rotation and accretion powered pulsars - pranab ghosh

Rotation and Accretion Powered Pulsars

WORLD SCIENTIFIC SERIES IN ASTRONOMY AND ASTROPHYSICS

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N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G TA I P E I C H E N N A I

World Scientic

Pranab GhoshTata Institute of Fundamental Research, India

World Scientic Series in Astronomy and Astrophysics Vol. 10

@]bObW]\O\R/QQ`SbW]\>]eS`SR>cZaO`a

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World Scientific Series in Astronomy and Astrophysics Vol. 10ROTATION AND ACCRETION POWERED PULSARS

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January 10, 2007 16:24 WSPC/Book Trim Size for 9in x 6in pranab

To the Memory of My Parents

Verba docent, exempla trahunt.

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February 13, 2007 15:0 WSPC/Book Trim Size for 9in x 6in pranab

Preface

The truth, the whole truth, and nothing but the truth, observes a char-

acter in Aldous Huxleys The Genius and the Goddess, All witnesses take

the same oath, and testify about the same events. The result, of course, is

fifty-seven varieties of fiction. We forget all too often that this describes

not only witnesses and artists, but scientists as well. We describe what is

piously believed to be objective truth, but the outcome is necessarily our

view of that truth. This book grew out of my desire to communicate to a

wide scientific readership in a simple, accessible way the unifying overview

we have begun to achieve in recent years of rotation- and accretion-powered

pulsars the phases that rotating, magnetic neutron stars go through dur-

ing their lives. The basic facts are scattered far and wide over professional

scientific literature, but the manner in which I collect and present them

reflects my own overview of the subject, as it must. For a long time, it has

been clear that, while there were classic, pioneering text-books on rotation-

powered pulsars, and excellent individual chapters on X-ray binaries within

collective volumes, the unified view was not really available as an advanced

text book. And yet, this overview is one of the milestones in modern high-

energy astrophysics. It is my hope that this book will fill that void.

It has been my pleasure and privilege to learn from many fine physicists

over the years, and their gifts to my mind will be felt throughout the book.

I am proud of this. Among them, the name of Prof. Frederick K. Lamb is

one that I always mention with great pleasure and honor. While writing the

book, I have marveled at how various branches of physics blend beautifully

to give us a comprehensive description of the physics of rotating, magnetic

neutron stars and their environments. While this synthesis may be found

to a certain extent in many parts of astrophysics, that of rotation- and

accretion-powered pulsars seems to be a remarkably strong example.

vii


viii Rotation and Accretion Powered Pulsars

For motivation and fruition of this project, appreciation and acknowl-edgment of a deep debt of gratitude go to my late parents, and to my wifeSurita. That thirst for knowledge which the former propagated into mehas been the ultimate motive power behind all my work. This book owes agreat deal to Suritas constant inspiration and encouragement, and to heruninching support and endless patience while it was being written.

Pranab Ghosh2006


Acknowledgments

Permissions for reproduction of previously-published gures have been ob-tained as follows. Permissions were obtained from sole/lead authors (and/orother authors where appropriate), the author names being given in the re-spective gure captions, and full references, e.g., journal/proceedings/booknames, volume and page numbers, given in the Bibliography. Permissionswere also obtained from the appropriate publishers/permission-grantingauthorities of these publications, as detailed in the following table. Incase none of the authors of a publication was available or alive, onlythe latter permission could be obtained. We thank all such authors andpublishers/permission-granting authorities for their kind permission, anddisplay the appropriate copyright sign and/or other material inrespective gure captions for those who require it as per termsand conditions. Those authors who showed further kindness by providingus with copies of their gures are much appreciated, and mentioned in therespective gure captions.

Table 1 PERMISSIONS

Figure(s) Publisher/Permission-granting authority

Book cover Chandra X-ray CenterEducation/Outreach Co-ordinator

2.1,2.4,7.13,7.15,7.16,13.4 Blackwell Publishing, Oxford3.1,4.18,4.19,6.1,9.1,9.12,9.13 Annual Reviews, Palo Alto9.23 Revista Mexicana de Astron. y Astros.2.3 Taylor & Francis Ltd, Oxford9.7,9.8,9.22,9.18 Astronomical Society of Japan4.9,4.10,4.11,4.14,7.19,B.1,D.1 Elsevier B.V., Amsterdam & Oxford

ix


x Rotation and Accretion Powered Pulsars

Table 1 PERMISSIONS (continued)

Figure(s) Publisher/Permission-granting authority

1.4,7.10 IEEE6.8 Astronomical Society of the Pacic6.6,11.2 International Astronomical Union2.5,4.8,4.12,4.15,4.16,5.5,5.7,5.9,6.4,7.23 American Physical society7.14,7.20,7.22,11.1 The Royal Society, London7.1,7.2,7.3 W.H. Freeman & Co., New York7.24,10.30 Societa` Astronomica Italiana8.1 Progress of Theoretical Physics9.26 M. Sako & G. Branduardi-Raymont14.1,8.3,8.4,8.5,5.1,5.3,10.6,10.165.6,10.20,9.19,9.21,7.6,5.11,5.125.13,9.11,10.5,10.15,10.17,5.14 American Astronomical Society10.21,11.4,6.3,5.10,7.12,1.5,9.105.2,5.4,5.15,1.10,1.9,11.8,10.2210.23,10.29,9.15,9.24,11.6,11.79.20,7.6,10.12,10.26,10.27,10.281.2,1.3,9.25,10.2,10.8,10.9,7.239.3,1.7,7.4,7.5,7.19,7.22,9.6,9.97.17,10.10,10.13,10.18,10.19,12.112.2,12.3,12.4,10.3,10.4,10.14,12.6

6.16 AAAS9.5,9.14,9.16,9.17,7.8,10.1,13.5,6.10 Cambridge University Press7.11 Koninklijke Nederlandse Akad. Wetensch.

6.2,6.7,6.9,6.11,6.12,6.13,9.2,8.2 Springer Science and Business Media10.24,11.5,10.3,10.4,10.14,12.6

6.14,6.15,13.3 MacMillan Publishers Ltd5.8 H. HeiselbergB.3 M. van der Sluys6.5 K. Kifonidis1.1 The Nobel Foundation11.3 The University of ChicagoC.1 ATNF Outreach12.5 Indian Acad. Sci.A.2 Princeton University Press


Contents

Preface vii

Acknowledgments ix

1. The Discovery of Pulsars 11.1 Rotation-Powered Pulsars: Radio Discovery . . . . . . . . 11.2 Accretion-Powered Pulsars: X-Ray Discovery . . . . . . . . 41.3 Pulsars as Neutron Stars . . . . . . . . . . . . . . . . . . . 10

1.3.1 Rotation-Powered Pulsars . . . . . . . . . . . . . . 101.3.2 Accretion-Powered Pulsars . . . . . . . . . . . . . 14

1.4 Powering Pulsars by Rotation and Accretion . . . . . . . . 171.4.1 Rotation Power . . . . . . . . . . . . . . . . . . . . 171.4.2 Accretion Power . . . . . . . . . . . . . . . . . . . 20

1.5 Galactic Distributions of Pulsars . . . . . . . . . . . . . . . 241.5.1 Rotation-Powered Pulsars . . . . . . . . . . . . . . 24

1.5.1.1 Pulsar distances . . . . . . . . . . . . . 261.5.1.2 Galactic electron distribution . . . . . . 271.5.1.3 Galactic pulsar distribution . . . . . . . 311.5.1.4 Selection eects . . . . . . . . . . . . . . 32

1.5.2 Accretion-Powered Pulsars . . . . . . . . . . . . . 341.6 Period Distributions of Pulsars . . . . . . . . . . . . . . . . 34

2. Physics of Neutron Stars I. Degenerate Stars 392.1 Historical Notes on Neutron Stars . . . . . . . . . . . . . . 392.2 Degenerate Stars . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.1 Degeneracy . . . . . . . . . . . . . . . . . . . . . . 472.2.2 Electron Degeneracy . . . . . . . . . . . . . . . . . 50

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xii Rotation and Accretion Powered Pulsars

2.2.3 Neutron Degeneracy . . . . . . . . . . . . . . . . . 512.2.4 Complete Degeneracy . . . . . . . . . . . . . . . . 51

2.3 The Landau Arguments . . . . . . . . . . . . . . . . . . . . 532.3.1 Degenerate Stars: Rough Mass Limits . . . . . . . 582.3.2 Degenerate Stars: Mass-Radius Relations . . . . . 59

2.4 White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . 602.4.1 The Stoner-Anderson Work . . . . . . . . . . . . . 612.4.2 Polytropes . . . . . . . . . . . . . . . . . . . . . . 652.4.3 Chandrasekhars Work . . . . . . . . . . . . . . . . 682.4.4 Modern Work . . . . . . . . . . . . . . . . . . . . . 75

2.5 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . 802.5.1 The Oppenheimer-Volko Work . . . . . . . . . . 812.5.2 A General-Relativistic Toy Neutron Star . . . . 88

2.6 Landau Arguments: General-Relativistic Modications . . 91

3. Physics of Neutron Stars II. Physics of Dense Matter-1 973.1 Matter at Low Densities: Electronic Energy . . . . . . . . 97

3.1.1 Wigner-Seitz Cells . . . . . . . . . . . . . . . . . . 983.1.2 The Thomas-Fermi Approximation . . . . . . . . . 101

3.2 Dense Matter Below Neutron-Drip Density: EquilibriumNuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.1 Eects of Lattice Energy . . . . . . . . . . . . . . 1103.2.2 Nuclear Shell Eects . . . . . . . . . . . . . . . . . 113

3.3 Neutron Drip . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.4 Matter Above Neutron-Drip Density: Nuclei Surrounded

by Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.4.1 Nuclear Surface Energy: A Primer . . . . . . . . . 1213.4.2 BBP Results . . . . . . . . . . . . . . . . . . . . . 1253.4.3 More Accurate Results . . . . . . . . . . . . . . . 1263.4.4 Simple Scalings . . . . . . . . . . . . . . . . . . . . 128

4. Physics of Neutron Stars III. Physics of Dense Matter-2 1314.1 Above Nuclear-Matter Density: Uniform Nuclear Matter . 131

