# Massive Quiescent Cores in Orion: VI. The Internal Structures and a
Candidate of Transiting Core in NGC 2024 Filament

Zhiyuan Ren11affiliation: National Astronomical Observatories, Chinese Academy
of Science, Chaoyang District Datun Rd A20, Beijing, China; Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>22affiliation: Key Laboratory of Radio
Astronomy, Chinese Academy of Science 33affiliation: The Department of
Astronomy, Peking University and Di Li11affiliation: National Astronomical
Observatories, Chinese Academy of Science, Chaoyang District Datun Rd A20,
Beijing, China; Email<EMAIL_ADDRESS><EMAIL_ADDRESS>22affiliation: Key
Laboratory of Radio Astronomy, Chinese Academy of Science

###### Abstract

We present a multi-wavelength observational study of the NGC 2024 filament
using infrared to sub-millimeter continuum and the ${\rm NH_{3}}$ $(1,1)$ and
$(2,2)$ inversion transitions centered on FIR-3, the most massive core
therein. FIR-3 is found to have no significant infrared point sources in the
Spitzer/IRAC bands. But the ${\rm NH_{3}}$ kinetic temperature map shows a
peak value at the core center with $T_{\rm k}=25$ K which is significantly
higher than the surrounding level ($T_{\rm k}=15-19$ K). Such internal heating
signature without an infrared source suggests an ongoing core collapse
possibly at a transition stage from first hydrostatic core (FHSC) to
protostar. The eight dense cores in the filament have dust temperatures
between 17.5 and 22 K. They are much cooler than the hot ridge ($T_{\rm d}=55$
K) around the central heating star IRS-2b. Comparison with a dust heating
model suggests that the filament should have a distance of $3-5$ pc from
IRS-2b. This value is much larger than the spatial extent of the hot ridge,
suggesting that the filament is spatially separated from the hot region along
the line of sight.

ISM:clouds – ISM: individual objects (NGC2024, Orion) – ISM: molecules –
stars: formation – stars: low-mass

##  1 Introduction

Revealing the transition stage between prestellar cores and protostars is
critical for understanding the entire star forming process. The low- and
intermediate-mass prestellar cores are supposed to initially stay in a
hydrostatic equilibrium, referred to as Bonnor-Ebert sphere (Ebert, 1955;
Bonnor, 1956). The density would gradually increase due to the self
contraction and external influences such as turbulence, compression flow, and
velocity perturbation which are usually presented in filamentary clouds (e.g.
Gómez et al., 2007; Hennebelle & Chabrier, 2011; Gong & Ostriker, 2015). As
the core becomes supercritical to the self gravity, the ”first collapse” would
occur and generate a much denser object called the first hydrostatic core
(FHSC, or first core). The FHSC has a density of $\sim 10^{13}$ ${\rm
cm^{-3}}$ so that the gas and dust become opaque to radiation (Larson, 1969).
The temperature in the FHSC would continuously increase due to the self-
gravity and possibly on-going accretion. Once the central temperature exceeds
2000 K, the heating would start to dissociate the ${\rm H_{2}}$ molecules. The
${\rm H_{2}}$ dissociation can provide an efficient coolant for the gas, and
largely reduce the thermal pressure support, thereby induce the ”second
collapse”. In this process the FHSC would evolve into a protostar (Masunaga &
Inutsuka, 2000; André, 2011).

Despite this delicately modelled evolutionary track, the observed examples for
the transition stage between pre- and protostellar cores are still scarce. Up
till now, only a few low-mass cores are suggested to be the FHSC candidates
(e.g. Boss & Yorke, 1995; Belloche et al., 2006; Chen et al., 2010; Pineda et
al., 2011; Pezzuto et al., 2012). And a few properties are expected for the
FHSCs based on their dust continuum emissions, including: (1) low luminosity
($<10^{-1}~{}L_{\odot}$) and temperature ($<20$ K); (2) no significant
emissions in far-infrared and shorter wavelengths ($\lambda<70~{}\rm\mu m$);
(3) dense and compact morphology in (sub)millimeter wavelengths.

At a distance of 415 pc (Menten et al., 2007; Sandstrom et al., 2007), Orion
molecular cloud is the closest and best-studied massive star-forming complex.
Besides the regions with bright young stars, Orion contains a huge amount of
cold, quiescent, and dense gas (e.g. Salji et al., 2015, and references
therein). In our previous studies (Li et al., 2007; Velusamy et al., 2008; Li
et al., 2013; Ren et al., 2014), the quiescent cores in Orion A and Orion
South have been selected and investigated. A large fraction of the cores were
found to be unstable to the self-gravity, and likely to have lower
temperatures than their surroundings (Li et al., 2013). The candidate cores
for the transition stage can be selected based on two requirements: the cores
should be supercritical to the self-gravity and meanwhile have no detectable
IR sources. Some likely candidates were examined but found to actually have
faint embedded IR sources and even the multiple stars which are associated
with the core fragmentation (Ren et al., 2014). Subsequent studies should be
performed over a larger field in order to enlarge the sample. And the
evolutionary stages should be evaluated based on more evidences.

Located $\sim 4$ degrees to the North of Orion A, NGC 2024 in Orion B cloud
contains extended gas structures with embedded cold dense cores. The major
fraction of the gas is assembled in a compact filamentary structure (Mezger et
al., 1992, also see Figure 1). A number of observations were performed to
examine the physical properties therein (Gaume et al., 1992; Mezger et al.,
1992; Mauersberger et al., 1992; Chandler & Carlstrom, 1996; Watanabe &
Mitchell, 2008; Alves et al., 2011; Choi et al., 2015). These observations
revealed that the dense filament is located behind the hot ionized gas and
have complex structures. But the specific properties and evolutionary state of
each core are still to be investigated. Gaume et al. (1992) presented Very
Large Array (VLA) observation in the ${\rm NH_{3}}$ lines, but only obtained
average physical parameters for several regions owing to the limited velocity
resolution and spectral sensitivity.

In this work, we presented new observational study for NGC 2024 using the
continuum emissions from mid-infrared to sub-millimeter bands. We examined the
mass and the temperature distributions of the entire filament. Moreover, to
study the star-forming properties in the most massive core FIR-3 therein, we
carried out new ${\rm NH_{3}}$ observation with the Karl G. Jansky Very Large
Array (JVLA). The results show that the dense cores on the filament tend to
have young evolutionary stages and should be spatially separated from the hot
gas and dust in the foreground. In Section 2, we described the observation. In
Section 3 and 4, we described the filament and dense core structures and their
dust temperatures. In Section 5, we presented the mass and temperature
distributions within FIR-3 based on the high-resolution ${\rm NH_{3}}$
spectral data. In Section 6, we discussed the core evolutionary state based on
their physical parameters. A summary is given in Section 7.

##  2 Observation and Data Reduction

The observation of NGC 2024 FIR-3 was carried out with the NRAO111The National
Radio Astronomy Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc. JVLA on
September 6, 2014. The antennae were in D configuration. One baseband (A0/C0)
was tuned at 23.9 GHz, in which two sub-bands were placed at the frequencies
of the ${\rm NH_{3}}$ $(1,1)$ and $(2,2)$ lines, respectively. Each subband
had a bandwidth of 8 MHz. All the hyperfine components (HFCs) of the ${\rm
NH_{3}}$$(1,1)$ lines are covered by the subband with a velocity resolution of
0.1 km s-1. 3C147, which has a flux density of 2.8 Jy in the K-band under the
D-configuration, was used as the bandpass and flux calibrator. The quasar
J0541-0541 was adopted as the antenna gain calibrator. The total on-source
integration time was 50 minutes. The CASA program222http://casa.nrao.edu and
were used for the data calibration analysis and imaging. Aladin sky
atlas333”Aladin sky atlas” is developed at CDS, Strasbourg Observatory, France
(Boch & Fernique, 2014) was also used for inspecting the images. The
synthesized beam size is $3.3^{\prime\prime}\times 2.8^{\prime\prime}$ and the
rms noise level is 7 mJy beam-1 (1.2 K) per 0.1 km s-1 channel.

In order to investigate the entire filament structure, the stellar emissions,
and the spectral energy distribution (SED), we also obtained mid- to far-
infrared images including (1) Spitzer/IRAC images from the Archive of the
Spitzer Enhanced Imaging Products (SEIP)
444http://sha.ipac.caltech.edu/applications/Spitzer/SHA/, (2) The Herschel
PACS 70, 100, and 160 $\rm\mu m$, SPIRE 250, 350, and 500 $\rm\mu m$ images
taken from the Herschel Science Archive555Herschel is an ESA space observatory
with science instruments provided by European-led Principal Investigator
consortia and with important participation from NASA.
http://www.cosmos.esa.int/web/herschel/science-archive. The Herschel
observations are part of the Gould Belt Survey (André et al., 2010). (3) The
MSX 12, 16.5 and 21 $\rm\mu m$ images from the MSX Image Server and Catalog
Overlays v 6.0666http://irsa.ipac.caltech.edu/applications/MSX/MSX/, and (4)
The JCMT/SCUBA 450 and 850 $\rm\mu m$ continuum maps from the JCMT science
archive777The James Clerk Maxwell Telescope is operated by the Joint Astronomy
Centre on behalf of the Science and Technology Facilities Council of the
United Kingdom, the Netherlands Organisation for Scientific Research, and the
National Research Council of Canada. http://www.cadc.hia.nrc.gc.ca/jcmt/.

The Herschel PACS and SPIRE images are Level-2.5 and Level-3 products,
respectively. The PACS 100 and 160 $\rm\mu m$ images are from the ”PACS-only”
data and were mapped at a scan speed of 20 arcsec s-1 and Repetition factor of
6. The PACS 70 $\rm\mu m$ and all the SPIRE images are in parallel mode and
have a scan speed of 60 arcsec s-1. The Herschel images have sensitivities
between 5 and 40 mJy pixel-1 (see Table A1), which are 2 to 3 orders of
magnitudes smaller than the filament emissions. Therefore, the flux
uncertainties should mainly come from the flux calibration and photometry
processes, as described in Appendix A.1..

##  3 The Mass Distribution of the Filament and the Ridge

###  3.1 Dust Continuum Emissions

The Spitzer, MSX, Herschel, and JCMT/SCUBA images from mid-infrared to sub-
millimeter bands are shown in Figure 1. Originally, the PACS 70, 100, and 160
$\rm\mu m$ images are in unit of Jy pixel-1, and the SPIRE and SCUBA images
are in Jy beam-1. In Figure 1, the intensity unit was scaled to Jy arcsec-2 by
dividing the original intensity scale with the pixel (PACS) or the beam (SPIRE
and SCUBA) areas, in order to provide a fair comparison for the intensity
scales throughout the wavelength range.

