DENSE PLASMAS, NUCLEAR REACTIONS AND ASTROPHYSICS

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DENSE PLASMAS, NUCLEAR REACTIONS AND ASTROPHYSICS

Evry Schatzman

To cite this version:

Evry Schatzman. DENSE PLASMAS, NUCLEAR REACTIONS AND ASTROPHYSICS. Journal de

Physique Colloques, 1980, 41 (C2), pp.C2-89-C2-96. �10.1051/jphyscol:1980216�. �jpa-00219808�

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DENSE PLASMAS, NUCLEAR REACTIONS AND ASTROPHYSICS

Abstract .- A certain number of questions are raised to physicists and astrophysi- cists concerning : (1) the conditions under which stellar matter becomes a highly correlated plasma ; (2) the role of mass loss in the formation of dense matter ;

(3) the evolution of white dwarfs accreting matter (with its consequences for the formation of compact X ray sources and pulsars) .

1 . Introduction.- The solution of a large number of astrophysical problems depend on the understanding of the physical properties of dense matter . However, the expression dense matter does not refer to a simple domain in the plane of the state variables

(T .p .) . (1) Dense matter can be defined as a highly correlated plasma . II we call a the radius of the ionic sphere,

a highly correlated plasma is defined by the condition

The classical domain of the solid state is defined by r - 160. The recent extension by Hansen and Mochkovitch (1979) to a lattice of Fermions and Bosons give for the density of melting at T = 0 , R^ = 100 for Bosons and R„ = 65 for Fermions, with Rc a/a , where a = (h /JXE e ) is the ionic Bohr radius and a is the ion sphere radius. This corresponds to densities

For oxygen 0, this gives log 1 Qp = 15 .38 ;

for 1 2C , this gives log 1 Qp = 14-16 ; and

for 1 6C , log 1 Qp = 14 .67 . It should be noti-

ced that these densities are far beyond the

threshold densities for inverse Beta reactions (Beta captures), and lay in the region of nuclear densities and beyond .

(Fig.l) : The (log T, log p) plane. The curves 1 12 16„ 16„ _ _.

H, C, C, 0 represent the melting curves for a one component plasma (OCP), according to the last results of Hansen and Mochkovitch (1979) . The boundary between the degenerate region (D) and the non degenerate region (ND) is plotted for

(A/Z) = 2. The level of electron capture for Car- bon and Oxygen is indicated by C e and 0 e . The nuclear density is p . The: region beyond 4p , where the difficulties begin is indicated by the dotted area. The line (1/9) p corresponds to the at which the baryon interactions are not negligi- ble.

(2) Dense matter can be defined as matter at sub-nuclear and nuclear density and beyond . The transition region begins some- where below nuclear density, in the domain where nuclear forces become important . The nuclear density, which is in number density 0.17 (fm) corresponds to p -, = _3

14 43 -3 nucx 10 g cm Nuclear forces are already

JOURNAL DE PHYSIQUE Colloque C2, supplément au n° 3, Tome 41, mars 1980, page C2-89

Evry Schatzman Observatoire de Nioe

Résumé .- Un certain nombre de questions sont posées aux physiciens et astrophysi- ciens concernant : (1) les conditions dans lesquelles la matière stellaire vient constituer un plasma fortement corrélé ; ( 2) le rôle de la perte de masse dans la formation de la matière dense ; (3) l'évolution des naines blanches sous l'effet de l'accrétion (avec ses implications pour la formation des sources X compactes et des pulsars) .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980216

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c2-90 JOURNAL DE PHYSIQUE

important at a number density of 0.02 (fm)-3 which correspongs to p = 10 13.48 -3

g cm

.

It seems that our inderstanding of physics

>

of neutron star matter at p % 4 p nucl

- -

-

³

1015 g cm is not satisfactory yet.

Matter at these densities can be studied in thermodynamic equilibrium. However, it is quite important to understand how these high density regions are reached. The kinematics and the thermodynamics of the transformation are important, and depend very much of the starting point in the (TI p) diagram. These properties of dense matter are relevant to ,

the structure of neutron stars, the formation of neutron stars, and the exploding process leading to supernovae.

2. Astrophysical considerations: stellar evolution at constant mass.- We shall consider, in the following, the conditions under which ordinary matter becomes dense matter.

