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Tao-Chung Ching

Publications and source records attributed to Tao-Chung Ching.

At least 19 recordsLinked to original sources

Measuring Magnetic Fields Near and Far with the SKA via the Zeeman Effect

Zeeman splitting in spectral lines -- both in emission and absorption -- provides direct estimates of magnetic field strength and direction in magnetized gas in our own Milky Way and in external galaxies. We discuss the potential for using the Square Kilometre Array (SKA) to measure the Zeeman effect in targets spanning an enormous range of distance: from cometary comas in the solar system, through Galactic molecular clouds, HI filaments in the cold neutral medium, high-velocity clouds, the Fermi Bubbles, and photodissociation regions (PDRs) traced by radio recombination lines, to OH masers and megamasers in nearby and distant starburst galaxies, and to cold neutral gas in damped Ly-alpha absorbing systems at cosmological redshifts. We update the sensitivity calculations of Robishaw et al. (2015) and indicate, for each science goal, whether it will be achievable with Array Assembly 4 (AA4) of SKA-Mid, with the staged delivery of AA*, or only with the full SKA buildout. Zeeman measurements will probe the magnetic field in situ in the warm and cold neutral interstellar medium, complementing SKA Faraday rotation programs; radio recombination lines, stackable across hundreds of transitions, extend this reach to HII regions and PDRs. In external galaxies, SKA-Mid will enable Zeeman studies of OH kilomasers in nearby starburst systems, substantially expand the census of megamaser Zeeman detections beyond the Arecibo sky, and probe magnetic fields in damped Lyman-alpha systems to field limits well below those currently achievable, opening a new window on the role of magnetic fields in galaxy formation and cosmic evolution.

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Measuring Magnetic Field Strengths in Galactic Star-forming Regions via the Zeeman Effect with the SKA

Magnetic fields thread the interstellar medium from the largest to the smallest scales and play an important role in molecular cloud evolution and star formation. Quantifying this requires measurements of the field strengths, and the most direct way to measure them is via the Zeeman effect in spectral lines. The effect is subtle for the typical field strengths expected from theory, from a few $\mu$G in diffuse molecular clouds to a few 10s of mG in dense star-forming regions, and detections are scarce. Existing measurements of magnetic field strength suggest dense clouds and cores are marginally supercritical (cannot prevent collapse, but can inhibit it), but may be biased due to small sample sizes. Zeeman effect measurements tracing different scales and densities within molecular clouds can reveal the variation of field strengths, providing critical measurements to address the question of whether star formation is primarily regulated by magnetic fields or turbulence on different scales. Observations with SKA precursors such MeerKAT and FAST are beginning to increase the number of Zeeman effect detections in nearby star-forming regions. The SKA will extend their reach to many regions within our Galaxy that are best representative of where most stars form, while zooming in on the densest star-forming regions, providing a statistical basis for the role of magnetic fields in molecular cloud evolution and star formation. We present predictions and plans for Zeeman effect observations with the SKA telescopes, demonstrating the significant advances they will provide for studies of magnetic fields in molecular clouds.

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Star Formation Drives Production of Low Energy Cosmic Rays

For over a century, the origin of low-energy cosmic rays (LECRs), the dominant heaters and ionizers of dense interstellar gas, remains elusive owing to solar modulation and uncertain transport processes. In this study, we introduce a new astrophysical approach based on HI Narrow Self-Absorption (HINSA) to obtain spatially resolved measurements of LECR ionization rates using high-fidelity HI observations toward the Orion region from the FAST telescope. The LECR ionization rate is found to scale with local star formation rate (SFR) as $log_{10}\zeta = (1.4\pm 0.70)log_{10}\mathrm{SFR} + (-10.5\pm 2.9)$. Moreover, it increases with visual extinction, and is found to exceed, toward active star-forming regions, the value predicted for diffuse regions based on \textit{Voyager} measurements and an external propagation model. These findings demonstrate that LECRs are generated in situ by star-forming activities rather than penetrating from the broader Galactic cosmic-ray population. This is further supported by \textit{Fermi}-LAT gamma-ray observations toward the Orion region. Together, these results resolve a key uncertainty in cosmic-ray origin and establish a new avenue for quantifying the energetic feedback that regulates the interstellar medium.

