# Cygnus A Obscuring Torus: Ionized, Atomic or Molecular? ††thanks: Herschel
is an ESA space observatory with science instruments provided by European-led
Principal Investigator consortia and with important participation from NASA.

Atharva Bagul,1 Patrick Ogle,2 Robert Antonucci,3 Philip Maloney,4 and Enrique
Lopez Rodriguez5
1Indian Institute of Science Education and Research Bhopal, 462066 Madhya
Pradesh, India
2Space Telescope Science Institute Baltimore, MD 21218, USA
3Department of Physics, University of California Santa Barbara, CA 93106, USA
4Private Astronomer, Boulder, CO 80301, USA
5Kavli Institute for Particle Astrophysics & Cosmology (KIPAC), Stanford
University, Stanford, CA 94305, USA<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The prototypical powerful FR 2 radio galaxy Cygnus A fits extremely well into
the quasar/radio galaxy unified model: high polarization with an angle almost
perpendicular to the radio jet and polarized flux showing broad permitted
lines. It has been claimed that ionized gas in the torus reveals a very clear
torus shape via Bremmstrahlung emission. We rule out the later with an
energetic argument, and we constrain the molecular and atomic gas properties
with existing observations. The atomic absorption against the core has been
shown to match the X-ray column only if the spin temperature is an implausible
$T_{\rm s}=1\times 10^{6}$ K. This points to a molecular medium for the X-ray
column $\log(N_{\rm H}~{}[\rm{cm^{-2}}])\sim 23.5$. Yet not low-J CO
absorption is detected to sensitive limits. The non-detection is surprising
given that this powerful radio galaxy hosts a luminous, dust-obscured active
nucleus and copious warm molecular hydrogen. These conditions suggest a
detectable level of emission. Furthermore, the torus X-ray column density
suggests detectable absorption. We explore various possibilities to explain
the lack of a signature from warm CO (200-250K). Specifically, that the
radiative excitation by the radio core renders low-J CO absorption below
current sensitivities, and that high-J levels are well populated and conducive
to producing absorption. We test this hypothesis using archival Hershel/SPIRE
FTS observations of Cygnus A of high-J CO lines ($14\geq J\geq 4$
transitions). Still high-J CO lines are not detected. We suggest that ALMA
observations near its high frequency limit can be critical to obtain the
signature of molecular line of the torus of Cygnus A.

###### keywords:

galaxies:individual:Cygnus A – galaxies:active – galaxies: Seyfert

††pubyear: 2022††pagerange: Cygnus A Obscuring Torus: Ionized, Atomic or
Molecular? ††thanks: Herschel is an ESA space observatory with science
instruments provided by European-led Principal Investigator consortia and with
important participation from NASA.–Cygnus A Obscuring Torus: Ionized, Atomic
or Molecular? ††thanks: Herschel is an ESA space observatory with science
instruments provided by European-led Principal Investigator consortia and with
important participation from NASA.

##  1 Introduction

The unification by orientation model of active galactic nuclei (AGN)
(Antonucci, 1993) posits that the appearance of optical total continuum
emission from the central engine (i.e. accretion disk and super massive black
hole) depends on the viewing angle with respect to the axis of an obscuring
region. This obscuring structure is commonly named as "the dusty torus", but
the specific morphology, size, and dynamics are still up for debate. AGN are
broadly classify as Type 1 and Type 2. For Type 1 AGN, our line-of-sight (LOS)
is close to an unobscured view of the optical continuum emission and broad
line region. Those AGN with their central engine emission obscured by an
optically thick, molecular, and dusty torus are called Type 2. The obscured
broad lines in Type 2 AGN can be seen in polarized flux via scattering by
ionized gas or dust across the ionization cones and narrow line regions
(Antonucci, 1984; Antonucci & Miller, 1985; Barvainis, 1987; Miller et al.,
1991).

While the exact configuration of the torus is still unclear, the torus must
provide a large obscuration with respect to its inner radius in order to block
roughly $50$ percent of the solid angle near the equatorial plane as seen by
the central continuum source (e.g., Elitzur, 2012). The torus also must be
roughly axisymmetric in order to produce a scattering quasi-conical narrow
line region that is photoionized by the central source of ultraviolet (UV)
continuum. Equatorial views within the obscured torus present a high column
density of gas, resulting in heavy and sometimes Compton-thick obscuration of
the central X-ray source (Ghisellini et al., 1994). The torus may be clumpy,
with dense clouds of self-shielding dust embedded in a medium of atomic and
ionized gas (Ramos Almeida & Ricci, 2017, for a review). Such an assemblage of
dusty molecular clouds might either be supported against collapse by velocity
dispersion (Krolik & Begelman, 1988) or in a molecular outflow ablated from
the inner edge of a molecular disk (Krolik & Begelman, 1986). One approach to
detect the signature of the obscuring torus is via CO in absorption or
emission. CO emission from a compact molecular torus may be undetectable
because of the small solid angle and the thermodynamic limit on the brightness
temperature. However, there are not constraints for the strength of CO
absorption against a radio core, and thus we can measure CO absorption
depending on the geometry, column density of molecular gas, and CO level
populations. An obscuring torus can also be detected by looking for signatures
of amorphous/crystalline silicate features (Spoon et al., 2022), PAHs (García-
Bernete et al., 2022), and other sub-mm dense molecular gas tracers like HCO
and rotational transitions of vibrationally excited HCN(Aalto et al., 2015;
Falstad et al., 2019).

