Astron. Nachr./AN 322 (2001) 5/6, 405–410

Helium core white dwarf evolution – including white dwarfcompanions to neutron stars

M.J. SARNA , E. ERGMA and J. GERSKEVITS

Nicolaus Copernicus Astronomical Center, Polsh Academy of Sciences, Bartycka Str. 18, 00–716 Warsaw, PolandPhysics Department, Tartu University, Ulikooli 18, EE2400 Tartu, Estonia

Received 2001 November 23; accepted 2001 November 26

Abstract. We present a detailed calculation of the evolution of low–mass helium white dwarfs. These white dwarfs areformed via long–term, low–mass binary evolution. After detachment from the Roche lobe, the hot helium cores have a ratherthick hydrogen layer with masses between 0.001 to 0.06 . We found that the majority of our computed models experienceone or two hydrogen shell flashes. The duration of the flashes is between a few y to a few y. In several flashesthe white dwarf radius will increase so much that it forces the model to fill its Roche lobe again. Our calculations show thatthe cooling history of the helium white dwarf depends dramatically on the thickness of the hydrogen layer. The presenceof low–mass helium white dwarf secondaries in millisecond pulsar binaries allows to determine the age of the systemsindependently of the rotational history of the pulsars. The same method may be applied to double degenerate systems. Wediscuss the cooling history of the low–mass, helium core white dwarfs in short orbital period millisecond pulsars.

Key words: stars: evolution – stars: white dwarfs

1. Introduction

Historically, the problem of the structure of single whitedwarfs (WDs) was solved essentially by Chandrasekhar(1936). His models for zero–temperature stars are based onthe equation of state for a Fermi–Dirac gas of non–interactingelectrons. He derived the mass–radius relation for whitedwarfs and an upper mass limit for such configurations. Mes-tel (1952) was the first to consider the rate of cooling of whitedwarfs. He assumed that the energy flow in the core is dueto the large conductivity of the degenerate electrons. Next,Hamada & Salpeter (1961) corrected the mass–radius rela-tion of Chandrasekhar including electrostatic interactions be-tween electrons and nuclei and inverse beta decays.

Kippenhahn, Khol & Weigert (1967) were the first whofollowed the formation of helium white dwarfs of low massin a binary system. These authors found that a hydrogen shellflash can be initiated near the base of the hydrogen–rich en-velope. Refsdal & Weigert (1970) have constructed linearseries of thermal equilibrium models of low mass into thewhite dwarf region, but only Kippenhahn, Thomas & Weigert(1968) and Refsdal & Weigert (1969) have followed the evo-lution of a close binary remnant beyond the onset of shelldegeneracy well into the white dwarf region. Those workshowed that helium white dwarfs of low mass are produced

only in low–mass close binary systems exchanging mass in“Case B” mass transfer (Refsdal & Weigert 1971).

Webbink (1975 – W75) considered the evolution of 0.1– 0.5 stars from the zero–age main sequence to the ex-tinction of the hydrogen–burning shell as a white dwarf. Thethickness of the hydrogen rich layers range from 2.1to 3.1 for 0.15–0.35 white dwarfs. He alsoshowed that the hydrogen thermal flashes cannot occur inWDs less massive than 0.2 . By postulating this con-straint, Alberts, Savonije & van den Heuvel (1996) suggestedthat the cooling age for WDs of mass can beconsiderably underestimated from the traditional WD cool-ing curves (Iben & Tutukov 1986 – IT86). Recently, Hansen& Phinney (1998a – HP98a) and Benvenuto & Althaus (1998– BA98) investigated the effect of different hydrogen layers( ) on the cooling evolution of

helium WDs.Driebe et al. (1998 – DSBH98) present a grid of evolu-

tionary tracks for low–mass WDs with helium core in themass range from 0.179 to 0.414 . The mass of the outerhydrogen layers at the beginning of the cooling phase in-creases from 1.4 to 25.5 for decreasing masses.According to Sarna, Ergma & Gerskevits–Antipova (2000 –SEG00) calculations for the hydro-gen layer left on the top of the helium core is much thicker( with ranging from 0.3 to 0.52)

© WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2001 0044-6337/01/5-612-03405 $ 17.50+.50/0

406 Astron. Nachr./AN 322 (2001) 5/6

Table 1. Parameters of binary pulsars in the Galactic disk (d)

