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Department of Physics College of Science
National Taiwan University Doctoral Dissertation
ଯ፦ໆࡡࢃԋୱޑᅶᄽϯ
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Evolution of Magnetic Fields in High Mass Star Formation
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Ya-Wen Tang
ࡰᏤ௲Ǻລමᐆ റγ
Advisor: Paul T. P. Ho, Ph.D.
ύ҇୯ 98 ԃ 6 Д June, 2009
All rights reserved.
In this thesis, I study the role of Magnetic (B) fields in the massive star forming process. It has been suggested that B field plays a key role in the star formation process - it sustains the molecular cloud from collapsing rapidly and helps to redis- tributes the flux density and angular momentum via ambipolar diffusion. However, there are limited measurements of B field strengths and also field morphologies, due to the weak signal and the limitation of instruments. The measurements of B-field morphologies associated with star-forming cores with high angular resolution,∼ a few arcseconds, are only available since recent decades. With the Submillimeter Array, the B field morphologies projected in the plane of sky (B⊥) are traced by mapping the thermal continuum emission of the dust grains at wavelengths of 870μm. I study four massive star star forming regions in various evolutionary stages: the collapsing core W51 e2/e8 and W51 North:dust, the Ultracompact H II (UC H II) region G5.89-0.39, and the closest massive star forming site Orion BN/KL. The source in the earliest evolutionary stage, the dense core MMS 6 in OMC-3, is also observed.
As inferred from the gas kinematics and the complicated B field morphology, the B field in G5.89 is most likely been overwhelmed by the stellar feedbacks, such as expansion of the UC H II region and the molecular outflows. While in the collapsing core W51 e2/e8, the hourglass-like B field associated with e2 seems to be located in a subcritical envelope at a scale of 0.5 pc, suggesting that the B field plays a dominant role in the formation process of the star-forming cores. The field geometry in W51 North:dust is complex but organized, correlated with the fragmentation and the rotation of the flattened structure. B field in Orion BN/KL shows a part of the larger scale (0.5 pc) hourglass morphology. In MMS 6, no smaller B field structure is detected, suggesting that the field is relatively uniform across the OMC-3 filament.
In this thesis, I conclude that the role of the B field varies with the evolutionary stages of the central stars. The high angular resolution B field map is crucial when study the role of the B field in the star forming region. To understand the role of the B field, kinematics of the molecular cloud and linking the field geometry with larger scale field are necessities.
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iV
Title Page . . . i
Abstract . . . iii
Table of Contents . . . v
List of Figures . . . viii
List of Tables . . . x
Citations to Previously Published Work . . . xi
Acknowledgments . . . xiii
1 Introduction 1 1.1 Star formation in molecular clouds . . . 2
1.1.1 Magnetic Field or Turbulence Dominates Star Formation? . . 2
1.1.2 Previous approach to the problem . . . 3
1.2 Massive Star Formation through Accretion? . . . 4
1.3 Approach to the Problems: B field morphologies in the plane of sky . 5 1.3.1 Alignment Mechanism . . . 5
1.3.2 Polarized Dust Emission . . . 5
1.3.3 The Chandrasekhar-Fermi Method . . . 6
1.4 Previous B⊥ measurements: far-IR to mm . . . 6
1.4.1 single dish polarization: SHARC, HERTZ and SCUBA . . . . 6
1.4.2 interferometer: BIMA and SMA . . . 7
1.5 Structure of the thesis . . . 7
2 Linking field geometry and collapse for the W51 e2/e8 cores 9 2.1 Introduction . . . 10
2.2 Observation . . . 12
2.3 Results . . . 14
2.3.1 Continuum Emission . . . 14
2.3.2 Dust Polarization . . . 16
2.4 Discussion . . . 18
2.4.1 Hourglass B field Morphology inside the e2 dust ridge? . . . . 18
2.4.2 Hourglass B field Morphology in the e8 dust ridge? . . . 19
2.4.3 Estimate of the Strength of the B field . . . 20 v
2.4.4 Characteristic Length Scales . . . 21
2.4.5 Role of B⊥ from Envelope (0.5 pc) to Collapsing Cores (0.02 pc) 24 2.4.6 Comparison with other star formation sites . . . 25
2.5 Conclusion and Summary . . . 26
3 field geometry in infall disk/ring system W51 North 43 3.1 Introduction . . . 44
3.2 Observation and Data Reduction . . . 46
3.3 Results . . . 47
3.3.1 Continuum emission . . . 47
3.3.2 Dust Polarization and the B Field . . . 48
3.4 Discussion . . . 49
3.4.1 Dust continuum in collapsing cores . . . 49
3.4.2 Fragmentation? . . . 50
3.4.3 B field dragged by the molecular ring? . . . 50
3.4.4 Correlation of outflow axes with B field direction and dust ridge 51 3.4.5 B field in massive collapsing cores . . . 51
3.5 Summary . . . 52
4 Remnant Disk in Massive Star Forming Region Orion BN/KL? 59 4.1 Introduction . . . 60
4.2 Observation . . . 62
4.3 Results . . . 62
4.3.1 Continuum Emission . . . 62
4.3.2 Polarization . . . 64
4.3.3 Comparison of polarization at 3 mm, 1 mm and 0.87 mm . . . 65
4.4 Discussion . . . 65
4.4.1 Dust grains being mechanically aligned? . . . 66
4.4.2 Remnant dusty disk? . . . 66
4.4.3 Polarization from entrained flows? . . . 68
4.5 Conclusion . . . 68
5 Submillimeter Array Dust Polarization Image of the Ultracompact H II Region G5.89-0.39 85 5.1 Introduction . . . 86
5.2 Source Description . . . 88
5.3 Observation and Data Analysis . . . 89
5.4 Results . . . 91
5.4.1 Continuum Emission . . . 91
5.4.2 Dust polarization . . . 93
5.4.3 C17O 3-2 emission line . . . 95
5.5 Discussion . . . 97
5.5.1 CO 3-2 polarization . . . 97
5.5.2 The kinematics traced by C17O 3-2 emission line . . . 98
5.5.3 Estimate of the B field strength . . . 100
5.5.4 Collapsing cloud or not? . . . 102
5.5.5 Compressed field? . . . 103
5.5.6 Comparison with Other Star Formation Sites . . . 105
5.6 Conclusions and Summary . . . 106
6 B field geometry in relatively quiescent core MMS 6 in OMC 2/3 123 6.1 Introduction . . . 123
6.2 Observations and Data Reduction . . . 124
6.3 Results . . . 125
6.3.1 Polarization . . . 125
6.4 Discussion . . . 125
7 Conclusion and Future Direction 129 7.1 Summary of Individual Source . . . 130
7.1.1 On-going projects . . . 131
7.2 Future Direction . . . 132
