The splashback radius of halos from particle dynamics. II. Dependence on
  mass, accretion rate, redshift, and cosmology
  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.
