Evolution of close binary stars with application to cataclysmic variables and Blue Stragglers.

Transcription

1 Evolution of close binary stars with application to cataclysmic variables and Blue Stragglers. DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Nikolay Andronov ***** The Ohio State University 2005 Dissertation Committee: Approved by Professor Marc Pinsonneault Professor Andrew Gould Professor Donald Terndrup Adviser Astronomy Graduate Program

2 ABSTRACT This work is dedicated to the study of the evolution of CVs and mergers of close binaries. In contrast to previous studies, this work uses modern empirical formulae for angular momentum loss by magnetized stellar winds. Two important differences with previous prescriptions include the saturation of angular momentum loss rate for fast rotators, and the existence of angular momentum loss in low mass, fully convective stars. Changes in the treatment of angular momentum loss rate have a dramatic effect on the understanding of the evolution of CVs. The timescale for angular momentum loss ( J) above the fully convective boundary is 2 orders of magnitude longer than inferred from the older model, and the observed angular momentum loss properties show no evidence for a change in a behavior at the fully convective boundary. This provides evidence against the hypothesis that the period gap is caused by an abrupt change in the angular momentum loss law when secondary becomes fully convective and implies that the timescale for CV evolution is much longer than was thought, comparable to a Hubble time. It is demonstrated that when evolved secondaries are included, a spread in the secondary mass-orbital period plane comparable to ii

3 that seen in the data is produced for either the saturated prescription for magnetic braking or the unsaturated model commonly used for CVs. It is argued that in order to explain this spread a considerable fraction of all CVs should have evolved stars as the secondaries. The predictions of my models are compared with diagnostics of the mass accretion rate in CVs and it is found that results are intermediate between the saturated and the older braking prescription. Taken together these suggest that either the angular momentum loss rate may be higher in CV secondaries than in single stars of the same rotation period, or that accretion happens in duty cycles. Possible origins of the Period Gap are discussed. It is demonstrated that main-sequence mergers can account for the observed number of single blue stragglers in M67. Applied to the blue straggler population as a whole, this implies that such mergers are responsible for about one quarter of the population of halo blue metal poor stars, and at least one third of the blue stragglers in open clusters for systems older than 1 Gyr. The observed trends as a function of age are consistent with a saturated angular momentum loss rate for rapidly rotating tidally synchronized systems. The predicted number of blue stragglers from main sequence mergers alone is comparable to the number observed in globular clusters. A population of subturnoff mergers of order 3-4% of the upper main sequence population is also predicted for stars older than 4 Gyr, which is roughly comparable to the small population of highly Li-depleted halo dwarfs. iii

4 Dedicated to all the little people... iv

5 ACKNOWLEDGMENTS First of all I wish to thank my adviser, Marc Pinsonneault, without who this work would not exist or would exist with completely different name on its title. I also want to express gratitude to people with who I worked on different projects during my stay here, in particular Andrew Gould and Donald Terndrup. I am grateful to teachers from who I learned during my study at OSU, especially Barbara Ryden and David Weinberg whose courses will probably remain in the my top-10 list of classes I ever took. I am in debt to the Department of Astronomy for friendly environment, hospitality and nice support during all these years. And finally, I want to thank Christopher Morgan, Joshua Pepper and Jerry Jaiyul as my regular companions in trips to gym. These work-outs considerably improved my physical fitness, which is a good non-documentary proof of my study at OSU (the origin of Buckeyes). v

6 VITA December 10, Born Ukhta, Russia Moscow Institute of Physics and Technology Graduate Teaching and Research Associate, The Ohio State University PUBLICATIONS Research Publications N. Andronov, and M. H. Pinsonneault Stellar Models Of Evolved Secondaries in CVs., ApJ, 614, 326, (2004) N. Andronov, M. H. Pinsonneault, and A. Sills, Cataclysmic Variables: An Empirical Angular Momentum Loss Prescription from Open Cluster Data, ApJ, 582, 358, (2003) A. Gould, and N. Andronov, ApJ, 516, 236, (1999) Complete Parallax and Proper-Motion Solutions for Halo Binary-Lens Microlensing Events., ApJ, 516, 236, (1999) FIELDS OF STUDY Major Field: Astronomy vi

7 Table of Contents Abstract Dedication Acknowledgments ii iv v Vita vi List of Tables List of Figures ix x Chapter 1 Introduction Binary stars Chapter 2 Magnetic braking Background Empirical angular momentum loss rate Tests of the saturated braking model Is magnetic braking different in single stars and stars in close binaries? 21 Chapter 3 Evolution of Cataclysmic Variables Background General Origin of CVs vii

8 3.1.3 Problems Frozen model of CVs Stellar model Angular momentum Angular momentum loss Mass - radius relation for the secondary Model and limitations Marginal contact Results Summary Full stellar models of evolved secondaries in CVs Model Results Summary Effects which are not included in the recent generation of models Starspots Tidal mixing The origin of the period gap Chapter 4 Mergers of close primordial binaries Overview Background on the population of blue stragglers Stragglers in different environments Prior theory viii

