# A Complete HCN Survey of the Perseus Molecular Cloud

T. M. Dame Center for Astrophysics $|$ Harvard & Smithsonian
60 Garden Street
Cambridge, MA 02138, USA  Charles J. Lada Center for Astrophysics $|$ Harvard
& Smithsonian
60 Garden Street
Cambridge, MA 02138, USA

###### Abstract

We present a survey of the Perseus molecular cloud in the J $=$
1$\rightarrow$0 transition of HCN, a widely used tracer of dense molecular
gas. The survey was conducted with the CfA 1.2 m telescope, which at 89 GHz
has a beam width of 11’ and a spectral resolution of 0.85 km s-1. A total of
8.1 deg2 was surveyed on a uniform 10’ grid to a sensitivity of 14 mK per
channel. The survey was compared with similar surveys of CO and dust in order
to study and calibrate the HCN line as a dense gas tracer. We find the HCN
emission to extend over a considerable fraction of the cloud. We show that the
HCN intensity remains linear with H2 column density well into the regime where
the CO line saturates. We use radiative transfer modeling to show that this
likely results from subthermal excitation of HCN in a cloud where the column
and volume densities of H2 are positively correlated. To match our HCN
observations the model requires an exponential decrease in HCN abundance with
increasing extinction, consistent with HCN depletion onto grains. The modeling
also reveals that the mean volume density of H2 in the HCN emitting regions is
$\sim$ 104 cm-3, well below the HCN critical density. For the first time, we
obtain a direct measurement of the ratio of dense gas mass to HCN luminosity
for an entire nearby molecular cloud: $\alpha$(HCN) $=$ 92 M⊙/(K km s-1 pc2).

Molecular clouds (1072); Interstellar molecules (849); Interstellar line
emission (844); Interstellar dust (836)

##  1 Introduction

The lowest rotational transition of CO at 115 GHz is widely acknowledged to be
the best general-purpose tracer of the total mass, distribution, and
kinematics of the largely unseen H2 gas in molecular clouds and it is for this
reason that the CfA 1.2 meter telescope has mainly remained tuned to CO for
over four decades. Surveys by this and other telescopes have taught us that
star formation takes place almost exclusively within molecular clouds, but
surprisingly the rate of star formation in a molecular cloud seems to depend
much less on its total mass than on the mass of gas it contains in the form of
dense cores and filaments (Lada et al. 2010). It has long been appreciated
that star formation occurs within such dense regions (e.g., Lada 1992), but
their high opacity in CO and limited spatial extents make them inconspicuous
in CO surveys. Instead, in Galactic clouds, rarer molecular species, such as
CS, HCN, NH3, and N2H+, with higher dipole moments than CO, have traditionally
been used to trace the densest gas in the clouds (e.g., Martin & Barrett 1978,
Benson & Myers 1989, Jackson et al. 2013, Pety et al. 2017 and many others)
since such molecules are not as easily collisionally excited to a state of
local thermodynamic equilibrium as CO. However, complexities of molecular
excitation, radiative transfer, and chemistry have long rendered direct,
quantitative interpretation of such subthermal molecular line emission
problematic.

As a result, measurements of infrared dust extinction (e.g., Lada et al. 2010,
2012) and dust continuum emission (e.g., Lombardi et al. 2014, Zari et al.
2016) have been used to more effectively trace the dense component of
molecular clouds. After H2, dust is the second most abundant constituent of
molecular clouds and is well mixed with the H2 gas. Because measurements of
the dust are not hampered by the factors (e.g., excitation, opacity, and
chemistry) that complicate the interpretation of molecular line observations,
dust provides more direct and reliable measurements of the total column
densities and masses of molecular clouds. Unfortunately, current observational
capabilities largely limit the use of dust as a dense gas tracer in distant
clouds and external galaxies and observations of molecular lines still remain
the best potential tracers of dense molecular material outside the Galaxy. To
increase the effectiveness of these molecular-line tracers it is essential to
calibrate them with dust measurements wherever possible.

