Table Of ContentLecture Notes
in Physics
Edited by J. Ehlers, Miinchen, K. Hepp, Zijrich
R. Kippenhahn, MOnchen, H. A. Weidenmtiller, Heidelberg
and J. Zittartz, Kijln
Managing Editor: W. BeiglbGck, Heidelberg
125
Nonradial and Nonlinear
Stellar Pulsation
Proceedings of a Workshop
Held at the University of Arizona
in Tucson, March 12 - 16,1979
Edited by
H. A. Hill and W. A. Dziembowski
Springer Werlag
Berlin Heidelberg New York 1980
Editors
Henry Allen Hill
Department of Physics, The University of Arizona, Tucson, AZ 85721/USA
Wojciech A. Dziembowski
Centrum Astronomiczne PAN, ul Bantycka 18, 00-716 Warszawa/Poland
ISBN 3-540-09994-8 Springer-Verlag Berlin Heidelberg New York
ISBN O-387-09994-8SpringerVerlag New York Heidelberg Berlin
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0 by Springer-Verlag Berlin Heidelberg 1980
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FOREWORD
A workshop was held in Tucson at the University of Arizona from March 12
through March 16, 1979 to furnish a forum for both theorists and observers to study
and exchange ideas on some of the current problems in nonradial and nonlinear stel lar
pulsation. This workshop, sponsored by the Department of Physics in conjunction with
Steward Observatory was organized into morning sessions of invited and contributed
papers and into afternoon sessions of discussions and informal contributions. This
structure was quite successful with many of the contributions to these workshop
proceedings motivated by interactions at these sessions. The scienti f ic success of
the workshop must in large measure be the result of the strong international
character of the l i s t of participants and the constructive mode in which they worked
together.
The idea for the work grew out of discussions with Drs. Robert ~ Rosenbaum,
former President of Wesleyan University and John P. Schaefer, President of the
University of Arizona, on the scientif ic program at SCLERA and on how the scholarship
and teaching in each of these two respective ins t i tu tes might be brought closer
together. The f inancial support which made the workshop and these proceedings
possible was furnished by President Schaefer. In the planning and execution stages
of the workshop, the advice and counsel of Dr. Robert H. Parmenter, Head of the
Department of Physics, was frequently sought.
The Scient i f ic Organizing Committee was Drs. W. Dziembowski, W. Fitch, H.
H i l l , E. Nather~ , S. Starrfield, H. Van Horn and R. White. The aim of the organizing
committee was to maintain a balance between theory and observation and through the
organization of the meeting, encourage the interaction of those working in these more
often than not isolated areas of science.
The local organizing committee consisted of Drs. T. Caudell, W. Dziembowski,
W. Fitch, H. Hi l l and R. White, and Mr. R. Bos. They were al l helped during the
running of the symposium by J. Logan and R. Rosenwald and by J. Brown, conference
coordinator.
I would also like to thank MacMillan Journals LTD for granting permission to
reprint the material in the Introduction from the workshop review by Douglas Gough in
Nature.
The scientif ic editing was performed at the University of Arizona and in part
at Wesleyan University in collaboration with visi t ing professor Dr. W. Dziembowski.
His broad knowledge of the f ield of stellar pulsation was invaluable. The editing of
the manuscript was performed pr imari ly by Ms. A. Whitehead with some of the
responsibility being shared by Dr. G. Harwood. The final production, supervised by
Dr. T. Caudell, was a collaborative ef for t of al l of us at SCLERA. In addition to
the SCLERA staff, a significant contribution was made to the final production by Ms.
J. Twehous. I express my sincere personal thanks to each person who helped complete
this project.
