ACCRETIONAL HEATING BY PERIODIC DWARF NOVA EVENTS

Transcription

1 THE ASTROPHYSICAL JOURNAL, 566:1084È1090, 2002 February 20 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. ACCRETIONAL HEATING BY PERIODIC DWARF NOVA EVENTS PATRICK GODON1 AND EDWARD M. SION Astronomy and Astrophysics, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085; patrick.godon=villanova.edu, edward.sion=villanova.edu Received 2001 August 1; accepted 2001 October 18 ABSTRACT We have carried out the Ðrst quasi-static evolutionary model sequence with time-variable accretion in which the combined heating e ect of boundary layer irradiation (including stellar rotation) and compressional heating on the underlying white dwarf accretor has been included. In the present study, we focus mainly on the behavior of the surface observables of the accreting white dwarf. This initial exploration followed the thermal evolution of a 0.6 M white dwarf in a dwarf nova over many dwarf nova accretion cycles. Accretion rates of B10~8 M _ yr~1, for outburst duration of days to weeks, are followed by a shuto of the radial infall during dwarf _ nova quiescence. The matter is assumed to accrete softly ÏÏ with the same entropy as the white dwarf outer layers, but a fraction of the energy liberated in the boundary layer is assumed to be absorbed by the outer layer of the star (boundary layer irradiation). Accretion is resumed and shut o repeatedly at intervals of months to simulate the thermal evolution of the white dwarf in typical dwarf novae. The timescale of the white dwarf cooling is such that after a complete (outburst ] quiescence) cycle, the surface of the white dwarf has not yet completely cooled down: the star has not yet readjusted and does not reach thermal equilibrium. When the evolution of the white dwarf is followed for 125 cycles (about 8 yr), the e ective temperature (T ) of the underlying eff white dwarf increases as a function of time (t) as log(t ) \ a log(t), where a [ 0. We Ðnd that a itself eff decreases with increasing temperature. The inclusion of the boundary layer irradiation has no detectable e ect on this particular result over the timescale studied here. When the boundary layer irradiation is taken into account, the e ective surface temperature of the accreting white dwarf can increase by more than B20,000 K (in addition to the increase due to compressional heating) during outburst. During quiescence the outermost layer of the star quickly radiates away the energy absorbed during outburst. The overall fraction of the outer layer of the star thermally a ected by the compressional heating is of the order of the mass accreted, namely, B10~8 M during a complete run. These results indicate that after _ a long evolution time of many accretion cycles, the e ective surface temperature of the white dwarf will increase substantially. We discuss the application of the sequence to HST studies of the observed heating of white dwarfs in dwarf novae. We also examine the e ect of compressional heating on the outer envelope structure, which directly demonstrates that there is a signiðcant heating e ect. We suggest that the envelope thermal structure resulting from compression and irradiation should be a crucial component in understanding the envelope structure of a prenova white dwarf. Subject headings: binaries: close È novae, cataclysmic variables È white dwarfs 1. INTRODUCTION White dwarfs in cataclysmic variables (CVs) accrete hydrogen-rich matter from a main-sequenceèlike star Ðlling its Roche lobe. The transferred gas with angular momentum forms an accretion disk, enshrouding the white dwarf. As a result of this disk accretion, the white dwarf in a CV experiences ongoing accretion at a low rate, interrupted every few weeks to months by intense accretion of days to weeks (dwarf nova accretion event, or nova-like high state), punctuated every few thousand years or more by a thermonuclear runaway (TNR) explosion (the classical nova). Observations of dwarf novae reveal far-ultraviolet Ñux declines during quiescence, which could be accounted for by cooling of the white dwarf in response to the heating it underwent in consequence of the dwarf nova outburst (Mateo & Szkody 1984; Kiplinger, Sion, & Szkody 1991; Pringle 1988; Long et al. 1993; Sion 1993). The two most important sources of heating of an accreting white dwarf during accretion of gas from a disk (or 1 Also at the Space Telescope Science Institute, Baltimore, MD magnetically controlled accretion column) are boundary layer (or column) irradiation and compressional heating. The former source is due to radiation directed downward during disk (or column) accretion, while the latter source is due to compression caused by the weight of the accreted mass as the white dwarf readjusts. These sources of heat modify and control the rate of cooling of the underlying white dwarf between classical nova outbursts, the temporal evolution of its surface observables (e ective surface temperature, luminosity, and radius), the envelope thermal structure, and hence the timescale to thermonuclear runaway and the nova outburst. These heating processes also bear directly upon the still-unanswered questions of how much disk mass is actually accreted during a dwarf nova outburst and what fraction of the total accretion energy goes into the kinetic energy needed to drive wind outñow during a dwarf nova outburst or prolonged states of high accretion (as in nova-like variables). Because of the compression exerted by the accretion of matter and the existence at the white dwarf surface of a nearly continuous heat source shining inward, the cooling rate and envelope thermal structure of a CV white dwarf should di er in the

