mean free path L = \frac{1}{π (2r)^{2} n}

before approach E = KE_{A} + KE_{B} + GP E = \frac{1}{2}mv^{2} + \frac{1}{2}mv^{2} − \frac{Gmm}{∞}

E_{initial} = mv^{2}

during approach E = KE_{A} + KE_{B} + GP E = \frac{1}{2}mv^{′2} + \frac{1}{2}mv^{′2} − \frac{Gmm}{R_{min}}

\frac{Gm^{2}}{R_{min}} = \frac{1}{2} E_{initial}

R_{min} = \frac{2Gm}{v^{2}}

duration of force Δt = \frac{b}{v}

size of force F = \frac{Gm_{1}m_{2}}{b^{2}}

impulse m_{1} Δv_{y} = F Δt = \frac{Gm_{1}m_{2}}{b^{2}} (\frac{b}{v} )

Δv_{y} = \frac{Gm_{2}}{b v}

distance travelled Δy = (velocity) (time) = (\frac{Gm_{2}}{b v} ) (\frac{b}{v} )

Δy = \frac{Gm_{2}}{v^{2}}

2 Δy ≪ b  ⇒  \frac{2 Gm_{2}}{v^{2}} ≪ b

volume V = (length ) (area of ring) = (v t) (2πb db)

N = V n = (2π vt b db) (n) contribution at impact parameter b = (number of encounters ) (Δv_{y}^{2})

= (2π vt n b db) (\frac{Gm}{vb})^{2}

total |(Δv_{y})^{2}| = ∫_{b_{min}}^{b_{max}} (2π vt n b db) (\frac{Gm}{vb})^{2}

total |(Δv_{y})^{2}| = (\frac{2πG^{2}m^{2}n}{v}) t ∫_{b_{min}}^{b_{max}} \frac{db}{b}

total |(Δv_{y})^{2}| = (\frac{2πG^{2}m^{2}n}{v}) t (ln \frac{b_{max}}{b_{min}} )

Λ ≡ (\frac{b_{max}}{b_{min}} )

ln Λ ≡ ln (\frac{b_{max}}{b_{min}} ) |(Δv_{y})^{2}| = |v^{2}|

v^{2} = (\frac{2πG^{2}m^{2}n}{v}) t ln Λ

t_{relax} = \frac{v^{3}}{2π G^{2}m^{2} n ln Λ}