Nov 19, 2010 - 1 Astronomy Department, University of Illinois, Urbana, IL 61801, USA ... Department, University of Kentucky, Lexington, KY 40506, USA.
The Astrophysical Journal, 725:466–479, 2010 December 10 C 2010.
doi:10.1088/0004-637X/725/1/466
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
MAGNETIC FIELDS IN INTERSTELLAR CLOUDS FROM ZEEMAN OBSERVATIONS: INFERENCE OF TOTAL FIELD STRENGTHS BY BAYESIAN ANALYSIS Richard M. Crutcher1 , Benjamin Wandelt2,3 , Carl Heiles4 , Edith Falgarone5 , and Thomas H. Troland6 2
1 Astronomy Department, University of Illinois, Urbana, IL 61801, USA UPMC Universit´e Paris 06, Institut d’Astrophysique de Paris, 98 bis, boulevard Arago, 75014 Paris, France 3 Departments of Physics and Astronomy, University of Illinois, Urbana, Il 61801, USA 4 Astronomy Department, University of California, Berkeley, CA 94720, USA 5 LRA/LERMA, CNRS UMR 8112, Ecole ´ Normale Sup`erieure & Observatoire de Paris, Paris, France 6 Physics and Astronomy Department, University of Kentucky, Lexington, KY 40506, USA Received 2009 October 1; accepted 2010 September 25; published 2010 November 19
ABSTRACT The only direct measurements of interstellar magnetic field strengths depend on the Zeeman effect, which samples the line-of-sight component Bz of the magnetic vector. In this paper, we use a Bayesian approach to analyze the observed probability density function (PDF) of Bz from Zeeman surveys of H i, OH, and CN spectral lines in order to infer a density-dependent stochastic model of the total field strength B in diffuse and molecular clouds. We find that at n < 300 cm−3 (in the diffuse interstellar medium sampled by H i lines), B does not scale with density. This suggests that diffuse clouds are assembled by flows along magnetic field lines, which would increase the density but not the magnetic field strength. We further find strong evidence for B in molecular clouds being randomly distributed between very small values and a maximum that scales with volume density n as B ∝ n0.65 for n > 300 cm−3 , with an uncertainty at the 50% level in the power-law exponent of about ±0.05. This break-point density could be interpreted as the average density at which parsec-scale clouds become self-gravitating. Both the uniform PDF of total field strengths and the scaling with density suggest that magnetic fields in molecular clouds are often too weak to dominate the star formation process. The stochasticity of the total field strength B implies that many fields are so weak that the mass/flux ratio in many clouds must be significantly supercritical. A two-thirds power law comes from isotropic contraction of gas too weakly magnetized for the magnetic field to affect the morphology of the collapse. On the other hand, our study does not rule out some clouds having strong magnetic fields with critical mass/flux ratios. Key words: ISM: magnetic fields – polarization – stars: formation Online-only material: color figures
function (PDF) of the magnitude of the total strength of the three-dimensional magnetic field and its relation to P (Bz ), the PDF of the observed Bz . Heiles & Crutcher (2005) and Heiles & Troland (2005) have discussed this assumption. They considered four analytic functions to describe P (B): a Dirac delta function, a flat or uniform PDF, a weighted Gaussian function, and a Gaussian function. The delta function is the form generally assumed (usually implicitly); this assumes that all clouds in a sample have the same B. Then both the mean and median values of P (Bz ) = 0.50 B. One simply finds the mean or median value of the set of observed Bz , and B equals twice this value. The other assumed possible forms for P (B) all yield mean and median values for Bz roughly equal to 0.5 B (Heiles & Crutcher 2005), so if one is only interested in inferring the approximate mean or median of B from a set of Bz measurements, the form of the P (B) for a set of clouds does not matter very much. This fact has made it possible to infer astrophysically meaningful results about interstellar magnetic fields from Zeeman observations. However, having only information about the mean or median value of Bz and their approximate relationship to B significantly limits our knowledge of interstellar magnetic fields. For testing theories of star formation, this limitation is very important. There are two opposite extreme theories for how interstellar matter collects into dense molecular clouds and how those clouds collapse to form stars, and these theories make very different predictions about P (B). We call these extremes the strong-field and the weak-field models. In the strong-field model, magnetic fields control the formation and evolution
1. INTRODUCTION Since their discovery 60 years ago, much work has been done to study the magnitude and morphology of interstellar magnetic fields. Observational techniques include linear polarization of thermal continuum emission from dust and of spectral lines, Faraday rotation of linearly polarized radiation as it traverses the interstellar plasma, and Zeeman observations of spectral lines. However, none of these observational techniques give full information about the total magnetic vector B. Linear polarization observations yield the morphology of the field in the plane of the sky and, by application of an imprecise statistical technique, an estimate of the mean field strength in the plane of the sky. Faraday rotation and Zeeman observations give precise information about the magnitude and direction, but only of the line-of-sight component Bz of B. Bz may be either negative or positive, corresponding to the line-of-sight component being directed toward or away for us, respectively. These observational limitations have significantly impacted our understanding of the role played by magnetic fields in the interstellar medium, including its role in the formation of molecular clouds from the atomic gas and the evolution of molecular clouds to form stars. It is possible to infer statistical information about the total magnetic field strength B in a sample of interstellar clouds by making assumptions. One assumption is that the direction of B is random from cloud to cloud, so that the set of possible observed magnitudes of Bz ranges from zero up to the full magnitude B of B. Another assumption concerns P (B), the probability density 466
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of the molecular clouds from which stars form, including the formation of cores and their gravitational collapse to form protostars. Detailed theoretical work has been carried out by a number of groups; Mouschovias & Ciolek (1999) have reviewed and summarized the state of this theory. The fundamental principle is that clouds are formed √ with sub-critical masses, M < MΦ , where MΦ = Φ/2π G, Φ is the magnetic flux, and the expression for the critical mass MΦ is from Nakano & Nakamura (1978). The exact numerical factor in the expression for MΦ may differ slightly from the Nakano & Nakamura (1978) result depending on cloud morphology. In this paper, we will use the symbol λ to refer to the ratio of the observed mass/ flux to the critical mass/flux; observationally, λ is proportional to the ratio of column density to magnetic field strength. So λ < 1 is a subcritical mass-to-flux ratio. The magnetic field is frozen only into the ionized gas and dust; neutral gas and dust contract gravitationally through the field and the ions, accumulating mass (but not flux) in the cloud cores. This process is known as ambipolar diffusion. When the core mass reaches and exceeds MΦ , the core becomes supercritical (λ > 1), collapses, and forms stars. The magnetic flux mostly remains behind in the envelope. In the weak-field model, molecular clouds are intermittent phenomena (Elmegreen 2000), with clouds forming at the intersection of turbulent supersonic and super-Alfv´enic flows in the interstellar medium. Generally such clouds are not gravitationally bound and dissipate. Star formation occurs only in the small fraction of clouds that are sufficiently dense to be self-gravitating. Magnetic fields may be present in this picture, but they are too weak to be energetically dominant. Without magnetic support, self-gravitating cores collapse rapidly to form stars. MacLow & Klessen (2004) have extensively reviewed arguments for the weak-field model in which supersonic turbulence controls star formation. The two theories have very different predictions for λ. The strong-field model requires that mass-to-flux ratios be subcritical in the lower-density interstellar gas, so the magnetic field dominates the energetics and (for self-gravitating clouds) supports clouds against gravity. In self-gravitating clouds and cores, the strong-field model predicts M/Φ to be weakly subcritical to slightly supercritical depending on the evolutionary state; the range in λ would be quite narrow, with λ ≈ 1. The weak-field model requires that turbulence dominate magnetic fields in the low-density gas and that motions be super-Alfv´enic. Simulations imply λ ∼ 1–10 or higher. Several Zeeman surveys of magnetic field strengths in diffuse and molecular clouds have been made. Crutcher (1999) summarized the molecular cloud Zeeman data available to that time and concluded that the mean M/Φ was approximately critical to slightly supercritical in molecular clouds. Bourke et al. (2001) extended the OH Zeeman work to the southern hemisphere with the Parkes telescope and found results for M/Φ that essentially agreed with the Crutcher (1999) conclusion. Heiles & Troland (2004) carried out an extensive survey in H i diffuse clouds with the Arecibo telescope, finding that the mean M/Φ was subcritical by about an order of magnitude (although these diffuse clouds are not self-gravitating). Troland & Crutcher (2008) carried out a very extensive OH Zeeman survey of dark cloud cores with the Arecibo telescope and found that the mean M/Φ was supercritical by about a factor of 2. Falgarone et al. (2008) carried out a CN Zeeman survey of molecular cores with the IRAM 30 m telescope and discussed earlier CN Zeeman results (Crutcher et al. 1996, 1999) together with the new data. CN samples higher densities than OH (∼105–6 cm−3 rather than
