0.91 | 0.2: | 0.040 | … | 0.61 | 0.20 | … | 0.61 | 0.23
3627 * | 1.05 | … | … | … |  0.88d | 0.28 | 0.3: | 0.038 | … | 0.68 | … | … | … | …
3628 | 1.05 | 1.34 | 1.44 | … |  2.78d | 0.64 | … | 0.037 | … | 0.46 | 0.27 | … | 0.62 | …
3690 | 1.44 | … | 2.99 | … | … | 0.36 | … | 0.048 | … | 0.27 | 0.10 | … | 0.63 | …
4030 | 0.51 | … | 0.45 | … | … | 0.53 | … | 0.105 | … | 0.29 | 0.27 | … | 0.27 | …
4038 | 1.37 | … | 1.68 | 1.52 | … | 0.44 | … | 0.072 | … | 0.29 | 0.34 | … | 0.31 | …
4102 * | 0.61 | … | 0.55 | … | … | 0.75 | … | 0.064 | … | 0.37 | … | … | 0.38 | …
4321 | 0.65 | … | 0.69 | … | … | 0.33 | … | 0.075 | … | 0.34 | 0.23 | … | 0.38 | …
4414 * | 0.71 | … | … | … | … | 0.27 | … | 0.091 | … | 0.22 | … | … | … | …
4527 | 0.90 | … | 0.98 | … | … | 0.45 | … | 0.046 | … | 0.30 | … | … | 0.35 | …
4569 * | 0.78 | … | 0.89 | … | … | 0.23 | … | 0.054 | 0.21 | 0.12 | 0.06 | … | 0.19 | …
4666 | 0.65 | … | 0.3: | … | … | 0.62 | … | 0.040 | … | 0.16 | 0.21 | … | 0.10 | …
4826 * | 0.62 | … | 0.75 | … | … | 0.42 | 0.68 | 0.099 | 0.28 | 0.22 | 0.15 | 0.36 | 0.13 | …
5033 * | 0.74 | … | … | 0.5: | … | 0.84 | … | 0.024 | … | 0.94 | … | … | 0.52 | …
5055 * | 0.55 | … | … | … | … | 0.27 | … | 0.046 | … | 0.24 | 0.12 | … | … | …
5194 * | 0.47 | … | 0.6: | … | … | 0.38 | 0.4: | 0.198 | 0.15 | 0.13 | 0.10 | … | 0.20 | …
5236 | 1.22 | … | … | … |  1.14c | 0.50 | 0.34 | 0.054 | … | 0.33 | … | … | … | 0.16
5775 | 0.94 | … | 0.5: | … | … | 0.19 | … | 0.032 | … | … | … | … | … | …
6240 | 1.68 | 1.36 | 0.92 | 0.4: |  1.65d | 0.21 | … | 0.080 | 1.00 | 0.75 | 0.58 | … | 0.14 | …
6701 * | 1.02 | … | 0.95 | 0.76 | … | 0.41 | … | 0.052 | … | 0.39 | 0.44 | 0.44 | 0.67 | …
6946 | 0.85 | … | 0.93 | 0.83 |  3.21d | 0.40 | 0.51 | 0.057 | 0.41 | 0.25 | 0.06 | 0.31 | 0.23 | …
6951 * | 0.74 | … | 1.13 | 1.15 | … | 0.56 | 0.59 | 0.071 | 0.87 | 0.28 | … | 0.30 | 0.45 | …
7469 * | 1.05 | … | 0.75 | 0.75 | … | 0.52 | 0.5: | 0.060 | … | 0.57 | 0.18 | … | 0.41 | …
7771 | 1.05 | … | 0.60 | 2.08 | … | 0.43 | 1.02 | 0.081 | … | 0.12 | … | 0.28 | 0.31 | …
A220 | 0.46 | … | 0.25 | … | 0.76 | 0.96 | … | 0.089 | … | 1.18 | 0.31 | … | 0.71 | 0.21
MWe | 0.60 | … | … | … | … | 0.30 | … | 0.10 | … | … | … | … | … | …
Ave | 0.90 | 1.1 | 0.95 | 1.06 | 1.42 | 0.46 | 0.52 | 0.046 | 0.49 | 0.41 | 0.27 | 0.34 | 0.42 | 0.29
|  $\pm$0.05 |  $\pm$0.2 |  $\pm$0.10 |  $\pm$0.11 |  $\pm$0.21 |  $\pm$0.03 |  $\pm$0.07 |  $\pm$0.006 |  $\pm$0.08 |  $\pm$0.04 |  $\pm$0.04 |  $\pm$0.05 |  $\pm$0.03 |  $\pm$0.07
MDf | 0.30 | 0.40 | 0.52 | 0.47 | 0.82 | 0.17 | 0.24 | 0.039 | 0.27 | 0.25 | 0.18 | 0.16 | 0.19 | 0.16
Errg | 0.05 | 0.15 | 0.15 | 0.2 | 0.3 | 0.05 | 0.15 | 0.15 | 0.3 | 0.2 | 0.3 | 0.2 | 0.15 | 0.3

Notes: a. Based on $APEX$ 12m observations by Zhou $\rm et\,al.\ $ (2022). b.
Based on $IRAM$ observatioins(HPBW $14"$) by Krips $\rm et\,al.\ $ (2008). c.
Includes $JCMT$ observations by Tan $\rm et\,al.\ $ (2018). d. From $APEX$ 12m
observations (HPBW $18"$) by F.P. Israel (unpublished) and by Zhang $\rm
et\,al.\ $ (2014). e. Milky Way Circumnuclear Molecular Zone values from
$MOPRA$ 22m observations (effective resolution $39"$) by Jones $\rm et\,al.\ $
(2012). f. Mean deviation of sample. g. Logarithm of typical observational
error of ratios plotted in Figures 3-11; individual errors vary depending on
observed intensity (cf. Table 3).

Table 7:  Line transition ratios as a function of CO luminosity

log $L$(CO)a |  HCNb |  $\,{\rm HCO^{+}}$ b
---|---|---
$\,{\rm{K\,km\,s^{-1}}}$ | (2-1)/(1-0) | (3-2)/(1-0) | (4-3)/(1-0) | (3-2)/(1-0)
(1) | (2) | (3) | (4) | (5)
7.4 | 0.31 (5) | 0.25 (9) | 0.16 (6) | 0.32 (6)
8.3 | 0.53 (5) | 0.41 (16) | 0.26 (9) | 0.37 (14)
8.8 | … | 0.68 (14) | 0.42 (4) | 0.45 (3)
9.3 | … | 0.35 (9) | 0.30 (4) | 0.45 (10)
9.9 | … | 0.67 (6) | … | 0.61 (7)
7.0-9.0c | 0.49 (12) | 0.40 (29) | 0.26 (19) | 0.37 (23)
7.0-10.0c | … | 0.43 (44) | 0.27 (24) | 0.43 (40)

Notes: a. Center of luminosity interval considered; b. Number of points in
interval is given between parentheses; entries for intervals with only one
point are deleted; c. Range considered for overall average.

##  3 Results

###  3.1 Galaxy sample

The new survey results are listed as velocity-integrated line intensities in
Tables 3 and 4. The headers identify molecular species, line transition,
telescope used, and the resolution at the observing frequency. Additional
$IRAM$ and $JCMT$ line intensities from the published literature complementing
the survey results are included in the tabulation, with the appropriate
reference at the bottom.

The database of 46 observed galaxies is further expanded by adding the ground-
state HCN, $\,{\rm HCO^{+}}$ and HNC intensities of galaxies .not included in
our observing program, extracted from published $IRAM$ 30m surveys. First, we
selected from the surveys by Costagliola $\rm et\,al.\ $ (2011), Jiang $\rm
et\,al.\ $ (2011) and Li $\rm et\,al.\ $ (2021) an additional 21 galaxies in
which the HCN(1-0), $\,{\rm HCO^{+}}$(1-0), and HNC(1-0) lines were all well-
detected, with associated $\,{\rm{}^{12}CO}$ and $\,{\rm{}^{13}CO}$ data from
Li $\rm et\,al.\ $ (2015) and Israel (2020). There are no comparable $IRAM$
30m surveys covering the higher transitions.

Secondly, we considered the luminous infrared galaxies specifically targeted
in large $IRAM$ 30m surveys by Graciá-Carpio $\rm et\,al.\ $ 2006, 2008,
García-Burillo $\rm et\,al.\ $ 2012, Privon $\rm et\,al.\ $ 2015, and Herrero-
Illana $\rm et\,al.\ $ 2019). We found 63 galaxies with well-measured
intensities of both HCN(1-0) and $\,{\rm HCO^{+}}$(1-0), 27 of which are also
detected in HNC(1-0) and 10 in HCN(3-2) and $\,{\rm HCO^{+}}$(3-2). About half
of these have corresponding $\,{\rm{}^{12}CO}$ and $\,{\rm{}^{13}CO}$ data,
published by Graciá-Carpio (2009), Li $\rm et\,al.\ $ (2015), and Herrero-
Illana $\rm et\,al.\ $ (2019).

In total, there are well-established intensities of the three $J$=1-0 HCN,
$\,{\rm HCO^{+}}$ and HNC lines for 94 galaxies covering a large luminosity
range ($9.2\leq logL_{FIR}\leq 12.3$). An additional 36 galaxies have good
HCN(1-0) and $\,{\rm HCO^{+}}$(1-0) line intensities only. About 50 galaxies
have good HCN(3-2) and $\,{\rm HCO^{+}}$(3-2) data, mostly provided by our new
survey that also contributes a smaller number of additional HNC(3-2), HCN(4-3)
and $\,{\rm HCO^{+}}$(4-3) data.

