arXiv subjects

Ian Stephens

Publications and source records attributed to Ian Stephens.

At least 19 recordsLinked to original sources

arXiv2026

Investigation of Hourglass-shaped Magnetic fields in the G35.20-0.74 Star-Forming Complex

To investigate the role of magnetic fields toward the G35N and G35S sub-regions in the G35.20-0.74 star-forming complex, we utilized multi-wavelength polarimetric observations from the SOFIA/HAWC+ at 154 $\mu$m and ACT at 220 GHz/1.3 mm. The ACT 220 GHz polarization data (resolution $\sim$1$'$) show an hourglass-shaped plane-of-sky magnetic field morphologies toward both the sub-regions, although with distinct symmetry axes. SOFIA/HAWC+ 154 $\mu$m data (resolution $\sim$13.6$''$) confirm an hourglass morphology in G35N, whereas G35S displays a different magnetic field configuration compared to the ACT observations. An hourglass morphology identified at clump scales ($\sim$pc) toward G35N is consistent with the previously reported B-field morphology at core scales ($\sim$0.05 pc), supporting the scenario of a magnetically regulated collapse. Using the SOFIA/HAWC+ data, we estimate magnetic field strengths of $\sim$600 $\pm$ 200 $\mu$G in G35N and $\sim$850 $\pm$ 310 $\mu$G in G35S. Energy balance analysis suggests that gravity and magnetic fields contribute comparably in G35N, while in G35S the gas dynamics are dominated by magnetic field, followed by gravity and turbulence. The higher field strength in G35S likely results from compression by the expanding HII region, highlighting the impact of stellar feedback. The derived magnetic field strengths and corresponding magnetic energies should be treated as upper limits due to unresolved beam-scale correlations and the limited fitting range of the polarization angle structure function. Overall, our results show that magnetic fields decisively regulate star formation, with G35N shaped by magnetically controlled collapse and G35S being strongly influenced by stellar feedback.

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arXiv2024

Applying the Velocity Gradient Technique in NGC 1333: Comparison with Dust Polarization Observations

Magnetic fields (B-fields) are ubiquitous in the interstellar medium (ISM), and they play an essential role in the formation of molecular clouds and subsequent star formation. However, B-fields in interstellar environments remain challenging to measure, and their properties typically need to be inferred from dust polarization observations over multiple physical scales. In this work, we seek to use a recently proposed approach called the Velocity Gradient Technique (VGT) to study B-fields in star-forming regions and compare the results with dust polarization observations in different wavelengths. The VGT is based on the anisotropic properties of eddies in magnetized turbulence to derive B-field properties in the ISM. We investigate that this technique is synergistic with dust polarimetry when applied to a turbulent diffused medium for the purpose of measuring its magnetization. Specifically, we use the VGT on molecular line data toward the NGC~1333 star-forming region ($\rm ^{12}CO$, $\rm ^{13}CO$, $\rm C^{18}O$, and $\rm N_{2}H^{+}$), and we compare the derived B-field properties with those inferred from 214 and 850~$\mu$m dust polarization observations of the region using SOFIA/HAWC+ and JCMT/POL-2, respectively. We estimate both the inclination angle and the 3D Alfv\'enic Mach Number $M_A$ from the molecular line gradients. Crucially, testing this technique on gravitationally bound, dynamic, and turbulent regions, and comparing the results with those obtained from polarization observations at different wavelength, such as the plane-of-the-sky field orientation, is an important test on the applicability of the VGT in various density regimes of the ISM.

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arXiv2024

Protoplanetary Disk Polarization at Multiple Wavelengths: Are Dust Populations Diverse?

Millimeter and sub-millimeter observations of continuum linear dust polarization provide insight into dust grain growth in protoplanetary disks, which are the progenitors of planetary systems. We present the results of the first survey of dust polarization in protoplanetary disks at 870 $\mu$m and 3 mm. We find that protoplanetary disks in the same molecular cloud at similar evolutionary stages can exhibit different correlations between observing wavelength and polarization morphology and fraction. We explore possible origins for these differences in polarization, including differences in dust populations and protostar properties. For RY Tau and MWC 480, which are consistent with scattering at both wavelengths, we present models of the scattering polarization from several dust grain size distributions. These models aim to reproduce two features of the observational results for these disks: (1) both disks have an observable degree of polarization at both wavelengths and (2) the polarization fraction is higher at 3 mm than at 870 $\mu$m in the centers of the disks. For both disks, these features can be reproduced by a power-law distribution of spherical dust grains with a maximum radius of 200 $\mu$m and high optical depth. In MWC 480, we can also reproduce features (1) and (2) with a model containing large grains ($a_{max}$ = 490 $\mu$m ) near the disk midplane and small grains ($a_{max}$ = 140 $\mu$m) above and below the midplane.

