# Revealing the diverse magnetic field morphologies in Taurus dense cores with
sensitive sub-millimeter polarimetry

Chakali Eswaraiah National Astronomical Observatories, Chinese Academy of
Sciences, A20 Datun Road, Chaoyang District, Beijing 100012, People’s Republic
of China  Di Li National Astronomical Observatories, Chinese Academy of
Sciences, A20 Datun Road, Chaoyang District, Beijing 100012, People’s Republic
of China  University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China  NAOC-UKZN Computational Astrophysics Centre, University of
KwaZulu-Natal, Durban 4000, South Africa  Ray S. Furuya Tokushima University,
Minami Jousanajima-machi 1-1, Tokushima 770-8502, Japan  Institute of Liberal
Arts and Sciences, Tokushima University, Minami Jousanajima-machi 1-1,
Tokushima 770-8502, Japan  Tetsuo Hasegawa National Astronomical Observatory
of Japan, National Institutes of Natural Sciences, Osawa, Mitaka, Tokyo
181-8588, Japan  Derek Ward-Thompson Jeremiah Horrocks Institute, University
of Central Lancashire, Preston PR1 2HE, United Kingdom  Keping Qiu School of
Astronomy and Space Science, Nanjing University, 163 Xianlin Avenue, Nanjing
210023, China  Key Laboratory of Modern Astronomy and Astrophysics (Nanjing
University), Ministry of Education, Nanjing 210023, China  Nagayoshi Ohashi
Academia Sinica Institute of Astronomy and Astrophysics, No.1, Sec. 4.,
Roosevelt Road, Taipei 10617, Taiwan  Kate Pattle Centre for Astronomy, School
of Physics, National University of Ireland Galway, University Road, Galway,
Ireland H91TK33  Sarah Sadavoy  Department for Physics, Engineering Physics
and Astrophysics, Queen’s University, Kingston, ON, K7L 3N6, Canada  Charles
L. H. Hull  National Astronomical Observatory of Japan, NAOJ Chile, Alonso de
Córdova 3788, Office 61B, 7630422, Vitacura, Santiago, Chile  Joint ALMA
Observatory, Alonso de Córdova 3107, Vitacura, Santiago, Chile  NAOJ Fellow
David Berry East Asian Observatory, 660 N. A’ohōkū Place, University Park,
Hilo, HI 96720, USA  Yasuo Doi Department of Earth Science and Astronomy,
Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba,
Meguro, Tokyo 153-8902, Japan  Tao-Chung Ching National Astronomical
Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District,
Beijing 100012, People’s Republic of China  Shih-Ping Lai Institute of
Astronomy and Department of Physics, National Tsing Hua University, Hsinchu
30013, Taiwan  Academia Sinica Institute of Astronomy and Astrophysics, P.O.
Box 23-141, Taipei 10617, Taiwan  Jia-Wei Wang Academia Sinica Institute of
Astronomy and Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan  Patrick M.
Koch Academia Sinica Institute of Astronomy and Astrophysics, No.1, Sec. 4.,
Roosevelt Road, Taipei 10617, Taiwan  Jungmi Kwon Department of Astronomy,
Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-0033, Japan  Woojin Kwon Department of Earth Science Education,
Seoul National University (SNU), 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic
of Korea  Pierre Bastien Centre de recherche en astrophysique du Québec &
département de physique, Université de Montréal, C.P. 6128 Succ. Centre-ville,
Montréal, QC, H3C 3J7, Canada  Doris Arzoumanian  Instituto de Astrofísica e
Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762
Porto, Portugal  Simon Coudé SOFIA Science Center, Universities Space Research
Association, NASA Ames Research Center, Moffett Field, California 94035, USA
Archana Soam SOFIA Science Center, Universities Space Research Association,
NASA Ames Research Center, Moffett Field, California 94035, USA  Lapo
Fanciullo Academia Sinica Institute of Astronomy and Astrophysics, No.1, Sec.
4., Roosevelt Road, Taipei 10617, Taiwan  Hsi-Wei Yen  Academia Sinica
Institute of Astronomy and Astrophysics, No.1, Sec. 4., Roosevelt Road, Taipei
10617, Taiwan  Junhao Liu School of Astronomy and Space Science, Nanjing
University, 163 Xianlin Avenue, Nanjing 210023, People’s Republic of China
Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210023, People’s Republic of China  Thiem Hoang
Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu,
Daejeon 34055, Republic of Korea  University of Science and Technology, Korea,
217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea  Wen Ping Chen
Institute of Astronomy, National Central University, Zhongli 32001, Taiwan
Yoshito Shimajiri National Astronomical Observatory of Japan, National
Institutes of Natural Sciences, Osawa, Mitaka, Tokyo 181-8588, Japan  Tie Liu
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, People’s Republic of China  Zhiwei Chen Purple Mountain
Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of
Sciences, 2 West Beijing Road, Nanjing 210008, People’s Republic of China
Hua-bai Li  Department of Physics, The Chinese University of Hong Kong,
Shatin, N.T., Hong Kong  A-Ran Lyo  Korea Astronomy and Space Science
Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea
Jihye Hwang Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro,
Yuseong-gu, Daejeon 34055, Republic of Korea  University of Science and
Technology, Korea, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of
Korea  Doug Johnstone NRC Herzberg Astronomy and Astrophysics, 5071 West
Saanich Road, Victoria, BC V9E 2E7, Canada  Department of Physics and
Astronomy, University of Victoria, Victoria, BC V8W 2Y2, Canada  Ramprasad Rao
Academia Sinica Institute of Astronomy and Astrophysics, No.1, Sec. 4.,
Roosevelt Road, Taipei 10617, Taiwan  Nguyen Bich Ngoc Vietnam National Space
Center, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi,
Vietnam  Pham Ngoc Diep Vietnam National Space Center, Vietnam Academy of
Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam  Steve Mairs East
Asian Observatory, 660 N. A’ohōkū Place, University Park, Hilo, HI 96720, USA
Harriet Parsons East Asian Observatory, 660 N. A’ohōkū Place, University Park,
Hilo, HI 96720, USA  Motohide Tamura National Astronomical Observatory of
Japan, National Institutes of Natural Sciences, Osawa, Mitaka, Tokyo 181-8588,
Japan  Department of Astronomy, Graduate School of Science, The University of
Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Astrobiology Center,
National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588,
Japan  Mehrnoosh Tahani Dominion Radio Astrophysical Observatory, Herzberg
Astronomy and Astrophysics Research Centre, National Research Council Canada,
P. O. Box 248, Penticton, BC V2A 6J9 Canada  Huei-Ru Vivien Chen Institute of
Astronomy and Department of Physics, National Tsing Hua University, Hsinchu
30013, Taiwan  Academia Sinica Institute of Astronomy and Astrophysics, No.1,
Sec. 4., Roosevelt Road, Taipei 10617, Taiwan  Fumitaka Nakamura  Division of
Theoretical Astronomy, National Astronomical Observatory of Japan, Mitaka,
Tokyo 181-8588, Japan  SOKENDAI (The Graduate University for Advanced
Studies), Hayama, Kanagawa 240-0193, Japan  Hiroko Shinnaga  Department of
Physics and Astronomy, Graduate School of Science and Engineering, Kagoshima
University, 1-21-35 Korimoto, Kagoshima, Kagoshima 890-0065, Japan  Ya-Wen
Tang  Academia Sinica Institute of Astronomy and Astrophysics, No.1, Sec. 4.,
Roosevelt Road, Taipei 10617, Taiwan  Jungyeon Cho  Department of Astronomy
and Space Science, Chungnam National University, 99 Daehak-ro, Yuseong-gu,
Daejeon 34134, Republic of Korea  Chang Won Lee Korea Astronomy and Space
Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of
Korea  University of Science and Technology, Korea, 217 Gajeong-ro, Yuseong-
gu, Daejeon 34113, Republic of Korea  Shu-ichiro Inutsuka Department of
Physics, Graduate School of Science, Nagoya University, Furo-cho, Chikusa-ku,
Nagoya 464-8602, Japan  Tsuyoshi Inoue  Department of Physics, Graduate School
of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan
Kazunari Iwasaki  Department of Environmental Systems Science, Doshisha
University, Tatara, Miyakodani 1-3, Kyotanabe, Kyoto 610-0394, Japan  Lei Qian
National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Road, Chaoyang District, Beijing 100012, People’s Republic of China  Jinjin
Xie National Astronomical Observatories, Chinese Academy of Sciences, A20
Datun Road, Chaoyang District, Beijing 100012, People’s Republic of China
Dalei Li  Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150
Science 1-Street, Urumqi 830011, Xinjiang, People’s Republic of China  Hong-Li
Liu Department of Astronomy, Yunnan University, Kunming, 650091, People’s
Republic of China  Chinese Academy of Sciences, South America Center for
Astrophysics, Camino El Observatorio #1515, Las Condes, Santiago, Chile
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, People’s Republic of China  Chuan-Peng Zhang  National
Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Road,
Chaoyang District, Beijing 100012, People’s Republic of China  Mike Chen
Department of Physics and Astronomy, University of Victoria, Victoria, BC V8W
2Y2, Canada  Guoyin Zhang  CAS Key Laboratory of FAST, National Astronomical
Observatories, Chinese Academy of Sciences, People’s Republic of China  Lei
Zhu  National Astronomical Observatories, Chinese Academy of Sciences, A20
Datun Road, Chaoyang District, Beijing 100012, People’s Republic of China
Jianjun Zhou Xinjiang Astronomical Observatory, Chinese Academy of Sciences,
150 Science 1-Street, Urumqi 830011, Xinjiang, People’s Republic of China
Philippe André  Laboratoire AIM CEA/DSM-CNRS-Université Paris Diderot,
IRFU/Service d’Astrophysique, CEA Saclay, F-91191 Gif-sur-Yvette, France
Sheng-Yuan Liu Academia Sinica Institute of Astronomy and Astrophysics, No.1,
Sec. 4., Roosevelt Road, Taipei 10617, Taiwan  Jinghua Yuan  National
Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Road,
Chaoyang District, Beijing 100012, People’s Republic of China  Xing Lu
National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
Nicolas Peretto  School of Physics and Astronomy, Cardiff University, The
Parade, Cardiff, CF24 3AA, UK  Tyler L. Bourke SKA Organisation, Jodrell Bank,
Lower Withington, Macclesfield, SK11 9FT, UK  Jodrell Bank Centre for
Astrophysics, School of Physics and Astronomy, University of Manchester,
Manchester, M13 9PL, UK  Do-Young Byun  Korea Astronomy and Space Science
Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea
University of Science and Technology, Korea, 217 Gajeong-ro, Yuseong-gu,
