Testing Molecular-Cloud Fragmentation Theories: Self-Consistent Analysis of OH Zeeman Observations
aa r X i v : . [ a s t r o - ph . GA ] S e p Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 30 October 2018 (MN L A TEX style file v2.2)
Testing Molecular-Cloud Fragmentation Theories: Self-ConsistentAnalysis of OH Zeeman Observations
Telemachos Ch. Mouschovias and Konstantinos Tassis Departments of Physics and Astronomy, University of Illinois at Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, 91109
ABSTRACT
The ambipolar-diffusion theory of star formation predicts the formation of fragments inmolecular clouds with mass-to-flux ratios greater than that of the parent-cloud envelope. Bycontrast, scenarios of turbulence-induced fragmentation do not yield such a robust prediction.Based on this property, Crutcher et al. (2009) proposed an observational test that could po-tentially discriminate between fragmentation theories. However, the analysis applied to thedata severely restricts the discriminative power of the test: the authors conclude that they canonly constrain what they refer to as the “idealized” ambipolar-diffusion theory that assumesinitially straight-parallel magnetic field lines in the parent cloud. We present an original, self-consistent analysis of the same data taking into account the nonuniformity of the magneticfield in the cloud envelopes, which is suggested by the data themselves, and we discuss impor-tant geometrical effects that must be accounted for in using this test. We show quantitativelythat the quality of current data does not allow for a strong conclusion about any fragmentationtheory. Given the discriminative potential of the test, we urge for more and better-quality data.
Key words: diffusion — ISM: clouds, magnetic fields — MHD — stars: formation — tur-bulence
The ratio of the mass and magnetic flux of interstellar molecularclouds has received well-deserved observational attention in re-cent years (e.g., Crutcher 1999; Heiles & Crutcher 2005). For acloud as a whole, the mass-to-flux ratio is an important input to theambipolar-diffusion theory of fragmentation (or core formation)in molecular clouds (e.g., see Fiedler & Mouschovias 1992, eq.[8]; 1993, eq. [1c] and associated discussion). What the ambipolar-diffusion theory predicts is the mass-to-flux ratio of fragments (or cores ) in molecular clouds and how this quantity evolves in timefrom typical densities ≃ cm − to densities ≃ cm − (Tassis & Mouschovias 2007). Observations have been in excel-lent quantitative agreement with the theoretical predictions in thatthe mass-to-flux ratio of cores is found to be supercritical by a fac-tor 1 - 4 (Crutcher et al. 1994; Crutcher 1999 and correction byShu et al. 1999, pp. 196 - 198; Ciolek & Basu 2000; Troland &Crutcher 2008; Falgarone et al. 2008). By contrast, simulationsof turbulence-driven fragmentation do not find cores with sys-tematically greater mass-to-flux ratios than those of their parentclouds (e.g., Lunttila et al. 2008). Therefore, the effort by Crutcheret al. (2009) (hereinafter CHT) to measure the variation of themass-to-flux ratio from the envelopes to the cores of four molec-ular clouds and thereby constrain cloud-fragmentation theories is amuch needed observational test.The effort by CHT to measure the magnetic field in four cloud envelopes yielded mostly nondetections, allowing only the place-ment of weak upper limits. Also, the data are suggestive of spatialvariations of the field in the cloud envelopes. This spatial varia-tion must be explicitly treated in the data analysis. Instead, CHTperformed an analysis based on the overly restrictive (and contra-dicted by the data) assumption of uniform magnetic field in the en-velope, which minimizes the potentially constraining power of theirobservations. CHT attempt to justify their restrictive assumptionby claiming that they are testing the “idealized ambipolar-diffusionmodel” that assumes initially straight-parallel field lines in the par-ent cloud. Thus, if the data and the data analysis in CHT are takenat face value, they at best test an input to a theory, not the predic-tion of the theory relating to the variation of the mass-to-flux ratiofrom a core to its envelope, given the field strength and its spatialvariation in the envelope. As we show below, the geometry of thefield lines in a parent cloud crucially affects the observed varia-tion of the mass-to-flux ratio from a core to the envelope while thefundamental prediction of the ambipolar-diffusion theory (that themass-to-flux ratio increases from the envelope to the core) remainsunchanged.In this letter, we present a novel analysis of the OH-Zeemandata, applicable also to other sets of data that show intrinsic varia-tion of the quantity being measured. c (cid:13) T. Ch. Mouschovias and K. Tassis
Table 1.
