A&A 544, A63 (2012) Astronomy DOI: 10.1051/0004-6361/201219067 & c ESO 2012 Astrophysics Mean motion resonances and the stability of a circumbinary disk in a triple stellar system R. C. Domingos, O. C. Winter, and V. Carruba UNESP, Univ. Estadual Paulista, Grupo de Dinâmica Orbital e Planetologia, Guaratinguetá, SP 12516-410, Brazil e-mail: [rcassia;vcarruba]@feg.unesp.br; [email protected] Received 17 February 2012 / Accepted 21 June 2012 ABSTRACT We numerically investigated the orbital stability of a circumbinary disk in a 3-D triple stellar system. We verified that there is a stable region (protected region) in which highly eccentric and/or inclined orbits can remain stable. In this paper we identify two-body mean motion resonances as a powerful mechanism to increase the eccentricity of the particle on short time-scales (10 kyr or less) and produce a longer-lived high-eccentricity population. These resonances are of high order and have not been previously considered in triple stellar systems. We show that this powerful mechanism can lead to regions containing instabilities and gaps. This process also produces a number of highly eccentric particles whose orbits remain stable. We show that there are limit values of eccentricity and semi-major axis for an orbit to be within the stability region. These quantities represent the dynamical effects of the inner binary and third star companion on a circumbinary disk. The value of this limit depends on the system parameters and should be considered in estimates of the stability, formation, and survival of bodies in triple systems. Key words. celestial mechanics – binaries: close – protoplanetary disks 1. Introduction binary orbital plane. The test particles are initially taken on placed in a cloud of orbits. Verrier & Evans (2008)showedgaps Over the past decade the Spitzer Space Telescope (Werner in the disk and three distinct stable populations of test particles: et al. 2004) has provided complementary information about a prograde disk, a retrograde disk, and a high-inclination halo. known debris disks in multiple-star systems. Observations of pairs of main sequence stars made by Spitzer revealed sta- It is important to study what happens in debris disks to un- derstand the role of stellar companions in planet formation. A ble circumbinary disks around pairs of stars with separations ff of 0.04–5.31 AU in 14 systems (Trilling et al. 2007,seeTable4 stellar companion within 100 AU probably a ects the formation of that paper). In addition, two of these binary systems have an of giant planets (e.g., Nelson 2000; Mayer et al. 2005; Thébault additional companion. et al. 2006; and others). Another example of a debris disk in a multiple star system In this work, we extend the investigations of debris disks per- is the circumbinary disk of HD 98800. This system is a young formed by Akeson et al. (2007) and Verrier & Evans (2008). We (<10 Myr) member of the TW Hydra association (Fekel & Bopp have considered a range of inclinations for the third star and for 1993). HD 98800 has four stars in two binaries, A and B, trav- the particle’s disk. We estimate the stable region in which par- elling around each other in highly inclined and eccentric orbits. ticles can survive despite the third star’s inclination and eccen- According to Furlan et al. (2007), the observations suggests the tricity. Our goal is not to reproduce the current configuration of existence of an optically thick wall at 5.9 AU around the B com- the debris disk of HD 98800, but to sample the range of pos- ponents and an inner, empty region. However, some optically sible outcomes for different initial inclinations of the disk and thin dust orbits the B binary in a ring between 1.5 and 2 AU. the third star. The astronomical motivation for our study is to In particular, Furlan et al. (2007) have argued that the peculiar investigate whether/where the orbital stability region of a debris structure and apparent lack of gas in the HD 98800 disk suggest disk is possible around the close binary, and to show how the that this system is likely already at the debris disk stage. disk structure can be shaped by its dynamical interaction with In later studies of debris disks in triple star systems (Akeson an eccentric and inclined third star. Investigating the details of et al. 2007; Verrier & Evans 2008), the quadruple stellar system these dynamical interactions is beyond the scope of the present HD 98800 was studied as a triple system consisting of the B in- work, therefore we will focus the possible impact of the mean ner binary orbited by a