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- ikfilicli-FORMATION AND EVOLUTION OF THE SUN . no f ER- -- - I . . . - ./ tr--

ROBERT R. BROWNLEE

University of California, Los Alamos Scientific LaboratoryLos Alamos, New Mexico

I. INTRODUCTION

In these days of renewed vnd vigorous interest in the origin,

structure, and past and future histories of the solar system, detailed

knowledge of the formation and evolution of the sun has become of.vital

importance. For example, knowledge of the rate at which radiant energy

was streaming from the solar surface, and the extent of that surface2

at the time of the formation of the planets is basic to any realistic

theory concerning their formation. Coupled with this new and wide-

spread interest in the time-history of the sun has been the advent of

the high-speed electronic computers and the development of techniques

which enable the astronomer to investigate many complicated relations

which heretofore had been considered beyond reach. It is my intention

to discuss briefly the present state of knowledge concerning the history

of the sun, what investigations are currently going on, and to indicate

some of the' fields of knowledge in which uncertainties and difficulties

continue to emist. - -LEGAL NOTICE

States, nor the Commission. nor any person acling on behalf of the Commus/M:

TWs report was prepared as an account of Government sponsored work. Neither the Un ted

- A. Makes any warranty or representation. expressed or implied. Mth respect to the accu-

 

As used in ure above, 'peraon a/Ung on behalf of the Commission' includes my em.

racy, completeness. or usefulness of the informaUon contatied in this report, or that the useof any inform.Uon. app....8, method, or pricess $../.led I 'W. report may oot Infringe

1privately owned righta; or

B. Assumes any ilibililes with respect to the use of, or for damages resulting from theuse of any blformation, apparatus, method, or process  sclosed in this report.

ployce or contractor of the Commission. or employee of such contractor, to the extent thatSuch employee or contractor of the Commlision, or employee of Buch conlactor prepares,disseminates, or provides access to. any information pursuant to 88 employment or contractwith the Commission, or hle employment with such contractor.

DISCLAIMER

This report was prepared as an account of work sponsored by anagency of the United States Government. Neither the United StatesGovernment nor any agency Thereof, nor any of their employees,makes any warranty, express or implied, or assumes any legalliability or responsibility for the accuracy, completeness, orusefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privatelyowned rights. Reference herein to any specific commercial product,process, or service by trade name, trademark, manufacturer, orotherwise does not necessarily constitute or imply its endorsement,recommendation, or favoring by the United States Government or anyagency thereof. The views and opinions of authors expressed hereindo not necessarily state or reflect those of the United StatesGovernment or any agency thereof.

DISCLAIMER

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.

4, .

KNOWLEDGE OF STELLAR FORMATION

A great variety of astronomical observations support the view

that stars are formed from the interstellar medium. As can easily

be shown  the most luminous stars are emitting radiant energy at such'.

a rate that they cannot possibly be very old, and these stars are

invariably found in intimate association with clouds of gas and dust.

' . This is true within our own and other galaxies. In some instances

the space motions of such stars can be measured and are found to be

too small relative to the surrounding clouds of gas to be mere pass-

ersby. The conclusion seems to be unmistakable that these stars were

born from the surrounding material.

There ts a class of stars known as the T Tauri stars whose

spectra and light variability suggest that they are passing through a

short lived phase of their evolution, and these stars are aleo

imbedded in clouds of interstellar gas and dust.

In a number of places in the sky, dark spheroidal globules can

be seen projected against luminous clouds and the obvious interpreta-

tion is that these are' protostars which will at some time in the future

contract sufficiently that they will begin to glow, and thus become

stars.

I hasten to point out that there have been many theoretical

difficulties in providing good quantitative explanations for the above

observations, but nevertheless the observations appear to give great

confidence to' our belief that stars are being born from the interstellar

medium.

2

BASIC SOLAR DATA

We may well ask if the sun contains information within it which

Will tell us anything of its birth and its aging processes. The

obvious fact about the sun, of couree, is that it is shining. An

inevitable result is chat it must evolve, for its supply of energy

can not be infinite. As this energy source must lie in the deep

interior, what can we say about it?

At first glance the deep interior appears to be less accessible

to scientific investigation than any region of space around us. How-

ever, if we ask the correct questions, we find that certain information

about this important region of the. sun is being communicated to us quite

directly. First, a gravitational field is emanating from the sun and

the motions of the planets and minor planets within this field allowI.

us to determine the solar mass. Secondly, the solar radiant energy

crossing a unit area of the earth's surface can be measured, and theluminosity of the sun, i.e., the rate at which energy is being emitted,

is easily derived. Thirdly, the diotance of the earth from the sun

,  . being known, the size of the sun can be determined quite easily. Thus

we determina the three basic ·stellar quantities for the sun .. the mass,.

the lurninoeity, and the radius.