Cygnus A (3C 405) (z=0.0562) is a classical luminous FR 2 galaxy based on its
radio morphology and it illustrates the AGN unified model very well. Cygnus A
contains a dust-obscured Type 1 AGN revealed by spectro-polarimetric
observations that show polarised broad emission lines (Ogle et al., 1997;
Antonucci et al., 1994). The central engine is obscured by a dusty, and
potentially clumpy, torus, which is also polarized by an ordered toroidal
B-field (Lopez-Rodriguez et al., 2018). Carilli & Barthel (1996) provided a
comprehensive review of the multiwavelength properties of Cygnus A. Early
interferometric observations revealed Cygnus A as a double radio source
(Jennison & Das Gupta, 1953). Cygnus A is ten times closer than other radio
galaxies of similar radio luminosity ($>10^{45}$ ergs sec-1), and contains an
AGN with a bolometric luminosity of order $10^{46}$ ergs sec-1 comparable to
high redshift quasars (Runnoe et al., 2012).

Barvainis & Antonucci (1994) put upper limits on the optical depth of CO (1 -
0) absorption against the radio core of Cygnus A. They suggest that the lack
of strong CO absorption may be due to the nuclear radio source being several
times larger in angular diameter than the individual clouds in the torus, but
smaller than the torus itself.

Maloney et al. (1994, hereafter MBR94) showed the undetected CO absorption can
be due to: a) atomic gas, or b) extremely high rotation temperatures and
depopulation of the low-J states due to radiative excitation by the bright
radio core. Verification of this model would strongly constrain the gas on
parsec (pc) scales and imply exotic level populations and absorption spectra.
In the region within tens of pc of the compact radio core, the solid-angle
averaged brightness temperature would be very high for the low-J transitions.
Then, the brightness temperature drops relatively suddenly to values lower
than the gas kinetic temperature at J $\sim 5-20$. Qualitatively, high-J
levels may be richly populated and thermalized, so that any absorption will
not be canceled by stimulated emission.

Ogle et al. (2010) used Spitzer IRS to detect emission from several $H_{2}$
pure rotational lines (0-0 S(1), S(3), S(5)-S(7)) in the central 3.7"x3.7"
region of Cygnus A. These $H_{2}$ emission lines allow to directly trace the
molecular gas emission with no $\alpha$CO factor. They use this to estimate
warm $H_{2}$ masses of $1\times 10^{7}M_{\odot}$ at 460 K and $5\times
10^{5}M_{\odot}$ at 1500 K. They put an upper limit of $<4\times
10^{9}M_{\odot}$ of $H_{2}$ at T=100 K. Other radio galaxies in their sample
have typical cool (50 K) molecular gas mass ratios of $~{}0.1$, based on CO
detections. Therefore, we might conservatively expect to find $>1.5\times
10^{8}M_{\odot}$ of molecular gas at 50 K in the central region of Cygnus A
that could potentially be detected in CO emission. Bellamy & Tadhunter (2004)
also detected hot molecular gas in the dusty structure using NIRSPEC, Keck 2
telescope from $H_{2}$ rotational lines (1-0 S(1), S(2), S(3), S(4)) in the
central aperture of PA $105^{\circ}$ data.

Archival Herschel Spectral and Photometric Imaging REceiver (SPIRE) (Griffin
et al., 2010) Fourier-transform spectrometer (FTS) observations provide a
unique data set for probing CO emission and absorption in the mid- and high-J
lines, and also examine what constraints can be placed on the radiative
excitation hypothesis.

The paper is organised as follows. In Section 2 we present the observational
details of Cygnus A using Herschel/SPIRE FTS. The overview of the results
obtained from the spectral analysis of the data collected is given in Section
3. In Section 4, 4.1 focuses on the failure of the Herschel/SPIRE to detect CO
emission, 4.2 discusses whether or not radiative excitation could be
important, and 4.3 comments on why we think the implied recombination
luminosity may be incorrect for the bremsstrahlung torus model. Section 4.4
comments about the failure of X-ray absorption in an atomic torus.

##  2 Archival data

In the present paper we analyze observations performed with the 3.5-m Herschel
Space Observatory (Pilbratt et al., 2010) using the SPIRE instrument. SPIRE is
the Herschel’s camera and spectrometer offering three-band imaging photometer
operating at $250$, $350$, and $500$ $\mu$m. The FTS uses two overlapping
bands to cover 191-671 $\mu$m, the Spectrometer Short Wavelength (SSW) band
covers 191-318 $\mu$m and the Spectrometer Long Wavelength (SLW) covers
294-671 $\mu$m.

We took the publicly available FTS observations from the Herschel Archive in
sparse sampling mode, with a single pointing of the two coaligned bolometer
arrays centered on the nucleus of Cygnus A. This mode is suitable for sources
that are smaller than the FTS beam (full width at half maximum, FWHM, $\sim
17\arcsec-42\arcsec$). For Cygnus A, $1\arcsec$ corresponds to $1$ kpc, so the
instrument beams of the central bolometers cover the bulk of the galaxy at all
wavelengths of interest. We retrieved the highly processed products. This
special processing, with manual intervention, was applied to isolated,
unresolved point sources to optimize background subtraction, utilizing the
median background signal from the off-source bolometers. The unapodized data
products from these two detectors (SSW and SLW) were used. These two spectra
were joined at $1090.64$ and $1090.94$ GHz frequency ($274.88$ $\mu$m and
$274.80$ $\mu$m wavelength). The SPIRE integration time was set to make a
$5\sigma$ detection at each wavelength in the RJ tail of the warm ($50-100$ K)
dust emission. A $3.9$h of exposure was scheduled providing $100$ repetitions,
so that the pattern noise limit was not reached.