9 4.3 The model Properties of stellar population Orbital decay of close binaries Mergers of contact systems Number of mergers Observational data Open clusters Globular clusters and Galactic Halo Subturnoff Mergers Results Model Properties and Parameter Variations Comparison of model predictions with data Summary and discussion Bibliography 181 ix

10 List of Tables 3.1 Model properties as a function of mass Parameters of model Properties of BMP stars x

11 List of Figures 2.1 Theoretical models for the upper envelope of rotation Surface velocity of a star with solar mass and composition as a function of age Orbital period distribution of CVs Angular momentum loss for a system The mass of the secondary - period relation Comparison of timescales Comparison of the time averaged mass accretion rate Comparison of the maximum possible accretion disk luminosity Orbital period as function of time Mass accretion rate as a function of period for the unsaturated prescription Boundaries of the period gap in the disrupted magnetic braking model The mass-period relation for models with saturated angular momentum loss law The mass-period relation for models with unsaturated angular momentum loss law Mass accretion rate as a function of period Relative change in radius and luminosity of a star with effective spot coverage 50% as a function of mass xi

12 3.14 Timescales as a function of orbital period Initial mass function of stars Orbital period as a function of age Phase space of binaries that produce mergers BSs in a CMD of three open clusters Theoretical selection of mergers Distribution of hydrogen abundance in mergers The predicted number of single BSs. Mixed vs unmixed, evolved vs unevolved products The predicted number of single BSs for different assumed bianary fraction and tidal locking The predicted number of single BSs for different assumed IMF The predicted number of single BSs for different assumed magnetic braking prescriptions The predicted number of single BSs. Standart model and theoretical 2σ deviations The predicted number of single blue stragglers normalized to horisontal branch stars The predicted fraction of sub turn-off mergers The relative number of single BSs to BSs found in binaries versus fraction of BSs in the BMP population xii

13 Chapter 1 Introduction About half of all stars are found in binaries, and about 30% of binary stars will interact at some point in their lifetime. Interacting binaries can therefore represent a significant fraction of all stars, and the consequences of interactions can be important for understanding stellar populations. In fact, studies of star clusters often reveal examples of stellar anomalies not explainable by single-star evolution, but which can be attributed to interacting binaries (Bailyn 1995). This work is dedicated to the study of two particular types of objects; Cataclysmic Variables and Blue Stragglers. Cataclysmic Variable stars (CVs) are mass-transferring binary systems with orbital periods between 1.3 and 10 hours (see Patterson 1984; Warner 1995 for reviews). The primary in a CV is a white dwarf, and the secondary is a low mass main sequence star (see Smith & Dhillon 1999) which is overfilling its Roche lobe and transferring mass onto the primary. As matter is accreted onto a primary, it forms an accretion disk, which has the luminosity in most cases exceeding the total luminocity of both components. In 1

14 the optical, CVs exibit variations with a range of different timescales. Some of this flickering is atributed to instabilities in the accretion disk (dwarf nova eruptions), the other are thought to be thermonuclear explosions of the accreted matter on the surface of the white dwarf (classical nova explosions). The picture of CV is therefore complicated; besides the quite poorly understood stellar physics of the secondary in the presence of a close companion, one must also include the barely better understood physics of accretion (again in the presence of a close companion), and the physics of thermonuclear processes that drive chemical evolution of media surrounding CVs. While CVs are not extremely luminous (their luminosity is comparable to solar), they are probably the most abundant accreting systems. It has been estimated that our galaxy may harbor about thousand CVs (Patterson 1998,for example). The study of CVs, therefore, may provide important constraints on areas as diverse as stellar astrophysics, accretion physics and models of galactic chemical evolution. The possible connection of CVs with type Ia supernova makes their study even more potentially important. Binary interactions or mergers may also be responsible for the existence of blue stragglers (BSs). They are stars that appear on the color-magnitude diagram of a cluster on a natural extension of the main sequence above the turn-off point. The existence of BSs was discovered by Sandage (1953) in the globular cluster M3. Over the years blue stragglers have been identified in many globular and open clusters, as reviewed by Stryker (1993), and they are numerous in some systems; the number of 2

15 BSs found in individual globular clusters ranges from 10 to 400 (Bailyn 1995; Davies, Piotto, & de Angeli 2004). The significant difference between BSs in globular clusters and in open clusters (or in the galactic halo) is that the production of BSs in the former could be significantly affected by the environment, while the production of BSs in the latter is intrinsically connected to properties of primordial binaries and the physics of the merger process alone. The study of BSs is useful because it can provide additional clues about the evolution of stars and stellar populations and may give important insights into a cluster of related fields in stellar astrophysics. Are star clusters valid templates for stellar population studies, or do dynamical effects significantly modify their global properties relative to field stars? How important are blue stragglers for the integrated light of stellar populations? In addition, the production rate for stellar mergers can potentially be used as a test of the angular momentum loss rate for tidally synchronized binaries. The study of close binary stars requires a lot of ingredients, many of which can be obtained from well-established studies in different fields. The application of well-founded theory or empirical data is, therefore, a necessity in such types of research. My work adopts such an approach. In particular, I use a state-of-the art stellar evolutionary code and recent empirical knowledge on distribution of properties of close binaries and angular momentum loss exhibited by main sequence stars. These angular momentum loss rates were inferred from studies of stellar rotations in 3