4.1.1 The Goldstone Expansion . . . . . . . . . . . . . . 1344.1.2 Goldstone Diagrams . . . . . . . . . . . . . . . . . 137

4.1.2.1 First-order diagrams . . . . . . . . . . . 1384.1.2.2 Second-order diagrams . . . . . . . . . . 139

4.1.3 Brueckners Reaction Matrix . . . . . . . . . . . . 1414.1.4 Correlations and Healing . . . . . . . . . . . . . 145


Contents xiii

4.1.5 Brueckner-Bethe-Goldstone (BBG) Theory . . . . 1504.1.6 The Variational Method . . . . . . . . . . . . . . . 154

4.1.6.1 Cluster diagrams . . . . . . . . . . . . . 1584.1.6.2 Modern calculations . . . . . . . . . . . 161

4.1.7 Recent Developments in BBG and VariationalApproaches . . . . . . . . . . . . . . . . . . . . . . 1644.1.7.1 Relativistic eects in Brueckner theory . 1684.1.7.2 Relativistic eects in variational

methods . . . . . . . . . . . . . . . . . 1704.1.7.3 Recent results . . . . . . . . . . . . . . . 171

5. Physics of Neutron Stars IV. Mass, Radius and Structure 1735.1 Masses and Radii . . . . . . . . . . . . . . . . . . . . . . . 173

5.1.1 Insensitivity of Radius to Mass . . . . . . . . . . . 1795.2 Internal Structure . . . . . . . . . . . . . . . . . . . . . . . 180

5.2.1 Crustal Properties: Mass and Radius . . . . . . . 1835.3 Non-Spherical Shapes: Rods and Plates . . . . . . . . . . . 186

5.3.1 Turning Nuclei Inside Out . . . . . . . . . . . . 1885.3.2 Physical Insights: Frustrated Fission . . . . . . . . 1895.3.3 Uncertainties . . . . . . . . . . . . . . . . . . . . . 190

5.4 The Maximum Mass of Neutron Stars . . . . . . . . . . . . 1925.4.1 The Maximum Compactness . . . . . . . . . . . . 1925.4.2 The Maximum Mass . . . . . . . . . . . . . . . . . 195

5.5 Rotating Neutron Stars . . . . . . . . . . . . . . . . . . . . 1985.5.1 The Hartle-Thorne Approximation . . . . . . . . . 2005.5.2 Arbitrary Rotation . . . . . . . . . . . . . . . . . . 2025.5.3 Maximum and Minimum Rotation . . . . . . . . . 205

5.5.3.1 The mass-shed limit . . . . . . . . . . . 2055.5.3.2 The gravitation-wave limit . . . . . . . . 2075.5.3.3 The r-mode . . . . . . . . . . . . . . . . 2095.5.3.4 The supramassive-collapse limit . . . . . 2105.5.3.5 Maximum angular velocity of neutron

stars . . . . . . . . . . . . . . . . . . . . 2125.5.4 Hartle-Thorne Approximation: Reprise . . . . . . 2145.5.5 Moments of Inertia . . . . . . . . . . . . . . . . . . 215

5.5.5.1 Crustal moment of inertia . . . . . . . . 218

6. Origin and Evolution of Neutron Stars 2216.1 Binary Stellar Evolution . . . . . . . . . . . . . . . . . . . 221


xiv Rotation and Accretion Powered Pulsars

6.1.1 Cases A, B, and C . . . . . . . . . . . . . . . . . . 2226.1.2 Orbital Changes . . . . . . . . . . . . . . . . . . . 225

6.1.2.1 Conservative mass transfer . . . . . . . 2256.1.2.2 Non-conservative mass transfer . . . . . 226

6.1.3 Stellar Evolution . . . . . . . . . . . . . . . . . . . 2286.1.3.1 Conservative evolution . . . . . . . . . . 2296.1.3.2 Common-envelope (CE) evolution . . . 231

6.2 Supernovae: Birth of Neutron Stars . . . . . . . . . . . . . 2346.2.1 Final Evolution of Helium Stars and Cores . . . . 2356.2.2 Core Collapse: Neutron Star Formation . . . . . . 2386.2.3 Core Bounce: Supernova Explosion . . . . . . . . 239

6.2.3.1 Explosion mechanisms . . . . . . . . . . 2416.2.3.2 Orbital changes due to supernova

explosions . . . . . . . . . . . . . . . . . 2446.2.4 Evolution of Proto-Neutron Stars . . . . . . . . . . 246

6.3 Rotation Power in Young Pulsars . . . . . . . . . . . . . . 2506.3.1 Missing Links: Ante-Deluvian Systems . . . . . . . 2526.3.2 Probes of Be-Star Outow: PSR B1259-63 . . . . 252

6.3.2.1 The Shvartsman surface . . . . . . . . . 2536.3.2.2 Propeller spindown . . . . . . . . . . . . 2556.3.2.3 Tilted Be-star disks . . . . . . . . . . . 2566.3.2.4 Recent work: further orbital dynamics . 258

6.4 Accretion Power in Middle-Aged Pulsars . . . . . . . . . . 2596.4.1 Evolution to Massive X-Ray Binaries . . . . . . . 2606.4.2 Evolution to Intermediate-Mass X-Ray Binaries . 2636.4.3 Evolution to Low-Mass X-Ray Binaries . . . . . . 264

6.4.3.1 CVs . . . . . . . . . . . . . . . . . . . . 2646.4.3.2 LMXBs . . . . . . . . . . . . . . . . . . 266

6.5 Rotation Power in Old, Recycled Pulsars . . . . . . . . . . 2726.5.1 Final Evolution of HMXBs and Recycling . . . . . 274

6.5.1.1 Recycled pulsars: single or withdegenerate companions . . . . . . . . . . 276

6.5.2 The Double Pulsar Binary J0737-3039 . . . . . . . 2806.5.3 Final Evolution of LMXBs and Recycling . . . . . 282

6.5.3.1 Gravitational radiation . . . . . . . . . . 2876.5.3.2 Magnetic braking . . . . . . . . . . . . . 2876.5.3.3 The minimum period and the period gap 2896.5.3.4 Donor evolution and expansion . . . . . 2916.5.3.5 Recycled pulsars from LMXBs . . . . . 294


Contents xv

6.5.4 Missing Links: Accreting Millisecond Pulsars . . . 2966.5.5 Irradiation of Low-Mass Companions . . . . . . . 3016.5.6 Missing Links: Black Widow Pulsars . . . . . . 3026.5.7 Pulsars with Planets . . . . . . . . . . . . . . . . . 304

7. Properties of Rotation Powered Pulsars 3077.1 Pulse Properties . . . . . . . . . . . . . . . . . . . . . . . . 307

7.1.1 Integrated Pulse Proles . . . . . . . . . . . . . . 3087.1.1.1 Pulse shapes . . . . . . . . . . . . . . . 3087.1.1.2 Interpulses . . . . . . . . . . . . . . . . 3087.1.1.3 Polarization . . . . . . . . . . . . . . . . 3107.1.1.4 The rotating vector model . . . . . . . . 3137.1.1.5 Frequency dependence and stability . . 313

7.1.2 Individual Pulses . . . . . . . . . . . . . . . . . . . 3157.1.2.1 Intensity variations and nulling . . . . . 3157.1.2.2 Subpulse drifting . . . . . . . . . . . . . 3187.1.2.3 Micropulses . . . . . . . . . . . . . . . . 3197.1.2.4 Giant pulses . . . . . . . . . . . . . . . . 321

7.2 Timing Properties . . . . . . . . . . . . . . . . . . . . . . . 3237.2.1 Pulsar Timing . . . . . . . . . . . . . . . . . . . . 3237.2.2 Secular Period Changes . . . . . . . . . . . . . . . 328

7.2.2.1 Characteristic age . . . . . . . . . . . . 3297.2.2.2 Braking index . . . . . . . . . . . . . . . 3317.2.2.3 Higher derivatives . . . . . . . . . . . . 332

7.2.3 Irregular Period Changes . . . . . . . . . . . . . . 3337.2.3.1 Timing noise . . . . . . . . . . . . . . . 3337.2.3.2 Millisecond pulsars as stable clocks . . . 3367.2.3.3 Limits on cosmic gravitational wave

background . . . . . . . . . . . . . . . . 3427.2.3.4 Glitches . . . . . . . . . . . . . . . . . . 3447.2.3.5 Glitches: starquakes . . . . . . . . . . . 349

7.2.4 Timing Rotation Powered Pulsars in Binaries . . . 3547.2.4.1 PSR 1913+16: historical notes . . . . . 3597.2.4.2 Binary rotation-powered pulsars:

general systematics . . . . . . . . . . . . 3617.2.4.3 PSR 1913+16: nursery of relativistic

gravity . . . . . . . . . . . . . . . . . . . 3657.2.4.4 PSR 1534+12 and similar relativity

laboratories . . . . . . . . . . . . . . . 371


xvi Rotation and Accretion Powered Pulsars

7.2.4.5 The double-pulsar binary as relativitylaboratory . . . . . . . . . . . . . . . . . 373

7.2.4.6 A related fundamental eect . . . . . 375

8. Superuidity in Neutron Stars and Glitch Diagnostics 3778.1 Superuidity in Neutron Stars . . . . . . . . . . . . . . . . 377

8.1.1 Pairing . . . . . . . . . . . . . . . . . . . . . . . . 3788.1.2 Gap Energy . . . . . . . . . . . . . . . . . . . . . . 3808.1.3 Rotating Superuids and Quantized Vortices . . . 382

8.2 Post-Glitch Relaxation: Two-Component Theory . . . . . 3868.3 Glitches: Vortex Pinning . . . . . . . . . . . . . . . . . . . 387

8.3.1 The Pinning Force . . . . . . . . . . . . . . . . . . 3898.3.2 Strong and Weak Pinning . . . . . . . . . . . . . . 3918.3.3 The Magnus Force . . . . . . . . . . . . . . . . . . 394

8.4 Post-Glitch Relaxation: Vortex Creep . . . . . . . . . . . . 3958.4.1 Rotational Dynamics: Steady State . . . . . . . . 3988.4.2 Approach to Steady State . . . . . . . . . . . . . . 400

