11institutetext: Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333
CA Leiden, Netherlands<EMAIL_ADDRESS>22institutetext: Instituto de
Física Fundamental, CSIC, Calle Serrano 121-123, 28006 Madrid, Spain
33institutetext: University of Vienna, Department of Astrophysics,
Türkenschanzstrasse 17, 1180 Vienna, Austria 44institutetext: Telespazio Vega
UK Ltd. for ESA/ESAC, Urbanizacion Villafranca del Castillo, 28691 Madrid,
Spain 55institutetext: IRAP, Université de Toulouse, CNRS, CNES, UPS, 9 Av.
colonel Roche, 31028 Toulouse Cedex 4, France 66institutetext: Department of
Astronomy, University of Maryland, College Park, MD 20742, USA
77institutetext: I. Physikalisches Institut der Universität zu Köln, Zülpicher
Strasse 77, 50937 Köln, Germany 88institutetext: USRA/SOFIA, NASA Ames
Research Center, Mail Stop 232-12, Building N232, P.O. Box 1, Moffett Field,
CA 94035-0001, USA 99institutetext: Max-Planck-Institut für Radioastronomie,
Auf dem Hügel 69, 53121 Bonn, Germany 1010institutetext: Institut de
Radioastronomie Millimétrique, 300 rue de la Piscine, 38406 Saint Martin
d’Hères, France

# [C ii] $158\,\mu\mathrm{m}$ line emission from Orion A.
II. Photodissociation region physics

C. H. M. Pabst [C ii] $158\,\mu\mathrm{m}$ line emission from Orion A. II.
Photodissociation region physics[C ii] $158\,\mu\mathrm{m}$ line emission from
Orion A. II. Photodissociation region physics J. R. Goicoechea [C ii]
$158\,\mu\mathrm{m}$ line emission from Orion A. II. Photodissociation region
physics[C ii] $158\,\mu\mathrm{m}$ line emission from Orion A. II.
Photodissociation region physics A. Hacar [C ii] $158\,\mu\mathrm{m}$ line
emission from Orion A. II. Photodissociation region physics[C ii]
$158\,\mu\mathrm{m}$ line emission from Orion A. II. Photodissociation region
physics[C ii] $158\,\mu\mathrm{m}$ line emission from Orion A. II.
Photodissociation region physics[C ii] $158\,\mu\mathrm{m}$ line emission from
Orion A. II. Photodissociation region physics D. Teyssier [C ii]
$158\,\mu\mathrm{m}$ line emission from Orion A. II. Photodissociation region
physics[C ii] $158\,\mu\mathrm{m}$ line emission from Orion A. II.
Photodissociation region physics O. Berné [C ii] $158\,\mu\mathrm{m}$ line
emission from Orion A. II. Photodissociation region physics[C ii]
$158\,\mu\mathrm{m}$ line emission from Orion A. II. Photodissociation region
physics M. G. Wolfire [C ii] $158\,\mu\mathrm{m}$ line emission from Orion A.
II. Photodissociation region physics[C ii] $158\,\mu\mathrm{m}$ line emission
from Orion A. II. Photodissociation region physics R. D. Higgins [C ii]
$158\,\mu\mathrm{m}$ line emission from Orion A. II. Photodissociation region
physics[C ii] $158\,\mu\mathrm{m}$ line emission from Orion A. II.
Photodissociation region physics E. T. Chambers [C ii] $158\,\mu\mathrm{m}$
line emission from Orion A. II. Photodissociation region physics[C ii]
$158\,\mu\mathrm{m}$ line emission from Orion A. II. Photodissociation region
physics S. Kabanovic [C ii] $158\,\mu\mathrm{m}$ line emission from Orion A.
II. Photodissociation region physics[C ii] $158\,\mu\mathrm{m}$ line emission
from Orion A. II. Photodissociation region physics R. Güsten [C ii]
$158\,\mu\mathrm{m}$ line emission from Orion A. II. Photodissociation region
physics[C ii] $158\,\mu\mathrm{m}$ line emission from Orion A. II.
Photodissociation region physics J. Stutzki [C ii] $158\,\mu\mathrm{m}$ line
emission from Orion A. II. Photodissociation region physics[C ii]
$158\,\mu\mathrm{m}$ line emission from Orion A. II. Photodissociation region
physics C. Kramer [C ii] $158\,\mu\mathrm{m}$ line emission from Orion A. II.
Photodissociation region physics[C ii] $158\,\mu\mathrm{m}$ line emission from
Orion A. II. Photodissociation region physics A. G. G. M. Tielens [C ii]
$158\,\mu\mathrm{m}$ line emission from Orion A. II. Photodissociation region
physics[C ii] $158\,\mu\mathrm{m}$ line emission from Orion A. II.
Photodissociation region physics[C ii] $158\,\mu\mathrm{m}$ line emission from
Orion A. II. Photodissociation region physics[C ii] $158\,\mu\mathrm{m}$ line
emission from Orion A. II. Photodissociation region physics

(Received 13 March 2021, Accepted 22 November 2021)

###### Abstract

Context. The [C ii] $158\,\mu\mathrm{m}$ fine-structure line is the dominant
cooling line of moderate-density photodissociation regions (PDRs) illuminated
by moderately bright far-ultraviolet (FUV) radiation fields. This makes this
line a prime diagnostic for extended regions illuminated by massive stars.

Aims. We aim to understand the origin of [C ii] emission and its relation to
other tracers of gas and dust in PDRs. One focus is a study of the heating
efficiency of interstellar gas as traced by the [C ii] line to test models of
the photoelectric heating of neutral gas by polycyclic aromatic hydrocarbon
(PAH) molecules and very small grains.

Methods. We make use of a one-square-degree map of velocity-resolved [C ii]
line emission toward the Orion Nebula complex, and split this out into the
individual spatial components, the expanding Veil Shell, the surface of OMC4,
and the PDRs associated with the compact H ii region of M43 and the reflection
nebula NGC 1977. We employed Herschel far-infrared photometric images to
determine dust properties. Moreover, we compared with Spitzer mid-infrared
photometry to trace hot dust and large molecules, and velocity-resolved IRAM
30m CO(2-1) observations of the molecular gas.

Results. The [C ii] intensity is tightly correlated with PAH emission in the
IRAC $8\,\mu\mathrm{m}$ band and far-infrared emission from warm dust, with
small variations between the four studied subregions (Veil Shell, OMC4, M43,
and NGC 1977). The correlation between [C ii] and CO(2-1) is very different in
the four subregions and is very sensitive to the detailed geometry of the
respective regions. Constant-density PDR models are able to reproduce the
observed [C ii], CO(2-1), and integrated far-infrared (FIR) intensities. The
physical conditions in the Veil Shell of the Orion Nebula, M43, and NGC 1977
reveal a constant ratio of thermal pressure $p_{\mathrm{th}}$ over incident
FUV radiation field measured by $G_{0}$. We observe strong variations in the
photoelectric heating efficiency in the Veil Shell behind the Orion Bar and
these variations are seemingly not related to the spectral properties of the
PAHs.

Conclusions. The [C ii] emission from the Orion Nebula complex stems mainly
from moderately illuminated PDR surfaces. The correlations of the different
tracers ([C ii], FIR, CO, $70\,\mu\mathrm{m}$, and $8\,\mu\mathrm{m}$
emission) show small variations that are not yet understood. Future
observations with the James Webb Space Telescope can shine light on the PAH
properties that may be linked to these variations.

##  1 Introduction

The [C ii] $158\,\mu\mathrm{m}$ fine-structure line is the dominant cooling
line of neutral atomic gas at moderate densities ($n_{\rm H}\lesssim
10^{4}\,\mathrm{cm^{-3}}$) and temperatures ($T_{\rm k}\lesssim
300\,\mathrm{K}$; Dalgarno & McCray 1972). This includes diffuse clouds in the
general interstellar medium (ISM) of galaxies as well as regions illuminated
by nearby, newly formed, massive stars (Hollenbach & Tielens 1999). In these
regions carbon atoms are rapidly ionized by penetrating far-ultraviolet (FUV)
photons $E<13.6$ eV) and molecules such as CO are quickly photodissociated.
Hence, most of the gaseous carbon is in the form of C+ ions. The ground state
of C+ has two fine structure levels separated by $\Delta E/k_{\rm
B}=91.2\,\mathrm{K}$. This transition has a critical density of
$(2\text{-}6)\times 10^{3}\,\mathrm{cm^{-3}}$ for collisions with atomic or
molecular hydrogen. Since the upper fine-structure level ${}^{2}P_{3/2}$ is
easy to excite collisionally at typical ISM densities and as the interstellar
carbon abundance is high, the [C ii] $158\,\mu\mathrm{m}$ line dominates the
cooling of the neutral ISM.

Extreme ultraviolet (EUV) radiation $E>13.6$ eV) from massive stars ionize
hydrogen atoms and create an H ii region in their environment. This ionized
gas is separated from the surrounding nascent molecular cloud by a
photodissociation region (PDR) where penetrating FUV photons dissociate
molecules and ionize low ionization potential atoms. These FUV photons
ultimately heat the PDR gas to temperatures of 200-1000 K. The neutral gas
(i.e., hydrogen in neutral form) couples to the FUV photon field through the
photoelectric effect on large polycyclic aromatic hydrocarbon molecules (PAHs)
and very small grains (VSGs; Bakes & Tielens 1994). This same process heats
the gas in diffuse interstellar clouds (Wolfire et al. 1995). Dense PDRs in
star-forming regions are bright in the far-infrared (FIR) atomic cooling
lines, e.g. the [C ii] $158\,\mu\mathrm{m}$ line at moderate densities and
warmish temperatures (Tielens & Hollenbach 1985; Hollenbach et al. 1991), as
well as the PAH emission features (Tielens 2008, and references therein),
fluorescent ro-vibrational lines (e.g., Sellgren 1986; Field et al. 1998;
Kaplan et al. 2017), pure rotational H2 lines (e.g., Sheffer & Wolfire 2013;
Habart et al. 2011; Allers et al. 2005), warm dust (e.g., Berné et al. 2007;
Arab et al. 2012), and a plethora of molecular radicals and reactive ions
(e.g., Pety et al. 2005; Fuente et al. 2003; Nagy et al. 2013; Goicoechea et
al. 2017). Their high surface brightness and compact size make dense PDRs very
appropriate targets for observational studies on the interaction of massive
stars with their interstellar environment. This allows detailed studies of the
physics and chemistry relevant to radiative feedback by massive stars. In
particular, observations of the dominant atomic fine structure lines, through
the energy balance, provide a direct probe of the heating processes of
interstellar neutral atomic gas (e.g., Tielens 2008; Okada et al. 2013; Pabst
et al. 2017; Salas et al. 2019).

The [C ii] $158\,\mu\mathrm{m}$ line is the brightest FIR line in the spectrum
of the Milky Way and, in general, of star-forming galaxies, typically
radiating about 0.3% of the IR dust continuum emission (e.g., Crawford et al.
1985; Bennett et al. 1994; Malhotra et al. 2001; Luhman et al. 2003; Stacey et
al. 2010; Díaz-Santos et al. 2013). As ionization of carbon requires FUV
photons with energies in excess of 11.2 eV, and such photons are only emitted
by short-lived massive stars, the [C ii] emission line is also considered a
tracer of the star formation rate (SFR; e.g., Pineda et al. 2014; De Looze et
al. 2011; Herrera-Camus et al. 2015, 2018). With ALMA and NOEMA, the use of
the [C ii] line as a SFR tracer has now been extended to the high redshift
universe (e.g., Walter et al. 2012; Venemans et al. 2012; Knudsen et al. 2016;
Bischetti et al. 2018; Khusanova et al. 2020). The use of the [C ii]
$158\,\mu\mathrm{m}$ line as a SFR indicator requires a reliable conversion
factor and extensive observational studies have focused on determining this
factor through detailed studies of the [C ii] line in samples of nearby star
forming galaxies (e.g., Malhotra et al. 2001; Herrera-Camus et al. 2015, 2018;
Chevance et al. 2016; Pineda et al. 2018). These observational studies have
revealed a so-called C+-deficit issue: the [C ii]/FIR luminosity ratio is
systematically lower in regions characterized by warm dust, possibly due to
the onset of other cooling processes of neutral dense gas (e.g., the [O i]
$63\,\mu\mathrm{m}$ line), changes in the coupling between the neutral atomic
gas and the FUV photons, reduced gas-heating efficiency when the strong FIR
field leads to positively charged PAHs and small grains making it more
difficult for photoelectrons to escape and heat the gas, or the presence of
energy sources other than FUV photons (e.g., AGNs or deeply embedded star
formation; Luhman et al. 2003; Abel et al. 2009; Graciá-Carpio et al. 2011).
Aside from the photoelectric effect heating the gas H2 formation heating and
collisional de-excitation of vibrationally excited H2 may play a role (e.g.,
Röllig et al. 2006). Besides this observational validation of the use of the
[C ii] line as a SFR indicator, we also need to develop a deep understanding
of the underlying physics involved in the heating and cooling of neutral
atomic gas if we want to confidently extrapolate the local results over
cosmological relevant timescales.

The recent rapid development of receiver technology has allowed the
construction of multi-element heterodyne arrays at FIR frequencies. In
particular, the upGREAT instrument has been specifically designed to map the
[C ii] $158\,\mu\mathrm{m}$ line (Risacher et al. 2016). This sensitive
instrument coupled with the nimble telescope of the Stratospheric Observatory
For Infrared Astronomy (SOFIA) allows for the first time wide-field [C ii]
observations of regions of massive star formation at sub-km s-1 spectral
resolution (Pabst et al. 2017; Bally et al. 2018). The C+SQUAD SOFIA Large
Program has surveyed one square degree of the molecular cloud cores in Orion
in the [C ii] $158\,\mu\mathrm{m}$ line to probe the radiative and mechanical
energy feedback of massive stars with their environment. The results revealed
the presence of a $\sim 2$ pc radius massive ($\sim 1500\,M_{\odot}$) shell of
neutral atomic gas expanding at 13 km s-1 (Pabst et al. 2019). The rapid
expansion of this bubble is driven by the hot plasma generated by the stellar
wind from the O7V star $\theta^{1}$ Ori C (Güdel et al. 2008) as envisioned by
Weaver et al. (1977). The data also revealed bubbles of neutral gas expanding
at a slower pace around the B1 stars powering the H ii regions M43 and NGC
1977 (Pabst et al. 2020). In this case, expansion is driven by the
overpressure of the ionized gas, a so-called Spitzer expansion. While these
studies focused on the kinematics of the region and the mechanical feedback by
massive stars, this data also allows an in depth study of the radiative
interaction of massive stars with neutral atomic gas on unprecedented spatial
scales.

In Paper I (Pabst et al. 2021), we used the C+SQUAD results to study the
global characteristics of the [C ii] emission in Orion as a template to
understand the use of the [C ii] emission as a SFR indicator and to
investigate the [C ii] deficit. In this paper, we separate the surveyed region
in its spatial components, compare the observations with specific PDR models,
and examine the radiative interaction of the powering stars with their
environment in detail. This paper is organized as follows: In Section 2, we
summarize the observations used in this study. Section 3 discusses the global
morphology of the [C ii] line emission and presents the correlations we find
of the [C ii] emission with other tracers of gas and dust in the Orion Nebula
(M42), M43, and NGC 1977, separately. In Section 4, we compare the
observations to PDR models, discuss the implications thereof, and examine the
heating efficiency of the [C ii]-emitting gas. We summarize our results in
Section 5.

