Non-conservative Mass Transfer Research Articles

The VLT-FLAMES survey of massive stars: NGC 346-013 as a test case for massive close binary evolution

NGC346-013 is a peculiar double-lined eclipsing binary in the Small Magellanic Cloud discovered by the VLT-FLAMES survey of massive stars. Spectra obtained with VLT-FLAMES are used to construct a radial velocity curve and photometry obtained with the Faulkes Telescope South is then used to derive orbital parameters, while spectra of the secondary are compared with synthetic spectra from TLUSTY model atmospheres. The orbital period is found to be 4.20381(12) days, with masses of 19.1+/-1.0 and 11.9+/-0.6 Msun. The primary is a rapidly rotating late-O dwarf while the secondary, an early-B giant, displays near-synchronous rotation and has filled its Roche lobe, implying that it was originally the more massive component with recent mass transfer `spinning up' the primary to near-critical rotation. Comparison with synthetic spectra finds temperatures of 34.5kK and 24.5kK for the primary and secondary respectively, with the nitrogen abundance of the secondary enhanced compared to baseline values for the SMC, consistent with the predictions of models of interacting binaries. NGC346-013 likely evolved via non-conservative mass transfer in a system with initial masses ~22+15Msun, with the well-constrained orbital solution and atmospheric parameters making it an excellent candidate for tailored modelling with binary evolution codes. This system will form a cornerstone in constraining the physics of thermal timescale mass transfer, and the associated mass transfer efficiency, in massive close binary systems.

ON THE FORMATION OF DOUBLE WHITE DWARFS THROUGH STABLE MASS TRANSFER AND A COMMON ENVELOPE

Although several dozen double white dwarfs (DWDs) have been observed, for many the exact nature of the evolutionary channel(s) by which they form remains uncertain. The canonical explanation calls for the progenitor binary system to undergo two subsequent mass-transfer events, both of which are unstable and lead to a common envelope (CE). However, it has been shown that if both CE events obey the standard alpha-prescription (parametrizing energy loss), it is not possible to reproduce all of the observed systems. The gamma-prescription was proposed as an alternative to this description, instead parametrizing the fraction of angular momentum carried away in dynamical-timescale mass loss. In this paper, we analyze simultaneous energy and angular-momentum conservation, and show that the gamma-prescription cannot adequately describe a CE event for an arbitrary binary, nor can the first phase of mass loss always be understood in general as a dynamical-timescale event. We consider in detail the first episode of mass transfer in binary systems with initially low companion masses, with a primary mass in the range 1.0--1.3 solar masses and an initial mass ratio between the secondary and primary stars of 0.83--0.92. In these systems, the first episode of dramatic mass loss may be stable, non-conservative mass transfer. This strips the donor's envelope and dramatically raises the mass ratio; the considered progenitor binary systems can then evolve into DWDs after passing through a single CE during the second episode of mass loss. We find that such a mechanism reproduces the properties of the observed DWD systems which have an older component with less than approximately 0.46 solar masses and mass ratios between the younger and older WDs greater than 1.

Wind Anisotropy and Angular Momentum Loss in a Massive Rotating Binary System

Abstract The model of anisotropic loss of the mass and angular momentum is constructed and the non-conservative evolution of the binary system is investigated in this paper. The joint effects of the centrifugal force and tidal force cause the configurations of two components to become triaxial ellipsoids. The high-order disturbing potential, which includes the rotational and tidal distortions, is applied to describe the local gravity in a close binary system. The $g_{\rm eff}$($\theta$, $\varphi$)-effect dominates the mass-loss distribution in the massive O-type star. Both the $g_{\rm eff}$($\theta$, $\varphi$)-effect and the $\kappa$-effect have an important influence on the equatorial ejection, and the Roche lobe overflow and the H-shell burning occur earlier in the rotational models. The rotation and tide can intensify the mass loss before mass overflow, and the rate of stellar wind goes down, resulting from a decrease of the luminosity in the subsequent stages. The high-order disturbing potential and other associated physical factors may significantly affect the Roche lobe and might be possible to drive the non-conservative mass transfer process when the stars approach the break-up rotation. Rotation and tide can allow the primary to shift towards the blue side of the HR diagram and modify the thermal relaxation time-scale in the slow phase of the mass transfer in Case A. The star attempts to attain thermal equilibrium and displays a slightly cyclical expansion and contraction. When stellar wind was taken into consideration in the model, the secondary star accreted less mass than the model without stellar wind.