8.5 Stellar Parameters from Glitch Data . . . . . . . . . . . . . 4058.6 Glitches: Recent Developments . . . . . . . . . . . . . . . . 409

9. Properties of Accretion Powered Pulsars 4139.1 Binary Characteristics . . . . . . . . . . . . . . . . . . . . 414

9.1.1 Displaying Binary Systematics . . . . . . . . . . . 4179.1.2 Newtonian Apsidal Motion . . . . . . . . . . . . . 4199.1.3 Orbital Period Changes . . . . . . . . . . . . . . . 4239.1.4 Neutron-Star and Companion Masses . . . . . . . 426

9.2 Pulse Proles . . . . . . . . . . . . . . . . . . . . . . . . . 4299.3 Secular Period Changes . . . . . . . . . . . . . . . . . . . . 4329.4 Timing Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 440

9.4.1 Power-Density Spectra . . . . . . . . . . . . . . . . 4439.4.1.1 Observed power spectra . . . . . . . . . 445

9.4.2 Time-Domain Analysis . . . . . . . . . . . . . . . 4499.5 Quasi-Periodic Oscillations (QPOs) . . . . . . . . . . . . . 451

9.5.1 QPOs in LMXBs . . . . . . . . . . . . . . . . . . . 4519.5.1.1 Low-frequency QPOs . . . . . . . . . . . 4519.5.1.2 High-frequency QPOs . . . . . . . . . . 4559.5.1.3 LF QPO diagnostics . . . . . . . . . . . 4569.5.1.4 HF QPO diagnostics . . . . . . . . . . . 458

9.5.2 QPOs in HMXBs . . . . . . . . . . . . . . . . . . . 464


Contents xvii

9.6 X-Ray Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 4659.6.1 Continuum Emission . . . . . . . . . . . . . . . . . 4669.6.2 Pulse-Phase Spectroscopy . . . . . . . . . . . . . . 4699.6.3 Cyclotron Features . . . . . . . . . . . . . . . . . . 4719.6.4 Emission Lines: Fluorescence, Recombination,

Resonance . . . . . . . . . . . . . . . . . . . . . . 4749.6.4.1 Fluorescence . . . . . . . . . . . . . . . 4749.6.4.2 X-ray ionization of stellar winds . . . . 4779.6.4.3 Observations . . . . . . . . . . . . . . . 4819.6.4.4 Recombination . . . . . . . . . . . . . . 4829.6.4.5 Recent observations . . . . . . . . . . . 4859.6.4.6 Wind diagnostics . . . . . . . . . . . . . 489

9.7 Mutants: Anomalous X-Ray Pulsars (AXPs) . . . . . . . . 490

10. Pulsar Magnetospheres 49510.1 Magnetospheres of Accretion-Powered Pulsars . . . . . . . 495

10.1.1 Exterior Flow and Plasma Capture . . . . . . . . . 49610.1.1.1 Capture from stellar winds . . . . . . . 49710.1.1.2 Fluctuations in stellar winds . . . . . . 50410.1.1.3 Capture from Roche-lobe overow . . . 505

10.1.2 Formation of Magnetospheres . . . . . . . . . . . . 50810.1.2.1 Lengthscales of magnetospheres . . . . . 511

10.1.3 Radial Flow . . . . . . . . . . . . . . . . . . . . . . 51310.1.3.1 Size of the magnetosphere . . . . . . . . 51510.1.3.2 The shock . . . . . . . . . . . . . . . . . 51710.1.3.3 Shape of the magnetosphere . . . . . . . 51810.1.3.4 Magnetospheric boundary with

accretion ow . . . . . . . . . . . . . . . 52410.1.4 Disk Flow . . . . . . . . . . . . . . . . . . . . . . . 52510.1.5 Thin Keplerian Accretion Disks . . . . . . . . . . 525

10.1.5.1 One temperature disks . . . . . . . . . . 52710.1.5.2 The -model of disk viscosity . . . . . . 52910.1.5.3 The nature of disk viscosity . . . . . . . 53110.1.5.4 The structure of -disks . . . . . . . . . 53410.1.5.5 Two temperature disks . . . . . . . . . . 538

10.1.6 The Disk-Magnetosphere Interaction . . . . . . . . 54210.1.6.1 Basic electrodynamic processes . . . . . 54310.1.6.2 Steady ow models . . . . . . . . . . . . 548


xviii Rotation and Accretion Powered Pulsars

10.1.7 Disk-Magnetosphere Boundary Layer . . . . . . . 55210.1.7.1 Boundary-layer behavior . . . . . . . . . 55610.1.7.2 Boundary-layer: nature and structure . 558

10.1.8 Inner Edge of the Disk . . . . . . . . . . . . . . . . 56010.1.9 Outer Transition Zone . . . . . . . . . . . . . . . . 56410.1.10 Further Work . . . . . . . . . . . . . . . . . . . . . 56610.1.11 Plasma Entry into Magnetospheres: Radial Flow . 569

10.1.11.1 Entry via Rayleigh-Taylor instability . . 57010.1.11.2 Cusp entry . . . . . . . . . . . . . . . . 57610.1.11.3 Entry by other modes . . . . . . . . . . 578

10.1.12 Plasma Entry into Magnetospheres: Disk Flow . . 58010.1.13 Accretion Flows Inside Magnetospheres . . . . . . 581

10.1.13.1 Field-aligned ow . . . . . . . . . . . . . 58210.1.13.2 Flow between eld lines . . . . . . . . 588

10.1.14 Accretion on Stellar Surface: StoppingMechanisms . . . . . . . . . . . . . . . . . . . . . . 59210.1.14.1 Radiative stopping . . . . . . . . . . . . 59210.1.14.2 Collisional stopping . . . . . . . . . . . 596

10.2 Magnetospheres of Rotation-Powered Pulsars . . . . . . . . 59710.2.1 The Goldreich-Julian Argument . . . . . . . . . . 59810.2.2 The Aligned Rotator . . . . . . . . . . . . . . . . . 602

10.2.2.1 The pulsar equation . . . . . . . . . . . 60510.2.2.2 Convergence . . . . . . . . . . . . . . . . 60610.2.2.3 Results . . . . . . . . . . . . . . . . . . 607

10.2.3 Problems with the Standard Model . . . . . . . . 61010.2.4 Vacuum Gaps . . . . . . . . . . . . . . . . . . . . 612

10.2.4.1 Ruderman-Sutherland gaps . . . . . . . 61610.2.5 The Oblique Rotator . . . . . . . . . . . . . . . . 62310.2.6 The Double-Pulsar Binary as Magnetospheric

Probe . . . . . . . . . . . . . . . . . . . . . . . . . 625

11. Pulsar Emission Mechanisms 62911.1 Emission by Rotation-Powered Pulsars . . . . . . . . . . . 629

11.1.1 Coherent Emission . . . . . . . . . . . . . . . . . . 63011.1.2 Emission by Bunches . . . . . . . . . . . . . . . . 63211.1.3 Maser Emission . . . . . . . . . . . . . . . . . . . 63311.1.4 Relativistic Plasma Emission . . . . . . . . . . . . 635


Contents xix

11.2 Emission by Accretion-Powered Pulsars . . . . . . . . . . . 63711.2.1 Radiation Transport in Strongly Magnetized

Plasmas . . . . . . . . . . . . . . . . . . . . . . . . 63811.2.1.1 Basic radiative processes . . . . . . . . . 64011.2.1.2 Transport calculations . . . . . . . . . . 64311.2.1.3 Results . . . . . . . . . . . . . . . . . . 645

11.2.2 Pulse Shapes and Spectra . . . . . . . . . . . . . . 645

12. Spin Evolution of Neutron Stars 64912.1 Spin Evolution of Rotation-Powered Pulsars . . . . . . . . 649

12.1.1 Electromagnetic Spindown . . . . . . . . . . . . . 64912.1.2 Propeller Spindown . . . . . . . . . . . . . . . . . 65012.1.3 Other Spindown Torques . . . . . . . . . . . . . . 65212.1.4 Spindown of PSR B1259-63 . . . . . . . . . . . . . 653

12.2 Spin Evolution of Accretion-Powered Pulsars . . . . . . . . 65612.2.1 Torques on Disk-Fed Pulsars . . . . . . . . . . . . 65612.2.2 Comparison with Observations . . . . . . . . . . . 66112.2.3 Torques on Wind-Fed Pulsars . . . . . . . . . . . . 665

12.3 Pulsar Period Magnetic Field Diagram . . . . . . . . . . 66612.3.1 The Spinup Line . . . . . . . . . . . . . . . . . . . 66912.3.2 Disk Diagnostics: Spinup Lines . . . . . . . . . . . 672

13. Neutron Star Magnetic Fields 67513.1 Exotic Atoms in Strong Magnetic Fields . . . . . . . . . . 675

13.1.1 Further Exotica: Molecular Chains . . . . . . . . 67913.2 Origin of Neutron-Star Magnetic Fields . . . . . . . . . . . 681

13.2.1 Fossil Fields . . . . . . . . . . . . . . . . . . . . . 68113.2.2 Thermo-Magnetic Eects . . . . . . . . . . . . . . 68113.2.3 Dynamos in Young Neutron Stars: Magnetars . . 682

13.3 Evolution of Neutron-Star Magnetic Fields . . . . . . . . . 68513.3.1 Accretion-Induced Field Decay . . . . . . . . . . . 686

13.3.1.1 Ohmic decay . . . . . . . . . . . . . . . 68713.3.2 Field Decay or Hiding/Burial? . . . . . . . . . . . 690

14. Strange Stars 69314.1 EOS of Strange Matter . . . . . . . . . . . . . . . . . . . . 69414.2 Structure of Strange Stars . . . . . . . . . . . . . . . . . . 69414.3 Search for Strange Stars . . . . . . . . . . . . . . . . . . . 696


xx Rotation and Accretion Powered Pulsars

Appendix A Astronomical Preliminaries 699A.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699A.2 Astronomical Co-Ordinate Systems . . . . . . . . . . . . . 700

