Astron. Astrophys. 350, 89–100 (1999) ASTRONOMY AND The...

Astron. Astrophys. 350, 89–100 (1999) ASTRONOMYAND

ASTROPHYSICS

The evolution of helium white dwarfs

II. Thermal instabilities

T. Driebe1, T. Blocker1, D. Schonberner2, and F. Herwig3

1 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, D-53121 Bonn, Germany (driebe,[email protected])2 Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany ([email protected])3 Universitat Potsdam, Institut fur Physik, Am Neuen Palais 10, D-14469 Potsdam, Germany ([email protected])

Received 12 May 1999 / Accepted 28 July 1999

Abstract. We calculated a grid of evolutionary models forwhite dwarfs with helium cores (He-WDs) and investigatedthe occurrence of hydrogen-shell flashes due to unstable hy-drogen burning via CNO cycling. Our calculations show thatsuch thermal instabilities are restricted to a certain mass range(M ≈ 0.21 . . . 0.30 M�), consistent with earlier studies. Mod-els within this mass range undergo the more hydrogen shellflashes the less massive they are. This is caused by the strongdependence of the envelope mass on the white dwarf core mass.The maximum luminosities from hydrogen burning during theflashes are of the order of105 L�. Because of the develop-ment of a pulse-driven convection zone whose upper boundarytemporarily reaches the surface layers, the envelope’s hydrogencontent decreases by∆X ≈ 0.06 (mass fractions) per flash.

Our study further shows that an additional high mass-lossepisode during a flash-driven Roche lobe overflow to the whitedwarf’s companion does not affect the final cooling behaviour ofthe models. Independent of hydrogen shell flashes the evolutionalong the final white dwarf cooling branch is determined by hy-drogen burning via pp-reactions down to effective temperaturesas low as≈ 8000 K.

Key words: stars: binaries: general – stars: evolution – stars:interiors – stars: white dwarfs

1. Introduction

In Driebe et al. (1998) (hereafter referred to as Paper I) we pre-sented a grid of evolutionary tracks for low-mass white dwarfswith helium cores (He-WDs) in the mass range from0.179 to0.414 M�. The lower masses allow applications to companionsof millisecond pulsars. As an example we derived a coolingage for the He-WD companion of the millisecond pulsar PSRJ1012+5307 of∼ 6 Gyr, which is in good agreement with thepulsar’s spin-down age of∼ 7 Gyr. The evolutionary tracks arebased on a1 M� model sequence extending from the pre-mainsequence stage through the red-giant branch (RGB) domain. Weforced the models to move off the giant branch and to evolve

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into the white dwarf regime by applying large mass-loss ratesat appropriate positions to take into account the binary natureof He-WDs (for details see Paper I).

As pointed out in Paper I one of the major results of ourstudy was the dominant contribution of hydrogen burning to theluminosity budget of the He-WDs. Therefore the final coolingevolution is slowed down, and the derived cooling ages are no-tably larger than those found in models which do not considernuclear burning or do not find hydrogen burning to be importantdue to much lower envelope masses.

In the present paper we will discuss the evolution of se-quences which undergo hydrogen shell flashes in detail. It isorganized as follows: In Sect. 2.1 we will briefly summarizethe main reasons for unstable nuclear burning, and report inSect. 2.2 on former studies concerning instabilities in He-WDs.In Sect. 3 we describe the evolutionary code used for our cal-culations. The main results of the calculations are discussed inSects. 4.1 to 4.4. Finally, conclusions are given in Sect. 5.

2. Unstable nuclear burning

2.1. The two general cases

There are two reasons for nuclear burning to become unstable:The first one is the decoupling of the thermal and mechanicalstructure of a star due to large electron degeneracy. If the equa-tion of state gets more and more independent of the temperaturea local increase in energy production cannot be stabilized by alocal expansion with following cooling of the affected layers.Therefore a further increase in energy production will lead to acorresponding increase in temperature which in turn will againraise the energy release. This phase of thermally unstable burn-ing will continue until the ongoing temperature increase resultsin an effective lifting of degeneracy in the burning zone. Thestronger coupling of thermal and mechanical structure can thenallow for an expansion and the transition to a phase of stablenuclear burning. This kind of instability is found, for instance,during the onset of central helium burning in low-mass stars(Minitial <∼ 2 M�,Mcore ≈ 0.47 M� for Z = 0.02) on the tipof the red giant branch (central helium flash).

90 T. Driebe et al.: The evolution of helium white dwarfs. II

The second reason for thermally unstable nuclear burning ismostly based on the geometry of the burning region and occursonly during shell burning. If the shell’s mass (its “thickness”)becomes too small compared to its radial extent, the expansionfollowing a local increase in temperature and energy productionis insufficient to cool the shell: The thin burning zone will insteadbe heated by expansion, and the energy production is further in-creased. The thermal runaway only stops when the thickness ofthe shell is large enough as to allow for a cooling expansion. Thiskind of unstable burning repeatedly takes place during the dou-ble shell-burning phase of AGB stars (Schwarzschild & Harm1965, Weigert 1966). There, these instabilities are known asthermal pulses or helium shell flashes caused by thermally un-stable helium burning.

Schwarzschild & Harm (1965) derived an instability crite-rion for non-degenerate matter by linear perturbation analysis.This criterion includes the shell thickness as well as the tem-perature exponentν of the energy generation rate (ε ∝ T ν) as ameasure of the temperature dependence of nuclear burning. Theassumption of non-degeneracy is justified for the helium layersof AGB models during most of the thermal pulse evolution. Ac-cording to the Schwarzschild & Harm criterion unstable burn-ing is favored when the shell becomes thinner and the burning isquite temperature sensitive, as it is the case for helium burning.A similar criterion for the study of thermal pulses was derivedby Sackmann (1977). In a recent study Frost et al. (1998) reportthat in advanced stages of thermal pulse evolution degeneracyin the region of the helium burning shell may become noticeableand can therefore lead to significantly stronger pulses (degen-erate pulses). Kippenhahn & Weigert (1990) give an instabilitycriterion which also takes degeneracy into account.

2.2. Unstable burning in helium white dwarfs

Unlike as in AGB stars, where the helium burning shell becomesthermally unstable, He-WDs show instabilities related to CNOcycling which dominates hydrogen burning in the lower, i.e.hotter part of the geometrically thin shell.

