result of the large thermal inertia of the convective INTRODUCTION

Transcription

1 THE ASTROPHYSICAL JOURNAL, 561:329È336, 2001 November 1 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. THE EVOLUTION OF CATACLYSMIC VARIABLE BINARY SYSTEMS WITH CIRCUMBINARY DISKS RONALD E. TAAM1 AND H. C. SPRUIT2 Received 2001 February 16; accepted 2001 May 22 ABSTRACT The e ect of a circumbinary (CB) disk on the evolution of a binary consisting of a low-mass secondary star with a white dwarf primary is investigated, taking into account the viscous spreading of matter within the circumbinary disk and the response of the secondary to mass loss. The CB disk is assumed to be fed a constant fraction d of the mass transfer from the secondary to the primary through a wind such as those observed in nova-like systems. The CB disk is e ective in draining orbital angular momentum from the system provided that d exceeds about In this case, the mass transfer rates are elevated and the evolution accelerated in comparison with an evolution with the standard angular momentum loss processes of magnetic braking and gravitational radiation. The mass transfer rates for a given system can vary by more than an order of magnitude during its evolution. With a CB disk, a binary can thus evolve between the various subclasses of cataclysmic variables. A large spread in transfer rates at a given orbital period results, reñecting a range of ages of the systems, the possible presence of a CB disk already at the beginning of the evolution of the binary, and the mass and evolutionary state of the donor. For high mass input rates into the circumbinary disk (d Z 0.015), the secondary can be completely dissolved in less than a Hubble time. Subject headings: binaries: close È novae, cataclysmic variables È stars: evolution 1. INTRODUCTION The evolution of cataclysmic variables (CVs) is intimately related to the angular momentum loss that drives mass transfer in these binaries. The identiðcation of gravitational radiation and of magnetic stellar winds as causes of this loss was made more than two decades ago in the pioneering works by Faulkner (1971; see also Paczyn ski 1967) and by Verbunt & Zwaan (1981), respectively. The inferred mass transfer rates of CVs pointed to a correlation of the mass transfer rate with orbital period, in the sense that systems at longer periods typically transfer mass at higher rates (Patterson 1984). This important observational result suggested that the evolution of long period systems (P Z 3 hr) was dominated by magnetic braking, while the shorter period systems were governed by gravitational radiation losses. Against this background, the existence of a dearth of CVs in the orbital-period range between 2.2 to 2.8 hr (known as the period gap) required explanation. It was shown by Rappaport, Joss, & Verbunt (1983) and Spruit & Ritter (1983) that this observational feature in the orbital period histogram could also be accommodated within the angular momentum loss paradigm, provided that the magnetic braking is suddenly reduced at the upper edge of the gap. In this picture, the main sequenceèlike secondary star is driven out of thermal equilibrium by the angular momentum losses (leading to mass transfer timescales shorter than the thermal timescale). Upon the reduction of magnetic braking, the secondary detaches from its Roche lobe and contracts to its thermal equilibrium state in the gap, reappearing as a mass-transferring CV when gravitational radiation losses shrink the orbit (and the Roche lobe) sufficiently to reinitiate mass transfer at a shorter orbital period. The abrupt change in the angular momentum loss rate required in this picture was attributed to the change in the donor starïs structure from one characterized by a radiative core and convective envelope to a fully convective conðguration. This transition to a fully convective structure was thought to be accompanied by either reduced magnetic activity (Rappaport et al. 1983; Spruit & Ritter 1983) or a reduced wind associated with rearrangement of the magnetic Ðeld (Taam & Spruit 1989) in response to a change in the character of the magnetic dynamo. Despite the general consensus about the essential role played by magnetic braking and gravitational radiation in CV evolution, there are a number of observational features that do not Ðt well. Among them are the absence of dwarf novae at the upper edge of the period gap, which shows that the mass transfer rate does not always decrease with orbital period (Shafter 1992). In addition, the mass transfer rates are not strictly a function of orbital period. They are observed to vary by over an order of magnitude at a given orbital period, from D10~10È10~9 M yr~1 in dwarf novae to D10~9È10~8 M yr~1 in nova-like variable sources (Patterson 1984; Warner 1987). The supersoft sources further aggravate the di erences between observation of evolutionary theory since the white dwarfs in these systems are thought to be accreting at high rates, M0 D 10~7 M yr~1 (van den Heuvel et al. 