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the ∼103–4 cm−3 sampled by OH), so the two surveys were complementary. The OH and CN Zeeman results were similar, however: a mean λ ∼ 2–3, with an uncertainty of order a factor of perhaps two. All of these results for cores are compatible with both models of star formation. Hence, observations having only mean (or median) values for M/Φ have been unable to discriminate decisively between the two extreme-case models for star formation. A distinguishing feature between the two models is the range of predicted values of λ. As outlined above, the strong-field model predicts λ ≈ 1, while the weak-field model predicts a broad range of λ from 1 to 10 or more. If we consider clouds with roughly comparable column densities, as observations suggest for molecular clouds, then the strong-field model implies a rather narrow range in B, while the weak-field model implies a much broader range in B. Knowledge of P (B) would provide this information. Another difference in the two theories is the predicted scaling of B with density. If magnetic fields are extremely weak, flows of matter to form clouds would be unaffected by B, the field would be pulled in by matter flowing perpendicular to field lines, and a scaling of B with density would be expected. At higher densities, Mestel (1966) showed that if magnetic field strengths were too weak to affect the morphology of clouds during gravitational contraction, B ∝ n2/3 ; he obtained this result analytically for the case of a spherical cloud, but for a more realistic non-spherical cloud whose shape may be dominated by turbulence, the scaling with density would be similar so long as the collapse was isotropic. In contrast, if B is sufficiently strong, cloud formation would be by flows along field lines, and at lower densities B would be independent of density. As ambipolar diffusion proceeded for gravitationally bound clouds, collapse will be impeded except along field lines, and the power-law coefficient will be 0.5. In this paper, we report the results of a Bayesian analysis of Zeeman data taken from the literature with the purpose of inferring information about P (B), the PDF of the total field strengths in diffuse and molecular clouds and cores, and the dependence of field strength on density. In Section 2, we discuss the Zeeman data used in the analysis. In Section 3, we describe the Bayesian analysis techniques that we employ. Our analysis started with two separate, simplified studies of (1) the lower-density gas sampled in H i and OH lines (Section 4) and (2) the higher-density gas sampled in OH and CN lines (Section 5). Although these studies and results are not the primary basis for the conclusions reached in this paper, they are presented here for completeness and because they may help readers better understand the analysis. Also, because these more restricted analyses used a somewhat different technique (model comparison) from our more comprehensive analysis (parameter estimation), they serve as some level of model checking for the PDF of B. Based on the success of these preliminary studies, we then carried out a third, comprehensive study of all of the Zeeman data (Section 6). In Sections 4–6, we discuss each of the three studies in succession and present the results. Finally, in Section 7, we discuss the implications of the studies and in Section 8 present our conclusions. 2. ZEEMAN DATA Our requirement for Zeeman survey data to be included in our analysis is that the data provide values for Bz , the uncertainty σ (Bz ), and the volume density n(H) in which the Zeeman effect has been measured. Here, n(H) is the density of hydrogen nuclei,
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n(H i) for atomic gas, and 2n(H2 ) for molecular gas. We choose to carry out our analysis in terms of n(H) rather than column density N (H) so we can study the scaling of magnetic field strength with volume density as well as the PDF of the total magnetic field strengths. We include H i data, because molecular clouds must form from atomic gas and we are interested in the evolution of the magnetic field in this process. We have considered data from the sources described below. Although this does not include all possible Zeeman data, leaving out most non-survey studies of single clouds, the OH survey of Bourke et al. (2001) that did not include estimates for n(H), and maser Zeeman results, which sample regions where very different astrophysics may dominate, it is a very comprehensive set of the available and relevant Zeeman data. Below we briefly describe the four Zeeman surveys and some of the astrophysical results obtained by each. The data from these four surveys form the data set that we analyze in this paper. 2.1. The 1999 Crutcher Study Crutcher (1999) assembled a set of 27 of the molecular cloud Zeeman results available at that time, estimated n(H2 ) for each position, and carried out a study of the role of magnetic fields in star formation. The set was very heterogeneous, including poor angular resolution (18 ) OH results for dark clouds to Very Large Array mapping studies of high-mass star formation regions; n(H2 ) ranged from about 500 cm−3 to about 6 × 106 cm−3 . That paper listed only upper limits for Bz in cases where it was believed that the Zeeman effect was not detected. However, the measured values (including non-detections) of Bz and their uncertainties are essential for our Bayesian analysis, so we have recovered those values from the literature cited by Crutcher (1999). The Crutcher (1999) analysis of these data was based on the 15 Zeeman detections only. The result was λ ≈ 2 and Alfv´enic turbulent Mach number MAlf,turb ≈ 0.8, where B = 2Bz was used to estimate the total field strength B from the measured Bz . The 15 detections were also fitted to a B ∝ nα power law, with the result α = 0.47 ± 0.08. However, these numbers may be biased in that the non-detections were ignored. 2.2. The H i Millennium Survey Heiles & Troland (2004) carried out a very extensive study of the four Stokes parameters in H i absorption toward continuum radio sources. They inferred H i line spin temperatures and optical depths from the data, from which they found the column densities N (H i) of each line component. They derived Bz toward 41 continuum sources, which have 136 Gaussian line components with meaningful Zeeman measurements. However, they discussed reasons for excluding a number of these results and ended up with a total of 66 statistically usable components. These Zeeman results were for the cold neutral medium (CNM)—that is, H i diffuse clouds. In addition to Bz and its uncertainty, which are provided by the Zeeman analysis, we need an estimate of n(H) for each line component. Heiles & Troland (2005) estimated TCNM ≈ 50 K and used the median pressure PCNM /k = 3000 cm−3 K (Jenkins & Tripp 2001; Wolfire et al. 2003) to estimate a median n(H)CNM ≈ 54 cm−3 for their CNM sample. We require an estimate of n(H) for each line component, which we obtain from the individual spin temperatures derived by Heiles & Troland (2004) and the assumption that PCNM /k = 3000 cm−3 K
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applies individually for each line component. For a very small number of cases, information about the spin temperature was not available; we used the median value for n(H)CNM in these cases. The results range between n(H) ≈ 3 cm−3 and 200 cm−3 , with the same median as reported by Heiles & Troland (2005). Although this procedure undoubtedly leads to significant errors in individual cases, it does not introduce systematic error. Our assumption (see below) of a factor of two uncertainty in all the n(H) is likely to cover the uncertainties in the diffuse-cloud n(H), which would be dominated by the conversion from spin temperature and (assumed) constant pressure. In any case, as we shall see from our Bayesian analysis, for the H i observations the magnetic field strengths have no density dependence, so the individual values of n(H) end up not impacting the results of our analysis. 2.3. The Arecibo OH Dark Cloud Zeeman Survey Troland & Crutcher (2008) carried out a sensitive survey of the OH Zeeman effect toward 34 dark cloud cores with the Arecibo telescope. They estimated densities from N (OH), [OH/H], and the assumption that the cloud dimension along the line of sight was equal to the mean size in the plane of the sky. The n(H2 ) ranged from about 200 cm−3 to about 6600 cm−3 with a median value of 1800 cm−3 . They reported nine probable detections, but carried out their analysis using mean and median values for all 34 positions, and not just for the detections. Again assuming B = 2Bz , they found λ ≈ 2 and MAlf,turb ≈ 1.5. 2.4. The CN Zeeman Survey Falgarone et al. (2008) used the IRAM 30m telescope to continue the earlier CN Zeeman work with the same telescope (Crutcher et al. 1996, 1999). Three results from the earlier CN Zeeman studies were listed both in the Crutcher (1999) compilation and in the Falgarone et al. (2008) complete listing of CN Zeeman results; without these duplicate listings, Falgarone et al. (2008) have 11 new results. They estimated densities from a combination of appealing to literature studies of densitysensitive molecular transitions and from N (CN), [CN/H], and the assumption that the dimension along the line of sight was equal to the mean size in the plane of the sky. The n(H2 ) ranged from about 1.5 × 105 cm−3 to about 1 × 106 cm−3 with a median value of 4×105 cm−3 . The result was then sensitive CN Zeeman measurements toward 14 positions, with eight detections of Bz . As for the Troland & Crutcher (2008) OH Zeeman study, the analysis was based on all 14 measurements and not just on the detections. Once more with the B = 2Bz assumption, they found λ ≈ 3 and MAlf,turb ≈ 1.7. 2.5. The Zeeman Data Set The Zeeman data set we analyzed, a superset formed by the four surveys described above, consists of 66 sensitive measurements of the Zeeman effect in H i diffuse clouds and 71 sensitive measurements of the Zeeman effect in molecular clouds. This set of 137 Zeeman measurements of Bz together with the uncertainties in Bz is collected from the published sources in Table 1 and are plotted against n(H) in Figure 1. The Bz from the four surveys described above are plotted with different symbols so the source of each point is clear. Table 1 identifies each measurement by the source name used in the original paper. In some cases, the same source name occurs multiple times. These do refer to independent measurements of Bz , either in distinct velocity components, or with different
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1000
|BZ ( G)|
100
10
1
0.1
101
102
103
104
105
106
107
nH (cm-3)
Figure 1. Set of diffuse cloud and molecular cloud Zeeman measurements of the magnitude of the line-of-sight component Bz of the magnetic vector B and their 1σ uncertainties, plotted against n(H) = n(H i) or 2n(H2 ) for H i and molecular clouds, respectively. Different symbols denote the nature of the cloud and source of the measurement: H i diffuse clouds, filled circles (Heiles & Troland 2004); dark clouds, open circles (Troland & Crutcher 2008); dark clouds, open squares (Crutcher et al. 1999), molecular clouds, filled squares (Crutcher et al. 1999); and molecular clouds, stars (Falgarone et al. 2008). Note that Zeeman measurements give the direction of the line-of-sight component as well as the magnitude. By convention, positive Bz denote fields pointing away from the observer and vice versa. Only the magnitudes |Bz | are plotted. The solid line segments show the most probable model from the comprehensive analysis, Section 6. Also shown (plotted as dotted line segments) are the ranges given by acceptable alternative model parameters to indicate on the Bz plane the uncertainty in the model.
