2 bility distribution function that we use is where Θ( x ) is the Heaviside step function. Here, the first term describes the smooth host halo component as having a Gaussian distribution in ln ρ with mean density ρ$_{h}$ and a Gaussian width in ln ρ of Δ. This distribution peaks at a density ρ$_{0}$ = e − 3Δ $^{2}$/ $^{2}$ρ$_{h}$ , a density slightly below ρ$_{h}$ . The second term in Eq. (2) is the high-density power- law tail due to substructures that remain from earlier generations in the structure-formation hierarchy. The parameter ρ$_{h}$ in Eq. (2) is in fact a function of r ; we will see below that ρ$_{h}$ ( r ) ≃ ¯ ρ to very good accuracy. There may also be further r dependence in P ( ρ ; r ) through an r dependence of the parameters Δ, α , and f$_{s}$ . As we will see, our simulations are not yet good enough to allow the r dependence of α to be resolved, and so we take it to be a constant. The simulations are, however, sufficiently resolved to see a strong dependence of f$_{s}$ ( r ) on r , which we detail below. The distribution P ( ρ ; r ) in Eq. (2) is normalized so that ∫ ∞ 0 P ( ρ ; r ) dρ = 1. It can be integrated to give the mean density, In this paper we re-visit and extend those calculations. We first describe in Section II the analytic model. We extend the model by including a finite width for the smoothly distributed component of dark matter in ad- dition to the high-density power-law tail. We then use in Section III state-of-the-art N-body simulations to mea- sure the width of the smooth component and the am- plitude of the power-law tail and thus calibrate the an- alytic model to current simulations. We moreover de- termine how the substructure distribution varies with Galactocentric radius in the halo. In Section IV we use the calibrated model to determine the boost factors for canonical WIMPs and for WIMPs with a Sommerfeld enhancement to the annihilation rate. Section V re- views the substructure-model parameters expected from WIMP models, and Section VI determines the angular dependence of the intensity of gamma-ray radiation both with and without the substructure boost factor we ob- tain. Section VII reviews the model and results and then provides some comments, caveats, and directions for fu- ture development of the model. in Ref. [15] the dependence of the boost factor assuming a canonical WIMP-annihilation rate (i.e., no Sommerfeld enhancement). A complementary analytical approach to this problem, based on the stable clustering hypothesis, was recently presented by [17]. P ( ρ ; r ) = f$_{s}$ √ 2 π Δ 2 1 ρ exp$^{{}$ − 1 2Δ 2 [ ln$^{(}$ ρ ρ$_{h}$ e Δ $^{2}$/ 2 )] 2 } +(1 − f$_{s}$ ) 1 + α ( r ) ρ$_{h}$ Θ ( ρ − ρ$_{h}$ )$^{(}$ ρ ρ$_{h}$$^{)}$$^{−}$$^{(2+}$$^{α}$$^{)}$ , (2) ¯ ρ ( r ) = ∫ ρ$_{max}$ 0 ρ P ( ρ ) dρ = f$_{s}$ρ$_{h}$ + (1 − f$_{s}$ ) ρ$_{h}$    1+ α α [ 1 − ( ρ$_{max}$ ρ$_{h}$ ) − α ] ; α = 0, (1 − f$_{s}$ ) ρ$_{h}$ ln ρ$_{max}$ ρ$_{h}$ ; α = 0 . (3) ¯ ρ ( r ) = 4 ρ$_{s}$ ( r/r$_{s}$ )(1 + r/r$_{s}$ ) $_{2}$, (1) This thus provides a relation between the mean density ρ$_{h}$ ( r ) of the smooth component and the canonical host halo density ¯ ρ . Strictly speaking, this relation requires the maximum density ρ$_{max}$ out to which the power-law tail extends. However, we will see that numerically 1 − f$_{s}$ is so small that ρ$_{h}$ ( r ) ≃ ¯ ρ ( r ) is a very good approximation for practical purposes. Before discussing our analytic model for substructure we begin by presenting the canonical smooth-halo model against which the substructure model will be compared. We take as the canonical model for the Galactic halo an NFW profile, as a function of Galactocentric radius r , with parameters ρ$_{s}$ = 0 . 051 GeV cm − 3 and r$_{s}$ = 21 . 7 kpc taken to provide a reasonable fit to the Milky Way rotation curve. If dark matter in the halo is clumped, then the densi- ties at all points with the same r will not necessarily be the same. Instead, there will be some probability distribution function P ( ρ, r ) defined so that P ( ρ, r ) dρ is the probability that a particular point in the Galac- tic halo (at some fixed Galactocentric radius r ) has a density between ρ and ρ + dρ . According to the argu- ments of Ref. [15], a fraction f$_{s}$ of the volume of the halo (in fact most of the volume; f$_{s}$ ≃ 1, as we will see) should be filled with a smooth dark-matter component with density ρ$_{h}$ , and a fraction 1 − f$_{s}$ ≪ 1 will consist of a high-density clumped component, with something like a power-law distribution of densities. Thus, the proba- We make use of the Milky Way–like halo in the Via Lactea II simulation [18] and calibrate our model to the P ( ρ ; r ) measured from the simulated particle distribu- tion. Fig. 1 shows the PDF P ( ρ ) derived from Via Lactea II for three Galactocentric radii, at 10 kpc (lowest solid line, red), at 100 kpc (middle line, blue), and at 300 kpc (highest line, cyan). To obtain this P ( ρ ), we consider an ellipsoidal shell that follows an iso-density contour in the halo. For each particle in that shell, we calculate δ$_{i}$ = ρ/ ¯ ρ where ρ is estimated in the usual manner from the nearest N neighbors using a symmetric SPH kernel III. CALIBRATION TO SIMULATIONS II. SUBSTRUCTURE MODEL