masses (Ezer and Cameron 1967) to the lower-mass range. The evolution of 0.1, 0.2, 0.3, and 0.4 solar mass stars has been followed, starting from the tllreshold of stability, the point a t ~vhich the gravitational energy of the contracting protostar becomes just greater than the s u l ~ ~ of the thermal, dis- sociation, and ionization energies of the gas, through the gravitational contraction phase, to the hydrogen-burning phase, if the star is able to burn hydrogen.

The general description of our stellar-structure computer program is given else~vhere (Ezer and Caineron 1965). This program \\-as modified before the evolution of these lo\\,-mass stars was studied. These modiiications, 11-hich are discussed below, included pressure ionization in the Saha cq~lation alld better treatment of the electroil screening factors in the thermonuclear reaction rates. We also discuss the equation of state in our stellar-structure prograin which takes into account the llonrelativistic partial and complete degeneracy of the electrons.

The chemical composition used in this study is the same as that used by us for the evolution of the higher-mass stars, that is, the solar composition, \\-here X = 0.739, Y = 0.240, and Z = 0.021 (Ezer and Cameron 1965). The opacity of the material \\-as calculated \\.it11 the latest version of the Los Alamos opacity code \\;hich contains line opacities. The transport of energy by convective Inass motions of the gas is treated by the mixing-length theory. The ratio of the nlixing length to the pressure scale height is talcen as 2.

In the present worl;, the main source of energy generation is gravitational contraction. But in the cases \\;here the interior temperatures become high enough to support nuclear burning, energy generation by the I-I(p, P + v ) ~ D , ?II(p, Y ) ~ I - I ~ , 31-Ie("-Ie, ~ P ) ~ I - I ~ , and %Ie(cr, Y ) ~ B ~ reactions and b~7 the CKO hi-cycle are included. The energy generation rates by these reactions are given in the above cited paper.

Eql~at ion of State For a rlonrelativistic gas the total pressure can be ~vritten as consisting of

three parts: the ion pressure Pi, the electron gas pressure PC, and the radiation pressure.

P = PI f P, f 4u+T4.

The pressure of the nondegenerate ion gas is given by

\\-here 9i is the gas constant per mole, and p 1 is the mean molecular \\eight of the ion gas which, for ions of atomic \\-eight A , and mass abundance X,, is given by

In the absence of electron degeneracy, the electron pressure P, is given by

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EZER X S D C.-iIvfEROiX: STARS OF LOW MASS

where pc is the mean molecular \\,eight per electron, given by

When the Fermi-Dirac distribution for the electrons starts to deviate signifi- cantly from the R/Iax\vell-Boltzmann distribution, the electron pressure of the gas niit11 a degeneracy parameter + is given, in general form, by

and

\\?here 0 = l / k T , nz, is the proton mass, and tlle kinetic energy E of an electron with mo~nentum p and mass m is

These expressions are valid for all densities and temperatures. They reduce to the perfect gas law for + < -4. For + > 20 the gas becomes completely degenerate and in this case the electron pressure of nonrelativistic gas is given by

where

In the transition region where the electron gas is partially degenerate but nonrelativistic, the ion and electron pressure and density are correctly represented b y the folio\\-ing expressions:

For a given pressure and temperature, the equation

is solved for (+) by an iterative procedure in order to determine the value of +. In the conlputing program the functions of Fllz(+) and $F312(+), which are

tabulated by AiZcDougall and Stoner, are expressed by the polynomials;

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34G4 C.\S.\DI \Y JOUIIS.\L OF PIIYSICS. VOL. 45. 1067

%F3/?.(#) and I;l12(#) \\-ere represented by a 29th-order polynomial for $ < 5.4. The order of the polynomial ?z is 17 for 5.4 < # < 20. The coefficients of these polynomials are given in the Appendix.