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Unveiling Dominant Toroidal Magnetic Fields in a Protostellar Outflow

Magnetic fields play a fundamental role in the formation of protostellar winds. In the magneto-centrifugal models, poloidal magnetic fields launch winds from accretion disks, and fast-rotating gas twists the fields into toroidal geometry that collimates and accelerates winds through magnetic hoop stress. However, toroidal fields in protostellar winds remain observationally unresolved. Here we report polarization observations of carbon monoxide emission toward the NGC1333 IRAS 4A protostellar outflow. The inferred magnetic fields are perpendicular to the outflow axis and aligned with the rotational structure of the outflow, indicating toroidal fields with strengths of a few milligauss, sufficient to collimate and accelerate the outflow at several hundred astronomical units from the protostar. A linear correlation is found between the curl of plane-of-the-sky magnetic field and the line-of-sight electric current density. Our analysis provides better constraints on ion-electron drift velocity in protostellar outflows and supports rotating outflows driven by the magneto-centrifugal mechanism.

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ALMA Polarization Study of the Magnetic Fields in Two Massive Clumps in the 20 km s$^{-1}$ Cloud of the Central Molecular Zone

We present the Atacama Large Millimeter/submillimeter Array (ALMA) observations of linearly polarized 870 $\mu$m continuum emission at a resolution of $\sim$0.2$^{\prime\prime}$ (2000 au) toward the two massive clumps, Clump 1 and Clump 4, in the 20 km s$^{-1}$ cloud. The derived magnetic field strengths for both clumps range from $\sim$0.3 to 3.1 mG using the Angular Dispersion Function (ADF) method. The magnetic field orientations across multiple scales suggests that the magnetic field dominates at the cloud scale, whereas gravity likely governs the structures at the core (0.01$-$0.1 pc) and condensation ($\le$ 0.01 pc) scales. Furthermore, the study on the angular difference between the orientations of the local gravity gradient and the magnetic field suggests that the gravity predominantly governs the dynamics in the diffuse regions, while both gravity and star formation feedback become increasingly significant within the dense regions. The ratio of the magnetic field tension force $F_\mathrm{B}$ to the gravitational force $F_\mathrm{G}$ suggests that the magnetic field may provide some support against gravity, but it is insufficient to prevent gas from infalling toward the dense cores.

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Pinched Magnetic Fields in the High-mass Protocluster W3 IRS5

We present polarization maps of dust emission at 340 GHz in the luminous high-mass protocluster, W3 IRS5, observed with the Submillimeter Array. The projected magnetic fields appear fairly organized with a pinched morphology in the northern part and a concave shape in the southern part. We fit the polarization maps with a two-component magnetic field model: an hourglass model centered at the continuum peak, SMM2, and an empirical sphere centered at the O-type star, IRS7. Using the Davis-Chandrasekhar-Fermi method, we calculate a projected field strength of $B_\mathrm{pos} = 1.4 \; \mathrm{mG}$. Along with the Zeeman measurement, a total magnetic field strength of $B_\mathrm{tot} = 1.6 \; \mathrm{mG}$ is obtained. We find that the gravitational energy is the most dominant, followed by magnetic energy, and then turbulent energy. Small values of the virial parameter, $α_\mathrm{vir} = 0.8$, and the ratio of timescales, $t_\mathrm{ff}/t_\mathrm{corss} = 0.6$, suggest an ongoing collapse. We also show collimated molecular outflows in the $\mathrm{CO \; (3-2)}$ and $\mathrm{SiO \; (8-7)}$ transitions. The morphology of magnetic fields and the surrounding \HII regions put forward a scenario for W3 IRS5. A gravitationally unstable dense core formed within a neutral gas ridge plowed by the expansions of W3 A and W3 B. The core began to contract, causing gravity to pull the magnetic field lines inward, which resulted in a pinched field morphology. Subsequent expansion of W3 F, ionized by IRS7, perturbed the magnetic field, creating concave patterns. The dynamical interactions among protostars led to misalignment of their outflows.