Henry A. Hi l l
Tucson, Arizona
March, 1980
TABLE OF CONTENTS
Introduction
D. Gough . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. a Scuti Stars
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Observational Evidence of Radial Mode Resonances and of
Nonspherical Symmetry in Some Variable Stars of a Scuti
and Related Types
W. Fitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Scuti Variables: The Link Between Giant- and
Dwarf-Type Pulsators
W. Dziembowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Nonlinear Calculat ionsfor Bump Cepheids
S.W. Hodson and A.N. Cox. . . . . . . . . . . . . . . . . . . . . . 34
A Nonlinear Study of AI Velorum
A.N. Cox and S.W. Hodson . . . . . . . . . . . . . . . . . . . . . . 41
Nonlinear 8 Scuti Models: The Main Sequence Catastrophe?
R.F. Stell ingwerf. . . . . . . . . . . . . . . . . . . . . . . . . . 50
A Program to Observe Very Low Amplitude Radial Velocity
Variations in a Scuti Stars
W.D. Heacox. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2. B Cephei and Other Early-Type Variables
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Pulsation Modes in B Stars With Variable L inePro f i les
M.A. Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
The Stab i l i t y of the B Cephei Stars
M.L. Aizenman. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Mul t iper iodic i ty and Nonradial Oscil lat ions of the
B Cephei Star 12 Lacertae
M. Jerzykiewicz . . . . . . . . . . . . . . . . . . . . . . . . . . 96
New B Cephei Stars and the B Cephei I ns tab i l i t y Strip
C. Sterken and M. Jerzykiewicz . . . . . . . . . . . . . . . . . . . 105
16 Lacertae: An Eclipsing System With a B Cephei Primary
M. Jerzykiewicz . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Linear, Nonadiabatic Pulsation Calculations For Models
of Upper Main Sequence and B Cephei Stars
H. Saio, J.P. Cox, C.J. Hansen and B.W. Carroll . . . . . . . . . . 135
Vl
Importance of Studying B Cephei Stars
in Open Clusters and Associations
S.M. Jakate and C. Sterken . . . . . . . . . . . . . . . . . . . . 147
Pulsation of High Luminosity Helium Stars
D.S. King, J.C. Wheeler, J.P. Cox, A.N. Cox and S.W. Hodson . . . 161
Some Comments About B Cephei Stars
H. Saio and J.P. Cox . . . . . . . . . . . . . . . . . . . . . . . 169
3. Solar Osci l lat ions
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 73
Observational Evidence
Observational Evidence for Global Osci l lat ions
of the Sun: ~ Review
H.A. H i l l . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 74
The Latest Results of the Velocity Spectroscopy
o f the Sun
A. Claverie, G.R. Isaak, C.P. McLeod,
H.B. van der Raay and T. Roca Cortes . . . . . . . . . . . . . . . 181
Implications of the Whole-Disk Doppler Observations
of the Sun
J. Christensen-Dalsgaard and D. Gough . . . . . . . . . . . . . . 184
Observations of Long Period Osci l lat ions in the
Solar Limb Darkening Function
R.T. Stebbins . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Solar Continuum Brightness Osci l la t ions: A Progress Report
T.M. Brown and R.L. Harrison . . . . . . . . . . . . . . . . . . . 200
Recent Observations of Solar Osci l lat ions at SCLERA
T.P. Caudell, J. Knapp, H.A. Hi l l and J.D. Logan . . . . . . . . . 206
Sources of Noise in Solar Limb Defini t ions
S.L. Keil and S.P. Worden . . . . . . . . . . . . . . . . . . . . 219
Observations with High Temporal Resolution
of the Solar Ca+ K Line
T. Duvall, W. Livingston and C. Mahaffey . . . . . . . . . . . . . 237
Excitation Mechanisms
Excitation of Solar g-Modes with Periods Near 160 Minutes
D. Keeley . . . . , . . . . . . . . . . . . . . . . . . . . . . . 245
The Collect ive Excitation of g-Modes in the Sun
C.L. Wolff . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 52
Comments on Gravity Mode Excitation Mechanisms
W. Dziembowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Vll
Theoretical Implications of Observation8
Some Theoretical Remarks on SolarOsci l la t ions
D. Gough . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Five Minute Osc i l la t ions as a Probe of the Solar I n t e r i o r
S.H. Lubow, E,J. Rhodes, Jr. and R.K. Ulr ich . . . . . . . . . . . 300
Sens i t i v i t y of Five Minute Eigenfrequencies
to the Structure of the Sun
G. Berthomieu, A.J. Cooper, D.O. Gough, Y. Osaki,
J. Provost and A. Rocca . . . . . . . . . . . . . . . . . . . . . . 307
How Deep Is the Solar Convection Zone?
