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THERMONUCLEAR RUNAWAY MODEL. .

WARREN M, SPARKS, X-3 , LOS ALAMOS NATIONAL LABORATORYG. SIEGFRIED KUTTER, NATIONAL SCIENCE FOUNDATIONSUMNER STARRFIELD, ARIZONA STATE UNIVERSITY ANI)

LOS ALAMOS NATIONAL LABORATORY, T-6JAMES W. TRURAN, LINIVERSITY OF ILLINOIS

PHYSICS OF CLASSICAL NOVAE: IAUMADRID, SPAIN. 27 to 30 JUNE 1989

1)1.SC”I.AIMER

COLLOQUIUM Nom 122

,... mf’”-,B ~,. t’. -

!=i~~~h~~ bsA,amos,New.ex,c08754~lasAlamos NationalLaborator

Dlsllllllljllflu ,,, ,,,,m m“”.,.. .

THERMONUCLEAR RUNAWAY MODEL

Warren M. SparksLos Ahmos N;~tirm:ll 14xmttory

G. Siegfried KutterNational Science Foundation

Sumner StamfleldArizona State University and

Los Aliimos National Laborato~

and

James W. TruranUniversity of Illinois

I. INTRODUCTION

The nova outburst mquims an energy source that is energetic enough to eject material and is able to

recur. The Thermonuclear Runaway (TNR) model, coupled with the binary nature of nova systems, satisfies

these conditions. The white dwarf/red dwarf binary nature of novae was first recognized as a necessary

condition by K.raft (1963, 1964, and these conference proceedings). The small separation characteristic of

novae systems al;ows the cool, red secondary to overflow its Roche lobe. In the absence of strong,

funneling magnetic fields, the angular momentum of this material prevents it from falling dkctly onto the

primary, and it fnt foxms a disk around the white dwarf, This material is eventually accrcted from the disk

onto the white dwarf. As the thickness of this hydrogen-rich layer increases, the degenerate matter at the

base reaches a temperature that is high enough to initiate thermonuclear fusion of hydrogen. Thermonuclear

energy mlcase incrc:scs the mnpcraturc which in turn increases the energy generation rate. Because the

material is degenerate, the pressure does not increase with temperature, which ncmnally allows a star to

adjust itwlf to a steady nuclear burning rate. Thus the tcmpcratum and nuclear energy gencmtion increase

and a TNR results. M $en the tcmpcratum reaches the Fetmi temperatum, degeneracy is lifted and the rapid

pmssurc incmasc causes material expansion. The hydrogen-rich material either is ejected or consumed by

nuclear burning, and the white dwarf returns to its pre-outburst state, TIMexternal source of hydrogen fuel

from the secondary allows the who!e process to repeat.

II . ‘TIIE DEVI1l.)PMENT OF A TNR

The temperature evolution provides significant infonmition about the development of a TNR, Figure

1 shows the temperature structure @’:\ie hydrogen-rich shell on a whiw dwarf at various times durir,g its

evolution It is from an equilibrium model with H initially in place (SW5111A), and its behavior is

charactcrist~c of all rnodcls, Curve 1 on the graph is the initial temperature distribution of the cnvclopc, II is

determined by t:lc intrinsic white dwurfs luminosity, Curve 2 rcprcscn[s lhc distribution when the pr~~t(m

capture onto CNO nuclei dominittcs over the p-p reactions ut -2x 10” K, TIc time between these firs[ IWIJ

CUIVCSis usually mtiny thoustinds of ycnrs nnd is very strongly depc Idcm on the whitr d~iirfs intrinsic

luminosity and weakly dependent on the CNO abundance of the envelope’s matter and [he white dwarf’s

mass. The next significant event, as [he lcmpcramns continues m rise, occurs at -3x107 K when convection

begins in the region above the maximum temperature (curve 3). This convective region continues to grow

cwtwa.rduntil it reaches the surface layers near the time of peak tempcratum. When the temperature (curve 4)

reaches 1@K nvo significant circumstances occur. Firs!, the temperature exceeds the Fermi temperature

and the electron degeneracy is lifted. AI this time the pressure increases rapid!y and hydrodynamics must be

includd in the calculations. SUond, the proton capIure rates on the CNO nuclei become shoner than the 11+

decay r~tes of lhc temporary reaction pnxluc[s of the CNO bicycle (1SN, 140, 150, and 1‘F). Thus, Ihe