Daejeon 34113, Republic of Korea  Sophia Dai  National Astronomical
Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District,
Beijing 100012, People’s Republic of China  Yan Duan  National Astronomical
Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District,
Beijing 100012, People’s Republic of China  Hao-Yuan Duan  Institute of
Astronomy and Department of Physics, National Tsing Hua University, Hsinchu
30013, Taiwan  David Eden  Astrophysics Research Institute, Liverpool John
Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool,
L3 5RF, UK  Brenda Matthews  NRC Herzberg Astronomy and Astrophysics, 5071
West Saanich Road, Victoria, BC V9E 2E7, Canada  Department of Physics and
Astronomy, University of Victoria, Victoria, BC V8W 2Y2, Canada  Jason Fiege
Department of Physics and Astronomy, The University of Manitoba, Winnipeg,
Manitoba R3T2N2, Canada  Laura M. Fissel Department for Physics, Engineering
Physics and Astrophysics, Queen’s University, Kingston, ON, K7L 3N6, Canada
Kee-Tae Kim Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro,
Yuseong-gu, Daejeon 34055, Republic of Korea  University of Science and
Technology, Korea, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of
Korea  Chin-Fei Lee  Academia Sinica Institute of Astronomy and Astrophysics,
No.1, Sec. 4., Roosevelt Road, Taipei 10617, Taiwan  Jongsoo Kim Korea
Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon
34055, Republic of Korea  University of Science and Technology, Korea, 217
Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea  Tae-Soo Pyo
SOKENDAI (The Graduate University for Advanced Studies), Hayama, Kanagawa
240-0193, Japan  Subaru Telescope, National Astronomical Observatory of Japan,
650 N. A’ohōkū Place, Hilo, HI 96720, USA  Yunhee Choi  Korea Astronomy and
Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055,
Republic of Korea  Minho Choi  Korea Astronomy and Space Science Institute,
776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea  Antonio
Chrysostomou  SKA Organisation, Jodrell Bank, Lower Withington, Macclesfield,
SK11 9FT, UK  Eun Jung Chung Department of Astronomy and Space Science,
Chungnam National University, 99 Daehak-ro, Yuseong-gu, Daejeon 34134,
Republic of Korea  Le Ngoc Tram University of Science and Technology of Hanoi,
Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam
Erica Franzmann  Department of Physics and Astronomy, The University of
Manitoba, Winnipeg, Manitoba R3T2N2, Canada  Per Friberg  Rachel Friesen
National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA
22903, USA  Gary Fuller  Jodrell Bank Centre for Astrophysics, School of
Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13
9PL, UK  Tim Gledhill School of Physics, Astronomy & Mathematics, University
of Hertfordshire, College Lane, Hatfield, Hertfordshire AL10 9AB, UK  Sarah
Graves  East Asian Observatory, 660 N. A’ohōkū Place, University Park, Hilo,
HI 96720, USA  Jane Greaves  School of Physics and Astronomy, Cardiff
University, The Parade, Cardiff, CF24 3AA, UK  Matt Griffin  School of Physics
and Astronomy, Cardiff University, The Parade, Cardiff, CF24 3AA, UK  Qilao Gu
Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong
Kong  Ilseung Han  Korea Astronomy and Space Science Institute, 776
Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea  University of
Science and Technology, Korea, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113,
Republic of Korea  Jennifer Hatchell  Physics and Astronomy, University of
Exeter, Stocker Road, Exeter EX4 4QL, UK  Saeko Hayashi  Subaru Telescope,
National Astronomical Observatory of Japan, 650 N. A’ohōkū Place, Hilo, HI
96720, USA  Martin Houde  Department of Physics and Astronomy, The University
of Western Ontario, 1151 Richmond Street, London N6A 3K7, Canada  Koji
Kawabata  Hiroshima Astrophysical Science Center, Hiroshima University,
Kagamiyama 1-3-1, Higashi-Hiroshima, Hiroshima 739-8526, Japan  Department of
Physics, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima, Hiroshima
739-8526, Japan  Core Research for Energetic Universe (CORE-U), Hiroshima
University, Kagamiyama 1-3-1, Higashi-Hiroshima, Hiroshima 739-8526, Japan
Il-Gyo Jeong Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro,
Yuseong-gu, Daejeon 34055, Republic of Korea  Ji-hyun Kang Korea Astronomy and
Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055,
Republic of Korea  Sung-ju Kang  Korea Astronomy and Space Science Institute,
776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea  Miju Kang
Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu,
Daejeon 34055, Republic of Korea  Akimasa Kataoka  Division of Theoretical
Astronomy, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588,
Japan  Francisca Kemper European Southern Observatory, Karl-Schwarzschild-Str.
2, 85748 Garching, Germany  Academia Sinica Institute of Astronomy and
Astrophysics, No.1, Sec. 4., Roosevelt Road, Taipei 10617, Taiwan  Mark
Rawlings East Asian Observatory, 660 N. A’ohōkū Place, University Park, Hilo,
HI 96720, USA  Jonathan Rawlings Department of Physics and Astronomy,
University College London, WC1E 6BT London, UK  Brendan Retter  School of
Physics and Astronomy, Cardiff University, The Parade, Cardiff, CF24 3AA, UK
John Richer  Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue,
Cambridge CB3 0HE, UK  Kavli Institute for Cosmology, Institute of Astronomy,
University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK  Andrew Rigby
School of Physics and Astronomy, Cardiff University, The Parade, Cardiff, CF24
3AA, UK  Hiro Saito  Faculty of Pure and Applied Sciences, University of
Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan  Giorgio Savini
OSL, Physics & Astronomy Dept., University College London, WC1E 6BT London, UK
Anna Scaife  Jodrell Bank Centre for Astrophysics, School of Physics and
Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, UK
Masumichi Seta  Department of Physics, School of Science and Technology,
Kwansei Gakuin University, 2-1 Gakuen, Sanda, Hyogo 669-1337, Japan  Gwanjeong
Kim Nobeyama Radio Observatory, National Astronomical Observatory of Japan,
National Institutes of Natural Sciences, Nobeyama, Minamimaki, Minamisaku,
Nagano 384-1305, Japan  Kyoung Hee Kim Korea Astronomy and Space Science
Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea
Mi-Ryang Kim  Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro,
Yuseong-gu, Daejeon 34055, Republic of Korea  Florian Kirchschlager Department
of Physics and Astronomy, University College London, WC1E 6BT London, UK
Jason Kirk  Jeremiah Horrocks Institute, University of Central Lancashire,
Preston PR1 2HE, UK  Masato I.N. Kobayashi Astronomical Institute, Graduate
School of Science, Tohoku University, Aoba-ku, Sendai, Miyagi 980-8578, Japan
Vera Konyves  Jeremiah Horrocks Institute, University of Central Lancashire,
Preston PR1 2HE, UK  Takayoshi Kusune  Kevin Lacaille  Department of Physics
and Astronomy, McMaster University, Hamilton, ON L8S 4M1 Canada  Department of
Physics and Atmospheric Science, Dalhousie University, Halifax B3H 4R2, Canada
Chi-Yan Law  Department of Physics, The Chinese University of Hong Kong,
Shatin, N.T., Hong Kong  Department of Space, Earth & Environment, Chalmers
University of Technology, SE-412 96 Gothenburg, Sweden  Sang-Sung Lee  Korea
Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon
34055, Republic of Korea  University of Science and Technology, Korea, 217
Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea  Yong-Hee Lee  School
of Space Research, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu,
Yongin-si, Gyeonggi-do 17104, Republic of Korea  Masafumi Matsumura Faculty of
Education & Center for Educational Development and Support, Kagawa University,
Saiwai-cho 1-1, Takamatsu, Kagawa, 760-8522, Japan  Gerald Moriarty-Schieven
NRC Herzberg Astronomy and Astrophysics, 5071 West Saanich Road, Victoria, BC
V9E 2E7, Canada  Tetsuya Nagata  Department of Astronomy, Graduate School of
Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan  Hiroyuki Nakanishi
Department of Physics and Astronomy, Graduate School of Science and
Engineering, Kagoshima University, 1-21-35 Korimoto, Kagoshima, Kagoshima
890-0065, Japan  Takashi Onaka Department of Physics, Faculty of Science and
Engineering, Meisei University, 2-1-1 Hodokubo, Hino, Tokyo 191-8506, Japan
Department of Astronomy, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Geumsook Park  Korea Astronomy
and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055,
Republic of Korea  Xindi Tang Xinjiang Astronomical Observatory, Chinese
Academy of Sciences, 150 Science 1-Street, Urumqi 830011, Xinjiang, People’s
Republic of China  Kohji Tomisaka  Division of Theoretical Astronomy, National
Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan  SOKENDAI
(The Graduate University for Advanced Studies), Hayama, Kanagawa 240-0193,
Japan  Yusuke Tsukamoto  Department of Physics and Astronomy, Graduate School
of Science and Engineering, Kagoshima University, 1-21-35 Korimoto, Kagoshima,
Kagoshima 890-0065, Japan  Serena Viti  Physics & Astronomy Dept., University
College London, WC1E 6BT London, UK  Hongchi Wang  Purple Mountain
Observatory, Chinese Academy of Sciences, 2 West Beijing Road, 210008 Nanjing,
People’s Republic of China  Anthony Whitworth School of Physics and Astronomy,
Cardiff University, The Parade, Cardiff, CF24 3AA, UK  Hyunju Yoo Korea
Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon
34055, Republic of Korea  Hyeong-Sik Yun  School of Space Research, Kyung Hee
University, 1732 Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 17104,
Republic of Korea  Tetsuya Zenko  Department of Astronomy, Graduate School of
Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan  Yapeng Zhang
Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong
Kong  Ilse de Looze  Physics & Astronomy Dept., University College London,
WC1E 6BT London, UK  C. Darren Dowell  Jet Propulsion Laboratory, M/S 169-506,
4800 Oak Grove Drive, Pasadena, CA 91109, USA  Stewart Eyres  University of
South Wales, Pontypridd, CF37 1DL, UK  Sam Falle Department of Applied
Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK  Jean-
François Robitaille Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
Sven van Loo  School of Physics and Astronomy, University of Leeds, Woodhouse
Lane, Leeds LS2 9JT, UK