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arXiv2022

(Sub)millimeter Dust Polarization of Protoplanetary Disks from Scattering by Large Millimeter-Sized Irregular Grains

The size of dust grains, $a$, is key to the physical and chemical processes in circumstellar disks, but observational constraints of grain size remain challenging. (Sub)millimeter continuum observations often show a percent-level polarization parallel to the disk minor axis, which is generally attributed to scattering by $\sim 100\mu$m-sized spherical grains (with a size parameter $x \equiv 2\pi a / \lambda < 1$, where $\lambda$ is the wavelength). Larger spherical grains (with $x$ greater than unity) would produce opposite polarization direction. However, the inferred size is in tension with the opacity index $\beta$ that points to larger mm/cm-sized grains. We investigate the scattering-produced polarization by large irregular grains with a range of $x$ greater than unity with optical properties obtained from laboratory experiments. Using the radiation transfer code, RADMC-3D, we find that large irregular grains still produce polarization parallel to the disk minor axis. If the original forsterite refractive index in the optical is adopted, then all samples can produce the typically observed level of polarization. Accounting for the more commonly adopted refractive index using the DSHARP dust model, only grains with $x$ of several (corresponding to $\sim$mm-sized grains) can reach the same polarization level. Our results suggest that grains in disks can have sizes in the millimeter regime, which may alleviate the tension between the grain sizes inferred from scattering and other means. Additionally, if large irregular grains are not settled to the midplane, their strong forward scattering can produce asymmetries between the near and far side of an inclined disk, which can be used to infer their presence.

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arXiv2021

Thermal Emission and Scattering by Aligned Grains: Plane-Parallel Model and Application to Multiwavelength Polarization of the HL Tau Disk

Telescopes are now able to resolve dust polarization across circumstellar disks at multiple wavelengths, allowing the study of the polarization spectrum. Most disks show clear evidence of dust scattering through their unidirectional polarization pattern typically at the shorter wavelength of $\sim 870 \mu$m. However, certain disks show an elliptical pattern at $\sim 3$mm, which is likely due to aligned grains. With HL Tau, its polarization pattern at $\sim 1.3$mm shows a transition between the two patterns making it the first example to reveal such transition. We use the T-matrix method to model elongated dust grains and properly treat scattering of aligned non-spherical grains with a plane-parallel slab model. We demonstrate that a change in optical depth can naturally explain the polarization transition of HL Tau. At low optical depths, the thermal polarization dominates, while at high optical depths, dichroic extinction effectively takes out the thermal polarization and scattering polarization dominates. Motivated by results from the plane-parallel slab, we develop a simple technique to disentangle thermal polarization of the aligned grains $T_{0}$ and polarization due to scattering $S$ using the azimuthal variation of the polarization fraction. We find that, with increasing wavelength, the fractional polarization spectrum of the scattering component $S$ decreases, while the thermal component $T_{0}$ increases, which is expected since the optical depth decreases. We find several other sources similar to HL Tau that can be explained by azimuthally aligned scattering prolate grains when including optical depth effects. In addition, we explore how spirally aligned grains with scattering can appear in polarization images.

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arXiv2021

The Transition of Polarized Dust Thermal Emission from the Protostellar Envelope to the Disk Scale