Kippenhahn et al. (1967) calculated a He-WD model withM = 0.264 M� which evolves through a phase of unstableburning. As a result the track in the Hertzsprung-Russell dia-gram (HRD) is reversed and the He-WD returns to the RGB do-main. The evolution of the white dwarf during the thermal insta-bility was followed in more detail by Kippenhahn et al. (1968).Giannone et al. (1970) found unstable burning in a0.268 M�He-WD model, whereas their models withM = 0.366 M� and0.426 M� did not show any sign of thermal instabilities. Unlikethe previously mentioned studies, Webbink (1975) found onlysmall hook-like excursions in his He-WD tracks close to thepoint of maximum effective temperature.

Iben & Tutukov (1986) calculated a0.298 M� He-WDmodel based on a1 M� model which suffered from high massloss episodes on the RGB, mimicking Roche-lobe overflow to acompanion. They found two strong thermal instabilities to oc-cur on the cooling branch resulting in a track similar to the oneof Kippenhahn et al. (1968).

Castellani et al. (1994) calculated a grid of He-WD modelsequences for different metallicitiesZ using the same tech-nique as Iben & Tutukov (1986). Like Giannone et al. (1970)they found thermal instabilities only below a certain mass limitwich depends onZ: For Z = 0.01 and 0.001 no flasheswere found forM >∼ 0.33 M� andM >∼ 0.35 M�, respec-tively. ForZ = 2 · 10−4 one hydrogen shell flash was foundfor M = 0.370 M� andM = 0.389 M�. In a recent studySarna et al. (1998) found instabilities in their cooling tracks forM < 0.2 M�, quite similar to those of Webbink (1975) who,however, used rather large time steps.

Due to large time steps and/or significantly smaller envelopemasses several studies did not find thermal instabilities at allduring the cooling of He-WDs, as e. g. Chin & Stothers (1971),Alberts et al. (1996), Althaus & Benvenuto (1997), Benvenuto& Althaus (1998) and Hansen & Phinney (1998).

3. The evolutionary calculations

Besides some minor modifications we used the evolutionarycode described by Blocker (1995). Nuclear burning is accountedfor via a nucleosynthesis network including 30 isotopes with allimportant reactions up to carbon burning similar as in El Eid(1994). The most recent radiative opacities by Iglesias et al.(1992) and Iglesias & Rogers (1996), supplemented by those ofAlexander & Ferguson (1994) in the low-temperature region,are employed. Diffusion is not considered. The initial compo-sition is (Y, Z) = (0.28, 0.02), the mixing length parameterα = 1.7 followed from calibrating a solar model. The Coulombcorrections to the equation of state are those given by Slatteryet al. (1982).

He-WDs are known to be components in binary systemswhere early case B mass transfer (Kippenhahn & Weigert 1967)must have taken place during RGB evolution which forces thestars to leave the RGB before the onset of helium burning andto evolve into the white dwarf regime. We did not calculatethe mass exchange phases during the RGB evolution in detailbecause we are primarily interested in the cooling properties ofthe white dwarf models themselves rather than in the generationof these models by binary star evolution. Hence we used anapproximate approach to get the pre-white dwarf models (seealso Iben & Tutukov 1986, Castellani et al. 1994, and Paper I):

We calculated a 1M� sequence from the pre-main sequencephase up to the tip of the RGB. Along the RGB we appliedmass-loss ratesMR according to Reimers (1975) withη = 0.5.At appropriate positions high mass loss rates,Mhigh, were in-voked in order to get models of desired final mass,M : 0.179,0.195, 0.234, 0.259, 0.300, 0.331 and0.414 M�. Fig. 1 dis-plays all He-WD sequences in the HRD. High mass loss variedfrom Mhigh ≈ 10−9 M� yr−1 for M ≈ 0.15 M� to about10−6 M� yr−1 for M ≈ 0.4 M�. These values were chosen toallow the models to hold thermal equilibrium during their furtherevolution (withM = Mhigh). For a more detailed descriptionwe refer to Paper I. Here, we only want to stress the importanceof sufficiently small time steps when the evolution through ther-

T. Driebe et al.: The evolution of helium white dwarfs. II 91

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3.43.63.844.24.44.64.85

log

(L

/Lsu

n)

log Teff [K]

Fig. 1. Hertzsprung-Russell diagram with evolutionary tracks for He-WD models. The tracks with solid lines belong to final white dwarfmasses of 0.179, 0.195, 0.234, 0.259, 0.300, 0.331 and 0.414M� (fromtop to bottom). The dotted lines show the flash phases forM = 0.234and0.259M�, the dashed line gives the1M� evolutionary path theHe-WD models are based upon.

Table 1. Total remnant massM , mass of the hydrogen-exhaustedcoreMc, total mass of the outer hydrogen layers (“thickness”)MH,envelope massMenv, and helium surface abundance by mass fractionY , atTeff = 5000 K forM > 0.2M� and at 10000 K forM ≤ 0.2M�after the end of RGB evolution.Mc is defined by the mass coordinatebelow whichX ≤ 0.35 with X being the mass fraction of hydrogen.

M/M� Mc/M� MH10−3M�

Menv10−3M�

Y

0.179 0.1693 5.061 10.211 0.4640.195 0.1859 4.937 9.598 0.4610.234 0.2220 8.118 13.098 0.3540.259 0.2524 4.771 7.232 0.3120.300 0.2960 3.189 4.746 0.3010.331 0.3281 2.509 3.744 0.3010.414 0.4116 1.446 2.175 0.301

mal instabilities should be followed properly. A brief discussionon this topic is given in the Appendix.

4. Results

4.1. General remarks

We briefly repeat the main results of our calculations which havebeen addressed in detail in Paper I. Due to the comparativelylarge remaining envelope masses after termination of the RGB

0.01

0.1

1

10

100

110100

LH

yd/L

g

Teff/1000 K

0.1790.1950.2340.2590.3000.3310.414

Fig. 2. Ratio of hydrogen (LHyd) to gravothermal luminosity (Lg) asa function ofTeff for He-WDs of different masses (inM�). For thesequences which undergo hydrogen flashes only the final evolutionalong the cooling branch is shown.