Zeeman tracers sensitive to different densities, or at different positions. Refer to the original papers for additional details about each observation. The Bz are inferred from linear least-squares fitting of observed Stokes V spectra by dI /dν spectra scaled by a parameter proportional to Bz , where I is the observed Stokes I spectrum and dI /dν is obtained by numerical differentiation of the Stokes I spectra. The procedure also yields the uncertainty σ in Bz ; these uncertainties are dominated by the stochastic noise in the observed spectra and are Gaussian-normal distributed. Note that Zeeman measurements give the direction of the line-of-sight component as well as the magnitude. By convention, positive Bz denote fields pointing away from the observer and vice versa. Hence, the PDF of a single Zeeman measurement is Gaussian, with the range perhaps distributed across both positive and negative values of Bz . The n(H) are inferred as described above. In no case does the inference n(H) depend on Bz , so the uncertainties in the n(H) are independent of the uncertainties in the Bz . The PDF of the uncertainties in the n(H) is unknown; in our analysis we make an assumption about this, as described below. The data are clearly not distributed uniformly in n(H), with significant gaps around n(H) ∼ 102.5 cm−3 and n(H) ∼ 104.5 cm−3 . This might suggest that there are selection effects or biases in the estimates of n(H) that may affect a statistical analysis. However, these gaps are purely due to the very limited number of species with significant Zeeman sensitivity, i.e., those with an unpaired electron. The only species with Zeeman detections in the interstellar medium are H i, OH, and CN. For reasons of abundance and excitation, these sample respectively the approximate density ranges 1 cm−3 < n(H) < 102 cm−3 , 103 cm−3 < n(H) < 104 cm−3 , and 105.3 cm−3 < n(H) < 106.3 cm−3 . In Figure 1, the effect of these different probes of Bz is apparent. At low densities, all of the data come from the Arecobo H i absorption-line study (Heiles & Troland 2004). The OH dark cloud Bz that fill the intermediate density
range of points in Figure 1 come mainly from the Arecibo OH emission-line survey (Troland & Crutcher 2008), but some from the heterogeneous (Crutcher et al. 1999) compilation. That compilation also included single-dish and interferometer studies of mainly OH but also H i lines that sampled higher densities; the highest density point, from an excited-state OH line, comes from this compilation. The CN survey (Falgarone et al. 2008) sampled the highest densities. The gaps in the distribution of the n(H) therefore reflect limitations in sampling of the full density range n(H) imposed by the very limited number of Zeeman-sensitive species and not any biases that would affect the statistical analysis. Finally, we should note that the Zeeman effect measures the line-intensity-weighted mean Bz in the gas that produces a single spectral line. When there are multiple line velocity components along the same line of sight, one can infer separate Bz for each velocity component independently. However, if magnetic fields were tangled within a cloud and only a single spectralline component were identified with that cloud, the Zeeman effect would recover only the net field and hence magnetic flux through the cloud. The oppositely directed line-of-sight components within the cloud would produce no net Zeeman effect. Hence, the total magnetic pressure within a cloud with a tangled magnetic filed would be higher than inferred from the Zeeman effect. Although such tangling would not occur if the magnetic energy dominated turbulent energy, for the case of weak magnetic fields the importance of magnetic pressure could be underestimated. 3. DATA ANALYSIS CONCEPTS 3.1. Obtaining the PDF for the Total Magnetic Field Strength Before describing the details of the statistical analysis that we carried out, we present a brief qualitative description of two PDFs that we employ, in order to clarify the concept of the
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Table 1 Zeeman Data
Table 1 (Continued)
Name
Species
Ref
nH (cm−3 )
BZ (μG)
σ (μG)
W3OH SgrB2(north) SgrB2(main) W3(main) S106 NGC 2024 S88B B1 W49B W22 W40 ρ Oph1 ρ Oph2 L1495W L1521 L1647 L889 Taurus16 TaurusG TMC1 L183 L134 TMC-1C 3C120 3C120 3C120 3C123 3C123 3C123 3C131 3C132 3C132 3C132 3C133 3C133 3C133 3C138 3C138 3C138 3C138 3C142.1 3C142.1 3C18 3C18 3C207 3C207 3C225a 3C225b 3C237 3C310 3C315 3C315 3C318 3C318 3C333 3C348 3C348 3C348 3C353 3C353 3C409 3C409 3C409 3C409 3C409 3C410
OH* Hi Hi Hi OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1.3 × 107 5.0 × 103 6.3 × 104 6.3 × 105 4.0 × 105 1.3 × 104 1.3 × 104 2.0 × 104 2.0 × 103 2.0 × 103 1.0 × 103 3.2 × 103 2.0 × 103 2.0 × 103 2.0 × 103 2.0 × 103 2.0 × 103 2.0 × 103 2.0 × 103 2.0 × 103 2.5 × 103 3.2 × 103 2.0 × 104 5.3 × 101 6.0 × 101 5.1 × 101 1.2 × 101 1.3 × 102 5.5 × 101 5.0 × 101 2.6 × 101 4.0 × 101 7.3 × 101 5.8 × 101 5.1 × 101 3.4 × 101 5.6 × 101 4.8 × 101 4.3 × 101 4.1 × 101 4.6 × 101 9.7 × 101 6.5 × 101 4.6 × 101 1.1 × 102 9.1 × 101 1.0 × 102 1.3 × 102 1.6 × 102 5.8 × 101 5.1 × 101 3.7 × 101 6.6 × 101 3.7 × 101 8.3 × 101 1.9 × 102 6.9 × 101 2.0 × 101 6.1 × 101 8.4 × 101 6.8 × 101 5.3 × 101 5.7 × 101 5.8 × 101 7.3 × 101 3.8 × 101
3100 −480 550 400 400 87.0 69 27.0 21.0 −18.0 −14.0 9.5 1.3 5.2 −0.4 −4.7 −0.6 0.6 7.5 2.0 −1.0 −2.6 1.4 −1.6 5.2 3.6 4.8 −2.6 1.2 0.7 9.7 4.2 −4.0 5.8 −0.3 −9.5 5.6 −5.6 7.3 10.6 −8.3 7.2 −1.2 12.8 −1.9 −3.2 −1.2 −1.3 −0.7 −2.7 −0.1 3.9 −0.2 −4.6 3.0 1.4 0.0 0.5 4.2 5.1 3.3 3.0 5.4 0.6 3.2 1.4
400 51 59 20 23 5.5 5 4.0 5.0 2.0 2.6 3.0 4.3 2.5 2.7 3.5 2.1 2.2 3.0 3.7 5.1 3.3 2.4 1.9 3.8 2.8 4.2 1.0 8.6 8.4 5.5 1.0 3.8 1.1 1.7 6.3 1.0 2.2 3.4 3.1 1.3 7.4 1.5 7.0 2.2 3.7 4.9 1.1 1.1 1.3 1.1 6.3 4.1 7.6 1.7 1.5 0.9 6.4 2.0 2.0 1.2 1.7 1.0 1.0 2.5 1.0
Name 3C410 3C410 3C410 3C433 3C433 3C433 3C454.3 3C454.3 3C454.3 3C454.3 3C75 3C75 3C78 3C98 3C98 4C13.65 4C13.65 4C13.67 4C13.67 Tau A Tau A Tau A Tau A L1457S L1457Sn L1448-CO L1448-Coe L1455-CO N1333-8 B5 L1495(6) IRAM04191 B217-2 L1521E L1521F L1524-2 L1524-4 L1551S1 B18-5 L1534 TMC1 L1507A1 CB23 L1544 L Ori 1 Ros 4 Mon16W Mon16 Mon16N L723 L771 L774w L774 L663 L694n L694s L810 NGC 2024 OMC1s OMC1n1 OMC1n4 S106CN S106OH M17SWHI M17SWCN S140
Species Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi Hi OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH OH CN CN CN CN CN CN CN CN CN
Ref
nH (cm−3 )
BZ (μG)
σ (μG)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
3.5 × 101
0.2 4.4 5.2 −0.7 −0.6 6.7 6.6 3.4 −2.1 0.8 5.0 −16.2 −4.6 −2.8 −2.4 5.2 −8.8 4.7 5.8 2.1 −3.1 7.2 2.1 −13.0 −21.3 −26.0 −20.6 −8.8 −4.0 −10.8 −5.1 3.3 13.5 5.7 −1.4 −5.4 −1.2 6.1 −3.6 1.2 9.1 −0.3 −6.7 10.8 −15.0 −4.4 7.2 51 4.2 3.0 −1.6 −9.1 −5.9 3.4 3.7 1.8 8.6 10 40 −360 80 −60 −520 140 −220 250
2.1 3.0 2.3 3.8 3.3 6.6 3.6 2.9 0.9 4.4 1.5 8.8 1.0 4.8 3.8 3.6 6.5 2.3 3.3 3.3 0.6 1.7 1.4 3.8 8.3 3.7 3.4 5.6 7.8 4.2 5.1 3.5 3.7 5.1 4.0 3.8 3.6 4.1 5.2 7.4 2.2 4.0 3.6 1.7 4.5 9.8 8.1 26 6.9 5.5 3.7 6.2 3.1 5.5 2.8 4.0 8.8 120 240 80 100 200 380 130 80 90
5.3 × 101 4.0 × 101 1.3 × 102 2.1 × 101 2.4 × 101 2.6 × 101 4.8 × 101 5.4 × 101 2.5 × 102 6.2 × 101 6.1 × 101 5.3 × 101 8.1 × 101 2.3 × 101 7.1 × 101 6.4 × 101 8.7 × 101 7.6 × 101 9.1 × 100 1.9 × 102 2.8 × 102 4.9 × 101 1.3 × 104 1.7 × 104 4.7 × 103 4.2 × 103 2.2 × 103 3.6 × 103 3.1 × 103 4.1 × 103 6.0 × 103 3.4 × 103 2.9 × 103 5.1 × 103 4.2 × 103 5.2 × 103 4.2 × 103 1.1 × 104 5.4 × 103 1.0 × 104 3.8 × 103 3.0 × 103 3.2 × 103 1.5 × 103 4.8 × 102 1.7 × 103 2.2 × 103 1.8 × 103 2.5 × 103 1.9 × 103 3.7 × 103 4.8 × 103 3.3 × 103 2.6 × 103 2.1 × 103 1.6 × 103 1.2 × 106 3.6 × 106 3.6 × 106 2.8 × 106 2.0 × 105 2.0 × 105 1.2 × 106 1.2 × 106 1.2 × 106
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MAGNETIC FIELDS IN INTERSTELLAR CLOUDS Table 1 (Continued)
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1
Species
Ref
nH (cm−3 )
BZ (μG)
DR21OH1 DR21OH2 W3OH G10.6 S255
CN CN CN CN CN
4 4 4 4 4
3.4 × 105 3.4 × 105 3.0 × 105 2.8 × 106 7.4 × 105
−360 −710 1100 740 −730
σ (μG) 100 120 330 270 340
P(|BTOT|)
Name
Notes. OH∗ is a rotational state of OH 120 K above the ground state, requiring high densities for excitation. References. (1) Crutcher 1999; (2) Heiles & Troland 2004; (3) Troland & Crutcher 2008; (4) Falgarone et al. 2008.
0
1
2
|BTOT|/|BTOT,1/2| 2
P(|BZ|)