After CO and its isotopologues, HCN is among the most readily detected
molecular species at millimeter wavelengths with the potential to trace dense
gas in molecular clouds. Knowledge of the dense gas contents of molecular
clouds is crucial for understanding star formation and galaxy evolution. This
is because local GMCs in the Milky Way are described by a star formation law
in which their star formation rates (SFRs) are both tightly and linearly
correlated with the masses of high column density gas in the clouds (Lada et
al. 2010, 2012). A similar linear star formation law also appears to
characterize entire galaxies where their total infrared luminosities (a
presumed proxy for SFRs) are tightly correlated with their HCN luminosities
(Gao & Solomon 2004, Shimajiri et al. 2017, Jiménez-Donaire et al. 2019). How
are these two star formation laws physically connected? Clearly, calibrated
HCN observations are of high astrophysical significance for ultimately
understanding the nature of this connection.

In their widely-cited paper, Gao & Solomon (2004) emphasized the need for
“systematic and unbiased large-scale observations” of HCN in Galactic
molecular clouds in order to calibrate HCN as a dense gas tracer and to better
understand the physical nature of the tight SFR-dense gas correlation. Yet to
date, owing primarily to the weakness of HCN emission relative to that of CO,
no more than one square degree of sky has been systematically surveyed in HCN
emission in any nearby cloud (e.g. Pety et al. 2017, Shimajiri et al. 2017).
Recently, Tafalla et al. (2021) mapped HCN (and other molecular species) in
Perseus using a stratified random sampling technique in which ten points were
randomly sampled at each of ten logarithmically-spaced levels of column
density. As we discuss below, this approach can be very effective at
characterizing the radiative transfer of molecular lines, but such a severely
under sampled study cannot be used to measure the total HCN luminosity of
Perseus nor its dense gas conversion factor (hereafter $\alpha({\rm HCN})$).
For that, a well-sampled survey is required.

Here we present the first complete HCN survey of a nearby molecular cloud,
carried out with the CfA 1.2 meter telescope in the 1$\rightarrow$0
transition. The primary goals of this survey were to: 1) accurately measure
the total HCN luminosity from an entire, spatially resolved, GMC in order to
make a robust determination of $\alpha$(HCN) for molecular clouds, and 2) to
determine the range of molecular hydrogen column and volume densities which
produce detectable HCN emission. We chose the Perseus cloud because of both
its typical level of star formation activity—not as flamboyant as the Orion
and Ophiuchus clouds nor as quiescent as the Pipe and California clouds—and
its ideal declination for mapping from the latitude of Cambridge. In addition,
the Perseus cloud has been fully mapped in both dust emission by Herschel
(Andre et al. 2010) and Planck (Planck Collaboration et al. 2014) and in near
infrared extinction (Lombardi et al. 2010), providing a well-calibrated, high-
resolution inventory of its dust content (Zari et al. 2016).

Figure 1: (a) HCN J=1-0 spectrum at the strongest point in the survey, $l$ =
158.333, $b$ = -20.5. (b) HCN spectrum at a typical position, $l$ = 160.0, $b$
= -19.167.

In the next section we describe our HCN survey in detail and show sample
spectra. In Section 3 we compare our HCN map to a previous CO map of Perseus
obtained with the same telescope and to the high resolution dust column
density map of Zari et al. (2016). The latter comparison is used to provide an
empirical conversion factor between HCN luminosity and total dense gas mass in
the Perseus cloud. In Section 4 we use our HCN survey, the Zari et al. dust
map, and radiative transfer modeling to investigate both the relation between
HCN emission and H2 column density and the range of gas volume densities
traced by HCN. In Section 5 we summarize our results.

##  2 Observations

The HCN ($J=1\rightarrow 0$) survey was conducted between February and April
2019 with the CfA 1.2 meter telescope, which is fully described in Dame et al.
(2001). Observations in the LSR frame were centered at a frequency of
88631.847 MHz, corresponding to the center line in a group of 3 HCN hyperfine
lines ($F=1-0,1-0,2-1$); the other lines are offset by $+$4.82
$\mathrm{km\,s^{-1}}$ and -7.07 $\mathrm{km\,s^{-1}}$ in radial velocity from
the central component. A 256 channel filter bank spectrometer with 0.25 MHz
channels provided a velocity resolution of 0.85 $\mathrm{km\,s^{-1}}$ and
coverage over 216 $\mathrm{km\,s^{-1}}$. We removed instrumental baseline
structure by frequency switching every 1 s by 10 MHz (34
$\mathrm{km\,s^{-1}}$) and subtracting the data from the two phases. We
subsequently removed a 5th order polynomial and folded the spectra by
averaging each spectral channel with its corresponding reference-phase mate 10
MHz higher in frequency. Figure 1 shows two of the final spectra from the
survey, one at the peak position and one at a typical position in the cloud.

