Free Access
Issue
A&A
Volume 571, November 2014
Article Number L3
Number of page(s) 6
Section Letters
DOI https://doi.org/10.1051/0004-6361/201424681
Published online 10 November 2014

© ESO, 2014

1. Introduction

In recent years, the number of detected low-mass (0.20 M) helium white dwarfs (He WDs) has increased dramatically, mainly as a result of multiple survey campaigns such as WASP, ELM, HVS, Kepler, and SDSS (Pollacco et al. 2006; Rowe et al. 2010; Brown et al. 2005, 2010, 2013; Silvotti et al. 2012; Kilic et al. 2012; Hermes et al. 2013; Maxted et al. 2014).

The existence of low-mass He WDs in close binaries with a radio millisecond pulsar (MSP), however, has been known for a few decades (e.g. van Kerkwijk et al. 2005, and references therein). Several attempts have been made to calibrate WD cooling models for such systems on the basis of the spin-down properties of the MSP (e.g. Alberts et al. 1996; Hansen & Phinney 1998; Driebe et al. 1998; Althaus et al. 2001; Panei et al. 2007). The idea is that the characteristic spin-down age of the MSP (τPSRP/ (2), where P is the spin period and is the period derivative) should be equivalent to the cooling age of the WD (τcool), assuming that the radio MSP is activated at the same time as the WD is formed, following an epoch of mass transfer in a low-mass X-ray binary (LMXB). Unfortunately, this method is highly problematic since τPSR generally is a poor true age estimator. It can easily be incorrect by a factor of 10 or more (Camilo et al. 1994; Lorimer et al. 1995; Tauris 2012; Tauris et al. 2012). Determining the true age of MSPs, however, is important for studying their spin evolution and constraining the physics of their previous recycling phase (Lazarus et al. 2014).

The discovery of the intriguing PSR J1012+5307 (Nicastro et al. 1995) sparked an intense discussion about WD cooling ages and MSP birthrates (Lorimer et al. 1995) given that τPSR> 20τcool. Soon thereafter, it was suggested (Alberts et al. 1996; Driebe et al. 1998; van Kerkwijk et al. 2005) that He WDs with a mass 0.20 M avoid hydrogen shell flashes, whereby their relatively thick (~10-2M) hydrogen envelope remains intact, causing residual hydrogen shell burning to continue on a very long timescale. Despite significant theoretical progress (e.g. Althaus et al. 2013, and references therein), our understanding of the thermal evolution of (proto) He WDs remains uncertain. In particular, a number of recent observations of apparently bloated WDs calls for an explanation.

In this Letter, we study the formation of a large number of low-mass He WDs by modelling close-orbit LMXBs. We carefully investigate the properties of the resulting proto-WDs and follow their evolution until and beyond settling on the WD cooling track. Finally, we compare our results with observations.

thumbnail Fig. 1

Evolutionary tracks in the (Teff,log  g)-diagram. The evolution from Roche-lobe detachment until settling on the WD cooling track and beyond is shown for a selection of our models. The colour scale represents the final WD mass. A few cases of vigorous hydrogen shell flashes explain the large (counterclockwise) loops in the diagram. Observed WDs are shown with symbols (stars: sdB+WD, double WDs (grey stars: WDs with poor mass constraints); triangles: pulsating WDs; squares: WD+MS; circles: WD+MSP – see Appendix A for references and data).

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2. Numerical methods and physical assumptions

Numerical calculations with a detailed stellar evolution code were used to trace the mass-transfer phase following the same prescription as outlined in Istrate et al. (2014). We investigated models with a metallicity of Z = 0.02, a mixing-length parameter α = l/Hp = 2.0, and a core convective overshooting parameter of δOV = 0.10. A wide range of LMXB systems were investigated with different initial values of donor star mass (M2), neutron star mass, orbital period, and the so-called γ-index of magnetic braking. The evolution of the low-mass (proto) He WD was calculated including chemical diffusion (mixing), hydrogen shell flashes (CNO burning), and residual shell hydrogen burning. Convective, semi-convective, and overshoot mixing processes were treated via diffusion. Thermohaline mixing was included as well, whereas gravitational settling and radiative levitation were neglected, as was stellar wind mass loss.