The Herschel/SPIRE observations were complemented using the following archival
observations. Dust continuum Atacama Large (sub-)Millimeter Array (ALMA)
observations were taken in four channels centered at 350, 348, 338, and 336
GHz with a 2 GHz bandwidth with a total on-source time of 28.42 minutes with
an angular resolution of $0.058$″, $0.053$″, and $0.089$″respectively using
ALMA band 7 (ID: 2018.1.01104.S, PI: Perley, Daniel). The phase calibrator was
J2007+4029 observed right after the science object. The final image was
computed using the standard ALMA pipeline using the Briggs weight and robust =
0.0. The final common beam size is measured to be $0.12\times 0.06$ mas2 with
a position angle of $-3^{\circ}$. The final images reached a sensitivity of
2.7 mJy/beam.111We do not have information about the sensitivity, except for
the comment in the QA2 report that the final sensitivity is higher than
requested by almost a factor $\sim 10$. As the core of Cygnus A is very bright
$(\sim 100mJy)$, this lost in sensitivity is still enough to obtain reliable
fluxes of the continuum emission. The ALMA data was reduced and analysed via
the Common Astronomy Software Applications (CASA) package (CASA Paper cite).
CASA is the primary data processing software for the ALMA and NSF’s Karl G.
Jansky Very Large Array (VLA), and is frequently used also for other radio
telescopes. This software can process data from both single-dish and aperture-
synthesis telescopes. We downloaded the offline version from their website
222https://casa.nrao.edu and did the analysis with the help of CASA
Documentation 333CASA Documentation to obtain the results plotted in Figure 2.

Dust polarization continuum observations using the Stratopsheric Observatory
for Infrared Astronomy (SOFIA) / High-resolution Wideband Camera Plus (HAWC+)
at $600$ and $856$ Hz with an angular resolution of $4.85$″and $7.80$″,
respectively (Lopez-Rodriguez et al., 2018). Dust polarization continuum
observations using the Gran Telescopio Canarias (GTC) / CanariCam at $1199.2$
and $1873.7$ Hz at an angular resolution of $0.3$″. For all these
observations, Cygnus A appears as a point-like source.

Figure 1: Herschel/SPIRE spectrum of CygnusA (3C 405). The $x$-axis is
observed frame frequency and the top $x$-axis is the rest frame frequency. The
expected frequencies of the C12O16 rotational ladder from $J=4$ to $J=14$ are
shown (red vertical lines). at the top x-label needs units. Figure 2: SED of
Cygnus A with the best fit model (red solid line) using a dust component (blue
solid line) a synchrotron component (black dashed line). The SED fluxes are
shown in Table 1.

##  3 Spectral Analysis

We show the SPIRE spectrum in Fig. 1, with the expected frequencies of the
C12O16 rotational ladder indicated. We found no emission or absorption lines
in the $v=0$ state arising from rotational levels from $J=4$ to as high as
$J=14$.

Figure 2 shows the $86-1900$ Hz spectral energy distribution (SED) of Cygnus
A. The flux densities of these observations are shown in Table 1 and discussed
in Section 2. The ALMA fluxes were estimated using the CASA tool 444More
information about the tool and its relevant API can be found at the CASA
Website (McMullin et al., 2007). For these SED values in Table 1, we use
different apertures. The entire FoV as mentioned in the 1 is not used as only
the nuclear flux is calculated. The flux is concentrated in the unresolved
several parsecs scale of the nucleus, so the aperture effects are minimized.
We fit the Herschel/SPIRE spectrum using a dust component and a synchrotron
component. Both components are assumed to be power-laws in the form

$f(x)=A\left(\frac{x}{x_{0}}\right)^{\alpha}$ (1)

where $A$ is the model amplitude, $x_{0}$ is the reference point and $\alpha$
is the power law index. The two power law functions had $\alpha$ values $2.39$
and $-1.09$ respectively. The final model follows remarkably well the SED of
Cygnus A.

Our measurements of the sub-mm continuum and upper limits to the CO line
fluxes from the Herschel SPIRE spectrum are given in Table 2. The expected CO
line frequencies are calculated using the rigid rotator approximation and
match well the values from Flower & Pineau Des Forêts (2010); Table 1A). The
density at which collisional de-excitation competes with radiative de-
excitation is called the critical density. The critical density for each CO
transitions is given by $n_{c}(H_{2})$. $n_{c}(H_{2})$ for CO transition (J to
J-1) is approximately $4\times 10^{3}*J^{3}$ (Kramer et al., 2004).
Collisional transitions with $\Delta J>1$ are less likely than $\Delta J=1$
and thus the critical density can be approximated as above. The continuum flux
values in column 4 (Table 2) were estimated by taking the mean flux around the
line center within a 50GHz-wide window. The standard deviation of the flux
within each window is shown in column 6. The instrumental resolution (km/s)
for the corresponding transition line center in the last column is calculated
from instrumental line width (GHz), that is $1.447$ GHz/$2.355$ as given in
the Guide to Herschel-SPIRE (version 1.0 Feb 8 2017). We also estimate the
$3\sigma$ upper-limit line fluxes as

$3\sigma_{\rm lf}=3\sqrt{2\pi}\sigma_{\rm noise}\theta_{inst,\nu},$ (2)

where $3\sigma_{\rm lf}$ is the 3$\sigma$ upper limit to the line flux,
$\theta_{inst,\nu}$ is the instrumental resolution for a corresponding line
center frequency. These values show the non-detected upper-limit CO absorption
or emission in Cygnus A corresponding to the pure rotational transitions.

