= Grazing impact parameter in gravitational focusing
{title2=$b_{\rm crit}=R\sqrt{1+2GM/(Rv_\infty^2)}$}
For a point projectile approaching an absorbing spherical body of radius $R$ in a prescribed exterior <inverse-square force>, <conservation of energy> and <angular momentum> give $b^2=r_p^2+2GMr_p/v_\infty^2$. This is increasing in the <periapsis> radius $r_p$, so $r_p=R$ gives the collision threshold displayed above. The <gravitational scattering angle> obeys $\tan(\chi/2)=GM/(bv_\infty^2)$. On the grazing family its deflection approaches zero at high asymptotic speed and $\pi$ as $v_\infty\to0^+$. Strict noncollision requires $b>b_{\rm crit}$; the equality is the tangency limit.
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