Pulsar name PPSR [ms] [d]

B1855+09 5.36 0.018 12.327 0.26J1804 27 9.34 0.042 11.129 0.24J2129 57 3.73 0.020 6.626 0.16J1603 72 14.84 0.014 6.309 0.33J0437 47 5.76 0.057 5.741 0.22J1045 45 7.47 0.017 4.084 0.19J1911 11 3.63 0.013 2.717 0.14J2317+14 3.45 0.0024 2.459 0.21J0218+42 2.32 0.080 2.029 0.20J0034 05 1.88 0.0051 1.589 0.17J0613 02 3.06 0.0096 1.199 0.15J1012+53 5.26 0.015 0.605 0.16J0751+18 3.48 0.008 0.263 0.15

– masses are calculated for assume value ofi= and

than calculated by HP98a, BA98 and DSBH98. The thick-ness of the hydrogen layers has significant importance for thecooling time–scale (nuclear burning can not be neglected).

2. Observational evidence

Low–mass helium white dwarfs are present in millisecondbinary pulsars (MBPs), double degenerate systems and cata-clysmic variables.

In Table 1 we present the observational data for MBPsin the Galactic disk (Camilo 1997; Lorimer et al. 1996). Ac-cording to Camilo’s latest compilation of the observationaldata of MBPs in the disk, 22 systems are in nearly circularorbits with companion masses 0.15 0.45 (he-lium white dwarfs). 13 of these systems have less than20 d ( 60%). If it is possible to build a “clear” evolutionarypicture for the disk systems, then we will discuss the statusof several of them (marked by boldface) later on as example.

3. The main aim

These data give a unique opportunity to test the evolutionaryage of the binary and, especially in the case of the MBPs,allows for age determination for neutron stars that are inde-pendent from their rotational history.

The age determination of neutron stars is quite sensi-tive to both the WD cooling and the evolution of the binary.Therefore it is necessary to integrate three independent fields:

WD cooling theory;

low–mass binary evolution calculations;

and the spin and magnetic field evolution of the NS.

4. Evolutionary scheme

The scenario for the formation of the low–mass X–ray bi-naries (LMXB’s) and next MBPs we adopted from Bhat-tacharya & van den Heuvel (1991).

The evolutionary sequences we have calculated comprisethree main phases:

detached evolution lasting until the companion fills itsRoche lobe;

semi–detached evolution on the time–scale (non–conservative in our calculations) – neutron star accreted thematter from Roche lobe filling companion (LMXB phase);

a cooling phase of the WD on the time–scale – binarysystems is detached, the final phase during which a systemwith a ms pulsar + low–mass helium WD is left behind (MBPphase).

5. Evolutionary calculations

SEG00 calculations were carried out under the following as-sumptions:

All evolutionary calculations were carried out using a stan-dard Henyey–type code, which has been adopted to low–massstars;

For radiative transport, we used the opacity tables of Igle-sias & Rogers (1996); for temperatures less than 6000 K weuse the opacity given by Alexander & Ferguson (1994);

In our calculations we assume that the semi–detached evo-lution of a binary system is non–conservative, i.e. the to-tal mass and angular momentum of the system are not con-served;

We perform our evolutionary calculations for binary sys-tems initially consisting of a 1.4 neutron star and aslightly evolved companion (subgiant) of two masses: 1and 1.5 ;

We have produced a number of evolutionary tracks corre-sponding to the different possible values of the initial orbitalperiod (ranging from 0.9 to 3.0 d).

6. The results

DSBH98 and SEG00 calculations show that the low–masshelium WDs (typically less massive than 0.25 ) have athickness of the hydrogen layer of usually one order of mag-nitude more massive as compared to previous calculations byHP98a and BA98 (where the mass of hydrogen envelope waschosen as free parameter). The donor star fills its Roche lobewhile it is evolving through the Hertzsprung gap, and there-fore, it transfers mass on its thermal time–scale. The resultsof DSBH98 and SEG00 evolutionary calculations differ alsofrom those by IT86 because of the different formation sce-nario for a low–mass helium WD. In IT86’s calculations adonor star fills its Roche lobe while it is on the red giantbranch. This implies that the mass transfer occurs on a dy-namical time–scale and that therefore a common envelopeforms.

