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Existence of the asymptotic velocities and the impact-parameter identity $J = μ b ∣ v_{-} ∣$

Statement

Let the hypotheses of Finite-range scattering: the simple outer turning point, the two global branches and the free tails hold, with finite-range potential $U$ , energy $E > 0$ , angular momentum $j = J e_{3}$ , $J > 0$ , outer turning point $r_{*}$ , and let $r : R \to R^{3}$ be the extension of the trajectory. Then:

the limits $v_{\pm} : = \lim_{t \to \pm \infty} r^{'} (t)$ exist and satisfy $∣ v_{+} ∣ = ∣ v_{-} ∣ = \sqrt{2 E / μ}$ ;
for every $t_{2} \leq t_{*}$ with $ρ (t_{2}) > R$ , the incoming free tail is the half-line ${r (t_{2}) + s v_{-} : s \leq 0}$ , whose affine extension is the straight line ${r (t_{2}) + s v_{-} : s \in R}$ ; its perpendicular distance from the origin is $b : = \frac{∣ r (t_{2}) \times v_{-} ∣}{∣ v_{-} ∣} = \frac{J}{μ ∣ v_{-} ∣}, i.e. J = μ b ∣ v_{-} ∣ .$

Facts & Assumptions

Given: The finite-range scattering orbit with its simple outer turning point $r_{*}$ and its two global branches, and times $t_{1} \geq t_{*}$ , $t_{2} \leq t_{*}$ with $ρ (t_{1}), ρ (t_{2}) > R$ .

[L1]

Free tails: for $t_{1} \geq t_{*}$ with $ρ (t_{1}) > R$ , $r (t) = r (t_{1}) + (t - t_{1}) r^{'} (t_{1})$ for all $t \geq t_{1}$ ; and symmetrically for $t_{2} \leq t_{*}$ with $ρ (t_{2}) > R$ , $r (t) = r (t_{2}) + (t - t_{2}) r^{'} (t_{2})$ for all $t \leq t_{2}$ (Finite-range scattering: the simple outer turning point, the two global branches and the free tails).

[L2]

On the free regime $ρ > R$ the potential vanishes identically, so the constant energy is $\frac{1}{2} μ ∥ r^{'} ∥_{2}^{2} = E$ there (Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential).

[L3]

Angular momentum: $j = μ r \times r^{'} = J e_{3}$ is constant with $∣ j ∣ = J$ (Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, Finite-range scattering: the simple outer turning point, the two global branches and the free tails).

[L4]

For nonzero $u, v \in R^{3}$ , the perpendicular distance from the origin to the line ${u + s v : s \in R}$ equals $∣ u \times v ∣ / ∣ v ∣$ : the closest point is $u - \frac{⟨ u, v ⟩}{∣ v ∣^{2}} v$ , whose squared norm is $∣ u ∣^{2} - \frac{⟨ u, v ⟩^{2}}{∣ v ∣^{2}} = \frac{∣ u \times v ∣^{2}}{∣ v ∣^{2}}$ by the Gram identity $∣ u \times v ∣^{2} = ∣ u ∣^{2} ∣ v ∣^{2} - ⟨ u, v ⟩^{2}$ (The squared cross-product norm is the Gram determinant of two vectors, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ ).

Proof

technique · direct

1.1givenL1

By [L1] the velocity is constant on the outgoing tail: $r^{'} (t) = r^{'} (t_{1})$ for all $t \geq t_{1}$ , so $v_{+} : = \lim_{t \to \infty} r^{'} (t) = r^{'} (t_{1})$ exists; symmetrically $v_{-} : = r^{'} (t_{2})$ is the constant velocity on the incoming tail $t \leq t_{2}$ .

2.1step 1.1L2algebra

By [L2] applied at $t_{1}$ and $t_{2}$ , $∥ r^{'} (t_{1}) ∥_{2}^{2} = ∥ r^{'} (t_{2}) ∥_{2}^{2} = 2 E / μ$ ; hence $∣ v_{+} ∣ = ∣ v_{-} ∣ = \sqrt{2 E / μ}$ . This is clause 1.

2.2step 1.1givenL1algebra

The incoming line: by [L1] all points $r (t)$ with $t \leq t_{2}$ lie on the set ${r (t_{2}) + s v_{-} : s \in R}$ , a line through $r (t_{2})$ with direction $v_{-} \neq 0$ (it has norm $\sqrt{2 E / μ} > 0$ ).

3.1step 1.1step 2.2L3L4algebra∎

Its perpendicular distance from the origin is, by [L4] applied to $u = r (t_{2})$ , $v = v_{-}$ , $\frac{∣ r (t_{2}) \times v_{-} ∣}{∣ v_{-} ∣} = \frac{∣ r (t_{2}) \times r^{'} (t_{2}) ∣}{∣ v_{-} ∣} = \frac{∣ j ∣ / μ}{∣ v_{-} ∣} = \frac{J}{μ ∣ v_{-} ∣},$ using that $j = μ r (t_{2}) \times r^{'} (t_{2})$ and $∣ j ∣ = J$ from [L3]. Defining $b$ to be this distance gives $J = μ b ∣ v_{-} ∣$ , which is clause 2.

Existence of the asymptotic velocities and the impact-parameter identity $J = μ b ∣ v_{-} ∣$

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Existence of the asymptotic velocities and the impact-parameter identity J=μb∣v−∣J=μb∣v−​∣

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Existence of the asymptotic velocities and the impact-parameter identity $J = μ b ∣ v_{-} ∣$