With the inclusion of the above equations in our stellar evolution program, the equation of state allows for partial and complete nonrelativistic degeneracy of the electrons. The use of this equation of state for the study of the internal structure of the low-mass stars under consideration is \\jell justified. Figure 1 sho~vs a temperature-density diagram for the equation of state. For tlle temperatures and densities above the dashed line corresponding to the degeneracy parameter $ = 20, the gas becomes co~npletely degenerate. Depending upon their thermal energy, the ions may form a uniform gas or an almost rigid lattice. When the ratio of the thermal energy of the ions to the "melting" energy of a lattice is smaller than unity, the ions form the lattice (Salpeter 1961), and conditions then approach the solid-state case. The solid curves in Fig. 1 represent the run of the central temperatures and densities for the indicated masses throughout the evolutionary study. I t can be seen that,

FIG. 1. Thc temperattrre-density diagram for the equation of state and the rnri of central ternperat~rres and dcnsities for the indicated masses throughout the evolutionary study.

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EZICR AND CAMEROX: STARS O F LOW MASS 3465

starting from the tliresliold of stability, the mate^-ial inside the stars is governed sufficiently \\.ell by the perfect gas la\\.. I t becomes increasingly degenerate as contraction proceeds. I t is only in the evolution of the O.llUo star that conditions approach the solid-state case.

The effect of pressure ionization in tlie Saha equation Iias been treated in the manner suggested by Rouse (1964). I-Ie postulates a modification of the Saha equation in order to remove the shift toward neutrality as density increases, and \\.rites the Saha equation as

\\-liere C,, is tlie concentration of the ptli ioilizatio~l stage, I<,(T) is a function of temperature, and (I, represents a density-dependent function that describes the probability that a pth ion can exist a t density p in a mean atomic v o l ~ ~ m e of radius ro. He finds cP, to be

\\-here y and m are constants to be determined, and a, is the classical Bohr orbit for electrons in the quantuni state with the principal quantun~ number n.

p and ro are related through

\\-here A is atomic \\-eight. Now a , = n2cro/Z, \\;liere (LO = fi2/p?, e is the elec- tronic charge, and Z is the atomic number of the element under discussion. Iiouse found m = 3 to be satisfactory for hydrogen and iron and m is assumed to he equal to 3 in our program. y has the follo\\;ing significance \\.it11 respect to the hydrogenic wave function: y = 1 corresponds to ro = no, y = 1.5 corresponds to the most probable position of the electron, and y > 1.5 indi- cates a position on tlie probability distribution beyond the nlaxiiiium. In Iiouse's calculation, y = S is used for iron with good results. I<no\\-ing y = 1.5 to be satisfactory for hydrogen, the range y = 1.5-3.0 \\;as investigated ~vitli the program for helium, and the value y = 2.8 \\?as found to be the most suitable; it is presently the value used in the prograiil for lielium.

In these high density, lo\\:-mass stars the electrostatic interaction energy bet\\-een neighboring nuclei beconles large compared to the tliermal energies. Tlie \\.eak screening correction to thermonuclear reaction rates \vhich usually gives a good approximation for higher-mass stars may not hold in tlie interior of tliese dense stars. Salpeter (1954) has sho\vn that tlie correct thermonuclear reaction rate due to electron screening is obtained by multiplying the unscreened tlierinonuclear rcaction rate by exp ( - z lo /kT) , \\,liere zlo is a potential energy term. We follow Reeves (1965) in evaluating the ( - z l o / k T ) term. If the screening is wealr, then -zlo/kT is given by

\vliere Z1, Z 2 are the charges of tlie two particles involved ( Z 1 >, Z z ) and [ is defined by

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3466 C.-INADIAK JOURSAL OF PIIYSICS. VOL. 45, 1967

where v i are the charges of the various nuclei, A , is the mass nurnber of species i and xi is the fractional density of that species.

If the electron gas is partly degenerate, then

deg.

