Where is the Second Planet in the HD 160691 Planetary System?

The set of radial velocity measurements of the HD 160691 has been recently published by Jones et.al 2002 (MNRAS). It reveals a linear trend that indicates a presence of the second planet in this system. The preliminary double-Keplerian orbital fit to the observations, announced by the discovery team, describes a highly unstable, self-disrupting configuration. Because the observational window of the HD 160691 system is narrow, the orbital parameters of the hypothetical second companion are unconstrained. In this paper we try to find out whether a second giant planet can exist up to the distance of Jupiter and search for the dynamical constraints on its orbital parameters. Our analysis employs a combination of fitting algorithms and simultaneous examination of the dynamical stability of the obtained orbital fits. It reveals that if the semi-major axis of the second planet is smaller than $simeq 5.2$ AU, the observations are consistent with quasi-periodic, regular motions of the system confined to the islands of various low-order mean motion resonances, e.g., 3:1, 7:2, 4:1, 5:1, or to their vicinity. In such cases the second planet has smaller eccentricity $simeq 0.2-0.5$ than estimated in the previous works. We show that the currently available Doppler data rather preclude the 2:1 mean motion resonance expected by some authors to be present in the HD 160691 planetary system. We also demonstrate that the MEGNO-penalty method, developed in this paper, which is a combination of the genetic minimization algorithm and the MEGNO analysis, can be efficiently used for predicting stable planetary configurations when only a limited number of observations is given or the data do not provide tight constraints on the orbital elements.Comment: 20 pages, 3 tables, 13 figures (in low resolution suitable for publishing in astro-ph), the preprint with high resolution images can be downloaded from http://www.astri.uni.torun.pl/~chris/hd160691.ps.gz (3.4MB). Submitted to Ap