The splashback radius $R_{rm sp}$, the apocentric radius of particles on their first orbit after falling into a dark matter halo, has recently been suggested as a physically motivated halo boundary that separates accreting from orbiting material. Using the SPARTA code presented in Paper I, we analyze the orbits of billions of particles in cosmological simulations of structure formation and measure $R_{rm sp}$ for a large sample of halos that span a mass range from dwarf galaxy to massive cluster halos, reach redshift 8, and include WMAP, Planck, and self-similar cosmologies. We analyze the dependence of $R_{rm sp}/R_{rm 200m}$ and $M_{rm sp}/M_{rm 200m}$ on the mass accretion rate $Gamma$, halo mass, redshift, and cosmology. The scatter in these relations varies between 0.02 and 0.1 dex. While we confirm the known trend that $R_{rm sp}/R_{rm 200m}$ decreases with $Gamma$, the relationships turn out to be more complex than previously thought, demonstrating that $R_{rm sp}$ is an independent definition of the halo boundary that cannot trivially be reconstructed from spherical overdensity definitions. We present fitting functions for $R_{rm sp}/R_{rm 200m}$ and $M_{rm sp}/M_{rm 200m}$ as a function of accretion rate, peak height, and redshift, achieving an accuracy of 5% or better everywhere in the parameter space explored. We discuss the physical meaning of the distribution of particle apocenters and show that the previously proposed definition of $R_{rm sp}$ as the radius of the steepest logarithmic density slope encloses roughly three-quarters of the apocenters. Finally, we conclude that no analytical model presented thus far can fully explain our results.
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