EARLY SCIENTIFIC ACTIVITIES OF JOHN PAUL COX Charles A. Whitney Harvard-Smithsonian Center for Astrophysics Cambridge, MA 02138 l, Introduction I eagerly accepted the invitation to speak about John Cox's scientific research, because I felt close,t o John after our early collaboration, and because I knew that the broader task of assessing his work and putting it into context must wait for a more objective historian. And the task of broadly summarizing the fields that interested him has already been undertaken by the organizing committee -- the program of this meeting is a list of the numerous topics John's research enriched. On the other hand, I doubt that I am in a position to discuss this remarkable body of work in a way that might be useful to this audience, which contains many who collaborated with John and who know the work more intimately than I. So I have chosen a more personal approach and I will confine myself to John's early pioneering papers on the cause of pulsational ins t ab i l i t y . 2. A Brief Vita John would have been 60 years old this autumn. He ~ras born on November 4, 1926, and he died on August 19, 1984, survived by his wife, Jane. He did his undergraduate and graduate work at Indiana U n i v e r s i t y , w h e r e he r e c e i v e d his Ph. D. in a s t r o n o m y in 1954 u n d e r t h e g u i d a n c e of Marsha l l Wrube l , The n e x t 8 y e a r s w e r e spent t e a c h i n g a t Cornell U n i v e r s i t y - - w i t h t i m e out for s u m m e r r e s e a r c h jobs. In t h e s u m m e r of 1957, he c a m e to Cambr idge . We w r o t e a j o in t p a p e r a n d I b e c a m e his f irst c o - a u t h o r . In 1963, he m o v e d to Boulder , Colorado, w h e r e he b e c a m e a Fellow of JILA a n d a professor a t t h e U n i v e r s i t y of Colorado. For n e a r l y 25 y e a r s he s e r v e d as a c o n s u l t a n t to t h e Los Alamos group t h a t is hos t ing this c o n f e r e n c e . In 1981, I w a s w o r k i n g on a h i s t o r y of t h e pu l sa t ion t h e o r y a n d I w r o t e J o h n asking abou t his e a r l y i n t e r e s t in s te l lar pulsa t ion . He sen t m e some l e t t e r s ( l a rge ly b e t w e e n t h e two of us) t h a t he h a d p r e s e r v e d in his files. These l e t t e r s p rov ide a n u n u s u a l gl impse a t J o h n ' s e a r l y c a r e e r , because he of ten w r o t e l e t t e r s to c l ea r his m i n d a n d l a y ou t t h e possible d i rec t ions for his work . Reading t h e m again has been an in t r igu ing lesson in h i s t o r y for m e . In fac t , t h r e e lessons e m e r g e . First , we h a d u n d e r e s t i m a t e d t h e d i f f i cu l ty of a p rope r n o n - a d i a b a t i c t r e a t m e n t ' , second, we w e r e a t f i rs t mis led by a too l i tera l a c c e p t a n c e of Eddington ' s idea t h a t t he pu lsa t iona l in s t ab i l i ty a n d t h e s u r f a c e phase lag of t he flux w e r e i n t i m a t e l y c o n n e c t e d ; th i rd , our e a r l y p e r i o d - l u m i n o s i t y re la t ion was r igh t for t h e w r o n g reason . 5, J o h n ' s Doctoral Thesis His Ph. D. thes is (1954) was a s t u d y of t he pu lsa t iona l d r iv ing force p r o d u c e d b y n u c l e a r sources in g ian t s ta r s . The resu l t w a s u n a m b i g u o u s a n d nega t ive . Using Epstein 's (1950) adiabatic pulsation solutions for a new, highly condensed red-giant star model, John was able to show that no nuclear processes, either at the center or in a she11, could account for the pulsation unless they had a temperature exponent of at least one hundred million. This was clearly impossible, and at the end of the paper, John pointed to the next region for the search -- the outer layers of the star. He said, In order to have sustained pulsations ... it appears to be a necessary condition that the contribution to [dissipation] from the outlying "non-adiabatic" region rrn~st be sufficiently negative to balance exactly the positive contributions from the adiabatic region. This implies that whatever is the cause of the pulsation phenomenon, it must be in the regions occupying, roughly, the outer 15 per cent of the stellar radius. It remains to be seen whether models with extensive hydrogen convection zones or different boundary conditions will remove these diff icul t ies . The mention of hydrogen convection zones is an allusion to Eddington's hypothesis that such a zone might behave as a heat valve producing a