The starting point of the discussion is given by the conventionnal stellar evolution at constant mass.

We shall be interested here by the evolutionnary tracks of the stellar center, for different values of the stellar mass. Such tracks havebeenplotted by Iben (1973a, 1974) and Shaviv (1967), Barkat (1975).

(Fig. 2) : The ( l o g T, log1 p) plane. Evolution a t constant mass. Evolutionnary t r a c k s of the c e n t e r of

is yon-degenerate. It is quite well known that the evolution of the star depends ve- very much whether the star reaches the helium ignition line in the degenerate region, or in the non degenerate region.

In the degenerate region, the equation of state is known as depending very little on the temperature. This has been recognized since Mestel (1952) as giving the possibility of a thermal runaway.

The mass in the hydrogen-exhausted core at the start of the thermal runaway appears to be relatively independant of the abundance of elements heavier than helium and also relatively independant of the toal mass outside the degenerate core. For va-

lues of initial hydrogen content X 0.6

-

0.7, the helium core mass is about 0.4 Mg. The thermal runaway produces a major change in the star. Core temperature ,rises until degeneracy is lifted. The star moves, in the Hertzsprung-Russell diagram,to the horizontal branch, where it starts evolving from the ZAHB (zero age horizontal branch). Stars of mass smaller than about 2.25 Mg reach the He ignition line in the degenerate region,whereas stars of mass larger than about 2.25 Mg reach the He ignition line in the non degenerate region. Helium burning begins and core temperature continues to increase, so that electron degeneracy never becomes ap- preciable in the core. Helium burning in the core leads to the production of carbon and oxygen. The stellar core moves towards the carbon ignition line, which is defined as the line where the rate of energy production by carbon burning is not balanced any more by the neutrino losses. The si- tuation again is quite different, depen-

a s t a r f o r d i f f e r e n t s t e l l a r mass a s given by IBEN ding on whether the stellar core is gene-

(1973a) ; the boundary between degenerate (D) and

non degenerate (ND) regions has been p l o t t e d f o r rate or not. I£ the stellar core is dege- (A/Z) = 2. The i g n i t i o n l i n e s f o r H, he and C have nerate, a thermal runaway can take place.

been p l o t t e d . The Carbon i g n i t i o n l i n e i s t h e igni-

t i o n l i n e of Paczvnski. If the stellar core is not degenerate, the

The numbers 1, 2, 3 , 7, 9 and 15 a r e t h e s t e l l a r temperature rise in the core prevents it

masses i n s o l a r u n i t s .

of ever' becoming strongly degenerate.

We have reproduced on figure 2 the evolu- The stellar core in which a thermal runa- tionnary tracks of Iben (1973a). Except for . , way due to carbon burning can take place, very low mass stars,. hydrogen ignition takes has a mass of the order of 1.4 Mg (the so- place in the region where the electron gas called Chandrasekhar limit), and this cor-

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responds to an initial stellar mass of about 7 Me. At the end of the helium burning phase, the chemical. composition of the core is a mixture of carbon and oxygen, and it depends on the stellar mass.

The exact place in the log p , log T plane atwhich the thermal runaway takes place depends mainly on the enhancement factor of thermonuclear reactions (Schatzman, 1978 ;

%las';uelr cnd Zs.ncovici, 1978 ;

Salpeter, 1954 ; Salpeter and Van Horn, 1964 De Witt, Graboske and Cooper, 1973 ; Mitler,

1977 ; Itoh, Totsuji and Ichimura, 1977 ;

Jancovici, 1977)

.

3. Problems of mass loss.- There are several ways of approaching the problem of mass loss :(1) the direct observationnal eyiden- ce ; (2) the determination of the mas4 of the parent stars of white dwarfs in gdlactic clusters ; (3) the abundance of iron in the Galaxy and the critical mass ; (4) the theoretical estimates of the rate of mass loss ;

(5) consequences of mass loss on stellar evolutions. We shall limit here ourselves to the empirical evidence, the theoretical ones being presently not sufficiently sure to give any reliable estimate of the rate of mass loss and we comment the consequences for stellar evolution. At most, the theoretical models of the rate of mass loss show the compatibility of the empirical evidence and of the theory. However, stellar evolution models with empirical rates of mass loss give interesting indications on the changes of stellar structure due to mass loss.