Magnetic Fields and Errors (in µ G) in Four Cloud Envelopes (datafrom CHT).Cloud B ± σ B ± σ B ± σ B ± σ L1448CO − ± − ± − ± ± B217-2 − ± ± ± ± L1544 − ± − ± ± ± B1 − ± ± − ± − ± The CHT data consist of existing OH Zeeman measurements infour molecular cloud cores and of four new measurements in the re-gion surrounding each of these cores (in the clouds L1448, B217-2,L1544, and B1). For each observation of an envelope’s line-of-sightmagnetic field B j , CHT quote an associated Gaussian uncertainty σ j . These four values for each cloud envelope are shown in Table1, col. 2 - 5 (taken from CHT Figs. 2 - 5). CHT assign a value to the magnetic field strength in each enve-lope, which is obtained from a simultaneous least-squares fit overthe 8 Stokes V spectra (2 spectral lines at each of 4 positions ineach envelope). The fit gives a single value of the line-of-sight fieldand a single value of its uncertainty in each envelope. The uncer-tainty was calculated under the assumption that there is no intrinsicspatial variation of the field strength in each cloud envelope and,therefore, any spread in the observed B j values is attributed to ob-servational errors. The CHT values for the envelope fields and theiruncertainties are shown in Table 2, column 2.Using this mean field, CHT calculate what they regard asthe magnetic flux of the envelope, which, combined with the fluxin the core, is used to obtain the quantity R defined by R =( I core ∆ V core /B core ) / ( I env ∆ V env /B env ) . The quantity I is thepeak intensity of the spectral line in degrees K, ∆ V is the FWHMin km s − , and B is the line-of-sight mean field in µ G. A value R = 1 would imply that the mass-to-flux ratio does not vary froman envelope to a core in the same cloud, while R > would im-ply a mass-to-flux ratio greater in the core than in the envelope.Since most of the CHT measurements of B in cloud envelopes arenondetections, the analysis relies sensitively on the treatment andpropagation of observational uncertainties to obtain limits on thederived quantity R .As mentioned above, CHT calculate a mean value of B env andan uncertainty on this mean under the explicit assumption that themagnetic field in the envelope can be described by a unique B env value, which their analysis seeks to constrain. However, the mag-netic field in the cloud envelope is not known a priori to have aunique uniform value. In fact, the data suggest the opposite (e.g.,observations 2 and 4 in L1448CO differ by more than 3 σ ; observa-tions 1 and 3 in B217-2 differ by more than σ ; observations 1 and3 in L1544 differ by more than 4 σ ; observations 1 and 2 in B1 differby more than σ ; see Table1). CHT justify this choice by restrict-ing their comparison to what they call the “idealized” ambipolar-diffusion theory, assuming that the field lines in the molecular cloudenvelope are straight and parallel.If, as the CHT data suggest, the assumption of zero-spread B env is relaxed, the uncertainties CHT calculate are not the relevant ones. A simple example will illustrate the point: Consider a cloudenvelope in which the magnetic field has a distribution of valueswith mean 10 µ G and spread 5 µ G. An observer makes only twomeasurements of the envelope field, each with uncertainty 0.1 µ G.The first measurement gives ± . µ G, and the second measure-ment gives ± . µ G (both very likely). Under the CHT assump-tion of zero spread, the mean and associated uncertainty are simplythe average, B mean = 12 µ G, and the propagated observationalerror, σ mean = ( P j =1 σ j ) / / . µ G. Clearly, however,this B mean differs from its true value by µ G, not by . µ G. Inother words, if there is significant spatial variation of B in a cloudenvelope, the CHT-kind of analysis grossly underestimates the un-certainty on the mean. A simple inspection of the CHT raw data, taken at face value, re-veals that these four clouds do not have straight-parallel field linesin their envelopes. But are such clouds expected on the basis oftheoretical considerations? Straight-parallel field lines in a parentcloud is an idealization in some