Figure 1: Dust temperature (color scale) and optical depth at 160 $\mu$m
(contours, from black to white: $3\times 10^{-3}$, $5\times 10^{-3}$, $1\times
10^{-2}$, $3\times 10^{-2}$, $5\times 10^{-2}$) from SED fit ($\beta=2$).
Figure 2: [C ii] line-integrated intensity
($v_{\mathrm{LSR}}=-10\,\text{-}\,+20\,\mathrm{km\,s^{-1}}$) from Orion A (at
a resolution of $16\arcsec$) with CO(2-1) line-integrated intensity
($v_{\mathrm{LSR}}=-10\,\text{-}\,+20\,\mathrm{km\,s^{-1}}$, coverage not
complete, cf. Paper I) in contours (from black to light gray: $1,2,3,4\times
10^{-6}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$). The red circles delineate
the three distinct shells of M42, M43, and NGC 1977. The stars indicate the
most massive stars within each region: $\theta^{1}$ Ori C (yellow) in M42, NU
Ori (pink) in M43, and 42 Orionis (purple) in NGC 1977. Arrows indicate the
positions of the four molecular cores OMC1-4 along the Integral-Shaped
Filament. Figure 3: [C ii] intensity per $1\,\mathrm{km\,s^{-1}}$ channel in
the range $v_{\mathrm{LSR}}=-9\,\text{-}\,19\,\mathrm{km\,s^{-1}}$. The
intensities in the last two channels
($v_{\mathrm{LSR}}=17\text{-}19\,\mathrm{km\,s^{-1}}$) contain [13C ii]
$F=2\text{-}1$ emission and are multiplied by 5.

##  2 Observations

We make use of velocity-resolved [C ii] line observations toward Orion A,
obtained by upGREAT onboard SOFIA, IRAM 30m velocity-resolved 12CO $J$ = 2-1
(230.5 GHz) and 13CO $J$ = 2-1 (220.4 GHz) line observations, Herschel dust
continuum photometry, Spitzer observations in the Multi-band Imaging
Photometer for Spitzer (MIPS) $24\,\mu\mathrm{m}$ band and the Infrared Array
Camera (IRAC) 8 $\mu$m band, ESO/Wide Field Imager (WFI) H$\alpha$
observations, and Digitized Sky Survey 2 (DSS-2) red-band observations
converted to H$\alpha$ intensity (by dividing by 81.3). The observations are
described in Paper I. We convolve all images to the same angular resolution of
$36\arcsec$ with a pixel size of $14\arcsec$. In the analysis of the [C ii]
data we include only points above the $3\sigma$ threshold, that is
$I_{\mathrm{[C\,\textsc{ii}]}}>5\times
10^{-5}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$ at $36\arcsec$ angular
resolution111The conversion factor from line-integrated brightness temperature
($W\;\mathrm{[K\,km\,s^{-1}]}$) to surface brightness
($I\;\mathrm{[erg\,s^{-1}\,cm^{-2}\,sr^{-1}]}$) is $2k_{\rm B}\nu^{3}/c^{3}$.
This is $I\mathrm{(erg\,s^{-1}\,cm^{-2}\,sr^{-1})}\simeq 7.0\times
10^{-6}\,W\mathrm{(K\,km\,s^{-1})}$ for the [C ii] line, and
$I\mathrm{(erg\,s^{-1}\,cm^{-2}\,sr^{-1})}\simeq 1.3\times
10^{-8}\,W\mathrm{(K\,km\,s^{-1})}$ for the CO(2-1) lines..

From the dust continuum photometry, we determine the dust effective
temperature and the dust optical depth by fitting a modified blackbody with
dust emissivity index $\beta=2$,

$\displaystyle I_{\lambda}=B(\lambda,T_{\rm
d})\,\left[1-\exp\left(-\tau_{160}\left(\frac{160\,\mu\mathrm{m}}{\lambda}\right)^{\beta}\right)\right],$
(1)

to the PACS $70\,\mu\mathrm{m}$, $100\,\mu\mathrm{m}$, and
$160\,\mu\mathrm{m}$ bands, and the SPIRE $250\,\mu\mathrm{m}$,
$350\,\mu\mathrm{m}$, and $500\,\mu\mathrm{m}$ bands (cf. Paper I and Pabst et
al. (2019) for details). Where the coverage of the PACS $100\,\mu\mathrm{m}$
band is not sufficient, we complete the SED output maps with an SED fit of the
remaining five PACS and SPIRE bands, which we use for illustrative purposes
only. The dust temperature in the five-band fit is slightly higher than in the
six-band fit (by about $0.3\text{-}0.5\,\mathrm{K}$), and the dust optical
depth is somewhat lower, but negligibly so. The differences are largest in the
low-intensity regions. The brightest regions in the BN/KL region and Orion S,
the star-forming cores close to the massive Trapezium stars, are saturated in
the Herschel images and hence excluded from our analysis. By integrating Eq. 1
from $\lambda=40\,\mu\mathrm{m}$ to 500 $\mu$m, we obtain the integrated far-
infrared (FIR) intensity.

Figure 1 shows the dust temperature and dust optical depth resulting from the
SED fits ($\beta=2$). The dust optical depth and dust temperature depend
sensitively on the exact value of $\beta$. In Appendix A and B we compare the
SED fits used here with the SED fits of Lombardi et al. (2014), who employ
$\beta\simeq 1.6$, and to SED fits, where we let $\beta$ be a free parameter.
The general behavior of the correlations identified in this study is not
affected.

##  3 Analysis

###  3.1 Global morphology and channel maps

Figure 2 shows the line-integrated [C ii] intensity from the Orion Nebula
complex in the Orion A molecular cloud. The mapped area comprises three
distinct regions: M42, M43, and NGC 1977. The most massive stars in the Orion
Nebula complex, the Trapezium stars, are found close to the surface of OMC1.
The heavily irradiated PDR at the surface of the molecular cloud radiates
bright [C ii] emission. Also the Veil Shell, the expanding shell that is
created by the stellar wind of the O7V star $\theta^{1}$ Ori C (the most
massive of the Trapezium stars), is readily observed in [C ii] emission. The H
ii region of M43 is encircled by a [C ii]-emitting shell, as well. In the
background of the B0.5V star NU Ori, M43 is bounded by the molecular cloud
with a [C ii]-emitting PDR at its surface. The shell surrounding the H ii
regions NGC 1973, 1975, and 1977 also emits substantially in the [C ii]
line222We refer to the region encompassing the three reflection nebulae NGC
1973, 1975, and 1977 as NGC 1977 for short. These regions are also
collectively known as Sharpless 279 (Sh2-279).. The brightest part associated
with NGC 1977 and irradiated by the B1V star 42 Orionis, however, is the PDR
at the surface of the molecular core OMC3. The three shells associated with
M42, M43, and NGC 1977 are subject to ongoing dynamical evolution due to over-
pressurized hot plasma (M42) and ionized gas (M43 and NGC 1977) created by the
central stars (Pabst et al. 2019, 2020).

Figure 3 shows the [C ii] intensity of the Orion Nebula complex per
$1\,\mathrm{km\,s^{-1}}$ channel in the range
$v_{\mathrm{LSR}}=-9\text{-}19\,\mathrm{km\,s^{-1}}$. Channel maps of
velocity-resolved observations allow to disentangle spatially distinct
structures within the line of sight and thus to obtain insight into the global
morphology of a region. We can distinguish several components: For
$v_{\mathrm{LSR}}<-2\,\mathrm{km\,s^{-1}}$, we observe [C ii] emission that
fills the space within the shell filaments of the Orion Nebula. This emission
is associated with the expanding Veil Shell, that is moving toward us (Pabst
et al. 2019). The edge-on shell filaments form a coherent structure and are
most distinctly visible in the velocity range
$v_{\mathrm{LSR}}=0\text{-}9\,\mathrm{km\,s^{-1}}$. In the
$v_{\mathrm{LSR}}=8\text{-}12\,\mathrm{km\,s^{-1}}$ range, the bright
OMC1/Huygens Region reveals its structure, studied in detail by Goicoechea et
al. (2015). M43 also becomes apparent in this velocity range. In
$v_{\mathrm{LSR}}=12\text{-}15\,\mathrm{km\,s^{-1}}$, a structure that seems
to be an extension of the Orion Bar becomes visible, that was also noted in
optical observations (Henney et al. 2007). Emission contributed by the [13C
ii] $F=2\text{-}1$ hyperfine component333The [13C ii] line splits into three
hyperfine components (the frequency of the [12C ii] line is 1900.537 GHz and
$\Delta v$ is given with respect to this frequency, Ossenkopf et al. (2013)):
component $\nu$ [GHz] $\Delta v$ [km s-1] rel. intensity $F=1\text{-}0$
1900.950 -65.2 0.250 $F=2\text{-}1$ 1900.466 +11.2 0.625 $F=1\text{-}1$
1900.136 +63.3 0.125 can be discerned in the last two channels,
$v_{\mathrm{LSR}}=17\text{-}19\,\mathrm{km\,s^{-1}}$, toward the brightest
part of the map, that is OMC1 and the bright eastern arm of the Veil Shell,
the Eastern Rim. The northern part of the map with NGC 1973, 1975, and 1977,
is visible in the range $v_{\mathrm{LSR}}=10\text{-}14\,\mathrm{km\,s^{-1}}$,
offset by $\sim 2\,\mathrm{km\,s^{-1}}$ from the emission peak of the OMC1
region. Many small-scale structures also possess distinct dynamic morphology,
that can be observed in the channel maps, but this is outside the scope of
this study.

Figure 4: PACS $70\,\mu\mathrm{m}$, PACS $160\,\mu\mathrm{m}$ and SPIRE
$250\,\mu\mathrm{m}$ bands in their respective native resolutions (see PACS
Handbook) and on logarithmic scale (units $\log_{10}\mathrm{MJy\,sr^{-1}}$).

Figure 4 shows the PACS $70\,\mu\mathrm{m}$, PACS $160\,\mu\mathrm{m}$ and
SPIRE $250\,\mu\mathrm{m}$ photometric images. We observe that colder
structures become more prominent in wavelengths longwards of
$160\,\mu\mathrm{m}$. At submillimeter wavelengths, the dust emission is
dominated by the so-called integral-shaped filament (ISF, Johnstone & Bally
1999). The ISF is a cold and dense structure that runs through the entire
Orion Nebula complex and has fragmented into molecular cores, OMC1, 2, 3, and
4, that are sites of active star formation.

On large scales ($\sim 25\,\mathrm{pc}$) along the Orion molecular cloud, a
global north-south velocity gradient was determined from observations of the
molecular gas (Bally et al. 1987). In the region observed in [C ii], we cannot
distinguish a significant global velocity gradient. Rather, the velocity
gradient is dominated by the separation of structures that move at distinct
velocities, the M42 Veil Shell in the south and the northern bubble of NGC
1977. The local velocity gradient between OMC1 and OMC3 is also observed in
molecular observations of Hacar et al. (2017). This abrupt velocity change, as
noted by Bally et al. (1987), is larger than the global velocity gradient
($\sim 0.5\,\mathrm{km\,s^{-1}\,pc^{-1}}$) from molecular observations would
suggest.

Figure 5: Regions in correlations plots discussed in Sections 3.3 to 3.8 on
IRAC $8\,\mu\mathrm{m}$ emission convolved to an angular resolution of
$36\arcsec$.

###  3.2 Definition of regions

In Paper I, we have discussed the global correlations. In this paper, we study
the correlations in more detail by dividing them into subregions. Figure 3 of
Paper I shows the correlation plots disregarding individual (sub)regions,
including all areas that fall outside of the later-defined regions, as point-
density plots. Table 1 of Paper I summarizes the power-law fits we find for
each correlation, using the ordinary least-squares (OLS) bisector method
(Isobe et al. 1990).

Because some of the spread in the correlation plots may stem from differences
in the characteristics of the sources and their individual components, we have
subdivided the survey area into four regions, that are further subdivided by
their emission components. Figure 5 gives an overview of the regions we define
in order to study the correlations in the following plot in greater detail. We
have divided the Orion Nebula into two regions, the region that is viewed edge
on, which comprises the Eastern Rim, the edge-on Veil Shell and the rim
separating M42 from M43, and the region that presumably has face-on geometry,
that is the surface of the ISF in the region of OMC4. We have defined the
latter as regions situated inside the Veil Shell with $\tau_{160}>8\times
10^{-3}$, which includes areas close to OMC1 (purple hatched areas in Fig. 5).
Those latter areas are defined by $I_{\mathrm{FIR}}>2\times
10^{-1}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$. Anticipating the discussion
on the [C ii]-FIR correlation in section 3.5, we have subdivided them into two
subregions, OMC1s and OMC1w. Points in OMC1s lie above the regression curve
computed from all OMC4 points, points in OMC1w lie below this average
regression curve. We encircle the points in OMC1w with black circles in the
correlation plots. In the panels showing the OMC4 correlation, we include the
points lying inside the OMC1 region in gray scale.

We have divided M43 into four subregions: The interior H ii region with the
molecular background and foreground expanding shell
($I_{\mathrm{H}\alpha}>4\times 10^{3}\,\mathrm{MJy\,sr^{-1}}$), the bright rim
around it, that is the shell, a region where [C ii] emission is faint
($I_{\mathrm{[C\,\textsc{ii}]}}<1.4\times
10^{-3}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$) but CO(2-1) emission is
still bright ($I_{\mathrm{CO(2\text{-}1)}}>1.5\times
10^{-6}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$), and a region where both [C
ii] and CO(2-1) emission are faint. NGC 1977 is divided into four subregions,
as well: The H ii region with the expanding shell
($I_{\mathrm{H}\alpha}>150\,\mathrm{MJy\,sr^{-1}}$), the shell surrounding it,
the region of OMC3, defined by $\tau_{160}>6\times 10^{-3}$ and
$I_{\mathrm{CO(2\text{-}1)}}/I_{\mathrm{FIR}}>3\times 10^{-5}$, and the
outward areas with faint H$\alpha$ emission
($I_{\mathrm{H}\alpha}<90\,\mathrm{MJy\,sr^{-1}}$).

###  3.3 FIR versus distance

Figure 6 shows the dependence of the FIR intensity on (projected) distance
from the respective central star in M42, M43 and NGC 1977. The FIR intensity
scales with the incident FUV radiation field. However, geometry and line-of-
sight effects are important. While, in principle, the incident radiation field
can be estimated from the stellar luminosity and the (true) distance from the
illuminating source, oftentimes only the projected distance is known.

Figure 6: FIR intensity versus (projected) distance from respective central
stars in M42 (top left: Veil Shell, bottom left: OMC4), M43 (top right) and
NGC 1977 (bottom right). The division of M43 and NGC 1977 into the regions
given in the legend is described in Section 3.2. The blue solid lines are the
fits summarized in Table 4. The blue dashed lines indicate a FIR intensity
scaling with $d^{-2}$.

We find that both in the limb-brightened edge of the Veil Shell and in OMC4 at
$d<1\,\mathrm{pc}$ the FIR intensity scales with $d^{-2}$ approximately. The
incident radiation field at a distance of $1\,\mathrm{pc}$ is $G_{0}\sim 500$,
derived from the presumed face-on geometry of the ISF in the OMC4 region using
$G_{0}\simeq I_{\mathrm{FIR}}/2/1.3\times
10^{-4}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$ (Hollenbach & Tielens 1999).
This is a third of the value derived from the total FIR luminosity from the
Orion Nebula (cf. Paper I) and reflects the geometry of the region. In fact,
this discrepancy corresponds to an angle of $70^{\circ}$ between the incident
radiation and the surface normal. At $d>1\,\mathrm{pc}$, the FIR intensity in
OMC4 scales with an exponent somewhat smaller than 2. This region is
characterized by large columns of cold dust that produce a FIR excess (cf.
Paper I).

