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Stability of Triple Star Systems with Highly Inclined Orbits S The Astrophysical Journal, 615:506–511, 2004 November 1 # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A. STABILITY OF TRIPLE STAR SYSTEMS WITH HIGHLY INCLINED ORBITS S. A. Khodykin Volgograd Pedagogical University, Volgograd 400074, Russia; [email protected] A. I. Zakharov Sternberg Astronomical Institute, Moscow University, Moscow 119899, Russia; [email protected] and W. L. Andersen Eastern New Mexico University, Portales, NM 88130; [email protected] Received 2003 February 16; accepted 2004 July 8 ABSTRACT It is well established that certain detached eclipsing binary stars exhibit apsidal motions whose values are in disagreement with calculated deviations from Keplerian motion based on tidal effects and the general theory of relativity. Although many theoretical scenarios have been demonstrated to bring calculations into line with observations, all have seemed unlikely for various reasons. In particular, it has been established that the hypothesis of a third star in an orbit almost perpendicular to the orbital plane of the close binary system can explain the anomalous motion at least in some cases. The stability of triple star systems with highly inclined orbits has been in doubt, however. We have found conditions that allow the long-term stability of such systems, so that the third- body hypothesis now seems a likely resolution of the apsidal motion problem. We apply our stability criteria to the cases of AS Cam and DI Her and recommend observations at the new Keck interferometer, which should be able to directly observe the third bodies in these systems. Subject headings: binaries: eclipsing — celestial mechanics — instabilities — stars: individual (AS Camelopardalis, DI Herculis) 1. INTRODUCTION Maloney et al. (1989), and Claret (1997, 1998). It is the purpose Discrepancy between observation and theoretical predictions of this paper to demonstrate the feasibility of one particular explanation that has been put forward in the literature. Our of the apsidal motion of certain detached binary stars has considerations may also find application in other triple star or remained an outstanding problem for two decades. In the cases suspected triple star systems such as in Coe et al. (2002), where of AS Cam and DI Her, for example, observed apsidal motion a triple star model for the X-ray pulsar AX J0051 733 is pro- rates are a fraction of the theoretical predictions based on stellar À posed to explain puzzling features of the spectrum of a candi- structure, tidal, and relativistic effects. date optical counterpart. As was first pointed out by Rudkjøbing (1959), the effect of It is well known that the hypothesis of third stars in outer relativistic gravity is significant in the case of a number of orbits of these close binary systems can bring theory into line detached binary stars. Although he considered only DI Her with observed apsidal periods (Khaliullin et al. 1991; Khodykin in detail, the number of interesting cases has grown to about half a dozen (Koch 1977; Moffat 1984), including AS Cam & Vedeneyev 1997), but the stability of such triple star systems has been in doubt (Harrington 1968). We show here that the (Maloney et al. 1989). inclusion of the apsidal motion as an additional perturba- The discovery of anomalous apsidal motion by Martynov & Khaliullin (1980) was initially considered a possible challenge tion leads to the conclusion that such triple star motions can be stable. In particular, the triple star models of Khodykin & to general relativity. Moffat (1984) invented an alternative grav- Vedeneyev (1997) and Khaliullin et al. (1991), which reconcile ity theory that harbored differing predictions for the apsidal the cases of AS Cam and DI Her with observations, are shown motion of binary stars and yet maintained agreement with other to be stable. Furthermore, we show that observations utilizing tests of general relativity. The predictive power of this theory the new Keck interferometer should be able to directly image is weakened by the existence of a new adjustable parameter the putative third bodies in these two systems. for each star. Moreover, increasingly severe tests of general relativity such as in Taylor & Weisberg (1989) make such large 2. THE LAGRANGE PLANETARY EQUATIONS