F = \frac{GMm}{r^{2}}

\frac{mv^{2}}{r} = \frac{GMm}{r^{2}}

v^{2} = \frac{GM}{r}

v = \sqrt{\frac{GM}{r}} = \sqrt{GM} r^{−1/2}

v^{2}(R) = \frac{GM_{enc}}{R}

M_{enc} = ∫_{0}^{R} 4πr^{2} dr ρ(r)

v ∝ R

ρ(r) = ρ_{0} r^{α}

M_{enc} = ∫_{0}^{R} 4πr^{2} dr ρ_{0} r^{α} = 4πρ_{0} ∫_{0}^{R} r^{2+α} dr

M_{enc} = (\frac{4πρ_{0}}{3 + α}) R^{3+α}

v^{2} = \frac{GM_{enc}}{R} ∝ \frac{R^{3+α}}{R} ∝ R^{2+α}

ρ(r) = ρ_{0} r^{−2.2}

μ = −k r

μ ∝ ln (I)

ln (I) ∝ −k r  ⇒  I(r) ∝ e^{−kr}