A dynamical analysis of the HD 37124 planetary system

In this paper we study the stability of the HD 37124 planetary system. Using initial conditions found by Butler et al. ([CITE]) we estimate the dynamical bounds on orbital parameters that provide stable (quasi-periodic) motions of the system. The stability analysis has been performed with the help of the MEGNO technique. This method, invented by Cincotta & Simó ([CITE]), makes it possible to distinguish efficiently between chaotic and regular dynamics of a dynamical system. The MEGNO analysis helps us to confirm the result of Butler et al. ([CITE]) who found that the critical factor for system stability is the eccentricity of the outer planet. This parameter is not well constrained by the current set of observations. For coplanar configurations, the limiting value of the eccentricity, providing quasi-periodic motion of the system, is roughly equal to 0.55. This value is typically slightly smaller when the system inclination increases (and companions' masses grow) but for $i \simeq 25^\circ$ it can be as low as $\simeq 0.25$. The system is located in a wide stable zone in the plane of eccentricities. The dynamics are insensitive to the initial phases of the planets. If the eccentricity of the outer planet is close to the current fit value $\simeq$0.4, then the system is regular over wide ranges of the relative inclination of the planets. We analyze whether a telluric planet can survive orbitally in the habitable zone of the system. This zone is covered by the investigated region of semi-major axes between 0.6 AU and 2.4 AU. The dynamics in this zone strongly depend on the eccentricity of the outer companion. For moderately low values of this parameter, $e_{\rm c} \simeq 0.1$–0.2, the habitable zone is a dynamical analogue of the Asteroid belt in the Solar system, and the habitable zone of the 47 UMa system. Moreover, in this instance, relatively wide stable areas exist. For increasing eccentricity of the outer planet the stable zones shrink rapidly, and if the parameter is larger than 0.4 they practically disappear. We describe an alternative integration of the MEGNO indicator in the planetary N-body problem, based on the symplectic leapfrog scheme and the tangent map by Mikkola & Innanen ([CITE])