MASS TRANSFER IN BINARY STARS USING SMOOTHED PARTICLE HYDRODYNAMICS. II. ECCENTRIC BINARIES

Despite numerous efforts to better understand binary star evolution, some aspects of it remain poorly constrained. In particular, the evolution of eccentric binaries has remained elusive mainly because the Roche lobe formalism derived for circular binaries does not apply. Here, we report the results of our Smoothed Particle Hydrodynamics simulations of mass transfer in eccentric binaries using an alternate method in which we model only the outermost layers of the stars with appropriate boundary conditions. Using this technique, along with properly relaxed model stars, we characterize the mass transfer episodes of binaries with various orbital parameters. In particular, we show that these episodes can be described by Gaussians with a FWMH of ~0.12 P_orb and that the peak rates occur after periastron, at an orbital phase of ~0.58, independently of the eccentricity and mass of the stars. The accreted material is observed to form a rather sparse envelope around either or both stars. Although the fate of this envelope is not modeled in our simulations, we show that a constant fraction (~5%) of the material transferred is ejected from the systems. We discuss this result in terms of the non-conservative mass transfer scenario. We suggest our results could be incorporated in analytical and binary population synthesis studies to help better understand the evolution of eccentric binaries and the formation of exotic stellar populations.

INTERACTING BINARIES WITH ECCENTRIC ORBITS. II. SECULAR ORBITAL EVOLUTION DUE TO NON-CONSERVATIVE MASS TRANSFER

We investigate the secular evolution of the orbital semi-major axis and eccentricity due to mass transfer in eccentric binaries, allowing for both mass and angular momentum loss from the system. Adopting a delta function mass transfer rate at the periastron of the binary orbit, we find that, depending on the initial binary properties at the onset of mass transfer, the orbital semi-major axis and eccentricity can either increase or decrease at a rate linearly proportional to the magnitude of the mass transfer rate at periastron. The range of initial binary mass ratios and eccentricities that leads to increasing orbital semi-major axes and eccentricities broadens with increasing degrees of mass loss from the system and narrows with increasing orbital angular momentum loss from the binary. Comparison with tidal evolution timescales shows that the usual assumption of rapid circularization at the onset of mass transfer in eccentric binaries is not justified, irrespective of the degree of systemic mass and angular momentum loss. This work extends our previous results for conservative mass transfer in eccentric binaries and can be incorporated into binary evolution and population synthesis codes to model non-conservative mass transfer in eccentric binaries.

Timing of the 2008 outburst of SAX J1808.4–3658 with XMM-Newton: a stable orbital-period derivative over ten years

We report on a timing analysis performed on a 62-ks longXMM-Newton observation of the accreting millisecond pulsar SAX J1808.4– 3658 during the latest X-ray outburst that started on September 21, 2008. By connecting the time of arrivals of the pulses observed during the XMM-Newton observation, we derived the best-fit orbital solution and a best-fit value of the spin period for the 2008 outburst. Comparing this new set of orbital parameters and, in particular, the value of the time of ascending-node passage with the orbital parameters derived for the previous four X-ray outbursts of SAX J1808.4–3658 observed by the PCA onboard RXTE ,w e fi nd an updated value of the orbital period derivative, which turns out to be u Porb = (3.89 ± 0.15) × 10 −12 s/s. This new value of the orbital period derivative agrees with the previously reported value, demonstrating that the orbital period derivative in this source has remained stable over the past ten years. Although this timespan is not sufficient yet for confirming the secular evolution of the system, we again propose an explanation of this behavior in terms of a highly non-conservative mass transfer in this system, where the accreted mass (as derived from the X-ray luminosity during outbursts) accounts for a mere 1% of the mass lost by the companion.

Orbital evolution of an accreting millisecond pulsar: witnessing the banquet of a hidden black widow?

