Put

x = r / r_{1}

. The NFW profile becomes

ρ_{D} = ρ_{1} / [x (1 + x)^{2}]

. Integrating spherical shells gives

M_{D} (r) = 4 π ρ_{1} r_{1}^{3} \int_{0}^{x} \frac{u}{( 1 + u ) ^{2}} d u = 4 π ρ_{1} r_{1}^{3} [lo g (1 + x) - \frac{x}{1 + x}] .

F (x) = [lo g (1 + x) - x / (1 + x)] / x

. The spherical circular speed is therefore

v_{NFW, c}^{2} (r) = \frac{G M _{D} ( r )}{r} = 4 π G ρ_{1} r_{1}^{2} F (x) .

At small

x

, the bracket is

x^{2} /2 - 2 x^{3} /3 + O (x^{4})

v_{NFW, c}^{2} ≃ 2 π G ρ_{1} r_{1} r, v_{NFW, c} \propto r^{1/2} .

At large

x

, the bracket is

lo g x - 1 + O (1/ x)

v_{NFW, c} ≃ 4 π G ρ_{1} r_{1}^{2} (\frac{l o g x - 1}{x})^{1/2} .

Thus the curve rises from zero, reaches one broad maximum, and eventually falls as

[lo g r / r]^{1/2}

. The untruncated NFW profile has logarithmically divergent mass, but its circular speed still tends to zero.

Differentiating

F

shows that its stationary point satisfies

\frac{x ^{2}}{( 1 + x ) ^{2}} = lo g (1 + x) - \frac{x}{1 + x} .

F^{'}

x (1 - x) / (1 + x)^{3}

: it increases up to one and decreases thereafter, crossing zero once at positive

x > 1

. Numerical bisection gives

x_{m a x} ≃ 2.16258

and

F (x_{m a x}) ≃ 0.2162166

. Hence

r_{m a x} ≃ 2.163 r_{1}, v_{NFW, c, max} ≃ 1.648 r_{1} G ρ_{1} .

Equivalently

v_{m a x} ≃ 0.4650 4 π G ρ_{1} r_{1}^{2}

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 Incoming links: NFW mass and circular-speed profile