Stability of planetary systems with respect to masses

The stability in the sense of Lagrange of the Sun–Jupiter–Saturn system and 47 UMa system with respect to masses on a time scale of 10⁶ years was studied using the method of averaging and numerical methods. When the masses of Jupiter and Saturn increase by 20 times (approximately, more accurate value depends on a time-scale of stable motion), these planets can have close approaches. Close approaches appear when analyzing osculating elements; they are absent in the mean elements. A similar situation takes place in the case of 47 UMa and other exoplanetary systems. The study of Lagrange stability with respect to masses allows us to obtain upper limits for masses of extrasolar planets.