0.1

1

10

100

3*108 109 1010 1011

LH

yd/L

g

Age [in yrs]

0.1790.1950.2340.2590.3000.3310.414

Fig. 3. Same as Fig. 2, but as a function oftpost−RGB. The age atthe very left of the diagram corresponds to effective temperatures ofaboutTeff ≈ 15000 . . . 18000K, the curves end at temperatures ofTeff ≈ 4000 . . . 5000K. We adoptedt = 0 when the models passTeff = 10000K for M < 0.2 M� andTeff = 5000K for M >0.2 M� after leaving the RGB.

evolution (see Table 11) one of the main characteristics of ourwhite dwarf models is that hydrogen burning due to pp-reactionsremains the main energy source down to effective temperatureswell below104 K (see Paper I and Fig. 2). This residual burningleads to a significant slow-down of the further evolution, result-ing in larger cooling ages (typically a few109 yr, see Fig. 3)than found for He-WD models where hydrogen burning is neg-ligible due to smaller, ad-hoc assumed2 envelope masses, oreven not considered at all. The implications of evolutionary en-

1 We note that the same table is shown in Paper I but the data forM =0.179 and0.195M� did not refer toTeff = 10000K as indicated inthe caption but erroneously toTeff = 5000K as for the larger masses.

2 In these calculations the envelope mass is taken as a free parameterand has not been computed from the mass-loss history of the He-WDprogenitor according to the binary nature of He-WDs.


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Lo

g (

L/L

sun)

Log Teff [K]

1. Subflash2. Subflash

Final flash-onset

2.Flash 1. Flash

Fig. 4. HRD with a complete evolutionary track for the He-WD se-quence withM = 0.234M�. The dashed line shows the1 M� trackwhich was used to generate the post-RGB model. Two main flashesoccurred, each followed by weaker subflashes (see also Fig. 8).

velope masses for the evolution of white dwarfs are discussedin Blocker et al. (1997).

Besides of this important property of He-WDs and its con-sequence for age determinations of, e.g., millisecond pulsarsystems as PSR J1012+5307 (cf. Paper I), another result ofour study concerns the occurrence of hydrogen shell flashes:We did not find any unstable hydrogen-burning in the se-quences with massesM = 0.179, 0.195 and0.414 M�. Onlyfor M = 0.234 and0.259 M� major hydrogen shell flashes(LHyd,max >∼ 105 L�) developed with concomitant extendedloops in the HRD. Only a temporal slight increase of the CNO-luminosity on the cooling branch was found in the sequenceswith M = 0.300 and0.331 M�.

The restricted mass range for the occurrence of hydro-gen flashes agrees with earlier results, e.g. Kippenhahn et al.(1968), Gianonne et al. (1970) or Castellani et al. (1994).We note that Webbink (1975) suggested a lower mass limitof M ≈ 0.206 M�, in good agreement to our findings. The0.298 M� model of Iben & Tututkov (1986) is fully unsta-ble, our0.300 M� model only marginally. The calculations ofCastellani et al. (1994) suggest that the upper boundary for theflash range is higher for lower metallicity. Therefore, their re-sult of Mflash

upper < 0.33 M� for Z = 0.01 is consistent withour calculations. Complementary calculations where equilib-rium rates for the pp-chains and the CNO-cycle were used in-stead of the nuclear network show three hydrogen shell flashesfor M = 0.227 M� and one strong hydrogen shell flash forM = 0.310 M�. Because we found no hydrogen flashes in ourstandard sequences withM = 0.300 M� andM = 0.195 M�we conclude that the mass range for the occurrence of instabili-

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B

F

E

F

DC

A

G

Fig. 5. Same as Fig. 4, but for theM = 0.259M� sequence. Hereonly one major flash was found. Properties of the models labeled bycapital letters are given in Table 2.

ties depends on uncertainties due to different input physics (e.g.diffusion as in Iben & Tutukov (1986)), and that the boundariesMflash

lower = 0.21 M� andMflashupper = 0.30 M� may be uncertain

by about0.01 M�.According to our calculations the number of hydrogen

flashes depends on the mass: We found one major hydrogen flashfor M >∼ 0.25 M�, two flashes for0.25 >∼ M/M� >∼ 0.23 andthree flashes for0.23 >∼ M/M� >∼ 0.21. Thus, we concludethat unstable hydrogen burning in He-WDs caused by the fadingCNO cycling (see next section) is restricted to a certain massrange ofM ≈ 0.21 . . . 0.30 M�. Furthermore, the number offlashes increases with decreasing white dwarf mass.

It is noteworthy that the cooling properties belowTeff <∼20000 K are independent on the occurrence of hydrogen shellflashes during the previous evolution. Hydrogen burning re-mains the main energy source along the cooling branch downto very low effective temperatures (see Sect. 4.3). Because He-WDs suffering from hydrogen flashes evolve back to the RGBregime one has to account for high mass-loss episodes due toRoche lobe overflow although the time spent away from thecooling branch is small compared to the cooling time itself. Ourcalculations show that the cooling is not affected by such highmass-loss episodes (see Sect. 4.4).

4.2. The main flashes

Figs. 4 and 5 show the complete evolutionary tracks for the se-quences withM = 0.234 M� and0.259 M�. While the evo-lution of theM = 0.259 M� model is characterized by onlyone major flash resulting in an extended loop in the HRD (seeFig. 5), the model withM = 0.234 M� experiences two strongflashes (see Fig. 4).


Table 2.Data for the marked points in Fig. 5. The age was set to zero atTeff = 10000K of the post-RGB evolution. The different evolutionarystages A-G are discussed in the text.

