Cooling curves and initial models for low-mass white dwarfs (<0.25 M_⊙) with helium cores

Abstract

We present a detailed calculation of the evolution of low-mass (<0.25 M_⊙) helium white dwarfs. These white dwarfs (the optical companions to binary millisecond pulsars) are formed via long-term, low-mass binary evolution. After detachment from the Roche lobe, the hot helium cores have a rather thick hydrogen layer with mass between 0.01 and 0.06 M_⊙. As a result of mixing between the core and outer envelope, the surface hydrogen content (X_surf) is 0.5−0.35, depending on the initial value of the heavy element Z and the initial secondary mass. We found that the majority of our computed models experience one or two hydrogen shell flashes. We found that the mass of the helium dwarf in which the hydrogen shell flash occurs depends on the chemical composition. The minimum helium white dwarf mass in which a hydrogen flash takes place is 0.213 M_⊙ (Z=0.003), 0.198 M_⊙ (Z=0.01), 0.192 M_⊙ (Z=0.02) or 0.183 M_⊙ (Z=0.03). The duration of the flashes (independent of chemical composition) is between a few ×10⁶ and a few ×10⁷ yr. In several flashes the white dwarf radius will increase so much that it forces the model to fill its Roche lobe again. Our calculations show that the cooling history of the helium white dwarf depends dramatically on the thickness of the hydrogen layer. We show that the transition from a cooling white dwarf with a temporarily stable hydrogen-burning shell to a cooling white dwarf in which almost all residual hydrogen is lost in a few thermal flashes (via Roche lobe overflow) occurs between 0.183 and 0.213 M_⊙ (depending on the heavy element value).

binaries: close, binaries: general, stars: evolution, stars: mass-loss, pulsars: individual: PSR J0437+4715, pulsars: individual: PSR J1012+5307

1 Introduction

Kippenhahn, Kohl & Weigert (1967) were the first who followed the formation of helium white dwarfs (WDs) of low mass in a binary system. The evolution of a helium WD of 0.26 M_⊙ (remnant) was investigated by Kippenhahn, Thomas & Weigert (1968), who found that a hydrogen flash can be initiated near the base of the hydrogen-rich envelope. The energy of the flash is sufficient to cause the envelope to expand to giant dimensions and hence it may be possible that another short-term Roche lobe filling can occur.

In Webbink (1975), models of a helium WD were constructed by formally evolving a model based on the homogeneous zero-age main sequence with the reduction of the mass of the hydrogen-rich envelope. When the mass of the envelope is less than some critical value, the model contracts adopting WD dimensions. Webbink found that thermal flashes do not occur for WDs less massive than 0.2 M_⊙. Alberts et al. (1996) have confirmed Webbink's finding that low-mass WDs do not show thermal flashes, and that the cooling age for WDs of mass M_WD0.20 M_⊙ can be considerably underestimated if the traditional WD cooling curves that were constructed for M_WD>0.3 M_⊙ are used (Iben & Tutukov 1986, hereafter IT86).

Recently, Hansen & Phinney (1998a, hereafter HP98) and Benvenuto & Althaus (1998, hereafter BA98) investigated the effect that different mass of the hydrogen layer (10⁻M_env/M_⊙4×10⁻³) has on the cooling evolution of 0.15M_He/M_⊙0.5 helium WDs. In both calculations (BA98 and HP98) the mass of the hydrogen envelope left on the top of the WD has been taken as a free parameter. BA98 found that thick envelopes appreciably modify the radii and surface gravities of models in which there was no hydrogen present (no-H models), especially in the case of low-mass helium WDs.

Driebe et al. (1998, hereafter DSBH98) present a grid of evolutionary tracks for low-mass WDs with helium cores in the mass range from 0.179 to 0.414 M_⊙. The tracks are based on a 1-M_⊙ model sequence extending from the pre-main sequence stage up to the tip of the red giant branch. Applying large mass-loss rates forced the models to move off the giant branch and evolve across the Hertzsprung—Russell diagram and down the cooling branch. They found that hydrogen flashes take place only for two model sequences, 0.234 and 0.259 M_⊙, and for very low-mass WDs the hydrogen shell burning remains dominant, even down to effective temperatures well below 10 000 K. According to our previous calculations (Ergma, Sarna & Antipova 1998) we find that for a low-mass WD with a helium core, which was formed during low-mass binary evolution (after detachment from the Roche lobe), the hydrogen layer left on the top of the helium core is much thicker (∼1–6×10⁻² M_⊙ with X_surf ranging from 0.3 to 0.52) than that used in the cooling calculations of HP98 and BA98. Also, in DSBH98 (see their table 1), for the two lowest total remnant masses, the envelope mass value is smaller than that obtained in our calculations.