(i) Direct observationnal evidence. All studied red giants and supergiants appear to have an expanding circumstellar envelope.

The analysis of the line profiles and thermodynamics of the envelope have been carried several times (for example, Sanner, 1976 ; Barnat, 1977).

It should be noticed that the interpreta- tion of the spectra is not easy and that the mass loss estimates vary greatly from one author to another. The rate of mass loss can be very large (lo5 Me and there is some evidence that it increases with the luminosity and decreases with the

kinectic energy of escape. A possible pa- rameter describing the rate of mass loss is the quantity (LR/M) (Reimers, 1975a, b) Theoretical estimates by Renzini et al.

(1977b), Ulmschneider et a1. (1977a, b) ba- sed on the old idea of Biermann (1946)

,

Schwarzchild (1948)

,

Schatzman (1949) of acoustic heating of the solar chromosphere and corona, adjusted to the rate of mass loss of the Sun are compatible with the mass loss rate of Reimers. We shall use,

in the following, the notation of Renzini

L R

-

¹

% _dt=

-

⁴^x^{10-l3 qR} ^{(M@ year} ⁾

where L, R and M are in solar units, and qR is an adjustable factor.

Mass loss in large mass stars takes place also at a considerable rate. Results obtained with Copernicus, according to Snow and Morton (1976) Show that for hot stars mass loss is a quite general phenome non : stars brighter than M = -6.0 are

B0l

losing mass. Estimates of mass loss, using infrared photometric results, have been given by Barlow and Cohen (1977). With a correction suggested by Lamers and Castor

(1978) concerning the velocity la^, the rates of mass loss of Barlow and Cohen should be doubled, with the following laws :

.

^for⁰^stars,

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- - -

- 13.6 x (L/Le) 1.10

-

¹

dt Me yr

.

^for^B^and^Asupergiants,

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% = - l o x 10-l3 (L/L@) 1.2 ~ ~ y r - l dt

(ii) White dwarfs. Different methods of analysis, as described by Sweeny (1976), and by Romanishin and Angel (i979) give estimates of the mass of parent stars of white dwarfs in globular clusters. Results

can be summarized in table I.

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JOURNAL DE PHYSIQUE

Table I

Mass of parents of white dwarfs.

: number of white dwarfs i n t h e c l u s t e r ;

Cluster

Hyades Pleiades Sirius groupe 61 CYg

y Leo NGC 2168

2287 2422 6633

PZyD min, max minimum and maximum mass of t h e p a r e n t s t a r s of white dwarfs i n t h e c l u s t e r

( a ) Sweeny ; (b) Romanishin and Angel.

It seems then that for an isolated star, the mass of the parent star can be as large as 7 Mg ; see also Weidemann (1977).

(iii) The critical mass. The concept of critical mass has been introduced in connection with the problem of supernovae ;

the critical mass is the mass above which a star, at a late stage of evolution, becomes a supernova.

Considering a given frequency of supernovae it is tempting to relate this frequency to the rate of stellar death. Tamman (1974) suggests, in our Galaxy, a birth rate of supernovae of 0.04 -yr-' (if it is an Sb galaxy); or perhaps of 0.1 yr-l (if it is an Sc galaxy). From the estimates of Ostriker et al. (1974), it is possible to give the total death rate for stars having a mass larger than M (table 11).

n~~

13 1 4 3

1

4 825 1 1025

Table I1 Total death rate

MWD

^min

2.7 5.25 4 1

-

5 4 6

z4

The,estimates of T for our Galaxy are quite unc rtain. The discussion is carried in the

£01 owing way :

(a) Assume stellar evolution without mass

i

loss. With a high birth rate of supernovae, it is necessary to assume that all stars having a mass larger than 3 to 4 Mg explode, For moderate mass stars, with

4 MO ^$,M $, 8 Me, the stellar core reaches the carbon ignition line in the degenerate region. The mass of the degenerate core is about 1.4 Mg. Ostriker et a1. (1974)

,

^consi-

(M/MO) larger than 2.5

2.9 5.1 7.4 9.1

MWD

^rnax

4.5 8 7 1.1

-

7 7

-

5

T (years) 6 10 25 50 100 Notes

a a a a a b b b b

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der the possibility that all moderate mass stars explode entirely, each ejecting a 1.4 Ma of iron-peak element at death. This, however, spreads in the Galaxy a larger amount of Iron, too large by a factor of the order of 60.