theoretical calculations that ren-ders a mathematically complicated multifluid, nonideal MHD sys-tem tractable while capturing all the essential physics of the coreformation and evolution problem. However, it has never been sug-gested that in a real cloud, which is an integral part of a dynamicISM, the envelope field lines will be straight and parallel. Distor-tions superimposed on the characteristic hour-glass morphology as-sociated with the compression of the field lines during gravitationalcore formation are routinely expected.Mouschovias & Morton (1985, Fig. 13) had sketched whatthey regarded as a more realistic field geometry in a molecularcloud in which there are several (in that case four) magneticallyconnected fragments. That figure is reproduced here as Figure 1a.This configuration can result from relative motion of the fragments(labeled A, B, C, and D) within the cloud, due to the cloud’s meangravitational field. The motion of a cloud as a whole relative tothe intercloud medium will also bend the magnetic field lines inan almost U-shape, as shown in Figure 1b. One can easily visualizelines of sight in Figures 1a and 1b (e.g., the line CC ′ ) along which ameasurement would yield B los , env > B los , core , although the actualfield strength in the core is greater than that in the envelope as evi-denced by the compressed field lines in the core. By contrast, alongAA ′ , almost the full strength of the core’s magnetic field will bemeasured, but only a fraction of the envelope’s field strength will bedetected. Altogether: (1) An idealization in a theoretical calculationshould not be mistaken for a prediction. (2) Observations that maypotentially reveal the geometry of the field lines can and should beused as input to build a particular model for the observed cloud (asdone in the case of B1 by Crutcher et al. 1994, and for L1544 byCiolek & Basu 2000). (3) The geometry of the field lines cannotbe ignored in analyzing data from observations that measure onlyone component of the magnetic field (e.g., Zeeman observations) ifthe purpose is to test a theory or discriminate between alternativetheories. The new analysis of the CHT data in § Unlike the CHT analysis, if the data show field reversals, the posi-tive and negative values of the measured B los , env must not be alge-braically averaged (which is what the CHT assumption of a single c (cid:13) , 000–000 elf-Consistent Analysis of OH Zeeman Observations L3 Figure 1.
Schematic diagrams: ( a, top ) A deformed flux tube that has frag-mented along its length in a molecular cloud (from Mouschovias & Morton1985). The deformation can be caused by the relative motion of the frag-ments. ( b, bottom ) Deformation of the field lines threading a cloud causedby its motion relative to the surrounding medium. The cloud is shown, forsimplicity, to contain only one fragment (or core), in the neighborhood ofwhich the hourglass shape of the field lines had been established duringcore formation but affected by the cloud’s motion.The dashed lines in bothfigures represent different lines of sight, whose significance is explained inthe text. magnetic field value in the envelope imposes on the data) and thenmultiplied by the plane of the sky area of the envelope in orderto obtain its magnetic flux. If the three (perhaps all four) of theobserved cloud envelopes exhibited true reversals in the field direc-tion (but see § only one algebraic sign ofthe magnetic field (the one corresponding to the greatest absolutevalues) should be considered in estimating the magnetic flux of theenvelope. Figure 2 and its caption clarify this point. Since both the data and theoretical considerations suggest that B env exhibits spatial variations, we reanalyze the CHT data prop-erly accounting for this effect and thus generalize the relevance ofthe data to realistic clouds (instead of idealized ones with straight-parallel field lines). High-quality data analyzed in this manner canpotentially discriminate between alternative fragmentation theo-ries, instead of just providing geometrical input to theories. NS LOS
Figure 2.