16 young open clusters. I will show that the angular momentum prescriptions used in earlier studies are inconsistent with the more recent measured rotation data in open clusters. The timescale for angular momentum loss ( J) above the fully convective boundary is 2 orders of magnitude longer than inferred from the older model, and the observed angular momentum loss properties show no evidence for a change in a behavior at the fully convective boundary. As will be demonstrated in the next chapters, this leads to dramatic change in our understanding of evolution of CVs and provides reasonable estimates of the production of BSs in low density populations of different age. This work is organized as follows; section 1.1 gives a short overview of binary interactions. Chapter 2 describes angular momentum loss rate due to magnetized winds, as one of the most important ingredients in the modeling of close binary stars. The 3rd chapter provides to description of our model the evolution of Cataclysmic Variables. The 4th chapter describes population synthesis of MS mergers of close binaries Binary stars Nature provides a variety of ways that binary stars can interact, with many possibilities depending upon the relative masses, evolutionary states of the components and their separation (see Paczynski 1971; Iben & Livio 1993). Binaries 4

17 with a wide range of orbital periods can either merge or experience mass transfer. Such interaction would produce a variety of unusual stellar objects (such as common envelope (CE) systems, Cataclysmic Variables (CVs), X-ray binaries, binary pulsars, Blue Stragglers (BSs) and possibly population of Blue Metal Poor (BMP) stars in the galactic halo. (Bailyn 1995). Furthermore, these interactions are not uncommon. The fraction of binaries in young open clusters with periods shorter than 5 days is 11% (Duquennoy & Mayor 1991). Such binaries may interact while both stars are on the main sequence. More widely separated binaries can interact when the primary leaves the MS and becomes large. The fraction of binaries that may experience interaction on the RGB of the primary (with assumed limiting separation of 0.5 AU) is 8.4% given the same initial distribution. Binaries with an initial separation between about 0.5 AU and 3 AU will interact when the primary is on AGB; about 11% of binaries fall into this category. In the next paragraphs we discuss each of these categories in turn. Binaries with a sufficiently small initial orbital period (below approximately 5 days) and sufficiently low masses (at least one of the components smaller than 1.2) can interact on the main sequence, and in this case complete mergers are more likely. Binary orbits decay from gravitational radiation. In addition, angular momentum loss from the magnetized winds of late-type stars (e.g., Weber & Davis 1967) results in a slowing of stellar rotation rates with time (Skumanich 1972). In tidally synchronized systems of short initial period, such angular momentum loss 5

18 would lead to a reduction of the size of the orbit and produce contact systems and eventual mergers. Although short-period binaries are rare in the field, they are substantially more common in young open clusters, which supports the idea that such systems eventually merge. Many of these mergers will manifets themselves as Blue Stragglers (stars that appear on the color-magnitude diagram of a cluster on the natural extension of main sequence above the turn-off). Not all stellar mergers or mass transfer events must manifest themselves as blue stragglers. Some merger products will appear below the main sequence turnoff. Such objects would be slightly less evolved than their single counterparts, but might have unusual rotation rates and surface abundances. There is an interesting subpopulation of highly overdepleted stars found in lithium studies in open clusters (e.g., Thorburn et al. 1993) and halo field samples (Thorburn 1994). Lithium can be used as a diagnostic of mixing in stellar interiors, and the primordial lithium abundance is an important test of Big Bang Nucleosynthesis. Because lithium is destroyed at low temperatures in stellar interiors, mergers would be likely to produce low surface lithium abundances. Ryan et al. (2001) have argued that these stars may have experienced mass transfer from a companion, and found that some such stars close to the turnoff rotate significantly more rapidly than their lithium-normal counterparts. If binaries are sufficiently wide (P 5 days) the timescale for orbital decay becomes too large for a main sequence merger. In this case mass transfer can occur 6

19 only when the primary leaves the main sequence. If the separation between stars is relatively small (a 0.5 AU) the primary will overfill its Roche lobe and accretion onto the secondary will occur on the red giant branch. Binaries that are even wider (0.5 AU a 3 AU), can also go through a similar accretion process when the primary is on the AGB (the red giant is not sufficiently large to overfill its Roche Lobe), or they can accrete a small amount of the substantial mass lost in the AGB wind phase. Mass transfer in such wider binaries has been studied by a number of investigators (e.g. de Kool 1990; Iben & Livio 1993). Binaries interact in the post-main sequence phase may form common envelope systems rather than simply transferring mass; furthermore, not all of the mass of the primary star will be donated to the secondary. A common-envelope phase occurs if the accretion is unstable (e.g. the Roche lobe shrinks faster than star loosing mass does, two possible sources of instability are discussed in detail by Hjellming & Webbnik 1987). The outcome of this phase may therefore (see de Kool & Ritter 1993) also be a secondary with a mass higher than its initial value, accompanied by a white dwarf (He white dwarf in the case of RGB accretion and CO white dwarf in the case of AGB accretion). Some of these post common-envelope systems are the precursors of Cataclysmic Variables. Interaction in such binaries can also explain the majority of Blue Metal Poor stars in the galactic halo. The picture of BMP stars provided by Preston & Sneden 7

