Does Kozai Resonance Drive CH Cygni?

THE ASTRONOMICAL JOURNAL, 116 : 444È450, 1998 July ( 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A. DOES KOZAI RESONANCE DRIVE CH CYGNI ? SEPPO MIKKOLA1,2 AND KIYOTAKA TANIKAWA2 Received 1997 November 20 ; revised 1998 March 23 ABSTRACT An analysis of the observed radial velocities in the system CH Cygni has been conducted in the framework of three-body dynamics. The observations and the fact that this star began symbiotic-like activity in the beginning of the 1960s can be explained in terms of a model consisting of a close binary of total mass D4 M that is orbited by a normal white dwarf of mass D1 M . The 4 M binary consists of a _ _ _ red giant of mass D3.5 M and a D0.5 M dim star of unknown type, which is probably difficult to _ _ detect observationally. The orbit of this inner binary has high inclination (more than 40¡) with respect to the orbit of the white dwarf. The dynamical phenomenon known as the Kozai resonance then causes large long-period eccentricity variations in the inner binary. When in the high-eccentricity state, the binary expels gas out of the red giant. Part of this ends up on the white dwarf, causing the activity. In our model, the eccentricity of the binary was low for a long time and has been growing steadily during this century. This explains the only recent symbiotic activity. Key words : binaries : symbiotic È celestial mechanics, stellar dynamics È stars : individual (CH Cygni) 1. The complex behavior of the spectrophotometric parameters led at Ðrst to a single-star model of CH Cyg (e.g., Faraggiana & Hack 1971). A binary model was suggested by measuring the radial velocities for both the cool and the hot components (e.g., Yamashita & Maehara 1979 ; Tomov & Luud 1984) and by the eclipse of the hot component by the red giant observed during 1985 MayÈOctober (Miko¡ajewski, Tomov, & Miko¡ajewska 1987). Recently, Hinkle et al. (1993) proposed a triple-star model for CH Cyg on the basis of precise measurements of radial velocities of high-resolution infrared spectra of the cool component. In their model, CH Cyg consists of a long-period (14.5 yr) outer binary comprising a GÈK dwarf and the short-period (756 day) inner binaryÈthe symbiotic pair. Hinkle et al. further suggest that CH Cyg cannot be an eclipsing system. However, Skopal et al. (1996a) noted the possible presence of eclipses in the light curve and suggested a model of CH Cyg in which an eclipsing triple system contains two red giants, one in the outer binary and a second one as the cool component in the symbiotic pair. Based on this model and on the multifrequency observations of CH Cyg during its recent (1992) active phase, Skopal et al. (1996b) further suggested that the outburst arises from accretion of material from the giant component onto a low-mass main-sequence dwarf in the symbiotic binary. In this paper, we consider the possibility of explaining the behavior of CH Cyg in terms of triple-star models. First, we discuss the Skopal-like model in which the hot component is located in the short-period binary, and second, the Hinkle-like model in which the hot component is the single star in the outer long-period binary. Because we are not astrophysicists but dynamicists, we discuss the astrophysics of the system at a rather general level and concentrate on the dynamics of the system. INTRODUCTION A principal problem in interpretation of the behavior of CH Cygni is the appearance of symbiotic activity in 1963 persisting to the present, after a long quiescence since the Ðrst observations made around 1885 (Miko¡ajewski, Miko¡ajewska, & Khudyakova 1990). The importance of this unsolved problem is elaborated by the symbiotic activity of CH Cyg, which strongly di†ers from that observed in other symbiotic stars. The blue continuum and spectral line proÐle development during active phases have no counterparts in other systems. Active phases in CH Cyg started with a brightening of 2È3 mag in the U band during a few monthsÏ time. Evolution continued with a gradual increase of activity for a number of years (1963È1965, 1967È1970, 1977È1984, 1992È 1995) and a Ðnal drop to quiescence. The end of the maximum brightness in mid-1984 was accompanied by the ejection of bipolar jets detected in radio (Taylor, Seaquist, & Mattei 1986). An irregular variability on a timescale of days to weeks and short-term Ñickering variations running from minutes to hours are characteristic. During quiescence, the symbiotic phenomenon practically disappears. The spectrum exhibits almost exclusively red giant