# Fault Localization in a Microfabricated Surface Ion Trap using Diamond
Nitrogen-Vacancy Center Magnetometry

Pauli Kehayias  Present address: MIT Lincoln Laboratory, Lexington,
Massachusetts, 02421, USA  Sandia National Laboratories, Albuquerque, New
Mexico 87185, USA  Matthew A. Delaney  Sandia National Laboratories,
Albuquerque, New Mexico 87185, USA  Raymond A. Haltli  Sandia National
Laboratories, Albuquerque, New Mexico 87185, USA  Susan M. Clark  Sandia
National Laboratories, Albuquerque, New Mexico 87185, USA  Melissa C. Revelle
Sandia National Laboratories, Albuquerque, New Mexico 87185, USA  Andrew M.
Mounce<EMAIL_ADDRESS>Center for Integrated Nanotechnologies, Sandia
National Laboratories, Albuquerque, New Mexico 87123, USA

###### Abstract

As quantum computing hardware becomes more complex with ongoing design
innovations and growing capabilities, the quantum computing community needs
increasingly powerful techniques for fabrication failure root-cause analysis.
This is especially true for trapped-ion quantum computing. As trapped-ion
quantum computing aims to scale to thousands of ions, the electrode numbers
are growing to several hundred with likely integrated-photonic components also
adding to the electrical and fabrication complexity, making faults even harder
to locate. In this work, we used a high-resolution quantum magnetic imaging
technique, based on nitrogen-vacancy (NV) centers in diamond, to investigate
short-circuit faults in an ion trap chip. We imaged currents from these short-
circuit faults to ground and compared to intentionally-created faults, finding
that the root-cause of the faults was failures in the on-chip trench
capacitors. This work, where we exploited the performance advantages of a
quantum magnetic sensing technique to troubleshoot a piece of quantum
computing hardware, is a unique example of the evolving synergy between
emerging quantum technologies to achieve capabilities that were previously
inaccessible.

##  I Introduction

Quantum computing is a promising field investigating new computing
capabilities using qubits and quantum algorithms. Superconducting qubits,
Rydberg atoms, quantum dots, solid-state defect centers, photons, and trapped
ions are among the candidate technologies being investigated for quantum
computing. Trapped-ion qubits are a leading technology for achieving a
scalable quantum computer, due to ion qubits having long coherence times,
optical and microwave state manipulation, controllable and reconfigurable
interactions between ions, and the possibility to use a room-temperature
apparatus. Microfabricated ion trap chips are an appealing approach for
creating the electrostatic and radiofrequency (RF) potentials needed for
scalable trapped-ion quantum computing, as they offer the possibility of
repeatable large-scale fabrication and the ability to route electronic and
photonic paths under the trap surface [1, 2]. As ion trap chips become more
sophisticated (with integrated photonics, photodiodes, electrodes, and
electrical routing), traditional electronics failure analysis (FA) techniques
have a growing challenge to keep up with their increasing complexity [3, 4, 5,
6, 7]. In particular, many FA techniques that rely on optical access may
struggle to detect faults deep within ion trap chips with high-density layouts
and a high metal density.

Magnetic field imaging (MFI) is an attractive nondestructive fault-
localization technique for locating and diagnosing short-circuit faults [8].
Using an MFI instrument, one maps the magnetic field emissions from currents
within a device, then infers the fault location by interpreting the current
paths from the resulting magnetic images. MFI is appealing because most
microfabrication materials (metals, semiconductors, and insulators) are
transparent to magnetic fields, making it possible to look deep into an
otherwise-opaque device independent of what materials are nearby. In addition,
MFI can yield information about internal current paths, which is not possible
with many other FA techniques. One instrument in particular, the quantum
diamond microscope (QDM) [9, 10, 11], has shown recent promise for high-
resolution MFI for electronics failure analysis and interrogation [12, 13, 14,
15, 16, 17, 18, 19]. By putting a diamond sample with a thin surface layer of
magnetically-sensitive nitrogen-vacancy (NV) color centers on top of a device
(in our case, an ion trap chip), illuminating the NVs with green laser light,
and imaging the NV fluorescence with an optical microscope, we can extract the
magnetic field in every pixel (Fig. 1a-b). Because QDM setups yield micron-
scale spatial resolution over a few-millimeter field of view, measure all
pixels simultaneously without scanning, and excel at detecting shallow
magnetic sources within a few microns of the diamond surface, they are an
ideal choice for electronics interrogation.

\begin{overpic}[width=411.93767pt]{fig1_v2b.pdf} \put(1.0,39.0){{\Large a}}
\put(1.0,17.0){{\Large b}} \put(27.0,39.0){{\Large c}} \end{overpic} Figure 1:
(a) Schematic drawing of the diamond on top of an ion trap chip in a QDM
setup. A 532 nm laser beam illuminates the NV layer, and the red NV
fluorescence is collected by a microscope objective and imaged onto a camera
with an optical microscope (not shown). (b) A photo of an example ion trap
chip in a QDM setup. (c) Cross-section drawing of the Roadrunner ion trap
chip, showing the metal and insulating layers, tungsten vias, trench
capacitors, and the device and handle silicon. For the MFI experiments, the
diamond sample (containing an NV layer) is mounted on top.

In this paper, we describe our work using a QDM to locate short-circuit faults
in an ion trap chip, leading to the discovery of the root-cause explanation
for these faults. We studied an ion trap chip of the Sandia Roadrunner design
[20] with several short-circuit faults (electrode shorts to ground). Initial
use of traditional test methods was unable to narrow down the root cause from
candidates such as electrostatic discharge (ESD), scratches at the bond pads,
scratches at the electrodes, conducting particles shorting the metal layers,
and others. After imaging the magnetic fields from probe currents applied to
the device, we found that every current path dissipated at a trench capacitor,
indicating that the integrated trench capacitors were most likely the cause.
To confirm this, we added intentional faults (overloaded capacitors, scratches
on the bond pads, and scratches on the electrodes) and compared the magnetic
images of the “natural” and “intentional” faults to validate this finding.

This work informs future ion trap chip designs and fabrication recipes, as we
now know to give extra scrutiny to the trench capacitor fabrication steps.
Additionally, we can avoid the potential problems introduced by trench
capacitors by replacing them with external chiplet trench capacitors that can
be tested separately before assembly. This result also confirms that the other
possible failure mechanisms (e.g. assembly) don’t need further scrutiny at
this time. Furthermore, as the QDM gains traction as a novel tool for
electronics interrogation, this work represents a unique application
direction. In addition to troubleshooting sophisticated next-generation
integrated circuits (ICs), this quantum sensing approach can also solve
problems for quantum computing hardware, supporting the development of
computing technology that can surpass the current classical-computing limits.

\begin{overpic}[width=411.93767pt]{nvBgFigV1b.pdf} \put(0.0,31.0){{\Large a}}
\put(43.0,31.0){{\Large b}} \end{overpic} Figure 2:  (a) An energy-level
drawing of the NV ground-state magnetic sublevels. Given a magnetic field
$B_{111}$ along the NV axis, the $m_{s}=\pm 1$ sublevels shift by $\pm\gamma
B_{111}$ in frequency. (b) An example single-pixel ODMR spectrum. We fit each
lineshape to extract its center frequency, which yields the measured $B_{111}$
for that pixel. The lineshapes are split into three peaks, due to the NV 14N
hyperfine interaction.