M. Sarna et al.: Helium core white dwarf evolution 407

Table 2. Comparison of Webbink and SEG00 models

Webbink SEG00

( ) ( ) (Gyr) ( ) (Gyr)

0.15 0.33 0.032 25.12 0.39 0.031 23.440.20 0.47 0.022 3.72 0.53 0.022 3.80

the cooling time, , is limited to the initial cooling stage duringwhich the WD cools until its central temperature has decreases by50% of its maximum value.

The results of this evolution is a close binary with a he-lium WD of 0.298 having a rather thin ( )hydrogen–rich ( =0.5) envelope. Note that in SEG00 cal-culations for all evolutionary sequences the accretion rateonto the NS never reaches the Eddington limit during thesemi-detached evolution. This is the reason why a muchthicker ( , with ranging from0.30 to 0.52) hydrogen–rich envelope is left on the donor starat the moment it no longer fills its Roche lobe.

It must be pointed out that, in terms of the hydrogen con-tent, the thickness of the hydrogen–rich envelope and coolingtime from SEG00 calculations agree very well with those ofW75 (see Table 2).

Another interesting result we have found is that the binarysystems we discuss in this paper show thermal flashes if themass of the helium WD is .

The nature of the flashes does not depend strongly onthe chemical composition. However, the nature of the flashesdoes depend on the binarity. To show the influence of it on thefinal fate of the WD cooling, we have computed sequenceswhere we did not take into account that the star is in a bi-nary system, e.g. during a hydrogen shell flash we do notallow Roche lobe overflow (RLOF). In a complete binarymodel calculation only one shell flash occurs accompaniedwith RLOF, whereas for the single star model calculation,four hydrogen shell flashes take place (see Fig. 1). However,the cooling time for helium WDs less massive than 0.2is not significantly changed. This is because the duration ofthe flash phase is very short in comparison with the normalcooling phase. However, the effect of binarity will be im-portant for the cooling history of more massive helium WDs( ).

In Table 3 we show the mass–radius relationship forSEG00, DSHB98 white dwarf models and for Hamada &Salpeter (1961) zero–temperature helium WD models calcu-lated for a surface temperature of 8500 K (as in van Kerk-wijk, Bergeron & Kulkarin 1996 for PSR1012+5307). Com-parison of the numbers demonstrates that for WD massesof 0.2 the results of DSHB98 and SEG00 calcula-tions differ significantly from a simple extrapolation obtainedfrom the cooling curves (Wood 1990) performed for carbonWDs with the thick hydrogen envelopes and from HP98a andBA98 (thin envelopes).

In addition, comparing the time–scales of HP98a andBA98 with the time–scales of our models and those of W75shows differences of an order of magnitude for WD massesof 0.25 .

7. Application to individual systems

Below we discuss the observational data for several systemsto which the results of SEG00 and DSHB98 calculations canbe applied, by taking into account the orbital parameters ofthe system, the pulsar spin–down time and the WD coolingtime–scale.

7.1. PSR J0437–47

The timing information for this millisecond binary system is:P=5.757 ms, =5.741 d, (intrinsic characteristic age ofpulsar) 5 Gyr, mass function f(M)= 1.239 (John-ston et al. 1993; Bell et al. 1995). Hansen & Phinney (1998b –HP98b ) have discussed the evolutionary stage of this systemusing their own cooling models described in HP98a. Theyfound a consistent solution for all masses in the range 0.15 –0.375 with thick (in the terminology of HP98a) hydrogenenvelopes of 3 .

Our calculations allow us to produce the orbital parame-ters and secondary mass for the PSR J0437–4715 system andfit its cooling age (2.5–5.3 Gyr, HP98b), and we find that thesecondary fills its Roche lobe when the initial orbital periodis 2.5 d. From our cooling tracks for a binary orbital periodof 5.741 d, the mass of the companion is 0.21 andits cooling age is 1.26–2.25 Gyr (for a Population I chemicalcomposition). These cooling models usually have one strong(with RLOF) hydrogen shell flash, after which the heliumWD enters the normal cooling phase.

7.2. PSR J1012+53

Lorimer et al. (1995) determined a characteristic age of theradio pulsar to be 7 Gyr, which could be even larger if thepulsar has a significant transversal velocity (HP98b). Usingthe IT86 cooling sequences, they estimated the companion tobe at most 0.3 Gyr old. HP98a models yield the followingresults for this system: the companion mass lies in the range0.13–0.21 and the WD age is 0.6 Gyr, the neutron starmass lies in the range 1.3–2.1 .