7.2.1 More polarization measurements toward earlier sources . . . . 132
7.2.2 Linking to larger scale B field . . . 133
7.2.3 Kinematics of the cores . . . 133
7.2.4 Simulate B⊥ of disks/flattened structures . . . 133
7.2.5 Higher sensitivity observations . . . 133
Bibliography 135
CURRICULUM VITAE 145
2.1 Schematic of structures in W51 e2/e8. . . 34
2.2 870 μm and 1.3 mm continuum and polarization maps obtained with the BIMA and SMA. . . 35
2.2 –continued.. . . 36
2.3 Polarization map of the SMA restored with the BIMA synthesized beam. 37 2.4 Plot of polarization percentage versis normalized intensity. . . 38
2.5 B field maps in W51 e2 and e8. . . 39
2.6 B field maps of the envelope and the collapsing cores in W51 e2/e8. . 40
2.7 Histogram of the differences between measured P.A. and a hypothetical radial field, and the cumulative distribution. . . 41
2.8 Plot of position angle versus differences. . . 42
3.1 Polarization map of W51 North:dust. . . 56
3.1 –continue.. . . 57
3.2 B field map of W51 North:dust . . . 58
4.1 SMA 870 μm continuum and its linear polarization maps. . . . 71
4.1 –continue . . . 72
4.1 –continue . . . 73
4.2 Map of dust continuum emission measured with BIMA and SMA . . 74
4.2 Polarization map of extended structures inferred 0.87 mm. . . 75
4.3 B field map inferred from 3 mm, 1 mm and 0.87 mm. . . 76
4.3 B field map of extended structures inferred 0.87 mm. . . 77
4.4 Continuum emission of the extended array data with uniform weighting. 78 4.4 B field map with extended array data with uniform weighting. . . 79
4.4 Retrieved NH3 map by Wilson et al. (2000). . . 80
4.5 B field map in Orion BN/KL with uniform weighting . . . 81
4.5 B field map in Orion BN/KL with uniform weighting. . . 82
4.6 Dust continuum and large scale CO outflow. . . 83
5.1 SMA 870 μm continuum and polarization maps. . . 109
5.1 –continue . . . 110 viii
5.2 Maps of polarized intensity and polarization vectors. . . 111
5.2 –continue . . . 112
5.3 Distribution plots of polarization. . . 113
5.4 Channel maps of C17O 3-2. . . 114
5.5 Moment maps of C17O 3-2. . . 115
5.5 –continue . . . 116
5.6 Position-Velocity diagrams of C17O 3-2. . . 117
5.7 Spectra of C17O. . . 118
5.8 Maps of derived mass to flux ratio and pressures. . . 119
5.8 –continue . . . 120
6.1 Map of dust continuum emission measured with the SMA in MMS6 . 126 6.2 Map of dust continuum emission at 1.3 mm. . . 127
2.1 SMA dust polarization at 870μm in e2 . . . . 30
2.1 SMA dust polarization at 870μm in e2 . . . . 31
2.2 SMA dust polarization at 870μm in e8 . . . . 32
2.3 Derived parameters in e2 and e8 . . . 33
3.1 SMA dust polarization at 870 μm in W51 North:dust . . . . 53
3.1 SMA dust polarization at 870 μm in W51 North:dust . . . . 54
3.1 SMA dust polarization at 870 μm in W51 North:dust . . . . 55
4.1 Observational parameters . . . 70
5.1 SMA dust polarization at 870 μm in G5.89-0.39 . . . 121
5.1 SMA dust polarization at 870 μm in G5.89-0.39 . . . 122
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Two chapters of the thesis have been published or accepted for publication in the Astrophysical Journal.
”Evolution of Magnetic Fields in High Mass Star Formation: SMA dust polarization image of the UCHII region G5.89-0.39” Tang, Ya-Wen, Ho, Paul T. P., Girart, Josep. M., Rao, Ramprasad, Koch, Patrick M., & Lai, S.-P. 2009, ApJ, 695, 1399 (chapter 5)
”Evolution of Magnetic Fields in High Mass Star Formation: Linking Field Geometry and Collapse for the W51 e2/e8 Cores” Tang, Ya-Wen, Ho, Paul T. P., Koch, Patrick M., Girart, Josep Miquel, Lai, Shih-Ping,
& Rao, Ramprasad 2009, ApJ, in press (chapter 2)
The following three chapters are in preparation to be submitted in ApJ.
Chapter 3: Field geometry in rotating disk/ring system W51 North: dust Chapter 4: Resolving the polarization in Orion BN/KL with 2 milli-parsec resolution
Chapter 6: B field geometry in relatively quiescent core MMS 6 in OMC 2/3
xi
I am most grateful to my thesis advisor Paul T. P. Ho for the support and inspiring discussions on both science and life. I thank Josep Miquel Girart, Ramprasad Rao and Shih-Ping Lai for the fruitful collaborations on the thesis projects. The visit to Stephane Guilloteau and Anne Dutrey in the summer of 2008 in Bordeaux is a very important and nice experience in the way I do research. I thank postdocs, students and assistants in IAA for all the good time we spent together and helpful discussions.
I appreciate the encouragement and support from Professor Hsiang-Kuang Chang and Yi-Jehng Kuan when I just started the graduate study. Finally, I thank Patrick Koch for the unconditionally mental support and also fruitful discussions.
xiii
Introduction
In star formation process, magnetic (B) field is expected to play an important role traditionally. It provides the mechanism to enable the material to accrete onto the central ”protostar” by removing the excess angular momentum. In larger scales, B field is thought to sustain the molecular cloud from collapse globally. However, there are limited observational results of the B field strength and morphology in the star forming regions, especially observations with higher angular resolution (a few arcseconds).
Toward low mass star forming regions, the B field morphology was revealed to be hourglass-like in NGC 1333 IRAS 4A (Girart, Rao & Marrone 2006), which is a consequence of the contraction and formation of a disk. Will high mass stars (M > 8M) form via an accretion disk as in the low mass case? Will interactions between group members and the much faster evolutionary time scales be the dominant criteria? Will the B field play a dominant role in the massive star formation process?
In the PhD study, I observe a list of massive star formation sites with high angular resolution (up to 0.7) to measure the B field morphologies using the thermal dust polarization and the gas kinematics using the molecular emission lines. I summarize current understanding of the process of star formation in molecular clouds, and also the control mechanism of the star formation process in §1.1. The method to probe the B field in star forming regions are described in §1.2.
1
1.1 Star formation in molecular clouds
It has been well known that the stars are formed inside the molecular cloud.