1992), such that hydrogen burning is stable (e.g., Paczyn ski & Zytkow 1978; Sienkiewicz 1980). A possible resolution for the range of mass transfer rates at a given period is the hypothesis that the system undergoes Ñuctuations in the mass transfer rate about an evolutionary mean, such that a given system behaves like a nova-like variable during the high state and a dwarf nova in the low state. It has been suggested that such cyclic behavior may be expected to arise as a result of the consequences of irradiation of the secondary component (King et al. 1 Department of Physics and Astronomy, Dearborn Observatory, Northwestern University, 2131 Sheridan Road, Evanston, IL ). However, such cycles are not found to develop for 2 Max-Planck-Institut fu r Astrophysik, Karl-Swarzschild-Strasse 1, low-mass stars less massive than about 0.65 M. This is a Postfach 1317, D5741 Garching, Germany. result of the large thermal inertia of the convective 329

2 330 TAAM & SPRUIT Vol. 561 envelope, which limits the ability of radiative e ects to expand the secondary (King et al. 1996). The cycle interpretation is also potentially in conñict with the observational evidence on the lack of infrared detectability of the secondaries in nova-like variables (Dhillon et al. 2000). Main-sequence stars were expected to be detected, and the nondetection suggests that these secondaries are cooler than main-sequence stars and hence overexpanded (and perhaps less massive than main sequenceèlike stars) at a given orbital period. This is most naturally explained if the inferred mass transfer rates in these systems are secular rather than large-scale Ñuctuations about a mean. Adopting the secular origin for the mass transfer rates, Pylyser & Savonije (1989) suggest that the spread in transfer rates reñects the di erences in the evolutionary state of the mass-losing star, with unevolved and evolved donors driving higher and lower rates of mass transfer, respectively. However, the detection of main sequenceèlike companions for dwarf novae above the period gap (see Dhillon 1998) suggests that these stars are not evolved. Finally, the high mass transfer rates for the short-period supersoft sources are difficult to understand within the context of the standard angular momentum loss processes alone. Hence it would appear that additional angular momentum loss processes are required to bring theory and observational data into agreement. In an attempt to address these issues, we have recently suggested (Spruit & Taam 2001) that a circumbinary (CB) disk could provide an e ective means for extracting orbital angular momentum from the binary system. In this picture, tidal torques are exerted on the CB disk, thereby removing angular momentum from the binary orbiting inside it. Presuming that some fraction of the mass lost from the donor star condenses into a CB disk, a feedback process can operate to elevate the transfer rates above that determined by magnetic braking or gravitational radiation losses acting alone. Meyer & Meyer-Hofmeister (2001) have already proposed that CB disks are present around black hole binaries. They argue that such disks would regulate the mass transfer rates in these binaries and cause them to have outbursts with consistent and long recurrence times. The authors suggest that CB disks could result as remnants of the common-envelope process in which these binaries developed their present close orbits. This possibility may also apply to the CVs studied here, and is discussed somewhat in 4. Observational support for the possible presence of a CB disk in a CV system has been summarized in Spruit & Taam (2001). SpeciÐc examples include the stationary low-velocity emission features seen in AM CVn by Solheim & Sion (1994) and in the supersoft source J0019.8]2156 by Deufel et al. (1999). In fact, Solheim & Sion explicitly attribute this emission to circumbinary material. In this paper, we relax some of the simpliðed assumptions adopted in Spruit & Taam (2001) and report on the results of detailed binary evolutionary calculations incorporating the presence of a CB disk. The assumptions underlying the model are described in the next section, and the numerical results are presented in 3. We summarize and discuss the implications of our results in the Ðnal section. 