A.2.1 Equatorial Co-Ordinates . . . . . . . . . . . . . . 700A.2.2 Precession of Equinoxes . . . . . . . . . . . . . . . 702A.2.3 Galactic Co-Ordinates . . . . . . . . . . . . . . . . 703A.2.4 Co-Ordinate Transformation . . . . . . . . . . . . 705A.2.5 Time Keeping . . . . . . . . . . . . . . . . . . . . 705A.2.6 Stellar Classication . . . . . . . . . . . . . . . . . 706

A.2.6.1 Color Index . . . . . . . . . . . . . . . . 706

Appendix B Binary Dynamics 709B.1 Binary Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . 709

B.1.1 Orbital Elements . . . . . . . . . . . . . . . . . . . 709B.1.2 Mass Function . . . . . . . . . . . . . . . . . . . . 710B.1.3 Position in Orbit . . . . . . . . . . . . . . . . . . . 711

B.2 Roche Lobes . . . . . . . . . . . . . . . . . . . . . . . . . . 712

Appendix C Single Star Evolution 717C.1 The Hertzsprung-Russell Diagram . . . . . . . . . . . . . . 717C.2 Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . 719

C.2.1 Hydrogen Burning . . . . . . . . . . . . . . . . . . 720C.2.2 Helium Burning . . . . . . . . . . . . . . . . . . . 722C.2.3 Advanced Burning Stages . . . . . . . . . . . . . . 722C.2.4 Essential Timescales . . . . . . . . . . . . . . . . . 722C.2.5 Mass-Radius Relations . . . . . . . . . . . . . . . . 723C.2.6 Brown Dwarfs . . . . . . . . . . . . . . . . . . . . 724

Appendix D The Two-Nucleon Potential 725D.1 Skyrme Interaction . . . . . . . . . . . . . . . . . . . . . . 728

Appendix E Tables of Pulsars 731E.1 Accretion Powered Pulsars and AXPs . . . . . . . . . . . . 731

Bibliography 735

Index 761


Chapter 1

The Discovery of Pulsars

1.1 Rotation-Powered Pulsars: Radio Discovery

There is no doubt today that the 1967 discovery of rotation-powered pulsarsat radio frequencies was the single most signicant event responsible for therecognition of neutron stars as a physical reality rather than a brainchildof brilliant physicists. The idea of neutron stars was conceived in the 1930sfollowing Chadwicks discovery of the neutron1, the possibility of their birthin supernovae proposed [Baade & Zwicky 1934a], and the rst calculationof their masses and radii performed [Oppenheimer & Volko 1939]. After1939, there was little interest in neutron stars for about two decades be-cause (a) the original motivation for the study of neutron cores of stars aspossible stellar energy sources had vanished with the acceptance of stellarthermonuclear reactions as this source of energy, and, (b) it was mistakenlythought (see Sec. 2.1) that neutron stars would not be able to form in thelast stages of stellar evolution, because of a paradox which was not resolveduntil 1959 [Cameron 1959b]. A revival of interest in neutron stars in the1960s was inspired largely by the discovery of discrete X-ray sources outsidethe solar system in 1962 [Giacconi et al. 1962], because these X-rays werethought at rst to be thermal emission from newborn, hot neutron stars.Even in 1965, however, physicists writing papers on neutron stars tended tobe somewhat skeptical at times, as illustrated by the following quote fromBahcall and Wolf (1965b): In order to prevent the investigation of neutronstars from degenerating into a philosophical discussion, it is necessary toconcentrate on those aspects of the theory which are at least in principleconnected with observation.

1For a brief historical account of neutron stars before 1967, see Sec. 2.1.

1


2 Rotation and Accretion Powered Pulsars

Then, in August-September 1967, a doctoral student named JocelynBell at the University of Cambridge, who had just started operating anewly-constructed large dipole array (2048 dipole antennae spread overan area of 4.5 acres, operating at 81.5 MHz) for studying interplanetaryscintillation of compact radio sources with her thesis supervisor AnthonyHewish, found a bit of scru on the records [Bell-Burnell 1977] of herchart recorder. Radio telescopes are extremely sensitive to man-made elec-tromagnetic interference (from, e.g., automobile ignition or electrical farmequipment), but this did not look like interference; in addition, it came re-peatedly from the same part of the sky [Bell-Burnell 1977]. On a recorderwith a short response time 0.05 seconds, Bells scru or uctuationscame out on November 28, 1967, to be a train of astounding, preciselyperiodic, pulses at a period of 1.337 seconds. Fig. 1.1 shows these pulses.

Bell and Hewish investigated the pulses, painstakingly eliminating onepossibility after another2. The pulses were not from radar beams reectedo the moon. They were not from man-made satellites in unusual orbits,or deep space probes. In fact, since the pulses came about 4 minutes earlier

Fig. 1.1 Pulses from PSR 1919+21, the rst rotation-powered pulsar, discovered in1967. Note that this pulsar is referred to as CP 1919 in the gure, as per the customs ofthe time: CP stands for Cambridge pulsar. Reproduced with permission by the NobelFoundation from Hewish (1975). c 1975 The Nobel Foundation.

2Bells foresight and perseverance was instrumental to this, as the custom at the timein radio-astronomy circles was to dismiss such signals as man-made [Lyne & Graham-Smith 1990]. Bells own delightful account of the discovery [Bell-Burnell 1977] containsthese words: I contacted Tony Hewish who was teaching in an undergraduate laboratoryin Cambridge, and his rst reaction was that they must be man-made. This was avery sensible response under the circumstances, but due to a truly remarkable depth ofignorance I did not see why they could not be from a star. We may well wonder wherewe would be today but for Bells depth of ignorance.


The Discovery of Pulsars 3

each night, i.e., kept to the sidereal time (see Appendix A), their source hadto be far outside the solar system. An immediate and interesting possibilitywas that of communication signals deliberately sent out by an extraterres-trial civilizationthe proverbial little green men of early science-ction.But this alternative became improbable when careful Doppler-eect analy-sis of the period of the pulses revealed only our Earths motion around ourSun, and no trace of the further Doppler shift expected from the motionof the planet inhabited by these distant intelligent beings around their sun[Bell-Burnell 1977]. By late December, 1967, another pulsing radio sourcewas discovered in a dierent part of the sky, making it clear that the phe-nomenon being observed was indeed natural radio emission. Studies of thedispersion of the radio signals established that the pulses were coming fromwithin our galaxy, but outside the solar system.

The rst pulsar results were announced to the world in the journalNature on February 24, 1968 [Hewish et al. 1968], and in a seminar thatHewish gave in Cambridge a few days before that. Every astronomer inCambridge, so it seemed, came to the seminar, recalled Bell later, andtheir interest and excitement gave me a rst appreciation of the revolutionwe had started [Bell-Burnell 1977]. The universal question was: what wasthe nature of these cosmic pulsing radio sources? Since a period 1.3seconds was too short to be associated with the oscillations of a large,low-density, normal star, Hewish et al. immediately called attention tocompact, high-density stars like white dwarfs and neutron stars, which areamong the end-points of stellar evolution (see Chapter 6 and Appendix C),and whose frequencies of radial oscillation ran from about ten seconds to asmall fraction of a second. A signicant historical point, to which we comeback in Sec. 2.1, is that the pulsar-discovery paper [Hewish et al. 1968] con-sidered oscillations, not rotations, of compact stars as the probable causeof radio pulses. Radio astronomers all over the world now directed theirtelescopes and their attention to the discovery of more of these new puls-ing radio sources, or pulsars, as they came to be called. New pulsarswere found in profusion, in particular two fast-rotating pulsars in super-nova remnants: the Crab pulsar (with a period of 33 milliseconds) and theVela pulsar (with a period of 89 milliseconds). This opened up a new, richeld of study in astronomy and astrophysics. Identication of pulsars withrotating neutron stars, which was established within about a year and a half(see Sec. 1.3) following the seminal paper by Gold (1968), led to torrentialtheoretical work in the 1970s on every conceivable aspect of neutron stars:internal structure, surface properties, origin, and evolution (see Chapters



2 and 3). In addition, attempts to understand the properties of rotation-and accretion-powered pulsars (see Chapters 7, 8 and 9) led to fundamen-tal work on the electrodynamics and plasma (i.e., ionized gas) dynamicsnear rotating, magnetic, compact stars (see Chapter 10), and on theoriesof gravitation . In recognition of the profound impact on physics of the dis-covery of rotation-powered pulsars, Hewish shared3 the 1974 Nobel Prizein Physics: he was cited for his pioneering research in radio astrophysics,particularly the discovery of pulsars.

More than 1700 rotation-powered pulsars are known today: as it is notpracticable to list their essential parameters in this book, we refer the readerto the Australia Telescope National Facility (ATNF) Pulsar Catalogue,available at the website: http://www.atnf.csiro.au/research/pulsar/psrcat.A uniform way of naming rotation-powered pulsars has emerged now. Eachpulsar is identied by its position on the sky in terms of two angles, rightascension and declination (see Appendix A). The sequence of numbers spec-ifying the angles is preceded by the letters PSR (for Pulsar, obviously, re-placing older prexes like CP for Cambridge Pulsar, etc). Thus, the rstpulsar discovered [Hewish et al. 1968] is PSR 1919+21, the Crab pulsar isPSR 0531+21, the (Hulse-Taylor) binary pulsar is PSR 1913+16, the rstmillisecond pulsar discovered is PSR 1937+21, and so on. A slight modi-cation has arisen in recent years because of the transition from the original1950 coordinates to the current 2000 coordinates, which is explained in Ap-pendix A. To distinguish between the two, the letter J is appended to theprex PSR for the 2000 coordinates, and the letter B for the old 1950 ones,the nomenclature and the transformation between the two being given inAppendix A.

1.2 Accretion-Powered Pulsars: X-Ray Discovery

X-ray astronomy is a product of the space age, as X-rays from celestialsources are absorbed by the Earths atmosphere, and the only way of ob-serving them is through detectors sent above the atmosphere in space ve-hicles4. The subject started in 1949 with the detection of X-ray emissionfrom the Sun by Geiger counters own on a V-2 rocket captured at theend of the Second World War. Solar X-ray ux was so weak, however,

3With Ryle, who won it also for achievements in radio astrophysics which lie outsidethe scope of this book.