analysis. Figure 2 (top) shows two PDFs for B (the total strength of the magnetic vector B), a delta-function PDF and a flat PDF. Below are shown the associated PDFs for Bz (the observed lineof-sight field strengths). Both of the P (B) shown in the figure are defined to have mean and median values of unity. P (Bz ) is what can be inferred from a set of Zeeman observations. Suppose that one finds that the Bz are distributed roughly equally between a minimum value of −B and a maximum value of B. Then these data would be consistent with the delta-function P (B), and one would conclude that all of the magnetic fields in the observed set have roughly the same magnitude. In such a case, the PDF of the observed Bz is due to the fact that only one of the three components of B is measured by the Zeeman effect, and there is a random distribution of the angles between B and the line of sight. On the other hand, if we observe many more small values of |Bz | than large values, with the PDF of the observed line-of-sight field strengths P (|Bz |) ∝ ln(B0 /|Bz |), then the observations suggest that the total strengths B in the sample are not all the same, but have a flat PDF between 0 and a maximum value B0 . There is an overabundance of small Bz both because some total field strengths in the sample are small and because all of the total field strengths are multiplied by cos(φ), where φ is the angle between the line of sight and each B. Hence, by a statistical analysis of the observed Bz , it will be possible to infer which of a set of assumed P (B) is the most probable, and by what factor in comparison with the other P (B)’s that are tested. The two PDFs we consider here do not correspond directly to any prediction from theory, so some explanation of why we focus on these two PDFs is worthwhile. The delta-function PDF is what in the past has been assumed implicitly in the analysis of Zeeman observations. It assumes that the total field strength B is the same in a presumably homogeneous sample of clouds. Its advantage is that it provides a simple way to analyze and discuss Zeeman data. Both the mean and median values of the observed distribution of Bz are equal to one-half of the total field strength. For a sample of molecular clouds, which generally have similar column densities, strong-field theories for clouds at similar evolutionary stages have λ’s within a narrow range, so the range of B is small. What might be described as an opposite extreme is to have the B uniformly distributed between 0 and some maximum value; this would allow for the large range in B’s in the weak-field theories. Of course, there are infinite possibilities. P (B) may be bimodal, or there may be more very small B than a uniform P (B) would give, or P (B) may be Gaussian or near Gaussian. In this first study of P (B) for Zeeman observations of diffuse and molecular clouds, we have chosen to focus our attention on these two simple forms of the PDF. Although they are unrealistically sharp, given
0
1
0
0
1
2
|BZ|/|BTOT,1/2| Figure 2. Plots of the PDFs for the observed line-of-sight field strengths Bz (bottom) and the associated PDFs for the total field strengths B (top). These are scaled such that the median values B1/2 = 1. The delta function for P (B) and its associated P (Bz ) are shown as thick lines, while the flat function for P (B) and its associated P (Bz ) are shown as thin lines.
the sparseness of the data and the fact that we have to infer P (B) from the measured P (Bz ), this simple approach seems best for this first study. Additional Zeeman measurements and specific predictions of P (B) from theory will enable future enhancements of these types of statistical studies. 3.2. Three Important PDFs So as to avoid notational clutter we will often simply use P (x) instead of the more complete Px (x) to write the PDF of x. That is P (x1 ) and P (x2 ) may have different functional forms, since x1 and x2 have different statistics. It will be clear from the context what is going on; when it is not we will add subscripts to P. Conditional probabilities are written P (x|y) and denote the probability density of x when y is given. Three forms of PDF arise frequently in our analysis, so we define abbreviated versions of them here. One is the Gaussian PDF, abbreviated as 1 2 G(x, m, V ) ≡ √ e−(x−m) /2V , 2π V
(1)
where m is the mean and V the variance of the distribution. We model the measurement errors of Bz as Gaussian. This is not a choice but a result of the way Bz is obtained from the original radio observations. The magnetic field component is determined using a linear least-squares fit to the Zeeman splitting. Since the uncertainty in the line profile arises from radiometer noise which is Gaussian, the uncertainty in Bz is a linear combination of Gaussians and hence is Gaussian. In fact, even if there were a small non-Gaussianity in the receiver noise its impact on the Gaussianity of the measurement error in Bz would be suppressed. Another is the PDF for a quantity distributed uniformly between two positive values, which we often use to describe the
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purported distribution of interstellar magnetic field strengths: ⎧ ⎨ 1 if a x < b, (2) U (x, a, b) = (a − b) ⎩ 0 otherwise. Finally, we have the top-hat PDF, which is uniform symmetrically around zero; this describes the distribution of the line-of-sight field component Bz for a given total field strength B: ⎧ ⎨ 1 if |x| xmax , (3) T (x, xmax ) = 2xmax ⎩0 otherwise. 3.3. Statistical Framework Our goal is to confront theories of star formation with an analysis of the Zeeman data that provide information about the total strength of the magnetic field, B, rather than just the measured line-of-sight component. We chose a Bayesian approach because Bayesian statistics provides a mathematically well-defined framework for the quantitative comparison of theories given data and an explicit set of assumptions. A Bayesian analysis provides a unique answer to the question “given the data and a detailed quantification of all prior uncertainties and assumptions, what is the relative probability of two models A and B?” These models A and B can be distinct models in the usual sense or can be instances of the same model but with different sets of model parameters. In the latter case, the analysis gives the PDF of each parameter. By contrast a frequentist analysis assumes that the only probabilities associated with models are 0 or 1, corresponding to the model being false or true, respectively. The data arise randomly within the true model. Hypothesis testing proceeds by inventing a test statistic, evaluating this statistic on the available data, and computing the distribution of the same statistic given the model being tested (the null hypothesis). The null hypothesis stands or falls based on an assessment of whether the test statistic takes a “typical” value on the data compared to the distribution predicted by the null-hypothesis. This approach is often referred to as a “Monte Carlo” analysis, although what is strictly true is that the analysis is a frequentist analysis using Monte Carlo techniques. 3.4. Using Bayesian Statistics In this section, we describe our Bayesian statistical framework for comparing different models of the interstellar magnetic field. A pedagogical introduction to Bayesian reasoning and a detailed probabilistic justification of Bayesian inference and model comparison can be found in Jaynes (2003). We proceed by parameterizing all the attributes and defining the relevant PDFs. 1. We denote a particular model by the letter M. This particular model is characterized by J parameters, which we denote by θj ; the set of θj constitutes the parameter vector θ . In this paper, we consider several models M for the magnetic field strength: a delta function with strength θ0 ; a flat distribution between two limits θ0 and θ1 ; and a more complicated PDF derived from numerical simulations. 2. We denote the data by the letter d. In this paper, the data (inferred from measurements of Stokes I and V spectra) consist of I measured line-of-sight field strengths Bˆ zi , where
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the hat denotes measured values (which have observational error), the subscript z indicates line-of-sight component, and the subscript i denotes the measurement number. 3. We specify the likelihood, which is the PDF of the data that would be obtained from model M and its set of parameters θ , by the function P (d|θ, M). 4. We specify the prior. The likelihood is unconstrained in the sense that any and all values for each θj are allowed and equally credible. We constrain the expected ranges of parameter values by assigning a PDF capturing the information about the parameters before the analysis is performed; this prior is denoted by P (θ, M). 5. We specify the evidence, which is the net probability that we obtain the data d from M and is denoted by P (d|M). This is related to the likelihood and prior: when we sum their product over all parameters θj , i.e., when we marginalize the parameters θ , we obtain the evidence: P (d|M) =
P (d|θj , M) P (θj |M).