However, the star formation efficiency inside the cloud is low, typically a few percent of the mass is converted into stellar mass (Zucherman & Evans 1974). One of the main questions in the star formation process is that how to sustain the molecular cloud from gravitational collapse. Both the B field and the turbulence are suggested to play the dominant role. In the classical scenario, the B field is suggested to play the important role in sustaining the cloud from collapse. The material starts to collapse toward a ”seed” through the ambipolar diffusion. However, this scenario can explain the isolated star formation and also only limited to the low mass case. In contrast, the turbulence dominant model seems to explain better the observed cluster formation, and also the diversity of the mass distribution of the stars. In the turbulent dominant theory, the molecular cloud is only a transient phenomena, which is formed by the compress of the turbulent flows. In this case, the B field is suggested to play a minor role in the evolution of the molecular cloud and therefore, the star formation process.
1.1.1 Magnetic Field or Turbulence Dominates Star Forma- tion?
To test these two extreme dominant models, the key parameter of the magnetic dominant model is the mass-to-flux ratioφ. The molecular cloud is original sustained by the magnetic field and is sub-critical, i.e. φ < 1. Because the ambipolar diffu- sion timescale τAD is ionization rate dependent: τAD ∝ χe ∝ n−0.5H2 (McKee 1989;
Mouschovias 1991). At denser region, τAD is smaller and therefore, the collapsing cores can be formed through ambipolar diffusion. One of the main prediction of the magnetic driven star formation is the existence sub-critical envelope and the super- critical cores.
In the turbulence driven model, the main argument is based on the ages of some star forming cores, typically only a few times of the free-fall time scale. In this sce- nario, the molecular cloud is supported by turbulence. Once the turbulence dissipates,
the molecular cloud will start to collapse, and therefore, the cloud is only a transient phenomena.
Why is it so important to study which mechanism is the dominant force? The observational constraint can provide the key to the theorists about the driving mech- anism. The prediction of the turbulent driven or magnetic driven star formation is different. The study of the driven mechanism is the fundamental question.
1.1.2 Previous approach to the problem
Mass to Flux ratio
The mass to flux ratio φM,B in the isolated low-mass star forming cores has been studied using the SCUBA in L183, L43, L1544 (Ward-Thompson et al. 2002; Crutcher et al. 2004), L1498 and L1517B (Kirk et al. 2006). The magnetic field and turbulence are reported to be equally important in these isolated and moderately quiescent cores (Ward-Thompson et al. 2006). As comparing the φM,B in four envelope and the corresponding cores, Crutcher, Hakobian, & Troland (2009) report that φM,B in the core does not larger than the one in the envelope, and suggest that the cores are not formed purely through ambipolar diffusion.
Estimate of the ages of star forming cores
The age of the intermediate cores, 14M < M < 80M, has been suggested to be in the order of million years, significantly longer than free fall or turbulent decay timescale (Netterfield et al. 2009). Proposed by the same authors, B field is more likely to be dominant. In contrast, Motte et al. (2007) suggested that high-mass pre- stellar and protostellar cores are in a high dynamic state, and therefore, turbulence seems to dominate in the molecular cloud.
In summary, there is no single answer to which force is dominant in the star formation processes in previous approaches.
1.2 Massive Star Formation through Accretion?
The formation of low mass stars (M ∼ 1M, L∼ 1L) are well characterized observationally. Infall via accretion disks, the removing of excess angular momentum via winds and outflows, as described by standard models (see review by Shu et al.
1987), appear to operate up to intermediate stellar masses (M ∼ 8 M, L ∼ 104 L). In particular, for stellar luminosity up to 104L, detected molecular outflows are highly collimated, circumstellar disks are found (Cesaroni et al. 2005), and their properties appear to be scaled-up versions of low mass cases (Beltr´an et al. 2008).
The formation process of the more massive stars could be fundamentally differ- ent. High mass star forming regions, because of their rarity, are usually at greater distances. They are also located always in dense and massive regions, as they are typ- ically formed in a group. Theoretically, the environment of high mass star forming region is very different from the low mass case because of the high radiation pressure and the stronger gravitational field. The observed molecular outflows appear poorly collimated. This may be because that the more massive stars will reach the main sequence stage very quickly, in 104 years. Accretion and molecular outflows are likely still in process when the central stars already begin nuclear burning and thereby ra- diate in the ultraviolet. Thus, the molecular material near the central star will be photoionized if the accretion pressure is not high enough. If the inner core is ionized, the high plasma pressure may affect the accretion and the outflow process (Shepherd 2003, K¨onigl 1999). Furthermore, observed outflows may contain the emission from multiple sources, since the massive stars are always formed in a dense cluster. Hence, both poor resolution and complexity have plagued observational studies. We thus need higher angular resolution observations to provide better physical constraints on the theoretic models of the massive star formation process.
1.3 Approach to the Problems: B field morpholo- gies in the plane of sky
In this thesis, I observe the B field morphology in the plane of sky (B⊥) in the massive star forming regions to analyze the problems listed before. The B⊥is derived from the observed polarized dust continuum emission. The method is explained here.
1.3.1 Alignment Mechanism
The polarized light of the dust continuum can provide the morphology of the B field on the plane of sky (B⊥). The dust grain is paramagnetic and elongated, and it is in the lowest energy state when aligned perpendicular to the magnetic field (Davis
& Greenstein (1951) for the original theory; Lazarian (2007) for the recent review of the alignment mechanisms). Although the alignment mechanism of the dust grains has been a difficult topic for decades, the radiation torques seem to be a promising mechanism to align the dust grains with the B field (e.g., Draine & Weingartner 1996;
Lazarian & Hoang 2007). Dust grains are thought to be aligned with their monor axes parallel to the B field in most of the cases, even if the alignment is not magnetic (Lazarian 2007).
1.3.2 Polarized Dust Emission
The polarized light can be from the absorption of the background stellar light, which is due to the difference of the fraction of the light being absorbed between the major and minor axes. Using this method, the detection of the polarization is restricted to the locations where the background stars can be seen. The polarization direction is parallel to the B field direction in this case. While in the star forming site, the extinction is so hight that most of the radiation will be absorbed. Observations at far-infrared and (sub-)millimeter, which trace the thermal dust emission from star forming sites directly are powerful to trace the dense regions. Again, in the case when dust grains are aligned with the B field, the polarized emission can be detected.
Due the difference in the emissivity along the major and minor axes, the emitted
light will appear to be linearly polarized. In this case, the polarized emission will be perpendicular to the B field direction.