Although the intensity of the HCN lines are typically 50-100 times weaker than
those of CO, the sky noise at 89 GHz is roughly 3 times lower than at 115 GHz,
owing mainly to a larger offset from a strong O2 telluric absorption line at
118 GHz. At a typical observing elevation of 40∘, the total system noise (sky
plus receiver plus side lobe leakage) was $\sim$240 K at the HCN (J $=$
1$\rightarrow$0) frequency, compared with $\sim$550 K at CO (J $=$
1$\rightarrow$0). The total integration per sky position was $\sim$40 minutes,
which yielded a uniform rms noise of 14 mK per channel. We note that the
integration times were not fixed but rather were adjusted for each observation
based on the instantaneous system noise to achieve that target rms. For
various practical reasons including calibration stability, we actually
observed the entire map 5 times over with an average integration time of
$\sim$8 minutes per point in each map.

Based on a measured beam width of 8.4’ at 115.3 GHz (see Appendix A in Dame et
al. 2001), the beam at 89 GHz is estimated to be 10.9’ (HPBW). Our
observations were guided by an existing 1.2 m CO survey of the Perseus cloud
from Dame et al. (2001), spatially smoothed to an angular resolution of 10.9’.
As shown in Figure 2b, we searched for HCN at all positions where the
integrated integrated intensity of the CO line, W(CO), was in excess of 13 K
km s-1, corresponding to an expected W(HCN)111Throughout this paper W(HCN)
corresponds to the intensity integrated over the three main hyperfine
components. The expected W(HCN) assumes W(CO)/W(HCN) = 71, a value we derived
by comparing CO (Wilson et al. 2005) and HCN (Pety et al. 2017) in Orion B.
The results presented below confirm that this ratio is approximately correct.
in excess of 3 times the target instrumental noise. A total of 292 positions
were observed on a 10’ grid in Galactic longitude and latitude, covering a
total area of 8.1 deg2.

##  3 Results and Discussion

###  3.1 Comparison of HCN, CO and Dust Extinction

A map of velocity-integrated HCN intensity derived from our new survey is
shown in Figure 2a, above a comparable CO map from the CfA survey in Figure 2b
and a map of K band dust extinction from Zari et al. in Figure 2c. The latter
two maps have been spatially smoothed to the intrinsic 11’ resolution of the
HCN map. The HCN surveyed area is outlined on all three maps. It is clear from
Figure 2b that our survey area covers most (78%) of the cloud’s CO luminosity,
and within that area we detect HCN at the 3-$\sigma$ level in 60% of the
positions observed. Using the dust map in Figure 2c to infer gas mass, we find
that the HCN detected positions contain 1.2 $\times$ 104 M⊙ of gas,
corresponding to 75% of the total gas mass in the survey area and 55% of all
the CO emitting gas in the Perseus cloud.

Figure 2: (a) HCN emission integrated over the full velocity range in which CO
and HCN are detected, -5 to 15 $\mathrm{km\,s^{-1}}$. (b) CO emission
integrated over the same velocity range and smoothed to an angular resolution
of 11’ to match that of the HCN survey. (c) The AK map of Zari et al. (2016)
likewise smoothed to an angular resolution of 11’. All three maps have the HCN
sampling boundary overlaid.  Figure 3:  The percentage of total cloud
luminosity contributed by positions below W(HCN). For example, $\sim$50% of
the total luminosity is contributed by positions with W(HCN) $<$ 0.4 K km s-1.
Along the top axis we used the linear fit in Figure 4d to convert W(HCN) to an
approximate value of AK.

