P1: MRM/SPH P2: MRM/UKS QC: MRM/UKS T1: MRM CB251-FM CB251/Tassoul December 8, 1999 15:50 Char Count= 0 STELLAR ROTATION JEAN-LOUIS TASSOUL Universite´ de Montreal´ v P1: MRM/SPH P2: MRM/UKS QC: MRM/UKS T1: MRM CB251-FM CB251/Tassoul December 8, 1999 15:50 Char Count= 0 PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http://www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain c Cambridge University Press 2000 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2000 Printed in the United States of America Typeface Times Roman 10.5/12.5 pt. and Gill Sans System LATEX2ε [TB] A catalog record for this book is available from the British Library. Library of Congress Cataloging in Publication Data Tassoul, Jean Louis. Stellar rotation / Jean-Louis Tassoul. p. cm. – (Cambridge astrophysics) ISBN 0-521-77218-4 1. Stars – Rotation. I. Title. II. Series. QB810.T36 2000 523.8 – dc21 99-16740 CIP ISBN 0 521 77218 4 hardback vi P1: MRM/SPH P2: MRM/UKS QC: MRM/UKS T1: MRM CB251-FM CB251/Tassoul December 8, 1999 15:50 Char Count= 0 Contents Preface page xiii 1 Observational basis 1 1.1 Historical development 1 1.2 The Sun 5 1.3 Single stars 11 1.4 Close binaries 16 1.5 Bibliographical notes 21 2 Rotating fluids 25 2.1 Introduction 25 2.2 The equations of fluid motion 25 2.3 The vorticity equation 30 2.4 Reynolds stresses and eddy viscosities 33 2.5 Applications to the Earth’s atmosphere 36 2.6 The wind-driven oceanic circulation 43 2.7 Barotropic and baroclinic instabilities 49 2.8 Self-gravitating fluid masses 55 2.9 Bibliographical notes 62 3 Rotating stars 65 3.1 Introduction 65 3.2 Basic concepts 66 3.3 Some tentative solutions 69 3.4 The dynamical instabilities 73 3.5 The thermal instabilities 82 3.6 The eddy–mean flow interaction 86 3.7 Bibliographical notes 89 4 Meridional circulation 93 4.1 Introduction 93 4.2 A frictionless solution 94 4.3 A consistent first-order solution 101 4.4 A consistent second-order solution 113 xi P1: MRM/SPH P2: MRM/UKS QC: MRM/UKS T1: MRM CB251-FM CB251/Tassoul December 8, 1999 15:50 Char Count= 0 xii Contents 4.5 Meridional circulation in a cooling white dwarf 118 4.6 Meridional circulation in a close-binary component 120 4.7 Meridional circulation in a magnetic star 126 4.8 Discussion 133 4.9 Bibliographical notes 135 5 Solar rotation 138 5.1 Introduction 138 5.2 Differential rotation in the convection zone 139 5.3 Meridional circulation in the radiative core 145 5.4 Spin-down of the solar interior 151 5.5 Discussion 158 5.6 Bibliographical notes 159 6 The early-type stars 162 6.1 Introduction 162 6.2 Main-sequence models 162 6.3 Axial rotation along the upper main sequence 172 6.4 Circulation, rotation, and diffusion 179 6.5 Rotation of evolved stars 182 6.6 Bibliographical notes 185 7 The late-type stars 190 7.1 Introduction 190 7.2 Schatzman’s braking mechanism 191 7.3 Rotation of T Tauri and cluster stars 194 7.4 Rotational evolution of low-mass stars 197 7.5 Bibliographical notes 204 8 Tidal interaction 207 8.1 Introduction 207 8.2 The tidal-torque mechanism 208 8.3 The resonance mechanism 214 8.4 The hydrodynamical mechanism 217 8.5 Contact binaries: The astrostrophic balance 230 8.6 Discussion 237 8.7 Bibliographical notes 240 Epilogue 245 Subject index 249 Author index 252 P1: MRM/SPH P2: MRM/UKS QC: MRM/UKS T1: MRM CB251-01 CB251/Tassoul November 13, 1999 16:27 Char Count= 0 1 Observational basis 1.1 Historical development The study of stellar rotation began at the turn of the seventeenth century, when sunspots were observed for the first time through a refracting telescope. Measurements of the westward motion of these spots across the solar disk were originally made by Johannes Fabricius, Galileo Galilei, Thomas Harriot, and Christopher Scheiner. The first public announcement of an observation came from Fabricius (1587–c. 1617), a 24-year old native of East Friesland, Germany. His pamphlet, De maculis in Sole observatis et apparente earum cum Sole conversione, bore the date of dedication June 13, 1611 and appeared in the Narratio in the fall of that year. Fabricius perceived that the changes in the motions of the spots across the solar disk might be the result of foreshortening, with the spots being situated on the surface of the rotating Sun. Unfortunately, from fear of adverse criticism, Fabricius expressed himself very timidly. His views opposed those of Scheiner, who suggested that the sunspots might be small planets revolving around an immaculate, nonrotating Sun. Galileo made public his own