J. Christensen-Dalsgaard, W. Dziembowski and D. Gough . . . . . 313
Nonradial Osc i l la t ions of Solar Models wi th an
I n i t i a l D iscont inu i ty in Hydrogen Abundance
A. Boury, R. Scuf la i re , A. Noels and M. Gabriel . . . . . . . . . 342
Nonradial Osc i l la t ions with High h-Values
C.A. Rouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
Time-Varying Gravi ta t ional Mul t ipo le Moments
Corresponding to Nonradial Solar Osc i l la t ions
W.W. Johnson, D.E. Winget, D.H. Douglass and H.M. Van Horn . . . . 357
Perturbations in Gravitational Potential Associated
with Solar Oscil lations
J. Christensen-Dalsgaard and D.O. Gough . . . . . . . . . . . . . 369
Outer Boundary Conditions Used in Pulsation Theory
Observational Tests of Pulsation Theory in the Solar Envelope
R.T. Stebbins, H.A. H i l l , R. Zanoni and R.Eo Davis . . . . . . . . 381
On the Study of Global Osc i l la t ions of the Sun Via
Fluctuat ions in the Solar Limb Darkening Function
J. Knapp, H.A. H i l l and T.P. Caudell . . . . . . . . . . . . . . . 394
Sensi t iv i ty to the Applied Boundary Conditions of the Solar
Eigenfrequency Spectrum with Periods Near Five Minutes
R.D. Rosenwald and H.A. Hi l l . . . . . . . . . . . . . . . . . . . 404
Solar Osc i l la t ions Interact ing Through a Mean Field
J.D. Logan, H.A. H i l l , P. Puccio and R. Rosenwald . . . . . . . 413
The Anisotropic Radiation Field Approximation and I t s
Effect on Wave Equation Solutions in the Solar Photosphere
H.A. H i l l , R.D. Rosenwald and R.S.~Robinson . . . . . . . . . . . 429
4. Degenerate Stars
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
The Observational Properties of the ZZ Ceti Stars
E.L. Robinson . . . . . . . . . . . . . . . . . . . . . . . . . . 444
VIII
Theoretical Aspects of Nonradial Osc i l la t ions
in White Dwarfs: A Summary
C.J. Hansen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
The Periods of ZZ Ceti Variables
A.N. Cox, S.W. Hodson and S.Go S ta r r f i e l d . . . . . . . . . . . . 458
Twelve Minute Light Variations in the Peculiar Star HD 101065
G. Wegner . . . . . . . . . . . . . . . . . . . . . . . . . ... 467
5. General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
On the Def in i t ion of Canonical Energy Density and of Canonical
Energy Flux for Linear and Adiabatic Osci l la t ions
of a Spherical Star
P. SmeYers and A. Weigert . . . . . . . . . . . . . . . . . . . . 473
Properties of Nonradial S te l la r Osci l la t ions
M. Gabriel and R. Scuf la i re . . . . . . . . . . . . . . . . . . . 478
On Mode Interact ion of Nonradial Osci l la t ions
M. Gabriel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
Keja~ord Index to Papers . . . . . . . . . . . . . . . . . . . . . . . . . . 496
INTRODUCTION*
D. Gough
University of Cambridg~
United Kingdom
In recent years considerable attention has been devoted to the observation
and theoretical description of variable stars. In this context the term "variable"
should be taken to mean pulsating or oscillating as opposed to cataclysmic or nova. The
distinction here is important; when a system is observed to oscillate, be i t a star
or a molecule, there exists then the potential of probing the workings of that system
in a quanti tat ive manner. I t is in th is sp i r i t that the fol lowing summary of the
workshop, both discussions and papers, is presented.