(4)
j
6. Finally, we specify the posterior probability of the set of parameters P (θ |d, M). This is what we really want to know: given a particular model M and the data d, what are the PDFs of the model’s parameters θ ? We calculate the posterior probability using Bayes’ theorem, which is a fundamental identity in probability theory: posterior × evidence = prior × likelihood
(5a)
or P (θ |d, M) P (d|M) = P (d|θ, M) P (θ |M).
(5b)
or, as usually written, P (θ |d, M) =
P (d|θ, M) P (θ |M) . P (d|M)
(5c)
This last form of the expression makes it clear that the evidence serves the function of a normalization constant in Bayes’ theorem so that, when summing or integrating over the parameters θ , the total probability on the left-hand side is unity. We are trying to make statements about the magnitude of the magnetic field vector. The probability distribution for the data given a model ranges from negative through positive values. Reversing this distribution to get information about the magnitude of the field vector using Equation (5c) gives a distribution that is only defined over positive values, again as it should, since the magnitude of a vector is always positive. For a single Bz observation, the inferred PDF for B is the same whether Bz or −Bz was observed. Essentially each Bz measurement adds a lower limit on the magnitude of B with larger values of B being disfavored, essentially as 1/B, though taking measurement noise on Bz into account modifies this behavior slightly. The 1/B behavior may be understood as follows: a model with large B allows for a broader range of Bz than a model with small B, which is wasteful if small Bz is observed. In Bayesian statistics, the notion that a model ought to be parsimonious arises automatically.
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MAGNETIC FIELDS IN INTERSTELLAR CLOUDS 3.5. Specifying the Prior
The ideal prior would be a product of delta functions that are centered on the actual parameter values θj . However, if we knew these actual values we would probably not be performing the observation. Rather, we usually have some expectation of the ranges in which the parameter values lie, and sometimes we even know that some portion of a particular range is more probable than another portion. This expectation is embodied in the prior P (θ, M). The usual case is that the prior is used to specify the exact nature of ignorance about the parameters before the observation is performed. Even if a range of values seem equally probable, it may be possible to specify plausible bounds on parameters before the observation is done. In specifying the range of parameter values for the prior, our strategy is to specify one of two PDFs. 1. A uniform prior, i.e., uniform probability per arithmetic interval, with the parameter θj lying between two limits θj,min and θj,max , so we have P (θj ) =
1 . θj,max − θj,min
(6)
1 . θj ln(θj,max /θj,min )
(7)
This is appropriate for quantities that are scale invariant. For example, for a range θj running from 1 to 100, the logarithmic prior has the same probability for the interval 1–2 as it does for 50–100; in contrast, the uniform prior has 50 times the probability for the latter range as for the former. Logarithmic priors are often appropriate for parameters measuring a physical scale, such as the magnetic field strengths in the interstellar medium. On the other hand, uniform priors are often preferred for parameters measuring location within a restricted range. 3.6. Parameter Inference Versus Model Comparison The evidence, P (d|M), is independent of the parameter values θ ; it has been marginalized with respect to these values. Because it is not dependent on θ it is often neglected when working entirely within a model space that is parameterized continuously. For example, in the process of parameter estimation, when we are comparing probabilities for different θ within a single model M, we can maximize with respect to all θj the function P (θ |d, M) ∝ P (d|θ, M) P (θ |M); (8a) and, in particular, for two sets of parameters θA and θB we need only consider the ratio P (d|θA , M) P (θA |M) P (θA |d, M) = × . P (θB |d, M) P (d|θB , M) P (θB |M)
using Bayes’ theorem for the models, the probability ratio of two models MA and MB is P (d|MA ) P (MA ) P (MA |d) = × . (9) P (MB |d) P (d|MB ) P (MB ) The importance of the evidence in Bayesian analysis derives from this fact: for a test where both models are judged equally probable before seeing the data, the relative probability of the two models given the data is the ratio of the evidences. It is clear from the preceding analysis that the evidence is dependent on the specification of the prior. Therefore, the value of a model comparison depends on faithful modeling of prior ignorance (or prior information). The situation is simplified if the models to be compared can be considered as special cases of an enlarged metamodel. In this case, model comparison amounts simply to parameter inference in the metamodel. The relative posterior probability of the two models is simply the ratio of the marginal posterior probability of the parameter of the metamodel which interpolates between them. 4. STUDY 1: LOW-DENSITY CLOUDS 4.1. Method
This is appropriate for quantities such as angles, for which the fractional interval of a range Δθj is independent of the value of θj itself. 2. A logarithmic prior, i.e., uniform probability per decade interval, with the parameter θj lying between two limits θj,min and θj,max , so we have P (θj ) =
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(8b)
However, when we compare discrete model classes, each with different parameterizations, the evidence becomes important:
As a first exercise, we analyze separately the observed Bˆ z data for H i (Heiles & Troland 2004) and for dark cloud OH (Troland & Crutcher 2008) in order to attempt to differentiate between three models which lead to different expected statistics of the observed magnetic fields in lower-density clouds. The three simple models are a fixed model, where all clouds have the same magnetic field B0 (a delta-function PDF); a flat model, where each cloud’s magnetic field is drawn from a uniform distribution between 0 and Bmax (U in Equation (2), with a = 0); and a numerical model taken from detailed computer simulations of magnetic fields in turbulence-driven structure formation. See Heiles & Crutcher (2005) for discussion of the delta function and flat models of P (B), including plots of P (B) and the associated P (Bz ) for each model. Figure 2 shows similar plots. We will evaluate the relative probabilities of these different models using just the evidence for each model. This expresses no prior preference of one model compared to another. Thus, for each model we must compute the evidence, which means evaluating the sum in Equation (4) or its integral equivalent. In detail, we model each measurement of the magnetic field component along the line of sight Bˆ zi (the hat indicates the observed quantity) as a true line-of-sight magnetic field value Bzi convolved with a Gaussian measurement error with variance Var(Bˆ zi ) = Vi . This Bzi in turn derives from the total strength B in the cloud, with Bzi being B times the cosine of the angle between B and the observed line of sight to the cloud. 4.2. Models for the PDF 4.2.1. Fixed Model
The fixed model assumes that the total magnetic field strength is everywhere the same, i.e., that Bi = B. Assuming that the B fields have random directions, uncorrelated with the other clouds, then Bzi has the top-hat PDF (T in Equation (3)) and the observed line-of-sight field Bˆ zi is given by the convolution ˆ P (Bzi |Bzi ) = G(Bˆ zi , Bzi , Vi ) T (Bzi , Bi ) dBzi . (10) However, since we are assuming the delta-function PDF, all total field strengths are assumed equal, although the field may point
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either toward or away from us. Given a set of magnetic field observations d, the likelihood is P (d|B) = P (Bˆ zi |B). (11) i
Here and below we omit the model label M, which appears in our definition P (d|B, M) of Section 3.4, to avoid clutter. Assuming a uniform prior over B (U in Equation (2)), the posterior density for B given the data is given by Bayes’ theorem of Equation (5), i.e., P (B|d) ∝ P (d|B). Similarly, we obtain the evidence for the fixed model P (Mfixed ) by integrating the likelihood over B, i.e., the integral form of Equation (4): (12) P (Mfixed |d) = P (d|B)dB. 4.2.2. Flat Model
In the flat model, the total magnetic field strength B of a cloud has equal probability of being in the interval 0 < B < Bmax . This is easily implemented by replacing P (Bˆ zi |Bi ) in Equation (10) by the appropriate one. Explicitly, the convolution becomes a pair of convolutions: ˆ G(Bˆ zi , Bzi , Vi ) T (Bzi , Bi ) P (Bzi |Bi ) = × U (Bi , 0, Bmax ) dBi dBzi .