In Figure 3 we show the cumulative fraction of total HCN luminosity as a
function of W(HCN) and extinction. Note that 60% of the total HCN luminosity
is emitted in regions of the cloud _below_ the standard dense-gas extinction
threshold of AK $=$ 0.8 mag. This result is both consistent with and
reinforces earlier suggestions of very extended HCN emission that were based
on more spatially limited HCN surveys within a few nearby clouds (Pety et al.
2017, Shimajiri et al. 2017, Kauffmann et al. 2017, Tafalla et al 2021). It is
becoming increasingly clear that comparable wide-field surveys of entire
molecular clouds will reveal similarly extensive HCN emission as we find in
Perseus.

Figure 4: Scatter plots derived from the maps in Figure 2, with AK converted
to H2 column density using the gas-to-dust ratio in Zari et al. (2016). (a)
Scatter plot derived from the CO and dust maps. (b) Scatter plot derived from
the HCN and dust maps. (c) The points from (a) averaged in bins 0.5 x 1021
cm-2 wide in N(H2). The line was fitted to points at N(H2) $\leq$ 4 x 1021
cm-2. (d) The points from (b) averaged in bins 3.5 x 1021 cm-2 wide in N(H2).
The line was fitted to all points and implies XHCN $=$ 102 x 1020 cm-2 K-1
km-1 s. In (c) and (d), the error bars indicate dispersion about the mean in
the bins; no dispersion is indicated for bins with only one data point. For a
standard gas-to-dust ratio, 1 AV $\approx$ 1 x 1021 cm-2, so the numbers on
the x axes approximately represent AV.

As noted above, the dust extinction map in Figure 2c can be linearly scaled to
a map of N(H2) by adopting a standard gas-to-dust ratio (see Zari et al. for
details), and we can use that map to test the optical depth properties of the
HCN and CO lines in Perseus. Scatter plots of HCN and CO emission as functions
of N(H2) are shown in the top two panels of Figure 4, while the corresponding
lower panels average the emissions in bins of N(H2). At N(H2) $<$ 4 x 1021
cm-2 the CO line does a good job of linearly tracing the molecular gas with a
corresponding X factor of 1.2 x 1020 cm-1 K-1 km-1 s, while at higher column
densities we see the well-known saturation of the CO line. In contrast, the
HCN plots in Figure 4 show no signs of saturation, the velocity integrated HCN
intensity remaining linear with N(H2) and yielding a corresponding HCN-to-H2
conversion factor, XHCN of 102 x 1020 cm-2 K-1 km-1 s, very close to the value
derived by Tafalla et al. (2021). Thus, even though its high dipole moment
suggests that, relative to CO, HCN should be a useful tracer of gas at high
_volume_ density, it is clearly also a good tracer of gas _column_ density
over the range in N(H2) of 0.2–2.3 x 1022 cm-2 (i.e., Av $\approx$ 2-20 mag),
at least in Perseus.

###  3.2 Calibrating HCN luminosity as a Tracer of Dense Gas Mass

Although the X-factor is useful for calculating molecular hydrogen column
densities from HCN integrated intensities in resolved nearby clouds, beam
dilution limits its use in external galaxies (and very distant clouds) where
the molecular gas is unresolved. Because for unresolved sources the line flux,
or equivalently line luminosity, is conserved with varying beam size, it is
more convenient to use a calibrated mass-luminosity relation to directly
extract dense gas masses.

Following Gao & Solomon (2004), we define the HCN mass-luminosity relation as:

$\alpha({\rm HCN})=M_{dg}/L({\rm HCN})$ (1)

where $M_{dg}$ is the mass (in units of M⊙) of the molecular gas at high
volume density and $L$(HCN) is the total HCN(1-0) luminosity (in units of K km
s-1 pc2) of the cloud.