Alberts et al. (1996) were the first to show that the cool-ing time–scale of a low–mass WD can be substantially largerif there are no thermal flashes which lead to RLOF and a re-duction of the hydrogen envelope mass. Our and DSBH98calculations confirmed their results that for low–mass heliumWDs ( 0.2 ) stationary hydrogen burning does indeedplay an important role. These new calculations of SEG00 aswell as of Ergma, Sarna & Greskevits–Antipova (2001) giveus the possibility of fitting the physical and orbital parame-ters of PSR J1012+5307, and allow an independent determi-nation of the physical age of the neutron star. In the case of

408 Astron. Nachr./AN 322 (2001) 5/6

Fig. 1. Hertzsprung–Russell diagram with evolutionary tracks. Evolutionary sequence 1+1.4 , Z=0.02, =2.0 d. Left panel RLOFis not allowed, right panel with RLOF.

SEG00 (Fig. 2) evolutionary calculations, this age is estimateto be between 7.2 and 8.6 Gyr, which is consistent with theestimated spin–down age of 7 Gyr for PSR J1012+5307 (vanKerkwijk et al. 1996, Callanan, Garnavich & Koester 1998).

7.3. PSR J0751+18

To explain the age of the first two MBPs, new cooling tracksare sufficient. However, to explain the age of PSR J0751+18,Ergma et al. (2001) proposed a strong wind generated bymagneto–dipole radiation from the pulsar magnetosphere.They included two effects: (i) illumination by X–ray, –rayand relativistic particles and (ii) irradiation–induced stellarwind.

The effective temperature of the companion duringillumination stage is determined from the relation:

(1)

where is the intrinsic luminosity corresponding to theradiation flux coming from the stellar interior, is the Stefan–Boltzmann constant and is the radius of the secondary.is the millisecond pulsar radiation that heats the photosphere.

For the stellar wind they used the standard van den Heuvel& van Paradijs model (1988):

(2)

where , and B, respectively, are the mass of thesecondary, the radius of the neutron star, and the magneticfield strength, is efficiency factor (from 0 to 1) and is theseparation between the stars in solar units. In this calculationsthey assume three values of : 0, 0.1 and 0.5 (very efficientirradiation).

The decreasing of the mass of the hydrogen envelopeleads to a significant decrease of the hydrogen shell–burningefficiency. Therefore, if compared to the evolution with sta-tionary shell burning, the cooling time is decreased. For ex-ample, when –2, the cooling age without massloss is 8.4 yr, with moderate mass loss 5.1 yr; withvery efficient mass loss 2.2 yr. The evolution of the massloss rate as a function of time is shown in Fig. 3.

M. Sarna et al.: Helium core white dwarf evolution 409

Table 3. M-R relation for a cooling low–mass WD with a helium core

log log logHamada SEG00 DSHB98 van Kerkwijk

0.155 0.0218 2.100 6.31 - - 1.351 6.690.180 0.0208 1.594 6.65 1.697 6.60 1.300 6.820.206 0.0198 1.469 6.83 1.485 6.82 1.236 7.000.296 0.0173 1.224 7.26 1.133 7.33 1.111 7.36

The first two columns present the zero–temperature M–R relation for a helium WD obtained by Hamada & Salpeter (1961). The third andfourth columns display SEG00 calculations of the stellar radius and gravity for parameters closest to the one of DSHB98, respectively.The last two columns illustrate the same quantities taken from the cooling tracks produced by Wood (1990) for carbon WDs with thickhydrogen envelopes. The stellar radius is calculated at T = 8500 K and is normalized by the zero–temperature radius.

the last two values in this row are taken from IT86.

Table 4. Age of selected MBPs

Nr PSRName [d] [ ] [Gyr] [Gyr] [Gyr] [Gyr]

1 J0437-47 5.741 0.22-0.32 5 2.5–5.3 6 1.26-2.252 J1012+53 0.605 0.16-0.18 7 1.4 0.4 7.2-8.63 J0751+18 0.263 0.15 6.3-8 0.8 7.3 2.2-8.44 B1855+09 12.327 0.25 0.01 4.95 0.5 1.4 4 25 J0034-05 1.589 0.15-0.54 4.4-9.2 4.4-15 10

Fig. 2. log g – log diagram with = 0.168 . The ar-row marks the position of the PSR J1012+5307 white dwarf. Twohorizontal regions are the gravity values inferred by Callanan et al.(1998) (upper) and van Kerkwijk et al. (1996) (lower). The verticallines show effective temperature constraints of Callanan et al. (1998)

7.4. PSR B1855+09

However, irradiation–induced wind is insufficient to explainthe age of PSR B1855+09. Also, standard cooling models byDSHB98 and SEG00 cannot explain the age of this pulsar.Using such models, the predicted age of the white dwarf istwice as long as the spin–down age of the pulsar.