3. Results

In Fig. 1 we have plotted a selection of our calculated evolutionary tracks, from the moment of Roche-lobe detachment until the end of our calculations, for (proto) He WDs with masses of 0.15−0.28 M. In general, our models fit the observations quite well. The few cases with discrepancies are sources with large uncertainties in the WD mass. Vigorous single or multiple cycle hydrogen shell flashes explain the large loops in the diagram, whereas mild thermal instabilities are seen e.g. for the 0.25 M proto-WDs at log g ≃ 4.5. It has been known for many years that a thermal runaway flash may develop through unstable CNO burning when a proto-WD evolves towards the cooling track (Kippenhahn & Weigert 1967; Webbink 1975; Iben & Tutukov 1986). During these flashes the luminosity becomes very high, whereby the rate of hydrogen burning is significantly increased (e.g. Nelson et al. 2004; Gautschy 2013, and references therein). Our models with strong flashes often experience an additional episode of mass loss via Roche-lobe overflow (RLO, see also Iben & Tutukov 1986; Sarna et al. 2000; Podsiadlowski et al. 2002; Nelson et al. 2004).

thumbnail Fig. 2

Contraction timescale, Δtproto of evolution from Roche-lobe detachment until settling on the WD cooling track, plotted as a function of WD mass, MWD. The initial ZAMS masses of the WD progenitors (the LMXB donor stars) are indicated with various symbols and colours. The red line marks Mflash ≃ 0.21 M for progenitor stars 1.5 M.

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For progenitor stars with M2 ≤ 1.5 M we find hydrogen shell flashes in WDs with masses of 0.21 ≤ MWD/M ≤ 0.28. Hence, the lowest mass for which flashes occur is Mflash = 0.21 M. However, we find a lower value of Mflash = 0.18 M for M2 = 1.6 M. It has been argued (e.g. van Kerkwijk et al. 2005) that the value of Mflash is important since it marks a dichotomy for the subsequent WD cooling such that WDs with a mass MWD<Mflash remain hot on a Gyr timescale as a result of continued residual hydrogen shell burning, whereas WDs with MWD>Mflash cool relatively fast as a result of the shell flashes that erode the hydrogen envelope. We find that this transition is smooth, however, and that the thermal evolution timescale mainly depends on the proto-He WD luminosity and not on the occurrence or absence of flashes.

In Fig. 2 we have plotted the time, Δtproto it takes from Roche-lobe detachment until the star reaches its highest value of Teff. (For WDs that undergo hydrogen shell flashes we used the time until the occurrence of highest Teff on their last loop in the HR-diagram.) The plot shows a very strong dependence on MWD. For very low-mass He WDs (i.e. those with MWD<Mflash, which therefore avoid hydrogen shell flashes), Δtproto may last up to 2 Gyr. This result has important consequences for their thermal evolution and contraction (see below). There is a well-known correlation between the degenerate core mass of an evolved low-mass star and its luminosity, L (Refsdal & Weigert 1971). After terminating the RLO, the star moves to the far left in the HR-diagram – (initially) roughly at constant L – while burning the residual hydrogen in the envelope at a rate proportional to L. We find that the total amount of hydrogen left in the envelope is always ~0.01 ± 0.005 M, in agreement with Sarna et al. (2000), and is correlated in a variable manner with MWD (especially for M2 ≥ 1.5 M, explaining the peak in Fig. 2). Therefore, the increase in Δtproto seen in Fig. 2 for decreasing values of MWD can simply be understood from their much lower luminosities following the Roche-lobe detachment (see also Figs. 5 and 10 in Istrate et al. 2014). Based on our calculated proto-He WD models, we find (see Appendix B) (1)The conclusion that Δtproto can reach ~Gyr was found previously for a few single models (e.g. Driebe et al. 1998; Sarna et al. 2000; Althaus et al. 2001). Here we show, for the first time, its systematic dependence on MWD.

Figure 3 shows the contraction phase for three proto-He WDs. The value of Δtproto increases significantly when MWD decreases from 0.24 to 0.17 M. Hence, low-mass (≲ 0.20 M) proto-He WDs can remain bloated on a very long timescale. It is important to notice that no pronounced discontinuity in Δtproto is seen at Mflash ≃ 0.21 M (cf. Figs. 2, 3, and C.1). Although the peak luminosity (and thus the rate of eroding hydrogen) is high during a flash, the star only spends a relatively short time (~103−106 yr) at high L when making a loop in the HR-diagram.

thumbnail Fig. 3

Radius as a function of stellar age for the progenitor stars of three He WDs of mass 0.17, 0.21 and 0.24 M. The most massive proto-WD evolves with hydrogen shell flashes – see inset. The epoch between the solid red star (RLO termination) and the open black star (max. Teff) marks the contraction (transition) timescale, Δtproto = 96−1910 Myr.