We have performed a timing analysis of all the four X-ray outbursts from the accreting millisecond pulsar SAX J1808.4-3658 observed so far by the PCA on board RXTE. For each of the outbursts we derived the best-fit value of the time of ascending node passage. We find that these times follow a parabolic trend, which gives an orbital period derivative $\dot P_{\rm orb} = (3.40 \pm 0.18) \times 10^{-12}$ s/s, and a refined estimate of the orbital period, $P_{\rm orb} = 7249.156499 \pm 1.8 \times 10^{-5}$ s (reference epoch $T_0 = 50914.8099$ MJD). This derivative is positive, suggesting a degenerate or fully convective companion star, but is more than one order of magnitude higher than what is expected from secular evolution driven by angular momentum losses caused by gravitational radiation under the hypothesis of conservative mass transfer. Using simple considerations on the angular momentum of the system, we propose an explanation of this puzzling result assuming that during X-ray quiescence the source is ejecting matter (and angular momentum) from the inner Lagrangian point. We have also verified that this behavior is in agreement with a possible secular evolution of the system under the hypothesis of highly non-conservative mass transfer. In this case, we find stringent constraints on the masses of the two components of the binary system and its inclination. The proposed orbital evolution indicates that in this kind of sources the neutron star is capable to efficiently ablate the companion star, suggesting that this kind of objects are part of the population of the so-called black widow pulsars, still visible in X-rays during transient mass accretion episodes.

Equipotential Surfaces and Lagrangian Points in Nonsynchronous, Eccentric Binary and Planetary Systems

We investigate the existence and properties of equipotential surfaces and Lagrangian points in nonsynchronous, eccentric binary star and planetary systems under the assumption of quasi-static equilibrium. We adopt a binary potential that accounts for nonsynchronous rotation and eccentric orbits and calculate the positions of the Lagrangian points as functions of the mass ratio, the degree of asynchronism, the orbital eccentricity, and the position of the stars or planets in their relative orbit. We find that the geometry of the equipotential surfaces may facilitate nonconservative mass transfer in nonsynchronous, eccentric binary star and planetary systems, especially if the component stars or planets are rotating supersynchronously at the periastron of their relative orbit. We also calculate the volume-equivalent radius of the Roche lobe as a function of the four parameters mentioned above. Contrary to common practice, we find that replacing the radius of a circular orbit in the fitting formula of Eggleton with the instantaneous distance between the components of eccentric binary or planetary systems does not always lead to a good approximation to the volume-equivalent radius of the Roche lobe. We therefore provide generalized analytic fitting formulae for the volume-equivalent Roche lobe radius appropriate for nonsynchronous, eccentric binary star and planetary systems. These formulae are accurate to better than 1% throughout the relevant two-dimensional parameter space that covers a dynamic range of 16 and 6 orders of magnitude in the two dimensions.

Binary star progenitors of long gamma-ray bursts

Context: The collapsar model for long gamma-ray bursts requires a rapidly rotating Wolf-Rayet star as progenitor. Aims: We test the idea of producing rapidly rotating Wolf-Rayet stars in massive close binaries through mass accretion and consecutive quasi-chemically homogeneous evolution - the latter had previously been shown to provide collapsars below a certain metallicity threshold. Methods: We use a 1D hydrodynamic binary evolution code to simulate the evolution of a 16+15 Mo binary model with an initial orbital period of 5 days and SMC metallicity (Z=0.004). Internal differential rotation, rotationally induced mixing and magnetic fields are included in both components, as well as non-conservative mass and angular momentum transfer, and tidal spin-orbit coupling. Results: The considered binary system undergoes early Case B mass transfer. The mass donor becomes a helium star and dies as a type Ib/c supernova. The mass gainer is spun-up, and internal magnetic fields efficiently transport accreted angular momentum into the stellar core. The orbital widening prevents subsequent tidal synchronization, and the mass gainer rejuvenates and evolves quasi-chemically homogeneously thereafter. The mass donor explodes 7 Myr before the collapse of the mass gainer. Assuming the binary to be broken-up by the supernova kick, the potential gamma-ray burst progenitor would become a runaway star with a space velocity of 27 km s-1, traveling about 200 pc during its remaining lifetime. Conclusions: .The binary channel presented here does not, as such, provide a new physical model for collapsar production, as the resulting stellar models are almost identical to quasi-chemically homogeneously evolving rapidly rotating single stars. However, it may provide a means for massive stars to obtain the required high rotation rates. Moreover, it suggests that a possibly large fraction of long gamma-ray bursts occurs in runaway stars.