B+- unstab!e nuclei and their rates must be included both in dw riuclear reaction network and in the energy

generation calculation (Starrfield er al. 1972). As the temperature increases, the CNO reactions are

controlled by the B+daay rates, and the energy gencmion rate becomes indepcnden[ of tempctature and

density and dependent only on the CNO isotopic abundances.

The last curve (#5) corttsponds to the tirm of peak temperature, which is reached only a few

seconds after cume 4. The dynamic time scale is now of the otrier of one second while the nuclear burning

time scale is given by cpT/qw. If the CNO abundances are solar then the nuclear burning time scale is also

of the otdcr of one second. This allows the envelope to expand before a strong TNR can develop. Under

these condhns little or no material is ejected by the expansion. I%c now rekindled hydrogen burning shell

some may eject tlw envelope by radiation pressure (or common envelope interaction) and produce a slow

nova (Sparks eI u/. 1978). If the initial CNO abundances ore increased above solar, the nuclear burning time

scale becomes shorter than the dynamic time scale. For large CNO overabundances the TNR becomes

strong enough to eject a portion of the hydrogen-rich envelope and produce a fast mwa (Sttileld er al.

1978). The peak temperature is also strongly dcpdent on the CNO abundances. At peak temperature the

convective time scale is of the order of 100 sec. This is approximately the decay time scale of the

B+- unstable nuclei. Therefore, a siz.eablc fraction of *C B+- unstable nuclei will be convected to the outer

layen before they decay, When they decay in the outer layers, they provide an additional energy source for

ejection at a relatively low gmvity, This complexity of physical phenomena shows why it is necessaty 10

treat hyclrmlynamics, time-dependent conv~tion, and nuclear physics correctly.

The energy mtcs are a useful diagnostic Im)l to undersutnd the TNR. The Imal energy conservation

quation can be wrhten in the form (Kut[er and Sparks 1974, lW!(I):

~ includes the transport ofenmgy by both radiative diffusion and convecti,))l, We lakl the terms on the

left side of equal ion ( ]) c)(]N,~RAV, ~]~, und (’Ml:(’~1~N[l(’is always p(~sltive●nd mp~~nts k

nuclear energy sm.tree. ‘I%cother terms of cquntion ( I) mi~y he cithm posilivr or negative, depending on

whether energy is being stored (sink of encr~y) or mlriIscd (soIIrrr of energy), rcspeclivcly. At 011times ;Iml

lmntions in the cnve.,qx, the sum (}f[hcsr terms must cqu;Il the nuclenr cnrrgy stmrcr Ily nmnit{wingthcw

tcmls, wc can de[enminc the accurncy [~fIhr cmlr.

Figure 2 shows these energy raIes as a function of mass during the early evolutionary stages of an

accreting 1.0 Me white dwarf (Kutter and Sparks 1980). ‘1’henuclear energy source is small and gravity is.the main energy source for the photon, mtemal and mechanical energy sinks. Basically, the accreting

hydrogen envelope is being compressed and heated under its own weight. Figure 3 shows a later stage of

evolution when the convective region is extending outward. Tne strength of the nuclear energy source has

incmascd and the resulting pressure increase has changed the motion within the envelope from slowly

contracting to slowly expanding. This is indicated by the positive sign for the gravitational energy

throughout the envelope. Convective flux is the main source of energy above the nuclear energy generation

region and mechanical flux the main source in the outer regions. These fluxes a.mcarrying energy from the

nuclear energy generation region and depositing it farther out. The leading edge of the convective region is

indicated by a sharp drop of both the convective flux (source) and the internal energy (sink).