deviations at AU scales seem unlikely. Several other, less ex- FOR THE HIERARCHICAL THREE-BODY SYSTEM otic, solutions have been proposed. In one scenario the rapid circularization of the orbit occurs because of dissipation of an- We have studied numerically the dynamical evolution of a gular momentum from stellar oscillations or a large amount of hierarchical triple system consisting of a massive CBS and a mass loss. Another reasonable guess is that the close binary third star of moderate mass. Figure 1 shows the notation used. system (CBS) orbit is surrounded by a resisting medium in the The calculations were done perturbatively using the disturbing form of gas clouds. The required density well exceeds ob- function method (Kopal 1978). We have assumed that the three servational limits in the case of DI Her, however. These and stars are pointlike and isolated from other stars. We ignore in- other alternatives are reviewed in Guinan & Maloney (1985), ternal dynamical exchanges such as synchronization, angular 506 TRIPLE STAR STABILITY 507 Since we are averaging over the mean anomaly, there is no Lagrange planetary equation for M, and furthermore the semi- major axis a has no time dependence. The planetary equations for the perturbation of the remaining CBS orbital elements by the third body are da ¼ 0; ð8Þ dt tb de 5 ¼ 5AeQS ¼ Ae sin2 sin 2; ð9Þ dt 2 tb d! ¼ A 2 À 5Q 2 À N 2 dt tb 1 þ 4e 2 À N cot i Q sin ! þ S cos ! ; ð10Þ 1 À e 2 di 1 þ 4e 2 ¼ AN Q cos ! À S sin ! ; ð11Þ dt 1 À e 2 tb d 1 þ 4e 2 ¼ AN csc iScos ! þ 2 Q sin ! ; ð12Þ dt tb 1 À e Fig. 1.—Kinematic variables describing the relative orientation of the orbits of the CBS and the third body. where 1=2 3ðÞ1 À e 2 Pq0 momentum exchange, and orbital precession of the CBS. A ¼ : ð13Þ Classical tidal effects and relativistic effects are assumed to be 21ðÞÀ e0 2 3=2P0 2(1 þ q þ q0) independent and additive. The disturbing functions for the CBS and the third body are The planetary equations for the perturbation of the third- adapted from Brown & Shook (1933, p. 14). They are, respectively, body orbital elements are X1 n n m3 r 3 m À (À m2) r nÀ1 R ¼ 4 2 1 P ; ð1Þ 0 r r0 (m m )n r0 nþ1 da n¼1 1 þ 2 ¼ 0; ð14Þ dt tb m1 þ m2 þ m3 m1m2 tb 0 R ¼ 2 R: ð2Þ de (m1 þ m2) m3 ¼ 0; ð15Þ dt tb We average R and R over the mean anomalies of the CBS 0 2 2 tb d! 3e þ 2 3 1 þ 4e 2 2 and third body (hence, twice averaged). The first-order terms ¼ B 2 À 2 Q þ S tb dt tb 2(1 À e ) 2 1 À e of the twice-averaged disturbing functions, R2 and R2 ,are 1 þ 4e 2 2 0 2 ÂÃÀÁ ÀÁ þ cot i0 QT þ SU ; ð16Þ q m1a 2 2 2 2 2 1 e 2 R2 ¼ 3N 1 À e À 15e Q þ 6e À 1 ; À 0 2 3=2 0 3 21ðÞÀ e a di0 1 þ 4e 2 ¼B QF þ SG ; ð17Þ ð3Þ dt 1 À e 2 tb 0 0 2 tb q(1 þ q þ q ) d 1 þ 4e R ¼ R2; ð4Þ 0 2 0 2 ¼B csc i 2 QT þ SU ; ð18Þ q (1 þ q) dt tb 1 À e where the masses are included via the ratios q ¼ m2=m1 and 0 0 q ¼ m =m1. We choose our units of measurement to be AU, where years, and solar masses, so that the Newtonian gravitational con- stant is G ¼ 4 2. The orientation of the third-body orbital plane 3qðÞ1 À e 2 P 4=3 B ¼ : ð19Þ with respect to the close binary orbital plane is described by the 21ðÞÀ e0 2 2P0 7=3(1 þ q)4=3(1 þ q þ q0)2=3 direction cosines (Q, S, N ) of the unit vector normal to the third- body orbital plane. We refer the direction cosines to the periastron, a perpendicular to the periastron, and the direction normal to the In writing equations (16)–( 18) we have introduced the close binary orbit. Let be the angle between the two orbital direction cosines of the nodal line of the third body referred to planes, and call the angle measured from the periastron to the line the same CBS axes by which Q, S,andN are defined above. of intersection of the orbital planes .Wethenhave These cosines are Q ¼ sin sin ; ð5Þ F ¼ sinðÞ 0 À cos i sin ! þ cosðÞ 0 À cos !; ð20Þ S ¼sin cos ; ð6Þ G ¼ sinðÞ 0 À cos i cos ! À cosðÞ 0 À sin !; ð21Þ N ¼ cos : ð7Þ H ¼ sinðÞ 0 À sin i: ð22Þ 508 KHODYKIN, ZAKHAROV, & ANDERSEN Vol. 615 Finally, we use cosines of the direction perpendicular to the third-body nodal line and behind the visual plane. These co- sines are T ¼ ½cosðÞ 0 À cos i cos i0 þ sin i sin i0 sin ! À sinðÞ 0 À cos i0 cos !; ð23Þ U ¼ ½cosðÞ 0 À cos i cos i0 þ sin i sin i0 cos ! þ sinðÞ 0 À cos i0 sin !; ð24Þ V ¼ cosðÞ 0 À sin i cos i0 À cos i sin i0: ð25Þ 3.
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