11institutetext:  Department of Physics, P.O.Box 64, FI-00014, University of
Helsinki, Finland, 11email<EMAIL_ADDRESS>
# Filaments in the OMC-3 cloud and uncertainties in estimates of filament
profiles ††thanks: Herschel is an ESA space observatory with science
instruments provided by European-led Principal Investigator consortia and with
important participation from NASA.

M. Juvela 11 E. Mannfors 11

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. Filamentary structures are an important part of star-forming
interstellar clouds. The properties of filaments hold clues to their formation
mechanisms and their role in the star-formation process.

Aims. We compare the properties of filaments in the Orion Molecular Cloud 3
(OMC-3), as seen in mid-infrared (MIR) absorption and far-infrared (FIR) dust
emission. We also wish to characterise some potential sources of systematic
errors in filament studies.

Methods. We calculated optical depth maps of the OMC-3 filaments based on the
MIR absorption seen in Spitzer data and FIR dust emission observed with
Herschel and the ArTéMiS instrument. We then compared the filament properties
extracted from the data. Potential sources of error were investigated more
generally with the help of radiative transfer models.

Results. The widths of the selected OMC-3 filament segments are in the range
0.03-0.1 pc, with similar average values seen in both MIR and FIR analyses.
Compared to the widths, the individual parameters of the fitted Plummer
functions are much more uncertain. The asymptotic power-law index has
typically values $p\sim 3$ but with a large scatter. Modelling shows that the
FIR observations can systematically overestimate the filament widths. The
effect is potentially tens of per cent at column densities above $N({\rm
H}_{2})\sim 10^{22}$ cm-2 but is reduced in more intense radiation fields,
such as the Orion region. Spatial variations in dust properties could cause
errors of similar magnitude. In the MIR analysis, dust scattering should
generally not be a significant factor, unless there are high-mass stars nearby
or the dust MIR scattering efficiency is higher than in the tested dust
models. Thermal MIR dust emission can be a more significant source of error,
especially close to embedded sources.

Conclusions. The analysis of interstellar filaments can be affected by several
sources of systematic error, but mainly at high column densities and, in the
case of FIR observations, in weak radiation fields. The widths of the OMC-3
filaments were consistent between the MIR and FIR analyses and did not reveal
any systematic dependence on the angular resolution of the observations.

###### Key Words.:

ISM: clouds – dust, extinction – ISM: structure – Stars: formation – Stars:
protostars

##  1 Introduction

Filaments are an important structural element of the interstellar medium (ISM)
and are intimately linked to the formation of new stars (André et al., 2010;
Hacar et al., 2022). Star-forming filaments have been investigated with
observations of thermal dust emission, recently with data from the Herschel
Space Observatory in particular (Pilbratt et al., 2010). The measurements have
been used to estimate the typical column-density and mass-per-length values of
star-forming filaments, the width of filaments, and parametric representations
for the filament profiles. The questions of the universality of filament
widths and the reliability of filament-property estimates are still topical
(Howard et al., 2021; Panopoulou et al., 2022). In Herschel studies, for
clouds within distances of 0.5 kpc, the typical filament full-width at half
maximum (FWHM) widths are of the order of 0.1 pc, which could point to a
common formation mechanism, either through a single event or as a more
dynamical accretion process (Arzoumanian et al., 2019; Hacar et al., 2022).
However, some higher FWHM values have also been found in Herschel studies,
often correlated with larger source distances and thus lower linear resolution
(Hennemann et al., 2012; Panopoulou et al., 2022). The estimated filament
widths appear to be tied to the range of angular scales that are probed by the
observations, which is qualitatively in agreement with the idea of a
hierarchical structure of the ISM, even in filaments. Indeed, Herschel
filaments themselves often reside in even larger and often elongated
structures. On the other hand, observations at higher angular resolution, such
as with the Atacama Large Millimeter/submillimeter Array (ALMA)
interferometer, have resulted in much smaller values and, especially in case
of line observations, in the detection of ‘fibres’ with sizes a factor of
several below the canonical 0.1 pc value (Hacar et al., 2018; Schmiedeke et
al., 2021). The full picture of ISM structure can only be obtained by covering
a large range of size scales, which may also require the combination of
different tracers and observational methods.

The mass-per-length values and the shapes of the filament profiles are central
observational parameters. Star formation is concentrated in the most massive
filaments, and a clear majority of young stellar objects are found in super-
critical filaments (André et al., 2010; Könyves et al., 2015). The critical
value is typically calculated as $M_{\rm crit}=2c_{\rm s}^{2}/G$, according to
the idea of filaments as isothermal infinite cylinders (Stodólkiewicz, 1963;
Ostriker, 1964). The same model predicts for the filament profiles an
asymptotic behaviour $\rho(r)\propto r^{-p}$ with $p=4$. The observed profiles
tend to be shallower, $p\sim$2 (Arzoumanian et al., 2011), which might be
explained by deviations from isothermal conditions and the influence of
magnetic fields, external pressure, or the dynamic growth of filaments
(Fischera & Martin, 2012; Kashiwagi & Tomisaka, 2021). To arrive at any firm
conclusions, the exponent $p$ should be measured to a precision of a few
tenths. So far, most estimates have been based on observations of dust
emission, especially from the large Herschel surveys, but might suffer from
some bias caused by radial variations in dust temperature and opacity.

At high column densities, the mid-infrared (MIR) absorption provides an
interesting alternative for estimating column densities (Butler & Tan, 2012;
Kainulainen & Tan, 2013). First, it is independent from dust temperature
variations, which can influence the analysis of dust emission in different
ways depending on the presence of internal and external radiation sources.
Second, the MIR absorption should be less affected by dust evolution when
compared to the sub-millimetre and far-infrared (FIR) dust emission (Roy et
al., 2013; Juvela et al., 2015b). Third, the existing MIR observations,
especially with the Spitzer (Werner et al., 2004) and Wide-field Infrared
Survey Explorer (WISE) (Wright et al., 2010) satellites, can provide an
angular resolution ($\sim 2-6\arcsec$) that is better than that of Herschel
($\sim 18\arcsec$ at 250 $\mu$m) or even large ground-based single-dish radio
telescopes. However, MIR observations have their own complications, such as
the unknown level of foreground emission and the potential effects of nearby
or embedded radiation sources.

In this paper we examine Orion Molecular Cloud 3 (OMC-3) and its filaments
using FIR and MIR observations. OMC-3 is located in the Orion A cloud, at the
northern end of the integral-shaped filament (e.g. Stutz & Kainulainen, 2015).
We adopt for the cloud a distance of $d=400$ pc, which is within the
uncertainties of the estimates that were given by Großschedl et al. (2018)
based on Gaia observations. Their Table 1 summarises earlier distance
estimates for different parts of the Orion A cloud.

Orion A was mapped by the Herschel satellite at 70-500 $\mu$m wavelengths.
These data have been used in many studies of the dust properties, structure,
and star formation in the region (e.g. Roy et al., 2013; Lombardi et al.,
2014; Stutz & Kainulainen, 2015; Furlan et al., 2016). Sadavoy et al. (2016)
estimated values $\beta\sim 1.7-1.8$ for the dust opacity spectral index in an
area that included OMC-3. Individual cores exhibited a larger dispersion in
$\beta$ values, which could be related to dust evolution or to observational
effects from temperature gradients (cf. Juvela & Ysard, 2012; Juvela et al.,
2015a), while $\beta$ appears to be systematically lower at millimetre
wavelengths (Schnee et al., 2014; Lowe et al., 2022).

The filamentary structure of Orion A has been studied extensively with
observations of the dust continuum emission, dust polarisation, and molecular
and atomic lines, from extended structures down to milliparsec scales (e.g.
Kainulainen et al., 2017; Pattle et al., 2017; Wu et al., 2018; Kong et al.,
2018; Tanabe et al., 2019; Goicoechea et al., 2020; Salas et al., 2021). Hacar
et al. (2018) identified in combined ALMA and IRAM N2H+ line data narrow
fibres with $FWHM\sim 0.035$ pc. These structures were thus narrower than
typical $\sim$0.1 pc filaments seen in Herschel studies with lower-resolution
($\sim 20-40\arcsec$) continuum observations (e.g. Arzoumanian et al., 2011;
Rivera-Ingraham et al., 2016). Schuller et al. (2021) combined Herschel data
with ground-based continuum observations from the ArTéMiS
instrument111http://www.apex-telescope.org/instruments/pi/artemis/
ARchitectures de bolomètres pour des TElescopes à grand champ de vue dans le
domaine sub-MIllimétrique au Sol. at the Atacama Pathfinder Experiment (APEX)
telescope (Revéret et al., 2014; Güsten et al., 2006). Although the angular
resolution of the combined data was $\sim 8\arcsec$, the estimated filament
widths in Orion A were 0.06-0.11 pc, close to the earlier Herschel results and
larger than the fibres. For the filaments in the OMC-3 region in particular
(in both its eastern and western parts), the widths were $\sim$0.06 pc. The
OMC-3 filaments are dense, with line masses $\sim 200M_{\sun}\,{\rm pc}^{-}1$
(Schuller et al., 2021). Based on dust polarisation data from the HAWC+
instrument of the Stratospheric Observatory for Infrared Astronomy (SOFIA), Li
et al. (2022) concluded that the filaments are also magnetically
supercritical.

The OMC-3 region was analysed further by Mannfors et al. (2022) using a
combination of Herschel data and an independent set of ArTéMiS observations.
In this paper we compare these FIR data with a new analysis of the MIR
absorption seen in Spitzer data. With the help of radiative transfer models,
we also more generally investigate potential sources of systematic errors that
could bias our estimates of the filament masses and profiles, in the case of
both MIR and FIR observations.

The contents of the paper are as follows. In Sect. 2 we present the methods
that are used to derive column densities from observations of dust extinction
and emission, and in Sect. 2.3 we describe the procedures we use to fit the
filament profiles with an analytical function. Section 2.4 explains the
radiative transfer modelling that is used to study potential sources of bias
in the filament analysis. In Sect. 3 we present the main results, including
the analysis of the OMC-3 filaments and, in Sect. 4, the analysis of synthetic
filament observations. In Sect. 5 we discuss the observational results and the
systematic errors that may affect the OMC-3 data and, more generally,
observations of similar high-column-density filaments. The final conclusions
are listed in Sect. 6.

##  2 Methods

###  2.1 Mid-infrared absorption

Mid-infrared observations provide a way to measure cloud mass distribution,
provided that there is sufficient background surface brightness and the column
density is high enough to result in measurable MIR extinction. The total
observed surface brightness, $I^{\rm obs}$, towards the cloud depends on its
optical depth, $\tau$, as

$I^{\rm obs}=I^{\rm true}+\Delta I=I^{\rm fg,true}+(I^{\rm ext,true}-I^{\rm
fg,true})\times e^{-\tau}+\Delta I.$ (1)

We have explicitly included a correction, $\Delta I$, which is needed if one
does not have absolute surface brightness measurements and the zero point of
the intensity scale is thus uncertain. In the equation, $I^{\rm fg,true}$ is
the amount of true emission that originates in front of the source, and
$I^{\rm ext,true}$ is true value of the extended background, against which the
cloud is seen in absorption. The corresponding observed value of $I^{\rm
ext,obs}$ can be estimated by interpolating the observed surface brightness
over the target. The optical-depth can be then calculated as

$\tau=-\ln\left(\frac{I^{\rm true}-I^{\rm fg,true}}{I^{\rm ext,true}-I^{\rm
fg,true}}\right)=-\ln\left(\frac{I^{\rm obs}-I^{\rm fg,obs}}{I^{\rm
ext,obs}-I^{\rm fg,obs}}\right).$ (2)

This requires a separate estimate of the foreground component. In Butler & Tan
(2012), $I^{\rm fg}$ was estimated by assuming that parts of the target cloud
are so optically thick that none of the background radiation comes through.
The minimum surface brightness is then also an estimate for $I^{\rm fg}$.
Because $I^{\rm obs}$ and $I^{\rm fg,obs}$ are affected by the same correction
$\Delta I$, this correction term disappears, and the optical depth can be
estimated even with an arbitrary zero-point offset in the data. However,
unless peak optical depths are very high, the minimum surface brightness
provides only an upper limit of the true foreground component. In real
observations, the non-constant background and the effect of local radiation
sources add to the overall uncertainty.