The energy rate distribution near peak nuclear burning is presented in Figure 4. The convective

region extends all the way to the sutfacc and the convective flux is the main source of energy to the outer

layers, The internal energy is now a source of energy because of the expansion. The rmclear energy source

now extends ail the way to the surface because the convective time scale is short enough so that some of the

13+-unstable nuclei reach the sutiace before they decay. Almost all of this energy is going into the lifting

(i.e., gravitational energy) of material. As the convective flux decreases the B+- unstable nuclei became an

important energy source for the outer layers.

111. COMPUTATIONAL MODELS

The main computational parameters that affect the strength of a TNR am the pressure in the energy

generation region and its composition, ~pecifically, the HCNO nuclei. The nuclear energy rate is

proportional to the product of the abundance of H and the CNO isotopes. Notmally the abundance of H is

huge and does not change over the critical stages of the TNR, Thus the CNO abundances are through their

catalytic effect the controlling factor in the energy generation rate, cspccirilly when the B+decay rates

dominate the burning cycle (SCCFig 5). The prcssurt confines the TNR allowing the temperature and energy

gcncraticm to reach high values, Analytical studies by Fujimoto (1982 a,b) give a critical pttssuru (log PC s

20,0) necessary for mass ejection, while numerical simulations indicate log FC z 19.3 (Truran and l.ivio

!986), The pressure is determined by the overlying envelope mass and the gravity, The gravity is specified

by the mass and radius of the white dww-f. Since the radius of,1 white dwarf depends mainly on its mass, it

follows that the gra”~itydepends only on the white dwarfs mass, We will now consider the various types of

nova models calculated to date to see what controls the strength of the ‘1’NR,In particular, we will

investigate what roles the input paritmetcrs nnd the physics pl:iy,

A.~” ‘mwml!hkls

The first realistic hydrodynamic studies of novae involved models with the hydrogen-nch envelope

initially in placr and in hydrosti~tit”iin(!thcmwl rql]ilibriltm (N1the white dw:irf (Stnwfkki rf al 1972;

Prinlnik et u1. 197t4; MclXmitld 1979), The ohscrvcd !enturrs of the classical nova--observed visuni light

curve, velocity CUIVC, c(}t~sti~nth~hmlctric luminosity :Ind shift of the wiivcl~ngth of peak emission--wer-r

Simuhttcd by these mmk]s (Sp:irks c1 (J/ 1°76; slii~fl[”](! (’( (J/ 1°7~), The m:l!,siin(j intrinsic luminosity ilf

the white dwarf and [he mass and composition of the envelope were input pammetcrs for these models.

“Ilwsc inpul purnnwtcrs dcwnninc(l IIwpIVSSIIWWI(Ic(~lllp~sili(mof Ihc nuclcur energy gcncrwion rcgitm

and, thus, the strength of the TNR. Whether or not energy is allowed to flow into the while dwarfs core

(below the hydrogen-rich envelope) has only a small effect on the depth and thus on the swcngth of the

TNR. For a specified white dwarf mass, the strength of the TNR depends mainly on the input envelope

mass and composition. The scleaion of the envelope mass was guided by hydrostatic accrcting white dwarf

models (Gia.nnone and Wcigert 1967; Taam and Faulkner 1975). At an early stage of modeling it was

discovered that the abundance of the CNO nuclei had to & enhanced above solar to get a strong TNR

(Starrfkld et al. 1972). Although it was suspected that this enhancement cam from a mixing up of white

dwti core material, the nature and efficiency of the mixing process were unknown and the CNO

enhancement was artificially set.

B. ~

The next logical improvement in nova calculations was an acmcting white dwarf rndcl (Nariai er al.

1980; Kutter and Sparks 1980; McDonald 1980; Prialnik er al. 1982). In these mdcls the white dwarf%

mass and intrinsic luminosity, the accmion rate and the internal energy and composition of the accrcting

material arc the input paramctm. Tlw envelope mass is not an input pammctcr but a complex function of the

urction rate and the internal energy of the accrcting material and the physics of radiation losses and

ccmpcssional heating during accrcticm. It is also weakly dependent on the composition of the accrcting

mttcr. This allows a mom realistic and satisfying method of determining the envelope mass. These malcls

nlso give general agreement with the sam observations as the H-in-place c-quilibrium models did. In

addition, w can make a more valid comparison of the calcukd ejected mass with observation. This also is

in god agm.cmcnt.