Figure 1a shows the SCUBA 450 $\rm\mu m$ emission (contours) overlaid on the
RGB image of the IRAC 3.6, 4.5 and 8.0 $\rm\mu m$ bands. The brightest star
and major heating source in this region is IRS-2b (Bik et al., 2003), which
was estimated to have a spectral type between O8V and B2V. It is located at a
projected distance of $30^{\prime\prime}$ (0.05 pc) from the filament. The FIR
cores (specified in Figure 2 and Section 3.2) are labeled with ’+’ symbols.
The 8 $\rm\mu m$ emission (red) shows the extended emission over the entire
region, and a more prominent ”hot ridge” structure going across FIR-5. The MSX
RGB image (12, 16.5, and 21 $\rm\mu m$ bands, shown in Figure 1b) also
evidently traces the hot ridge. In the MIPS 24 $\rm\mu m$ band (Figure 1c),
the extended emission region is largely saturated over a spatial extent of
100-200 arcsec (0.2-0.4 pc), and its southern boundary is almost in parallel
with the hot ridge. The extended emission and the ridge should represent the
Hii and the photo dissociation region (PDR) around IRS-2b, respectively (Roshi
et al., 2014); the ridge might be formed as the Hii region is expanding and
compressing the surrounding medium.

The PACS 70 $\rm\mu m$ image (Figure 1d) is also dominated by the hot-ridge
emission. Among the seven major dense cores identified in Mezger et al.
(1992), only FIR-4 is significantly seen. FIR-5 is totally blended with the
ridge emission and cannot be separated. The PACS 100 $\rm\mu m$ image (Figure
1e) exhibits similar features but with the filament more evidently seen (in
particular FIR-2 and 3), while the ridge becomes less intense. In PACS 160
$\rm\mu m$, SPIRE 250 $\rm\mu m$, 350 $\rm\mu m$, and 500 $\rm\mu m$, and
SCUBA 450 $\rm\mu m$ and 850 $\rm\mu m$ bands (Figure 1f to 1k), the emissions
are dominated by the filament and exhibits similar morphologies, while the
ridge is much weaker in those bands.

###  3.2 Core Identification and Flux Measurement

The filament and FIR cores are best revealed in the SCUBA 450 $\rm\mu m$ band.
On the 450 $\micron$ image, the dense cores were identified using the IDL
routine Hyper (Traficante et al., 2015). Hyper will extract local emission
peaks, identify the cores, fit the angular size and surface brightness of each
core with 2D Gaussian profile, and estimate the flux density within the core
area. In this process, the background emission is subtracted through a 2D
polynomial fitting. The best-fit core radii for the 450 $\rm\mu m$ cores are
shown in Figure 2a. Using Hyper, the seven FIR cores (Mezger et al., 1992)
were all clearly extracted. In addition, FIR-5 was resolved into two objects
(denoted as FIR-5a and 5b), which are consistent with the previous
observations (Wiesemeyer et al., 1997; Lai et al., 2002). FIR-5a was further
resolved into at least seven condensations in Lai et al. (2002).

As shown in Table 1, FIR-2, 6, 7 are fitted to have large ratios of $r_{\rm
maj}/r_{\rm min}$. Such elongated core areas should be a result of confusion
with the extended filamentary structures. As shown in Figure 2a, the cores are
evidently seen only above the 30% contour, while bellow this level, the
emission is mainly from the filament. Figure 2 also shows that the elongations
of the cores are altogether reasonably along the filament (both at 160 and 450
$\micron$). To eliminate the confusion, we performed another fitting assuming
each core to have a circular shape. The measured core radii are shown in Table
1, and the circular-shaped core areas are shown in Figure 2b. It is within our
expectation that the radius of the circular fitting is closer to the minor
axis of the elliptical core area.

In Figure 2b, the 450 $\rm\mu m$ emission is overlaid on the IRAC RGB image,
and the IRAC point sources in the Orion protostar catalogue (Megeath et al.,
2012) are also plotted with yellow stars. Only FIR-1, 2, and 4 coincide with
the IRAC sources, with the magnitudes of $M_{3.6\rm\mu m}=8.69$ to 9.09, as
listed in Table 2. The brightest IR source coincides with FIR-4. Based on the
detection limit, the FIR cores absent of IRAC sources should have
$M_{3.6\rm\mu m}>14.6$.

The source extraction with Hyper was also applied for the PACS 70 $\rm\mu m$
and 160 $\rm\mu m$ images (Figure 2c and 2d, respectively). The 70 $\rm\mu m$
image was examined for potential additional hot-dust sources, while the 160
$\rm\mu m$ image was adopted to compare with the 450-$\rm\mu m$ cores. As a
result, FIR-4, 6 and other two sources were identified at PACS 70 $\rm\mu m$.
But the two sources on the ridge should represent the emission peaks of the
hot dust rather than dense cores. On the PACS 160 $\rm\mu m$ image, all the
dense cores were identified, with the central positions consistent with the
450-$\rm\mu m$ values within $7^{\prime\prime}$ (see Table 1).

We adopted the 450-$\rm\mu m$ core areas (in the circular case) to measure the
flux densities of all the cores except FIR-5a and 5b. At 70 $\micron$, the
emission intensities within the FIR cores (except FIR-4 and 5) range from 1 to
3 Jy arcsec-2 which are much higher than the detection limit, but are
comparable with the extended emission level ($1.5\pm 0.9$ Jy arcsec-2). We
thus suggested a marginal detection for these cores above the extended
emission level. The flux measurement is described in more detail in Appendix
A.1. We also selected five positions on the hot gas ridge including one at
FIR-5 (Figure 3b), and measured the flux density at each position within a
square region that covers the ridge width ($d=25$ arcsec). On the Spitzer and
MSX images (3.6 to 21 $\rm\mu m$), there are no structures likely associated
with the filament. The measured flux densities within the core areas should
represent upper limits for the FIR cores. The flux densities of the cores in
the IRAC and MSX bands are listed in Table 2, and the values of Hershel and
SCUBA bands are shown in Table 3.

##  4 Dust Temperature Distribution

###  4.1 The SED fitting for the cores and the ridge

We estimated the dust temperature in the FIR cores and the ridge from their
spectral energy distributions (SEDs) throughout the MSX, Herschel and JCMT
bands. The SED is fitted using a grey-body emission model (Hildebrand, 1983).
Based on the radiative transfer, the flux density of the dust core is

$S_{\nu}=\Omega B_{\nu}(T_{\rm d})[1-\exp(-\tau_{\nu})],$ (1)

wherein $S_{\nu}$ is the flux density at the frequency $\nu$. $\Omega$ is the
solid angle of the core or the selected area. $B_{\nu}(T_{\rm d})$ is the
Planck function of the dust temperature $T_{\rm d}$. The optical depth
$\tau_{\nu}$ is determined by other parameters as

$\tau_{\nu}=\kappa_{\nu}\mu m_{\rm H}N_{\rm tot}/g,$ (2)

wherein $N_{\rm tot}$ is the gas column density (mostly HI+${\rm H_{2}}$),
$\mu=2.33$ is the mean molecular weight (Myers, 1983), $m_{\rm H}$ is the mass
of the hydrogen atom, and $g=100$ is the gas-to-dust mass ratio.
$\kappa_{\nu}$ is the dust opacity, and is expected to vary with the frequency
in the form $\kappa_{\nu}=\kappa_{\rm 230GHz}(\nu/{\rm 230GHz})^{\beta}$, with
the reference value of $\kappa_{\rm 230GHz}=0.9$ cm2 g-1, as adopted from dust
model for the grains with coagulation for $10^{5}$ years with accreted ice
mantles at a density of $10^{6}$ ${\rm cm^{-3}}$ (Ossenkopf & Henning, 1994).

The best-fit SEDs for the FIR cores are shown in Figure 3a (FIR-5 is put in
Figure 3b to compare with the ridge). The physical parameters ($\beta$ and
$T_{\rm d}$) are presented in Table 4. The cores were measured to have a small
variation both in $T_{\rm d}$ and $\beta$. FIR-4 has the highest temperature
of $T_{\rm d}=22$ K, which is in agreement with the brightest infrared source
therein. The total luminosities of the cores were estimated using $L_{\rm
core}=\int 4\pi D^{2}S_{\nu}d\nu$, and are also presented in Table 4. We note
that FIR-3 was also observed in 1.2 mm continuum (Hill et al., 2005) with the
flux density measured to be $S_{\rm 1.2mm}=10$ Jy. In comparison, by
extrapolating the SED curve, we obtained a much lower value of $S_{\rm
1.2mm}=6$ Jy. The difference is probably due to the low resolution
($24^{\prime\prime}$) and large mapping step ($44^{\prime\prime}$) in the
previous observation. The mapping step is larger than distances of FIR-1 and 4
from FIR-3, thus the cores would be poorly resolved and emissions from FIR-1
and 4 might be largely included in the measured flux density. The $\beta$
values in the filament are comparable to the highest $\beta$ values among the
other Orion regions (Goldsmith et al., 1997).

The SED within FIR-5 can be well fitted using two temperature components with
$T_{\rm d}=17.5$ and 55 K respectively (Figure 3b). The cold component is
comparable to the other FIR cores and should represent the SED of FIR-5
itself. The hot component ($T_{\rm d}=55$ K) should represent the contribution
from the hot ridge. The average flux densities of the four other positions on
the ridge are also shown in Figure 3b. The SED fitting resulted in $T_{\rm
d}=56$ K and $\beta\simeq 1.6$. We note that the $\beta$ value for the hot
component at FIR-5 is not well constrained because the Rayleigh-Jeans tail of
the SED is dominated by the cold component. We expect it to have a similar
value of $\beta\simeq 1.6$.

