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2

Kodels a r e cons t r cc t ed for vzxy xi,asr;lve sta;-s ir, :;IS

t o exaiaine the role of convect ive i n s c a b l l l t y . '212 smi-

convec t ive zone shows a maxinun growth ( i n nzss .Ezdctior~)

a t about 60 wi th a chosen i n i c i a l kiyerogcr, a h n d a c c e of

X = 0.70. S i x t y Y i i s also the nzximx, xass fcu,id by

t i o n s , for n e a r l y the same chemical composition. Radla'tio2

pressure h a s a c r i t i c a l destabilizing i n f l u e x e h e r e , Ir.

the l i m i t of t h e h i g h e s t nasses, the semiconvectivd zone

d i s a p p e a r s toward t h e s u r f a c e , b u t the s t a r s t i l l evo lves

inhomqeneously. It i s s'r,own t 'na t a s p 2 t o t i c limits a t

hydrogen exhaus t ion e x i s t for the n a s s f r a c t i o n 3f tke COT--

v e c t i v e c o r e , t h e mean hydrogen c o n t e n t , anG t h e hydroger;-

burning l i fe t ime.

z .

3

I. INTROJUCTION

T h e o r e t i c a l unders tanding O € v e r y rr,;jssi-;c. s.-azs Cepe.:

- critically ori a Xnowi t:*-l<3?.&:.:: z:-.<* 3;.iCln - I > ;

i n t e r n a l mixing.

v e c t i v e i n s t a b i l i t y .

s t r u c t e d for s t a r s which a r e s t r o n g l y unsta3l.e LO convectioy-

o u t s i d e the normal convect ive core, bi;t whicf! a r c nos-

This mixing may be ciue t10 roza-lion or ccc-

i n t h e p re s sn t paper , zadels ari3 coz-

rotatir-g. T h e assumptior, of c o n v a x i v a neuczz,1~y - . iri t k e

unstable p a r t of the envelope p e r n i t s calculation of rhe

e x t e n t of z i x i n g phencrnenalogicaliy, without refcrezce t o i:, -

d e t a i l s of convect ion theory. k b e s c r i p i o n r,f t h e so-

c a l l e d semiconvective zone which i s forned h a s been giver, ::

Paper I ( S t o t h e r s 1963) . “‘e pxesene p a p z investigaTes

the e x t e n t of t h e seaiconvective zom arid convect ive c o r e

a s a f u n c t i o n of s t e l l a r mass and e v o l u t i o n d u r i n g hydroScr,

burning. I

11. ASSUX?TIOXS

The g e n e r a l s t r u c t u r e of a very massive s t a r 3 a s b e e n

desc r ibed i n Paper I. W e adopt t h e same assumptions, d e f i -

?

A l l c r i s (1963) va lues of solar l~,: . i ias~ ty and r G d i . ; s

were used as normalizing quantities, w3er5zs . in Papsr I,

I * - . - .

I

5

Chandrasekha.r ' s (1935) v a l u e s w e r e used .4 A s u b s c r i p t

r e f e r s to the boundary of the convec t jve c o r c z t the e v o h -

t i o n a r y s taGe when hydrogen i s just exhauste2 i n the core.

111. MOT)ZLS OF VERY Y J S S T V E STAXS

The inain c h a r a c t e r i s t i c s of models fcjr s ix very zas s iv - .

s t a r s d u r i n g hydrogen bcrning are presented ia Ta31e 1.

computer, w i t h t h e sen iconvec t ive zone t a k e n e x F l i c i c l y L n - ~ c

account , 26 desc r ibed i n an e a r l i e r 2aper (S to thers 1966:.

This metkod of t r ea tmen t of the semiconvective zone, while

precise, g i v e s r e s u l t s r.early identical w i t h t h c a e calcul;; __ .

on the basis of t h e zone assixved r a d i a t i v e (Sakcsk i t a an6

Hayashi 1561; S t o t h e r s 1963, 1965). However, tte l a t c e r

method does not g i v e correct r e s u l t s when the zone exceees

an extent o f I\q = 0.2 (Stothers 1966).

2) Iiomoqeneous Kodels

- > Were t h e models t o form a homologous sequence, we s:xL.-.-

have e x a c t l y t h a t L - N3 and q = c o n s t a ~ t . However, t h e

models a r e f a r from homologous because or' the wide var ia -

4

tion i n r a d i a t i o n p res su re , On the b a s i s of the s tandard

. .