###  3.2 Normalized intensities

Because the $J$=1-0 line frequencies of HCN, HNC and $\,{\rm HCO^{+}}$ are
very close together, their intensities measured with the $IRAM$ 30m telescope
are directly comparable. This not true, however, for measurements in other
transitions, for measurements with a different telescope aperture, and for
measurements of species such as CO, which are all observed at a different
spatial resolution. Molecular line intensities observed at different
resolutions must be reduced (normalized) to the same resolution before they
can be compared in a meaningful way. The resolution of $22"$ is conveniently
intermediate between the resolutions of the $IRAM$ $J$=1-0 ($27"$) and $JCMT$
$J$=3-2 ($19"$) lines, and it is also the reference for the CO transition
ladders published by Israel (2020). Only minor extrapolations are required to
bring the observed $J$=1-0 and $J$=3-2 intensities to values corresponding to
this resolution.

For $J$=1-0 HCN we used the intensities of 36 sample galaxies measured in
larger beams (NRAO $\theta\,=\,72"$, 25 galaxies; SEST $\theta\,=\,57"$, 14
galaxies; FCRAO $\theta\,=\,50"$, 6 galaxies; OSO $\theta\,=\,44"$, 4
galaxies) in combination with the IRAM measurements ($\theta\,=\,27.6"$) of
the present survey to extrapolate the latter to the equivalent flux at $22"$.
On average, the extrapolation is almost linear with the resolution,
intermediate between the extremes expected for infinitely large (extended) and
infinitely small (pointlike) sources. The average extrapolation factor
$f_{10}\,=\,1.2$ (corresponding to a power law extrapolation with index 0.8)
was used for the remaining galaxies lacking observations at other resolutions.
As the modest change of resolution limits possible multiplication factors to
$1.00\leq\,f_{10}\,\leq 1.57$, extrapolation uncertainties add relatively
little to the overall errors.

We applied the HCN(1-0) normalization factors to the $\,{\rm HCO^{+}}$(1-0)
and HNC(1-0) intensities because these species have, unlike HCN, few
observations at other resolutions. The implicit assumption of identical
spatial emission distributions likewise does not introduce more than small
errors.

The paucity of published HCN, $\,{\rm HCO^{+}}$, or HNC measurements in higher
transitions at other resolutions than presented here was the rationale for
observing $J$=3-2 HCN and $\,{\rm HCO^{+}}$ transitions with both the JCMT 15m
and $IRAM$ 30m facilities. Their resolutions differ by a factor of two ($9.6"$
and $19"$), from which the normalized intensity at $22"$ is once again derived
by a modest extrapolation, with correction factors limited to
$1.00\,\geq\,f_{32}\,\geq 0.75$. The actual average reduction factors derived
from the data in Table 3 are $f_{HCN32}\,=\,0.91$ (33 galaxies) and
$f_{HCO32}\,=\,0.94$ (14 galaxies). The difference is not significant and
justifies the assumption of very similar HCN and HCO+ emission distributions.
No duplicate $J$3-2 data were obtained for the HNC lines but the spatial
distributions of HNC and HCN are even more likely to be identical so that the
same factors apply. Finally, we assumed the resolution dependence of the
$J$=4-3 HCN and $\,{\rm HCO^{+}}$ intensities to be the same as that of the
$J$=3-2 lines and interpolated the latter to obtain the normalization factors
for the $J$=4-3 intensities, with resulting averages $f_{HCN43}\,=\,0.65$ (24
galaxies) and $f_{HCO4-3}\,=\,0.74$ (8 galaxies). In Table 5 all intensities
are normalized to resolutions of $22"$ by the appropriate factors. This table
also includes $IRAM$ $J$=2-1 HCN intensities ($14"$) from Krips $\rm et\,al.\
$ 2008.

###  3.3 Line ratios and transition ladders

In columns 2 through 7 of Table 6, intensities of $\,{\rm HCO^{+}}$ and HNC
relative to HCN are listed. The resolutions change with each transition, but
they are near-identical for lines observed in the same transition. The $IRAM$
HCO+/HCN and HNC/HCN intensity ratios are each determined from lines in the
same spectrum and are thus quite accurate. The JCMT-derived line ratios are
less accurate as they are determined from separate single-line $JCMT$
observations made on different occasions. The HCN/CO intensity ratios in
Column 8 are based on the normalized $J$=1-0 HCN intensities from Table 5 and
the $J$=1-0 CO intensities from Israel(2020) and refer to identical beams of
$22"$. The transition ladder ratios of HCN, HNC and HCO+ in columns 9 through
14 are likewise derived from the normalized values in Table 5.

##  4 Analysis

This section identifies what observational patterns exist for the various
galaxies and in particular which luminosities and luminosity ratios, if any,
show a degree of correlation. Possible explanations for the resulting findings
and their meaning are discussed in the following Section 5.

The $IRAM30m$ line measurements of both HCN(1-0) and $\,{\rm HCO^{+}}$(1-0) in
130 galaxies constitute a sample with a size and homogeneity that renders it
highly suitable to the large-scale investigation of the molecular gas in
galaxies of a diverse nature and wide range of luminosities. Such
investigations have been carried out before, first by Gao $\&$ Solomon
(2004b), by Baan $\rm et\,al.\ $(2008), and most recently by Li $\rm et\,al.\
$ (2021) but there are major differences between their surveys and ours. The
Gao $\&$ Solomon data set is half the size, has a poorer resolution ($NRAO$
12m versus $IRAM$ 30m), contains only ground-state HCN and CO transitions, but
attempts to cover entire galaxies instead of just central regions. The data
set used by Baan $\rm et\,al.\ $ covers the HCN, $\,{\rm HCO^{+}}$ and HNC
ground state only, but includes CN and CS lines, it is smaller, less complete,
and combines data from telescopes with different and generally larger beams
making it very inhomogeneous and less accurate. The Li $\rm et\,al.\ $ $IRAM$
30m data set contains a similar number of detected galaxies (124 versus 130
with both HCN and $\,{\rm HCO^{+}}$, and 85 versus 94 with also HNC) but it
also includes only ground state transitions.

In the following, molecular line luminosities333Throughout this paper, line
luminosities are defined as the product of line flux in K km/s and beam
surface area in pc2 and intensities all refer to the normalized ($22"$)
central aperture discussed above. The far-infrared intensities and
luminosities, however, refer to the $IRAS$ aperture which is a factor of 100
greater and should be considered more representative for the galaxy as a
whole.

###  4.1 HCN(1-0) and $\,{\rm HCO^{+}}$(1-0) versus FIR and CO(1-0)
luminosities

Comparison shows that the normalized central HCN(1-0), $\,{\rm HCO^{+}}$(1-0),
and HNC(1-0) luminosities are all linearly correlated with each other, as well
as the CO luminosity, with slopes corresponding to average ratios HCN/CO =
0.046, $\,{\rm HCO^{+}}$/CO = 0.041, and HNC/CO = 0.021 (Fig 3). In each
relation, the dispersion does not change with luminosity. It exceeds the
observational uncertainty and reflects variation between individual galaxies.
Solomon $\rm et\,al.\ $ (1992) were the first to draw attention to the tight
relation between global HCN and FIR luminosities, further elaborated by Gao
$\&$ Solomon (2004a), and rediscovered or confirmed by almost all later
authors for HCN(1-0), $\,{\rm HCO^{+}}$(1-0), HNC(1-0), CS(3-2), and far-
infrared luminosity in matched beams.

The CO line and FIR continuum luminosities are also linearly related
(rightmost panel of Fig. 3) and it makes little difference whether the other
luminosities are plotted as a function of CO or of FIR luminosity. This is
illustrated by the similarity of the CO-FIR, HCN-CO and HCN-FIR diagrams in
Fig. 3 (top right, top left and bottom left panels).

In the bottom row of Fig. 3 we further investigate the relations between the
derived luminosity $L$, the observed surface brightness $I$, and the assumed
distance $D$. Measured in the same normalized beam, luminosities vary with
surface brightness and distance ($L\,\propto\,I\times D^{2}$) only. The
luminosity $L$(HCN) is strongly correlated with distance $D^{2}$ (center left
panel) and very weakly or not at all with observed intensity $I$(HCN) (center
right panel). Thus, the luminosities are essentially determined by the
distance and not by the intensity of the observed galaxies. The intensity
$I$(HCN) and distance $D^{2}$ are weakly anti-correlated (rightmost panel).
The surface brightness goes down with increasing distance as the constant
aperture covers an increasingly larger part of the galaxy. As Gao $\&$ Solomon
(2004b) already surmised, the HCN surface brightness peaks at the nucleus and
drops when it is averaged over increasingly larger surface area. This is well-
illustrated by HCN line maps (Green $\rm et\,al.\ $ 2016, Tan $\rm et\,al.\ $
2018, Jiménez-Donaire $\rm et\,al.\ $ 2019) that include several of the
galaxies in our sample. Because of the anticorrelation between $I$(HCN) and
$D^{2}$, the relation between luminosity $L$(HCN) and distance $D^{2}$ is
modestly sub-linear. The smaller departure from linearity, and the larger
dispersion in the $L$(CO)-$L$(FIR) and $L$(HCN)-$L$(FIR) diagrams is due to
the aperture mismatch between the FIR continuum and the molecular line
measurements. In matched apertures, the discrepancy disappears (Li $\rm
et\,al.\ $ 2021). Nevertheless, comparison of the results by Gao $\&$ Solomon
(2004b) and Li $\rm et\,al.\ $ (2021) shows that the use of global instead of
central luminosities leads to similar results.