Model tpost−RGB/107 yr log(Teff/K) log(L/L�)

A 1.469119 4.2865 -0.5726B 2.803540 4.3678 -0.0754C 2.803641 3.9532 -1.4070D 2.803643 3.8488 -1.0786E 2.803658 3.8413 -0.2007F 2.803744 4.3272 2.0633G 13.003598 4.2859 -0.8025

After mass loss terminates the RGB evolution, the remnantsevolve at almost constant surface luminosity to higher effec-tive temperatures towards the white dwarf cooling branch. Inthis phase the total luminosity is almost entirely supplied bynuclear burning due to CNO cycling (L ≈ LCNO). The contri-bution due to contraction is negligible. The situation changeswhen the star reaches the cooling branch. Now, the temper-ature within the hydrogen burning shell becomes too low tosupport further CNO cycling.LCNO drops significantly, andcontraction sets in until pp-burning takes over the main nuclearenergy production. StillLHyd � Lg holds in this early cool-ing period (see e.g. Fig. 11 and Fig. 5 in Paper I). The phaseof unstable burning starts at typical effective temperatures ofTeff ≈ 20000 . . . 25000 K on the cooling branch (close to pointA in Fig. 5). At this stage of evolution the energy productiondue to CNO cycling rises again and finally exceeds the pp-contribution by large amounts. In Fig. 5 this part roughly co-incides with the loop between point A and B in the HRD. Theduration for this period of evolution (∆tonset) depends on the re-maining envelope mass and the thermal timescale of the burningshell, and ranges from∆tonset ≈ 4·106 yr forM ≈ 0.23 M� to∆tonset ≈ 4·107 yr forM ≈ 0.30 M�. For example, the modelwith M = 0.259 M� gives∆tonset ≈ 1.7 · 107 yr, that withM = 0.234 M� only ∆tonset ≈ 5 · 106 yr. At the onset of thesecond flash forM = 0.234 M� one gets∆tonset ≈ 1.2·107 yrdue to the further reduced envelope mass.

The increasing energy release during the flash developmentcauses a steep temperature gradient in the vicinity of maximumenergy production and the formation of a pulse-driven convec-tion zone well inside the hydrogen burning shell (beyond pointB). The situation is displayed in Fig. 6: Att ≈ 2.80357 · 107 yrthe convective shell establishes right above the locus of maxi-mum energy production due to hydrogen burning. The degener-acy inside the burning shell is quite moderate as can be seen fromthe degeneracy parameterψ at maximum energy generation (forthe defintion ofψ see e.g. Kippenhahn & Weigert 1990). Duringthe onset of the unstable burning,ψ ≈ −0.5 . . .− 1.

Once the convection zone is fully established, the evolutionis rapidly accelerated (roughly between point B and D). The typ-ical time scale is now of the order of a few decades. The luminos-ity contribution due to nuclear burning increases several ordersof magnitude and the convection zone grows until it extends to

0.254

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0.259

2.8033 2.8034 2.8035 2.8036 2.8037-4

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Mr/M

sun

Ψ

Age [in 107 yr]

Ψ

max

convection

ε

Fig. 6. Development of the pulse-driven convection zone (interme-diate grey shaded region) during the hydrogen-flash phase of theM = 0.259M� sequence as a function of time. Point B from Fig. 5is located att ≈ 2.80354 · 107 yr. Also shown are the lines of max-imum energy generation due to hydrogen burning,εmax, the regionswhereεmax has dropped to1% (dark grey shaded) and0.1% (light greyshaded) respectively, and the degeneracy parameterψ (right y-axis) atthe point withε = εmax.

the stellar surface (see Fig. 6,t ≈ 2.80354 . . . 2.80364 ·107 yr).AlthoughLHyd increases to about105 L� in this phase (nearpoint C in Fig. 5), the surface luminosity drops by almost 2 or-ders of magnitude because the increased energy production islargely overcompensated by the energy loss due to the expan-sion of the envelope. This expansion roughly doubles the stellarradius and leads to a complete lifting of degeneracy in the shell(sharp drop ofψ). For instance, model B hasR ≈ 0.05 R�, andbeyond point C one findsR ≈ 0.1 R�.

The maximum hydrogen luminosity reached during the flashis supplied by pp-burning, although the onset of the instabil-ity is triggered by CNO cycling (see below, Fig. 7). ForM =0.259 M�,LHyd,max ≈ 1.3 · 105 L�. ForM = 0.234 M� bothflashes are of comparable strength withLHyd,max ≈ 2.7·105 L�for the first andLHyd,max ≈ 2.3 · 105 L� for the second flash.The time span between both flashes amounts to1.8 ·107 yr (seealso Fig. 7).

Beyond point C the most luminous part of the flash insta-bility has passed andLHyd decreases whileLg increases. Thesurface luminosity is again enhanced due to the increase ofLg. At point D (Teff ≈ 7000 K, see Fig. 5) the upper bound-ary of the pulse-driven convection zone reaches the surface (att ≈ 2.80364 · 107 yr in Fig. 6). Due to the convective transportof hydrogen from the surface layers down to the hydrogen burn-ing region the surface hydrogen abundance is reduced by about∆X ≈ 0.06 when surface convection establishes. The smallblueward evolution in the evolutionary track close to point D isrelated to these composition changes in the envelope (see Iben &Tutukov 1986). The compositional change is consistent with theresults of the flash model of Kippenhahn et al. (1968) who found∆X ≈ 0.08. Iben & Tutukov (1986) found a larger amount of


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3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

Log

Li/L

sun

Age [in 107 yrs]

LppLCNOLHyd

L

Fig. 7. Evolution of surface and hydrogen luminosity,L andLHyd,and the contributions due to CNO cycling and pp-burning as a functionof cooling age forM = 0.234M�. The figure shows the age rangewhere the hydrogen shell flashes develop. Aget = 0 is given by thefirst turning point after the track has entered the cooling branch (seepoint A in Fig. 5) and corresponds to a post-RGB agetpost−AGB ≈4.62 · 107 yr.

∆X ≈ 0.2, probably due to their consideration of gravitationaland chemical diffusion leading to different chemical profiles.

Between point D and E in Fig. 5 the lower boundary ofthe pulse-driven convection moves upwards, and at point E theconvection zone (and therefore surface convection) vanishes atall. Overall, the convection zone exists for≈ 1200 yrs, and for≈ 150 yrs it extends up to the surface. Beyond point E thecontraction of the inner regions of the hydrogen shell resumes,while the surface layers react by expansion, resulting in a red-ward motion in the HRD bringing the He-WD almost back to theRGB region. Finally, contraction seizes the surface layers andthe star evolves back to higher effective temperatures towardsthe cooling branch again.

Around point F just before re-entering the cooling branch,the so-called subflashes develop. Here, contraction initiates an-other small increase ofLHyd to ≈ 30 . . . 200 L� (compared to≈ 105 L� during the major flash) leading to a temporary ex-pansion of the outer layers of the star and only minor circle-likeexcursions in the HRD.