Knowledge of the value of $\alpha$(HCN) has been sparse, and until recently
there were no directly calibrated measurements. Gao & Solomon (2004) were the
first to estimate a value for $\alpha$(HCN). From theoretical considerations
they indirectly determined a value of 10 by assuming virial equilibrium, a
uniform density of $n_{{\rm H}_{2}}$ = 3 $\times 10^{4}$ cm-3, and a line
brightness temperature of 35 K for the HCN emitting gas. Among the first
attempts to empirically determine $\alpha$(HCN) in the Milky Way was that of
Wu et al. (2010), who obtained HCN measurements toward a sample of massive,
dense cores. Similar to Gao & Solomon, they used the virial theorem to
estimate masses of the HCN emitting gas and derived a value of $\alpha$(HCN) =
20 for the dense cores. The availability of extensive and complete surveys of
dust extinction and emission in GMCs (e.g., Lombardi et al. 2006, 2010, 2011,
2014) along with limited HCN mapping (e.g., Pety et al. 2017; Shimajiri et al.
2017, Kauffmann et al 2017) have recently enabled more direct measurements of
$\alpha$(HCN) by using high extinction as a proxy for high volume density.
Using Herschel dust maps, Shimajiri et al. (2017) derived values for
$\alpha$(HCN) in the range 35–454 in ten heavily extincted (AV $\gtrsim$ 8)
portions of three nearby GMCs. Collectively, however, these regions covered a
total of only 0.65 deg2, a small fraction the total cloud extents ($\sim$ 10
deg2). To facilitate a more appropriate comparison to extragalactic
observations where global cloud measurements will also include HCN emitted at
lower extinctions/densities, Shimajiri et al. (2017) argued that their
measurements of $\alpha$(HCN) should be scaled down by a factor of 10, roughly
the ratio of gas at AV $>$ 2 to that at AV $>$ 8 in typical GMCs. Kauffmann et
al.(2017) used infrared extinction maps in a section of the Orion A GMC to
measure the dense gas mass (at extinctions AV $>$ 7) and together with their
estimates of LHCN for that region found $\alpha$(HCN) $\leq$ 20 after making
approximate adjustments to account for HCN emission in gas with lower
extinctions. In the end these studies find results similar to that of Gao &
Solomon but only after uncertain corrections for unobserved HCN gas are
included.

Figure 5:  (a) Map of K band extinction from Zari et al. (2016). (b) Same,
except only emission at AK $>$ 0.8 mag is shown.

With the data presented here we can, for the first time, directly calibrate
$\alpha$(HCN) for an entire nearby molecular cloud. By simply summing over all
pixels in our HCN map (Figure 2a) and multiplying by the square of the cloud’s
distance we obtain L(HCN) = 55.3 K km s-1 pc2. The distance to Perseus has
recently been measured to high accuracy by Zucker et al. (2019) based on Gaia
DR2 data (294 $\pm$ 15 pc). The rms noise in the HCN spectra propagates to an
uncertainty on L(HCN) of only 1%, so a survey with higher angular resolution
or sensitivity would scarcely improve on our value. We estimated the amount of
HCN luminosity that might lie outside our sampling boundary by scaling the
amount of CO emission seen there (Figure 2b) by the mean HCN/CO intensity
ratio, which we measured to be $\sim$0.008 near the edges of the cloud. Such
weak, unobserved HCN might increase L(HCN) by $\sim$21%, to 66.9 K km s-1 pc2,
but this is probably an upper limit, since the HCN/CO ratio likely
falls—perhaps even to zero—toward the cloud edges. As an aside, we note that
all of the HCN lines that we estimate might lie outside our sampling boundary
would not be detectable at our current sensitivity.

