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Probability of Collisional Capture of Irregular Satellites Around the Gas Giant Planets and Mass Constraints for the Solar Nebula

We investigated the probability that an inelastic collision of planetesimals within the Hill sphere of the Jovian planets could explain the presence and orbits of observed irregular satellites. Capture of these satellites via this mechanism is highly dependent on not only the mass of the protoplanetary disk, but also the shape of the planetesimal size distribution. We performed 2000 simulations for integrated time intervals $\sim 2$ Myr and found that given the currently accepted value for the minimum mass solar nebula and planetesimal number density based upon the \citet{Bottke2010b} size distribution $dN \sim D^{-1.8} dD$, the collision rates for the different Jovian planets range between $\sim 60$ and $\gtrsim 10^3 \, \Myr^{-1}$ Additionally we found that the probability that these collisions remove enough orbital energy to yield a bound orbit was $\lesssim 10^{-5}$ and had very little dependence on the relative size of the planetesimals. Of these collisions, the collision energy between two masses were $\gtrsim 10^3$ times the gravitational binding energy for masses with radii $\sim 100$ km. Additionally, the combination that enough energy is dissipated by the collision is $\lesssim 10^{-6}$ and the ratio of collision energy to gravitational binding energy is sufficiently large enough to break apart, or completely obliterate, the colliding planetesimals, it is likely that for irregular satellites to be captured as a result of the collision of planetesimals, only a small fraction of the colliding material remained bound to the planet and would most likely be produced by a large mass with a radius $\sim 100$ km colliding with a smaller mass with radius $\lesssim 20$ km.