Polarized dust continuum emission has been observed with ALMA in an increasing number of deeply embedded protostellar systems. It generally shows a sharp transition going from the protostellar envelope to the disk scale, with the polarization fraction typically dropping from ${\sim} 5\%$ to ${\sim} 1\%$ and the inferred magnetic field orientations becoming more aligned with the major axis of the system. We quantitatively investigate these observational trends using a sample of protostars in the Perseus molecular cloud and compare these features with a non-ideal MHD disk formation simulation. We find that the gas density increases faster than the magnetic field strength in the transition from the envelope to the disk scale, which makes it more difficult to magnetically align the grains on the disk scale. Specifically, to produce the observed ${\sim} 1\%$ polarization at ${\sim} 100\,\mathrm{au}$ scale via grains aligned with the B-field, even relatively small grains of $1\,\mathrm{\mu m}$ in size need to have their magnetic susceptibilities significantly enhanced (by a factor of ${\sim} 20$) over the standard value, potentially through superparamagnetic inclusions. This requirement is more stringent for larger grains, with the enhancement factor increasing linearly with the grain size, reaching ${\sim} 2\times 10^4$ for millimeter-sized grains. Even if the required enhancement can be achieved, the resulting inferred magnetic field orientation in the simulation does not show a preference for the major axis, which is inconsistent with the observed pattern. We thus conclude that the observed trends are best described by the model where the polarization on the envelope scale is dominated by magnetically aligned grains and that on the disk scale by scattering.

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arXiv2021

HAWC+/SOFIA Polarimetry in L1688: Relative Orientation of Magnetic Field and Elongated Cloud Structure

We present a study of the relative orientation between the magnetic field and elongated cloud structures for the $\rho$ Oph A and $\rho$ Oph E regions in L1688 in the Ophiuchus molecular cloud. Combining inferred magnetic field orientation from HAWC+ 154 $\mu$m observations of polarized thermal emission with column density maps created using Herschel submillimeter observations, we find consistent perpendicular relative alignment at scales of $0.02$ pc ($33.6"$ at $d \approx 137$ pc) using the histogram of relative orientations (HRO) technique. This supports the conclusions of previous work using Planck polarimetry and extends the results to higher column densities. Combining this HAWC+ HRO analysis with a new Planck HRO analysis of L1688, the transition from parallel to perpendicular alignment in L1688 is observed to occur at a molecular hydrogen column density of approximately $10^{21.7}$ cm$^{-2}$. This value for the alignment transition column density agrees well with values found for nearby clouds via previous studies using only Planck observations. Using existing turbulent, magnetohydrodynamic simulations of molecular clouds formed by colliding flows as a model for L1688, we conclude that the molecular hydrogen volume density associated with this transition is approximately $\sim10^{4}$ cm$^{-3}$. We discuss the limitations of our analysis, including incomplete sampling of the dense regions in L1688 by HAWC+.

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arXiv2020

Probing the Temperature Structure of Optically Thick Disks Using Polarized Emission of Aligned Grains

Polarized continuum emission from aligned grains in disks around young stellar objects can be used to probe the magnetic field, radiation anisotropy, or drift between dust and gas, depending on whether the non-spherical grains are aligned magnetically, radiatively or mechanically. We show that it can also be used to probe another key disk property -- the temperature gradient -- along sight lines that are optically thick, independent of the grain alignment mechanism. We first illustrate the technique analytically using a simple 1D slab model, which yields an approximate formula that relates the polarization fraction to the temperature gradient with respect to the optical depth tau at the tau=1 surface. The formula is then validated using models of stellar irradiated disks with and without accretion heating. The promises and challenges of the technique are illustrated with a number of Class 0 and I disks with ALMA dust polarization data, including NGC 1333 IRAS4A1, IRAS 16293B, BHB 07-11, L1527, HH 212 and HH 111. We find, in particular, that the sight lines passing through the near-side of a highly inclined disk trace different temperature gradient directions than those through the far-side, which can lead to a polarization orientation on the near-side that is orthogonal to that on the far-side, and that the HH 111 disk may be such a case. Our technique for probing the disk temperature gradient through dust polarization can complement other methods, particularly those using molecular lines.

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arXiv2020

Detection of Irregular, Sub-mm Opaque Structures in the Orion Molecular Clouds: Protostars within 10000 years of formation?

