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Superfluid Hydrodynamics in Neutron Stars By Superfluid hydrodynamics in neutron stars by Gregory Allen Mendell A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Montana State University © Copyright by Gregory Allen Mendell (1991) Abstract: Superfluidity is predicted to exist in neutron stars. Superfluid effects on the dynamics of these stars have not been investigated in depth in the past. In this thesis, superfluid hydrodynamics in neutron stars is developed extensively. It is shown that superfluidity has important effects on the oscillation modes; dissipative properties, and stability of these stars. Very general hydrodynamic equations are derived which describe superfluid mixtures. The fluid equations are coupled to the electromagnetic and gravitational fields. Forces due to the quantized vortices of the superfluids are also included. It is shown that new vorticity-preserving forces can be introduced into the superfluid-mixture equations. The equations are then adapted to describe neutron stars composed primarily of superfluid neutrons, superconducting protons, and degenerate electrons and muons. The set of equations is closed by constructing a model of the total energy density and using it to express the dependent variables in terms of the independent variables. The low-frequency long-wavelength limit of the equations is determined. The results can be used to study superfluid effects on the global oscillations of neutron stars. The equations are generalized further to include dissipative effects. Most important is a form of dissipation known as mutual friction, which occurs only in superfluids. In neutron stars, mutual friction is due to electron scattering off the neutron and proton vortices. An energy functional is constructed which determines the damping time of a mode due to the various forms of dissipation, including mutual friction. Plane-wave solutions are found to the equations. Mutual friction is shown to be the dominant form of dissipation in neutron stars for sufficiently large angular velocities. Gravitational radiation tends to make all rotating stars unstable, while internal dissipation tends to counteract this instability. Thus, gravitational radiation can limit the maximum angular velocity of neutron stars. The most important conclusion of this thesis is that mutual friction completely suppresses the gravitational-radiation instability in rotating neutron stars cooler than the superfluid-transition temperature. SUPERFLUID HYDRODYNAMICS IN NEUTRON STARS by Gregory Allen Mendell A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy m Physics MONTANA STATE UNIVERSITY Bozeman, Montana August 1991 p-yl? 11 APPROVAL of a thesis submitted by Gregory Allen Mendell This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. I?- fluyst IcHl jLa SUU eVK Date Chairperson, Graduate Committee Approved for the Major Department ^ " I Z - ^ Date Approved for the College of Graduate Studies 9 Date ite DeGraduate Dean iii STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfillment of the requirements for a doc­ toral degree at Montana State University, I agree that the Library shall make it available to borrowers under rules of the Library. I further agree that copying of this thesis is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Requests for extensive copying or repro­ duction of this thesis should be referred to University Microfilms International, 300 North Zeeb Road, Ann Arbor, Michigan 48106, to whom I have granted “the exclusive right to reproduce and distribute copies of the dissertation in and from microfilm and the right to reproduce and distribute by abstract in any format.” Signature. ACKNOWLEDGMENTS I wish to thank my adviser, Dr. Lee Lindhlom, for -many useful discussions and suggestions during the course of this work, and for providing me with financial support from his National Science Foundation and MONTS grants. I also wish to thank my wife, Jeanette, for her support during my years in graduate school. TABLE OF CONTENTS Page 1. IN T R O D U C T IO N ...................................: ..............................: ......................... I Purpose and Plan of This Thesis ; ..................; . ; . I N eutron S t a r s .................................................................................................... : 10 Observational Properties ....................................... : ..................................10 Evidence of Superfluidity .......................... , ...........................................16 Equilibrium N eutron Star Models . : ...........................: * . 19 How Fast Can a Neutron Star R o t a t e ? ....................................................24 Superfluidity ............................................................................................................ 25 Observational Properties : . : ............................................ : . 25 Cooper P a irs .................................. 28 BCS T h e o r y .......................................................................... ; ; 31 Gap in Neutron Stars . .....................................................; 35 2. SUPERFLUID H Y D R O D Y N A M IC S .......................... ; ..................................37 London Theory . : ............................................................. 37 Landau E q u a t i o n s .............................................................; ..................................43 Smooth-averaged Superfluid Hydrodynamics 50 3. GENERAL THEORY OF SUPERFLUID M IXTURES ......................: . 55 Introduction ................. .... ..............................................................; . 55 Uncharged Superfluid Mixtures .................................................... ; . 58 Hamiltonian Form ulation ..................... : . : : ............................... : . ; 64 Electromagnetic CoupHng ............................. ..... ; ...................... : . 70 G ravitational Coupling . : ...............................: .......................................79 4. A DAPTING TH E EQUATIONS TO NEUTRON S T A R S ..............................81 In tro d u c tio n .................................................................................................................81 Nonlinear Hydrodynamic Equations .................................................................82 Energy Density ........................................................................................................ 87 Dependent V a r ia b le s ....................................................................................... 90 Vortex Energy Per Unit L e n g th ....................................... : ..........................93 I n t r o d u c t i o n ............................................; ...........................................; 93 Neutron V ortex ....................................... : ..................... 94 Proton Vortex ................. : ..............................; ......................: . 96 Equilibrium Conditions and Linear Perturbation Equations . : : ; ; 98 C haracteristic Frequencies 100 5. LOW -FREQUENCY-LONG-W AVELENGTH L I M I T ............................... 105 Introduction . ; ..................................................................: . 105 Reducing the Number of Independent Velocities ; . 106 Cyclotron-vortex F re q u e n c y .......................................................................... HO Ignoring Electromagnetic and Vortex Forces ................................... ; . 113 6. GENERAL THEORY OF MUTUAL FRICTION ...............................: . 116 In tro d u c tio n ...............................: ..................... ; ........................................... 116 M utual Friction Forces : ................................................; ..................... : 117 Mutual Friction Coefficients ......................: . ; . 121 7. DISSIPATIVE EFFECTS IN SUPERFLUID NEUTRON STARS . 128 Introduction ................................................................. , .................................. 128 Dissipative Effects . : .........................................................................................130 Dissipative Forces, Including Mutual Friction 130 LFLW l i m i t ......................................................................................................135 Mutual Friction Coefficients . ; ....................................... 136 LFLW Dissipative E quations ............................................................. 138 Effect of M utual Friction on The Damping Times ............................... 139 8. PLANE SYM M ETRIC M O D E S ...................................................................... 146 In tro d u c tio n ................................................................................................ : . 146 Axial Modes : ...................................: .......................................: . ; : 150 Inplane M o d e s .......................................................................................................... 155 9. STABILITY OF ROTATING SUPERFLUID NEUTRON STARS . 161 Introduction ................................................; ...........................................; .
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