###  2.2 Far-infrared dust emission

The observed FIR emission from the target cloud is

$I_{\nu}=\int B_{\nu}(T)e^{-\tau_{\nu}}d\tau_{\nu}$ (3)

and thus depends via the Planck function, $B_{\nu}$, on the dust temperature,
$T$, and optical depth, $\tau_{\nu}$, along the line of sight. In most
practical work, the medium is assumed to be homogeneous and optically thin,
which gives the simple relationship

$I_{\nu}=B_{\nu}(T)(1-e^{\tau_{\nu}})\approx B_{\nu}(T)\tau_{\nu}.$ (4)

The optical depth can be calculated from the modified blackbody (MBB) function
above and with further assumptions of the dust absorption coefficient
$\kappa_{\nu}$, optical depths can be converted to column density. Optical
depth is obtained from Eq. (4), once the dust temperature $T$ has been first
calculated by fitting the same Eq. (4) to multi-frequency observations. In the
optically thin case, the temperature determination does not depend on the
absolute value of the opacity $\kappa_{\nu}$, only on its frequency
dependence. At FIR wavelengths, this is usually written assuming a power-law
dependence,

$\kappa(\nu)=\kappa(\nu_{0})\,\,(\nu/\nu_{0})^{\beta}.$ (5)

We used either $\beta=1.8$ (OMC-3 observations) or $\beta=2.0$ (radiative
transfer models).

Column densities are estimated based on Herschel 160-500 $\mu$m data. In
OMC-3, with the assumed dust opacity, the column density reaches maximum
values above $N({\rm H}_{2})=10^{23}$ cm-2, which corresponds to a 160 $\mu$m
optical depth of $\tau=0.14$. The assumption of optically thin emission
therefore seems valid. There can be some optically thick regions, but only at
scales below the Herschel resolution. Saturation of short-wavelength emission
could lower the estimated colour temperatures and lead to higher column-
density estimates. However, even when optical depths are not negligible, the
use of the approximate form of Eq. (4) may still be preferred over the full
equation.

We calculated a low-resolution map (LR map) by convolving the Herschel data to
a common 41$\arcsec$ resolution and deriving a column-density map at the same
resolution. An alternative high-resolution map (HR map) is calculated
following Palmeirim et al. (2013). We use the convolution kernels presented in
Aniano et al. (2011), and the nominal resolution of the resulting column-
density map is 20$\arcsec$.

Mannfors et al. (2022) presented ArTéMiS observations of the OMC-3 field. The
350 $\mu$m map is centred at RA=5h35m20s Dec=-5$\degr$1$\arcmin$31$\arcsec$
(J2000), and it covers an area slightly larger than $9.3\arcmin\times
11.7\arcmin$ at an angular resolution of $8.5\arcsec$. The noise level is
$\sim$0.2 Jy beam-1, and the signal-to-noise ratio exceeds 100 in the
brightest part of the filament. The main filament is equally well visible in
the simultaneously observed ArTéMiS 450 $\mu$m map. However, because the 450
$\mu$m map is slightly smaller (not covering the filament segment A) and there
are no Herschel observations at this wavelength, the 450 $\mu$m data are not
used in this paper. The third column-density map is based on combined
(feathered222https://github.com/radio-astro-tools/uvcombine) Herschel and
ArTéMiS 350 $\mu$m surface brightness map. The map has a nominal resolution of
10$\arcsec$, although the temperature information is available only at a lower
resolution. In the following, we refer to this as the AR map.

The analysis with a single MBB is common, but an observed spectrum is never
going to precisely match any single MBB function, because $\beta$ and $T$ are
not constant in the clouds. The effects of the line-of-sight temperature
variations are well known (Shetty et al., 2009; Malinen et al., 2011; Juvela &
Ysard, 2012), and we return to these questions in Sect. 4.1 and in a future
paper. With a sufficient number of frequency channels and data with high
signal-to-noise ratio, the observations can be modelled as a sum of several
temperature components, thus reducing the bias associated with the single-
temperature assumption. Such an analysis could be more sensitive to other
assumptions, such as the dust opacity spectral indices. Nevertheless, methods
such as point process mapping (PPMAP) (Marsh et al., 2015) and inverse Abel
transform (Roy et al., 2014; Bracco et al., 2017) have been successfully
applied to many dust continuum observations. Of these, PPMAP sets fewer
requirements on the symmetry of the modelled object but is computationally
more expensive. Howard et al. (2019) compared the MBB and PPMAP methods for
filaments in the Taurus molecular cloud. The PPMAP method resulted in some 30%
decrease in the estimated filament FWHM values (Gaussian fits) and, perhaps
surprisingly, in a significant reduction in the estimated line masses.
However, that analysis also made use of the longer-wavelength SCUBA-2
observations, in addition to the Herschel 160-500 $\mu$m data.

###  2.3 Filament profile fitting

We fit filament profiles with a Plummer-type function:

$P(r;N_{0},R,p,\Delta r)=N_{0}[1+((r-\Delta r)/R)^{2}]^{(1-p)/2}$ (6)

(Whitworth & Ward-Thompson, 2001; Arzoumanian et al., 2011). Here $r$ is the
distance from the filament centre, in the direction perpendicular to the
filament path. In the case of OMC-3 data, the paths are defined by parametric
splines through a number of hand-picked points. The free parameters of the fit
are the peak column density, $N_{0}$, the size of the central flat part of the
profile, $R$, the power-law index, $p$, and the sideways adjustment, $\Delta
r$. The parameter $N_{0}$ is related to the central density of the filament
but also depends on the values of $R$ and $p$ as well as the scaling between
column density and mass (Arzoumanian et al., 2011). The parameter $\Delta r$
allows a shift if the local filament centre does not perfectly align with the
spline description of the filament. By allowing the shift $\Delta r$ in
individual profiles, one also avoids the artificial widening that would result
from imperfect alignment of the profiles.

The actual model fitted to OMC-3 data includes a linear background and the
final convolution of the model to the resolution of fitted column-density
data,

$F(r;N_{0},R,p,\Delta r,A,B)=\mathrm{Con}(P(r;N_{0},R,p,\Delta r))+A+B\cdot
r.$ (7)

The fit was done independently for each extracted 1D profile. This includes
the replacement of the 2D-convolution of an image with a 1D-convolution of
individual profiles. This is exact only if the 2D filament does not change
along its length, at the scale of a single beam. This also assumes that the
filament shape is close to Gaussian, because only in the case of a Gaussian
filament convolved with a Gaussian beam are the 1D and 2D results identical.
Ideally, one would build a 2D model of the entire region (with a global model
for the background as well) that would be convolved in 2D during the fitting.
However, the 1D approximation of Eq. (7) is sufficiently accurate and, of
course, is much faster to calculate (cf. Mannfors et al., 2022).

The fitting of Eq. (7) with six free parameters was done both with a normal
least-squares routine and with a Markov chain Monte Carlo (MCMC) routine, the
latter providing the full posterior probability distributions for the
estimated parameters. The MCMC fitting uses uninformative priors, except for
forcing positive values for $N_{0}$ and $R$. We also required $p>1$, since
$p=1$ corresponds to a flat column-density profile. Appendix A shows one test
on the expected accuracy of the parameter estimates as a function of the noise
level.

###  2.4 Radiative transfer models

We used a simple cloud model to study how the extracted filament parameters
might be affected by different error sources. In the case of FIR emission,
these include especially the line-of-sight temperature variations that cause
radially varying bias in the column-density maps. In the case of MIR analysis,
errors may be introduced by dust scattering and local thermal dust emission.

The cloud model was discretised onto a Cartesian grid of 2003 cells, with a
cell size of 0.0116 pc. The size of the cloud model is thus 2.32 pc or
20$\arcmin$ for a distance of 400 pc. The model consists of a single linear
filament, with a density profile matching the Plummer function with $R$=0.0696
pc and $p$=3. These correspond to a filament FWHM value of 0.14 pc or
72$\arcsec$ at the 400 pc distance. Tests are carried out with different
values of the filament column density, and observing the filament mostly in a
direction perpendicular to its main axis.

We used different dust models for the emission and scattering calculations.
FIR emission is calculated using the core-mantle-mantle (CMM) dust that is
included in the the Heterogeneous dust Evolution Model for Interstellar Solids
(THEMIS) model (Jones et al., 2013; Köhler et al., 2015; Ysard et al., 2016).
Although the dust model does not contain large aggregates or ice mantles, it
is appropriate for dense environments. As an example of a more evolved dust
populations, we use in some tests the THEMIS AMMI model, aggregate grains with
ice mantles.

Because the CMM model does not include small grains, the calculations of MIR
thermal dust emission were performed with the Compiègne et al. (2011) dust
model (in the following, COM), which is appropriate for a diffuse medium. The
thermal emission should come mainly from outer filament layers that consist of
relatively pristine material. This is not necessarily true for the MIR
scattering, which originates in regions where the optical depth at MIR
wavelengths, rather than at optical wavelengths reaches unity. Dust scattering
is therefore also calculated using the CMM dust model. Differences in the
scattering properties of different dust models are discussed, for example, in
Ysard et al. (2016) and Juvela et al. (2020).

The model filament is illuminated by an isotropic background according to the
Mathis et al. (1983) model of the interstellar radiation field (ISRF). In part
of the calculations, we included an additional point source, which was
modelled as a $T=15700$ K blackbody with a total luminosity of 590 solar
luminosities (similar to a B5V star). The source was placed at distance of
0.23 pc from the top end of the filament (in the orientation used in the
figures) and at distance of 0.93 pc from the filament axis. The viewing angle
was varied such that the source is in front of the filament, behind the
filament, or to the left of the filament. Since the amount of scattered light
is directly proportional to the illumination, these results can be easily
scaled for any source luminosity.

The radiative transfer calculations were performed with the SOC program
(Juvela, 2019), which gives 200$\times$200 pixel maps of the dust emission at
8, 160, 250, 350, and 500 $\mu$m and maps of scattered light at 8 $\mu$m. The
pixel size matches the model resolution and is 0.0116 pc or 6$\arcsec$ for the
assumed 400 pc distance. Since the effect of stochastic grain heating is small
at long wavelengths, the 160-500 $\mu$m dust emission was calculated assuming
equilibrium between the radiation field and the grain temperatures. The
stochastic nature of the grain temperatures was naturally taken into account
in the calculations of the MIR dust emission. The SOC program is based on
Monte Carlo simulations. The number of simulated photon packages was selected
so that the noise is a fraction of one per cent in the FIR maps and $\sim 1\%$
or less in the computed maps of scattered light (pixel-to-pixel).

The FIR surface brightness maps were further convolved with Gaussian beams. In
most tests, this was set to $FWHM$=24$\arcsec$ and the column-density maps
were calculated at the same resolution. The simulated MIR data were used at
the full model resolution, because the model pixels are already larger than,
for example, the Spitzer resolution at distances below 1.2 kpc.