We report ALMA and VLA continuum observations that potentially identify the four youngest protostars in the Orion Molecular Clouds taken as part of the Orion VANDAM program. These are distinguished by bright, extended, irregular emission at 0.87 mm and 8 mm and are optically thick at 0.87 mm. These structures are distinct from the disk or point-like morphologies seen toward the other Orion protostars. The 0.87 mm emission implies temperatures of 41-170 K, requiring internal heating. The bright 8 mm emission implies masses of 0.5 to 1.2 M_sun assuming standard dust opacity models. One source has a Class 0 companion, while another exhibits substructure indicating a companion-candidate. Three compact outflows are detected, two of which may be driven by companions, with dynamical times of ~300 to ~400 years. The slowest outflow may be driven by a first hydrostatic core. These protostars appear to trace an early phase when the centers of collapsing fragments become optically thick to their own radiation and compression raises the gas temperature. This phase is thought to accompany the formation of hydrostatic cores. A key question is whether these structures are evolving on free fall times of ~100 years, or whether they are evolving on Kelvin-Helmholtz times of several thousand years. The number of these sources imply a lifetime of ~6000 years, in closer agreement with the Kelvin-Helmholtz time. In this case, rotational and/or magnetic support could be slowing the collapse.

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arXiv2020

The VLA/ALMA Nascent Disk and Multiplicity (VANDAM) Survey of Orion Protostars. A Statistical Characterization of Class 0 and I Protostellar Disks

We have conducted a survey of 328 protostars in the Orion molecular clouds with ALMA at 0.87 mm at a resolution of $\sim$0.1" (40 au), including observations with the VLA at 9 mm toward 148 protostars at a resolution of $\sim$0.08" (32 au). This is the largest multi-wavelength survey of protostars at this resolution by an order of magnitude. We use the dust continuum emission at 0.87 mm and 9 mm to measure the dust disk radii and masses toward the Class 0, Class I, and Flat Spectrum protostars, characterizing the evolution of these disk properties in the protostellar phase. The mean dust disk radii for the Class 0, Class I, and Flat Spectrum protostars are 44.9$^{+5.8}_{-3.4}$, 37.0$^{+4.9}_{-3.0}$, and 28.5$^{+3.7}_{-2.3}$ au, respectively, and the mean protostellar dust disk masses are 25.9$^{+7.7}_{-4.0}$, 14.9$^{+3.8}_{-2.2}$, 11.6$^{+3.5}_{-1.9}$ Earth masses, respectively. The decrease in dust disk masses is expected from disk evolution and accretion, but the decrease in disk radii may point to the initial conditions of star formation not leading to the systematic growth of disk radii or that radial drift is keeping the dust disk sizes small. At least 146 protostellar disks (35% out of 379 detected 0.87 mm continuum sources plus 42 non-detections) have disk radii greater than 50 au in our sample. These properties are not found to vary significantly between different regions within Orion. The protostellar dust disk mass distributions are systematically larger than that of Class II disks by a factor of $>$4, providing evidence that the cores of giant planets may need to at least begin their formation during the protostellar phase.

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arXiv2019

Validating Scattering-Induced (Sub)millimeter Disk Polarization through the Spectral Index, Wavelength-Dependent Polarization Pattern, and Polarization Spectrum: The Case of HD 163296

An increasing number of young circumstellar disks show strikingly ordered (sub)millimeter polarization orientations along the minor axis, which is strong evidence for polarization due to scattering by ~0.1 mm sized grains. To test this mechanism further, we model the dust continuum and polarization data of HD 163296, one of the best observed disks with prominent rings and gaps, using the RADMC-3D radiative transfer code. We find that scattering by grains with a maximum size of 90$\mu$m can simultaneously reproduce the polarization observed at 0.87 mm (ALMA Band 7) and the unusually low spectral index of $\alpha$ ~ 1.5 between 0.87 and 1.25 mm (ALMA Band 6) in the optically thick inner disk as a result of more efficient scattering at a shorter wavelength. The relatively low spectral index of $\alpha$ ~ 2.5 inferred for the optically thin gaps is also reproduced by the same (relatively small) grains, as a result of telescope beam averaging of the gaps (with an intrinsic $\alpha$ ~ 4) and their adjacent optically thick rings (where $\alpha$ << 2). In this case, the long-standing tension between the grain sizes inferred from polarization and spectral index disappears because the relatively low $\alpha$ values are illusory and do not require large mm-sized grains. In addition, the polarization fraction has a unique pattern of azimuthal variation: higher along the major axis than the minor axis in the gaps but higher along the minor axis than the major axis in the rings. We find a rapidly declining polarization spectrum (with the fraction $p \propto \lambda^{-3}$ approximately) in the gaps, which becomes flattened or even inverted towards short wavelengths in the optically thick rings. These contrasting behaviors in the rings and gaps provide further tests of scattering-induced polarization that can be tested via multi-wavelength observations that resolve the disk substructure.

