Polytropes Astrophysics and Space Science Library

POLYTROPES ASTROPHYSICS AND SPACE SCIENCE LIBRARY

VOLUME 306

EDITORIAL BOARD Chairman W.B. BURTON, National Radio Astronomy Observatory, Charlottesville, Virginia, U.S.A. ([email protected]); University of Leiden, The Netherlands ([email protected])

Executive Committee J. M. E. KUIJPERS, Faculty of Science, Nijmegen, The Netherlands E. P. J. VAN DEN HEUVEL, Astronomical Institute, University of Amsterdam, The Netherlands H. VAN DER LAAN, Astronomical Institute, University of Utrecht, The Netherlands

MEMBERS I. APPENZELLER, Landessternwarte Heidelberg-Königstuhl, Germany J. N. BAHCALL, The Institute for Advanced Study, Princeton, U.S.A. F. BERTOLA, Universitá di Padova, Italy J. P. CASSINELLI, University of Wisconsin, Madison, U.S.A. C. J. CESARSKY, Centre d'Etudes de Saclay, Gif-sur-Yvette Cedex, France O. ENGVOLD, Institute of Theoretical Astrophysics, University of Oslo, Norway R. McCRAY, University of Colorado, JILA, Boulder, U.S.A. P. G. MURDIN, Institute of Astronomy, Cambridge, U.K. F. PACINI, Istituto Astronomia Arcetri, Firenze, Italy V. RADHAKRISHNAN, Raman Research Institute, Bangalore, India K. SATO, School of Science, The University of Tokyo, Japan F. H. SHU, University of California, Berkeley, U.S.A. B. V. SOMOV, Astronomical Institute, Moscow State University, Russia R. A. SUNYAEV, Space Research Institute, Moscow, Russia Y. TANAKA, Institute of Space & Astronautical Science, Kanagawa, Japan S. TREMAINE, CITA, Princeton University, U.S.A. N. O. WEISS, University of Cambridge, U.K. POLYTROPES

Applications in Astrophysics and Related Fields

G.P. HOREDT Deutsches Zentrum für Luft- und Raumfahrt DLR, Wessling, Germany

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: 1-4020-2351-0 Print ISBN: 1-4020-2350-2

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Visit Springer's eBookstore at: http://www.ebooks.kluweronline.com and the Springer Global Website Online at: http://www.springeronline.com TABLE OF CONTENTS

1 Polytropic and Adiabatic Processes 1 1.1BasicConcepts...... 1 1.2 Polytropic and Adiabatic Processes in a Perfect Gas ...... 2 1.3 Polytropic Processes for a General Equation of State ...... 7 1.4 Adiabatic Processes in a Mixture of Black Body Radiation and Perfect Gas ...... 10 1.5 Adiabatic Processes in a Mixture of Electron-Positron Pairs and Black Body Radiation . 11 1.6 Adiabatic Processes in a Completely Degenerate Electron or Neutron Gas ...... 13 1.7 Numerical Survey of Equations of State, Adiabatic Exponents, and Polytropic Indices . . 15 1.8Emden’sTheorem...... 24