To identify regions of high gas volume density, we adopt a dust column density
threshold of AK $>$ 0.8 mag because, as previously mentioned, the gas mass
above that threshold has been shown to be most closely correlated with the
star formation rates in local clouds (i.e., Lada et al. 2010; Evans et al.
2014; Heiderman et al. 2010). The premise that our extinction threshold also
corresponds to a high volume density threshold follows naturally from the fact
that molecular clouds are filamentary in geometry rather than sheet-like, with
inwardly increasing column density and volume density gradients. This is
perhaps best demonstrated by recent 3D extinction maps of the outer regions of
local clouds (Zucker et al. 2021). This idea also derives support from early
models of filamentary clouds showing that radial column density gradients
likely correspond to radial volume density gradients (e.g. Lada, Alves & Lada
1999, Bergin et al. 2001). Indeed, Bergin et al. (2001; see their Figure 10)
derived a relation between n(H2) and visual extinction in the IC 5146 cloud
indicating that n(H2) $>$ 104 cm-3 corresponded to Av $>$ 7 mag (i.e., AK $>$
0.8 mag)222We note that AV is related to AK through the reddening law: AK $=$
0.112 AV (Rieke & Lebofsky 1985).  for that cloud. More recently a number of
different turbulent MHD simulations of cloud formation and evolution made over
a wide variety of physical scales were all found to show clear correlations
between extinction and hydrogen volume density. These derived correlations are
quantitatively very similar to each other (Bisbas et al. 2019 and references
therein; see their Figure B1) and to the early cloud modeling mentioned above.
These various models and simulations consistently find that AK $>$ 0.8 mag
corresponds to n(H2) $>$ 104 cm-3.

The dense gas mass of the Perseus cloud can be derived from the map of dust
extinction shown in Figure 5. Adopting the gas-to-dust ratio given in Zari et
al. and defining dense gas as that characterized by extinctions AK $>$ 0.8
mag, we derive from Figure 5b a dense gas mass of $M_{dg}=0.51\times 10^{4}$
M⊙ and $\alpha$(HCN) $=$ 92 M⊙ $(\rm{K\ km\ s^{-1}pc^{2}})^{-1}$. Accounting
for HCN outside our sampling boundary might decrease this value to 76 M⊙
$(\rm{K\ km\ s^{-1}pc^{2}})^{-1}$. In any case, we find a value of
$\alpha$(HCN) in Perseus that is substantially higher than the value of 10
estimated by Gao & Solomon. However, assuming virialized clouds, Gao & Solomon
(2004) derived the value of 10 for the dense gas conversion factor using the
equation $\alpha$(HCN) $=$ 2.1 $\times$ n(H2)0.5(Tb)-1 where Tb is the
intrinsic brightness temperature of the HCN line. They assumed Tb $=$ 35 K and
n(H2) $=$ 3$\times 10^{4}$ cm-3 to get the value of 10. Using values more
appropriate for Galactic GMCs (e.g., Tb $\lesssim$ 1 K and n(H2) $\sim$ 104
cm-3) would give $\alpha$(HCN) $\gtrsim$ 210 using Gao & Solomon’s formula.

It’s worth noting here that another value of $\alpha$ can be calculated
directly from the value of X(HCN) derived in the previous section. In fact,
for any molecular tracer T, it can be shown that $\alpha$(T)X $=$ 2.15 $\times
10^{-20}$ X(T). For Perseus this yields $\alpha$(HCN)X $=$ 215\. This value is
larger than the dense-gas alpha derived above because it applies to _all_ of
the HCN-luminous gas, not just the dense gas at AK $>$ 0.8 mag. Multiplying
$\alpha$(HCN)X by L(HCN) yields an HCN-luminous mass of 1.2 $\times 10^{4}$
M⊙, in agreement with our direct measurement of this quantity in Section 3.1.

##  4 The Column and Volume Densities Traced by HCN

The HCN molecule has long been considered a dense gas tracer. Yet, in the
Perseus cloud both our observations and those of Tafalla et al. (2021) show
that HCN is a good tracer of hydrogen column density as well. This may seem
surprising given that the HCN emission is also characterized by very high
optical depths ($\tau$ $\sim$ 2-25) under the conditions typical of this and
other clouds. Tafalla et al. (2021) posited that the observed increase in
W(HCN) with extinction could be a consequence of the subthermal excitation of
the lines coupled with density gradients in the cloud. Consider that the HCN
brightness is given by the standard radiative transfer equation,
$T_{b}=(J(T_{ex})-J(T_{bg}))\times(1-e^{-\tau})$ which for $\tau>>1$ is
dominated by the first term. In this circumstance the observed brightness will
depend on this excitation term, which will increase as $T_{ex}$ increases with
volume density. If, in addition, the extinction (or column density) is
correlated with the volume density, as we argued earlier, the brightness of
the lines and W(HCN) will also increase with extinction (Tafalla et al. 2021).