(13)
All other aspects of the model remain the same as for the fixed model. The evidence calculation proceeds analogously, except here the integration in Equation (4) is over Bmax , the parameter in this model. 4.2.3. Numerical Model
In this model, we confront the data with the output from a numerical simulation of turbulence-driven cloud formation described in Falceta-Gon¸calves et al. (2008). The simulation output we had at our disposal was a data cube of magnetic field values b in simulation units, measured for all cells in the simulation, from which we derived a histogram h(b). This histogram or P (b) is similar in form to the weighted Gaussian P (B) discussed by Heiles & Crutcher (2005). We model the data as a random sample from a linearly scaled version of this numerically determined distribution. The likelihood for this model therefore becomes P (d|B0 ) = P (Bˆ zi |B0 ), (14) i
where P (Bˆ zi |B0 ) =
G(Bˆ zi , Bzi , Vi ) T (Bzi , Bi )
h(Bi /B0 ) dBi dBzi . B0 (15)
4.3. Results This first study was to test whether the H i and OH data (analyzed separately) for lower-density clouds could distinguish between three forms for P (B), a Dirac delta function or fixed B distribution, a flat distribution, and a numerical distribution based on a simulation. The delta function has every total field strength B within the data set equal to the same value, while the flat distribution has B ranging uniformly between zero
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and a maximum value (U in Equation (2)). The simulation (Falceta-Gon¸calves et al. 2008) used ideal MHD and did not include gravity. Use of this simulation is appropriate for study of the H i data since those clouds are not self-gravitating. It is less appropriate for the OH dark cloud results. The simulation had sonic and Alfv´enic Mach numbers Ms = 7 and MAlf = 2; the ratio of the thermal to magnetic pressures β = 0.1. At the end of the simulation, the result that we use, there was a volume density contrast of ∼107 and a column density contrast of ∼103 over the 5123 cells within the simulated cube. We take B and n for each cell in order to compute P (B) for this simulation. Only the H i and the OH dark cloud data sets were considered separately. Examination of Figure 1 shows that the upper envelopes of the Bz for H i (n(H) < 200 cm−3 ) and for OH dark clouds (400 cm−3 < n(H) < 1.3 × 104 cm−3 ) do not appear to have a very strong dependence on n(H), so for this initial study of these two data sets no dependence on n(H) was considered. For the H i data, the probability ratios of the likelihood that each of the assumed P (B) fits the data are delta function:numerical:flat ≈1:0.33:1.5. Although the flat P (B) has the highest probability, all three are roughly the same. The conclusion is that the H i data cannot distinguish between these P (B) with any confidence. This is similar to the conclusion reached by Heiles & Troland (2005) using a frequentist Monte Carlo method for the same data. For the flat P (B), we find the median B1/2 ≈ 6 ± 1 μG, in agreement with the Heiles & Troland (2005) result. For the OH dark cloud data, the probability ratios are delta function:numerical:flat ≈1:103 :104 . Hence, the delta function does not fit the data, and the flat distribution is favored over the numerical distribution. For the flat distribution, Bmax ≈ 29 ± 5 μG; this means that the total magnetic field strengths would be uniformly distributed between 0 and ∼30 μG, with a median value of ∼15 μG. The Bayesian result does not significantly change the mean or median value of the total field strength B as inferred by Troland & Crutcher (2008); it adds the information that there is a large range in the total field strength B in the OH dark clouds. 5. STUDY 2: HIGH-DENSITY CLOUDS 5.1. Method Our second study is of the molecular cloud observations, OH (Troland & Crutcher 2008) and CN (Falgarone et al. 2008). For n(H) 103 cm−3 we expect that self-gravity will lead the magnetic field strengths to scale with density; such a scaling of the upper envelope of the observed Bz is apparent in Figure 1. We can therefore improve the comparison by including information about the number density n(H). We assume B(n) = Cn1/2 and Bmax (n) = Cn1/2 for the fixed and flat models, respectively. We chose a square root dependence based on the result of Crutcher et al. (1999). Since we do not have information linking densities and magnetic fields for individual clouds from the numerical simulation, and since the numerical simulation we have does not include gravity (which must be included for these self-gravitating clouds), we only compare the fixed and the flat models. We cannot directly measure the value of ni for each cloud. Instead, we adopt a value nˆ i that is inferred from other measurements. These estimates are generally either from column densities inferred from the Zeeman species and the radii of the clouds inferred from maps of the plane-of-sky distribution or
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MAGNETIC FIELDS IN INTERSTELLAR CLOUDS
from molecular excitation. Based on our experience in making such estimates, we choose to trust the inferred value within a factor of two, although the actual degree of uncertainty is not precisely known. In this case, the actual value is related to the inferred one by a PDF. We adopt the logarithmic PDF of Equation (7): ⎧ 1 ⎨ if 12 < nnˆ ii < 2 P (nˆ i /ni ) = ln(4) nˆ i /ni (16) ⎩0 otherwise. Putting these together we can write for the likelihood of each measurement P (Bˆ zi , nˆ i |ni , C) = P (Bˆ zi |Bi (ni )) P (nˆ i |ni ),
(17)
where P (Bˆ zi |Bi (ni )) was defined as it was in Equations (10) and (13), and the dependence on C enters only in the function Bi (ni ). We further assume that each measurement is independent of all the other measurements, so the likelihood for the full data d, given the set of observed/inferred pairs Bˆ i and nˆ i , is P (d|{ni }, C) = P (Bˆ zi , nˆ i |ni , C). (18) i
To complete the probabilistic model, we need to assume a prior distribution for each ni and for C. There are few constraints on these quantities, so they can lie within broad ranges. For this case, the logarithmic prior of Equation (7) is appropriate. One difference here, however, is that we do not specify the limits. This makes the prior distributions improper because they are not normalizable, which is a consequence of trying to represent complete ignorance using a probability distribution. Our choice of this ignorance prior is being conservative to the extreme. However, all other probability distributions are well behaved in spite of our choice of an infinitely broad prior. These choices allow us to write down the unnormalized posterior density for C given I elements in the data set,
P (d|{ni }, C) P (ni ) P (C) P (C|d) = dn1 ...dnI . (19) P (d|M) Integrating over C, we obtain the expression for the evidence P (d|M) = P (d|{ni }, C) P (ni ) P (C) dn1 ...dnI dC. (20) 5.2. Results This study used only the molecular line data, OH and CN, and assumed a square root dependence of the upper bound of B on n(H). The two models tested were the fixed model (for the same n(H) all of the total field strengths B have the same value, i.e., a delta-function PDF) and the flat distribution (for the same n(H) the total field strengths B are distributed uniformly between 0 and a maximum value). Our analysis found an evidence ratio fixed:flat ≈1:105 . That is, between these two models, the data favor the total magnetic field strength B at a given density being uniformly distributed between 0 and a maximum value Bmax by a factor of ∼105 over the model with all B at a given n(H) having the same value. Both models assumed a square root dependence of Bmax with density n(H); therefore, our analysis provided no test of the validity of the assumed density dependence.
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6. STUDY 3: ALL CLOUDS 6.1. Method Finally, after the above limited model comparisons, we perform a more comprehensive exploration of model space over all three Zeeman tracers of magnetic field strength (H i, OH, and CN) by introducing parameters that both interpolate between the extreme cases we considered above as well as extend the model space. The full Zeeman data set described in Section 2 is used in this study. For our generalized model, we take the relationship between the cloud density ni and the maximum magnetic field in a cloud to be B0 , n < n0 α Bmax (n, θ ) = . (21) B0 nn0 , n > n0 Our use of this model was suggested by the data shown in Figure 1. The upper envelope of the Bz in Figure 1 appears to suggest that B is approximately constant at some B0 for low n(H) and that it increases above some density n0 (H) as B ∝ nα . This model is consistent with astrophysical expectations, cf. Mouschovias & Ciolek (1999); MacLow & Klessen (2004), if clouds assemble from lower-density material by motions predominately along the field. However, if the field is so weak that it does not influence cloud formation, a dependence of field strength on density is expected. Our model allows for this possibility, for a n0 smaller than the n(H) sampled by the measurements would yield B scaling with n(H) at all densities. Once self-gravity is attained, however, the field strength will increase with density as contracting clouds draw field lines together. We reiterate the definitions of the above three parameters: B0 is the maximum magnetic field in a low-density cloud from Equation (21); n0 is the crossover density between the Bmax = B0 regime and the power-law regime; and α is the power law in that relationship. Finally, we introduce a fourth parameter f, where 0 f 1, that allows for a flexible functional form for P (B). Above we had considered two extreme cases, a delta function and a flat function. Rather than the flat function PDF with B extending from 0 to a maximum value of B, we consider a flat PDF that excludes the smallest and largest values (see Figure 3). The largest values are excluded simply to keep the median value B1/2 = 1; the effect is for the PDF to be flat between a minimum value greater than 0 and a maximum value. If f = 0, we have the flat PDF and if f = 1 we have the deltafunction PDF of Figure 2. The delta-function and flat models we considered before are then special cases of this more general model. We collect the parameters together into the vector θ , which has four elements: θ = [B0 , f, n0 , α].