The formation of a helium white dwarf in a close binary system with diffusion

We study the evolution of a system composed of a 1.4-M⊙ neutron star and a normal, solar composition star of 2 M⊙ in orbit with a period of 1 d. Calculations were performed employing the binary hydro code presented by Benvenuto & De Vito that handle the mass transfer rate in a fully implicit way. We then included the main standard physical ingredients together with the diffusion processes and a proper outer boundary condition. We have assumed fully non-conservative mass transfer episodes. In order to study the interplay of mass loss episodes and diffusion we considered evolutionary sequences with and without diffusion in which all Roche lobe overflows (RLOFs) produce mass transfer. Another two sequences in which thermonuclearly driven RLOFs were not allowed to drive mass transfer have been computed with and without diffusion. As far as we are aware, this study represents the first binary evolution calculations in which diffusion is considered. The system produces a helium white dwarf of ∼0.21 M⊙ in an orbit with a period of ∼4.3 d for the four cases. We find that mass transfer episodes induced by hydrogen thermonuclear flashes drive a tiny amount of mass transfer. As diffusion produces stronger flashes, the amount of hydrogen-rich matter transferred is slightly higher than in the models without diffusion. We find that diffusion is the main agent in determining the evolutionary time-scale of low-mass white dwarfs even in the presence of mass transfer episodes.

From the Heart of the Ghoul: C and N Abundances in the Corona of Algol B

Chandra Low Energy Transmission Grating Spectrograph observations of Algol have been used to determine the abundances of C and N in the secondary star for the first time. In order to minimize errors arising from an uncertain coronal differential emission measure as a function of temperature, the analysis was performed relative to similar observations of an adopted "standard" star, HR 1099. It is demonstrated that HR 1099 and Algol are coronal twins in many respects and that their X-ray spectra are very similar in nearly all details, except for the observed strengths of C and N lines. The H-like 2p2P3/2,1/2 → 1s2S1/2 transitions of C and N in the coronae of Algol and HR 1099 demonstrate that the surface abundances of Algol B have been strongly modified by CN processing, as shown earlier by Schmitt & Ness. It is found that N is enhanced in Algol B by a factor of 3 compared with HR 1099. No C lines are detected in the Algol spectrum, indicating a C depletion relative to HR 1099 by a factor of 10 or more. These C and N abundances indicate that Algol B must have lost at least half of its initial mass and are consistent with predictions of evolutionary models that include nonconservative mass transfer and angular momentum loss through magnetic activity. Based on H-like and He-like transitions in O and Ne, it is estimated that Algol is slightly metal poor, by 0.2 dex in terms of the coronal abundances of light elements relative to HR 1099, while the Fe XVII 2p53d1P1 → 2p6 1S0 transition indicates a very similar Fe abundance. In reviewing coronal abundance results for active stars in the literature and drawing on an earlier Chandra study of the coronal abundances of HR 1099, it is concluded that Fe is very likely depleted in the coronae of both Algol and HR 1099 by 0.5 dex relative to their photospheric compositions, but that Ne is enhanced by a similar magnitude. Light elements such as C, N, and O are likely depleted in both stars by of order 0.3 dex. The similarities in these large-abundance anomalies in HR 1099 and Algol are notable. Despite such compositional fractionation in these coronae, the relative C and N abundances in HR 1099, determined by comparing observed line strengths with theoretical C/N line ratios, are consistent with recent solar values, indicating that differential fractionation between these elements is not significant and that little or no dredge-up of material subjected to CN processing has occurred on the subgiant component.

The impact of radiation and wind momenta on mass transfer in massive close binary systems

We investigate to what extent the radiation and stellar wind momenta in a massive close binary system can remove part of the matter flowing from one towards the other star during a mass transfer phase. We perform radiation-hydrodynamics simulations in the co-rotating frame of a binary system made-up of two main sequence stars of 27 and 26 in a 4 day orbit. We study the interaction of the winds of both stars, and of their photons, with the accretion stream originating from the Roche-lobe filling component. For our simulation, we adopt a mass transfer rate of 5 10-6 yr-1, a mid-point in the range of values during massive binary star evolution. Our simulations show that even for such moderate mass transfer rates, the wind and radiative momenta do not alter the dynamics of the accretion stream which is observed to follow essentially ballistic trajectories. Such a conclusion is reinforced for higher mass transfer rates because of the increased stream density and the correspondingly reduced radiation force. We anticipate that the radiation and wind momenta will affect the accretion stream only when its density is comparable to the wind's, a situation wherein the mass transfer rate is vanishingly small and irrelevant for binary star evolution. Alternatively, such reduced accretion stream density could be obtained from spatial dilution in wider systems, potentially leading to non-conservative mass transfer.