##  3 Results from OMC-3 observations

###  3.1 OMC-3 filament in MIR absorption

We used Eq.(2) and Spitzer data to calculate an 8 $\mu$m optical-depth map for
the OMC-3 field. We started by creating a mask for those filament regions that
are clearly visible in absorption (Fig. 1a). The extended component $I^{\rm
ext}$ was calculated by replacing the masked pixels with interpolated values.
In practice, this was done by convolving the map with a Gaussian beam with
$FWHM$=40$\arcsec$, where the convolution ignored as inputs all pixels inside
the filament mask or inside the manually created masks for point sources (Fig.
1b-c). Comparison of MIR and Herschel data showed that the MIR surface
brightness rises towards the north, and the MIR data are affected by local
radiation sources, whose effect changes rapidly as a function of the sky
position. In Fig. 1b, the masks have already been reduced to an area, where
the relationship between the MIR absorption and the FIR-based column densities
appeared consistent. The increased MIR emission towards the equatorial north
is caused mainly by the NGC 1977 open cluster (its northern sub-cluster), but
the closest B stars are still some 4 arcmin or 0.5 pc (projected distance)
north of the area included in Fig. 1 (Getman et al., 2019; Megeath et al.,
2022).

The level of foreground emission is not known. Butler & Tan (2012) estimated
the quantity corresponding to $I^{\rm fg}+\Delta I$ statistically as

$I^{\rm fg}+\Delta I=\langle I^{\rm obs}(I^{\rm obs}<I^{\rm
obs,min}+2\sigma)\rangle-2\sigma,$ (8)

where $I^{\rm obs,min}$ is the minimum observed surface brightness and
$\sigma$ is the noise. In the OMC-3 region, the extended surface brightness,
and therefore probably also the foreground component, increases strongly
towards both north and south. We set $I^{\rm fg}$ equal to the minimum
observed surface brightness, which is found close to the centre of the area
shown in Fig. 1. This may lead to some overestimation of the $\tau$ values at
that position (and a few undefined values), which needs to be taken into
account in the subsequent analysis. On the other hand, the selected value may
underestimate the foregrounds further in the south and the north.

The calculated map of the 8 $\mu$m optical depth is shown in Fig. 1d. Our main
interest is in the shape of the filaments, and scaling of $\tau$ to column
density is not needed. However, with a 8-$\mu$m dust opacity of 7.5 cm2 g-1,
the values obtained for the clean parts of the filaments (without strong
point-source contamination) are within a factor of $\sim$2 of those derived
from the Herschel data (when compared at 20$\arcsec$ resolution). This scaling
is used in some subsequent plots where column-density units are used.

Figure 1:  MIR observations of the OMC-3 field. Frame (a) shows the observed 8
$\mu$m surface brightness. Frame (b) shows the masks used, where the value 1
corresponds to the chosen filament region (shown with dashed contours in the
other frames) and the value -1 corresponds to masked stars. Frame (c) shows
the estimated extended emission, $I^{\rm ext}$, and frame (d) the resulting 8
$\mu$m optical depth. Frame (a) also indicates the four filament fragments
that were chosen for further analysis (solid white lines with labels A-D).
Figure 2:  Two-dimensional images of the four OMC-3 filament segments that are
marked in Fig. 1a. Each frame shows one segment that is extracted from the
column-density map so that the filament runs vertically at the centre of each
frame. The green areas correspond to pixels that are masked because of point
sources.

Figure 1 shows the MIR data and the derived 8 $\mu$m optical depth map. The
strongest MIR absorption is found within a y-shaped region, which can be
interpreted as one main filament on the west side and one side filament
extending towards the north-east. For further analysis, we selected from this
area four isolated filament segments that show the deepest MIR absorption
(i.e. correspond to the highest column densities) and are not significantly
contaminated by emission from embedded or nearby stars. The segments are
labelled with letters A-D in Fig 2a.

The column-density maps of the segments were then extracted as 2D images,
where the filaments are aligned vertically. We fitted each individual row as
well as the median profile of each filament segment with Eq. (7). The model
includes a linear background, the Plummer profile with a possible shift in the
direction perpendicular to the filament length, and convolution with a
Gaussian beam with FWHM=2.0$\arcsec$ (approximating the Spitzer beam size).
The results are shown in Fig. 3 for the filament segment A, based on data
within [-90$\arcsec$, +90$\arcsec$] of the filament centre. Corresponding
plots for the three other segments can be found in Appendix B.

Figure 3:  Results of the Plummer fits to OMC-3 filament segment A that was
measured using MIR extinction. Frame (c) shows the filament column densities
(units $10^{21}$ cm-2) as a 2D image, the filament running vertically in the
plot. The top row contains the median profile. Frame (a) shows the fit
residuals. Frame (d) shows the parameter estimates along the filament. The
shaded regions correspond to different percentile ranges of MCMC samples: [1,
99]% in dark grey, [10, 90]% in light grey, and [25, 75]% in red. The MCMC
median values are plotted as solid white curves. The values from separate
$\chi^{2}$ minimisation are plotted with dashed black curves. The vertical
dashed light green lines (frame b and frame d) show the median of the
parameter values in the individual least-squares fits, while the white half-
circles at the top of frame (d) correspond to the fit to the median profile.
Frame (b) shows histograms of the parameter distributions based on MCMC
samples over all profiles.

The figures also show the filament FWHM values that were calculated from the
parameters of the Plummer fits. The $FWHM$ values are usually much more robust
than the values of the individual parameters $R$ and $p_{0}$ (Suri et al.,
2019). There is no significant correlation between the fitted filament column
density (parameter $N$) and the FWHM. If the filament is not well defined (or
well matched by the assumed functional form), the parameter $N_{0}$ does not
accurately represent the central column density, because of partial degeneracy
with the fitted linear background. The median FWHM value of the filament is
$\sim$0.05 pc.

###  3.2 OMC-3 filaments in FIR emission

For comparison with the MIR results, we analysed the column-density maps
obtained from Herschel and the combined Herschel and ArTéMiS data. We
extracted column densities for the same short filament segments as in the MIR
analysis and fitted these with the model of Eq. (7).

We used the LR and HR column-density maps estimated with Herschel 160-500
$\mu$m data and the AR map that also makes use of ArTéMiS data. Figures 4 and
5 show the results for the first filament segment A, using the HR column-
density map (angular resolution 20$\arcsec$) and the AR map (angular
resolution 10$\arcsec$). The corresponding plots for the other filament
segments are shown in Appendix C. The values of $p$ are of the order of
$\sim$3, but with significant scatter. The higher-resolution maps result in
lower FWHM values, with the exception of segment A.

Figure 4:  Plummer fits of OMC-3 filament segment A, using the HR column-
density map (angular resolution 20$\arcsec$). Frame (c) shows the filament
segment as a 2D image (the top row containing the median profile), frame (a)
the fit residuals, frame (d) the parameter estimates along the filament, and
frame (b) the parameter histograms (cf. description in Fig. 3). The column-
density map has an angular resolution of 20$\arcsec$, and the fitted area is
[-90$\arcsec$, +90$\arcsec$] in the cross-filament direction.  Figure 5:
Plummer fits of OMC-3 filament segment A. The figure is the same as Fig. 4 but
uses the AR column-density map that is based on combined Herschel and ArTéMiS
data and has an angular resolution of 10$\arcsec$.

The above results apply to fits to data within $|r|<90\arcsec$ of the filament
centre. To test the sensitivity to the extent of the fitted region, the
analysis was repeated by varying the maximum distance between $r_{\rm
max}=60\arcsec$ to $r_{\rm max}=210\arcsec$. Figure 6 shows the resulting
median parameter values for the four filament segments. In addition to the HR
and AR data (20$\arcsec$ and 10$\arcsec$ resolutions), we include here the
results using the LR (41$\arcsec$ resolution) column-density map.

The FWHM values obtained from different column-density maps are relatively
consistent. The results are similar for the LR and AR maps, and HR data result
in only slightly lower values. We also repeated the LR and HR analysis using
the column-density maps provided by the Gould Belt Survey, where the maps have
resolutions of 36.3$\arcsec$ (Roy et al., 2013) and 18.2$\arcsec$ (Polychroni
et al., 2013). The 36.3$\arcsec$ resolution maps showed no noticeable
differences to our results with the LR map. The 18.2$\arcsec$ maps resulted in
slightly higher FWHM values that match our LR and AR results more closely.
These small differences could be caused by differences in the background
subtraction (which in the case of the HR map was done close to the filament),
the convolution kernels, and even the assumed $\beta$ values ($\beta=2$ in
Polychroni et al., 2013).

One clear outlier is segment A in the AR map (Fig. 6g), where the $FWHM$ value
increases with increasing $r_{\rm max}$. However, the values appears to be
affected by the masked area, which corresponds to the edge of the ArTéMiS
coverage (Fig. 5). In the better defined end of the segment, the FWHM values
also drop in segment A close to the general $\sim 0.05$ pc level. Segment B
also shows larger FWHMs in the LR and AR maps than in the HR map. The widths
are smaller in the high-density end and larger in the low-density end, the
median in this case picking the larger value.

The MIR data result in FWHM estimates that are even surprisingly close to the
values derived from dust emission. However, while all emission maps give
values $p\sim 3$ for the power-law index, the MIR results show a much larger
scatter. The values are high ($p>4$) for the B and C segments, and, as shown
in Figs. 20\- 21, the values are consistently high along the entire filament
segments.

Figure 6:  Parameters $p$ (upper frames) and FWHM (lower frames) in fits to
OMC-3 LR, HR, and AR maps (based on dust emission) and the MIR maps. Each
frame shows results for the four filament segments, A-D, the symbols
corresponding to median and the inter-quartile range of the least-squares
parameters over the filament length.

##  4 Analysis of synthetic filament observations

In this section we use the cloud model of Sect. 2.4 to examine sources of
potential bias in the dust emission and extinction observations. With the
parameters used in the simulations (0.0116 pc pixels, $R=6$ pixels, and
$p=3.0$), the model filament has a FWHM size of 0.14 pc or some 72$\arcsec$ at
a distance of 400 pc (Sect. 2.4). The synthetic observations were made using a
Gaussian beam with $FWHM=24\arcsec$. The beam size and filament properties are
roughly similar to those found in some Herschel studies (e.g. Arzoumanian et
al., 2011; Rivera-Ingraham et al., 2016; Panopoulou et al., 2022) but the
simulations are not intended to directly replicate the OMC-3 observations. In
particular, we examine a wider range of models with maximum column densities
ranging from $N({\rm H}_{2})=10^{21}$ cm-2 to $N({\rm H}_{2})=10^{24}$ cm-2.

###  4.1 Bias in FIR observations

We look first at FIR observations of a filament illuminated by an isotropic
external radiation field. Figure 7 shows the true optical depth profiles and
the profiles estimated from synthetic surface brightness maps for filaments of
different column density.

The results are almost correct up to $N({\rm H}_{2})\sim 10^{22}$ cm-2,
although $\tau$ is increasingly underestimated. When the column density
reaches $N({\rm H}_{2})=3\cdot 10^{22}$ cm-2, the estimated peak $\tau$ is
some 75% of the correct value and $p$ is overestimated by $\sim$4%. The
filament FWHM is overestimated by $\sim$30%, and the effect increases with
increasing column density. The results were not sensitive to the assumed beam
size.

The MBB fits were made with $\beta=2.0$, which is close to the actual value in
the dust model. At $N({\rm H}_{2})=10^{22}$ cm-2, the use of $\beta=1.8$ or
$\beta=2.2$ would change the column-density estimates by -22% or +27%,
respectively, while the FWHM is affected only at the $\sim$1% level.

Figure 7:  Profiles of optical depth, $\tau$, and dust temperature in the case
of model filaments illuminated by an isotropic external field. The black
curves show the true optical-depth profiles and the dashed red curves (fully
overlapping the black curves) the Plummer functions fitted to those profiles.
The blue and dashed cyan lines are, respectively, the optical depth profile
derived from synthetic 160-500 $\mu$m surface brightness observations and the
Plummer fit to those data. The values of $p$ and FWHM (in parsecs, in the plot
marked as $\theta$) are shown for both the true profile (left side, black
font) and the estimated (right side, blue font) $\tau$ profiles. The analysis
assumes $\beta=2.0$, but FWHM values for $\beta=1.8$ and $\beta=2.2$ are also
shown in cyan and magenta, respectively. The assumed beam size is 24$\arcsec$.
The dust temperature profiles (solid red curves) are cross-sections of the 3D
model and are shown at full model resolution.