As long as the derived luminosities are dominated by the distance factor
$D^{2}$ and no large changes occur in the underlying line intensities, the
luminosity plots will show (almost) linear relationships. As we will show, the
line intensities in this paper tend to be characterized by constant ratios.
The observed linearity of luminosity-versus-luminosity plots thus reflects
geometric factors unrelated to intrinsic galaxy properties.

###  4.2 $J$=1-0 HCN/CO and FIR/CO as a function of luminosity

Whereas absolute luminosities are dominated by galaxy distances and shed
little light on the physical conditions in the central regions of galaxies,
this may be different for luminosity444In identical apertures, there is no
meaningful distinction between intensity ratios and luminosity ratios, and we
will use both terms interchangeably ratios. First, we turn to relations
between the HCN(1-0)/CO(1-0) ratio and the CO, HCN, $\,{\rm HCO^{+}}$, and FIR
luminosities. With some reservations, Gao $\&$ Solomon (2004a) interpreted the
ratio $L$(HCN)/$L$(CO) as an indicator for the fraction of dense molecular gas
in galaxies. They found it to be rather low ($\leq 0.06$) in low-luminosity
normal spiral galaxies but dramatically increasing to $L$(HCN)/$L$(CO)$\sim
0.25$ in high luminosity galaxies $L$(FIR)$\geq 11$), all galaxies with
$L$(HCN)/$L$(CO)$\geq 0.06$ being (ultra)luminous (Gao $\&$ Solomon 2004b). We
do not confirm these findings. The more accurate HCN/CO ratios at galaxy
central positions shown in Fig. 4 (top left panel) do not exhibit such a jump
at high values of $L$(FIR) and show the same dispersion at low and high
luminosities. The behaviour of HCN/CO as a function of $L$(CO) is no different
and this is also true if we substitute $\,{\rm HCO^{+}}$(1-0) for HCN(1-0). In
Fig. 4, there is no systematic correlation between the HCN/CO ratio and either
$L$(FIR) (top left) or $L$(CO) (top right). Similar plots of infrared and
molecular line measurements in matching beams presented by Li $\rm et\,al.\ $
(2021 likewise fail to reveal correlations between the infrared luminosity and
the HCN/IR or $\,{\rm HCO^{+}}$/IR ratios. The center left panel of Fig 4
shows at best a weak correlation between $L$(HCN)/$L$(CO) and $L$(HCN), with
$L$(HCN)/$L$(CO)$\,\propto\,L$(HCN)0.1. Substitution of $\,{\rm HCO^{+}}$ for
HCN yields the same result with a slightly lower dispersion. A direct
comparison of the present results and those by e.g. Li $\rm et\,al.\ $ (2021)
with the results published by Gao $\&$ Solomon (2004a, b) is not easy because
the latter constructed spatially integrated luminosities from heterogeneous
data obtained at different resolutions, at lower sensitivities in much
narrower bands. It is not entirely clear how they constructed luminosity
ratios such as those of HCN to CO. In the combined database presented here,
luminous galaxies have on average significantly lower HCN/CO ratios than those
listed by Gao $\&$ Solomon (2004b), whereas many lower-luminosity galaxies
have significantly higher HCN/CO ratios. The two data sets have 39 galaxies in
common. For seven (18$\%$) galaxies Gao $\&$ Solomon list lower HCN/CO ratios,
but for twenty-four (62$\%$) they find (significantly) higher ratios. It is
noteworthy that almost all of these refer to (luminous) galaxies with wide
lines of low-amplitude, filling most of the observed spectral window and
leaving little room for accurate baseline-fitting. We suspect that, as a
result, Gao $\&$ Solomon have overestimated the corresponding HCN
luminosities, and that the more recently obtained data supersede these older
values.

In a similar way, the ratio $L$(FIR)/$L$(CO) is often presented as a proxy for
the star formation efficiency (more correctly: the inverse of the molecular
gas depletion rate), under the assumption that $L$(FIR) is proportional to the
star formation rate. We find, however, that the ratio $L$(FIR)/$L$(CO) is
uncorrelated with the luminosities $L$(CO), $L$(HCN) or $L(\,{\rm HCO^{+}})$,
and only shows a weak correlation with $L$(FIR) with a modest logarithmic
slope $\sim$0.13 (Fig. 4, center right). It is not clear whether this is a
slow increase over the full range of log $L$(FIR) or a rise only at values log
$L$(FIR)$\geq 11$ of the luminous galaxies. Gracía-Carpio $\rm et\,al.\ $
(2009) found a similar weak correlation between $L$(FIR)/$L$(HCN) and $L$(FIR)
with a logarithmic slope 0.24 for (ultra)luminous galaxies. The larger sample
studied here does not show such a correlation between either $L$(FIR)/$L$(HCN)
or $L$(FIR)/$L$($\,{\rm HCO^{+}}$) and $L$(FIR).

Gao $\&$ Solomon (2004b) found the ‘star formation efficiency’
$L$(FIR)/$L$(CO) to be strongly correlated with the ‘dense gas fraction’
$L$(HCN)/$L$(CO). We find a similar but weaker relation in the bottom left
panel of Fig. 4. Gao $\&$ Solomon did not find a relation between
$L$(FIR)/$L$(CO) and $L$(FIR)/$L$(HCN), but our data reveal these quantities
to be well-correlated (Fig. 4, bottom right), even though $L$(FIR)/$L$(HCN)
and $L$(CO) are not. In these bottom panels there is no systematic difference
between normal galaxies, starburst galaxies and AGNs.

Neither $L$(HCN)/$L$(CO) nor $L$(FIR)/$L$(CO) is correlated with distance $D$.
Hence, they also do not depend on the extent of the central area sampled.

###  4.3 Dense and very dense molecular gas: $\,{\rm HCO^{+}}$(1-0) and
HCN(1-0)

The relation between the dense gas supposedly traced by $\,{\rm HCO^{+}}$ and
the very dense gas traced by HCN is of further interest, especially since
their ground state intensities are derived from the same observed profiles so
that their ratio is free of systematic uncertainties and thus quite accurate.
Despite critical densities an order of magnitude apart, the $J$=1-0 $\,{\rm
HCO^{+}}$ and HCN luminosities are closely related and practically
interchangeable. Gracía-Carpio $\rm et\,al.\ $ (2008) claimed a correlation
between the ground-state HCN-to-$\,{\rm HCO^{+}}$ ratios and the infrared
luminositiies of (U)LIRGs but Privon $\rm et\,al.\ $ (2015) could not confirm
this, nor can we. The present data extending over a three times larger
luminosity range clearly establish that HCN/$\,{\rm HCO^{+}}$ ratios are not
related to either FIR or CO luminosities (Fig. 5 left panel), nor to CO
intensities. The average HCN-to-$\,{\rm HCO^{+}}$ ratio of 1.1 is constant and
slightly above unity. The same behavior is also found on smaller scales, such
as the central kilo-parsec of NGC 253 (Knudsen $\rm et\,al.\ $ 2007). In the
$J$=2-1 (Zhou $\rm et\,al.\ $ 2022), $J$=3-2 and $J$=4-3 (This paper, Zhang
$\rm et\,al.\ $ 2014, Imanishi $\rm et\,al.\ $ 2018) transitions the average
ratios are 0.9, 1.0, and 0.7 , respectively. The ratios of the first three
transitions are within each others errors but the $J$=4-3 ratio is
significantly lower. When the luminosity distance bias is eliminated, for
instance by weighting both HCN and $\,{\rm HCO^{+}}$ by their corresponding CO
luminosities, $\,{\rm HCO^{+}}$/CO increases somewhat more slowly than HCN/CO
with a logarithmic slope $\sim 0.6$ as shown in Fig 5 (center left panel).
There, low-luminosity normal galaxies and high-luminosity LIRGs do not define
distinct or separate groups, in line with the absence of correlations between
CO luminosity and the ratios of HCN to CO and $\,{\rm HCO^{+}}$ to CO in Fig.
4. Finally, the center right panel shows that the ‘very-dense-to-dense gas’
ratio $J$=1-0 HCN/$\,{\rm HCO^{+}}$ is unrelated to the ‘dense-to-modestly-
dense gas’ ratio $\,{\rm HCO^{+}}$/CO, and the rightmost panel shows that
there is no correlation between the $\,{\rm HCO^{+}}$/HCN and FIR/CO either.

###  4.4 HCN, $\,{\rm HCO^{+}}$, and the $\,{\rm{}^{12}CO}$/$\,{\rm{}^{13}CO}$
ground-state ratio

The observed isotopological ratio $I(\,{\rm{}^{12}CO}$)/I($\,{\rm{}^{13}CO}$)
depends on both the intrinsic isotopic abundance ratio and the CO optical
depth. The magnitude of the isotopic abundance ratio in galaxy centers is
poorly known and estimates range from very low values of 20-40 to high values
of 70-150 (see, for instance, the compilation by Viti $\rm et\,al.\ $ 2020).
Moreover, they may vary from galaxy to galaxy. The present database contains
measurements of the isotopological ratio for the majority of galaxies (see
Section 3.1). $\,{\rm{}^{12}CO}$(1-0)/$\,{\rm{}^{13}CO}$(1-0) ratios increase
both with the CO luminosity $L$(CO) (Fig 6 leftmost panel) and with the far-
infrared luminosity $L$(FIR). Ratios range from 6-14 at the lowest
luminosities and from 10-50 at the highest luminosities. Fig. 6 illustrates
that the ratio of HCN-to-$\,{\rm HCO^{+}}$ decreases as a function of
$\,{\rm{}^{12}CO}$/$\,{\rm{}^{13}CO}$ (rightmost panel) whereas the ‘dense-gas
ratio’ HCN/CO does not (center left panel) nor does the ratio $\,{\rm
HCO^{+}}$/CO. Likewise, no relation is found between the ‘star-formation-
efficiency’ FIR/CO and the isotopological ratio (center right panel).