phase lag in the emitted flux and causing the star to act as a heat engine. John's thesis provided a proof that such a valve mechanism was needed. Looking back in 1981, John wrote, "I became interested in the basic problem of the cause of cepheid pulsations, I think, only when I realized how inadequate nuclear sources were. " And in the abstract of his 1955 paper, we find the statement that the "cause of the pulsations must be sought ... where many of the usual approximations are not va}id. " This turned out to be prophetic of John's later work, which often involved careful formulations at the analytical boundary of current pulsation theory. ~4. A o D r o x i m a t e T r e a t m e n t of L i n e a r n o n - a d i a b a t i c P u l s a t i o n At Cornell during 1955 and 1956, John started looking for the Eddington heat valve. He wrote in retrospect, I remember during the early days (mostly while I was at Cornell) I was quite struck by the very small amount of work (in fact, essentially none) that had been done, or that was then being done, on the basic problem of the cause of the pulsations. It seems that Eddington, in his 1941-42 papers [pointing to the hydrogen ionization zone as the direct cause of heat-valve effect] was about the only person who had addressed that question. Yet i remember seeing quite a few papers on details of the shapes of light and velocity curves, etc. I found this quite an amazing fact. John adopted an iterative approach to the linear non-adiabatic equation for radial pulsation starting from the quasi-adiabatic approximation. Progress was slow because of his teaching load, but he began obtaining results in mid-1956. That year, I returned from a Post-Doc with Ledoux and %vrote John summarizing our work. We did not plan to publish because it seemed so tentative and we knew that Evry Schatzman was in the process of publishing independent work that was quite similar. None of us in the United States were aware of the seminal work then being done by Zhevakin in the Soviet Union, and we all still focussed on the ionization of hydrogen as the critical process, and | told John that | felt that the best way to attack the non-adiabatic pulsations would be "by setting up a discrete-shell model for the star and using electronic computers. " But John preferred an analytical and more general approach, and he soon outstripped us all in his understanding of the essential process. During the summer and fall of 1956, I received a series of letters describing his steady progress with an iterative approach to non-adiabaticity based on the formulations of Woltjer and of Schwarzschild, and John analytically developed the relationship between the run of £~amma in a stellar envelope and the phase lag of the emitted flux during pulsation. • . . J - 5. The Theoretical Penod-Lummosltv Relation Joh n ' s f ocus a t th i s t i m e w a s on t h e p h a s e lag of t h e o b s e r v e d f lux t h a t w e all t h o u g h t w a s d i r e c t l y r e s p o n s i b l e for i n s t a b i l i t y . In a l e t t e r w r i t t e n to m e on A u g u s t 22, ! 9 5 6 , he said: My own work seems to be ~oin~ rather well now, I've succeeded in generalizin£~ the treatment somewhat, and it now appears possible to prove that a phase lag in the emitted flux is a necessary condition for instability, without making any assumptions regarding the non-adiabaticity in the region under consideration. But it soon became clear that the relationship between the envelope structure and the instability was more complex than he had first judged. On November 2, 1956, he wrote that the phase lag for maximum instability depended on the detailed run of the wave function in the outer layers, so the purely schematic models he had been using would not be adequate to answer the question of cepheid instability. Then he added an exciting development: If it should turn out to be possible to say that sustained pulsation may in general exist only if the phase lag is near a quarter period, then a basis for a period luminosity relation seems to exist somewhere in the present theory, but I haven't yet been able to pin it down precisely. Eight d a y s l a t e r , h e w r o t e w i t h his u s u a l c a u t i o n , "I s e e m to h a v e f o u n d a p e r i o d - l u m i n o s i t y r e l a t i o n w h i c h a p p e a r s to a g r e e r e a s o n a b l y we l l w i t h o b s e r v a t i o n . " In e f f ec t , h e s h o w e d t h a t if, a long t h e c e p h e i d s