Schematic diagram of a star (e.g., Sun) that has a dipolar mag-netic field and is observed (for simplicity) along lines of sight parallel toits equatorial plane. To calculate the magnetic flux threading the star by ob-serving its surface field B surf , only the mean value of B surf in either thenorthern or in the southern hemisphere should be used. If both values areaveraged algebraically (as done by CHT), an erroneous flux value of zerowill be obtained. When intrinsic variation of B los in a cloud’s envelope exists,the spread in the observed values is the convolution of the measure-ment error and of the intrinsic spread of B los . To account for spatialvariation of B los , a likelihood analysis is needed (see Wall & Jenk-ins 2003; Lyons 1992; Lee 2004). We assume that the “true” B los follows a Gaussian distribution with mean B and intrinsic spread σ . This distribution is then “sampled” with N measurements B j ,each carrying a (Gaussian) error measurement σ j .At any specific envelope location, there is a probability exp ˆ − ( B − B ) / σ ˜ / √ πσ for the magnetic field to havea true value B . If the error of measurement at this same location is σ j , then the probability of observing a value B j of the field, given that its true value is B , is exp ˆ − ( B − B j ) / σ j ˜ / √ πσ j . How-ever, this is not the only way we could get an observed field value B j , since there are many different true values of the field that mightyield an observation B j due to measurement errors. To find the to-tal probability for a single observation of B j , we integrate over allpossible “true” values of the magnetic field at a single location toget (the likelihood for a single observation B j with observationaluncertainty σ j ): l j = Z ∞−∞ dB exp ˆ − ( B − B j ) / (2 σ j ) ˜ √ πσ j exp ˆ − ( B − B ) / (2 σ ) ˜ √ πσ . (1)The likelihood L for N observations of B j with individualuncertainties σ j to come from an intrinsic probability distributionwith mean B and spread σ is the product of the individual like-lihoods, L = Q Nj =1 l j which, after performing the integration inequation (1) and some algebraic manipulations, yields (see Venters& Pavlidou 2007) L ( B , σ ) = N Y j =1 q σ + σ j exp " − N X j =1 ( B j − B ) σ + σ j . (2)Any parameters that are not of direct interest (such as the in-trinsic spread σ in this case), can be integrated out of the likeli-hood. In this way, we can derive the probability distribution of theparameter of interest ( B ) while still allowing for all possible val-ues in σ , rather than arbitrarily demanding that σ = 0 (as in theCHT analysis). The integrated likelihood is called the marginalizedlikelihood , L m ; this probability distribution can then be used to de-rive confidence intervals and upper limits where appropriate. The(unnormalized) L m for the four clouds is shown in Figures 3a - 3d. c (cid:13) , 000–000 T. Ch. Mouschovias and K. Tassis
Table 2.
Magnetic Fields and Their Uncertainties (in µ G ), and Upper Limits on the Envelope Field (in µ G ) and Ratio R Cloud B mean ± σ mean B max L ± σ L | B env | ( σ ) | R | ( σ ) L1448CO ± − +9 −
27 2 . B217-2 +2 ± +7 −
22 2 . L1544 +2 ± +10 −
29 5 . B1 − ± − +5 −
20 1 . Col. 1 : The four observed clouds.
Col. 2 & : Mean field and its uncertainty as given by CHT and by the likelihood data analysis, respectively. Col. 4 & :Upper limits on the envelope magnetic field and on the ratio R , from the likelihood analysis. L m is derived by numerically integrating equation (2) over σ fordifferent values of B , and is shown as a solid curve in the four fig-ures; the location of the maximum-likelihood estimate for the mean B is marked by a heavy vertical line in each figure.The maximum-likelihood estimates and associated uncertain-ties of B for the four CHT clouds are shown in Table 2. The uncer-tainties are systematically greater than those quoted by CHT. The σ uncertainties are represented by the widths of the dark shaded(solid blue) boxes in Figures 3a - 3d. For comparison we show, ascross-hatched boxes, the σ spreads of the values of B that CHTquote for the same clouds, based on the same data.Upper limits for | B | can also be calculated using L m . The σ upper limits for the envelope magnetic field are given in Table2. (The σ upper limit is that value of | B | for which a fractionalarea equivalent to the Gaussian σ ( . ) is included under themarginalized likelihood curve between −| B | and | B | . Note thatthis is not the maximum-likelihood value of B plus two times theerror. In addition, L m has much longer tails than a gaussian, andhence σ , σ, ... values do not scale linearly.) The values of | B | and −| B | which include between them a fractional area of σ ofthe marginalized likelihood for the four clouds are marked withheavy down arrows in Figures 3a - 3d.To correctly propagate uncertainties onto the derived quantity R , we do a full Monte-Carlo calculation to derive the probabilitydistribution for the values of R as follows. We repeat the follow-ing experiment times: we draw I core , I env , ∆ V core , ∆ V env and B core from gaussian distributions with mean and spread equal tothe measurement and uncertainty quoted in CHT; we draw a meanvalue of B env from the marginalized likelihood of the previous sec-tion; we combine all the “mock observations” of these numbers toproduce one value of R . We use the values of R produced inthis way to numerically calculate the probability distribution for R . We then calculate the σ upper limit on | R | by requiring thatthe fractional integral of this distribution between − R and + R be . . The σ upper limits for | R | are given in Table 2, lastcolumn. These limits are not very strong: | R | is constrained to besmaller than a few, and for no cloud is the upper limit smaller than − in sharp contrast to the CHT conclusion, that R is in the range0.02 - 0.42 in the four observed clouds.In our analysis, we relaxed only one of the CHT assumptions(that of lack of spatial variation of B env , which is not consistentwith the data). We have retained the implicit assumption of similarorientations of the net B env and B core ( vectors ), because the datado not suggest any particular relative orientation of the two vec-tors. A more general analysis that would also relax this assumption -30 -20 -10 0 10 20 3000.0050.010.0150.02 -30 -20 -10 0 10 20 3000.010.020.030.04-30 -20 -10 0 10 20 30 B( µ G) M a r g i n a li ze d L i k e li hood ( a r b it r a r y un it s ) -30 -20 -10 0 10 20 3000.020.040.060.080.1 B217-2L1448COL1544 B1 Figure 3.