' There are yet three additional bits of information which should

be mentioned. First, through spectroscopy, something of the chemical

composition of the outermost layers can be determined. This problem is

fraught with difficulties but even so such derived information is veryimportant to our understanding of what has gone on in the interior.

3

4 *

-

Secondly, sonie kind of lower limit can be Bet to the age of the sun,

for certainly it cannot be younger than the earth. Furthermore the

surfacd of the earth contains information which suggests that the

luminosity of the sun has been relatively constant for very long periods

of time . The third obseryation ie 80 obvious that it can easily be

overlooked, but it is of basic importance to the theory of stellar

interiors. That is the fact that the sun is essentially spherically

symmatric and in hydrostatic equilibrium -. it is neither pulsating

mor blowing up, at least not on a very fast time scale.

The combination of all of the above information leads to many

interesting conclusions. For example, from the known size, i.e.,

volume, and mass, the mean density of the sun 18 immediately obtained.

The fact that the sun is in' hydrostatic equilibrium means that the total

pressure ac any point muat be just suffidient to support the weight of

the cverlying material. Making various assumptions about the density

distribution, we can regulate the pressure at every point by adjusting.

the terapsrature at that point until hydrostatic equilibrium is achieved.

Naturally, an arbitrarily chosen density distribution will undoubtedly

result in a temperature distribution which will in no way correspond

to .8 realistic flow of the energy such that a correct luminosity will

..be achieved. Obviously there needs to be a simultaneous solution for

all parameters of the problem 16 order to match what ia so "casually"

observed:

·· 4

A

Perhaps the most striking conclusion concerns the energy, for the

average rate of energy releaBe is about 2 ergs per gram per second,

and apparently there have been no marked changes in this rate for four6, I

or five billion years.-'.,7.

The proposed sources for this energy have been varied. Perhaps

4.the oldest suggested source is that of combustion. But proportioning

the mass in the most ideal way into carbon and oxygenD the maximum

age of the sun would only be about three thousand years. Clearly,

another source must be sought. It has been suggested that solar material

might be supplemented by an infall of meteoric -material, but the density

of the material in the vicinity of the sun is quite insufficient to be

held responsible for any significant contribution to the solar energy.

A little more than a century ago H. von Helmholts proposed that

gravitational contraction was the source of energy for the sun and stars.

The gravitational energy stored in the sun is approximately equal to

155 x 10 ergs/gram and if contraction were responsible for the luminosity

of the sun, the solar diameter would be decreasing at about the rate

of 40 meters/year, or about 4 kilometers in a century -- a rate which

is not observable. (Good angular diameter meaeurements of the sun can

be obtained only for about the last two centuries.) Howevers the length

of time for the solar material to contract from an infinitely large

size to its present dimension would be only a few tens of milliond of

years. This is also an interval of time which is much too short.

D

5

Finally, increased knowledge of the atom has led to the relation

20between mass and energy, 2 e wt2. or approximately 9 x 10 ergs/gram.'

.  This is a source so plentiful that in four billion years and at the

present lumingsity, the eun would have decreaaed in mass only about

. 0.02 per cent. It is clear that the sun is feeding upon its mass and

at., the present ratd of consumption, apparently has a long life ahead.

II. BASIC DIFFERENTIAL EQUATIONS OF STELLAR STRUCTURE

From the above information, and with the assumption of spherical

Symrne try, the following differential equations can be derived.

1. i&41 = .G *(r) 9(r)dr 2 Equation of Hydrostaticr Equilibrium

dM (r) 22. ----- = 4-1 r p (r) dr Continuity Equationdr

Conservation of M:ass

dL(r) 23. - = 4u r p (r) e Luminosity Equationdr

1

&I =- .3 K o (r) L(r) Energy Transport4. a. Radiationdr 4ac 3 4 2 EquationsT. (r) 41-  r

b dI = 13-1 I dE'dr r P dr Conduction

' '

6

:.1

5. a. P = k. P(r) T(r) +   T4 Equation of State H -

. .5/3 or 4/3b. P p Const ( 2, i

electron W /

I .