##  II Materials and methods

###  II.1 Ion trap chip specifications

The ion trap chip tested in this work is the latest device from the Quantum
Scientific Computing Open User Testbed (QSCOUT) program at Sandia National
Laboratories, the Roadrunner trap [20]. Similar to other recently
microfabricated traps from Sandia (such as the Peregrine and Phoenix traps
[21]), the Roadrunner ion trap chip has six metal layers for routing control
electrode signals. It also has integrated trench capacitors to shield the
applied DC voltages from the RF trapping potential, gold coating for improved
ion qubit performance, and a junction to allow for ion re-ordering (Fig. 1c).
The trap has a bowtie shape to improve the optical access for lasers
addressing the ions, which constrains all of the bond pads to the edge of the
trap [22]. On each edge of the trap there are 50 control electrode wire bonds,
along with connections for the RF trapping voltage and ground. The control
electrode routing to the edge of the trap (for wire bonding) occurs through
the lower metal layers. Additionally, this routing connects to trench
capacitors located in the device silicon. The device chip is attached to a
custom 30 mm 102-connection bowtie package, where each electrode is linked to
a pin in a 13$\times$13 array.

We detected nine electrode shorts to ground in the device, with resistances
ranging from 60-160 $\ohm$. These shorts were discovered electrically, but the
root-cause explanation remained unknown. Furthermore, the faults could be
located anywhere along the paths between the bond pads and the electrodes, and
these paths span millimeters in length and several conducting layers in the
device. However, using a QDM to map the internal current paths with high
spatial resolution and comparing to the device layout geometry, we can locate
the faults, collect statistics on what components are failing, identify which
elements in the fabrication process and/or design are responsible, and
implement future design changes to mitigate these failure mechanisms.

###  II.2 NV magnetic microscopy apparatus

Our QDM setup is a home-built optical fluorescence microscope, with a
kinematic sample stage to mount devices for MFI measurements. The ion trap
chip was mounted in a zero insertion force (ZIF) electrical socket on the
sample stage, through which a probe current (typically $\pm$20 mA) was applied
to the ion trap chip. We placed a $4\times 4\times 0.5$ mm3 diamond sample
with a 4 $\upmu$m sheet of magnetically-sensitive NV centers on top of the
device (Fig. 1). The NV electronic ground-state magnetic sublevels are
sensitive to the magnetic field $B_{111}$ along the N-V axis (the [111]
crystallographic direction) through the Zeeman effect. This leads to the
$m_{s}=0$ to $m_{s}=\pm 1$ transition frequencies being shifted by $\pm\gamma
B_{111}$, where $\gamma\approx 28$ GHz/T is the NV gyromagnetic ratio (Fig.
2a). We used 200-400 mW of 532 nm laser light to illuminate the NV layer from
the side of the microscope objective, which pumps the NVs into the $m_{s}=0$
sublevel. A resonant microwave field from a nearby wire loop (not shown in
Fig. 1) transfers NV population from the $m_{s}=0$ “bright state” to the
$m_{s}=\pm 1$ “dark state” sublevels, reducing the NV fluorescence intensity.
Scanning the microwave frequency across the NV resonance frequencies and
imaging the NV fluorescence with the microscope, we collect an optically-
detected magnetic resonance (ODMR) spectrum for each camera pixel (Fig. 2b).
We fit the ODMR spectra for each pixel, extract the lineshape center
frequencies, and obtain a resulting $B_{111}$ map detected by the NV layer [9,
10].

We used a pair of permanent magnets (not shown in Fig. 1) to apply a
$1.4~{}\mathrm{mT}$ bias magnetic field along the N-V axis. For each field of
view, we measured the magnetic field maps for equal and opposite probe
currents ($\pm$20 mA), then subtracted the results. This yields the magnetic
field map from the probe current only, excluding the bias magnetic field
contribution. Multiple magnetic images were stitched together after measuring
the device in several locations. Our magnetic images are 2.03 $\times$ 1.27
mm2 in size, and typically took 20 minutes to acquire.

\begin{overpic}[width=411.93767pt]{dataFig1_v2c.pdf} \end{overpic} Figure 3:
Magnetic field maps (measured along the NV [111] direction) for natural shorts
in the ion trap chip, superimposed on the top metal layer. The magenta traces
show the paths between the bond pads and the trapping-region electrodes, which
the applied currents follow until they short to ground. We include three
examples faults (one for each category): one near the bond pads (Pad 10), one
in the center isthmus (Pad 90), and one at a capacitor bank (Pad 32). The
insets show the $1\times 1~{}\mathrm{mm}^{2}$ fault locations enlarged,
indicating that the faults occur at trench capacitor edges (black).
\begin{overpic}[width=411.93767pt]{dataFig2_v2c.pdf} \end{overpic} Figure 4:
Magnetic field maps for intentionally overloaded capacitors in the ion trap
chip, superimposed on the top metal layer. The magenta traces show the paths
between the bond pads and the trapping-region electrodes, which the applied
currents follow until they short to ground. We include three example faults
(one for each category): one near the bond pads (Pad 54), one in center
isthmus (Pad 83), and one at a capacitor bank (Pad 24). The similarities here
to Fig. 3 supports the hypothesis that trench capacitors (black) are the
culprit.  \begin{overpic}[width=411.93767pt]{dataFig3_v2c.pdf} \end{overpic}
Figure 5:  Magnetic field maps for intentional scratches on the top metal
layer of the ion trap chip, superimposed on the top metal layer. The magenta
traces show the paths between the bond pads and the trapping-region
electrodes, which the applied currents follow until they short to ground. We
include two examples: a bond-pad scratch (Pad 80, shown with a zoomed-in $\pm
10~{}\upmu$T colorbar) and an electrode scratch (Pad 63). The inset photos
show the scratches, including one on Pad 78 that was not measured. Comparing
to the characteristics seen in Fig. 3, these magnetic images indicate that the
natural faults were probably not caused by scratches on the top metal layer.

##  III Results

To diagnose the shorts in our ion trap devices, we used our QDM instrument to
magnetically image the currents flowing through the device, mapping the
current flow from the bond pads to the current termination points. After
magnetically imaging the “natural” shorts (i.e shorts caused during
manufacturing), we found that all of them were located at trench capacitors,
suggesting trench capacitors as the failure point. We confirmed this
hypothesis by creating “intentional” shorts, either by overloading previously
non-shorted capacitors or by scratching bond pads and electrodes to cause
shorts to ground. Comparing the magnetic images of natural and intentional
shorts, we confirmed trench-capacitor failures as the root cause.