2 The Main Aim

Low-mass helium WDs are present in millisecond binary pulsars and double degenerate systems. This gives a unique opportunity to test the cooling age of the WD in a binary and, especially in the case of millisecond binary pulsars, enables age determinations of neutron stars that are independent of their rotational history to be performed.

3 The Evolutionary Code

The evolutionary sequences we have calculated comprises three main phases:

• (i)

  detached evolution lasting until the companion fills its Roche lobe on the time-scale t_d;

• (ii)

  semi-detached evolution (non-conservative in our calculations) on the time-scale t_sd−t₀t_d+t_sd;

• (iii)

  a cooling phase of the WD on the time-scale t_cool (the final phase during which a system with a millisecond pulsar and a low-mass helium WD is left behind) — the total evolutionary time is t_evolt₀+t_cool.

The duration of the detached phase is somewhat uncertain; it may be determined either by the nuclear time-scale or by the much shorter time-scale of the orbital angular momentum loss owing to the magnetized stellar wind.

In our calculations we assume that the semi-detached evolution of a binary system is non-conservative, i.e. the total mass and angular momentum of the system are not conserved. We can express the total orbital angular momentum (J) of a binary system as  

(1)

where the terms on the right-hand side are owing to: stellar-mass angular momentum loss from the system; magnetic stellar wind braking; gravitational wave radiation.

3.1 Stellar-mass angular momentum loss

The formalism that we have adopted is described in Muslimov & Sarna (1993). We introduce the parameter f₁ characterizing the loss of mass from the binary system and defined by the relations,  

(2)

where is the mass-loss rate of the system, ₂ is the mass-loss rate of the donor (secondary) star and ₁ is the mass accretion rate of mass on to the neutron star (primary). The matter leaving the system will carry off its intrinsic angular momentum in accordance with the formula  

(3)

where M₁ and M₂ are the masses of the neutron star and donor star, respectively, and MM₁+M₂. Here, we have introduced the additional parameter f₂, which describes the efficiency of the orbital angular momentum loss from the system owing to a stellar wind (Tout & Hall 1991). In our calculations we have f₂f₁=1; we calculate the fully non-conservative case, although additional calculations with f₁=0.9 and 0.5 (with f₂=1) give similar results. A result similar to ours was found by Tauris (1996), who showed that the change in orbital separation resulting from mass transfer in LMXBs (low-mass X-ray binaries) as a function of the fraction of exchanged matter f₁ that is lost from system is small (for 0.5f₁1). To understand whether the system evolution is conservative or non-conservative is not easy in the case of a rapidly rotating neutron star; no easy solution can be found. We propose as one possibility a factor that may help us to distinguish between the two cases — the surface magnetic field of the neutron star and its evolution during the accretion.

3.2 Magnetic stellar wind braking

We also assume that the donor star, possessing a convective envelope, experiences magnetic braking (Mestel 1968; Mestel & Spruit 1987; Muslimov & Sarna 1995) and, as a consequence, the system loses orbital angular momentum. For a magnetic stellar wind we used the formula for the orbital angular momentum loss  

(4)

where a and R₂ are the separation of the components and the radius of the donor star in solar units.

3.3 Gravitational wave radiation

For systems with very short orbital periods, during the final stages of their evolution we also take into account the loss of orbital angular momentum owing to emission of gravitational radiation (Landau & Lifshitz 1971):  

(5)

The mass and accompanying orbital angular momentum loss from these system are poorly understood problems in the evolution of binary stars. As is well known, the variation of the angular momentum depends critically on the assumed model (Ergma et al. 1998). In the case of binary systems with a millisecond pulsar, typically, two different models concerning the mass ejection and angular momentum loss can be adopted. The first is that the amount of angular momentum lost per 1 g of ejected matter is equal to the average orbital angular momentum of 1 g of the binary. The second is that the matter that flows from the companion star onto the neutron star (after accretion) is ejected isotropically with the specific angular momentum of the neutron star. In this paper, for our non-conservative approach we have adopted the first model. This significantly affects our results for semi-detached evolution (see fig. 2 in Ergma et al. 1998), but changes the cooling time-scale of the helium WD by very little.