The definitions of the variables in the above equations are:

r radial coordinate P total pressure

G gravitational constant p density

, M(r) mass interior to a sphere L(r) energy crossing the

of radius r surface of radius r

  energy generated ,gram-lsecond-f T temperature

a radiative constant c velocity of light.,

K Rosseland mean opacity B mean molecular weight

Ik Boltzmann' s constant a average mass/electron

r adiabatic exponent H mass of unit atomic

weight

Three quantities, u, the mean molecular weight, K, the Rosseland mean

opacity and 9, the energy generated per gram per second, have been l

introduced in the above equations. Each of these ia a function of the

chemical composition, the density and the temperature, and all are presumed

to be known. The calculations of the latter two quantities, while quite

feasible with electronic computers, are enormously complicated and require

elaborate machine codes and experimental results. However, assuming

them to be known, the object is to determine the temperature, pressure,

density, mass interior and the energy crossing the sphere at every point

' 7

.

: r within the star. Since there are, in effect, five equations in

five unknowns, a model for the sun or a star is obtainable.

The boundary equations for theoe equations are as follows:

At r = R= the radius of the stm, the pressure, the density and the

temperature are presumably knowa. At r = 0, the mass interior,

M(r) and the net flow of energy across thaC Bphere, L(r), must equal

zero. These ' conditions are two too many, and some freedom ofI .

choice in their use is poasible. In practice, the two are veed as a

cheek upon the calculations. In general, the history of the e'ntire

i problem has been the changing application of these boundary conditions.' To obtain a model of the sun, or any star, one needs the mass ,

luminosity, radius, and the chemical composition. In general these

.are a function 02 time, hence a model of the sun represents the sun

for only one moaent in time. However, since the. temperature, density

and chemical composition are finally known for every point within the

model, the rates of conversion of mass to energy are known for every

Point, and therefore the rate of change of the chemical composition

is also knewa. Such a change will alter the molecular weight, the

opacity and the energy generation and these will in turn alter the

structure of the model. Thus, having a model for one moment in time,

themodel for a subsequent time is determined.

The calculation of a model on a desk calculator is an exceedingly

laborious and time-consuming effort. The wide-spread use of electronict'

8

..

..

computers has made possible the calculaticn of such a model in· 8

few seconds and a number of different techniques have been developed. '

Because of the great interest in the sun, models for stars of one

solar mass have race ived a great deal of attention.

III. EVOLUTIONARY SEQUENCES FOR MODELS OF ONE SOLAR MASS

EARLIER RESULTS

The first detailed calculations of a vequence of models which

represented the contraction phase of the sun were made by Henyey,

Le Levier, and Levee*. Somewhat later, using a different cortiputational

technique, Brownlee and Coxi calculated evolutionary sequences uaing'

idproved opacities and considering the role of an outer convection zone

and all sourcea of nuclear energy generation. Among the latter, the

possible role played by deuterium was investigated. The results are

shown in the theoretical Hertzprung-Russell diagram in Figure 1.

Prom tile initial model for these tracke, when the radius was approxi-

Istately thirty times that of the present auni the contraction was traced

ubtil the transmutation of hydrogen luto helium in the center of the

motlel yielded enough energy to cause the rapid contraction to cease.

At the outset, about 97 percent of the gravitational energy was still

present. Thus calculations ranged over a tine during which almost all

*· Henyey, L. G., Le Levier, R., and Levee, R. D. 1955, Pub. Astronom.Soc. Pacific, 67, 154.

1 Brownlee, R. R. and Cox, A.N.,·1961, Sky mid Telescope, 22, 252.

1.

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6 1 «,4 7 -Ill 01. 'i

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LOS ALAMOSPTI' '.,0 LABORATORY

A- 627931ELE.iSE RE-ORDERC Ai,OVE NUMBER

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of the potential enargy was converted into internal energy Bad

radiated energy.. When no 8euterium was presents this period of

time was greater than 100 million yearm. When deuterium vac added

iri an ame',mt eqi:al to that found in the earth and in metooxiteso

about 2 huadredths of a percent by mass, chere was a significant

pause in the *entraction at the moment the de:itral temperature 'and

density Were sufficient for the proton-deuteron reaction tc occur.

Thia caused a total increase in the Contraction time' of a factor of

about two. An additional calculation was made on the assumption that

the deuterium was constantly mixed throughout the model, and again

the contraction time was increased.

Figure 2 shows plotted the radius of the model va time. The

momant the deuterium began to burn is quite apparent, as wall aeI.

the moment in which the resulting enargy release stopped supperting

the outermost layers.

Figure 3 shoes plotted the Ener#y Balance as a function of time.