The quantity f ' ( ~ ) / f ( ~ ) is a function of the degeneracy parameter D (the ratio of the Fernli energy to the mean thermal energy). The value of D is given by

D = 0.3 (p/p)213 l / T 6 for nonrelativistic cases.

p = ( C x,(Zi/Ai))-I is the number of nucleons per electron in the gas. We obtained the following polynomial fit:

When (-uo/kT) (77/22) < 1, then the strong screening correction should be used :

(zt,o/kT),., = 0.20[(21 + Z2)"3 - 2 l 5 I 3 - 22513] (P/p)113 (l/T6).

This formula is valid if

and

If these conditions are violated, then we have intermediate screening, which requires a more involved procedure. In our computing prograin we proceed in the following way. \Ve calculate the strong and weak electron-screening corrections for a set of temperature and density combinations and use which- ever has the smaller value. The calculated values of the electron-screening factor for the '1-1 (p, T ) ~ D and 3He(31-Ie, 2p)*He reactions are given in Table I

TABLE I Electron screening correction factors

Temperature Density f 1 . 1 f 3 . 3

for the temperature and density combinations representing the run of the central values of an evolving low-mass star.

RESULTS

The results of the evolutionary study have been presented in Tables 11, 111, IV, and V for the stars of 0.1, 0.2, 0.3 and 0.4 solar masses. The models

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EZER AND CAMERON: STARS OF LOW MASS

m m ~ a ~ . ? m m 1 . ? m m ~ . ? b m ~ . ? ~ ~ ~ m c o r n c . 1 i m - m t , a + i b 1 . 1 . b m m m ~ t , ~ a 0 0 3 - ~ c . 1 @ 1 @ 1 m m 0 1 0 1 c . 1 0 1 3 3 i 3 0 + SI V + + + + + + L.? LL? L? 1 4 L.? LO LO L t L? l-? L? L t 1 3 10 L? I.? LC) lo L t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m m m m c o m m m m m m m m m m m m m m m m m m ~ o m m m c ~ m

X X X X X X X X w m * X b b b m i m + m a m + o m b b ~ ~ ) m b m b m o 1 - m m t , m c , m i m o a i a m + ~ @ > m + m m w w ~ ~ ~ o ~ + , L ~ w b c ~ ~ ~ C I . . . . . . . . . . . . . . m i m m m 3 m w i m L ? 3 * m + w - m m 3 3 - - - m * a r 7 -

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CANADIAN JOURSAL O F PHYSICS. VOL. 4.5. 1967

X X X X X X - N + ~ L Q + ~ O ~ ~ ~ ? I ~ C C ~ + D , ~ N ~ I O ~ I C I - ? N ~ C C C ~ ~ N ~ ~ ~ C ~ M I ~ ~ U ) ~ C . ~ + , ~ ~ ~ ~ C O U ) ~ I ~ L " . + . . . . . . . . . . . . . . . . . . . . . i i i i i i N + i i N b i + i I . i L O i i @ l N i i i 3 3

~ + + ~ o a + ~ c + ~ ~ ~ ~ - ~ ~ ~ ~ ~ e w m ~ ) ~ ) r n a ~ ) r n ~ ~ ~ w ~ N M @ l 0 1 m m w + + + + m m + + + q + + q + + q w ~ Q ~ Q LQ Ih LC- Lq LC LO ID LQ 1-9 l C LQ LC LO LQ LO LQ LC 10 IT) LQ L O LQ L3 LC! LC-. LC LQ . . . . . . . . . . . . . . . . . . . . . . . . . . .

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EZER AND CAMERON: STrlRS O F LOIV Ivl.\SS 3 4 0

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CAXADI~IM JOUIIXAL OF PIIYSICS. VOL. 48, 10G7

O O C O O C O O O O O O O C O O O O O C C O O O O O O C O C O

X X X X X X X X X X X b d C l L C m O m m r 3 0 0 0 + L Q W ~ O m m W a m O N b W Q , b b C M w d L C C l b ~ O O O W ~ ~ ~ O ~ ~ W ~ : ~ O ~ ~ ~ ~ ~ ~ ~ ~ O L ~ O ~ O ] . . . . . . . . m o o ~ + o m + m i w d o . r 3 + d w ~ ~ ~ i m ~ ~ d w d + w m r 3 d f l m