3.4 Illumination of the donor star

In all cases we have included the effect of illumination of the donor star by the millisecond pulsar. In our calculations we assume that illumination of the component by the hard (X-ray and γ-ray) radiation from the millisecond pulsar leads to additional heating of its photosphere (Muslimov & Sarna 1993). The effective temperature T_eff of the companion during the illumination stage is determined from the relation  

(6)

where L_in is the intrinsic luminosity corresponding to the radiation flux coming from the stellar interior and σ is the Stefan—Boltzmann constant.

P_ill is the millisecond pulsar radiation that heats the photosphere, which is determined by  

(7)

and L_rot is the ‘rotational luminosity’ of the neutron star resulting from magnetodipole radiation (plus a wind of relativistic particles).

 

(8)

where R_NS is the neutron star radius, B is the value of the magnetic field strength at the neutron star and P_p is the pulsar period. f₃ is the factor that characterizes the efficiency of the transformation of irradiation flux into thermal energy (in our case we take f₃=2×10⁻³). Note that in our calculations the effect of irradiation is formally treated by means of modification of the outer boundary condition, according to equation (6).

In this paper we do not follow the magnetic field and pulsar period (P_p) evolution, as we did in our earlier papers (Muslimov & Sarna 1993; Ergma & Sarna 1996). We were mainly interested in finding initial models for low-mass helium WDs and in investigating the initial cooling phase of these low-mass helium WDs. From earlier calculations we know that if the magnetic field strength is greater than about 10⁹ G, the neutron star spins-up to tens and hundreds of ms, rather than several ms. This leads to a situation where the pressure of the magnetodipole radiation is insufficient to eject matter from the system. Also, from our previous calculations (see for example Ergma & Sarna 1996) we find that after accretion of a maximum of about 0.2 M_⊙, the neutron star has spun-up to ms periods if B<10⁹ G. Therefore, in this paper we accept that after an accretion of 0.2 M_⊙ the neutron star spins-up to about 2 ms. After spin-up the pulsar irradiation is strong enough to prevent accretion, and at this moment we include non-conservative mass loss from the system as described previously.

During the initial high mass accretion phase t_acc∼10⁷−10⁸ yr) the system may be observed as a bright LMXB. It is necessary to point out that the majority of LMXBs for which orbital period determinations are available (21 systems out of 24 according to the van Paradijs catalogue 1995) have orbital periods of less than one day. These systems, therefore, cannot be the progenitors of the majority of low-mass helium WD plus millisecond pulsar binary systems. A lack of LMXB systems with orbital periods between 1 and 3 d does not allow us to make a direct comparison between the observational data and the results of our calculations.

3.5 The code

The models of the stars filling their Roche lobes were computed using a standard stellar evolution code based on the henyey-type code of Paczyński (1970), which has been adapted to low-mass stars. The Henyey method involves iteratively improving a trail solution for the whole star. During each iteration, corrections to all variables at all mesh points in the star are evaluated using the Newton—Raphson method for linearized algebraic equations (see for example Hansen & Kawaler 1994). The Henyey method extended to calculate stellar evolution with mass loss, as adopted here, is well explain by Ziólkowski (1970). We note here that our code makes use of the stationary envelope technique, which was developed early in the life of our code in order to save disk space (Paczyński 1969). This method makes the assumption that the surface 0.5–5 per cent (by mass) of the star is not significantly affected by nuclear processes, such that it can be treated to good approximation as a homogeneous region (in composition) throughout the whole evolutionary calculation. During the cooling phase we assume that the static envelope is the surface 0.5 per cent of the star. This assumption is valid during the flashes because the time-scale is longer than the thermal time-scale of the envelope. We tested the possibility that the algorithm for redistributing mesh points introduces numerical diffusion into the composition profile. We find that if such numerical diffusion is real, it has only a marginal influence on the hydrogen profile. We would also like to note that in the heat equation we neglect the derivative with respect to molecular weight, since its effect is small. Convection is treated with the mixing-length algorithm proposed by Paczyński (1969). We solve the problem of radiative transport by employing the opacity tables of Iglesias & Rogers (1996). Where the Iglesias & Rogers (1996) tables are incomplete we have filled the gaps using the opacity tables of Huebner et al. (1977). For temperatures less than 6000 K we use the opacities given by Alexander & Ferguson (1994) and Alexander (private communication). The contribution from conduction present in the opacity tables of Huebner et al. (1977), has been included by us in the other tables since Iglesias & Rogers have not included it in theirs (Haensel, private communication). The equation of state (EOS) includes radiation and gas pressure, which is composed of the ion and electron pressure. Contributions to the EOS owing to the non-ideal effects of Coulomb interaction and pressure ionization, as discussed by Pols et al. (1995), have not been included in our program, and for this reason we stopped our cooling calculations before these effects become important. During the initial phase of cooling, the physical conditions in the hot WDs are such that these effects are usually small.