1.3.3 The Chandrasekhar-Fermi Method
The field strength in dense clouds can be estimated by using the Chandrasekhar- Fermi (CF) method (Chandrasekhar & Fermi 1953). Assuming the dispersion of the position angles of the detected polarization is caused by the perturbation by Alfv´en waves or turbulence in the magnetic field lines, the strength of the B field projected in the plane of sky (B⊥) is given by:
B⊥= Q 4π¯ρδV
δφ(mGauss). (1.1)
Here, the factor Q is a dimensionless parameter < 1, which depends on the cloud structure, the ¯ρ is the mean gas density in g cm−3,δV is the velocity dispersion along the line of sight in km s−1,δφ is the dispersion of the polarization position angles in radian. This CF method can be applied in a strong field condition (δφ < 25◦) with Q of 0.5, according to Ostriker, Stone, & Gammie (2001). The resultant polarized dust emission has been studied successfully using the SMA and BIMA, such as in the case of NGC1333 IRS 4A (SMA), W51 e1/e2 cores (BIMA), DR(21) (BIMA), NGC2071 (BIMA) and NGC2024 IRS5 (BIMA). In this thesis, I use the thermal dust polarized emission to map the B field in the massive star formation sites.
1.4 Previous B
⊥measurements: far-IR to mm
1.4.1 single dish polarization: SHARC, HERTZ and SCUBA
At 100 μm, observations done with Stokes on the KAO have angular resolutions of 35. At 350μm, observations obtained with Hertz in CSO are typically with the resolution of 18. The observed B⊥ morphologies tend to be uniform at pc scale, such as M17 (Dotson 1996), OMC-1 (Schleuning 1998), W3 (Dotson et al. 2000), and DR 21 MAIN (Kirby 2009). At 850 μm, the angular resolution of polarization
observations can reach to 14 with SCUBA in JCMT. The revealed polarization in the OMC-2 and OMC-3 filaments (Matthews, Wilson & Fiege 2001) and in OMC-1 (Vallee & Fiege 2007) are also uniform. There are also cases where the revealed B⊥ are complex, such as NGC 7538 (Dotson et al. 2000) and W51 A (Chrysostomou et al. 2002).
These observations are measuring the B field morphology in the scale of the molec- ular cloud. A decrease of polarization percentage is commonly seen toward the star forming cores (denser regions), where the field direction shows an abrupt change in some cases. The star forming cores can only be resolved in the interferometry obser- vations.
1.4.2 interferometer: BIMA and SMA
The interferometric observations at (sub-)millimeter carried out prior this thesis work are typically at the angular resolution of 3 to 10. Both uniform and complex field field geometries have been detected. Due to the sensitivity and the low polariza- tion percentage, most of the observations are toward high mass star forming region.
Complex field geometry has been detected in Orion BN/KL (Rao et al. 1998). Uni- form field geometries which suggestive of hourglass-like morphologies are seen toward NGC 2024 (Lai et al. 2002), NGC 1333 IRAS 4A (Girart, Rao & Marrone 2006), and G30.79 FIR 10 (Cortes & Crutcher 2006). At this scale, the decrease of polarized percentage toward the emission peak is also seen.
In interferometric observations, the star forming regions can be resolved with the angular resolution of Jeans lengths scales in the massive star forming regions. The B⊥ morphologies linking the cores with the molecular cloud can be resolved.
1.5 Structure of the thesis
To understand the roles of B field in the star formation regions, I have studied the B field morphology and the kinematics of the molecular clouds in different evo- lutionary stages. Because the polarization percentage is low, typically ∼ 5% in the
star forming regions, we choose to the star forming regions which are strong in the thermal dust emission. The stages of the source observed are in the earlier stage where collapse is still dominant, to the more evolved stage, where the stellar feedback start to influence the surrounding material. The angular resolution reached ranges from 0.7 to 3.
In Chapter 2 to 6, I present results of the case studies on five sources at different evolutionary stages. The conclusion and future direction are presented in Chapter 7.
Linking field geometry and collapse for the W51 e2/e8 cores
Ya-Wen Tang, Paul T. P. Ho, Patrick M. Koch, Josep M. Girart, Shih-Ping Lai, &
Ramprasad Rao 2009 ApJ, in press
ABSTRACT
We report our observational results of 870 μm continuum emission and its linear polarization in the massive star formation site W51 e2/e8. Inferred from the linear polarization maps, the magnetic field in the plane of sky (B⊥) is traced with an angular resolution of 0.7 with the Submillimeter Array (SMA). Whereas previous BIMA observations with an angular resolution of 3 (0.1 pc) showed a uniform B field, our revealed B⊥ morphology is hourglass-like in the collapsing core near the Ultracompact H II region e2 and also possibly in e8. The decrease in polarization near the continuum peak seen at lower angular resolution is apparently due to the more complex structures at smaller scales. In e2, the pinched direction of the hourglass- like B field morphology is parallel to the plane of the ionized accretion flow traced by H53α, suggesting that the massive stars are formed via processes similar to the low mass stars, i.e. accretion through a disk, except that the mass involved is much larger.
9
Furthermore, our finding that the resolved collapsing cores in e2 and e8 lie within one subcritical 0.5 pc envelope supports the scenario of magnetic fragmentation via ambipolar diffusion. We therefore suggest that magnetic fields control the dynamical evolution of the envelope and cores in W51 e2 and e8.
ISM: individual (W51 e2/e8) — ISM: magnetic fields — polarization — stars:
formation
2.1 Introduction
The magnetic (B) field has been suggested to play an important role in the star formation process. While the B field flux density is eventually redistributed via ambipolar diffusion (Mestel & Spitzer 1956; Mouschovias 1978), the collapse itself is slowed sufficiently to explain the low star formation rate observed in molecular clouds. Alternative support via turbulence (cf. Mac Low & Klessen 2004) seems less important on parsec (pc) scales since the B fields in the plane of sky (B⊥) are often observed to be organized and uniform across the cloud, such as in M17 (Dotson 1996), OMC-1 (Schleuning 1998) and DR21 MAIN (Kirby 2009). One key question is at which sizescale will the magnetic support be overcome by gravity. The morphology of the B field at that point may reveal the details of the contraction process such as geometry and timescale. The Submillimeter Array (SMA) can be used to address this question by resolving the B⊥ structures via dust polarization studies with high angular resolutions at typically a few arcseconds.
The B field is traced by the dust continuum emission. The dust grains are most likely not spherical in shape, but somewhat elongated. They are thought to be aligned with their minor axes parallel to the B field in most of the cases (Lazarian 2007).