###  III.1 Initial Testing: Natural Shorts

Figure 3 shows magnetic images from three faults, overlaid on the ion trap
chip. The magnetic images show the applied probe currents traveling along
internal traces (magenta) towards their respective electrodes (brown) before
shorting to ground and dissipating. Measuring magnetic images for the nine
faults, we found these faults fell into three categories:

  * •

Type I: Five faults near the bond pads (e.g. Pad 10). These occur at trench
capacitors (black) next to the electrodes, as illustrated by the inset.

  * •

Type II: Three faults in the center isthmus (e.g. Pad 90). These occur at
trench capacitors (black) in the center isthmus, as illustrated by the inset.

  * •

Type III: One fault $\sim$2 mm from the bond pads (Pad 32). This occurs at a
trench capacitor bank (black) along the path between the bond pad and the
electrode.

As indicated by the insets in Fig. 3, the locations where the probe currents
dissipate to ground suggest that trench capacitors are the culprit. To further
investigate this finding, we introduced intentional short-circuit faults into
the device, then measured their corresponding magnetic field images.

###  III.2 Hypothesis Testing: Intentional Shorts (Overloaded Capacitors)

To validate the hypothesis that trench capacitors are the root-cause
explanation for the faults, we overloaded several other capacitors (connected
to previously intact pins), causing them to short to ground, then measured
their magnetic images for comparison. As seen in Fig. 4, the capacitor-
overload intentional-short magnetic images resemble those of the natural
shorts, with Pads 54, 83, and 24 resembling faults of Types I, II, and III,
respectively. The similarities between Fig. 3 and Fig. 4 supports the trench
capacitor failure root-cause explanation.

###  III.3 Hypothesis Testing: Intentional Shorts (Scratches)

To rule out the other fault root-cause explanation possibilities, we also
added additional faults by scratching the top metal layer with a probe tip
(Fig. 5), again on previously intact pins. MFI above the bond pads was not
attempted because of the bond wires in the way [13]. However, after scratching
Pad 80, the resulting magnetic field map shows the fringe of a magnetic
feature to the right of the field of view. This differs from the natural Type
I faults, where current dissipates more inland with stronger and more compact
magnetic features. Conversely, for the scratched electrode in the center
isthmus (connected to Pad 63), the probe current traces the entire path before
dissipating at the electrode itself. This differs from the natural Type II
faults, where the current dissipates before reaching the electrode. Since
there are no bond pads or electrodes $\sim$2 mm from the bond pads, Type III
faults are unlikely to be reproduced by these intentional scratches. Finally,
we note that scratches on the top metal layer should be visible with an
optical microscope, while shorts in a trench capacitor (in the deeper layers)
should be invisible. While it is difficult to create intentional shorts to
ground due to conducting particles between the metal layers, these
hypothetical shorts should occur in random locations, which is inconsistent
with the fault/trench capacitor co-location. These arguments provide a
convincing case that trench capacitor faults are responsible for the shorts in
this ion trap chip.

##  IV Discussion

The above results lead us to conclude that the faults are due to the trench
capacitors, ruling out the other possible explanations (e.g. wire bond faults,
packaging, and metal particle contamination during fabrication) and saving us
from further analysis efforts pursuing solutions to those potential problems.
We identified several methods by which the trench capacitors could fail. One
possibility is ESD in the packaging lab or silicon fabrication facility.
Alternatively, the trench capacitor faults could be due to conducting
particles landing on the device during the relevant trench capacitor
fabrication steps. The investigation into distinguishing these possible
explanations is ongoing.

Measuring a second ion trap chip of a different design (a Peregrine trap [21])
with natural shorts spanning $\sim$100 $\ohm$ to $\sim$1 M$\ohm$, we also
found that its faults were also located at its trench capacitors [23]. Since
two different ion trap chip designs made in different fabrication runs having
different short resistances have the same fault root-cause explanations, this
suggests a possible common underlying mechanism. Furthermore, the combination
of these results informs future ion trap chip designs to either avoid trench
capacitors (replacing them with external chiplet trench capacitors that can be
tested separately before assembly) or to spend additional effort
troubleshooting the fabrication process. Overall, these results inform future
designs and fabrication attempts, enabling a more robust fabrication process
for better yield.

Other attempts to pinpoint the exact fault locations, such as cross-sectioning
the device using a focused ion beam (FIB) and looking for anomalies with a
scanning electron microscope (SEM), were unsuccessful. This can be interpreted
as an advantage of our high-resolution QDM MFI approach, since MFI was able to
locate the faults when cross-sectioning could not (even when the locations
were already known).

##  V Conclusions

In this work, we used a QDM apparatus for high-resolution MFI to locate short-
circuit faults in an ion trap chip, leading to the conclusion that the faults
were caused by trench capacitors. By intentionally overloading other
capacitors in the same device, and by scratching other bond pads and
electrodes, we were able to confirm that trench capacitors were the root
cause. This rules out other possible causes, such as mechanical damage in the
post-fabrication assembly processes. More generally, this work shows that, in
addition to troubleshooting multilayer and multi-die ICs for sophisticated
next-generation electronic components, a QDM can also apply its quantum
sensing advantages to solve problems for quantum computing hardware as well
[6, 7].

Solving FA problems is especially important for next-generation ion trap chips
with increasingly sophisticated designs, such as on-chip single-photon
avalanche diodes (SPADs) and on-chip photonic waveguides for laser routing
[24, 25, 26, 27, 28, 29, 30]. We note that our ion trap chip fabrication is on
a small production scale, where five good devices are sufficient for the
intended experiments. This means that, unless the failure rate is substantial
(approaching $100\%$), we can usually tolerate some failed devices, and we can
sometimes avoid having to solve a low-yield problem with a brute-force
fabrication approach. However, if this were a large-scale commercial
fabrication effort, then our QDM failure analysis method is even more useful
for locating where faults occur in the devices. In fact, commercial trapped-
ion quantum computing entities are more motivated to use this approach to
maximize their yield. Just as FA development is needed to match increasingly-
complex electronics fabrication, the same is also true for quantum computing
hardware if this technology is to be realistically scalable.

Quantum mechanics is transitioning from being an observed physics phenomenon
(the “first quantum revolution”) to being technologies where we use quantum
systems to achieve outcomes that are impossible classically (the “second
quantum revolution”) [31]. Both the QDM and ion trap chip are examples of such
quantum technologies. We note that this work, where we exploit the advantages
of a quantum sensor to troubleshoot a piece of quantum computing hardware, is
a unique example of one quantum technology supporting another. Finally, in
addition to being useful for FA, we anticipate that a QDM can provide
additional 2D and 3D electromagnetic characterization information for ion trap
chips that may be otherwise unavailable. These include mapping the RF trapping
potential ($\sim$50 MHz) and/or microwave gates fields (GHz-frequency [32,
33], to validate that these agree with what is expected from simulation,
ensure that other nearby components in the vacuum apparatus (such as the ion
trap chip socket) are not significantly altering the ion trapping environment,
measure the DC magnetic field gradients caused by magnetic components in the
ion trap chip (e.g. Kovar pins), and study electrical noise due to time-
varying patch potentials on the electrode surfaces [34].