We estimate the radio core fraction that contributes to the continuum at each
CO line frequency (4th column of Table 2 using the two power-law fit shown in
1. The lower frequency power-law component comes from the core of the radio
jet and that the higher frequency power-law component comes from dust
emission. The observed upper-limits to any CO absorption line fluxes and
optical depths depend on the covering fractions of these regions by any
foreground molecular clouds, which may be different. For example, if the dust
emission is extended over the galaxy, there may be no significant absorption
of this component, except by any molecular clouds that lie in dust lanes
crossing the nucleus. Similarly, the absorption of continuum from the radio
core depends on its angular extent, which has been resolved at cm wavelengths
by the VLA (Carilli et al. (2019a); Perley et al. (2017)), and the CO cloud
covering fraction. Since these continuum sources are not resolved by Herschel,
we consider several cases: 1) both radio and dust emitting sources are compact
and covered equally by CO clouds, 2) only the radio core is compact, 3) only
the dust emission is compact, 4) both of the sources are extended and have
different covering factors.

We can constrain the geometries of the continuum sources at low frequency
using continuum observations retrieved from the ALMA archive (Table 1, Section
2). Only the highest frequency (ALMA band 7) overlaps with our Herschel
observations. The continuum observed by ALMA is compact with a size of
$0.1$″555We acknowledge that, given the unresolved nature of the ALMA core of
Cygnus A, there is potential for a more precise estimation of its size through
the application of advanced deconvolution techniques. These techniques aim to
enhance the spatial resolution of the observations, allowing for a more
accurate determination of the core’s true angular extent. We thus propose that
further ALMA observations would be useful.. The core in ALMA is blended with
the jet but it does not probe the dust component and thus we can modify our
upper-limits on CO absorption by taking new ALMA observations.

| Frequency
---
[GHz]
| Flux Density
---
[Jy]
Aperture/FOV | References
86 | 1.28 | - | Agudo et al. (2014)
97.5 | 0.67 (0.22) | 59.732" | ALMA SED (This work)
145 | 0.75 (0.25) | 40.163" | ALMA SED (This work)
150 | 0.89 | 19" | Monfardini et al. (2011)
220 | 0.76 | 19" | Monfardini et al. (2011)
229 | 0.88 | - | Agudo et al. (2014)
230 | 0.48 | 1.10" $\times$ 0.91" | Wright & Birkinshaw (2004)
273 | 0.58 | 19" (65mm aperture) | Eales et al. (1989)
343.5 | 0.45 (0.16) | 16.953" | ALMA SED (This work)
375 | 0.56 | 19" | Eales et al. (1989)
600 | 0.36 (0.11) | 36".29 | Lopez-Rodriguez et al. (2018)
856.5 | 0.39 (0.09) | 25".67 | Lopez-Rodriguez et al. (2018)
1199.2 | 0.56 (0.04) | 18".59 | Lopez-Rodriguez et al. (2018)
1873.7 | 1.26 (0.02) | 10".62 | Lopez-Rodriguez et al. (2018)

Table 1: Nuclear fluxes for Cygnus A (3C 405).

CO J Transition | Continuum | Gaussian Fit Profile |  $\theta_{inst,\nu}$ [km/s]
---|---|---|---
J |  | Line Center
---
[GHz]
| $n_{c}(H_{2})$
---
[$cm^{-3}$]
| Mean Flux
---
(in-band continuum)
[Jy]
| Radio Core
---
Fraction
|  $\sigma$ noise
---
(mean flux)
[Jy]
|  $3\sigma$ CO
---
Line Flux U.L
[Jy*km/s]
|  $3\sigma$ CO
---
Line Flux U.L $\times 10^{-18}$
[W/m^2]
|  $3\sigma$ UL
---
to line EW
[km/s]
| $L_{CO}$
---
[$L_{\odot}$]
5-4 | 545.67 | $5\times 10^{5}$ | 0.523 | 0.798 | 0.183 | 450 | 8.18 | 860 | <$1.47\times 10^{7}$ | 796
6-5 | 654.7467 | $8.6\times 10^{5}$ | 0.47 | 0.677 | 0.1 | 210 | 4.59 | 447 | <$8.24\times 10^{6}$ | 663
7-6 | 763.8121 | $1.4\times 10^{6}$ | 0.462 | 0.551 | 0.087 | 158 | 4.02 | 342 | <$7.23\times 10^{6}$ | 568
8-7 | 872.8426 | $2\times 10^{6}$ | 0.44 | 0.435 | 0.124 | 196 | 5.72 | 445 | <$1.02\times 10^{7}$ | 497
9-8 | 981.8392 | $2.9\times 10^{6}$ | 0.523 | 0.338 | 0.173 | 236 | 7.74 | 451 | <$1.39\times 10^{7}$ | 442
10-9 | 1090.7974 | $4\times 10^{6}$ | 0.499 | 0.262 | 0.112 | 142 | 5.15 | 285 | <$9.28\times 10^{6}$ | 398
11-10 | 1199.713 | $5.3\times 10^{6}$ | 0.593 | 0.203 | 0.115 | 133 | 5.31 | 224 | <$9.56\times 10^{6}$ | 362
12-11 | 1308.581 | $6.9\times 10^{6}$ | 0.657 | 0.158 | 0.114 | 119 | 5.21 | 181 | <$9.33\times 10^{6}$ | 332
13-12 | 1417.399 | $8.8\times 10^{6}$ | 0.755 | 0.125 | 0.12 | 117 | 5.54 | 155 | <$9.93\times 10^{6}$ | 306
14-13 | 1526.161 | $1.1\times 10^{7}$ | 0.809 | 0.0992 | 0.182 | 164 | 8.35 | 203 | <$1.50\times 10^{7}$ | 284
|  |  |  |  |  |  |  |  |  | <$1.08\times 10^{8}$
---
(Total)