F t

- b *

6

model, Eddington (1926) showed t h a t the r e l a z i v e r a d i a t i o n

pressure, 1 - 3 , i n c r e a s e s with t o t a l s t e l l a r xass ( L F Z G L ~ ~ ~ I

the well-known q u a r t i c eqi ia t ion) . S i r z e 2 1~ : rqe Z = E : C ; L ~ ~ G I ; 7 .

a d i a b a t i c tempera ture g rad ien t , t h e i i l ~ ss fraction cor,zai;lz<

w i t h i n the convect ive c o r e w i l l ~ l s o :be l a r g e r sc the hi??,e:

masses (F ig . 1). This may be seen preclsely fron the cond iz i c r

d e f i n i n g t'ne convec t ive c o r e boundary, v i z . e q c a t i n g -;he

. - . r a d i a t i v e and a d i a b a t i c effective p ly t ro , s i c ;.?c;ces ZL

q4 and i n t r o d u c i n g t h e approximate mass- luninos i ty r e l a t i c :

~ - 3 .

- r r ius a sufficiently l a r g e nass Secozes conFie te ly

convec t ive . Tne photon flux escap i rq fron the s m f a c e i s L:-- ::.

determined e n t i r e l y Sy t;ie r a d i a t i v e s u r f a c e c o r d i t i o n ,

QL/M = 4 ' ~ ~ ( 1 - F0). \ --

I n th is csse, the mass-luminosity r e l a t i o n apprcjaches L Y .

- b) InhomGqeneous Models

For the lower-mass s t a r s i n w h i c h r a d i a t i o n p r e s s u r c

i s n o t dominant, the luminos i ty i n c r e a s e s du r ing e v o l u t i o n

i n accordance wi th tSle trend! i n d i c a t e d Sy t'le homololjocs

r e l a t i o n L - P4. ' Howaver, a s 1 - so approaches u n i t y i n C-ie

highel; masses, t h e luminos i ty remains c o n s t a n t i f t h e s t a r

does not evolve fully mixed (such mixing would reduce X, 1

I

7

and hence x e i n eq. [27_),

It may be simply shown t h a t s t a r s of even the h i g h e s t

mass must evolve inhomogeneously . Ccxsider the c o r e S o u ? , C a ~ - ~

for a s t a r in w h i c h 3 G throuc$mut. Z c p z t L ~ g --2, &.LE & L . U L . a L * I .

and a d i a b a t i c e f f e c t i v e p o l y t r o p i c i n d i c e s , w e can o b t a i n

t h a t 44 = K 4 / " e = (1 + X p J / ( l + Xe) . exhausted i n the convect ive core , the c o r e bouzdary r u s t

have shrunk t o a mass f r a c t i o n qf = (i + Xe)-'.

a g r a d i e n t of chemical camposi t ion left bei2inC m d sets zn

upper l i m i t on t h e composi t ion exponent, 1, i n t h e i n t e r -

media te zone. Hence t h e amount of hydrogen Zep le t ion i s

f i x e d for very massive s t a r s , Tae minimum mean hydrogen

c o n t e n t remaining a f t e r hydrogen burn ing w i l l be X

Mr.en hydrogen h a s beec

T h i s i rnp l i ss

G ~ / Z i: -

Thus any s t a r camposed i n i t i a l l y of pure hydrogen has L

For our chosen i n i t i a l coz- - > f i n a l qf 5 0.50 and X - 0.25. p o s i t i o n , t h e l i m i t i n g va lues a r e qf ,< 0.59 and

F i g u r e 1 shows t h e asymptot ic approach of the core boundax;,

toward a l i m i t i n g va lue ( s l i g h t l y i n e x c e s s of qf s i n c e

+ = 0.05 i n t h e f i g u r e ) a s t h e s t e l l a r inass i s inc reased .

2 0.14.

A

Sakash i t a , Ono, and Hayashi (1959) have shown t h a t sexi-

convec t ive mixing cannot extend t o t h e convec t ive core:

the steeper g r a d i e n t of mean molecular weight i n t h e deeper

- layers of the envelope i n h i b i t s t h e convective motions.