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EZPR :IND CAMERON: STARS OF LOW M I ~ S S 3471

are representative ones selected along the evolutionary tracks. The first colunln gives the evolution time, taking the time as zero a t the threshold of stability. The second and third coluinns give the radius and luminosity of the models in solar units and the fifth coluilln gives the corresponding effective temperature. The rest of the columns represent, in turn, the pl~otospl~eric density pDh, the central density p,, the central temperature T,, the hydrogen abundance, x,, by mass, a t the center of the star, and the relative contribution of the gravitational energy to the total energy generated by the star, L,/L. The evolutionary paths followed by stars in the mass range O.1Mo-0.4Mo are shown in the theoretical Hertzsprung-Russell diagram in Fig. 2. The stars are ml~olly convective froin the threshold of stability over the almost vertical part of the track. The 0.4Mo and 0.3Mo stars cease to be fully convective after 2.7 X lo7 and 5 X lo7 years, respectively. The evolution of the 0.4Mo star is similar to the evolution of the 0.5Mo and 0.7Mo stars which have already been described (Ezer and Cameron 1967). The fully convective phase is followed by a radiative phase. When the star reaches the

- 4 I O L I 1

3 80 370 3 60 3.50 3 40

log 5, ( O K )

FIG. 2. The theoretical Hertzsprung-Iiussell diagram for low-mass stars.

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3472 C.ASADIAS J0URX:lL O F PIIYSICS. VOL. 45, 106i

main sequence in 4 X lo8 years, the radiative core covers only about llalf of the star's mass. Figure 3 sho~vs the variation with time of the radius (R/Ro) in terms of solar units, the central temperature (Tc/l',), the central density (pc/ps), and the internal structure of the 0.4Mo star. The scaling factors are T, = 1.7 X lo7 'I<, and p, = 2.8 X 10"/cm3. Figure 4 shows the change of the luininosit)~ (LIL,), the relative contribution of the gravitational contraction energy to the luminosity (L,/L), and energy produced by the 'I-I(p, P+v)

Time ( y e a r s )

FIG. 3. The variation of the radius (R/Ro), central temperature (T,/T), ce~ltral detlsity (p,/p,), and the internal structure of the 0.4Mo star, with time.

2D (p, Y ) ~ I - I ~ , 31-Ie("-Ie, 2p).'I-Ie, and "-Ie(a, ?)'Be reactions. The scaling factors for the quantities are L, = 1.4 X erg/s, E I , ~ = 1.01 erg/g s, E3.3 = 2.24 erg/g s , and E ~ , . L = 3.09 erg/g s. We can see froin Fig. 3 that the internal structure of the 0.4l%fo star is fully convective for about 27 inillioll years. With further contraction the radiative core develops and, as this core covers about half of the star's mass, the energy generated by the nuclear burning replaces the gravitational contraction energy. The energy generation by nuclear burning is strong enough to develop a convective core a t the center of the star, covering a maximum 12% of the star's mass.

The size of this convective core is larger than that in a s tar of one solar mass. The reason for this is probably the higher temperature exponent of the fi-fi reactions a t lo\\-er temperatures. The inner layers expand slightly due to the lowering of the polytropic index for the inner convective region. The energy generation by 31-Ie-31-Ie burning increases as the convective core expands, since the 31-Ie abundance increases \\-it11 decreasing temperature. I t sholvs a decline ~vhen the convective core recedes. The second increase is due to the rising central temperatures and densities. 31-Ie reaches its equilibrium value,

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EZER A S D C.\MICIIOS: ST.\RS OF LOW IvlASS 34'73

10' lo8 I o9 1o1O 10" lo1" T i m e ( y e a r s )

FIG. 4. T h e t ime variation of the luminosity (LIL,), the relative con t r i bu t i o~~ of t he gravitational contraction energy t o the luminosity (L,/L) and energy prod~cced by different n ~ ~ c l e a r reactions, for the 0.41160 star.

