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ADAPTIVE CALCULATION OF A COLLAPSING MOLECULAR CLOUD CORE: THE JEANS CONDITION Leonardo Di G. Sigalotti Instituto Venezolano de Investigaciones Científicas (IVIC) Carretera Panamericana Km. 11, Altos de Pipe, Estado Miranda, Venezuela Jaime Klapp Instituto Nacional de Investigaciones Nucleares Ocoyoacac 52045, Estado de México, México Abstract In 1997 Truelove et al. introduced the Jeans condition to determine what level of spatial resolution is needed to avoid artificial fragmentation during protostellar collapse calculations. They first found using a Cartesian code based on an adaptive mesh refinement (AMR) technique that a Gaussian cloud model collapsed isothermally to form a singular filament rather than a binary or quadruple protostellar system as predicted by previous calculations. Recently Boss et al. in 2000 using a different hydrodynamics code with high spatial resolution reproduced the filamentary collapse solution of Truelove et al., implying that high resolution coupled with the Jeans condition is necessary to perform reliable calculations of the isothermal protostellar collapse. Here we recalculate the isothermal Gaussian cloud model of Truelove et al. and Boss et al. using a completely different code based on zooming coordinates to achieve the required high spatial resolution. We follow the collapse through 7 orders of magnitude increase in density and reproduce the filamentary solution. With the zooming coordinates, we are allowed to perform an adaptive calculation with a much lower computational cost than the AMR technique and other grid redefinition methods. Keywords: Molecular collapse, Newtonian codes, Jeans condition. 1. INTRODUCTION Observations of young stars have revealed a frequency of binary companions that is at least comparable to that found for pre–main–sequence [1] and main–sequence [2] stars. In addition, the detection of multiplicity Exact Solutions and Scalar Fields in Gravity: Recent Developments Edited by Macias et al., Kluwer Academic/Plenum Publishers, New York, 2001 223
224 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY among protostellar objects [3, 4, 5] suggests that most binary stars may have formed during the earliest phases of stellar evolution, probably as a consequence of protostellar collapse. In the last decade, fragmentation during the dynamic collapse of molecular cloud cores has emerged as the likely mechanism for explaining the detected binary frequency [6, 7, 8, 9]. However, studying the non–linear process of fragmentation unavoidably requires the use of three-dimensional hydrodynamical collapse calculations, which must rely upon convergence testing and intercode comparisons to validate their results. The reliability of isothermal collapse calculations has recently been questioned by the results of Ref. [10] where it was performed calculations of a particular Gaussian cloud model employing an adaptive mesh refinement (AMR) code based on Cartesian coordinates. They found that the cloud did not fragment into a binary or quadruple system as in previous calculations [11, 12, 13, 14] but, rather, collapsed into a singular filament consistent with the self-similar solution derived in Ref. [15]. In Ref. [10] it was also found that perturbations arising from the finite-difference discretization of the gravitohydrodynamic equations may induce "artificial" fragmentation in isothermal collapse calculations for which the Cartesian cell size exceeds one-fourth of the local Jeans length In other words, artificial fragmentation is avoided if the mass within a cell never exceeds 1/64 of the Jeans mass While these findings specialize to locally uniform Cartesian grids, in Ref. [16] it was found that for a non-uniform spherical coordinate grid the Jeans mass constraint translates into requiring that any of the four lengths do not exceed More recent intercomparison calculations [17] have shown that satisfying the Jeans condition at the location of the maximum density combined with sufficiently high spatial resolution is necessary to change the nature of the solution from a binary to a singular filament and achieve agreement with the AMR code results of Ref. [10]. High resolution is therefore necessary to produce reliable calculations of the collapse and fragmentation of molecular cloud cores. Here we present a further intercode comparison for the same Gaussian cloud model of Refs. [10] and [17] Boss. The calculation was made using a modified version of the spherical-coordinate based code described in [18], which adopts zooming coordinates to achieve the required high spatial resolution. These coordinates were previously used in Ref. [19] to study disk and bar formation during the isothermal collapse phase of a molecular cloud core up to extremely high density contrasts. With the zooming coordinates, the computational cells shrink adaptively in accordance with the cloud collapse and the spatial resolution increases
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EXACT SOLUTIONS AND SCALAR FIELDS I
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EXACT SOLUTIONS AND SCALAR FIELDS I
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To the 65th birthday of Heinz Dehne
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CONTENTS Preface Contributing Autho
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Contents ix 5 The case of 84 Part I
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Contents xi Luis O. Pimentel, Cesar
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Contents xiii 2.2 String theory ind
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PREFACE This book is dedicated to h
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CONTRIBUTING AUTHORS Eloy Ayón-Bea
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Contributing Authors xix Jaime Klap
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Contributing Authors xxi A.P. 70-54
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Contributing Authors xxiii L. Artur
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I EXACT SOLUTIONS
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SELF-GRAVITATING STATIONARY AXI- SY
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Self-gravitating stationary axisymm
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REFERENCES 13 be shown that the ana
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NEW DIRECTIONS FOR THE NEW MILLENNI
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New Directions for the New Millenni