The optical depth from the for the hot component is optically thin throughout
the observed wavelengths, with $\tau\simeq 0.1$ at $\lambda=70~{}\rm\mu m$,
while the cold dust becomes moderately optically thick at
$\lambda<100~{}\rm\mu m$. At low optical depth, Equation (1) can be
approximated as

$\displaystyle S_{\nu}$ $\displaystyle=\kappa_{\nu}B_{\nu}(T_{\rm d})\Omega\mu
m_{\rm H}N_{\rm tot}/g$ (3) $\displaystyle=\frac{\kappa_{\nu}B_{\nu}(T_{\rm
d})M_{\Omega}}{gD^{2}},$

wherein $D=415$ pc is the source distance, $M_{\Omega}$ is the gas mass within
$\Omega$. $N_{\rm tot}$ was estimated from the 450 $\micron$ peak intensity
(in Jy arcsec-2), and the masses of FIR cores and the ridge (within the square
aperture region) are calculated from their total $S_{450\micron}$ values. We
note that FIR-5a and 5b are resolved at 450 $\micron$, so their masses can be
separately obtained using the average $T_{\rm d}$ and $\beta$ values for the
two cores. The volume number density can be estimated from $N_{\rm tot}$ as
$n=N_{\rm tot}/2R_{\rm core}$. This value represents an average along the line
of sight. The results are also presented in Table 4. We note that if
$\kappa_{\rm 230GHz}$ value is different, the core mass would vary with
$\kappa_{\rm 230GHz}$ proportionally, while the temperature fit remains
unchanged. For example, if adopting $\kappa_{\rm 230GHz}=0.5$ cm2 g-1
(Preibisch et al., 1993), the core masses would increase by a factor of
$0.9/0.5=1.8$. The SED fitting is described in more details in Appendix A.2..

For the hot component, the IRAC 8 $\rm\mu m$ and MSX 12.5 $\rm\mu m$
intensities are above the SED curve, suggesting a dust component with even
higher temperature. Mezger et al. (1992) fitted the SED of the entire region
and identified two cold components with $T_{\rm d}=19$ and 22 K, which are
consistent with the temperature range for the FIR cores. But the hot component
was measured to have $T_{\rm d}=45$ K, which is cooler than the hot ridge
observed here. The difference is probably because the flux densities at
shorter wavelengths ($\lambda<350~{}\rm\mu m$) were not included in their
estimation.

It is also noteworthy that the ridge has a much lower $\beta$ than the values
in the FIR cores. To examine the validity of this discrepancy, we attempted to
fit the SED using a fixed value ($\beta=2.7$) comparable to the FIR cores. In
this case, the Rayleigh-Jeans tail ($\lambda=200$ to 1000 $\rm\mu m$) of the
SED largely deviates from the observed values and cannot be reconciled by
adjusting $\tau$ and $T_{\rm d}$. The difference in $\beta$ may indicate
different dust properties in the two components, and support the trend that
$\beta$ decreases with $T_{\rm d}$ (Arab et al., 2012).

##  5 The ${\rm NH_{3}}$ Gas Structure and Temperature Distribution

###  5.1 The Gas Distribution

The ${\rm NH_{3}}$ line spectra at the center of FIR-3 are shown in Figure 4,
with the hyperfine structures (HFS) labelled on the lines. The ${\rm NH_{3}}$
maps around FIR-3 are shown in Figure 5. The ${\rm NH_{3}}$ (1,1) inner
satellite group (isg) has lower opacities than the main group (mg) thus would
better reveal the dense gas structures. Figure 5a shows the integrated
emission of the isg, which is overlaid on the IRAC image. The ${\rm NH_{3}}$
gas is located around the FIR-3 core center and aligned in parallel with the
450 $\rm\mu m$ filament, suggesting that ${\rm NH_{3}}$ traces the densest gas
component in the core.

Figure 5b, 5c, and 5d show the integrated line emissions of the ${\rm NH_{3}}$
(1,1)-mg, -isg and (2,2)-mg. The ${\rm NH_{3}}$ $(1,1)$-mg and -isg show
similar morphologies, except that the mg emission is somewhat saturated due to
its higher optical depth. We also used Hyper to extract the compact sources in
the ${\rm NH_{3}}$ emission region (Figure 5b). As a result, four gas
condensations were found in FIR-3, which are labeled as Cd-1, 2, 3, 4. Another
two condensations in FIR-2 are labeled as Cd-n1 and Cd-n2. The physical
parameters of the condensations are shown in Table 5. Wherein the ${\rm
NH_{3}}$ column densities are estimated using Equation (A4).

Figure 5d shows that the (2,2) emission peak does not coincide with the (1,1)
peak but has a noticeable offset of $\simeq 3^{\prime\prime}$ to the
northwest. Since $(J,K)=(2,2)$ level has a much higher excitation energy
($E_{\rm u,(2,2)}$=81 K) than (1,1) ($E_{\rm u,(1,1)}$=24 K), the $(2,2)$
emission peak may indicate a higher temperature at its emission peak.

###  5.2 The Temperature Distribution from ${\rm NH_{3}}$

To investigate the temperature distribution from the ${\rm NH_{3}}$ lines, we
fitted the line profile using the fiducial radiative transfer model (e.g.
Friesen et al., 2009), which is specified in Appendix A.3. The best-fit line
profiles for the spectra at the core center are shown in Figure 4. For the
${\rm NH_{3}}$ $(2,2)$ line, we report the detection of the satellite
$F_{1}=3-2$ line ($f_{0}=23.721336$ GHz, as labelled on the figure), which has
a brightness temperature of $T_{\rm b}=4\pm 1$ K. The $(1,1)$ and $(2,2)$
lines are found to have similar excitation temperatures ($T_{\rm ex}=23$ K)
and line widths ($\simeq 1.0$ km s-1), suggesting them tracing the same gas
component.

The ${\rm NH_{3}}$ rotational temperature can then be estimated from the
$(1,1)$ and $(2,2)$ emissions using the method in Li et al. (2013). From the
observed spectra we calculated the intensity ratio between isg and mg, $R_{\rm
sm}=[\int T_{\rm b,isg}(1,1)d\nu]/[\int T_{\rm b,mg}(1,1)d\nu]$, and the
intensity ratio between the two transitions, $R_{12}=[\int T_{\rm
b}(1,1)d\nu]/[\int T_{\rm b}(2,2)d\nu]$. The rotational temperature is
calculated from the two ratios as

$T_{\rm rot}=41.5~{}{\rm K}/\ln[1.06\times C(1,1)\times R_{12}],$ (4)

wherein $C(1,1)=0.003+2.26R_{\rm sm}+0.00032\exp(5.38R_{\rm sm})$ is a
correction factor for the (1,1) optical depth. The uncertainty of the $T_{\rm
rot}$ is estimated to be $\delta T_{\rm rot}\simeq 1.5$ K based on the noise
level of the ${\rm NH_{3}}$ lines and the method of Busquet et al. (2009,
Equation A.7 therein). We estimated the relation between $T_{\rm rot}$ and the
gas kinetic temperature $T_{\rm k}$ based on the physical conditions in FIR-3
and using the RADEX program (van der Tak et al., 2007). The calculation is
specified in Appendix A.3. It resulted in a nearly thermalized population for
$(1,1)$ and $(2,2)$ levels as mainly due to the high density in FIR-3. As a
result $T_{\rm rot}$ is close to $T_{\rm k}$.

As shown in Figure 5e, the $T_{\rm k}$ map has a noticeable peak with $T_{\rm
k}\simeq 25$ K with the position coincident with the (2,2) emission peak. Away
from the peak, $T_{\rm k}$ varies between 15 and 19 K which are significantly
lower than the peak value, but similar with the dust temperature in FIR-3. The
temperature peak suggests an embedded YSO in condensation Cd-2. Since FIR-3
has no detectable IR point source, it is possible that the YSO is younger than
Class-0 and the heating is mainly due to the accretion or core collapse. In
this case the $T_{\rm k,peak}$ value may provide some constraint on accretion
rate and core evolutionary state.

We assumed that the central region originally had a similar temperature with
its surroundings ($T_{\rm k}=T_{\rm k,avg}=18$ K) in the starless stage, and
was then heated to the current value ($T_{\rm k}=T_{\rm k,peak}=25$ K) after
the central object started to accrete mass. The luminosity increase at the
second stage is then

$\Delta L=(4\pi/g)M_{\rm peak}\int k_{\nu}[B_{\nu}(T_{\rm
k,peak})-B_{\nu}(T_{\rm k,avg})]{\rm d}\nu,$ (5)

wherein $M_{\rm peak}$ is the hot-gas mass associated with $T_{\rm k}$ peak.
In equation (5) we also assumed that $T_{\rm k}$ can approximate the dust
temperature. The $T_{\rm k}$ peak is not resolved thus would represent an
average value within one beam, we thus consider the gas mass also within one-
beam area, that is $M_{\rm peak}=\pi R_{\rm beam}^{2}N({\rm H_{2}})\mu m_{\rm
H}=0.3~{}M_{\odot}$. As the result the luminosity increase is estimated to be
$\Delta L\simeq 50~{}L_{\odot}$.

On the other hand, the theoretical accretion luminosity is (Stahler et al.,
1980)

$L_{\rm acc}=\frac{GM_{\ast}\dot{M}_{\rm acc}}{R_{\ast}},$ (6)

where $M_{\ast}$, $R_{\ast}$, and $\dot{M}_{\rm acc}$ are the mass, radius,
and accretion rate for the central object, respectively. The stellar mass can
be estimated from its natal core mass assuming a star-forming efficiency of
$\epsilon=0.3$ (Alves et al., 2007). Adopting Cd-2 as the natal gas
condensation, we can obtain $M_{\ast}=\epsilon M_{\rm Cd-2}\simeq
0.3M_{\odot}$. We note that the spatial extent of the $T_{\rm k}$ peak may
only reflect the spatial range being heated, but is not to define a gas
condensation, thus was not used to estimate $M_{\ast}$. To estimate the
accretion rate, we also assume that the accretion energy is eventually
released mainly through the dust continuum emission so that $L_{\rm
acc}\simeq\Delta L$.