3.2 X in about 3.2 X lo8 years, \\-hen thc central temperat~~re and density of the 0.41160 star attain the values of 8.4 X 10°K and 90.9 g/cm3, respectively. ?'he energy generation by the 31-Ie-"I-Ie reaction increases as the helium abundance a t the center of the star rises during the evolution and shows a sharp decline wit11 the disappenrarlce of the co~lvective core.

Figure 5 gives the distribution of temperature, density, luminosity, radius, nuclear (Ex) and gravitational (E,) energies, and abundance (X3), by mass, a t the center of the 0.4Ado star, with the mass fraction, just before reaching the main sequence. The scaling factors are I,, = 8.52 X 1031 erg/s, T, = 8.43 X loG OK, p , = 10.0 g/cm3, X3 = 2.0 X 10-5 arid EN = 1.88 erg/g s. Figure G sho~vs tlle variations of the same quantities for the model a t which the central convective core starts to recede after it attains its maxinium size. The scaling factors are L, = 9.06 X lo3' erg/s, T , = 8.26 X loG "I<, p , = 83.1 g/cm3, X 3 = 9.07 X and E N = 2.98 erg/g s.

In the evolution of the 0.31160 star, the radiative core develops after 50 niillion years. When this radiative core covers about 5 % of the star's mass, the energy generation produced by nuclear burning becomes great enough to cause the ce~ltral regions to develop convective instability again. Therefore the 0.3Mo star reaches the main sequence as a \\.holly convective star in 8.3 X 108 years. During the hydrogen-burning phase, convective mixing brings hydrogen-rich surface material to the interior. This lceeps the energy generation a t a level high cnough to maintain the 0.3Mo star in a \vholly co~lvective state for about 8 X 10" )ears until 23% of the original hydrogen has bee11 consumed.

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3474 CAXADIAN JOURKAL OF PIIX'SICS. VOL. 46. 1967

FIG. 5. The distribution with mass fraction of temperature (T/T,), density luminosity (L/L,), nuclear (EN) and gravitational (E,) energies, ancl 31-Ie abundance ( X 3 ) by mass for the 0.4Mo star a t the model just before reaching the main sequence.

On the other hand, the 0.2Mo star is unable to develop a radiative core and reaclies the zero-age main sequence as a fully convective star in 1.7 X 10" years, staying fully convective during the hydrogen-burning phase for the time during which the evolution has been follo~ved. In the evolution of the O.1Mo star, the central temperature and density increase simultaneously as long as the gravitational potential energy released during the contraction goes into the thermal energy of the material. When the star contracts to about O.lRo, the material inside the star becomes degenerate with a degeneracy parameter of about 4. The central temperature remains almost constant for a while, then decreases as contraction continues. The maximum temperature that the O.1Mo can attain is about 4 X lo6 "I<. I-Iydrogen burning occurs only by the 1H(p, vP)" (p, y )We reactions. The maximum energy generation by these reactions is 0.146 erg/g s , while the gravitational contraction energy of the star is 0.017 erg/g s. But the four million degree maximum temperature of the O.1Mo star is just not high enough to stabilize the star on the hydrogen- burning main sequence. The evolution of the O.1Mo star has been followed for 1.23 X 10"ears, a t which time the central temperature has decreased to 2.8 X loG degrees. Further evolution only increased degeneracy and the star goes toward the white dwarf configuration.

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EZER A N D CAMERON: STARS OF LOW MASS

1 .o

FIG. 6. The variation with mass fraction of the variables defined for Fig. 6 for the model a t which the central convective core starts to recede.

The present study of the evolution of stars of low mass is still affected by uncertainties and assumptions in spite of some improvements over previous work. Vl'e are still neglecting the sphericity in the extended atmospheres of these stars and using Prandtl's mixing-length theory for calculating the flux of energy by the convective motion of the gases. In the study of the internal structure of the stars, it is custoinary to use crude inodels for the outer layers of the stars and usually the photosphere is defined as the point of optical depth 2/3. But in stars with low mass, convection might reach above the photosphere and energy might be partly carried by the convective motion of the gases. I-Iayashi and Nalcano (1963) have investigated this problem for a O.lMo star. They concluded that this representation of the photosphere is satisfactory.