4 Evolutionary Calculations

We perform our evolutionary calculations for binary systems initially consisting of a 1.4-M_⊙ neutron star (NS) and a slightly evolved companion (subgiant) of two masses, 1 and 1.5 M_⊙, and four chemical compositions (Z=0.003, 0.01, 0.02, 0.03). We have produced (Table 1¹) a number of evolutionary tracks corresponding to the different possible values of the initial orbital period (ranging from 0.7 to 3.0 d) at the beginning of the mass transfer phase.

Table 1.

Cooling track characteristics

5 The Results

In Table 1 we list the characteristic cooling time of the cooling phase of the WD t_cool, and the maximum possible evolution time of a system t_evol, which is a sum of the times of detached (determined by nuclear evolution), semi-detached and cooling phases. The cooling phase is the last phase of evolution of the WD, and in our calculations starts at the end of Roche lobe overflow (RLOF). The cooling time t_cool is limited to an initial cooling stage during which the WD cools until its central temperature has decreased by 50 per cent of its maximum value. From Table 1 it is clear that to produce short orbital period systems in a time-scale shorter than the Hubble time it is necessary either to have low Z or a more massive secondary.

In our calculations the donor star fills its Roche lobe during its evolution through the Hertzsprung gap, and therefore it transfers mass to its companion in a thermal time-scale.

Fig. 1 shows the evolutionary cooling sequences for models 20 and 22 (more details in Table 1). Model 20 presents the case of stable hydrogen burning. Model 22 presents the case in which there is thermal instability of the hydrogen-burning shell. The first flash is not strong enough to allow the star to overflow its Roche lobe, but during the second flash the radius of the secondary increases to fill its Roche lobe and short-time RLOF occurs.

Figure 1.

Hertzsprung—Russell diagram with evolutionary tracks. Evolutionary sequence (model 20) that undergoes long-term stable hydrogen burning is shown by the solid line. The dashed line indicates the same for model 22 that shows one weak (without RLOF) and one strong (with RLOF) hydrogen flash. Circles and triangles mark the cooling ages of 1 and 3 Gyr, respectively.

In Table 2² we present the mass—radius relationship for WDs from our calculations, DSBH98, the Wood models (Wood 1990), and the Hamada & Salpeter (1961) zero-temperature helium WD models calculated for a surface temperature of 8500 K (as in van Kerkwijk, Bergeron & Kulkarini 1996, for PSR 1012+5307). Comparison of the numbers demonstrates that for WD masses of <0.25 M_⊙, the results of our calculations differ significantly from a simple extrapolation obtained from the cooling curves performed for carbon WDs with thick hydrogen envelopes (Wood 1990). In addition, comparing the cooling time-scales of HP98 and BA98 with the time-scales of our models and those in Webbink (1975), shows differences of an order of magnitude (Table 3³) for WD masses of <0.25 M_⊙.

Table 2.

The mass—radius relation for a cooling low-mass WD with a helium core.

Table 3.

Comparison of the cooling time-scales of HP98 and BA98 with the cooling time-scales of Webbink and ours models.

6 Hydrogen flash burning

The problem of unstable hydrogen shell burning in low-mass helium WDs was first discussed in the literature more than 30 years ago (Kippenhahn, Thomas & Weigert 1968; Refsdal & Weigert 1971). Recently, Alberts et al. (1996) have claimed that they do not see any thermal flashes resulting from thermally unstable shell burning, as reported in IT86 and Kippenhahn, Thomas & Weigert (1968). Webbink (1975) found that such a severe thermal runaway as described by Kippenhahn et al. (1968) was not found in any of his model sequences, although mild flashes for M>0.2 M_⊙ did take place. Alberts et al. found that even reducing the time step to 50–100 yr would not lead to thermally unstable shell burning for M_WD<0.25 M_⊙. In DSBH98, thermal instabilities of the hydrogen-burning shell occur in their two models with 0.234 and 0.259 M_⊙. They concluded that hydrogen flashes take place only in the mass interval 0.21M/M_⊙0.3.