In M43 and NGC 1977, we do not find a strong dependence of the FIR intensity
on distance. We attribute this to the geometry of these regions. M43 is a
half-spherical shell situated in front of a dense background cloud close to
the ISF. NGC 1977 is an expanding H ii region (Pabst et al. 2020), whose limb-
brightened shell is readily observed in surface tracers. Considering only
points in the NGC 1977 shell with $d>1.5\,\mathrm{pc}$, the FIR intensity
drops with $d^{-2}$ approximately. However, this fit also describes points
with $d<1.5\,\mathrm{pc}$ to a certain extent.

Table 1: Summary of FIR-distance correlation in Fig. 6.

Region | $a$ | $b$ | $\rho$ | $rms\,\mathrm{[dex]}$
---|---|---|---|---
Veil Shell | $0.32\pm 0.07$ | $-2.08\pm 0.31$ | $-0.94$ | 0.19
OMC4 | $0.17\pm 0.04$ | $-1.65\pm 0.27$ | $-0.88$ | 0.19
M43 | – | – | $-0.41$ | 0.27
NGC 1977 | $0.07\pm 0.05$ | $-2.04\pm 1.3$ | $-0.68$ | 0.13
444$I_{\mathrm{FIR}}=a(d/1\,\mathrm{pc})^{b}$. In NGC 1977 fit only for
$d>1.5\,\mathrm{pc}$. $\rho$ is the Pearson correlation coefficient, $rms$ is
the root-mean-square of the residual of the least-squares fit.

In summary, we observe geometric dilution of the radiation field in M42, but
not in M43 and in NGC 1977 only partly. Table 4 summarizes the FIR-distance
correlations discussed above. The Veil Shell and OMC4 exhibit a strong
correlation, measured by the correlation coefficient $\rho$, while this is
less pronounced in M43 and NGC 1977. The spread in the correlations is non-
negligible, with $0.19\,\mathrm{dex}$ in the Orion Nebula being the smallest
($\rho=-0.69$, $rms=0.44\,\mathrm{dex}$ for the entire of NGC 1977). The
scatter around the fits can be due to geometry effects and the deviation of
the real distance from the projected distance.

The intensity of the incident radiation field, $G_{0}$ in Habing units, can be
expressed as

$\displaystyle G_{0}\simeq 200\chi\frac{L_{\star}}{1\times
10^{4}\,L_{\sun}}\left(\frac{1\,\mathrm{pc}}{d}\right)^{2},$ (2)

where $\chi=L_{\mathrm{FUV}}/L_{\star}$ (cf. eq. 5.43 in Tielens 2010). For
early B stars and late O stars ($T_{\mathrm{eff}}\simeq 2.5\text{-}4.0\times
10^{4}\,\mathrm{K}$), $\chi\simeq 0.5$ (Spaans et al. 1994). We estimate the
central star’s luminosity assuming it is re-radiated in dust continuum
emission. In M42, we have $L_{\mathrm{FIR}}\simeq 1.5\times 10^{5}\,L_{\sun}$,
hence theory predicts $G_{0}\simeq 1.5\times 10^{3}(1\,\mathrm{pc}/d)^{2}$.
From the data associated with OMC4, we estimate $G_{0}\simeq
500(1\,\mathrm{pc}/d)^{2}$ (Fig. 6 and Table 4). As we have shown in Paper I,
from the data for the edge-on PDR in the Eastern Rim we also estimate
$G_{0}\simeq 500(1\,\mathrm{pc}/d)^{2}$. This discrepancy between the
theoretical and observed relations is likely caused by the geometry of the
Orion Nebula. The close adherence to a $d^{-2}$ law illustrates that the
continuum emission is due to dust predominately heated by the central massive
stars.

In M43, we obtain $L_{\mathrm{FIR}}\simeq 1.4\times 10^{4}\,L_{\sun}$, hence
we expect $G_{0}\simeq 1.0\text{-}1.5\times 10^{3}$ in the shell at $d\simeq
0.3\text{-}0.4\,\mathrm{pc}$ from NU Ori. In NCG 1977, $L_{\mathrm{FIR}}\simeq
1.8\times 10^{4}\,L_{\sun}$. For OMC3 at a distance of $d\simeq
0.4\,\mathrm{pc}$ from 42 Orionis, we compute $G_{0}\simeq 1.1\times 10^{3}$.
In the shell at a distance of $d\simeq 1.0\text{-}1.5\,\mathrm{pc}$, we
estimate $G_{0}\simeq 100\text{-}300$.

###  3.4 [C ii] versus $70\,\mu\mathrm{m}$

The observed [C ii] intensity is compared to the $70\,\mu\mathrm{m}$ intensity
in Fig. 7. All four regions show a tight correlation of these two quantities.
These correlations are summarized in Table 5.

Table 2: Summary of [C ii]-$70\,\mu\mathrm{m}$ correlation in Fig. 7.

Region | $a$ | $b$ | $\rho$ | $rms\,\mathrm{[dex]}$
---|---|---|---|---
Veil Shell | $0.57\pm 0.23$ | $-5.07\pm 0.76$ | 0.94 | 0.10
OMC4 | $0.60\pm 0.17$ | $-5.27\pm 0.61$ | 0.91 | 0.13
M43 | $0.60\pm 0.19$ | $-5.24\pm 0.77$ | 0.94 | 0.06
NGC 1977 | $0.60\pm 0.62$ | $-5.10\pm 1.86$ | 0.77 | 0.10
555$\log_{10}I_{\mathrm{[C\,\textsc{ii}]}}=a\log_{10}I_{\mathrm{70\,\mu\mathrm{m}}}+b$.
Fit for $I_{70\,\mu\mathrm{m}}>5\times 10^{2}\,\mathrm{MJy\,sr^{-1}}$. $\rho$
is the Pearson correlation coefficient, $rms$ is the root-mean-square of the
residual of the least-squares fit.

As observed in Paper I, the correlations exhibit two regimes, with decreasing
slope from low to high $70\,\mu\mathrm{m}$ intensity. We fit and discuss only
the regime with $I_{70\,\mu\mathrm{m}}>5\times 10^{2}\,\mathrm{MJy\,sr^{-1}}$,
since the low-intensity regime is problematic because of calibration
uncertainties in the PACS bands. The [C ii] intensity scales with the
$70\,\mu\mathrm{m}$ intensity with a power of $0.57\text{-}0.60$ (for
$I_{70\,\mu\mathrm{m}}>5\times 10^{2}\,\mathrm{MJy\,sr^{-1}}$). The largest
scatter about the regression curve (and the lowest correlation coefficient) is
observed in NGC1977. Also, the spread is higher in OMC4 as compared to the
Veil Shell. The [C ii] intensity tends to turn off the regression curve toward
lower values at the high-intensity end. As bright-emission regions tend to be
optically thick in the [C ii] line (while cooling mainly through the front
side viewed edge-on), and [O i] $63\,\mu\mathrm{m}$ and $145\,\mu\mathrm{m}$
line cooling becomes significant, we expect less [C ii] emission as compared
to the intensity in the $70\,\mu\mathrm{m}$ band. In M43, the high-intensity
points (toward the shell and the H ii region and background cloud) also tend
to follow a slightly shallower slope, likely because of the same effect.

Figure 7: [C ii] intensity versus PACS $70\,\mu\mathrm{m}$ intensity in M42
(top left: Veil Shell, bottom left: OMC4 with OMC1 (in gray scale; black-edged
red points lie to the west of Orion S)), M43 (top right) and NGC 1977 (bottom
right). The color scale in M42 indicates the distance from $\theta^{1}$ Ori C.
The division of M43 and NGC 1977 into the regions given in the legend is
described in Section 3.2. The blue solid lines are the fits summarized in
Table 5.

###  3.5 [C ii] versus FIR

The observed [C ii] intensity is compared to the FIR intensity666Again, we
exclude the very bright (in FIR emission) regions of the BN/KL and Orion S
cores, that are saturated in our SEDs. in Fig. 8. All four regions show a
tight correlation of these two quantities. These correlations are summarized
in Table 7.

Table 3: Summary of [C ii]-FIR correlation in Fig. 8.

Region | $a$ | $b$ | $\rho$ | $rms\,\mathrm{[dex]}$
---|---|---|---|---
Veil Shell | $0.60\pm 0.27$ | $-2.56\pm 0.26$ | 0.94 | 0.09
OMC4 | $0.88\pm 0.33$ | $-2.59\pm 0.29$ | 0.92 | 0.17
M43 | $0.61\pm 0.21$ | $-2.63\pm 0.06$ | 0.92 | 0.07
NGC 1977 | $0.53\pm 0.39$ | $-2.62\pm 0.48$ | 0.79 | 0.08
777$\log_{10}I_{\mathrm{[C\,\textsc{ii}]}}=a\log_{10}I_{\mathrm{FIR}}+b$. Fit
for $I_{\mathrm{FIR}}>3\times
10^{-2}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$. $\rho$ is the Pearson
correlation coefficient, $rms$ is the root-mean-square of the residual of the
least-squares fit.

Similar to the [C ii]-$70\,\mu\mathrm{m}$ correlation, the [C ii]-FIR
correlation exhibits two regimes, but we fit only the regime with
$I_{\mathrm{FIR}}>3\times 10^{-2}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$. In
this regime, the [C ii] intensity depends less than linear on the FIR
intensity, the exponent lying in the range $0.53\text{-}0.88$ (for
$I_{\mathrm{FIR}}>3\times 10^{-2}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$).
The correlation coefficient in M42 and M43 is larger than 0.9, in NGC 1977 it
is significantly lower (0.79).

The dust temperature in OMC1w is somewhat lower than in the region to the
southeast of the Orion Bar. The dust optical depth at 160 $\mu$m lies between
$8\times 10^{-3}$ and $1\times 10^{-2}$. Thus, although this region is
associated with OMC1, it is included in our OMC4 correlation. Points in the
region south of OMC1 and north of OMC4, OMC1s, are also included in the OMC4
correlation. Those points have higher [C ii] intensity at the same FIR
intensity as points in OMC1w.

Most scatter about the regression curve is observed in OMC4, which also
possesses a larger average slope. Also the bright-CO and faint-[C ii] regions
in M43, lying close to the spine of the ISF, deviate from the regression
curve, which might be due to the FIR emission stemming from the colder
molecular background. The same is true for OMC3, as part of the ISF. As the
FIR intensity is dominated by the PACS $70\,\mu\mathrm{m}$ band, the [C
ii]-FIR correlation is very similar to the [C ii]-$70\,\mu\mathrm{m}$
correlation. The low-intensity regime ($I_{\mathrm{FIR}}<3\times
10^{-2}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$) is again fitted with
exponents larger than unity, but with large standard errors of the fit
parameters and lower correlation coefficients ($\rho\simeq 0.8$).

The points in the cavity to the west of the molecular ridge of Orion S/BN-KL,
that is in OMC1w, lie below the average regression of the OMC4 correlation.
This region is just to the north of the extension of the Orion Bar and is
characterized by a higher [C ii] centroid velocity as compared to the average
molecular background velocity. This region is subject to large-scale shocks
identified by Henney et al. (2007).

Figure 8: [C ii] intensity versus FIR intensity in M42 (top left: Veil Shell,
bottom left: OMC4 with OMC1 (in gray scale; black-edged red points lie to the
west of Orion S)), M43 (top right) and NGC 1977 (bottom right). The color
scale in M42 indicates the distance from $\theta^{1}$ Ori C. The division of
M43 and NGC 1977 into the regions given in the legend is described in Section
3.2. The blue solid lines are the fits summarized in Table 7. Dashed lines are
edge-on model outputs with the parameters ($G_{0}$, $n$, $A_{\mathrm{V,los}}$)
given in the legend. Dashed dotted lines in the OMC4 and the M43 panels are
face-on model outputs for the densities given in the legend. gray points in
M43 correspond to the estimated intensities of the background PDR toward the
center of M43 (cf. Sec. 4.2).

###  3.6 [C ii] versus $8\,\mu\mathrm{m}$

The [C ii] intensity is compared to the $8\,\mu\mathrm{m}$ intensity in Fig.
9. As this figure demonstrates, these intensities are tightly correlated.
Table 8 summarizes the [C ii]-$8\,\mu\mathrm{m}$ correlation in the four
regions.

Table 4: Summary of [C ii]-$8\,\mu\mathrm{m}$ correlation in Fig. 9.

Region | $a$ | $b$ | $\rho$ | $rms\,\mathrm{[dex]}$
---|---|---|---|---
Veil Shell | $0.67\pm 0.27$ | $-1.74\pm 0.42$ | 0.97 | 0.09
OMC4 | $0.56\pm 0.10$ | $-1.96\pm 0.29$ | 0.98 | 0.09
M43 | $0.68\pm 0.20$ | $-1.85\pm 0.28$ | 0.93 | 0.06
NGC 1977 | $0.87\pm 0.45$ | $-1.31\pm 1.18$ | 0.92 | 0.11
888$\log_{10}I_{\mathrm{[C\,\textsc{ii}]}}=a\log_{10}I_{\mathrm{8\,\mu m}}+b$.
Fit for $I_{8\,\mu\mathrm{m}}>3\times
10^{-4}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$. $\rho$ is the Pearson
correlation coefficient, $rms$ is the root-mean-square of the residual of the
least-squares fit.

The overall [C ii]-$8\,\mu\mathrm{m}$ relation is similar in all four regions.
The [C ii] intensity depends less than linear on the $8\,\mu\mathrm{m}$
intensity, with an exponent of $0.56\text{-}0.87$. The correlation coefficient
for each region is larger than 0.9, and the scatter around the regression is
usually small ($rms<0.11\,\mathrm{dex}$). The [C ii] intensity tends to turn
off the regression curve toward lower values at the high-intensity end. In the
correlation plot of OMC4, points in OMC1w have lower [C ii] intensity at the
same $8\,\mu\mathrm{m}$ intensity compared to points in OMC1s. Points within
OMC1 continue the general [C ii]-$8\,\mu\mathrm{m}$ trend.

One difficulty in the interpretation of the correlation is the unknown
additive intensity offset in the IRAC $8\,\mu\mathrm{m}$ band, a known IRAC
calibration issue for extended emission (IRAC Instrument Handbood Sec. 8.2).
We estimated the offset at $2\times
10^{-3}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$, but it might also be half as
large. This does not affect the regression greatly if we restrict ourselves to
the high-intensity end. It will affect, however, the estimate of the [C ii]
intensity from the $8\,\mu\mathrm{m}$ intensity at low intensities.

Figure 9: [C ii] intensity versus IRAC $8\,\mu\mathrm{m}$ intensity in M42
(top left: Veil Shell, bottom left: OMC4 with OMC1 (in gray scale; black-edged
red points lie to the west of Orion S)), M43 (top right) and NGC 1977 (bottom
right). The color scale in M42 indicates the distance from $\theta^{1}$ Ori C.
The division of M43 and NGC 1977 into the regions given in the legend is
described in Section 3.2. The blue solid lines are the fits summarized in
Table 8.