Three problems arc e- ident in ihecalculations. First, the mhanism of the enhancement of the CNO

nuclei was not understood. The second is shown in a plot of accretion rate verses white dwarf mass (Figure

6). 7?te location on this graph of the obscmul novae (Paucrson 1984) arc indicated by asterisks, while the

models that eject nova-like material (I%alnik er al, 1982; Starrficla et al. 1985) arc indicated by dots. nest

models am rtmrc massive and accrcte at a slower rate than the obsctvcd values, Models which have values

similar to those observed produce weak TNRs that do not eject materia! rapidly or look like a nova, They

tend to expand and then settle down to a m-dgiant configuration or to appear like a symbiotic novae (Livio et

d. 1989). Third, most accretion models have assumed thal the accrcted material has the same internal energy

as the outer Iaycrs when it comes to rcsI. Recently, Shtiviv and Starrficld (1987) and Rcgev and Shara

(1989) have shown that using mm realistic boundary conditions leads to a higher accretion energy and an

earlier and weaker TNR, This will cause the viable models in Figure 6 to mtwc downwad to even lower

accretion raws and thus, aggravate Ihe second problem.

c. “AmmwAMmklin.iu

Obscrvatitms of nova cjccta offer ample evidcncc for the mixing of whi[c clwti material with the

accmw.i material (Tmrtin and Livio 19X6;Spilr!:scl d. 19X8). In m a[[empt 10SOIVCIhc problems of [hc

cnhancemem of CNO nuclei in mwc cjrcfn nnd ICSSmassive white dwarfs producing novae, accretion

models with mixing hnvc been simulattd We nrhitrmily r!ividc these models into the categories of Iong-tcml

and short-term mixing. Long-term mixing includes those mixing processes which persist throughout the

:lu~.rrlifmprtwcss. Sh[-!l’l”ll) Illixillg in(’lllllus IINv4= pr(~”rsscs wlli~-h (KcIIr when ~-(mvrtli(m Illrns (m

1. Lcmg-Tetm Mixing

a) Diffusion

Diffusion allows the hydrogen-rich accreted material to penetrate into the white dwarf%core

material and vice versa. Prialnik and Kove[z (1984) and Kovetz and Prialnik (1985) have evolved

hydrostatic 0.9 and 1.25 ~ mexlels with fine-meshed zones to accurately determine the diffusion.

Figure 7 shows the resulting composition distribution for a 1.25 Me white dwti model with an

accretion rate of 10 lzhw. They have found that the TNR occurs below the initial core-envelope

interfaee and that the CNO abundance in the envelope can be increased up to 40%. Because of the

use of a hydrostatic code, it is not known which of their models would produce a nova. The tange

of acctetion rates chosen for these diffusion models are much slower than for the non-diffusion

accretion models. ‘l%isis because of the long time scale of diffusion. This separates the models

even further from the observations (see Figu~ 6). Shara ef al. (1986) and Ptialnik and Shara (1986)

have suggested that the white dwarf spends a considerable mount of time in a “hibernation” state.

Thus, the actual average accredon would be much lower than the obsemed accretion rate in

agreement with simulations.

b) Acctttion-Induced Shear Mixing

As disk material with high angular momentum is aemeted onto the white dwarf that initially

has little or no rotation, a shear instability develops, Kippenhahn and Thomas (1978) and Kutter and

Sparks (1987) show that under the condition of marginid stability a component of the angtili~

momentum gradient is in the radial direction. The acctmed material is shear mixed with the white

dwarf material and the composition disrnbution, to first order, is a function of the accrmed mass

only. The consetwtion of angular momentum causes rotational energy to lx converted into internal

energy, and the TNR occurs tcmsoon to produce mass ejection for a 1,0 ~ mmicl (Sparks and

Kutter 1987). Additional shtm-tetm shear mixing effects are discussed in ~1112c).