For illustration, Fig. 8 shows the evolution (arbitrary zeropoint) of the luminosity contribution due to hydrogen burning,LHyd, and the gravothermal luminosity,Lg, during the flashphase of theM = 0.234 M� sequence. The large plot showsthe evolution of the first subflash episode of the first hydrogenflash phase and the small inlet that of the second phase. Thepeak att ≈ 0.4512 · 107 yr which is not resolved within theplot range comes from the major flash, the second peak att ≈0.4518 · 107 yr with LHyd,max ≈ 250 L� followed by a shortperiod withLg < 0, i.e. expansion, marks the subflash. This firstsubflash is then followed by another slight increase inLHyd att ≈ 0.4528 · 107 yr with LHyd,max ≈ 20 L� causing the littlecircle in the track atlogL/L� ≈ 1.2 (see Fig. 4).

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0.45 0.451 0.452 0.453 0.454 0.455

LH

yd/L

sun,

Lg/

Lsu

n

Age [107 yr]

LgLHyd

-50

0

50

2.273 2.277

subflashmain flash

subflashmain flash

Fig. 8. Evolution ofLHyd (dashed line) andLg (solid line) as a functionof time for the0.234M� sequence during flash phase: Large plot:First flash phase; inlet: second flash phase (same plot labels). For thedefinition oft = 0 see Fig. 7.

After the second major flash (again unresolved in the timerange of this diagram att ≈ 2.2757 · 107 yr) a local maximumin LHyd with LHyd,max ≈ 30 L� and slight expansion is foundat t ≈ 2.2762 · 107 yr (i.e. about5 · 103 yr after the main flash)corresponding to the small loop in the evolutionary track atlogL/L� ≈ 1.5.

As previously mentioned the instability in He-WDs iscaused by the fading CNO-luminosity on the cooling brancharoundTeff ≈ 22000 . . . 25000 K. After pp-burning becamethe dominant contribution of hydrogen burning,LCNO againincreases on a typical timescale of a few106 to 107 yr. Fig. 7shows the situation for the two hydrogen flashes occurring forM = 0.234 M�. Beyondt ≈ 4.6 · 107 yr andt ≈ 5.6 · 107 yrLCNO, rises again fromLCNO ≈ 0.1 to a few102 L�. With on-going unstable burning the outer layers of the He-WD expandand slightly cool the shell so thatLCNO drops again. The peaksin LHyd (≈ 105 L�) are due to pp-burning.

To investigate the dependence of the mass on the occurrenceof hydrogen flashes we applied the criteria for unstable burningdiscussed in Sect. 2.1. As already seen in Fig. 6 the degeneracyin the hydrogen shell is quite moderate but far from being neg-ligible. We used the criterion of Kippenhahn & Weigert (1990)to account for the possible effect of degeneracy. The criterionstates that nuclear burning in a shell at radiusRshell and ofthickness∆R becomes unstable if

1 − ∇ad · 4δ4α− Rshell

∆R

> 0 . (1)

∇ad is the adiabatic temperature gradient andα andδ are given

by α =(∂ ln ρ∂ lnP

)T

andδ = −(∂ ln ρ∂ lnT

)P

. For comparison, we

also used the criterion of Schwarzschild & Harm (1965) whichpredicts unstable burning if

∆TTshell

>4ν

and∆RRshell

<52

· |Q| (2)

with Q ≈ −8 . . .− 4 and∆T being the temperature differencebetween lower und upper shell boundary. A main problem when


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Age [in yr]

0.4140.2590.195

Fig. 9. Different quantities as a function oftpost−RGB for the se-quences withM = 0.195 and 0.414M� (no flashes) andM =0.259M� (one flash): Upper left: Temperature at the maximum ofenergy generation,εmax, in the hydrogen burning shell. Upper right:Radial extent of the hydrogen burning shell. Lower left: Degeneracyparameterψ at ε = εmax. Lower right: Temperature derivative of theenergy generation rate,εT, atε = εmax.

dealing with these criteria is the definition of quantities as, forexample, representative temperatures and pressures in the shellor the typical radial extent of the shell.

However, although we find the instability criterions to befulfilled in several models the strict application of the aforemen-tioned criteria alone cannot explain the restricted mass range forthe occurrence of hydrogen flashes. Some general aspects onthe model properties important for the study of unstable burn-ing can be seen from Fig. 9 where different quantities of layersat maximum energy generation are shown along the coolingbranch evolution for the non-flash sequences withM = 0.195and 0.414 M� and the flash sequence withM = 0.259 M�covering most of the He-WD mass range.

As seen before forM = 0.234 M�, degeneracy of the shelllayers is moderate (lower left panel) in the upper part of thecooling branch evolution. ForM = 0.259 M� , ψ ≈ −0.5 . . . 0at the beginning of unstable burning (t ≈ 2 · 107 yr). ForM =0.414 M� degeneracy is slightly lower (ψ ≈ −1, t ≈ 2 ·105 yr)at the corresponding evolutionary stage, and forM = 0.195 M�slightly higher (ψ ≈ 0 . . . + 0.5, t ≈ 5 · 108 yr). For an ideal,non-relativistic gas one obtains for the non-degenerate limitingcaseα = δ = 1, whereas strong degeneracy(ψ � 1) leads

to α ≈ 35 (1 + µe

µ0

1ψ ) and δ ≈ 3

2µeµ0

1ψ with µ0 andµe being

the mean molecular weights per ion and electron, resp., (seeKippenhahn & Weigert 1990)3. Thus, inspecting Eq. (1), fromψ alone we would conclude that, if at all, degeneracy favourshydrogen shell flashes for lower He-WD masses.

The temperature sensitivity of nuclear burning alongthe cooling branch can be seen from the quantityεT =∂ ln ε/∂ lnT (= ν for ε ∝ T ν) which is shown in the lower rightpanel of Fig. 9. When the flash develops forM = 0.259 M�,one findsεT ≈ 12.5. For the other masses we haveεT ≈ 15(M = 0.414 M�) andεT ≈ 11 (M = 0.195 M�). Because un-stable nuclear burning is favoured for higher temperature sen-sitivity of the burning (see Eq. (2)) the occurrence of flashes isfavoured for heavier white dwarfs. Thus, combining the effectsof degeneracy and large temperature exponents in the energygeneration rate might, in principle, explain why high and low-mass He-WDs do not suffer from hydrogen flashes.