2. MODEL The evolution of the binary system is calculated following (mostly) the formulation in McDermott & Taam (1989). The response of a low-mass star to mass loss is calculated using a modiðed version of the stellar evolutionary code developed by Eggleton (1971, 1972). A model star is divided into 199 computational zones and is assumed to have a population I composition (X \ 0.7, Y \ 0.28, and Z \ 0.02). The mass transfer is assumed to be conservative even though a fraction, d, is assumed to be deposited into the CB disk. This neglect of the mass loss from the system is not signiðcant, however, since the values of d used are small (see 3). Furthermore, the conservative approximation is adopted in order to isolate the e ects of the CB disk on the binary without the additional e ects associated with nonconservative evolution. Angular momentum loss rates associated with gravitational radiation and magnetic braking are included following the prescription proposed by Verbunt & Zwaan (1981). In the latter case, the angular momentum loss is given as J0 \[0.5 ] 10~28f ~2k2M R4 ) 3, (1) d d 0 where M and R are the mass and radius of the donor star, respectively, d and d k2 is its moment of inertia in units of M R2. Here f is obtained from a Ðt to the equatorial rota- d d tion velocities of G and K type stars and is taken to be equal to 1.78 (Smith 1979), and ) is the angular frequency of the 0 binary system. Assuming a thin, Keplerian CB disk, the diskïs structure and evolution is described by a di usion equation of the form L& Lt \ 3 r L Lr C r1@2 L Lr (r1@2l&)d, (2) where & is the surface mass density and l is the viscosity. It is assumed that the disk is in both thermal and hydrostatic equilibrium. The calculation of the local structure is based on the one-zone approximation following the a viscosity description pioneered by Shakura & Sunyaev (1973). In this approximation, the e ect of advective energy transport in the radial direction on the disk structure is neglected. The viscous torque in the disk is given by J0 \[2nr2l&r L) Lr, (3) where ) is the local orbital frequency [) \ (GM/r3)1@2], with M the total mass of the binary system. The angular momentum loss rate from the binary system due to the CB disk is calculated by evaluating equation (3) at the inner disk radius. The work of Artymowicz & Lubow (1994) shows that the inner radius of the CB disk is approximately equal to 1.7 times the orbital separation and that this factor is relatively insensitive to the mass ratio of the binary system. Assuming that the torque is concentrated near the inner edge of the CB disk, the angular momentum loss rate from the binary can be written as J0 \ 3n(r /a)1@2) a2l &, (4) i 0 i i where a is the binary separation and the subscript i indicates the inner edge of the CB disk. The orbit of the secondary is assumed to instantaneously respond directly to angular momentum removed from it by the CB torque. Though the secondary Ðlls its Roche lobe exactly, the mass transfer rate actually follows the changing orbit with a small delay, since the Ðnite density scale height H in the atmosphere of the secondary softens ÏÏ its response. This delay is of the order H/Rt, where R is the evol

3 No. 1, 2001 EVOLUTION OF CV BINARY SYSTEMS WITH CB DISKS 331 secondaryïs radius and t is the evolution timescale of the orbit (Ritter 1988). The time evol steps we use for the evolution of the binary are generally 10È100 times larger than this, so we neglect this delay in the calculations. Since the mass and orbital separation of short period CVs are comparable to the mass and radius of T Tauri stars, the conditions in the CB disk are similar to the circumstellar disks about these preèmain-sequence stars. Hence, we adopt the opacities for dust grains and gas as described in Bell & Lin (1994) and Bell et al. (1997) for the determination of thermal equilibrium in the CB disks. As an illustration of the variation of viscosity as a function of column density at a given radius, see Figure 1. It is evident that the viscosity is a nonmonotonic function of the column density. In particular, the disk is locally unstable (where the derivative of the stress integral l& with respect to column density is negative) whenever the physical conditions in the disk are such that H-scattering dominates the opacity. We point out that we do not include the e ects of convection in the energy transport in this exploratory study. Its inclusion would shift the curve in the viscosity column density plane to higher densities (see Bell & Lin 1994), but the existence of an unstable regime is preserved. If there are places in the disk where the slope of the &-l& relation is negative, it is likely that the disk will be time dependent on a viscous timescale, oscillating about a range in viscosities corresponding to the upper turning point on the lower stable branch and the lower turning point on the upper stable branch (see Fig. 1). Since the viscous timescale is generally quite short compared with the binary evolution time, it would be prohibitive to follow such viscous time dependence. We avoid this problem by modifying the surface density dependence of the viscosity somewhat. For column densities above the upper turning point on the lower stable branch, we replace the dependence by a monotonic one (see Fig. 1). This functional choice facilitates con FIG. 1.