In theory (e.g. Young & Evans, 2005; Tomida et al., 2010), the YSO is expected
to have $R_{\ast}=10^{2}-10^{3}~{}R_{\odot}$ at the FHSC stage and then
collapse into $R_{\odot}$-scale during the protostar formation (second
collapse). We derived $\dot{M}_{\rm acc}$ for these two cases. First, if the
central object is a first core with $R_{\ast}=10^{2}~{}R_{\odot}$, the
expected accretion rate would be $\dot{M}_{\rm acc}\gtrsim 6\times
10^{-4}~{}M_{\odot}$ year-1. Alternatively, if the YSO has already collapsed
into $R_{\odot}$-scale, it only needs to have $\dot{M}_{\rm acc}\simeq 6\times
10^{-6}~{}M_{\odot}$ in order to maintain the observed $\Delta L$. The second
$\dot{M}_{\rm acc}$ value is comparable to the average $\dot{M}_{\rm acc}$ for
the low-mass protostars (Young & Evans, 2005) while the first one is two
orders of magnitude higher. As an indication, the central YSO would either
have a small radius ($\sim 1.0~{}R_{\odot}$) or a high accretion rate ($\sim
10^{-4}~{}M_{\odot}$ year-1) in order to generate the observed $T_{\rm
k,peak}$ feature. Correspondingly, the YSO should be either in a rapidly
collapsing phase or have lately collapsed into an $R_{\odot}$-scale protostar.

###  5.3 The Gravitational Instability

The possibility for the core collapse can also be evaluated through the Jeans
mass, which is

$\displaystyle M_{\rm J}$
$\displaystyle=\frac{\pi}{6}\frac{c_{s}^{3}}{G^{3/2}\rho^{1/2}}$ (7)
$\displaystyle=2M_{\odot}\left(\frac{\sigma}{0.2{\rm
km~{}s^{-1}}}\right)^{3}\left(\frac{n}{\rm
10^{3}~{}cm^{-}3}\right)^{-\frac{1}{2}}.$

$M_{\rm J}$ represents the highest mass that can be sustained by the internal
pressure related with the velocity dispersion $\sigma$. FIR-3 has $n\geq
10^{6}$ ${\rm cm^{-3}}$, and the velocity dispersion $\sigma_{\rm obs}=\Delta
V/\sqrt{8\ln(2)}\simeq 0.42$ km s-1. These parameters leads to $M_{\rm J}\leq
0.5~{}M_{\odot}$ which is smaller than the Cd-2 mass, suggesting that the
collapse of Cd-2 is possible unless there is additional support such as
magnetic field. Besides FIR-3, FIR-4, 5, and 6 were also suggested to be
unstable based on the observed H2CO line widths (Watanabe & Mitchell, 2008).

Using the ${\rm NH_{3}}$ kinetic temperature, the velocity dispersion due to
the thermal motion is estimated to be $\sigma_{\rm th}=k_{B}T_{k}/\mu m_{\rm
H}=0.25-0.33$ km s-1 as $T_{k}$ varies between 19 and 27 K. The non-thermal
contribution is $\sigma_{\rm nt}=\sqrt{\sigma_{\rm obs}^{2}-\sigma_{\rm
th}^{2}}\simeq 0.2-0.3$ km s-1. The $\sigma_{\rm nt}$ value is in the subsonic
rage but tend to be higher than the values in the cold dense gas (e.g.
$\sigma=0.1-0.2$ km s-1, Pineda et al., 2010). Based on the current data, it
is uncertain whether the FIR cores are experiencing a turbulence decay or
oppositely, becoming more turbulent due to the ongoing star-forming activities
(e.g. Fontani et al., 2012). We examined the ${\rm NH_{3}}$ line width over
emission region, and found no significant variation. A velocity field over a
larger area mapped by single dish may help better reveal the origin of the
turbulence.

##  6 The Evolutionary State of the FIR Cores

###  6.1 Mass-and-Luminosity Relation

The relation between the luminosity and mass can also provide an estimate for
the core evolutionary state (Molinari et al., 2008; Elia et al., 2013). The
Mass-Luminosity diagram is shown in Figure 6. The cores except FIR-4 are quite
closely aligned in a power law of $L\propto M^{p}$ with the index $p=1.0$. The
deviation of FIR-4 is expected due to its higher dust temperature. Besides the
IRAC source, active star formation in FIR-4 is also indicated by the radio
continuum and maser emissions therein (Choi et al., 2015). FIR-1, 2, and 6 may
also have protostars as due to the presence of IRAC sources or outflows
(Chandler & Carlstrom, 1996). But for these three cores, the stellar emissions
should still be too weak to significantly increase the total luminosities. It
is also possible that the IR sources at FIR-1 and 2 are just foreground stars.

Figure 6 also shows the $M-L$ distribution for the other four samples with
specified evolutionary stages, including Class-0 protostars (Andre et al.,
2000), prestellar cores (Elia et al., 2013), starless cores (Strafella et al.,
2015), and recently identified FHSC candidates (with respective references
shown in the figure caption). As the figure shows, the more evolved samples do
not necessarily have higher absolute luminosities, but instead show steeper
slopes for their $M-L$ relations, namely higher $p$ values. The NGC 2024 cores
have a similar slope with the prestellar sample ($p=1.0$).

From integration of Equation (1) we can have $L_{\rm core}\propto M_{\rm
core}T_{\rm d}^{4+\beta}$. It indicates that luminosity $L_{\rm core}$ can be
sensitively affected by $T_{\rm d}$. A sample would have $L\propto M^{1.0}$
only if the cores have quite similar temperatures. If $T_{\rm d}$ varies with
the core mass, the $L\propto M$ relation would be different. Assuming a power
law of $T_{\rm d}\propto M^{q}$, we can have $L_{\rm core}\propto M_{\rm
core}^{1+q(4+\beta)}$. The starless and prestellar samples exhibit relatively
flat power laws, probably because the more massive cores are more shielded
from the external heating, leading to $q\leq 0$. In contrast, the protostellar
cores might have higher $T_{\rm d}$ in more massive cores ($q>0$) because the
massive ones would preferentially become the birthplace for the high-mass
stars and/or multiple system. As a result, the relation would become steeper
than $L_{\rm core}\propto M_{\rm core}^{1.0}$. The core sample collected from
different regions might also be affected by the variation in $\beta$, which
could be more complicated and are not discussed here.

In fact, NGC 2024 would be distinguished from other samples by the presence of
strong external heating which have regulated cores into the similar
temperatures, and moreover, increased their absolute luminosities to be even
higher than the Class-0 sample. In such case, the absolute luminosities of the
individual cores would not effectively trace the evolutionary stages.
Similarly, the 70 $\micron$ emission would not indicate embedded protostars as
it works in cold isolated regions (e.g. Pineda et al., 2011). In particular,
the hot ridge has $T_{\rm d}\simeq 55$ K and strong 70 $\micron$ emission, but
should be solely illuminated by the external radiation and have no star
formation at all. In comparison, as a confirmed protostellar core, FIR-4 only
have $T_{\rm d}=22$ K. It could be heated to much higher temperature if
located closer to IRS-2b. In this case, the internal emission can be hardly
discerned.

###  6.2 The Dependence of Dust Temperature on the Stellar Radiation

We made a semi-quantitative estimation for the dependence of $T_{\rm dust}$ on
the stellar radiation. If IRS-2b has a spectral type of B0V, it would provides
an ultraviolet (UV) radiation with photon flux of $N_{FUV}=10^{48}$ s-1
(Stahler & Palla, 2005, Chapter 15 therein). If there is no strong absorbers
around the star, the UV radiation at a distance $r$ can be approximated as

$\chi(r)=\frac{\overline{E}N_{FUV}}{4\pi r^{2}}.$ (8)

We adopted $\overline{E}=9$ eV as the average photon energy for the UV
radiation field. For a dust core with high opacity in optical ($A_{V}\gg 1$)
in thermal equilibrium, the dust temperature would be (Li et al., 2003)

$T_{\rm d}=12.0\times\left[\frac{\chi(r)}{\chi_{0}}\right]^{1/6}~{}{\rm K},$
(9)

where $\chi_{0}=2.0\times 10^{-4}$ ergs s-1 cm-2 sr-1 is the unit intensity
for the interstellar radiation field (Draine, 1978). With the two equations
joined together, $T_{\rm d}$ would be related with $N_{FUV}$ and $r$ in the
form

$\displaystyle T_{\rm d}(r)$
$\displaystyle=12.0\left(\frac{\overline{E}N_{FUV}}{4\pi\chi_{0}}\right)^{1/6}r^{-1/3}$
(10) $\displaystyle=110~{}{\rm K}\times\left(\frac{\overline{E}}{9~{}{\rm
eV}}\right)^{\frac{1}{6}}\times\left(\frac{N_{FUV}}{10^{48}~{}{\rm
s^{-1}}}\right)^{\frac{1}{6}}\times\left(\frac{r}{\rm
0.1~{}pc}\right)^{-\frac{1}{3}}.$

The modelled $T_{\rm d}(r)$ function is shown in Figure 7. The $T_{\rm d}(r)$
curves for B1 and O9 stars are also presented for comparison, with the range
between them highlighted in yellow area. The data for the FIR cores and the
five positions on the ridge are presented on the diagram. Their $r$ values
represent the projected distances from IRS-2b.

At the observed distances, the model predicts a range of $T_{\rm d}\simeq
55-70$ K, which reasonably agrees with the temperature on the hot ridge, but
is completely deviated from the core temperatures (17-22 K). The low
temperatures in the FIR cores may indicate the actual distances from IRS-2b
being much larger than the projected values as seen on the image. In
comparison, the hot ridge should have a small uncertainty in its distance
because it mainly traces the PDR near IRS-2b. The data points for the ridge
suggests that the modeled $T_{\rm d}(r)$ function for B0 star can reasonably
describe the dust heating in NGC 2024. Based on the model, the temperatures of
FIR cores would indicate a distance of $r\simeq 5$ pc. As a result, the
filament should be well separated from the hot region around IRS-2b,
especially considering the fact that the hot gas nebula only has a spatial
extent of 0.2-0.4 pc as shown in the MIPS 24 $\rm\mu m$ emission.

In the case that the intervening absorption between IRS-2b and the filament is
not negligible, the U-V radiation would be transferred to longer wavelengths.
In this case the $T_{\rm d}(r)$ profile can be evaluated using another dust
heating model (Scoville & Kwan, 1976; Garay & Lizano, 1999; Wang et al.,
2012):

$T_{\rm d}(r)=71\left(\frac{\rm
0.1~{}pc}{r}\right)^{2/(4+\beta)}\left(\frac{L_{\rm
star}}{10^{5}~{}L_{\odot}}\right)^{1/(4+\beta)}\left(\frac{0.1}{f}\right)^{1/(4+\beta)},$
(11)

where the dust opacity index is adopted as the value for the hot component
($\beta=1.6$). The luminosity is adopted as the total luminosity of the main-
sequence B0 star that is $L=5\times 10^{5}~{}L_{\odot}$ (Stahler & Palla,
2005). $f=0.08$ cm g-1 is the dust emissivity at $\lambda=50~{}\micron$ for a
condition of $\tau_{100\rm\mu m}=0.1$ and $n\sim 10^{4}$ ${\rm cm^{-3}}$
(Scoville & Kwan, 1976). The derived $T_{\rm d}(r)$ relation is plotted with
thick gray line in Figure 7. This $T_{\rm d}(r)$ curve is only slightly bellow
the UV radiation model for the B0 star, and is also suggestive of a large
distance for the FIR cores, i.e. 2 to 3 pc from IRS-2b.