(22)
The cloud’s total magnetic field B is then modeled as uniformly distributed between f Bmax and Bmax (U in Equation (2)), P (B|n, θ ) = U (B, f Bmax (n, θ ), Bmax (n, θ )).
(23)
Of course, as always a random angle between the line of sight and the direction of the vector B is assumed for each Bˆ zi . Putting these together we can write for the likelihood of each measurement Bˆ zi P (Bˆ zi , nˆ i |ni , θ ) = P (Bˆ zi |Bi (ni , θ )) Pn (nˆ i |ni ), with P (Bˆ z i|Bi (ni , θ )) as in Equations (10) and (13).
(24)
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we remove a generous 1000 samples from the beginning of each chain to avoid dependence on our choice of initial condition for the chain. The correlation length along the chain was estimated to be 20 samples which gives us 400 uncorrelated samples from the joint posterior density.
1
P(|B|)
f
6.2. Results
0
0
1
2
|B|/|B1/2| Figure 3. Plot of the PDFs for the total field strengths B used in the comprehensive modeling of H i, OH, and CN. It is a flat distribution with mean and median at 1. However, there is a free parameter f to be determined by the Bayesian analysis. If f = 0, the PDF is the unaltered flat PDF with the total B uniformly distributed from 0 to a maximum value. If f = 1, the PDF becomes the Dirac delta function. For 0 < f < 1, the PDF is intermediate between the two extremes, more like what real simulations of the formation of clouds and cores produces (although these are generally not flat at the top). The specific example shown here has f = 1/3.
The likelihood for the entire data set can then be written just as in the previous section. Our choices of priors for the different parameters are as follows: P (ni ) ∝ 1/ni P (n0 ) ∝ 1/n0 1/α 0 < α < 3/4 P (α) ∝ 0 otherwise P (f ) = U (f, 0, 1) P (B0 ) = const.
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We chose the upper limit for α of 3/4 to restrict the exploration of parameter space to an astrophysically meaningful range. We explore the posterior density over the enlarged model space as a function of all parameters θ . This has the advantage over the discrete model comparison we performed above of computing the relative probabilities of all models in the full parameter space. However, since we now vary four parameters we need to choose a method for exploring the parameter space, because gridding in each parameter direction quickly leads to an unmanageably large number of sampling points. We choose to explore the space by constructing a random walk that is designed to follow the posterior PDF. This can be achieved using Markov Chain Monte Carlo methods; we use the well-known Metropolis–Hastings (MH) algorithm (Hastings 1970). Briefly, the MH algorithm proceeds as follows. Given our posterior density P (θ |d) and any starting position θ in the parameter space, the step to the next position θ in the random walk is obtained from a proposal distribution q(θ |θ ). Assuming q to be symmetric in θ and θ , the requirement of detailed balance leads to the following rule: accept the proposed move to θ if the acceptance ratio a 1, where a ≡ P (θ |d)/P (θ |d). If a < 1, remain at θ . This sequence of proposing new steps and accepting or rejecting these steps is then iterated until the samples have converged to the target distribution. We use this algorithm to generate chains of 5000 samples exploring the four-dimensional parameter space. We assess convergence by running a second chain from a different initial parameter combination and checking that the results obtained in this second run are the same as for the first run. In addition
Results of the Bayesian analysis are shown in Figure 4. The free parameters determined by the analysis are B0 , n0 , α, and f. The median values for the free parameters are: B0 ≈ 10 μG, n0 ≈ 300 cm−3 , α ≈ 0.65, and f ≈ 0.03. These parameters define the solid lines that were plotted in Figure 1; dotted lines show the outer ranges of acceptable models. Note that one observation, at n(H) ∼ 103.7 cm−3 , is well above the upper limit to B inferred in our analysis. There are several factors that could contribute to this. One is that this point is for the GMC Sgr B2 (north), which may be unique among the clouds observed. It is in the Galactic center region, where magnetic field strengths are much higher than in the outer regions of the Galaxy near the Sun, where the vast majority of the clouds with Zeeman measurements are located. It is probably the most massive GMC in the Galaxy, with extremely active star formation and unusual chemical properties, since it is the strongest source of emission from complex molecules. It is also possible that the volume density n(H) in the region where magnetic fields have been measured is significantly higher than the value used here. The Zeeman measurements were in the H i line, which generally samples lower densities. However, in the Sgr B2 environment the H i may be intermixed with much higher-density molecular gas known to exist in this cloud. If so, the H i Zeeman measurements would be sampling the magnetic field strength at a higher density than has been inferred. In order to test whether this cloud has significantly affected our results, we re-ran the analysis omitting this datum from the sample. We also more carefully studied the convergence of the chains for this new analysis; about 28,000 steps were performed, with rapid convergence. Omitting the Sgr B2 (north) measurement resulted in only small effects on n0 and α, while values for B0 and f were essentially unaffected. The results were that both n0 and α became slightly larger. Because we have no definite justification for rejecting this measurement, and no significant change in the astrophysical implications are implied, our discussion will be in terms of the parameters inferred by including it. Figure 4 shows the range of uncertainty in each of these parameters. Figure 5 shows the degree of correlation between the four parameters inferred in the Bayesian analysis. Values of n0 and α are highly correlated, such that a smaller value of n0 requires a smaller value of α. This correlation leads to a fairly large range in the possible values of α. Nonetheless, α > 0.6 is well established; the astrophysically important values α 0.5 (see Section 1 and discussion below) are excluded. Similarly, values of B0 and n0 are correlated. However, it is clear that there is a significant range in low n(H) over which B does not scale with n(H). There is also a very small anticorrelation between f and n0 . There is no significant correlations between the other parameters. Finally, f ≈ 0. This means that a distribution of the total strength of the magnetic field (B) ranging between essentially zero and a maximum value at each density is implied by the data.
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7. DISCUSSION Except for an attempt to infer P (B) for diffuse clouds by a frequentist Monte Carlo analysis (Heiles & Crutcher 2005), no previous attempt has been made to infer P (B) for interstellar Zeeman data. Previous studies all simply assumed
that B = 2Bz and discussed the astrophysical implications of the Zeeman measurements purely in terms of mean and/or median results. This Bayesian study of Zeeman observations of magnetic fields in H i and molecular clouds therefore provides new insights into the strengths of magnetic fields in interstellar clouds.