According to our computations, low-mass helium WDs with masses more than 0.183 M_⊙ (Z=0.03), 0.192 M_⊙ (Z=0.02), 0.198 M_⊙ (Z=0.01) and 0.213 M_⊙ (Z=0.003), may experience up to several hydrogen flashes before they enter the cooling stage. In Table 4⁴ we present several characteristics of the computed flashes. We discuss two kinds of flashes: in the first case (in Table 4, case 1) the secondary does not fill its Roche lobe during the flash, i.e. the mass of the WD does not change, and in the second case (case 2) the secondary fills its Roche lobe during the unstable hydrogen-burning phase and the system again enters into a very short duration accretion phase (see Table 4). We introduce four time-scales to describe the flash behaviour: (i) the flash rise time-scale Δt₁, the time for the luminosity to increase from minimum to maximum value (typically this value is between a few ×10⁶ and a few ×10⁷ yr — third column in the Table 4); (ii) the flash decay time-scale Δt₂, the time for the luminosity to decrease to the initial value (typically from a few hundred thousand to few tens of million years); (iii) ΔT is the recurrence time between two successive flashes and (iv) Δt_acc is the duration of the accretion phase when the secondary fills its Roche lobe during a hydrogen shell flash.

Table 4.

Flash characteristics.

For all sequences with several unstable hydrogen shell burning stages (usually for case 1), the first flash is the weakest. In the majority of cases when the flash forces the star to fill its Roche lobe, only one flash takes place. For four cases we found two successive flashes with RLOF (models 17, 23, 24 and 31), and for another two cases (models 47 and 53) the first flash is not powerful enough to force the secondary to fill its Roche lobe, but during the second flash it is.

How does the hydrogen flash burning influence the cooling time-scale? In Fig. 2, the luminosity and nuclear energy production rates versus the cooling time for models 20 and 22 are shown. Model 20 shows stable hydrogen burning and model 22 shows hydrogen flash burning. Although model 22 was more luminous than model 20 before the flash, afterwards the situation is reversed. After the flash, the burning mass of the hydrogen-rich envelope in model 22 has decreased to 0.0116 M_⊙, whereas the mass of the hydrogen envelope in model 20, in which stable hydrogen burning occurs, is almost twice as large (0.0241 M_⊙). If we look at how the maximum nuclear energy rate behaves with cooling time, we can see that after the flash in model 22, the maximum energy production rate is less than in model 20 (stable hydrogen burning).

Figure 2.

The surface effective temperature (upper panel), the nuclear energy production (middle panel) and the surface luminosity (lower panel) plotted as a function of the cooling time t_cool that is the time elapsed from t₀. Model 20 with stable hydrogen shell burning and t₀=7.9×10⁹ yr — thick line; model 22 with unstable hydrogen shell burning and t₀=7.8×10⁹ yr — dashed line. In the first flash RLOF does not occur, the second flash is accompanied by RLOF. For all figures, the cooling time t_cool is the time elapsed from t₀.

In Fig. 3 we present the behaviour of log T_eff, log ɛ_nuc and log L/L_⊙, and in Fig.4 log R_WD, M_env and M_f/M_⊙ are presented as a function of cooling time for model 7. Before the helium WD enters the final cooling phase, four unstable hydrogen flash burnings occur. The same parameters for model 17 (with RLOF) are shown in Figs 5 and 6.

Figure 3.

Hydrogen flashes on a helium WD of mass 0.213 M_⊙ (model 7) that show four flashes without RLOF. The curves present the effective temperature (upper panel), nuclear energy production in the hydrogen-burning shell (middle panel) and the luminosity (lower panel) as a function of cooling time, t₀=5.2×10⁹ yr.

Figure 4.

Hydrogen flashes on a helium WD of mass 0.213 M_⊙ (model 7). The curves present the WD radius (upper panel), the envelope mass (middle panel) and the mass of the helium core (lower panel) as a function of the cooling time, t₀=5.2×10⁹ yr.

Figure 5.

Same as in Fig. 3 but for model 17. During the first flash the secondary does not fill its Roche lobe, but during the second and third flashes RLOF occurs and the total mass of the WD decreases (t₀=1.4×10⁹ yr).

Figure 6.

Same as in Fig. 4 but for model 17. During the first flash the secondary does not fill its Roche lobe but during the second and third flashes RLOF occurs and the total mass of the WD decreases (t₀=1.4×10⁹ yr).

