fraction is well approximated by Note that at radii less than ∼ 20 kpc, 1 − f$_{s}$ ( r ) drops faster than Eq. (4); for example, 1 − f$_{s}$ (10 kpc) = 4 × 10 − 4 ≈ 1 . 5 × 10 − $^{3}$(¯ ρ (10 kpc) / ¯ ρ (100 kpc)) − 0 . $^{26}$. This close to the center, however, the clumpiness of the simulated halo is likely artifically suppressed due to finite resolu- tion effects. The best-fit values of α are 0 . 0 ± 0 . 1 at all radii greater than 20 kpc. In the following, we implicitly assume α = 0 and the radial dependence in f$_{s}$ given by Eq. (4). and the median density ¯ ρ is obtained from all the parti- cles in that ellipsoidal shell. These δ$_{i}$ are then binned in equally-spaced bins in log$_{10}$( δ ). In each of these bins, we calculate P (log$_{10}$ δ ) = ∑ i δ − 1 i where the sum is over all particles in that bin; the δ − 1 i weighting gives a volume- fraction distribution. The distribution in log$_{10}$ δ is then converted to a distribution in δ and normalized. The simulation results shown in Fig. 1 show a finite width Δ for the smooth component. However, care must be taken as Poisson fluctuations due to the finite num- ber N of nearest neighbors in the density estimator will also contribute to the width. In Fig. 2 we show P ( δ ) at 100 kpc for densities determined with N = 16, 32, 64, 128, and 1024. The dotted curves indicated the ex- pected contribution to the width from Poisson fluctua- tions (and note that the true and Poisson widths should add in quadrature), which we obtained by running the density estimator on a randomly distributed sample of 10 6 particles. As N is increased, the width of the smooth component decreases, but not quite as fast as the Poisson fluctuations, and by N = 1024 it is clear that the true width has been resolved to be about Δ ≃ 0 . 2. At 10 kpc The central features of Fig. 1 relevant here are the high-density power-law tails predicted by Ref. [15] (and seen already in simulations [16]). The figure shows that the amplitude of the high-density power-law tail is larger at larger radii. This can be attributed largely to the fact that the mean density ¯ ρ is ∼ 175 times lower at 100 kpc than at 10 kpc, and another factor ∼ 30 times lower at 300 kpc, and so the ratio of the density in substructures to the mean density is higher at larger radii. We now use this simulation to calibrate the analytic model at a variety of radii r , from 4 to 300 kpc. At each radius we fit for the power law parameters α and f$_{s}$ . We find that at radii greater than ∼ 20 kpc, the smooth-halo B. Finite Width of the Smooth Component A. Power-Law Tail FIG. 2. The probability distribution function P ( δ ) at 100 kpc for particle densities estimated from the nearest N = (16 , 32 , 64 , 128 , 1024) neighbors. FIG. 1. The probability distribution function P ( δ ) obtained from simulations. The solid curves are the simulation results at r = 300 , 100 and 10 kpc (from top to bottom). The dashed curves show our analytic approximations to the power law tail. The dotted curve indicates the contribution to the finite width of the smooth component at 10 kpc from Poisson fluctuations due to the use of N = 32 neighbors in the density estimator. Note that the mean host halo density ρ$_{h}$ to which the x axis is normalized is ∼ 175 × ( ∼ 4900 × ) smaller for the 100-kpc (300-kpc) curve than in the 10-kpc curve. 1 − f$_{s}$ ( r ) = 7 × 10 − 3 ( ¯ ρ ( r ) ¯ ρ ( r = 100 kpc)$^{)}$$^{−}$$^{0}$$^{.}$$^{26}$ . (4) 3