characteristics, and the V light curve often displays variations at a level of a few times 0.1 mag with a D100 day period (see, e.g., Muciek & Miko¡ajewski 1989). These variations were ascribed to pulsations of the giant star (Skopal 1997). Evolution in the line spectrum during di†erent outbursts is complex. For example, the hydrogen Balmer lines exhibited pronounced variable inverse P Cygni proÐles during the whole 1981È1984 maximum, which corresponded to variable, approximately ]40 km s~1 infall of the material to the accreting star. On the other hand, during the 1992È1995 active phase, a complicated emission/absorption structure with a P Cygni absorption component in the hydrogen line proÐles indicated an irregular, high-velocity (D1000 km s~1) outÑow of gas. 2. TRIPLE MODEL As noted above, the main problem is why CH Cyg was inactive up to about 1963 but has been observed after that signiÐcant activity. Since this is difficult to explain by the astrophysics of the stars alone, we study the possibility of a dynamical explanation. It seems that a triple system is complex enough to produce phenomena capable of igniting activity. ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ 1 National Astronomical Observatory, Osawa 2-21-1, Mitaka, Tokyo 181, Japan. 2 Tuorla Observatory, University of Turku, 21500 PiikkioŽ, Finland. 444 KOZAI RESONANCE It is worth mentioning that if this star really is a triple, it is the only known symbiotic triple system. In this case it is also one of the most compact known triples ; in fact, it is near the known stability limits for triple systems (e.g., Harrington 1975), especially if SkopalÏs model is correct. The study reported here began based on the published (Skopal 1995, 1997 ; Skopal et al. 1996a, 1996b) triple model, in which a heavy (near 4 M ) red giant (the ““ cool ÏÏ _ component) was orbited by the ““ hot ÏÏ entity that was a symbiotic binary consisting of a red giant with mass D1 M and a small main-sequence dwarf (the actual ““ hot ÏÏ _ component), which glowed as a result of mass Ñow from the giant. Numerical stability studies of this model (Mikkola, Tanikawa, & Skopal 1997) revealed that it seems impossible to construct a reasonable stable three-body model from these ingredients, especially as the outer orbit had to be taken as nearly circular to ensure long-term stability, while the radial velocity observations suggest a signiÐcant eccentricity. In fact, when we used SkopalÏs (1997) published orbital elements, P \ 756 days , P \ 5298 days , 1 2 e \ 0.323 , e \ 0.22 , 1 2 i \ 90¡ , i \ 90¡ , 1 2 u \ 233¡.3 , u \ 227¡.7 , 1 2 T [ JD 2,440,000 \ 47,302 days , 1 T [ JD 2,440,000 \ 45,813 days , 2 m \ 0.93 M , m \ 0.12 M , m \ 3.82 M , 1 _ 2 _ 3 _ binary \ (m , m ) , m \ hot component , 1 2 2 we got a collision of stars (very high eccentricity of the inner binary) in a timescale of decades only. When we used our Ðtting code (see ° 3.3) and allowed masses to be adjustable variables, we got an answer similar to SkopalÏs model, but the role of the components in the inner binary reversed, i.e., the code claimed that the hot-component observations would be related to the giant star (of mass D1 M ) in the inner binary. This conclusion seemed unavoidable, _ and thus we began searching for other explanations for the observations. Following A. SkopalÏs suggestion (1997, private communication) that the cool-component spectrum is a composite of two giant star spectra, we also constructed a model in which the cool componentÏs (observed) radial velocity is a linear combination of the calculated velocities of the two supposed red giants, and the observed hotcomponent data were Ðtted with the center-of-mass motion of the inner binary. A consequence of this model is, however, that the the relative speed in the long-period motion is also a†ected : the observed result is unremovably reduced, and the true speeds would then be much larger. Because of this the model can be made to Ðt the coolcomponent data only with masses rather di†erent from those originally suggested by Skopal (see Fig. 1). However, if this model is in any way correct, we must conclude that the hot-component data are not due to the actual hot component in the system, as the observations clearly do not agree with the prediction of computations given in the Ðgure. (The computed amplitude of the radial velocities of the hot component, with respect to the center of mass of the binary, is essentially larger than the observed scatter in the velocity values.) In the original Skopal model 445 the masses were di†erent and the