###  3.7 FIR versus $8\,\mu\mathrm{m}$

The FIR intensity is compared to the $8\,\mu\mathrm{m}$ intensity in Fig. 10.
As this figure shows, these intensities are tightly correlated. Table 9
summarizes the FIR-$8\,\mu\mathrm{m}$ correlations in the four regions. We
consider only the high-intensity regime ($I_{\mathrm{FIR}}>3\times
10^{-2}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$).

Table 5: Summary of FIR-$8\,\mu\mathrm{m}$ correlation in Fig. 10.

Region | $a$ | $b$ | $\rho$ | $rms\,\mathrm{[dex]}$
---|---|---|---|---
Veil Shell | $1.06\pm 0.44$ | $1.26\pm 0.92$ | 0.95 | 0.14
OMC4 | $0.70\pm 0.24$ | $0.84\pm 0.56$ | 0.95 | 0.15
M43 | $1.08\pm 0.36$ | $1.25\pm 0.51$ | 0.92 | 0.11
NGC 1977 | $1.00\pm 0.50$ | $1.00\pm 1.12$ | 0.83 | 0.15
999$\log_{10}I_{\mathrm{FIR}}=a\log_{10}I_{\mathrm{8\,\mu m}}+b$. Fit for
$I_{\mathrm{FIR}}>3\times 10^{-2}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$.
$\rho$ is the Pearson correlation coefficient, $rms$ is the root-mean-square
of the residual of the least-squares fit.

While both the [C ii]-FIR and the [C ii]-$8\,\mu\mathrm{m}$ relations are far
from linear, the $8\,\mu\mathrm{m}$-FIR relation is approximately linear. The
correlation coefficient is larger than 0.9 in M42 and M43, and 0.83 in NGC
1977. Scatter about the regression curve is significant, though, with
$0.11\text{-}0.15\,\mathrm{dex}$. In OMC4, the FIR-$8\,\mu\mathrm{m}$ relation
differs slightly from the relations found in the other regions, with a power-
law exponent smaller than unity. Points in OMC1w have higher FIR intensity
than others with the same $8\,\mu\mathrm{m}$ intensity. Points inside OMC1
continue the trend of those high-FIR points.

As the FIR measures the incident (FUV) radiation field, the good correlation
between the $8\,\mu\mathrm{m}$ and FIR intensities implies that the PAH flux
is proportional to the incident FUV field. Given that the IRAC
$8\,\mu\mathrm{m}$ band measures the emission by PAH cations (Peeters et al.
2002), this implies that there is little variation in the PAH ionization
balance within each of these regions and between the regions. Conversely, this
implies that the observed $8\,\mu\mathrm{m}$ intensity is a good measure of
the incident FUV field. Overall, we find that
$I_{\mathrm{FIR}}/I_{\mathrm{8\,\mu m}}\simeq 10$.

Figure 10: FIR intensity versus IRAC $8\,\mu\mathrm{m}$ in M42 (top left: Veil
Shell, bottom left: OMC4 with OMC1 (in gray scale; black-edged red points lie
to the west of Orion S)), M43 (top right) and NGC 1977 (bottom right). The
color scale in M42 indicates the distance from $\theta^{1}$ Ori C. The
division of M43 and NGC 1977 into the regions given in the legend is described
in Section 3.2. The blue solid lines are the fits summarized in Table 9.

###  3.8 [C ii] versus CO(2-1)

Figure 11: [C ii] versus CO(2-1) intensity in M42 (top left: Veil Shell,
bottom left: OMC4 with OMC1 (in gray scale; black-edged red points lie to the
west of Orion S)), M43 (top right) and NGC 1977 (bottom right). The color
scale in M42 indicates the distance from $\theta^{1}$ Ori C. The division of
M43 and NGC 1977 into the regions given in the legend is described in Section
3.2. Dashed lines are edge-on model outputs with the parameters ($G_{0}$, $n$,
$A_{\mathrm{V,los}}$) given in the legend. Dashed dotted lines in the OMC4 and
the M43 panels are face-on model outputs for the densities given in the
legend. gray points in M43 correspond to the estimated intensities of the
background PDR toward the center of M43.

The [C ii] intensity is compared to the CO (J=2-1) intensity in Fig. 11. While
for the Veil Shell and OMC4, there are clear trends – albeit with large
scatter – there is little correlation for M43 and NGC 1977. Specifically, for
M43, the correlation coefficient is only 0.37. The global correlation
coefficient for NGC 1977 is only 0.01. Limiting ourselves to the OMC 3 region
in this source results in a much better correlation coefficient ($\rho=0.67$).
toward OMC4, [C ii] and CO(2-1) intensities also exhibit a correlation, with
less scatter than in the Veil Shell ($\rho=0.93$). Despite the fact that the
Veil Shell is a PDR viewed edge-on and we expect a spatial offset between the
[C ii]- and the CO-emitting gas layers, [C ii] and CO(2-1) intensities show a
correlation, albeit with large scatter ($\rho=0.79$). The Veil Shell itself is
mostly CO-dark, however containing several CO-emitting globules (Goicoechea et
al. 2020). CO emission is mostly stemming from the background molecular cloud
and not the shell itself. Only in the Eastern Rim there is corresponding
CO(2-1) emission at a small spatial offset.

In the center of M43, CO(2-1) emission stems mainly from the molecular cloud
behind the H ii region, but has an additional broad, weak component of yet
undetermined origin (Pabst et al. 2020). The [C ii]-bright edge is also bright
in CO(2-1), albeit with a slight spatial separation. The shell of NGC 1977 has
nearly constant [C ii] intensity, but the CO(2-1) varies by a factor of about
30. In contrast, the CO(2-1) intensity is toward the OMC3 region varies
little, while [C ii] varies by two orders of magnitude.

Figure 12: [C ii]/FIR versus CO(2-1)/FIR in M42 (top left: Veil Shell, bottom
left: OMC4 with OMC1 (in gray scale; black-edged red points lie to the west of
Orion S)), M43 (top right) and NGC 1977 (bottom right). The color scale in M42
indicates the distance from $\theta^{1}$ Ori C. The division of M43 and NGC
1977 into the regions given in the legend is described in Section 3.2. Dashed
lines are edge-on model outputs with the parameters ($G_{0}$, $n$,
$A_{\mathrm{V,los}}$) given in the legend. Dashed dotted lines in the OMC4 and
the M43 panels are face-on model outputs for the densities given in the
legend. gray points in M43 correspond to the estimated intensities of the
background PDR toward the center of M43.

The [C ii]/FIR-CO(2-1)/FIR correlations, shown in Fig. 12, exhibit somewhat
different behavior than the [C ii]-CO(2-1) correlations. The normalized
intensities are generally less correlated than the absolute intensities
($\rho=0.53$ in the Veil Shell, $\rho=0.77$ in OMC4, $\rho=0.22$ in M43, but
$\rho=0.17$ in NGC 1977 and $\rho=0.67$ in OMC3 only). If one were to fit
regression curves to the data (both absolute and normalized intensities), no
consistent picture would emerge, the exponents of the power laws differing
widely. This is reflected in the global correlations discussed in Paper I.

###  3.9 Summary of the correlation studies

In summary, the correlations of the [C ii] intensity with the
$70\,\mu\mathrm{m}$, FIR, and $8\,\mu\mathrm{m}$ intensity, respectively, are
tight. The four regions, the M42 Veil Shell, OMC4, M43, and NGC 1977, behave
in a similar manner, and the spread we observe is minor. The spread may be due
to the dependency of the gas heating efficiency on the physical conditions and
possibly subtle variations of the abundance of PAHs and VSGs through the
regions. Most important to note is that the correlation of the [C ii]
intensity with each of the $70\,\mu\mathrm{m}$, FIR, and $8\,\mu\mathrm{m}$
intensity is not linear, but scales with those intensities to a power of less
than unity. The FIR-$8\,\mu\mathrm{m}$ correlation deviates only slightly from
linearity. In contrast, the [C ii]-CO(2-1) correlations reveal a more complex
behavior. This requires a more in-depth study which we pursue in section 4.

##  4 Discussion

###  4.1 Edge-on PDR models

We complement the correlations of the [C ii] emission with gas and dust
tracers in the following with new PDR model runs that are based on the models
of Tielens & Hollenbach (1985) with updates of Wolfire et al. (2010) and
Hollenbach et al. (2012). Pabst et al. (2017) describe the procedure by which
the emission from an edge-on PDR of a given line-of-sight depth
($A_{\mathrm{V,los}}$) is computed from the output of a 1D face-on PDR model.
Compared to those earlier models, we have updated the collisional excitation
rates and chemical rates in the models. In particular, we adopt the [O i]
collisional rates with H and H2 from Lique et al. (2018) (and F. Lique priv.
comm.). Updates to chemical rates are noted in Neufeld & Wolfire (2016). More
recently, we updated the photorates to those from Heays et al. (2017) and
adopted the exponential integral formalism for the depth dependence due to
dust. Additional changes to the chemical rates include oxygen chemistry rates
from Kovalenko et al. (2018) and Tran et al. (2018), and carbon chemistry
rates from Dagdigian (2019). In order to compute synthetic line emission
intensities we assume a Doppler line width of $\Delta
v=1.5\,\mathrm{km\,s}^{-1}$.

We compute models for several FUV intensities, gas densities, and line-of-
sight visual extinctions $A_{\mathrm{V,los}}$ appropriate for different
regions in the Orion Nebula, M43 and NGC 1977 (cf. Section 4.2). In Fig. 13,
we present the results of the constant-density PDR models for incident FUV
intensities of $G_{0}=100,500,\mbox{and }1000$, appropriate for selected
regions in the Orion Nebula, versus physical scale. The $x$-axes share the
same range of visual extinction, $A_{\mathrm{V}}=0\text{-}10\,\mathrm{mag}$
normal to the PDR surface.The three models in Fig. 13 qualitatively produce
similar results: A warm surface gas layer cools through the [C ii] and [O i]
lines. The FIR dust emission also peaks at the surface. The colder gas deeper
in the cloud emits mainly in low-J CO lines. Deep in the cloud, CO and water
molecules freeze out, while cosmic ray ionization enhances the abundance of
atomic carbon. The line-of-sight depth of the cloud slightly affects the
ratios of FIR, [C ii]-, and CO-line intensities. The [O i]
$145\,\mu\mathrm{m}$ intensity is significantly increased compared to face-on
models, while [O i] $63\,\mu\mathrm{m}$ is optically thick in both geometries.
For higher radiation field and density, the contribution of the [O i] lines to
the cooling overtakes that of the [C ii] line.

Figure 13: Results of our edge-on models described in Section 4.1 for incident
FUV intensities of $G_{0}=100,\;500,\,\mbox{ and }1000$ on physical scale,
illuminated from the right. The panels show the gas temperature $T$ (upper
panels), $I_{\mathrm{FIR}}$, $I_{\mathrm{[C\,\textsc{ii}]}}$,
$I_{\mathrm{[C\,\textsc{ii}]}}/I_{\mathrm{FIR,}}$,
$I_{\mathrm{[O\,\textsc{i}]\,63\,\mu m}}$,
$I_{\mathrm{[O\,\textsc{i}]\,145\,\mu m}}$, and
$I_{\mathrm{CO\,(2\text{-}1)}}$ (middle panels) and C+, C, O, and CO
fractional abundances (lower panels) versus physical scale. The gas densities
are $n_{\mathrm{H}}=2.0\times 10^{3}\,\mathrm{cm^{-3}},\;1.0\times
10^{4}\,\mathrm{cm^{-3}},\;1.0\times 10^{4}\,\mathrm{cm^{-3}}$ (left to right
panels), and the respective line-of-sight depths are
$A_{\mathrm{V,los}}=3,10,10$. We note that
$I_{\mathrm{[CII]}}/I_{\mathrm{FIR}}$ is a unitless ratio, whose value is
given by the y-axis.

###  4.2 Comparison of PDR models with observations

For each region, we estimated the input parameters of our PDR models – the gas
density $n$, the incident radiation field $G_{0}$, and the line-of-sight depth
$A_{\mathrm{V,los}}$ – from the SED results and the [C ii] and CO data. For
the Eastern Rim of the Orion Nebula, Pabst et al. (2020) report a gas density,
as estimated from the spatial separation of peak [C ii] and CO(2-1) emission,
of $n\simeq 1\times 10^{4}\,\mathrm{cm^{-3}}$, while the density drops to
$n\simeq 2\times 10^{3}\,\mathrm{cm^{-3}}$ in the southern Veil Shell,
estimated from the dust optical depth using the synthetic extinction curves of
Weingartner & Draine (2001) with $R_{\mathrm{V}}=5.5$. In the rim of M43,
Pabst et al. (2020) find $n\simeq 1\times 10^{4}\,\mathrm{cm^{-3}}$ from the
[C ii] and CO peak separation, as well. In NGC 1977, the dust optical depth
indicates a density of $n\simeq 2\text{-}5\times 10^{3}\,\mathrm{cm^{-3}}$,
using a line-of-sight length of 1 pc. The radiation field in the Orion Nebula
is estimated from the FIR-distance relation, that is $G_{0}\simeq
500(1\,\mathrm{pc}/d)^{2}$ (cf. Fig 6). The FIR emission toward the molecular
background of M43 suggests $G_{0}\simeq 2000$. In case of NGC 1977, we have
used the stellar parameters of 42 Orionis to estimate the radiation field. In
most of the shell, $G_{0}\simeq 100\text{-}200$, in concordance with Kim et
al. (2016). The radiation field in OMC3 may be slightly higher, $G_{0}\sim
400\text{-}1000$; Howe et al. (1991) estimate $G_{0}\simeq 360$.

The depth of the edge-on PDR along the line of sight has been estimated from
the dust optical depth. In the Eastern Rim, $\tau_{160}\simeq 1\times
10^{-2}$, which corresponds to $A_{\mathrm{V,los}}\simeq 30$ using
$A_{\mathrm{V}}\simeq N_{\mathrm{H}}/2\times 10^{21}\,\mathrm{cm}^{-2}$. In
the southern part of the Eastern Rim, $\tau_{160}\simeq 5\times 10^{-3}$,
hence $A_{\mathrm{V,los}}\simeq 15$. In the Southern Shell, we find
$\tau_{160}\simeq 1\times 10^{-3}$, that is $A_{\mathrm{V,los}}\simeq 3$. In
the east of the M43 shell, $\tau_{160}\simeq 1\times 10^{-2}$, and in the
western NGC 1977 shell $\tau_{160}\simeq 3\times 10^{-3}$, that is
$A_{\mathrm{V,los}}\simeq 10$. We produce PDR models for the parameters
estimated here, but let them vary around those values. This way we are able to
reject some combinations of input parameters. In the correlation plots, we
plot the edge-on model outputs only for $x>0.07\,\mathrm{pc}$, which
corresponds to the spatial resolution ($36\arcsec$) of the data.

We observe that the edge-on PDR models with the estimated parameters generally
match the observations very well. As suggested by the geometry of an edge-on
shell, the models require a larger column density than face-on models would
provide. Regions in the Veil Shell with $d<1.0\,\mathrm{pc}$ tend to emit
stronger in [C ii] than accounted for by the models with
$G_{0}=500\text{-}700$, $n=1\times 10^{4}\,\mathrm{cm^{-3}}$, and
$A_{\mathrm{V,los}}=30$. In the region of OMC4 and the ISF in the background
of the Orion Nebula, the deviation of the expected [C ii] emission from face-
on models from the observations is even larger. While the face-on PDR models
do not exceed an intensity of $1\times
10^{-3}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$, the observations emit as
much as $4\times 10^{-3}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$ outside the
Huygens Region/OMC1 and up to $6\times
10^{-3}\,\mathrm{erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$ toward the background PDR in
OMC1. Perhaps in OMC1 and OMC4, edge-on models would be more appropriate. In
summary, the edge-on models agree well with the observations if we allow the
edge-on column to vary between 3 and 30 times that of the face-on column.