##  VI Acknowledgements

We thank D. Stick and C. Nordquist for helpful discussions, R. Johnson for the
drawing in Fig. 1a, and L. Basso for the photograph in Fig. 1b. Sandia
National Laboratories is a multi-mission laboratory managed and operated by
National Technology and Engineering Solutions of Sandia, LLC, a wholly owned
subsidiary of Honeywell International, Inc., for the DOE’s National Nuclear
Security Administration under contract DE-NA0003525. This work was funded, in
part, by the Laboratory Directed Research and Development Program and
performed, in part, at the Center for Integrated Nanotechnologies, an Office
of Science User Facility operated for the U.S. Department of Energy (DOE)
Office of Science. This material is also funded, in part, by the U.S.
Department of Energy, Office of Science, Office of Advanced Scientific
Computing Research Quantum Testbed Program. This paper describes objective
technical results and analysis. Any subjective views or opinions that might be
expressed in the paper do not necessarily represent the views of the U.S.
Department of Energy or the United States Government.

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See pages 1 of suppl_v1h_arXiv.pdf See pages 2 of suppl_v1h_arXiv.pdf


1Indian Institute of Science Education and Research, Campus Road, Mohanpur,
Kolkata 741246, West Bengal, India
2Indian Institute of Astrophysics, II Block, Koramangala, Bengaluru 560034,
India
3National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, PR China
4Korea Astronomy and Space Science Institute (KASI) 776 Daedeokdae-ro,
Yuseong-gu, Daejeon 34055, Republic of Korea

#  $Herschel$ investigation of cores and filamentary structures in L1251
located in the Cepheus flare

Divyansh Dewan1,2,* Archana Soam2 Guo-Yin Zhang3 Akhil Lasrado2 Saikhom
Pravash Singh2 and Chang Won Lee4

###### Abstract

Context: Molecular clouds are the prime locations of star formation. These
clouds contain filamentary structures and cores which are crucial in the
formation of young stars.
Aims: In this work, we aim to quantify the physical properties of structural
characteristics within the molecular cloud L1251 to better understand the
initial conditions for star formation.
Methods: We applied the _getsf_ algorithm to identify cores and filaments
within the molecular cloud L1251 using the _Herschel_ multiband dust continuum
image, enabling us to measure their respective physical properties.
Additionally, we utilized an enhanced differential term algorithm to produce
high-resolution temperature maps and column density maps with a resolution of
${13.5}^{\prime\prime}$.
Results: We identified 122 cores in the region. Out of them, 23 are
protostellar cores, 13 are robust prestellar cores, 32 are candidate
prestellar cores (including 13 robust prestellar cores and 19 strictly
candidate prestellar cores), and 67 are unbound starless cores. _getsf_ also
found 147 filament structures in the region. Statistical analysis of the
physical properties (mass ($M$), temperature ($T$), size, and core brightness
(hereafter, we are using the word luminosity ($L$)) for the core brightness)
of obtained cores shows a negative correlation between core mass and
temperature and a positive correlation between ($M$/$L$) and ($M$/$T$).
Analysis of the filaments gives a median width of 0.14 pc and no correlation
between width and length. Out of those 122 cores, 92 are present in filaments
( 75.4%) and the remaining were outside them. Out of the cores present in
filaments, 57 ( 62%) cores are present in supercritical filaments ($M_{\rm
line}>16M_{\odot}/{\rm pc}$)

###### keywords:

ISM: Clouds – Stars: Formation – ISM: individual objects: Cepheus Cloud

<EMAIL_ADDRESS>
12.3456/s78910-011-012-3 #### 000 0000 1– 1

##  1 Introduction

Stars are found to be forming in the dense parts of the interstellar medium
(ISM) called molecular clouds. Inside these clouds, star formation is found to
occur in the large filamentary structures (André $et~{}al$. 2010; Molinari
$et~{}al$. 2010). Using _Herschel_ data, it has been found that these dusty
structures are hierarchical in nature (Men’shchikov 2021; Pokhrel $et~{}al$.
2017) and contain smaller units such as clumps, cores, filaments, and hub-
filament structures. The investigation of the statistics of these structures
can provide us with clues on the formation and evolution of star-forming
regions.

_Herschel_ dust maps at sub-millimeter wavelengths have played a crucial role
in showing the ubiquity of filamentary networks present in the cold ISM and
their importance in star formation in molecular clouds. Surveys of multiple
molecular clouds reveal a striking property of these filaments in that they
tend to have a characteristic width of 0.1 pc (Arzoumanian $et~{}al$. 2011).
The origin of this characteristic width is not yet clear with multiple
explanations claiming the width is a result of the thermodynamics of ISM gas
(Larson 2005), to the magneto-sonic scale below which the turbulence becomes
subsonic in diffuse, non-star-forming molecular gases (Padoan $et~{}al$.
2001). It has also been observed that the majority of prestellar cores form in
“supercritical” filaments where the mass per unit length exceeds the critical
line mass. This is calculated by modeling the filaments as nearly isothermal
long cylinders at a temperature of 10 K and is found to be $M_{\rm
line}>16M_{\odot}/{\rm pc}$ (Inutsuka & Miyama 1997). These supercritical
filaments are prone to gravitational instability, and as a result, they can
undergo fragmentation (Inutsuka & Miyama 1997). This fragmentation process
leads to the formation of denser regions called cores within the filaments
(Zhang $et~{}al$. 2020).

Recent research into filaments has revealed more details regarding their
structure and how it may impact core formation inside them. These include the
existence of velocity gradients roughly perpendicular to the axis of the
filament. These indicate the accretion of gas from the molecular cloud into
the filaments (Shimajiri $et~{}al$. 2019, Chen $et~{}al$. 2020). This
accretion flows along the filament in radial direction as shown in Hacar
$et~{}al$. (2022); Zhang $et~{}al$. (2020). The relation of mass growth of
cores within these filaments with this accretion was proposed by Gehman
$et~{}al$. (1996) and has been shown in works like Banerjee $et~{}al$. (2006)
and Gómez $et~{}al$. (2018)

There is a strong link present between the formation of prestellar cores and
filaments to star formation in these dense molecular clouds. These _Herschel_
survey results are in remarkable agreement with theory and numerical
simulations consistently showing that the ISM is highly filamentary on all
scales and star formation is intimately connected with self-gravitating
filaments (Hennebelle 2013). These simulations now successfully include
turbulence, gravity, various cooling processes, magnetohydrodynamics, and
radiation from massive stars. These numerical advances have provided us with
new and valuable insights into the physics of filaments and star formation,
including the formation, fragmentation, and further evolution of filaments
through accretion, and the central role of filaments in the rapid gathering of
gas into cluster-forming, dense regions (Könyves $et~{}al$. 2015; Li
$et~{}al$. 2023).

