On collisional capture rates of irregular satellites around the gas-giant planets and the minimum mass of 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 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{Nesvorny2003} and \citet{Charnoz2003} size distribution $dN\sim D^{-3.5}dD$, the collision rates for the different Jovian planets range between $\sim 0.6$ and $\gtrsim 170{\text{Myr}}^{-1}$ for objects with radii, $1\text{km}\le r\le 10\text{km}$. 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 objects was $\gtrsim {10}^3$ times the gravitational binding energy for objects with radii $\sim 100$ km. We find that, capturing irregular satellites via collisions between unbound objects can only account for $\sim 0.1%$ of the observed population, hence can this not be the sole method of producing irregular satellites.