Among different alignment mechanisms, radiation torques seem to be a promising mechanism to align the dust grains with the B field (Draine & Weingartner 1996;
Lazarian & Hoang 2007). Due to the differences in emissivity perpendicular and par- allel to the direction of alignment, the observed thermal dust emission will be linearly polarized. The direction of the linear polarization is therefore perpendicular to the B field. With the SMA, we are able to detect the polarized component of the thermal
dust emission at sub-millimeter (sub-mm) wavelengths in order to trace the B field within the dense cores, where stars are formed. Compared to the polarization studies via absorption and scattering of stellar light in the optical or near infrared (Goodman et al. 1995), sub-mm polarization, being derived directly from dust emission, does not suffer from a limited range in grain size and the possible contamination from the more diffuse emission and absorption along the line of sight.
In this paper, we present SMA observational results with an angular resolution of 0.7 (0.02 pc) of the massive star forming site W51 e2 and e8 in W51 MAIN. The dust continuum at a wavelength of 870μm, and the B⊥field inferred from its linearly polarized component are presented. The W51 MAIN is on the eastern edge of W51.
It is at a distance of 7.0±1.5 kpc (Genzel et al. 1981) or 6.1±1.3 kpc (Imai et al.
2002). Here, we adopt a distance of 7 kpc. There is a group of Ultracompact H II (UCHII) regions in W51 MAIN, and many H2O, OH and NH3 maser spots have been identified (Genzel et al. 1981; Gaume & Mutel 1987; Pratap et al. 1991) to be associated with the e2 and e8 regions. The terminology of the structures discussed in this paper is shown in the schematics in Figure 1. The radio continuum sources e2, e4, e8, e1 and e3 are UCHII regions (Gaume & Johnston 1993; Zhang & Ho 1997), and their locations are labelled in Figure 2. Hereafter, e2, e4, e8, e1 and e3 (when in italic) refer to their corresponding UCHII regions. The infall signatures toward the e2 and e8 regions (i.e., e2 and e8 collapsing cores) have been detected clearly in NH3(Ho
& Young 1996; Zhang & Ho 1997) and in CS (Zhang, Ho, & Ohashi 1998), indicating that they are in an early evolutionary stage. Furthermore, the total luminosity of the W51 MAIN is 2×106 L (Jaffe et al. 1987), indicating that it is a massive star forming site.
The polarized dust emissions associated with the envelope of the W51 e2 and e8 regions have been previously observed at 1.3 mm and 850μm. The B⊥field structure varies with different size scales. Chrysostomou et al. (2002) has shown that the morphology of the field on the very large scale observed with SCUBA with an angular resolution of ∼ 10 (0.5 pc) appears more complex, possibly because of projection effects from several clouds along the line of sight. With an angular resolution of ∼3
(0.1 pc) with the BIMA, Lai et al. (2001) found that the position angles (P.A.s)
of the polarization vectors vary smoothly across the e2 and e8 cores (Figure 2(a)), suggesting that the B field dominates over the turbulent motions in the envelope. At which scales will the B field lose its dominance over turbulence and gravity?
2.2 Observation
The observations were carried out on 2008 July 13 using the SMA (Ho, Moran, &
Lo 2004) 1 in the extended configuration, with seven of the eight antennas available.
The projected lengths of baselines ranged from 30 to 262 kλ. The largest size scale which could be sampled in this observation was ∼8 (0.3 pc). The local oscillator frequency was tuned to 341.482 GHz. With the 2 GHz bandwidth in each sideband, we were able to cover the frequency ranging from 345.5 to 347.5 GHz and from 335.5 to 337.5 GHz in the upper and lower sidebands, respectively. The phase center is near e2 at Right Ascension (J2000) = 19h23m43s.95, Declination (J2000)=14◦3034.00. e8 is ∼7 south of the phase center. The primary beam (field of view) of the SMA at 345 GHz is ∼30.
Linear polarization (LP) observations using interferometer arrays are best ob- tained using receivers which detect both orthogonal circular polarizations (CP) si- multaneously. However, the SMA receivers are intrinsically linearly polarized and only one polarization is available currently. Thus, quarter-wave plates were installed in order to convert the LP to CP. Detailed information of the design of the quarter- wave plates and how the quarter-wave plates were controlled is described in Marrone et al. (2006) and Marrone & Rao (2008). We assume that the smearing due to the change of the P.A.s on the time scale of 5 minutes in one cycle of polarization measurement is negligible.
The conversion of the LP to CP is not perfect. This instrumental polarization (also called the leakage terms) (see Sault et al. 1996) and the bandpass were calibrated by observing 3c454.3 for 2 hours while it was transiting in order to get the best coverage
1The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.
of parallactic angles. The instrumental polarization is ∼1% for the upper sideband and ∼3% for the lower sideband before calibration, and ∼0.6% after calibration in both sidebands. The complex gains were calibrated every 12 minutes by observing 1751+096 until it set, followed by 1925+211 for the last 3.5 hours. The absolute flux scale was calibrated using Titan.
The data were calibrated and analyzed using the MIRIAD package. After the standard gain calibration, self-calibration was also performed by selecting the visibil- ities with uv distances longer than 40 kλ. In order to Fourier transform the measured visibilities to the image, the task INVERT in MIRIAD was used with natural weight- ing. The Stokes Q and U maps are crucial for the derivation of the polarization. We use the dirty maps ofQ and U to derive the polarization in order to avoid a possible bias introduced from the CLEAN process. We applied CLEAN to the StokesI (total intensity) map in order to reduce the sidelobes. The presented SMA images have all been corrected for the primary beam attenuation. The synthesized beam of the presented maps is 0.7×0.6 with a P.A. of −58◦. The presented polarization vectors are gridded to a 0.3 spacing - which is about half of the synthesized beam FWHM - in order to show the curvature of the B field morphology. Therefore, adjacent polariza- tion vectors are not formally independent within one synthesized beam. However, as usual, relative information can be extracted at under the synthesized beam resolution.
The Stokes I, Q and U images of the continuum are constructed with natural weighting in order to get a better S/N ratio for the polarization. The noise levels of the I, Q and U images are ∼ 60, 4 and 4 mJy Beam−1, respectively. The strength (Ip) and percentage (P (%)) of the linearly polarized emission are calculated from:
Ip2=Q2+U2− σQ,U2 and P (%) = Ip/I, respectively. The term σQ,U is the noise level of the StokesQ and U images, and it is the bias correction due to the positive measure of Ip (Leahy 1989; Wardle & Kronberg 1974). The σIp is thus 4 mJy beam−1. To derive the polarization, the MIRIAD task IMPOL was used. The SMA polarization vectors presented are above 3σIp in red segments and between 2 to 3 σIp in black segments.
2.3 Results
The 870 μm continuum emission and its polarized components were detected (Figure 2; Table 1 & 2). The results are presented in this section.
2.3.1 Continuum Emission
In e2, a compact 870 μm continuum emission structure with a radius of ∼ 1
(0.03 pc) is centered at ∼0.7 east of e2. Extending to the north-west of this compact emission, a fainter structure with an overall length of ∼ 2 (0.07 pc) is detected.