ÈVariation of the viscosity, l, in the CB disk as a function of column density, &, at a radius of 1011 cm for a central binary mass of 1 M. The a viscosity parameter is equal to The dash-dotted curve corresponds to the monotonic form adopted in this work (see 2). vergence of the numerical scheme for the evolution of the disk, and should provide a reasonable representation of the average behavior of the disk on timescales long compared with the viscous timescale. Alternatively, it is conceivable that a smoothed ÏÏ version of the S curve would be more realistic (so that the transition takes place somewhere in the middle between the two turning points; F. Meyer 2000, private communication). A more detailed time-dependent analysis of transition waves in the unstable part of the disk will be needed to obtain a better description of their average e ect Numerical Treatment of the CB Disk The viscous evolution of the CB disk takes place on a much shorter timescale than the evolution of the binary. For efficient computing, it is therefore necessary to use an implicit integrator to time step the disk in equation (2). A spatial coordinate that is more suitable than the radius r is x \ r1@2. (5) The spatial discretization is implemented by simple centered di erencing on an equidistant grid in x. For the time stepping we use centered di erencing as well, i.e., a Crank- Nicholson scheme. Accuracy is controlled by choosing time steps such that the relative change in surface density between steps is kept near a Ðxed value (typically 0.05). An exception to this are the Ðrst few steps, since the initial disk is assumed empty and enforcing relative accuracy would lead to vanishingly small time steps. Since the di usion equation is numerically quite benign, smearing out inaccuracies rather than amplifying them, the result is independent of the initial development, which takes place at very low surface density. The dependence of viscosity on surface density implemented here is quite nonlinear. A fully implicit method would therefore require a rather time-consuming iteration at each time step. Instead of this, we have done the changes in viscosity between time steps explicitly rather than implicitly. That is to say, only the di usion equation itself is stepped implicitly. The resulting loss of accuracy is made up for by choosing a smaller time step. The dependence of viscosity on surface density has a positive slope at low surface density. A consequence of this is that the disk has an edge at a Ðnite distance. This is a general result (ZelÏdovich & Raizer 1986). This is not the case if the viscosity is just a function of position only; the disk then has a tail of low surface density that spreads quite far. For our present purposes, it is both sufficient and economical to make the grid just somewhat larger than the disk (we use a factor of D1.5). As the disk spreads, this is accomplished by doubling the grid in size (but with the same number of grid points) whenever its outer edge gets too close to the grid boundary. This reduces the resolution near the inner edge (where resolution matters most), but this has no important consequences, since the gradient in surface density in the inner regions of the disk also becomes smaller as the disk spreads. Finally, a certain minimum surface density (around 10~2 g cm~2) is maintained artiðcially, since the expressions for the viscosity cause numerical problems when extrapolated too far. This minimum value also provides the outer boundary condition. The inner boundary condition is determined by the mass Ñux into the disk. The time stepping in the binary evolution part of the code is done independently from that of the viscous disk. Time

4 332 TAAM & SPRUIT Vol. 561 steps passed to the disk evolution routine are broken up into smaller ones as needed for accuracy. 3. BINARY EVOLUTIONARY RESULTS Binary evolutionary sequences were calculated for model CV systems consisting of secondary components of varying mass and evolutionary state and a primary white dwarf component of 0.8 M. For a given secondary, computations were performed for a range in the fractional mass input rate into the CB disk, d [ 0.03, and for an a viscosity parameter typically equal to Main-Sequence Secondaries The evolution of a 0.6 M main-sequence star characterized by a radius of 0.55 R and a luminosity of L in a binary of orbital period 4.6 hr was followed for d ranging from 0 to As d was increased, mass transfer rates in the system generally increased. For example, for small d ([0.005) the binary evolution primarily follows that driven by magnetic braking, with the orbital period monotonically decreasing with time (see Fig. 2). The average rate over the period interval from 4.6 to 3 hr was D10~9 M yr~1. On the other hand, an increase in d by a factor of 3 to leads to a rapid acceleration of mass transfer to rates as high as 1.1 ] 10~7 M yr~1 and to a prolonged phase where the orbital period increased with time. The acceleration phase is not immediate, since the CB disk is initially empty and builds up as a function of time. In addition, d had to be increased slowly to the desired value to facilitate convergence in the disk computation. For example, the mass transfer increased to 10~8 M yr~1 after D107 yr, followed by a more rapid increase to 10~7 M yr~1 within an additional 5 ] 106 yr. In this case, the angular momen- tum loss rate due to the CB disk exceeded that due to magnetic