* ~ * * *
Over the past two decades the theory of the pulsation of the Classical
Cepheid and RR Lyrae stars has reached a high degree of sophistication. This kind of
pulsation is the simplest mode of oscillation a star can undergo: i t is periodic and
the motion is purely in the radial direction. Althought here remain some niggling
discrepancies between theory and observation, the agreement is sufficient to convince
most astrophysicists that our ideas are basically correct. Thus we seem to have a
firm foundation from which to extend our studies to stars whose osci l la t ions are
nonradial and nonperiodic.
Walter Fitch (Steward Observatory) reviewed observations of variable stars of
Scuti and related type. These stars pulsate at low amplitude, and many exhibit two
characteristic frequencies which bear a ratio close to that of two small integers.
Consequently much of the subsequent discussion concerned direct resonances between
two modes of oscillation. Arthur Cox (Los Alamos Scientific Laboratories) reported
his failure to reproduce this behavior theoretically from in i t ia l value integrations
even when the resonance conditions had been careful ly engineered, and Robert
Stellingwerf (Rutgers University), who had analyzed the stabi l i ty of pairs of singly
periodic l i m i t cycles of potential double mode Cepheid models, never found both
cycles to be simultaneously unstable to l inear perturbations. The double mode
behavior remains unexplained. During discussions of parametric resonances, i t was
pointed out that many of these stars seem to exhibi t characterist ics of strange
*Reprinted from Nature, 278, 685(1979)
a t t r a c t o r s . Some a Scut i s tars appear to pu lsate in many modes s imu l taneous l y ;
Wojciech Dziembowski (Copernicus Astronomical Center) estimated that non l inear i t i es
developed by any one mode alone were too small to l i m i t i t s amplitude to a value as
low as those observed, and concluded tha t mode i n t e r a c t i o n s must be respons ib le .
D e f i n i t i v e c a l c u l a t i o n s have not yet been performed, but techniques developed f o r
studying oceanic grav i ty wave in teract ions and plasma turbulence are avai lable for
tack l ing problems of th is kind. We can therefore ant ic ipate considerable advances in
th is study in the near future.
A c lass of s t a r s f o r wh ich t he re i s no c o n v i n c i n g e x p l a n a t i o n i s
characterized by B Cephei. Myron Smith (Universi ty of Texas at Austin) reviewed the
observat ions. The s tars are of spect ra l type B, and l i e in a s t r i p in the
Her tzsprung-Russel l diagram a l i t t l e above the main sequence. Mike Je rzyk iew icz
(Wroclaw U n i v e r s i t y Observatory) and C. Sterken (Free U n i v e r s i t y of Brussels)
reported new observations and pointed out that almost a l l stars in the B Cephei s t r i p
show signs of v a r i a b i l i t y . They appear to pu lsate in both rad ia l and nonradia l
modes. As wi th the B Scuti stars simple period rat ios are found, which suggest again
t ha t resonance mechanisms are operat ing . Frequency s p l i t t i n g , presumably by
ro ta t ion, has been measured, which of fers the exc i t ing prospect of t r y ing to i n fe r
the angular ve loc i t y w i th in these stars. Unfortunately we are not yet i n a p o s i t i o n to
do t h i s because the modes of o s c i l l a t i o n have not been unambiguously i d e n t i f i e d ;
d i f f e ren t workers expressed contradic tory opinions about the modes they thought were
responsible for the v a r i a b i l i t y .
An i n t r i g u i n g proper ty of some B Cephei s tars is t ha t they sw i tch from one
mode of o s c i l l a t i o n to another on a t imesca le of no longer than about I0 per iods.
Smith argued tha t t h i s is evidence t ha t the o s c i l l a t i o n s are conf ined to a ra the r
th in outer layer of the star, because otherwise i t would be d i f f i c u l t to envision how
d i s s i p a t i v e processes could e f f e c t the enormous energy t r a n s f e r tha t would be
involved. Others contradicted that claim, c i t i ng simple nonl inear osc i l l a to rs that
can a l t e r t he i r character on a dynamical timescale. I t is un l i ke ly that i t w i l l be
es tab l i shed in the near f u t u r e whether such o s c i l l a t o r s a c t u a l l y represent the
behavior of B Cephei stars in any way, because the basic physics of the v a r i a b i l i t y
is not yet understood.