4High-altitude balloons can also be used to detect hard X-rays.



that observable X-ray emission from sources outside the solar system wasnot expected, based on the assumption that sun-like stars are the domi-nant X-ray sources in the universe. The absurdity of this assumption maybe obvious to us today, but the atmosphere of skepticism was so persua-sive in 1962 that the avowed purpose of the rocket ight of Giacconi andco-workers [Giacconi et al. 1962] had to be the detection of solar X-raysbouncing o the moon, while the real purpose was a search for extrasolarX-ray sources [Rossi 1973]. That Aerobee rocket ight became history: thecounters on board detected a very bright X-ray source in the constellationof Scorpius (in addition to the expected X-ray uorescence from the moon),giving birth to extrasolar X-ray astronomy, and showing, in Rossis (1973)words, the boundless wealth and complexity of nature. Between 1962 and1970, about 50 cosmic X-ray sources were discovered by rocket and balloonights: most of these were inside our Galaxy, but the rst extragalacticX-ray sourcethe giant elliptical galaxy M87 in the Virgo clusterwasalso discovered in the same era. It became customary to name the galacticX-ray sources by the constellation they were in, and the order in which theywere discovered. Thus the rst extrasolar source ever detected was calledScorpius X-1, or Sco X-1 for short, and so on.

Accretion is a word which etymologically means the process of growthor enlargement by various means, in particular by external addition oraccumulation, as by external parts or particles5. This meaning is slightlyadapted in astrophysics to signify the gravitational capture by a star ofthe matter surrounding it (either matter through which it is passing, suchas the interstellar medium or moecular clouds, or matter which is owingtoward it, such as that coming from its companion in a binary system),leading to an increase in the stars mass. Study of accretion by the Sunand similar stars during their passage through diuse gas, in contexts whichare no longer of interest to us, goes back to Eddington (1930), who mistak-enly thought that the geometrical size of the star was the capture radius,inside which all particles of matter (following trajectories somewhat fo-cused toward the star by its gravitational attraction) were intercepted bythe star and accreted by it. Hoyle and Lyttleton (1939a,b) introduced theessential physics, namely that at all gas densities of astrophysical interest,the collisions between particles greatly reduced their transverse momenta,much enhancing the eective capture radius, which depended basically onthe stellar mass M and the relative velocity v between the star and the

5From the Latin accretus, past participle of the verb accrescere, to increase.



streaming matter, scaling as GM/v2, and usually exceeding the stellar ra-dius by a large factor. This radius, now called the accretion radius, is akey parameter in the theory of accretion. The eects of the thermal energycontent of the streaming gas on the accretion radius were included later.

The potential signicance of accretion by compact stars was emphasizedin 1965 by Zeldovich and Novikov, before the discovery of either rotation-or accretion-powered pulsars. They clearly expressed their appreciation ofthe crucial point about accretion by neutron stars as follows: However, themain stimulus for the study of accretion lies in the energy released duringaccretion. Particles incident on the surface of a neutron star give up to (0.20.3) c2 of energy per gram, which is much more than can be obtained fromnuclear reactions. Zeldovich and Novikov (1965) then went on to makethe prophetic remark that Connected with the accretion phenomenon isthe very possibility of observing neutron and cooled stars... (cooled starswas the name these authors used for black holes). As we know today, the1962 discovery of Sco X-1 had, in fact, been the rst clear detection ofa neutron star powered by accretion. But this was not realized in 1965.The 1967 discovery of radio pulsars, powered by rotation, not accretion,and their rapid identication with neutron stars led to the rst acceptanceof the reality of these stars (see Sec. 1.1). The 1971 discovery of binaryX-ray pulsars (see below) and their identication with accreting neutronstars nally made us aware of the abundant occurrence in nature of thephenomenon that Zeldovich and Novikov had speculated on in 1965. Forobjects which are candidates for being black holes, emission powered byaccretion remains our best observational probe to this day.

An optical counterpart for Sco X-1 was found in 1966 [Sandage etal. 1966], a faint, blue star with excess ultraviolet emission, somewhat re-sembling an old nova. Novae were well-known to be binary stars, withevidence for gas streams owing between the two stars. In 1967, Shklovskiiproposed that Sco X-1 is a neutron star forming a comparatively massivecomponent of a close binary system, and that a stream of gas owingout of the second component is permanently incident on the neutron star,X-ray emission occurring from the hot gas in the stream. This pioneering6

6Curiously, in a brief 1966 note, Zeldovich and Guseynov had advocated the use ofknown single-line spectroscopic binaries to search for black holes and neutron stars, argu-ing that the optically-unseen companions in some of these binaries could be such objects.They had also made a casual comment that gas moving in the strong gravitational eldof black holes could show spectral features in the X-rays. But these authors had notdescribed sources powered by accretion, emitting primarily X-rays. This is amazing inthe context of the 1965 Zeldovich-Novikov work on accretion described in the previousparagraph.

February 8, 2007 15:39 WSPC/Book Trim Size for 9in x 6in pranab


scenario contained most of the essential ingredients of what later becamethe standard model of compact X-ray sources and accretion-powered pul-sars, namely, (1) a neutron star or a black hole forming a binary stellarsystem with a normal companion star still burning thermonuclear fuel,(2) the companion transferring mass to the compact object by one or moreof several possible mechanisms, and, (3) the gravitational energy releasedby matter accreting onto the compact object converted into electromagneticradiation, primarily at X-ray wavelengths.

In 1969, Zeldovich and Shakura published the rst quantitative, de-tailed calculation of the process of spherical accretion by neutron stars, andthe spectrum of the X-rays emitted in the process, neglecting the magneticeld of the star. These authors started their paper with the remarkablestatement Neutron stars were born at the tip of the theoreticians pen over30 years ago7, but a convincing identication of such a star has yet to bemade, although their paper was published in Russian in March-April 1969,by which time the rotating neutron-star model of radio pulsars was becom-ing accepted, as we described in Sec. 1.1. Zeldovich and Shakura consideredvarious mechanisms for deceleration and stopping of matter near the stellarsurface, showing that the emergent X-ray spectrum depended both on thismechanism and on the accretion rate. The calculated spectra, which dif-fered considerably from the Planck spectrum of a black body (particularlywhen the deceleration mechanism was collective plasma oscillations), wascompared by them with the X-ray spectrum of Sco X-1 known at the time,after which they stated that only through such comparisons might one beable to ascertain whether the observed point X-ray sources are in fact neu-tron stars, and whether neutron stars are entitled to pass from the realm ofhypothetical objects into the class of reliably identied stars. Apparently,these authors did not believe that rotation-powered pulsars already entitledneutron stars to join the class of reliably identied stars; the verdict ofhistory turned out to be exactly contrary.

X-ray astronomy really came into its own in 1970 with the launch ofNASAs rst astronomy satellite Uhuru8 totally dedicated to surveying theX-ray sky in the energy range 220 keV, obtaining accurate positions (for itstime) of the X-ray sources, and studying their spectra and time-variability.Uhuru discovered some 300 X-ray sources during its twenty-seven month

7See footnote 1.8A Swahili word meaning freedom: the satellite was launched o the coast of Kenya,

on the independence day of that country.



life-span, establishing a new branch of astronomy which was subsequentlybrought to its full glory of abundance and maturity by successive genera-tions of X-ray satellites. In January 1971, about a month into the Uhurumission, Giacconi and his co-authors noticed that a previously known X-raysource in the constellation of Centaurus, Cen X-3, was exhibiting periodicpulses. Based on the Uhuru data obtained in Januaray and April, 1971,these authors obtained a pulse period of 4.84 seconds, and published theirresult in July, 1971 [Giacconi et al. 1971]. This was the discovery of the rstX-ray pulsar: Fig. 1.2 shows the Cen X-3 pulses. A second X-ray pulsar,Her X-1, was discovered in the constellation of Hercules by the end of 1971:it had a period of 1.24 second, and became one of the most widely studiedaccretion-powered pulsars [Tananbaum et al. 1972].

The binary nature of Cen X-3 soon became clear from the X-ray datagathered by Uhuru between January and December, 1971: the source un-derwent regular changes in intensity between two distinct levels with aperiod of 2 days, and its pulse-period underwent sinusoidal variationswith the same period, correlated with the intensity variations [Schreier etal. 1972], as shown in Fig. 1.3.

Schreier et al. wrote: We interpret this eect as due to an occultingbinary system. The changes in intensity are then due to occultation of theX-ray source by a large massive companion, and the sinusoidal variationsin the period of the 4.8 s pulsations are due to Doppler eect. The orbitalperiod of the X-ray source is 2.08712 0.00004 days.

This interpretation became the standard picture of binary X-ray pul-sars. The large massive companion in Cen X-3 was optically identiedby Krzeminski (1973) to be an evolved supergiant star of spectral class lateO (see Appendix A): it is now called Krzeminskis star, or V779 Cen. Sim-ilar results followed quickly for Her X-1, with a binary period of 1.7 days

Fig. 1.2 Pulses from Cen X-3, the rst accretion-powered pulsar, discovered in 1971.Reproduced by permission of the AAS from Schreier et al. (1972): see Bibliography.



Fig. 1.3 Orbital variation of the pulse-period and intensity of Cen X-3, indicating itsbinary nature. Reproduced by permission of the AAS from Schreier et al. (1972): seeBibliography.

[Tananbaum et al. 1972], and a relatively low-mass evolved (spectral classlate A / early F) optical companion named HZ Her.

Within about a year of their discovery, the X-ray emitting stars in thesebinary X-ray pulsars were argued to be neutron stars (see Sec. 1.3.2 fordetails), largely carrying over the expertise obtained a few years earlier indebating the case of rotation-powered pulsars. The idea that they werepowered by accretion onto the neutron stars also came naturally [Pringle &Rees 1972; Davidson & Ostriker 1973; Lamb et al. 1973] in the wake of thediscussion of accreting neutron stars which we have summarized above, andwhich had existed in the scientic literature for about seven years by then.Thus, the pioneering Shklovskii (1967) scenario for Sco X-1 blossomed intoa full-edged model for accretion-powered X-ray pulsars, which has becomeuniversally accepted today. Ironically, the binary nature of Sco X-1 itselfwas not established until 1975, and no X-ray pulses have yet been detectedfrom this source.