c) Compa.tison of Diffusion and Shear Mixing

For an accreting white dwarf Imh diffusion and shear mixing take place simultaneously, In

order to determine the regimes w,Iere each dominates. wc have done the fol!nwing: A 1,25M0

ecemting shear mixing while dwarf was e .olvcd until its composition distribution was similar 10the

D 12diffusion model (see Figui 17) of Priidnik m.! Kovctz ( I!M4), The composition distribution

due m shear mixing is to first order dctctmincd by thr amount of i~ccrctcdmass while that due to the

diffusion is to first order dctcrmincd by Ihc accruion time. If wc Ii]kc [his accreted mass and divided

it by the time then wc hiivc iJ Cri[it.ii! MUCIhII rii[~ where the two pmccsses produce similitr

composition distrihutiws. For ticcrcti(m riitcs nhovc [his ~iriti~;llViil(]c, shear mixing (.h]min;ltcs

while below it diffusion (h)tl]inil[(ms. I:or this ~om~)i!rison,” Ihc ctiti~ill ii~c~tiotl rate is 1()-ITMO~yr.

If the unglllm nwnwrmln~c~mkmtof the mhcrctingtl]iil~riiil is R(Iuw(I I)y ii fii~tor, then this ~ritk.iil

accretion rate is incrcwxl hy Ih:II S:IIIIC fncuw. Ii)r ~xilll)pl~, if the ~ng(lliir m(mwnltm~per gr;~m

accremd is only I().201 the Kq)kriiln disk ittlguliir nwnwnturn per grilin, then this cri[ic[il itccmtion

rate is raised [o 10-ISMf~yr. ‘1%(1S i! iippti~rs [hilt \hciir mixing dominums cxccp[ for lhc ~iisc of polw

accretion in DQ Her and AM Hcr Stilrs. Livio and Trurnn (1987) point out sonte obscrvatiomd

difficulties for ho[h [hc diffusion :mtl sh~ilr mixing nwch;inisms,

2. Short-Tetm Mixing

a) Convective Overshooting

Convective mixing is an extremely important mechanism in nova modeling and is included in

all of the previously discussed mwiels. It carries B+- unstable nuclei to the cooler outer regions and

brings fresh nuclear fuel down to the TNR region (Statileld er al. 1977). In its normal formulation

convection does not allow material from the TNR region to be mixed with deeper layers and vice

versa. However, convective overshooting will mix these two regions. A number of years, ago two

of us (W.M.S. and G. S. K.) simulated convective overshooting in a 1.0 ~ nova model by

allowing the convective elements to moss the eonvectivehon-convdve boundary whh the

calculated convective velocity. mese elements then decelerated due to the difference in den~ity

between them and their surroundings. This allowed us to estimate how much energy is deposited in

the lower regions. It turned out that the energy deposition was not enough to move the TNR region

inward appreciably. However, Woosley (1986) evolved a 1.2 Me CO white dwarf accreting solar

composition material with a diffusion coefficient prescription for overshooting. He found that the

CNO abundance of the ejected material - 10% Obviously convective overshooting must be

investigated further.

b) Flame Ropagation

The TNR region das not develop at the deepest point of hydrogen penetration into the com

material for either the diffusion or the accretion-induced shear mixing models. Thus, there is

hydrogen fuel below the TNR region. Therefore, in order for the TNR region to move deeper, it is

only necessary [o igni!e the dee~r layers. ‘IW mathematical formulation of such a process, possibly

modeled after flame propagation, is needed. f“~oosley (1986) suggesitd this could be in the form of

turbulent diffusion similar to the overshoot diffusion.

c) Convection-Induced Shear Mixing

The angular momentum disrnbution of accretion-induced shear mixing models increases

monotonically with radius its shown schemwici.dly in Figure fb. Convection tends to flatten out the

angular momentum distribution (see Figure W), just as it flattens out the composition disrnbution,

creating a steep anguhr momentum gradient at the inner and outer boundaries of the convective

region. This, in turn, his to convection-induced shear mixing which causes angular momentum

and accreted mittcrial to be mnspcmed from the outer layers to the deeper layers (see Figure flc),

Exploratory studies by Kutter and Spitrks (1989) of this mechanism suggest that it is capnble of

gmntly enhancing the TNR.

