= Solution
\b[Classification of attractive inverse-square orbits.] Write the <potential energy> as $U=-\kappa/r$, with $\kappa=GMm>0$ and zero at infinity. For nonzero <angular momentum> $J$, the <Kepler orbit> is a conic,
$$
r=\frac{J^2/(m\kappa)}{1+e\cos\phi},\qquad
e^2=1+\frac{2EJ^2}{m\kappa^2}.
$$
Negative total <orbital energy> gives an <elliptic orbit>, $0\le e<1$. Its special case $e=0$ is the <circular Kepler orbit>, at the minimum <energy> permitted for that $J$, $E=-m\kappa^2/(2J^2)$. Zero energy gives a <parabolic Kepler orbit>, $e=1$, and positive energy a <hyperbolic Kepler orbit>, $e>1$. For $J=0$, the motion is radial: these are degenerate limiting trajectories and may strike the centre or the finite planetary surface.
\b[The scattering geometry.] The <impact parameter> $b$ is the perpendicular offset of the undeflected incoming asymptote from the centre, not the actual <periapsis> radius $r_{\min}$. Gravity bends the incoming path toward the planet.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2019/paper-1-c10-grazing-orbit.png]
{title=Original calculated grazing hyperbola with its incoming asymptote, impact parameter and periapsis; the planet and trajectory use a dimensionless illustrative choice}
\b[The grazing threshold.] At infinity, <conservation of energy> and <angular momentum> give $E=mv_\infty^2/2$ and $J=mbv_\infty$. At closest approach the radial speed is zero, so the velocity is tangential and
$$
J=mr_{\min}v_p,\qquad
\frac12mv_p^2-\frac{GMm}{r_{\min}}=\frac12mv_\infty^2.
$$
Eliminating $v_p$ gives
$$
b^2=r_{\min}^2+\frac{2GM}{v_\infty^2}r_{\min}.
$$
This increases monotonically with $r_{\min}>0$. Setting $r_{\min}=R$ gives the <grazing impact parameter in gravitational focusing>,
$$
\boxed{b_{\min}=R\sqrt{1+\frac{2GM}{Rv_\infty^2}}
=R\sqrt{1+\frac{v_{\rm esc}^2}{v_\infty^2}}.}
$$
Here $v_{\rm esc}=\sqrt{2GM/R}$ is the surface <escape speed>. The expression has units of length. Equality is the tangency limit; a strictly noncolliding trajectory requires $b>b_{\min}$. At high speed, $b_{\min}\to R$, while at low speed $b_{\min}\sim\sqrt{2GMR}/v_\infty$. Thus <gravitational focusing> enlarges the collision cross-section to $\pi b_{\min}^2$.
\b[Deflection angle.] For this hyperbola,
$$
e=\sqrt{1+\frac{b^2v_\infty^4}{G^2M^2}}.
$$
The asymptotes occur at $\cos\phi_\infty=-1/e$. The change of velocity direction is $\chi=2\phi_\infty-\pi$, giving $\sin(\chi/2)=1/e$. Hence the <gravitational scattering angle> satisfies
$$
\tan\frac\chi2=\frac1{\sqrt{e^2-1}}=\frac{GM}{bv_\infty^2}.
$$
For the grazing trajectory this is
$$
\boxed{\tan\frac\chi2=\frac{GM}{b_{\min}v_\infty^2}.}
$$
It is the attractive counterpart of the <Rutherford impact-parameter relation>; the sign of the transverse bending changes, but the positive deflection magnitude has this same form.
\b[Range of angles.] Along the grazing family, varying $v_\infty>0$ gives
$$
\boxed{0<\chi<\pi.}
$$
The limits are $\chi\to0$ as $v_\infty\to\infty$ and $\chi\to\pi$ as $v_\infty\to0^+$. Neither endpoint is reached at finite positive speed. The low-speed limiting orbit still has a finite periapsis, but an increasingly distant incoming asymptote.
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