Figure 4:  Top row: the J=1-0 HCN/CO ratio as a function of $IRAS$ FIR
luminosity (left) and J=1-0 CO luminosity (right). The horizontal line in the
panel at left marks the ratio separating normal and luminous galaxies in the
paper by Gao $\&$ Solomon (2004b). Neither panel shows any increase in HCN/CO
at (U)LIRG luminosities, nor any change in the dispersion of the ratio. Center
row: The HCN/CO ratio as a function of L(HCN) (left) and the FIR/CO ratio as a
function of L(FIR) (right). Bottom row: The FIR/CO ratio versus HCN/CO ratio
(left); the FIR/HCN ratio versus the FIR/CO ratio (right). In both panels,
straight lines mark unity slopes. Fit parameters are listed in Table 8.

Figure 5: Left: The HCN-to-$\,{\rm HCO^{+}}$ versus the FIR-to-CO luminosity
ratio. The horizontal line marks equal HCN and $\,{\rm HCO^{+}}$ luminosities.
Center: The $\,{\rm HCO^{+}}$/CO versus the HCN/CO luminosity ratio. Right:
The $\,{\rm HCO^{+}}$/HCN versus the HCN/CO luminosity ratio. The solid line
has slope -0.55. Fit parameters are listed in Table 8.

Figure 6: From left to right, the ground-state isotopologue
$\,{\rm{}^{12}CO}$/$\,{\rm{}^{13}CO}$ ratio as a function CO luminosity and
the HCN/CO, FIR/CO, and HCN/$\,{\rm HCO^{+}}$ ratios as a function of the
$\,{\rm{}^{12}CO}$/$\,{\rm{}^{13}CO}$ ratio. Fit parameters are listed in
Table 8.

Figure 7: Top: the ground-state HCN/HNC isomer ratio as a function of FIR
luminosity (left) and the $\,{\rm{}^{12}CO}$/$\,{\rm{}^{13}CO}$ isotopologue
ratio (right). Center: the ground-state HCN/CO (left) and $\,{\rm
HCO^{+}}$/HCN (right) ratios as a function of the (inverse) HNC/HCN isomer
rasio. Bottom: the ground-state $\,{\rm HCO^{+}}$/HCN (left) and HNC/HCN
(right) ratios as a function of the HNC/$\,{\rm HCO^{+}}$ ratio. Solid lines
mark unity isomer ratios. Fit parameters are listed in Table 8.

Figure 8: Molecular line ladders: the HCN and $\,{\rm HCO^{+}}$ line intensity
ratios as a function of transition in normalized (equal) beams. Left to right:
the HCN $J$=2-1/$J$=1-0, $J$=3-2/$J$=1-0, $J$=4-3/$J$-1-0, and the HCO+
$J$=3-2/$J$=1-0 ratios as a function of $\,{\rm{}^{12}CO}$ $J$=1-0 luminosity.
Solid lines mark ratios of unity. Symbols are as in Fig.3 but almost all data
are from this paper. Fit parameters are listed in Table 8.

Figure 9: The leftmost two panels ratios illustrate the similarity of the
$\,{\rm HCO^{+}}$ and HCN ladders. The rightmost panels show that the $\,{\rm
HCO^{+}}$/HCN intensity ratios remain constant over the full range of $J$=1-0
CO luminosities. The $J$=4-3 data shown include data from APEX and ALMA in
addition to the IRAM data from this paper. Straight lines mark ratios of
unity. Fit parameters are listed in Table 8.

Figure 10: Comparison of the $J$=3-2/$J$=1-0 HCN ratio as a function of the
isotopological ratio $\,{\rm{}^{12}CO}$/$\,{\rm{}^{13}CO}$ (leftmost panel)
and FIR/CO (rightmost panel). Solid lines mark ratios of unity. Symbols are as
in Fig.3. Fit parameters are listed in Table 8.

Figure 11: Left: the HNC(3-2)/HNC(1-0) excitation ratio as a function of the
HCN(1-0) luminosity. The ground-state HCN/HNC ratio as a of the
HCN(3-2)/HCN(1-0) excitation ratio. Solid lines mark ratios of unit. Symbols
are as in Fig.3; most data are from this paper. Fit parameters are listed in
Table 8.  Table 8: Linear regression fits to luminosities and ratios plotted
in Figs. 3 through 11

y | x | a | b | rms | sfcc | Reference
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) |
L(CO) |  L(HCO+) | 1.03 | -1.60 | 0.21 | 0.95 | Fig. 3
L(CO) | L(HCN) | 0.99 | -1.16 | 0.25 | 0.92 |
L(CO) | L(HNC) | 1.02 | -1.83 | 0.29 | 0.94 |
L(FIR) | L(HCN) | 0.90 | -2.26 | 0.28 | 0.92 |
L(FIR) | L(CO) | 0.88 | -0.72 | 0.25 | 0.93 |
L(HCN) |  L(HCO+) | 1.01 | -0.11 | 0.18 | 0.97 |
L(HNC) | L(HCN) | 0.95 | 0.71 | 0.15 | 0.98 |
I(HNC) | I(HCN) | 1.97 | 0.43 | 1.87 | 0.91 |
I(HCN) |  I(HCO+) | 0.86 | 0.15 | 2.24 | 0.88 |
D2 | L(HCN) | 0.71 | 5.23 | 0.34 | 0.91 |
D2 | I(HCN) | -0.29 | 1.28 | 0.34 | \- 0.61 |
L(HCN) | I(HCN) | -0.14 | 1.39 | 0.43 | \- 0.27 |
L(CO) | L(HCN)/L(CO) | -0.02 | -1.08 | 0.25 | -0.06 | Fig. 4
L(HCN) | L(HCN)/L(CO) | 0.09 | -1.92 | 0.24 | 0.27 |
L(FIR) | L(HCN)/L(CO) | 0.02 | -1.54 | 0.25 | 0.06 |
L(FIR) | L(FIR)/L(CO) | 0.13 | 0.72 | 0.25 | 0.39 |
L(HCN)/L(CO) | L(FIR)/L(CO) | 0.43 | 2.63 | 0.25 | 0.38 |
L(FIR)/L(CO) | L(FIR)/L(HCN) | 0.64 | 2.03 | 0.23 | 0.53 |
L(CO) |  L(HCN)/L(HCO+) | -0.05 | 0.48 | 0.17 | -0.22 | Fig. 5
L(HCN)/L(CO) |  L(HCO+)/L(CO) | 0.59 | -0.51 | 0.15 | 0.69 |
L(HCO+)/L(CO) | L(HCN)/L(CO) | -0.14 | -0.15 | 0.18 | -0.17 |
L(FIR)/L(CO) |  L(HCN)/L(HCO+) | -0.00 | 0.04 | 0.18 | -0.01 |
L(CO) |  12CO/13CO | 0.12 | 0.08 | 0.14 | 0.58 | Fig. 6
12CO/13CO | L(HCN)/L(CO) | -0.29 | -0.98 | 0.25 | -0.12 |
12CO/13CO | L(FIR)/L(CO) | 0.18 | 1.85 | 0.25 | 0.15 |
12CO/13CO |  L(HCN)/L(HCO+) | -0.55 | 0.66 | 0.15 | -0.43 |
12CO/13CO | L(HCN)/L(HNC) | 0.06 | 2.26 | 0.87 | -0.07 | Fig. 7
L(FIR) | L(HCN)/L(HNC) | -0.08 | 3.20 | 0.87 | -0.14 |
L(CO) |  L(HNC)/L(HCO+) | -0.06 | 2.82 | 0.87 | -0.13 |
L(HNC)/L(CO) | L(HCN)/L(HNC) | -0.26 | 0.08 | 0.14 | -0.47 |
L(HNC)/L(HCN) |  L(HCO+)/L(HCN) | -0.05 | -0.06 | 0.17 | -0.02 |
L(HNC)/L(HCO+ |  L(HCO+)/L(HCN) | -0.54 | -0.20 | 0.12 | -0.74 |
L(HNC)/L(HCO+ | L(HNC)/L(HCN) | 0.46 | -0.20 | 0.12 | 0.66 |
L(CO) |  HCO+(3-2)/(1-0) | 0.12 | -1.37 | 0.25 | 0.32 | Fig. 8
L(CO) | HCN(2-1)/(1-0) | 0.21 | -2.08 | 0.18 | 0.64 |
L(CO) | HCN(3-2)/(1-0) | 0.17 | -1.81 | 0.21 | 0.46 |
L(CO) | HCN(4-3)/(1-0)) | 0.23 | -2.61 | 0.25 | 0.47 |
HCN(3-2)/(1-0) |  HCO+(3-2)/(1-0) | 0.57 | -0.14 | 0.17 | 0.68 | Fig. 9
HCN(4-3)/(3-2) |  HCO+(4-3)/(3-2) | 0.70 | 0.03 | 0.11 | 0.70 |
L(CO) |  $J$=3-2 HCO+/HCN | 0.14 | -0.15 | 0.24 | 0.17 |
L(CO) |  $J$=4-3 HCO+/HCN | 0.08 | -0.68 | 0.32 | 0.13 |
12CO/13CO | HCN(3-2)/(1-0) | 0.32 | -0.71 | 0.23 | 0.16 | Fig. 10
L(FIR)/L(CO) | HCN(3-2)/1-0) | -0.34 | 0.33 | 0.22 | -0.38 |
L(HCN) | HNC(3-2)/(1-0) | 0.07 | -1.02 | 0.24 | 0.18 | Fig. 11
HCN(3-2)/(1-0) | HCN/HNC(1-0) | -0.16 | 0.30 | 0.15 | -0.18 |

Notes: a. The regression fits are of the form log(y) = a log(x) + b. Columns 1
and 2 identify the data used in the fit. Column 3 and 4 list the slope a and
y-axis intercept of the fitted relation. Column 5 lists the r.m.s. residuals
of the fit; the observed dispersion exceeds the observational error, and the
physical range of the y-parameter is about four times the quoted r.m.s. Column
6 gives the Spearman correlation coefficient of the fit, ranging from -1.00
(perfect anti-correlation) to 1.00 (perfect correlation). Coefficients in the
range -0.35 to +0.35 should be interpreted as indicating a lack of
correlation, especially when slope a is also close to zero.