Figure 4, ie addition to showing the radius vs time plot and

the change of the potential and internal energies with times shows

the temperature changee vB times as well. Of pmrticular interest in

thie figure is the abrupt halt in the increase of the temperature in

the center indicated as Bone one. A abort time later the temperature

Changes £811 to zero throughout tlie modelp. and this is. said to be tile i· 1time of the primordial solar model cr the initial sun.

,\

11

...

2.0

SOLAR GRAVITATIONAL CONTRACTIONZONE INTERFACE POSITIONS VERSUS TIME

1.5 DEUTERIUM BURNINGNO DEUTERIUM BURNING

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LOS ALAMOSPHOTO LABOR,1 TORY

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Ul

80- - . - il ---) TOTALW - ..

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0 \- \

-4 - Fig. 4. Changes in the solar model --4\originally containing 0.02 1,er cent 'g POTENTIAL

-6 - deuterium. Zone 30 is the outside - -6

surface. Some of the gravitational po-tential energy during the contraction

14 - is converted to internal heat energy,ZONE I - 14

and the rest is radiated. Teml,eraturesare those midway between the bound- ZONE 6

12 - - 12

z aries of each shell, except for zone 15 which gives the temperature at a pointJ 10 - - 10w midway between the solar center andf the first interface. The letters at the ZONE 1408- top indicate four el)ochs for which de- -8

W I.-W tailed information is given on page 256.H 6- -6

80

4- -4(D

o TEMPERATURE ZONE 22

- f2- -2

ZONE 30==-- --- -

0 0

200 180 160 140 120 100 80 60 40 20 0

MILLIONS OF YEARS

'-9

LOS ALAMOSPHOTO LABORATORY

5 6 1 Al 9 31;0.

L U .6.*v.U; RD ORDERD C A*.01'E NUMBER

,.

Aft€r the wadel has reached the main sequence of stars its

path in the Solowatrio Magnitude 400 Effective Temperature diagrad

ts drastically aleer*d. Changes in structure still occur, but on a:5

much longer time voi.le. As ths hydrogen in the central regions i a

0942,tedd to helium, the mean molecular weight increases, the central

regions collapse slightly, the opacity decraaeee,.and the density

and temperatitre are increased. The energy preduction is thus enhanced  

and the luwinosity of the modul rises. The oVeer rezions actually '

ex'patid during elds phase, so that the radius and lominosity of the

.7todal slowly increase. (The decre*Be in mass dwing this phnse is

easeotially negligible.) After sote period of tima the model should

- pass through or nttar the positioa Of the present sua, depending upon

ita re*emblance to the sun.

All of the above mentioned calculations were sade with an assumed

chemical composition of 74.4 percent by weight of liydrogen, 23.6 percent

by weight of helium and 2 pescent by weight of the, heavier elegents.

It vaa fowld that the model took entirely too much ti*re to reach the'

I

lumineaity and radi.us of the present sun--a period of about 11 billion

years. An altered chemical composition waa suggested, since a decrease

in the initial abundance of hydrogen and a corresponding inorease in

the abundance of hellwa would serve to shorten the time betwe*n ·'the

initial and present solar donditions.

15

RECENT RESULTS

In the earliest stellar models, the simplest possible surface

boundary conditions were used, i.e., that the pressure, temperature ·

and denoity all vanished at the surface. While these conditions

were quite adequate for making preliminary and general calculations

of atellar interiors. experience has show'n a need for greatly refined

boundary conditions. In fact, it is now clear that nothing short of

a *11 determined model stellar at; aesphere will suffice to establish

a proper surface boundary condition. This is proving tolbe especially'

true for the study of calittracting solar models.

HayaShi* wae the first to point out that the calculations de-

sci'ibed above for the earliest phases of the contraction would have

resulted id an entirely different history had tHe convection in the,

outermost layers been handled dorrectly. Hle prediction was th# result

of his study of the importance of convection in the outer envelopes of

the late-type giant stars. In the models just desc:sibed the boundary

condition used was derived from atmospheric madels by Swihart and Fisehel t,r

and with an assumption of constant gamma between the point of the onset

of convection and tile outermost mass point of the interior. The assumed

value of.gaewa was apparently toe large and the envelope structure and

.,

* Hayaahi, C., 19610 Pub. Astran. Sos. Japan.131 4501

t.' Swihart, T. L., antd Fischel, D., 1961, Ap.J..Suppl. 5, 291

.

. I

.

16

f

..