2 ACCRETIONAL HEATING 1085 long term from those of a single nonaccreting degenerate. However, the actual di erence in the cooling rate and envelope thermal structure of a prenova white dwarf, or what values the surface variables have for any given M0 and M, has never been quantiðed. Up to now, there have been no studies of the heating of accreting white dwarfs under conditions of variable accretion rate. All studies have assumed that the rate of accretion is constant (Giannone & Weigert 1967; Shaviv & StarrÐeld 1987; Regev & Shara 1989; Prialnik & Shara 1986; Sparks & Kutter 1987; Prialnik & Kovetz 1995; Pringle 1988). In all but the last study (Pringle 1988), the e ect on the timescale to, and the violence of, a thermonuclear runaway were the objectives, with no attention given to what is observed at the surface of the accretor. The study by Pringle was semianalytic, with the goal of determining whether the irradiation of a white dwarf by the hot boundary layer during a dwarf nova outburst (and subsequent cooling) could explain the far-uv Ñux declines observed during dwarf nova quiescence. Recent work by Townsley & Bildsten (2002, in preparation) and by Bildsten (2002, in preparation) treat the problem of the e ect of long-term accretion semianalytically and explore the heating due to compression and decompression in the interval preceding and just following a nova. These studies, however, did not utilize full quasi-static evolutionary models with time-variable accretion. Their objective was to determine the heating e ect of thousands of classical nova explosions. The e ect of compressional heating on the observed T eff of CV white dwarfs and their cooling response to compressional heating by the added disk matter were explored by Sion (1995). His results were compared with the results of irradiation obtained by Pringle (1988), but the two e ects were not incorporated together. Sion (1995) examined compressional heating in the light of time-variable accretion but omitted boundary layer irradiation, and OPAL opacities were not incorporated. In the present simulations, we include for the Ðrst time the e ect of both sources of heating and of the stellar rotational velocity during the evolution of an accreting white dwarf with time-variable accretion and OPAL opacities. On the timescale studied here (3000 days, or about 8 yr), the overall e ect of the boundary layer irradiation on the outer envelope of the white dwarf is negligible, because during quiescence the star quickly radiates away the energy absorbed during outburst. Compressional heating is the main cause for a steady increase in the temperature of the outer stellar envelope. Details of the numerical modeling and the physical assumptions are given in the next section. The results are presented and discussed in TIME-VARIABLE ACCRETION EVOLUTIONARY CODE AND INPUT PHYSICS The long timescales we explore require the use of a onedimensional quasi-static evolutionary code without hydrodynamics. It is uncertain what the timescale is for spreading of the accreted material and heat over the surface of the white dwarf. However, Pringle (1988) argues that it should simply be a dynamical time. Some explicit two-dimensional hydrodynamic studies (Kley 1989) suggest that the assumption of spherical symmetry of the accretion process may be a reasonable approximation, if the material spreads rapidly with latitude. Initial exploration by Godon & Sion (2002, in preparation) conðrm this to be the case. Hence, the assumption of spherical accretion is a good approximation, and our one-dimensional quasi-static treatment is adequate for the heating and cooling timescales we are studying. Moreover, the timescales are prohibitively long to explore with multidimensional hydrodynamic codes Numerical Modeling We use an improved version of the quasi-static stellar evolution code of Sion (1995). The code includes timevariable accretion, OPAL opacities, and boundary layer irradiation, taking into account the stellar rotational velocity. The code also incorporates gravitational di usion in a manner identical to that of Iben, Fujimoto, & MacDonald (1992). All other details of the code can be found in Sion (1995) and references therein. Accretion rates of B10~8 M yr~1, for outburst duration of days to weeks, are followed _ by a shuto of the radial infall during dwarf nova quiescence. The matter is assumed to accrete softly ÏÏ with the same entropy as the white dwarf outer layers, but a fraction of the energy liberated in the boundary layer (L ) is assumed to be absorbed by the outer layer of the BL star (boundary layer irradiation; it is included as a source in the energy equation). And the energy advected by the Ñow into the star is negligible (see 2.2). In cyclic dwarf nova outburst and quiescence intervals, the white dwarf undergoes secularly increasing heating. Numerical simulations are carried out by switching on accretion for the duration