As another possibility for the low temperature, the filament can have a strong
self-shielding due to its higher opacity, thus can also maintain a low
temperature even if located close to IRS-2b. However, if the filament is near
the ridge, its outer layer would also be heated to a similar temperature with
the ridge, and then exhibit a hot component in its SED in addition to the cold
one. In fact, the FIR cores are much fainter than the ridge at $\lambda\leq
100~{}\rm\mu m$, and their SEDs are completely dominated by the cold
component. We thus conclude that the filament should indeed have a large
distance from IRS-2b. The previous studies showed that the dense molecular gas
should be located behind the Hii region and PDR (Roshi et al., 2014). The
current data further shows that the hot and cold components are actually
spatially separated rather next to each other. The expected spatial layout for
the filament and hot gas around IRS-2b are shown in Figure 7b.

##  7 Summary

We present an observational study with dust continuum and high-resolution
${\rm NH_{3}}$ line emissions to reveal the internal structures and physical
properties of the FIR cores in NGC 2024 filament. A particular focus of this
paper is the most massive core FIR-3 therein. The main results are:

1\. All the FIR cores in the filament are found to have low dust temperatures
between 17.5 and 22 K. FIR-3 has a compact morphology, dust temperature of
17.5 K, and no significant IR sources. The physical properties are suggestive
of an evolutionary stage younger than Class-0.

2\. In FIR-3, the ${\rm NH_{3}}$ line emissions exhibit a compact temperature
rise feature with a peak value of $T_{\rm k}=25$ K above the surrounding level
of $T_{\rm k}=15-19$ K. From the $T_{\rm k}$ peak, the central YSO is
estimated to have an accretion luminosity of $\Delta L\sim 50~{}L_{\odot}$.
The temperature rise without infrared counterpart suggests an ongoing or
lately occurred collapse from FHSC to Class-0 protostar.

3\. The FIR cores in NGC 2024 filament have comparable or even higher
luminosities than the Class-0 cores in the same mass range, but mostly due to
the external heating from IRS-2b. On the mass-luminosity diagram, the FIR
cores (except FIR-4) exhibit a correlation with the power law index ($p=1.0$)
comparable to the prestellar sample. For each core, the internal stellar
heating appears weak and has no significant influence on temperature or
luminosity.

4\. The FIR cores have much lower dust temperatures than the hot ridge (56 K).
The difference in their $T_{\rm d}$ suggests that the filament should be much
more distant from the central heating star IRS-2b. Comparison with the dust
heating models shows that the filament should be separated from the hot ridge
by $D=5$ pc along the line of sight. The filament and ridge also have distinct
dust opacity indices of $\beta\simeq 2.7$ and 1.6, respectively.

The observed physical properties suggest an overall cold dense state in the
filament. And FIR-3 could be an intermediate-mass core with an YSO possibly at
a transition stage from the FHSC to Class-0. Further high-resolution
observations (e.g. with ALMA) are expected to resolve its internal structures
and reveal the details of the collapse and the accretion state.

## Acknowledgment

We express our gratitude to Dr. A. Traficante for kindly providing a detailed
instruction for the usage of Hyper. We thank the anonymous referee for the
careful inspection of the manuscript and constructive comments. This work is
supported by the China Ministry of Science and Technology under State Key
Development Program for Basic Research (973 program) No. 2012CB821802, the
National Natural Science Foundation of China No. 11403041, No. 11373038, No.
11373045, Strategic Priority Research Program of the Chinese Academy of
Sciences, Grant No. XDB09010302, and the Young Researcher Grant of National
Astronomical Observatories, Chinese Academy of Sciences. This research has
made use of ”Aladin sky atlas” developed at CDS, Strasbourg Observatory,
France.

Table 1: The positions and sizes of the of the cores as fitted by Hyper.

Object |  position at 450 $\rm\mu m$ |  $70\rm\mu m$ offseta |  $160\rm\mu m$ offseta | $r_{\rm maj}^{b}$ | $r_{\rm min}^{b}$ |  pab | $r_{\rm core}^{c}$
---|---|---|---|---|---|---|---
| (RA,Dec) | (arcsec) | (arcsec) | (arcsec) | (arcsec) | (degree) | (arcsec)
FIR-1 | 05:41:41.40 -01:53:47.0 | unidentified | (1.4,-3.6) | 12 | 11 | 126 | 10/8.9
FIR-2 | 05:41:42.50 -01:54:06.0 | unidentified | (0.7,-1.8) | 16 | 9 | 157 | 9/7.8
FIR-3 | 05:41:43.00 -01:54:24.0 | unidentified | (-0.4,-2.4) | 15 | 10 | 146 | 10/8.9
FIR-4 | 05:41:44.00 -01:54:42.0 | (0.36,-4.68) | (0.4, -4.3) | 13 | 9 | 147 | 9/7.8
FIR-5a | 05:41:44.10 -01:55:40.0 | (1.8,8.64) | (2.9,-2.5) | 14 | 10 | 269 | 11/10.0
FIR-5b | 05:41:44.80 -01:55:32.0 | (0.36,-2.16) | (0.4,-4.7) | 13 | 10 | 107 | 12/11.1
FIR-6 | 05:41:45.00 -01:56:01.0 | (1.44,-2.16) | (1.8,-2.4) | 16 | 11 | 157 | 12/11.1
FIR-7 | 05:41:44.90 -01:56:15.0 | unidentified | (2.5,6.5) | 16 | 9 | 184 | 10/8.9

$a.$ The RA and Dec offset at 70 and 160 $\rm\mu m$ bands relative to the core
center positions at 450 $\rm\mu m$. The cores unidentified by Hyper are not
shown.

$b.$ The major, minor axes, and position angles (pa). The pa values are
counter-clockwise to the north.

$c.$ The best-fit core radius assuming a circular shape for the core area at
450 $\rm\mu m$. The second value is the radius deconvolved with the beam size,
i.e. $r_{\rm core}^{2}=r_{\rm obs}^{2}-(\theta_{\rm FWHM}/2)^{2}$, wherein
$\theta_{FWHM}=9^{\prime\prime}$ is the FWHM beam size.

Table 2: The maginitude of the IR sources associated with the cores in IRAC
bands and the flux densities of the cores in MSX bands.

Object | $M_{\rm 3.6\rm\mu m}^{a}$ | $M_{\rm 4.5\rm\mu m}^{a}$ | $M_{\rm 8.0\rm\mu m}^{a}$ | $S_{8\rm\mu m}$ | $S_{12\rm\mu m}$ | $S_{15\rm\mu m}$ | $S_{21\rm\mu m}$
---|---|---|---|---|---|---|---
| (mag) | (mag) | (mag) | (Jy) | (Jy) | (Jy) | (Jy)
FIR-1 | 9.09 | 8.48 | N | $<0.2$ | $<0.5$ | $<0.6$ | $<3.0$
FIR-2 | 9.90 | 9.17 | N | $<0.2$ | $<0.5$ | $<0.6$ | $<3.0$
FIR-3 | N | N | N | $<0.2$ | $<0.5$ | $<0.6$ | $<3.0$
FIR-4 | 8.69 | 7.09 | N | $<0.2$ | $<0.5$ | $<0.6$ | $<3.0$
FIR-5b | N | N | N | 43 | 50 | 65 | 325
FIR-6 | N | N | N | $<0.2$ | $<0.5$ | $<0.6$ | $<3.0$
FIR-7 | N | N | N | $<0.2$ | $<0.5$ | $<0.6$ | $<3.0$
Ridgec | N | N | N | 37 | 32 | 35 | 307

$a.$ The IRAC magnitudes in 3.6, 4.5 and 8.0 $\rm\mu m$ bands for the point
source possibly associated with the FIR cores, from Megeath et al. (2012). The
label ’N’ means non-detection, and would indicate $M_{3.6}<14.5$.

$b.$ FIR-5a and 5b are blended, thus the measured flux density should
represent a total value of the two. The emission from the hot ridge is also
included.

$c.$ In the IRAC 8 $\rm\mu m$ and all the MSX bands, the flux density is an
averaged value for the four box regions shown in Figure 3b. The four regions
are individually examined and found to have small differences in $T_{\rm d}$
($<\pm 2$ K).

Table 3: The continuum flux densities (in unit of Jy) of the cores and the
ridge in Herschel and SCUBA wavebands.

Object | $S_{70\rm\mu m}$ | $S_{100\rm\mu m}$ | $S_{160\rm\mu m}$ | $S_{250\rm\mu m}$ | $S_{350\rm\mu m}$ | $S_{500\rm\mu m}$ | $S_{450\rm\mu m}$ | $S_{850\rm\mu m}$
---|---|---|---|---|---|---|---|---
FIR-1 | 130 | 285 | 683 | 295 | 121 | 49 | 65 | 3.5
FIR-2 | 304 | 650 | 873 | 472 | 205 | 58 | 101 | 5.5
FIR-3 | 330 | 743 | 1282 | 694 | 298 | 80 | 156 | 7.0
FIR-4 | 1133 | 1620 | 1260 | 577 | 232 | 76 | 107 | 4.5
FIR-5a | 3571 | 3655 | 3951 | 1918 | 625 | 125 | 125/97 | 12.5
FIR-6 | 313 | 660 | 922 | 646 | 268 | 58 | 111 | 6.0
FIR-7 | 165 | 442 | 804 | 479 | 168 | 49 | 75 | 4.0
Ridgeb | 2376 | 1182 | 497 | 141 | 37 | 10 | 15 | 1.7

$a.$ FIR-5a and 5b are blended except at 450 $\micron$. The $S_{450\rm\mu m}$
values for the two cores are separately given, while $S_{\lambda}$ at other
bands represent the total value of the two. At $\lambda<100~{}\rm\mu m$, the
emission from the hot ridge is significant and would also contribute to the
total $S_{\lambda}$.