The Recurrent Nova U Scorpii in the 1999 Outburst: the First Detection of a Significant Orbital-Period Change

Abstract In this paper we present and discuss our time-resolved photometry of an eclipsing recurrent nova, U Sco, during an outburst in 1999, which was conducted from immediately after the optical maximum to the final fading toward the quiescence. In the first-ever complete light-curve, a few primary and secondary eclipses of the binary system were detected, and the timings of the minima were determined. We found that the eclipses showed no totality during the outburst. The depth of the primary eclipses was 0.4–0.8 mag, much shallower than that in quiescence. In the plateau phase, very little irradiation ($\leq 0.1 \,\mathrm{mag}$) was observed in the orbital light curve, which implies the existence and a large flaring rim of the accretion disk during the outburst. The minima of the eclipses were detected at earlier orbital phases for the predicted ephemerides. Thus, we obtained an orbital period change of the binary system as $\dot{P}/P =-1.7 (\pm 0.7) \times 10^{-6} \,\mathrm{yr}^{-1}$ from the $O-C$. Assuming that this period change is a result of the conservative mass transfer between the component stars, its mass-transfer rate reaches $\dot{M} = 2.4 (\pm 1.0) \times 10^{-6} \,{{{M}_{\odot}}} \,\mathrm{yr}^{-1}$ for a $1.37 \,{{{M}_{\odot}}}$ white dwarf and a $2.0 \,{{{M}_{\odot}}}$ mass-donor companion, which is too high to cause shell flashes, even on a massive white dwarf. Therefore, this large rate of the period change strongly indicates a non-conservative mass transfer in the binary system.

The population of wide binary millisecond pulsars

We present the results of a binary population synthesis study of wide binary millisecond pulsars (BMSPs) forming through four different evolutionary channels. The majority of binary millisecond pulsars are found to descend from systems in which the most massive component undergoes a common-envelope phase prior to the supernova explosion leading to the birth of the neutron star. The orbital period distributions of simulated samples of wide BMSPs are compared with the observed distribution of Galactic BMSPs for a variety of parameters describing the formation and evolution of binaries. The distribution functions typically show a short-period peak below 10 days and a long-period peak around 100 days. The observed distribution is best reproduced by models with highly non-conservative mass transfer, common-envelope efficiencies equal to or larger than unity, and no or moderate supernova kicks at the birth of the neutron star. Only few systems are found with orbital periods longer than 200 days, irrespective of the accretion efficiency of neutron stars. This occurs as a result of the upper limit on the initial orbital periods beyond which the binary avoids the common-envelope phase prior to the supernova explosion of the primary.

On the formation of Super-AGB stars in intermediate mass close binary systems

The evolution of a star of initial mass 9 , and metallicity in a Close Binary System (CBS) is followed in the presence of different mass companions in order to study their influence on the final evolutionary stages and, in particular, on the structure and composition of the remnant components. In order to do that, we study two extreme cases. In the first one the mass of the secondary is 8 , whereas in the second one the mass was assumed to be 1 . For the first of those cases we have also explored the possible outcomes of both conservative and non-conservative mass-loss episodes. During the first mass transfer episode, several differences arise between the models. The system with the more extreme mass ratio () is not able to survive the first Roche lobe overflow (RLOF) as a binary, but instead, spiral-in of the secondary onto the envelope of the primary star is most likely. The system formed by two stars of comparable mass undergoes two mass transfer episodes in which the primary is the donor. We have performed two sets of calculations corresponding to this case in order to account for conservative and non-conservative mass transfer during the first mass loss episode. One of our main results is that for the non-conservative case the secondary becomes a Super–AGB star in a binary system. Such a star undergoes a final dredge-up episode, similar to that of a single star of comparable mass. The primary components do not undergo a Super–AGB phase, but instead a carbon-oxygen white dwarf is formed in both cases (conservative and non-conservative), before reversal mass transfer occurs. However, given the extreme mass ratios at this stage between the components of the binary system, especially for the conservative case, the possibility of merger episodes remains likely. We also discuss the presumable final outcomes of the system and possible observational counterparts.