b) A way out of this difficulty, which has been discussed for example by Renzini (19771, by Schramm (1977) is the following. Assume that mass loss is a very efficient process, such that all stars with a mass smaller than

Merit

2 8 Mg become white dwarfs.Then, the problem of the iron excess is so1ved:I'his however implies (1) that the supernovae birth rate is of the order of (50)-l yr-l;

(2) that the efficiency of the mass loss is very high, increasing with the stellar mass, up to

nR

2 4

.

(defined in equ

.

⁴⁾

.

^Renzini

(1977) notices that such a high efficiency cannot apply to lower mass stars.

(c) On the other hand, consider the energy available during explosive carbon burning.

For a degenerate core of 1.4 Ma, close to the instability limit, the energy available, of the order of 0.54 MeV per nucleon, can be much larger than the binding energy of the star. Disruption of the star takes place only in the case of a detonation wave, where 2 large fraction of the available energy is transformed into kinetic energy.

If, instead of a detonation, there is a de- flagration, the energy available is transformed into thermal energy, producing a pressure excess which is a small fraction of the degenerate pressure. Mazurek et al.

(19 77) have suggested that the pressure excess due to carbon burning is so small, for densities larger the 5 x 10 7 g that no detonation wave can be produced.

(iv) Theoretical estimates. we shall not insist on this aspect of the problem, as the theoretical models are yet quite unsa- tisfactory, whether considering the mass loss due to radiation pressure or the mass 1.0spdue to the existence of a heating me-

chanism related to the presence of a deep hydrogen convective zone. In this connection, it is worth mentionning the results obtained with the satellite HEAO I, showing the presence of a corona in A s#ars, the

difficulties .met with the balance between the radiative emission of the chromosphere and the divergence of the mechanical flux, as estimated by Athay and White (1978).

(see' also Durrant, 1978). We can only say that the mass Cluxes ~::~icli can be derived from theoretical arguments are compatible with the observationg, but that this cannot yet be considerep as a full theory of the mechanism of mas's loss.

(v) Stellar evolution with mass loss. A large number of papers have been devoted recently to stellar evolution with mass low ; for moderate-mass stars by Fusi- Pecci and Renzini (1975a, b ; 1976 ; 1978), Renzini (1977, 1978), for a 5 M star by

0

Forbes (1968), for large mass stars by a number of authors, ~ h i o s i and Nasi (1974), Dearborn, Blake, H&inebach and Schramm

(1978), Sreenivasan and Wilson (1978a, b), Chiosi, Nasi and Sreenivasan (1978), Stothers and Chin (1978), Chiosi, Nasi and Bertelli (1979), de Loore, de Greve and Lamers (1977), Dearborn and Eggleton (1977) see also the review paper by Conti (1978).

We are concerned here with the changes in central density and temperature which are produced by mass loss. As an example, consider the evolution of a 5 Me star calcu- lated by Forbes (1968). The mass loss is considerable, leading to a final mass of 0.928 Ma. However, during the helium burning phase, the differences in central temperature and central density for the star evolving at constant mass, or with an heavy mass loss, are very small. Neverthe- less, the core of the star evolving at constant mass will move towards the carbon ignition line, whereas the small mass remnant will end as a white dwarf with log

p o = 7.32 and a low temperature ( = lo7 K) (Fig.3).

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JOURNAL DE PHYSIQUE

carbon c o n c e n t r a t i o n XE = 33 % i n number.

Consider now t h e e v e n t s which t a k e - p l a c e d u r i n g c o o l i n g . I f X < XE, we f i r s t d e p o s i t oxygen. The d e n s i t y of s o l i d oxygen, a t To = 1.615 Tc (Tc i s t h e f r e e z i n g p o i n t of pure carbon) i s l a r g e r t h a n t h e d e n s i t y of t h e remaining mixture of carbon and oxygen.

We a c t u a l l y have, from Pollock and Hansen (1973)

(Fig.3) : The (log T, log p) plane. Evolution with mass loss. The evolutionnary tracks for a 5 Me star are given at constant mass, and with mass loss (ac- cording to Forbes, 1968). The points labelled EE'FGH have the following characteristics.