braking by factors as large as 500. As a result of the high mass transfer rates, the secondary was found to expand during the high mass transfer phase, when its mass decreased to 0.33 M (see Fig. 3). Continued mass loss from the secondary component led to the near cessation of nuclear burning in its interior, with the central temperature declining to less than 2 ] 106 K. Eventually, the secondary contracted when its mass declined to 0.11 M. The fractional mass input rate into the CB disk distinguishing an orbital evolution to a longer period from an orbital evolution to a shorter period lies in the range 0.01 \d\ To determine the inñuence of the magnitude of the viscosity on the evolutionary results, a was increased to 0.01 for d \ and We adopt these choices for the viscosity parameter somewhat guided by the inferred values for dwarf novae in the quiescent state (a D 0.05; see Ludwig & Meyer 1998) and for FU OrionisÈtype phenomena (a [ 0.001; see Bell & Lin 1994). At d \ the evolution was still similar to the case without a CB disk, as in the case a \ 0.001, but the evolution was signiðcantly modiðed for d \ In the latter case, the peak mass transfer rates were increased by more than an order of magnitude, from 1.5 ] 10~8 M yr~1 for a \ to 1.8 ] 10~7 M yr~1 for a \ 0.01, leading to a phase where the orbital period increased (see Fig. 4). The evolution is similar to the case where d \ for a \ Hence a factor of 3 change in d was approximately equivalent to a factor of 10 change in a for the chosen parameters, indicating that the evolution is much more sensitive to fractional mass input rates into the CB disk than to the magnitude of its viscosity for the above range of parameters. The mass transfer rates and the evolution do not linearly scale with the viscosity since the torque depends on the product of the column density and viscosity in the CB disk. Since the column density decreases for an increase in the viscosity, it partially o sets the e ect of an increased a parameter FIG. 2.ÈVariation of mass transfer rate, M0, with orbital period, P, ina system containing a 0.6 M secondary main-sequence star for various fractional input rates into the CB disk. The viscosity parameter in the disk is taken to be The solid, dotted, dashed, and dash-dotted lines correspond to evolution for which d \ 0, 0.005, 0.01, and 0.015, respectively FIG. 3.ÈVariation of radius, R, in units of 1011 cm as a function of mass M in for the secondary star for the sequences shown in Fig. 2. M M

5 No. 1, 2001 EVOLUTION OF CV BINARY SYSTEMS WITH CB DISKS FIG. 4.ÈSame as Fig. 2, but for a \ The solid, dotted, and dotdashed curves correspond to d \ 0, 0.005, and 0.01, respectively FIG. 5.ÈSame as Fig. 2, but for a 0.2 M secondary main-sequence star. The dashed, solid, and dot-dashed lines correspond to evolution for which d \ 0, 0.01, and 0.02, respectively. To examine the evolution of systems at shorter orbital periods, a binary system consisting of a lower mass companion was considered. For a 0.2 M main-sequence star with a radius of 0.23 R, the orbital period of the system is 2.2 hr. In this case, angular momentum losses by gravitational radiation were included but magnetic braking was assumed negligible. In addition, a \ was adopted for the CB disk. As for the more massive main-sequence secondaries, the CB disk became e ective for d Z For the case d \ 0.01, the mass transfer rates were enhanced by a factor of 2 at an orbital period of 2 hr as compared to a sequence where the CB disk was absent (see Fig. 5). This increase reñected the fact that the contribution to the angular momentum loss rate from the CB disk exceeded that due to gravitational radiation by about 50%. After an initial adjustment phase (see above), the orbital period decreased monotonically. With increasing mass input rates in the CB disk (d D 0.02), the evolution rapidly accelerated to high mass transfer rates and to longer orbital periods, reaching a maximum of D2 ] 10~8 M yr~1 at an orbital period of 4.25 hr for d \ The critical fractional mass input rate into the CB disk distinguishing evolution to a longer orbital period from evolution to a shorter orbital period is similar to the range found for more massive mainsequence secondary components (i.e., between 0.01 and 0.015). The existence of such a critical rate is expected following the analytical description presented in Spruit & Taam (2001), when the timescale for angular momentum loss via the CB disk becomes shorter than that associated with magnetic braking or gravitational radiation. For rates above the critical value, the acceleration of the evolution becomes exponential Evolved Secondaries Recent work by Bara e & Kolb (2000) suggests that evolved secondary components in systems with orbital periods greater than 6 hr are required to reproduce the observed spectral types in these systems. As these systems may evolve to shorter periods, we have expanded our investigation to include such secondaries. SpeciÐcally, we have considered a binary evolution for an evolved