(Received 2020; Revised 2020)

###### Abstract

We have obtained sensitive dust continuum polarization observations at 850
$\mu$m in the B213 region of Taurus using POL-2 on SCUBA-2 at the James Clerk
Maxwell Telescope (JCMT), as part of the BISTRO (B-fields in STar-forming
Region Observations) survey. These observations allow us to probe magnetic
field (B-field) at high spatial resolution ($\sim$2000 au or $\sim$0.01 pc at
140 pc) in two protostellar cores (K04166 and K04169) and one prestellar core
(Miz-8b) that lie within the B213 filament. Using the Davis-Chandrasekhar-
Fermi method, we estimate the B-field strengths in K04166, K04169, and Miz-8b
to be 38$\pm$14 $\mu$G, 44$\pm$16 $\mu$G, and 12$\pm$5 $\mu$G, respectively.
These cores show distinct mean B-field orientations. B-field in K04166 is well
ordered and aligned parallel to the orientations of the core minor axis,
outflows, core rotation axis, and large-scale uniform B-field, in accordance
with magnetically regulated star formation via ambipolar diffusion taking
place in K04166. B-field in K04169 is found to be ordered but oriented nearly
perpendicular to the core minor axis and large-scale B-field, and not well-
correlated with other axes. In contrast, Miz-8b exhibits disordered B-field
which show no preferred alignment with the core minor axis or large-scale
field. We found that only one core, K04166, retains a memory of the large-
scale uniform B-field. The other two cores, K04169 and Miz-8b, are decoupled
from the large-scale field. Such a complex B-field configuration could be
caused by gas inflow onto the filament, even in the presence of a substantial
magnetic flux.

ISM: clouds – ISM: individual objects (B213) – ISM: magnetic fields –
polarization – techniques: polarimetric stars: formation

††facilities: JCMT

=500

##  1 Introduction

According to the filamentary paradigm of star-formation, low-mass stars
predominantly form in dense cores which are distributed in a chain-like
fashion along gravitationally unstable filamentary clouds (Hartmann, 2002;
Tafalla & Hacar, 2015; Marsh et al., 2016; André et al., 2014). Magnetic field
(B-field) is important at all scales during this process (Shu et al., 1987;
McKee & Ostriker, 2007; Crutcher, 2012; Ward-Thompson et al., 2020).
Nevertheless, the interplay between B-field, gravity, and turbulence in the
formation of cores and their collapse to form stars is still a subject of
investigation.

Studies of B-field on cloud scales with Planck 850 $\mu$m low-resolution
($\sim$5 arcmin or $\sim$0.2 at 140 pc) polarization observations and optical
and near-infrared (NIR) polarimetry of background stars have revealed that
low-density gas striations are mostly aligned to B-field and high-density
filamentary structures are oriented perpendicular to B-field (Chapman et al.,
2011; Alves et al., 2008; Sugitani et al., 2010; Planck Collaboration et al.,
2016a; Wang et al., 2020). These observations imply that material can
accumulate along field lines and aid in the assembly of dense structures
perpendicular to the B-field as a result of gravitational collapse and/or
converging flows (see Ballesteros-Paredes et al., 1999a; Hartmann et al.,
2001; Soler & Hennebelle, 2017).

If the large-scale, uniform B-field is inherited down to core scale ($<$ 0.1
pc), they govern not only the contraction, stability, and collapse of the core
(Mestel & Spitzer, 1956; Mouschovias & Spitzer, 1976), but also the properties
of circumstellar disk by help removing angular momentum via magnetic braking
(Mouschovias, 1991; Allen et al., 2003; Li et al., 2014). According to the
theory of isolated, low-mass star-formation via ambipolar diffusion
(Mouschovias, 1991; Mouschovias et al., 2006), the gravitational collapse of a
dense core is regulated by strong, ordered B-field such that the core
preferentially contracts along field lines. As a result, the core acquires an
oblate-like structure over 10 000 au scales. After gaining sufficient mass via
B-field-mediated contraction, initially the subcritical core becomes
supercritical and eventually collapses under its own gravity. At this stage,
the flux-freezing condition will no longer be valid due to efficient neutral-
ion decoupling. As a result of this ambipolar diffusion, the B-field will
acquire an hour-glass morphology on protostellar envelope scales, $<$ 1000 au
(e.g., Galli & Shu, 1993; Girart et al., 2006; Stephens et al., 2013). This
model predicts a positive correlation between angle of the mean B-field and
that of the minor axes of the filament and core, and the axes of both
pseudodisk symmetry and of the bipolar outflow (Fiedler & Mouschovias, 1992,
1993; Galli & Shu, 1993; Mocz et al., 2017; Hull & Zhang, 2019).

Evidence for magnetically regulated star-formation through observations of
coherent B-field across orders of magnitude in size scale (e.g., Li et al.
2006, Li et al. 2009, Hull et al. 2014) is not always the norm. Departure from
coherency, especially at smaller scales, can occur in regions dominated by
turbulence (e.g., Hull et al., 2017a), shocks from outflows (e.g., Hull et
al., 2017b), gravity-driven gas flows (e.g., Pillai et al., 2020), stellar
feedback driven by expanding ionization fronts from H ii regions (Arthur et
al., 2011; Pattle et al., 2018; Eswaraiah et al., 2020), or gas dynamics arise
from gravitational collapse (Ching et al., 2017, 2018). These observations
suggest that the very local environment can determine the morphology and role
of B-field.

We emphasize here that B-field observations of low-mass dense cores (i) formed
out of a single natal filament, (ii) characterized with ordered B-field at
larger scales (sub-pc to several pc; see Figure 1(a)), (iii) having signpost
of accretion flows (Palmeirim et al., 2013; Shimajiri et al., 2019), and (iv)
hosting pristine physical conditions, unaffected by any disruption by strong
stellar feedback are sparse. The Taurus B213 is one of these rare regions,
making the B213 cores the ideal laboratories to understand the role of B-field
in the star formation process.

We conduct sensitive dust polarization observations at 850 $\mu$m towards
B213, as part of the B-fields In STar-forming Region Observations (BISTRO;
Ward-Thompson et al. 2017) Survey, to resolve its B-field. BISTRO is a large
program on 15-m James Clerk Maxwell Telescope (JCMT), making use of its
SCUBA-2 camera and POL-2 polarimeter. B213 is a nearby (distance $\sim$140 pc;
Elias 1978) well-studied filament, which is part of the $\sim$10 pc filament
LDN 1495, as shown in Figure 1(a). B213 is fragmented into a chain of cores
which are in the early evolutionary stages of low-mass star-formation (Figure
1(b)). These include three prestellar cores, namely, Miz-8b, Miz-2, and HGBS-1
(Mizuno et al., 1994; Marsh et al., 2016); two Class 0/I protostellar cores,
IRAS 04166+2706 and IRAS 04169+2702 (Ohashi et al. 1997; Takakuwa et al. 2018;
Tafalla et al. 2010); and one evolved object, J04194148+2716070, classified as
a Class II T Tauri star (Davis et al., 2010). We hereafter refer to IRAS
04166+2706 and IRAS 04169+2702 as K01466 and K04169, respectively (cf., Kenyon
et al., 1990, 1993), adopting the core nomenclature of Bracco et al. (2017).

Here, for the first time, we resolve the B-field in the three cores of B213 on
0.01 – 0.1 pc spatial scales. In this letter our key aims are to examine
whether (i) B-field at scales $<$ 0.1 pc are coherent with, or decoupled from,
the uniform large-scale B-field, and (ii) the paradigm of magnetically
regulated, isolated low-mass star-formation holds in these cores. This paper
is organized as follows: Section 2 describes the observations and data
reduction. Sections 3 and 4 present the results and discussion, respectively;
and Section 5 summarizes our main findings.

Figure 1: (a) The overall structure of the B213 filament is shown using a
high-resolution (18$\farcs$2) column density map taken from the Herschel Gould
Belt Survey (HGBS) archive (Palmeirim et al., 2013). B213 is a part of the
10-pc-long large-scale filament LDN 1495 in the Taurus molecular cloud.
Measurements of the large-scale B-field orientation are overlaid: optical
polarimetry measurements are shown as magenta segments (Heyer et al., 1987;
Heiles, 2000; Goodman et al., 1990); NIR polarimetry measurements, as cyan
segments (Goodman et al., 1992; Chapman et al., 2011); and Planck 850 $\mu$m
dust emission polarimetry measurements, as yellow segments. (b) Our inferred
core-scale B-field geometry (red segments) superimposed on the area of our
total intensity (Stokes I) map which contains the fragmented cores of B213.
The extent of the map is shown as a white box in panel (a). Plotted segments
correspond to a detection at a minimum of 3$\sigma$ in polarization fraction
and 10$\sigma$ in total-intensity. Note that all the segments are shown with
equal lengths to better display the B-field morphology. The white contour
marks a column density of 1$\times 10^{22}$ cm-2 (Palmeirim et al., 2013), as
measured in the Herschel data, which outlines the structure of the parental
B213 filament, which has fragmented into the cores shown. Red contours drawn
in both panels mark 10$\sigma$ total intensity (Stokes $I$) values, where
1$\sigma$ $=$ 1.3 mJy beam-1 is the rms noise. Note that the apparent
$>$10$\sigma$ intensity values seen in low-column-density regions in panel (a)
result from low exposure times at the edges of the POL-2 map, and so delineate
the extent of our mapped area. Reference scale and beam size
($\sim$14$\farcs$1) are shown.