2 Undistorted Polytropes 27 2.1GeneralDiﬀerentialEquations...... 27 2.2 The Homology Theorem and Transformations of the Lane-Emden Equation ...... 33 2.2.1TheHomologyTheorem...... 33 2.2.2 Milne Homology Invariant Variables ...... 34 2.2.3EmdenVariables...... 35 2.2.4 Kelvin Variables ...... 37 2.3ExactAnalyticalSolutionsoftheLane-EmdenEquation...... 38 2.3.1 Ritter’s First Integral n =0...... 38 2.3.2 Ritter’s Second Integral n =1...... 38 2.3.3 Schuster-Emden Integral n =(N +2)/(N − 2)...... 40 2.3.4SingularSolution...... 44 2.4ApproximateAnalyticalSolutions...... 46 2.4.1 Series Expansion of Lane-Emden Functions near an Interior Point ...... 46 2.4.2 Expansion of Lane-Emden Functions near the Finite Boundary ...... 51 2.4.3AsymptoticExpansionsofLane-EmdenFunctions...... 53 2.4.4 Pad´eApproximantsforLane-EmdenFunctions...... 57 2.4.5AnalyticalSolutionsClosetoanExactSolution...... 60 2.4.6 Approximate Solutions by the Method of Multiple Scales ...... 64 2.5ExactNumericalSolutions...... 67 2.6PhysicalCharacteristicsofUndistortedPolytropes...... 79 2.6.1Radius...... 79 2.6.2 Density, Pressure, and Temperature ...... 79 2.6.3Volume,Surface,andMass...... 80 2.6.4MeanDensity...... 82 2.6.5 Gravitational Acceleration and Gravitational Potential ...... 82 2.6.6TheVirialTheorem...... 84 2.6.7GravitationalPotentialEnergyandInternalEnergy...... 92 2.6.8 Behaviour of Physical Characteristics as the Radius Increases ...... 97 2.6.9 Uniform Contraction or Expansion of N-dimensionalPolytropes...... 99 2.7TopologyoftheLane-EmdenEquation...... 102 2.7.1 Polytropic Indices n = −1, ±∞ ...... 106 2.7.2 Polytropic Index n = ±∞ ...... 117 2.8 Composite and Other Spherical Polytropes ...... 122 2.8.1CompositePolytropes...... 122 2.8.2Two-componentPolytropes...... 129 2.8.3 Loaded Polytropes ...... 132 2.8.4BoltzmannFactorPolytropes...... 134 2.8.5TheEmden-Fowlerequation...... 135

v vi Table of Contents

3 Distorted Polytropes 137 3.1Introduction...... 137 3.2 Chandrasekhar’s First Order Theory of Rotationally Distorted Spheres ...... 150 3.3 Chandrasekhar’s First Order Theory of Tidally Distorted Polytropes ...... 163 3.4Chandrasekhar’sDoubleStarProblem...... 170 3.5 Second Order Extension of Chandrasekhar’s Theory to Diﬀerentially Rotating Polytropes 182 3.6 Double Approximation Method for Rotationally and Tidally Distorted Polytropic Spheres 191 3.7 Second Order Level Surface Theory of Rotationally Distorted Polytropes ...... 199 3.8 Numerical and Seminumerical Methods Concerning Distorted Polytropic Spheres . . . . . 205 3.8.1 James’ Calculations on Rotating Polytropes ...... 205 3.8.2 Stoeckly’s Method of Linearized Diﬀerences ...... 209 3.8.3 Williams’ Optimal Matching Method ...... 213 3.8.4 The Self-Consistent Field Method of Ostriker and Mark ...... 214 3.8.5 The Ellipsoidal Method of Roberts ...... 221 3.8.6Miketinac’sMethod...... 224 3.8.7Eriguchi’sComplex-PlaneStrategy...... 226 3.8.8 Tabulation of Numerical Results for Critically Rotating Spheroids ...... 230 3.9RotatingPolytropicCylindersandPolytropicRings...... 235 3.9.1RotatingPolytropicCylinders...... 235 3.9.2PolytropicRings...... 238 3.10Magnetopolytropes...... 253 3.10.1Introduction...... 253 3.10.2 Roxburgh’s Theory of Spherical Magnetopolytropes ...... 255 3.10.3 Monaghan’s Magnetopolytropes with Poloidal Fields ...... 259 3.10.4PerturbationMethodsforMagnetopolytropes...... 264 3.10.5MagneticFieldsinCylindricalPolytropes...... 280