About 110 accretion-powered pulsars are known now, their periods rang-ing from 1.7 milliseconds to 9860 seconds: some of their essential param-eters are lised in Table E.1. These form a subset of binary, accretion-powered, compact X-ray sources known today from several generation of



X-ray satellites over the last three decades, starting with Uhuru, and com-ing all the way to the current satellites RXTE, BeppoSAX , Chandra, andXMM-Newton. It is no longer convenient to list them by the constellations,and the modern nomenclature system for X-ray sources closely follows thatfor rotation-powered pulsars (see Sec. 1.1). The position on the sky is spec-ied by right ascension and declination, and these numbers are precededby an abbreviation for (usually) the name of the satellite that generatedthe catalog of X-ray sources (and, occasionally, the version of that catalog).Thus, Cen X-3, the rst accretion-powered pulsar discovered, was named 4U1119-60 in the 4th Uhuru catalog, and the recently-discovered 2.5 ms pulsar(which we discuss later at length because of its importance as an evolution-ary link; see Sec. 6.5.4) was named SAX J1808.4-3658 or XTE J1808-359after the satellites BeppoSAX and RXTE respectively. As position deter-minations become more accurate with succeeding generations of satellites,the numbers change slightly, and more signicant digits are added. An al-ternative earlier system of using galactic co-ordintes (see Appendix A), inwhich the numbers were galactic longitude and latitude, preceded by theletters GX, has also been largely superseded. We may use it in this bookoccasionally for historical reasons.

In recognition of his pioneering eorts in X-ray astronomy, his monu-mental eorts in developing this eld to its present level of advancement,and his direct involvement in many of the nest discoveries in this eld (in-cluding that of accretion-powered pulsars, which we are discussing in thisbook), Giacconi shared9 the 2002 Nobel Prize in Physics: he was cited forpioneering contributions to astrophysics, which have led to the discoveryof cosmic X-ray sources.

1.3 Pulsars as Neutron Stars

1.3.1 Rotation-Powered Pulsars

In a Nature paper published almost exactly three months after the Hewishet al. (1968) discovery paper of radio pulsars, Gold (1968) made the pio-neering, denitive suggestion that these objects were rotating neutron stars,the pulse period being the rotation period of the neutron star10. Over a

9With Davis and Koshiba, who won it for their work in neutrino astrophysics.10In 1967, before the discovery of pulsars, Pacini had already described the emission

of electromagnetic radiation from rotating, magnetized neutron stars in a related butdierent context, which is discussed in Sec. 2.1.



duration of about a year, this suggestion took hold, grew in detail as moreproperties of radio pulsars were established, and eclipsed all other sugges-tions to become what is now the universally-accepted physical model ofpulsars. Gold initially (1968) appealed to the extreme stability of the peri-ods (1 in 108) to argue the case for a massive, compact, rotating object (asopposed to stellar vibrations, or oscillations of plasma congurations), andpredicted a slowing down of the observed repetition frequencies, whichwas subsequently conrmed. The fact that the rate of loss of rotationalenergy of the Crab pulsar was roughly the same as that required to powerthe Crab nebula (essentially repeating the 1967 scenario of Pacini), as Goldargued in a subsequent (1969) paper, clinched the issue.

We now summarize the arguments in favor of the rotating neutron-starmodel and against the competing models, as they stand today, rather thanfollowing the historical development. The crucial observational point usedin these arguments are that (1) the periods of the radio pulsars are so short( milliseconds to several seconds), (2) radio pulsars are exremely accurateclocks (measurements upto precisions 1 part in 1014 have been possible),and (3) the periods of radio pulsars always increase slowly and gradually,except for occasional, discontinuous decreases or glitches (which we treatin some detail in Chapters 7 and 8). The rst point implies that rotationsor vibrations of compact, dense stars, e.g., white dwarfs or neutron stars,(or orbital motions around such stars) must be involved rather than those oflarge, low-density, normal stars, which occur on much longer timescales,since all these characteristic times scale roughly as 1/2 (see below), where is the average density of the star. Further, since the shortest known periodamong radio pulsars is 1.6 ms (see Sec. (1.6)), and light travels 500 kmin this time, this distance must be an upper limit to the size of the emittingregion. Because of the great accuracy of the pulsar clock mechanism, wecan argue that this must also be a rough upper limit to the size of the staritself, since it is basically impossible to justify such accuracy for an emittingregion much smaller than the star and unrelated to the size of the latter:the two must be closely coupled.

Of the various compact-star phenomena, consider rst oscillations ofwhite dwarfs or neutron stars. Periods of fundamental modes of oscillationsof white dwarfs are in the range 210 s, and those of neutron stars are inthe range 110 ms [Meltzer & Thorne 1966]. It is thus clear that a singleclass of objects will not account for the entire range of pulsar periods.White-dwarf oscillations at higher harmonics will, of course, have shorterperiods than above, but it requires special circumstances to excite these



harmonics, and the sharpness of the pulses (one of their major observedcharacteristics) will then be destroyed by mode-mixing due to any smallnonlinearity (always present in real stars). In any case, the periods of well-known pulsars like Crab (33 ms) and Vela (89 ms), which lie in the gapbetween the above ranges for white dwarfs and neutron stars, are almostimpossible to account for in this way. Finally, if pulses were emitted bythis mechanism, the energy-loss from the oscillating system would lead to adecrease in the pulse period, the exact opposite of what is observed. Thus,oscillations are ruled out.

Consider next rotating white dwarfs. The shortest period P , or thehighest angular velocity , at which a star of mass M and radius R canrotate without being torn apart by centrifugal forces is given by

2R GMR2

. (1.1)

In terms of the average density of the star, Eq. (1.1) for the breakupangular velocity can be recast in the approximate form

(G)1/2, (1.2)

which is a very useful, general result for a variety of dynamical timescalesassociated with a star. White dwarfs with 107108 g cm3 thereforehave breakup rotation periods P = 2/ 110 s, a result which rules outrotating white dwarfs. Note also the similarity between this range of periodsand that given above for white-dwarf oscillations. This is a consequence ofthe virial theorem [Chandrasekhar 1935], according to which the period ofthe fundamental oscillation mode of a star is of the same order of magnitudeas its breakup period11 [Gold 1968], and so given roughly by Eq. (1.2), aswe anticipated above.

Now consider orbital motion around compact stars, either (a) of a smallobject in a close orbit around the star, or, (b) of a close binary systemof compact stars around each other. In the former case, balancing thecentrifugal force against the gravitational attraction of the star again yieldsan equation like Eq. (1.1) with the orbital radius r replacing the stellarradius R, and since r R for a close orbit, we recover Eq. (1.2). In thelatter case, we again get an equation of the form 1.1, with M replaced bythe total mass (= 2M for two identical stars of mass M) of the system, and

11Elastic restoring forces, which augment the usual gravitational restoring forces insolid or partly-solid stars like white dwarfs and neutron stars, shorten the oscillationperiods somewhat, but the scaling with density in Eq. (1.2) is still roughly valid.



the orbital radius r replacing the stellar radius R as before. With r 2Rfor two identical stars of radius R, we again roughly recover Eq. (1.2).This further illustrates the universal signicance of the dynamical timescale.The earlier arguments given for the range of periods thus roughly apply toorbital motion as well. Additional, severe diculties arise. First, such aclose binary system containing a neutron star would be a copious emitterof gravitational waves, the two objects spiralling into each other in a veryshort time due to energy-loss through these waves. This timescale for abinary system of two neutron stars, each of mass M, and a binary periodof P is roughly

Tspiral 103(

P

1 s

)8/3yr, (1.3)

an absurdly short lifetime. In reality, pulsar periods are observed to changetypically on timescales 107 yr. Furthermore, orbit decay due to gravi-tational radiation would cause the period to decrease, contrary to what isobserved12. For a planet or satellite of mass M in a close orbit around aneutron star of mass M ( 1), the spiral-in timescale is again given byEq. (1.3), but with an extra factor of 1 included in its right-hand side.Observed period-change timescales of pulsars would thus imply tiny masses( 1010) of the satellite. Even under the unlikely assumption that suchan insignicant object could produce the observed radio pulses, we imme-diately encounter the problem that such a satellite would be pulverized bythe enormous tidal forces in the strong gravitational eld close to a neutronstar. The high radiation eld of the pulsar only adds to the diculty, asit would tend to melt or evaporate the satellite13. This rules out orbitalperiods.

Finally, consider rotation periods of neutron stars. As applications ofEq. (1.2) show, neutron stars with 1014 g cm3 have breakup rota-tion periods 1 ms. Thus, rotation periods > 1 ms are all possible forneutron stars; indeed, the shortest pulse period 1.6 ms known for ra-dio pulsars is rather close to (but, of course, longer than) the numerical

12Double neutron-star binaries are known, of course, the classic example being PSR1913+16, the famous Hulse-Taylor pulsar discovered in 1974. The pulse period of thispulsar is P 59 ms, the orbital period is Porb 7.7 hours, and the rate of decrease ofthe latter due to gravitational radiation has been measured accurately. See Sec. 7.2.4for a full discussion.

13Indeed, the radiation eld of the binary pulsar PSR 1957+20, the so-called blackwidow pulsar, is believed to be slowly evaporating its low-mass white dwarf companion.See Sec. 6.5.5 for details.



value of the breakup period calculated from detailed neutron-star models.All the other objections against the alternative scenarios detailed abovealso do not apply to the rotating neutron-star scenario. Thus the rotatingneutron-star model survives as the only viable scenario. The accuracy ofthe pulsar clock relates well to the rotation of a massive, compact star witha solid surface, the lengthening of the pulse period to the slowing down ofthe rotation (or spindown, as it is called) due to loss of rotational energythrough electromagnetic emission at the rate expected for canonical mag-netic elds ( 1012 G) of neutron stars, and the above energy loss-rate tothe (apparent) energy supply-rate to canonical supernova remnants like theCrab nebula. Many details of the observed features of pulsars, which wedescribe in later chapters, have generally found satisfactory explanations inthe rotating neutron-star model, although our understanding of the pulse-formation mechanism is still not as good as it could be (see Chapter 11).Because of the universal agreement today that the pulsars originally (andstill being) discovered at radio frequencies are powered by the rotationalenergy of neutron stars, we shall refer to these as rotation-powered pulsarsthroughout the rest of the book. As these pulsars sometimes have emis-sion in other wavebands (optical, X-ray, -ray,...) as well, a categorizationwhich is more meaningful physically is in terms of their source of energy,which sets them apart from the other class of pulsars with which we areconcerned in this book.

