Orbital stability of systems of closelyspaced planets, II: configurations with coorbital planets
 Andrew W. Smith,
 Jack J. Lissauer
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We numerically investigate the stability of systems of 1 \({{\rm M}_{\oplus}}\) planets orbiting a solarmass star. The systems studied have either 2 or 42 planets per occupied semimajor axis, for a total of 6, 10, 126, or 210 planets, and the planets were started on coplanar, circular orbits with the semimajor axes of the innermost planets at 1 AU. For systems with two planets per occupied orbit, the longitudinal initial locations of planets on a given orbit were separated by either 60° (Trojan planets) or 180°. With 42 planets per semimajor axis, initial longitudes were uniformly spaced. The ratio of the semimajor axes of consecutive coorbital groups in each system was approximately uniform. The instability time for a system was taken to be the first time at which the orbits of two planets with different initial orbital distances crossed. Simulations spanned virtual times of up to 1 × 10^{8}, 5 × 10^{5}, and 2 × 10^{5} years for the 6 and 10planet, 126planet, and 210planet systems, respectively. Our results show that, for a given class of system (e.g., five pairs of Trojan planets orbiting in the same direction), the relationship between orbit crossing times and planetary spacing is well fit by the functional form log(t _{ c }/t _{0}) = b β + c, where t _{ c } is the crossing time, t _{0} = 1 year, β is the separation in initial orbital semimajor axis (in terms of the mutual Hill radii of the planets), and b and c are fitting constants. The same functional form was observed in the previous studies of single planets on nested orbits (Smith and Lissauer 2009). Pairs of Trojan planets are more stable than pairs initially separated by 180°. Systems with retrograde planets (i.e., some planets orbiting in the opposite sense from others) can be packed substantially more closely than can systems with all planets orbiting in the same sense. To have the same characteristic lifetime, systems with 2 or 42 planets per orbit typically need to have about 1.5 or 2 times the orbital separation as orbits occupied by single planets, respectively.
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 Title
 Orbital stability of systems of closelyspaced planets, II: configurations with coorbital planets
 Journal

Celestial Mechanics and Dynamical Astronomy
Volume 107, Issue 4 , pp 487500
 Cover Date
 20100801
 DOI
 10.1007/s1056901092880
 Print ISSN
 09232958
 Online ISSN
 15729478
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Stability
 Planetary systems
 Coorbital planets
 Industry Sectors
 Authors

 Andrew W. Smith ^{(1)}
 Jack J. Lissauer ^{(2)}
 Author Affiliations

 1. Deptartment of Mechanical Engineering, Stanford University, Stanford, CA, 94305, USA
 2. Space Science and Astrobiology Division, MS 2453, National Aeronautics and Space Administration, Ames Research Center, Moffett Field, CA, 94035, USA