of a dwarf nova outburst interval and then shutting o accretion and following the cooling of the white dwarf until the onset of the next outburst. In this way the e ects of compressional heating and irradiation can be assessed quantitatively. The simulations assume uniform heating of the white dwarf, rather than an equatorial accretion belt (corresponding to tangential accretion with angular momentum). Thus, the transfer of angular momentum (by shear mixing) into the white dwarf is neglected Heating of the W hite Dwarf Surface In this subsection we discuss the heating of the white dwarf surface due to accretion. The irradiation of the boundary layer is included as a source in the energy equation. However, we neglect the advection of energy by assuming that the accreted material added to the outer envelope of the star has the same temperature as the outer stellar envelope. This assumption is justiðed when the mass accretion rate is lower than the Eddington limit, which is the case in the present work. Compressional heating is considered in detail in 3. Boundary L ayer Irradiation.ÈThe energy liberated in the boundary layer (the so-called boundary layer luminosity) is given by L \ 1 BL 2 accc L ) D2 1 [ * (1) ) (R ) K * (Kluz niak 1987), where L acc \ GM * M0 /R * (2) is the total accretion energy, G is the gravitational constant, M is the mass of the star, R is the radius of the star, M0 is the * mass accretion rate, ) * is the angular rotation rate of the star, and ) (R ) is the Keplerian * angular velocity at one K * stellar radius.

3 1086 GODON & SION Vol. 566 In the present work we assume that only a fraction of the boundary layer luminosity is irradiating the star, namely, L \ a(l /2), (3) I BL and we choose a \ 0.5, which is twice as large as the value used in the work of Shaviv & StarrÐeld (1987) and about half the estimated value of Regev & Shara (1989), who obtained B0.2L. We then assume di erent rotation rates acc g \ ) /[) (R )], (4) * K * ranging from g \ 1 (for no boundary layer irradiation) to g \ 0.3. Advection of Energy.ÈThe fraction of energy advected into the star is expected to be very small (Godon 1997), namely, f \ L adv B 2.5 AH B2 * (5) L R acc * (Popham 1997), where H \ ) /c is the half-thickness of the disk in the vertical (z-) dimension, K s c is the sound speed in the disk, and the asterisk refers s to the value in the inner disk where r \ R. In cataclysmic variables one has * H/r B 0.05 and thus fb0.006, and it is the reason why we neglect the energy advected by the disk into the outer stellar envelope. 3. RESULTS AND DISCUSSIONS We follow the thermal evolution of a 0.6 M white dwarf _ over many dwarf nova accretion cycles (outburst ] quiescence), for a total period of about 8 yr (3000 days). All the details of the model runs are given in Table 1. The initial surface temperature is varied from 10,000 K up to 50,000 K. In some models, boundary layer irradiation is not included, i.e., ) \ ), while in other models the irradiation due to the boundary * K layer is included as described by equation (1), assuming ) /) \ 1. Two kinds of outburst cycles are considered: short-outburst, * K lasting 3 days and followed by 21 days of quiescence, and long-outburst, lasting 7 days and followed by 90 days of quiescence. These are typical of CV systems below and above the binary period gap, respectively. Not all initial values of the surface parameters (T, M, and R for a given M0 ) are numerically stable. First eff T and M were chosen, then di erent values of R were tried. eff Then models were run for the values of R leading to the most stable simulations. This explains the di erence in radii found for di erent T in Table 1. The e ect of the radius on the results is discussed eff in detail in this section. In the present work, we are keeping M0 and M constant, namely, M \ 0.6 M and M0 \ 10~8 M yr~1 during outburst (M0 \ 0 during _ quiescence). The e ect _ of M and M0 will be studied in a forthcoming paper. A typical run is shown for model 1 in Figure 1. In this speciðc model the outburst phase lasts three days and it is followed by a quiescence phase of 21 days. For clarity, no boundary layer irradiation is included in that particular model. As accretion processes onto the star, the outer stellar layer of the star is compressed by the weight of the freshly accreted matter. In response, the temperature of the outer layer increases during accretion. As accretion ceases and the TABLE 1 WHITE DWARF MODELS T a Cycleb T max log(r ) eff eff Model (K) ) /) (days) a ac (K) (cm) * K , (3, 21) , l... 10, (7, 90) , , (3, 21) , h... 14, (3, 21) , l... 14, (7, 90) , lh... 14, (7, 90) , , (3, 21) , h... 21, (3, 21) , l... 21, (7, 90) , lh... 21, (7, 90) , , (3, 21) , h... 29, (3, 21) , l... 29, (7, 90) , lh... 29, (7, 90) , , (3, 21) , h... 39, (3, 21) , l... 39, (7, 90) , lh... 39, (7, 90) , , (3, 21) , h... 42, (3, 21) , l... 42, (7, 90) , lh... 42, (7, 90) , , (3, 21) , h... 51, (3, 21) , l... 51, (7, 90) , lh... 51, (7, 90) , a E ective temperature is the initial temperature of the model. b The duration of the outburst (q ) and the duration of the quiescence (q ) phases are given as (q, q ). c Parameter a has been normalized o ÏÏ to ar, in order to account for q the change in the accretional o q energy as the radius varies from model to model.