Figure 6:  The predicted W(HCN)-AV relation for a model cloud where column
density and volume density are positively correlated. The points are the model
predictions for undepleted (circles) and depleted (triangles) gas and the
dashed lines are linear fits to the respective points above AV $=$ 4 mag.
Vertical dashed lines correspond to values of n(H2) in units of cm-3. These
models demonstrate that optically thick HCN emission can effectively trace the
total column density in a cloud provided the volume and column density
structures of the cloud are correlated.

To test this idea we used the radiative transfer program RADEX (van der Tak et
al. 2007) to predict W(HCN) as a function of AV. In brief, RADEX can calculate
W(HCN) for a uniform medium given N(HCN), n(H2), $T_{\rm K}$, and the observed
linewidth $\Delta$v. We adopted $T_{\rm K}$ $=$ 11 K, based on an ammonia
study of 193 dense cores in Perseus (Rosolowsky et al. 2008) and $\Delta$v $=$
2.3 km s-1, based on Gaussian fits to the central hyperfine components in
several regions with strong lines in our survey. For a set of $A_{V}$ values
between 0 and 15 mag we calculated: (1) the corresponding N(H2) from the gas-
to-dust ratio of Zari et al; (2) the corresponding N(HCN) by assuming an HCN
abundance relative to H2 of 3.1 $\times 10^{-9}$ (e.g., Tafalla et al. 2021);
and (3) the corresponding n(H2) value from the N(H2)-n(H2) relation of Bisbas
et al. (2019).

The predicted values of W(HCN) as a function of AV are shown by the red
circles in Figure 6. Although simplistic, the model provides some interesting
insights. First, it predicts that HCN lines only reach our 3$\sigma$ detection
limit of 0.17 K km s-1 at AV $\approx$ 3.5 mag. Below that extinction HCN
emission is predicted to sharply decrease, with W(HCN) falling to values of
only 0.0015 to 0.014 K km s-1 between AV = 1 and 2 mag, respectively. Such low
integrated intensities would present a significant challenge for observations
with even the most sensitive receivers, especially since the models do not
include the expected UV photodissociaton of HCN near the cloud edges (e.g.,
van Dishoeck & Black 1988). Second, the model demonstrates that above AV $=$ 4
mag, the W(HCN)-AV relation is indeed linear, the straight line fit to those
points in Figure 6 having a correlation coefficient of 0.99. We conclude that
our model based on the conjecture of Tafalla et al. can explain the curious
fact that HCN emission traces total gas column density surprisingly well
despite its high opacity.

We note however that the slope of the red line fit to the model points in
Figure 6 corresponds to an HCN-to-N(H2) conversion factor of XHCN $=$ 45
$\times$ 1020 cm-2 K-1 km-1 s, about a factor of 2 below our observed value
(Figure 4d). Moreover above AV $\approx$ 5 mag the model systematically over
predicts W(HCN) with increasing extinction. Tafalla et al (2021) found a
similar result in their modelling of HCN emission in Perseus and attributed it
to freeze out or depletion of HCN onto grains. To illustrate the possible
effects of depletion on our data we calculated a simple extinction-dependent
correction to our model. We assumed no depletion below 4 mag and above 4 mag
we required the W(HCN)-AV relation to remain linear with a slope that matched
the observed one as closely as possible and then calculate the corresponding
values of N(HCN).

The depletion-corrected model points and linear fit are shown by the blue
triangles and blue dashed line in Figure 6. As expected, with this adjustment
the model predictions much more closely match the observed data, yielding an
X(HCN) $=$ 107 $\times 10^{20}$ cm-2 K-1 km-1 s. Moreover, the model shows
that N(HCN) is very weakly dependent on AV, with N(HCN) $\sim$
A${}_{V}^{0.14}$ resulting in an exponentially decreasing HCN abundance with
cloud depth (i.e., N(HCN)/N(H2) $\sim$ e${}^{-({A_{V}\over{9.8}})}$) and
supporting the depletion hypothesis. At AV $=$ 15 mag, N(HCN) is only 20%
higher than it is at 4 mag and is a factor of 3 lower than its undepleted
value predicted by the model.