t

F * * -&&EL-

I 8

~

I

~

8

Thus t h e envelope can be o n l y p a r t i a l l y mixed, The u n s t a b l e

I i n t e r m e d i a t e zone g r G W S , however, as ;Ihe i u n i n o s i t y incrczse.; I

d u r i n g evo lu t ion . T n i s r e s u l t s 4ron -&e proport i o n a t e in-

which h a s crease in relative r a d i a t i o n pressure (eq. ~ 2 _ ) , r- -

a d e s t a b i l i z i n g i n f l u e n c e on t h e gas. I

I The semiconvect ive zone has a naxinun srowth i n s t a r s

w i t h a c r i t i c a l mass n e a r 60 YQ (Fig. 1). S t a r s of smallex

mass feel s t r o n g l y the d e s t a b i l i z i n g e f f e c t of i n c r e a s e d

r a d i a t i o n p r e s s u r e a s the mass is i n c r e a s e d , Hcwever, aSovc

i 60 M, f u r t h e r i n c r e a s e of r a d i a t i o n p r e s s u r e does n o t lowe:

I

I I

t h e a d i a b a t i c tempera ture g r a d i e n t s i g n i f i c a n e i y ( s i n c e 2

approaches 3 asympto t i ca l ly ) . Moreover, the l m i n o s i t y

b r i g h t e n s very l i t t l e d a r i n g e v o l u t i o n of the more nass:v2

s t a r s , caus ing near cons tancy of the con2ition.s i n ths ozic:

envelope. I n f a c t , d u r i n g evo lu t ion , t h e outward growch C L

the semiconvect ive zone i s s e r i o u s l y l i m i t e d s i n c e it i s

~i

, I

- formed i n i t i o close t o t h e s t e l l a r s u r f a c e , and i ts

'

I

~

inward growth i s i n h i b i t e d by t h e much s t e e p e r cpad ien t of

mean molecular weight occur r ing i n the r a d i a t i v e interned:-

- a t e zone of t h e more massive s t a r s .

The i n c r e a s e of X wi th s t e l l a r mass, however, masks

the a c t u a l i n c r e a s i n g ex ten t of t h e r a d i a t i v e zone. The

growing importance of t h i s zone is due t o t h e s m a l l e r amount

I

of semiconvection a s a+ approaches t h e s u r f a c e for higher

masses, and to t h e a s p p t o t i c 1ixi.t ta the i n c r z a s e of t h e

convec t ive core. Hence the g r a d i e a t af hycz~ge:? aS.;:-,Sz.r.cc

.,---I - w i t h raspcct to W B S f r a c t i o n beso.xes gcncler a:< t h e b C L A A L sc

mass i s inc reased , reaching , l i k e 1, an asyrr,pzutic l i m i t

( F i g 2 ) -

The t i m e s c a l e of evo lu t ion of very massive stars i s

I t Eoes riot change very on the order of a m i l l i o n years .

much with i n c r e a s i n g mass, because, a l though the l u r i n o s i t l r

is i n c r e a s i n g l y b r i g h t e r , a l a r g e r f r a c t i o n of the initial

hydrogen content ge ts consuiied. The t i m e s c z l e i s given by

J Z

where E = 6.0 x 10l8 gm/erg.

t r eme ly massive s t a r s , X a t co re hydrogen exhaustion

approaches a c o n s t a n t va lue and L - M, it i s c l e a r t h a t 7

Since , i n the l i r f i t of ex- -

. has a l o w e r bound, v i z .

E xeXe(2 + &) . = - 8 m G I + &

For an i n i t i a l composi t ion of pure hydrosen, T = 2.16

m i l l i o n yea r s , and for o u r chosen i n i t i a l conpos i t i on ,

10

, T = 1.36 m i l l i o n yea r s .

I

~

The e v o l u t i o n a r y t r a c k s of OUT rnGdelS on tha 3-R d i a - i I gram a r e shown i n Ficjure 3. The l o c u ~ of po in t s repl-csent- ,

ing modcis at tine saze cvolutionzxy sLacz tenz'l~; ;o CL:TVE:

o v e r toward c o o l e r er ' feckive tenperat ; res a t -,he :?ic;kest

masses, f o r t h e fo l lowing reason. L e t L - M". ,Then sincc.

R - M, w e have t h a t T, - M 'ff-2)'4 fron the Slack-body sur-

I "r

f a c e r e l a t i o n . I n t h e l i m i t of very l o w nasses, CI = 3 arid

so T, - K%; the e f f e c t i v e t e m p e r a t u e i n c r e a s a s wi th mass

(and l u m i n o s i t y ) . I n t h e l i m i t of very h i g h masses, CL = 1 I

I and so T, - M-*; t h e n t h e e f f e c t i v e teiriperature decreases

w i t h i n c r e a s i n g mass.