To investigate in more detail how the flashes develop, we show in Fig. 7 the evolution of the WD radius (upper panel), nuclear energy generation rate (upper-middle panel), maximum shell temperature and central temperature (lower-middle panel) and the surface luminosity (lower panel) as a function of the computed model number. In Fig. 7 we marked several time-scales that characterize the flash behaviour as vertical dashed lines (for numbers see Table 4). Δt₁ and Δt₂ describe the rise and decay times: the first characterizes the nuclear shell burning time-scale the second the Kelvin—Helmholtz (thermal) envelope time-scale modified by nuclear shell burning The accretion time (Δt_acc) is described by the square of the Kelvin—Helmholtz time-scale. The radiative diffusion time is defined as the Kelvin—Helmholtz time-scale of the extended envelope above the shell The shape of the first flash in Fig. 7 shows some characteristic changes that are connected with physical processes in the stellar interior. At the beginning of the flash the luminosity increases owing to the more effective hydrogen burning in the shell source. After reaching a local maximum, the luminosity then decreases while the nuclear energy generation rate is still increasing rapidly. This decrease of the surface luminosity is a result of temperature inversion forming below the hydrogen shell. The energy generated in the hydrogen shell splits into two fluxes: outwards and inwards. The helium core is heated effectively by the shell nuclear source — the central temperature increases by 2 per cent. In Fig. 8 the evolution of the luminosity and temperature profiles during the Δt₁ and Δt₂ phases are shown. We clearly see the inversion profile evolving and that the luminosity wave moves into the surface.

Figure 7.

Hydrogen flashes on a helium WD of model 17. Upper panel — the WD radius (solid line) together with the Roche lobe radius (dashed line); upper-middle panel — the nuclear energy production in the hydrogen-burning shell; lower-middle panel — the maximum shell temperature (solid line) and central temperature (dashed line); lower panel — the surface luminosity; all as a function of model number are shown. The vertical lines define different time-scales during the flashes.

Figure 8.

The evolution of temperature inversion layers and the luminosity profile during a hydrogen flash. The evolutionary sequences are as follows: solid line — local luminosity minimum; dashed line — maximum temperature of the hydrogen shell; short-dashed line — luminosity front moves outwards; long-dashed line — maximum luminosity; dashed—dotted line — the decline of luminosity in which the heated core is cooling effectively.

The nuclear energy generation rate in the shell has a maximum value far away from maximum surface luminosity. This is because the luminosity front is moving towards the stellar surface in a time-scale described by radiative diffusion (Δt_rd). After reaching a maximum value, the luminosity starts to decrease and the energy generation rate also declines in the hydrogen shell, over a time-scale Δt₂ (for a contracting envelope) the luminosity decreases to the minimum value. During the first flash, the stellar radius does not fill the inner Roche lobe. In the second and third flashes we have short episodes of super-Eddington mass transfer (see Table 4). During the RLOF phase, the orbital period slightly increases and the subgiant companion evolves quickly from spectral type F0 to A0.

As already pointed out, for several cases the secondary fills its Roche lobe and the system enters an accretion phase. During RLOF, the mass accretion rate is about three orders of magnitude greater than the Eddington limit (Fig. 9). All the accreted matter will be lost from the system (ΔM_acc∼0.0001−0.001 M_⊙). The accretion phase is very short, usually less than 1000 yr (ranging from 160 to 2500 yr — see Table 4). During the short super-Eddington accretion phase the system is a very bright X-ray source, with orbital period between 2 and 8 d.

Figure 9.

Mass accretion rate (model 42) versus time during a hydrogen shell flash with RLOF.

We notice that during the flash the evolutionary time-step strongly decreases and may be as short as several years.

7 Role of Binarity in the Cooling History of the Low-Mass WD

DSBH98 modelled single star evolution and produced WDs with various masses by applying large mass-loss rates at appropriate positions in the red giant branch to force the models to move off the giant branch. To show the influence of binarity on the final fate of the WD cooling, we have computed extra sequences (1.0 and 1.4 M_⊙, Z=0.02, P_i=2.0 d) where we did not take into account that the star is in a binary system e.g. during a hydrogen shell flash we do not allow RLOF. In a complete binary model calculation only one shell flash occurs accompanied with RLOF, whereas for the single star model calculation four hydrogen shell flashes take place. Owing to RLOF, the duration of the flash phase is 2.7×10⁶ yr; if we do not include binarity, the duration of the flash phase is 1.8×10⁸ yr. However, the cooling time for helium WDs less massive than 0.2 M_⊙ is not significantly changed. This is because the duration of the flash phase is very short in comparison with the normal cooling phase (towards the WD region). However, the effect of binarity will be important for the cooling history of more massive helium WDs. In Fig. 10 both cases of evolution on the Hertzsprung—Russell diagram are shown — in the left-hand panel RLOF is not allowed, whereas in right-hand panel RLOF takes place.

Figure 10.

Hertzsprung—Russell diagram with evolutionary tracks. Evolutionary sequence 1+1.4 M_⊙, Z=0.02, P_i=2.0 d. In the left-hand panel RLOF is not allowed, and in the right-hand panel RLOF is present.