20 (2000) and Carney et al. (2001) would be consistent with accretion happening on the AGB of the primary. In this case we would expect BMP stars to be accompanied by CO white dwarfs which have mass M. As it seems from low-statistics studies of BMP stars (Preston & Sneden 2000, for example), most BMP stars are found as members of relatively wide binaries, with companion consistent with white dwarf of mass about 0.6M. We, therefore may expect that such systems originate from accretion that happens during the AGB phase of the primary, with the final result being a BSs with a CO white dwarf companion. 8

21 Chapter 2 Magnetic braking Low mass main sequence stars lose angular momentum from a magnetized wind. In the presence of a magnetosphere, charged particles tend to move along magnetic field lines. As a result, the mass lost in stellar winds corotates with the star up to about an Alfven radius, which is much larger than the stellar radius. The net effect is that stars will lose angular momentum and spin down. This angular momentum loss is highly efficient. For example the average surface rotational velocity of stars in α P ersei (age 60 Myr) is about 200 km/s, in the Hyades (age 600 Myr) it is already about 6 km/s, while the surface rotational velocity of the sun (age 4.5 Gyr) is only about 2 km/s. As was pointed in the previous section, the angular momentum loss prescription is an important ingredient in the modeling of evolution of close binary systems. On the other hand, the most precise data related to the angular momentum loss comes probably from study of rotation of young low mass stars. The angular momentum evolution of single MS stars therefore can place strong constraints on theoretical models of close binary stars. The rest of this chapter is dedicated to a summary 9

22 of our empirical knowledge on magnetic braking, while chapters 3 and 4 describe application to CVs and BSs Background There has been significant progress in the modeling of angular momentum loss from a magnetized stellar wind (hereafter the magnetic braking, see Appendix A of Keppens, MacGregor & Charbonneau 1995). In early models of the solar wind (Weber & Davis 1967) solutions of the MHD equations were obtained in the equatorial plane. More recent calculations solve the full MHD equations in three dimensions. One prediction of the early models was that at low rotation rates the mean surface magnetic field strength would scale as the rotation rate ω and the angular momentum loss rate would scale as B 2 ω or ω 3. Such a scaling was consistent with early data on the spindown of stars (Skumanich 1972). However, at high rotation rates the angular momentum loss rate grows at a slower rate even with the same underlying dynamo model (Keppens et al. 1995), and there is the possibility that the strength of the magnetic field would saturate when the surface became filled with spots. Proxies for the magnetic field strength, such as X-ray fluxes and chromospheric activity indicators, exhibit a mass-dependent saturation at high rotation rates (Stauffer et al. 1997). 10

23 Both the morphology of stellar magnetic field and the properties of stellar coronae are difficult to infer and complicate estimates of angular momentum loss. There are also uncertainties related to the nature of the dynamo mechanism itself. The most popular model for the solar cycle is the interface dynamo (Parker 1993; MacGregor & Charbonneau 1997; Charbonneau & MacGregor 1997; Montesinos et al. 2001). In this model the toroidal magnetic fields are generated in a shear layer below a surface convection zone from a poloidal field (the ω component of an α ω dynamo), while the poloidal field is regenerated from the toroidal field in a nearby but different layer (the α component.) Such models are more successful in reproducing the properties of the solar cycle than the classical thin-shell dynamo in which both effects take place in the same region. Montesinos et al. (2001) extended such models to other stars. However, neither a thin shell dynamo nor an interface dynamo would operate in a fully convective star. Instead the generation of magnetic fields would require a distributed or turbulent dynamo that would be less effective (see Lanza et al for a discussion of potential implications of the interface dynamo for CVs). However, it is not clear that the theoretical models have sufficient predictive power to infer the absolute efficiency of different dynamo mechanisms. This induces a further model dependence in the angular momentum loss rates as a function of mass predicted by theory. 11

24 We have therefore chosen to use open cluster stars with a range of mass and age to empirically measure the angular momentum loss rates as a function of mass and rotation rate, as discusses in the next sections Empirical angular momentum loss rate Weber & Davis (1967) predicted an angular momentum loss rate proportional to ω 3 based upon a study of the solar wind; this is consistent with the time dependence of rotation inferred from early studies of solar-type stars in open clusters (Skumanich 1972). The strong dependence of the angular momentum loss rate on the angular velocity results primarily from scaling the mean magnetic field strength to the rotation rate. Rappaport, Verbunt, & Joss (1983) developed an empirical prescription that is commonly used in studies of cataclysmic variables; their relationship is given by: ( ) dj M R 4 mr γ ω 3 dyn cm. (2.1) dt wind where γ is a dimensionless parameter in the range from 0 to 4. There are serious difficulties with applying any angular momentum loss model which scales as ω 3 to observations of young low mass stars. The spindown of rapid rotators is predicted to be extremely fast (see for example Pinsonneault, Kawaler, & Demarque 1990), but rapidly rotating low mass stars are observed in young 12

