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Correlation Analysis of the Taurus Molecular Cloud Complex

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Correlation Analysis of the Taurus Molecular Cloud Complex University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations 1896 - February 2014 1-1-1985 Correlation analysis of the Taurus molecular cloud complex. Steven C. Kleiner University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_1 Recommended Citation Kleiner, Steven C., "Correlation analysis of the Taurus molecular cloud complex." (1985). Doctoral Dissertations 1896 - February 2014. 1860. https://scholarworks.umass.edu/dissertations_1/1860 This Open Access Dissertation is brought to you for free and open access by ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations 1896 - February 2014 by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. CORRELATION ANALYSIS OF THE TAURUS MOLECULAR CLOUD COI'IPLEX A Dissertation Presented By STEVEN CARL KLEINER Submitted to the Graduate School of the University of Massachusetts in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 1985 Department of Physics and Astronomy CORRELATION ANALYSIS OF THE TAURUS MOLECULAR CLOUD COMPLEX A Dissertation Presented By STEVEN CARL KLEINER Approved as to style and content by: Robert L. Dickman, Chairman Paul F. Goldsmith, /leraber Robert V. Krotkov, Member J^n Y. Kwan, Member Department of Physics and Astronomy 11 Steven Carl Kleiner 1985 © All Rights Reserved ill This work is dedicated to my parents. iv "...we were able to discover in the general wave record a very weak low frequency peak which would surely have escaped our attention without spectral analysis. This peak, it turns out, is almost cer- tainly due to a swell from the Indian Ocean, 10,000 miles distant. Physical dimensions are: 1 mm high, a kilometer long." W. E. Munk, quoted in Blackman and Tukey (1958) "Demere Rebus Tumultum Seneca, Epistles V I am indebted to my advisor, Robert L. Dickman, not only for his constant scientific and editorial guidance, but for his unswerving encouragement and support during my scientific appren- ticeship. I am very grateful to Paul Goldsmith for his support throughout my stay at the Five College Radio Astronomy Observatory and for his insightful criticisms of this and other manuscripts. I wish to acknowledge the generosity of Ron Snell and Peter Schloerb in pro- viding their Heiles 2 data for autocorrelation analysis. I thank John Kwan, Bob Krotkov, Nick Scoville, Tom Arny and John Scalo for many useful comments. The Five College Radio Astronomy Observatory is operated with support from the National Science Foundation under grant AST 82-12252 and with permission of the Metropolitan District Commission of the Commonwealth of Massachusetts. vi ABSTRACT Correlation Analysis of The Taurus Molecular Cloud Complex (May 1985) Steven Carl Kleiner, B.S., The Cooper Union, New York M.S., Ph.D., University of Massachusetts, Amherst Directed by: Professor Robert L. Dickman Autocorrelation and power spectrum methods have been applied to the analysis of the density and velocity structure of the Taurus Complex and Heiles Cloud 2 as traced out by ^-^CO J = 1 -» 0 molecular line observations obtained with the 14m antenna of the Five College Radio Astronomy Observatory. Statistically significant correlations in the spacing of den- sity fluctuations within the Taurus Complex and Heiles 2 were unco- vered. The length scales of the observed correlations correspond in magnitude to the Jeans wavelengths characterizing gravitational insta- bilities within i) interstellar atomic hydrogen gas for the case of the Taurus complex, and ii) molecular hydrogen for Heiles 2. The ob- served correlations may be the signatures of past and current gravita- tional instabilities frozen into the structure of the molecular gas. A velocity gradient of 0.25 km s~^ is detected across the core of the Complex, which as a whole appears to be in virial equili- brium. Autocorrelation analysis of the remaining, non-systematic, gas vii motions sets an upper limit of 0.6 pc to the correlation length of turbulence within the Taurus Complex. This limit is shown to be incon- sistent with suggestions that a lossless cascade (i.e. a Kolmogorov spectrum) of turbulent motions pervades the dense interstellar medium. The correlation length of turbulence within Heiles 2 is found to be 0.1 pc. This result represents the first measurement of the length scale of turbulence within a molecular cloud. The line-of-sight rms turbulent velocity is determined to be Vj- = 0.72 km s~^. The maxi- mum turbulent energy dissipation rate is calculated to be ^^^x ^ ^'5 X 10~2A ergs cm~-^ s~^. It is shown that the maximum dissipation of energy should occur on length scales close to the correlation length. The appendicies provide a comprehensive description of the analytical and numerical methods developed for the correlation analy- sis of molecular clouds. iix TABLE OF CONTENTS Acknowledgements Chapter I. INTRODUCTION 