8 Application to individual systems

Below we discuss the observational data for several systems to which the results of our calculations can be applied, by taking into account the orbital parameters of the system, the pulsar spin-down time and the WD cooling time-scale.

8.1 PSR J0437-4715

Timing information for this millisecond binary system: P_p=5.757 ms, P_orb=5.741 d, t=4.4−4.91 Gyr (the intrinsic characteristic age of the pulsar) and the mass function f(M)=1.239×10⁻³ M_⊙ (Johnston et al. 1993; Bell et al. 1995). Hansen & Phinney (1998b) have discussed the evolutionary stage of this system using their own cooling models described in HP98. They found consistent solutions for all masses in the range 0.15−0.375 M_⊙ with thick (in the terminology of HP98) hydrogen envelopes of 3×10⁻⁴ M_⊙.

Timing measurements by Sandhu et al. (1997) have detected a rate of change in the projected orbital separation a sin i, which they interpret as a change in i calculated for an upper limit of i<43° and a new lower limit for the mass of the companion of M∼0.22 M_⊙. Our calculations also allow us to produce the orbital parameters and secondary mass for the PSR J0437-4715 system and to fit its cooling age (2.5−5.3 Gyr, Hansen & Phinney, 1998b), and we find that the secondary fills its Roche lobe when the orbital period P_i is ∼2.5 d (Tables 1 and 4). From our cooling tracks for a binary orbital period of 5.741 d, the mass of the companion is 0.21±0.01 M_⊙ and its cooling age is 1.26−2.25 Gyr (for a Population I chemical composition). These cooling models usually have one strong (with RLOF) hydrogen shell flash, after which the helium WD enters the normal cooling phase.

8.2 PSR J1012+5307

Lorimer et al. (1995) determined a characteristic age of the radio pulsar to be 7 Gyr, which could be even larger if the pulsar has a significant transverse velocity (Hansen & Phinney 1998b). Using the IT86 cooling sequences, they estimated the companion to be at most 0.3 Gyr old. HP98 models yield the following results for this system: the companion mass lies in the range 0.13−0.21 M_⊙ and the WD age is <0.6 Gyr, the NS mass lies in the range 1.3−2.1 M_⊙.

Alberts et al. (1996) were the first to show that the cooling time-scale of a low-mass WD can be substantially larger if there are no thermal flashes which lead to RLOF and a reduction of the hydrogen envelope mass. Our and DSBH98's calculations confirmed their results that for low-mass helium WDs (<0.2 M_⊙) stationary hydrogen burning plays an important role. To produce short (<1 d) orbital period systems with a low-mass helium WD and a millisecond pulsar it is necessary that the secondary fills its Roche lobe between P_bif and P_b (Ergma et al. 1998). If the initial orbital period P_i (at RLOF) is less than P_bif, the binary system evolves towards short orbital periods. P_b is another critical orbital period value. If P_b<P_i(RLOF)<P_bif, then a short orbital period (<1 d) millisecond binary pulsar with low-mass helium WD may form. So the initial conditions of the formation of such systems are rather important. We calculated one extra sequence to produce a binary system with orbital parameters similar to PSR J1012+5307. The initial system: 1 plus 1.4 M_⊙, P_i(RLOF)=1.35 d, Z=0.01. The final system: M_f=0.168 M_⊙, P_f=0.605 d, M_env=0.041 M_⊙. In Fig. 11 in the effective temperature and gravity diagram we show the cooling history of this WD after detachment of the Roche lobe. The two horizontal regions are the gravity values inferred by van Kerkwijk et al. (1996) (lower, between long-dashed lines) and Callanan, Garnavich & Koester (1998) (upper, between short-dashed lines). Our results are consistent with the Callanan et al. (1998) estimates. It should be noted that the outer envelope is rather helium-rich after detachment from its Roche lobe. Bergeron, Weselmael & Fontaine (1991) have shown that a small amount of helium in a hydrogen-dominated envelope can mimic the effect of larger gravity.

Figure 11.

log g—log T_eff diagram with M_WD=0.168 M_⊙. The arrow marks the position of the PSR J1012+5307 WD. The upper horizontal region (between short-dashed lines) shows the gravity values inferred by Callanan et al. (1998) and the lower horizontal region (between long-dashed lines) shows values inferred by van Kerkwijk et al. (1996). The vertical lines show the effective temperature constraints of Callanan et al. (1998).