4 No. 2, 2002 ACCRETIONAL HEATING 1087 FIG. 1.ÈLog of the e ective surface temperature (in K) as a function of the log of the time (in days), for a T \ 10,000 K accreting white dwarf eff (model 1). The relation log(t ) \ a log(t) for the surface of the white dwarf is apparent in this model. No eff boundary layer irradiation is included. Every few cycles, the next outburst starts randomly,ïï i.e., after less than 21 days of quiescence. This has no e ect on the results, but it is the cause for the high-temperature peaks observed in the Ðgure (when two outbursts follow each other on a short timescale). outburst stage is followed by a quiescence stage, the outer stellar envelope cools down to adjust itself to a new hydrostatic equilibrium. However, for the cycles of interest studied here, the timescale of the white dwarf cooling is such that after a complete (outburst ] quiescence) cycle, the surface of the white dwarf is still cooling down: its temperature does not reach a plateau and is still decreasing just before the next outburst starts. The radius and the luminosity of the white dwarf are also still decreasing when the next outburst occurs. The star has not yet readjusted completely. This is shown clearly in Figure 2, where the surface parameters (T, R, and L ) are shown for one outburst eff only. By t \ 24 days (3 ] 21), the surface parameters are still decreasing and the star has not yet completely readjusted. It seems that, even if the star returns to hydrostatic equilibrium after outburst, its radius and temperature remain inñated, because the outer layer of the white dwarf is badly out of thermal equilibrium. This accounts for the compressional heating observed in the present work. In model 1, the temperature di erence between the outburst phase and the quiescence phase is about 2000 K, while the e ective surface temperature increases from its initial value of 10,000 K up to 16,700 K at the end of the run (after 8.21 yr, or 125 cycles). On a timescale of many cycles, the e ective surface temperature (T ) of the underlying white dwarf increases for all models as eff a function of time (t) as log(t ) \ a log(t), where a [ 0 (or equivalently T P ta; see Fig. eff 1). We Ðnd that a itself decreases with increasing eff initial temperature, namely, from ab0.1 at T \10,000 K to a B at T \ 50,000 K (see Table eff 1). However, model 2 (for example), eff which has an initial e ective surface temperature (at t \ 0) of almost 15,000 K, cannot be Ðtted on the graph to overlap model 1 at 15,000 K, because in model 1 the star is still readjusting, while model 2 is in an initial (thermal and hydrostatic) equilibrium state at that same temperature. Therefore, Table 1 cannot be used to create an evolutionary sequence of overlapping models; rather, for each model one can extrapolate to obtain T at any time log(t ) \ log(t ) ] a log (t), where T \ T (t \ 1) and t is given in days for a i particular model of Table i 1. When the boundary layer irradiation is taken into account during outburst, the e ective surface temperature of the accreting white dwarf increases, in addition to the increase due to compressional heating. During quiescence the outermost layer of the star quickly radiates away the energy absorbed during outburst, pointing to the fact that the inclusion of the boundary layer irradiation has no detectable e ect on the temperature of the underlying white dwarf (Fig. 3). Low-temperature models (1È3) were very sensitive (numerically speaking) to the inclusion of the boundary layer irradiation. The models were run for higher values of the stellar rotation rate (B0.6È1). The other models were run for lower values (down to B0.3). The inclusion of the boundary layer irradiation produced very sharp temperature peaks during outburst (Fig. 3; see also Table 1, where some models reach T B 70,000 K). Longer outbursts (7 days) eff are followed by a longer quiescence phase (90 days), and the average mass deposited is only B0.58 the mass accreted for the short-outburst case. Consequently, the compressional heating of the underlying white dwarf is less pronounced and is characterized by a smaller value of the parameter a (Table 1, except for model 1). On the other side, the longer outburst models have a larger outburst temperature increase, which compensates for the lower compressional heating. The net e ect is that about the same peak temperature is reached for the two di erent outburst cycles. Because the accretion energy is inversely proportional to the radius of the accreting star, one