Table 2: Measured upper-limit fluxes of the CO12O16 rotational levels from
J=$4$ to J=$14$ from the Herschel/Spire spectrum. Columns 1 and 2 are the
frequencies in the observed frame corresponding to the J transitions (Flower &
Pineau Des Forêts, 2010). Column 3 gives the critical density for each of the
transitions. Column 4 shows the measured mean continuum fluxes around the
corresponding line center in a $50$GHz-wide window. Radio core fraction
calculated for the particular frequency is given by column 5. The features of
the Gaussian fit profile are given in columns 6-9. Column 10 gives the CO
luminosity in terms of $L_{\odot}$. Rest of the details are given in Section 3

##  4 Discussion

###  4.1 Non-Detection of CO by Herschel/SPIRE

As discussed in Section 3, the Herschel/SPIRE observations failed to detect
any CO lines in emission and absorption.

####  4.1.1 Non-Detection of CO in Absorption

The $3\sigma$ upper limits are expressed as solar luminosities in Table 2,
calculated as

$L_{\rm CO}=1.04\times 10^{-3}\frac{S_{\rm CO}\nu_{\rm rest}D_{\rm
L}^{2}}{(1+z)}~{}[L_{\odot}]$ (3)

where $S_{\rm CO}$ is measured line flux in units of Jy km s-1, $D_{\rm
L}=240$ Mpc is the luminosity distance in units of Mpc, and $z$ is the
redshift. For all the lines in the Herschel data, the upper-limit luminosities
are $\lesssim 1\times 10^{7}L_{\odot}$. We estimate a total added upper-limit
luminosity of $<10^{8}L_{\odot}$. This upper-limit luminosity is estimated as
the sum of the individual line upper-limits. Undoubtedly overestimates the
actual upper-limit to the total CO line luminosity. A more careful analysis
taking into account that the fluxes in adjacent lines will be correlated (in a
manner that depends on the physical conditions in the emitting regions), and
the varying noise levels across the Herschel spectrum is required.

Relative to the $J=8-7$ line luminosity upper limit, the individual line upper
limits for Cygnus A range between 0.7 and 1.5. For comparison, in the flux-
limited ULIRG sample of Pearson et al. (2016), for the subsample in which all
lines from $J=5-4$ to $J=13-12$ were detected, these ratios range between 0.2
and 1.6. In comparison, our estimated upper limit to the luminosity over that
same ranfe is approximately 10.6 times higher than the upper limit to the J =
8 - 7 line luminosity. However it is important to note that deriving ratios
from upper limits can be inconclusive. Given the current data, we made the
assumption that all lines within this range possess equal strength, even
though this may not be entirely accurate. Co-adding all lines in this manner
would yield an upper limit to the flux in any line that would be
$\sim\sqrt{10}$ times lower than our present value. Although this approach
doesn’t account for the variations in noise at different frequencies, it
provides a reasonably acceptable estimation.

We then considered an alternative method to calculate the total luminosity by
multiplying this improved line flux by $\sim 7$, which is roughly the median
ratio observed in the Pearson et al. (2016) subsample. The resulting value is
about $2.2\times 10^{7}$, $\sim 5$ times smaller than the estimate obtained
from simply co-adding the observed fluxes. It is essential to acknowledge that
there is an inherent uncertainty of at least a factor of 2-3 in this number.
Specifically, the estimate could potentially be around 25% lower or 50%
higher, depending on which lines in Cygnus A are utilized as the reference (in
this case, we employed the 8-7 line as the reference for the Pearson et al.
(2016) subsample). Nevertheless, despite the uncertainties, we believe that
this approach represents a reasonably sensible and calculated
estimation666Note that any such attempt to refine the estimate of the
luminosity would be model-dependent, as noted above, and as can be seen
empirically from inspection of the variability in the CO SLEDs shown in
Pearson et al. (2016)’s Figure 16..

####  4.1.2 Non-Detection of CO in Emission

Is the failure to detect any emission lines surprising, given the expectation
based on the presence of a significant amount of hot mass in H2? To answer
this question, we need a comparison sample of galaxies with well-characterized
CO emission lines in these same mid-$J$ range. The best sample currently
available is the set of Herschel/SPIRE galaxies analyzed by Kamenetzky et al.
(2014, hereafter K14). This work established two significant results:

  * •

The CO pure rotation lines that fall within the SPIRE passband (effectively,
$J=5\rightarrow 4$ to $J=14\rightarrow 13$) are produced in a ‘warm’ (more
accurately, high-pressure) molecular gas component that contains only a small
fraction of the total molecular gas mass but completely dominates the CO line
luminosity; and

  * •

The ratio of the total warm CO luminosity to the far-infrared luminosity (the
latter defined as the $8-1000$ $\mu$m luminosity) is constant at about
$4\times 10^{-4}$, with no discernible dependence on luminosity or galaxy
classification (i.e., AGN or starburst).

This sample is admittedly limited — there are only 17 galaxies — but
represents the best currently available. From the obsrved $60~{}\mu$m flux of
Cygnus A, its far-infrared luminosity ($\nu L_{\nu}$) is $L_{\rm IR}\sim
10^{45}$ erg s-1. The K14 result then implies a warm CO luminosity $L_{\rm
CO}\sim 10^{8}L_{\odot}$. In other words, if Cygnus A resembles the K14 sample
in its ratio of warm CO to far-infrared luminosity (which does not imply that
it resembles this rather diverse set of galaxies in any other way), its
predicted SPIRE-band CO emission is comparable to the observed $3\sigma$ upper
limit. This strongly suggests that the failure to detect any CO emission line
from Cygnus A with Herschel/SPIRE is not unexpected. This result does not
indicate that there is anything necessarily anomalous about Cygnus A in this
respect. Based on the observed far-infrared luminosity, the SPIRE observation
is simply not deep enough to have unambiguously detected the emission.

