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Adaptive Calculation of a Collapsing Molecular Cloud Core 227 where denotes the gradient operator in terms of the zooming coordinates. The mass-density the fluid velocity and the gravitational potential transform according to the following relations respectively. Note that d, u and are non-dimensional quantities. in the transformed zooming space In the zooming coordinates, the radial length and time are proportional to for fixed and This allows the numerical resolution to change adaptively in accordance with the density increase since The equations of continuity (5) and momentum (6) are almost invariant under the transformation, except for the source terms on the right-hand side, whereas the Poisson equation (7) retains essentially the same form. Therefore, the spherical coordinate code described in Ref. [18] can be used for solving equations (5)-(7) with only minor changes. As in the previous version, the code employs a multi-step solution procedure to advance the fluid variables d and u through the use of spatially second-order accurate finite-difference methods. Advection of mass is performed using a generalization to spherical coordinates of the van Leer monotonic interpolation scheme. Temporal second-order accuracy is achieved by means of a predictor-corrector treatment of the convective and source hydrodynamical terms. The Poisson equation is solved by a spherical harmonic expansion, including terms up to This procedure allows for a separation of the radial and angular variables such that the three-dimensional equation can be replaced by a set of ordinary differential equations for the amplitudes which are efficiently solved by means of a tridiagonal matrix algorithm. The grid is uniformly spaced and kept fixed during the calculation. Setting a radial resolution consisting of points (including the origin ) is enough to guarantee satisfaction of the Jeans conditions (1) and (2) through 7 orders of magnitude increase in density. The grid has fairly equidistant points for with symmetry through the cloud’s equatorial plane being used to represent the bottom hemisphere of the cloud. The grid is also uniformly spaced with for corresponding to effective points due to the assumption of Thus, in a Fourier decomposition only even modes can grow. At the external surface of the
228 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY grid we impose an outflow boundary condition such that no disturbances can propagate into the domain of computation from the boundary surface. This boundary condition is naturally implemented because the gas flows out supersonically from the domain of computation. The supersonic outflow is due to the apparent motion with respect to the zooming coordinates. The present calculation provides a further intercomparison test at high resolution for the isothermal Gaussian cloud using a numerical framework that is completely different to those employed in Refs. [10, 17]. With the present zooming grid resolution, the calculation took about timesteps to complete the evolution, corresponding to about 30 hr of CPU time on an Origin 2000 machine. 3. RESULTS The Gaussian cloud model can increase in density by about 4 orders of magnitude before the isothermal approximation begins to fail. In real molecular cloud cores significant non-isothermal heating will take place once rendering the isothermal approximation invalid. However, for the purposes of this paper we ignore the effects
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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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Scalar field dark matter 173 growin
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Scalar field dark matter 175 This s
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Page 208 and 209: REFERENCES 183 The hypothesis of th
Page 210 and 211: INFLATION WITH A BLUE EIGENVALUE SP
Page 212 and 213: Inflation with a blue eigenvalue sp
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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
Page 240 and 241: The influence of scalar fields in p
Page 242 and 243: The influence of scalar fields in p
Page 244 and 245: EXACT SOLUTIONS AND SCALAR FIELDS I
Page 246 and 247: REFERENCES 221 [3] by R.C. Kennicut
Page 248 and 249: ADAPTIVE CALCULATION OF A COLLAPSIN
Page 250 and 251: Adaptive Calculation of a Collapsin
Page 258 and 259: REFERENCES 233 [5] [6] [7] [8] [9]
Page 260 and 261: REVISITING THE CALCULATION OF INFLA
Page 262 and 263: Revisiting the calculation of infla
Page 272 and 273: CONFORMAL SYMMETRY AND DEFLATIONARY
Page 274 and 275: Conformal symmetry and deflationary
Page 284 and 285: REFERENCES 259 [16] R. Maartens, D.
Page 286 and 287: IV EXPERIMENTS AND OTHER TOPICS
Page 288 and 289: STATICITY THEOREM FOR NON-ROTATING
Page 290 and 291: Staticity Theorem for Non-Rotating
Page 292 and 293: Staticity Theorem for Non-Rotating
Page 294 and 295: REFERENCES 269 The boundary is cons
Page 296 and 297: QUANTUM NONDEMOLITION MEASUREMENTS
Page 298 and 299: Quantum nondemolition measurements
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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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