The [C ii] emission toward the center of M43 consists of three components: the
blue-shifted expanding shell, the H ii region, and the PDR on the surface of
the background molecular cloud. According to the velocity profile in (Pabst et
al. 2020, Fig. 21), only 30% of [C ii] emission toward the center of M43 stem
from the background cloud, with 65% arising in the H ii region and 5% from the
foreground shell. The gray points in Fig. 8c show to the emission from the
background PDR, where we have assumed 50% of the FIR emission to stem from the
background PDR. This emission from the M43 PDR background can be fitted by a
face-on PDR model with $n\simeq 1\times 10^{4}\,\mathrm{cm^{-3}}$.

The shell of M43 emits slightly more [C ii] intensity than predicted by the
edge-on models with $G_{0}=1000$, $n=1\times 10^{4}\,\mathrm{cm^{-3}}$, and
$A_{\mathrm{V,los}}=10$. A larger column, that is higher $A_{\mathrm{V,los}}$,
mainly increases the FIR intensity, as the [C ii] line is already optically
thick. The [C ii] may be matched by models with lower radiation field, but
larger column, that is $G_{0}=500$ and $A_{\mathrm{V,los}}=20\text{-}30$. The
lower-$G_{0}$ models with higher $A_{\mathrm{V,los}}$ reproduce the [C ii]
versus CO(2-1) correlation in the shell more accurately.

The NGC 1977 shell seems to be characterized by
$A_{\mathrm{V,los}}=3\text{-}10$, while the PDR model output is less sensitive
to the exact density and radiation field. Also the correlations of the [C ii]
intensity with CO(2-1) intensity in the shell can be described by edge-on
models with $G_{0}=100\text{-}200$ and $n=1\text{-}5\times
10^{3}\,\mathrm{cm^{-3}}$. The PDR associated with OMC3 will be analyzed in
more detail by Kabanovic et al. (in prep.).

The edge-on PDR models with the same parameter as in the [C ii]-FIR
correlation seem to be able to explain the [C ii]/FIR-CO(2-1)/FIR correlation
(Fig. 12) in the Veil Shell. However, the behavior of the [C ii]-CO(2-1)
correlation (Fig. 11) reflects the presence of uncorrelated emission, as we
discuss below. The points in OMC4 lie close to the face-on model with
$n=1\times 10^{4}\,\mathrm{cm^{-3}}$ and varying radiation field ($G_{0}\simeq
200\text{-}2000$). Points in OMC1 continue the trend toward lower CO(2-1)/FIR
ratio, but seem to deviate toward lower-density models. The neutral gas
density in OMC1 is expected to be higher, however, $n\simeq 1\times
10^{5}\,\mathrm{cm^{-3}}$. Compared to uniform face-on models with adequate
incident radiation field ($G_{0}\simeq 2\times 10^{4}$), OMC1 emits too much
[C ii] intensity. Invoking a clumpy PDR structure generally increases the
amount of [C ii] emission from the models (e.g., Cubick et al. 2008). For M43,
all edge-on models produce similar [C ii]/FIR ratios at similar CO(2-1)/FIR
ratios, while the background PDR intensities scatter around the face-on model
prediction, indicating either varying density or varying geometry. Points in
NGC 1977 neatly follow the edge-on model predictions.

Figure 14: Correlation plot of [C ii]/FIR with dust optical depth $\tau_{160}$
as histogram. Colors indicate the density of data points in each bin. The line
indicates the inversely linear relation expected for a cloud slab of constant
dust temperature (Goicoechea et al. 2015), running through the median values
of [C ii]/FIR and $\tau_{160}$.

Figure 14 shows the global dependence of the [C ii]/FIR ratio on the dust
optical depth $\tau_{160}$. As opposed to points lying in OMC1, studied in
(Goicoechea et al. 2015, Fig. 15), we do not observe a linear reciprocal trend
in the [C ii]/FIR ratio with $\tau_{160}$ on larger scales. Goicoechea et al.
(2015) expect
$\log_{10}I_{\mathrm{[C\,\textsc{ii}]}}/I_{\mathrm{FIR}}\simeq-\log_{10}\tau_{160}+C$
for a cloud slab of constant dust temperature viewed in face-on geometry. The
fact that we do not observe this behavior globally indicates that, unlike for
OMC1, the temperature varies on large scales. However, there seems to be a
decreasing trend of [C ii]/FIR with increasing $\tau_{160}$. Low [C ii]/FIR
ratios are found toward the OMC4/ISF region, where $\tau_{160}$ is large due
to a large column of cooler dust. This cool dust likely emanates from deeper
layers in the cloud that are not exposed to FUV radiation.

Figure 15: Positions of spectra (green and white circles: spectra in M42,
blue: M43, red: NGC 1977) and line cut in Fig. 23 (red line) on [C ii] line-
integrated emission at its native resolution. The stars indicate the most
massive stars within each region: $\theta^{1}$ Ori C (yellow) in M42, NU Ori
(pink) in M43, and 42 Orionis (purple) in NGC 1977.

Figure 16: Spectra toward high-intensity regions in M42, indicated by green
circles with radius $40\arcsec$ in Fig. 15. Blue: [C ii], yellow: 12CO(2-1),
brown: 13CO(2-1) (multiplied by 3). Additionally, the [13C ii] $F=2\text{-}1$
hyperfine component is plotted (gray) in its systemic velocity (multiplied by
20) in spectra 1-4.

Figure 17: Spectra toward six regions of different geometry, indicated by
white circles with radius $40\arcsec$ in Fig. 15 (OMC1, behind the Orion Bar,
Eastern Rim, Western Edge, M42 cloud background, southern Veil Shell). Blue:
[C ii], Yellow: 12CO(2-1), Brown: 13CO(2-1) (not multiplied by 3).

Figure 18: Spectra toward the M43 shell, indicated by blue circles with radius
$20\arcsec$ in Fig. 15. Blue: [C ii], yellow: 12CO(2-1), brown: 13CO(2-1)
(multiplied by 3). Additionally, the [13C ii] $F=2\text{-}1$ hyperfine
component is plotted (gray) in its systemic velocity (multiplied by 20).

Figure 19: Spectra toward the NGC 1977 shell, indicated by red circles with
radius $40\arcsec$ in Fig. 15. Blue: [C ii], yellow: 12CO(2-1), brown:
13CO(2-1) (multiplied by the factor indicated in the respective panel).
Additionally, the [13C ii] $F=2\text{-}1$ hyperfine component is plotted
(gray) in its systemic velocity (multiplied by 20) in spectrum 1.

We follow up this discussion with a closer look at the spectra in these
high-[C ii] intensity regions. Fig. 15 shows a map of the positions toward
which the spectra are taken are indicated. The [C ii] spectra along with
CO(2-1) spectra are shown in Fig. 19. We conclude that all but one spectra
exhibit multiple components, which may explain the enhanced [C ii] emission
from these high-intensity regions. toward OMC1w (spectrum 4 in Fig. 19), that
has relatively less [C ii] intensity, we observe a single line component. A
detailed discussion of these spectra can be found in Appendix D. We remark
here that the analysis of the [13C ii] $F=2\text{-}1$ line, where possible,
indicates that only about 20-35% of the gas column estimated from the dust
optical depth is traced by [C ii] (a more thorough analysis of the [13C ii]
emission from Orion A will be done by Kabanovic et al., in prep.) and that the
power-law behavior of the [C ii]-FIR and [C ii]-$8\,\mu\mathrm{m}$
correlations does not seem to be caused by increasing optical depth with
increasing intensity. In addition, in some regions we see the effects of local
sources on the gas kinematics.

Figure 19 shows [C ii] and CO(2-1) spectra toward regions of different
geometry. In OMC1, [C ii] and CO(2-1) are slightly shifted from each other.
Spectra 3, 4 and 6 show the velocity shift between [C ii] and CO(2-1) emission
toward regions where [C ii] emission is arising in an edge-on geometry. This
indicates that the [C ii] and CO(2-1) emission arise in different regions of
the cloud. In contrast, spectra taken toward the molecular background in face-
on geometry, such as spectra 2 and 5, exhibit little systematic velocity
shift. However, even spectra 2 and 5 show substructure that deviates from a
Gaussian spectral form. Comparing the line widths of the [C ii] and CO spectra
would yield insight into the turbulent motions of the respective gas layers,
but this is outside the scope of this study and will be subject of a follow-up
study.

In general, the PDR models also reproduce the [C ii]-CO relationship, but the
scatter in those correlations is generally large. From examination of the
spectra, the relation of [C ii] with CO(2-1) emission is not immediately
clear. In most cases we find a velocity offset between the [C ii] and the
CO(2-1) lines. Also the global morphology is very different. While [C ii]
emission traces the edge-on shells and illuminated face-on PDR surfaces,
CO(2-1) emission traces the deeper molecular layers of a PDR and the
background molecular cloud, that might not be co-located with the shell. In
case of face-on PDRs, low-J CO emission arises predominantly in deeper layers
of the PDR, and the velocity difference is generally less than toward edge-on
structures.

In M43, models with $G_{0}=2000$, $n=1\times 10^{4}\,\mathrm{cm^{-3}}$, and
$A_{\mathrm{V,los}}=10$ clearly produce too much FIR emission. The
observations agree better with models with $G_{0}=700\text{-}1000$, however [C
ii] emission is somewhat enhanced compared to the model predictions. As in the
shell of M42, spectra toward the edge of M43 consists of at least two
components, shown in Fig. 19. Again, a detailed description and discussion of
the spectra is provided in Appendix D. The [C ii] column, computed from the
[13C ii] $F=2\text{-}1$ line, where possible, traces only a fraction (30-60%)
of the total gas column estimated from the dust optical depth. It is evident,
however, that the C+ column (with $A_{\mathrm{V}}\simeq 15$) is substantially
larger than expected for a face-on PDR with similar conditions. The edge-on
geometry of the M43 rim becomes very pronounced. The face-on model with
$n=1\times 10^{4}\,\mathrm{cm^{-3}}$ also reproduces the [C ii] and FIR
emission from the background molecular cloud, that is after subtraction of the
contribution from the foreground shell (gray points).

The [C ii] and CO(2-1) spectra toward OMC3 and the NGC 1977 shell are shown in
Fig. 19. FIR line emission toward OMC3 at the edge of NGC 1977 has been
subject to modeling efforts previously. Howe et al. (1991) conclude that the
spatial distribution of [C ii] emission can be explained by a clumpy medium.
Spectrum 1 toward OMC3 can be fitted better by an emitting and a (strong)
absorbing component than by two emitting components. The [13C ii] line is
slightly asymmetric, as well, but this is hard to quantify due to noise. It
seems to peak between the two emission peaks of the [12C ii] component, which
signifies self-absorption (Kabanovic et al. in prep.). In a clumpy medium, the
strong emission and absorption may be easily explained by temperature
gradients between the clump and inter-clump medium. While the [C ii] spectrum
may be fitted by one emitting and one absorbing component, both the 12CO and
the 13CO spectrum comprise two emitting components. It is unclear how the CO
components are related to the [C ii] components. As in other spectra, we find
a significant velocity offset.

While CO emission in OMC3 is strong, in the rest of the shell that is covered
by the CO observations it is very weak. 13CO is detected, but at low
temperatures the 12CO line in FUV-illuminated gas can still be optically thin
with enhanced 13CO emission due to chemical fractionation (e.g., Goicoechea et
al. 2020). In the case of optically thick 12CO emission, the excitation
temperature is very low ($T_{\mathrm{ex}}\simeq 4\text{-}5\,\mathrm{K}$).
Possibly CO emission stems from a molecular background, that is not associated
with the [C ii]-emitting shell, and that is only faintly illuminated. Another
possibility is that the CO-emitting gas originates from two slightly velocity-
shifted parts of the expanding gas around the NGC 1977 H ii region.

Summarizing, the spectra shed light on why [C ii] emission may be
underestimated by the standard PDR models we employ. First of all, virtually
all spectra consist of more than one component. One needs geometrically more
accurate and multi-component models to understand the precise emission
characteristics of both edge-on and face-on geometries. Edge-on models should
incorporate temperature gradients, that can lead to self-absorption (Kabanovic
et al. in prep.). In some cases, clumpy PDR models might be a good choice.
Also dynamics of the gas should be taken into account. Those can lead to
pressure variations within the gas by shear or compression. Furthermore, as we
see velocity shifts in the [C ii] and CO(2-1) line emission, photoevaporation
may be important and this can alter the emission from a PDR (Bertoldi 1989;
Störzer & Hollenbach 1998; Bron et al. 2018). Most of our discussed lines can
be fitted by Gaussians, but the spatial relationship of the emission
components is often unclear. We notice that in regions where we attribute the
emission to edge-on PDRs, the velocity of the CO-emitting gas is generally
offset from the velocity of the [C ii]-emitting gas. Either, this means that
the gas is subject to shear, or that [C ii] and CO trace different, possibly
unrelated, layers of gas. In edge-on (half-)shells such as the Veil Shell,
CO(2-1) emission arises in the molecular background, while [C ii] emission
stems from the edge-on shell. Compared to M42 and M43, NGC 1977 is evolving
rather slowly at its advanced age and tends to emit narrower lines. This might
reflect the mild stellar feedback it is experiencing at this stage. At the
same time, M42 and M43 are still heavily affected by driving stellar feedback
and lines tend to be broader, hinting at increased turbulence.

###  4.3 The $G_{0}$-$p_{\mathrm{th}}$ relationship

We estimate the thermal pressure $p_{\mathrm{th}}=nT$ in the PDR surfaces from
the gas density estimated above and the temperature predicted by the
respective PDR model at the estimated $G_{0}$. Fig. 20 shows the thermal
pressure versus incident radiation field in a number of regions from our study
and from the literature. Our data show a clear trend of increasing thermal
pressure with increasing strength of the radiation field. A similar trend is
apparent in studies of fine-structure line emission from PDRs (Young Owl et
al. 2002) as well as in studies of the CO ladder in PDRs (Joblin et al. 2018),
albeit that those relations are displaced somewhat. For the study of Young Owl
et al. (2002), this displacement reflects an overestimate of the incident FUV
field in several of the sources. Specifically, they adopted a $G_{0}$ of 5000
for NGC 1977S (the bright region in the southwest of NGC 1977), while we
derive a value of 1100. After correction, the data for this particular source
agrees, not surprisingly as both analysis are based upon fine-structure line
studies. Likewise, the $G_{0}$ value adopted for the Orion Bar by Young Owl et
al. (2002) is somewhat higher than current estimates (Salgado et al. 2016;
Joblin et al. 2018). The results of the studies by Joblin et al. (2018) and Wu
et al. (2018) are displaced toward higher pressures. Both these studies rely
on high-J CO transitions with high critical densities. Consequently, clumps or
gas density enhancements dominate the emission in these lines, whereas the
fine-structure lines measure the interclump gas. The offset of these studies
might be due to the importance of self-gravity in the clumps, which will
dominate the pressure distribution. Hence, the high-J CO lines may not be good
measures of the pressure in the [C ii]-emitting PDR layers. As a corollary, we
do not expect that constant-pressure PDR models will be applicable to these
environments. Joblin et al. (2018) note that specifically the [C ii] emission
in their sources (bright PDRs associated with massive star formation)
predominantly comes from lower-density interclump gas, that fills the beam.