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Joint Analysis of HI Absorption Zeeman Measurements and the Morphology of Filamentary HI Emission

We present a joint analysis of HI absorption Zeeman measurements and the morphology of filamentary HI emission to investigate the three-dimensional structure of the magnetic field in the diffuse neutral interstellar medium (ISM). Our analysis is based on the Arecibo Millennium Survey and new data from the Five-hundred-meter Aperture Spherical radio Telescope (FAST) toward radio sources 3C 75, 3C 207, and 3C 409. Toward 3C 409, we make a 4$σ$ Zeeman detection and infer $B_{LOS}$ = 9.1 +/- 1.9$μ$G, in agreement with Arecibo results. We quantify the dispersion of HI filaments at the locations and velocities of Zeeman components using GALFA-HI narrow-channel emission maps. Focusing on a subsample of 42 spectrally distinct components, we find a weak but statistically significant positive correlation (Spearman r = 0.3, $p = 0.01$) between $|B_{LOS}|$ and the circular variance of HI filament orientation angles. To examine its origin, we characterize the environments probed by HI absorption using dust emission, 3D dust maps, OH absorption, and CO emission. We find evidence that existing HI absorption Zeeman measurements trace magnetic fields that are coherent on parsec scales, probe primarily local gas ($100$-$500$ pc, often at distances consistent with the Local Bubble wall), and exhibit systematic differences in the magnitude of $B_{LOS}$. We attribute the correlation between Zeeman measurements and filamentary HI morphology to large-scale variations in magnetic field strength and/or inclination angle across different Galactic environments, which could arise due to the Local Bubble geometry or enhanced total field strength in denser regions.

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Impact of gravity on changing magnetic field orientations in a sample of massive protostellar clusters observed with ALMA

The magnetic field is integral to our understanding of the formation and dynamical evolution of molecular clouds and star formation within. We present a polarimetric survey of 17 massive protostellar cluster forming clumps, covered in 34 pointings in the 230-GHz window using the Atacama Large Millimeter/submillimeter Array (ALMA). The two array configurations, C43-1 and C43-4, probe linearly polarized dust emission, hence the plane-of-the-sky orientation of magnetic fields, at resolutions of 1\arcsec\ and 0\arcsec.4 that correspond to approximately 0.01pc core and $10^3$ au envelope scales, respectively. The relative orientations (ROs) of the magnetic field probed at two spatial scales are analyzed for the entire protostellar cluster sample and for a subset of objects in NGC 6334. We found a bimodal distribution of ROs with peaks at 0° (parallel) and 90°(orthogonal) for the entire sample combined as well as for NGC 6334. We investigate the physical origin of this bimodal distribution through a projected Rayleigh statistic (PRS) analysis in relation to column densities and local gravity in NGC 6334. We found an excess of parallel magnetic fields at column densities $> 10^{23}$ \cmm. The underlying cause of the RO distribution of the magnetic field is gravitational collapse at higher gas densities, which drags and reorients the magnetic field as shown in the alignment between the magnetic field and the direction of gravitational forces. The distribution of ROs observed here is consistent with the evolution of relative orientations of an initially sub-Alvénic cloud that becomes magnetically super-critical and super-Alvénic as the cloud collapses to form stars.

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Imprints of Stellar Feedback on Magnetic Fields in the Iris Nebula NGC 7023

We present 850 $μ$m polarized continuum observations carried out with the POL-2 polarimeter mounted on the James Clerk Maxwell Telescope (JCMT) towards NGC 7023 located in the Cepheus Flare region. NGC 7023 is a reflection nebula powered by a Herbig Ae Be star HD 200775 and also identified as a hub in the hub-filament cloud, LDN 1172/1174. We detect submillimetre emission well towards the northern (identified as C1) and the eastern region of the reflection nebula. We investigated the polarization structure and the magnetic field (B-field) morphology, which is found to be curved and follows the clump morphology. The comparison of the B-field morphology at the clump scales ($\sim$0.02 pc) with that of the envelope ($\sim$0.5 pc) suggests that the field lines are not preserved from envelope to clump scales, implying that an external factor may be responsible for disturbing the B-field structure. We estimated a magnetic field strength of 179$\pm$50 $μ$G in the starless core, 121$\pm$34 $μ$G in the protostellar core with a class I source and 150$\pm$42 $μ$G in the protostellar core with a class II source using the $N_{2}H^{+}$ (1-0) line observed with 13.7 m single dish radio telescope at Taeduk Radio Astronomy Observatory (TRAO). The stability analysis using these B-field strengths gives magnetically sub-critical values, while the magnetic, gravitational, and outflow kinetic energies are roughly balanced. We also suggest that the reordering of the magnetic field lines could be due to the interaction with the already evolved high-velocity outflow gas around the central star, which hints at the presence of outflow feedback.