The H2O (Genzel et al. 1981) and (J,K)=(9,6) NH3 (Pratap et al. 1991) masers are located in this north-west extension,∼ 2away from the continuum peak. Associated with the continuum peak, there are OH masers detected within 0.5 to the east and
∼1 to the south of e2 (Gaume & Mustel 1987; Fish et al. 2006), suggesting that it is an active star forming site.
In e8, the 870μm continuum peak is centered at 0.3 west of e8. e4, e1 and e3 are at the periphery of the 870 μm continuum emission. There is an extension toward the south-west with an overall length of∼3. Associated with e8, an NH3 maser spot was detected 0.8 south of the 870 μm continuum peak by Pratap et al. (1991). The OH (Gaume & Mustel 1987) and H2O (Genzel et al. 1981) masers are also associated with e8 and the 870 μm continuum peak, suggesting again that this is an active star formation site.
When fitted with a Gaussian, the deconvolved size of the 870 μm emission in e2 is 0.9×0.8, slightly larger than the synthesized beam, and therefore, the e2 core has been resolved. For e8, the deconvolved size is 0.9×0.3 with a P.A. of 12◦. Therefore, e8 has been resolved along the major axis of the dust ridge but not along the minor axis. In both e2 and e8, the 870 μm continuum emissions are associated with the NH3 cores (Ho et al. 1983; Zhang & Ho 1997), suggesting that they are also tracing the dense regions.
The measured 870 μm flux densities within the upper and lower boxes in Figure 2(b), associated with e2 and e8, are 9.3 and 4.0 Jy, respectively. The flux densi-
ties of the free-free continuum Ff f at 1.3 cm in e2 and e8 are 300 mJy (Gaume &
Johnston 1993) and 17 mJy (Zhang & Ho 1997), respectively. In order to estimate theFf f contribution at 870 μm, we extrapolate from 1.3 cm, assuming Ff f ∝ ν−0.1. Although this assumption of optically thin emission is crude, it has been shown that the resultant Ff f roughly agrees (within a factor of 3) with the estimate from the radio recombination line at 2 mm (Zhang, Ho, & Ohashi 1998), suggesting that the assumed Ff f ∝ ν−0.1 is reasonable. The extrapolated Ff f at 870 μm is ∼230 and 13 mJy for e2 and e8, respectively. As compared to the 870 μm flux densities, Ff f
contributes∼2% for the e2 region and 0.3% for the e8 region. Therefore, the 870 μm continuum is dominated by dust emission. Hereafter, the structures traced by the 870 μm emission in the e2 and e8 regions are named as e2 dust ridge and e8 dust ridge, respectively.
Assuming a dust temperature of 100 K (Zhang, Ho, & Ohashi 1998), a dust grain emissivity Q(λ) ∝ λ−β with β = 1, and the normal gas to dust ratio of 100, we estimate gas massesMgasof 245 and 106 Mfor the e2 and e8 dust ridges, respectively (cf. Tang et al. 2009). Note that theMgasgiven here is highly affected by the assumed β. If the assumed β is 2, the estimated Mgas will be 14 times larger. Assuming the extents along the line of sight are equal to the diameters of the emission area in the e2 and e8 dust ridges, the average gas number densitiesnH2 are 3.4×106 and 2.2×106 cm−3, respectively. By using the same equation and the same assumed values of β and dust temperature, the Mgas estimated from the 2 mm dust continuum (Zhang, Ho, & Ohashi 1998) for the e2 and e8 dust ridges are 1100 and 590 M, respectively.
The difference in the estimated Mgas at 2 mm and 870 μm is most likely due to the missing flux from the extended component, which is not recovered with our SMA observations. In comparison, with the same assumptions, theMgas of the envelope is 1834 M as traced at 1.3 mm by BIMA (Lai et al. 2001). The Mgas associated with the e2 and e8 dust ridges recovered with our SMA observations is∼ 19% of the Mgas
in the envelope.
The main conclusion from the dust continuum data is that the associated mass is large. The morphology of the dust continuum is elongated. The positional offsets between the various embedded sources are significant, such as between the positions
of the 870 μm peaks and the UCHII regions. These results are consistent with the formation of a cluster of stars.
2.3.2 Dust Polarization
The polarization in the e2 and e8 dust ridges is detected and resolved (Figure 2 (c) and (d)). Throughout the paper, P.A. is defined from the north to the east. In the e2 dust ridge, the bulk of the polarization vectors form a ring around the 870μm peak with a radius of ∼1 and with the geometric center near the continuum peak instead of e2. In the north-west extension of the dust ridge, the polarization appears to be perpendicular to the major axis of the extension.
The e8 dust ridge is ∼7 away from the phase center. Even though the antenna response is 15% less efficient than at the phase center, the polarization revealed is clearly also not as uniform as previously seen with BIMA. The polarization vectors again form a ring like structure around the continuum peak. The polarization is weaker in e8 with more vectors between 2 to 3 σIp.
In comparison, the polarization in the envelope of the e2 and e8 regions, as revealed with an angular resolution of 3 (0.1 pc) with BIMA, shows a relatively uniform distribution in P.A. and therefore, a fairly uniform B⊥ field at 1.3 mm (Figure 2(a);
Lai et al. 2001). In their results, the polarization in the e2 region is weak and resolved into e2 main and e2 pol NW, named in the same paper, according to the P.A. of the polarization vectors. The component e2 pol NW is at 3 to the north-west of e2.
There is a gap where no polarized emission is detected between e2 and e2 pol NW. In the e8 region, the polarization in the BIMA results is nearly uniform with a decrease in polarization percentage near the peak position.
In order to test if the differences in polarization properties from SMA and BIMA are due to their different angular resolutions, we smoothed our SMA results to the BIMA resolution, as shown in Figure 3. Wherever the polarized emission was both detected at 1.3 mm and 870 μm, the resultant P.A.s of the polarization differed by
∼ 30◦ on average. This significant difference can be due to the different sampling of the visibilities, which are in the range of 6 to 170 kλ (λ=1.3 mm) for the BIMA
and in the range of 30 to 262 kλ (λ=870 μm) for the SMA. Specifically, the SMA filtered out the more extended and uniform component which is larger than 8. At the same angular resolution, the derived global B⊥field directions in e2 and in e8 are therefore consistent in the regions where both the SMA and BIMA have polarization detections. Most importantly, the smoothed SMA polarization map shows that the polarization percentage has decreased significantly, especially near the continuum peak positions, where the field geometry is more complex at the resolution of 0.7.