25 open clusters (Stauffer & Hartmann 1988). Both chromospheric activity indicators and X-ray studies, furthermore, can be used as proxies for measurements of the strength of stellar magnetic fields. There is now extensive empirical evidence that both chromospheric and coronal indicators become independent of the rotation rate above a mass-dependent critical angular velocity (e.g. Patten & Simon 1996); this would imply a much lower angular momentum loss rate for rapid rotators. We note that this does not contradict the overall Weber & Davis (1967) model in the case of slow rotators; however all of the direct and indirect observational tests in low mass stars and open clusters of different ages indicates that it overestimates the angular momentum loss rate for rotation periods shorter than days. A number of different theoretical groups have investigated the spindown of low mass stars (Queloz et al. 1998; Collier-Cameron & Jianke 1994; Keppens, MacGregor, & Charbonneau, 1995; Krishnamurthi et al. 1997; Sills, Pinsonneault, & Terndrup, 2000). In all cases the survival of rapid rotation in young open clusters required a modification of the angular momentum loss law at high rotation rates. We therefore use an angular momentum loss prescription with the same function form as that of Sills, Pinsonneault, & Terndrup (2000): ( ) dj r = K w dt m wind ω 3 for ω ω crit ωω 2 crit for ω > ω crit (2.2) 13

26 Here ω crit is the critical angular frequency at which the angular momentum loss rate enters into the saturated regime. The constant K w g cm s and is calibrated to reproduce the known solar rotation period at the age of the Sun (see Kawaler 1988 for a discussion of the ingredients and uncertainties.) The value of ω crit can be inferred empirically by reproducing the observed spindown of low mass stars as a function of mass and age (the values we use for different masses are presented in the description of stellar model in chapter 3). There is therefore one clear implication from the large database of observational and theoretical work in the study of rotation in low mass open cluster stars: angular momentum loss prescriptions of the form used by Rappaport, Verbunt & Joss (1983) greatly overestimate angular momentum loss rates for secondary stars in the period range of interest for the study of CVs. Solokani, Motamen & Keppens (1997) propose an alternate physical mechanism (concentration of magnetic fields close to the pole for rapid rotators), but their model predicts a very similar angular momentum loss rate to the saturated wind model. The second important result in the context of CV studies is the observed mass dependence of the angular momentum loss rate. It has long been known that the observed saturation threshold for chromospheric and coronal activity indicators is mass dependent (see Noyes et al. 1984; Patten & Simon 1996). Krishnamurthi et al. (1997) found that a scaling of ω crit with the inverse Rossby number (the ratio of the rotation period to the convective overturn timescale) gave a reasonable fit to the 14

27 observed timescale for spindown as a function of mass from solar masses. Similar Rossby number scalings are also found to describe the saturation of activity indicators (Krishnamurthi et al. 1998) and are expected on general theoretical grounds for a shell dynamo (e.g. Durney & Latour 1978). In other words, the timescale for stars to spin down increases as the mass decreases. Observations of both activity indicators and rotation velocities down to very low masses have been obtained in open clusters with a range of ages (Jones, Fischer, & Stauffer 1996; Stauffer et al. 1997; Terndrup et al. 2000; Reid & Mahoney 2000). There is no observational support for an abrupt change in angular momentum loss properties at the point at which stars become fully convective (around 0.3 solar masses.) Sills, Pinsonneault, & Terndrup (2000) found that the mass dependence of ω crit could no longer be fit by a Rossby scaling below 0.5 solar masses; this suggests that the transition from a shell to a distributed dynamo occurs well above the fully convective boundary. However, some angular momentum loss was needed even below the fully convective boundary in order to explain the observed spindown of Hyades stars relative to Pleiades stars. Hawley (1999) and Hawley et al. (1999) also found that the timescale for high activity levels to survive was a smooth function of mass. We compare the predicted stellar rotation rates using the Sills, Pinsonneault, & Terndrup (2000) empirical angular momentum loss law as a function of mass and the Rappaport, Verbunt, & Joss (1983) angular momentum loss law with Pleiades and Hyades data in Figure 2. For these models we chose an initial rotation period 15

28 at the deuterium-burning birthline of 10 days (corresponding to the average rotation period inferred for T Tauri stars from Choi & Herbst 1996). Because of the strong feedback in the unsaturated angular momentum loss law, choosing a much shorter initial rotation period would yield very similar results for the Rappaport et al. loss rates. This choice of initial conditions corresponds to the expected upper envelope of rotation rates that are observed in open clusters. We note that because the most rapid rotators are rare (roughly 3% of the total population) the upper envelope as a function of effective temperature is subject to Poisson noise. However, the existence of stars rotating more rapidly than 10 km/s in the Pleiades is a direct contradiction of the predictions of an unsaturated angular momentum loss law. We do see evidence of a transition away from a pure Rossby scaling of the saturation threshold in the coolest Hyades stars. The upturn in rotation at the low mass end indicates a change in the efficiency of angular momentum loss (a pure Rossby scaling would reduce to the older angular momentum loss rates at this age.) However, this transition occurs at 0.6 solar masses, well above the fully convective boundary. Furthermore, there is clear evidence for spindown even in the lowest mass stars for which we have data. This extends into the very low mass regime from field star data (Hawley et al ) The stellar data becomes sparse below about 0.2 solar masses, and further observational data on very low mass stars would be useful to constrain the empirical angular momentum loss law at the bottom of the main sequence. It is also possible 16

