High-Mass, Four-Planet Models for HR 8799: Constraining the Orbital   Inclination and Age of the System
  Debates regarding the age and inclination of the planetary system orbiting HR 8799, and the release of additional astrometric data following the discovery of the fourth planet prompted us to examine the possibility of constraining these two quantities by studying the long-term stability of this system at different orbital inclinations and in its high-mass configuration (7-10-10-10 MJup). We carried out ~1.5 million N-body integrations for different combinations of orbital elements of the four planets. The most dynamically stable combinations survived less than ~5 Myr at inclinations of 0{\deg} and 13{\deg}, and 41, 46, and 31 Myr at 18{\deg}, 23{\deg}, and 30{\deg}, respectively. Given such short lifetimes and the location of the system on the age-luminosity diagram for low-mass objects, the most reasonable conclusion of our study is that the planetary masses are less than 7-10-10-10 MJup and the system is quite young. Two trends to note from our work are as follows. (1) In the most stable systems, the higher the inclination, the more the coordinates for planets b and c diverge from the oldest archival astrometric data (released after we completed our N-body integrations), suggesting that either these planets are in eccentric orbits or have lower orbital inclinations than that of planet d. (2) The most stable systems place planet e closer to the central star than is observed, supporting the conclusion that the planets are more massive and the system is young. We present the details of our simulations and discuss the implications of the results. 
Jeffrey J. Sudol and Nader Haghighipour
astro-ph.EP astro-ph.SR