This demonstrates that the low polarization percentage at the emission peaks is due to the limited angular resolution when a more complex underlying B field morphology has not been resolved. This effect can also be due to the decrease of the alignment efficiency of the dust grains in denser regions (Lazarian & Hoang 2007) or due to geometrical effects, such as the differences in the viewing angles (Gon¸calves et al.
2005). However, in this case, the complex B field structure is the dominant effect.
The polarization percentage P (%) decreases with increasing continuum intensity I in both e2 and e8 even for the higher resolution SMA results (Figure 4). Since the BIMA results come from a resolution effect, the same might be true for the SMA results at the emission peaks. Away from the emission peaks, the general increase in P (%) is somewhat misleading. Figure 2(a) shows that this effect is not symmetrical on either side of the elongated envelope, i.e. the P (%) differs with positions on the same contour level of I. This is reflected by the large dispersion in P (%) at any value of I/Imax. Several effects, including B field geometry related to the line of sight, need to be disentangled. That the P (%) ranges mainly between 1% to 10%
(Figure 4), seems to agree with the model of grain growth in the dense regions where grain alignments are via radiative torques (see Figure 11 in Pelkonen et al. 2009).
However, based on our results, the effects of angular resolution and geometry must first be taken into account.
2.4 Discussion
2.4.1 Hourglass B field Morphology inside the e2 dust ridge?
The inferred B field in the e2 dust ridge exhibits a complex but organized mor- phology (Figure 5). We have tested the hypothesis of the measured B field being radial, and have shown quantitatively the preference of a non-radial field at a high significance level (see Appendix). There are positions where no polarized emission (depolarization) is detected, extending along a P.A. ∼ 60◦ across the 870 μm peak (color scale in Figure 2(c)). The existence of non-radial field lines together with the depolarized zones are in favor of an hourglass field morphology. Along the extension of the dust ridge toward the north-west, the B field lines are approximately paral- lel to the major axis, which is consistent with the BIMA measurement at 1.3 mm.
These lines are radial-like, but the complex structure in the north-west could belong to another embedded source as possibly indicated by the masers.
Associated with e2, organized motions in the ionized gas have been revealed with the H53α radio recombination line (Keto & Klaassen 2008), with the maximum veloc- ity gradient along P.A. ∼ 60◦. These authors interpret this gradient as a supporting evidence for an accretion flow along a dense flattened structure, where the detected motion tracks the ionized particles on the surface of the dense midplane. Both the infall and rotation near e2 have also been detected in several molecular lines (Ho &
Young 1996; Zhang & Ho 1997; Zhang, Ho & Ohashi 1998). As discussed in Keto &
Klaassen (2008), this H53α accretion flow in the direction of P.A. ∼ 60◦ might drive the molecular outflow at P.A. ∼ − 20◦ as traced by the CO 2-1 line. The argument is based on the hypothesis that if the massive star formation process is similar to the low mass case, the bipolar outflow should be along the rotation axis. The linearly distributed H2O and OH masers in the W51 e2 region could trace an outflow (Fig- ure 19(c) in De Buizer et al. 2005), as identified with the CO 2-1 line. Although the determined direction may be highly uncertain, the rotation in NH3 (3,3) is more clearly revealed along PA=135◦ ( Figure 7 in Zhang & Ho 1997) and in CH3CN along PA=110◦ (Zhang, Ho & Ohashi 1998), which seems inconsistent with the gradient de-
tected with the H53α ionized flow. This might indicate that the revealed kinematics based on different lines may be from multiple embedded sources. Higher spatial reso- lution kinematic studies with hot core molecular lines will be helpful for deciphering the underlying structures.
The B field appears to be hourglass-like near e2, with the field lines pinched along the plane of the proposed H53α accretion disk. If the B field lines are frozen into the ionized material, the field lines will be tangled along with the rotation and infall motions. The revealed depolarization might then result from the more complex underlying B field. We note that the field lines seem to go to the core with an essentially radial pattern, and therefore, leading to a sharp pinched angle in the hourglass. In contrast, the low mass case (Girart, Rao, & Marrone 2006) shows a wider and smoother pinched angle. We speculate that a larger infall momentum and a larger differential rotation (Zhang, Ho, & Ohashi 1998) might drag the field lines along and result in a narrower pinched angle in the projected plane. Projection of a nearly pole-on hourglass-like morphology possibly also leads to similar signatures. In any case, the scenario of material accreting through a disk as proposed by Keto and Klaassen (2008) is supported by our inferred B field morphology.
2.4.2 Hourglass B field Morphology in the e8 dust ridge?
Along the e8 dust ridge, the B field also shows a systematic deviation from the larger scale (0.5 pc) B field revealed by BIMA. This can again be explained by the field lines being dragged along with the accretion toward e8. In this case, the revealed B field appears to be part of an hourglass structure on a larger scale of 4(∼0.08 pc) (Figure 5(b)), with its pinched direction parallel to the dust ridge. Centered on the e8 continuum peak, a compact hourglass structure would be more convincing except for the field lines to the north. There are H2O masers north of the e8 continuum peak, and another embedded source may be indicated. This could explain the incomplete hourglass structure here.
As in the case of e2, a zone of depolarization seems to be present at the continuum peak, along the north-south direction. This is consistent with the pinch direction of
the hourglass-like morphology being along the elongated e8 dust ridge. Rotation associated with the e8 collapsing core was detected in the direction of P.A. ∼ 156◦ (−24◦) with CH3CN (Zhang, Ho, & Ohashi 1998). In this scenario, the pinch direction of the hourglass-like B field is parallel to the plane of rotation. The rotation axis of the e8 collapsing core is then almost parallel to the B field threading the 870μm dust ridge. Note that the rotation direction as traced in CH3CN is still uncertain (Zhang, Ho, & Ohashi 1998). An accurate determination of the plane of rotation associated with the e8 collapsing core is needed to test if the larger scale B field controls the direction of accretion.
Although the plane of accretion (or the pinched angle of the hourglass) cannot be determined with certainty, the collapse signature was detected toward e8 (Ho &
Young 1996; Zhang & Ho 1997; Zhang, Ho & Ohashi 1998), consistent with the pos- sible hourglass-like B field morphology. Furthermore, this collapsing core is inside the 0.08 pc scale dust ridge, as revealed with the 0.7 angular resolution B field morphol- ogy. Based on this morphology and the presence of e4, e8, e1, and e3, we suggest that the star formation process involves different stages of fragmentation, proceeding at different evolutionary timescales.