However, it is most likely the radial thickness of the shell(see upper right panel in Fig. 9) which has the most importanteffect on the occurrence of hydrogen flashes. At the onset ofunstable burning we found∆R ≈ 10−2 R� forM = 0.259 M�(the shell’s borders are taken at the point withε = 10−3 · εmax).ForM = 0.195 M�,∆R is approximately a factor of two largerand forM = 0.414 M� a factor of two smaller. For low-massHe-WDs degeneracy of the shell is of minor importance. Also,the thickness of the shell is too large as to allow hydrogen shellflashes. On the other hand, for high-mass He-WDs the shell isso thin that the thermal cooling time is smaller than the typicaltime scale for the onset of the instability. Hence, for He-WDswith M >∼ 0.3 M� an instability might be initiated but the fastcooling of the shell prevents a hydrogen flash (see also nextsection).

4.3. Final cooling

After the He-WD models withM = 0.234 M� and0.259 M�have completed their major hydrogen flashes a final onset ofa flash results in a local hook in the cooling track (see Figs. 4and 5, point G). Again this increase of luminosity due to hy-drogen burning is caused by CNO cycling. This can be seenfrom Figs. 10 and 11 where the evolution of different lumi-nosity contributions for the two flash sequences are plottedas a function oftpost−RGB for the whole post-RGB evolution.LCNO shows a local maximum attpost−RGB ≈ 1.23 ·108 yr forM = 0.234 M�, approximately5 · 107 yr after the last majorflash. The corresponding point forM = 0.259 M� is at an age oftpost−RGB ≈ 1.40·108 yr, which is about108 yr after the majorflash. At this point of evolution hydrogen shell burning is alreadydominated by pp-burning (Lpp ≈ 20·LCNO), but the differencebetween both contributions is rather small compared to the laterevolution (see Figs. 10 and 11 fort >∼ 1 Gyr). Because the char-acteristic cooling time is now smaller than the typical timescalefor the onset of unstable burning another strong hydrogen flash is

3 Note that radiation pressure is negligible and∇ad does not dependonψ in the non-relativistic regime.


-8

-6

-4

-2

0

2

4

6

107 108 109 1010

Log

Li/L

sun

Age [in yrs]

Lpp LCNO LHyd L Lg(> 0)|LN|

Fig. 10. Evolution of different luminosity contributions (see plotlabels) as a function oftpost−RGB for the sequence withM =0.234M�: Lpp: luminosity due to pp-burning;LCNO: luminosity dueto CNO cycling;LHyd: complete luminosity due to hydrogen burn-ing; L: surface luminosity;Lg: contribution of gravothermal lumi-nosity due to contraction (Lg > 0); LN luminosity contribution dueto neutrino losses.t = 0 is adopted as in Fig. 3 (see also Fig. 7 fort = 3 . . . 8 · 107yr).

-8

-6

-4

-2

0

2

4

6

106 107 108 109 1010

Log

Li/L

sun

Age [in yrs]

Lpp LCNO LHyd L Lg(> 0)|LN|

Fig. 11. Same as Fig. 10, but for theM = 0.259M� sequence.

prevented. Such onsets of unstable CNO-burning have also beenfound in the sequences with0.300 M� and0.331 M�, resultingin similar hooks on the cooling track (see Fig. 1 in Paper I).

The further evolution in both flash sequences is comparableto the one without hydrogen flashes (see for instance Fig. 4 inPaper I). It is characterized by quiescent hydrogen shell burningvia the pp-chains until an age of about 10 Gyr. Then hydrogenburning terminates and contraction determines the surface lu-minosity evolution and final cooling (see Figs. 10 and 11).

Finally, we like to note that the mass-radius-relation of He-WDs is only affected by hydrogen shell flashes forTeff ≈20000 . . . 25000 K: The high burning rates associated with theflashes lead to a higher envelope consumption compared to se-quences without flashes, resulting in slightly smaller radii andenhanced evolutionary speeds on the upper part of the coolingbranch. Besides, since the timescale of the instabilities is short

compared to the characteristic cooling time (a few107 yr com-pared to Gyr) and most of the time during a flash is spent closeto the cooling branch, the influence of hydrogen shell flashes onthe mass-radius-relation is restricted to the onset phase wherethe changes in radius are moderate.

4.4. Roche lobe overflow during the flashes

During the hydrogen shell flashes the expansion of the outer-most layers (from a few 0.01 solar radii to a few solar radii)forces the He-WDs to evolve back into the RGB domain in theHRD (see Figs. 4 and 5). Due to the binary nature of He-WDs, inprinciple, mass transfer to a companion (in most cases anotherwhite dwarf or a pulsar) should be taken into account in thisparticular evolutionary phase. This has been considered, for in-stance, in the calculations of Iben & Tutukov (1986) leading toa significant removal of envelope matter and a faster exhaustionof hydrogen burning.

Besides the sequences already discussed we thus also in-vestigated the evolution of the flash model sequence withM = 0.259 M� considering high mass-loss episodes due toRoche lobe overflow. Because we were only interested in thegeneral influence on the final cooling properties of our modelswe used a rather simple algorithm to account for Roche lobeoverflow from the He-WD to a companion: For a given RocheradiusRRoche (in our case 2, 3 and 5R�) and actual modelradiusR, we increasedη in the Reimers mass loss formula bya factor of2 ·R/RRoche for R > RRoche. Otherwise,η was setto its standard value of 0.5 used for all post-RGB calculations.

Fig. 12 presents the evolutionary tracks in the HRD dur-ing the flash forM = 0.259 M� and different assumptionsof the Roche radius. The models quickly reach the line withR = RRoche. For comparison, also the track without Roche-lobe overflow is shown. During the adjustment phase, after themodels have passedR = RRoche, mass-loss rapidly rises up toMhigh ≈ 10−3 M�/yr and forces the models to evolve back toR ≈ RRoche. Finally, M ≈ 5 · 10−6 M�/yr when the modelsevolve along atR ≈ RRoche. When Roche lobe overflow ends(R < RRoche) mass loss has dropped toM = MReimers ≈10−11 M�/yr.

After the end of Roche lobe overflow and the subflash phasethe tracks rapidly merge with the one calculated without Rochelobe overflow at the beginning of final cooling evolution. AsFig. 12 shows the subflash excursion is the less pronounced thesmallerRRoche.