##  2 Observations and data reduction

POL-2 observations of two fields in B213 were carried out as part of the JCMT
BISTRO Survey (JCMT project code M16AL004) between 2017 November 05 and 2019
January 08. The two fields, shown in Figure 5, have a center-to-center angular
separation of $\sim$5′. The fields were each observed 20 times using the POL-2
DAISY mapping mode (Holland et al., 2013; Friberg et al., 2016). This mode
results in maps with a 12-arcminute diameter, of which the central $\sim$7
arcmin represents usable coverage, and so these two pointings represent a
tightly-spaced mosaic. The observations were made in JCMT weather bands 1 and
2, with 225 GHz atmospheric opacity ($\tau_{225}$) varying between 0.02 and
0.06. The total exposure time for the two fields is $\sim$28 hrs (14 hrs in
each of the two overlapping fields), resulting in one of the deepest
observations yet made by the BISTRO Survey.

The 850 $\mu$m POL-2 data were reduced using the pol2map routine recently
added to SMURF (Berry et al., 2005; Chapin et al., 2013)111
http://starlink.eao.hawaii.edu/docs/sun258.htx/sun258ss73.html. The final
mosaiced maps, calibrated in mJy beam-1, are produced from co-added Stokes
$I$, $Q$, and $U$ maps with pixel size of 4$\arcsec$, while the final debiased
polarization vector catalog is binned to 12$\arcsec$ to achieve better
sensitivity. The rms noise values in our Stokes $I$, $Q$, and $U$, and $PI$
maps, binned to a pixel size of 12$\arcsec$, are $\sim$1.3, $\sim$0.9,
$\sim$0.9, and $\sim$1.0 mJy beam-1, respectively. $PI$ represents polarized
intensity of the dust emission, debiased using the asymptotic estimator
method; our $PI$ map is shown in Figure 5. The instrumental polarization (IP)
of POL-2 was corrected for using the ‘August 2019’ IP model (Friberg et al.,
2018). The POL-2 data reduction process is described in detail by Doi et al.
(2020) and Pattle et al. (2020).

##  3 Results

###  3.1 B-field on small scales

We present the data of 28 polarization measurements satisfying the criteria:
(i) the ratio of intensity to its uncertainty $I/\sigma_{I}>$ 10 and (ii)
degree of polarization to its uncertainty $P/\sigma_{P}>$ 3, where $P$ $=$
$PI$/$I$. These measurements are listed in Table 1. The resulting $PI$ within
the core boundaries (see Appendix A) ranges from $\sim$2 to $\sim$4 mJy beam-1
with a median uncertainty in polarized intensity, $\sigma_{PI}$, of 0.64 mJy
beam-1. The polarization fraction ranges from $\sim$0.8 to $\sim$18% with a
median value of $\sim$7%. The B213 cores are characterized by weak dust
emission ($\sim$12 – 318 mJy beam-1) as well as weak polarized emission in
comparison to the other regions studied regions by the BISTRO program (Ward-
Thompson et al., 2017; Kwon et al., 2018; Soam et al., 2018; Wang et al.,
2019; Coudé et al., 2019; Liu et al., 2019; Pattle et al., 2019; Doi et al.,
2020).

Assuming a distance to Taurus of 140 pc, our observations allow us to
delineate B-field in B213 on scales ranging from $\sim$2000 au ($\sim$0.01 pc)
up to $\sim$0.25 pc, the length over which the cores K04166, Miz-8b and K04169
are distributed. The resulting B-field geometry, based on the 28 polarization
measurements (cf., Table 1), is shown in Figure 1(b). Since the three cores,
T-Tauri, Miz-2, and HGBS-1, have only a single measurement each (and also
because the background noise dominates at the locations of these cores – cf.,
Appendix A), we exclude them from further analysis and discussion. The overall
B-field morphology appears to be uniform within K04166 and K04169, but the
mean field directions are offset by $\sim$90$\arcdeg$ from one another. In
contrast, the B-field morphology in Miz-8b is complex.

We compute the weighted mean position angle of the core B-field,
$\bar{\theta}_{\mathrm{core,B}}$, using uncertainties in polarization angle as
weights. These values are given in Table 2. The
$\bar{\theta}_{\mathrm{core,B}}$ along with the low-resolution B-field
morphology based on Planck 850 $\mu$m polarization data, is shown in Figure 2.
Table 2 lists the offset between $\bar{\theta}_{\mathrm{core,B}}$ and the
large scale mean B-field orientation
($\theta_{\mathrm{B}}^{\mathrm{largescale}}$; see Appendix B) based on multi-
wavelength polarimetry. Also listed are the the offset between
$\bar{\theta}_{\mathrm{core,B}}$ and the position angle of each core’s major
axis ($\theta_{\mathrm{core}}$; see Appendix C).

Interestingly, we see completely different B-field geometry in each of the
three cores. The B-field in K04166 lies roughly parallel to the large-scale
field (or perpendicular to the filament), while that in K04169 lies roughly
perpendicular to the large-scale field (or roughly parallel to the filament).
The field direction in Miz-8b lies roughly half-way between the other two,
albeit with a larger standard deviation in B-field orientations (35$\arcdeg$;
see Table 2). Hence, we see that the core-scale B-field, in a set of cores
spanning $\sim 6\arcmin$, or $\sim$0.25 pc, appear to be rather complex.

Furthermore, we observe a good alignment between core B-field
($\sim$48$\arcdeg$) and outflows ($\sim$33$\arcdeg$) in K04166 (Figures 2 and
3(d)), consistent with studies by Davidson et al. (2011); Chapman et al.
(2011). In contrast, we see a misalignment between core mean B-field
($\sim$121$\arcdeg$) and outflows ($\sim$58$\arcdeg$) in K04169 (Figures 2 and
3(e)), in accordance with studies by Hull et al. (2013, 2014); Hull & Zhang
(2019); Yen et al. (2020). Despite the fact that these two cores lie within
$\sim$0.25 pc of each other, they exhibit different B-field/outflow
orientations.

Figure 2: Same as Figure 1(b) but now we only show the mean B-field
orientation in the three cores K04166, Miz-8b, and K04169, based on the
weighted mean of the position angles which we measure. The blue and red dashed
arrows denote the protostellar outflows (lengths are not to scale) emanated
from K04166 and K04169. The red contour around each core is drawn at $I$ $=$
13 mJy beam-1, corresponding to 10$\sigma$ in total intensity. The large-scale
B-field morphology, as determined from the over-sampled Planck 850 $\mu$m
polarization data (pixel size 1$\arcmin$), is shown as yellow segments. The
white contour is as described in Figure 1(b).

###  3.2 B-field strength

Based on the assumption that turbulence-induced Alfvén waves can distort
B-field orientations, the plane-of-sky component of the B-field strength
($B_{pos}$) can be estimated using the Davis-Chandrasekhar-Fermi relation
(Davis, 1951; Chandrasekhar & Fermi, 1953, DCF method):

$B_{\mathrm{pos}}\simeq Q\sqrt{4\pi\,\mu
m_{\mathrm{H}}\,n_{\mathrm{H_{2}}}}\,\left(\frac{\delta_{\mathrm{V_{NT}}}}{\delta_{\mathrm{\theta}}}\right),$
(1)

where $n_{\mathrm{H_{2}}}$ is the gas number density,
$\delta_{\mathrm{V_{NT}}}$ is the non-thermal gas velocity dispersion, and
$\delta_{\theta}$ is the dispersion in polarization angles about the mean
B-field orientation. $Q$ is a factor accounting for line-of-sight and beam
dilution effects, which we take as 0.5 based on studies using synthetic
polarization maps generated from numerically simulated clouds (Ostriker et
al., 2001), which suggest that without this correction factor, DCF-measured
B-field strength is overestimated by a factor of 2 when the angular dispersion
in the B-field is $\leqslant$25$\arcdeg$.

As illustrated in Appendix C, we have used the 850-$\mu$m Stokes I map to
extract core dimensions, column and number densities, and masses. To quantify
the non-thermal velocity dispersion induced by the turbulence, we estimated
the average velocity dispersion ($\delta_{\mathrm{V_{LSR}}}$) from archival
N2H+ (1 – 0) data (Punanova et al., 2018)222Data can be found at
http://cdsarc.u-strasbg.fr/ftp/J/A+A/617/A27/fits/, obtained using the IRAM
30-m telescope. The spatial and velocity resolutions of the N2H+ data are
26$\farcs$5 and 0.063 km s-1, respectively. The thermal contributions to the
observed velocity dispersions ($\delta_{\mathrm{V_{T}}}$) are estimated (based
on the mean dust temperatures of the cores given in Table 2). These components
are quadratically subtracted from the observed velocity dispersions
($\delta_{\mathrm{V_{LSR}}}$) to obtain non-thermal velocity dispersions
($\delta_{\mathrm{V_{NT}}}$). The angular dispersion in the B-field is
calculated using the relation for the inverse-variance-weighted standard
deviation of B-field (e.g, Wang et al., 2020). These estimated parameters are
listed in Table 2.

Using equation 1 and the parameters listed above, the B-field strength is
estimated to be 38$\pm$14 $\mu$G for K04166, 44$\pm$16 $\mu$G for K04169, and
12$\pm$5 $\mu$G for Miz-8b. Since the majority of the B-field segments in
K04166 and K04169 are confined to the core radii $\sim$20 – 50$\arcsec$, the
B-field strengths in these cores are mainly valid to the core-envelopes.
Further, we caution here that the B-field strength of Miz-8b could be highly
uncertain because of the limited number of B-field segments, and hence the
larger angular dispersion, used in the DCF method. The current estimations are
similar to the B-field strengths of $\sim$10 – 100 $\mu$G estimated in
relatively unperturbed low-mass star-forming regions (Crutcher et al., 2004;
Chapman et al., 2011; Crutcher, 2012), and two orders of magnitude less than
the $\sim$1 mG values estimated in massive star forming regions (eg., Curran &
Chrysostomou, 2007; Hildebrand et al., 2009; Pattle et al., 2017; Liu et al.,
2020).