29 that for the faintest stars the timescale for angular momentum loss could become much longer (see Basri & Marcy 1995 for a discussion of this point.) Any such effect, however, occurs at a significantly lower mass than the fully convective boundary and therefore has no direct effect on the existence of the period gap. This change in the angular momentum loss law has important consequences for the predicted time evolution of CVs which we present in the next section. The overall trends of this prescription for magnetic braking are: 1. It saturates at a level that scales inversely with the convective overturn timescale for masses greater than 0.6 M (see Krishnamurthi et al. 1997). This suggests consistency with either an interface or thin shell dynamo. It is possible to construct models of rapid rotators that do not have a saturation threshold for magnetic activity (Solanki, Motamen & Keppens 1997), but the torque is comparable to that obtained with saturated models. 2. It drops at a faster rate for lower mass stars (see Sills, Pinsonneault & Terndrup 2000), but with no abrupt change at the fully convective boundary; this suggests that the transition to a different dynamo mechanism is gradual rather than abrupt Tests of the saturated braking model The application of magnetic braking in the form that I have used has a dramatic impact on the predicted properties of CVs, as we will see in chapter 3. In particular, 17

30 the mass accretion rates that are inferred are about order of magnitude lower than the mass accretion rates that are inferred from observations (Paterson 1984 for example). This stimulated researchers to look into the angular momentum loss rate, to find alternative approaches which would simulteneously satisfy the observed angular momentum loss rates in single stars and the inferred accretion rates in CVs. One such proposal is considered in this section. Historically an angular momentum loss rate was derived using an average rotational velocities (V sin(i)) of stars (or rather distribution of rotations)in populations of different ages. An angular momentum loss rate is assumed to have some functional dependence of mass, radius, effective temperature and rotational rate of a star, which is motivated by theoretical considerations and then calibrated using rotational data. Besides the observational errors, there are two main ingredients that determine the precision with which the angular momentum loss rate is constrained: 1. The assumed initial conditions. 2. Ages of the stellar populations that are used. There are other ways of testing models of magnetized stellar winds. The X-ray luminosity of a star is one of the fingerprints of magnetic activity, and therefore it can be used to constrain the properties of stellar magnetic fields. The observed saturation of L x /L bol at high rotation rates has been used as evidence for a saturation in angular momentum loss at high rotation rates (MacGregor & Brenner 1991). Recently Ivanova & Taam (2003) used this method to demonstrate that an alternate 18

31 functional form for angular momentum loss rate that rises more steeply with increasing rotation rate is consistent with X-ray data. This is a valuable test for the idea that magnetic field can not increase indefinitely with the increased rotation rate. Using the magnetic wind model of Mestel & Spruit (1987), they suggested a braking law in a different form. Based on X-ray luminosity data for fast rotators from Pizzolato et al. (2003), they adopted the following functional dependence; dj dt ω 3 for ω ω crit ω 1.3 ωcrit 1.7 for ω > ω crit (2.3) To test this prescription they compared the prediction of their model to the rotational velocities of stars in the Pleiades, the Hyades and the Sun. Given the assumed values for the initial rotation rate of a solar mass star (close to break-up on ZAMS) and a Pleiades age of 70 Myrs, they claimed that their braking prescription provides a better fit to the data than the prescription of the form (2.2). Such an angular momentum loss formula would predict time averaged mass accretion rates an order of magnitude higher than which we could expect from saturated law, but lower than the unsaturated prescription by a comparable factor. In this section we demonstrate that it is difficult to match the suggested braking law with the value of maximum rotational velocity of solar mass stars in open clusters of different age. We used rotation data for stars in four clusters (instead 19

32 of the two used by Ivanova & Taam 2003). Furthermore, we assumed a much more realistic initial rotation rate, close to the fastest rotating ZAMS stars and the initial angular momentum from protostars as discusses by Tinker et al. (2002). In addition we adopted a more recent age estimate for of the Pleiades of 130 Myrs (Stauffer et al. 1998, Martin et al. 1998). The equatorial rotational velocity of a single star with solar mass and composition as a function of age for three different prescriptions is shown in the figure 6. The initial pre-ms star was assumed to have a rotation period of 3 days, which would produce a ZAMS star close to the fastest observed rotators and which corresponds to the upper envelope of observed pre-ms rotation rates. Starting in the pre-ms, a star is allowed to spin up as it shrinks and spin down as it loses angular momentum. All magnetic braking laws were calibrated to produce the solar rotation rate at the age of the Sun. The data points are maximum observed rotation velocities for solar mass stars in 4 different clusters; the Hyades (age 600 Myrs), the Pleiades (age 130 Myrs), α Persei (age 60 Myrs), and combined data for the young clusters IC2391 and IC2602 (age 30 Myrs). The open data point denote the age of the Pleiades of 70 Myrs, used by Ivanova & Taam (2003), which was used until recently in many spindown studies. However, recent brown dwarf lithium age estimates require an older age of about Myr (Stauffer et al. 1998, Martin et al. 1998). 20