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Beam Measurements of Full Stokes Parameters for the FAST L-band 19-beam Receiver

The Five-hundred-meter Aperture Spherical radio Telescope (FAST) has been fully operational since 11 January 2020. We present a comprehensive analysis of the beam structure for each of the 19 feed horns on FAST's L-band receiver across the Stokes I, Q, U, and V parameters. Using an on-the-fly mapping pattern, we conducted simultaneous sky mapping using all 19 beams directed towards polarization calibrators J1407+2827 and J0854+2006 from 2020 to 2022. Electromagnetic simulations were also performed to model the telescope's beam patterns in all Stokes parameters. Our findings reveal a symmetrical Gaussian pattern in the Stokes I parameter of the central beam without strong sidelobes, while the off-center beams exhibit significant asymmetrical shapes that can be fitted using a combination of log-normal and Gaussian distributions. The inner beams have higher relative beam efficiencies and smaller beam sizes compared to those of the outer beams. The sidelobes of the inner beams contribute approximately 2% of the total flux in the main lobe, increasing to 5% for outer beams, with a peak at 6.8%. In Stokes U, a distinct four-lobed cloverleaf beam squash structure is observed, with similar intensity levels in both inner and outer beams. In Stokes V, a two-lobed beam squint structure is observed in the central beam, along with a secondary eight-lobed structure. The highest squint peak in Stokes V is about 0.3% of the Stokes I in the outer beams. These results align closely with the simulations, providing valuable insights for the design of radio multi-beam observations.

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A Tale of Three: Magnetic Fields along the Orion Integral-Shaped Filament as Revealed by JCMT BISTRO survey

As part of the BISTRO survey, we present JCMT 850 $μ$m polarimetric observations towards the Orion Integral-Shaped Filament (ISF) that covers three portions known as OMC-1, OMC-2, and OMC-3. The magnetic field threading the ISF seen in the JCMT POL-2 map appears as a tale of three: pinched for OMC-1, twisted for OMC-2, and nearly uniform for OMC-3. A multi-scale analysis shows that the magnetic field structure in OMC-3 is very consistent at all the scales, whereas the field structure in OMC-2 shows no correlation across different scales. In OMC-1, the field retains its mean orientation from large to small scales, but shows some deviations at small scales. Histograms of relative orientations between the magnetic field and filaments reveal a bimodal distribution for OMC-1, a relatively random distribution for OMC-2, and a distribution with a predominant peak at 90$^\circ$ for OMC-3. Furthermore, the magnetic fields in OMC-1 and OMC-3 both appear to be aligned perpendicular to the fibers, which are denser structures within the filament, but the field in OMC-2 is aligned along with the fibers. All these suggest that gravity, turbulence, and magnetic field are each playing a leading role in OMC-1, 2, and 3, respectively. While OMC-2 and 3 have almost the same gas mass, density, and non-thermal velocity dispersion, there are on average younger and fewer young stellar objects in OMC-3, providing evidence that a stronger magnetic field will induce slower and less efficient star formation in molecular clouds.

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Polarization Calibration of the FAST L-band 19-beam Receiver: I. On-axis Mueller Matrix Parameters

We present the polarization calibration of the 19-beam receiver at 1420 MHz within the full illumination of the Five-hundred-meter Aperture Spherical Telescope from October 2018 to March 2023. We perform spider observations to characterize the on-axis Mueller matrix of the central beam. The calibrated polarization percentage and polarization angle of a source with strong linear polarization emission are about 0.2\% and 0.5$^{\circ}$. Several parameters of the central-beam Mueller matrix show time variability from months to years, suggesting relatively frequent polarization calibrations are needed. We obtain the Mueller matrix parameters of the 18 off-center beams with the combination of on-the-fly observations and spider observations. The polarization calibration provides consistent fractional Stokes parameters of the 19 beams, although the Mueller matrix parameters of the off-center beams are not as accurate as those of the central beam. The Mueller matrix parameters of the central beam do not show a strong dependence on the reflector surface. However, we notice different off-center Mueller matrix parameters between the eastern and western sides of the reflector surface. We provide average parameters of the 19-beam Mueller matrices which should be applicable to observations from 2020 to 2022 with several caveats. After applying the average parameters, on-axis fractional linear polarization measurements $\gtrsim$ 10\% and on-axis fractional circular polarization measurements $\gtrsim$ 1.5\% can be considered high-confidence detections. For sources with weak polarization, timely polarization calibrations using spider observations are required.