Recently two of us (G,S. K. :Ind W,M.S, h:~vcevolved:1 I.() Me CO white dwarf accre[ing

4.23 x I(Flo MOJyr, I’his tmmlclimludcd ;lccrcti(~ll-il~tlt]cq(ishciu mixing similar to previous mmlcls

(Spwks ;Illd Kuttcr l(~N7). Ill ilthliti[~ll,they :ISSIII}I(VI thilt ~-oltvl~c.tioll-it~(!llcc(i” Shcilr mixing renchcd

marginal st:lhility inslilrlt:lllc(}tlsly, “I”twresulting “!’NRwas cxtrcmcly violent with the petik energy

generation reitching 2,4 x 1016crg/(g-src). This lllod~l cjcctcd ilhollt 3 times its much mutcridl aS WilS

mxre~cd, wi[h vckwilics up m MN)() liIII/scc. l;{ r iJf’f’rf’fi,)n- iluf14r cfl ShCilr mixing, ttw itssumption of

reaching marginal stability instantaneously is reasonable because of [he long time scale for [he

ilccrc[ion process. I Iowcvcr, for f:(Jttvt:f:li(JIJ-ittll/1(’f’~fshcw mixing, this nssumpion is prolx~hly not

valid because of Ihe much shorter convective time scale. Ilewfore, this model is probably [(m

violent, but it does show the potential enhancement of the TNR by this mechanism. A hme-

dependent convection-induced shear mixing method is being ~ormulated.

IV. OIISERVATIONAL EVIDENCE OF TNR

Although the TNR model has predicted and reproductxl many observational nova features (Sparks

er tzl.1976), obsewarional confirmation of a TNR is more difficult. Since C is more abundant than N in a

solar mix (and presumably in the donor red star companion ), then C should be more abundant in the nova

ejea.a unless there has been proton capture onto the CNO nuclei. This becorms an even stronger statement if

material is mixed up fmm a CO white dwarfcom. l%e CO white dwarfs that novae initially had roughly

50% C and 50% O (Sparks et al. 1988). For the CNO nuclei, C captures protons most rcadil y and N the

least, so that the higher abundance of N thart C in all well-obsetvcd nova ejects (Truran and Livio 1986) is a

signature of CNO burning. However, CNO burning does not necessarily mean that a TNR occurred.

Sneden and Lambert (1975) analyzd the CN molecular bands ~]ear4215Awhich appcam-1in the

specmtm of DQ Her near maximum light. They found up~r limits on 1%2/Cador 15N/N which indicates

that the CNO bumirlg did not only &pcnd upon the proton capnme rates, but also upon the G+decay mtes

(Caughlan and Fowler 1962, 1972). This means that the temperature was at least l@ K. A TNR is the only

known way of reaching such a high temperature in the outer layers of a white dwarf. Unfotiunately, lower

limits on these ratios were not given. These ratios should be recalculated with symhetic spectra fkom

modem codes.

Additional evidence of a TNR would be the observation of Y-lineemission from the

0+- unstable nuclei in nova ejects (Truran et uf. 1978). Hoffman and Woosley ( 1986) calculated that the

decay signal from the z2Na Y-linemay be visible to tenrestrhd deiectors from novae within one kiloparsec.

The positron from the &cay of 13+-unstable nuclei will also prcducc Y-rays from the electron-positron

annihikt[ion. The prospects of detecting these with satclli[e Y–ray spectrometers are discussed by Lcising

and Clayton (1987).

The TNR model has been univcrmlly accepuxl as the cm.tseof the nova outburst hciIuse of its

agreement with observations. The mechanism for mixing white dwarf core materhtl with accreted matcriitl is

still under investigation.

We would like m express our Ihanks for many useful discussions :() Drs, A, Kovetz, R. Kraft, M.