$b.$ In each band, the flux density represents an average value for the four
box regions as shown in Figure 3b. The four regions are individually examined
and found to have a small difference of $\Delta T_{\rm d}$ ($<2$ K).

Table 4: The physical parameters of the cores and the ridge.

Object | $T_{\rm dust}$ | $\beta$ | $N_{\rm tot}$ | $n^{c}$ | mass | $L_{\rm bol}$
---|---|---|---|---|---|---
| (K) |  |  (cm-2) |  (cm-3) |  ($M_{\odot}$) |  ($L_{\odot}$)
FIR-1 | $18.5(1.0)$ | $2.5(0.3)$ | $3.5(0.7)\times 10^{23}$ | $3.1(0.9)\times 10^{6}$ | $4.5(0.9)$ | 80 (40)
FIR-2 | $18.0(1.0)$ | $2.7(0.3)$ | $4.3(0.8)\times 10^{23}$ | $2.4(0.7)\times 10^{6}$ | $5.2(3.0)$ | 130(70)
FIR-3 | $17.5(1.0)$ | $2.6(0.3)$ | $7.4(0.9)\times 10^{23}$ | $1.2(0.3)\times 10^{7}$ | $10.0(4.0)$ | 220(80)
FIR-4 | $22.0(1.0)$ | $2.9(0.3)$ | $3.6(0.5)\times 10^{23}$ | $6.0(1.8)\times 10^{6}$ | $5.0(2.0)$ | 570(100)
FIR-5aa | $17.5(1.0)$ | $2.7(0.3)$ | $4.0(0.6)\times 10^{23}$ | $3.2(0.9)\times 10^{6}$ | $5.9(3.0)$ | 120(50)
FIR-5b | $17.5(1.0)$ | $2.7(0.3)$ | $3.7(0.5)\times 10^{23}$ | $2.6(0.7)\times 10^{6}$ | $4.5(2.0)$ | 90 (40)
FIR-6 | $18.5(1.0)$ | $2.7(0.3)$ | $3.8(0.5)\times 10^{23}$ | $1.8(0.4)\times 10^{6}$ | $5.8(2.0)$ | 160(40)
FIR-7 | $18.0(1.0)$ | $2.6(0.3)$ | $3.6(0.5)\times 10^{23}$ | $1.8(0.4)\times 10^{6}$ | $4.9(1.5)$ | 110(30)
ridgeb | $56(1.0)$ | $1.6(0.2)$ | $3.1(0.6)\times 10^{22}$ | $2.5(0.8)\times 10^{5}$ | $0.5(0.2)$ | 600(300)

$a.$ The dust temperature is fitted from the total flux densities for FIR-5a
and 5b.

$b.$ Each physical parameter represents an average for the four square regions
as shown in Figure 3b.

$c.$ For the dense cores, the density is derived from the peak total column
density and the mean radius. For the ridge, the density is derived from the
column density assuming the thickness is equal to the average width.

Table 5: The physical parameters of the ${\rm NH_{3}}$ gas condensations.

Object |  offseta | $T_{\rm k}$ | radius |  massb |  $N$(${\rm NH_{3}}$)
---|---|---|---|---|---
| (arcsec) |  | (arcsec) |  ($M_{\odot}$) |  ($10^{15}$ cm-2)
Cd-1 | (-3.0,7.0) | 18(2) | 3.5(0.5) | 1.2(0.8) | 3.0
Cd-2 | (-1.5,4.0) | 25(2) | 3.5(0.5) | 1.0(0.7) | 2.9
Cd-3 | (0,0) | 19(2) | 4.0(0.5) | 2.0(1.5) | 3.8
Cd-4 | (2.5,-3.5) | 17(2) | 4.0(0.5) | 0.8(0.5) | 2.7
Cd-n1 | (-7.5,20.5) | 16(2) | 3.5(0.5) | 0.6(0.4) | 2.3
Cd-n2 | (-6.5,14.5) | 14(2) | 3.5(0.5) | 0.3(0.2) | 1.7

$a.$ The RA and DEC offset from the Cd-3. Cd-3 is centered at RA=05:35:26.53,
Dec=-05:04:00.6. The condensations are shown in Figure 5b.

$b.$ The Cd-3 mass is calculated assuming its average column density is equal
to the peak total column density in FIR-3 (Table 4), i.e. $M=\pi R^{2}\mu
m_{0}N_{\rm tot,FIR3}$. The masses of the other condensations are calculated
by comparing with Cd-3 and assuming that mass is proportional to ${\rm
NH_{3}}$ (1,1)-isg integrated intensity, i.e. $M_{\rm Cd\textendash i}=M_{\rm
Cd\textendash 3}(S_{\rm Cd\textendash i}/S_{\rm Cd\textendash 3})$.

Figure 1: The continuum emissions in the NGC 2024 filament region. (a) The RGB
image of the IRAC 3.6, 4.5, and 8.0 $\micron$ bands. (b) The RGB image of the
MSX 12, 16.5, and 21 $\micron$ bands. (c) The MIPS 24 $\micron$ image, wherein
the blank white region is saturated. (d)-(k) the images in Herschel PACS 70,
100, and 160 $\rm\mu m$ bands, SPIRE 250, 350, and 500 $\rm\mu m$ bands,
JCMT/SCUBA 450 and 850 $\micron$ bands. For the Herschel and SCUBA images, the
intensity units are scaled to Jy arcsec-2 without changing the image
resolution or pixel size. In each panel, the SCUBA 450 $\rm\mu m$ emission is
plotted in contours. The contour levels are 10, 50, 90% of the peak value
(0.81 Jy arcsec-2). And the locations of IRS-2b and hot gas ridge are labeled.
Figure 2: The dense core extraction on the (a)-(b) 450, (c) 70, and (d) 160
$\rm\mu m$ images using the Hyper routine. The 450 $\micron$ contours are 10,
30, 50, 70, 90% of the peak value. In panel (a), each ellipse indicates the
core area fitted by Hyper, with the semi-axes equal to the FWHMs in 2D
Gaussian fit. In panel (b), the fitting is performed assuming each core to
have a circular shape. The 450 emission contours and the cores overlaid on the
IRAC RGB image. The yellow stars indicate the protostars identified in the
IRAC bands (Megeath et al., 2012). We note that at 70 $\rm\mu m$, the Core
FIR-5 is blended with the hot dust emission, thus the two sources should
represent the emission peaks rather than dense cores.
Figure 3: (a) The spectral energy distributions (SEDs) for the dense core
FIR-1, 2, 3, 4, 6, 7. In each panel, the black square with error bars are the
observed values, and the dashed line is the best-fit SED profile. The cores
are not clearly detected above the surrounding extended emissions in the IRAC
and MSX bands, the data points are thus shown with downward arrows to indicate
upper limits. (b) The SEDs for the five areas on the ridge. At each position,
the flux densities are measured within a $25^{\prime\prime}$-wide square. The
upper-right panel shows the SED at the central position that contains the
blended emission from FIR-5. And an SED with two components are used to fit
the data. The lower panel shows the averaged flux densities of the other four
positions. The dotted lines show the best-fit SED for 20 to 100 $\micron$ data
using a fixed value of $\beta=2.7$, in order to exhibit the deviation from the
best-fit SED at longer wavelengths. The MSX 16.5 and 21 $\micron$ data are
also adopted in fitting the hot dust component. The MSX 12 $\micron$ intensity
is above the SED curve and may indicate component with even higher
temperature, while the IRAC 8 $\micron$ emission may largely represent the PAH
component.
Figure 4: The VLA ${\rm NH_{3}}$ (1,1) and (2,2) spectra at the FIR-3 center.
In each panel, the black solid line and the red dashed line represent the
observed spectrum and the HFS fitting, respectively. The HFS is denoted on
each emission peak, wherein ”mg”, ”isg”, and ”osg” represent the main, inner-
satellite, and outer-satellite groups, respectively.
Figure 5: (a) The integrated ${\rm NH_{3}}$ (1,1) isg emission (black
contours) overlaid on the IRAC RGB image; the SCUBA 450 $\rm\mu m$ emission is
also presented in white contours. The levels are 10, 30, 50, 70, 90% of the
maximum. The yellow dashed ellipses denote the FIR core areas fitted by Hyper.
(b) The integrated ${\rm NH_{3}}$ (1,1)-isg emission (false color image and
contours). The contour levels are 4, 8, 12, 16, 20 times the rms level (0.6 K
km s-1). The white circles indicate the condensation areas fitted by Hyper.
(c) The (1,1)-isg emission (contours) overlaid on the (1,1)-mg emission
(false-color image). (d) The (1,1)-isg emission (contours) overlaid on the
integrated (2,2) emission (false-color image). (e) The $T_{\rm k}$ map
estimated from the (1,1) and (2,2) emissions. The contours are also (1,1)-isg
emission.
Figure 6: The bolometric luminosity as a function of core mass for the NGC
2024 cores (green triangles), Class-0 protostars (cyan squares, André 2000),
FHSC candidates (purple squares), prestellar cores (orange dots, Elia et al.
2013), and star less cores (red circles, Strafella et al. 2015). The dashed
lines represent the best-fit power-law relation for each sample. For NGC 2024,
FIR-4 is excluded from the relation of other cores. The solid and solid-dotted
grey lines represent the evolutionary tracks for the low- and high-mass YSOs,
with the arrow indicating the evolution direction. The FHSC candidates are
from: 1Belloche et al. (2006), 2Chen et al. (2010), 3Pineda et al. (2011),
4Enoch et al. (2010), 5Dunham et al. (2011), 6Pezzuto et al. (2012, two
cores), 7Friesen et al. (2014). The numbers for the literatures are labelled
with the data points.
Figure 7: (a) the dust temperature $T_{\rm d}$ under external radiation field
of a central heating star as a function of the stellar distance $r$ (Li et
al., 2003). The $T_{\rm d}(r)$ curves corresponding to the three different
stellar types are plotted with different lines for comparison. The ”second
model” refers to that of Scoville & Kwan (1976), with the relation expressed
as Equation (11). The $(T_{\rm d},r)$ data measured for the hot ridge and the
FIR cores in the filament are shown. For the data points, $r$ represents the
projected distance from IRS-2b. (b) A schematic view showing the spatial
layout of the hot-gas/ridge and the filament.