secondary characterized by a central hydrogen content of 0.3, a mass of M, and a radius of 0.55 R. Such a model was constructed by removing mass from an evolved 1 M star. As in the binary systems described above, it was assumed that the companion was a 0.8 M white dwarf. Binary sequences were calculated for d ranging from 0 to 0.02, with the mass transfer initiated at an orbital period of 4.63 hr. The results of the mass transfer calculations are illustrated as a function of orbital period in Figure 6. It can be seen that the mass transfer rates increase for larger fractional mass input rates into the CB disk. For example, at an orbital period of 4 hr, the transfer rates increased by a factor of 2.25 from 4 ] 10~10 M yr~1, for an increase in d from 0 to 0.01, and by a factor of 10, for an increase to d \ For this higher value of d, we note that a period bounce occurred at 3.4 hr. At this point, the secondary has decreased to a mass of M, and its thermal structure is described by convective equilibrium throughout. A further increase in d to 0.02 led to an evolution where the mass transfer rate rapidly increased to D10~7 M yr~1. In this case a period bounce occurred at an orbital period of 4.3 hr, when the mass had been reduced to 0.32 M. To determine the dependence and generality of the results for a more evolved secondary component, a secondary star was considered with a mass of M and a radius of R in which hydrogen is depleted at the center. In this case, the orbital period of the system was 6.2 hr. For all of the calculated evolutionary sequences, the orbital period decreased monotonically with time. This trend was con- Ðrmed even for large values of d, in contrast to previous sequences. Of particular interest is the evolution at large d. For example, for d \ 0.02, the mass transfer rates varied in a narrow range between (4È7) ] 10~9 yr~1 (see Fig. 7) M

6 334 TAAM & SPRUIT Vol FIG. 6.ÈSame as Fig. 2, but for a M evolved secondary star. The solid, short-dashed, dashed, and dash-dotted lines correspond to evolution for which d \ 0, 0.01, 0.015, and 0.02, respectively. for D8 ] 107 yr. At a higher input rate (d \ 0.03), the mass transfer rates continually increased to values as high as 4 ] 10~7 M yr~1 for a mass of 0.12 M at the end of the evolutionary sequence. Since the response of an evolved star to mass loss is to shrink as a result of the inñuence of the radiative interior, there is no tendency for the star to expand with increasing mass transfer rates, in contrast to the situation for main sequenceèlike stars, where a signiðcant mass fraction of the star can become convective CB Disk Evolution In many of the binary sequences described above, the CB disk has a profound e ect on the evolution of the system. To elucidate the properties of this disk, we focus on them in this subsection. As an illustration of the evolution, consider the CB disk formed about the particular system for which the main-sequence component is 0.6 M, with the CB disk characterized by a relative mass input rate, d, equal to 0.01, and a viscosity parameter, a, equal to (see 3.1). Such a system lies in the parameter range distinguishing evolutions to shorter orbital periods from evolutions to longer orbital periods. The column density & and mid-plane temperature T are shown as a function of radius R and time in Figures 8 and 9, respectively. It can be seen that the CB disk has column densities and temperatures similar to those found for circumstellar disks surrounding young stellar objects (see Bell & Lin 1994). In particular, the column densities are [104 g cm~2 and except for a small inner region decrease outward. Similarly, the mid-plane temperatures lie in the range from 100 K at radii of several AU to D104 K in the innermost regions. The CB disk remains geometrically thin throughout, with the ratio of scale height to radius typically [0.03. We note that, in contrast to a CB disk characterized by a viscosity that is a function of position (e.g., Spruit & Taam 2001), the disk has a Ðnite edge (see 2.1). It is evident from Figure 8 that the disk expands outward as a function of time, extending to radii exceeding about 10 AU after a time corresponding to about 4.6 ] 107 yr. In contrast to accretion disks, the mass in the CB disk continually increases, amounting to 6.1 ] 10~4 M after 1.5 ] 107 yr and increasing to 2.8 ] 10~3 M within an additional 3.1 ] 107 yr of evolution. At both of these points in time, the CB disk is e ective in promoting mass transfer at rates of 1.6 ] 10~8 and 3 ] 10~9 M yr~1 since the angular FIG. 7.ÈSame as Fig. 2, but for a M evolved secondary star. The dashed, dot-dashed, dotted, and solid curves correspond to evolution for which d \ 0, 0.01, 0.02, and 0.03, respectively log R FIG. 8.ÈVariation of column density & ingcm~2, with radius R in cm, as a function of time for the evolutionary sequence characterized by a 0.6 M main-sequence star with d \ 0.01 and a \ The dashed and solid curves correspond to an evolution time of 1.5 ] 107 and 4.6 ] 107 yr, respectively.