1 A. Density Fluctuations 5 B. Turbulence 7 C. Stochastic Processes 9 D. Stochastic Analysis 10 II. LARGE SCALE DENSITY STRUCTURE 16 A. Colunm Density Autocorrelation Functions 16 B . Observations 19 C. Results of Correlation Analysis 24 1. Estimation of the ACF 24 2. Results of Estimation 27 D. Discussion 42 1. Corrections to Observed Length Scale 42 2. Interpretation of Length Scale 44 III. LARGE SCALE VELOCITY STRUCTURE 50 A. Characterization of Turbulence 52 1. The Autocorrelation Function 52 2. The Centroid Velocity ACF 54 3. Statistical Measures 63 B. Observations and Analysis 66 1. Statistical Measures 66 2. Autocorrelation Analysis 69 3. Structure at Large Lags 73 4. Structure at Small Lags 78 C. Discussion 86 1. Core Velocity Gradient 86 2. Virial Equilibrium 88 3. Small Scale Correlation Structure 90 ix Chapter IV. SMALL SCALE STRUCTURE (HEILES CLOUD 2) 96 A. Introduction 96 B. Observations 97 C. Density Structure 93 D. Statistical Measures 104 E. Heiles 2 Velocity ACF 107 F. Turbulence within Heiles 2 HI G. Velocity vs_. Scale Size 131 H. Magnitude of the Turbulent Velocity 141 I. Turbulent Energy Dissipation 151 1. Energy Loss Rates 152 2. Sources of Turbulent Energy 161 V. SUMMARY 167 A. Structure of Taurus Complex 168 1. Density Structure 168 2. Velocity Structure 170 B. Structure of Heiles 2 173 1. Density Structure 173 2. Velocity Structure 174 C. Correlation Techniques 179 D. What is to be Done? 180 REFERENCES 185 Appendix A. ANALYSIS OF VELOCITY FLUCTUATIONS 195 1. Magnitude of Velocity Fluctuations 195 2. ACF of Centroid Velocity Fluctuations 203 B. EFFECT OF INSTRUMENTAL NOISE 206 1. Degradation of the Observed ACF 206 2. Estimation of 208 3. Effect of Noise upon Power Spectrum 216 X Appendix C. NUMERICAL TECHNIQUES 219 1. Autocorrelation Calculations 219 2. Smoothing 226 3. Filtering 229 4. Suggested References 234 D. STATIONARITY AND THE STRUCTURE FUNCTION 236 1. Stationary Stochastic Processes 236 2. The Structure Function 240 3. Choice of the ACF 242 xi LIST OF FIGURES 2.1 Map of integrated ^3^0 J=1>0 radiation temperature observed from the Taurus molecular cloud complex 22 2.2 Biased autocorrelation function (ACF) of large-scale column density fluctuations within the Taurus clouds 29 2.3 Intermediate ACF estimator of column density fluc- tuations 31 2.4 Unbiased large-scale column density ACF 33 2.5 Biased column-density ACF after application of a digi- tal differencing filter to the data 36 2.6 Intermediate correlation estimator of differenced column density data 38 2.7 Biased ACF of column density variations within the Taurus clouds as derived from cloud parameters listed in the Lynds Catalog of Dark Nebulae 41 3.1 Comparision of model velocity autocorrelation functions describing gas motions within molecular clouds and the resulting ACFs of centroid velocity fluctuations 3.1a 60 3.1b 61 3.1c 62 3.2 Biased autocorelation function of ^-^CO J=l->'0 molecular line centroid-velocity fluctuations observed within the Taurus dark clouds 72 3.3 Map of ^^co J=l->-0 radiation temperature integrated over each of three velocity intervals, showing the syste- matic progression of gas velocities along the core of the complex 77 ACFs plotted as a function of tj , 3.4 Biased Taurus velocity | 3.4a uncorrected ACF 81 3.4b corrected for instrumental noise 82 3.4c reference ACF of random velocity field 83 xii 4.1 Map of integrated ^^CO J=l-»-0 radiation temperature from Heiles Cloud 2 portion of the Taurus complex 100 4.2 Unbiased autocorrelation estimator of column density fluctuations within Heiles 2 103 4.3 Biased ACF of centroid velocity fluctuations within Heiles 2 -^QS 4.4 Contour map of model velocity field containing a rotating ring 4.5 Biased ACF of model velocity field of Fig. 4.4 115 4.6 Biased ACF of centroid velocity fluctuations observed from Heiles 2 plotted as a function x of | only, uncorrected ( for instrumental noise and foreshortening 118 4.7 Biased Heiles 2 velocity ACF after correction for noise and foreshortening 122 4.8 Biased Heiles 2 velocity ACF after application of a simple digital differencing filter, plotted as a function of] T I 125 4.9 Biased centroid-velocity ACF after application of a digital Parzen high pass filter to the data, plotted as a function of T only 130 I j 4.10 Variation of o2(L), the rms magnitude of turbulent motions as a function of scale size, within the Heiles 2 cloud 4.10a a2(L) versus L 136 4.10b log[a2(L)] versus log(L) 137 4.11 Variation of e(L), the turbulent energy transfer rate as a function of spatial scale size, within Heiles 2 159 iixv CHAPTER I INTRODUCTION The formation, structure and evolution of molecular clouds are topics of fundamental interest to modern astronomy. It is now recognized, for example, that molecular clouds are a major component of the galactic disk, that they dominate the structure of the interstellar medium within the solar circle, and that they are the principle sites of star formation in the Milky Way (cf_.
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