## References

  * Alves et al. (2011) Alves, F. O., Girart, J. M., Lai, S.-P., Rao, R., & Zhang, Q. 2011, ApJ, 726, 63
  * Alves et al. (2007) Alves, J., Lombardi, M., & Lada, C. J. 2007, A&A, 462, L17
  * André (2011) André, P. 2011, Structure Formation in Astrophysics
  * Andre et al. (2000) Andre, P., Ward-Thompson, D., & Barsony, M. 2000, Protostars and Planets IV, 59
  * André et al. (2010) André, P., Men’shchikov, A., Bontemps, S., et al. 2010, A&A, 518, L102
  * Arab et al. (2012) Arab, H., Abergel, A., Habart, E., et al. 2012, A&A, 541, A19
  * Belloche et al. (2006) Belloche, A., Parise, B., van der Tak, F. F. S., et al. 2006, A&A, 454, L51
  * Bik et al. (2003) Bik, A., Lenorzer, A., Kaper, L., et al. 2003, A&A, 404, 249
  * Boch & Fernique (2014) Boch, T., & Fernique, P. 2014, 485, 277
  * Bonnor (1956) Bonnor, W. B. 1956, MNRAS, 116, 351
  * Boss & Yorke (1995) Boss, A. P., & Yorke, H. W. 1995, ApJ, 439, L55
  * Bray (2014) Bray, E. M. 2014, in Astronomical Society of the Pacific Conference Series, Vol. 485, Astronomical Data Analysis Software and Systems XXIII, ed. N. Manset & P. Forshay, 343
  * Busquet et al. (2009) Busquet, G., Palau, A., Estalella, R., et al. 2009, A&A, 506, 1183
  * Chandler & Carlstrom (1996) Chandler, C. J., & Carlstrom, J. E. 1996, ApJ, 466, 338
  * Chen et al. (2010) Chen, X., Arce, H. G., Zhang, Q., et al. 2010, ApJ, 715, 1344
  * Choi et al. (2015) Choi, M., Kang, M., & Lee, J.-E. 2015, AJ, 150, 29
  * Draine (1978) Draine, B. T. 1978, ApJS, 36, 595
  * Dunham et al. (2011) Dunham, M. M., Chen, X., Arce, H. G., et al. 2011, ApJ, 742, 1
  * Ebert (1955) Ebert, R. 1955, Z. Astrophys., 36, 222
  * Elia et al. (2013) Elia, D., Molinari, S., Fukui, Y., et al. 2013, ApJ, 772, 45
  * Enoch et al. (2010) Enoch, M. L., Lee, J.-E., Harvey, P., Dunham, M. M., & Schnee, S. 2010, ApJ, 722, L33
  * Fontani et al. (2012) Fontani, F., Caselli, P., Zhang, Q., et al. 2012, A&A, 541, A32
  * Friesen et al. (2014) Friesen, R. K., Di Francesco, J., Bourke, T. L., et al. 2014, ApJ, 797, 27
  * Friesen et al. (2009) Friesen, R. K., Di Francesco, J., Shirley, Y. L., & Myers, P. C. 2009, ApJ, 697, 1457
  * Garay & Lizano (1999) Garay, G., & Lizano, S. 1999, PASP, 111, 1049
  * Gaume et al. (1992) Gaume, R. A., Johnston, K. J., & Wilson, T. L. 1992, ApJ, 388, 489
  * Goldsmith et al. (1997) Goldsmith, P. F., Bergin, E. A., & Lis, D. C. 1997, ApJ, 491, 615
  * Gómez et al. (2007) Gómez, G. C., Vázquez-Semadeni, E., Shadmehri, M., & Ballesteros-Paredes, J. 2007, ApJ, 669, 1042
  * Gong & Ostriker (2015) Gong, M., & Ostriker, E. C. 2015, ApJ, 806, 31
  * Hennebelle & Chabrier (2011) Hennebelle, P., & Chabrier, G. 2011, ApJ, 743, L29
  * Hildebrand (1983) Hildebrand, R. H. 1983, QJRAS, 24, 267
  * Hill et al. (2005) Hill, T., Burton, M. G., Minier, V., et al. 2005, MNRAS, 363, 405
  * Johnstone & Bally (1999) Johnstone, D., & Bally, J. 1999, ApJ, 510, L49
  * Kukolich (1967) Kukolich, S. G. 1967, Phys. Rev., 156, 83
  * Lai et al. (2002) Lai, S.-P., Crutcher, R. M., Girart, J. M., & Rao, R. 2002, ApJ, 566, 925
  * Larson (1969) Larson, R. B. 1969, MNRAS, 145, 271
  * Li et al. (2003) Li, D., Goldsmith, P. F., & Menten, K. 2003, ApJ, 587, 262 (Paper I)
  * Li et al. (2013) Li, D., Kauffmann, J., Zhang, Q., & Chen, W. 2013, ApJ, 768, L5
  * Li et al. (2007) Li, D., Velusamy, T., Goldsmith, P. F., & Langer, W. D. 2007, ApJ, 655, 351 (Paper II)
  * Masunaga & Inutsuka (2000) Masunaga, H., & Inutsuka, S.-i. 2000, ApJ, 531, 350
  * Mauersberger et al. (1992) Mauersberger, R., Wilson, T. L., Mezger, P. G., Gaume, R., & Johnston, K. J. 1992, A&A, 256, 640
  * Megeath et al. (2012) Megeath, S. T., Gutermuth, R., Muzerolle, J., et al. 2012, AJ, 144, 192
  * Menten et al. (2007) Menten, K. M., Reid, M. J., Forbrich, J., & Brunthaler, A. 2007, A&A, 474, 515
  * Mezger et al. (1992) Mezger, P. G., Sievers, A. W., Haslam, C. G. T., et al. 1992, A&A, 256, 631
  * Molinari et al. (2008) Molinari, S., Pezzuto, S., Cesaroni, R., et al. 2008, A&A, 481, 345
  * Myers (1983) Myers, P. C. 1983, ApJ, 270, 105
  * Ossenkopf & Henning (1994) Ossenkopf, V., & Henning, T. 1994, A&A, 291, 943
  * Pezzuto et al. (2012) Pezzuto, S., Elia, D., Schisano, E., et al. 2012, A&A, 547, A54
  * Pineda et al. (2010) Pineda, J. E., Goodman, A. A., Arce, H. G., et al. 2010, ApJ, 712, L116
  * Pineda et al. (2011) Pineda, J. E., Arce, H. G., Schnee, S., et al. 2011, ApJ, 743, 201
  * Preibisch et al. (1993) Preibisch, T., Ossenkopf, V., Yorke, H. W., & Henning, T. 1993, A&A, 279, 577
  * Ren et al. (2014) Ren, Z., Li, D., & Chapman, N. 2014, ApJ, 788, 172
  * Roshi et al. (2014) Roshi, D. A., Goss, W. M., & Jeyakumar, S. 2014, ApJ, 793, 83
  * Rosolowsky et al. (2008) Rosolowsky, E. W., Pineda, J. E., Foster, J. B., et al. 2008, ApJS, 175, 509
  * Salji et al. (2015) Salji, C. J., Richer, J. S., Buckle, J. V., et al. 2015, MNRAS, 449, 1769
  * Sandstrom et al. (2007) Sandstrom, K. M., Peek, J. E. G., Bower, G. C., Bolatto, A. D., & Plambeck, R. L. 2007, ApJ, 667, 1161
  * Scoville & Kwan (1976) Scoville, N. Z., & Kwan, J. 1976, ApJ, 206, 718
  * Stahler & Palla (2005) Stahler, S. W., & Palla, F. 2005, The Formation of Stars
  * Stahler et al. (1980) Stahler, S. W., Shu, F. H., & Taam, R. E. 1980, ApJ, 241, 637
  * Strafella et al. (2015) Strafella, F., Lorenzetti, D., Giannini, T., et al. 2015, ApJ, 798, 104
  * Tafalla et al. (2004) Tafalla, M., Myers, P. C., Caselli, P., & Walmsley, C. M. 2004, A&A, 416, 191
  * Tomida et al. (2010) Tomida, K., Machida, M. N., Saigo, K., Tomisaka, K., & Matsumoto, T. 2010, ApJ, 725, L239
  * Traficante et al. (2015) Traficante, A., Fuller, G. A., Pineda, J. E., & Pezzuto, S. 2015, A&A, 574, A119
  * van der Tak et al. (2007) van der Tak, F. F. S., Black, J. H., Schöier, F. L., Jansen, D. J., & van Dishoeck, E. F. 2007, A&A, 468, 627
  * Velusamy et al. (2008) Velusamy, T., Peng, R., Li, D., Goldsmith, P. F., & Langer, W. D. 2008, ApJ, 688, L87
  * Wang et al. (2012) Wang, K., Zhang, Q., Wu, Y., Li, H.-b., & Zhang, H. 2012, ApJ, 745, L30
  * Watanabe & Mitchell (2008) Watanabe, T., & Mitchell, G. F. 2008, AJ, 136, 1947
  * Wiesemeyer et al. (1997) Wiesemeyer, H., Guesten, R., Wink, J. E., & Yorke, H. W. 1997, A&A, 320, 287
  * Young & Evans (2005) Young, C. H., & Evans, II, N. J. 2005, ApJ, 627, 293

##  Appendix A Appendix material

###  A.1 Using Hyper to measure the flux densities

The flux densities of the FIR cores are measured using Hyper (Traficante et
al., 2015). In each band, the core radius is convolved with the beam size,
namely $r_{\rm conv}=\sqrt{r_{\rm core}^{2}+r_{\rm beam}^{2}}$, in order to
rightly cover the observed core areas. These parameters are adopted from the
450-$\rm\mu m$ fitting results because the 450-$\rm\mu m$ band ideally trace
the cold dense cores. Hyper can subtract the background emission and separate
the overlapped cores using multi-gaussian fitting.

The flux errors are mainly determined by the three factors: (1) flux
calibration, (2) background emission subtraction, and (3) core area, i.e.
aperture size. The flux calibration has an uncertainty of 10% for PACS888see
PACS observer’s manual: http://herschel.esac.esa.int
/Docs/PACS/html/pacs_om.html and 5% for SPIRE999see SPIRE observer’s manual:
http://herschel.esac.esa.int/twiki/
bin/view/Public/SpireCalibrationWeb, 30% for SCUBA 450 $\rm\mu m$ and 10% for
SCUBA 850 $\rm\mu m$ (Johnstone & Bally, 1999). The background fitting depends
on its polynomial order. Varying the order from 0 to 3 would cause an
uncertainty of $\sim 10\%$ for the flux density. The core area depends the
sigma threshold adopted in the fitting. As it varies from 4 to 6 $\sigma$, the
core size would vary by $\sim 30\%$, causing the flux density to vary by $\sim
40\%$. Considering these factors, the flux error would be $\sim 70\%$ for
PACS, $\sim 60\%$ for SPIRE, $\sim 100\%$ for SCUBA 450 $\rm\mu m$ and $\sim
70\%$ for SCUBA 850 $\rm\mu m$. The flux densities in the Herschel bands are
also corrected for the photometric colour-correction factors (CC).