How are the o s c i l l a t i o n s dr iven? Morr is Aizenman (Nat ional Science
Foundation) reviewed the many ingenious ideas that have been proposed in the past,
a l l of which have fa i led . A recent suggestion by Ste l l ingwer f that the mechanism is
no more than the Eddington valve that drives c lassical Cepheids and RR Lyrae stars
was debated. He proposed that in te rpo la t ion in opacity tables on too coarse a gr id
had led to an underestimate of the ef f icacy of the valve, but Arthur Cox, one of the
ch ie f suppliers of opacity data to the astrophysical community, claimed that though
f i n e r r e s o l u t i o n decreased the s t a b i l i t y of the t h e o r e t i c a l models i t was never
s u f f i c i e n t to render them unstable. The suggestion was made that i f a l l the
mechanisms reviewed by Aizenman were operating in unison the sum of t h e i r
contr ibut ions might be su f f i c i en t to maintain the pulsations against dissipation.
This idea, which is reminiscent of an early suggestion for solving the solar neutrino
problem, was unenthusiastically received.
A signif icant fraction of the workshop was devoted to discussing osci l lat ions
of the sun. F i r s t , Henry H i l l (Univers i ty of Arizona) reviewed the observations
spanning the period range 3min-16omin and discussed the evidence already in the
l i terature for the osci l lat ions in the data being of solar origin. There followed a
sequence of presentations of new observations and discussions of the longer period
osci l lat ions in the upper atmosphere and the evidence for phase coherence.
By comparing d i f f e ren t diameter measurements made at the Santa Catal ina
Laboratory for Experimental Relat iv i ty by Astrometry (SCLERA), Hi l l argued that the
amplitudes of osci l la t ion rise more steeply with height above the photosphere than
can be comfortably accommodated wi th in the framework of l i near theory. At f i r s t
sight this should be no cause for alarm, because the amplitudes thus inferred in the
low chromosphere are so great that non l i near i t i es must surely be important.
Nonetheless several par t ic ipants were uneasy with the resu l t , because in the
photospheric regions, where presumably l i near theory is va l i d , the eigenfunctions
have the appearance of waves penetrating an evanescent region from above yet having
frequencies charac te r i s t i c of the resonating cav i ty beneath. Tuck Stebbins
(Sacramento Peak Observatory) presented measurements of r e l a t i v e o s c i l l a t i o n
amplitudes at d i f f e ren t posit ions in the wings of a spectrum l i ne which support
H i l l ' s conclusion, and Timothy Brown (High Alt i tude Observatory) found contradictory
evidence from shape changes of the limb darkening function. The issue is currently
unresolved.
For many people, the most convincing evidence that the osci l la t ion data arise
from genuine dynamical solar v ib ra t ions is t h e i r phase coherence. Peter Worden
(Sacramento Peak Observatory) c r i t i c i z e d ear ly analyses of the SCLERA data, but
Thomas Caudell (University of Arizona) presented new measurements of solar equatorial
diameter variations which, i t was f i na l l y agreed, convincingly maintained phase over
an interval of 23 days. This did not terminate the discussion; however, because as
Maurice Gabriel (Universite de Liege) pointed out, unless there were a considerable
difference in the amplitudes of eastward and westward propagating waves one would not
expect phase coherence over a period longer than half the mean rotation of the sun.
The detai ls of the phase data may indeed be consistent with beating phenomena induced
by rotation, but further analysis must be done. Polar diameter measurements, which
have not yet been analyzed, are not subject to this cr i t ic ism.
One of the most exciting aspects of solar osci l lat ions is the i r potential for
providing diagnostics of the solar i n t e r i o r . To rea l i ze th i s potent ia l i t is