The convergence of the tracks shows that the cooling prop-erties of He-WD models are not seriously affected by the Rochelobe events because similar tracks imply similar radii and thussimilar envelope masses which determine the cooling evolu-tion due to hydrogen burning. This is confirmed by Figs. 13, 14and 15. In Fig. 13 the evolution of the envelope mass,Menv,is given as a function ofTeff for the tracks from Fig. 12, inFig. 15 the corresponding evolution of surface luminosity as afunction of tpost−RGB is plotted. Fig. 14 shows the reductionof envelope mass for all sequences as a function oftpost−RGBclose toR = RRoche and beyond. Figs. 13 and 14 indicates that


-5

-4

-3

-2

-1

0

1

2

3

3.63.844.24.44.6

Log

L/L

sun

Log Teff [K]

No Roche lobetrack with RRoche=2 Rsuntrack with RRoche=3 Rsuntrack with RRoche=5 Rsun

R=RRoche=2 RsunR=RRoche=3 RsunR=RRoche=5 Rsun 1

1.2

1.4

1.6

1.8

2

2.2

2.4

3.73.83.944.14.2

Fig. 12. HRD with tracks for M =0.259M� and different Roche radii (2, 3 and5R�). ForR ≥ RRoche mass loss was ad-justed as to provideR ≈ RRoche as long asthe star does not evolve back to the whitedwarf domain. Lines of constant radius arealso plotted. The arrows indicate the courseof evolution. The inlet in the lower right cor-ner is a blow-up of the part close to the Rocheradii.

0.5

1

1.5

2

2.5

3

5101520253035

Men

v/10

-3M

sun

Teff [in 1000 K]

No Roche lobeRRoche = 2 RsunRRoche = 3 RsunRRoche = 5 Rsun

Fig. 13. Evolution ofMenv as a function ofTeff for tracks withM =0.259M� and different Roche radii (see labels). The smallerRRoche

the fasterMenv is reduced, but when the star is back on the coolingbranch the residual envelope mass is almost the same for all Rochetracks as it is for the non-Roche lobe track.

∼ 6 · 10−4 M� of the envelope mass is lost in the non-Rochemodel during the flash phase (taken from the first local maxi-mum inTeff to the last one). Note that the temporal increase ofMenv in Fig. 13 comes from the pulse driven convection zoneby mixing hydrogen-rich material somewhat below the layerswith X = 0.35 which determine the border between core andenvelope (cf. Fig. 6).

A significant reduction ofMenv sets in when dominant hy-drogen burning has again established at the end of the sub-flash phase (see Fig. 13 atTeff ≈ 27000 K and Fig. 14 for2.81 <∼ t/107 yr <∼ 2.95). This envelope reduction is sloweddown when the model enters the cooling branch (see Fig. 13at Teff,max ≈ 33000 K and Fig. 14 fort >∼ 2.95 · 107 yr).The envelope is eroded by mass loss and shell burning. This

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.8 2.85 2.9 2.95 3

Men

v/10

-3M

sun

Age/107 yr


1.61.71.81.9

22.12.2

5 10 15 20 25 30

Men

v/10

-3M

sun

Age/107 yr

Fig. 14. Evolution of envelope mass as a function oftpost−RGB fortracks withM = 0.259M� and different Roche radii (see labels).The plot shows the reduction ofMenv during and immediately afterRoche-lobe overflow. The inlet shows the evolution ofMenv on a largerscale.

can be written asMenv = Mwind + Mcore with the mass lossrateMwind and the core growth rateMcore. The core growthrate is given byMcore = LHyd

X·EHwith the hydrogen contentX

and the gain of energy due to hydrogen burning per mass unit,EH ≈ 6.3 · 1018 erg g−1.

For sequences without Roche-lobe overflowMenv is mainlydetermined by shell burning, i. e.Menv ≈ Mcore. Witha mean value ofLHyd ≈ 30 L� and X ≈ 0.7 it isMcore ≈ 4.3 · 10−10 M� yr−1. With ∆t ≈ 1.3 · 106 yr (seeFig. 14) one obtains∆Menv ≈ Mcore · ∆t ≈ 5.6 · 10−4 M� ingood agreement with the value obtained from Figs. 13 and 14.

For sequences with Roche-lobe overflow the situation is dif-ferent. Here, the consumption of envelope mass occurs pref-


-3

-2.5

-2

-1.5

-1

-0.5

0

108 109 1010

Log

(L

/Lsu

n)

Age [yr]


0

0.5

1

1.5

2

2.5

2.8036 2.8040 2.8044

Log

(L/L

sun)

Age/107 yr

Fig. 15. Evolution of surface luminosityL as a function oftpost−RGB

for tracks withM = 0.259M� and different Roche radii (see labels).The models suffering a high mass-loss episode show the same coolingbehaviour as the non-Roche sequence. The inlet shows the evolutionof L during the subflash phase.

erentielly due to the high mass-loss phase when the modelsare close to their Roche limit (Teff ≈ 8000 . . . 15000 K, seeFig. 13). For instance, for the sequence withRRoche = 3 R�it is ∆Menv ≈ 4.5 · 10−4 M�, corresponding to a mean massloss ofMwind ≈ 10−5 M� yr−1 lasting for∆t ≈ 40 yr. After-wardsMenv is further reduced by∆Menv ≈ 2 · 10−4 M� dueto nuclear burning (see Fig. 14).

In the end, the envelope masses are almost the same whenthe different models enter the final cooling branch (Menv ≈2.05 · 10−3 M�, see Fig. 14), i.e. the residual envelope massand the corresponding cooling evolution are rather independentfrom the evolutionary history.

On one hand, the latter result is consistent with the resultsof Iben & Tutukov (1986). In the case of Roche lobe overflowthe high mass loss determines the evolutionary speed, whereasotherwise the evolution is characterized by the thermal timescaleof the envelope. On the other hand, in their0.298 M� model thehigh mass loss causes a considerable loss of envelope mass (≈1.5 · 10−3 M�) and prevents hydrogen burning from becomingever dominant again. Consequently, their model cools downquite rapidly. The cooling time is of the order of a few108 yrwhereas our models are controlled by residual hydrogen burningleading to cooling times of the order of Gyr (see Figs. 10 and11).