disagreement even more pronounced. Also, when full three-body dynamics was included, we were unable to Ðnd orbital elements Ðtting the timings of the (claimed) observed eclipses. Thus, we must remain skeptical about the correctness of SkopalÏs model. After this failed attempt the observations were reexamined in detail in the hope of Ðnding an alternative explanation for the known facts. At this point it was noticed that the model of Hinkle et al. (1993) is likely to be correct in its basic setup, although details had to be adjusted. The description of the new model is as follows : The system consists of three stars, a red giant of mass m B 3.5 1 M or somewhat less ; a dwarf star of mass m B 0.5 M or _ 2 _ in so orbits the star m . The period of this binary is a little 1 excess of 2 yr. A third star of mass m B 0.9 M (probably a 3 _ white dwarf ) orbits the binary (m , m ) in a long-period 1 2 (14.5È16 yr) orbit. The orbital planes of the associated twobody orbits have a rather large mutual inclination of roughly i D 60¡ at present. In a system like this, the incli1,2 eccentricity cannot remain constant, but cycle nation and with a long period and large amplitude. (For example, had the inclination had the initial value of 90¡, then the maximum eccentricity would be 1.) This phenomenon is known as the Kozai resonance. In our model the eccentricity of the binary has had a low or moderate value until the beginning of the 1960s. Now the eccentricity is in the highvalue phase, and the star m comes very close to the red 2 giant at periastron. It may even be possible that it actually dives inside the red giantÏs atmosphere at this point. This interaction expels gas out of the big star m . Part of that gas 1 m , but a permust then fall to the smaller component 2 manent accretion disk cannot persist because the strong interactions repeat every 2 yr and destroy it (this may explain the invisibility of that component). However, some of the expelled gas Ðnally Ðnds its way to the more distant star m , causing pronounced activity. The radiation pressure of3the red giant may also enhance this process. It is worth noting here that the use of a Roche lobe theory in the explanation of the gas Ñow may not work for eccentric binaries. The stars simply do not have enough time to adjust to the potential during the fast periastron passage. Also, the gas Ñow cannot happen (instantaneously), even if the potential level surfaces would allow it to happen. Thus, it may be necessary that the star m actually dives inside m 2 1 in the gas-expelling interaction. However, a red giant does not really have a surface but just an extended thin atmosphere. In light of this, the above model seems physically plausible. Although the details of the mass transfer mechanism remain speculative, there is a recent discovery (Waters et al. 1998) that may support our model : in the so-called Red Rectangle Nebula, a binary is found to be surrounded by a dust disk that supposedly originates from the gas expelled by the binary while one of its components went through the red giant phase of evolution. Thus, the physical mechanism that we are proposing for the activity of CH Cyg may be similar to that explaining the Red Rectangle Nebula. 3. THE EVIDENCE In this section the observational evidence for the suggested model is discussed, and Ðtting of radial velocity data is carried out. It is necessary to stress that the computations and numerical values presented here are not to be taken as an attempt to give precise orbital elements for the object, 446 MIKKOLA & TANIKAWA but rather to obtain insight into the ability of the suggested model to explain the phenomena. 3.1. Radial V elocities First we consider the periodicities in the observed radial velocities that were collected from the literature (Hinkle et al. 1993 ; Hack et al. 1982, 1986, 1988 ; Kotnik-Karuza, Jurdana, & Hack 1992 ; Hack & Aydin 1992 ; Luud & Tomov 1984 ; Yoo & Yamashita 1982, 1984 ; Faraggiana & Hack 1971 ; Skopal, Miko¡ajewski, & Biernikowicz 1989 ; Skopal 1986 ; Smith 1979 ; Wallerstein 1981, 1983 ; Wallerstein et al. 1986 ; Yamashita & Maehara 1979). In what follows, the observational material discussed is the collection of data from all of the above sources (hereafter referred to as ““ collected data ÏÏ). We demonstrate that these can be reasonably Ðtted with the suggested model. From these data (see Fig. 2) it is clear that the star contains at least two objects, components of a long-period binary. We will refer to these as the ““ hot ÏÏ and the ““ cool ÏÏ components, corresponding to the model stars m and the 3 binary (m , m ), respectively. 