2.4.3 Estimate of the Strength of the B field
The B field strength can be estimated by comparing the gravitational force fG with the B field tensionfBfollowing Dotson (1996) and Schleuning (1998). The value of fG at a distance RG away from the center is given by
fG = G MRρ RG2
= 5× 10−26 MR
100M nH2
105cm−3( RG
0.1 pc)−2 dyne
cm3 , (2.1) where MR refers to the gas mass enclosed within a radius RG, ρ is the mass density at RG, andnH2 is the gas volume number density. The fB can be given by
fB= 1
4π B · B ∼ B2 4πRB
= 5× 10−26( B
mG)2( RB
0.5 pc)−1 dyne
cm3 , (2.2) where RB is the radius of a magnetic flux tube, and B is the B field strength. Since the e2 and e8 cores are known to be in a collapse stage, we conclude that fG > fB.
An upper limit ofB can then be derived. For the e2 collapsing core, MR is estimated to be 220M based on the 870μm flux density within a radius of 1of the continuum peak. This is consistent with the proposed self gravitating mass of>160 M (Ho &
Young 1996) based on the kinematics of the NH3 lines. RG is 1 (0.034 pc), and the mean nH2 within RG is 2.7×107 cm−3. Assuming RG = RB 0.034 pc, the B field strength in the e2 core is therefore< 19 mG. In the e8 collapsing core, MR is 94 M, and nH2 is 1.2×107 cm−3 within a radius of 1 centered on the peak position. The B field strength in the e8 core is therefore < 8 mG. Both of the upper limits of B are consistent with the lower limit of the larger scale B⊥ field strength of 1 mG (Lai et al. 2001) estimated from the method proposed by Chandrasekhar & Fermi (1953).
2.4.4 Characteristic Length Scales
To analyze the interactions between B field, gravitational force and thermal force in star forming sites, we further calculate the following three length scales following Mouschovias (1991): First, the interplay between ambipolar diffusion and Alfv´en waves is characterized by the Alfv´en length scale λA. Second, the interplay between gravitational and thermal pressure forces is characterized by the thermal Jeans length scale λT,cr, following Bonnor (1956) and Ebert (1955; 1957). Third, the interplay between magnetic and gravitational forces is characterized by the critical magnetic length scale λM,cr. They can be calculated using the following equations:
λA= 8 B
mG( nH2
106cm−3)−1( K
3× 10−3)−1 mpc, (2.3) λT,cr= 31
T
100 K( nH2
106cm−3)−1 mpc, (2.4) and
λM,cr= 36 B
mG( nH2
106cm−3)−1 mpc. (2.5)
Here, the parameter k (Eq. (6f) in Mouschovias 1991), related to the mean col- lision time between an ionized and a neutral particle, is assumed to be 0.5 when we derive Eq. (3), which is within the most likely range given in the reference in their paper. The factor K is related to the cosmic ionization rate. We assume K
= 3×10−3 (Mouschovias & Morton 1991) following the ionization rate calculated by Nakano (1979). With the assumed k and K, the estimated fractional ionization rate is 3×10−9 for a number density of 107 cm−3, which seems to be reasonable. T is the gas temperature, and nH2 is the gas volume number density. B is the B field strength.
Note that T is assumed to be 100 K in both the e2 and e8 dust ridges based on the analysis of the hot core lines by Zhang, Ho & Ohashi (1997). Since these natu- ral length scales depend on nH2 and B, they are calculated separately based on the detected continuum emission with the same assumption as in §3.1. In the 1.3 mm envelope, the nH2 is the mean number density within a best-fit Gaussian centered on the peak, and B is the lower limit of 1 mG. In the 870 μm dust ridges, nH2 is calculated within a radius of 1, and B is the upper limit calculated in §4.3. The calculated natural length scales in the e2 and e8 cores are listed in Table 3.
The physical meaning of these length scales are explained clearly in Mouschovias (1991) and references therein. λAgives the lower limit of the scale at which the B field can sustain the structure. At the scale R< λA, the ambipolar diffusion between neu- tral and ionized particles is more efficient and the Alfv´enic motion is less important.
λT,cr gives the scale where the gravitational force is equal to the thermal pressure. If an object has a size scale R> λT,cr, gravity can overwhelm the thermal pressure and collapse will start. λM,cr gives the upper limit of the scale where the cloud can be magnetically supported along the B field direction. In a region R> λM,cr, there will be enough mass and therefore the material can collapse if there are no other supporting forces.
Correlation with the SMA e2 dust ridge
In the e2 dust ridge, the revealed B⊥ morphology is clearly pinched with a radius of ∼ 0.8 near e2, comparable to the radius of the proposed rotating flattened struc- ture of 1 (Zhang & Ho 1997; Zhang, Ho, & Ohashi 1998) and the proposed ionized accretion disk of∼ 0.5 (Keto & Klaassen 2008). The derivedλT,cris∼ 0.2, suggesting that at∼ 1, gravity will easily overcome the thermal pressure support if there are no other supporting forces. The scale where ambipolar diffusion starts to take place (λA)
is ∼ 0.2 for the e2 dust ridge, consistent with the observed pinched B⊥ field lines.
Note that the revealed width of the depolarization zone near e2 is narrow (< 0.5), which is smaller than our synthesized beam. Higher angular resolution measurements with at least 0.3 resolution are needed to discriminate whether the depolarization is due to ambipolar diffusion, inefficient grain alignment or other mechanisms, such as geometrical effects. The calculated scale where the B field can sustain the structure against gravitational collapsing (λM,cr) is 0.8 along the field line. However, it is diffi- cult to compareλM,cr with the scale associated with the dust ridge, because the large scale (0.5 pc) B⊥field is twisted by∼45◦ (Figure 2(a)) at 3 resolution. Observations with visibilities at both shorter and longer uv ranges are needed in order to link the B⊥ in the core with the field in the envelope at the same wavelength. Note that the weak constraints onλA andλM,crresult from the large range of possible B field values.
Correlation with the SMA e8 dust ridge
The dust continuum emission appears to be ridge-like, and the minor axis of the e8 dust ridge is approximately parallel to the 0.5 pc B⊥field direction. The deconvolved length of the e8 dust ridge along the minor axis is barely resolved, and we adopt an upper limit to the radius of 0.3 along the minor axis. This is consistent with the estimated λM,cr < 0.7, and the estimated λT,cr of 0.3. This suggests that thermal pressure is significant as compared to gravity and field tension at this scale. Along the major axis, the deconvolved size is 0.9, which is larger than λT,cr. Furthermore, ambipolar diffusion is expected to dominate at the scale λA < 0.2 in e8. Hence we expect collapse and fragmentation to occur along the ridge. This is consistent with the revealed hourglass-like B field morphology associated with the e8 collapsing core at 0.7 resolution and the smooth B field morphology in the envelope at 3resolution.
These results in the e8 core seem to be consistent with the ambipolar diffusion model (Mouschovias & Morton 1991) and suggest that the formation of the dust ridge is influenced by the B field in the envelope.