As is clear from Figure 3 and the discussion above, the high critical density
of the HCN J$=$ 1$\rightarrow$ 0 line, 5 $\times 10^{5}$ cm-3 (Shirley 2015),
does not necessarily mean that most of the HCN emission from a molecular cloud
is emitted by gas at or above that density. This would certainly be the
inference one would draw from our observation that the HCN emission in Perseus
covers a significant fraction of the CO emitting area of the cloud, including
large portions where AK $<$ 0.8 mag and where the volume densities are likely
below 104 cm-3. As mentioned above, this circumstance has been hinted at in
recent wide-field observations of limited portions of a few nearby molecular
clouds (e.g., Pety et al. 2017, Shimajiri et al., 2017, Kauffmann et al.,
2017) and even some more distant GMCs (Evans et al. 2020, Patra et al. 2022).
The mean density of the gas emitting HCN emission in a particular cloud will
depend on such factors as the HCN abundance, gas kinetic temperature, mean
optical depth and the fraction of cloud mass below the critical density which
can still emit HCN sub-thermally.

Figure 7: The distribution of H2 volume densities at all observed positions,
inferred from RADEX modeling of the HCN line intensities.

To make a more quantitative assessment of the mean H2 density exciting the HCN
lines, we once more used the radiative transfer program RADEX (van der Tak et
al. 2007). However this time we solved for n(H2) using the observed W(HCN) as
an input to the calculation. As additional inputs to RADEX we obtained N(HCN)
at each position from the dust-derived N(H2), adopting the HCN abundance of
3.1 $\times 10^{-9}$ as above, and again adopted $T_{\rm K}$ $=$ 11 K and
$\Delta$v $=$ 2.3 km s-1. The volume density, n(H2), was left as the free
parameter, which was varied at each position until W(HCN) matched the
observations. The distribution of n(H2) values derived in this way is shown in
Figure 7. We find a mean n(H2) of $1.1\pm 0.6\times 10^{4}$ cm-3, similar to
values found by other recent studies cited above.

##  5 Summary

We have obtained a complete survey of HCN (J=1-0) emission from the nearby
Perseus molecular cloud using the CfA 1.2 meter telescope. We have compared
our survey with previous complete surveys of CO and dust emission with the
following results.

1\. The HCN (J=1-0) emission is detected over much of the cloud as defined by
complete CO and dust extinction maps. Specifically, HCN is detected at the
3-sigma level over 60% of the surveyed area. Within this area 71% of the mass
is found to be HCN luminous.

2\. Whereas the CO line clearly saturates at N(H2) $\geq$ 4 x 1021 cm-2, we
find that the HCN line remains linear with N(H2) throughout the cloud (up to
N(H2) $\sim$ 23 x 1021 cm-2) and this results in an HCN-to-H2 conversion
factor, XHCN, of 102 x 1020 cm-2 K-1 km-1 s.

3\. We use a simple RADEX radiative transfer model to demonstrate that,
despite its large line optical depths ($\tau$ $=$ 2-25), HCN emission will
linearly scale with extinction in filamentary structured molecular clouds
where a positive correlation exists between N(H2) and n(H2). To match our HCN
observations the model also requires a systematic (exponential) decrease in
HCN abundance with increasing extinction, likely due to HCN depletion onto
grains. Similar RADEX modeling of the HCN line intensities imply that the mean
H2 volume density exciting the HCN lines is $1.1\pm 0.6\times 10^{4}$ cm-3, a
factor of $\sim$50 less than the line’s critical density.

4\. We obtained the first robust calibration of the dense gas to HCN
luminosity ratio in a nearby molecular cloud, $\alpha$(HCN) $=$ 92 M⊙ $(\rm{K\
km\ s^{-1}pc^{2}})^{-1}$. This is $\sim$9 times higher than the widely-used
estimate of Gao & Solomon.

We gratefully acknowledge financial support from the Scholarly Studies Program
of the Smithsonian Institution. We thank the anonymous referee for thoughtful
comments that led to improvements in both the clarity and scope of the paper.
We also thank Neal Evans for useful suggestions on an early version of this
paper.

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