Figure 8 shows how the bias in the $p$, $R$, and FWHM estimates increases with
column density. For the normal ISRF and fits to the $|r|<0.58$ pc area, the
FWHM is at $N({\rm H}_{2})=10^{23}$ cm-2 overestimated by a factor of two. The
bias in $p$ is 20% at $N({\rm H}_{2})=10^{23}$ cm-2 and more than 50% at
$N({\rm H}_{2})=3\cdot 10^{23}$ cm-2. The fractional errors in $R$ are larger,
but even there become significant only beyond $N({\rm H}_{2})=3\cdot 10^{22}$
cm-2.

Figure 8 also shows results for a radiation field that is a factor of
$\chi=10$ or $\chi=100$ stronger at all frequencies.333More luminous sources
would intrinsically have higher UV luminosity, but the UV-to-IR ratio of the
radiation field at the filament location can still be much lower, depending on
the intervening extinction. This increases the temperature contrast in the
filament but also the average temperature ($\sim$7 K for the $N({\rm
H}_{2})=10^{22}$ cm-2 and $\chi=10$ case). Therefore, a change from $\chi=1$
to $\chi=10$ decreases the systematic errors by a factor of two, and for
$\chi=100$ the FWHM estimates remain accurate up to $N({\rm H}_{2})=10^{23}$
cm-2.

In many observations, fits can only be done using a limited sky area. Figure 8
also shows results for fits within the smaller $r_{\rm max}=0.35$ pc area.
This has no effect on the FWHM values, but the errors in $p$ increase. While
the observed column-density profile can be fitted with a Plummer function
(with parameters different from the true values), it does not match the
Plummer profile exactly, since the result depends on $r_{\rm max}$. In the
case of noiseless synthetic observations, a wider area always results in more
accurate estimates. The situation can be different in real observations if the
signal in the filament wings is dominated by noise and emission from unrelated
structures.

A change in inclination changes the line-of-sight optical depths but, unlike a
true increase of the filament column density, does not affect the
temperatures. Appendix D confirms that the inclination has only a minor effect
on the extracted filament parameters.

Figure 8:  Estimated $p$ and $R$ values and the corresponding FWHM as a
function of the peak column density of isotropically illuminated model
filaments. The black, blue, and red curves correspond, respectively, to
observations of model filaments in a normal ISRF, $\chi=1$, and in stronger
fields with $\chi=10$ and $\chi=100$. The solid lines with circles are fits to
the model profile at $|r|<0.58$ pc, while the dashed lines with squares are
fits to a narrower region, $|r|<0.35$ pc. The horizontal dotted lines indicate
the true parameter values in the models.

When the model includes a discrete radiation source, the parameter estimates
depend on the distance to the source. Figure 9 shows the true and the
recovered profiles at four positions along a model filament. The point source
is located behind the filament but, because the filament is optically thin for
FIR emission, the results are similar if the source were in front of the
filament. At $N({\rm H}_{2})=10^{22}$ cm-2 and $N({\rm H}_{2})=3\cdot 10^{22}$
cm-2, the FWHM errors increase towards the point source location but the
parameter $p$ is much less affected. Figure 9b demonstrates the central
flattening of the recovered $\tau$ profiles, and, as suggested by Fig. 8, the
Plummer fits do not follow the actual profile. Schuller et al. (2021) reached
qualitatively similar results, although in their case the effects were smaller
because of the stronger radiation field ($G_{0}$=1000).

Figure 9:  Selected optical-depth profiles along model filaments that are
illuminated by a point source. The filament column density is $N({\rm
H}_{2})=3\cdot 10^{22}$ cm-2 (frame a) or $N({\rm H}_{2})=1\cdot 10^{23}$ cm-2
(frame b). The black curves show the true optical depths, the blue curves the
optical depths estimated from FIR observations, and the dashed red curves the
Plummer fits to those optical depths. The cross-sections are selected from
positions $\Delta y=$0.46, 0.93, 1.39, and 1.86 pc along the filament, when
the point source is located at the position $\Delta y$=2.09 pc along the
filament and a distance 0.93 pc behind the filament. The curves from top to
bottom are in order of decreasing distance to the point source (increasing
order of $\Delta y$), and the parameters are listed in the same order (FWHM in
units of parsec).

Figure 10 shows how the parameter estimates vary along the filament in the
cases of isotropic illumination ($\chi=1$ or $\chi=10$) and the sum of an
isotropic field ($\chi=1$) and a point source. The figure confirms the rapid
increase of bias at column densities above $N({\rm H}_{2})=10^{22}$ cm-2. The
$p$ values are more sensitive to point-source illumination from one side,
while FWHM is more affected when the source is along the line of sight towards
the filament. The small bias at the filament ends is caused by these being
subjected to the full unattenuated external field.

Figure 10:  Variation in FIR-estimated parameters along model filaments in
cases of: isotropic illumination (frames a-d); a 590 $L_{\sun}$ point source
at $\Delta y=2.09$ pc and 0.93 pc behind the filament (frames e-h); and to one
side of the filament (frames i-l). The plotted quantities are the ratio
between the estimated and true FIR optical depths ($\tau/\tau_{0}$), the
Plummer parameters ($R$ and $p$), and the filament FWHM calculated based on
these. Each frame shows results for three model filaments with peak column
densities $N({\rm H}_{2})=10^{22}$ cm-2 (blue lines), $N({\rm H}_{2})=3\cdot
10^{22}$ cm-2 (cyan lines), and $N({\rm H}_{2})=10^{23}$ cm-2 (red lines). The
dashed lines in frames (a)-(d) correspond to cases with a higher isotropic
radiation field ($\chi=10$); all other cases include an isotropic field with
$\chi=1$. True values are plotted with dashed black lines.

The FIR results can also be affected by changes in dust properties. These
could be related to changes in the grain sizes and optical properties,
following the formation of larger aggregates and ice mantles (Ossenkopf &
Henning, 1994; Ormel et al., 2011; Jones et al., 2016). Figure 11 compares
calculations with uniform dust properties to with two two-component models.
The single-component models consist of COM, CMM, or AMMI dust. The AMMI model
tends to result in the largest systematic errors, especially in the FWHM. The
250 $\mu$m opacity of AMMI is five times higher than for COM, and a similar
difference also exists at shorter wavelengths, where dust absorbs energy. The
differences are thus caused mainly by changes in the optical depth, since
spectral index of AMMI dust ($\beta\sim 2.02$) is similar to the value
$\beta=2.0$ that was used in the MBB analysis.

We tested two cases with spatially varying dust properties. The first two-
component model consists of CMM dust and a modified CMM, where $\beta$ is
decreased from the original 160-500 $\mu$m spectral index $\beta\sim 1.97$
down to $\beta=1.5$. The abundance of the modified dust is calculated as
$\tanh(n/[10^{5}\,{\rm cm}^{-3}])$, so that the filament centre consists
entirely of the modified dust. This shows the effect of a change in the
spectral index, without a net change in the opacity. The results show
systematic but relatively minor variations in the filament parameters (Fig.
11).

The second two-component model consists of COM dust in the outer parts and
AMMI dust in the inner part. The relative abundance of the AMMI component
follows the same density dependence as above. The spectral indices are
different ($\sim$2.02 and $\sim$1.83 for AMMI and COM, respectively) but there
is a larger difference in the absolute opacities. While the pure AMMI model
led to the largest $R$ and FWHM estimates, the COM+AMMI combination leads to
the largest $p$ values. This illustrates qualitatively the potential effects
from spatial dust property variations. However, the quantitative results will
also be sensitive to the radial position and steepness of the transition in
dust properties.

Figure 11:  FIR-estimated filament parameters for single-dust models (COM,
CMM, or AMMI) and two models with spatial dust-property variations. In $\rm
CMM(2\beta)$, the transition is from normal CMM dust in the outer parts to
modified dust with $\beta=1.5$ in the inner part. For COM/AMMI, the transition
is from COM dust to AMMI dust. The filament column density is $N({\rm
H}_{2})=3\cdot 10^{22}$ cm-2, and the filament is illuminated by an isotropic
radiation field (frames a-d) or with a 590 $L_{\sun}$ point source at $\Delta
y$=2.09 pc and 0.93 pc to one side of the filament (frames e-h). The plotted
parameters are the estimated optical depth relative to its true value
($\tau/\tau_{0}$), the parameters $R$ and $p$ of the fitted Plummer functions,
and the resulting filament FWHM estimate.

###  4.2 Bias in MIR observations

The filament models of Sect. 4.1 were also used to examine how the MIR
analysis is affected by in situ dust scattering and emission. Radiative-
transfer calculations provide the surface brightness due to 8 $\mu$m dust
scattering with the CMM dust model and the thermal emission from
stochastically heated grains with the COM dust model. We concentrate here on
the systematic effects. Appendix E examines further some effects related to
observational noise.

####  4.2.1 Effect of MIR scattering

For a filament with $N({\rm H}_{2})=3\cdot 10^{23}$ cm-2, the scattering
results in errors of less than $\sim$1% in the estimated $\tau(8\,\mu{\rm
m})$. This remains true even if the isotropic radiation field is increased to
$\chi=10$ or the default point-source luminosity is increased by a factor of
50\. The calculations assume an intensity $I^{\rm bg}$=10 MJy sr-1 for the
background sky. This is similar to the OMC-3 field but still a relatively low
value. If the background is higher, the effects from scattering in the cloud
would be further reduced.

The importance of scattering increases with increasing column density. For
$\tau(8\,\mu{\rm m})<1$, the intensity of the scattered light follows the
column-density profile. If the filament is optically thick, the scattered
light will peak on either side of the column-density peak, with potentially
larger impact of the parameter estimates. However, Fig. 12 shows that for a
filament with $N({\rm H}_{2})=10^{24}$ cm-2, the maximum optical-depth errors
remain below 10%, both for an isotropic field $\chi=10$ or a point source with
luminosity 50 times the default value. If the isotropic radiation field is
increased to $\chi=50$, the errors exceed 20% for a $N({\rm H}_{2})=10^{24}$
cm-2 filament. If the column density is increased further by a factor of three
(to rather extreme values), the errors would exceed 60%. The bias is
determined mainly by the optical depth. The optical depth depends on the
column density but also the dust properties and would be more than two times
higher for the AMMI dust than for the CMM dust was used in Fig. 12.

Figure 12:  Modelled effect of scattering on the MIR optical-depth estimates
towards filament centre. Frame (a) shows the model optical depth, which
corresponds to a peak column density of $N({\rm H}_{2})=10^{24}$ cm-2 with the
CMM dust model. The other frames show the relative error in $\tau(8\mu{\rm
m})$ estimates. Frame (b) includes only an isotropic radiation field with
$\chi=10$. Frames (c)-(e) show the errors, when a point source, with
luminosity 50 times the default value, is included at $\Delta y=2.09$ pc and
0.93 pc behind the filament (frame c), in front of the filament (frame d), or
to one side.

Figure 13 shows the effect of scattering on the filament parameter in the case
of a $N({\rm H}_{2})=10^{24}$ cm-2 filament illuminated by an isotropic field
and a foreground point source. For the default radiation-field values, the
bias caused by light scattering is $\sim$1% or less. When the isotropic field
is increased to $\chi=20$, the errors in $p$ and FWHM rise to a few per cent.
If the point source is made 50 times stronger, the errors exceed $\sim$15%,
but only in a small area closest to the point source.

In summary, the scattering in the filament has only a minor effect on the
parameter estimation. The effect become visible only if the local radiation
field is very strong, the filament has a very high column density, and the
background surface brightness is low.

Figure 13:  Modelled effect of scattered light on the filament parameters
derived from MIR observations. The filament column density is $N({\rm
H}_{2})=10^{24}$ cm-2, and it is illuminated by an isotropic radiation field
and a foreground point source. The blue curves correspond to the case of
$\chi=1$ for the isotropic component and the point source with the nominal
luminosity. The cyan and red curves show, respectively, the results when both
radiation-field components are scaled by a factor of 20 or 100. The dashed
black lines show the correct values of the parameters.