L1251 is a molecular cloud present in Cepheus Flare at a distance of $\sim$
340 $\pm$7 pc, extending from $l\sim 100^{\circ}-116^{\circ},b\sim
9^{\circ}-25^{\circ}$ (Right Ascension$\sim 19^{h}-23^{h}$, Declination $\sim
68^{\circ}-80^{\circ}$) (Sharma $et~{}al$. 2022) as shown in Fig 1 It has an
elongated structure with a cometary morphology, which is believed to have been
formed as a consequence of its interaction with a supernova bubble (Grenier
$et~{}al$. 1989; Sharma $et~{}al$. 2022). The structure of the cloud, shown
clearly in Fig 2, alongside the elongated structures mentioned above is
similar to a body flying at hypersonic speed across an ambient medium (Balázs
$et~{}al$. 2004). The position angle of the elongated cloud is found to be
$107^{\circ}$.

The objective of this paper is to study the structure of this cloud by
identifying the cores and filaments present and studying their physical
properties. Our method in extracting cores and filaments is very similar to
the work done in Zhang $et~{}al$. (2022) where they studied the Perseus
molecular cloud. We used multi-wavelength _Herschel_ data (2.1) to create
high-resolution ($13.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$) column
density and temperature maps using an improved difference-term algorithm. We
then extracted cores and filaments using _getsf_ along with their physical
properties. Understanding the formation of prestellar cores is crucial for
unraveling the early stages of star formation. By analyzing the properties of
cores and filaments in L1251 using the new tool _getsf_ , we may gain insights
into the role of these structures in facilitating star formation. This can
contribute to broader discussions on the influence of cloud morphology on star
formation efficiency. We also get to compare the results of different software
used to detect prestellar cores in the region This work is similar to existing
work, particularly with Di Francesco $et~{}al$. (2020) and Pattle $et~{}al$.
(2016), but while the former study employs other algorithms _getsources_
(Men’Shchikov $et~{}al$. 2012) and _getfilaments_ (Men’shchikov 2013) for core
and filament extraction, respectively, using _Herschel_ images of Cepheus
Flare in 70, 160, 250, 350, and 500 $\mu$m with pixel sizes of the images set
as $3.2\text{\,}\mathrm{\SIUnitSymbolArcsecond}$,
$3.2\text{\,}\mathrm{\SIUnitSymbolArcsecond}$,
$6\text{\,}\mathrm{\SIUnitSymbolArcsecond}$,
$10\text{\,}\mathrm{\SIUnitSymbolArcsecond}$, and
$14\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ respectively, we use _getsf_
(Men’shchikov 2021), which handles core and filament extraction
simultaneously. By comparing our results with theirs, we can see the
difference in the results of these algorithms, especially since _getsf_
doesn’t take any input from the user. Meanwhile, the latter study uses SCUBA-2
observations in 450 and 850 $\mu$m wavelengths with maps on to
$6\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ pixels and CSAR algorithm (Kirk
$et~{}al$. 2013); while we use _Herschel_ dust maps at 160, 250, 350, and 500
$\mu$m. We also focus exclusively on L1251 and find correlations between the
mass, luminosity, size, and temperature of detected cores.

In Sect 2, we discuss the data we have used in our analysis and the algorithm
we use for identifying cores and filaments. We then discuss the results
including the cores and filaments we extracted, their properties, along with
the correlations between them in Sect 3 We also do a small comparison with the
findings of Di Francesco $et~{}al$. (2020). Finally, in Sect 4, we conclude
our work and list our findings.

##  2 Data, Observation, and Techniques

###  2.1 _Herschel_ Data

_Herschel_ PACS images in 70 and 160 $\mu$m (Poglitsch, A. $et~{}al$. 2010)
and _Herschel_ SPIRE images in 250, 350 and 500 $\mu$m (Griffin, M. J.
$et~{}al$. 2010) were used in our analysis (Fig 2). The SPIRE/PACS parallel-
mode was used to make scan maps with a speed of
$60\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ s-1. The beam sizes for
_Herschel_ observations in 70, 160, 250, 350, and 500 $\mu$m are 8.4, 13.5,
18.2, 24.9, and $36.3\text{\,}\mathrm{\SIUnitSymbolArcsecond}$, respectively.
The data was downloaded from ESA _Herschel_ Science Archive
111http://archives.esac.esa.int/hsa/whsa/. The observation id’s for the images
used are 1342188654, 1342188655, 1342189663, and 1342189664.

We assume that each band of $\it Herschel$ satisfies optically thin dust
emission (Planck Collaboration $et~{}al$. 2014)

$I_{\nu}=\kappa_{0}\left(\nu/\nu_{0}\right)^{\beta}r\mu m_{\rm H}N_{{\rm
H_{2}}}B_{\nu}(T),$ (1)

where $I_{\nu}$ is the specific intensity, $\kappa_{0}$ is the emission cross-
section at a reference frequency $\nu_{0}$, $r$ is the dust-to-gas mass ratio,
$\mu$ is the mean molecular weight, $m_{\mathrm{H}}$ is the mass of a hydrogen
atom, $N_{\mathrm{H_{2}}}$ is the gas surface density, $B_{\nu}(T)$ is the
Planck function for dust at temperature $T$.

Figure 1: L1251 highlighted in the 857 GHz Plank map of the Cepheus flare
along with another molecular cloud L1228 Figure 2: _Herschel_ SPIRE 350 $\mu$m
image. The elongated structures are highlighted in the picture

A power-law form was assumed for the dust opacity(Roy $et~{}al$., 2014),
$\kappa_{\lambda}=0.1\times(\lambda/300\,\mu\rm{m})^{-\beta}\ cm^{2}\,g^{-1}$.
$\beta$ could vary with frequency, grain size, and grain temperature, but
using a fixed value of $\beta=2$ the accuracy of surface density will not
deviate by more than 50% (Roy $et~{}al$., 2014). We use Planck data as a
reference benchmark and convolve each band of Herschel to the $\it Planck$
resolution. The offset in flux density of each band of $\it Herschel$ can be
obtained by linear comparison (Molinari $et~{}al$., 2010). They are 42.9,
11.3, 3.9, and 0.8 $\rm MJy\ sr^{-1}$ at 160, 250, 350, and 500 $\mu$m,
respectively.

We also used these multi-wavelength dust maps to obtain high-resolution
($13.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$) column density and
temperature maps of the region using pixel-by-pixel SED fitting of the data to
a modified blackbody function. Instead of convolving all images to the lowest
resolution and using that directly for fitting, we used _getsf_ , which works
using the method described in Men’shchikov (2021) which employs a difference
term algorithm to increase the resolution from
$36.3\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ as we would have obtained from
the former method to $13.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$. The
obtained density and temperature maps via _hires_ are shown in Fig 3 and 4
respectively.