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The JCMT BISTRO Survey: The Magnetic Fields of the IC 348 Star-forming Region

We present 850 $μ$m polarization observations of the IC 348 star-forming region in the Perseus molecular cloud as part of the B-fields In STar-forming Region Observation (BISTRO) survey. We study the magnetic properties of two cores (HH 211 MMS and IC 348 MMS) and a filamentary structure of IC 348. We find that the overall field tends to be more perpendicular than parallel to the filamentary structure of the region. The polarization fraction decreases with intensity, and we estimate the trend by power-law and the mean of the Rice distribution fittings. The power indices for the cores are much smaller than 1, indicative of possible grain growth to micron size in the cores. We also measure the magnetic field strengths of the two cores and the filamentary area separately by applying the Davis-Chandrasekhar-Fermi method and its alternative version for compressed medium. The estimated mass-to-flux ratios are 0.45-2.20 and 0.63-2.76 for HH 211 MMS and IC 348 MMS, respectively, while the ratios for the filament is 0.33-1.50. This result may suggest that the transition from subcritical to supercritical conditions occurs at the core scale ($\sim$ 0.05 pc) in the region. In addition, we study the energy balance of the cores and find that the relative strength of turbulence to the magnetic field tends to be stronger for IC 348 MMS than HH 211 MMS. The result could potentially explain the different configurations inside the two cores: a single protostellar system in HH 211 MMS and multiple protostars in IC 348 MMS.

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Polarization Substructure in the Spiral-Dominated HH 111 Disk: Evidence for Grain Growth

The HH 111 protostellar disk has recently been found to host a pair of spiral arms. Here we report the dust polarization results in the disk as well as the inner envelope around it, obtained with the Atacama Large Millimeter/submillimeter Array in continuum at lambda ~ 870 micron and ~ 0. 05" resolution. In the inner envelope, polarization is detected with a polarization degree of ~ 6% and an orientation almost everywhere parallel to the minor axis of the disk, and thus likely to be due to the dust grains magnetically aligned mainly by toroidal fields. In the disk, the polarization orientation is roughly azimuthal on the far side and becomes parallel to the minor axis on the near side, with a polarization gap in between on the far side near the central protostar. The disk polarization degree is ~ 2%. The polarized intensity is higher on the near side than the far side, showing a near-far side asymmetry. More importantly, the polarized intensity and thus polarization degree are lower in the spiral arms, but higher in between the arms, showing an anticorrelation of the polarized intensity with the spiral arms. Our modeling results indicate that this anticorrelation is useful for constraining the polarization mechanism and is consistent with the dust self-scattering by the grains that have grown to a size of ~ 150 micron. The interarms are sandwiched and illuminated by two brighter spiral arms and thus have higher polarized intensity. Our dust self-scattering model can also reproduce the observed polarization orientation parallel to the minor axis on the near side and the observed azimuthal polarization orientation at the two disk edges in the major axis.

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Dark Dragon Breaks Magnetic Chain: Dynamical Substructures of IRDC G28.34 Form in Supported Environments

We have comprehensively studied the multi-scale physical properties of the infrared dark cloud (IRDC) G28.34 (the Dragon cloud) with dust polarization and molecular line data from Planck, FCRAO-14m, JCMT, and ALMA. We find that the averaged magnetic fields of clumps tend to be either parallel with or perpendicular to the cloud-scale magnetic fields, while the cores in clump MM4 tend to have magnetic fields aligned with the clump fields. Implementing the relative orientation analysis (for magnetic fields, column density gradients, and local gravity), Velocity Gradient Technique (VGT), and modified Davis-Chandrasekhar-Fermi (DCF) analysis, we find that: G28.34 is located in a trans-to-sub-Alfvénic environment ($\mathcal{M}_{A}=0.74$ within $r=15$ pc); the magnetic field is effectively resisting gravitational collapse in large-scale diffuse gas, but is distorted by gravity within the cloud and affected by star formation activities in high-density regions; and the normalized mass-to-flux ratio tends to increase with increasing density and decreasing radius. Considering the thermal, turbulent, and magnetic supports, we find that the environmental gas of G28.34 is in a super-virial (supported) state, the infrared dark clumps may be in a near-equilibrium state, and core MM4-core4 is in a sub-virial (gravity-dominant) state. In summary, we suggest that magnetic fields dominate gravity and turbulence in the cloud environment at large scales, resulting in relatively slow cloud formation and evolution processes. Within the cloud, gravity could overwhelm magnetic fields and turbulence, allowing local dynamical star formation to happen.