1 2 One should remember that the radial velocity data of the hot component refer to gas moving near this object, and thus the rather large scatter is not surprising. The coolcomponent data, however, are for a red giant, and the scatter is considerably smaller here. The observers see only one object ; the di†erent stars cannot be seen separately. So they have one, and only one spectrum, which contains (1) features (spectral lines) that are typical for a red giant and (2) features (hydrogen absorption lines) that are ““ obviously ÏÏ due to a hot source. They measure (a) the red giant spectral lines, obtaining radial velocity values ; these are the ““ cool ÏÏ-component data plotted in the Ðgures. They measure (b) the hydrogen absorption lines, obtaining radial velocities that are di†erent from those for the red giant lines. These form the ““ hot ÏÏ-component data plotted in the Ðgures. One can see that the radial velocities have an average value of about [60 km s~1. Thus the system (its center of 0 small * RV -20 ’HOT’ ’COOL’ -40 RV km/s -60 heavy * RV cm-RV -80 -100 1960 1965 1970 1975 1980 1985 1990 Years 1995 2000 FIG. 1.ÈComparison of radial velocities of a Skopal-like model with observations (the collected data). The fast-Ñuctuating curve (““ small*RV ÏÏ) is the radial velocity of the small star in the binary (the assumed hot component), while the smooth (thin) curve (““ cm-RV ÏÏ) is the radial velocity of the center of mass of the binary. The cool-component data, indicated by the thick curve with small amplitude Ñuctuations (““ heavy*RV ÏÏ), are produced assuming blended spectra of the two giants in this model. The observed result was assumed to be a linear combination of the true radial velocities. The weights in this combination were adjustable constants in the Ðtting procedure. As the spectra cannot be disentangled in this model, it follows that the mass values suggested by the model are rather large (1.9, 1.0, 5.5 M ). Note that the hot-component data do not agree with the _ observations. Vol. 116 -20 ’HOT’ ’COOL’ -30 -40 -50 small * RV RV km/s -60 heavy * RV -70 -80 1960 1965 1970 1975 1980 1985 1990 Years 1995 2000 FIG. 2.ÈComparison of the collected observational data of the hot and cool components with a theoretical computation assuming a simple binary model. The y-coordinate is the radial velocity (RV) in km s~1. This Ðt demonstrates that the long-term behavior of the observed radial velocities is consistent with an outer binary of eccentricity D0.2 and a mass ratio D4. The cool-component data (plus signs) are connected by dashes to make clear the 2 yr wave superposed on the long (D15 yr) period. In the hot-component data it is difficult to see any short periodicity, but only random variation. mass) moves toward us with this speed, and the variations around the mean reÑect the internal motions of the stars in the system. Apparently, the observers can only see the luminous red giantÏs features in the spectrum and the hotcomponent feature during active periods. The small star m 2 (revolving around the red giant in the new model) cannot be seen, but this is not surprising, as the red giant is so much more luminous, and a persistent accretion disk cannot form. Figure 2 illustrates the observations and a Ðt of them with a long-period (15.7 yr) binary. Thus, the system contains a binary of two entities with a mass ratio of B1 . 4 Further analysis of the data suggests a periodicity near 2 yr (Hinkle et al. 1993). Figure 3 illustrates the di†erences between the data and the long-period Ðt. The times were taken modulo period (here 2.074 yr was used). These plots reveal the fact that such a periodicity seems to exist in the cool-component data, while the hot-component data do not show any clearly recognizable wave. In the top panel of the Ðgure we also include a smooth curve calculated from the model of Skopal (1997). This theoretical curve actually has much larger amplitude than the observations and is thus incompatible with them. We also carried out a Fourier analysis of the hotcomponent observations, but the only signal that we found was that due to the 1 year periodicity in observing activity. The 2 year sidelobe of that was also visible, giving warning that simple ways of analyzing the data may lead to spurious conclusions. 