e q u e n c e , t h e l a y e r of p a r t i a l i o n i z a t i o n of hydrogen occurred at a depth corresponding to a constant phase lag, then he cou ld derive a relation among period, luminosity, and mass. He had estimated the phase lag as the ratio of heat capacity to flux emitted in a full cycle -- essentially the thermal time to the surface. I suggested that he come to Cambridge the following summer so we could work out the numerical details on a "larI{e machine" that was being installed. John presented a brief description of his work at the Christmas, 1956, meeting of the American Astronomical Society. Although limited to first-order non-adiabatic terms, it constituted a mathematical statement of the hypothesis that had been rather intuitively expressed by Eddington. During the spring of 1957 John wrote a detailed discussion of his iterative treatment of non-adiabaticity in the schematic models. The paper (Cox 1958) was received by the Astrophysical Journal on April 29. The gist of that paper was that no particular phase lag was a necessary condition for instability, although an abrupt drop in the radiative flux probably was. He also broadened the search for the Eddington valve beyond the hydrogen zone and stressed the weakness of the first order theory. During John's visit to Cambridge in the summer of 1957, we performed some homology calculations based on his recent paper, and we made his theory of the period-luminosity relation quantitative, This was done without being able to calculate the actual net dissipation. John had merely considered the condition for minimum dissipation, and we were still unable to show the net dissipation was negative, because we had not done the full pulsation calculation. According to our formulation, the hydrogen and neutral helium ionizations occurred too close to the surface to explain the phase lag of classical cepheids, but the second ionization of helium appeared to occur at the correct depth. At the time, we still felt that Eddington's description was correct, and the instability ought to be related to the phase lag of the observed flux. At the end of the summer, l received a translation of Zhevakin's work on the non-adiabatic oscillation of discrete zone stars and I sent a copy to John. It, too, pointed to the second ionization of helium and it contained many important results on pulsational instability. We emended our manuscript (Cox and Whitney 1958) and added the references. Zhevakin had insisted that the phase lag of the surface flux was not related in a simple way to the pulsational instability. What counted was the degree of non-adiabaticity at the level of partial ionization. As it turned out, he was correct in this. Ironically, the heat-capacity function that John and I used to estimate the phase lag was actually a measure of the degree of non-adiabaticity in the critical region, so we got the right p e r i o d - l u m i n o s i t y re la t ion , bu t looking back, it s e e m s t h a t Z h e v a k i n w a s m o r e n e a r l y co r r ec t t h a n w e w e r e a t t h e t ime , because he h a d a l r e a d y r e j e c t e d Eddington 's s imple r e l a t i on b e t w e e n ins t ab i l i ty and s u r f a c e phase - l ag . 5_. Stems Toward The Exact Linear Non-adiabatic Treatment This was my last substantial collaboration with John, although we corresponded regularly for the next few years, and he visited Cambridge several times to use our computational facilities. Dur ing t h e w i n t e r of 1957-58 , J o h n felt doubts abou t t h e a d e q u a c y of his f i r s t - o r d e r t r e a t m e n t a n d he s t a r t e d w o r k i n g w i t h t he Wol t j e r v - e q u a t i o n - - a m o r e complex bu t , he hoped , also m o r e a c c u r a t e p r o c e d u r e t h a n t h e S c h w a r z s c h i l d technique~ (In t h e course of this work , he also publ ished a pape r ex t end ing his ana lys i s of t h e p e r i o d - l u m i n o s i t y r e l a t i on to r a d i a t i v e envelopes.) Finally, on January 29, 1960, after a year and a half, his paper on the approximate analysis was received at the Astrophysical Journal. It was titled '~A Preliminary Analysis of the Effectiveness of Second Helium Ionization in Inducing Cepheid Instability in Stars" (Cox 1960). Even the title of that paper reveals that John was not Convinced by this approximate treatment either, and in a letter dated Jan. 12, 1960 -- before he had submitted the paper -- he ~¢rote that he was