The CC is a function of the dust temperature 101010For PACS, see Müller,
Okumura & Klaas, PACS Photometer - Colour Corrections,
http://herschel.esac.esa.int/twiki/pub/Public/PacsCalibrationWeb
/cc_report_v1.pdf. For SPIRE, see SPIRE hand book,
http://herschel.esac.esa.int/Docs/SPIRE/html/spire_om.html. . The correction
includes two steps, which are performed iteratively: 1) doing the SED fitting
to determine the dust temperature for each core (see Section 4) and the
related CC; 2) using CC to correct the flux density of each core. The
temperature variation due to CC is small. The $T_{\rm d}$ values become
converged within two iterations. As shown in Section 4.1, the SED fitting
reveals two components with $T_{\rm d}\simeq 55$ K and $T_{\rm d}\simeq 17.5$
K. The cold component (FIR cores except FIR-5) has CC$\\_70=1.269$,
CC$\\_100=1.051$ and CC$\\_160=0.964$, and the hot component (ridge) has
CC$\\_70=0.982$, CC$\\_100=0.982$, and CC$\\_160=1.010$. The SPIRE bands only
trace the cold dust emissions and have CC$\\_250=0.94$, CC$\\_350=0.92$, and
CC$\\_500=0.89$.

For each band, we also measure the random noise level from the emission-free
region. The results are shown in Table A.1..

Table A1: The properties of the continuum data.

image data | rms noise | Absolute | resolution | pixel size |  peak intensitya |  average intensitya
---|---|---|---|---|---|---
|  mJy pix-1 | flux error | (arcsec) | (arcsec) |  mJy arcsec-2 |  mJy arcsec-2
IRAC $8~{}\rm\mu m$ | 0.2 | 3% | 2.0 | 0.6 | 489 | 150
MSX $12~{}\rm\mu m$ | 0.5 | 3% | 20 | 6 | 82 | 40
MSX $16.5~{}\rm\mu m$ | 0.4 | 5% | 20 | 6 | 87 | 52
MSX $21~{}\rm\mu m$ | 3.0 | 5% | 20 | 6 | 87 | 52
PACS $70~{}\rm\mu m$ | 5 | 10% | 9 | 2 | $12\times 10^{3}$ | $5.3\times 10^{3}$
PACS $100~{}\rm\mu m$ | 5 | 10% | 6.7 | 1.6 | $11\times 10^{3}$ | $2.0\times 10^{3}$
PACS $160~{}\rm\mu m$ | 6 | 10% | 11 | 3 | $6\times 10^{3}$ | $1.5\times 10^{3}$
SPIRE $250~{}\rm\mu m$ | 10 | 5% | 18 | 6 | $3\times 10^{3}$ | $0.8\times 10^{3}$
SPIRE $350~{}\rm\mu m$ | 10 | 5% | 25 | 10 | $1\times 10^{3}$ | $0.3\times 10^{3}$
SPIRE $500~{}\rm\mu m$ | 5 | 5% | 36 | 14 | $0.1\times 10^{3}$ | $0.2\times 10^{3}$
SCUBA $450~{}\rm\mu m$ | 10 | 30% | 9 | 2 | 810 | 400
SCUBA $850~{}\rm\mu m$ | 0.1 | 10% | 14 | 1 | 50 | 20

$a.$ The intensity units are coverted into mJy arcsec-2 in order to provide a
fair comparison for all the bands. The images are kept in original appearance
and not smoothed, re-gridded, or performed with any other computation. The
peak and average intensities represented the values estimated within the
observed region as shown in Figure 1.

###  A.2 The SED fitting

The SED fitting is based on the least square fit of the observed flux
densities, that is to look for minimum of

$\chi^{2}=\sum_{\lambda}[\log S_{\lambda}(T_{\rm d})-\log S_{\lambda,{\rm
obs}}]^{2},$ (A1)

$S_{\lambda}(T_{\rm d})$ is defined in Equation (1), and the free parameters
to fit are $T_{\rm d}$, $\beta$, and $M_{\rm core}$. The fitting is
implemented using the NonLinearLSQFitter in the Astropy package 111111Astropy
is a community Python package for astronomy (Bray, 2014).. The errors in
$\beta$, $T_{\rm d}$, and $M_{\rm core}$ are estimated by varying the data
points within the uncertainty range for the fitting. a $\delta T_{\rm
d}\simeq\pm 1$ K, $\delta\beta\simeq\pm 0.3$, and $50-60\%$ for $M_{\rm
core}$. In the case that the SED contains two temperature components with
comparable intensities, the wavelength range of $\lambda=160$ to 850 $\rm\mu
m$ is found to be still dominated by the cold component, while the
$\lambda=20$ to 70 $\rm\mu m$ is dominated by the hot component. In fact, the
two components are only blended around FIR-5. The other cores are dominated by
the cold dust while the emission out of the filament is dominated by the hot
dust.

###  A.3 The fitting of the ${\rm NH_{3}}$ hyperfine structures

In modelling the ${\rm NH_{3}}$ spectra, we first calculate the ${\rm NH_{3}}$
optical depth over the spectral frequency range

$\tau(\nu)=\tau_{0}\sum_{j=1}^{N}a_{j}\exp[-4\ln
2(\frac{\nu-\nu_{0}-\nu_{j}}{\Delta\nu})^{2}],$ (A2)

wherein $\nu_{j}$ and $a_{j}$ are the rest frequency and relative intensity
for each hyperfine component, respectively. $N$ is the total number of the
hyperfine transitions, and $\Delta\nu$ is the FWHM line width. The values of
$a_{j}$ and $N$ for (1,1) and (2,2) lines can be obtained from Kukolich
(1967). $\tau_{0}$ is the integrated optical depth of all the components. For
a given $(J,K)$ inversion transition, the modelled brightness temperature
$T_{\rm b}$ is (Friesen et al., 2009)

$T_{\rm b}(J,K)=\Phi\eta[J(T_{\rm ex})-J(T_{\rm bg})][1-\exp(-\tau_{\nu})],$
(A3)

where $\Phi$ is the beam filling factor for the emission region which is
adopted as $\Phi=1$. $\eta$ is the mean-beam efficiency which is also adopted
to be 1.0 if there is a flux calibrator. $T_{\rm ex}$ is the line excitation
temperature. $T_{\rm bg}=2.73$ K is the temperature of the cosmic microwave
background, $J(T)=(h\nu/k)[\exp(h\nu/kT)-1]^{-1}$ is the Planck-corrected
brightness temperature. Frequency $\nu$ is related to the radial velocity $v$
as $(\nu-\nu_{0})/\nu=(v-v_{\rm sys})/c$. The free parameters for the fitting
are $T_{\rm ex}$, $\nu_{0}$, $\Delta\nu$, and $\tau_{0}$. The best-fit
spectrum is obtained also using NonLinearLSQFitter.

The column density of the $(J,K)$ level is calculated following Equation (13)
in Rosolowsky et al. (2008):

$\displaystyle N(J,K)=$
$\displaystyle\frac{8\pi\nu_{0}^{2}}{c^{2}}\frac{g_{1}}{g_{2}}\frac{1}{A(J,K)}$
(A4) $\displaystyle\times\frac{1+\exp(-h\nu_{0}/kT_{\rm
ex})}{1-\exp(-h\nu_{0}/kT_{\rm ex})}\int\tau(\nu)d\nu,$

wherein $A(J,K)$ is the Einstein A coefficient. Using the partition function
Z,

$Z=\sum_{J}(2J+1)S(J)\exp\frac{-h[BJ(J+1)+(C-B)J^{2}]}{kT_{\rm rot}},$ (A5)

the total ${\rm NH_{3}}$ column density is $N(1,1)\times Z/Z(1,1)$, wherein
$B=298117$ MHz and $C=186726$ MHz are the rotational constants. The function
$S(J)$ accounts for the extra statistical weight of the ortho- over para-${\rm
NH_{3}}$ states, with $S(J)=2$ for $J=3$, 6, 9, … and $S(J)=1$ for all other J
values.

The previous studies (e.g. Tafalla et al., 2004) presented an empirical
formula for $T_{\rm k}$ as a function of $T_{\rm rot}$. But this relation
might be inaccurate at high temperatures ($T_{\rm k}>20$ K) and densities. To
accurately determine $T_{\rm k}$, we calculated the energy level distributions
using the Non-LTE radiative transfer model RADEX (van der Tak et al., 2007).
The input parameters includes $T_{\rm k}$, $n_{\rm tot}$, $N$(${\rm NH_{3}}$)
and $\Delta V$. We considered the $T_{\rm k}$ range from 15 to 30 K. The other
parameters are set to be the values measured in Cd-2. With the input
parameters, RADEX will estimate energy level population, and the theoretical
$T_{\rm rot}$ is determined by the ratio of $N(2,2)/N(1,1)$ assuming a
Boltzmann distribution:

$\displaystyle\frac{N(2,2)}{N(1,1)}=$
$\displaystyle\frac{g(2,2)}{g(1,1)}\exp\left(\frac{-\Delta E}{T_{\rm
rot}}\right),$ (A6)

The $T_{\rm rot}-T_{\rm k}$ relation can then be numerically sampled for a
series of $T_{\rm k}$ values in our range. The result is shown in Figure A1.
It shows that the derived relation for FIR-3 is very close to $T_{\rm
rot}=T_{\rm k}$. This is within our expectation since at such high densities
($10^{7}$ ${\rm cm^{-3}}$) the level population would be nearly thermalized.

Figure A1: The relation between the gas kinematic temperature $(T_{\rm k})$
and the rotational temperature $(T_{\rm rot})$ for the ${\rm NH_{3}}$ (1,1)
and (2,2) levels as numerically sampled by the RADEX program. The reference
line of $T_{\rm rot}=T_{\rm k}$ is also plotted.