3.2. T he Astrophysical Constraints and Kozai Resonance The feature that is most difficult to explain is the appearance of activity in the beginning of 1960s, combined with the fact that the star did not show such activity in its previous observational history, which extends back to 1885. If the system is a triple and the solution for this riddle is dynamical, then the only possibility is the Kozai resonance (Kozai 1962 ; Harrington 1968 ; Kinoshita & Nakai 1991 ; Lidov & Ziglin 1976 ; Marchal 1990 ; Sidlichovsky 1983 ; Innanen et al. 1997) : if the mutual inclination of orbital planes in a hierarchical triple system is large, then the eccentricity and inclination of the inner binary cycle with large No. 1, 1998 KOZAI RESONANCE in which b is the semiminor axis of the outer orbit, a is the semimajor axis of the inner binary, and m is the total mass b of the (inner) binary. The gravitational constant G \ 1 here ; thus, if astronomical units are used for the quantities present in the expression, the result must be divided by 2n to get this timescale in years. For wide outer orbits the unit of q is thus large, and so is the timescale of the Kozai phenomenon. Note, however, that the timescale for evolution of eccentricity depends on initial conditions also. We conclude from the above that to have de/dt [ 0 presently, we must have sin 2u [ 0. Also, it is possible to show that for the large-amplitude variation of eccentricity to occur, we must choose rather large inclination i [ 40¡. These are the conditions that the Kozai phenomenon and the observations of activity give as prerequisite for the Ðtting of the radial velocities. Figure 4 illustrates the evolution of the eccentricity and mutual inclination of the orbital planes in a model that is discussed in more detail in the ° 3.3. Finally, it must be emphasized that the tidal e†ects tend to increase the speed of u-precession. This lowers the amplitude of the e-cycling and shortens the period. Thus, the tidal interaction e†ectively protects the binary from developing too high an eccentricity, in this way avoiding any possibility of star merger. 15 HOT Skopal 10 5 0 RV km/s -5 -10 -15 0 0.5 1 Phase in years 1.5 2 15 COOL 10 5 0 RV km/s -5 -10 -15 0 0.5 1 Phase in years 1.5 2 FIG. 3.ÈResiduals of the collected data plotted in Fig. 2 with a longperiod binary. The results are presented as a function of the phase of the period of the inner binary. This period is also obtained from Fourier analysis for the cool-component data. T op : Residuals for the hot component from the binary model. Smooth curve : Theoretical data from the model published by Skopal (1997). Fourier analysis signals only a 1 yr periodicity, presumably due to the variation in observing activity. Bottom : Residuals for the cool component. One notes that it is not possible to see any trace of the period in the top panel, while the bottom panel shows clear, although still rather scattered, wavelike behavior. amplitude in long period. Such a variability in eccentricity could cause strong tidal interaction of the stars during the high-eccentricity era. This interaction may lead to gas eruptions, and thus the ingredients for the activity are available. The long period of the Kozai phenomenon explains the previous nonactive phase. If we take, as is natural and convenient, the initial orbital plane of the outer orbit as the (x, y)-plane and apply the quadrupole approximation for the perturbing potential, then the secular motions can be obtained from the Lagrangian equations. The following equations are valid for the restricted problem, but are still useful for qualitative considerations for our model (instead of the more complicated complete equations) : 15 e2 sin 2u sin i cos i di , \[ 8 dq J1 [ e2 de 15 \ eJ1 [ e2 sin 2u sin2 i , 8 dq du 3 \ J1 [ e2[2(1 [ e2) ] 5 sin2 u(e2 [ sin2 i)] , dq 4 (1) cos i (3 ] 12e2 [ 15e2 cos2 u) d) , \[ 4 dq J1 [ e2 where the time variable q has the unit b3Jm b, T \ evol a1.5m 3 447 (2) 3.3. Fitting the Radial V elocities We used a nonlinear optimization code to Ðt the data. First, an orbit of the triple system is tabulated, then a weighted square sum of observed and calculated radial velocities is formed to obtain the value of the object function, which the code then attempts to minimize. The algorithm uses function values only and thus is able to Ðnd acceptable solutions for ill-determined problems also. It is possible to Ðx some of the variables and let others vary. In principle, the orbit could be determined without ambiguity due to the three-body interactions, which change the approximating two-body orbits. In practice, however, the observations cover too short an interval of time and are too noisy for a reliable determination of the orbit. Thus, we cannot do much more than check if there are orbital elements and star masses giving a model that explains the observational facts. 