####  4.2.2 Effects of MIR dust emission

The effects of the 8 $\mu$m thermal emission from stochastically heated grains
($I^{\rm shg}$) was examined using the COM dust model. Figure 14 shows maps of
the dust emission for a $N({\rm H}_{2})=3\cdot 10^{23}$ cm-2 filament. An
isotropic radiation field ($\chi$=1) results in emission at a level of
$I_{\nu}(8\,\mu{\rm m})\sim 0.5$ MJy sr-1. This is not completely negligible
if the background sky brightness is low. The 590 $L_{\sun}$ point source has a
larger effect, which ranges from less than 1 MJy sr-1 far from the source to
$\sim$100 MJy sr-1 close to the source. For the column density of $N({\rm
H}_{2})=10^{23}$ cm-2, the surface brightness remains at a similar level, but
the morphology is different. The surface brightness follows more closely the
column-density distribution, and, for a source behind the filament (Fig. 14
frame c), the intensity peaks towards the centre of the filament, with only a
minor dip at $|\Delta x|<0.05$ pc.

Figure 14:  Maps of thermal dust emission $I^{\rm shg}$ calculated for the
$N({\rm H}_{2})=3\cdot 10^{23}$ cm-2 filament model. Frame (a) corresponds to
isotropic illumination with $\chi=1$, but the surface-brightness values are
multiplied by 100 just for plotting (the same colour bar applies to all
frames). The other frames include both the isotropic field and a point source
located at $\Delta y=2.09$ pc and at a distance of 0.93 pc in front, behind,
or to the left of the filament centre axis (frames b, c, and d, respectively).

Because the thermal emission is stronger and more extended than the scattered
light, it could even affect the observer’s estimate of $I^{\rm bg}$. We
calculated alternative $\tau$ maps, where the median value of the thermal
emission $I^{\rm shg}$ from stochastically heated grains (in the area visible
in the Fig. 14) was added to the original $I^{\rm bg}$. The added component is
not truly part of the sky background, because it originates within the source
itself and preferentially on the observer’s side of the source.

Figure 15 shows the results for $N({\rm H}_{2})=3\cdot 10^{23}$ cm-2, when the
point source is towards one side of the filament. Because the observed sky
brightness varies along the $\Delta y$ coordinate, the $\tau$ profiles do not
drop to zero in the filament wings. This effect is mostly eliminated by the
linear background component that is part of the fitted profile function (Eq.
(7)). Nevertheless, the parameter estimates vary by up to 50% with the
distance to the point source. Both $p$ and FWHM are more overestimated near
the point source, although $p$ drops sharply at the position closest to the
point source.

Figure 15 also shows results for a stronger isotropic field with $\chi=10$,
where $I^{\rm bg}$ has to take into account the extended emission. The optical
depths are now underestimated more, and the especially $p$ shows large bias.
Nevertheless, the filament FWHM is overestimated only by some 25%.

Other cases with $N({\rm H}_{2})=3\cdot 10^{23}$ cm-2 but different radiation
fields are shown in Appendix F. If the point source source is directly in
front of or behind the filament, its effect is amplified, as more line-of-
sight material is heated. The parameter values are then lower at $\Delta y\ga
1$, and the filament disappears close to the projected point-source location,
when the dip caused by the background extinction is filled by thermal
emission. This happens even earlier for models of lower column density,
because the lower opacity reduces the background absorption more than it
reduced the thermal emission.

In Appendix F we examine a model with lower column density, $N({\rm
H}_{2})=3\cdot 10^{22}$ cm-2, and higher background intensity, $I^{\rm
bg}=100$ MJy sr-1. Filament parameters are there generally accurately
recovered. However, the errors in $p$ and FWHM still reach 30% close to the
point source and, if the source is along the line of sight, the filament
disappears as an absorption feature.

Figure 15:  Estimated filament parameters along the $N({\rm H}_{2})=3\cdot
10^{23}$ cm-2 model filament, when the surface brightness includes 8 $\mu$m
thermal dust emission. The filament is illuminated by an isotropic background
($\chi=1$) and a point source at $\Delta y=2.09$ pc and a distance 0.94 pc to
one side of the filament (cf. Fig. 14d). Red curves correspond to calculations
with the true value of $I^{\rm bg}=10$ MJy sr-1 and blue curves to a case
where the median surface brightness from Fig. 14d is added to the estimate of
$I^{\rm bg}$. The black curves are for the case of a $\chi=10$ isotropic
radiation field (no point source), with a similarly adjusted $I^{\rm bg}$
estimate.

##  5 Discussion

We have examined the estimation of the filament properties with observations
of MIR extinction and FIR dust emission. In Sect. 5.1, we discuss the
observational results on the OMC-3 field. In Sect. 5.2, we concentrate on the
radiative transfer models and, based on the models, the systematic errors that
may affect the filament observations.

###  5.1 OMC-3 filament parameters

We analysed four filament segments, named A-D, in the OMC-3 cloud. Based on
MIR absorption, the median FWHM widths are $\sim$0.04 pc, with little
dependence on the fitted cross-filament extent $r_{\rm max}$. The values were
on average consistent between the four segments, but there were also
significant variations along the filaments. These are discussed further in
Sect. 5.3. The analysis of FIR emission also gave median FWHM$\sim$ 0.03-0.05
pc, but, in the case of segment B, up to $\sim$0.1 pc. FIR emission was
analysed using column-density maps with angular resolutions from 10$\arcsec$
to 41$\arcsec$. There were only small differences between the different map
versions, the HR version (20$\arcsec$ resolution) resulting in the smallest
values, with median values of 0.02-0.04 pc in Fig. 6. Fits to the median
profiles gave values that were similar to the median parameter values along
the filaments.

Previous Herschel studies have typically found filament widths of $\sim$0.1
pc. Arzoumanian et al. (2011) reported a narrow distribution of $0.10\pm 0.03$
pc in the cloud IC5146, at a distance of 460 pc. These (deconvolved) widths
were based on Gaussian fits, and column-density maps at $\sim 37\arcsec$
resolution and 250 $\mu$m surface brightness maps at $\sim 18\arcsec$
resolution gave similar results. Rivera-Ingraham et al. (2016) analysed 29
filaments in 13 separate fields at $d=100-500$ pc distances, the analysis of
column-density maps of 41$\arcsec$ resolution resulting in widths of $0.13\pm
0.05$ pc. In the above papers, the fits were done to the mean (or median)
radial profile of an entire filament that was detected with automated methods,
using DisPerSE (Sousbie, 2011) in the case of Arzoumanian et al. (2011) and
getfilaments (Men’shchikov, 2013) in the case of Rivera-Ingraham et al.
(2016). Arzoumanian et al. (2011) extended Herschel studies to 599 filaments
in eight regions at 140-460 pc distances, the distribution again peaking at
$\sim 0.1$ pc with an interquartile range of 0.07 pc. However, based on
Herschel data in Arzoumanian et al. (2019), Panopoulou et al. (2022) concluded
that the filament width estimates also depend on the source distance and
appear to be 4-5 times the beam size. Thus, the widths would increase from
less than 0.1 pc in the closest fields (e.g. Taurus and Ophiuchus) to almost
0.3 pc at $\sim$800 pc (the IC5146 cloud). This scaling appeared to hold for
regions of very different types, from the low-density Polaris field to the
Orion-B cloud with active star formation. Some distance dependence was also
noted in Rivera-Ingraham et al. (2016).

Although deconvolution by a larger telescope beam increases uncertainties,
with perfect observations of perfect Plummer profiles, the FWHM estimates
should not depend on the resolution of observations. Indeed, in our analysis,
the factor of 20 range in the angular scales is not associated with
corresponding systematic changes in the estimated filament widths. The average
FWHM were at or below 0.05 pc for all maps, although Herschel data could in
some cases lead to higher values FWHM$\sim$0.1 pc (segments A and B, Fig. 6).
The values are lower than those reported in Panopoulou et al. (2022) for
fields at similar distances (including Orion B with FWHM$\ga$0.15 pc), and do
not match distance/resolution dependence reported there. In all the above
studies, the column densities were based on the modelling of dust emission
with a single-component MBB. Howard et al. (2019) and Howard et al. (2021)
used the PPMAP method (Marsh et al., 2015) to analyse Herschel and SCUBA-2
observations of filaments in the Taurus and Ophiuchus clouds. They noted that
the use of the PPMAP method, which takes temperature variations into account,
resulted in a reduction in the estimated filament widths. Our results were
also roughly constant with respect to the size of the fitted cross-filament
area. Therefore, even if the extent of the fitted area typically increases
with the target distance, that should not necessarily lead to larger FWHM
estimates.

Filaments in the OMC-3 region were already investigated by Schuller et al.
(2021), who used Herschel 160-500 $\mu$m and ArTéMiS 350 $\mu$m and 450 $\mu$m
data (different observations from the ArTéMiS data analysed in Mannfors et al.
(2022)). The ArTéMiS surface-brightness data and Herschel temperature
information at 18.2$\arcsec$ resolution were combined to a column-density map
with 8$\arcsec$ resolution. This resulted in filament FWHM estimates of
0.06$\pm$0.02 pc. These values are similar to our AR results, where (ignoring
the outlying values of segment A) the median values range from 0.04 pc to
$\sim$0.1 pc (Fig. 6g). While Schuller et al. (2021) studied long,
automatically detected filaments, our analysis is limited to short segments at
the highest column densities.

In addition to FWHM, the values of the asymptotic power-law index, $p$, of the
fitted Plummer function are of interest. The Herschel and combined Herschel
and ArTéMiS data gave typically values $p=2-5$, with some variations depending
on the extent of the fitted area. These suggest that nearby cloud structures
are affecting the fits in the tails of the profile function (cf. Sect. 5.3).
Compared to FWHM, the individual $p$ and $R$ values could be measured less
precisely. Especially the MIR $p$ values showed a large scatter, which could
be caused in part by changes in the local MIR radiation field. The MIR $p$
estimates are similar for different values of $r_{\rm max}$, which suggests
that individual point sources within that area do not have a strong effect on
the results.

###  5.2 Systematic errors in filament parameters

We used radiative transfer simulations to study error sources in the analysis
of MIR and FIR observations. In the following we discuss their relative
importance and the potential effects on the OMC-3 results.

####  5.2.1 MIR observations

Dust scattering and thermal dust emission are potential error sources in the
analysis of MIR extinction. They depend on the local radiation field and are
significant at high column densities. We assumed in the tests a background
level of 10 MJy sr-1. However, if the background level is higher, the effects
of local emission and scattering will be reduced.

For dust scattering, errors reached tens of per cent only if the column
density is $N({\rm H}_{2})\sim 10^{24}$ cm-2 or higher. The intensity of the
isotropic background also had to be a factor of $\chi=100$ above the normal
ISRF or the luminosity of a point source at $\sim$1 pc distance had to be of
the order of $\sim 10^{4}$L☉. Thus, scattering should not be a significant
source of errors in the OMC-3 field. Scattering would tend to decrease the
$\tau$ estimates, increase the $p$ estimates, and lead to overestimations of
the filament FWHM (Fig. 13). These effects are more pronounced close to point
sources, especially if the source is on the line of sight towards the
filament. The presence of such point sources would be evident in the observed
maps, although their effect can extend over distances of several parsecs.

