tokens: the analytical solution without collisional damping is in good agreement with the results of the simulations. The timescale of dynamical friction is estimated from Equations ( 14 ) and ( 15 ). If and ≫ for m p,1 ≫ m p,2 , the damping timescale due to dynamical friction has the relation as (Ohtsuki et al. 2002 )where M disk is the total mass of a planetesimal disk, is the width of the disk, and is the Hill radius for protoplanets. Note that P DF is estimated to be ∼10 for m p,1 ∼ M ⊕ and ≈ ∼ 0.1. This estimate implies a planetesimal disk with M disk ≪ M ⊕ leads to eccentricity damping for Earth-sized protoplanets in a long timescale ≫10 4 yr via dynamical friction. In addition, dynamical friction increases of planetesimals. The increase in makes τ df longer. On the other hand, the decreasing timescale for M disk due to the collisional cascade, τ cc , is estimated from Equation ( 5 ) to be (Kobayashi Tanaka 2010 )For 100 km sized planetesimals ( m c ≈ 10 22 g) with ∼ 0.1, τ cc ≲ τ df . It should be noted that dynamical friction increases during the damping, which shortens τ cc and elongates τ df as discussed above. Therefore, even if τ cc ∼ τ df initially, τ cc eventually becomes much shorter than τ df . Equations ( 17 ) and ( 18 ) show τ df / τ cc is independent of M disk and Δ a but depends on , , and m c . For = = 0.03, P DF ≈ 60 so that τ cc ≪ τ df for g, while τ df ∼ τ cc for g. Therefore, as shown in Figures 4 and 5 , e and i of protoplanets insignificantly decrease for m c = 10 16 g because of τ cc ≪ τ df , while those are moderately damped for τ cc ∼ τ df with m c = 10 19 g. Therefore, the condition with t cc ≫ τ df is required for the e damping of protoplanets. Planets formed via giant impacts have high eccentricities and inclinations. For Earth-sized planets formed in the giant impact stages, the mean eccentricity is ∼0.1 (Chambers 2001 ; Kokubo et al. 2006 ), which is much larger than the current eccentricities of Earth and Venus. The depletion timescale of a planetesimal disk with is shorter than the e damping timescale via dynamical friction (see Equations ( 17 ) and ( 18 )). Therefore, larger planetesimals are required for e damping of Earth-sized planets. We consider such large planetesimals as being produced from a giant impact. The mass ratios of the largest giant impact ejecta to parent protoplanets are ∼0.01 (Genda et al. 2015 ). Therefore, we consider g, resulting in τ cc ≫ τ df . We perform the simulation for the orbital evolution of a high-eccentricity planet with mass M ⊕ in a swarm of giant impact ejecta. Giant impact ejecta initially have similar orbits to parent planets. However, the orbital phases of giant impact ejecta, π and Ω, are eventually distributed uniformly due to perturbation from other planets. The timescale to achieve a uniform distribution for π and Ω, which is roughly given by the precession rates for π and Ω obtained from the secular perturbation theory (Murray Dermott 1999 ), is estimated as ∼10 5
ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O O B-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O B-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model B-Citation I-Citation I-Citation O O O O O O