We can use our estimated B-field strength to infer the dynamic state and
physical properties of the cores (see Table 2). First, we estimate the
magnetic and turbulent pressures using the relations $P_{B}$ $=$ $B^{2}/8\pi$
and $P_{\mathrm{turb}}$ $=$ $\rho{\delta_{\mathrm{NT}}}^{2}$, respectively.
Second, we estimate the Alfvénic Mach number using the relation $M_{A}$ $=$
$\sqrt{3}(\frac{\delta_{\mathrm{NT}}}{V_{\mathrm{A}}})$, where Alfvén velocity
$V_{\mathrm{A}}$ $=$ $\frac{B_{\mathrm{los}}}{\sqrt{4\pi\,\rho}}$ (where
$\rho$ $=$ $n_{H_{\mathrm{2}}}\,\mu\,m_{\mathrm{H}}$). Third, we use the mass-
to-magnetic flux ratio to infer how important is the B-field in comparison to
gravity. We measure the mass-to-flux ratio in units of the critical value, as
described in Appendix D. Finally, we estimate the rotational energy of each
core to determine how rotation may influence the B-field in Appendix E. The
derived energy values, along with all other parameters, are listed in Table 2.

##  4 Discussion

Since the two protostellar cores, K04166 and K04169, are at a similar
evolutionary stage and share similar characteristics (see Table 2), we discuss
their energy parameters and gas kinematics with reference to the differences
in B-field morphology in Section 4.1. These aspects for the prestellar core
Miz-8b are addressed in section 4.2.

###  4.1 K04166 and K04169

Magnetic-to-turbulent pressure ratio is seen to be $\sim$1 in both cores (see
Table 2). This suggests that B-field and turbulence are near equilibrium with
each other. Equivalently, the Alfvénic Mach number ($\sim$1) suggests that
turbulent motions are trans-Alfvénic. Therefore turbulent motions are not
dominant over, and so do not shape the morphology of, the B-field in these
cores. The mean mass-to-flux ratio criticality of the cores, $\lambda$, is
found to be $\sim$1, suggesting that the core-envelopes may be magnetically
critical and marginally supported by B-field. The ratio of rotational to
magnetic energy (cf., Appendix E), $E_{\mathrm{rot}}/E_{\mathrm{mag}}$ $\ll$
1, which infers that the core rotational energy is too weak to alter the
B-field orientation.

Our analysis indicates that there is an equipartition among magnetic,
turbulent, and gravitational energies in the core-envelopes of K04166 and
K04169. Then the question arises as to why the mean B-field orientation in the
two cores are different from each other. We use the morphological
correspondence between N2H+ velocity gradients and B-field, as shown in
Figures 3(a) & (b), to shed light on this.

The velocity field in K04166 as inferred from velocity gradient map is well
defined, fairly uniform, and is almost perpendicular to the B-field segments.
This could be interpreted as bulk core rotation, with the angular momentum (or
core rotation-axis with PA $\sim$11$\arcdeg$) being parallel to the B-field
direction. In addition, the outflow is well-collimated and exhibits extremely
high velocity (EHV) components (Wang et al., 2014), suggesting a possible role
of B-field in channeling the outflow and transporting energy and angular
momentum away from the rotating circumstellar disk. The position angles of the
core (and filament) minor axis ($\sim$37$\arcdeg$), core rotation-axis
($\sim$11$\arcdeg$), and bipolar outflows ($\sim$33$\arcdeg$) are all roughly
aligned with both $\bar{\theta}_{\mathrm{core,B}}$ (48$\arcdeg$) and
$\theta_{\mathrm{B}}^{\mathrm{largescale}}$ (29$\arcdeg$) to within
$\sim$30$\arcdeg$, as shown in Figure 3(d). This strong geometrical
correspondence suggests that B-field, which is inherited from the large-scale
uniform B-field, have played a significant role in core evolution by allowing
gas contraction along field lines to form the core, subsequently governing its
collapse via ambipolar diffusion, and finally collimating the outflows. These
signatures are in accordance with paradigm of low-mass star-formation process
driven by ambipolar diffusion in K04166. However, we could not trace an hour-
glass morphology in the inner core (radii $<$ 20$\arcsec$ or $<$ 2800 au) due
to limited resolution ($14\farcs 1\sim$ 2000 au).

On the other hand, the velocity gradient map in K04169 appears to be rather
complex, which displays two converging flows, from the northeast and the
southwest (Figure 3(b)). Counter-rotation between the disk and the envelope in
K04169 is also reported (Takakuwa et al., 2018). We see that core mean B-field
($\bar{\theta}_{\mathrm{core,B}}$ $\sim$121$\arcdeg$) is nearly aligned
parallel to the mean orientation of velocity gradient ($\theta_{\mathrm{G}}$
$\sim$126$\arcdeg$; cf., Table 2). We suggest that this complex gas flows
might have altered the B-field from being parallel to the core minor axis in
the earlier stage to current perpendicular configuration in K04169. This might
have also caused the misalignment of outflows (PA $\sim$58$\arcdeg$), core
rotation-axis (PA $\sim$36$\arcdeg$), and core minor axis (PA
$\sim$36$\arcdeg$) with $\bar{\theta}_{\mathrm{core,B}}$ as shown in Figure
3(e) (see Table 2 for more details).

Figure 3: (Top panels) Velocity gradient (VG) maps for (a) K04166, (b) K04169,
and (c) Miz-8b. The length and angle of each black arrow corresponds to the
magnitude and direction, respectively, of the local VG. A reference arrow with
the units of km s-1 pc-1 is shown on each plot. Red segments show the B-field
orientations from Figure 1(b). (Bottom panels) The weighted mean B-field
geometry (thick red segments; same as Figure 2) in the cores of (d) K04166,
(e) K04169, and (f) Miz-8b overlaid on the Stokes I map (color scale). The
large-scale Planck mean B-field direction is shown as a yellow segment on each
core. Red and blue contours show the red-shifted and blue-shifted 12CO (2–1)
emission tracing the bipolar outflow in the center of each core. These were
obtained with the ALMA 7-m array by Tokuda et al. (2020). The cyan line
denotes the mean position angle of each outflow (33$\arcdeg$ for K04166 and
58$\arcdeg$ for K04169). Thin black line represents the position angle of the
core rotation-axis. The integrated velocity ranges are $-$10 to 0 and 10 to 20
km s-1 for the blue- and red-shifted outflows of K04166, and $-$5 to 4 and 11
to 17 km s-1 for the blue- and red-shifted outflows of K04169. The lowest and
subsequent contour steps are 0.4 and 1.2 K km s-1 , respectively. The angular
resolution of the ALMA data is $6.8\arcsec\times 6.5\arcsec$.

###  4.2 Miz-8b

Unlike those in K04166 and K04169, Miz-8b has a disordered B-field. It has a
magnetic-to-turbulent pressure ratio of $\sim$0.3 and a Alfvénic Mach number
of $\sim$2 (cf., Table 2), which suggests that turbulence is super-Alfvénic
and dominates over B-field. Cores formed in a weakly magnetized, turbulent
cloud would have a chaotic B-field configuration, because of the dominance of
turbulent eddies over structural dynamics and field lines (Stone et al., 1998;
Ballesteros-Paredes et al., 1999b; Mac Low & Klessen, 2004; Li et al., 2014).
We suggest that B-field in Miz-8b are complex because of the dominance of
turbulent flows (Figure 3(c)). As a result B-field is decoupled from large-
scale ordered field (Figure 3(f)). Since the inferred B-field strength in
Miz-8b is weaker (12$\pm$5 $\mu$G), it will not support the core against
gravity as the mass-to-flux ratio is found to be supercritical
($\lambda\sim$3).

##  5 Summary

We have performed deep dust polarization observations towards the Taurus B213
filament at 850 $\mu$m using SCUBA-2 and POL-2 on the JCMT as part of the
BISTRO survey. We successfully detected polarized signal in, and studied in
detail the B-field of, two proto-stellar cores (K04166 and K04169) and one
prestellar core (Miz-8b) on scales from 2000 au to 0.25 pc.The main findings
of this work are as follows:

  1. 1.

Despite having (i) ordered B-field on large scales, (ii) quiescent physical
conditions, and (iii) formed out of the same natal filament, the three B213
cores exhibit diverse magnetic field properties.

  2. 2.

Among the three cores, only one, K04166, retains a memory of the large-scale
B-field, with a field orientation consistent with those seen on larger scales.
The other two cores appear to have decoupled from the large-scale field.

  3. 3.

Using the Davis-Chandrasekhar-Fermi method, we estimate the B-field strengths
in K04166, K04169, and Miz-8b to be 38$\pm$14 $\mu$G, 44$\pm$16 $\mu$G, and
12$\pm$5 $\mu$G, respectively. The associated magnetic energies are in
equipartition with both turbulent and gravitational energies in the core
envelopes of K04166 and K04169, while being much smaller than the turbulent
energy in the core Miz-8b.

  4. 4.

Based on the correlation between the position angle of the core-scale B-field
and the large-scale field, the minor axis of the core, outflows, and the core
rotation axis, we suggest that the formation and evolution of K04166 are
regulated by B-field, consistent with the paradigm of low-mass star-formation
via ambipolar diffusion. However, as revealed by their complex velocity
fields, the evolution of the other two cores, K04169 and Miz-8b, could be
regulated by converging accretion flows and turbulent motions respectively.

We suggest that cores formed in a magnetically regulated molecular cloud may
not necessarily retain a memory of the large-scale B-field of the cloud in
which they form. Instead, localized differences in gas kinematics, which
probably arise due to gas inflows onto the filament, can affect the role of
B-field in the star formation process, and the subsequent properties of the
forming systems.