Carilli et al. (2022) detected CO (2-1) emission within -400km/s to +250km/s
velocity range, spanning about 4.4kpc in a north-south orientation, closely
following the dust lane in HST’s (Hubble Space Telescope) I-band imaging
(Figure 2 of Carilli et al. (2022)). The emission appears clumpy, with two
dominant regions approximately 1.5" in size, located northeast and southwest
of the nucleus, and faint emission across the galactic center, exhibiting a
significant asymmetry between the bright, double-peaked emission in the south
and diffuse emission in the north.

###  4.2 Radiative Excitation Hypothesis

To explain the failure of Barvainis & Antonucci (1994) to detect absorption in
the $J=1\rightarrow 0$ line against the radio core of Cygnus A, Maloney et al.
(1994) suggested that radiative excitation by the nonthermal continuum could
increase the excitation temperature of the CO molecules substantially.
Thereby, reducing the line optical depths and making the CO column
undetectable. Is this suggestion still a viable possibility?, and could
radiative excitation play a role in the failure to detect any lines in the
SPIRE band?

Thanks to the VLBI study by Boccardi et al. (2016), we have a much better
understanding of the morphology of the continuum emission at $3$mm than was
available in 1994. Most of the flux density, $S_{\nu}\approx 1$ Jy, arises
within a radius of 0.1 pc, which is much smaller than any size scale likely to
be relevant for even a nuclear molecular gas component. We can thus write the
mean intensity at the line frequency as

$J_{\nu}\approx 4.6\times 10^{-8}\;{\rm r_{\rm pc}^{-2}\
\;erg\,cm^{-2}\,s^{-1}\,Hz^{-1}\,sr^{-1}}$ (4)

where $r_{\rm pc}$ is the distance from the source in pc The Einstein
$A$-coefficient for the $1\rightarrow 0$ line is $A_{10}=7.165\times
10^{-8}\,{\rm s}^{-1}$. Using the relation between the $A$ and $B$
coefficients, the stimulated emission rate $B_{10}J_{\nu}$ will exceed the
spontaneous emission for any radius $r_{\rm pc}<1400$ pc, with the ratio
scaling as $1/r^{2}$. Hence even for size scales of 100 – 200 pc, stimulated
emission will completely dominate over spontaneous emission. More importantly,
for gas on these size scales the stimulated emission and photon absorption
rates will be far larger than the collisional de-excitation rates; e.g., at
$r=200$ pc and gas density $n=10^{4}$ cm-3, the stimulated emission rate is an
order of magnitude larger than the collisional de-excitation rate for any
plausible temperature for molecular gas. Hence it is quite possible that the
level populations and thus optical depths in the $1\rightarrow 0$ are
significantly affected by radiative excitation by the non-thermal continuum.
However, it is possible that radiative excitation could be dominant for the
$1\rightarrow 0$ line while having no significant effect on the CO rotation
lines within the SPIRE spectrum. This depends crucially on the size scale
characterizing the molecular gas.

Implicit in the analysis of MBR94 is the assumption that the solid angle-
averaged brightness temperature of the non-thermal continuum is much greater
than the gas kinetic temperature at all frequencies of interest. This was in
keeping with the then-current paradigm of $r\sim$ pc scales for the obscuring
‘tori’ that block our view of the central engine in Type 2 objects. However,
there is abundant evidence for obscuring gas (atomic and molecular) on much
larger scales in AGN. In Cygnus A, in particular, the VLBA HI absorption
observations of Struve & Conway (2010) and the bremsstrahlung torus suggested
by Carilli et al. (2019b) indicate size scales of $\sim 100-200$ pc for the
observed nuclear gas (but see below). This value is in agreement with those
derived from the IR SED modelling with CLUMPY (Nenkova et al., 2008) torus
models (e.g.: Privon et al., 2012). If this same scale applies to the nuclear
molecular gas, then radiative excitation will not be important in the SPIRE
spectrum. The reason is because the solid angle-averaged brightness
temperatures of the non-thermal continuum will be too low for this process to
be important for the warm molecular gas components observable in the SPIRE
spectrum.

The slope of the mm/submm-wavelength continuum in Cygnus A is not well
determined. However, the slope has only a secondary impact, as we will see.
For simplicity, assume the CO rotational line frequencies scale precisely as
$J$. Then, we can write the angle-averaged brightness temperature as

$T_{\rm b}=\frac{5.53J}{\ln(1+4.9\times 10^{-7}J^{3+\alpha}r^{2}_{\rm pc})}$
(5)

where the non-thermal continuum flux density goes as $\nu^{-\alpha}$. For the
$J=5\rightarrow 4$ line, the predicted mean brightness temperature is $T_{\rm
b}=4.5\times 10^{5}$ K and $9.0\times 10^{4}$ K for $\alpha=0$ and 1,
respectively, for $r_{\rm pc}=1$. However, if $r_{\rm pc}=100$, these numbers
become 58 K and 20 K. Hence radiative excitation by the non-thermal continuum
will not be important on $r\sim 100$ pc or larger size scales.

It is important to note that the scale of the gas producing the large X-ray
absorption column has not yet been established. As pointed out by Struve &
Conway (2010), although they detect strong, broad ($FWHM=231\pm 20$ km/s) HI
absorption against the counter-jet, which they argue arises in gas at $r\sim
80$ pc, against the core the absorption is very weak. One simple way to
resolve this discrepancy, as they also note, is if the X-ray absorbing gas
occurs on scales $\ll 100$ pc in a spatially compact torus. In such a torus,
the radiative excitation by the continuum could be important for most, if not
all, of the CO lines falling within the SPIRE spectrum. This is largely a
subject to debate because the mass of molecular gas in the torus would be much
too small to have been detected by SPIRE.