0.8 eccentricity 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Years 65 deg 60 55 50 45 40 inclination 35 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Years FIG. 4.ÈKozai resonance in action. T op : Evolution of eccentricity in the suggested model. Bottom : Evolution of mutual inclination of the two orbital planes. 448 MIKKOLA & TANIKAWA The object function to be minimized was chosen as 0.8 F \ ; W min [(O[C)2, V 2 ] , (3) k k crit k where (O[C) is the di†erence of observation and theory, k while W stands for the weight of the observation. As the k ““ critical ÏÏ error V , we used 5 km s~1, and this formulacrit tion was adopted to e†ectively ignore some very large deviations (which may not necessarily be random in character). In most computations we assume (rather arbitrarily) that the outer orbit is seen edge-on, and the (initial) plane of motion of the outer orbit is taken as the (x, y)-plane. In the coordinate system used this means that the inner binaryÏs inclination directly equals (initially) the mutual inclination of the orbit. We found that there is a large, complicated region in the parameter space in which the Ðt of the data is essentially equally good. Thus, some of the parameters can be chosen to Ðt the ““ astrophysical constraints.ÏÏ However, this does not mean that the parameters could be varied independently. The observations deÐne a complex relation between them. The hot-component data represent the hydrogen absorption component reÑecting best the hot-component orbital motion. These data are, however, a†ected by the circumstellar material in the system during the active phase. Also, it is possible that some contamination e†ect is present in the cool-component data, especially during the most active phase. Thus, in the Ðtting procedure the cool-component data were given a weight twice as large as that of the hotcomponent data. The results of this Ðtting are illustrated in Figure 5. This Ðgure compares the radial velocities with the collection of observations. The hot-component theoretical data are plotted as a thin line, while the cool-component velocity values are illustrated using a heavy line. The 2 yr Ñuctuation, due to the motion of the (observationally invisible) small companion star, is clearly visible and Ðts the observations reasonably well. In Figure 6, the evolution of eccentricity of the inner binary is illustrated as it happens in this model. One notes the low-eccentricity (e ¹ 0.2) phase lasting for nearly 200 yr -20 ’HOT’ ’COOL’ -30 -40 -50 WD RV RV km/s -60 Giant RV -70 -80 1960 1965 1970 1975 1980 1985 1990 Years 1995 Vol. 116 2000 FIG. 5.ÈComparison of the new triple-model radial velocities with the collection of observations. T hin curve, white dwarf (““ WD RV) data ; heavy line, giant star (cool-component) data. The 2 yr Ñuctuation is caused by the motion of the (observationally invisible) small companion star. Note that the model agrees well with the infrared data of Hinkle et al. (1993) ; these are connected with heavy lines for easy comparison with the similar smooth theoretical curve. Cool-component measurements other than HinkleÏs are marked with plus signs. eccentricity evolution 0.7 0.6 0.5 0.4 ecc(t) 0.3 0.2 0.1 0 1800 1850 1900 1950 2000 2050 Years 2100 FIG. 6.ÈEvolution of eccentricity of the inner binary in the (Ðrst) new model. The long period of low eccentricity explains the nonactivity of CH Cyg prior to 1963. This behavior of eccentricity is due to the Kozai resonance, and its phase depends critically on inclination and longitude of perihelion. before the 1960s. Thus it seems possible to associate the activity of CH Cyg with this e-cycle. The orbital elements and masses used in producing the above Ðgures were P \ 2.074 days , P \ 15.75 days , 1 2 e \ 0.315 , e \ 0.217 , 1 2 i \ 60¡ , i \ 0¡ , 1 2 ) \ 145¡ , ) \ 0¡ , 1 2 u \ 230¡ , u \ 325¡.25 , 1 2 " \ 304¡.13 , " \ 216¡.74 , 2 2 m \ 3.51 M , m \ 0.501 M , m \ 0.909 M , 1 _ 2 _ 3 _ epoch \ 1980 . Here P is the period in years and e, ), and u are the standard orbital elements, while " \ M ] ) ] u is the mean longitude. The coordinate system used di†ers from that customary in presenting binary star elements. Our x-axis is directed toward the observer, which we found convenient, especially for the consideration of the mutual inclination of the two orbits (because we assumed the outer orbit to be seen nearly edge-on). Because the observations near the most active phase may be more a†ected by the difficulties due to gas Ñow in the system, we produced other Ðts with less weight given to data near the year 1983. An example of the results is illustrated in Figures 7 and 8. The orbital elements and masses were quite di†erent from those above : P \ 2.052 days , P \ 14.98 days , 1 2 e \ 0.53 , e \ 0.27 , 1 2 i \ 52¡.6 , i \ 0¡ , 1 2 ) \ 115¡.56 , ) \ 0¡ , 1 2 u \ 230¡ , u \ 308¡.4 , 1 2 " \ 273¡.47 , " \ 210¡.0 , 1 2 m \ 3.00 M , m \ 0.336 M , m \ 0.845 M . 