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Filamentary Network and Magnetic Field Structures Revealed with BISTRO in the High-Mass Star-Forming Region NGC2264 : Global Properties and Local Magnetogravitational Configurations

We report 850 $μ$m continuum polarization observations toward the filamentary high-mass star-forming region NGC 2264, taken as part of the B-fields In STar forming Regions Observations (BISTRO) large program on the James Clerk Maxwell Telescope (JCMT). These data reveal a well-structured non-uniform magnetic field in the NGC 2264C and 2264D regions with a prevailing orientation around 30 deg from north to east. Field strengths estimates and a virial analysis for the major clumps indicate that NGC 2264C is globally dominated by gravity while in 2264D magnetic, gravitational, and kinetic energies are roughly balanced. We present an analysis scheme that utilizes the locally resolved magnetic field structures, together with the locally measured gravitational vector field and the extracted filamentary network. From this, we infer statistical trends showing that this network consists of two main groups of filaments oriented approximately perpendicular to one another. Additionally, gravity shows one dominating converging direction that is roughly perpendicular to one of the filament orientations, which is suggestive of mass accretion along this direction. Beyond these statistical trends, we identify two types of filaments. The type-I filament is perpendicular to the magnetic field with local gravity transitioning from parallel to perpendicular to the magnetic field from the outside to the filament ridge. The type-II filament is parallel to the magnetic field and local gravity. We interpret these two types of filaments as originating from the competition between radial collapsing, driven by filament self-gravity, and the longitudinal collapsing, driven by the region's global gravity.

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Magnetic fields of the starless core L 1512

We present JCMT POL-2 850 um dust polarization observations and Mimir H band stellar polarization observations toward the starless core L1512. We detect the highly-ordered core-scale magnetic field traced by the POL-2 data, of which the field orientation is consistent with the parsec-scale magnetic fields traced by Planck data, suggesting the large-scale fields thread from the low-density region to the dense core region in this cloud. The surrounding magnetic field traced by the Mimir data shows a wider variation in the field orientation, suggesting there could be a transition of magnetic field morphology at the envelope scale. L1512 was suggested to be presumably older than 1.4 Myr in a previous study via time-dependent chemical analysis, hinting that the magnetic field could be strong enough to slow the collapse of L1512. In this study, we use the Davis-Chandrasekhar-Fermi method to derive a plane-of-sky magnetic field strength ($B_{pos}$) of 18$\pm$7 uG and an observed mass-to-flux ratio ($λ_{obs}$) of 3.5$\pm$2.4, suggesting that L1512 is magnetically supercritical. However, the absence of significant infall motion and the presence of an oscillating envelope are inconsistent with the magnetically supercritical condition. Using a Virial analysis, we suggest the presence of a hitherto hidden line-of-sight magnetic field strength of ~27 uG with a mass-to-flux ratio ($λ_{tot}$) of ~1.6, in which case both magnetic and kinetic pressures are important in supporting the L1512 core. On the other hand, L1512 may have just reached supercriticality and will collapse at any time.

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Submillimeter Observations of Magnetic Fields in Massive Star-forming Region W75N

This paper presents the results of full polarization observations of the massive star-forming region W75N, conducted with 3 arcsec spatial resolutions at 345 GHz using the Submillimeter Array (SMA). The magnetic field structures in the dense cores of the region are derived using the linearly polarized continuum emission. The overall magnetic field strength and orientation are found to agree with those from the previous observations. The plane-of-sky (POS) component of the magnetic field in the region was calculated to be 0.8 \pm 0.1 mG using the angular dispersion function (ADF) method. Further analyses involving the polarization-intensity gradient-local gravity method and H13CO+ (4-3) line data indicated that the cloud is undergoing global gravitational collapse and the magnetic field is shaped by gravity and outflows in the dense core regions.

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