Following Young Owl et al. (2002), using the model of an H ii region bordering
on a PDR in pressure equilibrium, Seo et al. (2019) derive an analytic
expression for the correlation between $p_{\mathrm{th}}$ and $G_{0}$ in the
PDR:

$\displaystyle p_{\mathrm{th}}\simeq 4.6\times
10^{4}f^{-1/2}(\frac{\Phi_{\mathrm{FUV}}}{10^{51}\,\mathrm{s^{-1}}})^{-1/4}G_{0}^{3/4}\,\mathrm{K\,cm^{-3}},$
(3)

where $f=\Phi_{\mathrm{FUV}}/\Phi_{\mathrm{EUV}}$ is the ratio of the FUV flux
to the H-ionizing flux. For relevant spectral types, this ratio is constant to
within a factor 1.5 (Young Owl et al. 2002). This relation is also shown in
Fig. 20. The theoretical relation is well above the empirical relation derived
from the fine-structure lines but below the data derived from the CO analysis.

We note that all our analysis and that of Young Owl et al. (2002) only
measures the thermal pressure in the PDR. The total pressure includes
contributions from turbulence and magnetic fields and should be balanced by
the thermal pressure from the ionized gas and radiation pressure. For the
three sources in Orion, we have compared these and concluded that there is
equipartition between thermal, turbulent and magnetic pressure in the PDR and
that radiation pressure is typically not very important (Pabst et al. 2020). A
similar conclusion was reached by Pellegrini et al. (2009) in their detailed
analysis of the PDR associated with the Orion Bar. Moreover, while there is
approximate pressure equilibrium between the plasma and the ionized gas within
the Veil Shell, there is a strong pressure gradient from the background
molecular cloud toward us, which drives the rapid expansion of the shell. In
the PDRs of M43 and NGC 1977, the pressure of the ionized gas exceeds the PDR
pressure, indicating that magnetic or turbulent pressure may be quite
important in these regions.

In a linear fit, we find $p_{\mathrm{th}}/G_{0}\simeq 5\times
10^{3}\,\mathrm{K\,cm^{-3}}$ for our data points. The data points of Joblin et
al. (2018) agree well with the empirical relation of Wu et al. (2018). We note
that the error bars of the data points are usually large, which is mostly due
to the uncertainty in the estimated gas density ($\sigma\sim
0.5\,\mathrm{dex}$). As discussed before, the detailed physics of the models
used to infer physical conditions strongly affect the resulting estimates.
This applies to our data points, as well, since we rely on the standard PDR
structure and dust properties to estimate the gas density in M42, M43, and NGC
1977. Besides the pressure of the H ii region acting on the PDR, the
correlation between thermal pressure or gas density and the incident radiation
field may also reflect the star-formation history of a region, as Young Owl et
al. (2002) comment.

Howe et al. (1991) demonstrated that [C ii] emission from OMC3 can be modelled
successfully employing a clumpy PDR model. Considering the geometry of a
source is certainly important in modeling efforts. A slight deviation from
pure edge-on or face-on geometry results in differing estimates of the
physical conditions. Considering the detailed line shape of the observed lines
is equally important. Often emission lines reveal several line components
mixed within the line of sight or foreground components by self-absorption.
Complex velocity profiles may also hint at the presence of a clumpy medium,
hydrodynamic instabilities and/or colliding filaments and gas flows. These
processes imply a less homogeneous ISM than usually accounted for in PDR
models.

While we recognize that real PDRs might contain clumps and gas-density
enhancements (Joblin et al. 2018; Goicoechea et al. 2016; Cubick et al. 2008)
and that this may influence the analysis, we want to emphasize that the
physical properties of PDRs derived from the analysis of fine-structure lines
are in good agreement with the simple theoretical model based upon the
Strömgren relation in combination with pressure equilibrium between the
ionized and PDR gas and taking equipartition of thermal, turbulent, and
magnetic pressures into account.

Figure 20: Thermal pressure versus incident radiation field in several PDRs.
Red points are from Young Owl et al. (2002) (based on FIR line analysis),
green points are from Joblin et al. (2018) (based on high-J CO line analysis),
dark blue points are from this study (M42 S is the southern Veil Shell, M42 SE
is the southeastern Veil Shell, M42 E1 and M42 E2 lie in the bright eastern
arm of the Veil Shell (cf. green circles 1 and 2 in Fig. 15)). The green
dashed line is the fit by Wu et al. (2018) (based on high-J CO line analysis).
The dashed gray line is the analytic solution of Seo et al. (2019), once
divided by 3 to account for pressure equipartition. Table 6: Luminosities and
masses of M42, M43 and NGC 1977 per subregion.

Region |  | $A$ | $L_{\mathrm{FIR}}$ | $L_{\mathrm{[C\,\textsc{ii}]}}$ | $L_{\mathrm{[C\,\textsc{ii}]}}/L_{\mathrm{FIR}}$ |  $M_{\mathrm{gas}}$aa$a$Gas mass derived from dust mass, using $N_{H}\simeq 6\times 10^{24}\,\mathrm{cm^{-2}}\,\tau_{160}$ (Weingartner & Draine 2001). |  $L_{\mathrm{CO(2\text{-}1)}}$bb$b$ The CO(2-1) map does not cover the entire NGC 1977 region (cf. Fig. 2 in Paper I). |  $M_{\mathrm{CO}}$cc$c$Gas mass from CO(2-1) intensity, using $X(\mathrm{CO})\simeq 2\times 10^{20}\,\mathrm{cm^{-2}}\,(\mathrm{K\,km\,s^{-1}})^{-1}$ (Bolatto et al. 2013). |  $L_{\mathrm{24\mu m}}$dd$d$The MIPS $24\,\mu\mathrm{m}$ image is saturated toward the Huygens/OMC1 region. |  $L_{\mathrm{H\alpha}}$ee$e$The H$\alpha$ surface brightness in the EON is largely due to scattered light from the bright Huygens Region (O’Dell & Harris 2010). Also H$\alpha$ emission in M43 has to be corrected for a contribution of scattered light from the Huygens Region (Simón-Díaz et al. 2011).
---|---|---|---|---|---|---|---|---|---|---
|  | $\mathrm{[pc^{2}]}$ | $\mathrm{[L_{\sun}]}$ | $\mathrm{[L_{\sun}]}$ |  | $\mathrm{[M_{\sun}]}$ | $\mathrm{[L_{\sun}]}$ | $\mathrm{[M_{\sun}]}$ | $\mathrm{[L_{\sun}]}$ | $\mathrm{[L_{\sun}]}$
M42 | all | 45 | $1.8\times 10^{5}$ | 510 | $2.8\times 10^{-3}$ | 10900 | 1.1 | 8800 | 4700 | 3500
| OMC1ff$f$The Bright Ridge contains the BN/KL region and Orion S and is part of OMC1. Values for OMC1 include the Bright Ridge, where we have estimated the FIR luminosity from an SED fit to the three PACS bands, that are only saturated at the core of BN/KL. | 0.31 | $8.2\times 10^{4}$ | 53 | $6.5\times 10^{-4}$ | 1500 | $8.9\times 10^{-2}$ | 720 | – | 1500
| Bright Ridgeff$f$The Bright Ridge contains the BN/KL region and Orion S and is part of OMC1. Values for OMC1 include the Bright Ridge, where we have estimated the FIR luminosity from an SED fit to the three PACS bands, that are only saturated at the core of BN/KL. | 0.07 | $3.7\times 10^{4}$ | 11 | $2.9\times 10^{-4}$ | 870 | $2.8\times 10^{-2}$ | 220 | – | 470
| Eastern Rim | 1.3 | $1.7\times 10^{4}$ | 59 | $3.5\times 10^{-3}$ | 510 | $9.2\times 10^{-2}$ | 740 | 810 | 260
| OMC4 | 2.5 | $1.6\times 10^{4}$ | 45 | $2.8\times 10^{-3}$ | 2100 | $1.5\times 10^{-1}$ | 1200 | 640 | 470
| Veil Shell | 8.7 | $3.5\times 10^{4}$ | 150 | $4.3\times 10^{-3}$ | 2500 | $2.3\times 10^{-1}$ | 1800 | 1200 | 590
M43 | all | 0.74 | $1.4\times 10^{4}$ | 38 | $2.7\times 10^{-3}$ | 530 | $3.0\times 10^{-2}$ | 240 | 740 | 190
| background, H ii | 0.13 | $3.1\times 10^{3}$ | 8.9gg$g$$2.4\,L_{\sun}$ from background PDR, $6.0\,L_{\sun}$ from ionized gas, $0.5\,L_{\sun}$ from foreground neutral shell. | $2.8\times 10^{-3}$ | 53 | $4.6\times 10^{-3}$ | 37 | 100 | 99
| faint CO, faint [C ii] | 0.14 | $8.8\times 10^{2}$ | 4.1 | $4.7\times 10^{-3}$ | 29 | $2.5\times 10^{-3}$ | 20 | 49 | 12
| bright CO, faint [C ii] | 0.16 | $2.0\times 10^{3}$ | 6.0 | $3.0\times 10^{-3}$ | 200 | $9.2\times 10^{-3}$ | 74 | 87 | 18
| shell | 0.31 | $8.2\times 10^{3}$ | 19 | $2.3\times 10^{-3}$ | 250 | $1.4\times 10^{-2}$ | 110 | 410 | 63
NGC | all | 17 | $1.8\times 10^{4}$ | 160 | $8.9\times 10^{-3}$ | 2600 | $1.2\times 10^{-1}$ | 950 | 1000 | 63
1977 | H ii | 2.9 | $4.7\times 10^{3}$ | 37hh$h$$6.2\,L_{\sun}$ from ionized gas, $31\,L_{\sun}$ from neutral shell. | $7.9\times 10^{-3}$ | 310 | $1.1\times 10^{-2}$ | 87 | 300 | 17
| OMC3 | 2.6 | $5.8\times 10^{3}$ | 34 | $5.9\times 10^{-3}$ | 1200 | $8.9\times 10^{-2}$ | 720 | 190 | 8.6
| faint H$\alpha$ | 6.9 | $3.2\times 10^{3}$ | 33 | $9.7\times 10^{-3}$ | 730 | $3.0\times 10^{-2}$ | 240 | 280 | 19
| shell | 6.2 | $5.4\times 10^{3}$ | 69 | $1.3\times 10^{-2}$ | 670 | $1.6\times 10^{-2}$ | 130 | 100 | 7.9

101010

###  4.4 The origin of [C ii] emission and its relationship to CO-dark gas

In this section, we discuss the morphology of the emission and the
implications thereof. This was done briefly in Paper I, but we continue with a
more detailed discussion in this paper. We explore the origin of the [C ii]
emission and the amount of CO-dark gas associated with it in the following.
Table 10 compiles the luminosities and masses toward the regions we have
defined in our analysis and summarizes the results.

As we have stated in Paper I, most of the [C ii] emission (about 50%) comes
from the limb-brightened shell edges in M42, M43, and NGC 1977, that is
moderately irradiated PDR surfaces111111From Table 10 we add the [C ii]
luminosities of the shells in M42, M43, and NGC 1977 plus the Eastern Rim in
M42. This adds up to $328\,L_{\sun}$, which is 46% of $708\,L_{\sun}$, the
total [C ii] luminosity., but a significant contribution (25%) also comes from
the more diffuse emission from weakly irradiated surfaces121212i.e., all
emission that is not captured in Table 10 in the defined regions. Although the
heavily irradiated PDRs close to the central stars emit very bright [C ii]
emission, their contribution to the total [C ii] emission is minor (10-20%)
compared to the large shells.

The massive shells comprise a significant amount of CO-dark gas, with only a
few solar masses contained in dense CO-emitting globules found in the Veil
Shell (Goicoechea et al. 2020). If we rely on the gas mass estimated from the
dust opacity, we find that 25% of the gas mass in the Orion Nebula is
contained in the expanding shell. However, the shell mass estimate might be
contaminated by the molecular background. In a conservative estimate, we
attribute only 10% of the total gas mass (i.e., $1000\,M_{\sun}$) to the
shell. The mass in the dense molecular core, OMC1 with BN/KL and Orion S, is
also 15% of the total gas mass within the mapped area. Another 20% are
comprised in the dense filament, the ISF. Both fractions agree with the mass
fractions traced by CO emission (8% and 14%, respectively). The CO emission in
the line of sight toward the expanding shell likely stems from the background
gas rather than the shell itself (as judged from the velocity structure of the
spectra). While most of the CO emission in the region of the Orion Nebula
originates from the extended background cloud, H$\alpha$ emission is
concentrated around OMC1, almost 50% stemming from the small OMC1 region.

Also 30% of the FIR emission stem from the OMC1 region, including the FIR
emission of the Bright Ridge comprising BN/KL and Orion S in the three PACS
bands, that accounts for 15%, that is half of the FIR emission from the OMC1
region (see Paper I for a discussion of the “[C ii] deficit” or “FIR excess”).
About 25% stem from the much larger shell. About 10% stem from the region of
the OMC4, that is not bright in surface tracers, but contributes by its large
area. The same amount arises from the bright Eastern Rim, that has half the
area of the OMC4 region. While FIR emission from the OMC1 region contributes
significantly to the total FIR emission from the Orion Nebula, only 10% of the
total [C ii] emission originate in OMC1. About 30% of the [C ii] emission stem
from the shell. Also OMC4 is a minor contributor with 10%, where we note that
most of the emission associated with OMC4 actually stems from the regions
close to OMC1, which can therefore be attributed to relatively warm, heavily
irradiated gas. The bright Eastern Rim, with its four times larger area
compared to OMC1, contributes with 10% as well. Adding the contributions from
the defined areas, leaves 40% of [C ii] emission that stems from less
pronounced features, that is the extended background and structures lying
outside the shell, which may be excited by local sources or bubbles spawning
off the large bubble blown by $\theta^{1}$ Ori C.

We now turn to the other two regions within the Orion Nebula complex. In M43,
50% of [C ii] emission arise in the region we have defined as the shell.
However, this emission is likely contributed to by a significant emitting
background. About 25% stem from the inner region. Of this emission, a spectral
analysis shows that only 25% arise from the background, while 5% stem from the
expanding shell and 70% from the H ii gas filling the bubble. Faint [C ii]
emission can be observed outwards of the shell, accounting for 25% of the
total emission. The percentages in the FIR emission are similar, slightly more
(60%) stemming from the shell region. H$\alpha$ emission is concentrated in
the inner H ii region. Emission at 24 $\mu$m is generally connected to either
temperature fluctuations in very small grains or to dust in the ionized gas
heated by, for instance, trapped Lyman $\alpha$ radiation (Desert et al. 1986;
Churchwell et al. 2006; Salgado et al. 2016). Also CO emission is dominated by
the shell region and follows its morphology. The face-on shell seen toward the
H ii region does not exhibit CO emission, hence we conclude that the expanding
shell is not very massive and that the edge-on shell structures are sculpted
within the molecular background. In Pabst et al. (2020) we estimated an
expanding shell mass of only $8\,M_{\sun}$, which amounts to 1.5% of the total
gas mass in M43. This compares to 5% of [C ii] emission stemming from the
face-on shell. If we adopt a shell mass of $1000\,M_{\sun}$ in M42, 10% of the
gas mass account for 25% of the [C ii] emission, which is an equal relative
mass to luminosity ratio as in M43.