33 While the prescription in form (2.3) works for the stated assumptions of Ivanova & Taam (2003), it can be seen that their suggested braking law provides a worse fit to all of the data. If we decrease the saturation threshold for a solar mass star from 10ω to 6ω, the prescription seems able to reproduce the angular momentum evolution for young clusters. However, it then significantly overestimates the rotation rate for stars at the age of the Hyades. We conclude that it is important to use all available data and proper initial conditions to constrain the spindown properties of fast rotators. However, even if we assume that the functional dependence on ω in the form (2.3) is correct and derive an apropriate saturation threshold, the torque predicted by (2.3) would not be far away from the one predicted by saturated braking in the range of rotations at which CVs exist. It should be noted, that X-ray luminosity of fast rotators provides an independent and valuable measure for the average magnetic fields, and this therefore could be a potentially very promising way to constrain the properties of the dynamo and magnetic braking Is magnetic braking different in single stars and stars in close binaries? One of the assumptions in our approach is that the empirical angular momentum loss rates derived for single stars can be applied to close binaries. We assume that 21

34 the gravitational field of a close companion does not affect the dynamo or the internal structure of the star and that it does not affect the properties of the stellar wind. Stars in close binaries show on average a higher level of magnetic activity than the single stars of the same spectral type (Simon and Fekel 1987; Schrijver and Zwaan 1991) However, this would be expected because of the higher (on average) rotation rates in tidally locked binaries, that complicates the question of whether there is some additional mechanism which enhances the generated magnetic field. Basri (1987) claimed that the differences vanished when this was taken into account. There are theoretical models of enhanced dynamo activity in tidally locked binaries (Zaqarashvili, Javakhishvili & Belvedere 2002 for example); however, there is no strong observational evidence for such effects. We therefore conclude that the question about the applicability of the empirical rules for single stars to the case of close binaries does not have an obvious answer. In particular, the saturation threshold could potentially be affected by the presence of a companion. However, the clear evidence for angular momentum loss in fully convective stars implies that there is no good physical basis for invoking a sharp decrease in magnetic braking as the explanation of the CV period gap. 22

35 100 v sin i (km/s) 10 Pleiades, 100 Myr Teff (K) 100 Hyades, 600 Myr v sin i (km/s) Teff (K) Fig Theoretical models for the upper envelope of rotation compared with data in the Pleiades (top) and Hyades (bottom). The solid line with the open boxes is the empirical model ( M in the 0.1 M increments). The group of lines below are the RVJ models with γ = 0, 1, 2, 3, 4. Pleiades data are taken from Terndrup et al. 2000, Stauffer et al. 1997, and Anderson et al Hyades data are taken from Radick et al. 1987, Terndrup et al. 2001, and Kraft

36 Fig Surface velocity of a star with solar mass and composition as a function of age for 3 different angular momentum loss laws. The models were started at the D-burning birthline with P 0 = 3 days and evolved to the age and equatorial rotation period of the sun. 24

37 Chapter 3 Evolution of Cataclysmic Variables 3.1. Background General As it was explained in chapter 1, Cataclysmic Variable stars (CVs) are mass-transferring binary systems. The evolution of a CV is driven by angular momentum loss; it has long been known that low mass stars lose angular momentum from a magnetic wind (e.g. Weber & Davis 1967). This angular momentum is removed from the orbit in a synchronized binary system, which causes the orbit to decay. Gravitational radiation is a second angular momentum loss mechanism, which is important at short periods. Mass accretion reduces the moment of inertia of the system, and these two effects together will determine the time evolution of CV systems. In general, a given CV will evolve from a higher mass secondary with a longer period to a lower mass secondary with a shorter period. 25

38 Following the work of Basri (1987), the angular momentum loss properties of secondaries in CVs are typically assumed to be the same as those for single stars or detached binary stars. There has been a dramatic increase in the amount and quality of rotation data available for low mass stars in open clusters and the field, from the important early work of Stauffer & Hartmann (1987) to the present (see Stauffer 1997, Krishnamurthi et al. 1997, Reid & Mahoney 2000 for reviews). However, many theoretical studies of CVs use angular momentum loss rates (e.g. Rappaport, Verbunt & Joss 1983) that precede these data. We will show that an empirical angular momentum loss law as a function of rotation rate and mass both poses a challenge to our understanding of some of the important ingredients in the study of CVs and also presents an opportunity to learn about some crucial components by reducing the number of degrees of theoretical freedom Origin of CVs The evolution of a close binary that eventually forms a CV can be described in 4 stages. 1. Main sequence evolution. The more massive (primary) star leaves the main sequence first and expands on the red giant branch. A common envelope (CE) system forms when unstable mass accretion onto the secondary sets in (De Kool 1990; Iben & Livio 1993). 2. CE stage of evolution (e.g. De Kool 1990). This phase is short ( 10 4 years); 26