An interesting contrast to Cygnus A is presented by the detection of strong,
narrow CO absorption in both the $1\rightarrow 0$ and $2\rightarrow 1$ lines
against the bright, compact mm core of the powerful radio galaxy Hydra A at
$z=0.05435$ (Rose et al., 2019, 2020). The absorption features are very
narrow, and appear to arise in a $\sim$ kpc-scale disk of molecular gas. This
is precisely the sort of absorption feature that might have been expected in
Cygnus A. It is important to note, however, that the integrated emission in
the CO lines in Hydra A is much stronger than the absorption, which raises the
question why the observations to date of Cygnus A in the $1\rightarrow 0$ line
have not detected any emission, either.

Does this indicate that there is something anomalous after all about the CO
emission in Cygnus A? Perhaps not. The most sensitive current upper limits
(Fuente et al., 2000; Evans et al., 2005) correspond to a $1\rightarrow 0$
luminosity upper limit of $L^{\prime}_{\rm CO}<2.2\times 10^{8}\;{\rm
K\,km\,s^{-1}\,pc^{2}}$, while the far-IR luminosity $L_{\rm IR}\sim 2-4\times
10^{11}L_{\odot}$. Plotting this data point in Figure 4 of Krips et al. (2012)
and Figure 8 of Chung et al. (2009), who present CO $1\rightarrow 0$
observations of low-redshift Type 2 quasars and ULIRGs, respectively, suggests
that Cygnus A may represent one end of a continuum of CO emission for a given
IR luminosity, i.e., it represents an extreme, but not an outlier.

###  4.3 Bremsstrahlung Torus Model

The bremsstrahlung torus proposed by Carilli et al. (2019a) has a radius
$R_{\rm t}\sim 264$ pc and a half-thickness $h_{rmt}\sim 143$ pc, yields a
total volume $V_{\rm t}=2\pi^{2}R_{\rm t}h_{\rm t}^{2}\sim 1.1\times 10^{8}$
pc3. Given the observed emission, they derive a mean electron density of
$\bar{n}_{\rm e}=490$ cm-3. However, Carilli et al. (2019a) pointed out that
this uniformly-filled torus would produce too much Thomson scattering, by at
least a factor of 8. Their solution is to make the torus clumpy. The
bremsstrahlung emission scales as $n_{\rm e}^{2}$ and the Thomson optical
depth is $\tau_{T}\propto n_{\rm e}$, then the density increases by some
factor $\rho$ above the mean value. Thus, the length scale $L$ decreases
through the torus by $\rho^{2}$, while keeping the emission measure
$n_{e}^{2}L$ constant, but $\tau_{T}$ decreases by a factor of $\rho$.

However, the total mass of ionized gas within the torus raises a serious
energetic problem. We first note that the total volume recombination rate
within the torus is independent of the clumps. This is most simply seen by
noting that the volume recombination rate can be calculated by multiplying the
torus area by the column recombination rate, where the latter is fixed by the
observations. In other words, we must have $\rho^{2}\bar{n}_{\rm e}^{2}d={\rm
constant}$, where $d$ is any convenient length scale through the torus. This
fixes the effective radius of the torus as $R_{\rm eff}=R_{\rm t}/\rho^{2}$,
and the volume recombination rate is

$N_{\rm rec}=2\pi^{2}R_{\rm eff}h_{\rm t}^{2}\alpha_{\rm B}n_{\rm
e}^{2}=2\pi^{2}(R_{\rm t}/\rho^{2})h_{\rm t}^{2}\alpha_{\rm
B}\rho^{2}\bar{n}_{\rm e}^{2}\\\ =\alpha_{B}\bar{n}_{\rm e}^{2}V_{\rm t}$ (6)

independent of the clumping factor. For $T_{\rm e}=10^{4}$ K, we then get
$N_{\rm rec}\sim 2\times 10^{56}$ s-1. Decreasing $T_{\rm e}$ to $8000$ K
raises this number by about 20%.

Every recombination within the torus releases about one Rydberg of energy
(i.e. $2.179\times 10^{-11}$ erg s-1); if the torus is dusty, all this energy
will (given the large torus column density) eventually be absorbed by the dust
and re-radiated by the torus. This is a luminosity $L_{\rm rec}\sim 4\times
10^{45}$ erg s-1, or $1.1\times 10^{12}L_{\odot}$.

This is a very large number that exceeds estimates of the total observed
infrared luminosity of Cygnus A by a factor of $3-5$. Note also that this
luminosity underestimates the amount of energy that must be radiated by the
torus, since it only takes the ionizing luminosity into account, and also
ignores the substantial fraction of ionizing photons that will be absorbed
directly by the dust rather than the gas. This poses a serious problem for the
bremsstrahlung torus interpretation, which can only be alleviated by reducing
the size of the torus significantly.

###  4.4 The X-Ray obscuration is not produced by an atomic torus

A possibility raised by MBR94 for the absence of detectable CO absorption is
that the torus is entirely atomic. This is entirely feasible in an X-ray
dominated region (XDR) even for such large column densities for the pressure
to be below a critical value (Maloney, 1996). However, subsequent observations
have ruled this out as a possibility.