###  4.5 The ground-state HNC/HCN isomer ratio

The HCN isomer HNC(1-0) was detected by Hüttemeister $\rm et\,al.\ $ (1995) in
fifteen normal galaxies and mapped in five with the $IRAM$ 30m telescope. They
determined the isomer ratios by using older HCN measurements. Aalto $\rm
et\,al.\ $ (2002) presented a complementary HNC(1-0) survey of nine luminous
infrared galaxies, obtained with the relatively large beams of the $OSO$
($42"$) and $SEST$ ($55"$) telescopes. The survey of 23 galaxies by
Costagliola $\rm et\,al.\ $ (2011), also incorporated into our database,
covered a variety of molecular lines in the 3mm window, including HCN(1-0) and
HNC(1-0). In all surveys, the ground-state isomer ratio HNC/HCN varies
significantly from galaxy to galaxy. A strong motivation for HCN and HNC
observations is the potential use of their ratio as a diagnostic for molecular
gas kinetic temperature and density (Schilke $\rm et\,al.\ $ 1992, Graninger
$\rm et\,al.\ $ 2014), specifically sensitive to UV but not X-ray irradiation
(Bublitz $\rm et\,al.\ $ (2019, 2022). As in the previous smaller samples, the
HNC/HCN ratios of the more than ninety galaxies presented here are all in the
range 0.2–1.1; the average is 0.46 (Table 6). There is no correlation with
global far-infrared luminosity (also noted by Aalto $\rm et\,al.\ $ 2002), nor
with CO luminosity. Hüttemeister $\rm et\,al.\ $ (1995) did not find a
correlation with putative tracers of star formation rate or dust temperature.
In our much larger sample, the HCN/HNC ratio is wholly unrelated to either the
HCN/CO ratio or the HCN/FIR ratio. We confirm the anti-correlation found by
Costagliola $\rm et\,al.\ $ (2011) between the $\,{\rm HCO^{+}}$/HCN and the
HNC/HCN ratios when they are plotted against the HNC/$\,{\rm HCO^{+}}$ ratio
(Fig. 7, bottom panels) but unlike these authors we see no relation when they
are plotted against each other or sorted by galaxy type (Fig. 7 center right
panel). We did not find any other correlation involving the isomer ratio.

###  4.6 The excitation ladders of HCN, $\,{\rm HCO^{+}}$ and HNC

The normalized $J$=3-2 HCN and $\,{\rm HCO^{+}}$ luminosities are, just like
the ground-state luminosities, well-correlated with the $J$=1-0 CO
luminosities. The super-unity slope of about 1.12 is identical to the slope of
1.11 found by Li$\eta$ (2020) for the relation between HCN(3-2) and FIR
luminosities in matched apertures, and less than the more poorly defined slope
of 1.26-1.35 found by Bussman $\rm et\,al.\ $ (2008) in the larger HHST beam.
In Fig. 8 the individual HCN(2-1) (taken from Krips $\rm et\,al.\ $ 2008),
HCN(3-2), HCN(4-3), and HNC(3-2) intensities relative to the ground-state are
plotted as a function of the CO(1-0) luminosity. With increasing transition
the average line intensities drop (columns 9-11 in Table 6, and Fig. 8). Using
that figure to remove the bias due to unequal sampling, we find that the
average intensities of the HCN excitation ladder are represented by the series
1.00:0.65:0.45:0.25 which is steeper than the comparable CO ladder of
1.00:0.92:0.70:0.57 (Israel 2020). The averages for $\,{\rm HCO^{+}}$ are
practically identical (Table 6) columns 13-14). In all $J$ transitions,
individual HCN and $\,{\rm HCO^{+}}$ line intensities are very similar (Fig.
9. Because of the paucity of especially JCMT $J$=4-3 $\,{\rm HCO^{+}}$
observations, we added to this figure line ratios from the $ALMA$ observations
of compact (ultra)luminous galaxies from Imanishi $\rm et\,al.\ $ (2018) and
the APEX observations of normal galaxies from Zhang $\rm et\,al.\ $ (2014).
The APEX $J$=4-3 and JCMT $J$=3-2 beams are almost identical facilitating the
construction of $J$=4-3/$J$=3-2 ratios.

With each additional step on the ladder, average HCN and $\,{\rm HCO^{+}}$
intensities decrease but within each step, individual ratios rise with
increasing CO luminosity (Table 7, Fig 8). The $J$=1-0, $J$=3-2, and $J$=4-3
intensities HCN and $\,{\rm HCO^{+}}$ increase in tandem (leftmost panels in
Fig. 9). In each transition, their ratio is a constant and unrelated to either
CO (rightmost panels in Fig. .9 or FIR luminosity (Tan $\rm et\,al.\ $ 2018).

A similar lack of correlation characterizes the HCN/HNC ratios in the two
transitions. Again the spread in the $J$=3-2 values is about double that in
the $J$=1-0 values but the small sample reveals no distinction between AGNs
and starburst. In Figs. 5 and 6 we found the $\,{\rm HCO^{+}}$(1-0)/HCN(1-0)
species ratio to be inversely correlated with both the ‘dense-gas fraction’
HCN(1-0)/CO and the isotopologue ratio $\,{\rm{}^{12}CO}$/$\,{\rm{}^{13}CO}$
but independent from the CO luminosity and the ‘star formation efficiency’
FIR/CO. The excitation of HCN (and $\,{\rm HCO^{+}}$) is also unrelated to any
of these quantities as illustrated by plots of the $J$=3-2/$J$=1-0 ratios
against them. Fig. 10 shows that the excitation of HCN is not meaningfully
related to the isotopogical $\,{\rm{}^{12}CO}/\,{\rm{}^{13}CO}$ ratio (left
panel) or to HCN/CO (not shown) and FIR/CO (right panel). Finally, Fig. 11
shows that the excitation of HNC is not related to the luminosities of HCN
(left panel) or CO, and that the HCN/HNC isomer and the HCN(3-2)/HCN(1-0)
excitation ratio are likewise unrelated.

##  5 Discussion

###  5.1 Luminosity plots provide little information

The results in this paper confirm that the linear relations between the
HCN(1-0), CO(1-0), and FIR luminosities of whole galaxies, first established
by Solomon $\rm et\,al.\ $ (1992) and Gao $\&$ Solomon (2004a), also hold for
constant aperture measurements of galaxy centers. They also show that the
surface brightness of CO, HCN, and $\,{\rm HCO^{+}}$ drops in similar ways
when the area covered by the aperture increases. Variations in surface
brightness and intensity ratios are almost negligible compared to the
variation in galaxy distances. Distance is the single factor dominating the
luminosities in fixed apertures. This is the key factor explaining the near-
linearity of luminosity-luminosity relations and it also explains why
luminosity correlations look progressively better as greater ranges in
distance are considered. As a result, the luminosity-luminosity relations
published in the literature, and also presented here, are mostly trivial and
provide very little information on the physical properties of the galaxies
sampled555see also Kennicutt (1990). Specifically, the observation that linear
luminosity relations hold over a wide range of luminosities does not provide
evidence that the extra-galactic star formation rate is directly proportional
to the mass of dense gas. The scatter in the various luminosity-luminosity
plots carries more information than their linearity. The non-negligible
dispersion involves factors of five or more and indicates significant
variation among individual galaxies even as a systematical change with
luminosity is absent. Intensity ratios provide a much better means of
evaluating the information conveyed by molecular line emission than
luminosities.

The luminosity plots do not reveal systematical differences between the
various galaxy types either. The only exception is the $\,{\rm HCO^{+}}$
intensity which is always below that of HCN in AGN galaxies. Over the observed
range $9.0\,\leq$ log $L(FIR)$ (L⊙) $\leq\,12.5$, the linearity of the various
CO, HCN, or $\,{\rm HCO^{+}}$ luminosity relations does, however, imply that
interstellar medium (ISM) properties do not change as a function of galaxy
luminosity.

###  5.2 HCN and HCO+ trace the same gas

The average $J$=1-0 $\,{\rm HCO^{+}}$ to HCN intensity ratio is about 0.9 but
individual values can be up to three times higher or lower. The behavior of
$\,{\rm HCO^{+}}$ and HCN is practically the same in all observed transitions
(Fig. 9) which implies that both molecules essentially trace the same gas,
even at the higher critical densities. Over a wide range of luminosities, the
central regions of other galaxies are thus much like the inner Milky Way,
where Evans $\rm et\,al.\ $ (2020) found that HCN(1-0) and $\,{\rm
HCO^{+}}$(1-0) both seem to probe the same molecular cloud material. When
large areas are sampled instantaneously, as in extra-galactic measurements,
the integrated HCN or $\,{\rm HCO^{+}}$ emission from ensembles of molecular
gas clouds is easily dominated by extended regions of low density, down to 102
$\,{\rm cm^{-3}}$ (Evans $\rm et\,al.\ $ 2020). This may go some way to
explain the similar behavior of the emission from HCN and $\,{\rm HCO^{+}}$,
as well as CO, notwithstanding the different critical densities sampled (Table
1). From a practical point of view, in most of the analysis HCN and $\,{\rm
HCO^{+}}$ are interchangeable molecular species.