7 No. 1, 2001 EVOLUTION OF CV BINARY SYSTEMS WITH CB DISKS log R FIG. 9.ÈSame as Fig. 8, but for a variation of mid-plane temperature T with radius R in cm. momentum loss rate associated with the disk is, respectively, 4.2 and 6.5 times larger than that associated with magnetic braking. 4. SUMMARY AND DISCUSSIONS The detailed modeling reported above shows that the evolution of a mass-transferring binary can vary over a wide range when the e ect of a CB disk is included. With increasing mass in the CB disk, the evolution of the system accelerates as a consequence of the elevated mass transfer rates. In all the calculations (as in Spruit & Taam 2001), we have assumed that a Ðxed fraction of the mass transfer in the binary ends up in the circumbinary disk (via an outñow or wind). At the moment, this is just an assumption. The fact that the clearest evidence for outñows is seen in systems with high mass transfer rates makes it plausible that the input into the CB disk increases with transfer rate, but the dependence could be shallower or steeper than the proportionality assumed here. The calculations show that the evolution of the system is signiðcantly a ected by the CB disk when the fractional mass input rate into the disk, d, reaches a value of Z0.01. For d [ 0.01, the binary evolves to decreasing orbital periods as in a standard magnetic braking scenario, albeit with somewhat enhanced mass transfer rates. On the other hand, for d Z the evolution is accelerated to such an extent that the mass transfer approaches rates of D10~7 M yr~1. For main-sequence or slightly evolved secondary components, the system then evolves to longer orbital periods. Although the properties of the CB disk depend on the unknown viscosity parameter a, the numerical results indicate that the mass transfer rates are much more sensitive to d. The results, taken collectively, show that the rate of mass transfer can vary by more than an order of magnitude at a given orbital period, reñecting the e ectiveness of the CB disk (i.e., d and a) as well as the mass and evolutionary state of the secondary component of the system. Hence, the inclusion of the CB disk is capable of providing an explanation for the observed range of mass transfer rates inferred in CVs. For sufficiently large fractional input rates into the disk, the evolution can lead to the complete dissolution of the secondary component in less than a Hubble time, con- Ðrming the suggestion of Spruit & Taam (2001). The CB disk scenario proposed here explains the range in mass transfer rates, but it also raises some issues relating to the period gap, since for some parameter combinations the period gap would be Ðlled in by systems evolving through it without detaching. For example, to prevent the evolution of systems from longer orbital periods into the period gap requires that the secondaries not be very evolved. Our results for secondaries with a helium core (which evolve into the period gap) and stars with a helium-rich core (which can undergo a period bounce) are consistent with those of Bara e & Kolb (2000). These authors Ðnd that evolved secondaries with central hydrogen content in the range of 0.05 to 0.5 can be consistent with the period gap, provided that the mass transfer rates are elevated above those driven for main-sequence secondaries. In the framework of a CB diskè dominated evolution, d[0.01 is required for the occurrence of a period bounce in these systems. Additional constraints on d can be obtained by requiring that systems do not evolve from shorter periods into the period gap. Our numerical results suggest that d for the short-period CV population must be less than In systems for which the CB disk is ine ective (d [ 0.01) the preservation of the period gap requires the operation of a mechanism in which the angular momentum loss rate is suddenly reduced in the system. A mechanism such as the disrupted magnetic braking model has been proposed (see 1). Such a scheme would not preserve the period gap for evolved secondaries, since these stars would become convective at orbital periods below the upper edge of the gap. Hence, to avoid populating the period gap region with these systems, there should be a tendency for the fractional rate of mass input into the CB disk to be higher for more evolved secondaries. It is hard to identify a plausible reason for this to be so, as long as the physics determining d is not known in detail. We venture the speculation, however, that it has to do with the fact that these systems had previously undergone thermally unstable mass transfer at a longer orbital period, when the secondary was more massive than its white dwarf companion. In this case, some fraction of the matter that could not be accreted by the white dwarf may have remained in the system. As suggested also by Meyer & Meyer-Hofmeister (2001), such a process could already form a fairly massive CB disk at an early stage. This could promote higher mass transfer after the thermally unstable phase and facilitate the operation of a wind from the disk surrounding the white dwarf to sustain the mass transfer rates at required elevated values. In this picture, a range in mass transfer rates above the period gap would be necessary to maintain the presence of the gap, allowing for a range in the evolutionary state of the secondary. This would be consistent with population synthesis models invoking disrupted magnetic braking, provided that the rates exceed a minimum value of about 1.5 ] 10~9 M yr~1 (see Bara e & Kolb 2000). We point out that the mass transfer rates inferred for nova-like variables and supersoft sources can be achieved in systems for which d is near the upper end of the range above which systems could evolve into the period gap from