1.3.2 Accretion-Powered Pulsars

Let us summarize the arguments identifying binary X-ray pulsars as ro-tating, accreting neutron stars, again as they would stand today, ratherthan historically. Arguments favoring rotating neutron stars follow muchthe same route as those used for rotation-powered pulsars, using the ob-servational facts that (1) the pulse periods range from 2 milliseconds to 104 seconds, and, (2) the periods are quite stable after correction for theDoppler eect due to binary motion [Davidson & Ostriker 1973].

The rst point implies that, again, the only kind of stars capable ofhaving dynamical phenomena over this whole range of periods are compactstarswhite dwarfs, neutron stars, and black holesand, of these, the onlysingle class able to account for the entire range is that of neutron stars. Thisfollows from the arguments about dynamical timescales in terms of the av-erage stellar density summarized in the previous subsection, particularlyEq. (1.2). Rotating black holes are ruled out because non-axisymmetric



black holes, such as would be required for producing pulses, can have onlya transient existence, if any. For neutron stars, only rotation can cover thisentire range of periods, oscillation periods being far too limited in range( 110 ms), as explained in the previous subsection. The second point,the stability of the pulses, also argues in favor of rotations rather than os-cillations as the basic cause of modulation at the pulse period. Again, theargument is that the rotation of a compact, massive body, isolated exceptfor the gravitational inuence of its binary companion, is far stabler by na-ture than a train of oscillations excited by any conceivable mechanism. Bythe same argument, plasma oscillations are ruled out even more strongly.Finally, orbital periods around neutron stars are untenable because (a) therange of observed periods cannot be accounted for, and, (b) such periodswould change on absurdly short timescales due to emission of gravitationalradiation (see Eq. (1.3)). Thus, we are left with rotation periods of neutronstars as the only viable model.

That the source of power for the X-ray emission from binary X-raypulsars could not be the rotational energy of the neutron stars was clearfrom the beginning, since the typical observed X-ray luminosities of thesesources ( 1037 erg s1) were far too high to be accounted for in thismanner, even if we assumed, for the sake of argument, that these sourceswere all spinning down. This is easily seen by noting that the slowing-down time of a binary X-ray pulsar of period P and luminosity L is [Lambet al. 1973]:

Tslow Erot/L 60I45(

P

1 s

)2L137 yr, (1.4)

where Erot = 12I2 is the rotational energy of the neutron star, I being

its moment of inertia. In Eq. (1.4), L37 is L in units of 1037 erg s1, andI45 is I in units of 1045 gm cm2: this convention of expressing a physicalvariable in units of its typical order of magnitude in a given problem isstandard in astrophysics, and we shall be using it throughout this book.This timescale is absurdly small for the vast majority of the sources withperiods in the range 11000 seconds. Actually, periods of binary X-raypulsars show a complex behavior with time, consisting of (a) a secularcomponent which is spinup (i.e., P decreasing with time) for some pulsars,spindown (i.e., P increasing with time) for some, and and no signicantchange for yet others, superposed on (b) a uctuating noise component.



So the spindown argument is irrelevant to these pulsars as a generalclass14.

Accretion remains the only viable source of power to the class of binaryX-ray pulsars as a whole [Pringle & Rees 1972; Davidson & Ostriker 1973;Lamb et al. 1973]. The vast quantity of energy released by matter fallinginto the deep gravitational potential wells of neutron stars of radii R 106cm and masses M M, amounting to = GM/R 1020 ergs per gram ofaccreted matter, i.e., about a tenth of its rest-mass energy ( 0.1c2), makesaccretion an ideal source of power, as was recognized by Zeldovich and co-authors long ago [Zeldovich & Novikov 1965; Zeldovich & Shakura 1969].To generate X-ray luminosities 1037 erg s1 typical of bright, binary X-ray pulsars, accretion rates 1017 gm s1 109 M yr1 are thereforerequired. The binary companion of the neutron star is a natural, and morethan adequate, source of supply of the matter to be accreted, as stars losemass through a variety of processes during the course of their evolution,particularly if they are in binary systems. Massive, young, O and B starsdrive sizable outows of matter called stellar winds, and rapidly-rotatingBe stars shed, in addition, rings of matter from time to time. At severalstages of their thermonuclear evolution, stars expand by large factors tobecome giants or supergiants (see Appendix C): if they are in closebinaries, they lose mass copiously during these stages as they attempt toexpand beyond their limiting gravitational equipotentials, which are calledRoche lobes (see Appendix B). Finally, the X-rays emitted by the neutronstar can heat the companions surface, causing mass loss in a so-called self-excited wind. All this matter is available for accretion by the neutron star,which the latter can do through several types of accretion ows, rangingfrom spherical inow (for matter with negligible angular momentum relativeto the neutron star) to inow through an accretion disk (for matter withso much angular momentum that it can go into orbits around the neutronstar obeying Keplers laws). We discuss these processes in detail in laterchapters.

14Historically, the prototype binary X-ray pulsars Her X-1 and Cen X-3, which wereused to start building models for these pulsars, were soon found to have secular spinup,at which point rotation-power for emission became irrelevant and was forgotten. In thelast few years, this chapter has been reopened for a small subclass of X-ray pulsars withperiods 512 seconds: these show secular spindown, and their binary nature has neverbeen really established. The idea that these Anomalous X-ray Pulsars (AXPs) couldbe rotating neutron stars with unusually, but not impossibly, high magnetic elds, andthat the observed spindown is by the same electromagnetic torques that brake rotation-powered pulsars, is being explored. We return to AXPs in Chapter 9.



There is universal agreement today that binary X-ray pulsars are pow-ered by accretion. Additional supporting evidence has accumulated overthe years, for example from rates of secular spinup of binary X-ray pulsars[Lamb et al. 1973; Rappaprt & Joss 1977], and from strong evidence for thepresence of accretion disks in these and similar binaries through probes atother wavelengths, e.g., optical. In close analogy with what we describedfor the radio pulsars in the previous subsection, all major details of theproperties of binary X-ray pulsars have found satisfactory explanation inthe framework of the accreting neutron-star model. A particular triumphof this model has been the natural explanation of the evolutionary link be-tween young, high magnetic-eld, rotation-powered pulsars (e.g., the Crabpulsar) and old, low magnetic-eld, recycled rotation-powered pulsarsspun up to millisecond periods, whose discovery started in 1982 with the 1.6 millisecond pulsar PSR 1937+21. As this is a dominant theme of thisbook, we shall return to evolutionary questions repeatedly, particularly inChapter 6. Finally, in analogy with the nomenclature introduced in theprevious subsection, we shall refer to binary X-ray pulsars as accretion-powered pulsars throughout the rest of the book, despite the fact that theywere, and are still being, discovered by X-ray satellites. The reason, again,is that they have emission at other wavelengthsoptical, UV, IR, radio,and the likeand studies at some of the other wavelengths is often crucial(such as optical studies for the clarication of binary properties), whereastheir source of energy is their key physical feature distinguishing them fromthe rotation-powered pulsars described earlier.

1.4 Powering Pulsars by Rotation and Accretion

1.4.1 Rotation Power

Consider a neutron star of moment of inertia I, rotating with an angularvelocity , so that its rotational energy is given by

Erot =12I2 2 1046I45P2(s) erg, (1.5)

where I45 is I in units of 1045 g cm2, as before, and P = 2/ is therotation period. If the stars spin is changing at a rate , its rotationalenergy is changing at a rate

Erot = I +12I2 I = 4 1032I45P3(s)P14 erg s1. (1.6)



Here, I is the rate of change, if any, of the stars moment of inertia duringthe spin-change process, due to a variety of possible reasons we discusslater, and the approximate version of the right-hand side of Eq. (1.6) isobtained by neglecting the inertia-change term, which we shall be usingmost frequently. Loss of rotational energy is then inferred from spindown, < 0, or, equivalently, P > 0, as is the case for rotation-powered pulsars.In Eq. (1.6), P14 is P in units of 1014 second per second (s s1), a typicalorder of magnitude for rotation-powered pulsars, roughly comparable to ananosecond per day (ns d1), which is also used as a unit for P .

How does such a pulsar convert its rotational energy into electromag-netic energy and radiate it? This was the problem addressed by Pacini(1967) and Gold (1969). Young neutron stars have very large magneticelds (B > 1012 G), the origins of which we discuss in Chapter 13. As arst approximation, we can represent this eld by that of a magnetic dipole,since any magnetic eld conguration in the star can be represented by acombination of magnetic multipoles (see, e.g., Jackson 1975), as seen by anoutside observer, and the dipole component is dominant when the observeris suciently far away from the star. Consider a neutron star of magneticdipole moment oriented at an angle with respect to its rotation axis.(The magnetic eld at the stellar surface of radius R is given roughly byBs /R3, and the eld at either of the magnetic poles is given exactlyby Bp = 2/R3, so that typical orders of magnitude for young pulsars areBs 1012 G and 1030 G cm3.) This rotating oblique dipole appearsas a time-varying one from the point of view of a distant observer [Jackson1975], and so radiates energy at a rate

Emdr =224 sin2

3c3 1031B212R66P4(s) sin2 erg s1, (1.7)

as magnetic dipole radiation at a frequency . Here, B12 is the polar eldBp in units of 1012 G, and R6 is R in units of 106 cm. A comparison ofEqs. (1.6) and (1.7) shows how readily magnetic dipole radiation accountsfor the spindown rates of rotation-powered pulsars. In fact, the observedspindown rate of a given pulsar can be used to estimate its magnetic eldthrough the relation we obtain by combining Eqs. (1.6) and (1.7), namely,

B12 6

I45P14P (s)

R36 sin, (1.8)



and is the standard method for doing so. The angle of obliquity is, ofcourse, not known from the spindown observations alone, but we can either(a) set sin = 1 in Eq. (1.8), and interpret the result as the magnetic elddue to that component of the magnetic moment which is perpendicular tothe rotation axis, or, (b) use a statistically averaged value of sin expectedfor a collection of pulsars, unless we have clues to the value of fromother observation, e.g., that of polarization. Note that the magnetic eldstrength inferred in this way scales as

PP . Thus, because of its short

period (P 33 ms) and high spindown rate (P14 42), the Crab pulsarhas a higher energy, Erot 2 1049 erg, and higher energy-loss rate,Erot 51038 erg s1, than the canonical values appearing in Eqs. (1.5)and (1.6), but the inferred magnetic eld strength, B12 7, is canonical,since the factor of increase in P largely cancels the factor of decrease in P .In this argument, we have used the canonical estimate 1 for I45 and R6, aswe shall do throughout this book. It was the remarkable agreement betweenthe above estimate of Erot for the Crab pulsar and the inferred energyrequirements of the Crab nebula that proved instrumental for identifyingrotation-powered pulsars with neutron stars [Gold 1969].