I

..

the depth 02 the conmection reston are very sensitive to the value

of ggmma. u,ed. The correct bcundary condicioe should consider the

variable gauna ond the eenvactive efficiency, as liayashi has done.The result ** a lower temperature .for the Itermbet maes point of the

. laterlor, a correspobding change .in strueeure, and a mueh deeper

cotivective resion.

Recent reaults verifying Slayashi' e prediction have been re-

ported by Eser and Cameron*. dad Weymana and Moore. f Ezer and.Cmneroa

calculated twenty-seven models by .a technique which did not yield

a st:lek evolutionary sequence but which ser· d to delineete such a

sequence. They found thit for all •aodels above 2.5 solar rsdii,

the convection w#Is cmmplete all the way to the center. The

efficieney in carrying anergy by convaction is 80 muth greater than that

by radiation flow chat tha luminoot tios of the models are very high

when compared with the previously reported results. Further, since theenergy im ewitted at auch 3 great rate, the collapse of th,3 model ia

greatly hastened and the corresponding conversion 09 gravitational

energy la dreatly adeelerated. The final result is a grgatly ohortened

calculated titi,e for the andel to reach the initidl r.ain ecquer.co of

stars, 8.e. , i t s prittordial 13**te.

8 Rzer, D.5 and Cameron, A. G.W., 1962, The Early Evolution of the Slm.

preprint

9 Reymanno A.9. and Moore, E., ·1962, The Abmidance of Lithium and theSthucture. of Gra*itatte:lalbt *mtract#49 Stars of ddeSolar Meas, Peepriat

\

13 7

)„

Recent work at Los Altmos is in qualitatite· agreement vith

that of Bayashi and Ezer and Cameron. Since the te¢liniques used

  and the various assllmptions necess&:y to the determination of

boundary conditions, the convective energy flow and die chemical

colzposition are B'imilar but not necessarily the sawe, exait sigiree-

neak is tert:einly neR to be expected. l

·The chealieal Composition for the re,ults described below was

59.6 percedt by WBight hydrogen, 3:.4 percent by weight helium, and

2.0 percent by weight heavier elements.

The epacideo for this eompcoiticn ae calculated, by Stewart amd

Cor :12 los Alazoa *re shown in Flattrcs 5 and 6. '1114 coda whiah

is used Co calculdee these opacities coasiders baued-boufd, bound-

free, and free -free absorption, electron statterillg End negative ion

absorption. For the data shown, the ranse in temperattlre is fron:

4.343 x 10-4 kilovolts (5040'K) to 10 kilovolts (1.1603 x 108.K).

The range in density is fro,·1 10-14 gm/cra3 to 103 dns/(2 •

The equation of state data used haze also bean Computed by Stewart

mid Com for the identical ranges 02 temperflt»re and density.

Soma of the .results of the most recent calculations, .are shown

in tha figureE which follow.

In Figure 7 eha luminonity an a function of r#diuM is Chaun fee a

particular model with R/tle = 24.2. As the boundary coaditions be*48

used are considered Preliminary, the actual value of tha luminoaity ia to

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8- f.- . - :Fe=

1: - - - - 't. 0 -1 1- 311...... ....

84 1 - - 1 +1. 0

.t' - _I.

1- -.-+

d-1.-.... -1- 1 .....$: 84 1.4 - \f #- E. . . . . . . \\\

 ----»_s. . . . . . . I

:

-* mw- 1 ' =. i i - 6-wrif-.=i- 1 Tr 1- * d 4 *  +- - -

0 - . - 6  O 0 0- - - -- 'O/ T

24Fig. 7. The Luminosity (in units of 10 ergs/sec) vs Fractional

Radium for a Model with a Radium Equal to 24.2 Times thePresent Solar Radius

./

 --  t.t ,S .'.1.A'•It)S . ,1 , . 1 , „. ':' 'ORY

6 z ·C u w D

i .,., 3,1_:.'li 'ILL__11

Figure 8 shows the pressure profile *or 4he oaze zelel. Thu

.16 3pressure is in units of LJ crsdicm. .

Figures 9 an€ 10 ahm, the temperature (in kilovolts) and the

log of the density va radius, respectively. These two fiBuree are to

be coipared with iisures 11 and 12 which show plotted the arme2

quantities, but for a model of radius R/11 u 1.29. The great increase

in temperatlize ard density throug)tout the itar during t:lia course of the

contvaction can easily be noted.