Epoch t (yr) ( M / M ~ ) E 66.96 (6) 3 -862 E' 69.95 (6) 0.972

F 83.43 (6) 0.928

G 85.20 (6) 0.928

H 85.67 (6) 0.928

Beyond point H the curve is a sketch of the cooling and contraction towards the white dwarf stage.

4 . White dwarfs, X r a y s o u r c e s , p u l s a r s . - The p o s s i b i l i t y t h a t a w h i t e dwarf, c l o s e t o t h e g r a v i t a t i o n a l i - n s t a b i l i t y l i m i t , might c o l l a p s e i n t o a neutron s t a r , has been considered by S c h a t z m n and Canal

(19761, i n connection w i t h t h e presence of a neutron s t a r i n b i n a r y X r a y s o u r c e s

( B a h c a l l , 1978). The s t a t i s t i c s on p u l s a r s , by Taylor and Manchester (1977a, b ) sugges- t i n g a b i r t h r a t e of 0.16 yr-' f o r t h e pul-

'\

s a r i n t h e galaxy, much h i g h e r t h a n t h e ra- t e of production o f supernovae, t h e kinema- t i c p r o p e r t i e s of t h e p u l s a r s i n t h e galaxy

(Taylor and Manchester, 1977b), s u g g e s t a mechanism of production d i f f e r e n t from t h e supernova mechanism, and a way of s h o o t i n g t h e p u l s a r s w i t h a l a r g e v e l o c i t y i n t h e galaxy. Non e x p l o s i v e c o l l a p s e of w h i t e dwarfs, l e a d i n g e i t h e r t o t h e production of a bound neutron s t a r o r a f r e e neutron s t a r could provide t h e e x t r a s o u r c e of X r a y b i - nary s o u r c e s o r p u l s a r s . Off hand, t h e si- t u a t i o n might d i f f e r , depending on t h e che- m i c a l composition of t h e w h i t e dwarf. Assu- me w i t h Stevenson (1 79') that we can have an e u t e c t i c of carbo and oxygen, w i t h , a

Whereas we have

p ( s o l i d 0 )

-

^p( l i q u i d mixture) = P

0.00480

-

^0.00508 ^X ^>^0.00312.

Oxygen'then s o l i d i f i e s near t h e c e n t e r , and f i n a l l y , when t h e e u t e c t i c s o l i d i f i e s , a mixture of carbon and oxygen c r y s t a l s s o l i - d i f i e s a t T = 0.628 Tc.

I f X > Xk, we f i r s t d e p o s i t carbon. We ac- t u a l l y have

p ( s o l i d C )

-

^p( l i q u i d mixture) = P

Carbon s o l i d i f i e s n e a r t h e c e n t e r and f i - n a l l y when t h e e u t e c t i c s o l i d i f i e s a mixtu- r e of carbon and oxygen c r y s t a l s s o l i d i f i e s a t T = 0.628 Tc.

Assume now t h a t a c c r e t i o n on a w h i t e dwarf t a k e s p l a c e , t h e mass growing w i t h time.

I n t h e f i r s t c a s e , X < XE, e l e c t r o n c a p t u r e t a k e s p l a c e i n t h e c e n t e r b e f o r e reaching t h e g r a v i t a t i o n a l i n s t a b i l i t y l i m i t . The s t a r c o n t r a c t s and enter.dynamica1 c o l l a p - s e . I t should be n o t i c e d t h a t t h e h e a t i n g which i s a s s o c i a t e d w i t h t h e e l e c t r o n cap- t u r e has v e r y l i t t l e i n f l u e n c e on t h e equa- t i o n of s t a t e .

The energy a v a i l a b l e p e r e l e c t r o n can be determined i n t h e f o l l o w i n g way. A t cons- t a n t p r e s s u r e , t h e energy gained i s

16 16

, - P d v + ^E( 0 0 ) .