We thank the referee for constructive suggestions which have improved the
content and flow of this paper. This work is supported by the National Natural
Science Foundation of China (NSFC) grant Nos. 11988101, 11725313, and
U1931117, and the International Partnership Program of Chinese Academy of
Sciences grant No. 114A11KYSB20160008. This work is also supported by Special
Funding for Advanced Users, budgeted and administrated by the Center for
Astronomical Mega-Science, Chinese Academy of Sciences (CAMS). C.E. thanks
Tokuda Kazuki for kindly providing us with the ALMA data on the protostellar
outflows. C.E. thanks Sihan Jiao and Yuehui Ma for help with the analyses.
K.Q. acknowledges the support from National Natural Science Foundation of
China (NSFC) through grant U1731237. C.W.L. is supported by the Basic Science
Research Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education, Science and Technology
(NRF-2019R1A2C1010851). N.O. is supported by Ministry of Science and
Technology (MOST) grant MOST 109-2112-M-001-051 in Taiwan. C.L.H.H.
acknowledges the support of the NAOJ Fellowship and JSPS KAKENHI grants
18K13586 and 20K14527. W.K. was supported by the New Faculty Startup Fund from
Seoul National University. D.J. is supported by the National Research Council
of Canada and by a Natural Sciences and Engineering Research Council of Canada
(NSERC) Discovery Grant. A.S. acknowledges support from the NSF through grant
AST-1715876. The James Clerk Maxwell Telescope is operated by the East Asian
Observatory on behalf of The National Astronomical Observatory of Japan;
Academia Sinica Institute of Astronomy and Astrophysics; the Korea Astronomy
and Space Science Institute; Center for Astronomical Mega-Science. Additional
funding support is provided by the Science and Technology Facilities Council
of the United Kingdom and participating universities in the United Kingdom,
Canada and Ireland.

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Table 1: Polarization data along with the celestial coordinates of the pixels. RA (J2000) | Dec (J2000) | $I\pm\sigma_{I}$ | $Q\pm\sigma_{Q}$ | $U\pm\sigma_{U}$ | $PI\pm\sigma_{PI}$ | $P\pm\sigma_{P}$ | $\theta_{\mathrm{core,B}}\pm\sigma_{\theta_{\mathrm{core,B}}}$
---|---|---|---|---|---|---|---
($\arcdeg$) | ($\arcdeg$) | (mJy beam-1) | (mJy beam-1) | (mJy beam-1) | (mJy beam-1) | (%) | ($\arcdeg$)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
64.982471 | 27.157917 | 25.0 $\pm$ 0.6 | -0.7 $\pm$ 0.6 | -2.5 $\pm$ 0.7 | 2.5 $\pm$ 0.7 | 9.8 $\pm$ 2.8 | 37 $\pm$ 7
65.004946 | 27.161250 | 16.9 $\pm$ 0.7 | 1.1 $\pm$ 0.7 | 2.7 $\pm$ 0.7 | 2.8 $\pm$ 0.7 | 16.8 $\pm$ 4.4 | 124 $\pm$ 7
64.989963 | 27.161250 | 33.0 $\pm$ 0.7 | 0.3 $\pm$ 0.6 | 3.0 $\pm$ 0.6 | 3.0 $\pm$ 0.6 | 9.0 $\pm$ 1.9 | 132 $\pm$ 6
64.986217 | 27.161250 | 37.3 $\pm$ 0.7 | -0.7 $\pm$ 0.6 | 3.1 $\pm$ 0.7 | 3.1 $\pm$ 0.7 | 8.3 $\pm$ 1.8 | 141 $\pm$ 5
65.004946 | 27.164583 | 35.7 $\pm$ 0.7 | 1.1 $\pm$ 0.7 | 2.7 $\pm$ 0.7 | 2.9 $\pm$ 0.7 | 8.1 $\pm$ 1.9 | 124 $\pm$ 7
65.001200 | 27.164583 | 78.6 $\pm$ 0.7 | -0.2 $\pm$ 0.7 | 2.7 $\pm$ 0.7 | 2.6 $\pm$ 0.7 | 3.3 $\pm$ 0.9 | 137 $\pm$ 7
64.989963 | 27.164583 | 113.9 $\pm$ 0.9 | 2.3 $\pm$ 0.6 | 0.3 $\pm$ 0.6 | 2.2 $\pm$ 0.6 | 1.9 $\pm$ 0.5 | 94 $\pm$ 8
65.008696 | 27.167917 | 20.3 $\pm$ 0.6 | -1.6 $\pm$ 0.7 | 1.6 $\pm$ 0.7 | 2.2 $\pm$ 0.7 | 10.8 $\pm$ 3.3 | 157 $\pm$ 8
64.993708 | 27.167917 | 318.4 $\pm$ 1.7 | 2.7 $\pm$ 0.7 | 0.2 $\pm$ 0.6 | 2.6 $\pm$ 0.7 | 0.8 $\pm$ 0.2 | 92 $\pm$ 7
64.989963 | 27.167917 | 140.9 $\pm$ 0.8 | 3.1 $\pm$ 0.6 | 1.6 $\pm$ 0.6 | 3.5 $\pm$ 0.6 | 2.5 $\pm$ 0.4 | 104 $\pm$ 5
64.986212 | 27.174583 | 45.4 $\pm$ 0.6 | 1.9 $\pm$ 0.6 | 0.9 $\pm$ 0.6 | 2.0 $\pm$ 0.6 | 4.4 $\pm$ 1.3 | 103 $\pm$ 8
64.967479 | 27.181247 | 12.2 $\pm$ 0.5 | -1.5 $\pm$ 0.6 | -1.6 $\pm$ 0.6 | 2.1 $\pm$ 0.6 | 17.6 $\pm$ 5.1 | 24 $\pm$ 8
64.971225 | 27.184581 | 15.8 $\pm$ 0.6 | 1.4 $\pm$ 0.6 | -1.9 $\pm$ 0.7 | 2.3 $\pm$ 0.6 | 14.4 $\pm$ 4.0 | 63 $\pm$ 7
64.959979 | 27.191247 | 46.1 $\pm$ 0.6 | 0.2 $\pm$ 0.6 | 2.0 $\pm$ 0.6 | 1.9 $\pm$ 0.6 | 4.1 $\pm$ 1.3 | 132 $\pm$ 9
64.971221 | 27.194583 | 26.1 $\pm$ 0.6 | 0.2 $\pm$ 0.6 | 2.9 $\pm$ 0.6 | 2.9 $\pm$ 0.6 | 11.0 $\pm$ 2.5 | 133 $\pm$ 6
64.967475 | 27.194581 | 48.4 $\pm$ 0.6 | -2.0 $\pm$ 0.6 | 0.4 $\pm$ 0.6 | 1.9 $\pm$ 0.6 | 4.0 $\pm$ 1.3 | 174 $\pm$ 9
64.963729 | 27.194581 | 58.6 $\pm$ 0.6 | -1.5 $\pm$ 0.6 | -1.8 $\pm$ 0.6 | 2.3 $\pm$ 0.6 | 3.8 $\pm$ 1.0 | 25 $\pm$ 7
64.933733 | 27.217906 | 54.7 $\pm$ 0.6 | -0.8 $\pm$ 0.6 | -2.4 $\pm$ 0.6 | 2.5 $\pm$ 0.6 | 4.6 $\pm$ 1.1 | 36 $\pm$ 6
64.926237 | 27.217903 | 25.0 $\pm$ 0.6 | -0.6 $\pm$ 0.5 | -2.7 $\pm$ 0.7 | 2.7 $\pm$ 0.7 | 10.8 $\pm$ 2.7 | 39 $\pm$ 6
64.926237 | 27.221236 | 72.0 $\pm$ 0.6 | 0.4 $\pm$ 0.6 | -2.9 $\pm$ 0.6 | 2.8 $\pm$ 0.6 | 3.9 $\pm$ 0.8 | 49 $\pm$ 6
64.937479 | 27.227906 | 37.7 $\pm$ 0.6 | 1.8 $\pm$ 0.6 | -1.9 $\pm$ 0.7 | 2.5 $\pm$ 0.6 | 6.7 $\pm$ 1.7 | 66 $\pm$ 7
64.933729 | 27.227906 | 113.4 $\pm$ 0.7 | 2.2 $\pm$ 0.6 | -0.0 $\pm$ 0.7 | 2.1 $\pm$ 0.6 | 1.8 $\pm$ 0.6 | 90 $\pm$ 9
64.929979 | 27.227903 | 290.0 $\pm$ 1.4 | 1.2 $\pm$ 0.6 | -2.3 $\pm$ 0.6 | 2.5 $\pm$ 0.6 | 0.9 $\pm$ 0.2 | 59 $\pm$ 7
64.926233 | 27.227903 | 253.9 $\pm$ 1.5 | -1.8 $\pm$ 0.6 | -2.2 $\pm$ 0.7 | 2.8 $\pm$ 0.6 | 1.1 $\pm$ 0.3 | 25 $\pm$ 6
64.933725 | 27.234572 | 14.6 $\pm$ 0.6 | 0.5 $\pm$ 0.7 | -2.2 $\pm$ 0.7 | 2.2 $\pm$ 0.7 | 14.9 $\pm$ 4.5 | 51 $\pm$ 9
64.854983 | 27.247850 | 29.3 $\pm$ 1.1 | 3.6 $\pm$ 1.0 | -2.0 $\pm$ 1.0 | 3.9 $\pm$ 1.0 | 13.4 $\pm$ 3.5 | 76 $\pm$ 7
64.907471 | 27.251225 | 22.1 $\pm$ 0.8 | 4.0 $\pm$ 0.8 | -0.5 $\pm$ 0.8 | 3.9 $\pm$ 0.8 | 17.8 $\pm$ 3.8 | 86 $\pm$ 6
64.922458 | 27.267900 | 91.9 $\pm$ 1.3 | 1.3 $\pm$ 0.9 | 3.7 $\pm$ 0.9 | 3.8 $\pm$ 0.9 | 4.2 $\pm$ 1.0 | 125 $\pm$ 7

Table 2: Various parameters for K04166, K04169, and Miz-8b.