The MIR scattering depends strongly on the grain properties. The so-called
coreshine, which is observed at 3-4 $\mu$m wavelengths towards many dense
cores, has indicated surprisingly strong MIR scattering. This requires strong
dust evolution relative to diffuse clouds (Steinacker et al., 2010; Pagani et
al., 2010; Juvela et al., 2012c; Steinacker et al., 2014b; Lefèvre et al.,
2014). Lefèvre et al. (2016) investigated the 8 $\mu$m scattering towards the
pre-stellar, high-column-density core of L 183, where the estimated intensity
of the scattered light was hundreds of kJy sr-1. This corresponds to the
scattering in a more or less normal ISRF, and the high scattering efficiency
could be explained by large aggregate grains. The L 183 observations (and the
possibility of very large aggregates) suggests that the scattered signal could
be stronger than in our simulations. If the scattered light is in L 183 at a
level of $\sim 0.1$ MJy sr-1, scattering could cause noticeable errors in MIR
observations of high-mass star-forming regions, where the radiation fields are
much stronger.

In our models, the local thermal dust emission was a more significant factor
than the scattering. For the high column density of $N({\rm H}_{2})\sim
10^{24}$ cm-2 filament, the emission effects were tens of per cent, both in
the case of the normal ISRF and in the case of the 590 $L_{\sun}$ point
source. This led to the filament optical depths being underestimated and the
individual filament parameters ($p$, $R$, FWHM) to be overestimated. The
magnitude of the effects depends strongly on the presence of local radiation
sources (Fig. 13). If there is a point source on the line of sight towards the
filament, the thermal emission can completely mask the MIR absorption
(Appendix Sect. F). Quantitatively, these effects are sensitive to the
filament column density, the level of the background surface brightness, and
the location of the point source relative to the filament (Fig. 14). In the
comparison to the models, one must also take into account that observations
tend to underestimate the true column densities.

Scattering is not only a source of errors but can itself be used to study
filaments at high angular resolution. At near-infrared wavelengths the
scattering is stronger (in absolute terms and relative to the in situ thermal
emission) but the larger optical depths complicate the analysis (Juvela et
al., 2012b; Malinen et al., 2013). Scattering may also be measurable at MIR
wavelengths, above the MIR absorption (e.g. Steinacker et al., 2014b; Lefèvre
et al., 2014). However, this requires a low background sky brightness, such as
found at high Galactic latitudes (Steinacker et al., 2014a).

####  5.2.2 FIR observations

Far-infrared analysis of Sect. 3.2 is affected especially by the bias of
column-density estimates, which is caused by temperature variations in the
source and the analysis using the single-temperature MBB model. Appendix G
shows that these effects can be easily demonstrated even without complex
modelling.

In the radiative-transfer simulations of Sect. 4, the column-density estimates
vary depending on the data resolution (beam sizes), the analysis method, the
temperatures, and dust optical properties. It can be instructive to compare
the peak optical depths obtained at different resolutions and with different
methods. Table 4 lists values for the $N({\rm H}_{2})=10^{22}\,{\rm cm}^{-2}$
model. Since the filament FWHM is quite large, $\sim 72\arcsec$, the peak
values of the 40$\arcsec$ and 20$\arcsec$ maps differ only little. Shorter
wavelengths are more sensitive warm dust, tend to bias the temperatures more
upwards, and result in lower $\tau$ values. In Table 4 the effect is the
opposite for $\beta=1.8$, because the simulation used a dust model with
$\beta\sim 1.95$. A modest increase in the assumed $\beta$ (to a value above
the $\beta$ in the simulations) can even negate the natural tendency to
underestimate the column densities. When real observations are analysed, the
precise value of $\beta$ is unknown. However, because $\beta$ affects all
column-density estimates similarly, its effect on the observed filament
profiles is limited - as long as $\beta$ does not change significantly as a
function of the filament radius.

Table 1: Comparison of peak 250 $\mu$m optical depths estimated with different
analysis methods. The estimates are based on synthetic surface-brightness maps
of the $N({\rm H}_{2})=10^{22}\,{\rm cm}^{-2}$ filament model.

Case | Beam | $\beta$ | $\tau^{\rm max}(250\,\mu{\rm m})$ | Rel. error
---|---|---|---|---
| [$\arcsec$] |  | $[10^{-3}]$ | [%]
true | 6 | – | 1.49 | 0.00
MBB | 6 | 1.8 | 1.11 | -25.77
MBB | 24 | 1.8 | 1.04 | -29.99
HR | 18 | 1.8 | 1.04 | -30.34
HR | 18 | 2.0 | 1.33 | -10.48
HR | 18 | 2.2 | 1.71 | 14.61
HR (w) | 18 | 1.8 | 1.00 | -32.54
HR (w) | 18 | 2.0 | 1.32 | -11.46
HR (w) | 18 | 2.2 | 1.73 | 16.05
LR | 40 | 1.8 | 0.98 | -33.98
LR | 40 | 2.0 | 1.22 | -17.85
LR | 40 | 2.2 | 1.51 | 1.64
LR (w) | 40 | 1.8 | 0.96 | -35.27
LR (w) | 40 | 2.0 | 1.22 | -18.42
LR (w) | 40 | 2.2 | 1.52 | 2.30
444Relative errors are with respect to the true values that are read from the
3D model and reported on the first row. MBB refers to modified blackbody fits
where all data at the same resolution, ‘HR’ to the higher-resolution versions
(Palmeirim et al., 2013), and ‘(w)’ to alternative fits where the 160 $\mu$m
data are given a lower weight. The third column refers to the constant value
of $\beta$ used in the SED fits.

The bias of the column-density estimates increase with column density and
exceeded a factor of two at $N({\rm H}_{2})=10^{22}\,{\rm cm}^{-2}$ (Fig. 7).
Since these errors are correlated with the column density, they also affect
the observed filament profiles. The parameters $p$, $R$, and FWHM are all
biased upwards. Unlike in MIR observations, a stronger isotropic radiation
field decreases the errors by reducing the amount of very cold dust. In the
normal ISRF, the systematic errors of filament FWHM reach 50% when the column
density exceeds $N({\rm H}_{2})=10^{23}\,{\rm cm}^{-2}$. In a $\chi=10$ field,
the errors are less than half of this, and are they are further halved in a
$\chi=100$ field. This means that the errors can be of similar magnitude in a
low-mass star-forming region (low column density and low radiation field) as
in a high-mass star-forming region (high column density and high radiation
field). The errors in $R$ and $p$ also depend on both the column density and
the radiation field, and can reach 50% for a $N({\rm H}_{2})=10^{23}\,{\rm
cm}^{-2}$ in the normal ISRF (Fig. 10). A point source has a similar effect,
the errors increasing close to the source and with only a small dependence on
the source location (on the line of sight vs to one side of the filament).

The maximum column density of the OMC-3 filaments is in MBB analysis $N({\rm
H}_{2})\sim 10^{23}\,{\rm cm}^{-2}$, but the true column density could be even
a few times higher. Because the quiescent parts of the filament are likely to
be subjected to a radiation field of at most $\chi\sim 100$ (Mannfors et al.,
2022), and the effects of MIR scattering are likely to be insignificant. The
models predict a more significant role for the MIR dust emission. Figure 15
indicated a $\sim$25% effect for an isotropic radiation field with $\chi=10$.
This requires the extended thermal emission to also be taken into account in
the $I^{\rm bg}$ estimates. In observations this happens automatically (to
some accuracy), and the local thermal emission need not be separated from
other contributions to $I^{\rm bg}$. The fact that the OMC-3 FIR and MIR
observations resulted in similar FWHM estimates also suggests that the
systematic errors of the MIR analysis are unlikely to amount to tens of per
cent.

Filaments are located inside dense clouds, which significantly reduces the UV
flux that reaches the filament. Most of the 8 $\mu$m emission would then
originate in extended regions, and, unlike in our simple models, the emission
would be mostly uncorrelated with the filament structure. This will reduce the
systematic errors in the filament parameters. A second important factor is the
abundance of polycyclic aromatic hydrocarbons and other very small grains
(Draine, 2003). If these have already partly disappeared within the filament
(e.g. by sticking onto larger grains), the MIR emission would be suppressed.
Based on our models, FIR analysis could overestimate the width of $N({\rm
H}_{2})\sim 10^{23}\,{\rm cm}^{-2}$ filaments by tens of per cent in the
normal ISRF. However, a stronger radiation field reduces the errors to
$\sim$10% level, which is within the uncertainties of the MIR versus FIR
comparison (Fig. 6). Therefore, although the models show the possibility of
significant systematic errors in all observations, there is no contradiction
in the approximate agreement between the OMC-3 MIR and FIR estimates.

###  5.3 Reliability of profile fits

Based on synthetic observations of magnetohydrodynamic cloud simulations,
Juvela et al. (2012a) concluded that, at the nominal Herschel resolution, the
filament parameters could be recovered reliably only up to $\sim$400 pc
(assuming $\sim$0.1 pc filaments). OMC-3 is at this limit and the filaments
are partly narrower than 0.1 pc. Nevertheless, the angular resolution of at
least the MIR and AR maps should be sufficient for the profile analysis.

In addition to systematic errors, the results show significant random
fluctuations, especially in the $p$ and $R$ estimates. Some fits resulted in
values $p\ga 5$ that are inconsistent with most physical filament models.
Individual $p$ and $R$ values are more difficult to measure because a larger
value of $p$ can in Eq. (6) be compensated with a larger value of $R$, the fit
still recovering the same FWHM and generally fitting the observed profile. The
parameter $R$ probes the structure in the inner part of the filament and is
dependent on the data being able to resolve those smaller scales. Conversely,
$p$ probes the asymptotic behaviour at large distances and is sensitive to
nearby cloud structures and the general background fluctuations.

The error distributions of the parameters (including FWHM) are often
asymmetric for the individual profiles, with the MCMC estimates showing a long
tail to high values (cf. Fig. 19). The overall dispersion along the segments
(as shown by the histograms e.g. in Fig. 3b) give an empirical estimate for
the total uncertainty. Our data also contain missing values, which affect the
reliability of MIR profiles (e.g. Fig. 3) and the analysis of the AR map of
the filament segment A (Fig. 5). The missing data also bias the median
profiles. Rather that rejecting all profiles that contained missing values
(which could be almost all of the data), the median values were calculated
over the remaining pixels. Therefore, at a given distance from the filament
centre, the median value is based on different sections along the length of
the filament, leading to random errors in the median profile.

Figure 16 shows examples of the fits to the OMC-3 filament segment A, based on
the MIR, HR, and AR column-density estimates. Most MIR profiles do not have
data around offset $\Delta x=30\arcsec$ but in this case this does not have a
major effect on the fits. MIR profiles also tend to show a dip around $\Delta
x=-30\arcsec$, which could be caused by imperfections in the column-density
estimation (e.g. dust to nearby sources), or random column-density
fluctuations. These fits are associated with high values of $p\sim 5$. In
comparison, the profile at offset $\Delta y=30\arcsec$ along the filament (red
curve) decreases rather than increases towards negative $\Delta x$, and has
$p\sim 1.6$. For the maps HR and AR based on dust FIR emission, the fitted
profiles match better the observed profiles. However, the effect of the
missing data is clear in the AR results, where the observations at $\Delta
y$=10 and 20 arcsec (i.e. the profiles most affected by missing values, cf.
Fig. 5c) provide only weak constraints at negative $\Delta x$ values. This
leads to degeneracy between the Plummer and the background parameters, the
background is underestimated at negative $\Delta x$ offsets, and the fit
results in abnormally large FWHM values.

Figure 16:  Plummer fits for selected cross-sections of the OMC-3 filament
segment A. The three frames correspond to the MIR, HR, and AR column-density
maps, respectively. Each frame shows individual profiles for the offsets
$\Delta y$=10, 20, 30, and 40 arcsec (cf. Fig. 3-5; blue, cyan, red, and grey
lines, respectively) and the median profile (thick black lines). The best-fit
Plummer profiles are plotted with dashed lines of the same colour, and $R$,
$p$, and FWHM values of the fits are listed in the frames.