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1 1 Per free electron, this is

16

sM

+

E (160160)

,

^{where cM}= 10.4 mev is the threshold energy for the capture of one

16 16

electron on 160. E( O 0) = 16.541 MeV, so that the energy available is, per free electron, 2.028 Mev. The excess pressure due to non-degeneracy is only 0.0775, and can hardly have any influence on the collapse. If oxygen did not have time to ignite, the excess pressure would be only 0.0229. In the second case, X > XE, gravi- tational collapse takes place first. The energy at thresholdis E~ = 13.37 MeV. The electron capture and the energy of the 12c12c reaction rN = 13.93 Mev gives E = 2.275 Mev per free electron. The final con- tribution to the pressure is 0.0646 ; if carbon did not have time to ignite, the excess would be only 0.0295.

In both cases, the temperature which is obtained in the center is very high, of the order of 10100~, and the thermonuclear reaction rate at such temperatures is ex- tremely high. Assuming however that the condition of Mazurek et al. (1977) for the formation of a detonation wave has to be fullfilled, we find that the detonation can take place in a larger mass fraction (1.3 %

of the mass of the star) for the solid oxygen core, than for the solid carbon core

(0.94 % of the mass of the star). Anyhow, it is impossible to give a correct estimate of the difference of behaviour of the two kinds of stars without a complete analysis of the collapse and bounce. It is neverthe- less tempting to relate the possibly different exchange of momentum taking place in a binary star to a difference in chemical composition of the accreting white dwarf.

The binary nature of the star might be con- served (leading to a compact X ray source),

correlated two components plasmas, enhancement factor of nuclear reactions in mixtu- res, neutrino losses etc.,

...

⁾ ^;^{or astro-}

physicists still have to carry the computa- tions, using the date provided by the physicists, to the point where it is actually possible to compare the theory with tt:e ob- servations.

Alastuey, A., Jancovici, B. 1978, Astrophys J. 226, 2034.

Athay, G.R., White, O.R. 1978, Astrophys.

J.

226,

^1135.

Bahcall, J.N. 1978, Annu. Rev. Astron. As- trophys.

s,

^241.

Barkat, Z. 1975, Annu. Rev. Astron. Astro- phys.

15,

45.

Barlow, M.J., Cohen, M. 1977, Astrophys.

J.

213,

737.

Barnat, A.P. 1977, Astrophys. J. 213, 756.

Biermann, L. 1946, Naturwissenschaft

3 ,

118.

Canal, R., Schatzman, E. 1976, Astron. As- trophys.

s,

^229.

Chiosi, C., Nasi, E. 1974, Astron. Astro- phys.

34,

355.

Chiosi, C., Nasi, E. 1979, Astron; Astro- phys.

74,

^62.

Chiosi, C., Nasi, E., Bertelli, G. 1979, Astron. Astrophys. 74, 62.

Chiosi, C., Nasi, E., Sreenivasan, S.R.

1978, Astron. Astrophys.

63,

^103.

Conti, P.S. 1978, Annu. Rev. Astron. Astro- phys.

16,

^371.

Dearborn, D.S.P., Blake, J.B., Hainbach, K.L., Schramm, D.N.1978, Astrophys. J. 223,

r c 1

-

Dearborn, D.S.P., Eggleton, P.P. 1977, AS- traphys. J.

=,

^448.

De Witt, H.E.,Grabosker H.C., Cooper, M.S.

1973, Astrophys. J. 181, 439. _{- -} or the neutron star can be launched in the

De Loore, C., De GrGve, J.P., Lamers, H.J.

galaxy, producing a high pulsar. G.L.M. 1977, Astron. Astrophys.

61,

^251.

5. Conclusion.- A large number of physical

Durrant, C.J. 1978, Q. J. R. Astron. Soc.

problems, of a great importance for astro- 19. 411.

- .

--

^{- .}

physicists, await a solution, or a greater

precision in the solution, especially in Forbes, J.E. 1968, Astrophys. J.

153,

495.

the field of thermodynamics of dense matter Fusi-Pecci, F., Renzini, A. 1975a, Astron.

(transport processes, equation of state Astrophys.

2,

413.

above nuclear density, properties of highly

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JOURNAL DE PHYSIQUE

Fusi-Pecci, F., Renzini, A. 1975b, Mem. SOC R. Sci. LiSge 6th Ser. 8 , 383.

Fusi-Pecci, F., Renzini, A. 1976, Astron.

Astrophys.

46,

^447.

Fusi-Pecci, F., Renzini, A. 1978. 1.A.U Symposium no 80, p. 225, p. 406.