Although Erot is the rate of loss of rotational energy from the neutronstar, sometimes called the spindown power, only a small fraction of it ap-pears in the radio pulses. For example, the fraction of the above spindownpower for the Crab pulsar that goes into radio pulses is 107, i.e., ab-solutely tiny, as is the typical value 105 characteristic of the knownpopulation of rotation-powered pulsars. Where does the bulk of the energygo? It is generally thought that much of this energy goes into acceleratingcharged particles15 to high energies, which then deposit their energy in thesurrounding nebula (supernova remnant) in the case of young pulsars likethe Crab pulsar to make the nebula a strong emitter of electromagneticradiation (radio, optical, X-ray...). Note also that the magnetic dipole ra-diation we discussed above is clearly not what we observe as pulses fromthe pulsars, since this radiation is emitted at the stellar rotation frequency( /2 11000 Hz), and has an approximately sinusoidal pulse shape.The observed pulses are modulations at radio frequencies ( 10 MHz10 GHz), or even higher frequencies of electromagnetic radiation, and theirshapes resemble narrow spikes, covering a small fraction ( 102101) ofthe total pulse period (see Chapetr 7). Thus, the magnetic dipole radiationis merely a process that taps the reservoir of the neutron stars rotational

15-ray pulses, which have been detected from a small number of rotation-poweredpulsars so far, have been observed to carry upto 0.3 of the spindown power.



energy; the pulse-generation process is far more complex, to which we re-turn in Chapter 11. Whatever this process is, it must be clear that it isvery strongly tied to the stellar rotation, since the pulses come with suchstable periodicity, and it produces electromagnetic radiation in extremelysharp, narrow beams (unlike magnetic dipole radiation). Charged particlesmoving along the stellar magnetic eld lines, which are corotating with thestar, are therefore an essential ingredient of any such process, as we shallsee. In the universally-accepted model of rotation-powered pulsars, such aprocess generates a very narrow beam of radiation rotating with the star, asshown schematically in Fig. 1.4, and when this beam sweeps by the earth,rather like a lighthouse beam, we see the pulses.

1.4.2 Accretion Power

Consider a neutron star of mass M and radius R accreting mass at arate M . Since each unit of accreted mass releases an amount of gravita-tional potential energy GM/R on reaching the stellar surface, the lumnosity(i.e., energy per unit time) generated by the accretion process is given by[Zeldovich & Shakura 1969]

Fig. 1.4 Basic model of rotation-powered pulsars. Reproduced by permission of IEEEfrom Taylor (1991), Proc. IEEE, 79, 1054. c 1991 IEEE.



L =GMM

R 1.3 1037M17(M/M)R16 erg s1, (1.9)

where M17 is M in units of 1017 g s1. Thus, accretion rates of brightaccretion-powered pulsars with L 1037 erg s1 are 1017 g s1 109M yr1, as we mentioned before, and the accreted mass comesfrom the binary companion of the neutron star.

How, again, is this energy converted into electromagnetic radiationprimarily X-raysthat we see in the pulses? The basic process is thermalemission from the heated stellar surface and the hot plasma (ionized mat-ter) close to it, the source of heat being the gravitational potential energyreleased as described above, as the accreting matter falling towards the stel-lar surface is brought to a halt by a variety of dissipative processes. As arst approximation, we can treat this as blackbody emission (complexitiesdue to radiative transfer in the presence of intense neutron-star magneticelds are discussed in Chapter 11) at an eective temperature T from thatpart of the stellar surface which receives the accreting matter, and whosearea is A. By Stefans law, the luminosity is then

L = AT 4, (1.10)

which must equal that given by Eq. (1.9) in a steady state. Here, is theStefan-Boltzmann constant. What is the accretion area A on the neu-tron star? Note rst that if we equated this to the entire stellar sur-face area, As = 4R2, we would obtain the case of spherical accretiononto the neutron star, such as was considered by Zeldovich and Shakura(1969). While interesting in its own right, such a situation does not pro-duce pulses (as must be obvious since the stellar rotation produces nomodulation in the outgoing radiation) and so is irrelevant for accretion-powered pulsars, in which the intense magnetic eld of the neutron starchannels the ow of the accreting matter (which, being completely ion-ized, must move along the magnetic eld lines, a situation which is calledux-frozen ow; see, e.g., Jackson 1975) towards the magnetic poles ofthe star [Davidson & Ostriker 1973; Lamb et al. 1973]. Matter thus ac-cretes on two polar caps at the stellar surface, producing two hot spotswhich emit radiation, as shown in Fig. 1.5: matter raining down on ahot spot forms a column-like structure called accretion column16. Ifthe magnetic axis of the star is oblique with respect to its rotation axis

16Also called accretion funnel because of its shape on a larger scale: see Fig. 1.5.



Fig. 1.5 Basic model of accretion-powered pulsars. Reproduced by permission of theAAS from Lamb et al. (1973): see Bibliography.

(as for rotation-powered pulsars), this is a natural mechanism for mod-ulating the emitted radiation at the stellar rotation frequency, as ob-served. The area of these polar caps, which is what enters into Eq. (1.10),can be estimated by the following method [Davidson & Ostriker 1973;Lamb et al. 1973].

Far away from the neutron star, the ow of the accreting matter is notinuenced by the stellar magnetic eld, and is determined, rather, by thedetails of how it is lost by the companion star, and by the dynamics ofthe binary system (see Chapter 10). Close to the neutron star, on theother hand, the ow is completely dominated by the stellar magnetic eld,matter falling onto the star along the eld lines: this region is called themagnetosphere of the neutron star. Of crucial importance is the transitionboundary between the magnetospneric ow and the outer ow: its locationcan be calculated from various physical arguments involving the accretingmatter and the magnetic eld, which we discuss in Chapter 10. All thesearguments give a boundary radius rm which depends basically on the accre-tion rate M and the surface magnetic eld Bs of the neutron star, as may



be expected: for typical values appropriate for bright accretion-poweredpulsars (M17 1, B12 1), rm 108 cm (see Chapter 10 for detailedexpressions for various estimates of rm). The size of the polar cap is thenroughly estimated by the simple, but useful, geometrical argument that theeld lines passing through the outer edge of this cap must be the the outer-most stellar eld lines that can possibly close within rm (see Fig. 1.5). As arst approximation, we assume a dipolar magnetic eld for the star (as wedid for rotation-powered pulsars, and as we shall do throughout this book):the equation for the eld lines of a magnetic dipole in polar coordinates is[Jackson 1975]

r = r0 sin2 , (1.11)

where r0 is a eld line label, identifying the particular eld line by itsvalue of r on the equatorial plane of the dipole ( = /2). Since r0 = rmfor the outermost eld line we are concerned with here, the value of forthe footpoint of this eld line on the stellar surface (r = R) given fromEq. (1.11) by a (R/rm)1/2, which is also the semi-angle subtended bythe polar cap at the center of the star (see Fig. 1.5). Thus the total areaof the two polar caps (one at each pole of the star) is given by

Ap 2R22a = 2R3/rm 6R36(rm/108 cm)1 km2, (1.12)

which is much smaller than (in fact, only about 0.5 % of) the total surfacearea, As 1260 R26 km2, of the neutron star. Note that even if the stellareld conguration was a perfect dipole to begin with, the accretion processitself is likely to distort it, so that the above picture is only approximate.Nevertheless, this simple geometrical estimate, which can be regarded as anupper limit to the size of the hot spots, makes a good beginning, which canbe improved upon by detailed considerations of the accretion ow pattern(see Chapter 10).

The blackbody emission temperature of the hot spots can now be ob-tained by combining Eqs. (1.9), (1.10), and (1.12). The result,

T 4 107M1/417 (M/M)1/4R16 (rm/108 cm)1/4 K, (1.13)

readily shows why the bulk of the emission is in the X-rays, since the meanenergy of the photons emitted by a blackbody of this temperature is 2.7kT 9 keV for canonical values of the parameters in Eq. (1.13), k being



Boltzmanns constant17. Furthermore, since the blackbody temperature isonly a lower bound on the actual emission temperature, the latter may behigher. In chapter 11, we discuss how additional eects may modify theshape of the emergent spectrum of an accretion-powered pulsar from thatof a blackbody.

While a rotating hot spot does generate periodic modulations due toa varying angle of viewing, further complications are introduced into theanisotropy of the emergent radiation due to the stellar magnetic eld, sincethe opacity of matter in such elds is highly energy-dependent. The studyof the latter eect was pioneered by Lamb et al. (1973), who identied threeregimes of magnetic eects, depending on whether the stellar magnetic eldwas above, below, or comparable to a critical eld strength 2 1011 G.Far below the critical value, the magnetic eld has little eect on the opac-ity of the matter. Since the integrated density of matter, and so its opacity,is much greater along the accretion column than across it, the radiationemerges preferentially through the sides of the column, producing a fan-shaped beam. Far above this value, the magnetic eld drastically reducesthe opacity of matter, essentially making the accretion column transparent,so that the viewing-angle eect mentioned above produces the modulation.Finally, when the magnetic eld is

rotation and accretion powered pulsars - pranab ghosh

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