Stability of the directly imaged multiplanet system HR 8799: resonance   and masses
  A new era of directly imaged extrasolar planets has produced a three-planet system (Marois et al. 2008), where the masses of the planets have been estimated by untested cooling models. We point out that the nominal circular, face-on orbits of the planets lead to a dynamical instability in ~10^5 yr, a factor of at least 100 shorter than the estimated age of the star. Reduced planetary masses produce stability only for unreasonably small planets (<~2 M_ Jup). Relaxing the face-on assumption, but still requiring circular orbits while fitting the observed positions, makes the instability time even shorter. A promising solution is that the inner two planets have a 2:1 commensurability between their periods, and they avoid close encounters with each other through this resonance. That the inner resonance has lasted until now, in spite of the perturbations of the outer planet, leads to a limit <~10 M_Jup on the masses unless the outer two planets are also engaged in a 2:1 mean-motion resonance. In a double resonance, which is consistent with the current data, the system could survive until now even if the planets have masses of ~20 M_Jup. Apsidal alignment can further enhance the stability of a mean-motion resonant system. A completely different dynamical configuration, with large eccentricities and large mutual inclinations among the planets, is possible but finely tuned. 
Daniel C. Fabrycky and Ruth A. Murray-Clay