Marginalized likelihood (not normalized, solid curve) for thefour cloud envelopes observed by CHT. In each figure: the location of themaximum-likelihood value of B is marked with a heavy vertical line; thewidth of the dark shaded (solid blue) box indicates the σ values of B ; thewidth of the cross-hatched box indicates the σ values for B according toCHT; heavy down arrows mark the σ upper limit on | B | . would increase the uncertainties on R (although not on B env ) andwould further part from the CHT conclusions. We have presented a self-consistent analysis of recent OH Zeemanobservations by Crutcher et al. (2009), and have shown how suchdata can be combined in a statistically robust manner to obtain con-straints on the mass-to-flux ratio contrast ( R ) between molecularcloud cores and envelopes. Our analysis extends the constrainingpower of such measurements to realistic clouds, beyond the overlyrestrictive assumption of straight-parallel field lines in cloud en-velopes, adopted by CHT. We have shown that the CHT data arenot of good enough quality to constrain the ratio R and thereby test c (cid:13) , 000–000 elf-Consistent Analysis of OH Zeeman Observations L5 molecular-cloud fragmentation theories: (i) more integration timeis needed to reduce measurement errors (now most measurementsof B env yield only upper limits); and (ii) more measurements (in-stead of only four) in each envelope are needed to better constrainthe intrinsic spatial variation of B env . This kind of observationscoupled with the method of data analysis we presented in this Let-ter has great promise and can lead to significant progress in thefield of ISM physics in general and in understanding the role ofmagnetic fields in molecular-cloud dynamics in particular.
ACKNOWLEDGEMENTS
We thank Robert Dickman, Paul Goldsmith, Mark Heyer, Dan Mar-rone, and Vasiliki Pavlidou for valuable discussions. TM acknowl-edges partial support by NSF under grant AST-07-09206, and KTby JPL/Caltech, under a contract with the National Aeronautics andSpace Administration. c (cid:13)
REFERENCES
Ciolek, G. E., Basu, S., 2000, ApJ, 529, 925Crutcher, R. M., 1999, ApJ, 520, 706Crutcher, R. M., Hakobian, N., & Troland, T. H. 2009, ApJ, 692, 844Crutcher, R. M., Mouschovias, T. Ch., Troland, T. H., Ciolek, G. E., 1994,ApJ, 427, 839Falgarone, E., Troland, T. H., Crutcher, R. M., Paubert, G., 2008, A&A,487, 247Fiedler, R. A., Mouschovias, T. Ch., 1992, ApJ, 391, 199Fiedler, R. A., Mouschovias, T. Ch., 1993, ApJ, 415, 680Heiles, C., Crutcher, R. M., 2005, in Cosmic Magnetic Fields, eds. R.Wielebinski & R. Beck (Berlin: Springer), 137Lee, P. M., 2004, Bayesian Statistics, Oxford University Press, New YorkLunttila, T., Padoan, P., Juvela, M., Nordlund, A. 2008, ApJ, 686, L91Lyons, L., 1992, Statistics for Nuclear and Particle Physicists, CambridgeUniversity Press, New YorkMouschovias, T. Ch., Morton, S. A., 1985, ApJ, 298, 205Shu, F. H., Allen, A., Shang, H., Ostriker, E. C., Li, Z.-Y., 1999, in TheOrigin of Stars and Planetary Systems, eds. C. J. Lada & N. D. Kylafis(Dordrecht: Kluwer), 193Tassis, K., Mouschovias, T. Ch., 2007, ApJ, 660, 388Troland, T. H., Crutcher, R. M., 2008, ApJ, 680, 457Venters, T. M., Pavlidou, V., 2007, ApJ, 666, 128.Wall, J. V, Jenkins, C. R., 2003, Practical Statistics for Astronomers, Cam-bridge University Press, Cambridge, UKc (cid:13)000