IV. FINAL RENARKS

Schwarzschild and H a m (1955), among ot'ners, have alse

, i n t e g r a t e d sequences of s t e l l a r models f o x very massive

s t a r s d u r i n g hydrogen burn ing . These ea r l i e r r e s u i t s , bas;<

on t r e a t m e n t s of t h e u n s t a b l e i n t e r m e d i a t e zone w h i c h wexz

n o t e n t i r e l y s e l f - c o n s i s t e n t , a r e n e v e r t h e i e s s very s i i a i l zz

to o u r s because of the i n s e n s i t i v e dependence of the s t e l l z = r

s t r u c t u r e on the composi t ion mod i f i ca t ions (Hoyle 1960; i I . Paper I).

(.

I

11

W e t h e r e f o r e have c o n f i d e x e i n some p rev ious c o n c 1 ~ -

0 - - - s i o n s reached by t h e s e workers. In p a r t l c u - a r , sc:w~rzsc:r:: i 2

and H a r m (1959) p e r f o r m d a s t a 3 i ; i t y a n a l y s i s ;3z t n c i r

models of inassive stars aa6 5 o t n 5 a T ax--z-x-. 2: L G !::

which would be s tab le a g a i n s t r a d i a l pulsations. I n an

ear l ier paper ( S t o t h e r s 19651, ic w a s shown t h a z a t leasr z i - . ~

p u l s a t i o n a l e igen f rcquezc ie s would bc ;;n.zZ'feczeJ for V X L O L S

t r e a t m e n t s of t h e s ex ic snvec t ive zone i n such szars . Ke

t h e r e f o r e suppose t h c t the p u l s a t i o n z l GrowtI-. rites w i ; l

a l s o not be l a r g e l y affected. I n view of the v io lence of L . ? ~ :

c a l c u l a t e d i n s t a b i l i t y d m v e 65 M, , it would s*cex ha^ z?.!

c r i t i c a l m a s s inust ir, arLy c a s e L e f a i r l y well d a t e r m i n e < -

Frox t h e p r e s e n t work, 60 X A i s c lose t o t h e mrass a r

which the semiconvect ivc zone hts its grea te sE excer.2. -

co inc idence i s no t fortuitous s i n c e , as w e have seer., 21:~

effec-c of r a d i a t i o n p r e s s u r e op. t h e convec t ive instakilit-

i n c r e a s e s r a p i d l y up t o 60 M, , wherea f t e r i t i n c r e a s e s n s z L

s lowly , a sympto t i ca l ly reaching a l i m i t . Radia t ion pressuzc.

h a s a s i m i l a r e f fec t on t h e pu l sac iona l i n s t a b i l i t y ( likei.jL.,L

enhanced a s 2 approaches 3 ) .

- . .

r

Boury (1963) obta ined a maxinu;n s taSle mass of 230 >:

fo r stars composed of pure hydrogen. S ince t h e models of

Schwarzschi ld and H a r m had a hydrogen c o n t e n t of 0.75, w e

might expect t h a t t h e c r i t i ca l mass for p u l s a t i o n a l sta-

.I I I I

t

bility in our models wauld be s l i g h t l y less t han 60 Na.

12

T h i s work was Yupipr ted by an Lis - 5X pcatCccto~al I

r e s e a r c h a s s o c i a t e s h i p under t h e National A e r o m n t i c s and

Space Administration. It is a pleasure to thack Dr. RG5erz

Jastrow €or h i s h o s p i t a l i t y a t t he Institute f o r Space

S t u d i e s .

e*--- . A W J V --- I

*.

13

RZFEREXCES

' Boury, A. 1963, Ann. Z ' A ? . , E , 3.54.

Chandrasekkar, S. 1933, Ar Iz t roZcct lc3 TO ,:-&e Scu5.v sf Steilar Struct-cre (Chicago: University O € Chicago Press) ..

Eddington, A. S. 1926, Tr5e Z n t e r c a l C c n s t i z s t i c n of t5e S t a r s (CaniSriZge: cambridge Unive r s i -~17 ?r 3ss)

Ezer, D., 2nd Caxeron , A.G.W. 1363,, unpublisked.

Koyle, F, 1960, M. N., 12s. 22.

Schwarzschild, M., and H 2 r m , R. 1958, Ap. J., 12, 348.

. 1959, i b i d . , 1 2 9 637.

Stothers, R. 1963, Ap. J., 13,' 1074.

. 1965, i b i d . , 1&1, 671.

. 1966, i b i d . , 143, ,-".-., i n press.

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