Althaus, Serenelli & Benvenuto (2001 – ASB01) findthat element diffusion induces thermonuclear hydrogen shellflashes. ASB01 suggested that the occurrence of thesediffusion–induced flashes leads to white dwarf models witha hydrogen envelope mass too small to support any furthernuclear burning. This implies much shorter cooling age than

Fig. 3. The dependence of the mass loss rate as a function of time intwo different phases of evolution: semi–detached evolution (phase(1–2) – solid line) and cooling with irradiation induced wind. Forthe wind phase we show two different efficiencies of the wind: f=0.1(dash line) and f=0.5 (dash–dotted line).

in the case where diffusion is neglected. For PSR B1855+09such models predict an age of 4 2Gyr.

ASB01 have also found that diffusion considerably af-fects the mass–radius relation for low–mass helium whitedwarfs. The result of ASB01 is very interesting, however theyneglected one fundamental fact: PSR B1855+09 is a binaryms pulsar. If we plot the relation for theradius of the WD during the flash ( ) ina H–R diagram, then it is clear that during the first diffusion–induced flash the WD is overfilling its Roche lobe. The masstransfer/accretion rate is three order of magnitude larger thanthe Eddington limit. Therefore, all the accreted matter will belost from the system – a few (SEG00). This num-

410 Astron. Nachr./AN 322 (2001) 5/6

ber is comparable with the mass of the hydrogen envelope fora 0.242 WD: 2 (see ASB01). Therefore, weconclude that the effect of binarity will be more important forthe cooling history of PSR B1855+09 than diffusion–inducedflashes. The diffusion will be important in much wider MBPs(P 20 d).

7.5. PSR J0034–05

HP98b estimated a very low temperature limit ( 3500 K)for the white dwarf, yielding ages of more than 10 Gyr if he-lium models are used. The pulsar’s characteristic age deter-mined by Bailes et al. (1994) is 6.8 2.4 Gyr. Schonberner,Driebe & Blocker (2000) suggested that, for consistency be-tween the pulsar’s and white dwarf age, a carbon/oxygenwhite dwarf of 0.5 must be assumed. However, the pho-tometric measurements of Lundgren, Zepka & Cordes (1996)give . The cooling models of HP98b predict

as upper limit. The discrepancy betweenthe above results might be related to the same problem asin the case of PSR B1855+09. If the companion is a heliumwhite dwarf with , it is suggestedthat helium flash and binarity play an important role in MBPevolution.

8. Conclusions and discussion

We have performed evolutionary calculations to produce aclose binary system consisting of a NS and a low–mass he-lium WD. For the first time (SEG00) calculations of this sortreveal that a helium WD of mass 0.18 , before it entersa cooling stage, can experience a series of hydrogen flashes.We found that the majority of computed models experienceone or two hydrogen flashes. We found that the strength ofthe flashes does not depend strongly on the chemical compo-sition.

We argue (DSBH98, SEG00) that thick hydrogen layersmay change dramatically the cooling time–scale of the he-lium white dwarfs ( ), compared to the previouscalculations (HP98a, BA98) where the mass of the hydrogenenvelope was chosen as free parameter and was usually oneorder of magnitude less than that obtained from real binaryevolution computations.

Also, we have demonstrated that using new cooling trackswe can consistently explain the evolutionary status of the bi-nary ms pulsars like PSR J0437–47 or J1012+53.

However, some additional mechanisms are necessaryto explain the evolutionary status of PSR J0751+18 –irradiation–induced wind and PSR B1855+09, PSR J0034–05 – Roche lobe overflow.

Acknowledgements. This work was supported in part by the PolishNational Committee for Scientific Research under grant 2–P03D–005–16 and 2–P03D–014–07, and by the Estonian SF grant 4338.

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