9 Discussion

The results of our evolutionary calculations differ from those of Iben & Tutukov (1986) and Driebe et al. (1998) because of the different formation scenarios for low-mass helium WDs. In IT86's calculations a donor star fills its Roche lobe while it is on the red giant branch (i.e. has a thick convective envelope) with a well developed helium core and a thin hydrogen-burning layer. They proposed that the mass transfer time-scale is so short that the companion will not be able to accrete the transferred matter and will itself expand and overflow its Roche lobe. The final output is the formation of a common envelope and the result of this evolution is a close binary with a helium WD of mass 0.298 M_⊙ having a rather thin (1.4×10⁻³ M_⊙) hydrogen-rich (X=0.5) envelope.

DSBH98 did not calculate the mass exchange phases during the red giant branch evolution in detail but they also simulated the mass-exchange episode by subjecting a red giant branch model to a sufficiently large mass-loss rate. In both cases (IT86 and DSBH98) mass loss starts when the star (with a well-developed helium core) is on the red giant branch.

In our calculations the Roche lobe overflow starts when the secondary has either almost exhausted the hydrogen in the centre of the star, or has a very small helium core with a thick hydrogen-burning layer. During the semi-detached evolution the mass of the helium core increases from almost nothing to the final value (for more details concerning the evolution of such systems, see Ergma et al. 1998). This is the reason that a much thicker (∼1.5−6×10⁻² M_⊙, with X ranging from 0.30 to 0.52) hydrogen-rich envelope is left on the donor star at the moment it shrinks within the Roche lobe.

The second important point where our results differ from the results of DSBH98, is that in our calculations we can produce (after the secondary detaches from its Roche lobe) final millisecond binary pulsar parameters that we compare with observational data (orbital period, spin period of millisecond pulsar and mass of the companion). It was shown by Joss, Rappaport & Lewis (1987), and more recently by Rappaport et al. (1995), that the evolution of a binary system initially comprising a neutron star and a low-mass giant will result in as a wide binary containing a radio pulsar and a WD in a nearly circular orbit. The relation between the WD mass and the orbital period [see equation (6) in Rappaport et al. 1995] shows that if the secondary fills its Roche lobe while on the red giant branch, then for M_WD≈0.19 M_⊙ the final orbital period would be ∼5 d, which is far greater than the observed orbital period of the binary pulsar PSR J1012+5307 (P_orb=0.6 d).

Alberts et al. (1996), DSBH98 and the results of our calculations demonstrate clearly that, especially for low-mass helium WDs (<0.2 M_⊙), stable hydrogen burning remains an important, if not the main, energy source. HP98 and BA98 did consider nuclear burning, but found it to be of little importance since their artificially chosen hydrogen-envelope mass was less than some critical value, disallowing significant hydrogen burning. If we now compare the cooling curves of HP98 and DSH98 with ours, then there is one very important difference: they did not model the evolution of the helium WD progenitor and all their cooling models (see for example figs 11 and 12 in HP98) start with a high T_eff. In our models, cooling of the helium WD starts after detachment of the secondary from its Roche lobe (DSBH98 mimic this situation with mass loss from the star). This time, the secondary (proto-WD) has a rather low effective temperature (see for example Fig. 1). During the evolution with L approximately constant, the effective temperature increases to a maximum value, after which it decreases while there still remains an active hydrogen shell burning source. The evolutionary time needed for the proto-WD to travel from the minimum T_eff (after detachment from the Roche lobe) to maximum T_eff depends strongly on the mass of the WD (a smaller mass has a longer evolutionary time-scale).

So for low-mass helium WDs the evolutionary pre-history plays a very important role in the cooling history of the WD.

10 Conclusion

We have performed comprehensive evolutionary calculations to produce a close binary system consisting of an NS and a low-mass helium WD.

We argue that the presence of a thick hydrogen layer dramatically changes the cooling time-scale of the helium WD (<0.25 M_⊙), compared with previous calculations (HP98, BA98) where the mass of the hydrogen envelope was chosen as a free parameter and was usually one order of magnitude less than that obtained from real binary evolution computations.

Also, we have demonstrated that by using new cooling tracks we can consistently explain the evolutionary status of the binary pulsar PSR J1012+53.

Tables with cooling curves are available at http://www.camk.edu.pl/,sarna/.

Acknowledgments

We would like to thank Dr Katrina M. Exter for help in improving the form and text of the paper. We would also like to thank our referee Dr Peter Eggleton for very useful suggestions. In Warsaw, this work is supported through grants 2∔P03D–014–07 and 2—P03D—005–16 of the Polish National Committee for Scientific Research. Also, JA and EE acknowledges support from an Estonian SF grant 2446.