  A steepening feature in the outer density profiles of dark matter halos
indicating the splashback radius has drawn much attention recently. Possible
observational detections have even been made for galaxy clusters.
Theoretically, Adhikari et al. have estimated the location of the splashback
radius by computing the secondary infall trajectory of a dark matter shell
through a growing dark matter halo with an NFW profile. However, since they
imposed a shape of the halo profile rather than computing it consistently from
the trajectories of the dark matter shells, they could not provide the full
shape of the dark matter profile around the splashback radius. We improve on
this by extending the self-similar spherical collapse model of Fillmore \&
Goldreich to a $\Lambda$CDM universe. This allows us to compute the dark matter
halo profile and the trajectories simultaneously from the mass accretion
history. Our results on the splashback location agree qualitatively with
Adhikari et al. but with small quantitative differences at large mass accretion
rates. We present new fitting formulae for the splashback radius $R_{\rm sp}$
in various forms, including the ratios of $R_{\rm sp} / R_{\rm 200c}$ and
$R_{\rm sp} / R_{\rm 200m}$. Numerical simulations have made the puzzling
discovery that the splashback radius scales well with $R_{\rm 200m}$ but not
with $R_{\rm 200c}$. We trace the origin of this to be the correlated increase
of $\Omega_{\rm m}$ and the average halo mass accretion rate with an increasing
redshift.