AGNs do have systematically higher HCN/$\,{\rm HCO^{+}}$ ratios than star-
burst galaxies (Kohno $\rm et\,al.\ $ 2001, 2008; Imanishi $\rm et\,al.\ $
2007; Krips $\rm et\,al.\ $ 2008, Gracía-Carpio $\rm et\,al.\ $ 2008, and
various later papers). As HCN/CO ratios also appeared to be higher in AGN-
dominated galaxes, these authors concluded to an HCN over-luminosity
reflecting an HCN overabundance. Figs. 4 and 5 confirm that ground-state HCN
intensities exceed those of $\,{\rm HCO^{+}}$ in all AGN galaxies (except NGC
4258, Li $\rm et\,al.\ $ 2019) unlike star-burst galaxies. The situation is
complicated by overlooked AGNs, embedded in ULIRGs (Imanishi $\rm et\,al.\ $
2007, Imanishi 2009, Li $\rm et\,al.\ $ 2021). High-resolution observations
such as provided by ALMA may bring to light significant small-scale variation
that is smoothed out in the larger-beam observations considered here. AGNs
such as those in e.g. NGC 7469 (Izumi $\rm et\,al.\ $ 2020) and NGC 1068
(Butterworth $\rm et\,al.\ $ 2022) become manifest by the change in line
behavior in their immediate ($\sim$100 pc) surroundings. Fig. 9 includes the
HCN/$\,{\rm HCO^{+}}$ ratios of distant ULIRGS (12.0$\leq$log $L$(FIR)$\leq
12.3$) in the $J$=3-2 and $J$=4-3 transitions observed with ALMA (Imanishi
$\rm et\,al.\ $ 2018). The distinction between AGN and starburst galaxies is
less obvious in the higher transitions. In these galaxies, however, the
compact nuclear regions (diameter $\leq$ 500 pc) have again higher HCN/HCO+
ratios than the surrounding extended regions.

Attempts to explain high HCN/HCO+ ratios by analyzing the few observed
optically thick molecular lines in terms of star formation or (X-ray)
chemistry with radiative transfer models failed to produce conclusive results
(cf. Costagliola $\rm et\,al.\ $ 2011, Izumi $\rm et\,al.\ $ 2016, Imanishi
$\rm et\,al.\ $ 2016, Privon $\rm et\,al.\ $ 2020). We expect such modelling
to be successful only when molecular species are selected specifically capable
of diagnostically distinguishing different excitation mechanisms and, as in
the case of CO (Israel 2020), only when multiple transitions, including
optically thin lines in addition to optically thick lines, are considered.

Although the AGNs in our large sample have systematically higher
HCN(1-0)/HCO+(1-0) ratios, they do not have systematically higher ratios
HCN(1-0)/CO(1-0) (Fig. 4), and Aladro $\rm et\,al.\ $ (2015) found that their
HCN(1-0)/CS(3-2) ratios are not systematically higher, either. This suggests
that in AGNs, $\,{\rm HCO^{+}}$ is suppressed rather than HCN enhanced. This
suggestion is shared, on different grounds, by Imanishi $\rm et\,al.\ $
(2022). Papadopoulos (2007) has argued that environmental conditions
differently affecting the chemistries of ions and neutral molecules naturally
lead to $\,{\rm HCO^{+}}$ suppression. For instance, in the turbulent
molecular clouds that dominate in galaxy centers (cf. Israel 2020), free
electrons from the ionized outer layers are transported inwards, effectively
suppressing the $\,{\rm HCO^{+}}$ ion in addition to exciting HCN. Better
understanding requires a greater variety of diagnostic line measurements to
constrain models. In the absence of independent evidence whether $\,{\rm
HCO^{+}}$ or HCN is primarily affected by AGN environments, any conclusion
obtained from modelling is bound to remain speculative.

###  5.3 Modest optical depths suggest large translucent fraction

Decreases in optical depth $\tau$ and in isotopic abundance ratio
[$\,{\rm{}^{13}CO}$]/[$\,{\rm{}^{12}CO}$] both express themselves through
increasing isotopologue intensity ratios
$I(\,{\rm{}^{12}CO})/I(\,{\rm{}^{13}CO}$). The observed isotopologue ratios
increase with CO luminosity (Fig. 6, left panel) as well as FIR luminosity.
This may be due to (a). significant carbon monoxide photo-dissociation, (b).
selective $\,{\rm{}^{12}CO}$ nucleosynthesis, or (c). both. (a). Photo-
dissociation of CO in the strong radiation fields of luminous galaxies would
diminish the optical depths of both $\,{\rm{}^{12}CO}$ ($\tau_{12CO}$) and
$\,{\rm{}^{13}CO}$ ($\tau_{13CO}$) and cause the observed line ratio to
approach the isotopic abundance ratio, which itself may increase with
luminosity (Visser $\rm et\,al.\ $ 2009). Because luminous galaxies are
actively forming stars, low optical depths $\tau_{12CO}$ signify low
abundances [$\,{\rm{}^{12}CO}$]/[$\,{\rm H_{2}}$] rather than low overall ISM
optical depths. (b) Whereas selective photo-dissociation of $\,{\rm{}^{13}CO}$
appears to be unimportant (Visser $\rm et\,al.\ $ 2009, Romano $\rm et\,al.\ $
2017), selective nucleosynthesis of $\,{\rm{}^{12}CO}$ may occur in star-
forming galaxies rendering $\,{\rm{}^{12}CO}$ overabundant (Sage $\rm et\,al.\
$ 1991; Wilson 1999; Romano 2017). Thus, $\tau_{12CO}$ would increase with
respect to $\tau_{13CO}$. The increased aperture filling fraction of optically
thick emission raises observed $\,{\rm{}^{12}CO}$ intensity relative to
$\,{\rm{}^{13}CO}$ intensity.

The observed HCN-to-CO or FIR-to-CO ratios do not correlate with the
$I(\,{\rm{}^{12}CO})/I(\,{\rm{}^{13}CO}$) isotopologue ratio. This argues
against (b). because enhanced CO intensities would depress these ratios, and
supports (a) where $\,{\rm{}^{12}CO}$ intensities do not fall with decreasing
optical depths as long as $\tau_{12CO}\,\geq\,1$. Inferred optical depths
decrease from $\tau_{13CO}\,\approx\,0.08-0.13$ for the low-luminosity
galaxies to $\tau_{13CO}\,\approx\,0.03-0.07$ for the luminous galaxies. For
isotopic abundance ratios of 40, typical for nearby galaxy centers (cf. Viti
$\rm et\,al.\ $ 2020) these correspond to $\tau_{12CO}\,\approx\,3-5$ (low
luminosity) and .$\tau_{12CO}\,\approx\,1-3$ (high luminosity). For typical
luminous galaxy isotopic abundances 100 (cf. Viti $\rm et\,al.\ $ 2020), the
corresponding $\tau_{12CO}\,\approx\,3-8$ is similar to that of low-luminosity
galaxies. It appears that the ground-state lines of our sample are just
optically thick with optical depths of a few. The modest CO optical depths
represent averages over relatively large surface areas and suggest that the
ensemble of molecular clouds sampled contains a significant fraction of
translucent clouds in addition to more dense clouds. In the environments
sampled, low CO optical depths do not imply equally low molecular hydrogen
column densities $N(\,{\rm H_{2}})$ (cf. Israel 2020). Optical depths of the
very weak $J$=1-0 H13CN, H$\,{\rm{}^{13}CO}^{+}$, and HN13C lines have been
determined for a small number of bright galaxies (Nguyen-Q-Rieu $\rm et\,al.\
$ 1992, Costagliola $\rm et\,al.\ $ 2011, Jiang $\rm et\,al.\ $ 2011, Aladro
$\rm et\,al.\ $ 2015, Jiménez-Donaire $\rm et\,al.\ $ 2017, Li $\rm et\,al.\ $
2019, 2020b). Optical depths $\tau_{H13CN}$ are typically 0.02-0.15 with
higher values 0.25-0.35 for some LIRGs such as NGC 1614 and UGC 05101. Optical
depths $\tau_{H13CO}$ are in the range 0.02-0.10, not very different from
either $\tau_{H13CN}$ or $\tau_{13CO}$. These low optical depths likewise
suggest that a non-negligible fraction of the emission in these lines comes
from translucent gas. As already referred to in the previous section, HCN(1-0)
emission from Milky Way molecular clouds is easily detected also in extended
translucent regions with densities $N(\,{\rm H_{2}})\sim 500\,\,{\rm cm^{-3}}$
(Pety $\rm et\,al.\ $ 2017, Evans $\rm et\,al.\ $ 2020). At low
$\,{\rm{}^{13}CO}$ optical depths, the HCN/$\,{\rm HCO^{+}}$ intensity ratio
drops to values below unity (Fig. 6, right panel). In that case, a larger
fraction of the molecular gas in luminous galaxies is at reduced (column)
densities better traced by $\,{\rm HCO^{+}}$ than by HCN.