Thus, from our calculations it appears that different massloss histories during the phase of the hydrogen shell flashes donot influence the final cooling of He-WDs significantly. This re-sult seems to be consistent with the calculations of Kippenhahnet al. (1968): TheirM = 0.264 M� model evolves throughone major hydrogen shell flash and one strong subflash, bothbringing the white dwarf radius above the critical Roche radius(points Q to R and T to U in their Fig. 1), resulting in a total massloss of about10−3 M� which is comparable to the6 ·10−4 M�of the present calculation with only one high mass-loss episode.

After a rapid evolution back to the cooling branch the evolutionis slowed down, and atlog(L/L�) ≈ −2 (point W in their cal-culation) a cooling age of approximately 2.55 Gyr is reached,comparable with ourM = 0.259 M� model which has at thesame positiontpost−RGB ≈ 1.85 Gyr. On the other hand, byassuming a linear correlation betweenlogL/L� and log tcoolas a first approximation, the cooling age of the model of Iben &Tutukov (1986) can be estimated to be onlytcool ≈ 2.3 · 108 yrat log(L/L�) ≈ −2 (using their points S and T, see their Fig. 1and Table 1). This value is almost one order of magnitude be-low the one given by models which allow hydrogen burning tocontinue.

5. Summary

We have calculated evolutionary models of low-mass whitedwarfs with helium cores to study in detail their cooling be-haviour and their evolution including phases of thermally un-stable hydrogen burning (hydrogen shell flashes). We found thatthe occurrence of these thermal instabilities is restricted to themass range0.21 <∼ M/M� <∼ 0.3. It is noteworthy that a suf-ficient temporal resolution is essential to avoid numerical fluc-tuations in the energy output of the shell during early coolingbranch evolution.

The flashes occur during the fast cooling of the shell at thebeginning of the cooling branch evolution. This especially af-fects the lower part of the shell region where CNO cycling isthe dominant contribution to hydrogen burning. CNO-burningis temperature sensitive enough to cause thermal instabilitiesif two conditions are fulfilled: The shell is thin enough (thisis obviously not the case forM/M� <∼ 0.21) and the coolingtime is large enough as to avoid an extinction of the shell be-fore the instability is fully established (this does not hold forM/M� >∼ 0.3). The final cooling is not affected from possi-ble flash events, i.e. hydrogen shell burning establishes again asdominant energy source at the end of the flash episode as in thecase of non-flash models.

This result also holds when Roche-lobe overflow of the He-WD to a companion is considered during the expansion intothe RGB regime. One of the main features related to hydrogenshell flashes is the change of envelope composition due to apulse-driven convection zone which temporarily extends to theoutermost layers of the star. The resulting drop in hydrogen is∆X ≈ 0.06.

While the typical duration of a hydrogen flash is short com-pared to the cooling time of He-WDs (107 yr compared to Gyr)and He-WDs evolve close to their cooling tracks for most ofthe time during a flash, the influence of these unstable phaseson the mass-radius-relation of helium white dwarfs is moderateand restricted to effective temperatures of20000 . . . 25000 K.

Acknowledgements.F.H. acknowledges funding by theDeutscheForschungsgemeinschaft(grant La 587/16).


4.25 4.2 4.15 4.1 4.05 4Log Teff [K]

∆tmax=5*105 yr

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

4.3 4.25 4.2 4.15 4.1 4.05

Log

(L/L

sun)

Log Teff [K]

∆tmax=1*105 yr

∆tmax=5*104 yr

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6Lo

g (L

/Lsu

n)

∆tmax=2*104 yr

Fig. A1. Hertzsprung-Russell diagram for an evolutionary track withM = 0.195M� and different temporal resolutions (see labels). Theupper left panel shows the final track with the appropriate resolutiongiving a smooth course of the hydrogen luminosity contributionLHyd

(see Fig. A2). For the other tracks deviations caused by local fluctua-tions inLHyd (see Fig. A2) increase with increasing maximum allowedtime step.

Appendix A: remarks on the appropriate temporalresolution

The question of sufficient resolution always arises when onedeals with instabilities in evolutionary models. As mentioned inSect. 2.2 thermal instabilities in He-WD model calculations canobviously be missed if too large time steps are used. Large timesteps may also cause fluctuations in the shell energy output.

The four panels in Fig. A1 show a part of the evolutionarytrack of our He-WD model withM = 0.195 M� in the vicinityof the turn-around point,Teff = Teff,max with different tempo-ral resolution. The corresponding evolution of the luminositycontribution due to hydrogen shell burning,LHyd, is shown inthe four panels of Fig. A2.

The track with sufficient temporal resolution (upper leftpanel in Fig. A1,∆tmax = 2 · 104yr) shows no perturbationsdue to fluctuations in the shell energy production (smooth curvein Figs. A1 and A2). When the maximum time step is increasedto ∆tmax = 5 · 104yr small perturbations occur inLHyd andin the track (for−0.4 <∼ logL/L� <∼ −0.25, see upper rightpanel in Fig. A1). When the maximum time step is further in-creased by a factor of 2 (lower left panel in Figs. A1 and A2)the fluctuations inLHyd and in the track are much more promi-nent but the evolution stabilizes after a final hook in the track

0.5 1.0 1.5 2.0 2.5 3.0

Age [ 108 yr]

∆tmax=5*105 yr

-0.6

-0.4

-0.2

0.0

0.2

0.4

0 0.5 1.0 1.5 2.0 2.5 3.0

Log

(LH

yd/L

sun)

Age [ 108 yr]

∆tmax=1*105 yr

∆tmax=5*104 yr

-0.6

-0.4

-0.2

0.0

0.2

0.4

Log

(LH

yd/L

sun)

∆tmax=2*104 yr

Fig. A2. Evolution of the luminosity contribution due to hydrogen shellburning (LHyd) forM = 0.195M� with different temporal resolution(see also Fig. A1).

at logL/L� ≈ −0.5. In this case the shape of the tracks corre-sponds with those of e.g. Webbink (1975) or Sarna et al. (1998).Finally, for∆tmax = 5 · 105yr the calculations became numer-ically unstable.

Therefore, we selected∆tmax <∼ 2 · 104yr as the maximumtime step for our calculations. Additionally, to handle the largelocal changes in the luminosity budget of the models we cou-pled the evolutionary time step to changes inLHyd by reducing∆tmax by a factor of 2 ifLHyd changes by more than 5%.

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Astron. Astrophys. 350, 89–100 (1999) ASTRONOMY AND The...

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