Large $p$ values are also observed for some other dust emission data,
typically in fainter and less clear parts of the filaments. They can be
related to interference from nearby cloud structures, which can be other
filaments or clumps (e.g. northern part of filament B, Fig. 23) or diffuse
emission that appear as an extension of the filament itself (e.g. segment C,
$\Delta x\sim 60\arcsec$, Fig. 24). However, the filament segment D is
associated with large values $p\geq 4$ over its full length. Figure 17 shows
examples of the individual profiles. The main feature (in HR and AR maps) is a
dip at negative $\Delta x$ values. This is similar to the one seen in Fig. 16
and similarly appears to be the origin of the large $p$ values. The situation
is worst for the $\Delta x=10\arcsec$ profile, where the background also
curves up at positive $\Delta x$ offsets, raising the $p$ value above nine.
This suggest that the background might need to be modelled using a second
order polynomial, although $p$ would be partly degenerate with an second order
background term.

Figure 17:  Selected cross-sections of the OMC3-filament segment D. These
correspond to HR and AR data (blue and red lines, respectively) and the
offsets $\Delta y$=10 and 25 arcsec, where the $\Delta y$=10 arcsec profiles
have the lower column density. The dashed lines are the best-fit Plummer
profiles with the parameters listed in the figure.

Figure 18 compares the results that are based on the MIR, HR, and AR data on
the filament segment D and six versions of the fitted profile function. The
row B corresponds to our default model in Eq. (7), which has six parameters:
three for the Plummer function itself, one allowing a shift along the $\Delta
x$ axis, and two parameters describing the linear background. In Fig. 18, one
of the alternative fits omits the shift (row A) and one adds the second order
term to the background component (row C). So far all fits assume that the
filament profiles are symmetric, which is only approximately true for the
selected segments. In Fig. 18, we also show results for asymmetric Plummer
functions (separate $R$ and $p$ parameters on each side of the peak) that are
combined with a background modelled as a linear first order (row D) or a
second order (row E) polynomial.

If the filament shifts in the $\Delta x$ direction at small scales, the
omission of the $\Delta r$ term in Eq. 6 should lead to larger FWHM values. No
such effect is seen in Fig. 18, although this could still play a role in fits
of longer and more fragmented filaments. The addition of the second order
background term (figure row C) reduces the $p$ values significantly from
$p\sim 5$ to $p\sim 2.5$. The $R$ values are also smaller, but the mean value
and the scatter of the FWHM values has increased. When the second order
background term is included in the asymmetric Plummer fits, the results are
much less affected and especially the FWHM values remain practically unchanged
(row E vs row B). With the exception of the row C, the FWHM values are thus
similar for all the alternative fits.

In Fig. 18, the fits have between five and nine free parameters. In these
calculations, we also added penalties for values $R<0.005$ pc and $p>8$. The
AR results were somewhat sensitive to the $R$ threshold, because the filament
widths are not much larger than the beam size and unresolved filaments can
result in small $R$ values. The FWHM estimates from the LR maps could thus be
even more sensitive to any priors that are used. Accurate beam models are also
important, since they are used to deconvolve the observations. The LR beam is
well defined, while the effective beams of the HR and AR maps depend on the
way these maps are constructed. The HR column density is based on a
combination of intermediate maps that have different angular resolutions and
different sensitivity to temperature variations. The AR map is the combination
of data from two instruments, and the effective beam could be affected by
calibration differences or imperfections of the feathering procedure.

Figure 18:  Alternative fits of the filament segment D. The histograms show
the distributions of the $R$, $p$, and FWHM parameters that are estimated from
the MIR, HR, and AR maps (cyan, blue, and red histograms, respectively). The
rows are for alternative models of the profile function, where row B
corresponds to our default model. In the other fits, row A omits the shift of
the filament centre, row C adds a second order term to the background, row D
fits two-sided Plummer functions with linear background, and row E fits two-
sided Plummer functions with a second order polynomial for the background.

##  6 Conclusions

We have studied four filament segments in the OMC-3 cloud in Orion, using
observations of MIR extinction and FIR dust emission and with the goal of
measuring the filament widths. The Herschel FIR data were converted to column-
density maps at 20$\arcsec$ (LR maps) and 41$\arcsec$ (HR maps) resolution, as
well as an additional map of 10$\arcsec$ resolution from combined Herschel and
ArTéMiS 350 $\mu m$ observations (AR map). In addition to using observational
results from the OMC-3 filaments, we performed radiative transfer simulations
to investigate sources of systematic errors that could affect the measurement
of filament profiles. The study led to the following conclusions:

  1. 1.

The OMC-3 filament segments have FWHM values of 0.03-0.05 pc. Similar values
are obtained with the three column-density maps derived from FIR observations
(10-41$\arcsec$ angular resolution) and based on the MIR extinction map ($\sim
2\arcsec$ resolution). In the LR and AR maps, the estimated width was only
higher in segment B and (partly) in segment A, with $FWHM\sim 0.1$ pc.

  2. 2.

The MIR results showed little dependence on the extent of the fitted area,
which was varied between 60$\arcsec$ and 210$\arcsec$ maximum distances from
the filament centre. Some variation was observed due to map edges and, to a
lesser extent, due to the influence of unrelated background structures. The
results are based on Plummer fits, where the model itself includes terms for a
linear background.

  3. 3.

When estimated from FIR data, the values of the asymptotic power-law index,
$p$, in the Plummer function were $\sim 2-5$, with an average value of $p\sim
3$. The estimates derived from MIR extinction had a large scatter, and the
median values of the four segments ranged from $p\sim 3$ to as high as $p\sim
8$.

  4. 4.

The FWHM estimates are quite robust, even when the individual Plummer
parameters, $p$ and $R$, show a large scatter. This applies to both MIR and
FIR analysis.

  5. 5.

Synthetic observations of model filaments were analysed. The MBB fits to
160-500 $\mu$m dust emission led to the expected underestimation of column
densities. The error exceeded $\sim 50$% above $N({\rm H}_{2})=3\times
10^{23}$ cm-2 but depended on the dust properties.

  6. 6.

A similar bias exists in the filament parameters derived from FIR dust
emission. For a $N({\rm H}_{2})=10^{23}$ cm-2 filament heated by the normal
ISRF, the FWHM is overestimated by more than 50%. However, the error decreases
by more than a factor of two if the radiation field is ten times stronger and
the dust temperatures correspondingly higher. The errors also decrease rapidly
at lower column densities.

  7. 7.

The effect of point sources on the FIR analysis is qualitatively similar to
the isotropic field, systematic errors increasing closer to the source. There
is only a minor dependence on the source location: a source along the line of
sight versus a source to one side of the filament.

  8. 8.

The accuracy of the MIR extinction analysis is affected by the uncertainty of
the foreground emission and the potential effects of MIR dust scattering and
emission.

  9. 9.

In the models, MIR scattering shows only minor effects. The errors in filament
parameters reach 10% only at very high column densities ($N({\rm H}_{2})\sim
10^{24}$ cm-2) and in strong radiation fields ($\chi>10$). Errors are larger
near luminous point sources, whose presence should be clearly visible in the
maps. Scattering is sensitive to dust properties, and strong grain growth
could increase its effects by a factor of several.

  10. 10.

Thermal MIR emission is, in our models, a more significant error source than
MIR scattering. In a $N({\rm H}_{2})=3\times 10^{23}$ cm-2 model filament,
errors already reach the 10% level in the normal ISRF or within a 1 pc
distance of a 590 L☉ point source. Errors are still of the same order of
magnitude even in a stronger isotropic radiation field of $\chi=10$. However,
if the diffuse UV field is attenuated by the surrounding cloud or the
abundance of very small grains is lower inside the filament, the significance
of MIR emission is correspondingly decreased.

The estimated widths of the OMC-3 filaments were roughly equal between
different tracers and observations of different angular resolution. This is
encouraging, considering the many potential sources of systematic errors.
However, it also further highlights the differences between the dust filaments
and the narrower fibres that are observed in spectral lines. High-resolution
comparisons of the dust and gas tracers, within the same sources, are needed
to understand these differences and the exact role of filaments in the star-
formation process.

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##  Appendix A Test of the fitting routines

The profile function of Eq. (7) was fitted with a least-squares routine and
with a MCMC routine. Figure 19 shows results for the analysis of a synthetic
dataset. The inputs consist of a set of ideal profiles, which match the model
setup, and white noise that is varied from 1% to 20% of the filament peak
value. For illustration, we fix the pixel size to 6$\arcsec$ and the data
resolution (beam size) to 24$\arcsec$. The profile parameters are varied in
the ranges $R=18-30\arcsec$ and $p=1.6-4.4$. The simulation is not concerned
with the way the synthetic column-density observations are obtained (emission
or extinction), only on the noise effects in the actual profile fitting.

The figure shows that the parameter uncertainty depends not only on the noise
level but also on the values of $R$ and $p$. For example, $p$ is correlated
with $R$, and its uncertainty is much higher when the true value of $p$ is
higher. At high noise levels, the MCMC mean value tends to be above the least-
squares solution, because of the skewed error distribution (and the used
$p\geq 1$ constraint). Compared to the individual parameters, the FWHM
estimate is generally more accurate, but its uncertainty still increases
significantly when the value of $p$ is low.

The results depend on the size of the analysed region, and fits to a narrower
area around the filament will naturally lead to higher errors (especially for
low $p$). However, the errors in real observations can be dominated by
structured background emission rather than observational noise.

Figure 19:  Fits of synthetic filament profiles. Frame (c) shows 256 generated
profiles, where the noise relative to the filament peak increases
logarithmically from 1% at the bottom (at $\Delta y$=0) to 20% at the top (at
$\Delta y$=255). The residuals of the least-squares fits are show in frame
(a). Frame (d) shows the recovered parameter values, where the light-green
lines correspond to the true input values, and the least-squares estimates are
plotted in blue. For the MCMC calculations, the black lines show the mean of
the MCMC samples and the red, light-grey, and dark-grey bands correspond to
the [25, 75], [10,90], and [1, 99] percentile intervals, respectively. Frame
(b) shows histograms corresponding to MCMC samples along the full length of
the filament.

##  Appendix B Additional figures of fits to OMC-3 MIR data

Section 3.1 discussed the filament properties based on OMC-3 MIR observations,
and Fig. 3 showed the estimated parameters for the filament segment A. Here
Figs. 20-22 show the corresponding results for the other segments B-D.

Figure 20:  Plummer fits of MIR absorption. As Fig. 3 but for OMC-3 filament
segment B.  Figure 21:  Plummer fits of MIR absorption. As Fig. 3 but for
OMC-3 filament segment C.  Figure 22:  Plummer fits of MIR absorption. As Fig.
3 but for OMC-3 filament segment D.

##  Appendix C Additional figures of fits to OMC-3 FIR data

Section 3.2 presented results of Plummer fits to column-density data that were
obtained from Herschel and ArTéMiS FIR measurements, including figures for the
filament segment A. We present here the corresponding figures for segments
B-D. Figures 23-25 show the results based on Herschel data ($20\arcsec$ map
resolution and fits to an area [-90$\arcsec$,+90$\arcsec$] around the filament
centre). The corresponding results for the combined Herschel and ArTéMiS data
are plotted in Figs. 26-28.

Figure 23:  Plummer fits of Herschel column-density data at 20$\arcsec$
resolution. As Fig. 4 but for OMC-3 filament segment B.  Figure 24:  Plummer
fits of Herschel column-density data at 20$\arcsec$ resolution. As Fig. 4 but
for OMC-3 filament segment C.  Figure 25:  Plummer fits of Herschel column-
density data at 20$\arcsec$ resolution. As Fig. 4 but for OMC-3 filament
segment D.  Figure 26:  Plummer fits of combined Herschel and ArTéMiS data at
10$\arcsec$ resolution. As Fig. 5 but for OMC-3 filament segment B.  Figure
27:  Plummer fits of combined Herschel and ArTéMiS data at 10$\arcsec$
resolution. As Fig. 5 but for OMC-3 filament segment C.  Figure 28:  Plummer
fits of combined Herschel and ArTéMiS data at 10$\arcsec$ resolution. As Fig.
5 but for OMC-3 filament segment D.

