Relativistic Lidov-Kozai resonance in binaries

In the recent sample of detected extrasolar planetary systems, some planets exhibit large eccentricities. It may be explained by the Lidov-Kozai resonance (LKR) acting in binary stellar systems (e.g., Innanen et al. 1997, Takeda and Rasio 2005, Verrier and Evans 2008). If the inner planetary orbit is inclined to the orbital plane of the binary, the exchange of the angular momentum between orbits may force large amplitude eccentricity oscillations of the planetary orbit, and simultaneously its argument of pericenter ω 1 librates around ±π/2. However, the LKR may be suppressed by the general relativity (GR) correction to the Newtonian gravity (NG) through changing frequencies of pericenters. Here, we focus on the non-restricted problem and relatively compact systems, and the dynamical effects of including the GR interactions in the model of motion.

We consider the hierarchical triple system. The Hamiltonian written with respect to canonical Poincaré variables (e.g., Laskar and Robutel 1995), H = H kepl + H pert , where

(2.1) describes perturbed Keplerian motions of the inner binary (the central mass m 0 and m 1 ), and the outer binary (m 0 and more distant point-mass m 2 ),

where k is the Gauss gravitational constant, β i = (1/m i + 1/m 0 ) -1 are the reduced masses, r 1,2 , are the radius vectors of m 1,2 relative to m 0 , p 1,2 stand for their conjugate momenta relative to the barycenter, and ∆ = r 1 -r 2 . H GR stands for GR correction to the Newtonian potential of m 0 and m 1 (see, e.g., Richardson and Kelly 1988). We assume that the ratio of semi-major axes α = a 1 /a 2 < 0.2, and H pert H kepl . It means that both m 1,2 are small (planetary regime) or one of m 1,2 ∼ m 0 is relatively large, and one of these bodies is enough distant from m 0 (binary regime).

We expand H NG with respect to α and the Hamiltonian is averaged out with respect to the mean longitudes (Migaszewski and Goździewski 2008a), that leads to the secular term H sec = H NG + H GR , where

2 ), (2.2)

I stands for the mutual inclination, ω 1,2 are the pericenter arguments, and perturbing terms R l are derived in (Migaszewski & Goździewski, in preparation). The averaged GR term is

, where c is the velocity of light. The expansion in Eq. 2.2 generalizes the octupole theory (e.g., Ford et al. 2000) and the coplanar model (Migaszewski and Goździewski 2008a). After the Jacobi’s elimination of nodes (∆Ω = ±π), we eliminate one degree of freedom thanks to the integral of the total angular momentum, C.

reduced to two degrees of freedom. For α = 0.1, the relative errors of H sec approximated by the 10-th order expansion do not exceed 10 -8 in the relative magnitude (see Fig. 1).

To study H sec , we apply the representative plane of initial conditions, Σ, introduced in (Michtchenko andMalhotra 2004, Michtchenko et al. 2006) which crosses all phasespace trajectories. Due to symmetries of H sec with respect to the apsidal and nodal lines:

), (0, ±π), (±π/2, ±π/2), (±π/2, ∓π/2) }, (2.3) and these conditions define the Σ-plane, Σ = {e 1 cos ∆ϖ, e 2 cos 2ω 1 }, ∆ϖ ≡ ϖ 1ϖ 2 , (e 1 , e 2 ) ∈ [0, 1), see the left-hand panel of Fig. 2 for an illustration. Restricting (ω 1 , ω 2 ) to the above set, we also define

revealing levels of H sec without discontinuities (Libert and Henrard 2007).

The equilibria of H sec provide much information on the structure of the phase space. In the Σ-planes, these equilibria appear as quasi-elliptic or quasi-hyperbolic (saddle) points of the levels of H sec , according with the equations of motion:

The stability and bifurcations of equilibria in the full and in the restricted three-body problem were studied in many works (see, e.g., Kozai 1962, Krasinsky 1972, Krasinsky 1974,

analytic ( 10) AMD=0.01

analytic ( 10 Fig. 1. A test of the relative accuracy of the 10-th order expansion of H sec , and levels of H sec in the Σ S -plane (see the text for details) for different values of AMD compared with the semi-analytical (exact) averaging (see Michtchenko andMalhotra 2004 or Migaszewski andGoździewski 2008b). Differences between the theories are expressed in terms of the relative log-scale. Lidov and Ziglin 1974, Féjoz 2002, Michtchenko et al. 2006, Libert and Henrard 2007, Migaszewski and Goździewski 2008b) regarding the NG model. Here, we investigate more closely the equilibrium at the origin (e 1 = e 2 = 0), which is well known since Poincaré, in the presence of the GR interactions. According with the terminology of Krasinsky 1974, that is the trivial space solution of the 3rd kind (e = 0, I = 0), see Fig. 1a.

The zero-eccentricity equilibrium (ZEE) is related to the maximum of H sec and is Lyapunov stable. For a given value of C, the mutual inclination of circular orbits, i 0 , is also a maximal mutual inclination if I 1,2 < π/2. Moreover, for some smaller C (larger AMD), the origin may change its stability due to bifurcations illustrated in the Σ S -plane (Figs. 1b,c).

For instance, Fig. 1b illustrates a saddle

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