4 Relativistic Polytropes 287 4.1 Undistorted Relativistic Polytropes ...... 287 4.1.1 Spherical Polytropes with Einstein’s General Relativity ...... 287 4.1.2 Composite Polytropes in General Relativity ...... 299 4.1.3 Spherical Polytropes with Variable Gravitational Constant ...... 301 4.1.4 Polytropic Spheres in Bimetric Gravitation Theory ...... 304 4.1.5 Relativistic Polytropic Slabs and Cylinders ...... 308 4.2 Rotationally Distorted Relativistic Polytropes ...... 309 4.2.1Introduction...... 309 4.2.2 Perturbation Theory in the Post Newtonian Approximation ...... 310 4.2.3 Stoeckly’s Method for Post Newtonian Rotating Polytropes ...... 319 4.2.4 Slowly Rotating, Fully Relativistic Polytropes ...... 320 4.2.5 Slowly Rotating and Fully Relativistic Polytropes in Bimetric Gravitation . . . . . 325 4.2.6 Rapidly Rotating and Fully Relativistic Polytropes ...... 325

5 Stability and Oscillations 341 5.1DeﬁnitionsandGeneralConsiderations...... 341 5.2BasicEquations...... 346 5.3 Radial Oscillations of Polytropic Spheres ...... 361 5.3.1 Approximate Values of the Fundamental Oscillation Frequency ...... 361 5.3.2 Linear Radial Oscillations of the Constant Density Polytrope n =0...... 363 5.3.3 Linear Radial Oscillations of the Roche Model ...... 364 5.3.4 The Schuster-Emden Polytrope n =5 ...... 365 5.3.5 Linear Radial Oscillations of Polytropes with 0 ≤ n<5 ...... 366 5.3.6 Nonlinear, Second Order Radial Oscillations of Polytropes ...... 368 5.3.7 Linear Radial Oscillations of Composite Polytropes ...... 370 5.3.8 Radial Oscillations of Isothermal Spheres n = ±∞ ...... 372 5.4 Instability of Truncated Polytropes ...... 373 Table of Contents vii

5.4.1 The Jeans Criterion for the Inﬁnite Homogeneous Medium ...... 373 5.4.2 Radial Gravitational Instability of Polytropes under External Pressure ...... 374 5.5 Nonradial Oscillations of Polytropic Spheres ...... 383 5.5.1 Nonradial Oscillations of the Homogeneous Polytrope n =0...... 383 5.5.2 Nonradial Oscillations of Polytropes if 0

6 Further Applications to Polytropes 549 6.1ApplicationstoStarsandStellarSystems...... 549 6.1.1 Eddington’s Standard Model ...... 549 6.1.2UniformEnergySourceModel...... 552 6.1.3PointSource(Cowling)Model...... 558 6.1.4 Convective Model with Radiation Pressure ...... 559 6.1.5 Convective Model with Negligible Radiation Pressure ...... 562 6.1.6ApplicationstoDiﬀerentiallyRotatingPolytropes...... 565 6.1.7PolytropicPlanets...... 569 6.1.8FissionHypothesisofRotatingPolytropes...... 571 6.1.9ApplicationstoStellarSystems...... 574 viii Table of Contents

6.2 Polytropic Atmospheres, Polytropic Clouds and Cores, Embedded Polytropes ...... 583 6.2.1 Instabilities in Polytropic Atmospheres ...... 583 6.2.2PolytropicInterstellarClouds...... 591 6.2.3 Collapsing Polytropic Stellar Cores and Expanding Polytropes ...... 597 6.2.4EmbeddedPolytropes...... 603 6.3PolytropicWinds...... 607 6.3.1Summers’GeneralSolution...... 607 6.3.2BipolytropicWinds...... 620 6.3.3MagnetopolytropicWinds...... 623 6.3.4InteractionofPolytropicWindswithOtherObjects...... 628 6.4PolytropicAccretionFlows,AccretionDisksandTori...... 639 6.4.1LineAccretionofPolytropicFlows...... 639 6.4.2 Spherically Symmetric Accretion of a Polytropic Gas ...... 648 6.4.3PolytropicAccretionDisksandTori...... 657 Acknowledgments...... 666 AppendixA...... 667 AppendixB...... 667 AppendixC...... 672 ReferencesandAuthorIndex...... 674 SubjectIndex...... 714