Hansen, J.P. Mochkovitch, R. 1979, Preprint Iben, J. 1973, Explosive Nucleo Synthesis, ed. Schramm, D.N., Arnett, W.D., p. 115, Univ. Texas, Austin.

Iben, J. 1974, Annu. Rev. Astron. Astrophys 12, 215.

-

Itoh, N., Totsuji, H., Ichimaru, S. 1977, Astrophys. J.

218,

^477.

Lamers, H.J.G.L.M., Castor, J.I. 1978, Quo- ted from Chiosi et al. 1979, Astron. Astro- phys.

74,

^62.

Ostriker, J.P., Richstone, D.O., Thuan, T.X. 1974, Astrophys. J., Lett.

188,

L. 87.

Mazurek, T.J., Meier, D.L., Wheeler, J.L.

1977, Astrophys. J. 213, 518.

Mestel, L. 1952, Mon. Not. R. Astron. Soc.

112, 598.

Mitler, H .E. 1977, Astrophys. J

.

^{212, 513.}

Pollock, E .L

.,

^Hansen,^J^.P

.

1973, Phys

.

Rev. A

8 ,

^3110.

Rakavy, G

.,

^{Shaviv, G}

.

1967, Astrophys

.

^J

.

148,

803.

Reimers, D

.

1975a, Mern. Soc

.

^{R. Sci}

.

^Lisge,

6th Ser

.

^{8, 369}

^.

Reimers, D. 1975b, Problems in Stellar AtmosphSres and Envelopes, ed. Baschek, R

.,

Kegel, N.H

.,

Traving, G., Springer Verlag, p. 229.

Renzini, A. 1977, Advanced Stages of Stel- lar Evolution, Saas Fee, Geneva Obs. CH 1 290 Sauverny, Switzerland Ed

.

^p

.

¹⁵¹

.

Renzini, A. 1978, Mem

.

^Soc

.

^Astron

.

^Italia-

na

49,

371.

Renzini, A., Cacciari, C., Ulmschneider, P., Schmitz, F. 1977, Astron. Astrophys.

61, 39.

-

Romanishin, W

.,

Angel, J .R.P. 1979, Preprint

.

Salpeter, E .E 1954, Australian J

.

^Phys

.

7, 373.

-

Salpeter, E.E., Van Horn, H a . 1969, Astro- phys. J

. ^155,

^183.

Schatzman, E

.

1949, Ann. Astrophys. 12, 203

.

Schatzman, E

.

1978, Astron. Astrophys

. 65,

L 17.

Schramm, D.N. 1977, Advanced Stages in Stel- lar Evolution, Saas Fee, Geneva Obs. CH 1290 Sauverny, Switzerland Ed

.,

^p.^285.

Schwarzschild, M. 1948, Astr0phys.J

107,

1 . Snow, T .P. Jr., Morton, D.L. 1976, Astro- phys

.

^J

.

^Suppl

^.

^Ser

^.

^{3 2 , 4}²⁹

^.

Sreenivasan, S .R

.,

^Wilson,^W^J

.

^F

.

^1978,

Astrophys

.

^{Space Sci}

. ^3,

¹⁹³

.

Stevenson, D J

.

¹⁹⁷9, This Colloquium.

Stothers, R., Chin, C .W. 1978, Astrophys.

J

. ^22.6,

^231.

Sweeney, M A

.

1976, Astron

.

^Astrophys

. ^49,

375.

Tammann, G.A. 1974, Supernovae and Supervae Remnants, Cosmovici, C .B. ed., Reidel, p. 155.

Taylor, J .H

.,

Manchester, R .N. 1977a, As- trophys'. J

. =,

⁸⁸⁵

^.

raylor, J .H

.,

Manchester, R.N. 1977b, Annu- Rev. Astron

.

^Astrophys

. ^15,

^19.

Ulmschneider, P., Schmitz, F

.,

^Renzini,

A., Cacciari, C

.,

Kalkofen, W., Kurucz, R

.

1977, Astron. Astrophys

. 61,

⁵¹⁵

.

Weidemann, V. 1977, Astron. Astrophys

. ³

411.

Sanner, F. 1976, Astrophys. J

.

Suppl. Ser.

32, 115.

-

Schatzman, E. 1948, J

.

^Phys

.

^Rad.

2 ,

^46.