expects the accretional heating and the boundary layer irradiation to increase with decreasing radius. This is clearly seen in Table 1, where the highest temperatures are reached in models from series 5 and 7 with the smallest radius. The parameter a is also relatively larger for those models. As a Ðrst-order approximation one can assume a P 1/R, and one can normalize ÏÏ a by multiplying it by R. This is done in the sixth column of Table 1. It is interesting to note that the new value of a is now steadily increasing with decreasing initial surface temperature. The overall fraction of the outer layer of the star thermally a ected by the compressional heating is of the order of the mass accreted, namely B10~8 M during a complete run (8 yr). For a T \ 10,000 K accreting _ white dwarf, a 15% increase in T eff is observed in the B10~8 M outer stellar envelope. The increase is only 5% for T \ 20,000 _ K and is B1% at T B 50,000 K (Fig. 4). eff In comparison eff with the results of Sion (1995), we Ðnd that the inclusion of OPAL opacities has no detectable e ects, underscoring that previously obtained results that did not include OPAL opacities are actually valid. The main di erence with Sion (1995) is the e ect of the boundary layer irradiation, which heats up the surface of the star to a very high temperature during outburst. In the present work, we did not check the e ect of the mass accretion rate M0 and the mass of the star M on accretional heating, as was done by Sion (1995), but we did carry out many more model simulations. We are planning to carry out more numerical simulations to assess quantitatively the e ect of M0 and M on accretional heating. Because the accretion energy increases with increasing M0 and M, we expect compressional

5 1088 GODON & SION FIG.2a FIG.2b FIG.2c FIG. 2.È(a) Log of the e ective surface temperature (in K) as a function of time (in days) for model 1. At t \ 24 days the temperature is still decreasing [log(t ) \ 4.04], and it drops to log(t ) \ 4.01 at t \ 100 days (a decrease of about 750 K). (b) Same as a, but for luminosity (in solar units). (c) Same as a, but for log of the radius (in cm). heating and boundary layer irradiation to be more pronounced for higher M0 and M. From our preliminary results, we anticipate that the phase of high mass accretion, with the accretional heating included, could (on a long timescale of thousands of years) produce a temperature inversion in the envelope (Shaviv & StarrÐeld 1987), though this was not observed in the present work. This increase in temperature might a ect the TNR when it is about to occur: the timescale to TNR and also the total amount of mass accreted before the runaway occurs might both be reduced. It will be a less degenerate runaway, as compared with a model where the compressional heating is ignored or underestimated. Our present results further indicate that after a long evolution time of many accretion cycles, the e ective surface temperature of the white dwarf will increase substantially, and this could therefore explain the upper end of the distribution of surface temperature of white dwarfs in dwarf novae, where T B 30,000È45,000 K (see Sion 1999 and references therein). eff We are grateful to Jim MacDonald for his invaluable help in updating the evolution code of EMS. This research was carried out with the support of NASA through ADP grant NAG , HST grant GO-8139, and NSF grant AST , all to Villanova University.

6 FIG.3a FIG.3b FIG. 3.È(a) Log of the e ective surface temperature as a function of time [in log (days)] for an initial T \ 42,500 K accreting white dwarf (model 6). The irradiation of the boundary layer is not included. (b): Same as a, but for model 6h. Here the irradiation of eff the boundary layer has been included, assuming g \ ) /) \ 0.3. The peak temperatures during outbursts are of larger amplitude than those in a, but during quiescence the temperature proðle is the same. * K FIG.4a FIG.4b FIG. 4.È(a) Log of the temperature as a function of the logarithmic mass fraction for a B20,000 K accreting white dwarf (model 3). Solid line: initial temperature proðle; dashed line: temperature proðle at the end of the run, after 3000 days. The mass fraction a ected by accretional heating is roughly equal to the mass accreted by the star during the evolution of the model (B10~8 M ). (b) Same as a, but deeper in the outer stellar shell, for a mass fraction larger than B10~8M. The e ect of accretional heating is less pronounced.

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