Population synthesis of wide binary millisecond pulsars

Four evolutionary channels leading to the formation of wide binary millisecond pulsars are investigated. The majority of binary millisecond pulsars are found to descend from systems in which the most massive component undergoes a common-envelope phase prior to the supernova explosion leading to the birth of the neutron star. The orbital period distribution of simulated samples of wide binary millisecond pulsars is compared with the observed distribution of Galactic binary millisecond pulsars for a variety of parameters describing the formation and evolution of binaries. The distribution functions typically show a short-period peak below 10 d and a long-period peak around 100 d. The observed distribution is best reproduced by models with highly non-conservative mass transfer, common-envelope efficiencies equal to or larger than unity, a critical mass ratio for the delayed dynamical instability larger than 3, and no or moderate supernova kicks at the birth of the neutron star. Few systems are found with orbital periods longer than 200 d, irrespective of the accretion efficiency of neutron stars. This occurs as a result of the upper limit on the initial orbital periods beyond which the binary avoids the common-envelope phase prior to the supernova explosion of the primary.

Nonconservative evolution of cataclysmic binaries

We present a mechanism to take into account angular-momentum loss in binary systems with non-conservative mass transfer. In a number of cases, mass loss in the system can increase the orbital angular momentum of the stars. Including this mechanism in evolutionary models substantially expands the domain of stable mass transfer in binary systems. All observed cataclysmic binaries with known component masses fall within the calculated area for stable mass transfer.

Discovery of a massive equatorial torus in the η Carinae stellar system

The enigmatic object η Carinae is believed to represent an important, but short-lived, unstable phase in the life of the most massive stars, occurring shortly before they explode as supernovae or collapse directly to black holes. The putative binary1,2 system believed to constitute η Carinae survived an outburst in the previous century that lasted 20 years; and which created a nebula with pronounced bipolar lobes that together contain about 2.5 solar masses of material. The nebula also exhibits an equatorial ‘waist’ containing about 0.5 solar masses3. The physical mechanisms responsible for the outburst and the bipolar geometry are not understood. Here we report infrared observations (spectroscopy and imaging) that reveal the presence of about 15 solar masses of material, located in an equatorial torus. The massive torus may have been created through highly non-conservative mass transfer, which removed the entire envelope of one of the stars, leaving an unstable core that erupted in the nineteenth century. The collision of the erupted material with the pre-existing torus provides a natural explanation for the bipolar shape of the nebula.

Analytical solution for the evolution of a binary with stable mass transfer from a giant

We derive a simple analytical solution for the evolution of a close binary with nuclear time-scale driven mass transfer from a giant. This solution is based on the well-known fact that the luminosity and the radius of a giant scale to a good approximation as simple power laws of the mass M_c of the degenerate helium core. Comparison with results of numerical calculations by Webbink, Rappaport & Savonije (1983) show the analytical solution and the power law approximation to be quite accurate. The analytical solution presented does also allow (in parametrized form) for non-conservative mass transfer. Furthermore it is shown that the near constancy of the mass transfer rate over most of the mass transfer phase seen in the results by Webbink, Rappaport & Savonije is not a generic feature of this type of evolution but rather a consequence of a particular choice of parameters. The analytical solution also demonstrates that the level of mass transfer is largely set by the core mass of the giant at the onset of mass transfer. Finally we show that the model is selfconsistent and discuss its applicability to low-mass X-ray binaries.

Evolutionary Scenarios for Double Degenerate Systems

There are presently eight double degenerate systems with well determined orbital parameters, their periods being either a few hours or a few days (Marsh, Dhillon & Duck 1995; Marsh 1995). The masses of the primaries and secondaries lie in the range 0.15… 0.45M⊙.We calculate two evolutionary scenarios (Sarna, Marks & Smith 1996); the first is Algol-type evolution with two phases of stable mass transfer, and the second involves first a stage of common envelope (CE) evolution followed by a stage of stable mass transfer. In both calculations we assume non-conservative mass transfer by which we mean that the total mass and angular momentum of the system are not conserved. For both scenarios we start our calculations after the first stage of mass transfer has finished. In all calculations the primary is the initially more massive star that filled its Roche lobe and transferred material to the secondary during the first phase of mass transfer, hence the secondary is the star that fills its Roche lobe in our calculations. The system’s orbital period decreases and then increases until the system detaches; we are left with a detached white dwarf/white dwarf binary with an orbital period of the order of hours or of days (see Table 1). There must exist some bifurcation period below which the systems evolve towards orbital periods of the order of hours and above which the systems evolve to periods of the order of several days.