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4. Comparison with observational data of He WDs

In Table A.1 we list observed low-mass He WDs included among the plotted data in Fig. 1. We now discuss recent interesting sources in view of our theoretical modelling.

4.1. MSPs with low-mass (proto) He WDs in tight orbits

The companion star to PSR J1816+4510 (Porb = 8.7 h) was recently observed by Kaplan et al. (2012, 2013). They assembled optical spectroscopy and found an effective temperature of Teff = 16 000 ± 500 K, a surface gravity of log g = 4.9 ± 0.3, and a companion mass of MWD sin3i = 0.193 ± 0.012 M, where i is the orbital inclination angle of the binary. They concluded that while the spectrum is rather similar to that of a low-mass He WD, it has a much lower surface gravity (i.e. larger radius) than a WD on the cooling track. They discussed that PSR J1816+4510 perhaps represents a redback system (cf. Chen et al. 2013, for a formation scenario) where pulsar irradiation of the hydrogen-rich, bloated companion causes evaporation of material, which can explain the observed eclipses of the radio pulses for ~10% of the orbit. However, the very hot surface temperature of this companion (16 000 K) cannot be explained from a redback scenario. Redbacks typically have illuminated dayside temperatures of only Teff ≃ 6000 K (Breton et al. 2013). Here we suggest that this companion star is simply a low-mass proto-He WD. As we have demonstrated, such a star takes several 100Myr to reach the cooling track, and our models match the observed values of Teff and log g. (Note, for Porb = 8.7 h one usually expects MWD ≲ 0.18 M, cf. Istrate et al. 2014.)

Another case is the triple system PSR J0337+1715 recently discovered by Ransom et al. (2014), which raises fundamental questions about its formation (Tauris & van den Heuvel 2014). One open question is the order of formation of the two WDs orbiting the MSP (with Porb = 1.6 and 327 days). Spectroscopy of the inner companion by Kaplan et al. (2014b) verified that this is a 0.197 M He WD, as known from pulsar timing. They measured a low surface gravity of log g = 5.82 ± 0.05 and noted that its very high surface temperature, Teff = 15 800 ± 100K, could indicate that it had just experienced a flash. This would suggest a surprisingly short lifetime for this object. However, our modelling of ~0.20 M He WDs shows that these stars avoid flashes. Instead we find that for such a star it takes 400−600Myr (Fig. 2) to reach the WD cooling track. Therefore, we conclude that it is reasonable to detect such a WD at an early, bloated stage of its evolution.

4.2. NLTT 11748 and other low-mass (proto) He WD binaries

A large number of low-mass proto-He WDs (also classified as sdB stars) are found in binaries with another WD. These systems probably formed via stable RLO in cataclysmic variable systems resembling our calculations, but with a ~0.7 M CO WD accretor instead of a NS. NLTT 11748 was discovered by Steinfadt et al. (2010), with follow-up observations made by Kaplan et al. (2014a). This eclipsing detached binary consists of a 0.71−0.74 M CO WD with a very low-mass He WD and Porb ≃ 5.6 h. Our evolutionary tracks for a 0.16 M He WD are indeed consistent with their observed values of log g = 6.35 and Teff = 7600 K (and their estimated mass of 0.136−0.162 M). Brown et al. (2013) recently detected four binaries with low-mass WDs having log g ≃ 5, in accordance with our modelling of proto-He WDs presented here (cf. Figs. 1 and A.1).

4.3. Bloated, hot, low-mass He WDs detected by Kepler

Four (proto) He WDs have been found with A-star companions in the combined transit and eclipse data from the Kepler mission (van Kerkwijk et al. 2010; Carter et al. 2011; Breton et al. 2012). Three of these He WDs (KOI-74, KIC 10657664, KOI–1224) have MWD ≲ 0.26 M and are also plotted in Fig. 1. The mass estimates of these WDs are not very precise. However, within 1–2σ, the characteristics of these objects also seem to match our evolutionary tracks reasonably well.

The question now is why we see all these bloated proto-WDs given that WDs spend significantly more time on the subsequent cooling tracks. This is simply a selection effect. The WDs are only seen to eclipse A-stars in the Kepler data as long as they are bloated proto-WDs (and thus also more luminous than ordinary WDs, which have already settled on the cooling track).