Most of the gas mass in NGC 1977 as traced by the dust opacity, almost 50%, is
concentrated in OMC3, a dense region where the ISF bends westward. 25% of the
gas mass is situated in the shell surrounding the H ii region. However, we
cannot exclude the possibility that not all the mass is part of the expanding
shell. As the expansion velocity is rather low, $1.5\,\mathrm{km\,s^{-1}}$
(Pabst et al. 2020), the expansion may have stalled in denser parts of the NGC
1977 region. Also we find significant dust emission outside of the shell,
accounting for 25% of the total mass. In the inner part of the shell, the
enclosed H ii region, we find about 10% of the mass, amounting to a similar
mass per area ratio as in the outward regions. H$\alpha$ and
$24\,\mu\mathrm{m}$ are distributed in a similar fashion, predominantly
arising from the H ii region. The total CO emission originates almost entirely
from OMC3, but this is due to the fact that the CO observations have
insufficient coverage in the less CO-bright areas. We can draw the preliminary
conclusion, however, that we find a significant amount of CO emission besides
OMC3, which may stem from a more extended molecular background or is
associated with the shell structure. The rather extended shell in NGC 1977
dominates the total [C ii] emission in this region, contributing 43%. The
dense and bright OMC3 contributes only about 20%. 25% originate in the [C
ii]-emitting shell toward the H ii region, with a minor contribution from the
ionzed gas (cf. Pabst et al. 2020). As before, a significant amount of [C ii]
emission stems from weakly emitting surfaces, in NGC 1977 this is about 20%
within the map coverage. Only 30% of the total FIR emission stem from the
shell. In NGC 1977, the total [C ii]/FIR ratio is significantly increased
compared to M42 and M43, at almost 1%, while M42 and M43 exhibit a [C ii]/FIR
ratio of $\sim$0.3%. We attribute this to the lower radiation field in NGC
1977.

In summary, when considering the individual regions (see Table 10), [C ii]
emission from PDRs is the main origin of [C ii] emission from the Orion Nebula
complex, with about 70%. The [C ii] emission from ionized gas is a minor
contributor, less than 5% in M42 and NGC 1977, but 15% in M43. The remainder,
CO-dark H2 gas that is not captured in bright PDR surfaces, contributes with
about 20% (cf. Table 10). These percentages elucidate the importance of
observations toward fainter extended regions, as we have seen that those can
carry a significant amount of [C ii] luminosity. This assessment is in
agreement with the conclusion reached in Paper I when considering the overall
[CII] emission observed in this survey.

The fraction of CO-dark molecular gas in these regions of Orion is much less
than found elsewhere in the LMC and Milky Way. A recent study by Lebouteiller
et al. (2019) concerning the origin of the [C ii] line emission and pertaining
to its utility as a star-formation tracer, concludes that in the giant H ii
region N11 in the LMC more than 90% of the [C ii] emission arises in CO-dark
H2 gas and that most of the molecular gas is CO-dark (above 40%). Dwarf
galaxies generally have lower metallicity and penetration of FUV photons, and
therefore photodissociation of CO is enhanced. The GOT C+ survey has shown
that 20% of the Galactic [C ii] emission arise from ionized gas (Pineda et al.
2014), while 30% stem from PDRs, and only 25% from CO-dark H2 gas. Pineda et
al. (2013) note that most of the [C ii] emission in the Milky Way arises in
moderately FUV-illuminated regions ($G_{0}\simeq 2\text{-}50$), that is large
faint surfaces rather than small dense and bright PDRs. Our observations
toward the Orion Nebula are in good agreement with this conclusion, the bright
inner OMC1 region being a minor contributor to the total [C ii] luminosity.
The [C ii] observations toward L1630, comprising the Horsehead Nebula,
predominantly highlight the PDR surfaces illuminated by $\sigma$ Ori, where
95% of the total [C ii] emission in the mapped area arise. Yet, regions that
are bright in [C ii] trace only 8% of the gas mass, while 85% of the mass is
associated with strong CO emission (Pabst et al. 2017). In the Orion Nebula
complex, we find that about 70% of the [C ii] emission arises in PDR surfaces
(i.e. massive shells and surfaces of cores as tabulated in Table 10). The [C
ii]-emitting regions we have defined contain 64% of the gas mass (traced by
dust). A portion of that gas mass, about 30-50%, is likely associated with the
molecular background rather than the PDR surface. Hence, bright [C ii]
emission, that traces the H/H2 transition in a PDR, traces about 30% of the
total gas mass in the Orion Nebula complex. On a larger scale, the Orion
molecular clouds and star-formation regions are embedded in the Orion-Eridanus
superbubble. In the Orion-Eridanus region, an area of $\sim
400\,\mathrm{pc}\times 400\,\mathrm{pc}$, [C ii] emission (observed with COBE)
is dominated by the emission from extended low-density, low-UV field molecular
cloud surfaces associated with the active regions of massive star formation
(Abdullah & Tielens 2020).

###  4.5 Photoelectric heating efficiency

In Paper I we have discussed the observed [C ii] deficit, that is the decrease
in the [C ii]/FIR intensity with increasing FIR intensity, and related it to a
decreased photoelectric heating efficiency in dense PDRs, in addition to the
importance of [O i] cooling. Here, we examine the heating efficiency of PDR
material in more detail. We estimate the photoelectric heating efficiency by
the observed cooling efficiency of the gas, which we approximate by the
observed [C ii]/FIR ratio. At intermediate temperatures and densities, the
cooling is dominated by cooling through the [C ii] line. At higher
temperatures and densities, like those in the Huygens Region, the [O i] 63
$\mu$m line becomes important (Hollenbach & Tielens 1999). However, in the
shell surrounding the Extended Orion Nebula, gas cooling is most likely
dominated by [C ii], as the temperatures and densities are rather moderate. In
M43 and the densest part of NGC 1977 (in OMC3), [O i] emission will contribute
significantly (Herrmann et al. 1997; Young Owl et al. 2002).

Theoretically, the photoelectric heating efficiency depends on the ionization
parameter $\gamma=G_{0}T_{\mathrm{gas}}^{1/2}/n_{\mathrm{e}}$ (Bakes & Tielens
1994, Eq. 43). In Paper I, we estimated $\gamma$ in each pixel of the map.
Here, we set the gas density in OMC4 so that it varies approximately equally
with distance from OMC1 as in the Veil Shell, $n\simeq
10^{4}(1\mathrm{pc}/(d+0.1\,\mathrm{pc}))$. The radiation field varies as
$G_{0}\simeq 500(1\mathrm{pc}/(d+0.1\,\mathrm{pc}))^{2}$ (cf. Paper I). The
gas density in the shell of M43 is set to $n\simeq 1\times
10^{4}\,\mathrm{cm^{-3}}$ (cf. Pabst et al. 2020). In NGC 1977, we approximate
the gas density, using a column of $1\,\mathrm{pc}$, by $n\sim 6\times
10^{24}\tau_{160}\,\mathrm{cm^{-2}}/1\,\mathrm{pc}$. We use $G_{0}\simeq
150(1\,\mathrm{pc}/d)^{2}$ in M43 and $G_{0}\simeq 200(1\,\mathrm{pc}/d)^{2}$
in NGC 1977, approximated from the total FIR luminosity of M43 and NGC 1977,
respectively. The correlations between the photoelectric heating efficiency
$\epsilon$ and the ionization parameter $\gamma$ in the Orion Nebula (the Veil
Shell and OMC4), the shell of M43, and the shell of NGC 1977 are shown in Fig.
21. For M42, the data reveals a similar trend with a decreased efficiency with
increased ionization parameter as the theoretical predictions of Bakes &
Tielens (1994). The cooling efficiency is generally lower in OMC4 compared to
the Veil Shell, M43, and NGC 1977, due to a “FIR excess” from dense gas behind
the [C ii]-emitting layers (cf. Fig. 2f in Paper I). M43 comprises too few
data points to reveal a clear trend. For NGC 1977, the observations reveal a
heating efficiency that is largely independent of the ionization parameter,
but this is complying with the near-constant theoretical heating efficiency
for low $\gamma$. However, virtually all points lie below the theoretical
heating curve. As a caveat, we note that it is hard to constrain $\gamma$
observationally, which is mainly due to the difficulty of obtaining a robust
gas density estimate, but we deem it unlikely that we overestimate the density
(or electron density, assuming all electrons stem from carbon ionization)
systematically by an order of magnitude. We observe this same behavior when
averaging over the regions and employing average values of $\gamma$.

Figure 21: Photoelectric heating efficiency $\epsilon$, equated as cooling
efficiency [C ii]/FIR, in M42 (top left: Veil Shell, bottom left: OMC4 with
OMC1 (in gray scale)), the shell of M43 (top right), and the shell of NGC 1977
(bottom right) as function of the ionization parameter
$\gamma=G_{0}T^{1/2}/n_{\mathrm{e}}$. Colors indicate the distance from
$\theta^{1}$ Ori C in the Orion Nebula, from NU Ori in M43, and from 42
Orionis in NGC 1977, respectively. Contours in the upper-left and lower-right
panel indicate the density levels above which 90, 70, 50, 30, and 10% of the
points lie. The red curve is the theoretical prediction of Eq. 43 by Bakes &
Tielens (1994). Figure 22: The average photoelectric heating efficiency
$\epsilon$, equated as cooling efficiency [C ii]/FIR, in M42, M43, and NGC
1977, divided into the previously defined subregions, as function of the
ionization parameter $\gamma=G_{0}T^{1/2}/n_{\mathrm{e}}$. OMC1 includes [O i]
cooling, but excludes the BN/KL region and Orion S. Parameters for Orion A are
given in Table 13. The photoelectric heating efficiency in PDRs and diffuse
sight lines from the literature is plotted as open blue squares (cf. Fig. 9 in
Paper I). Data for the diffuse ISM are taken from Gry et al. (1992) and van
Dishoeck & Black (1986), data of NGC 2023 are from Hollenbach & Tielens
(1999), data for the Horsehead and L1630 are from Pabst et al. (2017), data
for IC 59 and IC 63 are from Andrews et al. (2018), data for the Orion Bar,
S6, M43 west, and OMC2 are from Herrmann et al. (1997), and data for the Orion
Veil are from Salas et al. (2019). The red curve is the theoretical prediction
of Eq. 43 by Bakes & Tielens (1994). Table 7: Cooling efficiency and estimated
physical parameters to compute $\gamma$ in Fig. 22.

Region |  | $L_{\mathrm{[C\,\textsc{ii}]}}/L_{\mathrm{FIR}}$ |  $G_{0}$aa$a$Estimated from average distance to respective central star with systematic uncertainty of $\pm 50\%$. |  $n$bb$b$Estimated from [C ii]-CO separation in Eastern Rim and M43 shell, otherwise $\tau_{160}$, with systematic uncertainty of $\pm 50\%$. |  $T$cc$c$Estimated from [C ii] excitation temperature (cf. Appendix A in Paper I). PDR models predict up to twice as large temperature for the [C ii]-emitting layers.
---|---|---|---|---|---
|  |  |  | $\mathrm{[cm^{-3}]}$ | $\mathrm{[K]}$
M42 | all | $3.6\times 10^{-3}$ | $1\times 10^{3}$ | $1\times 10^{4}$ | 200
| OMC1dd$d$[O i] $63\,\mu\mathrm{m}$ cooling contributes significantly in OMC1. We find $L_{\mathrm{[O\,\textsc{i}]}}\simeq 320\,L_{\sun}$ (Higgins et al. in prep.), hence $(L_{\mathrm{[C\,\textsc{ii}]}}+L_{\mathrm{[O\,\textsc{i}]}})/L_{\mathrm{FIR}}\simeq 4.5\times 10^{-3}$. The Bright Ridge with BN/KL and Orion S accounts for $L_{\mathrm{[O\,\textsc{i}]}}\simeq 90\,L_{\sun}$, hence including this area yields a cooling efficiency of $(L_{\mathrm{[C\,\textsc{ii}]}}+L_{\mathrm{[O\,\textsc{i}]}})/L_{\mathrm{FIR}}\simeq 5.1\times 10^{-3}$ for the PDR. | $6.5\times 10^{-4}$ | $2\times 10^{4}$ | $1\times 10^{5}$ | 300
| Eastern Rim | $3.5\times 10^{-3}$ | 500 | $1\times 10^{4}$ | 100
| OMC4 | $2.8\times 10^{-3}$ | 500 | $1\times 10^{4}$ | 100
| Veil Shell | $4.3\times 10^{-3}$ | 200 | $3\times 10^{3}$ | 100
M43 | all | $2.7\times 10^{-3}$ | $1\times 10^{3}$ | $1\times 10^{4}$ | 120
| background, H ii | $2.8\times 10^{-3}$ | $1.5\times 10^{3}$ | $1\times 10^{4}$ | 120
| faint CO, faint [C ii] | $4.7\times 10^{-3}$ | 300 | $1\times 10^{3}$ | 100
| bright CO, faint [C ii] | $3.0\times 10^{-3}$ | 300 | $1\times 10^{4}$ | 100
| shell | $2.3\times 10^{-3}$ | $1\times 10^{3}$ | $1\times 10^{4}$ | 120
NGC | all | $8.9\times 10^{-3}$ | 100 | $5\times 10^{3}$ | 90
1977 | H ii | $7.9\times 10^{-3}$ | 200 | $5\times 10^{3}$ | 90
| OMC3 | $5.9\times 10^{-3}$ | 200 | $2\times 10^{4}$ | 100
| faint H$\alpha$ | $9.7\times 10^{-3}$ | 100 | $1\times 10^{3}$ | 50
| shell | $1.3\times 10^{-2}$ | 100 | $5\times 10^{3}$ | 70
131313

Figure 22 shows the average cooling rate as a function of $\gamma$ averaged
over the regions defined in M42, M43, and NGC 1977, together with data from
the literature. The estimated parameters for M42, M43, and NGC 1977 employed
in this figure are given in Table 13. We include [O i] $63\,\mu\mathrm{m}$
cooling from Higgins et al. (in prep.) in OMC1. The [O i] $63\,\mu\mathrm{m}$
line is expected to be important in the Eastern Rim and M43, as well, with
$I_{\mathrm{[O\,\textsc{i}]}}/I_{\mathrm{[C\,\textsc{ii}]}}\sim 3$ (Pound &
Wolfire 2008; Kaufman et al. 2006), but will be of little significance in the
extended shell of NGC 1977. Observations of [O i] $63\,\mu\mathrm{m}$ and
$145\,\mu\mathrm{m}$ line emission in the Orion Nebula would provide the means
to establish better estimates of the gas temperature and gas density, besides
the fact that it substantially contributes to the gas cooling.

These average heating efficiencies show the same trend as the overall data in
Fig. 21 with a decrease with increasing ionization parameter but the spread is
rather large. The Orion data also agrees well with previous studies, both for
bright PDRs using fine-structure line measurements as also for diffuse clouds
from UV absorption line studies. This earlier data set also shows a large
spread. Comparing the data to the theoretical results of Bakes & Tielens
(1994), we see a similar behavior of decreasing heating efficiency with
increasing $\gamma$, consistently in Orion and earlier studies, but displaced
by a factor of $\sim 5$ to lower $\epsilon$. There seems to be an additional
factor, besides the ionization parameter, that affects the coupling of FUV
photons to the neutral atomic gas. This factor could be a variable abundance
of PAHs and VSGs in different regions within Orion (and other lines of sight).
A lower abundance of PAHs and VSGs diminishes the heating efficiency
systematically, while a variable composition of the PAH family or a varying
ratio of PAHs to VSGs can cause variation in the heating efficiency. As shown
by Berné et al. (2015) and Croiset et al. (2016) in the case of NGC 7023, the
population of PAHs is very sensitive to the physical conditions ($G_{0}$ and
$n$). Furthermore, the recombination rates of PAHs, employed in the
theoretical model, are not well-constrained experimentally.