##  Appendix D Effect of filament inclination

We tested the effect of different viewing angles using filament models with
column densities $N({\rm H}_{2})=3\cdot 10^{22}$, $10^{23}$, and $3\cdot
10^{23}$ cm-2 perpendicular to the filament axis. By increasing the
inclination from zero to 70.5 degrees, the line-of-sight column density
increases by a factor of three. Therefore, the comparison to the $N({\rm
H}_{2})=10^{23}-10^{24}$ cm-2 models shows the relative importance of the true
versus line-of-sight column densities.

Figure 29 shows that the effect of inclination is much smaller than that of
the actual filament column density. At higher inclinations, $R_{\rm 0}$ and
FWHM show no indication of rising close to the values of the filaments with
the same line-of-sight column density observed at zero inclination. This is
not surprising, given that the relative amounts of dust at different
temperatures is not affected by the inclination, and the 160 $\mu$m emission
remains optically thin ($\tau\sim 0.3$ at $N({\rm H}_{2})=10^{24}$ cm-2). At
the highest inclinations, the results may also be affected by edge effects, as
the line of sight enters and leaves the model volume (on the front and back
border of the model volume) relatively close to the filament axis, where the
density is not negligible.

Figure 29:  Dependence of filament parameters on the column density and
inclination of model filaments. The cyan, blue, and red curves and symbols
show, respectively, parameter estimates at column densities $N({\rm
H}_{2})=3\cdot 10^{22}$ cm-2, $N({\rm H}_{2})=10^{23}$ cm-2, and $N({\rm
H}_{2})=3\cdot 10^{23}$ cm-2 as functions of the filament inclination. Zero
inclination corresponds to the perpendicular view direction and 70.5 degrees
to a factor of three higher line-of-sight column density. For reference, the
horizontal black dashed lines show the zero-inclination values for the three
column densities and additionally for $N({\rm H}_{2})=10^{24}$ cm-2 (the
uppermost lines).

##  Appendix E Tests of fitting mid-infrared filament profiles

Errors in the determination of the foreground surface-brightness component
$I^{\rm fg}$ will affect not only the level of optical-depth estimates but
also cause uncertainty in the filament profiles. We examined these effects
with simulated observations, where the optical depth follows a Plummer
profile. The dimensionless parameters of the Plummer model are $\tau_{\rm
0}$=4.0 for the peak optical depth, $R_{\rm 0}$=3.0 for the central flat
region, and $p_{\rm 0}$=2.5 for the asymptotic power law. The filament is seen
against extended background $I^{\rm bg}$=10.0 units and the foreground is set
to $I^{\rm fg}$=2.0 units. We simulate a piece of a filament where the ridge
optical depth is further modified by a Gaussian function, dropping to 40% of
the maximum at both ends.

The simulation consists of synthetic observations with two sources of error.
The first is the observational noise and the second an error in the estimated
level of the extended emission component $I^{\rm ext}$. For simplicity, we use
the same noise value for both components. Although $I^{\rm ext}$ can often be
estimated based on a large number of pixels, the interpolation across the
extincted regions is uncertain, and the uncertainty of $I^{\rm ext}$ could be
larger or smaller than the uncertainty of a single surface brightness
measurement. The error of the $I^{\rm fg}$ estimate follows from the
observational noise, because it is here estimated from Eq. (8). To keep the
statistics similar to observations, $I^{\rm fg}$ is estimated using only 64
points along the filament length. However, to gather more statistics for the
following plots, the rest of the simulation is done using 1024 independent
profiles along the filament.

First, Fig. 30 shows examples of filament profiles that are derived from noisy
observations with 0.2 units of noise in both surface brightness and $I^{\rm
ext}$ values. Since Eq. (8) is used, the optical depth estimate is always
defined. On the other hand, there is a small bias whereby the optical depth at
the filament centre tends to be underestimated. Because this bias is largest
when the observed surface brightness approaches the assumed $I^{\rm fg}$
value, the recovered profiles tend to be flatter in the centre and thus also
have FWHM values that are slightly larger than for the true filament profile.

Figure 30:  Example simulated MIR observations of a model filament. Frames (a)
and (b) show the ideal absorption and the corresponding true optical depth
image. Frames (c) and (d) are the corresponding noisy realisations, with noise
0.2 units for both surface brightness observations and for the determination
of the level of $I^{\rm ext}$, and frames (e) and (f) show the differences
between frames (a) and ()b and between frames (b) and (d). The black curves in
frame g show examples of profiles, from the end of the filament (bottom curve)
to the centre of the filament length (uppermost curve). The curves are offset
along the y axis for better readability. For each ideal profile, there are ten
realisations of the profiles derived from noisy observations.

To study the accuracy of the recovered filament parameters, we simulated
further four situations. First, the noise values were set to either 0.2 or
0.55 units. Second, in addition to direct application of Eq. (8), we also
tested a case where $I^{\rm fg}$ is set based on the minimum observed surface
brightness value. The results in Fig. 31 suggest that the method used for
$I^{\rm ext}$ estimation has only little effect. On the other hand, with
increasing noise the central optical depth becomes significantly
underestimated while the computed FWHM values are biased upwards.

Figure 31:  Histograms of extracted filament parameters for simulated
observations. The first three columns show the recovered central optical depth
relative to the true value, $\tau/\tau_{0}$, and the parameters $R$ and $p$.
The last column contains the FWHM values computed from the Plummer parameters.
In frames (a)-(h), $I^{\rm fg}$ is estimated with Eq. (8), in frames (i)-(p)
simply based on the minimum observed value. The noise values are 0.2 in frames
(a)-(d) and (i)-(l) and 0.75 in frames (e)-(h) and m-p. In each frame, we
overplot ten histograms. Each of the individual histograms corresponds one
random error in the $I^{\rm ext}$ value and contains 640 realisations of
filament profiles with random observational errors.

##  Appendix F Further images related to dust emission

Related to the discussion in Sect. 4.1, Fig. 32 illustrates further the bias
in the optical depths derived from FIR dust emission. The figure shows the
ratio of estimated and true optical depths for the $N({\rm
H}_{2})=10^{21}-10^{23}$ cm-2 models and the alternative descriptions of the
radiation field. The optical-depth ratios are shown along the filament centre
line and, at two positions, across the filament.

Figure 32:  Ratio of estimated and true optical depths in case of simulated
FIR observations of dust emission. The rows correspond different filament
column densities. The columns correspond to different cases of the radiation
field, with isotropic illumination or, alternatively, also including a point
source at $\Delta y=2.09$ pc and at a distance of 0.93 pc from the filament
centre, either along the line of sight towards the filament or to the left of
the filament ($\Delta x$=-0.93 pc). The ratios are plotted along the filament
centre line (red curves, as a function of $\Delta y$) or across the filament
at $\Delta y=1.4$ pc (blue line) or $\Delta y=0.7$ pc (dashed cyan line).

Results on MIR dust emission were shown in Sect. 4.2.2. Similar to Fig. 15, we
include below figures for the other radiation field configurations. Figure 33
shows parameter estimates along a $N({\rm H}_{2})=3\cdot 10^{23}$ cm-2
filament that is illuminated by an isotropic radiation field with $\chi=1$. In
Figs. 34 and 35, the model also includes the 590 $L_{\sun}$ radiation source
that is located 0.94 pc in front of or behind the filament.

Figure 33:  As Fig. 15, but showing the effect of dust emission on the MIR
filament parameters in the case of an isotropic external radiation field only
($\chi$=1). The simulated surface brightness is assumed to consist of
extincted background radiation and thermal emission from stochastically heated
grains.  Figure 34:  As Fig. 33 but with the point source located 0.93 pc in
front of the model filament. Because of thermal emission from stochastically
heated grains, the filament is not visible as an MIR absorption feature at
$\Delta y>1.2$ pc and the corresponding data should be ignored.  Figure 35:
As Fig. 33 but with the point source located 0.93 pc behind the model
filament. The filament is not visible as an absorption feature at $\Delta
y>1.6$ pc, and the corresponding data should be ignored.

Figures 36-39 show the results for a less massive filament. The filament has
$N({\rm H}_{2})=3\cdot 10^{22}$ cm-2 but the assumed background sky brightness
is higher, 100 MJy sr-1.

Figure 36:  As Fig. 15 but for the case of lower column density $N({\rm
H}_{2})=3\cdot 10^{22}$ cm-2, higher background surface brightness $I^{\rm
bg}$=100 MJy sr-1, and the model filament illumination by an isotropic
radiation field only ($\chi$=1). The observed surface brightness is assumed to
consist only of the extincted background and thermal dust emission from
stochastically heated grains.  Figure 37:  As Fig. 36 but including a point
source at $\Delta y=2.09$ pc and 0.93 pc in front of the model filament.
Around $\Delta y\approx 2-2.2$ pc, the filament is not seen in absorption
because of thermal emission from stochastically heated grains.  Figure 38:  As
Fig. 36 but including a point source at $\Delta y=2.09$ pc and 0.93 pc behind
the model filament. Around $\Delta y\approx 2-2.2$ pc, the filament is not
seen in absorption because of thermal emission from stochastically heated
grains.  Figure 39:  As Fig. 36 but including a point source at $\Delta
y=2.09$ pc and 0.93 pc to one side of the model filament.

##  Appendix G Toy model of a filament with two temperature components

Figure 40 show toy models of a core and a filament, both with a Gaussian
optical depth profile with $FWHM(\tau)=20\arcsec$. If the dust properties and
especially the dust temperature were constant, one could precisely recover the
expected convolved optical depth profiles with
$FWHM=\sqrt{FWHM(\tau)^{2}+FWHM({\rm beam})^{2}}$, where we now assume
$FWHM({\rm beam})=20\arcsec$ and $40\arcsec$ for the HR and LR maps,
respectively.

The model of Fig. 40 consists however of two dust components at 10 K and 20 K
temperatures. The fractional abundance of one component increases
proportionally to $\tau$, and the abundance of the other component decreases,
keeping their sum equal to $\tau$. Thus, at the distance from the filament
centre where the optical depth has dropped to half of its maximum value, the
optical depths of the warm and cold components are equal. Both methods (HR and
LR) now give biased estimates of the filament width. If the filament centre is
colder, the filament width is overestimated, and if the centre is warmer, the
width is underestimated. The relative errors are larger for HR maps and,
unlike in the isothermal cases, the errors are not identical in spherical and
cylindrical geometries. In the filament case of Fig. 40b, the relative errors
are +29% and -10% for the HR map and -3% and +5% for the LR map. After
deconvolution555The deconvolved width is $\sqrt{FWHM({\rm
observed})^{2}-FWHM({\rm beam})^{2}}$. The formula is in this case exact
because the filament has a Gaussian profile. the estimated width of a filament
with a cold centre is 30.6$\arcsec$ based on the HR map and 24.8$\arcsec$
based on the LR map. Relative differences are larger for the HR map, which
results in deconvolved sizes of 30.6$\arcsec$ and 15.8$\arcsec$ for the cold-
core and the hot-core filaments, respectively. Quantitatively, the errors
depend on the average temperature, the temperature contrast, and steepness and
radial location of the temperature change. The above example with 20 K versus
10 K components may exaggerate the typical effects, but nevertheless suggests
that the errors can be significant.

Figure 40:  Optical-depth profiles of toy models consisting of 10 K and 20 K
dust. The model in the left frame is a Gaussian sphere, and the model on the
right a cylinder with the same Gaussian cross-section with $FWHM=20\arcsec$.
The black curves are the expected profiles observed at 20$\arcsec$ and
40$\arcsec$ resolution. The actually profiles (from HR and LR maps) are
plotted in red (models with warm centre) and blue (models with cold centre),
where solid lines show the HR and the dashed lines the LR results. The
estimated and expected FWHM values are quoted in the same colours (HR on left,
and LR on right side).