We now use the derived column density maps to obtain the total mass ($M$) of
L1251. We use the formula

$M=N(H_{2})\mu m_{H}A,$ (2)

where $N(H_{2})$ is the derived column density value from the column density
map Fig 3, $\mu$ is the mean weight of molecular gas taken to be 2.8 assuming
that the gas is 70% molecular hydrogen by mass (Ward-Thompson $et~{}al$.
2010), $m_{H}$ is mass of hydrogen atom, and A is projected area. The boundary
of the map was chosen by an observational mask where the usable part of the
map that contained the cloud was selected. This selection was made
empirically. The remaining part was considered background and was excluded
from measuring mass. While making the map, _getsf_ separates the background
image. This way, the mass of the background component is subtracted from the
mass of the cloud. Using this, we find the mass to be $\approx
2000\;M_{\odot}$.

Figure 3: $13.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ resolution column
density map of L1251 made using _getsf_ Figure 4:
$13.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ resolution temperature map of
L1251 made using _getsf_

###  2.2 Getsf

We use _getsf_ , a method for extracting sources and filaments in astronomical
images using separation of their structural components (Men’shchikov 2021), to
find, measure, and extract cores and filamentary structures in L1251. _getsf_
identifies structures in astronomical images by separating their structural
components. The basic steps involved in using _getsf_ and its image processing
are described below:

  1. 1.

First, all the different images need to be resampled to images with the same
pixel size and number. This is done via the built-in script - _resample_. An
observational mask may also be needed to reduce computational time.

  2. 2.

These resampled images are then used to obtain column density and temperature
maps using the built-in script - _hires_

  3. 3.

Then, _getsf_ uses structural decomposition into single-scale images to
isolate core and filamentary structures from their background. Noise due to
local fluctuations is removed via image flattening.

  4. 4.

These cleaned single-scale images are then combined over all the wavelengths
given, cores and filaments are identified, and their physical properties are
measured and cataloged using scripts _fmeasure_ and _smeasure_.

  5. 5.

The location and properties of detected structures are obtained in an
extraction catalog.

_getsf_ is a nearly fully autonomous algorithm with the only user input needed
for _hires_ being the offset in flux density (in MJy/sr) for the image,
correcting the image absolute calibration; and the maximum size of the
structure (in arcsecs) for detecting cores and filaments. More details on the
working of _getsf_ are in A

##  3 Results and Analysis

###  3.1 Core selection and classification

The obtained cores via _getsf_ are selected by the criteria described in B
This selection is based on benchmark tests given in Men’shchikov (2021) and is
based on properties like their size, detection significance, intensities, etc.
122 cores were identified by this method. Properties like their mass and
bolometric luminosity can be found by pixel-by-pixel SED fitting the
integrated flux measured at each wavelength by _getsf_ to a modified blackbody
function. The core size is defined as the mean deconvolved FWHM diameter at
the resolution of $13.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ of an
equivalent elliptical Gaussian source: $R=\sqrt{AFWHM\times BFWHM-13.5^{2}}$,
where AFWHM is the size of the major axis at half maxima and BFWHM is the size
of the minor axis at half maxima of the detected core. Using these parameters,
we classify the cores into different categories.

The classification is done by comparing the cores to a Bonnor-Ebert sphere
(Bonnor 1956; Ebert 1955). The critical mass of a Bonnor Ebert sphere - the
maximum mass that can support itself against gravitational attraction is given
by,

$M_{\rm crit}=\frac{2.4Rc_{\rm s}^{2}}{G}$ (3)

from Bonnor (1956); Ebert (1955), where R is the radius of the sphere and
$c_{\rm s}$ is the isothermal speed of sound, which is calculated using the
SED temperature of each core seperately, and G is the gravitational constant.

  1. 1.

The protostars in the region will have infrared emissions. The 70 $\mu$m dust
map is better suited to find regions of high temperature caused by these
emissions. Hence, our first step is to do a separate _getsf_ run with just the
70 $\mu$m image to locate the protostars in the cloud. These cores are then
checked via the criteria in B. All those who satisfy the criteria are termed
protostellar cores. As an additional measure, we have also cross-checked these
locations with the SIMBAD database to see if there are any known protostars in
the vicinity of identified protostellar cores and also to remove any galaxies
erroneously detected. The details of this are given in C.

  2. 2.

If the ratio $\alpha_{\rm BE}=M_{\rm crit}/M_{\rm core}\leq$ 2, the core is
deemed self-gravitating. This is analogous to the criteria used to select
self-gravitating objects based on the virial mass ratio (Bertoldi & McKee
1992) Cores meeting this criterion are called robust prestellar cores

  3. 3.

An empirical condition $\alpha_{BE}\leq 5\times(\sqrt{(AFWHM\times
BFWHM)}/$13.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$)^{0.4}$ obtained by
Monte Carlo simulations in Könyves $et~{}al$. (2015) is used to select
candidate prestellar cores. The simulated sources in Könyves $et~{}al$. (2015)
are all gravitationally bound, but the above-mentioned mass criteria cannot
fully select them, while this empirical formula can screen out 95% of the
gravitationally bound sources. We have modified the denominator in the formula
to 13.5 to account for the resolution of our column density map.

  4. 4.

The remaining cores are termed as unbound starless.

This methodology of the classification is described in Könyves $et~{}al$.
(2015). Out of 122 total cores, we found that 23 are protostellar, 13 are
robust prestellar cores, 32 are candidate prestellar (including 13 robust
prestellar cores and 19 strictly candidate prestellar cores), and 67 are
unbound starless cores. All of the selected robust prestellar cores also
belong to the broader criteria of candidate prestellar cores and hence are
also counted with them. The 19 strictly candidate cores are those that do not
fit the criteria for robust cores. The obtained cores are marked on the 250
$\mu$m dust map of L1251 in Fig. 5.

Figure 5: Positions of 122 cores identified via _getsf_ overlaid on 250 $\mu$m
dust map. 19 strictly candidate prestellar cores are shown in black, 23
protostellar cores are shown in green, 13 robust prestellar cores are shown in
red, and 67 unbound starless cores are shown in purple.

###  3.2 Statistical analysis of core properties

Using the data in the _getsf_ extraction catalog, the temperature and column
density maps, and equation 2 we obtained the mass, size, temperature, and
brightness of cores. These properties are shown in the histograms below (Fig
6). We obtained 0.2 $M_{\odot}$ as the median mass,
$46.3\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ (corresponding to 0.08 pc) as
the median radius, 13.9 K as the median temperature, and 0.05 $L_{\odot}$ as
the median luminosity.