No | Parameter | K04166 | K04169 | Miz-8b
---|---|---|---|---
| Weighted mean B-field orientation along with various offset position angles. |  |  |
1 | No. of B-field segments | 8 | 10 | 4
2 |  Weighted mean B-field orientation ($\bar{\theta}_{\mathrm{core,B}}$ $\pm\epsilon_{\mathrm{\bar{\theta}_{\mathrm{core,B}}}};\arcdeg$)a,b |  48 $\pm$ 2 |  121 $\pm$ 2 |  158 $\pm$ 4
3 |  Angular dispersion ($\delta_{\theta}$, $\arcdeg$) |  18$\pm$4 |  20$\pm$3 |  35$\pm$7
4 |  Standard deviation in ${\theta_{\mathrm{core,B}}}~{}~{}($°$)^{b}$ | 20 | 33 | 35
5 |  Position angle of core major axis ($\theta_{\mathrm{core}}$; $\arcdeg$)c |  127$\pm$20 |  126$\pm$16 |  119$\pm$30
6 |  $|\bar{\theta}_{\mathrm{core,B}}-\theta_{\mathrm{B}}^{\mathrm{largescale}}|$ ($\arcdeg$)d | 19 | 88 | 51
7 |  $|\bar{\theta}_{\mathrm{core,B}}-\theta_{\mathrm{core}}|$ ($\arcdeg$) | 79 | 5 | 39
| Core dimensions, mass, and column and number densities, etc. |  |  |
1 | Semi-major axis (a) (pc) |  0.032$\pm$0.005 |  0.036$\pm$0.006 |  0.029 $\pm$0.010
2 | Semi-minor axis (b) (pc) |  0.024$\pm$0.003 |  0.025$\pm$0.005 |  0.023 $\pm$ 0.005
3 |  Effective radius Reff (pc)e |  0.028 $\pm$ 0.003 |  0.030 $\pm$ 0.004 |  0.026 $\pm$ 0.005
4 |  Median dust temperature (Td) (K)f |  12.2 $\pm$ 0.4 |  12.2 $\pm$ 0.8 |  10.9 $\pm$ 0.2
5 |  Integrated flux ($F_{\mathrm{\nu}}$) (mJy) | 1564 | 1969 | 516
6 |  Mass (M) (M☉) |  0.55$\pm$0.28 |  0.69$\pm$0.36 |  0.22 $\pm$ 0.11
7 |  Column density ($N(\mathrm{H_{2}})$) $\times$1021 (cm-2) |  10 $\pm$ 6 |  11 $\pm$ 6 |  5$\pm$3
8 |  Number density ($n(\mathrm{H_{2}})$) $\times$104 (cm-3) |  9 $\pm$ 5 |  9 $\pm$ 6 |  5$\pm$4
| B-field strength, and magnetic and turbulent pressures, etc. |  |  |
1 |  B-field strength (using DCF method) ($\mu$G) |  38$\pm$14 |  44$\pm$16 |  12$\pm$5
2 |  B-field pressure ($P_{\mathrm{B}}$; $\times$10-10 dyn cm-2) |  0.6$\pm$0.4 |  0.8$\pm$0.6 |  0.06$\pm$0.05
3 |  Turbulent pressure ($P_{\mathrm{turb}}$; $\times$10-10 dyn cm-2) |  0.4$\pm$0.3 |  0.7$\pm$0.5 |  0.2$\pm$0.1
4 | $P_{\mathrm{B}}/P_{\mathrm{turb}}$ |  1.3$\pm$1.3 |  1.0$\pm$1.0 |  0.3$\pm$0.4
5 |  Mass-to-flux ratio criticality ($\lambda$ $=$ $(M/\phi)$/$(M/\phi)_{\mathrm{cri}}$) |  0.7$\pm$0.4 |  1.4$\pm$1.0 |  3$\pm$2
6 |  Alfvén velocity ($V_{\mathrm{A}}$; km s-1) |  0.17$\pm$0.09 |  0.21$\pm$0.12 |  0.07$\pm$0.04
7 |  Alfvénic Mach number ($M_{\mathrm{A}}$) |  1.1$\pm$0.6 |  1.2$\pm$0.7 |  2$\pm$1
| Energy parameters |  |  |
1 | $p$ | 0.27 | 0.27 | 0.27
2 |  Rotational energy $E_{\mathrm{rot}}$ ($\times 10^{41}$ erg) | 0.02 | 0.1 | 0.01
3 |  Magnetic energy $E_{\mathrm{mag}}$ ($\times 10^{41}$ erg) | 2 | 3 | 0.1
4 | $E_{\mathrm{rot}}/E_{\mathrm{mag}}$ | 0.01 | 0.04 | 0.05
| Various position angles (PAs) |  |  |
1 |  Core minor axis PA ($\arcdeg$) |  $\sim$37 |  $\sim$36 |  $\sim$29
2 |  Outflows PA ($\arcdeg$)g |  $\sim$33 |  $\sim$58 | –
3 |  Core ${\theta_{\mathrm{G}}}$ ($\arcdeg$; from W to N)h |  $-$169 |  $-$144 | 93
4 |  Core $\theta_{\mathrm{G}}$ ($\arcdeg$; from N to E)i | 101 | 126 | 3
5 |  PA of the core rotation-axis ($\arcdeg$; from N to E)j | 11 | 36 | 93

.

##  Appendix A Polarization properties: detection of weakly polarized dust
emission

Figure 4 plots $PI$ versus $I$ for each core, using the selection criterion
$I/\sigma_{I}$ $>$ 10 (gray filled circles). In at least three cores, K04166,
Miz-8b, and K04169, and also in the plot showing all of the cores, a slowly
increasing trend in $PI$ can be seen up to I $\sim$100 mJy beam-1, beyond
which $PI$ remains approximately constant, although there exist fewer data
points.

To extract the reliable data from our POL-2 measurements of the B213 cores, we
adopt the selection criterion $I/\sigma_{I}$ $>$ 10 and $P/\sigma_{P}$ $>$ 3,
which yield 28 polarization measurements (black circles in Figure 4). The
resulting median $\sigma_{PI}$ is 0.64 mJy beam-1. The $PI$ values lie in the
range 1.88 and 3.94 mJy beam-1, with a median of 2.60 mJy beam-1, whereas $I$
ranges from 13 to 318 mJy beam-1 as shown in Figure 4. The $P$ values lie in
the range 0.82 – 17.8% with a minimum $\sim$1%, median $\sim$7% and standard
deviation $\sim$5%. Above the 3$\sigma$ level in $PI$, a clear detection of
polarized intensity (yellow contours) within the core boundaries determined
from Stokes I map (red contour at 13 mJy beam-1) can be seen in the PI map, as
shown in Figure 5.

Figure 4: Polarized intensity ($PI$) versus intensity ($I$) plots for each
core, and all of the cores combined. The name of the core is stated in each
panel. Gray filled circles represent the data satisfying the criterion
$I/\sigma_{I}$ $>$ 10, whereas the black filled circles denote those
satisfying both the criteria $I/\sigma_{I}$ $>$ 10 and $P/\sigma_{P}$ $\geq$
3\. The dashed line represents the median $\sigma_{PI}$ $=$ 0.64 mJy/beam
determined from the filled black points.

Figure 5: Debiased polarized intensity (PI) map produced using our POL-2
Stokes $Q$ and $U$ maps of the B213 region. Non-smoothed PI contours are drawn
at [2, 3, 4]$\times\sigma_{PI}$, where $\sigma_{PI}$ is the rms noise, $\sim$1
mJy beam-1 (estimated using the pixels in a signal-free region of PI map). Red
contour corresponds to a POL-2 total intensity of 13 mJy beam-1. The $PI$ is
nearly zero in the area surrounding the cores but within the cores themselves
a clear detection can be seen. Cyan dashed circles mark the areas with
diameters of $3\arcmin$ and $7\arcmin$ around the central positions of the two
observed fields. Polarization measurements within the smaller circles as well
as the common area covered by both larger circles should be useful. Therefore,
the measurements of the three cores T-Tauri, Miz-2, and HGBS-1 may not be
reliable due to the dominance of background noise at their locations. Each of
the six cores are labelled.

##  Appendix B B-field at larger scales ($\sim$0.2 – 2.4 pc) determined from
optical, near-infrared (NIR), and Planck polarization data

In order to compare core-scale B-field (c.f. Section 3.1) with that in the
large-scale, low-density surrounding region, we make use of archival optical,
NIR, and Planck/850 $\mu$m low-resolution dust polarization data333Planck 353
GHz (850$\mu$m) dust continuum polarization data, comprising Stokes $I$, $Q$,
and $U$ maps for the B213 region, have been extracted from the Planck Public
Data Release 2 of Multiple Frequency Cutout Visualization (PR2 Full Mission
Map with PCCS2 Catalog: https://irsa.ipac.caltech.edu/applications/planck/).
Data have been reduced using the standard procedures described by Planck
Collaboration et al. (2015), Planck Collaboration et al. (2016b), Soler et al.
(2016), Baug et al. (2020) and references therein.; and the B-field
morphologies inferred from these data sets are shown in Figure 1. We select
the data within 1$\arcdeg$ of B213; the resulting values of the Gaussian mean
and standard deviation in B-field orientation
($\theta_{B}^{\mathrm{largescale}}\pm\sigma$) are given in Table 3. There
exist a significant number of optical/NIR B-field segments around B213;
however, NIR polarization measurements are confined to an area to the west of
B213 and therefore may not reveal the local B-field of B213. Visual inspection
suggests that optical- and Planck-inferred B-field is ordered. This is
confirmed by their mean B-field orientations which are respectively found to
be 29$\pm$14$\arcdeg$ and 29$\pm$17$\arcdeg$. These values are nearly
identical, with slightly different standard deviations (cf., Table 3), whereas
NIR polarization data show a curved morphology, which follows the compressed
and curved shell of LDN 1495, with a slightly different mean B-field
orientation of 37$\pm$17$\arcdeg$. Therefore, to delineate the mean B-field in
and around B213, we select the optical and Planck polarization data within
1$\arcdeg$ of B213, which yielded a mean orientation of 29$\arcdeg$. This
large-scale, coherent B-field ($\theta_{\mathrm{B}}^{\mathrm{largescale}}$)
with a mean orientation of 29$\arcdeg$, span spatial scales from $\sim$0.2 pc
($\sim$5$\arcmin$ resolution of Planck) to $\sim$2.4 pc (1$\arcdeg$ area
around B213).