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arXiv2019

The VLA/ALMA Nascent Disk and Multiplicity (VANDAM) Survey of Orion Protostars I. Identifying and Characterizing the Protostellar Content of the OMC2-FIR4 and OMC2-FIR3 Regions

We present ALMA (0.87~mm) and VLA (9~mm) observations toward OMC2-FIR4 and OMC2-FIR3 within the Orion integral-shaped filament that are thought to be the nearest regions of intermediate mass star formation. We characterize the continuum sources within these regions on $\sim$40~AU (0\farcs1) scales and associated molecular line emission at a factor of $\sim$30 better resolution than previous observations at similar wavelengths. We identify six compact continuum sources within OMC2-FIR4, four in OMC2-FIR3, and one additional source just outside OMC2-FIR4. This continuum emission is tracing the inner envelope and/or disk emission on less than 100~AU scales. HOPS-108 is the only protostar in OMC2-FIR4 that exhibits emission from high-excitation transitions of complex organic molecules (e.g., methanol and other lines) coincident with the continuum emission. HOPS-370 in OMC2-FIR3 with L~$\sim$~360~\lsun, also exhibits emission from high-excitation methanol and other lines. The methanol emission toward these two protostars is indicative of temperatures high enough to thermally evaporate methanol from icy dust grains; overall these protostars have characteristics similar to hot corinos. We do not identify a clear outflow from HOPS-108 in \twco, but find evidence of interaction between the outflow/jet from HOPS-370 and the OMC2-FIR4 region. The multitude of observational constraints indicate that HOPS-108 is likely a low to intermediate-mass protostar in its main mass accretion phase and it is the most luminous protostar in OMC2-FIR4. The high resolution data presented here are essential for disentangling the embedded protostars from their surrounding dusty environments and characterizing them.

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arXiv2019

KFPA Examinations of Young STellar Object Natal Environments (KEYSTONE): Hierarchical Ammonia Structures in Galactic Giant Molecular Clouds

We present initial results from the K-band focal plane array Examinations of Young STellar Object Natal Environments (KEYSTONE) survey, a large project on the 100-m Green Bank Telescope mapping ammonia emission across eleven giant molecular clouds at distances of $0.9-3.0$ kpc (Cygnus X North, Cygnus X South, M16, M17, MonR1, MonR2, NGC2264, NGC7538, Rosette, W3, and W48). This data release includes the NH$_3$ (1,1) and (2,2) maps for each cloud, which are modeled to produce maps of kinetic temperature, centroid velocity, velocity dispersion, and ammonia column density. Median cloud kinetic temperatures range from $11.4\pm2.2$ K in the coldest cloud (MonR1) to $23.0\pm6.5$ K in the warmest cloud (M17). Using dendrograms on the NH$_3$ (1,1) integrated intensity maps, we identify 856 dense gas clumps across the eleven clouds. Depending on the cloud observed, $40-100\%$ of the clumps are aligned spatially with filaments identified in H$_2$ column density maps derived from SED-fitting of dust continuum emission. A virial analysis reveals that 523 of the 835 clumps ($\sim63\%$) with mass estimates are bound by gravity alone. We find no significant difference between the virial parameter distributions for clumps aligned with the dust-continuum filaments and those unaligned with filaments. In some clouds, however, hubs or ridges of dense gas with unusually high mass and low virial parameters are located within a single filament or at the intersection of multiple filaments. These hubs and ridges tend to host water maser emission, multiple 70$\mu$m-detected protostars, and have masses and radii above an empirical threshold for forming massive stars.

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arXiv2019

The Life Cycle of Dust

Dust offers a unique probe of the interstellar medium (ISM) across multiple size, density, and temperature scales. Dust is detected in outflows of evolved stars, star-forming molecular clouds, planet-forming disks, and even in galaxies at the dawn of the Universe. These grains also have a profound effect on various astrophysical phenomena from thermal balance and extinction in galaxies to the building blocks for planets, and changes in dust grain properties will affect all of these phenomena. A full understanding of dust in all of its forms and stages requires a multi-disciplinary investigation of the dust life cycle. Such an investigation can be achieved with a statistical study of dust properties across stellar evolution, star and planet formation, and redshift. Current and future instrumentation will enable this investigation through fast and sensitive observations in dust continuum, polarization, and spectroscopy from near-infrared to millimeter wavelengths.