Livio, D. Prialnik, M. Shtirii, G, Sh:iviv, E, Sion, R. Williams. and S. Wooslcy. W, Sparks is grateful to

the IUE Observatory and the IAU Colloquium No. 122 org:inizing rmmmittce for financial aid. S, S[w-rlleld

is grateful to Drs. S. Colgn[c, A. N. Cox, C. }:, Keller, M, Hemlerson, and K. Meyer for the h~spiliili[y of

the I-OS Alamos Nati(m:~lI.dmrtilory ilfl(i ;~gcwrous nli(mncnt of c(m]pulcr till K*, “I”his w(wk ~ils suppoflc(!

in part by NSF Grants AS’1”H5-16173 um! AS”I”RH-IX215 t~~Ariz(mn Stillc llllivcrsity itml AS’IW- 11500 to

k University of Illinois, by the Institute of Geophysics and Planetary Physics at Los Alamos, by NASA

gnln!s I(JArizon;l SI;IIC (Inivcrsi!y :111(1[() II)(. [Inivctsity (d’ (’olo~il(lo, :IIMt hy IIN. DOII.

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1978, Ap. J., 220, 1063.Sparks. W. S., Sttileld, S, G., Truran, J, W, , and Kutter, G. S. 1988, in ~

,.

Springer-Verlag), p. 234,ed. K. Nomoto (Berlin:

Starrfie!d, S., Sparks, W. M., and Truran, J, W. 1985, Ap, J., 291, 136.Stti]eld, S., Trumn, J,, and Sparks, W, M. 1977, ir] CNO 1 o~s s) ‘ <, ed. J. Audouze

(Dordrecht: Reidel), p. 49,1978, Ap. J., 226, 186.

~turrtleld, S;, Truran, J. W,, Sparks, W, M., and Kuttcr, G, S. 1972, Ap. J,, 176, 169.Taam, R., and i:aulkner, J, 1975, Ap. J., 198, 435.Truran, J. W., and Livio, M. 1986, Ap, J,, 308, 721,Tr~ran, J, W., Stitrrtleki, S,, and Spiirks, W. M, 1978, ic Gi\nlmi~ R*ctros~, .“. .

ed. T. L. Cline and R. Ramtity, (NASA ‘T’ec!lnicfilMem(mmdurn 79619), p. 315.Woosley, S. E, 1986, in hth AdviirIccL! ~~lrsc of the Swiss ~

~, ed. D. }]ii~]~k, A. Mwxh?r, and CI. Mcync] (Geneva: Geneva observatory), p, 1.

—.,

“<.Curve 54 ,

k, “\*I >, >

Curve 44\ ●\.

\\ \*

I \\ \*

I \ \

.

,.-3 10”4 ,.-5 ,.-6

q=(l-M, /M.)Fig. 1. Log temperature versus mass at various times

during the TNR.

r

w

10’

‘;/INTERNAL (+)

!J, I,.3 I 1 J

3.m I 10”5 10.310 ‘o 10“’

O-l-r WMo

10’

1

10= J1 1 , 4

s.mulo4 104 104 10-7

Q.1-ti.

Fig. 3. Energy rates versus mass after theformation of a convective’ region onan accrcting white dwarf.

~’vNUCLEAR

90

-wME+ (+]

-=-=

.-=

Iu

: [1

,0!2 L ‘1

~ + (.)

2X1O”1 I I

see a 10-’ 10 “’ 10.s

10-7

o.lr n/M.

17

F14

13 7

NIi

/73 +,%

“c ck. 0-

~#---

P (PIY)

t- ~ (P) a)Fig. 5. The n~clei and reactions includ-d in the network.

,~ 4

,~ 4

=

g

“=,..!O

,..l!

,0 1?

●

●

✍ ✎✍14

MWDWO)

I“IK, (,. Altrl’tioll r.It II v~II.s IIs wllitt” {I W,II. I 111.IS:+.

olIs(”t.vI’11 III~\’iI(’ iIt. I’ illclll,lft’11 I)v ,t:; lt’t. i:. k:.

;111(1 Ill(lll! ’1’: I)v {I{)ts.

,4.15 .10 .05 0

x

I I I

Fig. 7. Composition

m(10-5 Me)

versus mass from

.05 .10 .15

Prialr.ik and Kovctz (1984).

/ (ii)

(1))

Mm.