We then looked for correlations between these properties. We found no
correlation in the way luminosity varied with mass or temperature (Fig 7(a)
and 7(b)) but obtained a negative correlation ($R^{2}=54.23\%$) of core mass
with core temperature (Fig 7(c)) and a positive correlation ($R^{2}=72.97\%$)
between ($M$/$L$) and ($M$/$T$) (Fig 7(d)).

Similiar correlations are also observed by Zhang $et~{}al$. (2022) and Marsh
$et~{}al$. (2014). The negative correlation of core mass with temperature is
consistent with our expectations. For a prestellar core, the primary heating
source is the Interstellar Radiation Field (ISRF). As the mass of the
prestellar core increases, the amount of dust shielding the interior of the
core from the radiation increases, and hence, the temperature decreases.

The positive correlation obtained between ($M$/$L$) and ($M$/$T$) of the cores
can be explained by considering the relation of the bolometric luminosity of
the cores with their temperature. This relation is shown explicitly in Zhang
$et~{}al$. (2022)

By making a histogram of the column densities of the positions of cores, we
see a sharp jump at a density of $\sim 10^{21}\mathrm{cm}^{-2}$ after which
cores start to form in the cloud. The maximum number of cores are formed just
after this density and their number decreases with increasing densities. The
histogram can be fitted by using a power-law of index $\sim$ 7.5 as shown in
Fig 6(e).

(a) Histogram of masses of detected cores

(b) Histogram of radii of detected cores

(c) Histogram of temperatures of detected cores

(d) Histogram of luminosities of detected cores

(e) Histogram of column density at locations of cores. The graph is fitted
with a power-law of index $\sim$ 7.5

Figure 6: Histograms of core properties

(a) No correlation between mass and luminosity of cores

(b) No correlation between luminosity and temperature of cores

(c) Correlation between mass and temperature of cores

(d) Correlation between ($M$/$L$) and ($M$/$T$) of cores

Figure 7: Correlation between core properties

###  3.3 Detected filaments

Filaments are complicated in their shapes and widths, often interconnected
with each other and various nearby branches, and have variable intensity along
their crests. The length of a filament is also not well defined, and neither
is it always clear which branch of the complex 3D structure of the filamentary
structure belongs to which filament. _getsf_ identifies and analyzes
filamentary structures by isolating a basic skeleton structure. The simplified
skeletons enable an easy selection and better measurements of only the well-
behaving, preferably isolated (not blended), and relatively straight parts of
the filaments. It identified 147 filament structures in the region. Using the
inbuilt script _fmeasure_ , we can measure the filamentary structure along its
skeleton and find its properties.

###  3.4 Statistical analysis of filament properties

Using _fmeasure_ to analyze filament properties on the
$13.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ column density map, we can find
physical properties like their lengths, widths, linear densities (same as line
mass), and masses. The working of _fmeasure_ is described in A. Altough the
length of a filament is not objectively well defined, it is clear to
understand that the longer a segment is, the more reliable it is as a
structure to measure filament properties. Out of the 147 identified filaments,
we selected the filaments for statistical analysis which had a length greater
than 0.2 pc (at least twice the accepted filament width) and detection signal-
to-noise ratio $>$2.5. This criteria leaves us with 70 filaments. Out of them,
11 have a length/width ratio $>$ 3\. For the selected filaments, we can see
the variation in linear densities of the filaments in the region in Fig 8.
From this picture, it is clear that the filaments near the center of the cloud
have a higher average density than those near the boundaries with clear
regions of high filament densities in the middle and left part of the region.
Other properties of the selected filaments are shown in the histograms in Fig
10, and 10(d). We note that the obtained median width of the identified
filaments is $\sim$0.14 pc, in good agreement with the universal
characteristic _Herschel_ filament width of 0.1 pc Herschel filaments
(Arzoumanian $et~{}al$. 2011). The universality of this characteristic width
has however been challenged in recent studies, which contend that filament
widths are a function of the resolution of the data used to derive them
(Panopoulou $et~{}al$. 2022).

We can now check the position of the cores and see how many of them are
located in filaments and the linear density of the filaments they are present
in. The methods used to measure linear densities by _getsf_ are discussed in
A. We can calculate the distance 1 pixel corresponds to as the distance to the
cloud multiplied by resolution (in radians). Taking the resolution as
${13.5}^{\prime\prime}$ and distance as 340 pc, we see that 1 pixel
corresponds to $\approx 8.9\times 10^{-9}$ pc. Therefore, 0.05 pc on both
sides of the 1-pixel thick filament skeleton (for a total width of 0.1 pc)
obtained via _getsf_ corresponds to 5.6 pixels on each side. With this, we
will claim a core to be present inside a filament if it is within 6 pixels of
a filament. We can see from Fig 9 that the majority of cores are present in
gravitationally supercritical filaments with linear densities
$>16M_{\odot}/\mathrm{pc}$. Their distribution with linear density is given in
Table 1.

###  3.5 Comparison with similar work

If we compare our results with those obtained in Di Francesco $et~{}al$.
(2020) who used _getsources_ , an older version of the program _getsf_ we
used, we see that they obtain a total of 187 cores. Out of them, 11 are
protostellar, 53 are robust prestellar, 86 are candidate prestellar and 90 are
unbound cores. We can see that using _getsf_ on the same region gives us a
lower number of cores but a higher number of identified protostellar cores
identified via 70 $\mu$m emission. The percentage of unbound cores obtained
via _getsf_ is 54.92 % while its 48.13% using _getsources_
They also found an extensive filamentary network in L1251 and observed 80-100%
of cores to be in filaments, in agreement with our findings

Linear Density (D) ($M_{\odot}$/pc) | Unbound Starless | Candidate Prestellar | Robust Prestellar | Protostellar
---|---|---|---|---
Not in Filament | 25 (37.31 %) | 0 (0 %) | 0 (0 %) | 5 (21.74 %)
D $<$ 8 | 18 (26.87 %) | 0 (0 %) | 0 (0 %) | 1 (4.35 %)
8 $<$ D $<$ 16 | 8 (11.94 %) | 7 (21.88 %) | 2 (15.38 %) | 1 (4.35 %)
D $>$ 16 | 16 (23.88 %) | 25 (78.13 %) | 11 (84.62 %) | 16 (69.56 %)

Table 1: Number of cores in filaments of varying linear densities Figure 8:
Linear densities of the 147 detected filaments in the region. Figure 9: The
detected cores overlaid on detected filaments along with their linear
densities

(a) Histogram of lengths of detected filaments

(b) Histogram of widths of detected filaments

(c) Histogram of densities of detected filaments

(d) Histogram of masses of detected filaments

Figure 10: Histograms of filament properties

##  4 Conclusion

We studied the dust properties of molecular cloud L1251 in the Cepheus flare
region using multiband _Herschel_ maps. We began with understanding the cloud
structure and other physical properties by making temperature and column
densities maps via the _hires_ script of _getsf_. _getsf_ was also utilised to
locate and classify prestellar cores and filaments in the cloud. Using this,
we identified 122 cores and 147 filaments. We then used _getsf_ ’s in-built
scripts _smeasure_ , _fitfluxes_ , and _fmeasure_ to inspect the physical
properties of cores, namely temperature, size, mass, and luminosity, as well
as filament masses, lengths, linear densities, and widths.