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New Directions for the New Millenni
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REFERENCES 21 Acknowledgments The r
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ON MAXIMALLY SYMMETRIC AND TOTALLY
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On maximally symmetric and totally
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DISCUSSION OF THE THETA FORMULA FOR
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Theta formula for the Ernst potenti
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REFERENCES 51 Rosenhain’s theta f
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THE SUPERPOSITION OF NULL DUST BEAM
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The superposition of null dustbeams
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REFERENCES 61 geodesic with respect
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SOLVING EQUILIBRIUM PROBLEM FOR THE
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Solving equilibrium problem for the
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REFERENCES 67 where the subindices
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ROTATING EQUILIBRIUM CONFIGURATIONS
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Rotating equilibrium configurations
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Rotating equilibrium configurations
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REFERENCES 75 5. DISCUSSION The ana
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INTEGRABILITY OF SDYM EQUATIONS FOR
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Integrability of SDYM Equations for
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REFERENCES 87 [18] [19] [20] [21] [
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II ALTERNATIVE THEORIES AND SCALAR
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THE FRW UNIVERSES WITH BAROTROPIC F
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The FRW Universes with Barotropic F
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REFERENCES 99 (1979) 515; H. Dehnen
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HIGGS-FIELD AND GRAVITY Heinz Dehne
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Higgs-Field and Gravity 103 is defi
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Higgs-Field and Gravity 105 Consequ
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Higgs-Field and Gravity 107 From th
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REFERENCES 109 Evidently by inserti
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THE ROAD TO GRAVITATIONAL S-DUALITY
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The road to Gravitational S-duality
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REFERENCES 121 The partition functi
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EXACT SOLUTIONS IN MULTIDIMENSIONAL
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Exact solutions in multidimensional
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REFERENCES 131 For possible physica
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EFFECTIVE FOUR-DIMENSIONAL DILATON
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Dilaton gravity from Chern-Simons g
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A PLANE-FRONTED WAVE SOLUTION IN ME
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A plane-fronted wave solution in me
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REFERENCES 6. SUMMARY 151 We invest
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III COSMOLOGY AND INFLATION
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NEW SOLUTIONS OF BIANCHI MODELS IN
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New solutions of Bianchi models in
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REFERENCES 163 Further solutions of
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SCALAR FIELD DARK MATTER Tonatiuh M
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Scalar field dark matter 167 tional
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Scalar field dark matter 169 being
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Scalar field dark matter 171 of dar
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Page 220 and 221: CLASSICAL AND QUANTUM COSMOLOGY WIT
Page 222 and 223: Quantum cosmology with self-interac
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Page 230 and 231: THE BIG BANG IN GOWDY COSMOLOGICAL
Page 232 and 233: The Big Bang in Gowdy Cosmological
Page 238 and 239: THE INFLUENCE OF SCALAR FIELDS IN P
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Page 244 and 245: EXACT SOLUTIONS AND SCALAR FIELDS I
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Page 260 and 261: REVISITING THE CALCULATION OF INFLA
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Page 272 and 273: CONFORMAL SYMMETRY AND DEFLATIONARY
Page 274 and 275: Conformal symmetry and deflationary
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Page 286 and 287: IV EXPERIMENTS AND OTHER TOPICS
Page 288 and 289: STATICITY THEOREM FOR NON-ROTATING
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Quantum nondemolition measurements
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REFERENCES 279 [11] M. Kasevich and
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ON ELECTROMAGNETIC THIRRING PROBLEM
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On Electromagnetic Thirring Problem
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REFERENCES 293 [8] [9] D. Brill and
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ON THE EXPERIMENTAL FOUNDATION OF M
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On the experimental foundation of M
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REFERENCES 309 [12] [13] [14] [15]
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LORENTZ FORCE FREE CHARGED FLUIDS I
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Lorentz force free charged fluids i
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REFERENCES 319 force can be foresee
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Index Axisymmetries and configurati
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INDEX 323 Theorem (cont.) staticity
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Prof. Heinz Dehnen: Brief Biography
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Prof. Dietrich Kramer: Brief Biogra
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