observations in Istoria e Dimostrazioni intorno alle Macchie Solari e loro Accidenti. In these three letters, written in 1612 and published in the following year, he presented a powerful case that sunspots must be dark markings on the surface of a rotating Sun. Foreshortening, he argued, caused these spots to appear to broaden and accelerate as they moved from the eastern side toward the disk center. The same effect made the sunspots seem to get thinner and slower as they moved toward the western side of the disk. Galileo also noticed that all spots moved across the solar disk at the same rate, making a crossing in about fourteen days, and that they all followed parallel paths. Obviously, these features would be highly improbable given the planetary hypothesis, which is also incompatible with the observed changes in the size and shape of sunspots. The planetary hypothesis, championed by Scheiner among others, was thus convinc- ingly refuted by Galileo. Eventually, Scheiner’s own observations led him to realize that the Sun rotates with an apparent period of about 27 days. To him also belongs the credit of determining with considerably more accuracy than Galileo the position of the Sun’s equatorial plane and the duration of its rotation. In particular, he showed that different sunspots gave different periods of rotation and, furthermore, that the spots farther from the Sun’s equator moved with a slower velocity. Scheiner published his collected ob- servations in 1630 in a volume entitled Rosa Ursina sive Sol, dedicated to the Duke of Orsini, who sponsored the work. (The title of the book derives from the badge of the Orsini family, which was a rose and a bear.) This was truly the first monograph on solar physics. 1 P1: MRM/SPH P2: MRM/UKS QC: MRM/UKS T1: MRM CB251-01 CB251/Tassoul November 13, 1999 16:27 Char Count= 0 2 Observational basis It is not until 1667 that any further significant discussion of stellar rotation was made. In that year the French astronomer Isma¨el Boulliaud (1605–1694) suggested that the variability in light of some stars (such as Mira Ceti) might be a direct consequence of axial rotation, with the rotating star showing alternately its bright (unspotted) and dark (spotted) hemispheres to the observer. This idea was popularized in Fontenelle’s Entretiens sur la pluralite´ des mondes – a highly successful introduction to astronomy that went through many revised editions during the period 1686–1742. To be specific, he noted “...that these fixed stars which have disappeared aren’t extinguished, that these are really only half-suns. In other words they have one half dark and the other lighted, and since they turn on themselves, they sometimes show us the luminous half and then we see them sometimes half dark, and then we don’t see them at all.” Although this explanation for the variable stars did not withstand the passage of time, it is nevertheless worth mentioning because it shows the interest that stellar rotation has aroused since its inception. As a matter of fact, nearly three centuries were to elapse before Boulliaud’s original idea was fully recognized as a useful method of measuring the axial rotation of certain classes of stars, that is, stars that exhibit a detectable rotational modulation of their light output due to starspots or stellar plages. For more than two centuries the problem of solar rotation was practically ignored, and it is not until the 1850s that any significant advance was made. Then, a long series of observations of the apparent motion of sunspots was undertaken by Richard Carrington and Gustav Sp¨orer. They confirmed, independently, that the outer visible envelope of the Sun does not rotate like a solid body; rather, its period of rotation varies as a function of heliocentric latitude. From his own observations made during the period 1853–1861, Carrington derived the following expression for the Sun’s rotation rate: (deg/day) = 1442 275 sin7/4 , (1.1) where is the heliocentric latitude. Somewhat later, Herv´e Faye found that the formula (deg/day) = 1437 310 sin2 (1.2) more satisfactorily represented the dependence of angular velocity on heliocentric lat- itude. Parenthetically, note that Carrington also found evidence for a mean meridional motion of sunspots. Convincing evidence was not found until 1942, however, when Jaakko Tuominen positively established the existence of an equatorward migration of sunspots at heliocentric latitudes lower that about 20 and a poleward migration at higher latitudes.