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For H i diffuse clouds (Section 4), there is no significant finding about the PDF of the total field strengths B. The data are consistent with all three of the model PDF’s that we investigated. Hence, the data are consistent with all field strengths in H i clouds identical, B ≈ 6 μG, or uniformly distributed between 0 and about 12 μG, or some intermediate distribution with a peak near the median. Our analysis agrees with a previous one (Heiles & Troland 2005) that used the same data but a different analysis technique. The mean or median total field strength of about 6 μG implies a mass-to-flux ratio that is subcritical by at least an order of magnitude, if all H i clouds had that total field strength. On the other hand, if the total field strength varies over an order of magnitude, it is possible that some H i clouds may be approximately critical or even supercritical. The inability of the data to definitively define the PDF of B makes it impossible to draw further conclusions. For OH in dark cloud cores (Section 4), without taking n(H) into account, there is a highly significant finding that the total field strengths distribute roughly uniformly between 0 and a maximum value Bmax ≈ 30 μG. Since this maximum value is about twice the mean or median value discussed by Troland & Crutcher (2008), who found that λ ≈ 2 (a factor of two supercritical), this suggests that some dark cloud cores may be critical or even subcritical. However, half of the total field strengths would be below the median; these would likely have λ’s more supercritical than the median value of 2. The inferred range in B with a flat PDF is very much greater than the range in the column densities N in the observed OH sample, so it is not possible that B and N are proportional such that all λ ≈ 2. There must be many highly supercritical dark cloud cores. Clouds with the highest B would have mass/flux ratios M/Φ that are mildly supercritical to perhaps mildly subcritical, while those with B ≈ 0 would have highly supercritical M/Φ. That is, λ must vary significantly from cloud to cloud, consistent with predictions of the weak-field theory. For the study of OH and CN together (Section 5) with an assumed n1/2 dependence of field strength on density, the finding is similar to the above one for OH alone with no dependence of the maximum value of B on density. That is, magnetic field strengths range from very small values to a maximum value, which by assumption scales as n1/2 . Those clouds with very weak magnetic fields would have highly supercritical M/Φ, while those with the strongest magnetic fields would have approximately critical M/Φ. The studies just discussed were exploratory and preliminary to our comprehensive analysis of all the data, which is the one on which our astrophysical conclusions are primarily based. For this comprehensive study (Section 6), we find that at densities less than about 300 cm−3 , the maximum value of B is density independent, while at n(H) > 300 cm−3 Bmax scales with density. The fact that we find no evidence for B scaling with density at low densities suggests that magnetic fields in the diffuse interstellar medium are sufficiently strong that cloud formation is primarily by accumulation of matter along magnetic field lines. Simulations (e.g., Hennebelle et al. 2008) have found that at low densities B does not increase with density. As discussed by Heiles & Troland (2005) and Heiles & Crutcher (2005), the field strengths in the diffuse interstellar medium are sufficient to play a very significant role in the dynamics of this medium. The inferred break-point density at which B does start to scale with n(H) could be explained as the approximate density at which clouds become self-gravitating, at least on average. As clouds contract gravitationally, the magnetic field
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will be dragged in with the matter, and the field strength will increase with increasing density. Just how B scales with n(H) is an important diagnostic of the astrophysics. For n(H) > 300 cm−3 , we find α ≈ 2/3 in the B ∝ nα parameterization, with α < 0.6 being very strongly excluded. Our result for α is therefore consistent with the α = 2/3 result of Mestel (1966), who studied spherical collapse with flux freezing. Spherical collapse requires a weak magnetic field, for a strong field would constrain collapse perpendicular to field lines and lead to clouds flattened along B and α 1/2 (Mouschovias & Ciolek 1999). For cloud contraction driven by ambipolar diffusion, α < 0.5, approaching 0.5 at late stages of the collapse. Hence, our result for the scaling of the field strength with density supports a generally weak magnetic field strength model—that is, one with the magnetic fields not sufficiently strong to dominate support and collapse. It should be noted that Crutcher (1999) found α = 0.47±0.08, a result in contradiction to the present result, in spite of the fact that the 15 Bz on which the Crutcher (1999) result was based are included in the data set we analyze here. Possible reasons for this difference are the much smaller size of the data set, the fact that no attempt was made to analyze P (B) (only a least-squares fit to the observed Bz was used), and the fact that only detections were analyzed—clearly an analysis technique biased toward larger values of B. For a subset of clouds with B near the maximum value of the PDF, it is reasonable that the magnetic field would be strong enough to affect the contraction and a smaller powerlaw exponent would be expected. Our Bayesian analysis has also given information about the PDF of the total magnetic field strengths B, namely, that the B in our sample of molecular clouds varies from essentially 0 to an upper envelope value. Before discussing this, however, we again note that the Zeeman effect measures the direction-averaged line-of-sight magnetic field strength. Therefore, if within a cloud there were two fields with oppositely directed but equally strong line-of-sight components within the same spectral-line velocity component, a Zeeman observation would measure Bz = 0. Thus, the very small fields implied toward some clouds by our Bayesian analysis could be the result, not of very weak uniform fields, but of oppositely directed fields which were in fact quite strong. Such a situation may occur for strong turbulence that dominates the magnetic field, or for a large-scale stretching and entrapment of field lines, both of which may cause field reversal. If so, one would then measure with the Zeeman effect only the net flux through the cloud and would not be sensitive to the higher pressure produced by the reversed magnetic field. If this were generally true, however, one might expect there to be many cases where one would directly observe fields that reverse in direction within single clouds, either from highresolution Zeeman mapping observations or from inference of Bz separately in two or more velocity components along the same line of sight. Such small-scale field reversals are extremely rare, however, occurring mainly in the H i absorptionline data (Heiles & Troland 2004). Even in these cases, it is not clear whether the multiple velocity components of the H i line arise within the same physical cloud or simply in different clouds along the line of sight to the background continuum source. Even if small-scale field reversals within molecular cores are responsible for the inferred P (B) that implies a large fraction of weak total field strengths in molecular clouds, such reversals within clouds are not compatible with the strong-field, ambipolar diffusion model of star formation, since such tangling of the field would be strongly suppressed by the strong fields.
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So our conclusion must be that all clouds cannot have strong magnetic fields. The previously inferred mean and median values for B and λ in molecular clouds (Crutcher 1999; Troland & Crutcher 2008; Falgarone et al. 2008) that implicitly assumed a delta-function PDF are not significantly changed by our analysis. Those earlier results found that λ ≈ 2–3, or slightly supercritical. Our inferred PDF means that many molecular cloud cores for which the Zeeman effect has been measured have field strengths significantly weaker than the mean value and hence a significantly supercritical mass-to-flux ratio—in at least a few cases, highly supercritical. Some clouds would of course have B stronger than the mean, and hence may be approximately magnetically critical or even subcritical. If ambipolar diffusion drives all star formation, magnetic fields in all clouds must be strong. Our analysis rules out this case. Turbulent driving of cloud and core formation can however start with weak fields and produce a wide range of field strengths in the clouds and cores that are formed. Examples of turbulent simulations that report this result are Tilley & Pudritz (2007) and Lunttila et al. (2009). We also note that this conclusion agrees with that reached by Crutcher et al. (2009) from a study of the ratio of λ between envelope and core in four dark clouds. 8. CONCLUSION Our Bayesian analysis of the line-of-sight component of the magnetic vector B provided by Zeeman observations was successful in providing information about the distribution of the total strengths B of magnetic fields in clouds. The lack of scaling of B with density for diffuse clouds suggests that clouds form by accumulation of matter along magnetic field lines. For molecular clouds the conclusion is that the distribution of the B range from very small values up to a Bmax that scales with density approximately as n2/3 . The power-law scaling is not consistent with a strong magnetic field model with magnetic support against gravity, but rather with one in which magnetic energy does not in general dominate gravity. Also, the fact that a significant population of molecular clouds must have very small magnetic field strengths implies that for many molecular
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clouds, magnetic fields do not dominate. Those clouds with field strengths near the maximum value in the distribution would however have magnetic energy generally comparable with gravitational and kinetic energies. The conclusion is that the role of magnetic fields in star formation is complicated and diverse, and that neither of the two opposite extreme-case theories of star formation (strong field and weak field) may fully describe how nature forms stars. We thank the reviewer for a very thorough and thoughtful review that significantly improved this paper. This research was partially supported by NSF grants AST 0440508, 0606822, and 0908841. REFERENCES Bourke, T. L., Myers, P. C., Robinson, G., & Hyland, A. R. 2001, ApJ, 554, 916 Crutcher, R. M. 1999, ApJ, 520, 706 Crutcher, R. M., Hakobian, N., & Troland, T. H. 2009, ApJ, 692, 844 Crutcher, R. M., Troland, T., Lazareff, B., & Kaz`es, I. 1996, ApJ, 456, 217 Crutcher, R. M., Troland, T., Lazareff, B., Paubert, G., & Kaz`es, I. 1999, ApJ, 514, L121 Elmegreen, B. G. 2000, ApJ, 530, 277 Falceta-Gon¸calves, D., Lazarian, A., & Kowal, G. 2008, ApJ, 679, 537 Falgarone, E., Troland, T. H., Crutcher, R. M., & Paubert, G. 2008, A&A, 487, 247 Hastings, W. K. 1970, Biometrika, 57, 97 Heiles, C., & Crutcher, R. M 2005, in Cosmic Magnetic Fields, ed. R. Wielebinski & R. Beck (Berlin: Springer) Heiles, C., & Troland, T. H. 2004, ApJS, 151, 271 Heiles, C., & Troland, T. H. 2005, ApJ, 624, 773 Hennebelle, P., Banerjee, R., Vazquez-Semadeni, E., Klessen, R., & Audit, E. 2008, A&A, 486, L43 Jaynes, E. T., & Bretthorst, G. L. 2003, Probability Theory: The Logic of Science (Cambridge: Cambridge Univ. Press) Jenkins, E. B., & Tripp, T. M. 2001, ApJS, 137, 297 Lunttila, T., Padoan, P., Juvela, M., & Nordlund, Å. 2009, ApJ, 702, L37 MacLow, M.-M., & Klessen, R. S. 2004, Rev. Mod. Phys., 76, 125 Mestel, L. 1966, MNRAS, 133, 265 Mouschovias, T. Ch., & Ciolek, G. E. 1999, in The Origin of Stars and Planetary Systems, ed. C. J. Lada & N. D. Kylafis (Dordrecht: Kluwer), 305 Nakano, T., & Nakamura, T. 1978, PASJ, 30, 681 Tilley, D. A., & Pudritz, R. E. 2007, MNRAS, 382, 73 Troland, T. H., & Crutcher, R. M. 2008, ApJ, 680, 457 Wolfire, M. G., McKee, C. F., Hollenbach, D., & Tielens, A. G. G. M. 2003, ApJ, 587, 278
