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 Incoming links: Collisional stellar system

During a weak encounter, approximate one star's trajectory by a straight line with speed

v

b

. At longitudinal coordinate

z = v t

, the transverse acceleration is

a_{⊥} = \frac{G mb}{( b ^{2} + v ^{2} t ^{2} ) ^{3/2}} .

Integrating from

t = - \infty

\infty

gives

δ v_{⊥} = \frac{2 G m}{b v} = \frac{2 G mb}{b ^{2} v} .

Here

G m / b^{2}

b / v

is the encounter duration, and the geometric projection produces the displayed transverse impulse.

In one crossing, the number of encounters with impact parameters in

(b, b + d b)

is, up to the system-geometry convention,

d n ≃ \frac{2 N b d b}{R ^{2}} .

Uncorrelated impulses add in mean square, so

⟨ (Δ v)^{2} ⟩_{cross} = \int_{b_{m i n}}^{b_{m a x}} (δ v)^{2} d n ≃ \frac{8 N G ^{2} m ^{2}}{R ^{2} v ^{2}} lo g \frac{b _{m a x}}{b _{m i n}} .

v^{2} \sim GN m / R

. Taking

b_{m a x} \sim R

and the strong-deflection scale

b_{m i n} \sim G m / v^{2} \sim R / N

lo g Λ \sim lo g N

and

\frac{⟨( Δ v ) ^{2} ⟩ _{cross}}{v ^{2}} ≃ \frac{8 l o g Λ}{N} .

Velocity memory is lost when the cumulative change reaches

v^{2}

, after

n_{cross} ≃ \frac{N}{8 lo g Λ} ≃ \frac{N}{8 lo g N} .

t_{rel} ≃ \frac{N}{8 lo g N} \frac{R}{v},

with an order-unity prefactor depending on density profile and convention. The system is a collisional stellar system when

t_{rel}

is shorter than its age or the evolutionary timescale being studied.

Solved by gpt-5.6-sol high.