39 during this stage the secondary spirals in towards the primary and the gravitational potential energy is absorbed by an envelope, which is subsequently ejected. After the envelope is ejected, the system consists of a white dwarf and a main sequence secondary star that does not necessarily overfill its Roche Lobe but might be close to it. The final separation of the main sequence star and the white dwarf should depend on the initial mass ratio of the stars in a binary and their initial separation. However, the current understanding of this phase of evolution is not adequate to predict the outcome of this phase (De Kool 1990). The only conclusion we can make is that at least some systems after this phase are close enough (with orbital periods below 5 days) to eventually form a CV. 3. Post-CE and Pre-CV evolution. During this phase stars in the binary get closer due to angular momentum loss from the system until the secondary overfills its Roche Lobe. 4. CV evolution. The outcome of the first 3 phases defines the starting point of the CV phase. The relevant ingredients are the masses of the white dwarf and the secondary (m wd and m 2 ) and the evolutionary state of the secondary which determines the M R relationship. We characterize the evolutionary state by its central hydrogen abundance (X c ). 27

40 Problems The study of CVs is a rich and complex one. Despite much work, there remain some significant unresolved problems in the field: 1. Classification of CVs. There are several observationally distinct classes of CVs. Nova like systems (NLs) show steady emission. Dwarf Nova are known for outbursts - sudden increases of their visual brightness by 2-5 magnitudes for periods of a few days (see Verbunt 1997, Warner 1995, Patterson 1984). Magnetic CVs (AM Her systems or mcvs) are discovered by their strong X-ray fluxes. VY Scl are the most mysterious variables - they spend most of their time at a relatively bright level, sometimes showing declining level of brightness. Verbunt (1997) and Hellier & Naylor (1998) have argued that they should be categorized either as NLs or DNs. The physical origin of these differences is not understood. 2 The period distribution of CVs and the origin of the period gap. There are only a few CVs observed with periods between 2 and 3 hours, while there are many systems above and below this period range. The lack of CVs in the 2-3 hour period range seems to be statistically significant (see Verbunt 1997, Hellier & Naylor 2000.) In Figure 1 we show the distribution of cataclysmic variables as a function of orbital period; the data are taken from Clemens, Reid & Gizis (1998). A successful theory 28

41 of CVs must explain the shape of this distribution and the origin of the period gap. The origin of the different classes of CVs necessarily involves a detailed consideration of the physics of the accretion disk and its interaction with the stars. This is beyond the scope of the current work. It is, however, possible to study the existence and origin of the period gap without knowledge of the detailed properties of accretion disks. A variety of models have been proposed attempting to explain the lack of systems in the period range between 2 and 3 hours (The Period Gap). Almost all of them are different implementations of the suggestion of Robinson et al. (1981) that some mechanism forces secondaries to suddenly shrink when they reach a mass of about 0.3 M. Due to the correlation between the mass of the secondary and the orbital period, this happens at about at 3 hours for typical white dwarf masses (see section 3.2.7). The mass transfer then stops until the secondary touches its Roche lobe again, reestablishing the contact and accretion. Robinson et al. (1981) noted that this is the characteristic mass at which the secondary becomes fully convective. Theorists have therefore focused on stellar properties that might change at the fully convective boundary. D Antona and Mazzitelli (1982) proposed that sudden mixing of 3 He when the star becomes fully convective could cause star to shrink. Later calculations, however, 29

42 showed that this effect is negligible (McDermott & Taam 1989.) Recent stellar evolution models do not predict any change of mass-radius relation at the point at which the star becomes fully convective (see table 1). Rappaport, Verbunt & Joss (1983) proposed the disrupted angular momentum loss model. The most recent description of this model can be found in Howell, Nelson & Rappaport (2000). The timescale for mass accretion is governed by the timescale for angular momentum loss. The Rappaport et al. (1983) angular momentum loss timescale for a magnetic stellar wind was shorter than the Kelvin-Helmholtz timescale. The secondary star, being out of thermal equilibrium, could then have a greater radius than a normal main sequence star of the same mass. Stellar magnetic fields are generated by a dynamo mechanism, and there are plausible theoretical grounds for believing that stellar magnetic field properties could be different in fully convective stars than in stars with radiative cores. As the total stellar mass decreases, the depth of the outer convection zone increases; at very low masses ( 0.3 solar masses) stellar models are fully convective. The solar dynamo is thought to be anchored at the interface between the radiative interior and the convective envelope; this mechanism will no longer operate in a fully convective star, which requires a distributed dynamo mechanism (Durney & Latour 1978). McDermot & Taam (1989) claimed that the net effect would be a drastic reduction in angular momentum loss when the secondary became fully convective. With a fully convective secondary, the timescale for angular momentum loss increases 30