third star. Akeson et al. (2007) investi- motion resonances on the disk because recent studies have men- gated the influence of a third star, with an orbital plane initially tioned them as a powerful mechanism for instabilities and gaps inclined at 10◦ with respect to the B inner binary orbital plane, in the disk, without investigating them in detail (Verrier & Evans on the HD 98800 disk. The authors found that the inclined stel- 2008; Farago & Laskar 2010). An analytical study of mean mo- lar orbit could produce a warp in the disk around the B inner tion resonances in this framework is intended to be carried out binary. Verrier & Evans (2008) considered the third star in three in the future. possible current orbital configurations of the HD 98800 system This paper is structured as follows. In Sect. 2, an overview of (Tokovinin 1999). The third star is considered to be in an or- the empirical expressions of the stability boundary applied here bital plane initially inclined at ∼143◦ with respect to the B inner is presented. In Sect. 3, we present the initial conditions for the Article published by EDP Sciences A63, page 1 of 8 A&A 544, A63 (2012) Table 1. Orbital elements of the B-A orbit of the HD 98800 system triple system. The empirical expressions for inner and outer bor- (Tokovinin 1999). ders are written as follows: a = 2.92 + 4.21e − 2.67e2 − 1.55μ a (1) Orbital Parameter I II III I B B B B a (UA) 61.9 67.6 78.6 a = 0.477 − 0.412μ − 0.708e + 0.794μ e + 0.276e2 A O A A A A A eA 0.3 0.5 0.6 ◦ − . μ 2 , ΩA ( ) 184.8 184.8 184.8 0 599 AeA aA (2) ◦ ωA ( ) 210.7 224.6 224.0 Table 2 gives the values of aO for each studied configuration. For the inner stability boundary, we found a value of ∼3.87 AU. Another approximation of the stability boundary for plane- Table 2. Outer critical semi-major axis, aO, for the three configurations tary orbits in binary systems can be obtained using the results investigated. from Domingos et al. (2006), who numerically studied the or- bital stability of a particle in the framework of the elliptic re- Authors I II III stricted three-body problem. Empirical expressions of the sta- Holman & Wiegert (1999) 10.97 7.96 7.04 bility boundaries for prograde and retrograde orbits were also Domingos et al. (2006) 11.40 8.83 8.08 derived. The authors have found aO values that agree well with Verrier & Evans (2007) 11.15 8.25 7.37 those from Holman & Wiegert (1999) and Verrier & Evans Numerical results 12.20 8.80 7.80 (2007). The results from Domingos et al. (2006)differ from pre- vious works in that the empirical expression for the outer critical Notes. The results are given in Astronomical Unit (AU). semi-major axis of the particle also depends on the eccentric- ity e of the particle. In the case of a prograde orbit, the outer semi-major axis is given by disk and the triple stellar system assumed here. In Sect. 4, we summarise the relevant results of our numerical simulations and aO = 0.4895(1 − 1.0305eA − 0.2738e)RH, (3) discuss the effects of the inner binary and third star perturba- where RH is the radius of the system’s B-A Hill’s sphere. The tions. In Sect. 5, we present a study on the identification of mean = motion resonances in the particle disk. Finally, our conclusions aO values for e 0 for each configuration of the third star are are presented in Sect. 6. shown in Table 2. Analysing Eqs. (2)and(3), the aO value strongly depends on the eA, aA and μA values when the particle is on circular orbit. 2. Stability boundary This relation may be a result of the fact that the third star’s gravi- tational perturbation on the particle can change significantly de- To study the stability boundary of a circumbinary disk in a pending on those parameters. This changes the minimum dis- triple stellar system, we have chosen the HD 98800 system. This tance between the third star and the inner binary. The third star’s system has been modelled as a triple stellar system in which gravitational perturbation on the components of the inner binary the B binary is orbited by a particle disk and a more distant third can also change, resulting in a perturbation on the particle. Thus, star. The A pair is treated as a single star disturber. the combination of these two effects must be important to the The B binary orbit has a semi-major axis of aB = 0.983 AU particle’s dynamical stability. According to Eq. (3), the critical and an eccentricity of eB = 0.7849. The solar masses of the semi-major axis aO beyond which the particle would not be sta- BpairareM1 = 0.699 and M2 = 0.582, and the mass ratio ble also depends on the e value.