1 _ 2 _ 3 _ No. 1, 1998 KOZAI RESONANCE -20 449 4 ’HOT’ ’COOL’ -30 3 R R 2 -40 -50 WD RV RV km/s -60 Giant RV 1965 1970 1975 1980 1985 1990 Years 1995 2000 FIG. 7.ÈComparison of the second triple-model radial velocities with observations. This plot is in most respects similar to Fig. 5. The two Ðts are not very much di†erent, in the sense that one cannot deem with any certainty which of the models gives the better Ðt. Thus, the determination of accurate elements cannot be taken as a goal of this investigation. The agreement with observations is, however, as good as one can expect, taking into account the scatter in the observations. 3.4. Eclipses We found that, in our model, the eclipse times (Skopal et al. 1996a) correlate strongly with the periastron passages in the inner binary (see Fig. 9) ; moreover, this seems to be a robust property in all acceptable Ðts of the radial velocities. In fact, the observed eclipses occur consistently somewhat after the periastron passage (this also applies to the closest projected approach). There is one exception to this rule : the eclipses near 1990. Because the red giant also oscillates, it is not always clear if the observed dip in luminosity is due to oscillation or to some other phenomenon. Thus, the conclusion follows that a likely explanation for the eclipses is a phenomenon excited by the tidal interactions, such as a large starspot. It is in fact difficult to understand otherwise how a practically invisible star could cause eclipses, especially because it is the small star m that could be eclipsed 2 (by the red giant). Recently, Mardling (1995) published a study on the interaction of stellar oscillations and the orbital motion in binary systems. The results showed that 0.8 0.7 0.6 0.5 0.4 ecc(t) 0.3 eccentricity evolution 0.2 0.1 0 1800 0 X -1 -70 -80 1960 X 1 1850 1900 1950 2000 2050 Years 2100 FIG. 8.ÈEvolution of eccentricity of the inner binary in the second model. The Kozai phenomenon starts here in a di†erent phase and has a di†erent period and amplitude. -2 -3 PROJ -4 1965 PROJ 1970 1975 1980 1985 1990 Years 1995 FIG. 9.ÈComparison of the observed eclipse times (vertical bars) with the pericenter passages (thick line ; distance) and the projected (apparent) distance of the centers of the stars (dotted line ; distance is plotted as negative when the small star is in front of the red giant and as positive when it is more distant than the giant). T hin line : The x-coordinate of the small star with respect to the red giant. One notes the strong correlation in all of these, but the lack of eclipses (1975È1984) cannot be directly explained with them. such an interaction can lead to chaotic oscillations and eccentricity variation. Since chaotic systems in general can jump from one type of behavior to another, it seems that this phenomenon (starspots, orbit oscillation interaction) may explain the long gap in the ““ eclipse ÏÏ observations. These arguments may not completely rule out true eclipses, but if they are real, then the 10 yr gap in eclipses is difficult to understand. The rate of change of the orientation of the orbit (especially )) as calculated from the Kozai resonance equations (or numerically) is much too slow, only B0¡.2 yr~1, so that viewing-condition evolution cannot explain the long period without eclipses. 4. DISCUSSION AND CONCLUSION CH Cygni is the only known (possibly) triple symbiotic star. Other symbiotic systems are mass exchange binaries with a period of about 2 yr. Thus one would, by analogy, expect the Skopal model to be correct. Our computations, however, showed that the model in which the hot component is a binary (component thereof) is dynamically heavily unstable, unless the orbits are almost circles. This, on the other hand, would make it difficult to understand the origin of the activity that began only a few decades ago. Also, if in this case the star sizes estimated from the (supposed) eclipses (Skopal 1997) were correct, then there would be no Roche Ñow in the system (owing to too large a distance). An alternative here would be that the giant (of mass D1 M ) would be in fact larger in size than