###  5.4 The dense molecular gas is lukewarm

The model calculations by Schilke $\rm et\,al.\ $ (1992) illustrate how the
HCN/HNC ratio increases with rising molecular gas kinetic temperature and
density, presumably caused by temperature-dependent destruction of HNC in
neutral-neutral reactions. Despite the degeneracy between temperature and
density, the analysis of observations in Orion allowed Hacar $\rm et\,al.\ $
(2020) to derive an empirical relation between the HCN/HNC ratio and the
independently determined kinematic gas temperature. The ratios
$I$(HCN)/$I$(HNC) = 1-5 observed by us are in the well-established part of the
relation in their Figure 3, where
$T_{kin}\,\approx\,10\,\times\,I$(HCN)/$I$(HNC). Thus, the kinetic
temperatures of the HCN gas in the sample galaxies are $T_{kin}$ = 10 \- 50 K,
with a mean of 20 K. These are beam-averaged temperatures: a temperature of 20
K might, for instance, also represent a mixture of 25 per cent gas at 50 K and
and 75 per cent gas at 10 K etc. The absence of ratios HCN/HNC$\geq$5,
however, rules out substantial contributions by shocked or high-temperature
gas (cf. Schilke $\rm et\,al.\ $ 1992; Hacar $\rm et\,al.\ $ 2020).

The analysis by Hacar $\rm et\,al.\ $ (2020) also reveals a strong dependence
of individual HCN and HNC intensities on gas temperature,
$I(HNC)\,\propto\,T_{\rm k}$ and $I(HCN)\,\propto\,T_{\rm k}^{2}$. At any HCN
intensity the corresponding molecular gas column density varies by up to an
order of magnitude depending on the actual temperature. The authors suggest
that these emissivity effects render extra-galactic HCN measurements
unreliable as tracers of dense molecular gas content. For the observed HNC/HCN
ratios, the PDR models by Meijerink $\rm et\,al.\ $ (2007) require molecular
gas column densities $N(H_{2})\,\lessapprox\,10^{22}$ cm-3.

###  5.5 Most of the lines are sub-thermally excited

In the models by Schilke $\rm et\,al.\ $ (1992), relatively modest typical gas
densities $n_{H}=n(HI)+2n(H_{2})\sim 10^{4}$ cm-3 accompany low HCN/HNC
ratios. The single-phase LVG models explored by Gracía-Carpio $\rm et\,al.\ $
(2008) to explain their $J$=1-0, $J$=3-2 HCN and $\,{\rm HCO^{+}}$
measurements also yield densities $n(\,{\rm H_{2}})\approx 6\times 10^{4}$
cm-3 as well as abundances [HCN]/[$\,{\rm HCO^{+}}$]$\approx$6\. We use the
Meijerink $\rm et\,al.\ $ (2007) grid of models of gas excitation by UV
photons (PDR) and by X-ray photons (XDR) to further investigate the gas
density. The observed HCN/$\,{\rm HCO^{+}}$ ratios fit the PDR excitation for
densities $n_{H}\,\approx\,(5-20)\times 10^{4}$cm-3 and UV irradiation
$G\,=\,(5-50)\times 10^{3}\,$G0666$G_{0}\,=\,1.6\times 10^{-3}$ erg cm-2 s-1,
Habing 1968. At the lowest CO luminosities ($L$(CO)=10${}^{7}-10^{8}$ K km/s
pc2), the implied densities are at the lower end with low implied irradiation
$G\,\leq 10^{2}\,$G0. The irradiation requirements are much less if other
heating mechanisms such as XDR or mechanical heating (Meijerink $\rm et\,al.\
$ 2006; Kazandjian $\rm et\,al.\ $ 2012, 2015) apply. Their work shows that
XDR excitation is unlikely, as also concluded by Pérez-Beapuits $\rm et\,al.\
$ (2007) who suggested similar densities for a small sample of Seyfert
galaxies.

Densities $n_{H}\,\approx\,5-20\times 10^{4}$ are above the critical density
for $\,{\rm HCO^{+}}$(1-0) but just below the critical density for HCN(1-0)
and HNC(1-0). They are well below the critical densities for all other
transitions of these molecules. All these lines are thus sub-thermally
excited. The HCN/CO model ratios from Meierink $\rm et\,al.\ $ (2007) suggest
that roughly between 5 and 20 per cent of the observed CO intensity originates
in the dense gas that radiates in HCN and $\,{\rm HCO^{+}}$ whereas the
remaining 80-95 per cent then comes, as found before, from less dense gas with
typical densities $n_{H}\,\approx\,10^{3}$ cm-3777These are emission
fractions, not mass fractions which may be different depending on the
excitation of the lines. The isotopologue ratio
$\,{\rm{}^{12}CO}$/$\,{\rm{}^{13}CO}$ is correlated with the dust temperature
derived from the far-infrared continuum (Costagliola $\rm et\,al.\ $ 2011, and
references therein) but not with the temperature of the dense gas derived from
the HCN/HNC isomer ratio (Fig. 7). This confirms that most of the CO-emitting
gas must be quite distinct from the HCN-emitting gas.

###  5.6 HCN/CO and FIR/CO do not trace dense gas ratio or star formation
efficiency

The ground-state HCN/CO and HCN/FIR intensity ratios are largely unrelated to
either distance or luminosity.

The intensity ratios FIR/HCN (or FIR/CO) and HCN/CO have been widely portrayed
as proxies for star formation efficiency888FIR/HCN and FIR/CO do not have the
dimension of an efficiency factor. Instead, they are the inverse of the time
in which the present amount of (dense) gas will be depleted at the present
rate of star formation – assuming that far-infrared and molecular line
intensities indeed measure what they are purported to do. respectively the
fraction of molecular gas that resides in dense gas (see e.g. Gao $\&$ Solomon
2004b, more recently Tan $\rm et\,al.\ $ 2018; Jiménez-Donaire $\rm et\,al.\ $
2019). This assumes that FIR continuum luminosities are directly proportional
to star formation rates, and that velocity-integrated CO and HCN line
intensities are directly proportional to the column densities of all molecular
gas and dense molecular gas, respectively.

Unfortunately, that is not exactly true for CO where factors such as
abundance, dynamics, and excitation all contribute to significant individual
as well as systematic variations in the relation between line intensity and
underlying gas column density (cf. Pety $\rm et\,al.\ $ 2017, Israel 2020).
Doubts have been expressed to various degrees also on the use of HCN and
$\,{\rm HCO^{+}}$ as reliable mass tracers by several authors (e.g.
Papadopoulos $\rm et\,al.\ $ 2007, 2012; Krips $\rm et\,al.\ $ 2008,
Costagliola $\rm et\,al.\ $ 2011, Privon $\rm et\,al.\ $ 2015, Mills $\&$
Battersby (2017), Pety $\rm et\,al.\ $ 2017, Graciá-Carpio $\rm et\,al.\ $
2020, Hacar $\rm et\,al.\ $ 2020), prompted by considerations of excitation,
chemistry, and radiative transfer. Recently, Li $\rm et\,al.\ $ (2020a)
specifically emphasized the large uncertainties in dense gas mass estimated
from a single line transition.

Single dish measurements of galaxies typically cover large areas sampling a
wide range of different ISM conditions. Studies simulating such measurements
with molecular line surveys of Milky Way areas (e.g. Stephens $\rm et\,al.\ $
2016, Shimajiri $\rm et\,al.\ $ 2017, Evans $\rm et\,al.\ $ 2020) show that
extended sub-thermal emission from marginally dense gas outshines the emission
from high-density filaments and clumps. In nearby galaxy centers (e.g.
Circinus galaxy, NGC 4945, M 51, NGC 253, NGC 1808), high $J$=1-0 HCN/CO
ratios occur that are directly ascribable to environmental conditions other
than mass (Curran $\rm et\,al.\ $ 2001, Matsushita $\rm et\,al.\ $ 2015,
Walter $\rm et\,al.\ $ 2017, Salak $\rm et\,al.\ $ 2018). Molecular line
intensities are a function of gas properties (chemical abundance, density,
opacity) and environmental conditions (irradiation, turbulence, shocks) that
dominate chemistry and excitation mechanism. These do not necessarily average
out and molecular gas masses are not simple, linear functions of measured line
luminosities.

The observational results in this paper emphasize the inconsistency of HCN/CO
ratios as an indicator of the dense gas fraction. The variation from galaxy to
galaxy is the same at any given intensity or luminosity ratio. No ratio of HCN
or $\,{\rm HCO^{+}}$ to CO or FIR intensity is systematically correlated with
the luminosity of a molecular line, with the FIR continuum luminosity, with
the observed isotopological ratio $I(^{12}$)CO/$I(^{13}$), or with distance.
Nor is there any systematic difference in HCN/CO ratio between normal
galaxies, starburst galaxies and AGNs. In the large number of central regions
sampled, the near-constancy of the various line ratio averages over a wide
range of luminosities is well-established. If, contrary to our assertion, the
HCN and CO luminosities were to reliably measure the respective masses of very
dense and modestly dense gas, there would still be no relation between the
’dense gas fraction’ HCN/CO and such quantities as total molecular gas mass or
CO optical depth. The purported dense gas fraction would also be wholly
unrelated to surface brightness or surface area covered.

Much the same can be said for the interpretation of the ratios FIR/CO,
FIR/HCN, and FIR/$\,{\rm HCO^{+}}$ as proxies for star formation efficiencies
(more correctly: inverse dense-gas-depletion times). The overwhelming lack of
correlation between tracers of different gas density and either the FIR
luminosity (also noted by Privon $\rm et\,al.\ $ 2015 and Li $\rm et\,al.\ $
2021), the FIR-to-HCN ratio, or the FIR-to-CO ratio is counter to what is
expected in the case of proxies for star formation rates and efficiencies. It
would mean that the star formation process is insensitive to the relative
amounts of dense ($\,{\rm HCO^{+}}$) and very dense gas (HCN). The star
formation efficiency would also be independent of surface area covered or
galaxy type, and there would be a surprising lack of any connection between
the rate and the efficiency of star formation. Specifically, it would imply
the same efficiency of star formation in local low-luminosity galaxies and in
very luminous galaxies (LIRGs and ULIRGs) where star formation rates are
usually considered to be one or two magnitudes higher. The overall lack of
significant correlations between the relevant data strongly implies that the
ground-state HCN-to-CO, FIR-to-CO, or FIR-to-HCN intensity ratios are not
meaningful as direct proxies for physical quantities such as the fraction of
dense gas in the ISM or the efficiency of star formation in galaxy centers.
Instead, especially Figs. 4 and 8 suggest that the line intensities of both CO
and HCN primarily reflect the excitation of the molecular gas.