8 336 TAAM & SPRUIT log Age FIG. 10.ÈVariation of mass transfer rate, M0, with age in years, for three binary evolutionary sequences. Secondaries characterized by a 0.6 M main-sequence star (d \ 0.015), a M evolved star (d \ 0.015), and a M evolved star (d \ 0.02) are illustrated as solid, dotted, and dashdotted curves, respectively. The viscosity parameter in the disk is taken to be shorter periods. In this case, the angular momentum loss timescale associated with the CB disk is less than that associated with magnetic braking early in the evolution, leading to immediate acceleration rather than delayed acceleration (see Spruit & Taam 2001) to high mass transfer rates. Our results indicate that CV systems with CB disks would be nova-like variables (M0 [ 10~8 M yr~1) for durations of more than 107 yr (see Fig. 10). For these systems the period derivative may either be positive or negative, depending on the evolutionary state of the secondary component. It is also evident that the peak mass transfer rates for some cases can attain values as high as 10~7 M yr~1, comparable to those inferred for supersoft sources. Hence a given system may make a transition from one CV class to another during the process of its evolution. For example, the early and late phases of the evolution of a 0.6 M secondary (with d \ 0.015) could be designated as nova-like, whereas the system in the intermediate phase could correspond to a supersoft source. For systems with such high transfer rates, our results indicate that their evolution drives them to increasing orbital periods, and hence that they formed at periods less than their present-day values. The short period system J0537.7[7304 at 3 hr (Orio et al. 1997) would, in the CB disk picture, have to have formed near the upper edge of the period gap. We thank Dr. K. Robbins Bell for the use of her opacity routines in the calculation of the CB disk structure. We also thank the referee for comments that led to improvements in the presentation of this study. This research was supported in part by the National Science Foundation under grant AST H. S. acknowledges support from the European Commission under TMR grant ERBFMRX-CT ( Accretion onto Black Holes, Compact Objects and Protostars ÏÏ). REFERENCES Artymowicz, P., & Lubow, S. H. 1994, ApJ, 421, 651 Orio, M., Della Valle, M., Massone, G., & O gelman, H. 1997, A&A, 325, Bara e, I., & Kolb, U. 2000, MNRAS, 318, 354 L1 Bell, K. R., Cassen, P. M., Klahr, H. H., & Henning, Th. 1997, ApJ, 486, Paczyn ski, B. 1967, Acta Astron., 17, Paczyn ski, B., & Zytkow, A. N. 1978, ApJ, 222, 604 Bell, K. R., & Lin, D. N. C. 1994, ApJ, 427, 987 Patterson, J. 1984, ApJS, 54, 443 Deufel, B., Barwig, H., Simic, D., Wolf, S., & Drory, N. 1999, A&A, 343, Pylyser, E. H. P., & Savonije, G. J. 1989, A&A, 208, Rappaport, S., Joss, P. C., & Verbunt, F. 1983, ApJ, 275, 713 Dhillon, V. S. 1998, in ASP Conf. Ser. 137, Wild Stars in the Old West: Ritter, H. 1988, A&A, 202, 93 13th North American Workshop on Cataclysmic Variables and Related Shafter, A. W. 1992, ApJ, 394, 268 Objects, ed. S. B. Howell, E. Kuulkers, & C. Woodward (San Francisco: Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 ASP), 23 Sienkiewicz, R. 1980, A&A, 85, 295 Dhillon, V. S., Littlefair, S. P., Howell, S. B., Ciardi, D. R., Harrop-Allin, Smith, M. A. 1979, PASP, 91, 737 M. K., & Marsh, T. R. 2000, MNRAS, 314, 826 Solheim, S.-E., & Sion, E. M. 1994, A&A, 287, 503 Eggleton, P. P. 1971, MNRAS, 151, 351 Spruit, H. C., & Ritter, H. 1983, A&A, 124, 267 ÈÈÈ. 1972, MNRAS, 156, 361 Spruit, H. C., & Taam, R. E. 2001, ApJ, 548, 900 Faulkner, J. 1971, ApJ, 170, L99 Taam, R. E., & Spruit, H. C. 1989, ApJ, 345, 972 King, A. R., Frank, J., Kolb, U., & Ritter, H. 1995, ApJ, 444, L37 van den Heuvel, E. P. J., Bhattacharya, D., Nomoto, K., & Rappaport, ÈÈÈ. 1996, ApJ, 467, 761 S. A. 1992, A&A, 262, 97 Ludwig, K., & Meyer, F. 1998, A&A, 329, 559 Verbunt, F., & Zwaan, C. 1981, A&A, 100, L7 McDermott, P. N., & Taam, R. E. 1989, ApJ, 342, 1019 Warner, B. 1987, MNRAS, 227, 23 Meyer, F., & Meyer-Hofmeister, E. 2001, in ASP Conf. Ser. 229, Evolution ZelÏdovich, Y. B., & Raizer, Y. B. 1986, Physics of Shock Waves and of Binary and Multiple Stars, ed. Ph. Podsiadlowski, S. Rappaport, A. R. High-Temperature Hydrodynamic Phenomena, ed. W. D. Hayes & R. F. King, F. DÏAntona, & L. Burderi (San Francisco: ASP), 167 Probstein (London: Acad. Press)