Conway & Blanco (1995) detected broad (FWHM $\sim 270$ km s-1) HI absorption
towards Cygnus A, and argued that the HI absorption could arise in an atomic
torus. A later study by Struve & Conway (2010) with much improved signal to
noise argued that this HI absorption arises in a disk with a radius $r\sim 80$
pc. Notably, strong HI absorption was detected only against the counter-jet;
the optical depth against the core is only $\tau=0.016$, whereas against the
counter-jet $\tau>3$. Such a low optical depth against the core can only be
associated with the obscuration measured against the X-ray continuum if the
spin temperature is high ($T_{\rm s}~{}10^{6}$ K) and the velocity dispersion
is very large ($\Delta V\sim 500$ km s-1). The former is a plausible value for
a nuclear atomic torus within a few hundred pc of the central engine, but the
latter requires that the gas is in very close proximity to the central super
massive black hole. However, Conway & Blanco (1995) pointed out that the radio
spectrum is flat down to $1.34$ GHz, implying the emission is still optically
thin at this frequency. This places strict limits on the pressure in the
torus, which must at minimum equal the absorbed radiation pressure; the
fraction of the incident bolometric luminosity that is absorbed can be
substantial ($f_{\rm abs}\sim 0.1-0.2$) because of the large column density.

This pressure constraint arises because the free-free absorption depth in an
atomic torus is non-negligible due to the significant electron fraction,
$x_{\rm e}\sim 3\times 10^{-2}$, maintained by the high X-ray ionization rate.
Scaled to parameters appropriate to Cygnus A (see section 3 of Maloney
(1996)), this constrains the torus pressure to $P/k\lessapprox 3\times 10^{8}$
cm-3 K. This pressure requires that the torus distance from the nucleus
$r\gtrapprox 55$ pc. This is comparable to the scale inferred by Struve &
Conway (2010) for their HI-absorbing gas, and is completely inconsistent with
a gas velocity dispersion large enough to be consistent with the weak HI
absorption measured towards the core. Hence we can be confident that the X-ray
absorption is not produced in an atomic torus, neither that imaged by Struve &
Conway (2010) or a putative $r\sim$ pc-scale torus.

##  5 Conclusion

We report the Herschel/SPIRE non-detection of CO emission and absorption in
Cygnus A, for the $\nu=0$ vibrational state arising from rotational levels
$14\geq J\geq 4$. MBR94 suggested that the optical depth for $J=1\rightarrow
0$ transitions is affected by radiative excitation by the non-thermal
continuum. However, assuming the $\sim 100-200$ pc scale of obscuring gas
found by Struve & Conway (2010) and Carilli et al. (2019b), we find that the
solid angle-averaged brightness temperature of the non-thermal continuum is
too low for radiative excitation to be significant in the Herschel/SPIRE
spectrum.

The $3\sigma$ upper limits to the line fluxes of the nuclear Cygnus A
Herschel/SPIRE spectrum imply a CO luminosity $L_{CO}\lesssim
10^{8}L_{\odot}$. While this is lower than expected from the median $L_{\rm
CO}-L_{\rm FIR}$ relation for galaxies, it falls within the expected range if
some of the FIR emission comes from the AGN. A more accurate estimate requires
the knowledge of the size scale of the X-ray absorbing gas, which is unknown.
Alternatively, more sensitive ALMA observations will lead to a detection of CO
in emission.

Contrary to the suggestion of Carilli et al. (2019b), the sub-mm continuum
emission from the torus cannot be attributed to bremsstrahlung. This
conclusion is based on the fact that the resulting recombination luminosity
exceeds the estimate of total infrared luminosity for Cygnus A by a factor of
3 to 5. This type of model would only work for sub-mm continuum emission at a
much smaller size scale, which may be probed by future ALMA observations of
Cygnus A.

## Acknowledgements

RA thanks Dr. David L. Meier and Dr. Masa Imanishi for their helpful
suggestions and valuable conversations which helped various parts of this
manuscript.

HIFI has been designed and built by a consortium of institutes and university
departments from across Europe, Canada and the United States under the
leadership of SRON Netherlands Institute for Space Research, Groningen, The
Netherlands and with major contributions from Germany, France and the US.
Consortium members are: Canada: CSA, U.Waterloo; France: CESR, LAB, LERMA,
IRAM; Germany: KOSMA, MPIfR, MPS; Ireland, NUI Maynooth; Italy: ASI, IFSI-
INAF, Osservatorio Astrofisico di Arcetri-INAF; Netherlands: SRON, TUD;
Poland: CAMK, CBK; Spain: Observatorio Astronómico Nacional (IGN), Centro de
Astrobiología (CSIC-INTA). Sweden: Chalmers University of Technology - MC2,
RSS & GARD; Onsala Space Observatory; Swedish National Space Board, Stockholm
University - Stockholm Observatory; Switzerland: ETH Zurich, FHNW; USA:
Caltech, JPL, NHSC.

SPIRE has been developed by a consortium of institutes led by Cardiff
University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA,
LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory
(Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK);
and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been
supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES,
CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and
NASA (USA).

Based on observations made with the NASA/DLR Stratospheric Observatory for
Infrared Astronomy (SOFIA) under the 05_0071 Program. SOFIA is jointly
operated by the Universities Space Research Association, Inc. (USRA), under
NASA contract NNA17BF53C, and the Deutsches SOFIA Institut (DSI) under DLR
contract 50 OK 0901 to the University of Stuttgart.

## Data Availability

We downloaded the data from Herschel/SPIRE data archive
777http://archives.esac.esa.int/hsa/whsa/, with the observation id 1342246994
(PI: Dr. Patrick Ogle) taken in the SpireSpectroPoint observing mode. In case
any reader has more queries regarding the data availability, they can further
contact the author.

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