Analysis of this data revealed that out of 122 cores, 92 are present in
filaments ( 75.4%) and the remaining do not have an identified parent
filament. Of the cores present in filaments, 57 ( 62%) are present in
supercritical filaments ($M_{\rm line}>16M_{\odot}/\mathrm{pc}$). We also find
a clear correlation between $M$ vs $T$ and ($M$/$L$) vs ($M$/$T$) of the
detected cores.

The former correlation is explained by considering the shielding effect of
dust grains on the core from interstellar radiation while the latter
correlation is a result of the relation of bolometric luminosity on
temperature via Wein’s Law.

Our results regarding core and filament properties are consistent with other
studies for _Herschel_ filaments of star-forming regions showing a
characteristic filament width of $\sim$ 0.1 pc and most prestellar cores
existing within dense filaments. We also see a stark column density threshold
above which prestellar cores are found (André 2017).

##  Appendix A Brief description of the working of _getsf_

We used the inbuilt script _hires_ of _getsf_ to obtain our column density and
temperature maps. To make column density and temperature maps, _getsf_ uses
pixel-by-pixel SED fitting to the Herschel data with a modified blackbody
function. It produces a base map of the region by convolving all the provided
_Herschel_ maps at 160, 250, 350, and 500 $\mu$m to the lowest resolution (In
our case, $36.3\text{\,}\mathrm{\SIUnitSymbolArcsecond}$) and using it to
create column density and temperature maps via fitting to the blackbody
function. Then the 160-350 $\mu$m maps are convolved and fitted to create a
less accurate map at $24.9\text{\,}\mathrm{\SIUnitSymbolArcsecond}$
resolution. This map is then convolved to the lower dimension
($36.3\text{\,}\mathrm{\SIUnitSymbolArcsecond}$) and the difference between
this map and the base map is found. This process is repeated for all images at
all resolutions. The difference terms are then added to the base density map.
To identify and select cores and filaments, _getsf_ performs the following
steps :

  1. 1.

Multi-wavelength images resampled to the same pixel size and the same grid of
pixels are taken as input.

  2. 2.

The images are spatially decomposed into single-scale images. Separation of
the structural components of sources and filaments from each other and from
their backgrounds happens from these spatially decomposed images.

  3. 3.

The residual noise and background fluctuations in the images of the separated
components of sources and filaments are removed via image flattening

  4. 4.

These cleaned single-scale images are then combined over all the wavelengths

  5. 5.

sources (positions) and filaments (skeletons) are detected in the combined
images of the components in their spatially decomposed images

The properties of cores and filaments are measured and multi-wavelength output
catalogs are formed using inbuilt scripts _smeasure_ and _fmeasure_. _getsf_
identifies several skeletal structures of the filamentary network and
considers the remaining network as its branches. _fmeasure_ takes in input as
the background-subtracted column density map and uses 2 methods to get the
linear densities of filaments, one is by directly integrating the area between
the skeleton and the largest extent on each side, and then dividing by length
to get density. The other method takes various sampling points along a crest
and considers its density. The median value of all sampling points is taken as
the density of this filamentary structure. The method is described in full in
Men’shchikov (2021)

##  Appendix B Selection criteria for source selection after _getsf_
extraction

The catalog obtained after a _getsf_ extraction contains information on the
detected courses. This information is used to select reliable cores from all
the detections. The information includes their detection significance and
goodness which are parameters relating to signal-to-noise ratio and source
reliability, the background subtracted peak intensities of the detected cores,
and the major size of half-maximum and the minor size of half-maximum of the
detections.

The selection criteria used to select reliable cores is based on benchmark
tests from Men’shchikov (2021):

  * •

$|$GOODM$|$ $>$ 1, where GOODM is monochromatic goodness

  * •

$|$SIGNM$|$ $>$ 1, where SIGNM is the detection significance from
monochromatic single scales.

  * •

$FXP_{BST}$/$FXP_{ERR}$ $>$ 2, where $FXP_{BST}$ is peak intensity and
$FXP_{ERR}$ is peak intensity error.

  * •

$FXT_{BST}$/$FXT_{ERR}$ $>$ 2, where $FXT_{BST}$ is total flux and $FXT_{ERR}$
is total flux error

  * •

AFWHM/BFWHM $<$ 2, where AFWHM and BFWHM are the major and minor axis full
widths at half-maximum of the elliptical approximations.

  * •

FOOA/AFWHM $>$ 1.15, where FOOA is full major axis of an elliptical footprint.

The first 4 conditions ensure that detected sources are well distinguished
from their background. The other 2 are conditions on their size and shape with
the 5th condition ensuring the core is circular or elliptical in shape, and
the last condition removing cores with unrealistically small ratios of their
footprint and half-maximum sizes

##  Appendix C Matches found by cross-referencing protostellar cores with
SIMBAD

We cross-referenced the positions of the protostellar cores detected via
_getsf_ with the SIMBAD database or young stellar objects or cores. Out of 23
protostellar detections, 14 were found to have corresponding SIMBAD entries
lying within $10\text{\,}\mathrm{\SIUnitSymbolArcsecond}$. Their positions and
SIMBAD identifiers are tabulated in Table 2:

No.. | RA (J2000) | DEC (J2000) | SIMBAD Identifier
---|---|---|---
1 | 338.8502558 | 75.2851396 | IRAS 22343+7501
2 | 337.4295695 | 75.2242914 | [KBB2008] L1251A C2
3 | 337.8064044 | 75.2133953 | [PWK2017] 62
4 | 337.844001 | 75.2057197 | [PWK2017] 63
5 | 339.8243047 | 75.1591029 | [PWK2017] 65
6 | 339.8120191 | 5.1800094 | [PWK2017] 72
7 | 339.0847361 | 75.3707067 | [PWK2017] 96
8 | 338.884616 | 75.2782663 | 2MASS J22352664+7516369
9 | 339.80096 | 75.2017797 | 2MASS J22391329+7512161
10 | 337.4995097 | 75.2344617 | IRAS 22290+7458
11 | 340.2499785 | 75.0652508 | IRAS 22398+7448
12 | 339.8728992 | 75.1822671 | JCMTSE J223931.5+751055
13 | 337.6335849 | 75.2360765 | NAME LDN 1251 A IRS 3
14 | 339.6975257 | 75.1921614 | SSTgbs J2238469+7511337

Table 2: SIMBAD objects corresponding to identified protostars

## Acknowledgements

We would like to thank Alexander Men’shchikov for his repeated assistance in
downloading and using _getsf_ which made most of this work possible. The
Herschel data was obtained from the ESA’s Herschel Science Archive (HSA). DD
and AS thank IIA VSP program for providing internship opportunity and
supporting this work.

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