Another uncertainty in our calculations arises from the fact that our sources of opacity do not contain water-vapor opacity nrhich might become inlportant for the evolutionary models for contracting stars of low inass (Aun~an 1966; Gingerich, Latham, Linslcy, and I<umar 1966). I-Iowever, Auinan and Boden- heimer (1966) have recently calculated models for late-type stars and they concluded that the results are more sensitive to the assumptions made on the energy transport by convection than to the inclusion of u7ater-vapor opacity. The existing studies determine the liiniting mass of the star which approaches the completely degenerate configuration as follo~vs, as shown in Table VI.

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C.AS.-\UI:\.l; JOURN:\L O F PIIYSICS. VOL. 45, 1967

TABLE V I Comparison of limiting masses for the lo~ver limit of the main sequence

--

Composition Lir~~i t ing mass

Author M/Mo X Y Z

Ezer and Cameron -0.10 0.739 0.240 0.021 I-Iayashi 0.12 0.90 0 . 0s 0 .02 Ha)lashi and Nalcano 0.08 0.61 0.37 0 .02 ICumar 0.07 0.62 0.35 0.02 ICtumar 0. (19 0.90 0.09 0.01

If allo~vance is made for the difference in coinposition, our mass limit is closer to Hayashi's and higher than the one given by Iiumar. A t present, it is difficult to give much \wight to any of these results until cool, nongray model atmosphere calculations are incorporated with the detailed stellar interior programs for the calculations of evolutionary sequences of stars of l o ~ v mass.

CONCLUSIONS

T h e results of this study confirm tile conclusion of I-Iayashi and Xalcano (1963) that the 0.3Mo star has developed a radiative core before reaching the main sequence bu t the 0.211fo s ta r reaches the main sequence as a wholly convective star. But , \\Tit11 the onset of thermonuclear reactions, the energy generation is strong enough to cause the central radiative regions of the 0.3Ado s tar to I~ecome convective again. Therefore, the stars in the mass range 0.2-0.311fo are probably evolving n ~ i t h complete mixing of the material. I t is found tha t the limiting inass for the stars to become conlpletely degenerate is 0.1 Afo for the composition under investigation. This inass liinit is lligller than the one given by I-Iayashi and Naltano and also by I<umar, ~ v h o studied the internal structure of the Ion?-mass stars assuming tha t the). can be represented by spheres of polytropic index 1.5.

\Ve wish to thanlr B. Sacltaroff and B. Goldstein for their invaluable assist- ance \\-it11 the development of the computer programs. The research has been supported in part by the National Aeronautics and Space Administration.

APPENDIX The coefficients of the polynomial representation of F1p(+) for + < 5.4

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EZER A S D CAMBROS: STARS O F LOW MASS

The coefficients of the polynoluial representation of FII?($) for 5.4 < g 20

The coefficiei~ts of the polynomial representatioi~ of F3/?($) for $ < 5.4

72 a,, I / 72 I1 72 a,,

The coefficiei~ts of the polynoll~ial representation of :FA,?($) for 5.4 < $ g 20

REFEIIENCES AUJIAX, J. I<. 1966. Astrophys. J. Suppl. XIV, 171. AUhlAS, J. I<. and BODENHEISIER, P. 1966. Preprint. BODEXHBI~IER, P. 1966. Astrophys. J. 144, 103. CHAXDRASEI~IIAI~, S. 1939. An introduction to the study of stellar structure (University

of Chicago Press, Chicago). EZEI~ , D. and CAJIEROS, A. G. W. 1965. Can. J. Phys. 43, 1497.

1967. Can. J. Phys. 45, 3429. GINGERICIT, O., LATIIASI, D., LINSI<Y, J., and I<c,\~AI~, S. S. 1966. Paper prese~lterl a t the

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