previously _ estimated. However, then it is difficult to understand the long period of noneclipsed activity. The model of a simple magnetic binary of period about 15 yr (Miko¡ajewski et al. 1995) requires a (too) heavy stellar wind (A. Skopal 1997, private communication) and pulsation of the giant star to explain the 2 yr periodicities. Also, it does not explain the recent beginning of activity (except as a chance occurrence). For our new model, which is actually just a modiÐed Hinkle model, we have demonstrated that the Kozai resonance is a possible dynamical explanation for the recent emergence of activity in CH Cyg. It appears that the radial velocity observations are reasonably compatible with the 450 MIKKOLA & TANIKAWA model. The recent discovery (Waters et al. 1998) that the Red Rectangle Nebula and dust disk therein have formed of gas ejected out of the red giant component in a binary star gives support to our physical model of the system. The idea of gas ejection out of a binary is clearly a possibility, and the mechanisms in the two systems may be similar. If the observed dips in the luminosity of the star are not actual eclipses but starspots, then our model is compatible with them too. The 10 yr gap in that phenomenon, however, may require more complex explanations, such as interaction of the binary dynamics and star pulsation (Mardling 1995). One problem with any triple-star model is that apparently only two entities are seen. In our model a plausible explanation is that the star (m ) close to the red giant is 2 invisible because it cannot collect a persistent accretion disk, owing to repeated dives to the giantÏs atmosphere and consequent disruption of the disk. When the eccentricity is small, there is no gas around for the disk to form. During our computations we found that the orbits and masses cannot be accurately determined ; even the outer period is uncertain by up to 10%. It seems actually nonsensical to quote the system parameters to many signiÐcant Ðgures, because very di†erent orbits Ðt the data equally well. Similarly, any least-squares Ðt will give underestimated error bounds, because the relatively large scatter in the data cannot really be represented by di†erential (i.e., small) parameter variations. Thus, the computations presented here are for obtaining insight rather than accurate numbers. Because the 2 yr periodicity is twice the period of Earth, there is always a chance that some observed phenomena, especially if the observing period is only a few years, are a†ected by observing-season periodicity. For example, in the hot-component observations, Fourier analysis produces a 1 yr and (as a side e†ect) a 2 yr signal that is due to the 1 yr periodicity in observing frequency. The suggestion by Miko¡ajewski, Tomov, & Miko¡ajewska (1997) that the CH Cyg activity is caused by stellar wind (plus a magnetic white dwarf ) may be partly right : the radiation pressure in the huge red giant and the pull of the companion star during its pericenter passage can, when combined, cause gas eruptions more easily than each of these phenomena individually. It is clear that, if this scenario is correct, the more or less irregular gas eruptions can be the cause of the complex observed spectral phenomena. If the single star is a magnetic white dwarf, then the gas falling into it can also explain the observed high-speed jet out of the hot component. It may also be worth noting that what is known about the stability of triple hierarchical systems in general (Harrington 1975) places CH CygÏs Skopal-like triple model just at the boundary of stability. If we speculate that in our model the accretion disk could extend up to the distances at which three-body dynamics restricts its stability, then it would be possible to see another 2 yr periodicity in the system : at least temporarily, maybe over a few years, a dense (density wave ?) gas cloud could be circling the white dwarf with a 2 yr period, giving rise to spectral lines originating from hot gas. Concerning the evolution of the system, if our model is correct, we expect that the present activity period may have a timescale of about 100 yr (as can be seen from Figs. 6 and 8). However, as the orbit could not be accurately determined, this estimate is uncertain. The maximum eccentricity may in fact be smaller than that in the sample computations ; the system may be closer to maximum eccentricity phase. Thus the 100 yr timescale is to be considered as an estimated maximum time for the duration of the activity. 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