The heating efficiency describes the energy input per FUV photon. When
equating the cooling efficiency [C ii]/FIR with the heating efficiency, we
assume that only FUV photons are re-radiated in FIR energy. We note that a
fraction of energy of O stars is emitted in the EUV and redistributed into
optical photons by the surrounding H ii region while cooler stars emit a
significant amount of their energy in the spectrum below 6 eV. These lower-
energy photons contribute to the heating of the large dust grains and increase
the FIR intensity emitted by those. Using CLOUDY (C17.02 Ferland et al. 2017)
models we find an average factor of $\sim 1.5$ increase depending on the
spectral type. Also the energy spectrum changes with spectral type, which
induces changes in the heating efficiency as derived by Bakes & Tielens (1994)
for the radiation field at $3\times 10^{4}\,\mathrm{K}$. For early B stars
($T_{\mathrm{eff}}\simeq 2.5\times 10^{4}\,\mathrm{K}$), Spaans et al. (1994)
derive a $\sim$15% lower heating efficiency. While those two corrections might
bridge the gap between the observations and predictions in NGC 1977 (except
OMC3), cooling efficiencies in regions in M42 and M43 are about 5 times
smaller than predicted by theory at the estimated $\gamma$. Another
complication in the interpretation of the observed [C ii]/FIR ratio is the
fact that most structures are observed in edge-on geometry. In the M42 and M43
edges, the [C ii] line tends to be optically thick toward the observer. The
real cooling efficiency might be higher (by a factor of $\sim 2$) than
observed if the shell structures cool mainly through the (unobservable) face-
on surfaces. Dust in H ii regions can also produce low [C ii]/FIR ratios. Only
OMC1 lies close to the theoretical curve, but we note that the cooling is
dominated by the [O i] $63\,\mu\mathrm{m}$ line141414The observed [C ii]
intensity in OMC1 is higher than predicted by standard face-on PDR models
(Pound & Wolfire 2008; Kaufman et al. 2006), but the [O i] $63\,\mu\mathrm{m}$
intensity is predicted accurately..

The Orion [C ii] survey also provides a way to link the observed heating
efficiency directly to the characteristics of the local PAH family. Rubin et
al. (2011) and Boersma et al. (2012) have analyzed a set of Spitzer/IRS
spectra that systematically explored the behavior with distance of PAHs in the
PDR associated with the Veil Shell. As Fig. 23 shows, the intensity of the [C
ii], FIR, and PAH bands all smoothly decrease with distance from the Orion Bar
(and hence the illuminating star). The gas heating efficiency, however, shows
a clear peak at about $10\arcmin$ from the Bar. In contrast, the ratio of the
IRAC 8 $\mu$m band – dominated by the 7.7 $\mu$m PAH feature – to the IRS 11.2
$\mu$m band – a blend of the PAH 11.0 $\mu$m and 11.25 $\mu$m features – is
largely constant over the full distance range probed. This band ratio is
generally thought to probe the degree of ionization of the emitting PAHs as
the former band traces emission by ionized PAHs while the latter is dominated
by neutral PAHs (Peeters et al. 2002; Boersma et al. 2012). Given the general
relation between $G_{0}$ and $n$ discussed in Sec. 4.3, we expect little
variation in the ionization parameter with distance and that is borne out by
the constant $8\,\mu\mathrm{m}$/$11.2\,\mu\mathrm{m}$ band ratio. As a
corollary, this again suggests that there is another factor that affects the
gas heating efficiency besides ionization. That could be abundance of PAHs and
VSGs. However, the intensity ratio
($I_{8\,\mu\mathrm{m}}+I_{11.2\,\mu\mathrm{m}})/I_{\mathrm{FIR}}$ – a good
proxy for the PAH abundance (Tielens 2008)– is constant along the line cut,
which hints that VSGs play an important role.

Figure 23: Line cut from the Orion Bar to the southwest, reaching the Eastern
Rim at $12\arcmin$, indicated in Fig. 15. The offset is given with respect to
$\theta^{1}$ Ori C. The IRAC 8 $\mu$m filter measures predominantly the
emission in the 7.7 $\mu$m PAH band. IRS has measured the strength of the 11.2
$\mu$m PAH feature. The $11.2\,\mu\mathrm{m}$ intensities are taken from Table
1 in Boersma et al. (2012).

Most of the observed [C ii] and FIR emission will stem from the background
PDR. The [C ii]/FIR increases with distance from the origin of the line cut,
until it hits the Eastern Rim, where it gradually decreases as the Rim piles
up along the line of sight. The decrease of the cooling efficiency in the Rim
may be due to increased $G_{0}$ (as a result of the geometry). Alternatively,
the edge-on geometry of the Rim increases the optical depth of the [C ii] line
compared to the FIR intensity, which leads to a diminished observed [C ii]/FIR
ratio. As we expect the [C ii] line to be optically thick in the Rim, the
latter will certainly be an important effect. Yet, we cannot exclude that
$G_{0}$ in the Rim is higher than in the molecular background, as reflected in
the higher dust temperature in the Rim (cf. Fig. 1).

Figure 24: [C ii] (blue), 12CO(2-1) (yellow), and 13CO (brown) spectra along
the line cut behind the Orion Bar in Fig. 23. The offset with respect to
$\theta^{1}$ Ori C is given in arcminutes. In addition, the [13C ii]
$F=2\text{-}1$ component is plotted (gray) in its systemic velocity and
multiplied by 20 in spectra 1, 2, and 9.

Spectra along the line cut, shown in Fig. 24, reveal complex dynamics of the
gas, and potentially foreground or self-absorption of the [C ii] line, that
can effect the cooling and heating efficiency in this region. The CO(2-1)
intensity peaks strongly between 8 and 10 arcmin. Those are the points
(positions V1 and V2) that show deviating characteristics in the study of
Boersma et al. (2012), that is a shift in peak position and line width.
Positions V1 and V2 positions also exhibit a higher fullerene-to-PAH ratio
than other positions. CO pv diagrams reveal dynamic structures in this region.
While those structures are more manifest in CO, spectra toward selected
regions reveal weaker corresponding [C ii] components (spectra 7 and 8 in Fig.
24). FIR emission seems to be dominated by re-radiation of FUV photons from
$\theta^{1}$ Ori C and/or $\theta^{2}$ Ori A, as concluded from the persistent
decline in FIR intensity.

##  5 Conclusion

In Paper I, we have studied the correlations of the [C ii] intensity with the
dust continuum intensity at 70 $\mu$m, the FIR intensity (40-500 $\mu$m), PAH
8 $\mu$m intensity, and CO(2-1) intensity in the Orion Nebula complex. Here,
we have focused on these same correlations divided into several subregions
(Veil Shell, OMC4, M43, and NGC 1977, further divided up into more
subregions). Typically, the observed correlations show only minor variations
between these regions. However, the [C ii]-CO(2-1) correlations are very
different between the regions as they are very sensitive to the geometry. The
geometric complexity of these regions is quite apparent when examining
individual spectra toward several representative regions. We have compared the
observations with constant-density PDR models and concluded that, in general,
the edge-on PDR models are able to reproduce the observed intensities, while
face-on PDR models cannot explain the observations. The estimated physical
parameters, that serve as model inputs, correspond to a constant
$p_{\mathrm{th}}/G_{0}$ ratio, which has been observed toward several other
PDRs, as well. We have explicated the origin of the [C ii] emission within the
mapped area in greater detail than in Paper I, focusing on the detailed PDR
processes and source properties. While the bulk of [C ii] emission from the
mapped area originates in PDR surfaces (only 7.5% from the bright OMC1), about
5% each in the Veil Shell and NGC 1977 and 15% in M43 stem from H ii gas.
Furthermore, we have compared the properties of PAHs in the southeast of the
Orion Bar (as measured by Spitzer) to the [C ii] cooling efficiency, but do
not find a correlation between the two: While the [C ii] cooling efficiency
varies with distance from the Orion Bar, the PAH charge does not do so. Future
studies may include the [O i] 63 $\mu$m and 145 $\mu$m cooling lines and alter
the picture obtained by the study of the [C ii] 158 $\mu$m line alone.
Moreover, the James Webb Space Telescope (JWST) will be able to measure
ISM/PAH properties with higher accuracy, and these results may shine further
light on the observed variations in the cooling efficiency.

###### Acknowledgements.

We thank the anonymous referee for valuable comments that helped improve the
manuscript. This work is based on observations made with the NASA/DLR
Stratospheric Observatory for Infrared Astronomy (SOFIA). 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. JRG thanks the Spanish
MCIU for funding support under grant AYA2017-85111-P and PID2019-106110GB-I00.
This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(Grant agreement No. 851435). Research on the interstellar medium at Leiden
Observatory is supported through a Spinoza award of the Dutch Science
Organisation (NWO). We thank C. R. O’Dell for constructive discussions and S.
T. Megeath for helpful advice. This work made use of the TOPCAT software
(Taylor 2005).

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##  Appendix A Comparison with Lombardi SED fits

Figure 25: Ratios of FIR intensity from different SED fits ($\beta=2$, $\beta$
free and the Lombardi et al. (2014) result). Figure 26: Ratios of the dust
temperature $T_{\mathrm{d}}$ from different SED fits ($\beta=2$, $\beta$ free
and the Lombardi et al. (2014) result).

As opposed to Lombardi et al. (2014), we include the shorter-wavelength PACS
$70\,\mu\mathrm{m}$ and $100\,\mu\mathrm{m}$ bands, since we do not expect
them to be opaque in the extended structures we are mainly interested in. In
fact, the extended shell of the Orion Nebula has $\tau_{160}<10^{-2}$, which
translates into $\tau_{70}<3\times 10^{-2}$; even the densest cores along the
ISF are described by $\tau_{160}\lesssim 10^{-1}$. Comparing our derived FIR
intensity, we note that it agrees with the FIR intensity computed from the SED
maps of Lombardi et al. (2014), except in the bright eastern arm, the Rim, and
the bubble edge of NGC 1973, 1975, and 1977. Here, we obtain significantly
lower FIR intensity than Lombardi et al. (2014) (cf. Fig. 25). These filaments
contain warmer dust that is only captured by the shorter-wavelength bands,
leading Lombardi et al. (2014) to derive too high a dust temperature and too
low a dust optical depth. Lombardi et al. (2014) obtain higher dust
temperatures in the Eastern Rim, yet significantly lower dust optical depth.

Figs. 25 and 26 show the ratios of the results of different SED fits performed
on the region of the EON. Fig. 25 shows the ratios of the FIR intensity of our
$\beta=2$ SED fit, an SED fit where we let $\beta$ a free parameter, and the
Lombardi et al. (2014) result. Fig. 26 shows the ratios of the dust
temperatures from these same fits. From the right-hand panel it becomes
obvious that the fit of Lombardi et al. (2014) yields significantly higher FIR
intensity and dust temperature in the brightest parts of the EON, especially
the eastern Rim, the OMC1 region and M43. This might be due to the fact that
Lombardi et al. (2014) uses only the longer wavelength bands, that do not
capture the peak of the SED of the warmer dust in these bright regions. This
apparently results in an overestimation of the dust temperature and
consequentially of the FIR intensity.

##  Appendix B Correlation plots from free-$\beta$ SED fit

Figure 27: Maps of the three SED fit output parameters $T_{\mathrm{d}}$,
$\tau_{160}$ and $\beta$.

In this section, we present the correlation plots of results of an SED fit
where we let the grain emissivity index $\beta$ be a free parameter. Fig. 27
shows the resulting maps of the three SED output parameters $T_{\mathrm{d}}$,
$\tau_{160}$ and $\beta$. From this figure, we notice that there seems to be a
rather tight correlation between the resulting $\beta$ and $\tau_{160}$, which
we quantify in Fig. 32.

Most of the pixels seem to be consistent with a $\beta$ of $\simeq 1.5$, as
suggested by Lombardi et al. (2014). The latter use low-resolution Planck
images to constrain $\beta$, which amounts to $\beta\simeq 1.6$ in the Orion
Nebula complex. The median of all pixels in our SED fit is $\beta$ of $\simeq
1.2$, but when including only pixels with $\tau_{160}>5\times 10^{-4}$, which
comprise most of the visible structure, the median shifts to $\beta$ of
$\simeq 1.4$. The peak values, around which the $\beta$ values are clustered,
is $\beta\simeq 1.6$. Standard models, however, suggest $\beta\simeq 2$
(Weingartner & Draine 2001). The study of Extaluze et al. (2013) suggests
varying $\beta$ according to the environment (radiation field). We note that
$\beta$ can be affected by the presence of an additional warm dust population
in the hot gas and we determine low $\beta$ in the EON cavity. $\beta$ may
vary with the grain composition, grain-size distribution, and with the
presence of ices on the grains (e.g., Ossenkopf & Henning 1994; Liseau et al.
2015). Extragalactic studies show systematic variations of $\beta$ with
galactocentric radius, which may be linked to metallicity gradients (e.g.,
Kramer et al. 2010; Tabatabaei et al. 2014).

In ratio plots of the dust temperature from the three different SED fits,
shown in Fig. 26, shell structures and interior show distinct deviations from
unity, seemingly confirming the assumption that dust properties and, hence,
$\beta$ indeed vary with environment (cf. Extaluze et al. 2013). Denser,
warmer structures close to the Trapezium stars seem to be characterized by
$\beta\gtrsim 2$, while the extended shell has $\beta\simeq 1\text{-}2$ and
the dilute shell interior $\beta\simeq 0\text{-}1$.

Figure 28: Histograms of the three SED fit output parameters $T_{\mathrm{d}}$,
$\tau_{160}$ and $\beta$. Dashed black lines indicate the median values
($\hat{T}_{\mathrm{d}}\simeq 31.7\,\mathrm{K}$, $\hat{\tau}_{160}\simeq
1.0\times 10^{-3}$, $\hat{\beta}\simeq 1.23$).

In histograms of the SED output parameters (Fig. 28), the dust temperature
lies mostly in the range of $T_{\mathrm{d}}\simeq 20\text{-}50\,\mathrm{K}$,
dust opacity can be mostly found in $\tau_{160}\simeq 10^{-4}\text{-}10^{-2}$,
and the grain emissivity index varies mostly between $\beta\simeq 0\text{-}2$,
yet with a significant tail of negative values. The latter correspond to areas
with little FIR emission and thus poor fit results.

Figure 29: Dust temperature $T_{\mathrm{d}}$ from an SED fit with $\beta$ a
free parameter versus distance from $\theta^{1}$ Ori C. Colors code the dust
optical depth for $\tau_{160}>5\times 10^{-4}$.

Figure 30: Dust temperature $T_{\mathrm{d}}$ versus dust optical depth
$\tau_{160}(>5\times 10^{-4})$ in a SED fit with $\beta$ a free parameter
($\rho\simeq 0.11$). Colors code the distance from $\theta^{1}$ Ori C.