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arXiv2019

Astro2020 Science White Paper: The Formation and Evolution of Multiple Star Systems

Significant advances have been made over the past decade in the characterization of multiple protostar systems, enabled by the Karl G. Jansky Very Large Array (VLA), high-resolution infrared observations with the Hubble Space Telescope, and ground-based facilities. To further understand the mechanism(s) of multiple star formation, a combination of statistics, high-angular resolution radio/millimeter continuum imaging, characterization of kinematic structure, magnetic fields via polarimetry, and comparison with numerical simulations are needed. Thus, understanding the origin of stellar multiplicity in different regimes of companion separation will soon be within reach. However, to overcome challenges that studies in this field are now confronted with, a range of new capabilities are required: a new millimeter/centimeter wave facility with 10 mas resolution at {\lambda}=1 cm, space-based near to far-infrared observatories, continued development of low to high resolution spectroscopy on 3m to 10m class telescopes, and an ELT-class telescope with near to mid-infrared imaging/spectroscopic capability.

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arXiv2019

Astro2020 Science White Paper: Measuring Protostar Masses: The Key to Protostellar Evolution

Knowledge of protostellar evolution has been revolutionized with the advent of surveys at near-infrared to submillimeter wavelengths. This has enabled the bolometric luminosities and bolometric temperatures (traditional protostellar evolution diagnostics) to be measured for large numbers of protostars. However, further progress is difficult without knowing the masses of the central protostars. Protostar masses can be most accurately determined via molecular line kinematics from millimeter interferometers (i.e., ALMA). Theoretical investigations have predicted the protostellar mass function (PMF) for various protostellar mass accretion models, and it is now imperative to observationally constrain its functional form. While ALMA has enabled protostellar mass measurements, samples approaching 100 sources are necessary to constrain the functional form of the PMF, and upgrades to ALMA and/or a new mm/cm facility will increase the feasibility of measuring such a large number of protostar masses. The masses of protostars will enable their stellar structure (radius and intrinsic luminosity), evolution, and accretion histories to be better understood. This is made more robust when effective temperatures and accretion rates can be measured via ground/space-based near to mid-infrared spectroscopy. Furthermore, access to supercomputing facilities is essential to fit the protostar masses via radiative transfer modeling and updated theoretical/numerical modeling of stellar structure may also be required.

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arXiv2019

Studying Magnetic Fields in Star Formation and the Turbulent Interstellar Medium

Understanding the physics of how stars form is a highly-prioritized goal of modern Astrophysics, in part because star formation is linked to both galactic dynamics on large scales and to the formation of planets on small scales. It is well-known that stars form from the gravitational collapse of molecular clouds, which are in turn formed out of the turbulent interstellar medium. Star formation is highly inefficient, with one of the likely culprits being the regulation against gravitational collapse provided by magnetic fields. Measurement of the polarized emission from interstellar dust grains, which are partially aligned with the magnetic field, provides a key tool for understanding the role these fields play in the star formation process. Over the past decade, much progress has been made by the most recent generation of polarimeters operating over a range of wavelengths (from the far-infrared through the millimeter part of the spectrum) and over a range of angular resolutions (from less than an arcsecond through fractions of a degree). Future developments in instrument sensitivity for ground-based, airborne, and space-borne polarimeters operating over range of spatial scales are critical for enabling revolutionary steps forward in our understanding of the magnetized turbulence from which stars are formed.

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arXiv2019

Time-Domain Photometry of Protostars at Far-Infrared and Submillimeter Wavelengths

The majority of the ultimate main-sequence mass of a star is assembled in the protostellar phase, where a forming star is embedded in an infalling envelope and encircled by a protoplanetary disk. Studying mass accretion in protostars is thus a key to understanding how stars gain their mass and ultimately how their disks and planets form and evolve. At this early stage, the dense envelope reprocesses most of the luminosity generated by accretion to far-infrared and submillimeter wavelengths. Time-domain photometry at these wavelengths is needed to probe the physics of accretion onto protostars, but variability studies have so far been limited, in large part because of the difficulty in accessing these wavelengths from the ground. We discuss the scientific progress that would be enabled with far-infrared and submillimeter programs to probe protostellar variability in the nearest kiloparsec.

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