5. Discussion and conclusions

We have demonstrated that low-mass (0.20 M), detached proto-He WDs may spend up to ~2 Gyr in the contraction (transition) phase from the Roche-lobe detachment until they reach the WD cooling track. This is important for an age determination of He WDs in general and for recycled MSPs in particular. We expect a fair number of He WDs to be observed in this (bloated) phase, in agreement with recent observations.

The duration of this contraction phase (Δtproto) decreases strongly with increasing mass of the proto-He WD, MWD. This can be understood from the well-known correlation between degenerate core mass and luminosity of an evolved low-mass star. Therefore, after Roche-lobe detachment, the rate at which the residual (0.01 ± 0.005 M) hydrogen in the envelope is consumed is directly proportional to the luminosity and thus MWD. The value of Δtproto is not particularly sensitive to the occurrence or absence of flashes.

Whether or not hydrogen shell flashes occur depends on the WD mass, its chemical composition, and the treatment of diffusion (mixing) of the chemical elements (e.g. Driebe et al. 1998; Sarna et al. 2000; Althaus et al. 2001; Nelson et al. 2004; Althaus et al. 2013). In general, we find flashes in our models with 0.21 ≤ MWD/M ≤ 0.28 for M2 ≤ 1.5 M. This result agrees excellently well agreement with the interval found by Nelson et al. (2004) for donors with solar metallicity, and also with the earlier work of Driebe et al. (1998). For M2 = 1.6 M WDs down to ~0.18 M are experiencing flashes.

Detailed studies by Althaus et al. (2001, 2013) found hydrogen shell flashes for a much broader range of final WD mass (0.18 <MWD/M< 0.41). However, as pointed out by Nelson et al. (2004), diffusion is an extremely fragile process, and turbulence can mitigate its effects. And more importantly, Nelson et al. (2004) find that both M2 and the mode of angular momentum losses may also affect the range for which hydrogen shell flashes occur. Indeed, we found a lower value of Mflash = 0.18 M for our models with M2 = 1.6 M. It has previously been shown that Mflash strongly increases with lower metallicity (e.g. Sarna et al. 2000; Nelson et al. 2004). The work of Althaus et al. (2013) was calculated for a constant M2 = 1.0 M (Z = 0.01). We have excluded such models with M2< 1.1 M (Z = 0.02) since these progenitor stars do not detach from their LMXB and evolve onto WD cooling track within a Hubble time.

Chemical diffusion via gravitational settling and radiative levitation was not included in this work. These effects seem to slightly increase Δtproto compared with models without diffusion (Nelson et al., in prep.). A systematic investigation of these and other effects on Δtproto and Mflash will be addressed in a future work.

Acknowledgments

A.G.I. acknowledges discussions with L. Nelson, P. Marchant, R. Stancliffe and L. Grassitelli. J.A. acknowledges financial support by the ERC Starting Grant No. 279702 (BEACON, led by P. Freire).

References

Online material

Appendix A: Observational data and time evolution in the (Teff, log g)-diagram

Table A.1

Observational data of a number of low-mass He WDs (and proto-He WDs), preferentially in tight binary systems.

thumbnail Fig. A.1

Selected tracks (Fig. 1) with a point marked for a time interval of 1 Myr (triangle), 50 Myr (square), 100 Myr (circle), and 1 Gyr (star).

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In Fig. A.1 we have plotted points for fixed time intervals of evolution along a number of selected tracks from Fig. 1. The density of points along these curves combined with the (proto) WD luminosities at these epochs can be used to evaluate the probability of detecting them. For a direct comparison with data population synthesis needs to be included to probe the distribution of WD masses. The observational data plotted in Fig. 1 were taken partly from the sources given in Table A.1 (primarily He WDs with MSP companions, main-sequence A-star companions, or He WDs that have been detected to show pulsations).Additional data for the plotted symbols can be found in Silvotti et al. (2012), Hermes et al. (2013), Brown et al. (2013).

Appendix B: The (proto) WD contraction phase

Figure C.1 shows the time Δtproto it takes from Roche-lobe detachment until the proto-He WD reaches its highest value of Teff and settles on the cooling track. Shown in this plot are all our calculated models for progenitor stars of 1.2 and 1.4 M (i.e. a subset of the models plotted in Fig. 2). The black line (Eq. (1)) is an analytical result obtained from a somewhat steep core mass-luminosity function () combined with the assumption (for simplicity) that in all cases 0.01 M of hydrogen is burned before reaching the highest Teff. The figure shows that this line also serves as a good approximate fit to our calculated models. For a given He WD mass, the fit to Δtproto calculated from our models is accurate to within 50%.