already at work on the "exact linear treatment but progress is temporarily slow because of a rush of other things to do and because the algebra is setting to be a real mess," The work went so well that he started an extensive numerical study that spring, and when the approximate paper came out, John had added a footnote in the proofs announcing the successful numerical integration of the eighth-order system of linearized pulsation equation. The work on the full set of linearized equations was finally submitted in July 1962. It was titled "On Second Helium Ionization as a Cause of Pulsational Instability in Stars, " and it ran 49 pages (Cox 1963). Here, at last, was a treatment that could evaluate the net dissipation quantitatively. The paper has a tone of authority that was lacking in the earlier "preliminary" papers. John concluded that helium second ionization "probably accounts for the instability in classical cepheids and 111% Lyrae variables and also (but less certainly) in W Virginis variables and dwarfs cepheids of the 6 Scuti type. " But the phase lag came out wrong. It was clear that his linear theory, which ignored the ionizations of hydrogen and neutral helium, was not giving the observed phase lag of the surface flux. This paper, with its mathematical rigor, ~vas a key to our understanding that the phase lag problem was quite distinct from the instability itself. Writing with D. S. King (King and Cox 1968), John later said, This study isolated the driving at small amplitudes, due to second helium ionization alone. It was possible, therefore, to obtain a clear picture of how this mechanism works and how it can lead to an instability strip which has the essential features of the observed strip. But I~etting the proper phase lag was another matter, and the clue came from independent, concurrent work of Baker and Kippenhahn (1962), who had included hydrogen and neutral 10 h e l i u m a n d t r e a t e d t h e l i nea r . n o n ' a d i a b a t i c p u l s a t i o n s of a c a r e f u l l y c o n s t r u c t e d c e p h e i d enve lope . T h e y f o u n d a la rge pos i t ive p h a s e lag p r o d u c e d b y n e u t r a l h e l i u m a n d h y d r o g e n . I t h i n k t h e s e p a p e r s w e r e t h e f i rs t c l ea r s igns c o n f i r m i n g Z h e v a k i n ' s conc lu s ion t h a t t h e i n s t a b i l i t y a n d t h e p h a s e lag w e r e two s e p a r a t e p r o b l e m s . In his classic r e v i e w p a p e r of i 9 7 4 , J o h n c lar i f ied t h e d i s t i nc t ion as follows: It is a r e m a r k a b l e fac t t h a t t h e c o n d i t i o n for t h e a p p e a r a n c e of t h e p h a s e lag a n d t h e n e c e s s a r y c o n d i t i o n for i n s t a b i l i t y a r e b o t h sa t is f ied . . . w h e r e m a n y c o m m o n t y p e s of p u l s a t i n g s t a r s a r e f o u n d . Because t h e s e t w o p h e n o m e n a ( i n s t ab i l i t y a n d p h a s e lag) a r e c a u s e d b y t h e ac t i on of t w o different ionization zones, it appears that the occurrence of the phase lag in pulsating stars is more or less an accident of nature; attributing both phenomena to a single physical mechanism, which was Eddington's view, is evidently not entirely correct. In 1960, John began a series of fruitful collaborations with the Los A l a m o s g roup , a t t h e i n s t i ga t ion of Ar t Cox, w h o h a d b e g u n n o n - l i n e a r n o n - a d i a b a t i c c o m p u t a t i o n s . J o h n ' s e a r l y a t t i t u d e t o w a r d t h e r e l a t i o n s h i p of l i nea r a n d n o n - l i n e a r m o d e l i n g w a s desc r ibed in s e v e r a l l e t t e r s : I feel t h a t th i s [ l inear] a p p r o a c h is still v a l u a b l e in v i e w of ou r p r e s e n t s t a t e of i g n o r a n c e r e g a r d i n g t h e c a u s e of t h e pu l sa t i ons . H o w e v e r , it will u l t i m a t e l y be n e c e s s a r y , of cour se , to go in to a n o n - l i n e a r t h e o r y before a r e a s o n a b l y c o m p l e t e U n d e r s t a n d i n g is possible. [To A . N . C . , Nov. 2, 1959] I ' m s o m e w h a t inc l ined to t h e v i e w p o i n t t h a t one shou ld a t t e m p t to e x h a u s t t h e possibili t ies of a l i n e a r n o n - a d i a b a t i c t h e o r y f i rs t . This shou ld , if n o t h i n g else, r e v e a l ~vhat f a c t o r s a r e l ikely to be i m p o r t a n t in a n o n - l i n e a r t r e a t m e n t . [To A . N . C . , Dec. 16, 19593 11