Figures 13 cid 14 show simi tar results derived from the earlier

work dene at Los Alamos. When theae arc cominred with tho four preceding

figures, thu different to=pereture and density distributions are quite

stribirt. The differencd ig wiraply that in (IR initial solar e:>del,r

the convection rosie;n does not extend throughout tlie star but is rather

gonfined to an cuter region and a swill central core. The model

whi<li is convective throushout shows quite flat profiles.

' It is of interast CO se,3 11JW the actual temparaturc gradient differs

from the adiabatic scadiant of the model with R/il' = 24.2. I'ltis caa beseen in Figure 15, vite:e the excess af the tesrerature gradient OVUr zhe

adiabatic is plotted versus the radius. As the successive models approach

th* radias of the preaent 340, this difference becomea less and less.

Eventually tho opacity in the central reglcus becomes aufficioutly low that

the cotriection ceases. In the outer regions the temperature graliaut

betomao that of the adiabat.

· 22

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-110-01•000-02' 4/ *01  •015 •02(  +02k •030 +03g +040 +049 +056 +055 +060 +065 +07(1 +075 +08  +085 +09  +09  7io/

Fig. 86 Log Pressure vs Fractional Radius for Model R • 24.2 (3.*

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 ' Fig. 9. - Temperature vs Fractional Radius for Model R = 2.4.8 0 . ,

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Fig710.-Log Dengity vs Fractional Radius for Model R   24.2.0.

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... F. . ,Fig: 11.- Temperature vs Fractional Radius for Model R = 1.29-0 -.  11 1-

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Fii. 124 Log Density vs Fractional Radius for Model R : 1.29 0 - « j

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  Fig. 13. Temperature vs Radius for a Model of the Initial Sun... --

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Fig. :14. Log Density vs Radius for a Model of the Initial Sun -·

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Fig. 15. The Excess of the Temperature Gradient over the AdiabaticGradient vs Fractional Radius for Model R 9 24.2 0 '.

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TRE, SOLAR FGTU94

·Finally., a tew egamvznte should be maile abouk the future/

..

bittory of  14 sua. From work' done by many workers' in the past and

iroe Calculations currently beins wade at #he Univarsity of Galifornia

by Manyey, at the California Institute of Tdghnoil*·Jy by R. 6. Saars,

at Priweat#a by SchwZI.rjechild, at the NABA Institute for Space Studies

6, Eze* and Coaeron, at the Los Alamos Scientifie Laboratory* and a

ninber of other places, it is clear that the amouat of information regard-

ins tile total life hiotory in the eun has jowt begun te acqpmulate.-

Even 60 we can state with come certainty whar the "imwidiate" ruture/ I

of the ste, will be .

Dy Cottlpering the "age" of the model wilen it *mt closely repressn#s

the sun with tlte age of the sun as derived by other muthods, it is

poseible to select a chemical compeaition which will give the "correct"

age. Thie compooition, depending on the opacltles one aess, etc., appears,

to be about 63 to 65 perceut by weight of hyflrogen, 2 to 3 pergoat by  1.weight of tile heavier clemente, the remainder beivg helium. Thuo an

\

1 additional "observed" fact is used to determine an additional model para- ,

6,2 ter.

Currently, the bent models seem to suggest that in the centleal

re$*ana of the Present sun about one halt. of the 'original hydrogen has

been converted to lielium. Thic proce3$ will continue until all of

hydrogen is illi:ned, at which time the cant:al regiums begin to contract

8081078 at mor e rapidly. Ths outer regiona gradually accelerate in their.. expansion and finally--curnnit modela buggest al ter gbout six or seven

*

31r

i.>t..1

litlizes *re yeare--the sun.e.it'l turn. a-*4 6:em the.main. eequeice...

rattler whruptly (16 1/*0-40 a giart gte.

She centrbl tekiperstars& a:0 6%*gities will h#*e l,Come

4.,Q p f E  4 fLY·:· 1.q 4·4· *4• P•:· -, - ·;i' helium vili iteelf bogin to be. tratle--w-:i -41--6-9 -*6* .*#w.. -na

mutod Lato carbon. Ther# gill follm a colowfbI voried of tbe sus,

as a giant Stag, making *sny ulttmente within ic. Unfor'*State ly,''

for 6% pirament the''theeretidal taile:totantjins of this epoell, cli 1

thes# fjo £0110W, bowevar encciti*g, te rather impes fectly held. The

'·i. abticip.fee.d iaeraaees of ot,e kz ladae in tim nqar futtlee. promiae to

meke thii field of Ateller mddeld n vety prefitkble ene.

1·

*

..

.

./

32