###  5.7 The ISM in galaxy centers.

The ISM in galaxy centers is poor in neutral atomic hydrogen ($\,{\rm HI}$)
and almost all gas is molecular hydrogen ($\,{\rm H_{2}}$). Analysis of the CO
and $\,{\rm{}^{13}CO}$ emission in various transitions reveals coexisting
molecular gas phases distinguished by density and temperature (cf. Israel
2020). If we assume PDR-excitation for the galaxies in this paper, the
emission from HCN and $\,{\rm HCO^{+}}$ originates in molecular gas at average
densities not exceeding $n_{H}$ = $5\times 10^{4}$ \- $2\times 10^{5}$ $\,{\rm
cm^{-3}}$, low average temperatures of $T_{k}$ = $10-50$ K, and very moderate
optical depths $\tau_{CO}$ = 1 - 5. The gas traced by HCN and $\,{\rm
HCO^{+}}$ is itself only a small fraction ($\leq 20\%$) of the molecular gas
traced by CO with substantially lower densities of about $n_{H}$ = $10^{3}$
$\,{\rm cm^{-3}}$, the precise fraction depending on the actual line ratios
and the assumed mode of excitation.

Loenen $\rm et\,al.\ $ (2008) used HCN, HNC, and $\,{\rm HCO^{+}}$ line ratios
to investigate the excitation mechanism. In addition to the PDR and XDR models
from Meijerink $\rm et\,al.\ $ (2007), they also considered models with
mechanical heating of cloud volumes in addition to PDR surface heating. They
found that most of the line ratios from the Baan $\rm et\,al.\ $ (2008) sample
require PDRs dominated by mechanical heating. The bottom panels in Fig. 7 are
the counterparts to Figs. 1 and 2 by Loenen $\rm et\,al.\ $ (2008). They
exhibit overall similarity but significantly tighter distributions, reflecting
the greater accuracy of our more homogeneous and sensitive database. The
combined relative intensities of the HCN, $\,{\rm HCO^{+}}$ and HNC lines
conclusively rule out XDR dominating the excitation in all galaxies as well as
pure PDR excitation in most galaxies (cf. Meijerink $\rm et\,al.\ $ 2007,
Loenen $\rm et\,al.\ $ 2008). The molecular gas excitation in our sample is
almost certainly dominated by mechanical heating. As a consequence, the
density and especially the irradiation requirements are relaxed to lower
values. Following Kazandjian $\rm et\,al.\ $ (2015), average densities are
$n_{H}\,=\,10^{4}-10^{5}\,\,{\rm cm^{-3}}$ and the implied average irradiation
drops from $G\,=3\times 10^{4}$ G0 to $G=3\times 10^{2}$ Go for mechanical
heating fractions $\alpha$ increasing from 0.1 to 0.5. The decrease by two
orders of magnitude reflects the greater efficiency of volume over surface
heating.

The relatively low density, low optical depth, and low irradiation of the
molecular gas in galaxy centers are characteristic of an ISM not actively
engaged in star formation. There is no independent evidence directly and
unambiguously linking molecular line intensities such as $J$=1-0 CO, HCN, or
$\,{\rm HCO^{+}}$ to gas column density or mass. In this paper, we also
observed an overall lack of significant correlations between line ratios and
properties that could be interpreted as related to large-scale physical
processes. Taken together, this forces the conclusion that, in spite of much
effort over the last few decades, single-dish molecular line observations
allow little more than assumption-driven speculation and are of little use in
the quantitative determination of star formation in galaxy centers.

The significant dispersion of measured line intensities and ratios around the
average values, typically a factor of three either way, implies different ISM
characteristics for different galaxies, including varying gas phase
combinations. Individual ISM descriptions surpassing the average treatment
given in this paper can be derived from detailed multi-transition, multiple-
species radiative transfer modelling. Preliminary examples have already been
published, such as M 82 (Loenen $\rm et\,al.\ $ 2010), NGC 253 (Rosenberg $\rm
et\,al.\ $ 2014a), and Arp 299 (Rosenberg $\rm et\,al.\ $, 2014b). The results
discussed in this paper show that only thus we may hope to identify the actual
ISM physics that underlies the observed patterns or lack thereof. Such a more
detailed treatment of the galaxies in the present sample is deferred to a
subsequent paper.

##  6 Conclusions

  1. 1.

This paper presents new $IRAM$ and $JCMT$ observations of 46 bright galaxies
in the $J$=1-0, $J$=3-2, and $J$4-3 transitions of HCN, $\,{\rm HCO^{+}}$ and
HNC. These are complemented by similar observations as well as
$\,{\rm{}^{12}CO}$(1-0) and $\,{\rm{}^{13}CO}$(1-0) from published IRAM and
JCMT surveys. The resulting extensive database covers 130 galaxies in HCN(1-0)
and $\,{\rm HCO^{+}}$(1-0) and 94 galaxies in HNC(1-0). In addition, it
includes 12 galaxies in HCN(2-1), about 50 galaxies in HCN(3-2) and $\,{\rm
HCO^{+}}$(3-2), 25 galaxies in HCN(4-3), and 18 galaxies in HNC(3-2) and
$\,{\rm HCO^{+}}$(4-3).

  2. 2.

The observed intensities were normalized to a common resolution of $22"$ in
order to produce meaningful line intensity ratios. The analysis systematically
explores luminosity-luminosity relations as well as relations between line
ratios, and relations between line ratios and luminosities.

  3. 3.

As expected from previous work, the $J$=1-0 HCN, $\,{\rm HCO^{+}}$, and HNC
luminosities are all linearly related to CO(1-0) and far-infrared (FIR)
luminosities. We also find that this is true for the luminosities in the
higher transitions of HCN(3-2), $\,{\rm HCO^{+}}$(3-2), HCN(3-2), HCN(4-3),
and $\,{\rm HCO^{+}}$(4-3). Very little can be concluded from this, however,
as the luminosity-luminostity relations are essentially trivial because the
luminosities are dominated by the variation in distance and not by the physics
of the galaxies sampled.

  4. 4.

Individual galaxy luminosities and line ratios show significant dispersion
around the mean in all comparisons, implying significant differences in
molecular gas properties between individual galaxies. The dispersion is
uncorrelated with luminosity or line ratio and more likely originates in the
detailed ISM physics than in systematic large-scale galaxy properties.
Analysis of the ISM in individual galaxies is deferred to a later paper.

  5. 5.

The average normalized HCN and $\,{\rm HCO^{+}}$ transitions ladders,
$J=$(n+1$\rightarrow$n)/$J$=(1-0), as well as the isotopological ratio $J$=1-0
$\,{\rm{}^{12}CO}$/$\,{\rm{}^{13}CO}$ are positively correlated with CO and
FIR luminosity. No other line ratio shows such a clear correlation with
luminosity.

  6. 6.

HCN and $\,{\rm HCO^{+}}$ have almost equal intensities and behave very
similarly across the entire sample. These two molecules trace the the same
gas, notwithstanding a significant difference in critical density. In AGN-
dominated galaxies, ground-state HCN intensities always exceed those of
$\,{\rm HCO^{+}}$. Suppression of $\,{\rm HCO^{+}}$ intensity is more likely
than the alternative of HCN-enhancement in starburst galaxies.

  7. 7.

Our radiative transfer models show that only $5-20\%$ of the observed CO
emission originates in the HCN/$\,{\rm HCO^{+}}$-emitting molecular gas. The
HCN and $\,{\rm HCO^{+}}$ emission represents a mixture of dense gas clouds
and an undetermined but significant amount of translucent molecular gas.
Except for $\,{\rm HCO^{+}}$(1-0), all observed lines are sub-thermally
excited.

  8. 8.

The observed CO, HCN, and $\,{\rm HCO^{+}}$ emission is not simply related to
molecular gas column density or mass. These lines reflect the excitation of
the gas, they are affected by the gas opacity and abundance, but they are not
reliable mass tracers.

  9. 9.

The HCN/CO line intensity ratio cannot be used as a proxy for the dense gas
fraction, and the FIR/HCN and FIR/CO intensity ratios are also meaningless as
proxies for star formation efficiencies or even molecular gas depletion times.
Because the molecular lines do not reliably trace mass, comprehensive
understanding of star formation requires a more appropriate determination of
gas mass.

  10. 10.

The molecular line emission from galaxy centers rules out a dominant heating
contribution by X-rays (XDRs) but is fully consistent with UV-photon heating
(PDRs) enhanced by a significant mechanical heating contribution due to
turbulence or shocks.

  11. 11.

The densest molecular gas in the galaxy centers sampled by ground-state HCN
and $\,{\rm HCO^{+}}$ lines has relatively low average kinetic temperatures
$T_{\rm kin}\,=\,10-50$ K, relatively low average densities $n_{\rm
H}\,=\,10^{4}-10^{5}\,\,{\rm cm^{-3}}$, and relatively low optical depths of
only a few. Most of the gas sampled by CO has densities $n_{\rm
H}\,\leq=\,10^{3}\,\,{\rm cm^{-3}}$. If the mechanical heating fraction is
$50\%$, the energy input required is only $G\,\sim\,300$ G0.

## Acknowledgements

Most of the JCMT observations were obtained in service mode. I am especially
indebted to the $JCMT$ operators and observers, who helped to collect the
large data base described in this paper.

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