A simple dark-matter halo model gives the mass density

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Astrophysics & Cosmology · Calculus of change and accumulation

Conditions of Validity, Ch Multiple integrals: Fubini, the change-of-variables theorem and integrals that do not exist

Problem

A simple dark-matter halo model gives the mass density ρ(r)=ρs(r/rs)−p\rho(r) = \rho_s (r/r_s)^{-p}, with rr the distance from the halo centre in three-dimensional space (n=3n=3) and rsr_s a fixed scale radius. The enclosed mass out to radius rsr_s is ∫r≤rsρ dV\int_{r\le r_s}\rho\,\mathrm{d}V, and the mass beyond rsr_s is ∫r≥rsρ dV\int_{r\ge r_s}\rho\,\mathrm{d}V. ((a)) The model's inner slope is p=1p=1. Decide whether the mass enclosed within rsr_s is finite, given that ρ\rho itself diverges at r=0r=0. ((b)) The model's outer slope is p=3p=3. Decide whether the mass from rsr_s to infinity is finite, and confirm the verdict by evaluating the integral directly. ((c)) A steeper alternative model has outer slope p=4p=4. Decide whether the mass from rsr_s to infinity is finite for this model.

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