Appendix C: Nuclear burning during flashes

thumbnail Fig. C.1

Calculated models of proto-He WDs from Fig. 2 for M2 = 1.2 M (orange) and M2 = 1.4 M (blue). The black line is a fit to the data. It can also be derived analytically using a modified core mass-luminosity relation for low-mass evolved stars, combined with an assumed fixed amount of residual hydrogen (0.01 M) to be burned. The red line separates models with and without flashes.

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Table C.1

Ages and remaining hydrogen of the two proto-He WDs shown in Fig. C.2.

thumbnail Fig. C.2

Evolutionary tracks in the HR-diagram for a 0.221 M proto-He WD with flashes (brown) and for a 0.212 M proto-He WD without flashes (blue). See Table C.1 for data.

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To compare the burning of residual envelope hydrogen for a case with and without large thermal instabilities (hydrogen shell flashes), we have plotted tracks in the HR-diagram shown in Fig. C.2. The age of the stars and the total amount of hydrogen remaining in their envelopes are given in Table C.1 for the points marked in the figure. These models were chosen very close to (but on each side of) Mflash ≃ 0.21 M, in both cases for a 1.3 M progenitor star. As discussed in the main text, although the peak luminosity is high during a flash (and thereby the rate at which hydrogen is burned), the star only spends a relatively short time (~106 yr) in this epoch. (For more massive He WDs it is even less time – for example, it only lasts ~103 yr for a 0.27 M He WD.) Therefore, the amount of additional hydrogen burned as a result of flashes is relatively small. In the example shown in Fig. C.2 it amounts to about 12% of the total amount of hydrogen at the point of Roche-lobe detachment. Hence, the flashes may appear to reduce Δtproto by ~100 Myr. However, one must bear in mind that the proto-WDs that experience

flashes are also the WDs with the least amount of hydrogen in their envelopes after RLO.

For a star that experiences flashes, the residual hydrogen present in the envelope following the LMXB-phase is processed roughly as follows: 70% during the epoch from Roche-lobe detachment until reaching highest Teff, 10% during the flashes, and 20% after finally settling on the WD cooling track.

All Tables

Table A.1

Observational data of a number of low-mass He WDs (and proto-He WDs), preferentially in tight binary systems.

Table C.1

Ages and remaining hydrogen of the two proto-He WDs shown in Fig. C.2.

All Figures

thumbnail Fig. 1

Evolutionary tracks in the (Teff,log  g)-diagram. The evolution from Roche-lobe detachment until settling on the WD cooling track and beyond is shown for a selection of our models. The colour scale represents the final WD mass. A few cases of vigorous hydrogen shell flashes explain the large (counterclockwise) loops in the diagram. Observed WDs are shown with symbols (stars: sdB+WD, double WDs (grey stars: WDs with poor mass constraints); triangles: pulsating WDs; squares: WD+MS; circles: WD+MSP – see Appendix A for references and data).

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In the text
thumbnail Fig. 2

Contraction timescale, Δtproto of evolution from Roche-lobe detachment until settling on the WD cooling track, plotted as a function of WD mass, MWD. The initial ZAMS masses of the WD progenitors (the LMXB donor stars) are indicated with various symbols and colours. The red line marks Mflash ≃ 0.21 M for progenitor stars 1.5 M.

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In the text
thumbnail Fig. 3

Radius as a function of stellar age for the progenitor stars of three He WDs of mass 0.17, 0.21 and 0.24 M. The most massive proto-WD evolves with hydrogen shell flashes – see inset. The epoch between the solid red star (RLO termination) and the open black star (max. Teff) marks the contraction (transition) timescale, Δtproto = 96−1910 Myr.

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In the text
thumbnail Fig. A.1

Selected tracks (Fig. 1) with a point marked for a time interval of 1 Myr (triangle), 50 Myr (square), 100 Myr (circle), and 1 Gyr (star).

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In the text
thumbnail Fig. C.1

Calculated models of proto-He WDs from Fig. 2 for M2 = 1.2 M (orange) and M2 = 1.4 M (blue). The black line is a fit to the data. It can also be derived analytically using a modified core mass-luminosity relation for low-mass evolved stars, combined with an assumed fixed amount of residual hydrogen (0.01 M) to be burned. The red line separates models with and without flashes.

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In the text
thumbnail Fig. C.2

Evolutionary tracks in the HR-diagram for a 0.221 M proto-He WD with flashes (brown) and for a 0.212 M proto-He WD without flashes (blue). See Table C.1 for data.

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In the text

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