Table 3: Mean B-field orientations, determined from optical, near-infrared, and low resolution sub-mm (Planck/850 $\mu$m) polarization observations. Wavelength | diameter (arcmin) | No of stars/segments |  $\theta_{B}^{\mathrm{largescale}}\pm\sigma$ ($\arcdeg$) | Offset PA$\ddagger$ ($\arcdeg$)
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
Opticala | 60 | 15 | 29$\pm$14 | 104
NIR | 60 | 42 | 37$\pm$17 | 96
Sub-mm (Planckb) | 60 | 445 | 29$\pm$17 | 104

##  Appendix C Geometries, effective radii, masses, and column and number
densities of B213 cores

To estimate various energy and pressure terms for the cores, we extract their
masses, column, and number densities from the POL-2 Stokes I map. For this,
core dimensions are obtained by fitting the ellipse function mpfitellipse.pro
from the Marquardt library to the 10$\sigma$ Stokes I contours (13 mJy beam-1)
of each core. The resulting core dimensions (a $=$ semi-major and b $=$ semi-
minor), effective radius ($R_{\mathrm{eff}}$ $=$ $\sqrt{ab}$), and position
angle in degrees east of north are given in Table 2.

The integrated fluxes ($F_{\mathrm{\nu}}$) and median dust temperatures
($T_{\mathrm{d}}$; from the HGBS temperature map) over the core are used to
estimate core masses using the relation (Hildebrand, 1983):

$M=\frac{F_{\nu}\,D^{2}}{B_{\mathrm{\nu}}(T_{\mathrm{d}})\,\kappa_{\nu}},$
(C1)

where $D$ = 140 pc, is the distance of B213, $\kappa_{\nu}$ $=$ 0.0125 cm2 g-1
(e.g., Johnstone et al., 2017) is the dust mass opacity, and
$B_{\mathrm{\nu}}(T_{\mathrm{d}})$ is Planck function for a blackbody at
temperature $T_{\mathrm{d}}$. The uncertainty in mass is estimated by
propagating the standard deviation in $T_{\mathrm{d}}$, 10% of the value of
$F_{\mathrm{\nu}}$ as the flux calibration uncertainty of SCUBA2 (Dempsey et
al., 2013), and a 50% uncertainty in dust mass opacity (eg., Roy et al.,
2014).

The column and number densities of the cores are estimated using the following
relations:

$N(\mathrm{H_{2}})=\frac{M}{\mu\,m_{\mathrm{H}}\,\pi\,R_{\mathrm{eff}}^{2}},\\\
$ (C2)

and

$n(\mathrm{H_{2}})=\frac{3\,M}{4\,\mu\,m_{\mathrm{H}}\,\pi\,R_{\mathrm{eff}}^{3}}.$
(C3)

Estimated masses and column and number densities and their corresponding
uncertainties are given along with $T_{\mathrm{d}}$ and $F_{\mathrm{\nu}}$
values in Table 2.

##  Appendix D Mass-to-flux ratio criticality

To infer the importance of B-field with respect to the gravity, we estimate
the mass-to-magnetic flux ratio in units of the critical value (hereafter
mass-to-flux ratio criticality) using the following relation (Crutcher et al.,
2004; Chapman et al., 2011),

$\lambda=\frac{(M/\phi)}{(M/\phi)_{\mathrm{crit}}}=7.6~{}N_{\mathrm{\parallel}}(\mathrm{H_{2}})/B_{\mathrm{tot}},$
(D1)

where $N_{\mathrm{\parallel}}(\mathrm{H_{2}})$ is the mean column density
(N(H2)POL2), in units of 1021 cm-2, along the magnetic flux tube and
$B_{\mathrm{tot}}$ is the total B-field strength in $\mu$G. The critical mass-
to-flux ratio, ($M/\phi$)crit $=$ 1/$\sqrt{(4\pi^{2}G)}$ (Nakano & Nakamura,
1978), corresponds to the stability criterion for an isothermal gaseous layer
threaded by perpendicular B-field. A cloud region with $(M/\phi)$ $>$
$(M/\phi_{\mathrm{crit}})$, i.e. $\lambda$ $>$ 1, will collapse under its own
gravity, and so such a cloud is considered to be supercritical. A cloud with
$\mu$ $<$ 1 will be in a subcritical state because of the significant support
rendered by B-field. Taking the mean $N(\mathrm{H_{2}})$ $=$
$N_{\mathrm{\parallel}}(\mathrm{H_{2}})$ as (10$\pm$6)$\times$1021,
(11$\pm$6)$\times$1021 cm-2, and (5$\pm$3)$\times$1021 cm-2 and $B$
$=~{}B_{\mathrm{tot}}$ a s 38$\pm$17, 48$\pm$22, and 12$\pm$5 $\mu$G, we
estimate $\mu$ values of 2$\pm$1, 2$\pm$1, and 3$\pm$2 for K04166, K04169, and
Miz-8b, respectively.

However, considering (a) the projection effects between
$N_{\mathrm{\parallel}}$(H2)/$B_{\mathrm{tot}}$ and the actual measured
N($\mathrm{H_{2}}$)/$B_{\mathrm{\parallel}}$ (where $B_{\parallel}$ is the
plane-of-the-sky B-field strength), (ii) B-field being perpendicular to the
core elongation in the case of oblate spheroid, or parallel to the core
elongation in the case of a prolate spheroid, and and (iii) the assumption
that B-field is randomly oriented with respect to the line of sight, the
actual value of $\mu$ becomes (1/3)$\lambda_{\mathrm{obs}}$ for K04166 as the
mean B-field is perpendicular to the core major axis; and
(3/4)$\lambda_{\mathrm{obs}}$ for K04169 as mean B-field is parallel to the
major axis (Planck Collaboration et al., 2016a, see their Appendix D.4444In
order to correct the estimated mass-to-flux ratio (in critical units) for
projection effects, a factor 1/2 is valid for a spheroid cloud, 1/3 for an
oblate spheroid flattened perpendicular to the B-field, and 3/4 for a prolate
spheroid elongated along the B-field ). No correction was applied on the
$\lambda$ value of Miz-8b because of the misalignment between mean B-field and
core major axis. Therefore, the resulting $\lambda$ values are 0.7$\pm$0.5,
1.4$\pm$1.0, and 3$\pm$1, which are given in Table 2.

##  Appendix E Ratio of magnetic-to-rotational energy

By assuming that the cores are uniform density spheres and we measure the
rotational and magnetic energies using the following relations (see Wurster &
Lewis 2020):

$E_{\mathrm{rot}}=\frac{pMR^{2}\Omega^{2}}{5},$ (E1)

and

$E_{\mathrm{mag}}=\frac{B^{2}V}{8\pi},$ (E2)

In Equation E1, the correction factor, $p$ $=$
$\frac{2(3-\mathcal{A})}{3(5-\mathcal{A})}$ $=$ 0.27 (where $\mathcal{A}$ is
the power-index in the density distribution of the form $\rho\propto
r^{-\mathcal{A}}$, here we we consider $\mathcal{A}$ $=$ 1.6), accounts for
the density distribution in the sphere (see Xu et al. 2020 for more details).
We use the effective radii $R$ $=$ $R_{\mathrm{eff}}$ $=$ $\sqrt{ab}$ (where
$a$ $=$ semi-major axis and $b$ $=$ semi-minor axis), and the volume of the
core $V$ $=$ (4/3)$\pi R_{\mathrm{eff}}^{3}$. $\Omega$ is the angular velocity
or magnitude of velocity gradient of the core, measured from the N2H+ data,
and is found to be 2.05$\pm$0.02 km s-1 pc-1 for K04166, 3.86$\pm$0.04 km s-1
pc-1 for K04169, and 1.88$\pm$0.02 km s-1 pc-1 for Miz-8b (Punanova et al.,
2018, see their Table B.2). $M$ is the mass of the cores (cf., Appendix C).The
derived energy values and their ratio are given in Table 2.

##  Appendix F Morphological correlation between B-field and the gradients of
velocity

We model the observed line-of-sight centroid velocities of N2H+
($V_{\mathrm{LSR}}$, km s-1) and the corresponding offset length scales in sky
coordinates (R.A ($\Delta_{\alpha}$ in pc) and Dec ($\Delta_{\delta}$ in pc))
around each pixel in terms of velocity gradients in R.A
($\nabla_{V_{\alpha}}$, km s-1 pc-1) and Dec ($\nabla_{V_{\delta}}$, km s-1
pc-1) and constant systematic velocity of that reference pixel ($V_{0}$, km
s-1) using the a first-degree bivariate polynomial of the form (Goodman et
al., 1993; Henshaw et al., 2016; Sokolov et al., 2018)

$V_{\mathrm{LSR}}=V_{\mathrm{0}}+\nabla_{V_{\alpha}}\Delta_{\alpha}+\nabla_{V_{\delta}}\Delta_{\delta}.$
(F1)

We have used the IDL algorithm mpfit to perform weighted, non-linear, minimum
chi-square fitting to constrain the velocity gradients $\nabla_{V_{\alpha}}$
and $\nabla_{V_{\delta}}$, and their corresponding uncertainties. These are
further used to derive the magnitude (${\mathcal{G}}$) and direction
($\Theta_{\mathcal{G}}$) of the velocity gradients using the following
relations

${\mathcal{G}}\equiv|\nabla_{V}|=\sqrt{{\nabla_{V_{\alpha}}}^{2}+{\nabla_{V_{\delta}}}^{2}}$
(F2)

and

$\Theta_{\mathcal{G}}=\arctan\left(\frac{{\nabla_{V_{\alpha}}}}{\nabla_{V_{\delta}}}\right).$
(F3)

We considered at least six adjacent pixels, lying within the beam size of IRAM
30-m telescope for N2H+ (1 – 0), 26$\farcs$5, around each pixel when fitting
was performed. In addition, we estimate the uncertainties in the VGs using
equation 2 of Punanova et al. (2018). These were used as weights while
performing the weighted fits. The top panels of Figure 3 show VGs superimposed
on the POL-2 Stokes I maps of K04166, K04169, and Miz-8b.